We should chiefly depend not upon that department of the soul which is most superficial and fallible (our reason), but upon that department that is deep and sure, which is instinct.

It is a common observation that a science first begins to be exact when it is quantitatively treated. What are called the exact sciences are no others than the mathematical ones.

Kepler's discovery would not have been possible without the doctrine of conics. Now contemporaries of Kepler-such penetrating minds as Descartes and Pascal-were abandoning the study of geometry ... because they said it was so UTTERLY USELESS. There was the future of the human race almost trembling in the balance; for had not the geometry of conic sections already been worked out in large measure, and had their opinion that only sciences apparently useful ought to be pursued, the nineteenth century would have had none of those characters which distinguish it from the ancien régime.

And what, then, is belief? It is the demi-cadence which closes a musical phrase in the symphony of our intellectual life.

We shall do better to abandon the whole attempt to learn the truthunless we can trust to the human mind's having such a powerof guessing right that before very many hypotheses shall have been tried, intelligent guessing may be expected to lead us to one which will support all tests, leaving the vast majority of possible hypotheses unexamined.

Among the minor, yet striking characteristics of mathematics, may be mentioned the fleshless and skeletal build of its propositions; the peculiar difficulty, complication, and stress of its reasonings; the perfect exactitude of its results; their broad universality; their practical infallibility.

We cannot begin with complete doubt.