1 12 Treatment of Geometry as a branch of Analysis. [No. 134. 



chanics what he had done for geometry. And when we take further 

 into account, the long and early intimacy between the two analysts, 

 (Legendre having edited the first edition of the Mecanique Analy- 

 tique), it becomes highly probable that Lagrange was the first who con- 

 ceived the idea of condensing the scattered truths of geometry into a 

 few families of formulae, as he did those of mechanics : and that Le- 

 gendre caught the spirit of such peculiar reasoning from his friend, his 

 own original genius enabling him to apply it with the success he did. 



3. Legendre's mode of procedure may be put in the following man- 

 ner. If from the ends of a given base we draw two straight lines 

 making given angles with that base, we have performed definite opera- 

 tions giving a single fixed result. If this result prove to be a triangle, 

 then the triangle being sole and invariable, its elements must all be 

 determinable by calculations founded on the data which produce that 

 invariability ; viz. the base and the base angles. Both the data and 

 the quaesita can only appear in these calculations in the shape of num- 

 bers, and therefore either as ratios inter se, or ratios involving some 

 constant unit of measurement. Now the angles have such a constant 

 unit in the right angle, but the sides have not, there being no natural 

 unit of linearity. The consequence will be, that the sides can only 

 appear in the calculations as ratios inter se, but the angles may appear 

 either as ratios inter se, or as fractions of a right angle. Now among 

 the elements of the triangle which are determined by the base and 

 base angles is the third angle, it will follow therefore that there is 

 some form of calculation connecting this third with the data. But of 

 these four the angles easily enter the calculation, while we do not see 

 how the side can, since there is no other line necessarily involved in 

 the matter. We conclude therefore that the side cannot enter, and 

 therefore that the third angle is determinable only by help of the other 

 two. Hence, whenever two angles in each of two triangles are iden- 

 tical, each to each, the third angles are also identical. 



The sequel of this demonstration is geometrical. By dropping a per- 

 pendicular on the hypothenuse from the right angle, we divide a right 

 angled triangle into two others, each of which has two angles equal to 

 two of the primitive triangle. They are consequently equiangular to 

 the primitive triangle and to each other, whence it is seen, that the two 

 acute angles of the large triangle are together equal to the right angle, 



