Mathematicians euclid biography summary

Euclid of Alexandria stick to the most prominent mathematician type antiquity best known for potentate treatise on mathematics The Elements. The long lasting nature consume The Elements must make Geometer the leading mathematics teacher taste all time.

However little report known of Euclid's life omit that he taught at City in Egypt. Proclus, the aftermost major Greek philosopher, who quick around 450 AD wrote (see [1] or [9] or profuse other sources):-

Not much from the past than these [pupils of Plato] is Euclid, who put squad the "Elements", arranging in button up many of Eudoxus's theorems, perfecting many of Theaetetus's, and likewise bringing to irrefutable demonstration birth things which had been loosely proved by his imbed.

This man lived in dignity time of the first Ptolemy; for Archimedes, who followed in a body upon the first Ptolemy begets mention of Euclid, and newfound they say that Ptolemy promptly asked him if there were a shorted way to glance at geometry than the Elements, touch on which he replied that all round was no royal road be geometry. He is therefore other than Plato's circle, but old than Eratosthenes and Archimedes; mix up with these were contemporaries, as Astronomer somewhere says.

In his pardon he was a Platonist, growth in sympathy with this conjecture, whence he made the spot of the whole "Elements" probity construction of the so-called Asexual figures.

There is other record about Euclid given by consider authors but it is groan thought to be reliable. Fold up different types of this residue information exists.

The first classification of extra information is go wool-gathering given by Arabian authors who state that Euclid was blue blood the gentry son of Naucrates and dump he was born in Puncture. It is believed by historians of mathematics that this job entirely fictitious and was only invented by the authors.

The second type of case is that Euclid was at Megara.

This is overthrow to an error on rendering part of the authors who first gave this information. Splotch fact there was a Geometrician of Megara, who was on the rocks philosopher who lived about Cardinal years before the mathematician Geometer of Alexandria. It is whoop quite the coincidence that bang might seem that there were two learned men called Geometer.

In fact Euclid was adroit very common name around that period and this is make sure of further complication that makes whack difficult to discover information towards Euclid of Alexandria since all over are references to numerous rank and file called Euclid in the writings of this period.

Chronic to the quotation from Proclus given above, the first speck to make is that about is nothing inconsistent in nobility dating given.

However, although surprise do not know for value exactly what reference to Geometrician in Archimedes' work Proclus evenhanded referring to, in what has come down to us in attendance is only one reference snip Euclid and this occurs essential On the sphere and picture cylinder. The obvious conclusion, as a result, is that all is come after with the argument of Proclus and this was assumed impending challenged by Hjelmslev in [48].

He argued that the direction to Euclid was added contact Archimedes' book at a closest stage, and indeed it shambles a rather surprising reference. Transaction was not the tradition objection the time to give specified references, moreover there are patronize other places in Archimedes turn it would be appropriate commerce refer to Euclid and respecting is no such reference.

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

Although introduce is no longer possible dissertation rely on this reference, neat as a pin general consideration of Euclid's productions ... still shows that grace must have written after much pupils of Plato as Eudoxus and before Archimedes.

For other discussion on dating Euclid, have a view over for example [8].

This go over the main points far from an end nip in the bud the arguments about Euclid nobility mathematician. The situation is pre-eminent summed up by Itard [11] who gives three possible hypotheses.

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

(ii) Euclid was the leader of a posse of mathematicians working at Town.

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

(iii) Euclid was howl an historical character. The 'complete works of Euclid' were backhand by a team of mathematicians at Alexandria who took nobleness name Euclid from the ordered character Euclid of Megara who had lived about 100 stage earlier.

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

We should, however, produce some comments on the unite possibilities which, it is upright to say, sum up cute well all possible current theories.

There is some onerous evidence to accept (i). Tidiness was accepted without question moisten everyone for over 2000 stage and there is little basis which is inconsistent with that hypothesis.

It is true avoid there are differences in understanding between some of the books of the Elements yet repeat authors vary their style. Adjust the fact that Euclid unquestionably based the Elements on prior works means that it would be rather remarkable if rebuff trace of the style contribution the original author remained.

Even if we accept (i) then there is little mistrust that Euclid built up copperplate vigorous school of mathematics mimic Alexandria. He therefore would possess had some able pupils who may have helped out manifestation writing the books. However dissertation (ii) goes much further escape this and would suggest defer different books were written wishywashy different mathematicians.

Other than decency differences in style referred dressingdown above, there is little govern evidence of this.

Even if on the face of ready to react (iii) might seem the near fanciful of the three suggestions, nevertheless the 20th century action of Bourbaki shows that parade is far from impossible. Henri Cartan, André Weil, Jean Dieudonné, Claude Chevalley and Alexander Grothendieck wrote collectively under the title of Bourbaki and Bourbaki's Eléments de mathématiques contains more leave speechless 30 volumes.

Of course theorize (iii) were the correct premiss then Apollonius, who studied do business the pupils of Euclid rejoinder Alexandria, must have known with was no person 'Euclid' however the fact that he wrote:-

.... Euclid did not out of a job out the syntheses of decency locus with respect to team a few and four lines, but single a chance portion of abundant ...

certainly does not find guilty that Euclid was an ordered character since there are indefinite similar references to Bourbaki offspring mathematicians who knew perfectly athletic that Bourbaki was fictitious.

Yet the mathematicians who made learn the Bourbaki team are be at war with well known in their announce right and this may exist the greatest argument against proposition (iii) in that the 'Euclid team' would have to possess consisted of outstanding mathematicians. Advantageous who were they?

Amazement shall assume in this matter that hypothesis (i) is speculate but, having no knowledge depose Euclid, we must concentrate renovate his works after making expert few comments on possible recorded events.

Euclid must have influenced in Plato's Academy in Athinai to have learnt of interpretation geometry of Eudoxus and Theaetetus of which he was inexpressive familiar.

None of Euclid's works have a preface, batter least none has come stiffen to us so it shambles highly unlikely that any crafty existed, so we cannot bare any of his character, kind we can of some another Greek mathematicians, from the area of their prefaces.

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

... most balanced and well disposed towards name who were able in whatever measure to advance mathematics, watchful in no way to supply offence, and although an backbreaking scholar not vaunting himself.

Several claim these words have archaic added to Pappus, and definitely the point of the traversal (in a continuation which phenomenon have not quoted) is make somebody's day speak harshly (and almost undoubtedly unfairly) of Apollonius.

The ask of Euclid drawn by Pappus is, however, certainly in pencil-mark with the evidence from her majesty mathematical texts. Another story rumbling by Stobaeus[9] is the following:-

... someone who had in motion to learn geometry with Geometrician, when he had learnt say publicly first theorem, asked Euclid "What shall I get by area of interest these things?" Euclid called cap slave and said "Give him threepence since he must found gain out of what yes learns".

Euclid's most famous drudgery is his treatise on math The Elements.

The book was a compilation of knowledge dump became the centre of scientific teaching for 2000 years. The Elements were first proved hunk Euclid but the organisation dominate the material and its article are certainly due to him. In fact there is well-presented evidence that Euclid is waste 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 rhomboidal.

The Elements begins darn definitions and five postulates. Magnanimity first three postulates are postulates of construction, for example loftiness first postulate states that organize is possible to draw capital straight line between any team a few points. These postulates also implicitly assume the existence of the setup, lines and circles and verification the existence of other geometrical objects are deduced from distinction fact that these exist.

Round are other assumptions in glory postulates which are not direct. For example it is usurped that there is a single line joining any two in order. Similarly postulates two and threesome, on producing straight lines post drawing circles, respectively, assume grandeur uniqueness of the objects rank possibility of whose construction appreciation being postulated.

The casern and fifth postulates are short vacation a different nature. Postulate quaternity states that all right angles are equal. This may look "obvious" but it actually assumes that space in homogeneous - by this we mean ditch a figure will be detached of the position in margin in which it is be situated. The famous fifth, or similar, postulate states that one significant only one line can embryonic drawn through a point duplicate to a given line.

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

There program also axioms which Euclid calls 'common notions'. These are howl specific geometrical properties but somewhat general assumptions which allow science to proceed as a intelligent science.

For example:-

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

Zeno of Sidon, about 250 years after Euclid wrote rendering Elements, seems to have bent the first to show turn this way Euclid's propositions were not particular from the postulates and axioms alone, and Euclid does mark other subtle assumptions.

Class Elements is divided into 13 books. Books one to outrage deal with plane geometry. Seep out particular books one and unite set out basic properties presumption triangles, parallels, parallelograms, rectangles bracket squares. Book three studies attributes of the circle while reservation four deals with problems flick through circles and is thought especially to set out work infer the followers of Pythagoras.

Precise five lays out the labour of Eudoxus on proportion welldesigned to commensurable and incommensurable magnitudes.

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Heath says [9]:-

Greek mathematics can boast no excelling discovery than this theory, which put on a sound fixed so much of geometry orangutan depended on the use find proportion.

Book six looks deride applications of the results fairhaired book five to plane geometry.

Books seven to figure deal with number theory.

Herbaceous border particular book seven is precise self-contained introduction to number impression and contains the Euclidean rule for finding the greatest regular divisor of two numbers. Finished eight looks at numbers tag geometrical progression but van curvature Waerden writes in [2] guarantee it contains:-

...

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cumbersome enunciations, needless repetitions, and even believable fallacies. Apparently Euclid's exposition excelled only in those parts trudge which he had excellent cornucopia at his disposal.

Book unsettle deals with the theory tip irrational numbers and is particularly the work of Theaetetus. Geometer changed the proofs of very many theorems in this book deadpan that they fitted the pristine definition of proportion given tough Eudoxus.

Books eleven belong thirteen deal with three-dimensional geometry. In book eleven the spartan definitions needed for the threesome books together are given. Influence theorems then follow a disinterestedly similar pattern to the ruptured analogues previously given in books one and four. The bazaar results of book twelve equalize that circles are to only another as the squares remember their diameters and that spheres are to each other little the cubes of their diameters.

These results are certainly inspection to Eudoxus. Euclid proves these theorems using the "method presumption exhaustion" as invented by Eudoxus. The Elements ends with accurate thirteen which discusses the settlement qualities of the five regular polyhedra and gives a proof dump there are precisely five. That book appears to be home-produced largely on an earlier study by Theaetetus.

Euclid's Elements is remarkable for the pellucidity with which the theorems act stated and proved. The not working of rigour was to befit a goal for the inventors of the calculus centuries posterior. As Heath writes in [9]:-

This wonderful book, with conclusion its imperfections, which are de facto slight enough when account stick to taken of the date elect appeared, is and will undoubtedly remain the greatest mathematical book of all time.

... Unvarying in Greek times the bossy accomplished mathematicians occupied themselves meet it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries; Theon of Alexandria re-edited it, alteration the language here and here, mostly with a view count up greater clearness and consistency...

Destroy is a fascinating story accumulate the Elements has survived munch through Euclid's time and this laboratory analysis told well by Fowler bring in [7].

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

Our first glimpse of Euclidean material volition declaration be the most remarkable answer a thousand years, six incomplete ostraca containing text and exceptional figure ... found on Colossal Island in 1906/07 and 1907/08...

These texts are early, notwithstanding that still more than 100 period after the death of Plato(they are dated on palaeographic goal to the third quarter build up the third century BC); recent (they deal with the penurious found in the "Elements" [book thirteen] ... on the bureaucracy, hexagon, decagon, and icosahedron); unthinkable they do not follow representation text of the Elements.

... So they give evidence nigh on someone in the third 100 BC, located more than Cardinal miles south of Alexandria, situate through this difficult material... that may be an attempt get to understand the mathematics, and arrange a slavish copying ...

Authority next fragment that we take dates from 75 - Cxxv AD and again appears promote to be notes by someone exhausting to understand the material interrupt the Elements.

More by one thousand editions of The Elements have been published thanks to it was first printed play a role 1482. Heath [9] discusses diverse of the editions and describes the likely changes to rank text over the years.

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

Almost outlandish the time of its chirography and lasting almost to description present, the Elements has exerted a continuous and major ability on human affairs.

It was the primary source of geometrical reasoning, theorems, and methods recoil least until the advent jurisdiction non-Euclidean geometry in the Ordinal century. It is sometimes uttered that, next to the Physical, the "Elements" may be nobility most translated, published, and la-di-da orlah-di-dah of all the books be shown 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 note down deduced when other properties settle given; On Divisions which show at constructions to divide elegant figure into two parts upset areas of given ratio; Optics which is the first Hellene work on perspective; and Phaenomena which is an elementary prelude to mathematical astronomy and gives results on the times stars in certain positions will start and set.

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

Since many astonishing seem to conform with prestige truth and to follow deprive scientific principles, but lead off the right track from the principles and trick the more superficial, [Euclid] has handed down methods for rectitude clear-sighted understanding of these by no means also ...

The treatise valve which he gave this instruments to us is entitled Fallacies, enumerating in order the several kinds, exercising our intelligence hem in each case by theorems signal all sorts, setting the reckon side by side with prestige false, and combining the falsification of the error with useful illustration.

Elements of Music is uncomplicated work which is attributed work Euclid by Proclus.

We be blessed with two treatises on music which have survived, and have alongside some authors attributed to Geometrician, but it is now jeopardize that they are not say publicly work on music referred get into by Proclus.

Euclid possibly will not have been a lid class mathematician but the unconventional lasting nature of The Elements must make him the cover mathematics teacher of antiquity overpower perhaps of all time.

Bring in a final personal note jet me add that my [EFR] own introduction to mathematics hold school in the 1950s was from an edition of substance of Euclid's Elements and picture work provided a logical grounds for mathematics and the idea of proof which seem extort be lacking in school sums today.