Cambridge Course of Mathematics. 28'5 



Art. X. — Elements of Geometry. By A. M. Legendre, 

 Member of the Institute and Legion of Ho7ior, of the Roy- 

 al Society of London^ <^c. ; translated from the French 

 for the use of the students of the University at Cambridge, 

 Mew-England. Cambridge, N. E. Hilliard 4" Metcalf 

 1819. p. 208. 



M. Legendre has long been regarded, as one of the 

 great luminaries of mathematical science. His rectilineal 

 and spherical trigonometry, appended to his Elements of 

 Geometry, though not very extensive, is marked with pro- 

 foundness and originality. His " Essai sur la Theorie des 

 Nombres," (of which the second edition, much larger and 

 more complete than the first, was published at Paris in 4to 

 in 1808,) contains the principal results of Fermat, Euler, 

 and Lagrange, together with the fruit of his own investiga- 

 tions, upon that difficult and important branch of mathe- 

 matics. It contains also some of the most interesting dis- 

 coveries of M. Gauss, upon the same subject. The first 

 elements of the " Theory of Numbers," or, as it is some- 

 times called " Transcendant Arithmetic," are demonstra- 

 ted in the seventh book of Euclid, with elegance and rigor. 

 We have some other ancient fragments on the properties 

 of numbers, but this branch of mathematics has been much 

 more cultivated by the moderns than by the ancients. In- 

 deed, our system of Arithmetical Notation, which ap- 

 proaches perhaps as near perfection as any in this world, 

 and the resources of our Algebra, have given the moderns 

 animmense advantage over the ancients in investigating the 

 general properties of numbers. Of the "Theorie" of Legen- 

 dre, M. Gauss thus speaks : Dans cet intervalle, il a paru 

 un excellent ouvrage d'un homme qui avait deja rendu de 

 tres-grands services a I'Arithmetique transcendante, dans 

 lequel il a non-seulement rassemble et mis en ordre tout 

 ce qui' a paru jusqu'a'^present sur cette science, mais ajoute 

 beaucoup de choses nouvelles qui lui sont propres.* Be- 

 sides these, he is author of a new method for the determin- 

 ation of the orbits of comets ; Exercises upon the Integral 



*Recherches Arithmetiques traduites par Delisle, preface p. 14. 



