MR EDWARD SANG ON THE THEORY OF COMMENSURABLES. 



755 



95. If a series of equal straight lines be drawn, making equal angles with 

 each other, the angle being the supplement of the double of a muarif angle of 

 any system, the distances of the extremities of the lines from each other are 

 commensurable with the lines. 



Let the equal lines AB, BC, CD, DE, EF, &c., make equal angles ABC, BCD, 

 BDE, DEF, &c., such that the 

 supplement of ABC is double 

 of a muarif angle of any 

 system ; then all the diagonals 

 AC, AD, AE, AF ; BD, BE, BF, 

 &c., are commensurable with AB. 



For the angle BAC being half the supplement of ABC is muarif; so is CDA 

 its double, DEA its triple, and so on ; wherefore, all the angles of the figure being 

 muarif, all the sides of the trigons are commensurable. 



If the angles belong to the tetragonal system, the sides are also commensur- 

 able with the radii of the circles described, one through the points A, B, C, D, E, 

 F, and the other to touch the lines AB, BC, CD, &c. ; and also the co-ordinates 

 of the points referred to any system of rectangular axes making muarif angles 

 with any of the lines may be obtained rational. 



96. To construct a trigon such that its three sides and the line joining the 

 middle of one side with the opposite corner may be all commensurable. 



Having bisected the side AB of the 

 trigon ABC in F, and joined CF, we 

 have AC'-hCB^ = 2AF^^-2FC^ so that 

 the problem becomes this : to find two 

 square numbers whose sum is double 

 of the sum of other two square numbers. 

 Put BC = «, CA = 5, AF = y^, FC = /, 

 then 



so that the sum of the squares of a and h is also the sum of the squares Qik^-l 

 and Ti—l; now we have seen that every number which can be divided into two 

 squares in more than one way is the product of two or more prime numbers of 

 the form 4w + l. Hence we have only to take any two or more prime numbers 

 a, ,5, 7, 5, of the form \.n + 1, and decompose their product into two squares in 

 two different ways, so as to obtain 



and then we have k = \[^V^-q), l=^[p—q), or doubling all in order to avoid 

 fractions, 



