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



Fermat 2 of Ch. XXII proved that the difference of two biquadrates is 

 never a square. Hence no congruent number is a square. 



L. Euler 33 noted (as had Leonardo) that p 2 5q* are both squares for 

 p = 41, 5 = 12; p 2 7g 2 both squares for p = 337, q = l20. He made p z aq* 

 squares also for a = 6, 14, 15, 30. The method is that used by him 76 for 

 concordant numbers. 



P. Cossali 34 undertook to reconstruct Leonardo's Liber Quadratorum, 

 then believed to be lost. A sufficient (adverse) report will be found under 

 Genocchi, 35 Woepcke 36 and Boncompagni. 37 



A. Genocchi 35 stated that Cossali 34 was wrong in believing that 

 Leonardo's method of making 2 d=a both squares is only special. While 

 indirect, it is general and succeeds when the problem is solvable. In fact, 

 it coincides exactly with the formulas obtained by Euler 76 after complicated 

 calculations. This coincidence escaped Cossali, who filled many pages 

 with useless calculations without discovering the general solution. 



F. Woepcke 36 noted that of the [26 distinct J congruent numbers in the 

 table of 29 lines by Cossali, 34 p. 126, only 12 are primitive, including all but 

 65 and 154 of those noted under Luca Paciuolo. 20 



B. Boncompagni 37 disagreed with the explanation by Cossali, 34 p. 132, 

 of Leonardo's method. The latter had remarked that h will be a con- 

 gruent number if its quotient by a given congruent number hi is a 

 square q 2 . According to Cossali's interpretation, q is rational only when 

 (/i 1 +2)(2/i 1 +2)(3/ii+4) is a rational square; while a more plausible inter- 

 pretation leads always to a rational q. 



L. Pisanus " 3S made n 2 13 and n~ all rational squares. Since 



" 



are the squares of 2d 2 +4d+3 and 2d 2 +4d+l, take d = 2 and we get 

 13-2536=D. In (a 2 +6 2 ) 2 4a6(a 2 -6 2 ) = (a 2 2a6-6 2 ) 2 , take a = ct 2 , 

 fe = s 2 . Then 



(c 2 Z 4 +s 4 ) 2 4c*V(d 2 +s 2 )(cZ 2 -s 2 ) = D. 



Take c = 13, Z 2 = 25, s 2 = 36. But (13-25) 2 -36 2 is the product of the squares 

 found before. Hence (c 2 4 +s 4 ) 2 /{4V(c 2 i 4 s 4 )} is the required square ra 2 . 

 J. Hartley 39 took z 2 -f-13=(z-|-2/) 2 , x*-lZ = (x-yz)\ and from the two 

 rational values of x got \f = 13(2 l)/{z(z+l) }. The latter is a square for 

 z = (r 2 +s 2 )/(2rs) if r 2 +s 2 =13, 2rs=D. Take r=-3-gt, s = 2-t. Then 

 r 2 +s 2 = 13 gives =(4-60)/(0 2 -f-l). Take = 2, whence r = l/5, s = 18/5, 



33 Algebra, 2, 1770, 226; French transl., Lyon, 2, 1774, p. 291; Opera Orania, (1), I, 459- 

 "Origine, trasporto in Italia, primi progressi in essa dell'algebra, 1, 1797, 115-172. Cf. G. 



Libri, Histoire des Sc. Math, en Italic, ed. 2, III, 1865, 139, 140, 265. 

 36 Comptes Rendus Paris, 40, 1855, 775-8. 

 36 Atti Accad. Pont. Nuovi Lincei, 14, 1860-1, 259. 

 87 Annali di Sc. Mat. e Fis., 6, 1855, 149-151. 



58 Ladies' Diary, 1803, p. 41, Quest. 1099; and Prize Prob. 1118, 1S04, pp. 44-6; Leybourn's 



Math. Quest. L. D., 4, 1817, 10-11, 31-33. The Prize problem stated that there are 

 rational squares x 2 , y z such that x 2 13 are squares, and 13# 2 is the area of a right triangle 

 whose sides are integers; 13 is a sum of two squares, double the product of whose roots 

 is a square, and if the latter square be added to and subtracted from 13 the results 

 are squares. 



59 The Diary Companion, Supplement to the Ladies' Diary, London, 1803, 45. 



