Euclid

Euclid of Alexandria: the author without a face

Timeline c. 530 BCPythagoras and his followers study numbers and figures in southern Italy. c. 387 BCPlato founds the Academy in Athens. Over the door, according to tradition: “Let no one ignorant of geometry enter”. 331 BCAlexander the Great founds Alexandria, in the Nile delta. c. 300 BCEuclid teaches in Alexandria, in the time of King Ptolemy I, and writes the Elements. c. 250 BCArchimedes, in Syracuse, cites the Elements. c. AD 100A papyrus found at Oxyrhynchus, in Egypt, preserves a passage from Book II. c. 370Theon of Alexandria prepares the edition that will be copied over the following centuries. c. 450Proclus writes his commentary on Book I, with the little news we have of Euclid. c. 800The Elements are translated into Arabic in Baghdad. c. 1120Adelard of Bath translates the Arabic text into Latin. 1482First printed edition, in Venice, by Erhard Ratdolt. ◀ Previous Follow the journey Next ▶ c. 387 a.C. AthensPlato founds the Academy. The geometry Euclid will organise grows up in the Greek schools. c. 300 a.C. AlexandriaEuclid teaches in the time of Ptolemy I and writes the thirteen books of the Elements. c. 250 a.C. SyracuseArchimedes already cites the Elements in his works. c. 100 OxyrhynchusA papyrus found in the town’s rubbish heaps preserves a passage from Book II, with a figure. c. 370 AlexandriaTheon of Alexandria prepares the edition that will be copied for a thousand years. c. 450 AthensProclus writes his commentary on Book I, with the little news we have about Euclid. c. 800 BaghdadIn the House of Wisdom the Elements are translated into Arabic. c. 1120 BathAdelard of Bath translates the Arabic text into Latin, and the book returns to Europe. 1482 VeniceErhard Ratdolt prints the first edition, with the figures in the margins. More than a thousand editions followed. Enable JavaScript to see the map. The stages are listed below. The journey of the Elements, from Athens and Alexandria to the first printed edition in Venice. Map: © OpenStreetMap contributors

October 14, 2026 Read

The Elements: definitions, postulates and common notions

A point is that which has no part. A line is breadthless length. — Euclid, Elements, Book I, Definitions 1 and 2 Imagine you want to convince someone, beyond any possible doubt, that the angles of a triangle always add up to 180°. Measuring is not enough: the protractor makes errors, and you can only measure a few triangles. You need an argument. But every argument starts from something. Where should you start? If every claim must be justified by another, and that one by another, the chain never ends.

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The first proposition: the equilateral triangle

On a given finite straight line, to construct an equilateral triangle. — Euclid, Elements, Proposition I.1 After the definitions, postulates and common notions, the Elements begin with a small and beautiful problem. You have a segment $AB$. You want to construct, with straightedge and compass only, a triangle whose three sides are equal to $AB$. It is the first of the book’s 465 propositions, and it shows in a few lines how all the others will work.

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Pythagoras's theorem, Euclid's way

In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. — Euclid, Elements, Proposition I.47 Long before the Greeks, the surveyors of Egypt, who re-marked the fields after each Nile flood, were called rope-stretchers. The story goes that they used a rope with 12 equally spaced knots: stretched into a triangle with sides 3, 4 and 5, it gave a right angle. The Babylonians knew much more: a clay tablet from around 1800 BC, Plimpton 322, lists right triangles with whole-number sides, some of them huge, such as 3,367, 3,456 and 4,825.

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Algebra done with squares and rectangles

If a straight line is cut at random, the square on the whole equals the squares on the segments plus twice the rectangle contained by the segments. — Euclid, Elements, Proposition II.4 The Greeks had no letters for unknowns, no plus or minus signs, no equals sign. When they wanted to speak of $a\cdot b$, they spoke of a rectangle with sides $a$ and $b$; when they wanted $a^2$, they spoke of a square with side $a$. The words “square” and “cube” that we use for powers come from this. Book II of the Elements is made entirely in this way: it consists of identities of algebra, stated and proved as facts about areas. Historians call it geometric algebra.

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Euclid's algorithm: the greatest common divisor

The Euclidean algorithm is the granddaddy of all algorithms, because it is the oldest nontrivial algorithm that has survived to the present day. — Donald Knuth, The Art of Computer Programming, vol. 2 (1969) You have two rods, one 252 cm long and the other 105 cm, and you want the longest ruler that measures both exactly, a whole number of times. In today’s language you want the greatest common divisor, $\gcd(252,105)$. You could list all the divisors of each number and look for the largest common one, but with big numbers that is impractical. In Book VII of the Elements, Euclid teaches a much faster way.

October 14, 2026 Read

The primes never run out

Prime numbers are more than any assigned multitude of prime numbers. — Euclid, Elements, Proposition IX.20 A prime number is an integer greater than 1 that can only be divided by 1 and by itself: 2, 3, 5, 7, 11, 13, … All the others, such as $12=2\cdot2\cdot3$, are products of primes. The primes are the atoms of arithmetic. The further you go, the rarer they become. Between 1 and 100 there are 25 primes; between 1,000,001 and 1,000,100, only 6. Do they run out at some point? Euclid showed that they do not, with a four-line argument that the English mathematician G. H. Hardy considered one of the most beautiful in all of mathematics.

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The fifth postulate and the other geometries

Out of nothing I have created a strange new universe. — János Bolyai, letter to his father (3 November 1823) Euclid’s first four postulates are short and obvious: draw a line, extend it, draw a circle, all right angles are equal. The fifth is different: If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side.

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The legacy of Euclid

He studied and nearly mastered the six books of Euclid, since he was a member of Congress. — Abraham Lincoln, autobiographical note (1860) After the Bible, the Elements are one of the most copied, translated and printed books in history. Since the first printed edition, in Venice in 1482, there have been more than a thousand editions. For more than two thousand years, “studying geometry” meant “studying Euclid”. The first printed edition of the Elements, made by Erhard Ratdolt in Venice in 1482, in a Latin translation. The figures are in the margins, beside the text; an early reader filled the pages with notes.Folger Shakespeare Library · Wikimedia Commons · CC BY-SA 4.0 The many faces of a book This site’s old trigonometry course gathered some editions that changed the way Euclid was presented:

October 14, 2026 Read

Image credits

Historical images All come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. No portrait of Euclid was made in his lifetime; the images of him in circulation are inventions by much later artists. Oxyrhynchus papyrus with Proposition II.5, c. AD 100, University of Pennsylvania Museum: source page · public domain. Detail of the School of Athens, by Raphael, 1509–1511: source page · public domain. Proposition I.47 in Oliver Byrne’s edition, London, 1847, scan by Mirko Junge: source page · public domain. First printed edition of the Elements, Erhard Ratdolt, Venice, 1482, Folger Shakespeare Library: source page · CC BY-SA 4.0. Arabic edition of the Elements, version of Nasir al-Din al-Tusi, Rome, 1594, Houghton Library: source page · public domain. Blog figures The cover, the construction of the equilateral triangle, the figure for Pythagoras’s theorem and the one for completing the square are original illustrations, with versions in each of the site’s languages. The completing-the-square example, $x^2+10x=39$, and the exercise $x^2+8x=65$ come from this site’s old course on the quadratic formula, whose figures were redrawn; the examples 3, 4, 5 and 6, 8, 10 and the table of editions come from the old geometry and trigonometry courses.

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