Mathematics reveals its secrets only to those who approach it with pure love, for its own beauty.

Eureka! (I have found it!).

Archimedes to Eratosthenes greeting. ... certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards because their investigation by the said method did not furnish an actual demonstration. But it is of course easier, when we have previously acquired by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge.

Two magnitudes whether commensurable or incommensurable, balance at distances reciprocally proportional to the magnitudes.

Those who claim to discover everything but produce no proofs of the same may be confuted as having actually pretended to discover the impossible.

The centre of gravity of any cylinder is the point of bisection of the axis.

Spoken of the young Archimedes: . . . [he] was as much enchanted by the rudiments of algebra as he would have been if I had given him an engine worked by steam, with a methylated spirit lamp to heat the boiler; more enchanted, perhaps for the engine would have got broken, and, remaining always itself, would in any case have lost its charm, while the rudiments of algebra continued to grow and blossom in his mind with an unfailing luxuriance. Every day he made the discovery of something which seemed to him exquisitely beautiful; the new toy was inexhaustible in its potentialities.