Euclid: Elements, Books I–IV

About 2,300 years ago, in the city of Alexandria in Egypt, a teacher named Euclid wrote a book about shapes. He started with a few things everyone agrees on, like you can draw a straight line from any point to any other point. Then, step by step, he proved hundreds of new facts. Each step used only the facts before it.

His book, the Elements, was used to teach geometry for more than 2,000 years. Some people say it has been copied and printed more than any other book except the Bible. Nobody can count that for sure, but it was the main math book in schools for a very long time.

Euclid's most famous proof is near the end of his first book. If a triangle has a square corner, the squares built on its two short sides add up to the square on its long side. A 3, 4, 5 triangle works: 9 + 16 = 25. Builders have used that trick to make square corners for a very long time.

Who, when and why

Euclid taught in Alexandria around 300 BC, in the years after Alexander the Great, when the new Greek kings of Egypt were filling the city with scholars. We know almost nothing else about his life. Later writers tell a few stories, but they wrote centuries afterward.

The problem Euclid faced was not a lack of geometry. Greek mathematicians had found a great many results over the previous 250 years, and Egyptian and Babylonian scribes had used practical rules for far longer. The problem was order. Which facts depend on which? What do you have to assume? Euclid gathered the known results into one chain, starting from a small set of definitions, postulates and common notions, and proved everything else from them.

Postulates 1 to 3 describe the two tools allowed: a straightedge (to draw and extend lines) and a compass (to draw circles). The fifth, the parallel postulate, is longer and less obvious than the rest. Euclid held it back until Proposition 29, as if he used it only when he had to.

The first proposition

Notice the pattern every proposition follows: a statement of what is to be done or proved, a figure with lettered points, the construction, the proof with a reason for each step, and the closing words: being what it was required to do (for a construction) or what it was required to prove (for a theorem). Later readers shortened the Latin form of the second to Q.E.D.

What Books I to IV contain

The theorem at the end of Book I

Euclid means real squares with real areas, not numbers. His proof draws a line from the right angle straight down through the big square, cutting it into two rectangles, and shows that each rectangle has the same area as one of the small squares. The visual below shows the same equality by sliding the squares, using two facts Euclid proves earlier: parallelograms on the same base and between the same parallels are equal in area (I.35), and a triangle is half of a parallelogram with the same base and height (I.41).

What later mathematics kept, and what it changed

• Kept: almost every result in Books I to IV, the idea of proof from stated assumptions, and the compass-and-straightedge constructions still taught in school.
• Repaired: Euclid sometimes relied on a figure. In I.1 he never proves that the two circles actually cross. In 1899 David Hilbert rewrote the foundations with a longer, complete list of axioms.
• Changed: in the 1820s and 1830s Nikolai Lobachevsky and János Bolyai showed that a consistent geometry can be built with the parallel postulate replaced (Gauss had reached the same idea privately). Einstein's general relativity (1915) describes space-time with such curved geometry.

Why the fifth postulate mattered

For two thousand years mathematicians tried to prove the parallel postulate from the other four, Ptolemy and Proclus among them, and later Islamic mathematicians such as Omar Khayyam and Nasir al-Din al-Tusi, and the Italian Girolamo Saccheri (1733). Each attempt either failed or quietly assumed something equal to it. The eventual answer was surprising: it can't be proved, because geometries where it is false are just as consistent.

This changed how mathematicians think about axioms. They are no longer "self-evident truths" about space but chosen starting points. Which geometry describes the real world is a question for measurement, not for logic alone.

How the Elements reached us

No copy from Euclid's time survives. Our text comes from medieval Greek manuscripts (the oldest dated one, at Oxford, was copied in AD 888), from Arabic translations made in Baghdad around AD 800, and from Latin translations made from the Arabic and the Greek in the 1100s. Small papyrus scraps from Roman Egypt show that the text was much the same in about AD 100. The first printed edition came out in Venice in 1482. The standard Greek text today is J. L. Heiberg's (1883–1888), which Heath translated.

Readers have argued about the story that King Ptolemy asked Euclid for an easier way and was told there is "no royal road" to geometry. It comes from Proclus, writing about 750 years later, and a nearly identical story is told of another mathematician and Alexander the Great. Treat it as a legend about the subject, not a fact about the man.

Big questions

• Is a proof that relies on a picture really a proof?
• Why does a list of five postulates produce so much? Could a different list produce a different world?
• Many people, from Abraham Lincoln to Bertrand Russell, said studying Euclid taught them what it means to prove something. What does a proof give you that a measurement can't?

Later books on this ladder answer Euclid directly. Spinoza wrote his Ethics in Euclid's form, with definitions, axioms and propositions. Descartes joined geometry to algebra. Newton set out the Principia as propositions with proofs, in Euclid's manner.