MR EDWARD SANG ON THE THEORY OF COMMENSURABLES. 



751 



which can be brought into the preceding forms by multiplying each member of 

 the fractions by ^ or by v. 



83. The tetragonal system has for its modulus the symbol Vi. 



84. The modulus of the trigonal system is a/3. 



85. If a trigon be constructed, having one angle muarif of one system, and 

 another muarif of another system, its sides are incommensurable. 



For if (f) and 6 be two muarif angles, the one involving the surd VJi and the 

 other the surd \/v , the sine of (p + would involve the product VJIu, but then the 

 cosine would involve the two surds Vf^, Vi^ separately, wherefore the side inter- 

 mediate between (p and would be incommensurable with either of the other two. 



86. Having given the three integers which represent the sides of a rational 

 sided trigon, to find the modulus of the muarif system to which its angles belong. 



Let a, b, c, be the three sides, then, the area of the trigon being s, we have, 



4 S = V{(« + & + c) ( — a + b + c) (a—b + c) (a+b — c)} 



wherefore, if we decompose each of the four numbers a + b + c, —a + b + c, a — b + c, 

 a + b — c, into its prime factors, and reject all those factors which occur twice, 

 the square root of the product of the remaining factors is the required modulus. 

 Thus we have the following cases : — 



a 



b 



c 



Modulus. 



a 



b 



c 



Modulus. 



2 



3 



4 



J15 



9 



10 



11 



x/2 



3 



4 



6 



Jl 











4 



5 



6 



V7 



3 



5 



7 



V3 



5 



6 



7 



V6 



5 



7 



9 



x/ll 



6 



7 



8 



J15 



7 



9 



11 



J195 



7 



8 



9 



V5 



9 



11 



13 



V35, &c. 



87. If m be prime, for every pair of values of cc and z/, we have two angles, 

 according as we combine m with ^^ or with t/'^ ; bat if w be composite, then for 

 every way in which m can^be represented as a product fxu, we have two angles. 



Section 6. — Miscellaneous Propositions. 



88. To construct a trigon, such that the three sides and the line bisecting one 

 of the angles may be all commensurable. 



Let the angle BAG of the trigon BAG be bisected by the line AD; it is 

 required to determine the dimensions, so that 

 all the lines be represented by integer numbers. 



From the principles of geometry we know 

 that BA : AG : : BD : DG, while the square of 

 AD is the difference between the two rectangles 

 BA.AG and BD.DG. If we denote the ratio of 

 BA to AG hj p : q, p and q being prime to each -^-^ 

 other, we may put BA = a;p, AG = a;q; BD=^j9, and 'DG = 7/q, whence BA.AG 



