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equalities, one of the apogee, the other of the nodes, and assigned the maximum value. Grant has shown the values of some of these as given by observation in the tables of Meyer and more modern tables, and has compared them with the values assigned by Newton from his theory; and the comparison is very remarkable.

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Apses. . . . . . . . *- - - - - - - - - - - - - 3.4. O Mean annual motion of nodes 19. 18. 1,23 19.2 1.22,50 Mean value of “variation ”.. 36. Io 35.47 Annual equation . . . . . . . . - - - - II.5 I II. I.4 Inequality of mean motion of

a POgee. . . . . . . . . . . . . . . . . . . . I9.43 22. I 7 Inequality of mean motion of

nodes. . . . . . . . . . . . . . . . . . . . . 9.24 9.o

The only serious discrepancy is the first, which has been already mentioned. Considering that some of these perturbations had never been discovered, that the cause of none of them had ever been known, and that he exhibited his results, if he did not also make the discoveries, by the synthetic methods of geometry, it is simply marvellous that he reached to such a degree of accuracy. He invented the infinitesimal calculus which is more suited for such calculations, but had he expressed his results in that language he would have been unintelligible to many.

Newton's method of calculating the precession of the equinoxes, already referred to, is as beau

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tiful as anything in the Principia. He had already proved the regression of the nodes of a satellite moving in an orbit inclined to the ecliptic. He now said that the nodes of a ring of satellites revolving round the earth's equator would consequently all regress. And if joined into a solid ring its node would regress; and it would do so, only more slowly, if encumbered by the spherical part of the earth's mass. Therefore the axis of the equatorial belt of the earth must revolve round the pole of the ecliptic. Then he set to work and found the amount due to the moon and that due to the sun, and so he solved the mystery of 2,000 years. When Newton applied his law of gravitation to an explanation of the tides he started a new field for the application of mathematics to physical problems; and there can be little doubt that, if he could have been furnished with complete tidal observations from different parts of the world, his extraordinary powers of analysis would have enabled him to reach a satisfactory theory. He certainly opened up many mines full of intellectual gems; and his successors have never ceased in their explorations. This has led to improved mathematical methods, which, combined with the greater accuracy of observation, have rendered physical astronomy of to-day the most exact of the sciences. Laplace only expressed the universal opinion of posterity when he said that to the Principia is assured “a pre-eminence above all the other productions of the human intellect.”

The name of Flamsteed, First Astronomer Royal, must here be mentioned as having supplied Newton with the accurate data required for completing the theory.

The name of Edmund Halley, Second Astronomer Royal, must ever be held in repute, not only for his own discoveries, but for the part he played in urging Newton to commit to writing, and present to the Royal Society, the results of his investigations. But for his friendly insistence it is possible that the Principia would never have been written; and but for his generosity in supplying the means the Royal Society could not have published the book.

Sir Isaac Newton died in 1727, at the age of eighty-five. His body lay in state in the Jerusalem Chamber, and was buried in Westminster Abbey.


Edmund Halley succeeded Flamsteed as Second Astronomer Royal in 1721. Although he did not contribute directly to the mathematical proofs of Newton's theory, yet his name is closely associated with some of its greatest successes. Photographed specially for this work from the original, by kind permission of the Royal Society, London.



He was the first to detect the acceleration of the moon's mean motion. Hipparchus, having compared his own observations with those of more ancient astronomers, supplied an accurate value of the moon's mean motion in his time. Halley similarly deduced a value for modern times, and found it sensibly greater. He announced this in 1693, but it was not until 1749 that Dunthorne used modern lunar tables to compute a lunar eclipse observed in Babylon 72 I B.C., another at Alexandria 201 B.C., a solar eclipse observed by Theon 360 A.D., and two later ones up to the tenth century. He found that to explain these eclipses Halley's suggestion must be adopted, the acceleration being Io" in one century. In 1757 Lalande again fixed it at I o”.

The Paris Academy, in 1770, offered their prize for an investigation to see if this could be explained by the theory of gravitation. Euler won the prize, but failed to explain the effect, and said: “It appears to be established by indisputable evidence that the secular inequality of the moon's mean motion cannot be produced by the forces of gravitation.”

The same subject was again proposed for a prize which was shared by Lagrange' and Euler, neither finding a solution, while the latter asserted the existence of a resisting medium in

space. -
* Born 1736; died 1813.

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