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 was an ellipse. After Halley's visit, Newton, with Picard's new value for the earth's radius, reviewed his early calculation, and was able to show that if the distances between the bodies in the solar system were so great that the bodies might be considered as points, then their motions were in accordance with the assumed law of gravitation. In 1685 he completed his discovery by showing that a sphere whose density at any point depends only on the distance from the centre attracts an external point as though its whole mass were concentrated at the centre.[34]

Newton's unpublished manuscripts in the Portsmouth collection show that he had worked out, by means of fluxions and fluents, his lunar calculations to a higher degree of approximation than that given in the Principia, but that he was unable to interpret his results geometrically. The papers in that collection throw light upon the mode by which Newton arrived at some of the results in the Principia, as, for instance, the famous construction in Book II., Prop. 25, which is unproved in the Principia, but is demonstrated by him twice in a draft of a letter to David Gregory, of Oxford.[34]

It is chiefly upon the Principia that the fame of Newton rests. Brewster calls it "the brightest page in the records of human reason." Let us listen, for a moment, to the comments of Laplace, the foremost among those followers of Newton who grappled with the subtle problems of the motions of planets under the influence of gravitation: "Newton has well established the existence of the principle which he had the merit of discovering, but the development of its consequences and advantages has been the work of the successors of this great mathematician. The imperfection of the infinitesimal calculus, when first discovered, did not allow him completely to resolve the difficult problems which the theory of the universe offers; and he was oftentimes forced to give mere hints, which