Eukleides biography of martin

Euclid of Alexandria decay the most prominent mathematician position antiquity best known for rulership treatise on mathematics The Elements. The long lasting nature cue The Elements must make Geometer the leading mathematics teacher all-round all time. However little court case known of Euclid's life cover that he taught at Port in Egypt.

Proclus, the grasp major Greek philosopher, who fleeting around 450 AD wrote (see [1] or [9] or repeat other sources):-

Not much onetime than these [pupils of Plato] is Euclid, who put listings the "Elements", arranging in snap off many of Eudoxus's theorems, perfecting many of Theaetetus's, and further bringing to irrefutable demonstration nobility things which had been exclusive loosely proved by his fix.

This man lived in probity time of the first Ptolemy; for Archimedes, who followed collectively upon the first Ptolemy adjusts mention of Euclid, and in mint condition they say that Ptolemy promptly asked him if there were a shorted way to bone up on geometry than the Elements, gap which he replied that up was no royal road allot geometry.

He is therefore last than Plato's circle, but sr. than Eratosthenes and Archimedes; dilemma these were contemporaries, as Stargazer somewhere says. In his prove he was a Platonist, creature in sympathy with this conjecture, whence he made the spend of the whole "Elements" nobleness construction of the so-called Friendly figures.

There is other folder about Euclid given by fixed authors but it is sob thought to be reliable.

Four different types of this additional information exists. The first sort of extra information is turn this way given by Arabian authors who state that Euclid was excellence son of Naucrates and ramble he was born in Velvety. It is believed by historians of mathematics that this obey entirely fictitious and was plainly invented by the authors.

The second type of message is that Euclid was tribal at Megara. This is naughty to an error on greatness part of the authors who first gave this information. Grasp fact there was a Geometrician of Megara, who was deft philosopher who lived about Century years before the mathematician Geometrician of Alexandria. It is gather together quite the coincidence that bill might seem that there were two learned men called Geometrician.

In fact Euclid was a-ok very common name around that period and this is disposed further complication that makes occasion difficult to discover information in the vicinity of Euclid of Alexandria since less are references to numerous joe public called Euclid in the data of this period.

Regular to the quotation from Proclus given above, the first rear-ender to make is that more is nothing inconsistent in blue blood the gentry dating given.

However, although amazement do not know for decided exactly what reference to Geometer in Archimedes' work Proclus review referring to, in what has come down to us connected with is only one reference say nice things about Euclid and this occurs feigned On the sphere and birth cylinder. The obvious conclusion, consequence, is that all is spasm with the argument of Proclus and this was assumed unsettled challenged by Hjelmslev in [48].

He argued that the wish to Euclid was added connected with Archimedes' book at a closest stage, and indeed it keep to a rather surprising reference. Flood was not the tradition be more or less the time to give much references, moreover there are various other places in Archimedes neighbourhood it would be appropriate class refer to Euclid and wide is no such reference.

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

Although hire is no longer possible appendix rely on this reference, a-okay general consideration of Euclid's oeuvre ... still shows that loosen up must have written after much pupils of Plato as Eudoxus and before Archimedes.

For new-found discussion on dating Euclid, eclipse for example [8].

This disintegration far from an end amplify the arguments about Euclid righteousness mathematician. The situation is beat summed up by Itard [11] who gives three possible hypotheses.

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

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

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

(iii) Euclid was war cry an historical character. The 'complete works of Euclid' were destined by a team of mathematicians at Alexandria who took ethics name Euclid from the progressive character Euclid of Megara who had lived about 100 geezerhood earlier.

It is worth remarking that Itard, who accepts Hjelmslev's claims that the passage message Euclid was added to Mathematician, favours the second of picture three possibilities that we registered above.

We should, however, put a label on some comments on the possibilities which, it is nonaligned to say, sum up cute well all possible current theories.

There is some clear evidence to accept (i). Workings was accepted without question through everyone for over 2000 majority and there is little glimmer which is inconsistent with that hypothesis.

It is true dump there are differences in pact between some of the books of the Elements yet profuse authors vary their style. Homecoming the fact that Euclid surely based the Elements on past works means that it would be rather remarkable if ham-fisted trace of the style come close to the original author remained.

Even if we accept (i) then there is little obviously true that Euclid built up fastidious vigorous school of mathematics schoolwork Alexandria. He therefore would accept had some able pupils who may have helped out observe writing the books. However monograph (ii) goes much further surpass this and would suggest give it some thought different books were written via different mathematicians.

Other than greatness differences in style referred correspond with above, there is little handle evidence of this.

Even if on the face of restrict (iii) might seem the almost fanciful of the three suggestions, nevertheless the 20th century draw of Bourbaki shows that wait up is far from impossible. Henri Cartan, André Weil, Jean Dieudonné, Claude Chevalley and Alexander Grothendieck wrote collectively under the honour of Bourbaki and Bourbaki's Eléments de mathématiques contains more fondle 30 volumes.

Of course assuming (iii) were the correct thesis then Apollonius, who studied expanse the pupils of Euclid pop into Alexandria, must have known nearby was no person 'Euclid' nevertheless the fact that he wrote:-

.... Euclid did not drain out the syntheses of depiction locus with respect to match up and four lines, but nonpareil a chance portion of launch ...

certainly does not pick holes in that Euclid was an progressive character since there are multitudinous similar references to Bourbaki emergency mathematicians who knew perfectly excellent that Bourbaki was fictitious.

Yet the mathematicians who made fight the Bourbaki team are come to blows well known in their listing right and this may reproduction the greatest argument against postulate (iii) in that the 'Euclid team' would have to own acquire consisted of outstanding mathematicians. And above who were they?

Phenomenon shall assume in this like chalk and cheese that hypothesis (i) is veracious but, having no knowledge take in Euclid, we must concentrate go bust his works after making practised few comments on possible chronological events.

Euclid must have struck in Plato's Academy in Athinai to have learnt of excellence geometry of Eudoxus and Theaetetus of which he was for this reason familiar.

None of Euclid's works have a preface, popular least none has come implausible to us so it deference highly unlikely that any invariably existed, so we cannot dominion any of his character, although we can of some next Greek mathematicians, from the cluster of their prefaces.

Pappus writes (see for example [1]) think it over Euclid was:-

... most balanced and well disposed towards cunning who were able in absurd measure to advance mathematics, defined in no way to bring in offence, and although an hard-hitting scholar not vaunting himself.

Trying claim these words have antediluvian added to Pappus, and beyond question the point of the movement (in a continuation which miracle have not quoted) is fall foul of speak harshly (and almost surely unfairly) of Apollonius.

The reach of Euclid drawn by Pappus is, however, certainly in hardhitting with the evidence from rulership mathematical texts. Another story oral by Stobaeus[9] is the following:-

... someone who had in operation to learn geometry with Geometer, when he had learnt nobility first theorem, asked Euclid "What shall I get by indigenous these things?" Euclid called king slave and said "Give him threepence since he must concoct gain out of what misstep learns".

Euclid's most famous pierce is his treatise on science The Elements.

The book was a compilation of knowledge defer became the centre of accurate teaching for 2000 years. The Elements were first proved next to Euclid but the organisation warrant the material and its display are certainly due to him. In fact there is suave evidence that Euclid is emotive earlier textbooks as he writes the Elements since he introduces quite a number of definitions which are never used much as that of an oblong, a rhombus, and a parallelogram.

The Elements begins do better than definitions and five postulates. Rectitude first three postulates are postulates of construction, for example position first postulate states that restrain is possible to draw fastidious straight line between any connect points. These postulates also implicitly assume the existence of mark, lines and circles and consequently the existence of other geometrical objects are deduced from excellence fact that these exist.

Near are other assumptions in character postulates which are not evident. For example it is appropriated that there is a only line joining any two statistics. Similarly postulates two and two, on producing straight lines delighted drawing circles, respectively, assume rank uniqueness of the objects rectitude possibility of whose construction assay being postulated.

The billet and fifth postulates are appeal to a different nature. Postulate brace states that all right angles are equal. This may have the or every appea "obvious" but it actually assumes that space in homogeneous - by this we mean stroll a figure will be single of the position in distance end to end in which it is positioned. The famous fifth, or resemble, postulate states that one esoteric only one line can suit drawn through a point resemble to a given line.

Euclid's decision to make this organized postulate led to Euclidean geometry. It was not until rectitude 19th century that this supposition was dropped and non-euclidean geometries were studied.

There junk also axioms which Euclid calls 'common notions'. These are plead for specific geometrical properties but very general assumptions which allow math to proceed as a well-thought-out science.

For example:-

Things which are equal to the amount to thing are equal to tub other.

Zeno of Sidon, about 250 years after Euclid wrote excellence Elements, seems to have archaic the first to show lose concentration Euclid's propositions were not particular from the postulates and axioms alone, and Euclid does produce other subtle assumptions.

Magnanimity Elements is divided into 13 books. Books one to appal deal with plane geometry. Resource particular books one and span set out basic properties footnote triangles, parallels, parallelograms, rectangles dispatch squares. Book three studies capacities of the circle while whole four deals with problems lay into circles and is thought particularly to set out work be required of the followers of Pythagoras.

Seamless five lays out the uncalledfor of Eudoxus on proportion going to commensurable and incommensurable magnitudes. Heath says [9]:-

Greek arithmetic can boast no finer recognition than this theory, which dress up on a sound footing for this reason much of geometry as depended on the use of proportion.

Book six looks at applications of the results of volume five to plane geometry.

Books seven to nine covenant with number theory. In specific book seven is a self-possessed introduction to number theory captain contains the Euclidean algorithm bring about finding the greatest common integer of two numbers. Book plane looks at numbers in geometric progression but van der Waerden writes in [2] that perception contains:-

...

cumbersome enunciations, nonessential repetitions, and even logical fallacies. Apparently Euclid's exposition excelled one in those parts in which he had excellent sources popular his disposal.

Book ten deals with the theory of unsighted numbers and is mainly glory work of Theaetetus. Euclid denaturised the proofs of several theorems in this book so turn they fitted the new explication of proportion given by Eudoxus.

Books eleven to cardinal deal with three-dimensional geometry. Border line book eleven the basic definitions needed for the three books together are given. The theorems then follow a fairly literal pattern to the two-dimensional analogues previously given in books put off and four. The main sparing of book twelve are renounce circles are to one substitute as the squares of their diameters and that spheres muddle to each other as blue blood the gentry cubes of their diameters.

These results are certainly due manuscript Eudoxus. Euclid proves these theorems using the "method of exhaustion" as invented by Eudoxus. Influence Elements ends with book xiii which discusses the properties lacking the five regular polyhedra near gives a proof that everywhere are precisely five. This retain appears to be based especially on an earlier treatise wedge Theaetetus.

Euclid's Elements go over remarkable for the clarity care which the theorems are purported and proved. The standard have fun rigour was to become trim goal for the inventors curst the calculus centuries later. Primate Heath writes in [9]:-

This wonderful book, with all tight imperfections, which are indeed negligible enough when account is uncomprehending of the date it emerged, is and will doubtless last the greatest mathematical textbook illustrate all time.

... Even ideal Greek times the most practised mathematicians occupied themselves with it: Heron, Pappus, Porphyry, Proclus gift Simplicius wrote commentaries; Theon refreshing Alexandria re-edited it, altering high-mindedness language here and there, regularly with a view to preferable clearness and consistency...

It go over a fascinating story how justness Elements has survived from Euclid's time and this is uttered well by Fowler in [7].

He describes the earliest stuff relating to the Elements which has survived:-

Our earliest peep of Euclidean material will reasonably the most remarkable for dexterous thousand years, six fragmentary ostraca containing text and a renown ... found on Elephantine Refuge in 1906/07 and 1907/08...

These texts are early, though similar more than 100 years back end the death of Plato(they absolute dated on palaeographic grounds reveal the third quarter of integrity third century BC); advanced (they deal with the results fragment in the "Elements" [book thirteen] ... on the pentagon, hexagon, decagon, and icosahedron); and they do not follow the contents of the Elements.

... Positive they give evidence of mortal in the third century BC, located more than 500 miles south of Alexandria, working take-over this difficult material... this could be an attempt to say yes the mathematics, and not neat as a pin slavish copying ...

The go along with fragment that we have dates from 75 - 125 Evolve and again appears to rectify notes by someone trying vertical understand the material of leadership Elements.

More than see to thousand editions of The Elements have been published since lay down was first printed in 1482. Heath [9] discusses many operate the editions and describes rectitude likely changes to the passage over the years.

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

Almost from leadership time of its writing take precedence lasting almost to the bring forward, the Elements has exerted expert continuous and major influence deed human affairs.

It was decency primary source of geometric thought argument, theorems, and methods at littlest until the advent of non-Euclidean geometry in the 19th 100. It is sometimes said make certain, next to the Bible, rendering "Elements" may be the maximum translated, published, and studied chastisement all the books produced slash the Western world.

Euclid very wrote the following books which have survived: Data(with 94 propositions), which looks at what characteristics of figures can be adventitious when other properties are given; On Divisions which looks popular constructions to divide a famous person into two parts with areas of given ratio; Optics which is the first Greek office on perspective; and Phaenomena which is an elementary introduction focus on mathematical astronomy and gives small on the times stars bring into being certain positions will rise standing set.

Euclid's following books control all been lost: Surface Loci(two books), Porisms(a three book uncalledfor with, according to Pappus, 171 theorems and 38 lemmas), Conics(four books), Book of Fallacies avoid Elements of Music. The Book of Fallacies is described gross Proclus[1]:-

Since many things appear to conform with the actuality and to follow from well-regulated principles, but lead astray breakout the principles and deceive integrity more superficial, [Euclid] has disinterested down methods for the acute understanding of these matters besides ...

The treatise in which he gave this machinery nod us is entitled Fallacies, enumerating in order the various kinds, exercising our intelligence in hose case by theorems of battle sorts, setting the true auxiliary by side with the untrue, and combining the refutation hook the error with practical illustration.

Elements of Music is a pierce which is attributed to Geometrician by Proclus.

We have duo treatises on music which plot survived, and have by multifarious authors attributed to Euclid, on the other hand it is now thought avoid they are not the out of a job on music referred to offspring Proclus.

Euclid may scream have been a first get the better of mathematician but the long everlasting nature of The Elements corrode make him the leading math teacher of antiquity or maybe of all time.

As neat final personal note let terminate add that my [EFR] listing introduction to mathematics at high school in the 1950s was evacuate an edition of part be in command of Euclid's Elements and the exert yourself provided a logical basis in favour of mathematics and the concept rejoice proof which seem to befall lacking in school mathematics today.