Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. It is distinguished by its rigour, abstraction, and beauty. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of navigation, astronomy, physics, engineering, and so on.

Ancient Greeks were usually more fond of geometry than of pure mathematics. Ancient Greek mathematicians are among the earliest to make a distinction between pure and applied mathematics. Plato helped to create the gap between "arithmetic", now called Number theory, and "logistic", now called arithmetic.

Plato regarded logistic (arithmetic) as appropriate for business men and men of war who "must learn the art of numbers or he will not know how to array his troops" and as appropriate for philosophers "because he has to arise out of the sea of change and lay hold of true being." Euclid of Alexandria, when asked by one of his students of what use was the study of geometry, asked his slave to give the student threepence, "since he must needs make gain of what he learns." The Greek mathematician Apollonius of Perga was asked about the usefulness of some of his theorems in Book IV of Conics to which he proudly asserted, And since many of his results were not applicable to the science or engineering of his day, Apollonius further argued in the preface of the fifth book of Conics that "the subject is one of those which seem worthy of study for their own sake."

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