PREFACE. The efforts of Cantor and Ms collaborators show that a chronological history of mathematics down to the nineteenth century can be written in four large volumes. To cover the last century with the same elaborateness, it has been estimated that about fifteen volumes would be required, so extensive is the mathematical literature of that period. But to retain the chronological order and hence devote a large volume to a period of at most seven years would defeat some of the chief purposes of a history, besides making it very inconvenient to find all of the material on a particular topic. In any event there is certainly need of histories which treat of particular branches of mathematics up to the present time. The theory of numbers is especially entitled to a separate history on account of the great interest which has been taken in it continuously through the centuries from the tune of Pythagoras, an interest shared on the one extreme by nearly every noted mathematician and on the other extreme by numerous amateurs attracted by no other part of mathematics. This history aims to give an adequate account of the entire literature of the theory of numbers. The first volume presents in twenty chapters the material relating to divisibility and primality. The concepts, results, and authors cited are so numerous that it seems appropriate to present here an introduction which gives for certain chapters an account in untechnical language of the main results in their historical setting, and for the remaining chapters the few remarks sufficient to clearly characterize the nature of their contents. Perfect numbers have engaged the attention of arithmeticians of every century of the Christian era. It was while investigating them that Fermat discovered the theorem which bears his name and which forms the basis of a large part of the theory of numbers. A perfect number is one, like 6 = 1+2+3, which equals the sum ot its divisors other than itself. Euclid proved that 2*>~"1(2P—1) is a perfect number if 2P—1 is a prime. For p = 2, 3, 5, 7, the values 3, 7, 31, 127 of 2P-1 are primes, so that 6, 28, 496, 8128 are perfect numbers, as noted by Nicomachus (about A. D. 100). A manuscript dated 1456 correctly gave 33550336 as the fifth perfect number; it corresponds to the value 13 of p. Very many early writers believed that 2P —1 is a prime for every odd value of p. But in 1536 Regius noted that 29-l = 511 -7-73, 2n-1=2047 = 23-89 v\ >&are not primes and gave the above fifth perfect number. Cataldi, who ^founded at Bologna the most ancient known academy of mathematics,