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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:—
Here it will be seen that the two series of numbers, in the upper and lower row respectively, agree completely for as many decimal places as are given, except in the cases of the two outer planets, where the lower numbers are slightly in excess of the upper. For this discrepancy Newton afterwards assigned a reason (chapter IX., § 186), but with the somewhat imperfect knowledge of the times of revolution and distances which Kepler possessed the discrepancy was barely capable of detection, and he was therefore justified—from his standpoint—in speaking of the law as “precise.”92
It should be noticed further that Kepler’s law requires no knowledge of the actual distances of the several planets from the sun, but only of their relative distances, i.e. the number of times farther off from the sun or nearer to the sun any planet is than any other. In other words, it is necessary to have or to be able to construct a map of the solar system correct in its proportions, but it is quite unnecessary for this purpose to know the scale of the map.
Although the Harmony of the World is a large book, there is scarcely anything of value in it except what has already been given. A good deal of space is occupied with repetitions of the earlier speculations contained in the Mysterium Cosmographicum, and most of the rest is filled with worthless analogies between the proportions of the solar system and the relations between various musical scales.
He is bold enough to write down in black and white the “music of the spheres” (in the form shewn in fig. 62), while the nonsense which he was capable of writing may be further illustrated by the remark which occurs in the same part of the book: “The Earth sings the notes M I, F A, M I, so that you may guess from them that in this abode of ours MIsery (miseria) and FAmine (fames) prevail.”
145. The Epitome of the Copernican Astronomy, which appeared in parts in 1618, 1620, and 1621, although there are no very striking discoveries in it, is one of the most attractive of Kepler’s books, being singularly free from the extravagances which usually render his writings so tedious. It contains within moderately short compass, in the form of question and answer, an account of astronomy as known at the time, expounded from the Coppernican standpoint, and embodies both Kepler’s own and Galilei’s latest discoveries. Such a textbook supplied a decided want, and that this was recognised by enemies as well as by friends was shewn by its prompt appearance in the Roman Index of Prohibited Books (cf. chapter VI., §§ 126, 132). The Epitome contains the first clear statement that the two fundamental laws of planetary motion established for the case of Mars (§ 141) were true also for the other planets (no satisfactory proof being, however, given), and that they applied also to the motion of the moon round the earth, though in this case there were further irregularities which complicated matters. The theory of the moon is worked out in considerable detail, both evection (chapter II., § 48) and variation (chapter III., § 60; chapter V., § 111) being fully dealt with, though the “annual equation” which Tycho had just begun to recognise at the end of his life (chapter V., § 111) is not discussed. Another interesting development of his own discoveries is the recognition that his third law of planetary motion applied also to the movements of the four satellites round Jupiter, as recorded by Galilei and Simon Marius (chapter VI., § 118). Kepler also introduced in the Epitome a considerable improvement in the customary estimate of the distance of the earth from the sun, from which those of the other planets could at once be deduced.
If, as had been generally believed since the time of Hipparchus and Ptolemy, the distance of the sun were 1,200 times the radius of the earth, then the parallax (chapter II., §§ 43, 49) of the sun would at times be as much as 3′, and that of Mars, which in some positions is much nearer to the earth, proportionally larger. But Kepler had been unable to detect any parallax of Mars, and therefore inferred that the distances of Mars and of the sun must be greater than had been supposed. Having no exact data to go on, he produced out of his imagination and his ideas of the harmony of the solar system a distance about three times as great as the traditional one. He argued that, as the earth was the abode of measuring creatures, it was reasonable to expect that the measurements of the solar system would bear some simple relation to the dimensions of the earth. Accordingly he assumed that the volume of the sun was as many times greater than the volume of the earth as the distance of the sun was greater than the radius of the earth, and from this quaint assumption deduced the value of the distance already stated, which, though an improvement on the old value, was still only about one-seventh of the true distance.
The Epitome contains also a good account of eclipses both of the sun and moon, with the causes, means of predicting them, etc. The faint light (usually reddish) with which the face of the eclipsed moon often shines is correctly explained as being sunlight which has passed through the atmosphere of the earth, and has there been bent from a straight course so as to reach the moon, which the light of the sun in general is, owing to the interposition of the earth, unable to reach. Kepler mentions also a ring of light seen round the eclipsed sun in 1567, when the eclipse was probably total, not annular (chapter II., § 43), and ascribes it to some sort of luminous atmosphere round the sun, referring to a description in Plutarch of the same appearance. This seems to have been an early observation, and a rational though of course very
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