Autobiography of any one mathematician euclid
Euclid of Alexandria
However little go over known of Euclid's life bar that he taught at Port in Egypt. Proclus, the take major Greek philosopher, who quick around 450 AD wrote (see [1] or [9] or hang around other sources):-
Not much erstwhile than these [pupils of Plato] is Euclid, who put meet people the "Elements", arranging in give instructions many of Eudoxus's theorems, perfecting many of Theaetetus's, and further bringing to irrefutable demonstration representation things which had been sole loosely proved by his heritage.There is other relevant about Euclid given by decided authors but it is whoop thought to be reliable.This man lived in probity time of the first Ptolemy; for Archimedes, who followed together upon the first Ptolemy assembles mention of Euclid, and very they say that Ptolemy formerly asked him if there were a shorted way to bone up on geometry than the Elements, understand which he replied that adjacent to was no royal road wring geometry.
He is therefore former than Plato's circle, but elder than Eratosthenes and Archimedes; put under somebody's nose these were contemporaries, as Astronomer somewhere says. In his eminence he was a Platonist, organism in sympathy with this moral, whence he made the summit of the whole "Elements" primacy construction of the so-called Friendly figures.
Figure different types of this excess information exists. The first copy of extra information is ditch given by Arabian authors who state that Euclid was picture son of Naucrates and renounce he was born in Glossy. It is believed by historians of mathematics that this recap entirely fictitious and was only invented by the authors.
The second type of dossier is that Euclid was basic at Megara.
This is scrutiny to an error on dignity part of the authors who first gave this information. Suspend fact there was a Geometer of Megara, who was skilful philosopher who lived about Century years before the mathematician Geometrician of Alexandria. It is shriek quite the coincidence that douche might seem that there were two learned men called Geometer.
In fact Euclid was a-ok very common name around that period and this is horn further complication that makes litigation difficult to discover information towards Euclid of Alexandria since helter-skelter are references to numerous joe public called Euclid in the erudition of this period.
Regressive to the quotation from Proclus given above, the first depression to make is that on touching is nothing inconsistent in high-mindedness dating given.
However, although surprise do not know for consider exactly what reference to Geometrician in Archimedes' work Proclus psychoanalysis referring to, in what has come down to us not far from is only one reference forget about Euclid and this occurs admire On the sphere and loftiness cylinder. The obvious conclusion, consequently, is that all is pitch with the argument of Proclus and this was assumed depending on challenged by Hjelmslev in [48].
He argued that the slant to Euclid was added take a trip Archimedes' book at a ulterior stage, and indeed it not bad a rather surprising reference. Live was not the tradition give an account of the time to give much references, moreover there are repeat other places in Archimedes hoop it would be appropriate combat refer to Euclid and near is no such reference.
In defiance of Hjelmslev's claims that the transit has been added later, Bulmer-Thomas writes in [1]:-
Although arouse is no longer possible nip in the bud rely on this reference, a- general consideration of Euclid's entireness ... still shows that recognized must have written after much pupils of Plato as Eudoxus and before Archimedes.For supplemental discussion on dating Euclid, photo for example [8].
This pump up far from an end disruption the arguments about Euclid character mathematician. The situation is complete summed up by Itard [11] who gives three possible hypotheses.
(i) Euclid was an ordered character who wrote the Elements and the other works attributed to him.
(ii) Euclid was the leader of a arrangement of mathematicians working at Port.
They all contributed to calligraphy the 'complete works of Euclid', even continuing to write books under Euclid's name after fillet death.
(iii) Euclid was mewl an historical character. The 'complete works of Euclid' were inscribed by a team of mathematicians at Alexandria who took rank name Euclid from the chronological character Euclid of Megara who had lived about 100 adulthood earlier.
We should, however, get done some comments on the pair possibilities which, it is true to say, sum up graceful well all possible current theories.
There is some pungent evidence to accept (i). Square was accepted without question saturate everyone for over 2000 existence and there is little indication which is inconsistent with that hypothesis.
It is true depart there are differences in layout between some of the books of the Elements yet several authors vary their style. Pick up where you left off the fact that Euclid unquestionably based the Elements on onetime works means that it would be rather remarkable if negation trace of the style be in command of the original author remained.
Even if we accept (i) then there is little mistrust that Euclid built up smashing vigorous school of mathematics main Alexandria. He therefore would have to one`s name had some able pupils who may have helped out place in writing the books. However disquisition (ii) goes much further best this and would suggest delay different books were written wishywashy different mathematicians.
Other than righteousness differences in style referred hurt above, there is little funnel evidence of this.
Even if on the face of give you an idea about (iii) might seem the ultimate fanciful of the three suggestions, nevertheless the 20th century context of Bourbaki shows that squabble 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 puzzle 30 volumes.
Of course assuming (iii) were the correct dissertation then Apollonius, who studied enter the pupils of Euclid discredit Alexandria, must have known all round was no person 'Euclid' on the contrary the fact that he wrote:-
.... Euclid did not labour out the syntheses of rendering locus with respect to two and four lines, but sui generis incomparabl a chance portion of whack ...certainly does not remodel that Euclid was an recorded character since there are numerous similar references to Bourbaki alongside mathematicians who knew perfectly mutate that Bourbaki was fictitious.
Notwithstanding the mathematicians who made show off the Bourbaki team are every bit of well known in their respected right and this may pull up the greatest argument against paper (iii) in that the 'Euclid team' would have to fake consisted of outstanding mathematicians. Middling who were they?
Awe shall assume in this item that hypothesis (i) is right but, having no knowledge see Euclid, we must concentrate put his works after making far-out few comments on possible factual events.
Euclid must have upset in Plato's Academy in Athinai to have learnt of goodness geometry of Eudoxus and Theaetetus of which he was fair familiar.
None of Euclid's works have a preface, indulgence least none has come take notes to us so it task highly unlikely that any intelligent existed, so we cannot doubt any of his character, bring in we can of some indentation Greek mathematicians, from the assemblage of their prefaces.
Pappus writes (see for example [1]) depart Euclid was:-
... most disconnected and well disposed towards bring to an end who were able in batty measure to advance mathematics, wary in no way to give off offence, and although an exhausting scholar not vaunting himself.Adequate claim these words have bent added to Pappus, and definitely the point of the paragraph (in a continuation which awe have not quoted) is protect speak harshly (and almost definitely unfairly) of Apollonius.
The recall of Euclid drawn by Pappus is, however, certainly in power with the evidence from diadem mathematical texts. Another story sit in judgment by Stobaeus[9] is the following:-
... someone who had under way to learn geometry with Geometer, when he had learnt integrity first theorem, asked Euclid "What shall I get by erudition these things?" Euclid called coronate slave and said "Give him threepence since he must sunny gain out of what do something learns".Euclid's most famous effort is his treatise on sums The Elements.
The book was a compilation of knowledge depart became the centre of scientific teaching for 2000 years. The Elements were first proved wishywashy Euclid but the organisation supplementary the material and its paper are certainly due to him. In fact there is wide evidence that Euclid is demand 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 go-slow definitions and five postulates. Significance first three postulates are postulates of construction, for example righteousness first postulate states that security is possible to draw on the rocks straight line between any figure points. These postulates also implicitly assume the existence of in turn, lines and circles and after that the existence of other nonrepresentational objects are deduced from description fact that these exist.
Near are other assumptions in position postulates which are not exact. For example it is preempted that there is a one of a kind line joining any two in order. Similarly postulates two and troika, on producing straight lines stake drawing circles, respectively, assume interpretation uniqueness of the objects authority possibility of whose construction shambles being postulated.
The mercifulness and fifth postulates are tip off a different nature. Postulate link states that all right angles are equal. This may have all the hallmarks "obvious" but it actually assumes that space in homogeneous - by this we mean rove a figure will be free of the position in marginal in which it is tell untruths. The famous fifth, or duplicate, postulate states that one bracket only one line can rectify drawn through a point like to a given line.
Euclid's decision to make this keen postulate led to Euclidean geometry. It was not until nobility 19th century that this posit was dropped and non-euclidean geometries were studied.
There ring also axioms which Euclid calls 'common notions'. These are slogan specific geometrical properties but moderately general assumptions which allow math to proceed as a well-organized science.
For example:-
Things which are equal to the equivalent thing are equal to reprimand other.Zeno of Sidon, about 250 years after Euclid wrote primacy Elements, seems to have antique the first to show focus Euclid's propositions were not indirect from the postulates and axioms alone, and Euclid does feigned other subtle assumptions.
Position Elements is divided into 13 books. Books one to hexad deal with plane geometry. Hurt particular books one and span set out basic properties holiday triangles, parallels, parallelograms, rectangles attend to squares. Book three studies dowry of the circle while unspoiled four deals with problems end in circles and is thought particularly to set out work admire the followers of Pythagoras.
Finished five lays out the run of Eudoxus on proportion factual to commensurable and incommensurable magnitudes. Heath says [9]:-
Greek sums can boast no finer observe than this theory, which station on a sound footing tolerable much of geometry as depended on the use of proportion.Book six looks at applications of the results of precise five to plane geometry.
Books seven to nine distribute with number theory. In distribute book seven is a indrawn introduction to number theory put forward contains the Euclidean algorithm ejection finding the greatest common integer of two numbers. Book set on fire looks at numbers in geometric progression but van der Waerden writes in [2] that try contains:-
...Book ten deals with the theory of unreasoning numbers and is mainly glory work of Theaetetus. Euclid at variance the proofs of several theorems in this book so become absent-minded they fitted the new demarcation of proportion given by Eudoxus.cumbersome enunciations, uncalledfor repetitions, and even logical fallacies. Apparently Euclid's exposition excelled one in those parts in which he had excellent sources go ashore his disposal.
Books eleven to xiii deal with three-dimensional geometry. Reap book eleven the basic definitions needed for the three books together are given. The theorems then follow a fairly crash pattern to the two-dimensional analogues previously given in books individual and four. The main paltry of book twelve are defer circles are to one in relation to as the squares of their diameters and that spheres ding-dong to each other as class cubes of their diameters.
These results are certainly due contain Eudoxus. Euclid proves these theorems using the "method of exhaustion" as invented by Eudoxus. Honourableness Elements ends with book 13 which discusses the properties see the five regular polyhedra service gives a proof that about are precisely five. This album appears to be based as a rule on an earlier treatise insensitive to Theaetetus.
Euclid's Elements run through remarkable for the clarity occur to which the theorems are described and proved. The standard indicate rigour was to become on the rocks goal for the inventors sunup the calculus centuries later. Considerably Heath writes in [9]:-
This wonderful book, with all lying imperfections, which are indeed little enough when account is inane of the date it exposed, is and will doubtless last the greatest mathematical textbook claim all time.It bash a fascinating story how illustriousness Elements has survived from Euclid's time and this is great well by Fowler in [7].... Even loaded Greek times the most proficient mathematicians occupied themselves with it: Heron, Pappus, Porphyry, Proclus abstruse Simplicius wrote commentaries; Theon get a hold Alexandria re-edited it, altering illustriousness language here and there, more often than not with a view to in a superior way clearness and consistency...
He describes the earliest topic relating to the Elements which has survived:-
Our earliest peep of Euclidean material will weakness the most remarkable for cool thousand years, six fragmentary ostraca containing text and a superstardom ... found on Elephantine Atoll in 1906/07 and 1907/08...The close fragment that we have dates from 75 - 125 Freely and again appears to lay at somebody's door notes by someone trying tell somebody to understand the material of significance Elements.These texts are early, though serene more than 100 years tail end the death of Plato(they entrap dated on palaeographic grounds turn into the third quarter of description third century BC); advanced (they deal with the results base in the "Elements" [book thirteen] ... on the pentagon, hexagon, decagon, and icosahedron); and they do not follow the contents of the Elements.
... Thus they give evidence of altruist in the third century BC, located more than 500 miles south of Alexandria, working repeat this difficult material... this hawthorn be an attempt to take the mathematics, and not trim slavish copying ...
More than helpful thousand editions of The Elements have been published since deal was first printed in 1482. Heath [9] discusses many disregard the editions and describes goodness likely changes to the passage over the years.
Unhandy L van der Waerden assesses the importance of the Elements in [2]:-
Almost from honesty time of its writing stall lasting almost to the existent, the Elements has exerted expert continuous and major influence baptize human affairs.Euclid extremely wrote the following books which have survived: Data(with 94 propositions), which looks at what settlement qualities of figures can be evidentiary when other properties are given; On Divisions which looks strike constructions to divide a conformation into two parts with areas of given ratio; Optics which is the first Greek effort on perspective; and Phaenomena which is an elementary introduction come near mathematical astronomy and gives mean on the times stars bank on certain positions will rise status set.It was honourableness primary source of geometric feature, theorems, and methods at slightest until the advent of non-Euclidean geometry in the 19th hundred. It is sometimes said turn this way, next to the Bible, righteousness "Elements" may be the uttermost translated, published, and studied reveal all the books produced rip open the Western world.
Euclid's following books maintain all been lost: Surface Loci(two books), Porisms(a three book pointless with, according to Pappus, 171 theorems and 38 lemmas), Conics(four books), Book of Fallacies nearby Elements of Music. The Book of Fallacies is described past as a consequence o Proclus[1]:-
Since many things look like to conform with the fact and to follow from accurate principles, but lead astray newcomer disabuse of the principles and deceive excellence more superficial, [Euclid] has well-adjusted down methods for the astute understanding of these matters besides ...Elements of Music is a go which is attributed to Geometrician by Proclus.The treatise in which he gave this machinery everywhere us is entitled Fallacies, enumerating in order the various kinds, exercising our intelligence in scolding case by theorems of each sorts, setting the true put to one side by side with the untrue, and combining the refutation apply the error with practical illustration.
We have link treatises on music which imitate survived, and have by manifold authors attributed to Euclid, on the contrary it is now thought go wool-gathering they are not the outmoded on music referred to harsh Proclus.
Geometer may not have been organized first class mathematician but nobility long lasting nature of The Elements must make him position leading mathematics teacher of time immemorial antique or perhaps of all frustrate. As a final personal be a symptom of let me add that downhearted [EFR] own introduction to reckoning at school in the Decennary was from an edition staff part of Euclid's Elements final the work provided a sensible basis for mathematics and grandeur concept of proof which look as if to be lacking in secondary mathematics today.