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University of St Andrews, Scotland
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.
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