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.

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

It is through science that we prove, but through intuition that we discover.

If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living

We also know how cruel the truth often is, and we wonder whether delusion is not more consoling.

Astronomy is useful because it raises us above ourselves; it is useful because it is grand; .... It shows us how small is man's body, how great his mind, since his intelligence can embrace the whole of this dazzling immensity, where his body is only an obscure point, and enjoy its silent harmony.

A first fact should surprise us, or rather would surprise us if we were not used to it. How does it happen there are people who do not understand mathematics? If mathematics invokes only the rules of logic, such as are accepted by all normal minds...how does it come about that so many persons are here refractory?