CHAPTER XVII. 



SYSTEMS OF TWO EQUATIONS OF DEGREE TWO. 



TWO QUADRATIC EQUATIONS IN TWO UNKNOWNS. 



Beha-Eddin 1 (1547-1622) included (as Prob. 3) among the 7 problems 

 remaining unsolved from former times that to make x 2 -\-y = W, y 2 -\-x = 5. 

 Nesselmann noted that there is no rational solution. Marre (p. 323) 

 noted that it leads to a: 4 20a; 2 ++95 = 0, having no rational solution. 



Cataldi 2 required x, y when x 2 + y 2 and xy/ (x y) 2 have given values, 

 and treated separately the case in which the values are 20 and 1. 



Fermat 3 treated the problem to find in how many ways a given number 

 m is the difference of two numbers whose product is a square. If m = 2 k p a q b , 

 where p and q are odd primes, the number of ways is 2ab-\-a+b. If there 

 is a third odd prime r with the exponent c, the number of ways is 

 4a6c+2a6+2ac+26c+a+6+c; etc. 



If 4 x 2 +y = y 2 +x, x 2 +y 2 =n, then x = y or x = l y. For the latter, 

 x 2 +y* = (ry-l) 2 gives y. 



A. Martin 5 found the rational solutions of x+y = x 2 +y 2 = D by setting 

 x = az, y = bz, 2 = (a+6)/(a 2 +6 2 ), where a 2 +& 2 =D, the last being satisfied 

 in the usual way. M. Brierley 6 took y = rx and found oj=(r+l)/(r 2 +l). 

 Then take r = 3/4, whence r 2 + 1 = D . 



J. Hammond, 7 to divide a product N of two unknown primes x, y into 

 two parts P, Q, each > 1, such that PQ= 1 (mod JV), tabulated for each m 

 (Km^l5) all solutions n, m, P, Q, N, x, y of P+Q = N, PQ+l=mN, 

 whence P = m-\-n, Q = m+n it N=2m+n+ni, rwi = wi 2 1. 



PROBLEMS OF HERON AND PLANUDE; GENERALIZATIONS. 



Heron of Alexandria 8 (first century B.C.) treated the two problems: 

 (I) Find two rectangles such that the area of the first is three times the 

 area of the second [and the perimeter of the second is three times the 

 perimeter of the first]. It is stated that the sides of the first are 3 3 -2 = 54 

 and 54-1 = 53; those of the second, 3 (53 +54) -3 = 318 and 3; the areas 

 are 2862 and 954 [semi-perimeters 107, 321]. 



1 Essenz der Rechenkunst von Mohammed Beha-eddin ben Alhossain aus Amul, ar ( abisch u. 



deutsch von G. H. F. Nesselmann, Berlin, 1843, p. 55. French transl. by A. Marre, 

 Nouv. Ann. Math., 5, 1846, 313. Cf. Genocchi, Annali di Sc. Mat. e Fis., 6, 1855, 297. 



2 Nuova Algebra Proportionale, Bologna, 1619, 51 pp. (chiefly on cubes and cube roots, 



pp. 42-43). 



3 Oeuvres, II, 216; letter from Fermat to Mersenne, Dec. 25, 1640. 



* The Gentleman's Math. Companion, London, 4, No. 20, 1817, 643-4. 

 6 Math. Quest. Educ. Times, 62, 1895, 70. 



6 Ibid., 67, 1897, 72. 



7 Math. Quest, and Solutions, 1, 1916, 18-19. 



8 Liber Geeponicus (ed., F. Hultsch), 218-9. H. Schone, Heronis Opera, III, Leipzig, 1903. 



485 



