The famous Greek scientist and mathematician Euclid (300 BC) is best known as the author of “The Elements”. This is a mathematical textbook focusing mainly on geometry, proportion and number theory that has been influential in teaching mathematics for over two thousand years.

### Historical Introduction:

Not much is known about Euclid personally but it is believed that he was born around 325 BC and that he taught at Alexandria in Egypt. There has been speculation whether he was a creative mathematician himself or, whether he collected the work of others or that his name represented a group of mathematicians. Information about Euclid is recounted by Proclus, a 5th-century-AD philosopher.

He stated that Euclid lived in the time of the Ptolemy I (323 to 285 BC) and constructed the “Elements”, arranging in order many of Eudoxus’s theorems and perfecting many of Theaetetus’s concepts. Proclus also stated that “The Elements” also proved concepts which had been only loosely established by his predecessors.

Euclid and Archimedes are often considered contemporaries. Euclid’s mathematical education is thought to have been obtained from students in Plato’s school in Athens.

### The Elements

“The Elements” is Euclid’s most famous work and includes a compilation of works from earlier Greek mathematicians such as Pythagoras, Hippocrates, Theaetetus and Eudoxus. The book is logically set out into thirteen books so that it can be used easily as a reference.

In Book 1 Euclid, lists twenty-three definitions, five postulates (or rules) and five common notions (assumptions) and uses them as building blocks; from these all other proofs and theorems would be derived. For example, the first postulate states that it is possible to draw a straight line between any two points.

Book I proves elementary theorems about plane geometry.

Book 2 deals with geometric algebra.

Book 3 investigates the properties of circles and this book is believed to be the work of Pythagoras and his students.

Book 4 concerns the construction of regular polygons, in particular the pentagon.

Book 5 establishes the arithmetic theory of proportion and ratio and is the work of Eudoxus.

Book 6 applies the theory of ratios in Book 5 to plane geometry.

Book 7 deals with elementary number theory including prime numbers and contains the Euclidean algorithm for finding the greatest common divisor of two numbers.

Book 8 looks at geometric series.

Book 9 concerns the application of results from Books 7 and Book 8.

Book 10 deals with the theory of irrational numbers and is mainly the work of Theaetetus and contains his “method of exhaustion”.

Book 11 examines three-dimensional geometry giving basic definitions.

Book 12 continues with three-dimensional geometry, calculating the relative volumes of cones, pyramids, cylinders, and spheres using “the method of exhaustion” as invented by Eudoxus.

Book 13 investigates the five Platonic solids (pyramid, cube, octahedron, dodecahedron, icosahedron) in a given sphere, based on a work by Theaetetus.

### Other Contributions and Accomplishments:

Five other works of Euclid have survived:

“The Data”, a work on geometrical problems.

“On Divisions of Figures” which concerns the division of geometrical figures into two or more equal parts or into various ratios.

“Catoptrics” which examines the mathematical theory of mirrors, especially images formed by plane and spherical concave mirrors.

“Optics” which is a Greek treatise on perspective and “Phaenomena”, a treatise on spherical astronomy.

A Latin translation of the Elements was made around 1120 AD by English monk Adelard of Bath, who had acquired a copy of an Arabic version in Spain and the first complete English translation of “The Elements” was made in 1570 by merchant Sir Henry Billingsley. The availability of Euclid’s work undoubtedly helped create the flourishing of mathematicians of the seventeenth century including Blaise Pascal, Johannes Kepler, Pierre De Fermat, René Descartes and Isaac Newton.

The developing predominance of the sciences and mathematics in the 18th and 19th centuries earned Euclid a crucial place in the curriculum of schools and universities throughout the Western world.