168 



G. B. AIRY, ESQ., ON THE SUBSTITUTION OF METHODS 



Construct the similar and equal rectangles DE, FG, with sides 6 and B ; and apply them 

 with their angles meeting at H, in such a manner that the side DH or B of one shall be in 

 the same straight line with HG or B of the other ; then will the side FH or b of one be in 

 the same straight line with HE or b of the other. In the right-angled triangles IDH, HFK, 

 the sides including the right angles are equal, therefore the angle FHKis equal to the angle 

 DIH, and is the complement of the angle DHI; therefore IH and HK are in the same 

 straight line. 



To DH apply the rectangle DM whose side DL or HM is equal to a ; the sides DL and 

 HM will be in the same straight lines with DI and HE. To HE apply the rectangle HO, 

 whose side HN or EO is equal to A ; the sides HN and EO will be in the same straight 

 lines with HD and EL Produce LM, ON, to intersect in P, and join KP. 



Then, because the rectangle LH, which is the rectangle contained under a and B, is equal 

 to HO, which is the rectangle contained under b and A ; by Proposition (A), LH and HO 

 are the complements of the parallelograms about the diameter of the rectangle LO ; therefore 

 /ZT and consequently IHK (which are in the same straight line) are in the diameter; therefore 

 LHKP is a straight line. 



In like manner, to HG apply the rectangle HQ whose side GQ or HR is equal to c; 

 and to HF a.^^\y the rectangle HS whose side FS or HT is equal to C; and produce ST 

 and QR to meet in V; and join LV. Then, proceeding from the hypothesis that the rect- 

 angle contained under c and B is equal to the rectangle contained under 6 and C, it will be 

 shewn in like manner that KHIV is a straight line. 



Therefore PKHIV is one straight line. 



Complete the rectangle WX. Then WH, HX, are complements of the parallelograms 

 about the diameter of WX, and are therefore equal. But WH is the rectangle contained 

 under a, G, and HX is the rectangle contained under c, A ; therefore the rectangle contained 

 under the lines a, C, is equal to the rectangle contained under the lines A, c. a.E.D. 



Corollary. By repeating the operation, the theorem may be extended to four or any 

 number of terms of comparison of rectangles, following in a similar order. 



Proposition (C). If two right-angled triangles are equiangular, and if a, A, are their 

 hypothenuses, and b, B, homonymous sides ; the rectangle contained under the lines a, B, is 

 equal to the rectangle contained under the lines A, b. - 



