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When Kepler had before him that wonderful series of observations of the planet, Mars, which had been accumulated by the extraordinary skill of Tycho, he proved, after much labour, that the movements of the planet refused to be represented in a circular form. Nor would it do to suppose that Mars revolved in one circle, the centre of which revolved in another circle. On no such supposition could the movements of the planets be made to tally with those which Tycho had actually observed. This led to the astonishing discovery of the true form of a planet's orbit. For the first time in the history of astronomy the principle was laid down that the movement of a planet could not be represented by a circle, nor even by combinations of circles, but that it could be represented by an elliptic path. In this path the sun is situated at one of those two points in the ellipse which are known as its foci.

Very simple apparatus is needed for the drawing of one of those ellipses which Kepler has shown to possess such astonishing astronomical significance. Two pins are stuck through a sheet of paper on a board, the point of a pencil is inserted in a loop of string which passes over the pins, and as the pencil is moved round in such a way as to keep the string stretched, that beautiful curve known as the ellipse is delineated, while the positions of the pins indicate the two foci of the curve. If the length of the loop of string is unchanged then the nearer the pins are together, the greater will be the resemblance between the ellipse and the circle, whereas the more the pins are separated the more elongated does the ellipse become. The orbit of a great planet is, in general, one of those ellipses which approaches a nearly circular form. It fortunately happens, however, that the orbit of Mars makes a wider departure from the circular form than any of the other important planets. It is, doubtless, to this circumstance that we must attribute the astonishing success of Kepler in detecting the true shape of a planetary orbit. Tycho's observations would not have been sufficiently accurate to have exhibited the elliptic nature of a planetary orbit which, like that of Venus, differed very little from a circle.

The more we ponder on this memorable achievement the more striking will it appear. It must be remembered that in these days we know of the physical necessity which requires that a planet shall revolve in an ellipse and not in any other curve. But Kepler had no such knowledge. Even to the last hour of his life he remained in ignorance of the existence of any natural cause which ordained that planets should follow those particular curves which geometers know so well. Kepler's assignment of the ellipse as the true form of the planetary orbit is to be regarded as a brilliant guess, the truth of which Tycho's observations enabled him to verify. Kepler also succeeded in pointing out the law according to which the velocity of a planet at different points of its path could be accurately specified. Here, again, we have to admire the sagacity with which this marvellously acute astronomer guessed the deep truth of nature. In this case also he was quite unprovided with any reason for expecting from physical principles that such a law as he discovered must be obeyed. It is quite true that Kepler had some slight knowledge of the existence of what we now know as gravitation. He had even enunciated the remarkable doctrine that the ebb and flow of the tide must be attributed to the attraction of the moon on the waters of the earth. He does not, however, appear to have had any anticipation of those wonderful discoveries which Newton was destined to make a little later, in which he demonstrated that the laws detected by Kepler's marvellous acumen were necessary consequences of the principle of universal gravitation.

[PLATE: SYMBOLICAL REPRESENTATION OF THE PLANETARY SYSTEM.]

To appreciate the relations of Kepler and Tycho it is necessary to note the very different way in which these illustrious astronomers viewed the system of the heavens. It should be observed that Copernicus had already expounded the true system, which located the sun at the centre of the planetary system. But in the days of Tycho Brahe this doctrine had not as yet commanded universal assent. In fact, the great observer himself did not accept the new views of Copernicus. It appeared to Tycho that the earth not only appeared to be the centre of things celestial, but that it actually was the centre. It is, indeed, not a little remarkable that a student of the heavens so accurate as Tycho should have deliberately rejected the Copernican doctrine in favour of the system which now seems so preposterous. Throughout his great career, Tycho steadily observed the places of the sun, the moon, and the planets, and as steadily maintained that all those bodies revolved around the earth fixed in the centre. Kepler, however, had the advantage of belonging to the new school. He utilised the observations of Tycho in developing the great Copernican theory whose teaching Tycho stoutly resisted.

Perhaps a chapter in modern science may illustrate the intellectual relation of these great men. The revolution produced by Copernicus in the doctrine of the heavens has often been likened to the revolution which the Darwinian theory produced in the views held by biologists as to life on this earth. The Darwinian theory did not at first command universal assent even among those naturalists whose lives had been devoted with the greatest success to the study of organisms. Take, for instance, that great naturalist, Professor Owen, by whose labours vast extension has been given to our knowledge of the fossil animals which dwelt on the earth in past ages. Now, though Owens researches were intimately connected with the great labours of Darwin, and afforded the latter material for his epoch-making generalization, yet Owen deliberately refused to accept the new doctrines. Like Tycho, he kept on rigidly accumulating his facts under the influence of a set of ideas as to the origin of living forms which are now universally admitted to be erroneous. If, therefore, we liken Darwin to Copernicus, and Owen to Tycho, we may liken the biologists of the present day to Kepler, who interpreted the results of accurate observation upon sound theoretical principles.

In reading the works of Kepler in the light of our modern knowledge we are often struck by the extent to which his perception of the sublimest truths in nature was associated with the most extravagant errors and absurdities. But, of course, it must be remembered that he wrote in an age in which even the rudiments of science, as we now understand it, were almost entirely unknown.

It may well be doubted whether any joy experienced by mortals is more genuine than that which rewards the successful searcher after natural truths. Every science-worker, be his efforts ever so humble, will be able to sympathise with the enthusiastic delight of Kepler when at last, after years of toil, the glorious light broke forth, and that which he considered to be the greatest of his astonishing laws first dawned upon him. Kepler rightly judged that the number of days which a planet required to perform its voyage round the sun must be connected in some manner with the distance from the planet to the sun; that is to say, with the radius of the planet's orbit, inasmuch as we may for our present object regard the planet's orbit as circular.

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