I entered an omnibus to go to some place or other. At that moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with non-Euclidean geometry.

If we ought not to fear mortal truth, still less should we dread scientific truth. In the first place it can not conflict with ethics? But if science is feared, it is above all because it can give no happiness? Man, then, can not be happy through science but today he can much less be happy without it.

But all of my efforts served only to make me better acquainted with the difficulty, which in itself was something.

It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better.

For a long time the objects that mathematicians dealt with were mostly ill-defined; one believed one knew them, but one represented them with the senses and imagination; but one had but a rough picture and not a precise idea on which reasoning could take hold.

There are no solved problems; there are only problems that are more or less solved.

[T]he different branches of Arithmetic - Ambition [G]eometry is not true, it is advantageous.