vi PREFACE. 



in many later books. This subject of the Chinese remainder problem found 

 application in questions on the calendar; for example, to find the year x 

 of the Julian period when the solar cycle, lunar cycle, and Roman indiction 

 are given numbers r : , r 2 , r 3 , we seek a number which has the remainders 

 T\ t r 2 , r z when divided by 28, 19, 15, respectively, these being the periods 

 of the solar, lunar, and indiction, cycles. 



The problem of finding the number of positive integral solutions of 

 ax-}-by = c, where a, 6, c are positive integers, was treated by Paoli in 1780, 

 Hermite in 1855-58, and many others. There is the corresponding question 

 for a system of such equations. 



Systems of equations of the type x-\-y-\-z = m, ax-\-by-\-cz = n, where 

 m, n, a, b, c are given positive integers and the unknowns are to have 

 positive integral values, occurred in Chinese and Arabic manuscripts of 

 the sixth and tenth centuries respectively, in Leonardo Pisano's writings, 

 and in many of the early printed books on algebra and arithmetic. The 

 usual method of solution, which began with the elimination of one unknown, 

 was called regula Coed, or the rule of the virgins, a term later applied to a 

 system of any number of linear equations in any number of unknowns with 

 positive integral coefficients. The most important papers on general sys- 

 tems of linear equations or congruences are those by Heger (1858), H. J. S. 

 Smith (1859, 1861, 1871), Weber (1872, 1896), Frobenius (1878-79), Kro- 

 necker (1886), and Steinitz (1896). 



Chapter II closes with a series of modern theorems, such as the fact that, 

 if co is irrational, there exist infinitely many pairs of integers x, y, for which 

 y ux is numerically less than the reciprocal of V5#; and Minkowski's 

 theorem (of prime importance for the theory of algebraic numbers) that ; 

 if /i, ...,/ are linear homogeneous functions of Xi, . . ., x n with any real 

 coefficients whose determinant is unity, we can assign integral values not 

 all zero to Xi, . . ., x n , such that each /,- taken positively does not exceed 

 unity. 



Chapter III treats of partitions, which have important applications 

 to symmetric functions and algebraic invariants. The first investigation 

 was that by Euler in 1741, who discussed the two problems of finding the 

 number of ways in which a number n (as 6) is a sum of a given number m 

 (as 2) of distinct parts (6 = 5+1 = 4+2), and the number of ways n is a 

 sum of m equal or distinct parts (so that also 6 = 3+3 is counted). The 

 numbers in question are the coefficients of x n in the expansions of x m(m+1) I2 JD 

 and x m /D, respectively, into series of powers of x, where 



Functions like these which serve to enumerate all the partitions of a specified 

 kind are now called generating functions. In his more attractive exposition 

 in his Introductio in Analysin Infinitorum of 1748, Euler noted that l/D is 

 the generating function giving the number of partitions of n into parts ^m 

 which need not be distinct. For n = 5, m = 3, these partitions are 3 + 2, 

 3+1 + 1, 2+2+1, 2+1 + 1 + 1, 1 + 1 + 1 + 1 + 1. Similarly, the reciprocal 



