Euclid

• 1. Introduction

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  Euclid is called the father of geometry, yet almost nothing about the man himself can be confirmed. We do not know where he was born. We do not know the year he died. We barely know the century in which he lived, beyond a rough placement around 300 BC. The earliest writers who tried to describe him, the scholars Proclus and Pappus of Alexandria, worked many centuries after his death. Later traditions only deepened the fog. Medieval Islamic writers invented a detailed life story for him. Byzantine and early Renaissance scholars confused him with an entirely different man, a philosopher named Euclid of Megara. How can someone leave behind a treatise that shaped mathematics for over two thousand years while leaving so little trace of who he was? What did he actually write, beyond the work that made his name a synonym for geometry itself? And why did so many people across so many cultures feel the need to fill in the blanks?

• 2. A Life In Shadow

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  Very little is known of Euclid's life, and the few firm anchors come from indirect reasoning rather than records. Proclus placed Euclid shortly after several of Plato's followers and before the mathematician Archimedes, who lived around 287 BC. He also placed Euclid during the rule of Ptolemy I. Some scholars estimate his birth around 330 or 325 BC, while others refuse to guess at all. His birthplace is unknown, though he is presumed to have been of Greek descent. It has been speculated that he died around 270 BC.

  Alexandria sits at the center of what historians can reconstruct. The city was founded by Alexander the Great in 331 BC, and the rule of Ptolemy I from 306 BC onward gave it a stability that was rare amid the wars over Alexander's divided empire. Ptolemy commissioned the massive Musaeum, a leading center of education, and Euclid is speculated to have been among its first scholars. In his Collection, Pappus notes that Apollonius studied with Euclid's students in Alexandria, which suggests Euclid worked and founded a mathematical tradition there.

  The name itself carries meaning. The English 'Euclid' is the anglicized form of the Ancient Greek Eukleides, built from 'eu', meaning well, and 'kles', meaning fame, with a suffix meaning son of. Together the parts read as renowned or glorious. By metonymy in English, the word Euclid can stand for his most famous work, or for a copy of it, and is sometimes treated as synonymous with geometry itself. The historian Thomas Heath argued that Euclid was probably educated by Plato's disciples at the Academy in Athens, since most capable geometers of the era lived there.

• 3. The Invented Biographies

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  Islamic biographical sources offer some of the most vivid accounts of Euclid, and almost none of them can be verified. One report preserved by Ali Ibn Yusuf al-Qifti calls Euclid "the engineer, the carpenter of Tyre, son of Naucrates son of Berenice, the one who manifested geometry and excelled in it, known as the lord of Geometry." The same account describes him as Greek by origin, Syrian in residence, Tyrian by city, and a carpenter by trade, supposedly domiciled at Damascus. Heath contended this fictionalization was meant to strengthen the connection between a revered mathematician and the Arab world.

  Confusion of identity haunted Euclid for centuries. He is often called 'Euclid of Alexandria' precisely to separate him from Euclid of Megara, an earlier philosopher and pupil of Socrates who appears in dialogues of Plato. The Roman compiler Valerius Maximus, writing in the 1st century AD, mistakenly put Euclid's name in place of Eudoxus as the mathematician Plato sent people to when they asked how to double the cube. That early misplacement helped Euclid the mathematician become tangled with Euclid of Megara in now-lost Byzantine sources. The Byzantine scholar Theodore Metochites, around 1300, explicitly merged the two men.

  The error spread through print. Erhard Ratdolt's 1482 first printed edition of Campanus of Novara's Latin translation of the Elements carried the conflation forward. A 1505 translation appended most surviving biographical fragments about either Euclid to its preface, and later publications passed the muddle along. One offshoot even claimed Euclid was born in Gela, Sicily, a detail borrowed from Euclid of Megara. The Renaissance scholar Peter Ramus eventually reevaluated the claim and proved it false through problems in chronology and contradictions in the early sources.

• 4. No Royal Road

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  Proclus preserved the most famous anecdote about Euclid, a scene that paints him as a teacher with little patience for shortcuts. According to the story, Ptolemy asked whether there was a quicker path to learning geometry than reading the Elements. Euclid replied that "there is no royal road to geometry." The other surviving stories picture him as a kindly and gentle old man, though all of them are of uncertain historicity.

  The royal road tale may not even be his. A very similar exchange between Menaechmus and Alexander the Great is recorded by Stobaeus. Both accounts were written in the 5th century AD, neither names its source, and neither appears in ancient Greek literature. That parallel casts doubt on whether the words were ever Euclid's at all.

  Dating Euclid's activity is harder than it first appears, because contemporary references are simply missing. The earliest original mention of him comes from Apollonius, in the prefatory letter to the Conics in the early 2nd century BC. Apollonius wrote that Euclid "had not made the synthesis of the locus on three and four lines but only an accidental fragment of it, and even that was not felicitously done." The oldest physical scraps of material from the Elements, dating from roughly 100 AD, were found on papyrus fragments in an ancient rubbish heap at Oxyrhynchus in Roman Egypt. The firmest direct citations only arrive in the 2nd century AD, by Galen and Alexander of Aphrodisias, by which point the work was a standard school text. Some medieval scholars went so far as to argue Euclid never existed, suggesting his name arose from a corruption of Greek mathematical terms.

• 5. Anatomy Of The Elements

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  The Elements runs to thirteen books, and Euclid's achievement was less invention than architecture. The classicist Markus Asper concluded that Euclid assembled accepted mathematical knowledge into a cogent order and added new proofs to fill the gaps. The historian Serafina Cuomo called it a reservoir of results. Much of its content traces back to earlier mathematicians, including Eudoxus, Hippocrates of Chios, Thales and Theaetetus, while other theorems appear in Plato and Aristotle. Even so, Michalis Sialaros argues that the tight structure reveals authorial control well beyond that of a mere editor.

  The work is traditionally divided into three topics: plane geometry in books 1 to 6, basic number theory in books 7 to 10, and solid geometry in books 11 to 13. Book 1 lays the foundation with 20 definitions, then 10 assumptions split into five postulates and five common notions. Four of the postulates are straightforward. The fifth, known as the parallel postulate, became famous in its own right. Book 1 also holds 48 propositions, and the last of them includes the earliest surviving proof of the Pythagorean theorem, which Sialaros described as remarkably delicate.

  The later books shift terrain. Book 2 has been read as geometric algebra, though critics since the 1970s call that label anachronistic. Book 5 presents the general theory of proportion, and from book 7 onward, as the source puts it, "Euclid starts afresh. Nothing from the preceding books is used." Book 7 contains the Euclidean algorithm for finding the greatest common divisor of two numbers. Book 9 carries Euclid's theorem, the proof that there are infinitely many prime numbers. Book 10, the largest and most complex, wrestles with irrational numbers, while book 11 opens the final stretch on solids with 37 definitions of its own.

• 6. Beyond Geometry

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  At least five works of Euclid besides the Elements have survived, each following the same logical structure of definitions and proved propositions. The Optics is the earliest surviving Greek treatise on perspective, opening with geometrical optics and the basic rules of perspective. The Data is a short text on the nature and implications of given information in geometrical problems. The Phaenomena is a treatise on spherical astronomy, similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC.

  Two surviving works carry doubts. Catoptrics concerns the mathematics of mirrors, including images formed in plane and spherical concave mirrors, though its attribution to Euclid is sometimes questioned. On Divisions survives only partially in Arabic translation, deals with dividing geometrical figures into equal parts or given ratios, and includes thirty-six propositions.

  Four more works are credibly attributed to Euclid but have been lost entirely. The Conics was a four-book survey of conic sections, later superseded by the more comprehensive treatment of the same name by Apollonius. The Pseudaria, according to Proclus, advised beginners on avoiding common fallacies in geometrical reasoning. The Porisms was probably a three-book treatise with roughly 200 propositions, and the mathematician Michel Chasles speculated that its lost results touched on transversals and projective geometry. The Surface Loci is nearly a blank, its contents guessed at only from its title.

• 7. The Long Afterlife

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  The Elements is often ranked, after the Bible, as the most frequently translated, published, and studied book in the history of the Western world. Alongside Aristotle's Metaphysics, it may be the most successful ancient Greek text, and it dominated mathematical teaching across the Medieval Arab and Latin worlds. The geometrical system it established held the field for so long that today it is called Euclidean geometry, a name coined to set it apart from the non-Euclidean geometries discovered in the early 19th century.

  The printed history of the work runs through named hands. The first English edition appeared in 1570, published by Henry Billingsley and John Dee. In 1847, Oliver Byrne released a celebrated version titled The First Six Books of the Elements of Euclid in Which Coloured Diagrams and Symbols Are Used Instead of Letters for the Greater Ease of Learners, using colored diagrams to aid teaching. Later, David Hilbert produced a modern axiomatization of the work.

  Euclid's name now travels far beyond the page. The European Space Agency launched a space telescope called Euclid. A lunar crater bears the name Euclides, as does the minor planet 4354 Euclides. The poet Edna St. Vincent Millay caught the strange purity of his legacy in a single line: "Euclid alone has looked on Beauty bare."