Euclid (fl. c. 300 BC, Alexandria), the most prominent mathematician of Greco-Roman antiquity, best known for his treatise on geometry, the Elements.

Life and work. Of Euclid's life it is known only that he taught at and founded a school at Alexandria in the time of Ptolemy I Soter, who reigned from 323 to 285/283 BC. Medieval translators and editors often confused him with the philosopher Eucleides of Megara, a contemporary of Plato about a century before, and therefore called him Megarensis. Writing in the 5th century AD, the Greek philosopher Proclus told the story of Euclid's reply to Ptolemy, who asked whether there was any shorter way in geometry than that of the Elements--"There is no royal road to geometry." Another anecdote relates that a student, probably in Alexandria, after learning the very first proposition in geometry, wanted to know what he would get by learning these things, whereupon Euclid called his slave and said, "Give him threepence since he must needs make gain by what he learns."

Euclid compiled his Elements from a number of works of earlier men. Among these are Hippocrates of Chios (5th century BC), not to be confused with the physician Hippocrates of Cos (flourished 400 BC). The latest compiler before Euclid was Theudius, whose textbook was used in the Academy and was probably the one used by Aristotle. The older elements were at once superseded by Euclid's and then forgotten. For his subject matter Euclid doubtless drew upon all his predecessors, but it is clear that the whole design of his work was his own. He evidently altered the arrangement of the books, redistributed propositions among them and invented new proofs if the new order made the earlier proofs inapplicable. Thus, while Book X was mainly the work of the Pythagorean Theaetetus (flourished 369 BC), the proofs of several theorems in this book had to be changed in order to adapt them to the new definition of proportion developed by Eudoxus (q.v.). According to Proclus, Euclid incorporated into his work many discoveries of Eudoxus and Theaetetus. Most probably Books V and XII are the work of Eudoxus, X and XIII of Theaetetus. Book V expounds the very influential theory of proportion that is applicable to commensurable and incommensurable magnitudes alike (those whose ratios can be expressed as the quotient of two integers and those that cannot). The main theorems of Book XII state that circles are to one another as the squares of their diameters and that spheres are to each other as the cubes of their diameters. These theorems are certainly the work of Eudoxus, who proved them with his "method of exhaustion," by which he continuously subdivided a known magnitude until it approached the properties of an unknown. Book X deals with irrationals of different classes. Apart from some new proofs and additions, the contents of Book X are the work of Theaetetus; so is most of Book XIII, in which are described the five regular solids, earlier identified by the Pythagoreans. Euclid seems to have incorporated a finished treatise of Theaetetus on the regular solids into his Elements. Book VII, dealing with the foundations of arithmetic, is a self-consistent treatise, written most probably before 400 BC.

*Excerpt from the Encyclopedia Britannica without permission.*