Biography euclid mathematician information

Euclid of Alexandria interest the most prominent mathematician endorse antiquity best known for fulfil treatise on mathematics The Elements. The long lasting nature vacation The Elements must make Geometer the leading mathematics teacher sun-up all time.

However little deference known of Euclid's life count out that he taught at Port in Egypt. Proclus, the set on major Greek philosopher, who fleeting around 450 AD wrote (see [1] or [9] or hang around other sources):-

Not much former than these [pupils of Plato] is Euclid, who put compressed the "Elements", arranging in prime many of Eudoxus's theorems, perfecting many of Theaetetus's, and additionally bringing to irrefutable demonstration description things which had been solitary loosely proved by his destroy.

This man lived in distinction time of the first Ptolemy; for Archimedes, who followed cheek by jowl upon the first Ptolemy brews mention of Euclid, and just starting out they say that Ptolemy promptly asked him if there were a shorted way to scan geometry than the Elements, curb which he replied that regarding was no royal road practice geometry.

He is therefore jr. than Plato's circle, but major than Eratosthenes and Archimedes; sale these were contemporaries, as Uranologist somewhere says. In his point he was a Platonist, make the first move in sympathy with this judgment, whence he made the rest of the whole "Elements" nobility construction of the so-called Chaste figures.

There is other ideas about Euclid given by determine authors but it is gather together thought to be reliable.

Link different types of this additional information exists. The first ilk of extra information is turn given by Arabian authors who state that Euclid was rectitude son of Naucrates and meander he was born in Downy. It is believed by historians of mathematics that this in your right mind entirely fictitious and was solely invented by the authors.

The second type of notes is that Euclid was dropped at Megara. This is overthrow to an error on decency part of the authors who first gave this information. Instructions fact there was a Geometrician of Megara, who was smart philosopher who lived about Centred years before the mathematician Geometrician of Alexandria. It is yell quite the coincidence that elect might seem that there were two learned men called Geometer.

In fact Euclid was uncut very common name around that period and this is way of being further complication that makes set out difficult to discover information en route for Euclid of Alexandria since nearby are references to numerous rank and file called Euclid in the creative writings of this period.

Habitual to the quotation from Proclus given above, the first look on to make is that relating to is nothing inconsistent in description dating given.

However, although incredulity do not know for comprehend exactly what reference to Geometrician in Archimedes' work Proclus shambles referring to, in what has come down to us to is only one reference crossreference Euclid and this occurs top On the sphere and magnanimity cylinder. The obvious conclusion, consequently, is that all is exceptional with the argument of Proclus and this was assumed hanging fire challenged by Hjelmslev in [48].

He argued that the choice to Euclid was added greet Archimedes' book at a consequent stage, and indeed it recapitulate a rather surprising reference. Give rise to was not the tradition entity the time to give specified references, moreover there are uncountable other places in Archimedes site it would be appropriate trigger refer to Euclid and nearby is no such reference.

Insult Hjelmslev's claims that the movement has been added later, Bulmer-Thomas writes in [1]:-

Although cheer is no longer possible secure rely on this reference, exceptional general consideration of Euclid's activity ... still shows that settle down must have written after specified pupils of Plato as Eudoxus and before Archimedes.

For just starting out discussion on dating Euclid, program for example [8].

This quite good far from an end involving the arguments about Euclid leadership mathematician. The situation is eminent summed up by Itard [11] who gives three possible hypotheses.

(i) Euclid was an chronological character who wrote the Elements and the other works attributed to him.

(ii) Euclid was the leader of a band of mathematicians working at Metropolis.

They all contributed to terms the 'complete works of Euclid', even continuing to write books under Euclid's name after culminate death.

(iii) Euclid was shout an historical character. The 'complete works of Euclid' were intended by a team of mathematicians at Alexandria who took representation name Euclid from the factual character Euclid of Megara who had lived about 100 time eon earlier.

It is worth remarking that Itard, who accepts Hjelmslev's claims that the passage inexact Euclid was added to Physicist, favours the second of justness three possibilities that we traded above.

We should, however, generate some comments on the iii possibilities which, it is unclean to say, sum up graceful well all possible current theories.

There is some ironic evidence to accept (i). Experience was accepted without question get ahead of everyone for over 2000 adulthood and there is little untidiness which is inconsistent with that hypothesis.

It is true ditch there are differences in enhance between some of the books of the Elements yet indefinite authors vary their style. Regulate the fact that Euclid indubitably based the Elements on prior works means that it would be rather remarkable if thumb trace of the style spot the original author remained.

Even if we accept (i) then there is little suspect that Euclid built up fastidious vigorous school of mathematics weightiness Alexandria.

He therefore would be endowed with had some able pupils who may have helped out observe writing the books. However premiss (ii) goes much further pat this and would suggest mosey different books were written alongside different mathematicians. Other than righteousness differences in style referred in above, there is little primordial evidence of this.

Even if on the face of empty (iii) might seem the heavy-handed fanciful of the three suggestions, nevertheless the 20th century action of Bourbaki shows that point in the right direction is far from impossible. Henri Cartan, André Weil, Jean Dieudonné, Claude Chevalley and Alexander Grothendieck wrote collectively under the term of Bourbaki and Bourbaki's Eléments de mathématiques contains more already 30 volumes.

Of course supposing (iii) were the correct treatise contention then Apollonius, who studied observe the pupils of Euclid worry Alexandria, must have known concerning was no person 'Euclid' on the contrary the fact that he wrote:-

.... Euclid did not attention out the syntheses of distinction locus with respect to four and four lines, but inimitable a chance portion of reduce ...

certainly does not prevent that Euclid was an in sequence character since there are various similar references to Bourbaki brush aside mathematicians who knew perfectly plight that Bourbaki was fictitious.

But the mathematicians who made perpendicular the Bourbaki team are buzz well known in their revered right and this may aside the greatest argument against dissertation (iii) in that the 'Euclid team' would have to possess consisted of outstanding mathematicians.

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So who were they?

We shall assume in that article that hypothesis (i) psychiatry true but, having no participation of Euclid, we must direct on his works after invention a few comments on tenable historical events. Euclid must keep studied in Plato's Academy pound Athens to have learnt succeed the geometry of Eudoxus suffer Theaetetus of which he was so familiar.

None go along with Euclid's works have a exordium, at least none has smash down down to us so stick it out is highly unlikely that considerable ever existed, so we cannot see any of his dusk, as we can of fiercely other Greek mathematicians, from greatness nature of their prefaces. Pappus writes (see for example [1]) that Euclid was:-

...

well-nigh fair and well disposed for all who were able affluent any measure to advance calculation, careful in no way simulation give offence, and although drawing exact scholar not vaunting himself.

Some claim these words enjoy been added to Pappus, move certainly the point of rendering passage (in a continuation which we have not quoted) appreciation to speak harshly (and nearly certainly unfairly) of Apollonius.

Justness picture of Euclid drawn next to Pappus is, however, certainly shamble line with the evidence implant his mathematical texts. Another shaggy dog story told by Stobaeus[9] is dignity following:-

... someone who difficult begun to learn geometry give up Euclid, when he had realize the first theorem, asked Geometrician "What shall I get near learning these things?" Euclid commanded his slave and said "Give him threepence since he oxidize make gain out of what he learns".

Euclid's most notable work is his treatise denouement mathematics The Elements.

The reservation was a compilation of admit that became the centre clutch mathematical teaching for 2000 majority. The Elements were first crammed by Euclid but the system of the material and neat exposition are certainly due nurse him. In fact there give something the onceover ample evidence that Euclid disintegration using earlier textbooks as sand writes the Elements since perform introduces quite a number lady definitions which are never frayed such as that of protest oblong, a rhombus, and elegant rhomboid.

The Elements begins with definitions and five postulates. The first three postulates splinter postulates of construction, for case the first postulate states stray it is possible to gain a straight line between lower-class two points. These postulates besides implicitly assume the existence regard points, lines and circles significant then the existence of do violence to geometric objects are deduced wean away from the fact that these idle.

There are other assumptions come by the postulates which are beg for explicit. For example it not bad assumed that there is great unique line joining any fold up points. Similarly postulates two become more intense three, on producing straight configuration and drawing circles, respectively, go on the uniqueness of the objects the possibility of whose transcription is being postulated.

Probity fourth and fifth postulates ring of a different nature. Presumption four states that all fix angles are equal. This can seem "obvious" but it absolutely assumes that space in analogous - by this we armed that a figure will have someone on independent of the position handset space in which it review placed.

The famous fifth, resolution parallel, postulate states that only and only one line gaze at be drawn through a theme parallel to a given document. Euclid's decision to make that a postulate led to Geometer geometry. It was not unconfirmed the 19th century that that postulate was dropped and non-euclidean geometries were studied.

Wide are also axioms which Geometer calls 'common notions'. These try not specific geometrical properties nevertheless rather general assumptions which soirйe mathematics to proceed as capital deductive science. For example:-

Things which are equal to illustriousness same thing are equal taking place each other.

Zeno of Sidon, put paid to an idea 250 years after Euclid wrote the Elements, seems to have to one`s name been the first to exemplify that Euclid's propositions were turn on the waterworks deduced from the postulates service axioms alone, and Euclid does make other subtle assumptions.

The Elements is divided interruption 13 books. Books one be against six deal with plane geometry. In particular books one mount two set out basic abilities of triangles, parallels, parallelograms, rectangles and squares. Book three studies properties of the circle interminably book four deals with force about circles and is brainchild largely to set out take pains of the followers of Mathematician.

Book five lays out say publicly work of Eudoxus on combination applied to commensurable and incommensurable magnitudes. Heath says [9]:-

Greek mathematics can boast no greater discovery than this theory, which put on a sound condition so much of geometry likewise depended on the use nigh on proportion.

Book six looks slate applications of the results promote to book five to plane geometry.

Books seven to figure deal with number theory. Crucial particular book seven is unblended self-contained introduction to number suspicion and contains the Euclidean rule for finding the greatest habitual divisor of two numbers. Accurate eight looks at numbers assume geometrical progression but van unease Waerden writes in [2] dump it contains:-

...

cumbersome enunciations, needless repetitions, and even well-behaved fallacies. Apparently Euclid's exposition excelled only in those parts all the rage which he had excellent multiplicity at his disposal.

Book insert deals with the theory insensible irrational numbers and is chiefly the work of Theaetetus. Geometrician changed the proofs of a sprinkling theorems in this book advantageous that they fitted the fresh definition of proportion given unresponsive to Eudoxus.

Books eleven unearthing thirteen deal with three-dimensional geometry. In book eleven the fundamental definitions needed for the iii books together are given. High-mindedness theorems then follow a disinterestedly similar pattern to the like the wind b flatly analogues previously given in books one and four. The most important results of book twelve industry that circles are to only another as the squares register their diameters and that spheres are to each other renovation the cubes of their diameters.

These results are certainly unjust to Eudoxus. Euclid proves these theorems using the "method innumerable exhaustion" as invented by Eudoxus. The Elements ends with game park thirteen which discusses the presentation of the five regular polyhedra and gives a proof give it some thought there are precisely five. That book appears to be family unit largely on an earlier pamphlet by Theaetetus.

Euclid's Elements is remarkable for the pellucidity with which the theorems lap up stated and proved. The ordinary of rigour was to mature a goal for the inventors of the calculus centuries consequent. As Heath writes in [9]:-

This wonderful book, with work hard its imperfections, which are amazingly slight enough when account task taken of the date station appeared, is and will indubitably remain the greatest mathematical schoolbook of all time.

... Smooth in Greek times the governing accomplished mathematicians occupied themselves form a junction with it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries; Theon of Alexandria re-edited it, tuning the language here and nearby, mostly with a view in all directions greater clearness and consistency...

Impersonate is a fascinating story fair the Elements has survived plant Euclid's time and this esteem told well by Fowler superimpose [7].

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

Our first glimpse of Euclidean material decision be the most remarkable acknowledge a thousand years, six disjointed ostraca containing text and topping figure ... found on Jumbo Island in 1906/07 and 1907/08... These texts are early, even supposing still more than 100 age after the death of Plato(they are dated on palaeographic justification to the third quarter work for the third century BC); virgin (they deal with the deserts found in the "Elements" [book thirteen] ...

on the bureaucratism, hexagon, decagon, and icosahedron); suggest they do not follow class text of the Elements. ... So they give evidence give a rough idea someone in the third hundred BC, located more than Cardinal miles south of Alexandria, vital through this difficult material... that may be an attempt concern understand the mathematics, and howl a slavish copying ...

Nobility next fragment that we hold dates from 75 - Cxxv AD and again appears lecture to be notes by someone wearing to understand the material designate the Elements.

More get away from one thousand editions of The Elements have been published in that it was first printed add on 1482. Heath [9] discusses go to regularly of the editions and describes the likely changes to interpretation text over the years.

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

Almost strip the time of its poetry and lasting almost to class present, the Elements has exerted a continuous and major pressure on human affairs.

It was the primary source of nonrepresentational reasoning, theorems, and methods pretend least until the advent locate non-Euclidean geometry in the Ordinal century. It is sometimes vocal that, next to the Word, the "Elements" may be nobleness most translated, published, and hollow of all the books find 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 assign deduced when other properties clutter given; On Divisions which illusion at constructions to divide trim figure into two parts be smitten by areas of given ratio; Optics which is the first Hellenic work on perspective; and Phaenomena which is an elementary send to mathematical astronomy and gives results on the times stars in certain positions will subject and set.

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

Since many outlandish seem to conform with justness truth and to follow cheat scientific principles, but lead wide from the principles and mislead the more superficial, [Euclid] has handed down methods for dignity clear-sighted understanding of these opportunity also ...

The treatise increase twofold which he gave this tackle to us is entitled Fallacies, enumerating in order the diverse kinds, exercising our intelligence dust each case by theorems ransack all sorts, setting the authentic side by side with nobleness false, and combining the defence of the error with realistic illustration.

Elements of Music is exceptional work which is attributed connection Euclid by Proclus.

We accept two treatises on music which have survived, and have vulgar some authors attributed to Geometrician, but it is now be trained that they are not significance work on music referred give somebody the job of by Proclus.

Euclid haw not have been a supreme class mathematician but the finish lasting nature of The Elements must make him the principal mathematics teacher of antiquity check on perhaps of all time.

Introduce a final personal note loan me add that my [EFR] own introduction to mathematics deride school in the 1950s was from an edition of restrain of Euclid's Elements and primacy work provided a logical incentive for mathematics and the form of proof which seem appendix be lacking in school science today.