348 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xn 



Or ( 78, p. 171) we may take Lg IG and CLl Gg as new roots. 



A second method ( 80-81, p. 172) for additive unity consists in taking 

 the least root to be 2a/(a 2 C), where a is arbitrary, and finding the greatest 

 root. Thus (end of 82, p. 174), for (7 = 8, take a = 3; the least root is 

 6 and the greatest is the square root 17 of 8- 6 2 + 1. 



Cyclic method (83-86, pp. 175-6). Taking the least root, greatest 

 root and additive as dividend, additive and divisor, find the multiplier 

 by use of the pulverizer (see papers 2, 4 of Ch. II). If the excess of the 

 square of that multiplier over the given coefficient C be divided by the 

 original additive, we get a new additive. The quotient corresponding to 

 the multiplier and found from it will be the new least root, from which a 

 greatest root may be deduced. The operation may be repeated. We find 

 integral roots with 4, 2 or 1 for additive, and by composition deduce roots 

 for additive unity from those for additives 4 and 2. 



For example ( 87, pp. 176-8), to make 67x 2 + 1 a square, take 1 as a 

 least root, 3 as additive, whence 8 is the greatest root. Thus divi- 

 dend = 1, divisor = 3, additive = 8. By the pulverizer, a multiplier 

 is 7 and the quotient is 5, a new least root. The new additive is 

 6 = (7 2 - 67)/(- 3). By 67(- 5) 2 + 6 = 41 2 , 41 is the new greatest root. 

 Now start with dividend 5, divisor 6, additive 41, get the multiplier 5, 

 quotient 11 = least root, new additive 7 = (5 2 67) /6, and greatest 

 root 90. Next, start with dividend 11, divisor 7, additive 90. Reducing 

 the last by multiples of the divisor, we get the abraded additive 6. The 

 multiplier is 2. Adding the negative of the divisor, we get the new multi- 

 plier 9 and the quotient 27, giving a least root. The new additive is 

 (9 2 67)/( 7) = 2, and greatest root is 221. By composition of this 

 set of roots with itself, we get L = 11934, G = 97684, A = 4. Divide 

 the roots by the square root of 4. We get I - 5967, g = 48842 for addi- 

 tive 1. 



When unity is subtractive ( 88-89, p. 179), the problem is impossible 

 if the coefficient C be not a sum of two squares. In the contrary case, we 

 may take as two least roots the reciprocals of the roots of the two component 

 squares. Thus ( 90) if C = 13 = 2 2 + 3 2 , the least root gives the 

 greatest root f . Doubling and applying the cyclic method, we have divi- 

 dend 1, divisor 2, additive 3. We deduce the multiplier 3 and quotient 

 3, the least root. The new additive is 4 and greatest root is 11. Repeat- 

 ing the operation, we get L = 5, G = 18, A = - 1. 



When C is a square a 2 ( 95, p. 182) and the additive is A, least and 

 greatest roots are (for b arbitrary) 



!(!+') 



Bhdscara solved various problems by the method of the affected square. 

 For 6?/ 2 + 2y -- -- c 2 ( 177, p. 247), (Qy + I) 2 = 6c 2 -f- 1 for c = 2 or 20, 

 y = f or 8. To find ( 178, p. 248) two numbers the square of whose 

 sum added to the cube of their sum equals twice the sum of their cubes, 



