460 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xvi 



start with any two numbers a, b and take k = ab(a+b}/(ab}. Then 



Or we may form the right triangle with the legs a 2 6 2 , 2ab and take as k 

 the double of their product. 



Alkarkhi 3 (beginning of the eleventh century), to make + 2 and 2 

 squares, began by solving the system y+x z =n,y z 2 = D. Set y = 2x+l, 

 so that ?/+a; 2 =n. Then y x 2 = 2x+l x 2 will be the square of 1 z if 

 x = 2. Then z 2 = 4, ?/ = 5 and = 4/5 [since the initial system is satisfied if 

 2 / = x 2 [y^\. The method is stated to be useful in the solution of z 2 + mx = D , 

 x 2 nz = D . Although this problem does not belong directly to the present 

 subject, it has been inserted here in view of the use by Leonardo of the 

 same method. 



Leonardo Pisano 4 mentioned about 1220 the problem, which had been 

 proposed to him by Johann Panormitanus of Palermo, to find a square 

 which when either increased or decreased by 5 gives a square. He stated 

 that the answer is the square of 3+J+ij C = TH; fo r > its square increased 

 by 5 gives the square of 4-^, and decreased by 5 gives the square of 

 2+ J+ J [ = fiH- He said that he would treat such questions in a work to 

 be entitled " liber quadratorum." The latter, 5 dated 1225, opened with 

 a bare mention of this special problem, but later 6 took up the general 

 problem: To find a number which added to a square and subtracted from 

 the same square gives squares ; or, what is equivalent, to find three squares 

 l and a number (congruum) y such that 



Since any square is the sum of consecutive odd numbers 1,3, , beginning 

 with unity, y must equal the sum of those odd numbers which enter the 

 sum for xl and not in x\, and again those in xl and not in x 2 2 . He proposed 

 to determine y so that the number of consecutive odd numbers whose sum is 

 xl xl shall bear to the number making up xlxl& given ratio a/6. Let first 



m -<^ 



b < a-b' 



To treat together 7 the two cases separated by Leonardo, let s and t represent 

 a and b when a +6 is even, but represent 2a and 2b when a-\-b is odd. Set 



, , m = s(a b}, n = t(a 'ti) J u = np, 



p = s(a-\-b}, q = t(a-\-b), v = mq, 



3 Extrait du Fakhri, French transl. by F. Woepcke, Paris, 1853, (28), p. 85; same in (27), 



pp. 111-2. 



4 At the beginning of his Opuscoli, published by B. Boncompagni in Tre Scritti Inediti di L. 



Pisano, Rome, 1854, 2, and in Scritti di L. Pisano, Rome, 2, 1862, 227. 



5 Tre Scritti, 55, seq. Scritti, II, 253-283. B. Boncompagni, Comptes Rendus Paris, 40, 



1855, 779, and R. B. McClenon, Amer. Math. Monthly, 26, 1919, 1-8, gave a summary 

 of the topics treated in the liber quadratorum. Cf. O. Terquem, Annali di Sc. Mat. 

 Fis., 7, 1856, 140-7; Nouv. Ann. Math., 15, 1856, Bull. Bibl. Hist,, 63-71. Xylander 

 wrongly said that Leonardo borrowed from Diophantus (cf. Libri, 24 II, 41). 



8 Invenire numerum, Tre Scritti, p. 83; Scritti, II, 265. 



7 A. Genocchi, Note analitiche sopra Tre Scritti . . . , Annali di Scienze Mat. e Fis., 6, 

 1855, 275-8. Cf. Leonardo" 5 of Ch. XIII. 



