For the same reason Goethe knew it would be historically unjustified to expect that Kepler could have conceived an aspect of the universe implicit in his own conception of nature. Hence it did not disturb him in his admiration for Kepler, that through him the Copernican aspect of the universe had become finally established in the modern mind - that is, an aspect which, as we have seen, is invalid as a means of forming a truly dynamic conception of the world.

In forming his picture of the universe, it is true, Copernicus was concerned with nothing but the spatial movements of the luminous entities discernible in the sky, without any regard to their actual nature and dynamic interrelationships. Hence his world-picture - as befits the spectator-form of human consciousness which was coming to birth in his own time - is a purely kinematic one. As such it has validity for a certain sphere of human observation.

When Kepler, against the hopes of his forerunner and friend, Tycho Brahe, accepted the heliocentric standpoint and made it the basis of his observations, he did so out of his understanding of what was the truth for his own time. Kepler's ideal was to seek after knowledge through pure observation. In this respect Goethe took him as his model. Kepler's discoveries were a proof that man's searching mind is given insight into great truths at any stage of its development, provided it keeps to the virtue of practising pure observation.

It has been the error of Newton and his successors up to our own day, to try to conceive the world dynamically within the limits of their spectator-consciousness and thus to form a dynamic interpretation of the universe based on its heliocentric aspect. This was just as repellent to Goethe as Kepler's attitude was attractive.

But by so sharply distinguishing between Newton and Kepler, do we not do injustice to the fact that, as the world believes, Kepler's third law is the parent of Newton's law of gravitation? The following will show that this belief is founded on an illusory conception of the kind we met before. As we shall see, Kepler's discovery, when treated in a Keplerian way, instead of leading to Newton, is found to be in full agreement with the very world-picture to which our own observations have led us.

It is an established conviction of the mathematical scientist that, once an observed regularity in nature has been expressed as a mathematical equation, this equation may be transformed in any mathematically valid way, and the resulting formula will still apply to some existing fact in the world. On innumerable occasions this principle has been used in the expectation of providing further insight into the secrets of nature. We came across a typical instance of this in discussing the basic theorem of kinematics and dynamics (Chapter VIII). Another example is Newton's treatment of Kepler's third law, or - more precisely - the way in which Newton's law of gravitation has been held to confirm Kepler's observations, and vice versa,

It will be our task to analyse the Kepler-Newton case on the very lines of our treatment of the two parallelogram theorems. This analysis will give us insight into a truth which we have to regard as one of the basic maxims of the new science. It says that whether a given formula, derived mathematically from one that was first read from nature, still expresses some fact of nature, cannot be decided by pure mathematical logic, but only by testing it against truly observable phenomena.

Through Kepler's third law a certain relation is expressed between the spatial dimensions of the different planetary spheres and the time needed by the relevant planet to circle once round the circumference of its own sphere. It says: 'The squares of the periodic times of the planets are always in the same proportion as the cubes of their mean distances from the sun.' In mathematical symbols this reads:
t₁² / t₂² = r₁³ / r₂³
We shall see later how Kepler arrived at this law. The point is that there is nothing in it which is not accessible to pure observation. Spatial distances and lengths of time are measured and the results compared. Nothing, for instance, is said about the dynamic cause of the movements. The assertion is restricted - and this is true also of the first and second law - to a purely kinematic content, and so precisely to what the earthly onlooker can apprehend. Now it is said that Kepler's third law is a necessary consequence of Newton's law of gravitation, and that - since it is based on pure observation - it therefore establishes the truth of Newton's conception. In this assertion we encounter a misconception exactly like the one in the statement that the theorem of the parallelogram of forces follows by logical necessity from the theorem of the parallelogram of velocities. For:

(a) The law of gravitation itself derives from Newton's formula for the centripetal force acting at a point which moves along a circle, this formula being itself the result of an amplification of the formula for centripetal acceleration by the factor 'mass' (as if the latter were a pure number):

Centripetal acceleration:
a = 4ϲr / t²