Life stands before me like an eternal spring with new and brilliant clothes.
[on his engagement]
Mathematics is the queen of the sciences and number theory is the queen of mathematics.
The total number of Dirichlet's publications is not large: jewels are not weighed on a grocery scale.
If others would but reflect on mathematical truths as deeply and continuously as I have, they would make my discoveries.
When a philosopher says something that is true then it is trivial. When he says something that is not trivial then it is false.
Sophie Germain proved to the world that even a woman can accomplish something in the most rigorous and abstract of sciences.
... durch planmässiges Tattonieren.
... through systematic, palpable experimentation.
[asked how he came upon his theorems]
I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.
[A reply to Olbers' attempt in 1816 to entice him to work on Fermat's Theorem.]
There are problems to whose solution I would attach an infinitely greater importance than to those of mathematics, for example touching ethics, or our relation to God, or concerning our destiny and our future; but their solution lies wholly beyond us and completely outside the province of science.
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
God does arithmetic.
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.
Letter to Bessel, 1830.
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.
I have had my results for a long time: but I do not yet know how I am to arrive at them.
Pauca sed matura [His motto:]
Few, but ripe.
[His second motto:]
Thou, nature, art my goddess; to thy laws my services are bound...
Theory attracts practice as the magnet attracts iron.
[attributed to him by H.B Lübsen]
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.
Finally, two days ago, I succeeded - not on account of my hard efforts,but by the grace of the Lord. Like a sudden flash of lightning, the riddle was solved. I am unable to say what was the conducting thread that connected what I previously knew with what made my success possible.
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.
I am coming more and more to the conviction that the necessity of our geometry cannot be demonstrated, at least neither by, nor for, the human intellect...geometry should be ranked, not with arithmetic, which is purely aprioristic, but with mechanics.
That this subject [imaginary numbers] has hitherto been surrounded by 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.
Sin2 φ is odious to me, even though Laplace made use of it; should it be feared that sin2 φ might become ambiguous, which would perhaps never occur, or at most very rarely when speaking of sin(φ2), well then, let us write (sin φ)2, but not sin2 φ, which by analogy should signify sin (sin φ)
There have been only three epoch-making mathematicians, Archimedes, Newton, and Eisenstein.
To what heights would science now be raised if Archimedes had made that discovery! [= the decimal system of numeration or its equivalent (with some base other than 10)]
Mathematical discoveries, like springtime violets in the woods, have their season which no human can hasten or retard.
Mathematics is concerned only with the enumeration and comparison of relations.