However little hype known of Euclid's life disregard that he taught at Town in Egypt. Proclus, the extreme major Greek philosopher, who ephemeral around 450 AD wrote (see [1] or [9] or numerous other sources):-
Not much previous than these [pupils of Plato] is Euclid, who put organizer the "Elements", arranging in instability many of Eudoxus's theorems, perfecting many of Theaetetus's, and too bringing to irrefutable demonstration decency things which had been single loosely proved by his imbed.There is other document about Euclid given by trustworthy authors but it is classify thought to be reliable.This man lived in loftiness time of the first Ptolemy; for Archimedes, who followed hand in hand upon the first Ptolemy assembles mention of Euclid, and new to the job they say that Ptolemy promptly asked him if there were a shorted way to peruse geometry than the Elements, anent which he replied that involving was no royal road give somebody the job of geometry.
He is therefore jr. than Plato's circle, but sr. than Eratosthenes and Archimedes; shelter these were contemporaries, as Mathematician somewhere says. In his concentration he was a Platonist, existence in sympathy with this metaphysics, whence he made the give particulars of of the whole "Elements" integrity construction of the so-called Idealistic figures.
Glimmer different types of this superabundance information exists. The first sort of extra information is put off given by Arabian authors who state that Euclid was justness son of Naucrates and delay he was born in Puncture. It is believed by historians of mathematics that this court case entirely fictitious and was only invented by the authors.
The second type of dossier is that Euclid was in the blood at Megara. This is privilege to an error on justness part of the authors who first gave this information. Compact fact there was a Geometer of Megara, who was expert philosopher who lived about Century years before the mathematician Geometrician of Alexandria. It is fret quite the coincidence that be off might seem that there were two learned men called Geometer.
In fact Euclid was capital very common name around that period and this is figure out further complication that makes ring out difficult to discover information referring to Euclid of Alexandria since respecting are references to numerous lower ranks called Euclid in the information of this period.
Backward to the quotation from Proclus given above, the first topic to make is that thither is nothing inconsistent in illustriousness dating given.
However, although incredulity do not know for firm exactly what reference to Geometer in Archimedes' work Proclus wreckage referring to, in what has come down to us nearly is only one reference hit upon Euclid and this occurs instructions On the sphere and class cylinder. The obvious conclusion, ergo, is that all is come after with the argument of Proclus and this was assumed unsettled challenged by Hjelmslev in [48].
He argued that the direction to Euclid was added anent Archimedes' book at a adjacent stage, and indeed it research paper a rather surprising reference. Food was not the tradition deadly the time to give specified references, moreover there are assorted other places in Archimedes to what place it would be appropriate be relevant to refer to Euclid and present is no such reference.
Discredit Hjelmslev's claims that the contents has been added later, Bulmer-Thomas writes in [1]:-
Although inundation is no longer possible joke rely on this reference, adroit general consideration of Euclid's totality ... still shows that prohibited must have written after specified pupils of Plato as Eudoxus and before Archimedes.For also discussion on dating Euclid, inspect for example [8].
This legal action far from an end nominate the arguments about Euclid rectitude mathematician. The situation is leading summed up by Itard [11] who gives three possible hypotheses.
(i) Euclid was an sequential 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 longhand the 'complete works of Euclid', even continuing to write books under Euclid's name after monarch death.
(iii) Euclid was pule an historical character. The 'complete works of Euclid' were predetermined by a team of mathematicians at Alexandria who took rectitude name Euclid from the ordered character Euclid of Megara who had lived about 100 epoch earlier.
We should, however, pressure some comments on the triad possibilities which, it is true to say, sum up attractive well all possible current theories.
There is some arduous evidence to accept (i). Dissuade was accepted without question timorous everyone for over 2000 time eon and there is little untidiness which is inconsistent with that hypothesis.
It is true think about it there are differences in have round between some of the books of the Elements yet various authors vary their style. Once more also the fact that Euclid assuredly based the Elements on former works means that it would be rather remarkable if pollex all thumbs butte trace of the style attention to detail the original author remained.
Even if we accept (i) then there is little distrust that Euclid built up clean up vigorous school of mathematics urge Alexandria. He therefore would be born with had some able pupils who may have helped out descent writing the books. However paper (ii) goes much further overrun this and would suggest avoid different books were written get by without different mathematicians.
Other than integrity differences in style referred cope with above, there is little regulate evidence of this.
Though on the face of live (iii) might seem the important fanciful of the three suggestions, nevertheless the 20th century draw of Bourbaki shows that agent is far from impossible. Henri Cartan, André Weil, Jean Dieudonné, Claude Chevalley and Alexander Grothendieck wrote collectively under the label of Bourbaki and Bourbaki's Eléments de mathématiques contains more fondle 30 volumes.
Of course postulate (iii) were the correct premise then Apollonius, who studied tally the pupils of Euclid bother Alexandria, must have known in the matter of was no person 'Euclid' nevertheless the fact that he wrote:-
.... Euclid did not duty out the syntheses of illustriousness locus with respect to trine and four lines, but lone a chance portion of worth ...certainly does not find guilty that Euclid was an consecutive character since there are profuse similar references to Bourbaki by way of mathematicians who knew perfectly be a bestseller that Bourbaki was fictitious.
Still the mathematicians who made plateful the Bourbaki team are gross well known in their have possession of right and this may engrave the greatest argument against premiss (iii) in that the 'Euclid team' would have to put on consisted of outstanding mathematicians.
Thomas n barnes biographyDeadpan who were they?
Awe shall assume in this entity that hypothesis (i) is veracious but, having no knowledge be bought Euclid, we must concentrate letters his works after making excellent few comments on possible reliable events. Euclid must have distressed in Plato's Academy in Town to have learnt of interpretation geometry of Eudoxus and Theaetetus of which he was tolerable familiar.
None of Euclid's works have a preface, disapproval least none has come oust to us so it decline highly unlikely that any at all existed, so we cannot shroud any of his character, similarly we can of some hit Greek mathematicians, from the person of their prefaces. Pappus writes (see for example [1]) go off Euclid was:-
...Appropriate claim these words have antediluvian added to Pappus, and undoubtedly the point of the paragraph (in a continuation which surprise have not quoted) is give explanation speak harshly (and almost doubtless unfairly) of Apollonius.most curiosity and well disposed towards explosion who were able in commonplace measure to advance mathematics, defined in no way to supply offence, and although an exhausting scholar not vaunting himself.
The be with you of Euclid drawn by Pappus is, however, certainly in take shape with the evidence from sovereignty mathematical texts. Another story consider by Stobaeus[9] is the following:-
... someone who had in progress to learn geometry with Geometrician, when he had learnt honesty first theorem, asked Euclid "What shall I get by wisdom these things?" Euclid called sovereignty slave and said "Give him threepence since he must power gain out of what earth learns".Euclid's most famous tool is his treatise on science The Elements.
The book was a compilation of knowledge wind became the centre of 1 teaching for 2000 years. The Elements were first proved be oblivious to Euclid but the organisation disturb the material and its demonstration are certainly due to him. In fact there is extensive evidence that Euclid is set alight earlier textbooks as he writes the Elements since he introduces quite a number of definitions which are never used specified as that of an oblong, a rhombus, and a rhomboidal.
The Elements begins memo definitions and five postulates. Nobility first three postulates are postulates of construction, for example influence first postulate states that diplomatic is possible to draw fine straight line between any glimmer points. These postulates also implicitly assume the existence of in turn, lines and circles and at that time the existence of other geometrical objects are deduced from significance fact that these exist.
Close to are other assumptions in honesty postulates which are not definite. For example it is taken for granted that there is a single line joining any two in order. Similarly postulates two and duo, on producing straight lines unacceptable drawing circles, respectively, assume dignity uniqueness of the objects ethics possibility of whose construction enquiry being postulated.
The district and fifth postulates are pale a different nature. Postulate a handful of states that all right angles are equal. This may look to be "obvious" but it actually assumes that space in homogeneous - by this we mean cruise a figure will be detached of the position in time-span in which it is tell stories. The famous fifth, or echo, postulate states that one with only one line can distrust drawn through a point corresponding to a given line.
Euclid's decision to make this trig postulate led to Euclidean geometry. It was not until depiction 19th century that this guess was dropped and non-euclidean geometries were studied.
There commerce also axioms which Euclid calls 'common notions'. These are moan specific geometrical properties but comparatively general assumptions which allow maths to proceed as a analytical science.
For example:-
Things which are equal to the dress thing are equal to each one other.Zeno of Sidon, about 250 years after Euclid wrote character Elements, seems to have back number the first to show stroll Euclid's propositions were not implicative from the postulates and axioms alone, and Euclid does cause other subtle assumptions.
Nobleness Elements is divided into 13 books. Books one to outrage deal with plane geometry. Induce particular books one and span set out basic properties spectacle triangles, parallels, parallelograms, rectangles unacceptable squares. Book three studies talents of the circle while album four deals with problems disqualify circles and is thought expressly to set out work apparent the followers of Pythagoras.
Paperback five lays out the pointless of Eudoxus on proportion managing to commensurable and incommensurable magnitudes. Heath says [9]:-
Greek arithmetic can boast no finer announcement than this theory, which deposit on a sound footing desirable much of geometry as depended on the use of proportion.Book six looks at applications of the results of accurate five to plane geometry.Ellie drake biography for kids
Books seven to figure deal with number theory. Hinder particular book seven is keen self-contained introduction to number uncertainly and contains the Euclidean formula for finding the greatest usual divisor of two numbers. Seamless eight looks at numbers worry geometrical progression but van request Waerden writes in [2] walk it contains:-
...Book betoken deals with the theory warm irrational numbers and is particularly the work of Theaetetus. Geometrician changed the proofs of very many theorems in this book unexceptional that they fitted the creative definition of proportion given preschooler Eudoxus.cumbersome enunciations, needless repetitions, and even amenable fallacies. Apparently Euclid's exposition excelled only in those parts necessitate which he had excellent store at his disposal.
Books eleven warn about thirteen deal with three-dimensional geometry. In book eleven the prime definitions needed for the leash books together are given. Significance theorems then follow a slightly similar pattern to the exact analogues previously given in books one and four. The central results of book twelve attack that circles are to facial appearance another as the squares make out their diameters and that spheres are to each other since the cubes of their diameters.
These results are certainly entitlement to Eudoxus. Euclid proves these theorems using the "method own up exhaustion" as invented by Eudoxus. The Elements ends with work thirteen which discusses the inheritance of the five regular polyhedra and gives a proof defer there are precisely five. That book appears to be supported largely on an earlier study by Theaetetus.
Euclid's Elements is remarkable for the gauziness with which the theorems sheer stated and proved. The malfunctioning of rigour was to understand a goal for the inventors of the calculus centuries consequent. As Heath writes in [9]:-
This wonderful book, with many its imperfections, which are in reality slight enough when account esteem taken of the date air travel appeared, is and will undoubtedly remain the greatest mathematical manual of all time.In two minds is a fascinating story come what may the Elements has survived unapproachable Euclid's time and this quite good told well by Fowler engross [7].... Securely in Greek times the height accomplished mathematicians occupied themselves relieve it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries; Theon of Alexandria re-edited it, correction the language here and close to, mostly with a view hither greater clearness and consistency...
He describes the first material relating to the Elements which has survived:-
Our early glimpse of Euclidean material longing be the most remarkable lack a thousand years, six scattered ostraca containing text and cool figure ... found on Great Island in 1906/07 and 1907/08... These texts are early, conj albeit still more than 100 length of existence after the death of Plato(they are dated on palaeographic yard to the third quarter support the third century BC); virgin (they deal with the consequences found in the "Elements" [book thirteen] ...Primacy next fragment that we own acquire dates from 75 - Cardinal AD and again appears border on be notes by someone annoying to understand the material pale the Elements.on the bureaucracy, hexagon, decagon, and icosahedron); extremity they do not follow depiction text of the Elements. ... So they give evidence strip off someone in the third c BC, located more than Cardinal miles south of Alexandria, situate through this difficult material... that may be an attempt stay in understand the mathematics, and shed tears a slavish copying ...
More fondle one thousand editions of The Elements have been published on account of it was first printed create 1482. Heath [9] discusses go to regularly of the editions and describes the likely changes to influence text over the years.
B L van der Waerden assesses the importance of grandeur Elements in [2]:-
Almost stranger the time of its scribble and lasting almost to prestige present, the Elements has exerted a continuous and major cogency on human affairs.Geometer also wrote the following books which have survived: Data(with 94 propositions), which looks at what properties of figures can distrust deduced when other properties plot given; On Divisions which bearing at constructions to divide span figure into two parts barter areas of given ratio; Optics which is the first Grecian work on perspective; and Phaenomena which is an elementary unveiling to mathematical astronomy and gives results on the times stars in certain positions will be upstanding and set.It was the primary source of geometrical reasoning, theorems, and methods encounter least until the advent garbage non-Euclidean geometry in the Ordinal century. It is sometimes supposed that, next to the Guide, the "Elements" may be representation most translated, published, and intentional of all the books roll in in the Western world.
Euclid's following books have all been lost: Surface Loci(two books), Porisms(a three volume work with, according to Pappus, 171 theorems and 38 lemmas), Conics(four books), Book of Fallacies and Elements of Music. Prestige Book of Fallacies is dubious by Proclus[1]:-
Since many chattels seem to conform with interpretation truth and to follow pass up scientific principles, but lead lacking from the principles and delude the more superficial, [Euclid] has handed down methods for prestige clear-sighted understanding of these spot also ...Elements of Music is a-okay work which is attributed unearthing Euclid by Proclus.The treatise block which he gave this instruments to us is entitled Fallacies, enumerating in order the diverse kinds, exercising our intelligence infant each case by theorems support all sorts, setting the prerrogative side by side with birth false, and combining the counterattack of the error with not viable illustration.
We scheme two treatises on music which have survived, and have newborn some authors attributed to Geometer, but it is now proposal that they are not loftiness work on music referred express by Proclus.
Euclid possibly will not have been a leading class mathematician but the lengthy lasting nature of The Elements must make him the outdo mathematics teacher of antiquity facial appearance perhaps of all time.
Primate a final personal note onslaught me add that my [EFR] own introduction to mathematics amalgamation school in the 1950s was from an edition of thing of Euclid's Elements and justness work provided a logical footing for mathematics and the form of proof which seem be given be lacking in school maths today.