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]
Fig. 62.—The “music of the spheres,” according to Kepler. From the Harmony of the World.
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 imperfect explanation, of that remarkable solar envelope known as the corona which has attracted so much attention in the last half-century (chapter XIII., [§ 301]).