[24] The spheroidal figure of the earth was established by the comparison of an arc of the meridian that had been measured in France, with a similar arc measured in Lapland, from which it appeared that the length of a degree of the meridian increases from the equator towards the poles, conformably to what ought to result upon the supposition of the earth having the figure of an oblate spheroid.  The length of the Lapland arc was determined by means of an expedition which the French Government had despatched to the North of Europe for that purpose.  A similar expedition had been despatched from France about the same time to Peru in South America, for the purpose of measuring an arc of the meridian under the equator, but the results had not been ascertained at the time to which the author alludes in the text.  The variation of gravity at the surface of the earth was established by Richer’s experiments with the pendulum at Cayenne, in South America (1673-4), from which it appeared that the pendulum oscillates more slowly—­and consequently the force of gravity is less intense—­under the equator than in the latitude of Paris.—­Translator.

[25] It may perhaps be asked why we place Lagrange among the French geometers?  This is our reply:  It appears to us that the individual who was named Lagrange Tournier, two of the most characteristic French names which it is possible to imagine, whose maternal grandfather was M. Gros, whose paternal great-grandfather was a French officer, a native of Paris, who never wrote except in French, and who was invested in our country with high honours during a period of nearly thirty years;—­ought to be regarded as a Frenchman although born at Turin.—­Author.

[26] The problem of three bodies was solved independently about the same time by Euler, D’Alembert, and Clairaut.  The two last-mentioned geometers communicated their solutions to the Academy of Sciences on the same day, November 15, 1747.  Euler had already in 1746 published tables of the moon, founded on his solution of the same problem, the details of which he subsequently published in 1753.—­Translator.

[27] It must be admitted that M. Arago has here imperfectly represented Newton’s labours on the great problem of the precession of the equinoxes.  The immortal author of the Principia did not merely conjecture that the conical motion of the earth’s axis is due to the disturbing action of the sun and moon upon the matter accumulated around the earth’s equator:  he demonstrated by a very beautiful and satisfactory process that the movement must necessarily arise from that cause; and although the means of investigation, in his time, were inadequate to a rigorous computation of the quantitative effect, still, his researches on the subject have been always regarded as affording one of the most striking proofs of sagacity which is to be found in all his works.—­Translator.