752 



ME EDWARD SANG ON THE THEORY OF COMMENSURABLE S. 



— x^pq and BD.DC = e/^^^^, so that AD^ = (i»^— ?/^) pq\ thus it seems that the 

 product (cc+p) {pc—y)pq must be a square ; now j9 and q are prime to each other, 

 Avherefore they must be found as factors in the product {x+p) (x—t/), the 

 remaining factors being in couples. Hence the following cases are possible, — 



Isl 

 2d. 

 3d. 

 4th. a! + y=a-qc 



oc + y = a^cpq; 

 x-^y — o^c ; 

 x+y^^a^pc ; 



x—y—h^G 

 x—y = 'b'^qc 

 x — y = h'^qc 

 x—y — lr'pc 



in all of which the eifect of 

 the c is only to augment the 

 numbers. 



Hence the solutions, — 



1. 2x=a^pq + h'^ ; 

 2x = a'^ + h'^pq ; 



2x-- 

 2x-- 



-a'^p + b^q ; 

 -a^q + b^p; 



2y — a^pq-h-\ 

 2y = a^-b'^pq; 

 2y = ar'p—b-q; 

 2y=a~q — b'^p : 



which give 



-QG = {a^-b''pq){p + q); 

 BG = {a^p-b^q)(p + q); 

 BC = (a^q-b-'p)(j> + g); 



CA = (<x^j35'+ b^)q ; 

 GA=(a^ + b'^pq)q; 



CA = {a^p + g'^q)q ; 

 CA = {d^q + Pp)q ; 



AB = (a^pq-\-b^)p ; 

 AB = {a'^ + b'^pq)p ; 

 AB = {a^p + b''q)p; 

 AB — {a^q + b^p)q : 



in which p, q, a, J, may be assumed at will, subject only to the restriction, that the 

 values of BC be not negative. These four solutions are complementary in pairs. 



A much more elegant solution of the problem is obtained from the properties 

 of muarif angles. Since the trigons BAD, DAC have their sides commensurable, 

 and their angles BAD, DAC alike, they must belong to the same muarif system. 

 Wherefore, in anyone system assume BAD = DAC = and ADB = (^, then we 

 have sin ABD = sin ((^ + ^), sin ACD = sin {(p—^) so that all the ratios are 

 rational, since all the sines involve the same irreducible surd. 



89. To construct a trigon, such that the three sides and the lines bisecting 

 two of the angles may be all commensurable. 



To construct a trigon ABC, such that 

 its three sides and the lines AD, CF bi- 

 secting two of its angles may be all com- 

 mensurable. 



In any system of muarif angles, as- 

 sume two, 6 and 0, of which the sum may 

 be less than a right angle ; then having 

 •^ taken any base AC, make at A, CAD, 

 DAB, each equal to 6, and at C, ACF, FCB, each equal to ^, then all the lines 

 and all the areas in the figure are commensurable. 



90. To construct a trigon of which the three sides and the three lines bisecting 

 the three angles may be all commensurable. 



