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.ens 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

In order to understand this we must go back to what we learnt in the course of our optical studies as to the two forms of vision arising from the activity of the eye's inner light - the dream-vision and the seeing of after-images. Of these two, seeing in dream is in a certain sense the purer form of inner seeing in that it arises without any outer stimulus exercised upon the physical organ of sight. On the other hand, it lacks that objective conformity to law characteristic of the after-images which mirror the order of the external world. There is an arbitrary, enigmatic element in dream-pictures, and their logic often seems to run counter to that of waking consciousness. A further characteristic of dream-perception is that we are tied to the level of consciousness prevailing in the dream. While we are dreaming we cannot awaken to the extent of being able to make the pictures the object of conscious observation.