OF ARTS AND SCIENCES. 



139 



" Prop. 3. All right-angled triangles which have one of their acute 

 angles common or equal will have equal values of n corresponding to 

 the common or equal angles. 



" Let AB E^ A B'E' be two right triangles, right-angled at E 

 and E', and having the common angle A the hypothenuse of the one 



and leg of the other (which include the common angle A) being in 

 the right lines A G^ A F, which include the angle A, that is common 

 to the two triangles ; that is, A B and A E' are \a A G, A B' and 

 A E in A E. When the angles are equal, but not common, we may- 

 imagine A BE to he one of the triangles, and we may suppose the 

 leg of the other triangle that is adjacent to the angle that is equal 

 to A to be applied to A G, so that the angle which equals A shall 

 coincide with A ; then will the hypothenuse of the applied triangle lie 

 on A E, and we may conceive that A B' E' represents the applied 

 triangle ; so that the case of equal angles is reduced to that of a com- 

 mon angle. he\EC = AE from E towards F, and E'C'^A E' 

 from E' towards G^ then draw right lines from C to U, and from 

 C to B' ; and it is evident by Sim., B. I., p. 4, that ABC, ABC 

 are isosceles triangles, B E, B'E' being the perpendiculars from their 

 vertical angles to their bases. Join the vertices of the isosceles tri- 

 angles by the right line BB' = c, and put .4 ^ = -B C= a, ^ C= h, 



b 



J = w ; also put A B' =^ B' C = a', AC = b', 



AE 

 AB 



CE 

 'CB 



= -7^ = V = w ; also 



