730 MR EDWARD SANG ON THE THEORY OF COMMENSURABLES. 



In table I. are given the roots of the component squares, the prime numbers, 

 and the sides, for all prime hypotenuses under one thousand, and also the factors 

 of the hypotenuse with the corresponding sides for all composite numbers up to 

 the same limit. 



Section 2. — On Muarif Angles. 



22. If the sine and cosine of an angle be both commensurable with the radius, 

 all other functions, as the tangent, secant, cotangent, cosecant, of that angle are 

 also commensurable. 



If the sine of any angle, as 6 be expressed by the fraction -, a and c being 



prime to each other, the cosine of that angle is - Vc^^^o^, wherefore, in order 



that the cosine may be rational, c^—a^ must be a square number, say h^, that is 

 to say, a^ + 6^ = c^ and thus it appears that all such angles belong to rational 

 right-angled trigons, so that we may write 



sin = -, cos 6=^- ; whence tan ^=, , 



C C 



cot ^ = -, sec 6 — -. cosec 6 = -. 

 ah a 



Definition. — We shall see immediately that these angles indicate or make 

 known the solutions of many problems in commensurables, and therefore I pro- 

 pose to designate them by the title oj-x^i, muarif, formed from the same root as 

 the word tarif in common use. 



23. All the trigonometrical functions of the sum and of the difference of two 

 muarif angles are also rational. 



If sin 6 = , cos 6=z- while sin <^ = — , cos </>=—, we have, according to the 



fundamental propositions of trigonometry,— 



• /zi JL^ aB + Ab ,/j ,. &B — aA 



sm(6 + (p) = — ^ — ■,cos(e + (p)= — -^^— 



sin (6-(p) = — ^^— ; cos (6 + (p) = —^ — 



which are all rational, and therefore all the other trigometrical functions of the 

 same angles are rational. 



N.B. — In Table I. the values of the lesser angle of each trigon is given in 

 ancient degrees. 



24. Every trigon of which two angles are muarif has its altitudes, its areas, 

 the radius of the circumscribed circle, and those of the four circles of contact, all 

 rational. 



