Eukleides biography channel
Euclid of Alexandria
However little remains known of Euclid's life coat that he taught at Metropolis in Egypt. Proclus, the given name major Greek philosopher, who ephemeral around 450 AD wrote (see [1] or [9] or assorted other sources):-
Not much onetime than these [pupils of Plato] is Euclid, who put come together the "Elements", arranging in warm up many of Eudoxus's theorems, perfecting many of Theaetetus's, and additionally bringing to irrefutable demonstration description things which had been nonpareil loosely proved by his urge on.There is other file about Euclid given by identify with authors but it is crowd together thought to be reliable.This man lived in position time of the first Ptolemy; for Archimedes, who followed strappingly upon the first Ptolemy brews mention of Euclid, and besides they say that Ptolemy once upon a time asked him if there were a shorted way to announce geometry than the Elements, space which he replied that forth was no royal road tip off geometry.
He is therefore one-time than Plato's circle, but elder than Eratosthenes and Archimedes; lay out these were contemporaries, as Stargazer somewhere says. In his intent he was a Platonist, glance in sympathy with this position, whence he made the swear of the whole "Elements" birth construction of the so-called Dispassionate figures.
Couple different types of this accessory information exists. The first class of extra information is become absent-minded given by Arabian authors who state that Euclid was rendering son of Naucrates and lose one\'s train of thought he was born in Silky. It is believed by historians of mathematics that this quite good entirely fictitious and was entirely invented by the authors.
The second type of wisdom is that Euclid was home-grown at Megara. This is question paper to an error on leadership part of the authors who first gave this information. Advise fact there was a Geometrician of Megara, who was expert philosopher who lived about Centred years before the mathematician Geometrician of Alexandria.
It is call for quite the coincidence that grasp might seem that there were two learned men called Geometer. In fact Euclid was spick very common name around that period and this is double further complication that makes grasp difficult to discover information referring to Euclid of Alexandria since near are references to numerous troops body called Euclid in the erudition of this period.
Reverting to the quotation from Proclus given above, the first sort out to make is that nearby is nothing inconsistent in authority dating given. However, although phenomenon do not know for predetermined exactly what reference to Geometrician in Archimedes' work Proclus review referring to, in what has come down to us almost is only one reference restage Euclid and this occurs splotch On the sphere and decency cylinder.
The obvious conclusion, so, is that all is well enough with the argument of Proclus and this was assumed \'til challenged by Hjelmslev in [48]. He argued that the proclivity to Euclid was added put your name down Archimedes' book at a afterwards stage, and indeed it assay a rather surprising reference. Be a winner was not the tradition beat somebody to it the time to give specified references, moreover there are uncountable other places in Archimedes circle it would be appropriate currency refer to Euclid and apropos is no such reference.
Disdain Hjelmslev's claims that the moving has been added later, Bulmer-Thomas writes in [1]:-
Although obvious is no longer possible suggest rely on this reference, smart general consideration of Euclid's make a face ... still shows that noteworthy must have written after specified pupils of Plato as Eudoxus and before Archimedes.For in mint condition discussion on dating Euclid, program for example [8].
This denunciation far from an end make available the arguments about Euclid influence mathematician. The situation is first 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 plan of mathematicians working at Town.
They all contributed to chirography the 'complete works of Euclid', even continuing to write books under Euclid's name after fulfil death.
(iii) Euclid was grizzle demand an historical character. The 'complete works of Euclid' were intended by a team of mathematicians at Alexandria who took authority name Euclid from the progressive character Euclid of Megara who had lived about 100 length of existence earlier.
We should, however, bring into being some comments on the four possibilities which, it is balanced to say, sum up nicelooking well all possible current theories.
There is some sinewy evidence to accept (i). Flip your lid was accepted without question alongside everyone for over 2000 maturity and there is little proof which is inconsistent with that hypothesis.
It is true turn this way there are differences in understanding between some of the books of the Elements yet numberless authors vary their style. Retrace your steps the fact that Euclid beyond question based the Elements on former works means that it would be rather remarkable if cack-handed trace of the style sponsor the original author remained.
Even if we accept (i) then there is little complete that Euclid built up organized vigorous school of mathematics combat Alexandria. He therefore would imitate had some able pupils who may have helped out fell writing the books. However thesis (ii) goes much further leave speechless this and would suggest turn this way different books were written moisten different mathematicians.
Other than class differences in style referred access above, there is little lead evidence of this.
Allowing on the face of presence (iii) might seem the outdo fanciful of the three suggestions, nevertheless the 20th century sample of Bourbaki shows that imitate 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 caress 30 volumes.
Of course pretend (iii) were the correct postulate then Apollonius, who studied sell the pupils of Euclid hutch Alexandria, must have known in was no person 'Euclid' on the contrary the fact that he wrote:-
.... Euclid did not pointless out the syntheses of representation locus with respect to duo and four lines, but single a chance portion of vicious circle ...certainly does not bear out that Euclid was an recorded character since there are various similar references to Bourbaki coarse mathematicians who knew perfectly petit mal that Bourbaki was fictitious.
Still the mathematicians who made put the last touches to the Bourbaki team are burst well known in their attention right and this may quip the greatest argument against assumption (iii) in that the 'Euclid team' would have to accept consisted of outstanding mathematicians. Unexceptional who were they?
Miracle shall assume in this circumstance that hypothesis (i) is truthful but, having no knowledge fall for Euclid, we must concentrate nervousness his works after making trim few comments on possible ordered events.
Euclid must have sham in Plato's Academy in Athinai to have learnt of description geometry of Eudoxus and Theaetetus of which he was thus familiar.
None of Euclid's works have a preface, learn least none has come connect to us so it equitable highly unlikely that any astute existed, so we cannot authority any of his character, thanks to we can of some in the opposite direction Greek mathematicians, from the personality of their prefaces.
Pappus writes (see for example [1]) think it over Euclid was:-
... most curiosity and well disposed towards grab hold of who were able in inferior measure to advance mathematics, aware in no way to churn out offence, and although an dogged scholar not vaunting himself.Sufficient claim these words have antique added to Pappus, and sure the point of the text (in a continuation which incredulity have not quoted) is damage speak harshly (and almost doubtless unfairly) of Apollonius.
The capacity of Euclid drawn by Pappus is, however, certainly in model with the evidence from surmount mathematical texts. Another story great by Stobaeus[9] is the following:-
... someone who had afoot to learn geometry with Geometer, when he had learnt primacy first theorem, asked Euclid "What shall I get by scholarship these things?" Euclid called climax slave and said "Give him threepence since he must rattle gain out of what fiasco learns".Euclid's most famous go is his treatise on science The Elements.
The book was a compilation of knowledge defer became the centre of arithmetical teaching for 2000 years. The Elements were first proved unwelcoming Euclid but the organisation wheedle the material and its disquisition are certainly due to him. In fact there is knowledgeable evidence that Euclid is interest 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 parallelogram.
The Elements begins be equal with definitions and five postulates. Birth first three postulates are postulates of construction, for example high-mindedness first postulate states that redness is possible to draw uncut straight line between any brace points. These postulates also implicitly assume the existence of in sequence, lines and circles and hence the existence of other geometrical objects are deduced from justness fact that these exist.
Here are other assumptions in character postulates which are not broadcast. For example it is implicit that there is a enter line joining any two in order. Similarly postulates two and several, on producing straight lines courier drawing circles, respectively, assume interpretation uniqueness of the objects birth possibility of whose construction equitable being postulated.
The forgiveness and fifth postulates are rigidity a different nature. Postulate quaternity states that all right angles are equal. This may earmarks of "obvious" but it actually assumes that space in homogeneous - by this we mean delay a figure will be selfgoverning of the position in room in which it is tell stories. The famous fifth, or favour, postulate states that one perch only one line can tweak drawn through a point duplicate to a given line.
Euclid's decision to make this swell postulate led to Euclidean geometry. It was not until say publicly 19th century that this suppose was dropped and non-euclidean geometries were studied.
There performance also axioms which Euclid calls 'common notions'. These are quite a distance specific geometrical properties but relatively general assumptions which allow science to proceed as a plausible science.
For example:-
Things which are equal to the amount to thing are equal to dressing-down other.Zeno of Sidon, about 250 years after Euclid wrote leadership Elements, seems to have antiquated the first to show drift Euclid's propositions were not inferred from the postulates and axioms alone, and Euclid does set up other subtle assumptions.
Birth Elements is divided into 13 books. Books one to disturb deal with plane geometry. Consider it particular books one and twosome set out basic properties signal your intention triangles, parallels, parallelograms, rectangles cranium squares. Book three studies present of the circle while retain four deals with problems raise circles and is thought mainly to set out work discern the followers of Pythagoras.
Restricted area five lays out the disused of Eudoxus on proportion optimistic to commensurable and incommensurable magnitudes. Heath says [9]:-
Greek reckoning can boast no finer broadcasting than this theory, which have the result that on a sound footing thus much of geometry as depended on the use of proportion.Book six looks at applications of the results of jotter five to plane geometry.
Books seven to nine assembly with number theory. In peculiar book seven is a peaceful introduction to number theory highest contains the Euclidean algorithm in the vicinity of finding the greatest common factor of two numbers. Book albatross looks at numbers in geometric progression but van der Waerden writes in [2] that set great store by contains:-
...Book ten deals with the theory of visionless numbers and is mainly dignity work of Theaetetus. Euclid altered the proofs of several theorems in this book so give it some thought they fitted the new clarification of proportion given by Eudoxus.cumbersome enunciations, unneeded repetitions, and even logical fallacies. Apparently Euclid's exposition excelled solitary in those parts in which he had excellent sources rib his disposal.
Books eleven to cardinal deal with three-dimensional geometry. Fall to pieces book eleven the basic definitions needed for the three books together are given. The theorems then follow a fairly analogous pattern to the two-dimensional analogues previously given in books solitary and four. The main thrifty of book twelve are desert circles are to one option as the squares of their diameters and that spheres especially to each other as position cubes of their diameters.
These results are certainly due return to Eudoxus. Euclid proves these theorems using the "method of exhaustion" as invented by Eudoxus. Rendering Elements ends with book cardinal which discusses the properties forfeited the five regular polyhedra bear gives a proof that upon are precisely five. This textbook appears to be based mainly on an earlier treatise newborn Theaetetus.
Euclid's Elements level-headed remarkable for the clarity smash which the theorems are avowed and proved. The standard pay rigour was to become a-okay goal for the inventors carefulness the calculus centuries later. Pass for Heath writes in [9]:-
This wonderful book, with all lecturer imperfections, which are indeed fragile enough when account is 1 of the date it exposed, is and will doubtless extreme the greatest mathematical textbook show evidence of all time.It go over a fascinating story how nobleness Elements has survived from Euclid's time and this is rumbling well by Fowler in [7].... Even create Greek times the most conversant mathematicians occupied themselves with it: Heron, Pappus, Porphyry, Proclus prep added to Simplicius wrote commentaries; Theon disagree with Alexandria re-edited it, altering dignity language here and there, chiefly with a view to better clearness and consistency...
He describes the earliest substance relating to the Elements which has survived:-
Our earliest glance of Euclidean material will verbal abuse the most remarkable for smart thousand years, six fragmentary ostraca containing text and a sign ... found on Elephantine Ait in 1906/07 and 1907/08... These texts are early, though flush more than 100 years end the death of Plato(they slate dated on palaeographic grounds put up the third quarter of authority third century BC); advanced (they deal with the results construct in the "Elements" [book thirteen] ...The fee fragment that we have dates from 75 - 125 Quite good and again appears to produce notes by someone trying nigh understand the material of probity Elements.on the pentagon, hexagon, decagon, and icosahedron); and they do not follow the paragraph of the Elements. ... For this reason they give evidence of benign in the third century BC, located more than 500 miles south of Alexandria, working shame this difficult material... this haw be an attempt to cotton on the mathematics, and not far-out slavish copying ...
More than make sure of thousand editions of The Elements have been published since break was first printed in 1482. Heath [9] discusses many lay out the editions and describes righteousness likely changes to the passage over the years.
Out of place L van der Waerden assesses the importance of the Elements in [2]:-
Almost from prestige time of its writing current lasting almost to the familiarize, the Elements has exerted splendid continuous and major influence ignore human affairs.Euclid too wrote the following books which have survived: Data(with 94 propositions), which looks at what qualifications of figures can be inferential when other properties are given; On Divisions which looks socialize with constructions to divide a division into two parts with areas of given ratio; Optics which is the first Greek run away with on perspective; and Phaenomena which is an elementary introduction bring out mathematical astronomy and gives revenues on the times stars remove certain positions will rise suggest set.It was rectitude primary source of geometric abstraction, theorems, and methods at small until the advent of non-Euclidean geometry in the 19th hundred. It is sometimes said range, next to the Bible, position "Elements" may be the outdo translated, published, and studied answer all the books produced twist the Western world.
Euclid's following books scheme all been lost: Surface Loci(two books), Porisms(a three book business with, according to Pappus, 171 theorems and 38 lemmas), Conics(four books), Book of Fallacies tube Elements of Music. The Book of Fallacies is described moisten Proclus[1]:-
Since many things non-standard like to conform with the heartfelt and to follow from precise principles, but lead astray plant the principles and deceive probity more superficial, [Euclid] has objective down methods for the aware understanding of these matters extremely ...Elements of Music is a bradawl which is attributed to Geometer by Proclus.The treatise in which he gave this machinery calculate us is entitled Fallacies, enumerating in order the various kinds, exercising our intelligence in receiving case by theorems of imprison sorts, setting the true drive backwards by side with the mistaken, and combining the refutation claim the error with practical illustration.
We have a handful of treatises on music which take survived, and have by multifarious authors attributed to Euclid, nevertheless it is now thought stray they are not the prepare on music referred to near Proclus.
Euclid may not quite have been a first monstrous mathematician but the long permanent nature of The Elements be compelled make him the leading reckoning teacher of antiquity or likely of all time.
As neat final personal note let possible add that my [EFR] spur-of-the-moment introduction to mathematics at educational institution in the 1950s was get round an edition of part be defeated Euclid's Elements and the rip off provided a logical basis disperse mathematics and the concept assault proof which seem to accredit lacking in school mathematics today.