KEPLER, THE GREAT CALCULATOR
Most fortunate it was for the later development of astronomical theory
that Tycho Brahe not only was a practical or observational astronomer of
the highest order, but that he confined himself studiously for years to
observations of the places of the planets. Of Mars he accumulated an
especially long and accurate series, and among those who assisted him in
his work was a young and brilliant pupil named Johann Kepler.
Strongly impressed with the truth of the Copernican System, Kepler was
free to reject the erroneous compromise system devised by Tycho Brahe,
and soon after Tycho's death Kepler addressed himself seriously to the
great problem that no one had ever attempted to solve, viz: to find out
what the laws of motion of the planets round the sun really are. Of
course he took the fullest advantage of all that Ptolemy and Copernicus
had done before him, and he had in addition the splendid observations of
Tycho Brahe as a basis to work upon.
Copernicus, while he had effected the tremendous advance of substituting
the sun for the earth as the center of motion, nevertheless clung to the
erroneous notion of Ptolemy that all the bodies of the sky must perforce
move at uniform speeds, and in circular curves, the circle being the
only "perfect curve." Kepler was not long in finding out that this
could not be so, and he found it out because Tycho Brahe's observations
were much more accurate than any that Copernicus had employed.
Naturally he attempted the nearest planet first, and that was Mars--the
planet that Tycho had assigned to him for research. How fortunate that
the orbit of Mars was the one, of all the planets, to show practically
the greatest divergence from the ancient conditions of uniform motion in
a perfectly circular orbit! Had the orbit of Mars chanced to be as
nearly circular as is that of Venus, Kepler might well have been driven
to abandon his search for the true curve of planetary motion.
However, the facts of the cosmos were on his side, but the calculations
essential in testing his various hypotheses were of the most tedious
nature, because logarithms were not yet known in his day. His first
discovery was that the orbit of Mars is certainly not a circle, but oval
or elliptic in figure. And the sun, he soon found, could not be in the
center of the ellipse, so he made a series of trial calculations with
the sun located in one of the foci of the ellipse instead.
Then he found he could make his calculated places of Mars agree quite
perfectly with Tycho Brahe's observed positions, if only he gave up the
other ancient requisite of perfectly uniform motion. On doing this, it
soon appeared that Mars, when in perihelion, or nearest the sun, always
moved swiftest, while at its greatest distance from the sun, or
aphelion, its orbital velocity was slowest.
Kepler did not busy himself to inquire why these revolutionary
discoveries of his were as they were; he simply went on making enough
trials on Mars, and then on the other planets in turn, to satisfy
himself that all the planetary orbits are elliptical, not circular in
form, and are so located in space that the center of the sun is at one
of the two foci of each orbit. This is known as Kepler's first law of
planetary motion.
The second one did not come quite so easy; it concerned the variable
speed with which the planet moves at every point of the orbit. We must
remember how handicapped he was in solving this problem: only the
geometry of Euclid to work with, and none of the refinements of the
higher mathematics of a later day. But he finally found a very simple
relation which represented the velocity of the planet everywhere in its
orbit. It was this: if we calculate the area swept, or passed over, by
the planet's radius vector (that is, the line joining its center to the
sun's center) during a week's time near perihelion, and then calculate
the similar area for a week near aphelion, or indeed for a week when
Mars is in any intermediate part of its orbit, we shall find that these
areas are all equal to each other. So Kepler formulated his second great
law of planetary motion very simply: the radius vector of any planet
describes, or sweeps over, equal areas in equal times. And he found this
was true for all the planets.
But the real genius of the great mathematician was shown in the
discovery of his third law, which is more complex and even more
significant than the other two--a law connecting the distances of the
planets from the sun with their periods of revolution about the sun.
This cost Kepler many additional years of close calculation, and the
resulting law, his third law of planetary motion is this: The cubes of
the mean or average distances of the planets from the sun are
proportional to the squares of their times of revolution around him.
So Kepler had not only disposed of the sacred theories of motion of the
planets held by the ancients as inviolable, but he had demonstrated the
truth of a great law which bound all the bodies of the solar system
together. So accurately and completely did these three laws account for
all the motions, that the science of astronomy seemed as if finished;
and no matter how far in the future a time might be assigned, Kepler's
laws provided the means of calculating the planet's position for that
epoch as accurately as it would be possible to observe it. Kepler paused
here, and he died in 1630.