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

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.

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

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

Response, when asked how he came upon his theorems.

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

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

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.

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.

It may be true that people who are merely mathematicians have certain specific shortcomings; however that is not the fault of mathematics, but is true of every exclusive occupation. Likewise a mere linguist, a mere jurist, a mere soldier, a mere merchant, and so forth. One could add such idle chatter that when a certain exclusive occupation is often connected with certain specific shortcomings, it is on the other hand always free of certain other shortcomings.

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.

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

Mathematicians stand on each other's shoulders.

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

Mathematics is the queen of the sciences and arithmetic [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 first rank.

Theory attracts practice as the magnet attracts iron.

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.

Sophie Germain proved to the world that even a woman can accomplish something in the most rigorous and abstract of sciences and for that reason would well have deserved an honorary degree.

I have a true aversion to teaching. The perennial business of a professor of mathematics is only to teach the ABC of his science; most of the few pupils who go a step further, and usually to keep the metaphor, remain in the process of gathering information, become only Halbwisser [one who has superficial knowledge of the subject], for the rarer talents do not want to have themselves educated by lecture courses, but train themselves. And with this thankless work the professor loses his precious time.

Complete knowledge of the nature of an analytic function must also include insight into its behavior for imaginary values of the arguments. Often the latter is indispensable even for a proper appreciation of the behavior of the function for real arguments. It is therefore essential that the original determination of the function concept be broadened to a domain of magnitudes which includes both the real and the imaginary quantities, on an equal footing, under the single designation complex numbers.

The higher arithmetic presents us with an inexhaustible store of interesting truths - of truths, too, which are not isolated, but stand in a close internal connection, and between which, as our knowledge increases, we are continually discovering new and sometimes wholly unexpected ties.

I believe you are more believing in the Bible than I. I am not, and, you are much happier than I.

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.

A great part of its [higher arithmetic] theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity on them, are often easily discovered by induction, and yet are of so profound a character that we cannot find the demonstrations till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simple methods may long remain concealed.

Mathematical discoveries, like springtime violets in the woods, have their season which no human can hasten or retard.

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