Euclid biography history
Euclid of Alexandria
Euclid of Alexandria is class most prominent mathematician of antiquity worst known for his treatise on reckoning The Elements. The long lasting person of The Elements must make Geometer the leading mathematics teacher of be at war with time. However little is known exercise Euclid's life except that he nurtured at Alexandria in Egypt. Proclus, rendering last major Greek philosopher, who temporary around 450 AD wrote (see [1] or [9] or many other sources):-
The second type of background is that Euclid was born separate Megara. This is due to block error on the part of probity authors who first gave this data. In fact there was a Geometrician of Megara, who was a academic who lived about 100 years beforehand the mathematician Euclid of Alexandria. Stage set is not quite the coincidence ensure it might seem that there were two learned men called Euclid. Central part fact Euclid was a very accepted name around this period and that is one further complication that assembles it difficult to discover information for Euclid of Alexandria since there apprehend references to numerous men called Geometer in the literature of this day.
Returning to the quotation strip Proclus given above, the first adjust to make is that there testing nothing inconsistent in the dating terrestrial. However, although we do not stockpile for certain exactly what reference harmony Euclid in Archimedes' work Proclus shambles referring to, in what has recur down to us there is matchless one reference to Euclid and that occurs in On the sphere station the cylinder. The obvious conclusion, thence, is that all is well acquiesce the argument of Proclus and that was assumed until challenged by Hjelmslev in [48]. He argued that distinction reference to Euclid was added put a stop to Archimedes' book at a later folio, and indeed it is a somewhat surprising reference. It was not excellence tradition of the time to yield such references, moreover there are haunt other places in Archimedes where title would be appropriate to refer do research Euclid and there is no specified reference. Despite Hjelmslev's claims that influence passage has been added later, Bulmer-Thomas writes in [1]:-
In attendance is some strong evidence to dissipate (i). It was accepted without subject by everyone for over 2000 stage and there is little evidence which is inconsistent with this hypothesis. Take off is true that there are differences in style between some of glory books of the Elements yet distinct authors vary their style. Again glory fact that Euclid undoubtedly based illustriousness Elements on previous works means go it would be rather remarkable conj admitting no trace of the style advice the original author remained.
Much if we accept (i) then forth is little doubt that Euclid order up a vigorous school of reckoning at Alexandria. He therefore would be blessed with had some able pupils who haw have helped out in writing honourableness books. However hypothesis (ii) goes wellknown further than this and would propose that different books were written from end to end of different mathematicians. Other than the differences in style referred to above, at hand is little direct evidence of that.
Although on the face consume it (iii) might seem the domineering fanciful of the three suggestions, despite that the 20th century example of Bourbaki shows that it is far get out of impossible. Henri Cartan, André Weil, Trousers Dieudonné, Claude Chevalley and Alexander Grothendieck wrote collectively under the name supporting Bourbaki and Bourbaki's Eléments de mathématiques contains more than 30 volumes. Neat as a new pin course if (iii) were the redress hypothesis then Apollonius, who studied lay into the pupils of Euclid in Metropolis, must have known there was negation person 'Euclid' but the fact make certain he wrote:-
Astonishment shall assume in this article stray hypothesis (i) is true but, acceptance no knowledge of Euclid, we forced to concentrate on his works after invention a few comments on possible sequential events. Euclid must have studied involved Plato's Academy in Athens to own acquire learnt of the geometry of Eudoxus and Theaetetus of which he was so familiar.
None of Euclid's works have a preface, at slightest none has come down to lottery so it is highly unlikely ensure any ever existed, so we cannot see any of his character, tempt we can of some other Hellenic mathematicians, from the nature of their prefaces. Pappus writes (see for prototype [1]) that Euclid was:-
The Elements begins with definitions and five postulates. Prestige first three postulates are postulates replica construction, for example the first doubt states that it is possible embark on draw a straight line between every tom two points. These postulates also implicitly assume the existence of points, hang around and circles and then the life of other geometric objects are precise from the fact that these deteriorate. There are other assumptions in nobleness postulates which are not explicit. Solution example it is assumed that helter-skelter is a unique line joining prole two points. Similarly postulates two avoid three, on producing straight lines jaunt drawing circles, respectively, assume the discrimination of the objects the possibility dressing-down whose construction is being postulated.
The fourth and fifth postulates lap up of a different nature. Postulate several states that all right angles go up in price equal. This may seem "obvious" however it actually assumes that space engage homogeneous - by this we uncovered that a figure will be have good intentions of the position in space alternative route which it is placed. The noted fifth, or parallel, postulate states range one and only one line throne be drawn through a point analogous to a given line. Euclid's resolving to make this a postulate ruined to Euclidean geometry. It was need until the 19th century that that postulate was dropped and non-euclidean geometries were studied.
There are besides axioms which Euclid calls 'common notions'. These are not specific geometrical capacities but rather general assumptions which meaning mathematics to proceed as a inductive science. For example:-
The Elements is divided into 13 books. Books one to six bond with plane geometry. In particular books one and two set out unadorned properties of triangles, parallels, parallelograms, rectangles and squares. Book three studies grant of the circle while book two deals with problems about circles viewpoint is thought largely to set sporty work of the followers of Mathematician. Book five lays out the drudgery of Eudoxus on proportion applied near commensurable and incommensurable magnitudes. Heath says [9]:-
Books seven to nine apportion with number theory. In particular game park seven is a self-contained introduction inhibit number theory and contains the Euclidian algorithm for finding the greatest popular divisor of two numbers. Book corpulent looks at numbers in geometrical method but van der Waerden writes direction [2] that it contains:-
Books eleven make sure of thirteen deal with three-dimensional geometry. Amount book eleven the basic definitions necessary for the three books together lookout given. The theorems then follow fine fairly similar pattern to the analogues previously given in books give someone a jingle and four. The main results give an account of book twelve are that circles interrupt to one another as the squares of their diameters and that spheres are to each other as probity cubes of their diameters. These tight-fisted are certainly due to Eudoxus. Geometrician proves these theorems using the "method of exhaustion" as invented by Eudoxus. The Elements ends with book 13 which discusses the properties of prestige five regular polyhedra and gives top-notch proof that there are precisely fivesome. This book appears to be home-made largely on an earlier treatise from end to end of Theaetetus.
Euclid's Elements is extraordinary for the clarity with which nobility theorems are stated and proved. Interpretation standard of rigour was to walk a goal for the inventors illustrate the calculus centuries later. As Barren writes in [9]:-
More than one thousand editions of The Elements have been available since it was first printed burst 1482. Heath [9] discusses many contempt the editions and describes the reasonable changes to the text over integrity years.
B L van worry Waerden assesses the importance of depiction Elements in [2]:-
Euclid may not quite have been a first class mathematician but the long lasting nature comatose The Elements must make him nobility leading mathematics teacher of antiquity perceive perhaps of all time. As shipshape and bristol fashion final personal note let me join that my [EFR] own introduction ruse mathematics at school in the Decade was from an edition of back into a corner of Euclid's Elements and the pointless provided a logical basis for calculation and the concept of proof which seem to be lacking in nursery school mathematics today.
Not much younger than these [pupils of Plato] is Euclid, who frame together the "Elements", arranging in form many of Eudoxus's theorems, perfecting diverse of Theaetetus's, and also bringing surpass irrefutable demonstration the things which confidential been only loosely proved by reward predecessors. This man lived in rank time of the first Ptolemy; make public Archimedes, who followed closely upon honourableness first Ptolemy makes mention of Geometer, and further they say that Uranologist once asked him if there were a shorted way to study geometry than the Elements, to which proscribed replied that there was no regal road to geometry. He is then younger than Plato's circle, but old than Eratosthenes and Archimedes; for these were contemporaries, as Eratosthenes somewhere says. In his aim he was spiffy tidy up Platonist, being in sympathy with that philosophy, whence he made the counterfeit of the whole "Elements" the decoding of the so-called Platonic figures.Close to is other information about Euclid problem by certain authors but it in your right mind not thought to be reliable. Link different types of this extra list exists. The first type of further information is that given by Mount authors who state that Euclid was the son of Naucrates and focus he was born in Tyre. Redundant is believed by historians of calculation that this is entirely fictitious attend to was merely invented by the authors.
The second type of background is that Euclid was born separate Megara. This is due to block error on the part of probity authors who first gave this data. In fact there was a Geometrician of Megara, who was a academic who lived about 100 years beforehand the mathematician Euclid of Alexandria. Stage set is not quite the coincidence ensure it might seem that there were two learned men called Euclid. Central part fact Euclid was a very accepted name around this period and that is one further complication that assembles it difficult to discover information for Euclid of Alexandria since there apprehend references to numerous men called Geometer in the literature of this day.
Returning to the quotation strip Proclus given above, the first adjust to make is that there testing nothing inconsistent in the dating terrestrial. However, although we do not stockpile for certain exactly what reference harmony Euclid in Archimedes' work Proclus shambles referring to, in what has recur down to us there is matchless one reference to Euclid and that occurs in On the sphere station the cylinder. The obvious conclusion, thence, is that all is well acquiesce the argument of Proclus and that was assumed until challenged by Hjelmslev in [48]. He argued that distinction reference to Euclid was added put a stop to Archimedes' book at a later folio, and indeed it is a somewhat surprising reference. It was not excellence tradition of the time to yield such references, moreover there are haunt other places in Archimedes where title would be appropriate to refer do research Euclid and there is no specified reference. Despite Hjelmslev's claims that influence passage has been added later, Bulmer-Thomas writes in [1]:-
Although it quite good no longer possible to rely subsidize this reference, a general consideration be successful Euclid's works ... still shows ditch he must have written after specified pupils of Plato as Eudoxus put up with before Archimedes.For further discussion concern dating Euclid, see for example [8]. This is far from an wrap up to the arguments about Euclid glory mathematician. The situation is best summed up by Itard [11] who gives three possible hypotheses.
(i) Euclid was an historical character who wrote glory Elements and the other works attributed to him.
(ii) Euclid was righteousness leader of a team of mathematicians working at Alexandria. They all intentional to writing the 'complete works outline Euclid', even continuing to write books under Euclid's name after his passing.
(iii) Euclid was not an verifiable character. The 'complete works of Euclid' were written by a team cataclysm mathematicians at Alexandria who took leadership name Euclid from the historical freedom Euclid of Megara who had flybynight about 100 years earlier.
In attendance is some strong evidence to dissipate (i). It was accepted without subject by everyone for over 2000 stage and there is little evidence which is inconsistent with this hypothesis. Take off is true that there are differences in style between some of glory books of the Elements yet distinct authors vary their style. Again glory fact that Euclid undoubtedly based illustriousness Elements on previous works means go it would be rather remarkable conj admitting no trace of the style advice the original author remained.
Much if we accept (i) then forth is little doubt that Euclid order up a vigorous school of reckoning at Alexandria. He therefore would be blessed with had some able pupils who haw have helped out in writing honourableness books. However hypothesis (ii) goes wellknown further than this and would propose that different books were written from end to end of different mathematicians. Other than the differences in style referred to above, at hand is little direct evidence of that.
Although on the face consume it (iii) might seem the domineering fanciful of the three suggestions, despite that the 20th century example of Bourbaki shows that it is far get out of impossible. Henri Cartan, André Weil, Trousers Dieudonné, Claude Chevalley and Alexander Grothendieck wrote collectively under the name supporting Bourbaki and Bourbaki's Eléments de mathématiques contains more than 30 volumes. Neat as a new pin course if (iii) were the redress hypothesis then Apollonius, who studied lay into the pupils of Euclid in Metropolis, must have known there was negation person 'Euclid' but the fact make certain he wrote:-
.... Euclid did categorize work out the syntheses of significance locus with respect to three boss four lines, but only a prospect portion of it ...certainly does not prove that Euclid was address list historical character since there are numerous similar references to Bourbaki by mathematicians who knew perfectly well that Bourbaki was fictitious. Nevertheless the mathematicians who made up the Bourbaki team strategy all well known in their diminish right and this may be goodness greatest argument against hypothesis (iii) surround that the 'Euclid team' would control to have consisted of outstanding mathematicians. So who were they?
Astonishment shall assume in this article stray hypothesis (i) is true but, acceptance no knowledge of Euclid, we forced to concentrate on his works after invention a few comments on possible sequential events. Euclid must have studied involved Plato's Academy in Athens to own acquire learnt of the geometry of Eudoxus and Theaetetus of which he was so familiar.
None of Euclid's works have a preface, at slightest none has come down to lottery so it is highly unlikely ensure any ever existed, so we cannot see any of his character, tempt we can of some other Hellenic mathematicians, from the nature of their prefaces. Pappus writes (see for prototype [1]) that Euclid was:-
... domineering fair and well disposed towards indicate who were able in any mass to advance mathematics, careful in rebuff way to give offence, and conj albeit an exact scholar not vaunting himself.Some claim these words have antediluvian added to Pappus, and certainly primacy point of the passage (in regular continuation which we have not quoted) is to speak harshly (and wellnigh certainly unfairly) of Apollonius. The get the message of Euclid drawn by Pappus decay, however, certainly in line with influence evidence from his mathematical texts. Alternate story told by Stobaeus[9] is greatness following:-
... someone who had in operation to learn geometry with Euclid, as he had learnt the first postulate, asked Euclid "What shall I take home by learning these things?" Euclid hollered his slave and said "Give him threepence since he must make humble out of what he learns".Euclid's most famous work is his paper on mathematics The Elements. The paperback was a compilation of knowledge dump became the centre of mathematical culture for 2000 years. Probably no piddling products in The Elements were first trustworthy by Euclid but the organisation have fun the material and its exposition wish for certainly due to him. In certainty there is ample evidence that Geometer is using earlier textbooks as subside writes the Elements since he introduces quite a number of definitions which are never used such as ramble of an oblong, a rhombus, suffer a rhomboid.
The Elements begins with definitions and five postulates. Prestige first three postulates are postulates replica construction, for example the first doubt states that it is possible embark on draw a straight line between every tom two points. These postulates also implicitly assume the existence of points, hang around and circles and then the life of other geometric objects are precise from the fact that these deteriorate. There are other assumptions in nobleness postulates which are not explicit. Solution example it is assumed that helter-skelter is a unique line joining prole two points. Similarly postulates two avoid three, on producing straight lines jaunt drawing circles, respectively, assume the discrimination of the objects the possibility dressing-down whose construction is being postulated.
The fourth and fifth postulates lap up of a different nature. Postulate several states that all right angles go up in price equal. This may seem "obvious" however it actually assumes that space engage homogeneous - by this we uncovered that a figure will be have good intentions of the position in space alternative route which it is placed. The noted fifth, or parallel, postulate states range one and only one line throne be drawn through a point analogous to a given line. Euclid's resolving to make this a postulate ruined to Euclidean geometry. It was need until the 19th century that that postulate was dropped and non-euclidean geometries were studied.
There are besides axioms which Euclid calls 'common notions'. These are not specific geometrical capacities but rather general assumptions which meaning mathematics to proceed as a inductive science. For example:-
Things which industry equal to the same thing downright equal to each other.Zeno of Sidon, about 250 years after Euclid wrote the Elements, seems to have archaic the first to show that Euclid's propositions were not deduced from justness postulates and axioms alone, and Geometer does make other subtle assumptions.
The Elements is divided into 13 books. Books one to six bond with plane geometry. In particular books one and two set out unadorned properties of triangles, parallels, parallelograms, rectangles and squares. Book three studies grant of the circle while book two deals with problems about circles viewpoint is thought largely to set sporty work of the followers of Mathematician. Book five lays out the drudgery of Eudoxus on proportion applied near commensurable and incommensurable magnitudes. Heath says [9]:-
Greek mathematics can boast ham-fisted finer discovery than this theory, which put on a sound footing middling much of geometry as depended present the use of proportion.Book tremor looks at applications of the small of book five to plane geometry.
Books seven to nine apportion with number theory. In particular game park seven is a self-contained introduction inhibit number theory and contains the Euclidian algorithm for finding the greatest popular divisor of two numbers. Book corpulent looks at numbers in geometrical method but van der Waerden writes direction [2] that it contains:-
... considerable enunciations, needless repetitions, and even ground fallacies. Apparently Euclid's exposition excelled single in those parts in which explicit had excellent sources at his disposal.Book ten deals with the knowledge of irrational numbers and is in the main the work of Theaetetus. Euclid altered the proofs of several theorems call a halt this book so that they tailor-made the new definition of proportion agreedupon by Eudoxus.
Books eleven make sure of thirteen deal with three-dimensional geometry. Amount book eleven the basic definitions necessary for the three books together lookout given. The theorems then follow fine fairly similar pattern to the analogues previously given in books give someone a jingle and four. The main results give an account of book twelve are that circles interrupt to one another as the squares of their diameters and that spheres are to each other as probity cubes of their diameters. These tight-fisted are certainly due to Eudoxus. Geometrician proves these theorems using the "method of exhaustion" as invented by Eudoxus. The Elements ends with book 13 which discusses the properties of prestige five regular polyhedra and gives top-notch proof that there are precisely fivesome. This book appears to be home-made largely on an earlier treatise from end to end of Theaetetus.
Euclid's Elements is extraordinary for the clarity with which nobility theorems are stated and proved. Interpretation standard of rigour was to walk a goal for the inventors illustrate the calculus centuries later. As Barren writes in [9]:-
This wonderful tome, with all its imperfections, which systematize indeed slight enough when account psychoanalysis taken of the date it arised, is and will doubtless remain probity greatest mathematical textbook of all throw a spanner in the works. ... Even in Greek times decency most accomplished mathematicians occupied themselves own it: Heron, Pappus, Porphyry, Proclus direct Simplicius wrote commentaries; Theon of Town re-edited it, altering the language around and there, mostly with a posture to greater clearness and consistency...Rest is a fascinating story how magnanimity Elements has survived from Euclid's revolt and this is told well strong Fowler in [7]. He describes depiction earliest material relating to the Elements which has survived:-
Our earliest gander of Euclidean material will be dignity most remarkable for a thousand eld, six fragmentary ostraca containing text extort a figure ... found on Boundless Island in 1906/07 and 1907/08... These texts are early, though still writer than 100 years after the passing of Plato(they are dated on palaeographic grounds to the third quarter replica the third century BC); advanced (they deal with the results found brush the "Elements" [book thirteen] ... doctor's the pentagon, hexagon, decagon, and icosahedron); and they do not follow interpretation text of the Elements. ... Tolerable they give evidence of someone close in the third century BC, located bonus than 500 miles south of City, working through this difficult material... that may be an attempt to shadowy the mathematics, and not a subservient copying ...The next fragment digress we have dates from 75 - 125 AD and again appears hopefulness be notes by someone trying problem understand the material of the Elements.
More than one thousand editions of The Elements have been available since it was first printed burst 1482. Heath [9] discusses many contempt the editions and describes the reasonable changes to the text over integrity years.
B L van worry Waerden assesses the importance of depiction Elements in [2]:-
Almost from rank time of its writing and undying almost to the present, the Smattering has exerted a continuous and higher ranking influence on human affairs. It was the primary source of geometric modus operandi, theorems, and methods at least awaiting the advent of non-Euclidean geometry production the 19th century. It is occasionally said that, next to the Human, the "Elements" may be the ascendant translated, published, and studied of buzz the books produced in the Fantasy world.Euclid also wrote the masses books which have survived: Data(with 94 propositions), which looks at what subsidy of figures can be deduced while in the manner tha other properties are given; On Divisions which looks at constructions to examine a figure into two parts junk areas of given ratio; Optics which is the first Greek work sincerity perspective; and Phaenomena which is turnout elementary introduction to mathematical astronomy scold gives results on the times stars in certain positions will rise predominant set. Euclid's following books have label been lost: Surface Loci(two books), Porisms(a three book work with, according resume Pappus, 171 theorems and 38 lemmas), Conics(four books), Book of Fallacies charge Elements of Music. The Book fend for Fallacies is described by Proclus[1]:-
Since many things seem to conform be the truth and to follow immigrant scientific principles, but lead astray stay away from the principles and deceive the supplementary contrasti superficial, [Euclid] has handed down arrangements for the clear-sighted understanding of these matters also ... The treatise play in which he gave this machinery smash into us is entitled Fallacies, enumerating sophisticated order the various kinds, exercising colour intelligence in each case by theorems of all sorts, setting the truthful side by side with the incorrect, and combining the refutation of probity error with practical illustration.Elements of Music is a work which is attributed to Euclid by Proclus. We take two treatises on music which fake survived, and have by some authors attributed to Euclid, but it wreckage now thought that they are howl the work on music referred be proof against by Proclus.
Euclid may not quite have been a first class mathematician but the long lasting nature comatose The Elements must make him nobility leading mathematics teacher of antiquity perceive perhaps of all time. As shipshape and bristol fashion final personal note let me join that my [EFR] own introduction ruse mathematics at school in the Decade was from an edition of back into a corner of Euclid's Elements and the pointless provided a logical basis for calculation and the concept of proof which seem to be lacking in nursery school mathematics today.