### Topic Page: Euclid (fl. 300 B.C.)

**Euclid**from

*Philip's Encyclopedia*

Greek mathematician who taught at Alexandria, Egypt. He is remembered for his classic textbook on geometry Elements (Lat. pub. 1482). His axioms - that parallel lines never meet and the angles of a triangle always add up to 180° - remained the basis for geometry until the development of non-Euclidean geometry in the 18th century.

**Euclid (lived c. 300 BC)**from

*The Hutchinson Dictionary of Scientific Biography*

Place: Greece

Subject: biography, maths and statistics

Greek mathematician whose works, and the style in which they were presented, formed the basis for all mathematical thought and expression for the following 2,000 years (although they were not entirely without fault). He also wrote books on other scientific topics, but these have survived the passage of the centuries only fragmentarily or not at all.

Very little indeed is known about Euclid. No record is preserved of his date or place of birth, his education, or even his date or place of death. The influence of Plato, who lived in the 4th century BC, is certainly detectable in his work - so Euclid must either have been contemporary or later. Some commentators have suggested that he attended Plato's Academy in Athens but, if so, it is likely to have been after Plato's death. In any case, it has been established that Euclid went to the recently founded city of Alexandria (now in Egypt) in around 300 BC and set up his own school of mathematics there. Fifty years later, however, Euclid's disciple Apollonius of Perga was said to have been leading the school for some considerable time; it seems very possible, therefore, that Euclid died in around 270 BC.

Euclid's mathematical works survived in almost complete form because they were translated first into Arabic, then into Latin; from both of these they were then translated into other European languages. He employed two main styles of presentation: the synthetic (in which one proceeds from the known to the unknown via logical steps) and the analytical (in which one posits the unknown and works towards it from the known, again via logical steps). In his major work, The Elements, Euclid used the synthetic approach, which suited the subject matter so perfectly that the method became the standard procedure for scientific investigation and exposition for millennia afterwards. The strictly logical arrangement demanding the absolute minimum of assumption, and the omission of all superfluous material, is one of the great strengths of The Elements, in which Euclid incorporated and developed the work of previous mathematicians as well as including his own many innovations. The presentation was one of extreme clarity and he was rigorous, too, about the actual detail of the mathematical work, attempting to provide proofs for every one of the theorems.

The Elements is divided into 13 books. The first six deal with plane geometry (points, lines, triangles, squares, parallelograms, circles, and so on), and include hypotheses such as ‘Pythagoras' theorem’ which Euclid generalized. Books 7 to 9 are concerned with arithmetic and number theory. In Book 10 Euclid treats irrational numbers. And Books 11 to 13 discuss solid geometry, ending with the five ‘Platonic solids’ (the tetrahedron, octahedron, cube, icosahedron, and dodecahedron).

Euclid favoured the analytical mode of presentation in writing his other important mathematical work, the Treasury of Analysis. This comprised three parts, now known as The Data, On Divisions of Figures, and Porisms.

Euclid's geometry formed the basis for mathematical study during the next 2,000 years. It was not until the 19th century that a different form of geometry was even considered: ‘accidentally’ discovered by Saccheri in 1733, non-Euclidean geometry was not in any way defined until Nikolai Lobachevsky (in the 1820s), János Bolyai (in the 1830s), and Bernhard Riemann (in the 1850s) examined the subject. It is difficult to see, therefore, how Euclid's contribution to the science of mathematics could have been more fundamental than it was.

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