All great progress takes place when two sciences come together, and when their resemblance proclaims itself, despite the apparent disparity of their substance.

To invent is to discern, to choose.

Mathematical discoveries, small or great are never born of spontaneous generation They always presuppose a soil seeded with preliminary knowledge and well prepared by labour, both conscious and subconscious.

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

In one word, to draw the rule from experience, one must generalize; this is a necessity that imposes itself on the most circumspect observer.

How is error possible in mathematics?