Euclid writes the Elements

Around 300 BCE a teacher named Euclid, working in Alexandria, the new Greek capital of Egypt, put together the Elements: a course of mathematics in thirteen 'books', each of them once a roll of papyrus. It starts from very little. Book I opens with 23 definitions, such as 'a point is that which has no part' and 'a line is a length without breadth', five postulates, among them that a straight line can be drawn between any two points and a circle drawn with any centre and radius, and five 'common notions', such as 'things equal to the same thing are equal to each other'. From these it proves, one step at a time, more than 460 propositions, each resting only on what has already been shown. Books I to VI are plane geometry, and Book I ends with the theorem we name after Pythagoras. Books VII to IX are about whole numbers: they contain the method for finding the greatest common divisor that is still called Euclid's algorithm, and the proof that the prime numbers never run out. Book X sorts out irrational lengths, and Books XI to XIII move into three dimensions and end by building the five regular solids and proving that there can be no others.

Of Euclid himself almost nothing is known. Nearly all of it comes from Proclus, a philosopher who wrote a commentary on Book I some 750 years later: Euclid lived in the time of the first Ptolemy, the Macedonian general who ruled Egypt from 323 BCE; he was younger than Plato's pupils and older than Archimedes; and when the king asked whether there was a shorter way into geometry than the Elements, he answered that there is no royal road. Nor was the book the first of its kind. Hippocrates of Chios had written an 'Elements' around 440 BCE, and Euclid took his theory of proportion (Book V) from Eudoxus and much of Books X and XIII from Theaetetus, mathematicians of the century before him. What was his own was the design: a single chain of proof from a few stated starting points. It drove the older textbooks out of use, and they are lost. Medieval and Renaissance scholars confused him with Euclid of Megara, a philosopher who lived a century earlier, so the first English translation still calls him 'Euclide of Megara'.

No copy from Euclid's lifetime survives. The earliest glimpse is a handful of potsherds from Elephantine, more than 800 kilometres up the Nile from Alexandria, on which someone in the third century BCE worked through problems from Book XIII; then come papyrus scraps from Roman Egypt. In the fourth century CE Theon of Alexandria, the father of Hypatia, re-edited the text, and his version pushed out all the others: only in 1808 was a manuscript in the Vatican recognised as older than Theon. The oldest complete copy with a date was finished in September 888, probably in Constantinople. Around 800 al-Hajjaj translated the Elements into Arabic for the caliphs in Baghdad, and the version of Ishaq ibn Hunayn, revised by Thabit ibn Qurra, became the standard text of the Islamic world. From Arabic, Adelard of Bath made the first Latin translation that survives, around 1120, and Campanus of Novara's Latin version was the first to be printed, by Erhard Ratdolt in Venice on 25 May 1482. More than a thousand editions have followed, among them Henry Billingsley's English one of 1570, Christopher Clavius's Latin textbook of 1574, which Matteo Ricci and Xu Guangqi used for the first Chinese translation in 1607, and Oliver Byrne's of 1847, which replaced letters with coloured shapes.

For two thousand years the Elements was how Europe, and much of the Islamic world, learned what a proof is. Britannica writes that the ideas of Kepler, Fermat, Descartes and Newton were rooted in it, and it is sometimes said that only the Bible has been more translated, printed and studied in the Western world. Its one weak-looking point turned out to be its deepest. The fifth postulate, which says when two straight lines will meet, seemed less obvious than the others, and for centuries mathematicians from Ptolemy and Proclus to Saccheri tried to prove it from the rest. In the early nineteenth century Gauss, Nikolai Lobachevsky and János Bolyai saw that it cannot be proved: a geometry in which it is false is just as consistent. Such non-Euclidean geometries later proved essential to Einstein's general theory of relativity. Yet the method of the Elements, to say exactly what you assume and then prove everything else, is still how mathematics is done.