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



or ! 2 +3 2 +5H ---- or /i 2 +(2/i) 2 4-(3/i)H ---- . For example, 



24(l 2 +3 2 +5 2 )=S40. 



To find 16 a congruent number whose fifth part is a square, take a = 5 

 and determine b so that 6, a-\-b, a b are all squares, say g z , h~, fc 2 , respec- 

 tively. Then 5 = # 2 +/c 2 . Either g = l, k = 2, whereas a+6=5+l is not a 

 square, or g = 2, k = l, whence 4a6(a 2 6 2 )=720 is the desired congruent 

 number. Returning to the earlier problem to make z 2 5 both squares, 

 and using the values a = 5, 6 = 4, just found, we have s = 10, t = S, and, 

 by (2), ra = 10, q = 72, whence z 2 =(S2/2) 2 , z 2 = 41. Since 720 = 5-12 2 , we 

 reduce the numbers in the ratio 1 : 12 and get the solution a; = 41/12. 



Leonardo 17 affirmed that no square can be a congruent number. This 

 proposition is of special historical importance since it implies that the 

 area of a rational right triangle is never a square and that the difference of 

 two biquadrates is not a square. Leonardo stated without proof 18 the 

 lemma that if a congruent number were a square there would exist integers 

 a } b for which a : 6=a+6 : a b (proved impossible earlier). 



Leonardo 19 noted that many numbers are not congruent; but any 

 number is a congruent if the quotient of any congruent number by it is a 

 square. A number is congruent if it equals one of the four numbers a, b, 

 a-\-b, a b, and if the remaining three are squares. For example, 16, 9, 

 16+9 are squares, so that 16 9 = 7 is a congruent number. To make 

 x i x both squares, let A; be a congruent number and g~ k=f 2 , 

 then we have the solution x = g' 2 /k since 



To make X z mX both squares, we set X = mx and are led to the preceding 

 problem, whence X = mg 2 /k. Leonardo considered the example with A: = 24, 

 g = 5. Cf . Alkarkhi, 3 and Ch. XVIII. 



Luca Paciuolo 20 reproduced part of Leonardo's Liber Quadratorum; he 

 gave as the first five " congruente " numbers 24, 120, 336, 720, 1320, their 

 corresponding squares (" congruo " 21 ) being 5 2 , 13 2 , 25 2 , 41 2 61 2 . From 

 n and n+1 he derived the congruent number 2n(n + l) }2(n+n+l) }, the cor- 

 responding square being |n 2 +(n+l) 2 } 2 . He made x 2 b fractional squares 

 for 6 = 5, 7, 13 ; and solved 2 + 10 = D , 2 1 1 = D . He gave a table of 52 

 congruent numbers, of which only 22 14 are primitive, the latter being all 

 in the table in the Arab MS. 1 (viz., the first six and 65, 70, 154, 210, 231, 

 330, 390, 546); the Arab had the advantage of excluding values a, b not 



18 Volo invenire, Tre Scritti, 95; Scritti, II, 271. Genocchi, 7 p. 288. 



17 Tre Scritti, 98; Scritti, II, 272. Cf. Ch. XXII. 



18 For a proof, with a historical discussion, see Genocchi, 7 pp. 293-310 (pp. 131-2). Cf. F. 



Woepcke, Jour, de Math., 20, 1855, 56; extract in Comptes Rendus Paris, 40, 1855, 781. 



19 Tre Scritti, 98; Scritti, II, 272. Genocchi, 7 pp. 310-3, 345-6. 



20 Luce de Burgo, Summa de arithmctica geometria, Venice, 1494; ed. 2, Toscolano, 1523, 



ff. 14-18. 



21 Thus interchanging Leonardo's two terms. Cf. Bibl. Math., (3), 3, 1902, 144. Also noted 



by Boncompagni. 8 



22 F. Woepcke, Annali di Mat., 3, I860, 20G; Atti Accad. Pont. Nuovi Lincei, 14, 1860-1, 259. 



