CHAP. I] POLYGONAL, PYRAMIDAL AND FIGURATE NUMBERS. 3 



i 



Plutarch, 6 a contemporary of Nicomachus, gave the theorem that if 

 we multiply a triangular number by 8 and add 1, we obtain a square: 



. 



This theorem was given by lamblichus 7 (about 283-330 A.D.), who 

 treated at length (pp. 82-176) polygonal and pyramidal numbers. 



Diophantus 8 (about 250 A.D.) generalized this theorem and proved by 

 a cumbersome geometric method that 

 (1) 8(m - 2) P ; + (w - 4) 2 = \(m - 2)(2r - 1) + 2} 2 , 



and spoke of this result as a new definition of p equivalent to that of 

 Hypsicles. Diophantus gave a rule for finding r, equivalent to the solution 

 of (1) for r, and a rule for finding p equivalent to 



, [( m -2)(2r-l)+2] 2 -(m-4) 2 

 Pm ' S(m - 2) 



but did not give the equal simpler expression 

 (3) Pl-lr{2+(m-2)(r-l)j. 



In fact, starting with (2), he gave a long geometric discussion to find the 

 number of ways a given number can be polygonal, but made little headway 

 before the abrupt termination of the fragment. G. Wertheim 9 gave a 

 lengthy continuation in the same geometric style which eventually leads 

 to the geometric equivalent to (3) and remarked that we can readily find 

 from (3) the ways in which a given number p can be polygonal: Express 

 2p as a product of two factors > 1 in all possible ways; call the smaller 

 factor r; subtract 2 from the larger factor and find whether or not the 

 difference is divisible by r 1; if it is, the quotient is m 2, and p is 

 a p r m . Since m 2 equals 2(p r)/[r(r 1)], the latter must be an 

 integer ^ 1, so that _ 



l - 1). 



For example, if p = 36, then r ^ 8. Since r divides 2p = 72, we have 

 r = 2, 4, 8, 3, 6, of which r = 4 is excluded. We get 



In the Roman Codex Arcerianus 10 (450 A.D.?) occur a number of 

 special cases of the remarkable formula for pyramidal numbers 





6 Platonicae quaestion., II, 1003. 



7 In Nicomachi Geraseni arith. introd., ed., S. Tennulius, 1668, 127. 



8 Polygonal Numbers. Greek text by P. Tannery, 1893, 1895. Engl. transl. by T. L. 



Heath, Cambridge, 1885, 1910; German transl. by F. T. Poselger, 1810, J. O. L. Schulz, 

 1822, and G. Wertheim, 1890; French transl. by G. Massoutie", Paris, 1911. Cf. Nessel- 

 mann, Algebra der Griechen, 1842, 462^76; M. Cantor, Geschichte Math., ed. 3, 1, 485-7. 



9 Zeitschrift fur Math. Physik, Hist. Lit. Abt. 1897, 121-6. Reproduced by T. L. Heath, 



Diophantus, ed. 2, 1910, 256, where doubt is expressed as to the validity of the restora- 

 tion in view of the ease with which the geometric equivalent of (3) can be derived geo- 

 metrically from that of (2). 



10 Cf. M. Cantor, Die Romischen Agrimensoren, Leipzig, 1875, 95-127. 



