CHAP. I] POLYGONAL, PYRAMIDAL AND FIGURATE NUMBERS. 9 



Hence we may assign any values to q, z 3 , n and find 6. Likewise for 

 Bachet's generalization, we may assign any values to all five products 

 other than the square 6 2 , and find 6. 



A. Ge"rardin 49 noted that the simplest solution of Diophantus' problem is 

 furnished by the three numbers (x 2 -j- l)/2, 2 , x, with a = x 2 , fi = x8, 7 = x, 



|(z 2 + 1) + 2 + x = x\ 



Set x = 2H + 1. Then 2 - 2# 2 = - 1, with the solutions (H, 6) = (1, 1), 

 (5, 7), (29, 41), etc., giving the numbers 5, 1, 3; 61, 49, 11: 1741, 1681, 59. 

 Rene" F. de Sluse 50 gave the table [cf. Stifel 24 ] 



in which the numbers (like 1, 3, 3, 1) hi a diagonal are binomial coefficients, 

 those in the third column are triangular numbers, those in the fourth 

 column are pyramidal numbers with triangular base, those in the fifth are 

 triangular pyramids of the second order. 



B. Pascal 51 gave the same table and noted that any number in it is the 

 sum of the numbers in the preceding column and hence (p. 504) is the sum 

 of the number above it and that immediately to its left. He noted (p. 533) 

 that n(n + !) -(n + k 1) is divisible by kl, the quotient being a 

 figurate number. 



G. W. Leibniz 52 gave a table formed by the diagonals (as 1, 2, 1) of the 

 above table. 



J. Ozanam 53 found pairs* of triangular numbers 15 and 21, 780 and 990, 

 1747515 and 2185095, whose sum and difference are triangular. Their 

 sides are 5 and 6, 39 and 44, 1869 and 2090. Polygonal numbers are treated 

 in the English translation by C. Hutton, London, 1803, pp. 40-47, p. 60. 



Pierre Esmond de Montmort 54 cited special cases of (1), due to Dio- 

 phantus. 



F. C. Mayer 55 defined "generalized figurate" numbers 



x(x + l)--(x + n - 1) ^x(x-i)--(x + n- 2) 



a - l^n - + d- a ) 1.2-.- (n-1) ' 



49 Sphinx-Oedipe, 6, 1911, 42. 



60 MS. 10248 du fonds latin, Bibliotheque Nationale de Paris, f. 187. 



51 Traite" du triangle arith., Paris, 1665 (written 1654); Oeuvres, III, 1908, 466-7. 



62 Leibniz Math. Schriften, ed., C. I. Gerhardt, VII, 101. 



* Others are 171 and 105, 3741 and 2145. Gerardin gave a general discussion in Sphinx- 

 Oedipe, 1914, 113. 



63 Recreations math, et phys., 1, 1696, 20; new eds., 1723, etc. 



64 M4m. Acad. Roy. Sc., 1701. Essai d'Analyse sur les Jeux de Hazards, 1708; ed. 2, 1713, 17. 

 66 Maiero, Comm. Acad. Petrop., 3, ad annum 1728 [1726], 52. 



