Great Mathematicians 4 – Euclid

Given that Euclid’s influence on mathematics, geometry in particular, has never diminished over two thousand years, it is extraordinary that we know so little about his life. He was born around 300BC, and was amongst the first teachers at the great university of Alexandria, founded by Ptolemy I, but it is likely that he studied mathematics in Athens with some of Plato’s students. Euclid wrote around a dozen influential mathematical books, but it is his 13 volume treatise The Elements that bestrides history.

Euclid never claimed to be a great original mathematician: Elements gathered all the existing knowledge of geometry and number theory, but was so colossal in scope it inevitably contained many new ideas and proofs. Books I to VI deal with what we would now regard as standard plane geometry (known as Euclidean geometry), and formed the basis of all geometry teaching and text books for 2000 years, only to be supplanted, in part, by the development of non-Euclidean geometry in the mid-19th century.

Books VII to X deal with number theory and the foundations of arithmetic. In particular, Book X deals with irrational numbers, which caused such consternation to earlier Greek mathematicians, especially the Pythagorean school. Books XI to XIII deal with three-dimensional figures, the culmination being the construction of the five Platonic solids – these are the only solid figures where all the faces are identical and equilateral, and are: tetrahedron (4 triangular faces), cube (6 squares), octahedron (8 triangles), dodecahedron (12 pentagons), icosahedron (20 triangles). He also proved that no other Platonic solids existed. Study of these solids was inextricably linked to the Golden Ratio (of which more in a later blog), which appears in several places in the Elements.

In the Elements, Euclid was the first to set out the basis for rigorous proof. Beginning with definitions and axioms, Euclid then goes on to develop many geometrical and arithmetic theorems using strict logic. For example, his axioms include:

• Given two points there is one straight line that joins them
• All right angles are equal
• Things equal to the same thing are equal
• The whole is greater than the part

From axioms such as these, examples of the proofs he constructed are:

• There are an infinite number of prime numbers
• The base angles of an isosceles triangle are equal
• Radius of a circle meets a tangent at a right angle

Euclid also developed many constructions, such as angle bisectors and accurate ways to draw a pentagon, which are still taught today.

From the title page of the first English translation

Although Europe in the dark ages lost sight of the Elements, the book had an immense influence on Islamic mathematicians and there were many translations from Greek to Arabic. It was from an Arabic version that the first translation was made into Latin (by Adelard of Bath, who obtained a copy in Spain whilst disguised as a Muslim student) and hence became known in 12th century Europe. The first direct translation into Latin from the original Greek appeared in Vienna in 1505, with the first English translation in 1570. The ideas and methods of mathematicians such as Kepler, Descartes, Fermat and Newton were deeply rooted in the Elements – the greatest and most influential mathematical textbook ever written. The logical approach, moving from the simplest axiomatic statements, through pure logic, to the deepest mathematical results, has proved to be the enduring basis of all mathematical development – and, presumably, always will.

Euclid – interesting facts:

1. Abraham Lincoln kept a copy of the Elements in his saddlebag, often reading it to make him a better lawyer by understanding the concept of proof.
2. One of Euclid’s other books, Optics, is the first surviving dissertation on perspective.
3. Did Euclid have another name? No, the Greeks didn’t use surnames: he was known as Euclid of Alexandria.
4. Until the 20th Century, only the Bible sold more books than the Elements.