PREFACE. v 



Pollock stated that 5, 7, 9, 13, 21, 11 terms are needed to express every 

 number as a sum of tetrahedral, octahedral, cubic, icosahedral, dodeca- 

 hedral, and squares of triangular, numbers, and related facts. In 1862-63 

 Liouville proved that the only linear combinations of three triangular 

 numbers A which represent all numbers are A+A'+cA" (c= 1, 2, 4, 5) and 

 A+2A'+dA" (d = 2, 3,4). 



Chapter II opens with an account of the method of solving ax+by = c 

 given by the Hindu Brahmegupta in the seventh century. It was based on 

 the mutual division of a and b, as in Euclid's process of finding their greatest 

 common divisor. Essentially the same method was rediscovered in Europe 

 by Bachet de Meziriac in 1612, and expressed in the convenient notation 

 of the development of a/6 into a continued fraction by Saunderson in 

 England in 1740 and by Lagrange in France in 1767. The simplest proof 

 that the equation is solvable when a and b are relatively prime is that given 

 by Euler in 1760, who noted that, on dividing c ax (x = 0, 1, . . ., 6 1) 

 by b, we obtain b distinct remainders which are therefore 0, 1, . . ., b 1 

 in some order, the remainder zero leading to a solution. Since the same 

 principle underlies the most elegant proof of Euler 's generalization a?=\ 

 (mod 6) for /3= <(&) of Fermat's theorem, it was a simple step to solve our 

 equation, or what is the same thing the congruence ax = c (mod 6), 

 by multiplying its members by a ~ l . This step was made about 1829 by 

 Binet, Libri, and Cauchy. Or we may evidently employ Wilson's gener- 

 alized theorem, which states that the product of the positive integers less 

 than and prune to b is = 1 (mod 6). In 1905 Lerch expressed the solution 

 of az= 1 (mod 6) as a sum involving the greatest integer function. 



In the Chinese arithmetic of Sun-Tsu, about the first century, occurs 

 the problem of finding a number having the remainders 2, 3, 2 when divided 

 by 3, 5, 7, respectively, with a rule leading to the answers 23 + 3 -5 -7n. 

 The same problem and answer 23 occur in the Greek arithmetic of Nico- 

 machus, about 100 A.D. The rule is essentially the following, given 

 centuries later by Beveridge, Euler, and Gauss: To obtain a number x 

 having the remainders n, r 2 , . . . when divided by m\, m 2 , . . ., respectively, 

 where mi, m 2 , . . . are relatively prime in pairs, find numbers <xi, 2 , 

 such that cti=l (mod m*), oa=0 (mod m/nti), where m is the product 

 mim 2 . . .; then ^ = airi+a 2 ^2+ ... is an answer. In the seventh cen- 

 tury, the Chinese priest Yih-hing extended this rule to the case in which 

 mi, m 2 , . . . are any integers: express the least common multiple of mi, m 2 , 

 . . . as a product m = juiju 2 . . .of factors relatively prime in pairs (some of 

 which may be unity), such that m divides m z , and find i, 2 , . . . such 

 that &i=l (mod ju t -), a t = (mod w/ju); then # = o:iri+a: 2 r 2 + .... 



The Hindus Brahmegupta and Bhascara found the correct answer 59 to 

 the " popular problem" of finding a number having the remainders 5, 4, 3, 2 

 when divided by 6, 5, 4, 3, respectively; Leonardo Pisano in 1202 added 

 the condition that the number be a multiple of 7. He treated the problem 

 of Ibn al-Haitam (about 1000 A.D.) of finding a multiple of 7 which has 

 the remainder unity when divided by 2, 3, 4, 5 or 6, a problem occurring 



