The second and most important of these, published in 1619, though the leading idea in it was discovered early in 1618, was regarded by Kepler as a development of his early Mysterium Cosmographicum (§ 136). His speculative and mystic temperament led him constantly to search for relations between the various numerical quantities occurring in the solar system; by a happy inspiration he thought of trying to get a relation connecting the sizes of the orbits of the various planets with their times of revolution round the sun, and after a number of unsuccessful attempts discovered a simple and important relation, commonly known as Kepler's third law:—
The squares of the times of revolution of any two planets (including the earth) about the sun are proportional to the cubes of their mean distances from the sun.
If, for example, we express the times of revolution of the various planets in terms of any one, which may be conveniently taken to be that of the earth, namely a year, and in the same way express the distances in terms of the distance of the earth from the sun as a unit, then the times of revolution of the several planets taken in the order Mercury, Venus, Earth, Mars, Jupiter, Saturn are approximately ⋅24, ⋅615, 1, 1⋅88, 11⋅86, 29⋅457, and their distances from the sun are respectively ⋅387, ⋅723, 1, 1⋅524, 5⋅203, 9⋅539; if now we take the squares of the first series of numbers (the square of a number being the number multiplied by itself) and the cubes of the second series (the cube of a number being the number multiplied by itself twice, or the square multiplied again by the number), we get the two series of numbers given approximately by the table:—
| Mercury. | Venus. | Earth. | Mars. | Jupiter. | Saturn. | ||
| Square of periodic time |
⋅058 | ⋅378 | 1 | 3⋅54 | 140⋅7 | 867⋅7 | |
| Cube of mean distance |
⋅058 | ⋅378 | 1 | 3⋅54 | 140⋅8 | 867⋅9 | |
Here it will be seen that the two series of numbers, in the
