Euclid of Alexandria

Born: approximately 325 BC

Died: approximately 265 BC in Alexandria, Egypt

Euclid of Alexandria is the most outstanding mathematician of antiquity best known for his treatise on mathematics The Elementss. The long permanent nature of The Elementss must do Euclid the taking mathematics instructor of all clip. However small is known of Euclid ‘s life except that he taught at Alexandria in Egypt. Proclus, the last major Grecian philosopher, who lived about 450 AD wrote ( see or or many other beginnings ) : –

Not much younger than these [ students of Plato ] is Euclid, who put together the “ Elementss ” , set uping in order many of Eudoxus ‘s theorems, honing many of Theaetetus ‘s, and besides conveying to incontrovertible presentation the things which had been merely slackly proved by his predecessors. This adult male lived in the clip of the first Ptolemy ; for Archimedes, who followed closely upon the first Ptolemy makes reference of Euclid, and further they say that Ptolemy one time asked him if there were a shorted manner to analyze geometry than the Elementss, to which he replied that there was no royal route to geometry. He is hence younger than Plato ‘s circle, but older than Eratosthenes and Archimedes ; for these were coevalss, as Eratosthenes someplace says. In his purpose he was a Platonist, being in understanding with this doctrine, whence he made the terminal of the whole “ Elementss ” the building of the alleged Platonic figures.

There is other information about Euclid given by certain writers but it is non thought to be dependable. Two different types of this excess information exists. The first type of excess information is that given by Arabian writers who province that Euclid was the boy of Naucrates and that he was born in Tyre. It is believed by historiographers of mathematics that this is wholly fabricated and was simply invented by the writers.

The 2nd type of information is that Euclid was born at Megara. This is due to an mistake on the portion of the writers who foremost gave this information. In fact there was a Euclid of Megara, who was a philosopher who lived about 100 old ages before the mathematician Euclid of Alexandria. It is non rather the happenstance that it might look that there were two learned work forces called Euclid. In fact Euclid was a really common name around this period and this is one farther complication that makes it hard to detect information refering Euclid of Alexandria since there are mentions to legion work forces called Euclid in the literature of this period.

Returning to the citation from Proclus given above, the first point to do is that there is nil inconsistent in the dating given. However, although we do non cognize for certain precisely what mention to Euclid in Archimedes ‘ work Proclus is mentioning to, in what has come down to us there is merely one mention to Euclid and this occurs in On the domain and the cylinder. The obvious decision, hence, is that all is good with the statement of Proclus and this was assumed until challenged by Hjelmslev in. He argued that the mention to Euclid was added to Archimedes book at a ulterior phase, and so it is a instead surprising mention. It was non the tradition of the clip to give such mentions, furthermore there are many other topographic points in Archimedes where it would be appropriate to mention to Euclid and there is no such mention. Despite Hjelmslev ‘s claims that the transition has been added subsequently, Bulmer-Thomas writes in: –

Although it is no longer possible to trust on this mention, a general consideration of Euclid ‘s plants… still shows that he must hold written after such students of Plato as Eudoxus and before Archimedes.

For farther treatment on dating Euclid, see for illustration. This is far from an terminal to the statements about Euclid the mathematician. The state of affairs is best summed up by Itard who gives three possible hypotheses.

( I ) Euclid was an historical character who wrote the Elementss and the other plants attributed to him.

( two ) Euclid was the leader of a squad of mathematicians working at Alexandria. They all contributed to composing the ‘complete plants of Euclid ‘ , even go oning to compose books under Euclid ‘s name after his decease.

( three ) Euclid was non an historical character. The ‘complete plants of Euclid ‘ were written by a squad of mathematicians at Alexandria who took the name Euclid from the historical character Euclid of Megara who had lived about 100 old ages earlier.

It is deserving noting that Itard, who accepts Hjelmslev ‘s claims that the transition about Euclid was added to Archimedes, favours the second of the three possibilities that we listed supra. We should, nevertheless, make some remarks on the three possibilities which, it is just to state, sum up reasonably good all possible current theories.

There is some strong grounds to accept ( I ) . It was accepted without inquiry by everyone for over 2000 old ages and there is small grounds which is inconsistent with this hypothesis. It is true that there are differences in manner between some of the books of the Elements yet many writers vary their manner. Again the fact that Euclid doubtless based the Elementss on old plants means that it would be instead singular if no hint of the manner of the original writer remained.

Even if we accept ( I ) so there is small uncertainty that Euclid built up a vigorous school of mathematics at Alexandria. He hence would hold had some able students who may hold helped out in composing the books. However hypothesis ( two ) goes much further than this and would propose that different books were written by different mathematicians. Other than the differences in manner referred to above, there is small direct grounds of this.

Although on the face of it ( three ) might look the most notional of the three suggestions, however the twentieth century illustration of Bourbaki shows that it is far from impossible. Henri Cartan, Andr & # 233 ; Weil, Jean Dieudonn & # 233 ; , Claude Chevalley, and Alexander Grothendieck wrote jointly under the name of Bourbaki and Bourbaki ‘s El & # 233 ; ments de math & # 233 ; matique contains more than 30 volumes. Of class if ( three ) were the right hypothesis so Apollonius, who studied with the students of Euclid in Alexandria, must hold known there was no individual ‘Euclid ‘ but the fact that he wrote: –

… . Euclid did non work out the syntheses of the venue with regard to three and four lines, but merely a opportunity part of it…

surely does non turn out that Euclid was an historical character since there are many similar mentions to Bourbaki by mathematicians who knew absolutely good that Bourbaki was fabricated. Nevertheless the mathematicians who made up the Bourbaki squad are all good known in their ain right and this may be the greatest statement against hypothesis ( three ) in that the ‘Euclid squad ‘ would hold to hold consisted of outstanding mathematicians. So who were they?

We shall presume in this article that hypothesis ( I ) is true but, holding no cognition of Euclid, we must concentrate on his plants after doing a few remarks on possible historical events. Euclid must hold studied in Plato ‘s Academy in Athens to hold learnt of the geometry of Eudoxus and Theaetetus of which he was so familiar.

None of Euclid ‘s plants have a foreword, at least none has come down to us so it is extremely improbable that any of all time existed, so we can non see any of his character, as we can of some other Grecian mathematicians, from the nature of their forewords. Pappus writes ( see for illustration ) that Euclid was: –

… most just and good fain towards all who were able in any step to progress mathematics, careful in no manner to give offense, and although an exact bookman non boasting himself.

Some claim these words have been added to Pappus, and surely the point of the transition ( in a continuance which we have non quoted ) is to talk harshly ( and about surely below the belt ) of Apollonius. The image of Euclid drawn by Pappus is, nevertheless, surely in line with the grounds from his mathematical texts. Another narrative told by Stobaeus is the followers: –

… person who had begun to larn geometry with Euclid, when he had learnt the first theorem, asked Euclid “ What shall I acquire by larning these things? ” Euclid called his slave and said “ Give him threepence since he must do addition out of what he learns ” .

Euclid ‘s most celebrated work is his treatise on mathematics The Elementss. The book was a digest of cognition that became the Centre of mathematical instruction for 2000 old ages. Probably no consequences in The Elementss were foremost proved by Euclid but the administration of the stuff and its expounding are surely due to him. In fact there is ample grounds that Euclid is utilizing earlier text editions as he writes the Elementss since he introduces rather a figure of definitions which are ne’er used such as that of an oblong, a rhombu

s, and a rhomboid.

The Elements begins with definitions and five posits. The first three posits are posits of building, for illustration the first posit provinces that it is possible to pull a consecutive line between any two points. These posits besides implicitly assume the being of points, lines and circles and so the being of other geometric objects are deduced from the fact that these exist. There are other premises in the posits which are non expressed. For illustration it is assumed that there is a alone line fall ining any two points. Similarly postulates two and three, on bring forthing consecutive lines and pulling circles, severally, assume the singularity of the objects the possibility of whose building is being postulated.

The 4th and 5th posits are of a different nature. Contend four provinces that all right angles are equal. This may look “ obvious ” but it really assumes that infinite in homogenous – by this we mean that a figure will be independent of the place in infinite in which it is placed. The celebrated fifth, or parallel, contend provinces that one and merely one line can be drawn through a point analogue to a given line. Euclid ‘s determination to do this a posit led to Euclidean geometry. It was non until the nineteenth century that this posit was dropped and non-euclidean geometries were studied.

There are besides maxims which Euclid calls ‘common impressions ‘ . These are non specific geometrical belongingss but instead general premises which allow mathematics to continue as a deductive scientific discipline. For illustration: –

Thingss which are equal to the same thing are equal to each other.

Zeno of Sidon, about 250 old ages after Euclid wrote the Elementss, seems to hold been the first to demo that Euclid ‘s propositions were non deduced from the posits and maxims entirely, and Euclid does do other elusive premises.

The Elements is divided into 13 books. Books one to six trade with plane geometry. In peculiar books one and two set out basic belongingss of trigons, analogues, parallelograms, rectangles and squares. Book three surveies belongingss of the circle while book four trades with jobs about circles and is thought mostly to put out work of the followings of Pythagoras. Book five lays out the work of Eudoxus on proportion applied to commensurable and incommensurable magnitudes. Heath says: –

Grecian mathematics can tout no finer discovery than this theory, which put on a sound picking so much of geometry as depended on the usage of proportion.

Book six looks at applications of the consequences of book five to shave geometry.

Books seven to nine trade with figure theory. In peculiar book seven is a self-contained debut to figure theory and contains the Euclidean algorithm for happening the greatest common factor of two Numberss. Book eight looks at Numberss in geometrical patterned advance but van der Waerden writes in that it contains: –

… cumbersome dictions, gratuitous repeats, and even logical false beliefs. Apparently Euclid ‘s expounding excelled merely in those parts in which he had first-class beginnings at his disposal.

Book 10 trades with the theory of irrational Numberss and is chiefly the work of Theaetetus. Euclid changed the cogent evidence of several theorems in this book so that they fitted the new definition of proportion given by Eudoxus.

Books eleven to thirteen trade with 3-dimensional geometry. In book thirteen the basic definitions needed for the three books together are given. The theorems so follow a reasonably similar form to the planar parallels antecedently given in books one and four. The chief consequences of book 12 are that circles are to one another as the squares of their diameters and that domains are to each other as the regular hexahedrons of their diameters. These consequences are surely due to Eudoxus. Euclid proves these theorems utilizing the “ method of exhaustion ” as invented by Eudoxus. The Elements ends with book 13 which discusses the belongingss of the five regular polyhedra and gives a cogent evidence that there are exactly five. This book appears to be based mostly on an earlier treatise by Theaetetus.

Euclid ‘s Elementss is singular for the lucidity with which the theorems are stated and proved. The criterion of cogency was to go a end for the discoverers of the concretion centuries subsequently. As Heath writes in: –

This fantastic book, with all its imperfectnesss, which are so little plenty when history is taken of the day of the month it appeared, is and will doubtless stay the greatest mathematical text edition of all clip. … Even in Greek times the most complete mathematicians occupied themselves with it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries ; Theon of Alexandria re-edited it, changing the linguistic communication here and at that place, largely with a position to greater clarity and consistence…

It is a absorbing narrative how the Elements has survived from Euclid ‘s clip and this is told good by Fowler in. He describes the earliest stuff associating to the Elementss which has survived: –

Our earliest glance of Euclidean stuff will be the most singular for a thousand old ages, six fragmental ostraca incorporating text and a figure… found on Elephantine Island in 1906/07 and 1907/08… These texts are early, though still more than 100 old ages after the decease of Plato ( they are dated on palaeographic evidences to the 3rd one-fourth of the 3rd century BC ) ; advanced ( they deal with the consequences found in the “ Elementss ” [ book 13 ] … on the Pentagon, hexagon, decagon, and icosahedron ) ; and they do non follow the text of the Elementss. … So they give grounds of person in the 3rd century BC, located more than 500 stat mis south of Alexandria, working through this hard stuff… this may be an effort to understand the mathematics, and non a slavish copying…

The following fragment that we have day of the months from 75 – 125 AD and once more appears to be notes by person seeking to understand the stuff of the Elementss.

More than one 1000 editions of The Elementss have been published since it was foremost printed in 1482. Heath discusses many of the editions and describes the likely alterations to the text over the old ages.

B L new wave der Waerden assesses the importance of the Elementss in: –

About from the clip of its authorship and enduring about to the present, the Elements has exerted a uninterrupted and major influence on human personal businesss. It was the primary beginning of geometric logical thinking, theorems, and methods at least until the coming of non-Euclidean geometry in the nineteenth century. It is sometimes said that, following to the Bible, the “ Elementss ” may be the most translated, published, and studied of all the books produced in the Western universe.

Euclid besides wrote the following books which have survived: Datas ( with 94 propositions ) , which looks at what belongingss of figures can be deduced when other belongingss are given ; On Divisions which looks at buildings to split a figure into two parts with countries of given ratio ; Optics which is the first Grecian work on position ; and Phaenomena which is an simple debut to mathematical uranology and gives consequences on the times stars in certain places will lift and put. Euclid ‘s following books have all been lost: Surface Loci ( two books ) , Porisms ( a three book work with, harmonizing to Pappus, 171 theorems and 38 lemmas ) , Conics ( four books ) , Book of Fallacies and Elementss of Music. The Book of Fallacies is described by Proclus: –

Since many things seem to conform with the truth and to follow from scientific rules, but lead astray from the rules and lead on the more superficial, [ Euclid ] has handed down methods for the clear-sighted apprehension of these affairs besides… The treatise in which he gave this machinery to us is entitled Fallacies, reciting in order the assorted sorts, exerting our intelligence in each instance by theorems of all kinds, puting the true side by side with the false, and uniting the defense of the mistake with practical illustration.

Elementss 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 writers attributed to Euclid, but it is now thought that they are non the work on music referred to by Proclus.

Euclid may non hold been a first category mathematician but the long permanent nature of The Elementss must do him the taking mathematics instructor of antiquity or possibly of all clip. As a concluding personal note allow me add that my [ EFR ] ain debut to mathematics at school in the 1950s was from an edition of portion of Euclid ‘s Elementss and the work provided a logical footing for mathematics and the construct of cogent evidence which seem to be missing in school mathematics today.

J J O’Connor and E F Robertson