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.

It is with logic that one proves; it is with intuition that one invents.

Geometry is not true, it is advantageous.

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.

But for harmony beautiful to contemplate, science would not be worth following.

Ideas rose in clouds; I felt them collide until pairs interlocked, so to speak, making a stable combination.

Every good mathematician should also be a good chess player and vice versa.