Thus, in a sense, mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs.

The teacher manages to get along still with the cumbersome algebraic analysis, in spite of its difficulties and imperfections, and avoids the smooth infinitesimal calculus, although the eighteenth century shyness toward it had long lost all point.

Projective geometry has opened up for us with the greatest facility new territories in our science, and has rightly been called the royal road to our particular field of knowledge.

The greatest mathematicians, as Archimedes, Newton, and Gauss, always united theory and applications in equal measure.

Mathematics in general is fundamentally the science of self-evident things.

Every one who understands the subject will agree that even the basis on which the scientific explanation of nature rests is intelligible only to those who have learned at least the elements of the differential and integral calculus, as well as analytical geometry.

Among mathematicians in general, three main categories may be distinguished and perhaps the names logicians, formalists, and intuitionists may serve to characterize them.