734 MR EDWARD SANG ON THE THEORY OF COMMENSURABLE S. 



The half of this angle, viz., IT 43' 43" has for its tangent -207 6118, to which 

 we have the successive approximations, — 



4 14 2 5 18 



I I I I T, ¥3 li. &e., which again give 



the trigons 17, 15, 8 ; 26, 24, 10; or 13, 12, 5; 601, 551, 240; 2930, 2688, 1166; 

 or 1465, 1344, 583 ; &c. 



33. To find a trigon having its sides rational and its area commensurable with 

 the squares of those sides, and which shall have, approximately, the shape of a 

 given trigon. 



The general solution of this problem follows at once from the preceding. We 

 must find two muarif angles approximating to two angles of the trigon, and with 

 these construct a rational trigon. 



But some of the special cases admit of peculiar solutions. 



Example 1. 



34. To construct a trigon of which the three sides may be represented by 

 three contiguous numbers while the area is a multiple of the square unit. 



Let a—1, a, a+l be the three sides, S being the area, then — 



16S2 = 3a(a-2)a(a + 2) = 3a2(a2-4) 



wherefore S{a^—4) must be a square number, or «^— 4 must be the triple of a 

 square. Let, then, «-— 4 = 3^:'^ This equation cannot be satisfied when x is odd, 

 for then a also would be odd, and the first member of the equation would be of 

 the form 4?z+ 1, while the second would be of the form 4?? + 3 ; hence both a and 

 o) must be even, or we may put — 



a=2a,a> =-2z, whence 



a2-l = 3^2; 02 = 3^2 + 1. 



Now, when we develope Vd in the form of a chain fraction we obtain the 

 approximations, — 



1; 1, 2; 1, 2; 1, 2; 1, 2; 

 01 1 2 57 19267197. 

 10 11 8 4 11 15 41 56' 



which belong, alternately, to the equations a- — 3z^—2 and a^ = 3z^+l; the values 

 of a thus obtained are 1, 2, 7, 26, 97, &c., the next term being four times the 

 last term less the penult. Hence the trigons are — 



3, 



13, 



51, 



193, 



723, 



2701, 



10083, ^ 





4, 



14, 



52, 



194, 



724, 



2702, 



10084, 



>&c 



5, 



15, 



53, 



195, 



725, 



2703, 



10085, J 





