CHAPTER I. 



POLYGONAL, PYRAMIDAL AND FIGURATE NUMBERS. 



The formation of triangular numbers 1, 1 + 2, 1 + 2 + 3, -, and of 

 square numbers 1, 1 + 3, 1 + 3 + 5, , by the successive addition of 

 numbers in arithmetical progression, called gnomons, is of geometric origin 

 and goes back to Pythagoras 1 (570-501 B.C.) : 9 



1 



If the gnomons added are 4, 7, 10, (of common difference 3), the 

 resulting numbers 1, 5, 12, 22, are pentagonal. If the common differ- 

 ence of the gnomons is m 2, we obtain m-gonal numbers or polygonal 

 numbers with m sides. 



In the cattle problem of Archimedes (third century B.C.), the sum of 

 two of the eight unknowns is to be a triangular number (see Ch. XII). 



Speusippus, 2 nephew of Plato, mentioned polygonal and pyramidal 

 numbers: 1 is point, 2 is line, 3 triangle, 4 pyramid, and each of these 

 numbers is the first of its kind; also, 1 + 2 + 3 + 4 = 10. 



About 175 B.C., Hypsicles gave a definition of polygonal numbers 

 which was quoted by Diophantus 8 in his Polygonal Numbers, "If there 

 are as many numbers as we please beginning with one and increasing by 

 the same common difference, then when the common difference is 1, the 

 sum of all the terms is a triangular number; when 2, a square; when 3, a 

 pentagonal number. And the number of the angles is called after the 

 number exceeding the common difference by 2, and the side after the 

 number of terms including 1." Given therefore an arithmetical progres- 

 sion with the first term 1 and common difference m 2, the sum of r 

 terms is the r-th m-gonal number 3 p r m . 



The arithmetic of Theon of Smyrna 4 (about 100 or 130 A.D.) contains 

 32 chapters. In Ch. 15, p. 41, the squares are obtained from 1 + 3 = 4, 



X F. Hoefer, Histoire des mathematiques, Paris, ed. 2, 1879, ed. 5, 1902, 96-121; W. W. R. 

 Ball, Math. Gazette, 8, 1915, 5-12; M. Cantor, Geschichte Math., 1, ed. 3, 1907, 160-3, 

 252. 



2 Theologumena arithmeticae, ed. by F. Ast, Leipzig, 1817, 61, 62. For a French transl. and 



notes, see P. Tannery, Pour 1'histoire de la science Hellene, Paris, 1887, 386-390 (374). 



3 Denoted by P (n ? in Encyc. Sc. Math., I, li, p. 30. 



4 Theonis Smyrnaei Platonici, Latin transl. by Ismael Bullialdi, 1644. Cf . Expositio rerum 



mathematicarum ad legendum Platonem utilium, ed., E. Hiller, pp. 31-40. 

 2 1 



