In anxious and uncertain times like ours, when it is difficult
to find pleasure in humanity and the course of human affairs, it is particularly consoling to think of such a supreme and quiet man as Kepler. Kepler lived in an age in which the reign of law in
nature was as yet by no means certain. How great must his faith in
the existence of natural law have been to give him the strength to
devote decades of hard and patient work to the empirical
investigation of planetary motion and the mathematical laws of
that motion, entirely on his own, supported by no one and
understood by very few. If we would honor his memory fittingly, we
must get as clear a picture as we can of his problem and the
stages of its solution.
On the occasion of the three hundredth anniversary of
Kepler's death. Published in the Frankfurter 'Zeiturig
(Germany), November 9,1930.
Copernicus had opened the eyes of the most intelligent to the fact that the best way to get a clear grasp of the apparent
movements of the planets in the heavens was to regard them as movements round the sun conceived as stationary. If the planets moved uniformly in a circle round the sun, it would have been comparatively easy to discover how these movements must look
from the earth. Since, however, the phenomena to be dealt with were much more complicated than that, the task
was far harder. First of all, these movements had to be deter-mined empirically from the observations of Tycho Brahe. Only then did it become possible to think about discovering the general laws which these movements satisfy.
To grasp how difficult a business it was even to determine the actual movements round the sun one has to realize the following. One can never see where a planet really is at any given moment, but only in what direction it can be seen just then from the earth, which is itself moving in an unknown manner round the sun. The difficulties thus seemed practically insurmountable.
Kepler had to discover a way of bringing order into this chaos. To start with, he saw that it was necessary first to try to find out
about the motion of the earth itself. This would simply have been
impossible if there existed only the sun, the earth, and the fixed
stars, but no other planets. For in that case one could ascertain
nothing empirically except how the direction of the straightsun-
earth line changes in the course of the year (apparent movement of
the sun with reference to the fixed stars). In this way it was
possible to discover that these sun-earth directions all lay in a
plane stationary with reference to the fixed stars, at least
according to the accuracy of observation achieved in those days,
when there were no telescopes. By this means it could also be
ascertained in what manner the line sun-earth revolves round the sun. It turned out that the angular velocity of this motion varied in a regular way in the course of the year. But this was not of much use, as it was still not known how the distance from the earth to the sun alters in the course of the year. Only when these changes were known, could the real shape of the earth's orbit and the manner in which it is described be ascertained.
Kepler found a marvelous way out of this dilemma. To begin
with it followed from observations of the sun that the apparent path of the sun against the background of the fixed stars differed in speed at different times of the year, but that the angular velocity of this movement was always the same at the same time of the astronomical year, and therefore that the speed of rotation of the straight line earth-sun was always the
same when it pointed to the same region of the fixed stars. It was
thus legitimate to suppose that the earth's orbit was closed,
described by the earth in the same way every year-which was by no
means obvious a priori. For the adherents of the Copernican
system it was thus as good as certain that this must also apply to
the orbits of the rest of the planets.
This certainly made things easier. But how to ascertain the real
shape of the earth's orbit? Imagine a brightly shining lantern M
somewhere in the plane of the orbit. Assume we know that this
lantern remains permanently in its place and thus forms a kind of
fixed triangulation point for determining
the earth's orbit, a point which the inhabitants of the earth
can take a sight on at any time of year. Let this lantern M be
further away from the sun than the earth. With the help of such a
lantern it was possible to determine the earth's orbit,
in the following way:
First of all, in every year there comes a moment when the earth E lies exactly on the line joining the sun B and the lantern M. If at this moment we look from the earth E at the lantern M, our line of sight will coincide with the line BM (sun-lantern). Suppose the latter to be marked in the heavens. Now imagine the earth in a different position and at a different time. Since the sun B and
the lantern M can both be seen from the earth, the angle at E in the triangle SEM is known. But
we also know the direction of SE in relation to the fixed stars
through direct solar observations, while the direction of the line
SM in relation to the fixed stars has previously been ascertained
once for all. In the triangle SEM we also know the angle at S.
Therefore, with the base SM arbitrarily laid down on a sheet of paper, we can, in virtue of our knowledge of the angles at E and S, construct the triangle SEM. We might do this at frequent intervals during the year; each time we should get on our piece of
paper a position of the earth E with a date attached to it and a
certain position in relation to the permanently fixed base SM. The
earth's orbit would thereby be empirically determined, apart from
its absolute size, of course.
But, you will say, where did Kepler get his lantern M? His
genius and nature, benevolent in this case, gave it to him.
There was, for example, the planet Mars; and the length of the
Martian year-i.e., one rotation of Mars round the sun-was known.
At one point, it may happen that the sun, the earth, and Mars lie
v.ery nearly on a straight line. This position of Mars regularly
recurs after one, two, etc., Martian years, as Mars moves in a
closed orbit. At these known moments, therefore, SM always
presents the same base, while the earth is always at a different
point in its orbit. The observations of the sun and Mars at these
moments thus constitute a means of determining the true orbit of
the earth, as Mars then plays the part of our imaginary lantern.
Thus it was that Kepler discovered the true shape of the earth's
orbit and the way in which the earth describes it, and we who come
after-Europeans, Germans, or even Swabians-may well admire and
honor him for it.
Now that the earth's orbit had been empirically determined,
the true position and length of the line SE at any moment was
known, and it was not so terribly difficult for Kepler to
calculate the orbits and motions of the rest of the planets, too, from observations-at least in principle. It was nevertheless an im-mense task, especially considering the state of mathematics at the time.
Now came the second and no less arduous part of Kepler's life
work. The orbits were empirically known, but their laws had to be
guessed from the empirical data. First he had to make a guess at
the mathematical nature of the curve described by the orbit, and
then try it out on a vast assemblage of figures. If it did not
fit, another hypothesis had to be devised and again tested. After
tremendous search, the conjecture that the orbit was an ellipse
with the sun at one of its foci was found to fit the facts. Kepler
also discovered the law governing the varia-tion in speed during
one revolution, which is that the line sun-planet sweeps out equal
areas in equal periods of time. Finally he also discovered that
the squares of the periods of revolution round the sun vary as the
cubes of the major axes of the ellipses.
Our admiration for this splendid man is accompanied by another
feeling of admiration and reverence, the object of which is no man
but the mysterious harmony of nature into which we are born. The
ancients already devised the lines exhibiting the simplest
conceivable form of regularity. Among these, next to the straight
line and the circle, the most important were the ellipse and the
hyperbola. We see the last two embodied-at least very nearly so-in the orbits of the heavenly bodies.
It seems that the human mind has first to construct forms
independently before we can find them in things: Kepler's mar-
velous achievement is a particularly fine example of the truth
that knowledge cannot spring from experience alone but only from
the comparison of the inventions of the intellect with observed