It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment.

Life stands before me like an eternal spring with new and brilliant clothes.

It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again; the never-satisfied man is so strange if he has completed a structure, then it is not in order to dwell in it peacefully,but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms for others.

You know that I write slowly. This is chiefly because I am never satisfied until I have said as much as possible in a few words, and writing briefly takes far more time than writing at length.

Mathematics is the queen of sciences and number theory is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank.

There have been only three epoch-making mathematicians, Archimedes, Newton, and Eisenstein.

Sin^{2} φ is odious to me, even though Laplace made use of it; should it be feared that sin^{2} φ might become ambiguous, which would perhaps never occur, or at most very rarely when speaking of sin(φ^{2}), well then, let us write (sin φ)^{2}, but not sin^{2} φ, which by analogy should signify sin (sin φ)

You have no idea, how much poetry there is in the calculation of a table of logarithms!

If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries.

We must admit with humility that, while number is purely a product of our minds, space has a reality outside our minds, so that we cannot completely prescribe its properties a priori.

Mathematicians stand on each other's shoulders.

Theory attracts practice as the magnet attracts iron.

Arc, amplitude, and curvature sustain a similar relation to each other as time, motion, and velocity, or as volume, mass, and density.

Mathematics is the queen of science, and arithmetic the queen of mathematics.

Does the pursuit of truth give you as much pleasure as before? Surely it is not the knowing but the learning, not the possessing but the acquiring, not the being-there but the getting there that afford the greatest satisfaction. If I have exhausted something, I leave it in order to go again into the dark. Thus is that insatiable man so strange: when he has completed a structure it is not in order to dwell in it comfortably, but to start another.

I have had my results for a long time: but I do not yet know how I am to arrive at them.

I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half proof = 0, and it is demanded for proof that every doubt becomes impossible.

[On Sophie Germain] When a person of the sex which, according to our customs and prejudices, must encounter infinitely more difficulties than men... succeeds nevertheless in surmounting these obstacles and penetrating the most obscure parts of [number theory], then without doubt she must have the noblest courage, quite extraordinary talents and superior genius.

Ask her to wait a moment - I am almost done.

The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic.

The enchanting charms of this sublime science reveal only to those who have the courage to go deeply into it.

I protest against the use of infinite magnitude ..., which is never permissible in mathematics.

When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again.

I am giving this winter two courses of lectures to three students, of which one is only moderately prepared, the other less than moderately, and the third lacks both preparation and ability. Such are the onera of a mathematical profession.

That this subject [of imaginary magnitudes] has hitherto been considered from the wrong point of view and surrounded by a mysterious obscurity, is to be attributed largely to an ill-adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question.

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