OF ARTS AND SCIENCES. 



141 



angle that lies in one of the lines A F, A G coincides with the leg 

 of another triangle that lies in the other of these two lines ; but this 

 exception is only apparent, for the value of n in these two triangles 

 is the same as that of n in the two triangles ABE, ABE', .*. when 

 the hypothenuse of one triangle coincides with a leg of another triangle, 

 the value of 7i, that corresponds to A in one of the triangles, is the 

 same as in the other triangle. 



" Cor. We can now easily find the value of m that corresponds to 

 the vertical angle of an isosceles triangle whose base-angles are rep- 

 resented by n (or to which n corresponds). 



" For let ^ / be drawn from the base-angle A of the isosceles tri- 

 angle ABC, perpendicular to the opposite side B C, meeting it in 

 /, then from what has been shown we get C I=nb, or (since b = 

 2na) C/=2«2a, .-. BI=a— CI=a{l —2n^), and (since by 

 (1) of prop. 2,m=l — 2 n^) we get m = — = jg, which can also 

 be easily obtained from other considerations. And since n corre- 

 sponds equally to the base-angles of all the isosceles triangles repre- 

 sented by ABC, A B'C, and since w =^ 1 — 2n^, it follows that 

 all isosceles triangles whose base-angles are equal will have equal 

 values of m corresponding to their vertical angles. 



" Prop. 4. If there are two right-angled triangles, such that a leg 

 of the one divided by its hypothenuse gives the same quotient as a leg 

 of the other divided by its hypothenuse, then shall the angle included 

 by the leg and hypothenuse of the one triangle be equal to the angle 

 included by the leg and hypothenuse of the other triangle. 



" Let ABC, D F E be two triangles right-angled at C and E, 

 such that TB = TTp ^^ ^* ' ^^^^^ ^^^^^ ^^^ angle B AC equal the angle 



A ■ 



FD E. For on the longer leg D E of the one take D C equal to 

 A C, and through C draw C G perpendicular to D E, meeting D F 

 in 6r, then shall the triangles ABC, D GC be identical ; for since 

 the right triangles D F E, D G C have the angle D common, we 



