Asking and Answering Quetions in the History
of Mathematics
Aarhus University, Denmark
Tinne Hoff Kjeldsen Henrik Kragh Sørensen
Summer 2012
Contents
Historiography of Mathematics 1
1 T. H. Kjeldsen, S. A. Pedersen, and L. M. Sonne-Hansen (2004). “Introduction”.
In: New Trends in the History and Philosophy of Mathematics. New
Trends in the History and Philosophy of Mathematics. Ed. by T. H. Kjeldsen,
S. A. Pedersen, and L. M. Sonne-Hansen. University of Southern Denmark
Studies in Philosophy 19. Odense: University Press of Southern
Denmark, pp. 11–25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 T. H. Kjeldsen (July 2012). “Uses of History for the Learning of and
about Mathematics. Towards a theoretical framework for integrating
history of mathematics in mathematics education”. Plenary address at
the HPM International Congress in Korea. . . . . . . . . . . . . . . . . . . 16
3 I. Grattan-Guinness (May 2004). “The mathematics of the past: distinguishing
its history from our heritage”. Historia Mathematica, vol. 31,
no. 2, pp. 163–185. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4 J. Lützen and W. Purkert (1989). “Conflicting Tendencies in the Historiography
of Mathematics: M. Cantor and H. G. Zeuthen”. In: The History
of Modern Mathematics. Ed. by D. E. Rowe, J. McCleary, and E. Knobloch.
Vol. 3. Proceedings of the Symposium on the History of Modern Mathematics,
Vassar College, Poughkeepsie, New York, June 20–24, 1989. 3 volumes.
Academic Press, pp. 1–42. . . . . . . . . . . . . . . . . . . . . . . . 61
5 M. Epple (2011). “Between Timelessness and Historiality: On the Dynamics
of the Epistemic Objects of Mathematics”. Isis, vol. 102, no. 3,
pp. 481–493. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
i
6 Anachronical and diachronical history. From H. Kragh (1994). An Introduction
to the Historiography of Science. First published 1987. Cambridge
etc.: Cambridge University Press, pp. 89–107. . . . . . . . . . . . . . . . . 96
7 Discussion of geometric algebra. From J. Lützen and K. Ramskov, eds.
(1999). Kilder til matematikkens historie. 2nd ed. København: Matematisk
Afdeling, Københavns Universitet, pp. 10–17. . . . . . . . . . . . . . . . . 115
Mini course on History of Analysis 123
8 The Method of Archimedes. From T. L. Heath, ed. (1953). The Works
of Archimedes with the Method of Archimedes. New York: Dover Publications,
pp. 12–21. Adopted from J. Lützen and K. Ramskov, eds. (1999).
Kilder til matematikkens historie. 2nd ed. København: Matematisk Afdeling,
Københavns Universitet, pp. 22–26. . . . . . . . . . . . . . . . . . . . 123
9 K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I.
Grattan-Guinness. Princeton and Oxford: Princeton University Press.
Chap. 1, pp. 10–48. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
10 T. H. Kjeldsen (2011). “Does history have a significant role to play for the
learning of mathematics? Multiple perspective approach to history, and
the learning of meta level rules of mathematical discourse”. In: History
and Epistemology in Mathematics Education. Proceedings of the Sixth European
Summer University ESU 6. Ed. by E. Barbin, M. Kronfellner, and C.
Tzanakis. Vienna: Verlag Holzhausen GmbH, pp. 51–62. . . . . . . . . . . 166
11 Fermat on maxima and minima. From D. J. Struik (1969). A Source
Book in Mathematics. 1200–1800. Cambridge (Mass.): Harvard University
Press, pp. 222–225. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
12 Fermat on maxima and minima. From J. Fauvel and J. Gray, eds. (1987).
The History of Mathematics: A Reader. London: Macmillan Press Ltd., pp. 359–
360. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
13 Wallis on interpolation. From D. J. Struik (1969). A Source Book in Mathematics.
1200–1800. Cambridge (Mass.): Harvard University Press, pp. 244–
246. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
14 Roberval on the quadrature of the parabola. From E. Walker (1932). A
Study of the Traité des Indivisibles of Gilles Persone de Roberval. New York,
pp. 181–182. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
15 H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In:
From the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press.
Chap. 2, pp. 49–93. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
ii
16 N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In:
A History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24.
Providence (Rhode Island): American Mathematical Society. Chap. 3,
pp. 73–103. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
17 Newton on fluxions and fluents. From M. E. Baron and H. J. M. Bos,
eds. (1974). Newton and Leibniz. History of Mathematics: Origins and
Development of the Calculus 3. The Open University Press, pp. 22–25. . 263
18 Newton on the method of drawing tangents. From D. T. Whiteside, ed.
(1964). The Mathematical Works of Isaac Newton. Vol. 1. Johnson Reprint
Corp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
19 Leibniz’ process of discovery. From M. E. Baron and H. J. M. Bos, eds.
(1974). Newton and Leibniz. History of Mathematics: Origins and Development
of the Calculus 3. The Open University Press, pp. 42–43. . . . . . 268
20 Bishop Berkeley’s The Analyst. From D. E. Smith (1959). A source book in
mathematics. 2nd ed. 2 vols. New York: Dover Publications, Inc., pp. 627–
634. Adopted from J. Lützen and K. Ramskov, eds. (1999). Kilder til matematikkens
historie. 2nd ed. København: Matematisk Afdeling, Københavns
Universitet, pp. 95–99. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
21 J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In:
A History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24.
Providence (Rhode Island): American Mathematical Society. Chap. 6,
pp. 155–195. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
22 I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis”
of the Early Nineteenth Century”. Archive for History of Exact Sciences,
vol. 6, no. 5, pp. 372–400. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
23 H. Freudenthal (1970–1971). “Did Cauchy Plagiarize Bolzano?” Archive
for History of Exact Sciences, vol. 7, no. 5, pp. 375–392. . . . . . . . . . . . . 344
Mini course on History of Algebra 362
24 Old Babylonian problems from BM 13901. From E. Robson (2007). “Mesopotamian
Mathematics”. In: The Mathematics of Egypt, Mesopotamia,
China, India, and Islam: A Sourcebook. Ed. by V. J. Katz. Princeton and
Oxford: Princeton University Press. Chap. 2, pp. 57–186, pp. 104–107. . . 362
25 Islamic algebra. From J. L. Berggren (2007). “Mathematics in Medieval
Islam”. In: The Mathematics of Egypt, Mesopotamia, China, India, and Islam:
A Sourcebook. Ed. by V. J. Katz. Princeton and Oxford: Princeton University
Press. Chap. 5, pp. 515–675, pp. 542-545. . . . . . . . . . . . . . . . . 366
26 E. Robson (2005). “Influence, ignorance, or indifference? Rethinking the
relationship between Babylonian and Greek mathematics”. Bulletin of
the British Society for the History of Mathematics, vol. 4, pp. 1–17. . . . . . . 370
iii
27 R. Franci (2010). “The history of algebra in Italy in the 14th and 15th
centuries. Some remarks on recent historiography”. Actes d’història de la
ciència i de la tècnica, vol. 3, no. 2, pp. 175–194. DOI: 10.2436/20.2006.
01.158. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
28 B. M. Kiernan (1971). “The Development of Galois Theory from Lagrange
to Artin”. Archive for History of Exact Sciences, vol. 8, no. 1–2,
pp. 40–154, pp. 40–66. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
29 Abel’s 1824 proof. From P. Pesic (2003). Abel’s Proof. An Essay on the
Sources and Meaning of Mathematical Unsolvability. Cambridge (MA)/London:
The MIT Press, pp. 155–169. . . . . . . . . . . . . . . . . . . . . . . . . . . 434
30 J. Lützen (2009). “Why was Wantzel Overlooked for a Century? The
Changing Importance of an Impossibility Result”. Historia Mathematica,
vol. 36, no. 3, pp. 374–394. . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
31 P. M. Neumann (1999). “What groups were: A study of the development
of the axiomatics of group theory”. Bull. Austral. Math. Soc. Vol. 60,
pp. 285–301. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
iv
Archimedes then describes some theorems that he has found and mentions
that he has included the proofs. He continues
Text 8: The Method of Archimedes. From T. L. Heath, ed. (1953). The Works of
Archimedes with the Method of Archimedes. New York: Dover Publications, pp. 12–21.
Adopted from J. Lützen and K. Ramskov, eds. (1999). Kilder til matematikkens historie.
2nd ed. København: Matematisk Afdeling, Københavns Universitet, pp. 22–26.
Summer University 2012: Asking and Answering Questions Page 123 of 479.
This is followed by some theorems about centers of gravity and the
argument for the above mentioned theorem. This argument concludes with
the following remark:
Text 8: The Method of Archimedes. From T. L. Heath, ed. (1953). The Works of
Archimedes with the Method of Archimedes. New York: Dover Publications, pp. 12–21.
Adopted from J. Lützen and K. Ramskov, eds. (1999). Kilder til matematikkens historie.
2nd ed. København: Matematisk Afdeling, Københavns Universitet, pp. 22–26.
Summer University 2012: Asking and Answering Questions Page 124 of 479.
Text 8: The Method of Archimedes. From T. L. Heath, ed. (1953). The Works of
Archimedes with the Method of Archimedes. New York: Dover Publications, pp. 12–21.
Adopted from J. Lützen and K. Ramskov, eds. (1999). Kilder til matematikkens historie.
2nd ed. København: Matematisk Afdeling, Københavns Universitet, pp. 22–26.
Summer University 2012: Asking and Answering Questions Page 125 of 479.
Text 8: The Method of Archimedes. From T. L. Heath, ed. (1953). The Works of
Archimedes with the Method of Archimedes. New York: Dover Publications, pp. 12–21.
Adopted from J. Lützen and K. Ramskov, eds. (1999). Kilder til matematikkens historie.
2nd ed. København: Matematisk Afdeling, Københavns Universitet, pp. 22–26.
Summer University 2012: Asking and Answering Questions Page 126 of 479.
Chapter 1
Techniques of the Calculus, 1630--1660
Kirsti M011er Pedersen
1.1. Introduction
During the first six decades of the 17th century mathematics was in a
state of rapid development. In this period ideas were born and developed
which were to be taken up later by Isaac Newton and G. W.
Leibniz. Many me.thods were developed to solve calculus problems;
common to most of them was their ad hoc character. It is possible to
find examples from the time before Newton and Leibniz which, when
translated into modern mathematical language, show that differentiation
and integration are inverse procedures; however, these examples are
all related to specific problems and not to general theories. The special
merit of Newton and Leibniz was that they both worked out a general
theory of the infinitesimal calculus. However, it cannot be said that
either Newton or Leibniz gave to his calculus a higher degree of mathematical
rigour than their predecessors had done.
As the ideas which were the basis of the methods preceding the work
of Newton and Leibniz came to bear fruit, the methods themselves fell
into oblivion. In this chapter, therefore, great importance will be
attached to the earlier ideas, and the methods will be illustrated by simple
examples. The picture of what the mathematicians of the time achieved
may thus appear somewhat distorted, but a rendering of the more
complicated examples would be all too easily submerged in calculations.
That it is possible to find simple problems is due to the fact that it was
the practice of the mathematicians of the time to verify their methods by
applying them to problems of which the solutions were known beforehand.
Then the next step was to find new results by means of these
methods.
It is impossible to deal comprehensively with this topic in a single
chapter. My approach will be to exemplify the calculus of the period
by relatively few methods, which are described in some detail. This
implies that the methods of many important mathematicians will have
10
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 127 of 479.
1.1. Introduction 11
to be left unmentioned. A more general survey giving a more profound
impression of the development of the calculus from 1630 to 1660 may
be found in the rich literature on this subject. 1 I have made my choice
on the assumption that to give even a tolerably satisfactory general survey
in a single chapter would mean listing names and outlining techniques
in a way which could not possibly give a proper impression of the methods
and style of the time to a reader who is not acquainted with the period.
One criterion for the selection of methods has been that they should
render a picture of the way in which the mathematicians of the time did
actually solve the problems with which they were most heavily engaged;
another has been that they should inform the reader of the ideas which
were to become sources of inspiration for later methods. Where different
methods are based on similar ideas, I have tried to select the
writer who first formulated the idea.
Of the period 1630-1660, no less than of all other periods, it holds
true that if you really want to set its mathematics into relief then you
must know the mathematics which preceded it. The mathematics of
the period in question were greatly influenced by classical Greek
mathematics2
and also by that of the previous period. The reason for
the importance of Greek mathematics was that during the 16th century
it had become usual for the mathematicians to acquire a knowledge of
this discipline, and it formed a basic element in the mathematical equipment
of most of them. Greek mathematics was especially admired for
its great stringency. But its methods were not heuristic; they were
not well-fitted to suggest ideas as to how to attack a new problem, a
fact which will be illustrated later in connection with quadratures and
cubatures.
It was natural, therefore, to search for other methods which, if they
could not live up to the Greek requirement of exactness, were at least
able to suggest ideas as to the solution of problems. The seeds of such
methods are to be found in the previous period, the end of the 16th
and the beginning of the 17th centuries, which was a fertile time for the
exact sciences as a whole. Astronomy made great progress through the
work of Johannes Kepler; Simon Stevin contributed much to statics
with his treatise De Beghinselen der Weeghconst (' The elements of the
art of weighing': 1586a). In mechanics Galileo Galilei's deduction of
the laws of freely falling bodies and of the parabolic paths of projectiles
meant a break with Aristotelian physics and the beginning of a new
epoch, where mathematics was to be extensively used in physics.
1 See, for example, Baron 1969a, Boyer 1939a and Whiteside 1961a, and their
bibliographies.
2 There are excellent bibliographies of Greek mathematics in Boyer 1968a and Kline
1972a.
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 128 of 479.
12 1. Techniques of the calculus, 1630-1660
Kepler made use of infinitesimal methods in his works. The
interest he took in estimating the volumes of wine casks resulted in the
book Nova stereometria doliorum vinariorum (' New measurement of
large wine casks': 161Sa). There he considered solids of revolution
as composed in various ways of infinitely many constituent solids.
For example, he regarded a sphere as made up of an infinite number of
cones with vertices at the centre and bases on the surface of the sphere.
This led to the result that the sphere is equal in volume to the cone which
has the radius of the sphere as altitude and as base a circle equal to the
surface of the sphere, that is, a circle with the diameter of the sphere as
radius (Kepler 161Sa, ,Prima Pars, Theorem 11; Worksl , vol. 4, 563, or
Works2, vol. 9, 23 f.).
Galileo planned to write a book on indivisibles, but this book never
appeared; however, his ideas had a great influence on his pupil
Cavalieri, with whose work we shall deal later.
1.2. Mathematicians and their society
A great many mathematicians of the 17th century were not mathematicians
by profession. This tendency was especially noticeable in
France; there only GiBes Personne de Roberval occupied a chair of
mathematics, while great mathematicians like Pierre de Fermat, Rent~
Descartes and Blaise Pascal worked without any official connection
with their discipline. Like the mathematician who inspired him,
Franc;ois Viete, Fermat was a lawyer, and worked as such in Toulouse
for most of his career. Descartes and Pascal were men of private means
and, apart from mathematics, were also occupied with physics and
philosophy. Descartes spent a large part of his time outside France,
living for long periods in Holland and elsewhere.
This stay of Descartes in Holland served to inspire several Dutch
mathematicians, among whom was Frans van Schooten. He was a
member of the School of Engineering at Leyden, while his more important
pupils, whose treatises he published along with his own, mostly
worked professionally outside mathematics. However, the most
illustrious of his pupils, Christiaan Huygens, devoted his whole life to
mathematics and physics. In 1666 the Academie des Sciences was
founded in Paris, and Huygens was offered a membership which he
accepted. As a member of the Academie he received an ample stipend.
In Italy, the most outstanding mathematicians and physicists, such as
Galileo Galilei, Bonaventura Cavalieri and Evangelista Torricelli, held
offices within their own fields, partly at universities and partly as court
mathematicians.
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 129 of 479.
1.3. Geometrical curves and associated problems 13
The development of that part of mathematics with which this chapter
is concerned started later in England than on the Continent. Hence
the only English mathematician with whom we shall deal in this chapter
is John Wallis, who was Savilian Professor of Geometry at Oxford from
1649. It should be mentioned that in Thomas Harriot England had a
brilliant scientist whose work both in algebra and the calculus preceded
some of the methods discussed in this chapter. But only his Artis
analyticae praxis (' Practice of the analytical art': 1631a), which con~
tains his less important work, was published (posthumously) at this
time; thus his unpublished results will not be considered.
The period provides several good examples of the independent and
almost simultaneous discovery of methods with striking resemblance,
which often gave rise to disputes about priority and charges of plagiarism.
Today, we are able to establish that as a rule these charges 'were unfounded;
but at the time this was not possible, since it was not common
to publish one's treatises. For this there were two principal reasons.
First, after 1640 publishers were reluctant to print mathematical litera~
ture, which was not very profitable; and second, mathematicians were
reticent about publishing their new methods, wanting to release the
results only. Many treatises had to wait a very long time for their
publication: several were left unprinted until the end of the 19th and
the beginning of the 20th centuries, and some remain unpublished to
this day.
Not until the last third of the 17th century did scientific periodicals
come into existence; before that time mathematicians communicated
by letter. Here the Frenchman Marin Mersenne played an important
part, for he kept in touch with many European scientists by correspondence
and meetings which he held at his convent in Paris. To the
mathematicians he sent the problems which he could not solve himself,
and took care that the results and manuscripts he received were circulated
among those interested in them.
1.3. Geometrical curves and associated problems
In the 17th century the calculus was closely bound up with the investigation
of curves, since there was as yet no explicit concept of the
variable or of functional relationships between variables. The first
curves to be dealt with were those inherited from the Greeks: the conic
sections, Hippias's quadratrix, the Archimedean spiral, the conchoid
of Nlcomedes, and the cissoid of Diocles. (For the definition and the
history of these and the following curves see, for example, Loria 1902a.)
As the century went on, these curves were augmented by, among
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 130 of 479.
14 1. Techniques of the calculus, 1630-1660
others, the cycloid, the higher parabolas and hyperbolas (ym = kxn and
kymxn = 1 respectively, m and n being natural numbers and k a constant),
the spiral of Galileo, and the conchoid to a circle, also termed 'the
limayon of [Etienne] Pascal', which is in turn a variant of the curves
called ' the ovals of Descartes '.
Next to the conic sections the cycloid, the curve traced by a point on
the circumference of a circle which rolls along a horizontal line, was the
curve most often investigated. Its early history is connected with a
problem called' Aristotle's wheel' (see Drabkin 1950a). When solving
this problem Roberval generalised the motion which generates the curve,
and considered the curtate and the prolate cycloid (which are traced by
points on a radius and respectively outside and inside the circle) as well
as the ordinary cycloid. In 1658 Blaise Pascal arranged a competition
designed to find the area of a section of the cycloid, its centre of gravity,
the volumes of solids obtained by revolving the section about certain axes,
and the centres of gravity of these volumes (Pascal 1658a and 1658b).
In La geometrie (1637a) Descartes introduced his oval as a curve
involved in the solution of various optical problems. One of these
problems was to determine the form of a lens which makes all the rays
that come from a single point or that are parallel converge at another
unique point, after having passed through the lens (Descartes 1637a,
362; 1925a, 135).
Similarly, Galileo's spiral was the attempted solution of a physical
problem concerning the path of a body which moves uniformly around a
centre and at the same time descends towards the centre with constant
acceleration. The recognition of the shape of another of Galileo's
curves, namely, the catenary, caused the mathematicians many difficulties.
This curve has the form of a chain suspended from two points
(see section 2.8).
The three last-mentioned curves are examples of an interplay between
physics and mathematics. Before discussing this topic further
we shall answer the question: what kind of problems concerning curves
did the mathematicians solve in the period before 1660 ?
Pascal's competition of 1658 relates to certain typical problems
which were solved. Other problems consisted in finding tangents,
surface areas and extreme values; furthermore, some inverse tangent
problems (that is, to find a curve which has tangents with a specific
property) were considered. Finally, about the middle of the century,
the rectification of arcs became a question of interest. Although there
are earlier examples of rectifications, Christopher Wren's rectification
of the cycloidal arc in the late 1650s was the first widely known one.
He sent the result to Pascal outside the competition (see Wren 1659a,
or Wallis Works, yoJ. 1, 532-541).
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 131 of 479.
1.4. Algebra and geometry 15
Even though the solutions to these problems could be applied both
to physics and to astronomy, their inspiration owed more to Greek
mathematics than to physics and astronomy. The Greeks had worked
on all the types of problem mentioned above; one may therefore
consider work on them as a continuation of the tradition of the Greek
mathematicians. This does not mean that there was no correlation
between mathematics and physics. This continued to happen, if for
no other reason than that in this period important physicists were often
also important mathematicians. It is nevertheless difficult to point
unambiguously to a concrete physical problem which inspired the
mathematicians to take up the above-mentioned problems. In the
late 1650s, however, a new mathematical problem cropped up which
sprang from physics, namely the study of evolutes, which was started
by Huygens in connection with his work on the pendulum clock.
1.4. Algebra and geometry
When the Greeks came to realise the exi~tence of incommensurable
magnitudes, which meant that the rational numbers are not sufficient
for purposes of measurement, they made geom~try the foundation of
that part of mathematics which was not number theory, the straight line
being a substitute for a continuous field of numbers. This attitude
resulted in the geometric algebra on which Euclid, Archimedes and
Apollonius based their calculations.
In the course of time the theory of equations became separated from
geometry, and a good deal of symbolism was gradually developed for this
discipline. Viete contributed much to the introduction of symbols
with his work In artem analyticen isagoge (' Introduction to the analytic
art': 1591a), in which he emphasised the advantage of using symbols
to indicate not only unknown but also known quantities (Viete 1591a,
ch. V, 5; Works, 8, or 1973a, 52). In this way he could deal with
equations in general.
Viete also connected algebra and geometry by determining the
equations which correspond to various geometrical constructions. He
only employed this technique when the geometrical problems were
determinate and led to determinate equations in one unknown quantity.
The next step was to use an indeterminate equation in two unknown
quantities when solving problems concerning geometriG loci. Fermat
and Descartes took this step almost simultaneously.
Fermat's treatise Ad locos pIanos et solidos isagoge (' Introduction
to plane and solid loci': 1637a) contains a pedagogic introduction
to analytic geometry and some of its applications. However, the
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 132 of 479.
16 1. Techniques of the calculus, 1630-1660
treatise did not have any great influence, for the simple reason that
Descartes's La geometrie Was published before it was generally known.
La geometrie treats many subjects with supreme skill, but it starts with
an introduction to analytic geometry that was not easy for the uninitiated
to follow. Notwithstanding this fact, the work had a tremendous influence,
especially after van Schooten had published it in Latin translation
and with commentaries in 1659. Its success was mainly due to
Descartes's notation, which bore the hallmark of genius. It will not
surprise the modern reader, as it is the beginning of the notation still
in use; but for the time it was revolutionary. There is no doubt
that the notation and the thoughts embodied in La geometrie had a
positive-if only indirect--influence on the development of the calculus.
1.5. Descartes's method of determining the normal, and Hudde's rule
In La geometrie Descartes described his technique of determining the
normal to an algebraic curve at any point. He attached great importance
to the method, as can be seen from the following introductory remarks
(1637a, 341; 1925a, 95) :
This is my reason for believing that I shall have given here a
sufficient introduction to the study of curves when I have given a
general method of drawing a straight line making right angles with
a curve at an arbitrarily chosen point upon it. And I dare say that
this is not only the most useful and most general problem in
geometry that I know, but even that I ever desired to know.
Let the algebraic curve ACE be given and let it be required to draw
the normal to the curve at C (see figure 1.5.1). Descartes supposed the
line CP to be the solution of the problem. Let CM x, AM= y,
Figure 1.5.1.
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 133 of 479.
1.5. Descartes on determining the normal, and Hudde's rule 17
AP=v and CP=s. Although he always used a particular example,
for the sake of convenience we &hall suppose the curve to have the
following equation:
x=f(y). (1.5.1)
We shall also modernise his notation to some extent.
Besides the curve, Descartes considered the circle Cl' with centre at P
and passing through C; that is, the circle with the equation
(1.5.2)
This circle will touch the curve CE at C without cutting it, whereas the
circle cQ
(1.5.3)
with centre at a point Q different from P and passing through C will
cut the curve not only at C but also in another point. Let this point
be E. This means that the equation obtained x from
(1.5.1) and (1.5.3),
(1.5.4)
has two distinct roots; 1 but' the more C and E approach each other,
the smaller the difference of the two roots, and at last, when the points
coincide, the roots are exactly equal, that is to say when the circle through
C touches the curve at the point C without cutting it ' (Descartes 1637a,
346-347; 1925a, 103-104).
Thus the analysis has brought Descartes to the conclusion that CP
will be a normal to the curve at C when P (that is, v) is so determined
that the equation
(f(y»2 +(v - y)2 - S2 = 0 (1.5.5)
has two roots equal to Yo (or the corresponding equation with y eliminated
has one pair of equal roots). With modern conceptions it is not
difficult to realise that this requirement gives the correct expression,
(1.5.6)
for the sub-normal MP.
Descartes illustrated his method by finding, among other things,
the normal to the ellipse (1637a, 347; 1925a,104). Putting its equation
in the form
r
x2=ry-- y2,
q
he found the equation corresponding to (1.5.5) to be
(1.5.7)
1 Descartes only considered curves for which (j(y»2 is a polynomial in y or y2 a
polynomial in x.
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 134 of 479.
18 1. Techniques of the calculus, .1630-.1660
(
rq - 2vq) qv2
-- qs2
y2+ y+ =0.
q-r q-r
(1.5.8)
This equation has two roots equal to Yo when
rq-2vq.. 2
q-r -:Yo and (1.5.9)
because the point C is given, the value Yo is known, and from (1.5.9)
the sub-normal v --Yo can be determined:
r r
v-Yo=:z-qYo. (1.5.10)
Although an indication, not to say a full account, of what happens
when the two points C and E coincide would involve limit-considerations,l
Descartes, by taking the double contact of the circle with the
curve as a characteristic of the normal, has avoided the use of infinitesimals
and obtained an algebraic method. His correspondence
indicates that in solving some of his problems he did employ methods
which involved the use of infinitesimals. However, he did not consider
them precise enough to be published.
In principle, Descartes's method is applicable to any algebraic curve.
But when the equation of the curve is not a simple algebraic equation,
the method becomes tedious because of the laborious calculations which
it is necessary to carry out in order to determine v by comparing the
coefficients.
The Dutch mathematician (later Burgomaster of Amsterdam)
Johann Hudde invented a rule for determining double roots. He
described his method in a letter to Frans van Schooten, who published
it in his 1659 Latin edition of Descartes's La geometrie (Hudde 1659a,
507) :
If in an equation two roots are equal, and if the equation is
multiplied by any arithmetical progression in such a way that the
first term of the equation is multiplied by the first term of the
progression and so on, I say that the product will be an equation
in which the given root is found again.
1 If we let the coordinates of E be (Yo+Ll.Y,!(Yo+Ll.y», then the requirement that
C and E be on the same circle with centre at Q on the axis gives us the condition:
AQ= +Ll.y + (!(yo+Ll.y)-!(yo») (!(yo+Ll.Y)+f(Yo»)
Yo 2 Ll.y 2 '
(To obtain this result, let F be the mid-point of CE and note that QF ..LCE.) P and v
are then determined by the coincidence of the points C and E, that is :
v =AP = lim A Q =!'(yo)f(yo) +Yo.
LJ.y-..O
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 135 of 479.
1.5. Descartes on determining the normal, and Hudde's rule 19
Fot this rule Hudde gave a proof which in modern notation may be
rendered as follows. Let x = Xo be a double root in the polynomial
p(x), that is,
n
p(x)=(X-XO)2 L Cl(iXi
;=0
n
L Cl(i(Xi+·2 - 2xOxi ..j-l.+ X0
2Xi ),
;=0
(1.5.11 )
and let a, a+d, ..., a+(n+2)d be an arbitrary arithmetical ",",,,,.,.,,,Q
We then multiply the constant term Cl(OX02 in p(x) by a, the term of the
first degree by a +d, and so on. Let the result of this procedure be
denoted by (p(x), a, d); that is,
n
(p(x), a, d)= L Cl(i{(a+(i-I-2)d)xi+2-2(a+(i+ l)d)xoxi +l
;=0
(Note that
(p(x), a, d) = ap(x) +dxp'(x), (1.5.13)
where p'(x) is the derivative of p(x) and ' dx ' means dx x.) If we put
Xo x, the expression in curled brackets in (1.5.12) vanishes. We
therefore have (p(xo), a, d) = O.
This necessary condition for a polynomial to have one pair of equal
roots made Descartes's method easier to apply, because one might so
arrange the arithmetical progression that a difficult term might be
multiplied by O. We see that in his studies in autumn 1664 Newton
found the sub-normal to a curve by using a combination of Descartes's
method and Hudde's rule (Newton Papers, vol. 1,217 H.).
Hudde applied his rule to the determination of extreme values,
acting on the assumption that if C(. is a value which makes p(x) extreme,
then the equation p(x) = p(C(.) has two equal roots (see Haas 1956a,
250-255). He also extended his procedure to a rule for determining
sub-tangents (1659b). He did not prove this rule, but it is interesting
because it is one of the first general rules. Let the equation of the
curve be p(x, y) = 0, where p is a polynomial in x and y; Hudde's rule
then states that the sub-tangent t to a point (x, y) is given by
-x(p(x,y), a, d)y
t= .
(p(x, y), a, d)x
(1.5.14)
The subscripts mean that in the numerator p(x, y) is to be considered
as a polynomial in y and in the denominator as a polynomial in x.
From (1.5.13) we have
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 136 of 479.
20 1. Techniques of the calculus, 1630-1660
t= -x(ap(x, y) +dypy'(x, y»
ap(x, y) +dxpx'(x, y)
(1.5.15 )
(where the prime indicates differentiation with respect to the subscript
variable), or, since p(x, y) =0,
t= -ypy'(x, y).
Px'(x, y)
(1.5.16)
Hudde's method was not forgotten after the introduction of the
differential calculus; for example, l'Hopital commented on it in his
1696a, ch. 10, para. 192 (see also section 2.5 below).
1.6. Roberval's method of tangents
In the late 1630s Gilles Personne de Roberval and Evangelista Torricelli
independently found a method of tangents which used arguments from
kinematics. In 1644, in his Opera geometrica, Torricelli published an
application of his method to the parabola (Torricelli 1644a, 119-121 ;
Works, vol. 2, 122-124). In the same year Mersenne, in his Cogitata
physico mathematica (' Physico-mathematical thoughts '), mentioned
Roberval's method and applied it also to the parabola (Mersenne 1644a,
115-116; see Jacoli 1875a). One of Roberval's pupils, Franyois du
Verdus, wrote a treatise on Roberval's method. It was eventually
published in 1693 (Roberval Observations) and became quite wellknown,
so the kinematic method came to bear Roberval's name.
The method rests on two basic ideas. The first is to consider a
curve as the path of a moving point which is simultaneously impressed
by two motions. The second is to consider the tangent at a given point
as the direction of motion at that very point. If the two generating
motions are independent, then the direction of the resultant motion is
found by the parallelogram law for compounding motions. However,
Roberval also applied his method to curves like the quadratrix and the
cissoid, where the generating motions which he considered were dependent.
He ingeniously compensated for the dependence when compounding
the motions, as we shall see.
Roberval succeeded in determining the correct tangents to all the
curves which were generally considered at his time. For the conic
sections, however, the tangents were not determined correctly, because
he took the generating motions to be the motions away from the foci or
from the focus and the directrix, and wrongly used the parallelogram
rule in compounding these motions (see Pedersen 1968a, 165 ff.).
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 137 of 479.
1.6. Roberval's method of tangents
H
Figure 1.6.1.
21
To illustrate the method, we shall first see how Roberval determined
the tangents to the cycloids (Roberval Works2a, 58-63). Let ABC be
a cycloid generated by the circle AD; that is, ABC is the path of the
point A when the circle makes one turn on the line AC (compare figure
1.6.1, where the ordinary cycloid is drawn). The motion of A is then
compounded of a uniform motion with direction AC or and a uniform
rotation about the centre of the generating circle, the direction of
this at a point E being the tangent to the generating circle at E or the
line FH. The ratio between the speeds of these motions is equal to
the ratio between AC and the perimeter ADA, so if the point H is
determined by
EF: FH =AC: perimeter ADA, (1.6.1)
then EH will be the tangent to the cycloid at E. For the ordinary
cycloid, the ratio on the right hand side is equal to unity, and Roberval
proved geometrically that EH is parallel to FB.
Thus the method is easily applied to the cycloid; but to see how
general it is, let us also consider Roberval's determination of the tangent
to the quadratrix. In figure 1.6.2 we let the two sides AD and CD
of a square ABCD move simultaneously, AD being rotated uniformly
about A and CD being paralleledly displaced in such a way that AD
and CD coincide with AB at the same time. The point of intersection
between the two lines will then describe a quadratrix DFH. Let F~
the point of intersection between IN and ADl~be one of the points of
the quadratrix and let us see how he determines the tangent at F.
(Actually he considers a point on DFH's prolongation, but the principle
is the same.)
Robcrval starts by letting the line FK represent the velocity of the
line IN. From the definition of the quadratrix follows that F describes
the line FK in the same time as D1 describes the arc DiB, whence arc
DIB represents the speed of D1's circular motion. As
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 138 of 479.
22 1. Techniques of the calculus, 1630-1660
R
s
Figure 1.6.2.
(the speed of the circular motion of F) : (the speed of the
circular motion of DI) = AF: ADI arc FG : arc D1B, (1.6.2)
the arc FG represents the speed of F's circular motion; and further, as
the direction of this latter motion is perpendicular to AF, the circular
motion of F will be represented by the line-segment FR on the perpendicular
with length equal to arc FG. To obtain F's direction of movement
he then draws the line RS through R parallel to AF and seeks the
point of intersection, M, between RS and AB (which is the line through
K parallel to IF) and connects F and M. FM will then be the tangent.
Roberval used this general approach in other cases too. His argument
for it is not quite clear, but it has a great deal in common with the
following. F's motion s;an be considered in two ways:
(1) F's motion on the quadratrix is compounded of the motion F
has by taking part in AF's motion (with the instantaneous velocity FR)
and the motion F has on AF because it has to be the point of intersection;
the direction of the last motion is AF or RS. By compounding
these two motions we see that the line of direction of the movement of F
starts at F and ends on the line RS.
(2) Similarly, it is realised, by compounding the motion F has when
it takes part in the motion of IF with its motion on IF, that its direction
of motion is a line starting at F and ending on AB.
As both the conclusion of {I) and (2) must be fulfilled, the above construction
follows.
By taking the instantaneous direction of motion as known, Roberval
and Torricelli had avoided the use of infinitesimals in their method.
Their method had the further advantage of being applicable to curves
which are not referred to a Cartesian coordinate system. The method,
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 139 of 479.
1.7. Fermat's method of maxima and minima 23
however, was not general as long as the velocities could not be generally
determined.
It is interesting to note that Newton's method of tangents from 1666
is inspired by the same ideas as Roberval's. For algebraic curves
Newton only had to use the method once to obtain the sub-tangent
expressed by a formula; but for transcendental curves like the quadratrix
he found the tangent in almost the same manner as had Roberval
(Newton Papers, vol. 1,416-418).
1.7. Fermat's method of maxima and minima
About 1636 there was circulated among the French mathematicians a
memoir of Fermat entitled Methodus ad disquirendam maximam et
minimam (' Method of investigating maxima and minima': Methodus).
It was remarkable, for it gave the first known general method of determining
extreme values. It contained another striking feature, namely,
the idea of giving an increment to a magnitude, which we might interpret
as the independent variable.
The memoir opens with the sentence: 'The entire theory of determining
maxima and minima is based on two positions expressed in
symbols and this single rule'. The rule is the following:
I. Let A be a term related to the problem;
11. The maximum or minimum quantity is expressed III terms
containing powers of A ;
II I. A is replaced by A +E, and the maximum or minimum is then
expressed in terms involving powers of A and E ;
IV. The two expressions of the maximum or minimum are· made
, adequal " which means something like 'as nearly equal as
possible' ;1
V. Common terms are removed;
VI. All terms are divided by a power of E, so that at least one term
does not contain E;
VII. The terms which still contain E are ignored;
VIII. The rest are made equal.
The solution of the last equation will give the value of A which
makes the expression take an extreme value. Fermat illustrated his
method by finding the point E on the line-segment AC which makes
the rectangle AE. EC a maximum. Let AC b and let us replace
Fermat's A by x (so that AE=x), and his E bye; we then have to
1 Fermat used the word' adaequo '. Mahoney has translated this as ' set adequal '
(1973a, 162). The idea of adequality derives from Diophantus (ibid., 163-165).
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 140 of 479.
24 1. Techniques of the calculus, 1630-1660
maximize the expression x(b- x). In accordance with the method, we
have
(x+e)(b-(x+e)) ~x(b-x), (1.7.1)
where ~ signifies the adequality. Removing common terms, we have
and dividing bye,
b ~2x+e.
Finally we ignore the term e and obtain b= 2x.
(1.7.2)
(1.7.3)
It is tempting to reproduce Fermat's method by letting A x,
E = !'ix, and the quantity = f(x); the rule then tells us
IV, V
VI
VII, VIII
f(x +!'ix) - f(x) ~ 0,
f(x +!'ix) - f(x) '"" 0
!'ix ~ ,
(
f(X+!'iX) f(X)) =0.
!'ix L>x =0
(1.7.4)
(1.7.5)
(1.7.6)
For differentiable functions this might be interpreted in modern terms
as if the x which makes f(x) a local extreme value is determined by the
equation
f(x)= lim {f(X+!'iX)--f(X~} =0.
L>x--+O !'ix
(1.7.7)
However, this would be to read too much into the method. Primarily,
Fermat did not think of a quantity as a function. Secondly, he did not
say anything about E being an infinitesimal, or even a small magnitude,
and the method does not involve any concept of limits; it is purely
algebraic. Thirdly, the statement in VI makes no sense in this interpretation,
as we always have to divide by E to the first degree.
Nevertheless, his examples show us that on occasion he divided by
higher powers of E than one. The reason for this is that, if the quantity
contained a square root, he squared the adequality before applying the
last steps of the rule. Note that he did not emphasise that his method
gave only a necessary condition.
Few results in the history of science have been so closely examined
as Fermat's method of maxima and minima. He wrote about a dozen
short memoirs where he explained and applied his method. Historians
have been puzzled by his very short descriptions, and disagree about the
dating of the memoirs and about the order of his ideas. To me it seems
probable that he developed his ideas in the way that he intimated in his
manuscript 'Syncriseos et anastrophes' (Syncriseos; see Mahoney
1973a, 145-165).
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 141 of 479.
1.7. Fermat's method of maxima and minima 25
Fermat says here that he got the idea of a process for determining
extreme values by studying Viete's theory of equations and combining it
with the expression' fLovrxX6, , used by Pappus to characterise a minimal
ratio (see Pappus Collections, book VII, theorem 61). Fermat takes
, fLovrxxo, , to mean' singular' in the sense of ' unique' (see his Works,
vol. 1, 142, 147), and gives an illustrative examplc of what he meant.
The line-segment of the length B has to be divided by a point so that
the product of the segments is maximum. The required point is the
midpoint which makes the maximum equal to B2/4. If Z < then
the equation
X(B--X)=' Z (1.7.8)
will have two roots. Let them be A and E. Following Viete, Fermat
obtains
A(B--A)=E(B E) (1.7.9)
or
(1.7.10)
By dividing by A -- E, it is seen that B = A + E. The closer that Z
approaches B2/4, the smaller will be the difference between A and E;
at last, when Z = B2/4, A will be equal to E, and B = 2A, which is the
unique solution leading to the maximum product. In other words, to
find the maximum you have to equate the two roots.
As it can be complicated to divide by the binomial A - E, Fermat
chose to let the two roots be A and A +E; then he divided by E,
and finally equated the two roots by putting E = O. After these considerations
he repeated his procedure from Methodus sketched in I-VIII
at the beginning of this section. In this procedure he did not put E = 0,
but ignored the terms still containing E. However, the process is the
same, and it became common practice to put E, or a corresponding
magnitude, equal to 0 when his method was applied.
Until it was realised that the important process is
lim {f(X +t.x) - f(X)},
L'.x--+O t.x
(1.7.11)
the procedure that involved dividing by E and putting E = 0 was a
thorn in the mathematicians' side. They were severely criticised for it,
and they admitted that it was unsatisfactory.
Huygens who knew, applied and simplified Fermat's method, tried
in vain to justify it logically (manuscript from 1652 printed in Huygens
Works, vol. 12, 61). Instead he found another method, and one of
which he could give a proof (ibid., 62 ff.). This method combined
Fermat's idea of an extreme value as unique with Descartes's idea of a
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 142 of 479.
26 1. Techniques of the calculus, 1630-1660
double-root which he used in his method of normals. Briefly and in
modern terms: Let p(x) be a polynomial and let p(xo) be a maximum;
when a <p(xo), the equation p(x)=a has two roots which will be equal
when a =P(xo). By a comparison of coefficients, Xo may then be
determined from the relation
(1.7.12)
where Pl(X) is again a polynomial. As the applicability of this method
is very limited, and as it is intricate to use, Huygens admitted that
Fermat's method was easier to operate, and he himself accepted it.
Among others, Pierre Brulart requested Fermat to give a proof of
his method. In his answer 1643a Fermat took another line, considering
the coefficients of the powers of E in the development of f(A ±E).
Although he could not prove it rigorously, he made it seem plausible
that a maximum or minimum can be determined from the equation
obtained by putting the coefficient of E equal to O. Further, he showed
that he understood that the coefficient of E2 must be smaller than 0 for a
maximum and greater for a minimum.
To Fermat it was more important to see that a method worked in
practice than to give an exact proof of it. The method of maxima and
minima had proved its value, for it gave the correct results when applied
to a series of problems. Among these was the determination of the
points of inflection of a curve in the manuscript' Doctrinam tangentium '
(Fermat Works, vol. 1, 166-167).
Fermat, however, did not stop at that; he extended the use of the
procedure II I-VI II from Methodus to other fields. This enabled
him to determine tangents to curves (as will be seen in the next section),
centres of gravity (1638a), and the sine law of refraction (1662a).
1.8. Fermat's method of tangents
In Methodus, Fermat made a determination of the tangent to the parabola,
and presented this as an application of his method of maxima and
mmIma. Before discussing the method we shall consider the example
(Fermat Works, vol. 1, 134-136). Let the parabola DB with axis DC
be given as in figure 1.8.1. Fermat wants to find the tangent at B ;
suppose it to be BE, and let the sub-tangent be EC. He takes an
arbitrary point 0 on BE and draws 10 parallel to the ordinate BC.
Let P be a point of intersection of 10 with the parabola.
From the inequality 10> lP, and from the property of the parabola
DC: DI=CB2: IP2, (1.8.1)
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 143 of 479.
1.8. Fermat's method of tangents
Figure 1.8.1.
it follows that
DC: D1> CB2: 102•
Since the triangles E10 and ECB are similar, we have
CB2 : 102 = EC2 : E12.
Thus
DC: D1> EC2: E12.
27
(1.8.2)
(1.8.3)
(1.8.4)
Let DC = x (x is known since the point B is given), EC = a (the unknown
quantity) and 1C=e. Then (1.8.4) becomes
x: (x-e»a2 : (a-e)2,
or
Fermat replaces this inequality by the adequality
(1.8.5)
(1.8.6)
(1.8.7)
By using the procedure of the method of maxima and minima he obtains
a = 2x, and thereby determines the tangent.
In a letter to Mersenne of January 1638 Descartes objected to this
determination, maintaining that it did not solve the problem of an
extreme value (see Fermat Works, vol. 2, 126-132, or Descartes Works,
vol. 1, 486-493). He also accused Fermat of not having used the
specific property of the curve, so that the determination would give the
same result for all curves. The last objection is clearly wrong, and may
be ascribed to the hostile attitude which Descartes took to Fermat after
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 144 of 479.
-
28 1. Techniques of the calculus, 1630-1660
Fermat had criticised his Dioptrique (1637a). The first objection,
however, is worth examining.
The inequality 10> IP holds for curves concave with respect to
the axis, and the inequality 10 < IP for convex curves. For curves
without points of inflection it is possible from these inequalities to find a
magnitude depending on a - e and x - e which has an extreme value for
x-e=x (see Itard 1947a, 597, and Mahoney 1973a, 167). As x( =DC)
is known, a may be determined from the requirement for an extreme
value. Neither in Methodus nor in Fermat's later writings, however,
is there any indication that this was the way he related his method of
tangents to his method of maxima and minima. In the memoir
1638b of June 1638, Fermat, after having explained his method, wanted
to show that there was a relation between the method of maxima and
minima and that of tangents. However, by solving a problem of
extrema he did not find the tangent to the curve, but rather the normal.
This gave an algorithm quite different from the one used in Methodus
and explained in the memoir. He is therefore not likely to have used
this relation when he established his method of tangents. (By the way,
the problem of extreme values which Fermat solved was suggested by
Descartes in his first attack on Fermat's method.) So Descartes was
right after all in raising the objection that the method of tangents was
not a direct application of the method of maxima and minima.
When, in the memoir just mentioned, Fermat explained his method of
tangents to Descartes, he clearly showed that he used only the procedure
drawn from the method of maxima and minima. Descartes thereafter
accepted the method. In modern notation Fermat's explanation can
be reproduced in the following way. Let B be the point (x, y) on the
curve f(x,y)=O and let DI=x-e (see figure 1.8.1). From the similar
triangles EOI and EBC we obtain
(1.8.8)
Since 10 is almost equal to PI, Fermat writes
(1.8.9)
This is the adequality to which he applied his procedure from the method
of maxima and minima. It is not difficult to see that it will lead to an
expression for a corresponding to
a = _yfll'
f:c' .
(1.8.10)
rb
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 145 of 479.
1.8. Fermat's method of tangents
If we have the parabola ax=y2, we obtain from (1.8.9)
y2(a-e)2
a(x e)---2--::::00,
a
or
and since y2 = ax, then
which is (1.8.7).
As the method requires a development of
(
y(a-'-.e))
f x--e, --a-- ,
29
(1.8.11)
(1.8.12)
(1.8.13)
it was in its original presentation only applicable to curves
(because in Fermat's time only algebraic functions were developed).
However, in 'Doctrinam tangentium' Fermat extended its field of
application to include some transcendental curves. He introduced two
principles (Fermat Works, vol. 1, 162), stating that it was allowed
(1) ... to replace the ordinates to the curves by the ordinates
to the tangents [already] found ...
(2) ... to replace the arc lengths of the curves by the corresponding
portions of tangents already found . .. .
These two principles enabled him to determine the tangent to the cycloid
(ibid., 163). Let HCG be a cycloid with vertex C and generating circle
CMF (figure 1.8.2), and RB be the tangent at an arbitrary point R.
For the sake of convenience we reproduce his analysis with use of some
modern symbols. Let CD=x, RD=f(x), MD =g(x), and the magnitude
to be investigated DB a. The specific property of the cycloid is the
following:
f(x)=RM+MD=arc CM+g(x). (1.8.14)
Let DE = e, and draw NE parallel to RD intersecting RB at N and the
circle at 0; as usual in the method of tangents, we have that
NE=f(x)(a e) ::::of(x e),
a
(1.8.15)
where
f(x-e)=arc CO+g(x-e)=arc CM--arc OM+g(x-e). (1.8.16)
Let MA be the tangent to the circle at M intersecting NE at V, and let
MA=d and AD=b.
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 146 of 479.
30 1. Techniques of the calculus, 1630-1660
A
Figure 1.8.2.
From the first principle Fermat obtains
(1.8.17)
and from the second
(1.8.18)
Thus
f( ) CM
de g(x)(b- e)
x e ::::oarc 7;+ b ' (1.8.19)
which together with (1.8.14) and (1.8.15) gives
(arc CM +g(x»(a - e) CM de g(x)(b - e)
a ::::0 arc -7;+ b . (1.8.20)
Hence, by the standard procedure,
arc CM +g(x) d +g(x)
a b
(1.8.21)
or
f(x) d+g(x)
a b
(1.8.22)
Geometrically it is seen that
iOn
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 147 of 479.
1.9. The method of exhaustion
d+g(x) g(x)
---b- =---;;so
that the tangent at R is parallel to MC.
1.9. The method of exhaustion
31
(1.8.23)
The method of geometrical integration which was considered in the
first part of the 17th century to be ideal was the exhaustion method,
which had been invented by Eudoxus and improved by Archimedes.
The name is unfortunate because the idea of the method is to avoid the
infinite, and the method therefore does not lead to an exhaustion of the
figure to be determined, as will be seen from the following outline of the
idea behind it (see Dijksterhuis 1956a, 130-132).
The method aims at showing that an area, a surface or a volume to
be investigated, X, is equal to a known magnitude of the same kind K
(for example, X may be the surface of a sphere and K four circles
on the sphere). A monotone ascending sequence In and a monotone
descending sequence Cn of, respectively, inscribed and circumscribed
figures to X are constructed. Thus we have the result :
for all n, In < X < Cn' (1.9.1)
It is then shown either that for any magnitude € > °there exists a number
N such that
(1.9.2)
or that for any two magnitudes of the same kind fk and v where fk > v> 0,
there exists a number N such that
and further that
for all n, In < K < Cn'
(1.9.3 )
(1.9.4)
From (1.9.1), (1.9.2) or (1.9.3), and (1.9.4), it follows by a reductio ad
absurdum that K = X.
This last demonstration always proceeds in the same manner, independent
as it is of the magnitudes in question. Nevertheless, whenever
applying the method, the Greek mathematicians wrote out the argument
down to the last detail. The reason may be that they did not have a
notation which made it easy for them to deal with the general case.
Furthermore, it is rather complicated to establish the basic inequalities
of the proof, especially (1.9.4), and the method can be used only if K
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 148 of 479.
32 1. Techniques of the calculus, 1630-1660
is knQwn in advance. This means that it needs to. be supplemented by
anQther methQd, if results are to. be prQduced.
AmQng the mathematicians Qf the early 17th century there was a
desire to. find such a methQd Qf Qbtaining results which, in CQntrast to.
the methQd Qf exhaustiQn, WQuld be direct. It WQuld be as well if the
new methQd, apart frQm giving results, CQuld be used to. prQve the
relatiQns achieved. Such a direct methQd might have been obtained
had it been realised that
lim Cn = lim In' (1.9.5)
n-+CiJ n-rOO
and had X been put equal to. that limit; hQwever, this was nQt within
the style Qf expressiQn and PQwer Qf abstractiQn Qf 17th-century mathe-
maticians.
The path which they fQllQwed was that Qf an intuitive understanding
Qf the geQmetric magnitudes. They imagined an area to. be filled up,
fQr example, by an infinite number Qf parallel lines. When, in 1906,
Heiberg fQund Archimedes's The method, it was discQvered that
Archimedes tQQ had adQpted this PQint Qf view in his search fQr results.
HQwever, he did nQt regard it as sufficiently rigorous to be applied in
proofs. Kepler, too., had used techniques invQlving such intuitive CQnsiderations,
and it was the purpose of the first systematic expositiQn of
the methQd Qf indivisibles to legitimise such techniques. This expositiQn,
Geometria indivisibilibus continuorum nova quadam ratione promota
(' GeQmetry by indivisibles of the cQntinua advanced by a new method' :
1635a, hereafter referred to. as Geometria), by Cavalieri, appeared in
1635, when he was a professor Qf mathematics at the University of
Bologna. The ideas that it contained were developed in 1627, as can
be seen in a letter from Cavalieri to. Galileo. (Galileo Works, vol. 13,
381 ).
The mathematicians differed on the importance to attach to a proof
by the methQd Qf indivisibles. Most of those who thought about the
matter regarded the method of indivisibles as heuristic, and thQught that
an exhaustion proof was still necessary. The exhaustion method was
therefore mQdified and extended during the 17th century (see Whiteside
1961a, 333-348). In many cases, hQwever, mathematicians confined
themselves to the remark that the results achieved by the method Qf
indivisibles could be easily demQnstrated by an exhaustion proof.
1.10. Cavalieri's method of indivisibles
Geometria, and Cavalieri's later wQrk Exercitationes geometricae sex
(' Six geometrical exercises': 1647a), became well-known among
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 149 of 479.
1.10. Cavalieri's method of indivisibles 33
mathematicians. The works inspired many of them to find their own
methods, whereas others like Fermat and Roberval found their integration
methods independently of Cavalieri.
Cavalieri presented two methods of indivisibles in his Geometria,
and called them the 'collective' and the 'distributive' methods respectively.
The first six of the seven books of Geometria embody the
collective method, and a summary of it is given in Exercitationes, Book I.
The framework of this section cannot possibly allow for a full account of
the wide spectrum of concepts and ideas which Cavalieri introduced
and developed in these six books, but th~ following outline gives a
rough idea of his approach.
Figure 1.10.1.
Let there be given a plane figure F = ABC limited by the curve
ABC, and the straight line AB, called the 'regula' (figure 1.10.1).
Cavalieri imagined that a straight line starting along AB is uniformly
displaced parallel to AB, and considered the bunch of parallel linesegments
which made up the section between F and the line during the
motion. He named these line-segments' all the lines of the given figure'
(' omnes lineae propositae figurae '), 'and sometimes referred to them as
, the indivisibles of the given figure'; let us denote them by (f)F(l).
Expressed in modern terms, Cavalieri constructed a mapping
(1.10.1 )
from the set of plane figures into a set consisting of bunches of parallel
line-segments. He then extended Eudoxus's theory of magnitudes (see
book V of Euclid's Elements) to include his new magnitudes {(f)F(l)}.
Thereafter he established-although not in a mathematically satisfactory
manner-the fundamental relation
(1.10.2)
between two plane figures (Cavalieri 1635a, Book Il, Theorem 3).
By letting the regula be a plane he obtained in a similar way the relation
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 150 of 479.
34 1. Techniques of the calculus, 1630-1660
SI: S2={9S,(P): (9S2(P), (1.10.3)
where Si is a solid and (9s,(p) all the planes belonging to it, i = 1, 2.
Cavalieri's aim was to find the ratio on the left hand side of (1.10.2)
by calculating the ratio on the right hand side. In doing so he was
greatly helped by a postulate which leads to 'Cavalieri's theorem'
(described below), a skilful use of previous results, theorems about
similar figures, and the concept of powers of line-segments.
The postulate (1635a, Corollarium to Theorem 4 of Book II) states
that if in two figures Fl and F2 with the same altitude every pair of corresponding
line-segments (that is, line-segments at equal distances from
the common regula) has the same ratio, then {9Ji'1 (I) and (9Ji',(l) have this
ratio too. In modern notation and using figure 1.10.2,
Figure 1.10.2.
if fl(x): f2(X)=b: c for all x O<x<a,
then (9Ji'JI): (9Ji',(l)=b: c. (1.10.4)
This, together with (1.10.2), immediately gives' Cavalieri's theorem':
If fl(x): f2(X)=b: c for all x O<x<a,
(1635a, Book II, Theorem 4).
Cavalieri's skilful employment of his previous results may be illustrated
by a simple example. It is easily realised from figure 1.10.3
that
From these relations follows the theorem that the parallelogram ACDF
is the double of each of the triangles ACF and CDF. However,
Cavalieri was capable of interpreting them in a more general way.
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 151 of 479.
1.10. Cavalieri's method of indivisibles 35
Figure 1.10.3.
By setting AC = CD and using concepts which we cannot go into here
he obtained a result which he could use every time he needed a proportion
corresponding to
a a
J X dx: J a dx = 1 : 2 (1.10.7)
o 0
(1635a, Corollarium I I to Theorem 19 of Book I I: compare figure
1.10.4).
Figure 1.10.4.
Cavalieri found an alternative to integrating x2 by introducing the
squares of line-segments. If, instead of considering the line-segments
of (l)F(I), we take their squares situated in parallel planes, we obtain
what he called' all the quadrates of the given figure' (' omnia quadrata
propositae figurae '); this aggregate will be denoted by (l)F( Ol).
Let us illustrate the use of this concept by an example. For each I
in the parallelogram ACGE in figure 1.10.5 we have
OR,T,+ OTIVI=20RIS,+20T1Sl, (1.10.S)
where OR,T, means the square on the side R1T1. From this relation
Cavalieri concluded that
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 152 of 479.
36 1. Techniques of the calculus, 1630--1660
r-______T-____~~C
Figure 1.10.5.
(9AEa( Of) +(9CEG( Of) =
2(9ABFE( Of) +2((911iEF'( Of) +(9CBM( Of». (1.10.9)
Since the triangles AEC and CEG are congruent, we have
and similarly
(1.10.10)
(1.10.11)
He further proved that, since the triangles CEG and MEF are similar,
the following relation holds:
(9CEG(OI): (9MEF(OI)=EG3: EF3=8: 1. (1.10.12)
In the same way he found that
(9ACGE(OI): (9ABlt'E(OI)=EG2: EF2=4: 1. (1.10.13)
From (1.10.9)-(1.10.13) it follows that
(9ACGE( 01) = 3(9CEG( 01) (1.10.14)
(1635a, Book II, Theorem 24). This result has as an immediate consequence
that a cylinder is three times the inscribed cone. Cavalieri
applied the relation (1.10.14) to a series of problems concerning conics,
interpreting it by analogy with (1.10.6) as a relation which was an
alternative to
(1.10.15)
The first six books of Geometria are in their general style a copy of
the Greek classical mathematical works, built up of definitions and
postulates from which the theorems are carefully deduced, all verbally.
Although Cavalieri ingeniously used his concepts to obtain many results,
this made the reading of the book rather tedious. Perhaps. he felt this
himself; at least, he wrote to Galileo in 1634 that he composed the
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 153 of 479.
1.11. Wallis's method of arithmetic integration 37
seventh book of Geometria to help those who found the concept of
'all the lines' too difficult (Galileo Works, vol. 16, 113). In this last
book, concerning the distributive method, he turned to a more intuitive
treatment of the indivisibles.
As we saw in the relation (1.10.2), by the collective method Cavalieri
found the ratio between two figures by comparing the aggregates of
indivisibles. In the distributive method, two figures with the same
altitude were compared by comparing corresponding indivisibles. The
basic relation in this theory was Cavalieri's theorem (1.10.5), for which
he gave a new proof without using the concepts from the collective
theory.
A part of the criticism to which Cavalieri's methods were exposed
was levelled against the nature of his indivisibles and the problem of the
structure of the continuum. Some mathematicians took him to mean
that a plane figure was made up of indivisibles and that were linesegments.
This was against the Aristotelian view of a continuum as
divisible into parts of the same kind as the original the parts
again being infinitely divisible. To avoid his seeming error of dimensionality,
they tried tentatively to conceive a plane figure as composed
of rectangles with infinitesimal breadth. But the distinction was of theoretical
interest only, for it remained usual to consider the ratio between
two areas, so that an eventually missing Llx was cancelled by the relation
A
B
Lan
r. bn '
(1.10.16)
where an and bn are the altitudes in the rectangles of which the areas A
and B are composed.
The conception of an area as a kind of a sum r. anLlx did not solve
the problem, because it was still uncertain what was meant by an infinitesimal
magnitude and by an infinite sum. Despite the lack of
rigour in their foundations, the methods were useful insofar as they
provided the mathematicians with new results.
1.11. Wallis's method of arithmetic integration
To determine the area under the spiral of Galileo, Fermat used an
arithmetic quadrature which he described in a letter to Mersenne in
1638 (Mersenne Correspondence, vol. 7, 377-380). In his Traite des
tndivisibles1
Roberval squared many figures on the basis of arithmetical
1 Traite uses a method of infinitesimals which Roberval worked out about 1630.
The date of the composition of the Traite is, however, unknown. It was first printed
in 1693 (Roberval Works!) and reprinted in 1730 (Works2).
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 154 of 479.
38 1. Techniques of the calculus, 1630-1660
considerations. Pascal observed in his treatise Potestatum numericarum
summa (' sum of numerical powers') that his results concerning the
m
sums ~ (A +id)n (where A, dand n are natural numbers) could be
;=0
applied to the quadratures of curves (Pascal Worksl , vol. 3, 364; Works2,
vol. 2, 1272). Using proofs by complete induction he also established
the rules for determining the binomial coefficients (1654a).
But most of the results based on a method of arithmetic integration
were achieved by John Wallis. His treatise on the subject, Arithmetica
infinitorum (' The arithmetic of infinites': 1655a), is not burdened
with proofs, for he relied boldly and confidently on his really astounding
intuition as to the correlation between the sums of different series. He
called his favourite method in the treatise' modus inductionis ': later it
was termed' incomplete induction'. One might also call it ' conclusion
by analogy'.
Wallis started the treatise by establishing by this method that
I
~ i
~-l
(1+ 1)Z- 2,
and similarly that
I
/
~ i2
;=0
(l + 1)12
I
~ i3
i=O _ 1 1
(l + 1)/3 - "4 + 4/'
~ in;=0 1 al an _ 1
(l +1)zn n + +1+' .. Zn-l'
(1.11.1)
(1.11.2)
where the a/s are rational numbers and n=4, 5, 6 (Wallis 1655a.
Propositions I, XIX and XXXIX; Works, vol. 1, 365, 373 and 382),
From this he concluded,that
{ ±in}lim i=O _ 1
l-+co (I+1)In - n +
(1.11.3)
(ibid., 384). This relation enabled him to square the curves y = xn in
figure 1.11.1 to obtain
a {(~)n (~)n (~)n}"~ y I + I +... I
~=lim
a 'n+ n+ n~ b /-rCO a a ... a
x=O
{
I}~ inl' ;=0 1
= l:~ (1+ l)zn = n+
(1.11.4)
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 155 of 479.
1.11. Wallis's method of arithmetic integration
- - - - - - - -- - a-- ---
-- fa-----
b,
v
Figure 1.11.1.
a result which corresponds to
a
J xn dx
o 1
n+
39
(1.11.5)
This result was not new, and indeed it had been found by many of
Wallis's predecessors; but he did not stop there. He extended the
range of n in (1.11.3) to include at least all rational numbers except - 1.
The foundation of his extension is an observation which he made in
connection with the formula (1.11.3), namely: If the numbers 1nl ,
In., .••, [nr are in geometric progression (where nI' n2, ••• , nr are nonnegative
whole numbers), then
{
/-1-1 }
tim i¥O In} , j = 1, 2, ... r,
/-'00 '\"n
1"., 1 }
;=0
(1.11.6)
will be in arithmetic progression (ibid., 387). Further, from the fact
that for O~p~q (q= 1,2,3, ... )
10, [l/q, 12/q, .•• [p/q, ••. [1 are in geometric progression,
11 112 1 P 2 . ' h ' ., +-, +-, . .. +-, . " are In ant metlc progressIOn,
q q q
and the first and last members of the latter sequence are the reciprocals
of the values of the right hand side of (1.11.3) for n = 0 and n = 1 respectively,
he concluded that
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 156 of 479.
40 1. Techniques of the calculus, 1630-1660
{
li1 iPlq
}
1
. ;=0 1lm ._--
I-HA) ±/plq - 1+p.
i=O q
(1.11.7)
(ibid., 390). He did not doubt that the relation (1.11.7) held good for
all plq ~ 0; he even said that it was valid for an irrational exponent,
such as )3 (ibid., 395), and as he extended the concept of power to
include negative powers he considered (1.11.7) to be valid for them too--
except --1 (ibid., 408). By means of (1.11.7), he was now able to
determine, whenPlq was a rational number different from -1, the
ratios between the areas under the curves y = xP Iq and the circumscribed
rectangles. He could also determine the ratios between the volumes
obtained by a revolution of these areas about an axis and the circumscribed
cylinders.
After that, Wallis proceeded to study polynomials in x; he applied
the formula (1.11.7) to binomial expansions of (xP(Dn xn))m when p, n
and m are small natural numbers and D is a constant, and by analogy
deduced that
D
J [xP(Dn_xn)]m dx
o
n. 2n . ... mn
(mp+ l)(mp+n+ 1)(mp+2n+ 1) ... (mp+mn+ 1)
(1.11.8)
(ibid., 419-420 and 425-430), a result which he put into various tables.
(For clarity I render the last of his sums as integrals.) He further
extended (1.11.8) to include the case where p and n are positive rational
numbers (ibid., 433).
One of Wallis's purposes was to square the circle; he stressed that
from (1.11.8) and its extension we know for m = 0, 1,2,3 ... the' sums'
and
and for m= 1,2,3, ... the' sum'
R
(1.11.9)
J (Rl/m_xllm)m dx
o (1.11.10)
where R is the radius and D the diameter of the circle. He wished to
h b
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 157 of 479.
1.11. WaUis's method of arithmetic integration 41
find the values of these' sums' for m = t, and he introduced the symbol
, D ' to signify the reciprocal of (1.11.10):.
D -- --------- =-R2 (4)-- 1(R2- X 2 )1/2 dx Tr'
(1.11.11)
a principle of interpolation which we cannot go into here, he sue-·
ceeded in establishing the formula
where
Rn--l
'R-------- = an
f (R2_ x2)<nI2H dx
o
for n 1, 2, 3, ...,
n 2,4,6, ...
4.6.8 ... (n+l)
an~2=3 D, n=3,5,7, ...
. .5.7. .. n
(1.11.12)
(1.11.13)
(see Prag 1929a, 389-392, and Whiteside 196Ja, 237-241). From the
fact that
for n=l, 2, 3 ..., (1.11.14)
he concluded that the ratio an+l/an is continuously decreasing, 1
so that
and hence
(1.11.16)
From the formulae (1.11.13) he obtained for odd n the inequalities
, Wallis was lucky that his sequence behaved in this way, for a sequence defined by
2n 2n+l
a, = k, a2 = 1, a2n+l = 2;;'1 a 2n -l> and a2n+2 = ~ a 2n
will not generally have an+1/an continuously decreasing.
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 158 of 479.
--
42 1. Techniques of the calculus, 1630-1660
3.3.5.5.7~7 ... (n 2).n.n J(n+2)<D
2 . 4 . 4 . 6 . 6 . 8 ... (n - 1)(n - 1)(n + 1) n + 1
3.3.5.5.7.7 ... (n-2).n.n J(n+l)
< 2 . 4 . 4 . 6 . 6 . 8 ... (n - 1)(n - 1)(n + 1) -n-'
(1.11.17)
In the limit as n -'TOO, these give a result now called' Wallis's product' :
~ = 0 = _3_._3_..,-5_'_..,.-5_.,.-7-::'7_-cc.9_'-,-9-:::-.___._. (1.11.18)
7T
(Wallis Works, vol. 1,469).
1.12. Other methods of integration
Most of the methods of integration in use before the time of Newton and
Leibniz made use of an equidistant sub-division of intervals and compared
the area or volume to be found with a known one, as we have seen
with Cavalieri and Wallis. However, Fermat had a method which
allowed him to make an absolute calculation of an area, employing a
sub-division which meant that the areas of the infinitesimal rectangles to
be summed were in a geometric progression with quotient less than unity.
We may illustrate this by means of an example from his treatise on
quadrature De aequationum, which he wrote about 1658 using ideas he
had already had in the 1640s (see Mahoney 1973a, 243 f.).
Fermat considered the hyperbolas
yxn=k, k is a constant, n=2, 3, 4, .... (1.12.1)
C F
81--_-+
A
--- a ---
---X1-----
Figure 1.12.1.
b
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 159 of 479.
-
1.12. Other methods of integration 43
For convenience I reproduce his arguments in modern terms. He
divided the x-axis to the right of the point G (see figure 1.12.1) in
intervals GH, HO, OM, of lengths Xl - a, X 2 - Xl' X3 - X 2, •••
(a=AG), so that
and hence
He then considered the circumscribed rectangles
R1=b(XI a),whereb=GE, 1
RI' =Y1'-I(Xr - X,._l)' r = 2, 3, ....f
From (1.12.1 )--(1.12.4) it follows that
RI=~
Yl(X2 Xl)
RI' Yr-l(Xr xr- 1 )
R1'+l Y1'(x1'+l ~- x1')
X na X na__1'__ =_1_=
Xr- 1n Xl anxl a
(1.12.2)
(1.12.3)
(1.12.4)
(1.12.5)
(1.12.6)
which means that the circumscribed rectangles are in a geometric progression
with quotient a/xn _ 1 •
To determine the sum S of a geometric progression with first term IX
and quotient u/v (u < v), Fermat used the following relation:
v-u IX
-U-=S-IX (1.12.7)
(this is equivalent to S=IX/(l-u/v». Hence, if S denotes the sum of
the rectangles R1', we have:
-a b(xl-a)
(1.12.8)
a S-b(x1-a)
or
xn_1-a ba
(1.12.9)
Xl a S b(xl-a)'
He then imagined the intervals Xl - a, X 2 - Xl' ... to be sufficiently
small and almost equal, and he concluded that the left hand side of
(1.12.9) by adequality is equal to n 1. Further, as the intervals are
small, he concluded that S-b(x1-a) in the relation (1.12.9) can be
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 160 of 479.
44 1. Techniques of the calculus, 1630-1660
set equal to the area ()' defined by the hyperbola and the lines GH and
GE. Hence
n 1= ba = AG . GE,
a a
(1.12.10)
and the quadrature is achieved.! We could have obtained (1.12.10)
by taking the limit of both sides of (1.12.9) for Xl approaching a, but he
did not use limits. He observed that his method could not be applied
when n ,= 1 as the rectangles will then be equal.
Figure 1.12.2.
1 Fermat called his method' logarithmic' (Works, Vo!. 1, 265). In his time the
word ' logarithmic' was used to characterise a connection between a geometric and an
arithmetic progression; hence' logarithmic' was also used at that time where today we
would say 'exponential'. Let us indicate in modern terms how his expression and
proof can be interpreted. If we let a=exp (to) and xr=exp (to+r~t), r=l, 2, 3, ...,
then we have a sub-division which is equivalent to (1.12.2). An easy calculation shows
that
Hence
and
Rr=k exp [ - (n-l)(to+(r-l)~t)](exp [~t]-l). (1)
00
S= 1: Rr=k exp [-(n-l)to](exp [At]-1) : (l-exp [-(n-1)~tD, (2)
r=1
lim S=(k exp [-(n-l)toD: (n-l)=(a. b) : (n-I) .
.1.t....,.O
(3)
b
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 161 of 479.
1.12. Other methods of integration 45
An ingenious use of geometrical considerations and arguments
from statics led mathematicians to many transformations corresponding
to transformations of integrals, which could be applied to find connec··
tions between various problems solved by quadratures and cubatures.
In his 1658c Pascal systematically drew up schedules in which appear
the sums necessary to determine areas and volumes as well as their
centres of gravity. He found a fundamental theorem for these connec··
tions by conceiving the volume KCAB (see figure 1.12.2) both as
composed of the rectangles FDOO' FD. DO and as composed of the
areas EGl' = ARI (Pascal 1658c, 'Lemme generel' in the section
, Traite des trilignes rectangles'; Works]> vol. 9, 3-5). That is,
I FD. DO= I EG/'. (1.12.11)
AB AC
If we put AB=a, AC=b, AD=x, FD==y f(x) and DO=z=g(x)
(both being monotone functions), the relation corresponds to
a
J f(x)g(x) dx
(}
b (f"(Y) )
J J get) dt dy,
() (}
(1.12.12)
which can be obtained by an integration by parts. Since f(a) 0 we have:
Jf(x)g(x) dx==·- J( j get) dt) f'(x) dx
(} (} ()
=, I(f-r)g(t) dt) dy. (1.12.13)
When g(x) = x, we obtain
a b x2
J xy dx = J - dy.
(} () 2
(1.12.14)
Roberval found the summation form of (1.12.14) in his Traite in a
way similar to that of Pascal (Roberval Works2a, 271), and it was used
by Fermat too (Works, vol. 1, 272). Among other things, it could be
applied to the determination of the centre of gravity of the ar~a i y dx.
()
Let the x-coordinate of this point be t; in modern notation the argument
is the following (see figure 1.12.3). If we consider a lever AC
a
and let the area Jy dx operate on the arm g on the one side, and at the
() a
other let all the rectangles y6.x of the area Jy dx or BDC operate each
(}
on the arm x, then there will be equilibrium. Hence we have
a a
t f y dx = f xy dx. (1.12.15)
(} (}
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 162 of 479.
46 1. Techniques of the calculus, 1630--1660
Figure 1.12.3.
Therefore, by (1.12.14),
b x2
f dy
~=
o
(1.12.16)- - -a
f Y dx
o
which gives the x-coordinate of the centre of gravity. The y-coordinate
can be found in a similar way.
(1.12.16) is equivalent to the relation
b a
7T f x2
dy=27T~ f y dx, (1.12.17)
o 0
which states that the volume obtained by revolving the area BCD
about the axis BD (compare figure 1.12.3) is equal to the product of the
area and the distance traversed by the centre of gravity. This is a
special case of the theorem now known as ' Pappus-Guldin's theorem',
formulated by Paul Guldin in Centrobaryca (1635-1641a, vol. 2, 147)
in the following way: 'the product of a rotating quantity and the path
of rotation [that is, the circumference of the circle traversed by the
centre of gravity], is equal to the quantity generated by the rotation'.
The theorem is also found in Book VII of Pappus's Collections, but it
may be a later addition (see, for example, Ver Eecke 1932a).
1.13. Concluding remarks
The examples given in sections 1.5-1.8 and 1.10-1.12 illustrate the
remark in the introductory section 1.1 about the special character of the
infinitesimal methods in the period 1630-1660. In the case of the
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 163 of 479.
1.13. Concluding remarks 47
methods of quadrature we saw that they were all naturally founded on
the conception of an area as an infinitesimal sum. However, mathematicians
differed in their ways of approaching the problems raised by
that concept. And not only were the methods of the various mathematicians
based on different ideas; some of them also developed
different methods, each one adapted to solve special problems of
quadrature.
Some of the methods of solving tangent or normal problems led to
fixed rules--of which the most general one was Hudde's rule for determining
the sub·,tangent to an algebraic curve---while others only
suggested a procedure. The ideas behind the methods differed widely.
Descartes used an argument about the number of points of intersection
between a cirele and the curve; Fermat employed similar triangles
and the concept of adequality; while Roberval's method was founded
on an intuitive conception of instantaneous velocity and the law of
parallelogram of velocities. The characteristic triangle (with sides ~x,
~y and ~s) did not explicitly play a part in the deduction of the tangent
methods. Nevertheless, it was applied by (for example) Pascal in
connection with a transformation of a sum (see section 2.3); but not
until Leibniz was the importance of this triangle fully recognised.
Thus the period did not in itself bring any perception of basic
concepts which were applicable to the determination of tangents as well
as to quadratures. An important reason why mathematicians failed to
see the general perspectives inherent in their various methods was
probably the fact that to a great extent they expressed themselves in
ordinary language without any special notation and so found it difficult
to formulate the connections between the problem they dealt with.
As an illustration we may consider one of the results achieved by the
different quadrature methods outlined in the preceding sections. This
result can be expressed in modern terms as
a an+1
J xn dx~~-­
o n+ l'
(1.13.1)
where n is a natural number different from - 1. The mathematicians
of that period, however, could not express their result so simply;
they had to refer to areas under special parabolas. Their terminology
did not prevent them from seeing connections such as that between the
rectification of the parabola and the quadrature of the hyperbola, or the
relation of certain inverse tangent problems to quadratures; but it may
have barred their way to a deeper insight into the meaning of these
connections.
These remarks are not to be taken in the negative sense at all. It
is not the task of a historian of mathematics to evaluate the work of
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 164 of 479.
48 1. Techniques of the calculus, 1630-1660
earlier mathematicians by present mathematical standards, nor to
emphasise the inadequacy of their conq~pts as compared to modern
ones. On the contrary, a historian of mathematics ought to enter into
the mode of thought of the period under consideration in order to bring
out the development of the mathematical ideas in its historical context.
Briefly, it may be said that the mathematicians in the period preceding
the invention of the calculus blazed the trail for its invention. They
did so by employing heuristic methods, by making the geometry analytical,
and by seeking methods for solving problems of quadratures and
tangents. 1
1 I am grateful to Dr. John North of Oxford University for correcting some of my
linguistic mistakes, and to Dr. D. T. Whiteside of Cambridge University for his valuable
comments on the manuscript.
Text 9: K. M. Pedersen (1980). “Techniques of the Calculus, 1630–1660”. In: From the
Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by I. Grattan-Guinness.
Princeton and Oxford: Princeton University Press. Chap. 1, pp. 10–48.
Summer University 2012: Asking and Answering Questions Page 165 of 479.
DOES HISTORY HAVE A SIGNIFICANT ROLE TO
PLAY FOR THE LEARNING OF MATHEMATICS?
Multiple perspective approach to history, and the learning of meta level
rules of mathematical discourse
Tinne Hoff KJELDSEN
IMFUFA, NSM, Roskilde University, PO-Box 260, Roskilde, Denmark
thk@ruc.dk
ABSTRACT
In the present paper it will be argued that and proposed how the history of mathematics can play a
significant role in mathematics education for the learning of meta rules of mathematical discourse. The
theoretical argument is based on Sfard’s theory of thinking as communicating. A multiple perspective
approach to history of mathematics from the practice of mathematics will be introduced along with the
notions of epistemic objects and techniques. It will be argued that by having students read and analyse
mathematical texts from the past within this methodology, the texts can function as “interlocutors”. In such
learning situations the sources can assist in revealing meta rules of (past) mathematical discourses, making
them explicit objects for students’ reflections. The proposed methodology and the potential of history for the
learning of meta-discursive rules of mathematical discourse is exemplified by analyses of four sources from
the 17th
century by Fermat and Newton belonging to the calculus, and it is demonstrated how meta level
rules can be made objects of students’ reflections. The paper ends with a proposal for a matrix-organised
design for how the introduced approach to history of mathematics for elucidating meta-discursive rules
might be implemented in upper secondary mathematics education.
1 Introduction
One can think of several purposes for using history in mathematics education: (1) For
pedagogical reasons; it is often argued that history motivates students to learn
mathematics by bringing in a human aspect. (2) As a didactical method for the learning
and teaching of the subject matter of mathematics. (3) For the development of students’
historical awareness and knowledge about the development of mathematics and its driving
forces. (4) For general educational goals, with respect to which the so called cultural
argument makes the strongest case for history, but history can also serve general
educational goals in mathematics education of developing interdisciplinary competences
as a counterpart to specialisation (Beckmann 2009). These purposes are not necessarily
mutually independent. In carefully designed teaching sessions all four of the above
mentioned purposes can be realized in varying degrees.1
Regarding the question whether history promotes students’ learning of mathematics I
have argued in (Kjeldsen 2011), that by adopting a multiple perspective approach to
history from the practice of mathematics, history has potentials in developing students’
mathematical competence while providing them with genuine historical insights. In the
present paper, I will go a step further and suggest that history might have a much more
1
See (Kjeldsen 2010) where it is shown how all these four purposes can be accomplished in problem
oriented and student directed project work. In (Jankvist and Kjeldsen 2011) two avenues for integrating
history in mathematics education are discussed with respect to the development of students’ mathematical
competence and historical awareness anchored in the subject matter of mathematics, respectively, both
within a scholarly approach to history. In (Kjeldsen forthcoming) a didactical transposition of history from
the academic research subject to history in mathematics education is proposed for developing a framework
for integrating history of mathematics in mathematics education.
In Evelyne Barbin, Manfred Kronfellner, and Constantinos Tzanakis, (eds.)
History and Epistemology in Mathematics Education Proceedings of the Sixth European Summer University ESU 6.
Vienna: Verlag Holzhausen GmbH, 2011, pp. 51-62.
Text 10: T. H. Kjeldsen (2011). “Does history have a significant role to play for the
learning of mathematics? Multiple perspective approach to history, and the learning
of meta level rules of mathematical discourse”. In: History and Epistemology in
Mathematics Education. Proceedings of the Sixth European Summer University ESU 6.
Ed. by E. Barbin, M. Kronfellner, and C. Tzanakis. Vienna: Verlag Holzhausen
GmbH, pp. 51–62.
Summer University 2012: Asking and Answering Questions Page 166 of 479.
profound role to play for the learning of mathematics. This suggestion is based on Sfard’s
(2008) theory of commognition.
In the following it will be argued that, and proposed how, the history of mathematics can
play a significant role in the teaching and learning of mathematics. The theoretical argument
is outlined in section 2. In section 3, the multiple perspective approach to history of
mathematics from its practice is presented along with some tools of historians’. The
adaptation for mathematics education is discussed in section 4. The potential of history for
the learning of meta-discursive rules of mathematical discourse is exemplified in section 5
through analyses of four sources from the 17th
century by Fermat and Newton belonging to
the calculus. In section 6 a proposal is outlined for a so called matrix-organised design for
how such an approach to history of mathematics for elucidating meta-discursive rules might
be implemented in upper secondary school. The paper ends with a concluding section 7.
2 The theoretical argument for the significance of history
In Sfard’s (2008, 129) theory of Thinking as Communicating mathematics is seen as a
discourse that is regulated by discursive rules, and where the objects of mathematics are
discursive constructs. There are two kinds of discursive rules both of which are important
for the learning of mathematics: object-level rules and meta-discursive rules.
The object-level rules have the content of the discourse as object. In mathematics they
regard the properties of mathematical objects. The meta-discursive rules have the
discourse itself as object. They govern proper communicative actions shaping the
discourse. The meta-discursive rules are often tacit. They are implicitly present in
discursive actions when we e.g. judge if a solution or proof of a mathematical problem or
statement can count as a proper solution or proof (Sfard 2000, 167). The meta-discursive
rules are not necessary; they are given historically.
The meta-discursive rules are connected to the object-level of the discourse and have an
impact on how participants in the discourse interpret its content. As a consequence,
developing proper meta-discursive rules are indispensable for the learning of mathematics
(Sfard 2008, 202). This means that designing learning situations where meta-discursive rules
are elucidated is an important aspect of mathematics education. History of mathematics is an
obvious method for illuminating meta-discursive rules. Because of the contingency of these
rules, they can be treated at the object level of history discourse, and thereby be made into
explicit objects of reflection. Hence, history might have a significant role to play for the
learning of mathematics, precisely because meta-discursive rules can be treated as objects of
historical investigations. By reading historical sources students can be acquainted with
episodes of past mathematics where other meta-discursive rules governed the discourse. If
students study original sources in their historical context, and try to understand the work of
past mathematicians, their views on mathematics, the way they formulated and argued for
mathematical statements etc. the historical texts can play the role as “interlocutors”, as
discussants acting according to meta rules that are different than the ones that govern the
discourse of our days mathematics and (maybe) of the students. By identifying meta rules
that governed past mathematics and comparing them with the rules that govern e.g. their
textbook, students can be engaged in learning processes where they can become aware of
their own meta rules. In case a student is acting according to non-proper meta rules he or she
might experience what Sfard calls a commognitive conflict, which is “a situation in which
different discursants are acting according to different metarules” (Sfard 2008, 256). Such
Text 10: T. H. Kjeldsen (2011). “Does history have a significant role to play for the
learning of mathematics? Multiple perspective approach to history, and the learning
of meta level rules of mathematical discourse”. In: History and Epistemology in
Mathematics Education. Proceedings of the Sixth European Summer University ESU 6.
Ed. by E. Barbin, M. Kronfellner, and C. Tzanakis. Vienna: Verlag Holzhausen
GmbH, pp. 51–62.
Summer University 2012: Asking and Answering Questions Page 167 of 479.
situations can initiate a metalevel change in the learner’s discourse.
This, of course, presupposes a genuine approach to history. In section 3 and 4 it will be
argued that within a multiple perspective approach to the history of the practice of
mathematics, and by using historian of mathematics’ tools such as the idea of epistemic
objects and techniques, original sources can be used in mathematics education to have
students investigate and reflect upon meta-discursive rules. For further discussion of this
see (Kjeldsen and Blomhøj 2011), where also some student directed problem oriented
project work performed by students at degree level mathematics are analysed with respect
to students’ reflections about meta-discursive rules to provide empirical evidence for the
theoretical claim. These projects will not be presented here. Instead I will present a
proposal (see section 6) for a so called matrix-organised design for how such an approach
to history of mathematics for investigating meta-discursive rules might be implemented in
upper secondary school.
3 A multiple perspective approach to history
The so called whig interpretation of history has been debated at length in the
historiography of mathematics.2
In mathematics education Schubring (2008) has pointed
out how translations of sources, due to an underlying whig interpretation of history, have
changed the mathematics of the source. In the whig interpretation history is written from
the point of view of the present, as explained by the British historian Herbert Butterfield,
who coined the term in the 1930s:
It is part and parcel of the whig interpretation of history that it studies the past with
reference to the present … The whig historian stand on the summit of the twentieth
century and organises his scheme of history from the point of view of his own day.
(Butterfield 1931, 13)
If we want to use history to throw light on changes in meta rules from episodes of past
mathematics to our days mathematics whig interpretations of history poses a problem,
because, as it has been pointed out by Wilson and Ashplant (1988, 11) history then
becomes “constrained by the perceptual and conceptual categories of the present, bound
within the framework of the present, deploying a perceptual ‘set’ derived from the
present”. In this quote, Wilson and Ashplant emphasis exactly why one cannot design
learning and teaching situations that focus on bringing out differences in meta rules of past
episodes in the history of mathematics and modern ones within a whig interpretation of
history. Historical sources cannot function as “interlocutors” that can be used to clarify
differences in meta rules if the sources is interpreted within the framework of how
mathematics is conceptualized and perceived of today.
The trap of whiggism can be avoided by investigating past mathematics as a historical
product from its practice. This implies to study the sources in their proper historical
context with respect to the intellectual workshop3
of their authors, the particular
mathematicians, to ask questions such as: how was mathematics viewed at the time? How
did the mathematician, who wrote the source, view mathematics? What was his/hers
2
Discussions of whig interpretations in the historiography of mathematics can be followed e.g. in the
following papers (Unguru 1975), (van der Waerden 1976), (Freudenthal 1977), (Unguru and Rowe,
1981/82), (Grattan-Guiness 2004).
3
See (Epple 2004).
Text 10: T. H. Kjeldsen (2011). “Does history have a significant role to play for the
learning of mathematics? Multiple perspective approach to history, and the learning
of meta level rules of mathematical discourse”. In: History and Epistemology in
Mathematics Education. Proceedings of the Sixth European Summer University ESU 6.
Ed. by E. Barbin, M. Kronfellner, and C. Tzanakis. Vienna: Verlag Holzhausen
GmbH, pp. 51–62.
Summer University 2012: Asking and Answering Questions Page 168 of 479.
intention? Why and how did mathematicians introduce certain concepts? How did they
use them and for what purposes? Why and how did they work on the problems they did?
Which kinds of tools were available for the mathematician (group of mathematicians)?
Why and how did they employ certain strategies of proofs? Such questions can reveal
underlying meta rules of the discourse at the time and place of the sources. By posing and
answering such questions to the sources, possibilities for identifying meta rules that
governed the mathematics of the source can emerge, and hereby also opportunities for
turning meta rules into explicit objects of reflection in a teaching and learning situation.
As explained by Kjeldsen (2009b, 2011) one way of answering such questions and to
provide explanations for historical processes of change is to adopt a multiple perspective
approach to the history of the practice of mathematics. I have taken the term “a multiple
perspective” approach from the Danish historian Jensen (2003). It signifies that episodes of
the past can be studied from several perspectives, several points of observation, depending
on which kind of insights into, or from, the past, we are searching for. Episodes in the
history of mathematics can e.g. be studied from the perspective of sub-disciplines within
mathematics to understand if, and if so, how other fields in mathematics have influenced the
emergence and/or the development of the episode under consideration. They can be studied
from an applied point of view to understand e.g. dynamics between pure and applied
mathematics, or the role of mathematical modelling in the production of mathematical
and/or scientific knowledge. They can be studied from a sociological perspective to
understand the institutionalization of mathematics, its funding etc. They can be studied from
a gender perspective, from a philosophical perspective and so on.
4 Adaptation for mathematics education
In mathematics education the above approach can be implemented on a small scale, by
focusing on a limited amount of perspectives that address the intended learning. In the
present context the purpose is to use past mathematics and history of mathematics as a
means for elucidating meta discursive rules and make them into explicit objects of students’
reflections. Hence, students should study the sources to answer clearly formulated historical
questions that concern the underlying meta rules of the mathematics in the source.
Theoretical constructs that have been developed by historians of mathematics and/or
science to investigate the history of scientific practices can be used to “open” the sources.
With respect to the purpose of the present paper of uses of history to reveal meta rules of a
(past) mathematical discourse by studying the history of mathematics from its practice, the
notions of epistemic objects and techniques are promising tools. The term epistemic object
refers to mathematical objects that are treated in a source, i.e. the object about which
mathematicians were searching for new knowledge or were trying to grasp. The term
epistemic technique refers to the methods employed in the source by the mathematicians
to investigate the epistemic objects.4
These theoretical constructs can give insights into the
dynamics of concrete productions of pieces of mathematical knowledge, since they are
constructed to distinguish between elements of the source that provide answers and
elements that generate mathematical questions.5
4
These notions have been adapted into the historiography of mathematics by Epple (2004) from
Rheinberger’s (1997) study of experimental science.
5
For examples of uses of this methodological tool see (Epple 2004) and (Kjeldsen 2009a).
Text 10: T. H. Kjeldsen (2011). “Does history have a significant role to play for the
learning of mathematics? Multiple perspective approach to history, and the learning
of meta level rules of mathematical discourse”. In: History and Epistemology in
Mathematics Education. Proceedings of the Sixth European Summer University ESU 6.
Ed. by E. Barbin, M. Kronfellner, and C. Tzanakis. Vienna: Verlag Holzhausen
GmbH, pp. 51–62.
Summer University 2012: Asking and Answering Questions Page 169 of 479.
The question is whether history dealt with in this way, where students study episodes
from the history of mathematics from perspectives that pertain to meta rules of (past)
discourses, ask historians’ questions to the sources concerning the practice of mathematics,
and answer them using theoretical constructs such as epistemic objects and techniques, can
facilitate meta level learning in mathematics education. In the following section four texts
from the 1600s will be analyzed to provide some answers to this question.
5 Analysis of four sources within the proposed methodology
Four texts from the 1600s will be used in the following; two by Pierre de Fermat (Fermat I
and Fermat II) and two by Isaac Newton (Newton I and Newton II). Fermat I is Fermat’s
text on maxima and minima taken from Struik’s (1969) A Source Book in Mathematics,
1200-1800, whereas Fermat II is called “A second method for finding maxima and
minima”, which is published in Fauvel’s and Gray’s (1988) reader in the history of
mathematics. Newton I is Newton’s demonstration of how he found a relation between the
fluxions of some fluent quantities from a given relation between these. This text is the one
prepared by Baron and Bos (1974), whereas Newton II is Newton’s method of tangent
taken from Whiteside’s (1967) The Mathematical Works of Isaac Newton. The quality of
these translations of sources can be criticised, and investigated for degrees of whiggism
(Schubring 2008), but this will not be done in the present paper. In a teaching situation the
students should work with the four texts, but in order to give the reader an impression of
the texts, summaries of the four texts are inserted here:
In Fermat I, Fermat stated a rule for the evaluation of maxima and minima and gave an
example. The text is summarised below in Box 1.
Fermat I: On a method for the evaluation of max. and min.
Rule: let a be any unknown of the problem
• Indicate the max or min in terms of a
• Replace the unknown a by a+e – express max./min. in terms of a and e
• “adequate” the two expressions for max./min. and remove common terms
• Both sides will contain terms with e – divide all terms by (powers of) e
• Suppress all terms in which e will still appear – and equate the others
• The solution of this equation will yield the value of a leading to max./min.
Example: To divide the segment AC at E so that AE x EC may be a maximum
aA E Cb - a
b
Max: a(b-a) = ab-aa
(a+e)b-(a+e)(a+e) = ab+eb-aa-2ae-ee
ab+eb-aa-2ae-ee ~ ab-aa “adequate”
eb ~ 2ae + ee remove common terms
b ~ 2a + e ; b=2a ; a=½b; divide, suppress, solve
Box 1
If the above procedure is translated into modern mathematics using functions and the
derivative it can be explained why Fermat reached the correct solution. But this does not
explain how Fermat was thinking, since he knew neither our concept of a function nor our
concept of derivatives. In Fermat II we can get a glimpse of how Fermat was thinking.
Text 10: T. H. Kjeldsen (2011). “Does history have a significant role to play for the
learning of mathematics? Multiple perspective approach to history, and the learning
of meta level rules of mathematical discourse”. In: History and Epistemology in
Mathematics Education. Proceedings of the Sixth European Summer University ESU 6.
Ed. by E. Barbin, M. Kronfellner, and C. Tzanakis. Vienna: Verlag Holzhausen
GmbH, pp. 51–62.
Summer University 2012: Asking and Answering Questions Page 170 of 479.
The text is summarised below in Box 2.
Fermat II: A second method for finding maxima and minima
• Here he explained why his “rule” leads to max./min.: correlative equations – Viete
• Resolving all the difficulties concerning limiting conditions
Example: To divide the line b such that the product of the segments shall be a max.
If one proposes to divide the line b in such a way that the product of the segments [a
and (b-a)] shall equal z’’ … there will be two points answering the question, and they
will be found situated on one side and the other of the point corresponding to the max.
Z’’
a e
ba-aa = z’’ and be-ee = z’’
ba-aa = be-ee ; ba-be = aa-ee
Divide by a-e
b = a + e
At the point of maximum we will have a = e, then
b = a +a = 2a, hence as before a=½b.
If we call the roots a and a+e (instead of a and e) the
procedure follows the rule from text I.
Box 2
In Newton I, Newton explained through an example, how, given a relation between
fluent quantities, a relation between the fluxions of these quantities can be found. In Box 3
his procedure is summarised and illustrated with an example of a second degree equation
instead of the third degree equation that Newton used in the text.
Newton I: Find relation between fluxions from fluents
Newton’s fluxions and fluents
• Curves are trajectories (paths) for motions
• Variables are entities that change with time – fluents x , y
• The speed with which fluents change – fluxions x’ , y’ (Newton: dots!)
• Newton: All problems relating to curves can be reduced to two problems:
1. Find the relation between the fluxions given the relation between the fluents.
2. The opposite.
ox’x
y
oy’
Example: axx+bx+c-y=0 substitute x, y with x+x’o, y+y’o
a(x+x’o)(x+x’o)+b(x+x’o)+c-y-y’o=0
axx+a2xx’o+ax’x’oo+bx+bx’o+c-y-y’o=0
a2xx’o+ax’x’oo+bx’o-y’o=0
a2xx’+ax’x’o+bx’-y’=0 divided by o; cast out terms with o
a2xx’+bx’-y’=0 hence y’/x’=2ax+b
Box 3
In Newtons’s terminology o denotes an infinitely small period of time, so ox’ [Newton
used a dot over x instead of x’ to designate the fluxions] is the infinitely small addition by
which x increases during the infinitely small interval of time.
Finally, in Newton II, Newton showed how to draw tangents to curves and illustrated it
with the same example as he used in the first text. In Box 4 below the example is
Text 10: T. H. Kjeldsen (2011). “Does history have a significant role to play for the
learning of mathematics? Multiple perspective approach to history, and the learning
of meta level rules of mathematical discourse”. In: History and Epistemology in
Mathematics Education. Proceedings of the Sixth European Summer University ESU 6.
Ed. by E. Barbin, M. Kronfellner, and C. Tzanakis. Vienna: Verlag Holzhausen
GmbH, pp. 51–62.
Summer University 2012: Asking and Answering Questions Page 171 of 479.
illustrated with reference to the example used in Box 3.
Newton II: To draw Tangents to Curves
ox’x
y
oy’
Example:
T A B b
c
d
D
Similar triangles: dcD and DBT
TB:BD = Dc:cd “infinitesimal triangle”
BT/y = x’o/y’o =x’/y’
x’/y’ can be found by the method from Newton I
Box 4
The suggestion made in this paper is that these four sources can be used to exhibit
changes in meta rules of mathematical discourse, if students read the sources from the
perspective of rigor, and focus on entities and arguments. The following worksheet (Box
5) can be used to guide the students work. It consists of two sets of questions. The first set
concerns questions that help the students to identify the epistemic objects and techniques
of the two texts. The students are asked to compare and contrast the answers they get from
studying Fermat, Newton, and their textbook, respectively.
Perspective
Rigor – entities, arguments
Worksheet: History from the practice of math. Compare/contrast Fermat and Newton
Questions:
What mathematical objects are Fermat/Newton dealing with? Compare/contrast
How do they perceive them? – compare with your textbook
What are the problems they are trying to solve?
What techniques are they using? – what do we do today?
How do they argue for their claims? – how do we argue today?
Can you find any changes in understandings of the involved mathematical concepts from
Fermat over Newton to today? Explain
Can you find any changes in the way of argumentation from Fermat over Newton to
today? Explain
What kind of objections do you think your math teacher would have to Fermat’s and
Newton’s texts?
Epistemic objects
and techniques
Meta-rules – explicit object of reflection
Opportunities provided by historyBox 5
The second set of questions refers directly to meta rules of the involved mathematical
discourses.
Text 10: T. H. Kjeldsen (2011). “Does history have a significant role to play for the
learning of mathematics? Multiple perspective approach to history, and the learning
of meta level rules of mathematical discourse”. In: History and Epistemology in
Mathematics Education. Proceedings of the Sixth European Summer University ESU 6.
Ed. by E. Barbin, M. Kronfellner, and C. Tzanakis. Vienna: Verlag Holzhausen
GmbH, pp. 51–62.
Summer University 2012: Asking and Answering Questions Page 172 of 479.
Regarding the first set of questions, an analysis of the four texts and the comparison
between the objects that Fermat and Newton investigated, how they perceived them, the
problems they tried to solve, the techniques they used and the arguments they employed
might be summarised in the following scheme (Box 6):
Fermat:
Objects:
curves - algebraic expressions
ex.: multiplication of line segments
Perceive:
Area; geometrical problems treated
by algebraic methods
Problem:
evaluate max/min
Techniques:
equations, roots, algebraic mani.
Argue:
Text 1: shows the method works on
an example
Text 2: heuristic arguments with
roots in equations given by
an example
Newton:
Objects:
any curve
variables that change in time
Perceive:
trajectories for moving particles
Problem:
relations between fluxions (velocities)
given relations between the fluents
Techniques:
algebraic mani; physics, geometry
Argue:
Physical arguments about distance
and velocity, algebraic arguments,
infinitesimal triangle, o-infinitely
small
Box 6
Regarding the second set of questions, which refers to meta rules of the discourse, the
following changes can be discussed (se Box 7):
Changes in understanding:
Fermat: curves; algebraic expressions
Newton: curves, traced by a moving point, variables change in time
Today: functions, correspondence between variables in domains
Changes in the way of argumentation:
Fermat: ad hoc; “it works – its true”; heuristic argument, no infinitely
small quantities
Newton: more general procedure, physical arguments, infinitesimal
triangle, infinitely small quantities (o)
Today: limit, the real numbers, epsilon-delta proofs
Box 7
In Kjeldsen and Blomhøj (2011) we have analysed some student directed problem
oriented project work conducted by students in a degree level university mathematics
programme. Here we were able to demonstrate that history, used within the framework of
a multiple perspective approach to the history of mathematics from its practice, can be
Text 10: T. H. Kjeldsen (2011). “Does history have a significant role to play for the
learning of mathematics? Multiple perspective approach to history, and the learning
of meta level rules of mathematical discourse”. In: History and Epistemology in
Mathematics Education. Proceedings of the Sixth European Summer University ESU 6.
Ed. by E. Barbin, M. Kronfellner, and C. Tzanakis. Vienna: Verlag Holzhausen
GmbH, pp. 51–62.
Summer University 2012: Asking and Answering Questions Page 173 of 479.
used in mathematics education to give students insights into how meta rules of a
mathematical discourse are established and why/how they change. These projects were
made in a rather unique educational setting and the question is whether this methodology
can be implemented in more traditional educational settings. The analyses of the sources
guided by the worksheet (Box 5) and presented in Box 6 and Box 7 suggest that this
approach can elucidate meta rules and turn them into explicit objects for students
reflections. In the following section I present an outline for a so called matrix-organised
design for how such a multiple perspective approach to history of mathematics from its
practice might be implemented in upper secondary mathematics education.
6 Implementation in upper secondary school: A proposal
In the Danish upper secondary school system history of mathematics is part of the
mathematics curriculum. The curriculum is comprised of a core curriculum which is
mandatory and is tested in the national final, and a supplementary part, which should take
up 1/3 of the teaching. History is mentioned explicitly in the supplementary part, which
means that all upper secondary students should be taught some aspects of history of
mathematics. The supplementary part of the curriculum is tested in an oral examination
together with the core curriculum. In Box 8 below an outline is presented for a matrix
organised design for how history could be (but has not yet been) implemented in a Danish
upper secondary school for elucidating meta rules within the theoretical framework of
section 2, 3 and 4, using the sources and the worksheet presented in section 5.
Implementation in a Danish high school: a proposal
Step 1: Six groups – basic groups (worksheets would have to be prepared for
each group with respect to the intended learning)
1. The mathematical community in the 17th century
2. The standard history of analysis
3. Who were Fermat and Newton?
4. The two texts of Fermat - the questions of the worksheet of Box 5
5. The two texts of Newton - the questions of the worksheet of Box 5
6. Berkeley’s critique of Newton
Step 2: Six groups – expert groups (each group consists of at least one member
from each of the basic groups)
The experts teach the other group members of what they learned in their
basic group. Each expert group write a common report/prepare an oral
presentation of the collected work from all six basic groups as it was
discussed in their expert groups
Step 3: A plenary discussion lead by the teacher focuses on methods of
argumentation, the development/changes in the perception of objects and
techniques, compared with the standards of today.
Box 8
This design follows a three step implementation. First six groups (so called basic
groups) are formed who look into some aspects of the historical episode in question. In
Box 8 it is suggested e.g. that group 1 investigates what the mathematical community of
the 17th
century looked like. Guided by a worksheet with questions relevant for the
intended learning, the work in this group will provide the students with a sociological
perspective on mathematics and its development. In step 2 new groups (so called expert
Text 10: T. H. Kjeldsen (2011). “Does history have a significant role to play for the
learning of mathematics? Multiple perspective approach to history, and the learning
of meta level rules of mathematical discourse”. In: History and Epistemology in
Mathematics Education. Proceedings of the Sixth European Summer University ESU 6.
Ed. by E. Barbin, M. Kronfellner, and C. Tzanakis. Vienna: Verlag Holzhausen
GmbH, pp. 51–62.
Summer University 2012: Asking and Answering Questions Page 174 of 479.
groups) are formed. They consist of at least one member from each of the six basic
groups. In this way each new group consists of individual experts. Each expert now
teaches the other members of the new group what he/she learned in his/hers basic group,
and based on their shared knowledge provided by the various experts they answer the
second set of questions of the worksheet in Box 5. The design is referred to as being
matrix organised because it can be illustrated with a matrix, where the members of basic
group 1 is listed in column 1, the members of basic group in column 2, etc. In step 2 the
expert groups are formed by taking the students in the rows, i.e. expert group 1 consists of
the students listed in row 1; expert group 2 of the students listed in row 2, etc. In this way
all expert groups consists of at least one member from each basic group. In such a set up it
is possible to create complex teaching and learning situations where students work
independently and autonomously in an inquire-like environment, developing general
educational skills as well.6
7 Discussion and conclusion
The main question in the present paper is whether working with sources in the spirit of the
worksheet of Box 5 within the methodology outlined in section 3 may give rise to
situations where meta rules of (past) mathematical discourses are made into explicit
objects of students’ reflections, and whether this can assist the development of students’
proper meta rules of mathematical discourse. As pointed out above, the analyses of the
sources guided by the questions of the worksheet in Box 5, and the suggestions for
answers outlined in Box 6 and 7, suggest that history and historical sources can be used
within the methodological framework of section 2, 3 and 4 to elucidate meta rules and
make them explicit objects for students reflections.
Regarding the second part of the question, whether such an approach to the use of
history and historical sources in mathematics education also can assist the development of
students’ proper meta rules of our days mathematics is a complex question which is much
more difficult to document. The framework and methodology outlined in this paper
provide a theoretical argument for the claim that history has the potential for playing such
a profound role for the learning of mathematics, but in order to realize this in practice
more research needs to be done, and methodological tools for detecting students’ meta
rules and for monitoring any changes towards developing proper meta rules need to be
developed.
REFERENCES
– Baron, M.E., Bos, H.J.M., 1974, “Newton and Leibniz”, History of Mathematics Origins and
Development of the Calculus 3, Milton Keynes: The Open University Press.
– Beckmann, A., 2009, “A Conceptual Framework for Cross-Curricular Teaching”, The Montana
Mathematics Enthusiast 6, Supplement 1.
6
Such a matrix organised design for using history in mathematics education to elucidate meta rules of past
and present mathematics, using sources from the history of the development of the concept of a function, to
have students reflect upon those, to develop students’ mathematical competence, and general educational
skills of independence and autonomy is being tried out in a pilot study in a Danish upper secondary class at
the moment. Preliminary results from this study indicate that some of the students act according to meta
discursive rules that coincide with Euler’s; and that reading some of Dirichlet’s text created obstacles for the
students, that can be referenced to the differences in meta discursive rules. Results from the study will be
published in forthcoming papers.
Text 10: T. H. Kjeldsen (2011). “Does history have a significant role to play for the
learning of mathematics? Multiple perspective approach to history, and the learning
of meta level rules of mathematical discourse”. In: History and Epistemology in
Mathematics Education. Proceedings of the Sixth European Summer University ESU 6.
Ed. by E. Barbin, M. Kronfellner, and C. Tzanakis. Vienna: Verlag Holzhausen
GmbH, pp. 51–62.
Summer University 2012: Asking and Answering Questions Page 175 of 479.
– Butterfield, H., 1931, The whig interpretation of history, London: G. Bell.
– Epple, M., 2004, “Knot Invariants in Vienna and Princeton during the 1920s: Epistemic Configurations of
Mathematical Research”, Science in Context 17(1/2), 131-164.
– Fauvel, J., Gray, J., 1988, The History of Mathematics: A Reader, Milton Keynes: The Open University
Press.
– Freudenthal, H., 1977, “What is Algebra and what has it been in History?” Archive for History of Exact
Sciences 16, 189-200.
– Grattan-Guiness, I., 2004, “The mathematics of the past: distinguishing its history from our heritage”,
Historia Mathematica 31, 163-185.
– Jankvist, U.T., Kjeldsen, T.H., 2011, “New Avenues for History in Mathematics Education: Mathematical
Competencies and Anchoring”, Science & Education, 20, 831-862.
– Jensen, B.E., 2003, Historie – livsverden og fag, Copenhagen: Gyldendal.
– Kjeldsen, T.H., 2009a, “Egg-forms and Measure-Bodies: Different Mathematical Practices in the Early
History of the Modern Theory of Convexity”, Science in Context 22, 1-29.
– Kjeldsen, T. H., 2009b, “Abstraction and application: new contexts, new interpretations in twentiethcentury
mathematics”, in The Oxford Handbook of the History of Mathematics, E. Robson & J. Stedall
(eds.), New York: Oxford University Press, pp. 755–780.
– Kjeldsen, T. H., 2010, “History in mathematics education - why bother? Interdisciplinarity, mathematical
competence and the learning of mathematics”, in Interdisciplinarity for the 21st Century: Proceedings of
the 3rd International Symposium on Mathematics and its Connections to Arts and Sciences, B. Sriraman
& V. Freiman (Eds.), Charlotte, NC: Information Age Publishing, pp. 17-48.
– Kjeldsen, T. H., 2011, “History in a competency based mathematics education: a means for the learning of
differential equations”, in Recent developments on introducing a historical dimension in mathematics
education, V. Katz & C. Tzanakis (eds.), Washington DC: The Mathematical Association of America,
chapter 15.
– Kjeldsen, T. H., forthcoming, “Uses of History in Mathematics Education: Development of Learning
Strategies and Historical Awareness”, to be published in CERME 7, Proceedings of the seventh Congress
of the European Society for Research in Mathematics Education
– Kjeldsen, T. H., Blomhøj, M., 2011, “Beyond Motivation – History as a method for the learning of metadiscursive
rules in mathematics”, Educational Studies of Mathematics, published on line first.
– Rheinberger, H-J., 1997, Towards a History of Epistemic Things: Synthesizing Proteins in the Test Tube,
Standford: Standford University Press.
– Schubring, G., 2008, “The debate on a “geometric algebra” and methodological implications”, Paper
presented at the HPM 2008 satellite meeting of ICME 11, Mexico.
– Sfard, A., 2000, “On reform movement and the limits of mathematical discourse”, Mathematical Thinking
and Learning 2(3), 157-189.
– Sfard, A., 2008, Thinking as Communicating, Cambridge: Cambridge University Press.
– Struik, D.J., 1969, A Source Book in Mathematics, 1200-1800, Cambridge, Massachusetts: Harvaard
University Press.
– Unguru, S., 1975, “On the Need to Rewrite the History of Greek Mathematics”, Archive for History of
Exact Sciences 15, 67-114.
– Unguru, S., Rowe, D., 1981/1982, “Does the Quadratic Equation Have Greek Roots? A Study of
Geometric Algebra, Application of Areas, and Related Problems”, Libertas Mathematica 1 1981, 2 1982.
– Van der Waerden, B.L., 1976, “Defence of a Shocking Point of View”, Archive for History of Exact
Sciences 15, 199-210.
– Whiteside, D.T., 1967, The Mathematical Works of Isaac Newton, volume 1, New York: Cambridge
University Press.
– Wilson, A., Ashplant, T.G., 1988, “Whig history and present-centred history” Historical Journal 31, 1-16.
Text 10: T. H. Kjeldsen (2011). “Does history have a significant role to play for the
learning of mathematics? Multiple perspective approach to history, and the learning
of meta level rules of mathematical discourse”. In: History and Epistemology in
Mathematics Education. Proceedings of the Sixth European Summer University ESU 6.
Ed. by E. Barbin, M. Kronfellner, and C. Tzanakis. Vienna: Verlag Holzhausen
GmbH, pp. 51–62.
Summer University 2012: Asking and Answering Questions Page 176 of 479.
IV ANAJ,YSIS BEFOltE NEWTON AND I,EIBNIZ
values of y corresponding to Xl' X2, X3' ... are lh ~= an, Y2 = a"rn, Ys = anr2n, .. " Then the
sum S of the rectangles 11Xl + 12x2 .+. lsxs + ... is
(1 r)an+1(1 + 1'n+l + r2n +2 + ...)
When r = 8
q
(s < 1) and n ¥ 1, then
J:xn dx = an
+1
lim
1 _. r'
p+ q n +
As we see, this procedure holds for n positive and negative, but it fails for n -1.
This method approaches our modern method of' limits; it uscs the concept of the limit of
an infinite geometric series.
8 1<'I<JRMAT. MAXIMA AND MINIMA
Modern textbooks on calculus take up first the differential and then the intcgral calculus. It
may therefore come as a surprise to find that up to the middle of the seventeenth century
the whole theory of infinitesimals concentrated on the computation of areas, volumes, and
centel's of gravity, that is, on what we now call the integral calculus. Tangent constructions
were, until that period, based on the property that the tangent has only one point in
common with the curve, as we can see in Euclid or Apollonius. Archimedes, in his book on
spirals, found tangents by a method that seems to have been inspircd by kinematic con·
siderations. Even Torrieelli, when determining the tangent at a point of the "hyperbola"
xmyn ,= k, still used the ancient method (A. Agostini, "Il metodo delle tangenti fondato
sopra la dottrina dei moti nelle opere di Torrioel1i," Periodico di matematica [4] 28 (1950),
141-158), and Descartes sought the normal prior to the tangent, and found it in some cases
of algebraic curves by asking for double roots of a certain equation that expresses the
abscissa of the intersections of thc curve with a circle.
The beginning of the differential calculus, in which the tangent appears as the limit of
a secant, can be studied in considerations concerning maxima and minima, as in Kepler's
Nova stereometria doliorum vinariorurn (Linz, 1615; see Selection IV.2). Here we read that
"near a maximum the decrements on both sides are in the beginning only imperceptible"
(deerementa habet insitio insensibilia; Opere, IV (1863), 612).
With Fermat we obtain an algorithm based on this fact. To understand his approach and
its subsequent development into the method of the" characteristic triangle" (dx, dy, ds) wc
must take notice of the fact that Fermat and Descartes were among the first to apply the
new algebra dcveloped by Cardan, Bombelli, and Viete to the geometry of thc ancients.
This was, as we have seen, the beginning of the coordinate method. Descartes published his
method in 1687, but Fermat's discovery was known only through his correspondence until
1679, the year of the publication of his works. Here is :Fermat's approach, from his Oeuvres,
In (1896), 121--123. It is followed by a paper in which he applied his method to the finding
of a center of gravity (Ibid., 124-126).
Text 11: Fermat on maxima and minima. From D. J. Struik (1969). A Source Book in
Mathematics. 1200–1800. Cambridge (Mass.): Harvard University Press, pp. 222–225.
Summer University 2012: Asking and Answering Questions Page 177 of 479.
FERMA'r. MAXIMA AND MINIMA
(1) ON A METHOD FOR THJ~ EVALUATION Ol" MAXIMA AND MINIMA'
The whole theory of evaluation of maxima and minima presupposes two un·
known quantities and the following rule:
Let a be any unknown of the problem (which is in one, two, or three dimensions,
depending on the formulation of the problem). Let us indicate the maximum
or minimum by a in terms which could be of any degree. "Ve shall now
replace the original unknown a by a +. e and we shall express thus the maximum
or minimum quantity in terms of a and e involving any We shaH
adequate [adegaZer], to use Diophantus' term,2 the two expressions of the
maximum or minimum quantity and we shall take out their common terms.
Now it turns out that both sides will contain terms in e or its powers. We shall
divide all terms bye, or by a higher power of e, so that e will be completely
removed from at least one of the terms. We suppress then all the terms in which
e or one of its powers wi1lstill appear, and we shall equate the others; or, if one
of the expressions vanishes, we shall equate, which is the same thing, the positive
and negative terms. The solution of this last equation wiH the value of a,
which will lead to the maximum or minimum, the original
expression.
Here is an example:
To divide the segment AO [Fig. 1] at E 80 that AE x JiJO may be a maxim1tm.
Fig. 1
We write AO = b; let a be one ofthe segments, so that the other will be b···· a,
and the product, the maximum of which is to be found, will be ba -- a2
• Let now
a + e be the first segment of b; the second will be b a e, and the product of
the segments, ba ..- a2
.+ be -- 2ae .-. e2 ; this must be adequateq"with the preeeding:
ba - a2 . Suppressing common terms: be ~ 2ae + e~ truppressing e:
b = 2a.3
To solve the problem we must consequently take the half of b.
We ean hardly expeet a more general method.
ON THE TANGENTS OF CURVES
We use the preceding method in order to find the tangent at a given point of a
curve.
Let us consider, for example, the parabola BDN [Fig. 2] with vertex D and
of diameter DO; let B be a point on it at which the line BE is to be drawn tangent
to the parabola and intersecting the diameter at E.
1 This paper was sent by Fermat to Pather Marin Mersenne, who forwarded it to Des·
cartes. Descartes received it in January 1638. It became the subject of a polemic discussion
between him and Fermat (Oeuvres, I, 133). On Mersenne, see Selection 1.6, note 1.
2 Soe Solection IV.7, note 5.
3 Our notation is modern. For instance, where we have written (following the French
translation in Oeuv"es, III,122) be ~ 2ae + e2
, Fermat wrote: Bin E adaequabitur A in E
bis + Eq (Eq standing for E quadratum). The symbol ~ is used for "adequatos."
Text 11: Fermat on maxima and minima. From D. J. Struik (1969). A Source Book in
Mathematics. 1200–1800. Cambridge (Mass.): Harvard University Press, pp. 222–225.
Summer University 2012: Asking and Answering Questions Page 178 of 479.
IV ANALYSIS BEFORE NEWTON AND LEIBNIZ
Fig. 2
N
We choose on the segment BE a point 0 at which we draw the ordinate OJ;
also we construct the ordinate BO of the point B. We have then: OD/DJ>
B02/0J2, since the point 0 is exterior to the parabola. But B02/0J2 = OE2/JE2,
in view of the similarity of triangles. Hence OD/DJ> OE2/JE2.
Now the point B is given, consequently the ordinate BO, consequently the
point 0, hence also OD. Let OD = d be this given quantity. Put OE a and
OJ = e; we obtain
Removing the fractions:
da2
+ de2
- 2dae > da2
- a2
e.
Let us then adequate, following the preceding method; by taking out the
common terms we find:
or, which is the same,
Let us divide all terms bye:
de + a2
'" 2da.
On taking out de, there remains a2
2da, consequently a = 2d.
Thus we have proved that OE is the double of OD-which is the result.
This method never fails and could be extended to a number of beautiful
problems; with its aid, we have found the centers of gravity of figures bounded
by straight lines or curves, as well as those of solids, and a number of other
results which we may treat elsewhere if we have time to do so.
I have previously discussed at length with M. de Hoberval 5 the quadrature
of areas bounded by curves and straight lines as well as the ratio that the solids
which they generate have to the cones of the same base and the same height.
4 Fm'mat wrote: D ad D _. E habebit majorem proportionem quam Ag. ad Aq. +
Eq. - A in E bis (D will have to D - E a larger ratio than A 2 to A 2 + E2 - 2AE).
5 See the letters from Fermat to Hoberval, written in 1636 (Oeuvres, HI, 292-294, 296-
297).
Text 11: Fermat on maxima and minima. From D. J. Struik (1969). A Source Book in
Mathematics. 1200–1800. Cambridge (Mass.): Harvard University Press, pp. 222–225.
Summer University 2012: Asking and Answering Questions Page 179 of 479.
}<'ERMAT. MAXIMA AND MINIMA 8 225
Now follows the second illustration of Fermat's "e-method," where Fermat's e =,
Newton's 0 = Leibniz' dx.6
(2) CEN'l'Im OF GRAVITY OF PAB.ABOLOID OF R1WOLUTION, USING
'.rIm SAME METHOD'I
Let aBA V (Fig. 3) be a paraboloid of revolution, having for its axis lA and for
its base a circle of diameter aTV. Let us find its center of gravity by using the
same method which we applied for maxima and minima and for the tangents of
curves; let us illustrate, with new examples and with new and brilliant applica.
tions of this method, how wrong those are who believe that it may fail.
Fig. 3
A
81-------1
C'-----"__r
In order to carry out this analysis, we write lA = b. Let; 0 be the center of
gravity, and a the unknown length of the segment AO; we intersect the axis lA
by any plane BN and put IN e, so that NA 0= b -- e.
It is clear that in this figure and in similar ones (parabolas and paraboloids)
the centers of gravity of segments cut off by parallels to the base divide the axis
in a constant proportion (indeed, the argument of Archimedes can be extended
by similar reasoning from the case of a parabola to all parabolas and paraboloids
of revolution8). Then the center of gravity of the segment of which N A is the
axis and BN the radius of the base will divide AN at a point E such that
NA/AE lA/AO, or, in formula, b/a = (b e)/AE.
6 The gist of this method is that we change the variable x inJ(x) to x + e, e small. Since
J(x) is stationary near a maximum or minimum (Kepler's remark), J(x + e) - J(x) goes to
zero faster than e does. Hence, if we divide bye, we obtain an expression that yields the
required values for x if we let e be zero. The legitimacy of this procedure remained, as we
shall see, a subject of sharp controversy for many years. Now we see in it a first appreach
to the modern formula: j'(x) = !im .f..53: + e) ,- J(x) , introduced by Cauchy (1820-21).
8-+0 e
7 This paper seems to have been sent in a letter to Mersenne written in April 1638, for
transmission to Roberval. Mersenne reported its contents to Descartes. Fermat used the
term "parabolic conoid" for what we call "paraboloid of revolution."
8 "All parabolas" means "parabolas of higher order," y = 7cxn, n > 2. Tho reference is to
Archimedes' On floating bodie8, n, Prop. 2 and following; see '1'. L. Heath, The works oJ
Archimede8 (Cambridge University Press, Cambridge, England, 1897; reprint, Dover, New
York),264ff.
Text 11: Fermat on maxima and minima. From D. J. Struik (1969). A Source Book in
Mathematics. 1200–1800. Cambridge (Mass.): Harvard University Press, pp. 222–225.
Summer University 2012: Asking and Answering Questions Page 180 of 479.
Descartes, Fermat and Their Contemporaries 359
c I D E
N
Removing the fractions:
da2
+ de2
- 2dae > da2
a2e.
Let us then adequate, following the preceding method; by taking out the common
terms we find:
or, which is the same,
Let us divide all terms bye:
de+a2~2da.
On taking out de, there remains a2 = 2da, consequently a = 2d.
Thus we have proved that CE is the double of CD-which is the result.
This method never fails and could be extended to a number of beautiful problems;
with its aid, we have found the centres ofgravity offigures bounded by straight lines or
curves, as well as those of solids, and a number of other results which we may treat
elsewhere if we have time to do so.
I have previously discussed at length with M. de Roberval the quadrature of areas
bounded by curves and straight lines as well as the ratio that the solids which they
generate have to the cones of the same base and the same height.
11.C2 A second method for finding maxima and minima
In studying the method of syncriseos and anastrophe of Viete, and carefully following
its application to the study of the nature of correlative equations, it occurred to me to
derive a process for finding maxima and minima and thus for resolving easily all the
Text 12: Fermat on maxima and minima. From J. Fauvel and J. Gray, eds. (1987). The
History of Mathematics: A Reader. London: Macmillan Press Ltd., pp. 359–360.
Summer University 2012: Asking and Answering Questions Page 181 of 479.
360 The History of Mathematics
difficulties concerning limiting conditions which have caused so many problems for
ancient and modern geometers.
Maxima and minima are in effect unique and singular, as Pappus said and as the
ancients already knew, although Commandino claimed not to know what the term
'singular' signified in Pappus. It follows from this that on one side and the other of the
point constituting the limit one can take an ambiguous equation, and that the two
ambiguous equations thus obtained are accordingly correlative, equal and similar.
For example, let it be proposed to divide the line b in such a way that the product of
the shall be a maximum. The point answering this question is evidently the
middle of the line, and the maximum product is equal to b2
/4; no other division
of this line a product equal to b2
/4.
But if one proposes to divide the same line b in such a way that the product of the
segments shall equal z" (this area being besides supposed to be less than b2
/4) there will
be two points answering the question, and they will be found situated on one side and
the other of the point corresponding to the maximum product.
In fact let a be one of the segments of the line h, one will have ha - a2
= z"; an
ambiguous equation, since for the segment a one can take each of the two roots.
Therefore let the correlative equation be he e2
z". Comparing the two equations
according to the method of Viete:
ba -- he a2
- e2. .
Dividing both sides by a -- e, one obtains
b a -+ e;
the lengths a and e will moreover be unequal.
in place of the area z", one takes another greater value, although always less than
h2
/4, the segments a and e will differ less from each other than the previous ones, the
points of division approaching closer to the point constituting the maximum of the
product. The more the product increases the more on the contrary diminishes the
difference between a and e until it will vanish exactly at the division corresponding to
the maximum product; in this case there will only be a unique and singular solution,
the two quantities a and e becoming equal.
Now the method ofViete applied to the two correlative equations above leads to the
equality b = a -+ e, therefore if e = a (which will always happen at the point
constituting the maximum or the minimum) one will have, in the case proposed,
b = 2a, which is to say that if one takes the middle of the segment b, the product of the
segments will be a maximum.
Let us take another example: to divide the segment b in such a way that the product
of the square of one of the segments with the other shall be a maximum.
Let a be one of the segments; one must have ba2
a3
maximum. The equal and
similar correlative equation is be2
- e3
. Comparing these two equations according to
the method of Viete:
dividing both sides by a - e one obtains
ba -+ be a2
-+ ae -+ e2
,
which gives the form of the correlative equations.
Text 12: Fermat on maxima and minima. From J. Fauvel and J. Gray, eds. (1987). The
History of Mathematics: A Reader. London: Macmillan Press Ltd., pp. 359–360.
Summer University 2012: Asking and Answering Questions Page 182 of 479.
244 IV ANALYSIS BEFORE NEWTON AND I,]':IBNIZ
In Proposition II,f(y) = y (the line AOJ( is a straight line), Ply) = b 2, and
Pascal has several more examples based on this change of variables.
It is here that we meet one of Pascal's references to a fourth dimension, when he generalizes
his triligne8 from plane to space and beyond: "l,a quatrieme dimension n'est point
contrc la pure geometrie" (The fourth dimension is not against pure geometry). See H.
Bosmans, "Sur l'intcrpretation geometriquc donnoe par Pascal al'cspace aquatre dimen..
sions," Annales de la Societe ScientiJique de Bruxelles 42 (1923), 337-345.
13 WALIAS. COMPUTATION OF 7T BY SUCCESSIVI<J INTEIWOLATIONS
After 1650, analytic methods began to receive morc attention and to replace geometric
methods based on the writings of tho ancients. This was duc partly to the acceptance into
geometry of thosc algebraic methods that Descartes and Fermat had introduced, and partly
to 1;he still very active intcrest in numerical work-·-interpolation, approximation, logarithms
·--a heritage of the sixteenth and early seventeenth centuries. This tradition was strong in
England, where Napier and Briggs had !abored.
This analytic method advanced rapidly through the efforts of ,John Wallis (1616-1703),
of Emmanuel College, Cambridge, who in 1649 became the Savilian professor of geometry
at Oxford. He was one of the founders of the goyal Society and, through his work, influenced
Newton, Gregory, and other mathematicians. In his Arithmetica in;finitorum
(Oxford, 1655), he led explorations into the realms of the infinite with daring analytic
methods, using interpolation and extrapolation to obtain new results. The title of the book
shows the difference between Wallis' method-he called it "arithmetica"; we would say
(with Newton) "analysis "-and the geometric method of Cavalieri. First Wallis derived
Cavalieri's integral in an original way. Thereupon, he plunged into a maelstrom of numerical
work and, with fine mathematical intuition to guide him in his interpolations, arrived at the
infinite product for 1T that bears his name. See J. F. Scott, The mathematical work of John
Wallis (Taylor and Francis, Oxford, 1938); also A. hag, "John Wallis," Quellen und
Studien zur Geschichte der Mathematik (B) I (1931), 381-412.
-----------------------.--.--------~-------
Proposition 39.1
Given a series of quantities that are the cubes of a series of
numbers continuously increasing in arithmetic proportion (like the series of
cubic numbers), which begin from a point or zero (say 0, 1,8,27,64, ... ); we
ask for the ratio of this series to the series of just as many numbers equal to the
highest number of the first series.
1 In previous propositions Wallis has derived the limit
n
L ik
Hm i;;'l =_~_
n.... conk
+ 1
k+l
for le = 1,2. This Proposition 39 prepares for the case le = 3; it shows Wallis's typical
inductive and analytic method.
Text 13: Wallis on interpolation. From D. J. Struik (1969). A Source Book in
Mathematics. 1200–1800. Cambridge (Mass.): Harvard University Press, pp. 244–246.
Summer University 2012: Asking and Answering Questions Page 183 of 479.
WAI,LIS. COMPUTA'fION 01<' 'I'C BY INTEItPOLATIONS 13 I 245
The investigation is carried out by the inductive method, as before. We have
2 1 1
4: 4: + 4;
3 1 1
-
8 4: + 8;
4 1
1"2 4: +
27 + 64 -- 10O 5 1
4: +
125 = 225 6 ] 1
4: +
7 1 1
-- 4: +
and so forth.
The ratio obtained is always greater than one-fourth, or i\;. But the excess
decreases constantly as the number of terms increases; it is i\;, "~, fo, to, .t-.;;, ...
There is no doubt that the denominator of the fraction increases with every
consecutive ratio by a multiple of 4" so that the excess of the resulting ratio over
i\; is the same as I :4 times the number of terms after 0, etc.
Proposition 40. Theorem. Given a serics of quantities that arc the cubes of a
series of numbers continuously increasing in arithmetic proportion bcginning,
for instance, with 0, then the ratio of this series to the series of just as many
numbers equal to the highest number of the first series will be greater than i\;.
The excess will bc 1 divided by four times the number of terms after 0, or the
cube root of the first term after °divided by four times the cube root of the
highest term.
The sum of the series 03 + 13 + ... + ZS is I + lZ3 + ~-t/ [3, or, if m is the
number of terms, ~ 13 + ~ 13 = ~ ml3
+ ~ ml2
• This is apparent from the previous
reasoning.
If, with increasing number of terms, this excess over i\; diminishes continuously,
so that it becomes smaller than any given number (as it; clearly does),
when it goes to infinity, then it must finally vanish. Therefore:
Proposition 41. '1'heorem. If an infinite series of quantities which are the cubes
of a series of continuously increasing numbers in arithmetic progression, beginning,
say, with 0, is divided by the sum of numbers all equal to the highest and
equal in number, then we obtain i\;. This follows from the preceding reasoning.
Proposition 42. Oorollary. The complement AD']' [Fig. I] of half the area of
the cubic parabola therefore is to the parallelogram '1'D over the same arbitrary
base and altitude as 1 to 4.
Indeed, let ADD be the area of half the parabola AD (its diameter AD, and
the corresponding ordinates DO, DO, ctc.) and let AD,]' be its complement.
Text 13: Wallis on interpolation. From D. J. Struik (1969). A Source Book in
Mathematics. 1200–1800. Cambridge (Mass.): Harvard University Press, pp. 244–246.
Summer University 2012: Asking and Answering Questions Page 184 of 479.
246 IV ANALYSIS BEFORE NEWTON AND I,EIBNIZ
Fig. 1
D o
Since the lines DO, DO, etc., or their equals AT, AT, etc. are the cube roots 2
of AD, AD, ... , or their equals 1'0, 1'0, ... , these TO, TO, etc. will be the
cubes of the lines AT, AT, ... The whole figure AOT therefore (consisting of
the infmite number of lines TO, TO, etc., which are the cubes of the arithmetically
progressing lines AT, Al', ... ) will be to the parallelogram ATD
(consisting of just as many lines, all equal to the greatest TO), as 1 to 4, according
to our previous theorem. And the half-segment AOD of the parabola (the
residuum of thc parallelogram) is to the parallclogram itself as 3 is to 4.
In Proposition 44 the result of these considerations on the quotient of thc two series
1 i
k
and 2: n
k
(n + 1 terms) is laid down in a table for k 0, 1,2.... , 10. Wallis discriminates
for ik between the series of equals (k = 0), of the first order (k = 1), of the second
order (k = 2), and so forth (series aequalium, primanorum, secundanorum, and so forth).
Proposition 54. Theorem.3
If we consider an infinite series of quantities begin.
ning with a point or °and increasing continuously as the square, cube, bi.
quadratic, etc. roots of numbers in an arithmetic progression (which I call the
series of order k = i, 1, 1, ... ), then the ratio of the whole series to the series
of all numbers equal to the highest number is expressed in the following table:
k Result
!
-1.
11
or as 1 to
2 Wallis uses the terms "atio subduplicata, 8ubtriplicata etc., to denoto square, cubic, etc.,
roots; the ratio 8ubduplicata of A 21B2 is AIB. These terms are not classical, and may be '
medieval: WaIlis uses them here and in his Mathesis univet'8ali8 (Oxford, 1657), chap. 30.
The term duplicate ratio is classical; see gucIid, Element8, Book V, Definition 9: if alb = blc,
then ale has the duplicate ratio of alb, hence ale = a21b2. Similarly, triplicate ratio in
Definition 10 means the ratio of cubes. See G. gnestrom, "Ueber den Ursprung des Termes
'ratio subdupIicata'," Bibliotheca mathematica [3] 4 (1903), 210-211; 6 (1905), 410; 12
(1911-12), 180-1S1.
3 Propositions 54 and 59 are supplementary, with the tabulation of g Xk dx = I/(k + I)
for all positive rational k.
Text 13: Wallis on interpolation. From D. J. Struik (1969). A Source Book in
Mathematics. 1200–1800. Cambridge (Mass.): Harvard University Press, pp. 244–246.
Summer University 2012: Asking and Answering Questions Page 185 of 479.
Text 14: Roberval on the quadrature of the parabola. From E. Walker (1932). A Study
of the Traité des Indivisibles of Gilles Persone de Roberval. New York, pp. 181–182.
Summer University 2012: Asking and Answering Questions Page 186 of 479.
Chapter 2
Newton,
H. J. M. Bos
2.1. Introduction and biographical summary
The starting-point of this chapter is the' invention', or rather' inventions',
of the calculus. Both Newton (in 1664-1666) and (in
1675) created, independently of each other, an calculus.
Their inventions were very different in concepts and but each
contains so much of what we now recognise as essential to the calculus
that the expression' invention of the calculus' is justified in both cases.
I go on to consider the subsequent development of the calculus till
about 1780. In this development the Leibnizian type of calculus with
differentials and integrals proved more successful than the Newtonian
fluxional calculus; therefore I concentrate on the former.
Many great and lesser mathematicians were involved in the development
of the calculus in the period covered by this chapter. I shall
restrict myself to those who played the prime roles in the story: Isaac
Newton, Lucasian professor of mathematics at Cambridge and later
Master of the Mint in London; Gottfried Wilhelm Leibniz, historian
and scientist at the ducal court of Hanover; Jakob. Bernoulli, professor
of mathematics at Basle; his brother Johann Bernoulli, younger by
thirteen years, who after a professorate at Groningen succeeded Jakob
in Basle in 1705; Guillaume Franc;ois Marquis de l' Hopital, a French
nobleman living by private means, and an able mathematician eagerly
interested in the new developments in infinitesimal methods; and finally
Leonhard Euler, who studied with Johann Bernoulli and then entered
a career in the typically 18th-century scientific institutions, the academies.
He was professor at the St. Petersburg (now Leningrad) Academy from
1730 to 1741 and from 1766 till his death; in the intervening years he
served the Berlin Academy as professor.
Many of the great ideas that were to make Isaac Newton famous in
mathematics and natural science came to him in the years 1664-1666.
49
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 187 of 479.
50 2. Newton, Leibniz and the Leibnizian tradition
At that time he was a graduate student at Trinity College, Cambridge,
but for some time during those two years he lived in Lincolnshire,
staying away from Cambridge for fear of the Plague (compare Whiteside
1966a). His ideas on gravity, which he was to work out later and present
to the world in his famous Principia (1687a), date from that period, as
well as his theory of colours, published in the treatise Opticks in 1704,
the binomial series theorem and his fluxional calculus, which we shall
discuss in more detail in section 2.2.
As with gravity and colours, publication of these mathematical ideas
in print was long delayed. Newton did compose several accounts of
his findings in infinitesimal calculus. In October 1666 he summarised
the discoveries of the fruitful two years in a tract on fluxions (1666a) ;
in 1669 he wrote a treatise on infinite series, the De analysi (1669a),
which circulated in manuscript form among members of the Royal
Society; from 1671 dates a treatise on the method of fluxions and
infinite series (1671a); and in about 1693 he composed a treatise on
the quadrature of eurves (1693a). However, the 1666 traet and the
treatise on the method of fluxions were not published in his lifetime,
the De analysi was published only in 1711, and the treatise on quadratures
of curves in 1704. Meanwhile the Principia of 1687 had brought
for the first time to the general public indications of his methods in
infinitesimal calculus, but these were not enough to show the scope and
power of his mathematical discoveries.
About the turn of the century a fair amount was published about
Leibniz's calculus (as we shall see in sections 2.5-2.8 below), and
sufficient information about Newton's calculus was available to show
that both men had found new methods in essentially the same mathematical
field. This caused a nasty quarrel over priority, in which feelings
of personal and national pride combined with insufficient insight in the
mathematics involved (at least in the case of the lesser participants in
the debate) to create a distasteful muddle of misunderstandings and
insinuations which has only been cleared up through patient historical
research in the present century. The net result of the historical research
is that Leibniz found his calculus later than Newton and independently
of him, and that he published it earlier.
In 1669 Newton had succeeded Isaac Barrow as Lucasian professor,
but in the 1690s he grew dissatisfied with his position at Cambridge.
He visited London often, to attend meetings of the Royal Society, of
which he was a fellow from 1672, and to be present at sessions of Parliament
as a member for the Cambridge University constituency. He
moved finally to London in 1696 when he was offered the office
of Warden of the Mint. In 1703 he became president of the Royal
Society, a post which he held till his death. His position as the most
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 188 of 479.
2.1. Introduction and biographical summary 51
eminent British scientist was further emphasised by a knighthood in 1705.
By the 1710s so much on the fluxional calculus was in print that the
method was taken up and applied by others. However, this further
development of the Newtonian type of calculus remained restricted to
Great Britain, and it did not achieve much. Reasons of the lack of
success lie in the isolation from the Continental developments in analysis
because of the priority dispute, in the lack of mathematicians in Britain
of sufficient stature to really develop Newton's calculus, and in an overstressed
loyalty to Newton's conception of the calculus and to his notations,
which were less versatile than Leibniz's.
On the Continent Leibniz's inventions gave rise to a much more
intense development, to whose origins in the 1670s we now turn.
Before Leibniz entered the service of the house of Hanover in 1676
he had spent four ~ars in Paris on a diplomatic mission, which left
him ample time to pursue his interest in mathematics, the sciences,
history, philosophy and many other things. He met many French
philosophers and made two visits to London to the Royal The
Paris years were his formative period. When he arrived in 1672 his
knowledge of mathematics was slight, despite the fact that he had published
a small tract on combinatorics. He was trained in law at the
university of his home town of Leipzig. In Paris Christiaan Huygens,
who lived there at that time, recognised Leibniz's mathematical abilities
and guided his first studies in the higher mathematics. Leibniz's
, growth to mathematical maturity' (see. Hofmann 1949a) was indeed
impressive; it led to his discovery of the calculus in 1675, the elaboration
of that calculus in the following years and its publication in 1684-
1686. He contributed to other branches of mathematics as well, for
instance to algebra (solvability of equations, determinants) and to nearly
all other fields of human learning, including religion, politics, history,
physics, mechanics, technology, mathematics, geology, linguistics and
natural history. Many of his results were not immediately published
and became known only gradually, through correspondence (from his
comparative intellectual isolation in Hanover Leibniz corresponded with
over a thousand scholars), through publication of short articles in journals
(he was one of the founders of the first scientific journal in Germany,
the Acta eruditorum), and later through the publication of his manuscripts,
most of which he kept and which are now stored at the Leibniz
archive in Hanover.
Leibniz's publication of his calculus in two articles in the Acta of
1684 and 1686 did not provoke great commotion in mathematical
circles. The articles were rather short, and they were marred by
misprints and in places deliberately obscure, so that it is in fact surprising
that in the following decade they were understood at all.
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 189 of 479.
52 2. Newton, Leibniz and the Leibnizian tradition
Jakob and Johann Bernoulli studied the articles from 1687, and by
1690 they showed, in articles published in the Acta, that they had
mastered the Leibnizian symbolism and its use. They both started a
correspondence with Leibniz; the contact between Johann and Leibniz
was especially intensive and productive. After 1690 a stream of articles
in the Acta and in other journals, written by the Bernoullis and Leibniz
and later joined by I'H6pital and others, showed the learned world that
the new calculus was something to be reckoned with.
However, for people of lesser mathematical calibre than the
Bernoullis, it would have been very difficult actually to learn the calculus
from these articles. What was wanted was a proper textbook of the
calculus. Such a textbook came, though only of the differential calculus,
in 1696 with l'H6pital's Analyse des infiniment petits pour l'intelligence
des lignes courbes (' Analysis of infinitely small quantities for the understanding
of curved lines': 1696a).
The Marquis de l'H6pital was introduced to the calculus by Johann
Bernoulli, who, after finishing his medical studies in 1690, had travelled
to Paris, where he impressed learned circles by a method to determine,
by means of differentials, the curvature of arbitrary curves-a problem
which by the methods of Cartesian analytic geometry was well nigh
unsolvable. l'H6pital was most impressed and asked Bernoulli to give
him, for a good fee, lectures on the new method. Bernoulli accepted
and the lectures were given, in Paris and at the country chateau of the
Marquis. They were written out and both men kept copies. After
about a year Bernoulli left Paris but agreed to continue instructing
I'H6pital by letter. In fact the agreement was that Bernoulli, for a
handsome monthly salary, would answer all I'H6pital's questions concerning
mathematics, would send him all his mathematical discoveries
and would give no one else access to these findings (see Bernoulli
Correspondence, 144); a most curious and hardly honourable agreement
which put Bernoulli's originality strictly in I'H6pital's service. From
the start Bernoulli did not quite keep to the letter of the contract, and
l'H6pital soon realised that he could not bind a brilliant mathematician
in this way. But when in 1696 l'H6pital published his textbook, and
Bernoulli saw that most of its content was taken from his lectures with
not more than a passing reference to the Marquis's indebtedness to
Bernoulli, he-could only be angry in silence, being bound by the contract.
Later, after l'H6pital's death, Johann Bernoulli did try to get his
part in the Analyse acknowledged, but by that time his credibility in
priority questions had become very low because of open quarrels on
such matters with his brother. Jakob Bernoulli was a rather introverted
personality, but he was sensitive to praise from members of the
mathematical community and he resented being overshadowed by ,his
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 190 of 479.
2.1. Introduction and biographical summary 53
brilliant younger brother. Johann, on the other hand, liked his own
success too much to spare his brother's feelings. So there appeared
insinuating remarks in articles, and later a quarrel exploded and went on
quite openly. Johann Bernoulli's claim to much of the content of the
Analyse was found to be justified only when in 1921 the manuscript of
his Paris lectures on the differential calculus was found (see Johann
Bernoulli 1924a).
However strained their mutual relations, through the writings of
these men the Leibnizian calculus became known and proved its power.
By the first decade of the 18th century other mathematicians devoted
themselves to the new calculus, such as Jakob Pierre Varignon,
Niklaus BernoulIi (a nephew) and Daniel Bernoulli (son of Johann).
The family Bernoulli continued to yield famous mathematicians throughout
the 18th century.
In these early days the new calculus consisted mainly of rather
loosely connected methods, and problems solved by these methods.
The man who reshaped the Leibnizian calculus into a soundly organised
body of mathematical knowledge was Leonhard Euler. Euler was the
central figure of continental mathematics in the middle years of the 18th
century. He published an enormous number of books and articles on
mathematics, mechanics, optics, astronomy, navigation, hydrodynamics,
technical matters such as artillery and shipbuilding, and very many
other topics. He maintained this impressive productivity despite losing
the sight of one eye in 1735 and becoming completely blind in 1766.
His position at the academies involved him in many other tasks besides
scientific research, such as advice on the performance of new inventions
as fire-engines and pumps, and on technological enterprises like canalbuilding
and the construction of water-works in the park of the royal
palace Sans Souci of Prussia's Frederick the Great.
Euler's greatest influence on the calculus and on analysis in general
was through his great textbooks, in which he gave analysis a definitive
form, which it was to keep until well into the 19th century. These
textbooks, written in Latin, were: Introductio ad analysin infinitorum
(' Introduction to the analysis of infinites ': 1748a), Institutiones calculi
differentialis (' Textbooks on the differential calculus': 1755b), and
lnstitutiones calculi integralis (' Textbooks on the integral calculus' :
1768-1770a).
These were the men who created the calculus and shaped the
Leibnizian tradition in analysis. In sections 2.3-2.8 I shall describe
the mathematics involved, but first I shall devote the next section to
an overview of the Newtonian calculus.
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 191 of 479.
54 2. Newton, Leibniz and the Leibnizian tradition
2.2. Newton's fluxional calculus
As was mentioned above, Newton's main mathematical discoveries in
the infinitesimal calculus date from 1664 to 1666. (For a detailed
account of his achievements in this period, see Newton Papers, vol. 1,
145--154, and Works2, vol. 1, viii-xiii.) Autodidactically he quickly
acquired adequate knowledge of existing theories in the field, benefitting
especially from reading Descartes's La geometrie in van Schooten's
edition with commentaries, and from the works of Wallis. Starting from
these studies he developed in these fruitful two years his fluxional calculus.
In Newton's discoveries, complex, deep and many-sided as they are,
a number of central themes may be distinguished. These are: series
expansions, algorithms, the inverse relationship of differentiation and
integration, the conception of variables as moving in time, and the doctrine
of prime and ultimate ratios. Although these themes are interconnectedly
present in almost all of his studies in the infinitesimal
calculus, I shall deal with them separately.
Newton valued power-series expansions very highly, because they
provide a means to reduce the analytical formulae of curves to a form
in which all terms simply consist of a constant times a power of the
variable. Thus transcendental curves (admitting no algebraic equation),
as well as algebraic curves with complicated equations, can be represented
by much simpler equations (be it with an infinite number of
terms). Newton saw that this has two great advantages. Firstly,
series expansion makes it possible to apply rules and algorithms which
are defined for simple equations only, to a much wider range of curves.
In particular, the relation
1
Jxn dx n+ 1 xn+1, (2.2.1)
which was known in various forms by the 1660s (see sections 1.10 and
1.11) can be used, in combination with power-series expansions, to
provide series expressions for the quadratures of almost all curves.
Secondly, series expansion provides a ready means for the approximation
and simplification of formulae through the discarding of higherorder
terms-a feature which he used with virtuosity in his applications
of his mathematical methods to physical problems.
Newton's most famous series expansion is the' binomial theorem',
which he found in the winter of 1664-1665 and which states that the
well-known binomial expansion for integer powers n,
(2.2.2)
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 192 of 479.
2.2. Newton's fluxional calculus 55
can be generalised for fractional powers 0( =plq, in which case the right
hand side of
0( 0« 0( - 1)
(a+x)a=a"+T aa-1x +-
1
T aa-2x2
+ . .. (2.2.3 )
is an infinite series. He found the theorem in connection with the
problem of squaring the circle y = (1 -- x2 )1 /2. He compared the formulae
(1-.x2)O, (1_x2)1/2, (1_x2)2/2, (1_x2)3/2, (1_x2)4/2, ....
third, fifth, ... formulae involve no root, and therefore the quadratures
of the corresponding curves are easily found:
quadrature of y = (1 - x2
)O is X, }
quadrature of y = (1 X2)2/2 is x Jix3,
quadrature of y = (1 - X 2)4 /2 is x- ix3 +!X5.
(2.2.4)
On examining the coefficients in these expansions, Newton noted that
the denominators are the odd numbers 1, 3, 5, 7, ... and that the
numerators are, in the successive expansions, {I}, {I, I}, {1, 2, 1},
{I, 3, 3, I}, ..., that is, the numbers in the ' Pascal triangle', which
he knew could be expressed for successive integral values of n as
{
1 n(n 1) n(n - 1)(n - 2) }
, n, 1. 2' 1 . 2 . 3 ,. .. .
He then guessed that, by analogy, the same expressions would apply
for fractional values of n. When n = t this yields:
quadrature of y=(1-x2)l/2 is
(2.2.5)
He then saw that this procedure of guessing, or ' interpolating " expansions
such as (2.2.5) from the scheme of the series (2.2.4) could be
applied to the equations of the curves as well as to their quadratures,
and in this way he found that
(2.2.6)
Not satisfied with the reliability of the interpolation procedure, he
checked (2.2.6) in two ways. He showed that the product of the right
hand side of (2.2.6) with itself yields 1- x2 (that is, all further coefficients
in the product series are zero), and he saw that a common method of
root extraction known as the 'galley method', applied formally to
1 - x2, yields the same series. In the same way as with root extraction,
he used the algorithm of long division to obtain series expansions, for
instance,
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 193 of 479.
56 2. Newton, Leibniz and the Leibnizian tradition
1
= 1- x +x2
x3
+X4 - ••• ,
+x
(2.2.7)
which provided the quadrature of the hyperbola y= 1/(1 +x). He
also obtained (2.2.7) by assuming that the binomial expansion applied
when n= --1.
In the De analysi (1669a), in which these methods of series expansions
are explained and used, Newton also provides a general rule to
compute, for a given polynomial equation
La.ijxiyi = 0
between x and y, the first coefficients of the pertaining series
y= Lbixi
(Papers, vol. 2, 222-247).
(2.2.8)
(2.2.9)
Both in the way that Newton found the binomial theorem and in the
application of series expansions in general, the relation, which we now
write as
1
f xn dx=-- xn+1
J n+l'
(2.2.10)
plays an important role. He mentioned this 'quadrature of simple
curves' at the outset of his De analysi: ' RULE 1. Ifaxm/n=y, then
will (na/(m+n))x(m+n)/n equal the area ABD' (ibid., 206-207; see
figure 2.2.1). Later in that treatise he gave a general procedure (of
which rule 1 is a direct consequence) for finding the relation between
the quadrature of a curve (as AD in figure 2.2.1) and its ordinate. The
procedure makes it clear that Newton recognised the inverse relationship
of integration and differentiation (although, of course, he did not use
these terms). He explains his method by means of an example, from
which, however, the generality of the procedure is quite clear. He
A
Figure 2.2.1.
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 194 of 479.
2.2. Newton's flux£onai calculus 57
proceeds as follows (ibid., 242-245). In figure 2.2.1 let area ABD = z,
BD =y and AB = x; let further BfJ = 0 and let BK = v be chosen such
that area BD8fJ = area BKHb:= ov. Consider, as example, the curve
for which
Z .JX3 /2, (2.2.11)
that is (removing roots to get a polynomial equation),
then also
(2.2.13 )
from which
(2.2.14)
Now by removing the terms without 0, which are equal on both sides
from (2.2.12), and dividing the remainder by 0, we obtain
2zv +ov2
= t(3x2
+3xo +0
2
). (2.2.15)
Now Newton takes BfJ ' infinitely small', in which case, as the
suggests, v =y and the terms with 0 vanish:
2zy=!X2. (2.2.16)
Inserting the value of z from (2.2.11), he obtains
y = X 1 /2• (2.2.17)
Clearly the procedure is applicable to all polynomial relations between
x and z. It consists in essence of calculating the derivative (in this
case the y) for any algebraic function z of x.
Newton saw clearly that the problem of quadratures was to be
approached in this inverse way: by calculating y for all manner of
algebraic z, he could find all manner of curves (y, x) which are quadrable.
Indeed, he calculated many such quadrable curves, writing them together
in extensive lists, which are thus nothing less than the first tables of
integrals (compare Papers, vol. 1,404-411).
The essential element in the foregoing procedure is the substitution
of ' small' corresponding increments 0 and ov for x and z in the equation.
In studies on the determination of maxima and minima, tangents
and curvature, Newton had extensively made use of this method, and
he had worked out various algorithms for these problems, by which he
could calculate the slope of the tangent or the curvature in any point of
an algebraic curve. (In modern terms, he had developed algorithms
to determine the derivative of any algebrai~ function.) Later he reformulated
these algorithms and their proofs in terms of fluents and
fluxions, and we shall come back to them after discussing these concepts. l
I Compare, for instance, Newton 1671a, in Papers, vol. 3, 72-73.
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 195 of 479.
58 2. Newton, Leibniz and the Leibnizian tradition
The terms' fluents' and' fluxions' indicate Newton's conception
of variable quantities in analytical geometry: he saw these as ' flowing
quantities', that is, quantities that change with respect to time. Thus,
when considering the curve of figure 2.2.1, he would conceive the point D
as moving along the curve, while correspondingly the ordinate y, the
abscissa x, the quadrature z or any other variable quantity connected
with the curve would increase or decrease, or in general change or
, flow'. He called these flowing quantities 'fluents' (as opposed to
the constant quantities occurring in the figure or in the problem at
hand), and he called their rate of change with respect to time their
, fluxion '. In his earlier researches he indicated fluxions by separate
letters; in 1671a he introduced the dot-notation, where the fluxions of
the fluents x, y, z are X, y, % respectively.
It should be remarked that the way in which the fluents vary with
time is arbitrary. Newton often makes, for simplicity, an additional
assumption about the movement of the variables, supposing that one
of the variables, say x, moves uniformly, so that x= 1. Such assumptions
can be made because the values of the fluxions themselves are not of
interest but rather their ratio, such as y/x, which gives the slope of the
tangent. By this conception of quantities moving in time Newton
thought himself able to solve the foundational difficulties inherent in
considering , small' ,corresponding increments of variables, which are
so small that we may discard them, and yet are not equal to zero, as we
want to divide through by them. In his approach to this problem,
his theory of prime and ultimate ratios, which we shall discuss in section
2.10, his conception of flowing quantities is essential; through this
conception he comes very near to a use of limits as foundation of the
calculus.
We now return to the algorithms mentioned above. The corresponding
increments of variables, can be expressed in terms of fluxions :
let 0 now be an infinitesimal element of time, then the corresponding
increments of the fluents x, y, z, ... are xo, yo, %0, ... respectively.
The· ratio of y to x can now be determined in a way which is evident
in the following example, which Newton gives himself in 1671a (Papers,
vol. 3, 79-81). Let a curve be given with equation
x3- ax2 +axy - y3 = o.
Substituting x + xo and y +yo for x and y respectively yields
(x3+ 3xox2
+3X2
0
2
X + X30
3) (ax2
+2axox +ax2
0
2
)
+(axy +axoy +ayox +axyo2)
- (y3 + 3yoy2 + 3y 20 2y +y30 3) = O.
(2.2.18)
(2.2.19)
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 196 of 479.
2.2. Newton's fluxional calculus 59
Deleting x3
- ax2 +axy - y3 as equal to zero from (2.2.18), dividing
through by 0 and discarding the terms in which 0 is left, yields
3xx2
- 2axx +axy +ayx - 3yy2 = 0,
from which the ratio of y and x is easily obtained:
y
x
(2.2.20)
(2.2.21 )
We note that the numerator and the denominator in the result are (apart
from a sign) the partial derivatives fx and Iy of f(x, y) = x3
ax2
+axy -- y3,
the left hand side of the equation of the curve. Thus
y
x
Ix
~.
(2.2.22)
Indeed, this relation is implicit in the algorithms which, as we mentioned
before, Newton worked out for problems of maxima and
minima, and curvature. He even at one time introduced special notations
in this connection (see Papers, vo!. 1,289-294), f![ for the
left hand side of the equation of the curve (with the right hand side zero).
He then wrote .f![ and f![. for what we would write as xix and yfy respectively
(the so-called 'homogeneous partial derivatives'), using
further symbols for homogeneous higher-order partial derivatives occurring
in connection with curvature. However, the connection of
Newton's .f![ and f![. with modem partial derivatives should not be
considered without some qualifications; he defined them formally as
modifications of the formula f![, and he did not explicitly view f![ as a
function of two variables which assumes also other values than the zero
in the equation.
With these algorithms, and further finesses which we cannot go
into here, Newton was able to solve what he formulated as one of the
two fundamental problems in infinitesimal calculus: given the fluents
and their relations, to find the fluxions.
The second problem is the converse of the first: given the relation
of the fluxions, to find the relation of the fluents. Transposed in
modern terminology, this means: given a differential equation, to find
its solution. This of course is a much harder problem than the first.
Newton did more about the problem than formulate it; his integral
tables, already mentioned, form a means toward its solution, and he also
studied various individual differential equations (or rather, fluxional
equations).
As we have seen in the previous section, Newton\ calculus was not
to have the influence which Leibniz's achieved. Therefore, within the
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 197 of 479.
60 2. Newton, Leibniz and the Leibnizian tradition
space and organisation of this chapter, we must leave it at this short
summary of the fluxional calculus and some more remarks on its foundations
in section 2.10, turning now our attention to the more successful
rival, the Leibnizian calculus.
2.3. The principal ideas in Leibniz's discovery
One of the most precious documents of the Leibniz archive at Hanover
is a set of mathematical manuscripts dated 25, 26 and 29 October, and
1 and 11 November, 1675. 1
On these sheets Leibniz wrote down his
thoughts, more or less as they came to him, during a study of that most
important problem of 17th-century mathematics: to find methods for
the quadrature of curves. In the course of these studies he came to
introduce the symbols 'J' and 'd', to explore the operational rules
which they obey in formulas, and to apply them in translating many
geometrical arguments about the quadrature of curves into symbols and
formulas. In short, these manuscripts contain the record of Leibniz's
, invention' of the calculus. We will discuss them in more detail
below, but first we will mention three principal ideas which guided him
in those fateful studies in 1675.
The first principal idea was a philosophical one, namely Leibniz's
idea of a characteristica generalis, a general symbolic language, through
which all processes of reason and argument could be written down in
symbols and formulas; the symbols would obey certain rules of combination
which would guarantee the correctness of the arguments.
This idea guided him in much of his philosophical thinking; it also
explains his great interest in notation and symbols in mathematics and
in general his endeavour to translate mathematical statements and
methods into formulas and algorithms. Thus, in studying the geometry
of curves, he was interested in methods rather than in results, and
especially in ways to transform these methods into algorithms performable
with formulas. In short, he was looking for a calculus for
infinitesimal-geometrical problems.
The second principal idea concerned difference sequences. In
studying sequences aI' a2, a3, ••• , and the pertaining difference sequences
bI = aI a2, b2 = a2 - aa, b3 = a3 a4, ••• , Leibniz had noted that
(2.3.1)
This means that difference sequences are easily summed, an insight
which he put to good use in solving a problem which Huygens suggested
1 They are discussed in Hofmann 1949a, and an English translation is given in
Child 1920a.
«
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 198 of 479.
2.3. The prlncipal ideas in Leibniz's discovery 61
to him in 1672: to sum the series t + -l + i + -lo +15 +..., the denominators
being the so-called 'triangular numbers' r(r + 1)/2. He
found that the terms can be written as differences,
2 2 2
(2.3.2)
~(r +1) = r - r +
and hence
n
---=2.- 2
n+
(2.3.3)
In particular, the series, when ~ 11 :nmed to infinity has sum 2. This
result motivated him to study a whole scheme of related sum and
difference sequences, which he put together in his so-called' harmonic
triangle' (figure 2.3.1), in which the oblique rows are successive difference
sequences, so that their sums can be easily read off from the
scheme (Leibniz Writings, vol. 5, 405: compare Hofmann 1949a, 12;
1974a, 20).
1
l ~.
t i t
!. 1
-& 1.
12 4,
i -Jo 1 1 1
1nl 20 1>
1
·10 io ..L .1.
i"6 60 30
.1 .1- ~.L. 1 1 ..L ,7 42 105 140 105 42
Figure 2.3.1.
Leibniz's ' harmonic triangle'. The numbers in the n··th row are
Surnmations can be read off from the scheme as, for example :
1 1 1 1 1 1
3+12+ 30 + 60 + 105+'"
These results were not exactly new, but they did make Leibniz
aware that the forming of difference sequences and of sum sequences
are mutually inverse operations. This principal idea became more
significant when he transposed it to geometry. The curve in figure
2.3.2 defines a sequence of equidistant ordinates y. If their distance
is 1, the sum of the y's is an approximation of the quadrature of the curve,
and the difference of two successive y's yields approximately the slope
of the pertaining tangent. Moreover, the smaller the unit 1 is chosen,
the better the approximation. Leibniz concluded that if the unit could
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 199 of 479.
62 2. Newton, Leibniz and the Leibnizian tradition
o 1 18
Figure 2.3.2.
be chosen infinitely small, the approximations would become exact: in
that case the quadrature would be equal to the sum of the ordinates,
and the slope of the tangent would be equal to the difference of the
ordinates. In this way, he concluded from the reciprocity of summing
and taking differences that the determination of quadratures and tangents
are also mutually inverse operations.
Thus Leibniz's second principal idea, however vague as it was in
about 1673, suggested already an infinitesimal calculus of sums and
differences of ordinates by which quadratures and tangents could be
determined, and in which these determinations would occur as inverse
processes. The idea~lso made plausible that, just as in sequences the
determination of differences is always possible but the determination
of sums is not, so in the case of curves the tangents are always easily to
be found, but not so the quadratures.
The third principal idea was the use of the ' characteristic triangle'
in transformations of quadratures. In studying the work of Pascal,
Leibniz noted the importance of the small triangle cc'd along the curve
in figure 2.3.3, for it was (approximately) similar to the triangles formed
Figure 2.3.3.
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 200 of 479.
g
2.3. The principal ideas in Leibniz's discovery 63
by ordinate, tangent and sub-tangent, or ordinate, normal and subnormal.
The configuration occurs in many 17th-century mathematical
works; Pascal's use of it concerned the circle. Leibniz saw its general
use in finding relations between quadratures of curves and other quantities
like moments and centres of gravity. For instance, the similarity
of the triangles yields cc' x Y = cd x n; hence
cc' x y Lcd x n. (2.3.4)
The left hand side can be interpreted as the total moment of the curve
arc with respect to the x-axis (the moment of a particle with respect to
an axis is its weight multiplied by its vertical distance to the axis),
whereas the right hand side can be interpreted as the area formed by
plotting the normals along the x-axis.
b
Figure 2.3.4.
As an example of Leibniz's use of the characteristic triangle, here is
his derivation of a special transformation of quadratures which he called
'the transmutation' and which, for good reasons, he valued highly
(compare Hofmann 1949a, 32-35 (1974a, 54-60), and Leibniz Writings,
vol. 5, 401-402). In figure 2.3.4 let the curve Occ'C be given, with
".,-..
characteristic triangle cdc' at c. Its quadrature f2 = OCB, the sum of
the strips bee'b', can also be considered as the sum of the triangles
Occ' supplemented by the triangle OBC :
f2 = L 6.0cc' +6.0BC. (2.3.5)
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 201 of 479.
64
Now
2. Newton, Leibniz and the Leibnizian tradition
/).Occ' = 1cc' x Op
= 1cd x Os
(since the characteristic triangle cdc' is similar to /).Osp)
=1bqq'b'. (2.3.6)
Now for each c on Occ'C we can find the corresponding q by drawing the
tangent, determining s and taking bq = Os. Thus we form a new curve
Oqq'Q, and we have from (2.3.5) :
.22=1 (quadrature Oqq'Q)-t-/).OCB. (2.3.7)
This is Leibniz's transmutation rule which, through the use of the
characteristic triangle, yields a transformation of the quadrature of a
curve into the quadrature of another curve, related to the original curve
through a process of taking tangents. It can be used in those cases
where the quadrature of the new curve is already known, or bears a
known relation to the original quadrature. Leibniz found this for
instance to be the case with the general parabolas and hyperbolas (see
section 1.3), for which the rule gives the quadratures very easily. He
also applied his transmutation rule to the quadrature of the circle, m
which investigation he found his famous arithmetical series for 7T :
(2.3.8)
The success of the transmutation rule also convinced him that the
analytical calculus for problems of quadratures which he was looking
for would have to cover transformations such as this one by appropriate
symbols and rules.
The transmutation rule as Leibniz discovered it in 1673 belongs to
the style of geometrical treatment of problems of quadrature which was
common in the second half of the 17th century. Similar rules and
methods can be found in the works of Huygens, Barrow, Gregory and
others. Barrow's Lectiones geometricae (1670a), for instance, contain a
great number of transformation rules for quadratures which, if translated
from his purely geometrical presentation into the symbolism and
notation of the calculus, appear as various standard alogrithms of the
differential and integral calculus. This has even been used (by J. M.
Child in his 1920a) as an argument to give to Barrow, rather than Newton
or Leibniz, the title of inventor of the calculus. However, this view can
be sustained only when one disregards completely the effect of the
translation of Barrow's geometrical text into analytical formulas. It
is the very possibility of the analytical expression of methods, and hence
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 202 of 479.
2.3. The principal ideas in Leibniz's discovery 65
the understanding of their logical coherence and generality, which was
the great advantage of Newton's and Leibniz's discoveries.
It is appropriate to illustrate this advantage by an example. To do
this, I shall give a translation, with comments, of Leibniz's transmutation
rule into analytical formulas.
The ordinate z of the curve Oqq'Q is, by construction,
z=y x (2.3.9)
(note the use of the characteristic triangle). The transmutation rule
states that, for OB = xo,
Xo Xo
Jy dx = t J z dx + txoyo· (2.3.10)
o 0
Inserting z from (2.3.9), we find
x,
Jy dx
o
x, x, dy
= t JY dx - t J x -d dx +txoyo'
o 0 x
Hence
x, x, dy
!Y dx+ !x dx dx=xoyo, (2.3.11 )
so that we recognise the rule as an instance of ' integration by parts'.
Apart from the indication of the limits of integration (0, xo) along
the J-sign, the symbolism used above was found by Leibniz in 1675.
The advantages of that symbolism over the geometrical deduction and
statement of the rule are evident: the geometrical construction of the
curve Oqq'Q is described by a simple formula (2.3.9), and the formalism
carries the proof of the rule with it, as it were. (2.3.11) follows immediately
from the rule
d(x y) =x dy+y dx. (2.3.12)
These advantages, manipulative ease and transparency through the rules
of the symbolism, formed the main factors in the success of Leibniz's
method over its geometrical predecessors.
But we have anticipated in our story. So we return to October 1675,
when the transmutation rule was already found but not yet the new
symbolism.
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 203 of 479.
66 2. Newton, Leibniz.and the Leibnizian tradition
2.4. Leibniz's creation of the calculus
In the manuscripts of 25 October-ll November 1675 we have a close
record of studies of Leibniz on the problem of quadratures. We find
him attacking the problem from several angles, one of these. being the
use of the Cavalierian symbolism' omn.' in finding, analytically (that is,
by manipulation of formulas) all sorts of relations between quadra··
tures. 'Omn.' is the abbreviation of 'omnes lineae', 'all lines';
in section 1.10 it was represented by the symbol ' (9 '.
A characteristic example of Leibniz's investigations here is the following.
In a diagram such as figure 2.4.1 he conceived a sequence of
--y-
S~
0
1
1
1
I
1
1
xl
w_-,~I
1
I
1
ult. x
1
~1
1 Y
_ I
,
i~
~
\
w
1\
B wC
Figure 2.4.1.
ordinates y of the curve OC; the distance between successive ordinates
is the (infinitely small) unit. The differences of the successive ordinates
are called w. OBC is then equal to the sum of the ordinates y. The
rectangles like w x x are interpreted as the moments of the differences w
with respect to the axis OD (moment = weight x distance to axis).
Hence the area OCD represents the total moment of the differences w.
~ ~
OCB is the complement of OCD within the rectangle ODCB, ·so that
Leibniz finds that ' The moments of the differences about a straight
line perpendicular to the axis are equal to the complement of the sum
of the terms' (Child 1920a, 20). The' terms' are the y. Now w is
the difference sequence of the sequence of ordinates y; hence, conversely,
y is the sum-sequence of the w's, so that we may eliminate y
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 204 of 479.
2.4. 'Leibniz's creation of the calculus 67
and consider only the sequence wand its sum-sequences, which yields:
, and the moments of the terms are equal to the complement of the sum
of the sums' (ibid.). Here the 'terms' are the w. Leibniz writes
this result in a formula using the symbol 'Omn. ' for what he calls
, a sum'. We give the formula as he gave it, and we add an explanation
under the accolades; n is his symbol for equality, ' ult. x ' stands
for ultimus x, the last of the x, that is, and he uses and
commas where we would use brackets (ibid.) :
omn. xw n
~
moments of
the terms w
total sum of sums
of the terms (2.4.1 )
the sums
(Compare the form of (2.4.1) with that of (2.3.11).) he
sees the possibility to obtain from this formula, by various substitutions,
other relations between quadratures. For instance, of
xw=a, w=a/x yields
a a
omn. a n ult. x, omn. - - omn. omn. -,
x x
(2.4.2)
which he interprets as an expression of the ' sum of the logarithms in
terms of the quadrature of the hyperbola' (ibid,. 71). Indeed, omn. a/x
is the quadrature of the hyperbola y = a/x, and this quadrature is a
logarithm, so that omn. omn. a/x is the sum of the logarithms.
We see in these studies an endeavour to deal analytically with problems
of quadrature through appropriate symbols and notations, as well
as a clear recognition and use of the reciprocity relation between difference
and sum sequences. In a manuscript of some days later,
these insights are pushed to a further consequence. Leibniz starts here
from the formula (2.4.1), now written as
omn. xl n x omn. 1- omn. omn. I. (2.4.3 )
He stresses the conception of the sequence of ordinates with infinitely
small distance: ' . .'. I is taken to be a term of the progression, and x is
the number which expresses the position or order of the I corresponding
to it; or x is the ordinal number and I is the ordered thing' (ibid., 80).
He now notes a rule concerning the dimensions in formulas like (2.4.3),
namely that omn., prefixed to a line, such as I, yields an area (the
qUildrature); omn., prefixed to an area, like xl, yields a solid, and so on.
Such a law of dimensional homogeneity was well-known from the
Cartesian analysis of curves, in which the formulas must consist of
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 205 of 479.
68 2. Newton, Leibniz and the Leibnizian tradition
terms all of the same dimension. (In (2.4.3) all terms are of thret:
dimensions, in x2 +y2 = a2 all terms are of two dimensions; an expression
like a2 +a is, if dimensionally interpreted, unacceptable, for it would
express the sym of an area and a line.)
This consideration of dimensional homogeneity seems to have suggested
to Leibniz to use a single letter instead of the symbol ' omlfl. "
for he goes on to write: 'It will be useful to write Jfor omn, so that
JI stands for omn. I or the sum of all l's ' (ibid.). Thus the J-sign is
introduced. 'J' is one of the forms of the letter's' as used in script
(or italics print) in Leibniz's time: it is the first letter of the word
summa, sum. He immediately writes (2.4.3) in the new formalism:
Jxl = x J1- JJI ;
he notes that
(2.4.4)
(2.4.5)
and he stresses that these rules apply for' series in which the differences
of the terms bear to the terms themselves a ratio that is less than any
assigned quantity' (ibid.), that is, series whose differences are infinitely
small.
Some lines further on we also find the introduction of the symbol' d '
for differentiating. It occurs in a brilliant argument which may be
rendered as follows: The problem of quadratures is a problem of
summing sequences, for which we have introduced the symbol 'J'
and for which we want to elaborate a calculus, a set of useful algorithms.
Now summing sequences, that is, finding a general expression for Jy
for given y, is usually not possible, but it is always possible to find
an expression for the differences of a given sequence. This finding of
differences is the reciprocal calculus of the calculus of sums, and therefore
we may hope to acquire insight in the calculus of sums by working
out the reciprocal calculus of differences. To quote Leibniz's own
words (ibid., 82) :
Given I, and its relation to x, to find fI. This is to be obtained
from the contrary calculus, that is to say, suppose that J1=ya.
Let 1=yald; then just as f will increase, so d will diminish the
dimensions. But Jmeans a sum, and d a difference. From the
given y, we can always find yid or I, that is, the difference of the y's.
Thus the 'd '-symbol (or rather the symbol 'lid') is introduced
Because Leibniz interprets f dimensionally, he has to write the 'd'
in the denominator: I is a line, JI is an area, say ya (note the role of
, a' to make it an area), the differences must again be lines, so we must
write' yajd '. In fact he soon becomes aware that this is a notational
disadvantage which is not outweighed by the advantage of dimensional
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 206 of 479.
2.4. Leibniz's creation of the calculus 69
interpretability of Jand d, so he soon writes' d(ya) , instead of 'yaJd'
and henceforth re-interprets 'd' and 'J' as dimensionless symbols.
Nevertheless, the consideration of dimension did guide the decisive
steps of choosing the new symbolism.
In the remainder of the manuscript Leibniz explores his new
symbolism, translates old results into it and investigates the operational
rules for Jand d. In these investigations he keeps for some to the
idea that d(uv) must be equal to du dv, but finally he finds the correct
rule
d(uv)=,u dv+v duo (2.4.6)
Another problem is that he still for a long time writes Jx, JX2, ••• for
what he is later to write consistently as Jx dx, Jx2
dx, . .. .
A lot of this straightening out of the calculus was still to be done
after 11 November 1675; it took Leibniz roughly two years to complete
it. Nevertheless, the Jllanuscripts which we discussed contain the
essential features of the new, the Leibnizian, calculus: the concepts of
the differential and the sum, the symbols d and $, their inverse relation
and most of the rules for their use in formulas.
Let us summarise shortly the main features of these Leibnizian
concepts (compare Bos 1974a, 12-35). The differential of a variable y
is the infinitely small difference of two successive values of y. That is,
Leibniz conceives corresponding sequences of variables such as y and x
in figure 2.4.2. The successive terms of these sequences lie infinitely
close. dy is the infinitely small difference of two successive ordinates y,
dx is the infinitely small difference of two successive abscissae x, which,
in this case, is equal to the infinitely small distance of two successive y's.
A sum (later termed 'integral' by the Bernoullis) like Jy dx is the
sum of the infinitely small rectangles y x dx. Hence the quadrature
of the curve is equal to Jy dx.
-----------x
Figure 2.4.2.
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 207 of 479.
70 2. Newton, Leibniz and the Leibnizian tradition
Leibniz was rather reluctant to present his new calculus to the general
mathematical public. When he eventually decided to do so, he faced
the problem that his calculus involved infinitely small quantities,
which were not rigorously defined and hence not quite acceptable in
mathematics. He therefore made the radical but rather unfortunate
decision to present a quite different concept of the differential which
was not infinitely small but which satisfied the same rules. Thus in
his first publication of the calculus, the article 'A new method for
c
,...£-._ _ _-+-c__x__,---"_ __
A 8
Figure 2.4.3.
maxima and minima as well as tangents' (1684a) in the issue for October
1684 of the Acta, he introduced a fixed finite line-segment (see figure
2.4.3) called dx, and he defined the dy at C as the line-segment satisfying
the proportionality
y: a=dy: dx,
a being the length of the sub-tangent, or
dy=!.. dx.
a
(2.4.7)
(2.4.8)
So defined, dy is also a finite line-segment. Leibniz presented the
rules of the calculus for these differentials, and indicated some applications.
In an article published two years later (1686a) he gave some
indications about the meaning and use of the J-symbol. This way of
publication of his new methods was not very favourable for a quick and
fruitful reception in the mathematical community. Nevertheless, the
calculus was accepted, as we shall see in the following sections.
2.5. l'Hopital's textbook version of the differential calculus
Leibniz's publications did not offer an easy access to the art of his new
calculus, and neither did the early articles of the Bernoullis. Still, a
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 208 of 479.
2.5. L'H6pital's textbook on the differential calculus 71
good introduction appeared surprisingly quickly, at least to the differential
calculus, namely l'H6pital's Analyse (1696a).
As a good textbook should, the Analyse starts with definitions, of
variables and their differentials, and with postulates about these differentials.
The definition of a differential is as follows: 'The infinitely
small part whereby a variable quantity is continually
or decreased, is called the differential of that quantity' (ch. 1). For
further explanation l'H6pital refers to a diagram (figure 2.5.1), in which,
01
M /1
;7L
iR
/s I
/ 1
/ i// I
/ I
/ 1
/ 1
/ i'/ 1
y 1
_ _1_ _ _ _ _ _ _ __
A P P
Figure 2.5.1.
with respect to a curve AMB, the following variables are indicated:
~
abscissa AP=x, ordinate PM=y,chord AM=z, arcAM=s and
~
quadrature AMP=f2. A second ordinate pm ' infinitely close' to PM
is drawn, and the differentials of the variables are seen to be: dx = Pp,
dy = mR, dz = Sm, ds =Mm (the chord Mm and the arc Mm are taken
to coincide) and df2 = MPpm. I'H6pital explains that the 'd' is a
special symbol, used only to denote the differential of the variable
written after it. The small lines Pp, mR, ... in the figure have to be
considered as ' infinitely small'. He does not enter into the question
whether such quantities exist, but he specifies, in the two postulates,
how they behave (ibid.) :
Postulate 1. Grant that two quantities, whose difference is an
infinitely small quantity, may be used indifferently for each other:
or (which is the same thing) that a quantity, which is increased or
decreased only by an infinitely smaller quantity, may be considered
as remaining the same.
This means that AP may be considered equal to Ap (or x = x +dx),
MP equal to mp (y=y+dy), and so on.
The second postulate claims that a curve may be considered as the
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 209 of 479.
72 2. Newton, Leibniz and the Leibnizian tradition
assemblage of an infinite number of infinitely small straight lines, or
equivalently as a polygon with an infinite number of sides. The first
postulate enables I'Hopital to derive the rules of the calculus, for
instance:
d(xy) = (x+ dx)(y +dy) - XY}
=X dy+y dx+dx dy
=X dy+y dx
(2.5.1 )
, because dx dy is a quantity infinitely small, in respect of the other
terms y dx and x dy: for if, for example, you divide y dx and dx dy
by dx, we shall have the quotients y. and dy, the latter of which is infinitely
less than the former' (ibid., ch. 1, para. 5). l'Hopital's concept
of differential differs somewhat from Leibniz's. Leibniz's differentials
are infinitely small differences between successive values of a variable.
I'Hopital does not conceive variables as ranging over a sequence of
infinitely close values, but rather as continually increasing or decreasing;
the differentials are the infinitely small parts by which they are increased
or decreased.
In the further chapters l'Hopital explains various uses of differentiation
in the geometry of curves: determination of tangents, extreme
values and radii of curvature, the study of caustics, envelopes and
various kinds of singularities in curves. For the determination of
tangents he remarks that postulate 2 implies that the infinitesimal part
Mm of the curve in figure 2.5.2, when prolonged, gives the tangent.
T A
y
x
m
I dy
---IR
dx I
I
I
I
I
I
I
I
I
I
I
p p
Figure 2.5.2.
Therefore Rm : RM, or dy : dx, is equal to y : PT, so that PT =y(dx/dy),
and the tangent can be constructed once we have determined y dx/dy
(ibid., ch. 2, para. 9) :
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 210 of 479.
2.6. Johann Bernoulli's lectures on integration 73
Now by means of the difference of the given equation you can
obtain a value of dx in terms which all contain dy, and if you
multiply by y and divide by dy you will obtain an expression for
the sub-tangent PT entirely in terms of known quantities and free
from differences, which will enable you to draw the required
tangent MT.
To explain this, consider for example the curve ay2 7_~ x8. The' difference
of the equation' is derived by taking differentials left and
2ay dy = 3x2
dx.
dx can now be expressed in terms of dy :
2ay
dx= 3x2 dy.
Hence
PT
dx = 2ay = 2ay2
y 3x2 3x2 '
which provides the construction of the tangent.
(2.5.2)
(2.5.3 )
(2.5.4)
The 'difference of the equation' is a true differential equation,
namely an equation between differentials. l'H6pital considers expressions
like 'dy/dx' actually as quotients of differentials, not as
single symbols for derivatives.
2.6. lohann Bernoulli's lectures on integration
In 1742, more than fifty years after they were written down, Johann
Bernoulli published his lectures to I'H6pital on ' the method of integrals'
in his collected works (Bernoulli 1691a), stating in a footnote that he
omitted his lectures on differential calculus as their contents were now
accessible to everyone in I'H6pital's Analyse. His lectures may be
considered as a good summary of the views on integrals and their use in
solving problems which were current around 1700.
Bernoulli starts with defining the integral as the inverse of the
differential: the integrals of differentials are those quantities from which
these differentials originate by differentiation. This conception of the
integral-the term, in fact, was introduced by the Bernoulli brothersdiffers
from Leibniz's, who considered it as a sum of infinitely small
quantItIes. Thus, in Leibniz's view, f y dx = fl means that the sum
of the infinitely small rectangles y x dx equals fl; for Berr.oulli it
means that dfl =y dx.
Bernoulli states that the integral ofaxp dx is (a/(p + 1))xP +l, and he
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 211 of 479.
74 2. Newton, Leibniz and the Leibnizian tradition
gives various methods usable in finqing integrals; among them is the
method of substitution, explained by several examples, such as the
following (1691a, lecture 1) :
Suppose that one is required to find the integral of
(ax+xx) dxJ(a+x).
Substituting J(a +x) ='y we shall obtain x =yy -- a, and thus
dx = 2y dy, and the whole quantity
(ax+xx) dxJ(a+x)=2y6 dy-2ay4 dy.
It is now easy and straightforward to integrate this expression;
its integral isb? - ! ay5 and, after substituting the value of y, we
find the integral to be ~(x+a)3J(x+a)-!a(x+a)2J(x+a).
The principal use of the integral calculus, Bemoulli goes on to
explain, is in the squaring of areas. For this the area has to be considered
as divided up into infinitely small parts (strips, triangles, or
quadrangles in general as in figure 2.6.1). These parts are the differentials
of the areas; one has to find an expression for them 'by
means of determined letters and only one kind of indeterminate'
(ibid., lecture 2), that is, an expression f(u) du for some variable u. The
required area is then equal to the integral Jf(u) duo
Figure 2.6.1.
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 212 of 479.
2.7. Euler's shaping of analysis 75
The further use of the method of integrals is in the so-called' inverse
method of tangents' (ibid., lecture 8). The method, or rather the type
of problem which Bernoulli has in mind here, originated in the 17th
century; it concerns the determination of a curve from a given property
of its tangents. He teaches that the given property of the tangents has
to be expressed as an equation involving differentials, that is, a differential
equation. From this differential equation the equation of the
curve itself has to be found by means of the method of integrals. His
first example is (ibid., lecture 8; see figure 2.6.2) :
y
Figure 2.6.2.
It is asked what kind of curve AB it is whose ordinate BD is
always the middle proportional between a given line E and the subtangent
CD (that is, E:BD=BD:CD). Let E=a, AD=x,
DB=y, then CD=yy: a. Now dy: dx=y: CD=yy/a (that is,
CD=yy/a); therefore we get the equation y dx=yy dy: a or
a dx =y dy; and after taking integrals on both sides, we get
ax = tyy or 2ax =yy; which shows that the required curve AB is
the parabola with parameter = 2a.
In the further lectures BernQulli solves many instances of inverse
tangent problems. He devotes considerable attention to the question
how to translate the geometrical or often mechanical data of the problem
into a treatable differential equation. The problems treated in his
lectures concern, among other things, the rectification (computation of
the arc-length) of curves, cycloids, logarithmic spirals, caustics (linear
foci occurring when light-rays reflect or refract on curved surfaces), the
catenary (see section 2.8 below), and the form of sails blown by the wind.
2.7. Euler's shaping of analysis
In the (about) 50 years after the first articles on the calculus appeared,
the Leibnizian calculus developed from a loose collection of methods
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 213 of 479.
76 2. Newton, Leibniz and the Leibnizian tradition
for problems about curves into a coherent mathematical discipline:
Analysis. Though many mathematicians, such as Jean le Rond
d'Alembert, Alexis Clairaut, the younger generation of Bernoullis, and
others, contributed to this development, it was in a large measure. the
work of one man: Leonhard Euler. Not only did Euler contribute
many new discoveries and methods to analysis, but he also unified and
codified the field by his three great textbooks mentioned already in
section 2.1.
Shaping analysis into a coherent branch of mathematics meant first
of all making clear what the subject was about. In the period of
Leibniz, the elder Bernoullis and l'H6pital, the calculus consisted of
analytical methods for the solution of problems about curves; the
principal objects were variable geometrical quantities as they occurred in
such problems. However, as the problems became more complex and
the manipulations with the formulas more intricate, the geometrical
origin of the variables became more remote and the calculus changed
into a discipline merely concerning formulas. Euler accentuated this'
transition by affirming explicitly that analysis is a branch of mathematics
which deals with analytical expressions, and especially with
functions, which he defined (following Johann Bernoulli) as follows:
, a function of a variable quantity is an analytical expression composed
in whatever way of that variable and of numbers and constant quantities'
(1748a, vol. 1, para. 4). Expressions like xn , (b+X)2ax (with constants
a and b) were functions of x. Algebraic expressions in general, and
also infinite series, were considered as functions. The constants and
the variable quantities could have imaginary or complex values.
Euler undertook the inventorisation and classification of that wide
realm of functions in the first part of his Introduction to the analysis of
infinites (1748a). The Introduction is meant as a survey of concepts
and methods in analysis and analytical geometry preliminary to the
study of the differential and integral calculus. He made of this survey
a masterly exercise in introducing as much as possible of analysis without
using differentiation or integration. In particular, he introduced the
elementary transcendental functions, the logarithm, the exponential
function, the trigonometric functions and their inverses without recourse
to integral calculus-which was no mean feat, as the logarithm
was traditionally linked to the quadrature of the hyperbola and the
trigonometric functions to the arc-length of the circle.
Euler had to use some sort of infinitesimal process in the Introduction,
namely, the expansion of functions in power-series (through long division,
binomial expansion or other methods) and the substitution of
infinitely large or infinitely small numbers in the formulas. A characteristic
example of this approach is the deduction of the series expansion
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 214 of 479.
....
2.7. Euler's shaping of analysis 77
for aZ
(1748a, vol. 1, paras. 114-116), where he proceeds as follows.
Let a> 1, and let w be an ' infinitely small number, or a fraction so
small that it is just not equal to zero '. Then
aw=l+if;
for some infinitely small number if;. Now put
if;=kw
in which k depends only on a; then
aW
=1+kw
and
w = log (1 +kw)
if the logarithm is taken to the base a.
(2.7.1)
(2.7.2)
(2.7.3)
(2.7.4)
Euler shows that for a = 10 the value of k can be found (approxima..
tely) from the common table of logarithms. He now writes
aiw =(l +kw)i (2.7.5)
for any (real) number i, so that by the binomial expansion
iw _ 1 ~ k i(i·- 1) k2 2 i(i - 1)(i - 2) k3 3 ___
a - + 1 w+ 1.2 w + 1.2.3 w { .... (2.7.6)
If z is any finite positive number, then i = z/w is infinitely large, and by
substituting w = z/i in (2.7.6) we obtain
aZ=aiw =1+{kz+ lii~2!) k2
z2
+ 1(~~ ~~(~;i2) k3
z3
+. ... (2.7.7)
But if i is infinitely large, (i-l)/i=l, (i-2)/i=1, and so on, and we
arrive at
(2.7.8)
The natural logarithms arise if a is chosen such that k 1. Euler gives
that value of a up to 23 decimals, introduces the now familiar notation e
for that number and writes (ibid., para. 123) :
(2.7.9)
In the next chapter Euler deals with trigonometric functions. He
writes down the various sum-formulas and adds: 'Because (sin. Z)2 +
(cos. z)2= 1, we have, by factorising, (cos. z+ .J-1 . sin. z)(cos. z
- .J-1 . sin. z) = 1, which factors, although imaginary, nevertheless
are of immense use in comparing and multiplying arcs' (ibid., para. 132).
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 215 of 479.
b
78 2. Newton, Leibniz and the Leibnizian tradition
He further finds that
(cos y ± J-1 sin y)(cos z ± J-:::t sin z) =
cos (y+z):±: sin (y+z), (2.7.10)
and hence
(cos z ± J-:::Y sin z)n = cos nz ± J- 1 sin nz, (2.7.11 )
a relation usually called ' de Moivre's formula' as it occurs already in
the work of Abraham de Moivre (see Schneider 1968a, 237-247).
By expanding (2.7.11) Euler obtains expressions for cos nz and sin nz.
Now taking z to be infinitely small (so that sin z = z and cos z """" 1),
nz = v finite and hence n infinitely large, he arrives, by methods similar
to those above, at
cos v= 1v2
v4 v6
(2.7.12)+ + ... ,
v3 'l)o v7
(2.7.13)SIn v=v- + + ...
(ibid., para. 134). Some paragraphs later (art. 138) we find, derived
by similar methods, the identities:
exp (± v J-1) =cos v + J-:::Y sin v, (2.7.14)
cosv=t(exp [vJ-l]+exp [-vJ-l]), (2.7.15)
sinv=2J_l (exp [vJ-l]-exp [-vJ-l]). (2.7.16)
Euler's Textbooks on the differential calculus (1755b) starts with two
chapters on the calculus of finite differences and then introduces the
differential calculus as a calculus of infinitely small differences, thus
returning to a conception more akin to Leibniz's than to l'Hopital's :
, The analysis of infinites ... will be nothing else than a special case
of the method of differences expounded in the first chapter, which
occurs, when the differences, which previously were supposed finite,
are taken infinitely small' (1755b, para. 114). He considers infinitely
small quantities as being in fact equal to zero, but capable of having
finite ratios; according to him, the equality 0 . n = 0 implies that 0/0
may in cases be equal to n. The differential calculus investigates the
values of such ratios of zeros. Euler proceeds to discuss the differentiation
of functions of one or several variables, higher-order differentiation
and differential equations. He also obtains the equality
02V 02V
- - --
oxoy ayox (2.7.17)
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 216 of 479.
2.8. The catenary and the brachistochrone 79
for a function V of x and y (though not using this notation, and without
obtaining a fully rigorous proof; 1755b, paras. 288 ff.).
In his discussion of higher-order differentiation Euler gives a
prominent role to the differential coefficients, p, q, r, defined, for a
function y =f(x), as follows;
dy dx (2.7.18)
(where p is the coefficient with which to multiply the constant dx in
order to obtain dy, so that p is again a function of x); and similarly,
dp=qdx (so that ddy=q(dx)2),
dq=r dx (so that dddy=r(dx)3), ...
(2.7.19)
(2.7.20)
These differential coefficients are, though differently defined, equal to
the first- and higher-order derivatives of the function f. In his textbook
on the integral calculus he treats higher-order differential equations in
terms of these differential coefficients, thus, in some measure, the
way for the replacement of the differential by the derivative as funda··
mental concept of the calculus.
The three-volume Textbooks on the integral calculus (1768--1770a)
give a magisterial close to the trilogy of textbooks. Here Euler gives a
nearly complete discussion of the integration of functions in terms of
algebraic and elementary transcendental functions, he discusses various
definite integrals (including those now called the beta and gamma
functions), and he gives a host of methods for the solution of ordinary
and partial differential equations.
Apart from determining, through these textbooks, the scope and style
of analysis for at least the next fifty years, Euler contributed to the
infinitesimal calculus in many other ways. Two of these contributions
are worth special emphasis. Firstly, he gave a thorough treatment of
the calculus of variations, whose beginnings lie in the studies by the
Bernoullis of the brachistochrone and of isoperimetric problems (see
section 2.8 below). Secondly, he applied analysis, and indeed worked
out many new analytical methods, in the context of studies in mechanics,
celestial mechanics, hydrodynamics and many other branches of natural
sciences, thus transforming these subjects into strongly mathematised
form. In the next section I shall describe one example of each of these
ways.
2.8. Two famous problems: the catenary and the brachi,ftochrone
In writing the history of the calculus, it is customary to devote much
attention to the fundamental concepts and methods. This tends to
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 217 of 479.
80 2. Newton, Leibniz and the Leibnizian tradition
obscure the fact that most mathematicians spend most of their time
not in contemplating these concepts and methods, but in using them to
solve problems. Indeed, in the 18th century the term' mathematics'
comprised much more than the calculus and analysis, for it ranged from
arithmetic, algebra and analysis through astronomy;, optics, mechanics
and hydrodynamics to such technological subject~ as artillery, shipbuilding
and navigation. In this section I discuss two famous problems
whose solution was made possible by the new methods of the differential
and integral calculus; in the next section I shall say something about
what more was made possible through these methods.
The catenary problem
The catenary is the form of a hanging fully flexible rope or chain
(the name comes from catena, which means' chain '), suspended on
two points (see figure 2.8.1). The interest in this curve originated with
B
A
Figure 2.8.1.
Galileo, who thought that it was a parabola. Young Christiaan Huygens
proved in 1646 that this cannot be the case. What the actual form was
remained an open question till 1691, when Leibniz, Johann Bernoulli
and the then much older Huygens sent solutions of the problem to the
Acta (Jakob Bernoulli, 1690a, Johann Bernoulli 1691b, Huygens 1691a
and Leibniz 1691a), in which the previous year Jakob Bernoulli had
challenged mathematicians to solve it. As published, the solutions did
not reveal the methods, but through later publications of manuscripts
these methods have become known. Huygens applied with great
virtuosity the by then classical methods of 17th-century infinitesimal
mathematics, and he needed all his ingenuity to reach a satisfactory
solution. Leibniz and Bernoulli, applying the new calculus, found the
solutions in a much more direct way. In fact, the catenary was a testcase
between the old and the new style in the study of curves, and only
because the champion of the old style was a giant like Huygens, the
test-case can formally be considered as ending in a draw.
A short summary of Johann Bernoulli's solution (he recapitulated it
in his 1691a, lectures 12 and 36), may provide an insight in how the
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 218 of 479.
2.8. The catenary and the brachistochrone
p
Figure 2.8.2.
new method was applied. In figure 2.8.2 let AB be part of the
81
Using arguments from mechanics, he inferred that the forces Fo and F l ,
applicable in B and A to keep the part AB of the chain in position, are
the same (in direction and quantity) as the forces required to keep the
weight P of the chain AB in position, suspended as a mass at E on
weightless cords AE and BE, which are tangent to the curve as in the
figure. Moreover, the force Fo at B does not depend 9n the choice of
the position of A along the chain. P may be. put equal to the length s
of the chain from B to A; Fo = a, a constant; and from composition
of forces we have
Hence
P: Fo=s: a=dx: dy.
dy a
dx s
(2.8.1 )
(2.8.2)
This is the differential equation of the curve, though in a rather
intractable form as x andy occur implicitly in the arc-length s. Through
skilful manipulation Bernoulli arrives at the equivalent differential
equation
dy (2.8.3 )
I shall not follow his argument here in detail, but the equivalence can
be seen by going backwards and calculating ds from (2.8.3) :
ds= J(dy2+dx2) = J(x2:a2+ 1)dx= J(:2d~a2r (2.8.4)
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 219 of 479.
82 2. Newton, Leibniz and the Leibnizian tradition
Hence by integration
xdx dx
s=J ----;--(2 2)= )(x2
_·a2
)=a-. (2.8.5)
V x -a dy
Through a substitution x -)- x +a Bernoulli reduces (2.8.3) to
adx
dY =")(x2 +2ax), (2.8.6)
This substitution is needed to move the origin to B. In the differential
equation (2.8.6) the variables are separated, so that the solution is
adx
y = J )(X2 +2ax)'
(2.8.7)
and the question is left to find out what the right hand side means.
At that time, in the early 1690s, Bernoulli had not yet the analytical
form of the logarithmic function at his disposal to express the integral
as we would (namely, as a log (a + x + )[x2
+ 2ax])). Instead he gave
geometrical interpretations of the integral, namely, as quadratures of
curves. He noted that the integral represents the area under the curve
a2
z - --:----:-.,---.,.
- ) (x2 +2ax)'
(2.8.8)
But he also interpreted (through transformations which again we shall
not present in detail) the integral as an area under a certain hyperbola
and even as an arc-length of a parabola. By these last two interpretations,
or ' constructions' as this procedure of interpreting integrals was
called, he proved that the form of the catenary' depended on the quadrature
of the hyperbola' (we would say: involves only the transcendental
function the logarithm) and with this proof the problem was, to the
standards of the end of the 17th century, adequately solved.
The brachistochrone problem
If a body moves under influence of gravity, without friction or air
resistance along a path y (see figure 2.8.3), then it will take a certain
time, say Ty, to move to B starting from rest in A. Ty depends on
the form of y. The brachistochrone (literally: shortest time) is the
curve Yo from A to B for which Ty is minimal. It can easily be seen
that the fall along a straight line from A to B does not take the minimal
time, so there is a problem: to determine the brachistochrone.
The problem was publicly proposed by Johann Bernoulli in the Acta
of June 1696 (Bernoulli 1696a) and later in a separate pamphlet. Several
solutions reached the Acta and were published in May 1697 (Johann
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 220 of 479.
2.8. The catenary and the brachistochrone
A
Figure 2.8.3.
83
Bernoulli 1697a, l'Hopital ,1697a, 1697a and Newton ;
see Hofmann 1956a, 35-36). Bernoulli's own solution used an analogy
argument: he saw that the problem could be reduced to the problem
of the refraction of a light-ray through a medium in which the density,
and hence the refraction index, is a function of the height Leibniz
and Jakob Bernoulli first considered the position of two consecutive
straight line-segments (see figure 2.8.4) such that from P to Q is
minimal. This is an extreme value problem depending on one variable
and therefore solvable. Extending this to three consecutive straight
segments and considering these as infinitely small, they arrived at a
differential equation for the curve, which they solved. They found,
as did Johann Bernoulli, that the brachistochrone is a cycloid (compare
section 1.8) through A and B with vertical tangent at A. Newton had
also reached this conclusion.
Figure 2.8.4.
The problem of the brachistochrone is very significant in the history
of mathematics, as it is an instance of a problem belonging to the calculus
of variations. It is an extreme value problem, but one in which the
quantity (Ty ), whose extreme value is sought, does not depend on one
or a finite number of independent variables but on the form. of a curve.
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 221 of 479.
84 2. Newton, Leibniz and the Leibnizian tradition
Jakob Bernoulli proposed, as a sequel to his solution of the brachistochrone
problem, further problems of this type, namely the so-called
isoperimetric problems. In the case of the brachistochrone, the class of
curves considered consists of the curves passing through A and B.
In isoperimetric problems one considers curves with prescribed length.
For instance, it could be asked to find the curve through A and B
with length I and comprising, together with the segment AB, the largest
area (see figure 2.8.5). Jakob Bernoulli made much progress in finding
methods to solve this type of problem. Euler unified and generalised
these methods in his treatise 1744a, thus shaping them into a separate
branch of analysis. Lagrange contributed to the further development
of the subject in his 1762a, in which he introduced the concept of variation
to which the subject owes its present name-the calculus of variatons.
On its history, see especially Woodhouse 1810a and Todhunter
1861a.
8
Figure 2.8.5.
2.9. Rational mechanics
The catenary and brachistochrone problems were two problems whose
solution was made possible by the new methods. There were many
more such problems, and their origins were diverse. The direct
observation of simple mechanical processes suggested the problems of
the form of an elastic beam under tension, the problem of the vibrating
string (which Taylor, Daniel Bernoulli, d'Alembert, Euler and many
others studied; see section 3.3) and the problem of the form of a sail
blown by the wind (discussed by the Bernoulli brothers in the early
1690s).
More technologically involved constructions suggested the study of
pendulum motion (which Huygens initiated), the path of projectiles,
and the flow of water through pipes. Astronomy and philosophy
suggested the motion of heavenly bodies as a subject for mathematical
treatment. MathemRtics itself suggested problems too: special difText
15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 222 of 479.
trn
2.9. Rational mechanics 85
ferendal equations were generalised, types of integrals were classified
(for example, elliptic integrals), and so on. Certain types of problems
began rather quickly to form coherent fields with a unified mathematical
approach: the calculus of variations, celestial mechanics, hydrodynamics,
and mechanics in general. Somewhat later, probability (on
which Jakob Bernoulli wrote a fundamental treatise Ars conjectandi
(' The art of guessing '), which was published posthumously as 1713a),
joined this group of mathematicised sciences, or sub-fields of mathe-
matics.
Something more should be said here about the new branches of
mechanics (or ' rational mechanics' as it was then called, to distinguish
it from the study of machines), which acquired its now familiar mathematicised
form in the 18th century. The basis for this mathematicisao.
tion was laid by Newton in his Philosophiae naturalis principia mathematica
(1687a), in which he formulated the Newtonian laws of motion
and showed that the supposition of a gravitational force inversely
proportional to the square of the distance yields an appropriate descripo
tion of the motion of planets as well as of the motion of falling and proo
jected bodies here on earth. He gave here (among many other things)
a full treatment of the motion of two bodies under influence of thei r
mutual gravitational forces, several important results on the 'threcbody
problem', and a theory of the motion of projectiles in a resistillf:
medium. However, a great deal in the way of mathematicisation of the:>;'
subjects still had to be done after the Principia. Though Newton made
full use of his new infinitesimal methods in the Principia, he found and
presented his results in a strongly geometrical style. Thus, although
implicitly he set up and solved many differential equations, exactly or
by approximation through series expansions, one rarely finds them
written out in formulas in the Principia. Neither are his laws of motion
expressed as fundamental differential equations to form the startingpoint
of studies in mechanics.
In the first half of the 18th century, through the efforts of men like
Jakob, Johann and Daniel Bernoulli, d'Alembert, Clairaut and Euler,
the style in this kind of study was further mathematicised~that is, the
methods were transformed into the analytical methods-and they were
unified through the formulation of basic laws expressed as mathematical
formulas, differential equations in particular. Other fields were also
tackled in this way, such as the mechanics of elastic bodies (on which
Jakob Bernoulli published a fundamental article 1694a) and hydrodynamics,
on which father and son Johann and Daniel Bernoulli wrote
early treatises (1743a and 1738a respectively).
Great textbooks of analytic mechanics, such as Euler's Mechanica
(1736a), d'Alembert's 1743a and Lagrange's 1788a, show a gradual
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 223 of 479.
86 2. Newton, Leibniz and the Leibnizian tradition
process of mathematicisation of mechanics. Though Euler's Mechanica
was strongly analytical, the formulation of Newton's laws in terms of
differential equations (now termed' Newton's equations ') occurred for
the first time only in a study of Euler published in 1752 (see Truesdell
1960a). These branches of rational mechanics were very abstract
fields in which highly simplified models of reality were studied. Therefore,
the results were less often applicable than one might have hoped.
These studies served to develop many new mathematical methods and
theoretical frameworks for natural science which were to prove fruitful
in a wider context only much later. Still, the interest in the problems
treated was not entirely internally derived. Thus the proj!;:ctiles of
artillery suggested the study of motion in a resisting medium, while the
three-body problem was studied by Newton, Euler and many others,
especially in connection with the motion of the moon under the influence
of the earth and the sun, a celestial phenomenon which was of the utmost
importance for navigation as good moon tables would solve the problem
of determining a ship's position at sea (the so-called 'longitudinal
problem '). Indeed, Euler's theoretical studies of this problem, combined
with the practical astronomical expertise of Johann Tobias Mayer,
gave navigation, in the 1760s, the first moon tables accurate enough to
yield a sufficiently reliable means for determining position at sea.
Central problems in hydrodynamics were the efflux of fluid from an
opening in a vessel, and the problem of the shape of the earth. The
latter problem was of philosophical as well as practical importance,
because Cartesian philosophy predicted a form of the earth elongated
along the axis, while Newtonian philosophy, considering the earth as a
fluid mass under the influence of its own gravity and centrifugal forces
through its rotation, concluded that the earth should be flattened at the
poles. In practice, the deviation of the surface of the earth from the
exact sphere form has to be known in order to calculate actual distances
from astronomically determined geographical latitude and longitude.
Several expeditions were held to measure one degree along a meridian
in different parts of the earth, and the findings of these expeditions
finally corroborated the Newtonian view.
2.10. What was left unsolved.' the foundational questions
The problem that was left unsolved throughout the 18th century was
that of the foundations of the calculus. That there was a problem was
well-known, and that is hardly surprising when one considers how
obviously self-contradictory properties were claimed for the fundamental
concept of the calculus, the differential. According to l'H6pital's
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 224 of 479.
...
2.10. What was left unsolved: the foundational questions 87
first postulate, a differential can increase a quantity without increasing
it. Nevertheless, this postulate is necessary for deriving the rules of
the calculus, where higher-order differentials (or powers or products of
differentials) have to be discarded with respect to ordinary differentials,
and similarly ordinary differentials have to be discarded with respect to
finite quantities (see (2.5.1)). Also, when Bernoulli takes the differential
of the area fl to be equal to y dx he discards the small triangle at the top
of the strip (like MmR in figure 2.5.2) because it is infinitely small with
respect to y dx. Thus the differentials have necessary but apparently
self-contradictory properties. This leads to the foundational question
of the calculus as many mathematicians since Leibniz saw it :
FQ 1: Do infinitely small quantities exist?
Most practitioners of the Leibnizian calculus convinced themselves
in some way or other that the answer to FQ 1 is ' yes', and thus they
considered the rules of the calculus sufficiently proved. There is,
however, a more sophisticated way of looking at the a way
which for instance Leibniz himself adopted (see Bos 53--66).
He had his doubts about the existence of infinitely small quantities, and
he therefore tried to prove that by using the differentials as possibly
meaningless symbols, and by applying the rules of the calculus, one would
arrive at correct results. So his foundational question was:
FQ 2: Is the use of infinitely small quantities in the calculus reliable?
He did not obtain a satisfactory answer.
In Newton's fluxional calculus (see section 2.2) there also was a
foundational problem. Newton claimed that his calculus was independent
of infinitely small quantities. His fundamental concept was
the fluxion, the velocity of change of a variable which may be considered
to increase or decrease with time. In the actual use of the fluxional
calculus, the fluxions themselves are not important (in fact they are
undetermined), but their ratios are. Thus the tangent of a curve is
found by the argument that the ratio of ordinate to sub-tangent is equal
to the ratio of the fluxions of the ordinate and the abscissa respectively:
y!a=y!i: (y is the fluxion of y, i: the fluxion of x; see figure 2.10.1).
-.---.~------ --_."'----
er
Figure 2.10.1.
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 225 of 479.
88 2. Newton, Leibniz and the Leibnizian tradition
He explains that the ratio of the fluxions :fIx is equal to the' prime'
or 'ultimate' ratio of the augments or decrements of y and x (see
Newton 1693a ; Works2, vol. 1,141). That is, he conceives corresponding
increments Bb of x and Ec of y, and he considers the ratio,EclCE for
Ec and CE both decreasing towards 0 or both increasing from O.
In the first case he speaks of their ultimate ratio which they have just
when they vanish into zero or nothingness; in the latter case he speaks
about their prime ratio, which they have when they come into being from
zero or nothingness. The ratio :fIx is precisely equal to this ultimate
ratio of evanescent augments, or equivalently to this prime ratio of
, nascent' augments.
Obviously there is a limit-concept implicit in this argument, but it is
also clear that the formulation as it stands leaves room for doubt. For
as long as the augments exist their ratio is not their ultimate ratio, and
when they have ceased to exist they have no ratio. So here too is a
foundational question, namely:
FQ 3: Do prime or ultimate ratios exist?
2.11. Berkeley's fundamental critique of the calculus
Most mathematicians who dealt with calculus techniques in the early
18th century did not worry overmuch about foundational questions.
Indeed, it is significant that the first intensive discussion on the foundations
of the calculus was not caused by difficulties encountered in working
out or applying the new techniques, but by the critique of an outsider
on the pretence of mathematicians that their science is based on secure
foundations and therefore attains truth. The outsider was Bishop
George Berkeley, the famous philosopher, and the target of his critique
is made quite clear in the title of his tract 1734a: 'The Analyst; or a
Discourse Addressed to an Infidel Mathematician Wherein It Is
Examined Whether the Object, Principles, and Inferences of the
Modern Analysis are More Distinctly Conceived, or More Evidently
Deduced, than Religious Mysteries and Points of Faith'.
As we have seen, Berkeley indeed had a point. In sharp but captivating
words he exposed the vagueness of infinitely small quantities,
evanescent increments and their ratios, higher-order differentials and
higher-order fluxions (1734a, para. 4) :
Now as our Sense is strained and puzzled with the perception of
Objects extremely minute, even so the Imagination, which Faculty
derives from Sense, is very much strained and puzzled to frame clear
Ideas of the least Particles of time) or the least Increments generated
therein: and much more so to comprehend the Moments, or those
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 226 of 479.
2.11. Berkeley's fundamental critique of the calculus 89
Increments of the flowing Quantities in statu nascenti, in their very
first origin or beginning to exist, before they become finite Particles.
And it seems still more difficult, to conceive the abstracted Velocities
of such nascent imperfect Entities. But the Velocities of the
Velocities, the second, third, fourth and fifth Velocities, &c. exceed,
if I mistake not, all Humane Understanding. The further the
Mind analyseth and pursueth these fugitive Ideas, the more it is
lost and bewildered; the Objects, at first fleeting and minute, soon
vanishing out of sight. Certainly in any Sense a second or third
Fluxion seems an obscure Mystery. The incipient Celerity of an
incipient Celerity, the nascent Augment of a nascent Augment i.e.
of a thing which hath no Magnitude: Take it in which light you
please, the clear Conception of it will, if I mistake not, be found
impossible, whether it be so or no I appeal to the trial of every
thinking Reader. And if a second Fluxion be inconceivable, what
are we to think of third, fourth, fifth Fluxions, and so onward
without end?
Further on comes the most famous quote from The analyst: 'And
what are these Fluxions? The Velocities of evanescent Increments?
And what are these same evanescent Increments? They are neither
finite Quantities, nor Quantities infinitely small nor yet nothing. May
we not call them the Ghosts of departed Quantities?' (para. 35).
Berkeley also criticised the logical inconsistency of working with small
increments which first are supposed unequal to zero in order to be
able to divide by them, and finally are considered to be equal to zero in
order to get rid of them.
Of course Berkeley knew. that the calculus, notwithstanding the
unclarities of its fundamental concepts, led, with great success, to
correct conclusions. He explained this success-which led mathematicians
to believe in the certainty of their science-by a compensation
of errors, implicit in the application of the rules of the calculus. For
instance, if one determines a tangent, one first supposes the characteristic
tria'ngle similar to the triangle of ordinate, sub-tangent and tangent,
which involves an error because these triangles are only approximately
similar. Subsequently one applies the rules of the calculus to find the
ratio dy/dx, which again involves an error as the rules are derived by
discarding higher-order differentials. These two errors compensate
each other, and thus the mathematicians arrive' though not at Science,
yet at Truth, For Science it cannot be called, when you proceed blindfold,
and arrive at the Truth not knowing how or by what means'
(1734a, para. 22).
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 227 of 479.
90 2. Newton, Leibniz and the Leibnizian tradition
2.12. Limits and other attempts to solve the foundational questions
Berkeley's critique started a long-lasting debate on the foundations of
the calculus. Before mentioning some arguments in this debate, it may
be useful to recall how in modern differential calculus the foundational
question is solved. Modern calculus concerns functions and relates to
a function f its derivative 1', which is again a function, defined by means
of the concept of limit:
f'(x)= lim (f(X+h)-f(X)). (2.12.1)
Df h-+O h
The preliminaries for this approach were worked out in the 18th and
19th centuries; they played different roles in the various approaches to
the foundational questions which were adopted in that period. It is
instructive to list the preliminaries. They are:
(1) the idea that the calculus concerns functions (rather than
variables) ;
(2) the choice of the derivati'oe as fundamental concept of the
differential calculus (rather than the differential) ;
(3) the conception of the derivative as a function; and
(4) the concept of limit, in particular the limit of a function for
explicitly indicated behaviour of the independent variable (thus explicitly
lim (p(h)), rather than merely the limit of the variable p).
k->O
Of the various approaches to the questions raised by Berkeley's
critique, we have already seen the one adopted by Euler: he did conceive
the calculus as concerning functions, but for him the prime concept
was still the differential, which he considered as equal to zero but capable
of having finite ratios to other differentials. Obviously this still leaves
the foundational question QF 3 of section 2.10 unanswered. In fact,
it does not seem that Euler was too much concerned about foundational
questions.
Berkeley's idea of compensating errors was used by others to show
that, rather than proceeding blindfold, the calculus precisely compensates
equal errors and thus arrives at truth along a sure and well-balanced
path. The idea was developed by Lazare Carnot among others.
Another approach was due to Joseph Louis Lagrange, who supposed
that for every function f and for every x one can expand f(x +h) in a
senes
(2.12.2)
So Lagrange defined the' derived function' f'(x) as equal to the coefficient
of h in this expansion. The idea, published first in 1772a, became
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 228 of 479.
2.12. Attempts to solve the foundational questions 91
somewhat influential later through Lagrange's Thiorie des fonctions
analytiques (Functions). As a solution of the foundational questions
the idea is unsound (not every f(x+h) can be so expanded, and even so
there would be the question of convergence), but in other ways this
approach was quite fruitful; it conceived the calculus as a theory about
functions and their derived functions, which are themselves again functions.
For more details on Carnot and Lagrange, see sections 3.3 and
3.4.
Eventually the most important approach towards solving the foundational
questions was the use of limits. This was advocated with respect
to the fluxional calculus by Benjamin Robins (see his 1761a, vot. 2, 49),
and with respect to the differential calculus by d'Alembert. Robins
and d'Alembert considered limits of variables as the limiting value which
these variables can approach as near as one wishes. Thus d'Alembert
explains the concept in an article 1765a on ( Limite ' in the Encyclopedie
which he edited with D. Diderot: 'One magnitude is said to be the
limit of another magnitude when the second may approach thc first
within any given magnitude, however small, though the first magnitude
may never exceed the magnitude it approaches'.
m
x
R o A p p
Figure 2.12.1.
In the Encyclopedie article ( Differentiel ' (1764a) d'Alembert gave
a lengthy explanation, with the parabola y2 =ax as example. His argument
can be summarised as follows. From figure 2.12.1 it follows that
MP/PQ is the limit of mO/OM. In formulae, mO/OM=a/(2y+z),
and algebraically the limit of a/(2y+z) is easily seen to be a/2y. One
variable can have only one limit, hence MP/PQ = a/2y. Furthermore,
the rules of the calculus also give dy/dx = a/2y, so that we must conceive
dy/dx not as a ratio of differentials or as 0/0, but as the limit of the
ratio of finite differences mO/OM.
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 229 of 479.
92 2. Newton, Leibniz and the Leibnizian tradition
Robins and d'Alembert were not the first to formulate the concept of
limit; in fact it occurs already implicitly in ancient Greek mathematics,
and later Simon Stevin for instance came very close to formulating it
(see his Works, vo!. 1, 229-231). For a very long time after Robins
and d'Alembert propagated the use of this concept to solve the founda~
tional questions, the limit approach was just one among many approaches
to the problem. The reason why it took so long until the value of the
limit approach was recognised lay in the fact that Robins and d'Alembert
considered limits of variables. In that way the concept still involves
much unclarity (for details, see Baron and Bos 1976a, unit 4) which
could only be removed once the limit concept was applied to functions
under explicitly specified behaviour of the independent variable.
2.13. In conclusion
In the century which followed Newton's and Leibniz's independent
discoveries of the calculus, analysis developed in a most impressive way,
despite its rather insecure foundations, thus making possible a mathematical
treatment of large parts of natural science. During these developments
analysis also underwent deep changes; for Newton and
Leibniz did not invent the modern calculus, nor did they invent the
same calculus. It will be useful to recall, in conclusion, the main
features of both systems, their mutual differences, and their differences
from the forms of calculus to which we are now used (compare Baron
and Bos 1976a, unit 3, 55-57).
Both Newton's and Leibniz's calculi were concerned with variable
quantztzes. However, Newton conceived these quantities as changing
in time, whereas Leibniz rather saw them as ranging over a sequence of
infinitely' close values. This yielded a difference in the fundamental
concepts of the two calculi; Newton's fundamental concept was the
fluxion, the finite velocity or rate of change (with respect to time) of the
variable, while Leibniz's fundamental concept was the differential, the
infinitely small difference between successive values in the sequence.
There was also a difference between the two calculi in the conception
of the integral, and in the role of the fundamental theorem. For Newton
integration was finding the fluent quantity of a given fluxion; in his
calculus, therefore, the fundamental theorem was implied in the definition
of integration. Leibniz saw integration as summation; hence for
him the fundamental theorem was not implied in the definition of integration,
but was a consequence of the inverse relationship between summing
and taking differences. However, the Bernoullis re-interpreted the
Leibnizian integral as the converse of differentiation, so that throughout
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 230 of 479.
2.13. In conclusion 93
the 18th century the fundamental theorem was implied in the definition
of integration.
Both Newton and Leibniz worked with infinitely small quantities and
were aware of the logical difficulties inherent in their use. Newton
claimed that his calculus could be given a rigorous foundation by means
of the concept of prime and ultimate ratio, a concept akin to (but
not the same as) the concept of limit.
Leibniz valued notation very much, and his choice of symbols for
the calculus proved to be a happier one than Newton's. His use of
separate letters, 'd' and ' J" indicated the role of differentiation and
integration as operators; moreover, his symbols were into
complicated formulas much more easily than were Newton's. In
general, Leibniz's calculus was the more analytical; Newton's was
nearer to the geometrical figures, with accompanying arguments in prose.
These are the principal differences between the two If we
compare them with the modern calculus, we note three further differences.
Firstly, whereas Newton's and Leibniz's were concerned
with variables, the modern calculus deals with functions.
Secondly, the operation of differentiation is defined in the modern
calculus differently from in the 18th century ; it relates to a function a
derived function, or derivative, defined by means of the concept of limit.
Thirdly, unlike 18th-century calculus, modern analysis has a generally
accepted approach to the problem of the foundation of the calculus
namely, through a definition of real numbers (instead of the vague
concept of quantity which had to serve as a basis for analysis before the
1870s) and through the use of a well-defined concept of limit. The next
chapter describes much of this future progress.
Text 15: H. J. M. Bos (1980). “Newton, Leibniz and the Leibnizian Tradition”. In: From
the Calculus to Set Theory, 1630–1910. An Introductory History. Ed. by
I. Grattan-Guinness. Princeton and Oxford: Princeton University Press. Chap. 2,
pp. 49–93.
Summer University 2012: Asking and Answering Questions Page 231 of 479.
CHAPTER 3
NICCOLO GurCCIARDINI
3.1. Introduction
From the J660s to the 1680s, Isaac Newton and Gottfried WilheJm Leibni7,
created what we nowadaY8 recognize as infinitesimal calculus. A st.udy of their
achievements reveals elements of continuity with previous work 2) as
well as peculiarities which distinguish their methods and concepts from those which
are accepted in present day mathematics. The statement it8elf tha.t "Ncwton and
Leibniz invented the calculus" is problematic. In the first placc, they two
different versions of calculus, and the problem of comparing the two, of establishing
equivalences and differences, arises (see Chapter 3.5). In the second place, what do
we mean by "inventing calculus" in this context?
The novelty of Newton's and Leibniz's contributions can be briefly characterized
by pointing out three aspects of their mathematical work: problem-reduction, the
calculation of areas by inversion of the process for calculating tangents, the creation
of an algorithm. The "invention of calculus" can thus be conceived as consisting of
these three contributions.
Newton and Leibniz realized that a whole variety of problems about the calculation
of centres of gravity, areas, volumes, tangents, arclengths, radii of curvature,
surfaces, etc., that had occupied mathematicians in the first half of the seventeenth
century, were instances of two basic problems. Furthermore, they fully realized
that these two problems were the inverse of each other (this is the "fundamental
theorem" of calculus). They thus understood that the solution of the former, and
easier, problem could be used to answer the latter. Last but not least, Newton
and Leibniz developed two efficient algorithms that can be applied in a systematic
and general way. It is thanks to these contributions that Newt.on and Leibniz
transformed mathematics.
The peculiarity of Newton's and Leibniz's algorithms is a fact that the historian
is sometimes led to forget. In fact, both, especially the latter, look very much the
same as the one we employ nowadays. We can thus be tempted to modernize their
calculi. As a matter of fact, their calculi are strongly embedded in the culture of
their own times. We make two major points. Neither Newton's nor Leibniz's calculi
are about "functions" (see (Bos 1980, 90).) The concept of function emerged only
later (see Chapter 4). Newton and Leibniz talk in terms of "quantities" rather
than "functions", and they refer to these quantities, their rates of change, their
differences, etc., related to specific geometric entities (typically a given curve). Thus
the reader will notice that in what follows I will always use the term "function"
in "quotation marks". Furthermore, while we are used to referring to calculus as
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 232 of 479.
'74 3. NEWTON'S METHOD AND LEIBNIZ'S CALCULUS
the continuum of the real numbers, the continuum to which Newton and Leibniz
refer is geometrical or kinematical. It is by referring to an intuitive geometric or
kinematic continuum that Newton and Leibniz develop their limit procedures (see
3.5.2).
3.2. Newton's method of series and ftuxions
3.2.1. A mathematician working in isolation. Isaac Newton was born
into Cl family of small landowners. After receiving an elementary education, he was
sent to Cambridge, where he matriculated as a sub-sizar in 1661. "Sub-sizars" were
poor students who worked as servants to the fellows and the rich students. Newton
raised himself from this condition to become Lucasian Professor, Warden of the
Mint, Cl member of Parliament and President of the Royal Society. His funeral was
described by Voltaire as being as full of pomp as those of a king. His success in
British society was determined by the high esteem which his published scientific
discoveries aroused. In his secret, unpublished, studies Newton cultivated interests
that would have ruined his public image. He was involved in alchemical studies,
and his theological interests, inspired by deep religious feelings, gave him strongly
critical attitude towards the established Church.
Some of Newton's greatest scientific discoveries were made during the years
1665-1667, when Cambridge university was closed because of the plague. During
these anni mirabiles Newton performed experiments with prisms, convincing
himself of the composite nature of white light, stated the binomial theorem for fractional
powers, discovered the calculus of fiuxions and speculated about the moon's
motion. For complicated reasons, he did not immediately share his mathematical
results with others. This is only explained in part by the cost of mathematical publications
at that time. More decisive was his introverted character that led him to
keep his thoughts to himself. Furthermore, he was not completely confident about
the conceptual foundation of his calculus. To these causes which may have hindered
Newton from publishing his discoveries on calculus, one can add that it was
a practice of some seventeenth century mathematicians to keep their mathematical
methods secret. The mathematical tools, which allowed the solution of problems,
were considered private property, not to be shared too generously with others. Very
much as painters kept the secrets for obtaining colours for themselves, the mathematicians
often gave the solution without revealing the demonstration. In 1676 the
secretary of the Royal Society, Henry Oldenburg, obtained from Newton two letters
in which some of his mathematical results were summarized. These two letters
were meant to inform a German philosopher, Gottfried Wilhelm Leibniz, about the
scope of Newton's achievements. The Philosophiae Naturalis Principia Mathematica
(1687), where Newton developed his theory of gravitation, also contained results
connected with calculus. It was only in 1704 that Newton published a systematic
treatise on calculus: the De quadratura curvaTwn. This was too late to prevent a
priority dispute with Leibniz, who had already published his differential calculus
in 1684. Leibniz was accused of plagiarism by Newton and by the British fellows
of the Royal Society. Actually he had discovered differential and integral calculus
in 1672-1676 independently. He therefore asked the Royal Society to withdraw the
accusation of plagiarism that was circulating in several papers. A committee of
the Royal Society, guided secretly by Newton, reported that Leibniz was guilty of
plagiarism. The Newtonian and the Leibnizian schools difFered strongly on a wide
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 233 of 479.
3.2. NEWTON'S METHOD OF SERIES AND FLUXIONS
range of issues. They maintained different cosmologies, different views on the relationships
between God and nature, different views on space and time and on the
conservation laws basic in physics. The priority dispute divided them mathematically.
This was a bitter outcome for Leibniz, who had always maintained that the
demonstrative power of mathematics could end all disputes and promote a more
harmonious world.
3.2.2. The binomial series (1664 to 1665). It appears that Newton's interest
in mathematics began in 1664, when he read Frangois Viete's works (1646),
Descartes's Geometric (1637) (the second Latin edition (1659-1661) with Frans van
Schooten's commentaries and Hudde's rule), William Oughtred's Clavis mathematicac
(1631), and Wallis's Ar'ithmctica Infinitorum (1656). It was from reading this
selected group of mathematical works in "modern analysis" that Newton learned
about the most exciting discoveries on analytic geometry, algebra, tangent problems,
quadratures and series. After a few months of self-instruction he was able,
in the winter 16641665, to make his first mathematical discovery: the "binomial
theorem" for fractional powers. In slightly modernized notation, he stated:
/ m/n m m/n--], 1 m (rn ) 'fn/n--2 2
(3.1) (a+x)mn a +-a x+ .- ---la x+···.
n 1.2 n n
Newton obtained this result generalising by Wallis's "inductive" method for squaring
the unit circle. The process of interpolation with which Newton determined the
binomial coefficients is too long to be described in detail here. A good presentation
of Newton's guesswork can be found in (Edwards 1979, 178-187). Here it will suffice
to say that Newton arrived at
(3.2)
113 11 5 11 7
Ix - -x - -_·x ----x
23 85 167
5 1 9
-x
1289
as a series for the area under the curve (1 - x2?/2, a result which allows one to
calculate the circle's area. He further noted that, since the area under xn and over
the interval [0, xl is /(n + 1), he could extend the result valid for the area to
the curve itself to obtain
(3.3) (1 .- x2)1/2 = 1 _ ~X2 _ ~.T4 __ 1
2 8 16
By working through similar examples, Newton guessed the general law of formation
of the binomial coefficients for fractional powers (see (3.1)). He further
extrapolated (3.1) to negative powers. The case n -1,
(3.4) (1 + x)-l = 1 - x + x2 - x3 + x4 - ....
is particularly relevant. Since the proof of the binomial series rested on shaky
"inductive" Wallisian procedures, Newton felt the need to verify the agreement
of the series obtained by applying (3.1) by algebraical and numerical procedures.
For instance, he applied standard techniques of root extraction to (1 - x2)1/2 and
standard techniques of "long division" to (1 + x)-l, and he was happy to see that
he obtained the series (3.3) and (3.4).
He also knew that the area under (1 +x)--l and over the interval [0, x], or the
negative of this area if -1 < x < 0, is In(l + x). He could thus express In(1 + x) as
a power series by termwise integration of (3.4):
(3.5)
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 234 of 479.
76 3. NEWTON'S METHOD AND LEIBNIZ'S CALCULUS
Actually the order of Newton's reasoning is quite unexpected: He first obtained
(3.5) via interpolation, and then he obtained (3.4) by differentiation. The series
(3.5) allowed Newton to calculate In(l +x), for x ~ O. He carried out his numerieal
calculations up to more than fifty decimal places!
We note three aspects of Newton's work on the binomial series. First of all
he introduced, following Wallis's suggestion, negative and fractional exponents.
Without this innovative notation (xa
/
b
for \/xa) it would not have been possible
to interpolate or extrapolate the binomial theorem from positive integers to the
rationals. Secondly, Newton obtained a method for representing a large class of
"curves" by a power series. For him curves are thus given not only by finite algebraical
equations (as for Descartes) but also by infinite series (preferably power
series) understood by Newton and by his contemporaries as infinite equations. In
1665 mathematicians had just begun to appreciate the usefulness of infinite series
as representations of "difficult" curves. Transcendental curves, such as the logarithmic
curve, can thus be given an "analytical" representation to which the rules
of algebra can be applied. Before the advent of infinite series, such "functions" had
no analytic representation, but they were generally defined in geometric terms. It
should be noted that Newton had a rather intuitive concept of convergence. For
instance he realized that the binomial series (3.1) can be applied when x is "small".
Newton developed no rigorous treatment of convergence.
3.2.3. The fundamental theorem, 1665 to 1669. Newton's first systematic
mathematical tract bears the title De analysi per aequationes nurnero terrninOr1Lrn
infinitas. Newton began this short summary of his discoveries with the
enunciation of three rules that can be rendered as follows (Newton 1669, 206 ff.):
Rule 1: If y axm/n , then the area under y is (an) / (n + m )xm/n+I.
Rule 2: If y is given by the sum of more terms (also an infinite number of
terms), Y YI +Y2 + ... ,then the area under Y is given by the sum of the areas of
the corresponding terms.
Rule 3: In order to calculate the area under a curve f(x, y) = 0, one must
expand Y as a sum of terms of the form axm
/
n
and apply Rule 1 and Rule 2. Rule
1 had been stated by Wallis. As we will see, Newton provided a proof of this rule
based on the fundamental theorem. The binomial series proved to be an important
tool implementing Rule 3. In several cases, however, the binomial series cannot be
applied. In the years from 1669 to 1671 Newton devised several clever techniques
for obtaining a series z = I: bixi , i rational, from an implicit "function" f(x, z) = o.
He also had a method for "reverting" series. That is, given z I: b;xi, he had a
method of successive approximations which led to x = I: aiz;. It is reverting the
power series expansion of z = In(l + x) (formula (3.5)) that he obtained the series
for x = eZ
(see (Edwards 1979, 204-205) and Chapter 4).
The most general result concerning the squaring of curves (i.e., "integration")
is the fundamental theorem of calculus which Newton discovered in 1665. Newton's
reasoning, which resembles Barrow's (see 2.2.4), refers to two particular curves (see
Fig. 3.1) z = x3
/ a and y = 3x2
/ a, but it is completely general: y is equal to the
slope of z and is defined as
(3.6)
m(3
bg = dh n(3'
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 235 of 479.
3.2. NEWTON'S METHOD OF SERIES AND FLUXIONS 77
'If
FIGURE 3.]
where bg is an ordinate of the curve y, and rn(3 and [2(3 are infinitesimal increments
of z and x, while dh is a unit length segment. It follows immediately that the area
bpsg (= [2(3 . bg) and the area /111,)..1/ (= rn(3 . dh) are equal. It was commonplace
in seventeenth century mathematics to consider the area subtended by a curve to
be equal to the sum of infinitely many infinitesimal strips such as bpsg. It follows
that the curvilinear area subtended by y, e.g., d1/m, is equal to the rectangular area
dhO'p. A knowledge of z then allows us to "square" y, since "the area under y (the
derivative curve) is proportional to the difference between corresponding ordinates
of z" (Westfall1980, 127). In Leibnizian terms, Newton proves that the integral of
the derivative of z is equal to z (see (Newton 1665)).
A proof of the fact that the derivative of the integral of y is equal to y was
given by Newton at the end of De analysi as a proof of Rule 1. He proceeded as
follows.
Newton considered a curve AD6 (see Fig. 3.2), where AB = x, BD = y and
the area ABD = z. He defined B (3 = 0 and B K = v such that "the rectangle
B(3HK ov) is equal to the space B{36D." Furthermore, Newton assumed that
B(3 is "infinitely small." With these definitions one has that A(3 = x + 0 and the
area A6(3 is equal to z + ov. At this point Newton wrote: "from any arbitrarily
assumed relationship between x and z I seek y." He noted that the increment of
the area OV, divided by the increment of the abscissa 0 is equal to v. But since one
can assume "B{) to be infinitely small, that is, 0 to be zero, v and y will be equal."
Therefore, the rate of increase of the area is equal to the ordinate (Newton 1669,
242244).
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 236 of 479.
78 3. NEWTON'S METfIOD AND LEIBNIZ'S CALCULUS
FIGURE 3.2
The fundamental theorem allowed Newton to reduce the probems of quadrature
to the search for primitive "functions". He actually calculated the tangent for
a great variety of "curves", so compiling what he called "tables of Huents" (in
Leibnizian terms "table of integrals"). We will see in the next section how he
deployed the fundamental theorem in order to square curves.
3.2.4. The method of fluents, fluxions and moments (1670 to 1671).
While the De analysi was devoted mainly to series expansions and the use of series
in quadratures, the De rnethodis serierurn et .fluxionv,rn written in 1670-1671 was
mainly devoted to the use of an algorithm that Newton had developed in the years
from 1665 to 1666. The objects to which this algorithm is applied are quantities
which "flow" in time. For instance the motion of a point generates a line and the
motion of a line generates a surface. The quantities generated by a "flow" are
called "Huents". Their instantaneous speeds are called "fluxions". The "moments"
of the fluent quantities are "the infinitely small additions by which those quantities
increase during each infinitely small interval of time" (Newton 16701671, 80).
Consider a point which flows with variable speed along a straight line. The distance
covered at time t is the fluent, the instantaneous speed is the fluxion, and the
"infinitely" (or "indefinitely") small increment acquired after an indefinitely small
period of time is the moment. Newton further observed that the moments "are as
their speeds of How", i.e., as the fluxions) (Newton 1670--1671, 78). His reasoning
is based on the idea that during an "infinitely small period of time" the Huxion
remains constant and so the moment is proportional to the fluxion. Newton warns
the reader not to identify the "time" of the f-luxional method with real time. Any
fluent quantity whose f-luxion is assumed constant plays the role of fluxional "time".
Newton did not develop a particularly handy notation in this context. He employed
a, b, c, d for constants, v, x, y, z for the f-luents and l, m, n, T for the respective
fluxions, so that, e.g., m is the fluxion of x. The "indefinitely" (or "infinitely")
small interval of time was denoted by o. Thus the moment of y is no. It was only
in the 1690s that Newton introduced the now standard notation where the fluxion
of x is denoted by x and the moment of ::r; by xo. The fluxions themselves can be
considered as fluent quantities so that one can seek for the f-luxion of n/m. In the
1690s Newton denoted the "second" f-luxion of x by x.
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 237 of 479.
3.2. NEWTON'S METHOD OF SERIES AND FLUXIONS '79
x
FIGURE ~).3
Newton did not use a single notation for the area undeI' a curve. Generally he
put words such as "the area of" or a capital Q before the analytical expression of
the curve. In some cases he used "[~jx'~" for "the area under the curve of equation
y = ajx2
" (in Leibnir,ian terms this would be J(ajx2
)dx). As we will see (3.2.6)
Newton also employed xto denote a Huent quantity whose Huxion is x. The limits
of integration were either understood by the context or explained by words.
In the De rnethodis Newton gives the solution of a series of problems. The
main problems are to find maxima and minima, tangents, curvatures, areas and
arclengths. The representation of quantities as generated by continuous How allows
all these problems to be reduced to the following Problems 1 and 2:
1) Given the length of the space continuously (that at every
time), find the speed of motion at any time proposed.
2) Given the speed of motion continuously, find the length of the
space described at any time proposed.
(Newton 16701671, 7071)
The problems of finding tangents, extremal points and curvatures are related to the
former, and the problems of finding areas and arclengths are related to the latter.
Imagine a plane curve f (x, y) = 0 to be generated by the continuous How of
a point P(t). If (x, y) are the Cartesian coordinates of the curve, yjx will be
equal to tawy, where r is the angle formed by the tangent in P(t) with the x-axis
(see Fig. 3.3). According to Newton's conception, the point will move during the
"indefinitely small period of time" with uniform rectilinear motion from P(t) to
P(t + 0). The infinitesimal triangle indicated in Fig. 3.3 has sides equal to yo and
.1:0 and so tan r = yojXo = yjx. An extremal point will have yjX = tall{ = O.
Newton showed that the radius of curvature is given by p (1 +(yjx)2)3/2 j(jjjx2).
The fact that the finding of areas can be reduced to Problem 2 is a consequence
of the fundamental theorem. Let z be the area generated by continuous uniform
How (x = 1) of ordinate y (see Fig. 3.2). The speed of motion is given continuously,
i.e., it is given by i. By the fundamental theorem y = i. In order to find the area,
a method is required for obtaining z from y = i. This is Problem 2. It should be
stressed how the conception of quantities as generated by continuous How allowed
to Newton to conceive the problem of determining the area under a curve as a
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 238 of 479.
80 cl. NEWTON'S METHOD AND LEIBNIZ'S CALCULUS
special example of Problem 2. The reduction of arclength problems to Problem 2
depends on the application of Pythagoras's theorem to the moment of arclength 8:
"50 V(xoP--+:-Cyo)2 (see Fig. :3.:-1). Therefore 8 =
L------------~c--
The basic algorithm for Problem 1 is given by Newton an example (Newton
16701671, '78-81). He considered the equation x3
-~ ax2
axy y3 =, O. He
substituted x+:i;o in place of x and y+yo in place of y. Deleting .<:3 -- ax2
+a:ry-y:'
as equal to zero and then dividing by 0, he obtained an equation from which he
cancelled the terms which had 0 as a factor. These terms have the property that
they "will be equivalent to nothing in respect to the others", since "0 is supposed
to be infinitely small." At last Newton arrived at
(3.7) 3xx2
2a:cx + axy + ay:r ~ 3yy2 O.
This result is achieved by employing a rule of cancellation of higher-order inflnitesimals
(equivalent to Leibniz's :r + dx x), according to which, if x is finite and 0
is an infinitesimal interval of time, then
(:3.8) x + xo = x.
Notice that the above example also contains the rules for the fluxiom; of a product
xy and of x"', respectively: xy + yx and nxn -- 1
x.
Newton dealt with irrational "functions" as follows. He considered y2 - a2
x,;a~~--x2
O. He set z and so obtained y2 a2 - z = 0 and
a2x2 - = O. Applying the direct algorithm, he determined 2yy - i = 0 and
2a2 x:r; -- 4xx" - 2iz O. He then eliminated i, restored z = X~X2, and thus
arrived at
2yy + (-a2x + 2xx2)/ vfaj----x2 = 0
as the relation sought between y and X.
Even though Newton presents his "direct" algorithm by applying it to particular
cases, his procedure can be generalized. Given a curve expressed by a function in
parametric form, f(x(t),y(t)) = 0, the relation between the fluxions x and y is
obtained by application of the equation
f(
. . ) of.x + xo, y + yo = ~xo
ox
of 2
~yo + 0 ( ... ) = o.
oy
After division by 0, the remaining terms in 0 are cancelled. Such a modern re··
construction clearly says more than what Newton could express. I used concepts
and notation, not a.vailable to Newton, for a function f(x(t), y(t)) and for partial
derivatives. However, with due caution, it can be used to highlight the following
points.
1) Newton assumes that, during the infinitesimal interval of time 0, the motion
is uniform, so that when x flows to x + XO, y flows to y + yo. Therefore, f(x, y) =
f(x + xo, y + yo).
2) Newton applies the principle of cancellation of infinitesimals, so in the last
step the terms in () are dropped.
Newton's justification for his algorithmic procedure is not much more rigorous
than those in the works of Pierre Fermat or Hudde. As we will see in the next
subsection, he was soon to face serious foundational questions.
Problem 2 is, of course, much more difficult. Given a "fluxional equation"
f(x, y, x, y) = 0, Newton seeks a relation g(x, y, c) 0 (c constant) such that the
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 239 of 479.
;),2, NEWTON'S METHOD OF SERIES AND FLUXJONS 81
o
FIGURE 3.4
application of the direct algorithm yields f(x, y, x, y) = 0, In [,eibnil':ian terms, he
poses the problem of integrating di{f'erential equations,
Newton has a very general strategy which allows him to solve a great variety
of such "inverse problems", His strategy is twofold, 1) Either he changes variable
in order to reduce to a known table of Huents (in Leibnizian terms, a "table of
integrals") or 2) he deploys series expansion techniques (termwise integration), His
strategy is a great improvement on the geometrical quadrature techniques of, e,g.,
Huygens, or the techniques of direct summation of, e,g" Wallis (see Chapter 2),
We can give some of the Havour of Newton's first strategy by looking again at
the quadrature of the cissoid which had occupied Huygens and Wallis in the late
1650s (see Chapter 2 and (van Maanen 1991)). Newton used y = as
the equation for the cissoid (see Fig. 2,21). Problem 2 is solved by the determination
of a z such that ijX = x2
/,rax~x2, For k = :1N2va - X,
(3.9) k = ~/~x _ x2
:1; 2
Rearranging, we get
(3.10)
In Leibnizian terms, z Joa
3Vax- x2dx 2[k(x)]o. The area under the cissoid
and over the interval [0, a] is therefore three times the area under the semicircle
with equation y vax - x2 . Notice that the second term on the right of (3.10)
vanishes when "integrated" over [0, a],
When the first strategy failed, Newton tried the second, He generally reduced
the quadrature to the area under the graph of a circular or a hyperbolic "function",
such as (a2
X2)±1/2 or a/(b+cx). These he could evaluate by binomial expansion
and termwise "integration". An example follows.
Consider a circle with unit length radius (see Fig. 3.4): The moment of the
arc eo is to the moment of the abscissa xo as 1 to . Applying the binomial
theorem to (1 x2 )-1/2 and "integrating" termwise, Newton obtained the arcsin
series
(3,11)
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 240 of 479.
82 3. NEWTON'S METHOD AND LETBNIZ'S CALCULUS
"R.everting" the above series by a process of successive approximations, he obtained
the power series for sin.
Newton was able to solve the inverse problem for a large class of fiuxional
equations. Had he published his tract in 1671, he would have aroused awe in all
the corners of Europe.
3.2.5. The geometry of and ultimate ratios (1671 to l'T04). As
we have seen, Newton employed methods characteristic of the seventeenth-century
"new analysis" in his early writings. He used series and infinitesimal quantities.
Infinitesimals entered mainly as moments, momentaneous increments of a "flowing"
variable quantity. The kinematical approach to the calculus was therefore prevalent
in Newton's work from the very beginning. For him, reference to our intuition of
continuous "flow" provided a means to "define" the reference objects of the calculus:
fiuents, fiuxions and moments (see 3.5.2).
Up to the composition of the De methodis, Newton described himself with
pride as a promoter of the seventeenth-century "new analysis". However, in the
1670s he abandoned the calculus of fluxions in favour of a geometry of fluxions
where infinitesimal quantities were not employed. He labelled this new method the
"synthetical method of fiuxions" as opposed to his earlier "analytical method of
fiuxions" (Newton 1967--1981, 8, 454-455). Some of the results on the synthetical
method were summarized in Section 1, Book 1 of Principia Mathematica entitled
"The method of prime and ultimate ratios". He wrote:
whenever in what follows I consider quantities as consisting of
particles or whenever I use curved line-elements [or minute curved
lines] in place of straight lines, I wish it always to be understood
that I have in mind not indivisibles but evanescent divisibles,
and not sums and ratios of definite parts but the limits of such
sums and ratios, and that the force of such proofs always rests
on the method of the preceding lemmas. (Newton 1687/1999,
441-442)
He also pointed out that the method of prime and ultimate ratios rested on the
following Lemma 1:
Quantities, and also ratios of quantities, which in any finite time
constantly tend to equality, and which before the end of that
time approach so close to one another that their difference is
less than any given quantity, become ultimately equal. (Newton
1687/1999, 433)
Newton's ad absurdum proof runs as follows:
If you deny this, let them become ultimately unequal, and let
their ultimate difference be D. Then they cannot approach so
close to equality that their difference is less than the given difference
D, contrary to the hypothesis. (Newton 1687/1999,433)
This principle might be regarded as an anticipation of Cauchy's theory of limits
(see Chapter 6), but this would certainly be a mistake, since Newton's theory of
limits is referred to as a geometrical rather than a numerical model.
The objects to which Newton applies his "synthetical method of fluxions" or
"method of prime and ultimate ratios" are geometrical quantities generated by
continuous flow (i.e., "fluents"). While in his early writings Newton represented
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 241 of 479.
3,2, NEWTON'S METHOD OF SERIES AND FLUXIONS
FIGURE 3,5
a~---,l f
A
n
~'-'--'
BF C D
FIGURE 3.6
E
83
the fiuents with algebraical symbols, in this new approach he referred directly to
geometrical figures. These figures, however, are not static, as in classic geometry:
they must be conceived as "in motion" .
A typical problem is the study of the limit to which the ratio of two geometrical
fiuents tends when they vanish simultaneously (Newton used the expression of the.
"limit of the ratio of two vanishing quantities"). For instance, in Lemma 7 Newton
shows that given a curveACB (see Fig. 3.5):
the ultimate ratio of the arc, the chord, and the tangent to one
another is a ratio of equality. (Newton 1687/1999, 436)
The proof, which rests on Lemma 1, is based on the fact that a difference between
~ ~
the arc ACB and the tangent AD, or the arc ACB and the chord AB, can be made
less than any assignable magnitude by taking B sufficiently close to A.
In Lemma 2 Newton shows that a curvilinear area AabcdE (see Fig. 3.6) can
be approached as the limit of the inscribed AKbLcMdD or the circumscribed
AalbmcndoE rectilinear areas. The proof is magisterial in its simplicity. Its structure
is still retained in present day calculus textbooks in the definition of the definite
integral. It consists in showing that the difference between the areas of the circumscribed
and the inscribed figures tends to zero, as the number of parallelograms
tends to infinity. In fact this difference is equal to the area of parallelogram ABla:
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 242 of 479.
81 3. NEWTON'S METHOD AND LEIBNIZ'S CALCULUS
"but this rectangle, because its width AB is diminished indefinitely, becomes less
than any given rectangle" (Newton 1687/1999, 43:3).
Notice how in Lemma 2 and Lemma 7 Newton gives a proof of two assumptions
that were made in the seventeenth-century "new analysis". The "new analysts"
(Newton himself in his early writings!) had assumed that a curve can be conceived
as a polygonal of infinitely many infinitesimal sides and that a curvilinear area can
be conceived as an infinite summation of infinitesimal strip (see Chapter 2). In the
Georne/;Tia cUTvilinea and in PTincipia, curves are smooth and curvilinear areas are
not resolved into infinite8imal elements. In the synthetical method of fluxions onc
always works with finite quantities and limi ts of ratios and sums of finite quantities.
In De q1Ladmlvxa C1LTVar1Lrn Newton presented a calculus version of the method
of prime and ultimate ratios (sce (Newton 16911692) and (Newton 1704)). However,
he made it clear that such symbolical demonstrations were safdy grounded in
geometry (sce 3.5.4). Newton began working on this treatise devoted to "integration"
in the early 1690s. It is opened by the declaration that calculus is referred
to as only finite flowing quantities: "Mathematical quantities I here consider not
as consisting of least possible parts, but as described by a continuous motion. [... ]
These geneses take place in the reality of physical nature and are daily witnessed
in the rnotion of bodies" (Newton 1704, 122).
For instance, in order to find the fluxion of y xn by the method of prime and
ultimate ratios, Newton proceeded as follows:
Let the q'uantity x fiow 1Lniformly and the fi'Uxion of the q1Lantity
:];11 needs to be f01Lnd. In the time that the quantity x comes in
its fiux to be x + 0, the quantity xn will come to be (x o)n,
that is [when expanded] by the method of infinite series
(3.12) x" + nox11 1
~ (n2
n) + ... ;
and so the augments 0 and n()Xn~l + ~(n2 - n)o2xn - 2 + .. .
are one to the other as 1 and nxn~l + ~(n2 n)oxn--2 + ... .
Now let those augments come to vanish and their last ratio will
be 1 to ; consequently the fiuxion of the quantity x is to
the fiuxion of the quantity :(;11 as 1 to nxn~l. (Newton 1704,
126128)
Notice that the increment 0 is finite and that the calculation aims at determining
the limit of the ratio [(x + 0)" x11
]/o as 0 tends to zero.
3.2.6. Higher-order ftuxions and the Taylor series (1687 to 1692). In
the 1690s Newton introduced a notation for fiuxions and higher-order fiuxions. He
wrote X, X, X, etc., for first, second, third, etc., fiuxions. He also used the notation
1; for the iiuent of .x. Dots and accents could be repeated to generate higher-order
fiuxions and higher-order fiuents. Newton also employed overindexes in order to
n
avoid the multiplication of dottl and accents: so he wrote Y for the nth fiuxion of y
(Newton 1967-1981, 7, H~18 and 162).
In discussing higher-order fluxions, Newton stated that every ordinate y of a
curve in the x-y plane can be expressed, assuming x 1, as a power series whose nth
11
term is equal to the nth fiuxion of y, i.e., y, divided by n! (see (Newton 1691-1692,
7, 9698)). He probably arrived at this statement by generalizing his experience
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 243 of 479.
:l.3. LElBNIZ'S DIFFERENTIAL AND INTEGRAL CALCULUS 85
with power series (see some examples in 3.2): For all of them this property holds.
On the other hand, if we assume that y is expressible as a power series such as
y = a + bx + cx2
+ dx3
+ e:r;4 + ... , one gets immediately that y(O) cc= a, y(O) b,
y(O) = 2c, etc.
Newton thus stated a theorem, nowadays called the TayJor theorem, which was
to play an important role in the development of eighteenth··century cakulus (sce
Chapter 4).
It should be noted that already in the PTincipia , Scholium to ProposiLion
93, Book 1, and Proposition 10, Book 2) Newton had come close to stating that
the nth term of a power series expansion is proportional to the nth fluxioH. He had
actually stated that the first ternl represents the ordinate, the second the tangent
(or the velocity), the third the curvature (or the acceleration), and so on. In Book
3 he had also solved the problem of determining "a parabolic curve that will pass
through any number of given points" by a procedure which is equivalent to the
so-called Gregory-Newton interpolation formula (a version of which he discovered
in about 1676). It is indeed remarkable to see how important power serje~ were in
the work of Newton. From his early research on tangent::; and quadratures to his
mature development of a theory of higher··order fluxions he used power series as a
major analytical tool.
3.3. Leibniz's differential and calculus
3.3.1. A mathematician and diplomat. GottfriedWilhclrn Leibniz was
born in Leip7.ig in 1646 from a Protestant family of distant Slavonic origins. His
father, a professor at Leip7.ig University, died in 1652, leaving CL rich library, where
the young Gottfried began his scholarly life. He studied philosophy and law in
the Universities of Leipzig, Jena and Altdorf. He also received some elementary
education in arithmetic and algebra. Early on he formulated a project for the
construction of a mathematical language with which deductive rea80ning eould be
conducted. His manuscripts related to symbolical reasoning reveal anticipation of
the nineteenth-century algebra of logic. Leibniz never abandoned hiH programme
of devising a "charaeteristica universalis". As we will see, he conceived his mathematical
research as part of this ambitious project. More specifically, his interest in
number sequences played a role in the discovery of differential and integral calculus.
After receiving his doctorate in 1666 from the University of Altdorf, he entered into
the service of the Elector of Mainz. From 1672 to 1676 he was in Paris on a diplomatic
mission. Here he met several distinguished scholars, most notably Christiaan
Huygens, who belonged to the recently established Academie Royale des Sciences.
It was in Paris, following Huygens's counsel, that Leibniz learned mathematics.
In a few months he had digested all the relevant contemporary literature and was
able to contribute original research. His discovery of calculus dates from the years
1675-1677. He published the rules of differential calculus in 1684 in the Acta CTuditOTUm,
a scientific journal that he had helped to found in 1682. In 1676 his seminal
period of study in Paris came to an end. After 1676 Leibniz worked in the service
of the Court of Hanover. He embarked on political projects, the most ambitious
of which was the reunification of the Christian churches. Leibniz was very good
in divulging his mathematical discoveries through scientific journalt: and learned
correspondence. While Newton kept his method secret, Leibni~ made great efforts
to promote the use of calculus. In Basel, Paris and Italy several mathematicians,
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 244 of 479.
86 3. NEWTON'S METIlOD AND LEIBNIZ'S CALCULUS
such as the Bernoulli brothers, I'Hopital, Varignon, Manfredi, and Riccati began
to use and defend the new calculus of sums and differences. A notable advance
occurred at the turn of the century when .lakob and .lohann Bernoulli extended
integral calculus and applied it to dynamics.
Leibniz died in 1716. His funeral was attended only by his relatives and by his
secretary. Leibniz's intellectual interests spanned from technology to mathematics,
from physics to logic, from politics to religion. He is remembered as one of the
profoundest philosophers and onc of the most creative mathematicians of all ages.
3.3.2. Infinite series (1672 to 1673). Leibniz's interests in combinatorics
led him to consider finite numerical sequences of differences such as
(3.13)
He noted that it is possible to obtain the sum b1 + /;2 + ... + bn as a difference,
al - an+l.When extrapolated to the infinite, this simple law led to interesting
results with infinite series. For instance, in order to flnd the sum of the series of
reciprocals of the triangular numbers
00
2
00
Lbn ,
n=l
(3.14)
n(n + 1)n=]
Leibniz noted that the terms of this series may be expressed using a difference
sequence by setting
(3.15) bn
2 2
--0 = an - an+l·
n n ..
Therefore
(:"\.16) Lbn
2
al - as+l = 2-
8+1
n=l
So, if we "sum" all the terms, we obtain 2.
Leibniz applied this procedure successfully to several other examples. }or instance
he considered the "harmonic triangle" (see Fig. 3.7). In the harmonic triangle
the nth oblique row is the difference sequence of the (n + l)th oblique row. It
follows, for instance, that
1 1 1 1 1
(3.17) 4 + 20 + 60 + 140 + " . = 3'
This research on inflnite series implies an idea that played a central role in Leibnizian
calculus (see (Bos 1980, 61)). The sum of an infinite number of terms bn can
be achieved via the difference sequence an.
3.3.3. The geometry of infinitesimals (1673 to 1674). In 1673 Leibniz
met with the idea of the so-called "characteristic triangle". He was reading Pascal's
Lettres de "A. Dettonville" (1659). Pascal, in dealing with quadrature problems,
had associated a point on a circumference with a triangle with inflnitesimal sides.
Leibniz generalized this idea. Given any curve (see Fig. 3.8) he associated an
inflnitesimal triangle to an arbitrary point P. One can think of the curve as a
polygonal constituted by infinitely many inflnitesimal sides. The prolongation of
one of the sides gives the tangent to the curve. A line at right angles with one of
the sides is the normal. Call t and n the length of the tangent and the normal,
respectively, intercepted between P and the x-axis. From the similarity of the three
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 245 of 479.
3.3. LBIIlNIZ'S DIFFERENTIAL AND INTEGRAL CALCULUS 87
J
6
1 1
12 12
1 J J I
2() 30 20 :5
I 1 1 1 J.
6 30 60 60 30
1 J... _L .L
42 105 105 42
FIGURE 3.7
y
(J
x
FIGURE 3.8
triangles shown in Fig. 3.8, Leibniz obtained several geometrical transformations
which allowed him to transform a problem of quadrature into another problem. He
stated equivalences which he would later write as Jkdx = Jydy, Jydx = J(J'dy,
Jyd8 Jtdy, Jyd8 Jndx (here n is the normal, t is the tangent, k is the
subnormal and (J' is the subtangent). The most useful transformation obtained by
Leibniz in 16731674, i.e., the years immediately preceding the invention of the
algorithm of calculus, is the "transmutation theorem" ((Hofmann 1949,3235) and
(Bos 1980, 62-64)).
Leibniz cOllsidered a smooth convex curve OAB (see Fig. 3.9). The problem is
to determine the area OABG. Let PQN be the characteristic triangle associated
to the point P. The area OABG can be seen either as the sum of infinitely many
strips RPQS or as the sum of the triangle OBG plus the sum of infinitely many
triangles OPQ. We can write
(3.18) OABG
1
RPQS = 20G . GB + L OPQ.
Let the prolongation of PQ (i.e., the tangent in P) meet the y-axis in T and let
OW be normal to the tangent. Triangle OTW is thus similar to the characteristic
triangle PQN; therefore,
(3.19)
PN
OW
PQ
OT
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 246 of 479.
88 3. NEWTON'S METHOD AND LEIBNIZ'S CALCULUS
B
w
FIGURE 3.9
The area of the infinitesimal triangle OPQ is thus
(3.20) OPQ = ~OW . PQ = ~OT . PN.
Leibniz defines a new curve OLlvI, related to the curve OAB through the process of
taking the tangent. The new curve has an ordinate in R equal to 01'. Geometrically
the construction is obtained by drawing the tangent in P and determining the
intersection l' between the tangent and y-axis. In symbols not yet available to
Leibniz, the ordinate z of the new curve OLM is z = y - ;rdy/d.T.
Leibniz has thus shown that
OABG
(3.21)
~OG. GB + l',OPQ
2
1 1
-OG· GB + l',~OT· PN
2 2
~OG. GB + ~OLMG
2 2 '
where OLMG is the area subtended by the new curve. In modern symbols, setting
y as the ordinate of the curve 0 AB (see (Bos 1980, 65)),
(3.22)
l
XO
1 1 iXO 1 1 j'XO 1 f'xO dy
ydx = --XoYo + - zdx = -XoYo + , ydx - x--d:J.:.
o 2 2 , 0 2 2 0 2 0 dx
Leibniz's geometrical "transmutation" is thus equivalent to integration by parts.
He was later (see, e.g., (Leibniz 1714, 408)) to express it as
(3.23) Jydx = xy - Jxdy.
Leibniz thus achieved, through the geometry of the infinitesimal characteristic triangle,
a reduction formula for integration. The integration of curve 0 AB is reduced
to the calculation of the area subtended to an auxiliary curve OLM related
to OAB through the process of taking the tangent, The relation of the tangent
and quadrature problem began thus to emerge in Leibniz's mind. This work with
the characteristic triangle also made him aware of the fruitfulness of dealing with
infinitesimal quantities,
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 247 of 479.
:3.3. LEIBNI?;'S DIFFERENTIAL AND INTEGRAL CALCULUS 89
y c
dx x
FIGURE 3.10
3.3.4. The calculus of infinitesimals to 1mb Lcib··
niz made the crucial steps which led him to forge the algorithm which is still
utilized, though in a revised form and in a different conceptual context. He be·
gan considering two geometric constructions which had played a relevant role in
seventeenth-century infinitesimal techniques: viz., the characteristic triangle and
the area subtended to a curve as the sum of infinitesimal strips.
Let us consider a curve C (see Fig. 3.10) in a Cartesian coordinate system. Leib··
niz imagines a subdivision of the :r:-axis into infinitely many infinitesimal intervals
with extremes Xl, X2, :r;3, etc. He further defines the differential dx =--' Xn-l-I - xn .
On the curve and on the y-axis one has the corresponding successions SI, s~, 83,
etc., and YI, Y2, Y3, etc. Therefore ds = Bn-l-l Sn and dy Yn-l-I Yn' 'rhe
characteristic triangle has sides d:r:, dB, dy. The tangent to the curve C forms an
angle, with the x-axis such that tan, ~-= dy/dx. The area subtended to the curve
is equal to the sum of infinitely many i:lLrips ydx. Leibniz initially employed Cavalieri's
symbol "omn.", but he soOD replaced this notation with the now familiar
.r ydx, where I is a long "s" for "sum of". The first publlshed occurrence of the
d-sign was in (Leibniz 1684), while the integral appeared in (Leibniz 1686). Three
aspects of Leibniz's representation of the curve C in termi:l of differentials should
be noted.
1) The symbols d and .r applied to a finite quantity x generate an infinitely
little and an infinitely great quantity, respectively. So, if x is a finite angle or a finite
line, dx and .rx are, respectively, an infinitely little and an infinitely great angle or
line. Thus the two symbols d and .r change the order of infinity but preserve the
geometrical dimensions. Notice that Newton's dot symbol does not do that. If x is
a finite fiowing line, :i; is a finite velocity.
2) Since geometrieal dimension is preserved, the symbols d and I can be iterated
to obtain higher-order infinitesimall:l and higher-order infinites. So ddx is infinitely
little compared to d2:, and I I 2; is infinitely great eompared to I x. A hierarchy of
infinitesimals and infinites is thus obtained. Higher-order differentials were denoted
by repeming the symbol d. It became usual, from the mid-1690i:l, to abbreviate
dd ... d (n times) by dn
, so that the nth differential of x is dn
x.
3) T'he representation of the curve C in terms of differentials can be achieved
in a variety of ways. One can chose the progressions of xn , Yn and Sn so that dx
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 248 of 479.
90 3. NEWTON'S METHOD AND LEIBNIZ'S CALCULUS
is constant or dy is constant or ds is constant. Or one can choose the three abovementioned
progressions such that dx, dy and ds are all variable. For instance, the
choice of dx constant (i.e., the Xn equidistant) generates successions of Yn and Sn
where ds and dy are not (generally) constant. As Bos has shown in (Bos 1974) the
choice of dx constant is equivalent to selecting x as the independent variable and s
and y as dependent variables. (The Newtonian equivalent is to choose i; constant,
i.e., x flowing with uniform velocity.)
Bos stresses, moreover, that the Leibnizian calculus is not concerned with "functions"
and "derivatives" but with progressions of variable quantities and their differences.
Therefore we should not read, for instance, dy/ dx as the derivative of
y(x) as a function of x but as a ratio between two differential quantities, dy and
dx. The conception of dy/dx as a ratio renders the algebraical manipulation of
differentials quite "natural". For instance, the chain rule is nothing more than a
compound ratio:
(3.24)
dy
dx
dy dw
dw dx
Selecting a variable x so that dx is constant simplifies the calculations since
ddx 0 and higher-order differentials of x are cancelled. There is another way for
cancelling higher-order differentials. When onc has a sum A + Cl: and (l; is infinitely
little in comparison to A, it can be stated that A +Cl: = A. This rule of cancellation
for higher-order infinitesimals can be stated as follows:
(3.25)
Leibniz calculated the difFerential of xy and xn as follows:
d(xy) = (x + dx)(y + dy) xy = xdy + ydx + dxdy xdy + ydx,
while
dxn = (x + dx)n - xn nxn~ldx + dx2 (- .. ) = nxn~ldx.
In fact, he assumed that dxdy cancels against xdy + ydx and that dx2
cancels
against dx (see 3.5.2 for Leibniz's attempts to justify this procedure).
Differentials of roots such as y .;jx a can be achieved by rewriting yb =
x a, taking the differentials, byb~ldy = axa~ldx, and rearranging so that difXCi
(a/b)dx\lxa~b. A similar reasoning leads to d(l/xa) -adx/xaH .
Leibniz was clearly proud of the extension of his calculus. In the predifferentiation
period (see 2.2) roots and fractions were difficult to handle. Leibniz published
the rules for differential calculus in 1684 in a short and difficult paper which bears
a title with the English translation A new method for maxima and minima as well
as tangents) which is neither impeded by fractional nor irrational quantities) and a
remarkable type of calculus for them.
Leibniz generally performed integration by reductions of Jydx through methods
of variable substitution or integration by parts. These methods could be worked
out in a purely analytical way. Instead of requiring complex geometrical constructions
of auxiliary curves (as in the method of transmutation), the new notation
allowed algebraical manipulations.
The most powerful method for performing integrations came from the understanding
of the fundamental theorem of calculus. The notation d and J, for difference
and sum, immediately suggests the inverse relationship of differentiation and
integration. Leibniz conceived Jydx as the "sum" of an infinite sequence of strips
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 249 of 479.
3.4. MATHEMATIZING FORCE 91
FIGURE 3.11
ydx. From his research on infinite series he knew that a sum of an infinite sequence
can be obtained from the difFerence sequence (see 3.3.2). In order to reduce f ydx
to a sum of differences, one must find a z such that dz = ydx. Thus, at once,
(3.26) Jydx Jdz = d Jz = z.
Once the inverse relation of differentiation and integration is understood, several
techniques of integration follow. For instance the rule of transmutation (integration
by parts) comes by inverting d(xy) xdy + ydx. Wc thus obtain xy = f d(xy)
f xdx + fydx.
As an example of Leibniz's inverse algorithm we can consider the applicaton of
the transmutation theorem to the quadrature of the cycloid generated by a circle
of radius a rolling along the vertical line x = 2a (sce Fig. 3.11). The ordinate BC
is equal to BE + EC = BE + AE, where AE is the length 8 of the circular arc.
Since d8/a = dxr/2ax - x 2 , it follows that 8 = f; adu/V2au~u2. (Nowadays
we have notation for the elementary transcendental functions and we would write
8 = a . arccos( (a - x) / a).) Thus the equation of the cycloid is
(3.27) y = \hax - x2 + lx
adu/V2au - u2
.
Since dy/dx (2a - x)/V2ax - x 2 , from (3.22),
(3.28) lxO
ydx xoYo - lxO
x2
dx.
If we take Xo = 2a and Yo = 7ra, formula (3.28) gives 37ra2
/2 for the area subtended
under the half-arch (see (Dupont and Roero 1991, 118-119)).
Leibniz was greatly interested in the applications of his calculus to geometry and
dynamics. In this applied context he wrote and solved several differential equations.
This very important subject entered into the world of continental mathematics
thanks to Leibniz's development of integration techniques (see 11.2.2).
3.4. Mathematizing force
The publication in 1687 of Newton's Principia was perhaps the major event of
seventeenth-century natural philosophy. The reaction of Leibniz to the Principia
is too complex a subject to be tackled here. To mention just a few points, Leibniz
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 250 of 479.
92 :3. NEWTON'S METHOD AND LEIBNIZ'S CALCULUS
disagreed with Newton's cosmology of universal gravitation, with his conceptions
of absolute time and space, with his dynamical principles, and with his theological
views (sec (Bertoloni Meli 1993a)). It is of interest for us that Leibniz and his
school were critical of Newton's mathematical methods in dynamics.
Even though Newton was one of the discoverers of calculus, he made explicit
use of it in only a few isolated propositions in the Pr-incipia. Instead he employed
the synthetical method of fiuxions, i.e., the method of prime and ultimate ratios
(3.2.5). Limits of ratios and limits of sums, as well as infinitesimals of various
orders, occur very often in his geomcLrical dynamics. A "translation" into the
language of calculus thus might appear trivial. However, the mathematicians who,
at the beginning of the eighteenth century, set themselves the task of applying the
calculus to Newton's dynamics (most notably Pierre Varignon, Jakob Hermann,
and Johann Bemoulli) had difficult problems to surmount. In some cases, the
geometrical demonstrations of the Pr-incipia can be translated almost at once into
calculus concepts; in other cases, this translation is complicated, unnatural, or even
problcmatic.
Today, we take it for granted that calculus is a better suited tool than geometry
for dealing with dynamics. But at the beginning of the eighteenth century,
the choice of mathematical methods to be applied to dynamics was problematic.
Newton's mathematization of dynamics was mainly, even though not exclusively,
geometrical and several members of the Newtonian school, up to Colin Maclaurin
and Matthew Stewart at the middle of the eighteenth century followed Newton
from this point of view (see (Guicciardini 1989)).
Before writing the Pr-incipia, Newton had already turned his attention toward
geometrical methods. In the 1670s he was led to distance himself from his early
highly analytical mathematical research. Newton began to criticize modern mathematicians:
He stressed the mechanical character of modern algebraical methods,
their utility only as heuristic tools and not as demonstrative techniques, and the
lack of referential clarity of the concepts employed. By contrast, he characterized
the "geometry of the Ancients" as simple, elegant, concise, adherent to the
problem posed, and always interpretable in terms of existing objects. Needless
to say, notwithstanding Newton's rhetorical declaration of continuity between his
method!:) and the methods of the "Ancients," his geometrical dynamics is a wholly
seventeenth-century affair.
The reasons that induced this champion of analytics, series, infinitesimals and
algebra to spurn his analytical research are complex. They have to do with foundational
worries about the nature of infinitesimal quantities as well as with his desire
to find in geometry a unifying principle of techniques which grew wildly in his early
writings. They also have to do with his dislike of Descartes, towards anything
Cartesian, and with his admiration for the geometrical methods of Huygens (see
(Westfall 1980, 377-381)).
But other factors combined to give to the Pr-incipia the geometrical form we
know. A sixteenth-century approach to natural philosophy, exemplified in the works
of Johannes Kepler and Galileo Galilei, saw the Book of Natur-e as written in circles
and triangles, not in equations. Furthermore, the community of natural philosophers
to which Newton addressed the Principia was trained in geometry, certainly
not in calculus: In 1687 almost a still unpublished discovery. It would have been
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 251 of 479.
3.4. M ATHEMATIZING FORCE
f
'.
c
"
"
"
"
----- -, -tI- ~.... ............ ..
5'
FIGURE 3.12
hopelessly difficult for them to understand a completely new dynamics expressed
into a completely new language.
Another important factor that led Newton to use geometry in dynamics has
to do with the relative weakness of calculus in 1687. Newton knew how to apply
calculus to the simplest problems. We have manuscripts in which he writes fiuxional
(i.e., differential) equations of motion for the one-body problem ((Newton 1691--
1692, 122-129) and (Guicciardini 1999)). However, universal gravitation allows
perturbed motions in planetary orbits. The possibility of mathematizing fine details
of planetary motions (such as the precession of equinoxes) or planetary shapes and
tides was crucial for Newton and his followers. The calculus was not yet powerful
enough to allow such dynamical studies. Geometry on the other hand offered a
means to tackle these problems, at least at a qualitative level (see (Greenberg
1995)).
Employing the geometry of prime and ultimate ratios, refusing the new analysis
in favour of the synthetical method of fiuxions, was not therefore a defensive,
backward move, but rather it was seen by Newton as a progressive move, a choice
of a more powerful method. Newton believed this method was better, both from a
foundational point of view and from a demonstrative point of view.
Let us consider, as an example of Newton's geometrical techniques in dynamics,
the treatment of Kepler's area law of planetary motions, i.e., Proposition 1 of Book
1 of the Principia. This proposition states that Kepler's area law holds for any
central force. Newton's geometric proof is based on an intuitive theory of limits.
In the Principia we read:
The areas which bodies made to move in orbits describe by radii
drawn to an unmoving centre of forces lie in unmoving planes
and are proportional to the times. (Newton 1687/1999, 444)
Newton's proof is as follows. Divide the time into equal and finite intervals,
6.tl, 6.t2 , 6.t3 , etc. At the end of each interval the force acts on the body "with
a single but great impulse" (ibid.) and the velocity of the body changes instantaneously.
The resulting trajectory (see Fig. 3.12) is a polygonal ABCDEF. The
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 252 of 479.
94 3. NEWTON'S METHOD AND LEIBNIZ'S CALCULUS
areas BAB, BBC, BCD, etc., are swept by the radius vector in equal times. Applying
the first two laws of motion, it is possible to show that they are equal. In fact,
if at the end of 6.t], when the body is at B, the centripetal force did not act, the
body would continue in a straight line with uniform velocity (because of the first
law of motion). This means that the body would reach c at the end of 6.t2 such
that AB Bc. Triangles BAB and BBc have equal areas. However, we know that
at the end of 6.tt , when the body is at B, the centripetal force acts. Where is the
body at the end 6.t2? In order to answer this question, one has to consider how
Newton, in Corollary 1 to the laws, defines the mode of action of two forces acting
"simultaneously": "A body, acted on by two forces simultaneously, will describe
the diagonal of a parallelogram in the same time as it would describe the sides by
those forces separately" (ibid., 417). Invoking the above corollary, Newton deduces
that the body will move along the diagonal of parallelogram BcCV and reaching
C at the cnd of 6.t2. Cc is parallel to VB, so that triangles BBc and BBC have
equal area8. It follows that triangles BAB and BBC have equal areas. One can
iterate this reasoning and construct points C, D, E, F. They all lie on a plane, since
the force is directed towards B, and the areas of triangles BCD, BDE, BEF, etc.,
arc equal to the area of triangle BAB. The body therefore describes a polygo..
nal trajectory which lies on a plane, and the radius vector BP sweeps equal areas
BAB, BBC, BCD, etc., in equal times. Newton passes from the polygonal to the
smooth trajectory by a limit procedure based on the method of prime and ultimate
ratios. He writes:
Now let the number of triangles be increased and their width decreased
indefinitely, and their ultimate perimeter ADF will [... ]
be a curved line; and thus the centripetal force by which the body
i8 continually drawn back from the tangent of this curve will act
continually, while any areas described, BADB and BAFB, which
are always proportional to the times of the description, will be
proportional to those times in this case. (Ibid., 145)
That is to tiay, since Kepler's area law always holds for any discrete model (polygonal
trajectory generated by an impulsive force) and since the continuous model (smooth
trajectory generated by a continuous force) is the limit of the discrete models for
6.t --J> 0, then the area law holds for the continuous model. The area swept by SP
is proportional to time.
The Leibnizians proceeded in a completely different way. They tackled Kepler's
area law from an analytical point of view. After partial results obtained by Jakob
Herrnann in 1716 (see (Guicciardini 1999)), they obtained the following analytical
representation for centripetal force.
The most natural choice is to use polar coordinates (r, e) so that the origin
co'incides with the centre of force. The radial and transversal acceleration are thus
expressed by the following two formulae:
(3.29) a = d2
r _ r (de) 2
r d{;2 dt
and
(3.30)
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 253 of 479.
3.5. NEWTON VERSUS LEIBNIZ 95
Let A be the area swept out by the radius vector. Then 2dA/dt T2
dO/dt and
2d2
A/dt2
= T2d2
0/ dt2
+ 2T(dT/ dt) (de/ dt) = Tat. 1<'or a central force, at is equal to
zero. By integrating (3.30), we obtain dA/dt k (i.e., the areal velocity is equal to
a constant k). Inversely, if dA/dt = k, it follows by differentiation that at is zero
(i.e., the force is central). Proposition 1 and its inverse are thus embedded in the
analytical formulation of tramlVersal and radial acceleration.
The above demonstration is quite 8traightforward: Mathematically speaking, it
requires only elementary calculus and the use of polar coordinates. However, such
a demonstration was only worked out in the 1740s in the works of Daniel Bernoulli,
Leonhard Euler and Alexis Claude Clairaut on constrained and planetary motion
(sec (Bertoloni Meli 1993b)).
This example shows how different the approach of the Leibnizian school was
to the mathematization of dynamics (sce (Whiteside 1970)). In the Leibnizian
approach the geometry of infinitesimals is the model from which one can work out
differential equations. The trajectory is represented locally in terms of differentials.
The study of the geometrical and dynamical relationships of infinitesimals leads to
differential equations which can be manipulated algebraically until the result sought
is achieved. During the algebraical manipulation the geometrical interpretability of
the symbols is not at issue. On the other hand, Newton adheres to geometry: The
symbols he employs are always interpreted in geometric terms, and they are actually
exhibited in the geometrical model, whose geometrical and dynamical properties
I1re central to the demonstration.
3.5. Newton versus Leibniz
3.5.1. "Not-equivalent in practice". It is not easy to establish a compari·son
between Leibniz's and Newton's calculi because Leibniz and Newton presented
several versions of their calculi. Leibniz never published a systematic treatise but
rather divulged the differential and integral calculus in a series of papers and letters.
He changed his mind quite often especially on foundational questions. Newton
abandoned his earlier version of calculus based on moments and opted for the
method of prime and ultimate ratios.
In my opinion, Leibniz's and Newton's calculi have sometimes been contrasted
too sharply. For instance, it has been said that in the Newtonian version variable
quantities are seen as varying continuously in time, while in the Leibnizian version
they are conceived as ranging over a sequence of infinitely close values (Dos 1980,
92). It has also been said that in the fiuxional calculus, "time", and in general
kinematical concepts such as "fluent" and "velocity", play a role which is not accorded
to them in differential calculus. It is often said that geometrical quantities
are seen in a different way by Leibniz and Newton. For instance, for Leibni2 a curve
is conceived as polygonal~with an infinite number of infinitesimal sides--while for
Newton curves are smooth (Bertoloni Meli 1993a, 61-73).
These sharp distinctions, which certainly help us to capture part of the truth,
are made possible only by simplifying the two calculi. As a matter of fact, they are
more applicable to a comparison between the simplified versions of the Leibniz;ian
and the Newtonian calculi codified in textbooks such as l'Hopital's Analyse des
infiniment petits (1696) and Sirnpson's The DoctTine and Application of Fluxions
(1750) rather than to a comparison between Newton and Leibniz. It seems to me
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 254 of 479.
96 3. NEWTON'S METIIOD AND LEIBNIZ'S CALCULUS
that important aspects of their mathematics are ignored in these historical interpretations.
For instance, one should not ignore Leibniz's highly skeptical attitude
towards the existence of infinitesimals: He would have agreed with Newton that
variables vary continuously and that curves are smooth. Leibniz explicitly employed
infinitesimals as heuristic devices. In much the same way Newton conceived
"moments" as useful abbreviations which can be eliminated by translating infinitesimalist
proofs into rigorous limit-based proofs. Furthermore, Newton's conception
of "time" as used in the fluxional calculus is highly abstract: He was quite careful
to avoid any identification of "fluxional time" with "real time". "Fluxional time" is
just a variable fluent with constant fluxion. So the fluxional calculus is not simply
founded on kinematics but rather of the abstract concept of continuous variation.
The diilenmces between the Leibnizian and the Newtonian calculi should not
be overstressed. In particular, as I shall argue in this section, the differences should
not be looked for at the syntactic or at the semantic level b'ut rather at the pragmatic
level. After all, the two calculi shared a great deal in common both at the
syntactic level of the algorithm and at the semantic level of the interpretation of
the algorithm's symbols and the justification of the algorithm's rules. It is possible
to translate between the Huxional and the differential calculus (through correspondences
between io and dx). The Leibnizian and the Newtonian mathematicians
made such translations: They were aware that there is not a single theorem which
can be proved in onc of the two calculi and which cannot have a counterpart in the
other. It was exactly this "equivalence" which gave rise to the quarrel over priority.
In discussing the question of equivalence, A. R. Hall writes quite appropriately:
Did Newton and Leibniz discover the same thing? Obviously,
in a straightforward mathematical sense they did: [Leibniz's]
calculus and [Newton's] fluxions are not identical, but they arc
certainly equivalent. [...] Yet one wonders whether some more
subtle element may not remain, concealed, for example, in that
word "equivalent". I hazard the guess that unless we obliterate
the distinction between "identity" and "equivalence", then if two
sets of propositions are logically equivalent, but not identical,
there must be some distinction between them of a more than
trivial symbolic character. (Hall 1980, 257-258)
In order to explore this more subtle and concealed level, where a comparison between
Newton's and Leibniz's calculi can be established, S. Sigurdsson has proposed
to use the category "not-equivalent in practice". Despite the equivalence of the two
calculi,
[this] equivalence breaks down once it is realized that competing
formalisms suggest separa.te directions for research and therefore
generate different kinds of knowledge. (Sigurdsson 1992, 110)
Similarly I. Schneider has remarked that "the starting point, the main emphasis and
the expectations of the two pioneers were not at all identical" (Schneider 1988, 142).
D. Bertoloni Meli has drawn a comparison between a Newtonian and a Leibnizian
mathematician and two programmers who use different computer languages:
Even if the two programmes are designed to perform the same
operations, the skills required to manipulate them may differ
considerably. Thus subsequent modifications and developments
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 255 of 479.
3.5. NEWTON VERSUS LEIBNIZ
may follow different routes, and this is precisely what happened
in Britain and on the Continent in the eighteenth century: de--
spite the initial "equivalence" of fluxions and differentials.
(Bertoloni Meli 1993a, 202)
97
I agree with the approach of the above--mentioned scholars. Rather than looking
for sharp distinctions between the two calculi, we should look for subtler, less
evident aspects. Newton and Leibniz had two "mathematically equivalent" symbol--
isms. At the syntactical level they could translate each other's results and, at the
semantieal level, they agreed on important foundational questions. Nonetheless,
at the pragmatic level, they oriented their research in different directions. Belonging
to the Newtonian or to the Leibnizian school meant having different skills and
different expectations. It meant stressing different lines of research and different
values. After all, it often happens in history of mathematics that the difference
between two schools does not lay in logical or conceptual incommemmrabilities but
rather in more pragmatic aspects: such as the teaching methods, the formation of
mathematicians, the expectations for future research, the of values which
support the view that a method of proof is preferable to another, etc.
In the following three sections, I will look for such a comparison between the
two schools focusing on three aspects: the conceptual foundations, the algorithms
and the role of geometry.
3.5.2. The problem of foundations. The problem of foundations did not
exists in the seventeenth century in the form which it took in the early nineteenth
century (see Chapter 6). One of the most important foundational questions faced
by seventeenth- and eighteenth-century mathematicians was a question concerning
the referential content of mathematical symbols (typically "do infinitesimals ex--
ists?"). This "ontological" question was followed by a "logical" question about the
legitimacy of the rules of demonstration of the new analysis (typically "is x -+dx = x
legitimate?"). To these two questions Newton and Leibniz gave similar answers.
They both stated that (a) actual infinitesimals do not exist; they ar-e useful
fictions employed to abbr-eviate pr-oofs, (b) injinitesimals .should be defi:ned mther- as
var-ying quantities in a .state of appr-oaching zcr-o, (c) infinilesimals can be completely
avoided by limit-based pr-oofs, which constitute the 7"igor-ou.s for-mulation of calculus,
(d) hmit-based pr-oofs ar-e a dir-ect ver-sion of and ar-e th1LS equivalent to the indir-ect,
ad absurdum Archimedean method of e.Tha1Lstion.
Once the calculus had been reduced to limit-based proofs, the logical question
took the form: "Are limit-based proofs legitimate?" In order to answer this question,
both Newton and Leibniz used the concept of continuity. However, the former
legitimated limits in terms of our intuition of continuous flow, while the latter
referred to a philosophical "principle of continuity".
To the question, "Do differentials exist?", many Leibnizians answered in the
affirmative. Leibniz did not..From his very early manuscripts (see (Leibniz 1993))
to his mature works, it is possible to infer that for him actual differentials were just
"fictions", symbols without referential content (sce (Knobloch 1994)).
Nonetheless the use of these symbols was justified, according to Leibniz, since
correct results could be derived by employing the algorithm of differentials. As
Leibniz said, differentials arc "fictions", but "well-founded fictions". Why "wellfounded"?
Leibniz seems to have had the following answer. He denies the actual
infinite and actual infinitesimal and conceives the differentials as "incomparable
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 256 of 479.
98 3. NEWTON'S METHOD AND LEIBNIZ'S CALCULUS
quantities": varying quantities which tend to zero. In his writings of the 1690s
Leibniz describes these "incomparables" as magnitudes in a fluid state which is
different from zero but which is not finite. These quantities would give a meaning
to dy/dx as a ratio between two quantities. In fact, if dy and dx are zero, we have
the problem of giving a value to 0/0, but if they are finite, they cannot be neglected
(thUS x + dx = x would be invalid).
However, in other later writings Leibniz stated that differentials are wellfounded,
since they are symbolic abbreviations for limit-procedures. From this
viewpoint, the calculus of differentials is a shorthand for a calculus of finite quantities
and limits, equivalent to Archimedean exhaustion. He wrote:
In fact, instead of the infinite or the infinitely small, one can
take magnitudes that are so large or so small that the error will
be less than the given error, so that one differs from the style of
Archimedes only in the expressions, which are, in our method,
more direct and more apt to the art of discovery. (Leibniz 1701,
350)
Newton's approach to the question of the existence of infinitesimals is similar.
Newton also spoke of infinitesimals ("moments" or "indefinitely little quantities")
as a shorthand for longer and more rigourous proof given in terms of limits. He
also speaks of infinitesimals as "vanishing quantities" in such a way that they seem
to be defined as something in between zero and finite, as quantities in the state
of disappearing, or coming to existence, in a fuzzy realm in between nothing and
finite. More often he makes clear that infinitesimals can be replaced by using limits.
There is not, therefore, a strong conceptual opposition between Leibniz and
Newton but rather a different attitude. Both agreed that limits provide a rigorous
foundation for the calculus, but for Leibniz this was more a rhetorical move in
defence of the legitimacy of the differential algorithm, while for Newton this was
a programme that should be implemented. While Newton explicitly developed a
theory of limits (see 3.2.5), Leibniz simply alluded to the possibility of building the
calculus based on such a theory. Leibniz could live with the infinitesimal quantities;
Newton made a serious effort in the Pr'incipia and De quadratura to eliminate them
(see (Lai 1975)), (Kitcher 1973) and (Guicciardini 1999)).
Leibniz often refers to the heuristic character of calculus in order to justify the
use of differentials. 1<'or him "metaphysical" questions on the foundations should
not interfere with the acceptance of calculus. Calculus, according to Leibniz, should
be seen also as an ars inveniendi: As such it should be valued by its fruitfulness,
more than by its referential content. According to Leibniz, we can calculate with
symbols devoid ofreferential content (for instance, with A) provided the calculus
is structured in such a way as to lead to correct results. Newton could not agree:
For him mathematics devoid of referential content could not be acceptable.
The argument of continuity with the "geometry of the Ancients" also played
a different role in Newton's and in Leibniz's conceptions. Fer Newton, showing
a continuity between his method and the methods of Archimedes was a crucial
step in guaranteeing the acceptability of the "new analysiD". Leibniz stressed this
continuity only in passing references deviced to reassure the dubious or to reply
to critics. He preferred to stress the novelty and revolutionary character of his
calculus.
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 257 of 479.
3.5. NEWTON VERSUS LEIBNIZ 99
The next foundational question concerns the Icgitimacy of proofs based on
limits. Newton in the Pr-incipia considers the objection that "there is no such thing
as an ultimate proportion of vanishing quantities, inasmuch as before vanishing the
proportion is not ultimate, and after vanishing it does not exist at all." However,
he observes that
by the same argument it could equally be contended that there
is no ultimate velocity of a body reaching a certain place at
which the motion ceases; for before the body arrives at this place,
the velocity is not the ultimate velocity, and when it arrives
there, there is no velocity at all. But the answer is easy; to
understand the ultimate velocity as that with which a body is
moving, neither before it arrives at its ultimate place and the
motion ceases, nor after it has arrived there, but at the very
instant when it arrives, that is, the very velocity with which the
body arrives at its ultimate place and with which the motion
ceases. (Newton 1687/1999, 442)
In order to demonstrate the existence of limits, Newton thus referred to the
intuition of continuous motion: We know by intuition that natural evolve
by continuous motion and that in every instant of time there is a velocity of flow.
Leibni?:, to the contrary, in order to justify the limiting procedures referred
to a metaphysical principle of continuity which he expressed in several forms and
contexts (see (Breger 1990).) The "law of continuity" pervades Leibniz's thought.
He made use of it in cosmology, in physics and in logic. Thus, invoking the law of
continuity, he affirmed that rest can be conceived as an infinitely little velocity or
that equality can be conceived as an infinitely little inequality. In 1687 he stated
this principle as follows in his difficult philosophical prose:
When the difFerence between two instances in a given series or
that which is presupposed [in datis] can be diminished until it
becomes smaller than any given quantity whatever, the corresponding
difference in what is sought [in quaesitis] (Leibniz 1687,
52)
In order to explain the meaning of this general principle, Leibniz refers to the
geometry of conic sections. An ellipse, he says, may approach a parabola a8 closely
as onc pleases, so that the difference between the ellipse and the parabola (the
difference between what "results") may become "less than any given difference",
provided that one of the foci (what is "posed") is removed far enough away from
the other. Consequently, the theorems valid for the ellipse can be extrapolated
to the parabola "considering the parabola as an ellipse when one of the foci is
infinitely distant, or (in order to avoid this expression) as a figure which differs
from a certain ellipse less than any given difi'erence" (ibid.). It is the continuous
dependence between what is "posed" and what "results" that justifies limit-based
reasonings in which one extrapolates to the parabola what has been proved of
the ellipses: "In continuous magnitudes the exclusive extremum can be treated as
inclusive" (Leibniz 1713, :385).
3.5.3. The two algorithms: Method versus calculus. Leibniz's and Newton's
algorithms are related through correspondences between io and dx. The two
schools could easily translate each other's results. The main advantage of Leibniz's
":c;t\~_·-_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 258 of 479.
100 :3. NEWTON'SIVIETHOD AND LEIBNIZ'S CALCULUS
algorithm conccrns the integral sign. With Leibniz's I ydx the integration-variable
x is explicitly indicated. Newton's [~], Qy and if need to be accompanied by verbal
statements. This has effects on integration techniques. In the Leibnizian calculus,
integration by substitutions and by parts can be performed in a more mechanical
way. This advantage was recognized by the Newtonians, who often employed hybrid
notations: E.g., Maclaurin wrote F, yx in (Maclaurin 1742, 665 ft'.).
I. Schneider remarks (Schneider 1988, 143) that in Leibniz's calculus the fun··
damental theorem is somehow "built into" the notation itself. Indeed, Leibniz's
symbols d and I that differentiation and integration arc operations and
that they are the inverses of each other.
As Scr'iba has observed (Scriba 196:)), Ncwton emphasized the use of infinite
series. He expanded fluents into infinite series and "integrated" termwise. Leibniz
also employed this technique. However, Leibniz preferred integration in "closed"
form: He looked for quadraturelS expressed not by infinite series but by a finite
combination of "functions". Newton also obtained "closed" integrations, but it is
certainly true that for him infinite series played a more prominent role than for
Leibniz. This "contrast" is thus a matter of emphasis; i.e., it is a contrast whicb
relates to the values which direct research along different lines.
Leibniz and Newton had equivalent symbolit:lm but different approaches to no·
tation. The former attached great importance to the construction of an efficient
algoritbm and chose symbols carefully. The latter was not particularly concerned
with notation. Leibniz thought of his calculus as part of a general programme
leading to the creation of a mathcsis universalis, a language in which all reasoning
could be framed. He often insisted on the advantages of symbolical reasoning as
a method of discovery. Nobody, according to Leibniz, could follow a long reasoning
without freeing the mind from the "effort of imagination". The calculus was
dcviced to favour this "blind reasoning" (cogitatio caeca) (sec (Pasini 1993, 205)).
Newton, on the other band, did not value mechanical algorithmic reasoning. He
always spoke of the geometrical demonstrations of Huygens in the highest terms and
contrasted the elegant geometrical methods of the "Ancients" with the mechanical
algebraic methods of Descartes (which "provoked to him nausea" (Newton 1967
]981,4, 277). He made clear that the symbols of the "analytical method of fluxions"
had to be interpreted in terms of the "synthetical method". It is this interplay
between algorithm and geometry that characterizes Newton's method.
Leibniz's concern with symbolism led him to develop an algebra of differentials
(sec 3.3.4). His main target was the construction of a set of algorithmic rules:
a calculv,s. The rules of calculus are instructions on how to manipulate the d's
and the 1"s, and they allow algorithmic procedures which are as much as possible
independent of the initial geometrical context. Leibniz even considered d"x for a
fractional a. We note that the chain rule in Leibnizian terms takes a form (sce
formula (3.24)) which is suggested by the notation itself. Everything can be done,
of course, also in Newton's notation. Newton, however, preferred to give examples
which show the rule rather than give the rule itself. For instance, he would introduce
the chain rule with an example, as a set of instructions applied to the solution of a
particular problem.
3.5.4. The role of geometry. Newton valued geometrical thinking very
highly. As we have seen in 3.2.5, he developed a geometrical version of the method
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 259 of 479.
3.5. NEWTON VERSUS LElDNIZ 101
of Huxion in the 1670s: He called it the "synthetical method of fIuxions" in opposition
to the "analytical method". Newton employed the synthetical method
especially in dynamics (see 3.4). He often affirmed that the synthetical method was
more rigorous and that it actually founded and justified the procedures employed in
the analytical method. This foundation and justification depended on two factors.
First of all the geometrical method of fluxions offered a model in which the
analytical method could be interpreted. In the geometrical method the fluents and
fluxions were exhibited to the eye, their existence in "rerum natura" proved ostcm··
sibly. In the second place, Newton conceived his geometrical method of fluxions as
a generalization of the method of exhaustion of the "Ancient Geometers".
The role given to geometry by Newton led him to underestimate the importance
of notation. If a demonstration is legitimated when each step of it is interpretable
in geometric terms, there is no motivation to develop the algorithm independently
from geometry.
The complexity of the relationship between calculus and should be
stressed here. Newton's method was concerned with "fluxions and series". His
treatment of series expansions remained a highly analytical in Newtonian
fluxional works, even when the interpretation of power series as Taylor expansions
paved the way for a geornctrical, or kincmatical, interpretation of the successive
terms (e.g., as position, velocity, acceleration, variation of acceleration, etc.).
On the other hand, Leibniz, notwithstanding his declarations in favour of a
calculus as "blind reasoning", always embedded his algorithm in a geometrical
interpretation. Leiblliz's differentials and integrals, as much as Newton's fluents
and fiuxions, were referred to as geometrical objects. It is revealing that Leibniz
always paid attention to the geometrical dimensions of the combination of symbols
occurring in a differential equation. It was by studying the geometry of difIerentials
(e.g., the characteristic triangle) that Leibniz and his immediate followers could
extract differential equations. Once a differential equation was obtained, it waR,
however, handled as much as possible as an algebraic object. From time to time,
it was necessary to use geometric thinking to interpret the model under study (see
4.2). Leibnizians had to do so since the rules of the calculus did not allow
the solution of the problems in geometry and dynamics that they faced (especially
when transcendental "functions" occurred). A complete algebraization of calculus
came only in the late eighteenth century. The calculus as "blind reasoning" was
thus more a. des'idemtum than a reality. Reinterpretation of the symbolism in
the geometric model was possible, and in many cases necessary, but, contrary to
Newton's approach, this reinterpretation was not seen as a value, as Cl strategy to
be pursued.
The stress on algorithmic improvements and 011 the idea that progress could be
obtained by symbolical manipulations had momentous consequences in the Leibnizian
school. Continental mathematicians felt that the differential and integral
calculus opened new field of research. In this field many new results could easily
be obtained by following as a guideline the analogies suggested by the calculus's
notation. New generalizations, new relations and formulas could be found. The
mechanization and standardi"mtion of mathematical research renderecl possible by
the stress over the algorithm rendered the Leibnizia.n school much more active and
open to innovation.
Text 16: N. Guicciardini (2003). “Newton’s Method and Leibniz’s Calculus”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 3, pp. 73–103.
Summer University 2012: Asking and Answering Questions Page 260 of 479.
102 3. NEWTON'S METHOD AND LEIDN1Z'S CALCULUS
Leibniz and the Leibnizian mathematicians looked at the geometrical proofs of
Newton's Pr'incivia with suspicion. One of their aims was to translate Newton's
geometrical proofs into the language of the differential and integral calculus. Indeed
mechanics proved to be a great source of inspiration for Leibnizians. It is
by trying to develop new mathematical tools for the mechanics of extended bodies
(rigid, elm.,tic and fluid) that mathematicians such as Varignon, Johann and Daniel
Bernoulli, Clairaut, Euler, d'Alembert, and Lagrange enriched calculus by developing
new concepts and techniques (see (Truesdell1968)). Such important resultf:l
of eighteenth-century calculus as trigonometric series, partial differential equations,
and the calculuf:l of variations were to a great extent motivated by the analytical
approach to dynamics that Leibni2 had sought to promote (sce Chapters 4, 11, and
12). The eighteenth century was thus characterized by the analytical programme
emphasized by the Leibnizian school, while the role attributed to geometry by
Newton and his followers faded away.
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Summer University 2012: Asking and Answering Questions Page 262 of 479.
2 The introduction of v (where v is ultimately to be put equal to y) you
may regard as something of a red herring! Newton was making the
assumption that v exists, where f(x) < v < f(x + 0), such that the rectangle
ov = curvilinear area Bf3(jD; since this is always possible for a
simply convex curve, the equation he formed was, in consequence, exact.
3 In modern notation, if LX y dx z, where z = f(x), then y dz/dx =
f'(x): in particular, if z = [n/(m + n)]ax(m+II)/II, then y axm1n•
4 Although, in earlier researches, Newton did sketch in the outline of a
geometrical proof of the fundamental theorem of the calculus (on the lines
of the proofs subsequently published by Barrow and Gregory) he seems
to have later preferred to rely on the reversibility of the operations, so that
differentiation and integration are regarded essentially as inverses, the
one of the other (i.e. if = f(x) fy dx, then ~: f'(x) = y, and conversely,
if y = f'(x) = ~: then z fy dx f(X)).
Exercise 7
Use Newton's method to show that, if z -.j(a2
+ x2
), y x/-.j(aZ
+ x2
).
SA 7
z = -.j(a2
+ x2
), Z2 a2
-+ x2
, (z + OV)2 a2 + (x + o?
+ 20vz + 02V2 = a2 + x2
-I- 20x + 0
2
20vz -+ 02V2 20x + 02
zv x = zy, (v y)
y = x/z =, x/-.j(a2 -I- x2)
C3.6 FLUXIONS AND
Even before writing the De Analysi Newton had experimented with other
types of notation and other forms of demonstration (see flow diagram,
p. 12). In the small tract written in 1666, he developed a fairly comprehensive
treatment of a whole range of calculus problems based on the generation
of curves by motion. These ideas, which constituted the foundation
of what he called his 'method of fluxions' were developed more fully in
the 1671 tract and it is from that that we will quote. The passage which we
have chosen conveys well the 'flavour' of Newton's fluxions and fluent~
and suggests clear links with mediaeval ideas on motion, developed by
Galileo, Torricelli and Barrow. Because of this, you may not find it easy
to follow.
It now remains, in illustration of this analytical art, to deliver some typical problems
and such especially as the nature of curves will present. But first of all I would observe
that difficulties of this sort may all be reduced to these two problems alone, which I
may be permitted to propose with regard to the space traversed by any local motion
however accelerated or retarded:
1 Given the length of the space continuously (that is, at every [instant of] time), to
find the speed of motion at any time proposed.
2 2 Given the speed of motion continuously, to find the length of the space described
at any time proposed.
22
So in the equation x 2
y, if y designates the length of the space described in any time
which is measured and represented by a second space x as it increases with uniform
Text 17: Newton on fluxions and fluents. From M. E. Baron and H. J. M. Bos, eds.
(1974). Newton and Leibniz. History of Mathematics: Origins and Development of the
Calculus 3. The Open University Press, pp. 22–25.
Summer University 2012: Asking and Answering Questions Page 263 of 479.
Extract from William lones' edition of
Newton's Fluxions, 1711 (Turner Collection,
University of Keele).
3 speed: then 2xx will designate the speed with which the space at the same moment of
time proceeds to be described. And hence it is that in the sequel I consider quantities
as though they were generated by continuous increase in the manner of a space which
a moving object describes in its course.
We can, however, have no estimate of time except in so far as it is expounded and
measured by an equable local motion, and furthermore quantities of the same kind
alone, and so also their speeds of increase and decrease, may be compared one with
another. For these reasons I shall, in what follows, have no regard to time, formally so
considered, but from quantities propounded which are of the same kind shall suppose
4 some one to increase with an equable flow: to this all the othcrs may be referred as
though it were time, and so by analogy the name of 'time' may not improperly be
conferred upon it. And so whenever in the following you meet with the word 'time'
(as I have, for clarity's and distinction's sake, on occasion woven it into my text), by
that name should be understood not time formally considered but that other quantity
through whose equable increase or flow time is expounded and measured.
But to distinguish the quantities which I consider as just perceptibly but indefinitely
growing from others which in any questions arc to be looked on as known and
determined and are designated by the initial letters a, b, c and so on, I will hereafter
call them fluents and designate them by the final letters v, x, y and z. And the speeds
with which they each flow and are increased by their generating motion (which I
23
Text 17: Newton on fluxions and fluents. From M. E. Baron and H. J. M. Bos, eds.
(1974). Newton and Leibniz. History of Mathematics: Origins and Development of the
Calculus 3. The Open University Press, pp. 22–25.
Summer University 2012: Asking and Answering Questions Page 264 of 479.
might more readily call fluxions or simple speeds) I will designate by the letters
5 v, X, yand z: namely, for the speed of the quantity v I shall putlj, and so for (he speeds
of the other quantities I shall put x,y and zrespectively.
Notes
Although it may appear to help if we express some of Newton's statements
in the notation of the calculus it should be borne in mind that, by doing
so, we risk distortion in that we may the work a of clarity and
I If s
ds
dt
v
which was absent.
where t is the time and s the to find the l.e.
2 If v q)(t), to find s, i.e. s r(p(t) dt. These arc the two
.10
inverse ,..H',·,hl"'11 from which Newton his calculus.
3 If y 2x Since IS
taken to be constant.
4 Since time can be measured
x x t. The
uniformly, can be used as a 'measure' of time.
5 ]f v, x, y, arefluents, variables
then D, x, y, i, represent the jluxions, or
of these
This may be an appropriate point to say about Newton's
'dot'-notation, particularly as you may ultimately want to compare it
with the notation developed by Leibniz. Newton with dot··
notation of one kind or another from 1665 onwards flow diagram,
p. 12). He did not settle on the 'standard' Newtonian form of doL-notation
until late 1691 and, in the original version of the 1671 traet, he used literal
symbols I, rn, n, r for the fiuxions of v, x, y, z. In 1710, William Jones made
a transcript of the 1671 treatise on fluxions and inserted the dot-notation
and this transcript was subsequently copied in all published editions. In
the translation we are using, Whitesidc has to adhere to the
'standard' dot-notation because it is a great aid to understanding. In
England, at any this notation, used to denote differentiation with
respeet to t (where t is the time), has become familiar and useful. In eomthe
Newtonian dot-notation with the notation developed by
Leibniz (dx, dy) we should bear in mind that Newton's decision to adhere
to a standard form of dot-notation and to use it consistently was certainly
made with knowledge of the existence of the Leibnizian notation in
Europe.
Exercise 8
If Y x 3
, what is the fluxion of x? What is the f1uxion of y? How is thc
f1uxion of y related to the f1uxion of x? What are x and y ealled? Which
variable is taken by Newton to move uniformly?
SA 8
.y; y ","0 3x2
x; x and y are ealledfiuents, x is taken to move uniformly so
that x k (k normally is taken to be 1).
Let us now consider how, given a relation between the fluent quantities
Newton set about finding a relation between thejluxions ofthese quantities.
24
Text 17: Newton on fluxions and fluents. From M. E. Baron and H. J. M. Bos, eds.
(1974). Newton and Leibniz. History of Mathematics: Origins and Development of the
Calculus 3. The Open University Press, pp. 22–25.
Summer University 2012: Asking and Answering Questions Page 265 of 479.
L
DEMONSTRATION
The moments ofthe 11uent quantities (that is, their indefinitely small parts, by addition
of which they increase during each infinitely small period of time) are as their speeds
of 11ow. Wherefore if the moment of any particular one, say x, be expressed by the
prodnct of its speed x and an infinitely small quantity 0 (that is, by xo), then the
moments of the others, v, y, z, [...], will be expressed by vo, yo, zo, [...] seeing that
vo, xo, yo and io are to one another as lj, x, y and z.
Now, since the moments (say, xo and yo) of 11uent quantities (x and y, say) are the
infinitely small additions by which those quantities increase during each infinitely
2 small interval of time, it follows that those quantities x anci y after any infinitely small
interval of time will become x + xo and y + yo. Consequently, an equation which
expresses a relationship of fluent quantities without variance at all times will express
that relationship equally between x xo and y -+ yo as between x and y; and so
3 x + .xo and y + yo may be substituted in place of the latter quantities, x and y, in
the said equation.
4
Let there be given, accordingly, any equation x3
ax2
+ axy -- l °and substitute
x + xo in place of x and y +- yo in place of y: there will emerge
(x3
+ 3xox2
+ 3X2
0
2
X + X3( 3) (ax2
+ 2axox +
+(axy +- axoy ayox + aX)J()2) (y3 + 3yo/ + + y3( 3) 0.
Now by hypothesis x3
-- ax2
+ axy y3 = 0, and when these terms arc erased and
the rest divided by 0 there will remain
3xx2
+ 3x2
0x + X\)2 - 2axx --- ax2
0 + axy + ayxl- axyo --
__ y3()2 0.
But further, since 0 is supposed to be infinitely small so that it be able to express the
moments of quantities, terms which have it as a factor will be equivalent to nothing
5 in respect of the others. I therefore cast them out and there remains 3xx2
2axx +
axy + ayx - 3;iyz = 0, as in Example 1 above.
Notes
It is accordingly to be observed that terms not multiplied by () will always vanish, as
also those multiplied by () of more than one dimension; and that the remaining terms
after division by 0 will always take on the form they should have according to the
rule. This is what I wanted to show.!
1 The little '0' which we saw as a general increment in the De Analysi has
now become an 'infinitely small period of time', say lit.
2 All variables arefluent quantities and their moments are correspondingly
expressed by the products of their respective velocities and the time '0',
We ean think of xo,yo, as (dx/dt)bt, (dy/dt)lit" ,.
3 Hf(x, y) 0 expresses a relationship between x and y which is valid at
all times, then
.f(x, y) = f(x + xo, y + yo)
f(x + (dx/dt)bt, y+ (dy/dt)bt)
4 ax2
+ axy o
(x + XO)3 - a(x + XO)2 + a(x + xo)(y + yo) - (y + yO)3
The steps followed are, successively: (i) expand, (ii) remove common terms
from both sides, (iii) divide by 0, (iv) delete terms containing 0, 'since 0 is
supposed to be infinitely small'.
5 The relation, 3xx2
- 2axx + axy + ayx - 3yy2 0, can be rewritten
in the form,y /x
3x2 - 2ax + ay dy
- ax dx
- fx!j~, wherej~ andj~ are the
partial derivatives of f (x, y) with respect to x and y respectively. (See
MlO02
, Unit 15.)
1 NMP, Ill, pp. 79-81.
2 The Open University (1971) MlOO Mathematics: A Foundation Course, The Open
University Press.
25
Text 17: Newton on fluxions and fluents. From M. E. Baron and H. J. M. Bos, eds.
(1974). Newton and Leibniz. History of Mathematics: Origins and Development of the
Calculus 3. The Open University Press, pp. 22–25.
Summer University 2012: Asking and Answering Questions Page 266 of 479.
Text 18: Newton on the method of drawing tangents. From D. T. Whiteside, ed.
(1964). The Mathematical Works of Isaac Newton. Vol. 1. Johnson Reprint Corp.
Summer University 2012: Asking and Answering Questions Page 267 of 479.
ult.x
B c
We have now mentioned the three important ideas which underlie
Leibniz's invention of the calculus:
1 Leibniz's interest in symbolism and notation in connection with his
idea of a general symbolic language;
The insight that their differences are
quadraturcs and
:; Thecharacteristic transformations
of the
]n the 25 October 11 November 1 Leibniz combincd these
ideas in a series of studies on the analytic treatment of infinitesimal
problems, which contain the invention of the calculus. They are known
to us becausc the manuscripts in which Leibnizjotted down his thoughts,
more or less as they came to are still extant. These manuscripts, dated
29 October and 1 and 11 November 1 form a most precious
record of a process of invention. It is not often that we are able to follow
the successive steps in a major mathematical discovery, and in this section
we will indicate these steps and illustrate them by fragments ofthe original
texts.
Leibniz's starting point was the study of relations between quadratures,
expressed analytically (in formulae) by means ofthe symbolism introduced
by Cavalieri (see Unit C2 pp. 13-8). That is, he wrote 'omn./' (abbreviation
for omnes I, 'all!'), for the quadrature of a curve whose ordinates are l.
To give you the flavour of this starting point of Leibniz's study, here is an
argument from the manuscript of 26 October. The text is very brief, it
consists only of the sentences we quotel
and a series of formulae, so we
have added some explanation.
Consider a sequence of equidistant ordinates y of a curve as in the figure
(which is an amplification of Leibniz's figure). The differences of the ys are
called w. The area OCD is the sum of all rectangles xw. Now x x w is the
statical moment of wwith respect to the horizontal axis. (Statical moment
= weight x distance to axis; in this case the weight of w is taken equal
to its length.) Therefore area OCD is the sum of the moments of the
differences w. Now area OCD is the complement of area OCB in the
rectangle OBCD, and the area OCB is, in Cavalieri's terminology, the sum
of all 'terms' y. Hence:
The moments of the differences about a straight line perpendicular to the axis are
equal to the complement of the sum of the terms.
Now the ws are the differences of the ys, so that conversely the 'terms' y
are the sums ofthe w. So if we take any sequence with terms wand replace
in the preceding sentence 'differences' by 'terms' and 'terms' by 'sum ofthe
terms' we have:
and the moments of the terms are equal to the complement of the sum of the sums.
Leibniz expresses this result in Cavalierian symbolism:
omn.xw n
'--~
moments of
the terms w
ult.x, omn.w., - oliin.omn~w
'-----v-----' ~--'
total sum of the sums
of the terms
complement of the sum
of the sums of the terms
1 Child, J. M. (1920) The Early Mathematical Manuscripts ofLeibniz, London.
42
Text 19: Leibniz’ process of discovery. From M. E. Baron and H. J. M. Bos, eds. (1974).
Newton and Leibniz. History of Mathematics: Origins and Development of the
Calculus 3. The Open University Press, pp. 42–43.
Summer University 2012: Asking and Answering Questions Page 268 of 479.
11 is Leibniz's symbol for equality; he llses overlining where we would
use brackets; the commas are separating symbols; ult. stands for ultimus
(last), meaning the last terms of the sequence. You should note the central
role of the theory of difference sequences in this see Section
C3. 12 p. 36.
Now Leibniz with this formula, and derives other formulae from
purely analytically, without use of a He does this
substituting variables in the of w, and he the results
as relations between In this way he finds for instance:
az az
omll.az n u]t.x, omn.· omn.omn
x x
substitution xw az, W and
a a
"omn.a n ult.x,om11.·· omn.OIl1I1.
x x
(bY substitution xw a, W
a
Leibniz "..j,F'rr,rpjiQ the last
the last theorem expresses the sum of the logarithms in terms of the known quadral ure
of the hyperbola.
y = f!. is the equation of the rectangular hyperbola, hence omn.
a
is the
x x
quadrature of the hyperbola. Now this quadrature is a logarithm
I
a~ . a
would say ~...~.. 0= log x for some base for the loganthm), so omn.omn.
x x
is the sum of the logarithms. So the equation indeed expresses the sum of
the logarithms in terms of the quadrature of the hyperbola.
You should compare this way of deriving transformations of quadratures
with Leibniz's study on the transmutation, and note the advantage of a
symbolism through which these transformations can be performed by
means of formulae instead of by inspection of complicated figures.
Exercise 19
Leibniz also derived from his basic formula the relation
a a a
omn.~ n x, omn. X2 ~ omn.omn.~
Could you imagine how?
SA 19
a
By using the substitution w =
Three days later (29 October) we find Leibniz exploring the operational
rules for the symbol omn., noting for instance that omn.yz is not equal to
omn.y x omn.z. In this investigation Leibniz suddenly chooses a new
symbol instead of omn. :
It will be useful to write ffor omn., so that fI = omn.l, or the sum of the Is.
Iis the long script s, it stands for summa, sum, so that the symbol is shorter
and applies better to Leibniz's conception of the quadrature: the sum of
the terms, rather than the Cavalierian 'all terms'. Leibniz writes IIfor
omn.omn., he stresses that the differences between the terms are infinitely
small and he writes simple quadrature relations in the new symbolism:
43
..'
Text 19: Leibniz’ process of discovery. From M. E. Baron and H. J. M. Bos, eds. (1974).
Newton and Leibniz. History of Mathematics: Origins and Development of the
Calculus 3. The Open University Press, pp. 42–43.
Summer University 2012: Asking and Answering Questions Page 269 of 479.
96 Tekst 26: Berkeley om analysens grundlag
d) Tror Berkeley p˚a de resultater, som man har opn˚aet med fluxionsregningen?
e) Diskuter forskellen mellem religiøs og matematisk viden ifølge Berkeley.
Berkeley
A Discourse Addressed to an Infidel Mathematician
Though I am a stranger to your person, yet I am not, Sir, a stranger to the
reputation you have acquired in that branch of learning which hath been your
peculiar study; nor to the authority that you therefore assume in things foreign
to your profession; nor to the abuse that you, and too many more of the like
character, are known to make of such undue authority, to the misleading of unwary
persons in matters of the highest concernment, and whereof your mathematical
knowledge can by no means qualify you to be a competent judge. [. . . ]
Whereas then it is supposed that you apprehend more distinctly, consider
more closely, infer more justly, and conclude more accurately than other men,
and that you are therefore less religious because more judicious, I shall claim
the privilege of a Freethinker; and take the liberty to inquire into the object,
principles, and method of demonstration admitted by the mathematicians of the
present age, with the same freedom that you presume to treat the principles and
mysteries of Religion; to the end that all men may see what right you have to
lead, or what encouragement others have to follow you. [. . . ]
The Method of Fluxions is the general key by help whereof the modern mathematicians
unlock the secrets of Geometry, and consequently of Nature. And,
as it is that which hath enabled them so remarkably to outgo the ancients in
discovering theorems and solving problems, the exercise and application thereof
is become the main if not the sole employment of all those who in this age pass
for profound geometers. But whether this method be clear or obscure, consistent
or repugnant, demonstrative or precarious, as I shall inquire with the utmost
impartiality, so I submit my inquiry to your own judgment, and that of every
candid reader. — Lines are supposed to be generated1
by the motion of points,
planes by the motion of lines, and solids by the motion of planes. And whereas
quantities generated in equal times are greater or lesser according to the greater
or lesser velocity wherewith they increase and are generated, a method hath been
found to determine quantities from the velocities of their generating motions.
And such velocities are called fluxions: and the quantities generated are called
flowing quantities. These fluxions are said to be nearly as the increments of the
flowing quantities, generated in the least equal particles of time; and to be accurately
in the first proportion of the nascent, or in the last of the evanescent
increments. Sometimes, instead of velocities, the momentaneous increments or
1
Introd. ad Quadraturam Curvarum.
Text 20: Bishop Berkeley’s The Analyst. From D. E. Smith (1959). A source book in
mathematics. 2nd ed. 2 vols. New York: Dover Publications, Inc., pp. 627–634.
Adopted from J. Lützen and K. Ramskov, eds. (1999). Kilder til matematikkens historie.
2nd ed. København: Matematisk Afdeling, Københavns Universitet, pp. 95–99.
Summer University 2012: Asking and Answering Questions Page 270 of 479.
Tekst 26: Berkeley om analysens grundlag 97
decrements of undetermined flowing quantities are considered, under the appellation
of moments.
By moments we are not to understand finite particles. These are said not
to be moments, but quantities generated from moments, which last are only the
nascent principles of finite quantities. It is said that the minutest errors are not
to be neglected in mathematics: that the fluxions are celerities, not proportional
to the finite increments, though ever so small; but only to the moments or nascent
increments, whereof the proportion alone, and not the magnitude, is considered.
And of the aforesaid fluxions there be other fluxions, which fluxions of fluxions
are called second fluxions. And the fluxions of these second fluxions are called
third fluxions: and so on, fourth, fifth, sixth, etc., ad infinitum. [. . . ] But the
velocities of the velocities — the second, third, fourth, and fifth velocities, etc.
— exceed, if I mistake not, all human understanding. [. . . ]
Berkeley diskuterer herefter konkrete eksempler og forskellige metoder til at finde
fluxionerne.
[. . . ] But whether this method be more legitimate and conclusive that the former,
I proceed now to examine; and in order thereto shall premise the following lemma:
— “If, with a view to demonstrate any proposition, a certain point is supposed,
by virtue of which certain other points are attained; and such supposed point
be itself afterwards destroyed or rejected by a contrary supposition; in that case,
all the other points attained thereby, and consequently thereupon, must also be
destroyed and rejected, so as from thenceforward to be no more supposed or
applied in the demonstration.”2
This is so plain as to need no proof.
Now, the other method of obtaining a rule to find the fluxion of any power
is as follows. Let the quantity x flow uniformly, and be it proposed to find the
fluxion of xn
. In the same time that x by flowing becomes x + o, the power xn
becomes x + o
n
, i.e., by the method of infinite series
xn
+ noxn−1
+
nn − n
2
ooxn−2
+ &c.,
and the increments
o and noxn−1
+
nn − n
2
ooxn−2
+ &c.
are one to another as
1 to nxn−1
+
nn − n
2
oxn−2
+ &c.
Let now the increments vanish, and their last proportion will be 1 to nxn−1
. But
it should seem that this reasoning is not fair or conclusive. For when it is said,
2
Berkeley’s lemma was rejected as invalid by James Jurin and some other mathematical
writers. The first mathematician to acknowledge openly the validity of Berkeley’s lemma was
Robert Woodhouse in 1803.
Text 20: Bishop Berkeley’s The Analyst. From D. E. Smith (1959). A source book in
mathematics. 2nd ed. 2 vols. New York: Dover Publications, Inc., pp. 627–634.
Adopted from J. Lützen and K. Ramskov, eds. (1999). Kilder til matematikkens historie.
2nd ed. København: Matematisk Afdeling, Københavns Universitet, pp. 95–99.
Berkeley then discusses specic examples and dierent ways of nding the uxions.
Summer University 2012: Asking and Answering Questions Page 271 of 479.
98 Tekst 26: Berkeley om analysens grundlag
let the increments vanish, i. e., let the increments be nothing, or let there be no
increments, the former supposition that the increments were something, or that
there were increments, is destroyed, and yet a consequence of that supposition,
i. e., an expression got by virtue thereof, is retained. Which by the foregoing
lemma, is a false way of reasoning. Certainly when we suppose the increments to
vanish, we must suppose their proportions, their expressions, and everything else
derived from the supposition of their existence, to vanish with them. [. . . ]
I have no controversy about your conclusions, but only about your logic and
method: how you demonstrate? what objects you are conversant with, and
whether you conceive them clearly? what principles you proceed upon; how
sound they may be; and how you apply them? [. . . ]
The great author of the metod of fluxions felt this difficulty, and therefore he
gave in to those nice abstractions and geometrical metaphysics without which he
saw nothing could be done on the received principles: and what in the way of
demonstration he hath done with them the reader will judge. It must, indeed, be
acknowledged that he used fluxions, like the scaffold of a building, as things to
be laid aside or got rid of as soon as finite lines were found proportional to them.
But then these finite exponents are found by the help of fluxions. Whatever
therefore is got by such exponents and proportions is to be ascribed to fluxions:
which must therefore be previously understood. And what are these fluxions?
The velocities of evanescent increments. And what are these same evanescent
increments? They are neither finite quantities, nor quantities infinitely small,
nor yet nothing. May we not call them the ghosts of departed quantities? [. . . ]
And, to the end that you may more clearly comprehend the force and design
of the foregoing remarks, and pursue them still farther in your own meditations,
I shall subjoin the following Queries: —
[. . . ]
Qu. 4. Whether men may properly be said to proceed in a scientific method,
without clearly conceiving the object they are conversant about, the end proposed,
and the method by which it is pursued? [. . . ]
Qu. 8. Whether the notions of absolute time, absolute place, and absolute
motion be not most abstractely metaphysical? Whether it be possible for us to
measure, compute, or know them?
[. . . ]
Qu. 16. Whether certain maxims do not pass current among analysts which
are shocking to good sense? And whether the common assumption, that a finite
quantity divided by nothing is infinite, be not of this number?3
[. . . ]
Qu. 31. Where there are no increments, whether there can be any ratio of in-
3
The earliest exclusion of division by zero in ordinary elementary algebra, on the ground
of its being a procedure that is inadmissible according to reasoning based on the fundamental
assumptions of this algebra, was made in 1828, by Martin Ohm, in his Versuch eines vollkommen
consequenten Systems der Mathematik, Vol. I, p. 112. In 1872, Robert Grassmann took the same
position. But not until about 1881 was the necessity of excluding division by zero explained in
elementary school books on algebra.
Text 20: Bishop Berkeley’s The Analyst. From D. E. Smith (1959). A source book in
mathematics. 2nd ed. 2 vols. New York: Dover Publications, Inc., pp. 627–634.
Adopted from J. Lützen and K. Ramskov, eds. (1999). Kilder til matematikkens historie.
2nd ed. København: Matematisk Afdeling, Københavns Universitet, pp. 95–99.
Summer University 2012: Asking and Answering Questions Page 272 of 479.
Tekst 27: Eulers formler 99
crements? Whether nothings can be considered as proportional to real quantities?
Or whether to talk of their proportions be not to talk nonsense? [. . . ]
Qu. 63. Whether such mathematician as cry out against mysteries have ever
examined their own principles?
Qu. 64. Whether mathematicians, who are so delicate in religious points, are
strictly scrupulous in their own science? Whether they do not submit to authority,
take things upon trust, and believe points inconceivable? Whether they have not
their mysteries, and what is more, their repugnances and contradictions? [. . . ]
Tekst 27: Eulers formler
I bind 1 af Introductio in analysin infinitorum fra 1748 behandlede Euler sammenhængen
mellem de trigonometriske funktioner og eksponentialfunktionen. Det var hans
konsekvente brug af betegnelserne sin x og cos x for sinus og cosinus, samt π for den
halve omkreds af enhedscirklen, der gjorde, at disse fik almindelig udbredelse i den
matematiske symbolik.
Nedenfor er gengivet uddrag af hans behandling i den engelske oversættelse i
[Fauvel & Gray 1987, pp. 449–51].
a) Gennemg˚a Eulers udledning af rækkeudviklingerne for sin x og cos x. Hvilken
formel baseres udledelsen p˚a?
b) Gennemg˚a udledningen af Eulers formler.
c) Kommenter Eulers brug af uendelig sm˚a og store størrelser.
Euler’s unification of the theory of elementary functions
126. After logarithms and exponential quantities we shall investigate circular
arcs and their sines and cosines, not only because they constitute another type of
transcendental quantity, but also because they can be obtained from these very
logarithms and exponentials when imaginary quantities are involved.
Let us therefore take the radius of the circle, or its sinus totus, = 1. Then
it is obvious that the circumference of this circle cannot be exactly expressed
in rational numbers, but it has been found that the semicircumference is by
approximation = 3.14159.26535.89793 . . . [127 decimal places are given] for which
number I would write for short π, so that π is the semicircumference of the circle
of which the radius = 1, or π is the length of the arc of 180 degrees.
Text 20: Bishop Berkeley’s The Analyst. From D. E. Smith (1959). A source book in
mathematics. 2nd ed. 2 vols. New York: Dover Publications, Inc., pp. 627–634.
Adopted from J. Lützen and K. Ramskov, eds. (1999). Kilder til matematikkens historie.
2nd ed. København: Matematisk Afdeling, Københavns Universitet, pp. 95–99.
Summer University 2012: Asking and Answering Questions Page 273 of 479.
Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
Summer University 2012: Asking and Answering Questions Page 274 of 479.
Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
Summer University 2012: Asking and Answering Questions Page 275 of 479.
Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
Summer University 2012: Asking and Answering Questions Page 276 of 479.
Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
Summer University 2012: Asking and Answering Questions Page 277 of 479.
Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
Summer University 2012: Asking and Answering Questions Page 278 of 479.
Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
Summer University 2012: Asking and Answering Questions Page 279 of 479.
Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
Summer University 2012: Asking and Answering Questions Page 280 of 479.
Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
Summer University 2012: Asking and Answering Questions Page 281 of 479.
Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
Summer University 2012: Asking and Answering Questions Page 282 of 479.
Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
Summer University 2012: Asking and Answering Questions Page 283 of 479.
Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
Summer University 2012: Asking and Answering Questions Page 284 of 479.
Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
Summer University 2012: Asking and Answering Questions Page 285 of 479.
Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
Summer University 2012: Asking and Answering Questions Page 286 of 479.
Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
Summer University 2012: Asking and Answering Questions Page 287 of 479.
Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
Summer University 2012: Asking and Answering Questions Page 288 of 479.
Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
Summer University 2012: Asking and Answering Questions Page 289 of 479.
Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
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Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
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Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
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Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
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Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
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Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
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Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
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Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
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Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
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Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
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Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
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Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
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Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
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Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
Summer University 2012: Asking and Answering Questions Page 303 of 479.
Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
Summer University 2012: Asking and Answering Questions Page 304 of 479.
Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
Summer University 2012: Asking and Answering Questions Page 305 of 479.
Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
Summer University 2012: Asking and Answering Questions Page 306 of 479.
Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
Summer University 2012: Asking and Answering Questions Page 307 of 479.
Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
Summer University 2012: Asking and Answering Questions Page 308 of 479.
Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
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Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
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Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
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Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
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Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
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Text 21: J. Lützen (2003). “The Foundation of Analysis in the 19th Century”. In: A
History of Analysis. Ed. by H. N. Jahnke. History of Mathematics 24. Providence
(Rhode Island): American Mathematical Society. Chap. 6, pp. 155–195.
Summer University 2012: Asking and Answering Questions Page 314 of 479.
Bolzano, Caucy and "New Ana sis"
of Ear Nineteent Century
I. GRATTAN-GuINNESS
Communicated by J. E. HOFMANN
Summary
Ill this paper 1 I discuss the development Of mathematical analysis during the
second and third decades of tile nineteenth century; and in particular I assert that
the well-known correspondence of:new ideas to be found in the writings of BOLZANO
and CAueHY is not a coincidence, but that CA~ICH¥, had read one particular paper of
BOLZANO and drew on its results without acknowledgement, The reasons for this
conjecture involve not only the texts ill question but also the state of development
of mathematical analysis itself, CAUClty both as personality and as mathematician,
and the rivalries which were prevalent in Paris at that time.
Contents
t. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
2. The Common Ideas in BOLZAI~O and C,~ucI~,Y . . . . . . . . . . . . . . 373
3. The New Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 378
4. The Old Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 38t
5. CAUCI~Y'S Originality as a Mathematician . . . . . . . . . . . . . . . . 384
6. The State of Parisian Mathematics . . . . . . . . . . . . . . . . . . . 387
7. CAtJCHY'S Personality . . . . . . . . . . . . . . . . . . . . . . . . . 393
8. The Availability and Familiarity of BOLZANO'S Work . . . . . . . . . . . 395
9- Tile Personal Relations between BOLZANO and CAUCHY . . . . . . . . . . 397
10. Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
Index of Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
1. Introduction
The central theme of this paper is an historical conjecture concerning the
development of mathematical analysis in the early nineteenth century. It is
well known that the major event was the publication in t821 of the Cours d'Ana-
1 This paper is a revised and greatly expanded version of a lecture entitled "Did
Cauchy read Bolzano before writing his Cours d'Analyse?" given at tile Problemgeschichte
der Mathematik seminar at Oberwolfach, West Germany, on the 26th November,
1969. I wish to thank Professors J. E. HOFMANN and C.J. SCRIBA for their
invitation to this seminar.
The text draws frequently on my history of The Development o/ the Foundations
o/ Mathematical Analysis /rom Euler to Riemann and Joseph Fourier 1768--1830,
which are both to be published by the M.I.T. Press and are referred to ill later footnotes
as Foundations and Fourier, respectively. Tile latter work was written with the
collaboration of Dr. J. R. RAVETZ, and the former with the help of his detailed
criticism: I wish to record here my indebtedness to his assistance.
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 315 of 479.
]3olzano, Cauchy and New Analysis 373
lyse of AUCUSTIN-LouIS CAUCHY(t789--t857), s in which CAUCHYpresented a
new type of analytical reasoning far superior to previous ideas for the development
of analysis -- limits, functions, the calculus, and so on. CAUCHY'S
achievement was the so-called "arithmeticisation" of analysis, a method whose
development and application has been a major interest for mathematicians ever
since.
It has been also well-known for some time that CAUCHYhad been anticipated
in his basic ideas of the new analysis by an obscure pamphlet published in Prague
in t 817 by BERNARDBOLZANO(t 78t --1848). In contrast to the broad programme
of CAUCHY'Sbook, BOLZANOdevoted his little work to the proof of a theorem
which he described in its title: "Purely analytical proof of the theorem, that
between any two values [of a function/(x)] which guarantee an opposing result
[in sign] lies at least one real root of the equation [/ (x) ----0]." s The "pure analysis"
which ]3OLZANOproduced in his proof is exactly that which we find greatly
developed and extended in CAUCHY'SCoufs d'Analyse and his later writings on
analysis.
I do not believe that we have here an example of a remarkable coincidence
of new ideas. Such occurrences are of course well-known in the history of science,
but I shall argue for the conjecture that in this case CAUCHYwas welt acquainted
with BOLZANO'Spaper and that he drew on its novelties without ever making
acknowledgement to him.
The argument for this thesis is not based on new documentary evidence:
there is no reference to BOLZANO'Swork among the scattered fragments of
CAUCHY'Spapers and letters, no library record of CAUCHY'Sreading or borrowing
BOLZANO'Spaper, no copy of it in his personal library (which in fact has been
dispersed). My reasons for the conjecture are circumstantial and related to
intellectual matters, and involve not only the general development of analysis
at that time but also that aspect of the growth of science which is ignored all too
often by its historians -- the social and educational situation of the period; and
the personalities of the principal characters.
2. The Common Ideas in Bolzano and Cauchy
We consider first the directly corresponding results in the two works, in each
case in its general historical setting.
2.1. Continuity of a Function. Normally the continuity of a function was then
identified with its description by a single algebraic expression, and the function
was usually thought to be differentiable: in fact, under EULER'S influence the
2A.-L. CAucI~Y,Cours d'Analyse de l'Ecole Royale Polytechnique. ITM Pattie: Analyse
Algdbrique (1821, Paris) = Oeuvres, (2) 3. No further parts of this work were
published: it is referred to in later footnotes as Cours.
3 ]3. BOLZANO, "Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey
Werthen, die ein entgegengesetztes Resultat gew~hren, wenigstens eine reelle Wurzel
der Gleichung liege," (18t7, Prague)= Abh. KSnigl. B6hm. Gesell. Wiss., (3) 5
(18t4--17: publ. 18t8), 60pp. = Ostwald's Klassiker, No. t53 (ed. P. JOURDAI~¢:
1905, Leipzig), 3--43. French trans, in Rev. d'Hist. Sci. Appl., 17 (1964), 136--164:
there have also been various other translations and issues. The paper is referred to
in later footnotes as Beweis.
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 316 of 479.
374 I. GRATTAN-GuINNESS:
term "continuous" was usually confined to functions which we now call "differentiable".
4 There were efforts to move away from this view, including by
EULER himself; but nobody had come at all close to the formulation of continuity
given by BOLZANOand CAUCHY:
BOLZANO: "A function /(x) varies according to the law of continuity for all
values of x which lie inside or outside certain limits, is nothing other than this:
if x is any such value, the difference /(x +co)--/(x) can be made smaller than
any given quantity, if one makes w as small as one ever wants to. ''~
CAUCHY:"The function /(x) will remain continuous with respect to x between
the given limits, if between these limits an infinitely small increase of the variable
always produces an infinitely small increase of the function itself".*
One of the most interesting and important features of this formulation of
continuity is that it extends the old formulation beyond that of differentiability,
for it also encompasses functions with corners. I think that BOLZANOwas aware
of the extension in t8t7, for in later manuscripts he studied the distinction
between the new continuity and differentiability to the extent of constructing
a continuous non-differentiable function of the type studied later only by the
school of WEIERSTRASSin the t870's. ~ But CAUC~IYseems to have seen the new
idea only as a reformulation of the old one when he wrote the Cours d'Analyse,
for the examples he gave there of continuous functions were all of standard
a
differentiable algebraic expressions, with the functions x~ for negative a, and x'
regarded as "discontinuous" at x----0 since they then became infinite) In fact,
he explicitly discussed the distinction only in a paper of 1844, and then in a
way which tried to give the impression that he had known it all along:
"In the works of Euler and Lagrange, a function is caned continuous or
discontinuous, according as the diverse values of that function, corresponding
to diverse values of the variable ... are or are not produced by one and the same
equation .... Nevertheless the definition that we have just recalled is far from
offering mathematical precision; for the analytical laws to which functions can
be subjected are generally expressed by algebraic or transcendental formulae
[that is, by the EULERIANrange of algebraic expressions~, and it can happen that
various formulae represent, for certain values of a variable x, the same function:
then, for other values of x, different functions."
He then quoted the example
(20
V~-- 2 f x~ / xifx=>0
~. t~fidt=t_xif x< O, (t)
0
EULER'Sclassic presentation of his theory of functions was given in the opening
sections of both volumes of his Introductio ad analysin in/initorum (2 vols: 1748,
Lausanne) = Opera Omnia, (1) 8--9.
5 B. BOLZANO, Beweis, preface, part IIa.
A.-L. CAUCI~Y,Cours, 34--35 = Oeuvres, (2) 3, 43.
7 See B. BOLZANO,Functionenlehre (ed. K. RYCHLIK), in his Schriflen, 1 (1930,
Prague), esp. pp. 66--70, 88--89.
8A.-L. CAvcI~Y,Cours, 36--37 = Oeuvres, (2) 3, 44--45.
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 317 of 479.
Bolzano, Cauchy and New Analysis 375
in which the first two forms are "continuous" in EULER'S sense while the third
is "discontinuous" ;
"... but the indeterminacy ceases if for Euler's definition we substitute that
which I have given [in the Cours d'Analysej-.9
2.2. Convergence of a Series. A major innovation of the new analysis was the
study of the convergence of a series (or of classes of series) as a general problem
separate from and indeed prior to that of its summation; but it would be wrong
to presume that the problem of convergence had previously been ignored or
taken for granted. 17 th and t8 *h century mathematicians were perfectly well
aware that a series was to be interpreted as a term-by-term addition of its
members, and that individual series (usually series of constant terms or certain
power series) could be shown to be convergent, especially if they were associated
with some geometrical limiting procedure such as the approximation to a curve
by a polygon. But this understanding had been endangered during the 18th
century, especially by EULER'S great ability to devise complicated new methods
of summation of series. Today we understand that some of these methods reduce
to orthodox smnmation for orthodox convergent series and some do not; but
EULER and his contemporaries seemed to have regarded all methods as legitimate,
giving "the" sum of the series rather than its sum relative to the method
of summation involved. This more sophisticated understanding began to develop
only in the t 890's, under the leadership of BOREL:1° until then, series considered
"divergent" (that is, oscillatory series as well as those with an infinite sum)
had been banished from analysis under the influence of CAUCHY'S work. But he
and BOLZANOwere not the first to consider the convergence of a series to be an
important property worthy of investigation of its own. GAUSS had even advanced
as far as a sophisticated convergence test by t 8t 211: FOUI~IER had already treated
the convergence of particular examples of his series in 1807, in his first paper on
the diffusion of heatl~: LAGRANGEhad tried to find expressions for the remainder
term of a TAYLOR series, in connection with his long held belief that the series
could serve as the foundation of the calculus; 13 and LACROIX was also aware of
the need for general formulation of convergence. ~4 Both BOLZANOand CAOCHY
also stressed that the convergence of a series is to be determined only by the
tendency of the nth partial sums to a limiting value s as n tended to infinity; ~5
9 A.-L. CAuci~¥, "M6moire sur les fonctions continues ou discontinues" C.R.
Acad. Roy. Sci., 18 (t844), 1t6--130 (see pp. t16--117) = Oeuvres, (t) 8, t45--160
(pp. 145--146).
10 For extended discussion, see my Foundations, ch. 4.
11 K. F. Gauss, "Disquisites generales ..." Comm. Soc. Reg. Sci. Gdltingen Rec.,
2 (18t1--13: publ. 1813), cl. math., 46pp. = Werke, 3, 123--t62: see art. 16. For a
history of convergence tests, see the appendix to my Foundations.
13 j. B. J. FOURIER, " Sur la propagation de la chaleur," MS. 1851, Ecole Nationale
des Ponts el Chaussdes, Paris: see arts. 42--43. The publication of this entire manuscript
constitutes the body of my Fourier: see there ch. 7 on this point.
13 See especially his Thdorie des/onctions analytiques .... (2nd edition: 1813, Paris)
= Oeuvres, 9: part 1, arts. 35--40.
i~ See especially his Traild du calcul diJ/drentiel et du calcul intdgral (Ist edition:
t 797--t800, Paris), 1, 4--9.
15 B. BoLzAxo, Beweis, art. 5. A.-L. CAIJCHY, Cours, t23--t25 = Oeuvres, (2) 3,
114--t15.
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 318 of 479.
376 I. GRATTAN-GUINNESS:
thus this correspondence is not so striking, although the idea was still then very
much a new one. But in both works we find a new type of result, not to be found
in any other contemporary writing. BOLZANO had defined a class of series:
"... which possess the property that the variation (increase or decrease) which
their value suffers through a prolongation Eof terms] as far as desired remains
always smaller than a certain value, which again can be taken as small as one
wishes, if one has already prolonged the series sufficiently far", 1~ and then he
proved that for series with this property,
"... there always exists a certain constant value, and certainly only one,
which the terms of this series always approach the more, and towards which they
can come as close as desired, if one prolongs the series sufficiently far." ~ CAUCHY
stated that :
"For the series 1u, to be convergent it is yet necessary that for increasing
values of n the different sums
u~ +u~+ 1+u~+~
~C . . . . .
... finish by constantly achieving numerical values smaller than any assignable
limit. Reciprocally, when these various conditions are fulfilled, the convergence
of the series is assured. ''18
In other words, they both found a general condition for convergence in terms
of the behaviour of (s,+~--s~) as n tended to infinity: a result of quite profound
originality. Contrary to general belief, BOLZANO in fact only asserted the sufficiency
of the condition in his paper; his proof is very difficult to follow even with
the ideas of his new analysis, and in fact is faulty. The necessity of the condition
is far easier to recognise and prove: CAUCltY did prove it, but then avoided difficulties
by hinting that sufficiency followed as a consequence (which it does not !) :
"the sums s~, s~+1.... differ from the limit s, and consequently among themsalves,
by infinitely small quantities. ''19
2.3. Bolzano's Main Theorem. The theorem which BOLZANO actually proved in
his paper was the following generalisation of the theorem of his title:
Let /:(x) and ]~(x) be continuous functions for which /: (~) < /2 (*¢) and
]1(/5)> ]~(/5): then ]1(a) = ]~ (a) for at least one value a of x between c~and/5. (The
basic theorem is the case where in (x) ~ 0.)
As a theorem it is most untypical of its time: that is, a general theorem concerning
the properties of functions was not the kind of result then being sought
in analysis. ]3OLZANOhimself saw it rather as a theorem in the theory of equations,
as a companion to GAUSS'S recent proofs of the decomposition of a polynomial
lS B. BOLZANO, Beweis, art. 5.
17 t3. BOLZANO, Beweis, art. 7.
is A.-L. CAUCHY, Gouts, t24--125 = Oeuvres, (2) 3, 115--116.
19 A.-L. CAOCHY, Cours, 125 =Oeuvres, (2) 3, 115. My italics.
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 319 of 479.
Bolzano, Cauchy and New Analysis 377
into linear and quadratic factors. ~° CAucltY saw it as a theorem of the new analysis,
and put it twice into the Cours d'Analyse (in its restricted form): firstly
with a naive geometrical argument, and later, in the part of his book reserved
for those with a special interest in analysis, with a condensation argument which
seems very much like an unrigorous version of the intricate proof developed in
BOLZANO'S paper. ~1
2.4. Bolzano's Lemma. A crucial lemma required by BOLZA?¢Oto establish the
existence of the real root was the following lemma:
"If a property M does not apply to all values of a variable quantity x, but
to all those which are smaller than a certain u: so there is always a quantity U
which is the largest of those of which it can be asserted that all smaller x possess
the property M." 32
With this extraordinary theorem came another new idea into analysis, completely
untypical of its time: the upper limit of a sequence of values. It is not to
be found explicitly in CAlJCltY'S Cours d'Analyse, but instead we have there a
frequent use of phrases like "... the largest value of the expression ..." when
calculating limiting values, especially in connection with the development of
tests for convergence of a series.~3 As with continuity of a function, CAUCltY was
revealingly only partially aware of the significance of the idea; for he used it
only as a tool for developing the proofs of his particular theorems and not as a
profound device for investigating more sophisticated properties of analysis.
Therefore it would be especially surprising if it were CAUCHY'S own invention:
not until the t860's was it introduced again and properly used, by the WEIERSTRASS
school of analysts. 2~
2.5. The Real Number System. Lastly, a point which is less striking than the
others but worth mentioning: the considerations given in both works to the real
numbers. In the course of proving his lemma as well as in other parts of his paper
BOLZANO had recourse to extended considerations of real numbers, especially
regarding the rational or irrational limiting values of sequences of certain finite
series of rationals. ~ In later manuscripts he extended these remarks into a full
theory of rational and irrational numbers of the type which, like continuous nondifferentiable
functions and the theorem on upper limits, was next investigated
s0 K. F. GAuss, "Demonstratio nova altera ..." and "Theorematis de resolubitate
•..", Comm. Soe. Reg. Sci. Gdttingen Rec., 3 (t814--15: publ. t816), cl. math., 107--134,
and t33--142 = Werke, 3, 31--56, and 57--64.
2~ A.-L. CAUCHY,Cours,43--44 and 46(>--462 = Oeuvres, (2) 3, 50--5 t and 378--380.
22 ]3. ]~OLZANO,Beweis, art. 12.
23 See especially the sections on convergence tests in chs. 6 and 9 of the Cours.
2~There is a distinction between ]3OLZANO'Sintroduction of an upper limit and
CAuc~Y'S "largest value of the expression ...", in that CAUCHY actually used the
Limes of a sequence (whose every neighbourhood contains members of the sequence),
while BOLZANOdefined the upper limit (which does not necessarily have this property) ;
but we cannot interpret this distinction as intentional in ]~OLZANO and CAUCHY'S
time and I do not know of any recorded awareness of it then. For a brief discussion
of the point, see P. E. ]3. JOURDAIN, "On the general theory of functions," fourn, rei.
ang. Math., 128 (1905), 169--210 (pp. t85--t88).
25 ]3. BOLZANO, Beweis, art. 8: see also art. t2.
26 Arch. Hist. Exact Sci,, Vol. 6
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 320 of 479.
378 I. GRATTAN-GUINNESS:
only by WEIERSTRASS and his followers.Is CAUCHY wrote just once on the real
number system: it was in the Cours d'Analyse, where he gave a superficial formal
exposition of the real number system. The initial stimulus for this work was
foundational questions concerning the representation of complex numbers; but
he took the development of the ideas well into BOLZANO'Sterritory, twice including
the remark that "when B is an irrational number, one can obtain it by
rational numbers with values which are brought nearer and nearer to it" 37 _
merely a remark on a property of the real numbers and not as a definition of the
irrational number in the sense of the later work, as has sometimes been thought.
Once again CAUCHYdid not fully appreciate the depth of BOLZANO'Sthought;
and yet it is clear from his partial success that he was aware of ]3OLZANO'Sideas,
rather than from his partial failure that he was ignorant of them. The striking
feature of this remark, as with his interpretation of continuity and his only
incomplete use of the upper limit, is that it is there at all, rather than that it appears
in a mutilated form.
3. The New Analysis
Thus we find a significant collection of unusual results in the two works: yet
there is a much stronger and more profound link between them, which cannot
be identified by means of precise quotations or references -- namely, a unity o/
approach. We have here a good example of the rule that the whole is greater than
the sum of the parts, for it is the homogeneity and general applicability of these
new ideas which is their most significant feature. The term "arithmeticisation of
analysis" is given to them, because they operate by means of arithmetical differences
and proofs within the analysis are based on the arithmetical manipulation
of them; but I do not favour this name, partly because it is identified
with the later WEIERSTRASSIANdevelopments of analysis but principally because
the arithmeticisation is only at the service of something more profound: the
theory of limit-avoidance.
When we speak of "introducing the concept of a limit" into analysis, we are
actually introducing limit-avoidance, where the limiting value is defined by the
property that the values in a sequence avoid that limit by an arbitrarily small
amount when the corresponding parameter (the index n for the sequence s~ of
nth partial sums, say, or the increment ~ in the difference (/(x +~)--/(x)) for
continuity) avoids its own limiting value (infinity and zero, in these examples).
The new analysis formed in ]3OLZANO'Spamphlet and developed in CAUCHY'S
text-books was nothing else than a complete reformulation of the whole of
analysis in limit-avoidance terms, terms which CAUCHYmade quite explicit in
the introduction to the Cours d'analyse:
"When the values successively attributed to a particular variable approach
indefinitely a fixed value, so as to finish by differing from it by as little as one
wishes, this latter is called the limit of all the others." 2s
2s These manuscripts were published in K. RYCHLIK(ed.), TheoriederreellenZahlen
im Bolzanos handschriitlichen Nachlasse (1962, Prague).
27A.-L. CAUCHY,Cours, 409 and 4t5 = Oeuvres, (2) 3, 337 and 341.
2s A.-L. CAUCHY,Cours, 4 ~-Oeuvres, (2) 3, 19.
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 321 of 479.
Bolzano, Cauchy and New Analysis 379
One important aspect of limit-avoidance is that it is independent of the continuum
of values over which the analysis is conducted. Limit-avoidance can be
developed whether an infinitesimal or non-infinitesimal field is being used: the
use of the WEIERSTRASSIANterm "arithmeticisation of analysis", applied to the
period when WEIERSTRASS excluded infinitesimals from analysis, has led us to
forget that its limit-avoiding character was shown also by the earlier period
instigated by BOLZANO,who used both types of continuum in his analysis, 29
and CAucrI¥, who practiced only infinitesimals throughout his mathematical
career. Since WEIERSTRASS'S time, we have held a fairly contemptuous view of the
infinitesimalists which I regard as unfair. A remarkable amount of pure and
applied analysis was developed from the time of NEWTON onwards with the aid
of infinitesimals; but there were important foundational difficulties involved in
their use, and in fact CAUCHYis a good example of them. These difficulties seem
to me to lie especially in the foundations of the calculus, which if we examine
from the point of view of limit-avoidance also reveal the attraction that infinitesimals
must have had to the founders of the algebraic calculus.
We make our point in the LEIBI~IZIAX notation, which not only became the
standard system but also contained a key to the difficulties that the infinitesimalists
faced. When we calculate the derivative by means of the definition
dxdY-=D~ "h~olim[ /-(x+h)--/(x) ' (2)
we may quite easily obtain the value of the derivative involved; but we are left
with the important foundational question of how that value is obtained in light
0
of the fact that the ratio on the right hand side of (2) becomes ~- when h = 0.
The virtue of infinitesimals, quantities which obeyed the law
a +h=a (3)
of addition to the "ordinary" numbers, was that, being non-zero they avoided
the limiting value and therefore the difficulty of 0~-; on the other hand, being
smaller than "any assignable quantity" (that is, any non-infinitesimal), they
effectively allowed the limit to be taken. This view was of course an inconsistent
one, but I think that it lay basically behind infinitesimalist reasoning and was
the source of its difficulties. The infinitesimal was either zero or non-zero,
according to the needs of the moment: thus it could be added to or withdrawn
from any quantity in an equation, with the presumed certainty of leaving the
mathematical situation described by that equation undisturbed. We may see
this as a double-interpretation for the infinitesimal -- a limit-avoiding interpretation
as a non-zero quantity, and what we may call by contrast a "limitachieving"
interpretation as an essentially zero quantity allowing the limit to
be taken. From this distinction there follows a corresponding double-inter-
29 In the Beweis ]3OLZANOdid not explicitly discuss the possible continua, and
seemed to have allowed the use of infinitesimals; but later in the year he published
another pamphlet, on Die drei Probleme der Recti/ication, der Complanation und die
Cubirung, ohne Betrachtung des unendlich Kleinen .... und ohne irgend eine nicht s~reng
erweisliche Voraussetzung gel6st; ... (t817, Prague)= Schri/ten, 5 (t948, Prague),
67--138.
26*
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 322 of 479.
380 I. GRATTAN-GUINNESS:
dy
pretation of dx' Let us take a specific example of a derivative, say for the
function
y = xL (3)
whose derivative
is calculated from
dy
dx = 3 x2 (4)
dy lim[ (X+h)8--x3 ]d. =Dr. • (5)
k~o t k
When h achieves its limiting value zero (4) gives us the value of the derivative,
dy
and so the denoting symbol ~ is in fact just a symbol and is not to be taken as
an arithmetical ratio "dy+dx". Thus it is not valid to multiply through (4) by
dx to obtain
dy = 3x2dx. (6)
(6) follows from (4) by turning from the limit-achieving interpretation of dd--~-Yxas
a whole symbol to its limit-avoiding interpretation, where it is the ratio" d'y+dx".
For if we avoid the limiting value by the non-zero infinitesimal quantity dx, then
we see from the right hand side of (5) that the situation for the increment dy
(=d(x~)) is given by
dy = 3x~dx + q,
where q is a second-order infinitesimal obeying the law
(7)
a+q=a (8)
,, dy in thisof addition to ordinary" or first-order infinitesimal quantities a. -d7
kind of situation, if we wish to consider it, could arise by dividing throughout (7)
to give:
dy q
a. - 3 x~ + d. ' (9)
a result of a [undamentally di//erent kind [rom (4). There is a difference between
the two far greater than the first order infinitesimal ~q~7.:we see a basic qualitative
dy appears in (4) as a limit-achieving symbol but in (9) as a limit-
~t9
difference, for dk-x
avoiding ratio. Further, the deduction of (9) from an infinitesimal equation (7)
is not necessary to the derivation of (4). For let us suppose that we change continua
so that in WEIERSTRASSIANstyle we reject the use of infinitesimals. Then
(4) and (5) still stand (with the limit now of course taken over the non-infinitesimal
field); but (7) and all its consequences, such as (9), disappear altogether
for (7) itself changes into tile identity
o=o, (1o)
whether or not it was true in the infinitesimal continuum.
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 323 of 479.
Bolzano, Cauchy and New Analysis 381
The ideas that I have presented here are essentially straightforward, and are
susceptible of considerable extension; but they are independent of the modern
interest in developing a consistent theory of infinitesimals. 8° They do not themselves
establish a consistent infinitesimalism but at least show that much can
be clarified in terms which could have been understood and developed in the
infinitesimalist period. Yet they were far from the considerations of the time: in
particular, CAUCHY'S treatment of the foundations of the calculus was as incoherent
and incompetent as any that were ever offered. In his Rdsumd des
le9ons ... sur le calcul in/initdsimal of 1823, the next instalment of his new analysis
after the Cours d'Analyse, he explicitly rejected LAGRANGE'S faith in
TAYLOR'Sseries, but he replaced it with an extraordinary theory of the derivative
which made simultaneous use of both LAGRANGE'Stheory of derived functions
/' (x),/"(x), ... and also of CARNOT'Stheory of differentials dx, ddx, ... : infinitesimals
not only achieved the limit in CAUCHY'Ssystem but they also avoided it,
at times by non-infinitesimal amounts, changing their role with every appearance
of new and usually unnecessary notation. 31 However, when CAUCHYcame to
integration he was wonderfully successful, laying out the whole basic structure
of the theory of the "CAucH¥ integral" (defined in terms of the area as the
limit of a sum) in a masterly display of the power of the new analysis of limit-
avoidance.
This is what the new analysis was: only in limit-avoidance terms can its full
power and subtlety be appreciated, and theorems such as the necessary and
sufficient condition for convergence in the diminishing of (s~+~- s,) -- where the
limit s is avoided altogether -- and BOLZANO'Stheorem on the existence of upper
limits, can be seen to their best advantage. Yet to understand BOLZANO and
CAUCH¥'S work we must look at the old as well as the new. What sort of analysis
had they replaced ?
4. The Old Analysis
We have referred earlier briefly to certain features of t8 th century analysis,
and it is appropriate now to make more detailed remarks about its character.
In speaking of the "old analysis", we are referring only to the subject immediately
prior to BOLZANOand CAUCHY'Swork; and we find that many of its features
were the result of problems in other areas of mathematics, especially in the
solution of difference and differential equations. Following the leadership of
EULER, his contemporaries (mainly D'ALEMBERT, DANIEL BERNOULLI and
LAGRANGE) and successors (mainly LAGRANGE, LAPLACE and MONGE) had
developed a wide range of solution methods. It is impossible to describe them all
in a sentence, but often they involved the construction of exact differentials
prior to integration to give functional solutions, or assumptions of particular
kinds of solution which led via the conditions of the problem to auxiliary equa-
30 See A. ROBINSON,Non-Standard Analysis (t966. Amsterdam); and also the
work initiated by C. SCHMEIDEN6: D. LAUGWITZ,"Eine Erweiterung tier Infinitesimalrechnung",
Math. Zeitschr., 69 (1958), 1--39.
31A.-L. CAucI-I¥, Rdsumd des lemons donndes ~ l'Ecole Royale Polytechnique sur le
calcul in]initdsimal. Tome premier (1823, Paris)= Oeuvres, (2) 4, 5--26t. No other
volumes were published: see here lecture 5.
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 324 of 479.
382 I. GRATTAI~-GuINNESS:
tions. The analytical techniques themselves -- which involved not only differentiation
and integration, but also summation and rearrangement of series
(especially power series), manipulations of algebraic expressions, the taking of
limiting cases (in moving from difference to differential equations for example),
and so on -- were normally used as required without consideration of their
validity. This is not intended as a criticism, but merely a general statement of
the situation: it led to an enormous range of results in pure and applied mathematics
which have remained important ever since. Further, there were cases when
questions of rigour and validity did arise, of which the most important was the
problem of the motion of the vibrating string; 32 but ill general the situation at
the beginning of the 19~ century was that not only were such considerations
relatively limited but the techniques themselves were susceptible of, and received,
plenty of further development without concern for the rigour involved. This is a
matter of great importance when considering the "new analysis" of BOLZANO
and CAUCHY. Their new foundations, based on limit avoidance, certainly swept
away the old foundations, founded largely on faith in the formal techniques; but
it would be a mistake of posterior wisdom to assume that old foundations had
been in a serious and comprehensive state of decay and were recognised as such
by those who were using them. Historians of science seem to be only too ready
to make assumptions of this kind when considering "revolutions" in science:
they also tend to identify anticipations of a new system in the old one with that
new system instead of what they probably were, something else in the old system
which was quite different and also interesting. The historiographical point here
is the danger of determinism; that because a body of knowledge developed in a
particular way, then it must be viewed historically as having been capable of
developing only that way, certainly from the intellectual point of view and perhaps
even chronologically. Yet in fact any situation is always open to a variety
of future developments: we must not allow the intermediate historical processes
that actually happened to distort our vision of the situation from which they
started.
I have already claimed that the new analysis replaced an old analysis which
does not seem to have needed such a radical replacement: from the point of view
of the BOLZANo-CAucHY question, it follows that it is all the more surprising that
exactly the same type of replacement began to emerge twice within four years.
But we must consider also the anticipations of the new system ill the old one.
The "new analysis" laid great stress on the rigour of processes: did no "old
analyst" try to do the same ? Yes, certainly, but not ill any way resembling the
comprehensive and homogeneous character of the new method: they had other
ideas which were quite different and also interesting. EULER tried hard, though
with little practical success, to produce a consistent infinitesimalism in his
"reckoning with zeros", including consideration of different orders of infinitesimal.
D'ALEMBERT tended to distrust infinitesimals altogether, while LAGRANGE
tried to avoid all limiting processes by defining the derivatives of a function in
82For a discussion of foundational questions in the light of this problem, see my
Foundations, Ch. t ; and for an extended account of the solution of differential equations
ill this period, see C. TRUESDELL,The rational mechanics o[ flexible or elastic
bodies 1638--1788, L. Euleri Opera Omnia, (t) 11, pt. 2 (196o, Zurich).
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 325 of 479.
Bolzano, Cauchy and New Analysis 383
terms of the coefficients of its expansion as a TAYLol~series. This was "limitavoidance"
of a completely different and considerably less successful kind, and
it won few supporters. One of them, however, was ARBOGAST, WhO tried towards
the end of the century to reduce the number of distinctions between types of
function to a group based on analytical rather than algebraic or mechanical
considerations. L'I-IuILIER offered a thoughtful essay on the taking of limits:
I am sure that CAISCHYread it, for he always used the notation "lim" for a
limiting value which L'HuILIER introduced there. But I doubt if he learnt much
more from it, for the results obtained are severely limited, being concentrated on
the derivative and often providing no more than a re-writing of known ideas.
L'HIJILIEI~ also criticised (with iustice) EULEI~'Suse of infinitesimals, and CARNOT
took it further into a profound essay on orders of the infinitely small and the
interpretation of the LEIBNIZIAN notations as infinitesimals. But perhaps the
best example, especially from the point of view of anticipations of BOLZAI~Oand
CAUCHY, iS LACROIX, the principal text-book writer of the day. He was not an
important creative mathematician, but he was capable of some measure of
appreciation of contemporary work and he read exhaustively among the earlier
literature. I referred earlier to his understanding of convergence of series as a
general problem, which he learnt from D'ALEMBERT'S vague warnings against
divergent series in the t760's: he also gave in t806 a formulation of continuity
vaguely similar to that of BOLZANOand CAucI~Y.3~Thus we may say that LACROlX
anticipated them if we wish; yet it would be more misleading than illuminating
to do so, not least to the understanding of LACROIX'S results. For one cannot
find in LACROIX'Swritings the general aim that BOLZANOand CAI~CI~Yachieved,
not even in the new editions of his works that continued to appear after CAUCI-IY'S
text-books were published.
What would have happened if CAUCH¥ had not read BOLZANO? Without
doubt, foundational questions would have received discussion, but it seems to
me most unlikely that the radical reform that in fact happened would have taken
place: rather only parts of that theory would probably have emerged, especially
in the convergence of series and the integral as the limit of a sum, while the rest,
apparently sound enough, would have received well-meaning but limited examination.
But in order to put the old and the new analyses into better perspective
we must describe some of the fundamental problems which were current before
BOLZANO'S paper; and at the same time we shall pass on to further aspects of
the CAUCHY-BoLZANOquestion, aspects which involve not only analysis itself
but also the Paris in which CAUCHYwas working and the way in which his
mathematical genius was inspired.
~ S. ~'. LACROIX,Traitd dldmentaire du calcul intdgral (2nd edition; 1806, Paris):
see art. 60. The other works to which we referred explicitly were L. F. A. ARBOGAST,
Mdmoire sur la nature des/onctions arbitraires qui ent~ent dam les intdgvales des dquations
aux diHdrentielles partielles (t 791, St. Petersburg) : S. L'HuILIER, Exposilion dldmentaire
des principes des calculs supdrieures (1786, Berlin), esp. chs. I and aI ; and L. N. M.
CARNOT, Re/lexions suv la mdtaphysique du calcul in/initdsimale (1st edition: 1797,
Paris. 2nd edition: 1813, Paris). On EULER'S and LAGRANGE'Sviews on analysis, see
A. P. JUSCI-IKEWITSCI-I,'" Euler and Lagrange ~ber die Grundlagen der Analysis," Sammelband
der zu Ehren des 250. Geburtstages Leonhard Eulers (ed. K. SCI-IRODER:1959,
Berlin), 224--244; and on all these and other developments, my Foundations, chs. 1and 3.
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 326 of 479.
384 I. GRATTAN-GUINNESS:
8. Cauchy's Originality as a Mathematician
If CAUCHYcame to his new ideas independently of BOLZANO,then he perceived
a completely novel approach to analysis and detected its superiority over known
techniques which themselves were not lacking in power or generality. This kind
of achievement is characteristic of certain mathematicians: it reflects their sensitive
"intuition for problems", their ability to see far beyond contemporary
work into totally new ways of solving current problems, or even of forming new
problems of which others were hardly aware. GAuss is a prime example of such
a thinker, with his notebooks already filled with the seeds of most t9 th century
mathematics within its first decade: ]3OLZANOshows this ability, too, and to
the extent that he was in fact extremely limited in ability at "orthodox" developments
of current and popular methods. Thus in i8t6, for example, before the
flood of his own new thinking, he published a treatise on the binomial series in
the style of the old analysis which is really quite remarkably uninteresting)4 But
CAUCHYis a good example of originality of another kind, lacking such sensitivity
and feeling for new problems but, when stimulated by the achievements or
especially lack o/success in some contemporary work, would expand the accomplished
fragments into immense generalisations and extensions within the same
field of research. His monument in mathematics in his theory of functions of a
complex variable and their integration, one of the great achievements of all 19th
century mathematics. Its origins are to be found in a large paper of t814 (his
25th year) on the validity of using complex numbers in the evaluation of definite
integrals. The technique had been used for decades from time to time, without
much consideration of its validity: in particular, in June 18t4, LEGENDI~E
published an instalment of the second volume of his Exercises du calcul intdgral,
a work containing various methods of evaluating definite integrals whose main
aim was towards the development of his theory of elliptic integrals. 35This instalment
concerned itself chiefly with integrals whose integrands were the product
of rational and trigonometric functions, and it provided the spark for CAUCI~Y'S
fire, for from LEGENDRE'Swork CAUCHYcame to the following generalised problem
concerning the evaluation of definite integrals: what are sufficient conditions for
the validity of using complex variables in such evaluations ? His solution was the
equality of two mixed partial differentials:
oxOy l(z)dz-- Oy~x /(z)dz, (1t)
where z is a complex function of x and y;
z=h(x, y) +ik(x, y) (12)
and thus
/(z) =~,(x, y)+iv(x, y). (13)
From this fruitful equation (t 1) stemmed a variety of general theorems (including
the "Cauchy-Riemann equations") and thence hosts of particular integrals,
34]~. :BOLZANO,Der binomische Lehrsatz and als Folgerung aus ihm der Polynomische
und die Reihen.... (1816, Prague). The most interesting section is on pp. 27--40.
85A.-M. LEGENDRE,Exercises du calcul inl@ral sur divers ordres des hombres transcendantes
el sur les quadratures (3 vols: 1811--t 7, Paris).
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 327 of 479.
]3olzano, Cauchy and New Analysis 385
including the evaluation of some of LEGENDRE'S. CAUCHYpresented his paper in
August (18t4) to the Institut de France, and LEGENDREwas one of its examiners:
he rightly praised its many important new results, but had a most interesting and
important dispute with CAUCHYover the evaluation of
oo
f xcosax dx (t4)
• si~b~ t + x~"
0
a
Put in modern terms, if we regard the integral as a function of ~- then it has a
discontinuity of magnitude ~ at the odd multiple values of its argument. CAVCHY
had by separate equations evaluated the left- and right-hand limiting values of
a a
the function for ~- < t ands- > t" but in the t814 instalment of his book LEGE~DI~E
had used a power series expansion method on a generalisation of (t4) to
produce in a limiting case the arithmetic mean of CAUCHY'Stwo evaluations for
a
b --t, and he could not understand that this new type of algebraic expression
- - the integral representation -- could in fact give a discontinuous function.
CAUCHY produced a spurious piece of infinitesimal reasoning to resolve the
situation to LEGENDRE'S satisfaction; 36 but it must have shown him that there
were foundational questions in real variable analysis apart from the use of
complex numbers with which he would have to deal.
Let us return, however, to the question of CAUCHY'Stype of mathematical
inspiration. We see in this episode that CAUCHYwas directly stimulated by
LEGENDRE'S attempts at integral evaluation to work in exactly the same field,
rather than to intuite from it some more general and abstract kind of problem
concerned with the use of functions of a complex variable. In the t814 paper for
example, the theory of singularities and residues which he was to produce in
later years was given in a real variable integral form, which we may write as:
Y~ ~¢2 X~ Y2
ff o, ff , f~7-x dxdy--. -~x dydx= . ES(X +p,Y +q) (t5)
Yl J('l X1 Yl 0
--S(X +p, Y--q) -- S(X--p, Y +q) + S(X--p, Y--q)]dp,
os
where ~ has an infinity at the point (X, Y) inside the rectangle bounded by
the sides, x = x1, x = x2, y = yl and y = Y2.3~ His later fine achievements in the
new analysis with the theory of integration may be traced in large part to the
issues involved in the profound result (15).
In the following year of 18t5 CAVC~IYhad another large paper ready, this
time on the propagation of water-waves.3s Complex variables were again present,
3nFor a full account of this episode see my Foundations, ch. 2.
a7CAucH¥'s paper was "M6moire sur les int6grales d6finies", Mdm. prds. A cad.
Roy. Sci. div. say., (2) 1 (1827), 60t--799 =Oeuvres, (1) 1, 31 9---506. LEaENI)RE'S
evaluation of tile integral (t4) is in his as, 2, t24.
~sA.-L. CAucI~¥, "Ttl~orie de la propagation des ondes ~ la surface d'nn fluide
pesant d'une profondeur ind6finie," Mdm. prds. Acad. Roy. Sci. div. say., (2) 1 (1827),
3--3t2 = Oeuvres, (1) 1, 4--318.
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 328 of 479.
386 I. GRATTAN-GUINNESS:
as they were to be in all of his mathematical output; and integrals were also to
be found, for the prominent new feature here was tile use of integral methods to
solve linear partial differential equations (and thus to use again the integral
representation of a function). The inspiration in this case is not so easy to trace,
as it is impossible to say how much of FOURIER'S then still unpublished work on
heat diffusion he had seen; but he knew of POlSSON'S (lesser) work in the same
field, and doubtless he was aware of some results of LAPLACE which we shall
discuss later. At all events, in 18t 7 his further researches brought him to "Fourier's
Integral Theorem":
oo oo
/(x) =
0 0
in a short paper whose rushed and excited tone suggests that he had really found
the result independently of FOURIER.39 FOURIER acquainted him with his own
prior discovery of the theorem, and then CAUCHY certainly did read his manuscripts:
not only did he publish an acknowledgement in 1818,40 but in all his
later work on integral solutions to partial differential equations there was a new
confidence and dexterity, and again -- extensions and generalisations (to multiple
integral solutions, and so on) of what FOURIER had already done. 41
And then we come to t 82t and tile Cours d'Anatyse: large numbers of theorems
on all aspects of real and complex variable function theory, based on the ideas
which we listed in our section 2. From where had the inspiration come this time ?
From within CAucI~= himself ? Perhaps; but it is so utterly untypical of his kind
of achievement whereas under the hypothesis of his prior reading of BOLZANOit
is SO perfect an example of it, that it seems difficult not to accept the latter
possibility. Perhaps I can best illustrate the force of this point by describing my
own researches into the development of the foundations of analysis during this
period. I had started naturally enough with CAUCHY'SCours d'Analyse and his
other contributions to analysis, and in the course of reading other of his writings
his need for an initial external stimulus to his genius had become clear to me.
Thus I wanted to find the source of the new ideas of the Cours d'Analyse, and so
I made a special search of all of CAOCHY'Swork written prior to 182t. I found
many important things, especially the 18t 4 integrals paper and the disagreement
over (t4) with LECENDRE, and the affair of 1817 over FOU~IER'S Integral Theorem
(16): there was clearly plenty of motivation for CAUCHYto try to improve
analytical techniques. But of the new ideas that were to achieve that aim -- of
them, to my great surprise, I could find nothing. Only later did I follow up my
knowledge that BOLZANOhad done "something" in analysis which no-one had
read (or so I thought) ; and I can remember quite clearly the extraordinary effect
of reading BOLZANO'S 18t7 pamphlet and seeing the Cours d'Analyse emerging
from its pages. I then re-read the Cours d'Analyse and found the fine details of
3. A.-L. CAUCHY,"Sur une loi de r6ciprocit6 qui existe entre certaines fonctions",
Bull. Sci. Soc. Philom. Paris (1817), 121--124 = Oeuvres, (2) 2, 223--227.
~0A.-L. CAucI~Y, "Second note sur les fonctions r6ciproques", Bull. Sci. Soc.
Philom. Paris (t8t8), 178--181 = Oeuvres, (2) 2, 228--232.
4, For discussion of these developments, see my Fourier, chs. 21 and 22; and
BURKHARDT 3a, chs. 8--11 passim.
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 329 of 479.
Bolzano, Cauchy and New Analysis 387
correspondence; but more than that, I could see CAlsCI~Y'Smind at work in its
own individual way, taking the fragments of BOLZANO'Sthought as he had taken
LEGENDRE'S morsels and FOURIER'S substantial achievements earlier, and producing
from them whole new systems of mathematical thought.
But if CAUCHYowed so much to BOLZANO,why did he not acknowledge him ?
To answer this question, we move more fully into the social situation of the time:
to Paris, the centre of the mathematical world.
6. The State of Parisian Mathematics
Almost every mathematician of note at this time either lived in or at ]east
visited Paris. One consequence of this galaxy of brilliance was that a state of
intense rivalry and sometimes bitter enmity existed almost continuously in the
Parisian scientific circles. Everybody was affected by it, although some less
than others; and the reasons were not always purely scientific. There were deep
and passionate political or religious disagreements, too, heightened by the Napoleonic
era and its violent end and brief resurrection in the mid-t810's. These
rivalries pose an exciting and difficult problem for the historian of the period,
for their detection and description calls for the most careful reading of even the
finest point in the most obscure paper, as well as reading between the lines of
all the scientific literature of the time. Very little work has been done on these
rivalries: indeed, most historians have failed to notice them altogether. 4~ But
perhaps I can give some idea of how they affected the situation and bore especially
upon CAUCHYand his Cours d'Analyse by describing two of the most important
controversies of the time -- as fully as I have been able to disclose them.
We have mentioned FOURIER'S name several times, and the first controversy
involved his work on heat diffusion. Like GAUSS and BOLZANO,he also had a
strong intuition for new problems, and seemingly from about 1802 he began
work on the then novel study of the mathematical description of the diffusion
of heat in continuous bodies. His early work on the problem proceeded by means
of a discrete n-body model, and though he achieved considerable mathematical
success a small hut vital error in the model itself brought failure to his efforts to
obtain a solution for the corresponding continuous bodies by taking n to infinity.
Then he had a slight CAocI~v-like inspiration from a small paper of 1804 by BlOT
on the propagation of heat in a bar 43 to start again by forming the partial differential
equation directly, and in the brief periods of leisure allowed him in the
next three years from his duties as Prefect of Is~re at Grenoble and from his
Egyptological researches he created a genuine revolution of his own: a revolution
in mathematical physics, which he took beyond the realm of NEWTONIANmechanics
into a new physical territory of heat diffusion, with its own equations and
physical constants and a fresh range of solution methods based on the use of
linear equations, the method of separation of variables (then mainly used in solving
42An exception is YI. ]3URKI-IARDT33" for scattered remarks, see ch. 8 pctssim.
See also my Foundations, esp. chs. 2--5; and Fourier, esp. chs. 21 and 22.
~3 j. B. BLOT, "M6moire sur la propagation de la chaleur," Bibl. Brit., 27
(1804), 3t0--329 = fourn. Mines, 17 (t804), 203--224. FOURIERnever acknowledged
BLOT'Spaper]
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 330 of 479.
388 I. GRATTAN-GUINNESS :
ordinary differential equations) and the superposition of special solutions. FOURIER
series were only one consequence of these new methods: another was his
creation of the basic theory of the misnamed "Bessel functions", and indeed it
was there that he showed his mathematical technique at its greatest. By 1807
he had progressed far; but he was unable to solve the problem of heat diffusiou
in an in/inite continuous body, and so he wrote up his theoretical achievements
and experimental results in a large monograph submitted to the Institut de
France in December ~4. LAGRANGEand LAPLACEwere the most important of the
examiners: for various conceptual reasons LAGRANGE was opposed to the whole
approach based on separation of variables, but LAPLACEwas very impressed and
began to take great interest in FOURIER'S work. So a struggle began over the
reception of FOURIER'S paper, with LAPLACE, FOURIER and MONGE (another
examiner, and personally close to FOURIER) in support, and opposition from
LAGRANGEand -- POISSON.
We must consider POISSON for a moment, for in him more than in any other
single person lies the key to the Parisian mathematical rivalries. He graduated
brilliantly from the Ecole Polytechnique in t803, and to the aging grand masters
of Parisian mathematics -- LAGRANGE, LAPLACE, LEGENDRE and MONGE -- he
must have seemed to be the only heir to their crown: FOURIER was so occupied
with administrative work at Grenoble that he could not be expected to be
achieving substantial mathematical work, while CAUCHY was still only in his
early teens. So Polssox was placed in a position of special favour from the
beginning of his career which he exploited to the full, especially by means of
influential positions on Parisian scientific journals; but over the next twenty
years he gradually but steadily lost favour and reputation to FOURIER and then
CAUCHY as they emerged and surpassed him in the quality of their work. The
t807 paper of FOURIER was crucial in this development. By 1805 or 1806 Polssox
was already aware of some of FOURIER'S results and the type of solution that he
was trying to develop: he replied not only by applying to FOURIER'S diffusion
equation in t806 the ideas of LAGRANGE and LAPLACE on solutions of partial
differential equations using power series of functions, 45 but also by publishing a
denigrating five-page review of FOURIER'S monograph in 1808 in a journal of
which he was mathematical editor. 4° However, LAPLACE, acting in his typical
political way, maintained his interest in POlSSON (and also in BLOT) while
gradually changing his interests towards FOURIER'S methods and results. In t809
he published a miscellany on analysis which -- without reference to FOURIER --
just happened to contain a treatment of the diffusion equation with initial condi~4For
the references of this manuscript, see 12; and for a detailed analysis of its
contents, see my "Joseph Fourier and the revolution in mathematical physics",
Journ. Inst. Maths. Applics., 5 (t 969), 230--253. Much new information on FOURIER'S
life and Prefectural responsibilities is contained in my Fourier, ch. t.
4~ S. D. POISSON, "M6moire snr les solutions particuli~res des 6quations diff6rentielles
et des 6quations aux diff6rences", Journ. Ec. Polyt., call. 13, 6 (1806), 60--116
(pp. 109--t11).
46 S. I). POlSSON,"M6moire sur la propagation de la chaleur darts les corps solides",
Nouv. Bull. Soc. Philom. Paris, 1 (1808), t t2--t t6 ~ FOURIER'SOeuvres, 2, 2t 3--221.
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 331 of 479.
]3olzano, Cauchy and New Analysis 389
tions over an infinite interval. His solution
oo
0
brought into mathematics a result which later was developed as the "Laplace
transform"; it may well have been CAUCH¥'S inspiration to try integral solutions
to partial differential equations. 4~ It was certainly FOURIER'S inspiration, for it
showed FOURIER that an integral, rather than a series, solution was applicable in
the case of an infinite interval and it led him to "'Fourier integrals" and thus to
his integral theorem (t6). Meanwhile, POlSSON had been opposing FOURIER'S
solution method in favour of functional solutions by means of indirect references
in the context of the vibration of elastic surfaces; 48 but FOURIER and his supporters
eventually managed to secure a prize problem for heat diffusion in the
Institut de France for January, 1812. To the revision of the manuscript of 1807
FOURIER added a new section on FOURIER integrals, and also two more new parts
on physical aspects of heat which were inspired by discussions with LAPLACE.
He won the prize, but the criticisms of LAGRANGE in the examiners' report hurt
him for the rest of his life:
"... This work contains the true differential equations of the transmission of
heat, both in the interior of the bodies and at their surface, and the novelty of
the purpose adjoined to its importance has determined the class [of the Institut]
to crown this work, observing, however, that the manner of arriving at its
equations is not free from difficulties and its analysis of integration still leaves
something to be desired, both relative to its generality and on the side of rigour. "4°
LAGRANGE died in 1813; but publication of this second paper was no more
likely than its predecessor and so FOURIER wrote his book on heat diffusion as
the third version of his work. It did not appear until 1822, 5o having been delayed
partly by FOURIER'S own difficulties in developing the physical aspects of heat
(which he eventually omitted and promised for a sequel which was never written) ;
and the t812 prize paper did not appear until still later. 51 By this time FOURIER
47 p. S. LAPLACE,"M6moire sur divers points d'analyse", Journ. Ec. Polyt., cab. t 5,
8 (1809), 229---265 (pp. 235--244) = Oeuvres, 14, 178--214 (pp. 184--193).
4s See especially the preamble to a prize problem on this topic in Hist. cl. sci.
math. phys. Inst. Fr. (1808: publ. 1809), 235--24t. Obviously written by POlSSON,
it extols the virtues of functional solutions to the wave equations -- in implied contrast
to FOURIER series solutions which were then available. In controversial circumstances
(described in my Fourier, ch. 2t), POlSSON read his own paper on the subject in 1814,
which was published as "M6moire sur les surfaces elastiques", Mdm. cL sci. math.
phys. Inst. Fr., (t812), pt. 2 (publ. t8t6), 167--225.
49 Published in FOURIER'S Oeuvres, 1, vii--viii. The manuscript is kept in the
Archives of the Acaddmie des Sciences, Paris.
FOURIER never allied himself closely to LAPLACE, and gave no acknowledgement
to LAPLACEin the prize paper. It may be that LAORANGE'Scontinued general opposition
was supplemented by LAPLACIAN annoyance: the remarkable story of the relations
between LAPLACEand FOURIER from 1807 until the 1820's is described in my Fourier,
chs. 21 and 22.
so j. ]3. J. FOURIER, Thdorie analytique de la chaleur (1822, Paris) = Oeuvres, 1.
51 j. ]3. J. FOtlRIER, "Th6orie du mouvement de la chaleur dans les corps solides",
Mdm. Acad. Roy. Sci., 4 (t819--20: publ. 1824), t85--555; and 5 (1821--22: publ.
t826), 153--246 = Oeuvres, 2, 3--94.
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 332 of 479.
390 I. GRATTAN-GuINNESS :
had risen to a strong political position, having been appointed sdcretaire perpdtuel
of the Acaddmie des Sciences in 182t; and then there developed the second of
our major controversies, which directly involved CAUCHY'SCours d'Analyse --
the convergence problem of FOURIERseries.
FOURIER series contain many of the problems which we tackle by means of
the new analysis, but we have not yet described any of FOURIER'S work in that
field. The reason is that, although he understood all the basic analytical problems
-- convergence, the possibility of discontinuous functions, the integral as an
area -- before both BOLZANOand CAUCHYhad begun their work, he was not
strongly attracted to pure analysis as a study and so did not develop his own
understanding to the extent of that which he was capable.~2 Doubtless CAUC~IY
was aware of this fact, for in the Cours d'Analyse he put the following theorem:
"When the different terms of the series [~ u~ are functions of the
P ~
same
[--r=l J
variable x, continuous with respect to that variable in the vicinity of a particular
value for which the series is convergent, the sum of the series is also a continuous
function of x in the vicinity of that particular value."sa
The theorem is remarkable for its falsehood: it was known in its day to be
false, and indeed CAUCHYknew it was refuted when he put it in his book. But to
find the reasons why it was included, we must examine the type of counterexamples
which were then known. They were in fact FOURIERseries:
/(x) = ½ao + ~. (a, cos rx +b, sin rx), (18)
r=l
where
2v~
Ifao=~- l(u)du,
+~
a, = ~ /(u) cos rudu,
+=
I f /(u) sinrudu,b,----
(19)
r = 1, 2.... (20)
r = 1, 2..... (2t)
The trigonometric functions are continuous, and so the series on the fight hand
side of (18) is covered by CAUCHY'Stheorem: thus if [(x) is discontinuous, tile
series cannot be convergent to it. But FOURIER had produced several series of
discontinuous functions, and had shown by direct consideration of their nth
partial sums that they were convergent; and since t8t 5 POlSSOXhad found that
he had had to abandon his belief in functional and power series solutions in favour
of FOURIERseries solutions, and he had found similar examples also. So what was
CAUCHY'Spurpose in stating his theorem ? There was of course an intellectual
aspect to it, for CAUCHYdid have a proof: suffice it to say for now that the
52 In the 1807 manuscript 1~, see arts. 42--43, 64--74: in the 1811 paper ~1, see
part 1, 269--273 and 304--316: in the book 50 (mostly written by 1815), see arts.
177--I 79 and 222--229.
5a A.-L. CAUCHY, Cours, 131--t32 = Oeuvres, (2) 3, 120.
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 333 of 479.
t3olzano, Cauchy and New Analysis 39t
distinctions between modes of uniform and non-uniform convergence which
resolve the difficulty were not noticed by anybody until the 1840's, that CAUCHY'S
theorem had some role to play in their development, and that shortly afterwards,
in his last years, he wrote a pathetic paper of his own on the subject
presenting the same type of idea without any reference to recent work. 5~ But
on the personal side, there was a message to FOURIER and POlSSON between
the lines of his theorem: "your trigonometric series may be very interesting, but
do you have a general convergence proof for them ? Do your series not affront
the results of the new analysis ?"
The later developments of this rivalry read almost like a novel. 55 Briefly,
POISSON had already published a general proof in 1820 based on rather crude
manipulations of the "Poisson integral"
+~
f (1__p2)I(~)1 -- 2p cos (x -- ~) +p2 d ~ ; 56 (22)
but, while he never abandoned it, it impressed few of his contemporaries. If
CAUCHY knew it when he wrote the Cours d'Analyse, then his theorem was
already a comment on it; but in a short paper of 1826 on the convergence problem
he certainly showed his awareness of it. For he began that paper with a version
of POISSON'S convergence proof based on (22) to produce the FOURIER series (18);
and then he remarked:
"The preceding series [(t8)] can be very usefully employed in many circumstances.
But it is important to show its convergence. ''5~
CAocltY'S own proof followed; and while it was of considerably better mathematical
calibre than POISSON'S, it contained one vital flaw -- the false assumption
that if u~--*v, as n tends to infinity, then 2 ur and 2 v~ converge together. That
r=l r=l
this assumption is false was pointed out in a paper of t829 on the convergence
problem by the young DIRICHLET. In this masterpiece DIRICHLET showed the
power of the new analysis in producing the famous sufficient "Dirichlet conditions"
for the convergence of a FOURIER series to its function: that it may have
5~A.-L. CAUCHY, "Note sur les s6ries convergentes ...", C. R. Acad. Roy. Sci.,
36 (1853), 454--459 = Oeuvres, (1) 12, 30--36. For a detailed account of the introduction
of modes of convergence, see my Foundations, ch. 6. The relevance of CAUCHY'S
theorem in the Cours is especially connected with one paper important in the development
of modes of convergence: P.L. SEIDlgL'S "Note tiber eine Eigenschaff der
t~eihen, welche discontinuirlichen Functionen darstellen", Abh. Akad. Wiss. Mi~nich,
7 (t 847--49), math:phys. KI., 381--393. This paper (by a pupil of DIRICHLET!) dealt
explicitly with that theorem in the light of discontinuous FOURIER series, and is more
than likely to have been the (unmentioned) inspiration of CAUCHY'Spaper of five
years later.
55A detailed description is given in my Foundations, eh. 5.
56 S.-D. PolssoN, "M6moire sur la mani~re d'exprimer les fonctions ..." Journ.
Ec. Polyt., cah. 16, 11 (1820), 417--489 (pp. 422--424).
57A.-L. CAUCHY,"M6moire sur les d6veloppements des fonctions en s6ries p6riodiques",
Mdm. Acad. Roy. Sci., 6 (1823: puN. 1827), 603--6t2 (p. 606) = Oeuvres, (1) 2,
12--t9 (p. 14).
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 334 of 479.
392 I. GRATTAN-GuINNESS:
a finite number of discontinuities and turning values in an otherwise continuous
and monotonic course.5s And his proof was a development of a sketched argument
in FOURIER'S book of 1822: he took an alternative form to FOURIER'S for the nth
partial sum of the series and applied to it a precise version of the proof that
FOURIER had outlined) 9 Yet there was more than mathematics in DIRICHLET'S
paper, too, for during his visit to Paris in 1826 he formed such a close personal
attachment to FOURIER that his work on the convergence problem was a personal
homage in FOORIER'S last years. However, he formed no close relationship
to CAUCHY: as well as pointing out the error in CAUCHY'S 1826 proof and finding
general convergence conditions which, in allowing discontinuities in the function
refuted CAOCHV'S 1821 theorem, he reported in his paper a presumably verbal
remark of CAUCHY'S on his t826 paper that:
"The author of this work himself acknowledges that his proof is defective for
certain functions for which, however, convergence is incontestable. ''8°
One can find CAUCHY'S reaction to DIRICHLET'S results if one looks carefully:
in 1833 CAUCHY published in French at Turin a summarised version of all his
t820's text-books (based on the lectures that he had been giving there ill Italian),
and was careful to include his theorem from the Cours d'Analyse word for
word. 61
And so we return to ]3OLZANOand his Prague pamphlet. Is it any wonder
that in an atmosphere like this CAUCHY made no acknowledgement to him ?
References were often not made (apart from honorific citations of the great
names of the past), either between members of the Paris cliques or outside them;
and even then they were some times double-meant. For example, when CAUCHY
finally managed to get his t8t4 paper on definite integrals and the 1815 paper
on water-waves published in 1827 he introduced in 1825 some extra notes and
footnotes to the texts and introduced fawning references to the powerful secrdtaire
b
perpdtuel (FouRIER), especially with regard to his invention of the notation f to
represent the definite integral; he also inserted attacks on the declining POlSSON.62
But there seems to me to be more specific reasons for CAUCtIY'S failure to acknowledge
]3OLZANO.He had appreciated the qualities of BOLZANO'Swork, and I think
that he deliberately excluded references to an obviously obscure work in order
to prevent its acquaintance by rivals such as Polsso~ and FOURIER (and perhaps
others such as A~IP~RE). This is perhaps not a nice remark to make about CAUCHY
but it is all too justified, and indeed CAUCHV'S personality is worth our separate
attention.
5s p. G. LEJEUNE-DIRICHLET, "Sur la convergence des s6ries trigonometriques ...",
Journ. rei. ang. Math., 4 (t829), t57--169 ----Werke, 1, 117--132. DIRICI-ILET'Scontributions
to the new analysis in this and other works (described in my Foundations,
ch. 5), surpass in my view any other of CAucnY'S successors -- including ABEL.
S9See J. B. J. FOURIER5°, esp. art. 423.
s0 See P. G. LEJEUNE-DIRICHLET 5s, 157 = Werke, 1, 119.
61 A.-L. CAUCHY,RdsumdsAnalytiques (t833, Turin), 46 = Oeuvres, (2) 10, 55--56.
6, For CAUCHY'Sacknowledgements to FOURIER, see 37, 623 = Oeuvres, (1) 1, 340;
and 30, 194 (omitted from Oeuvres, (1) 1, t97). For the attacks in 3s on PolssoN, see
pp. 187--188 = Oeuvres, (1) 1, 189--191.
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 335 of 479.
Bolzano, Cauchy and New Analysis 393
7. Cauchy's Personality
If CAUCHYwas one of the greatest mathematicians of his time, he was one
of the most unpleasant personalities of all time: a fanatic for Catholic and Bourbonist
causes to the point of perversion, he had to prove his superiority at all
times over even the weakest of his contemporaries and to publish a virtually
continuous stream of work. He also wrote articles on education, the rights of the
Catholic and Bourbon causes, and the reform of criminals, to supplement his
mathematical output; but he never helped and even at times hindered his
younger colleagues in their careers and work. A good example of this concerns a
young man who wrote the following of him:
"Cauchy is a fool, and one can't find any understanding with him, although
he is the mathematician who at this time knows how mathematics should be
treated ... he is extremely catholic and bigoted .... "
The writer was ABEL, in a letter sent to his friend HOLMBOE when, like
DIRICHLET, he visited Paris in October, t826. ea Poor ABEL: he cannot have
known how right he was, just as he did not understand the Parisian political
situation. While in Berlin during the previous January, he had written a paper
on convergence tests and their application to the binomial series which made
important use of the new analysis: he had also spotted the weakness in CAUCHY'S
theorem of the Cours d'Analyse and made the first public mention of the point
in a footnote to the paper. 84 Later in the same letter to HOLMBOEhe remarked:
"I have worked out a large paper on a certain class of transcendental functions
to present to the Institut. I am doing it on Monday. I showed it to Cauchy: but
he would hardly glance at it. And I can say without bragging that it is good.
I am very curious to hear the judgement of the Institut .... "~
This was the paper which ushered in the transformation of LEGENDRE'S theory
of elliptic integrals into his own theory of elliptic functions; and the story of its
fate is only too characteristic of Parisian science and of CAUCtIY. CAUCIIY and
LEGENDRE were the examiners: CAUCHYtook it and, perhaps because of ABEL'S
footnote against his theorem, ignored it entirely: only after ABEL'S death in
t829 did he fulfil a request to return it to the Acaddmie des Sciences. It was
finally published in 184t, when the manuscript vanished in sensational circumstances,
to be rediscovered only in the t950's. This story is well-known ;~ however,
there is one aspect of it which has been little remarked upon but which shows
the depths to which CAUCH¥ could sink. When ABEL'S paper was in the press
another Norwegian mathematician presented a paper to the Acaddmie des Sciences
6s Niels Hendrik Abel. Mdmorial publid h l'occasion du centenaire de sa naissance
(t 902, Christiana), Correspondance d'Abel .... 135 pp. (pp. 45 and 46) = Texte original
des lettres .... 61 pp. (pp. 41 and 42). Also in Oeuvres (ed. L. SYLOW& S. LIE), 2, 259.
6~lxT.H. ABEL, "Untersuchungen fiber die Reihe ...", Journ. rei. ang. Math., 1
(t826), 311--329 (p. 316) =Oeuvres (ed. B. HOLMBOE), 1, 66--92 (p. 71) =Oeuvres
(ed. L. SYLOW& S. LIE), 1, 219--250 (p. 225).
,5 In addition to the references in ,3, we may add for this passage ABEL'S Oeuvres
(ed. B. HOLMBOE), 2, 269--270.
~ For a detailed account of this affair, see O. ORE,Niels Hendrik Abel -- mathematician
extraordinary (t957, Minneapolis), 246--261.
27 Arch. Hist. Exact Sci., Vol. 6
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 336 of 479.
394 I. GI~A:rTAN-GuINNESS:
on elliptic functions. CAUCHYwas again an examiner, and his report contains the
following words:
"Geometers know the beautiful works of Abel and of Mr. Jacobi on the
theory of elliptic transcendentals. One knows that of the important papers ...
one of them in particular was approved by the Acaddmie in t829, on the report
of a commission of which Mr. Legendre was a part ECAucHYhimself having been
the other!l, then crowned by the Institut in 1830, and that the value of the prize
was remitted to Abel's mother. In fact this illustrious Norwegian, whom a project
of marriage had determined to undertake a voyage in the depth of winter,
unfortunately fell ill towards the middle of January t829 and, in spite of the
care that had been lavished on him by his fianc6e's family, he died of phthisis
on the 6th April, having been confined to bed for three months ....
"Before completing this report where we have often had to recall the works
of Abel, it appears to us proper to dispel an error which is already quite widespread.
It has been supposed that Abel died in misery, and this supposition has
been the occasion for violent attacks directed against scholars from Sweden and
from other parts of Europe. We would want to believe that the authors of these
attacks will regret that they expressed themselves with such vehemence, when
they read the Preface of the ... Oeuvres d'Abel, recently published in Norway
by Mr. Holmboe, the teacher and friend of the illustrious geometer. They will
see there with interest the flattering encouragements, the expressions of esteem
and admiration that Abel received from scholars during his life, particularly
from those who occupied themselves at the same time as he with the theory of
elliptic transcendentals .... -67
In fact CAUCH¥must have known that, while preparing his t839 edition of
ABEL'S works, HOLMBOEhad tried without success to obtain the 1826 manuscript
from the Acaddmie des Sciences and that its publication in t84t was due only to
the fact that he had raised the matter to governmental level. Anyone capable of
writing in this manner, knowing the negative role played by himself in the matter
under discussion, would hardly think twice about borrowing from an unknown
paper published in Prague without acknowledgement.
But how unknown was BOLZANO'Spaper ?
e~ A.-L. CAI:CH¥, "Rapport sur un m6moire de M. Broch, relatif ~ une certaine
classe d'int6grales," C.R. Acad. Roy. Sci., 12 (184t), 847--850 = Oeuvres, (t) 6,
146---t49. ABEL'Spaper was then appearing as "M6moire Bur une propri6t6 g6n6rale
d'une classe tr~s-6tendue de fonctions transcendantes", Mdm. prds. Acad. Roy. Sci.
div. say., (2) 7 (184t), 175--254 =Oeuvres (ed. L. SYLOW & S. LIE), 1, 145--211.
BRocI~'Spaper appeared as "M6moire sur les fonctions de la forme
$
+--
f xs-ye-l /(xo)R (xQ) "o Ox",
Journ. rei. ang. Math., 23 (1846), 145--t95 and 20t--242: we note the five-year
delay, and the fact that its publication was not in the journal of the Acaddmie to
which it had been assigned. CAUCHY'Sreport (with LIOUVlLLEas co-signatory but
certainly not author!) prefaced the paper on pp. 145--147: he was referring in the
above quotation to the "Notice Bur la vie de l'auteur" that HOLMBOEput in his
edition of ABEL'SOeuvres, 1, v--xiv. At the end of that edition HOLMBOEincluded
a selection of his letters from ABEL,and we note from s3 and s5 that he did not include
ABEL'Sremark on CAUCHY.
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 337 of 479.
]3olzano, Cauchy and New Analysis 395
8. The Availability and Familiarity of Bolzano's Work
We have mentioned several times that BOLZANO'Sachievements anticipated
specifically the work of the WEIERST~SS school in the t860's, and it was they
who first brought BOLZANO'Smathematical publications 6s to general attention
at that time. Du Bols R~YMOND,CANTO~,HA~KEL, HAR~ACK,H~IZ~E, SC~WA~Z,
STOLZ -- they formed perhaps the most talented group ever to work on foundational
problems in analysis, and they all had a deep interest in the history of
their subject. I do not know which of them first came across BOLZANO'Swritings :
the first to make a reference in print was HA~KEL in t871, 69 but SCHWARZwas
the one most interested in these questions and it was he who around that time
named WEIERSTaASS'Stheorem on the existence of a limiting value of an infinite
closed sequence of values the "BoLzAZ~o-W~I~RSTRASS theorem", in view of
BOLZA~O'S theorem on the existence of an upper limit in his t817 pamphlet
which we quoted in section 2.4.7° WEIERSTRASS'S group were then studying
continuous non-differentiable functions, rational and irrational numbers, and
the early ideas of set theory, on all of which BOLZANOhad preceded them; and
so it had tended to be assumed (posterior wisdom again) that in his own day
BOLZANO was not read at all. Without any doubt his works were not widely
available -- for proof of this, we need only mention that it is today extremely
difficult to find copies of any of them. But it would be a mistake to assume that
because they appeared as pamphlets they could not have become widely familiar.
On the contrary, at that time the publication of pamphlets was a common
method of issuing scientific literature and indeed avoided the notorious delays
of academy journals: CAUCHYfor example, always anxious for rapid publication,
put some very important work into pamphlets and lithographs, and even published
his own mathematical journal during two periods of his life. 7a There seems to
have been a well organised trade in the sale of such material, based on the catalognes
of book shops designed especially for scientific and intellectual circles:
it was by these means, for example, that BOLZANOin Bohemia managed to learn
of and obtain the current literature. So we may presume that the work was in
reasonably fluid circulation -- and surely especially in Paris, the scientific centre
of the age. CAucHY himself reveals this in his own writings. Although his refer-
6s Apart from the Beweis and the works listed in 35and 8~, BOLZANOalso published
Betrachtungen iiber einige Gegenstdnde der Elementargeometrie (1804, Prague) =
Schri[ten, 5 (1948, Prague), 9--49; and Beitrdge zu einer begri~ndeten Darstellung der
Mathematik. 1. Lie[erung (t8t0, Prague) = (t926, Paderborn). (No other parts published.)
BOLZANO'Sfriend F. PRIHONSK~"posthumously published his Paradoxien des
Unendlichen (1851, Leipzig): there have been various re-issues and translations of
this work, including an English edition (1950, London).
65H. HANKEL,"' Grenze", Allg. Eric. Wiss. Kiinste, sect. 1, pt. 90 (1871, Leipzig),
t 85--21 t : see pp. t 89, 209---210. The first major study was by STOLZ,as " B. Bolzanos
Bedeutung in der Geschichte der Infinitesimalrechnung", Math. Ann., 18 (1881),
255--279 (and corrections in 22 (1883), 5t8--519).
70 See K. SCHWARZ, "Zur Integration der partiel Differentialgleichung
~u ~2u
ax, + ~- =0", Journ. rei. ang. Math., 74 (t872), 218--253 (p. 221) = Abhandlungen,
2, t75--210 (p. 178).
~ See his Exercises des Mathdmatiques (4 vols. and I instalment: 1826---30, Paris),
and Exercises d'Analyse et de Physique Mathdmatique (4 vols: 1840---47, Paris). They
appear respectively in his Oeuvres, (2) 6---9; and (2) 11--14.
27*
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 338 of 479.
396 I. GRATTAN-GUINN]~SS:
ences were often not always given, they show that he was abreast of current
writings in all European languages, and not only the most prominent authors,
books and journals: there are also references to little known material. In the
Cours d'Analyse, for example, he referred to a pompous little tract of t820
published in London on rules of signs in the theory of equations, 7°' which was at
least as obscure as BOLZANO'S pamphlet. In fact, ]3OLZANO had given his paper
two opportunities for publication, for not only did he issue it as a pamphlet in
1817, but -- with the same printing -- inserted it into the 18t8 volume of the
Prague Academy Abhandlungen. ~3 That journal was available in Paris: indeed,
the Biblioth~que Impdriale (now the Biblioth~que Nationale) began to take it
with precisely the volume containing Bolzano's pamphlet. 74 So here is at least one
plausible possibility for CAUCHY to have found a copy of BOLZANO'S paper, quite
apart from the book-trade: he could have noticed a new journal in the library's
stock and examined it as a possible course of interesting research.
We turn now from the availability to the familiarity of BOLZANO'S WOrkS.
We have seen that they were not widely circulated, although probably more so
than might be imagined; but apart from that I feel that all important factor in
the apparent indifference of his contemporaries was a lack o[ understanding of
what he had achieved. Since his important results were so far ahead of its time,
only a genius of CAUCHY'S type and magnitude could bring them to the realisation
they deserved (and of which their creator was probably incapable). We
can appreciate this point better if we return to ABEL. There is no reference to
any of ]3OLZANO'S works in ABEL'S writings, and seemingly no direct influence
either, even though they had both written on the binomial series; but ABEL had
certainly- read some BOLZANO, for he expressed great admiration for him in a
notebook and hoped to meet him in Prague during his European tour. v5 1 suspect
that several mathematicians were in ABEL'S position: impressed by BOLZANO'S
work, but unable to take it further themselves. 7e But without doubt there were,
unfortunately, many who never discovered it at all. This, therefore, is a situation
in marked contrast to CAUCI~Y'S works, which were read by everybody -- including
BOLZANO.
72 p. NICHOLSON,Essay on involution and evolution: containing a new accurate
and general method o[ ascertaining the numerical value o] any Junction ... (t 820, London).
CAUCH¥'S reference is in the Cours, 500 = Oeuvres, (2) 3, 409: he also wrote a number
of papers on this subject in the 18t0's, but with an interest towards structural properties
(permutations, etc.) rather than in the foundations of analysis. For commentary,
see H. WusslNG, Die Genesis des abstrakten Gruppenbegri[/es (1969, Berlin) esp.
pp. 61--66.
~3 See the references in a
7~The present call mark of this volume is R. 15 200 in the Ddpartment des Imprimds.
There is no record of its readers, neither does it contain any annotated
markings or corrections. The only other copy of the work known to me in Paris is
in the holding of the journal by the Musdum Nationale d'Histoire Naturelle -- a source
hardly likely to have been used by CAUCHY.The copy has no revealing annotations on it.
75 See L. SYLOW, "Les 6tudes d'Abel et ses d6couvertes," s2, 59 PP. (pP. 6 and t 3);
and K. RYCHLIK, "Niels Hendrik Abel a ~echy", Pok. mat. [ys. astron.;9 (1964),
317--319.
~6LOBACHEWSKYalso knew BOLZANO'S18t 7 pamphlet on the roots of a continuous
function: see B. L. LAPTIEV,"0 ~H6HHoTeqHBIX8an~eax RH~Ir~ImypHazXOB,~bl~aHm,xx
H. H. JIo6aqeBcKoMy", VClI. MaT. HayIL 14 (t959), pt. 5, t53--t55.
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 339 of 479.
]3olzano, Cauchy and New Analysis 397
9. The Personal Relations between Bolzano and Cauchy
That CAOCHYread BOLZANO'S18t7 pamphlet is the subject of our conjecture;
but that BOLZANOread CAOCHY'SCours d'AnaIyse is beyond question, for in an
important manuscript of the 1830's on analysis he referred to CAUCHY'Sas one
of the recent formulations of continuity in his own style.7~ By then of course,
BOLZANO'S ideas had gained much publicity through CAUCHY'S book, which
itself had been published at KSnigsberg in t 828 in a German translation which
may well have been the version that BOLZANOread. Yet there was never a priority
row between the two over their common ideas. This is, however, not surprising.
In the first place, BOLZANOwas no CAUCHY,incessantly anxious for publication
and his "rights"; and in addition he was already a controversial figure in Bohemia
on account of his progressive views on society and religion. Thus, even if
he had wanted to stage a priority row from Bohemia against the great CAUCHY
in Paris, he would have found it especially difficult. But I would suggest that
there is still another reason why BOLZANOdid not promote such a row; namely,
that he probably never noticed the correspondence of ideas -- or at least their
significance -- when he read the Cours d'Analyse. For the Cours is a large
book, nearly 600 pages in length; and almost all of it is CAUCHY,applying BOLZANO'S
germinal ideas to one analytical problem after another. But the ideas
themselves and the direct points of correspondence appear only here and there
in its course, and could easily be missed in the general context.
This view is strengthened when we consider their personal relations. There
was no meeting between the two in the t810's or 1820% for CAUCHYwas in
France and BOLZANOin Bohemia; but after the fall of the Bourbons in 1830
CAUCHY exiled himself, firstly to Italy, and then, between 1833 and t835, to
Prague to assist in the education of the son of the dethroned King CHARLESX.
The tone of BOLZANO'Sreaction to CAUCHY'Svisit to Prague, in a letter he sent
to his friend PRiHONSK~ in August, t833, indicates quite clearly that he had had
no contact with CAUCHYof any sort and that he suspected no direct use of his
results by CAUCHY:
"The news of Cauchy's presence Ein PragueJ is uncommonly interesting for
me. Among all living mathematicians today he is the one whom I esteem the
most and to whom I feel the most akin; I owe to his inventive spirit some of the
most important proofs. I ask you very much to recommend me to him and to
say that I would have travelled now straight to Prague to make his personal
acquaintance, if I -- after what you tell me of his appointment -- could not
hope for certain that I will meet him at the end of September ..... ,,Ts
There were in fact a few meetings, for BOLZANOdescribed them in a letter of
December, t843 to FESL:
"Cauchy, the mathematician, was ... in the years t834 or t835 ... in Prague,
where we met a few times during the few days that I was accustomed to spend
at that time (at Easter and in the autumn) in Prague. After my departure I let
72]3. BOLZANO 7, in Schri/ten, 1 (1930, Prague), t5: see also p. 94.
~s See E. WINTER (ed.), "Der b6hmische VormArz in Briefen ]3. Bolzanos an
F. Pr~honsk3~ (t824--t848)", Ver6][. Inst. Slav., Dtsch. Akad. Wiss. Berlin., 11 (1958),
306 pp. (p. 156).
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 340 of 479.
398 I. GRATTAN-GuINNESS:
Kulik deliver to him (1834) an essay filling a single quarto sheet which I had
drafted for Cauchy sometime in French, on the famous mathematical problem
of the rectification of curves, because I rightly feared that he would find the
"Paper on the three problems of rectification, planing and cubing" published
in 18t779 too comprehensive and difficult. Early last year, as I was looking
through some issues of Cauchy's writings 8° bound with the usual coloured wrappers,
and [turned to I the lists of works announced on the back, I noticed with
astonishment a small note by him on the same subject, that he had published
as a lithograph in Paris in t834 (therefore presumably only after he had read my
little essay). Naturally I would be very eager to read the note .... ,,81
Eventually BOLZANOmanaged to obtain a copy of the paper: in fact it came
through FESL who pointed out to him that it had been written in 1832 rather
than 1834 and so could not be related to his essay, and that it treated the subject
in a quite different way. BOLZANOadmitted this in an acknowledgement to
FESL in May, t 844,82 and it is quite clear that in this case there was not even a
correspondence of ideas; but on the foundations of analysis a very different
situation seems to have applied. One would dearly like to know the content of
their conversations; but if BOLZANO ever wondered even for a moment that
CAUCHYhad read his 18t7 paper before writing the tours d'Analyse, I imagine
that he would have been pleased rather than annoyed. For when he wrote that
paper, he had known then that it was a significant work which would probably
not reach the audience that it deserved; and so he had ended its preface with a
plea to the scientific community which I believe CAUCHYaccepted:
"... I must request ... that one does not overlook this particular paper because
of its limited size, but rather examine it with all possible strictness and
make known publicly the results of this examination, in order to explain more
clearly what is perhaps unclear, to revoke what is quite incorrect, but to let
succeed to general acceptance, the sooner the better, what is true and right. ''s8
10. Epilogue
My conjecture has aroused considerable adverse criticism before publication,
and will doubtless receive much more now: thus to minimise the possibility of
misunderstandings of this paper, a few points may be worth stressing.
t. Part of my purpose has been to describe some of the extra-intellectual
aspects of Parisian mathematics; and whether or not my conjecture is correct
~9Tile reference for this work is given in 29
80Presumably the Exercises d'Analyse listed in 71
sl See I. SEIDERLOVA,"]3emerkung zu den UmgAngen zwischen B. Bolzano und
A. Cauchy," ~as. pdst. mat., 87 (t962), 225--226.
82 See sl. CAUCHY'Spaper, read to the Acaddmie des Sciences on the 22nd October,
1832, was the "M6moire sur la rectification des courbes et la quadrature des surfaces
courbes", Mdm. Acad. Roy. Sci., 22 (1850), 3--15 = Oeuvres, (1) 2, 167--t77; but in
the publisher's lists in the Exercises d'Analyse it is described as an t l-page lithograph
of t 832, which was its first publication. I do not understand why ]3OLZANOthought
that it had been published in t 834.
s8 ]3. ]3OLZANO,Beweis, end of preface.
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 341 of 479.
Bolzano, Cauchy and New Analysis 399
I am firmly convinced that rivalries of the type of which I have given some
examples played an important role in Parisian mathematics, and so I have tried
to bring to the attention of historians of this period the kinds of historical problem
that they will have to face in interpreting its literature. In addition, the theory
of "limit-avoidance" is an historical tool which appears to be some use in one
form or another in investigating the development of analysis and the calculus
in this and other periods.
2. I cannot stress too strongly that in characterising Cauchy's genius as responsive
to exterior stimuli I am trying to describe rather than decry the depth and
extent of his originality. Without any question he and GAUSS were the major
mathematicians of the first decades of the nineteenth century: thus his work
has to be given especial attention by historians. It is of course not my position
that CAUCHYwould never give references without intending a double meaning,
but I do think that in his writings, and equally in those o[ his" colleagues", questions
of this type do need to be borne very carefully in mind. With regard to ]3OLZANO'S
pamphlet, it is possible that CAUCHY,the busy and active researchmathematician
and professor at three Paris colleges, simply did not bother to mention it or
even forgot that he had read it (though personally I would not regard this
explanation as sufficient). My case would be much strengthened by documentary
evidence of some kind: CAUCHYdid leave a Nachlass containing mathematical
manuscripts and correspondence, for it was used by VALSON when preparing
his excessively admiring biography of CAUCHY,s4 but unfortunately it was kept
in the family and there is reason to think that, like his library, it has now been lost.
3. I remarked that CAUCHYwas familiar with European languages: in the
case of German, it is perhaps worth mentioning explicitly (from a number of
examples) that he examined in t817 a manuscript in German sent in to the
Acaddmie des Sciences,s5 and that he reviewed MCBIUS'SDer barycentrische Calcul
in t828.s~ We may also record another "coincidence of ideas" with obscure
German writing strikingly similar to the case of t3OLZANO'Spamphlet. In April
t847, GRASSMANN,then a schoolmaster at Stettin, sent to CAUCHYtWO copies
of his t844 Ausdehnungslehre, but he never received any acknowledgement; however
between late 1847 and t853 CAUCHYpublished a number of papers on a
theory of "clefs algCbriques" which basically used the same sort of ideas and
even some almost identical notation, s7 I offer no judgement here on the matter:
84 C.-A. VALSON, La vie et les travaux du Baron Cauchy (2 vols.: 1868, Paris):
see esp. vol. 2, viii--x.
s5 See Proc~s-Verbaux des sdances de l'Acaddmie tenues depuis la fondation jusqu'au
mois d'ao~t, 1835 (10 vols: 1910--22, Hendaye), 6, 210. I may remark here that these
volumes are an invaluable source of historical insight into the period 1795--1835,
when the rivalries were at their height. They give the minutes of all the private
meetings of the A caddmie des Sciences, which the participants can hardly have expected
to be published!
s6 A.-L. CAUCHY,Bull. Univ. Sci. Ind. [Ferrusac~, Sci. math. phys. chim., 9 (1828),
77--80. Not in the Oeuvres.
sTFor the references and some discussion of the affair, see M. J. CROWE,A history
of vector analysis (1967, Notre Dame and London), 82--85 and 106. CROWE'Slast
reference in his 63is inaccurate and ill fact misleading; it should be "MCmoire sur les
clefs algCbriques", Exercises d'Analyse et de Physique Mathdmatique, 4 (t847, Paris),
356---400 = Oeuvres, (2) 14, 4t7--460.
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 342 of 479.
400 I. GRATTAN-GUINNESS: Bolzano, Cauchy and New Analysis
I merely record it as another example of the kind of historical problem which
surrounds the great achievements of the Parisian mathematicians of the time,
when Paris was the centre of the scientific world and CAUCHY'$ achievements
among its principal adornments.
Index of Names
We list here the names and dates of persons mentioned in the main text.
D'ALEMBERT, JEANLE ROND (t 7t 7--1783)
AMPERE, ADRIENMARIE (1775--1836)
ARBOGAST,LOUISFRANCOISANTOINE(1759--1803)
BERNOULLI,DANIEL (1700--t 782)
BESSEL, FRIEDRICHWILHELM(1784--1846)
BLOT, JEAN BAPTISTE(t 774--1862)
DU BOIS REYMOND,PAUL DAVID GUSTAV(t83t--1889)
BOLZANO,BERNARDPLACIDUSJOHANNNEPOMUK(t781--1848)
BOREL, EMILE FELIX EDOUARD JUSTIN (187t--t959)
CANTOR, GEORGFERDINANDLUDWIGPHILIPP (1845--t 918)
CARNOT,LAZARENICOLASMARGUERITE(1753--t823)
CAUCHY,AUGUSTIN-LouIs (t 789--1857)
CHARLESX, KING (t 757--1836)
DIRICHLET,PETER GUSTAVLEJEUNE- (1805--t859)
EULER, LEONHARD (I707--t 783)
FESL, MICHAEL JOSEPH (1786--1864)
FOURIER, JEAN BAPTISTEJOSEPH (I 768--t830)
GAUSS, KARLFRIEDRICH(1777--t855)
GRASSMANN,HERMANNG/JNTHER(1809---t877)
HANKEL,HERMANN(t839--1873)
HARNACK,CARLGUSTAVAXEL (I851--1888)
HEINE, EDUARD HEINRICH (I821--I881)
L'HUILIER, SIMONANTOINEJEAN (1750--1840)
HOLMBOE,BERNTMICHAEL(1795--1850)
JACOBI, CARLGUSTAVJACOB(t804--1851)
KULIK, JAKOBPHILIPP (1793--1863)
LACROIX, SYLVESTREFRAN9OIS(1765--1843)
LAGRANGE,JOSEPHLOUIS (1736--18t 3)
LAPLACE,PII~RRESIMON(1749---1827)
LEGENDRE, ADRIEN MARIE (1752--t833)
LEIBNIZ, GOTTFRIED WILHELM (1646--1716)
M6BIUS, AUGUSTFERDINAND(t 790--1868)
MONGE, GASPARD(! 746--1818)
NEWTON, ISAAC (1642--1727)
POISSON, SIMI~ONDENIS (1781--1840)
PR~HONSK~',FRANZ(1788--t859)
RIEMANN, GEORG FRIEDRICH BERNHARD (1826--1866)
SCHWARZ, KARLHERMANN AMANDUS (1843--192t)
STOLZ,OTTO(1842--1905)
TAYLOR,BROOKE(1685--1731)
VALSON,CLAUDEALPHONSE(1826-- ? )
WEIERSTRASS, KARLTHEODORWILHELM(t 8t 5--1897)
Enfield College of Technology
Middlesex
England
(Received February 17, 1970.)
Text 22: I. Grattan-Guinness (1970). “Bolzano, Cauchy and the “New Analysis” of the
Early Nineteenth Century”. Archive for History of Exact Sciences, vol. 6, no. 5,
pp. 372–400.
Summer University 2012: Asking and Answering Questions Page 343 of 479.
Did Cau@ Plagiarizegolzano?
H. FREUDENTHAL
1. Introduction
t. In an elaborate erudite paper* I. GRATTAN-GUINNESShas put forward a case
that CAUCHYplagiarized BOLZANO:
In Section 2, he discusses why i] CAUCHYplagiarized BOLZAXO,he did it so
badly,
In Section 3, he presents a new limit concept which he calls "limit avoidance",
In Section 4, he mentions some facts from analysis before CAUCHY'Stime,
In Section 5 he claims that CAUCHYcould not have written a so "utterly
untypical" work as his Cours d'Analyse of 182t without having been inspired
by somebody else,
In Section 6-7 he analyzes the quarrels among French mathematicians
around 1800 and CAUCHY'S bad character so as to explain psychologically why
CAUCHY plagiarized BOLZANO,
In Section 8 he discusses whether CAUCHYcould have read BOLZANO,
In Section 9 he deals with the personal relations between CAUCHYand BOLZANO.
Here I wish to discuss the specific question set as the title of this paper,
whether CAUCHY plagiarized BOLZANO, a question not considered directly by
GRATTAN-GuINNESS.
I have to apologize that I am not well enough acquainted with the chronique
scandateuse of the French Academy to follow GRATTAN-GuINNESSthere. On the
other hand I entirely agree with him that a historian is obliged to read between
the lines**, though I think it just as important to read the lines themselves.
In history of mathematics it is also a good idea to understand the mathematics
involved.
The question set as the title of the present paper can be put more precisely
by asking
whether CAUCHYread BOLZANO,
whether CAUCHYcould have learned new things from BOLZANO,
whether these things were so important that he should have cited BOLZANO.
* I. G~ATTA~-GI3INNBSS,"Bolzano, Cauchy and the New Analysis of the Early
Nineteenth Century", Archive for History of Exact Sciences 6 (1970), 372-400.
** p. 387, t 7.
27*
Text 23: H. Freudenthal (1970–1971). “Did Cauchy Plagiarize Bolzano?” Archive for
History of Exact Sciences, vol. 7, no. 5, pp. 375–392.
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376 H. FREUDENTHAL:
It is no sacrilege to ask such questions, even the last one. False ascriptions are
a tradition in mathematics; twice I have met opposition when I refuted such
ascriptions*.
2. The Style of Cauchy's Text-Books on Calculus**
CAUCHY is credited with having laid the first solid foundations of what is
now called Analysis or Calculus. Though this is true, it is not the whole truth,
and in a certain sense it is a misleading statement. It is true that mathematicians
learned from CAUCHY'S Cours d'Analyse and other text-books what continuity
and convergence were and how to test for them, how to be careful with TAYLO~
series and how to estimate their remainders, how to avoid pitfalls when multiplying
and rearranging series, how to deal with multivalued functions, how to define
differential quotients and integrals, how to be careful with improper and singular
integrals, and that they found there the first example of the powerful method
that later became standard in analysis and recently has come to be called "epsi-
lontics".
To know what was new in CAUCHY'S textbooks on Calculus, we had better
listen to his own words, in the Introduction to his Cours d'Aualyse***:
Quant aux m6thodes, j'ai cherch6 ~ leur donner route la rigueur qu'on
exige en g~orn~trie, de mani~re ~ ne jarnais recourir aux raisons tir~es de la
g~n~ralit~ de l'alg~bre. Les raisons de cette esp~ce, quoique assez cornmun6rnent
admises, surtout dans le passage des s6ries convergentes aux s~ries
divergentes, et des quantit~s r~elles aux expressions imaginaires, ne peuvent
~tre consid~rdes, ce rne sernble, que comme des inductions propres ~ faire
pressentir quelquefois la v~rit6, mais qui s'accordent peu avec l'exactitude si
vant~e des sciences math~rnatiques. On doit m~me observer qu'elles tendent
faire attribuer aux formules alg~briques une ~tendue ind~finie, tandis que,
dans la r6alit~, la plupart de ces formules subsistent uniquernent sous certaines
conditions, et pour certaines valeurs des quantit~s qu'elles renferrnent. En
ddterrninant ces conditions et ces valeurs, et en fixant d'une mani~re precise
le sens des notations dont je me sers, je fais disparaltre toute incertitude;
et alors les diff~rentes formules ne pr~sentent plus que des relations entre
les quantit~s r6elles, relations qu'il est toujours facile de v~rifier par la substitution
des nornbres aux quantit~s elles-m~mes. I1 est vrai que, pour rester
constamrnent fiddle ~ ces principes, je me suis vu forc~ d'admettre plusieurs
propositions qui parMtront peut-~tre un peu dures au premier abord. Par
exemple, j'~nonce dans le chapitre VI, qu'une sdrie divergente n'a pas de somme/
dans le chapitre VII, qu'une dquation imaginaire est seulement la reprdsentation
symbolique de deux dquations entre quantitds rdelles; dans le chapitre IX, que,
si des constantes ou des variables comprises dam une ]onction, apr~s avoir dtd
supposdes rdelles, deviennent imaginaires, la notation ~ l'aide de laquelle la ]onc*
GR.~.TTAN-GuINNESSremarks (p. 398, 5f.b.) that his "conjecture has aroused
considerable adverse criticism before publication". In his lecture on this subject
before an audience of mathematicians rather than historians that I attended, it was
his mathematics rather than his thesis on CAuctiY that aroused opposition.
** CAUCHY,Oeuvres (2) 3~S.
*** CAUCHY,Oeuvres (2) 3.
Text 23: H. Freudenthal (1970–1971). “Did Cauchy Plagiarize Bolzano?” Archive for
History of Exact Sciences, vol. 7, no. 5, pp. 375–392.
Summer University 2012: Asking and Answering Questions Page 345 of 479.
Did Cauchy Plagiarize Bolzano ? 377
tion se trouvait exprimde, ne peut gtre conservde dans le calcul qu'en vertu d'une
convention nouvelle propre d~ ]ixer le sens de cette notation dam la derni~re
hypoth~se; & c. Mais ceux qui liront mon ouvrage reconnaltront, je l'espfire,
que les propositions de cette nature, entralnant l'heureuse nfcessit6 de mettre
plus de pr&ision dans les tlifories, et d'apporter des restrictions utiles ~ des
assertions trop ~tendues, tournent au profit de l'analyse, et fournissent plusieurs
sujets de recherches qui ne sont pas sans importance. Ainsi, avant d'effectuer
la sommation d'aucune sfrie, j'ai dr examiner dans quels cas les sfries peuvent
6tre somm6es, ou, en d'autres termes, quelles sont les conditions de leur
convergence; et j'ai, ~ ce sujet, 6tabli des r~gles g6n6rales qui me paraissent
m6riter quelque attention.
The "generality of algebra" meant that what was true for real numbers,
was true for complex numbers, too, what was true for convergent series, was
true for divergent ones, what was true for finite magnitudes, held also for infinitesimal
ones. Today it is hard to believe that mathematics ever relied on such
principles, and since differentials now are only an uneasy remainder of the preCAUCH¥
period, we readily identify CAUCI~Y'Srenovation with the progress from
"infinitesimal" methods to epsilontics, in spite of CAUCHY'Sown, much broader,
appreciation, by which all metaphysics was barred from mathematics. The next
generation of mathematicians, who had been brought up with the Cours d'Analyse,
and the generations after WEIERSTRASS,CANTOR and DEDEKIND, who knew
which course the development of analysis was due to take after CAUCHY, put
the stress differently than CAUCItYand his generation would have done; at that
time, and even more today, people would not properly understand what it meant
if you told them that CAUCHYabolished "the generality of algebra" as a foundation
stone of mathematics.
I. GRATTAN-GuINNESShas been puzzled by the "untypical" character of
CAUCHY'S work on Calculus as compared to his production before t821. It is
indeed puzzling. But GRATTAN-GuINNESSmight have added that it is untypical
even if compared with CAUCHY'Swork after t82t. The strange thing is that in
his research papers CAucttY never lived up to the standards he had set in his
Cours d'Analyse. Though he had given a definition of continuity, he never proved
formally the continuity of any particular function. Though he had stressed tile
importance of convergence, he operated on series, on FOURIER transforms, on
improper and multiple integrals, as though he had never raised problems of
rigor. In spite of the stress he had laid on the limit origin of the differential
quotient, he developed also a formal approach to differential quotients like
LAGRANGE'S. He admitted semi-convergent series and rearrangements of conditionally
convergent series if he could use them. He formally restricted multivalued
complex functions of x as logx, Vx, and so on, to the upper half plane,
but if he could use them in the lower half plane, he easily forgot about this
prescription. CAUCHYlooks self-contradictory, but he was simply an opportunist
in mathematics, notwithstanding his dogmatism in religious and political affairs.
He could afford this opportunism because, with the background of a vast experience,
he had a sure feeling for what was true, even if it was not formulated or
proved according to the standards of the Cours d'Analyse.
Text 23: H. Freudenthal (1970–1971). “Did Cauchy Plagiarize Bolzano?” Archive for
History of Exact Sciences, vol. 7, no. 5, pp. 375–392.
Summer University 2012: Asking and Answering Questions Page 346 of 479.
378 H. FREUDENTHAL:
Why, then, was the Cours d'Analyse so different from his other work ? Not
because it was more fundamental, but because it was a textbook, in which he
not only communicated his results but also made explicit his background experience.
CAUCHYwas not a lover of foundational research like BOLZANO,but to teach
mathematics to beginners, he had to analyze and to present the techniques
implicit in his background. A similar situation is common today, when a modern
teacher of mathematics will make explicit his logical habits, even though he is
not a logician.
There is at least one work of CAOCH¥, his theory of determinants of 18t2%
which shows the same "untypical" features; it is not to be wondered at that
for a long time this was the only textbook on determinants. The most "untypical"
CAIJC~IY of all, however, is found in his marvellous first communication on
Elasticity of t822"*, which by its conceptual style towers high above the usual
algorithmic swamp in which he moves.
Certainly, one has to be careful with stylistic arguments. If CAUCHY'Swork
had come down to us anonymously, by stylistic arguments we might attribute
the Cours d'Analyse, the introduction to elasticity, and the remainder of his
scientific work to at least three different CAUCHYS; on account of content we
might even attribute his work on complex functions also to at least three CAOC~IYS,
so as to account for the strange phenomenon of periodic amnesia: often he asserts
propositions he had recognized as wrong a short time before*** and for
26 years he seems to have forgotten the most important paper he wrote in this
field****.
CAUCHYdid not live in vacuo. He was moved by work of others, and though
he made lavish acknowledgements to work of others, we can never be sure whether
he cited all sources of his inspiration. By his own testimony we know that LEIBNIZ
was inspired to his discoveries in Calculus by work of PASCAL which actually
was only weakly related to what LEIBNIZ himself finally achieved; even according
to modern standards LEIBNIZ could hardly have been obliged to cite PASCALon
these grounds. In any case from LEIB~IZ' publications we could not guess who
among LEIBNIZ' predecessors was the most influential.
To tell from mere stylistic arguments that CAUCRY'S Cours d'Analyse must
have been inspired by essentially other sources than those on complex functions
or hydrodynamics, is an utterly dangerous conclusion. I have spent so much
time on it because the difference of style between the Cours d'Analyse and other
work of CAUCHYis indeed striking, and because I. GRATTAN-GUINNESSconfesses
that this feature was the starting point of his investigation.
* CAUCHY, Oeuvres (2) 1, 9t-t69. (M6moire sur les fonctions qui ne peuvent
obtenir que deux valeurs...) See also Oeuvres (2) 1 64-90. (M6moire sur le hombre
de valeurs qu'une fonction peut acqu6rir.)
** CAUCHY, Oeuvres (2) 2, 300-304.
*** E.g. the conditions for development into a series of partial fractions in
CAUCHV,Oeuvres (2) 7, 324-362, and (i) 8, 55-64, or multivalued functions in CAt3CmZ,
Oeuvres (l), 8, 156-160 and (!) 8, 264.
**** A. L. CAIYCI~Y,M6moire sur les int6grales d6finies prises entre des limites
imaginaires, Paris 1825, 4°, 68 pages. Reprinted in Bull. sci. math. 7 (t874), 265-304;
8 (1875), 43-55, 148-159; due to be reprinted in CAtYCnY,Oeuvres (2) 15.
Text 23: H. Freudenthal (1970–1971). “Did Cauchy Plagiarize Bolzano?” Archive for
History of Exact Sciences, vol. 7, no. 5, pp. 375–392.
Summer University 2012: Asking and Answering Questions Page 347 of 479.
Did Cauchy Plagiarize Bolzano ? 379
3. Bolzano's Pamphlet of 1817
The first theorem of BOLZANO'Spamphlet* is what is now called CAUCHY'S
convergence theorem; since a theory of real numbers is lacking, its proof can be
nothing but a sham. We will come back to this point.
The next theorem is usually described as the theorem on the existence of the
lowest upper bound of a bounded set of real numbers; in fact the only bounded
sets considered are lower classes as used in DEDEKIND cuts, SO that it would
be better to term it the theorem on the existence of the cut number. From old
times this existence has been used implicitly or explicitly. It was BOLZANO'S
great idea to prove it. The proof, using a sequence of dichotomies and the "Cauchy
convergence criterion", is correct.
The third theorem is about continuous functions f and q5 with [(c~)< ~b(~)
and f(/~)> $ (~); it states the existence of an intermediate x where f(x)= $ (x).
Continuity had been defined in the preface in a perfectly modern way. The theorem
is derived by considering the subset of y such that [ (x) < $ (x) for all x ~y and
by applying the preceding theorem to it. Again it is a merit of BOLZANOto have
recognized the idea to prove it.
The last theorem asserts the existence of a real root of a polynomial between
two points where its values are of opposite sign.
As compared to CAUCHY'Swork, BOLZANO'Spamphlet is clumsily written and
partially confused. ]3OLZANOhas no term for convergence, and none for the limit
of a sequence; he always circumscribes the convergence to a certain limit by the
sentence that defines this property. Of course he has no term for lowest upper
bound either. His terminology is unusual; a sequence of functions is called a
ver~nderliche Gr6sse, and a single function a best~ndige Gr6sse. The CAUCHYconvergence
criterion is formulated for a sequence, not of numbers, but of functions,
and the property that is formulated, is, ill fact, uniform convergence although
BOLZANO draws no conclusion from it (e.g. with respect to continuity); the
criterion is actually applied to numerical sequences only**. The proof of this
criterion is worse than faulty, it is utterly confused and not at all related to the
thing to be proved. At that time it was, indeed, hard to understand that such
a theorem could not be proved without an underlying theory of real numbers;
recently published papers of BOLZANO show that later he became aware of
this fact.
This failure does not prevent the pamphlet from being a marvellous piece
of work; the proofs of the other theorems are correct.
4. The Common Ideas in Bolzano and Cauchy
I am borrowing the titles of this section and of the subsections t-5 from
I. GRATTAN-GUINNESS; his remarks in the corresponding section will be analyzed
here.
* B. BOLZANO, Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey
Werthen, die ein entgegengesetztes Resultat gewdhren, wenigstens eine reelle Wurzel der
Gleichung liege (18t7), Prague = Abh. K6nigh B6hm. Gesell. Wiss. (3) 5 (1814-1817;
publ. 1818), 60 p.- Also in: OSTWALD'SKlassiker No. 153, ed. Ph. E. B. JOURDAIN.
** This is dissimulated in I. GRATTAN-GUINNESS'quotation, where the hypothesis
of the theorem is replaced with a provisional announcement taken from another
section of the pamphlet.
Text 23: H. Freudenthal (1970–1971). “Did Cauchy Plagiarize Bolzano?” Archive for
History of Exact Sciences, vol. 7, no. 5, pp. 375–392.
Summer University 2012: Asking and Answering Questions Page 348 of 479.
380 H. FREUDENTHAL"
4.1. Continuity o/a Function. BOLZAXO'Sand CAUCHY'Sdefinitions are equivalent.
BOLZANO'SiS far better; it is modern (though instead of ~ and e he uses
coand f2) ; the succession of the quantifiers is correct and clear. CAUCHY'Sdefinition
uses the language of infinitesimals (an infinitely small increase of the variable
produces an infinitely small increase of the functions); even the succession of
the quantifiers is not clear in this formulation.
It is hard to explain how CAUCHY, if borrowing the definition of continuity
from ]3OLZANO, could have presented it in deteriorated form; later on such
occurrences are explained by I. GRATTAN-GuINNESS as instances of CAUCHY'S
failure to fathom the depth of ]3OLZANO'Sthought. There is, however, not the
slightest reason to assume that CAUCHYlearned tile concept of continuous function
from BOLZANO,since it was already instrumental in CAUCHY'S* treatise of t8t4
on complex functions (the Cauchy integral theorem):
Solution. -- Si la fonction 9 (z) croit ou d6croit d'une mani~re continue
entre les limites z =b', z =b", la valeur de l'int6grale sera repr6sent6e,
l'ordinaire, par
(b") -- 9 (b').
Mais, si, pour une certaine valeur de z repr6sent6e par Z et comprise entre
les limites de l'int6gration, la fonction ~ (z) passe subitement d'une valeur
d6terminde ~ une valeur sensiblement diffdrente de la premiere, en sorte qu'en
d6signant par ~ une quantit6 tr~s petite, on ait
(Z +~) -- 9 (Z --~) =A,
alors la valeur ordinaire de l'int6grale d6finie, savoir,
(b") -- ~ (b')
devra ~tre diminu6e de la quantit6 A, comme on peut ais6ment s'en assurer.
To within a formal definition the full-fledged idea of continuity is presented
not only here; it is also the main idea underlying the introduction of the CAUC~IY
principal value of singular integrals, which provided CAUC~IY'Sapproach to his
integral theorem. There can be little doubt that here was CAUCHY'S point of
departure to continuity.
I. GRATTAN-GuINNESSclaims that in t 82t CAUCHYdid not know that continuity
did not imply differentiability, while BOLZANOknew it. There is no proof for the
second claim, and in the light of the role continuity plays in CAUC~IY'Streatise of
18t4, the first claim is ridiculous.
4.2. Convergence o[ a Series. In the case of the Cauchy convergence criterion
CAUCHY'S formulation is much better than BOLZANO'S. If CAUCHY ever read
BOLZANO,and even if he did not understand his confused exposition, the possibility
can hardly be excluded that he guessed what BOLZANOmeant and consequently
arrived at an improved version. Of course, this is no proof that it really happened
this way. CAUCHYprepares tile announcement of his criterion by a fine heuristic
approach which, undoubtedly, is his own**; when reading his exposition, one can
* C~,UCH¥,Oeuvres (1) 1, 402-403.
•• CAUCHY,Oeuvres (2) 3, I t 5-t 16.
Text 23: H. Freudenthal (1970–1971). “Did Cauchy Plagiarize Bolzano?” Archive for
History of Exact Sciences, vol. 7, no. 5, pp. 375–392.
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Did Cauchy Plagiarize Bolzano ? 381
imagine him standing at the blackboard, explaining that for a sum Y, u. to converge,
it does not suffice that the u, converge to 0, nor does it suffice that the
u. +u.+l converge to 0, nor does it suffice that the u,+u,+ 1+u,+ 2 converge
to 0, and so on, and that in order to get convergence of the sum you have rather
to make all these expressions arbitrarily small by choosing n large.
In today's mathematics this is so natural an approach that one feels little
need to ask who invented it, yet in the historical setting the CAUCHYconvergence
criterion looks like a premature discovery. In fact, if we expect a great many
applications of the CAUCHY convergence criterion in CAOCH¥'S work, we are
likely to be disappointed. It is applied at essentially two places:
First, to justify the majorant method of convergence proofs (if [a~[<]c. I for
almost all n, and if Y, [c,] converges, then Y, a~ converges), which in the particular
case of a geometrical series as a majorant, is the foundation of CAUCHY'Sfamous
"Calcul des limites" in power series and differential equations,
Second, to prove the convergence criterion on alternating series (if the [a~[
are such that a~a~+1~ O, [a~] ~[ a~+l [, and lira a~ = 0, then • a~ converges).
As soon as these two criteria have been established, the reader of tile Cours
d'Analyse may readily forget about the CAUCHYconvergence criterion.
This is not to be wondered at since there was not any other essential use of
the CAUCHYconvergence criterion up to the rise of the direct methods of the
variational calculus at the turn of the t9 th century. The majorant method and
the criterion on alternating series as algorithmic tools were just what mathematicians
in CAUCHY'Stime, and even later, needed. The CAUCHYconvergence criterion
with its much more involved logical structure, lacked this algorithmic appeal.
CAUCHY'S work in analysis would not have looked different if he had never
formulated the CAUCHYconvergence criterion and, instead, had accepted the
principle of the majorant method and the criterion on alternating series as obvious
truths which did not need a proof, just as, for instance, he accepted without
argument that the endpoints of a nested sequence of intervals, shrinking to zero,
had a limit*.
From CAUCHY'Stime up to the end of the {9th century the CAUCHYconvergence
criterion was an expression of logical profundity rather than a practical tool.
This is what I meant when I characterized the CAUCHYconvergence criterion
as a "premature discovery"--a characterization which at the same time means
a praise of its discoverers.
I. CvRATTAN-GUINNESScould have made a relatively strong point against
CAUCHYout of the argument that the CAUCHYconvergence criterion fitted less
into CAUCHY'Swork than anything else. Strangely enough he did not. Though
he challenged CAUCHY'S originality in much weaker cases, he did not do so in
this one, which would have been the strongest.
Though I cannot exclude the possibility that CAUCI-IYborrowed his convergence
criterion from BOLZANO, I stress that i do not see any indication that he
actually did so.
* CAucI~¥, Oeuvres (2) 3, 379; ill the proof of the theorem of the intermediate
zero of a continuous function.
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382 H. FREUDENTHAL:
4.3. Bohano's Main Theorem. The theorem on the vanishing of a continuous
function between two points where its values are of opposite sign is still less
fundamental to CAUCHY'SCalculus. It is almost self-evident that such a pure
existence theorem did not mean much at that time. In CAVCHY'SCours d'Analyse
it stands in the classical constructive context of solving numerical equations,
particularly in connection with a method of LEGENDRE*, cited by CAUCHY**.
The theorem itself had long been known. BOLZANO'S and CAUCHY'S merit
is to have proved it. I. GRATTAN-GuINNESS'statement that CAUCHY'Sproof uses
a condensation argument is far off the mark if by "condensation argument"
he means what is usually understood by this term. His claim that CAUC~IY'S
proof
seems very much like an unrigorous version of the intricate proof developed
in BOLZANO'Spaper
is as wrong as can be. The most convincing though somewhat lengthy way to
refute this claim is to quote CAUCI~Yhimself***:
Th6or6me I. -- Soit /(x) une /onction rdelle de la variable x, qui demeure
continue par rapport ~ cette variable entre les limites x = xo, x = X. Si les deux
quantitds /(xo),/(X) sont de signes contraires, on pourra satis]aire ~ l'dquation
(~) l(x) =0
par une ou plusieurs valeurs rdelles de x comprises entre xo et X.
Ddmonstration. -- Soit x0 la plus petite des deux quantit6s x0, X. Faisons
X--x o =h,
et d6siguons par m un nombre entier quelconque sup6fieur ~ l'unit6. Comme
des deux quantit6s /(xo),/(X), l'une est positive, l'autre n6gative, si Yon
forme la suite
2 h
et que, dans cette suite, on compare successivement le premier terme avec
le second, le second avec le troisi6me, le troisi6me avec le quatri6me, etc.,
on finira n6cessairement par trouver une ou plusieurs lois deux termes cons6cutifs
qui seront de sigues contraires. Soient
t (xl), !(X')
deux termes de cette esp6ce, xI 6tant la plus petite des deux valeurs correspondantes
de x. On aura 6videmment
Xo<x~<X' <X
et
h I (X-xo).
X'--xl-- m -- m
* M.-A. LEGENDRE, Essai sur la th6orie des hombres. Suppl6Inent, f6vrier
4816, § III.
** CAI~CHY,Oeuvres (2) 3, 38t.
*** CAUCHV,Oeuvres (2) 3, 378-380.
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Did Cauchy Plagiarize 13olzano ? 383
Ayant ddtermin6 x, et X' comme on vient de le dire, on pourra de m~me,
entre ces deux nouvelles valeurs de x, en placer deux autres x=, X" qui, substitu6es
dans f(x), donnent des r6sultats de signes contraires, et qui soient
propres g v6rifier les conditions
x1< x~ < X" < X',
t 1
x" - x~ = ~ (x' - xl) = ~ (x - x0).
En continuant ainsi, on obtiendra: t ° une s6rie de valeurs croissantes de x,
savoir
(2) x0, x1, x,~.... ;
2° une s6rie de valeurs ddcroissantes
(3) x, x', x", ...,
qui, surpassant les premi6res de quantit6s respectivement 6gales aux pro-
duits
t 1
I x (X-x0), ~- x (X-Xo), ~ x (X-Xo) .... ,
finiront par diff6rer de ces premieres valeurs aussi peu que l'on voudra.
On doit en conclure que les termes g6n6raux des s6ries (2) et (3) convergeront
vers une limite commune. Soit a cette limite. Puisque la fonction ](x)
reste continue depuis x =x o jusqu'~ x =X, les termes g6n6raux des s6ries
suivantes
l(Xo), l(xl), /(x2) .....
t(x), !(x'), l(X") ....
convergeront 6galement vers la limite commune /(a); et, comme en s'approchant
de cette limite ils resteront toujours de signes contraires, il est clair
que la quantit6 /(a), n6cessairement finie, ne pourra diffdrer de z6ro. Par
cons6quent on v6rifiera l'6quation
(1) /(x) =0,
en attribuant ~ la variable x la valeur particnli&re a comprise entre x0 et X.
En d'autres termes,
(4) x =a
sera une racine de l'6quation (t).
CAUCH¥'S proof is simply a faithful description of the naive procedure for
solving equations numerically (the title of this Note is" Sur la r&olution nurndrique
des dquations"). The only sophistication is that the length of the unit interval
is replaced by a more general h, and the 10 of our decimal system by a general
basis m.
The proof is not a version of BoLzAzvo's and it is as rigorous as a proof can be.
The only correct remark I. GRATTAN-GuINNESS made is that BOLZANO'S proof is
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384 H. FREUDENTHAL:
intricate; it goes by way of the existence of the least upper bound of a bounded
set (or rather the existence of the cut number); once this existence is presumed,
BOLZANO'Sproof is more elegant than CAUCHY'S.
Anyhow there is not the slightest need to suppose that CAUCHY took his
proof from BOLZANO. The idea, however, that such a theorem needed a proof
and could be proved, may well have come from BOLZANO.The title of BOLZANO'S
pamphlet could have been enough to inspire CAUCHYto prove the theorem even
if he never read tile pamphlet itself.
Of course this does not prove that CAUCHYever saw BOLZANO'Spamphlet.
4.4. Bolzano's Lemma. The corner stone in I. GRATTAN-GUINNESS'case that
CAUCItYplagiarized ]3OLZANO,is the following argument: In his Cours d'Analyse,
instead of the limit concept, which would have been sufficient, CAUCHYused
the concept of upper limit, which was not needed, simply because he found it
in ]3OLZANO'Spamphlet. If this were true, it would, indeed, prove convincingly
that CAUCHYknew BOLZANO'Spamphlet.
It was pointed out to I. GRATTAN-GUINNESSthat his statement here rests on
a few mathematical errors. In I. GRATTAN-GUINNESS'paper we now find a text
(section 2.4), which, mathematically and historically, is wrong, as I will show
in all details; further, attached to this text, footnote 24, which in fact invalidates
the main text, and which is wrong in itself. I will now analyze this paragon of
confusion.
As I explained, ][3OLZANOproved in his pamphlet the existence of the least
upper bound of bounded sets of a special kind (DEDEKINDlower classes). I. GRATTAN-GUINNESSquotes
BOLZANO'Stext and then continues:
with this extraordinary theorem came another new idea into analysis, completely
untypical of its time : the upper limit of a sequence of values.
Speaking of upper limit rather than of least upper bound could be a terminological
deviation, since for a long time usage here was unsettled. It is certain,
however, that I. GRATTAN-GUINNESSmeans "upper limit" since he refers to a
sequence rather than to a set or a lower class, and since he continues with a
reference to a convergence test of CAUCHY,the V~-criterion for the convergence
of ~, u~ (with positive u,). Here, indeed, the upper limit (that is, in modern terms,
the largest accumulation value) is needed and is used. I. GRATTAN-GuIz~NESSsays
that the term of upper limit is
... not to be found explicitly in Cauchy's Cours d'Analyse, but instead
we have there a frequent use of phrases like "...the largest value of the ex-
pression..."
This is entirely wrong. At one of the places alluded to by I. GRATTAN-GuINNESS
we read*
Cherchez la limite ou les limites vers lesquelles converge, tandis que n
croit ind6finiment, l'expression (u,,)1/~ et d6signez par k la plus grande de ces
limites, ou, en d'autres termes la limite des plus grandes valeurs de l'expression
* CAucnY, Oeuvres (2) 3, t21.
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Did Cauchy Plagiarize Bolzano ? 385
dont il s'agit. La s6rie (1) sera convergente si l'on a k< 1, et divergente si
l'on a k> 1.
At another place*:
Cherchez la limite ou les limites vers lesquelles converge, tandis que n
crolt ind6finiment, l'expression (0~)1/.. Suivant que ]a plus grande de ces
limites sera inf6rieure ou sup6rieure ~ l'unit6, la s6rie (3) sera convergente
ou divergente.
The alternative definition is here repeated in the proof of the theorem:
Considdrons d'abord le cas off les plus grandes valeurs de l'expression (0~)1/~
convergent...
It is difficult to say which one of the two definitions was operative, since
the proofs do not use the explicit value of the upper limit but only its being < t
(or > t), that is, the existence of an U such that (u~)l/~< U < 1 for almost all n
((u.)l/~> U > t for infinitely many n). Contrary to I. GRATTAN-GuINNESS' statement
the term of upper limit (la plus grande de ces lirnites) is explicit in CAUCHY'S
text. On the other hand the plural form and the context "la lirnite des plus grandes
valeurs de l'expressions" clearly show that this is not CAUCHY'Sterminology for
the upper limit as suggested by I. GRATTAN-GuINNESS' quotation "the largest
value of the expression..." Cut out this way from CAUCHY'Stext by I. GRATTANGUI]qNESS,
it is meaningless because it does not allow the hidden quantifiers to
be traced.
It does not matter too much what artificially isolated pieces of a text mean
if the text is globally clear; in the present case it is not far-fetched, and it is in
agreement with the global text to assume that "la plus grande valeur" applies
to a finite set, to wit the set of (u,)1/~, .... (u.+k)1/~+k,and the plural is to indicate
that all such sets are considered.
I. GRATTAN-GUINNESScontinues:
As with continuity of a function, CAUCIIY was revealingly only partially
aware of the significance of the idea; for he used it only as a tool for developing
the proofs of his particular theorems and not as a profound device for investigating
more sophisticated properties of analysis. Therefore it would be
especially surprising if it were CAUCHY'Sown invention...
Everybody who is not a stranger to calculus knows that there is no other use of
upper limits than just those theorems where CAUCI-IYused them. Even today
they provide an unusual and ineffective device. The conclusion that it was not
CAUCtIY'S invention because he used it too little is consequently mistaken.
I. GRATTAN-GuINNESS still suggests that CAUCHY took this tool from BOLZANO.
When he wrote that sentence, he certainly believed that this tool was in BOLZANO'S
pamphlet. Probably he was misled by the so-called BOLZANO-WEIERSTRASS
Theorem on the existence of an accumulation point for an infinite bounded set
of numbers, which can be proved by showing the existence of the upper limit.
* CAucI~Y, Oeuvres (2) 3, 235.
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386 H. FREUDENTHAL :
BOLZANO'Sname in this context, however, is an honorific rather than an historic
epithet as is HEINE'S name in "HEINE-BOREL theorem"*.
CAUCHYdid not use the notion of upper limit more often than he did, because
he could not**, and he did not take it from BOLZANO,because it was not in
BOLZANO'Spamphlet. There is no doubt that I. GRATTAN-GUINNESSnow knows
these facts, but instead of cancelling the whole section, he has nullified it in a
footnote:
There is a distinction between BOLZANO'Sintroduction of an upper limit
and CAUCHY'S"largest value of the expression..." in that CAUCHYactually
used the Limes of a sequence.., while BOLZANOdefined the upper limit...
but we cannot interpret this distinction as intentional in BOLZANO'Sand
CAUCHY'Stime...
First, neither did CAUCHYuse the term "largest value of the expression" nor
did BOLZANOspeak of upper limits. According to modern terminology the terms
are upper limit (orlimit superior) and least upper bound (orcut number), respectively.
Second, CAUCHYdoesnot use the limit but the upper limit--I. GRATTAN-GUINNESS
seems still not to grant that these are different things. Third: Both BOLZANO'S
and CAUCHY'Sconcepts of least upper bound and upper limit, respectively, were
introduced on purpose because in the given context neither of them could use
any other concept.
The fact that at first I. GRATTAN-GUINNESSdid not notice this distinction,
does not entitle him to claim that BOLZANOand CAUCHYcould not make it.
They did not have to, because they were confronted with different situations,
and it is no use asking whether they would have made the distinction if there
had been some need to do so.
To summarize, at this point there is no influence of BOLZANOon CAUCHY
visible.
4.5. The Real Number System. I. GRATTAN-GUINNESSsays:
In the course of proving this Lemma as well as in other parts of his paper
BOLZANOhad recourse to extended considerations of real numbers regarding
the rational or irrational limiting values of sequences of certain finite series
of rationals...
On the contrary:
CAUCHY wrote just once on the real number system: it was in the Cours
d'Analyse, where he gave a superficial exposition of the real number system.
The initial stimulus for this work was foundational questions concerning the
representation of complex numbers; but he took the development of the ideas
well into BOLZANO'Sterritory, twice including the remark that "when B is
* HEINE first recognized the importance of uniform convergence, but he did not
formulate covering properties.
** Even a concept like the least upper bound was not of any importance for the
mathematics of the CAUCHYera. Such concepts become instrumental only with the
direct methods of the variational calculus at the end of the 19m century, in particular
after HILBEgT'S salvation of DIEICHLET'Sprinciple.
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Did Cauchy Plagiarize Bolzano ? 387
an irrational number one can obtain it by rational numbers with values
which are brought nearer and nearer to it"--merely a remark on a property
of the real numbers and not as a definition of the irrational number... Once
again CAUCHY did not fully appreciate the depth of BoLzAxo's thought;
and yet it is clear from his partial success that he was aware of BOLZANO'S
ideas rather than from his partial failure that he was ignorant of them.
It is hard to believe, but the truth is just the other way round. It is true that
neither BOLZANO nor CAUCHY defined real numbers (in later investigations
]3OLZANO tried to do so). There is, however, nothing in BOLZANO'S pamphlet
that justifies the sentence quoted. There are no "extended considerations on
real numbers...", there is not any consideration of real numbers and not even
anything that could be misunderstood as such by somebody unaccustomed to
reading mathematics. What I. GRATTAN-CjuINNESS writes is a pure invention.
The terms "rational" and "irrational" do occur once, in § 8, when, using as an
example the decimal development of 1 ]3OLZANOwarns the reader against believing
that the limit of a sequence of different rational numbers must be irra-
tional.
On the contrary, CAUCH¥'Soccupation with real numbers in the Cours d'Analyse
is hatefully misrepresented. CAUCHY, though not defining real numbers, at least
defines the algebraic and exponential operations on real numbers; starting from
the rational numbers, where they had been defined directly, he extends the
definitions to the real numbers by continuity. In this context he twice uses the
fact that real numbers can be obtained as limits of rational ones. These are not
isolated remarks as I. GRATTAN-GuINNESS claimed, but rather a deliberate use
of this property in a meaningful context.
In any case CAUCHYwrote in the Cours d'Analyse much more on real numbers
than BOLZAI~Odid in his pamphlet (which was nothing). What could CAUCHY
learn at this point from BOLZANO? What was the "depth of ]3OLZANO'Sthought"
that CAUCHYcould not fathom? The bare Nothing or the fact that 0.ttt ... is
rational ? Where did he trespass into BOLZANO'Sterritory, if this territory consisted
of Nothing or of the fact that 0.ttl... was rational?
4.6. Summary as to the Common Ideas in Bolzano and Cauchy.
1. The idea of continuity, common to them both, was arrived at by each of
them independently.
2. The CAUCHYconvergence criterion was formulated by each of them; it
is possible that CAUCH¥took it from ]~OLZANO,though it can easily be explained
as an original invention of CAocltY'S.
3. The theorem on the intermediate value of a continuous function had long
been known as a more or less obvious proposition. The idea to prove it may
have come to CAUCHYwhen he read the title of BOLZANO'Spamphlet if he ever
did. His proof is different from BOLZAI~O'S.
4. As regards upper limits and least upper bounds, there is no common element.
5. On real numbers ]3OLZANO'S pamphlet contains nothing, while CAUC~IY
in his Cours d'Analyse developed a theory of operations with real numbers.
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388 H. FREUDENTHAL:
In section 2 I explained how the Cours d'Analyse rested on a much broader
basis of ideas than the few CAUCHYcould have borrowed from BOLZANO'Spamphlet.
Therefore I. GRATTAN-GUINNESS'insinuating question*
What would have happened if CAUCHY had not read BOLZANO.~
is irrelevant. The present section shows that there is even little if any cause
to ask the other insinuating question**
But if CAUCHYowed so much to BOLZANO,why did he not acknowledge him ?
Before analyzing his answer on this question, we shall cast a glance at his section
3.
5. Limit-Avoidance
I quote I. GRATTAN-GUINNESS' new limit definition***:
When we speak of "introducing the concept of a limit" into analysis, we are
actually introducing limit-avoidance, where the limiting value is defined by
the property that the values in a sequence avoid that limit by an arbitrarily
small amount when the corresponding parameter [the index n or the sequence
s, of n-th partial sums, say, or the increment c~in the difference (] (x + ~) -- ] (x))
for continuity I avoids its own limiting value (infinity and zero in these examples).
The new analysis of BOLZANO'S pamphlet and developed in CAUCHY'S
text-books was nothing else than a complete reformulation of the whole of
analysis in limit-avoidance terms...
No, no, and no. BOLZANOand CAUCHYknew better than I. GRATTAN-GUINNESS
what was convergence and what was continuity. It is true there are bad 19thcentury
textbooks where you can find such silly definitions, but this was neither BOLZANO'S
fault nor CAUCHY'S****
6. Cauchy's Character
To explain why CAUCHY plagiarized BOLZANO, I. GRATTAN-GuINNESS writes
a story about what he calls the Paris clique of mathematicians. No doubt he
has studied that chronique scandaleuse better than anybody else. But if the
secrets of that society are as relevant to understanding the history of mathematics
as he suggests, why does he wrap himself in veils of mystery rather than disclose
them ? Why does he concoct a pompous story from plain historical facts and
unfathomable allusions ?
Whoever has studied CAUCHY'S work knows how chaotic it is. A proposition
is stated, then refuted, only to be stated once more; a procedure is severely
criticized, only to be applied successfully at the next opportunity; for no reason
* p. 383, 12 f.b.
** p. 387, 5.
*** p. 378, t3 f.b. -- 5 f.b.
**** When I. GRATTA•-GUI•NESS lectured at the Utrecht Mathematical Colloquium
everybody protested. An hour later people thought they had convinced him. It is
a pity they had not done so.
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Did Cauchy Plagiarize Bolzano ? 389
notations are changed back and forth. No, I. GRATTA~-GuINNESS says, stating
a certain apparently wrong theorem was a strategic move in the secret game of
the Paris clique. As long as I do not know the secret information on which such
conclusions must be based, I cannot challenge them*.
A critic is on a safer ground when I. GRATTAN-GuINNESS gives his sources.
To prove that CAUCHY took sides in the quarrels of the "Paris clique" (which
is utterly improbable) he mentions, in the same work, "fawning references to
the powerful secrdtaire perpftuel (FouRIER)" and "attacks on the declining
POISSON"**. Any one who checks the sources will find that neither is the reference
to FOURIER fawning nor is Polsso~ attacked. The first reads
X'P
si l'on ddsigne avec M. FOURIER avec f ] (x)d x l'int6grale d6finie, prise entre
les limites x = x', x = x" . . . ~"
and it is the style in which such acknowledgements have been made a thousand
times by mathematicians. At the second place quoted we find CAUCHY, rather
than attacking POlSSON, explaining why he had overlooked certain consequences
of his theory which had meanwhile been discovered by •OISSON.
To understand what citations mean for mathematicians, it would be worthwhile
to make a statistical study of them, say around CAUCHY. Isolated examples
are of little value. At the very period when, according to I. GRATTAN-GUINNESS,
CAUCtIY had reasons to fawn FOURIER and to attack PoISSON, he used the introduction
to his Cours d'Analyse to extend his thanks to LAPLACE and POISSON,
who had advised him to publish his courses, and at the end of the same introduction
he acknowledged the good counsel he had received from Polsso~, AMPERE
and CORIOLIS. Should we interpret these acknowledgments, too, as attacks ?
It is well known that CAUCHYwas a strange fellow, and to prove it, there is
no need to invent strange stories about him. The strangest is his quixotic conduct
after the July revolution of 1830, when as a lone paladine he followed his king
to his exile court in Prague. He was a religious and political dogmatic who often
exhibited an appalling lack of human relations.
* A characteristic pomposity is the remark in footnote 85 that the Proc~s verbaux
des sdanees de l'Acaddmie tenues depuis la ]ondation jusqu'au mois d'aoC~t 1835 (t0 vols;
19t0-22, Hendaye) "are an invaluable source of historical insight into the period
1795-1835, when the rivalries were at their height. They give the minutes of all the
private meetings of the Acaddmie des Sciences, which the participants can hardly
have expected to be published!"
In fact, there is little that might be regarded as sensational to be found in the
Proc~s verbaux. The style is the same as that of the later Comptes Rendus; the greater
part is routine business. The meetings were not private but public. All spontaneous
remarks were afterwards carefully edited or omitted; the oral text is better reflected
by the newspaper reports.
** CAUCHY, Oeuvres (1) 1, 340 and 189-191; another source mentioned is not
accessible to me.
The adjectives "powerful" and "declining" are melodramatic stereotypes. There
has never been any secrdtaire perpdtuel who was not powerful, but I doubt whether
FOURIER was more so than his predecessors or successors. Facing a powerful secrdtaire
perpdtuel, POISSON, too, needed an adjective though it is a pity that I. GRATTANGUINNESShit
on one that is so trivially mistaken as is "declining".
28a Arch. Hist. Exact Sci., Vol. 7
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390 H. FREUDENTHAL:
There is a story about CAUCHYand a manuscript of ABEL. In t826, when his
first important work had yet to appear, ABEL visited Paris. A few times he met
CAUCHY, who at that period was interested only in mathematical physics. In
Paris ABEL wrote the famous work he presented to the French Academy in October
t826. In t829 he died. In the late thirties the editor of his Oeuvres, who knew
about the manuscript, tried to get it back from the Academy, but it could not
be found. Suddenly, in t841, the text of the manuscript appeared in print in a
publication of the Academy, though, strangely enough, the manuscript itself
was still lost.
This trackless manuscript has always been an exciting feature in the melodramatic
life of ABEL,who according to the stories died in misery, oblivion, and
disappointment. (It has long been known that this story is untrue*.)
In such a story a villain is needed. According to old LEGENDRE,ABEL'S
paper was illegible, so the referees, CAUCHYand himself, could not read it. Even
today it is commonly believed that the manuscript was lost by CAUCHY'Sneglect.
In t922 a copy of CAUCHY and LEGENDRE'S report on ABEL'S paper, dated
29 June t829, was discovered**; it proved that CAUCHY'Saccount of his role in
the story was correct. It is obvious that CAUCHY had no further business
with ABEL'S manuscript, since after the July revolution of 1830 he went abroad
and did not return before t838. The academician LIBRI, however, who to annoy
other people, had invented the main facts in ABEL'S melodramatic life, got some
business with ABEL'S paper; in any case he read the proofs, though according
to him without the manuscript. LIBRI was a mediocre mathematician who became
famous by his sudden departure to London in t848, when he was accused of
having over many years stolen from the French public libraries a million's worth
of rare books and manuscripts. Thus it was not too far-fetched to look into LIBRI'S
estate in the Moreniana library in Florence. Finally, in 1952, VIGGOBI~UN did
so, and he found ABEL'S manuscript***. A written explanation of it by LEGENDRE
had been published in World War II**** but had not been noticed. It readst:
Ce M6moire a 6t6 mis d'abord entre les mains de M. Le Gendre qui l'a
parcouru, mais voyant que l'6criture 6toit peu lisible et les caract~res alg~briques
souvent real form6s, il le remit entre les mains de son confrere, M. Cauchy
avec pri~re de se charger du rapport. M. Cauchy distrait par d'autres affaires
et n'ayant re~u nulle provocation pour s'occuper du M6moire de M. Abel,
attendu que celui-ci n'6tait rest6 que peu de jours ~ Paris apr~s la pr~sentation
de son M6moire ~t l'Acad6mie, et n'avait charg6 personne de suivre cette
affaire auprfis des cornmissaires, M. Cauchy, dis-je, a oubli6 pendant tr~s
long temps le M6moire de M. Abel dont il 6toit d@ositaire. Ce n'est que vers
* Read VIGC-OBRUN'S debunking paper in Journal r. u. angew. Math. 193
(1954), 239-249.
** D. E. SMITH,Amer. Math. Monthly 29 (1922), 394-5. Among my autographs,
29. Legendre and Cauchy sponsor Abel. -- It is in agreement with the Proems verbaux
(el. footnote*, p. 389).
*** See footnote *.
**** G. CANDIDE, Sulla mancata pubblieazione, nel t 826 delia celebre Memoria
di Abel. Tip. Editr. "Marra" di G. Bellone, Galatina t942, XX.
t Journ. r. u. angew. Math. 193, 244-245.
Text 23: H. Freudenthal (1970–1971). “Did Cauchy Plagiarize Bolzano?” Archive for
History of Exact Sciences, vol. 7, no. 5, pp. 375–392.
Summer University 2012: Asking and Answering Questions Page 359 of 479.
Did Cauchy Plagiarize Bolzano ? 39t
le mois de mars 1829, que les deux Commissaires apprirent, par l'avis que
l'un d'eux r6~ut** d'un savant d'Allemagne, que le M6moire de M. Abel, qui
avait 6t6 present6 ~ l'Acad6mie, contenait ou devait contenir des r6sultats
d'analyse fort interessants, et qu'il 6tait 6tonnant qu'on n'en efit pas fair
de rapport ~t l'Acad6mie. Sur cet avis M. Cauchy rechercha le M6moire, le
trouva et se disposait ~t en faire son rapport; mais les Commissaires furent
retenus par la consid6ration que M. Abel avait d6j~t publi6 dans le Journal
de Crelle une pattie de son M6moire pr6sent6 ~ l'Aead6mie, qu'il continuerait
probablement ~t faire paraitre la suite, et qu'alors le rapport de l'Acad6mie,
qui ne pouvait ~tre que verbal, deviendrait intempestif*.
Dans cet 6tat de choses nous apprenons subitement la mort de M. Abel,
perte tr~s fAcheuse pour les sciences, et qui parait maintenant rendre le rapport
n6cessaire pour conserver s'il y a lieu, dans le receuil des savants 6trangers,
un des principaux titres de gloire de son auctor**.
This unveils the mystery around ABEL'S manuscript. It is not unusual that
referees neglect their task, in particular, if they are not interested in the subject
or if it is the work of a virtually unknown author, though I agree that CAUCHY
was usually more careful. Delays of 10-t 5 years in printing treatises accepted
by the French Academy were not unusual either; every publication needed a
royal authorization. In ABEL'S case it may have played a role that the essential
part of the manuscript had already been published in "Crelle's Journal".
I. GRATTAN-GUINNESS'report on this event is a distortion of the story as it
is known now. He omits all evidence that is in favour of CAUCHY,and he falsifies
two points***:
First he claims that the neglected manuscript
... was the paper which ushered in the transformation of LEGENDRE'S theory
of elliptic integrals into his own theory of elliptic functions...
to add one more melodramatic feature. The paper on elliptic functions was
published in Crelle's Journal. The manuscript in question was about "ABEL'S
theorem"; an extract also appeared in Crelle's Journal.
Second, he claims:
CAUCHYtook it and, perhaps because of ABEL'S footnote against him, ignored
it entirely: only after ABEL'S death in t829 did he fulfil a request to return
it to the Acaddmie des Sciences.
The reader can check that this is in all essentials contrary to LEGENDRE'S
report. If I. GRATTAN-GuINNESSis in the possession of secret information that
refutes LEGENDRE'Sreport, he should reveal his sources. Meanwhile I am entitled
to consider LEGENDRE'S report as correct.
* The procedure of a formal report was applied only to manuscripts; printed
pieces submitted to the Academy were given a rapport verbal.
** Sic.
*** p. 393.
28b Arch. Hist. Exact Sci., Vol. 7
Text 23: H. Freudenthal (1970–1971). “Did Cauchy Plagiarize Bolzano?” Archive for
History of Exact Sciences, vol. 7, no. 5, pp. 375–392.
Summer University 2012: Asking and Answering Questions Page 360 of 479.
392 H. FREUDENTHAL:Did Cauchy Plagiarize Bolzano ?
I. GRATTAN-GUINNESScontinues:
...there is one aspect of it which has been little remarked upon but which
shows the depths to which CAUCHYcould sink.
The evidence I. GRATTAN-GuINNESSproduces for CAUCHY'Smoral downfall is an
expos6 of 1841, where CAUCHYfirst praises ABEL and then refutes the story that
ABEL died in misery. We now know that CAUCHY'Sexpos6 is correct.
I. GRATTAN-GUINNESSdoes not explain in what CAUCH¥'Sdownfall consisted,
but anyhow it was a downfall and
...anyone capable of writing in this manner, knowing the negative role
played by himself in the matter under discussion, would hardly think twice
about borrowing from an unknown paper published in Prague without acknow-
ledgment.
Anyone ? Maybe. But CAUCHYwas someone.
Mathematical Institute
Rijksuniversiteit
Utrecht
(ReceivedFebruary 1, 1971)
Text 23: H. Freudenthal (1970–1971). “Did Cauchy Plagiarize Bolzano?” Archive for
History of Exact Sciences, vol. 7, no. 5, pp. 375–392.
Summer University 2012: Asking and Answering Questions Page 361 of 479.