CHAP. I] POLYGONAL, PYRAMIDAL AND FIGURATE NUMBERS. 5 



polygonal numbers; for example, Boethius, 19 G. Valla, 20 Martinus, 21 

 Cardan, 22 J. de Muris, 22a Willichius, 23 Michael Stifel, 24 who gave a table 

 of figurate numbers (binomial coefficients), Faber Stapulensis, 25 and F. 

 Maurolycus, 26 who gave 



Pi = 3?;-' + r, p r 6 = 2p r ~ l + r 2 , 



p; + p;~ l = P:, PI = PI + 2PT 1 , PI = PI + PT\ 



and treated (pp. 32-74) polygonal numbers of the second order or central 

 polygonal numbers (the pentagonal being 1, 6, 16, 31, 51, 76, , when 

 in the second are counted the vertices and center of a pentagon), as well as 

 central pyramidal numbers (the pentagonal being 1, 7, 23, 54, 105, ). 

 Also I. Unicornus, 27 and G. Henischiib. 28 



Johann Faulhaber 29 treated polygonal and pyramidal numbers. 



Johann Benzius 30 devoted twenty chapters to these and figurate numbers. 



J. Rudolff von Graffenried 31 noted that 



the final number being 666 for r = 6. 



C. G. Bachet 32 wrote a supplement of two books to the Polygonal Num- 

 bers of Diophantus. The most important ones of his theorems (when ex- 

 pressed as formulas) are as follows: 



I, 10. p^ +r = p k m + p r m + kr(m 2), p r m = p r z + (n 



II, 18. p r m + Pn H h p = p r m pl + r 2 (m 2) (pj + p 



II, 21. 3(p; + pl r + h p") = p r m pl + (n + 



II, 25. 1' + 2 3 + + n 3 



II, 28. n s + Qp n 3 + I = (n + I) 3 . 



II, 31, 32. k 3 + (2k) 3 +-+ (nk) 3 = k z (pl) 2 = k(k + 2k + - + nk}\ 



19 Arithmetica boetij, 1488, etc., Lib. 2, Caps. 7-17. 



20 De expetendis et fvgiendis rebvs opvs, Aldus, 1501, Lib. III. 



21 Ars Arithmetica, 1513, 1514; Arithmetica, 1519, 15-18. 



22 Practica Arith., 1537, etc. 



22a Arith. Speculativae, 1538, 53-62. 



23 1. Vvillichii Reselliani, Arith. libri tres, 1540, 95-111. 



24 Arith. Integra, 1544. See references 16-18, 50-52. 



26 Stapulensis, Jacobi Fabri, Arith. Boe'thi epitome, 1553, 54-65. 



26 Arith. libri dvo, 1575, 6-8, 14-21. Historical remarks on same by M. Fontana, Memorie 



dell' Istituto Nazionale Ital., Mat., 2, Pt. 1, 1808, 275-296. 



27 De rArithmetica Vniversale, 1598, 67-70. 



28 Arith. Perfecta et Demonstrata [1605], 1609, 133. 



29 Cubicoss Lustgarten, 1604 (also in part 2 of Petrum Rothen, Arithmetica Philosophica, 



Niirnberg, 1608); Neuer Math. Kunstspiegel, Ulm, 1612, which notes that 1335 (men- 

 tioned in the Bible, Daniel, XII, 12) is a pentagonal number whose root 30 is a pronic 35 

 number with the pentagonal root 5 whose root 2 is pronic, while 2300 (Daniel, VIII, 14) 

 is tetradecagonal whose root 20 is pronic, etc.; Numerus Figuratus, 1614, 24 pp.; 

 Miracula Arithmetica, Augspurg, 1622, a book chiefly on arithmetical combinations 

 giving the "Wunder Zahl " 666, the Apocalyptic number mentioned in the Bible, 

 Revelations, XIII, 18; cf. Remmelin, 35 A. G. Kastner, Geschichte Math., Ill, 111-52. 

 80 Manuductio ad Nvmervm Geometricvm, Kempten, 1621. 



31 Arith. Logistica Popularis, 1618, 238, 627. 



32 Diophanti Alex. Arith., 1621. 



