The ancient Greeks recognised two branches of arithmetic. One of them, the philosophical study, expounded by Euclid in Books VII-IX, developed as a logical science the properties of whole numbers. It was not so much concerned with addition, multiplication and the manipulation of fractions, in which the notation developed in the middle ages has greatly improved the methods of the ancients, as with prime, odd and even numbers, with digressions to triangular, polygonal and perfect numbers. The Greeks were familiar with the facts (1) that when a prime number is a factor of the product of two whole numbers ab it must be a factor of either a or 6, and (2) that every whole number can be expressed, in only one way, as a product of prime factors. A late offshoot of this branch of the subject, destined to develop into one division of the modern Theory of Numbers, was considered by Diophantos of Alexandria (English edition by Sir T. L. Heath), who was mainly concerned with finding solutions in rational numbers of indeterminate equations such as

yz + a=u2, zx + a = v2, xy + a=w2.

The second branch of arithmetic known to the Greeks (Aoyto-riK?/) was concerned with its applications to measurements, to monetary calculations and to various problems in which the science was used in practical life. No exposition of this is known to have survived the ages.