142. Although the two laws of planetary motion just given were only fully established for the case of Mars, Kepler stated that the earth’s path also must be an oval of some kind, and was evidently already convinced—aided by his firm belief in the harmony of Nature—that all the planets moved in accordance with the same laws. This view is indicated in the dedication of the book to the Emperor Rudolph, which gives a fanciful account of the work as a struggle against the rebellious War-God Mars, as the result of which he is finally brought captive to the feet of the Emperor and undertakes to live for the future as a loyal subject. As, however, he has many relations in the ethereal spaces—his father Jupiter, his grandfather Saturn, his dear sister Venus, his faithful brother Mercury—and he yearns for them and they for him on account of the similarity of their habits, he entreats the Emperor to send out an expedition as soon as possible to capture them also, and with that object to provide Kepler with the “sinews of war” in order that he may equip a suitable army.
Although the money thus delicately asked for was only supplied very irregularly, Kepler kept steadily in view the expedition for which it was to be used, or, in plainer words, he worked steadily at the problem of extending his elliptic theory to the other planets, and constructing the tables of the planetary motions, based on Tycho’s observations, at which he had so long been engaged.
143. In 1611 his patron Rudolph was forced to abdicate the imperial crown in favour of his brother Matthias, who had little interest in astronomy, or even in astrology; and as Kepler’s position was thus rendered more insecure than ever, he opened negotiations with the Estates of Upper Austria, as the result of which he was promised a small salary, on condition of undertaking the somewhat varied duties of teaching mathematics at the high school of Linz, the capital, of constructing a new map of the province, and of completing his planetary tables. For the present, however, he decided to stay with Rudolph.
In the same year Kepler lost his wife, who had long been in weak bodily and mental health.
In the following year (1612) Rudolph died, and Kepler then moved to Linz and took up his new duties there, though still holding the appointment of mathematician to the Emperor and occasionally even receiving some portion of the salary of the office. In 1613 he married again, after a careful consideration, recorded in an extraordinary but very characteristic letter to one of his friends, of the relative merits of eleven ladies whom he regarded as possible; and the provision of a proper supply of wine for his new household led to the publication of a pamphlet, of some mathematical interest, dealing with the proper way of measuring the contents of a cask with curved sides.[90]
144. In the years 1618-1621, although in some ways the most disturbed years of his life, he published three books of importance—an Epitome of the Copernican Astronomy, the Harmony of the World,[91] and a treatise on Comets.
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 |