Euclid

A point is that which has no part. A line is breadthless length. These definitions appear modest until they begin to govern an entire world of proof. Geometry teaches a severe politeness: grant the premises, follow the construction, and accept only what necessity compels.

Things equal to the same thing are equal to one another. Add equals to equals and the wholes are equal. Such common notions seem almost too obvious to mention, which is precisely why they are powerful. Proof is built from statements no one notices until the mind needs a bridge.

There are more prime numbers than any given finite number. Suppose you have a list of primes, multiply them all together and add one. That result is either prime itself or divisible by a prime not on your list. Either way, your list is incomplete. No list can ever be complete. The primes are infinite, and I can prove it without counting a single one.

If a straight line falling on two straight lines makes interior angles on the same side less than two right angles, the two straight lines will meet if produced indefinitely on that side. For two thousand years, geometers suspected this postulate was too complex to be truly foundational. They spent centuries trying to prove it from the other four....

When Ptolemy I asked if there was a shorter way to geometry than through the Elements, I told him there is no royal road to geometry. The story may be apocryphal. The sentiment is not. Mathematics has no rank system. The King's intuition is not worth more than a shepherd's proof. A valid demonstration is valid whoever gives it. This is the most democratic...

The diagonal of a square is incommensurable with the side — there is no common unit of measurement that divides both exactly. This was a genuine scandal in Greek mathematics. It meant the world of number and the world of geometry were not perfectly aligned. The Pythagoreans reportedly tried to suppress the discovery. The proof is in my Book X, across 115...

Magnitudes are in the same ratio if, when equimultiples are taken, the greater multiple of either exceeds, equals, or falls short of the other in the same way. This definition of proportion — due largely to Eudoxus — handles irrational ratios without using them directly. It is perhaps the most sophisticated definition in all of ancient mathematics, and it...

In a right-angled triangle, the square on the side subtending the right angle equals the sum of the squares on the sides containing the right angle. My proof uses the construction of squares on each side and shows they can be cut and rearranged. Pythagoras may have known this result. I proved it in a form that holds for every right triangle that will ever...

The Elements is not a collection of new discoveries. Most of what is in it was known before me — Pythagoras, Eudoxus, Theaetetus had established many of the results. My contribution was the architecture: the decision to begin from five postulates and derive everything else by logical deduction alone, with no appeal to intuition or observation beyond the...