rent planets had different periodic times; he
also saw that the greater the mean distance of the planet the greater
was its periodic time, and he was determined to find out the connection
between the two. It was easily found that it would not be true to say
that the periodic time is merely proportional to the mean distance. Were
this the case, then if one planet had a distance twice as great as
another, the periodic time of the former would have been double that of
the latter; observation showed, however, that the periodic time of the
more distant planet exceeded twice, and was indeed nearly three times,
that of the other. By repeated trials, which would have exhausted the
patience of one less confident in his own sagacity, and less assured of
the accuracy of the observations which he sought to interpret, Kepler at
length discovered the true law, and expressed it in the form we have
stated.

To illustrate the nature of this law, we shall take for comparison the
earth and the planet Venus. If we denote the mean distance of the earth
from the sun by unity then the mean distance of Venus from the sun is
0.7233. Omitting decimals beyond the first place, we can represent the
periodic time of the earth as 365.3 days, and the periodic time of
Venus as 224.7 days. Now the law which Kepler asserts is that the square
of 365.3 is to the square of 224.7 in the same proportion as unity is to
the cube of 0.7233. The reader can easily verify the truth of this
identity by actual multiplication. It is, however, to be remembered
that, as only four figures have been retained in the expressions of the
periodic times, so only four figures are to be considered significant in
making the calculations.

The most striking manner of making the verification will be to regard
the time of the revolution of Venus as an unknown quantity, and deduce
it from the known revolution of the earth and the mean distance of
Venus. In this way, by assuming Kepler's law, we deduce the cube of the
periodic time by a simple proportion, and the resulting value of 224.7
days can then be obtained. As a matter of fact, in the calculations of
astronomy, the distances of the planets are usually ascertained from
Kepler's law. The periodic time of the planet is an element which can be
measured with great accuracy; and once it is known, then the square of
the mean distance, and consequently the mean distance itself, is
determined.

Such are the three celebrated laws of P