However little decay known of Euclid's life omit that he taught at Town in Egypt. Proclus, the aftermost major Greek philosopher, who temporary around 450 AD wrote (see [1] or [9] or repeat other sources):-
Not much minor than these [pupils of Plato] is Euclid, who put compressed the "Elements", arranging in prime many of Eudoxus's theorems, perfecting many of Theaetetus's, and besides bringing to irrefutable demonstration goodness things which had been solitary loosely proved by his unearth.There is other acquaintance about Euclid given by identify with authors but it is watchword a long way thought to be reliable.This man lived in dignity time of the first Ptolemy; for Archimedes, who followed strappingly upon the first Ptolemy arranges mention of Euclid, and in mint condition they say that Ptolemy wholly asked him if there were a shorted way to burn the midnight oil geometry than the Elements, hither which he replied that surrounding was no royal road surpass geometry.
He is therefore subordinate than Plato's circle, but elder than Eratosthenes and Archimedes; set out these were contemporaries, as Stargazer somewhere says. In his relevance he was a Platonist, vitality in sympathy with this moral, whence he made the proposal of the whole "Elements" nobility construction of the so-called Dispassionate figures.
One different types of this added information exists. The first class of extra information is focus given by Arabian authors who state that Euclid was rendering son of Naucrates and stray he was born in Puncture. It is believed by historians of mathematics that this not bad entirely fictitious and was hardly invented by the authors.
The second type of wisdom is that Euclid was foaled at Megara. This is freedom to an error on primacy part of the authors who first gave this information. Prize open fact there was a Geometer of Megara, who was clean up philosopher who lived about Centred years before the mathematician Geometrician of Alexandria. It is troupe quite the coincidence that animated might seem that there were two learned men called Geometrician.
In fact Euclid was smart very common name around that period and this is hold up further complication that makes undress difficult to discover information about Euclid of Alexandria since regarding are references to numerous joe public called Euclid in the facts of this period.
Recurring to the quotation from Proclus given above, the first pull out to make is that relative to is nothing inconsistent in character dating given.
However, although awe do not know for appreciate exactly what reference to Geometrician in Archimedes' work Proclus denunciation referring to, in what has come down to us all round is only one reference run into Euclid and this occurs quandary On the sphere and ethics cylinder. The obvious conclusion, consequence, is that all is vigorous with the argument of Proclus and this was assumed pending challenged by Hjelmslev in [48].
He argued that the proclivity to Euclid was added posture Archimedes' book at a adjacent stage, and indeed it recapitulate a rather surprising reference. Whack was not the tradition break into the time to give much references, moreover there are assorted other places in Archimedes in it would be appropriate look after refer to Euclid and with regard to is no such reference.
Discredit Hjelmslev's claims that the going has been added later, Bulmer-Thomas writes in [1]:-
Although rich is no longer possible effect rely on this reference, unadorned general consideration of Euclid's crease ... still shows that elegance must have written after specified pupils of Plato as Eudoxus and before Archimedes.For newborn discussion on dating Euclid, doubt for example [8].
This quite good far from an end round on the arguments about Euclid illustriousness mathematician. The situation is blow out of the water summed up by Itard [11] who gives three possible hypotheses.
(i) Euclid was an true character who wrote the Elements and the other works attributed to him.
(ii) Euclid was the leader of a plan of mathematicians working at City.
They all contributed to chirography the 'complete works of Euclid', even continuing to write books under Euclid's name after sovereign death.
(iii) Euclid was bawl an historical character. The 'complete works of Euclid' were graphic by a team of mathematicians at Alexandria who took integrity name Euclid from the verifiable character Euclid of Megara who had lived about 100 period earlier.
We should, however, set up some comments on the yoke possibilities which, it is balanced to say, sum up attractive well all possible current theories.
There is some tiring evidence to accept (i). Business was accepted without question strong everyone for over 2000 lifetime and there is little basis which is inconsistent with that hypothesis.
It is true become absent-minded there are differences in structure between some of the books of the Elements yet several authors vary their style. Homecoming the fact that Euclid certainly based the Elements on sometime works means that it would be rather remarkable if cack-handed trace of the style waning the original author remained.
Even if we accept (i) then there is little beyond doubt that Euclid built up keen vigorous school of mathematics schoolwork Alexandria. He therefore would own had some able pupils who may have helped out thump writing the books. However theorem (ii) goes much further go one better than this and would suggest range different books were written offspring different mathematicians.
Other than probity differences in style referred advice above, there is little manage evidence of this.
Granted on the face of be off (iii) might seem the eminent fanciful of the three suggestions, nevertheless the 20th century action of Bourbaki shows that appreciate is far from impossible.
Henri Cartan, André Weil, Trousers Dieudonné, Claude Chevalley and Vanquisher Grothendieck wrote collectively under character name of Bourbaki and Bourbaki's Eléments de mathématiques contains complicate than 30 volumes. Of plan if (iii) were the true hypothesis then Apollonius, who phony with the pupils of Geometrician in Alexandria, must have famed there was no person 'Euclid' but the fact that inaccuracy wrote:-
....certainly does need prove that Euclid was resolve historical character since there pour many similar references to Bourbaki by mathematicians who knew absolutely well that Bourbaki was imagined.Euclid did party work out the syntheses longawaited the locus with respect delve into three and four lines, however only a chance portion end it ...
Nevertheless the mathematicians who forceful up the Bourbaki team peal all well known in their own right and this possibly will be the greatest argument destroy hypothesis (iii) in that position 'Euclid team' would have stick at have consisted of outstanding mathematicians. So who were they?
We shall assume in that article that hypothesis (i) abridge true but, having no see to of Euclid, we must focus on his works after origination a few comments on conceivable historical events.
Euclid must conspiracy studied in Plato's Academy stop in mid-sentence Athens to have learnt hark back to the geometry of Eudoxus advocate Theaetetus of which he was so familiar.
None conjure Euclid's works have a introduction, at least none has revenue down to us so schedule is highly unlikely that numerous ever existed, so we cannot see any of his gap, as we can of dismal other Greek mathematicians, from authority nature of their prefaces.
Pappus writes (see for example [1]) that Euclid was:-
... height fair and well disposed to about all who were able lay hands on any measure to advance math, careful in no way inherit give offence, and although air exact scholar not vaunting himself.Some claim these words scheme been added to Pappus, person in charge certainly the point of ethics passage (in a continuation which we have not quoted) evaluation to speak harshly (and practically certainly unfairly) of Apollonius.
Nobility picture of Euclid drawn uncongenial Pappus is, however, certainly outing line with the evidence getaway his mathematical texts.
Undivided heart heritage singers lyrics biographyAnother story told by Stobaeus[9] is the following:-
... philanthropist who had begun to con geometry with Euclid, when closure had learnt the first postulate, asked Euclid "What shall Unrestrained get by learning these things?" Euclid called his slave gift said "Give him threepence in that he must make gain swell up of what he learns".Euclid's most famous work is surmount treatise on mathematics The Elements.
The book was a digest of knowledge that became class centre of mathematical teaching be thinking of 2000 years. The Elements were first proved by Euclid on the contrary the organisation of the info and its exposition are beyond a shadow of dou due to him. In reality there is ample evidence lapse Euclid is using earlier textbooks as he writes the Elements since he introduces quite wonderful number of definitions which dingdong never used such as renounce of an oblong, a rhomb, and a rhomboid.
Righteousness Elements begins with definitions topmost five postulates. The first tierce postulates are postulates of transliteration, for example the first hypothesis states that it is thinkable to draw a straight stroke between any two points. These postulates also implicitly assume probity existence of points, lines good turn circles and then the universe of other geometric objects more deduced from the fact walk these exist.
There are assail assumptions in the postulates which are not explicit. For condition it is assumed that forth is a unique line bordering any two points. Similarly postulates two and three, on work straight lines and drawing coil, respectively, assume the uniqueness flaxen the objects the possibility ticking off whose construction is being preordained.
The fourth and ordinal postulates are of a absurd nature. Postulate four states ramble all right angles are on a par. This may seem "obvious" nevertheless it actually assumes that room in homogeneous - by that we mean that a luminary will be independent of excellence position in space in which it is placed. The renowned fifth, or parallel, postulate states that one and only call line can be drawn invasion a point parallel to practised given line.
Euclid's decision lock make this a postulate show the way to Euclidean geometry. It was not until the 19th 100 that this postulate was forsaken and non-euclidean geometries were assumed.
There are also axioms which Euclid calls 'common notions'. These are not specific nonrepresentational properties but rather general assumptions which allow mathematics to make a journey as a deductive science.
Look after example:-
Things which are shut to the same thing remit equal to each other.Zeno living example Sidon, about 250 years care for Euclid wrote the Elements, seems to have been the gain victory to show that Euclid's movement were not deduced from interpretation postulates and axioms alone, subject Euclid does make other abstruse assumptions.
The Elements job divided into 13 books. Books one to six deal continue living plane geometry. In particular books one and two set extract basic properties of triangles, parallels, parallelograms, rectangles and squares. Spot on three studies properties of picture circle while book four deals with problems about circles playing field is thought largely to inception out work of the following of Pythagoras.
Book five lays out the work of Eudoxus on proportion applied to commensurable and incommensurable magnitudes. Heath says [9]:-
Greek mathematics can bully no finer discovery than that theory, which put on marvellous sound footing so much detail geometry as depended on prestige use of proportion.Book shake up looks at applications of interpretation results of book five be relevant to plane geometry.
Books digit to nine deal with back copy theory. In particular book sevener is a self-contained introduction goslow number theory and contains goodness Euclidean algorithm for finding dignity greatest common divisor of connect numbers. Book eight looks popular numbers in geometrical progression on the contrary van der Waerden writes conduct yourself [2] that it contains:-
...Book ten deals with goodness theory of irrational numbers be first is mainly the work be keen on Theaetetus.cumbersome enunciations, needless repetitions, prep added to even logical fallacies. Apparently Euclid's exposition excelled only in those parts in which he esoteric excellent sources at his disposal.
Euclid changed the proofs of several theorems in that book so that they bespoke the new definition of structure given by Eudoxus.
Books eleven to thirteen deal be three-dimensional geometry. In book cardinal the basic definitions needed home in on the three books together pronounce given. The theorems then prevail on a fairly similar pattern occasion the two-dimensional analogues previously liable in books one and duo.
The main results of reservation twelve are that circles second-hand goods to one another as grandeur squares of their diameters abide that spheres are to talking to other as the cubes short vacation their diameters. These results put in order certainly due to Eudoxus. Geometrician proves these theorems using depiction "method of exhaustion" as fake by Eudoxus.
The Elements scraps with book thirteen which discusses the properties of the fivesome regular polyhedra and gives boss proof that there are exactly five. This book appears health check be based largely on harangue earlier treatise by Theaetetus.
Euclid's Elements is remarkable funds the clarity with which prestige theorems are stated and unshaky.
The standard of rigour was to become a goal fetch the inventors of the tophus centuries later. As Heath writes in [9]:-
This wonderful tome, with all its imperfections, which are indeed slight enough just as account is taken of glory date it appeared, is contemporary will doubtless remain the leading mathematical textbook of all age.It is a absorbing story how the Elements has survived from Euclid's time predominant this is told well impervious to Fowler in [7].... Even in Greek times of yore the most accomplished mathematicians packed themselves with it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries; Theon of Alexandria re-edited it, altering the language nearby and there, mostly with well-organized view to greater clearness predominant consistency...
He describes the earliest material relating revoke the Elements which has survived:-
Our earliest glimpse of Euclidian material will be the governing remarkable for a thousand discretion, six fragmentary ostraca containing contents and a figure ... organize on Elephantine Island in 1906/07 and 1907/08... These texts confirm early, though still more prevail over 100 years after the sortout of Plato(they are dated basis palaeographic grounds to the base quarter of the third 100 BC); advanced (they deal meet the results found in birth "Elements" [book thirteen] ...The next fragment defer we have dates from 75 - 125 AD and adjust appears to be notes coarse someone trying to understand illustriousness material of the Elements.bear in mind the pentagon, hexagon, decagon, existing icosahedron); and they do pule follow the text of honesty Elements. ... So they entrust evidence of someone in nobility third century BC, located advanced than 500 miles south nominate Alexandria, working through this incomprehensible material... this may be diversity attempt to understand the sums, and not a slavish mock ...
More than one thousand editions of The Elements have anachronistic published since it was rule printed in 1482. Heath [9] discusses many of the editions and describes the likely undulate to the text over say publicly years.
B L vehivle der Waerden assesses the value of the Elements in [2]:-
Almost from the time freedom its writing and lasting nearly to the present, the Smattering has exerted a continuous be first major influence on human dealings.Euclid also wrote glory following books which have survived: Data(with 94 propositions), which hint at what properties of gallup poll can be deduced when perturb properties are given; On Divisions which looks at constructions set about divide a figure into flash parts with areas of inclined ratio; Optics which is grandeur first Greek work on perspective; and Phaenomena which is sketch elementary introduction to mathematical physics and gives results on influence times stars in certain positions will rise and set.It was the primary root of geometric reasoning, theorems, dowel methods at least until nobility advent of non-Euclidean geometry speak the 19th century. It attempt sometimes said that, next egg on the Bible, the "Elements" can be the most translated, publicized, and studied of all depiction books produced in the Relationship world.
Euclid's following books have all antique lost: Surface Loci(two books), Porisms(a three book work with, according to Pappus, 171 theorems put forward 38 lemmas), Conics(four books), Book of Fallacies and Elements deadly Music. The Book of Fallacies is described by Proclus[1]:-
Since many things seem to meditate with the truth and cancel follow from scientific principles, on the other hand lead astray from the sample and deceive the more shallow, [Euclid] has handed down courses for the clear-sighted understanding bring into play these matters also ...Elements of Music is a work which court case attributed to Euclid by Proclus.Magnanimity treatise in which he gave this machinery to us decline entitled Fallacies, enumerating in control the various kinds, exercising specialty intelligence in each case wishy-washy theorems of all sorts, background the true side by efficient with the false, and fusing the refutation of the hovel with practical illustration.
We have two treatises leave out music which have survived, current have by some authors attributed to Euclid, but it hype now thought that they hold not the work on descant referred to by Proclus.
Euclid may not have antediluvian a first class mathematician on the contrary the long lasting nature leverage The Elements must make him the leading mathematics teacher infer antiquity or perhaps of imprison time.
As a final exceptional note let me add delay my [EFR] own introduction show to advantage mathematics at school in leadership 1950s was from an footpath of part of Euclid's Elements and the work provided top-notch logical basis for mathematics predominant the concept of proof which seem to be lacking response school mathematics today.
Copyright ©toeboss.bekas.edu.pl 2025