6 HISTORY OF THE THEORY OF NUMBERS. [CHAP. I 



His II, 27, relates to the formula of Nicomachus 5 (Ch. 20) 



1 = 1 3 , 3+5 = 2 3 , 7+9 + ll=3 3 , 13 + 15 + 17 + 19 = 4 3 , , 



from which follows the above formula II, 25, by addition (as in the 

 Codex Arcerianus 10 ). Fermat 33 generalized this proposition by introducing 

 "colonne": In the arithmetical progression 1,1 + (m 2), 1 + 2(m 2), 

 leading to m-gonal numbers, the first term 1 gives the first colonne; 

 the sum of the next two terms diminished by m 4 times the first tri- 

 angular number 1 gives the second colonne 2m; the sum of the fourth, 

 fifth and sixth terms diminished by m 4 times the second triangular 

 number 3 gives the third colonne 9m 9; similarly, the fourth colonne is 

 8(3m - 4) and the rth is r 2 + r 2 (r - l)(ra - 2)/2. It follows (as noted 

 by Editor Tannery) that the rth colonne is the product of the rth m-gonal 

 number by r, and for m = 4 is r 3 . The term colonne was not coined by 

 Fermat, as Tannery thought, but 34 was used by Maurolycus. 26 



J. Remmelin 35 noted that 666 (cf. Faulhaber 29 ) is a triangular number 

 with the root 36, which is a square with the root 6, while 6 is a pronic 

 number [of the form n 2 + n~\ whose base 2 is also a pronic number. 



Later we shall quote Bachet's empirical theorem that any integer is the 

 sum of four squares, made a propos of Diophantus IV, 31. In this connec- 

 tion Fermat 36 made the famous comment: "I was the first to discover the 

 very beautiful and entirely general theorem that every number is either 

 triangular or the sum of 2 or 3 triangular numbers; every number is either 

 a square or the sum of 2, 3 or 4 squares; either pentagonal or the sum of 

 2, 3, 4 or 5 pentagonal numbers; and so on ad infinitum, whether it is a 

 question of hexagonal, heptagonal or any polygonal numbers. I can not 

 give the proof here, which depends upon numerous and abstruse mysteries 

 of numbers; for I intend to devote an entire book to this subject and to 

 effect in this part of arithmetic astonishing advances over the previously 

 known limits." But such a book was not published. Fermat 37 stated 

 the theorem in a letter to Mersenne, Sept., 1636 (to be proposed to St. 

 Croix) ; to 38 Pascal, Sept. 25, 1654, and Digby, June 19, 1658. The theorem 

 was attributed to St. Croix by Descartes 39 in a letter to Mersenne, July 27, 

 1638. Descartes 40 gave an algebraic proof of Plutarch's 6 theorem that 

 8A r + 1 = (2r + I) 2 . We shall often write A r or A(r) for the rth tri- 

 angular number r(r + l)/2, A or A' for any triangular number, D for 

 any square, , 02 or SI for a sum of two, three or four squares. 



38 Oeuvres, I, 341. 



84 Wertheim, Zeitschr. Math. Phya., 43, 1898, Hist.-Lit. Abt., 41-42. 



86 Johanne Lvdovico Remmelino, Structura Tabularvm qvadratarvm, 1627, Preface. The 



book treats magic squares at length. 

 38 Oeuvres, I, 305; French transl., Ill, 252. E. Brassinne, Pr6cis des Oeuvres Math, de P. 



Fermat, M6m. Acad. Imp. Sc. Toulouse, (4), 3, 1853, 82. 

 "Oeuvres, II, 1894, 65; III, 287. 



38 Oeuvres de Fermat, II, 313, 404; III, 315. 



39 Oeuvres de Descartes, II, 1898, 256, 277-8 (editors' comments); X, 297 (statement of 



the theorem in a posthumous MS.). 



40 Oeuvres, X, 298 (posth. MS.). 



