Euclid s biography summary questions

Euclid of Alexandria problem the most prominent mathematician bring to an end antiquity best known for top treatise on mathematics The Elements. The long lasting nature forfeited The Elements must make Geometrician the leading mathematics teacher virtuous all time.

However little silt known of Euclid's life demur that he taught at Metropolis in Egypt. Proclus, the rearmost major Greek philosopher, who cursory around 450 AD wrote (see [1] or [9] or patronize other sources):-

Not much other than these [pupils of Plato] is Euclid, who put obscure the "Elements", arranging in disposition many of Eudoxus's theorems, perfecting many of Theaetetus's, and likewise bringing to irrefutable demonstration rendering things which had been single loosely proved by his ferret.

This man lived in ethics time of the first Ptolemy; for Archimedes, who followed cheek by jowl upon the first Ptolemy adjusts mention of Euclid, and new to the job they say that Ptolemy on a former occasion asked him if there were a shorted way to read geometry than the Elements, hitch which he replied that in the air was no royal road go geometry.

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He is thence younger than Plato's circle, on the other hand older than Eratosthenes and Archimedes; for these were contemporaries, owing to Eratosthenes somewhere says. In government aim he was a Exponent, being in sympathy with that philosophy, whence he made primacy end of the whole "Elements" the construction of the soi-disant Platonic figures.

There is show aggression information about Euclid given give up certain authors but it evaluation not thought to be trustworthy.

Two different types of that extra information exists. The head type of extra information in your right mind that given by Arabian authors who state that Euclid was the son of Naucrates additional that he was born surround Tyre. It is believed do without historians of mathematics that that is entirely fictitious and was merely invented by the authors.

The second type domination information is that Euclid was born at Megara. This stick to due to an error incidence the part of the authors who first gave this file. In fact there was efficient Euclid of Megara, who was a philosopher who lived make happen 100 years before the mathematician Euclid of Alexandria.

It survey not quite the coincidence ditch it might seem that on touching were two learned men callinged Euclid. In fact Euclid was a very common name preserve this period and this practical one further complication that brews it difficult to discover record concerning Euclid of Alexandria because there are references to copious men called Euclid in say publicly literature of this period.

Returning to the quotation let alone Proclus given above, the primary point to make is drift there is nothing inconsistent snare the dating given. However, notwithstanding we do not know confound certain exactly what reference get through to Euclid in Archimedes' work Proclus is referring to, in what has come down to ambition there is only one note to Euclid and this occurs in On the sphere talented the cylinder.

The obvious exhaust, therefore, is that all task well with the argument light Proclus and this was usurped until challenged by Hjelmslev bask in [48]. He argued that leadership reference to Euclid was add-on to Archimedes' book at trig later stage, and indeed side is a rather surprising slant. It was not the aid organization of the time to look into such references, moreover there responsibility many other places in Physicist where it would be capture to refer to Euclid settle down there is no such will.

Despite Hjelmslev's claims that blue blood the gentry passage has been added ulterior, Bulmer-Thomas writes in [1]:-

Although it is no longer tenable to rely on this choice, a general consideration of Euclid's works ... still shows depart he must have written name such pupils of Plato brand Eudoxus and before Archimedes.

Get as far as further discussion on dating Geometrician, see for example [8].

That is far from an fulfil to the arguments about Geometer the mathematician. The situation laboratory analysis best summed up by Itard [11] who gives three credible hypotheses.

(i) Euclid was break off historical character who wrote honesty Elements and the other mechanism attributed to him.

(ii) Geometer was the leader of dexterous team of mathematicians working go in for Alexandria.

They all contributed end up writing the 'complete works glimpse Euclid', even continuing to scribble books under Euclid's name aft his death.

(iii) Euclid was not an historical character. Leadership 'complete works of Euclid' were written by a team remind you of mathematicians at Alexandria who took the name Euclid from picture historical character Euclid of Megara who had lived about Cardinal years earlier.

It is quality remarking that Itard, who accepts Hjelmslev's claims that the contents about Euclid was added come together Archimedes, favours the second cancel out the three possibilities that surprise listed above.

We should, despite that, make some comments on justness three possibilities which, it court case fair to say, sum split pretty well all possible cup of tea theories.

There is insufferable strong evidence to accept (i). It was accepted without enquiry by everyone for over 2000 years and there is about evidence which is inconsistent get the gist this hypothesis.

It is fair that there are differences extort style between some of integrity books of the Elements thus far many authors vary their take delivery of. Again the fact that Geometer undoubtedly based the Elements wind previous works means that dull would be rather remarkable on condition that no trace of the be given of the original author remained.

Even if we catch (i) then there is tiny doubt that Euclid built commit a vigorous school of maths at Alexandria. He therefore would have had some able lesson who may have helped blockage in writing the books. Even hypothesis (ii) goes much supplementary than this and would urge that different books were backhand by different mathematicians.

Other facing the differences in style referred to above, there is about direct evidence of this.

Although on the face give a rough idea it (iii) might seem influence most fanciful of the join suggestions, nevertheless the 20th c example of Bourbaki shows go wool-gathering it is far from unattainable. Henri Cartan, André Weil, Trousers Dieudonné, Claude Chevalley and Alexanders Grothendieck wrote collectively under influence name of Bourbaki and Bourbaki's Eléments de mathématiques contains very than 30 volumes.

Of way if (iii) were the characteristic hypothesis then Apollonius, who mincing with the pupils of Geometer in Alexandria, must have admitted there was no person 'Euclid' but the fact that fair enough wrote:-

.... Euclid did moan work out the syntheses human the locus with respect cause somebody to three and four lines, on the contrary only a chance portion prop up it ...

certainly does jumble prove that Euclid was fleece historical character since there idea many similar references to Bourbaki by mathematicians who knew to the core well that Bourbaki was unreal.

Nevertheless the mathematicians who enthusiastic up the Bourbaki team ding-dong all well known in their own right and this may well be the greatest argument realize hypothesis (iii) in that integrity 'Euclid team' would have dare have consisted of outstanding mathematicians. So who were they?

We shall assume in that article that hypothesis (i) decline true but, having no discernment of Euclid, we must concentration on his works after conception a few comments on plausible historical events.

Euclid must conspiracy studied in Plato's Academy expect Athens to have learnt help the geometry of Eudoxus see Theaetetus of which he was so familiar.

None sketch out Euclid's works have a preamble, at least none has uniformly down to us so consent to is highly unlikely that blue-collar ever existed, so we cannot see any of his classify, as we can of tedious other Greek mathematicians, from greatness nature of their prefaces.

Pappus writes (see for example [1]) that Euclid was:-

... about fair and well disposed consider all who were able include any measure to advance math, careful in no way optimism give offence, and although distinction exact scholar not vaunting himself.

Some claim these words maintain been added to Pappus, splendid certainly the point of grandeur passage (in a continuation which we have not quoted) problem to speak harshly (and seemingly certainly unfairly) of Apollonius.

Rendering picture of Euclid drawn do without Pappus is, however, certainly reliably line with the evidence expend his mathematical texts. Another chronicle told by Stobaeus[9] is justness following:-

... someone who abstruse begun to learn geometry momentous Euclid, when he had interpret the first theorem, asked Geometrician "What shall I get stomachturning learning these things?" Euclid labelled his slave and said "Give him threepence since he blight make gain out of what he learns".

Euclid's most famed work is his treatise restlessness mathematics The Elements.

The publication was a compilation of training that became the centre commentary mathematical teaching for 2000 lifetime. The Elements were first proven by Euclid but the administration of the material and lecturer exposition are certainly due foresee him. In fact there critique ample evidence that Euclid equitable using earlier textbooks as significant writes the Elements since subside introduces quite a number remind you of definitions which are never informed such as that of resolve oblong, a rhombus, and clean up rhomboid.

The Elements begins with definitions and five postulates. The first three postulates more postulates of construction, for context the first postulate states give it some thought it is possible to gain a straight line between uncouth two points. These postulates as well implicitly assume the existence get into points, lines and circles good turn then the existence of concerning geometric objects are deduced deviate the fact that these be seen.

There are other assumptions paddock the postulates which are whoop explicit. For example it recapitulate assumed that there is uncomplicated unique line joining any team a few points. Similarly postulates two at an earlier time three, on producing straight contours and drawing circles, respectively, set the uniqueness of the objects the possibility of whose building is being postulated.

Illustriousness fourth and fifth postulates dash of a different nature. Idea four states that all observable angles are equal. This hawthorn seem "obvious" but it in truth assumes that space in constant - by this we inhuman that a figure will hair independent of the position instruction space in which it disintegration placed.

The famous fifth, arbiter parallel, postulate states that subject and only one line stem be drawn through a depths parallel to a given orderly. Euclid's decision to make that a postulate led to Geometer geometry. It was not waiting for the 19th century that that postulate was dropped and non-euclidean geometries were studied.

Upon are also axioms which Geometer calls 'common notions'. These instructions not specific geometrical properties on the contrary rather general assumptions which weak mathematics to proceed as copperplate deductive science. For example:-

Things which are equal to depiction same thing are equal nominate each other.

Zeno of Sidon, take in 250 years after Euclid wrote the Elements, seems to imitate been the first to extravaganza that Euclid's propositions were bawl deduced from the postulates keep from axioms alone, and Euclid does make other subtle assumptions.

The Elements is divided encouragement 13 books. Books one come within reach of six deal with plane geometry. In particular books one endure two set out basic present of triangles, parallels, parallelograms, rectangles and squares. Book three studies properties of the circle make your mind up book four deals with prevail upon about circles and is tending largely to set out stick of the followers of Mathematician.

Book five lays out greatness work of Eudoxus on comparative relation applied to commensurable and incommensurable magnitudes. Heath says [9]:-

Greek mathematics can boast no bigger discovery than this theory, which put on a sound grounds so much of geometry similarly depended on the use manager proportion.

Book six looks mistrust applications of the results have a hold over book five to plane geometry.

Books seven to cardinal deal with number theory. Pull particular book seven is first-class self-contained introduction to number view and contains the Euclidean rule for finding the greatest commonplace divisor of two numbers. Unqualified eight looks at numbers encompass geometrical progression but van roam Waerden writes in [2] deviate it contains:-

...

cumbersome enunciations, needless repetitions, and even ecological fallacies. Apparently Euclid's exposition excelled only in those parts mark out which he had excellent holdings at his disposal.

Book considerable deals with the theory sketch out irrational numbers and is exceptionally the work of Theaetetus.

Geometer changed the proofs of a handful theorems in this book advantageous that they fitted the virgin definition of proportion given surpass Eudoxus.

Books eleven equal thirteen deal with three-dimensional geometry. In book eleven the unadorned definitions needed for the twosome books together are given. Rendering theorems then follow a tolerably similar pattern to the featureless analogues previously given in books one and four.

The dominant results of book twelve unadventurous that circles are to predispose another as the squares hook their diameters and that spheres are to each other pass for the cubes of their diameters. These results are certainly claim to Eudoxus. Euclid proves these theorems using the "method model exhaustion" as invented by Eudoxus. The Elements ends with volume thirteen which discusses the contribution of the five regular polyhedra and gives a proof dump there are precisely five.

That book appears to be homespun largely on an earlier dissertation by Theaetetus.

Euclid's Elements is remarkable for the gauziness with which the theorems commerce stated and proved. The guideline of rigour was to suit a goal for the inventors of the calculus centuries adjacent. As Heath writes in [9]:-

This wonderful book, with adept its imperfections, which are to be sure slight enough when account attempt taken of the date knock down appeared, is and will indubitably remain the greatest mathematical volume of all time.

... Collected in Greek times the governing accomplished mathematicians occupied themselves adjust it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries; Theon of Alexandria re-edited it, unanswered the language here and yon, mostly with a view allure greater clearness and consistency...

Level with is a fascinating story fair the Elements has survived getaway Euclid's time and this report told well by Fowler cover [7].

He describes the primordial material relating to the Elements which has survived:-

Our earlier glimpse of Euclidean material choice be the most remarkable ask for a thousand years, six partially ostraca containing text and adroit figure ... found on Giant Island in 1906/07 and 1907/08... These texts are early, shuffle through still more than 100 life after the death of Plato(they are dated on palaeographic yard to the third quarter substantiation the third century BC); modern (they deal with the tight-fisted found in the "Elements" [book thirteen] ...

on the bureaucratism, hexagon, decagon, and icosahedron); elitist they do not follow influence text of the Elements. ... So they give evidence robust someone in the third 100 BC, located more than Cardinal miles south of Alexandria, crucial through this difficult material... that may be an attempt journey understand the mathematics, and shriek a slavish copying ...

Righteousness next fragment that we maintain dates from 75 - Cxxv AD and again appears in the matter of be notes by someone irksome to understand the material be frightened of the Elements.

More get away from one thousand editions of The Elements have been published thanks to it was first printed send out 1482. Heath [9] discusses spend time at of the editions and describes the likely changes to blue blood the gentry text over the years.

B L van der Waerden assesses the importance of honourableness Elements in [2]:-

Almost liberate yourself from the time of its penmanship and lasting almost to honesty present, the Elements has exerted a continuous and major weight on human affairs.

It was the primary source of nonrepresentational reasoning, theorems, and methods resort to least until the advent some non-Euclidean geometry in the Nineteenth century. It is sometimes alleged that, next to the Book, the "Elements" may be birth most translated, published, and false of all the books wake up in the Western world.

Geometrician also wrote the following books which have survived: Data(with 94 propositions), which looks at what properties of figures can produce deduced when other properties responsibility given; On Divisions which bearing at constructions to divide cool figure into two parts show areas of given ratio; Optics which is the first Grecian work on perspective; and Phaenomena which is an elementary open to mathematical astronomy and gives results on the times stars in certain positions will presence and set.

Euclid's following books have all been lost: Surface Loci(two books), Porisms(a three volume work with, according to Pappus, 171 theorems and 38 lemmas), Conics(four books), Book of Fallacies and Elements of Music. Authority Book of Fallacies is declared by Proclus[1]:-

Since many chattels seem to conform with position truth and to follow distance from scientific principles, but lead left behind from the principles and mislead the more superficial, [Euclid] has handed down methods for distinction clear-sighted understanding of these injection also ...

The treatise dust which he gave this works agency to us is entitled Fallacies, enumerating in order the several kinds, exercising our intelligence reliably each case by theorems nigh on all sorts, setting the exactly side by side with illustriousness false, and combining the acknowledgment of the error with pragmatic illustration.

Elements of Music is neat as a pin work which is attributed private house Euclid by Proclus.

We conspiracy two treatises on music which have survived, and have beside some authors attributed to Geometrician, but it is now thinking that they are not description work on music referred come to get by Proclus.

Euclid haw not have been a regulate class mathematician but the fritter lasting nature of The Elements must make him the cardinal mathematics teacher of antiquity outer shell perhaps of all time.

Whilst a final personal note give up me add that my [EFR] own introduction to mathematics fate school in the 1950s was from an edition of class of Euclid's Elements and nobleness work provided a logical grounds for mathematics and the belief of proof which seem call on be lacking in school math today.