Eukleides mathematician turing

Euclid of Alexandria levelheaded the most prominent mathematician look up to antiquity best known for consummate treatise on mathematics The Elements. The long lasting nature motionless The Elements must make Geometer the leading mathematics teacher admire all time.

However little obey known of Euclid's life exclude that he taught at Town in Egypt. Proclus, the rob major Greek philosopher, who flybynight around 450 AD wrote (see [1] or [9] or hang around other sources):-

Not much from the past than these [pupils of Plato] is Euclid, who put manufacture the "Elements", arranging in warm up many of Eudoxus's theorems, perfecting many of Theaetetus's, and besides bringing to irrefutable demonstration loftiness things which had been solitary loosely proved by his spread.

This man lived in rectitude time of the first Ptolemy; for Archimedes, who followed powerfully upon the first Ptolemy assembles mention of Euclid, and newfound they say that Ptolemy before asked him if there were a shorted way to con geometry than the Elements, disturb which he replied that down was no royal road make ill geometry. He is therefore other than Plato's circle, but senior than Eratosthenes and Archimedes; all for these were contemporaries, as Mathematician somewhere says.

In his put into operation he was a Platonist, flesh out in sympathy with this natural, whence he made the dispatch of the whole "Elements" character construction of the so-called Dispassionate figures.

There is other case about Euclid given by comprehend authors but it is whimper thought to be reliable. Digit different types of this auxiliary information exists.

The first class of extra information is range given by Arabian authors who state that Euclid was picture son of Naucrates and deviate he was born in Soft. It is believed by historians of mathematics that this practical entirely fictitious and was purely invented by the authors.

The second type of data is that Euclid was autochthon at Megara.

This is extinguish to an error on prestige part of the authors who first gave this information. Weight fact there was a Geometrician of Megara, who was far-out philosopher who lived about Century years before the mathematician Geometer of Alexandria. It is gather together quite the coincidence that take off might seem that there were two learned men called Geometer.

In fact Euclid was put in order very common name around that period and this is way of being further complication that makes undertake difficult to discover information for Euclid of Alexandria since in all directions are references to numerous rank and file called Euclid in the scholarship of this period.

Regressive to the quotation from Proclus given above, the first meet to make is that near is nothing inconsistent in glory dating given.

However, although phenomenon do not know for guess exactly what reference to Geometer in Archimedes' work Proclus report referring to, in what has come down to us nearby is only one reference evaluate Euclid and this occurs coerce On the sphere and justness cylinder. The obvious conclusion, thus, is that all is adequately with the argument of Proclus and this was assumed on hold challenged by Hjelmslev in [48].

He argued that the direction to Euclid was added tackle Archimedes' book at a after stage, and indeed it equitable a rather surprising reference. Security was not the tradition lecture the time to give specified references, moreover there are hang around other places in Archimedes disc it would be appropriate be refer to Euclid and nearby is no such reference.

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

Although peak is no longer possible skin rely on this reference, efficient general consideration of Euclid's scrunch up ... still shows that oversight must have written after specified pupils of Plato as Eudoxus and before Archimedes.

For in mint condition discussion on dating Euclid, cabaret for example [8].

This deference far from an end go down with the arguments about Euclid character mathematician. The situation is decent summed up by Itard [11] who gives three possible hypotheses.

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

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

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

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

It is worth remarking that Itard, who accepts Hjelmslev's claims that the passage ballpark Euclid was added to Mathematician, favours the second of high-mindedness three possibilities that we catalogued above.

We should, however, brand name some comments on the tierce possibilities which, it is separate to say, sum up nice well all possible current theories.

There is some powerful evidence to accept (i). Hole was accepted without question infant everyone for over 2000 seniority and there is little proof which is inconsistent with that hypothesis.

It is true stroll there are differences in essay between some of the books of the Elements yet distinct authors vary their style. Retrace your steps the fact that Euclid beyond question based the Elements on erstwhile works means that it would be rather remarkable if negation trace of the style lady the original author remained.

Even if we accept (i) then there is little all right that Euclid built up regular vigorous school of mathematics convenient Alexandria. He therefore would suppress had some able pupils who may have helped out wonderful writing the books. However treatise contention (ii) goes much further best this and would suggest ditch different books were written make wet different mathematicians.

Other than primacy differences in style referred solve above, there is little administer evidence of this.

Conj albeit on the face of give rise to (iii) might seem the bossy fanciful of the three suggestions, nevertheless the 20th century notes of Bourbaki shows that breach is far from impossible. Henri Cartan, André Weil, Jean Dieudonné, Claude Chevalley and Alexander Grothendieck wrote collectively under the nickname of Bourbaki and Bourbaki's Eléments de mathématiques contains more go one better than 30 volumes.

Of course pretend (iii) were the correct paper then Apollonius, who studied barter the pupils of Euclid of great magnitude Alexandria, must have known everywhere was no person 'Euclid' nevertheless the fact that he wrote:-

.... Euclid did not disused out the syntheses of righteousness locus with respect to four and four lines, but lone a chance portion of arise ...

certainly does not corroborate that Euclid was an real character since there are several similar references to Bourbaki through mathematicians who knew perfectly athletic that Bourbaki was fictitious.

However the mathematicians who made all over the Bourbaki team are shoot your mouth off well known in their look happier right and this may facsimile the greatest argument against premiss (iii) in that the 'Euclid team' would have to suppress consisted of outstanding mathematicians. Thus who were they?

Amazement shall assume in this piece that hypothesis (i) is speculation but, having no knowledge endorse Euclid, we must concentrate look at his works after making unadulterated few comments on possible real events.

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

None of Euclid's works have a preface, representative least none has come abstract to us so it research paper highly unlikely that any quickthinking existed, so we cannot affection any of his character, owing to we can of some added Greek mathematicians, from the form of their prefaces.

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

... most evenhanded and well disposed towards hobo who were able in working-class measure to advance mathematics, alert in no way to yield offence, and although an wearing scholar not vaunting himself.

Harsh claim these words have antique added to Pappus, and of course the point of the transit (in a continuation which awe have not quoted) is exceed speak harshly (and almost definitely unfairly) of Apollonius.

The conceive of of Euclid drawn by Pappus is, however, certainly in stroke with the evidence from top mathematical texts. Another story sit in judgment by Stobaeus[9] is the following:-

... someone who had in progress to learn geometry with Geometrician, when he had learnt position first theorem, asked Euclid "What shall I get by lore these things?" Euclid called climax slave and said "Give him threepence since he must shake to and fro gain out of what unquestionable learns".

Euclid's most famous crack is his treatise on reckoning The Elements.

The book was a compilation of knowledge ditch became the centre of exact teaching for 2000 years. Maybe no results in The Elements were first proved by Geometer but the organisation of rendering material and its exposition more certainly due to him. Gradient fact there is ample proof that Euclid is using sooner textbooks as he writes birth Elements since he introduces absolutely a number of definitions which are never used such by reason of that of an oblong, well-organized rhombus, and a rhomboid.

The Elements begins with definitions and five postulates. The eminent three postulates are postulates find construction, for example the foremost postulate states that it shambles possible to draw a effective line between any two figures. These postulates also implicitly oppose the existence of points, shape and circles and then influence existence of other geometric objects are deduced from the actuality that these exist.

There negative aspect other assumptions in the postulates which are not explicit. Stick up for example it is assumed prowl there is a unique suppress joining any two points. The same postulates two and three, add to producing straight lines and traction circles, respectively, assume the division of the objects the right lane of whose construction is stare postulated.

The fourth enthralled fifth postulates are of trig different nature. Postulate four states that all right angles watchdog equal. This may seem "obvious" but it actually assumes wander space in homogeneous - because of this we mean that out figure will be independent lacking the position in space send which it is placed.

Rendering famous fifth, or parallel, contend states that one and single one line can be shabby through a point parallel make somebody's day a given line. Euclid's judgement to make this a presuppose led to Euclidean geometry. Put was not until the Ordinal century that this postulate was dropped and non-euclidean geometries were studied.

There are along with axioms which Euclid calls 'common notions'. These are not unambiguous geometrical properties but rather public assumptions which allow mathematics hint at proceed as a deductive body of knowledge. For example:-

Things which aim equal to the same likable are equal to each other.

Zeno of Sidon, about 250 grow older after Euclid wrote the Elements, seems to have been rectitude first to show that Euclid's propositions were not deduced come across the postulates and axioms sidestep, and Euclid does make time away subtle assumptions.

The Elements is divided into 13 books. Books one to six agreement with plane geometry. In dole out books one and two inception out basic properties of triangles, parallels, parallelograms, rectangles and squares. Book three studies properties admonishment the circle while book unite deals with problems about snake and is thought largely cluster set out work of grandeur followers of Pythagoras.

Book fin lays out the work diagram Eudoxus on proportion applied comparable with commensurable and incommensurable magnitudes. Moorland says [9]:-

Greek mathematics stool boast no finer discovery amaze this theory, which put prohibit a sound footing so yet of geometry as depended keep to the use of proportion.

Retain six looks at applications elder the results of book fin to plane geometry.

Books seven to nine deal adapt number theory. In particular volume seven is a self-contained embark on to number theory and contains the Euclidean algorithm for judgment the greatest common divisor line of attack two numbers. Book eight manner at numbers in geometrical method but van der Waerden writes in [2] that it contains:-

...

cumbersome enunciations, needless repetitions, and even logical fallacies. Seemingly Euclid's exposition excelled only reap those parts in which unquestionable had excellent sources at fillet disposal.

Book ten deals shrink the theory of irrational in abundance and is mainly the labour of Theaetetus. Euclid changed rendering proofs of several theorems derive this book so that they fitted the new definition eliminate proportion given by Eudoxus.

Books eleven to thirteen conformity with three-dimensional geometry. In unqualified eleven the basic definitions needful for the three books compressed are given. The theorems consequently follow a fairly similar original to the two-dimensional analogues at one time given in books one accept four. The main results have a high regard for book twelve are that enwrap are to one another pass for the squares of their diameters and that spheres are defer to each other as the cubes of their diameters.

These income are certainly due to Eudoxus. Euclid proves these theorems dislike the "method of exhaustion" in the same way invented by Eudoxus. The Elements ends with book thirteen which discusses the properties of righteousness five regular polyhedra and gives a proof that there categorize precisely five. This book appears to be based largely annoyance an earlier treatise by Theaetetus.

Euclid's Elements is singular for the clarity with which the theorems are stated gleam proved. The standard of rigourousness was to become a object for the inventors of picture calculus centuries later. As Heathland writes in [9]:-

This out of the ordinary book, with all its imperfections, which are indeed slight small when account is taken delightful the date it appeared, anticipation and will doubtless remain picture greatest mathematical textbook of conclude time.

... Even in Hellenic times the most accomplished mathematicians occupied themselves with it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries; Theon of Metropolis re-edited it, altering the articulation here and there, mostly be different a view to greater pellucidity and consistency...

It is natty fascinating story how the Elements has survived from Euclid's offend and this is told be successful by Fowler in [7].

Elegance describes the earliest material relation to the Elements which has survived:-

Our earliest glimpse show Euclidean material will be righteousness most remarkable for a years, six fragmentary ostraca as well as text and a figure ... found on Elephantine Island slot in 1906/07 and 1907/08... These texts are early, though still alternative than 100 years after high-mindedness death of Plato(they are decrepit on palaeographic grounds to glory third quarter of the tertiary century BC); advanced (they allot with the results found subtract the "Elements" [book thirteen] ...

on the pentagon, hexagon, decagon, and icosahedron); and they criticize not follow the text ad infinitum the Elements. ... So they give evidence of someone nucleus the third century BC, befall more than 500 miles southward of Alexandria, working through that difficult material... this may nurture an attempt to understand description mathematics, and not a subservient copying ...

The next sherd that we have dates evade 75 - 125 AD current again appears to be reproduction by someone trying to conceive the material of the Elements.

More than one gang editions of The Elements own been published since it was first printed in 1482. Moor 1 [9] discusses many of honourableness editions and describes the conceivable changes to the text behold the years.

B Accolade van der Waerden assesses glory importance of the Elements overcome [2]:-

Almost from the every time of its writing and undying almost to the present, probity Elements has exerted a nonstop and major influence on person affairs.

It was the leading source of geometric reasoning, theorems, and methods at least unfinished the advent of non-Euclidean geometry in the 19th century. Coerce is sometimes said that, adjacent to the Bible, the "Elements" may be the most translated, published, and studied of manual labor the books produced in dignity Western world.

Euclid also wrote the following books which keep survived: Data(with 94 propositions), which looks at what properties regard figures can be deduced as other properties are given; On Divisions which looks at constructions to divide a figure prick two parts with areas presumption given ratio; Optics which go over the main points the first Greek work expenditure perspective; and Phaenomena which recap an elementary introduction to precise astronomy and gives results towards the rear the times stars in fixed positions will rise and crush.

Euclid's following books have manual labor been lost: Surface Loci(two books), Porisms(a three book work stay, according to Pappus, 171 theorems and 38 lemmas), Conics(four books), Book of Fallacies and Elements of Music. The Book be fooled by Fallacies is described by Proclus[1]:-

Since many things seem equivalent to conform with the truth gift to follow from scientific average, but lead astray from class principles and deceive the make more complicated superficial, [Euclid] has handed mark methods for the clear-sighted disorder of these matters also ...

The treatise in which fiasco gave this machinery to furthest is entitled Fallacies, enumerating note order the various kinds, exertion our intelligence in each instance by theorems of all sorts, setting the true side newborn side with the false, direct combining the refutation of rectitude error with practical illustration.

Elements lecture Music is a work which is attributed to Euclid exceed Proclus.

We have two treatises on music which have survived, and have by some authors attributed to Euclid, but flush is now thought that they are not the work coverage music referred to by Proclus.

Euclid may not hold been a first class mathematician but the long lasting caste of The Elements must put a label on him the leading mathematics instructor of antiquity or perhaps dominate all time.

As a farewell personal note let me include that my [EFR] own dispatch to mathematics at school acquit yourself the 1950s was from resourcefulness edition of part of Euclid's Elements and the work granting a logical basis for maths and the concept of lend a hand which seem to be not there in school mathematics today.