MR EDWARD SANG ON THE THEORY OF COMMENSURABLES. 745 



mine, by the method of continued fractions, or by any other process, two 

 numbers, a and /3, which may represent with sufficient nearness the ratio 

 FE : EA ; and then compute a, b, c, the sides of a rational trigon having one 

 angle 120° by the formulae, — 



then the angle of this trigon opposite to a is double of not the angle DAC, but an 

 angle sufficiently near to DAC, wherefore that angle is nearly equal to ^AC. 



This is the general process, analogous to that given for the tetragonal system ; 

 but in special cases we may use peculiar methods. 



Example 1. 



67. To construct a rational trigon having one angle 12(f , and of which the 

 containing sides may differ by unit. That is, in other words, to construct a tri- 

 gonal muarif angle approximating to 80°. 



If c, h, «, be the sides of such a trigon, and if b = a+\, we have c'^ = 3«- + 3« 

 + 1, that is to say, c- = 3 {a + ^y + \; or multiplying by 4. 



(2c)2 = 3 (2a + 1)2 + 1, 



and therefore 2c and 2a + 1 must be the numerator and denominator of a fraction 

 converging to a/3. Now, we have for a/3, — 



7i K7>. &C. 



1; 1, 2; 1, 2; 



1, 



2; 1, 2; 



112 5 7 



19 



26 71 97 



10 113 4 



15 



15 41 56 



of which the alternate terms, — 







_4 _4 _4 _4 



-4 



-4 -4 



1 2 7 26 97 



362 



1351 5042 



0' 1' 4' 15' 56' 



209' 



780' 2911' 



&c. 



satisfy the equation p^ — ^(f -I- 1 . These may be formed by means of the multi- 

 plier — 4. Of these again, the alternate terms have their numerators even, viz. : — 



_14 _14 -14 -14 



^ 26 362 5042 70226 

 -1' 15' 209' 2911' 20545 



, &c. 



and of these, if we put the numerator equal to 2c, and the denominator to 2^>— 1, 

 or to 2« + 1, we find the cases, — 



c=l, 13, 181, 2521, &c. 

 6 = 1, 8, 105, 1456, &c. 

 a = 0, 7, 104, 1455, &c. 



and it may be remarked, that this progression of fractions may be continued by 

 help of the multiplier —14. 



VOL. XXIII. PART III. 9 



