A Biography Made of Two Anecdotes
Euclid is thought to have worked at Alexandria around 300 BCE, under the first of the Ptolemies, at the institution that would grow into the Museum and its library. That is close to the sum of reliable knowledge. The main source is Proclus, writing some seven centuries later, and even he hedges. Two stories cling to the name: that Euclid told King Ptolemy there is no royal road to geometry, and that he ordered a coin given to a student who asked what profit there was in learning proofs. Both are almost certainly later inventions.
The vacuum has invited theories. Some scholars have proposed that Euclid was less an individual than a name attached to the collective output of a school of Alexandrian mathematicians, in the way that Bourbaki functioned in the twentieth century. The suggestion is a minority position and hard to test, but it makes a real point. The Elements is visibly a compilation, drawing on Eudoxus for proportion, Theaetetus for irrationals and solids, and the earlier Pythagorean tradition for much of the number theory.
The Machine of Proof
What the Elements contributed was not new theorems so much as architecture. Thirteen books begin from definitions, five postulates and a set of common notions, then build every subsequent result by deduction from what has already been established. Plane geometry comes first, then a theory of proportion that handles incommensurable magnitudes without a concept of real numbers, then number theory including the algorithm still named for Euclid and a proof that the primes are infinite, and finally solid geometry ending with the five regular polyhedra.
The fifth postulate, about parallel lines, is longer and clumsier than the rest, and readers noticed almost immediately. For two millennia mathematicians in Greek, Arabic and Latin tried to derive it from the other four. Every attempt failed, and in the nineteenth century Bolyai, Lobachevsky and Riemann showed why: replace it and you obtain consistent geometries of curved space. Euclid's apparent flaw turned out to mark the boundary of a whole continent of mathematics, and Einstein's account of gravitation would eventually be written in the geometry that lay beyond it.
How the Text Travelled
No manuscript from Euclid's own hand exists. The text we read descends through late antique recensions, notably that of Theon of Alexandria in the fourth century CE, and modern editions rest heavily on a Vatican manuscript that appears to preserve a version earlier than Theon's. Arabic translators in ninth-century Baghdad produced their own tradition, and scholars from al-Nayrizi to Omar Khayyam and Nasir al-Din al-Tusi commented on it and probed the parallel postulate long before Latin Europe rejoined the conversation.
Europe received the Elements largely through those Arabic intermediaries, beginning with the twelfth-century translations associated with Adelard of Bath and Gerard of Cremona. The first printed edition appeared at Venice in 1482, and by the nineteenth century the book had been through more editions than almost any work except the Bible. Its influence spread beyond mathematics. Spinoza wrote ethics in geometrical order, Newton cast the Principia in Euclidean form, and the American Declaration of Independence borrowed the language of self-evident truths.
