MR EDWAED SANG ON THE THEORY OF COMMENSURABLES. 



731 



If the angles at A and at B be both muarif, the sides of the trigons AFC, BFC 

 are commensurable ; wherefore the area of the 

 trigon is commensurable with the squares of 

 any of the lines AF, FC, FB, BC, CA, also the 

 trigons ADB, AEB, are similar to CFB and AFC 

 respectively ; wherefore their sides also are com- 

 mensurable. 



Or in general, all the angles shown in the 

 figure are muarif, and therefore all the trigons 

 have commensurable sides and areas. 



If a, h, G, be the three sides of any trigon, and S the surface, the radius of the 

 2S 



a + ft + c' 

 2S 



those of the three circles of external contact are 

 -, and that of the circumscribed circle is -r-^ and those, 



inscribed circle is 



2S 2S 



a + & — c' a — h + c' —a + h + c 



obviously, are all rational if S, a, h, and c be so. 



25. If any straight line be assumed as a base, and if, at each of its extremities 

 any number of muarif angles be made, the sides of these being indefinitely 

 extended, all the distances intercepted on them are commensurable with the base, 

 and all the included areas are commensurable with the square of the base. 



It is evident that all the angles obtained by this construction are muarif, and 

 that, therefore, the areas of all the trigons formed on the base are commensurable 

 with its square, while their sides are all commensurable with the base itself. Now, 

 all the segments are differences or sums of the sides of these trigons, and all the 

 areas are differences or sums of their areas ; wherefore, all of these are commen- 

 sm-able with the square of the base, and all of those with the base itself 



26. If, at any of the points of intersection of the preceding article, muarif 

 angles be made, the segments and areas so obtained are all commensurable with 

 the base, and with its square. 



The truth of this assertion follows at 

 once from that of the preceding. 



27. If, at the point of contact of a straight 

 line with a circle muarif angles be made, if 

 the extremities of the chords so obtained be 

 joined and continued indefinitely, and if 

 tangents be applied at their extremities, 

 all the intercepted straight lines are com- 

 mensurable with the diameter, and all the 

 areas with the square of the diameter. 



Since the angles which a chord makes 

 with a tangent at its extremity are equal to those in the alternate segments of the 



