ELEMENTS
OF
PHYSICAL BIOLOGY
BT
ALFRED J. LOTKA, M.A., D.Sc.
"Voil& un homme qui a fait son miens pour _enniwer
deux ou trois cents de ses concitoyens; mais son intention
6tait bonne: il n'y a pas de quoi dftniire PersepolM.
Yoltovre
BALTIMORE
WILLIAMS & WLKINS COMPANY
1925
Composed and Printed at the
WAVERLf PRESS
BT THE VYlLLIAMS & WlLKINS
Baltimore, Md., U. S. A.
T
COPTEIQHT 1925
WILLIAMS & WILKINS COMPANY
Made in United States of America
Published February, 1925
ALL BIGHTS KESERVIpD
v^
DEDICATED
TO THE MEMORY
OF
JOHN HENBY POYNTING
PREFACE
The preface is that part of a book which is written last, placed
first, and read least. As I approach my concluding task I am moved
to reflect why a preface should be written at all. This question, if
followed into all the intricacies of which it holds potentiality, should
apparently result in a composition new in literature, a Preface to the
Preface. Such precedent should not be lightly established, for it
suggests a vista of future degenerations after the pattern of Josiah
Royce's infinite succession of maps, each containing within Itself its
own replica on a reduced scale. But without going to such lengths
as this, the philosophy of the preface may perhaps briefly be summarized
to this effect, that It is the author's subjective introduction ,
to the more objective matter that should follow. Here he may, if
this is deemed of any interest, say something regarding the circumstances
that gave origin to the w-ork, and the conditions under which
it came into being. He may express his feelings as to its alleged purpose,
and may follow custom by giving voice to pious wishes as to the
function which the product of his presumptive mind may fulfill in an
Universe in which no event, however trivial be it no more than the
addition of one more book to the groaning library shelves is without
distant reverberations.
As to origin, the first plan of the work was laid about 1902, in the
author's student days In Leipzig. The development of the topic is
recorded, in outline, in various publications, of which the first
appeared in 1907 in the American Journal of Science. Eeference to
this and to its various sequels will be found in pertinent places in
the text that follows. The last stage of the work, arrangement of
the matter in collected form, and filling in the flesh about the skeleton
framework elaborated in the journal literature, was carried out
at the Johns Hopkins University upon the invitation of the Department
of Biometry and Vital Statistics. For the courtesies so
extended to him the author wishes here to express his thanks, as
well as for the interest shown in the progress of the work by Dr.
Raymond Pearl and the members of the Department, notably Drs.
W. T. Howard, L. J. Reed and J. R. Miner. Outside the walls of
VU1 PREFACE
this University I think with very particular appreciation of the
never-failing succor in times of mathematical trouble, which I
found at the hands of Prof. F. R. Sharpe of Cornell University; also
of the patient assistance, upon more than one occasion, from Prof.
W. B. Fite of Columbia University. And I gratefully recall encouragement
received from Dr. G. K. Burgess, Director of the Bureau
of Standards, especially in the earlier stages of the work, when
encouragement was most needed.
Acknowledgment has been made in the text for numerous quotations.
The somewhat extended excerpts from certain articles published
in the Scientific Monthly call for special notice here, and I
wish to express my thanks both to the author, Prof. G. W. Martin,
and to the Editor of the Monthly, for permission to quote thus at
length from its pages. I am similarly indebted to the Editor of
Harpers Magazine for permission to reproduce here certain portions
of an article from my pen, entitled "Biassed Evolution", which
originally appeared in the May issue (1924) of that publication.
Toward the publishers, Messrs. Williams and Wilkins and in
particular Mr. C. C. Thomas, I have every occasion to entertain
feelings of the most cordial appreciation. Through their courteous
attentions the business of bookmaking was made a pleasure.
My greatest debt is acknowledged in the dedication. Whatever
merits this book possesses may well be credited to the influence
and teaching of Poynting. There is little danger that its faults
shall be charged to his account.
As to the topic of the work it seems unnecessary to say many words
here, inasmuch as a delineation of this has been made the subject of a
special chapter on The Program of Physical Biology. Only this
explanation it may be well to offer here, that, as proposed in Chapter
V} the term Physical Biology has been employed to denote the broad
application of physical principles and methods in the contemplation
of biological systems, whereas Biophysics, in common parlance,
relates rather to the special field of certain physical aspects of the
life processes of the individual. With this terminology, Physical
Biology would comprehend Biophysics within its scope.
The writer cannot in reason expect to have produced a work
without blemish. Even an approach to such absolute perfection is
the rare privilege of a few. He would, however, be unjustified in
addressing the reading public at all if he did not entertain the hope
PEEFACE IX
that, despite shortcomings, these pages may bring to the reader
new assets, here and there a new piece for his mental furniture, now
and again a new perspective, a new comprehensive outlook over a
body of facts and relations in themselves perhaps familiar.
The work has been largely one of systematization, and of development
of method. Factual material has been introduced essentially
for the purpose of illustrating the point of view to be set forth. There
seems therefore hardly any occasion for apologetic explanations that
anything of the nature of completeness in the presentation of pertinent
facts was in nowise armed at. Indeed, it must be obvious upon
most casual reflection that such completeness, in a subject of the
amplitude of that here taken in view, could be achieved only in a
cyclopedic work of several tiers of volumes.
Considerable care has been taken to cite in detail the sources consulted.
It was felt that, on account of the wide dispersal of these
citations over a broad field of scientific literature, few readers could
be expected to be familiar with all the branches of pertinent library
lore, and for this reason a collation of such references should have a
value of its own, even apart from the text. At the same time the
compilation of anything like a complete bibliography could not be
undertaken on the present occasion.
It is hoped that the mathematical mien of certain pages will not
deter biologists and others, who may be disposed to look askance at
symbols of the art, from acquiring an interest in other portions of the
book. Biometricians will, presumably, not shrink on this score; to
them, and to physicists, (whomI should greatly wish to numberamong
my readers) I may perhaps confess that I have striven to infuse the
mathematical spirit also into those pages on which symbols do not
present themselves to the eye. For this I offer no apology.
For the sake of space economy recapitulary paragraphs have, as a
rule, not been given a place in the text. An exception has however
been made in Chapters XX, XXXIII and XXIV, the last of
which, in particular resumes and amplifies somewhat certain phases
of the topics discussed in earlier chapters. The reader who may
wish briefly to review the substance of his reading as he proceeds,
should find suitable assistance in the rather detailed Analytical
Synopsis of Chapters that has been placed immediately after the
Table of Contents. And finally, a bird's eye survey of the general
X PREFACE
field covered in this work can be obtained by consulting the Tabular
Synopsis at the end.
Here, then, I make my exit from the prefatory stage and commend
my work to the tender mercies of the reader; not without some trepidation,
for I recall how Voltaire said of one: "II fit une philosophic
comrne on fait un bon roman; tout parut vraiseniblable, et rien ne
fut vrai;" and there comes to mind the language still plainer of du
Maupassant "Depuis qu'ils ont appris a lire et a ecrire, la betise
latente se degage." I trust that the reader's response to these pages
may not be too fervent an Amen to the prayer of The Sceptical
Chymisi "It is to be hoped that these men, finding that they can
not longer write impertinently and absurdly .... will be
reduced either to write nothing, or books that may teach us something
. , . .
;
and so, ceasing to trouble the world with riddles or impertineneies,
we shall either by their books receive an advantage, or
by their silence escape an inconvenience."
ALFEED J. LOTSA.
Johns Hopkins University, May, 1924-
CONTENTS
PART I. GENERAL PRINCIPLES
CHAPTER I
Regarding definitions 3
CHAPTER II
Evolution Defined : The History of a System in the Course of Irreversible
Transformation 20
CHAPTER III
The Statistical Meaning of Irreversibility 30
CHAPTER IV
Evolution Conceived as a Redistribution 41
CHAPTER V
The Program of Physical Biology 49
PART n. KINETICS
CHAPTER VI
The Fundamental Equations of the Kinetics of Evolving Systems. General
Case 57
CHAPTER VII
Special Case :
Single Dependent Variable 64
CHAPTER VIII
Special Cases : Two and Three Dependent Variables 77
CHAPTER IX
Analysis of the Growth Function 100
CHAPTER X
Further Analysis of the Growth Function 128
PART III. STATICS
CHAPTER XI
General Principles of Equilibrium 143
CHAPTER XII
Chemical Equilibrium, as an Example of Evolution under a Known Law. . 152
xi
Xll CONTENTS
CHAPTER XIII
Inter-species Equilibrium. . ,
161
CHAPTER XIV
Inter-species Equilibrium :
Aquatic Life. 171
CHAPTER XV
The Stage of the Life Drama 185
CHAPTER XVI
The Circulation of the Elements. The Water Cycle 209'
CHAPTER XVII
The Carbon Dioxide Cycle 218
CHAPTER XVIII
The Nitrogen Cycle 229
CHAPTER XIX
The Phosphorus Cycle 246
CHAPTER XX
Cycles : Conclusion and Summary 252
CHAPTER XXI
Moving Equilibria. . . 259
CHAPTER XXII
Displacement of Equilibrium 280
CHAPTER XXIII
The Parameters of State 300
PART IV. DYNAMICS
CHAPTER XXIV
The Energy Transformers of Nature 325
CHAPTER XXV
The Helation of the Transformer to Available Sources 336
CHAPTER XXVI
The Correlating Apparatus 362
CHAPTER XXVII
Extension of the Sensuous World Picture 371
CHAPTER XXVIII
The Adjusters. 381
CONTENTS Xlll
CHAPTER XXIX
Consciousness 388
CHAPTER XXX
The Function of Consciousness 394
CHAPTER XXXI
The Origin of Consciousness in the Living Organism 402
CHAPTER XXXII
Energy Relations of Consciousness 406
CHAPTER XXXIII
Review of the Correlating Apparatus 410
CHAPTER XXXIV
Conclusions ] Retrospect and Prospect 417
Synoptic chart of physical biology 435
LIST OF ILLUSTRATIONS
FIG. 1. Graph of model process illustrating the statistical meaning of
irreversibility. Reproduced from A. J. Lotka, Two Models in Statistical
Mechanics, Am. Math. Monthly, vol. 31, 1924, p. 122 31
FIG. 2. Graph of second model process illustrating the statistical meaning
of irreversibility. Reproduced from A. J. Lotka, Two Models in
Statistical Mechanics, Am. Math. Monthly, vol. 31, 1924, p. 124 33
FIG. 3. Frequency diagram for the deviations from the mean appearing in
figure 2. Reproduced from A. J. Lotka, Two Models in Statistical
Mechanics, Am. Math. Monthly, vol. 31, 1924, p. 125 34
FIG 4. The law of population growth for the United States according to
Pearl and Reed 68
FIG. 5. Growth of a population of Drosophila (fruit flies) undsr controlled
experimental conditions, according to Pearl and Parker. 69
FIG. 6. Growth of a bacterial colony (B. dendroides). according to
H. G. Thornton 71
FIG. 7. Growth of rat according to H. H. Donaldson and T. B. Robertson. 73
FIG. 8. Growth of sunflower seedlings according to H. S. Reed and R. H.
Holland; computed curve by L. J. Reed 74
FIG. 9. Growth of sunflower seedlings. The same data as in figure 8, but
plotted in logarithmic diagram 75
FIG. 10. Presumptive curve of growth of endemic malaria according to
the Ross Equation. Reproduced from A. J. Lotka, Am. Jour. Hygiene,
January Supplement, 1923 84
FIG. 11. Course of parasitic invasion of insect species according to W. R.
Thompson 85
FIG. 12. Increasing diffusion-in-time of successive generations in the
progeny of a population element 86
FIG. 13. Course of parasitic invasion of insect species, according to Lotka;
elementary treatment 90
FIG. 14. Course of parasitic invasion of insect species, according to Lotka;
more exact treatment 91
FIG. 15. Some historical human survival curves, exhibiting an evolutionary
trend toward longer average duration of life 103
FiGi 16. Survival curves for the State of Massachusetts, for the three
decades 1890-1900-1910. After Glover 104
FIG. 17. Survival curves for different countries. After Glover 105
FIG. 18. Survival curve plotted on logarithmic scale. II. S. 1910 107
FIG. 19. Logarithmic survival curves for man, Drosophila, and Proales decipiens.
Plotted according to centiles of life-span. After R. Pearl. . 109
FIG. 20. Diagrams to illustrate proof of stability of normal age distribution.
Reproduced from A. J. Lotka, Proc. Natl. Acad. Sci., vol. 8,
1922, p. 339 , 113
xv
^ LIST OP ILLUSTRATIONS
FIG. 21. "Stable" age distribution, as exemplified by the population of
England and Wales in the decade 1871-1880........................... 114
FIG, 22. Diagram of relation between birth rate per head 6 and death rate
per head d in population with stable age distribution ................ 117
FIG. 23. Diagrammatic illustration of influence of random and of selective
slaughtering upon survival curve of biological species .............. 120
FIG. 24. Growth of favored type in mixed population of two phenotypes
After J. B. S. Haldane............................................. 124
FIG. 25. Effect of selection on population comprising two phenotypes with
Mendelian inheritance. After J. B. S. Haldane ..................... 126
FIG. 26. Feed consumed, and increase in live weight of steers at several
ages. After Moulton, Trowbridge and Haigh....................... 133
io. 27. Some fundamental types of equilibrium, in a system with two
dependent variables............................ j4g
FIG 28. Map of integral curves for the Ross malaria equations.' Repro-
1923
A' J" L tka'
Am' JoUr- Hygiene> January Supplement,
FIG 29 Model of surface corresponding to figure 28 '."Reproduced from
ff^ A ?'
JoUr' Hygiene> January Supplement, 1923 ......... 150G. dO. Age distribution in population of molecules of two substances in
monomolecular chemical equilibrium. Reproduced from A. J. Lotka,Am. Jour. Science, 1907, p. 208.......
155
n
contents
"
of
''
* of srasshoppe
FIG 34. Seasonal food habits of the Meadow" Lark.' After H.c'.B^ant* '
'
^
of
FIG. 39. Key to figure 38 .............................................. 177
205
LIST OF ILLUSTRATIONS
FIG. 45. Periodic classification of the elements, showing division into
petrogenic elements and metallogenic elements. After H. S.
Washington ^
7
FIG. 46. Circulation of the elements in nature. The water cycle 215
FIG. 47. Circulation of the elements in nature. The carbon cycle 226
FIG. 48. Circulation of the elements in nature. The nitrogen cycle 230
OQTJ
FIG. 49. Organic nitrogen circulation
*0i
FIG. 50. The rise of the saltpeter industry
235
FIG. 51. The rise of the fixed nitrogen industry
24
FIG. 52. Circulation of the elements in nature. The phosphorus cycle. . . 247
FIG. 53. The Soxhlet extraction apparatus
253
FIG. 54. Circulation of the elements in nature. The sodium chloride
cycle ;
m
FIG. 55. Uranium and its products of radioactive disintegration 263
FIG. 56. Equilibrium polygon for Radon (radium emanation) in contact
with its disintegration products
266
FIG. 57. Relative abundance of the elements. Diagram according to
W. D. Harkins, based on analysis of meteorites. (Jour. Am. Chem.
Soc., 1916, p. 863)
271
FIG. 58. Equilibrium polygon for the human species and some of the species
on which it depends for its food supply 277
FIG. 59. The Yellow Shark and some of his relatives, as an example of
possibly orthogenetic development 296
FIG. 60. Law of urban concentration 308
FIG. 61. Relation between rate of reproduction in Drosophila (fruit fly)
and density of mated population. After Pearl and Parker 309
FIG. 82. Relation between mean length of life and population density in
Drosophila. After Pearl and Parker 310
FIG. 63. Hyperbolic curves obtained by plotting as ordinates the number
of genera 1, 2, 3, . . . n species, and as abscissae the number n of such
species. After J. G. Willis 314
FIG. 64. Relation between number and size of genera of all flowering
plants, plotted logarithmically. After J. C. Willis 315
FIG. 65. Relation between number and size of Rubiaceae, plotted logarithmically.
After J. C. Willis 315
FIG. 66. Relation between number and size of genera of chrysomelid beetles,
plotted logarithmically. After J. C. Willis 316
FIG. 67. Solubility of carbon dioxide in water, expressed in volumes of COj
measured at normal temperature and pressure, per volume of water.
After G. W. Martin 318
FIG. 68. The mill-wheel of life 334
FIG. 69. Mechanical walking beetle, exhibiting the several characteristic
elements of the correlating apparatus 341
FIG. 70. The evolution of man's means of transportation 367
FIG. 71. Growth of American Railways 369
FIG. 72. How the future enters into the determination of the motion of the
walking mechanical beetle, thus imitating purposive action (teleology)
382
LIST OF TABLES
Table 1. The program of physical biology 53
Table 2. Growth of the population of the United States 67
Table 3. Growth of bacterial colony. After H. G. Thornton 70
Table 4. Growth of sunflower seedlings. After H. S. Reed and R. H.
Holland 74
Table 5. Course of parasitic invasion of insect species, according to W. R.
Thompson 86
Table 6. Tabular survey of different modes of interdependence of biological
species 98
Table 7. Example of normal age distribution 113
Table 8. Relation between birth rate per head b and death rate per head d
in a population with normal age distribution 117
Table 9. Growth of steers, and feed consumed 134
Table 10. Survey of methods of marine biological census 178
Table 11. Principal components of the earth's surface crust 185
Table 12. Distribution of gases in the atmosphere at different levels 188
Table 13. Constituents of the atmosphere, at earth's surface and in toto . . 190
Table 14. Composition of the Ocean 191
Table 15. Comparison of air and aquatic atmosphere 192
Table 16. Composition of lithosphere 194
Table 17. Composition of human body 197
Table 18. Composition of living organisms. After H. F. Osborne 198
Table 19. Comparison of composition of blood serum and sea water 201
Table 20. Moisture contents of foods 211
Table 21. Discharge of the World's rivers 214
Table 22. Progress of by-product coke ovens in the United States 233
Table 23. Growth of the saltpeter industry 237
Table 24. Uses of saltpeter 237
Table 25. Meteoric rise of the nitrogen fixation industry 242
Table 26. Supply of plant foods in the soil 257
Table 27. Rate of participation of the elements in the cycle of nature. . . 258
Table 28. Radioactive equilibrium of radium in contact with its disintegration
products 265
Table 29. Radioactive equilibrium of radon (radium emanation) in contact
with its disintegration products 268
Table 30. Geological time table. After C. Schuchert 269
Table 31. World's production of population and its accessories. After
R. Pearl 278
Table 32. Association of high blood pressure with over-weight 295
Table 33. Distance travelled in one hour by different modes of conveyance 368
Table 34. Man's correlating apparatus, native and artificial contrasted. . 411
Table 35. Man's correlating apparatus (continued). The Adjusters 413
Table 36. Classification of the sciences 423
ANALYTICAL SYNOPSIS OF CHAPTERS
PART I. GENERAL PRINCIPLE
Chapter I. Regarding Definitions. Definitions are arbitrary, 3 But are
governed by considerations of expediency, 3 The problem of expediency in
framing definitions is not always a simple one, 3 Pseudo-problems arising
from failure to recognize the arbitrary character of definitions: Hunting the
Jabberwock, 4 Subjective discontinuities introduced by the senses, 5
Examples: Colors, e.g. blue, green; light and heat waves; distinction between
animals and plants; between biological species; between living and non-living
matter, 5 To such abrupt subjective divisions there may correspond no objective
discontinuity in nature, 5 Definitions in Biology, 5 Vitalism versus
Mechanism; merely a question of terms, 7 Herbert Spencer's "proximate
definition" of life, 7 Inadequacy of Spencer's definition, 7 Sir Edward
Schaefer's standpoint, 8 Line of division recedes with increasing knowledge
8 Alleged characteristics of living matter: Growth from within 8 Chemical
growth as distinguished from physical growth of crystals, 10 Growth from
unsaturated solution, as distinguished from growth of crystals from supersaturated
solution, 10 "Selective" growth, 10 Reproduction, 11 Vital
Force, 13 Physical chemistry of structured systems, 13 Geometrical element
lacking in physical chemistry of today, 13 Systems ordinarily considered are
either structureless or of simple structure, 14 This absence of structural
features in physico-chemical systems is due to subjective, not objective reasons,
14 Due in part to convenient arbitrary restrictions, 15 Relation
between growth, environment and structure, 15 The laws of chemical dynamics
in structured systems will be the laws that govern the evolution of a
system comprising living organisms, 16 Application to Biology: The organism
as a structured physico-chemical system, 16 The travelling environment,
milieu inte"rieur, 17 Increasing independence of organism of its remote environment,
18 The policy of resignation: Abandoning the attempt to define
life, 18 Parallels in the history of science : Abandonment of the attempt to
prove Euclid's twelfth postulate led to new systems of geometry, 18 Abandonment
of attempts to build perpetual motion machines was equivalent to
recognizing the law of conservation of energy, 18 Abandonment of the attempt
to detect the earth's motion through the ether is the foundation of the
modern theory of relativity, 18 The ideal definition is quantitative, 19
Desirability of establishing a quantitative definition and conception of evolution,
19.
Chapter II. Evolution Defined. Definition should conform as far as possible
to common usage of the term, 20 Analysis of common conception of evolution,
20 Evolution is history, but not all history is evolution, 20 Systems in
purely periodic motion would not be said to evolve. They repeat in endless
xxi
XXil ANALYTICAL SYNOPSIS OF CHAPTERS
succession the same series of events. In an evolving system each day is unlike
any other day, 21 Evolution not a mere changeful sequence, 21 Abortive
attempts to formulate the direction of evolution, 21 These attempt definition
in terms of a single component, 22 Such definitions are foredoomed to
failure, a successful definition must be framed in terms of the evolving system
as a whole, 22 Evolution is the history of a system in the course of irreversible
transformation, 24 Scope of this definition: What it excludes; what it
includes 24 The line of division depends on the nature and extent of our
knowledge regarding the system, 25 This is in harmony with the fact that
problems of evolution are largely problems of probability, 25 All real transformations
are irreversible, hence all real history is evolution, 26 What
then is gained by the definition? 26 It indicates the direction of evolution as
the direction of irreversible transformations, the direction of increasing
entropy, 26 Example of pendulum. Irreversible feature introduced by frictional
force, 27 Inertia-free or completely damped systems, 28 Accelerations
vanish with velocities, 29 Velocities are single-valued functions of configuration,
29.
Chapter III. The Statistical Meaning of Irreversibility., Apparent irreversibility
(progressiveness in time) of certain theoretically periodic processes,
30 Their periodicity, with eventual return to initial state, never observed in
practice, 30 Explanation of this discrepancy: Macroscopic model illustrative
of irreversibility of gaseous diffusion, 30 Return to initial state possible, but
exceedingly rare (highly improbable), 31 The model is competent to illustrate
also the highly improbable event of return to initial state, 32 Dynamical
theory indicates not only occasional but periodic return to initial state, 32 A
second model illustrates this also, 32 Evolution aspassage from less probable to
more probable states, 35 Inadequacy of this "principle" : it is indefinite in failing
to specify the characteristic with respect to which probability is reckoned;
and it is incomplete in failing to draw attention to certain energy relations,
35 Irreversibility is relative, depending upon the means naturally available
or arbitrarily permitted to operate upon the system, 35 Significance of this
m organic world: Macroscopic irreversibility of diffusion processes in nature,
36 Need of a method of mathematical analysis to deal with cases intermediate,
in specified degree, between the following two extremes: (a) Wholly indiscriminate
(pure chance) operation upon material in bulk, (b) Wholly
determinate operation, with nothing left to chance, upon materials discriminated
and acted upon in detail, piece by piece, and circumstance by circum-
stance,
_36--This
method must take account of degree of perfection of the
mechanical and psychic equipment by which each organism reacts upon its
environment, 37-Senses as a means of overcoming chance, 37-PhysicaI significance
of our subjective sense of forward direction in time, which finds no
expression in the differential equations of pure dynamics, 37-This subjectivetune sense may be related to the influence of initial conditions in dynamics,38-But the direction of evolution seems related rather to that directedness in
Zdltl
18
f+rCt
TtiC f aperi dic r S6emingly aPeriodic ?**, 38Inadequacy
of thermodynamic method, 39-The linking of evolution with the
ANALYTICAL SYNOPSIS OF CHAPTERS XX111
concepts of thermodynamics and statistical mechanics is instructive as suggesting
a conception of the direction of evolution, the direction of increasing
entropy, increasing probability, 39 This point of view, however, is inadequate
for application to concrete cases of organic evolution, because data are furnished
in terms unsuited to the methods of thermodynamics, 39 Neither are
existing methods of statistical mechanics, as applied to molecules and the
like, helpful; the instrument is ill adapted to the scale of the object, 39 New
method needed, that shall accept its problems in terms of biological data, as
thermodynamics accepts its problems in terms of physical data; a General
Theory of State, an "Allgemeine Zustandslehre "
39.
Chapter IV. Evolution Conceived as a Redistribution. Evolution viewed
as a redistribution of matter among the components of a system, 41 System
described by statement of mass of each component, and indication of value of
certain parameters, 41 Analytical expression of history of system given by
relations or equations established between the variables and parameters denning
the state of the system, 41 Fundamental equations usually simplest
in form of differential equations, 42 Particular form of equations of evolving
systems, 42 General form of equations of evolving system, 43 Equations
as applied to life-bearing system, 43 Definition of the components arbitrary
but conclusions relate to the components as defined, 44 Relation of evolution,
as here conceived, to the problem of the origin of species, 44 Inter-group
and intra-group evolution, 44 Analytical indication of intra-group evolution,
45 Fundamental equations resemble in form the equations for an inertia-free
or completely damped system, 47 Fundamental equations, as here given,
may not cover all cases, but are at any rate of very wide scope, 47 Equations
interpreted to include possible lag or lead effects, 47 Singular implications of
lag and lead effects; possible relation to phenomena of memory and will, 48
Appearance of lag and lead effects in equations may, however, be spurious, 48.
Chapter V. The Program of Physical Biology. Systematization and
division of subject, 49 General mechanics of evolution, 49 Macro-mechanics
and micro-mechanics, 50 Statistical mechanics as the connecting link, 50
Stoichiometry, the study of mass relations (material transformations), 50
Energetics or Dynamics, the study of the energy transformations, 50 Kinetics
and Statics, 51 Equilibrium and steady states, 51 Moving equilibria, 51
Displacement of equilibrium; Le Chatelier's principle, 52 Sociological
analogues of forces and "quasi-dynamics" (economics), 52 The term Physical
Biology to be used to cover the territory indicated in this chapter, 52 Methods
of obtaining data, 52 Chart of Program of Physical Biology, 53 Methods of
elaborating data, 54.
PART II. KINETICS
Chapter VI. The Fundamental Equations of Kinetics of Evolving Systems.
General case, 57 Some implications of the fundamental equations in their
general form, 57 Equations of constraint, 58 Elimination of variables.
Introduction of constants A, 58 Evolution with parameters P, Q and A
constant, 58 Equilibria or steady states, 59 Number and character of
XXIV ANALYTICAL SYNOPSIS OF CHAPTEES
equilibria. Example: Fly population, 59 Fundamental equations transformed
by introduction of excess of actual masses over equilibrium masses in
place of the former, 60 Expansion in Taylor's series, and solution in exponential
series, 60 Characteristic equation for exponential coefficients, 60 Significance
of sign and character of roots X of characteristic equation, 61 Stability
of equilibrium, 61 Mode of approach to equilibrium: Aperiodic and
periodic (oscillatory) type, 61 Analytical confirmation and extension of a
passage in Herbert Spencer's First Principles, 61 Zero or negative roots of the
equilibrium equation: Unfit species, 62.
Chapter VII. Fundamental Equations of Kinetics (Continued). Special
Case: Single Dependent Variable. Law of population growth, 64 Population
of United States, 66 Stability of equilibrium, 67 Experimental populations,
69 Diminishing population, 70 Growth of individual organism, 71
Autocatakinesis, 76.
Chapter VTII. Fundamental Equations of Kinetics (Continued). Special
Cases: Two and Three Dependent Variables. Interdependence of species,
77 Several types of interdependence, 77 Analytical characteristics of those
types, 78 Effect of these characteristics upon the nature of the solution of the
equation of Kinetics for two variables, 79 Concrete examples, 5 Martini's
equation for immunizing diseases, 79 A special case noted by Watson, 81
The Ross malaria equations, 81 Example in parasitology, 83 Thompson's
treatment of the case, 83-Objections to this, 87-Treatment of case by general
equation of kinetics, 88-Nature of the solution: it represents a periodic or
oscillatory process, 90-Comparison with observed facts according to L O
Howard^
90 Annihilation of one species by another, 92-Case of three depen-dent variables, 94-Relation of this to a practical problem in sea fisheries, 95Replaceable
and irreplaceable components, 95-Limiting factors, 97-Liebig
s law, 97 Chart of types of interdependence of biological species 98
* 2!r
K' A*dysis of the Growth Unction. The form of thi function
A+ ? T l
aggregates 100-Demographic functions, 101-Survival
M?^'^ i T
r
,
mal Ege diataibu*^ HO-DemograPhic relations in
normal population, 115-Rate of increase per generation, 118-Effect of
selects slaughtering 119-Intelligence as a discriminating agency 20
(With MendeH J B. S
Chapter X. Further Analysis of the Growth Function. Adjustment of the
e
teful
produce, 132-nrnHnna 1QO T?~i j.- j
' -^ vvu^o uiv, muuiaiS ICept tor
p^ss^s^it^s^SrPS-^;-- -:s-i;
ANALYTICAL SYNOPSIS OF CHAPTEBS XXV
from standpoint of domestic species is high efficiency from human standpoint,
135 Network of chains of interrelated species, 136 Transformation factors
and their economic significance, 137.
PART ill STATICS
Chapter XI. General Principles of Equilibrium. Equilibria and stationary
states, 143 Scope of Statics, 143 Kinetic, dynamic, and energetic definition
of equilibrium, 143 Our chief interest here in stationary states not true
equilibria, 145 General equilibrium condition, 145 Different types of equilibria,
146 Illustration: Malaria equilibrium according to Ross, 147 Metastable
equilibrium, 151 Exceptional cases, 151.
Chapter XII. Chemical Equilibrium, as an Example of Evolution under a
Known Law. Case of simple balanced reaction, 152 Relation between
"birth" rate and "death" rate of molecules, 153 The survival factor p (a)
and life curve of molecules, 153 Reaction constant as a a force of mortality,
154 Chemical reaction as a case of survival of the fittest, 154 Analogy of
chemical reaction to course of events in population of mixed biological species,
155 Presumptive mechanism of chemical reaction (Baly), 155 The fugaceous
transitional state intermediate between chemical compounds (Schonbein),
156 Laws of thermodynamics, as determinants of the end state in an equilibrium
reaction, are, in this case, the Law of Evolution of the system, 157Natural
laws conveniently expressed as maximum or minimum principles, 157Law
of organic evolution may be expected to take this form, 158 Law of
chemical evolution is framed in terms of the system as a whole ;
law of organic
evolution must undoubtedly also be thus framed, 158 Law of equilibrium
expressed in form of a minimum principle, 158.
Chapter XIII. Inter-Species Equilibrium. Equilibrium condition in more
particular form, 161 Numerical illustration, 162 Economic relationship of
coefficients appearing in equilibrium condition, 163 Sources of information
regarding biological equilibrium :
Biological surveys, 164 Analysis of stomach,
contents, 166 Intra-species equilibrium (with Mendelian inheritance), 170.
Chapter XIV. Inter-species Equilibrium: Aquatic Life. Special occasion
for demological studies of aquatic population, 171 Fishes as natural dragnets,
171 Aquatic food chains in relation to human food, 2 Importance of aquiculture
for the future, 172 Loss of fertilizer through modern methods of
sewage disposal, 172 Partial restoration of such material to human food supply
through fisheries, 173 Different methods of census of marine population :
dragnet; bottom samplers; recatching of marked catches; centrifuge; dilution
culture, 173 The nannoplankton, 173 Work of Petersen in Kattegatt, 174
Quantitative estimates of principal groups of the marine population of Kattegatt,
and their interrelation, 175 Summary of methods of marine biological
census (Table), 176 Food chains, 176 Necessity of occupying ourselves with
the more remote links of our food chain, 177 "Feeding our foods,
" a species
of symbiosis, 180 Agriculture and aquiculture, as mining and as manufacturing
industries, 180 Resources but recently tapped, and others still untouched,
180 Food chains in aquatic species, 180 Wastefulness of long food chains,
XXVI ANALYTICAL SYNOPSIS OF CHAPTEES
181 Economic value of shell fisheries owing to simple food chain coverting
vegetation directly into human food (oysters, etc.); 181 Primary, secondary
and tertiarv foods, 171 Cycles and the circulation of the elements in Nature.
183.
'
Chapter XV. The Stage of the Life Drama. The tripartite world, 185
Atmosphere, 185 How the earth holds her atmosphere, 185 Cosmic losses
from the atmosphere negligible, 188 Cosmic accessions to the atmosphere,
192Hydrosphere, 192 Aquatic atmosphere, 193 Lithosphere, 193 Cosmic
accessions to lithosphere, 195 Composition of earth's crust, 195 Relation to
composition of organism, 197 Similarity in composition of sea water and
blood serum, 201 Significance of this as regards aquatic origin of terrestrial
fauna, 202 Chemical correlation in soil and in organism, 204 Accessibility
of valuable earth constituents, 206 Accessibility not in any direct relation to
abundance, 206 H. S. Washington's classification of petrogenic (rock-forming)
and metallogenic (ore-forming) elements of the periodic table, 207
Dissipating and concentrating processes in Nature, 208.
Chapter XVI. The Circulation of the Elements : The Water Cycle. Circulation
of elements vaguely realized by ancients, 209 Water requirements of
human body, 210Water requirements of plants; rainfall as limiting factor,
211 Sources of supply, 213 Quantitative estimates of several items in water
cycle, 213 Water cycle diagram, 215 Fraction of total water circulation taking
part in life cycle, 216.
Chapter XVII. The Carbon Dioxide Cycle. Combustion as an essential
feature of the life process, 218 Analogy of flame arising from spark and life
arising from germ, 218 Rarity of ignition except from pre-existing flame analogous
to apparent impossibility of life originating except from preexisting life,
218 Oxygen as an inorganic food, 219 Plant nature of man in his attitude
toward this inorganic food, 219-The carbon cycle, 220-Carbon as the organic
element, 220 Source and gate of entry into the organic carbon cycle, 220
Some estimates of quantities involved in the carbon cycle, 220 Interrelation
of green plants and animals, 221 Carbonization of dead vegetable matter and
its significance for the industrial civilization of the present era, 221The present
an atypical epoch: man living on his capital, 222-Attempts to establish
the balance sheet of the earth's carbon economy, 222 Absorption of C02 in
weathering of rocks, 223-Formation of CO2 by burning of coal, 224-The ocean
as an equalizer regulating the C02 content of the air, 224-Unccrtainty as to
net loss or gam of total C02 in atmosphere, 225 The oxygen cycle, 225-Oriinof atmospheric carbon dioxide and oxygen, 225-Carbon cycle diagram 226Loss
of oxygen from the atmosphere, 228.
Chapter XVIII. The Nitrogen Cycle. Natural demand and supply 229Seemmg
abundance of nitrogen illusory, 229-Nitrogen cycle diagram,' 230-Cate of entry into nitrogen cycle, 232-Leak of nitrogen out of circulation
2ipLoss
of nitrogen in
combustion, distillation and coking of coal 233-Sk-mficance and rapid increase of by-product recovery type of coke ovens, 233-Other losses of combined nitrogen, 234-Accessory sources of combined nitronSJ
f ^'
234
rHuman ^erference in nitrogen cycle, 236-Exploitation
of guano deposits, 236-CMlean nitre beds, 236-Consumption of
ANALYTICAL SYNOPSIS OF CHAPTERS XXV11
saltpeter as fertilizer and otherwise, 237 Origin of nitre beds, 238 Industrial
nitrogen fixation processes, 239 Birkeland and Eyde nitric acid process.
Cyanamide process. Haber ammonia process. Biicher cyanide process,
239 Combination of Haber process with Solvay soda process, 239 Ostwald
oxidation process, 241 Meteoric rise of nitrogen fixation industry, 241 Its
ethnological significance, 241 Economic and energetic significance of concentration,
e.g. of supply of combined nitrogen, 243 Localized sources of concentrated
supplies as centers of attraction in man's economic activities, 244
Total circulation tends to increase, 245.
Chapter XIX. The Phosphorus Cycle, Immobile elements, 246 Natural
phosphorus supply of soils, 246 Phosphorus cycle diagram, 247 Leakage of
phosphorus from circulation, 248 Phosphate rock and the migration of phosphorus,
248 The role of fish and birds in the phosphorus cycle, 249 Soil losses
of phosphorus, 250 Phosphate slag as fertilizer, 251.
Chapter XX. Cycles: Conclusion and Summary. Circulation of chlorine
and the alkalis, 252 The sea and the sun as a Soxhlet extractor, 252 Sodium
chloride cycle diagram, 254 Differential behavior of sodium and potassium
in the process of extraction, 255 Comparative rarity of potash in soils, 255
Effect of World War on potash market, 256 Circulation of sulphur, 256
Circulation of iron, 256 Summary of cycles, 257 Supply of plant food in soil,
257 Rate of participation of elements in cycles of nature, 258.
Chapter XXI. Moving Equilibria. Principle of continuity, 259 Equation
of slowly moving equilibrium; first approximation, 259 Higher approximations,
260 Special case :
pace set by slowest member in a chain, 261 Radioactive
equilibrium, 261 Equilibrium polygon, 266 Extinction of unadapted
species, 266 Example of inaccuracy of first approximation, 268 Radioactive
chains as cosmic clocks, 268 Geological time table, 269 Origin of elements
and ultimate genesis of organisms, 269 Relative abundance of the elements,
271 Terminal stages of the earth's evolution: Geophysics and geochemistry,
273 Joly's theory of periodic melting of the earth's crust, 275 Organic
moving equilibria, 276 Equilibrium polygon, 277.
Chapter XXII. Displacement of Equilibrium. Perfectly general case of
influence of change in parameters will not here be considered, 280 Principle
of Le Chatelier, 281 Some common misstatements of the principle, 282
Early essayed application to biology, 283 Conditions of validity of the principle,
284 Extension of scope of rigorous applicability, 286 Area and rent,
288 Discussion of displacement of equilibrium independently of Le Chatelier's
principle, 289 Case I: Displacement of equilibrium between food and feeding
species, 289 Case II: Change of circulation through moving cycles, 292
Some significant cases of instability, 294 Vicious circles, 294 Cumulative
cycles simulating orthogenesis, 296 Benign cycles, 297.
Chapter XXIII. The Parameters of State. Topographic parameters, 300
Intensity factor of energy, 303 Simpliest examples of topographic parameters
:
volume, area, 301 Complexity of topography in organic evolution,
301 Simplification of problem by "substituting ideal upon which it is possible
to operate, for intractable reality", 302 Empirical study of biogeography and
ecology, 302 Conjugate parameters, 303 The intensity law in organic and
xxviii ANALYTICAL SYNOPSIS OF CHAPTERS
economic systems, 303-Rent as a measure or index of population pressure
304 But population pressure exists independently of rent, e.g., in species
other than man, 304-Distant analogy of law of population pressure to gas law,
305-Law of urban concentration, 306-Biological background of population
pressure, 307 Influence of population density on rate of reproduction (Pearl
and Parker), 308 Influence of population density on duration of life (Pearl
and Parker), 309 Topographic parameters during period of diffusion, 311
Willis' theory of Age and Area, 311 Climatic parameters, 317 Their laboratory
investigation (Pearl and Parker), 319 Parameters of state and the
analytical condition for equilibrium, 319 Thermodynamic analogy, 320
Inversion of typical problem of thermodynamics, 321 Systems of QuasiDynamics,
321.
PART IV. DYNAMICS
Chapter XXIV. The Energy Transformers of Nature. The fundamental
equations of Kinetics do not exhibit any explicit reference to dynamical or
energetic relations, 325 But certain of the components S are energy transformers,
325 Fundamental characteristics of energy transformers, 325
Cyclic working; output and efficiency, 326 Thermodynamic law of maximum
output, 326 Reversible and irreversible processes, 327 Composite and
coupled transformers, 327 Accumulators, 328 Chemical accumulators, 328
Growth, 328 Law of growth, 328 Anabions and catabions, 329 Systems of
transformers, 329 Plant and animal as coupled transformer, 330 The World
Engine, 331 Share of sun's energy that falls to different constituents of world,
331 Share falling to organic circulation, 331 Relation of transformer cycle to
circulation of the elements, 334 Influence of limiting factors upon working of
world engine, 334 Evolution of the World Engine, 335.
Chapter XXV. Relation of the Transformer to Available Sources. Distributed
and localized sources of energy, 336 Random and aimed collisions, 337
Negative correlation, 337 The correlating apparatus, 338 Component elements
of correlating apparatus: Depictors, Receptors, Elaborators, Adjusters,
339 Receptor-effector circuit begins and ends in environment, 340
Significance of this, 340 Correlating apparatus not peculiar to living organisms,
340 Mechanical imitations of living beings (automatons) 341 Chess
as a conventional model of the battlefield of life, 343 The biological contest
considered in the light of the chess analogy, 343 Topographic map, centers of
mobility and centers of influence as the elements of the game, 343 Zones of
influence, 344 Collisions or encounters, 344 Zones of mobility, 345 Analytical
statement of problem of organic conflict, 345 The behavior schedxile,
346 Specific productivity, 347 Effect of change in zone pattern, (Intraspecies
evolution), 348 Biologic relation of economic value, 350 Effect of
change in behavior schedule, 350 Rigid or automaton type and elastic or
free-choice type of behavior schedule, 350 Relation between ideal and
actual organism, 352 Effect of small departure from perfect adjustment,
353 Relation of economic value to physical energy, 354 Economic conversion
factors of energy, 355 General or aggregate effect of individual
ANALYTICAL SYNOPSIS OF CHAPTERS XXIX
struggles for energy capture, 356 The law of evolution adumbrated as a law
of maximum energy flux, 357- Statistical mechanics of a system of organisms,
358 Mean free path, 358 Frequency of collision and capture, 359
Influence of size of organism, 359 Curves of pursuit, 360 Random motion
under a bias, 360 Use of models, 360.
Chapter XXVI. The Correlating Apparatus, 362 Receptors, 363- Artificial
receptors, 364 Significance of these in evolution of modern man, 364 Effectors,
366 Artificial effectors; industrial evolution, 367 Singular effects of industrial
evolution, 368.
Chapter XXVII. Extension of the Sensuous World Picture. The Elaborators,
371 The scientific world picture :
Systems of coordinates, 372 The ego
as a coordinate reference frame, 372 The ego immaterial, 373 Interpenetration
of the egos, 374 Where is Mind?, 375 Fundamental premises and implicit
assumptions, 376 Difficulty of shaking off preconceived premises, 377 Obstacle
which this raises to understanding, 377 The communicators, 378
Orthogenesis in human evolution, 378 Orthogenesis does not suspend selection,
380 Importance of curiosity in evolution, 380.
Chapter XXVIII. The Adjusters. Mechanistic and teleological interpretation
of adjusters, 381 Significance of the future in the operation of adjustors,
382 The future that may be and the future that will be, 382 The
doubtful cases, 383 Final causes or purposes not usually postulated when
sufficient account can be given of events in terms of efficient causes (e.g., in
terms of mechanistic explanation), 384 Adaptive adjustment of tastes, 385
Spencer's hedonistic principle, 386 Genuine utility for social service, 386.
Chapter XXIX. Consciousness. Relation of consciousness to physical
conditions, 388 Conditional relations, 388 A fundamental hypothesis admitted,
389 Consciousness is closely bound up with life processes and structures,
390 Consciousness dependent on metabolism. The personal element,
391 Consciousness possibly a general property of matter, 392.
Chapter XXX. The Function of Consciousness. The contents of consciousness
determined by past and present bodily states, 394 Operative relations of
consciousness to physical conditions, 394 Not only knowledge but motive
required in the working of the human organism, 394 Motivation in lower
organisms appears fatalistic (simple tropisms), 395 Purposive action, 395
Dynamic psychology, instinctive drives to action, 396 Individual traits. Instinct
of workmanship and self-expression, 396 Influence of special aptitudes,
399 The industrial and the personal problem of satisfying instincts, 400.
Chapter XXXI. The Origin of Consciousness in Living Organisms. Why
has nature resorted to consciousness as means for effecting adaptive reactions
of organisms? 402 The problem of psycho-physical parallelism, 402 The
double aspect theory of consciousness, 403 Physical analogies, 403 Physicochemical
theory of consciousness as a state related to the transitional state of
molecules in chemical reaction, 403 Argument of simplicity of structure
may be invoked in favor of this theory, 404 Origin of consciousness, 404
Some elementary forms of consciousness perhaps a general property of all
matter, 404 In that case not consciousness has been evolved, but only a particular
type of consciousness, a consciousness integrated around an ego, 404.
XXX ANALYTICAL SYNOPSIS OF CHAPTEBS
Chapter XXXII, Energy Relations of Consciousness. Seeming conflict
between directive power of consciousness and determinate character of physical
events, 406 Possible explanations : First alternative possible inaccuracy
of laws of dynamics, 406 Second alternative singular orbits with indeterminate
motion, 407 Conception of Clerk Maxwell and of J. Boussinesq, 407
Third alternative possible influence of factors eliminated from equations of
dynamics, 408.
Chapter XXXIII. Review of the Correlating Apparatus. Tabular synopsis
and classification of Depictors or Informants, 410 Epictors or Transformants,
411 Adjusters, 412 Internal adjusters, 412 Tabular synopsis of adjusters,
413 External adjustors, 414 Anatomical and physiological adjustment of
social activities, 414 Economic adjustment of social activities, 415 Disadvantages
of latter form of adjustment, 415 Its advantages or necessity in the
human community, 415.
Chapter XXXIV. Conclusion; Retrospect and Prospect, The life struggle
in the modern community, 417 New character of this struggle :
competition is
principally within the species, 417 Organization of motives lags behind industrial
organization, 418 Philosophy as a necessary part of scientific enquiry,
418 Classification of Sciences in relation to Self and External World, 419
Bertrand Russell's viewpoint regarding object and subject, 420 This leads to
a natural division and classification of the sciences, 421 The ego as at once a
Knower and a Wilier; hence the reaction of knowledge upon the emotions, 421,
424 The poetry of science, 425 Significance of emotional reaction of knowledge
for our future evolution, 425 Evolution of the Self may have overshot
the mark, 426 Man's opportunity to influence the destiny of his species by
his own initiative, 427 Desirability of concerted action, 427 Of constructive
optimism, 428 Evolutionary value of nurture and tradition, 428 Fallacy
of cynical deprecation of potency of nurture, 429 Extension of Spencer's
hedonistic principle, 430 How it explains the appearance of design in nature,
430 Such design can be neither proved nor disproved, and may therefore be
made an article of faith, 430 Man may embrace the World Purpose for his
own, 430 Survival value of such attitude, 430 Orthogenesis in field of ethics,
430 Once more the reaction of knowledge upon the emotion, 432 Evolution
is not achieved by struggling against cosmic forces, but by adaptation to them
(Claude Bernard, Sir Charles Sherrington), 433 He that survives must be in
some measure a collaborator with Nature, 433 Evolution towards fusion of
personal will with natural law, 433.
I
GENERAL PRINCIPLES
CHAPTER I
REGARDING DEFINITIONS
Truth comes out of error more readily than out of confusion.
Bacon.
A definition is a purely arbitrary thing. If I choose to define a
triangle as a plane figure bounded by four sides and having four
angles; and if, also, I define a quadrilateral as a plane figure bounded
by three sides and having three angles, I shall run into no logical
conflicts; my geometry need in no wise depart from that of Euclid;
I shall need to make no changes in existing works on geometry,
beyond that of substituting throughout the word triangle for the
word quadrilateral, and vice versa.
But while a definition is in this sense, from the point of view of
logic, a purely arbitrary thing, while my definition of a triangle as a
four-sided figure may be admissible, it is by no means expedient.
Thus the definition of terms, which naturally forms one of the first
steps in the systematic treatment of any subject, may present no
particular problems of logic, but it does present certain problems of
expediency.
In the geometrical example cited, the unusual definitions given,
though quite permissible, are inexpedient for simple etymological
reasons. Such a choice of terms would be misleading, and, instead of
assisting the memory, would impose upon it an unnecessary burden.
In this case the application of the principle of expediency is obvious
to the point of being grotesque, the example having purposely been
chosen to illustrate the principle in drastic fashion.
But the framing of definitions at times involves more subtle considerations
of expediency, so subtle in fact, that they may be overlooked,
or misunderstood, and a problem which is, in truth, a problem
of definition, falsely masquerades as a problem of fact. Certain
pseudo-problems of science have owed their origin to a failure to
realize this circumstance. 1
1
On the other hand, some very fundamental advances of science are, upon
critical examination, found to rest essentially upon the establishment of a
3
4 ELEMENTS OF PHYSICAL BIOLOGY
The writer of the book of Genesis shows good judgment. Our
legendary forebear, the originator of the first biological system of
nomenclature, sees each creature first, and thereupon names it.
We have not always been equally wise. Sometimes we have tried to
invert the method; we have found or made a name, and then gaily
set forth on an expedition to discover the thing that should answer
to that name; we have hunted the Jabberwock. Forgetful of the
wisdom of Mephistopheles :
Derm eben wo Gedanken fehlen
Da stellt ein Wort zur rechten Zeit sich ein
we have given way to an inherent bias of the human ,mind described
in characteristic fashion by H. G. Wells2
:
. . . . when we have a name we are predisposed and sometimes
it is a very vicious predisposition to imagine forthwith something answering
to the name If I say Wodget or Crump, you find yourself passing
over the fact that these are nothings, .... and trying to think what
sort of a thing a Wodget or a Crump may be. You find yourself insensibly, by
subtle associations of sound and ideas, giving these blank terms attributes. 8
So the biologist of the past generation, finding in his native
vocabulary the words animal and plant, forthwith proceeded in an
effort to establish precise distinctions between animals and plants,
never giving any thought, it would seem, to the fact that these names
had already been parceled out generations ago, by "popular"
consent, by unscientific persons without any regard to fine distinctions.
There is clearly, here, the tacit assumption that because two
distinct words are found in the vocabulary, therefore two correspondingly
distinct things exist in nature. In point of fact, we know well
enough (though we may not at all times have this knowledge clearly
in the focus of our consciousness) that in nature many things form
finely graded series, with extremes at the two ends, extremes to which
judicious definition. A notable instance of this is the enunciation of the principle
of the survival of the fittest, which is essentially of the nature of a definition,
since the fit is that which survives. Regarding the epistemological
significance of definitions compare A. N. Whitehead and B. Russell, Principia
Mathematica 1910, vol. 1, p. 12.
*
H. G. Wells, First and Last Things, 1908, p. 32.
8
"Gewohnlich glaubt der Mensch, wenn er nur Worte
Es mtisse sich dabei auch etwas denken lassen."
REGARDING DEFINITIONS 5
^ our vocabulary has lent more or less definitely associated names, but
with no definite line of demarcation between. Examples of this are
\ innumerable. We speak of objects as being red, orange, yellow,
green, blue, violet, etc. There is nothing in nature to correspond to
such staccato classification of colors: the visible spectrum runs continuously
from a wavelength of about 8 x 10 ~*
mm. (extreme red) to
about 4 X 10~4
mm. (extreme violet). Cases therefore must necessarily
arise when we are in doubt whether to call a thing blue, or
green, for example; and such doubt can be resolved, if at all, only
by arbitrary definition. The question is not "what is green, and
what is blue," but, at best, "what shall we agree to call green, and
what blue."
It lies in the nature of the mechanism by which we enter into
possession of our knowledge, that problems of definition of this kind
*- arise. We are equipped with two separate and distinct senses, the
one responding to electromagnetic waves ranging from about 4 x 10
~4
to 8 x 10
~4
mm., light waves; the other to somewhat longer waves
otherwise of the same character, heat waves. Accordingly we have
two separate terms in our language light and heat, to denote two
phenomena which, objectively considered, are not separated by any
line of division, but merge into one another by gradual transition.
Here the question might be raised whether an electromagnetic wave of
a length of 9 x 10
~ 4
mm. is a light wave or a heat wave. The answer
is obvious: Call it what you please, it is merely a question of arbitrary
definition. We must beware of
.... that false secondary power
'*
By which we multiply distinctions, then
Deem that our puny boundaries are things
That we perceive, and not that we have made.
Wordsworth.
Definitions in Biology. The attempt to establish a rigorous distinction
between "animals" and "plants" may be similarly regarded.
Expediency demands that if these terms are appropriated for
exact scientific use, their sense, when so used, shall, if possible, be
reasonably near akin to the sense commonly associated with these
words. The difficulties encountered in seeking to establish a satisfactory
line of division between animals and plants were long regarded
as difficulties in a problem of fact. It was thought that some biologi*
cal principle must be sought which divided animals from plants.
6 ELEMENTS OF PHYSICAL BIOLOGY
The truth is, of course, that we may define "animals" and "plants"
any way we please- as for instance by reserving the term, plant for an
organism possessing cellulose' but whether such definition is "correct"
or "satisfactory" is not a question of biological fact, it is a
question of expediency. It is not a question whether there is any
definable difference between animals as a class and plants as a class,
nor what this difference is, but whether it is expedient to retain for
purposes of strict scientific classification the popular terms "animals"
and "plants," which were not originally founded upon any rigorous
examination of facts; and if so, where we should, by definition, draw
the line of separation.
When the problem is viewed in this way the difficulty of distinguishing
between animals and plants vanishes. In the case of the
higher forms of life it is easy to establish biological distinctions that
do not conflict with the popularly drawn lines of division. In the
case of certain lowly forms of life popular distinctions cannot exist,
since these forms are not known to the public except through biological
publications. And the biological line of demarcation we can,
by definition, draw arbitarily where we choose, or, better perhaps,
we may say that the terms "animal," "plant," do not correspond to
any fundamental objective distinction and, though conveniently
applied to certain common forms of living matter, are entirely
unnecessary
4
and only introduce difficulties of definition and classification
when applied to certain simple organisms. What difference does
it make whether we call Volvox a plant or an animal? Whether it
is a plant or an animal is merely a matter of definition, not a question
of biological fact.
Somewhat similar remarks apply to the narrower divisions into
which the biologist divides the world of living organisms. Disputes
as to what constitutes a species are fruitless. "A species is a thing
described as such." This is simply a matter of definition. If on
grounds of expediency one definition is preferable to another, it may
be well to urge its general adoption. But its adoption or rejection will
neither add nor subtract one jot from our stock of ascertained facts.
It is necessary to guard against the error of disputing about mere
words. Not always does this error strut about in such blatant form
as in the example quoted by Fechner: S. Sachs, in a book published
4
R. W. Glaser, Science, 1918, vol. 48, pp. 301-302: "We are justified at
present in not classifying viruses either wit|i plants or animals."
KEGARDING DEFINITIONS 7
in 1850, takes the astronomers to task for their presumptuous speculations:
"How do they know that the star they call Uranus is Uranus?"
If any one should think that in our day it is no longer necessary to
guard against errors of this kind (though less gross, perhaps), let
him consider such a question as this: Is not the perennial debate
between vitalism and mechanism a quibble about words? Is not the
whole situation summed up accurately in the words of L. J. Briggs:
5
"The mechanism of plant processes not at present explainable on a
physico-chemical basis would be termed by the vitalistic school
"vital," by the physico-chemical school "unknown"?
And in searching for the essential characteristics of life, those that
should finally and conclusively distinguish the living from the nonliving,
are we not just searching for the thing in nature that should
correspond to a word in our vocabulary? Are we not hunting the
Jabberwock?
Definitions of Life. The difficulty of giving a precise meaning to
the word life has been realized probably by everyone who has ever
seriously attempted a definition. Herbert Spencer remarks:
Classifications are subjective concepts, which have no absolute demarcations
in Nature corresponding to them .... Consequently, when we
attempt to define anything complex .... we can scarcely ever avoid
including more than we intended, or leaving out something that should be
taken in. Thus it happens that on seeking a definition of life, we have great
difficulty in finding one that is neither more nor less than sufficient.
Nevertheless he proceeds to establish his definition of life: "The
continuous adjustment of internal relations to external relations." 8
It cannot be said that Spencer has been very happy in this choice
of a definition or that he has been at all successful in avoiding the very
pitfalls which he himself so clearly points out. For obviously many
purely mechanical systems fall under this definition. It would, for
example, include a windmill provided with a device automatically
turning its arms into the most favorable plane according to the
direction of the wind. 7
Indeed, in a sense it is true of every physical
6
L. J. Briggs, Jour. Washington Acad. Sci., 1917, vol. 7, p. 89; compare also
E. M. East, Mankind at the Crossroads, 1923, p. 21.
6
Herbert Spencer, Principles of Biology, section 30.
7
Compare the following : "No one has yet succeeded in formulating a cleancut
definition of the limits of the reflex either at its lower or its higher extreme,
and perhaps no one ever will; for the whole list of behavior types, from machines
to men, probably form a closely graded series." C. J. Herrick: The
Evolution of Intelligence and Its Organs. Science, 1910, vol. 31, p. 18.
8 ELEMENTS OF PHYSICAL BIOLOGY
system that it "adjusts its internal relations to external relations."
For this statement simply implies that there is a tendency for the
establishment of equilibrium between a selected portion of a physical
system, and the remainder, the environment. Thus, for example, if
the system 2H2 + 2 is left to itself hi a suitable vessel at 1480C.8
H2
and one atmosphere pressure, the ratio
^rp:
which we may term an
"internal relation" of the system, assumes the value 0.0002. If
now the external conditions of temperature and pressure are changed
"FT
to 2929C. and one atmosphere pressure, the internal relation rrr-z
adjusts itself to the new external condition and acquires the value
0.11.
With better judgment than Herbert Spencer, Sir Edward Schafer9
frankly evades the definition of life. He remarks :
The ordinary dictionary definition of life is "the state of living." Dastre,
following Claude Bernard, defines it as "the sum total of the phenomena common
to all living beings." Both these definitions are, however, of the same
character as Sidney Smith's definition of an Archdeacon as "a person who
performs archidiaconal functions." I am not myself proposing to grapple
with a task that has proved too great for the intellectual giants of philosophy,
and I have the less inclination to do so because recent advances in knowledge
have suggested the probability that the dividing line between animate and
inanimate matter is less sharp than it has hitherto been regarded, so that the
difficulty of finding an inclusive definition is correspondingly increased.
It is, indeed, an elementary historical fact that, as knowledge has
advanced, the scope embraced in the term "vital" processes has
continually decreased, since Wohler took the first cut out of it in
1828 by the synthesis of a "vital product" (urea) in the laboratory;
and the field of known physico-chemical processes going on in
living organisms has correspondingly increased. For the rest, the
most uncompromising vitalist does not deny that some, at least, of
the processes going on in living matter are physico-chemical. Even
so fundamentally biological a process as the stimulation of an ovum
to development we have learnt to effect by purely physical means.
Alleged Characteristics of Living Matter. On the other hand
some of the features commonly ascribed to living matter as its peculiar
8
W. Nernst, Theoretische Chemie, 1913, p. 713.
9
E. A. Schaefer, Presidential Address at Dundee Meeting of Brit Assoc
Adv. Sci. 1912.
EEGAEDING DEFINITIONS
and characteristic attributes seem Irrelevant to the point of triviality.
"^his.remark applies particularly to the distinction sometimes claimed
for living matter, that it grows "from within," as distinguished from
crystals, which, in a suitable mother liquor, "grow from without."
There may or may not be many and profound differences between
a bacterial colony growing in a culture medium, on the one hand,
and on the other hand a mass of crystals growing in a supersaturated
solution. But whether the growth takes place from within or without
is merely an accident of structure. If a droplet of chloroform is
brought near to a glass particle coated with shellac, the drop flows
around the particle, engulfs it, absorbs the shellac coating and
finally rejects the "undigested" glass particle.
10
The droplet thus
grows "from within."
In point of fact "growth from within" is the rule and not the
exception in chemical systems. For what do we mean by growth?
We mean the increase of the mass of one component of a system at the
expense of another. It is precisely the same thing as that which
occupies the center of attention of the physical chemist, though he
does not ordinarily call it growth. In fact, he does not find it necessary
to give it any particular name, for, being accustomed to the use
Ci'YYI
of mathematical methods and symbols, he simply writes it -=- ,
rate
Cit
d /m\of increase of mass with time, or, more often, T.\ I,
rate of increase of
concentration (mass/volume) with time. And in homogenous
systems, at least, which (on account of their comparative ease of
theoretical and experimental treatment) figure prominently in the
physical chemistry of today, growth is necessarily from within.
Some writers (J. Loeb, The Organism as a Whole, 1916, p. 28)
have seen a characteristic feature, peculiar to living organisms, as
distinguished, for example from crystals growing out of a solution,
10 "Let it be clearly understood that this illustration is here quoted, not as
an example of life-like analogies in the world of non-living matter; nor as a
veiled suggestion that such a drop of chloroform represents even a modest
degree of success in the artificial imitation of life; nor yet again as an argument
that the conduct of amoeba can today be fully accounted for on a physicochemical
basis; this example was cited merely to show that "growth from
within" cannot be claimed as a distinguishing characteristic of living matter.
For further discussion of so-called simulacra vitae see McClendon, Physical
Chemistry of Vital Phenomena; Burns and Paton, Biophysics, 1921, p. 403.
10 ELEMENTS OF PHYSICAL BIOLOGY
in the fact that the latter grow by a physical process, the former by
chemical processes. Leaving aside the question as to whether there
exists any fundamental distinction between physical and chemical
processes, at most the point to which attention is drawn by these
authors would class living organisms with chemical, as distinguished
from physical systems, but would furnish no basis whatever for
separating organisms in a class by themselves from other chemical
systems. This is not saying that they are not in a class by themselves,
but only that the distinction suggested fails in effect.
It has similarly been urged, as a distinction between the growth of
a crystal and that of an organism, that the former will grow only in
a supersaturated solution of its own substance, while the latter extracts
from an unsaturated solution the substance needed for its
anabolism.
This is really the same distinction in another form. It may
distinguish the organism from the growing crystal, but leaves it m...
one class with any chemically reacting system whatever, since in the
case of the latter also there is "growth," i.e., formation of one or
more products of reaction, in a system which need not be physically
supersaturated in the narrow sense in which the crystallizing solution
is. In a wider sense11
the system may indeed be said to be supersaturated
with regard to a chemical substance that is formed within
it but in the same sense a system can probably be said to be supersaturated
with regard to the substance of a bacterial colony growing
therein.
Neither can we subscribe to the view set forth by J. Loeb (The
Organism as a Whole, 1906, p. 29), that the synthesis of specific
materials from simple compounds of non-specific character distinguishes
living from non-living matter. In every chemical reaction
specific materials are formed. In a mixture of hydrogen, chlorine,
and nitrogen, the hydrogen and the chlorine unite, leaving the nitrogen
on one side unchanged. This is merely a brutally simple example
of a universal fact. Chemical reaction is always selective. And if
"complexity" is to be made the characteristic of life
processes, then
the question immediately arises, what degree of complexity is required
to place a given process in the category of life processes?
"Namely in the sense that it is
metastable, that is, its thermodynamic
potential is not at a minimum.
EEGARDING DEFINITIONS 11
Reproduction. Another characteristic that has been cited by some
as exclusively peculiar to living organisms is the power of reproducing
their kind. "How, says Driesch in effect, can a mechanism provide
for its own reconstitution? No machine known to us is able to construct
another like itself, nor can it repair its own parts."
12
Undue emphasis on this alleged distinction between living and nonliving
machines seems ill advised, for two reasons. In the first place,
though it may be true that no man-made engine exists that performs
the functions of self-repair and self-reproduction, no one has ever
attempted, so far as I know, to demonstrate that no such engine can
be built. Anyone who should be disposed to regard this objection
as specious should reflect for a moment on the amazing development
in technical arts within the last thirty or forty years. Half a century
ago one might with equal justice have pronounced flight a fundamental,
essentially biological characteristic of birds, incapable of
duplication by man-made engines.
But in another, perhaps more significant respect, we must regard
as misplaced the emphasis sometimes laid on the power of reproduction
in organisms, and its absence in human artefacts. It is based on
an exaggerated conception of the part played by the parent in the
making of the offspring. This probably has its origin in the instance
of reproduction that to us is naturally of supreme interest, the reproduction
of man. As a mammal, the young human organism grows
within the parent body, and seems to us to be in some way fashioned
by the parent; this conception must be at the basis ofThe alleged !
distinction between organic reproduction and the incapacity of ;
'.
;
non-living engines to reproduce their kind, for without such conception
the comparison would lack all parallel. Now, in point of
fact, we need but call to mind the familiar hatching of a chick to
realize that the part necessarily played by the parent in the formation
of the young individual is really very restricted. The process in this
case goes on, for the most part, in complete isolation from the parent.
13
12
H. C. Warren, Jour. Philos., Psychol. and Scientific Method, vol. 13, 1916,
p. 36.
13
Compare E. G. Conklin, Heredity and Environment, 1918, pp. 99,45,
109: "The hen does not produce the egg, but the egg produces the hen
and also other eggs We know that the child comes from
the germ cells and not from the highly differentiated bodies of the parents,
and furthermore that these cells are not made by the parents' bodies but
IZ ELEMENTS OF .PJtilKlUAJLi
As for the initiation of cell division of the ovum, we now know that,
in some cases at least, this can be effected by ordinary physical
means.
Recent development in experimental embryology suggest a more
rational view of this process of self-reproduction of the living engine,
a view which strips it of at least some of its mystery, and which
certainly takes from it any force it might otherwise have had as a
basis for distinction between living and non-living matter. If, after
the first division of the ovum of a frog, the two cells are separated,
each will under suitable conditions develop into a separate and complete,
normal organism. These two organisms A and B are, in fact
twin brothers or sisters. No one would for a moment entertain the
thought that in this case A reproduces B, or vice versa. Now suppose
that in some way, after the first division, A alone grows into a complete
mature organism, while the single cell B remains attached to
it, say for six months. At the end of this time it is separated, and
stimulated to start its growth into a frog. We would ordinarily
describe this state of affairs by saying that A reproduced B as
offspring, that B was the child of A. In point of fact it is merely a
delayed twin brother or sister of its elder brother or sister A. u A
had little or nothing to do with the production of B; the latter grew,
very much in the same way as A grew in its own time. That nature
has evolved, in surviving races, this method of delayed development,
so as to stretch out the totality of living organisms in a long chain, a
succession in tune, is of course a fact of most fundamental importance,
the significance of which will deserve our profound contemplation.
One of its consequences has been to render possible a practically
infinite number of organisms, built from a finite and quite restricted
amount of matter, the same substance being used over and over
again, for it is literally true that we live on our forefathers. Had all
these cells have arisen by the division of antecedent germ cells
Parents da not transmit their characters to their offspring, but these germ cells
in the course of long development give rise to adult characters similar to those
of the parent."
14
The perhaps somewhat doubtfully authenticated cases of fetus in fetu,
"those strange instances in which one might almost say that a man may be
pregnant with his brother or sister," add a touch of realism to the discussion
here presented. For further data on this singular subject see G. M. Gould
and W. L. Pyle, Anomalies and Curiosities of Medicine, 1897, pp. 199 et seq.
Compare also in this connection, the phenomenon of pedogenesis; see for
example, G. H. Parker, Psyche, 1922, vol. 29, p. 127. ,
these organisms sought to grow simultaneously, their career would
have been stopped by lack of material. ,
If anyone should object that these reflections leave out of account
entirely the r61e of sex in reproduction, with all the complex phenomena
of the fusion of gametes, the mingling of chromosomes, and
biparental inheritance, the obvious reply is that these phenomena
are
now known to be less fundamental than they formerly appeared;
that
reproduction of an organism can very well take place without them;
and that therefore they may at most serve to distinguish certain forms
of life from non-living matter, but they cannot possibly be made the
basis of a distinction between living matter in general and that which
we commonly describe as non-h'ving. ,
Vital Force. If we have cause to hesitate in defining hie,_
still
more is it the part of wisdom to be very conservative in the coming
and use of such phrases as vital force, nerve energy, and the like.
Shall we not do well to follow the biblical example, and wait, to name
the animal, until it is physically present to our senses? Or, to pass
from legend to the world of scientific fact, let us borrow, if
we^
can,
the method of the physicist: He discovers that a quantity f mv
possesses certain important properties. Then, he proceeds to name
it: Energy, in particular, kinetic energy. But biologists have been
disposed sometimes to adopt the reverse procedure: they have
named a vital force, a nerve energy, a mental energy, and what not,
and now they entertain the pious hope that in due time they may
discover these "things." That there is something radically at
fault with such terms is evident from the fact that forces and energy
are magnitudes, and "to define a magnitude and to say how it is
measured are one and the same thing."
15 But who has ever told us
how to measure vital force18
and such like?
Physical Chemistry of Structured Systems. In the physical
chemistry of today structure, that is to say, geometrical configuration,
plays a subordinate rdle. For obvious reasons the theory of
chemical reaction in homogeneous, or in heterogeneous systems of
comparatively simple form, is more approachable than that of systems
which possess intricate structure, resulting in complicated mechanical
18
Nature, September 25, 1922, p. 405.
18
G. Bunge, in his Physiologic and Pathologic Chemistry, 1902, p. 1, remarks
: "I regard vital force as a convenient resting place where, to quote
Kant, 'reason can repose on the pillow of obscure relations.'" Curiously
enough this damning admission is made by an advocate of vitalism.
14 ELEMENTS OF PHYSICAL BIOLOGY
interactions of their parts, in accompaniment of chemical reaction.
In technical practise, too, reactions in homogeneous systems (solution,
gas) are common, and where there is heterogeneous structure,
this is usually of a form very simple as compared with the complex
biological structures.
But this comparative absence, from physico-chemical discussion,
of reference to structure, to geometrical features, is not due to any
inherent characteristic property of chemical systems, as contrasted
with the structurally complex organic systems: the reason for the
simplicity is to be found in ourselves. It is not a physical phenomenon
of the thing observed, but a psychological phenomenon in the
observer. Physical chemistry is still a comparatively young science,
and naturally the simpler phenomena have been sought out for first
attention. This is not because complex physico-chemical structures
do not exist, nor even because they are unimportant. On the contrary,
it is to be expected that the future will bring important developments
in this direction, as followed, for example by Sir William
Bayliss in his work Interfacial Forces in Physiology,
The rate of formation, the rate of growth, of a chemical substance,
is a definite function of its environment. In a structureless system
the nature and state of this environment is defined in comparatively
simple terms (e.g., by stating the concentration of each of the reacting
substances) .
But in a system possessing structure, the environment of a given
portion of the system depends on the structure, the topography of the
system, which, in general, will be variable with the time. In particular,
the structure may be such that a given substance or complex of
substances carries its own immediate environment around with it.
The rate of formation (growth) of that substance will then depend
largely upon the mechanical properties of those portions of the
system which accompany this substance or complex in its travels
through the system.
The complete discussion of a system of this kind may well fall
outside the scope of present day physical chemistry, not because it
is inherently foreign to that branch of science, but because no case
of this kind, sufficiently simple to invite discussion on a mathematical
and physico-chemical basis, has clearly presented itself.17
^Compare
W. M. Bayliss, Physiology, 1915, p. XI, "All that we are justified
in stating is that, up to the present, no physico-chemical system has beea-**l||
EBGARDING DEFINITIONS JLO
Yet there is absolutely nothing in such a case that in principle
places it outside the pale of physico-chemical science. It is largely
as the result of intentional selection of simple conditions that the
systems with which the chemist ordinarily deals (outside of biological
chemistry) are comparatively structureless.
We can, in fact, even now lay down certain general observations
with regard to structured physico-chemical systems.
Let us consider a system of this kind in which local conditions are
subject to variation from point to point and from instant to instant.
We fix our attention on some one component which requires for its
growth certain definite conditions of its immediate environment.
If this component is associated with a structure whose geometrical
and mechanical properties secure and maintain for it a comparatively
constant suitable environment amid the changing conditions of the
system, then that component will grow.
Furthermore, the several components will compete with greater or
less success for the material available for their growth, in proportion
as their structure is more or less perfectly adapted to secure and
maintain for them a suitable environment.
The chemical dynamics of such a system, that is to say, the laws
governing the distribution of matter among its several components,
may evidently assume a fundamentally different character from that
to which we are accustomed from our study of ordinary structureless
systems. For in these latter the arrangement and rearrangement of
matter within the system depends chiefly on chemical coefficients
(affinity coefficients), and scarcely at all on geometrical features.
In structured systems, on the other hand, there is the possibility
that geometrical and mechanical features may play the dominant
role. This possibility will present itself particularly in those systems
which receive a continuous or periodic supply of free energy, for
instance in the form of illumination. Here the advantage will go to
those structures that are adapted to direct available energy into such
channels as lead to the maintenance of the environment required for their
growth.
1*
But a little reflection shows that this is precisely the princimet
with having the same properties as those known as vital; in other words,
none have, as yet, been prepared of similar complexity and internal
coordination.
18 It should be observed that nothing has been said of life in describing the
system. The system may or may not comprise living organisms, the argument
pie which governs survival in the struggle for existence among living
organisms. Hence we may say:
The laws of the chemical dynamics of a structured system of the kind^
described will be precisely those laws, or at least a very important section
of those laws, which govern the evolution of a system comprising living
organisms.
For it is precisely structured systems of the kind considered above
that are presented to us in living organisms growing in an "environ-
ment."
Application to Biology. The several organisms that make up the
earth's living population, together with their environment, constitute
one system,
19
which receives a d
:1
y supply of available
energy from the sun.
Each Individual is composed of various chemical substances assembled
into a definite structure and capable of growth, i.e., of accretion
out of the environment by chemical reaction1
provided a suitable
medium or environment is offered.
Moreover, each mobile organism carries with it a travelling environment,
suitable for the growth of its substance. It maintains
this environment by virtue of the peculiar mechanical properties
associated with its structure, whereby it is enabled to turn to this
use, directly or indirectly, the available energy of the sun's light.
And while the travelling environment may not be absolutely constant,
remains the same. This suggests that a term, such as life, so vague that it
defies definition, is perhaps not likely to play an important part in any exact
argument; -we may, indeed, find it wholly unnecessary. It may, in time, in the
literature of exact science, meet with the fate of the word cause: a term of
rare and at best incidental occurrence in records of exact investigations.
19
This fact deserves emphasis. It is customary to discuss the "evolution
of a species of organisms.
3 '
As we proceed we shall see many reasons why we
should constantly take in view the evolution, as a whole, of the system [organism
plus environment]. It may appear at first sight as if this should prove a
more complicated problem than the consideration of the evolution of a part
only of the system. But it will become apparent, as we proceed, that the
physical laws governing evolution in all probability take on a simpler form
when referred to the system as a whole than to any portion thereof.
It is not so much the organism or the species that evolves, but the entire
system, species and environment. The two are inseparable.
"The organism, as Uexkull teaches us, must be studied, not as a congeries
of anatomical and physiological abstraction, but as a piece of machinery, at
work among external conditions," 0, C. Glaser, Science, vol. 21, 1910, p. 303.
it is more nearly so than the more remote portions of the system,
and keeps within such limits of variation as are compatible with the
survival of the organism or its species. A concrete illustration may
help to make this point clear. Many aquatic forms of life are constantly
bathed in a saline solution sea water. Their body fluids are
accordingly in equilibrium with this environment. Variations in the
salinity of their environment, if they exceed certain comparatively
narrow bounds, are apt to be fatal to such organisms.
The higher organisms have made themselves (largely) independent
of their immediate environment. Their tissues are bathed from
within by a fluid (the blood) which they carry around with them, a
sort of "internal environment." 20
The degree of perfection with which this constancy of the internal
or traveling environment, independently of the external environment,
is developed, increases as we ascend the biologic scale. This is
lucidly set forth, for example, by Claude Bernard:21
Chez tous les 6tres vivants le milieu inte'rieur qui est un produit de 1'organisme,
conserve les rapports ne'ce'ssaires d'Schange avec le milieu e'xte'rieur ;
mais
& mesure que 1'organisme devient plus parfait, le milieu organique se sp< cifie et
s'isole en quelque sorte de plus en plus du milieu ambiant.
It is the peculiar structure and the mechanical properties of the
organism that enable it to secure and maintain the required environment
(including the milieu interieur). The higher animals, in
particular, are provided with an intricate apparatus, comprising many
members, for securing food (internal environment) as well as for
warding off hostile influences.
20 "Etant donne* que 1'eau de mer a un contact si intime avec les organismes
de la mer et que non seulement elle les entoure de ses flots, mais qu'elle traverse
leurs branchies et impregne en partie les corps des inverte"bre"s, il semble assez
justind de la placer dans la m&ne cate"gorie que les autres liquides physiologiques."
S. Palitzsch, Comptes Rendus de Carlsberg, vol. 10, part 1, 1911,
p. 93. Compare also the following:
"Not only do the body fluids of the lower forms of marine life correspond
exactly with sea water in their composition, but there are at least strong
indications that the fluids of the highest animals are really descended from
sea water .... the same substances are present in both cases, and
in both cases sodium chloride largely predominates." L. J. Henderson, The
Fitness of the Environment, 1913, pp. 187-188. See also ibid., pp. 116 and 153;
H. F. Osborn, The Evolution and Origin of Life, 1917, p. 37 ; D'Arcy W. Thompson,
Growth and Form, 1917, p. 127.
21
Introduction a l'6tude de la me'decine expe"rimentelle, 1885, p. 110.
The increasing independence, as we ascend the biological;
~
jale,
which the organism displays toward its more remote environment, is
thus accompanied by a parallel increase in the perfection of the
apparatus by which this independence is earned. Here again we
may quote Claude Bernard :
22
A mesure que 1'ori s'eleve dans l'e"chelle des etres, ces appareils deviennent
plus parfaits et plus eomplique's; ils tendent & affranchir completement
1'organisme des influences et des changements survenus dans le milieu exte"rieur.
Chez les animaux inverte'bre's, au contraire, cette independence visa-vis
du milieu exte"rieur n' est que relative.
The Policy of Resignation : Its Parallel in Other Sciences. Whatever
may be our ultimate conclusions, we may do well to adopt
at least as a temporary expedient the policy of resignation; with
Sir Edward Schafer we may abandon the attempt to define life.
Perhaps, in doing this, we are following historical precedents : Geometers
have had to resign themselves to the fact that Euclid's
parallel axiom cannot be proved. But as the reward of this
resignation came the new geometries of Bolyai, Lobatchewski and
Riernann. Enlightened inventors have abandoned the attempt to
build a perpetual motion machine; but again, resignation is rewarded
with the recognition of a fundamental law, the law of conservation
of energy. Physicists, following Einstein, have abandoned, for the
time being at any rate, the attempt to determine experimentally the
earth's absolute motion through space. The reward has been the
theory of relativity, one of the greatest events in the history of
science.
The whole development of science, especially in recent years, is
a record of tearing down barriers between separate fields of knowledge
and investigation. Little harm, and perhaps much gain, can come
from a frank avowal that we are unable to state clearly the difference ;'
between living and non-living matter. This does not in any way
'
commit us to the view that no such difference exists. l
For the present, then, we shall adopt the position that the problem
is essentially one of definition. The question is not so much "What is
life," but rather, "What shall we agree to call life?" And the
answer, for the present at any rate, seems to be that it is immaterial
how we define life; that the progress of science and our understanding
of natural phenomena is quite independent of such a definition.
22
Ibid.
We shall, wherever convenient, continue to employ the terms life,
living organism, merely as a matter of convenience. This use of the
terms does not imply or presuppose any precise distinction between
living and non-living matter; it merely rests upon the fact that in
most cases ordinarily met there is essentially universal agreement as
to whether a portion of matter is to be classed in the first or in the
second category. We will adopt the policy of Sir William Bayliss :
If asked to define life I should be inclined to do as Poinsot, the mathematician
did, as related by Claude Bernard: "If anyone asked me to define
time, I should reply: Do you know what it is that you speak of? If he said
Yes, I should say, Very well, let us talk about it. If he said No, I should say,
Very well, let us talk about something else."
The ideal definition is, undoubtedly, the quantitative definition,
one that tells us how to measure the thing defined; or, at the least,
one that furnishes a basis for the quantitative treatment of the subject
to which it relates. We have already spoken of evolution.
Most of what follows will relate directly or indirectly to evolution.
It will be well here, while discussing definitions, to establish a definition,
a conception, of evolution that shall, as far as may be, have the
quantitative stamp.
CHAPTER II
EVOLUTION DEFINED
Nature must be considered as a whole if she is to be understood in detail.
Bunge.
As has been abundantly made plain, the choice of a definition is
a matter of expediency. In adopting, for special use in exact science,
a term already in general use, we must seek, so far as possible, to
embody in our definition the fundamental and essential features of
the concept denoted by the term as used popularly and by the best
workers, thinkers and writers. 1
In so far as there is divergence in
the use of the term, it may be well to frame the definition broadly,
so as to cover a wide range of phenomena and lead to a comprehensive
view of natural events, corresponding to the essential unity
of nature. In this way we shall be most likely to see the facts of
nature arraying themselves in a natural order, and to achieve that
economy of thought which is secured by a well devised system of
classification. Facts which naturally belong together will, then, be
found together, in our system, in the same or in neighboring pigeon-
holes.
Now if we seek to analyze what is in our minds when we speak of
the evolution of a given system, we find' and on this probably all are
agreed that the fundamental, the central thought, is that of the
history of the system. But the concepts of the history and of the
evolution of a system, though related, are not identical if they
were, one word would suffice to denote the single concept. The
popular and also the scientific conception of evolution contains as an
essential feature the element of progress, of development. We would
not ordinarily class as evolution the history of such a system as a
swinging pendulum, or a celestial body circling in its orbit, in so far
1
Compare Bertrand Russell, Analysis of Mind, 1921, p. 197: "The use of the
word comes first, and the meaning is to be distilled out of it by observation and
analysis." "In each case the work consists chiefly in making explicit processes
which are instinctive," as J. W. N. Sullivan (Aspects of Science, 1923,
p. 24) remarks apropos of certain other matters.
20
EVOLUTION DEFINED 21
as these motions are purely periodic or cyclic. In the history of such
systems the element of progression in time, of development, is lacking.
They repeat in endless succession the same series of events. The
hand of the clock, like a symbol of perpetual youth, goes through
its daily double cycle, making no distinction between yesterday, today
and tomorrow. It is the calendar that reminds us we grow older
year by year, the calendar that turns a new and different leaf each
o!ay. "The book of Nature is the book of Fate. She turns the
gigantic pages leaf after leaf, never returning one. . . . .
"2
But, to characterize the kind of history we speak of as evolution,
it is not enough that each day be unlike every other; it is not merely
that a system never passes twice through the same state;
3
not merely
that a biological species never retraces its steps,
4
or that "when a
race has lived its term it comes no more again."
5
Evolution not a Mere "Changeful Sequence." Such a statement
as those cited in the preceding paragraph alone is insufficient to distinguish
evolution as a progress from merely a changeful sequence',
it is insufficient to define the direction of evolution.6
For if the
world's events taken in historical order A, B, C . . . are a
changeful sequence, the same is also true of the inverted series
. . .
C, B, A, Mere unlikeness of two days is insufficient to
tell us which is antecedent to the other. To determine this we
must know something regarding the character of the unlikeness.
In a vague way this character is indicated by the term progress,
which, as already remarked, is closely associated, in popular
conception, with evolution. And the more rigorous scientific disciplines
of biology, too, leave us with a not very clearly defined
idea of progression as one of the fundamental characteristics of
those changes which are embraced by the term evolution. Such
phrases as "the passage from lower to higher forms" which are
often used to describe the direction of evolution, are vague, and
2
Emerson, Conduct of Life, Everyman's Library Edition, 1915, p. 157. Compare
also Lee Wilson Dodd's lines:
" . . . . Nor do the stars retrace
their glistening snail marks of slow destiny."
3
J. Perrin, Traite" de Chimie Physique, 1903, vol. 1, p. 142.
4
Petronievics, Science Progress, 1919, p. 406.
6
Emerson, Conduct of Life, p. 158.
6
For this reason the characterization of the trend of evolution given by
Petronievics, loc. cit., is inadequate.
22 ELEMENTS OP PHYSICAL BIOLOGY
undoubtedly contain an anthropomorphic element. 7
At best they
give every opportunity for divergence of opinion as to what constitutes
a "higher form." If, on the other hand it is stated that
evolution proceeds from simpler to more complex forms, or from
less specialized to more specialized forms, then the direction of
evolution is but poorly defined, for the rule is at best one with
many exceptions. It should be particularly noted that all these
efforts to specify the direction of evolution attempt to do so in terms
of a single component of the evolving system. Such definitions of
the direction of evolution are foredoomed to failure. It is the system
as a whole that evolves, and we can hope to establish a definition of
the direction of evolution only in terms of the system as a whole.
Evidently, we must seek a more precise indication of the direction
of evolution if our definition is to be truly expedient. We must
analyze further the contents of our mind when it contemplates the
concept of evolution. We return to our examples of the pendulum,
or of the earth in its orbit. When frictional resistances are neglibible,
or are disregarded, the periodic series of events in the system may be
history, but seems hardly worthy of the name evolution. In actual
fact the motion of the pendulum bob gradually dies down, owing to
friction and other dissipative forces. The motion is not strictly
periodic. The pendulum does not, actually, count out similar
seconds, unidentified, but marks, by its greater amplitude, an earlier
vibration as distinguished from a later. So also, the earth in its
motion is slightly delayed by frictional forces introduced by the tides;
it slows down a little as the centuries pass. The strictly periodic process
is changed into one in which successive days differ by a trifle
in length. The process has a definite direction in time. We feel
justified in speaking of the system as "evolving." Now the thing to
mark is that what has imparted to the process its directed character
is frictional resistance, dissipative forces, typical irreversible effects,
to speak in the language of the physicist.
7
"Evolution is thus almost synonymous with progress, though the latter
term is usually confined to processes of development in the moral, as distinguished
from the physical world. Further, this idea, as Mr. Spencer remarks,has rather a subjective value in existence, as judged by our feelings" (Encycl'
Brit., 9th edition, vol. 8, p. 751). Compare also Bertrand Russell Our Knowledge
of the Eternal World, 1914, p. 12. "A process which led from amoeba to
man appeared to the philosophers to be obviously a progress though whether
the amoeba would agree with this opinion is not known."
EVOLUTION DEFINED 23
Again, consider a typical example of what we are all agreed to
speak of as evolution: the history of the earth and its living inhabitants.
The readjustments, the re-adaptations of life-forms
which have here taken place, were undoubtedly due in part to changes
in external conditions, such as climate, geographic distribution of
land and sea, etc. In part, also, such changes have gone on and are
going on before our eyes independently of any external changes,
and under approximately constant conditions. Organic evolution
being a slow process, it takes a certain time, when equilibrium or
near-equilibrium is disturbed, for a new equilibrium or near-equilibrium
to become established. There is therefore a tendency for
internal readjustments or changes to lag behind the external changes
by which they are conditioned. As a special case, if an external
change is followed by constant external conditions, internal changes
may continue to proceed under constant external conditions.
Now such internal changes in a material system, which lag behind
the determining external changes, or which go on under constant
external conditions, are typically irreversible processes.*
8
A. process is said to take place reversibly, if the direction of the change is
reversed by a suitable alteration, however small, of the (generalized) force
applied to produce the change. For example, if two equal weights are suspended
from the ends of a string passed over a simple pulley, then, the weights
being initially at rest, any weight however small, added on one side of the system
will produce motion downward on that side, provided there is no friction at the
pulley and no stiffness in the string. If, on the contrary, a weight, however
small, is lifted off from the same side of the system, motion will be initiated
in the opposite direction. Note that if there is friction at the pulley, these
statements are no longer true. It will now require a weight of definite size,
perhaps a decigram, or a milligram, to start or reverse the motion.
In the first instance the change is reversible, in the second it is said to be
irreversible. Note that in this example the circumstance that imparts to the
process an irreversible character is the presence of friction, which causes the
dissipation of energy, that is to say, its conversion into heat at the temperature
of the surroundings.
Again, consider a vessel containing water at a temperature Ti in a room at
temperature Tj. If TI is ever so slightly greater than T2,
heat passes from the
vessel to the surroundings, and vice versa. When, therefore, TI and Ts are
very nearly equal, the passage of heat from the vessel to the surroundings is
essentially reversible. If there is a material difference between TI and T8 ,
the
heat transference is irreversible. For example, if the vessel is at 50C. and the
room at 20C., heat will pass from the vessel to the room. And the direction
of this heat transfer will remain unchanged if the temperature of the vessel is
24 ELEMENTS OF PHYSICAL BIOLOGY
We are thus led, from two slightly different points of view to the
following definition of evolution: Evolution is the history of a system
undergoing irreversible changes.
Scope of Definition, It is worth while at this point to consider
briefly what kind of history this definition excludes and what it
includes. It has already been noted that we have excluded certain
purely mechanical systems of periodic habit, such as the frictionless
pendulum and the planet circling in its orbit through empty space,
in absence of tidal effects. 9
It is not the case, however, that all
purely mechanical systems are excluded, that is to say, all systems
in which all energy is either kinetic or potential (configurational),
all forces either inertia forces or positional forces. If our knowledge
of such a system is statistical in character, if we know only averages
of certain of the variables defining the state of the system, it may
happen that certain changes therein appear to us irreversible, and
would accordingly be classed, by our definition, among processes of
evolution.10
This leads to the seemingly embarrassing conclusion
that a process is or is not a process of evolution, according to the
reduced 1, 2 or even 10. Not until the vessel is cooled by more than 30 will
the stream of heat be reversed. The passage of heat in such case, from a body
at one temperature to another at essentially lower temperature, is irreversible
in this sense, the sense in which the term is employed by the physicist in discussions
of this kind.
In the case in which internal readjustments lag behind changes in external
conditions, there is necessarily a finite difference between the applied (generalized)
force, and the opposing resistance. Such processes are, therefore, of
necessity, irreversible.
From the examples given, it will be seen that during a reversible change a
system is at all times (very nearly) in equilibrium. It can therefore be said
that a reversible change is one in which the system passes through a continuous
succession of equilibria. In fact, the change is strictly reversible only
if the difference in the applied (generalized) force and the resistance is infinitesimal,
and the change is infinitely slow.
6
Such tidal effects act as brakes and destroy the exact periodicity of the
motion.
1S
The irreversibility also of those changes occurring in a system whose
internal adjustments lag behind changes in the applied forces, may be apparent,
and may disappear when detailed knowledge of the individual parts of the
system takes the place of statistical data. The reason for this is that when the
reaction or readjustment is expressed in a statistical way, an average of individual
reactions may show a lag, although each individual reaction itself maybe immediate.
EVOLUTION DEFINED 25
nature and extent of our knowledge regarding the system. So, for
example, the establishment of thermal equilibrium in a body of gas
initially at non-uniform temperature is evolution if we merely know
its total mass, composition, volume, pressure and initial temperature
distribution. But should we be informed of the exact initial state
of each molecule, then the process by which thermal equilibrium is
established (if this does occur) would be classed, together with the
journey of the earth in its orbit, among the cases excluded, as mere
history, from our definition.
This, upon reflection is neither as strange, nor as embarrassing as
it may at first sight appear. For problems of evolution are in large
measure problems of probabilities, statistical problems. Incidentally,
this reflection disposes of the rather foolish objection sometimes
raised against the theory of evolution, that it ascribes the course of
events in an evolving system to chance. When we describe a phenomenon
as being governed by chance, we do not, of course, mean
that there are no definite causes (determining factors) at work; we
merely state in these terms that the causes are complex and not
known to us in detail.
Practically there is no cause for embarrassment, since we never do
know material systems in sufficient detail to compute their state at
every instant from the initial state, except in terms of averages.
In principle, however, it is necessary to make the admission that, in
the last analysis, whether we class the history of a system as evolution
or not must depend on the extent and detail of our knowledge
of that system.
11
It will thus be seen that the line of division between reversible
(purely mechanical) and irreversible (dissipative) processes is not
11
To be quite exact, evolution, according to this, should be defined in terms
of a point of view, say about as follows :
Evolution is the history of a system, regarded as a progressive change or
development, to which its unidirectional character is imparted by irreversible
changes going on in the system.
That a point of view is involved is also implied in the following definition
given by Karl Pearson:
"A causal description of the appearance of successive stages in the history
of a system forms a theory of the evolution of that system.
"If the theory be so satisfactory that it resumes in some simple statement
the whole range of organic change, we term it the law of evolution," (Grammar
of Science, 1900, p. 375).
26 ELEMENTS OF PHYSICAL BIOLOGY
so very sharply drawn. Furthermore, the cases excluded are, in
point of fact, ideal cases. Real processes are always irreversible.
Hence, after all, history, real history, is always evolution, and, though f
.
in principle the two concepts may be distinct, in practice they "-('
coincide in scope.
What then is gained by our definition of evolution?
This is the gain: Having analyzed the submerged implications of
the term evolution as commonly used, so as to bring them into the
focus of our consciousness, and having recognized that evolution, so
understood, is the history of a system in the course of irreversible
transformation; we at once recognize also that the law of evolution
is the law of irreversible transformations; that the direction of evolution
(which, we saw, had baffled description or definition in ordinary
biological terms), is the direction of irreversible transformations.
'
.
And this direction the physicist can define or describe in exact terms. Jl
For an isolated system, it is the direction of increasing entropy.
12
The law of evolution is, hi this sense, the second law of thermo-
dynamics.
13
12
More generally, it is the direction of decreasing thermodynamic potential,
this potential being variously defined, according to the conditions of
transformation.
13
"The second law (of thermodynamics) is the law of evolution of the world
accessible to our observation" (Chwolson, Lehrbuch der Physik, 1905, vol. iii,
p. 499; Scientia, 1910, vol. iii, p. 51.
the second law of the theory of energy is now generally
regarded as essentially a statistical law. So viewed, the second law of energy
becomes a principle stated wholly in terms of the theory of probability. It
is the law that the physical world tends, in each of its parts, to pass from certain -
-v
less probable to certain more probable configurations of its moving particles.
As thus stated the second principle .... becomes a law of evolution"
(Josiah Royee, Science, 1914, vol. xxxix, p. 551.)
"Tin systeme isole ne passe jamais deux fois par le me'me e"tat.
"Le second principe affirme un ordre nece"ssaire dans la succession de deux
phenomenes, sans retour possible aux etats deja traverse's. C'est pourquoi
j'ai cru expressif d' appeler ce principe un principe devolution. II se trouve
qu'en proposant ce nom je suis fidele a la pens^e de Clausius, car le mot
farpomft, d'oti ii a tire"
entropie, signifie precisement Evolution." (J.Perrin,
Trait6 de Chimie Physique, 1903, vol. 1, pp. 142-143.)
"II est hautement improbable qu'un systeme isole" passe deux fois par le
mgme e"tat; cela est d'autant plus improbable que la complication du systeme
est plus grande, et pratiquement il serait insensS de se placer dans cette
hypothec d'un retour a I'Stat initial." (J. Perrin, loc. cit., p. 146) . A
EVOLUTION DEFINED 27
Simple Mechanical Example. It will be desirable, at this point,
to consider, by the aid of a simple example, the manner in which
some of the facts considered in the preceding pages find expression
in the analytical formulation of the behavior of mechanical systems.
Take the example of the simple pendulum. For small vibrations the
restoring force, tending to draw back the bob to its lowest position,
3/
is easily shown to be mg -, where x is the horizontal displacement,
6
m the mass of the bob, and g the acceleration of gravity. This force
is expended upon two items, first, in overcoming the inertia of the
bob, and producing an acceleration a. The force so expended is
cc
measured by ma. Second, a part of the force mg - is expended in
b
overcoming the resistance of the air. If v is the velocity of the bob,
this part of the force is measured (for ordinary velocities) by kv,
where k is a constant depending on the shape of the bob, etc. We
have then
ma + kv = mg (1)
/Jj*
or, since the velocity v is the rate of change of x with time, i.e., =->
GlV
fiir nD (i y
and a is the rate of change of with time, i.e. a = = ,
at at ctt
d?x dx x . .
Now there are certain general characteristics to be observed in
this equation, characteristics which are typical of the equations of
motion of mechanical systems. The equation contains the first and
second derivatives of x with regard to t and no higher derivatives,
The first derivative is introduced by the frictional force, and disappears
if this force is zero, Le., if the coefficient k in (2) is zero.
The equation then takes the simpler form
Now in this simplified form the equation has the following peculiarity:
It is indifferent to the sign of t.
For, in differentiating twice
in succession with regard to t, the positive sign of the second deriva-
28 ELEMENTS OP PHYSICAL BIOLOGY
tive is restored. This is the analytical symptom, as it were, of the
reversibility of the process.
14
It should be noted that this peculiarity
disappears at once if the frictional term k ~ is present, for a single
differentiation with regard to-* yields a result with sign opposite
to that of differentiation with regard to t. The presence of a frictional
force, therefore, imparts to the process an irreversible character,
it establishes a distinction between t and-i; it singles out one direction
in time as a peculiar direction, the forward direction, the direction
of progression.
Now, in point of fact, in the equations of motion of all real systems
j
the frictional term (viscosity term) k
j or its equivalent is present,
d?x
though it may be small as compared with the inertia term
*-^,
The reversible system, in which this term is wholly absent (zero)
is an ideal case, it represents a limit towards which real systems may
approach; an abstraction.
Inertia-Free or Completely Damped Systems. There is
another such ideal limiting case, another abstraction, which is of
much interest because certain important classes of real systems
approach it very closely. This is the case in which the inertia term
m is negligible, so that in the case of the pendulum,
difor
example, the equation representing the history of the system
reduces to
, dx x /, N
k = -mg r (4)
at I
The history of such inertia-free systems is typically of the irreversible
kind. They have, furthermore, a property illustrated by certain
features in the equation (4) above :
dx
It will be observed that if cc is zero, then --=- also is zero, or, as we
may put it, the velocity vanishes with the displacement from equilibrium.
Moreover, differentiation of (4) gives
di* I dt
u Compare H. PoincarS, Thermodynamique, 1908, p. 441.
EVOLUTION DEFINED 29
from which it is seen that the acceleration also vanishes with the
velocity.
15
This implies that when the system is in its equilibrium
position, it is also actually at r$si, unlike the pendulum, which swings
twice through its equilibrium position in each vibration. The former
property, the vanishing of the accelerations with the velocities,
so that the equilibrium position is necessarily the position of rest,
is characteristic of an important class of systems, including those
with which we shall here be chiefly concerned.
Another important characteristic of such systems, which is also
exemplified by equation (4), is that the velocity is uniquely determined
for every value of x. This is not the case in the motion
represented by (3). This latter equation gives, upon integration,
--.C-* (6)
dx
so that for every value of x there are two possible values of .
(M>
W.e have, then, at the one extreme the "purely mechanical" system
free from frictional (Viscosity) effects, and, in its most typical form,
periodic in habit.
As an intermediate link we have systems exhibiting both inertia
and frictional effects. Their action may resemble that of a pendulum
swinging in air; typically the history of such a system exhibits the
phenomenon of damped oscillations, a periodicity over which there
is superimposed the dying away of the motion. The damping is
introduced by the frictional effects.
At the other extreme we have inertia-free, or, as we might say,
completely damped systems/
6
typically irreversible in their history.
The system and processes with which we shall largely be concerned
here seem to belong essentially to this third type, as will be seen in
the development of the theme.
Compare E. Buckingham, Theory of Thermodynamics, 1900, p. 33.
16
This does not, however, preclude the possibility of oscillations. More
will be said on this point later.
IISC Lib B'lore
574 N25
2544
CHAPTER III
THE STATISTICAL MEANING OF IRREVERSIBILITY
Supposons que nous voulions placer un grain d'avoine au milieu d'un tas
de hie": cella sera facile; supposons que nous voulions ensuite 1'y retrouver et
1'en retirer; nous ne pourrons pas y parvenir. Tous les phe*nom6nes irr6versibles,
d'apres certains physiciens, seraient construits sur ce modele.
JET. Poincari.
One point, to which, allusion has been made incidentally, calls
for comment. Many processes which, viewed in the gross, present
the appearance of typically dissipative, irreversible phenomena, have
long been suspected, and have in recent years been fully demonstrated
to be, in fact, of the reversible type, "purely mechanical"
processes, the details of which are merely hidden from our view
owing to the diminutive dimensions (and correspondingly immense
number) of the units at play. So, for example, consider the case
of two vessels A and B at equal pressures, communicating by a tube
that can be closed by means of a turncock. Let the vessel A contain
1 gram of nitrogen gas, and let B contain 1 gram of oxygen
gas, the communication between A and B being closed. It is a
matter of common knowledge that if the stopcock is now opened, the
gas from the A will flow over into the vessel B and vice versa, and in
a short time an equilibrium is reached in which each vessel contains
0.5 gram of each gas. Now, in point of fact, the molecules
of the gas behave (approximately) like a number of elastic spheres,
their equations of motion contain no dissipative term, but are of
the type (3) (Chapter II). We should therefore expect the system
to exhibit periodic motion, we should expect that after a certain
lapse of time the initial condition should return, and that all
the nitrogen should once more be contained in the vesel A, all
the oxygen in B. In actuality, such a thing is never observed.
How is this discrepancy to be explained?
Let us replace the two vessels and the gas molecules by some
simple analogues of dimensions readily accessible to our senses, and
let us., watch a process analogous to the diffusion of the gas from
one vessel into the other. We provide ourselves with two boxes
30
STATISTICAL MEANING OF IRREVERSIBILITY 31
or urns.1
In one of these, A we place 50 black balls; in the other
B, we place 50 white balls. We shuffle both boxes thoroughly, and
then draw blindly a ball from A, and one from B, and we return
them to opposite urns. We continue this as long as desired. The
more lightly drawn curve in figure 1 shows the graphic record of
an actual series of drafts of this kind. The stair-case-like line
shows how in successive drafts the number of black balls in box A
gradually diminished until at last there remained about 25, onehalf
of the original number, in box A. But note that there are
fluctuations, sometimes the box contains 26, 27, 28, then again
3O 4O
NUMBEff Cl
9060
(DOUBLE.)
Fia. 1. GRAPH OF MODEL PROCESS ILLUSTRATING THE STATISTICAL MEANING
OF IRREVERSTBILITY
The more lightly drawn curve records the number of black tickets remaining
in urn A after successive drafts. The heavier curve records the previous and
ensuing history of 50 tickets found in urn A at the end of the fiftieth draft,
(Reproduced from A. J. Lotka, Two Models in Statistical Mechanics, Am.
Math. Monthly, vol. 31, 1924, p. 122.)
27, 26, etc. of the original balls. It is nowise impossible that, if
we continue the drafts for a long time, some time or other all the
original 50 black balls will be back in box A; but it is highly improbable
that this should happen within any reasonable time. Curiously
enough, the urn model is competent to illustrate also this
highly improbable course of events. For this purpose, instead of
starting with 50 black and 50 white balls, we start with the balls,
or in this case more conveniently tickets, all white, and numbered
from 1 to 50 in urn A, and from 51 to 100 in urn B. After a suit-
1
A. J. Lotka, Am. Math. Monthly, March, 1924; Science, 1924, vol. 59,
p. 532.
32 ELEMENTS OP PHYSICAL BIOLOGY
able number of drafts, say 50 double drafts, in which a record is
kept of all the numbers drawn, the urns are opened, and the tickets
in the second urn are now blackened. The drawing is then continued,
for, say, another 50 drafts, recording each time the numbers
drawn. The numbers on the tickets enable us to trace the previous
history of the 50 black tickets, before they were blackened. In
an experiment actually carried out it was found that these 50 black
tickets were originally distributed essentially evenly in the two urns.
The curve representing the first 50 drafts is an ascending curve,
the system passed, during this stage of the process, from more probable
to less probable states, as shown in the first, ascending portion
of the more heavily drawn curve in figure 1. In the second
series of fifty drafts the curve descends in normal fashion, with
increasing probability of the successive states of the system. It
may seem like a contradiction of terms that what amounts practically
to an infinitely improbable series of drafts should be capable
of actual realization at will. But if the series of drafts described
were extended to great length in both directions, say one
million drafts before blackening the tickets, and one million after,
it would be seen that the peak on the curve is indeed a very exceptional
feature. It is a perfectly safe bet that in two million
drafts not more than one such peak, going up to 50 black balls in
one urn, would be encountered.
The model described exemplifies among others the fact that in
an exceedingly long lapse of time it may some time occur that the
system will return to its original state. This is quite in accord with
the laws of mechanics; in fact, as already noted, these laws actually
demand that every mechanical system of finite dimensions must,,
ultimately return to its initial state, and must do this not onee
only, but in everlasting reiteration at regular intervals: the motion
is periodic. This property also is capable of illustration by
a simple model, such as the following: Twenty-six pendulums of
periods T = 0.5, 0.6, . . .
2.9, 3.0 seconds are started simultaneously
to the left from their equilibrium position, and are then allowed
to oscillate undisturbed. Count is then made, at the end of
every tenth of a second, of the number of pendulums on the left
of
^
the median. In this way the staircase curve figure 2 was obtained
(computation here taking the place of actual observation).
It will be seen that in the brief fragment of a period covered by the
STATISTICAL MEANING OF IKREVEBSIBILITY 33
record, this exhibits all the characteristics of a "passage from a
less probable to a more probable distribution," though, in point of
fact, we know that the system has a perfectly definite period of
7385 years. The appearance of chance in this wholly determinate
mechanical process is brought into still greater prominence if we
plot the deviations, from the mean, of the number of pendulums
on the left of the median position, at successive counts. We thus
obtain the points indicated by small circles in figure 3. These
group themselves very obviously about a typical Gaussian curve
of random distribution, namely one having a standard deviation of
y2L',
this curve has been drawn in the diagram, and, as will be
seen, the agreement is good, considering the smallness of the sample
FIG. 2. GBAPH OF SECOND MODEL PEOCESS ILLUSTRATING THE STATISTICAL
MEANING OF IEEEVEESIBILITY
Number of pendulums found on left of median position at successive epochs.
(Reproduced from A. J. Lotka, Two Models in Statistical Mechanics, Am.
Math. Monthly, vol. 31, 1924, p. 124.)
(412 observations, extending over 41.2 seconds, out of a total period
of 7385 years). Thus for long stretches of time the periodicity of
the motion of the system of pendulums is very effectively masked
under an aspect of "chance."
These simple models illustrate very clearly how the seeming conflict
between the periodicity of all mechanical motions and the apparently
one-sided course of events, directed toward one definite
end state, is resolved. The actual process of isothermal gaseous
diffusion is, in fact, periodic, but with a period so long that humanly
speaking, the return to the initial state never occurs at all. For all
stretches of time that can have any real significance in human
thought (and this includes the vast historical ranges of all geology
and astronomy), it may therefore be said in a certain sense that evo-
34 ELEMENTS OF PHYSICAL BIOLOGY
lution proceeds, in all but a vanisMngly small class of- exceptional
cases, from less probable states (e.g., uneven distribution of the 50
black balls in the two urns) to more probable states, tending ultimately
toward a most probable state. This statement cannot
however be allowed to pass without a word of caution. It is mean-
-/ft
83
FIG. 3. FREQUENCY DIAGRAM FOE THE DEVIATIONS FROM THE MEAN
APPEARING IN FIGURE 2
Abscissae represent deviations from the mean (13) in number of pendulums
on left of median; ordinates represent corresponding frequencies,
among the observations recorded in figure 2. (Reproduced from A. J. Lotka,
Two Models in Statistical Mechanics, Am. Math. Monthly, vol. 31, 1924 t>
125.)
'*'
STATISTICAL MEANING OF IEREVEESIBILITY 35
ingless unless the characteristic with regard to which probability
is reckoned is explicitly or implicitly indicated. Probability is essentially
a matter of classification. An improbable event is one
that is a member of a small class, and whether it is so or not depends,
clearly, on our system of classification. For this reason the broad
statement which has sometimes been made,
2
that the direction of
evolution is from less probable to more probable states, is not only
inadequate, but is really meaningless. It is indefinite In failing to
specify with regard to what characteristic probability is to be reckoned;
and it is incomplete in failing to call attention to the fundamentally
important connection between the particular probabilities
in question and available energy.
Another point, which has not hitherto perhaps received its deserved
attention is clearly brought out by the two models described,
namely, that irreversibility is a relative term. For, obviously,
if we use our visual discrimination in selecting the balls
drawn from the boxes (instead of drawing blindly), we can easily
bring it about that in short order all the black balls are back in
box A. Thus a process may be reversible or not, according to the
means that are naturally available or arbitrarily permitted in operating
upon the system under consideration; somewhat as the trisection
of an angle is or is not an impossible geometric construction,
according as we are or are not forbidden the use of instruments
other than ruler and compass. In the case of molecular
aggregates this fact has long been duly appreciated, having been
first pointed out by Clerk Maxwell, who remarked that a demon
capable of dealing with individual molecules would be able to cheat
the second law of thermodynamics. But it seems to have been
pretty generally overlooked that the relative character of irreversibility
has an important significance in certain natural processes
taking place on a macroscopic scale. In point of fact this is a matter
whose importance in the world of living organisms can hardly
be rated too high. For there are certain diffusive processes going
on in nature which, from the standpoint of thermodynamics, are
not of the irreversible type; but which might as well be, so far as
any benefit derived from their reversibility by the organism (and,
in particular, by man) is concerned. If I should be the fortunate
2
See for example, J. Royce, Science, 1914, vol. 34, p. 551.
36 ELEMENTS OF PHYSICAL BIOLOGY
possessor of a pound of gold dust, and some malicious person
should take it and scatter it far and wide, so that it became hopelessly
diluted with dust and refuse, it would be a small comfort
to me to know that it were merely mechanically commingled with
such foreign matters, that it had not irreversibly undergone solution
or chemical transformation, and that therefore it could theoretically
be recovered without the expenditure of work. In practice
its recovery might entail the expenditure of far more energy
than if the gold were present in reasonably concentrated solution.
The point is that in practice I am restricted to operations in bulk
upon reasonably large quantities, in reasonably concentrated form,
otherwise the theoretical ideal of recovery without work is very
far from being attained. And with this restriction placed upon my
operations, certain processes acquire an irreversibility which they
do not possess apart from that restriction. The illustration of the
pound of gold has the advantage of simplicity and cogency. But
if by any chance it has conveyed the impression that only in peculiar
and far-fetched cases does this kind of irreversibility enter into
play, then it is an unfortunate example indeed, for nothing could
be farther from the truth. The fact is that nature abounds in
just such dissipative processes as the scattering to the four winds, to
utter inutility, of materials of the highest importance to life; and
one of the central problems which the organism has to solve in the
struggle for existence, is the reconcentration, into his immediate
environment and into his body, of valuable materials that have
become scattered by agencies beyond his control. It is not the
least of the triumphs which have made man the lord of creation,
that he has learnt, beyond all comparison more effectively than any
of his competitors, to carry out this process of reconcentration to
satisfy his needs. So a fleet of ships, year by year, bear a burden of
saltpeter from Chile to all civilized countries, to balance the losses
from our depleted fields. So, in his most recent technical achievement,
man has learnt to draw from the air a supply that will continue
unfailing, long after the Chilean nitre beds are exhausted.
The fact is, in dealing with the physics of such macroscopically
irreversible effects, it will ultimately be necessary to develop a
method of mathematical analysis that shall be competent to distinguish
and handle not only the extremes the case of a primitive
organism that can deal only in the gross, without intelligent or
STATISTICAL MEANING OF IKREVEBSIBILITY 37
other discrimination, with, the matter and situations presented to
it; and an ideally perfect organism, that should expend just the
minimum of effort, directed with absolute precision toward the
attainment of its ends. A method must be devised that shall duly
take account of, and use as a fundamental datum for its deductions,
the particular character, the particular degree of perfection
of the mechanical and psychic equipment or organization by which
each organism reacts more or less selectively upon its environment.
We shall have occasion to refer to this matter again in greater
detail in a~ later section. But here it is well to note that our two
models are suggestive also with regard to this aspect of the subject.
For it appears at first sight as if there were a fundamental difference
in character between the first and the second model, since it
is essential for the operation of the urn model that the drawing be
done blindly, so as to give chance a part in the process, as we would
say; whereas the pendulum model we operate with our eyes open,
apparently in full consciousness of what is going on. Chance
seems to play no part here, the system is mechanically determinate.
But there is a blindness which is not of the eye, and there is a
vision that surpasses optical vision. The same struggle for existence
which has developed in man the organ of sight, to depict
for him the external world, to furnish him with a map on which
to base his plan of campaign, has also, in latter days, developed
his internal vision, whereby he extends his world-picture beyond
the powers of the bodily eye. It is immaterial by which process
his map is drawn 'its function is the same; whether I peep into
the urn and manipulate the drafts by the light of my eyes; or
whether, in the light of my knowledge of mechanics, I adjust the pendulums
to equal lengths and phases; or again, whether, in the more
serious affairs of life I employ these same faculties to diverse ends,
the effect is the same: In greater measure or less these organs and
faculties emancipate me from the bonds of the fortuitous and make
me a controller of events. Their function is to substitute choice for
chance, to introduce aimed collisions in place of random encounters.
Origin of Subjective Sense of Direction-in-Time. The failure of
the differential equations of dynamics to discriminate between t
and t raises the question as to the physical significance and
origin of our subjective conviction of a fundamental difference
between the forward and the backward direction in time, a con-
38 ELEMENTS OP PHYSICAL BIOLOGY
viction that is intimately bound up with the concept of evolution,
for, whatever may ultimately be found to be the law of evolution,
it is plain that no trend of any kind can be denned or even described
without reference to a favored direction in time.
One view which suggests itself is that this conviction is our subjective
appreciation of the trend from less probable to more probable
states recognized in statistical mechanics. But this does not
seem very satisfying, for we somehow feel that our conviction must
rest on something more fundamental than this somewhat accidental
circumstance, which, as the models described clearly show, is fundamentally
incompetent to distinguish between the forward and the
backward direction in time. For the pr ^in figure 1, for example,
may indifferently be traversed from lefTto right or vice versa, it
presents the same general character in either sense.
Another alternative is to suppose that the differential equations
of dynamics, as formulated by us today, are either an incorrect, or
else an incomplete statement of facts. The latter view is, indeed,
upon reflection, found to have a certain warrant. For the differential
equations of motion alone do not fully determine the actual
course of events; this depends further on the value of certain arbitrary
constants of integration; or, to speak in terms of physical
entities, upon the initial velocities of the particle of which the system
is composed. Strictly speaking it is only when the initial
velocities are zero, that the equations of motion, considered in their
totality, are indifferent to the substitution t' = t. From this
point of view our sense of the forward direction in time would appear
as our subjective appreciation of the fact that, once a material
system has been started on a certain course, with certain initial
velocities, there then remains no further freedom; its history must
continue to unfold in the direction determined by the initial veloc-
ities.
It seems, however, that it is not with this perfectly general type
of irreversibility of the course of events that we are chiefly concerned
in the study of evolution. The concept of evolution, according
to the analysis which has been made of it in preceding
pages, applies principally, if not exclusively, to systems that outwardly
at least affect the aperiodic habit, systems that do not
return periodically to their initial state, but show a definite trend,
whereby yesterday and tomorrow are never alike, and differ more-
STATISTICAL MEANING OF IEEEVEESIBILITY 39
over in some definite and characteristic fashion, even though we
may not be fully competent, at the present epoch of science, to
specify exactly wherein lies the characteristic difference.3
Inadequacy of Thermodynamic Method. Our reflections so far
have linked the fundamental problem of the direction, the trend
of evolution, with the disciplines of thermodynamics and statistical
mechanics. From this point of view the direction of evolution
is identified with the direction of the unfolding of irreversible processes,
the direction of increase of entropy (in thermodynamics) or
of increasing probability (in statistical mechanics).
A certain mental satisfaction may be derived from, this conclusion.
It gives us, in principle at least, an answer to our question
"Quo vadis?" But practically the answer is very inadequate. If
the conclusions, the methods of thermodynamics, or of statistical
mechanics, are to be applied to a concrete case, the data of the
problem must be presented in a veiy particular form. So long as
we deal with volumes, pressures, temperatures, etc., our thermodynamics
serve us well. But the variables in terms of which we
find it convenient to define the state of biological (life-bearing)
systems are other than these. We may have little doubt that the
principles of thermodynamics or of statistical mechanics do actually
control the processes occurring in systems in the course of
organic evolution. But if we seek to make concrete application we
find that the systems under consideration are far too complicated to
yield fruitfully to thermodynamic reasoning; and such phases of
statistical mechanics as relate to aggregation of atoms or molecules,
seem no better adapted for the task. To attempt application
of these methods to the prime problems of organic evolution
is much like attempting to study the habits of an elephant by means
of a microscope. It is not that the substance of the elephant is inherently
unfitted to be viewed with the microscope; the instrument
is ill adapted to the scale of the object and the investigation.
It would seem, then, that what is needed is an altogether new
instrument; one that shall envisage the units of a biological population
as the established statistical mechanics envisage molecules,
atoms and electrons; that shall deal with such average effects as
3
Perhaps the objective interpretation of our subjective sense of direction
in time must be sought in quantum mechanics. Cf. A. J. Lotka, loc. cit., p,
126, and, W. S. Franklin, Science, 1924, vol. 60, p. 258.
40 ELEMENTS OF PHYSICAL BIOLOGY
population density, population pressure, and the like, after the manner
in which thermodynamics deal with the average effects of gas
concentration, gas pressures, etc.; that shall accept its problems in
terms of common biological data, as thermodynamics accepts problems
stated in terms of physical data; and that shall give the answer
to the problem in the terms in which it was presented. What is
needed, in brief, is
something of the nature of what has been termed
"Allgemeine Zustandslehre,"
4
a general method or Theory of State.
It is somewhat along these lines that the system now to be
sketched is conceived.
A term introduced by J. R. Rydberg, quoted by C. Benedicks, Zeitschr. f.
phys. Chemie, 1922, vol. 100, p. 42.
CHAPTER IV
EVOLUTION CONCEIVED AS A REDISTRIBUTION
Toutes ces choses ne peuvent se determiner surement que par des mesures
precises que nous chercherons plus tard; mais auparavant il fallait au moins
sentir ie besoin de les chercher. /. B. Biot.
It now behooves us to establish, with respect to the problem of
evolution, a viewpoint, a perspective, a method of approach, which
has hitherto received its_ principal development and application
outside the boundaries of biological science. Such prior development
and applications, however extraneous to our chief line of
interest here, may well serve us in our present interrogations, since
we shall be in a position to profit by the precedents established in
methods, in conclusions, and, most particularly, in habit of thought.
This perspective is that which contemplates an evolving system
as an aggregation of numbered or measured components of several
specified kinds, and which observes and enregisters the history of
that system as a record of progressive changes taking place in the
distribution, among those components, of the material of which
the system is built up.
It is thus that physical chemistry views the progressive changes
in a system comprising several chemical species, that is to say
elements, compounds, phases, etc. It describes the system by
enumerating these components, by stating their character and extent
(mass); and by further indicating the values of certain quantities
or parameters, such as volume or pressure, temperature, etc.,
which, together with the masses of the components, are found
experimentally to be both necessary and sufficient, for the purposes
in view, to define the state of the system. With the instantaneous
state of the system thus defined, physical chemistry investigates by
observation and by deductive reasoning (theory) the history, the
evolution of the system, and gives analytical expression to that
history, by establishing relations, or equations, between the variables
denning these states (after the manner set forth above), and
the time.
It is commonly found that these fundamental equations assume
the simplest, the most perspicuous form, when they are written
41
42 ELEMENTS OF PHYSICAL BIOLOGY
relative to rates of change of the state of the system, rather than
relative to this state itself. That is to say, it is found that the
expressions for the rate of increase in mass, the velocity of growth,
of the several components, are simpler, more primitive in form, than
the expressions giving directly the mass of each component as a
function of the time. In the language of the calculus, the differential
equations display a certain simplicity in form, and are therefore,
in the handling of the theory at least, taken as the starting
point, from which the equations relating to the progressive states
themselves, as functions of the time, are then derived by integration.
1
So, for example, a simple system may be defined as comprising
4 grarn-rnoleeules of hydrogen, 2 gram-molecules of oxygen, and
100 gram-molecules of steam, at one atmosphere pressure, and at
1800C. The fundamental relation expressing the law of evolution,
the historical pattern, of the system, is in this case given by
the law of mass action:
.
/{^ (_-! }
v dt v 3
v2
where v is the volume, mi is the mass of steam, m2 the mass of hydrogen,
and ms,
the mass of oxygen (all expressed in gram-molecules).
The coefficients h, h, are functions of the temperature,
or, for a given temperature, are characteristic constants of the re-
action.
We are not, here, interested in the particular form of the law
of mass action. What does interest us is the general form of the
equation (1). It states that the rate of increase in mass, the velocity
of growth of one component, steam (mass mi), is a function of
the masses m*, nis, of the other components, as well as of the mass
mi itself, and, besides, of the parameters v (volume) and T (temperature),
the latter being contained in the coefficients fc x ,
/c2 . This
statement, in its more general form, is written, according to established
notation2
1
In experimental observation usually (though not always) the reverse
attitude is adopted.
2
For the benefit of the non-mathematical reader it may here be explained
that aquation (2) is merely a short-hand expression, so to speak, of the simple
statement: The rate of increase of mass with time of the component Si is
dt
& junction of, or is determined by, the masses mi, m2 , ma, of the components
Si, S%, Ss,
as well as by the volume v and the temperature T. Precisely similar
ia the construction to be placed on the equations (3).
EVOLUTION CONCEIVED AS REDISTRIBUTION 43
dni-i
= F (nn, mz, ms ; v, T) (2)
at
Now it is this habit of thought, expressed in equation (2) ,
that
is to be transplanted into the contemplation of problems of evolution
in general, and organic evolution in particular; this point of
view, this perspective, which regards evolution as a process of redistribution
of matter among the several components of a system,
under specified conditions.3
Having thus passed from the specific to the general from the
case of physico-chemical systems to a general formulation -we now
retrace our steps to the particular, but in a new direction. We
now contemplate the kind of systems that form the object of study
of the biological sciences.
With the outlook gained in our preceding reflections we envisage
the life-bearing system, in the progress of evolution, as an assembly
of a number of components: Biological species; collections or aggregations
of certain inorganic materials such as water, oxygen,
carbon dioxide, nitrogen, free and in various combinations, phosphorus,
sulphur, etc.
These components are placed in various relations of mutual interaction
under specific conditions of area, topography, climate, etc.
Under these conditions each may grow, decay, or maintain equilibrium.
In general the rate of growth ~7T~of any one of these components
will depend upon, will be a function of, the abundance
in which it and each of the others is presented; this rate of
growth will also be a function of the topography,
4
climate, etc.
If these latter features are defined in terms of a set of parameters
Pi P2 . . .
PJ, we may write, in the same sense as equation (2)
3
Compare F. B. Jevons, "Evolution," Macmillan, 1902, Chapter VI, p. 72.
4
Terrestial species have an essentially two-dimensional distribution, so
that area functions here in a manner somewhat analogous to that in which
volume enters into physico-chemical relations. Aquatic life, with its threedimensional
sphere of activity, is enacted in systems whose extension is
described in terms of volume. More detailed topographic parameters may be
required to define in sufficient completeness the configuration, the structure
of these systems. The absence of such detail from the more familiar formulations
of chemical dynamics is due to the purely accidental circumstance
that the systems commonly dealt with in that branch of science are either
homogeneous, or of comparatively simple heterogeneous structure.
44 ELEMENTS OF PHYSICAL BIOLOGY
~ = Fx (Z 1; Z,, . . .
XnJ Pi, Pt, . . . Pi)
CW
j~v
= Ft (Xi, Xz, . . .
Xn', Pi, Pa, PI) (3)
at
j-y=
Fn (Xi, X^ . . . XR ; PI, Pj, . . .
Pj)
at
In general there will be n such equations, one for each of the n
components.
The purport of these equations (3) remains uncertain so long as
the components (e.g., biological species, etc.) Si Sz . . $n are
undefined. What definitions we may adopt for these is, in accordance
with the principles discussed anteriorly, a matter for arbitrary
disposition; though we may be guided in our choice by considerations
of expediency. These may advise different policies from case
to case, according to the particular phase of the problem taken in
view. The conclusions reached will, of course, depend upon the
particular definitions chosen. This is as it should be; the conclusions
apply to the components as defined, and, in general, to no
other. This seems clear enough, but if any further exposition is
needed, it will be found In examples shortly to be considered.
Intra-Group Evolution. While the precise definition of the components
Si S2 . . ,
SB may, and indeed must, be held over for
determination as each separate phase of the general problem is
singled out for treatment, yet there is one class of cases regarding
which it is appropriate to set forth certain reflections at this
juncture.
It may have been observed that so far nothing has been said
regarding one aspect of or/ganic evolution which, in the history of
the subject in the past, and hi the minds and writings of its exponents
and students today, occupies a dominating position
namely, the relation of evolution to the modification of species
with the passage of time, and, in particular, to the origin of new
species.
Now a biological species, however defined, is not a homogenous
group. It comprises portions (individuals) varying more or less
widely with regard to numerous features, such as stature, weight, etc.
If our description of the distribution of matter in the system is
to be at all exhaustive, we shall need to know, not only the extent
EVOLUTION CONCEIVED AS EEDISTEIBUTION 45
(total mass) of each species, but also its constitution, as expressed
by the frequency, the relative abundance, of each statistical type
within the species. In the case of man, for example, we may wish
to know the fraction F (56) of the total population whose height
is comprised within the limits 58 and 57 inches at a given instant.
And in observing the evolution of the system of which this population
forms part, we shall be interested, not only in the growth
(or decay) of the population as a whole, but also in the rate of
change of the abundance (frequency) of each statistical type. This
phase of the problem does not differ essentially, in character, firom
that first considered : it is essentially a question of distribution and
changes in distribution of mass in the system among its several
cornponent; only now we have fixed our attention on a different
set of components, components defined in a different way, on a
finer scale. In the first instance we had taken in view the distribution
and changes in distribution of the matter of the system
among the several major groups or species; now we are considering
the distribution within each such group. This division of the general
problem of organic evolution into two aspects has certain practical
advantages, and it will be convenient to have names to designate
the two separate aspects or domains of evolution. We shall
accordingly speak of inter-group evolution on the one hand, when
referring to changes in the distribution of the matter of the system
among several component groups; and we shall speak of intra-group
evolution when referring to changes in the distribution of matter
within the group, among its statistical types, however defined.
It is possible to set up equations relating to intra-group evolution,
similar in general character to those set forth above relative
to inter-group evolution. However, this phase of the problem is
probably treated more satisfactorily in other ways, of which some
examples will arise in due course.5
It is nevertheless, desirable, to indicate in the system, of equations
(3) the incidence of intra-group evolution.
Conveniently this may be done by writing the ith equation, for
example, in more detailed form:
= Fi (X1} Z,, . . . Xn ; Pi, PSj
. . .
PI; Qi, Q, . . . Qk) (4)
at
5
See Chapter IX, p. 122; Chapter XIII, p. 170; Chapter XXV, pp. 348 et seq.
46 ELEMENTS OF PHYSICAL BIOLOGY
where Qi, Q2) , . .
Qk are parameters defining the character of
several components (e.g., biological species) S; such definition may
take the form of a set of characteristic frequency functions, or
some other form. These parameters will, in general, be functions
of the time, that is to say, each component may, in general, be
variable in character through the occurrence of intra-group evolution.
Whether this variability is limited, or constrained to follow
a certain course (orthogenesis), whether variations take place in
continuously graded series or per saltum, or in any or all of these
ways, are questions which will not be discussed at this point. At
present all that need be said is that the origination of a new species
in any of these ways falls within the scheme of our description of
evolution as a change in the distribution of matter among the components
of the system.
6
It may be remarked, in passing, that in general a complete definition
of the system may require an infinite number of parameters
P and Q; this does not necessarily cause any undue complication
in practise, since in many cases certain of the parameters P, Q either
remain constant, or change so slowly that, in discussing changes in
the variables X, we may treat these parameters as if they were
constant.
The parameters Q, defining the character of the species, are in
general functions of the time, as has already been remarked. In
this respect organic evolution exhibits a very important distinction
as compared with chemical evolution, i.e., the evolution of a
system in the course of chemical transformation. In such a system
the character of the components is usually fixed once for all.
Water is H2 for all time, unlike a species of organism which is
subject to change in character. One important result of this is
that, so far as we know, organic evolution is a process without end,
for there seems to be no limit to the variety of forms of living matter,
as there is no limit to the variety of geometric configurations
and mechanical systems that can be formed from a given portion
of matter. Chemical evolution, on the contrary, terminates, under
constant condition, in a definite equilibrium, determined once for
all, by those conditions.
8
This is not a new definition of evolution, it is a conception of evolution
wholly compatible with the definition that has been laid down in
preceding pages.
EVOLUTION CONCEIVED AS BEJHSTRIBTJTION 47
It will be observed that the fundamental equations (3) resemble
in form the equation (4) of Chapter II, which was given as a
typical example of the equation for an inertia-free or completely
JV
damped system. The velocities -37 are single-valued functions F
of the variables X. It is the single-valued character of the functions
F that gives the system its stamp as an inertia-free or
completely damped system; a system in the course of typically
irreversible transformation.
It is not maintained that these equations cover all cases that
may be brought within the purview of the present study; nor shall
we, in all cases, be tied down to the scope of these equations. They
are, however, of very broad scope, and, upon reflection, will be
found to cover at least a large and significant portion of the field
of our interests here.
To one point, however, it may be well to draw attention. To
read these equations in their broadest interpretation we must be
prepared to consider cases in which the phenomenon of lag or lead
enters. Perhaps the terms lag and lead require explanation. In
some cases the course of events today depends on certain features
in the state of the world at a previous date. So, for example, the
number of persons of age 50 in the year 1924 depends (among other
things) on the birthrate in the year 1874. Or, to quote another
instance, the number of new cases of scarlet fever today depends
on the number of infections a week ago. There is thus a lag in
the appearance of the observable effect in the system. In other
cases there may be a lead. The price of land on Church Street
today may suffer an increase or perhaps decrease because it becomes
known that in a year's time a railway station is to be built
nearby.
Since effects of this kind must be contemplated as a possibility,
we must be prepared to read our equations in the following sense:
The rate of increase ~T7~of the mass of the component Si at the
at
present instant, is a function of the masses Xi} Xz, etc., at some
other instant of time, say Xi at (t pi), X2 at (t p2), etc., where
some of the p's may be negative (corresponding to a lead).
48 ELEMENTS OF PHYSICAL BIOLOGY
We must be prepared to consider our equations in this interpretation.
Illustrative examples will not be offered here, as the mathematical
treatment of these cases is somewhat troublesome. The
interested reader will find an illustration in the author's monograph
on the Ross Malaria equations, Am. Jour. Hygiene, vol. 3,
January Supplement, 1923. Only a reflection of general character
shall find its place here. It is characteristic of systems whose
history is defined by equations thus involving a lag, that, in general,
the course of events at a v a instant is dependent upon the previous
history of the system over a certain finite range of time. The
consequences of this feature are somewhat singular. If the world's
events followed a system of equations of this kind, we might have two
worlds, in every respect identical today, but each with a different
past, and, in consequence each with a different future. And a similar
reflection necessarily applies in the case of lead.
This conclusion is perhaps not in harmony with a mechanistic
conception of the universe. But the phenomena of memory and of
will are of precisely such character as to introduce lags and leads
into the world's equation, and we may be well advised to keep our
minds open as to the possible effects of this circumstance upon the
course of events.
It should be observed, that the appearance of a lag or lead in
our equation may be spurious. It may be due to a species of mathematical
shorthand. It is easier to describe a person as having become
infected with scarlet fever three days ago, than to describe
precisely his present state today ensuant upon that infection. Hence
we may prefer to, or, for lack of detailed knowledge we may be
forced to write our equations in terms of (t p), although, if all
the facts were known, we could, were we so disposed, write them in
terms of t. And the same reflection applies to the appearance of
a lead in our equations. Whether, with complete knowledge of all
circumstances bearing on the situation, there should still remain
a residuum of influences that could find expression only in terms
of a lag or lead, this, perhaps, is fundamentally the nature of the
problem of the influence of consciousness upon the course of the
world's historv.
CHAPTER V
THE PROGRAM OF PHYSICAL BIOLOGY
. . . . It will be the function of this new branch of science to investigate
biological phenomena as regards their physical aspects, just as Physical
Chomistry has treated the physical aspects of chemical phenomena. Because
thiy field has not yet been systematically explored .... the individual
data of Physical Biology appear, as yet, as more or less disconnected facts, or
as regularities for which no proper place is found in the existing scheme of
pre,Bent-day science; and the investigations of isolated problems in this field
are as yet carried on as something of a scientific hobby by amateurs, with the
result that they are guided by chance rather than by plan. . . . and are
often totally lacking in any fundamental guiding principles or connecting
theo.'y. As results gathered in this disconnected fashion accumulate, the need
of their unification into a harmonious whole, into a distinct discipline of
science, becomes more and more acutely felt. Such unification necessarily
involves the working out of a viewpoint that shall make the several facts and
relations fall in line naturally in an orderly system; in other words, what is
needed is a labor of organisation. In the course of this, new and unforeseen
problems will inevitably arise, and a fruitful field of scientific endeavor should
thus be opened for the investigator. Porstmann.
A first use to which we may with advantage put the results of
the preceding analysis, is to systematise the subject here treated;
there will thus be gained a general plan of work and a division of
the topic into natural sections, upon which the arrangement of
the succeeding chapters will, in the main, be based.
Physical Biology,
1
as here conceived and discussed, is essentially
a branch of the greater discipline of the General Mechanics oj Evolution,
the mechanics of systems undergoing irreversible changes in
the distribution of matter among the several components of such
system.
1
In introducing the term Physical Biology the writer would suggest that the
term Biophysics be employed (as hitherto) to denote that branch of science
which treats of the physics of individual life processes, as exhibited in the individual
organism (e.g., conduction of an impulse along nerve or muscle); and
that the term Physical Biology be reserved to denote the broader field of the
application of physical principles in the study of life-bearing systems as a
whole. Physical biology would, in this terminology, include biophysics as a
subordinate province.
For a summary statement of what might be termed the program of biophysics
see A. Forbes, Science, 1920, vol. 52, p. 331.
49
50 ELEMENTS OF PHYSICAL BIOLOGY
It so happens that many of the components that play an
tant role In nature, both organic and inorganic, are built
large numbers of individuals, themselves very small as compe The
with the aggregations which they form. Accordingly the stud^ono-
S3
r
stenis of this kind can be taken up in two separate aspects, nam x- 3,
first with the attention centered upon the phenomena displaceby
the component aggregates in bulk; we may speak of this as hose
Bulk Mechanics or Macro-Mechanics of the evolving system. A^al,
secondly, the study of such systems may be conducted with f
eviattention
centered primarily upon the phenomena displayed The
the individuals of which the aggregates are composed. This brand's
of the subject may suitably be termed the Micro-Mechanics^wo
the evolving system. It is evident that between these two branc'ent
or aspects of the general discipline there is an inherent relat-ilar
arising from the fact that the bulk effects observed are of the na
of a statistical manifestation or resultant of the detail workiiistic
the micro-individuals. The study of this inherent connected of
accordingly, the special concern of a separate branch which w^ads
speak of as Statistical Mechanics, This terminology is i]P our
coincident with accepted usage, but in part must be unden the
to refer to an expansion of the subject beyond the bounds h
covered, whereby its scope shall be extended so as to inched in
statistical treatment of the dynamical problems presented &thegregates
of living organisms; that is to say, aggregates of ? betransformers
possessing certain significant special properties sribe
Each of the branches of the mechanics of evolution enur'SAce
so far naturally splits up into two subdivisions, according
v be
devote our attention to the material changes or to the energy ch all
involved. By an extension of prior usage in physical chemistr^n
may employ the term Stoichiometry to denote that branch of
"
science which concerns itself with the material transformations.,
with the relations between the masses of the components. The
discussion of this branch presents itself as the more elementary
task, and will therefore be taken up first; after this has been disposed
of we shall be better prepared to discuss the second aspect,
the Energetics or Dynamics of Evolution.
Taking now a survey of the Stoichiometry of systems in evolutionary
transformation, we can hardly find a better guide, for
the organization of this subject, than the fundamental equations
PROGRAM OF PHYSICAL BIOLOGY
^7 = F, (X, X,, . . .
Xn; P, Q)
at
which we may speak of as the fundamental equations of the Kinetics
of Evolution, since they furnish expressions for the velocities of transformation
and exhibit the relations between these velocities and
the masses of the several components, as well as the parameters
P, Q.
The very form of these equations suggests, as the first and most
elementary problem, the treatment of the case of evolution under
constant conditions, as defined by constant P's and Q's. This
will, accordingly be the coarse here adopted, treating first the general
case of n variables Xi, X2 ,
. . . Xn ,
and then some special cases
in which the number of variables is restricted to 1, 2 and 3.
The perfectly general case, of evolution under conditions of
wholly unrestrained variability of the P's and Q's, is mathematically
uninviting (though not wholly intractable), and is also of minor
interest in practice. Little will therefore be said of this. Certain
special types of changes in the P's and Q's (e.g., slow changes) will
find a place in the next subdivision of the general subject, the Statics
of evolving systems. This branch is, in a sense, a special division
of the Kinetics of Evolution, namely that which concerns itself
with systems in which the velocities of transformation are zero,
so that there is Equilibrium, or, to be more exact, a Steady State.
This, of course, implies, strictly speaking, constancy of the parameters
P, Q. But something very much like equilibrium presents
itself in certain cases when these parameters change slowly. There
may then arise what has been termed a Moving Equilibrium. In
view of the important role which such moving equilibria play in
nature, their discussion must form a part of the program of Physical
Biology.
A second special problem of evolution under changing conditions
(changing parameters P) that lends itself to treatment with comparative
ease, is that which concerns itself with initial and end states
(equilibria), as influenced by changing conditions, without demanding
any information regarding the intermediate steps passed through
by the evolving system. So, in physical chemistry, we may enquire
what will be the effect, upon equilibrium, of a change in pressure
or in temperature. Similar questions may be raised regarding
52 ELEMENTS OP PHYSICAL BIOLOGY
equilibria or steady states in life-bearing systems, and the matter
calls for at least passing notice. This leads to the consideration of
the Principle of Le Chatelier and, as a natural sequel of the train of
thought thus started, to the examination of relations which may
exist between certain of the parameters P, somewhat as, in physical
chemistry, significant relations exist between pressure and volume,
for example. It is found that some, at any rate, of the parameters
P present analogy to the intensity and the capacity factors of an
energy; out of this have arisen in the past sporadic efforts to establish
systems of social dynamics and the like, which, however,
have been based upon an acceptance as identity of what is only
analogy. Those who have followed this road have been led, not so
much perhaps to erroneous conclusions, as to blind alleys, to barren
fields. True progress can be expected only by retracing one's
steps from such tentative excursions and striking out in a new
direction; forsaking the way of quasi-dynamics, and breaking a
trail toward a system of true dynamics, both of the individual
(micro-dynamics) and of the system as a whole (macro-dynamics).
Of intra-species evolution, as expressed in changes in the parameters
Q that define the character of the several species, little will
be said here. The reasons and justification for this step-fatherly
treatment of so important an aspect of our topic have been set
forth in the preceding text. A discussion of at least one phase of
intra-species evolution, falling under the head of dynamics, will,
however, be presented when dealing with that phase of the subject;
and we shall briefly note, in due course, some aspects of intraspecies
evolution, as discussed more particularly by J. B. S.
Haldane.
These, in broad outline, are some of the principal land-marks
in the territory ultimately to be covered by Physical Biology, and to
be given a preliminary survey here. In concentrated form the
lay of the land, as set forth above, is sketched in the diagram or
scheme table 1, which should be found helpful both in presenting to
the eye the salient features of the field of investigation, and als'i in
furnishing a logical basis for the systematic arrangement of the
subject in the ensuing chapters.
It remains, in this chapter, to enumerate the methods by which
Physical Biology may be expected to develop. For the gathering
of data two methods are available: observation in natural condi-
1
.2 -S
Q,
1H
O
"
54 ELEMENTS OP PHYSICAL BIOLOGY
tions, and observation under experimental (laboratory) conditions.
Examples of both these methods will be noted.
For the elaboration of data, the establishment of regularities
(Jaws/, there is available in this field, as everywhere in science, the
method of induction, aided, if need be, by statistical technique.
In this volume, however, emphasis will be laid upon deductive
methods of mathematical analysis, as applied either to data furnished
by observation, or to "unknown" quantities, blanks, as it
were, in our formulae, ready for numerical substitution whenever
concrete data become available.
The principal subdivisions of our topic, and the relations between
them, as outlined above, are summarised graphically in table 1.
KINETICS
CHAPTER VI
THE FUNDAMENTAL EQUATIONS OF KINETICS OF EVOLVING
SYSTEMS
L'emploi des signea mathe'matiques est chose naturelle toutes les fois qu'il
s'agit de discuter des relations entre des grandeurs. A. Cournot.
General Case. We now proceed to the systematic study of
the subject in accordance with the general plan laid down in the
preceding survey of the Program of Physical Biology. According
to this schedule (table l)we approach first of all the section of Macromechanics
(that is to say, the mechanics of evolving systems
regarded as bujlt up of component species in the gross), without
carrying the analysis down to the finer details of individual organisms.
And, of the field of Macromechanics, we shall here take up
the section of Stoichiometry, that is to say, we shall for the present
confine our attention to the relations between the masses involved,
leaving aside for a later section the associated energy changes.
Of the general field of Stoichiometry the first division to be taken
up, according to our schedule, is the Kinetics of Evolution, and we
shall here begin with a brief consideration of the Fundamental
Equations in their general form
= Fi (* Xtl . . . XK ; P, Q) (1)
at
One phase of the general problem before us must evidently be
the study (by observation, experiment, or any other method
available) of the character or form of the functions F which tell us
how the growth of each component is dependent upon the other
components and the parameters P and Q. It might be supposed,
indeed, that until this phase of the problem had received consideration,
the system of equations (1), would be at best a barren expression
of facts. But this is a misconception. Without knowing
anything regarding the precise form of these functions, a good deal
of information of considerable interest can be derived from these
equations; and before proceeding to the consideration of concrete
57
58 ELEMENTS OF PHYSICAL BIOLOGY
cases, in which something is known regarding the particular form of
the functions F, it is proper, at this point, to extract from the fundamental
equations (1) all the information that we can.
This requires a little mathematical manipulation of comparatively
simple character,
The variables X, the masses of the several components of the
system, are not, in general, wholly independent. They are subject
to certain constraining relations. For example, for any self-contained
system, we shall have an equation of the form
X, Xa + . ,
.+Xn = SZ = A = const. (2)
expressing the constancy of the total mass of the system; and a
similar equation holds separately for every chemical element.
1
These equations prohibit certain changes of mass in the system.
They are analogous to equations introduced in mechanical problems^^,^ /by
the geometric constraints that limit possible displacements
(as in the case of a ball rolling down an inclined surface, and prevented,
by the resistance of the surface, from falling vertically).
They are commonly spoken of as equations of constraint.
Equations of constraint such as
2Z = A = const.
will in general enable us to eliminate certain of the variables X,
expressing them in terms of the other X's and of certain constants A.
If, by the aid of m such equations of constraint, m variables have
been eliminated, the system of equations (1) can be reduced to a
simpler one, of identical form, but containing only (n m) variables ^.. (
and the same number of equations. .-. {
In all that follows we shall assume that this has been done, that
we now read equations (1) in this sense : the n variables Xi X% . . . .
Xn are those left over after eliminating as many of the X's as the
equations of constraint permit. The functions F will in general,
after this elimination, contain constants A introduced by the equations
of constraint.
We shall now, except where otherwise stated, restrict our considerations
to the case in which the parameters P and Q and A
are either constant, or change so slowly that we may disregard ff
/"'
1
Except in those rare cases in which radioactive or other atomic disintegra- /'
ticms occur. / *_
FUNDAMENTAL EQUATIONS OF KINETICS 59
their variation. This means that we restrict ourselves to the consideration
of simple inter-group evolution under constant condi-
tions.
With this understanding, the first point we may observe about
the system of equations (1) is that they define certain conditions of
equilibrium, or, to be more precise, of steady states. For such a
state ensues whenever all the velocities vanish, that is to say,
according to (1) whenever
Fi = F, = . . .
= Fn = (3)
We thus have n equations between the n variables Xi, XZi
. . .
Xn, which, in general, determine certain values
(4)
such that when masses of the several components have these values,
they persist in these values; the system is at rest, as regards changes
in the distribution of matter among its components Si, S2 ,
. . . Sn .
In general there will be a number of such possible equilibria,
some of which will be stable, some unstable. The determination
of their number and character is a technical point in the theory of
equations, for the general treatment of which the reader must be
referred to the pertinent literature (see, for example, Picard, Traite"
d' Analyse, vol. 1, 1891, pp. 83, 123; vol. 2, 1893, pp. 183, 193, 196,
footnote) .
To give a touch of concreteness to the discussion at this stage a
very simple example may be given to illustrate how several equilibria
may occur in a life-bearing system. A perfectly screened
dwelling may be kept indefinitely free from flies. This is a condition
of equilibrium, but of unstable equilibrium; for if only a few
flies gain access, presently these will breed, and the room will
become inhabited by a population of flies whose number will depend
on the amount of food present, on the measures taken to combat the
pest, etc. Unless these measures are v,ery active, the flies will not
be wholly exterminated, but the population will attain some approximately
steady number (for a given season) . There are, then, in this
case, two possible equilibria; one with a zero population of flies, the
other with some positive number of fly population.
60 ELEMENTS OF PHYSICAL BIOLOGY
The equilibria in nature, involving countless species, are of course
much more complicated in character, but the general principle is
the same; and we must expect that in general a variety of different
equilibria are possible, some unstable and some stable.
For the further discussion of the equations (1) it is now desirable
to express these relations, not in terms of the masses X}
but in
terms of the excess of each mass Xi over its corresponding equilibrium
value,
xi = Xi-d (5)
The equation (1) then takes the form
,
~r- = / i
at
(6)
the parameters P, Q being omitted, for the sake of brevity. Expanding
the right hand member by Taylor's theorem we obtain
the system of equations
dt
dt
+ + +
. -f
dt
A general solution of this system of equation is
(7)
(8)
where the (?'s are constants, of which n are arbitrary, and Xi, X2
. . . Xn are the n roots of the equation
2
for X
Oil X 0,ii . . . dm
- 2)(X) = (9)
2
This equation is commonly spoken of as the characteristic equation. For
greater detail see A. J. Lotka, Proc. Am. Acad. Sci., vol. 55, 1920, pp. 137-153.
FUNDAMENTAL EQUATIONS OF EJNETIC3 61
It is seen by inspection of the solution (8) that if all the X's are
real and negative, xi}
x2 ,
. . , xn all approach zero as t increases
toward infinity, since e~m = 0. But according to (5)., as the
re's approach zero, the X's approach their equilibrium valu_es C.
In this case, obviously, equilibrium is stable, since, by allowing
sufficient time to elapse, we can always make the system approach
as near as we please to Xi = Ci, for all values of the subscript i.
Precisely similar conclusions hold if some or all of the X's are complex,
and the real parts of all the X's are negative.
If all the roots X ar(
e real (and negative),
3
each term in the
solution (8) diminishes continually and approaches zero asymptotically
as t approaches infinity.
If all the constants G are of the same sign, the sum. of the series
also will, evidently have a similar type of approach to equilibrium.
If some of the G's are positive, others negative, there may be a
species of irregular oscillations, the mass X of the component rising
sometimes above its equilibrium value (7, sometimes falling below
it. Ultimately, however, the equilibrium is approached from one
side only, s,ince ultimately tbe term containing the numerically
smallest X will predominate over all other terms.
If some of the roots X are complex, the solution will contain
truly oscillatory terms, since the exponential function, for complex
exponents, assumes the trigonometric form
g ft _}_ i gin ft) (10)
In this case there will be regular oscillations about the equilibrium
position; in general these oscillations will be damped, that is to say,
their amplitude will diminish, so that equilibrium is approached
more and more closely, but always with oscillation the equilibrium
is approached from both sides at once, so to speak, the oscillations
persisting forever, though on a diminishing and ultimately vanishing
scajle.
These conclusions are the analytical confirmation and extension
of an inference drawn by Herbert Spencer
4
on qualitative grounds :
3
The analytical condition that the real parts of all the roots X shall be negative
is given by Hurwitz, Math. Ann., vol. 46, 1895, p. 521. See also Blondel,
Ann. de Physique, 1919, pp. 117, 153. It might be noted here that a necessary
though by no means sufficient condition, evidently is that the absolute term
in D (X), that is to say D (0), shall be positive when n is even, and negative
when n is odd; for this absolute term is equal to the product of all the roots X.
*
Herbert Spencer, First Principles, Chapter 22, Section 173.
62 ELEMENTS OF PHYSICAL BIOLOGY
Every species of plant and animal is perpetually undergoing a rhythmical
variation in number now from abundance of food and absence of enemies
rising above its average, and then by a consequent scarcity of food and abundance
of enemies being depressed below its average .... amid these
oscillations produced by their conflict, lies that average number of the species
at which its expansive tendency is in equilibrium with surrounding repressive
tendencies. Nor can it be questioned that this balancing of the preservative
and destructive forces which we see going on in every race must necessarily
go on. Since increase of numbers cannot but continue until increase of mortality
stops it: and decrease of number cannot but continue until it is either
arrested by fertility or extinguishes the race entirely.
It will be observed, however, that our analysis enables us to be
considerably more specific in distinguishing several modes of approach
to equilibrium, and in indicating the particular conditions
under which each occurs. A point that here deserves particular
emphasis is that, in order to make these distinctions and indications,
it is by no means necessary to have a complete knowledge, or even
very extensive information regarding the functions F. Only the
coefficients of the first (linear) terms in the Taylor expansion of
these functions enters into the determinant D(X),, and only these
need therefore be known to draw the requisite conclusions regarding
the stability and mode of approach of equilibrium. Furthermore,
Spencer's commentary relates specifically to the case of species
related to each other in certain particular ways (food, enemies,
etc.): the analysis here given is framed on perfectly general lines
and covers any sort of interrelation, interdependence of the components
Si, S . . . Sn . Certain particular interrelations will
be duly considered in the course of the further development of the
theme. Here it may be well to draw attention once for all to the
fact that there is nothing whatever to restrict the application of
the principles and methods set forth to systems in organic evolution.
Indeed, a typical example to which these reflections apply
is the case of a chain of elements in the course of radioactive trans-
formation.5
It will be seen later, in considering a concrete example, that the
equilibrium equation (12) may yield zero or negative roots for C.
A zero root is simply interpreted; if the equilibrium to which it
relates is a stable one, this means that the species in question will
5
A. J. Lotka, Proc. Am. Acad. Sci., vol. 55, 1920, p. 148; Proc. Natl. Acad.
Sci., vol. 7, 1921, p. 168; Phil. Mag., August, 1912, p. 353.
FUNDAMENTAL EQUATIONS OF KINETICS 63
become extinct
1
'. It is unfit, is unadapted ultimately
7
to survive
under the existing conditions8
(as defined by the parameters P, etc.).
A negative value of C may also signify that the species is incapable
of survival under the existing conditions. Masses cannot assume
negative values. As soon as any component passes through zero
it ceases to function in the system, whose history is henceforth
represented by a new set of equations in which this component
does not appear.
These conclusions must in one respect be accepted with a certain
caution. Since there may be several equilibria, a species may be
incapable of existence in the neighborhood of one such equilibrium,
but might nevertheless succeed in maintaining itself in the neighborhood
of another equilibrium. Whether such cases occur in practise
may be left an open question. Our analysis suggests this
possibility.
e
This conclusion does not apply, of course, if the equilibrium with X = O
is unstable, as in the example of a fly population cited above.
7
It maj*, however, persist for a time, and, it may be, for a long time. An
instance in point is furnished, outside the field of organic evolution, by a
chain of elements in radioactive transformation. Although here the ultimate
equilibrium is one with a single survivor, namely the last link in the chain, yet
for millions of years the several products exist side by side in constant ratio
though in slowly diminishing amount. At the head of such a chain is always
found a "parent substance" which is the longest-lived in the chain. This is
not accident. It is easily shown that if at some prior period this substance
was preceded by a more rapidly decaying pre-parent, this latter must have
disappeared in the course of the ages. It is totally unfit, unadapted, even for
a temporary equilibrium, under present conditions. (See A. J. Lotka, Phil.
Mag., August, 1911, p. 354.)
8
It should hardly be necessary to point out that adaptation is purely
relative.
CHAPTER VII
FUNDAMENTAL EQUATIONS OF KINETICS (CONTINUED)- SPECIAL
CASE: SINGLE DEPENDENT VARIABLE
If arithmetic, mensuration and weighing be taken away from any art, that
which remains will not be much Plato.
Law of Population Growth. It will add concreteness to the present
exposition to consider at this point a numerical illustration.
The simplest possible example of numerical application of the
equations set forth in the preceding pages will be one in which there
is only a single variable X. The fundamental system of equations
then reduces to a single equation
f=F(JT) (1)
at
A. case in point arises when for any reason one particular biological
species or group grows actively, while conditions otherwise remain
substantial!}" constant.
This seems to be essentially what has occurred in certain human
populations. It is true that, in their growth, they have carried along
with them a complicated industrial system or group, comprising both
living and non-living elements. We may, however, look upon the
number of the human population itself as a sort of single index or
measure of the growth of the group as a whole.1
1
This is an example in which certain of the variables X are connected by
equations of constraint. So, for example, the number of head of cattle Nc has
for many years past, in the United States, been about six tenths of that of the
human population, Nh so that we have an equation of constraint
or, putting the average mass of cattle at 1000 pounds per head, that of a human
being at 100 pounds (an average to include all ages)
Xe =
Similar equations of constraint apply to the other species of domesticated
animals and plants, so that the mass of each can be (approximately) expressed /
in terms of the single variable Xh.
64
FUNDAMENTAL EQUATIONS OF KINETICS 65
Applying to this case the equation (1) and the general method set
forth in the preceding pages, we are led to consider first of all the equilibrium
equation
^ - F OT) = (2)
at
This equation, obviously, has a root at X = 0, for at least one
female is required to start the growth of a population. Expanding
by Taylor's theorem we shall therefore have for F a series lacking the
absolute term,
2
thus
= F (X) = aX + 6X> + cX3
+ . . . (3)
dt
Furthermore, the equation (2) will have at least one other root,
since there must be some upper limit to the growth of the population.
The simplest case satisfying this condition is that in which the right
hand member of (3) terminates at the second degree term, i.e.,
^f = oZ + &Z (4)
at
The characteristic equation
3
for X is in this case simply
A - a = (5)
and the solution of (4) therefore is
X = (?*' + Sue20 '
Substituting this in (4) and equating coefficients of homologous
terms we find
Gn =-<?! (7)
a
<?m = -!<?'x (8)
a2
so that (6) is a simple geometric series. Its sum is
Z- f - (9)
o a
1 --f>at -p at _ 1
dX2
Otherwise would not vanish, with X
dt
3
Corresponding to equation (9), Chapter VI.
4
The series (6) is divergent for large values of t if a is positive. But the
expression (9) for the sum of (6) remains a solution of (4).
66 ELEMENTS OF PHYSICAL BIOLOGY
In formula (9) either Gi or the origin of time is arbitrary. A simplification
can be effected by so adjusting the origin of time that
(10)
This fixes the value of the constant Gi
(?:=-- (11)
and the formula (9) becomes
The equation (9) can also be written in another form. If we denote
by X the value of X when t = o, and write x = X + 7, then (9)
becomes
a
X = --- (13)
"'- 1
Population of United States. Formula (12) has been applied by
Pearl and Reed6
to the population growth of the United States.
7
The calculated curve for the number TV of the population fits the observed
data over a long period of years (1790 to 1910) with remark-
5
This result can also be obtained by direct integration of (4) in finite form.
The process given above has here been followed to illustrate the general method
of the solution (S), (9), of Chapter VI.
6
P. F. Verhuist, Mem. Acad. Roy. Bruxelles, 1844, vol. 18, p. 1; 1846, vol.
20, p. 1; R. Pearl and L. J. Reed, Proc. Natl. Acad. ScL, vol. 6, 1920, p. 275;
Scientific Monthly, 1921, p. 194; R. Pearl, The Biology of Death, 1922, p. 250.
The last-mentioned work, especially, should be consulted for a detailed discussion.
For a general treatment of the population problem see also particularity
E. M. East, Mankind at the Crossroads, 1923 (Scribners).
7
Measured, in this case, by the increase in the number N of persons. This
Is evidently, in first approximation at any rate, proportional to the total
mass X of the population.
FUNDAMENTAL EQUATIONS OF KINETICS 67
able faithfulness, as will be seen from table 2 and the graph shown
in figure 4. Numerically the formula (12) here takes the form
N = 197,273,000
1+e-,-0. 03134 1'
(14)
and the time t' (in years) is dated from April 1, 1914 (t
1
, being negative
for dates anterior to this). This epoch is one of peculiar interest.
It represents the turning point when the population passed from a
progressively increasing to a progressively diminishing rate of growth.
Incidentally it is interesting to note that if the population of the
TABLE 2
Results of fitting United States population data 1790 to 1910 by equation (14)
United States continues to follow this growth curve in future years,
it will reach a maximum of some 197 million souls, about double its
present population, by the year 2060 or so. Such a forecast as this,
based on a rather heroic extrapolation, and made in ignorance of the
physical factors that impose the limit, must, of course, be accepted
with reserve.
Stability of Equilibrium. The equilibrium at X = 0, i.e., with
total absence of human population, is evidently unstable, since A = a
and a is an essentially positive quantity, its numerical value being,
for the population of the United States, 0.0313395. The second
equilibrium, corresponding to the saturation point, is evidently at
88 ELEMENTS OF PHYSICAL BIOLOGY
X = r ,
and here it is easily found by the substitution x = X +
that X = a, the equilibrium is stable.
FIG. 4. THE LAW OF POPULATION GROWTH FOR THE UNITED STATES ACCORDING
TO PEARL AND REED
The lower S-shaped limb corresponds to the actual approach to equilibrium
from below. The upper limb represents the presumptive course of
events for a diminishing population approaching equilibrium from above.
FUNDAMENTAL EQUATIONS OF KINETICS
Experimental Populations. Pearl has also fitted the same formula
to the population of a number of countries, but the range covered by
the available observation in these other cases is less extended, so that
there is less opportunity for comparison with observed figures. Of
particular interest is an application of the same formula, also by
Pearl, to an experimental population of fruit flies (Drosophila). In
this case practically the entire range of the S-shaped curve defined
by equation (12) is realized, and a glance at the plot in figure 5
shows that the agreement of the observed figures (represented by
small circles) and the calculated curve is exceedingly satisfactory.
Still closer is the agreement in the case of bacterial cultures studied
aso
ss-s
125
JOO
GftOWH Of DKOSOPHILA
z
Days.
FIG. 5. GROWTH OF A POPULATION OF DROSOPHILA (FRUIT FLIES) TJNDEB
CONTROLLED EXPERIMENTAL CONDITIONS, ACCORDING TO PEARL AND
PARKER
by H. G. Thornton8
(Annals of Appl. Biology, 1922, p. 265), whose
observations are set forth in table 3 and are shown by the small
circles in figure 6 ;
the theoretical curve to fit these points, as computed
here in the laboratory, is shown in the fully drawn line. As
will be seen the agreement is excellent. This is due in part to the
fact that the figures plotted represent the means of a number of
individual cultures.
A veiy particular interest attaches to this example, inasmuch as
it forms, as it were, a connecting link between the law of growth of a
8
Similar results have been obtained by A. G. McKendrick and M.
Kesava Pai, Proc. Roy. Soc. Edin., vol. 31, 1911; for a study of the growth of
yeast, see A. Slator, Trans. Chem. Soc., 1921, vol. 119, p. 126.
70 ELEMENTS OF PHYSICAL
population, and the law of growth of the individual. A colony of
unicellular organisms, regarded as a whole, is analogous to the body of
a multicellular organism .
Or, to put the matter the other way about,
a man, for example, may be regarded as a population of cells. We
need not, therefore, be greatly surprised, if the growth of the multicellular
organism should be found to follow a law similar to that
exhibited by populations. And in point of fact, as will be shown a
little further on, this expectation is in not a few instances fulfilled.
TABLE 3
ifrowth of bacterial colony, according to H. G. Thornton (Ann. Appl, Biol.,
p. 865)
For graph see figure 6
*According to the equation y = 0.2524
e +0.005125
Diminishing Population. It may be noted here that in one respect
the formula (12), while particularly simple in form, is of more
restricted scope than (13). The former defines the characteristic Sshaped
curve that appears in the graphs of actual populations shown
in figure 4. But formula (13) gives a more complete definition of a
curve composed of two limbs; one of these is the S-shaped curve
already considered. The other is a steeply descending arc, shown in
the upper portion of figure 4. This portion of the curve has not been
4
FUNDAMENTAL EQUATIONS OF KINETICS 71
realized in any recorded population. It represents the computed
course of events if the population initially exceeds its equilibrium
strength, i.e., if is positive.
so
43
Pays.
FIG. 6. GEOWTH OF A BACTEBIAL COLONY (B. DENDEOIDES)
Observations by H. G. Thornton
Growth of Individual Organism. Although, strictly speaking, the
growth of the individual organism is a subject properly belonging to
the field that has here been termed "micromechanics" (Chapter V),
yet, in view of the very close analogy which has been found to exist,
in certain cases, between the law of growth of a population, and that
of the individual, it may be noted here that the formula (9) has also
72 ELEMENTS OF PHYSICAL BIOLOGY
been applied by T. B, Robertson9
and by Wo. Ostwald10
and others,
to the growth of the individual organism. An example of such application
Is shown in figure 1, which exhibits the growth day by day, of
male white rats according to observations by H. H. Donaldson"
and computations by T. B . Robertson. The fully drawn curve represents
the calculated values of the mass of the rats (in grams) at different
ages. The circles indicate selected observed values, namely,
those which diverge most widely from the computed values. It
will be seen that up to about the one hundredth day the agreement is
good. Above this there is no agreement worthy of the name.
Another example,, and one in which the computed curve fits the
observed values with remarkable agreement, is the growth (in height)
of sunflower plants as studied by H. S. Reed and R. H. Holland.12
The curve to fit these observations has been recomputed by the
method of least squares by Dr. L. J. Reed, who has very kindly placed
his results at the author's disposal. They are shown in figure 8. The
observations on which they are based are shown in table 4. It will
be seen that the fit is practically perfect through the whole range of
observations.
In practice we are not usually given the differential equation (4),
but data corresponding to points on the integral curve (12). We
then have the problem of determining from these points the characteristic
constants of the curve. The detailed working out of theprob-
9
Archiv liir die Entwiekelungsmeehanik der Organismen, 1907, vol. 25,
p. 4; 190S, vol. 26, p. 108; The Chemical Basis of Growth and Senescence,
publ. Lippincott, 1923. T. B. Robertson also quotes A. Monnier, Publications
of Irist. of Botany, Univ. Geneva, 1905, which in turn refers to Chodat as having
recognized the analogy of organic to autocatalytic growth. The idea is
rather an obvious one, which probably has occurred to many. The earliest
reference noted by the writer is L. Errera, Revue de 1'Univeriste de Bruxelles,
1899-1900, May issue. Something very similar is found in Ostwald, Vorlesungen
Ober Naturphilosophie, 1902, p. 342. These last-mentioned lectures were
published in 1902. but were actually delivered somewhat prior to that date.
The writer recalls that Ostwald referred to the matter in his lectures on General
Chemistry in 1902, and probably others will recall similar references on
earlier occasions.
10
Die Zeitlichen Eigenschaften der Entwickelungsvorgange, Leipzig, 1908.
11
Boas Memorial Volume, 1906, p. 5; see also the same author's book
The Rat, 1915.
12
H. S. Reed and R. H. Holland, Proc. Natl. Acad. Sci., vol. 5, 1919, pp.
135-144.
FUNDAMENTAL EQUATIONS OP KINETICS 73
lem may well be left to the reader, after pointing out the following
interesting property of the curve (12). Taking reciprocals and
writing A for a/b, we have
_ _3 a_/o)
A"
log (A% -1) = - o(f-* )
(15)
(16)
oo
O SO 100 (SO ZOO ZSQ 3CO
Ays. in days.
FIG. 7. GROWTH OF RAT ACCORDING TO H. H. DONALDSON AND T. B.
ROBERTSON
Circles indicate only those observations that diverge most widely from the
calculated curve.
where t is the time corresponding to the point of inflection of the Scurve.
Thus if we plot A 1 against t on logarithmic curve paper,
we shall obtain a straight line diagram. This is shown in figure 9 for
the same data (growth of sunflower) which have already been
74 ELEMENTS OP PHYSICAL BIOLOGY
00
y-
Z6/.I
O 7 H 2/ 28 35 -?2 +9 ?& 63 TO 77" &<f
Age in days.
FIG. 8. GHOWTH OP SUNFLOWER SEEDLINGS ACCORDING TO H. S. REED AND
R. H. HOLLAND; COMPUTED CURVE BY L. J. REED
TABLE
Growth in height of sunflower plants (H. S. Reed and R. H. Holland;* computed
values by L. J". Reed)
MEAN HEIGHT
Troc. Natl. Acad. Sci., vol. 5, 1919, p. 140.
10.0
7 63 7O 77 84- SI 38
Age. m day-s,
FIG. 9. GEOWTH OF STJNFLOWEB SEEDLINGS
The same data as in figure 8, but plotted in logarithmic diagram, with
ordinates as indicated in the legend.
75
exhibited by another method in figure 8 .
Taking now three equidistant
points of time ^ t is and the corresponding ordinates &, 2, &, it
is readily shown that
ta-ft ^gLlLg; (17)'
(t + &)-2& TM-&
where 3/ denotes the geometric mean of 1 and |3 and while mdenotes
their arithmetic mean.
Thus the constant A can be determined from three suitably chosen
points on the curve (after smoothing graphically if need be). The
other constants then follow easily.
Autocatakinesis. Both Robertson and Wo. Ostwald .draw attention
to the similarity between the law of growth (12) and the law of
formation of a chemical substance by autocatalysis under certain conditions.
The analogy is interesting, but must not be taken too
seriously, inasmuch as in the one case the rate of growth is determined
by ordinary chemical influence, in the other (organic growth)
by a complicated combination of factors both of chemical and of
physical character. Rather more to the point seems to be the suggestion
made above in connection with the law of growth of bacterial
colonies, that, the body of a multicellular organism being a "population"
of cells, it is not altogether surprising that it should be found to
follow the law of growth of a population.
A suggestion of terminology by Wo. Ostwald is worth noting. He
proposes that growth of any kind, in which the substance or structure
itself acts as nucleus for the formation about it of further quantities
of the same substance or structure, be broadly termed autocatakinetic
growth, the narrower terms autocatalysis of autocatalytic
growth being reserved for that particular kind of autocatakinesis
which is chemical in character. This is a useful suggestion, as it
leaves the term autocatakinesis quite neutral and free from any implication
or restrictions as to the mechanism by which the growth takes
place.
CHAPTER VIII
FUNDAMENTAL EQUATIONS OF KINETICS (CONTINUED) SPECIAL
CASES: TWO AND THREE DEPENDENT VARIABLES
Life is a system of relations rather than a positive and independent existence
G. A. Sola.
Interdependence of Species. The case of two dependent variables,
X\, Xz, which we now approach, is of interest as the simplest
example exhibiting the relation between interdependent species.
This relation can take on a variety of forms. Most fundamental,
perhaps, is that form of interdependence (1) in which one species
Si serves as food to another species SZ) so that, in this sense, Si
becomes transformed into S2} thus
Si > $2
All animal species, and many plants also, thus derive their substance
from other species on which they feed. And several different
types of this form of interdependence are observed. In
the first type, (a) the organism $2 kills Si outright in the process
of feeding upon it. We might term S2 in such a case an episite
of Si, in contradistinction from the second type, (&) in which 2
lives on Si without killing it outright, being parasitic upon Si.
The host is in most cases more or less injured by the parasite, and
all pathogenic organisms fall into this class. For this reason quantitative
epidemiology appears as one of the special branches of the general
subject under consideration here.
A third type (c) of interdependence, is that in which S2 is saprophagous
or saprophytic, feeding upon the cadavers of Si after death
from other causes; or, /S2 may live on waste products discharged by
Si. In contrast to episites and parasites, saprophagous species are
presumably beneficial rather than injurious to the host species,
since they function as scavengers. Still another type, (d), of interdependence
is that of symbiosis, in which Si and *S2 live in partnership
which, as a rule, is in some degree mutually beneficial. Man and
his domesticated animals and plants are obvious examples of this
type.
77
7S ELEMENTS OF PHYSICAL BIOLOGY
In addition to these types (Id) to (Id), another large group of
cases (2) are those in which two or more species compete for a common
food supply.
In their general form the fundamental equations of kinetics
for the case of two dependent variables are
dt
dl
-j- AuXi -r AitXz + .
+ AtiXi -r AttKi T .
(1)
or, after the transformation (5), of Chapter VI
= anXi -f-
at
dxs
~dt
= CsjXj + QssZs 4- . . .
(2)
We may note, first of all, as a general rule, the following observations
regarding the coefficients a in the several types of interdependence
enumerated above:
la. S2 lives on Si by killing Si outright (episitic type). In this
case, evidently S2 unfavorably influences the growth of Si} while,
on the contrary, Si is advantageous to the growth of &, so that
5 dZi
Z2 dt
< 0, i.e., an <
t
~ > O, i.e., an >
oXi at
(3)
(4)
lb. 82 parasitic upon Si. Here aiz < 0, 021 > 0, as in case la.
Ic. S2 saprophytic upon Si, or living upon waste products of Si.
Here we may expect that aiz ^ 0, a2i > 0.
Id. S2 feeds on Si, but at the same time cultivates it in symbiosis.
Here aiz > 0, a2i > 0.
The characteristic equation for two variables is
X
X
= (5)
FUNDAMENTAL EQUATIONS OF KINETICS 79
or, In expanded form,
X2
(an + o^X + (andn 021^:2)
= (6)
X = I {
(au + a22)
=*=
V(au - a22 )
2
+ 4a2 iai2 }
(7)
Certain general conclusions follow immediately. So, for example,
if ai, ais are both of the same sign, as in the case of saprophytes
and of symbiosis, the quantity under the radical is necessarily positive,
and hence both roots for X are real. The oscillatory type of approach
to equilibrium is here excluded. In the case of two species of the
type (la) or (16), the episitic or parasitic type, on the contrary,
the possibility of oscillations is indicated, under conditions where
the equations (1) are applicable.
CONCRETE EXAMPLES
Martini's Equations for Immtinizing Diseases. Numerical applications
of the case of two dependent variables are not easily obtained.
Of concrete examples in general terms (with unknown or
very imperfectly known values of the constants involved) several
are to be found in the literature. The simplest of these is a case
for which the equations are given, without solution, by Martini
in his Berechnungen und Beobachtungen zur Epidemiologie der Malaria
(Gente, Hamburg, 1921, p. 70), namely, the case of the growth of
an endemic disease of the type that confers acquired immunity
upon persons who recover from it (e.g., measles, scarlet fever, etc.)
Martini writes
u = fraction of the population affected and infective
i = fraction of the population not available for new infection (i.e., immune
or already affected)
(1
_ i )
= fraction of the population available for new infection
p fraction of the population newly affected per unit of time
q = fraction of the population of affected population that ceases to be so,
per unit of time, by recovery or by death
m = fraction of "unavailable" population that loses immunity or dies
per unit of time
a = infe.otivity (a proportionality factor)
Martini puts the newly affected population, per unit of time, jointly
proportional to the infective population u and to the population
available for new infection, (1 i), so that
p = a u(l t)
80 ELEMENTS OF PHYSICAL BIOLOGY
Then, obviously,
= a u(l - - qu = (a - q}u
- a ui
di
I- (9)
= a u(l i) mi = a u mi a ui
di j
The characteristic equation for X here reduces, near the origin,
simply to l
|
(a _ q]
_ A )
(m - X) = (10)
and the solution, near the origin, is
(7, ^-mt 4- . . 1
(11)
from which it is seen that the equilibrium near the origin is stable
if and only if a < q
There is a second equilibrium at
in (a q)
= I
(12)
Since, however, i can in reality assume only positive values, this
equilibrium has a meaning only if a > q, i.e., just in the case in
which the equilibrium at the origin is unstable. Hence we conclude
that if a. < q the equilibrium at the origin is the only possible one,
and is stable, so that the disease will die out.
In the other alternative, a > q, the second equilibrium has a real
meaning, and we can develop a solution in series
- = ' Xlt '
X!t 1
(13)
i - I = (?'SlleXj
*+ (?' 2 , 2 e
X2t
+ . . .
where X3 ,
X2 are given by
q-
1
A. J. Lotka, Nature, vol. Ill, 1923, p. 633.
gi
FUNDAMENTAL EQUATIONS OF KINETICS Oi
We need not here consider the case (a
- q) < 0, for, as pointed out
above, in this case the second equilibrium is meaningless. But
when a > q, it is seen from (14) that the real parts of the two
roots will then in any case be negative, so that the equilibrium,
if
it exists at all, is stable. Furthermore, there will be two real and
distinct, two real and coincident, or two complex roots, according as
a- q 1 m ~> _ ft K)
2.
-1
- \i"J
a" a 4 q
<
In the last-mentioned case equilibrium will be approached by a
series of oscillations above and below the final state of equilibrium,
so that a series of "epidemic" waves will appear, a feature which
has an obvious interest in connection with the type of disease here
discussed.2
The oscillations thus occasioned by the factors duly
taken into account in this elementary analysis may in practise be
enhanced by factors here neglected, such as varying virulence of
the disease, exhaustion of susceptible population, seasonal influences,
etc. The last-mentioned, however, will in general tend to produce
a separate series of waves whose period will have no relation to that
of the oscillations derived above.
For the special case = 1, Prof. G. N. Watson has given a complete
solution, which may be consulted in the original.
3
It is to be noted
that this case cannot, according to the analysis here presented,
lead to oscillations, since the condition for oscillations according
to (15) reduces to
^"
:o (16)
and cannot be satisfied, as the square of a real quantity is necessarily
positive.
The Ross Malaria Equations. A system of equations has been
established by Sir Ronald Ross4
to represent, under certain
2
Compare J. Brownlee, Investigation into the Periodicity of Infectious
Diseases, Public Health, vol. 25, 1915, p. 125.
3
G. N. Watson, Nature, vol. Ill, 1923, p. 808.
4
Sir Ronald Ross, The Prevention of Malaria, second edition, 1911, p. 679.
This volume also contains a bibliography. For a detaile ddiscussion of the
Ross malaria equations see A. J. Lotka, Am. Jour. Hygiene, January
Supplement, 1923.
82 ELEMENTS OF PHYSICAL BIOLOGY
conditions, the course of events in the spread of malaria in a human
population by the bites of certain breeds of mosquitoes infected
with the malaria parasite. These equations are of the same form
as Martini's equations discussed above, and inasmuch as Ross's
malaria equations have been very fully treated by the writer in
a separate monograph,
5
their detailed study may here be omitted.
It will suffice to reproduce from this monograph one of the curves
representing the course of events, the presumptive growth of malaria
in a human population, as defined by the differential equations of
Sir Ronald Ross.
-4
~
t
years
FIG. 10. CUHVE OF GROWTH OF ENDEMIC MALARIA ACCORDING TO SIB RONALD
Ross's EQUATIONS
The upper (S-shaped) curve relates to the particular case in which the
initial malaria rates in the human and the mosquito population stand
in the ratio which they have at equilibrium or are both small. The lower
curve represents the course of events when the initial malaria rate is 4.2 per
cent in the human population, and 1.4 per cent in the mosquito population.
(The zero of the time scale is arbitrary.) Ordinates are malaria rates (human)
expressed as fraction of unity. (Reproduced from A. J. Lotka, Am. Jour, of
Hygiene, January Supplement, 1923.)
5 A. J. Lotka, loc. cit.
FUNDAMENTAL EQUATIONS OP KINETICS 83
It will be observed (fig. 10) that the curve showi consists of two
parts, the one an Srshaped limb which, in point of fact, is very
nearly identical in type with the Verhulst-Pearl population curve.
The second limb ascends very steeply, almost vertically, and finally
bends to join the S-shaped limb.
The meaning of these curves is as follows: If a small nucleus
of malarial infection is introduced into a (constant) population
of human beings and mosquitoes, both being previously free from
such infection, then the growth of malaria in the human population
will follow the course represented by the S-shaped limb. It will
be seen that the process is a rather leisurely one, its essential completion
occupying about ten years, according to Ross's figures. (Strictly
speaking it is never quite complete in a finite time.) Seasonal effects
are here disregarded.
On the other hand, if at the start there is already present a certain
malarial rate in the human population, and also in the mosquito
population, then, in the case here depicted, there is for a time a
very rapid increase of malaria in the human population, until, in
the brief space of about two months, the S-shaped curve is reached.
After that the course of events is the same as in the first case.
In practise, in temperate climes, we can expect only short sections
of the S-shaped curve to be realized, owing to the interruptions
of the seasons. No data are available for a numerical comparison
of these results with observed conditions. Close agreement is not
to be expected, as the Ross equations refer to a rather highly idealised
case, a constant population both of men and mosquitoes. The
latter could be even distantly approached only in the tropics. There
is room here for further analysis along more realistic lines. It
must be admitted that this may lead to considerable mathematical
difficulties. The case of periodic seasonal influences is perhaps
the one that promises to yield most readily to mathematical
treatment.
The Ross malaria equations are a typical example of equations
affected with a lag, owing to the period of incubation. For a detailed
discussion of this feature the reader must be referred to the author's
monograph published by the American Journal of Hygiene.
An Example in Parasitology. An interesting and practically
significant case of inter-group evolution, of conflict between two
species, has been made the subject of an analytical study by W. R.
84 ELEMENTS OF PHYSICAL BIOLOGY
Thompson.
6
He considers a host species^ numbering initially n individuals,
and a parasite species, initially p individuals. On a number
of simple assumptions, for which the reader must be referred to
the original papers, he develops the following formula for the fraction
a of the host population attacked, in the t
ih
generation, by parasites
pa*
a =
(a* - a)
n -p a 1
where a is the ratio of the "reproductive power" of the parasite
to that of the host, the reproductive power being measured by the
number of eggs deposited per female. It is assumed that only
one egg is deposited in each host.
Putting
a1 = ert
(18)
and
X (19)
P P
Thompson's formula becomes, after a simple transformation,
-^r-i (20)
which will be recognized, once more, as the equation of the law
of simple autocatakinetic growth. According to the value of a
and K several different cases may arise, whose graphs are shown in
figure 11. It should be observed that
K = 1 according as a= 1 (21)
r = according as a= 1 (22)
< >
In the special case that a = 1 the formula (20) becomes indeterminate
and a is then given by
P
n-p(t-l)
a hyperbolic relation (see fig. 11, the last diagram).
(23)
W. R. Thompson, Comptes Rendus Acad. Sci., vol. 174, 1922, pp. 201, 1443,
vol. 175, p. 65. See also R. A. Wardle and P. Buckle, The Principles of Insect
Control, Longmans, Green & Co., 1924.
FUNDAMENTAL EQUATIONS OF KINETICS 85
Thompson gives a number of numerical examples exhibited
in table 5 . From these examples and from this formulae he concludes
that the invasion of the host species by the parasite may at first
Fia. 11. COTJESE OF PAKASITIC INVASION OF INSECT SPECIES ACCOBDINQ TO
W. R, THOMPSON
proceed only very slowly, and that nevertheless, after a certain
time, the increase may become very rapid. From a practical standpoint
this is important to observe, since it implies that the first
effect of "sowing" the parasite among a species of insect hosts,
ELEMENTS OF PHYSICAL BIOLOGY
with a view to destroying them, may be quite discouraging, and
that this must not be taken as indicative of ultimate failure. This
TABLE 5
Percentage (IQOa) of host species attacked by parasite in t
lh
generation, according
to W. R. Thompson
p= initial number of parasite species.
7i=initial number of host species.
parasite
host
a=ratio of reproductive powers
The figures in the columns headed 1, 2, 3, etc., denote the values of
100 a in the 1st, 2d, 3d, etc., generation.
7i'm& in
J>OO
\
\
\
\
FIG. 12. INCREASING DIBTUSION-IN-TIME OF STJCCBSSIVH GENERATIONS IN
THE PEOGENY OF A POPULATION ELEMENT
observation is especially significant since in practise one must
often be satisfied with the introduction of a relatively very small
number of parasites. So, for example, in using the parasite species
FUNDAMENTAL EQUATIONS OF KINETICS 87
Liparis di&par, commonly about one thousand individuals have
been sown. Supposing that there were one thousand million hosts,
and that the parasite reproduced twice as fast as the host, it would
require, according to Thompson's calculations, about 19 generations
to exterminate the host; and then, even to the sixteenth generation,
only 10 per cent or less of the host species would be attacked,
Thompson's formula is open to certain objections. Its derivation
seems to involve the assumption that each generation of the parasite
is coextensive in time with the corresponding generation of the host.
Furthermore, the use of a generation as a sort of time unit is unsatisfactory,
because a generation is a very diffuse thing, spread
out over varying lengths of time. This is easily seen by considering
the progeny of a batch of individuals all born at the same time
t 0. If we call this the th
generation, and if fli, ao are respectively
the lower and the upper limit of the reproductive period, then it
is clear that the next generation, the first, will extend over the
interval of time from aL to a2 ,
the second from 2a^ to 2a2 ,
the nth
from nai to naz ,
an interval which will ultimately become very large
as n increases, as shown by successive intercepts between the two
sloping lines in figure 12.
Less objectionable, perhaps, is the fact that Thompson's formula
is expressed in terms of the "rate of multiplication per generation"
of the two species. This term is not as clear as it might be. From
the context it appears that it refers to the number of eggs deposited
per female. This number is closely and simply related to the
ratio 12 of the total births in two successive generations. If an
individual of age a reproduces, on an average /3 (a) individuals per
unit of time, the ratio R is evidently given by
f*
=
j
Jo
(a)p(a)do
The relation of this R to the rate of increase r per head of the population
is not altogether obvious and cannot be expressed in simple
form. For a population with fixed age distribution, it will be
shown in Chapter IX that r is given by
= e
-ra
p(a)p(a)fa
|"SO
/
= I
@(a)p(d)da r I a^(a}
Jo Jo
+
(25)
88 ELEMENTS OF PHYSICAL BIOLOGY
Hence
Jra
aj3(a)p(a)da + . . . (26)
In view of the doubtful features in Thompson's formula which
have been indicated above, it appears desirable to attack his problem
in quantitative parasitology from another angle. We may do
this by following the general method which has here been set forth
and exemplified.
Treatment of the Problem by the Method of Kinetics. Let
NI be the number of the host population, &i its birthrate per
head, (the deposition of an egg being counted a birth), and di its
death rate per head from causes other than invasion by the
parasite. Let kNiN* be the death rate per head due to invasion by the
parasite, in the host population, the coefficient k being, in general,
a function of both jVi and JV2 ,
the latter symbol designating the
number of the parasite population.
The birth of a parasite is contingent upon the laying of an egg
in a host, and the ultimate killing of the host thereby. To simplify
matters we will consider the case in which only one egg is hatched
from any invaded host. If an egg is hatched from every host
killed by the invasion, then the total birthrate in the parasite population
is evidently kNiN*. If only a fraction k' of the eggs hatch,
then the total birthrate in the parasite population is evidently
kk' NiN, which we will denote briefly by KNiN2 .
Lastly, let the
deathrate per head among the parasites be dz . Then we have,
evidently
dt
(27)
dt
~ i 2
- - where
rx has been written for (b{ dt).
Regarding the function k, we shall now make the very broad
assumption that it can be expanded as power series in NI and Nz ,
thus
k = a. + &Ni + yN-t + . . . (28)
FUNDAMENTAL EQUATIONS OP KINETICS
It will be convenient first of all to consider an approximation.
7
If the coefficients 3, 7, etc., are sufficiently small, we shall have,
for values of A'i, X* not too large, essentially
at
(29)
(30)
Integrating, and putting
k'a
TI
(^21
a.
we obtain
d2 Iog(x + p) + n \og(y + q)
- h'ax - ay = M (33)
where M is an arbitrary constant of integration. Expanding by
Taylor's theorem we find
M' (34)
By giving successively different values to the arbitrary constant
Mr
a family of closed curves is obtained for the plot of (34) in rectangular
coordinates, as indicated in figure 13. In the neighborhood
of the origin, where terms of higher than second degree are negligible,
(34) reduces simply to
(-^
= constant (35)
P~ <f
and the integral curves are small ellipses.
7
It will be observed that the "diagonal" terms in the linear part of (27),
(29) are lacking, i.e., there is no linear term containing 2V2 in the first equation
of (27) and none containing NI in the second. This gives rise to an exceptional
case to which the general solution given in Chapter VI is not applicable.
90 ELEMENTS OF PHYSICAL BIOLOGY
The course of events represented by these curves is
evidently
a cyclic or periodic process, corresponding to a circulation around
the closed curves. The period of oscillation, near the origin,
8
is
given by
T = 27r/Vr1 d2 (36)
FIG. 13. COURSE OF PARASITIC INVASION op INSECT SPECIES, ACCOEDINQ TO
LOTKA; ELEMENTARY TREATMENT
This finding accords well with the observation made by L. 0. Howard :
With all very Injurious lepidopterous larvae .... we constantly see
a great fluctuation in numbers, the parasite rapidly increasing immediately
after the increase of the host species, overtaking it numerically, and. reducing
it to the bottom of another ascending period of development.
8
The purely periodic solutions have been discussed by the author in Proe.
Natl. Acad. Sci., 1920, vol. 7, p. 410. The writer, however, at that time overlooked
the existence also of the other types of solution, and also stated that
the period of oscillation is independent of initial conditions. This is an error
which he takes the present opportunity to correct. The expression given
by him loc. cit. for the period of oscillation holds only in the neighborhood of
x = y = 0. See also note 10 below.
FU^TDAMEXTAL EQUATIONS OF KINETICS 91
The remarks of W. R. Thompson relative to this may also be
quoted :
Recent studies oa the utilization of entomophagous parasites seem to
show that the role of these auxiliaries of man finds its maximum effectiven^==
when the noxious insect has increased in numbers to the point of a
ula-me, one or more of the factors of natural equilibrium having somehow
failed
'
to act as a check. The expansion of the noxious species then
automatically produces an increase in the number of parasites; generation
FIG. 14. COURSE OF PABASITIC INVASION OP INSECT SPECIES, ACCORDING TO
LOTKA; MORE EXACT TREATMENT
after generation this number increases at the expense of the host, until it first
equals, and presently surpasses the number of the host species, and finally
almost annihilates the host; but then comes a moment when the parasite population,
having grown to excess, largely disappears for the lack of food
supply.
9
9
L. O. Howard, Bureau of Entomology, Technical Series No. 5, 1897, p. 48.
It will be noted that the analysis given by W. R. Thompson fails to give any
indication of this oscillatory process.
92 ELEMENTS OF PHYSICAL BIOLOGY
In this elementary discussion the terms of second and higher
degree in Ar
i and JV2 in equations (27) have been neglected, as in
(29). When they are taken into account it is found that, in
general, the system of closed curves in figure 13 is replaced by a
spiral winding about the second equilibrium point (see fig, 14).
The process is still oscillatory, but, in general, it is the nature of a
damped oscillation. For detail of this more exact treatment the
reader must be referred to the original literature.
10
Annihilation of One Species by Another. The preceding example
was suggested by W. H. Thompson's paper in the Comptes Rendus.
Another case, leading to equations of the same form, had been
previously treated by the writer, namely the following :
A species S* feeds on a species Si, which, in turn feeds on some
source presented in such large excess that the mass of this source
may be considered constant during the period of time under consideration.
Then we have the obvious relation
(37)
The first term in the right hand member, Mass of newly formed
Si per unit of time, evidently must vanish with Xi}
and we shall
therefore write this term biXi, where 61 will, in general, be a function
of both Xi and X2 .
Similarly, and for the same reasons we
shall write for the third term, excretory matter, etc., eliminated
from Si per unit of time, diXi. Contracting these two terms into
one we shall write for biXi diXi the single term riXi. The
middle term, mass or Si destroyed per unit of time by 82, must
evidently vanish with either Xi or X2 separately. We shall accordingly
write this term kXiX2 ,
so that the equation now appears
at
(38)
10 H. Poincare, Journal de Mathem., 4th ser., vol. 1, p. 172. (See also E.
Picard, Traite d' Analyse, vol. Ill, Chapter IX, p. 214); A. J. Lotka, Jour.
Washington Acad. Sei., 1923, vol. 13, p. 152.
FUNDAMENTAL EQUATIONS OF KINETICS 93
(39)
For the species S which feeds on Si we have a corresponding
relation
'Mass of
newly]
formed 2 per Mass of S-.
I unit of time I
destroyed or
I (derived from] eliminated per
S\ as food in- unit of time
.gested) j
which, for reasons precisely analogous to those set forth above with
reference to equation (38), we shall write
at
(40)
It will be seen that these two equations are identical in form with
those discussed in the preceding example, and it is therefore unnecessary
to repeat the analysis there given. But one point invites
attention, and will naturallylead us on presently to the consideration
of a case in three dependent variables, which offers certain features
of special interest.
Let us leave aside all other considerations, and restrict our enquiry
here to the following question : Is it possible, under the circumstances
represented by our equations, for the hostile species $2 to exterminate
completely the species Si upon which it feeds? This would,
of course bring in its train also the extermination of Sz .
To answer this question we note that
Xi (n-
X,dX,
is satisfied by the particular solution
i
= for all values of
(41)
(42)
Hence if we plot the integral curves of (41) in rectangular coordinates,
the axis of X% is itself one of the integral curves. Now if rx , k,
K, and c?2 are single-valued functions of Xi}
and Xz ,
as it is reasonable
to suppose that they are, then anypoint in the XiX2 plane is traversed
by only one integral curve, since the derivative -=? is uniquely
ClX 2
determined by (41). It follows that no integral curve can cross
the axis of Xz and therefore Xi can never fall to the value zero,
94 ELEMENTS OF PHYSICAL BIOLOGY
it can be zero only if it had that value from the beginning. The
food species cannot, therefore be exterminated by the predatory
species, under the conditions to which our equations refer.
This argument fails, however, at the origin, where the derivative
dXi/dXz assumes the indeterminate form 0/0. But in the neighborhood
of the origin, where second degree terms are negligible, we
have, essentially,
dXi Xi TI
dXz Xy dz
from which it is seen that in the positive quadrant the integral
curves always slope downward from left to right near the origin.
They cannot, therefore, in the positive quadrant, cross the axis
of Xz at the origin, any more than at any other point, and the conclusion
is now fully established, that the species Sz cannot exterminate
Si under the conditions here considered.
A word of caution, however, is perhaps in order. Although
82 cannot exterminate Si, it may so reduce the latter in numbers as to
render it very vulnerable, and liable to extinction from those other
influences which have deliberately been ignored in the development
of the equations.
Case of Three Dependable Variables. A singularly interesting
conclusion is reached when we enquire what may be the effect
of introducing, into the system discussed in the preceding section,
a third species, which also serves as food for Sz, so that this now has
two sources to draw upon, after the pattern
One would perhaps naturally suppose that this alternative and
additional source would in some measure protect Si from the attacks
of Si. But we shall see presently that this is not necessarily the
case, that, in fact, in certain circumstances, the introduction of the
third species 83 may bring about the extermination of Si, from which,
as we have seen, Si is naturally protected by the diminution of
growth forced upon $2 when this species too greedily consumes its
sole source of food.
FUNDAMENTAL EQUATIONS OF KINETICS 95
The equations for this case are similar to those for the preceding,
except that we must add a third equation for the species $3
~= r,Xs
- hXzX* (44)
at
and add a term to the equation for S2 to represent the consumption
of the species 83 as food by S2 ,
so that this equation becomes
dXz
: JK.Xi.X.2 -j-
at
+ HX3
- d2 ) (45)
We now have, near the axis of Xs>
_ . fAC\
J-tr V TJ"V~ ^ V*Q)
and, if Xz is sufficiently large, the projection of the integral curves
upon the XiX plane will slope upward from left to right in the
positive quadrant. Such an integral curve may therefore cut
through the X%X3 plane, that is to say, the species Si may be reduced
to zero.
Similarly, near the axis of Xi
_J = !
r
J.
(47)
j~Y V K'Tf J ^
tt-A. 2 "- 2 ^Vwti. I
~~"
ti2
and hence an integral curve may cut through the XsXi, plane, thus
reducing X3 to zero.
This observation is of practical interest. It has been pointed
out that in sea fisheries the accompanying presence of a common
fish may cause the extermination of a rarer species which, were
it present alone, would be protected by its very scarcity, since this
would make fishing unprofitable. But the more abundant fish
continues to render a balance of profit from the trawling operations,
and thus the rarer species, so long as any of it remains, is gathered
in with the same net that is cast primarily for common species.
Replaceable and Irreplaceable or Indispensable Components. The
last two examples present an illustration of a point which calls
for brief comment. So long as the species Sz has only one source
jy
of food, Si} it is to be observed that becomes negative as soon
dt
98 ELEMENTS OF PHYSICAL BIOLOGY
as Xi is zero (see equation 40). In this sense Si is an essential
component of the system, relatively to S, i.e., it is indispensable
for the growth and even the mere continued existence of S. On
the other hand, when two or more sources, such as Si and Ss are
provided, the vanishing of either Xi or Xs singly does not bring
j~y~
with it a negative value of . The components Si and $3 can
more or less effectively replace, act as substitutes for, each other.
When the feeding species ($2 in the example) is the human species,
the facts indicated above find their expression in economic terms.
It is an elementary fact of common knowledge that among the
varied materials which the human race requires for its growth and
sustenance are many that are more or less readily interchangeable.
So, for example, a deficiency in the wheat crop may be in some
degree met by increased supply of potatoes or other starchy food;
or a deficiency in beef may be compensated by increased production
in pork. On the other hand, there are certain requisites that are
irreplaceable, and therefore absolutely essential. The most obvious
example of such is our supply of oxygen.
In terms of our general analysis these facts would be expressed
somewhat as follows. If we survey the various components whose
masses Xi,Xz . . . Xn appear in the function FI
= Fi (Xi, Xa, . . . -X"n)
at
we may effect a classification by first of all dividing them into two
classes, namely those that adversely affect the rate of growth of
Xi, that is to say, those for which
JL^< O
dZj dt
and those that promote the rate of growth of Xi, those for which
dt
Among the latter components, those favorable to the growth
of X{, there is a special class distinguished by the following property :
If X is the mass of a component of this special class, then
dXi .
~j7
is invariably negative as soon as X^ is zero, no matter what
FUNDAMENTAL EQUATIONS OP KINETICS ~
97
may be the masses Xi, X , . . of the other components. Xk
is indispensable for the growth and the sustenance of Xi] X& is
an essential of irreplaceable component, relatively to Xi.
The ground upon which we are here treading is evidently close
to the biological basis of economics. A detailed analysis of -the
relations involved belongs to the domain of the dynamics of lifebearing
systems, and will in due course be considered in its natural
place.
Limiting Factors. In general the several components that promote
the growth of the component of Si will be presented in varied abundance.
If one essential component (or a group of components which
jointly are essential) is presented in limited amount, any moderate
increase or decrease in the ample supply of the other components
will have little or no observable influence upon the rate" of growth
Fi of Si. An essential component presented in limited supply thus
acts as a check or brake, as a limiting factor, upon the growth of Si.
The significance of such limiting factors seems to have been first
pointed out by J. Liebig:
11
"Der Ertrag (des Bodens) ist von dem
im minima in ihm enthaltenen Nahrstoff abhangig." And again :
Fur die Wiederherstellung der Ertrage der durch die Cultur erschopften
Felder durch Stallmistdiinguug ist die Zufuhr von alien den Nahrstoffen
welche das Feld im tJberschuss enthalt, vollkommen gleichgiiltig, und es
wirken nur diejenigen Bestandteile desselben gtinstig, durch. welche ein im
Boden enstandener Mangel an einem oder zwei Nahrstoffen beseitigt wird.
Limiting factors not only set certain bounds to the growth of the
components to which they are thus related, but are competent
also to give rise to the phenomenon of "moving equilibrium" the
discussion of which is reserved for a later section dealing with
equilibria generally, under the heading of Statics.
This chapter may fittingly be concluded with table 6, which exhibits
the principal modes of interdependence of biological species, and
summarizes the analytical characters of these, as discussed above.
11
J. Liebig, Die Chemie in ihrer Anwendung auf Agricultur, 1876, p. 334.
See also ibid., pp. 332, 333, 381, 382.
O
O
Q! S s -
i
- .S
ic "x ^ 'S
H s
? -p
S -^
H C S2
,? Q
-C T3
-J g
ft _ -i-i
.9 .2
T':
.2 S
v= ^3
>
'42 Ml
o a
T3
g fcO
- " HI
as 'S -*S
1 S>
& &o
CQ L.2
^'
W cu 6
fl
^
^3 tb
'
ft CO fl
5,'x
53
r-s
i
o
, 03 O
L.g ^
a
L-i
c3
Pn
99
CHAPTER IX
ANALYSIS OF THE GROWTH FUNCTION
Elegant intellects which despise the theory of quantity are but half developed.
A. N. Whitehead.
The Form of the Growth Function F. The fundamental equations
(1) of Chapter VI express in a very general, and for that reason
somewhat colorless way, the interdependence of the several components
of evolving systems of the kind here under discussion. *
In the special cases with one, two and three dependent varial^teg
that have been presented as examples, the particular form affif the ^&^
concrete meaning of the functions F appearing in these equations -*
has been illustrated for these specific instances, without any attempt
at systematization from a general standpoint. It is desirable now to
make a somewhat detailed analysis of the functions F in their more
general aspect.
A natural step to take is, first of all, to split up the function F
into a positive and a negative term; that is to say, to express the
rate of increase of the mass Xi of the component & as the balance,
the surplus, of the mass Ui added to that component per unit of
time, over the mass Vi eliminated therefrom per unit of time. Thus
dXi
-~rt .i\-rt (i)
Growth of Aggregates. Among the components of systems of
the kind in which we are here mainly interested, a particularly
important type are those built up of a large number of essentially
similar units or individuals. Such are the aggregates of molecules
that constitute the components (chemical elements and compounds)
of the systems with which physical chemistry is
concerned; such,
also, are the aggregates of individual organisms that constitute
the biological species, the component population groups, of which
are built up the systems in which organic evolution takes its course.
In the case of an aggregate of this kind, if N is the number of
individuals, and m their average mass per head, we have
100
ANALYSIS OF GROWTH FUNCTION 101
Xi = miNi (2)
dXi dNi dnii . .
= mi + Nl
- (3)
If the average mass per individual, nii, is constant, the second term
of the right hand member in (8) drops out, and we have simply,
This holds strictly for aggregates of similar molecules, for example,
whose masses are all equal and constant. It will often hold with
close approximation (as will be set forth in greater detail shortly)
for populations of living organisms of one species.
dNi
Now -J7 1
the rate of increase in numbers of the aggregate Si,
naturally appears, after the manner indicated above, as the balance
of the number of newly formed individuals Bi per unit of time, and
the number Di eliminated per unit of time. When these symbols
refer to a population of living organism, Bi is the (total) birth rate
and Di the (total) death rate per unit of time. We have, then,
f-^-A (5)
at
which it is often convenient to write
at
the lower case letters &, d denoting birth rate and death rate per
head per unit of time.
Combining (4) and (5) we have
^-(Ifc-DOmi (7)
at
Demographic Functions. The quantities B, D, or 6, d lend
themselves to further analysis in terms of more fundamental characteristics
of the aggregate. In a qualitative way everyone is
familiar with the manner of the elimination of individuals from a
population by death: some are carried off in infancy, some in child-
102 ELEMENTS OF PHYSICAL BIOLOGY
hood, adolescence, and maturity, until the remnant is finally called
in old age. Quantitatively this fact finds expression in an actuarian's
life table, or the corresponding life curve, of which some
examples are shown in figures 15 to 17. Starting with some large
number,, say a million, of newly born individuals, counted at birth,
if we follow this sample batch of population through life, we find
them thinning out, at first rather rapidly in infancy and childhood
(steep part of curve on left) : then more slowly (more gentle
slope) in mid-life; and faster again as the natural term is approached.
At any particular age a there are thus left, out of the original batch,
a fraction p(a) of survivors. The fraction p(a), which we may speak
of as the survival factor,
1
is a measure of the probability, at birth,
that a random individual of the batch shall reach age a, under the
conditions under which the data assembled in the life table were
collected. The value of p(a) for every age of life depends, of course^"
on the general conditions of life in the population. It therefore
varies in different localities and at different epochs, as illustrated
in figures 15, 16, 17. When it is desired to bring out the fact that p(a)
depends on the time, it may be written p(a, f). But as a rule the
change with time is not very rapid and we may often consider p(a)
as a function of the age a alone. We shall do this in the present
analysis^ which relates to a population in a selected locality under
essentially constant conditions of life.
If we denote by Na the survivors to age a, out of an original
batch Ar
counted at birth, we have
JV = N p(a) (8)
We will -write
dNa
/1A\
\LO)'
or
dNa d
> r , j
Nada da
The coefficient jua thus defined is termed the force of mortality at
age a. From its definition it is clear that it measures the death
rate per head in a population composed entirely of individuals of
age a.
1
The function p (a) is that commonly denoted by lx in actuarial notation,
p.nd tabulated in the principal column of a "Life Table,"
SSfSS]
\
\
\
\
\
\
\
s.ooo
70,000
64,000
FIG. 15. SOME HISTORICAL HUMAN SURVIVAL CURVES, EXHIBITING AN EVOLUTIONARY
TREND TOWARD LONGBB AVERAGE DURATION OF LIFE
The improvement is probably rather one of general hygiene than of man's
physiological constitution. The earlier figures are of very doubtful accuracy.
103
104 ELEMENTS OF PHYSICAL BIOLOGY
AGE IN YEARS
65,000
-45,000
40,000
35,000
30,000
25,000
20/500
15,000
IO.OOO.
5.000
x:
X 4\
\
il
60 70
ASE IN YEARS
30,000
25,000
2O.OOO
I5.OOO
FIG. 16. STJEVIVAI, CUEVES FOB THE STATE OF MASSACHUSETTS, FOB THE
THBEE DECADES 1890-1900-1910. AFTEB GLOYEB
The data on -which these curves are based are naturally more reliable than
those of the older life tables shown in figure 14, and exhibit very clearly the
upward trend of the average length of human life in recent decades.
ANALYSIS OF GROWTH FUNCTION 105
AEE M YEAJIS
FIG. 17. SURVIVAL CURVES FOE DIFFERENT COUNTRIES, SHOWING INFLUENCE
OP LOCAL CONDITIONS UPON LENGTH OF HUMAN LIFE. AFTER GLOVER
A particularly simple case is that in which \ia is independent of
a, the force of mortality is independent of the age, or is the same at
all ages, We then have by integration of (10)
and
p(a)
(11)
(12)
106 ELEMENTS OF PHYSICAL BIOLOGY
Such a simple life curve as this is not to be expected in a species
of living organisms. It implies that the individual does not age, that
his chance of living another year is just as good at ninety years of
age as at fifty or at ten or at five; he can die, as it were, only by accident;
he is perpetually young. Survival curves of this form do
occur and play a significant role in the aggregates of atoms and molecules
which the chemist and the physicist make it their province
to study. The atoms of an element in radioactive transformation,
for example, are picked off, one by one, according to a law of
this form, and so are the molecules of a chemical compound decomposing
by a monomolecular reaction. If, in such a case, the plot
the function p(a) is drawn on logarithmic paper, evidently a straight
line is obtained, since
logep(a)
= jua (13)
(14)
da
The force of mortality is here seen as the (constant) slope of the
curve representing loge p(a).
In the more general case it is also often convenient to plot p(a)
on a logarithmic scale. This has been done in figure 18 for the
United States survival curve shown in figure 17. It is seen that the
curve thus obtained is at first convex toward the axis of a, but soon
becomes concave toward that axis and then remains so to the end.
The significance of this is, of course, that the force of mortality
is very high in infancy, decreases in early childhood, until it reaches
a minimum about the twelfth year of life; and finally increases continually
to the end of the life span.
The detailed analysis of the human survival curve is a matter
of interest not only to the student of evolution, but also to the guardian
of public health and to the insurer of human life, the actuarian.
The essentially practical requirements of these has led to a highly
developed technique, amounting virtuaUy to a separate branch of
science, in the preparation and analysis of life tables. Of this phase
of the subject no more needs to be said here, since there is a voluminous
special literature available. Only the more strictly biological
aspect of the matter is for us here of immediate interest, and of this
more will be said later, in discussing the physical basis underlying
the survival curve.
ANALYSIS OF GROWTH FUNCTION 107
Survival Curve Data. Without having recourse to any refinements
of mathematical analysis it is clear that a close relation exists
between the survival curve of a given species and its rate of increase,
its fate in the straggle for existence and for dominance. It might
10,000
/o
K
O -? 8 IS. 16 SO & S3 X 36 -fO 44 -fff .52 S6 60 64 68 7S 76 80 ff* 88 9 -36 /OO
Ag<s in Years.
FIG. 18. SURVIVAL CURVE PLOTTED ON LOGARITHMIC SCALE.
UNITED STATES 1910
The slope of the curve thus plotted measures the Force of Mortality, This
slope is at first very steep, decreases to about the twelfth year of life, when it
reaches a minimum (point of inflection), and then increases again continuously
to the end of life.
be expected, therefore, that the study of such survival curves for
different species of organisms should have formed an essential part
of quantitative studies in evolution. As a matter of fact, however,
data for survival curves of organisms in their natural environment
108 ELEMENTS OF PHYSICAL BIOLOGY
are, for obvious reasons not easily obtained. It is difficult
enough
to make an accurate census of ages in a supposedly intelligent and
responsible human community, where the cooperation of the Individual
can at least in a measure be engaged. A census of ages in the
population of field, forest and stream must indeed be a severe test
of the ingenuity of any investigator that should approach such an
enterprise. Not that it is utterly hopeless. Isolated data can and
have been gathered. Every layman knows that the age of trees, for
example, can be estimated at least approximately by the number of
rings in a cross section of the trunk. The age of certain fishes can be
gaged from tell-tale marking on their scales. The age of certain birds
can be told from their plumage. An estimate of the average age of
herring gulls has been made, on this basis, by J. T. Nichols.2
But
data of this kind are at best scant. The problem is more manageable
in the case of domestic species, or of species in captivity. The
most complete study of this character is probably the work of R.
Pearl and S. L. Parker with colonies of fruit flies
(Drosophila)
grown in an experimental "universe," a glass bottle containing a
suitable quantity of banana juice.
3
The survival curves thus obtained
are singularly like those for the human species, when allowance
is made for the difference in the total life span. The fruit
fly's allotted days, in the most favorable case, number about ninety,
or, say, one day for each year of human life. In figure 19 are shown,
plotted on the same diagram, three survival curves; one of these is
that of the human population of the United States in 1910; the
second is the survival curve for one of the numerous varieties of
fruit flies.
Por only one other organism, Proales decipiens (a rotifer), a life
table and curve has been prepared, on the basis of observations
by B. Noyes.
4
The life curve has been computed by Pearl and Doe-
ring
5
and compared with that of man, on the basis of the life span
2
J. T. Nichols, in The Condor, vol. 17, 1915, p. 181; for other data on the
longevity of animals see C. S. Minot, The Problem of Age, Growth and Death,
1908, pp. 226, 266; A. Weismann, tJber die Dauer des Lebens, Jena, 1882. See
also A. T. Masterman, Report on the scales of freshwater fish in relation to
age determination. H. M. Stationery, Office 1924.
3
R. Pearl and S. L. Parker, American Naturalist, vol. 55, 1921, p. 503; vol.
56, 1922, p. 403.
*
B. Noyes, Jour. Exp. Zool., vol. 35, 1922, p. 225.
*
R. Pearl and C. R. Doering, Science, vol. 57, 1923, p. 211.
ANALYSIS OF GHOWTH FUNCTION" 109
measured from the point at which, the force of mortality is a mimimura
to the extreme end of life. It is the curves thus obtained that
are shown in figure 19. As will be seen, they exhibit a close analogy.
30 4O SO 60 70
CENTILES OF LIFE SPAN
8O 90 fOQ
FIG. 19. LOGARITHMIC SURVIVAL CURVES FOR MAN, DROSOPHILA, AND
PBOALES DECIPIENS
Plotted according to centiles of life-span; the human life curve is plotted
with the twelfth year of life (point of inflection, see figure 18) as zero of the
time scale. After R. Pearl.
Influence of Age Distribution Upon Rate of Growth of Population.
For our present purposes it is not sufficient to fix our attention on a
group of individuals at the moment of their birth, and to observe
the gradual dirninution of this group by deaths; we must take in
110 ELEMENTS OF PHYSICAL BIOLOGY
view an actual population, comprising individuals of all ages, and
we must enquire into the effect of deaths upon such a group as this.
If the survival curve for such a population were of the simple
form
p(a) = e-v (12)
with n, the force of mortality, independent of a, it is evident that
the age distribution in the population would have no influence upon
the total death rate. For if individuals of all ages are equally
susceptible to death, it is evidently immaterial how the population
is constituted as regards age. The death rate per head, in such a
population would, in fact, be simply equal to the force of mortality.
The Stable or "Normal" Age Distribution. But in the populations
with which the biologist and the vital statistitian deals, the
force of mortality varies very decidedly with the age, and it might
therefore be supposed that any discussion of the rate of increase of
a population of organisms must fully take into account the age distribution.
This supposition, however, involves an assumption,
namely, the assumption that the age distribution itself is variable.
Now, in point of fact, the age distribution is indeed variable, but only
within somewhat restricted limits. Certain age distributions will
practically never occur, and if by arbitrary interference, or by a
castastrophe of nature, some altogether unusual form were impressed
upon the age distribution of an isolated population, the "irregularities'"''
would tend shortly to become smoothed over. There is,
in fact a certain stable type of age distribution about which the actual
age distribution varies, and toward which it returns if through
any agency disturbed therefrom.5
The form of this distribution,
in an isolated population (i.e., with immigration and emigration
negligible) is easily deduced, as follows:
If we denote by N (f) the total number of the population at time
t,
and by N ($) c(\a} da that fraction of the total whose age is comprised,
at time t, within the limits a and a + da, it is evident that
the individuals of this element of the population are the survivors
of the B(t-a)da persons born at times (<-o). Hence if p(a)
is the survival factor, we have
s
For a formal proof of this see F. R. Sharpe and A. J. Lotka, Phil. Mag.,
April, 1911, p. 435; A. J. Lotka, Proc. Natl. Acad., vol. 8, 1922, p. 339.
ANALYSIS OF GKOWTH FUNCTION 111
B (t
- o) p(o) da = tfft) c(o) da (15)
If we denote by D (t) the total death rate at time t, we have, evi-
dently
'*' (17)
We are supposing that we are dealing with a population in which
the survival factor is constant, i.e., independent of t. If c(a) also is
independent of tf,
the integral in (17) is evidently a constant and
we have
.
)
= const. = d (18)
where d is the death rate per head.
On the other hand, since c(w) is independent of the time, we have
c(0) = const. = b (19)
Brief reflection shows that the value of c(o) for zero age is the birth
rate per head. Thus we have
6 d = r - const. (20)
where r is the "natural rate of increase" of the population.
We have further
B(t - a} = bN(t - a) (21)
= bN(tie~ra
(22)
since the natural rate of increase r is constant. Thus by (16)
c(a) = be-fa
p(a) (28)
The normal age distribution (23) is of fixed form in th'e sense
that, once established, it perpetuates itself. More tlikn this it is
also>
stable>
in the sense that, if disturbed by a temporary changem the conditions of life (e.g., war), it will
spontaneously return uponrestoration of normal conditions.
112 ELEMENTS OF PHYSICAL BIOLOGY
A rigorous proof of this stability of the age distribution (23)
cannot be briefly given, and for details of such proof the reader must
be referred to the author's publications in the journal literature.8
The general character of the proof may, however, be indicated.
If the original population has any arbitrary age distribution, such
as that represented by the more heavily drawn curve in figure 20
by judicious trimming we could reduce the population to the normal
distribution represented by the lower curve tangent to the heavy
curve; or, we could, by filling in gaps, supplement the population
to fit the upper tangent normal distribution curve. The trimmed
down population, having the normal age distribution, would always
retain it. The same is true of the supplemented population. The
actual population will therefore always lie between the trimmed and
the supplemented. Moreover, it can be shown that at intervals
of about one hundred years (the span of life) new tangent curves can
be drawn to the actual distribution curve, the new tangents lying
between the old. Thus the two tangent curves ultimately approach
until they coincide, and then, necessarily, the actual distribution
curve lying between them coincides with them also. That is to
say, the actual conforms with the normal age distribution.7
It must be understood that the normal or stable form of age distribution
represents merely a broad type, toward which actual
age distributions will tend. However, the approach seems to be
at times very close, as is shown by the figures in table 7 giving the
observed and the calculated age distribution for England and Wales
in decennium 1871-1880. A graphic representation appears in
figure 21, which shows the observational data, plotted (in dotted
lines) as a stepped curve. The corresponding figures calculated by
the formula (13) are plotted in two ways, namely, first as a stepped
curve (drawn in full), for comparison with the observational data;
and also as a continuous curve. The latter brings out a feature that
may be noted in passing, namely the fact that the curve of age distribution
is very flat, roughly linear, over a wide range, from the
7
An exceedingly interesting effort of early date to demonstrate the ultimate
approach to geometric increase of the birthrate, independently of initial
conditions (e.g. starting with a single pair of parents) is to be found in
L. Euler, Recherches g6n<rales sur la mortalite". See also E. T. Gumbel,
Jahresber. Deutsch Math, verein., vol. 25, p. 251.
ANALYSIS OF GROWTH FUNCTION 113
FIG. 20. DIAGRAMS TO ILLUSTRATE PROOF OP STABILITY OF NORMAL AGE
DISTRIBUTION
Reproduced from A. J. Lotka, Proc. Natl. Acad. Sci. ;
vol. 8, 1922, p. 339
TABLE 7
Population of England and Wales 1871-1880, as an example of "normal" age
distribution
114 ELEMENTS OP PHYSICAL BIOLOGY
Si,
.C3S
30
.028
.026
.OSf
.022
420
,0/8
.0/6
.O/-f
MS
.010
.008
.006
.OO$
.002
.OOO
\^
.-=.03373 &-
10 -30 -50 60 70 SO SO /<>
a =
age,
FIG. 21. "STABLE" AGE DISTRIBUTION, AS EXEMPLIFIED BY THE POPULATION
OF ENGLAND AND WALES IN THE DECADE 1871-1880
I IK
ANALYSIS OF GHOWTH FUNCTION 1JLtJ
fifth to the eightieth year of life.
5
This, of course, is a special feature
of a human population when the natural rate of increase r
lies in the neighborhood of a certain value.
Minor fluctuations of the age distribution will not greatly affect
the birth rate and the death rate. Since actual populations approximate
the normal age distribution, i.e., that defined by equation (23),
it seems permissible and it is certainly expedient to assume, in further
discussion, that the normal age distribution is actually established;
we may then virtually disregard the influence of the age distribution
upon the rate of increase of the component. This element is,
as it were, automatically ruled out of further discussion, by the natural
establishment of the normal age distribution;
9
a circumstance
which is in so far fortunate, as we are here interested in the relation
of the rate of growth to the more fundamental biological characteristics;
the age-distribution appearing merely as an adventitious
element complicating the relation, without being essential to the
fundamental characterization of the species.
Demographic Relations in "Normal" Population, It is worth
while to note briefly in passing that in the case of a population in
normal age distribution, many demographic relations assume a
simple form. So, for example, the relation between birth rate per
head 6 and death rate per head d is here given by the formula10
1/b =
g-rp( ) d (25)
Jo
which, to second order approximation, is reducible to the simple
form
1 - U U____ + j=*L (26)
where L is the mean length of life, defined as
Compare J.
Brownlee, The Use of Death Rates as a Measure of Hygienic
Conditions, Report to Medical Research Council, London, 1922, pp 36-378
Of. G. Eijkman, Onthr
"
J.--.-TTI .. .. ~ .
>j.r!,
p. 269;
10
A. J. Lotka, Quart. Publ. Am. Statist. Assoc., 1918, p. 121; 1921, p. 998.
116 ELEMENTS OF PHYSICAL BIOLOGY
Ceo
p(a) da (27)
while U is defined11
by
/"6co
L '
=
ap(a) da (28)
<Jo
L'
U =
Z"' (29)
As an illustration it may be mentioned that in the saniv, population
(England and Wales, 1871-1880) which as already been referred
to, the relation (26) is found by actual computation to be, numerically
0.767 0.233
___ + ___ = 41 _35 (maies )
^
0.7369 0.2G31
. _--j
-- . _ 44,62 (females) (31)Oi
The observed values of 6 and d, together with those computed by
the formulae (22), (23), are shown in table 8. It will be seen that
the agreement is good.
The relation (26) between b and d lends itself readily to graphic
representation by means of a diagram involving only straight lines.
This is shown in figure 22. The method is as follows.
_ r
Along OX mark off a length OP =
L
U
Along OY mark off a length OQ ==
L
Complete the rectangle QOPM.
Suppose we are now given
6 - 0.0337
It is required to find d
Along OX mark off 0V = 0.0337
Join VM and produce it to meet OY in W
Read off at W on the scale of OY
d = 0.0200
11
U is proper fraction. It then follows from (26) that L is intermediate
, 1,1between - and -; this circumstance has been noted by Bortkewitch, Mitt-
lere-Lebensdauer.
ANALYSIS OF GROWTH FUNCTION 117
A single equation, such as (25), connecting the two quantities
b and d, is of course insufficient to determine their value. To do
this requires a second independent relation. Such a relation Is
furnished by the following considerations:
The law of the fixed age distribution holds separately for each
Thus, if we denote JV/ the total number of the female popula-sex.
TABLE 8
b-,0337
.008 .016
b Bir1~h rafs pe.r head per annum.
FIG. 22. DIAGRAM OF RELATION BETWEEN BIRTH RATE PER HEAD 6 AND
DEATH RATE PEE HEAD d IN POPULATION WITH STABLE
AGE DISTRIBUTION
tion} by 6/ the number of female births per unit of time and per head
of female population, and finally by pj- (a) the survival factor in the
female population, then the number of females of an age between
a and a + da is given by Njbje~
n
pf (a) da. If these reproduce, on
an average, 0/ (a,) females per unit of time, then the total number of
female births per unit of time is evidently given by
118 ELEMENTS OF PHYSICAL BIOLOGY
Ca
I
t/oi
(32)
where ai} a are the lower and upper age limits of the female reproduction
period. But Nfy=
Bf ;
hence
(a)8f (a} da (34)
This is the required second relation12
between 6 and d. The birth
rate and death rate in a population of fixed age distribution are
thus completely determined as soon as the functions pf (a) and
/?/ (a) are given.
Aside from its direct interest as the second relation required to
complete the determination of 5 and d in a population with fixed age
distribution, equation (34) is also of value in enabling us to deduce
certain secondary results, of which two will here be noted.
Rate of Increase per Generation. To derive our first conclusion
from (25), we expand e~
ra
under the integral sign. We find
1 =
/3(o) p(o) da - r aj8(a) p(a) da + ... 35
t/oj Jai
A. little reflection shows that the first integral measures the ratio
between the total number of births in two successive generations.
Denoting this by R, and the second integral by S, we have
22 = 1 + rS - . . . (36)
a formula wjhich connects the rate of increase per head per annum
r and the rate of increase R in the total number of births in one
generation, per generation. It will be seen that this relation is
not one that could very well be foretold by any simple elementary
or common sense principle. It is this relation that is implicitly
involved in W. R. Thompson's treatment of a problem in parasitology
cited in Chapter VIII (page 87) ;
at least, we must know this
relation if we are to interpret his results in terms of the usual tune
unit, such as the year, whereas Thompson reckons in generations.
13 A. J. Lotka, Jour. Washington Acad. Sci., 1913, p. 293.
ANALYSIS OF GROWTH FUNCTION 119
Effect of Selective Slaughtering. A second deduction from equation
',34} follows from the fact that the integral extends only between
the age limits of the reproductive period. In consequence of this
the rate of increase, per head, of the population is independent of
the form of the life curve (the survival factor) outside this period;
thus the slaughtering of the "superannuated" members of a herd
has no effect upon its rate of increase, though both death rate and
birth rate are augmented (by equal amounts).
12
The economic
significance of this, as a means for raising the productive capacity
of the herd, considered as a food factory, is serf-evident.
It is true, as previously noted, that generally the influence of
age distribution upon the rate of growth of a species is eliminated
from discussion by the spontaneous establishment of the fixed or
normal distribution. But the subject referred to in the preceding
paragraph clearly shows that there are exceptions to this general
rule. We have here an instructive example in which the influence
of the age distribution upon the course of events is so fundamental
that a discussion of the case which should fail to cover this feature
would be lacking in an essential particular. Indiscriminate killing
of one species by another, as practised by the untutored savage or
the dumb animals, has the effect of reducing the ordinates of the life
curve of the food species more or less along its whole extent. This
is illustrated in figure 23. We may suppose that the life curve of
some species, wild cattle, for example, is that shown in a full line,
curve I, and that curve II represents the life curve for the same species
after the habitat of the species has been invaded by a tribe of primitive
hunters, too thoughtless or too unintelligent to regulate their
depredations in accordance with anything of the nature of game laws.
Curve III is the type of curve that would be produced by selective
killing guided by intelligent control, as distinguished from random
lolling. The verticals erected at a\ and a2 represent the limits of
the period of reproduction for the cattle. The slaughtering, in the
case of curve III, takes place exclusively after the end of the period
of reproduction. This is an extreme case, which, in practice, for
obvious reasons, would be only distantly approached. For both
in game preservation and in animal husbandry on the farm many
factors have to be considered; not only the quality of the meat of
animals at different ages, but the cost of raising and maintenance.
It is out of the question to restrict the slaughtering to the post-
120 ELEMENTS OF PHYSICAL BIOLOGY
reproductive period. The unnecessary male may, in fact, be slaughtered
as soon as the cost of his upkeep exceeds the corresponding gain
hi marketable value of his carcass. Just what is the most advantageous
tune for his slaughter is a question in agricultural economics which
has been discussed, among others by A. Gouin and P. Andouard.13
These authors compute that to bring up three calves, from weaning
to the age of three and one-half years, requires about 33,000 kgm.
of hay (or their equivalent); this quantity would suffice to bring
FIG. 23. DIAGRAMMATIC ILLUSTRATION OF INFLUENCE OF RANDOM AND OP
SF.LECTIVE SLAUGHTERING UPON SURVIVAL CURVE OF BIOLOGICAL SPECIES
seven head of cattle to the end of their second year, which would
give a gain, in meat, of 40 per cent, as compared with three head
brought to the age of three and one-half years.
Intelligence as a Discriminating Agency. It is particularly worth
while to note how in this connection human intelligence exerts
its influence upon the course of physical events by substituting
systematic selection in the place of the more haphazard, more random
actions characteristic of the mentally less developed species.
13
Bull. Soc. Natl. d'Agriculture, Paris, 1910, p. 695.
ANALYSIS OF GROWTH FUNCTION 121
These latter must. In many ways, remain, dependent upon certain
average manifestation, whereas man, in a corresponding situation,
is able to discriminate and address himself directly to individual
elements of which these averages are the expression in the gross.
In the manifold interactions on a macroscopic scale that constitute
the perennial struggle for existence on nature's battlefield,
the lower organisms are in many respects situated in a position
analogous to man's relation toward the ultra-microscopic elements
of his environment, the molecules and atoms. These he can handle
only in bulk, unable to avail himself of the distinctive features of
this or that particular atom or molecule.
Just as our senses and our bodily members are inadequate for
the task of handling individual atoms, so the mental or other discriminating
powers of the lower organism are often inadequate to
lead it toward any other than a rather random selection, and by
no means an optimum selection, among the somewhat varied opportunities
of self-service presented to it.
However, though man does far excel the other creatures in this
respect, the difference is, after all, one in degree and not in kind.
Many, if not all organisms, possess in some degree the power of selection,
are in some measure independent of pure haphazard. This
introduces an altogether peculiar complication into the dynamics
of systems comprising living organisms, a complication of which the
statistical dynamics of molecular physics are free. Not only
is the living organism capable of performing, on a macroscopic
scale, exploits analogous to those which in the world of molecules
are permitted only to such figments of the imagination as Maxwell's
demon; but this power of "cheating chance," as it were, is
possessed in different degree by the several living organisms,
and a dynamics of systems comprising living matter must necessarily
take account not only of this faculty, but of the gradations in this
faculty which have a large part in assigning to the several biological
species their place in the scale of evolution. This will require the
development of special methods. It is essential that we bear in mind
constantly the ultimate aim of our reflections. The transformations
of matter, the change in its distribution among the components of
the physical system in the course of evolution, are the first to strike
the eye, and are properly the first to receive our systematic consideration,
just because they are of more obvious and elementary character.
122 ELEMENTS OF PHYSICAL BIOLOGY
But our fundamental aim must ultimately be to gain an enlarged
understanding of the dynamic relation involved, of the play of forces,
the transformations of energy,
Kinetics of Intra-Group Evolution. It is not intended to attempt
here a systematic discussion in any sense exhaustive of the course of
events in ultra-group evolution, that is to say, in the redistribution
of matter among the several sub-types of which a biological species
is composed. We shall however briefly note an enterprise, led more
particularly by J. B. S. Haldane/
4
to investigate the trend of
evolution in a population in which selection is operating upon a
character subject to Mendelian inheritance.
Casel. Haldane considers first of all the simple case of a species of constant
total population, consisting of two types (phenotypes) A and B that do not
interbreed, but react upon each other merely through competition in the same
environment.
Haldane treats the case as follows :
Let the nth generation, counted immediately after fertilization,
15
consist of
types A and B in the ratio ua :
1, and let the coefficient of selection be k, i.e.,
let (1fc) of B survive (to breeding age)
16
for every one of A. Then the survivors
(to breeding age) of the nth generation, and hence the first numbers17
of the (n-j-l)th, will be in the ratio
14
J. B. S. Haldane, Trans. Cambridge Phil. Soc., January, 1924, vol. xxiii,
pp. 19-41. Part II, read July 14, 1924, has not yet reached the author.
15
The words in italics are here added to Haldane's text, in accordance with
the first paragraph of his paper, page 20.
18
The words in italics are here added to Haldane's text, in accordance with
his paper, page 23, line 6.
17 The meaning of this term is not altogether clear. It seems to refer to the
total births in the (n + l)th generation, or, in other words, to the (n+l)th
generation, counted at birth. This does not seem altogether consistent with
the rest of the argument, notably the wording referred to in footnote 15. On
the other hand the terms referred to in footnote 16 are vague unless reproduction
takes place once and once only in the life of the individual, since in general
there is not only one single age of reproduction, so that the fraction surviving
to breed cannot be represented by any single number. It seems that it would
be better to base the argument on the total number born in each generation,
i.e., on the generation, counted at birth; and to regard the ratio R between
the births in two successive generations as a function of /3 (a) and p (a), as set
forth in Chapter VIII equation (24) and Chapter IX equations (35), (36). We
would then start with equation (38) of page 123 as a fundamental assumption;
beyond this the argument would remain essentially unchanged.
ANALYSIS OF GROWTH FUNCTION 123
wn :l-A = Un + i:l (37)
Ua + i
"rr& (38)
If u be the original ratio of A:B in the Oth generation, then
ua
= (1
- Ar)-
n
o (39)
If we write yn. for the proportion of 2?'s to the total population of A and
B in the nth generation, then
(40)
1 -f un 1 4- (1 k)~nu
* =
y + (1
_ ^1, (1
_ yo)
^^
If we start with an equal number of births of A and B
2/o
= i (43)
1
i i f~t 7.^1 n1 -f- (I K)
If h is very small, i.e., if selection is very slow, then approximately
1
(44)
(45)
fcn = log
^ (46)
Vn
In equation (45) we recognize once more the Verhulst-Pearl relation. It
is seen that in the circumstances to which the discussion relates, the better
adapted of the two types grows, along the typical S-shaped curve, at the expense
of the less well adapted, ultimately displacing it entirely. This is quite
in accord with what we should expect, since the total population A + B is
constant, that is to say, it just holding its own. Any constituent of the population
that falls below the average in adaptation, must therefore be unable to
hold its own, and diminishes continually.
The curve representing graphically the change of the population increase
of type A at the expense of B, which is ultimately displaced entirely is shown
in figure 24 for the case k = 0.001, i.e., that 999 B's survive to reproduce,
for every 1000 A's. In these circumstances 9184 generations are needed for
the proportion of A'B to increase from 1 per cent to 99 per cent.
124 ELEMENTS OF PHYSICAL BIOLOGY
Case 2. Selection of a Simple Mendelian Character, with intermingling
of dominant and recessive type. Kaldane next takes the following case:
Consider the case of a population containing two, one, or no "doses" of a
completely dominant Mendelian factor A. Mating is at random and selection
acts in equal degree in both sexes upon the character produced by the factor.
Pearson18
and Hardy19
have shown that in a population mating at random,
the square of the number of lieterozygotes is four times the product of the
numbers of the two iiomozygous classes. Let un A. : la be the proportion of the
types of gametes produced in the (n 1)
<A
generation. Then in the nth generation
the initial proportion of the two classes of zygotes will be
:2un Aa:la (47)
iGOr
A
-5COO +5000
FIG. 24. GROWTH OP FAVOHED TYPE IN MIXED POPULATION OF
Two PHBNOTYPES
Constant total population. Abscissae = number of generations; ordinates
= percentage of favored type in total population. After J. B. S. Haldane.
The proportion of recessives to the whole population is
(48)
Now only (1 k) of the recessives survive to breed, so that the survivors
are in the proportion
a Aa: (1 k) aa (49)
18
K. Pearson, Phil. Trans. Boy. Soc., 1904, A 203, p. 53.
18 G. H. Hardy. Science, 1908, vol. 28, p. 49.
ANALYSIS OF GROWTH FUNCTION 125
The numbers of the next generation can most easily be calculated from the
new gai:;etic ratio iir~ i. This is immediately obvious in the case of aquatic
orga-uisfiia 7,iio shed their gametes into the water. If each zygote produces IV
giiiietes v/aieh conjugate, the numbers of gametes of type A and a respectively,
are. clearly.
CVtin
2
T Ar
w) A and (Nua + N{
1 - k } )o (50)
so thut
(51)
1 + n /C
Xow if vre know the original proportion of recessives y ,
we start with a
population
u -AA:2uoAa:!Aa (52)
where
^o = y
~* - 1 (53)
and we can at once calculate
1 + MO &
and thence u^ and so on.
For complete selection we have k = 1 (recessives all die). Then
n = n + M (56)
t/n = y (l+ Wy J)-2 (57)
If we start with a population containing J recessives, the second generation
will contain the third ^, the nth Thus
999 generations will be
(n + IP
required to reduce the proportion to 1: 1,000,000, and we need not wonder that
recessive sports still occur in most of our domestic breed of animals.
When selection is not very intense, we can proceed as follows :
(58)
(59)
126 ELEMENTS OP PHYSICAL BIOLOGY
When k is small
(60)
100
IK,
5
= ua u + log
(61)
(62)
(63)
1
7
-5000
Generations.
+5QOQ 10000
FIG. 25. EFFECT OP SELECTION ON POPTJLATION COMPRISING Two PHENOTYPES
WITH MENBELIAN INHEEITANCE
Upper curve, dominants favored; lower curve, recessives favored. Abscissae
= generations; ordinates = percentage of population with the favored character.
After J. B. S. Haldane.
If we start from a population containing 25 per cent recessives UQ = 1,
kn = n + iogen - 1 (64)
In figure 25 is shown the growth curve for the dominants when k 4- 0.001
(upper curve) and for the recessives when k = 0.001 (lower curve), i.e.,
the favored type has an advantage of 1 in 1000, as in figure 24. In each case
16,582 generations are required to increase the proportion of the favored type
ANALYSIS OF GROWTH FUNCTION 127
front 1 per cent to 99 per cent, but dominants increase more rapidly than recessives
when they are few, and more slowly when they are numerous. The change
occurs mci=t rapidly when ya ,
the proportion of recessives to the total popuHaldane
deals also with several other cases, for the details of
which the reader must be referred to the original. A continuation
of Haldane's article is anticipated at the time of this writing.
CHAPTER X
FURTHER ANALYSIS OF THE GROWTH FUNCTION F
We fat all creatures else to fat us, and we fat ourselves for maggots.
Shakespeare,
Adjustment of Birth Rate to Optimum. In the preceding discussion
of the equations (25) and (34) of chapter IX it has been
supposed that the form not only of p(a), but also of /3(a), the rate of
procreation at different ages, is fixed. It is only on this supposition
that the two equations (25) and (34) of Chapter IX together determine
both 6 and d. In point of fact the function (3 (a) is
undoubtedly,
for most species of organsms, very elastic (much more so than p(a),
the survival factor), and capable of adapting itself to varying circumstances.
This is especially so in the case of man, who exhibits
in particularly high degree the rather astonishing phenomenon of
a portion of matter whose growth is at least partially under the control
of a will in some manner associated with it. But, in the organic
world at large alsoj there is presumably at least some tendency for
the adjustment of the procreation factor so to take place as to make
the rate of increase r a maximum under the existing conditions.
Too high a procreation factor would lead to excessive sacrifices in
progeny that could not be raised to maturity, and would increase
the death rate more than the birth rate. On the other hand too
small a procreation factor would obviously fail to give the maximum
attainable rate of increase. Somewhere between the two extremes
a certain optimum procreation factor will make r a maximum.
From this point of view the two relations that effectively determine
the actual values of the two variables 6 and d are
i f*<o
-= I
e-ra(a) da (l)
and
r = maximum compatible with
e~ra
{$(a} p(a) da
128
(2)
ANALYSIS OF GROWTH FUNCTION 129
The view that the rate of procreation thus expands or contracts
in sympathy with the expanding or contracting food supply (or
economic conditions generally) has been developed in detail (for
the human species) by R. Lascaux in his work La Production et la
Population (1921), That some such adjustment occurs with many
biological species is a very plausible, one might say an inevitable
supposition; but the approach to the ideal optimum is probably
often only very imperfect, if only because the nature of the case calls
for a large "factor of safety."''
In any event it is true that the birth rate does not play so unqualifiedly
a dominant role in determining the rate of growth of a species
as might appear on cursory reflection. The equation (5) of Chapter
IX, expressing the rate of growth of the species in terms of the
birth rate and death rate, while it renders correctly the quantitative
relations to which it refers, is only a partial or one-sided representation
of the facts, and is even open to misinterpretation. Incautiously
construed it might be taken to imply that growth of an aggregate
of living organisms takes place by births of new individuals into
the aggregate. This, of course, is not the case. The new material
enters the aggregate in another way, namely in the form of food consumed
by the existing organisms. Births and the preliminaries of
procreation do not in themselves add anything to the aggregate, but
are merely of directing or catalyzing influences, initiating growth, and
guiding material into so many avenues of entrance (mouths) of the
aggregate, provided that the requisite food supplies are presented,
provided that the system is, in a sense, "supersaturated" with
regard to the species seeking to grow therein. The final result
may not depend very greatly on the number of births, somewhat
as the final state of a crystallizing solution is independent of the
number of crystal germs initially sown therein.
It will be desirable to develop our analysis in such manner as
to bring out the relations thus involved.
Aggregates of Constant Units. In aggregates of a simpler kind, as
presented to our view in ordinary physico-chemical systems, each
individual unit retains its identical substance unchanged throughout
its period of existence as such unit. So, for example, a molecule of
water consists of two particular atoms of hydrogen united to one
specific atom of oxygen; and these same atoms continue to exist
together as the building stones of the molecule as long as this continues
in existence as a water molecule.
130 ELEMENTS OF PHYSICAL BIOLOGY
In these circumstances the mass of each unit is obviously constant
throughout its period of existence as such: and furthermore, addition
to the component, or elimination therefrom, can take place only by
the actual entry or departure of a complete unit. If, therefore, the
mass of each unit is rtii, and if Bi new units are added to the aggregate
per unit of time, while Dj are eliminated, we have, in this case,
very simply,
^=(Bi->i)mi (3)
at
Aggregates of Variable Units. For aggregates of living organisms
we can also write an equation identical in form with (3), as has
already been noted in Chapter IX; Bi is in this case the total birth
rate, DI the total death rate, and m^ the average mass per head of the
living population. But this is an inadequate representation of the
significant facts. The equation, thus written, glosses over certain
important characteristics of living organisms. Unlike molecules in a
system in the course of chemical transformation, each unit in an
aggregate of living organisms does not retain its substance unchanged
in identity or in total mass. In fact, each unit is itself an aggregate
within the larger aggregate that constitutes the species or biological
group, and for each individual unit (organism) separately we can
write an equation analogous to equation (1) of Chapter IX
where U'i is the total mass taken up (ingested) per unit of time by the
unit organism, and Vi is the total mass eliminated therefrom per unit
of time. So, for example, in the course of one year a boy ten years
old and weighing 32.5 kgm. may consume about 600 kgm. of food
(inclusive of water and oxygen), may eliminate about 599 kgm of
wastes, and will grow in actual mass by about 1 kgm., so that we have
~ = 600 - 599 = 1 = O.OSmi (5)
at
The Stream of Substance Through, the Form of the Organism.
It will be observed that a portion of the intake, but a portion only, is
expended in adding to the total mass of the unit. The remainder
ANALYSIS OF GROWTH FUNCTION 131
B,-17',-^ (6)
at
is expended without, apparently,
1 any
resulting increase in the total
mass of the unit. This constant expenditure of substance, and the
equally constant intake required to balance it, is a fundamental characteristic
of the units here under discussion. In the adult, whose
mass is (on an average) approximately constant, we have simply
and the entire intake goes to meet the requirements of maintaining
the mass of the unit at constant level. 2
Turning now from the consideration of the individual unit to that
of the aggregate of N such units, evidently, if the average intake per
unit of time per individual is U'i, and if the average elimination is
Vi, then we shall have for the rate of increase of the total mass Xi of
the aggregate.
cLTi _= NU'i- NV'i - Nm'idi (9)
at
where di is the death rate per head per unit of time and m'i is the average
mass of a unit (organism) at death.
Two Types of Organisms: Economical and Lavish Birth Rate.
The relative importance of the second and the third term in the right
hand member of the equation (9) differs greatly in different biological
species. At the one extreme we have a type of which perhaps the
most characteristic representative is man. With a mean length
of life of about fifty years, his body must be replaced about twice
in a century to maintain the population equilibrium. If we assume
(as a rough but sufficient approximation) that the average weight of
man at death is 50 kgm., this means that the third term,
1
Indirectly a part of the excess Ri of the mass intake over the mass increase
may contribute to that increase, namely by furnishing some of the energy required
for anabolism. But we are here discussing mass relations only. The
energy relations are reserved for separate consideration later.
Except
during gestation, if the mass of the fetus is reckoned in with that
of the mother.
132 ELEMENTS OF PHYSICAL BIOLOGY
vy 50
in a stationary population, is about ^p kgrn. for the entire popu-
ou
lation, or just about 1 kgm., or say 2 pounds per head per annum. To
put it crudely, of the food consumed by each human individual in a
year, 2 pounds go, on an average, to replace the bodies of his fellows
departed that year. This, it will be seen, is an insignificant, almost
wholly negligible fraction of the 1000 pounds
3
or more than he consumes,
in all, in a year. Of the total food consumed by the human
race, then, about 0.2 per cent3
goes to replace the bodies eliminated
by death. The remainder is for current maintenance of the living.
And the total food consumption
3
may be of the order of 7 to 10 times
the mass of the population per annum.4
But, as already stated, man represents an extreme type, the extreme
economy of life, with low death rate and correspondingly low
birth rate. Of the opposite extreme, lavish, seemingly wasteful
extravagance, examples are exceedingly common, though it may not
be easy to give full quantitative detail. Among the most wasteful
breeders are, no doubt many aquatic species, including fish, since
their young are ill protected and become ready victims of other
species. So a ling weighing 54 pounds was found to be carrying
twenty-eight million eggs.
5
An oyster may have sixty million eggs.
But some familiar land animals are prolific enough, even if they do
fall far behind the standards just exemplified. The brown rat may
have five or six litters averaging about eight or ten each, in a year.
6
Domestic Animals Kept for Produce. Accurate figures can be
obtained in case of domestic animals. While these do not represent
so extreme an example, a special interest attaches to them owing to
their direct relation to human food economics. The most prolific
among domestic animals is the pig. In reasonably good farm conditions
a sow should average three litters in two years, each of seven
farrows, of which five are successfully raised and marketed.
Even with the high mortality artificially induced by man in his domestic
stock the item of running expenditure in feed for mere maintenance
is far in excess of the replacement cost, that is to say, the feed
3
Exclusive of oxygen.
4
For quantitative data on growth in man see C. S. Minot, The Problem of
Age, Growth and Death.
6
J. A. Thomson, The Wonders of Life, 1914, p. 130.
s
H. H. Donaldson, The Rat, 1915, p. 190.
ANALYSIS OF GROWTH FUNCTION 133
stored up and finally utilized in the carcass of the slaughtered animal.
From a detailed study of the vital economics of beef production made
at the University of Missouri7
figure 20 is reproduced here to show
these relations. The convex curve shows the average growth per
head in a group of steers fed with a ration regulated to secure a maxi-
MOOO
/DO SOO 30O -90O 00 600 700 goo SCO /OOO //OO /200 t30O HOO OOO
FIG. 26. FEED CONSUMED, AND INCREASE IN LIVE WEIGHT OF
STEERS AT SEVERAL AGES
Dry matter consumed is represented on one-tenth the scale of the live
weight. After Moulton, Trowbridge and Haigh.
mum of growth, without storage of surplus fat; the approximately
straight line mounting upward shows the steadily increasing integrated
amount of feed consumed since birth. Only the dry weight
of the feed is plotted, and the scale employed is ten times more
7
University of Missouri, College of Agriculture, Bulletins 43, 54, 55 (E. C.
Trowbridge and L. D. Haigh).
134 ELEMENTS OP PHYSICAL BIOLOGY
condensed than that used for the live weight else the second
would rise too steeply as to lie for the most part far beyond the HnvT
of the page. Thus is shown the great disproportion between the feed
TABLE 9
Average Yearly Gains of Steers*
Srowth
not
body of the growing steer-
and
ceraM ;
' S &r M the interest of the P^uoer is concerned,
in mere
mamtenance, for the private satisfaction and benefit
ANALYSIS OF GROWTH FUNCTION 135
of the animal, so to speak. The numerical data on which figure 26
is based are exhibited in table 9, together with the corresponding
figures observed when the animals are somewhat underfed and overfed
respectively.
The instances that have been cited man on the one hand, and the
highly prolific species, both feral and captive, on the other are eloquent
illustrations of the elasticity of adaptation. Clearly, a species
may hold its own, in the straggle for existence, either by the aid of
well-developed protective devices resulting in a low death, rate, and
requiring only a correspondingly low birth rate; or, a less well protected
species may balance a high death rate by an equally high birth
rate. Which of these two methods would be chosen in the natural
course of events is a question that it might be difficult to answer on
any general a priori principle, so long as attention remained fixed on
a single species. Perhaps one would have expected evolution to turn
in a favor of the more economical method of meeting a low death rate
with a low birth rate. In point of fact both types of organism the
economical type (as judged by its own standard) with low death rate,
and the wasteful with high death rate exist side by side in abundance.
This is a good example to illustrate the purely relative character
of fitness, and to remind us once more that we cannot expect any
success in attempts to define the direction of evolution in terms of a
single species. It is not the individual species, the individual components
of the system, that evolve, but the system as a whole, comprising
all the species and their environment. The species of the
economical type, with low death rate, are largely dependent for their
subsistence on the presence of species of the opposite type; we must
think here of a competition, not between individual species, but between
groups of species, groups consisting, in the simplest case, of
two species each, a food species or prey, and a feeding or predatory
species. Of two such groups, that one will, other things equal, have
the advantage in the struggle, in which high productivity of the food
species is accompanied by economy of life on the part of the feeding
species. From the point of view of the hog, so to speak, the high mortality
in the pen is a disastrous inefficiency and maladaptation, a
misfortune to be borne, as best it may, with porcine philosophy
1
.
From the point of view of the consumer on the other hand, this high
mortality is, quite on the contrary, a measure of the efficiency, the
eminent fitness of swine as producers of pork; and his only regret
136 ELEMENTS OF PHYSICAL BIOLOGY
is that so much of the feed placed in the trough goes merely to carry on
"what may be called the personal activities of the animals themselves."
8
It is to be noted, however, that only a part of the material
accountable as waste from the standpoint of the food species is gain
for the feeding species. Deaths from disease are a pure loss to both
species.
Similar reflections, of course, apply, mutatis mutandis, to those cases
in which the feeding species derives its nourishment from some current
product of the life activity of the food species or host, instead of
from its carcass. The most notable example of this in the food economy
of man is his exploitation of the milch cow, who is a far more
efficient producer than the beef steer. The latter at best consumes
over 6 pounds of nutriment for every pound of product.
9
According
to the investigations of the National Research Council about 18 per
cent of the energy of grain fed to cattle is recovered for human consumption
in milk, but only about 3.5 per cent in beef. Similarly,
crops on a given area will yield about four to five times as much protein
and energy when fed to dahy COWT
S as when used for beef production.
In providing mineral substances and vitamines the milk of
cows contrasts even more favorably with the beef animal. The
vitamines and calcium salts contained in hay and grain are stored in
the muscular tissue only to a slight extent, but are in relative abundance
in milk.10
From the standpoint of the dairyman a thoroughbred
prize cow, such as Glista Ernestine (a Holstein), which gave in one
year 833 pounds of butter fat, and in one hundred days 10,000 pounds
of milk, is a very model of efficiency, producing more than her own
weight in milk each month. But from the point of view of the bovine
species such record performances are gross inefficiency, approaching
in some cases perilously near to total biological unfitness, for some
of the record Jersey cows are probably unable, under the conditions
of the stable at any rate, to raise their own calves the over-rich
milk would probably kill the young animal.
Network of Chains of Interrelated Species. The relation between
man and the domesticated species of animals and plants on which he
8
I have here borrowed a felicitous phrase from an anonymous writer on
the editorial page of the New York Times, February 10, 1921.
9
University of Minnesota, Agr. Exp. Station Bulletin 193, pp. 68, 69
(T. L. Haecker).
10
Jour. Franklin Inst., vol. 190, 1920, p. 155.
ANALYSIS OF GROWTH FUNCTION 137
EO largely depends for food, in the present state of civilization, is only
a particularly tangible, a particularly accessible example of an intricate
network of relationships that connect more or less closely all
living species. In this network each species or component is interlaced,
like a link in a meshed coat of mail, with other species, which
in turn connect with still others, and so forth. In our effort to
get some sort of mental grasp of the complicated interlocking of these
elements we seize upon some one link, some one species or component,
and we note, first of all, that whatever is eliminated from one component
of a self-contained system must pass into one or more other
components of the system. So, for example, the component Si
may be a herd of cattle. The matter eliminated from this component
goes in part as food to build up or sustain a human population; in
part it goes as fertilizer on the fields to furnish nutriment for crops;
still other parts are worked up into various industrial products, such
as leather, glue, etc. We thus have, in schematic representation,
Cattle
\
Human
population
\
Leather
s
SS
Fertilizer
L
\
\
sk
On the other hand, the substance of the herd itself is recruited from
certain other components of the system, grass, clover, corn, etc., so
that we may further develop the scheme
Clover
Grass
\
Corn
/ \
Cattle
\
Human
population
\\
Leather
\N
Fertilizer
Transformation Factors and Their Economic Significance. In
general any one component thus appears as a link in a complicated
chain or rather network of chains; the component St , for example
138 ELEMENTS OP PHYSICAL BIOLOGY
receives a certain fraction at, i of the mass VtXt eliminated per unit
of time from the component St', it passes on to the component S&
a certain fraction ik of the mass ViXi eliminated from Xi itself.
/l~y
The rate of growth n2
of Xi is the balance of the sum Ui of all contributions
it receives over and above the sum V\ of all the contributions
which it makes to other components, thus
dXi = mXi - viXi = Ui - Vi (10)
at
= ZatiVtXi - S/3iki>iXi (11)
the first summation being extended over all those components St
which contribute to Xi and the second over all those components
8k to which Si contributes.
But we may also analyse the contributions to and from the component
Si in another way. We may say that, of the total contributions
per unit of time UiXi to the mass Xi, a certain fraction yuUiXi
is derived from Sf. Then
(12)
= vfXf (13)
7it
and, substituting (13) in (10),
Lastly, if the system is not self-contained, we must add a term Ji
for "imports" per unit of time, and a term EI for "exports" per
unit of time, that is to say,
-^ = vtXf
- ViXi + Ii-Ei (15)
dt 7n
When Si is the human species, the coefficients a, jS, 7 have an obvious
economic significance. The restriction of this remark to the human
species must not be taken to imply that there is in this feature something
wholly peculiar to man, but rather, that underlying our economic
manifestations are biological phenomena which we share in common
with other species; and that the laying bare and clearly formulating
ANALYSIS OF GROWTH FUNCTION 139
of the relations thus involved In other words, the analysis of the biophysical
foundations of economics is one of the problems coming
\ within the program of physical biology. Hints as to the direction
in which we may or must look for light on this phase of our problem
have now been noted upon several occasions. So it was observed
that the components of a life-bearing system can be divided into two
classes, relative to the component Si, namely, on the one hand those
components S-}
for which -^ ~^T
was positive, or, as we may say,
those useful to the species Si, those having for it a positive value,
and. on the other hand, components 8% for which ^ 3i
l
<0, comV.A
ic at
ponents harmful to Si, or having for it a negative value. Elsewhere
we have noted the classification of components into replace,
able and indispensable components, a classification that at once
recalls elementary economic reflections.
These hints we note in passing. They may serve to put our minds
in a state of preparedness for the more formal and decisive attack of
the problem, to which we shall be led in the last division of our enquiry,
dealing with the dynamics of life-bearing systems.
Jf
STATICS
CHAPTER XI
GENERAL PRINCIPLES OF EQUILIBRIUM
Repeatedly, in preceding chapters, occasion has arisen to refer to
stationary states or equilibria. Inevitably, in the discussion of the
kinetics of evolution, one is led to consider incidentally certain
conditions and special cases in which the velocities of the changes in
the evolving system are zero; when, that is to say, the system under
discussion is in a steady or stationary state, in equilibrium. Viewed
from this avenue of approach equilibrium presents itself as a special
case of motion or change, namely motion or change with zero velocity.
Indeed, something very like equilibrium occurs also with velocities
that are merely small, not vanishingly small. In such case the
phenomenon of moving equilibrium may present itself, as we shall
have occasion to observe in greater detail in due course.
Stationary states equilibria and near-equilibria play an important
role in nature, and it is desirable at this point to give them
something more than incidental consideration; to sketch, at least in
outline, their systematic study; to stake out, in the rough, that field
which, in our survey of the Program of Physical Biology (Chapter V)
was designated as the Statics of Evolution, and was systematized
according to the schedule
Statics
"I
Equilibria Moving equilibria Displacement of
(steady states) equilibrium
It will be convenient, in the development of the subject, to follow,
in the main, the schedule thus set forth.
Kinetic, Dynamic, and Energetic Conceptions of Equilbrium.
While we shall, in this section, conceive a stationaiy state
from the standpoint of kinetics, defining it as a state in which certain
velocities vanish, it must be noted that there are also other conceptions
of equilibrium. Etymologically the word equilibrium is tied,
in stricter usage, to a dynamic conception: Aequa libra, the poised
balance, is symbolic of a state in which forces are balanced, in which
the resultant force vanishes,
143
144 ELEMENTS OF PHYSICAL BIOLOGY
A third conception of equilibrium, differing from the second, the
dynamic, only in point of view, not in scope, is derived from a consideration
of energy relations. A system in dynamic equilibrium is
found to be characterized by the attainment of a minimum (or sometimes
a maximum) of certain functions having the dimensions of
energy; a state in which the virtual work done in any very small
displacement compatible with the constraints vanishes.1
So, for
example, a ball placed hi a hemispherical cup, is in equilibrium when
its potential energy is a minimum compatible with the geometry of
the system. More generally, equilibrium is, according to this view,
defined as a state in which certain potentials have a minimum (or
a maximum).
Pedantic usage would demand that the term equilibrium be reserved
for states satisfying the dynamic and energetic conditions of
rest or invariability in time. It would deny the appellation equilibrium
to certain states commonly so designated. Metabolic equilibrium,
population equilibrium, and the like, are not true equilibria,
in this narrower sense, but are steady states maintained with a constant
expenditure, a constant dissipation, of energy. It is not necessary,
however, at present, to lay any stress on this distinction. The
occasional use of the word equilibrium in speaking of what is merely
a steady state maintained with a continuous expenditure of free
energy is not likely to cause any serious confusion; and we may as
well take the usual liberties in the matter, whenever this course is
dictated by convenience and does not offend against essential principles.
Where express distinction becomes necessary, we may speak
in specific terms of true equilibrium and quasi-equilibrium, respectively,
to denote the two separate types included hi the generic term
"stationary state" or "steady state."
A complete treatment of the entire field of the statics of evolving
systems should, to be entirely systematic, cover both types of
stationary states. There are, however, two reasons for departing
somewhat from such strictly systematic arrangement. The first
is that the statics of true equilibria have been developed to a high
degree in the discipline of thermodynamics, so that an exposition of
the pertinent principles and conclusions would be little more than
1
Stability of equilibrium demands, further, that the work done on the
system in any small, but finite, displacement, be positive, that the potential
energy be a minimum (maximum being, in this case, excluded).
GENERAL PRINCIPLES OF EQUILIBRIUM 145
a transcription into these pages of what can be found abundantly set
forth elsewhere in the standard literature. However much one might
be tempted, in the interest of a well rounded presentation, to sketch
at this point at least an outline of the relevant chapters of thermodynamics,
economy of space dictates the briefer expedient of referring
the reader to the existing literature, so abundant that it seems superfluous
to mention titles.
A second reason for passing lightly over true equilibria at this
point, is that the steady states with which we are most frequently
and most closely concerned in the field of organic evolution (our
main topic here), are of the second class; not true, equilibria in the
dynamic sense, equilibria in which all forces are balanced; but
what we have termed above quasi-equilibria, states maintained
constant or approximately so with a continual expenditure, a continual
dissipation or degradation of available energy. To such as
these we shall give our chief attention, though in part our discussion
will be framed broadly to cover indifferently either type of steady
state. For the sake of example, too, reference will be made, on
occasion, to systems evolving toward a true equilibrium; systems for
which the law of evolution is capable of direct expression In comparatively
simple thermodynamical terms; systems which, by that
very fact, are peculiarly adapted to serve as paradigms exhibiting
the characteristic form of a law of evolution.
General Equilibrium Condition. As has already been noted incidentally,
the general condition for equilibrium, or, to be more
precise, for a stationary state, is obtained by equating to zero the
velocity of growth of each component of the system, thus
= Fi(Xi, Xz, . . . Xn) (1)
at
F! = Fz
= . . .
= Fn = (2)
This condition, in general, furnishes n independent equations,
which determine one or more sets of values of the variables X, thus
Xz ~ * '
(3)
146 ELEMENTS OP PHYSICAL BIOLOGY
If the values C thus determined are real and positive
2
they evidently
define an equilibrium or a steady state, the character
(stability
mode of approach) of which depends upon the nature of the roots
X of a certain characteristic equation, as has been indicated on an
earlier occasion.
Different Types of Equilibrium. Graphic Representation. A
particularly graphic representation of the different types of equilibrium
is obtained if, instead of seeking solutions of the fundamental
equations (1) expressing Xi, X2 . . . Xn in terms of t, we on the
contrary eliminate t from this system of equations. This is
very
readily effected by division, which leads to the new system
dXi dXz dXn
T!
=
~F~*
= ' ' '
"
71 (4)
This system of equations defines a family of curves passing through
the equilibrium points, which here appear as singular points. The
situation is particularly transparent in the case of two variables
Xi, X2,
since this readily permits of plotting the integral curves in
rectangular coordinates in the plane of the paper. We have already
had occasion incidentally to employ this method of treatment in an
example in Chapter VIII, in which the conflict between a host species
and a parasite species was examined analytically. Without going
into extensive technical details it is advisable now at least to enumerate
and briefly describe the several types of equilibria and the
topography, characteristic of each type, presented by the integral
curves in and about a singular point. These types are somewhat
numerous, even if we restrict ourselves to the case of two variables,
and brevity is therefore imperative.
Type 1. Roots Xi and X2 real and negative. Equilibrium is stable; integral
curves run directly into singular point as in figure 27, A.
Type 2. Roots Xi and X2 real and positive. The topography is similar to that
of type 1, but integral curves are traversed outward from singular point.
Unstable equilibrium, figure 27, B.
Type 3. Roots Xi and X2 real and of opposite sign. Integral curves in general
do not pass through singular points, but curve away from it. Unstable equilibrium,
figure 27, C.
2
Masses cannot assume negative or imaginary values. Hence negative
roots may fail to define equilibria; a similar statement holds regarding
complex roots.
GENERAL PKDSTCIPLES OF EQUILIBRIUM 147
Type 4. Roots Xi and X2 complex, real parts negative. The integral curves
are spirals winding into the origin, forever approaching it without ever reaching
it. Stable equilibrium, figure 27, D,
Type 5. Boots Xi and X2 complex, real parts positive. The topography is
similar to that of type 4, but integral curves are traversed outward from
singular point. Unstable equilibrium, figure 27, E.
Type 6. Roots Xi and X pure imaginaries. This gives rise to several distinct
subtypes.
Subtype F. Integral curves are closed loops enclosing the origin. Process
is purely periodic. Figure 27, F.
Subtype G. Integral curves are spirals winding inward. Stable equilibrium.
This is the case treated in Chapter VIII, where a representative diagram
will be found. Figure 27, G. Another subtype is similar to G but spiral
winds outward. Unstable equilibrium.
Subtype H. Integral curves are spirals winding about a closed loop.
Types I AND J occur when \i = X2 .
Figure 27, 1, J.
As an example illustrating the occurrence of two types of equilibrium,
two types of singular points, the topographic chart of the
integral curves defined by the Ross equations for the spread of malaria
under certain conditions is shown in figure 28. It will be seen that
there are two singular points, one at the origin 0, unstable, of type
C; the other at T, stable, of type A. This chart obviously suggests
"stream lines" and a three dimensional model. Such a model
(purely qualitative) is shown in figure 29. The feature of interest is
that a singular point like 0, of type C, is represented by a col
("notch") in the landscape; whereas the stable equilibrium of type
A is represented by a pit, as at the point T.
While this model refers to a very particular case, it serves to bring
out a noteworthy fact, namely, that there are necessarily certain
regularities in the occurrence of the various types of equilibria. So,
for example, it is clear that two pits of the character of the point T
cannot occur without some other type of singular point between
them, just as it is physically impossible for two mountains to rise
from a landscape without some kind of a valley between. For a
detailed study of this phase of the subject the reader must be referred
to the mathematical literature.
3
3
For further discussion of the various types of singular points that may
occur the reader is referred to the mathematical literature, of which the
following may be mentioned: E. Picard, Trait<5 d'Analyse, 1891, vol. 1, pp. 83,
123; 1893, vol. 2, pp. 183, 193, 196 (footnote); 1896, vol. 3, pp. 228, 238; v. Dyk,
Sitzungsber Bayer. Akad. Wissensch. Miinchen, March 6, 1909,^AbhandL 15;
Abhandlungen derKgl. Bayer. Akad. Wissensch., March, 1913, vol. 26,Abhandl.
10; Sitzungsber, 1891, p. 23; 1892, p. 101; H. Liebmann, Lehrbuch der Differential
gleichungen, 1901, pp. 101, 102, 134.
ELEMENTS OF PHYSICAL BIOLOGY
FIG. 27. SOME FUNDAMENTAL TYPES OP EQUILIBRIUM, IN A SYSTEM WITH Two
DEPENDENT VAEIABLES
GENERAL PRINCIPLES OF EQUILIBRIUM 149
FIG. 28. MAP OF INTEGRAL CURVES FOR THE Ross MALARIA EQUATIONS, AS AN
EXAMPLE EXHIBITING Two SINGULAR POINTS, OP TYPE 1 AND 3 (SEE
FIGURE 27, A AND C)
The heavy lines are integral curves; the lighter lines are auxiliaries
(isoclines) employed in constructing the graphic solution of the differential
equations. (Reproduced from A. J. Lotka, Am. Jour. Hygiene, January
Supplement, 1923.)
150 ELEMENTS OP PHYSICAL BIOLOGY
.
,JH'/!' <>';''
X" \' A >' "
"ft *M'ft' "'" s. '"-'
;
GENERAL PRINCIPLES OF EQUILIBRIUM 151
Metastable Equilibrium. The graphic representations of the
malaria equilibrium furnish the occasion for another remark of general
character regarding certain equilibria. The circumstance that gives
rise to the first malaria equilibrium, the one in which the malaria
rate is zero, (point in figs. 28 and 29) is the autocatakinetic character
of the growth, of a malaria endemic . This is a common characteristic
of the growth of living systems; growth is initiated by a nucleus
of the same species of matter that is added by the growth. Conversely,
in the entire absence of any nucleus of a particular species of
living matter, growth of that species cannot take place, even though
all other conditions for such growth may be satisfied, even though
the system may he, as it were, supersaturated with regard to that
species of matter. In these circumstances an equilibrium may be
presented which is unstable in the sense that, upon the introduction
of a suitable nucleus, growth immediately sets in.
4
Equilibria of
this type, which are stable in the absence of a suitable "nucleus"
but in which change is immediately initiated upon introduction of
such a nucleus, have been termed "metastable" equilibria.
Exceptional Cases: A brief reference must suffice regarding
certain exceptional cases that may arise. So it may happen tbat one
of the roots X of the characteristic equation vanishes. An example
of this was encountered in dealing with the Ross malaria equations.
It was found that as the number of mosquitoes per head of the
human population approaches a certain critical value, two singular
points approach each other, and finally fuse, giving one "double"
point.
5
Another special case that may arise, and whose mention must here
suffice, is that of multiple roots of the characteristic equation, the
case in which two or more of the roots are equal.
6
4
In inorganic systems an analogous state of affairs is observed in supersaturated
solutions or vapors which, are brought to crystallization or to condensation
by the introduction of a suitable nucleus. Dynamically the characteristic
of a metastable equilibrium is that the thermodynamic potential of the
system, though a minimum, is not an absolute minimum.
6
See A. J. Lotka, Am. Jour. Hygiene, 1923, vol. 3, January supplement,
p. 12.
6
Compare H. Liebmann, loe. cit., pp. 102, 134; A. J. Lotka, Zeitschr. f.
physikal. Chemie, 1912, vol. 80, p. 16.
CHAPTER XII
CHEMICAL EQUILIBRIUM AS AN EXAMPLE OF EVOLUTION UNDER
A KNOWN LAW
I wanted to remind the biologists that in the early stages of life what they
are accustomed to speak of as natural selection passes over into what might
be described as a mere physical selection of stabler compounds. K. Pearson.
One of the simplest examples of equilibria in systems of the type
that interests us here systems composed of several groups each
consisting of numerous similar individuals as units is the equilibrium
resulting from a pair of balanced or opposing chemical
reactions.
This case illustrates so well, in their simplest form, a number of
typical traits of the phenomena here under discussion, that it will
pay to give it brief consideration.
We shall select for this purpose the simplest possible type of
balanced chemical reaction at constant volume and temperature,
namely a reaction which is monomolecular in both directions. A
substance Si undergoes a transformation into S%, and Sz in turn is
converted back into Si, one molecule alone taking part, in each case,
in the transformation. If Xi and xz are the respective concentrations
of Si and $2, we have, at a given temperature, by the law of mass
action, the rate of decomposition of Si and Sz respectively.
(Ztei)
= - faxi (1)
(D^) = - fax* (2)
Or, since at constant volume concentrations x are proportional to
numbers n of molecules
(Dm) = - fciin (3)
(Dnz) font (4)
where fci, k2 are coefficients (functions of the temperature) characteristic
of the reaction.
152
CHEMICAL BQUILIBBIUM 153
The rate of increase of the substance S is the excess of its rate of
formation 6 over its rate of decomposition, in strict analogy to the
birth rate and death rate in a human population
(fti-fcOm (5)
at
(6)
at
In a population of living organisms the material for the formation
of new individuals must ultimately be derived from the bodies of
those that have died. But the connection is a complicated one involving
many steps. In the population of molecules here under
consideration the relation between birth rate and death rate is of the
simplest possible form. Each molecule of Si that "dies" becomes a
molecule of 82, and vice versa. Thus equations (5), (6) assume the
form
and in equilibrium
, ,
kiUi (7)
at
- = kiHi ^22 (8)
at
K\
If we fix our attention upon HI molecules of Si at the moment of their
formation, we can apply to these particular molecules the equation
(3), from which we have, by integration
1 *
(10)
similarly for S2
= naCO)*-*** (ll)
\
But 77:7 is the probability pi(a), at the moment of its formation,
that a molecule of Si picked at random at such moment, will reach
age o. The life curve for the molecules of Si is thus defined by
Pi(a)
= <rkia
(12)
154 ELEMENTS OF PHYSICAL BIOLOGY
and for Sz
pj (a)
= e-^ (13)
while the mean lengths of life are
.,/T
= I
Jo
and in equilibrium
^ (16)
ft 2 La
the molecules are present in amounts proportional to their respective
mean lengths of life, although, they are "born" in equal numbers,
since fan* = AVZ.I. The significance of this is brought out in the
diagram figure 30, in which the population of molecules is plotted
"in age groups," for the substance Si and S2 separately. Since the
birth rate is the same for both populations, they begin at a common
ordinate; but the curve for 82, the substance with greater k, greater
force of mortality,
lies entirely below that for Si; the areas of the two
curves are, in fact, proportional to the mean lengths of life of the
molecules of the corresponding substances. Thus, in the struggle
for existence the stabler (fitter) molecules of Si have the advantage,
being, on an average, longer-lived.
There is thus an obvious analogy between the course of events in
such a population of different species of molecules, on the one hand,
and a mixed population of different species of organism on the other,
an analogy which extends into details for the exposition of which
space is lacking here. 1
The analogy is not a meaningless accidental
circumstance, but depends on identity of type in the two cases. It
can be said quite generally (so as to apply to either case) , that in a
material system in which physical conditions vary from instant to
instant and from point to point, certain individual constituents
(molecules, organisms) may have a transitory existence as such,
each lasting just so long as its conditions and those of its neighborhood
continue within certain limits. Although the "life period" of each
individual constituent may be thus limited, an aggregate of a number
1
For such details see A. J. Lotka, Am. Jour. Sci., vol. 24, 1907, pp. 199, 375.
CHEMICAL EQUILIBRIUM 155
of such individuals may nevertheless have prolonged existence, provided
that the fluctuations in the conditions of the system, from point
to point and from instant to instant, do not exceed certain limits,
and that by some process or other new individuals are formed as the
old are eliminated. Of the character of the fluctuations, and their
relation to the "length of life" in the case of living organisms, more
will be said in a later chapter. As to the circumstances, the fluctuations,
that lead to the translation of a molecule in chemical reaction
from one state into another, we may with advantage adopt a view-
3OOO
FIG. 30. AGE DISTRIBUTION IN POPULATION OF MOLECULES OF Two SUBSTANCES
IN MONOMOLECULAR CHEMICAL EQUILIBRIUM
The birth rate per head, of both species, is the same as indicated by their
common zero ordinate; but the species having the lesser force of mortality
(reaction constant) predominates, as shown by the greater area under the
corresponding curve. (Reproduced from A. J. Lotka, Am. Jour. Science,
1907, p. 208.)
point set forth in some detail by the writer on an earlier occasion,
2
and expressed more recently by Professor Baly
3
in these terms :
Every complete reaction consists of three separate stages, with each of
which is associated its characteristic energy change. In general, molecules
in the free state exist in a phase which is non-reactive, and in order to carry out
any reaction it is first of all necessary to bring them into a reactive phase.
This, which is the first stage of the reaction, requires that a definite amount
2
A. J. Lotka, Am. Jour. Sci., 1907, p. 213 et seq.
3
E. C. Baly, Photosynthesis. Nature, 1922, p. 344.
j.uu ELEMENTS OP PHYSICAL BIOLOGY
of energy should be supplied to each molecule, the amount necessary being
the difference in energy contents of the initial phase and the particular phase
necessary for the reaction in question.
The second stage of the reaction is the atomic rearrangement whereby
new molecules are produced, and it is this stage, and this stage alone, which
is represented by the equation of the reaction.
The third and final stage is the change in phase of the newly synthesised
molecules, whereby they pass into their normal and non-reactive phases.
These last two stages are both accompanied by an escape of energy. If the
sum of the amounts of energy evolved in the second and third stages is greater
than that absorbed in the first stage, the reaction is exothermic; whilst an
endothermic reaction is one in which the energy necessary for the first stage
is greater than the total amount evolved in the second and third stages.
It should be remarked that the second stage is, apparently, passed
through in an exceedingly brief space of time, so that at any instant
only an imperceptibly small amount of substance exists in the transitional
state. We are, in fact, almost wholly devoid of any information
regarding matter in this state, and the words of Schonbein4
hold true in almost their full force today: "Presumably, between the
state in which two portions of matter exist after completion of chemical
combination, and the state in which they previously existed
separately, there is a series of transition states of which the chemistry
of today knows nothing." Probably the only positive and direct
experimental evidence we have of matter in this intermediate state
between two compounds is furnished by the superlatively refined
methods of Sir J. J. Thomson and Dr. F. W. Aston, which not only
reveal but actually weigh such decapitated molecules as CHs, whose
length of life is measured in ten-mil lionths of a second.5
As to the agencies, the "fluctuations" that provide, every now and
again, the requisite energy to carry a transforming molecule "over
the crest of the hill," there is first the thermal agitation of the molecules,
second the influence of incident light in photochemical reactions,
and third the influence of catalysts, whose action probably
depends on a flattening of the path over the hill crest, the point of
4
Jour. Prakt. Chem., vol. 55, I, p. 152.
B
Compare A. J. Lotta, loc. cit., p. 214. F. W. Aston, Science Progress,
1912; I. Langmuir, Jour. Am. Chem. Soc., 1920, vol. 42, p. 2190. Perhaps in
this connection should be mentioned also recent studies on the duration of
atoms in their several quantum states. See E. C. Tolman, Proc. Natl.
Acad. Sci., 1924, vol. 10, p. 85; L. A. Turner, Phys. Rev., 1924, vol. 23, .>
p. 464; F. M. Kannenstine, Astrophys. JL, 1924, vol. 59, p. 13. ^
CHEMICAL EQUILIBBIUM 157
departure and the final state remaining unchanged. For discussions
of these technical details the reader must be referred to the literature,
a few of the more recent publications being noted in a footnote below.6
While the details of the manner of the "birth" and "death" of the
molecules in chemical transformation are, as yet beyond the range
of the observation of the physicist, the fundamental laws of energetics,
which hold true generally, and independently of particular
features of mechanism, are competent to give substantial information
as to the end product, at any rate, of the evolution of such a system
as considered in the simple example above. The final equilibrium
must accord, as regards its dependence on temperature, pressure and
other factors, with the second law of thermodynamics, which may
thus be said to function as a law of evolution for a system of this kind.
This is a point worth dwelling on a little at length, inasmuch as our
knowledge of the form and character of the law of evolution for this
special type of system may be expected to serve as a guide in the
search for the laws of evolution in the more complicated systems,
belonging to an essentially different type, which confront us in the
study of organic evolution. The second law of thermodynamics can
be expressed in various ways, but the form in which it serves our
present purpose best is that which states that the system evolves
toward a state in which certain functions (thermodynamic potentials)
of the variables defining its condition are at a mrnimum, somewhat
as a ball placed in a hemispherical bowl ultimately comes to rest in
the position in which its (gravitational) potential is a minimum,
namely, at the lowest point of the bowl. Mary laws of nature are
conveniently
7
expressed in this form, as minimum (or maximum)
8
Regarding the r61e played by thermal agitation and by radiation see G.
W. Todd and S. P. Owen, Phil. Mag., vol. 37, 1919, p. 224; I. Langmuir, Jour.
Am. Chem. Soc., 1920, vol. 42, p. 2190; W. H. Rodebush, Jour. Am. Chem.
Soc., vol. 45, 1923, p. 606; J. M. Lowry, Trans. Faraday Soc., vol. 17, 1922, p.
596; J. A. Christiansen, Zeitschr. phys. chem., vol. 103, 1922, p. 91. Regarding
the influence of radiation see especially the publications of Professor Baly.
See also J. Mellor, Chemical Statics and Dynamics, 1904, pp. 394, 414, 415;
and Perrin and Hammick, Atoms, 1923, p. 168. The literature on catalysis
is so extensive that no attempt is made here to give even a key to it.
7
Fundamentally this is a matter of convenience, and does not predicate
anything narrowly characteristic of natural laws. The fact that the course
of events is uniquely determined implies that the laws which determine that
course can be expressed in the manner referred to. For a discussion of this
question see J. Petzoldt, Maxima and Minima und Okonomie, Altenburg,
1891, pp. 17 et seq.
158 ELEMENTS OF PHYSICAL BIOLOGY
laws, and it is to be expected that the law of evolution in life-bearing
systems also, (where, as we shall see later, mechanism cannot be
lightly waved aside into the convenient catch-all of the laws of
thermodynamics), will be found to receive its most convenient
expression in this form. In another respect the case of chemical
evolution may confidently be expected to be found a good model
in the treatment of the broader problem of evolution. It is to be
noted that the law of chemical evolution is expressed in terms of the
system as a whole. It is the thermodynamic potential of the entire
system that approaches a minimum. Biologists have rather been
in the habit of reflecting upon the evolution of individual species.
This point of view does not bear the promise of success, if our aim
is to find expression for the fundamental law of evolution. We shall
probably fare better if we constantly recall that the physical object
before us is an undivided system, that the divisions we make therein
are more or less arbitrary importations, psychological rather than
physical, and as such, are likely to introduce complications into the
expression of natural laws operating upon the system as a whole.
As regards the formulation of the laws of evolution in form of a
maximum or minimum principle, it should be remarked that one such
principle follows directly from the fundamental equations of kinetics
as set forth in Chapter VI.
If we multiply the first of these equations by Xi, the second by X2,
and so on, we obtain
at
-2 .
~TT ^ ^1X1X2 + 022X2- +...-}- a2n.Xn.X2 +at (17)
dt
K n "" "
nn- n.
Hence by addition
A
-S & = 2Q(xh xt, . . . xn) + . . .
(18)
where Q represents a quadratic form. The relation thus obtained is
not of general utility in this form. However, by a linear substitution
CHEMICAL EQUILIBRIUM 159
& = NlXl + N,X, + . . . + 2STna;n (19)
the equation (18) can be transformed into
~
at
(20)
where Xi, X2 ,
. . . Xn are the n roots of the characteristic equation for
X, the same X's that function as exponents in the series solution of the
original system of equations.
8
Now it will be recalled that the condition
for stability at the origin is that all the real parts of the roots
shall be negative. But in that case the quadratic form Q' is definite
and negative. Hence the condition for stability at the origin can
be expressed by saying that the quadratic form Q' must be definite
and negative; or, by saying that Q' must have a minimum at the
origin. And the law of evolution, near the origin, evidently is,
according to (20), that S 2
continually decreases. (At points remote
from the origin the terms of higher order, which have here been
omitted, may cause increases in S.)
The chief interest of the minimum principle here indicated lies in
its analogy
9
to certain theorems in dynamics and thermodynamics,
for which reference must be made to the literature, in particular to
P. Duhem, Traite d'Energetique, 1911, vol. 1, pp. 460 et seq.; F.
Michaud, Ann. de Phys., 1921, vol. 16, pp. 148 et seq.
8
For the sake of simplicity the argument has here been presented in the form
in which it appears when all the roots X are distinct and real. For a detailed
discussion of the conditions of stability when some of the roots X are multiple
or complex see E. Goursat, Cours d' Analyse, 1915, vol. 3, pp. 31-43.
9
The analogy to the dynamical cases treated in the reference cited becomes
particularly plain if we bear in mind that
= when i =j= j
= 2Xi when i j
so that the quadratic form Q' can be written
'
dQ' 5Q'
160 ELEMENTS OP PHYSICAL BIOLOGY
where the bracketed exponent (2) denotes the symbolic square, in which
'-
}
is>
replaced by
\JTC w\x W vjj
-, and the product =-r-
>T~J
ig replaced by -^ ^--.
\
The condition that the form so defined shall be negative, is that the deter-
minant
shall be negative, and also all determinants derived from it by striking out
the last p lines and the last p columns.
In the present case the same condition, can be expressed in simpler form
to the effect that Xi, \^, . . . \a must all be negative. But it is worth
while, in order to bring out the analogy, to note also the more complicated
general form of the condition.
CHAPTER XIII
INTER-SPECIES EQUILIBRIUM
Since the struggle for existence is chiefly a struggle for subsistence, a careful
comparative account of the food of various competing species and genera at
different places and seasons and at all ages of the individual .... cannot
fail to throw much light upon the details, causes and effects of the
struggle. Forbes.
Equilibrium Condition in More Particular Form. The fundamental
relations of statics are derived immediately from the corresponding
equations of kinetics by substituting in the latter the value zero
for the several velocities. This has already been noted with regard to
the equations of kinetics in their most general form. In somewhat
more particular form, useful in common numerical applications,
we have a condition for equilibrium derived from the system of
equations (14) of Chapter X
.
Xt
- ViXi (1)
dt 7if
(2)
7if
Xf JIM
i (3)
It should be noted that these formulae hold equally well if the
masses are measured in ordinary units (e.g., pounds) or if they are
measured in "head of population," with the proviso, of course, that
the coefficients ui, Vi, are in each case expressed in corresponding
units.
The equation (1) expresses the fact that, for each component, the
total inflow is just balanced by the total outflow, so that nowhere in
the system is any accumulation of mass going on. This clearly
implies that, unless there is complete equilibrium, the matter in the
system must be in circulation, it must be going through one or more
cycles. Such cycles are, indeed, very characteristic features in the
scheme of nature.
161
162 ELEMENTS OF PHYSICAL BIOLOGY
Numerical Illustration. We may, by the way of illustration, apply
the formula (3) to the equilibrium between the several biological
species comprised in a life-bearing system. For obvious reasons
numerical data are most readily available for man and the species
directly under his control. So, for example, we may let Xt. represent
the mass (or number) of a human population, and Xt the mass (or
number) of a population of sheep serving as food for that human
population. In the United States in 1918 the consumption of mutton
(or lamb) per head of the population per annum was 5.417 pounds.
This is not strictly an equilibrium ration, since our population is
increasing. However, the difference between this and the equilibrium
ration is probably small. In our example we will therefore
put
jitui
= 5.417 pounds = 0.1096 sheep
1
(4)
Again in the United States in 1918 the number of sheep slaughtered
per annum was 23.22 per cent of the standing herd. Hence
afit>i
= 0.2322 (5)
so that
Xf = "^Xi (6)f
0.2322
= 0.4718 Xi W
In 1918
Xi = 103,587,955 head (8)
Hence
Xt = 48,873,000 head (^
i
According to the Year Book of the Department of Agriculture, 1920, p.
759 the number of sheep slaughtered under Federal inspection m 1918 was
8 769,498. According to R. Pearl, The Nation's Food, 1920, p. 61, this represented
77 per cent of all the sheep slaughtered in that year, so that the total
number slaughtered was 1 1,370,000. The total dressed weight of these, according
to the Year Book, p. 826, was 502,214,000 pounds, which makes the average
of one Bhoop carcass 49.45 pounds.
The standing herd of sheep in 1918, according to the Year Book, p. 701,
was 48,C03,000 on farms, or, adding a correction for animals not on farms!
say
48,963,000.
The percentage of animals slaughtered in a year out of the standing
herd, was therefore 23.2216 per cent.
For a review of various estimates of the output of herds of cattle, sheep
swine, etc., the reader ia referred to a paper by R. H. Rew in the Journal of
the Royal Statistical Society, 1902, vol. 65, p. 666.
INTER-SPECIES EQUILIBRIUM 163
The actual standing herd of sheep in 1918 was 48,963,000 head of
which 90,000 head furnished mutton for export, the United States
not being a self-contained system.
It is to be noted that in this example the products 7jf U[ and
oifi vi are more easily ascertained than the individual coefficients <y
it, a, v. Inasmuch as these coefficients, in the formula (3), appear only
in these products, it is not necessary, for the purposes of this example,
to ascertain the values of the coefficients separately.
The example of the equilibrium between a human population and
the national herd of sheep, cited primarily for the purpose of illustrating
the equilibrium equation (3), incidentally brings out some other
points that may be noted in passing. We meet here a pointed suggestion
of economic factors entering into play in the processes which
we are studying. For the coefficients a 7, u, v, have obvious economic
relationship. So, for example, Vf, the proportion of sheep slaughtered
per annum (in a stationary state of the system), is of the nature of
interest on the standing herd, which latter, in turn, is of the nature of
capital. The gross interest rate of 23.2 per cent is, of course, greatly
diminished, in effect, by the extensive accessories, representing a
further investment of capital, and by the general running expenses
required, in addition to the mere herd, to produce, transport, and
market the ware. On the other hand, certain secondary products
(e.g., wool) add their quota to the returns on the invested capital.
Again, the coefficient Aif, which measures what fraction of the total
consumption by the component Si (human population) is derived
from St (sheep), is clearly a factor of economic significance. Mutton
.is typically a commodity of the replaceable type beef, pork, fish,
etc., furnishing ready substitutes. In such case as this the factor 7 if
will be elastic, capable of assuming, in ready response to slight
changes in general conditions, a whole range of values from zero up.
In the case of less readily replaceable commodities the coefficient 7
wih
1
be of more rigid habit.
The full significance and the precise nature of the relation between
the biological and the economic characteristics of a system must form
the subject of special considerations to which we shall find ourselves
led inevitably later, in our efforts to gain insight into the dynamics of
life-bearing systems. At this juncture it may not be amiss to indicate
in preparation of a viewpoint to be more fully developed later, that
if economic factors force themselves upon our notice primarily in the
164 ELEMENTS OF PHYSICAL BIOLOGY
consideration of systems comprising a human population, this is riot
because the operation of economic stresses is peculiar to human
aggregations, but only because these stresses find their ready numerical
expression and measure in such communities; owing to the development
of a system of social cooperation and division of labor, coupled
with a very special mechanism of adjustment by "economic
exchange/' which is peculiar to man. For though not a few other
species, bees, ants, etc., display a social organization in some respects
perhaps superior to ours, their organizations make use of other
expedients than the transfer of ownership through a universal
medium of exchange, in bringing about the allocation, to each individual,
of his share in productive effort and in product.
For obvious reasons our information regarding the interdependence
of the several biological species and other components of our lifebearing
system is most complete and most exact in so far as it relates
to species under human cultivation, species that contribute, as producers,
to our political economy. However, this does not mean that
we arc wholly cut off from all information regarding the life balance of
other species. Two sources, two methods of observation furnish us
with data on this subject, namely, first, biological surveys, and second
analyses of the stomachs of sample specimens. A third method would
coiiHisli in establishing experimental systems comprising several
HpocioH of organisms and making periodic censuses by direct count,
after tho manner of the work of Pearl and Parker with a single
tuxu'ioB (Drosophila). There is here an attractive field open for
wViirch Perhaps the readiest, though not the most interesting
approach to this problem would be the study of mixed bacterial
crowbhH ay along the lines followed, for a single species, by H. G.
Thornton, Annals of Applied Biology, vol. 9, 1922, p. 241.
Biological Surveys. Biological surveys, supplemented by estimate*
depending more or less on personal judgment, are aimed to
rivo UB Jmo degree of quantitative description of our world by mvesoranisms.
A
the number and the variety of living organisms.
biolORioal survey would enumerate the several general and
ound in the locality examined, and would furthermore give
mowuro of tho extent of each species, either in numbers or msome
1 "u tohte tonne. It would give us a species of "General Demolv''
cor globe. Needless to say, in this matter we are very far
lined perfection.
The best that can be done is to
INTER-SPECIES EQUILIBRIUM 165
give rather crude estimates, based, in the most favorable instances, on
counts or observations made with some degree of care, but without
pretense of great precision.
As to the number of species, some interesting figures are given by
J, A. Thomson.2
On the small island of Britain alone 462 different
birds have been observed; the total number of living species of birds
he estimates at not less than ten thousand. Of vertebrates he quotes
an estimate by H. Gadow.3
The total number of recent species this
author puts at 24,241, as follows:
Mammals 2,702
Birds 9,818
Reptiles 3,441
Amphibians 925
Fishes 7,328
Primitive vertebrates 27
24,241
The vertebrate elite, however, forms but a small minority in the
scheme of nature. It has been intimated that, if the present order
of things should come to a term, the supremacy would, as likely as
not, pass from the crowned vertebrate Homo sapiens to the now
despised, presently perhaps to be feared, creeping thing, the insect.
Indeed, it has been pointed out that were it not for the relentless
internecine warfare which its members carry on among themselves,
we should very soon find ourselves driven out of house, home and
granary by the insect pest. Even as it is, though the largest insects
barely exceed, individually, the size of some of the smallest vertebrates,
yet, as D. Sharp remarks, "the larger part of the animal
matter existing on the lands of the globe is in all probability locked
up in the form of insects." He estimates th,e number of insect species
that have been definitely named, at 250,000, and suggests that this
is only about a tenth of the total. The number of plant species has
been estimated at 200,000. Darwin records the finding of 20 species
in a patch of turf four feet by three.
As to the numerical strength of the several species, here again some
telling figures are given by J. A. Thomson.4
2
J. A. Thomson, The Wonder of Life, 1914, p. 11.
3
See also H. Gadow, The Wandering of Animals, 1913, p. 74.
*
J. A. Thomson, loe. cit., pp. 9-10.
166 ELEMENTS OF PHYSICAL BIOLOGY
_
At the spring maximum of the Rotifer Synchoeta there may be about three
millions to a square yard of lake. At the summer maximum of the slimy Alga
Clathrocystis ceruginosa there may be 500 millions to the square yard; at the
autumn maximum of a well-known diatom Melosira varians, which has a summer
maximum as well, there are about 7000 millions to the square yard, so that
the waters of the lake form a veritable living soup In an ordinary
sample from a warm part of the Atlantic and from a depth of 500 metres
(which is the most densely populated as far as plants go), there are likely
to be about 5,000 plant-cells to a liter; but there may be as many as a
quarter of a million
Elsewhere Thomson tells us that in the midst of a swarm of fish at
spawning time in the Norwegian fjords a boat may be so lensely
packed in among the mass of fish that an oar stuck upright into the
swarm remains standing for an appreciable time after the hand relinquishes
its hold.5
Examination of Stomach. Contents. The second method by which
information has been gathered regarding the interdependence of biological
species consists, as already stated, in examining the contents of
trie stomachs of sample specimens. This method has been applied
particularly to birds and fishes. The results of such an analysis of
the feeding habits of the common crow and of the starling are strikingly
brought to view in the accompanying charts figures 31 and 32,
reproduced by courtesy of the Department of Agriculture from Bulletins
868 and 1102. Such a chart does not, of course, yield any direct
information regarding the relative abundance of the several species
upon which the crow feeds, but it does give us at least an indication of
a resultant compounded of that relative abundance and a number of
other factors, such as the preference or selective tastes of the crow,
the greater or less degree of protective characters with which nature
has endowed the various species exposed to attack, etc.
The converse problem, also, has been investigated, namely the
extent to which different species of birds participate in the destruction
of one selected noxious insect. So, for example, Bulletin 107 (1914)
of the Department of Agriculture lists 45 different species of birds
that were found to have fed upon the alfalfa weevil. A similar study
by H. C. Bryant
8
was carried out during a grasshopper outbreak in
B
For an account of bird censuses in the United States see M. T. Cooke,
Bulletin No. 1165, Bureau of Biological Survey, U. S. Department of
Agriculture.
6
University of California Publications in Zoology, 1912, no. 1, vol. 11.
INTER-SPECIES EQUILIBRIUM 167
California, when "in the infected areas the grasshoppers were computed
to number from 20 to 30 per square yard." Bryant's results
are shown, in part, in tabular and diagrammatic form in figures
33 and 343 reproduced from his original publication.
. 31. SEASONAL ANALYSIS OF THE STOMACH CONTENTS OF CROW
^
*'
;
;
'^jfiftitAr.^'i
"~\^^/&8SS8
V^&3PI
FIG. 3:2. SEASONAL ANALYSIS OF THE STOMACH CONTENTS OF STARLING
The same author has also given us a classic in his extended study
"A Determination of the Economic Status of the Western Meadow
Lark in California." This paper contains among other things a
1(58 .ELEMENTS OF PHYSICAL BIOLOGY
detailed bibliography up to 1913, and a historical
survey of the
methods employed and the investigators who have labored in the
TOTAL DESTRUCTION OF GRASSHOPPERS PEP SQUARE MILE DAILY, IS0.453
FlG. 33. COMPARATIVE DAILY DESTRUCTION OF GRASSHOPPERS BY SEVERAL
SPECIES oir BIRDS
After H. C Bryant
ANIMAL FOOD VEGETABLE FOOD
Jan.
Feb.
Mar.
Apr.
May
June
July
Aug.
Sept.
Oct.
Nov.
Dec.
TOTAL
VM/M/////////A,
Fia. 34. SEASONAL FOOD HABITS OP THE MEADOW LARK
After H. C. Bryant
field. The results obtained by Bryant exemplify the proverbial
voracity of birds. Young birds require about one-half their own
weight in food each day. In the course of a year the average meadow
IXTEK-SPECIES EQUILIBHIUM 169
lark, weighing say, 4 ounces, consumes about the following quantities
of food.
pounds
% ft
FIG. 35. STOMACH CONTENTS OP A BREWER'S BLACKBIRD
Since an adult bird weighs about 4 ounces, this means that it consumes
on an average about 24 times its weight of food in a year.
Dr. Bryant remarks: "If we consider that there is an average of one
meadow lark to eveiy two acres of land available for cultivation
(11 million acres) in the Sacramento and San Joaquin Valleys, and
that each pair of birds raises an average of four young, it takes over
343| tons of insects each day to feed the young birds in the valley
alone." His findings regarding the seasonal changes in the food of
the meadow lark are summarized in a number of charts of which one
is reproduced in figure 38. An illustration of the voracity of birds,
and their destructiveness to insects is also seen in figure 35 (from Bulle-
170 ELEMENTS OP PHYSICAL BIOLOGY
tin 107 of the Department of Agriculture) showing the stomach contents
of a Brewer's Blackbird, for this bird was found to have gorged
upon 374 larvae, 65 pupae and 3 adults of the alfalfa weevil. Observations
on the feeding habits of young English sparrows are recorded
by E. R. Kalmbach in Bulletin 107 of the United States Department
of Agriculture. This author remarks (p. 54) :
From a series of five observations it appears that the parent English sparrows
visited their nest on an average about once every 5 minuted, or a little
more than 11 trips an hour. The four adults captured had as food for tlieir
young 2 kernels of wheat; 17 alfalfa weevil larvae; 1 ground beetle, 9 weevil
larvae and a caterpillar; and 23 weevil larvae, respectively. Three other
adults taken in the fields had food for nestlings in their bills. This amounted
to IS weevil larvae and an aphid in the first, 5 larvae in the second, and 3
coccinellid larvae. 13 weevil larvae, and 2 pupae in the third.
Though this is rather heterogeneous assortment, it would appear that 15
larvae of the weevil or their equivalent in bulk of other insects would be a fair
estimate of an average amount of food brought in at each trip by adult birds.
In fact, it is certain that the material brought in frequently greatly exceeded
this amount.
Allowing 15 larvae at each trip and 11 trips per hour, these birds would
bring in 165 larvae per hour. Then, assuming that the young were being fed
for 12 hours each day, a conservative estimate, we would have a total of 1980
larvae consumed by one brood in one day. Straw-thatched sheds containing
upward of 100 nest holes, both old and new, are frequent, and it is not uncommon
to find farmyards where this number of nests are occupied. There are
also ample nesting sites about the other buildings and in the ever-present
Lombardy poplar, cottonwood, or box elder. Such a colony of birds would
devour a daily total of 198,000 larvae, or an equivalent bulk in other food. As
the young birds remain in the nest for at least 10 days and are probably fed
several days longer by the adults, they will have eaten food equivalent to the
bulk of 1,980,000 larvae during their nestling life.
Intra-Species Equilibrium. As has been remarked on a previous
occasion, it is not intended, in this volume, to take up the discussion
of the evolutionary changes within the confines of a species. Passing
notice may, however, be given to the fact that the equilibrium within
a Mendelian population has been discussed by G. H. Hardy7
and by
R. C. Punnett8
and latterly by J. B. S. Haldane.9
(See p. 122.)
7
G. H. Hardy, Science, 1908, vol. 28, p. 49.
8
R. C. Punnett, Mimicry in Butterflies.
8
J. B. S. Haldane, Cambridge Philosophical Society, 1923.
CHAPTER XIV
INTER-SPECIES EQUILIBRIUM AQUATIC LIFE
Aquiculture is as susceptible to scientific treatment as agriculture; and the
fisherman, who has been in the past too much the hunter, if not the devastating
raider, must become in future the settled farmer of the sea, if his
harvest is to be less precarious. W. A. Herdman.
From what has already been set forth the direct economic importance
of studies in general demology should be sufficiently clear,
if this term be used to denote the quantitative study of the population
of the several species of organisms living together in mutual
interdependence through their food requirements, feeding habits,
and in other ways.
In no other field, perhaps, has the study, from this angle, and
under this economic impetus, been so systematically undertaken,
as in the biology of aquatic species. On the one hand the threedimensional
extension of the systems here involved (as distinguished
from the essentially two-dimensional spread of land species over the
earth's surface'), facilitates, in certain respects, the operation of
sampling (by the use of the dragnet) and counting; on the other,
the close relation of such investigations to the practical problems
of our inland and our ocean fisheries has furnished alike the economic
occasion and the financial support for work on an extended scale.
The methods employed have by this time developed into a more or
less standardized technique. The dragnet, already referred to,
and the stomach and gills of fish, acting, as it were, as natural dragnets,
themselves caught within the collector's man-made dragnet,
are among the principal accessories in this 'field of investigation.
L. H. Tiffany recommends particularly the gizzard shad as a convenient
collector and sampler of Algae.
1
Contrasting these natural
samplers with man-made contrivances he remarks:
These living tow nets (i.e., gizzard shad) do not get caught on snags and
roots, the string does not break, and the algal collection is very representative
of the body of water from which the fish were taken. It is only neces-
1
Science, 1922, vol. 56, p. 285.
171
172 ELEMENTS OF PHYSICAL BIOLOGY
sary to catch the young fish and examine their stomachic and intestinal
content to secure a proportionate sample of the plankton.
Tiffany examined specimens from streams and ponds in Illinois,
and also from Ohio. He points out that the gizzard shad fulfills
an important rule as an intermediary link in the food chain: algae,
shad, game-fishes, man.
Thus, the gizzard shad is making useful for man the energy stored in plant
forms which occupy no land areas, which do not interfere with the ordinary
disposition or utilization of bodies of water ('except the occasional contamination
of water for drinking purposes by some algae), which involve no
labor of cultivation on the part of man, and which are of no value for direct
humaa consumption.
The world's population in the last hundred years has increased about
150 per cent. Along with this increase has had., to corne a corresponding
increase in the world's food supply. One of the ways in which this necessityhas
been met is the securing of new acres of soil in which to grow crops. It
is easily seen, however, that there is a limit to new acreage. In the future,
therefore, we may have to turn more of our attention to the cultivation of
the waters for food supplies. We may have to develop an industry of aquiculture
as we have developed an industry of agriculture. The time is
rapidly approaching when fish will be more highly prized as food and more
extensively used than now. As that time comes, the cultivation of algae
will be a first step toward greater fish production. A second step may be
the introduction of fish like the gizzard shad into fish ponds and lakes to
make more readily available the phytoplankton for fish food.
In a summary survey of the Food Resources of the Sea,
2
G. W.
Martin makes similar observations. He remarks: "So far as the
actual cultivation of the sea's resources, as distinguished from their
mere exploitation, is concerned, we have made only the feeblest
beginnings." Somewhat in the same vein is W. A. Herdman's
comment: "Aquiculture is as susceptible to scientific treatment
as agriculture; and the fisherman who has been in the past too
much the hunter, if not the devastating raider, must become in the
future the settled farmer of the sea, if the harvest is to be less precarious."
Viewing the matter from a slightly different angle W. F.
Wells of the New York Conservation Commission draws attention
to the fact that in our modern great cities, with the widespread
adoption of the water system of sewage disposal, valuable fertilizer
material is lost from its natural place in the fields. By enriching
2
Scientific Monthly, 1922, p. 456.
INTER-SPECIES EQUILIBRIUM 173
the vegetation in the water and furnishing abundant life thereto,
it may again be restored to the people as fish and shell fish in the
place
^of
beef and mutton. And the exchange is not such a bad
bargain. For it has been found that in carp ponds, for example,
the production of market ware was 95 pounds per acre, as contrasted
with 73 pounds of beef per acre of farmland.3
Perhaps one of the
most telling illustrations of the economic importance of pisciculture
is presented to us in the Alaska purchase: Within fifty years after
the acquisition of our Northern province, it had yielded, in its salmon
fisheries alone, seven and a half times its purchase price.
The quantitative study of ocean life (on which must be based an
intelligent system of marine aquiculture) may be said to elate from
1880, when Henson introduced the use of the dragnet. To this
has since been added the use of bottom samplers or grabs; also an
ingenious, if less
trustworthy method of making a census of the
marine population, which consists in catching a number of live fish,
marking them, throwing them back in the water, and then noting
the^ percentage of marked fish in the fishermen's catch during the
period that follows. In this way it has been estimated, for example,
that the North Sea contains about fifteen hundred million plaice,
a figure about equal to the earth's human population.
The field of utility of the dragnet and most of the other methods
described is evidently limited to the larger denizens, such as are
effectively held in the meshes of a net. Even a fine silk net will
fail to hold
^a
very numerous constituent of the population of the
sea, a constituent that is highly important not only on account of
its great extent, but also because of the role it plays in the food
traffic of marine life. Lohmann showed (1911) how these fine
organisms (the nannoplankton') can be collected by the use of a
centrifuge.
4
Allen has developed a special dilution culture method
of count for the nannoplankton organisms, which is modelled after
the pattern of bacterial count technique. He showed sea water
to contain 464,000 organisms per liter (exclusive of bacteria). Al-
*
It is true that beef is much superior in food value, pound for pound, but
it is also much more costly to produce. It should also be noted that the yield
from carp ponds is very high as compared with that of marine fisheries E
J. Allen (Food from the Sea, 1917, quoted by W. F. Thompson, Scientific
Monthly, 1922, p. 546) estimates the yield in the North Sea at 15 pounds per
acre.
4
G. W. Martin, loc. cit., p. 461.
174 ELEMENTS OF PHYSICAL BIOLOGY
lowing for systematic errors Allen thinks a population of one million
organisms per liter (one per cubic millimeter) to be a conservative
estimate. This is not excessive crowding, in view of the size of
these organisms, as illustrated by the diagram, figure 36, reproduced
from G. W. Martin's article. Before the extent and significance
of this nannoplankton was realized, the amount of food required
by the animals of the sea seemed so much in excess of the amounts
\
FIG. 36. DENSITY OF MICROORGANISMS (EXCLUSIVE OF BACTERIA) IN SEA
WATER
About one organism, measuring 6 microns in diameter, is found per cubic
millimeter of water. The relative dimensions are here shown. After G. W.
Martin.
revealed by the earlier methods of collection, as to give rise to the
suggestion (Putter 1907-1909)
that the nutrition of marine animals was on an entirely different plane from
that of land animals, and that a large number of them, especially the smaller
ones, absorbed dissolved organic matter directly from the water, without
the mediation of plants. Putter's arguments have not been generally accepted,
and more recent studies have invalidated many of them. Nevertheless,
it is possible that something of this sort is more general than we
realize. Mitchell (1917) reported an experiment strongly indicating that an
oyster can utilize dextrose dissolved in sea water.
The most elaborate attempts to calculate the production of the sea have
been those of the Danish biologist Petersen and his associates. As a result
of their studies, these workers have come to the conclusion that the plankton
plays a very small part in the nutrition of the animals of the sea, and
INTEK-SPECIES EQUILIBRIUM 175
176 ELEMENTS OF PHYSICAL BIOLOGY
that the fundamental food of all marine forms in northern waters at any
rate is the "dust fine detritus" of the sea bottom, derived primarily from
the eel grass, Zostera.
These investigators have studied in particular the conditions in
the Kattegat, a rather shallow arm of the sea between Denmark
and Sweden, about 150 miles in extreme breadth and 90 miles in
extreme width. Their principal conclusions are exhibited in the
diagram figure 37, and are as follows:
It is assumed that about half of the total amount of Zostera annually
produced in this area is washed elsewhere by the currents. The balance,
estimated at 24,000,000 tons, serves as the basis for the animal life of the
area. The useless animals, that is, those that are of no value to man and
do not serve as food for fish, feeding directly on the Zostera, amount to about
5,000,000 tons. Useful animals, mainly those capable of serving as food
for fish, are estimated at 1,000,000 tons. These are not all utilized by food
fish, however. Starfish account for perhaps 200,000 tons; 500,000 tons are
eaten by the larger gastropods and crustaceans, of which only a part are
consumed by fish; while plaice and other flatfish consume about 50,000 tons
producing 5000 tons of human food annually. Cod are much less economical,
since they get their food at third hand, so to say, and each ton of the 6000
tons produced annually represents about one hundred times as much of the
original synthesized organic food. On the other hand, the cod help to keep
down the predatory gastropods and crustaceans (see figs. 38 and 39). The
herring is the most important food fish feeding on the plankton, (mainly on
copepods) and it in turn is eaten by the cod. Perhaps the most striking
feature brought out by these figures is the comparatively trifling amount
of human food finally produced from such a large amount of organic material.
Summary of Methods. A tabular summary of some of the principal
methods by which data have been secured regarding relative
and absolute frequency of organisms and species is given in table 10.
Food Chains. It has already been remarked, in dealing with the
general kinetics of the type of systems here under consideration,
that each component of the system appears as a link in a chain or
a network of chains, receiving contributions from components
(sources') above, and discharging material into other components
(sinks') below. Food chains, such as spoken of by Tiffany in the
passage quoted on page 172, are a particular example of such chains
of components. The study of food chains is one of the important
tasks of the economic biologist. For we cannot afford to restrict
our attention to the immediate source from which we draw our
INTER-SPECIES EQUILIBRIUM 177
By courtesy of the Illustrated London News
FIG. 38. THE VABIED DIET OF THE CODFISH
178 ELEMENTS OF PHYSICAL BIOLOGY
-a .f
CO
a
o
ft
he O
o> ri
o
I---
S.2
!-
ft
02 .5
INTER-SPECIES EQUILIBRIUM 179
supplies of food for the human population. The sources of these
sources also demand attention. The problem forces itself upon
our notice primarily (in the present state of society) in connection
with agriculture. The fields cannot continue indefinitely 'to yield
undiminished annual crops if the materials drawn from them are
not in some way replenished. One important constituent needs
no human intervention: carbon dioxide, owing to its gaseous form,
automatically seeps in by diffusion as fast as it is absorbed by the
FIG. 30. KEY TO FIGUKK 3
green plants. The same is true in some degree of free nitrogen ,
though the capacity of plants to assimilate this element is
^
strictly
limited.
5
Water, also, is, in most agricultural areas, provided by
the automatic meteorological processes of evaporation and condensation
in rainfall. But as to certain other essential materials,
notably combined nitrogen, phosphorus, potash and sulphur, the
inherently immoUW constituents of the fertile soil, for these auto-
*
Science. November 24, 1922, p. 605.
"Fiir sicli micht beweglich," Liebig, Die Chemie in ihrer Anwendung
auf Agricultur, 1876, p. 382.
180 ELEMENTS OF PHYSICAL BIOLOGY
matio replacement does not occur in sufficient measure
^
to satisfy
the agricultural needs of the densely populated countries of this
age. It becomes necessary for man to feed his food. Early man
and primitive man, may reap where he has not sown. BuMong
ago our tribe turned from the life of a nomad and hunter to tilling
the soil and to animal husbandry. Thus was established a system
of symbiosis with the links next above us in the food chain : the
harvest ripening on the plain; and the cattle, grazing upon the
pasture in summer, fed from the crib in winter. But early agriculture
was essentially of the nature of a mining industry. It drew
from the soil as from a bottomless well, without thought of a possible
exhaustion of the source, or of any feasible replenishment to diminishing
resources. Except that, by a semi-automatic, semi-empirical
process, natural fertilizer was allowed to restore to the soil at least
a part of its strength to bring forth a crop. One more step forward
and man graduated from mining farmer into manufacturing farmer.
The field became a factory fed with raw materials in the form of
saltpeter, potash salts and phosphate fertilizer, imported, if need
be, from afar; and producing its output of agricultural flora arid
fauna. Lastly, in our own generation, we have learned to divert
into the life stream the sluggish element, so essential to life, so illnamed
by the French -azote; to make ourselves independent of the
saltpeter beds, to assure our future against a nitre famine by opening
the inexhaustible mine of the atmosphere. It is a singular thing
that this element, so accessible, so abundant, in which we arc
literally bathed within and without, every instant of our life, should
so long have remained foreign to our industrial economy. Strange
circumstances, yet not without close parallel. For even now we
are powerless to avail ourselves effectively of the golden flood of
energy that daily pours upon us without limit from above while
we turn earthward to dig laboriously for the plainly exhaustible
supply of coal to supplement our limited bodily energies.
Food Chains in Aquatic Species. The principle that long food
chains are essentially wasteful finds particular application also in
the practical problem of the economic and rational utilization of
marine organisms for human sustenance. As Professor Martin
observes, the most economical course would be to utilize marine
vegetation directly as food for man and his domestic animals. The
INTER-SPECIES EQUILIBRIUM 181
use of algae as food for the table can hardly be expected to become
an item of any consequence. Its use as cattle fodder presents
better prospects. But our chief reliance will no doubt continue
to be on the assimilation of marine plants by fish. To quote again
Professor Martin:7
Since man prefers to harvest the plant life of the sea indirectly, those
animals which feed directly on the plants are able to increase with less waste
and at a more rapid rate, considered in total populations, than those which
feed on other animals. Most of our food fish, for example, feed on smaller
fish; these in turn feed upon small crustaceans and the latter eat the microscopic
plants and detritus, so that in many instances the fish we eat are removed
three or four steps, perhaps more, from the original food source. This
is more significant than may seem apparent at first glance, since it involves
an enormous waste. Before any organism can grow, the energy needed
merely to live must be supplied, and by the time a crustacean is eaten by
a minnow, or a minnow by a food fish, it will, on the average, have consumed
a quantity of food several times its own weight. These facts are well brought
out in the statistics of Petersen, and the diagram figure 37 based thereon.
The edible shellfish, however oysters, clams, mussels and the like feed
for the most part directly on the marine plants and this is one reason why
the extension of the shell fisheries represents so much promise.
Another advantage of this branch of aquiculture is also to be noted,
namely, that
most shellfish, like land crops, stay where they are planted. Even the scallop,
which can swim about after a fashion, is restricted in its movement,
and could readily be controlled. Oyster culture is already a great and important
industry, but it has not nearly approached its possibilities. Clam
culture is still in an embryonic state, and scallop culture has yet merely
been suggested. When some of the problems confronting the establishment
of these industries have been solved, we may hope to have acquired additional
information concerning the ecology of the sea, which will help us in
our approach to the more difficult problems of the future.
Primary, Secondary and Tertiary Foods. It is often convenient
to classify the foods for human consumption according' to their
relative position in that portion of the food chain which is under
human control. Commonly the meats of domestic animals are
7
For some further bibliographic indications regarding the food consumed
by fishes see A. S. Pearce, Ecology, vol. 1924, p, 258. See also W. A, Herdman,
Founders of Oceanography, 1923; J. Johnstone, Introduction to Oceanography,
1923.
182 ELEMENTS OF PHYSICAL BIOLOGY
classed as secondary foods, since the production of these meats
takes plate in two steps, first the growing of the fodder (for the
most part materials not comestible by man), and second the transformation
of a part of this fodder, in the animal economy, into food
adapted for human consumption. On the other hand the fisherman's
haul is for us primary food, growing feral, without human
intervention.
This classification must not be pressed. Where fields are supplied
with fertilizer it might well be maintained that the cr^ps of wheat,
potatoes, etc., commonly classed as primary, are secondary foods,
while butchers' meats are tertiary. Even the catch of fish, and
huntsman's quarry may not be strictly primary, in so far as game
laws and regulations regarding the pollution of lakes and rivers
represent some degree of symbiotic intervention on the part of man.
Fine points apart, the distinction between the primary foods (crops
and fish) and the secondary foods (butchers' meats) is economically
most significant, for the consumption of fodder by farm animals
is an item not merely comparable with human food consumption,
but exceeding this latter manyfold. The fact, of course is, that
farm animals are far from being economical and efficient converters
of raw materials into food for human consumption. They represent
a luxury, a humoring of the tastes of men at the expense of
their purses. With the present density of population we can afford
the luxury. Presumably the future will see retrenchments, with
pastures and cornfields converted to wheat. Still tastes arc not
accidental thiogs. Allowing for vagaries and exceptions, the
things we like are, on the whole, good for us and for the species.
Whether man can maintain his present status with a materially
abridged meat ration is perhaps an open question. Should the
answer be in the negative, the conclusion would seem to be forced
upon us that an overcrowding of the earth would react unfavorably
upon the vigor of the race; quality would be sacrificed to quantity.
How this might affect the ultimate fate of our species is a subject
for speculation. The pessimist might take a cue from palaeontology,
recalling that the extinction of a species seems to follow, not infrequently,
close upon its period of greatest development. The
optimist, on the other hand, might perhaps extend the suggestion
that when overcrowding does come, the ones to survive most surely,
if not most abundantly, will be those whose superior qualities
INTER-SPECIES EQUILIBRIUM 183
will enable them, in spite of intensified competition, to draw to
themselves a sufficiency of the more desirable, though perhaps not
absolutely essential articles of consumption. Natural selection
would thus operate by the preferential survival of an aristocracy,
while a submerged tenth would furnish a drain for the discharge of
the unfit. Certainly, in the interests of the species, it were better
that the inferior constituents be purged from the system than that
they should drag down the general level to mediocrity and perhaps
below the line of viability. But these are speculative reflections.
Cycles. Food chains, were we able to trace them through their
entire course, would undoubtedly be found to form closed cycles
or a network of cycles. This is indeed a practical necessity for the
continued performance of the processes or organic nature, processes
that have gone on essentially unchanged in their general character,
however modified in detail, for many millions of years.
A few of the simpler food chains we may be able to follow with
something approaching completeness through their cycle. For
the most part, however, the system of interlocking cycles in nature
is complex beyond all reasonable hope of detailed analysis in its
entirety.
If we are satisfied to omit innumerable details, we can trace, for
each of the most important chemical elements7
concerned, the broad
outline of its cycle in nature. The elements and simple compounds
principally concerned are
Carbon (dioxide) COz
Oxygen O2
Nitrogen free N, NH3 ,
nitrites and nitrates
Water H2 O
Phosphorus (phosphates, etc. )
Brief consideration will presently be given to each of these cycles
in turn. First, however, it will be well to review some of the essential
facts regarding the occurrence of the chemical elements,
generally, in nature. For the drama of life is like a puppet show in
which stage, scenery, actors and all are made of the same stuff. The
players, indeed, "have their exits and their entrances," but the
exit is by way of translation into the substance of the stage; and
each entrance is a transformation scene. So stage and players are
T}ound together in the close partnership of an intimate comedy; and
184 ELEMENTS OF PHYSICAL BIOLOGY
if we would catch the spirit of the piece, our attention must not all
be absorbed in the characters alone, but must be extended also to
the scene, of which they are born, on which they play their part,
and with which, in a little while, they merge again.
8
8
Since the words above were written I have run across the following singularly
apposite passage in- John Morley's Introduction to Wordsworth Collected
Poetic Works: "Wordsworth's claim, his special gift, his lasting contribution,
lies in the extraordinary stremxousness, sincerity and insight with
which he first idealizes and glorifies the vast universe around us, and then
makes of it, not a theatre on which men play their parts, but an animate
presence, intermingling with our works, pouring its companionable spirit
about us, and 'breathing grandeur upon the very humblest face of human
life,'"
CHAPTER XV
THE STAGE OF THE LIFE DRAMA
When the elements have been mingled in the fashion of a man, and come
to the light of day, or in the fashion of the race of wild beasts or plants or
birds, then men say that these come into being', and when they are separated,
they call that in common parlance, death .... let not the error prevail
over the mind that there is any other source of all the perishable creatures
that appear in countless numbers. Empedocles.
Our stage is a tripartite world: The heavens above, the waters
of the sea, and the solid ground beneath our feet; the atmosphere,
TABLE 11
Principal components of earth's surface
the hydrosphere and the lithosphere. The total mass of the earth is
about 6.5 X 1021
tons. But it is only the outer crust that interests
us here, for the deeper layers have little or no part in terrestrial life.
If we arbitrarily take a layer ten miles thick for the crust, the distribution
of the material among the three main divisions is, according to
F. W. Clarke, about as shown in table 11.
The Atmosphere. There are two ways of confining a gas. The one
most familiar in the laboratory and in products of human workmanship
generally, is to enclose the gas in a suitable envelope, such as a
glass vessel, or the cylinder of an engine; to put something around
the body of gas to be confined. The other way, nature's way on a
large scale, is just the opposite, and consists in putting something
into the gas, or putting the gas around something. It is so the earth
holds her atmosphere by gravitational attraction. Her hold is not
impartial. She draws closest to her the densest constituents, and
185
186 ELEMENTS OF PHYSICAL BIOLOGY
gives longer leash to the lighter. The atmosphere, in consequence,
is not a homogeneous body, but varies in composition with altitude.
At the same time, owing to its elasticity, the air in its lower strata
is compressed by the weight of the overlying atmosphere, so that 99
per cent of the whole is contained within a shell 30 km. (18| miles)
thick. The remaining 1 per cent extends out into space without any
Of CLOUDS * QROlMAW OUST J
FIG. 40. CROSS SECTION OF THB ATMOSPHERE, SHOWING SOME FEATURES
CHARACTERISTIC OF DIFFERENT ALTITUDES
After W. J. Humphreys
assignable limit, but at any rate to a height of some 300 km. (185
miles), as evidenced by the aurora. A graphic representation of the
broad divisions of the atmosphere is shown in figure 40, adapted from
Wegener and Humphreys.
1
A more detailed and exact statement of
1
Physikalische Zeitschrift, 1911, vol. 12, p. 172. See also W. J. Humphreys'
work, The Physics of the Air (Lippincott, 1920), pp. 68, 69. Also, the
same author, Bulletin Mt. Wilson Weather Observatory, vol. 2, 1909, p. 67;
STAGE OF THE LIFE DRAMA 187
OF CONSTANT TffMPfMTVfie.
FIG. 41. COMPOSITION OF THE ATMOSPHERE AT DIFFERENT LEVELS
After W. J. Humphreys
Journ. Franklin Institute, vol. 175, 1913, p. 208, 212. A singular view has
lately been put forward by L. Ve"gard. He identified certain green lines in
the auroral spectrum with lines observed when solid nitrogen is rendered
phosphorescent by x-rays. He concludes that the upper atmosphere contains
solid nitrogen. (See Nature 1924 vol.113 p. 716.) Ve"gard's view has been opposed
by J. C. McLennan, Roy. Soc., June 19, 1924.
188 ELEMENTS OF PHYSICAL BIOLOGY
its composition at different altitudes, up to 140 kin., is given in
table 12 and illustrated graphically in figure 41, derived from
Humphrey's work.
A complete discussion of the r61e played by the atmosphere in the
round of terrestrial life would amount to nothing less than a treatise on
meteorology, such as forms no part of the present project. What
TABLE 12
Percentage distribution of gases in the almosphere
important factors weather and climate are in the business of providing
the sustenance of life is a matter of common knowledge, and a
very particular concern of the farmer. Yet we must here be satisfied
with little more than a passing reference to meteorology, noting
only a few elementary facts which bear directly upon the subject in
hand, the circulation of the chemical elements in nature.
Losses from the Atmosphere. A first question that suggests itself
in this discussion of the economy of nature is this: Since the atmosphere
is "open at the top," so to speak, is there not a loss, a constant
leakage of gas out into space? The answer to this question must be
STAGE OF THE LIFE DRAMA 189
sought in terms of the molecular constitution of the gases of our
atmosphere. A cubic centimeter of air contains (at 0C.) about 3.15
X 1019
molecules. These are in continuous agitation, somewhat
after the manner of a swarm of gnats, except that they flit about with
speeds comparable with that of a rifle bullet (about 500 meters per
second) rather than with the leisurely flight of an insect. At a temperature
of 0C. a molecule of nitrogen has, on an average, a velocity
of 492 meters per second. A molecule of hydrogen, under the same
conditions, would have an average velocity of 1839 meters per second.
It must be understood that these figures represent, in each case, a
mean about which the velocities of individual molecules cluster, so
that a certain proportion of them will fall below and others will exceed
the figures stated. At the earth's surface the average distance
travelled between two successive collisions is about ^iroTolTo cmBut
in the upper ranges of the atmosphere conditions are very
different. If we follow the estimates and computations of J. H. Jeans,
we find that at an altitude of 3200 km. the atmospheric pressure is reduced
to about 1/10
14
of its value at sea level; but even at this low
pressure there are still about 300,000 hydrogen molecules per cubic
centimetre. (At this altitude all other gases except hydrogen are
practically absent) . The mean free path between collisions is now
10,000 km. or about If times the earth's radius. In such circumstances
collisions between molecules are rare, and for the most part the
molecules move freely through space in parabolic or elliptical orbits,
and become virtually diminutive satellites of the earth. Since the
day of Jules Verne's story From the Earth to the Moon it has been a
matter of popular knowledge that a body projected from the earth
with a velocity exceeding 7 miles per second will go off in a hyperbolic
orbit, never to return. This applies to the molecules of a gas.
Any of them that may be travelling outward with such a speed in
the region where collisions are so rare as to be negligible, will leave
the earth for good and will thus be lost to our atmosphere. The
rate of leakage from the atmosphere thus depends on the number of
molecules per unit of time that acquire the limiting (outward) velocity
of 7 miles per second. This number, in turn, depends on the
temperature in the region under consideration, a point regarding
which our information is very uncertain. But an exact knowledge
of this temperature is not needed to compute a major limit, a maximum
figure which the rate of escape certainly cannot exceed. It
190 ELEMENTS OF PHYSICAL BIOLOGY
is thus found that under present conditions2
the earth holds her
atmosphere so effectively that there cannot be any appreciable
leak even in many millions of years.
Cosmic losses from the atmosphere then, are, for nil practical purposes;
wholly negligible. Certain other subtractions from and accessions
to the atmosphere we shall have occasion to note as we consider
the circulation of the several elements. As a matter of fact the composition
of the atmosphere in the region in which living organism
have their habitation is very nearly uniform3
and constant, except
TABLE 13
Constituents of the, atmosphere
Earth's surface = 1,97 X 10 8
square iniloH ~ 5.5 X 1018
square foot.
One square mile = 2.79 X 10 7
square fcofc.
One metric ton = 2205 pounds.
s
It may be noted in passing that in the earth's pant history conditions
may have been different. If at any time the temperature of the upper atmosphere
was about 750C., then there must have been a very distinct loss
of hydrogen by leakage into space. The moon, and certain of the planets
having a lesser gravitational pull or a higher temperature (Mercury), have
probably lost in this way any atmosphere that they may have had. For a
detailed discussion of this and other points in connection with the escape
of gases from the atmosphere the reader may be referred to J. H. Jeans's
Dynamical Theory of Gases, 1921, Chapter XV. See also K A. Milne, Trans.
Cambr. Phil. Soc., vol. 22, 1923, p. 483; J. E. Jones, ibid., p. 535.
8
Except in the neighborhood of volcanoes, and in lesser degree, in or near
large cities or manufacturing centers, where large amounts of waste gases
may be discharged into the atmosphere.
STAGE OF THE LIFE DRAMA 191
192 ELEMENTS OF PHYSICAL BIOLOGY
as regards its moisture content, and wo may accept for the composition
of the atmosphere at the earth's surface, moisture excluded, the
figures shown in the first two columns of table 13. The third column
shows the total amounts of the several constituents of the entire
atmosphere, according to W. J. Humphreys (Monthly Weather
Review, vol. 49, June, 1921, p. 341).
Cosmic Accessions to the Atmosphere. Meteorites falling upon
the earth from space bring with them certain o entities of entangled
or occluded gases. While this contribution tt the atmosphere is at
the present time, presumably, of negligible dimrnsions (see also page
195, Cosmic Accessions to the Lilhospfwrc), yet in the course of the
TABLE 15
Comparison of air and aqiutlic atmosphere
*
Measured at 0G. and 760 nun. Ug.
f42.7ingm.; according to F. W. Clarke, Data of CicocluMnistry, Geological
Survey Bulletin. 491, 1920, p. 142; G. Linck, KroiHlaufvorgiitige in dor ErdgescMchte,
Jena, 1912, p. 6.
long procession of ages past this source may not have been wholly
insignificant. Data on this question, are at best very uncertain,
and a mere passing reference must suffice.
4
The Hydrosphere. In comparison with the ocean alt other aggregations
of water upon the earth are insignificant in aimmnt. The
bald statement of the total volume of the ocean 302 million cubic
miles 'conveys but little to the mind. More impressive it is to
recall that the average depth of the sea is 2$ miles, and that, even if
these waters were spread over the whole earth, leaving no continents,
the average depths would still be If miles.
The average composition of the ocean is shown in table 14. Here
again, the figure for the volume of total dissolved solids, 4.8 million
cubic miles, is made more readily comprehensible by a graphic illustration.
The salts of the ocean, made into one solid block, would
*
F. W. Clarke, Data of Geochemistry, U, S. Geological Survey Bulletin
695, 1920, pp. 58, 269, 282.
STAGE OF THE LIFE DRAMA 193
cover the entire United States and Alaska to a depth, of Ixfr miles;
or according to J. Joly, they would encrust the whole earth to a depth
of 112 feet.
5
The Aquatic Atmosphere. Aquatic species perform their respiration
in contact with an atmosphere of gases held in solution in the
water that surrounds them. This atmosphere is very different both
in concentration and also in composition from, that in which we live,
as is apparent from table 15.
A comparison of the several columns in table 15 is an object lesson
on the adaptability of living organisms to varied conditions.
The atmosphere in which fish and other marine animals live in comfort
would not only drown us with its principal constituent, water,
but, even if this were removed, the residual gases would suffocate us
for lack of oxygen; and if the deficiency in this gas were made up by
the addition of the amount required to bring the percentage up to
that to which we are accustomed, we would still be choked by the
excessively high percentage of carbon dioxide.6
The Lithosphere. Immense as the ocean appears to us, with its
average depth of 2| miles, yet it constitutes less than vsV<r of the
total mass of the earth, whose bulk is thus concentrated chiefly in
lithosphere. Of the deeper layers of this lithosphere we have but
scant and indirect knowledge. Earthquakes give evidence of some
change in constitution about half way down to the center of the globe.
Conditions and occurrences at such depth as this would seem, in the
present state of our knowledge, to have little bearing upon the life
at the surface. Other indirect evidence regarding the earth's interior
is derived as follows : The volume of the globe being known from
triangulation, and its weight from direct determination with a bal-
ance7
or in other ways, the mean density of the earth is found to be
6
F. W. Clarke, ioc. cit., p. 24; Sci. Trans. Roy. Soc. Dublin, 1899, vol. 7,
p. 30.
6
It must be admitted, however, that much of this COg in sea water is partially
neutralized by alkali.
7
A concise survey of the principal determinations of the mass of the earth
will be found in J. H. Poynting's little book (in the Cambridge Manuals series)
The Earth (1913) It may add interest to the bald figures to note hero in
passing that the density of the earth's crust (2.7) is not very widely different
from that of the moon (3.46). This fact has a certain significance in connection
with Sir Charles Darwin's theory of the origin of the moon, according to
which our satellite originally formed part of the earth and was thrown off
by a species of tidal disruption. The bulk of the moon, then, would be formed
of material derived from the outer layers of the earth.
194 ELEMENTS OF PHYSICAL BIOLOGY
TABLE 16
Average composition of terrestrial matter*
LITHO8PHERB
(93 PERCENT)
HYDBOSPHBBB
(7 PEE CENT)
AVEBAGK
INCLUDING
ATMOSPHERE
Oxygen 49. 19
Silicon 25 . 71
Aluminum 7 . 50
Iron 4.68
Calcium 3.37
Sodium 2.61
Potassium 2 . 38
Magnesium 1 .94
Hydrogen ."': 0.872
Titanium 0.648
Chlorine .228
Bromine
Phosphorus . 142
Carbon . 139
Manganese .108
Sulphur .093
Barium .075
Chromium .062
Zirconium .048
Vanadium .038
Strontium .032
Fluorine .030
Nickel 0.030
Nitrogen 0.030
Cerium, yttrium .019
Copper .010
Lithium 0.005
Zinc 0.004
Cobalt 0.003
Lead 0.002
Boron .001
Glucinum .001
100.000
85.79
0.05
-1-14
0.4
0.14
10.67
2.07
0.008
0.002
0.09
100.000
46.68
27.60
8.05
03
63
72
2.56
2.07
0.145
0.696
0.095
0.152
0.149
0.116
0.100
0.079
0.066
0.052
0.041
0.034
0.030
0.031
0.020
0.010
0.005
0.004
0.003
0.002
0.001
0.001
100.172
*
Adapted from F. W. Clarke, Data of Geochemistry, 1921, p. 35 ;
Clarke and
Washington, Proc. Natl. Acad. ScL, 1922, vol. 8, p. 114.
5.5. But the mean density of the rocks outcropping at the surface8
is only 2.7. Whatever may be the character of the earth's interior,
8
Compare also E. D. Williamson and L. H. Adams, Jour. Washington
Acad. Sci., 1923, vol. 13, p. 413; H. S. Washington, ibid., p. 453.
STAGE OF THE LIFE DRAMA 195
it is thus evident that it is composed of/lenser material than we find
at the surface. In point of fact, comparison with meteorites, and
other evidence, make it appear likely that the earth's interior is virtually
a metallic regulus (chiefly iron), encrusted with a slag not unlike
that which separates out and floats on the molten mass of metal in
a blast furnace. Such, essentially, is the raw material of our landscape,
such our habitation, such the ground on which we tread and
from which we draw the substance of our body. For the fertile soil
in the plain is but the weathered variant of the granite strength of
the hills, and we ourselves but a strangely metamorphosed portion
of the world-stuff, the slag coating of a metal core.
Cosmic Accessions to the Lithosphere. The earth receives a constant
shower of meteorites from interstellar space, at an annual rate
of some 20,000 tons. 9
This seems a large amount. As a matter of
fact, spread over the surface of the globe (197 million square miles),
the accumulated meteoric material of a thousand million years would
make a layer only 1 inch thick, if its density were that of water, or a
correspondingly thinner layer of denser material.
Composition of the Earth's Crust. Table 16 adapted from F. W.
Clarke's Data of Geochemistry and a more recent publication by
Clarke and Washington, shows the average composition of the known
terrestrial matter. The first column, in particular, gives the figures
for the solid crust. This table exhibits a number of facts and relations
of interest. Perhaps the first significant circumstance that
strikes the eye, in glancing at the table, is the very unequal deal
with which nature has distributed matter among the ninety odd10
known elements. One-half the lithosphere, and one-quarter of the
atmosphere are made up of the element oxygen.
The eight most abundant elements of the earth's crust (oxygen, silicon,
aluminum, iron, calcium, sodium, potassium and magnesium) the only ones
whose amounts are over 1 per cent constitute together over 98 per cent of
the earth's crust. These, with hydrogen, titanium, carbon and chlorine
twelve in all make up 99.5 per cent; thus leaving only one-half of one per
9
S. Arrhenius, Worlds in the Making, 1908, p. 108. See also Young,
Astronomy, 1904, p. 475.
10 Uncountable isotopes aside.
196 ELEMENTS OF PHYSICAL BIOLOGY
Ba
Cr 0.08
tt
LJL_
S 0-18
O.ll a
d2\g.$8
IS
Aly
o.or
Ca
3.ZZ
Si
Sff.80
o
SO.OS.
Compos.'/Van of arf/>'o
At
7.30
A/a
Z.S8
a p ra?o r F Mn
/y n 0.11 ato o.og
s
O.I1-
Fe
O.Ol 63.0
0.10 O.H
Z.S
Compost ft'an a'f Human Boc/y.
FIG. 42. COMPARISON OF COMPOSITION OF EARTH'S CRUST AND HUMAN BODY
Figures indicate percentages
STAGE OF THE LIFE DRAMA 197
cent for all the other elements, among them some quite indispensible for
our existing civilization. 11
Relation to Composition of the Organism. Of the elements specifically
significant for the living organism, only one, oxygen, is present
in great abundance. Carbon, hydrogen and nitrogen, the principal
"organic elements/' are among the less abundant constituents
of the globe. On the whole it may be said the living organisms are
composed of comparatively rare elements. We are, indeed, earthborn,
but yet not altogether common clay.
12
This is well brought out
TABLE 17
Average composition of human body
in the chart figure 42, which shows, side by side, the average composition
of the known terrestrial matter, and, in comparison, the approximate
composition of the human body. This latter is not exactly a
representative sample of the totality of living matter (see tables 17
11
H. S. Washington, Jour. Franklin Institute (1920), vol. 190, p. 7. Tho
figures have been slightly modified in accordance with the latest data of
F. W. Clarke and H. S. Washington, Jour. Natl. Acad. ScL, 1922, vol. 8 p.
114.
12
Indeed, taken literally the expression "common clay," as applied to
man, is an extreme case of poetic license; for aluminum and silicon the
chief constituents of clay, and taking second and third place in rank of abundance
among the components of the earth's crust, are both present only in
tracea in the human body.
198 ELEMENTS OP PHYSICAL BIOLOGY
^
STAGE OF THE LIFE DRAMA 199
d
ft C
03 CQ
200 ELEMENTS OF rriYBTCAL BTOLOftY
.a
.a
Ol VCJ
m
S
M.O O
tCJ
JD S
ft
cu q H
cu . y
w l>> 2
ti .in a
-fl
STAGE OF THE LIFE DKAMA 201
and 18) but will serve well enough for the present pupose. The chart
brings out very pointedly the selective character of the organism's
activity in gathering to itself the substance of its body. Thus carbon,
which both in function and in relative quantity figures so prominently
in living matter, appears as an insignificant lifctle block in the
chart of the earth's crust. A similar contrast, if not quite so extreme,
is seen in the case of nitrogen. With aluminum and silicon the comparison
works the other way about very plentiful in the earth's
crust, these elements are practically absent from the human body.
13
Taking the mother earth as a whole, and the organism as a whole,
it certainly cannot be said that there is much evidence of "inheritance
TABLE 19
Composition of the salts in sea water and in blood serum in per cent*
*
Maccallum, Trans. Koy. Soc. Canada, 1908, II, p. 145.
of parental characters" in their respective compositions. Still, resemblance
is not wholly lacking. This becomes evident if we compare,
not the entire organism with the whole of the earth's crust, but the
blood of a mammalian, for example, with the water of the ocean. This
comparison is made in table 19 and figure 43. The likeness thus seen,
imperfect as it is, can hardly be ascribed to accident. The fact is,
13
Silica furnishes, however, the skeletal support in a variety of living
forms (radiolaria, sponges, plants). Cf. G. Bunge, Physiological and Pathological
Chemistry, 1902, p. 23-24. For a detailed discussion of the distribution
of the chemical elements in organic nature see Vernadsky, Revue Ge"n.
Sci. The reader interested in this phase of the subject should not fail to
acquaint himself with this article, which came to the writer's attention too
late to be given more than this passing note here.
202 ELEMENTS OF PHYSICAL BIOLOGY
Cl
55.3
Na
30.6
SoJJds
in
7.66
3.79
KU
Br
O.f3 O.O
0.4
1.0
2.7 0.4
Soh'ds
in
Blood Serum
No
39 an a dry fi
Cl
45
. 43. COMPARISON OF SEA WATER AND BLOOD SERUM
STAGE OF THE LIFE DKAMA 203
as pointed out by Palitzsch,
14
that aquatic species are in such intimate
contact, both at their body surface, and more particularly in their
gills, with the surrounding water, that the latter might almost be
considered continuous with their bloody fluids, so that sea water may
justly be "placed in the same category as the other physiological
fluids." "Frederiq
15
has shown that the amount of sodium chloride
in the blood of Crustacea varies, and all but corresponds, with the
density of the water in which the creature has been kept." More
highly organized aquatic species have made themselves in greater
degree independent of the salinity of their environment;
18
and finally,
it would appear, when the marine ancestors of terrestrial vertebrates
emerged from the sea and adventured life on dry land, they packed,
as it were, a portion of their saline environment in their baggage, and
took it along with them on their excursion as an essential part of their
milieu interieur. And to this day, according to this view, we ourselves
cany about with us in our arteries and veins, if not a portion
of the actual ocean, at least a roughly approximate replica of its
brine. For, as L. J. Henderson17
remarks: "Not only do the body
fluids of the lower forms of marine life correspond with sea water in
their composition, but there are at least strong indications that the
fluids of the highest animals are really descended from sea water."
Some such indications may be seen in the reflections (conceived from
a slightly different point of view) of G. Bunge,
18
"I am convinced that
the remarkably high percentage of salt in vertebrate animals, as well
as the desire to take salt with our food, can be satisfactorily explained
only by the theory of evolution." In support of this consideration
Bunge points out that in the weathering of rocks by the action of
rain-water charged with carbonic acid, the sodium is dissolved and
carried off as carbonate, while the potassium largely remains behind
in combination with silica. The sodium carbonate being washed to
14
Comptes-rendus, Laboratoire de Carlsberg, 1911, vol. 10, part I, p. 93.
See Chapter I, footnote 19.
16
Arch, de Zool. Exp. et Gen., 1885, Ser. 2, vol. 3, p. XXXV; see also D'Arcy
Thompson, Growth and Form, 1917, p. 127.
16
Compare D'Arcy Thompson, loo. cit., p. 127-130; Claude Bernard, Introd.
a 1' etude de la me"decme exp., 1855, p. 110 as quoted on p. 17 (and footnote
20) of Chapter I.
17
The Fitness of the Environment, 1913, p. 187.
18
Loc. cit., 1902, p. 101-103,
204 ELEMENTS OF PHYSICAL BIOLOGY
the sea undergoes double decomposition with the alkali earth chlorides,
these latter being deposited as carbonates (limestone and dolomite)
while the sodium remains in solution as salt. Thus sea water
is rich in sodium chloride and poor in potassium, while on dry land
the balance is essentially reversed.19
Plants and invertebrate animals,
Bunge points out, contain little sodium, unless they live in a
highly saline habitat, in or near the sea, or on salt steppes. Yet the
land vertebrates are all remarkably rich (comparatively) in salt, in
spite of the scanty supply around them.
Is not the large amount of sodium chloride found in the present inhabitants
of dry land another proof of the genealogical connection which we
are forced to accept from morphological facts? There is no doubt that each
of us in his individual development has come through a stage in which he
still possessed the chorda dorsatis and the branchial arches of his sea-dwelling
ancestors. Why may not the high average of salt in our tissues be also
inherited from them?
Support for the supposition thus suggested is seen by Bunge in the
fact that the younger a vertebrate is in its individual development,
the more salt does it contain. Furthermore, cartilage contains the
highest percentage of sodium of all the tissues of our body, and is
also the tissue of greatest antiquity. The human skeleton is originally
composed of cartilage, which is replaced, for the most part, by
bone as the individual matures.
These are facts which lead most readily to the interpretation th?
*
the
vertebrates living on dry land originally came from the sea, and aw still
continuing to adapt themselves to their present surroundings, where they
can get but little salt. We prolong this process of acclimation by taking
advantage of the salt strata which have been left on the land by our primeval
element, the salt flood.
Chemical Correlation in Soil and in Organism. Our ancestral
resemblance to the soil from which we spring is also exhibited by
evidence converging from a different source. II. S. Washington,
19
This observation must be accepted with some caution. Compare
Whitney, Science, 1922, vol. 56, p. 218. "Until we determine the actual loss,
through chemical denudation, of silica, alumina, iron, potash and other
electrolytes in the colloidal state, carried by rivers, we are in no position
to even speculate as to whether erosion is a selective process which might
change the chemical composition of the soil."
STAGE OF THE LIFE DEAMA 205
in his studies on The Chemistry of the Earth's Crust, already cited,
draws attention to the fact that in the rocks soda and iron tend to be
associated together as a pair on the one hand, and potash and magnesia
on the other. This is well brought out in the diagram figure
44, reproduced from Washington's memoir, in which it is seen that
the points representing the analysis of a number of rock samples tend
to group themselves about the diagonal of the square, indicating that
FeO
FlG. 44. COKBELATION IN THE OCCURRENCE OF Na, Fe AND ~K, Mg IN ROCKS
A large number of analyses here plotted show a marked tendency to array
themselves along the diagonal, showing that high percentage of Na is commonly
associated with high percentage of Fe; K and Mg follow a similar
relation. After H. S. Washington.
high content of soda goes together with high content of iron, but with
low potash and low magnesia; and vice versa. The point of special
interest to us here in the present connection is that to which Dr.
Washington draws attention in the words:
Curiously enough, the same correlation between these two pairs of elements,
soda and iron, and potassium and magnesium, seems to hold good in
the organic world. This is apparently shown by the following facts: In
206 ELEMENTS OF PHYSICAL BIOLOGY
autotrophic plant metabolism potash is an essential element, as is also magnesium,
in that chlorophyll (which in the leaves acts as the carbon-transferring
substance) is a magnesium salt of a complex organic acid, while sodium
and iron are generally toxic toward (at least the higher, gymnospermous
and angiospermous) plants. On the other hand, sodium, rather than potassium,
is the alkali metal essential to the higher animals, salt being a very
necessary article of diet (in part because of its chlorine, and in part because
of its sodium, content), and sodium chloride is present in the blood plasma;
and at the same time, hemoglobin and its derivatives (which act as oxygen
carriers, and are analogous to chlorophyll in plants) are iron salts of organic
acids closely related to that of chlorophyll; while, similarly potassium and
magnesium are more toxic toward the higher animals than are the other
pair.
This singular parallel must not, of course, be looked upon as an
instance of resemblance due to anything of the nature of inheritance
of ancestral traits in the biological sense. Rather must it be connected
in our minds with the fact that systems composed of the same
fundamental substances, will display certain analogies through interplay,
in them, of the same chemical affinities.
Accessibility of Valuable Earth-Constituents. Such a comparison
as has been made above of the relative abundance of the several elements
in the living organism and in the environment from which it
draws its supplies, would be misleading if attention were not drawn
to another factor aside from abundance, which enters strongly into
play in the quest for the necessities of life. More important than
mere abundance is accessibility. For, a substance may be present in
comparatively large quantities, and yet be difficult to lay hold of,
either on account of its wide dispersal in dilute form, or for other
reasons. On the contrary, a comparatively rare substance may be
procurable with relative ease, if it occurs segregated in concentrated
or otherwise readily accessible form. Perhaps the most telling illustrations
of this are to be found in industry. The element copper for
2
example is found only to the extent of about -r per cent in the
earth's crust. Yet it is one of the most important metals in the arts,
and is not ordinarily thought of as particularly rare. This is because,
in those regions where it does occur, it is found in concentrated form,
either as native metal, or as rich ore. Other instances are readily
cited. Tin, lead and zinc are all rarer than copper, and each rarer
than its precursor, in the order named. Still rarer are silver, tungsten,
gold, bromine and platinum, all of which find important use in
STAGE OF THE LIFE DRAMA 207
^
O r-l (-H r-t
WrH
M
Si
r-i CM CO
42803 22 19
iO CO t- 00 CJ
1SI
n S
02
WO *z
W ffl
02 o ,_,
O CU
-> J _O
pa 5
R H
-<1
^
O
p
OM
W
fM
208 ELEMENTS OF PHYSICAL BIOLOGY
the arts. But the mode of occurrence of these substances is such that
they can be gathered or rained with comparative ease. It has been
pointed out by H. S. Washington that the elements, as arranged in
Mendeleeff's table, naturally fall into two groups divided by a zigzag
line, as shown in figure 45. Above this line are the rock elements
or petrogenic elements, that enter into the principal rock-forming
minerals (and also, the gases of the atmosphere). Below the line
are the ore elements or metallogenic elements, which commonly occur
in concentrated form as ores and as native metal.
Now man's industrial activities are merely a highly specialized
and greatly developed form of the general biological struggle for
existence ;
and this same feature of accessibility and of concentration
in segregated supplies (ores and the like), which is a prime condition
for the very existence of some of our industries, is also involved,
in closely analogous manner, in the more primitive life processes.
Our fields demand fertilizers bringing ammonia, nitrates, potash,
phosphates, etc., in suitably concentrated form, if they are to bear a
harvest commensurate with the needs of a modern community.
And this again is merely an accentuated example of the still more
primitive needs of the unsophisticated flora and fauna of virgin nature.
Of scattering, dissipating processes there are plenty. Rain and snow
wash most of what is soluble, and much that is not, into the rivers
and^ut to sea. Our own activities in modern intensive agriculture
briiig each year to the land a highly concentrated diet of fertilizers,
which in the very act of cultivation are scattered and diluted many
thousand
times. And our modern sewage system is deliberately
wasteful of vital substances, which it discharges into streams and
out TO sea. All this dissipation must in some way be balanced if the
regime
is to continue. Thus the circulation of matter in nature must
no/fc only provide for the mere presence of certain substances on which
^che maintenance of life depends, but it must furnish them in suitable
'
concentration and, generally, in available form. It must, therefore,
in many cases include, as a definite step, a segregating or concentrating
process
20
as well as simple motion through a cycle.
We shall have occasion to note concrete illustrations of this in the
separate consideration of the circulation of the several elements, to
which we now proceed.
20
The significance of this from the point of view of energetics will engage
our special consideration in a later section.
CHAPTER XVI
THE CIRCULATION OF THE ELEMENTS -THE WATEH CYCLE
The great Sea-water finds its way
Through long, long windings of the hills;
And drinks up all the pretty rills
And rivers large and strong:
Then hurries back the road it came
Returns on errand still the same;
This did it when the earth was new;
And this for evermore will do
As long as earth shall last.
Wordsworth.
The ancients, totally blind as they were perforce to the fine details
of material transformations revealed by the search light of modern
chemistry, nevertheless recognized in its broad features the cycle of
life, the circulation of the elements in nature. "Dust thou art,
and unto dust shalt thou return," we read in an old book of wisdom.
Heracleitus (536 to 470 B.C.), promulgator of the famous doctrine
Tr&vra /Set has a more detailed, if not more accurate conception of the
cycle of Nature, which he formulates in these terms:
Earth
S \
Fire Water
Vapor
(Air)
The human mind was not yet schooled, then, to polish the facets of
this rough gem, and bring out the sharp-edged truth as we see it
today
Solid
\
Liquid
Vapor
209
210 ELEMENTS OF PHYSICAL BIOLOGY
But all honor to the minds that discerned through the mists of dawn
the bold features of the landscape to be revealed in the sunlight of
later day. Today we recognize not four elements, but over ninety,
not counting those modern variants, the isotopes. And we follow in
much detail not one cycle, but, as particularly pertinent to life, five
major cycles the circulation of water, carbon, oxygen, nitrogen
and phosphorus. This was noted already at the conclusion of
Chapter XIV, but the discussion of the cycles was deferred to give
space to a preliminary survey of the scene in which these cycles churn
the planet's surface in their age-long duty.
We are now prepared to take up the thread where we broke off;
we turn our attention first to the water cycle.
Water Requirements of Human Body. We do not ordinarily
class water as a food, though we partake of it by the same channel ..
|
that gives entrance to the materials commonly so classed, and
although the lack of water, if we are by any circumstance deprived
of this substance, is felt even more acutely than an interruption in
the adequate supply of food. The fundamental basis for this distinction,
its origin in the unsophisticated mind, is undoubtedly the fact
that we have a separate sense of thirst, distinct from the signals of
hunger and appetite originating from nutritive demands of the body.
And this naive, unsophisticated distinction is entirely in accord with
the reasoned analysis of the respective functions of water and of
food in the narrower sense. It is undoubtedly just because of this
difference in function that thirst and hunger have been developed as
separately recognized sensations. Water acts merely as a vehicle;
unlike the food, which undergoes extensive and complicated reactions
within the economy, water leaves the body essentially as it
enters it, unchanged chemically, though charged (in part) with substances
in solution.
It would be a gross error, however, to suppose that water, because
it functions thus in accessory capacity, and escapes the more intimate
transmutations of metabolism, can be lightly regarded in making a
survey of the participation of the several elements in the cycle of
nature. It must be remembered that water constitutes as much as
60 per cent of the total mass of the human body, for example, and a
still greater proportion of the substance of most of our food-stuffs,
as shown in table 20. _
.,*f?
WATER CYCLE 211
Thus an adult human being consumes per diem about 3 liters of
water of which about 1 liter is contained in his solid food. In point
of fact he consumes about 5 pounds of water for every pound of dry
solid matter ingested. It is thus seen what an important item water
is in the daily economy of the human organism.
The excretion of water by the kidneys, lungs, intestine and skin
is somewhat in excess of the intake. The excess of the outgo over
the intake is formed in the body by the oxidation of hydrogen
organically combined.1
Water Requirements of Plants. Rainfall as a Limiting Factor.
In the economy of most plants the traffic of water is of even
greater importance
2
than in man and land animals generally, and
TABLE 20
Moisture content of some common foods
indirectly the moisture needs of plants are, of course, of fundamental
importance also to the animal population feeding on the
vegetable growth of the soil, so that the water supply of a territory
1
L. J. Henderson (The Fitness of the Environment, p. 133) estimates that
a man weighing 60-70 kgm. excretes daily:
grams
Water 2500-3500
Carbon dioxide 750- 900
All other substances 60- 125
and that water is, accordingly, three-fourths and carbon dioxide one-fifth
of the total excreted. The proportion of water excreted from kidneys, skin,
lungs and intestines, is given by Kirk (Physiology, p. 208) as 1:0.5:0.22:0.09,
or 11.5:5.75:2.5:1.
2
L. J. Henderson, loc. cit., estimates that of the materials ingested by
ordinary green plants, more than nine-tenths is water, and carbon dioxide
at least five sixteenths of the remaining tenth.
212 ELEMENTS OF PHYSICAL BIOLOGY
may function as the basic limiting factor of the total life which that
territory is able to support. A study of these relations with particular
reference to the human population of the United States, has
been made by W. J. McGee, who remarks :
3
Hellriegel in Germany and King in this country have shown that crop
plants require for their growth_a quantity of water, measured by transpiration,
averaging from 300 to 600 (with a mean of about 450) times the weight
of the plants after drying; and common field experience indicates that, in
addition to the moisture passing through the plants, the soil requires an even
larger quantity to maintain a texture suitable for crop growth much of
which passes away through evaporation and seepage. On this basis "the
agricultural duty of water" in this country has been formulated as the production
of one-thousandth part of its weight in average plant crop. Reckoning
human food and drink on this basis, and assuming that meats require
(chiefly in the growth of plants used as feed for the animals) ten times the
quantity of water represented in vegetal food, it appears that the adult who
eats 200 pounds each of bread and beef in a year consumes something like
1 ton of water in drink and the equivalents of 400 tons in bread and 4000 tons
in meat, or 4401 tons in all figures corresponding fairly with the results of
intensive agriculture in arid districts. Accordingly, the "duty of water"
considered in relation to human population may be stated roughly as the
maintenance of a human life a year for each 5 acre-feet used effectively in
agriculture.
Now mainland United States (i.e., the chief body of our territory, exclusive
of Alaska and the insular possessions) comprises something over
3,000,000 square miles, or somewhat less than 2,000,000,000 acres of land; yet
the annual rainfall the sole original source of fresh water averages barely
2* feet (30 inches), or hardly 5,000,000,000 acre-feet. So while the land area,
if peopled to the density of Belgium (over 640 per square mile,) would carry
a population of 2,000,000,000, the water supply suffices for only 1,000,000,000.
The conclusions of McGee may have to be modified in point of
detail, and some of his figures may perhaps have to be revised; but
the general principle underlying his reflections attract our attention.
The moisture needs of the living population (all species included)
are a large and fundamental item in biological economy; whether
this item proves the ultimate limiting factor of population growth,
as McGee suggests, is a question whose answer must be sought in
3
Science, 1911, vol. 34, p. 429; Yearbook of the Department of Agriculture,
1910, pp. 169-176; Bureau of Soils Bull. 71 (1911), pp. 7-14; World's
Work, 1912, vol.. 23, p. 443. Compare also L. J. Briggs, The Water Requirements
of Plants, U. S. Department of Agric., Bureau of Plant Ind Bulletins
284, 285.
WATER CYCLE 213
terms of Liebig
?
s Law of the Minimum; a dearth of other essentials
may make itself felt before the limit of available moisture is reached.
The Sources of Supply. Such, then, in broad outline, are the
moisture needs of organic nature; as such they have existed, in greater
or less degree, for millions of years, and have been satisfied, and will
continue long to be satisfied, from a source essentially inexhaustible
because constantly replenished by the return flow : The great reservoir
is the ocean, with its 302 million cubic miles of water, and an
evaporating surface of over 144 million square miles. Annually
there rise from this into the atmosphere about 63,300 cubic miles of
water, to which some 22,800 more are added by evaporation from the
land, making a total of 86,100 cubic miles. This figure also represents
the total precipitation in rain, snow, etc., but of the total
about 56,700 fall back directly into the ocean, and only the balance,
29,400 is available for the needs of the land. It has been noted that
the evaporation from the land is about 22,800 cubic miles. The
difference between this and the precipitation on land, the balance of
6500 cubic miles, is the drainage from the land to the ocean by rivers.
No attempt will be made to estimate what proportion of the precipitation
on land is derived from evaporation over the sea, and what
proportion comes from the land itself. Some idea of the relation,
however, can be formed from a consideration of the difference or the
ratio of the rainfall within the area drained by rivers and the amount
actually discharged by them into the sea. John Murray
4
has collected
information on this subject, with the result shown in table 21,
which covers 33 of the world's principal rivers. It will be seen that
the total rainfall in the area of these rivers is 10,186 cubic miles,
the discharge to the sea is 2182 cubic miles, leaving a difference of
8004 cubic miles unaccounted for. A certain portion of this perhaps
represents seepage, but the bulk must correspond to water reevaporated
from the land and from inland waters. It will be observed
that for the 33 rivers combined the proportion of the discharge
to sea to the total rainfall is about one-fifth. For individual
rivers the ratio varies widely, between the extreme of 0.58 (Rhone)
and 0.027 (Nile). Naturally the climate very materially affects this
figure.
4 Scottish Geographical Magazine, 1887, vol. 3, p. 76.
214 ELEMENTS OF PHYSICAL BIOLOGY
There is also a circulation of waters of the sea in very large dimensions.
According to L. J. Henderson5
the Gulf Stream in the
Straits of Yucatan carries 200 million tons per second,
6
travelling
TABLE 21
Showing the drainage area, annual rainfall, annual discharge, and ratio of
discharge to rainfall, of S3 rivers in different parts of the world
5
The Fitness of the Environment, 1913, p. 182.
8
Across any cross-section of the stream, presumably; the statement is
not clear on this point.
WATER CYCLE 215
with, a mean velocity of about 80 miles per day. This circulation in
the ocean has of course no direct part in the water cycle of the organic
world, but indirectly is most important on account of its climatic
effects.
m\
r
Oceo/%
///&./m.
mill $f mi.
figures \rifhout denomination afie cubic
FIG. 46. CIRCULATION OP THE ELEMENTS IN NATURE. THE WATEE CTCLEI
Water Cycle Diagram. The principal figures relating to the
circulation of water on the globe are exhibited in diagrammatic form
in figure 46, which tells the story more effectively than words. It
may here be added in explanation that the 3000 cubic miles of
216 ELEMENTS OF PHYSICAL BIOLOGY
moisture contained in the air are practically restricted to the lower
6 miles or so of the atmosphere. For the rest, this moisture is unevenly
distributed, as everybody knows from personal weather
observation. Water vapor and condensed water differs in this
respect from the other permanently, gaseous constituents of the
atmosphere. (See table 12 and figures 40, 41.)
Fraction of Total Water Circulation Taking Part in Life Cycle.
Only a fraction of the total circulation of water actually passes
through the organic cycle. We may make an attempt as follows to
obtain a rough idea of the order of magnitude of the fraction
thus concerned.
If the entire land surface were cultivated to produce crops at the
rate adopted as standard by W. J. McGee, the growth produced
(figured in dry weight) would be T^u~ff
of the rainfall. Furthermore,
this growth would evaporate, by transpiration, about 500 times its
own weight of water, (this is assuming one crop per year). It would
therefore evaporate just about one-half the annual rainfall. But the
22 800
total evaporation on the land is
OQ'QQQ
= tnrce fourths of the annual
rainfall on land, the remaining fourth being drained to sea by
rivers. Hence, of the water evaporated on land, one-half times
four-thirds = two-thirds is evaporated by plants and thus takes
direct part in the organic cycle. In comparison with the evaporation
by plants, that from animals is undoubtedly negligible, especially
in view of the coarseness of our data. If we put the evaporation
on land as one-fourth of the total evaporation, we finally arrive
at the value one-sixth as that fraction of the total water in circulation,
which takes actual part in the organic circulation.
Desert areas cannot materially alter this estimate, since they contribute
but little to either side of the account, both evaporation and
life being meagre or absent. In some measure this remark also applies
to frigid wastes of the polar regions, where the low temperature
makes for comparatively low evaporation (a factor counter-balanced,
it is true, in some degree, by the extensive cover of ice and snow) .
For temperate zones McGee's figure for cultivated fields is undoubtedly
too high to apply as an average for the entire land. In the tropics,
on the other hand, it is perhaps not excessive. On the whole the
fraction one-sixth computed as above is probably too high, but perhaps
it serves to give us an idea of the order of magnitude involved.
WATER CYCLE 217
Another estimate leading to a materially lower result, is obtained
as follows: If we accept Engler's estimate that one-fiftieth of the
atmospheric carbon dioxide, that is to say, 4.4 X 1010
metric tons,
takes part in the organic cycle; and we adopt the figure given by
L. J. Henderson2
that the water taken up by plants, is about 11 times
the C02 which they absorb, we obtain, as a very rough estimate of
the water engaged in the organic cycle, an amount of 5 X 10U
metric tons, or, in round numbers, 120 cubic miles. This, then, is
about vfar of the total annual circulation of water, or about $%-$ of the
total rainfall on land.
CHAPTER XVII
THE CARBON DIOXIDE CYCLE
Behold how great a matter a little fire kindleth. St. James.
If the lamp of life is a poetic symbol, it is an image essentially
true to fact. Not only is life, in particular animal life, largely a
combustion process: like the flame, life reaches out for fuel, and
with the power gained, strains again for more. Like the flame it
consumes, and it spreads. And as the fire sends out sparks, of which
many die, but a few, falling upon favorable ground, flare up as a
second generation, in reproduction of the parent flarne; so the living
creature scatters its seed, some to die, but some also to live again
the life of the parent. "But," someone perhaps will remark, "a
fire may start without preexisting flame; whereas all life is itself
begotten of life." Is this distinction really so fundamental? In
nature undisturbed by man the starting of a fire spontaneously is
a rare event; and that, after all, is the most that we can say positively
regarding the origination of life from the non-living it is
either so rare or so unobtrusive1
an event as to have escaped our
observation. No doubt it took man many thousands of years to
acquire the art of lighting a fire, may not in the lapse of time a
second Prometheus arise to teach us also how to kindle the torch
of life? Let us not delay his coming by closing our minds to the
possibility.
2
1
Compare F. J. Allen's view, as presented by L. L. Woodruff in The Evolution
of the Earth and its Inhabitants, 1919, p. 102: "Life at this stage was
of the humblest kind, since there were no definite organisms, only diffuse
substances trading in energy, and between this stage and the evolution of
cellular organisms an immense period elapsed." If this picture of the beginning
of life is true to fact, the process was unobtrusive ; probably, if we
were shown a specimen of such elementary "living" matter, we should not
recognize it as such. All this is in accord with what has been said in an
earlier
^
chapter regarding the definition of life. If we continue to use the
word life, this is merely a matter of convenience and does not imply any
departure from the point of view set forth in the opening chapters. See
also p. 19.
218
CARBON DIOXIDE CYCLE 219
Be that as it may, the fundamental fact remains that slow combustion,
oxidation as the chemist calls it, is a dominant feature in
the physiology of animal life, and that the leading roles in this
action are played by the elements carbon and oxygen.
In our method of securing our supply of these elements there is
a certain dyssymmetry. Carbon we eat in our meals; oxygen we
breathe in in respiration. But in function the two elements stand
in essentially symmetrical relation; the two together and impartially
furnish us with the requisite energy for our life activities.
^
Thus
we must regard oxygen as food as much as carbon. This fact
deserves a passing note, since it is sometimes stated that assimilation
of inorganic food is a characteristic of plants, as distinguished
from animals. The statement rests on an arbitrary and wholly
gratuitous exclusion of oxygen from our list of foods. It just so
happens that there is one item on the animal's menu, namely, oxygen
that is gaseous and is spread broadcast; does not therefore have to
be hunted and captured. Toward this the animal assumes the
same attitude which, presumably, plants adopt toward all their
foods: he takes it in unconsiously. This touch of plant nature
which we recognize in ourselves should serve to give us a sympathetic
insight into the "psychology" of plants. At the same time it reminds
us once again of the esentially arbitrary character of the
division of organisms into two classes, animals and plants. One
and the same organism possesses both animal and plant characteristics,
and this is true even of that most highly specialized of
^all
animals, the human being. In view of the symmetry in function
that exists between carbon and oxygen, and the inseparable relation
2
It is easy to strike a match, but this is merely a way of borrowing tho
efforts of others, those who have made the match possible, and those who
have manufactured it. How many city dwellers could, by their unaided efforts,
start a fire where none was before? Anyone who, in an emergency,
may have been forced to attempt the feat will appreciate how among tho
ancients "the spark of fire was zealously guarded and soon invested with
sacred attributes .... The chief function of the vestal virgins in Homo
was to keep the perpetual fire; and in the Catholic church today with its
never extinguished light we have the last survival of what was once a social
custom "
Another curious survival of this custom is quoted
by E. A. Seligman (Principle of Economics, 1908, p. 69): "Whenever the
location of gas works is changed the fire is transferred by a brand from tho
old to the new building. Under no consideration would a new fire bo started."
220 ELEMENTS OP PHYSICAL BIOLOGY
in which, they stand in the life processes of the organism, we shall
consider jointly, in this chapter, the circulation of carbon on the
one hand, and of oxygen on the other.
The Carbon Cycle. A very particular interest attaches to the
carbon cycle. Carbon is the organic element par excellence, whose
absence from any chemical substance stamps this forthwith, by
common if somewhat arbitrary consent, as inorganic: whose presence
affords the soil ? ,, \ season for the growth of what might be
termed the tropical jungle in the domain of chemistry. For in the
compounds of carbon nature seems to have run riot, in a revel of
creative versatility, as if vying to set a record unapproached elsewhere
in all the realm of chemistry, for number, variety and complexity
of her children. Other elements 'oxygen, nitrogen, phosphorus,
sulphur, iron, indeed play a significant role in life processes;
but the indispensable bond that ever links all other ingredients in
organic unity is carbon. Furthermore, carbon is preeminently the
energy carrier, the standard coin of the organic real, in which both
the first cost of installation, of anabolic tissue building, and also
the running cost of operation, of metabolism, is defrayed.
Indescribably complex far beyond the understanding of the
organic chemist of today as are the metamorphoses that carbon
undergoes in the economy of the organism, its source and its gate of
entry into the organic cycle are comparatively simple. Some two
and a half million million tons (2.2 X 1015
kgm.) of carbon dioxide,
in the air, and perhaps twenty to twenty-five times this amount
contained in the waters of ocean, lakes and rivers, these constitute
the store from which all life ultimately draws its supply.
Of this vast store, according to an estimate made by C. Engler,
3
about one-thousandth part actually takes part in the cycle of life.
The total carbon locked up in living organisms, which would be
a measure, in a way, of the spread of life on our globe, is difficult
to gage even roughly. One hardly knows how much or how little
significance to attach to an estimate by A. G. Hogbom4
that the
total quantity of carbon in all living matter is of about the same
order as that contained in the atmosphere, namely, 6 X 1011
metric
3
Linck, Kreislaufvorgaxige in der Erdgescliichte, 1912, p. 6; Engler, tJber
Zerfalls-prozesse in der Natur.
4
Clarke, Data of Geochemistry, 1920, p. 49.
CAKBON DIOXIDE CYCLE 221
tons; an amount which, spread over the entire surface of the globe,
would cover it with a film of carbon 1 mm. thick, or with a film of
living matter about | inch thick.
The organic carbon cycle, reduced to its simplest terms, is a
closed chain of three links.
Atmosphere
/ Green
Animals* Plants
Green plants, under the influence of sun light, absorb C02 from the
atmosphere and convert it, with elimination of oxygen, into the many
and complex compounds of the plant substance.5
Animals consume
plants (directly or indirectly) as food, and in the course of the
operations of their typically active lives (as compared with the
typically passive, vegetative existence of the majority of plants)
they reoxidize the carbon reduced in the photosynthetic plant
processes, and return C02 to the atmosphere, thus completing the
cycle.
In actual fact this simple fundamental cycle is complicated by a
number of influences. The decay of dead animals and plants adds
a comparatively small item to the discharge of C02 into the atmosphere.
Plants are somewhat mor-e resistant to complete decay
than animals, and one result of this is the accumulation of notable
6
To discuss here the chemistry of photosynthesis in plants would lead
us too far afield. Very important advances have been made recently in
this field of biochemistry. It must suffice here to refer to the original literature
of which the following articles may be mentioned: E. C. Baly, Photosynthesis,
Nature, March 16, 1922, p. 344. Report of discussion on photosynthesis
at the British Association Meeting, Nature, December 23, 1922,
p. 856. I. M. Heilbron, The Photosynthesis of Plant Products, Nature,
April 14, 1923, p. 502. 0. Baudisch, On the Formation of Organic Compounds
from Inorganic by the Influence of Light, Science, April 20, 1923, p. 451.
0. Baudisch, The Influence of Light on Inorganic Matter and Life Processes,
Jour. Industrial and Eng. Chemistry, May, 1923, p. 451. J. C. Bose, Effect
of Infinitesimal Traces of Chemical Substances on Photosynthesis, Nature,
July 21, 1922, p. 95. Baly, Heilbron and Parker, Photochemical Production
of Formaldehyde, Nature, September 1, 1923, p. 323. J. H. Mathews,
Trends in Photochemical Research, Jour. Ind. and Eng. Chem., September,
1923, p. 885.
222 ELEMENTS OF PHYSICAL BIOLOGY
quantities of reduced carbon in the form of peat and coal. This
process of fossilization is slow, and would not in itself, in any short
period, materially affect the carbon cycle. It has, however, fur- /\
nished the occasion for a phenomenon which, judged in a cosmic ,
'
perspective, represents a purely ephemeral flare, such as must ultimately
appear utterly insignificant in the geological calendar, if
duration alone is considered; but which to us, the human race in
the twentieth century is of altogether transcendent importance:
The great industrial era is founded upon, and at the present day inexorably
dependent upon, the exploitation of the fossil fuel accumulated
in past geological ages.
We have every reason to be optimistic; to believe that we shall
be found, ultimately, to have taken at the flood this great tide in
the affairs of men; and that we shall presently be carried on
the crest of the wave into a safer harbor. There we shall view r\
with even mind the exhaustion of the fuel that took us into port,
knowing that practically imperishable resources have in the meanwhile
been unlocked, abundantly sufficient for all our journeys to
the end of time. But whatever may be the ultimate course of
events, the present is an eminently atypical epoch. Economically
we are living on our capital; biologically we are changing radically
the complexion of our share in the carbon cycle by throwing into
the atmosphere, from coal fires and metallurgical furnaces, ten
times as much carbon dioxide as in the natural biological process
of breathing. How large a single item this represents will be realized
when attention is drawn to the fact that these human agencies
alone would, in the course of about five hundred years, double '-r
the amount of carbon dioxide in the entire atmosphere, if no compensating
influences entered into play. In point of fact the per- f
centage of carbon dioxide in the atmosphere exhibits remarkable I
constancy and there are several very large items, in addition to
those already touched upon, both on the ingoing and outgoing side
'
of the account. The case of man has been singled out for mention
here merely because our knowledge of the human population and
economy enables us to make a reasonably close estimate of
his contributions. The quota supplied by the remaining animal
species can hardly even be guessed at. But probably the greatest
source of atmosphere C02 are volcanoes and mineral springs. Cotopaxi
alone has been credited with an annual discharge of two million 4
tons of the gas. *-
*
[
CAEBON DIOXIDE CYCLE 223
On the debit side there is first of all the item of consumption by
plants. E. H. Cook,
6
from very uncertain data, computes that
leaf action alone more than compensates for the production of
carbon dioxide, consuming about one-hundredth of the total atmospheric
oxygen in a year. Yost7
computes that if the entire land
area were planted with sun flowers, about 6.5 X 1011
tons of CO2
would be absorbed per annum. Forests would be considerably
less efficient, and would take care of about 2.8 X 1010
tons per annum,
or, say in round numbers one-hundredth of the atmospheric
carbon dioxide. An older estimate, by Liebig, puts the annual
output of the soil of Central Europe at 2.5 tons of dry organic matter
per hectar, or, say 1 ton per acre. Allowing 40 per cent carbon
in such organic matter we find for [the total annual production
of carbon in plants 13,000 million (1.3 X 1010
) tons. This is
about ten times the world's annual coal consumption, and about
one-fiftieth of the total carbon in atmospheric carbon dioxide.
Arrhenius points out that if all this carbon fixed by plants were
deposited in peat bogs, the atmosphere would be depleted in half
a century. But, of course, only a small proportion of the bodies
of plants are thus "horded" and removed from the organic life cycle.
A figure of perhaps greater interest, because based upon observations
of processes actually going on in a selected portion of the
universe, is given by W. R. G. Atkins. From the observed change in
the hydrogen ion concentration in the water of the English Channel,
this author has calculated that 250 metric tons of organic carbon
(figured as hexose) were produced per square kilometer between
July and December. From similar observations made at Port
Erin, Moore found a production of 300 metric tons per square
kilometer during the six months that included the vernal maximum,
of diatom production.
8
An important item among the withdrawals from the atmosphere
is the absorption of carbon dioxide in the weathering of rocks, with
replacement of silicates by carbonates. A. G. Hogborn reckons this
item as about balancing the reproduction of carbon dioxide in the
F. W. Clarke, loc. oit., p. 48; Phil. Mag., 5th ser., vol. 1882, p. 387.
7
Pflanzenphysiologie, 1913, p. 151.
8
Journal of Marine Biol. Assoc. October, 1922, vol. 12, no. 4, Nature,
January 27, 1923, p. 132.
224 ELEMENTS OP PHYSICAL BIOLOGY
combustion of coal. Its accumulated record is seen in the sedimentary
rocks, which, according to F. W. Clarke,
9
contain 30,000
times as much as C02 as are today present in the atmosphere.
10
Sterry Hunt "illustrates the effect of weathering by the statement
that the production from orthoclase of a layer of kaolin (china clay)
500 meters thick and completely enveloping the globe would consume
21 times the amount of C02 now present in the atmosphere."
Chamberlin and others estimate that it would take about 10,000
years to consume the present amount of atmospheric C02 by the
weathering of rocks. Loss of C02 by peat formation may be estimated
at the same figure. The formation of COa by the burning
of coal would, according to these estimates, cover the loss by weathering
and peat formation combined, seven times over.
Finally, there is the great reservoir of the ocea, in equilibrium
with the atmosphere, and absorbing, under present conditions, some
18 to 25 times as much C02 as it leaves in the air above it. Thus
the sea acts as a vast equalizer; of every ton of C02 thrown into
the atmosphere by volcanoes, or by coal fires, for example, the
ocean ultimately receives directly or indirectly, about 1900 pounds,
only the balance of 100 pounds remaining in the atmosphere. It
is thus seen that even extensive contributions from the lithosphore
have but a slight effect upon the atmospheric store, and fluctuations
are in this way ironed out and moderated. Arrhenius11
points
out, moreover, that at the present time the carbon dioxide content
of the air over the ocean is on an average 10 per cent lower than
over land. From this, and the fact that generation of C02 by coal
(and probably also from volcanoes) has in late years been increasing,
he concludes that the air is, at the present epoch, becoming
richer in this gas.
8
Loc. cit., p. 48. See also C. Shuchert, The Evolution of the Earth and
its Inhabitants (Yale Press, 1919), p. 52.
10
Hogbom's figure, quoted by Arrhenius (Worlds in the Making, 1908,
p. 54), is 25,000 for limestones and dolomites. Am. Jour. ScL, 3d ser, 1880,
vol. 19, p. 349. Clarke, loc. cit, p. 48.
11
Loc. cit., p. 54. Arrhenius' figure differs somewhat from the one given
above. He supposes that the sea takes up five-sixths of the COZ thrown
into the air, i.e., 1 ton would yield up 1667 pounds to the ocean, leaving 333
pounds in the air.
ft
CARBON DIOXIDE CYCLE 225
As to which side of the account shows a net balance, in the carbon
cycle, we have no certain knowledge. Arrhenius builds his conception
of the future industrial development of our race on the
expectation that the atmosphere is gaming in carbon dioxide, under
the present regime of "evaporating" our coal mines, as it were
into the air. On the other hand, if our atmospheric C02 is of
volcanic origin, and the balance is maintained today with the aid
of discharge from the lithosphere, then the ultimate extinction of
the earth's plutonic fires would bring in its train the depletion of the
atmosphere and secondarily the extinction of life. "The cessation
of volcanism would signify the end of life on the globe." A similar
position is taken by C. Schuchert,
12
who further remarks:
We should add that if there were again as much life as there is at present,
all the carbon of the atmosphere would be in the living plants and animals,
and, if such a condition were possible death would come to them all ....
Life and its abundance at any time are conditioned by the amount of this
gas (COt) present in the atmosphere.
This remark of Schuchert's is suggestive as illustrating in concrete
manner the relative amounts of carbon concerned in the life balance.
It is, perhaps, somewhat misleading in making no mention of the
equalizing influence of the ocean which has been noted above. In
point of fact, if all the C02 in the air were withdrawn, a nearly equal
amount would rise from the ocean to take its place.
A summary, in graphic form, of the principal relations in the
carbon cycle noted in the preceding paragraphs, will be found in
figure 47, which should aid in giving a comprehensive picture of
the situation. <
;
The Oxygen Cycle. The organic oxygen cycle is, of course,,
directly related to the carbon cycle, although other features also
enter into operation in regulating the oxygen balance of the atmosphere.
The complementary relation between animals (essentially
oxidizors of carbon) and plants (essentially reducers of carbon dioxide)
is indeed a biological fact of fundamental importance at the
present stage of evolution. But if we look back through the vista
of ages, to the time before the advent of life, such as
we^know it,
our curiosity is aroused as to the origin of the atmospheric carbon
i
The Evolution of the Earth and its Inhabitants, 1919, p. 52.
226 ELEMENTS OP PHYSICAL BIOLOGY
H
O
o
NA
B
W
w
tJ
W
ft
o
o
e
!
t-3
P
O
M
O
(JABEON DIOXIDE CYCLE 227
dioxide and oxygen. Various views have been upheld on this
subject. F. W. Clarke13
remarks:
It is likely that carbon dioxide has been added to the atmosphere by volcanic
agency, in some such manner as this: Primitive carbon, like the graphite
found in meteorites, a,t temperature no greater than that of molten lava,
reduced the magnetite of igneous rocks to metallic iron, such as is found in
many basalts, and was itself thereby oxidized. Then, discharged into the
atmosphere as dioxide,, it became subject to the familiar reactions which
restored it to the lithowphere a,s coal or limestone.
Arrhonius,
14
referring to Koehno's reflections on this subject,
points out that the atmosphere contains about 1.2 X 101
!
5
tons of
oxygen, an amount which roughly*
6
corresponds with the mass of
fossil coal in the sedimentary rocks. "The supposition appears
natural, therefore, that all the oxygen of the air may have been
formed at the expense of atmospheric carbon dioxide. Probably
all the oxygen of the air owes its existence to plant life."
F. W. Clarke resumes the views of a number of investigators as
follows:
10
C. J. Koehne assumed that the primitive atmosphere contained no free
oxygon, and he has boon followed by T. L. Plupson,
17
J. Lemberg,
18
J. Steven-
son,
10 and Lord Kelvin." Lomberg and Kelvin, however, do not go to extremes,
but admit that possibly Homo free oxygen was present even in the
earliest times. Lemberg argued that the primeval atmosphere contained
chiefly hydrogen, nitrogen, volatile chlorides, and carbon compounds; the
oxygen which is now free, being Mum united with carbon and iron. The
13
F. W. Clarke, loc. oil., p. 55.
"Worlds iu the Making, 1908, p. 58.
111
The correspondence is rather distant if we accept Engler's estimate of
the world's coal reserves, namely, !J X 10 1S
tons, containing 75 per cent carbon..
Hoc Lin ok, loo. oil;., p. 37, Kngler'H estimate is probably low. The
World Altnanao, 1921, p. 201., given 7.5 X H) n!
tons. This would correspond
to 1.5 X 10 13
tons oxygon, as against 1.2 X ID 11'
tons in the atmosphere. It
is true that these ostinmteM of coal reserves cover only such coal as it would
pay to mine.
F. W. Clarke, loc. oil;., 1021, p. 50.
17 Chom. News, ISO.'J, vol. 07, p. i:?5. Also several notes in vols. 68, 69,
and 70. For Koohno'n work wee PhipHou'n papers, 1893-1894.
"Zeitsohr. Deutsoh, gool. (lenell, 1888, vol. 40, pp. 030-634.
I'PhiloH. Mag., 1900, 5th ser., vol. 50, pp. 312-,'JOO; Oth ser., 1902, vol. 4;
p. 435; 1905, vol. 9, p. MS; 1900, vol. 11, p. 220.
*
Ibid., 1899, 5th Her., vol. 47, pp. 85 89.
228 ELEMENTS OF PHYSICAL BIOLOGY
liberation of oxygen began with the appearance of low forms of plant life,
possibly reached a maximum in Carboniferous time, and has since diminished.
Stevenson's argument is much more elaborate, and starts with an estimate
of the uncombined carbon now existent in the sedimentary formations. In
the deposition of that carbon, oxygen was liberated, and from data of this
kind it is argued that the atmospheric supply of oxygen is steadily increasing,
while that of carbon dioxide diminishes. The statement that no oxygen has
been found in the gases extracted from rocks is also adduced in favor of the
theory. First, an oxidized crust and no free oxygen in the air; then processes
of reduction coming into play ;
and at last the appearance of lower
forms of plants, which prepared the atmosphere to sustain animal life. The
arguments are ingenious, but to my mind they exemplify the result of attaching
excessive importance to one set of phenomena alone. It is not clear
that due account has been taken of the checks and balances which are actually
observed. At present the known losses of oxygen seem to exceed the
gains. For example, C. H. Smyth21
has estimated that the oxygen withdrawn
from the air by the change of ferrous to ferric compounds, and so
locked up in the sedimentary rocks, is equal to 68.8 per cent of the quantity
now present in the atmosphere.
G. Bunge
22
also supports the view that atmospheric oxygen is
continually diminishing, becoming bound by the ferrous oxide
resulting from the decomposition of silicates. Accumulation of
carbon dioxide in the atmosphere, at any rate under present conditions,
is also assumed by S. Arrhenius, as has already been remarked.23
21
Jour. Geology, 1905, vol. 13, p. 319.
22
Text book of Physiological and Pathological Chemistry, 1902, pp. 16-17.
23
See Jour. Franklin Inst., 1920. vol. 190, pp. 114-121.
CHAPTER XVIII
THE NITROGEN CYCLE
If the denuuul becomes inHiHteut enouKh, wo cannot doubt that methods
will be dovimul which mil Kiv us the desired results. To question that would
bo to admit thai; man haw nearod the culmination of his
evolutionary career
and is preparing to bequeath the iwwtery of the earth to hia successor whoever
that may be.' (7. W. Martin,
'
Natural Demand and Supply. The proportion of nitrogen to
carbon in the human body IB 1 :3, in the atmosphere it is 5,500: 1. A
human adult contains in IUH body about 42 pounds of nitrogen, and
over 50 pounds of curium. Over every square foot of the earth's
surface rifles a column containing Homo- .1500 pounds of nitrogen, and
only about. 1/4 pound of carbon. The demand and the supply of
these two element appear, therefore, at first sight, to be altogether
out of all proportion favorable to nitrogen. Yet, in point of fact, the
practical problem of Heeuring an adequate supply for the substance
and expansion of life w incomparably more complex in the case of
nitrogen than in the cawo of carbon. The reason for this somewhat
remarkable, inversion in to bo noon in the fact that nitrogen is readily
aecoHsibio an food for living organisms only when it occurs in certain
chemical combinations, and nitrogen thus combined is far from
plentiful. It has I een estimated by T. II. Norton that this available
or "nomadic." nitrogen -i.e., that which taken part in the migration
through the organic, cycle -amounts to only about two one-millionths
of the total nif.rogen of the atmosphere, or, way, to about 8 X 109
tons. In fact, njUrogon w today probably the chief of those limiting
factors' which, in accordance with Liobig's law of the minimum,
establish the bounds for the extreme expansion of living matter upon
the earth. At the name time the circumstance of the chemical
idiosyncrasies of the element nitrogen introduces a certain complexity
into the nitrogen cycle, which strikes the eye at a glance in
the charts, figures -18 and 40, exhibiting the essentials of the nitrogen
1
Compare ("}. Bunge, I'liymolotfioal and Pathological Chemistry, 1902,
p, 17; Grinmsll J<uu, Qu. Jour. lOconomioH, 1920, vol. 34, p. 394.
220
230 ELEMENTS OF PHYSICAL BIOLOGY
NITEOGEN CYCLE
231
'Nitrogenous
Vegetable.
\tmosf>harit\
,
Nftrogan
Nitrogenous
Animal
Compounds!
Urea
Ammonia
Kl<!, '!!), OltOANIC NlTHOUIW CntOULATION
". -18 H!H>WH in hrondor outline the main feature of the
oycki, while figure -ID cx!nl)i(,,s in greater detail especially those stages
in the eyde UuU. am muni, iudijuately aKsociated with life agencies.
2
''
For dcituilH (in UUH ])linH<> of the milijocfc the reader must be i-oferred
to Mio Hpocutl liUu'iil.iinv A. good Hiinunary, fairly detailed and complete,
yot coticiHO, will IHI found in [{ llubor, Zur Sfcickstoff-Frage, Born 1908
pp. 1-10.
232 ELEMENTS OF PHYSICAL BIOLOGY
Gate of Entry into Nitrogen Cycle, The natural gateway for
the entry of atmospheric, elementary nitrogen, into the organic
cycle is a narrow one. So far as at present known, only a limited
class of organisms possess the faculty of "fixing" this element, that
is, taking it in its gaseous state from the atmosphere and converting
it into the condensed (liquid or solid) form, in which only it has common
acceptation as coin of the organic realm. The organisms known
to take part in this natural process of nitrogen fixation are three,
namely certain bacteria having their habitat in the soil; certain
leguminous plants (peas, beans, clover, alfalfa), working in conjunction,
in symbiosis, with nitrogen-fixing bacteria lodged in tubercles
upon their roots; and, thirdly, the wheat plant has recently been
shown to possess the independent faculty of assimilating nitrogen
from the air. This recent demonstration,
3
of course, suggests the
ready question whether after all, a number of other plants may not be
similarly endowed. This remains as a matter for further investigation,
but it is improbable that we shall have occasion to change
materially our present impression, namely, that the natural avenues
by which elementary nitrogen gains admission from the atmosphere
into the cycle of life are rather narrowly restricted. As to other
avenues opened up by the man, these will be considered presently.
Leak of Nitrogen out of Circulation. While there is thus a
narrowly restricted class of vegetable organism through which
nitrogen trickles in a thin stream from the elementary supply in the
atmosphere into the life cycle, the majority of plants derive the
supply for their biological needs from ready formed nomadic nitrogen
in the soil, that is to say nitrogen combined in the form of
ammonium salts, nitrites and nitrates. These substances are subject
to oxidation and reduction in the soil under the influence of
various bacteria, as indicated in the chart figure 49. The result of
these changes is that a certain fraction of the nomadic nitrogen is
continually leaking out of the circulation and joins the general
reservoir of free nitrogen in the atmosphere. This is only one of a
number of items on the losing side of the balance sheet. Other items
will be found in following up the details of the two charts already
3
C. B. Lipman and J. K". Taylor, Science, November 24, 1922, p. 605. These
authors report that in their experiments wheat plants assimilated 13 to 21
per cent of their nitrogen content from the air.
NITROGEN CYCLE 233
referred to. There is loss in the autumnal leaf fall of deciduous
plants; in the decay of dead plants, or in their fossilization as peat,
lignite and coal. Forest fires, and the burning of wood and coal;
the distillation of coal for illuminating gas; the oxidation of coal in
metallurgical furnaces, the coking of coal in ovens of the so-called
beehive type; all these are operations in which a greater or less proportion
of the combined nitrogen in the coal is liberated into the air
in the free, unavailable form. In view of the great loss which this
represents in the economy of our food resources, the highest importance
attaches to the modern drift away from the beehive coke oven
to the by-product oven, in which the major part of the nitrogen in
the coal is recovered as ammonia.
TABLE 22
Fraction, of total output of coke in the United States, produced in by-product ovens
These by-product ovens were introduced in 1893. It is estimated
that during the period from 1893 to 1910 alone, through the cont
Limed use of the old beehive type coke oven, over 9,300,000 tons of
ammonium sulphate were wasted, representing, at the prices then
prevailing, a value of 553 million dollars. In addition to this must
be reckoned a further loss in the resulting field crops. Had all the
nitrogen wasted in the beehive ovens been spread as fertilizer on the
field, this would have increased the crops some 20 per cent.4
An
idea of the extent arid significance of the healthy modern drift towards
replacement of the beehive by the recovery coke oven may be
gathered from table 22, reproduced from an article by Grinnell Jones
in the Quarterly Journal of Economics, 1920, vol. 34, p. 402. H. E.
Fischer, writing in the Journal of the Franklin Institute, 1920, vol.
190, p. 191, remarks that if all the coal in the United States were used
as coke, and the ammonia recovered in the process, this alone would
4
This and other data regarding nitrogen losses here set forth are drawn,
largely, from an article by J. D. Pemiock in the Journal of Industrial and
Engineering Chemistry, 1911.
234 ELEMENTS OF PHYSICAL BIOLOGY
furnish one million tons of ammonia, which corresponds to about
one-half the world's total production of nitrogen compounds.
Returning to the chart (fig. 48) and continuing to trace the progress
of nitrogen in the organic cycle, we note next that combined
nitrogen is absorbed from plants by animals in their food. It is
rejected from the animal economy in part as excretory matter
(manure, etc.), in part in the bodies of dead animals, in so far as these
are not themselves consumed as food. A large item here, in the
economics of the human community, is the refuse from slaughter
houses. Certain portions of this are recovered for various uses
(glue, leather, etc.). Some is made into fertilizer. Much of it
goes to waste, and thus gives opportunity for another leak of nitrogen
out of the life cycle to the elementary form in the atmosphere. It is
difficult to form any estimate of the extent of this loss, but there can
be no doubt that it represents a waste of hundreds of tons of nitrogen
daily. Much also is lost from the other item of animal waste materials,
not a little of the loss being occasioned by modern methods of
sewage disposal in large cities. Such methods represent, from the
standpoint of agricultural economy, a luxury, which however, will
be thought worth the price if the means are at hand to make good
the loss from other sources. Those portions of animal refuse which
are placed on the soil of crop-bearing fields and pastures return, at
least in part, into the organic cycle.
Accessory Sources of Combined Nitrogen. It is of course absolutely
essential for the continuance of the life cycle that the losses
of combined nitrogen which have been noted should in some way be
compensated by equal or greater accessions to the total amount of
nomadic nitrogen. One source of such compensating revenue has
already been noted, namely the direct assimilation of elementary
nitrogen from the atmosphere by a narrowly restricted class of plants.
There are two other natural sources. Volcanoes and furnaroles
belch notable quantities of ammonium chloride into the air; nitric
acid is also formed by the action of lightning, while ammonia is
produced by the passage of the silent electric discharge (aurora)
through the atmosphere. Arrhenius5
estimates that the amount of
nitrogen annually bound in this way amounts to about 1.4 X 109
B
S. Arrhenius, Worlds in the Making, 1908, p. 144. See also Haber,
Zeitschr. f.
Angev. Chemie, 1910, p. 685.
NITEOGEN CYCLE 235
metric tons, or one part in 3 millions of the total atmospheric nitrogen.
The products are washetl into the soil by the descending rain, together
with more or less of the same substances that have escaped
into the atmosphere from the ground and are thus restored to the
soil. Estimates which have been made of the quantities involved
3.6 I
.8
O
oo
IB30 6O TO WOO /OSO 90
YEAR.
FIG. 50. THE RISE OF THE SALTPETER INDUSTKY
SO 30
are somewhat conflicting, but on the whole the gains of the soil in
this way are held to exceed its losses.6
The stages and agencies so far reviewed may be collectively designated
as those constituting the "natural nitrogen cycle," as distinguished
from a group now to be considered, which are charac-
6
F. W. Clarke, loc. cit., 1920, p. 52; Linck, Kreislaufvorgange, 1912, pp.
6-7. It has also been put forward (Sch.dn.bcin) that a certain amount of
nitric acid is formed in the evaporation of moisture from the earth (Bunge
Physiological Chemistry, 1902, p. 11). But this is doubted by Ostwald.
(Grundlinien der Anorganischen Chemie, 1912, p. 384.)
236 ELEMENTS OF PHYSICAL BIOLOGY
terized by human interference with the course of nature. It Is
hardly necessary to point out that such a distinction between natural
and artificial agencies is merely a convenient use of brief terms; the
fact must never be lost sight of that man himself is very essentially
part of nature, and that his development, whether physiological,
psychological, sociological, economic, or what not, is part of the
great process of nature.
Human Interference in Nitrogen Cycle. Man's earliest conscious,
purposive intervention in the nitrogen cycle dates from
antiquity, and primarily consisted merely in taking more or less
pains, in an empirical way, that the nitrogenous waste material of
animal economy be, as far as possible, restored to the soil. Perhaps
the first recorded use of fertilizers not derived from current wastes
of domestic animals is the exploitation of guano by the Incas, which
dates from antiquity, and was brought to the notice of Europe by
de la Vega in 1604. It seems to have aroused no interest until
attention was again drawn to it two hundred years later by von
Humboldt and by Justus Liebig, and large scale importations of
guano into Europe began soon after this. A greater event in the
history of agriculture was the opening up, in 1831, of the Chilean
nitre beds. Without this source of saltpeter the modern development
of intensive agriculture, and the consequent growth of population
in all civilized countries, would have been at the least greatly
hampered. The rapid rise of the saltpeter industry is clearly exhibited
in table 23 and the corresponding graph figure 50.
While our chief interest here is in the agricultural use of Chile
saltpeter, its consumption in the industries is altogether too extensive
to be passed by without mention. J. D. Pennock7
gives the figures
shown in table 24 for the relative amounts of saltpeter consumed in
different uses.
It should be observed that Pennock's figures relate to peace time
conditions. Even so, nearly one-half the consumption is taken up
in the manufacture of explosives. Nitrogen thus employed is, of
course, lost to the life cycle. A certain loss also concurs in the refining
of the caliche (native saltpeter), and in the production of
nitric acid and sulphuric acid8
therefrom.
7
J. D. Pennock, Jour. Industr. and Eng. Chem., 1911, p. 172.
8
When manufactured by the Chamber process.
NITROGEN CYCLE 237
TABLE 23
Growth of the saltpeter industry
(III)
Nitrogon on basin of
15.05 por con I
17.5
3,498
40,230
156,339
233,334
287,207
410,095
478,230
424,900
302,850
502,370
519,570
496, 120
The figures in column I are the Chilean nitrate production, as given by Parsons
and Petit, Brokers. The figures in oolxims 11 and III, from 1831 to 19 11
inclusive, represent the World's consumption of saltpeter according to C!(5ni
Civil, vol. 62, p. 192. The figures from 1913 to 1918 in oolums II and II I ropi'eserit
the total production of Chilean and Indian nitrate, according to Griunell
Jones, Jour. Frankl. Inst., 1920, vol. 134, p. 398. The precipitous drop in the
Chilean production in 1915 was due to a blockade established by the GermariH
during the early stages of the, World War. See Fig. 50, in which circles indicate
production, the drawn out curve consumption.
TABLE 2t
Distribution of Chili saltpeter consumption in the United States in 1010 among
different uses
In manufacture of fertilizer.
In manufacture of dyestuffs.
In general chemistry
In glass
In explosives
In nitric acid
In sulphuric acid
Unaccounted for
13
12
10
4
41
9
100
238 ELEMENTS OF PHYSICAL BIOLOGY
Origin of Nitre Beds. The origin of the nitre beds is uncertain.
The presence of boron in the deposits, and the association of this
element with ammonia in volcanic emanations, have been regarded
by some as evidence that the saltpeter is of ultimately volcanic origin.
Others have ascribed it to organic sources, such as altered guano
deposits. But whatever be their origin, this is certain, that the saltpeter
beds represent an accumulation of ages, and that the present
rapid rate of consumption is out of all proportion with the rate of
formation of the deposits. In other words, here, as in the case of
coal, we are living on our capital, and must prepare ourselves for its
impending exhaustion.9
It is true that our other sources of combined
nitrogen notably ammonia from coke ovens supplement our
drafts upon the nitre beds, and thus help to defer the day of scarcity.
But coal itself is a limited stock, and other sources of combined
nitrogen seem quite inadequate for the needs which we have developed
under the stimulation of temporarily bountiful supplies.
It is a peculiarity of living substance that in times of plenty it tends
to grow beyond the bounds compatible with ultimate stability; it
overshoots the mark so to speak; the curve along which it approaches
its equilibrium is very apt to be humpbacked, or it may be oscil-
latory.
10
The prospect of a period of actual diminution (not mere
marking time) to follow upon a period of exuberant prosperity, is
one that an organism gifted with foresight must look upon with
disquietude. Such foresight may, then, lead an organism so gifted,
to make efforts to provide for untoward future exigencies, either by
laying by supplies, in times of plenty, for times of stress; or by
devising means, if possible, to increase, by new measures, the supplies
which, under the old regime, would presently fall short of requirements.
Man has not always made a display of brilliant foresight,
but in this instance, in making ready for the exhaustion of the nitrate
supply, he has taken time by the forelock, and all indications are
that long before the emergency arises he will have made himself
_
9
The probable date of this exhaustion has been variously estimated.
Little value can be attached to positive statements. More significant, perhaps,
is the negative report made in 1913 to the Chilean Government by the
Inspector General of Nitrate Deposits: "There is no fear of the Chilean
nitrate deposits being exhausted for two hundred years" (Grinncll Jones
loc. cit., p. 401).
'
"Compare what has been said on this subject in Chapter XI.
NITROGEN CYCLE 239
ready to meet it. For the last decade has seen the development, to
full industrial capacity, of several processes for the fixation of atmospheric
nitrogen, its conversion into compounds directly or indirectly
adapted to enter the cycle of nomadic nitrogen. For
details regarding these modern industrial developments the reader
must be referred to the technical literature.
11
It must suffice to
indicate here very briefly the nature of the several processes.
1. The Birkeland and Eyde Process, is essentially man's
imitation of the production of nitric acid by lightning. Air is
passed through an electric arc fanned out into a broad disc by a
magnetic field. The process is commercially viable only where
very cheap power is available, and has been developed mainly in
Scandinavia, with the use of water power.
2. The Cyanamide (Frank and Caro) Process effects the absorption
of atmospheric nitrogen by calcium carbide in the electric
furnace. The product can be employed directly as a fertilizer, or can
be made to yield ammonia and other nitrogen compounds.
3. The Haber Process effects the synthesis of ammonia from
nitrogen and hydrogen under pressure (100 to 200 atmospheres)
in the presence of a catalyst. An allied process is that of Claude,
which works at very high pressure (1000 atmospheres).
ia
4. The Biicher Process, which has not yet passed beyond the
experimental stage, yields cyanides; these can also, if the market
warrants it, be made a source of ammonia.
A highly significant development is the union of the Habcr
ammonia process with the Solvay process, whereby the carbon
dioxide obtained as a waste product in the manufacture of the
hydrogen for ammonia synthesis, is utilized in the production of
sodium carbonate; while, on the other hand, the formation of large
quantities of the nearly worthless calcium chloride waste of the
Solvay process, as ordinarily conducted, is avoided. Inasmuch us
"soda ranks second only to sulphuric acid among all chemicals in
magnitude of output (1,390,628 short tons in the United States in
1918) and fundamental importance, .... this (combination
of the Haber and the Solvay processes) may well prove to be the most
11
Sec for example the articles by G. H. Fischer and Grinnell Jones already
cited.
12
H. E. Fisher, Jour. Franklin lust., 1920, vol. 190, p. 201.
240 ELEMENTS OF PHYSICAL BIOLOGY
1,6
FIG. 51. THE RISE OF THE FIXED NITROGEN INDUSTRY
NITROGEN CYCLE 241
significant development in industrial chemistry of the present
decade."13
5. The Ostwald Process, A subsidiary process bridging the
gap from the product (ammonia) of the processes of the Haber and
Frank and Garo types, to the market requirements of nitric, acid
(nitrate and nitrites), is the Ostwald process for the catalytic oxidation
of ammonia.
The Meteoric Rise of Nitrogen Fixation Industries. In the period
during arid immediately following the World War the situation
in the nitrogen industries was abnormal, production being temporarily
activated to fever heat, inasmuch as nitrogen compounds
are among the most indispensable of war materials. Tims it came
about that the development of nitrogen fixation had for its immediate
motive not so much the constructive spirit of the arts of
peace, providing for the future needs of men, as the malice and
forethought of the conspirators of war. The conflict left on our
hands, upon the conclusion of the armistice, both completed and
unfinished manufacturing plants in excess of immediate needs in
times of peace. Legislative difficulties also hampered well-designed
efforts to convert these plants to industrial use. These arc temporary
conditions; although one may not be able to foresee exactly
how, in detail, these industries will finally adjust themselves, there,
can be little doubt that from now on synthetic nitrogen compounds
will continue to be drawn in increasing amounts from the atmosphere
into the life cycle. The phenomenal growth of this infant industry
within the past ten or twelve years is forcibly brought out in the
graph (fig. 51) and the corresponding table 25. It will bo observed
that in 1909 only about 1 per cent of the world's needs in combined
nitrogen were satisfied from the new-born industry. In .1.917 its
contribution had swollen to 30 per cent, and by 1920 the capacity of
existing plants was adequate to furnish 43 per cent of the world'n
requirements.
This extraordinary development is something much more than a
fundamental new departure in industry. It represents nothing lens
than the ushering in of a new ethnological era in the history of the
human race, a new cosmic epoch. In the short span of a dozen
years 'geologically speaking in an instant- man has initiated
transformations literally comparable in magnitude with cosmic
13
Grinnell Jones, loc. cit., p. 414.
242 ELEMENTS OF PHYSICAL BIOLOGY
NITEOGBN CYCLE 243
processes. Accepting Arrhenius' liberal estimate of the total quantity
of combined nitrogen washed down to the soil in the annual
rain fall over all continents of the globe, namely 400 million tons,
it is seen that the new industry even now is capable of furnishing a
supplementary supply equal to one six-hundredth of this prodigious
quantity.
14
Economic and Energetic Significance of Concentration. We are,
of course, greatly more interested in this six-hundredth which is
under our control 'and of which a due proportion falls, in consequence,
upon our fields in concentrated form 'than in the very
much larger quantity that nature scatters with sublime indifference
on stony places and good soil alike. This is just one of those cases
to which reference has already been made in a general way. It is
not so much the quantity of the material provided by nature that
counts, as its accessibility; and accessibility here means, among other
things, suitable concentration. This question did not so obviously
project itself into the discussion of the carbon cycle, because the
__
natural source of carbon is atmospheric carbon dioxide, which, being
a gas, in the very nature of things spreads evenly, and presents itself
unsought, by a spontaneous process, at the mouth of the hungry
plant. But the combined forms of nitrogen are for the most part
solids or solutions, occurring in definitely localized amounts of greatly
varying concentration. The labors forced upon us in our efforts to
satisfy the nitrogen needs of our fields are, to be precise, not primarily
work of production, but virtually work of concentration; or to be more
exact, work of bringing about concentration at the particular locality
where it is wanted by transportation if need be. It is only because
we find k easier, in some instances, to produce than to concentrate
existing supplies, that we elect the former expedient; just as we may
prefer to feed a boiler with a fresh supply of water, rather than to
u For land and sea together Arrhenius estimates 1500 million tons of nitrogen
in the rain fall.
Note added in correcting
1
proof.
Since the writing of this chapter there have become available Trade Information
Bulletins No. 220 and 240 of the U. S. Department of Commerce,
which give further and more recent data. In Bulletin 240 J. M. Braham
estimates the capacity of the world's nitrogen fixation plants at 490,000
tons for 1923. The reader interested in this topic may also consult J. R.
Partington, The Nitrogen Industry, publiBhcd by Constable, 1924.
244 ELEMENTS OF PHYSICAL BIOLOGY
return to it the condensed exhaust from the engine. That the mere
concentration of existing supplies should at all require the doing of
physical work is a circumstance of particular interest, not only in its
economic relations,
13
but also, and quite particularly, from the standpoint
of energetics. This is a matter that will duly engage our
attention in a later section, devoted to the energetics of the several
processes that have here been considered in their purely material
or stoiehiometi'ic aspect.
Meanwhile it is interesting to observe that such localized sources
of concentrated supplies as those presented in the Chilean nitre
beds virtually function as centers of attraction toward which gravitates
a stream of human beings 'Or their representatives in the form
of ships and other conveyances arriving in search of cargo and going
out laden with material. To an ultramundane observer who should
survey the scene in suitable perspective, the activities around the
nitre beds must appear very like the busy swarming of a colony of
ants around the treasure trove of some silvan inhabitant departed
this life; who, having completed his earthly career, is now yielding up,
in the dissolution of death, such energies as still remain locked up
in the carcass. Attractions such as this are, in a sense, merely
apparent; they are the outward symptoms of a complicated chain of
cause and effect10
characteristic of the behavior of living organisms.
Yet they are often so consistent in their action that it would not be
unreasonable to essay a systematic treatment of the movements in
a world comprising such centers of attraction and such moving pawns,
on the basis of brute tropisms unanalyaed into their ultimate component
agencies. Here it must suffice to have pointed out that our
highly complex industrial system, our far-flung intricate network
of lines of traffic by land and sea, is but a sublimated copy, on a
heroic scale, of the hustle and bustle that is going on all around us in
nature, in response to attractions, tropisms, determinants of the
15
A striking illustration of this is cited by Haber (Zeitschr. f Angew
Chemie, 1910, I, p. 685). If the gold in sea water were extracted and" apportioned
evenly to all the human inhabitants of the globe, we should all
be millionaires three times over; yet it does not pay to as much as begin this
extraction.
_
15
Commonly accompanied by that anticipatory inversion of the sequencein time, of effect and cause, which is the earmark of purposive action.
NITHOGEN CYCLE 245
moves of an army of checkers over the mosaic of the earth's
topography.
17
Total Circulation Tends to Increase. The study of the
nitrogen cycle furnishes us with a first occasion to take note of a
phenomenon the full significance of which will become apparent in
dealing with the dynamics of evolving systems. It is to be observed
that the general trend of man's effort, especially in this new epoch
of nitrogen fixation, has been towards drawing into the organic
circulation a greater amount of matter, enlarging the wheel of the
mill of life, so to speak. There can be little doubt that this trend
will continue and increase in the future, and that it is the expression
of one aspect of a general law;
18
the other aspect of this law, and its
full significance, must be reserved for later discussion, as already
intimated.
17
For a discussion of the natural concentrating processes the reader may
be referred to the following articles: A. C. Lane, Nature's Coneentraiors,
Engineering and Mining Journal, 1897, vol. 03, p. 54-2. ,1. C. Russell, Concentration
as a Geological Principle, Bulletin Ccol. 8oc. An., 1907, vol. IS,
pp. 1-28. E. Blackweldcr, The Geologic Role of Phosphorus, Aia. Jour.
Sci., 1910, vol. 62, p. 285. W. Lindgren, Concentration and Circulation of
the Elements from the Standpoint of Economic Geology, Economic Geology,
1923, vol. IS, pp. 419-44.2.
18
The phenomenon is, of course, closely related to that In which R. Luscaux
refers as "unc propriete particuliere do la vie: cclle do rextentiion"
(La Production ct la Population 1921, p. 37). ftcc also A. J. Lotka, I'roc.
Nat 'I Acad. Sci., 1922, p. 47.
CHAPTER XIX
THE PHOSPHORUS CYCLE
There is no coming into being of aught that perishes, nor any end for it
.... but only mingling, and separation of what has been mingled.
Empedocles.
Immobile Elements. The circulation of water, carbon dioxide
and oxygen in nature is greatly assisted by the freely occurring
processes of evaporation, condensation (rainfall) and diffusion. In
the case of nitrogen these processes still give some aid, as in the distribution
broadcast of atmospheric nitrogen, and in the formation j.
and precipitation of nitrogen "fixed" by lightning, etc. In phosphorus
we have, on the contrary, a typical example of the inherently
immobile elements needful to the living organism to adopt Liebig's
phrase (fur sich nicht beweglicli) . The successive steps in the concentration,
diffusion, and reconcentration of this element, as available
for the substance of the organism, accordingly display a characteristic
complexity. Some of the principal items and steps in the phosphorus
cycle are set forth in diagrammatic form in figure 52.
Natural Phosphorus Supply of Soils. The virgin soil contains,
in general, a certain natural supply of phosphates. So, for example,
Van Hise1
reports that the virgin soil of Ohio, Illinois and Wisconsin
contained, to a depth of 8 inches, 2077 pounds of PaOg per acre.
/
After fifty-five years of cultivation this figure had sunk to 1813 pounds
per acre, a loss of 36 per cent. These figures show the extreme importance
of a conservation of our resources of phosphorus. The loss
probably occurs partly through erosion by rain water charged with
carbonic acid, which dissolves phosphates in the soil and in rocks,
and ultimately "washes a certain proportion out to sea. To this
unavoidable loss, however, is added a large item of preventable loss
through our failure to ensure that the phosphorus of animal wastes
be returned to the soil. So, for example, farmyard manure contains
over three-fourths of the phosphorus in the feed and bedding supplied
to the animals. It is therefore very essential that this be returned
1
Ann. Acad. Polit. Soc. Sci., 1909, vol. 33. . ,
246
PHOSPHORUS CYCLE 247
o
o
W
FM
248 ELEMENTS OF PHYSICAL BIOLOGY
to the soil as completely as possible. Of the one-fourth of the phosphorus
fed to the animals, and not accounted for in the manure, a
considerable fraction appears in the bodies of these animals, especially i
in the bones. These, then, also should find their way back to the
*
soil, as they do to some extent through the practice of using bone
meal, either as such, or after conversion into superphosphate,
2
as fertilizer. This practice is materially assisted by the modern
methods of meat production on a large scale, with very complete
utilization of by-products.
Leakage of Phosphorus from Circulation. The human cadaver
is in the great majority of cases returned to the soil in the
regular course of events, although under conditions which, obviously
do not render it very readily available for crop production. An
adult contains about 1| pounds of phosphorus, or 3.4 pounds IW
If we allow one-half of this for a "unit of population," i.e., 1.7 pounds 4
P2 6 ,
with a death rate of 1.3 per cent per annum, we find that the
^
amount of P2 5 annually committed to the cemeteries of the United
States is about 1105 tons. This is about the equivalent of 3300
tons of phosphate rock, or about one-thousandth of the annual
production of that material in the United States.
But a very much larger amount of waste is occasioned from the
human population by the practice of running the sewage from cities
into rivers and thus to sea. Van Hise estimates that annually
400,000 tons P2 5 or the equivalent of 1,200,000 tons of phosphate
rock are thus ran to waste. He remarks: "The wide dispersal of
the vast quantities of phosphorus which it took the process of nature
an indefinite period to segregate, must cease. The loss is irrop- Jj
arable." Much has been done in recent years to comply with this
demand.
Phosphate Rock and the Migration of Phosphorus. In the
meantime, in our phosphorus economy also, as in the case of the
combined nitrogen of the Chilean nitre beds, we are living on our
capital. For, as our fields tend to become depleted of phosphorus
under intensive agriculture, we restore some of the rarefied clement
to the tired soil by drawing upon the accumulations of ages, in the
form of phosphate rock. As a matter of fact, in this we are welding
2
By treatment with sulphuric acid, which renders the phosphorus more
readily available to plants. These are technical details that cannot be ontered
into here. The reader must be referred to the pertinent agricultural A-and technological literature.
PHOSPHOEUS CYCLE 249
the closing link in a very remarkable endless chain of nature. For
the phosphates washed by the rivers3
into the sea serve as food to the
marine vegetation and indirectly to the fishes and other aquatic
species.
4
These act in this case as concentrating agents, the bones
and teeth of fish, and some shells of Crustacea and molluscs, being
comparatively rich in phosphorus.
5
In its further migration the
3
For details the reader may be referred to an article by E. Blackwclder
in the Am. Jour. Sci., 1916, vol. 62, p. 285, from which the following particularly
pertinent passage may here be noted: "Of the vast quantity of dissolved
mineral matter annually delivered to the sea by the run-off, it is estimated
that about 0.45 per cent consists of phosphorus pentoxide. Using
the best available figures for the amount of water thus brought to the ocean
annually, it is calculated that if the phosphatic material in the form of solid
tri-calcium phosphate were loaded into standard railroad cars it would fill
a train stretching continuously from Boston to Seattle and would be 7 to 12
times as great as the world's total production of phosphate rock in 1911.
Nevertheless, so great is the volume of the oceans, and so vast the area of
their floors, that if all this material were deposited in solid form uniformly
over the bottom of the sea, it would build annually a layer less than 0.2 mm.
thick. Of the phosphorus poured into the sea, so large a proportion is utilized
by living beings that the net working balance dissolved in oceanic water
constantly averages less than 0.005 per cent, expressed as PzOs, or, in other
words about 0.18 per cent of the dissolved salts. In this solution, phosphorus
seems to have reached the most dilute state in which it exists during
the course of its complex migrations. Its subsequent transformations generally
tend to ever greater concentration, almost until the cycle is closed
upon itself.
4
Compare also W. Lindgren, Concentration and Circulation of the Elements
from the Standpoint of Economic Geology: "In the sea water the bluegreen
algae concentrate phosphorus, certain molluslcs or crustaceans feed
on the algae, and other meat-eating mollusks devour the vegetarians. Small
fishes eat the molluslcs, large fishes eat the small, finally seals and birds swallow
the fishes, and so in about six transformations the phosphorus originally
contained in the sea-water may come to rest in deposits of guano on desert
islands or in accumulations of bones of vertebrate denizens of the sea" (Economic
Geology, 1923, vol. 18, p. 431).
6
In this connection may be noted again a passage from Blackwelder's
article, p. 289 (see footnote 3): "As phosphorus ascends in the evolutionary
scale of animals, its concentration tends to increase, although irregularly.
The protozoan, air dried, contains less than 0.6 per cent PaOc. According
to Juday quantities of minute crustaceans from Lake Mendota contain in
the air-dried condition 1.8 to 2.4 per cent of PaGc, or several times that of
the protozoans. A Russian biochemist, Sempelovski, found in entire fresh
specimens of a cartilaginous fish (the common skate) 0.91 per cent PaOs,
whereas the average for eight Teleostean fishes with well-developed bones
was about 1.5 per cent. Certain brachiopods, such as those of the family
Lingulidae form shells of fibro-crystalline tricalcium phosphate probably
either the mineral dahllite or staffelite."
250 ELEMENTS OF PHYSICAL BIOLOGY
phosphorus here splits into two streams. On the one hand the hard
parts of dead fish and other sea animals fall to the bottom and form
a phosphatic deposit. So, for example, it is reported that in certain
localities a single draft of the dredge has brought up 1500 shark's
teeth from the sea bottom. These deposits become further enriched
through the replacement of their calcium carbonate by calcium
phosphate under the action of the sea water.6
Subsequently some
of the deposits so formed have been raised, in a crust upheaval,
above the sea level, so as to form sedimentary strata from which
we now derive some of our supplies of phosphate rock.
The other division of the stream in the flow of phosphorus is perhaps
one of the most remarkable examples of a cycle in the economy
of nature. The fish of the sea are eaten by birds, who flock in great
hordes and have their nesting places upon rocky islands and shores.7
There an accumulation of immense amounts of guano has taken
place in the course of centuries and ages. Of this guano some has
been returned directly to the land by the agency of man. Other
portions, of more ancient origin,, have undergone transformation,
and have passed into fossil form by reaction with the rock base on
which they were in the first instance deposited. This is the origin
of a second class of (metamorphic) phosphate rock, which also wo
mine and spread on our field, so that this loop in the chtiin also is
closed.
Soil Losses of Phosphorus. From this sketch of the migration
of phosphorus in nature it is seen that qualitatively the path of tho
element is a closed circulation. Unfortunately the cycle is quantitatively
quite incomplete. Van Hise, quoting Whitson's invest! gations,
shows that, on a very conservative estimate, tho soil of the
United States loses annually some 2 million tons of P2 G ,
the equivalent
of 6 million tons of phosphate rock. This is about double the
total output of our phosphate quarries, and about four times our
domestic consumption of that output. It is thus scon that tho
situation with regard to our supply of phosphatic fertilizers is an
exceedingly serious one it would perhaps not be out of place to say
For a detailed discussion of some of these concentrating processes and
related matters the reader is referred to F. W. Clarke, loc. cit., 1921, pp. 132
495, 502. See also Chapter XVIII, footnote 17.
^
7
For an excellent illustrated account of the part played by the Guanay
bird in this cycle see R. C. Murphy, Natl. Gcogr. Mag. Sept. 4, 1924, p. 279.
PHOSPHORUS CYCLE 251
ao alarming one. For here we have no reserve which we may hope
to find means of tapping in the future, such as is presented to us, in
the case of nitrogen, by the inexhaustible supply in the atmosphere.
Phosphorus is at best a comparatively rare element, constituting
only about 0.14 per cent of the earth's crust (see table 16,
Chapter XV).
The general and alarming decrease in the crop yield per acre in various
states so well described by Mr. James J. Hill,
8
is largely due to the depletion
of the soil in phosphorus The work (at the Ohio Agricultural
Experiment Station9
) upon different fertilizers shows that for the soils tested
in their experiments phosphorus was the controlling element in producing
an increase in the cereal crops.
10
And elsewhere11
Van Ilise remarks :
The average rock contains twenty times as much potassium as phosphorus.
Therefore, looking toward the distant future, if we consider ratios, we may
unhesitatingly assert that the problem of maintaining the fertility of the
soil in phosphorus will be twenty times as difficult as for potassium; but
this ratio by no means measures the real difference, for when a deposit contains
a moderate percentage of a substance it may be possible to utilize it
commercially, whereas, if the percentage falls below this amount it is without
value.
Phospbtatic Slag as Fertilizer. To the consideration of the
migration of the element potassium thus referred to by Van Hise,
we shall presently turn our attention. Before leaving the subject
of the migration of phosphorus, one secondaiy source of phosphate
fertilizer remains to be noted here, namely, the by-product (slag)
obtained in certain processes of steel manufacture, in which the
phosphorus contained in the ore (and very objectionable as a constituent
in steel) becomes segregated, and is thus eliminated, in the
slag. This latter, by suitable processes, is converted into a very
serviceable fertilizer.12
8
The Natural Wealth of the Land, and its Conservation. Paper given
at the White House Conservation Conference, May 13, 1908.
9
Ohio State Agr. Coll. Bull. 141, 182.
10
Van Hise, loc. cit., p. 704.
11
Idem, loc. cit., p. 701.
12
For details regarding the use of phosphatic slags as fertilizer the reader
may be referred to G. S. Eobertson, Basic Slag and Rock Phosphates, Cambridge
University Press, 1922.
CHAPTER XX
CYCLES: CONCLUSION AND SUMMARY
The saltness of the sea is due to the numerous springs of water, which in
penetrating the earth, find salt mines, and dissolving parts of these carry
them away with them to the ocean and to the other seas, from whence they
are never lifted by the clouds that produce the rivers. Leonardo da Vinci.
The Circulation of Chlorine and the Alkalis. It will be convenient
to consider jointly the migration of the elements chlorine,
sodium, and potassium in nature, inasmuch as they are closely
connected.
There is a piece of laboratory apparatus known as the Soxhlet
Extractor, of which the chemist makes use when he wants to prepare
a solution, an extract, of one of the constituents of a mixture
of substances. This apparatus, as represented in figure 53, operates
on a simple principle. The solvent (water, ether, petroleum spirit,
etc.) is placed in the flask A heated by a bimsen flame B, so as to
drive vapors of the solvent up through the tube C to the top of a
tubular vessel D containing the material M to be extracted. The
whole apparatus is open at the top, but escape of vapor is prevented
by a condenser tube E cooled by a water jacket F. The Vapor
condensing in E drips down upon the material M, and dissolves
out the substance to be extracted. The condensate accumulates in
the vessel D and rises in the syphon tube I until it reaches the top
of the syphon, whereupon it drains back into the flask A and is
reevaporated, this cycle being repeated indefinitely as long as desired.
Since the vapor of a liquid containing a non-volatile substance in
solution is pure solvent, the action continues until practically all
the soluble substance has been extracted.
It is almost literally true that we pass our lives in the midst of a
gigantic Soxhlet apparatus. The flask is the sea basin; the solvent
is the water of the ocean, the rain, and our rivers and lakes. The
material extracted is the earth's crust (rocks, soil, etc.). The place
of the Bunsen burner is taken by the sun which raises water vapor
from the ocean's surface, into clouds which drift over the land.
252
CYCLES: CONCLUSION AND SUMMARY 253
FIG. 53. THE SOXHLET EXTRACTION APPARATUS
The action of this apparatus is closely analogous to the natural extraction
of soluble constituents from the earth's crust by the water in circulation
through clouds, rain, rivers and the sea, under the influence of the
sun's heat.
254 ELEMENTS OF PHYSICAL BIOLOGY
CYCLES: CONCLUSION AND SXJMMAEY 255
The cold upper atmosphere acts as a condenser (see fig. 54), beyond
which no clouds can pass out above. Presently the moisture of the
clouds is precipitated as rain over the face of the earth. It drains
into rivers and lakes back into the sea, charged now with the soluble
constituents of rock and soil. This operation has been going on
over and over for ages, with the result that the greater part of the
more soluble constituents of the rocks is by this time collected
in the ocean, imparting to its water its characteristic salt taste.
It is worthy of more than passing note that these relations, in all
essentials, were recognized by that universal genius, Leonardo da
Vinci, whose remarks on this subject appear at the head of this
chapter.
The principal soluble constituents of the earth's crust are the
carbonates and chlorides of the alkali metals, sodium and potassium.
These, then, mainly take part in the extraction process described.
There is, however, an important difference in the behavior of sodium
and potassium in this process. In the igneous rocks sodium and
potassium are present in very nearly equal proportions (see table
16). Yet the ocean is very much richer in sodium than in potassium
(see table 14). What becomes of the potassium?
1
It is a circumstance
highly significant for terrestrial life that potassium salts seem
to be largely absorbed from their solutions on their passage through
soil and clay. Thus the soil would retain a supply of the element
so essential for plant growth, while the less vitally important sodium
'in other respects so similar to its next kin potassium has in
large part passed on into the ocean. 2
Nevertheless, potassium is not present in most soils in profusion;
under intensive agriculture the soil becomes impoverished in this
constituent also, and recourse must be had to sources of potassium
salts in concentrated form, the deposits left by the drying up of
ancient seas, to make up the deficiency. In this field, also, the
World War has materially affected the complexion of agricultural
economics. Its influence has been twofold. In the first place, since
1
In this connection, and for details regarding the circulation of Na, K,Cl,
and the related question of the age of the ocean, the reader may be referred
to F. W. Clarke, Data of Geochemistry, pp. 136, 137, 145, et seq.
2
These suppositions must be viewed with a certain caution, as has already
been pointed out; see p. 204, footnote 19.
256 ELEMENTS OF PHYSICAL BIOLOGY
the only areas highly productive of potassium salts were contained
within the domains controlled at that time by Germany, it became
necessary for the Allies to find other sources of the needed element.
One outcome of this was a temporary development of potash recovery
from the waste of cement works and other materials. Unfortunately
much of the commerce thus started had to be abandoned
again when the customary sources of potash became once more
available after the close of the war.
The second effect of the war upon the potash situation arises
from the political changes which it brought about. The Alsatian
potash deposits, formerly controlled by the same monopoly as the
German Stassfurt deposits, are now in French domains, and the
monopoly is broken.3
Some of the quantitative aspects of the migration of chlorine
and sodium are shown in figure 54.
The Circulation of Sulphur. Sulphur occurs in abundance in
inorganic nature in the form of the sulphates of the alkalies and
alkaline earths. Plants assimilate these compounds directly
and then form proteids containing about 0.3 to 2 per cent of sulphur.
In this form it is assimilated by animals, who excrete the
element chiefly in the form of sulphates. These, returned to the
soil, complete the cycle.
Tlie Circulation of Iron. The importance of iron in the economy
of the living organism is out of proportion to the comparatively
small amount of this element actually present in the body. Thus
the body of a human adult holds only about 4.5 grams of iron,
contained, for the most part, in the red blood corpuscles. The
importance of this comparatively small amount of the metal arises
out of the fact that it fulfills the essential function of an oxygen
carrier, a catalyst as it were, mediating the transfer of oxygen from
the air in the lungs to the tissues of the body through the blood
stream. Similarly, in plants, the iron is contained chiefly in the
chlorophyll, whose catalytic action is a fundamental condition for
the assimilation of carbon dioxide from the air. This catalytic
action of iron is attributable to the ease with which it passes from
ferrous to the ferric condition and vice versa, and plays a significant
3
G. Jones, Q. Jour. Econ., 1920, vol. 34, p. 392,
CYCLES: CONCLUSION AND SUMMARY 257
idle not only within the body of the organism, but also In the soil,
where it hastens the oxidation of carbonaceous matter, thus rendering
carbon once more available for the organic cycle.
4
Summary of Cycles. In conclusion of this chapter a few remarks
and tables regarding the circulation of the elements in general
may be offered.
In retrospect we may observe that alcharacteristic stamp is
placed on certain of the elementary cycles by the form in which
TABLE 26
Supply of plant foods in soil
Number of years the supply of several elements would last, utilizing soil
to a depth of 7 inches, and producing annually a crop of 100 bushels of com.
Average compositon of soil assumed equal to that of 2110 samples of common
rocks in the United States. (C. G. Hopkins, Annual Acad. Sci. 1909, vol. 33,
p. 638).
*
Assuming stalks returned to the land.
each element occurs or takes part in the circulation. Thus, the
gases oxygen and carbon dioxide occur in nearly uniform distribution,
so that their migration is free from certain complications that
arise in the case of the other elements. Water occupies a position
intermediate between the gaseous 'elementsjand^those which like
phosphorus, potash, etc., as solids, are subject to local segregation,
and thus introduce problems of transportation in one form or another.
As vapor, water drifts with air currents. But owing tn the phenomenon
of precipitation, water, unlike the permanent gases, is very
unevenly distributed, the supply available for life processes being
strictly a matter of climatic conditions. Thus, in desert regions
water functions as the limiting factor of life.
Nitrogen, although
4
G. Bunge, Physiological and Pathological Chemistry, 1902, p. 21.
258 ELEMENTS OF PHYSICAL BIOLOGY
gaseous in the elementary state, is chiefly operative in combined
forms, so that its distribution in available form, is also a locally
varied phenomenon.
All these facts have their influence not only on the primitive
flora and fauna as a function of geographic site, but play also an
important role in those secondary life phenomena which we commonly
describe as commerce and trade:.
Two tables 26, and 27, are, finally appended, the one giving certain
data of interest regarding the Supply of Plant Foods in the
TABLE 27
Rate of participation of the elements in cycles of nature
*
Authorities : A. = Arrhenius, Worlds in the Making, 1908 ;
C. = Clarke,
Data of Geochemistry, 1921; L = Linck, Kreislaufvorgange, 1912; M. = McGee,
Science, 1911, vol. 134. The numbers in the column "Authority" refer to
pages of the works cited.
Soil, according to C. G. Hopkins; the other giving estimates of the
Rate of Participation of the Elements in the Cycle of Nature.
It seems hardly necessary to point out that the quantitative
estimates cited in these chapters on the circulation of the elements
in nature represent only very rough approximations, the best perhaps
that can be attained in the present state of our knowledge.
As F. W. Clarke3
remarks, "Such estimates may have slight numerical
value, but they serve to show how vast and how important the
processes under consideration are." Rough as the data are, they
give us, presumably, at least an idea of the order of magnitudes
involved. The least that can be claimed for them is, in the words
of Clarke5
once more: "In calculations of this sort there is a certain
fascination, but their chief merit seems to lie in their suggestiveness."
6
F. W. Clarke, Data of Geochemistry, 1920, p. 48.
CHAPTER XXI
MOVING EQUILIBRIA
Aus dieser Untersuchung wird kein Dualismua hervorgehen, sondern eine
Wissenschaft, welche Organisches und Anorganisches umfasst, und die den
beiden Gebieten gemeinsamen Thatsachen darstellt. E. Mack.
In preceding pages we have considered as examples of biological
"equilibria," states that quite obviously can be regarded only as
rough approximations to equilibria or steady states; and we have not,
so far, examined critically the justification for this attitude. It is
desirable to give at least brief consideration to this matter.
It is common custom, in dealing with the relatively simple systems
studied in physical chemistry, to assume that a sufficiently slow change
in one parameter (e.g., volume) defining the state of the system, brings
in its train a succession of states each of which is essentially equilibrium.
So, for example, if we slowly raise the piston in a gas-tight
cylinder containing X grams of water and Y grams of water vapor at a
temperature 6, it is commonly assumed that at every instant the
quantities X, Y are such as correspond to equilibrium at the temperature
8.
The Principle of Continuity. The basis of the assumption referred
to in the preceding paragraph is rarely if ever discussed. Obviously
it is to be sought in the principle of continuity. If the parameter P
is constant, at the value P ,
the variables X, Y, . . .
defining the
state of the system have certain values X , YQ, . . . We tacitly
assume that if the parameter PO is nearly constant at the value P
(that is to say, passes through P in very slow change) then the variables
X, Y will have nearly the value of Xo, Y ,
. .
Or, in the
notation of an earlier section, if
^-i^tt:,,*,, . . .P) (i)
at
and if, with P = P = constant
^ = pz
= . .
= pi
= . . .
= o (2)
260 ELEMENTS OF PHYSICAL BIOLOGY
gives
Xi = (?i
= constant (3)
then we assume that with
P = P() (4)
where P(t) is a slowly changing function of t,
we shall have
Xi = d(t) (5)
where Ci(t") is a root of the system of equations
Fi(i)
= Fz (ti = . . .
= Fi(t) . . .
= (6)
(for all values of
It should be noted that, strictly speaking, this involves a contradiction.
For if the velocities F are zero, the variable X cannot be
changing. And, in point of fact, the result (5) represents a first
approximation which is not in all cases free from significant error.
Higher Approximation. It is possible, in certain cases, to proceed
to second, third and higher approximations by successive steps, or,
as will be shown, by a single formula. So, for example, for a system
in two variables X, Y, we may write, first of all
d
-j
= F^X, Y, i) (7)
^ = F(X, Y, t) (8)
'The first approximation here is
F! = Fz
= (9)
X = X1 (t) (10)
Y = Fi(<) (11)
The second approximation we obtain by differentiating (10),
(11), so as to obtain the derivatives Xi, YI, which, although not
zero, are nevertheless small, according to our supposition of a slow
change. Substituting these in (7), (8) we find
Fi - Xi'(t) (12)
F, = F,'(0 , (13)
MOVING BQUILIBEIA 261
Hence
X = Xt(t) (14)
F = F2 (*) (15)
And so on, for successive higher approximations. But this process
can be contracted into a compact expression. We have
^J = ?f1
,
^Oj d
JE *>1}*Y
dt
~
di" bX dt bY dt
'
^! = 5f> _)_ ^J^I L ^.2
^I
d d* d,Y di
+ dF d<
(
If we substitute in (16) (17)
f-X/> (18)
dF
Tt
= YS0 (19)
the right hand member must vanish, in view of (9), (10), (11) and
we have
+*'+*from^which
it is seen that X2 (t), F2 (0 can be expressed directly as the
solution of the system of equations
Similarly the (n + i)
th
approximation is found by writing
_' = _ = o
(24)
Special Case. Returning to the general case of n variables, suppose
that for all values of Xl Xz . . .
differing appreciably from
262 ELEMENTS OF PHYSICAL BIOLOGY
the equilibrium values, the velocities
]
...
are negligible
(Z6 QJ(I
dXr
as compared with some one of them, say -j-. Then we can write,
at
practically
Xt,
. . . Xn)
= (25)
Fn (Xi, Xz, . , . Xr,
. . . Xa)
where ^r is excluded from the system (25) . This defines
X2
= Cz (Xr) (26)
etc.
In addition to this we then have the equation
TTLT
*
= Ft (Ci, C*, . . . Xt,
. . . Cn ) (27)
at
= Fr (Xr) (28)
Hence
which is directly integrable
? ,v x (30)
In such case as this, then, that particular change which is much
slower than all the others, sets the pace and controls the whole process.
It acts as a brake, as a limiting factor.
RADIOACTIVE EQUILIBRIUM
Moving equilibria play an important role in evolutionary processes
of the most varied type, as emphasized almost ad nauseam by Herbert
Spencer,
1
though some of the most typical and at the same time
most fundamental examples were unknown to him. For, the most
exact, quantitatively precise illustrations of moving equilibria are
to be seen in the evolution of chemical elements by successive steps
1
First Principles, Chapter XXII.
MOVING EQUILIBRIA 263
ggsiggaga^^
FIG. 55. URANIUM AND ITS PRODUCTS OF RADIOACTIVE DISINTEGRATION
(U = uranium; To = ionium; Ra = radium; Rn = radon == radium emanation;
Po = polonium; Pb = lead.) Each element in the chain is produced
from its predecessor either by the emission of an alpha particle, i.e., a doubly
charged helium atom, in which case the atomic number is decreased by two
units; or by the emission of a beta particle, i.e., an electron, in which case
the atomic number increases by one unit.
264 ELEMENTS OF PHYSICAL BIOLOGY
of atomic disintegration, accompanied, in most cases known to us,
by radioactive manifestations. It is not within the plan or compass
of this work, to give a detailed account of what has by this time grown
into an extensive special field of physical science. It must suffice
to refer to the chart (fig. 55) of one of the typical series of radioactive
transformation chains, and to state briefly the simple law of transformation
of such elements by spontaneous atomic disintegration : The
amount of a substance transformed per unit of time is directly proportional
to the amount of that substance present, so that if Si is the
ith
substance in a transformation chain
and if we denote by Xt the mass of Si, then we have a system of
equation
"V T7" "\ T7" /fjrO
-Ai-^-x-AA (32)
where the coefficients X are constants, invariable under all conditions
to which observation to the present date has extended.
It will be seen that the system of equations (32) is a simple special
case of the general form discussed in Chapter VI. Its solution2
is of the form there indicated, but the simplicity of the differential
equations is reflected in the integrals, which here appear as finite
series, the expression for the mass of z
th
substance, being
Xt = a,-,ie~
Xlt
+ ai,2e~
X4t
4- . . .
+ ai,t<T"V (33)
The series contains only linear terms, and breaks off at the term in a#.
If XA is (numerically) the least of the X's, then, evidently, after a
sufficient lapse of time the term in X& outweighs all other terms, which
thus become negligible, so that
Xi ctik
Yk
= (34)
The coefficients a are easily determined3
as functions of the X's. It
is thus found that
^ "
(Am - X*) (X*+2
- X,) . . . Ow_! - X*) (X, - A*)
if * > * (35)
am
- if i < k (36)
2
For a somewhat remarkable method of integration (by a multiple
integral), see A. Debierne, Les ide"es modernes de la Matiere, 1913, p. 328.
8
See, for example, E. Rutherford, Radioactive Substances, 1913, pp. 422,
423.
MOVING EQUILIBEIA 265
Thus, after a sufficient lapse of time, the substances Si, S2 ,
. . .
Sk are always present together in constant proportion, so that we
have a moving equilibrium of a veiy simple type, illustrated graphically
in figure 56, in accordance with some of the constants given in
tables 28 and 29. The most slowly decaying substance here acts as
TABLE 28
Radioactive equilibrium of radium in contact with its disintegration products
on the basis oj data in Jour. Am. Chem. Soc., 1923, vol. 45, pp. 872-873
*Half-decay period = = 190 years.
fPolonium.
the controlling, slowly variable parameter, and sets the pace with
which all the subsequent members in the transformation chain keep
step, so that the polygons representing the system in its successive
stages are geometrically similar. (Compare fig. 58 on p. 277.)
The second of the two expressions for the ratio aifi: /akk calls for
brief comment. If i < /c, that is to say if the substance Si precedes,
in the transformation chain, the substance Sk which has its lowest
disintegration rate, then Si does not appear at all in the equilibrium.
Hence when an aggregation of substances in radioactive equilibrium
is found in nature, the substance at the head of the chain (the "parent
substance") is always the one of slowest disintegration rate. But,
obviously we cannot from this draw any conclusion as to whether or
not it is itself a product of disintegration of a pre-parent of more rapid
Oi-% A
decay .ate. Aay such pre-parei*
OP PHYSICAL BIOLOGY
as it were, have been
CONTACT WITH ITS DISINM
davs (half-decay period)
torn a geometric
series
Tte different equations
transformation
"dily
to deterge the
. H. Mitchell, PhU. Mag.,
23, p. 353.
. , ^rairw of radioactive
f^ation> and there is
rf OTCCessive approxima.
However, the first
, P- '
A'
MOVING EQUILIBBIA 267
approximation is so simple, that it is commonly applied. Equating
the right hand member of (32) to zero we find immediately
-j
= *LI
(37)
and by an obvious extension
- ~* (38)
Afc Xi
It is easily shown that Li}
the reciprocal of X;, is the "mean length of
life" of an atom of the substance $,> We may write (38)
~ - ~ (39)
which expresses the fact that, in first approximation, the amounts of
several substances present together in radioactive equilibrium arc
in the ratio of the respective mean lengths of life.
5
This result can
also be read out of (35) if \\h \,
the least of the |X|'s, is negligible in
comparison with all the other |X|'s, so that the denominator reduces
to the product
It is thus seen that the closeness of the first" approximation depends
on the relative magnitude of X& and the remaining X's. In many
chains of radioactive transformation the parent substance is very
slow in its disintegration, and the first approximation (giving what
Rutherford has termed the secular equilibrium) is exact within the
limits of experimental error. But if one of the more rapidly decaying
members is isolated and is then allowed to come into equilibrium with
5
This is a special case of a general law that if all the exponents \ arc real
and negative, the final stages of the process of evolution are characterized by
constancy in the ratios of the Variables .T. Compare p. 261, Special <7a.svj;
also, Lotka, A. J., Proc. Am. Acad. Sci., vol. 55, 1920. Ifc should be noted,
however, that in the general case, xt denotes not mass of Si, but CXCGHB of
that mass over the equilibrium mass of Si. In the radioactive equilibrium
there is no distinction between X and x, since the ultimate value of both
is zero.
For the second and higher approximation, applied to the radioactive
equilibrium, the reader may be referred to A. J. Lotka, Proceedings Natl.
Acad. Sci., 1921, vol. 7, p. 170.
268 ELEMENTS OF PHYSICAL BIOLOGY
its own products of disintegration, the error of the first approximation
may become appreciable.
7
This is shown, for example in
table 29, which exhibits the amounts of radon gas (radium emanation)
and its several products of disintegration in radioactive equilibrium.
It will be observed that in the case of radium C there is a discrepancy
of about 1 per cent between the amount computed by first approximation
and the true amount.
Radioactive Chains as Cosmic Clocks. It must be noted that all
that has been said above regarding the amounts of the substances
present in radioactive equilibrium does not apply to the last link in
the chain, the end product. This does not, of course, take part in
the equilibrium, but accumulates, if non-volatile, as in the case of lead,
or, it may in part escape and be lost to observation, as in the case of
helium.
TABLE 29
Radioactive equilibrium of radon (radium emanation) in contact with its disintegration
-products, on basis of data in Jour. Am. Chem. Soc.,
1923, vol. 45, pp. 872-873
*Half~decay period = 0.69315 L = 3.85 days.
If the amounts of parent substance and end product are large as
compared with the amount of intermediates, the amount of any one
end product formed is evidently simply proportional to the amount
of parent substance lost by disintegration in a given time. The
chain of substances in transformation behaves, in fact, much like a
sandglass clock having a number of bulbs and from the accumulation
in the end bulb we can obtain an indication of the age of
the system, on the assumption that originally all was in the top bulb,
that initially only the parent substance was present. The application
of this principle to radioactive mineral deposits has given us a
quantitative time scale in historical geology where before we had to
7
Compare E. Rutherford, Radioactive Substances, 1913, p. 430.
MOVING EQUILIBRIA 269
rest satisfied with a crude chronology recognizing only order of precedence,
or at best dealing in exceedingly uncertain jjestimates of lapse
of time. Thus the investigation of radioactivity, remote as it seems
from the field of life phenomena, has nevertheless contributed to
biology essential information regarding the time that has been available
for the evolution of the earth and its inhabitants. The estimate
reached upon this basis is that the age of the radium-bearing rocks
(uranium ore) examined is at least eight million years, and at most
seventeen hundred million years old. For a resume" of various estimates
of the age of the earth the reader may be referred to G.
TABLE 30
Geologic time table
After Sclmchert
Schuehert, The Evolution of the Earth and its Inhabitants, 1919'
pp. 56 et seq.; 80; and to the Proceedings of the American Philosophical
Society, 1922, vol. 61, pp. 247-288. See also E. Rutherford,
Radioactive Substances, 1913. Schuchert's estimate is that
"geologic time endured about eight hundred million years," distributed
among the several geological eras as indicated in table 30.
The Origin of the Elements and the Ultimate Genesis of the
Organism. The case of radioactive equilibrium has here been introduced
primarily by the way of illustration, as probably the most
typical example in nature of a moving equilibrium in a system in the
course of evolution. But the matter is also of more material interest
to us in our survey of the evolution of the earth as the abode of life.
For, as has already been emphasized, we are not only on the earth but
of it; we have thus a two-fold interest in the evolution of its substance
first, as providing the stage upon which our life drama is set; and
second, as furnishing the material of our bodies. Of these same ele-
270 ELEMENTS OF PHYSICAL BIOLOGY
ments that make up the earth's crust we also are composed: their
genesis is therefore also the first, remote chapter in the genesis of
our own bodies. Through the discovery of radioactive chains of elements
we hold a clue regarding the fundamental influences that
have determined the quantitative chemical composition of our world,
and have thus appointed the measure of the supplies available for
our needs. Those elements whose genesis is known to us came into
being in perfectly definite proportions. Presumably the same is
true also of those whose precise mode origin is still unknown. There
is good evidence to support this view. We have at present no detailed
quantitative knowledge of the laws which determine the value of the
decay coefficients X of the radioactive elements, and which thus ultimately
fix their relative abundance in equilibrium. But a significant
qualitative relation has been pointed out by W. D. Harkins. 8
When
the elements in a radioactive chain are arranged in order of their
atomic numbers,
9
and are separated into two groups, those of odd
and those of even number, it is found that each even-numbered element
is more abundant than the adjacent odd-numbered elements.
And, what is of particular significance, the law of relative abundance
of odd and even-numbered elements extends also to those elements,
regarding whose precise mode of origin we have not, as yet, that sure
knowledge which is gained by direct observation within the four
walls of the physical laboratory. (See fig. 57.)
8
Jour. Am. Chem. Soc., 1916, vol. 38, pp. 863, 869; 1923, vol. 45, pp. 1420-
1433. Compare also F. W. Aston, Nature, March 15, 1924, p. 394. For
other regularities observed in the length of life of radioactive elements see
K. Fajans, Radioaktivitat (Sainmlung Vieweg Heft 45) 1921.
9
The chemical elements, arranged in ascending order of atomic weights,
beginning at hydrogen = 1, may be given ordinal numbers 1, 2, 3, etc., indicating
their position in the series. These ordinal numbers have been found
to have important relation to the atomic architecture. They have been
termed the atomic numbers. The definition here given is not quite exact;
in certain places allowance must be made, gaps left for unknown elements,
and the several isotopes of one element receive the same atomic number,
though differing in their atomic weights. A more precise definition is the
following: The atomic number of an element represents the excess of positive
over negative charges in the constitution of the atomic nucleus. Each
atomic number also represents the place occupied by the element in Mendeleef's
table (Jour. Am. Chem. Soc., 1923, vol. 45, p. 868). For further information
the reader must be referred to the special literature; of comprehensive
works the following may be mentioned: Bragg, X-rays and Crystals;
F. W. Aston, Isotopes.
FIG. 57. RELATIVE ABUNDANCE OF THE ELEMENTS
Each element of even atomic number is more plentiful than the adjacent
elements of odd stomic numbers. Diagram according to W. D. Harking,
based on analysis of meteorites. (Jour. Am. Chem. Soc., 1916, p. 863.)
271
272 ELEMENTS OF PHYSICAL BIOLOGY
But the laboratory is not a prison, and the eye of the physicist is
free to sweep the sky, where nature's great smelteries gleam at night.
With the aid of the spectroscope he has studied the multitudes of the
stars, and has recognized in them a number of distinct stages of evolution.
Life's day is far too short to give the observer any opportunity
to study directly the evolutionary changes in any one star.
But by piecing together the observations made upon the mixed population
of stars of different ages, it has been possible to construct with
considerable certainty the main stages in stellar evolution, just as the
stages of human life could be gathered from a single observation of
a mixed population comprising persons of all ages. The evidence
points clearly that the elements, such as we know them, are the
product of "the general brewing of material which occurs under the
intense heat in the interior of the stars." Out of such foundry came
our own abode, if we accept the well-considered views of Eddingtor..
'"
"I do not say that the earth was a gaseous body when it first became
recognizable as an independent planet, but I am convinced that its
material was at one time merged in a completely gaseous sun/'
And since we are of earth, ours also is the same origin. The hand that
writes these words and the eye that reads them alike are composed
of the selfsame atoms that came into being, ages and ages ago, in
the young sun. Far, far more wonderful than any dream of old
10
A. S. Eddington, The Borderland of Astronomy and Geology, Nature,
1923, p. 18, also, The Interior of a Star, Supplt. to Nature, May 12, 1923. The
Tder
who wishes to acquaint himself in greater detail on this subject may
refer to Eddingtons's -work Stellar Evolution. In the interest of unbiassed
presentation it must be noted here that T. C. Chamberlin (The Origin of
the Earth, 1916) has put forward a theory of the origin of the earth and the
planets which is at variance with that sustained by Eddington. On the
other hand it is also proper to mention a fundamental objection to theories
of cosmogony of the type of that of Chamberlin and Moulton, which is based
on the supposition that the luminous stars are formed by the collision of
dead suns. "The distances separating the stars are enormous compared
with their own dimensions. Sir Frank Dyson once used the illustration of
twenty tennis balls distributed at random throughout the whole interior of
the earth, to give a model of the density of distribution of the stars
Taking a very liberal view of the kind of approach that can be held to constitute
a collision, it is estimated that a star would suffer a collision about
once in a hundred million million years" (Eddington). For a survey of
the modern views on this subject see J. Barrell, The Evolution of the Earth
(Yale University Press, 1919).
MOVING EQUILIBBIA 273
mythology is the story of our creation. Thus was the birth of man
prepared in the grey dawn of time ;
thus the metal of his frame compounded
in the flaming furnace of a star. ....
TERMINAL STAGES OF THE EARTH'S EVOLUTION
Geophysics and Geochemistry. For the last stages in the evolution
of the elements and their chemical combinations we do not look
to the stars. We can study them at close quarters in the field and in
the laboratory. In this way, with the application of physics and
chemistry to general problems of geology, have grown up the sciences
of geophysics and geochemisty. Indeed, it naturally might be
supposed that on the terrestrial phases of inorganic evolution we
should be altogether better informed than on those prior stages, far
remote in time and space, which run their course in distant suns.
But this is true, at best, only in restricted measure. It is a singular
circumstance that, in some ways, we are better informed regarding the
physics and chemistry of the stars, of which the nearest, outside the
solar system, is twenty-five million million miles distant, than regarding
that of our own planet. To say that the earth's surface layers
accessible to our direct observation are comparable, scale for scale,
to the shell of an egg, is to err on the side of liberality. The deepest
burrow into the earth made by human agency, the mine shaft at
Morro Velho, Brazil,
11
is 1-|- miles (6400 feet) deep or only about
frsT of the earth's diameter. Direct observation can therefore give
us at best only the most uncertain information regarding theconditions
at even moderate depths. Where the crust has been creased
and thrown into folds, subsequent denundation may have exposed
layers of some 50 or 60 miles aggregated thickness.12
But, though
this gives us an invaluable record of some of the most significant
chapters in the earth's history, it adds little or nothing to our knowledge
of conditions of temperature and pressure even at comparatively
trivial depths, and regarding chemical composition also it gives us,
after all mere surface indications. Indeed, our information on all
these points is very largely of a negative character. We know from
the earth's average density that the composition of the core must be
very different from that of the shell. We know that the temperature
u Sir Charles Parsons, Nature, February 19, 1920, p. 677.
12
G. Schuchert, The Evolution of the Earth, Yale University Press, 1919,
p. 67.
274 ELEMENTS OF PHYSICAL BIOLOGY
gradient observed in boreholes near the surface averages an increase
of about 1F. for every 60 feet (1C. for every 35 meters) descent, but
of the further course of temperature at greater depths we know with
reasonable certainty only that it cannot continue at this gradient,
which would give a phantastic temperature of 300,000F. (1 80,000C.)
at the center. Lord Kelvin's calculations, based on the rate of cooling
of the earth, and the more recent figures of the same character
given by Van Orstrand, are rendered uncertain in their application
owing to the presence of radium, in unknown amounts, evolving heat
in its distiiitegration. Strutt has made the tentative estimate that
the temperature rises uniformly to a depth of about 30 miles, and
after that remains sensibly constant at 2700F.
A little more definite is our information regarding the pressure in
the earth's interior. A first approximation of its value is found by
considering the earth as a fluid. It is thus found13
that the pressure
at the center would be three million atmospheres. In any case there
can be no doubt whatever that the pressures reached vastly exceed
anything at our command in the laboratory, where a pressure of
twenty-four thousand atmospheres, employed by P. W. Bridgman
in his researches, stands out as a record achievement, though it coiv
responds to a depth of rock of only 56 miles.14
It must be clear from what has been said above, that all conjectures
as to the physical, chemical and subatomic transformations going on
in the earth's interior are subject to a very large margin of uncertainty.
In this connection it is well to recall the words of F. W. Clarke:15
The chemistry of great pressures and concurrently high temperatures is
entirely unknown, and its problems are not likely to be unravelled by any
experiments within the range of our resources. The temperatures we can
command, but the pressures are beyond our reach We may
devise mathematical formulae to fit determinable conditions; but the moment
we seek to apply them to the phenomena displayed at great depths, we are
forced to employ the dangerous method of extrapolation, and our conclusions
are not verified.
13
This supposition, in calculating pressures, probably does not err far
from the truth. The researches of P. D. Adams (Journal of Geology, February,
1912), P. W. Bridgeman and others have shown that at depths of some
30 miles rocks probably give way like butter to the pressure of the layers
above them.
" W. D. Lambert, Jour. Washington Acad. Soi., 1920, p. 126. Sir Charles
Parsons, loc. cit.
15
Data of Geochemistry, p. 271.
MOVING EQUILIBRIA 275
In these circumstances we can feel but little confidence in inferences
based upon our laboratory observations, relating to radioactive ami
other possible atomic transformations going on at greater depths
within the earth. If the degree arid character of radioactivity which
we observe in the accessible surface layers were to continue throughout
the mass of the earth, the amount of heat developed would be
much in excess of the observed losses by radiation.10
Unless therefore,
we are to draw the highly improbable inference that the temperature
of the earth's mass is steadily rising, we are forced to one of
two assumptions. Either the radioactive elements arc segregated
chiefly in the earth's crust; or, the present rate of heat loss by radiation
from the earth does not represent its average rate. This latter
is the alternative elected by J. Joly
17
in an original conception. According
to this there are alternate periods of accumulation of heat in
the solid rocks, followed by periods in which these rocks, having finally
become melted, well to the surface in a death-dealing flood of fire.
Thus, by convection, a process far speedier than conduction, through
the solid rock mass, heat is dissipated until, after sufficient cooling,
a second period of quiescence, with a solid earth's crust, is ushered hi.
And so, in long waves of perhaps some thirty million years duration,
the planet alternates between periods of hospitable clemency and
periods intolerant of life.
There is, for us living inhabitants of this globe, a certain wildly
romantic element, a feature of calamitous tragedy, in the hypothetical
picture of the world's history thus summoned up before our
imagination.
In its biological aspect how great and wonderful it all is I The living being
working out his destiny on this poor raft, unknowing of the fiery ocean upon
which this world is floating: unknowing of the inevitable sinking and uplifting
which in truth largely controls the destinies of his race. .Deathdealing
forces all around, and yet the light of life shining age after age upon
the earth.
16
Compare V. Moritz, Dor Stoffwechsel der Erde, Zeitschr. f. Eloldrochemie,
1922, p. 421.
17
J. Joly, Movements of the Earth's Crust; lecture under the auwpicoH of
the Royal Dublin Society, published in Nature, 1923, p. 003. A third possibility
is that under the extreme conditions of temperature and pressure
prevailing in the earth's interior reversal of the familiar radioactive disintegrations,
or similar endothermie processes may go on. (Jour Natl
Acad. Sci., 1924, p. 89.
276 ELEMENTS OF PHYSICAL BIOLOGY
Such conceptions as this, stimulating as they are to the imagination
in reconstructing for us an image of the remote past and distant
future, must be entertained with reserve, remembering the words of
caution quoted above from Clarke's classic work. We must be
prepared to consider the possibility that under the extreme conditions
of temperature and pressure prevailing at great depths other subatomic
transformations than those known to us in the laboratory may occur.
Perhaps some evidence of this is seen in the evolution of helium as a
component of natural gas, in amounts (up to 1.5 per cent18
) in excess
of anything readily accounted for on the basis of the observed radioactivity
of the rocks. And, while the subatomic transformations
known to us are exothermic, accompanied by liberation of heat, others
undoubtedly are endothermic, associated with absorption of heat.
We seem to have carte blanche, in the present state of knowledge, in
our speculations regarding the net heat balance of the elemental
transformations that may be going on under our feet.
On surer ground rest our conceptions regarding the organization of
matter, especially in the more superficial layers of the earth, under the
action of ordinary physical and chemical influences. That the prime
factor effecting the first and fundamental segregation of the lighter
elements is flotation under gravity can hardly be doubted; this
statement would in fact, be little more than a platitude if we were
assured that the elements themselves remain unchanged under the
extreme conditions of temperature and pressure prevailing in the earth's
interior. Beyond this prime factor the study of mineral and rock
formation becomes a complex chapter in applied physical chemistry,
the consideration of which is not within the plan of this work. The
reader who wishes to follow out further this phase of the subject will
find a comprehensive survey of the field in an article Dcr Stoffwechsal
der Erde, by V. Moritz, in the Zeitschrift fur Elektrochemic,
1922, pp. 411-421; and in the memoir The Chemistry of the Earth's
Crust by H. C. Washington, which has already been quoted.
Organic Moving Equilibria. Of moving equilibria in the organic
world, data are most readily available for the system comprising man
and his domestic animals. Here the human race acts as the controlling
factor, drawing its dependents after it in its growth. The equi-
18 R. B. Moore, Nature, 1923, p. 91; Cady and McFarland have reported
one instance of 1.84 per cent. J, Joly, Radioactivity and Geology, 1909,
p. 218.
MOVING EQUILIBEIA 277
librium polygon for the principal items of animal husbandly in the
United States is shown in figure 58. The geometric similarity of
successive polygons is in this case only approximate, the proportion of
the several components varies somewhat; except in the case of the
sheep population, however, the variation is moderate over the halfcentury
from 1871 to 1921. (Compare fig. 56 on p. 265.)
Aside from the features for the express illustration of which the
diagram figure 58 was drawn, it also serves to point once more to the
-70
FIG. 58. EouiLiBEitnvi POLYGON .FOE THE HUMAN SPECIES AND SOME OF THE
SPECIES ON WHICH IT DEPENDS FOU ITS FOOD SUPPLY. SCALES
READ IN MILLIONS
fact already emphasized, that the concept of evolution, to serve us in
its full utility, must be applied, not to an individual species, but to
groups of species which evolve in mutual interdependence; and
further, to the system as a whole, of which such groups form
inseparable part.
It would be conveying a false impression in a very essential respect,
to exhibit the example illustrated in figure 58, without a comment
in emphatic reservation. Although, in a roughly approximateway,
it is true, as shown by the polygon diagram, that in its relation
278 ELEMENTS OF PHYSICAL BIOLOGY
\?
^
MOVING EQUILIBRIA 270
to certain staples of agricultural production, our population lias
advanced in a succession of moving equilibria; yet, tbo progress of
modern industrial civilization on the whole is essentially the very
antithesis of a moving equilibrium conditioned by and following upon
the changes of a slowly varying parameter. Quito on the contrary,
the development of this age is rather of the nature of a rocket-like
ascent with a speed altogether unparalleled in all previous history of
organic evolution, and at the cost of rapid depletion of capital
resources. Certain aspects of this phenomenon are reserved for
consideration in a later chapter. Here it will bo sudiciently to the
point to draw attention to table 31, reproduced from H. Pearl's
essay on The Population Problem, which shows the altogether disproportionate
increase in the growth of our material accessories iu
recent years, as compared with that of the population itself.
The human species, considered in broad perspective, a,s auui,
t inc.liK ling
its economic and industrial accessories, has swiftly and radically
changed its character during the epoch in which, our life has been laid.
In this sense we are far removed from equilibrium
......
a fact which is of
the highest practical significance, since it implies that a period of
adjustment to equilibrium conditions lies before us, and ho would be
an extreme optimist who should expect that such adjustment can be
reached without labor and travail. We can only hope that our race
may be spared a decline as precipitous as is the upward slope; along
which we have been carried, heedless, for the most part, both of our
privileges and of the threatened privation ahead. While such sudden
decline might, from a detached standpoint, appear as in accord with
the eternal equities, since previous gains would in cold terms balance
the losses, yet it would be felt as a superlative catastrophy. Our
descendants, if such as this should be their fate, will see poor compensation
for their ills in the fact that we did live in abundance and
.
Pearl, Geographical lioviow, 1922, vol. 12, p. 038,
MOVING EQUILIimrA
< 1
J
to certain staples of agricultural production, our population
advanced in a succession of moving equilibria; yet the p roarer
modern industrial civilization on the whole is essentially tinantithesis
of a moving equilibrium conditioned by and follmvim'; upo
the changes of a slowly varying parameter. Quite on the cmtinin ,
the development of this age is rather of the nature of a rurkef -Hie
ascent with a speed altogether unparalleled in all previous hi:>f urv nt"
organic evolution, and at the cost of rapid depletion of capital
resources. Certain aspects of tins phonomo.ii.on are reserved l<uconsideration
in a later chapter. Here it will bo NudioirnMy 1" the
point to draw attention to table 31, reproduced from \\. ('earl's
essay on The Population Problem, which shows the altotfof her drproportionate
increase in the growth of our material arrejisorie;, hi
recent years, as compared with that of the population if self,
The human species, considered in broad porHpeetivo, aH.'iimit inelud'
ing its economic and industrial accessories, ban .swiftly and rndirally
changed its character during the epoch in which our life has been laid.
In this sense we are far removed from equilibrium -a facf which n < it"
the highest practical significance, since it implies thai- a period 1*1'
adjustment to equilibrium conditions lieH before UH, n,nd he would be
an extreme optimist who should expect that such adjiiHtmenJ can be
reached without labor and travail. We can only hope I Imi our rnee
may be spared a decline as precipitous a,s I'H the upward nlupe ult.nf,
which we have been carried, heedless, for the mowt part, both <.f .ur
privileges and of the threatened privation ahead. While mich iujd lea
decline might, from a detached standpoint, appear JIM hi aivnnl with
the eternal equities, since previous gains would in c.old l.musk'tlain-f
the losses, yet it would be felt as a superlativo (!ji,l ( /i.sl (
nphv. Mmdescendants,
if such as this should be their fate, will HOC poor rum
pensation for their ills in the fact that we did live hi abundance ;i nd
luxury.
)!1
R. Pearl, Geographical Review, 1922, vol. 12, p. (;JH
MOVING EQUILIBRIA 279
to certain staples of agricultural production, our population has
advanced in a succession of moving equilibria; yet the progress of
modern industrial civilization on the whole is essentially the very
antithesis of a moving equilibrium conditioned by and following upon
the changes of a slowly varying parameter. Quite on the contrary,
the development of this age is rather of the nature of a rocket-like
ascent with a speed altogether unparalleled in all previous history of
organic evolution, and at the cost of rapid depletion of capital
resources. Certain aspects of this phenomenon are reserved for
consideration in a later chapter. Here it will be sufficiently to the
point to draw attention to table 31, reproduced from R. Pearl's
essay on The Population Problem, which shows the altogether disproportionate
increase in the growth of our material accessories in
recent years, as compared with that of the population itself.
The human species, considered in broad perspective, as a unit including
its economic and industrial accessories, has swiftly and radically
changed its character during the epoch in which our life has been laid.
In this sense we are far removed from equilibrium a fact which is of
the highest practical significance, since it implies that a period of
adjustment to equilibrium conditions lies before us, and he would be
an extreme optimist who should expect that such adjustment can be
reached without labor and travail. We can only hope that our race
may be spared a decline as precipitous as is the upward slope along
which we have been carried, heedless, for the most part, both of our
privileges and of the threatened privation ahead. While such sudden
decline might, from a detached standpoint, appear as in accord with
the eternal equities, since previous gains would in cold terms balance
the losses, yet it would be felt as a superlative catastrophy. Our
descendants, if such as this should be their fate, will see poor compensation
for their ills in the fact that we did live in abundance and
luxury.
1(1
R. Pearl, Geographical Review, 1922, vol. 12, p. 638.
CHAPTER XXII
DISPLACEMENT OF EQUILIBRIUM
Die Physik wird aus dem Studium des Organischcn an sicli noch schr vio.l
neue Einsichten scliopfen miissen, bcvor sie auch das Organischo bowiiltigcn
kann. E. Mack.
In preceding pages we have passed in review some of the principal
features of interest presented by systems maintained constantly at or
near equilibrium, while one of more of the parameters determining
such equilibrium were slowly changing, thus engendering a moving
equilibrium.
One might proceed to a consideration, on a more general basis,
of the changes brought about in an evolving system through changes
of any kind, including rapid ones, in the parameters. In the most
general case this would amount to the discussion of a system of
differential equations of the form.
Xn,
t)
in which the time t entered explicitly into the function F.
It is not proposed to take up the study of this perfectly general
case; it must suffice to point to the mathematical literature regarding
equations of this form.1
^
But there is another special phase of the general problem which,
like the case of slow changes, yields with comparative ease to
analytical. treatment; namely, that special phase which enquires
only into the ultimate effect, upon equilibrium, of a given total
change in a parameter, leaving aside all questions relating to the
path by which the displacement of equilibrium takes place. Such
a separate consideration of this special and restricted phase of the
general problem is rendered possible by the fact that, in certain
cases at any rate, the displacement of the equilibrium is independent
of the path of the change, and depends only on the given initial and
m,
lE
!S
f r
!f
ample>
E" Picard>
Tra^ d'Analyse, vol. 3, 1908, pp. 187, 188,
194, 197; E. Goursat, Cours d'Analyse, vol. 2, 1918, pp. 482, 498.
280
DISPLACEMENT OF EQUILIBRIUM 281
final values of the parameters whose modification provokes or is
associated with the change. So, in physico-chemical transformations
("changes of state") the principle of Le Chatelier enables us
to predicate, within certain limits, the sign of the displacement of
equilibrium conditioned by a change in certain of the parameters
upon whichjthe equilibrium depends.
THE PRINCIPLE OF LE CHATELIER
The principle of Le Chatelier is best illustrated by a simple
example. Consider the simple chemical reaction
2H2 + Oa =2HaO + 58.3 oal.
At high temperatures this reaction is reversible; that is to say, it
takes place to some extent in the direction of the upper arrow, but
also to some extent in the direction of the lower arrow, and an
equilibrium is finally established between these two opposing reactions.
Now this is what the Le Chatelier principle tells us:
If we add either H alone or alone to the system, the equilibrium
is shifted in the direction of the upper arrow, that is to say, in
such direction as to absorb some of the added constituent.
Similarly, if we heat the system, the equilibrium is shifted in the
direction of the lower arrow, that is to say, in the direction of the
reaction which absorbs heat. The principle, as enunciated by
Le Chatelierz
himself, is:
Every system in chemical equilibrium, under the influence of a change of any
single one of the factors of equilibrium,
3
undergoes a transformation in such
direction that, if this transformation took place alone, it would produce a
change in the opposite direction of the factor in question.
The factors of equilibrium are temperature, pressure, and electromotive force,
corresponding to three forms of energy heat, electricity and mechanical
energy.
The second paragraph of the principle as quoted above, requires
special emphasis. It is often omitted, even by authors of the highest
2
Recherches sur les Bquilibres Chimique, 1888, pp. 48, 210; Comptes Rendus,
1884, vol. 99, p. 786; Mellor, Chemical Statics and Dynamics, 1904, pp.
435H136.
8
It appears that some French writers employ the term "facteur d'e'quilibre"
as synonymous with "intensity factor of an energy." (Cf. F. Michaud, Ann.
de Phys., vol. 16, 1921, p. 132.)
282 ELEMENTS OP PHYSICAL BIOLOGY
repute/ with the result that a vagueness is introduced for which
Le Chatelier himself cannot justly be made responsible. This
vagueness is then often rendered still worse by departures from, the
original wording, aimed at an extension of the scope of the hiw to
all conceivable systems and "factors," an extension which is gained
with a total sacrifice of all validity of the principle. So, for example,
if we seek to apply the principle as quoted above, but omitting
the restriction of the second paragraph, to the water equilibrium
already mentioned, and if we select as "factor" of equilibrium not
pressure but volume, the principle would lead us to reason as
follows: On diminishing the volume of the system, that transformation
will take place which, did it take place alone (i.e., at
constant pressure), would be accompanied by increase in volume;
a conclusion which is false. As has been shown by Ehrcnfest/
1
the error arises through failure to discriminate, in the application
of the principle, between the intensity factor (e.g., pressure) and
the capacity factor (e.g., volume) of an energy.
It must appear singular that so obvious a defect of the principle,
as commonly quoted, should so generally have escaped attention,
and should for example, have passed unnoted through seven cditionw
of so excellent a work as Nernst's Theoretische Chemie. Khronfost
points out that the explanation lies in the very vagueness of the
principles, which permits it to be construed in each case to suit
circumstances. The principle is commonly applied fix post facto,
and its competence to predict thus escapes any serious test. This,
however, is only a partial explanation. After all, the fundamental
reason for the tardy recognition, and the still more tardy admission
in the general literature, of the weakness of tho principle, an commonly
quoted, must be sought in an inherent weakness of the human
mind: by a curious inversion of what might be expected in logical
sequence, the last things to receive critical scrutiny are always the
fundamental premises of our arguments. This 'is true both as
regards the judgment of the average individual, of the people ut
large, and often even of the man of very superior intellect. One
recalls, in this connection, MacAuley's remarks regarding Dr.
Johnson: "How it chancedJhat a man who reasoned upon his
4
See, for example, W. Nernst, Theoretische Chemie, 1913, p 60S
' f'
Sf !
iem" 19U'
V L 77) P' 735 ' Cf' also
e, 1911, vol. 1, p. 467.
DISPLACEMENT OF EQUILIBRIUM 283
premises so ably should assume his premises so foolishly is one of
the great mysteries of human nature."
If such an outwardly slight departure from Le Chatelier's original
enunciation as the omission of his second
'
'explanatory" paragraph,
thus completely destroys the validity of his principle, what is to
be said of such sweepingly vague settings as in the followingexamples
:
The broadest definition of the principle of Le Ghaielier is that a system tends to
change so as to minimize an external disturbance (W. D. Bancroft, Journal of
the American Chemical Society, 1911, p. 92).
Every external action produces in, a body or system changes in such direction,
that in consequence of this change the resistance of the body or system against the
external action is increased.
If we regard the faculty of adaptation of animals and plants from the point
of view that the organisms undergo, under the influence of external actions,
changes which render them more resistant to those actions, then the property
of non-living matter which is expressed by the principle of Le Chatelier-Braun
may be regarded as a sort of adaptation of such non-living matter (Chwolson,
Trait6 de Physique, 1909, vol. 3, p. 547).
If the equilibrium of a natural complex (system of masses, organism, system
of ideas') is disturbed, it adapts itself to the stimulus (Reiz] which causes the disturbance,
in such manner that the said stimulus continually diminishes until
finally the original or a new equilibrium is again established (J. Lowy, Jvosmos,
1911, p. 331).
The last two examples are of particular interest to us here as
suggesting application of the principle to biological systems. As a
matter of fact, such application of the vaguely formulated principle
(in a form in which it would be injustice to link it with the name of
Le Chatelier) antedates by many years its enunciation by the
French physicist. The following passages in Herbert Spencer's
First Principles a're pertinent :
Among the involved rhythmical changes constituting organic life, any disturbing
force that works an excess of change in some direction is gradually
diminished and finally neutralized by antagonistic forces, which thereupon
work a compensating change in the opposite direction, and so, after more or
less of oscillation, restore the medium condition. And this process it is which
constitutes what physicians call the vis mcdicatrix naturae,
This is a conclusion which we may safely draw without knowing the special
re-arrangements that effect the equilibration: If we see that a different mode
of life is followed after a period of functional derangement by some altered
condition of the system if we see that this altered condition, becoming by
284 ELEMENTS OF PHYSICAL BIOLOGY
and by established, continues without, further change, wo havo no altornafcivo
but to say that the now forces brought to boar on the wyHtom have boon compensated
by the opposing forces they Imvo evoked (I<!irnt Principles, (Chapter
XXII, EquiUbriatwnt 173).
Almost simultaneous with Lo Ohatelicr's publication (1884) is
the following pronouncement.
L'4tre vivant cst agonc6 do to Ho mauioro quo ohaquo influence povturbatrice
provoque d'elle memo la miso on activity do I'apparoll oomponsateur qui doit
neutraliser et rcparer lo dominate (I/sou Frdddricq, A roll I von do Zoolome
Exp. et G<Sn., ser. 2, vol. 3, 1885, p, xxxv).
Now
it is not denied that such oxproHHionn U,B thin have, a co.rtain
utility, as describing with, fair accniracy a goodly proportion of a,
class of phenomena to which they rolat<\ ]tii/. <.o
'A'cwi^iiak) such
statements, as "Lo Giiatelier's l'rin<tiplo," w^tfffotty nuKloadtng.
That principle, in its exact and narrower formulation IB
rigorously
true, as much so as the laws of thonrvodyjianiioH froni which it c,tm
be deduced; it has no exceptions, any more than tluvre Ls any
exception to the law that heat flown by simple* oonduction from the
hotter of two bodies to the colder.
The alleged "principle," an applied to biological H.y.stc.mH, lacks
the sureness which the true Lo Chatdicr principle POHHWHOH, in its
stricter
formulations, in physical choiniHtry. An org/infom may,
by exposure to a certain influence A, Ixujomo moro rosiHtaut to thd
influence, as in the caso of acquired immunity after an attack of
infectious disease, or after habitation to Hiicl/a poiMou as aracmic:.
But, by exposure to another influence B it may boeomo USHH rowiHtant
to B, as in the case of cumulative poiaoriH, or of a-naphylaxiH. The
Le Chatelier principle does not enable us horo to pmUct in which
direction the effect will take place in a new and untried wiwo of some
influence C.
Conditions of Validity of Le Omtelier's Principle. The question
arises why the principle thus breaks down in ita aj^licsation to
biological cases of the kind cited. The answer is found by examining
the basis on which the proof of tho principle rests.
Such an examination brings out the fact that one of tho nocoasary
conditions for the applicability of the principle is stability of the
ZT V^11 aPPlication made. Now the equilibria
commonly contemplated in physical chemistry are stable, so that
DISPLACEMENT OF EQUILIBEIUM 285
this condition is satisfied. But it is not always satisfied in the
equilibrium of the living organism. The organism is, indeed,
stable with regard to many of the commonly occurring attacks of
its environment. But it is of little consequence to the species
whether, for example, the individual organism is stable with regard
to the ingestion of a large dose of strychnine, for in nature such
ingestion will occur so rarely, if at all, as to influence in no appreciable
degree the life of the species. It is not necessary for the
stability of the species, that the individual be stable at all times.6
In point of fact, we know perfectly well that sooner or later each
individual finds itself in a condition of instability, by "accident"
or sickness, and dies. An analysis
7
of the basis of the principle of
Le Chatelier reveals the fact, among others, that all demonstrations
of this principle postulate, as a fundamental characteristic of the
systems to which it applies, that they be in stable equilibrium. The
principle can, therefore be applied at best only with cautious
reservation to living organisms, reservation such as, for example,
Le Dantec8
makes: "In studying as closely as possible the consequences
of disease in living organisms, when they survive such
diseases, I have drawn attention .... to the fact that all
these consequences, such as acquired immunity and the production
of antitoxic sera, can be summarized in the principle of Le
Chatelier." But with such reservation the principle loses its chief
utility, which consists in its power to predict the course of events.
Indeed, it might be accused, in such restricted form, of being little
more than a tautological platitude, which tells us that if the system
or unit in question is stable, then it is stable. This is not quite
such a damning accusation as may at first sight appear, for the
same can be brought against the principle of the survival of the
fittest, which nevertheless has proved supremely fertile in biological
research. In point of fact there is a close relationship between
6
Compare what has been said in the discussion of chemical equilibrium
regarding the stability of aggregates composed of individuals, themselves of
limited stability, of limited life period (Chapter XII),
7
Such an analysis, carried out in considerable detail, has been given by the
writer in Proc. Am. Acad. Arts and Sci., 1922, vol. 57, pp. 21-37. The importance
of the restriction to stable equilibria, in connection with biological systems,
has also been pointed out by C. Benedicks, Zeitschr. f. phys. Cheinic,
1922, vol. 100, pp. 42-51. A. J. Lotka, Am. Jour. Hygiene, 1923, p. 375.
8
La Stability de la Vie, 1910, p. 24.
286 ELEMENTS OF PHYSICAL BIOLOGY
the two principles. But it is important to note that the principle
of the survival of the fittest is avowedly statistical in character, and
is to be applied to organisms in the gross. This is true, also, of the
principle of Le Chatelier in physico-chemical systems; its field of
application is to aggregates of molecules, not to the individual.
But the applications that have been essayed in biology have been
made to the individual; such application can at the best yield a
judgment of probabilities. In physical chemistry we deal for the
most part with stable equilibria. But in biology, as has already
been pointed out, though the races that come under our observation
possess stability as races (else they would not have survived
to be our contemporaries), it does not follow at all, that each and
every individual is at all times in a state of steady equilibrium.
Aside from the limitation in the applicability of th&-prin"ciple to
stable systems, other limitations appear in such an analysis of its
foundations as has been referred to above. So, for example, loose
analogy to the physico-chemical equilibrium, as affected by the
addition of a quantity of one of the reacting substances, might lead
one to draw the erroneous inference that in a community infected
with malaria, the introduction of additional malaria parasites would
shift the equilibrium in the direction of a higher malaria rate.
But there is every reason to expect, on the contrary, that the
equilibrium remains unchanged by such addition. For a close
analysis of the reason for this divergence in the two cases the
reader must be referred to the original paper already cited. It must
suffice here to state briefly that this reason is to be found in the
existence in the physico-chemical case of equations of constraint,
relations between certain variables, and in the absence of analogous
relations in the case of malaria.
Extension of Scope of Rigorous Applicability. While the prime
result of a searching analysis of the foundations of the Le
Chatelier principle is to emphasize rather the restrictions of its
scope, yet in certain respects such analysis does furnish a rigorous
basis for a certain generalization of its applicability beyond those
bounds where its warrant rests on the firm ground of thermodynamics.
And this extension of the strict
applicability of the
principle takes place essentially in two directions. On the one hand
the thermodynamic justification, at any rate in the form commonly
presented, covers only true equilibria, and does not extend to
DISPLACEMENT OF EQUILIBRIUM 287
steady states maintained with constant dissipation of energy.
This restriction does not appear in the demonstration of the principle
on the broadest grounds that suffice for its establishment,
9
Le Chatelier's principle applies, in certain cases, to steady states
of the more general type, as well as to true equilibria.
The second direction in which the analysis, on general grounds, of
the principle, enlarges its field of warrant, is in the matter of the
kinds of "factors" to which it is properly applicable. It has already
been pointed out that, in its physico-chemical application, it must
be used with proper discrimination as to the distinction between the
capacity and the intensity factor of an energy, as, for example,
volume and pressure. It is found, upon analysis, that the applicability
of the principle to the effect of a change in pressure, for
example rests upon the following fundamental property of the
pressure and volume of a system in stable equilibrium.
1. For every value of v, the volume of the system, there is a
definite value of p i}
the pressure which it exerts, the internal pressure,
as we may term it.
2. The volume v increases or decreases according as the internal
pressure p\ is greater or less than the external pressure p upon the
enclosure, that is to say,
dv > . >= according as pi =; p6 .
3. It can be shown that, given (1) and (2), stability demands that
the curves representing the relative between p (ordinates) and v
(abscissae) must slope from left to right downwards. For if such
a curve slopes in the opposite direction, then the slightest displacement
from equilibrium will immediately cause the system to travel
with cumulative effect, avalanche-like, along the pv curve further
and further away from the starting point.
10
9
For justification of this and other statements made in those paragraphs
the reader is referred to the author's paper already cited. It may be remarked
that Ehrenfest (loc. cit.) expresses the belief that such broader scope belongs
to the principle, but he does not support his impression with proof.
10 It is interesting to note that an upward slope, from left to right,, occurs
in the middle limb of the van der Waals' pv curve of a gas. But this limb
represents an unstable state which is never realized, the gas, instead of following
this part of the curve, partially condenses and traces a horizontal straight
line for the pv relation.
288 ELEMENTS OF PHYSICAL BIOLOGY
Now these fundamental properties (1), (2) and (3), of a capacity
and an intensity factor of an energy
11
are shared by certain parameters
that have no direct or simple relation to energy whatsoever;
and since the applicability of the principle depends upon these
properties, it will extend to such other parameters possessing them.
As an example may be mentioned the relation between area a
occupied by a population, and the rent per unit area RI that an
(average) individual is willing to pay. If .Ri is greater than Re ,
the rent at market rate, the individual will move into a more
spacious apartment, and a will increase, and vice versa; so that
==
according as Ri = RB
at ~^~ <~On
the other hand the curves representing, in rectangular coordinates,
the relation between rent and area available per head, necessarily
slope from left to right downward. If it were true, as sometimes
stated, that the more a man has, the more he wants, economic
equilibrium would be an unstable condition.
This example is presented here with reservation. There may be
various complications in practice that may form obstacles to the
simple application of the principle indicated. But it will serve
to show how a perfectly rigorous justification may exist for the
application of the principle of Le Chatelier outside the field of plain
energetics and thermodynamics. Where, and only where such
justification can be clearly shown to exist, there it will be permissible
and useful to apply the principle. Applications made broadcast,
without prior examination of the parameters involved, perhaps
without any thought at all of reasonable parameters, are of little
if any worth.
One other word of caution must be said, for which the example of
area and rent will furnish a suitable illustration. Before we apply
11
Owing to the custom of counting heat absorbed by a system as positive,
but work done upon it as negative, the relation analogous to that of (2) takes
the form, in the case of heat energy,
dQ <- n ,. <
^ according as 9 j =0,Ql ^
where Q is the quantity of heat absorbed by the system at a temperature from
a source at the temperature 0. Here the Qd curves slope upward from left to
right. Cf. A. J. Lotka, loc. cit., p. 36.
DISPLACEMENT OF EQUILIBBIUM 289
the principle to any particular parameter, we must be sure that the
contemplated change will modify this parameter alone, and not
also at the same time others that are in principle, if not in physical
fact, to be regarded as independent. So, for example, one reason
why the example of area and rent was presented above with express
reservation is that, ordinarily at any rate, it may be difficult or impossible
to modify the area of a population without modifying at the
same time certain other features, such as the supply of nutriments
furnished in the soil, etc.
On the whole, so far, it must be said that the result of a careful
analysis of the principle of Le Chatelier yields negative results, so
far as practical application to biological systems is concerned. The
chief conclusion is that great caution must be exercised in employing
the principle. This result may be somewhat disappointing, but
it is for that none the less important. Facts are stubborn things;
it seems a pity to demolish the idol of a pretty generalization, but
in such things we cannot permit the wish to be father to the thought.
And the idol is not wholly demolished in fact his hitherto doubtful
title to certain domains has been established on a clear basis. But
his province must be recognized as very definitely bounded.
DISCUSSION OF DISPLACEMENT OF EQUILIBRIUM INDEPENDENTLY
OF LE CHATELIER'S PRINCIPLE
In view of the limitations in the field of strict applicability of the
principle of Le Chatelier, we are in general forced to consider
separately each particular case of displacement of equilibrium.
How such cases may be treated may be exemplified by the following
two instances.
Case 1. Displacement of Equilibrium between Food and Feeding
Species. Consider a species $2 of mass Xz ,
which requires
for its (equilibrium) sustenance of a mass k2Xz of food. Let this
food be derived exclusively from the slain bodies, of total mass
di XT,, of a species /Si. Let a fraction e of all the deaths in Si be
those caused by S2 feeding upon Si. Then
k2X2
= ediXi (1)
--
290 ELEMENTS OF PHYSICAL BIOLOGY
It is generally in the interest of the species $2 to reduce this ratio
to a minimum, especially in such a case as that of a domestic
species Si, kept by man ($2) to provide him with flesh food. For
the species Si itself consumes food, and is thus directly or indirectly
a tax upon the system. In fact, the species Si is merely a sort of
food factory for S2 ,
and the less of Si is required to produce the
requisite amount of food kzX2 ,
the more efficient is *Si as a food
factory.
We may therefore enquire what the formula (2) tells us regarding
the efficiency of Si as a food producer.
Xi h
It will be observed that the ratio a = = may be reduced
Jiz etti
in two ways by operating upon the species Si (operating on S2 ,
it
might be reduced by diminishing 7e2 ;
but we will exclude this from
consideration) ;
an increase in either e or in di will bring about this
result of reducing a. Now e would be increased if the species $2
helped to protect S\ from its other enemies. This, of course, is one
of the obvious expedients employed by man toward his domesticated
sources of sustenance.
XBut the species $2 may also operate to reduce the ratio a. =
As
A*
by increasing di, and it may do this in several ways,
etti
We may write
(3)
(4)
where Ni is the number of the population of Si} mi the mass per
head of this living population, TO/ the mass per head of the individuals
slain by SZ) and j is a factor, namely -, Evidently dij is the
TTt/]^
death rate per head in the population Si; to simplify matters
we may assume that j is (nearly) unity, so that d\ represents directly
the death rate per head in Si.
It is evidently possible to increase di} the death rate per head in
Si, without disturbing the equilibrium, provided that the birth rate
61 is increased in equal amount. There are several ways of accomplishing
this. The most obvious is systematic breeding. We
may briefly consider the analysis of an ideally simple case in point.
DISPLACEMENT OF EQUILIBRIUM 291
Let bi be the natural birth rate per head and di the death rate
per head, in a population of NI individuals of a food species Si.
Of the total deaths, let pNi occur through various other causes,
while qNiN2 are clue to the destruction of Si by the species S2 that
feeds upon Si.
In equilibrium, then, we have
&i
- di = bi
- qN'2
- p = (5)
N2
= b
^2 (6)
9
Furthermore, let the species $2 ,
when in equilibrium, consume
fNz individuals of species NI, so that
#2 = qNiNz (7)
Ni = ^ (8)
Now let $2 "cultivate" the species $1, so that the birth rate of the
latter is raised from bi to bi + aN2 .
The conditions for equilibrium now are
&i + ffNt
' - qNa' -p = (9)
Nz
'
= ^ (10)
q-aNI
= '
as before (llj
The effect of this cultivation, then has been, in this case, to leave
the population of Si, the food species, unchanged. But the feeding
species S2 has increased in the ratio
q-aThis
result could hardly have been foreseen by the aid of the
principle of Le Chatelier.
In this argument it has been supposed, as a first approximation,
that q is a constant. In point of fact q will no doubt be somewhat
modified when the species Si is "cultivated" by Sa. The effect of
such modification of q would then be superimposed upon the effect
derived in the argument set forth above.
292 ELEMENTS OF PHYSICAL BIOLOGY
Case 2. Change of Circulation through Moving Cycles. Among
the moving equilibria in nature an important class are those which
arise in systems traversed by matter in cyclic transformations.
Consider a very simple example of a cyclic transformation chain,
such as that in which, of three components Si, Sz ,
S3 ,
the first
becomes converted into the second, the second into the third and
the third returns to the first, after the pattern:
7 \$3 < -'
$2
We may write the equations of the transformation
_ _ _ ci (~y- y y \ n T" /-. Tf
/'iv-ii, A, AS; ga^s g\A
at
dZ2
=
dt
' "'
dt
' '
where in the most general case gi, g2 , gs are each of them a function
of Xi} X%} X$.
When a steady state is established, so that the derivatives -7-
Cvt
vanish we have, evidently
7? ~Y -V /1 o\
A-i-^-z-^-s (Id;
from which it is seen that, in the steady state, that component is
most abundant, which has the slowest proportional rate of decomposition,
the smallest g. This smaller g acts as a "bottle neck"12
in the cycle, causing material to accumulate in front of it. It acts
as a brake, as a limiting factor, upon the rate of circulation through
the system.
12
1 am borrowing this term from the language of efficiency engineers, who
employ it to denote a point, in a consecutive series in industrial operations, at
which the progress of work is arrested by a local "limiting capacity."
DISPLACEMENT OF EQUILIBEIUM 293
If the total mass of Si, $ and Ss is in some way fixed, so that we
put Xi + ^2 + Xa = M = const., we have, evidently, for a steady
state
jA. I
~~ . , ., , . _. .
*' * *
.
~~~
IrL \ 14:7
Similarly
Xz = *W M.
(15)
X9 --M (16)
f/l ~|- (72 + f/8
The circulation I, i.e., the mass circulating through the system
per unit of time, is evidently given by
M (17)
tfi -I- f/a + f/a
and
.^/^.ff^lM.)
OHi (r/i H- r/z -I- <j*r
'-"-'--
M approx,, if f/i IB smtill (19)
Furthermore,
Hence
Hence, to increase the circulation, the best effect, other things
equal, is obtained by seeking to increase the smallest g.
This result, also, the Lo Chatelier's principle seems incompetent
to predict,
294 ELEMENTS OF PHYSICAL BIOLOGY
Some Significant Cases of Instability. It lies in the nature
of things that a special interest attaches to stable states and stable
systems. They represent the lasting features in the changeful
landscape of nature that is what we mean by stability. They are
the survivors in the struggle for existence.
But the class of unstable states and systems is not without a
special interest of its own. Departures from stability, so far from
forming insignificant exceptions, are found to play an important
role both in normal and in pathological life processes.
The body does not always react* in the direction of a restored
equilibrium when exposed to a disturbing influence. The opposite
type of reaction is sufficiently frequent to give occupation and means
of livelihood to a distinct profession, whose business it is to prevent
this adverse type of reaction from proceeding to the point where it
sets a limit to life. A particularly pernicious form of this adverse
reaction to influence tending to disturb the life equilibrium is that
known among medical men as the vicious circle. A. departure from
equilibrium, instead of stimulating a compensating response, provokes
a further departure in the same direction, with cumulative
effect. If this process goes on only to a certain point and then
stops, there may be little or no damage sustained. But conditions
may arise which, by their very nature, produce a continued accumulation
of deviations from the stable equilibrium position, until the
limits compatible with the continua,tion of organized life processes
are exceeded. So, for example, a person exposed to hardships
through adverse economic conditions, suffers from malnutrition;
this lowers his resistance to bacterial infection; he contracts tuberculosis;
there is a loss of appetite, and malnutrition not only is
accentuated, but may become fixed even if better economic conditions
are provided. And now a closed cycle, a "vicious circle,"
is established, and the disease grows like an avalanche tumbling
down a slope,
13
gathering weight at each revolution of the cycle, on
the downhill path to dissolution, thus
13
A condition analogous to that represented by the middle (ascending) limb
of the van der Waals' curve for the relation between pressure and volume of
gas.
niSPL.AOKMMNT Ol'
1
KQ'U I LI BRtUM 205
r
loss of .'ippetile
---
I
lowered reKistance tc
infection
1 , 035
0.803
1 . 477
*
SU(,inl,ie,al Build-in Md-ropolK-tiu'Lifo Jnauriuicc Company, July, .1923,
p. 8 and a forllioominff Htudy by Prof. I. V. Iliacook, Yule University.
f Five per con I, above or below average weight for age and height.
296
KLEM1SNT8 OF
Cumulative
>f
><Z Mf#*a
^3f
.
59.
of aa
',,,-v
: .^-^v v h i ST *
>
'
(/<
l
.
'"*"''" co'"- rol
reptiles
DISPLACEMENT OF IIQUILIBRrUM 207
were swept on upon a tidal wave of unremitting growth, until their
cost of living exceeded their earning capacity, until their very (strength
proved their fatal weakness; unable to gather, in a day's run, sufficient
food to fill their .monstrous paunch, they became the victims
of their colossal ambition; their carcases remained enshrouded in
the rocks, monumental wrecks by the wayside, where, the caravan of
evolution has passed on. . . . .
But there is another view which may account for this chapter in
Natural History, as lias boon pointed out to me by Mr. ,1. !. S.
Haldanc in correspondence the gist of which is, briefly, as follows:
The large reptiles of the secondary age had large, pituitary glands.
It was probably the secretion of these that determined their large
size. Now a large animal has a high blood pressure- -a fact which
is exemplified, in a way, in statistics of clinical observations on
human material, witness table 32 p. 295. With high blood pressure
there would be a tendency for the capillaries to leak. The thing
that stops them leaking is pituitrin. Thus selection will tend to
increase the pituitary gland in large animals. Unless it is possible
for variations to arise increasing the output of pituitrin but not that
of the anterior lobe, successive generations will tend to become bigger
and bigger, till they ultimately perish of hyperpituitarism.
1
'
1
Sometimes, it seems, the trend of evolution by a cumulative cydc
may be grotesque rather than pernicious. This might well be the
explanation of such singular vagaries as those observed, for example,
in a group of fishes related to the shark. This series (fig. />{)) (to
which my attention was drawn by Mr. J. T. Nichols of the American
Museum of Natural History) exhibits progressive flattening of the
body accompanied by thinning out of the tail, until the latter I'M
reduced to a more lash.. Hero some gland controlling growth may
have become increasingly active in response to selection operating
on some useful quality, and meanwhile some secondary effect had
to be taken into the bargain, regardless of utility.
But cumulative cycles do not always work toward destruction or
toward mere caprices devoid of utility. The effect of gathering
momentum is equally potent in the constructive sphere.
'
Perhaps'
the most striking examples of this have occurred in the realm of
14
Somewhat similar views have boon oxprosHod by A. I I. Hturtcvant. Science
1924, vol. 59, p. 579.
'
2gg ELEMENTS OP PHYSICAL BIOLOGY
mental phenomena. The oyc
lfi
and the hand have probably contributed
more than any other single cireumstancn to the evolution of
the human mind up to its present level. The possession of tin
agile
member gave opportunity for exercise of the menial faculties, this
in turn reacted towards increased development, of the tactile sense
and manipulative skill of the hand, and so on in a cumulative cycle.
A similar "cumulative cycle" has probably had a,
large part, in
developing our faculty of speech. In the present stage of our
development, we find it almost indispensable, in Ihinkiny, to use
language, a vehicle whose primary function would seem to bo the
transmission of thought from one individual to another, and which
would seem wholly superfluous in the traffic of thought within the
precincts of one mind, as money currency is needless in the giveand-take
within the same household. Which, then came first,
thought, or language? Neither, of course, can claim dear precedence.
They must have developed together, in mutual stimulation.
The habit of communicating thoughts to others- must have
reacted upon the thinker and made him more perfect, first as a
thinker, and then again, in turn, as a communicator of thought, as
a speaker; and so, in a species of cycle, not vicious but benign,
thought promoted speech arid speech furthered thought,, in an
endless chain of cause and effect, such as thai; which we witness in
the somewhat useless but rather entertaining spectacle of a wit
chasing its tail, or (to turn from, the fine to the useful arts), in the
economically more significant performance of the donkey urged to
unwonted productive effort by the hope of (witching up with that
elusive wisp of hay dangled by the driven' before the poor beast's
nose. Cause and effect are so intermingled in a chain of alternations
that they have become indistinguishable,
To cite in third place a more modern instance, the mutually
fertilizing influence upon each other of pure and applied science
falls into the same class of benign cycles. Here also cau.se and
effect are so intermingled that the relative merits of the two not
very clearly separated branches of scientific endeavor are hardly
1B
Of. G. H. Parker, Proc. Am. Phil. Soc-.., 1022, vol. 01, p. 107; alno (1. Elliot,
Nature, 1923, vol. 112, p. 443.
DISPLACEMENT OF EQUILIBRIUM 299
a subject for profitable debate. If, as some18
bold, "the final justification
of science is the power it creates for the use of mankind/'
then we must call to remembrance that this "power must be
created before it is used."17
The instances cited should be sufficient to demonstrate how
effectively resourceful nature makes use, in her economy, of instability,
with its cumulative potency, as a progressive force; as
well as of stability, the essentially conservative element in evolution.
Indeed, we, of the human race, have good reason to be mindful
of this fact, for the wholly unparalleled rapidity of our scientific and
industrial evolution in past decades is itself the most brilliant
example of instability and its cumulative power as a factor in
evolution.
16
The writer is not among these, if by final justification is meant the only
sufficient justification. No one would think of demanding such justification
for art. Why require it of science? Such an attitude towards her is like that
of the man who, having received repeated favors from his fellow, begins to
acquire the habit, and to look upon such favors as due to him, and as the sole
justification for the other's existence.
17
C. S. Minot, Science, 1911, p. 119.
CHAPTER
THE PARAMETERS OF STATE
In dealing with any natural phenomenon especially one of a vital nature,
with all the complexity of living organisms in type and habit the mathematician
has to simplify the conditions until they reach the attenuated character
which lies within the power of his analysis. Karl Pearson.
Little has been said, so far, of the parameters P employed to
define the state of the systems under consideration. In the earlier
chapters these parameters have, in fact, been largely eliminated
from discussion by restricting the treatment to the case of evolution
under constant parameters P; subsequently the special cases
of evolution with slowly changing P's, and the influence upon *|
equilibrium alone of changes of unrestricted kind in the P's have
been discussed; but all of this from a general standpoint, without
giving much thought to the particular nature and properties of
these parameters. It is desirable now to give some attention to
this hitherto neglected phase of the subject.
The simplest, and in many respects a very illuminating example
of the nature and function of the parameters P is furnished in the
thermodynamic treatment of physical systems. Here we are accustomed
to the use of such parameters as pressure, temperature,
surface tension, etc., to define the state of the systems under con-
sideration.
Topographic Parameters. Obviously there is much latitude
in the choice of such parameters; for if any parameter P can be
employed to define the state of a given system, any single-valued
function F (P) of that parameter will also serve, though certain
selections of parameters may be found much more advantageous
in practice than others. In particular, it is found, in systems
amenable to thermodynamic treatment, that P's can be so selected
that they appear as the intensity factors of an energy.
1
This
selection has actually been made in the example cited above. Or,
alternatively, any one of the P's so selected, can be replaced by
1
Helmholtz, Die Thermodynamisch-chemischen Vorgange 1882 (Ges. <
Abh., vol. 3, p. 958) ; P. Ehrenfest, Zeitschr. f.
phys. Chemie, 1911, p. 234.
\
300
PAEAMETEES OF STATE 301
the extensity factor of an energy. So, in place of the pressure p
we can introduce the volume v, these two parameters being connected
by a functional relation of the form.
v = <p(p, Zi, Za,
. . . Zn> (1)
The parameter v (volume) is almost the simplest type imaginable
of a topographic parameter. In many systems commonly considered
in physical chemistry the only way in which the topography
of the system plays any appreciable role in the processes going on
therein is through the volume defined by the boundaries of the
system. Even the shape of the boundary is in most cases immaterial.
In the systems in which organic evolution is proceeding, the
situation is very different. In one respect the topographic parameters
are often, in this case, even simpler than in the physicochemical
example, namely in this, that living organisms (except
aquatic species) make their excursions, extend their activities, essentially
in a space of two dimensions the earth's surface, or at
least a rather thin shell near that surface. Hence we are interested
in areas rather than volumes; in place of a parameter v, volume, we
may expect to find figuring in the discussion a parameter a, area.
But aside from this slight simplification (which does not always
apply), the influence of topography in systems in the course of
organic evolution is immeasurably more complex than in the simple
physico-chemical systems that form the chief subjects of study
in the laboratory and in theory. Indeed, the conditions presented
in nature are so complex that we can hardly hope to construct any
systematic mathematical analysis of this phase of the subject, except
by the expedient of dealing in somewhat radical abstractions,
such as evolution "in a uniform environment" or, perhaps, in an
environment reproducing in very greatly simplified form some of
those principal geographic features that are typical of our globe.
There is something unsatisfactory in such abstractions that seem
rather far remote from conditions actually met in nature. But it
must be remembered that such abstractions are a necessary, and,
as experience has abundantly shown, a very effective aid to our
limited mental powers, which are incompetent to deal directly with
unexpurgated nature in all its complexity. Neither should it be
forgotten that the worker in the laboratory, though he may seem
to be nearer to nature, himself is dealing essentially in abstractions.
302 ELEMENTS OF PHYSICAL BIOLOGY
When the physical chemist investigates a chemical reaction in a
constant temperature bath, he is not copying nature, where constant
temperature is an exception, but is deliberately establishing
an "unnatural" situation. He does this in order to separate the
influence of one factor upon the course of events from that of a
multitude of others; feeling confident that when he has gained an
insight into the workings of such a simplified and, in a sense, unnatural
system, he will be the better equipped to understand, or at
least to make a further study of more complicated systems, approaching
more and more nearly those occurring in nature. It is precisely
the same principle which justifies us, in "substituting an ideal, upon
which it is possible to operate, for intractable reality,"
2
when we
essay the systematic treatment of natural processes by mathematical
analysis. So, for example, Karl Pearson in his memoir on Random
Migration treats among others the case in which "breeding grounds
and food supply are supposed to have an average uniform, distribution
over the district under consideration;"
3
and the simple case of
"migration into a cleared rectangular area," etc. Somewhat similar
topographic simplicity is assumed as the basis of studies of Brownlee
on the Mathematical Theory of Random Migration and Epidemic
Distribution. We shall have occasion to refer to these studies again
in another connection. It is not intended to follow up this phase
of the subject here.
Neither will any attempt be made to sketch here even in outline
the empirical side of the subject, our observational knowledge regarding
the dependence of life in its various forms upon the parameters
of state. There is a ripe and extensive literature available
on this special phase of biology, which it is unnecessary to duplicate
here. It will suffice to refer to standard works on geographical
biology and to general ecology.
4
2
Nature, 1922, p. 764.
3
Draper's Company Memoirs, Biometric Series III, 1906.
4
The following may be mentioned: A. Engler and 0. Drude, Die Vegetation
der Erde, Sammlung pflanzengeographischer Monographien (a cyclopedic
work in many volumes). A. F. W. Schimper, Clarendon Press, 1903, Plant
Geography. E. Warming, Clarendon Press, 1909, Oecology of Plants. A. R.
Wallace, 1876, The Geographical Distribution of Animals. F. E. Beddard,
1895, Textbook of Zoogeography. H. Gadow, Cambridge University Press,
1913, The Wandering of Animals. E. L, Trouessart, 1922, La Distribution
Geographique des Animaux,
PARAMETERS OF STATE 303
In physico-chemical systems the topographic parameter v (volume)
is the capacity factor of an energy, as had already been noted; and
associated with v is what may be termed a conjugate parameter
Pi (pressure), which is the intensity factor of the energy in question,
i.e., that factor which determines the direction of any change in
the capacity factor v, according to the scheme
= according as pi
= pe (2)
dt
< ^
where pe is the external pressure.
5
The Intensity Law in Organic and Economic Systems. Is
there anything corresponding to the relation (2), the Intensity Law,
as it has been termed, in the case of the topographic parameter a
(area) which enters into the definition of the state of a system in
the course of organic evolution? This matter has already been
referred to, in a way, in discussing the Principle of Le Chatelier.
It was there noted that, in the case of human population area and
rent are related to each other in accordance with a scheme of the
type (2). More generally, supply and demand in economics stand
in a relation of this type, and accordingly present a certain analogy
to the capacity and intensity factors of an energy an analogy which,
by some writers, has been construed as actual identity in kind,
prices having, by these writers, been identified with the intensity
factor of an "economic energy." Now energy is a perfectly definite,
measurable thing, of definite dimensions. Those who thus speak of
a special form of "economic energy" should be prepared to give us
at least some indication how this energy is to be measured, in the
customary units of energy. No such indication is forthcoming. On
6
The relation (2) is essentially the Helm-Ostwald Intensity Law. Although
this law is not as universal as its sponsors would make it appear, yet it has a
certain field of utility. For a critique of this law see M. Planck, Eight Lectures
on Theoretical Physics, Columbia University Press, 1915, p. 11. The
form of the relation (2) may be taken as the definition of a pair of conjugate
parameters. However, the definition must be made a little more general to
cover certain cases. We shall say that G, g are conjugate parameters if either
dG ^ n >
-5- < according as gt
< ge (1)
dG > . .. <
-j- according as g.
= g, (2)
304 ELEMENTS OF PHYSICAL BIOLOGY
the contrary, as we shall see in dealing with the dynamics of evolution,
the economic equivalent of a form of energy (which is something
quite different from its mechanical equivalent), is not constant
but variable though it tends to approach, or to fluctuate
about, a certain value.
On the mistaken identification of prices and related economic
quantities with the intensity factor of an energy, some authors have
sought to build a system of biodynamics (social dynamics).
6
The
analogy which certain conjugate parameters of the perfectly general
kind bear to intensity and capacity factors of an energy present
the opportunity for such efforts, which are, in themselves, well
worth while. But it must not be forgotten that the result of such
efforts can be only a species of quasi-dynamics, something analogous
to, but not identical with, the dynamics of physical forces.
Just what the relation between such quasi-dynamics and true dynamics
may be, is a separate problem, to which we shall have occasion
to give some attention in the section devoted specifically to the
dynamics of life-bearing systems.
The paying of rent in coin of the realm is, of course, a phenomenon
peculiar to the human species. But the peculiarity is one of
mode of manifestation rather than of inherent quality. We may
speak of the rent per unit area that the representative individual
is willing to pay as a measure or at least an index of the "population
pressure." Now this population pressure this willingness to
sacrifice effort for the sake of gaining elbowroom is present quite
independently of our peculiar method of expressing it in terms of
rent. It exists also among other species, though we may lack so
convenient a gauge for it as we have, in our own case, in rent. We
shall see later how at least a quantitative conception of such biophysical
(economic) entities as population pressure and the like can
be gained on a general basis, which applies to species other than
human.
6
Compare G. Helm, Die Lehre von der Energie, 1887, pp. 72 et seq.; Ostwald,
W., Energetische Grundlagen der Kulturwissenschaften, Leipsic, 1909,
p. 155. Die Philosophic der Werte, Leipsic, 1913, pp. 260, 314-317, 326, 328.
Among other writers who touch on the subject of the relation of economic value
and price to energy are :
Budde, Energie und Recht, Leipsic, 1902, p. 56; Winiarski,
"Essai sur la Mecanique Sociale," Revue Philosophique, 1900, vol. 49,
p. 113. See also J.
Davidson, Qu. Jour. Economics, August, 1919, p. 717.
PARAMETERS OF STATE 305
For the present, without making here a closer analysis of the
conjugate parameters a, p, (area, rent or population pressure) it
will suffice to point out that the mere existence of the relation
= according as p\
= pe (3)
at < *^
may give into our hand the means of drawing certain conclusions
regarding the behavior of the system, Quite independently of the
intimate -nature of these quantities, a, p. In illustration of this it is
only necessary to refer once more to the discussion of the principle
of Le Chatelier in Chapter XXII.
Distant Analogy to Gas Law. Again, still without any searching
analysis of all the physical implications of the parameters a
and p, we may note certain facts regarding the relation.
a = $(P) Zlt Xt,
. . . Xn) (4)
which connects the conjugate parameters a, p, just as pressure p
and volume v, in physico-chemical systems, are connected by the
relation
v = p(p, XL, Xa> . . .) (5)
This latter, in the simple case of a gas takes the form
v = m (6)
P
m. ,, , ., ,
or, since is the density d
T = RT (7)
d
constant, at constant temperature
Let N be the total number of a (human) population, a the area
occupied by it, and Ni the total income of the population. Let p
be the rent per unit area; and let the population spend a fraction
R' of its income on rent. Then evidently
pa = NR'i (8)
306 ELEMENTS OF PHYSICAL BIOLOGY
A.r
Putting = population density = d' we haveto
a
~^R'i (9)
a
It will be seen that there is a certain analogy between the formulae
(8), (9) as applied to the relation between rent (population pressure)
and area occupied by a population, on the one hand, and the formulae
(6), (7) as applied to the relation between the pressure p and
the volume v of a gas. The analogy is particularly close if the income
per head i, and the factor Rr
are constant as a changes. The
former may be true, approximately, if the inhabitants of area a
derive their income from some extraneous source quite independent
of this area. It will not be true, even in rough approximation, if
the inhabitants derive their income from the produce of the area a
itself. It is not the author's intention to emphasize unduly mere
analogies. Nevertheless, the one here presented seems worthy of
passing notice. Compare also E. Woodruff, Expansion of Races.
The actual relation between population density and rent or land
value is a subject of great economic interest. Research in this
subject is in progress under the auspices of the Institute for Research
in Land Economics, Madison, Wis., but has not, at this
time of writing, matured to published results.
Law of Urban Concentration. An empirical law of urban concentration
was pointed out some years ago by F. Auerbach (Petermanns
Mittheilungen, 1923, p. 74). Arranging in order of magnitude
the cities of a given country, he found that the product of
population and ordinal number (rank) was approximately constant.
Thus, plotting rank against population, he obtained a
hyperbolic curve, or, plotting the product of these two quantities,
he obtained, roughly speaking, a straight line. For some of his
curves the reader must be referred to the original publication.
The illustration (fig. 60) shows the graph obtained by plotting the
logarithm of the population of the cities of the United States (1920)
against the logarithm of their respective rank. In the higher ranks
only every fifth city has been plotted. It will be seen that, excepting
cities of rank 4, 5 and 6, the plot does approximate quite
closely to a straight line. The slope of this line, however, is not
exactly unity, as demanded by Auerbach's law, but 0.93, so that the
PARAMETERS OF STATE 307
actual law of urban concentration in the United States, in 1920,
was, within the limits indicated, given by
where P denoted the population of the city of rank R.
^ rOPULATiCFi
(Tens of thoussndsj
3 4- 5 6 T 9 9 10
RANK
CO UO 40 50 GO TO 09(5 SM
FIG. 60. LAW OF URBAN CONCENTRATION
Graph obtained by plotting as ordinates, on logarithnic scale, Population of
United States Cities, and as abscissae the corresponding Rank (in order of
magnitude), also on logarithmic scale.
It may be left an open question how much significance is to be
attached to this empirical formula. We shall meet with a similar
relation in Chapter XXIII in dealing with Willis' theory of Age and
Area.
308 ELEMENTS OP PHYSICAL BIOLOGY
The Biological Background of Population Pressure. In the
preceding paragraphs we have accepted it as a fact that there is
at any rate some degree of relation between population density and
the "desire for expansion" which finds expression, in the human
species, in the willingness to spend a certain fraction of the income
upon ground rent or its equivalent in interest and taxes upon real
estate. It is not proposed to attempt here any searching analysis
of the precise physical significance of this desire for expansion. But
that it is ultimately referable to very cogent biological and physical
factors is self-evident. The degree of crowding of a group of organisms
affects both its death rate (mean length of life) and also
its rate of reproduction, as well as, no doubt other significant vital
fimctions. As Pearl and Parker7
remark, in their paper on the
influence of population density upon rate of reproduction in Drosophila
:
It has long been known that degree of crowding of organisms in a given
space, or the density of the population, has an influence upon various vital
processes of the individuals composing the population. In the matter of
growth Semper8
and before him Jabez Hogg showed that volume of water
apart from food and other conditions has an influence upon the rate. This
subject has again been studied recently by Bilski. 10
Farr11
showed that there
is in man a definite relation between population and death rate. This old
work of Farr has recently been gone over carefully and confirmed by Brownlee.
12
Drzwina13
and Bohn show that a particular concentration of a toxic
substance, just lethal for a single individual in a given volume of water (working
with such organisms as infusoria, planarians, hydra, tadpoles, etc.), will
be sub-lethal if several individuals are present in the same fixed volume of
water.
Influence of Population Density on Rate of Reproduction. Pearl
and Parker14
have determined experimentally the relation between
rate of reproduction and the density of a population of Drosophila
(fruit flies) in a universe of constant volume (glass bottle). This
7
R. Pearl, and S. Parker, Proc. Natl. Acad. Sci., vol. 8, 1922, p. 212.
8
K. Semper, Animal life as affected by the natural conditions of existence.
Fourth Edition, London, 1890.
9
Cited from Semper, loc. cit,
10
F. Bilski, Pfliiger's Arch., vol. 188, 1921, p. 254.
11
W. Farr, Decenn. Suppl. Reg. Gen., 1861-1870.
12
J. Brownlee, Jour. Roy. Stat. Boo., vol. 82, 1919, pp. 34-77; vol. 83, 1920,
pp. 280-283.
13
A. Drzwina and G. Bohn, C. R. Soc. Biol. Paris, vol. 84, 1921, pp. 917-919.
R. Pearl and S. Parker, Proc. Natl. Acad. Sci., vol. 8, 1922, p. 212.
PARAMETERS OF STATE 309
relation is found to resemble in form Farr's law for the death rate
D in a human population of density d:
log D = log a + k log d (10)
in which a and k are constants. Pearl and Parker found
log y = 1.54 - O.OOSz - 0.658 log x (11)
16
cS
Of
I
10
10 SO 3O ^0 &0 SO 70 ffO
fl/fS PER BOTTLE
FIG. 61. RELATION BETWEEN RATE OF REPRODUCTION IN DROSOPHILA (FRUIT
FLY) AND DENSITY OF MATED POPULATION
The circles indicate observed values; the drawn-out curve is computed
according to equation (11). After Pearl and Parker.
310 ELEMENTS OF PHYSICAL BIOLOGY
where y denotes the number of imagoes per mated female per \
and x denotes the mean density of the mated population (measured a s
flies per bottle) over a sixteen-day period. The results obtainea
by Pearl and Parker are exhibited graphically in figure 61, the small
circles denoting observed values, and the drawn-out curve values
computed by the formula cited above. It should be noted, however,
that in Farr's law the coefficient k of log d is positive, whereas
in the Pearl and Parker formula it is negative; death rate in-
40 "60 ~60 100 /10" 14$ 160 180
OF FLIES IN BOTTLE AT BBSfNMtf&
FIG. 62. RELATION BETWEEN MEAN LENGTH OP LIFE AND POPULATION DENSITY
IN DROSOPHILA
The curve is a freehand smoothing of the observations indicated by the
small circles. After Pearl and Parker.
creases with population density, whereas rate of reproduction, in
this series of experiments, was found to decrease as the density in-
creases.
Influence of Population Density on Duration of Life. The same
authors have also investigated the relation between population
density and duration of life in Drosophila.
15
Their results are
shown graphically in figure 62. The smoothed curve exhibits a
very interesting feature. While crowding unmistakably diminishes
1S
R. Pearl and S. Parker, Am. Jour. Hygiene, vol. 3, 1922, p. 94; Amer.
Naturalist, vol. 56, 1922, p. 312.
PARAMETERS OF STATE 311
the average length of life, as one would naturally expect, the curve
does not fall continuously from left to right, but has a maximum.
The flies do not thrive best when they have the most ample space
per individual; there is an optimum density which is best suited to
their needs for company or for some other obscure factor supplied
by a certain moderate amount of crowding.
TOPOGRAPHIC PARAMETERS DURING PERIOD OF DIFFUSION
Willis' Theory of Age and Area. As regards the influence of
those topographic parameters which define the boundaries of the
system, a special case arises during that period in the life of a body
of organisms, when its spread has not yet extended to those boundaries.
Certain aspects of the phenomena presented during this
period of diffusion have been made the subject of a painstaking
study, conducted with much originality of view, by C. J. Willis.
16
One may not agree in all points with Dr. Willis' conclusions, but
the material of fact collated by him is in itself significant and of
value. As Prof. W. Bateson in a review of this book remarks : "To
have hit on a new method of investigating even a part of the theory of
evolution is no common achievement, and that the author has done
this cannot in fairness be denied." Dr. Willis' principal thesis is
essentially this, that the area occupied by a biological species is a
measure of its antiquity in evolution. To be more precise, and to
quote the author's own words:
'
The area occupied at any given time, in any given country, by any group of
allied species at least ten in number, depends chiefly, so long as conditions
remain reasonably constant, upon the ages of the species of that group in that
country, but may be enormously modified by the presence of barriers, such as
seas, rivers, mountains, changes of climates from one region to the next, or
other ecological boundaries, and the like, also by the action of man and other
The thesis, thus stated, is "well hedged" with qualifications.
Professor Bateson remarks: "Every evolutionist agrees that, apart
16
Age and Area: A Study in Geographical Distribution and Origin of Species;
Cambridge University Press, 1922. For review see Nature, 1923, p. 39;
Science Progress, January, 1923, p. 474.
312 ELEMENTS OF PHYSICAL BIOLOGY
from disturbing elements, area is a measure of age.
17
.... We
are, however, asked to believe that in practice this mode of estimating
the age of a species is, on the whole, trustworthy; that
endemic species and varieties in general can and must be for the
most part accepted as new starters in evolution, and not as
(remnants of) survivors."
Dr. Willis supports his views by observations gathered chiefly in
Ceylon and New Zealand. His evidence is not altogether convincing.
Professor Bateson in the review already quoted remarks pointedly:
"On any theory of evolution endemics (and rare species) must
be in part novelties and in part relics; but why, apart from the
theory of Age and Area, we should believe that endemics are in
such great majority novelties I do not clearly understand, for though
we know little of origins, we are certain that myriads of species
have become extinct. It is surely contrary to all expectation that
the process of extinction should be in general so rapid, and the final
endemic phase so short that the number of species in that final
stage should be so insignificant." Referring to the same point A.
G. Thacker remarks: "Species and genera do die out, and therefore
there are diminishing ranges as well as expanding ranges. .
. , . Dr. Willis thinks that veiy few of the small range species
are, in fact, decreasing.
55
For a further critique of the theory of
Age and Area the reader may well be referred to the two reviews
already cited, while Dr. Willis
5
own presentation of his case will be
found expanded in detail in his book Age and Area, Cambridge
University Press, 1922. Quite recently (I add this in correcting
proof) Prof. G. V. Yule has lent his able advocacy to Willis
5
theory in an extensive paper published in the Phil. Trans. Roy.
Soc., 1924, vol. 213, Series B, pp. 21-87, Some of the assumptions
underlying Professor Yule's argument do not seem to commend
themselves to the critical reader, in particular his supposition that
17
'"'Tor this and other reasons Dr. Willis
5
findings seem hardly competent
to furnish a basis of attack against any otherwise established or suggested
theory of evolution. On this point Dr. Willis himself does not seem wholly
consistent, for though he advises in one place that it would be 'wiser to abandon
natural selection
5
as the general principle that has guided evolution, in
another place he admits that 'nothing can come into lasting existence without
its permission.' This admission is all that any thoughtful adherent of any
theory of evolution asks.
55
A. G. Thacker, The Dynamics of Distribution,
Science Progress, 1923, vol. 17, p. 474.
PARAMETERS OF STATE 313
the number of new species thrown is independent of the number of
individuals.
Turning from the theoretical and debatable portion of Dr. Willis'
contribution, to the factual material presented by him, we are confronted
by a number of very remarkable relations, such as those
represented graphically in figure 63. In this drawing ordinates
represent the number of genera in the several natural families of
plants and animals indicated, and the corresponding abscissae
represent the number of species in each genus plotted. So, for example
(fig. 63), of the family Cornpositae 1143 genera were noted,
as follows:
446 genera of 1 species
140 genera of 2 species
97 genera of 3 species
43 genera of 4 species
55 genera of 5 species
etc.
An examination of these drawings clearly brings out the following
facts: The rnonotypic genera, with one species each, are always the
most numerous, commonly forming about one third of the whole
group; the ditypics, with two species each, are next in frequency,
genera with higher numbers of species becoming successively fewer.
Set out graphically as in figure 63, the genera exhibit what Dr.
Willis calls a "hollow curve" of frequency (in point of fact a hyperbola
of the generalized type), and, as Professor Bateson remarks,
there is no gainsaying the fact that these curves, though collected
from miscellaneous sources, have a remarkable similarity. Perhaps
more striking still is the relation established in figures 64, 65, and
66. The quantities plotted here are of the same character as in
figure 63, but they are plotted on a doubly logarithmic scale, with
the remarkable result that the graphs obtained are in close approximation
straight lines. Thus if x denotes the number of species
in a genus, and y denotes the number of genera comprising x species,
we have
log x + a log y b = (12)
or
xy = const* (13)
314 ELEMENTS OF PHYSICAL BIOLOGY
that is to say, the variables x and y are connected by a hyperbolic
18
relation. It should be noted that this relation covers a wide variety
of cases, including both plants and animals.
Monospecific Genenaal this end of curvs
o
Linepf GeneraJ
of
5spp.
MIXED
CURVES
The
large
dots represent the
Origins
By courtesy of Nature
dumber of species for size of area.)
Mi
T
d h "OT CUrVeS '
^^"-^" !?"? ^ Beginning of each are thenumbers of monotypes.
FIG. 63. HYPEHBOLIC CORVES OBTAINED BY PLOTTING AS ORDINATES THE-\USIBBB OF GBKEBA HAvmc 1, 2, 3, . . . n SPECIES, AND AS ABSCXSSAE
THE N0MBEE n OP SUCH SPECIES
The last curve above shows as ordinates the number of species of endemic
compositae in the Galapagos Islands, as abscissae the corresponding areaso\er which such species have spread. After J. C. Willis.
**** **" ^h^erbola *V = const, falls as a
PARAMETERS OF STATE
315
Mumber of spacfes
10 20 30 40 50 100
1-0 1-2 1-4 16 1-8 2-0
log (N 9 of species)
-2 -4
JS# courtesy of Nature
FIG. 64. RELATION BETWEEN NUMBER AND SIZE OF GENERA op ALL
PLANTS, PLOTTED LOGARITHMICALLY
After J. C. Willis
NS of species
10 30
10.000
100
8 "Tb"~" 1-2
log fN? of species)
By eourttty of Natws
FIG. 65. RELATION BETWEEN NTTMBEE AND SIZE OF RTOIACBAK Pr.OTran
LOGAEITHMICALLT
After J. C. Willis
316 ELEMENTS OP PHYSICAL BIOLOGY
Dr. Willis' interpretation of the remarkable curves obtained by
him may be quoted in his own words (Nature, February 9, 1924,
p. 178):
If species of very limited area and genera of one species (which also have
usually small areas) are, with comparatively few exceptions, the young beginners
in the race of life, and are descended in general from the species of wider
dispersal and the larger genera, and if the number of species in a genus is,
broadly speaking, a measure of its age, the idea at once suggests itself that a
Number of species
FIG. 68. RELATION BETWEEN NUMBER AND SIZE OF GENBEA OF
BEETLES, PLOTTED LOGARITHMICALLY
After J. C. Willis
given stock may be regarded as "throwing" generic variations much as it
throws offspring, so that the number of genera descended from one prime ancestor
may be expected to increase in geometric ratio or according to the law of
compound interest. The number of species descended from one ancestor
might be expected to follow the same form of law with a more rapid rate of
growth. On such a very rough conception it is found that the form of frequency
distribution for sizes of genera should follow the rule that the logarithm
of the number of genera plotted to the logarithm of the number of species gives
a straight line.
PAEAMETBBS OF STATE 317
It follows from the conception stated that the excess of the slope of the line
over unity should measure the ratio of the rate of increase of genera to that of
species. The slope should always, therefore, lie between the limits 1 and 2, for
a slope of less than unity would have no meaning, and a slope exceeding 2
would imply that generic variations were more frequent than specific variations.
Hitherto no exception has been found to the required rule. One group
of fungi tested (Hymenomycetineae) gave a line with a slope very little
exceeding unity (1.08), but the figures found for flowering plants lie between
the narrow limits 1.38 and 1.64, with an average of about 1.43. Snakes and
lizards both give a figure very near 1.50, and the Chrysomelidae about 1.37.
For further details the reader must be referred to Dr. Willis'
book Age and Area, and to a paper by E. S. Pearson, in Biometrika,
August, 1923, vol. 15, of which the following passage, part of that
author's conclusions, may be quoted:
There is no doubt that these principles represent a certain aspect of the
process of evolution, but I believe that Dr. Willis has stressed their importance
beyond the limits which the evidence of observation will bear. They cannot
explain everything, and we have seen that in many cases results which we are
led to predict with their assistance are scarcely borne out, while in other cases
recurrent distributions can also be accounted for on different hypotheses.
Climatic Parameters. The state of a physico-chemical system
is commonly described in terms of its temperature (in addition
to pressure, etc.). Without attaching any deep significance to the
analogy, it may be remarked that systems in which organic evolution
is under way commonly require, as essential parameters to
define their state, the statement of sundry quantities that describe
climatic conditions, such as temperature, humidity, precipitation,
light, etc. The influence of these upon the course of events must
form an essential part of the study of evolution in life-bearing
systems.
The investigations of the influence of temperature upon metabolic
processes an approximate doubling of reaction velocities for
every 10C. rise in temperature we shall here note only very briefly
as belonging rather to the field of biochemistry than to the more
strictly ecological studies in which we are here interested. More
in the line of our immediate interests are certain data gathered in
oceanographic researches, which have already furnished us with
much illustrative material. The influence of light, temperature, and
318 ELEMENTS OF PHYSICAL BIOLOGY
C02 concentration is discussed by G. W. Martin in his paper already
cited.
19
He remarks:
Plants on land receive the full benefit of the sun's rays as we know them.
Plants living under water receive only a portion of the rays that reach the land.
Part of the light that strikes the water is reflected, and the part that penetrates
the water is gradually absorbed in passing through that medium, the red and
yellow rays first, the blue and violet last. This differential absorption is
reflected in the curious and well-known distribution of marine algae according
to color the green kind growing in shallow water, the brown in an intermediate
zone, and the red in the deepest water, although there are, of course,
numerous exceptions to this general rule of distribution. Another property
FIG. 67. SOLUBILITY OP CARBON DIOXIDE IN WATER, EXPRESSED IN VOLUMES
OF C02 MEASURED AT NORMAL TEMPERATURE AND PRESSURE,
PER VOLUME OF WATER
After G. W. Martin
of light is that it is refracted by water, and the greater the angle at which the
rays strike the water, the greater will be the refraction. In the tropics, where
the rays are practically vertical, the amount of refraction is
insignificant, but
in high latitude, where the rays strike the water at a sharp angle, the refraction
is
marked, as a result of which the rays are bent into a more nearly vertical
direction, thus increasing their penetration in depth, and partly compensatingfor the unfavorable angle at which they strike the water. Helland Hansen
was able to show that in the Atlantic Ocean south of the Azores, on a brightsummer's day, light is abundant at a depth of 100 meters, still including at that
depth a few red rays. At 500 meters the red rays have completely disappearedbut blue and ultra-violet rays are still plentiful, and may be detected at 1000
19
G. W. Martin, Scientific Monthly, 1922, p. 456.
PARAMETEKS OF STATE 319
meters, but have completely disappeared at 1700 meters. It is not probable,
however, that under the most favorable conditions photosynthesis may be
carried on at depths greater than 200 meters.
Temperature is less directly important in the sea than on the land since
there is no great danger of injurious extremes being reached. Indirectly, its
importance lies in the fact that carbon dioxide is much more soluble in cold
water than in warm (see fig. 67) and it is probably this, rather than the direct
influence of temperature, which accounts for the fact that the most luxurious
development of plant life is in the colder waters of the earth.
Among laboratory investigations of the influence of "climatic parameters,"
under controlled conditions, may be reckoned the work of
R. Pearl and S. Parker in their studies "On the Influence of Certain
Environmental Factors on the Duration of Life in Drosophila."
20
In these experiments it was found, for example, that certain species
of flies, kept in bottles closed by a single layer of silk bolting cloth
("ventilated bottles"), had 10 per cent longer life, on an average,
than similar flies kept in bottles whose neck was plugged with cotton
wool.
PARAMETERS OF STATE AND THE ANALYTICAL CONDITION FOR
EQUILIBRIUM
In the fundamental equations both of the Kinetics and of the
Statics of material transformations, as set forth in earlier chapters,
the coefficients are in general functions of the parameters of state,
and it is only on the supposition that evolution is proceeding under
essentially constant conditions of topography, climate, etc., that
these coefficients could be treated as constants.
Furthermore, since these same coefficients enter into the analytical
conditions for equilibrium, as set forth in Chapter XII, these
conditions must be read in the sense that they hold true when certain
specified parameters of state are held constant. If another set
of
parameters, instead, is held constant, the equilibrium conditions
will retain the same form, but the values of the coefficients will
change accordingly. This is
precisely analogous to the state of
affairs regarding the thermodynamic conditions for equilibrium
Generally it can be said that in equilibrium the thermodynamic
potential is a minimum, but the expression for this potential will
80
Arner. Naturalist, vol. 56, 1922, p. 385.
320 ELEMENTS OF PHYSICAL BIOLOGY
vary according as pressure and temperature, or volume and temperature,
for example, are held constant.
In concluding this section it is desirable to call attention to a
modification, in outward form, of which the analytical condition for
equilibrium,
Q' = minimum, dQ' = (14)
(see Chapter XII) is susceptible. Since certain parameters pi,
j?2, . . . are to be held constant in the application of this condition,
the addition of a set of terms PI dpi + P2 dpz -f . . .to
the expression 8Q' for a small virtual displacement will in nowise
alter its value. We may, then replace the condition (14) by the
fully equivalent one
dO' dQ'
= (d$)p
=
^ dXi -f
^dXs + . . . -f Pidpj. + P2dp, + . . . (15)
where the subscript p denotes that the parameters pi, p2,
. . .
,
conjugate to the parameters P1}
P2,
, . . . are to be held constant
in forming the expression (15). This statement of the condition
(15) adds, of course, nothing new to the case. It is mentioned
here only on account of its formal agreement
21
with the similar conditions
for equilibrium which, as already pointed out, play an important
role in thermodynamics. However, in the analogous equations
erf thermodynamics the expression d $ is a complete differential.
In the present instance we have no basis for the supposition that
(15) is the true differential of a function $ (Zi, Z2 , PI, P2 ,
. .
.) The question may, indeed be raised, whether by a suitable
choice of parameters and variables it can be achieved that
(15), in the case here under consideration, is such a complete differ-
21
Compare, for example, Van Laar, Seehs Vorlesungen iiber das thermodynamische
Potential, 1906, p. 43.
This formal agreement seems to extend also to another feature The thermodynamic
condition for stable equilibrium demands that the second differential
(d**) p shall be positive, and this in turn demands that ^shall be negative.
(See M. Planck, Thermodynamik, 1905, pp. 134, 190; Duhem, Traite d'Energ^tique
1911 vo .
1, p. 466; A. Winkelmann, Handbuch der Physik, 1906, vol.
d, p. o90.) Similarly, in the general case, stability demands that if Q, g are
conjugate parameters of the type (1) of footnote 5, then ^- < 0.
PARAMETERS OF STATE 321
ential. But this is a separate problem, on which we shall not here
expend further effort. Only this shall be noted in passing: Whereas,
in the thermodynamical treatment of physico-chemical phenomena
a function <3? is given (essentially as the expression of the laws of
thermodynamics), and whereas certain consequences are derived
from this known function, the type of problems with which we
are here concerned is of inverse nature. We are given certain data
regarding the behavior of these systems, for example, the fact that
their evolution follows more or less closely a system of equations of
the type of the general equations (1) (Chapter VI) of the Kinetics
of material transformation; and the problem may be raised, as to
whether there exist functions $ analogous to the functions known as
thennodynamic potentials, in terms of which the behavior of the
system can be concisely epitomized, after the manner of thermodynamics.
If such a plan could be successfully carried out, the
result would be a species of quasi-dynamics of evolving systems, in
which certain parameters P played a rdle analogous to forces, without
being in any sense identical with forces (or even with generalized
forces); certain other conjugate parameters p would play a role
analogous to displacements, and certain functions <& would resemble
in their relations to certain events in the system, the energy functions
$ (free energy, thermodjrnamic potentials) of thermodynamics.
That certain isolated portions of such a general system of quasidynamics
have some degree of viability seems probable. Whether
the general system is capable of development in a form possessing
any considerable utility shall here be left an open question. For
at this point we shall leave the path followed so far, and shall strike
out in a new direction, with a view to sketching, not a system of
quasi-dynamics or quasi-energetics, but the dynamics and energetics,
in the strict sense, as ordinarily understood, of life-bearing
systems in the course of evolution.
PART IV
DYNAMICS
UK*.
CHAPTER XXW
THE ENERGY TRANSFORMERS OF NATURE
Die Natur hat sich die Aufgabe gestellt das der Erde zustromende Licht im
Fluge zu erhasehen und die beweglichste aller Krafte, in die starre Form
verwandelt, aufzuspeiehern. Zur Erreichung dieses Zweckes hat sie die Erdkruste
mit Organismen iiberzogen,welche iebend das Sonnenlicht in sich aufnehmen
und unter Verwendung dieser Kraft eine fortlaufende Summe chemischer
Differenzen erzeugen. Diese Organismen sind die Pflanzen. Die
Pflanzenwelt bildet ein Reservoir, in welchem die fliichtigen Sonnenstrahlen
fbdert und zur Nutzniessung geschickt, niedergelegt werden. /. R. Mayer.
We approach now the third and last stage in our enquiry, toward
which all that has gone before may be said, in a way, to have been in
the nature of preparation.
The fundamental equations of kinetics
d
^=Fi (X1 ,Xi ,
. .
.;P,Q) (1)
dt
may appear at first sight to contain no hint of dynamical, of energetic
implications. These can be read into the equations only by calling
to mind the physical nature of certain of the components whose masses
X appear in the equations : These components aggregates of living
organisms are, in their physical relations, energy transformers.
The evolution which we have been considering, and shall continue
in this last phase to consider, is, then, essentially the evolution of a
system of energy transformers; the progressive redistribution of the
matter of the system among these transformers. The dynamics
which we must develop is the dynamics of a system of energy transformers,
or engines.
1
Fundamental Characteristics of Energy Transformers. We shall
do well to begin by calling to mind some of the fundamental elements
or characteristics of energy transformers or engines, and of the manner
of their working. An engine, such as a steam engine, for example,
receives energy from a source such as a coal fire. This energy is
absorbed by a working substance (water or steam), which, in the proc-
1
See also A. J. Lotka, Jl. Wash. Acad. Sci., 1924, p. 352.
325
326 ELEMENTS OF PHYSICAL BIOLOGY
ess, undergoes modification or change of state (in a general sense of
the term) ;
the working substance, at some stage in the operation of
the engine, again gives out energy, of which a part in engines of
human construction, commonly appears in a particular, selected
form adapted to some end useful to and purposed by the maker or
owner. Another fraction of the energy discharged by the working
substance, is passed on to a sink or absorber of energy, which may be
simply the surrounding air, or in the case of a naval engine it may be
the sea water employed to cool the condensed steam before it returns
to the boiler. This discharge of a portion of the energy from the
source into a sink is practised, not designedly because any useful
purpose is served thereby, but unavoidably because, in the case of
all forms of heat engines, the second law of thermodynamics inexorably
demands this payment of a tax to nature, as it were.
Cyclic Working; Output and Efficiency. A finite change of the
working substance, performed just once, can yield only a finite
amount of work. Hence an engine of this type, in order to operate
continuously so as to furnish a steady supply of energy of indefinite
amount, must of necessity work in a cycle, returning periodically to
its initial state many times. For a given engine, working under
given conditions, the total output W/t per unit of time is proportional
to the quantity M (mass) of working substance and its frequency
of circulation, n, per unit of time, through the cycle; thus
W
~~kMn (2)
Regarding the variation in the output for different engines, and for
operation under different conditions, two fundamental laws of the
greatest theoretical and practical importance, the very corner-stones
of the edifice of
thermodynamics, inform us that
1. The maximum output of which a heat engine is capable under
ideal conditions of working is independent of the nature of the working
substance, and of the details of mechanism and construction of the
engine.
2. This maximum output obtainable under ideal conditions of
operation depends solely upon the temperature of the source and that of
the sink; with a suitably chosen temperature scale the law of the maximum
output W can be put in the extremely simple form
W=Q~~ (3)
-i a
327
ENERGY TRANSFORMERS OF NATURE
where Q is the energy drawn from the source, T the (at)..
^
temperature of the source, and ^ that of the sink. The ratio
-Q
which measures the fraction of the energy Q converted into woi ^
spoken of as the efficiency of the transformer. ^ j
The actual performance of a heat engine always falls shoit -
^
usually far short of the theoretical maximum (3) attainable
ideal conditions of reversible operation, whereas all real pio
"~ '
'77 I llO
as has been pointed out in an earlier chapter, are irreversible.
first service rendered by the laws of thermodynamics is thus a neg
"lIlMPV
live one, to save us from vain efforts to achieve the impossible,
tell us what we cannot do; they give us no guarantee as to what we c(
do, in this matter of engine efficiency. In other fields these same
principles are, indeed, found competent to yield us information
most positive character, as the physicist and physical chemist know>
from boundless wealth of example; the very fact that they hold independently
of substance and form lends to their application a catholicity
hardly equalled elsewhere in science, and at the same time givfv
into our hands an instrument of the most extreme economy of thought,
since we are relieved, in such application, of the necessity of treating
each particular case, with all its complication of detail, on its own
merits, but can deal with it by the short cut of a general formula.
Still, the austere virtue of this impartiality with respect to substance
and form becomes something of a vice when information is sought'
regarding certain systems in which mechanism plays, not an incidental,
but the leading role. Here thermodynamics may be found powerless
to assist us greatly, and the need for new methods may bo folk
The significance of this in our present concerns will be seen as the
topic develops.
Composite and Coupled Transformers. The simplest typo of
transformer of the kind that here chiefly interests us would comprise
one working substance fed from one source and discharging to omi
sink.
Two or more such transformers may, however, work in paralln!
from one source, thus forming in the aggregate one composite trariK"
former. Or, two or more may be coupled in series or cascade, tho,
sink of one functioning as the source for the next of the series. Ho,
for example, W. L. R. Emmett has constructed a composite engine,
consisting of two separate engines, the first operating with mercury
328 ELEMENTS OF PHYSICAL BIOLOGY
for its working substance, at a higher temperature, and the second
operating with water at a lower temperature. It is to be observed
that two such "coupled'
7
transformers again constitute a transformer,
a compound transformer, which may possess certain special virtues,
from the standpoint of the engineer, or in other respects.
Accumulators. A special type of transformers is that in which the
energy is transformed into a latent form, and is thus stored up for
future use. A great variety of accumulators are in technical use.
In the simplest case such an accumulator may consist of empounded
water or a raised storage tank, ready upon the opening of a sluice
or a valve to discharge its stored up energy. More closely akin to
the systems in which we are here primarily interested is the lead
accumulator or secondary battery, in which electrical energy is
transformed into and stored as chemical energy, somewhat as the
energy of sunlight is, in the leaves of plants, transformed into chemical
energy and stored up in the form of starch.
This type of chemical storage is of very particular interest because
of the remarkable phenomena to which it is competent to give rise
through the circumstance that the substance in which the energy
is stored in chemical form is itself the working substance of a transformer.
For in that case, if a mass TiM stores an amount of energy
IT
7
,
we have, according to (2) for a small interval of time dt
dW hdM T
IF" IT-**71
(4)
If the transformer functions at a constant rate (i.e., with a fixed
number of cycles per unit of time) and if the coefficient k is independent
of the size of the transformer, we have by integration of (4)
kn
M = MQe~h (5)
The transformer under these conditions, grows according to the law
of compound interest.
2
For small ranges of size the assumption of a sensibly constant k
is
reasonable, and the law thus deduced may be expected to represent
the facts tolerably well. For greater range we must regard k as a
function of If and write
k = a + b M + c M2
+ . . .
(6)
2
Compare L. J. Briggs, The Living Plant as a Physical System Jour
Wash. Acad. Sc., 1917, vol. 7, p. 95.
ENERGY TRANSFORMERS OF NATURE 329
so that (4) becomes
^=n(a+6M+. .
.) (7)
Mdt
In second approximation, therefore, (breaking off the bracketed series
at the second term) we find for the law of growth of the transformer
the Verhulst-Pearl law (see Chapter VII).
where m = M + a/6 and the subscript zero denotes the value of the
variable at the instant t = 0.
Anabions and Catabions. The living organism partakes of the
functions both of an energy accumulator and of an energy dissipator.
The former function is especially marked in plants and in the young
growing organism. Biological terminology speaks of the process of
energy accumulation by the growth (synthesis) of the working substance
as anabolism, and of the liberation of the stored energy with
conversion into other forms as catabolism. Organisms in which
anabolic processes predominate are conveniently classed together as
anabions (plants), those in which catabolic processes predominate,
as catabions (animals). The line of division cannot be sharply
d) awn, a fact which was commented upon in some detail in the first
chapter. But in the majority of cases organisms have a pronounced
bias toward one or the other of the two forms, and no difficulty arises
in classifying them.
We may form the conception of a system of transformers comprising,
in the most general case, individual single transformers, aggregates
of composite transformers, and coupled transformers; some or all of
which may partake in greater or less degree of the nature of
accumulators.
It is precisely such a system of transformers that is presented to us,
on a vast scale, in nature, by the earth with its population of living
organisms. Each individual organism is of the type of the simple
transformer, though it does not operate with a single working substance,
but with a complex variety of such substances, a fact which has
certain important consequences.
330 ELEMENTS OF PHYSICAL BIOLOGY
Plant and Animal as Coupled Transformers .
Coupled transformers
are presented to us in profuse abundance, wherever one
species feeds on another, so that the energy sink of the one is the
energy source of the other,
A compound transformer of this kind which is of very special interest
is that composed of a plant species and an animal species feeding
upon the former. The special virtue of this combination is as follows.
The animal (catabiotic) species alone could not exist at all, since
animals cannot anabolise inorganic food. The plant species alone,
on the other hand, would have a very slow working cycle, because the
decomposition of dead plant matter, and its reconstitution into C02 ,
completing the cycle of its transformations, is very slow in the absence
of animals, or at any rate very much slower than when the plant is
consumed by animals and oxidized in their bodies. Thus the compound
transformer (plant and animal) is very much more effective
than the plant alone. We shall have occasion to refer to this matter
again.
It is, of course, conceivable that the anabolic and catabolic functions
should, in their entirety of a complete cycle, be combined in one
structure, one organism. Physically there is no reason why this
should not be, and, in fact, nature has made some abortive attempts
to develop the plant-animal type of organism; there are a limited
number of plants that assimilate animal food, and there are a few
animals, such as Hydra viridis, that assimilate carbon dioxide from
the air by the aid of chlorophyll.
3
But these are exceptions, freaks of
nature, so to speak. For some reason these mixed types have not
gained for themselves a significant position in the scheme of nature.
Selection, evolution, has altogether favored the compound type of
transformer, splitting the anabolic and the catabolic functions, and
assigning the major share of each to a separate organism.
The several individual organisms of one species form in the aggregate
one large transformer built up of many units functioning in
parallel.
3
Hydra viridis, however, is probably not a single organism, but an organism
of the animal type harboring in its body separate plant-like organisms
with, which it lives in symbiosis.
ENERGY TRANSFORMERS OF NATURE 331
And lastly, the entire body of all these species of organisms,
together with certain inorganic structures, constitute one great
world-wide transformer. It is well to accustom the mind to think
of this as one vast unit, one great empire.
The "World Engine. The great world engine in which each of
us is a most insignificant little wheel has its energy source, its
firebox, so to speak in the sun,
4
ninety-eight million miles away from
the working substance (the "boiler"). From the engineer's standpoint
this would be an execrably bad design, if a high efficiency
alone were the aim in view. For of the five hundred thousand million
million million horsepower which the fiery orb radiates into space
year in, year out, a ridiculously small fraction
2
is
intercepted by the earth. It would take more than two billion,
earths placed side by side to form a continuous shell around our sun
at the earth's distance, and thus to receive the total output of solar
heat. The other planets receive corresponding amounts. The
remainder of the sun's disbursements sweeps past us into the depths
of space, to unknown destiny.
Of the energy that reaches the earth, 35 per cent is reflected
(principally from the clouds), and 65 per cent is absorbed. The
surface of the solid globe receives on an average
5
not quite 2 gramcalories
(1.94) per square centimeter, placed normal to the beam, per
minute, or enough heat to melt a layer of ice 424 feet thick every
year. Arrhenius8
quotes Schroeder to the effect that about 0.12 per
cent of this energy is absorbed by the green vegetation, the gate of
entrance through which practically
7
all the energy taking part in the
4
The recognition of this fact is credited by Herbert Spencer (First Principles,
172, footnote) to Herschel (Outlines of Astronomy, 1833).
5
C. G. Abbot, The Sun, 1911, p. 298. In the tropics, at noon, a plot of 250
acres receives energy at the rate of one million horsepower (W. W. Campbell,
Science, 1920, vol. 52, p. 548).
6
Jour. Franklin Inst., 1920, p. 118. Compare also G. Ciamician, Die
Photochemie der Zukunft (Sammlung Chemischer Vortrage, 1922, p. 429).
Asstiming an area of one hundred twenty-eight million square kilometers as
inhabited by plants, Ciamician computes that thirty-two billion tons of dry
matter per annum is produced, the equivalent of 17 times the world's annual
coal production.
7
Certain bacteria whose metabolism is based on iron, sulphur or selenium
derive their energy from other sources. They are thus independent of sun-
332 ELEMENTS OF PHYSICAL BIOLOGY
life cycle must pass. And of this last amount only 24 per cent falls
to plants cultivated for human needs.8
The forests take the major
share, 67 per cent; 7 per cent falls on the grass steppes, and 2 per
cent on desert plants. If these figures leave the mind somewhat
confused with detail, it may assist the imagination to form an adequate
picture of the life cycle in its totality if we reflect that the total
energy thus corn-sing through the system every year is of the order of
22 times'5
the world's annual coal production. Conversely this
statistical fact may serve to form for us a correct estimate of the really
cosmic magnitude of human interference with the course of nature.
The organic circulation, the living part of the world engine, though
to us of most direct interest, is quantitatively speaking only a small
part of the whole. If the organic cycle gives occupation to an amount
of energy of the order of 20 times the world's coal consumption,
light a fact of the greatest significance in connection with the problem of the
origin of terrestial life as we know it today. For green plants carry on their
life business by the aid of chlorophyll, a substance representing a high degree
of specialization, such ae could not very well be supposed to exist in the most
primitive life forms.
8
H. A. Spoehr, Jour. Ind. and Eng. Chem., 1922, vol. 14, p. 1144. Regarding
the efficiency of cultivated plants in recovering solar energy for the use
of man, the calculations of H. A. Spoehr are of interest. On the basis of 1.5
gram calories per square centimeter per minute for the value of the solar
radiation received at the earth's surface, he computes the daily energy income
per square meter (six hours insolation) as 5400 kilogram calories. Figuring the
heat of combustion of coal at 8000 kilogram calories, this gives the equivalent
of 0.675 kilograms of coal per square meter, or 16.4 tons of coal per acre. For
ninety days of insolation this represents the equivalent of 1476 . 63 tons of coal.
Spoehr then proceeds to obtain a figure for the efficiency of a wheat crop
in the utilization of this energy. Assuming a large yield of 50 bushels or 17 . 619
hectoliters per acre, and considering this entirely as starch, we find an energy
equivalent of 0.623 ton of coal. The efficiency here, then, is measured by the
,
- 0.623
very low figure
- = 0.0004 = 0.04 per cent.
14/0
It should be noted, however, that not all the heat absorbed by the plant
appears stored up in the body of the plant. A large amount is used up in the
work of evaporation (transpiration). According to L. J. Briggs (Jour. Washington
Acad., 1917, p. 92; Journal Agr. Research, 1914, pp. 1-63), the
energy stored by the plant represents from 1 to 5 per cent of the energy dissipated
during the growth of the plant. See also C. L. Holsberg, Jour. Ind. Eng
Chem., 1924, vol. 6, pp. 524-525: Progress in Chemistry and the Theory of
Population.
ENERGY TBANSFORMERS OP NATURE 333
the winds represent some 5000 times that amount of coal.
9
Ocean
currents are another large item. Some idea of the magnitude of the
energy here involved may be gathered from an estimate given by
L. J. Henderson according to which the gulf stream alone conveys
10
two-hundred million tons of water per second through the straits of
Yukatan. If this body of water were cooled to arctic temperature we
should have a transfer of energy at the rate of eight and a half billion
horsepower. Most important of all, in the inorganic cycle, is
the circulation of water by evaporation, precipitation, and river flow
(including waterfalls) back to the ocean. Of the masses involved
a picture had been presented in Chapter XVI. As to the energy
involved, Henderson estimates the horsepower of evaporation from.
100 square kilometers of tropical ocean at over one-hundred million
horsepower.
C. P. Steinmetz11
has calculated that if eveiy raindrop falling in
the United States could be collected, and all the power recovered
which it could produce in its descent to the ocean, this would yield
about three-hundred million horsepower. G, Ciamician12
quotes an
estimate by Engler of the world's total water power as the equivalent
of seventy billions of tons of coal. According to C. G. Gilbert
and J. E. Pogue
13
the production of hydroelectricity in the United
States in 1910 was the equivalent of forty-million tons of coal,
whereas nearly ten times that amount went into the production of
steam and carboelectric power. These authors further estimated
that the water power developed at the date indicated represented
about 10 per cent of that readily available, and 3 per cent of the total
that might be open to development under elaborate arrangements
for storage.
14
9
For a discussion of the Atmosphere considered as an engine see Sir Napier
Shaw's Rede Lecture, published in Nature, 1921, p. 653. This author arrives
at the estimate that "the best you can expect from the steam-laden air of the
equatorial region working between the surface and the stratosphere, under
favorable conditions, is a brake-horsepower efficiency of 25 per cent."
10
L. J. Henderson, The Fitness of the Environment, p. 182.
11
Survey Graphic, 1922, vol. 1, p. 1035 (cited by H. A. Spoehr, Jour. Ind.
Eng. Chem., 1922, vol. 14, p. 1143).
12
Die Photochemie der Zukunft; Samml. Techn. Vortrage, 1914, p. 431.
13
Power, its significance and needs; Smithsonian Institution Bulletin 102,
Part 5.
14
For a comprehensive survey of power development actual and potential
see F. G. Baum, II. S. A. Power Industry. Also W. S. Murray, A Superpower
System for the Region between Boston and Washington. United States
Geological Survey Professional Paper 123 of 1921.
334 ELEMENTS OF PHYSICAL BIOLOGY
Relation of Transformer Cycle to Circulation of the Elements.
The circulation of substance in the organic world and its inorganic
background, which was considered in an earlier chapter in its purely
material relations, now acquires a new significance. We recognize
in it now a typical characteristic of the great world engine which,
for continued operation, must of necessity work thus in cycles. The
picture presented to our minds is that of a gigantic overshot mill
wheel, receiving from above the stream of sunlight with its two
hundred twenty-seven million gross horsepower though much of
this is spilt without effect and discharging below its dissipated
energy in the form of heat at the general temperature level. The
FIG. 68. THE MILL-WHEEL OF LIFE
main outstanding features of the wheel are represented diagrammatically
in figure 68. But in detail the engine is infinitely
complex, and the main cycle contains within itself a maze of subsidiary
cycles. And, since the parts of the engine are all
interrelated,
it may happen that the output of the great wheel is
limited, or at
least hampered, by the performance of one or more of the wheels
within the wheel. For it must be remembered that the output of
each transformer is determined both by its mass and by its rate of
revolution. Hence if the working substance, or any ingredient of the
working substance of any of the subsidiary transformers, reaches its
limits, a limit may at the same time be set for the performance of the
great transformer as a whole. Conversely, if any one of the subsidi-
ENEBGY THANSFOBMEBS OF NATUBE 335
ary transformers develops new activity, either by acquiring new
resources of working substance, or by accelerating its rate of revolution,
the output of the entire system may be reflexly stimulated.
As to the significance of this for the evolution of the system as a whole
more will be said later, in the discussion of certain phases of the
evolution of the human species in particular; for it is hardly necessary
to remark that the case of man presents features of so remarkable
character that it calls for special consideration, quite aside from the
pardonable excess of interest which we personally feel in the
creature.
Evolution of the World Engine; The picture we must keep before
us, then, is that of a great world engine or energy transformer composed
of a multitude of subsidiary units, each separately, and all
together as a whole, working in a cycle. It seems, in a way, a singularly
futile engine, which, with a seriousness strangely out of keeping
with the absurdity of the performance, carefully and thoroughly
churns up ah
1
the energy gathered from the source. It spends all its
work feeding itself and keeping itself in repair, so that no balance is
left over for any imaginable residual purpose. Still, it accomplishes
one very remarkable thing; it improves itself as it goes along, if we
may employ this term to describe those progressive changes in its
composition and construction which constitute the evolution of the
system. For the statement will bear reiteration and emphasis
this is the conception we must form of organic evolution: the evolution
of the great world engine as a whole, not merely that of any
single species of organisms considered separately. What is the trend
of this development? Toward what end does the great transformer
shape and reshape itself? A provisional answer to the question will
be suggested in due course. For a time we must now abandon our
broad viewpoint, and turn from the consideration of the great transformer
as a whole, to a discussion of certain of its subsidiary engines
which present points of special interest and importance.
CHAPTER XXV
RELATION OF THE TRANSFORMER TO AVAILABLE SOURCES
As an enterprise, mathematics is characterized by its aim, and its aim is to
think rigorously whatever is rigorously thinkable or whatever may become
rigorously thinkable in course of the approved striving and refining evolution
of ideas. C. J. Keyser.
Distributed and Localized Sources of Energy. In Garnet's
classical analysis of the operation of a heat engine the source of
energy is taken for granted as one of the fundamental data of the
problem.
In nature sources of energy are not thus supplied unconditionally,
and for our present purposes it becomes necessary to extend the
analysis of transformer operation so as to take into its scope also
some of the significant characteristics of the sources from which the
engines of nature derive their supplies. And here a fundamental
distinction is to be made between two lands of sources, namely, (1)
evenly, or at least continuously distributed sources, and (2) localized
sources, heterogeneously distributed.
If the transformer draws its energy supply from a source uniformly
distributed over a region R, at any point of which it can make contact
with the source, then, evidently, within the region R the performance
of the transformer is independent of its location. So, for example,
plants derive their energy from sunlight falling upon them gratuitously,
and draw their supplies of material partly from atmospheric carbon
dioxide and oxygen diffusing to them by a spontaneous process,
and partly from dissolved salts seeping to their roots automatically,
that is to say, by a process essentially independent of any intervention
on the part of the plant. And quite in accord with this general distribution
of plant food, the typical plant is a sessile, passive organism.
If, on the contrary, the transformer draws its supply from discontinuous,
heterogeneously distributed sources, then continued
operation demands at least some degree of relative motion between
the transformer and the sources, so that the occasional collisions
may occur between the transformer and a source.
336
RELATION OF TRANSFORMER TO AVAILABLE SOURCES 337
Random and Aimed Collisions. Purely random collisions, such
as those contemplated in the kinetic theory of gases, may suffice to
bring an adequate supply to the transformer.1
But evidently the
output of the transformer will be enhanced if, instead of relying upon
a precarious supply gleaned in fortuitous encounters, a suitable
correlation is established between the motion of the transformer and
the location of the sources. This may be accomplished in two ways,
as follows:
1. There may be actual mechanical union2
positively connecting
transformer and source, so that there is a functional relation (in the
mathematical sense) between the motion of the transformer and the
topography of the source. A simple instance in point is a trolley
car. Here there is a definite relation between the topography of the
system (track), the reaction of the transformer up it, and the distribution
of the source. The car is not free to move except along the track
and along the trolley wire.
2. Contact with the source may not be positively secured, but
merely rendered more probably than in purely random collisions, by
the occurrence of more or less accurately aimed collisions. Source and
transformer are in this case mechanically independent, the motion of
the source is not fully determined when the topography of the system
is given; a certain freedom remains. There is, in this case, not functional
relation, but only correlation between the motion of the transformer
and the topography of the system: no specific motion is
determined, only certain motions are rendered more probable than
others.
Negative Correlation. It is to be noted, of course, that such correlation
between the motion of a transformer and the location of features
1
For an experimental investigation of the movements of lower organisms
(Paramecium, Colpidium, Trachelomonas) see Przibram, Pftiigers Archlv,
1913, vol. 153, pp. 401-405. The movements were found to follow the law deduced
by Einstein and Smoluchowski for Brownian movement (which, of
course is random), namely that the mean square of the displacements of a
particle in any direction in equal intervals of time t is proportional to t. The
order of magnitude of the movements of the organisms, however, and the
influence of temperature, were quite different in the case of Brownian
movement. (For an account of the Einstein-Smoluchowski law see, for example,
C. Schaefer, Einfiihrung in die theoretische Physik, 1921, vol. 2, p. 487.)
2
Compare L. T. y Quevedo, Essai sur 1'automatique, Revue Ge'ne'rale des
Sciences, 1915, vol. 26, p. 601.
338 ELEMENTS OF PHYSICAL BIOLOGY
of its environment is competent to bring other benefits aside from a
a supply of energy and material (food). All transformers are more or
less vulnerable. Exposed to an environment varying from point to
point and from instant to instant, a transformer will in general sooner
or later meet with an injurious stress, that is to say, a stress that will
change its structure or constitution to a point where effective operation
is impaired or altogether abolished. If it is desirable, in the
interest of increased output, that collisions with suitable energy
sources be rendered more probable than in purely random motion, it is
evidently equally desirable, in the interest of continued operation of
the transformer, that collision with harmful features in the environment
be rendered less probable. In other words, in addition to
apparatus establishing a positive correlation between the motion of the
transformer and the location of sources, it is desirable that there be
also provided apparatus establishing negative correlation between
such motion and the location of injurious features of the environment.
Collisions should, as far as possible, be aimed toward sources, and
away from points of danger. The fate, the success of the transformer,
will evidently depend both on the versatility of the aim, and on its
accuracy; on the number and character of targets picked out for aim,
and on the closeness with which the hits upon the target cluster around
the bull's eye.
The Correlating Apparatus. It is on the general plan indicated in
the preceding paragraph that nature's mobile transformers, especially
the typical animal organisms, operate. In the competition among
these, for food and for safety, the accuracy and the versatility of
aim characteristic of each species will evidently be most important
determinants of relative success or failure, and hence of the trend of
evolution. The dynamics of evolution thus appears essentially as
the statistical dynamics of a system of energy transformers, each
having a characteristic vulnerability, a characteristic versatility and
accuracy of aim. It is here that the method of thermodynamics is
inadequate. Its austere virtue of impartiality toward different
mechanisms becomes a vice when information is sought regarding
systems in which mechanism plays a leading r61e. The mechanism,
or, to use a somewhat broader term free from mechanistic implications,
the apparatus, by which the correlation is established between motion
and environment, by which behavior is adapted to circumstances, is
here not an incidental detail to be lightly dismissed as of secondary
RELATION" OP TRANSFOEMEB TO AVAILABLE SOTJBCES 339
importance, but must occupy the very center of attention. To a
somewhat detailed consideration of this apparatus we now proceed;
in the interest of vivid, realistic presentation of the subject it will,
however, be desirable to abandon from this point on the veiy general
treatment and, to speak now specifically in terms of biological units,
organisms, rather than in terms of the broader physical concept of
energy transformers. It should be constantly borne in mind, however,
that this change in attitude, or, it were better to say, in terminology,
is chiefly a matter of convenience and effectiveness of presentation,
and the fundamental physical principles involved, as set
forth in the more general terms, should never be allowed to sink far
below the surface of our immediate thought.
The Component Elements of the correlating Apparatus. The
continued existence of the organism, toward which his actions are
aimed, demands that he shall direct his energies, his activities in
accordance with the state of his environment, of the external world,
avoiding unfavorable conditions, and seeking out those favorable or
necessary to his maintenance. This includes the locating and seizing
of food.
But as a material, physical system, his actions are primarily determined
by his own state. Hence, in order that his actions, determined
immediately by the state of the organism himself, may be mediately
determined by the state of the external world, apparatus must be
provided whereby the state of the organism becomes in a certain
suitable manner a function of the state of the external world. The
external world is depicted in the organism by a certain apparatus,
a set of organs and faculties, which we may appropriately term the
Depictors.
The depictors include first the Receptors or Organs of Special Sense
(eyes, ears, nose, etc.) ;
and second the Elaborators, whose function is
to combine and further elaborate the crude information furnished by
the senses. The physical location and structure and mode of operation
of the elaborators is much less obvious than that of the receptors.
In fact, we ordinarily recognize them rather as faculties (Memory,
Reason) than as organs.
Another set of organs and faculties, the Adjusters, determine the
particular reactions, the behavior of the organism, in the light of the
information brought in by the receptors and further elaborated by the
elaborators. So the hungry bird, sighting a worm on the lawn, flies
340 ELEMENTS OF PHYSICAL BIOLOGY
down to the spot from the tree on which it is perched, and secures its
prey. The sight of the worm, together perhaps with the memory of
earlier meals collected near the same spot, acts as a stimulus or
Drive to responsive action. This last step, action, commonly involves
the use of members, or motor organs, Effectors, such as wings,
feet, hands, etc. In complicated cases, as in human behavior, the
elaborators may also play an important role in the effector step of
the process by which motion is correlated to environment, behavior
adapted to circumstance. So, for example, the traveller, before
setting out on a journey, plans his itinerary, perhaps months or years
in advance.
Receptor-Effector Circuit Begins and Ends in Environment. It is
a noteworthy fact that this process of correlating action to conditions
is essentially cyclic in character: It has its origin in the external
world, which becomes depicted in the organism, provokes a response,
the terminal step of which is usually, if not always, a reaction upon
the external world. The net result is, in a sense, that the external
world has reacted upon itself. The organism has acted as an intermediary.
There is something more than a mere surface significance
in this fact. For it is true generally that animals function essentially
as catalysers, as agents assisting in a change that is "trying" to take
place on its own account. So the green grass is, in a sense, hungry
for oxygen, namely in the sense that its oxidation is accompanied by
a diminution of free energy. The animal consuming the grass and
deriving from its oxidation the requisite energy for further activity
has not initiated any revolutionary process, but has merely helped
nature in its course, has merely rolled the ball downhill, so to speak.
This is all that the animal organism is competent to do, and man is
not exempt from this restriction: In all our doings, whether we will
it or not, we are assisting in a fundamental natural process, we are
obeying an inevitable law of energetics.
Correlating Apparatus Not Peculiar to Living Organisms . It must
not be supposed that the typical elements of the correlating apparatus,
the receptors, adjusters and effectors, are wholly peculiar to
living organisms. They can be very clearly recognized also in certain
mechanisms of human construction. In fact, owing to the
circumstance that the operation of such man-made mechanism is
fully known to us, that they harbor no mysterious "vital" principle
or ill-understood element of consciousness, such purely mechanical
EELATION OP TKANSFOEMEB TO AVAILABLE SOUECES 341
contrivances furnish particularly apt illustrations of the principles
involved in the operation of the correlating apparatus. It is therefore
well worth while to consider here a simple example of this kind.
Some time ago there appeared on the market an ingenious toy,
primarily designed, no doubt, merely to amuse; but, in point of fact,
highly instructive. Its general appearance and simple mechanism
are illustrated in figure 69. The beetle "walks" on two toothed
wheels, of which one is an idler, while the other is rotated by a spring
whose gradual release is ensured by a simple escapement device.
At its forward end reckoning in the direction of motion (at the
"head") the toy is provided with a pair of antennae, of which one is
a dummy, and rises clear of the table upon which the beetle is placed
to exhibit its talents. The other antenna is operative and is so bent
downward as to glide along the table top, in contact with it. A little
FIG. 69. MECHANICAL WALKING BEETLE, EXHIBITING THE SEVERAL CHARACTEEISTIC
ELEMENTS OP THE COERELATING APPARATUS
in advance of the propelling wheel is another smaller toothed wheel,
running idle, and disposed transversely to the direction of the driving
wheel. This transverse wheel clears the table without contact in the
normal working position of the beetle. The animal, if placed somewhere
near the center of the table, makes a straight track, apparently
intent upon reaching the edge and seeking destruction in a species
of mechanical suicide. But the moment the operative antenna
clears the edge of the table, the body of the toy, till then held up by
the contact of the antenna with the table surface, sinks down a fraction
of an inch, and the transverse wheel now contacts with the
table. In consequence the toy rotates until the running wheel
is parallel with the table edge, and the insect continues its peregrinations
with the operative antenna hugging the side of the table top.
Clearly here the antenna is a receptor, which "apprises" the insect
of certain features in its environment, which depicts, in a crude but
342 ELEMENTS OF PHYSICAL BIOLOGY
sufficient manner the environment in the toy. The law of depiction
is here extremely simple; a depression in the external world (table
top) is translated into a downward tilt in the angle of repose of the
toy.
The adjuster, in this case, is the transverse wheel, about as simple
an example of an adjuster as can well be imagined. It "construes"
the information furnished by the receptor antenna, and modifies in
accordance with this information the law of motion of the toy, in
such manner as to preserve the beetle from a fall which might destroy
that stability of form on which the continued operation, according to
schedule, of the mechanism depends.
It would be easy to cite a number of other examples of devices constructed
either as toys, scientific curiosities, or for actual technical
use, which exhibit more or less prominently the typical correlating
apparatus, with receptor, adjuster and effector. By far the most
highly perfected of such automatic devices is the modern machineswitching
apparatus for telephones, which eliminates the "operator"
at the central offices. This device, which fulfills an amazing
multiplicity of functions, will be found briefly described in non-technical
language in the April number 1923 of The Bell System Technical
Journal.
Perhaps more directly in line with our present interest here is a
mechanical chess player that was designed some years ago by L.
Torres y Quevedo;
3
this device successfully counters any move (with
a limited number of pieces, merely as a matter of simplicity) that a
living opponent may choose to make upon the board. The game of
chess itself is so well conceived a conventionalization of the battle
of life
4
that it is well worth the while to make a seeming digression to
analyze the fundamental elements of this remarkable game. The
bearing of this analysis upon certain problems of biological evolution
will then become apparent.
8
A description will be found in the Scientific American Supplement,
November 6, 1915, p. 296.
4
The aptness of this illustration has no doubt been remarked by many.
I have recently noted the following pertinent references : F. L. Wells, Mental
Adjustments, 1917, p. 6. Eddington, Time Space and Gravitation, 1920, p.
184; T. H. Huxley, A Liberal Education and Where to Find it. Collected
Essays, 1894, vol. 3, p. 81. A bibliography of the mathematical treatment of
chess will be found in W. W. R. Ball, Mathematical Recreations, 1911,
Chapter VI, p. 109.
RELATION OF TRANSFORMER TO AVAILABLE SOURCES 343
Chess as a Conventional Model of the Battlefield of Life. A
game of chess is a succession of physical events. How is its course
determined?
The elements that determine this course are as follows :
1. A topographic map, a chart of geometric constraints, the chess
board.
2. Movable upon this chart, a number of movable points (chessmen),
each the center of & field of influence, defined for each movable
point in relation to the geometric constraints. So, for example, the
field of influence of a pawn extends to the two squares diagonally
in front of the pawn.
3. A law restricting the time-rate of advance of each moving point
(moves alternate from white to black).
4. A law defining the influence upon each other of two points in
collision, i.e., two points whose fields of influence have interpenetrated
to a prescribed extent. An example of this is the rule that a chessman
arriving upon a square occupied by a hostile piece, throws the latter
off the board.
5. A law restricting the movements of the points when not in
collision, i.e., when outside one another's field of influence. So, for
example, a bishop may move only diagonally.
6. The elements enumerated so far place restrictions upon permissible
changes (moves). These elements alone cannot, evidently,
determine any occurrence of any kind: Absolute immobility, for
example, or any random move that did not violate the rules of the
game, would equally satisfy the conditions enumerated.
7. In addition to the elements, 1, 2, 3, 4, 5, there must therefore
be in operation some positive principle (tropism) which not merely
restricts possible occurrences, but which determines actual events.
In chess this principle is furnished by the effort of each player to
bring about checkmate. Each move is so aimed (with greater or less
accuracy and breadth of view, versatility, according to the skill of the
player) as ultimately to force a checkmate.
From the battlefield of chess we now turn our eyes on the scene of
the great biological contest: Before us is a topographic map, over
which move those organisms that are by nature gifted with motion.
We may think of each such organism as a moving point, the center
of a field of influence. As the chessplayer must accustom his mind's
eye to see, radiating out from each chessman, its field of influence-
344 ELEMENTS OF PHYSICAL BIOLOGY
upon the board, so we, in envisaging the battleground of organic
evolution, must see each organism carrying around with it, as if
rigidly attached to its body, a field, or a target, of zones, of the following
character.
A. Zones of Influence, In general the motion of the individual
will be determined by laws too complicated to be readily analyzed, and
therefore will be described as random. But there will upon occasion
be a rather abrupt break away from such random movement, according
as a certain feature of the environment lies without or within certain
zones. For example, the movements of a fly wandering about on a
window pane are, presumably, in all cases physically determinate.
But in a homogeneous field (uniform illumination etc.), the motion
will assume, on the whole, a random character. We may suppose
that, in first approximation at any rate, the migrations of the individual
will follow some such law as those developed, for example by
Sir Ronald Ross,
5
by Pearson and Blakeman,
6
or by Brownlee7
for
random migration. But, bring some particle of food within the field
of sensuous observation of the fly, and the law of motion instantly
changes from random to more or less clearly directed.
We may, then, construct about each individual a sort of target of
zones of influence. The ideal would be to draw this target on a
quantitative plan, according as a stimulus of strength s exerts a
directing influence d at a distance r. In practice there may be difficulty
in constructing these zones, but we may at least conceive them
as drawn.
We may say that a given organism is "in encounter" with a given
point (e.g., a feature of the topographic chart) when that point falls
within its field of influence. Similarly we may say that two organisms
are in encounter when the one falls in the field of influence of the other.
This encounter is mutual or one-sided according as each is within the
other's field}
or as only one is in the other's field, but not conversely,
for example if A sees B, but B does not see A; or, to take an example
5
Sir Ronald Ross, Prevention of Malaria, 1922, second edition, pp. 179, 700.
6
K. Pearson and J. Blakeman, Drapers' Company Research Memoirs,
III: XV, 1906.
7
J. Brownlee, Proc. Roy. Soc. Edin., 1910-1911, vol. 31, pp. 262-289; of other
references related to this subject the following have been noted: F. Y. Edgeworth,
Entomological Statistics, Metron, 1920, vol. 1, p. 75; W. H. Cole, Science,
1922, vol. 55, p. 678; W. B. Hardy, Nature, December 30, 1922, p. 866
(Twelfth Report of the Development Commissions 1922).
345
RELATION OP TRANSFORMER TO AVAILABLE SOUBCES
from chess, a bishop may threaten a pawn, though the pawn does not
threaten the bishop. Zones of influence may extend over millions of
miles, as in the case of a traveller steering his course by the stars.
B. Zones of Mobility. We may stake out around each organism
a target of zones indicating the distance which it is physically capable
of travelling in 1, 2, , . .n units of time. These zones also we shall
think of as attached to the organism and carried round with it in
its wanderings through the landscape.
It is clear that the fate of the organism, and the history, the evolution
of the system as a whole, will depend, first, on the character of
the zones of influence and the zones of mobility; and second, on the
nature of the correlation, the law of the aimed movements,, established
through these zones. We may seek to establish analytical expressions
for this dependence.
Let g be a parameter defining the character or "pattern" of a target
of zones of influence or of mobility of the organisms of species S.
Thus, for example, q might be parameter, or one of a set of parameters,
defining visual acuity, measured on some suitable scale, at a distance
of 5, 10, 15, . . .
feet, under standard conditions. Or, g might be a
parameter defining the minimum time required for the organism to
reach a point 5, 10, 15, ... feet from his actual position, under
standard conditions. 8
Analytical Statement of Problem. We may now enquire:
1. What will be the effect upon the rate of growth of the species
if the parameter g is increased by a (small) amount dqf If r is the
fractional rate of increase of the species S, can we establish an expres-
d?*
sion for the partial derivative ;?
A glance at the chess analogy will help to make clear the nature of
the question thus raised. In chess we might ask: What would be the
effect upon the course of the game if, other things equal, we were to
modify in some stated particular the rules limiting the permitted
moves of a given piece, for instance by allowing a pawn to move two
squares, instead of the conventional one?
2, A second enquiry of peculiar interest relates, not to the character
(pattern) of the zones of influence and mobility, but to the form of
8
Isochrone charts of essentially this character, relating to travelling facilities,
were, according to Darmstaedter, first suggested by K. Bichter in 1833 arid
actually prepared by Sir Francis Galton in 1881 (L. Darmstaedter, Handbuch
zur Geschichte der Naturwissenschaften und der Technik, 1908, p, 792).
346 ELEMENTS OF PHYSICAL BIOLOGY
relation established, through these zones, between the action of the
organism and his environment. For it is hardly necessary for us to
be reminded that two individuals or species with the same visual
acuity, for example, may react in very different manner on seeing the
same thing.
Here again the chess analogy is helpful. The corresponding enquiry
with regard to chess is: What would be the effect upon the
course of the game, if, with unchanged rules as to the moves of the
pieces, a given change were made in the method, or the ability, of one
of the players?
To deal with these problems it is desirable to introduce two concepts,
that of the Behavior Schedule, and that of Specific Productivity
in a given activity.
THE BEHAVIOR SCHEDULE
It has been remarked9
that "a living organism is both cause and
effect of itself." We may say in somewhat more detailed statement,
that the organism goes through a certain routine of motions or activities
which are rendered possible by ^& structure, and which, in turn,
are a necessary condition for the continued existence of that structure.
These activities in general involve the expenditure of certain quantities
of free energy, and a part of the energy so expended necessarily
is spent in collecting (earning} a "replacement" amount equal to the
total expenditure, to balance the account, to cover the cost of living.
While this phenomenon is, in a general way, characteristic of all
mobile forms of life, the particular method followed in this cyclic
activity of gathering and spending free energy varies in the most
multiform manner from one species or type of organism to another.
Each type of organism may thus be said to possess a characteristic
Behavior Schedule, which may be defined in terms of certain coefficients
as follows: Of its total expenditure E per unit of time, a representative
individual of the population will spend, on an average a
fraction Xj in a particular activity Aj, which may be defined as the
maintaining of a parameter Uj at the value u$. So, for example,
10
a human being may expend on an average, per day,
9
Kant, Kritik of Judgment. Transl. Bernard, London, 1892, p. 274.
10
The figures in this example are chosen arbitrarily, although an effort
has been made to make them reasonably realistic. In view of the wide variations
in standards and cost of living in different countries, different social
RELATION OF TRANSFORMER TO AVAILABLE SOURCES 347
number of calories
ISO external work (services sold) in
maintaining his daily food supply
at 3,000 cal.
2,500 internal work (physiological
work) in maintaining his body
temperature at 9SF.
130 external work (services sold) in
maintaining Ms house rent at,
daily 2.00
60 external work (services sold) in
maintaining his dairy supply of
clothing at SO , 75
30 external work (services sold) in
maintaining his daily supply of
sundries at SO. 25
100 external work (services sold) in
maintaining the rate of increase
of the population at 1 per cent per annum
3,000 calories total expenditure and total earnings
(Note: 1 calorie = 30S6 foot pounds.)
SPECIFIC PRODUCTIVITY
Consider some particular activity A j
which results in maintaining
a parameter Uj (e.g., food capture per head per unit of time) at the
value Uj, then we will define P-, 3
the Specific Productivity of energy
Ej spent in activity Aj by
Pj = ''
(cj
a constant) (1)
and we note that
strata, and at different epochs, close figuring, in such an example as this, would
be out of place. Numerical data pertinent to this example will be found
scattered widely in various sources, of which the following may here be mentioned:
J. Amar, Le Moteur Humain, 1914, p. 254; R. Hutchison, Food and
Dietetics, 1902, p. 37:46; J. LeFevre, La Chaleur Animale, 1911; F. H. Streightoff,
The Standard of Living, 1911.
348 ELEMENTS OF PHYSICAL BIOLOGY
EFFECT OP CHANGE IN ZONE PATTERN
(Intra-Species Evolution)
We are now prepared to consider the analytical representation of
the influence of changes in the pattern of the zones of influence and
the zones of mobility, on the one hand, and, on the other, of a change
in the behavior schedule, on the proportional rate of increase r of
the species of organisms under discussion.
This proportional rate of increase, r = -777 ,
is in general a func-
.Arft
tion of the parameters U (food capture, shelter, etc.), so that we may
write
r = r(ui, u2 ,
. . .
Uj, . . . )
(3 )
and
-^L^-^L^i (4)
Now the specific productivity P-t
in activity A$ itself depends upon
the character of the zone pattern. For, the more perfectly the individual
is apprised of the relevant features of its environment (i.e.,
the more perfectly developed its zones of influence), the better, other
things, equal, will it be able to direct its activities to the ends defined
by the parameters U; and a similar remark evidently applies
to the zones of mobility. If, then, q is a parameter denning the
character of these zones, we may write
Pi = Pj(g) (6)
and
|
r
JL |p (7)
-1^*1 (8)
Ouj Oq
or, more generally, since a change in q may affect not only a single
productivity Pj} but also others, PI, P2,
. . . Pn
RELATION OF TRANSFORMER TO AVAILABLE SOURCES 349
the summation being extended over all the activities AI, Az ,
. .
A-} ,
. . , Aa in so far as they are affected by the parameter q.
(It is immaterial whether those not so affected are included in the
summation or not, since they will contribute a zero term.)
It lends a certain interest to the relation (9) if we observe that such
dr dr
partial derivatives as ;r ,
^ possess a concrete signification, as
follows :
Consider two small increments Agi and Aqz in two parameters
QI and go. (To make matters concrete, suppose qi measures visual
acuity, qz auditory acuity.) If Aqi and Ag-2 are such that
dr . dr ,
_A3l
-g--Ag
2
= (10)
i.e., such that
Agi _ dr / dr
A^
=
d^/d^
(11)
then it will be indifferent for the rate of the increase of the species
whether visual acuity is increased by Agi or auditory acuity by Ag2 .
We might say, in this sense, that the increments Agi and Ag2 are, in
this event, equivalent, or that that they have the same total value
(in exchange against each other) for the species. Moreover, from
(10) it is evident, on this same understanding, that the partial deriva-
dr
tive
^
measures the value (in exchange) per unit,
11
to the species, of
dr
the parameters gi, and similarly measures the value (in exchange)
per unit of the parameters gz . We may symbolize these facts by
writing
dr
dS-** <12)
dr
d^.
=
^ <*
The relation (9) then appears in the form
y-i
u An arbitrary proportionality factor enters, which is
conveniently made
unity by suitable choice of units.
350 ELEMENTS OF PHYSICAL BIOLOGY
Certain steps in the development set forth above are reminiscent of
the hedonistic calculus of Jevons and his school of economists. One
is thus naturally led to look for a relation between value (in exchange)
as here defined, and economic value in exchange, as conceived by those
authors. It should be expressly noted, however, that the reasoning
here followed, and the conclusions reached, are quite independent of
any economic theory. As for the relation of the present reflections
to economic theory, this will become apparent in the paragraphs that
follow.
EFFECT OF CHANGE IF BEHAVIOR SCHEDULE
We may note, first of all, that
drj _ dr C)j _ 5?- ,
dlpd^jdipd^
j
where p-}
= ^77defines what we may term the marginal productivity
OIL j
of energy expended upon the parameter Uj. This marginal productivity
p; will, in general, differ from the total productivity Pj
defined by equation (1).
It is, however, desirable, to express the relation between r and the
behavior schedule in another way, with the following considerations
in mind.
Rigid or Automaton Type and Elastic Type of Behavior
Schedule. The apparatus by which the coefficients X defining the
behavior pattern are determined varies widely in different species
of organisms. At one extreme we may suppose that we have an
organism, with a rigid, inelastic behavior schedule, the coefficients X
being determined explicitly, once for all, by the properties of the
individual. The extreme case of this kind is to be found, presumably,
in plants, where, for example, the amount of energy expended in
anabolism to replace wear and tear (e.g., annual leaf-fall) may be
taken as a comparatively simple function of the leaf area. Many
of the lowest forms of animals, actuated by simple tropisms, no doubt
also approximate closely to such a rigid behavior schedule, what might
be termed the automaton type of behavior schedule.
But in the higher animals, and most particularly in man, we have
an elastic behavior-schedule. Here the A's are not fixed in simple
explicit manner by the physical character of the organism. We
BELATIOX OF TBANSFOKMER TO AVAILABLE SOURCES 351
encounter here the phenomenon which we experience in ourselves
subjectively as free choice between alternative courses of action,
alternative values of the X's open to us to choose from.
This cannot mean, of course, that the X's are wholly arbitrary, or
physically indeterminate. Some action is and must be taken. But
the natural principle which operates in the determination of the X's,
of the behavior schedule, is not immediately obvious.
The avenue of approach which seems to give most promise of
lending us an insight into the relations here involved, is the following:
The elastic type of behavior schedule, the free-choice schedule, as
distinguished from the automaton type, is essentially a characteristic
of the more highly organized among the mobile (animal) organisms.
This fact is so prominently displayed to us in our own selves, that
the adaptive superiority of the elastic type over the automaton type
is commonly (and perhaps in a measure unjustly) taken for granted.
If this assumption of such superiority be true, then the principle
which operates in determining the coefficients X's must be that they
tend, on the whole, to be adjusted, in the operation of free choice,
toward values favorable to the growth of the species. In the ideal
case of perfect adaptation the X's would, according to this view, be
such as make r, the proportional rate of increase of the species, a
maximum. Let us see what this would imply.
In reacting upon its environment to influence the parameter Uj,
the organism itself necessarily undergoes some modification, since
it gives up energy Ej, (subjectively this modification is commonly
felt as fatigue). In other words, if the state of the individual is
defined by the values of certain (internal) parameters /i, /a . >
then the expenditure of an element of energy 8 E-} by the individual,
if not accompanied by other effects, is accompanied by a change
5/i,5/2 . . . in the parameters /. At the same time the (external)
parameter Uj is modified by ou-}
.
The effect of these modifications upon r is given by
where, for brevity, the contracted notation has been employed
* *'
drd/k
352 ELEMENTS OF PHYSICAL BIOLOGY
If r is to be a maximum, (5r) must vanish for any arbitrary small
value of 5 Ej, so that we must have for every subscript j, according
to (16)
dr dj dr d/ .
(18j
or, adopting the notation of (12), (13), (15)
t'uiPj
+ t'fpt
= (19)
Relation between Ideal and Actual Organism. We have thus
far considered an ideal type of organism of the free choice type of
behavior schedule, constructed on the principle that the X's shall be
so chosen as to make the proportional rate of increase r a maximum.
It remains to consider the relation between this ideal type of organism
and the actual organism. The actual organism is not consciously
guided by any consideration of the effect of his actions upon
the rate of increase of his species. At least the instances in which
such considerations are operative are so exceptional that we may well
leave them out of account. What guides a human being, for example,
in the selection of his activities, are his tastes, his desires, his pleasures
and pains, actual or prospective. This is true, at least, of some of
his actions, those which are embraced in his free-choice type of behavior
schedule. That the human behavior schedule12
also contains
an element of the non-elastic (automaton) type may be admitted in
deference to those who have leveled their destructive criticism at the
hedonistic account of human behavior. We may, however, restrict
our discussion here to that portion or phase of conduct which is
determined by hedonistic influences. In this case, then, we are dealing
with an organism which seeks to make, not r, but 0, its total pleasure
("ophelimity" in Pareto's terminology) a maximum. Argument
precisely similar to that developed above here leads to the condition.
dQ 5uj dO
r
d/
+ = C20)
12
In a modern civilized community we cannot very well speak of one typical
behavior schedule of the individual, owing to the division of labor, with
specialization of individuals in different pursuits. This matter will be found
discussed more particularly, in Chapter XXVIII dealing with the adjusters
HELATION OP TRANSFORMER TO AVAILABLE SOURCES 353
ft is immediately seen that (18) and (20) will lead to the same
adjustment of the activities of the individual if, and only if the
50
marginal ophelmiities r are proportional to the corresponding
A *'
'
derivatives ;r~ i.e.
OUj
-__ _ c
OUj UMj
We see then, that an organism actuated by pleasure and pain will
so distribute its activities as to make r a maximum, if, and only if, its
marginal ophelimities are proportional to the corresponding deriva-
br
tives r~ .
OUj
Effect of Small Departure from Perfect Adjustment. If we
look upon the sense of pleasure and pain as an adjunct serving the
express purpose of directing the activities of the organism towards
ends beneficial to the growth of the species, then a species for which
condition (21) were satisfied would represent perfect adaptation in
this respect.
This leads us to a somewhat different setting of our problem regarding
the influence of a change in the behavior pattern upon the rate of
or
increase of the species. Instead of enquiring after :^7 ,
we may seek
OHij
information regarding the influence, upon r, of a change in the tastes
50
or desires of the species as defined by the derivatives r . AnJ
OUj
answer can be given at any rate in the neighborhood of the "perfect"
adjustment of tastes defined by (21), i.e., for a species whose behavior
schedule does not depart materially from that defined by (18).
Consider a species for which the ideal (perfect) adjustment is given by
ujpj
+ fpf
= (22)
Suppose that this species actually adjusts its activities according to
the plan (23) departing slightly (by an "error of valuation" 5e) from
the perfect adjustment, namely
(uj + 8e)pj + vfpt
= (23)
354 ELEMENTS OF PHYSICAL BIOLOGY
Then it can readily be shown13
that
<.
/ ?>
7^=
~ V;Pi ?T- (SujPj + WfPf) (24)
O6j y ouj
j
which is the required analytical expression for the influence of a small
"error of valuation" upon the rate of increase of the species.
The utility which such formulae as here developed may possess
must be sought, not so much in their application to numerical
examples data for this are now and may long remain unavailable
as in the light they throw, quantitatively, upon the biological
foundations of economics, in the relations which they reveal between
certain biological and certain economic quantities. It must be
remembered that the mathematical method is concerned, not only,
and indeed not primarily, with the calculation of numbers, but also,
and more particularly, with establishment of relations between magnitudes.
14
Relation of Economic Value to Physical Energy. The behavior
schedule has been quantitatively defined in terms of energy.
This, if not the only possible definition, is at any rate a convenient
one, and has also the advantage of emphasizing the important relation
of the organism to the energy sources of his environment. His
correlating apparatus is primarily an energy capturing device its
other functions are undoubtedly secondary. Evidence of this is
manifold. The close association of the principal sense organs, eyes,
ears, nose, taste buds, tactile papillae of the finger tips, with the
anterior (head) end of the body, the mouth end, all point the same
lesson,
15
which is further confirmed by the absence of any well
developed sense organs in plants.
16
Exceptions here do indeed prove
the rule, for sensitive plants, with a well-defined correlating apparatus,
are just those which have departed so far from norm as to
consume flesh food. And contrariwise, we ourselves are "blind"
toward the one food that is omnipresent and which we consume by
" A. J. Lotka, Jour. Washington Acad. ScL, 1915, vol. 5, p. 397.
14
Compare A. Cournot, Researches into the Mathematical Theory of
Wealth, English translation by Irving Fisher, 1897, p. 3.
16
Herbert Spencer, Principles of Biology, vol. 2, p. 166; E. H. Starling, Science,
1909, p. 394; A. J. Lotka, Annalen der Naturphilosophie, 1910, p. 67.
16
For a resume of "plant psychology," see C. H. Farr, Atlantic Monthly,
December, 1922; also Sir Frederick Keehle, The Plant Commonwealth and
Its Mode of Government, Nature, 1924, vol. 144, pp. 13, 55.
RELATION OF TRANSFORMER TO AVAILABLE SOURCES 355
an almost unconscious, vegetative process, namely oxygen. If we
seek an insight into the "psychology of plants/' it may be well to
begin by imagining what our mental state would be if, in all our food
quest, we remained as passive and indifferent as in the function of
breathing.
The life contest, then is primarily a competition for available
energy, as has been pointed out by Boltzmann.17
Energy in this
sense and for this reason has value for the organism which is a very
different thing from saying (as some have said or implied) that
economic value is a form of energy. It is true that different
kinds of energy are in a certain sense interconvertible into each
other at fairly definite rates by exchange upon the market, in a
human population. But the conversion factors here involved are of
a totally different character from those that enter into the analytical
expression of the law of conservation of energy.
This must be immediately apparent from the fact alone that the
'mechanical equivalents" of the several forms of energy are absolute
constants, whereas the economic conversion factors are somewhat
variable, though they have often a species of approximate constancy,
a fact which calls for explanation.
Economic Conversion Factors of Energy, A simple example
may help to clarify the view; the case of the automatic vending
machine, the penny-in-the-slot chocolate dispenser, for instance.
The salient facts here are :
1. A definite amount of money brings in exchange a definite amount
of commodity (and of energy).
2. The physical process is a typical case of "trigger action," in
which the ratio of energy set free to energy applied is subject to no
restricting general law whatever (e.g., a touch of the finger upon a
switch may set off tons of dynamite).
3. In contrast with the case of thermodynamic conversion factors,
the proportionality factor is here determined by the particular
mechanism employed.
Reflection shows that all transformation of money or of economic
assets of any kind into energy by exchange upon the market is of
17 Der zweite Hauptsatz der mechanischen Warmetheorie, 1886 (Gerold,
Vienna), p. 210; Populate Schriften, No. 3, Leipsic, 1905; Nernst, Theoretische
Chemie, 1913, p. 819; Burns and Paton, Biophysics, 1921, p. 8; H. F. Osborn,
The Origin and Evolution of Life, 1918, p. XV.
356 ELEMENTS OP PHYSICAL BIOLOGY
this character. It is always a case of trigger action. Somewhere
there is a store of available energy, which can be tapped with an
expenditure of greater or less effort. The payment of the price sets
in motion the requisite machinery for the release of that energy (or
for its transfer of ownership, the release being delayed at the discretion
of the buyer).
In view of the entire absence of any general law regulating the
ratio of energy released to energy applied in such cases of trigger
action, we may ask the question, how does it come about that economic
conversion factors, economic ratios-in-exchange of different
forms of energy, display any regularity whatever? The answer is
not far to seek. The approximate constancy of the economic conversion
factors is traceable to the approximate constancy in type
of the mechanism involved, namely the human organism and its
social aggregations. Just as one particular slot machine will always
deliver a certain package of chocolate, so a certain social organization
under similar conditions will render (approximately) the same
amount of selected form of energy in return for a stated sum of money.
As to the circumstances that quantitatively determine these economic
conversion factors, for a discussion of these the reader must be referred
to the literature;
18
only this may be remarked here, that the
conception advanced by Ostwald,
19
for example, that the determining
feature is the (physical) availability of the particular form of energy,
is inadequate.
Collective Effect of Individual Struggle for Energy Capture. Our
reflections so far have been directed to the selfish efforts of each
organism and species to divert to itself as much as possible of the
stream of available energy. But if we recall once more the admonition
of Bunge Nature must be considered as a whole if she is to be
understood in detail we shall be led to enquire: What must be the
18
A. J. Lotka, Proc. Natl. Acad. Sei., 1921, vol. 7, p. 192.
19
W. Ostwald, Die Energie, 1908, p. 164. Of other literature more or less
pertinent to the subject the following may be mentioned: G. Helm, Die Lehre
von der Energie, Leipsic, 1887, pp. 72 et seq. W. Ostwald, Energetische
Grundlagen der Kulturwissenschaften, Leipsic, 1909, p. 155. Die Philosophic
der Werte, Leipsic, 1913, pp. 260; 314-317, 326, 328; Budde., Energie und Recht,
Leipsic, 1902, p. 56; Winiarski, Essai sur la M<eanique Sociale, Revue Philosophique,
1900, vol. 49, p. 113; J. Davidson, Qu. Jour. Economics, August, 1919,
p. 717.
RELATION OF TRANSFORMER TO AVAILABLE SOURCES 357
effect, upon the world-as-a-whole, of the general scrimmage for
available energy?
It has already been pointed out that the operation of the correlating
apparatus is of the nature of a cycle beginning and terminating in the
external world; that the grazing cow is able to subsist because the
grass is "hungry for oxygen," that animals are essentially catalysers,
oiling the machinery, as it were and assisting energy in its downhill
path to levels of lower availability (higher entropy). If we had only
the animal kingdom to consider we should in the first instance be
disposed to conclude that the cosmic effect of the scrimmage for
available energy would be to increase the total energy flux, the rate of
degradation of the energy received from the sun. But plants work
in the opposite direction. And even among animals, greater
efficiency in utilizing energy, a better husbanding of resources, and
hence a less rapid drain upon them, must work to the advantage of
a species talented in that direction.ao
There are thus two opposing
tendencies in operation, and it is difficult to see how any general
principle can be applied to determine just where the balance will be
struck.
The Law of Evolution Adumbrated as a Law of Maximum
Energy Flux. This at least seems probable, that so long as there is
an abundant surplus of available energy running "to waste" over the
sides of the mill wheel, so to speak, so long will a marked advantage be
gained by any species that may develop talents to utilize this "lost
portion of the stream." Such a species will therefore, other things
equal, tend to grow in extent (numbers) and this growth will further
increase the flux of energy through the system. It is to be observed
that in this argument the principle of the survival of the fittest yields
us information beyond that attainable by the reasoning of thermo-
dynamics.
21
As to the other aspect of the matter, the problem of economy in
husbanding resources will not rise to its full importance until the
available resources are more completely tapped than they are today.
Every indication is that man will learn to utilize some of the sunlight
that now goes to waste. The general effect will be to increase the rate
20
Compare JrJohnstone, The Mechanism of Life, 1921, p. 220.
21
This fact has been recognized independently by the writer and also by
H. Guilleminot. For details see A. J. Lotka, Proc. Natl. Acad. Sci., 1922, p.
153.
508 ELEMENTS OF PHYSICAL BIOLOGY
of energy flux through the system of organic nature, with a parallel
increase in the total mass of the great world transformer, of its rate of
circulation, or both.22
One is tempted to see in this one of those maximum laws which are
so commonly found to be apt expressions of the course of nature.23
But historical recollections here bid us to exercise caution; a prematurely
enunciated maximum principle is liable to share the fate of
Thomsen and Berthelot's chemical "principle of maximum work."
STATISTICAL MECHANICS OF SYSTEMS OF ORGANISMS
Let us call to mind once more the picture of the life conflict viewed
as the interplay of organisms moving over a topographic chart and
suffering a succession of collisions with each other and with features
of their environment.
We might seek to develop, for such a system, a discipline of
statistical mechanics similar to that which the physicist has developed
to deal with the kinetic theory of gases and allied problems.
No attempt will be made here to carry out this project in any degree
of completeness, or to carry it to such ultimate conclusions as it may
be competent to furnish. But it may not be out of place to indicate
at least a few points, in addition to the pertinent matter contained
in the preceding paragraph, that can be offered as first steps toward
the development of such a discipline.
Mean Free Path. In the elementary kinetic theory of gases it
is shown that the mean free path I of a molecule in a gas is given by24
(25)
Mrs*
For a more detailed discussion of this point see A. J. Lotka, Proc. Natl.
Acad. SeL, 1922, p. 147.
23
Compare J. Larmor, Proc. London Math. Soc., 1883-1884, vol. 15, p. 159;
Petzoldt, Maxima and Minima und Okonomie.
2 *
See, for example, Winkelmann, Handbuch der Physik, 1906, vol. 3, p. 696.
This elementary result is based on the simplifying assumption that the velocity
of all the moving molecules is the same. Taking into account the velocities
ranging according to the Maxwellian law, it is found that I = < ,,
~"
But
for our purposes it is quite sufficient to accept the simple elementary method
and its result.
RELATION OP TRANSFORMER TO AVAILABLE SOURCES 359
where Ar
is the number of molecule per unit volume, and s their
diameter. This, of course, relates to motion in three dimensions.
We may apply similar methods to the case of two species, Ni
individuals of Si and AT
2 individuals of S, operating on a square
mile, say, of ground. If the field of influence of Si is a circular field of
diameter s it is easily shown that (according to the elementary
theory) the mean free path for the individuals of Si is, expressed in
miles,
Z=- (26)
Nzs
Frequency of Collisions and of Capture. If v is the average
velocity of the individuals of Si, the frequency collision will be,
approximately at any rate,
F = vNiNz s (27)
In general not all collisions, but only a certain fraction c, will result
in capture. The total captures per unit of time, per square mile, will
then be
C = cvNiNts (28)
It may be noted in passing that the expression thus found represents,
in certain cases at any rate, the term k Ni Nz which appears in
equation (38) of Chapter VIII, where it was introduced without
analyzing its precise physical significance.
Influence of Size of Organism. It is interesting to observe the
influence of size of the predatory organism on the frequency of
capture. Suppose, leaving all other factors the same, we halve the
size of the predatory organisms, so that the new value of Ni, which
we may designate by a prime, is given by
Ni' = mi (29)
If the velocity of the smaller organisms is unchanged, we shall have
for the new frequency of capture
Fr
= 2NiNicvs (30)
More appropriate, perhaps, is the supposition that the new velocity
v
f
is related to the old v in the proportion of the linear dimensions,
i.e., as W' = -\/2 = 1.26 We should then have
360 ELEMENTS OF PHYSICAL BIOLOGY
F'
=
= 1.6 NiN2cvs (32)
= 1.6 F (33)
There is then, other things equal, an advantage for predatory species
in small size. In nature, presumably, there is a tendency to strike
a compromise between the advantage thus gained, and certain disadvantages,
such as relative defencelessness, incurred by decreased
stature.
Curves of Pursuit. The coefficient c which occurs in (28) is
open to further discussion and analysis. A collision is characterized
by the fact that through its duration the law of motion differs essentially
and discontinuous^ from the law of motion between collisions.
The coefficient c evidently depends on the law of motion during
collision. During collision the two individuals follow a course of
the type known as a curve of pursuit. For a discussion of these
curves so far as they have been studied, the reader must be referred
to the pertinent literature.25
The mathematical theory of problems
in pursuit and conflict generally has hitherto been developed principally,
if not exclusively, in connection with warfare. In any effort
to deal with the more general theory of conflict between biological
species it may be well to have an eye open to the methods followed
in the treatment of problems of military and naval tactics. Here,
however, this mere hint must suffice.
Random Motion under a Bias. Various problems relating to
the motion of a point following in part a random course, but at the
same time controlled by some kind of directing influence, have been
dealt with by L. Baehelier in his Calcul des Probabilites, 1912, to which
the reader may be referred. Chapter XX, on Probabilites Cinematigues,
may be found suggestive in connection with the type of motion
presented by living organisms, a motion that can be regarded as
containing both a systematically directed and also a random element.
Use of Models. Another point shall be passed over here with a
mere suggestion. The mathematical treatment of the statistical
mechanics of the kind of systems here taken in view may appear to
threaten formidable difficulties. It is to be hoped that this will not
25
See, for example, Boole, Differential Equations, fourth edition, p. 251.
BELATIOX OF TBANSFORMEB TO AVAILABLE SOURCES 361
altogether prevent its attack, even if at first sweeping simplifications
must be made in the fundamental assumptions. But the writer
wishes to draw particular attention to one method that in the past
has not been used to any considerable extent, and which may be
found serviceable where ordinary analytical methods become forbidding.
The method to which I refer is a special form of the graphic
method, namely the method of working models. It is well worth
considering whether interesting light may not be thrown on various
problems of biological conflict, by the use of models designed to imitate
the biological warfare somewhat after the manner in which the
war game imitates the armed conflict of nations.
CHAPTER XXVI
THE CORRELATING APPARATUS
Let us now consider the character of the material Nature whose necessary
results have been made available .... for a final cause. Aristotle,
In preceding pages we have dealt with the correlating apparatus,
and its functions, in the gross, without paying more than passing
attention to the details of its constitution and operation. It is
desirable now to fill in some of these omitted details; we shall consider
in turn the depictors, elaborators, effectors and adjusters, with
especial attention to the case of the human species. For it goes
without saying that a particular interest attaches to the study of the
correlating apparatus in man, if only because in no other creature
has it developed such singular excrescences as that which prompts, for
example, the writing of this book and furnishes the means for the
accomplishment of the task. Man thus stands out, if not as a
superior being, at any rate as a highly peculiar creature. One might
perhaps dispute the propriety of the study so unhesitatingly advocated
by Pope; but one cannot truthfully deny the fundamental
fact of man's inordinate interest in himself. Moreover, there are
certain very important phenomena that play a prominent rdle
in the operation of the correlating apparatus, phenomena for the
study of which our chief source, indeed our only altogether indisputable
source of data, lies in our own self: that group of phenomena
briefly summarized under the term consciousness. Inevitably
therefore, a study of the correlating apparatus, if carried out
in any detail,, will center largely about the manifestation of this
apparatus in the human species.
In dealing with the several elements of the correlating apparatus
it will be convenient to depart somewhat from what might appear the
most natural order. Receptors and effectors, though functionally
separated at the two extreme ends of the operative cycle, are in certain
respects rather closely related in practice. Similarly there is in
certain respects an approach between the elaborators and the adjusters.
We shall, accordingly, consider the several elements in
the order Receptors, Effectors, Elaborators, Adjusters.
362
CORRELATING APPARATUS 363
THE RECEPTORS
The receptors, as lias already been noted, are one of the two classes
of organs or faculties concerned with the depiction of the environment
in the organism (the other class being the elaborators) . A typical
receptor is the eye, which, in the most literal sense, depicts the
external world upon the retina. The law of depiction, in this case,
is given by the principles of geometrical optics, and is most easily
expressed in terms of two systems or reference frames of coordinates,
the one fixed with regard to suitably chosen features in the external
world, the other fixed with regard to the eye that forms the image.
So, for example, the impression my eye received at noon today was
determined by the fact that it was located at the N.W. corner of
Monument Square and was pointed upward at an angle of about 45
degrees, and turned in the E. S.E. direction. The tip of the Washington
Monument, whose coordinates in the "external world"
system are x, y z, say, was thus depicted upon my retina3
by a point
with the coordinates x', y' ,
z'. This appearance in the analytical,
expression of the depicting process, of two coordinate reference
frames, or at least something equivalent, is very characteristic and
should be particularly noted. We shah
1
have occasion to refer to
this matter again.
But the picture formed by the eye, typical as it is of the process
of depiction, is nevertheless essentially incomplete. We know our
own world picture as something much more expansive, much more
intimate and profuse in detail. When I view a landscape I have not
merely in my eye a dead photograph of the rolling hills and placid
valleys, in their garb of green, and stirring with life. These things
I see; but at the same time I hear the bells of the grazing herd, and
all those familiar noises that belong to the tout ensemble of the rustic
scene; while the fragrance of the sun-scorched woods, or perhaps the
perfume of flowers, enters to fill out another aspect of the picture.
So all our receptors or sense organs are, in a larger sense, depicting
organs, each supplying its special share toward that composite world
picture into which their several contributions are united and further
1
Strictly speaking this is an inverted statement of the facts. We are
directly cognizant of sense impressions x' y' z', as our prime data, and we
infer, or hypothecate, as secondary data, corresponding coordinates x, y, z,
of an external world. See Chapter XXXIV, footnote 2. But for our present
purpose the more usual ("naive) viewpoint will serve.
364 ELEMENTS OF PHYSICAL BIOLOGY
developed by certain special faculties, the Elaborators, notably the
Memory and Imagination.
Artificial Receptors, A detailed discussion of the natural
receptors with which our body is supplied by the physiological
processes of embryological development and growth is
unnecessary
here, since these things are fully discussed in special works devoted
to these subjects, such as for example, Ladd and Woodworth's Elements
of Physiological Psychology (Scribner's, 1915). But a special
development of the receptors in man, which has had, and is destined
still to have, an altogether superlative importance in the evolution
of the world is passed over in total silence in most works of the
character cited, and calls for discussion here in some proportion to
the importance of this phase of the subject: the artificial aids and
adjuncts to our senses which the ingenuity of man has pressed into
his service. With prophetic eye Robert Hooke in 1665 foresaw no
doubt very imperfectly what these aids were destined to accomplish
for the human race.
The next care to be taken, in respect to the senses, is a supplying of their
infirmities with instruments, and, as it were, the adding of artificial organs to
the natural; this .... has been of late years accomplished with prodigious
benefit to all sorts of useful knowledge It seems not improbable,
but that by these helps the subtility of the composition of bodies,
the structure of their parts, the various texture of their matter, the instruments
and manner of their inward motions, and all the other possible appearances
of things, may come to be more fully discovered
That the artificial adjuncts to our correlation apparatus are subject
to a process of evolution by selection, by trial and error, was clearly
recognized by David Hume,
2
who in this, as in other important points,
anticipated the conceptions of Darwin and Spencer. The last
mentioned was the first to realize fully and to point out clearly the
role of man-made contrivances, as artificial sense organs on the one
hand, and as artificial members (effectors) on the other.3
As we
Dialogues
concerning Natural Religion, 1757, Edition of 1907, pp. 189-190.
Hume's reflections relate particularly to the artificial effectors (as represented
by^a ship, for example), but the underlying principle is, of course, the same.
3
Compare Herbert Spencer, Principle of Psychology, Chapter VII, Section
164; Emerson, The Conduct of Life, Everyman's Library Edition, pp. 159-190
(original edition, 1860); Wiener, Die Erweiterung unserer Sinne, Leipzig 1900;
Lehmann, Die Kinematographie, Leipzig, 1911, p. 1.
CORRELATING APPARATUS 365
look back today upon the progress In recent centuries, it is plain
beyond ail possible misunderstanding that the ushering in of the
era of the man-made adjuncts to his natural body has given not only
a new direction to the process of evolution, but has speeded up its progress
to an extent without the remotest parallel in the history of our
globe. How long it may have taken man to develop his organ of sight
by the slow processes of physiological evolution we are quite unable to
say, but this is certain, that the time is reckoned in many millions of
years. Contrast with this the following brief historical record: The
use of spectacles was introduced about 1350.4
The invention of the
microscope is credited to the Dutch optician Janssen, 1590. A modern
microscope is capable of a magnification of several thousand diameters.
(This, however, does not indicate its "separating'' power, which
in point of fact, is only about 200 times greater than that of the
naked eye.) Particles too small to be formed into a distinct image
by the microscope can still be detected by the ultra microscope,
invented by Zsigmondy in 1903. By the method of X-ray photography
developed chiefly by Laue 1912 and Bragg 1915, direct
optical evidence of the arrangements of atoms in a crystal is obtained,
and the distance between the layers, of the order of
T,u~o,tfoTr of an inch, is measured. This represents, in effect, a
million-fold improvement on the separating power of the eye. The
method of C. T. R. Wilson5
1912 renders visible to the eye the track
of an electron, whose diameter is of the order of one ten-millionniillionth
of an inch. The power of the natural vision is, in such case
as this, virtually multiplied by two hundred billion. The step from
the use of a crude pair of spectacles in 1300 to this was taken in the
space of about six centuries and much of the progress is condensed
within a space of less than three decades. The evolution of man's
artincal sense organs has proceeded at a pace so utterly out of scale
with that of his natural equipment that the aid of diagrammatic
representation quite forsakes us here. Any attempt to plot such.
progress as this on a single scale in rectangular coordinates could
4
Darmstaedter in his Handbueh der Gechichte der Naturwissensehaftem
states that the Emperor Nero in A.D. 63 employed a cut emerald to view the
gladiator contests. Roger Bacon in A.D. 1250 recommended the use of lenses
for persons of weak sight. The chronicle of St. Catherine's Convent at Pisa
mentions Alessandro de Soina as maker of spectacles.
s
Proc. Roy. Soc. Ser. A., vol. 87 (1912), p. 277.
306 ELEMENTS OF PHYSICAL BIOLOGY
merely result in two limbs of a curve, the first essentially coinciding
with the time axis, the second rising abruptly, at an angle indistinguishable
from a right angle.
8
The historical sketch thus drawn represents but a diminutive
fragment of the total development of man's artificial correlating
apparatus. It is out of the question to cover here in any exhaustive
manner even the aids to vision alone. We must pass by with a mere
mention the astronomical telescope, which opens the pupils of our
eyes one hundred inches wide;
7
or that fantastic application of the
stereoscope to astronomical objects, which enables us to see the
universe as it would appear to a gigantic being with his eyes many
millions of miles apart. As for a general survey of the entire field of
our artificial sense organs, to give this would amount to nothing less
than the writing of a work, in several volumes, on general observational
methodology. For our purpose here these hints must suffice.
THE EFFECTORS
The natural effectors, like the natural receptors, have received
their full share of attention, and it is not proposed to add here to the
existing literature
8
on this special phase of the subject. As to the
artificial aids to the effectors, these are plainly evident on every
hand, and do indeed give a stamp altogether its own to this unprecedented
industrial era in which we live. Gilbert and Pogue in
their memoir on Power9
estimate that the use of power derived from
coal and other extraneous sources (i.e., not from the human body)
gives to each man, woman and child the service equivalent of 30
servants. But in point of fact this figure, based merely on the total
horsepower developed, gives an altogether inadequate picture of the
real facts. Machinery has not only increased our energy output,
it has immensely multiplied the speed of production. One extreme
s
Compare also H. Heath Balden, The Evolution of Behavior, Psychol.
Eev., 1919, p. 247.
7
Or more than double this, with the aid of Michelson's interference device.
s
The following may here be mentioned: J. Amar, Le Moteur Humain, 1914;
A. Keith, Engines of the Human Body, 1920; 0. Fisher (Teubner), 1906
Grundlagen fiir eine Mechanik der lebenden K5rper; C. Bell, Animal Mechanics,
1838 (Library of Useful Knowledge); W. M. Feldmann, Biomathematics,
Lippincott 1923: L. L. Burlingame, General Biology, (Jonathan Cape) 1923.
*
Smithsonian Institution Bulletin, 102, Part 5.
CORRELATING APPARATUS 367
example of this is seen in the printing plant. A modern newspaper
press, with a crew of 6 to 10 men, can turn out 80,000 complete
16-page papers per hour, all folded, counted and delivered ready for
the carrier boys. It may be left to the reader to make some kind of
estimate of the time that would be required to write the same matter
out in longhand, to say nothing of the folding and counting.
SOO miles.
FIG. 70. THE EVOLUTION OF MAN'S MEANS OF TRANSPORTATION
Distance travelled in one nour by different means of conveyance
Artificial Effectors; Industrial Evolution. In the development
of our artificial effectors, as with the receptors, the progress
of evolution has been a rocket-like ascent. To take a simple and
very moderate example, the evolution of means of personal transportation
is exhibited diagrammatically in figure 70, which, is in
368 ELEMENTS OP PHYSICAL BIOLOGY
essence a target of zones of operation of the kind referred to in
Chapter XXV, The figures on which the diagram, are based, some
of which may very well have been superseded, are the following:
TABLE 33
Distance travelled in one hour by different modes of conveyance
It has already been noted that the evolution of man's artificial
aids to his effectors, by a process of "survival of the fittest" was recognized
as early as 1757 by David Hume. We can now add, from the
modern quantitative viewpoint, that this resemblance between the
development of our inorganic accessories and that of organic populations
extends also, in a number of instances, to the growth curves.
So, for example, figure 71 exhibits the curve of growth of American
Railways (United States).
19
It will be recognized as the typical
S-shaped Yerhulst-Pearl curve of growth.
Singular Effects of Industrial Evolution. Aside from its obvious
result of greatly enhancing the effectiveness of the correlating
apparatus, the modern development of its artificial aids has
had certain other less obvious and in some respects rather singular
effects. For the artificial portion of our correlating apparatus differs
in several important respects from our native endowment. My
microscope does not die with my body, but passes on to my heirs.
There is thus a certain permanence about many portions at least of
the artificial apparatus. And for that very reason the development
of this artificial equipment of human society has a cumulative force
that is unparalleled in ordinary organic evolution. This cumulative
effect is most of all marked in the artificial aids to our elaborators.
"Figures for 1840 Sci. Am. Reference Book, 1914, p. 235; 1850 to 1910;
World Almanac. 1921, p. 277; figure for 1918, Statistical Abstracts, 1920, p. 814.
Compare also E. Pearl. The Population Problem, Geogr. Rev., October, 1922
p. 83S.
CORRELATING APPARATUS 369
The most singular feature of the artificial extension of our natural
body is that they are shared in common by a number of individuals.
When the sick man consults the physician, who, we will say, makes
a microscopic examination, for example, the patient is virtually hiring
a pair of high-power eyes. When you drop a nickel into the telephone
box, you are hiring the use of an ear to listen to your friend's voice
five or ten miles distant. When the workingman accepts a wage
of forty dollars for his weekly labor, he is in fact paying to his em-
1830 10 I9SO
FIG. 71. GEOWTH OF AMEBICAN RAILWAYS
plovers an undetermined amount for the privilege of using his
machines as artificial members to manufacture marketable wares.11
The modern development of artificial aids to our organs and
faculties has exerted twr
o opposing influences.
On the one hand it has in a most real way bound men together into
one body: so very real and material is the bond that modern society
might aptly be described as one huge multiple Siamese Twin.
On the other hand, since the control over certain portions of this
common body is unevenly distributed among the separate individ-
11
In certain, branches of industry (e.g., shoemaking) a rental is actually paid
by the worker for the use of machines.
370 ELEMENTS OP PHYSICAL BIOLOGY
uals, certain of them may be said in a measure to own parts of the
bodies of others, holding them, in a species of refined slavery; and
though neither of the two parties concerned may be clearly conscious
of the fact, it is often resented in a more or less vague way by the one
less favored. Herein lies one source of much of the social unrest
that has accompanied the development of modern industrialism.
The more optimistic among us may entertain a hope that in tune the
unifying influence of our ever growing common body may outweigh
the disruptive forces that ever and again manifest themselves too
plainly for our comfort. That a species of "slavery," that is to say
of ownership of one person's body by another or by others, should
prevail, is in the last analysis an absolutely unavoidable situation,
once we recognize that no sharp lines are drawn to separate the
individual from his fellow; willy-nilly we must accept the fact. We
ma}', however, seek to control the distribution of this ownership in
the way most advantageous to the general welfare. That is the
pin-pose of our property laws, such as they are and such as they
will be.
CHAPTER XXVII
EXTENSION OF THE SENSUOUS WORLD PICTURE
. . .
bodily eyes
Were utterly forgotten, and what I saw
Appeared like something in myself, a dream,
A prospect of the mind."
Wordsworth.
The Elaborators. While the sense organs (receptors) are the
prime agents in the formation of our world picture, they are not the
sole agents that contribute. The further elaboration of this picture
is largely the work of our mental faculties: memory, which stores
impressions and makes our picture a chronological album or file rather
than a single snapshot : and imagination,
1
which fills in those features
of the landscape that are hidden from view by intervening obstacles.
This process of filling in hidden features may be either arbitrary, as
in the demonistic interpretation of nature current among savages
and surviving among us in many superstitions, in the play of the
child, and in the fancy of the poet; or, the process of filling in may
be more or less rigorously directed by certain rules which long experience
has taught us to be productive of a realistic picture, a picture
which, when comparison with actuality becomes possible, is found to
be "in accord with facts;" one that may safely be made a basis for
zwechmassig action.2
The realistic type of thinking, the kind which
"works," has perhaps been evolved by a process of survival of the
fittest from the other kind (termed autistic thinking).
8
The body
of rules which realistic thinking follows constitutes the science of
logic. The constructive exercise of the imagination within the limits
prescribed by logic is what we call the process of reasoning. When
1
It should be noted in passing that the word imagination itself means
image-formation or depiction.
2
It is deserving of emphasis that the function of imagination is not merely
the conception of mythical creations, but also, and quite particularly, the
presentation, to the mind, of realities. Hence imagination plays an important
role in the exact Sciences.
3
See F. L. Wells, Mental Adjustments, 1917, p. 46.
371
372 ELEMENTS OF PHYSICAL BIOLOGY
its premises are suitably chosen, for instance, on the basis of sense
perceptions, the product is a contribution to a realistic world picture.
The Scientific "World Picture: Coordinate Systems. The last
refinement in world depiction is the scientific world picture. In
the exact sciences this assumes the numerical form. So, for example,
the position of a particular object in space is identified, in the simplest
case (namely, if the object is sufficiently described as a point) by
the statement of three numbers, the space coordinates of that point.
The motion of the point is then described or represented (pictured)
by a set of equations
=/.(*)
where x, y, z are the coordinates of the moving point at instant t,
this last being also identified by means of a number.
Now an infinite number of equally appropriate pictures of this
kind can be given for the motion of the point, for the coordinates
z, y, z can be chosen in an infinite number of ways. They may be
rectangular coordinates, or polar coordinates; their reference frames
may be fixed on the earth, or on the sun, or in any other suitable
manner. But each choice of a system of coordinates will furnish its
own particular set of functions /; each picture will in this respect
differ from every other such picture. Not a little of the success of
the scientific investigator depends on his judgment in choosing a
suitable system of coordinates, such as will furnish the simplest
and most convenient picture, the most convenient functions /. So,
for example, the motion of the planets is most simply described
with reference to a frame of coordinates fixed to the sun, and not,
for example, one fixed to the earth. The discovery of a particularly
convenient reference frame, though in a sense it adds nothing to
our knowledge of concrete facts, may nevertheless constitute one of
the major events in the progress of human knowledge. For, such
discoveries "'make the Universe anew within the minds of men."4
The Ego as a Coordinate Reference Frame. It is a matter
of very particular interest that the first rudiments of this numerical
world picture, in terms of coordinates, is to be discerned in one of
4
W. C. Curtis, Science and Human Affairs, 1922, p. 186. ^'
EXTENSION OF SENSUOUS WORLD PICTURE 373
the most fundamental traits of the human mind, one of the most
elementary, irresistible, na'ive intuitions to which we are all subject,
namely, the distinction between a self and an external world. If
we wished to express mathematically the relations between the appearance
upon the ground glass screen of a photographic camera,
and the "external world" about the camera moving through that
world, we should undoubtedly find that the simplest expression
would be obtained in terms of a reference frame fixedly attached
to the more solid features of the external world, and a second reference
frame fixedly attached to the camera.
So we find it most convenient to express the infinite variety of
sense perceptions (the totality of our world-picture) and the relations
between them in terms of a self and an external world. This
naive intuitional view may be taken to be the rough unanalyzed raw
material out of which, by a process of attenuation and abstraction,
the scientific method has finally peeled the kernel of the two systems
of coordinates the one in which the external world is described, and
the second, privately owned by the individual as it were, in which
his world picture is recorded. The self thus appears as something of
the nature of a (somewhat blurred) reference frame of coordinates,
or rather, a set of several such frames, for the individual is
quite accustomed to a multiple entry system of bookkeeping, in
which the same object is entered both as seen and as felt, for
example.
5
The Ego Immaterial. This conception of the self as something
of the nature of a system of coordinate reference frames will be found
rather helpful in several connections. So, for example, it instantly
makes clear why we cannot conceive of the self as something material.
5
It is no doubt this double entry system that is largely responsible for our
construct of an external world compounded of what are at bottom nothing more
than complexes of sensations. As E. B: Holt remarks, "even two reflexes
acting within one organism bring it about that the organism's behavior is no
longer describable in terms of the immediate sensory stimulus, but as a function
of objects and situations in the environment" (The Freudian Wish,
1916, p. 76). Those complexes of sensations which we refer or ascribe to
objects are characterized by a greater or less degree of permanence. It is this
permanence of association of certain sensations that induces us to postulate
an object as their carrier. Such relative permanence lends a peculiar interest
to the sensation complex, so that it has truly been said that the aim of science
is the search for the invariants of Nature (E. W. Brown, Scientific Monthly,
1921, p. 408).
374 ELEMENTS OF PHYSICAL BIOLOGY
When I say that / (the ego) was at Grand Central Station yesterday
at noon, it is true that from the nature of things a certain object which
I speak of as my body, i.e., the body of the ego, was there at the
time stated. But this is merely an implication, it is not a simple
identification of the ego with the body, otherwise the phrase my
body would be pointless. Once I become accustomed to think of
the ego as something of the nature of a coordinate reference frame,
the matter is perfectly clear. The thing that counts, in the depiction
of the world in me, is the position of my reference frame relative
to the external world. To say that J was at the station yesterday
at noon is to say that this reference frame was thus situated. The
body attached to this reference frame is in that sense my body.
And clearly, the ego so denned is something immaterial.
Interpenetration of Egos. Then again, it is a notorious fact
that the boundaries between the self and the external world cannot
be clearly drawn.6
The objective significance of this has already
been noted, in the discussion of our artificial receptors and effectors,
which are largely '''shared'
3
by a number of persons. The conception
of the self indicated above makes it immediately clear that
any attempt to establish boundaries between the self and the external
world, or, for the matter of that, between two selves, is not only useless
but meaningless. Coordinate reference frames have no boundaries,
and freely interpenetrate each other, being merely immaterial
aids to fix our ideas. So the overlapping of many egos in fields common
to them, their essential unity with one another and with the
universe, ceases to appear as a strange thought entertained by
peculiarly minded people, and becomes an obvious truth.
6
Compare F. B. Sumner, Scientific Monthly, 1922, vol. 14, p. 233. "The
organism and the environment interpenetrate one another through and
through the distinction between them is only a matter of convenience."
Also C. J. Iveyser, Mathematical Philosophy, "How blind our familiar assumptions
make us! Among the animals, man, at least, has long been wont to
regard himself as a being quite apart from and not as part of the cosmos round
about him. From this he has detached himself in thought, he has estranged
and objectified the world, and lost the sense that he is of it. And this age-long
habit and point of view, which has fashioned his life and controlled his thought,
lending its characteristic mark and color to his whole philosophy and art and
learning, is still maintained, partly because of its convenience, no doubt, and
partly by force of inertia and sheer conservatism, in the very teeth of the
strongest probabilities of biological science. Probably no other single hypothesis
has less to recommend it, and yet no other so completely dominates
the human mind."
orrK
EXTENSION OP SENSUOUS WORLD PICTUBE
Where Is Mind?7
The question which has sometimes been raised,
as to location of the mind, is also seen in a better light. The thing
that counts, in the affairs of the individual, is the relative position
of the several reference frames, his own and that of the external
world. Where the adjusters, even their concrete,
^
material
apparatus, are located, is inconsequential. We may bo in doubt
whether the expression "the location of the mind" has any meaning;
but there can be no doubt that such location is of no practical
consequence.
If it can in any sense be said that mind perceives the world
of physical phenomena, this at least is certain, that it does
^
so
exclusively through the channels of our sense organs. The mind
looks into the physical world through the pupils of our eyes as t.ii
small boy watches the ball game through a knothole in the fence;;
the position of the knothole determines how much he shall sec or
the game, and in what perspective. And the mind plays upon
the physical events in our world as the organist plays upon
the pipes of a modern organ, from a keyboard whose location in
immaterial. The question "Where is the mind?" or "Whom in
the ego?" might, if we rightly understood all the pertinent facts,
appear as absurd as the tourist's request to be shown the equator
These terms are merely symbols of reference by the aid of which wo
describe relations between things.
8
And quite in accord with this, we see in death not so much ovon
as the dissolution of a system of coordinates, but merely the IONH of
their pragmatic significance:
7
Compare Bertrand Russell, The Analysis of Mind, 1921, pp. 141-MB.
"The subject appears to be a logical fiction, like mathematical poinfcH and
instants. It is introduced, not because observation reveals it, but bocauw) it in
linguistically convenient, and apparently demanded by grammar. Nominal
entities of this sort may or may not exist, but there is no good ground for
assuming that they do.
"If we are to avoid a perfectly gratuitous assumption we must diupcmw
with the subject as one of the actual ingredients of the world."
s
From this point of view one is scarcely disposed to agree altogether with
the proposition of Sir C. S. Sherrington in his Providential Address before Urn
British Association at Hull (1922) : "The how of the mind's oonnnctiou \vilh Itn
bodily place seems still utterly enigma." Nature, 1922, vol. 110, p. ,'Hil. Fur
a presentation of a point of view somewhat opposed to the one hero Hot forth,
the reader may be referred to an article The Group Mind and the Mineral ll'ill
by J. Laird, in the Moaist, 1923, p. 453.
376 ELEMENTS OF PHYSICAL BIOLOGY
And then so slight, so delicate is death
That there is but the end of a leaf's fall,
A moment of no consequence at all.
Mark Swann.
Fundamental Premises and Implicit Assumptions. It has been
noted in passing that the practical adequacy (Zweckmassigkeit) of
the world picture formed by the aid of the elaborators depends on
the judicious choice of the premises upon which the further elaboration
is based. This, of course, is a very important condition. In so far
as the premises are direct data of observation, the condition resolves
itself into a demand for care and accuracy in taking observation,
and for judgment in weighing evidence, especially where the data are
presented in statistical form, with perhaps a considerable margin of
doubt attached to each individual report.
But there are other less obvious cases, in which a scrutiny of the
fundamental assumptions is needful and presents no little clifficulty.
For some of these premises relate, not to observational data
but to conceptual structures peculiar to the world picture into which
these data are fitted. Such premises are the fundamental postulates
and axioms of geometry. By a stroke of genial intuition the founder
of Euclidian geometry set down among his fundamental premises
the parallel axiom, which states, in effect, that through a given
point one and only one straight line can be drawn parallel to a given
straight line. The true character of this fundamental premise remained
hidden from the understanding of geometricians for many
centuries. So difficult is it sometimes for us to become clearly aware
of the nature of fundamental assumptions underlying our reasoning.
Not till the end of the eighteenth century (Gauss 1792) was it realized
that the parallel axiom, so far from being a "necessary truth," is essentially
of the nature of an arbitrary assumption, and is only one of
several alternatives each of which can claim equal legitimacy, each
of which leads to a separate system of geometry. The Euclidian
system was, until recently, the one in terms of which the world picture
of the physicist found most convenient description. But in
recent years, observations of electrons moving with velocities approaching
that of light, and a number of other phenomena harmoniously
comprehended in Einstein's theory of relatively, have given
us a new world picture that finds its most convenient representation
in terms of non-Euclidian geometries.
EXTENSION OF SENSUOUS WOBLD PICTURE 377
Difficulty of Shaking off Preconceived Premises. The examination,
and, where need be, revision of our fundamental premises is a task
of a wholly different order from that of rearing upon these premises
a structure of logical argumentation. It is a task that often demands
the efforts of giant intellects, of men of altogether unusual independence
of thought. Most of us are held back by our preconceived,
intuitive judgments, which, blindly entertained, blind us also against
the recognition of possible alternatives. "The main reason for
the painfully slow progress of the human race is to be found in the
inability of the great mass of people to establish correctly the premises
of an argument." Nor is it only the great mass of the people, the
average minds, that suffer from this ineptitude. Of Dr. Johnson,
MacAuley remarks
How it chanced that one who reasoned on his premises so ably should assume
his premises so foolishly is one of the great mysteries of human nature. The
same inconsistency may be observed in the schoolmen of the middle ages.
These writers show so much acuteness and force of mind that the modern
reader is perpetually at a loss to comprehend how such minds came by such
data.
As a matter of fact our innate perversity in this matter, our inveterate
conservatism in all that concerns some of those "implicit
and unrecognized assumptions out of which sophistry is bred,"
is not due to negative influences, to sloth of mind, alone. There is
a strong positive misguiding influence at work also: The wish
is father to the thought.
10
The fact is, we are confronted here with a species of aberration
of our adjusters; an aberration which our race must outgrow if it
is to come into its full inheritance under the sway of evolution.
For, as Simpson remarks, "the stabilization of our institutions
rests ultimately upon our ability to know and to test assumptions,
8
Elliott, Am. Math. Monthly, 1922, p. 331.
10
There is, fortunately, a natural corrective for our inclination to allow
likes and dislikes to influence our reason. This corrective is found in the
instinct of curiosity, the faculty that impels men to seek the truth, even if it be
unpalatable. In fact, a certain type of mind seems to take a particular satisfaction
in digging up such otherwise displeasing revelations; the cynic and
even the muckraker thus has his useful function. It is probably safe to say,
too, that curiosity a desire to know life in all its phases, to "experience reality"
is fundamentally the motive that impels some individuals to taste of the
less savory phases of life.
378 ELFMENTS OF PHYSICAL BIOLOGY
and upon a willingness to revise them without partisanship, or bitterness,
or distress." 11
One direction in which we may be called upon to make such a revision
is our conception of the self or ego} along the lines already
indicated. The old intuitive conception of the self, which narrows
it clown and fences it off rigorously from the rest of the world, may
have to give place to a broader conception. This may require the
breaking through of inhibitions that obstruct our view and prevent
us from gaining a full and lively realization of our essential unity with
the universe; it may involve in some degree a retracing of our steps
in past evolution; a retrenchment of our overdeveloped self-consciousness,
to make room for a more comprehensive world-conscious-
ness.
THE COMMUNICATORS
Aside from the ordinary receptors and elaborators, there is another
avenue, a highly specialized type of receptors, by which the
world picture acquires further detail and extension, namely by communication
from, one individual to another. How far the faculty
of such communication may be developed in species other than our
own is today an open question; nor is this of much consequence for
us here, as we are concerned with principles rather than with concrete
examples, and a single indubitable instance, such as we have
in ourselves, suffices for the establishment and exemplification of
the principle.
Orthogenesis in Human Evolution. In the human species the
communication of information from one to another takes place
chiefly through speech, tradition and carved, written or printed
records. The incalculable significance of these aids to the individual's
elaboration of his world-picture needs no emphasis. In
a recent number of Nature there appeared Professor Bohr's address
on the structure of the atom, delivered on the occasion of the award
to him of the Nobel prize for 1922. In this historical survey of the
development of his theory he mentions nearly fifty names of investigators
who directly or indirectly contributed to this part of our
world-picture. A person intelligently reading this lecture, making
the picture part of his own mental stock-in-trade, is thus virtually
11
Simpson, Am. Math. Monthly, 1922, p. 331.
EXTENSION OP SENSUOUS WORLD PICTURE 379
endowed with fifty pairs of eyes and hands, and has the benefit of
the workings of fifty brains, for the most part master brains of the
first rank, for the list includes such names as, Faraday, Maxwell,
J. J. Thomson, Rutherford, Hertz, Lorentz, Planck, Einstein, to
mention only a few. It is this thought-transmitting propensity of
the human species, more than any other, that gives it a superlative
lead over all the other creatures of the globe.
12
Man is the only
animal who in any considerable measure bequeaths to his descendants
the accumulated wisdom of past generations. Evolution in this
case proceeds not merely by the slow process of selection, but is
immensely hastened by the cumulative and continuous growth of
a body of knowledge exempt from those laws of mortality which set a
term to the life of the individual. Such evolution by a process not
directly dependent upon (although subject to) selection is what biologists
have termed orthogenesis the direct genesis of a trait or species
by the pursuit of an inherent trend or bias, irrespective of selective
influences. Such orthogenesis (of the functional, not the physiological
kind noted in Chapter XXII) is exhibited not only in the cumulative
effect of tradition and printed records, but even more strikingly
in the basic process itself by which the scientific world picture
is developed. This development has not been wholly an evolution by
selection, by survival of the fittest, at least not if we take the word survival
in its literal sense. On the contrary, this is perhaps the clearest
example extant of evolution b}" orthogenesis, by the unfolding of an inherent
trend independently of any selecting influences. Our Galileos,
our Newtons, our Thomsons, our Einsteins, have not been singled out
by a process of lethal selection from among others less fit to survive.
The process by which viable, pragmatically competent systems of
thought (or world-depiction) are evolved is quite other than this.
The Copernican system has survived over the Ptolemaic, not because
the originators and the supporters of the former were better
adapted to life under the then existing conditions rather the reverse
was true. It is not even a mere principle of economy of thought
that gave victory to the Copernican system. The decisive factor
12
The importance of the accumulation of human knowledge through tradition,
permanent records, and lasting technical installations has been emphasized
by A. Korzibski (Manhood of Humanity, Dutton, 1921) to whose central
thesis the existence of a fundamental hiatus between man and the rest of
creation I cannot, however, subscribe.
380 ELEMENTS OF PHYSICAL BIOLOGY
was that the comparative simplicity of the Copernican system eased
the further advance of knowledge, which had been very effectively
checked by the intricacies of the epicycles. It is not so much that
the human niind has a bias for simplicity -though that may be true
as that, with its finite limitations, the mind progresses faster and
farther as soon as a simpler system is substituted in place of a more
complicated, even when both systems are otherwise competent truly
to represent facts.
Orthogenesis does not, of course, suspend selection. Of two
species, that one will most survive, whose orthogenesis, if any,
leads it in a favorable direction. Presumably, destructive orthogenesis
may occur, as well as constructive. Indeed, it has been suggested
that some of the extinct monster species were thus drawn
to their doom by an orthogenesis, controlled perhaps by endocrine
glands, that swelled their dimensions beyond all bounds of propriety.
What is most significant is that orthogenesis, whether constructive
or destructive, must accelerate the pace of evolution. No more
telling demonstration of this could be asked than the prodigious
advance made by man in recent centuries and decades, advances
traceable directly to the orthogenetic character of the evolution of
his mental furniture. The provocative agent in this orthogenesis
is curiosity, the faculty that anticipates needs, that solves problems
before they become burning issues; that inspires research in pure
science, and throws in for good measure practical applications, as
gifts that are often only too truly gratuitous. And coupled with
this curiosity, a native impulse that induces men to impart their
findings to others.
Meanwhile orthogenesis continues.13
Man has travelled far,
from the crude intuitive recognition of the ego and the non-ego, the
first dim realization of a depiction of one system of coordinates upon
another; to our latest conception of the part played by coordinate
reference frames in our intercourse with nature. The most modern
world-picture, as drawn for us by Einstein and his followers, seems
to outstrip, for the time being, not only our needs, but our immediate
opportunities for practical application. But that is the virtue of
orthogenesis.
13
For a somewhat detailed presentation of the case of orthogenesis in man
see the author's article Biassed Evolution in Harper's Magazine, May, 1924.
CHAPTER XXVIII
THE ADJUSTORS
Was wir Willen nennen 1st niehts anderes als die Gesammtheit der teilweise
bewussten und mit Voraussicht des Erfolges verbundenen Bedingungen einer
Bewegung In der bewussten Willenshandlung fallen Ursache und
Zweck Zusammen. E. Mack.
From the nature of things the adjusters are more recondite, both
in their material substance and in their operation, than either the
receptors or the effectors. It has already been remarked that the
location of the adjusters is inconsequential; partly for this reason,
and partly because their concrete material apparatus does not need
to be external (as must be that of the receptors and effectors), we
have at most a very imperfect consciousness of the location of the
adjusters. It is chiefly from hearsay that the man in the street
associates the brain with mental processes.
It is, then, not greatly surprising, that the functioning of the adjustors
should be shrouded in a good deal of mystery, which even
introspection into our own experience does not by any means dispel.
If unsatisfied curiosity is the fans et origo of human interest, such
interest should certainly not be lacking in this phase of our present
enquiry.
Mechanistic and Teleological Interpretation of Adjusters. We
have seen, from the example of a simple toy, that typical and fully
competent adjusters can very well be provided in and by purely
mechanical structures. In the toy beetle anticipatory correlation
between the reaction of the beetle and untoward variations of the
environment is established as follows: The beetle is progressing
along the straight line AB, (see fig. 72) . Its law of motion is that
of uniform progression along this straight line. Suppose a scale of
centimeters is laid along AB. Successively higher scale divisions
along A B are reached at successively later intervals. In fact, this
scale, with the beetle moving along it (at constant velocity, we may
suppose, to simplify the argument) constitutes a clock. If at the
time t the driving wheel of the beetle is at the zero mark, scale
381
382 ELEMENTS OF PHYSICAL BIOLOGY
divisions to the left correspond to and represent past instants, and
those to the right represent future instants. Suppose the antenna
is 5 divisions long, and that the table edge is at division mark 15.
If for any reason the adjuster apparatus failed to function, at time
t = 15 the driving wheel of the beetle would pass the table edge and
fall over. When the adjuster apparatus is functioning, five time
units in anticipation of the threatened catastrophe the antenna
"senses" the danger, and the creature turns aside into the path of
safety. Note that this anticipatory reaction depends upon the
correspondence between points forward upon the line of advance, and
future instants of time. A supposititious future, & future that may be,
is depicted, instant by instant, by successive points in the line of
advance of the beetle on the supposition that its law of motion
i
I
J Q J
J U ' t
~^
J
-_-_j-.^__^
I
,1
I
FIG. 72. How THE FUTURE ENTERS INTO THE DETERMINATION OF THE MOTION
op THE WALKING MECHANICAL BEETLE, THUS IMITATING PURPOSIVE
ACTION (TELEOLOGY).
continues unchanged. The behavior of the beetle is determined in
terms of this depiction of a supposititious future.1
1
This conception of a future that may be is found both in A. N. Whitehead,
Principles of Natural Knowledge, and also in E. Mach's Analyse der Empfindungen
(1903, p. 78). The expression seems to raise a question as to just
what meaning may justly be ascribed to it in a determinate universe. One
admissible meaning is indicated in the test above: The future that may be
is that computed by extrapolation according to a law of motion that in fact
does not hold up to the instant under consideration. Another construction
that suggests itself is that the future that may be is one compatible with certain
differential equations, but not necessarily with the integration constants.
The toy beetle shows very clearly how a future that may be can, seemingly,
influence the course of events. It is interesting to note that there is another
circumstance which also is competent to explain the apparent influence of this
fictitious future upon the present course of events. Mach points out, with
especial reference to the seemingly miraculous foresight displayed by certain
THE ADJUSTOES 383
The depiction in this case is plainly mechanical or geometric.
We, as living, conscious organisms, in certain circumstances exhibit
a precisely analogous behavior; our action is determined by a picture
(psychic in this case) of the future that we seek to avoid or to attain. 2
"In human purpose the result to be attained is first pictured in
consciousness, and the thinker then proceeds by a series of acts to
fulfill his preconceived aim." The law of "preconceiving" Is precisely
the same in both cases, at any rate in the simple case in which
the reaction is one of avoidance of an unfavorable condition. An
image of the future of the system is constructed on the supposition
that the mechanism or organism does not interfere. This is the future
that may be. Interference, financed from the fund of free energy,
then takes place accordingly, thus modifying the course of events
and determining the future that will be.
Little doubt enters our mind in construing the course of events in
these two cases. We are directly conscious of our own volition
(whatever its precise physical significance may be.) We hesitate
not at all in describing our action as purposive, as directed to and
determined by an end, by a final cause. As to the tin beetle, we have
dissected him and fully understand his mechanism. We would think
it foolish, with our peep behind the scenes, to impute to him volition
or purpose; we describe his action as mechanical, as fully determined
by an efficient cause.
The Doubtful Cases. But what shall we do when confronted with
a case that falls into neither of these categories? An amoeba, for
example? We cannot enter the amoeba in spirit and become parties
to its conscious experience; we do not even know whether it has any
such experience. On the other hand its mechanism is not completely
known to us. To class it among purposive, teleological
beings so long as we are ignorant of its working, and to be prepared
instinctive actions, that the life process is essentially periodic in character,
repeated over and over in successive generations, and that it may not be so
much the future of individual A that influences its action, as the past of its
progenitors A', A", . . . who were placed in similar circumstances on
earlier occasions. "Es ist dann micht eine mogliche Zukunft die wirken
konnte, sondern eine unzahlige Mai dagewesene Vergangenheit, die gewiss
gewirkt hat."
2
H. C. Warren, Journal of Philos. Psych, and Sci. Method, 1916, vol. 13,
p. 5.
384 ELEMENTS OF PHYSICAL BIOLOGY
to reclassify it among "purely mechanical" structures3
us buun as we
come to understand its physical operation, seenis hardly a very
commendable way to marshal our mental stock-in-trade. There is
here involved something of those mysteries to which reference has
3
"Die Vitalisten begehen nun den Fehler, die Zielstrebigheit dort zu leugnen,
wo ikre Ursache durch physikalische Gesetze erklarbar ist, da sie eben
diese Zielstrebigkeit als rait den physikalischen Gesetzen nicht zusarnmenhangend
betrachten." (C. Doelter, Aus dem Grenzgebiete des Organischen
und Anorganisehen, 1906, p. 13.) It is a singular fact possessing a certain psychological
interest, that as soon as we understand the modus operandi of a teleological
mechanism we are disposed to reject its interpretation in terms of
"final causes." Why this preference for the mechanistic view? There is another
closely analogous case. So long as we remain in ignorance of the precise
working of our nervous system in the phenomenon of memory, we are forced
to contemplate this faculty as being p ,rhaps wholly psychic in character. To
quote Bertrand Russell (Analysis of Mind, 1921, p. 92), "I am inclined to . . . .
hold that past experience only affects present behavior through modifications
of physiological structure. But the evidence seems so far from conclusive
that I do not think we ought to reject entirely the possibility that mnemic
causation may be the ultimate explanation of mneinic phenomena.'' Here
also we feel that just as soon as a physical basis of memory were made manifest,
we should discard the hypothesis of a purely psychic mneme. A certain
light is perhaps thrown on this curious psychological bias when we consider
the matter from the point of view of the equations of motion of a mechanical
system. We noted in Chapter IV (pp. 47, 48) that conceivably the equations
representing the course of events might contain a lag or a lead, that is to say,
the motion at time t
might, for example, depend explicitly upon the condition
of the system at time (t
- a) or perhaps at time ( + &), or both. We also
noted in passing that something of this sort seems actually to be the case in
systems comprising living organisms, since the reaction of these upon their
environment is a function of their previous history through the intervention
of memory, and a function of the future through the intervention of volition.
But we also noted particularly that the appearance of a lag or lead in the
equations might be spurious. So for example the rate of incidence of new malaria
cases today may be expressed as a function of the number of persons
bitten by infected mosquitoes nine days ago (period of incubation). But
this is merely a short way of describing the state of these persons today;
so that
the equation could also be written without the lag term. Why do we, in
mechanical systems, give preference to an equation free from a lag term, whenever
such can be framed? The first thought that suggests itself, perhaps, is
that we do this from a bias for simplicity. But this is probably not the true
explanation. More likely we make this selection because there is something
arbitrary about the lag term. It does not give a unique representation of the
process. The differential equation free from a lag or lead, on the contrary, is
more free from arbitrary features, and is presumably in some sense unique.
THE ADJUSTORS 385
been made. The mystery may be in part of our own making. The
difficulty in answering a question sometimes arises from the fact
that the question has been badly put. Certainly no harm can come
from an effort to make a survey of some of the relevant facts and
their relations. To such a survey we shall proceed forthwith. Here
it may be well to summarize briefly three cardinal points in our
observations so far:
1. Mechanisms teleological in their operation can be constructed,
which we would not in any ordinary sense of the word describe as
conscious.
2. The active types of teleological mechanisms in nature (animals)
impress us as being in some sense conscious, though in the case of the
lower rungs on the scale we feel very doubtful as to just what meaning
to assign to this statement.
3. In our own selves we feel that consciousness (volition) plays a
dominant role in the teleological operation of our bodies, and, in
particular, in the operation of the adjusters.
Adaptive Adjustment of Tastes. In all that has been said so far
the individualistic character of tastes has been emphasized. A
man's likes and dislikes are essentially his own personal affair.
As Pareto4
remarks, if a man dislikes spinach, it is useless trying to
prove to him, as one would demonstrate a proposition in geometry,
that spinach tastes good. Judgments of this kind are typically not
of that class in which "universal assent" can be attained, the class
with which the worker in physical science is mainly concerned.5
Now this does not mean, as might perhaps at first sight appear,
that tastes of different individuals are wholly random collections of
likes and dislikes, dealt out purely haphazard, like the cards from a
well shuffled pack. However erratic human desires may appear in
detail, in the gross they display a species of uniformity, of law, of
constancy; a fact recognized long ago by Adam Smith, who "considered
a science of economics possible because of a few outstanding
traits of man which guaranteed self-preservation, while also promoting
the welfare of society at large."
6
The same constancy in the
4
Manuel d'Economie Politique, 1909, p. 62.
5
Compare N. Campbell, The Elements of Physics, J. W. N. Sullivan,
Aspects of Science, 1923, p. 30.
6
0. F. Boucke, Am. EC. Review, 1922, p. 599.
386 ELEMENTS OF PHYSICAL BIOLOGY
average performance (symptomatic of desires) of men was again noted
by Quetelet, and more particularly by Herbert Spencer,
7
who was
presumably the first to recognize the full significance, in the evolution
of the race, of that "adjustment of feelings to actions" which has
made "pains .... the correlatives of actions injurious to
the organism, and pleasures .... the correlatives of actions
conducive to its welfare." This, as Spencer points out, is the
inevitable outcome of natural selection, since "there must ever have
been, other things equal, the most numerous and long-continued
survivals among those races in which these adjustments of feelings
to actions were the best, tending ever to bring about perfect adjust-
ment.
3 '
Genuine Utility for Social Service. And what is the "perfect
adjustment?" A hint of the quantitative method of approach to
this question has been given in Chapter XXV. The argument there
set forth suggests something of the nature of an absolute standard
of value, to which actual standards established in the community
must approach, however imperfectly. Without attempting any
quantitative definition, or, indeed, any definition at all, such an
absolute standard has been directly or implicitly mooted by many.
Irving Fisher8
thus speaks of a genuine utility for social service, a
concept which is evidently intended to be purified from the caprice
of the individual. The question is, remarks T. N. Carver,
9
"not
what men are actually like, but what men fit best into the cosmos.
What are the earmarks of a good man, that is, of a man who adds
strength to the community or nation? It is not enough if we study
the variations of human institutions, habits, morals, etc. We want
to know what institutions, habits and moral systems work well,
what kind of a nation or social organization fits into the cosmos and
grows strong under the conditions of the Universe. Similarly, as to
; individual motives, it is not simply a question as to what motives
'/_
'
actually govern human behavior, though it is important that we
f should know that. It is of equal importance that we know what
motives or combinations of motives work well." Jusfc as man has
7
Herbert Spencer, Principles of Psychology, section 124; Data of Ethics,
section 34.
8
Irving Fisher, Am. Econ. Review, June, 1918, vol. 8.
9
T. N. Carver, Qu. Jour. Econ., November, 1918, p. 197.
THE ADJUSTOES 387
learned, in the progress of ages, to think logically, to think in accord
with reality, so he must yet learn to will rightly, that is, in harmony
with Nature's scheme. We have here a thought that seems fundamental
for a natural system of ethics. I will not follow it up further
at this point, but shall take occasion to return to it in the closing
chapter.
CHAPTER XXIX
CONSCIOUSNESS
Vous connalssez, n'est-ce pas, cette jolie griserie de I'&me? On ne pense
pas, on ne reve pas non plus. Tout votre etre vous echappe, s'envole, s'eparpille.
On est la mouette qui plonge, la poussiere d'e"cume qui flotte au soleil entre
deux vagues, la fumee blanche de ce paquetbot qui s'eloigne, ce petit corailleur
a voile rouge, cette perle d'eau, ce flocon de brume, tout excepte soi meme.
Alphonse Daudet.
Intimately involved, both, in the process of world depiction, and in
the operation of the adjusters, as we know them in ourselves, is the
phenomenon of conscious-ness. In preceding pages this has been
taken for granted. But our study would be incomplete indeed if we
did not give some separate consideration to a phenomenon that is of
so superlative importance in the shaping of the world's events.
Consciousness is a natural phenomenon which we know directly
in ourselves, and whose existence we infer in others from their behavior
in view of its greater or less resemblance, in type, to ours.
Relation of Consciousness to Physical Conditions. What mainly
interests us here, regarding this phenomenon, is its relation to physical
processes and structures.
These relations are presented to us in two aspects, and we may
accordingly distinguish them as (1) conditional, and (2) operative
relations; that is to say, first relations between consciousness and the
physical conditions necessary for its manifestation; and second,
relations between consciousness and physical events that seem to be
dependent upon consciousness for their occurrence, in the operation
of the correlating apparatus. The enquiry into these latter relations
must cover (a) the function of consciousness in directing the course of
events, and (5) the origin of conscious mechanism or apparatus which
so directs that course; the reasons that can be assigned, if any, why
organic evolution should have seized upon consciousness as a tool to
secure adaptive (zweckmassig) behavior in the organism.
CONDITIONAL RELATIONS
It is frequently pointed out with critical emphasis that mechanistic
attempts to explain consciousness are philosophically unsound.
388
CONSCIOUSNESS 389
Such, strictures are based upon a misconception of the function of
science, mechanistic or other. Science does not explain anything,
consciousness occupies in this respect a position in no wise peculiar.
Science does not explain electricity, for example. Science is less
pretentious. All that falls within its mission is to observe phenomena
and to describe them and the relations between them. It is true that
in loose parlance such statements are commonly made as: The
inertia of matter is explained by the electron theory. But all this
means is that certain relations have been established between the
properties of an electric charge, and those of amass of gross "matter;"
that the laws of motion of matter can be comprehended in the laws of
motion of an electron. If explanation means the making comprehensible
in this sense, then this is explanation. But who should say that
the attempt to establish relations between consciousness and other
phenomena is philosophically unsound? One would be disposed to
retaliate by questioning the soundness of his philosophy.
Quite on the contrary, the study of the relations between consciousness
and other phenomena is not only legitimate, but altogether alluring
and full of promise.
Here, however, a difficulty confronts us at the outset. Strictly
speaking, the only consciousness I can ever know (unless altogether
revolutionary developments in our sources of knowledge should
follow) is my own. This seems to impose most serious restrictions
upon the investigations.
A Fundamental Hypothesis Admitted. However, there is general
assent that, in this study, we shall admit the fundamental hypothesis
that the consciousness of my fellowmen exists and is sufficiently like
my own to constitute a proper subject for study as a type phenomenon.
We even go farther, and quite willingly admit the propriety
of investigations regarding the consciousness of dogs, cats, apes, mice,
sparrows, and so forth, as manifested by their behavior. All this is
well enough, until some one begins to ask impertinent questions.
Just where we are to stop in hypothecating consciousness? And
what do you mean by saying that Jones's consciousness is like Smith's?
or like that of a dog? or of an amoeba? These questions are embarrassing.
But so long as we deal with restricted material and restricted
enquiries regarding the relations between consciousness and other
phenomena, we find in practice we can ignore these fine points. And
then certain basic propositions find at least provisional acceptance.
The first of these we may enunciate as follows:
390 ELEMENTS OF PHYSICAL BIOLOGY
Consciousness is Closely Bound Up with Life Processes and
Structures. The statement hardly requires exemplification. Starve
a man, and he becomes unconscious. The same may happen after
a blow on the head, and so on.
Unfortunately, however, our proposition is not so proof against
criticism as it appears. For it is admitted that we observe consciousness
only by its manifestations similar to those which we know
directly in ourselves. The inference that consciousness is absent in
non-living matter reminds one somewhat of the assumption commonly
made by ignorant persons that insects or similar voiceless creatures
feel no pain. Since non-living matter lacks the familiar means of manifesting
any consciousness which it might possess, it is evidently not
permissible to base, upon the absence of such familiar manifestation,
any conclusion as to the absence of consciousness. We are really
arguing in a circle: First we agree to postulate consciousness where
certain manifestations are observed. Then we turn around and say
that the presense of these manifestations is characteristic of consciousness,
and their absence of its absence.
The truth is, all we can state with any degree of confidence is that,
if non-living matter possesses any kind of consciousness, this must be
of a character so radically different from our own as wholly to transcend
our powers of imagination. This may appear at first sight a
rather pointless observation. But it must be remembered that the
statement holds true with almost equal force with regard to living
matter in the case of such elementary forms as amoeba, for example.
The importance of such reflections as this is that it draws our attention
to the significance of forms or modes of consciousness. The general
scheme of nature is more readily understood if we contemplate
the several material objects as gifted with graded modes of consciousness,
than if we suppose them sharply divided into conscious and
unconscious. It is true that this point of view commits us to
admit the existence of modes of consciousness utterly inaccessible to
human experience. But this admission is not as damaging as it may
appear, for, in a restricted measure at least, we are forced to make it
in any case as soon as we hypothecate consciousness in a dog or a
flea, for example.
Allowing, however, for these finer distinctions, the statement may
stand that consciousness of the character familiar to us in ourselves,
or of a character approaching to this, is closely bound up with life
processes, in the way indicated.
CONSCIOUSNESS 391
Consciousness Dependent on Metabolism. The Personal Element.
This suggests that the continued conscious state of matter requires
constant excitation by the metabolic processes, somewhat as the continued
maintenance of a magnetic field in the neighborhood of a
conductor requires constant excitation by the passage of a current.
And since metabolism is essentially a state of chemical flux, or chemical
reaction in progress, one is led to suspect that the conscious state
may be in some way correlated to that, transitional state through
which matter must pass on its way from one stable molecular combination
to another1
a state regarding which our knowledge today is
extremely fragmentary. Almost the only direct evidence we have of
matter actually in that state is the observation by Sir J. J. Thomson
and by F. W. Aston of such molecular debris as CH3 and the like, by
a method capable of detecting these fugaceous aggregations of atoms
even though their life period be only a few millionths of a second.
Meanwhile it must be borne in mind that in the case of the highly
complex and bulky molecules characteristic of organic matter, the
"intermediate" state between two compounds may be something
more lasting. It is even conceivable that "open" molecules may, in
this realm of chemistry, be the more stable configuration; the molecules
may perhaps be more or less in a state of oscillation, in a species
of tautomerism, between two "closed" compounds. This would,
in a way, harmonize with the continuity of consciousness, as known
to us, if consciousness be indeed typically associated with the "open"
state of molecules. On the other hand it may give us at least some
distant idea of the meaning of consciousness as applied to so-called
non-living inorganic matter. Such consciousness as may here occur
would be, it seems, of the nature of flashes of almost infinitesimally
short duration. If this conception appears fantastic, it must be borne
in mind that brevity of time is altogether a relative concept; by geological
standards human life itself is merely a flash of lightning in the
eternal darkness; and though to most of us the altogether impersonal
species of consciousness which seems to be here implied must appear
inconceivable, those who have observantly come through the experience
of syncope may not find the thought so unreasonable.
Pendant la syncope, dit un auteur qui a pu e'tudisur sur lui m&rae ce phe"nomene,
c'est le n6ant psychique absolu, 1' absence de toute conscience, puis on
1
Compare Sehonbein, Jl. f.
prakt. Chemie, vol. 40, p. 152.
gg2 ELEMENTS OF PHYSICAL BIOLOGY
commence a avoir un sentiment vague, illimite, infini, un sentiment d'existence
generate sans ancune delimitation, sans la moindre trace de distinction
entre le moi et le non moi; on est alors une partie organique de la nature ayant
conscience du fait de son existence, mais n'en ayant aucune du fait de son
unite organique; on a, en deux mots, une conscience impersonelle. . . .
On a des sensations stupides, si je puis m'exprimer ainsi, c'est a dire, des
sensations qui, justement parcequ'elles restent isolees, ne peuvent pas etre
connues, mais seulement senties.
2
In an entirely normal state, too, certain persons seem to have
realized the experience of a species of impersonal consciousness. An
example of this is to be seen in the quotation from Daudet's Lettres de
man Moulin that has been placed at the head of this chapter.
3
Thus
the phenomenon of the ego is neither as fundamental nor as simple
as it may appear to the naive observer. The occurrence of primitive
consciousness of a kind, which does not recognize an ego, is not only
-
possible,
for ought we know, but should be regarded as decidedly
probable. It may indeed, be argued with much plausibility (as we
have already had occasion to note) that the ego is a mere artifice, an
aid to thought (just as a frame of rectangular coordinates is a mere
figment, convenient for the purpose of defining the position and configuration
of geometrical structures), but has no objective existence,
and forms no indispensable element of the more general phenomenon
of consciousness.
The separation of "chemical" processes from other physical processes
is almost certainly merely a matter of convenience. If, then
we tentatively associate consciousness with certain states of chemical
strain in molecules, we are forced to contemplate the possible extension
of out- conception to matter under physical strain generally. Space
does not permit us to follow up in detail the implications of such a
point of view. It must suffice to indicate that it leads us to question
Herzen,
Le cerveau et Pactivite c6re"brale, 1887, p. 236; quoted by Janet,
L'automatisme psychologique, 1889, p. 43.
3
A very similar thought is expressed by Theodor Dreiser in Proteus (American
Mercury, 1924, vol. 1, p. 9) :
"And I am the birds flying in the air over the river,
The sun, the shade,
The warmth, the grass,
And myself
And not myself
Dreaming in the grass."
CONSCIOUSNESS 393
whether a much frowned upon species of anthropomorphism may not,
after all, be in some sense legitimate. When we say that a soap bubble,
for example, tends to contract under surface tension, or perhaps when
we use even less guarded language and say that it is trying to contract,
our terms are commonly thought reprehensible as being more
picturesque than scientific. Yet we ought to be prepared for the
conception that the straining of the bubble to contract may not be
so fundamentally different a thing from the straining of an amoeba
to engulf a food particle, or the straining of a Newton to assimilate
a new conception or to solve a problem in philosophy. The two
phenomena may be far separated, indeed, upon the scale of evolution,
yet they may be two rungs upon the same scale.
CHAPTER XXX
THE FUNCTION OF CONSCIOUSNESS
In the last chapter we have considered some of the reflections
that suggest themselves In connection with the first fundamental
proposition, namely that consciousness, as we know it in ourselves,
is closely bound up with life processes and structures.
A second basic proposition regarding consciousness, which will
find at least provisional acceptance, is that consciousness, as known
to ourselves, is of variable content, and that its content is a function
1
of our past and present bodily states. The form of the functional
relation2
is in part known, but only in part. It is, of course, by
virtue of this functional relation that consciousness plays a part
in the receptor-adjustor-effector apparatus.
This leads us to the second aspect of the relation between consciousness
and physical structures and events, namely, the function
which consciousness fulfills in the organism, and the problem as to
how it has come about that living organisms have appropriated to
themselves, in the course of evolution, the property of consciousness,
or, to be more precise, of that particular type of consciousness
which is characteristic of them.
OPERATIVE RELATIONS
One function of consciousness in the operation of the receptoradjustor-effector
apparatus has already been considered, namely
the part which it plays in elaborating further the crude data for the
world-picture supplied by the special senses.
But consciousness plays an equally fundamental role in the
operation of the adjusters. It is quite evident that no amount of
cold, intellectual information, pure and simple, can in itself ever
1
In the mathematical sense.
2
For a discussion of the metaphysical aspect of this functional relation
the reader must be referred to the special literature. Attention is particularly
directed to a paper by L. T. Troland in the Jour. Washington Academy, 1922,
vol. 12, p. 141. See also H. C. Warren, Jour. Phil. Psych, and Sci. Method'
pp. 20-21, footnote.
'
394
FUNCTION OF CONSCIOUSNESS 395
determine an action. If I am standing at a street crossing, and my
senses inform me that, should I start to walk across at this moment,
I should be killed by a passing car, this information in itself can
" 4
nave no effect one way or the other upon my action. In addition
to the cold knowledge, there must be something of the nature of
motive, which gives me an interest, on the one hand in the act of
crossing (as for example to reach the restaurant across the street
and satisfy my hunger), and on the other in avoiding collision with
with the street car (the anticipatory image in my mind, of the
suffering which this would cause).
In the lower organisms motivation appears to us almost wholly
mechanical and fatalistic. The tropisms of a moth apparently
draw it toward a light with the same mechanical inevitableness as
the gears of the toy beetle constrain it to follow the table ,edge.
The opposite extreme, the highest refinement and complexity in
motivation, we observe in our own selves.
In human purpose the result to be attained is first pictured in consciousness,
and the thinker then proceeds by a series of acts to fulfill his preconceived
aim The typical purposive experience consists in a thought of
some future occurrence followed by a series of actions -which culminate in the
very situation which the original idea represented.
3
But what determines "the result to be attained," which is thus
pictured in the consciousness? The matter has been well expressed
by Veblen in a passage which at the same time brings out clearly the
relation of the purely intellectual functions of consciousness to the
emotional (motivating) functions:
The ends of life, the purposes to be achieved, are assigned by man's instinctive
proclivities; but the ways and means of accomplishing those things which
the instinctive proclivities make worth while are a matter of intelligence.
Men take thought, but the human spirit, that is to say,
the racial endowment of instinctive proclivities, decides what they shall take
thought of, and how, and to what effect.
3
H. C. Warren, Jour. Phil. Psych, and Scientific Method, 1916, vol. 13,
p. 5. Compare also F. L. Wells, Mental Adjustments, 1917, p. 11. "The free
imagination of wished-for things results well for the mind through painting
in more glowing colors the excellence of what is wished for, and firing the ambition
to strive for it more intensely." For a rather different viewpoint see
B. Russell, Analysis of Mind, 1921, pp. 75-76.
ELEMENTS OF PHYSICAL BIOLOGY
Dynamic Psychology; Instinctive Drives to Action. Thus, in
our analysis of the operation of the correlating apparatus, we are
led to a consideration of the fundamental instincts that furnish
the driving power behind the activities of the organism. The study*'
of these instincts has been organized into a definite branch
Dynamic Psychology of which we can here note only a few of the
outstanding features. We are treading upon ground not altogether
cleared, at this date, from controversial entanglements.
Certain psychologists, notably Freud and his followers, have
taken the view that the really fundamental drives for action can
be reduced to a very few, such as hunger, the sex call, and the
gregarious instinct.4
From these the less elementary motives are,
according to this school, derived by a process of "sublimation."
In contrast with this R. S. Woodworth regards as separate and
independent sources of action those drives which are operative in v
the course of creative activities:
The various creative proclivities which (Freud) refers to the redirected
energy of the sex instinct as their sole driving force, have in Professor Woodworth's
opinion driving forces of their own. Of sublimation he holds that when
an intellectual interest, say, is made to supplant a sex impulse, the latter is
not drawn into service, but resisted. 5
Individual Traits. Instinct of Workmanship and Self-Expression.
We may well leave it to the psychologists to settle their
4
McDougall in his Introduction to Social Psychology, lists twelve "simple
instincts" as follows:
Such lists as this seem rather arbitrary, and in some degree dependent on the
whim of their author. Compare for example F. W. Taussig, Inventors and
Money-Makers, 1915, p. 7; Irving Fisher, The Survey, March, 1919, p. 937.
6
M. F. Washburn reviewing Woodworth's Dynamic Psychology, 1918, in
Science, 1918, vol. 48, p. 373.
FUNCTION OF CONSCIOUSNESS 397
difference of opinion in this field. For our purposes it is
enough to know that these drives to action are operative, that some
ends to men are "instinctively worth while." And if we ask:
What are those ends that are instinctively worth while, we find
that a mere list of primitive instincts is at most only a very inadequate
answer to our question. In their broad fundamental
instincts men are, indeed, essentially alike. But that adjustment
of the relative intensity of their several instinctive proclivities,
which finds its material expression in the behavior-schedule, in
the coefficients X (Chapter XXV) is capable of infinite variation.
What is "worth while" is after all a matter of taste, and de gustibus
non est disputandum "people of radically different temperaments
cannot come to any understanding by intellectual means/'
6
their
vain efforts to do so lead only to mutual repugnance, for "differences
of taste produce greater exasperation than differences on points of
Science." 7
This divergence of tastes among men, responsible as
we must no doubt hold it for many bitter feuds between individuals
and between nations, is not without its useful aspect. In the present
state of the human race, with its divisions of labor and its specialization,
there is room for all sorts and conditions of men, and this
statement applies not only as regards their abilities, but also, and
perhaps more particularly, as regards the channels into which
each seeks by preference to divert the stream of his energies, the
sphere in which he most willingly applies his abilities. Much could
be said in support of the proposition that differences in temperament,
rather than differences in intellectual endowment, determine the
place that each man is fitted to occupy in the social scheme. For,
as Young has said, "With the talents of an angel, a man may be a
fool." It is not enough to be able, there must be a strong impulse
to do. "Not only to know, but according to thy knowledge to do,
is thy vocation," says Fichte. A study of motivation in exceptional
men, men of genius, is both instructive and inspiring. The dominat-
6
Rosa Mayreder, quoted in the The Lancet 1913, p. 1006. Compare also
William James, On a Certain Blindness in Human Beings.
7
MacAuley, Critical Essays, p. 501. The contrast has been drawn particularly
between introverts and extroverts "who find it so difficult to understand
each other, and so easy to despise," Compare also Nature, July 21, 1923, p.
88: "In actual life the want of rapport between these types is a matter of
daily observation."
398 ELEMENTS OF PHYSICAL BIOLOGY
ing drive in the life of such a man is undoubtedly the imperative
impulse to translate into outward expression, into concrete reality,
that inward image which, as Warren very aptly observes, is the
first essential element in human purposive action. Creative minds
have borne their own eloquent testimony:
As the sculptor must dream the statue prisoned in the marble, as the artist
must dream the picture to come from the brilliant unmeaning of his palette
. . . .
,
so he who writes must have a vision of his finished work8
. . .
An attribute which may be taken for granted in every artist is passionate
intensity of vision. Unless vision is passionately intense, the artist will not
be moved to transmit it, . . . .
9
Lowell, consulted by a young author as to the royal road to good
style, advises him that the first requirement is to have something
that "will not stay unsaid." The same intentness upon outward
expression of the inner vision speaks in the words of Carlyle:
A certain inarticulate self-consciousness dwells dimly in us, which only our
works can render articulate and decisively discernible. Our works are the
mirror wherein the spirit first sees its natural lineaments. Hence, too, the
fallacy of that impossible precept, Know thyself; till it can be translated into
this partially possible one : Know what thou canst work at.
Blessed is the man who has found his work: let him ask no other blessedness.
Know thy work, and do it; and work at it like Hercules.
This imperative impulse to materialize a mental conception is
by no means the monopoly of the artist. In this bantering style
Arnold Bennett says of the amateur inventors "They have glimpsed
perfection; they have the gleam of perfection in their souls." This
is the goad that drives them on to unrelenting effort: the vision of
their finished work.
The race of contrivers and inventors does obey an inborn and irresistible
impulse. Schemes and experiments begin in childhood and persist as long as
life and strength hold. It matters not whether a fortune is made or pecuniary
distress is chronic : there is increasing interest in new dodges, unceasing trial
of new devices And it would seem that no satisfaction from
pecuniary success or worldly recognition equals the absorbed interest of trial,
experiment, novel problems, happy solutions. 10
8
M. K,. S. Andrews in The Perfect Tribute, 1907, p. 6.
9
Arnold Bennett.
10
Taussig, Inventors and Money-Makers, 1915, p. 21 ;
see also A. J. Lotka,
Independent, July 12, 1919, p. 54.
FUNCTION OF CONSCIOUSNESS 399
Influence of Special Attitudes. There is undoubtedy a relation
between the intensity of development of the instinct of workmanship
and the natural endowment, the talents of the individual.
The two things do not run altogether parallel, unfortunately, as
evidenced by the well-known crank type of individual, who has
something of the enthusiasm of genius, but lacks judgment in
directing his energies into worth while channels. There are
too, men lacking the impulse to put to competent use great
natural gifts that they possess. But the general rule holds that:
"Special aptitudes clamor for the opportunity of asserting themselves.
The tasks which are their fit occasion of self-expression are
the supreme joy of the man of genius, who will suffer every earthly
privation rather than brook the thwarting of his talents."11
It might be supposed that such exceptional development of the
instinct of workmanship and self-expression as speaks in the words
cited can play but a subordinate r61e in a world peopled for the most
part with "ordinary" individuals. But this is a misconception on
several counts. How effective may be the inspiration which we can
all draw from the testimony of our betters is perhaps an open question.
But it must be remembered that in the shaping of the world's
events men of genius play a part proportionate to their greatness.
It has been remarked that the world's history could be written as a
series of biographies of great men. The general average of excellence
in a community is upheld and advanced by the exceptional few, the
leaders in thought and deed.
But perhaps our chief interest in the manifestation of the instinct
of workmanship and self-expression in its superlative measure, as
we see it in the life and works of men of genius, lies in the fact that,
after all, these men were human, even as we are; and that the trait
which in them is developed to this high degree, is shared in some
measure also by the rest of mankind. Here the instinct may not
rise to such pitch as to be clearly felt and recognized by the individual;
he may be only dimly conscious of a vague unsatisfied want
when he fails to find a normal outlet for it. Psychiatrists and
economists tell us that the individual thus thwarted is apt to ascribe
his vague feeling of discomfort to any but the true cause, and that
social unrest in our present industrial system is due in no small
11
H. T. Moore, The Sense of Pain and Pleasure.
400 ELEMENTS OP PHYSICAL BIOLOGY
measure to the want of an adequate outlet for the instinct of workmanship
and self-expression. "A human being whose instincts are
balked becomes tin enemy of society/' for "primitive instincts can
be guided but not suppressed, If they become pent up the danger
of unrestrained outbreak is great."
12
And referring to the same
matter Taussig remarks:
It Is obvious that the sum of human happiness would be greater if
. . . , all commonly took direct satisfaction in the activities of earning a
living. , . , The satisfaction of instinct conduces pro tanto to happiness,
the balking of it to unhappiness Among those instincts to which
it seems possible to give wide scope, without danger of satiation or remorse,
is that of contrivance (workmanship). And yet the modern organization
of industry smothers it in a great and probably growing proportion of men.
If it is indeed true that the proportion of men so cramped in their
creative efforts is growing, we have here a most regrettable condition,
for the words of Emerson remain true:
In every variety of human employment .... there are among the
numbers that do their task perfunctorily, as we say, or just to pass, as badly
as they dare there are the working men on whom the burden of the busines
falls those who love the work and love to see it rightly done, who finish their
task for its own. sake; and the state and the world is happy that has most of such
finishers.
The Industrial and the Social Problem. The instinct of workmanship
thus gives rise to a twofold problem in human affairs.
On the one hand, considering the modern human community in the
gross, so far as the development of social conditions are or can be
brought under intelligent control, it is evidently desirable to provide
for the healthy satisfaction of this eminently desirable trait of human
character: "We have here too valuable and creative a tendency to
allow it to be longer neglected, thwarted and dissipated."
13
On the other hand each individual, if life is to bring him a good
share of that satisfaction which it potentially holds for him., must
order his affairs (so far as circumstance permit) with his eyes open
to the demand of this instinct of workmanship. Failure to do this
may rob him of his just heritage; he may find, too late, that he has
sold his birthright for a mess of pottage, that, in the words of Arnold
12
Irving Fisher.
13
Ordway Tead.
FUNCTION OP CONSCIOUSNESS 401
Bennett, "his existence is a vast and poisonous regret." But even
if he is spared this extreme, it holds true in any case that "better
self-understanding means better self-control, and a wiser ordering
of one's actions along the normal paths of happiness."
14
14
F. L. Wells, Mental Adjustments, 1917, p. vii.
CHAPTER XXXI
THE ORIGIN OF CONSCIOUSNESS IN LIVING ORGANISMS
Die eigenartigen Ziige der Organismen sind als provisorische Leitfaden aufzufassen.
E. Mack.
We have had occasion to note that "behavior" very similar, in
certain characteristics, to that of living organisms can be secured
in a purely mechanical structure, such as that of the toy beetle.
This raises the question why nature should have resorted to consciousness
as a means for bringing about those reactions so characteristic
of the Irving organism. The most obvious answer that suggests
itself is that, for some reason beyond our present ken, the
conscious organism is simpler than any purely mechanical structure
that could be built to perform even approximately the same diversity
of tasks with the same degree of effectiveness. But certain difficulties
arise if we accept (as is commonly done) the plausible
hypothesis that to eveiy state of consciousness there corresponds
a definite state of the material structure of the conscious organism.
The Problem of Psycho-physical Parallelism. There is first
of all the classical problem of psycho-physical parallelism. If the
events in the physical world are wholly determinate, and if every
conscious experience is in turn determined point by point by the
physical substratum of the organism, what can be the utility of
consciousness? For on this supposition my willing to perform a
certain act is a mere incidental accompaniment of the physical
circumstances that inevitably must bring about the object of iny
act. To quote H. C. Warren:1
In denying directive selection to forethought we reduce consciousness to
the role of an epiphenomenon. If purposive thought is not effective in producing
mental or muscular activity, of what value is consciousness in the universe?
Is it anything more than a spectator of the physical changes which
constitute real activity and form the basis of history?
1
Jour. Phil. Psych. andSei. Method, 1916, v. 13, p. 20.
402
OEIGIN OF CONSCIOUSNESS IN LIVING OHGANISMS 403
Professor Warren finds the answer to this question in the "doubleaspect"
theory of consciousness.2
,
The objection (stated in the preceding paragraph) may hold against the
traditional parallelistic world-view, but it loses force if we adopt the doubleaspect
standpoint. According to this interpretation our thoughts and purposes
are only our way of experiencing what an independent observer might
perceive as physiological activity. One set of occurrences is as "real" as
the other.
So, for example, my feeling of "hunger," would appear to a suitably
equipped outside observer as a contraction of my stomach, the presence
in it of an accumulation of certain digestive fluids; a particular
disposition of the molecules of certain nerves and certain portions of
the brain, etc.
Physical Analogies . Such dual aspect of a phenomenon is known
to us also outside of the particular case of consciousness. So, for
example, the magnetic force (intensity) at a certain point is one aspect
of the same phenomenon which could also be described by a statement
of the position of the molecules of the permanent magnet, say,
to which the field is due. The "problem" of psycho-physical parallelism
is probably due to an inadequate statement of the case. It is
probably, in this sense, of the nature of a pseudo-problem. To say
that a necessary condition for the wilting of these words is the willing
of the author to write them, and to say that a necessary condition for
the writing of them is a certain state and configuration of the material
of his brain, these two statements are probably merely two ways
of saying the same thing. A state of consciousness can be described
either in terms of its "contents," or in terms of the disposition of the
molecules, etc. of the brain, just as a magnetic field might be described
either in terms of an intensity chart or in terms of the position of a
number of magnets.
But, evidently, the double-aspect conception of consciousness helps
us not at all to recognize, in the intervention of consciousness, any
plausible reason for simplification of mechanisms to be gained by this
means. For fruitful suggestions we must look elsewhere, and it is in
2
For a discussion of this theory, originated independently by a number of
authors (Fechner, 1863; Clifford, 1878; Bourrat, 1883; Prince, 1885; Strong,
1903; Heymans, 1905) see Tolman, Jour. Washington Acad. Sci., 1922, vol.
12, p. 153.
404 ELEMENTS OF PHYSICAL BIOLOGY
this difficulty that the theory of consciousness sketched in Chapter
XXIX gives promise of assistance. For on the one hand the appearance
upon the field of action of molecules in the
a
opened-up" state
furnishes a whole range of states of matter that do not play a significant
role in the
'
operation of ordinary mechanism. With this extra
material the conscious mechanisms may conceivably accomplish
what the mechanic, working with matter in its ordinary states, is
powerless to do without complications of structure altogether prohibitive.
Again, our conception of the physics of conscious matter reveals
to our view an interplay of forces, to effect purposive adjustment,
within the molecule; whereas the mechanic, in his constructions, must
bring contending forces to bear and to produce their resultants through
separate material members, gross masses. Here, then, may be found
the opportunity for that economy of parts which is characteristic of
living, conscious mechanisms.3
In some such manner as this it seems possible to account with
reasonable plausibility at least for the fact that in the evolution of
the animal type of organism, the type equipped with a correlating
apparatus, consciousness has been seized upon as an effective means
to secure adaptive behavior.
Origin of Consciousness. One question remains as yet unanswered.
Whence did the organisms derive this consciousness? Where and how
did consciousness come into being: In a living organism? Then was
the organism unconscious prior to the event? Or else, did consciousness
arise outside the organism? Then consciousness would not be
tied inseparably to life.
The answer to these questions has already been foreshadowed.
It is not consciousness that has been evolved an elementary flash
of consciousness may be a native property of matter but a particular
kind of integrated consciousness, a consciousness spun into a continuous
thread by a faculty of memory, a consciousness embroidered as
upon a canvas, whose function is to hold in place and in their proper
relation the components of the picture. This background, this reference
frame, in the state of development in which we observe it in
ourselves, is the ego, to whom all experiences are referred. The material
organs to which this integrating function is entrusted is the nervous
system, including the brain.
3
For a somewhat detailed discussion of this and related phases of the
subject the author's article The Intervention of Consciousness in Mechanics
maybe consulted. See Science Progress, 1924, p. 407.
OEIGIN OF CONSCIOUSNESS IN LIVING ORGANISMS 405
The nervous system is that bodily system, the special office of which, from
its earliest appearance onward throughout evolutionary history, has been
more and more to weld together the body's component parts into one con-
a
-=^solidated mechanism reacting as a unity to the world about it. More than any
other system it has constructed out of a collection of organs an individual of
unified act and experience. It represents the acme of integration of the animal
organism As such it has spelt biological success to its possessor.
4
4
Sir C. S. Sherrington, Presidential Address at British Association Meeting,
1922. Nature, 1922, vol. 110, p. 350. Compare also A. G. Tansley, The New
Psychology, 1916, p. 20, referring to Holt, The Freudian Wish, 1916, pp. 76-94.
Tansley summarizes Holt's view in the words: "Professor Holt very clearly
expounds the view that mind is merely the 'integration' of the organism's
motor responses to stimuli. Professor Holt's position is that 'even two reflexes
acting within one organism bring it about that the organism's behavior is no
longer describable in terms of the immediate sensory stimulus, but as a function
of objects and situations in the environment.' The secret of the connection
>-,
of mind, and brain remains as dark as ever. Professor Holt does, indeed,
admit that mind is a 'synthetic novelty' 'the advent of specific response
. . . . is the "birth of awareness and therefore of psychology itself.' But
even if the integration of 'reflex responses' to become 'specific response' (i.e.
response to an object or situation rather than to a mere stimulus) is rightly
described as 'awareness,' and this is by no means self-evident, we are not
thereby in the least degree helped to understand awareness or cognition in
terms of anything else. We are still absolutely bound to interpret mind in terms
of our own mind the only mind of which we have direct knowledge, though we may
learn much about the conditions of its evolution."
"The behavior of an organism adapted to its surroundings is related rather
to the objects and situations of those surroundings than to physical and
chemical stimuli as such, and this takes place by integration (i.e., the putting
together to make a new whole) of simple motor responses to form complex
ones. Thus the specific responses of an organism may be regarded as 'functions'
(in the mathematical sense) of the objects and situations of its environment,
and the history of the evolution of response to environment, i.e., of
behavior, and of mind itself, is the history of successive integrations of these
'functions' to more and more complex purposes."
ENERGY RELATIONS OF CONSCIOUSNESS
Die Physik wird in der Biologie viel mehr leisten wenn sie erst noch durch
die letztere gewachsen sein wird. E. Mach.
To the naive observer, at any rate, consciousness appears to exert
a directive action upon the course of events. If we regard the
physical world as a determinate system, the events in which are completely
determined by the physical laws to which matter and energy
are subject, a question thus arises: Where, in such a scheme as this,
is there any opportunity for the agency of consciousness to bring its
influence to bear? Several alternative answers to this question seem
compatible with our limited knowledge.
First Alternative: Possible Inaccuracy of Laws of Dynamics.
First, the laws of physics as known to us may be an inaccurate representation
of facts. Indeed, we know that they must be thus inaccurate,
for we are far from having completely solved the problem of the
Universe. It is therefore possible that something omitted from
oui' formulation of dynamics and energetics is the origin of our perplexity.
That such an eventuality as this cannot be wholly ignored
is demonstrated, for example, by certain features of the Bohr theoiy
of the atom. "The quantum, conditions determining the permissible
Bohr orbits can be explained physically only by attributing to the
electrons a knowledge of the future." 1
This is a case in which the
equations of motion contain a term with a lead, not a spurious lead
that can be eliminated by suitable substitutes, but a real, essential
lead that must of necessity appear in the equations. Let there be
no misunderstanding. It is not intended to suggest that there is a
direct relation between this circumstance and consciousness. The
example is cited merely to show that the conceptions of classical dynamics
are far from exhausting the types of conceptions, in such
matters, that we must be prepared to contemplate. At the same
time it is only fair to observe that it seems hardly logical for a
being composed of electrons to affect great surprise at the fact
1
C. G. Darwin, Nature, 1923, vol. Ill, p. 771, vol. 112, p. 279.
406
ENERGY RELATIONS OF CONSCIOUSNESS 407
that these electrons display certain properties reminiscent of
consciousness.
Second Alternative :
Singular Orbits with Indeterminate Motion.
~But, in point of fact, without going so far afield, we can
see in the domain of classical dynamics opportunity for the
entrance of consciousness as a directive agent upon the field of
physical events. For there are certain cases in which the course of
these events is not fully determinate in classical dynamics. Every
case of unstable equilibrium is an instance in point. What do we
know regarding the future fate of a cone set up and exactly balanced
on its point? Any infinitesimally small force applied to it at any
time will decide its fall in one direction or another. The history of
such a system as this depends on infinitesimals, that is to say, on
data of an order of magnitude that must escape the observation of
our senses. To state the matter in more general terms, the orbits of a
system moving in accordance with the laws of classical dynamics are
of two kinds. Stable orbits are characterized by the fact that a small
change in the conditions of the systems will bring about a small
change in the orbit. Such is the orbit of a ball rolling down an
inclined plane, after being started at an angle 6 with a velocity v.
If we slightly change the angle 6 or the velocity v, or both, the orbit
also will be but slightly changed.
But there 'are other, unstable orbits, such as that of a bah
1
rolling
along the ridge of a straight watershed. Such orbits correspond to
singular integrals, or contain singular points. Here, if the angle of
the initial velocity deviates ever so little from the orientation of
the ridge, the ball will proceed along an orbit totally different from
that following the ridge it will descend into one or other of the
valleys on the two sides of the ridge.
Here again, infinitesimal interference will produce finite, and, it
may be, very fundamental changes in the result.
This, essentially, is the nature of the conception suggested years
ago by Clerk Maxwell,
2
and independently, it seems, by J.
Boussinesq.
3
It is to be noted that such unstable equilibria and
2
Life of Clerk Maxwell, by Lewis Campbell and William Garnett, 1882,
p. 434. L. J. Henderson, The Order of Nature, 1917, p. 213; J. W. N. Sullivan,
Aspects of Science, 1923, p. 156.
^
3
Cours de Physique Mathematique ;
Conciliation du veritable de'terminisme
me'ehanique avec 1'existence de la vie, etc. Originally published 1S78, republished
1922, Gauthier-Villars, Paris.
408 ELEMENTS OF PHYSICAL BIOLOGY
orbits as exemplified by the cone precariously balanced on its point
and by the ball performing its tight-rope trick along the ridge of a
watershed are typical of systems disposing of a fund of ''available"
energy. Such a fund of free energy, in turn, is typical of living*-,
organisms. To quote again Clerk Maxwell:
In all such cases there is one common circumstance, the system has a
quantity of potential energy, which is being transformed into motion, but
which cannot begin to be so transformed until the system has reached a certain
configuration, to attain which requires an expenditure of work which in certain
cases may be infinitesimally small, and in general bears no definite proportion
to the energy developed in consequence thereof Every
existence above a certain rank has its singular points, the higher the rank,
the more of them. At these points influences too small to be taken into account
by a finite being may produce results of the greatest importance. All great
results produced by human endeavor depend on taking advantage of these
singular states when they occur. 4
In the course of this our mortal life we more or less frequently find ourselves
on a physical or moral watershed, where an imperceptible deviation
is sufficient to determine into which of two valleys we shall descend. 5
This conception makes the interference of consciousness (will)
in physical events an exceptional occurrence:
It appears that in our own nature there are more singular points' where
prediction, except from absolutely perfect data, and guided by omniscience of
contingency, becomes impossible than there are in any lower organization.
But singular points are by their very nature isolated and form no appreciable
fraction of the continuous course of our existence. 6
Third Alternative: Possible Influence of Factors Eliminated
from Equations of Dynamics. As to the last quotation, perhaps,
one feels disposed to hesitate in adopting the standpoint suggested
by the great physicist. For it seems that even the most trivial
voluntary action involves the interference of consciousness in the
course of physical events, and of such more or less trivial voluntary
acts our waking consciousness is filled to the brim. A certain
interest may therefore attach to an alternative point of view which
has been developed by the writer elsewhere, and which is based on
the following consideration:
4
Clerk Maxwell, loc. cit., p. 443.
B
Clerk Maxwell, loc. cit., p. 441.
6
Clerk Maxwell, loc. cit., p. 444.
ENERGY EELATIONS OF CONSCIOUSNESS 409
A. quantity which does not appear in the working equation describing
the laws of action of a physical system may nevertheless
play a significant role in the world's events. So, for example, a
mathematical theory of wealth, covering at any rate certain aspects
of economics, can be built up in terms of prices and sales alone,
without pushing the analysis of fundamentals beyond this point;
that is to say, without examining the human emotions and motives
that, presumably, find their numerical expression in prices. On such
basis as this, for example, Cournot founded his admirable "Researches
into the Mathematical Theory of Wealth."
But to most of us it will appear quite evident that such a treatment
as this is necessarily a very incomplete presentation of the
actual events, however exactly it may represent the resultant effects
observed. For it wholly ignores our desires and purposes, which to
us appear very real and important constituents of the course of
nature.
This example should open our eyes to the possibility, with which
we must be prepared to reckon, that the equations of dynamics,
however perfectly they may picture the course of certain physical
events, may fail entirely to reveal or to give expression to underlying
agency that may, in fact, be of fundamental significance. The
interference of consciousness in mechanics may be very real, and
yet the course of events may appear fully determined by the laws
of dynamics.
7
Energetics of Aimed Collisions. Aimed collisions imply a
correlation between a present state or event and a future occurrence
or eventuality (a future that will be or a future that may be).
Psychologically this correlation is apprehended as purposive forethought.
Physically it implies the disposal of a fund of free energy,
since the energy for bringing about (or for avoiding) a future encounter
cannot itself be derived from that encounter. There is thus
of necessity a fundamental connection between purposive action and
the disposal of a fund of free energy. Purposive behavior can and
does occur only in material structures disposing of such a fund.
7
For a more detailed presentation of this viewpoint the reader must be
referred to ths author's article The Intervention of Consciousness in Mechanics,
Science Progress, January, 1924.
1""
CHAPTER
REVIEW OP THE CORRELATING APPARATUS
Man. is the arch machine, of which all these shifts drawn from himself are
toy models. Emerson.
Some of the outstanding features of our reflections and observations
in the chapters immediately preceding are summarized in tables
34 and 35, which at the same time bring out a number of other
significant facts.
Table 34 sets forth systematically the main facts regarding the
receptors and effectors, as exhibited particularly in the case of man.
(A few entries shown in parentheses relate to species other than man,
in cases in which man lacks the feature to be exemplified).
In this tabulation Depictors or Informants are shown as divided up
into Receptors, Elaborators, Relators, and Communicators. The
Receptors are further subdivided into Internal Ceptors, Contact
Ceptors and Distance Ceptors. These terms hardly require explanation,
especially in view of the other entries on the table, which will
serve to elucidate them. In every case represented, the name of
the corresponding faculty, the native or natural organs, and the
artificial organ or organs are shown. The list of the latter is far from
exhaustive.
The imagination, the faculty by the aid of which we hold before our
mind's eye the material upon which we operate in the process of
thinking, is shown divided into autistic and realistic imagination.
The term autistic thinking has been introduced by psychologists
to denote that type of thought in which the fancy is given free reins,
unchecked by any demand for correspondence with reality; the thought
of the savage, the child, the dreamer, the poet.
1
Realistic thinking,
on the contrary, is constrained by the laws of Logic, which assure a
due correspondence between the products of cogitation and the
features of the external world to which they relate.
1
"Les poetes se consolent, comme les enfants, avec des images." Anatole
Prance , Le Jardin d'JEpicure.
410
EEVIEW OF COEEELATENG APPAEATUS 411
The item "semi-realistic imagination" calls for special notice.
It was suggested primarily by the case of the so-called realistic novel.
Such a novel is realistic as to the types presented, though the specific
. instances may have no counterpart in reality. Such semi-realistic
thinking plays an important part in Science. Every physical
formula is, in a sense, an instance of this type of thinking. When
I say that the area of a rectangular surface measuring 2 by 4 feet is
8 square feet, this does not impty that any such rectangle exists
in reality. It may exist or not, the statement has a certain quality of
truth independent of the reality of such a rectangle; the statement
relates to a type rather than to a specific instance.
The terms "Sense of Time" and "Spatial Sense" are self-explanatory.
They are suggested without prejudice as to their metaphysical
significance. We must accept it as a.
fact that we distinguish experience
as having a definite sequence in time, and that we are able, unaided
by artificial adjuncts, to gage with very fair accuracy equality of
time intervals within certain somewhat narrow limits. Similar
remarks apply to the entry Spatial Sense. Such entries are needed
to make the table complete. If the reader, on metaphysical or other
grounds dislikes these terms, others may be substituted at his
discretion.
The item Communicators calls for little comment. These are, in a
sense, receptors and effectors set apart for or employed in a particular,
and a highly important use.
Just as the Depictors or Informants throw a picture of the external
world upon or into the organism, so the Epictors or Transformants
translate into material reality the plans conceived, the pictures
formed in the mind. The Brooklyn Bridge, for example, is a material
representation or picture of the plans that once were in the designer's
mind.
These transformants include once again the elaborators, relators
and communicators, for these fulfill a double function; the transformants
further include the Internal and the External Effectors,
whose nature and significance is apparent from the table.
If time had permitted, the writer would much have liked to give
to this table a quantitative cast by adding columns showing persons
employed, production, consumption, imports, exports, and capitals
invested in the arts and industries corresponding to the several items
shown in the table, adding, in other words, a quantitative descrip-
412 ELEMEXTS OF PHYSICAL BIOLOGY
tion of the behavior schedule of human society. The table would
thus give a coherent, biologically founded, picture of the life activities
of the Body Politic. The behavior of this body as a whole, no less
than that of its constituent organisms, is conditioned by its anatomical
constitution and its biological needs. The substitution of
artificial for natural organs in nowise alters this fact.
The adjusters could not very conveniently be accommodated in
the scheme of table 34. They have been separately systematized in
table 35,
The adjusters are here shown divided into two groups. The
Internal Adjusters are those that control the distribution of effort in
different pursuits in one organism, within the organism, so to speak.
The External Adjusters, on the other hand, control the distribution
of effort among the several organisms or groups of organisms comprised
in one species.
Internal Adjusters. The internal adjusters operate through the
faculty of choice (wish, will, etc.). Such choice may be made without
any conscious reference to any objective principle, purely according
to the dictates of instinctive impulse or tastes. No attempt is
made to enumerate such instincts in any degree of completeness.
They are shown classified according to the principal beneficiary into
egoistic and altruistic instincts. The agent is not always conscious
of the relation of his actions to the beneficiary. Thus, for example,
the scientific investigator, working under the stress of the instincts
of curiosity, workmanship, and self-expression, may have little or
no thought of conferring a benefit on society. Such benefit nevertheless
follows.
The choice may not be made simply in response to instinctive
impulse. It may be more or less consciously guided by "principles,"
such as may be given either empirically, on the word of an accepted
authority, the Church for example; or as worked out systematically
by the agent himself (philosophy of conduct, ethics).
These two types of choice may be respectively termed instinctive
and reflective, or, in analogy with the terms employed with regard
to thought, the instinctive may also be classed ag autistic, the
reflective as realistic, since the former seeks no basis outside the
individual himself, the latter tends to seek an objective basis. The
discipline of Ethics here appears as a regulator of conduct in close
analogy to the manner in which logic functions as a regulator of
thought.
REVIEW OF CORKELATING APPARATUS 413
i
s .5
>3 ^ffi 02 PH
fl.Sla'1
i 6 S fe
rt o
p< q
s'
o
bO
H
1
O
.= .-1*5
a.1.8
1
"
oo
W
a
o .3
11 oo a)
^H 53
eg O
o
O
a a
.2 Q
N
3 lr^
414 ELEMENTS OF PHYSICAL BIOLOGY
This analogy is brought out with the greatest clarity in the following
quotations, which should be read in parallel:
Logic: C. J. Keyser, Mathematical Ethics: Clerk Maxwell
Philosophy. 1922, p. 136.
"Logic is the muse of thought, ". . . . an abandonment of
When I violate it I am erratic; if I wilfulness without extinction of will,
hate it, I am licentious or dissolute; but rather by means of a great develif
I love it, I am free the highest opment of will, whereby, instead of
blessing the austere rouse can give." being consciously free and really in
Compare also the saying of Seneca: subjection to unknown laws, it be"Si
tibi vis oinnia subjicere, te subjice comes consciously acting by law, and
rationi." really free from the interference of
unknown laws."
In the development of ethics, as in that of logic, the influence of
natural selection must be to favor those habits of thought and action
which are conducive to the welfare of the species. And. just as we
have reached the point where at least the rudiments of logic are
unassailably fixed in the healthy adult mind, so we may expect that
socially sound principles of conduct will in time be more and more
accepted as inevitable truths, their converse as "unthinkable," at
least to the naive mind. Indeed, in very appreciable measure this
is the case even now. Theft, for example, is to the normal person
utterly unthinkable. The philosophic mind will of course, still, at
all times, be able to conceive of unethical conduct as a possible
alternative, just as today the philosopher can, as a sort of tour de
force, of mental gymnastics, overcome his native faith in an external
world, and assume temporarily the role of the solipsist.
External Adjusters. The external adjusters, those that govern
the distribution of effort among the several individuals of the species,
present several points of much interest. Their operation is, of
course, essentially restricted to species living in organized communities,
such as the bee, the ant, and, quite particularly, man.
Xature has developed two entirely distinct methods of adjusting
the distribution of activities among the several members of such a
social group. The individual may be born into the world or nurtured
to adolescence with a definitely specialized endowment of anatomical
equipment and physiological and psychological faculties and predilections.
So the queen bee is fitted for her very particular part, the
drone for his, and the working bee for its task. Each is capable only
BEVIEW OF COKKELATING APPAEATUS 415
of playing the part assigned, and presumably each is perfectly content,
wholly innocent of the joys and sorrows characteristic of the
other's calling. In our own human species this condition is nearly
approached as regards the special tasks of sex, at least in their
primary forms.
The second method adopted by nature is really a very singular,
and one feels tempted to say, a highly ingenious one. Here the
individual may have little or no natural disposition to specialize in
a blind obedience to the call of social expediency. Nature, as it
were, gives the reins into his hands, and only demands the right to
impose a tax on all his profits. Smith and Jones get together,
Smith offers Jones a pound of bread, and Jones accepts it in return
for six ounces of beef. Each is satisfied. Each has driven a purely
selfish bargain, without the least thought of the good of the community.
But the community has already collected its tax the
exchange, on the whole, is to its benefit.
Now this method has its advantages and disadvantages. One
feels an unpleasant suspicion that c
if Smith and Jones, instead of
driving a mutually selfish bargain, instead of working in a measure
against each other, could be brought to combine their forces in the
definite purpose of serving the community, the latter would collect,
not only a residual tax, but a more complete benefit. In the simple
example of the loaf and the steak this is hardly apparent, but there
can be little doubt that in human affairs much is lost by internal
friction, by men pulling this way and that, instead of pulling together.
The other disadvantage is that this method of division of labor
frequently must fail to satisfy those working instincts which man still
possesses, useless as they may appear in the circumstances. Now
unsatisfied instincts are painful things.
But this method, in spite of such disadvantages as it may have, has
established itself very firmly among us. It must possess fundamental
advantages. And it is not difficult to point out where these enter
into the scheme of things. The tasks of the beehive, the anthill,
are comparatively simple and few. Two, three, perhaps four
sets of instincts serve the required ends. But to quote Veblen,
2
<f
the higher the degree of intelligence and the larger the available
body of knowledge current in any given community, .... the
2
T. Veblen, The Instinct of Workmanship, 1914, p. 6.
416 ELEMENTS OP PHYSICAL BIOLOGY
more complicated will be the apparatus of expedients and resources
employed to compass those ends that are instinctively worth while/'
So, in our own community, the list of different gainful occupations
takes up over ten closely printed pages of the Statistical Abstracts
of the United States. A behavior schedule of such complexity as
this can hardly be taken care of by an assortment of as many instinctive
proclivities. What is happening today is that the instinct
of workmanship, so far as it plays a part in the working day of
mankind, is sufficiently broad and plastic to adapt itself, with
reasonable success, to a variety of tasks. Misfits there are, and some
degree of heartache in consequence. But perfect adaptation is
found in no department of life, neither can we expect it here.
CHAPTER XXXIV
CONCLUSION RETROSPECT AND PROSPECT
Esperons dans ces etres inconcevables qui sortiront de 1'homme, comme
1'homme est sorti de la brute. Anatole France.
The Life Straggle in the Modern Community, It is not wholly
from, partiality, nor from self-complacency alone, that we are led to
view the evolution of man, and especially of modern man, with a
peculiar interest. An impartial judge, if such could be produced,
would doubtless concede to the'human species, as it stands today, an
unique and predominant position in the scheme of Nature. For,
civilized man has achieved the distinction of practically clearing the
board of all foes of a stature in any way comparable with his own.
This has resulted for him in a very special form of the struggle for
existence. With the conflict against other species relegated to the
background, man's combat with his own kind has been forced to the
center of the stage. Increasing population pressure and continued
success in the control of disease can only add to this effect, which is
furthermore enhanced, while the struggle is at the same time forced
into a very particular mold, by the industrial regime which has
bonded the body politic into one organic whole. Under this regime
we enter the battle of life en bloc. The activities of any one single
individual are as a rule wholly incompetent to keep him or anyone else
alive. The product of a day's work of a stenographer, for example,
is a pile of papers covered with black marks. Taken out of its setting
this is an entirely useless product, totally unfit to support the life
of the producer. And much the same is true of the work of the
bookkeeper, the draftsman, even the skiEed engineer or chemist.
The race as a whole, indeed, still contends with external opposing
forces. But the individual, in a large proportion of cases, is conscious
only of competition with his fellows. This competition takes
the form of an intricate system of bargains to purchase the least
intolerable form of dependence that the individual can secure, or to
achieve such approach to independence as may be had. Whoever
fails entirely in this struggle is Nature's convicted delinquent, the
417
4jg ELEMENTS OP PHYSICAL BIOLOGY
unfit unable to draw to himself the needful share of the necessities
of life, out of the general fund from which the body politic as a whole
is supplied.
A new and characteristic form of the struggle for existence thus
arises the trend and presumptive outcome of which is for us a matter
of evident interest.
Organization of Motive Lags behind Industrial Organization.
While the human species, as a mechanical going concern, has become
organized into a social whole, the motivation that keeps it
going
has not undergone the same thoroughgoing organization, but continues
to be in great measure individualistic in type. Social ends are
achieved through appeal to individualistic instincts. Our present
industrial system operates by way of the mutually selfish bargain, in
which each party to the transaction seeks his own advantage, regard"ass
of the gain or loss to society as a whole. The system works tolerably
well, beyond reasonable expectation perhaps; at least so it seems
to those accustomed to the system. The competitive element which
it introduces is not without salutory action. But, making proper
allowance for this, one is left to ponder whether there may not, in
due course, be evolved a superior system, that shall secure the interests
of the community more directly, and with less loss by internal
friction in the social machinery. There are certain difficulties in the
way of achieving this in man by those instinctive methods which
operate effectively in insect societies, as has already been pointed
out. Human activities are too multifarious to permit of simple
adjustment by a stereotyped set of instinctive proclivities that predestine
each infant at birth for his own particular sphere of activity
through life. Nevertheless, we are not left wholly without encouragement
that the future evolution of our race may proceed in a
direction that shall ultimately ease the conflict between man and
man, and between man and the world at large.
Philosophy as a Necessary Part of Scientific Enquiry, We have
already had occasion to refer to the importance of attaining a clear
realization of the nature of our fundamental assumptions. The
scientific investigator has, in the past, not given to this phase, the
philosophic phase, of human enquiry, the attention which it sooner
or later must demand as a necessary condition of unimpeded progress.
For if we define philosophy as the critical examination of the funda-
CONCLUSION 419
mental data of experience,
1
it is seen that philosophy is not something
apart from science, but must form, an integral and essential part of
every science. No science can be said to have reached the adult
stage until critical examination of its fundamental concepts has at
least begun.
Classification of the Sciences in Relation to Self and External
World. Of all fundamental assumptions in our thinking there is
probably none so basic, so all pervading, as that of the division of the
universe (that is to say, of the totality of experience) into an ego
and a non-ego, a self and an external world, a knower and the known,
an observer and a thing observed, a subject and an object, mind and
matter; all of these pairs of terms are, as commonly employed, more
or less close synonyms.
Both unsophisticated thought and scientific thought accepts this
division as axiomatic, and the physical sciences propose to study the
external world, while psychology proposes to study mind, or the ego.
Philosophy does not accept this division as axiomatic, but proposes,
among other things, to inquire into the nature and significance
of the distinction. The philosopher asks : What is ego, and what is
non-ego? Where is the line drawn, and how do I come by this
distinction? Thus philosophy studies the relations between the ego and
the non-ego? According to the point of view that has been here
1
This definition contains an implicit assumption, namely, that there are
fundamental data of experience. This is not necessarily true. It may be that,
no matter how far the analysis be carried, the data arrived at as the most fundamental
can always be resolved by further analysis into others more fundamental.
Thus the characteristic feature of philosophic enquiry, which distinguishes
it from scientific enquiry, is perhaps rather one of direction than
one of result. "While science builds up from axiomatic data, philosophy works
from these axiomatic data downward in the opposite direction. This is the
view proposed by Bertrand Russell in his work Our Knowledge of the External
World.
2
This statement evidently holds true quite independently of the reality or
objective existence of the ego and the non-ego or external world. Disputes as
to such objective existence seem rather pointless. The fundamental data of
'"- our experience are sense impressions. When certain relations obtain between
certain sense impressions, we ascribe the latter to an external object. Thus,
if I see a cat, and proceed to go through certain movements which I speak of
as stroking the cat, I may presently have the sensation of soft fur against my
hand, and a purring noise at my ear. In that case I say that the cat is real.
Or, going through the same motions, I may feel the impact of my hand against
420 ELEMENTS OF PHYSICAL BIOLOGY
presented the ego is not a concrete thing, but is of the nature of a
system of reference, in relation to which experience is
described,
somewhat as the geographical location of a city is defined in terms of
longitude and latitude. Just as longitude may be reckoned from the
meridian of Paris or from that of Greenwich or from any other convenient
datum circle, so experience is describable with reference to a
number of different egos.
The relation between the ego and the external object is brought
out with great clarity by Bertrand Russell in his Analysis of Mind
(1921, p. 100), after this manner: Sets of photographs of the stars
might be made, first by taking all kinds of views, in different directions,
from one point in space. The collection thus brought together
would give the appearance of different stars in a certain place,
or, as we may say, would be a catalogue of the world of stars as it
appears from a certain center of perspective. But another collection
might be made by photographing the same star from all kinds of
different points in space. TKs collection would give us all the apa
hard surface, etc. In that case I say that the supposed cat is merely the
image, in a mirror, of a cat. Fundamentally the so-called real cat is merely a
convenient hypothetical construct, convenient because it greatly simplifies
language and thought. It would be possible to express in terms of sense
impressions alone everything that is ordinarily expressed in terms of an
external world. For that very reason it is quite impossible to prove that
this external world has a real or objective existence, or, for the matter of
that, that it has not. That the external world is a hypothetical construct is
not ordinarily realised, because that construct is formed by an instinctive
and unconscious process, so far as ordinary experience is concerned. In extraordinary
circumstances, that call for the exercise of intense conscious
reflection, the hypothetical nature of the construct is immediately apparent,
as for example in the case of the Bohr atom, or the Einstein space-time
continuum. But at bottom the real cat is just as hypothetical as the theoretical
Bohr atom, although perhaps less subject to revision as to its precise
description.
If we continue to employ the terms external world, object, etc., this must not
be taken to imply that we are oblivious of the hypothetical character of these
constructs, but only that we resort to the usual expedient, for the sake of
grammatical simplicity. No harm will result from this use of the conventional
terminology so long as we bear in mind its character, and do not allow ourselves
to be drawn into the discussion of pseudo-problems arising out of that
terminology and not out of the fundamental facts. Thus, for instance, to
enquire whether mind has an objective existence is a question based on a misconception,
not only of the term mind, but of the general character of all the
data of our knowledge.
CONCLUSION
pearances of a certain star in different places, or, as we may say,
they would be a catalogue of the different aspects of one object.
3
The essential difference between the ego and the non-ego is, in
this sense, merely a difference in the system of filing, so to speak, a
collection of data. Arranged as a collection of observations from
one perspective, they are the experiences of an ego. Arranged as a
collection of aspects of the world from different perspectives, they
are, in their totality, the appearances of the external world.
This double system of filing or cataloguing our experiences is
responsible for certain of the divisions of science. This is indicated
diagrammatically in the chart table 36, which exhibits, for example,
the position of the Physical Sciences, in the general scheme of human
enquiries, as the study of the external world, or of that set of data
which by common consent we ascribe to an external world; certain
other data we associate more particularly with the self, whom we
regard as equipped with a body, and gifted with consciousness. This
twofold aspect of the self gives rise to a corresponding twofold
development of the science of living beings. Biology, with its
branches, studies more particularly the body of the organism, while
Psychology busies itself with the phenomena of consciousness, as
such. The relation of consciousness to the prevailing state of the
body is perhaps to be regarded as par excellence the sphere of study
of Psychophysics.
With regard to the present and past the Self is a Knower. With
regard to the future he is a Witter. That is to say, he has direct
cognizance of certain past and present events through sensation
and memory; he has direct cognizance of certain future events through
his will.
4
In accordance with this twofold aspect of the Self, psychol-
3
Compare also the ontograms and phanograms of K. Gerhards, Naturwissenschaften,
1922, pp. 423-430, 446-452.
4
Indirectly, i.e., by inference, he may have other knowledge regarding past,
present and future. But as to direct knowledge the statement made above
applies. It is very commonly overlooked that we have knowledge of the
future through our will. This oversight is partly due to the fact that we sense
this kind of future in a particular way, unlike our sensing of the past and
present. It is partly also due to the masking of this kind of knowledge of the
future through uncertainties surrounding the actual realization of our intentions.
Upon reflection it will be found not unnatural that there should be an
appearance of fundamental distinction between our knowledge of the past
through memory, and our knowledge of the future through will. For, if for
422 ELEMENTS OF PHYSICAL BIOLOGY
ogy may be regarded as divided into two main branches, dealing
respectively with the Self as a Knower and with the Self as a Wilier.
Some of the principal subdivisions of these two main branches are
indicated on the chart. Thus the first branch comprises first of all
the study of the avenues by which we enter into possession of knowledge,
i.e., the study of the special senses, including both the natural
senses, and their artificial auxiliaries; furthermore the general theory
of knowledge (Episternology, Ei'kenntnisstheorie, Scientific Methodology);
and the sciences of thought, both autistic and realistic or
logical.
The second major branch is shown as divided into Dynamic
Psychology or Psychology of Motivation (including, for example,
the study of instincts); Esthetics and the Theory of Values; and
Ethics.
brevity we may be allowed to speak in the customary terms of unsophisticated
speech, we are, through memory, cognizant of the past state of the external
world, in so far as there are effects (engrams) in our bodies of those past states;
whereas, conversely, we are, through will cognizant of the future state of the
external world, in so far as there are causes in us of those future states. This
consciousness of causes within us of certain future (external) states, is just
what we call will. Will is our subjective realization of what to an objective
observer would appear as (plrreical) causes of the events "willed." Bertrand
Russell's provisional definition of memory as "that way of knowing about the
past, which has no analogue in our knowledge of the future," seems, in this light,
unwarranted (Analysis of Mind, 1923, p. 165). We have here oily one particular
phase of the double-aspect theory of consciousness that has been noted
ia^Chapter
XXXI. From this point of view the question that is sometimes
raised, as to whether our sense of willing (free will), our desires, are illusions,
logical constructs, fictions, or the like (see B. Russell, Analysis of Mind, 1923,
p. 32) seems pointless. The question whether we strike a man because we are
angry, or whether as William James has put it, we are angry because we strike
a man, seems to fall into this category. There are certain conditions in us that
are tending to bring about, and that presently do bring about, the blow.
Our own particular way of sensing these conditions is that which we express by
saying that we entertain the emotion of anger, that we have forethought and
intent to hit the man. From this "double-aspect" point of view there seems no
room for any essential conflict between the standpoint taken, for example,
by fiertrand Russell (Analysis of Mind, pp. 32, 280) that desires like forces are
fictitious; and, on the other hand, the standpoint taken of the present writer,
in Science Progress, January, 1924, pp. 417, 418, that forces, like desires, may,after all, not be purely fictitious. They may be the subj eetive, i.e., immediate
or directly sensed aspect of certain states that can also become topics of
objective, Le., indirect observation.
a
53 o o
CC rC! W
&q EH H
3 .9
>*"i
b) CQ
a o .g
J
& o
a3 rt
-2 B
1-8 o
H
to
"S
"^
'o 03
"S *
423
424 ELEMENTS OF PHYSICAL BIOLOGY
This chart is not of course put forward as in any sense complete,
but rather as suggestive of a natural classification, and as setting
forth certain relations of somewhat fundamental character.
REACTION OF KNOWLEDGE UPON EMOTIONS
The feature in this chart that chiefly interests us in our present
considerations is that division of the topics which arises from the
twofold aspect of the Self, the Knower and the Wilier.
Fundamentally the world of knowledge and the world of will, of
desire, are wholly apart, as we have already had occasion to observe.
Logic has nothing whatever to do with motives as such, though it
may, of course, busy -itself with the consequences of motives. Contrariwise,
motives are in themselves unrelated to or independent of
knowledge. The knowledge that if I cross the street at this moment
I shall be killed or maimed does not in itself determine my action nor
even my intent; this depends on my emotional bias, whether I am
bent on suicide, or desirous of maintaining as nearly as possible a
pleasant existence.
Nevertheless a connection is established between knowledge and
will, though the fact that the Knower and the Wilier are united in
one physical body, so that physical reactions do occur between
knowing and willing. That the recognition of a truth is attended
with an emotional affect we well know, not only from our own experience,
but also from the fervid expression of some of the great pioneers
of science. So Kepler, upon completion of the evidence establishing
his third law of planetary motion, exclaims jubilantly: "Nothing
holds me; I will indulge in my sacred fury." Or we may recall the
words of Poincare:
Science puts us in constant relation with something greater than ourselves;
it presents to us a spectacle always renewed and always more vast. Behind
what it shows us so grand it makes us divine something still more grand;
this spectacle is a joy for us, but it is a joy in which we forget ourselves, and
this is morally healthful.
Quotations exhibiting this spirit, this connection in men's minds
between the contemplation of the intellect and the affections of the
emotional sphere, could be multiplied almost without limit.
What result may we, then, expect to flow in the emotional sphere,
from that phenomenal expansion of man's intellectual horizon which
CONCLUSION 425
modern science has brought, and the still greater revelations that are
undoubtdly-yet to come? Men have gained much knowledge, but
as yet they have scarcely had time to realize what they know, and the
grandeur of the new truths. It is not enough to know; there must be
a vivid imagination presenting to the mind's eye for contemplation
at once the many facets of the glittering gem. When men shall have
learned not only to know but to appraise the esthetic value of their
knowledge, when they shall be filled with the glory of it all, surely a
new era must dawn for the poetic arts. Wordsworth says :
Poetry is the breath and finer spirit of all knowledge ;
it is the impassioned
expression which is in the countenance of all Science If the time
should ever corne when what is now called Science .... shall be ready
to put on, as it were, a form of flesh and blood, the poet will lend his divine
spirit to aid the transfiguration, and will welcome the Being thus produced
as a dear and genuine inmate of the household of man.
It is, presumably, with such thoughts as this, that he writes in
his Prelude
. . . . The song would speak
Of that interminable building reared
By observation of affinities
In objects where no brotherhood exists
To passive minds ....
And Renan, in L'Avenir de la Science, shows similar sentiment:
Disons done sans crainte que, si le merveilleux de la fiction a pu jusqu'ici
sembler necessaire a la poesie, le merveilleux de la nature, quand il sera devoile'
dans toute sa splendeur, constituera une poesie mille fois plus sublime, une
poe"sie qui sera la reality meme, qui sera fi, la fois science et poesie.
We may well ask : If the simple Hebraic myth was competent to
inspire a Haydn to compose an Oratorio of the Creation, what tone
poem shall adequately celebrate the new meaning, in the mind of the
modern astronomer, of the words
The Heavens declare the Glory of God
The Wonders of His power proclaims the firmament.
Thus we are not left wholly without indication as to what may be
the fundamental trend of the future evolution of our race. That the
evolution of our intellectual capital is subject to a certain particular
kind of orthogenesis we have observed in an earlier chapter. The
426 ELEMENTS OF PHYSICAL BIOLOGY
esthetic reaction of the new knowledge upon the race should bring
with it a corresponding quasi-orthogenetic development in the
sphere of esthetics, and through its mediation, also in the closely
allied sphere of ethics. "When souls reach a certain clearness of
perception, they accept a knowledge and motive above selfishness." We
have noted elsewhere that, physically, no clear line of division can be
drawn between the body and the environment. Psychologically too,
we saw that the ego is not something that divides the world into
separate fields, but is rather of the nature of a standard of reference
in terms of which we find it most convenient to describe experience.
But our system of coordinate reference frames, from a good servant,
has threatened to become a bad master. As Keyser remarks,
we have "estranged and objectified the world, and lost the sense that
we are of it." It is as if evolution had overshot the mark, as if the
race must in some degree retrace its step, and regain something of
that impersonal consciousness that now seems to be only the occasional
property of a few, who, like Wordsworth, are at times "unable
to think of external things as having external existence, and who
commune with all that they see as something not apart from, but
inherent in their own immaterial nature."5
Perhaps this transfiguration
cannot be achieved but by the race passing through some great
purging cataclysm, out of which a remnant may evolve toward a
higher goal. It is familiar fact in geology that the species which pass
on the stock to later eras of evolution are commonly not the main
branches nor the most highly developed members of the evolutionary
tree. So, also, it may not be the descendants of the now dominant
divisions of our species that shall carry on the torch to light the new
era, when "the world shall no longer be beheld as an alien thing,
beheld by eyes that are not its own."5
But this uncertainty cannot
be allowed to deter us in such efforts as we may see fit to make to
further by our own initiative the progress of the species, according
_
*
Compare E. Maeh, Die Analyse der Empfindungen, 1903, p. 24; "An
einem lieiteren Sommertage im Freien ersehien mir einmal die Welt sammt
meinem leh als eine zusammenhangende Masse von Empfindungen, nur im
Ich starker zusammenMngend." Also Emerson: "... the sense of
being which in calm hours rises, we know not how, in the soul, is not diverse
from things, but one with them .... We first share the life by which
things exist, and afterward see them as appearances in nature, and forget
that we have shared their cause."
8
C. J. Keyser, The Human Worth of Rigorous Thinking, 1916, p. 126.
CONCLUSION 427
to our best lights. It is not the least of our privileges that there seem
to have been given into our hands the means whereby we may in a
measure ourselves influence and control the fate and future of our
species. It must be admitted that the fraction of the population is
small that recognizes or cares anything about this great and golden
opportunity for man to have a significant voice in the shaping of the
world's destiny. But history has given us cause to be optimistic
as to the possibilities of achievement by a small minority of perfervid
men. "A little leaven leaveneth the whole lump." In our own day
we have seen this principle only too effective in various propaganda
not of the most desirable type. If men respond thus readily to
guidance of very doubtful competence, those better qualified have
at least no prima fade grounds for despairing utterly of establishing
a body of followers. If enlightened men refuse to exert their influence
toward the guidance of affairs according to their best knowledge, they
certainly have no right to complain when they see the crowds following
after leaders less hesitant, if also less competent. Even sheep
have a bellwether. Leadership does not necessarily require intelligence
of a high type. To say that the populace does not want
the highest type of leader is beside the point. The populace does
not pick its leader; it is the leader that picks the populace. However,
the writer does not mean to suggest that the man of science, should
at the present epoch, proffer his services as a leader in spiritual
affairs. His reluctance to do anything of this kind is wholly to be
commended. But it would seem that the time is ripe for scientific
men at least to consider some degree of concerted action, some degree
of systematic communion among those who continue to feel a proper
interest in the ethical issue, now that the mythological trappings
with which this issue is commonly encumbered, 'have been fairly
effectively relegated to the background in well appointed minds.
Such concerted action seems needful, not so much for the virtue of
that strength which is in union, as on account of that weakness which
is the failing of the isolated individual:
"Tis meet
That noble minds keep ever with their likes
For who so firm that cannot be seduced?
The writer believes that he is expressing a feeling entertained by
many, though not perhaps always clearly apprehended, if he states
428 ELEMENTS OF PHYSICAL BIOLOGY
that the world is ripe for some sort of concerted effort, a binding
together in one form or other, of men possessing the scientific outlook
and method of thought, combined with a sincere interest in the fulfilment
of the great World Purpose. It is perhaps well to call to
mind that the face of science is not turned squarely and unhesitatingly
in this direction. Not only has the World War shown us all too
clearly that the weapons of science are as keen for internecine warfare
as for the campaign in the conquest of truth; more than this, there is a
certain fashionable cynicism abroad which affects a scientific pose.
There are those who, having hitched their wagon to a hog, fare forth
proclaiming that the world is nothing but a dung heap; and those who
advocate a humoring of the elementary man in us as a health measure.
From the standpoint of mental hygiene, stupidity, too, is a rather
healthy condition. Yet somehow, even for the sake of our health
most of us would hardly elect to be stupid that is, if we had any
choice in the matter. Cynicism has its uses. But in the end no
one, not even the cynic himself, takes it very seriously. The pessimist
spends his energy in jeremiads while the optimist is
covering
the ground with his forward stride. Let us endorse the stand taken
by L. Witmer:7
What the world needs today is more of the optimism of the progressive and
a little less of the pathological fear of the standpatter, more faith in creative
evolution, more hope of reaching yet higher levels of achievement, and more
of that freedom from prejudice called charity, another name for love the
productive passion.
Evolutionary Value of Nurture and Tradition. One form in
which the demon of pessimism puts forth its head is in an overloud
emphasis on the futility of the effects of nurture as opposed to the
elemental force of nature* Without committing ourselves to indefensible
extremes in the opposite direction, we may put in a strong
plea in support of the merits of nurture, merits so plain that their
very obviousness has made us in some degree blind to them, and has
thus given a weapon into the hands of the type of pessimist of which
7
What is
Intelligence, Scientific Monthly, 1922, p. 67.
iQol To
a
/
e
^' Jeimillgs>
Hered% and Environment," Sci. Monthly, Sept.,
19.4, p. 225: What has gotten into the popular consciousness as Mendelism
still presented in the conventional biological gospels-has been grotesquely
inadequate and misleading; its seeming implications as to the trivial r61e of
the environment has become null and void."
CONCLUSION 429
we speak. Those who so glibly discourse on the futility of efforts to
improve man's nature, would they be satisfied to spend their lives
among untutored savages? Would they relish the company of
unclean slatterns and the vermin-ridden denizen of the slums?
Would they care to live under the same roof with persons innocent
of the rudiments of propriety in ministering to the natural needs
of their bodies? If the benefits of nurture in these lesser occasions
are so welcome, why make light of them in their relation to the more
serious affairs of life? Such inconsistency is surely inconsistent also
with the scientific spirit. In point of fact, tradition, so far from
being a negligible factor in shaping the world of men, is one of the
most powerful influences for evil, as for good, known to history.
The tradition most easily within control is family tradition. In it
we have a powerful lever competent to stir great masses into motion
slow, perhaps, but none the less effective.
Another reflection which is submitted to the attention of those who
make light of the potency of nurture is this: Suppose we took this
attitude of laisser faire in the realm, of science. Suppose we said :
"What is the use of teaching the young generation the accumulated
wisdom of scientific lore at bottom they will still remain essentially
savages." Can we picture to ourselves with any degree of satisfaction
the inevitable effect of such a policy upon the advance of science?
The fact is, there is a fundamental error in this emphasis upon the
alleged futility of nurture. It is based upon a disregard of the
organic unity of the social body. It narrowly views the evolution
of man as that of an individual "contained in his skin." The evolution
of our race today is something very different from this, as we
have had ample opportunity to observe in preceding chapters. It
is man plus his "artificial" aids to his life activities that evolves as
one unit. These artificial aids most assuredly include traditions of
all sorts, whether handed down by word of mouth, deposited in the
archives of learned societies, or perpetuated in any other way. And
there is no reason that can claim even a show of validity why we
should attempt to draw a line of division, and say: "Here, in the
province of science, industry and so forth, tradition is one of the
processes essentially forming part of the evolution of man; but here,
in ethics, tradition merely lends him a superficial veneer at bottom
he remains a savage." Our varied institutions of industry, commerce,
law, etc., are no doubt subject to change and even to occa-
430 ELEMENTS OF PHYSICAL BIOLOGY
sional upheaval. Yet they have a considerable degree of stability,
quite comparable with that of the somatic substance of our race; it
also is neither unalterable nor wholly immune from danger of extinction
in some world-wide cataclysm.
We have every reason, then, not only to anticipate, but to encourage
a development of man's emotional and ethical being alongside
with and in the light of the advances made in the intellectual
sphere. And indications are not lacking that in the emotional sphere,
as in the intellectual, an orthogenetic bias is ready to anticipate
selective evolution. The trend of selection in the realm of emotions,
of instinctive proclivities, of tastes we have already''noted: it follows
Spencer's hedonistic principle, according to which those races are
best adapted for survival, in whom adjustment of agreeable
feelings to beneficial action is most perfect. And the principle
admits of extension. Not only is it advantageous that we should
desire those things that profit us and our species, but it is evidently
equally essential that we should not set our hearts upon things
impossible of attainment. The man who is forever crying for the
moon is not well adapted for existence in this practical world. Thus
the things we purpose, we who have stood the test of survival, must,
on the whole, or at least in reasonable net balance, be things that
corne to pass. Evidently the statement can be turned about: the
things that come to pass are, in many instances at any rate, things
we purpose; what is more important, they are the general type of
things that we purpose. They have, accordingly, to us a characteristic
appearance of purposefulness. And the human mind, contemplating
the spectacle of the world's events, is impressed with this
appearance of purposefulness, and finds itself constrained, by an
inborn bias, by an instinctive intuition, to construe this appearance
as the outcome of design.
The fact seems to be that the operation of a fundamental purpose
or design in Nature is one of those things that can neither be proved
nor disproved. We are, therefore, at liberty, if we so choose to,
believe in such a purpose. This is an occasion for the legitimate
exercise of faith.
We may, if we will, embrace this purpose for our own. Such will
spells ultimate survival. No better guarantee for the welfare of the
race could be furnished, than its essential harmony with Nature.
Selection, then, would seem to point the way toward a will in con-
CONCLUSION 431
formity with, that general principle which, for want of a better term,
we may describe as the Supreme Purpose of the Universe.
But selection alone does not determine the path of evolution. Just
as the purchaser in a store is dependent upon the bias of the storekeeper
who lays in a stock of assorted goods, so evolution must humor
any bias there may be in the variety of types presented for selection.
Are men's wills changing? Is there a drift in the general average?
If so, whither does it tend? Toward a merging with the Supreme
Purpose, or away from it? Have we any instrument competent to
discover as much as a hint of an answer to these questions?
Perhaps we have. It is true that the recent world cataclysm has
reminded us all too clearly of the potency of knowledge to destroy;
of the danger that our orthogenesis be of the fatal type; that man,,
having grown too clever, may perish by the very perfection of his own
weapons. But if the proverbial cat has nine lives, the human race
at present has some seventeen hundred million, and the presumption
is that even the most disastrous conflict would leave some remnant
to carry on. Meanwhile there are not lacking signs on which the
optimist may hang his hopes of a happier issue of our orthogenesis.
He will point out that since our evolution has been upward in thepast,
it is reasonable to expect it to continue so in the future. He will
point to the rude ancestors, more beast than man, from whom we have
ascended. Then, he will say, if you wish to form a conception of tha
future of our race, consider the foremost, the most enlightened spirits
of today, and reflect that these will represent the average of a clay
that is coming. These men, looking out upon the world, are impressed
above all with the essential unity of Nature, and of man with
her. For "man is part of Nature, the part that studies the whole."
Therefore, as man's eyes are opened by modern science to view the
intricacies of atomic architecture, or to fathom the secret places of
the stars, what is it but the Universe awakening, like some great
giant, to a consciousness of his remoter members?9
To the cold
intellectual realization of this unity of man with the Universe there
is an emotional counterpart. A full and intensely vivid realization
9
This -was sensed by Hegel, who held that "the course of history is the process,
not simply by which man comes to a consciousness of God, but that by
which God comes to a consciousness of himself." (A. K. Rogers, A Student's
History of Philosophy, 1921, p. 448). Compare also H. G. Wells, God the
Invisible King; and M. Lembat, Science Progress, 1923, p. 112.
432 ELEMENTS OF PHYSICAL BIOLOGY
of the Inspiring truth is undoubtedly accompanied by feeling of
high exaltation. To this we have, among many others, the testimony
of Wordsworth
I felt the sentiment of Being spread
O'er all that moves and all that seemeth still;
O'er all that, lost beyond the reach of thought
And human knowledge; to the human eye
Invisible, yet liveth to the heart.
. . . . Wonder not
If high, the transport, great the joy I felt
Communing of this sort through earth and heaven
With every sort of creature.
It seems inevitable that a lively sense of this merging of the Self with
the Universe shall hold as one of its constituents a fusion of personal
desires with the Supreme Purpose of the Universe. "No man,"
says Emerson "has a right perception of a truth, who has not been
reacted on by it, so as to be ready to be its martyr."
The relation of knowledge and of ignorance to will is discussed,
from a somewhat different angle, by Bertrand Russell in his book
Our Knowledge of the External World (1914, p. 234). He remarks:
The apparent indeterminateness of the future .... is merely a
result of our ignorance. It is plain that no desirable kind of free will can be
dependent simply upon ignorance Let us therefore imagine a set
of beings who know the whole future with absolute certainty, and let us ask
ourselves whether they could have anything that we should call free will.
. . . . The beings we are imagining would easily come to know the causal
connections of volitions and therefore their volitions would be better calculated
to satisfy their desires than ours are.
If the extension of Spencer's hedonistic principle, as sketched
above, applies, and if through orthogenetic bias or otherwise the
human race shall provide the requisite material for selection to
operate upon, then these beings which Bertrand Russell "imagines"
for the sake of his argument, will become something of an actuality
as our evolution culminates. To them the words of Emerson will
appry, that they will be "made of the same stuff of which events are
made The mind that is parallel with the laws of nature
will be in the current of events, and strong with their strength."
Their attitude will be biologically sound, for, as Claude Bernard has
said:
CONCLUSION 433
It is not by struggling against cosmic conditions that the organism develops
and maintains its place ;
on the contrary, it is by an adaptation to, and agreement
with, these conditions. So, the living being does not form an exception
to the great natural harmony which makes things adapt themselves to one
another: it breaks no concord; it is neither in contradiction to nor struggling
against general cosmic forces ;
far from that, it forms a member of the universal
concert of things, and the life of the animal, for example, is only a fragment of
the total life of the universe.
Or, to quote a spokesman of our own generation, the closing words
of Sir Charles Sherrington's Presidential Address still ring in our
ears :
One privilege open to the human intellect is to attempt to comprehend
.... the how of the living creature as a whole. In the biological synthesis
of the individual this problem is concerned with mind. It includes examination
of man himself as acting under a biological trend and process which is
combining individuals into a multi-individual organization, a social organism
surely new to the world. Man, viewing this great supra-individual process,
can shape his course conformably with it even as an individual, feeling that
. . . . to rebel would be to sink lower rather to continue his own evolution
upward.
Thus, in the light of modern knowledge, man is beginning to discern
more clearly what wise men of all ages have intuitively felt his essential
unity with the Universe; and the unity of his puny efforts with the
great trend of all Nature. A race with desires all opposed to Nature
could not long endure; he that survives must, for that very fact, be
in some measure a collaborator with Nature. With extending
knowledge must come awakening consciousness of active partnership
with the Cosmos "When souls reach a certain clearness of perception,
they accept a knowledge above selfishness;" and "he that
sees through the design must will that which must be." This is no
mere resignation of a man to his fate, though the saying of Anatole
France be true "Les grandes arnes se resignent avec une sainte joie."
Not even joyful resignation is adequate; the state of the fully
awakened consciousness is better described by the great physicist
Clerk Maxwell, as "an abandoment of wilfulness without extinction
of will, but rather by a great development of will whereby, instead
of being consciously free and really in subjection to an unknown law,
it becomes consciously acting by law, and really free from interference
of unrecognized laws."
434 ELEMENTS OF PHYSICAL BIOLOGY
Such. Is the outlook to which the development of modern Science
seems inevitably to be leading the thoughts of men. This is the goal
of evolution, the perfect adjustment of feelings to actions, which
guarantees survival: To say with the great Stoic "0 Universe,
whatsoever is in harmony with thee, is in harmony with me." The
being whose will is so adjusted is Fortune's favorite; all things must
bend to his will as they bend to Nature's law. For his will is
Nature's law.
SYNOPTIC CHART
OF PHYSICAL BIOLOGY AND ALLIED DISCIPLINES AS TREATED IN
THIS WORK
"Ce mage divisa en plusieurs parties ce qui n'avait pas besom d'etre divise";
il prouva m^thodiquement tout ce qui e"tait *clair; il enseigna tout ce qu'on
savait; il se passionna froidement, et sortit suant et hors d'haleine. Toute
1'assemble'e alors se r6veilla et crut avoir assist^ a une instruction."
Voltaire.
435
SYNOPTIC CHARTS
Guide to References. The numerals prefixed to the several items on
Charts I, II. Ill, IV refer to the following list, which indicates some of the
principal references to such items in this book. In a few instances of points
not otherwise covered in this book supplementary references to the literature
are given.
436
SYNOPTIC CHAETS 437
Am. Jl. Hygiene vol. 3 Jan
Suppl. 1923 p. 25.
See items 64 to 76.
64 et seq.
152 et seq.
64 et seq.
77 et seq.
79, 82; Am. Jl. Hygiene
loc. cit.
83 et seq., 88 et seq.
94.
266, 296, 297; see also items
72 to 76.
61, 146 to 149.
Not treated in this work.
63.
63.
Not treated in this work.
CHART III
59, 145.
59.
146, 148, 149.
See items 81 to 125.
See items 82 and 84.
See item 83.
152 et seq.; 155, 157.
144.
280 et seq.
259 et seq.
276 et seq.
161 et seq.
163.
95, 163.
See items 92 to 96.
See items 93, 94.
164, 171 et seq., 175, 177
Not treated in this work.
166 et seq.
See items 97 to 122.
277; see also items 98, 99.
173 et seq.; 181.
181.
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
12Q
130
131
132
133
134
135
136
Not specifically treated in
this work. For incidental
references see items 105,
106, 107, 117 to 119.
See items 102 to 119.
209, 215, 333.
218, 226, 334.
225; see also item 103
229 to 232, 236.
246, 247.
252 et seq. ; 254.
256.
See items 110 to 119.
188.
225 to 228.
See item 111.
232 et seq.
248 et seq.; 250.
255.
Not treated in this work.
232 to 255.
234, 248.
239 to 242.
Not treated in this work.
See item 122.
180, 238, 279; see also item
119.
280 et seq.
300 et seq.
321.
CHAKT IV
331 et seq.
See items 128 and 129.
345 et seq., 358.
Not treated in this work.
See items 131, 132.
336 to 345.
Not treated in this work.
See items 134 et seq.
328 to 330, 336.
336 et seq.
See item 137 et seq.
438 ELEMENTS OP PHYSICAL BIOLOGY
137
138
139
140
141
142
143
144
145
146
147
148
See Item 138 et seq.
317,
300, 311.
363, 370.
45, 46, 122, 345, 348.
Jl. Washington Acad. Sci.
vol. 2 1912 p. 69.
17, 336 et seq.; 362 et seq.;
410 et seq.
339, 371 et seq.; 410 et seq.
339, 347 et teq. ;
363 et seq.
371 et seq.; 410 et seq.
Indirectly referred to in the
substance of pages 383,
394, 395, 398, 402, 407.
For the general energetics
of nutrition and related
149
150
151
152
153
154
155
156
to
163
topics see special works on
the subject, as, for example,
Lefevre, La
Chaleur Animale et Bioenerge'tique,
1911.
410 (Table); see also item
ISO.
340, 347, 366, 411.
See items 153, 154.
339, 346, 381, 412, 414, 415,
472.
350.
See item 155.
350, 353, 416.
See Table 34, page 412, and
text relating thereto.
ADDENDA
The following references and notes, which have been omitted
from the text through inadvertence or because they came to the
author's notice too late, are here appended as a supplement. As
stated in the Preface, no attempt at completeness of bibliography
has been made. Perhaps, however, the material presented in these
pages may be found, by anyone sufficiently interested, to furnish a
serviceable base of departure for the gathering of a more exhaustive
collation.
Chapter I, page 9, footnote 10. See also L. A. Herrera, Royal Academy of
the Lincei, June 15, 1924. Imitation of nervous and cellular tissue, by means
of potassium hydroxide, silica and alcohol.
Chapter II, page 36. Compare also A. J. Lotka, Irreversibility, Cosmic and
Microcosmic, Jour. Washington Acad. Sci., August, 1924, pp. 352-3. It should
perhaps be pointed out that the dissipative processes of nature, as discussed
in this chapter, include, of course, both processes irreversible with regard to
macroscopic operations, and also processes irreversible in the usual sense,
that is, thermodynamically irreversible.
Chapter II, page 44. Intra-group Evolution. If the composition or constitution
of a group or species of organisms is described in terms of a frequency
function, intra-group evolution is exhibited by the fact that the characteristic
coefficients in such a function are functions of the time. This point has been
referred to by Arne Fisher in Amer. Math. Monthly. March-April issue,
1923, p. 97. Fisher thus writes the normal frequency function in the form
(t)
Chapter V, page 49 Compare also G. Bohn, La Forme et le Mouvement,
Flammarion, 1921, p. 161.
Chapter VII, Growth of Individual Organism, page 71. In this connection
reference may also be made to the work of Lecomte de Nouy, of the Rockefeller
Institute for Medical Research, on the rate of cicatrization of wounds.
Page 72, footnote 9, add the following reference: H. C. Bastian, Th.3 Nature
and Origin of Living Matter, p. 47.
Chapter VIII. For further discussion of the mathematical analysis of
epidemiology and related topics see alsoR. Ross, Proc. Royal Soc., Ser. A,
vol. 92, 1916, p. 204; vol. 93, 1917, pp. 212, 225, also; J. Brownlee, Proc. Royal
Medical Soc., May, 1918, p. 115.
439
440 ADDENDA
Chapter XV, Composition of Terrestrial Matter, page 194. See also V. M.
Goldschrnidt, Geochemische Verteilungsgesetze der Elemente, Vitenskapsseiskapets
Skrifter I Mat.- Naturv. Klasse, 1923, no. 3, 1924, nos. 4 and 5; H. S.
Washington, Jour. Washington Acad. Sci., 1924, p. 435; L. H. Adams, ibid.,
p. 459; J. H. Jeans, Nature, vol. 114, 1924, p. 828; H. Jeffreys, The Earth,
Cambridge University Press, 1924.
Chapter XV, page 203. Reference should have been made to Macallum's
paper in the Trans. Roy. Soc. Canada, 1908, ii, p. 145. The marine origin of
terrestrial animals was taught in remote antiquity, by Anaximander (610-
540 B.C.)
Chapter XXII, page 298. The role of the hands in the evolution of man's
intelligence seems to have been clearly recognized by the Greek philosopher
Anaxagoras (500-428 B.C.) "He explained man's intelligence by the power
of manipulation that came when the forelimbs were freed from the tasks of
locomotion" (W. Durant).
Chapter XXIII, page 300. Compare also 0. Matousek, Geological Laws of
Population, Amer. Jour. Physical Anthropology, 1924, vol. 7, p. 389.
Chapter XXV, page 342. For further details regarding automatic telphone
exchanges see also F. A. Ellson, Automatic Telephones (Pitman's Technical
Primers, London, 1924); W. Aitken, Automatic Telephone Systems (Ernst
Benn, London, 1924)
Page 354. The influence of diet upon the mode of life was remarked by
Aristotle (384-322 B.C.): "Of beasts some are gregarious, and others are
solitary they live in the way which is best adapted to obtain the food of their
choice." Very clearly the characteristics of animals and plants are referred
to their respective food sources by La Mettrie in his little book L'Homme
Plante (1748). He remarks very pointedly: "Les etres sans besoins sont
aussisans esprit." (See .-.].-.;, ]v!:
jo-: 211) M -id 301 in Chapters XVII and XXVI.)
Chapter XXVI, pages >';.".. <-.- .!.,
:
5:<vir :; l
data on the introduction of
eyeglasses see, for example, R. Greef, Die Erfindung der Augenglaser, (A
Ehrlich, Berlin, 1921).
Page 366. Regarding the modern industrial development, with the use of
machinery and, incidentally, with reference to the phonograph and radio
concerts of today, the following words of Aristotle have acquired a strangely
prophetic significance: "If every instrument should accomplish its own work,
obeying or anticipating the will of others, . . . . if the shuttle should
weave, or the plectrum touch the lyre, without a hand to guide them, then
chief workmen would not need assistants, nor masters slaves."
Chapter XXVII, page 374. Regarding the delimitation of the self, and
related matters see also E. Mach, Analyse der Empfindungen; K. Pearson,
Grammar of Science, 1900, p. 63; B. Bavink, Ergebnisse und Probleme der
Naturwissenschaften, pp. 313, 320 et seq.
Page 375. Add to footnote 7: Compare also the saying of Lamartine:
"It is not I who think, but my ideas which think for me." And a modern
writer has remarked that "it is not we who think, thinking is rather something
that happens to us." Accordingly it would be grammatically better advised
to employ an impersonal form of verb, and to say, instead of I think, rather
ADDENDA 441
it thinks, just as we say it thunders. Language is not altogether devoid of
evidence of a vague realization of this. Especially in describing dreams and
visions, with their characteristic submergence of the ego, we use such terms
as methought, es dunkte mich, etc.
Chapter XXX, page 395. Poets are good psychologists, in part, no doubt,
because by nature introspective; in part, also, because keenly interested in
human nature. The relation of emotions and of reason to action is well stated
by Pope
"On life's vast ocean diversely we sail,
Reason the card, but passion is the gale."
And La Fontaine has it
"O douce volupte sans qui des notre enfance
Le vivre et le mourir nous deviendraient egaux
"
General references. The following references, some of them guides to
bibliographic sources, may be regarded essentially as an extension of the
footnote 8 on page 366.
Chapter XXVI. M. Schips, Mathematik und Biologie, Teubner, 1922.
Contains a bibliography.
H. Przibram, Anwendung Elementarer Mathematik auf Biologische Probleme,
Engelmann, Leipzig, 1908. Contains a bibliography.
G. Bohn, La forme et le mouvement, essai de dynamique de la vie. Flammarion,
Paris, 1921. Contains a bibliography.
O. Fischer, Physiologische Mechanik, in Encyklopaedie der Mathematischen
Wissenschaften, vol. IV, I, II, Heft I, pp. 62-126. The index on the
literature of the subject contains a list of 21 textbooks and 421 monographs.
OF
Abbott, C. G., 331.
Adams, L. H., 194.
Adams, F. D., 274.
Aitken, W., 440.
Allen, E. J., 173.
Amar, J., 347, 366.
Anaxagoras, 440.
Anaximander, 440.
Andrews, M. R. S., 397.
Aristotle, 362, 440.
Arrhenius, S., 224, 227, 228, 234, 243,
331.
Aston, F. W., 156, 270, 391.
Atkins, W. R. G., 223.
Auerbach, F., 306.
Aurelius, Marcus, 434.
B
Bachelier, L., 360.
Bacon, R., 3, 365.
Ball, W. W. R., 342.
Baly, E. C., 155, 157, 221.
Bancroft, W. D., 283.
Barrell, J., 272.
Bateson, W., 311.
Baudisch, 0., 221.
Baum, F. G., 333.
Bavink, B., 440.
Bawden, H. Heath, 366.
Bayliss, W., 14, 19.
Bell, C., 366.
Benedicks, G., 40, 285.
Bennett, A., 398.
Bernard, G., 8, 17, 18, 203, 432.
Bilski, F., 307.
Biot, J. B., 41.
Birkeland and Eyde, 239.
Blackwelder, E., 245, 249.
Blakeman, J., 344.
Blondel, 61.
Boas, 72.
Bohn, G., 307, 439.
Bohr, N., 378, 406.
Boltzmann, L., 355.
Bolyai, 18.
Boole, 360.
Bortkiewitch, 116.
Bose, J. G., 221.
Boucke, 0. F., 385.
Bourrat, 403.
Boussinesq., 407.
Braham, J. M., 243.
Bragg, 270, 365.
Bridgeman, P. W., 274.
Briggs, L. J., 7, 212, 332.
Braun (and Le Chatelier), 283.
Brown, E. W., 373.
Brownlee, J., 81, 115, 307, 344, 439.
Bucher, 239.
Buckingham, E., 29.
Buckle, P., 83.
Budde, 304, 356.
Bunge, G., 13, 201, 203, 204, 228, 257.
Burlingame, L. L., 366.
Burns, D., 9.
Burns and Paton, 355.
G
Campbell, L., 407.
Campbell, N., 385.
Campbell, W. W., 331.
Carlyle, 398.
Carver, T. N., 386.
Chamberlin, 272.
Chatelier, Le, 281, 305.
Chodat, 72.
Christiansen, J. A., 157.
Chwolson, 26, 283.
443
444 INDEX OF NAMES
Ciamician, G., 331, 333.
Clarke, F. W., 185, 191, 192, 197,
220, 223, 224, 226, 227, 235, 250,
255, 258, 274.
Clerk, Maxwell, 35, 379, 407, 433.
Clifford, 403.
Cole, W. H., 344.
Conklin, E. G., 11.
Cook, E. H., 223.
Cooke, M. T., 166.
Copernicus, 379.
Cournot, A., 57, 409.
Curtis, W. C., 372.
D
Dantec, Le, 285.
Darmstaedter, L., 345, 365.
Darwin, C., 165.
Darwin, C. G., 193, 406.
Dastre, 8.
Daudefc, A., 388, 392.
Davidson, L, 304, 356.
Debierne, 264.
Dodd, L. W., 21.
Doelter, C., 384.
Doering, C. R., 108.
Donaldson, H. H., 72, 73, 132.
Dreiser, T., 392.
Driesch, 11.
Drzwina, A., 307.
Duhem, P., 160, 282, 320.
Durant, W., 440.
Dyk, v., 147.
Dyson, F., 272.
E
East, E. M., 7, 66.
Eddington, A. 8., 272, 342.
Edgeworth, F. Y., 344.
Ehrenfest, 282, 287, 300.
Eijkman, C., 115.
Einstein, A., 18, 337, 376, 379.
Elliot, G., 298, 377.
Ellson, F. A., 440.
Emerson, R. W., 21, 364, 400, 426, 432.
Emmetfc. W. L. R., 327.
Empedocles, 185, 246.
Encyclopedia Britanniea, 22.
Engler, 227.
Errera, L., 72.
Euclid, 376.
Euler, L., 112.
Fajans, 270.
Faraday, M., 379.
Farr, G. H., 354.
Farr, W., 115, 307.
Fechner, 6, 403.
Feldmann, W. M., 366.
Fevre, J. Le, 347.
Fiehte, 397.
Fischer, 0., 441.
Fisher, A., 439.
Fisher, H. E., 239.
Fisher, I., 386, 400.
Fisher, O., 366.
Forbes, A., 49.
France, A., 410.
Franck and Caro, 239.
Franklin, W. S., 39.
Fredericq, L., 203, 284.
G
Gadow, H., 165.
Galton, F., 345.
Garnett, W., 407.
Gauss, 376.
Gilbert, C. G., 333, 366.
Glaser, O. C., 16.
Glaser, R. W., 6.
Glover, 103, 104, 105.
Goethe, 4.
Goldschmidt, V. M., 440.
Gould, G. M., 12.
Goursat, E., 159, 280.
Greef, R.;
440.
Grinnell, Jones, 229, 237, 238, 239,
241.
Guilleminot, 357.
Gumbel, E. T., 112.
INDEX OF NAMES 445
H
Haber, 234, 239.
Haecker, L., 136.
Haigh, L. D., 133.
Haldane, J. B. S., 52, 122 et seq.,
170, 297.
Hammick, 157.
Hardy, G. H,, 124, 170.
Hardy, W. B., 344.
Harkins, W. D., 270, 271.
Haydn, 425.
Heath, Bawden H., 366.
Hegel, 431.
Heilbron, I. M., 221.
Helm, G., 303, 304, 356.
Helmholtz, 300.
Henderson, L. J., 17, 203, 211, 217,
333.
Henson, 173.
Heracleitus, 209.
Herdman, W. A., 171, 181.
Herrera, 439.
Herrick, G. J., 7.
Herschel, J., 331.
Hertz, 379.
Herzen, 391, 392.
Heymans, 403.
Hill, J. J., 251.
Hiscock, I. V., 295.
Hise, van, 251.
Hogbom, A. G., 223, 224.
Hogg, J., 307.
Holland, B. H., 72, 74, 75.
Hopkins, C. J., 257, 258.
Holsberg, C. L., 332.
Holt, E. B., 373.
Hooke, E., 364.
Howard, L. O., 90, 91.
Huber, B., 231.
Hume, D., 364.
Humphreys, W. J., 186, 192.
Hunt, S., 224.
Hurwitz, 61.
Hutchison, B., 347.
Huxley, T. H., 342.
James, (St.), 218.
Janet, 392.
Janssen, 365.
Jeans, J. H., 189, 190.
Jennings, H. S., 428.
Jevons, F. B., 43.
Johnson, 282, 377.
Johnstone, J., 181, 357.
Joly, J., 275.
Jones, J. E., 190.
K
Kalmbach, E. E., 170.
Kant, 13, 346.
Keehle, F., 354.
Keith, A., 366.
Kelvin, 227, 274.
Kepler, 424.
Kesava, Pai M., 69.
Keyser, C. J., 336, 374, 414, 426.
Kirk, 211.
Koehne, C. J., 227.
Korzibski, 379.
Laar, Van, 320.
La Fontaine, 441.
Laird, J., 375.
Lamartine, 440.
Lambert, W. D,, 274.
Lane, A. C., 245, 365.
Langmuir, I., 156, 157.
Lascaux, E., 245.
Le Chatelier, 52.
Le Fevre, J., 347.
Lehmann, 364.
Lembat, M., 431.
Lemberg, J., 227.
Liebig, J., 97, 179, 223, 236, 246.
Liebmann, H., 147, 151.
Linck, 191, 220, 227, 235.
Lindgren, W., 245, 249.
Lipman, C. B., 232.
446 INDEX OF NAMES
Lobatchewski, 78.
Loeb, J., 9, 10.
Lohmann, 173,
Lorentz, 379.
Lotka, A. J., 31, 33, 34, 39, 48, 60,
62, 63, SO, 81, 82, 90, 91, 92, 110,
115, 149, 150, 151, 154, 155, 245,
268, 267, 285, 288, 325, 354, 356,
357^ 3S03 398, 404, 409, 439.
Lowell, 398.
Louy, J., 283.
Lowry, 157.
M
Maeallum, 440.
MaeAuley, 282, 377, 397.
MeOiendon, 9.
McGee, W. J., 212, 216.
McKendrick, A. G., 69.
McLennan, J, G., 187.
Mach. E., 259, 280, 381, 382, 383, 402,
406, 426, 440.
Marcus Aurelius, 434.
Martin, G. W., 172, 173, 181, 229, 318.
Martini, 79.
Mathews, J. H., 221.
Matousek, 0., 440.
Maxwell, Clerk, 35, 379, 407, 414,
433.
Mayer, J. H., 325.
Mayreder, R., 397.
Meilor, J., 157, 281.
Mettrie, La., 440.
Miehaud, F., 160, 281.
Michelson, 366.
Milne, E. A., 190.
Minot, C. S., 132, 299.
Mitchell, 175, 266.
Monnier, A., 72.
Moore, 223.
Moore, H. T., 399.
Moritz, 275.
Morley, J., 184.
Moulton, 134, 272.
Murphy, R. C., 250.
Murray, J., 213.
Murray, W. S., 333.
N
National Research Council, 136.
Nernst, W., 8, 282, 355.
Nero, 365.
Newton, I., 379.
Nichols, J. T., 108, 297.
Norton, T. H., 229.
Nouy, L. de, 439.
Noyes, B., 108.
Osborn, H. F., 17, 355.
Ostwald, W., 72, 76, 235, 240, 274,
303, 304, 356.
Owen, S. P., 157.
Pai, M. K, 69.
Palitzsch, S., 17, 203.
Pareto, W., 385.
Parker, G. H., 12, 298.
Parker, S. L., 69, 108, 164, 307, 308,
319.
Parsons, C., 273.
Parsons and Petit, 237.
Partington, J. E., 243.
Paton, D. N., 9, 355.
Pearce, A. S., 181.
Pearl, E., 66, 68, 69, 108, 109, 162,
278, 279, 307, 308, 319, 329, 368.
Pearson, K., 25, 124, 300, 317, 344,
440.
Pennock, J. D., 233, 236.
Perrin, J., 21, 26, 157.
Pfeffer, 226.
Phipson, T. L., 227.
Picard, E., 59, 147, 280.
Petersen, 175, 176, 181.
Petronievics, 21.
Petzoldt, 157.
Planck, M., 303, 320, 379.
Plato, 64.
Pogue, J. E., 333, 366.
Poincare", H., 28, 30, 92, 424.
Poinsot, 19.
INDEX OF NAMES 447
Pope, 441.
Porstmann, 49.
Poynting, J. H., 193.
Prince, M., 403.
Przibram, 337, 441.
Ptolemy, 379.
Punnett, E. C., 170.
Putter, 174.
Pyle, W. L., 12.
Q
Queredo, L. J., 337, 342.
R
Reeds H. S., 74.
Reed, L. J., 66, 68, 72, 74, 75.
Renan, 425.
Rew, R. H., 162.
Richter, K., 345.
Riemann, 18.
Robertson, G. S., 251.
Robertson, T. B., 72, 73.
Rodebush, W. H., 157.Rogers,
A. K., 331.
Ross, R., 48, 81, 84, 147, 149, 150, 344,
439.
Royce, J., 26, 35.
Russell, B., 4, 20, 22, 375, 395, 419,
420, 422, 432.
Russell, J. C., 245.
Rutherford, E., 268, 269.
Rydberg, J. R., 40.
S
Sachs, S., 6.
Sala, G. A., 77.
Sehaefer, C., 337.
Schafer, E., 8.
Schips, M., 441.
'
Sehonbein, 156, 235, 391.
Schroeder, 331.
Schuchert, C., 224, 225, 269, 273.
Seligmann, R. A., 219.
Sempelowski, 249.
Semper, K., 307.
Seneca, 414.
Shakespeare, W., 128, 427.
Sharpe, F. R., 110.
Sharp, D., 165.
Shaw, N., 333.
Sherrington, C. S., 375, 405, 433.
Simpson, 377, 378.
Slator, A., 69.
Smith, A., 385.
Smith, S., 8.
Smoluchowski, 337.
Smyth, C. H., 228.
Soina, A. de, 365.
Solvay, 239.
Spencer, H., 7, 61, 283, 331, 354, 364,
386.
Spoehr, H. A,., 332, 333.
Steinmetz, C., 333.
Stevenson, J., 227.
Streightoff, F. H., 347.
Strong, 403.
Strutt, 274.
Sturtevant, A. H., 297.
Sullivan, J. W. N., 20.
Sumner, F. B., 374.
Swann, M., 376.
T
Taussig, F. W., 396, 398.
Taylor, J. T., 232.
Tead, 0., 400.
Thacker, G., 312.
Tiffany, L. H., 171.
Thomson, J. A., 132, 165.
Thomson, J. J., 156, 379, 391.
Thompson, D'Arcy W., 17, 203.
Thompson, W. F., 173,
Thompson, W. R., 83, 85, 86,
91, 118.
Thornton, H. G., 69, 70, 71, 164.
Todd, G. W., 157.
Tolman, R. C., 156, 403.
Trowbridge, R. C., 133.
U
Uexktill, 16.
INDEX OP NAMES
Van Orsfcrand, 274,
Veblen, T,, 395, 415.
Vega, 236.
Vegard, 187,
Verhulst, 66, 368.
Vernadsky, 201.
Verne, J., 189.
Vinci, Leonardo da, 252, 255.
Voltaire, 435.
W
Waals, van der, 287, 294.
Warren, H. C., 11, 383, 394; 395, 398,
402.
Washburn, M. F., 396.
Washington, H. S., 197, 205, 206, 207.
Watson, G. I\
T
.,
81.
Wegener, 186.
Wells, H. G., 4, 431.
Wells, F. L., 342, 371, 395, 401.
Whitehead, A, N., 4, 100, 382.
Whitney, M., 204, 255.
Wiener, 364.
Wilson, C. T. K., 365.
Williamson, E. D., 194.
Willis, C. J., 311 et seq.
Willis, J. C., 307.
Winiarski, 304, 356.
Winkelmann, A., 320.
Witmer, L., 428.
Wohler, 8.
Woodruff, L. L., 218.
Wordsworth, 5, 184, 209, 371, 425,
426, 432.
Yost, 223.
Young, 397.
Yule, G. V., 312,
Zsigmondy, 365
OF SUBJECTS
A
Abbildung, 363.
Accelerations vanishing with velocities,
29.
Accessibility of raw materials, 206-
207.
Accumulators, 328.
Adaptation purely relative, 63.
Adjusters, 339, 381 ;
evolution of, 346;
location of, 381; chart, 412; physiological
or anatomical, 414; external,
414; economic, 415; internal,
472.
Age and Area theory, 311.
Age distribution, influence on growth,
109; "normal", 110, 114; "stable",
110, 112, 113, 114.
Agriculture, evolution of, 180.
Aggregates, growth, 100; of constant
units, 129; of variable units, 130.
Aim, accuracy and versatility of, 338.
Aimed collisions, 337, 409.
Alfalfa weevil, 166.
Anabions, 329.
Animal and plant, distinction, 4-5.
Animals characteristically active,
mobile, 336; as catabions, 329.
Annihilation, see extermination, also,
extinction of species.
Approximate expression for equilibrium
proportions, 267-268.
Aptitudes, influence of special, upon
motivation, 399.
Aquatic, life, interspecies equilibrium,
171; aquatic atmosphere, 193.
Aquiculture, 173-181.
Area and rent, 288-305; area as
topographic parameter, 301; area
and age, see age.
Artificial receptors, 364, 440, correlating
apparatus, 410, 440.
Assumptions, implicit, 376-377.
Asymmetry of time, 30 et seq., 37.
Atmosphere, 185; losses from, 188;
accessions to, 192; aquatic, 193.
Autistic thinking, 410.
Autocatalysis, 72.
Automatic telephone exchange, 342,
440.
Automaton type of behavior
schedule, 350.
B
Bacteria, growth of, 69.
Bargain, mutually selfish, 415.
Beef production, 132 et seq; efficiency,
136.
Behavior schedule, 346; rigid or
automaton type, 350; elastic type,
350; effect of change in, 350; regulation
according to maximum principle,
351; of ideal organism, relation
to that of actual, 352; effect of small
departure from perfect adjustment,
353; gainful occupations, 416.
Behavior types, 7.
Benign circles, 298, 440.
Biassed movement, exemplified by
chess, 343; biassed evolution, 380.
Biochemistry, 317.
Biological basis of economics, 163,
164; surveys, 164.
Biophysics, program of, 49.
Birthrate and deathrate, relation
between, 115; in "normal" population,
115; adjustment to optimum,
128; economical and lavish, 131, 132
et seq., 135.
449
450 INDEX OF SUBJECTS
Births, ratio of total, in successive
generations, 87.
Blood pressure and size of organism,
295, 297.
Blood serum, composition, compared
with sea water, 201, 202.
Body limits not clearly drawn, 374.
Body, relation to ego, 374.
Body fluids, 17.
Body politic, essential unity of, 369.
Body politic, 412.
Boreholes, deep, 273.
"Bottlenecks" in cycles of nature,
292.
Brownian movement, 337.
Bulk mechanics, 50.
C
Capacity factors of energy, 282-303.
Capture, frequency of, 359.
Carbon dioxide, absorption in
weathering of rocks, 223-224.
Carbon (dioxide) cycle, 218-226;
energetics of, 334.
Carbon dioxide, origin of atmospheric,
227; solubility at different
temperatures, 318.
Carp, productivity, 173.
Cartilage, sodium chloride content,
in relation to marine origin of
terrestrial species, 204.
Catabions, 329.
Catastrophic theory of earth's crust
(Joly), 275.
Cause and effect intermingled, 298.
Cause, effective and final (Mach),
381.
Census of flora and fauna, 165-166; of
aquatic species, 173.
Centrifuge, use of, in making biological
survery, 173.
Changeful Sequence, 21.
Characteristic Equation, 60.
Chatelier, Le (principle of), 281, 293,
305.
Chemical reaction as an example of
evolution, 152; theory of, 155, 156.
Chemical Elements. See Elements.
Chess, 342-343 et seq.
Chess player, Mechanical, 342.
Chili saltpeter, 235 et seq., 237.
Choice, instinctive and reflective,
412.
Circulation of the elements in nature,
209; tends to increase, 245.
Circulation through moving cycles,
change of, 292.
Circulation, increase of, and law of
evolution, 357-358.
Cities, law of rank in size, 306.
Classification of sciences, 419.
Climatic parameters, 317.
Clocks, radioactive chains as cosmic,
268.
Coal, exhaustion of, 222.
Coal consumption, compared with
solar energy, 332.
Codfish, varied diet of, 177.
Coefficients of transformation and
their economic significance, 137 et
seq.
Coke ovens, losses of nitrogen from,
233; by-product, replacing beehive
type, 233.
Collision frequency, 359.
Collisions, random and aimed, 337.
Collisions, energetics of, 409.
Communicators, 378, 411.
Competition of man with man, 417.
Complete differential, 320.
Completely damped systems, 28;
characterized by single-valued
growth function, 47.
Composite transformers, 327.
Concentration of natural resources
208.
Concentration, processes in nature,
245-250; economic and energetic
significance, 243; law of urban, 306.
Conclusion, 417.
Conduct, regulators of, 412.
INDEX OP SUBJECTS 451
Conjugate parameters, 287-288, 303,
320; See also parameters, topo-
graphic.
Consciousness, 388 et seq. ;
mechanistic
explanations, 388; relation to
physical conditions, 388; conditional
relations to physical conditions,
388; fundamental hypothesis
of consciousness other than my
own, 389; difficulty of attaching
definite meaning to term consciousness
as applied to lower
organisms, etc., 389; bound up
with life processes and structures,
390; not separated sharply from
unconsciousness, 390; forms or
modes, of 390; impersonal, 388
(Daudet), 391, 392; personal, 391;
""-dependent on metabolism, 391;
physico-chemical theory of, 391;
evolution of, 393; operative relations
of, 394; function of, 394; a
function of bodily state, 394; origin
of, 402 et seq. 404; physical analogy,
403; double aspect theory of, 403;
physico-chemical theory of, 404;
intervention in mechanics, 404;
possible bearing of quantum theory
on, 406; energy relations of, 406 et
seq. ;
intervention of, in mechanics,
404, 407, 408; universe awakening
to consciousness of itself, 431.
Constraints, 58, 64.
Continuity, principle of, 259.
Convenience, in selection of coordinate
reference frames, 372.
Conversion factors of energy,
economic, 355.
Coordinate systems, 372; reference
frames, 372.
Correlating apparatus, 17, 336 et seq.
338, 362 et seq.; elements of, 339;
not peculiar to living organisms,
340; evolution of, 345; perfect
adjustment, 353; review, 410; chart,
410; see also under its separate
elements, such as Receptors, Effectors,
Adjusters, Elaborators,
Communicators, Depictors, Informants,
Transformants, etc.
Correlation, between movement of
organism and the environment, 336
et seq.; negative, between movement
of organism and danger
points, 337.
Cosmic consciousness, 431.
Cosmic irreversibility, 36.
Cosmic mind, 375.
Cosmogony, 272,
Coupled transformers, 327.
Cumulative cycles simulating orthogenesis,
296.
Curves of pursuit, 360.
Cycles, benign, 298.
Cycles in nature, 183, 209, 252;
summary, 257; rate of participation
of elements in cycles of nature, 258;
law of increasing circulation, 357,
358.
Cycle of life, 334.
Cyclic working of transformers, 326.
Cyanamide process, 239.
D
Damped systems, 28.
Data, methods of gathering, 52.
Death, 185; Deathrate, see also birth-
rate.
"Death" of molecules in chemical
reaction, 157.
Definite quadratic form, condition
for, 159.
Definitions, 3; in biology, 5; of life, 7,
18, 19; quantitative, 19; of evolution,
19, 20.
Delayed development theory of reproduction,
12.
Delayed twin, 12.
Demographic functions, 101.
Demographic relations in "normal"
population, 115.
Demology, general, 164.
Density of aquatic population, 174.
INDEX OF SUBJECTS
Depiction (Abbildung) 339, 363;
realistic, 371.
Depictors, 339,
Determinism and free will, seeming
conflict, 406, 407.
Dextrose, assimilated by oyster, 175.
Diffusion of species, 311.
Dilution culture method of making
census of aquatic species, 173.
Direction of evolution, 21 et seq. 26,
37.
Direction in time, 37.
Discontinuities introduced by the
senses, 5.
Displacement of equilibrium, 280.
Displacement of cycles, change in
circulation resulting from, 292.
Dissipative effects, 22, 24.
Domestic animals kept for produce,
1S2.
Double aspect theory of consciousness,
403.
Double entry bookkeeping, in world
depiction, 373.
Drama of life, 183.
Drives (motives), 396, 399.
Drosophila, 69, 1083 307, 309, 310, 319.
Dynamic psychology, 396; as applied
to industrial and social problem,
400.
Dynamics of evolving systems, 325.
Dynamics of life-bearing systems,
121, 122.
E
Earth's crust, composition, 194, 195,
196, 439; thickness, 273; periodic
melting of (Joly's theory), 275.
Economic systems, intensity law applied
to, 303.
Economic energy (alleged) 303.
Economic significance of transformation
factorSj 137; conversion factors
"of energy, 355; value and general
utility for social service, 386.
Economics, biological basis, 163, 164,
244, 304; relation to energetics, 303,
304; system of, developed without
explicit reference to human desires,
409.
Eel grass as fundamental food of
aquatic life, 176.
Effectors, 340; evolution of, 347; artificial,
366; internal and external,
411.
Efficiency of milk and beef production,
136; of plants in utilizing
sunlight, 332.
Ego, as coordinate reference frame,
372; immaterial, 373; boundaries of,
374; need of broader conception,
378; as a particular development in
general consciousness, 404; as a
product of integration of consciousness,
405; classification of
sciences in relation to distinction
between ego and external world,
419; as ICnower and as Wilier, 421.
Egos, interpenetration of, 374, 440.
Elaborators, 339, 370.
Elements, relative abundance, 195,
204, 206, 207, 270, 271; correlation
of occurrence, 205; metallogenic
and petrogenic, 207-208; circulation
tends to increase, 245; immobile,
246; circulation in nature, rate of,
258; origin of, 269.
Emotions, reaction of knowledge
upon, 424.
Encounters, see collisions.
Energetics of aimed collisions and
purposive behavior, 409.
Energy, See also waterpower, windpower,
sun.
Energy transformers of nature, 325 et
seq. ; transformers, fundamental
characteristics, 325; sources, distributed
and localized, 336; relation
of economic value to, 354, 355;
economic conversion factors of,
355; law of evolution, 357; relations
of consciousness, 406 et seq.
INDEX OF SUBJECTS
Engine, the world as an, 331.
3pictors or transformants, 411.
Spidemiology, 77, 79, 439.
Equation, fundamental, of kinetics
of evolving systems, 57, 77; characteristic,
60; of constraint, 58, 64; of
kinetics, exceptional case, 89; case
of three dependent variables, 94; of
dynamics, possible significance of
factors eliminated therefrom, 408.
Eqiiilibria, number and character of,
59; relations between several
equilibria, 147.
Equilibrium, 51; moving 51; displacement
of 51, 143, 280, 289; conditions
for, 59; equation, 59; mode of approach,
61; equation, zero roots, 62;
roots, 63; population, stability, 67;
moving, 143; kinetic, dynamic and
energetic conception, 143; general
principles, 143; condition for stability,
144; general condition for,
145; equations, 145; character of
146; different types, 146, 148;
malaria, 147, 149, 150; model surface
147, 149, 150; metastable, 151;
chemical,, 154, 155; chemical, determined
in accordance with laws of
thermodynamics, 157; condition in
particular form, 161; interspecies,
161, 171; first approximate expression
for, 259, 267, 268; moving, 259;
second and higher approximations,
260; radioactive, 262; polygon, 266,
277; displacement of, between food
and feeding species, 289; influence
of parameters of state, 319; thermodynamic
conditions, 319; defined
in terms of minimum law, 320;
metastable and voluntary action,
408.
Equivalent increments, 349.
Esthetics, 422.
Ethics, 387; as regulator of conduct
along "realistic" lines, 412.
Ethnological bearing of advances in
agriculture, 180.
Euclidian and non-Euclidian geometry,
in the world picture, 376.
Evolution of system as a whole, 16;
definition, 19, 20, 24, 25; and history,
20; and progress, 21 ;
direction
of, 21, 22; and probability, 25; and
irreversibility, 26; second law of
thermodynamics as the law of
evolution, 26; passage to more
probable state, 34, 35; redistribution
41 et seq.; intra-group and inter-group,
44, 45; general mechanics
of, 49; macro- or bulk mechanics,
and micro mechanics, 50; dynamics
or energetics, 50; second law of
thermodynamics as the law of
physico-chemical evolution, 157;
of man's exploitation of natural
resources, 180; of elements, 269;
under changing parameters, 280;
law of maximum energy flux, 357;
industrial, 367; of the ego, 378;
biassed (orthogenesis), 380; value
of nurture and tradition for evolution,
428.
Exhaustion of coal resources, 222.
Expediency, in definitions, 3, 5.
Exponential series, 60.
Extermination of one species by
another, 87, 92.
External world, 373; as hypothetical
construct, 419-420; see also ego.
Extinct species, 266, 296, 297, see also
extermination.
Extraction of rocks and soil by rainfall,
252 et seq.
F
Farr's rule connecting birthrate,
deathrate and mean length of life,
115.
Fatigue, 351.
Feeding our foods, 180.
Fetus in fetu, 12.
Fitness a relative term, 63.
454 INDEX OF SUBJECTS
Flame, analogy to living organism,
218.
Fly population, equilibrium, 59.
Food chains, 136; 176 aquatic, 175,
180.
Food, influence on psychology, 219,
354, 440.
Food and Feeding species, displacements
of equilibrium between, 289.
Foods, primary, secondary and
tertiary, 181.
Foods, moisture contents, 211.
Foods, supply of plant foods in soil,
257.
Force of mortality, 102.
Force of mortality in chemical reactions,
153, 155.
Fossil fuel, exhaustion, 222.
Free energy, 321; relation to voluntary
action, 408.
Frequency of collision and capture,
359.
Friction, 22, 24.
Fruit fly, see Drosophila,
Fundamental equations of kinetics of
evolving systems, 57 et seq.;
general case, 57 et seq.; solution,
60; case of single dependent variable,
64; two dependent variables,
77.
Future "that may be," 382; influence
of, on course of events, 382, 383;
role of, in purposive action, 395,
398, 409; knowledge of the, through
will, 421.
G
Gas law, analogy of population
pressure, 305.
Genera and species, Willis' theory
(age and area), 313.
Generation, rate of increase of population
per generation, 118.
Generations, diffusion in time of successive,
86; ratio of successive, 87.
Genesis, ultimate of organism, 269.
Genius, motivation in men of, 399.
Genuine utility for social service,
386.
Geochemistry, 274. See also geophysics,
and numerous references
under Clarke, F. W.
Geophysics, 273. See also geo-
chemistry.
Grasshoppers destroyed by birds
168.
Growth, 9, 14, 43; law of population,
64; in diminishing population, 70;
determination of constants, 73.
Growth function, single-valued, 47.
Growth, of bacteria 69; of individual,
71, 439 ;
determination of constants,
73.
Growth of aggregates, 100.
Growth of population, influence of
age distribution, 109.
Growth in population with Mendeliam
inheritance, 122.
Growth in man, 132; of industries,
278; of world's population, 278; of
transformers, 328.
Growth, Verhulst-Pearl law, 368.
Guanay (bird), 250.
Guano, 236.
H
Half-decay period (of radio-active
substance), 268.
Hand, r61e of, in evolution, 298, 440.
Hedonistic principle of Spencer,
extension of 430, 432.
Higher forms, 21.
Human body, composition, compared
with earth's crust, 196, 197.
Hydrosphere, 192.
Ideal Organism, relation to actual,
352.
Imagination, 371.
Immobile constituents of soil, 179.
INDEX OF SUBJECTS
Immunising diseases, Martini's equations,
79.
Indifferent increments, 349.
Indispensable components, 95.
Industrial evolution, 307, 368.
Industrial psychology, 400.
Industrial organization, imperfection
of adjustment of human motives to,
418.
Inertia-free systems, 28 et soq.
Informants, 339; 410.
Instability, significant cases of, 294.
Instinctive arid reflective choice, 412.
Instinct as determinant of action,
395.
Instinct of workmanship and selfexpression,
396; 416.
Instincts, list of, 396; danger of
suppressed, 400; egoistic and
altruistic, 412; inadequacy as external
adjusters in human community,
415.
Integration of nervous system, 405.
Intelligence as discriminating agency,
120.
Intensity factor of energy, 282, 303.
Intensity law (of energy), 303.
Interpendence of species, 77; chart, 98,
99; network of chains, 136; methods
of gathering data, 164.
Interfacial forces, 14.
Intra-group evolution, 44; kinetics,
122, 439.
Inter-group evolution, see intraspecie
evolution.
Interspecies equilibrium, 161.
Intra-species evolution, 44, 122, 348;
analytical statement of problem,
345.
Invariants of nature, 373.
Iron, circulation in nature, 256.
Irreplaceable components, 65; see
also replaceable components.
Irreversibility, 21 et seq. 23; statistical
meaning, 30; macroscopic, 36,
439.
Irreversibility and evolution, 26.
Jabberwock, 4, 7.
K
Kinetics of Evolution, fundamental
equation, 44, 51.
Kinetics of evolving systems, 57 et
seq.
Knowledge, reaction of, upon emotions,
424.
L
Lag, 23.
Lag and lead in equations of kinetics,
47; lag and lead, spurious, 48.
Law of urban concentration, 306
Le Chatelier's principle, 281; conditions
of validity, 284, 286.
Length of life of molecules in chemical
reaction, 155.
Liebig's law of limiting factor, 97;
229.
Life, definition, 7, 18; origin of, 218;
cycle of, 334; curve, 102 et seq.;
table, 102 et seq.; life and oxidation,
218; life struggle in modern
community, 417.
Light, influence on biochemical reactions,
317, 318; influence on marine
life, 318; penetration into sea water,
318.
Limiting factors, 97, 229; water, 212.
Liparis dispar, 87.
Lithosphere, 193; composition, 194;
accessions to, 195.
Living matter, characteristics, 8;
origin of, 218.
Logic, 371; 410.
Logic and ethics, parallelism, 414.
Losses from atmosphere, 188.
M
Machines, 7, 11; rent on, paid by
worker, 369.
Macro-mechanics, 50.
456 INDEX OF SUBJECTS
Malaria, 79, 81 ; equilibrium, 147, 149,
150.
Marine origin of terrestrial species,
203.
Mass action, law, and its generalization,
42.
Maximum andminimum laws, 157.
Maximum energy flux, law of evolution,
357.
Maxwell's demon, 35, 121.
Meadowlark in relation to species on
which it feeds, 167.
Mean free path, 358.
Mean length of life, 114, 115, 267; of
molecules in chemical reaction, 155.
Mechanical imitations of correlating
apparatus, 340, 342.
Mechanical model of function of
correlating apparatus, 381.
Mechanical properties in relation to
growth, 12.
Mechanics, macro- and micro-, 50;
bulk, 50; statistical, 50.
Mechanistic conception, 48.
Mechanistic and teleological interpretation
of adjusters, 381, 382, 383,
384.
Memory, 48; 339.
Mendelian inheritance, 122.
Metastable equilibrium, 10, 151,
Metastability and voluntary action,
407.
Methods of biological census of
aquatic species, 173; summary, 176.
Methods of gathering data regarding
interdependence of species, 164.
Micro-mechanics, 50.
Migration, see random.
Milieu interieur, 17, 203.
Milk Production, record, 136.
Mill wheel of life, 334.
Mind, cosmic, 375; group mind, 375;
location of, 375.
Mind merely a symbol t>f reference.
375.
Minimum law of equilibrium, 320.
Minimum laws. 157.
Models, use of, for attacking refractory
problems in mathematical
analysis, 360.
Moisture contents of foods, 211.
Monomolecular reactions, 106.
Mortality and survival, 103 et seq.
Mortality, force of in chemical reaction,
153, 155.
Motion of organisms, laws of, as
exemplified by chess, 343.
Motivation, 395; in genius, 399.
Movement, see also migration.
Moving equilibria, 143, 259; organic,
276.
Moving equilibrium, special case;
pace set by the slowest transformation
in a chain, 261, 265; 267 (footnote
5) ; example of radioactive chain
of transformations, 262.
Mutually selfish bargain, 415.
N
Nannoplankton, 173.
Nerve energy, 13.
Nitrebeds, origin, 238; exhaustion
238.
Nitrogen, nomadic, 229.
Nitrogen cycle, 229, 236; diagram, 230,
231; gate of entry, 232; human interference,
236; as limiting factor, 229;
losses, 232; direct assimilation by
plants, 233; accessory sources of
234; oxidation process, 241; fixation
processes, 239; industry, 240, 241,
242.
Nomadic nitrogen, 229.
"Normal" population, 115.
Nurture and tradition, evolutionary
value of, 428.
Nutrition of marine animals from
dissolved organic matter, 175.
Object and subject, see ego.
Objective world, origin of concept,
405.
INDEX OF SUBJECTS 457
Ocean, composition, 191; ocean, 192;
origin of salt in, 255.
Oceanography, 181.
Ontograms and phanograms, 421.
Optimum birthrate, 128.
Organic cycle, energetics of, 334.
Organic matter, composition, 197,
198, 199, 200.
Organisms as composite transformers,
329; as coupled transformers, 330.
Orthogenesis, 296.
Orthogenesis in human evolution,
378.
Oscillations, 61.
Oscillatory approach to equilibrium,
79.
Oxidation as typical life process, 218.
Oxygen as food, 219.
Oxygen cycle, 225.
Oyster culture, 181.
Parameters of state P, 43; 300; influence
on equilibrium, 319.
Parameters Q defining character of
species, 46.
Parameters, evolution under changing,
280; conjugate, 287, 288, 303;
topographic 300, 440; relations between,
301 ; climatic, 317.
Parasitology, 83.
Passive character of plants, 336.
Peat formation, 224.
Pedogenesis, 12.
Pendulum as example of simple
mechanical motion, 27.
Perfect adjustment of correlating
apparatus, 353.
Period of Diffusion, 311.
Periodic processes, 22, 24.
Periodicity in seemingly random
series, 32.
Phanograms and ontograms, 421.
Phenotypes (Mendelian) 122.
Philosophy as part of scientific
enquiry, 418.
Phosphate rock, 248.
Phosphatic slag (as fertilizer) 251.
Phosphorus, natural supply in soils,
246; migration of, 248; losses of,
248, 250.
Phosphorus cycle, 246; diagram, 247.
Physical Biology, Program of, 49.
Physical chemistry of structured
systems, 13.
Physico-chemical evolution?
law of,
14, 15, 16; physico-chemical evolution,
152, 155; second law of thermodynamics
as the law of evolution,
157.
Physico-chemical systems, structured,
15.
Plaice, 173.
Plant and animal, distinction, 4.
Plant "psychology," 219, 355.
Plant characters, in man, 219, 355.
Plant foods in soil, supply of 257.
Plant and animal as coupled transformers,
330.
Plants as accumulators (anabions),
329.
Plants, efficiency in utilization of
sunlight, 332; absence of correlating
apparatus in, 355.
Plants, characteristically passive,
sessile 336.
Population, law of growth, 64;
United States,
5
66; experimental,
69; Drosophila 69; diminishing, 70;
bacterial colony, 71 ;
determination
of constants, 73; saturation point,
67; rate of increase per generation
118; census of flora and fauna,
165, 166; growth of world's, 278;
law of urban concentration, 306.
Population equilibrium, stability, 67.
Population element, progeny of, 86.
Population with "normal'
1
age distribution,
110.
Population density, 307, 308, 309;
equatic, 174.
Population pressure, 288, 304, 305,
307.
458 INDEX OF SUBJECTS
Potassium in relation to life, 255,
Potential) 10, 157; thermodynamic,
319.
Power, see energy, also waterpower,
windpower, sun, coal.
Preconceived premises, difficulty of
shaking off, 377.
Premises, fundamental, 376.
Pressure-volume relation, 287.
Prices, analogy to intensity factor of
energy, 303.
Primary foods, 181.
Principle of continuity, 259.
Principle of Le Ghatelier, 281; conditions
for validity, 284.
Proales decipiens, 108.
Product I
vit}* of carp, 173.
Productivity, specific, 347.
Program of physical biology, chart,
53.
Progress and evolution, 21.
Pseudo-problems, 3, 4, 7.
Psychology, dynamic, 396.
Psycho-physical parallelism, 402.
Purposive action, 395; relation to free
energy, 409.
Pursuit, curves of, 360.
Q
Quadratic forms, 158.
Quantum theory and the asymmetry
of time, 39.
Quantum theory, possible relation to
phenomena of will and consciousness,
406.
Quasi-dynamics, 52.
R
Radioactive transformations 58, 62,
83, 106.
Radioactive equilibrium, 262; approximate
expression for, 267, 268.
Radioactive substances as cosmic
clocks, 208.
Radioactivity, periodic melting of
earth's crust (Joly's theory), 275.
Radium, 263, 265.
Railways, growth of American, 36
Rainfall, 214.
Random collisions, 337.
Random migration, 337, 344; undt
bias, exemplified by chess, 34?
random movement under a bias
360.
Reaction constant regarded as fore
of mortality, 153.
Realistic world picture, 371 et seq.
Reasoning, 371 et seq.
Realistic thinking, 410.
Reason, 339.
Receptor-effector circuit, 340.
Receptors, 339, 363 et seq.; evolutioi
of 347, 348; artificial, 364.
Record milk production, 136.
Reflective and instinctive choice, 412,
Rent and area, 288, 305.
Rent on machinery, paid by worker,
369.
Replaceable components, 95; 163.
Representation (Abbildung), 363.
Reproduction, 8, 11, 12; extravagance
in 132, 135.
Resignation, policy of, in science, 19.
Rigid type of behavior schedule, 350.
Rivers, discharge of world's, 214.
Rocks, weathering, 223, 224.
S
Salt of ocean, origin, 255.
Saltpeter industry, 235, 236.
Saturation point of population, 67.
Science, pure and applied, mutual
stimulation of, 298, 299.
Sciences, classification 419, 421, 422;
chart, 423.
Scientific world picture, 372.
Sea water and body fluids, 17; composition,
191; composition compared
with blood serum, 201, 202.
Secondary foods, 181.
Selective slaughtering, effect on rate
of increase of species, 119.
458 INDEX OF SUBJECTS
Potassium in relation to life, 255.
Potential, 10, 157; thermodynamic,
319.
Power, see energy, also waterpower,
windpower, sun, coal.
Preconceived premises, difficulty of
shaking off, 377.
Premises, fundamental, 376.
Pressure-volume relation, 287.
Prices, analogy to intensity factor of
energy, 303.
Primary foods, 181.
Principle of continuity, 259.
Principle of Le Ghatelier, 2S1; conditions
for validity, 284.
Proales decipiens, 108.
Productivity of carp, 173.
Productivity, specific, 347.
Program of physical biology, chart,
53.
Progress and evolution, 21.
Pseudo-problems, 3, 4, 7.
Psychology, dynamic, 396.
Psycho-physical parallelism, 402.
Purposive action, 395; relation to free
energy, 409.
Pursuit, curves of, 360.
Quadratic forms, 158.
Quantum theory and the asymmetry
of time, 39.
Quantum theory, possible relation to
phenomena of will and consciousness,
406.
Quasi-dynamics, 52.
R
Eadioactive transformations 58, 62,
63, 106.
Radioactive equilibrium, 262; approximate
expression for, 267, 268.
Radioactive substances as cosmic
clocks, 268.
Radioactivity, periodic melting of
earth's crust (Joly's theory), 275.
Radium, 263, 265.
Railways, growth of American, 369.
Rainfall, 214.
Random collisions, 337.
Random migration, 337, 344; under
bias, exemplified by chess, 343;
random movement under a bias,
360.
Reaction constant regarded as force
of mortality, 153.
Realistic world picture, 371 et seq.
Reasoning, 371 et seq.
Realistic thinking, 410.
Reason, 339.
Receptor-effector circuit, 340.
Receptors, 339, 363 et seq. ;
evolution
of 347, 348; artificial, 364.
Record milk production, 136.
Reflective and instinctive choice, 412.
Rent and area, 288, 305.
Rent on machinery, paid by worker,
369.
Replaceable components, 95; 163.
Representation (Abbildung), 363.
Reproduction, 8, 11, 12; extravagance
in 132, 135.
Resignation, policy of, in science, 19.
Rigid type of behavior schedule, 350.
Rivers, discharge of world's, 214.
Rocks, weathering, 223, 224.
Salt of ocean, origin, 255.
Saltpeter industry, 235, 236.
Saturation point of population, 67.
Science, pure and applied, mutual
stimulation of, 298, 299.
Sciences, classification 419, 421, 422;
chart, 423.
Scientific world picture, 372.
Sea water and body fluids, 17; composition,
191; composition compared
with blood serum, 201, 202.
Secondary foods, 181.
Selective slaughtering, effect on rate
of increase of species, 119.
INDEX OF SUBJECTS 459
Self consciousness as a particular
mode of general consciousness, 404,
Self-expression, instinct, of, 396.
Self, see ego.
Series solution of fundamental equation
of kinetics, 60.
Sex, in reproduction, 13.
Shark series of fishes, possible orthogenesis
in, 296.
Siamese twin, the body politic as a
multiple, 369.
Simulacra vitae, 9, 439.
Single-valued growth function, 47.
Singular orbits, and the intervention
of consciousness in mechanics, 407.
Slavery, 370.
Social psychology, 400.
Social utility, 386.
Sodium chloride cycle, 252; diagram,
254.
Soil, supply of plant foods in, 257.
Solar energy, See Sun.
Sophistry, bred out of unrecognized
assumptions, 377.
Soxhlet extractor, 253.
Spacial sense, 411.
Specific productivity, 347.
Speech and thought, cumulative
evolution of, 298.
Spencer's hedonistic principle, an
extension of, 430, 432.
Spread of species, 311.
Stability of age distribution, 11, 113.
Stage of the life drama, 183, 185.
Stars, evolution, 272.
Statics, 51.
Statistical mechanics, 24; of lifebearing
systems, 121 ;
of systems of
energy transformers, 325, 345, 358;
of systems of organisms, 358,
Statistical models of irreversible
processes, 30.
Steady states, 51, 59, 143.
Steers, growth of, 133 et seq.
Stoichiometry, 50.
Stomach contents, analysis of, 166.
Stream of substance through the form
of the organism, 130.
Structured systems, 13.
Subject, see ego.
Subjective sense of direction in time,
37.
Sulphur, circulation in nature, 256.
Sun, energy intercepted by plants,
331 ; energy received by earth from,
331; energy radiated by, 331.
Sun's energy, compared with coal
consumption, 332.
Surveys, biological, 164; surveys of
marine life, 173.
Survival factor, 102.
Survival curve, 103, 104, 105, 107.
Survival curves for experimental
populations, 108.
Tastes, adaptive adjustments of 385;
divergence of, 397.
Teleological interpretation, and
mechanistic, of adjustors, 381, 382,
383, 384.
Teleological mechanisms, 342, 440.
Temperature, influence on biochemical
reactions, 317.
Terrestrial species, marine origin,
203.
Tertiary foods, 181.
Thermodynamic potential, 10, 157.
Thermodynamics, second law, and
evolution, 26; second law of, as law
of evolution in physico-chemical
transformations, 157.
Thinking: autistic and realistic, 410;
semi-realistic, 411.
Tidal dissipation, 22, 24.
Time, 22; positive and negative in
mechanics, 27, 37; subjective sense
of, 37; direction in, 37, 411; geologic,
as indicated by radioactive substances,
268, 269.
Topographic parameters, 300, 311.
Toy beetle, 341, 381.
Tradition, evolutionary value of, 428.
Transformants or epictors, 411.
Transformation factors, their economic
signficance, 137.
OF SUBJECTS
Transformer cycles, 326,
Transformers, composite and
coupled, 327, 330; growth, of, 328.
Transportation, evolution of 367.
Travelling environment, 16.
Twin, 12.
Types of organisms, economical and
lavish birthrate, 131, 132 et seq.
U
Unadapted species, 266.
Unfit species, 266.
Unity of man with cosmos, 426.
Universe awakening to consciousness
of itself, 431.
Universe, man's unity with 433.
Uranium chain of radioactive elements,
263.
Urban concentration, law of, 306.
Urea, 8.
Utility, genuine, for social service,
386.
Value in exchange (of small increments
of two sensori-motor parameters),
349,
Value, economic, relation to physical
energy, 354, 355; absolute standard
of 353, 386; and genuine utility for
social service, 386; theory of, 422.
Verbal problems, 4, 7.
Verfmlst-Pearl law, 329, 36S.
Vestal flame, 219.
Vicious circles, 294, 295.
Vital processes, 8.
Vital force, 13.
Volcanic discharges, 222.
Voluntary action and determinism,
406 et seq.
Volvox, 6.
W
War game, 361.
Water circulation, fraction taking
part in life cycle, 216.
Water cycle, 209; diagram, 215;
energetics of, 333; water requirements
of organisms, 210, 211, 212;
supply, 213.
Waterpower, 333.
Watershed, moral, and voluntary
action, 408.
Wealth, mathematical theory of, 409.
Weathering of rocks, 223, 224.
Windpower, 333.
Will, 48; general, 375.
Working substance, 325 et seq.
Workmanship, instinct of, 396.
World Engine, 331; evolution of, 335.
World picture, scientific, 372, double
entry bookkeeping, 373.
World Purpose, 428, 430.
Z
Zero roots of equilibrium equation,
62.
Zone pattern, effect of change upon
rate of increase of species, 348.
Zones of influence, 344.
Zones of mobility, 345.
Zostera (eel grass), 176.
Zweckmassig action, 371.
WN THE "elder days of art" each artist or craftsman
I enjoyed the privilege of independent creation.
j| He carried through a process of manufacture from
beginning to end. The scribe of the days before the
printing press was such a craftsman. So was the
printer in the days before the machine process. He
stood or fell, as a craftsman, by the merit or demerit
of his finished product.
Modern machine production has added much to the
worker's productivity and to his material welfare; but
it has deprived him of the old creative distinctiveness.
His work is merged in the work of the team,
and lost sight of as something representing him and
his personality.
Many hands and minds contribute to the manufacture
of a book, in this day of specialization. There are
seven distinct major processes in the making of a book:
The type must first be set; by the monotype method,
there are two processes, the "keyboarding" of the MS
and the casting of the type from the perforated paper
rolls thus produced. Formulas and other intricate
work must be hand-set; then the whole brought together
("composed") in its true order, made into pages
and forms. The results must be checked by proof
reading at each stage. Then comes the "make-ready"
and press-run and finally the binding into volumes.
All of these processes, except that of binding into cloth
or leather covers, are carried on under our roof.
The motto of the Williams & Wiikins Company is
Sans Tache. Our ideal is to publish books "without
blemish" worthy books, worthily printed, with worthy
typography books to which we shall be proud to
attach our imprint, made by craftsmen who are willing
to accept open responsibility for their work?
and who
are entitled to credit for creditable performance.
The printing craftsman of today is quite as much a
craftsman as his predecessor. There is quite as much
discrimination between poor work and good. We
are of the opinion that the individuality of the worker
should not be wholly lost. The members of our staff
who have contributed their skill of hand and brain to
this volume are:
Composing Room: Ernest Salgado, Henry Shea, Andrew Rassa,
William Sanders, George Behr, Zeddie Breithaupt, Ray Kauffman,
Benjamin Armiger, Harry Harmeyer, James Jackson, Benjamin
Hatcher, William Kidner, Herbert Leitch, William Fite, Harry
LaMotte, John Dotterwich, William Koch, William Harrison, Jr.,
George Moss, Walter Phillips, Arthur Childress, Richard King,
Edward Rice.
Folder: Krug.
Press Room: August Hildebrand, R. S. Gallagher, William Harrison,
Jr., Henry Augsburg.
Culler: Murphy.
Keyboard: Vera Taylor, Harry Susemihl, Anna Kelly, Catherine
Kocent, Eleanor Luecke, Hannah Scott, Anna Thomas, Katharine
Wilson, Minnie Foard<
Proof Room: Sarah Katzin, Mary Reed, Alice Reuter, Lucille Bull,
Ethel Strasinger, Ruth Treischman, Edna Clark, Audrey Tanner,
Dorothy Strasinger, Lillian Gilland, Angeline Eifert, Arthur Baker.
Casters: Martine Griffen, Ernest Wann, Charles Aher, Kenneth
Brown, Mahlon Robinson, Frank Malonosky, George Smith, Henry
Lee.
Some Recent Books
ABILITY TO SELL
What sales-manager wouldn't like to spot it ? What enterprising
individual who hasn't asked himself whether or not
he has it? In these days of competition, when salesmanship
means so much to business, the ability to sell is an
important qualification. Dr. Merrill Jay Ream's little
book under this title tells a big story worthy of careful
study. Based on certain experimental tests in actual
selling.
Cloth, $1.25 postpaid
POPULAR RESEARCH NARRATIVES
A group of fifty five-minute stories of research, invention
or discovery, directly from the "men who did it," pithily
told in language for laymen, young and old. They were
collected by the Engineering Foundation, of New York
City. Give the busy man a better idea of what's doing
in the scientific world than hours spent in diligent digging.
Admirable adventure stories for the youth in school or
industry. A special arrangement with Engineering Foundation
makes possible a
Price of 50c. 160 pages} cloth bound
COMMON SENSE OF THE
THEORY OF RELATIVITY
Yes, it has common sense to it. Dr. Paul Heyl's little
book brings it out, and by apt and vivid illustration enables
any mind of intelligence to get the fundamental idea
Einstein presents. A little book which you can read in
half an hour, but one which you will want to read many
times.
Cloth, $1.00 postpaid