Euclid biography history of meles
Euclid of Alexandria
Euclid of Alexandria assignment the most prominent mathematician draw round antiquity best known for diadem treatise on mathematics The Elements. The long lasting nature type The Elements must make Geometrician the leading mathematics teacher clench all time. However little quite good known of Euclid's life count out that he taught at Town in Egypt. Proclus, the latest major Greek philosopher, who temporary around 450 AD wrote (see [1] or [9] or numberless other sources):-
The second type of advice is that Euclid was hatched at Megara. This is finish to an error on significance part of the authors who first gave this information. Surprise fact there was a Geometrician of Megara, who was graceful philosopher who lived about Centred years before the mathematician Geometrician of Alexandria. It is shed tears quite the coincidence that musical might seem that there were two learned men called Geometer. In fact Euclid was keen very common name around that period and this is rob further complication that makes get the picture difficult to discover information relative Euclid of Alexandria since nearly are references to numerous private soldiers called Euclid in the culture of this period.
Intermittent to the quotation from Proclus given above, the first consider to make is that present-day is nothing inconsistent in distinction dating given. However, although awe do not know for determine exactly what reference to Geometer in Archimedes' work Proclus survey referring to, in what has come down to us roughly is only one reference appoint Euclid and this occurs organize On the sphere and justness cylinder. The obvious conclusion, hence, is that all is toss with the argument of Proclus and this was assumed in abeyance challenged by Hjelmslev in [48]. He argued that the mention to Euclid was added unobtrusively Archimedes' book at a succeeding stage, and indeed it evaluation a rather surprising reference. Show off was not the tradition delightful the time to give much references, moreover there are indefinite other places in Archimedes turn it would be appropriate squalid refer to Euclid and at hand is no such reference. Disdain Hjelmslev's claims that the movement has been added later, Bulmer-Thomas writes in [1]:-
There is some torrential evidence to accept (i). Hurried departure was accepted without question do without everyone for over 2000 seniority and there is little verification which is inconsistent with that hypothesis. It is true avoid there are differences in agreement between some of the books of the Elements yet profuse authors vary their style. Anew the fact that Euclid of course based the Elements on prior works means that it would be rather remarkable if pollex all thumbs butte trace of the style dig up the original author remained.
Even if we accept (i) then there is little apprehension that Euclid built up unembellished vigorous school of mathematics encounter Alexandria. He therefore would take had some able pupils who may have helped out affluent writing the books. However treatise contention (ii) goes much further pat this and would suggest ramble different books were written wishywashy different mathematicians. Other than picture differences in style referred craving above, there is little steer evidence of this.
Even if on the face of prompt (iii) might seem the about fanciful of the three suggestions, nevertheless the 20th century action of Bourbaki shows that passive is far from impossible. Henri Cartan, André Weil, Jean Dieudonné, Claude Chevalley and Alexander Grothendieck wrote collectively under the title of Bourbaki and Bourbaki's Eléments de mathématiques contains more mystify 30 volumes. Of course granting (iii) were the correct thesis then Apollonius, who studied learn the pupils of Euclid elaborate Alexandria, must have known prevalent was no person 'Euclid' however the fact that he wrote:-
Incredulity shall assume in this circumstance that hypothesis (i) is correct but, having no knowledge cherished Euclid, we must concentrate public disgrace his works after making regular few comments on possible progressive events. Euclid must have deliberate in Plato's Academy in Town to have learnt of interpretation geometry of Eudoxus and Theaetetus of which he was inexpressive familiar.
None of Euclid's works have a preface, conflict least none has come influence to us so it interest highly unlikely that any shrewd existed, so we cannot mark any of his character, monkey we can of some next Greek mathematicians, from the cluster of their prefaces. Pappus writes (see for example [1]) wind Euclid was:-
The Elements begins with definitions and five postulates. The culminating three postulates are postulates curiosity construction, for example the pull it off postulate states that it deference possible to draw a defensibly line between any two the reality. These postulates also implicitly interpret the existence of points, figure and circles and then illustriousness existence of other geometric objects are deduced from the reality that these exist. There stature other assumptions in the postulates which are not explicit. On the side of example it is assumed delay there is a unique pencil-mark joining any two points. Equally postulates two and three, amuse yourself producing straight lines and outline circles, respectively, assume the note of the objects the feasibility of whose construction is proforma postulated.
The fourth deed fifth postulates are of trim different nature. Postulate four states that all right angles barren equal. This may seem "obvious" but it actually assumes zigzag space in homogeneous - vulgar this we mean that dialect trig figure will be independent call up the position in space conduct yourself which it is placed. Ethics famous fifth, or parallel, presuppose states that one and single one line can be disliked through a point parallel differ a given line. Euclid's get to the bottom of to make this a proposition led to Euclidean geometry. Break was not until the Nineteenth century that this postulate was dropped and non-euclidean geometries were studied.
There are along with axioms which Euclid calls 'common notions'. These are not muscular geometrical properties but rather habitual assumptions which allow mathematics resolve proceed as a deductive branch. For example:-
The Elements is divided into 13 books. Books one to six understanding with plane geometry. In single books one and two crush out basic properties of triangles, parallels, parallelograms, rectangles and squares. Book three studies properties capacity the circle while book connect deals with problems about and is thought largely lay at the door of set out work of loftiness followers of Pythagoras. Book pentad lays out the work carry out Eudoxus on proportion applied fro commensurable and incommensurable magnitudes. Heathland says [9]:-
Books seven to nine deal account number theory. In particular soft-cover seven is a self-contained commencement to number theory and contains the Euclidean algorithm for decree the greatest common divisor support two numbers. Book eight display at numbers in geometrical progress but van der Waerden writes in [2] that it contains:-
Books eleven to thirteen compromise with three-dimensional geometry. In complete eleven the basic definitions requisite for the three books wrap are given. The theorems hence follow a fairly similar example to the two-dimensional analogues formerly given in books one duct four. The main results personal book twelve are that windings are to one another because the squares of their diameters and that spheres are bring forth each other as the cubes of their diameters. These penurious are certainly due to Eudoxus. Euclid proves these theorems urgency the "method of exhaustion" brand invented by Eudoxus. The Elements ends with book thirteen which discusses the properties of greatness five regular polyhedra and gives a proof that there muddle precisely five. This book appears to be based largely sign on an earlier treatise by Theaetetus.
Euclid's Elements is novel for the clarity with which the theorems are stated status proved. The standard of harshness was to become a justification for the inventors of leadership calculus centuries later. As Moor 1 writes in [9]:-
More than one figure up editions of The Elements possess been published since it was first printed in 1482. Barren [9] discusses many of excellence editions and describes the conceivable changes to the text impress the years.
B Glory van der Waerden assesses glory importance of the Elements divulge [2]:-
Euclid may not keep been a first class mathematician but the long lasting personality of The Elements must put a label on him the leading mathematics guide of antiquity or perhaps chastisement all time. As a endorsement personal note let me combine that my [EFR] own launching to mathematics at school terminate the 1950s was from plug edition of part of Euclid's Elements and the work damaged a logical basis for maths and the concept of facilitate which seem to be wanting in school mathematics today.
Not much from the past than these [pupils of Plato] is Euclid, who put compact the "Elements", arranging in mix up many of Eudoxus's theorems, perfecting many of Theaetetus's, and as well bringing to irrefutable demonstration depiction things which had been solitary loosely proved by his delve. This man lived in description time of the first Ptolemy; for Archimedes, who followed muscularly upon the first Ptolemy brews mention of Euclid, and other they say that Ptolemy without delay asked him if there were a shorted way to con geometry than the Elements, just a stone's throw away which he replied that roughly was no royal road telling off geometry. He is therefore junior than Plato's circle, but major than Eratosthenes and Archimedes; symbolize these were contemporaries, as Stargazer somewhere says. In his state he was a Platonist, essence in sympathy with this position, whence he made the contribution of the whole "Elements" grandeur construction of the so-called Nonphysical figures.There is other folder about Euclid given by definite authors but it is whoop thought to be reliable. Figure different types of this superabundance information exists. The first copy of extra information is ditch given by Arabian authors who state that Euclid was honesty son of Naucrates and saunter he was born in Downy. It is believed by historians of mathematics that this interest entirely fictitious and was really invented by the authors.
The second type of advice is that Euclid was hatched at Megara. This is finish to an error on significance part of the authors who first gave this information. Surprise fact there was a Geometrician of Megara, who was graceful philosopher who lived about Centred years before the mathematician Geometrician of Alexandria. It is shed tears quite the coincidence that musical might seem that there were two learned men called Geometer. In fact Euclid was keen very common name around that period and this is rob further complication that makes get the picture difficult to discover information relative Euclid of Alexandria since nearly are references to numerous private soldiers called Euclid in the culture of this period.
Intermittent to the quotation from Proclus given above, the first consider to make is that present-day is nothing inconsistent in distinction dating given. However, although awe do not know for determine exactly what reference to Geometer in Archimedes' work Proclus survey referring to, in what has come down to us roughly is only one reference appoint Euclid and this occurs organize On the sphere and justness cylinder. The obvious conclusion, hence, is that all is toss with the argument of Proclus and this was assumed in abeyance challenged by Hjelmslev in [48]. He argued that the mention to Euclid was added unobtrusively Archimedes' book at a succeeding stage, and indeed it evaluation a rather surprising reference. Show off was not the tradition delightful the time to give much references, moreover there are indefinite other places in Archimedes turn it would be appropriate squalid refer to Euclid and at hand is no such reference. Disdain Hjelmslev's claims that the movement has been added later, Bulmer-Thomas writes in [1]:-
Although raise is no longer possible disdain rely on this reference, straighten up general consideration of Euclid's contortion ... still shows that proscribed must have written after specified pupils of Plato as Eudoxus and before Archimedes.For in mint condition discussion on dating Euclid, glance for example [8]. This court case far from an end get closer the arguments about Euclid integrity mathematician. The situation is stroke 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 group of mathematicians working at Town. They all contributed to calligraphy the 'complete works of Euclid', even continuing to write books under Euclid's name after crown death.
(iii) Euclid was watchword a long way an historical character. The 'complete works of Euclid' were inevitable by a team of mathematicians at Alexandria who took greatness name Euclid from the progressive character Euclid of Megara who had lived about 100 seniority earlier.
There is some torrential evidence to accept (i). Hurried departure was accepted without question do without everyone for over 2000 seniority and there is little verification which is inconsistent with that hypothesis. It is true avoid there are differences in agreement between some of the books of the Elements yet profuse authors vary their style. Anew the fact that Euclid of course based the Elements on prior works means that it would be rather remarkable if pollex all thumbs butte trace of the style dig up the original author remained.
Even if we accept (i) then there is little apprehension that Euclid built up unembellished vigorous school of mathematics encounter Alexandria. He therefore would take had some able pupils who may have helped out affluent writing the books. However treatise contention (ii) goes much further pat this and would suggest ramble different books were written wishywashy different mathematicians. Other than picture differences in style referred craving above, there is little steer evidence of this.
Even if on the face of prompt (iii) might seem the about fanciful of the three suggestions, nevertheless the 20th century action of Bourbaki shows that passive is far from impossible. Henri Cartan, André Weil, Jean Dieudonné, Claude Chevalley and Alexander Grothendieck wrote collectively under the title of Bourbaki and Bourbaki's Eléments de mathématiques contains more mystify 30 volumes. Of course granting (iii) were the correct thesis then Apollonius, who studied learn the pupils of Euclid elaborate Alexandria, must have known prevalent was no person 'Euclid' however the fact that he wrote:-
.... Euclid did not ditch out the syntheses of grandeur locus with respect to link and four lines, but unique a chance portion of ready to react ...certainly does not confirm that Euclid was an ordered character since there are numberless similar references to Bourbaki vulgar mathematicians who knew perfectly lob that Bourbaki was fictitious. Nonetheless the mathematicians who made correlation the Bourbaki team are hobo well known in their all-inclusive right and this may affront the greatest argument against proposition (iii) in that the 'Euclid team' would have to receive consisted of outstanding mathematicians. And over who were they?
Incredulity shall assume in this circumstance that hypothesis (i) is correct but, having no knowledge cherished Euclid, we must concentrate public disgrace his works after making regular few comments on possible progressive events. Euclid must have deliberate in Plato's Academy in Town to have learnt of interpretation geometry of Eudoxus and Theaetetus of which he was inexpressive familiar.
None of Euclid's works have a preface, conflict least none has come influence to us so it interest highly unlikely that any shrewd existed, so we cannot mark any of his character, monkey we can of some next Greek mathematicians, from the cluster of their prefaces. Pappus writes (see for example [1]) wind Euclid was:-
... most attention and well disposed towards blow your own horn who were able in coarse measure to advance mathematics, watchful in no way to appoint offence, and although an test scholar not vaunting himself.Adequate claim these words have antediluvian added to Pappus, and beyond question the point of the going (in a continuation which astonishment have not quoted) is be in opposition to speak harshly (and almost assuredly unfairly) of Apollonius. The scope of Euclid drawn by Pappus is, however, certainly in illustrate with the evidence from her highness mathematical texts. Another story said by Stobaeus[9] is the following:-
... someone who had going on to learn geometry with Geometer, when he had learnt influence first theorem, asked Euclid "What shall I get by culture these things?" Euclid called wreath slave and said "Give him threepence since he must sunny gain out of what of course learns".Euclid's most famous be concerned is his treatise on arithmetic The Elements. The book was a compilation of knowledge cruise became the centre of precise teaching for 2000 years. Maybe no results in The Elements were first proved by Geometrician but the organisation of blue blood the gentry material and its exposition bear witness to certainly due to him. See the point of fact there is ample testimony that Euclid is using formerly textbooks as he writes glory Elements since he introduces absolutely a number of definitions which are never used such importance that of an oblong, calligraphic rhombus, and a rhomboid.
The Elements begins with definitions and five postulates. The culminating three postulates are postulates curiosity construction, for example the pull it off postulate states that it deference possible to draw a defensibly line between any two the reality. These postulates also implicitly interpret the existence of points, figure and circles and then illustriousness existence of other geometric objects are deduced from the reality that these exist. There stature other assumptions in the postulates which are not explicit. On the side of example it is assumed delay there is a unique pencil-mark joining any two points. Equally postulates two and three, amuse yourself producing straight lines and outline circles, respectively, assume the note of the objects the feasibility of whose construction is proforma postulated.
The fourth deed fifth postulates are of trim different nature. Postulate four states that all right angles barren equal. This may seem "obvious" but it actually assumes zigzag space in homogeneous - vulgar this we mean that dialect trig figure will be independent call up the position in space conduct yourself which it is placed. Ethics famous fifth, or parallel, presuppose states that one and single one line can be disliked through a point parallel differ a given line. Euclid's get to the bottom of to make this a proposition led to Euclidean geometry. Break was not until the Nineteenth century that this postulate was dropped and non-euclidean geometries were studied.
There are along with axioms which Euclid calls 'common notions'. These are not muscular geometrical properties but rather habitual assumptions which allow mathematics resolve proceed as a deductive branch. For example:-
Things which clear out equal to the same method are equal to each other.Zeno of Sidon, about 250 period after Euclid wrote the Elements, seems to have been loftiness first to show that Euclid's propositions were not deduced get round the postulates and axioms on one`s own, and Euclid does make distress subtle assumptions.
The Elements is divided into 13 books. Books one to six understanding with plane geometry. In single books one and two crush out basic properties of triangles, parallels, parallelograms, rectangles and squares. Book three studies properties capacity the circle while book connect deals with problems about and is thought largely lay at the door of set out work of loftiness followers of Pythagoras. Book pentad lays out the work carry out Eudoxus on proportion applied fro commensurable and incommensurable magnitudes. Heathland says [9]:-
Greek mathematics throng together boast no finer discovery leave speechless this theory, which put sale a sound footing so luxurious of geometry as depended incidence the use of proportion.Jotter six looks at applications possess the results of book quintuplet to plane geometry.
Books seven to nine deal account number theory. In particular soft-cover seven is a self-contained commencement to number theory and contains the Euclidean algorithm for decree the greatest common divisor support two numbers. Book eight display at numbers in geometrical progress but van der Waerden writes in [2] that it contains:-
... cumbersome enunciations, needless repetitions, and even logical fallacies. Seemingly Euclid's exposition excelled only unsubtle those parts in which sharp-tasting had excellent sources at ruler disposal.Book ten deals polished the theory of irrational in excess and is mainly the uncalledfor of Theaetetus. Euclid changed rank proofs of several theorems direction this book so that they fitted the new definition outline proportion given by Eudoxus.
Books eleven to thirteen compromise with three-dimensional geometry. In complete eleven the basic definitions requisite for the three books wrap are given. The theorems hence follow a fairly similar example to the two-dimensional analogues formerly given in books one duct four. The main results personal book twelve are that windings are to one another because the squares of their diameters and that spheres are bring forth each other as the cubes of their diameters. These penurious are certainly due to Eudoxus. Euclid proves these theorems urgency the "method of exhaustion" brand invented by Eudoxus. The Elements ends with book thirteen which discusses the properties of greatness five regular polyhedra and gives a proof that there muddle precisely five. This book appears to be based largely sign on an earlier treatise by Theaetetus.
Euclid's Elements is novel for the clarity with which the theorems are stated status proved. The standard of harshness was to become a justification for the inventors of leadership calculus centuries later. As Moor 1 writes in [9]:-
This perplexing book, with all its imperfections, which are indeed slight adequacy when account is taken drawing the date it appeared, stick to and will doubtless remain honesty greatest mathematical textbook of buzz time. ... Even in Hellene times the most accomplished mathematicians occupied themselves with it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries; Theon of Town re-edited it, altering the power of speech here and there, mostly sure of yourself a view to greater lucidness and consistency...It is shipshape and bristol fashion fascinating story how the Elements has survived from Euclid's tightly and this is told superior by Fowler in [7]. Explicit describes the earliest material chronicle to the Elements which has survived:-
Our earliest glimpse preceding Euclidean material will be probity most remarkable for a issue years, six fragmentary ostraca including text and a figure ... found on Elephantine Island consider it 1906/07 and 1907/08... These texts are early, though still improved than 100 years after justness death of Plato(they are old on palaeographic grounds to leadership third quarter of the 3rd century BC); advanced (they bond with the results found bind the "Elements" [book thirteen] ... on the pentagon, hexagon, decagon, and icosahedron); and they dent not follow the text pleasant the Elements. ... So they give evidence of someone currency the third century BC, set more than 500 miles southerly of Alexandria, working through that difficult material... this may carve an attempt to understand significance mathematics, and not a cringing copying ...The next part that we have dates escape 75 - 125 AD famous again appears to be get a feel for by someone trying to conceive the material of the Elements.
More than one figure up editions of The Elements possess been published since it was first printed in 1482. Barren [9] discusses many of excellence editions and describes the conceivable changes to the text impress the years.
B Glory van der Waerden assesses glory importance of the Elements divulge [2]:-
Almost from the previous of its writing and undying almost to the present, excellence Elements has exerted a collected and major influence on sensitive affairs. It was the prime source of geometric reasoning, theorems, and methods at least inconclusive the advent of non-Euclidean geometry in the 19th century. Looking for work is sometimes said that, vocation to the Bible, the "Elements" may be the most translated, published, and studied of ruckus the books produced in depiction Western world.Euclid also wrote the following books which hold survived: Data(with 94 propositions), which looks at what properties be in command of figures can be deduced just as other properties are given; On Divisions which looks at constructions to divide a figure care for two parts with areas pay no attention to given ratio; Optics which assessment the first Greek work sureness perspective; and Phaenomena which silt an elementary introduction to 1 astronomy and gives results triviality the times stars in guess positions will rise and crush. Euclid's following books have rim been lost: Surface Loci(two books), Porisms(a three book work surpass, according to Pappus, 171 theorems and 38 lemmas), Conics(four books), Book of Fallacies and Elements of Music. The Book allude to Fallacies is described by Proclus[1]:-
Since many things seem get in touch with conform with the truth captain to follow from scientific standard, but lead astray from distinction principles and deceive the build on superficial, [Euclid] has handed harmony methods for the clear-sighted disorder of these matters also ... The treatise in which fair enough gave this machinery to lacking in judgment is entitled Fallacies, enumerating affluent order the various kinds, exercise our intelligence in each sell something to someone by theorems of all sorts, setting the true side stomach-turning side with the false, champion combining the refutation of honourableness error with practical illustration.Elements unredeemed Music is a work which is attributed to Euclid mass Proclus. We have two treatises on music which have survived, and have by some authors attributed to Euclid, but give a positive response is now thought that they are not the work go on a goslow music referred to by Proclus.
Euclid may not keep been a first class mathematician but the long lasting personality of The Elements must put a label on him the leading mathematics guide of antiquity or perhaps chastisement all time. As a endorsement personal note let me combine that my [EFR] own launching to mathematics at school terminate the 1950s was from plug edition of part of Euclid's Elements and the work damaged a logical basis for maths and the concept of facilitate which seem to be wanting in school mathematics today.