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Again, in 1774, the Academy submitted the same subject, a third time, for the prize; and again Lagrange failed to detect a cause in gravitation.

Laplace" now took the matter in hand. He tried the effect of a non-instantaneous action of gravity, to no purpose But in 1787 he gave the true explanation. The principal effect of the sun on the moon's orbit is to diminish the earth's influence, thus lengthening the period to a new value generally taken as constant. But Laplace's calculations showed the new value to depend upon the excentricity of the earth's orbit, which, according to theory, has a periodical variation of enormous period, and has been continually diminishing for thousands of years. Thus the solar influence has been diminishing, and the moon's mean motion increased. Laplace computed the amount at Io" in one century, agreeing with observation. (Later on Adams showed that Laplace's calculation was wrong, and that the value he found was too large; so, part of the acceleration is now attributed by some astronomers to a lengthening of the day by tidal friction.)

Another contribution by Halley to the verification of Newton's law was made when he went to St. Helena to catalogue the southern stars. He measured the change in length of the sec

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ond's pendulum in different latitudes due to the changes in gravity foretold by Newton. Furthermore, he discovered the long inequality of Jupiter and Saturn, whose period is 929 years. For an investigation of this also the Academy of Sciences offered their prize. This led Euler to write a valuable essay disclosing a new method of computing perturbations, called the instantaneous ellipse with variable elements. The method was much developed by Lagrange. But again it was Laplace who solved the problem of the inequalities of Jupiter and Saturn by the theory of gravitation, reducing the errors of the tables from 20' down to 12", thus abolishing the use of empirical corrections to the planetary tables, and providing another glorious triumph for the law of gravitation. As Laplace justly said: “These inequalities appeared formerly to be inexplicable by the law of gravitation —they now form one of its most striking proofs.” Let us take one more discovery of Halley, furnishing directly a new triumph for the theory. He noticed that Newton ascribed parabolic orbits to the comets which he studied, so that they come from infinity, sweep round the sun, and go off to infinity for ever, after having been visible a few weeks or months. He collected all the reliable observations of comets he could find, to the number of twenty-four, and computed their parabolic orbits by the rules laid down by Newton. His object was to find out if any of them really travelled in elongated ellipses, practically undistinguishable, in the visible part of their paths, from parabolae, in which case they would be seen more than once. He found two old comets whose orbits, in shape and position, resembled the orbit of a comet observed by himself in 1682. Apian observed one in 1531; Kepler the other in 1607. The intervals between these appearances is seventyfive or seventy-six years. He then examined and found old records of similar appearance in 1456, 1380, and 1305. It is true, he noticed, that the intervals varied by a year and a half, and the inclination of the orbit to the ecliptic diminished with successive apparitions. But he knew from previous calculations that this might easily be due to planetary perturbations. Finally, he arrived at the conclusion that all of these comets were identical, travelling in an ellipse so elongated that the part where the comet was seen seemed to be part of a parabolic orbit. He then predicted its return at the end of 1758 or beginning of 1759, when he should be dead; but, as he said, “if it should return, according to our prediction, about the year 1758, impartial posterity will not refuse to acknowledge that this was first discovered by an Englishman.” [Synopsis Astronomiae Cometica, 1749.] Once again Halley's suggestion became an inspiration for the mathematical astronomer. Clairaut, assisted by Lalande, found that Saturn would retard the comet Ioo days, Jupiter 518 days, and predicted its return to perihelion on April 13th, 1759. In his communication to the French Academy, he said that a comet travelling into such distant regions might be exposed to the influence of forces totally unknown, and “even of some planet too far removed from the sun to be ever perceived.” The excitement of astronomers towards the end of 1758 became intense; and the honour of first catching sight of the traveller fell to an amateur in Saxony, George Palitsch, on Christmas Day, 1758. It reached perihelion on March 13th, 1759. This fact was a startling confirmation of the Newtonian theory, because it was a new kind of calculation of perturbations, and also it added a new member to the solar system, and gave a prospect of adding many more. When Halley's comet reappeared in 1835, Pontecoulant's computations for the date of perihelion passage were very exact, and afterwards * This sentence does not appear in the original he showed that, with more exact values of the masses of Jupiter and Saturn, his prediction was correct within two days, after an invisible voyage of seventy-five years!

memoir communicated to the Royal Society, but was first published in a posthumous reprint.

Hind afterwards searched out many old appearances of this comet, going back to 1 I B.C., and most of these have been identified as being really Halley's comet by the calculations of Cowell and Cromellin' (of Greenwich Observatory), who have also predicted its next perihelion passage for April 8th to 16th, 19 Io, and have traced back its history still farther, to 240 B.C.

Already, in November, 1907, the Astronomer Royal was trying to catch it by the aid of photography.

9. Discovery of NEw PLANETs — HERscHEL, PIAzzi, ADAMS, AND LE VERRIER.

It would be very interesting, but quite impossible in these pages, to discuss all the exquisite researches of the mathematical astronomers, and to inspire a reverence for the names connected with these researches, which for two hundred years have been establishing the universality of Newton's law. The lunar and planetary theories, the beautiful theory of Jupiter's satellites, the figure of the earth, and the tides, were mathematically treated by Maclaurin, D'Alembert,

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