AL L V A R G U L L S T R A N D
How I found the mechanism of intracapsular
accommodation
Nobel Lecture, December 11, 1911
With the normal eye, except in old age, one can see clearly at different distances.
To permit this, the lens alters its shape, so that the refractive power
of the whole of the eye's optical system changes, in a way which resembles
most closely that which takes place in a camera when one objective lens is
exchanged for another without altering the extension of the bellows. This
alteration of the eye's optical system is known as accommodation, and its
mechanism has for long been an object of interest for research. In this connection,
however, research has exclusively dealt with the way in which the alteration
in the shape of the lens is brought about, and what alteration takes place
in the eye apart from the lens. Now since the lens is enclosed in a capsule,
one can say, in brief, that it is the extracapsular accommodation-mechanism
which has so far been investigated and discovered. But if we ask what takes
place within the lens-capsule during accommodation, we then pose the problem
of the intracapsular accommodation-mechanism.
Without any knowledge of the structure of the lens, one might perhaps be
inclined to think that this is a very simple problem. If one alters the shape of
a water-filled bladder, some of the water particles will be displaced in relation
to the others, and the streaming motion will not be of any particular significance.
And the behaviour would be identical, if the bladder were filled, not
with water, but with a substance of the consistency of the lens, in which the
particles were freely movable in relation to one another. But it is just this
respect in which the lens differs. It consists throughout of an infinite number
of skilfully arranged, microscopically fine fibres, which terminate at various
depths below both surfaces of the lens, and which run from one end to the
other in coils lying in the direction of its edges. Since, on account of the speed
with which accommodation takes place, we can exclude the possibility of
fluid flowing through the fibre walls, and since the lens substance, as one can
easily convince oneself, is clearly lacking in elasticity, it is obvious that on
change in the shape of the lens during accommodation the various lens-fibres
must remain of constant volume, and that a change of shape can only come
M E C H A N I S M O F I N T R A C A P S U L A R A C C O M M O D A T I O N 415
about through the displacement of the lens fibres relative to one another. But
the fibres are fastened at both ends to the immediate surroundings, and it is
therefore only in their central part that displacement of the individual lens
fibres in relation to one another can take place, as a result of changes in the
shape of the coils. It should now be obvious that to a given change in the
shape of the lens there must correspond a definite displacement of the fibres
within the lens-substance depending on their arrangement. Thus, there must
exist a definite mechanism for accommodation within the lens capsule.
I shall deal later with the question of how the most essential feature of this
mechanism could have been predicted on the basis of the anatomical structure
of the lens. That the most important aspects could have been predicted occurred
to me first, when I found the mechanism during the research I undertook
to gain accurate knowledge concerning image-formation in the eye.
These researches proved difficult, because the refractive power of the lens is
by no means that of an ordinary homogeneous medium, the same throughout,
with no change from point to point, since the lens substance consists of
a heterogeneous medium with, at least in the young, a continually varying
refractive index. The laws governing optical image-formation in such media
were completely unknown, and much of what was thought to be known,
proved to be wrong.
Even if it had been simply my intention to find these laws, I was nevertheless
obliged in order to do this to realize the reformation of the theory of optical
image-formation in general, which I had started and also have carried through,
before daring to apply myself to the study of the problem in heterogeneous
media. Essentially, therefore, the way in which I came to an understanding
of the mechanism of intracapsular accomodation can best be described by
giving a survey of my more important work on optical image-formation in
general, and that in the eye in particular.
Most people even now think of optical image-formation as being a focusing
of light rays issuing from one point on the object, to produce a corresponding
point in the image. Most people would also agree that the impression one
has from the picture which a good photographic objective projects on a
screen, strengthens this idea. But it is obvious here that if the laws of optical
image-formation were generally valid one should obtain an equally good
image with other optical systems. However, since this is by no means the case,
it follows that this theory of optical image-formation is not applicable here;
thus a general law has to be found that does not simply relate to individual
cases where the idea of a point is not to be taken in too rigid a sense.
416 1911 A.GULLSTRAND
A special case is found in the immediate neighbourhood of the axis of a
centrally-aligned optical system consisting of rotation-areas. If one cuts off
such an optical system with a small cylinder whose axis coincides with that
of the optical system, one can say, with enough accuracy for practical purposes,
that an object enclosed within this cylinder is reproduced through
the system, point for point, and the accuracy of this statement increases in
proportion as the diameter of the cylinder is reduced. But this is not strictly
accurate, until the cylinder becomes so narrow that it coincides with the axis.
One can express this in another way, by saying that point-by-point imageformation,
with complete focusing, takes place in the paraxial region, but
it should be noted that this region, if taken as being finite, is to be considered
as a thread-like, enclosed axis, and that only the parts of the refracting
surfaces of the optical systems that are cut off by this axis, come into consider-
ation.
In order to represent this case approximately by physical means we must
narrow the optical system as much as possible, without producing disturbing
diffraction phenomena. If we do this with an ordinary biconvex lens, which
we then place in the objective of a photographic camera, we find that a tolerable
image can be produced in the neighbourhood of the point at which the
optical axis of the lens intersects the plate; but if, without using the focusingscreen,
we examine under high magnification the image now floating in the
air, we also find, even with monochromatic light, that the mechanism of
image-formation here differs from that in which every light ray arising from
a point on the object strikes the same point in the image.
With a small aperture, and therefore, a small central field, one can say that
the present theory of image-formation corresponds to reality with sufficient
accuracy for practical purposes, so far as axially-symmetrical systems are concerned.
In this way, a straight line in the object is always reproduced by a
straight line in the image. This image-formation is therefore termed collinear.
An additional feature of the collinear image-formation of an object-area at
right angles to the axis, is conformity of the image with the object.
It follows clearly from what I have said, that when we consider the actual
circumstances of optical image-formation within a system with a large aperture
and a large image-field, the law of collinear imagery is not even approximately
valid for the general case. Thus, when the general laws of optical
image-formation were unknown, there was no alternative except to define
the realities as deviations from the supposed ideal of collinear image-formation.
Abbe preceded most others in this connexion and deserves most praise
M E C H A N I S M O F I N T R A C A P S U L A R A C C O M M O D A T I O N 4 1 7
for the present upsurge in optical techniques. The fact that this splendid progress
was achieved without knowledge of the general laws of image-formation,
is a result of the fact that what investigators were trying to achieve was precisely
that axially-centred system which comes as close to the ideal of collinear
imagery as possible, and that this can be solved by trigonometric methods,
which work independently of the laws of image-formation. In other words,
specifically-designed optical instruments, such as the modern photographic
objective, come very close to the ideal of collinear imagery. But as yet we
had no precise knowledge of the working of an optical system not built according
to these construction principles. Moreover, for technical purposes it
is sufficient to say that a good instrument behaves with a greater or lesser
degree of accuracy, according to the laws of collinear image-formation and
this is enough from the technical viewpoint. And science often draws its impetus
from man's practical needs, and if technical optics had not required
general laws of image-formation, it seems clear that science would not have
found these laws. Such a need was strongly felt in opthalmology and physiological
optics.
Here the first requirement was for a knowledge of the general laws governing
the focusing of light rays. When monochromatic light originating
from a point and passing through an optical system, is not focused again to
a point, how is the refracted ray built up, since it is not homocentric? Geometrical
optics had answered this question from its own viewpoint, for the
special case where the light-point is situated on the axis of an axially-centred
system with spherical surfaces. The central ray of the refracted bundle coinciding
with the axis is intersected by the adjoining rays, whatever their direction
of travel, in one point, the focal point, and the course of the remaining
rays is determined by the so-called spherical aberration. But already, in the
case where the light-point does not lie on the axis of the system, this theory is
erroneous. If we depict the rays which, in a normally centred system such as
a telescope, pass from an eccentric object-point through the centre of the effective
aperture, that is to say through the centre of the objective in the astronomical
telescope and in the comparable and now common prismatic telescope,
then we find that, after passing through the instrument, these rays
generally intersect the adjoining rays in two separate focal points. The phenomenon
known as astigmatism follows, and it will be readily understood
that astigmatism represents the general case, and that its absence - anastigmatism
- constitutes the special case, which is characterized by the fact that
both focal points coincide.
418 1 9 1 1 A . G U L L S T R A N D
The quest for knowledge of the general laws governing the focusing of
light rays led us to a direct investigation of the constitution of astigmatic
bundles of rays. As with anastigmatic bundles, the more nearly they can be
considered to be homocentric, the smaller the iris aperture is relative to the
distance of the focal point from the iris, so a type, the so-called Sturm conoid
had been discovered, which was considered to represent an astigmatic bundle
with a small aperture. It was consistently forgotten, however (and this omission
gives a most striking example of the peculiarities of scientific psychology)
that the Sturm conoid was derived under conditions where the iris aperture
is infinitely small, not only compared with its distance from the two focal
points, but also compared with that separating these points, the so-called focal
range. But whereas in the case of astigmatism, in a usual centred optical
system near the focal point the refracted ray bundle lies outside the axis, in an
astigmatic eye the aperture of the ray bundle is as a rule greater than the focal
range, practically never small in comparison with it. A simple investigation
shows in confirmation of this, that the Sturm conoid cannot be used as a
model for the narrow astigmatic ray bundles, in the case in which technical
and physiological optics are most interested. Its role is in practice restricted to
representing ray bundles of the type found in the laboratory.
A bundle which radiates from an object-point and is refracted in an optical
system with single redacting media separated by continuous surfaces, has the
property that one can construct a plane from any one point to any other
(the so-called plane of balance) which is at right angles to the combined rays
of the bundle. The investigation of the structure of ray bundles is thus the
same as that of the structure of standard bundles. Such an investigation comes
under the heading of differential geometry, in that the properties of the standard
bundle are studied under the nearest conditions to a standard environment,
just as in general, when the equations of surfaces cannot be expressed
explicitly, the surfaces are studied under the conditions approximating most
closely to a certain set or arbitrarily chosen point. In this study a closer knowledge
of the bundle is obtained by taking differential quotients of a higher
order in the equation of the surface. Now the Sturm conoid was derived by
ignoring differential quotients of a higher order than the second, and therefore
the first task I set myself was the investigation by differential geometry of
astigmatic ray bundles, taking into account all the differential quotients of
the third order, in the balance-plane equation. I found the quantities determining
the structure of the bundle, and I also derived the formulae for calculating
these quantities in optical systems of all types. By simple verifying
M E C H A N I S M O F I N T R A C A P S U L A R A C C O M M O D A T I O N 4 1 9
experiments I showed that the knowledge gained in this way concerning the
structure of astigmatic rays is sufficient for practical purposes, with the apertures
normally found in optical instruments. Laws were thus also obtained
governing the occurrence, but not the essentials, of the so-called coma error.
Up to now we had only been able to make calculations for the paraxial region,
and thus not even for a ray bundle issuing from a point located a finite distance
from the axis of a centred system; in other words in a centred system of
spherical surfaces means were known of calculating coma error with an infinitely
small angle of incidence, but not with a finite angle of incidence.
In this, the simplest case, when the optical system consists of centred spherical
surfaces, and the equal rays of the ray bundle intersect the optical axis
at a finite angle, the problem should also be of interest in design optics. Here
the bundle is determined by two asymmetry values, i.e. to use the language
of technical optics, there are two types of coma error in such a system. But
when the representatives of geometrical optics started their investigations they
were unacquainted with mine - they first of all studied the first type of
coma error, and then discovered one too many. The reason for this is that in
geometrical optics it is generally considered that it is possible to proceed from
a two-dimensional concept, while the problem here is three-dimensional and
must be treated as such.
In considering simpler questions the normal eye can justifiably be considered
as a centred optical system, but the ray that in the case of sharp reproduction
goes through the centre point of the pupil, does not coincide with the axis
but makes a finite angle with it. Relative to this ray, the so-called sight line,
the refracted bundle is astigmatic with finite asymmetry values, or, putting
it another way, it is subject to coma error. On revising Helmholtz' eye diagram
in the light of my formulae, I obtained a figure for these asymmetry
values, with the result that they appeared to be unharmful to vision; but the
knowledge obtained in this way concerning the structure of the ray bundle
refracted in the eye was insufficient, for without a knowledge of the laws
governing aberration in the eye we cannot expect in physiological optics to
use this knowledge in determining the nature of the accommodation mech-
anism.
I have just said that physiological optics occupied itself with the focusing
of rays along the axis of a centred system, and expressed the deviations from
homocentricity that occur with monochromatic light, as spherical aberration.
This unfortunate expression was based on the fact that spherical surfaces only
had been studied, and the aberration depends on the shape of the surfaces.
420 1 9 1 1 A . G U L L S T R A N D
Helmholtz had already substituted for this term that of monochromatic aberration.
The value for this, applicable in cases where the angular aperture is
not too great, had long since been calculated along the axis in a centred instrument
with spherical surfaces, and was given in textbooks on geometrical optics.
But this particular case is inapplicable in relation to the eye, and it was
thus necessary to derive a general theory of monochromatic aberrations, in
order to know the basic characteristics of the focusing of the rays in the eye.
To do this, it was necessary to take all the differential quotients up to the
fourth power in the normal plane equation. I carried out this mathematical
investigation in its entirety for the astigmatic ray bundle.
The difficulties were greater in the case of anastigmatic ray bundles, and I
was unable in my early work to make a complete study of their structure
and of the geometrical significance of the asymmetry values. This study in
fact necessitates a detailed examination of a field of mathematics that has so
far hardly been investigated, involving the problems of the so-called lines of
principal curvature of single points, and the relation between the two caustic
surfaces at common points of contact. It was only after carrying out such an
investigation that I was able to find the knowledge necessary for dioptrics in
Fig. I
M E C H A N I S M O F I N T R A C A P S U L A R A C C O M M O D A T I O N 4 2 1
general and in particular for the dioptrics of the eye, concerning anastigmatic
ray bundles, and astigmatic ray bundles in which the focal range is small in
relation to the aperture; and only then was it possible to carry out a physiological
examination of optical image-formation in the eye. The methods I developed
for this purpose showed that on the one hand the aberrations in the
ray bundle refracted in the eye are of a very complex and still unknown type,
and on the other the radiant emission seen around light-points, e.g. fixed stars,
is an expression of the individual character of the aberration. Thus one can
reproduce experimentally this radiant emission round images oflight-points,
by designing the optical system so that the refracted bundle is characterized
by the aberration values required therefor. Fig. 1 shows such an image of a
light-point. I obtained it by photographing a greatly reduced image of the
sun with an objective consisting simply of spherical and cylindrical glass lenses.
The picture was produced by photomechanical reproduction straight
from the original negative.
The physiological investigations on the eye carried out by means of the
above-mentioned methods which I developed showed clearly that the importance
usually ascribed to diffusion circles in the case of optical imageformation
is exaggerated, because otherwise the eye, with the large disc of
confusion found in it, would have nothing like the sharpness of vision that it
in fact has. It has been supposed that the very thin cross-section of a ray bundle
is of major importance in image-formation, but it is clear, both from physiological
investigations carried out on the eye, and from physical experiments,
that in image-formation the light distribution over the cross-section
of the ray bundle is of primary importance and the size of the cross-section
is only of secondary importance. Now, since the light intensity is greatest
when rays running very close to each other converge together, and since the
points in which such rays converge are located on the so-called caustic surfaces,
it is clear that it is these surfaces, and the geometrical values which, on
the basis of my previous investigations, characterize them, that are of central
importance for the problem of image-formation.
At the same time as the diffusion circles lost their importance, it became
necessary to study the effect of diaphragms from a new viewpoint. This had
been studied mainly in relation to restriction of the ray bundle, an investigation
which sufficed for the optical instruments where collinear imagery was
realized more or less ideally. But in the case of image-formation in general
another aspect of the function of the diaphragm comes to the fore, viz. the
optical projection through the diaphragm centre or another selected projec-
422 1 9 1 1 A . G U L L S T R A N D
tion centre in it. Just as it is possible to produce a more or less geometrical
projection in a camera without a lens or by shadowplay, so an optical system
can be inserted during the tests so as to preclude sharp image-formation approximating
to optical projection. If we draw the rays that in the diaphragm
of the optical system pass from the different points of an object through the
projection centre, and study these rays on a screen, we find that each point
of the object surface corresponds to a point on the screen; thus a point-bypoint
correspondence is produced through the optical projection. And if in
addition the opening of the diaphragm is reduced so far as is possible without
producing interference through diffraction, then, just as in a camera without
a lens and with shadowplay, we get a picture which, while it is lacking in
sharpness and does not geometrically resemble the object, does however give
a recognizable image of the object.
Now optical image-formation can be distinguished from optical projection
in that it involves focusing of the rays, and it can also be characterized mathematically
on the basis that the convergence of the rays must be total to at
least the first order. We already knew that optical point-by-point imagery
does not take place in the general case, and that it does not occur even in a
centred optical system in the case of excentrically-located object points. The
Abbe school of investigators had overcome this difficulty by assuming two
different collinear image-formations, but in this way they were only able to
take into account the rays that run in the meridianal plane and in the plane
perpendicular to this, rays which together constitute a very small fraction of
the light concerned in image-formation. Thus this fiction could have no justification
except that the general laws were still unknown.
These general laws followed from my basic equation of optical imageformation,
and they stated first and foremost that optical image-formation
of lines generally exists. In any optical system, through the constants of the
system and the position and shape of the object surface, there are two systems
of reproducible lines running along it, which intersect each other at various
points at finite angles and are reproduced with total focusing of the rays to at
least the first order, each system being on a separate picture surface. No other
optical image-formation takes place, and the mutual points of contact between
the two surfaces of the picture are only point-by-point pictures of the
individual points of the reproducible line system. Just as ideally we can popularize
theories of collinear image-formation by saying that in the case of narrow
diaphragm each ray emitted from an object passes through a corresponding
picture-point, thus analogously in reality we can popularize theories
M E C H A N I S M O F I N T R A C A P S U L A R A C C O M M O D A T I O N 4 2 3
of optical image-formation by saying that in the case of narrow diaphragm
each ray emitted from a reproducible object line passes through a corresponding
picture line although with these lines point-by-point imagery does not
take place. In a plane that lies perpendicular to the axis of a centred system,
the reproducible lines comprise meridians and parallel circles, and the two
surfaces of the picture meet at the points of intersection with the axis. In a
plane passing through this point perpendicular to the axis, a more or less
point-by-point imagery is found in the centre, while generally at the periphery
optical projection only takes place and a reproduction of the one line
system, usually meridian lines, may be found in the intermediate zone.
Only after this study of the general optical image-formation had enabled
me to ascertain its character and fundamental equation was it possible with
any prospect of progress to come to grips with the problem of image-formation
in heterogeneous media and particularly the eye lens. The refractive index,
i.e. the factor which at a certain angle of incidence determines the direction
of the light when it passes from one medium to another, varies in heterogeneous
media from one point to the next and the result of this is primarily
that the path of the light in such a medium is not a straight line. Instead of
travelling in straight lines in common media (the light rays), light follows
curvilinear paths (trajectories as they are termed) in heterogeneous media of
continually variable refractive index; the shadow thus never lies in the same
straight line as the object and the light source. One such medium is the earth's
atmosphere and as a consequence a star that has just appeared to rise above the
horizon has in fact still not passed it. Astronomers and physicists had accordingly
studied the phenomenon known as terrestrial or astronomical refraction
but confined themselves to determining the shape of the trajectory so that the
problem ofimage-formation starts at the very point where these studies cease.
On the other hand the situation in the eye-lens had given rise to physical and
physiological research which has however, owing to the wrong methods
being used, yielded mainly incorrect results.
I first showed by means of a general mathematical study of such media that
the fundamental equation and hence the laws of image-formation governing
normal homogeneous media also apply in media with a continuously variable
refractive index and for the passage of light between different such media as
well as between one such medium on the one hand and a homogeneous
medium on the other. After I had also deduced the formulae required for
calculating the image-formation and aberration, I was in possession of the
mathematical tools which are necessary and, in conjunction with physiolog-
424 1 9 1 1 A . G U L L S T R A N D
ical examinations, adequate for gaining a detailed knowledge of the image
formation in the eye and of the intracapsular accommodation mechanism.
For these physiological examinations, however, an accurate knowledge
was required of the surfaces and thickness of the cornea. By and large, research
workers had been content to ignore the effect of the posterior surface since
the difference in refractive index between the corneal substance and the aqueous
humour is relatively small. They then had only the anterior surface to
consider and this was sufficiently well known. However, the radius of the
posterior corneal surface had been determined, albeit by unreliable methods,
and was at least enough to show that for certain examinations it was inadmissible
to neglect the refraction at this surface. I had therefore to repeat these
examinations with new, precise methods before I could proceed further. The
extent to which this was necessary can be appreciated most readily from
the fact that the thickness of the cornea at its top has generally been estimated
as about I mm, whereas my studies showed that in round figures it is
only half a millimetre, a result moreover, which had been obtained earlier by
Blix and to which inadequate attention was paid.
To compute the refraction of a lens which is a heterogeneous medium, it
is advisable to take special account of the change in convergence of the light
which occurs during passage through the heterogeneous medium. This change
is wholly analogous to the diffraction in an optical system, and therefore the
substance of the lens can be treated as an optical system in its own right, the
"core lens" as it is known. To calculate its effect on the transient light the
law governing the variations in refractive index needs to be known - in other
words an indicial equation is required. Such an equation has been established
by the physicist Matthiessen and in deriving it he only took account of up to
second-order differential quotients. The refractive-index determinations that
have been carried out have shown this law to be sufficiently accurate in itself,
provided however that it does not clash with known mathematical or physical
facts. It has been used for integration in the same way as if completely
accurate and has yielded, inter alia, a simple formula for the so-called total
index of the lens. This latter is an imaginary refractive index for a homogeneous
lens which, for the same surfaces, has the same refraction as the eye-lens.
According to Matthiessen's law the total index exceeds the refractive index
at the centre by the same amount as the latter exceeds the refractive index at
the surface.
Yet this Matthiessen indicial equation is so hopelessly at variance with facts
that it immediately proved necessary to include all the differential quotients
M E C H A N I S M O F I N T R A C A P S U L A R A C C O M M O D A T I O N 4 2 5
of up to the fourth order to obtain a sufficiently exact indicial equation. This
then contains seven constants which remained to be determined. Were the
eye-lens of the same size as the object-lens in a telescope, these constants could
be found by determining the refractive index at a suitable number of points
using the refractometer. But owing to the small distances in the eye-lens and
the relatively insignificant differences in the refractive indices, not more than
three sufficiently reliable equations could be derived in this way: it could be
assumed that the values at the centre, the poles and the edges had been determined
with sufficient accuracy. I derived a further two equations from the
curvature of the surfaces. If all the points in the lens substance are found at
which the refractive index has the same value, together they form an isoindicial
surface. The anatomical structure of the lens lends definite support to
the assumption that at both poles the isoindicial surfaces have the same radius
of curvature as the lens surfaces, and from there these equations follow. One
equation is also derived from the loss of refraction sustained by the eye on
extraction of the lens and here too the exact values for the cornea1 system
have to be used and the aberration along the eye's axis as measured by direct
experiment has also to be included in the calculation. The refraction of the
"core lens" from which the total index of the lens follows, is determined in
this way. This is not determined by Matthiessen's law when differential quotients
of higher than the second order are included in the indicial equation.
I was unable to find any reliable physiological method to derive the seventh
and last equation but I have shown that if this equation is derived by assuming
Matthiessen's indicial equation to be valid along the axis, no inadmissible
approximation is made. On the one hand it has previously been found that
this equation agrees so closely with the observations that it can be applied in
every case where it does not conflict with fact (which does not apply here),
and on the other hand I have been able to demonstrate that discrepancies as
regards the variation in index along the axis have no significant effect on the
result, even when they become so large that they certainly lie outside the
bounds of possibility.
After thus finding the indicial equation for the lens at accommodation rest I
was able, using the formulae I had previously derived, to work out fully its
optical system. The same task then remained in respect of the accommodating
lens. Following Helmholtz' example, repeated measurements had been made
of the changes undergone by the lens during accommodation but precise
details were lacking of the degree ofaccommodation corresponding to a given
degree of change in the shape of the lens. It is well known that it is mainly the
426 1 9 1 1 A . G U L L S T R A N D
anterior lens surface which increases in curvature during accomodation and
at the same time moves slightly forward, although so far nothing is known for
certain about an accommodative dislocation of the posterior lens surface, the
curvature of which changes only slightly during accommodation. Hence
there was a need to measure by means of a new and exact method the radius
of the anterior lens surface in the same eye, both at rest and during accommodation
for an accurately measured minimum distance. In this way the refraction
of the core lens is obtained for the state of accommodation which is
determined by the radius of the anterior lens surface. Furthermore I introduced
the two conditions which follow from the anatomical structure of the
lens, namely that no compression can occur in the centre when the shape
changes, and that the volume limited by the largest closed isoindicial surface
cannot undergo any change either. I then had all the data which, over and
above knowing the thickness of the lens and the radius of curvature of its
surface, are required to find the indicial equation of the accommodating lens
and then calculate from that its optical system.
Here I have a diagrammatic meridional section of the lens firstly at rest and
secondly at maximum accommodation, at about the age of twenty (Figs.2
and 3). In both diagrams there are three continuous lines, the outermost of
which represents the section through the lens surface, the other two being
meridional sections of two isoindicial surfaces. The refractive index of the section
passing through the poles of the lens is 1.386, so falling short of the reFigs.2
and 3
M E C H A N I S M O F I N T R A C A P S U L A R A C C O M M O D A T I O N 4 2 7
fractive index in the centre, 1.406, by two units of the second decimal. The
line at the centre represents the isoindicial surface where the refractive index
is only two units of the third decimal less than in the centre. The distance
between these curves illustrates the fact that from the surface towards the
centre the refractive index increases initially at a relatively fast rate which rate
then decreases progressively towards the centre.
I obtained the cross-section of the isoindicial surfaces by computing the
coordinates point by point from the indicial equations.
Their shapes are therefore quite correct, with the sole reservation that it is
possible, although so far unproved, that the accommodating core lens may
exhibit a slight degree of asymmetry. I only wish to add that provided an
arbitrary difference between the radii of curvature within the limits specified
by measurements carried out hitherto is introduced into the calculation, the
resulting asymmetry of the core lens will not be significant and in any case
will have absolutely no effect on the result I will outline.
Owing to our defective knowledge, the shape of the cross-section of the
lens surfaces is drawn in a more arbitrarily schematized manner. As far as
possible I have drawn them as parabolas - which seems most closely to agree
with the present state of our knowledge - and connected the ends of the parabolas
by arbitrary curves, approximately ensuring that the volume contained
between the lens surfaces and the largest continuous isoindicial surface remains
practically unchanged during accommodation. For simplicity's sake I thus
drew the whole accommodating lens symmetrically round the equatrial
plane, notwithstanding that very probably such is not the case. For example,
the anterior lens surface could be hyperbolic with constant top radius and thus
the lens would be of asymmetrical shape with symmetrical isoindicial surfaces.
The manner in which I have drawn the cross-section of these lens surfaces
thus makes no claim to represent the actual situation, all the less so as this
situation is unknown to us in such detail, and it only seeks to illustrate schematically
the essential features of the change in lens shape during accommodation.
The conclusions I draw regarding the intracapsular accommodation
mechanism will therefore not differ from those obtainable solely from the
exactly calculated change in shape of the isoindicial surfaces as well as from
the opthalmometrically determined changes in lens thickness and in the top
radii of the lens surfaces. The former increases by about 11% during accommodation,
while the curvature of the anterior lens surface increases by 87%
in round numbers, against which that of the posterior lens surface increases
by only 12.5%.
428 1 9 1 1 A . G U L L S T R A N D
The most notable feature to emerge from a comparison of these two diagrams
is the strikingly large change in shape undergone by the innermost of
the two drawn isoindicial surfaces. These two surfaces change and tend to
become spherical, but the inner surface changes most. As shown by a comparison
of the two indicial equations, this phenomenon in the drawings is an
expression of the fact that the accommodative change in shape of the isoindicial
surfaces increases with their proximity to the centre of the lens. On
the other hand it follows from the mathematical analysis that this condition
corresponds to an accommodative increase in refraction greater than the one
that would be brought about by the change in shape itself if the lens were
homogeneous. Since the refraction of a homogeneous lens of particular shape
is given by the refractive index, it follows that if a lens is made having the
same shape as that of the eye, firstly at rest and secondly at maximum accommodation
with the refractive index in both cases chosen so that the refraction
of the homogeneous lens is the same as that of the eye lens, the refractive index
of the homogeneous lens corresponding to the state of accommodation must
be made larger than that of the lens corresponding to rest. But it is this same
imaginary refractive index which, as I have previously mentioned, has been
termed the total index of the eye lens. Thus in the eye the total index of the
lens increases during accommodation, a truly remarkable situation although
as we have seen, the total index is not a physical refractive index but an imaginary
concept. It is not necessary to know the dioptrics of the lens to prove
this. The length of the eye and the refraction of the lens during accommodation
can be found from the constants of the corneal system, which can be
measured directly, and from the loss of refraction when the lens is extracted,
this factor being known from satisfactorily precise examinations. Together
with the change in refraction observed for a given measured change in the
curvature of the anterior lens surface these values indicate the contribution of
the lens to the refraction during accommodation and hence the accommodative
change of the total index. I introduced these values of the lens refraction
when calculating the indicial equations and it was only by means of these
equations that the accommodative change in the total index of the lens was
proved to be governed anatomically by the fact that the accommodative
change in shape of the isoindicial surfaces of the lens increases with their proximity
to the centre of the lens. I purposely say anatomically because the displacement
liable to occur between adjacent lens fibres can only be measured
on the rnicroscopid scale. It thus follows directly from these two diagrams
that within the closed isoindicial surfaces the axipetal displacement of the lens
M E C H A N I S M O F I N T R A C A P S U L A R A C C O M M O D A T I O N 4 2 9
tissue constituents in the equatorial plane which occurs during accommodation
increases with their proximity to the axis. This condition is one of the
essential characteristics of the intracapsular accommodation mechanism. The
importance of the accommodative change in the total index of the lens for
the dioptrics of the eye is best brought out by the fact that unless it is known
it is simply impossible to derive a scheme for the accommodating eye that is
not in sharp contrast with the remaining known facts.
On the lens at rest I have marked by means of a rounded comer the point
where the suspensory ligament of the lens, the zonule, is attached at the front,
not far from the equator. A glance at the two figures is sufficient to realize
that the lens particles within the largest closed isoindicial surface nearest this
point are displaced towards the lens axis during accommodation. This is the
second important characteristic of the intracapsular accommodation mechanism.
Owing to the nearness to the zonule attachment it hence follows that
the anterior zonule attachment must also approach the axis during accommodation.
In other words, the intracapsular accommodation mechanism
leads of necessity to the requirement that the extracapsular accommodation
mechanism must result in an accommodatively reduced tension of the zonule.
This was Helmholtz' opinion of the extracapsular accommodation mechanism
but this opinion has been exposed to such unbridled attacks that the proof
of its mathematical necessity as supplied by the intracapsular mechanism ought
not to be underrated, notwithstanding that Hess in Würzburg, who was unaware
of the intracapsular mechanism, proved the correctness of Helmholtz'
view in such a way that anyone willing and able to see the truth in physiological
experiments had to acknowledge the proof as conclusive.
One further conclusion can be drawn from these figures and the fact that
the lens particles are neither compressible nor freely displaceable. It follows
that when a change in shape occurs the volume enclosed within an isoindicial
surface must remain approximately constant. But during the considerable
changes of shape which the isoindicial surface undergoes, their area cannot
remain constant provided they remain surfaces of rotation. A glance at the
inner line in both figures is sufficient to prove this. The result once again is
that some process similar to a radial folding of the isoindicial surfaces must
follow the change in shape. I have previously pointed out that the rays which
we see round a luminous point or a fixed star are determined by the nature
of the aberration. The corresponding property in the case of a wave surface
is obtained during the passage of the light through the lens. As revealed by
mathematical analysis, this characteristic of the wave surface can only occur
430 1 9 1 1 A . G U L L S T R A N D
owing to a structure of the isoindicial surfaces that can most readily be visualized
as shallow, radial, folded structures becoming more indefinite towards
the centre, and such structures also occur in the normal eye at rest. Tests using
the subjective stigmatoscope developed by me also show plainly that the visible
ray formations alter their character during accommodation which signifies
a change in the radial folded structures of the isoindicial surfaces. It will perhaps
be objected here that in the presence of such structures of the isoindicial
surfaces it was inadmissible to treat them as surfaces of rotation in the mathematical
examination in the way I did. The fact is, however, from what is to
be inferred from the results obtained using the subjective stigmatoscope, that
the broken wave surface of the fascicle in the eye - here I must use mathematical
language - has a complete contact of the fourth order with a surface
of rotation and it is this surface which is the object of the mathematical examinations
of the lens dioptrics.
We have thus found the intracapsular accommodation mechanism characterized
mainly by the fact that during the accommodation the particles in
the equatorial plane are displaced in the axipetal direction, the extent increasing
with their proximity to the axis, further by the fact that the zonule
attachment on the anterior lens surface is displaced in the axipetal direction,
and finally by the fact that displacement occurs to different extents along
various radii of the lens. I shall now demonstrate that this mechanism corresponds
so accurately to what we know of the lens structure that it should have
been possible to forecast it completely. I have mentioned that the lens consists
throughout ofmicroscopically fine fibres in the form of loops aligned towards
the equator. Consequently, the equatorial plane contains only cross-sections
of these fibres. They are intersected at almost a right angle and are so tightly
packed that they have assumed hexagonal shape. It is now quite obvious that
when a centripetal movement occurs where during unit time the same surface
area of the cross-section must pass through the circles having the lens
centre as their centre, the amount of displacement must increase with decreasing
size of the circle's periphery, i.e. with proximity to the axis. Furthermore,
it is also quite clear that in the more central parts of the equatorial plane,
where the displacement is large, not all the cross-sections in the circle can
move towards the centre to the same extent because they do not have the
room within the smaller circle that they occupy after complete displacement.
The displacement must therefore necessarily take place to varying degrees
according to the radii in the equatorial plane. And finally: because the whole
lens consists of packed fibres, which intersect the equatorial plane at almost
M E C H A N I S M O F I N T R A C A P S U L A R A C C O M M O D A T I O N 4 3 1
a right angle to form a loop aligned towards the equator while they are attached
at the front and back and cannot be displaced towards each other, the
lens cannot become thicker in any other way than that these loops stretch,
whereupon some part of them must move axipetally. Since furthermore when
the lens thickens, the curvature of the anterior lens surface greatly increases
at its pole, this can occur in no other way than that the peripheral parts of the
fibres running along the anterior lens surface participate in this axipetal movement.
For the same reasons it is very probable that identical behaviour applies
to the parts of the lens fibre loops lying in the equatorial plane and along the
peripheral parts of the posterior lens surface, but this could not be concluded
with certainty since the increase in its curvature at the pole is so insignificant
and since it is not known whether this can be succeeded by an accommodative
flattening. It need only be known that during accommodation the anterior
zonule attachement moves towards the axis.
It will be appreciated that in view of the lens structure and of the way in
which the lens alters shape during accommodation, the intracapsular accommodation
mechanism simply cannot be any other than the one which follows
from my studies of the lens dioptrics. It ought therefore to have been possible
to find it from what we knew before I initiated my studies, it ought to have
been possible to foresee, inter alia, that the total index of the lens must increase
during accommodation, and that the Helmholtz accommodation theory
must essentially be correct, even though the physiological experiments contradicted
the fact. But no one thought of it. It must also be admitted that it
would have been a bold thought before definite proof was available. And
hence the intracapsular accommodation mechanism would still have remained
unknown had it not been necessary to study the dioptrics of heterogeneous
media to discover precise details of image-formation in the eye.