138 PROCEEDINGS OF THE AMERICAN ACADEMY 



it is evident by (4) that n cannot pass the limits -|- 1 and — 1, and 

 that n depends on the angle A ; also that n = 1 corresponds to ^ =0, 

 and n = — 1 to .4 = two right angles ; so that a is expressed in (5) 

 as required. 



" Prop. 2. To find the value of n, that corresponds to the base- 

 angles of any isosceles triangle. 



" Let AB C he any isosceles triangle ; having A B = CB = a, 

 for its sides, and A C=h 

 for its base, and let the 

 base be produced in the 

 direction A C to any point, 

 D, then, Sim., B. I., p. 13, 

 the angle BCD is the 



supplement of the angle -j^ ■ — —^ — j ^ 



B C A. Bisect the base 

 of the triangle in E, then draw the right line BE from the vertex 

 B to JE, and the triangles ABE, CBE are identical (Sim., B. I., 

 pp. 8 and 4) ; so that the angle A E B equals the angle C E B, and 

 these angles are right, Sim., B. I., def. 10, and BE is perpendicular 

 to the base of the triangle. By (5) of prop. 1, we get ar = 



^2 _|_ j2 — 2nh a, or by reduc. J = 2na, or n=^ -^ — -j^ = -^, 



a 



as required. Also, if we use m instead of n, for the vertical angle (jB), we 

 have b^ = a^-\- a^ — 2ma a = 2{l — 7/2) a~, or 1 — m= ^r,, or since 



h = 2na,we get ^^ =2n% .-. 2n^ = l — m,orn= ± v'iEl?, (1), 

 which is another form of n ; and it is manifest that if we take the 

 upper sign before the radical for the value of n that corresponds to 

 the acute angle B C A, we must take the lower sign before the radi- 

 cal in order to get the value of n that corresponds to the obtuse 

 angle BCD, which (as before noticed) is the supplement o( B C A 

 {=BAC). 



" Cor. 1. By what has been done it is evident, that, if we divide 

 one of the legs of a right-angled triangle by the hypothenuse, we get 

 the value of n that corresponds to the included angle ; for evidently 

 the same value of n corresponds to the isosceles triangle ABC, and 

 to the identical right triangles into which it is divided by the perpen- 

 dicular B E from its vertical angle. 



" Cor. 2. It is manifest from (1), that all those isosceles triangles 

 which have equal values of n for their base angles also have equal 

 values of m for their vertical angles. 



