The creative act always requires a stepping back. It's called the incubation period. The incubation period - one of the four phases of creativity - is when you're not consciously thinking of a problem, and you're letting it marinate. So this is why you hear time and again, people saying they had that "Eureka" moment in the bath, like Archimedes, or in the shower, or while going for a walk or in a coffeehouse.
It may be said that the conceptions of differential quotient and integral, which in their origin certainly go back to Archimedes, were introduced into science by the investigations of Kepler, Descartes, Cavalieri, Fermat and Wallis. . . .
[The works of Archimedes] are without exception, monuments of mathematical exposition; the gradual revelation of the plan of attack, the masterly ordering of the propositions, the stern elimination of everything not immediately relevant to the purpose, the finish of the whole, are so impressive in their perfection as to create a feeling akin to awe in the mind of the reader.
What exactly is mathematics? Many have tried but nobody has really succeeded in defining mathematics; it is always something else. Roughly speaking, people know that it deals with numbers, figures, with relations, operations, and that its formal procedures involving axioms, proofs, lemmas, theorems have not changed since the time of Archimedes.
Surely with as good reason as had Archimedes to have the cylinder, cone and sphere engraved on his tombstone might our distinguished countrymen leave testamentary directions for the cubic eikosiheptagram to be engraved on theirs. Spirit of the Universe! wither are we drifting, and when, where, and how is all this to end?
You're alive!" Percy said to the others. "The giants said you were captured. What happened?" Leo shrugged. "Oh, just another brilliant plan by Leo Valdez. You'd be amazed what you can do with an Archimedes sphere, a girl who can sense stuff underground, and a weasel." "I was the weasel," Frank said glumly.
I could see why Archimedes got all excited. There was nothing finer than the feeling that came rushing through you when it clicked and you suddenly understood something that had puzzled you. It made you think it just might be possible to get a handle on this old world after all.