'fl 
i 
^ 
r^ 
, _ &!£:: steering* 
'^vi-^ r7' Mathenrstical 
2^H-j Sciences 
:D/^ifii 
PREFACE. 
libraiy 
The efforts of Cantor and his 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 time 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 
v 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 
Jl 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 
v^, contents. 
J' ' ' 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 of its divisors other than itself . Euclid 
,. proved that 2^~'^{2^ — \) is a perfect numbeflf 2^ — 1 is a prime. For p = 2, 
3, 5, 7, the values 3, 7, 31, 127 of 2''-l are primes, so that 6, 28, 496, 8128 
are perfect numbers, as noted by Nicomachus (about A. D. 100). A manu- 
script dated 1456 correctly gave 33550336 as the fifth perfect number; it cor- 
* ! responds to the value 13 of p. Very many early writers believed that 2^ — 1 
I is a prime for every odd value of p. But in 1536 Regius noted that 
2^-1 = 511 = 7-73, 211-1=2047 = 23-89 
are not primes and gave the above fifth perfect number. Cataldi, who 
founded at Bologna the most ancient known academy of mathematics, 
