This declaration of principles met, at the time, with the hostility of all astronomers of any reputation, but it has become the starting-point of the discovery of the laws on which the whole edifice of astronomy reposes. Had Kepler, however, been left to depend entirely on his own resources, he might, perhaps, have never completed his task. A fortunate circumstance brought him an unexpected ally. Tycho, having taken refuge in Bohemia, sent for the young astronomer (Kepler then was but twenty-nine years old), to assist him in the composition of the "Rudolphine Tables."[74]
"This," says Kepler, "was a providential interposition. I repaired to Bohemia early in the year 1600, in the hope of learning the correction of the eccentricities of the planets. Perceiving that Tycho made use of a mixed system (which made Mercury and Venus revolve around the sun, and all these planets, with their companions, around the earth), I asked his permission to follow out my own ideas. It was the will of Providence again, that we should occupy ourselves with Mars. My whole attention, therefore, was directed to this planet: and it is through the movements of Mars we must obtain our insight into the secrets of astronomy, or remain ignorant of them for ever (ex Martis motibus omnino necesse est nos in cognitionem astronomiæ arcanorum venire aut ea perpetuo nescire)."[75]
Why this preference given to Mars? In the first place, because, among all the planets then known, it was Mars which, in its movement round the sun, departed most from the circle; next, its orbit approaches nearest to the earth's; the earth is very near to Mars when she passes between that planet and the sun,—that is to say, when she is in opposition, while she retires from it triple the distance when in conjunction,—that is, when the sun is between her and Mars. Hence arise certain variations of aspect, particularly adapted to make manifest the form of the orbit, and the law of the real movement of the "red planet, Mars." As for the other planets, as far as they were then known, their orbits differ so little from the circle, that the nature of the curve which they describe in reality would never have been exactly recognised by any inexperienced star-gazer.
For these reasons Kepler regarded as providential the choice he had been led to make of Mars at the outset of his astronomical career. Before the close of 1601, Tycho died, bequeathing to his young fellow-worker a treasury of observation. Thenceforth Kepler undertook to finish without assistance the famous Rudolphine Tables. They cost him five-and-twenty years of assiduous labour. Looking upon Tycho's observations, because of their exactness, as "a gift from the Divine Goodness," he employed them, in the first place, as a test of the old hypotheses of planetary orbits and movements. Let us do our best to grasp the range and bearing of this part of his work.
In the system of Copernicus, which Kepler ardently adopted, the earth revolves around the sun. Now, observation having shown that the sun remains seven or eight days longer in the northern than in the southern signs of the Zodiac, we must of necessity admit that the sun, instead of being situated in the centre of the terrestrial orbit, occupies a point outside that centre, in such a manner that the earth must sometimes be nearer to, and sometimes farther from, the sun. The distance by which it departs from the centre of its orbit, which Copernicus, like the ancients, supposed to be circular, is called its eccentricity.
Astronomers were long preoccupied with the idea of seeking in this eccentricity a point where the movements should appear equal. This point was the centre of the equant,—a name given to the eccentric circle described from the point of equality or from the centre of the mean movements.
Now, let us recall the principal condition of the problem which Kepler had undertaken to solve. This condition required that the straight line drawn from the centre of our globe to the centre of the sun,—in a word, that the vector radius, as it is called, should describe around the sun certain angles, whose variability should agree with the results of observation.
Starting from this point, Kepler found that, for certain positions of Mars (in the aphelion and perihelion, corresponding to the minimum and maximum of velocity), the centre of the orbit, always supposing it to be circular, divided into two equal parts (or bisected) the total eccentricity: in other words, that it exactly occupied the middle between the centre of the eccentric and the equant of Ptolemæus; but it did not appear to him necessary to bisect it in other positions, intermediate between those of the aphelion and the perihelion. He established that the differences in longitude amounted to eight or nine minutes. Now, observations so exact as those of Tycho were altogether incompatible with such great error.[76] Therefore, the geometrical hypothesis which gave these errors was false; the orbit of Mars could not be a circle, and to save these eight or nine minutes, furnished by observation but in disaccord with theory, it would be needful to recommence all the calculations of astronomy. This conclusion, not less legitimate than daring, supplied Kepler with the first decisive step in the task he had undertaken.
This is not the place to relate all the essays and miscarriages through which this man of genius passed before finally completing his discovery of the rules that bear his name. But we may put before the reader the construction which led to them.
On a sheet of paper let us mark down by a point (Fig. 68) the place occupied by the earth in relation to the sun.[77] From this point o, we draw a right line terminating at a, the sun's noon-day position (for example, on the 1st of January); the succeeding lines shall touch upon a´ a´´, which the sun occupies successively after the same interval of time (twenty-four hours, or the exact duration of the earth's rotation on its axis);—and let us continue after this mode until the sun has accomplished, by its own proper movement from west to east, the whole circuit of the heavens, traversing 360 degrees in the space of a year. If we ascribe to the radius o a a certain length, corresponding to a definite solar diameter, the lengths of all the others, corresponding to the variations of the same diameter, will depend upon that of the first, which, for facility of calculation, we suppose to be divided into one thousand parts.