CHAP. I] POLYGONAL, PYKAMIDAL AND FIGURATE NUMBERS. 39 



A. Ge*rardin, 21, 1914, 133-5, considered numbers expressible simul- 

 taneously in the form (x -f l}(x + 2) -(a; + p)/p\ for p = 2, 3, 

 He, 22, 1915, 203-5, considered the representation of numbers by x z + y z 

 + 2 2 + w, where w is polygonal. 



The question [Meyl 136 ] of tetrahedral numbers which are squares 

 reduces to N 3 N = 6n 2 , which was treated incompletely by L. Aubry, 

 26, 1919, 85-87. 



A. Boutin, 26, 1919, 35, 123, proved that no number is simultaneously 

 triangular, hexagonal, and a square. 



For minor remarks on triangular numbers, see Glaisher 88 of Ch. Ill; 

 Euler 12 of Ch. VII; Realis 53 and paper 8 of Ch. XIII; Pepin, 193 and Cunning- 

 ham 282 (on A, = cA y ) of Ch. XXI; Mathieu 282 of Ch. XXII. 



In Vol. I of this History were quoted theorems on triangular numbers 

 by G. W. Leibniz, p. 59; V. Bouniakowsky, pp. 283-4; R. Lipschitz, pp. 

 291-2; E. Barbette, p. 373; H. Brocard, p. 425; and P. Jolivald, p. 427. 



PAPERS ON POLYGONAL OR FIGURATE NUMBERS NOT AVAILABLE FOR REPORT. 



G. U. A. Vieth, Ueber fig. Zahlen, Progr., Dessau, 1817. 



J. P. L. A. Roche, D6m. nouv. des formules des piles de boulets, Toulon, 1827. 



H. Anton, Arith. Reihen hoh. Ord. u. die fig. Z., Progr. Ols, 1850. 



A. Wiegand, Trigonaltriaden in arith. Progres., Halle, 1850. 



J. Van Cleeff, Verhandeling over de polygonaal of veelhoekige getallen, Groningen, 1855. 



N. Nicolaides, Les Mondes, 7, 1865, 693; 8, 1865, 615, 708. 



J. L. A. Le Cointe, Les Mondes, 8, 1865, 707. 



Soufflet, Les Mondes, 13, 1867, 336 [last 3 papers on fig. numbers]. 



J. Talir, Arith. Reihe hoh. Ord. u. fig. Z., Progr., Waidhofen, 1872. 



G. de Rocquigny-Adanson, Les nombres triang., Moulins, 1896. 



