MR EDWARD SANG ON THE THEORY OF COMMENSURABLES. 749 



cepted distances are commensurable with the base, and all the areas with the 

 rhombus constructed on the base with the angle 0. 



75. If at any of the intersections of the preceding article muarif angles of the 

 same system be made, the segment and areas so intercepted are all commensurable. 



76. If at the point of contact of a straight line with a circle any number of 

 muarif angles of the system 6 be made, all the lines joining the extremities of the 

 chords so formed, and all the tangents there applied continued indefinitely, 

 intercept segments and areas which are commensurable with each other. 



77. If one muarif angle of the system 6 be equal to an angle of the system 0, 

 the two systems are identic. 



Hence it follows that the determination of a system by the angle is not quite 

 appropriate ; that is to say, the angle 6, no more than the angle (p, can be re- 

 garded as the modulus or mastar of the system. 



78. The numerical expressions for the sines of all muarif angles of the same 

 system involve the same irreducible surd. Or, 



The numerical expressions for the areas of all muarif figures of the same 

 system, when expressed in squares of the linear unit, involve the same irreducible 

 surd. Let — and ^p be the expressions for the cosines of two angles 6 and ©, then 

 have we 



sin 6 = ^-^ '-, sin © = ^-^ , 



V V 



and 



cos{e + &)=:^~^ ^-j^ ^; 



now if the two angles and © be both muarif of one system, their sum + @ 

 must also be so ; that is to say, the cosine of ^+ © must be rational, and for this 

 it is necessary that the product of 



V{tP-s^) by \/(T2-S2; 



be also rational, and this can only be when f—s'-' and T^— S'-^ involve the same 

 unsquare factors. 



Since the area of any trigon expressed in squares of the linear unit is half the 

 product of the two containing sides multiplied by the sine of the included angle, 

 the same irreducible surd must enter into the expression for the area. 



79. All muarif angles of which the sines involve the same irreducible surd, 

 belong to the same system. 



Let Q and <p be two muarif angles, of which the sines are respectively 

 sin Q=-.\/m and sin (p—^s/m, m being a number which has no square divisor; 

 then, if the angles be muarif of any systems, their cohines must be rational. 

 Now 



' cos (0 + <^) = cos 6 . COS — sin Q . sin ^ 



= cos Q . cos (b — -^m 



VOL. XXIII. PART III. 9 P 



