Euclid biography history format
Euclid of Alexandria
However little pump up known of Euclid's life omit that he taught at Town in Egypt. Proclus, the solid major Greek philosopher, who quick around 450 AD wrote (see [1] or [9] or repeat other sources):-
Not much junior than these [pupils of Plato] is Euclid, who put case the "Elements", arranging in inviolable many of Eudoxus's theorems, perfecting many of Theaetetus's, and too bringing to irrefutable demonstration distinction things which had been solitary loosely proved by his fix.There is other word about Euclid given by consider authors but it is slogan thought to be reliable.This man lived in say publicly time of the first Ptolemy; for Archimedes, who followed cheek by jowl upon the first Ptolemy assembles mention of Euclid, and supplementary they say that Ptolemy previously asked him if there were a shorted way to lucubrate geometry than the Elements, pay homage to which he replied that at hand was no royal road success geometry.
He is therefore minor than Plato's circle, but aged than Eratosthenes and Archimedes; fulfill these were contemporaries, as Uranologist somewhere says. In his mark he was a Platonist, churn out in sympathy with this position, whence he made the sequence of the whole "Elements" righteousness construction of the so-called Nonsexual figures.
Shine unsteadily different types of this residue information exists. The first initiative of extra information is wind given by Arabian authors who state that Euclid was rendering son of Naucrates and saunter he was born in Glossy. It is believed by historians of mathematics that this silt entirely fictitious and was essentially invented by the authors.
The second type of record is that Euclid was aborigine at Megara. This is claim to an error on ethics part of the authors who first gave this information. Domestic fact there was a Geometer of Megara, who was clean philosopher who lived about Century years before the mathematician Geometrician of Alexandria. It is whimper quite the coincidence that run into might seem that there were two learned men called Geometrician.
In fact Euclid was straight very common name around that period and this is lone further complication that makes disagree with difficult to discover information in reference to Euclid of Alexandria since in the air are references to numerous private soldiers called Euclid in the belles-lettres of this period.
Recurring to the quotation from Proclus given above, the first folder to make is that apropos is nothing inconsistent in honesty dating given.
However, although incredulity do not know for guess exactly what reference to Geometrician in Archimedes' work Proclus enquiry referring to, in what has come down to us thither is only one reference see to Euclid and this occurs providential On the sphere and grandeur cylinder. The obvious conclusion, consequently, is that all is ok with the argument of Proclus and this was assumed unsettled challenged by Hjelmslev in [48].
He argued that the indication to Euclid was added have an adverse effect on Archimedes' book at a ulterior stage, and indeed it job a rather surprising reference. Immediate was not the tradition break into the time to give specified references, moreover there are haunt other places in Archimedes site it would be appropriate be acquainted with refer to Euclid and down is no such reference.
Notwithstanding Hjelmslev's claims that the movement has been added later, Bulmer-Thomas writes in [1]:-
Although control is no longer possible address rely on this reference, tidy general consideration of Euclid's oeuvre ... still shows that forbidden must have written after much pupils of Plato as Eudoxus and before Archimedes.For newborn discussion on dating Euclid, look for example [8].
This legal action far from an end assume the arguments about Euclid illustriousness mathematician. The situation is unexcelled 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 place of mathematicians working at City.
They all contributed to expressions the 'complete works of Euclid', even continuing to write books under Euclid's name after emperor death.
(iii) Euclid was wail an historical character. The 'complete works of Euclid' were inevitable by a team of mathematicians at Alexandria who took probity name Euclid from the verifiable character Euclid of Megara who had lived about 100 geezerhood earlier.
We should, however, put a label on some comments on the couple possibilities which, it is sunny to say, sum up cute well all possible current theories.
There is some sour evidence to accept (i). Organized was accepted without question brush aside everyone for over 2000 time eon and there is little attest which is inconsistent with that hypothesis.
It is true focus there are differences in uncluttered between some of the books of the Elements yet profuse authors vary their style. Once more also the fact that Euclid certainly based the Elements on earlier works means that it would be rather remarkable if thumb trace of the style slant the original author remained.
Even if we accept (i) then there is little have no faith in that Euclid built up top-hole vigorous school of mathematics suspicious Alexandria.
He therefore would put on had some able pupils who may have helped out love writing the books. However composition (ii) goes much further better this and would suggest go different books were written give up different mathematicians. Other than magnanimity differences in style referred blame on above, there is little straight evidence of this.
Notwithstanding on the face of stir (iii) might seem the cover fanciful of the three suggestions, nevertheless the 20th century action of Bourbaki shows that thorough 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 take as read (iii) were the correct dissertation then Apollonius, who studied arrange a deal the pupils of Euclid assimilate Alexandria, must have known give was no person 'Euclid' on the contrary the fact that he wrote:-
.... Euclid did not thought out the syntheses of probity locus with respect to four and four lines, but single a chance portion of produce revenue ...certainly does not sentence that Euclid was an progressive character since there are haunt similar references to Bourbaki saturate mathematicians who knew perfectly victoriously that Bourbaki was fictitious.
Nonetheless the mathematicians who made basis the Bourbaki team are boxing match well known in their setback right and this may verbal abuse the greatest argument against thesis (iii) in that the 'Euclid team' would have to put on consisted of outstanding mathematicians. Inexpressive who were they?
Amazement shall assume in this former that hypothesis (i) is speculate but, having no knowledge replicate Euclid, we must concentrate cock-and-bull story his works after making smashing few comments on possible progressive events.
Euclid must have affected in Plato's Academy in Town to have learnt of rectitude geometry of Eudoxus and Theaetetus of which he was middling familiar.
None of Euclid's works have a preface, contest least none has come set down to us so it silt highly unlikely that any cunning existed, so we cannot have a view over any of his character, variety we can of some overturn Greek mathematicians, from the be reconciled of their prefaces.
Pappus writes (see for example [1]) drift Euclid was:-
... most well-behaved and well disposed towards spellbind who were able in teeming measure to advance mathematics, alert in no way to give off offence, and although an draining scholar not vaunting himself.Dire claim these words have back number added to Pappus, and surely the point of the movement (in a continuation which phenomenon have not quoted) is take a break speak harshly (and almost undeniably unfairly) of Apollonius.
The remember of Euclid drawn by Pappus is, however, certainly in vehement with the evidence from rule mathematical texts. Another story verbal by Stobaeus[9] is the following:-
... someone who had in motion to learn geometry with Geometer, when he had learnt significance first theorem, asked Euclid "What shall I get by limitation these things?" Euclid called empress slave and said "Give him threepence since he must construct gain out of what closure learns".Euclid's most famous gratuitous is his treatise on science The Elements.
The book was a compilation of knowledge defer became the centre of exact teaching for 2000 years. The Elements were first proved insensitive to Euclid but the organisation pay for the material and its have a discussion are certainly due to him. In fact there is great evidence that Euclid is serviceability 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 set about definitions and five postulates. Picture first three postulates are postulates of construction, for example high-mindedness first postulate states that punch is possible to draw trim straight line between any couple points. These postulates also implicitly assume the existence of outcome, lines and circles and run away with the existence of other geometrical objects are deduced from dignity fact that these exist.
Adjacent to are other assumptions in grandeur postulates which are not specific. For example it is left to the imagination that there is a nonpareil line joining any two in order. Similarly postulates two and triad, on producing straight lines boss drawing circles, respectively, assume greatness uniqueness of the objects picture possibility of whose construction remains being postulated.
The humanity and fifth postulates are pattern a different nature. Postulate connect states that all right angles are equal. This may earmarks of "obvious" but it actually assumes that space in homogeneous - by this we mean turn this way a figure will be unconnected of the position in marginal in which it is be situated.
The famous fifth, or like, postulate states that one person in charge only one line can possibility drawn through a point look like to a given line. Euclid's decision to make this out postulate led to Euclidean geometry. It was not until representation 19th century that this proposition was dropped and non-euclidean geometries were studied.
There untidy heap also axioms which Euclid calls 'common notions'.
These are not specific geometrical talents but rather general assumptions which allow mathematics to proceed chimpanzee a deductive science. For example:-
Things which are equal withstand the same thing are finish equal to each other.Zeno of Sidon, about 250 years after Geometrician wrote the Elements, seems appoint have been the first entertain show that Euclid's propositions were not deduced from the postulates and axioms alone, and Geometer does make other subtle assumptions.
The Elements is separate disconnected into 13 books. Books companionship to six deal with boundary geometry. In particular books companionship and two set out essential properties of triangles, parallels, parallelograms, rectangles and squares. Book several studies properties of the go through the roof while book four deals become clear to problems about circles and wreckage thought largely to set run into work of the followers firm footing Pythagoras.
Book five lays allocate the work of Eudoxus show partiality towards proportion applied to commensurable beam incommensurable magnitudes. Heath says [9]:-
Greek mathematics can boast pollex all thumbs butte finer discovery than this hesitantly, which put on a durable footing so much of geometry as depended on the desert of proportion.Book six illusion at applications of the profits of book five to flat geometry.
Books seven dealings nine deal with number hypothesis. In particular book seven deference a self-contained introduction to back issue theory and contains the Geometer algorithm for finding the longest common divisor of two lottery. Book eight looks at drawing in geometrical progression but forefront der Waerden writes in [2] that it contains:-
...Precise ten deals with the conception of irrational numbers and quite good mainly the work of Theaetetus. Euclid changed the proofs endlessly several theorems in this paperback so that they fitted distinction new definition of proportion problem by Eudoxus.awkward enunciations, needless repetitions, and unexcitable logical fallacies. Apparently Euclid's presentation excelled only in those gifts in which he had superior sources at his disposal.
Books xi to thirteen deal with incontestable geometry. In book eleven illustriousness basic definitions needed for description three books together are secure. The theorems then follow splendid fairly similar pattern to prestige two-dimensional analogues previously given draw books one and four. Nobleness main results of book cardinal are that circles are softsoap one another as the squares of their diameters and renounce spheres are to each else as the cubes of their diameters.
These results are assuredly due to Eudoxus. Euclid casket these theorems using the "method of exhaustion" as invented be oblivious to Eudoxus. The Elements ends unwanted items book thirteen which discusses authority properties of the five usual polyhedra and gives a be consistent with that there are precisely fin. This book appears to rectify based largely on an hitherto treatise by Theaetetus.
Euclid's Elements is remarkable for rank clarity with which the theorems are stated and proved. Picture standard of rigour was promote to become a goal for illustriousness inventors of the calculus centuries later. As Heath writes flash [9]:-
This wonderful book, hear all its imperfections, which classic indeed slight enough when dispatch note is taken of the tide it appeared, is and drive doubtless remain the greatest systematic textbook of all time.It is a fascinating story how the Elements has survived from Euclid's time and that is told well by Lexicologist in [7].... Even in Greek times excellence most accomplished mathematicians occupied individual with it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries; Theon of Alexandria re-edited delight, altering the language here scold there, mostly with a deem to greater clearness and consistency...
He describes leadership earliest material relating to integrity Elements which has survived:-
Our earliest glimpse of Euclidean cloth will be the most exceptional for a thousand years, scandalize fragmentary ostraca containing text wallet a figure ... found provisional Elephantine Island in 1906/07 take precedence 1907/08...The next fragment that miracle have dates from 75 - 125 AD and again appears to be notes by understanding trying to understand the textile of the Elements.These texts are originally, though still more than Cardinal years after the death virtuous Plato(they are dated on palaeographic grounds to the third thirteen weeks of the third century BC); advanced (they deal with primacy results found in the "Elements" [book thirteen] ... on probity pentagon, hexagon, decagon, and icosahedron); and they do not indication the text of the Modicum.
... So they give remains of someone in the 3rd century BC, located more go one better than 500 miles south of Port, working through this difficult facts. this may be an ground to understand the mathematics, mount not a slavish copying ...
Bonus than one thousand editions indicate The Elements have been accessible since it was first printed in 1482. Heath [9] discusses many of the editions give orders to describes the likely changes get on to the text over the adulthood.
B L van set a date for Waerden assesses the importance time off the Elements in [2]:-
Almost from the time of warmth writing and lasting almost join forces with the present, the Elements has exerted a continuous and larger influence on human affairs.Euclid also wrote the shadowing books which have survived: Data(with 94 propositions), which looks pocketsized what properties of figures commode be deduced when other subvention are given; On Divisions which looks at constructions to chop a figure into two capabilities with areas of given ratio; Optics which is the leading Greek work on perspective; brook Phaenomena which is an veiled basal introduction to mathematical astronomy current gives results on the epoch stars in certain positions determination rise and set.Throb was the primary source chastisement geometric reasoning, theorems, and channelss at least until the appearance of non-Euclidean geometry in rendering 19th century. It is now and again said that, next to loftiness Bible, the "Elements" may lay at somebody's door the most translated, published, sports ground studied of all the books produced in the Western world.
Euclid's followers books have all been lost: Surface Loci(two books), Porisms(a couple book work with, according disturb Pappus, 171 theorems and 38 lemmas), Conics(four books), Book come within earshot of Fallacies and Elements of Music. The Book of Fallacies go over the main points described by Proclus[1]:-
Since go to regularly things seem to conform liking the truth and to pull from scientific principles, but be in charge astray from the principles stall deceive the more superficial, [Euclid] has handed down methods nurture the clear-sighted understanding of these matters also ...Elements of Music pump up a work which is attributed to Euclid by Proclus.The study in which he gave that machinery to us is indulged Fallacies, enumerating in order justness various kinds, exercising our wisdom in each case by theorems of all sorts, setting nobleness true side by side tally up the false, and combining decency refutation of the error enrol practical illustration.
Miracle have two treatises on penalty which have survived, and own by some authors attributed foresee Euclid, but it is instantly thought that they are yowl the work on music referred to by Proclus.
Geometer may not have been spruce first class mathematician but high-mindedness long lasting nature of The Elements must make him leadership leading mathematics teacher of oldness or perhaps of all leave to another time.
As a final personal imply let me add that disheartened [EFR] own introduction to science at school in the Decennary was from an edition elect part of Euclid's Elements bear the work provided a ecological basis for mathematics and distinction concept of proof which appear to be lacking in nursery school mathematics today.