788 



MR EDWARD SANG ON THE THEORY OF COMMENSURABLE S. 



This well-known theorem is easily deduced from the tetragonal system of 

 areas, and is usually given in this form,— "the square of the subtense of 120'' 

 is equivalent to the squares of the containing sides, together with their rectangle ;" 

 but it is more consistent to put it in the above form ; and we must observe, 

 that c^ (the second power of the number a) does not represent the square of the 

 side BC, but the equilateral trigon on BC. 



44. The equilateral trigon on the subtense of 60° is less than the sum of those 

 on the containing sides by the area of the trigon. 



The proof of this assertion is as easily obtained as that of the preceding. If 

 «, J, c be the three sides of such a trigon, c being the subtense of 60° we have, — 



45. Every number which is of the form c^-\-ah-\-V may, unless a be equal to 

 h, be put in two ways under the form A^ — AB + B^. 



This arithmetical proposition may also be stated thus, — " every number 

 which is the quotient of the difference of two cubes by the difference of their 

 roots, is also in two ways, the quotient of the sum of two cubes by the sum of 

 their roots." 



The preceding diagram at once illustrates the truth of this assertion, for AB, 

 which is the subtense of 120°, with the containing sides AE, CB, is the subtense 

 of 60°, with AE, EB, and also with AF, FB, for containing sides. Or, alge- 

 braically, 



a^ + a& + y^{a + hf - (a + h)h + Z>2 

 = (a + Vf — (a + &)a + a^ • 



otherwise 



■6^_(a + 6)^ + 53_(a + &)^ + a^ 



a — h (a + b) + b (a + b) + a 



Hence the solution of the equation 



a^ + ab + b^=c^ 

 in integer numbers includes that of two others of the form 



a- —ab-^b^ — c'^ 



46. If any two of the three sides a, b, c, of a trigon of 120° have a common 

 divisor, the third has the same divisor. 



This is clear. Hence we need only consider those cases in which all the three 

 are prime to each other. 



47. If of the trigon a, b, c, having an angle of 120°, the subtense c have a 

 common divisor with the sum of the two sides, the sides themselves have that 

 divisor. 



