According to most accounts, geometry was first discovered among the Egyptians, taking its origin from the measurement of areas. For they found it necessary by reason of the flooding of the Nile, which wiped out everybody's proper boundaries. Nor is there anything surprising in that the discovery both of this and of the other sciences should have had its origin in a practical need, since everything which is in process of becoming progresses from the imperfect to the perfect.

The Pythagoreans considered all mathematical science to be divided into four parts: one half they marked off as concerned with quantity, the other half with magnitude; and each of these they posited as twofold. A quantity can be considered in regard to its character by itself or in relation to another quantity, magnitudes as either stationary or in motion. Arithmetic, then, studies quantity as such, music the relations between quantities, geometry magnitude at rest, spherics magnitude inherently moving.

*A Commentary on the First Book of Euclid's Elements*

Wherever there is number, there is beauty.

Quoted in M Kline, *Mathematical Thought from Ancient to Modern Times*

This, therefore, is mathematics:

she reminds you of the invisible form of the soul;

she gives light to her own discoveries;

she awakens the mind and purifies the intellect;

she brings light to our intrinsic ideas;

she abolishes oblivion and ignorance which are ours by birth.

Quoted in M Kline, *Mathematical Thought from Ancient to Modern Times*

It is well known that the man who first made public the theory of irrationals perished in a shipwreck in order that the inexpressible and unimaginable should ever remain veiled. And so the guilty man, who fortuitously touched on and revealed this aspect of living things, was taken to the place where he began and there is for ever beaten by the waves.

Scholium to Book X of Euclid V.

JOC/EFR February 2006

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