CHAPTER XIII. 



FURTHER SINGLE EQUATIONS OF THE SECOND DEGREE. 



EQUATION LINEAR IN ONE UNKNOWN. 

 Brahmegupta 1 (born 598 A.D.) solved axy = bx-{-cy+d. Let e be an 



arbitrary number and set q=(ad-\-bc)le. To the greatest and least of e, q 

 add the least and greatest of b, c, and divide the sums by a. We get the 

 values of x, y (that of x on adding to c and vice versa). Thus, if 

 xy = 3x+4y+9Q, take e = 17, whence g = 6, j/ = 17+3, z = 6+4. Another 

 method is to give a special value to one of the unknowns. 



Bhdscara 2 (born 1114) gave a like rule for a I, but added e and q 

 to (or subtracted them from) b and c in either order, and gave both geom- 

 etric and algebraic proofs of the rule. Thus for xy = 4x- J r3y+2, take 

 e = l, whence <? = 14; adding 4, 3 to 1, 14 in both orders, we get 17, 5 and 4, 

 18 as sets of values of x, y; taking e = 2, we get 5, 11 and 10, 6. The same 

 example was treated in 209, p. 269, by assigning any value as 5 to y 

 and deducing x = 17. 



On axy+bx+cy+d = Q see Wezel 86 , and papers 121-141 (on optic for- 

 mula) of Ch. XXIII; also, Bervi 61 of Vol. I, p. 451; and *P. von Schaewen. 2 " 



L. Euler 3 noted that 4.mn mn is never a square since 



is impossible; also ^pmnmn is not a square if m is of the form 4n 2 g n. 



Euler 4 proved that no number of the form 4mn m n or Smn 3m 3n 

 can be a square, and many such propositions. 



Euler 5 stated without proof that 4mnzmn= D is impossible. This 

 arises from the fact that the divisors of mx z +y z are of the form 4mz+l, so 

 that d = 4mz 1 is not a divisor, whence dn^m+y z . He 6 treated similarly 

 the case m = l, and proved that 4ran m n+ D. 



P. Bedos 7 erred in his proof that 4mn m 1 4= D. 



Several 8 proved that 4mn m n is never a square or triangular number. 



S. Giinther 9 solved y-ax* = bz by use of the continued fraction 



a _o_ a_ 



1 Brahme-sphut'a-sidd'hanta, Ch. 18 (Algebra), 61-34. Algebra, with arith. and mensura- 



tion, from the Sanscrit of Brahmegupta and Bhdscara, transl. by Colebrooke, 1817, 

 pp. 361-2. 



2 Vija-gamta, 212-4; Colebrooke, 1 pp. 270-2. 



2a Zeitschrift fiir d. Realschulwesen, 38, 1913, 141-6. 



3 Corresp. Math. Phys., (ed., Fuss), 1, 1843, 191, 202 (180, 259, 260); letters to Goldbach, 

 Jan. 19, and Feb., 1743. 



4 Comm. Acad. Petrop., 14, 1744-6, 151; Comm. Arith., I, 48-49; Op. Om., (1), II, 220. 



5 Opera postuma, 1, 1862, 220 (about 1778). 



6 Corresp. Math. Phys., (ed., Fuss), 1, 1843, 114-7; letter to Goldbach, Mar. 6, 1742. 



7 Nouv. Ann. Math., 11, 1852, 278 (Euler's correct proof, p. 279). 



8 Math. Quest. Educ. Time's, 70, 1899, 73. 



9 Jour, de Math., (3), 2, 1876, 331-340. 



27 401 



