OF SPACE. 27 



BC (120). But FAE=:AFB (117), and EAC=:EBC-}-EAB=EDC 

 +EFB=BFC, therefore FAE : EACrzAFB : BFC, and FE : EC 

 =:AB : BC; but FEzrDB (114). In the same manner it may be 

 shown that the other homologous sides are proportional. 

 Scholium. Hence equiangular triangles are also called similar. 



122. Theorem. Equal and equiangular parallelo- 

 grams have their sides reciprocally proportional. 



If ABz=BC then DB : BEzzBF : BG. For 



DB : BF=AB : GF(120)=BC : GF=BE: BG 



(120); or DB ; BE=BF : BG. 



E C 



123. Theorem. Equiangular parallelograms, having 

 their sides reciprocally proportional, are equal. 



For they may be placed as in the last proposition, and tlie demon- 

 stration will be exactly similar. 



Scholium. Hence is derived the common method of finding the 

 contents of rectangles ; let a and h be the sides of a rectangle, then 

 1 '. a',\h ', ahj and the rectangle is equal to that of which the sides are 

 1 and ah, or to ah square units. The rectangle contained by two lines 

 is therefore equivalent to the product of their numeral representa- 

 tives. 



124. Theorem. Equiangular parallelograms are to 

 each other in the ratio compounded of the ratios of their 

 sides. 



Or in the ratio of the rectangles or numeral D 



products of their sides. For since AB : BCzz / / / 



AD : DC (120), and DC : CEziDB : BE,mul- A B/ / C 



tiplying the former equation by the members of E 



the latter, AB.DB : BC.BE=AD.CE. 



125. Theorem. Similar triangles, and figures com- 

 posed of similar triangles, are in the ratio of the squares 

 of their homologous sides. 



Since similar triangles are the halves of q^ 



equiangular parallelograms, which are in the \ 7\ 5 



ratio compounded of the ratios of their sides \/^ \ \/\ 



(124), the triangles are in the same ratio, or A B D E 



