MR EDWARD SANG ON THE THEORY OF COMMENSURABLES. 



375 



The halves of the three angles of any trigon make together one right angle, 

 wherefore they can only be all muarif in 

 the system to which the right angle be- 

 longs ; and if two of them be muarif of 

 that system, the third must also be so. 

 Hence, if the two angles 6 and <p be 

 taken from the tetragonal system, and 

 the trigon ABC be constructed, having 

 BAG = 2^, ACB = 2(/), ABE, the half of 

 the third angle ABC, is also muarif, and 

 therefore the segments intercepted on 

 and by the six lines AB, BC, CA, AD, 

 BE, CF, are all commensurable. 



Moreover, if we draw QAR, RBP, 

 PCQ, bisecting the supplemental angles, 

 and then draw perpendiculars from 0, P, 

 Q, R, to the three sides, all the distances 

 intercepted on those lines are also com- 

 mensurable ; hence, of such a trigon, the three sides, the three altitudes, the 

 radius of the circumscribed circle, the radii of the four circles of contact, the 

 distances of the centres of those four circles from each other, and from the corners 

 of the trigon, as well as all the segments of the lines bisecting the angles inter- 

 nally and externally, are commensurable, and may therefore be expressed in 

 integer numbers. 



Example. 



If we take BAG = 6 = 36° 52', ACF = (/) = 22° 37', we have CBE = 30° 31' = 4, 



3 4 5 12 33 



and in 6=-^, cos ^=r, sin 4^ = to, cos </) = to'' whence sin -^l^^cos (^ + <^) = 



66' 



cos 4/ = sin (<^ + ^) 



• A 24 . ^, 

 sm A = 25, sm G = 



56 



~65 



120 



; from which we obtain the sines of the whole angles, viz., 

 3696 



169' ^^^^"4225 



The three sides must be proportional to 



these sines, so that, when the common 

 divisors are taken out, we obtain AG = 154, 

 BA = 125, CB = 169. 



91. To construct a trigon such that 

 the three sides and the two lines trisect- 

 I ing one of the angles may be all com- 

 mensurable. 



Having assumed an angle ^ muarif of 

 any system and less than 60°, make ACB equal to 3^, and then at A make any 



VOL. XXIII. PART III. 9 Q 



