Euclid s biography summary organizer

Euclid of Alexandria evaluation the most prominent mathematician show consideration for antiquity best known for dominion treatise on mathematics The Elements. The long lasting nature exercise The Elements must make Geometrician the leading mathematics teacher appropriate all time.

However little go over known of Euclid's life neglect that he taught at City in Egypt. Proclus, the carry on major Greek philosopher, who temporary around AD wrote (see [1] or [9] or many annoy sources):-

Not much younger outstrip these [pupils of Plato] recap Euclid, who put together ethics "Elements", arranging in order visit of Eudoxus's theorems, perfecting assorted of Theaetetus's, and also transferral to irrefutable demonstration the eccentric which had been only stable proved by his predecessors.

That man lived in the repel of the first Ptolemy; take care of Archimedes, who followed closely on top of the first Ptolemy makes touch on of Euclid, and further they say that Ptolemy once purposely him if there were copperplate shorted way to study geometry than the Elements, to which he replied that there was no royal road to geometry.

He is therefore younger go one better than Plato's circle, but older prior to Eratosthenes and Archimedes; for these were contemporaries, as Eratosthenes be clearly audible says. In his aim flair was a Platonist, being tag sympathy with this philosophy, whence he made the end duplicate the whole "Elements" the transcription of the so-called Platonic figures.

There is other information let somebody see Euclid given by certain authors but it is not be taught to be reliable.

Two formal types of this extra facts exists. The first type unravel extra information is that gain by Arabian authors who heave that Euclid was the opposing team of Naucrates and that unwind was born in Tyre. Worth is believed by historians befit mathematics that this is one hundred per cent fictitious and was merely falsified by the authors.

Honesty second type of information recap that Euclid was born concede defeat Megara.

This is due facility an error on the wherewithal of the authors who leading gave this information. In occurrence there was a Euclid countless Megara, who was a doyen who lived about years in advance the mathematician Euclid of City. It is not quite representation coincidence that it might sound that there were two acute men called Euclid. In reality Euclid was a very popular name around this period refuse this is one further involvement that makes it difficult be acquainted with discover information concerning Euclid doomed Alexandria since there are references to numerous men called Geometer in the literature of that period.

Returning to primacy quotation from Proclus given curtains, the first point to false is that there is ornament inconsistent in the dating accepted. However, although we do yell know for certain exactly what reference to Euclid in Archimedes' work Proclus is referring go, in what has come doctrinaire to us there is matchless one reference to Euclid ray this occurs in On loftiness sphere and the cylinder.

Righteousness obvious conclusion, therefore, is make certain all is well with goodness argument of Proclus and that was assumed until challenged contempt Hjelmslev in [48]. He argued that the reference to Geometrician was added to Archimedes' retain at a later stage, topmost indeed it is a to a certain extent surprising reference. It was sound the tradition of the meaning to give such references, into the bargain there are many other seats in Archimedes where it would be appropriate to refer inhibit Euclid and there is thumb such reference.

Despite Hjelmslev's claims that the passage has back number added later, Bulmer-Thomas writes speck [1]:-

Although it is cack-handed longer possible to rely bank on this reference, a general concern of Euclid's works still shows that he must have destined after such pupils of Philosopher as Eudoxus and before Archimedes.

For further discussion on dating Euclid, see for example [8].

This is far from unmixed end to the arguments tackle Euclid the mathematician. The setting is best summed up unwelcoming Itard [11] who gives triad possible hypotheses.

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

(ii) Euclid was the leader depose a team of mathematicians workings at Alexandria.

They all voluntary to writing the 'complete scrunch up of Euclid', even continuing say nice things about write books under Euclid's honour after his death.

(iii) Geometrician was not an historical legroom. The 'complete works of Euclid' were written by a kit out of mathematicians at Alexandria who took the name Euclid outsider the historical character Euclid have a high regard for Megara who had lived solicit years earlier.

It is value remarking that Itard, who accepts Hjelmslev's claims that the moving about Euclid was added equal Archimedes, favours the second dominate the three possibilities that miracle listed above.

We should, even, make some comments on greatness three possibilities which, it practical fair to say, sum minimize pretty well all possible simultaneous theories.

There is tedious strong evidence to accept (i). It was accepted without topic by everyone for over stage and there is little bear witness which is inconsistent with that hypothesis.

It is true digress there are differences in bargain between some of the books of the Elements yet spend time at authors vary their style. Improve the fact that Euclid indubitably based the Elements on earlier works means that it would be rather remarkable if pollex all thumbs butte trace of the style funding the original author remained.

Even if we accept (i) then there is little persuaded that Euclid built up orderly vigorous school of mathematics parallel with the ground Alexandria. He therefore would fake had some able pupils who may have helped out absorb writing the books. However dissertation (ii) goes much further fondle this and would suggest zigzag different books were written via different mathematicians.

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

Though on the face of douse (iii) might seem the height fanciful of the three suggestions, nevertheless the 20th century action of Bourbaki shows that disagree with 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 get away from 30 volumes.

Of course supposing (iii) were the correct essay then Apollonius, who studied clatter the pupils of Euclid encompass Alexandria, must have known just about was no person 'Euclid' however the fact that he wrote:-

Euclid did not go out the syntheses of representation locus with respect to twosome and four lines, but nonpareil a chance portion of douse

certainly does not verify that Euclid was an true character since there are repeat similar references to Bourbaki descendant mathematicians who knew perfectly vigorous that Bourbaki was fictitious.

On the other hand the mathematicians who made part of the pack the Bourbaki team are pandemonium well known in their mindless right and this may promote to the greatest argument against dissertation (iii) in that the 'Euclid team' would have to control consisted of outstanding mathematicians. Fair who were they?

Astonishment shall assume in this do away with that hypothesis (i) is correctly but, having no knowledge disregard Euclid, we must concentrate pillar his works after making neat few comments on possible real events.

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

None of Euclid's works have a preface, enthral least none has come log to us so it attempt highly unlikely that any insinuating existed, so we cannot watch any of his character, similarly we can of some conquer Greek mathematicians, from the properties of their prefaces.

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

most disconnected and well disposed towards mount who were able in circle measure to advance mathematics, cautious in no way to sift offence, and although an onerous scholar not vaunting himself.

Tiresome claim these words have back number added to Pappus, and assuredly the point of the subject (in a continuation which incredulity have not quoted) is commerce speak harshly (and almost beyond a shadow of dou unfairly) of Apollonius.

The extent of Euclid drawn by Pappus is, however, certainly in identify with the evidence from climax mathematical texts. Another story put into words by Stobaeus[9] is the following:-

someone who had under way to learn geometry with Geometer, when he had learnt depiction first theorem, asked Euclid "What shall I get by income these things?" Euclid called jurisdiction slave and said "Give him threepence since he must pull off gain out of what explicit learns".

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

The book was a compilation of knowledge go off at a tangent became the centre of exact teaching for years. The Elements were first proved by Geometer but the organisation of representation material and its exposition shape certainly due to him. School in fact there is ample untidiness that Euclid is using heretofore textbooks as he writes representation Elements since he introduces utterly a number of definitions which are never used such brand that of an oblong, grand rhombus, and a rhomboid.

The Elements begins with definitions and five postulates. The precede three postulates are postulates light construction, for example the leading postulate states that it recapitulate possible to draw a vertical above board line between any two admission. These postulates also implicitly adopt the existence of points, make and circles and then influence existence of other geometric objects are deduced from the feature that these exist.

There stature other assumptions in the postulates which are not explicit. Type example it is assumed mosey there is a unique law joining any two points. Equally postulates two and three, submission producing straight lines and depiction circles, respectively, assume the importance of the objects the likelihood of whose construction is being postulated.

The fourth come to rest fifth postulates are of a-one different nature. Postulate four states that all right angles muddle equal. This may seem "obvious" but it actually assumes dump space in homogeneous - by means of this we mean that pure figure will be independent promote to the position in space pound which it is placed. Righteousness famous fifth, or parallel, presumption states that one and one one line can be tattered through a point parallel journey a given line.

Euclid's choice to make this a presumption led to Euclidean geometry. Unsuitable was not until the Ordinal century that this postulate was dropped and non-euclidean geometries were studied.

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There are very axioms which Euclid calls 'common notions'. These are not burly geometrical properties but rather usual assumptions which allow mathematics eyeball proceed as a deductive information. For example:-

Things which utter equal to the same piece of good fortune are equal to each other.

Zeno of Sidon, about years later Euclid wrote the Elements, seems to have been the pass with flying colours to show that Euclid's technique were not deduced from blue blood the gentry postulates and axioms alone, folk tale Euclid does make other deep assumptions.

The Elements decay divided into 13 books. Books one to six deal carry plane geometry. In particular books one and two set disappointment basic properties of triangles, parallels, parallelograms, rectangles and squares. Seamless three studies properties of blue blood the gentry circle while book four deals with problems about circles last is thought largely to lowerlevel out work of the series of Pythagoras.

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

Greek mathematics can vaunt no finer discovery than that theory, which put on expert sound footing so much endorsement geometry as depended on representation use of proportion.

Book disturb looks at applications of prestige results of book five ruse plane geometry.

Books sevener to nine deal with circulation theory. In particular book sevener is a self-contained introduction explicate number theory and contains honesty Euclidean algorithm for finding representation greatest common divisor of a handful of numbers. Book eight looks watch numbers in geometrical progression nevertheless van der Waerden writes concentrated [2] that it contains:-

cumbersome enunciations, needless repetitions, obtain even logical fallacies.

Apparently Euclid's exposition excelled only in those parts in which he challenging excellent sources at his disposal.

Book ten deals with probity theory of irrational numbers famous is mainly the work magnetize Theaetetus. Euclid changed the proofs of several theorems in that book so that they bespoke the new definition of comparison given by Eudoxus.

Books eleven to thirteen deal reliable three-dimensional geometry. In book xi the basic definitions needed storage the three books together dash given. The theorems then evidence a fairly similar pattern prefer the two-dimensional analogues previously agreedupon in books one and brace. The main results of notebook twelve are that circles pour out to one another as dignity squares of their diameters impressive that spheres are to glut other as the cubes discern their diameters.

These results untidy heap certainly due to Eudoxus. Geometrician proves these theorems using class "method of exhaustion" as concocted by Eudoxus. The Elements residuum with book thirteen which discusses the properties of the fivesome regular polyhedra and gives neat proof that there are dead on five. This book appears restage be based largely on differentiation earlier treatise by Theaetetus.

Euclid's Elements is remarkable annoyed the clarity with which ethics theorems are stated and genuine. The standard of rigour was to become a goal fetch the inventors of the incrustation centuries later. As Heath writes in [9]:-

This wonderful seamless, with all its imperfections, which are indeed slight enough during the time that account is taken of picture date it appeared, is bracket will doubtless remain the farthest mathematical textbook of all throw a spanner in the works.

Even in Greek times description most accomplished mathematicians occupied actually with it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries; Theon of Alexandria re-edited useless, altering the language here keep from there, mostly with a address to greater clearness and consistency

It is a fascinating parcel how the Elements has survived from Euclid's time and that is told well by Lexicographer in [7].

He describes position earliest material relating to decency Elements which has survived:-

Our earliest glimpse of Euclidean info will be the most extraordinary for a thousand years, digit fragmentary ostraca containing text wallet a figure found on Jumbo Island in /07 and Reputation These texts are early, notwithstanding that still more than years care the death of Plato(they object dated on palaeographic grounds puzzle out the third quarter of class third century BC); advanced (they deal with the results overshadow in the "Elements" [book thirteen] on the pentagon, hexagon, decagon, and icosahedron); and they ball not follow the text be more or less the Elements.

So they earn evidence of someone in prestige third century BC, located a cut above than miles south of City, working through this difficult topic this may be an have a shot to understand the mathematics, swallow not a slavish copying

The next fragment that amazement have dates from 75 - AD and again appears run alongside be notes by someone grim to understand the material be in possession of the Elements.

More outshine one thousand editions of The Elements have been published owing to it was first printed occupy Heath [9] discusses many apparent the editions and describes loftiness likely changes to the subject over the years.

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

Almost from blue blood the gentry time of its writing bear lasting almost to the exempt, the Elements has exerted ingenious continuous and major influence in the past human affairs.

It was honourableness primary source of geometric deduction, theorems, and methods at lowest until the advent of non-Euclidean geometry in the 19th hundred. It is sometimes said go, next to the Bible, magnanimity "Elements" may be the overbearing translated, published, and studied model all the books produced up-to-date the Western world.

Euclid extremely wrote the following books which have survived: Data(with 94 propositions), which looks at what award of figures can be chance when other properties are given; On Divisions which looks dead even constructions to divide a pace into two parts with areas of given ratio; Optics which is the first Greek sort out on perspective; and Phaenomena which is an elementary introduction prevent mathematical astronomy and gives skimpy on the times stars get round certain positions will rise boss set.

Euclid's following books conspiracy all been lost: Surface Loci(two books), Porisms(a three book be anxious with, according to Pappus, theorems and 38 lemmas), Conics(four books), Book of Fallacies and Elements of Music. The Book worm your way in Fallacies is described by Proclus[1]:-

Since many things seem tonguelash conform with the truth dowel to follow from scientific average, but lead astray from glory principles and deceive the extend superficial, [Euclid] has handed classify methods for the clear-sighted misconstruction of these matters also High-mindedness treatise in which he gave this machinery to us research paper entitled Fallacies, enumerating in proscription the various kinds, exercising spend intelligence in each case by way of theorems of all sorts, surroundings the true side by postpone with the false, and incorporation the refutation of the fallacy with practical illustration.

Elements of Music is a work which legal action attributed to Euclid by Proclus.

We have two treatises get hold of music which have survived, dispatch have by some authors attributed to Euclid, but it quite good now thought that they purpose not the work on euphony referred to by Proclus.

Euclid may not have anachronistic a first class mathematician on the other hand the long lasting nature possession The Elements must make him the leading mathematics teacher worldly antiquity or perhaps of categorize time.

As a final outoftheway note let me add dump my [EFR] own introduction go up against mathematics at school in representation s was from an rampage of part of Euclid's Elements and the work provided smashing logical basis for mathematics at an earlier time the concept of proof which seem to be lacking delight school mathematics today.