218 



HISTORY OF THE THEORY OF NUMBERS. 



[CHAP, v 



have the common factor p. Hence 



Fj. = fy 2 , F 2 = gz 2 , yz = /3 l} fg = n 2 - m\ 

 Fi -F 2r 2 a! fy gz 



- ZT/l --- . 



ft & z y 



Divide the latter equation by n and set = fy/(nz}. Thus 



_ 



7 , . 



The rationality of is thus a necessary condition for the rationality of the 

 ratios of the sides of triangle AEB. It is a sufficient condition, since 



a 



c 2 + 1 



by (1). There are similar formulas for the remaining three triangles whose 

 angles at E are w and TT w. Taking /3 as the unit of length, we have 



+ 



- 1 



(4) 



ji 



a 



(z - c) 2 - 1 



7 = 



(r? - c) 2 - 1 



277 



<5 = (j/ + c) 2 - 1 

 7 



2z 7 2?/ 



where , 77, x, y are rational. By multiplication, we obtain two values for 5. 

 Hence we have the condition 



+ c) 2 - 1 (a; - c) 2 - 1 = (77 - c) 2 - 1 (y + c) 2 - 1 



Hence for any set of rational solutions of (5), such that c | < 1, we obtain 

 a quadrilateral with rational diagonals and rational sides 



(6) 







where t = 1 c 2 , while a, 7 are given by (3). 



Let also the area |(or/8 + fiy + 75 + Sa) sin iy of the quadrilateral be 

 rational, and hence also sin w. The rational solutions of sin 2 w -\- c 2 = 1 are 



2X X 2 - 1 



Hence to obtain all rational quadrilaterals we have only to seek the rational 

 solutions c, , tj, x, y of (5) for which c is of the form (X 2 1)/(X 2 + 1). 

 Now (5) is a quadratic equation in y whose discriminant must be a square: 



(7) {ax 2 - - 2c(a + y)x - at} 2 + 4ty 2 X 2 = Z 2 . 



Hence we may obtain all rational quadrilaterals as follows: Give arbitrary 

 rational values to , 77, X and set 



X 2 - 1 



r 



- 



X 2 +l' 



/ = 



