Euclid

Euclid was a Greek mathematician who worked in Alexandria around 300 BCE. His Elements, a thirteen-book account of geometry and number theory built up step by step from a handful of definitions and assumptions, became the most influential mathematics textbook ever written. For more than two thousand years it was the model of what a rigorous argument should look like, and the geometry of flat space is still called Euclidean after him.

Almost nothing certain is known about Euclid the man. He left no letters or prefaces, and no writer of his own time describes him. The main account comes from Proclus, a philosopher writing in the fifth century CE, more than seven hundred years later. Proclus placed him after the circle of Plato and before Archimedes, in the reign of Ptolemy I, the Macedonian general who took Egypt after the death of Alexander the Great and founded the Ptolemaic Kingdom. That gives a working life around 300 BCE. Dates of birth and death, usually put at about 325 and 265 BCE, are estimates, and his birthplace is unknown.

He is generally believed to have taught in Alexandria, the new Greek capital on the Nile delta, where the early Ptolemies were gathering scholars at the Museum. The later writer Pappus says that the mathematician Apollonius studied there with Euclid's pupils, which suggests that Euclid founded a school. Some scholars have wondered whether "Euclid" might have been the name of a team, or even a collective pen name, but most accept that a single author wrote the works under his name.

In the Middle Ages he was often confused with Euclid of Megara, a pupil of Socrates who lived a century earlier, and the mistake was still being printed in the sixteenth century. Medieval Arabic biographies gave him a father and a birthplace in Tyre, details historians treat as invented.

Two stories about him survive, both late. In one, Ptolemy asks whether there is a shorter way to learn geometry than the Elements, and Euclid replies that there is no royal road to geometry. A nearly identical story is told of the mathematician Menaechmus and Alexander, so it may be a stock tale attached to more than one teacher. In the other, told by Stobaeus, a student asks what he will gain by learning geometry, and Euclid tells a slave to give him a coin, since he must profit from what he learns. Neither can be checked, and both are best read as illustrations of how later Greeks imagined him.

The Elements is not a collection of new discoveries. Much of its content came from earlier Greek mathematicians, including Pythagoras and his followers, Eudoxus, and Theaetetus. Euclid's achievement was to arrange this knowledge into a single logical structure in which each result is proved from what came before.

Book 1 opens with definitions (a point is that which has no part), five common notions (things equal to the same thing are equal to each other) and five postulates. The first four are simple: a straight line can be drawn between any two points, a line can be extended, a circle can be drawn with any centre and radius, and all right angles are equal. The fifth, the parallel postulate, is longer and less obvious. It says that if a line crosses two others and makes the inner angles on one side add up to less than two right angles, those two lines will meet on that side if extended far enough. From these starting points Euclid proves, among much else, that the angles of a triangle add up to two right angles and, at the end of Book 1, the theorem of Pythagoras.

Books 1 to 6 treat plane geometry, including circles, regular polygons, and a general theory of ratio and proportion, probably taken from Eudoxus, that could handle lengths which cannot be measured by whole numbers. Books 7 to 9 deal with whole numbers. Here Euclid gives the method now called the Euclidean algorithm, which finds the greatest common divisor of two numbers by repeatedly dividing and taking the remainder, still used in computing today. Book 9 proves that there are infinitely many prime numbers: take any finite list of primes, multiply them together and add one, and the result must have a prime factor that is not on the list. Book 10, the longest, classifies irrational lengths, and Books 11 to 13 move into three dimensions, ending with the proof that there are exactly five regular solids.

Modern mathematicians have noted that Euclid sometimes relied on assumptions he did not state, for example about points lying between others. The structure nonetheless set the pattern of axioms, definitions and proofs that mathematics still follows.

Several other works survive under his name. The Data deals with what can be deduced when certain parts of a figure are given. The Optics is the earliest surviving Greek treatise on perspective, treating vision as straight lines from the eye. The Phaenomena applies spherical geometry to astronomy, and On Divisions, preserved partly in Arabic, concerns cutting figures into parts with given ratios. Works on conic sections, porisms, surface loci and fallacies are known only by title or from later descriptions. A treatise on mirrors attributed to him is probably by someone else.

The Elements was copied, edited and commented on throughout antiquity. A papyrus fragment of around 100 CE survives from Oxyrhynchus in Egypt, and the edition by Theon of Alexandria in the fourth century CE became the basis of most later manuscripts. In the Islamic Golden Age, the work was translated into Arabic in Baghdad, first by al-Hajjaj ibn Yusuf around 800 and later in a version revised by Thabit ibn Qurra, and Arabic scholars wrote extensive commentaries, especially on the parallel postulate. Latin Europe recovered it mainly through translations from Arabic, notably by Adelard of Bath in the twelfth century and Campanus of Novara in the thirteenth. The first printed edition appeared in Venice in 1482, and more than a thousand editions have followed.

For centuries educated people learned to reason from Euclid. Isaac Newton set out the proofs of his Principia in Euclidean geometric form, and Abraham Lincoln said he studied the first six books to learn what it meant to demonstrate something.

The parallel postulate had the most surprising afterlife. Mathematicians from antiquity onward tried and failed to prove it from the other four. In the early nineteenth century Gauss, Lobachevsky and Bolyai showed that it could not be proved, because consistent geometries exist in which it is false. These non-Euclidean geometries later became the mathematical language of Albert Einstein's general theory of relativity, in which space itself is curved. At the end of the nineteenth century David Hilbert rewrote Euclid's system with a complete set of axioms. Euclidean geometry remains the geometry of everyday space taught in schools.