The Life of Galileo Galilei, with Illustrations of the Advancement of Experimental Philosophy / Life of Kepler

by John Elliot Drinkwater Bethune

This passage sets in the strongest light one of the most notorious errors of the older philosophy, to which Kepler himself was remarkably addicted. If Gilbert had asserted, in direct terms, that the moon attracted the water, it is certain that the notion would have been stigmatized (as it was for a long time in Newton's hands) as arbitrary, occult, and unphilosophical: the idea of these subterranean humours was likely to be treated with much more indulgence. A simple statement, that when the moon was over the water the latter had a tendency to rise towards it, was thought to convey no instruction; but the assertion that the moon draws out subterranean spirits by sympathy, carried with it a more imposing appearance of theory. The farther removed these humours were from common experience, the easier it became to discuss them in vague and general language; and those who called themselves philosophers could endure to hear attributes bestowed on these fictitious elements which revolted their imaginations when applied to things of whose reality at least some evidence existed.

It is not necessary to dwell upon the system of Tycho Brahe, which was identical, as we have said, with one rejected by Copernicus, and consisted in making the sun revolve about the earth, carrying with it all the other planets revolving about him. Tycho went so far as to deny the rotation of the earth to explain the vicissitudes of day and night, but even his favourite assistant Longomontanus differed from him in this part of his theory. The great merit of Tycho Brahe, and the service he rendered to astronomy, was entirely independent of any theory; consisting in the vast accumulation of observations made by him during a residence of fifteen years at Uraniburg, with the assistance of instruments, and with a degree of care, very far superior to anything known before his time in practical astronomy. Kepler is careful repeatedly to remind us, that without Tycho's observations he could have done nothing. The degree of reliance that might be placed on the results obtained by observers who acknowledged their inferiority to Tycho Brahe, maybe gathered from an incidental remark of Kepler to Longomontanus. He had been examining Tycho's registers, and had occasionally found a difference amounting sometimes to 4´ in the right ascensions of the same planet, deduced from different stars on the same night. Longomontanus could not deny the fact, but declared that it was impossible to be always correct within such limits. The reader should never lose sight of this uncertainty in the observations, when endeavouring to estimate the difficulty of finding a theory that would properly represent them.

When Kepler first joined Tycho Brahe at Prague, he found him and Longomontanus very busily engaged in correcting the theory of Mars, and accordingly it was this planet to which he also first directed his attention. They had formed a catalogue of the mean oppositions of Mars during twenty years, and had discovered a position of the equant, which (as they said) represented them with tolerable exactness. On the other hand, they were much embarrassed by the unexpected difficulties they met in applying a system which seemed on the one hand so accurate, to the determination of the latitudes, with which it could in no way be made to agree. Kepler had already suspected the cause of this imperfection, and was confirmed in the view he took of their theory, when, on a more careful examination, he found that they overrated the accuracy even of their longitudes. The errors in these, instead of amounting as they said, nearly to 2´, rose sometimes above 21´. In fact they had reasoned ill on their own principles, and even if the foundations of their theory had been correctly laid, could not have arrived at true results. But Kepler had satisfied himself of the contrary, and the following diagram shews the nature of the first alteration he introduced, not perhaps so celebrated as some of his later discoveries, but at least of equal consequence to astronomy, which could never have been extricated from the confusion into which it had fallen, till this important change had been effected.

The practice of Tycho Brahe, indeed of all astronomers till the time of Kepler, had been to fix the position of the planet's orbit and equant from observations on its mean oppositions, that is to say, on the times when it was precisely six signs or half a circle distant from the mean place of the sun. In the annexed figure, let S represent the sun, C the centre of the earth's orbit, Tt. Tycho Brahe's practice amounted to this, that if Q were supposed the place of the centre of the planet's equant, the centre of Pp its orbit was taken in QC, and not in QS, as Kepler suggested that it ought to be taken. The consequence of this erroneous practice was, that the observations were deprived of the character for which oppositions were selected, of being entirely free from the second inequalities. It followed therefore that as part of the second inequalities were made conducive towards fixing the relative position of the orbit and equant, to which they did not naturally belong, there was an additional perplexity in accounting for the remainder of them by the size and motion of the epicycle. As the line of nodes of every planet was also made to pass through C instead of S, there could not fail to be corresponding errors in the latitudes. It would only be in the rare case of an opposition of the planet in the line CS, that the time of its taking place would be the same, whether O, the centre of the orbit, was placed in CQ or SQ. Every other opposition would involve an error, so much the greater as it was observed at a greater distance from the line CS.

It was long however before Tycho Brahe could be made to acquiesce in the propriety of the proposed alteration; and, in order to remove his doubts as to the possibility that a method could be erroneous which, as he still thought, had given him such accurate longitudes, Kepler undertook the ungrateful labour of the first part of his "Commentaries." He there shewed, in the three systems of Copernicus, Tycho Brahe, and Ptolemy, and in both the concentric and excentric theories, that though a false position were given to the orbit, the longitudes of a planet might be so represented, by a proper position of the centre of the equant, as never to err in oppositions above 5´ from those given by observation; though the second inequalities and the latitudes would thereby be very greatly deranged.

The change Kepler introduced, of observing apparent instead of mean oppositions, made it necessary to be very accurate in his reductions of the planet's place to the ecliptic; and in order to be able to do this, a previous knowledge of the parallax of Mars became indispensable. His next labour was therefore directed to this point; and finding that the assistants to whom Tycho Brahe had previously committed this labour had performed it in a negligent and imperfect manner, he began afresh with Tycho's original observations. Having satisfied himself as to the probable limits of his errors in the parallax on which he finally fixed, he proceeded to determine the inclination of the orbit and the position of the line of nodes. In all these operations his talent for astronomical inquiries appeared pre-eminent in a variety of new methods by which he combined and availed himself of the observations; but it must be sufficient merely to mention this fact, without entering into any detail. One important result may be mentioned, at which he arrived in the course of them, the constancy of the inclination of the planet's orbit, which naturally strengthened him in his new theory.

Having gone through these preliminary inquiries, he came at last to fix the proportions of the orbit; and, in doing so, he determined, in the first instance, not to assume, as Ptolemy appeared to have done arbitrarily, the bisection of the excentricity, but to investigate its proportion along with the other elements of the orbit, which resolution involved him in much more laborious calculations. After he had gone over all the steps of his theory no less than seventy times—an appalling labour, especially if we remember that logarithms were not then invented—his final result was, that in 1587, on the 6th of March, at 7^h 23´, the longitude of the aphelion of Mars was 4^s 28° 48´ 55´´; that the planet's mean longitude was 6^s 0° 51´ 35´´; that if the semidiameter of the orbit was taken at 100000, the excentricity was 11332; and the excentricity of the equant 18564. He fixed the radius of the greater epicycle at 14988, and that of the smaller at 3628.

When he came to compare the longitudes as given by this, which he afterwards called the vicarious theory, with the observations at opposition, the result seemed to promise him the most brilliant success. His greatest error did not exceed 2´; but, notwithstanding these flattering anticipations, he soon found by a comparison of longitudes out of opposition and of latitudes, that it was yet far from being so complete as he had imagined, and to his infinite vexation he soon found that the labour of four years, which he had expended on this theory, must be considered almost entirely fruitless. Even his favourite principle of dividing the excentricity in a different ratio from Ptolemy, was found to lead him into greater error than if he had retained the old bisection. By restoring that, he made his latitudes more accurate, but produced a corresponding change for the worse in his longitudes; and although the errors of 8´, to which they now amounted, would probably have been disregarded by former theorists, Kepler could not remain satisfied till they were accounted for. Accordingly he found himself forced to the conclusion that one of the two principles on which this theory rested must be erroneous; either the orbit of the planet is not a perfect circle, or there is no fixed point within it round which it moves with an uniform angular motion. He had once before admitted the possibility of the former of these facts, conceiving it possible that the motion of the planets is not at all curvilinear, but that they move in polygons round the sun, a notion to which he probably inclined in consequence of his favourite harmonics and geometrical figures.

In consequence of the failure of a theory conducted with such care in all its practical details, Kepler determined that his next trial should be of an entirely different complexion. Instead of first satisfying the first inequalities of the planet, and then endeavouring to account for the second inequalities, he resolved to reverse the process, or, in other words, to ascertain as accurately as possible what part of the planet's apparent motion should be referred solely to the optical illusion produced by the motion of the earth, before proceeding to any inquiry of the real inequality of the planet's proper motion. It had been hitherto taken for granted, that the earth moved equably round the centre of its orbit; but Kepler, on resuming the consideration of it, recurred to an opinion he had entertained very early in his astronomical career (rather from his conviction of the existence of general laws, than that he had then felt the want of such a supposition), that it required an equant distinct from its orbit no less than the other planets. He now saw, that if this were admitted, the changes it would everywhere introduce in the optical part of the planet's irregularities might perhaps relieve him from the perplexity in which the vicarious theory had involved him. Accordingly he applied himself with renewed assiduity to the examination of this important question, and the result of his calculations (founded principally on observations of Mars' parallax) soon satisfied him not only that the earth's orbit does require such an equant, but that its centre is placed according to the general law of the bisection of the excentricity which he had previously found indispensable in the other planets. This was an innovation of the first magnitude, and accordingly Kepler did not venture to proceed farther in his theory, till by evidence of the most varied and satisfactory nature, he had established it beyond the possibility of cavil.