nearly everything useful that was done in mathematics before the seventeenth century has suffered one of two fates: either it has been so greatly simplified that it is now part of every regular school course, or it was long since absorbed as a detail in work of greater generality.

The most remarkable thing about all of these profound utterances is that human beings no stupider than ourselves once thought they made sense.

Bertrand Russell said about a quarter of a century ago: “Mathematics, rightly viewed, possesses not only truth but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.”

he may repeat Poincaré’s “Mathematics for mathematics’ sake. People have been shocked by this formula and yet it is as good as life for life’s sake, if life is but misery.”

According to Leibniz, of all mathematics up to the time of Newton, the more important half is due to Newton.

the Principia is still rated as the most massive addition to scientific thought ever made by one man.

More important than the technical algebra of these ancient Babylonians is their recognition—as shown by their work—of the necessity for proof in mathematics. Until recently it had been supposed that the Greeks were the first to recognize that proof is demanded for mathematical propositions. This was one of the most important steps ever taken by human beings.

Before Pythagoras it had not been clearly realized that proof must proceed from assumptions. Pythagoras, according to persistent tradition, was the first European to insist that the axioms, the postulates, be set down first in developing geometry and that the entire development thereafter shall proceed by applications of close deductive reasoning to the axioms.

Pythagoras then imported proof into mathematics. This is his greatest achievement.

We can easily construct the diagonal geometrically, but we cannot measure it in any finite number of steps.

it cannot be denied that he did irritate many infinitely better mathematicians than himself into creating some real mathematics.

All constructions effected with other implements were dubbed “mechanical” and, as such, for some mystical reason known only to Plato and his geometrizing God, were considered shockingly vulgar and were rigidly taboo in respectable geometry.

If anyone who has not studied the calculus imagines Archimedes’ problem an easy one he may time himself doing it.

According to one account the soldier had stepped on the diagram, angering Archimedes to exclaim sharply, “Don’t disturb my circles!” Another states that Archimedes refused to obey the soldier’s order that he accompany him to Marcellus until he had worked out his problem.

Descartes’ distinctive talent had made itself evident long before he left school. As early as the age of fourteen, lying meditating in bed, he had begun to suspect that the “humanities” he was mastering were comparatively barren of human significance and certainly not the sort of learning to enable human beings to control their environment and direct their own destiny.

This phase did not last long. Tiring of his bawdy companions, Descartes gave them the slip and took up his quarters in plain, comfortable lodgings in what is now the suburb of Saint-Germain where, for two years, he buried himself in incessant mathematical investigation. His gay deeds at last found him out, however, and his hare-brained friends descended whooping upon him. The studious young man looked up, recognized his friends, and saw that they were one and all intolerable bores.

Descartes settled down to three years of meditation. In spite of his lofty thoughts he was no gray-bearded savant in a dirty smock, but a dapper, well dressed man of the world, clad in fashionable taffeta and sporting a sword as befitted his gentlemanly rank. To put the finishing touch to his elegance he crowned himself with a sweeping, broad-brimmed, ostrich-plumed hat.

Possibly his reason for never marrying may have been, as he informed one expectant lady, that he preferred truth to beauty; but it seems more probable that he was too shrewd to mortgage his tranquillity and repose to some fat, rich, Dutch widow. Descartes was only moderately well off, but he knew when he had enough. For this he has been called cold and selfish. It seems juster to say that he knew where he was going and that he realized the importance of his goal.

as he dryly remarks, he soon discovered that the number of those who understand man is negligible in comparison with the number of those who think they understand geometry.

He was now (1628) thirty two, and only his miraculous luck had preserved his body from destruction and his mind from oblivion. A stray bullet at La Rochelle might easily have deprived Descartes of all claim to remembrance

To say that fear alone stopped Descartes from publishing Le Monde is to miss the more important part of the truth. He was not only afraid—as any sane man might well have been; he was deeply hurt. He was as convinced of the truth of the Copernican system as he was of his own existence. But he was also convinced of the infallibility of the Pope. Here now was the Pope making a silly ass of himself by contradicting Copernicus.

As a sop to his subconscious self-respect he decided that Le Monde should be published after his death. By that time perhaps the Pope too would be dead and the contradiction would have resolved itself.

the bones of Descartes were returned to France (all except those of the right hand, which were retained by the French Treasurer-General as a souvenir for his skill in engineering the transaction)

the bones of Descartes were returned to France (all except those of the right hand, which were retained by the French Treasurer-General as a souvenir for his skill in engineering the transaction)

Commenting on the return of Descartes’ remains to his native France, Jacobi remarks that “It is often more convenient to possess the ashes of great men than to possess the men themselves during their lifetime.”