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



It gave p r 5 = |(3r 2 + r), p r 6 = ^(4r 2 -f- 2r), where the plus signs should be 

 minus. M. Cantor 11 suggested the following probable derivation. By 

 factoring the numerator of (2), we obtain 



r (m - 2) (m-4) 



As known by Archimedes (b. Syracuse about 287 B.C.), 



1+2 + , '(r + D r . y , ... . . r(r + l)(2r + l) 



2 6 



Hence 



1 [2(m - 2) . 2(m - 4) r + 1 r 



-y- -r 2 r -r + rj = - -(2p; + r). 



6 

 The Hindu Aryabhatta 12 (b. 476 A.D.) gave the formula 



. r(r + 1) r(r + l)(r + 2) (r + I) 3 - 



- 



for the number of spheres in a triangular pile, and hence for the rth 

 pyramidal number PI of order 3, called also a tetrahedral number. The 

 Hindus of his time knew 13 also that P\ P r z + PS" 1 , whence 



6P; = r(r + l)(2r + 1). 



The above general formulas relating to polygonal and pyramidal numbers 

 were collected about 983 A.D. by Gerbert 14 (Pope Sylvester II). 

 Yang Hui 16 gave in his Suan-fa, 1261, the formulas 



1 + (1 + 2) + (1 + 2 + 3)+ + (1 + 2 + - . . + n) = n(n + l)(n + 2)/6, 



I 2 + 2 2 + + n 2 = Jn(n + *)(n + 1) 



for the sums of triangular numbers and squares. 



Chu Shih-chieh, 16 in 1303, tabulated in the form of a triangle the 

 binomial coefficients as far as those for eighth powers. This arithmetical 

 triangle was known 17 to the Arabs at the end of the eleventh century. 

 Such a triangle was published by Petrus Apianus. 18 



Many of the early arithmetics mentioned (some with fuller titles) in 

 Vol. I, Ch. I, of this History, gave definitions and simple properties of 



11 Die Romischen Agrimensoren, 1875, 122; Geschichte der Math., 1, ed. 2, 519; ed. 3, 558. 



Cf. H. G. Zeuthen, Bibliotheca Mathematica, (3), 5, 1904, 103. 

 "French transl. by L. Rodet, Jour. Asiatique, 13, 1879; Lecons de calcul d'Aryabhatta, 



p. 13, p. 35. 

 13 E. Lucas, La Nature (Revue des Sciences), 14, 1886, II, 282-6: L'Arithm&ique en Batons 



dans 1'Inde au temps de Clovis. 

 " Geometric, Chs. 55-65. 



1S Y. Mikami, Abh. Geschichte Math. Wiss., 30, 1912, 85. 

 18 Ibid., 90. Cf. K. L. Biernatzki, Jour, fur Math., 52, 1856, 87; Stifel. 24 



17 M. Cantor, Geschichte der Math., 1, ed. 3, 1907, 687. 



18 Ein newe . . . Kauffmans Rechnung . . . , Ingolstadt, 1527, title page. The latter was 



reproduced by D. E. Smith, Rara Arith., 1908, 156, who remarked that he knew of no 

 earlier publication of this Pascal triangle. 



