732 



MR EDWARD SANG ON THE THEORY OF COMMENSURABLES. 



circle, every angle obtained in this way is muarif, and therefore the truth of the 

 proposition follows. 



28. The double of any angle of which the tangent is rational, is a muarif angle. 

 If the two sides AB, BC, of the right angle ABC be commensurable ; that is, 



if the tangent of the angle BAG be rational, BAD 

 double of BAG is muarif. 



Since AC bisects the angle BAD, we have BA : 

 AD : : BC : CD and AG' = BA . AD - BC. CD, whence 

 easily BA' - AG' : BA' + AC' : : BA : AD : : BC : CD 

 wherefore AD and CD are both rational, and conse- 

 quently BD is rational, that is to say, the angle BAD 

 is muarif. 



Otherwise if tan 6 = -, whence sin 6 = 



a 



cos 



d: 



and sin 26 = 



2ah 



a2 + 62 ' 



cos 26 = 



a" 



b' 



+ b' 



(a2 + fe2)' 

 that is both sin 26 and cos 



x/(a2 + 6') 



26 are rational. 



29. To find a muarif angle which may approximate as closely as may be 

 desired to a given angle. 



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

 theorem; we have to compute the series of fractions which approach to the 

 tangent of the half of the given angle, and from these to deduce the corresponding 

 muarif angles. 



In particular cases, however, special solutions may be obtained. 



Example 1. 



30. To construct a rational right-angled trigon of which the angle may be 

 nearly half a right angle. 



Having constructed the right-angled isoskeles tri- 

 gon ABC and bisected the angle at A by the line AD, 

 draw DE perpendicular to the hypotenuse AB, then, 

 as is easily shown, CD = DE = EB, and CD : DB : : 1 

 : a/2; or, CD : AG : : 1 : l-l-\/2, that is tan CAD= 



T 7^, the Broun ckerian approximations to which are 



i + Y'^ 



02222 2 2 2 

 10j.2_5 12 29 70 

 1 2 5 12 29 70 169 



which, by help of the formula A = 2ab, B = «'— 5', C = a'-\-b\ 



5, 4, 3, 



29. 20, 21, 



169, 119, 120, 



985, 607, 696, 



5741, 4059, 4060, &c. 



2 



169 

 408 



408 



, &c. 



985 

 give the cases, — 



