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Euclid

fl. c. 300 BC · Greek (worked in Alexandria)

Mathematics — geometry

Euclid was a Greek mathematician who taught in Alexandria, probably in the reign of Ptolemy I, about whose life almost nothing is known. His Elements, thirteen books systematising geometry and number theory from definitions, postulates and common notions, became the most influential mathematics textbook ever written and was used for over two thousand years. Geometry built on his axioms is still called Euclidean geometry.

How to spot them

▸The Elements in thirteen books; 'the father of geometry'
▸The parallel (fifth) postulate, whose denial led to non-Euclidean geometry
▸Euclid's algorithm for the greatest common divisor; proof that there are infinitely many primes
▸Alexandria under Ptolemy I; 'there is no royal road to geometry'

Key discoveries & works

The Elements (c. 300 BC)

Thirteen books of geometry and number theory built from definitions, five postulates and common notions

The parallel postulate (c. 300 BC)

The fifth postulate of the Elements; attempts to prove it from the others eventually produced non-Euclidean geometry

Proof that there are infinitely many prime numbers (c. 300 BC)

Elements Book IX, Proposition 20

Algorithm for finding the greatest common divisor of two numbers (c. 300 BC)

Given in Elements Book VII; one of the oldest algorithms still in use

Optics (early geometrical treatise on vision) (c. 300 BC)

Treats vision geometrically using straight-line rays from the eye

Data and Phaenomena (c. 300 BC)

Other surviving works: a companion to the Elements on what is 'given', and spherical astronomy

Key ideas

Axiomatic method: deriving a whole body of results by strict proof from a small set of definitions, postulates and common notions
The fifth postulate is independent of the others; replacing it gives the hyperbolic and elliptic geometries later developed by Lobachevsky, Bolyai and Riemann
Book XIII constructs the five regular (Platonic) solids and shows there are no others

Quick-fire facts

•Taught at Alexandria, probably in the time of Ptolemy I (reigned 323–283 BC)
•Proclus tells that when Ptolemy asked for a shorter way to learn geometry, Euclid replied that there is no royal road to geometry
•Elements Book I, Proposition 5 (the base angles of an isosceles triangle are equal) is known as the pons asinorum, 'bridge of asses'
•Elements Book I, Proposition 47 is Pythagoras' theorem
•The first printed edition of the Elements was published in Venice by Erhard Ratdolt in 1482
•The Elements is often said to be second only to the Bible in number of editions
•Oliver Byrne's 1847 edition presented the first six books with coloured diagrams in place of letters

Drill it — 8 cards

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Sources

MacTutor History of Mathematics, University of St Andrews — Euclid of Alexandria
Thomas L. Heath — A History of Greek Mathematics