50 Fun Facts About Euclid

Euclid is traditionally known by what title?

The father of geometry set out a system that went unchallenged until the early 19th century; only when rival geometries appeared did his need a name of its own, Euclidean.

What is the Greek title of Euclid's great treatise, the Elements?

Stoicheia was also the Greek word for the letters of the alphabet, and Proclus said its leading theorems serve the rest of geometry as letters serve a language.

How many books make up the Elements?

Thirteen books are Euclid's own, yet for centuries editions ran to fifteen: two extra books by later hands, the first of them by Hypsicles, travelled with the text.

In which city is Euclid generally accepted to have spent his career?

Alexandria: Pappus records that a later geometer spent years there with Euclid's pupils, the main evidence that Euclid taught in the city and founded its school of mathematics.

Under which ruler's reign did Proclus place Euclid's activity?

Ptolemy I was one of Alexander the Great's generals and made himself a king when the empire was carved up; Proclus, writing seven centuries later, dated Euclid by his reign.

Which famous line is Euclid said to have spoken?

"There is no royal road" to geometry, he reportedly told a king who wanted a shortcut. The tale is doubtful: the same story is told of Menaechmus and Alexander the Great.

Which US president wrote that he had 'nearly mastered' the first six books of Euclid after serving in Congress?

Abraham Lincoln had less than a year of schooling in all, and wrote that he took up Euclid because he regretted his want of education and did what he could to supply it.

Scholars long confused Euclid with a namesake from Megara who reportedly sneaked into Athens dressed as a woman to hear whom?

Socrates was the draw: as Aulus Gellius tells it, Athens had banned Megarians on pain of death, so this other Euclid walked some twenty miles by night in a woman's tunic and veil.

What do we now call the division of a line that Euclid named 'extreme and mean ratio'?

The golden ratio first appears in Book 2, where Euclid makes the cut before he has even defined what a ratio is, and he later uses it to build a regular pentagon.

Medieval Arabic sources, considered unreliable, claimed Euclid was born in which city?

Tyre was his birthplace and Damascus his home in the Arabic accounts, which also name his father as Naucrates; Thomas Heath judged the cities were added to tie a famous Greek to the East.

Besides a compass, which drawing tool do Euclid's first three postulates allow?

Straightedge: the postulates let you draw a line, extend it and draw a circle, nothing more. It carries no markings, so nothing in a construction is ever measured.

Which of Euclid's postulates is the famous parallel postulate?

The fifth postulate is far less obvious than the rest, and for more than two thousand years geometers tried to prove it from them; their failure led to non-Euclidean geometry.

Euclid's parallel postulate says two lines meet on the side where the interior angles add up to less than what?

Two right angles, or 180 degrees: if the inside angles total less, the two lines lean toward each other and must cross. Change that rule and you get the hyperbolic and elliptic geometries.

What does the first definition in Euclid's Elements define?

A point comes first, as 'that which has no part'; modern geometry has stopped trying to define it and leaves 'point' as a primitive term whose meaning comes only from the axioms.

Which famous theorem does Euclid prove at the climax of Book 1 of the Elements?

The Pythagorean theorem is Proposition 47, proved with no numbers at all: Euclid cuts the big square into two rectangles and shows each equals one of the smaller squares.

Which result is the pons asinorum, Proposition 5 of Book 1 of Euclid's Elements?

The isosceles triangle theorem, that the base angles are equal, became the 'bridge of asses' either for its bridge-like diagram or as the first hurdle slower students could not cross; nobody is sure which.

What does the Euclidean algorithm, set out at the start of Book 7 of the Elements, find?

The greatest common divisor of two numbers falls out by subtraction alone: keep taking the smaller from the larger until one divides the other. It is among the oldest algorithms still in common use.

What does Euclid's theorem, in Book 9, prove?

There are infinitely many prime numbers because any finite list can be beaten: multiply its primes together, add one, and the result has a prime factor the list left out.

Book 10 of Euclid's Elements, its longest, deals in modern terms with which kind of number?

Irrational numbers, handled as lengths with no common measure, fill about a quarter of the Elements, a bulk historians find out of proportion to what the book delivers.

The final book of the Elements is devoted to which five shapes?

The Platonic solids end the Elements with a proof that no sixth can exist, and Proclus claimed that building them was the goal of the whole work.

Which is the smallest 'perfect number', a term Euclid defines in Book 7 of the Elements?

6 equals the sum of its divisors 1, 2 and 3. Euclid went on to give a recipe for building such numbers, and two thousand years later Euler proved it catches every even one.

Which of Euclid's surviving works is the earliest Greek treatise on perspective?

Optics treats sight as straight rays fanning out from the eye in a cone, and starts from the assumption that an object looks smaller the narrower the angle it fills.

Which lost four-book work of Euclid was superseded by a fuller treatise of the same name from Perga?

Conics is known mainly through Pappus, who says the geometer from Perga built the first four books of his own treatise on Euclid's.

How did ancient Greek mathematicians usually refer to Euclid?

As "the author of Elements" he was little more than a byline; some medieval Arabic writers even explained the name 'Euclid' as a compound meaning 'key of geometry'.

A papyrus scrap of Euclid's Elements, among the oldest known, was dug from the ancient rubbish dumps of which Egyptian town?

Oxyrhynchus gave up the scrap, a Book 2 proposition with its diagram, in 1897. Its finders dated it to about AD 300, but later study moved it back some two centuries.

The edition of Euclid's Elements that the Greek world copied for centuries was made by Theon, the father of which philosopher?

Hypatia's father Theon tidied and corrected the text, and his version was the only Greek one anyone knew until a copy his editing had not touched turned up in the Vatican.

Which Jesuit missionary worked with the scholar Xu Guangqi to publish the first Chinese edition of Euclid's Elements, in 1607?

Matteo Ricci and Xu stopped after the first six books, and Chinese readers waited more than two centuries for the rest.

Erhard Ratdolt's 1482 Euclid, the first in print, solved which printing problem?

Diagrams had defeated earlier printers, Ratdolt wrote, which is why almost no mathematics was in print; he boasted that his figures could be set as easily as letters.

Euclid ends his proofs with a Greek phrase that Latin turned into Q.E.D.; what does it mean?

"Which was to be demonstrated" became a badge of rigour beyond mathematics: Spinoza closed the proofs of his Ethics with Q.E.D., a work of philosophy laid out in geometrical order on Euclid's model.

Which adviser to Elizabeth I wrote the celebrated 'Mathematical Preface' to the first English edition of Euclid's Elements?

John Dee, the queen's astronomer and astrologer, aimed it at readers outside the universities, and it became his most reprinted work; the translator, Henry Billingsley, ended up Lord Mayor of London.

What was distinctive about Oliver Byrne's 1847 Euclid?

It used coloured diagrams instead of letters 'for the greater ease of learners', as its title promised: a proof points to the red triangle or the yellow square, never to triangle ABC.

Which mathematician published a description of hyperbolic geometry in 1829?

Nikolai Lobachevsky's was the first account of a geometry that breaks Euclid's parallel rule to reach print; it appeared in a provincial Kazan journal, and the St Petersburg Academy turned the work down.

When János Bolyai published his non-Euclidean geometry, which famous mathematician replied that praising it 'would amount to praising myself'?

Carl Friedrich Gauss told Bolyai's father he had reached the same results some thirty years before and never printed them. Bolyai, who had boasted of creating a new world out of nothing, published no more.

Which German mathematician produced a modern axiomatisation of Euclidean geometry around 1899?

David Hilbert's Foundations of Geometry rebuilt the subject on axioms that fixed weaknesses in Euclid's own, and its axiomatic method became a hallmark of 20th-century mathematics.

Which Victorian author defended Euclid's textbook in the 1879 book Euclid and His Modern Rivals?

Lewis Carroll, an Oxford mathematics lecturer, wrote it as a comic play under his real name, Charles Dodgson: Euclid's ghost visits a sleeping examiner, and each rival textbook is tried and found wanting.

Which Renaissance painter showed Euclid stooping over a slate to teach his pupils in The School of Athens?

Raphael is said to have given the stooping geometer the face of his friend Bramante, architect of the new St Peter's, and he painted his own portrait a few figures away.

What figure does the very first proposition of Euclid's Elements show how to construct?

An equilateral triangle, drawn where two circles cross. Critics later found a hole in this very first proof: nothing Euclid has assumed guarantees that the circles meet at all.

In an old tale, what did Euclid have his slave give a student who asked what geometry would gain him?

A few coins, threepence in the usual translation, 'since he must make gain out of what he learns': Euclid's dry verdict, in a story preserved by Stobaeus, on studying geometry for profit.

Which space agency built and operates the Euclid space telescope, launched in July 2023?

Europe's ESA had booked a Russian Soyuz rocket; when Russia halted those launches in 2022, the telescope flew from Florida on a SpaceX Falcon 9 instead.

What is the Euclid spacecraft's mission?

To study dark energy and dark matter, Euclid maps billions of galaxies; it carries his name because the density of matter and energy is tied to the geometry of the universe.

Besides a spacecraft, which two celestial objects carry Euclid's name?

A lunar crater and a minor planet: the crater Euclides is small, but a skirt of pale ejecta around it catches the high Sun and makes it one of the brightest spots on the Moon.

Book 6 of Euclid's Elements opens by proving that triangles of equal height have areas in the same ratio as what?

Their bases: with the height fixed, a triangle on twice the base has twice the area. Euclid uses that link between lengths and areas to carry his theory of ratios into the study of similar figures.

The Elements became a standard university text as part of which medieval curriculum?

The quadrivium covered the four mathematical arts, arithmetic, geometry, astronomy and music, the last because it was studied as the theory of ratios between numbers; students reached it after the trivium.

Which lost work of Euclid dealt with a kind of proposition midway between a theorem and a problem?

The Porisms are known mainly from the hints Pappus left, and restoring them became a long-running puzzle that drew in Robert Simson and Michel Chasles.

What did Euclid's surviving work On Divisions deal with?

Cutting figures into parts in given ratios fills its 36 problems, but the book survives only in an Arabic version whose translator left out all but four of the proofs as too easy.

Book 2 of the Elements, on rectangles and squares, is traditionally given what debated label?

Geometric algebra reads the book's rectangles and squares as algebraic identities in disguise; since the 1970s critics have called that anachronistic, because algebra itself lay many centuries ahead.

What are assumptions such as 'the whole is greater than the part' called in Euclid's Elements?

Common notions are general truths about size that any proof needs before it can compare one thing with another; the postulates, by contrast, belong to geometry alone.

Euclid's surviving work the Phaenomena is a treatise on which subject?

Spherical astronomy: the Phaenomena treats the sky as a turning sphere and uses geometry to work out when stars rise and set, much as an earlier book by Autolycus had done.

According to Proclus, Euclid's lost Pseudaria was written to help beginners avoid what?

Common fallacies were its whole subject: Proclus says Euclid set true proofs beside false ones so that beginners could practise finding the flaw. Almost nothing else is known of the book.

Which English philosopher is said to have fallen in love with geometry at forty on finding Euclid open in a library?

Thomas Hobbes, John Aubrey says, read one proposition, swore 'By God, this is impossible!', then chased the proof back through the earlier ones step by step until he was convinced.