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delight, was a display of pure fancy. His demonstration of the three fundamental laws of planetary motion was the most strict and complete theory that had ever been attempted. It has been often suggested that the revival by Copernicus of the notion of a moving earth was a help to Kepler. No one who reads Kepler's great book could hold such an opinion for a moment. In fact, the excellence of Copernicus's book helped to prolong the life of the epicyclical theories in opposition to Kepler's teaching. All of the best theories were compared by him with Observation. These were the Ptolemaic, the Copernican, and the Tychonic. The two latter placed all of the planetary orbits concentric with one another, the sun being placed a little away from their common centre, and having no apparent relation to them, and being actually outside the planes in which they In OVe. Kepler's first great discovery was that the planes of all the orbits pass through the sun; his second was that the line of apses of each planet passes through the sun; both were contradictory to the Copernican theory. He proceeds cautiously with his propositions until he arrives at his great laws, and he concludes his book by comparing observations of Mars, of all dates, with his theory.

His first law states that the planets describe ellipses with the sun at a focus of each ellipse. His second law (a far more difficult one to prove) states that a line drawn from a planet to the sun sweeps over equal areas in equal times. These two laws were published in his great work, Astronomia Nova, seu Physica Caelestis tradita commentariis de Motibus Stellae Martis, Prague, I6o9. It took him nine years more to discover his third law, that the squares of the periodic times are proportional to the cubes of the mean distances from the sun. These three laws contain implicitly the law of universal gravitation. They are simply an alternative way of expressing that law in dealing with planets, not particles. Only, the power of the greatest human intellect is so utterly feeble that the meaning of the words in Kepler's three laws could not be understood until expounded by the logic of Newton's dynamics. The joy with which Kepler contemplated the final demonstration of these laws, the evolution of which had occupied twenty years, can hardly be imagined by us. He has given some idea of it in a passage in his work on Harmonics, which * The ruled logarithmic paper we now use was not is not now quoted, only lest someone might say it was egotistical — a term which is simply grotesque when applied to such a man with such a life's work accomplished. The whole book, Astronomia Nova, is a pleasure to read; the mass of observations that are used, and the ingenuity of the propositions, contrast strongly with the loose and imperfectly supported explanations of all his predecessors; and the indulgent reader will excuse the devotion of a few lines to an example of the ingenuity and beauty of his methods. It may seem a hopeless task to find out the true paths of Mars and the earth (at that time when their shape even was not known) from the observations giving only the relative direction from night to night. Now, Kepler had twenty years of observations of Mars to deal with. This enabled him to use a new method, to find the earth's orbit. Observe the date at any time when Mars is in opposition. The earth's position E at that date gives the longitude of Mars M. His period is 687 days. Now choose dates before and after the principal date at intervals of 687 days and its multiples. Mars is in each case in the same position. Now for any date when Mars is at M and the earth at E, the date of the year gives the angle E,SM. And the observation of Tycho gives the direction of Mars compared with the sun, SEAM. So all the angles of the triangle SEM in any of these positions of E are known, and also the ratios of SE, SE, SE, SE, to SM and to each other. For the orbit of Mars observations were chosen at intervals of a year, when the earth was always in the same place. But Kepler saw much farther than the geometrical facts. He realised that the orbits are followed owing to a force directed to the sun; and he guessed that this is the same force as the gravity that makes a stone fall. He saw the difficulty of gravitation acting through the void space. He compared universal gravitation to magnetism, and speaks of the work of Gilbert of Colchester. (Gilbert's book, De Mundo Nostro Sublunari, Philosophia Nova, Amstelodami, 1651, containing similar views, was published forty-eight years after Gilbert's death, and forty-two years after Kepler's book and reference. His book De Magnete was published in 16oo.) A few of Kepler's views on gravitation, extracted from the Introduction to his Astronomia Nova, may now be mentioned: — 1. Every body at rest remains at rest if outside the attractive power of other bodies.

then to be had by going into a stationer's shop. Else he would have accomplished this in five minutes.

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2. Gravity is a property of masses mutually attracting in such manner that the earth attracts a stone much more than a stone attracts the earth. 3. Bodies are attracted to the earth's centre, not because it is the centre of the universe, but because it is the centre of the attracting particles of the earth. 4. If the earth be not round (but spheroidal?), then bodies at different latitudes will not be attracted to its centre, but to different points in the neighbourhood of that centre. 5. If the earth and moon were not retained in their orbits by vital force (aut alia aliqua aequipollenti), the earth and moon would come together. 6. If the earth were to cease to attract its waters, the oceans would all rise and flow to the In OOn . 7. He attributes the tides to lunar attraction. Kepler had been appointed Imperial Astronomer with a handsome salary (on paper), a fraction of which was doled out to him very irregularly. He was led to miserable makeshifts to earn enough to keep his family from starvation; and proceeded to Ratisbon in 1630 to represent his claims to the Diet. He arrived worn out and debilitated; he failed in his appeal, and died from fever, contracted under, and fed upon, disappointment and exhaustion. Those were not

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