Optics of the Doer
Three basic concepts form the foundation for the present-day scientific
description of a vast field of optical phenomena, among them the
occurrence of the spectral colours as a result of light passing through
a transparent medium of prismatic shape. They are: 'optical
refraction', 'light-ray', and 'light-velocity' - the latter two serving
to explain the first. In a science of optics which seeks its foundation
in the intercourse between man's own visual activity and the doings and
sufferings of light, these three concepts must needs undergo a decisive
change, both in their meaning and in their value for the description of
the relevant optical phenomena. For they are all purely kinematic
concepts typical of the onlooker-way of conceiving things - concepts,
that is, to which nothing corresponds in the realm of the actual
phenomena.
Our next task, therefore, will be, where possible, to fill these
concepts with new meaning, or else to replace them by other concepts
read from the actual phenomena. Once this is done the way will be free
for the development of the picture of the spectrum phenomenon which is
in true accord with the Goethean conception of Light and Colour.
*
The first to be brought in this sense under our examination is the
concept of the 'light-ray'.
In present-day optics this concept signifies a geometrical line of
infinitely small width drawn, as it were, by the light in space, while
the cone or cylinder of light actually filling the space is described
as being composed of innumerable such rays. In the same way the object
producing or reflecting light is thought of as composed of innumerable
single points from which the light-rays emerge. All descriptions of
optical processes are based upon this conception.
Obviously, we cannot be satisfied with such a reduction of wholes into
single geometrically describable parts, followed by a reassembling of
these parts into a whole. For in reality we have to do with realms of
space uniformly filled with light, whether conical or cylindrical in
form, which arise through certain boundaries being set to the light. In
optical research we have therefore always to do with pictures,
spatially bounded. Thus what comes before our consciousness is
determined equally by the light calling forth the picture, and by the
unlit space bordering it.
Remembering the results of our earlier study, we must say further of
such a light-filled realm that it lacks the quality of visibility and
therefore has no colour, not even white. Goethe and other 'readers',
such as Reid and Ruskin, tried continually to visualize what such a
light-filled space represents in reality. Hence they directed their
attention first to those spheres where light manifests its
form-creative activity, as in the moulding of the organ of sight in
animal or man, or in the creation of the many forms of the plant
kingdom - and only then gave their mind to the purely physical
light-phenomena. Let us use the same method to form a picture of a
light-filled space, and to connect this with the ideas we have
previously gained on the co-operation in space of levity and gravity.
Suppose we have two similar plant-seeds in germ; and let one lie in a
space filled with light, the other in an unlit space. From the
different behaviour of the two seeds we can observe certain differences
between the two regions of space. We note that within the light-filled
region the spiritual archetype of the plant belonging to the seed is
helped to manifest itself physically in space, whereas in the dark
region it receives no such aid. For in the latter the physical plant,
even if it grows, does not develop its proper forms. This tells us, in
accordance with what we have learnt earlier, that in the two cases
there is a different relation of space to the cosmically distant,
all-embracing plane. Thus inside and outside the light-region there
exists a quite different relation of levity and gravity - and this
relation changes abruptly at the boundaries of the region. (This fact
will be of especial importance for us when we come to examine the
arising of colours at the boundary of Light and Dark, when light passes
through a prism.)
*
After having replaced the customary concept of the light-bundle
composed of single rays by the conception of two dynamically polar
realms of space bordering each other, we turn to the examination of
what is going on dynamically inside these realms. This will help us to
gain a proper concept of the propagation of light through space.
In an age when the existence of a measurable light-velocity seems to
belong to the realm of facts long since experimentally proved; when
science has begun to measure the universe, using the magnitude of this
velocity as a constant, valid for the whole cosmos; and when entire
branches of science have been founded on results thus gained, it is not
easy, and yet it cannot be avoided, to proclaim that neither has an
actual velocity of light ever been measured, nor can light as such ever
be made subject to such measurement by optical means - and that,
moreover, light, by its very nature, forbids us to conceive of it as
possessing any finite velocity.
With the last assertion we do not mean to say that there is nothing
going on in connexion with the appearance of optical phenomena to which
the concept of a finite velocity is applicable. Only, what is
propagated in this way is not the entity we comprise under the concept
of 'light'. Our next task, therefore, will be to create a proper
distinction between what moves and what does not move spatially when
light is active in the physical world. Once more an historical
retrospect will help us to establish our own standpoint with regard to
the existing theories.
The first to think of light as possessing a finite velocity was
Galileo, who also made the first, though unsuccessful, attempt to
measure it. Equally unsuccessful were attempts of a similar nature made
soon afterwards by members of the Accademia del Cimento. In both cases
the obvious procedure was to produce regular flashes of light and to
try to measure the time which elapsed between their production and
their observation by some more or less distant observer. Still, the
conviction of the existence of such a velocity was so deeply ingrained
in the minds of men that, when later observations succeeded in
establishing a finite magnitude for what seemed to be the rate of the
light's movement through space, these observations were hailed much
more as the quantitative value of this movement than as proof of its
existence, which was already taken for granted.
A clear indication of man's state of mind in regard to this question is
given in the following passage from Huygens's famous Traité de la Lumière, by
which the world was first made acquainted with the concept of light as
a sort of undulatory movement.
'One cannot doubt that light consists in the movement of a certain
substance. For if one considers its production one finds that here on
the earth it is chiefly produced by fire and flame, which without doubt
contain bodies in rapid motion, for they dissolve and melt numberless
other bodies. Or, if one considers its effects, one sees that light
collected, for instance, by a concave mirror has the power to heat like
fire, i.e. to separate the parts of the bodies; this assuredly points
to movement, at least in true philosophy in which one traces all
natural activity to mechanical causes. In my opinion one must do this,
or quite give up all hope of ever grasping anything in physics.'
In these words of Huygens it must strike us how he first provides an
explanation for a series of phenomena as if this explanation were
induced from the phenomena themselves. After he has drawn quite
definite conclusions from it, he then derives its necessity from quite
other principles - namely, from a certain method of thinking, accepting
this as it is, unquestioned and unalterably established. We are here
confronted with an 'unlogic' characteristic of human thinking during
its state of isolation from the dynamic substratum of the world of the
senses, an unlogic which one encounters repeatedly in scientific
argumentation once one has grown aware of it. In circles of modern
thinkers where such awareness prevails (and they are growing rapidly
to-day) the term 'proof of a foregone conclusion' has been coined to
describe this fact.1
'Proof of a foregone conclusion' is indeed the verdict at which one
arrives in respect of all the observations concerned with the velocity
of light - whether of existing phenomena detectable in the sky or of
terrestrial phenomena produced artificially - if one studies them with
the attitude of mind represented by the child in Hans Andersen's story.
In view of the seriousness of the matter it will not be out of place if
we discuss them here as briefly as possible, one by one.2
The relevant observations fall into two categories: observations of
certain astronomical facts from which the existence of a finite
velocity of light and its magnitude as an absolute property of it has
been inferred; and terrestrial experiments which permitted direct
observation of a process of propagation connected with the
establishment of light in space resulting in the measurement of its
speed. To the latter category belong the experiments of Fizeau (1849)
and Foucault (1850) as well as the Michelson-Morley experiment with its
implications for Einstein's Theory of Relativity. The former category
is represented by Roemer's observations of certain apparent
irregularities in the times of revolution of one of Jupiter's moons
(1676), and by Bradley's investigation into the reason for the apparent
rhythmic changes of the positions of the fixed stars (1728).
We shall start with the terrestrial observations, because in their case
alone is the entire path of the light surveyable, and what is measured
therefore is something appertaining with certainty to every point of
the space which spreads between the source of the light and the
observer. For this reason textbooks quite rightly say that only the
results drawn from these terrestrial observations have the value of
empirically observed facts. (The interpretation given to these facts is
another question.)
Now, it is a common feature of all these experiments that by necessity
they are based on an arrangement whereby a light-beam can be made to
appear and disappear alternately. In this respect there is no
difference between the first primitive attempts made by Galileo and the
Academicians, and the ingeniously devised experiments of the later
observers, whether they operate with a toothed wheel or a rotating
mirror. It is always a flash of light - and how could it be otherwise?
- which is produced at certain regular intervals and used for
determining the speed of propagation.
Evidently what in all these cases is measured is the speed with which a
beam of light establishes itself in space. Of what happens within the
beam, once it is established, these observations tell nothing at all.
The proof they are held to give of the existence of a finite speed of
light, as such, is a 'proof of a foregone conclusion'. All they tell us
is that the beam's front, at the moment when this beam is first
established, travels through space with a finite velocity and that the
rate of this movement is such and such. And they tell us nothing at all
about other regions of the cosmos.
That we have to do in these observations with the speed of the
light-front only, and not of the light itself, is a fact fully
acknowledged by modern physical optics. Since Lord Rayleigh first
discussed this matter in the eighties of the last century, physicists
have learnt to distinguish between the 'wave-velocity' of the light
itself and the velocity of an 'impressed peculiarity', the so-called
'group-velocity', and it has been acknowledged that only the latter has
been, and can be, directly measured. There is no possibility of
inferring from it the value of the 'wave-velocity' unless one has a
complete knowledge of the properties of the medium through which the
'groups' travel. Nevertheless, the modern mind allows itself to be
convinced that light possesses a finite velocity and that this has been
established by actual measurement. We feel reminded here of Eddington's
comment on Newton's famous observations: 'Such is the glamour of a
historical experiment.' (Chapter XIV.)3
Let us now turn to Roemer and Bradley. In a certain sense Roemer's
observations and even those of Bradley rank together with the
terrestrial measurements. For Roemer used as optical signals the
appearance and disappearance of one of Jupiter's moons in the course of
its revolution round the planet; thus he worked with light-flashes, as
the experimental investigations do. Hence, also, his measurements were
concerned - as optical science acknowledges - with group-velocity only.
In fact, even Bradley's observations, although he was the only one who
operated with continuous light-phenomena, are exposed to the charge
that they give information of the group-velocity of light, and not of
its wave-velocity. However, we shall ignore these limitations in both
cases, because there are quite other factors which invalidate the
proofs they are held to give, and to gain a clear insight into these
factors is of special importance for us.
Roemer observed a difference in the length of time during which a
certain moon of Jupiter was occulted by the planet's body, and found
that this difference underwent regular changes coincident with the
changes in the earth's position in relation to Jupiter and the sun.
Seen from the sun, the earth is once a year in conjunction with
Jupiter, once in opposition to it. It seemed obvious to explain the
time-lag in the moon's reappearance, when the earth was on the far side
of the sun, by the time the light from the moon needed to cover the
distance marked by the two extreme positions of the earth - that is, a
distance equal to the diameter of the earth's orbit. On dividing the
observed interval of time by the accepted value of this distance,
Roemer obtained for the velocity of light a figure not far from the one
found later by terrestrial measurements.
We can here leave out of account the fact that Roemer's reasoning is
based on the assumption that the Copernican conception of the relative
movements of the members of our solar system is the valid conception,
an assumption which, as later considerations will show, cannot be
upheld in a science which strives for a truly dynamic understanding of
the world. For the change of aspect which becomes necessary in this way
does not invalidate Roemer's observation as such; it rules out only the
customary interpretation of it. Freed from all hypothetical by-thought,
Roemer's observation tells us, first, that the time taken by a flash of
light travelling from a cosmic light-source to reach the earth varies
to a measurable extent, and, secondly, that this difference is bound up
with the yearly changes of the earth's position in relation to the sun
and the relevant planetary body.
We leave equally out of account the fact that our considerations of the
nature of space in Chapter XII render it impermissible to conceive of
cosmic space as something 'across' which light (or any other entity)
can be regarded as travelling this or that distance in this or that
time. What matters to us here is the validity of the conclusions drawn
from Roemer's discovery within the framework of thought in which they
were made.
Boiled down to its purely empirical content, Roemer's observation tells
us solely and simply that within the earth's cosmic orbit light-flashes
travel with a certain measurable speed. To regard this information as
automatically valid, firstly for light which is continuously present,
and secondly for everywhere in the universe, rests again on nothing but
a foregone conclusion.
Precisely the same criticism applies to Bradley's observation, and to
an even higher degree. What Bradley discovered is the fact that the
apparent direction in which we see a fixed star is dependent on the
direction in which the earth moves relatively to the star, a phenomenon
known under the name of 'aberration of light'. This phenomenon is
frequently brought to students' understanding by means of the following
or some similar analogy.
Imagine that a machine-gun in a fixed position has sent its projectile
right across a railway-carriage so that both the latter's walls are
pierced. If the train is at rest, the position of the gun could be
determined by sighting through the shot-holes made by the entrance and
exit of the bullet. If, however, the train is moving at high speed, it
will have advanced a certain distance during the time taken by the
projectile to cross the carriage, and the point of exit will be nearer
the rear of the carriage than in the previous case. Let us now think of
an observer in the train who, while ignorant of the train's movement,
undertook to determine the gun's position by considering the direction
of the line connecting the two holes. He would necessarily locate the
gun in a position which, compared with its true position, would seem to
have shifted by some distance in the direction of the train's motion.
On the other hand, given the speed of the train, the angle which the
line connecting the two holes forms with the true direction of the
course of the projectile - the so-called angle of aberration - provides
a measure of the speed of the projectile.
Under the foregone conclusion that light itself has a definite
velocity, and that this velocity is the same throughout the universe,
Bradley's observation of the aberration of the stars seemed indeed to
make it possible to calculate this velocity from the knowledge of the
earth's own speed and the angle of aberration. This angle could be
established by comparing the different directions into which a
telescope has to be turned at different times of the year in order to
focus a particular star. But what does Bradley's observation tell us,
once we exclude all foregone conclusions?
As the above analogy helps towards an understanding of the concept of
aberration, it will be helpful also to determine the limits up to which
we are allowed to draw valid conclusions from the supposed occurrence
itself. A mind which is free from all preconceived ideas will not
ignore the fact that the projectile, by being forced to pierce the wall
of the carriage, suffers a considerable diminution of its speed. The
projectile, therefore, passes through the carriage with a speed
different from its speed outside. Since, however, it is the speed from
hole to hole which determines the angle of aberration, no conclusion
can be drawn from the latter as to the original velocity of the
projectile. Let us assume the imaginary case that the projectile was
shot forth from the gun with infinite velocity, and that the
slowing-down effect of the wall was great enough to produce a finite
speed of the usual magnitude, then the effect on the position of the
exit hole would be precisely the same as if the projectile had moved
all the time ' with this speed and not been slowed down at all.
Seeing things in this light, the scientific Andersen child in us is
roused to exclaim: 'But all that Bradley's observation informs us of ,
with certainty is a finite velocity of the optical process going on
inside the telescope!' Indeed, if someone should claim with good reason
(as we shall do later on) that light's own velocity is infinite, and
(as we shall not do) that the dynamic situation set up in the telescope
had the effect of slowing down the light to the measured velocity -
there is nothing in Bradley's observation which could disprove these
assertions.
*
Having thus disposed of the false conclusions drawn by a kinematically
orientated thinking from the various observations and measurements of
the velocity which appears in connexion with light, we can carry on our
own studies undisturbed. Two observations stand before us representing
empirically established facts: one, that in so far as a finite velocity
has been measured or calculated from other observations, nothing is
known about the existence or magnitude of such a velocity except within
the boundaries of the dynamic realm constituted by the earth's presence
in the universe; the other, that this velocity is a 'group'-velocity,
that is, the velocity of the front of a light-beam in process of
establishment. Let us see what these two facts have to tell us when we
regard them as letters of the 'word' which light inscribes into the
phenomenal world as an indication of its own nature.
Taking the last-named fact first, we shall make use of the following
comparison to help us realize how little we are justified in drawing
from observations of the front speed of a light-beam any conclusions
concerning the kinematic conditions prevailing in the interior of the
beam itself. Imagine the process of constructing a tunnel, with all the
efforts and time needed for cutting its passage through the resisting
rock. When the tunnel is finished the activities necessary to its
production are at an end. Whereas these continue for a limited time
only, they leave behind them permanent traces in the existence of the
tunnel, which one can describe dynamically as a definite alteration in
the local conditions of the earth's gravity. Now, it would occur to no
one to ascribe to the tunnel itself, as a lasting quality, the speed
with which it had been constructed. Yet something similar happens when,
after observing the velocity required by light to lay hold on space,
this velocity is then attributed to the light as a quality of its own.
It was reserved for a mode of thought that could form no concept of the
real dynamic of Light and Dark, to draw conclusions as to the qualities
of light from experiences obtained through observing its original
spreading out into space.
To speak of an independently existing space within which light could
move forward like a physical body, is, after what we have learnt about
space, altogether forbidden. For space in its relevant structure is
itself but a result of a particular co-ordination of levity and gravity
or, in other words, of Light and Dark. What we found earlier about the
qualities of the two polar spaces now leads us to conceive of them as
representative of two limiting conditions of velocity: absolute
contraction representing zero velocity; absolute expansion, infinite
velocity (each in its own way a state of 'rest'). Thus any motion with
finite velocity is a mean between these two extremes, and as such the
result of a particular co-ordination of levity and gravity. This makes
it evident that to speak of a velocity taking its course in space,
whether with reference to light or to a physical body in motion, is
something entirely unreal.
Let us now see what we are really told by the number 186,000 miles a
second, as the measure of the speed with which a light-impulse
establishes itself spatially. In the preceding chapter we learnt that
the earth's field of gravity offers a definite resistance to our visual
ray. What is true for the inner light holds good equally for the outer
light. Using an image from another dynamic stratum of nature we can say
that light, while appearing within the field of gravity, 'rubs' itself
on this. On the magnitude of this friction depends the velocity with
which a light-impulse establishes itself in the medium of the resisting
gravity. Whereas light itself as a manifestation of levity possesses
infinite velocity, this is forced down to the known finite measure by
the resistance of the earth's field of gravity. Thus the speed of light
which has been measured by observers such as Fizeau and Foucault
reveals itself as a function of the gravitational constant of the
earth, and hence has validity for this sphere only.1 The same is true
for Roemer's and Bradley's observations, none of which, after what we
have stated earlier, contradicts this result. On the contrary, seen
from this viewpoint, Roemer's discovery of the light's travelling with
finite speed within the cosmic realm marked by the earth's orbit
provides an important insight into the dynamic conditions of this
realm.
*
Among the experiments undertaken with the aim of establishing the
properties of the propagation of light by direct measurements, quoted
earlier, we mentioned the Michelson-Morley experiment as having a
special bearing on Einstein's conceptual edifice. It is the one which
has formed the foundation of that (earlier) part of Einstein's theory
which he himself called the Special Theory of Relativity. Let us see
what becomes of this foundation - and with it the conceptual edifice
erected upon it - when we examine it against the background of what we
have found to be the true nature of the so-called velocity of light.
It is generally known that modern ideas of light seemed to call for
something (Huygens's 'certain substance') to act as bearer of the
movement attributed to light. This led to the conception of an
imponderable agency capable of certain movements, and to denote this
agency the Greek word ether was borrowed. (How this word can be used
again to-day in conformity with its actual significance will be shown
in the further course of our discussions.) Nevertheless, all endeavours
to find in the existence of such an ether a means of explaining wide
fields of natural phenomena were disappointed. For the more exact
concepts one tried to form of the characteristics of this ether, the
greater the contradictions became.
One such decisive contradiction arose when optical means were used to
discover whether the ether was something absolutely at rest in space,
through which physical bodies moved freely, or whether it shared in
their movement. Experiments made by Fizeau with running water seemed to
prove the one view, those of Michelson and Morley, involving the
movement of the earth, the other view. In the celebrated
Michelson-Morley experiment the velocity of light was shown to be the
same, in whatever direction, relative to the earth's own motion, it was
measured. This apparent proof of the absolute constancy of
light-velocity - which seemed, however, to contradict other
observations - induced Einstein to do away with the whole assumption of
a bearer of the movement underlying light, whether the bearer were
supposed to be at rest or itself in motion. Instead, he divested the
concepts of space and time, from which that of velocity is usually
derived, of the absoluteness hitherto attributed to them, with the
result that in his theory time has come to be conceived as part of a
four-dimensional 'space-time continuum'.
In reality the Michelson-Morley experiment presents no problem
requiring such labours as those of Einstein for its solution. For by
this experiment nothing is proved beyond what can in any event be known
- namely, that the velocity of the propagation of a light-impulse is
constant in all directions, so long as the measuring is confined to
regions where the density of terrestrial space is more or less the
same. With the realization of this truth, however, Einstein's Special
Theory loses its entire foundation. All that remains to be said about
it is that it was a splendid endeavour to solve a problem which,
rightly considered, does not exist.1
*
Now that we have realized that it is inadmissible to speak of light as
consisting of single rays, or to ascribe to it a finite velocity, the
concept of the refraction of light, as understood by optics to-day and
employed for the explanation of the spectrum, also becomes untenable.
Let us find out what we must put in its place.
The phenomenon which led the onlooker-consciousness to form the idea of
optical refraction has been known since early times. It
consists in the fact, surprising at first sight, that an object, such
as a coin, which lies at the bottom of a vessel hidden from an observer
by the rim, becomes visible when the vessel is filled with water.
Modern optics has explained this by assuming that from the separate
points of the floor of the vessel light-rays go out to all sides, one
ray falling in the direction of the eye of the observer. Hence, because
of the positions of eye and intercepting rim there are a number of
points from which no rays can reach the eye. One such point is
represented by the coin (P in Fig. 12a). Now if the vessel is filled
with water, light-rays emerging from it are held to be refracted, so
that rays from the points hitherto invisible also meet the eye, which
is still in its original position. The eye itself is not conscious of
this 'break' in the light-rays, because it is accustomed to 'project'
all light impressions rectilinearly out into space (Fig. 12b.). Hence,
it sees P in the position of P'. This is thought to be the origin of
the impression that the whole bottom of the vessel is raised.
This kind of explanation is quite in line with the peculiarity of the
onlooker-consciousness, noted earlier, to attribute an optical illusion
to the eye's way of working, while charging the mind with the task of
clearing up the illusion. In reality it is just the reverse. Since the
intellect can form no other idea of the act of seeing than that this is
a passive process taking place solely within the eye, it falls, itself,
into illusion. How great is this illusion we see from the fact that the
intellect is finally obliged to make the eye somehow or other 'project'
into space the impressions it receives - a process lacking any concrete
dynamic content.
Once more, it is not our task to replace this way of 'explaining' the
phenomenon by any other, but rather to combine the phenomenon given
here with others of kindred nature so that the theory contained in them
can be read from them direct. One other such phenomenon is that of
so-called apparent optical depth, which an observer encounters when
looking through transparent media of varying optical density. What
connects the two is the fact that the rate of the alteration of depth,
and the rate of change of the direction of light, are the same for the
same media.
In present-day optics this phenomenon is explained with reference to
the former. In proceeding like this, optical science makes the very
mistake which Goethe condemned in Newton, saying that a complicated
phenomenon was made the basis, and the simpler derived from the
complex. For of these two phenomena, the simpler, since it is
independent of any secondary condition, is the one showing that our
experience of depth is dependent on the density of the optical medium.
The latter phenomenon we met once before, though without reference to
its quantitative side, when in looking at a landscape we found how our
experiences of depth change in conformity with alterations in
atmospheric conditions. This, then, served to make us aware that the
way we apprehend things optically is the result of an interplay between
our visual ray and the medium outside us which it meets.
It is exactly the same when we look through a vessel filled with water
and see the bottom of it as if raised in level. This is in no sense an
optical illusion; it is the result of what takes place objectively and
dynamically within the medium, when our eye-ray passes through it. Only
our intellect is under an illusion when, in the case of the coin
becoming visible at the bottom of the vessel, it deals with the coin as
if it were a point from which an individual ray of light went out.. ..
etc., instead of conceiving the phenomenon of the raising of the
vessel's bottom as one indivisible whole, wherein the coin serves only
to link our attention to it.
*
Having thus cleared away the kinematic interpretation of the
coin-in-the-bowl phenomenon, we may pass on to discuss the optical
effect through which the so-called law of refraction was first
established in science. Instead of picturing to ourselves, as is
usually done, light-rays which are shifted away from or towards the
perpendicular at the border-plane between two media of different
optical properties, we shall rather build up the picture as light
itself designs it into space.
We have seen that our inner light, as well as the outer light, suffers
a certain hindrance in passing through a physical medium - even such as
the earth's gravity-field. Whilst we may not describe this retardation,
as is usually done, in terms of a smaller velocity of light itself
within the denser medium, we may rightly say that density has the
effect of lessening the intensity of the light. (It is the time
required for the initial establishment of a light-filled realm which is
greater within such a medium than outside it.) Now by its very nature
the intensity of light cannot be measured in spatial terms. Yet there
is a phenomenon by which the decrease of the inner intensity of the
light becomes spatially apparent and thus spatially measurable. It
consists in the alteration undergone by the aperture of a cone of light
when passing from one optical medium to another.
If one sets in the path of a luminous cone a glass-walled trough filled
with water, then, if both water and surrounding air are slightly
clouded, the cone is seen to make a more acute angle within the water
than outside it (Fig. 13). Here in an external phenomenon we meet the
same weakening in the light's tendency to expand that we recognized in
the shortening of our experience of depth on looking through a dense
medium. Obviously, we expect the externally observable narrowing of the
light-cone and the subjectively experienced change of optical depth to
show the same ratio.
In order to compare the rate of expansion of a luminous cone inside and
outside water, we must measure by how much less the width of the cone
increases within the water than it does outside. (To be comparable, the
measurements must be based upon the same distances on the edge of the
cone, because this is the length of the way the light actually
travels.) In Fig. 13 this is shown by the two distances, a-b and a'-b'.
Their ratio is the same as that by which the bottom of a vessel appears
to be raised when the vessel is filled with water (4:3).
Thus by means of pure observation we have arrived at nothing less than
what is known to physical optics as Snell's Law of Refraction. This law
was itself the result of pure observation, but was clothed in a
conceptual form devoid of reality. In this form it states that a ray of
light in transition between two media of different densities is
refracted at their boundary surface so that the ratio of the angle
which is formed by the ray in either medium with a line at right angles
to the boundary surface is such that the quotient of the sines of both
angles is for these media a constant factor. In symbols
sin α / sin β = c.
It will be clear to the reader familiar with trigonometry that this
ratio of the two sines is nothing else but the ratio of the two
distances which served us as a measure for the respective apertures of
the cone. But whereas the measurement of these two distances is
concerned with something quite real (since they express an actual
dynamic alteration of the light), the measuring of the angle between
the ray of light and the perpendicular is founded on nothing real. It
is now clear that the concept of the ray, as it figures in the usual
picture of refraction, is in reality the boundary between the luminous
space and its surroundings. Evidently the concept of the perpendicular
on the boundary between the two media is in itself a complete
abstraction, since nothing happens dynamically in its direction.
To a normal human understanding it is incomprehensible why a ray of
light should be related to an external geometrical line, as stated by
the law of refraction in its usual form. Physical optics, in order to
explain refraction, had therefore to resort to light-bundles spatially
diffused, and by use of sundry purely kinematic concepts, to read into
these light-bundles certain processes of motion, which are not in the
least shown by the phenomenon itself. In contrast to this, the idea
that the boundary of a luminous cone is spatially displaced when its
expansion is hindered by an optical medium of some density, and that
the measure of this displacement is equal to the shortening of depth
which we experience in looking through this medium, is directly
evident, since all its elements are taken from observation.
*
From what we have here found we may expect that in order to explain the
numerical relationships between natural phenomena (with which science
in the past has been solely concerned), we by no means require the
artificial theories to which the onlooker in man, confined as he is to
abstract thinking, has been unavoidably driven. Indeed, to an observer
who trains himself on the lines indicated in this book, even the
quantitative secrets of nature will become objects of intuitive
judgment, just as Goethe, by developing this organ of understanding,
first found access to nature's qualitative secrets. (The change in our
conception of number which this entails will be shown at a later stage
of our discussions.)
1 Compare with this our account in Chapter X of the rise of the
atomistic-kinematic interpretation of heat.
2 The following critical study leaves, of course, completely untouched
our recognition of the devotion which guided the respective observers
in their work, and of the ingenuity with which some of their
observations were devised and carried out.
3 The assumption is that the wave-velocity differs from the
group-velocity, if at all, by a negligible amount.
4 Once this is realized there can be no doubt that with the aid of an
adequate mathematical calculus (which would have to be established on a
realistic understanding of the respective properties of the fields of
force coming into play) it will become possible to derive by
calculation the speed of the establishment of light within physical
space from the gravitational constant of the earth.
5 The grounds of Einstein's General Theory were dealt with in our
earlier discussions.