CHAP. XV] LINEAR FUNCTIONS MADE SQUARES. 447 



conditions are satisfied since 168+120+1 = 17 2 , 168-120-J-1 = 7 2 . To 

 make 168i 2 +l and 12QI- + 1 squares, we have a double equality, satisfied 

 by x= -1648825564/1242622079. 



G. W. Leibniz 17 discussed the problem to find three numbers the sum 

 and difference of every pair of which are squares. 



M. Rolle 18 found four numbers the difference of any two of which is 

 a square, and the sum of any two of the first three is a square : 



z 2 , D = A+B+C. 



For y = l, 2 = 2, 4 = 2399057, 5 = 2288168, C= 1873432, D = 6560657. 



T. F. de Lagny 19 solved 4z+6 = 2/ 2 , 9z+13=2 2 by a "new method." 

 Eliminating x, we have 9?/ 2 /4 1/2 = z 2 . Hence 9i/ 2 2 = D, say the square 

 of 3y a. Thus y is found in terms of a. 



P. Halcke 20 divided 6 into two parts such that each increased by 6 

 gives a square, and made 6+#, 12 x both squares. 



Mal6zieux 21 proposed the first problem of Fermat. 9 It is a question of 

 finding three equal sums of two squares. 



The problem to find three numbers the sum and difference of any two 

 of which are squares received at the time of its proposal no comment 

 except the mere statement by C. Bumpkin- 2 that 1873432, 2399057, 2288168 

 furnish an answer. 



J. Landen 23 took as the numbers 



y, 2 = -0 



Then xy, xz, yz are squares. It remains to make 



a square. Set g=f+r. Then E= D if 



1 , 2/V (/ 6 -3/ 2 )r 22 



- l1 "-< 



which gives r and hence 0=/(/ 8 +6/ 4 -3)/(l+6/ 4 -3/ 8 ). The case /=2 

 gives Bumpkin's 22 answer. Or we may take f z g-+l, / 2 +<? 2 , 2fg as the 

 numbers, whence xz, yz are squares. For the preceding value of g it 

 is verified that / 2 </ 2 (<7 2 +/ 2 ) + l are squares, whence their product E is a 

 square. Or we may make E = (f 4 l)(g 4 1) a square by equating it to 

 (/ 4 -l)V+l) 2 , whence =/ 2 / V2^/ 4 . Set/=l-d; then 2-/ 4 becomes a 



17 MS. dated Apr. 1, 1676, in Bibliothek Hannover. D. Mahnke, Bibliotheca Math., (3), 



13, 1912-3, 39. Cf. Euler. 28 



18 Journal des Savans, Aug. 31, 1682; Sphinx-Oedipe, 1906-7, 61-2. Cf. Coccoz, 74 Rignaux. 89 



19 Nouv. Elemens d'Arith. et d'Algebre, Paris, 1697, 451-5. 



20 Deliciae Math., oder Math. Sinnen-Confect, Hamburg, 1719, 235. 



21 Elements de Geome'trie de M. le Due de Bourgogne, par de Mal6zieux, 1722; Sphinx-Oedipe, 



1906-7, 4-5, 45. 



22 Ladies' Diary, 1750, p. 21, Quest. 311. Cf. Euler. 28 



23 C. Button's Diarian Miscellany, extracted from Ladies' Diary, 3, 1775, 398-401, Appendix. 



Leybourn's Math. Quest, proposed in Ladies' Diary, 2, 1817, 19-22. Cf. Euler. 28 



