Cambridge Course of Mathematics. 283 
_ 
ART. MiasElenests of Geometry. By A .M. os ape 
Member of the Institute and Legion of Honon, of the R 
al Society of London, &c.; translated from the eck 
for the use of the students of the University at Cambridge, 
New-England. macehrle, NE. Hiltiard & Metcalf, 
1819, p. 208. >» 
M. Legendre has long been regarded, as one of the 
sreat 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 a 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 —- 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 ees has been much 
more cultivated by the moderns t y the ancients. In- 
deed, our system of Arithmetical Notation, which ap- 
proaches erhaps as near perfection as any in this world, 
and the resources of our Algebra, aa given the moderns 
pospoie ne advantage over the ancients in investigating the 
ea ofnumbers. Of the “Theorie” of Legen- 
dre, M. Gauss thus speaks : : Dans cet intervalle, il a paru 
un excellent ouvrage dun homme qui avait déja rendu de 
trés-grands services a roa tr ———— dans 
lequel il a non-sewlement rassemblé et mis en ordre tout 
ce qui a paru jusqu’a‘présent sur sua science, mais ajouté 
beaucoup de choses nouvelles qui lui sont propres.* Be- 
sides these, he is author of anew method for the determin- 
ation of the orbits of comets; Exercises upon the Integral 
*Recherches Arithmétiques traduiteg par Delisle, preface p. 14, 
