50 Fun Facts About Euclid

Euclid is traditionally known by what title?

His system dominated the field until the early 19th century and is still called Euclidean geometry.

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

By metonymy "Euclid" can simply mean the book itself, or geometry in general.

How many books make up the Elements?

Books 1 to 6 cover plane geometry, 7 to 10 number theory and 11 to 13 solid geometry, roughly speaking.

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

He may have been among the first scholars of Ptolemy I's Musaeum, and Apollonius later studied with his pupils there.

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

Ptolemy ruled from about 305 to 282 BC and built the Musaeum, Alexandria's great centre of learning.

What is Euclid's famous reply when the king asked for a quicker way to learn his subject?

The anecdote is doubtful; a near-identical story is told about Menaechmus and Alexander the Great.

Which two ancient scholars are the main sources for the traditional account of Euclid's life?

Both wrote many centuries after Euclid; Proclus's commentary on Book I dates from the 5th century AD.

With which earlier thinker was Euclid long confused by medieval and Renaissance scholars?

The Socratic philosopher reportedly sneaked into Athens dressed as a woman to hear Socrates; the mix-up put "Megarensis" on early printed editions.

What does the Greek name Eukleides mean?

It combines eu- (well), klês (fame) and the suffix -ides (son of).

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

They made him a Tyre-born Greek living in Damascus, son of Naucrates; historians think this was invented to link him to the Arab world.

How many postulates does Book 1 of the Elements set out?

Together with five common notions they form the axiomatic basis for everything that follows.

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

Attempts to prove it from the other four failed for two thousand years and eventually produced non-Euclidean geometry.

The parallel postulate says two lines meet on the side where the two interior ones sum to less than what?

In hyperbolic geometry infinitely many lines through a point miss a given line; in elliptic geometry every line meets it.

How many definitions open Book 1 of the Elements?

They cover lines, angles and regular polygons; Book 1 then has 48 propositions ending with the Pythagorean theorem.

Which famous result about right-angled triangles closes Book 1, in the earliest surviving proof?

Propositions 46 to 48 deliver it; medieval students nicknamed proposition 47 Dulcarnon, "the two-horned".

What is the pons asinorum, Proposition 5 of Book 1?

Latin for "bridge of asses", it was the point where weak students stumbled; another medieval name was Elefuga, "flight of the wretches".

Which book of Euclid's Elements contains the algorithm for finding a greatest common divisor?

It is one of the oldest algorithms still in everyday use, and works by repeatedly replacing the larger number with the difference.

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

At least 200 different proofs of it now exist.

Which numbered book of the Elements is by far the largest and deals with irrational magnitudes?

Book 5, the general theory of proportion, is considered among the work's most important sections.

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

Theaetetus described the tetrahedron, cube, octahedron, dodecahedron and icosahedron, and Euclid gave them a complete treatment.

Book 7 of the Elements defines a "perfect number"; which is the smallest?

Its proper divisors 1, 2 and 3 add up to 6; the next perfect numbers are 28, 496 and 8,128.

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

The Phaenomena covers spherical astronomy, and Catoptrics, on mirrors, may not be his at all.

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

The Pseudaria warned beginners against fallacies, and the Porisms may have anticipated projective geometry.

How did ancient Greek mathematicians usually refer to Euclid?

The Greek is ho stoicheiotes; some medieval scholars even doubted he was a real person.

Where were the oldest physical fragments of the Elements, dating from around 100 AD, found?

Papyrus Oxyrhynchus 29 preserves Book 2, Proposition 5 with its diagram.

Whose 4th-century edition of Euclid was for centuries the only surviving Greek source?

A manuscript not derived from Theon's was only found by François Peyrard in the Vatican in 1808.

Which English monk translated the Elements into Latin from Arabic around 1120?

Western Europe had lost the text apart from fragments of Boethius's translation until then.

In which year did Erhard Ratdolt publish the first printed edition of the Elements?

It was based on Campanus of Novara's Latin version and repeated the confusion with Euclid of Megara.

The Elements is often said to be second only to which book in number of editions published?

The count runs well over a thousand editions since 1482.

Who published the first English edition of the Elements in 1570?

John Dee supplied its famous "Mathematical Preface" and copious notes.

What was distinctive about Oliver Byrne's 1847 edition of the first six books?

Its bold red, yellow and blue shapes look startlingly like modernist art.

Which mathematician published a description of hyperbolic geometry in 1829?

János Bolyai published independently in 1832, and Gauss claimed to have got there first without publishing.

Which Hungarian mathematician independently published a treatise on hyperbolic geometry in 1832?

He argued that only the physical sciences, not pure reasoning, could decide whether the universe is Euclidean.

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

Gaps had been found in Euclid's reasoning in the late 19th century; Tarski later offered another axiom system.

Which author of Alice in Wonderland defended Euclid in the 1879 book Euclid and His Modern Rivals?

The Oxford mathematics don objected to Euclid being replaced by newer textbooks in schools.

Which two ancient mathematicians are usually ranked alongside Euclid as the greatest of antiquity?

The Elements was already circulating by the 3rd century BC, since both take several of its propositions for granted.

Which earlier mathematicians' work did Euclid synthesise in the Elements?

Classicist Markus Asper says his achievement was assembling accepted knowledge in a cogent order and adding proofs to fill gaps.

The earliest surviving reference to Euclid appears in whose prefatory letter to a treatise on conics?

Writing in the early 2nd century BC, he complained that Euclid had not properly synthesised the locus on three and four lines.

Which agency launched the Euclid telescope in July 2023?

It rode a Falcon 9 to the Sun-Earth L2 point to study dark energy and dark matter.

What is the Euclid spacecraft's mission?

It measures the accelerating expansion of the universe from a halo orbit 1.5 million km beyond Earth.

Besides a spacecraft, Euclid's name is carried by which two celestial objects?

The crater Euclides is on the Moon and 4354 Euclides orbits in the asteroid belt.

Book 6 of Euclid's Elements rests on its first proposition, comparing triangles of the same height to what?

It applies the theory of ratios from Book 5 to plane figures.

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

The quadrivium of arithmetic, geometry, music and astronomy followed the trivium of grammar, logic and rhetoric.

Which lost work of Euclid held about 200 propositions of a type between a theorem and a problem?

Michel Chasles speculated they contained ideas related to transversals and projective geometry.

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

It survives only partially, in Arabic translation, and runs to thirty-six propositions.

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

Critics since the 1970s call the label anachronistic, since algebra proper came many centuries later.

Alongside its postulates, how many 'common notions' does Book 1 of the Elements state?

The common notions all concern comparing magnitudes, and together with the postulates give ten assumptions underpinning 48 propositions.

Which surviving work of Euclid is a treatise on spherical astronomy?

It resembles On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC.

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

Its Greek title literally means 'Fallacies'; almost nothing survives beyond its scope and a few lines.

Book 8 of the Elements chiefly discusses what?

From Book 7 onwards Euclid 'starts afresh', as Benno Artmann puts it, using nothing from the earlier books as he turns to number theory.