...aptly illustrated by Aristotle. In his Analytics he observes that, if we could perceive that the sum of the angles of a triangle is equal to two right angles, we should be forced to seek proof, or could not be said to have knowledge. Nonsense! Is a proof needed...

...tiles used in paving floors may have afforded an experimental demonstration of the fact that the sum of the angles of a triangle is equal to two right angles. We know from Eudemus, who was a pupil of Aristotle, that the first geometricians proved the general property...

...paving doors may have afforded an experimental demonstration of the latter result, 'namely, that the sum of the angles of a triangle is equal to two right angles. We know from Eudemus that the first geometers proved the general property separately for three species...

...possible, and contingent truths, whose opposite is possible.2 The truths of geometry — for example, the theorem that the sum of the angles of a triangle is equal to two right angles or the theorem that alternate internal angles are equal — are examples of necessary truths...

...this is the sum for every triangle. The existence of the one triangle that would complete the proof of the theorem that the sum of the angles of a triangle is equal to two right angles and hence the proof of the equivalent theorem, viz. the "parallel-postulate," he could not establish....

...postulates), the definitions and the judgments of direct perception. When we demonstrate, for instance, the theorem that the sum of the angles of a triangle is equal to two right angles, we conclude that this proposition must be true, because the denial of it would contradict the axiom that...

...apply to it the most rigorous tests. For example, take the simple proposition in geometry, the sum of the angles of a triangle is equal to two right angles. We know that this is true.1 It is thoroughly tested knowledge. We know that if there is at any time a...

...reasoning consists. Consider now as another example the reasoning used in demonstrating that the sum of the angles of a triangle is equal to two right angles. We could measure them one after the other separately with the help of a goniometer and then add them up....

...This certainty is what many have sought and found in mathematics. The Greeks had the idea that this theorem, that the sum of the angles of a triangle is equal to two right angles, is true, period. They felt that it is an absolute, objective truth — a property of the natural...

...alternative method would be to adopt as an experimental fact, which it may seem to many to be, that the sum of the angles of a triangle is equal to two right angles. We do not wish to deny that an account of mensuration may be constructed on these lines ; but we do not...