754 



MR EDWARD SANG ON THE THEORY OF COMMENSURABLES. 



angle 6 muarif of the same system, and less than the supplement of S(p, and 

 then the lines AC, CB, BA, CH, CI, are all commensurable. 



92. To construct a trigon such that the three sides and the four lines tri- 

 secting two of the angles may be all commensurable. 



Assume two angles 6 and (p, the sum 

 of which must be less than 60°, and make 

 CAB equal to 3d, ACB equal to Sep, then 

 all the lines AB, BC, CA, AD, AE, CH, 

 CI, and all the segments of those lines, 

 are commensurable. 



93. To construct a trigon of which 

 A ^c the three sides and the six lines trisect- 



ing the three angles may be all commensurable. 



The third parts of the three angles A, B, C, make together 60°, wherefore if 



these third parts be muarif of any system, 

 60° must belong to that system. Such 

 a figure, then, can onl}'^ belong to the 

 trigonal system of which the modulus 

 is VS. 



Assume then from the trigonal sys- 

 tem two angles, 6 and cp, of which the 

 A o F ^c sum is less than 60°, then the defect of 



their sum from 60° is also muarif of that system. Make BAC equal to 3^, ACB 

 equal to 30, and trisect the angles ; these angles are all muarif, and consequently 

 all the lines and all their segments are commensurable, while all the areas are 

 commensurable with equilateral trigon s constructed on any of the lines or parts. 

 94. It is impossible to construct a trigon such that its sides and also the lines 

 dividing its angles into more than three equal parts may be all commensurable. 



If it were proposed to construct a trigon such that its sides and the lines 

 dividing each of two of its angles into n equal parts may be all commensurable, 

 we should only have to assume 6 and (p muarif angles of any system, but such 

 that n{6 + (p) may be less than 180° ; and then to make the angles at A and C 

 equal to nd and to n<p respectively. But the n'^' part of the remaining angle would 

 not be muarif, unless, in the system \/r, n were 2, or in the system \/3, n were 3 ; 

 for no submultiple of 180°, excepting 90° and 60°, can be muarif of any system. 



Here it is worthy of remark that the tetragonal and trigonal systems are 

 founded on the divisions of the whole revolution into 4 and into 6 equal parts ; 

 while 4 and 6 are the only two divisors which separate the prime numbers 

 greater than themselves into two groups ; those groups being, for 4, of the two 

 forms 4^-1-1 and 4w— 1 ; while for 6 they are classed as of the forms 6w-l- 1 and 



