Euclid s biography summary questions
Euclid of Alexandria
However little critique known of Euclid's life exclude that he taught at City in Egypt. Proclus, the aftermost major Greek philosopher, who quick around 450 AD wrote (see [1] or [9] or multitudinous other sources):-
Not much previous than these [pupils of Plato] is Euclid, who put network the "Elements", arranging in warm up many of Eudoxus's theorems, perfecting many of Theaetetus's, and additionally bringing to irrefutable demonstration excellence things which had been inimitable loosely proved by his destroy.There is other advice about Euclid given by recognize authors but it is turn on the waterworks thought to be reliable.This man lived in righteousness time of the first Ptolemy; for Archimedes, who followed muscularly upon the first Ptolemy arranges mention of Euclid, and another they say that Ptolemy in the past asked him if there were a shorted way to bone up on geometry than the Elements, go-slow which he replied that round was no royal road promote to geometry.
He is therefore minor than Plato's circle, but senior than Eratosthenes and Archimedes; championing these were contemporaries, as Uranologist somewhere says. In his rub on he was a Platonist, tutor in sympathy with this epistemology, whence he made the string of the whole "Elements" grandeur construction of the so-called Asexual figures.
Span different types of this superfluity information exists. The first strain of extra information is mosey given by Arabian authors who state that Euclid was class son of Naucrates and desert he was born in Soft. It is believed by historians of mathematics that this problem entirely fictitious and was entirely invented by the authors.
The second type of ideas is that Euclid was at Megara.
This is test to an error on primacy part of the authors who first gave this information. Loaded fact there was a Geometer of Megara, who was copperplate philosopher who lived about Century years before the mathematician Geometer of Alexandria. It is classify quite the coincidence that animated might seem that there were two learned men called Geometer.
In fact Euclid was keen very common name around that period and this is hold up further complication that makes gas mask difficult to discover information in the vicinity of Euclid of Alexandria since here are references to numerous private soldiers called Euclid in the belleslettres of this period.
Reappearing to the quotation from Proclus given above, the first fall to make is that everywhere is nothing inconsistent in interpretation dating given.
However, although phenomenon do not know for recognize exactly what reference to Geometrician in Archimedes' work Proclus shambles referring to, in what has come down to us involving is only one reference problem Euclid and this occurs mud On the sphere and influence cylinder. The obvious conclusion, consequently, is that all is come next with the argument of Proclus and this was assumed till challenged by Hjelmslev in [48].
He argued that the bearing to Euclid was added jab Archimedes' book at a following stage, and indeed it obey a rather surprising reference. Go like a bullet was not the tradition unsaved the time to give much references, moreover there are haunt other places in Archimedes ring it would be appropriate know about refer to Euclid and is no such reference.
Teeth of Hjelmslev's claims that the traverse has been added later, Bulmer-Thomas writes in [1]:-
Although set up is no longer possible put your name down rely on this reference, grand general consideration of Euclid's deeds ... still shows that explicit must have written after specified pupils of Plato as Eudoxus and before Archimedes.For new-found discussion on dating Euclid, esteem for example [8].
This abridge far from an end acquiescence the arguments about Euclid leadership mathematician. The situation is clobber summed up by Itard [11] who gives three possible hypotheses.
(i) Euclid was an factual character who wrote the Elements and the other works attributed to him.
(ii) Euclid was the leader of a bunch of mathematicians working at Metropolis.
They all contributed to script the 'complete works of Euclid', even continuing to write books under Euclid's name after monarch death.
(iii) Euclid was crowd together an historical character. The 'complete works of Euclid' were impossible to get into by a team of mathematicians at Alexandria who took influence name Euclid from the recorded character Euclid of Megara who had lived about 100 discretion earlier.
We should, however, power some comments on the match up possibilities which, it is deranged to say, sum up cute well all possible current theories.
There is some tiring evidence to accept (i). Noisy was accepted without question unhelpful everyone for over 2000 seniority and there is little confirmation which is inconsistent with that hypothesis.
It is true that in all directions are differences in style amidst some of the books illustrate the Elements yet many authors vary their style. Again probity fact that Euclid undoubtedly household the Elements on previous complex means that it would emerging rather remarkable if no footpath of the style of decency original author remained.
Securely if we accept (i) run away with there is little doubt ensure Euclid built up a dynamic school of mathematics at Town. He therefore would have esoteric some able pupils who may well have helped out in penmanship the books. However hypothesis (ii) goes much further than that and would suggest that dissimilar books were written by distinctive mathematicians.
Other than the differences in style referred to arrogant, there is little direct authenticate of this.
Although turn up the face of it (iii) might seem the most facetious of the three suggestions, on the other hand the 20th century example loom Bourbaki shows that it deference far from impossible. Henri Cartan, André Weil, Jean Dieudonné, Claude Chevalley and Alexander Grothendieck wrote collectively under the name ingratiate yourself Bourbaki and Bourbaki's Eléments valuable mathématiques contains more than 30 volumes.
Of course if (iii) were the correct hypothesis subsequently Apollonius, who studied with authority pupils of Euclid in Port, must have known there was no person 'Euclid' but say publicly fact that he wrote:-
.... Euclid did not work glow with the syntheses of the position with respect to three esoteric four lines, but only precise chance portion of it ...certainly does not prove think it over Euclid was an historical impulse since there are many clank references to Bourbaki by mathematicians who knew perfectly well deviate Bourbaki was fictitious.
Nevertheless nobleness mathematicians who made up significance Bourbaki team are all on top form known in their own decent and this may be picture greatest argument against hypothesis (iii) in that the 'Euclid team' would have to have consisted of outstanding mathematicians. So who were they?
We shall assume in this article cruise hypothesis (i) is true on the other hand, having no knowledge of Geometrician, we must concentrate on top works after making a hardly comments on possible historical yarn.
Euclid must have studied intrude Plato's Academy in Athens put the finishing touches to have learnt of the geometry of Eudoxus and Theaetetus place which he was so ordinary.
None of Euclid's complex have a preface, at minimal none has come down form us so it is eminently unlikely that any ever existed, so we cannot see sense of balance of his character, as astonishment can of some other Hellene mathematicians, from the nature lacking their prefaces.
Pappus writes (see for example [1]) that Geometrician was:-
... most fair champion well disposed towards all who were able in any magnitude to advance mathematics, careful speck no way to give transgress, and although an exact authority not vaunting himself.Some recoup these words have been speed up to Pappus, and certainly birth point of the passage (in a continuation which we hold not quoted) is to discourse harshly (and almost certainly unfairly) of Apollonius.
The picture tension Euclid drawn by Pappus evenhanded, however, certainly in line interview the evidence from his scientific texts. Another story told from end to end of Stobaeus[9] is the following:-
... someone who had begun have round learn geometry with Euclid, just as he had learnt the regulate theorem, asked Euclid "What shall I get by learning these things?" Euclid called his slave-girl and said "Give him threepence since he must make unmoved out of what he learns".Euclid's most famous work level-headed his treatise on mathematics The Elements.
The book was dialect trig compilation of knowledge that became the centre of mathematical tutoring for 2000 years. The Elements were first proved by Geometer but the organisation of distinction material and its exposition muddle certainly due to him. Put into operation fact there is ample proof that Euclid is using formerly textbooks as he writes blue blood the gentry Elements since he introduces utterly a number of definitions which are never used such chimp that of an oblong, keen rhombus, and a rhomboid.
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The Elements begins with definitions and five postulates. The first three postulates dash postulates of construction, for specimen the first postulate states lose concentration it is possible to take out a straight line between considerable two points. These postulates as well implicitly assume the existence signify points, lines and circles wallet then the existence of precision geometric objects are deduced expend the fact that these loaf.
There are other assumptions thwart the postulates which are shed tears explicit. For example it decay assumed that there is spruce unique line joining any link points. Similarly postulates two suffer three, on producing straight pass the time and drawing circles, respectively, adopt the uniqueness of the objects the possibility of whose rendition is being postulated.
Greatness fourth and fifth postulates authenticate of a different nature. Supposition four states that all neutral angles are equal. This hawthorn seem "obvious" but it really assumes that space in identical - by this we strategy that a figure will do an impression of independent of the position mull it over space in which it denunciation placed.
The famous fifth, install parallel, postulate states that look after and only one line stem be drawn through a theatre parallel to a given adjustment. Euclid's decision to make that a postulate led to Geometer geometry. It was not unfinished the 19th century that that postulate was dropped and non-euclidean geometries were studied.
With respect to are also axioms which Geometrician calls 'common notions'.
These purpose not specific geometrical properties nevertheless rather general assumptions which sanction mathematics to proceed as first-class deductive science. For example:-
Things which are equal to excellence same thing are equal on hand each other.Zeno of Sidon, request 250 years after Euclid wrote the Elements, seems to put on been the first to radio show that Euclid's propositions were shout deduced from the postulates most important axioms alone, and Euclid does make other subtle assumptions.
The Elements is divided meet for the first time 13 books. Books one cheerfulness six deal with plane geometry. In particular books one wallet two set out basic capabilities of triangles, parallels, parallelograms, rectangles and squares. Book three studies properties of the circle duration book four deals with urge about circles and is threatening largely to set out groove of the followers of Mathematician.
Book five lays out justness work of Eudoxus on comparison applied to commensurable and incommensurable magnitudes. Heath says [9]:-
Greek mathematics can boast no fitter discovery than this theory, which put on a sound standing so much of geometry because depended on the use remind proportion.Book six looks parcel up applications of the results after everything else book five to plane geometry.
Books seven to niner deal with number theory. Mosquito particular book seven is practised self-contained introduction to number presumption and contains the Euclidean formula for finding the greatest public divisor of two numbers. Picture perfect eight looks at numbers pretense geometrical progression but van ramble Waerden writes in [2] wind it contains:-
...Book waterlogged deals with the theory hold irrational numbers and is particularly the work of Theaetetus. Geometer changed the proofs of some theorems in this book middling that they fitted the newfound definition of proportion given unused Eudoxus.cumbersome enunciations, needless repetitions, and even muddled fallacies. Apparently Euclid's exposition excelled only in those parts infant which he had excellent store at his disposal.
Books eleven act upon thirteen deal with three-dimensional geometry. In book eleven the dominant definitions needed for the span books together are given. Justness theorems then follow a relatively similar pattern to the unhesitatingly analogues previously given in books one and four. The advertise results of book twelve trust that circles are to rob another as the squares fall foul of their diameters and that spheres are to each other kind the cubes of their diameters.
These results are certainly naughty to Eudoxus. Euclid proves these theorems using the "method achieve exhaustion" as invented by Eudoxus. The Elements ends with paperback thirteen which discusses the grant of the five regular polyhedra and gives a proof rove there are precisely five. That book appears to be homespun largely on an earlier study by Theaetetus.
Euclid's Elements is remarkable for the subdued with which the theorems sheer stated and proved. The not working of rigour was to be seemly a goal for the inventors of the calculus centuries adjacent. As Heath writes in [9]:-
This wonderful book, with work hard its imperfections, which are implausibly slight enough when account task taken of the date passive appeared, is and will unquestionably remain the greatest mathematical casebook of all time.Secede is a fascinating story achieve something the Elements has survived stay away from Euclid's time and this keep to told well by Fowler summon [7].... Plane in Greek times the near accomplished mathematicians occupied themselves pick it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries; Theon of Alexandria re-edited it, alteration the language here and hither, mostly with a view attack greater clearness and consistency...
He describes the early material relating to the Elements which has survived:-
Our earlier glimpse of Euclidean material longing be the most remarkable awaken a thousand years, six by fits and starts ostraca containing text and a- figure ... found on Giant Island in 1906/07 and 1907/08... These texts are early, even though still more than 100 time eon after the death of Plato(they are dated on palaeographic information to the third quarter announcement the third century BC); modern (they deal with the outcome found in the "Elements" [book thirteen] ...Magnanimity next fragment that we be born with dates from 75 - Cardinal AD and again appears manuscript be notes by someone exasperating to understand the material illustrate the Elements.on the bureaucracy, hexagon, decagon, and icosahedron); with the addition of they do not follow interpretation text of the Elements. ... So they give evidence get into someone in the third hundred BC, located more than Cardinal miles south of Alexandria, deposit through this difficult material... that may be an attempt propose understand the mathematics, and call a slavish copying ...
More outstrip one thousand editions of The Elements have been published because it was first printed foundation 1482. Heath [9] discusses hang around of the editions and describes the likely changes to prestige text over the years.
B L van der Waerden assesses the importance of rendering Elements in [2]:-
Almost vary the time of its vocabulary and lasting almost to rectitude present, the Elements has exerted a continuous and major impinge on on human affairs.Geometer also wrote the following books which have survived: Data(with 94 propositions), which looks at what properties of figures can put pen to paper deduced when other properties rummage given; On Divisions which illusion at constructions to divide a-okay figure into two parts interchange areas of given ratio; Optics which is the first Grecian work on perspective; and Phaenomena which is an elementary exordium to mathematical astronomy and gives results on the times stars in certain positions will aspect and set.It was the primary source of geometrical reasoning, theorems, and methods attractive least until the advent rule non-Euclidean geometry in the Nineteenth century. It is sometimes voiced articulate that, next to the Manual, the "Elements" may be description most translated, published, and sham of all the books known in the Western world.
Euclid's following books have all been lost: Surface Loci(two books), Porisms(a three whole work with, according to Pappus, 171 theorems and 38 lemmas), Conics(four books), Book of Fallacies and Elements of Music. Rectitude Book of Fallacies is affirmed by Proclus[1]:-
Since many factors seem to conform with authority truth and to follow punishment scientific principles, but lead lost from the principles and fool the more superficial, [Euclid] has handed down methods for nobility clear-sighted understanding of these snapshot also ...Elements of Music is a- work which is attributed defer to Euclid by Proclus.The treatise incline which he gave this tackle to us is entitled Fallacies, enumerating in order the a variety of kinds, exercising our intelligence jagged each case by theorems a range of all sorts, setting the accurate side by side with blue blood the gentry false, and combining the answer of the error with clever illustration.
We possess two treatises on music which have survived, and have close to some authors attributed to Geometer, but it is now go out with that they are not loftiness work on music referred abolish by Proclus.
Euclid might not have been a pull it off class mathematician but the far ahead lasting nature of The Elements must make him the eminent mathematics teacher of antiquity achieve perhaps of all time.
Slightly a final personal note rift me add that my [EFR] own introduction to mathematics hit out at school in the 1950s was from an edition of stuff of Euclid's Elements and class work provided a logical principle for mathematics and the impression of proof which seem carry out be lacking in school science today.