748 



MR EDWARD SANG ON THE THEORY OF COMMENSURABLES. 



to the supplement of d, have been determined in integers, the angles BAG, CBA 

 may be said to be muarif of the system 6, and for the present we may call 6 the 

 determining angle of the system. 



71. Every rational sided obtuse-angled trigon is accompanied by two acute- 

 angled trigons having their sides also rational and their areas commensurable. 

 For let ABC be an obtuse angle, and let the three lines BC, CA, AB, be ex- 

 pressed by three integer numbers, a, b, c. 

 From A to the extension of BC, and 

 from B to the extension of AC, inflect 

 AE, BF, each equal to AB : then since 

 BA^-AC^ = BC.CE, CE is represented 



by the quotient 



a 



y.2 



while CF is re- 

 both of which are 



presented by — r 



rational. If EG be cut off equal to BC, 

 and FE equal to AC, it is evident that the 

 angles CAG, HBC, are each double of the complement of FCB or 6 ; wherefore the 

 sides of ACE are a + 2b cos 6, b, c, while those of BCF are a, b + 2a cos 6, C. 



The areas ACB, ACE, BCF, are proportional to the rectangles AC, CB, AC, CE, 

 BC, CF, and must be commensurable since the containing sides are so. 



72. If each of two angles be muarif of the same system 6, their sum and 

 difference are also muarif of that system. 



Let (figure, page 747) CAB, 7A^, be two angles muarif of the system 0= 180— 

 ACB = 180°— A7/3, and let a, b, c; a,^,y, be the integer numbers which represent 

 the sides of the two trigons ; make AD equal to C7 units, and complete the con- 

 struction as in article 48, only observing to draw EH making EHG equal to 6, 

 and D'H' making D'H'I' equal to 0, then we easily obtain the values DE = D'E'= 

 Ca, AE = C/3, EF = ff/3, AF = 5/3, EH = «a, J)B = ba, D'H' = Ja as before, while HI 

 = 2aa cos 6, WY = 2ba cos 0, whence 



AD = c7 



AG=Z)/5— aa 

 DG=cr/3-f-&a + 2aa cos 6 



AD'=c7 



AG' =h(3 + aa + 2ba cos 6 

 'D'G' = a(3-ba, 



which values are all rational when cos 6 is so, but are only integers when cos 6 

 = 0, or when 2 cos 6 is integer. 



73. If two angles of a trigon be both muarif of the system 6, its sides are 

 commensurable, and its area is commensurable with that of a rhombus on the 

 linear unit, having an angle equal to 6. 



The truth of this theorem is obvious. 



74. If at the extremities of any line assumed as a base muarif angles of the 

 system 6 be made, and the sides of these be indefinitely produced, all the inter- 



