**Euclid Of Alexandria** was a Greek mathematician best known for his treatise on geometry The Elements. This document influenced the development of Western mathematics for more than 2000 years.

- The long lasting nature of The Elements must make Euclid the leading mathematics teacher of all time.
- However little is known of Euclid's life except that he taught at Alexandria in Egypt.
- This man lived in the time of the first Ptolemy; for Archimedes, who followed closely upon the first Ptolemy makes mention of Euclid, and further they say that Ptolemy once asked him if there were a shorted way to study geometry than the Elements, to which he replied that there was no royal road to geometry.
- There is other information about Euclid given by certain authors but it is not thought to be reliable.
- The second type of information is that Euclid was born at Megara.
- There was a Euclid of Megara, who was a philosopher who lived about 100 years before the mathematician Euclid of Alexandria.
- It is not quite the coincidence that it might seem that there were two learned men called Euclid.
- Euclid was a very common name around this period and this is one further complication that makes it difficult to discover information concerning Euclid of Alexandria since there are references to numerous men called Euclid in the literature of this period.
- However, although we do not know for certain exactly what reference to Euclid in Archimedes' work Proclus is referring to, in what has come down to us there is only one reference to Euclid and this occurs in On the sphere and the cylinder.
- He argued that the reference to Euclid was added to Archimedes' book at a later stage, and indeed it is a rather surprising reference.
- It was not the tradition of the time to give such references, moreover there are many other places in Archimedes where it would be appropriate to refer to Euclid and there is no such reference.
- This is far from an end to the arguments about Euclid the mathematician.
- (i) Euclid was an historical character who wrote the Elements and the other works attributed to him.
- (ii) Euclid was the leader of a team of mathematicians working at Alexandria.
- They all contributed to writing the 'complete works of Euclid', even continuing to write books under Euclid's name after his death.
- (iii) Euclid was not an historical character.
- The 'complete works of Euclid' were written by a team of mathematicians at Alexandria who took the name Euclid from the historical character Euclid of Megara who had lived about 100 years earlier.
- It is worth remarking that Itard, who accepts Hjelmslev's claims that the passage about Euclid was added to Archimedes, favours the second of the three possibilities that we listed above.
- Again the fact that Euclid undoubtedly based the Elements on previous works means that it would be rather remarkable if no trace of the style of the original author remained.
- Even if we accept (i) then there is little doubt that Euclid built up a vigorous school of mathematics at Alexandria.
- Euclid did not work out the syntheses of the locus with respect to three and four lines, but only a chance portion of it ...
- It is unknown if Euclid was an historical character. Similarly, there are many similar references to Bourbaki by mathematicians who knew perfectly well that Bourbaki was fictitious.
- Nevertheless, the mathematicians who made up the Bourbaki team are all well known in their own right and this may be the greatest argument against hypothesis (iii) in that the 'Euclid team' would have to have consisted of outstanding mathematicians.
- We shall assume in this article that hypothesis (i) is true but, having no knowledge of Euclid, we must concentrate on his works after making a few comments on possible historical events.
- Euclid must have studied in Plato's Academy in Athens to have learnt of the geometry of Eudoxus and Theaetetus of which he was so familiar.
- None of Euclid's works have a preface, at least none has come down to us so it is highly unlikely that any ever existed, so we cannot see any of his character, as we can of some other Greek mathematicians, from the nature of their prefaces.
- The picture of Euclid drawn by Pappus is, however, certainly in line with the evidence from his mathematical texts.
- Euclid's most famous work is his treatise on mathematics The Elements.
- Probably no results in The Elements were first proved by Euclid but the organisation of the material and its exposition are certainly due to him.
- There is ample evidence that Euclid is using earlier textbooks as he writes the Elements since he introduces quite a number of definitions which are never used such as that of an oblong, a rhombus, and a rhomboid.
- Euclid's decision to make this a postulate led to Euclidean geometry.
- It was not until the 19th century that this postulate was dropped and non-euclidean geometries were studied.
- There are also axioms which Euclid calls 'common notions'.
- Zeno of Sidon, about 250 years after Euclid wrote the Elements, seems to have been the first to show that Euclid's propositions were not deduced from the postulates and axioms alone, and Euclid does make other subtle assumptions.
- In particular book seven is a self-contained introduction to number theory and contains the Euclidean algorithm for finding the greatest common divisor of two numbers.
- Apparently Euclid's exposition excelled only in those parts in which he had excellent sources at his disposal.
- Euclid changed the proofs of several theorems in this book so that they fitted the new definition of proportion given by Eudoxus.
- Euclid proves these theorems using the "method of exhaustion" as invented by Eudoxus.
- Euclid's Elements is remarkable for the clarity with which the theorems are stated and proved.
- It was the primary source of geometric reasoning, theorems, and methods at least until the advent of non-Euclidean geometry in the 19th century.
- Euclid also wrote the following books which have survived: Data (with 94 propositions), which looks at what properties of figures can be deduced when other properties are given; On Divisions which looks at constructions to divide a figure into two parts with areas of given ratio; Optics which is the first Greek work on perspective; and Phaenomena which is an elementary introduction to mathematical astronomy and gives results on the times stars in certain positions will rise and set.
- Euclid's following books have all been lost: Surface Loci (two books), Porisms (a three book work with, according to Pappus, 171 theorems and 38 lemmas), Conics (four books), Book of Fallacies and Elements of Music.
- Elements of Music is a work which is attributed to Euclid by Proclus.
- We have two treatises on music which have survived, and have by some authors attributed to Euclid, but it is now thought that they are not the work on music referred to by Proclus.
- Euclid may not have been a first class mathematician but the long lasting nature of The Elements must make him the leading mathematics teacher of antiquity or perhaps of all time.

Born about 325 BC, (probably) Alexandria, Egypt. Died about 265 BC, Alexandria, Egypt.

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**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive