29/ 



Legendrc's Geometry, 29!S^ 



tering into long discussions and giving numerous details^ 

 which must "be imperfectly understood without diagrams 

 and without a minute comparison of the arrangement adopt- 

 ed by the writers of whom we are speaking. This is a 

 particular upon which a sound opinion cannot be formed 

 without personal inspection. We think that the conclusion 

 at which we have arrived, and which we have stated at the 

 beginning of this paragraph will be inevitable in the mind 

 of every one who will be at the pains of a comparison some- 

 what extensive and elaborate. All that we ask oCour read- 

 ers on the point now under consideration, is, that they will 

 not conclude us to be entirely and necessarily wrong, until 

 they have given the subject an attentive examination. 

 "In hisjudices desidero, qui tractarunt in sua amplitudine.* 

 We trust that this will not be considered an unreasonable 

 claim upon their candor. 



On the geometry of solids or volumes, also, the elements 

 of Legendre and Lacroix are very much more complete, 

 than those of Euclid. On this point, it is impossible to 

 convey an adequate idea to those who are not, to a consid- 

 erable extent, acquainted with the subject. At the time of 

 Euclid, the geometry of solids appears to have been quite 

 imperfectly investigated. It is true, that before this period, 

 the five regular bodies had been studied in the celebrated 

 school of Plato; and Archimedeshad made hisbrillianf discov- 

 eries in relation to the properties of the sphere and cylinder. 

 But the properties of Polyedrons in general, and their meas- 

 ure, have not received, until within a short period, the at- 

 tention which their importance merited ; and M. Legendre 

 in particular, has contributed much to the elucidation of the 

 subject. M. Cauchy, also, has done considerable towards 

 the perfection of this part of elementary geometry. 



We are now sufficiently prepared to enter with advan- 

 tage upon an examination of the work before us. The 

 definitions and axioms are laid down very much in the usual 

 style. The latter are nearly the same in substance with 

 those of Euclid, and differ from them principally in the cir- 

 cumstance, that the idea of equality is not drawn out into 

 particulars. They are but five in number. It is evident 

 he does not attempt a complete enumeration of them, a 

 thing which no geometer has accomplished. A straight 

 line is defined to be, "the shortest way from one point to 



* Valckenaer ad Herodotum, p. 585= 



