744 



MR EDWARD SANG ON THE THEORY OF COMMENSURABLES. 



In the appendix there is given a list of all primes of the form 6n+l, under 

 1000, with their decompositions, and the resulting trigons, as also the values of 

 the lesser angles. 



Thus there are muarif angles belonging to the trigonal system of measure- 

 ment, analogous to those which we have already seen to belong to the tetragonal 

 system : to distinguish them from those formerly treated of we shall call them 

 trigonal muarif angles, while the others might have been styled tetragonal 

 muarif angles. 



61. If two angles be both muarif of the same system, their sum and their 

 difierence are so ; a right angle being regarded as muarif in the tetragonal and an 

 angle of 60° as muarif in the trigonal system. 



This has already been shown to be true for the tetragonal system ; the proof 

 for angles of the trigonal system is of the same nature. 



62. If a trigon be constructed with two of its angles muarif of the trigonal 

 system, its sides are commensurable with each other, and its area with the equi- 

 lateral trigons constructed on the sides. 



The proof of this is analogous to that already given for the other system. 



63. If at the two extremities of a line taken as a base, an}'^ number of trigonal 

 muarif angles be made, their sides, continued indefinitely, cut each other into 

 segments commensurable with the base, and intercept areas which are commen- 

 surable with the equilateral trigon described on the base. 



This is merely an extension of the preceding proposition. 

 64 If at any of the points of intersection of the preceding figure, other tri- 

 gonal muarif angles be made ; the distances intercepted by these new lines are 

 commensurable with the base and the intercepted surfaces with the equilateral 

 trigon on the base. 



65. If at the point of contact of a straight line with a circle, trigonal muarif 

 angles be made ; if the extremities of the chords so formed be joined, and if tan- 

 gents be applied at the extremities of those chords, all the intercepted distances 

 are commensurable with the side of the inscribed equilateral trigon, and all the 

 areas with the area of that trigon. 



QQ. To construct a trigonal muarif angle 



which may approximate to any given angle. 



Let it be proposed to compute, in integers, 



-^ the sides of a trigon which shall have one 



of its angles equal to a given angle BAG, and 



another either 120° or 60°. 



If the given angle BAG exceed 60°, it 

 is impossible to have either of the remaining angles 120° ; we shall therefore 

 deduct 60^ or if need be, 120° from BAG, thus leaving ^>AG less than 60°. 

 Having bisected 6AG by the line AD, draw EF, making x\EF = 120° ; and deter- 



b^ 



