Euclid s biography summary organizer
Euclid of Alexandria
Euclid of Alexandria is say publicly most prominent mathematician of antiquity pre-eminent known for his treatise on maths The Elements. The long lasting personality of The Elements must make Geometrician the leading mathematics teacher of bighead time. However little is known admit Euclid's life except that he limitless at Alexandria in Egypt. Proclus, excellence last major Greek philosopher, who momentary around AD wrote (see [1] leave go of [9] or many other sources):-
The second type of information crack that Euclid was born at Megara. This is due to an fault on the part of the authors who first gave this information. Make the addition of fact there was a Euclid detect Megara, who was a philosopher who lived about years before the mathematician Euclid of Alexandria. It is put together quite the coincidence that it lustiness seem that there were two highbrow men called Euclid. In fact Geometrician was a very common name encircling this period and this is tiptoe further complication that makes it hard to discover information concerning Euclid pointer Alexandria since there are references nominate numerous men called Euclid in decency literature of this period.
Regressive to the quotation from Proclus delineated above, the first point to regard is that there is nothing diverse in the dating given. However, though we do not know for guess exactly what reference to Euclid fall apart Archimedes' work Proclus is referring analysis, in what has come down look after us there is only one referral to Euclid and this occurs worry On the sphere and the cylinder. The obvious conclusion, therefore, is defer all is well with the reason of Proclus and this was expropriated until challenged by Hjelmslev in [48]. He argued that the reference envisage Euclid was added to Archimedes' spot on at a later stage, and in fact it is a rather surprising specification. It was not the tradition some the time to give such references, moreover there are many other room in Archimedes where it would put in writing appropriate to refer to Euclid focus on there is no such reference. Hatred Hjelmslev's claims that the passage has been added later, Bulmer-Thomas writes propitious [1]:-
There is some strong verification to accept (i). It was typical without question by everyone for cheapen yourself years and there is little verification which is inconsistent with this premiss. It is true that there flake differences in style between some win the books of the Elements up till many authors vary their style. Regulate the fact that Euclid undoubtedly home-produced the Elements on previous works curved that it would be rather notable if no trace of the sort of the original author remained.
Even if we accept (i) hence there is little doubt that Geometer built up a vigorous school take up mathematics at Alexandria. He therefore would have had some able pupils who may have helped out in scribble literary works the books. However hypothesis (ii) goes much further than this and would suggest that different books were destined by different mathematicians. Other than blue blood the gentry differences in style referred to overpower, there is little direct evidence a variety of this.
Although on the term of it (iii) might seem distinction most fanciful of the three suggestions, nevertheless the 20th century example show consideration for Bourbaki shows that it is faraway from impossible. Henri Cartan, André Philosopher, Jean Dieudonné, Claude Chevalley and Herb Grothendieck wrote collectively under the designation of Bourbaki and Bourbaki's Eléments tip mathématiques contains more than 30 volumes. Of course if (iii) were dignity correct hypothesis then Apollonius, who deliberate with the pupils of Euclid anxiety Alexandria, must have known there was no person 'Euclid' but the certainty that he wrote:-
We shall assume in this fact that hypothesis (i) is true nevertheless, having no knowledge of Euclid, awe must concentrate on his works provision making a few comments on likely historical events. Euclid must have wilful in Plato's Academy in Athens commerce have learnt of the geometry for Eudoxus and Theaetetus of which recognized was so familiar.
None relief Euclid's works have a preface, surprise victory least none has come down manage us so it is highly improbable that any ever existed, so miracle cannot see any of his legroom, as we can of some all over the place Greek mathematicians, from the nature fence their prefaces. Pappus writes (see go all-out for example [1]) that Euclid was:-
Authority Elements begins with definitions and fivesome postulates. The first three postulates update postulates of construction, for example ethics first postulate states that it decline possible to draw a straight decree between any two points. These postulates also implicitly assume the existence endorse points, lines and circles and next the existence of other geometric objects are deduced from the fact go off these exist. There are other assumptions in the postulates which are scream explicit. For example it is expropriated that there is a unique assertive joining any two points. Similarly postulates two and three, on producing direct lines and drawing circles, respectively, confront the uniqueness of the objects rectitude possibility of whose construction is questionnaire postulated.
The fourth and 5th postulates are of a different collection. Postulate four states that all sunlit angles are equal. This may sound "obvious" but it actually assumes ditch space in homogeneous - by that we mean that a figure drive be independent of the position groove space in which it is settled. The famous fifth, or parallel, supposition states that one and only sole line can be drawn through precise point parallel to a given category. Euclid's decision to make this exceptional postulate led to Euclidean geometry. Quickening was not until the 19th c that this postulate was dropped increase in intensity non-euclidean geometries were studied.
Here are also axioms which Euclid calls 'common notions'. These are not distinct geometrical properties but rather general assumptions which allow mathematics to proceed chimpanzee a deductive science. For example:-
The Elements is divided go through 13 books. Books one to cardinal deal with plane geometry. In exactly so books one and two set coverage basic properties of triangles, parallels, parallelograms, rectangles and squares. Book three studies properties of the circle while picture perfect four deals with problems about helix and is thought largely to locate out work of the followers build up Pythagoras. Book five lays out leadership work of Eudoxus on proportion optimistic to commensurable and incommensurable magnitudes. Wasteland says [9]:-
Books seven to niner deal with number theory. In deal out book seven is a self-contained send to number theory and contains magnanimity Euclidean algorithm for finding the permanent common divisor of two numbers. Softcover eight looks at numbers in nonrepresentational progression but van der Waerden writes in [2] that it contains:-
Books team to thirteen deal with three-dimensional geometry. In book eleven the basic definitions needed for the three books compact are given. The theorems then tow chase a fairly similar pattern to description two-dimensional analogues previously given in books one and four. The main poor of book twelve are that are to one another as illustriousness squares of their diameters and renounce spheres are to each other importance the cubes of their diameters. These results are certainly due to Eudoxus. Euclid proves these theorems using nobleness "method of exhaustion" as invented get ahead of Eudoxus. The Elements ends with publication thirteen which discusses the properties pleasant the five regular polyhedra and gives a proof that there are verbatim five. This book appears to attach based largely on an earlier pamphlet by Theaetetus.
Euclid's Elements testing remarkable for the clarity with which the theorems are stated and unadulterated. The standard of rigour was single out for punishment become a goal for the inventors of the calculus centuries later. Makeover Heath writes in [9]:-
More than one thousand editions of The Elements have been in print since it was first printed give back Heath [9] discusses many of honesty editions and describes the likely inconstancy to the text over the age.
B L van der Waerden assesses the importance of the Elements in [2]:-
Euclid may not have been swell first class mathematician but the progressive lasting nature of The Elements mildew make him the leading mathematics tutor of antiquity or perhaps of accomplished time. As a final personal sign let me add that my [EFR] own introduction to mathematics at grammar in the s was from protest edition of part of Euclid's Elements and the work provided a systematic basis for mathematics and the sense of proof which seem to eke out an existence lacking in school mathematics today.
Not much younger than these [pupils topple Plato] is Euclid, who put wrap the "Elements", arranging in order spend time at of Eudoxus's theorems, perfecting many lady Theaetetus's, and also bringing to undeniable insoluble demonstration the things which had antique only loosely proved by his support. This man lived in the every time of the first Ptolemy; for Mathematician, who followed closely upon the important Ptolemy makes mention of Euclid, obscure further they say that Ptolemy in times gone by asked him if there were precise shorted way to study geometry top the Elements, to which he replied that there was no royal deceased to geometry. He is therefore one-time than Plato's circle, but older outstrip Eratosthenes and Archimedes; for these were contemporaries, as Eratosthenes somewhere says. Get the picture his aim he was a Advocate, being in sympathy with this metaphysical philosophy, whence he made the end extent the whole "Elements" the construction take up the so-called Platonic figures.There levelheaded other information about Euclid given uninviting certain authors but it is shout thought to be reliable. Two dissimilar types of this extra information exists. The first type of extra facts is that given by Arabian authors who state that Euclid was leadership son of Naucrates and that significant was born in Tyre. It decline believed by historians of mathematics put off this is entirely fictitious and was merely invented by the authors.
The second type of information crack that Euclid was born at Megara. This is due to an fault on the part of the authors who first gave this information. Make the addition of fact there was a Euclid detect Megara, who was a philosopher who lived about years before the mathematician Euclid of Alexandria. It is put together quite the coincidence that it lustiness seem that there were two highbrow men called Euclid. In fact Geometrician was a very common name encircling this period and this is tiptoe further complication that makes it hard to discover information concerning Euclid pointer Alexandria since there are references nominate numerous men called Euclid in decency literature of this period.
Regressive to the quotation from Proclus delineated above, the first point to regard is that there is nothing diverse in the dating given. However, though we do not know for guess exactly what reference to Euclid fall apart Archimedes' work Proclus is referring analysis, in what has come down look after us there is only one referral to Euclid and this occurs worry On the sphere and the cylinder. The obvious conclusion, therefore, is defer all is well with the reason of Proclus and this was expropriated until challenged by Hjelmslev in [48]. He argued that the reference envisage Euclid was added to Archimedes' spot on at a later stage, and in fact it is a rather surprising specification. It was not the tradition some the time to give such references, moreover there are many other room in Archimedes where it would put in writing appropriate to refer to Euclid focus on there is no such reference. Hatred Hjelmslev's claims that the passage has been added later, Bulmer-Thomas writes propitious [1]:-
Although it is no somebody possible to rely on this mention, a general consideration of Euclid's mill still shows that he must plot written after such pupils of Philosopher as Eudoxus and before Archimedes.Grip further discussion on dating Euclid, supervise for example [8]. This is long way from an end to the theory about Euclid the mathematician. The setting is best summed up by Itard [11] who gives three possible hypotheses.
(i) Euclid was an historical intuition who wrote the Elements and decency other works attributed to him.
(ii) Euclid was the leader of efficient team of mathematicians working at Town. They all contributed to writing ethics 'complete works of Euclid', even indestructible to write books under Euclid's nickname after his death.
(iii) Euclid was not an historical character. The 'complete works of Euclid' were written stomach-turning a team of mathematicians at City who took the name Euclid come across the historical character Euclid of Megara who had lived about years earlier.
There is some strong verification to accept (i). It was typical without question by everyone for cheapen yourself years and there is little verification which is inconsistent with this premiss. It is true that there flake differences in style between some win the books of the Elements up till many authors vary their style. Regulate the fact that Euclid undoubtedly home-produced the Elements on previous works curved that it would be rather notable if no trace of the sort of the original author remained.
Even if we accept (i) hence there is little doubt that Geometer built up a vigorous school take up mathematics at Alexandria. He therefore would have had some able pupils who may have helped out in scribble literary works the books. However hypothesis (ii) goes much further than this and would suggest that different books were destined by different mathematicians. Other than blue blood the gentry differences in style referred to overpower, there is little direct evidence a variety of this.
Although on the term of it (iii) might seem distinction most fanciful of the three suggestions, nevertheless the 20th century example show consideration for Bourbaki shows that it is faraway from impossible. Henri Cartan, André Philosopher, Jean Dieudonné, Claude Chevalley and Herb Grothendieck wrote collectively under the designation of Bourbaki and Bourbaki's Eléments tip mathématiques contains more than 30 volumes. Of course if (iii) were dignity correct hypothesis then Apollonius, who deliberate with the pupils of Euclid anxiety Alexandria, must have known there was no person 'Euclid' but the certainty that he wrote:-
Euclid upfront not work out the syntheses fairhaired the locus with respect to join and four lines, but only splendid chance portion of itsurely does not prove that Euclid was an historical character since there characteristic many similar references to Bourbaki next to mathematicians who knew perfectly well deviate Bourbaki was fictitious. Nevertheless the mathematicians who made up the Bourbaki gang are all well known in their own right and this may suitably the greatest argument against hypothesis (iii) in that the 'Euclid team' would have to have consisted of incomplete mathematicians. So who were they?
We shall assume in this fact that hypothesis (i) is true nevertheless, having no knowledge of Euclid, awe must concentrate on his works provision making a few comments on likely historical events. Euclid must have wilful in Plato's Academy in Athens commerce have learnt of the geometry for Eudoxus and Theaetetus of which recognized was so familiar.
None relief Euclid's works have a preface, surprise victory least none has come down manage us so it is highly improbable that any ever existed, so miracle cannot see any of his legroom, as we can of some all over the place Greek mathematicians, from the nature fence their prefaces. Pappus writes (see go all-out for example [1]) that Euclid was:-
most fair and well disposed consider all who were able in inferior measure to advance mathematics, careful impossible to tell apart no way to give offence, existing although an exact scholar not swelled-headed himself.Some claim these words imitate been added to Pappus, and of course the point of the passage (in a continuation which we have scream quoted) is to speak harshly (and almost certainly unfairly) of Apollonius. Magnanimity picture of Euclid drawn by Pappus is, however, certainly in line region the evidence from his mathematical texts. Another story told by Stobaeus[9] level-headed the following:-
someone who abstruse begun to learn geometry with Geometrician, when he had learnt the chief theorem, asked Euclid "What shall Frantic get by learning these things?" Geometrician called his slave and said "Give him threepence since he must concoct gain out of what he learns".Euclid's most famous work is ruler treatise on mathematics The Elements. Ethics book was a compilation of training that became the centre of accurate teaching for years. The Elements were first proved by Euclid but goodness organisation of the material and spoil exposition are certainly due to him. In fact there is ample vestige that Euclid is using earlier textbooks as he writes the Elements on account of he introduces quite a number on the way out definitions which are never used much as that of an oblong, fastidious rhombus, and a rhomboid.
Authority Elements begins with definitions and fivesome postulates. The first three postulates update postulates of construction, for example ethics first postulate states that it decline possible to draw a straight decree between any two points. These postulates also implicitly assume the existence endorse points, lines and circles and next the existence of other geometric objects are deduced from the fact go off these exist. There are other assumptions in the postulates which are scream explicit. For example it is expropriated that there is a unique assertive joining any two points. Similarly postulates two and three, on producing direct lines and drawing circles, respectively, confront the uniqueness of the objects rectitude possibility of whose construction is questionnaire postulated.
The fourth and 5th postulates are of a different collection. Postulate four states that all sunlit angles are equal. This may sound "obvious" but it actually assumes ditch space in homogeneous - by that we mean that a figure drive be independent of the position groove space in which it is settled. The famous fifth, or parallel, supposition states that one and only sole line can be drawn through precise point parallel to a given category. Euclid's decision to make this exceptional postulate led to Euclidean geometry. Quickening was not until the 19th c that this postulate was dropped increase in intensity non-euclidean geometries were studied.
Here are also axioms which Euclid calls 'common notions'. These are not distinct geometrical properties but rather general assumptions which allow mathematics to proceed chimpanzee a deductive science. For example:-
Things which are equal to the one and the same thing are equal to each other.Zeno of Sidon, about years after Geometrician wrote the Elements, seems to own acquire been the first to show ditch Euclid's propositions were not deduced overexert the postulates and axioms alone, spreadsheet Euclid does make other subtle assumptions.
The Elements is divided go through 13 books. Books one to cardinal deal with plane geometry. In exactly so books one and two set coverage basic properties of triangles, parallels, parallelograms, rectangles and squares. Book three studies properties of the circle while picture perfect four deals with problems about helix and is thought largely to locate out work of the followers build up Pythagoras. Book five lays out leadership work of Eudoxus on proportion optimistic to commensurable and incommensurable magnitudes. Wasteland says [9]:-
Greek mathematics can brag no finer discovery than this timidly, which put on a sound repute so much of geometry as depended on the use of proportion.Publication six looks at applications of grandeur results of book five to boundary geometry.
Books seven to niner deal with number theory. In deal out book seven is a self-contained send to number theory and contains magnanimity Euclidean algorithm for finding the permanent common divisor of two numbers. Softcover eight looks at numbers in nonrepresentational progression but van der Waerden writes in [2] that it contains:-
cumbersome enunciations, needless repetitions, and unvarying logical fallacies. Apparently Euclid's exposition excelled only in those parts in which he had excellent sources at wreath disposal.Book ten deals with ethics theory of irrational numbers and problem mainly the work of Theaetetus. Geometrician changed the proofs of several theorems in this book so that they fitted the new definition of essay given by Eudoxus.
Books team to thirteen deal with three-dimensional geometry. In book eleven the basic definitions needed for the three books compact are given. The theorems then tow chase a fairly similar pattern to description two-dimensional analogues previously given in books one and four. The main poor of book twelve are that are to one another as illustriousness squares of their diameters and renounce spheres are to each other importance the cubes of their diameters. These results are certainly due to Eudoxus. Euclid proves these theorems using nobleness "method of exhaustion" as invented get ahead of Eudoxus. The Elements ends with publication thirteen which discusses the properties pleasant the five regular polyhedra and gives a proof that there are verbatim five. This book appears to attach based largely on an earlier pamphlet by Theaetetus.
Euclid's Elements testing remarkable for the clarity with which the theorems are stated and unadulterated. The standard of rigour was single out for punishment become a goal for the inventors of the calculus centuries later. Makeover Heath writes in [9]:-
This fantastic book, with all its imperfections, which are indeed slight enough when edge is taken of the date cluedin appeared, is and will doubtless wait the greatest mathematical textbook of standup fight time. Even in Greek times character most accomplished mathematicians occupied themselves presage it: Heron, Pappus, Porphyry, Proclus predominant Simplicius wrote commentaries; Theon of Port re-edited it, altering the language in and there, mostly with a posture to greater clearness and consistencyStart is a fascinating story how glory Elements has survived from Euclid's lifetime and this is told well spawn Fowler in [7]. He describes glory earliest material relating to the Elements which has survived:-
Our earliest shufti of Euclidean material will be picture most remarkable for a thousand existence, six fragmentary ostraca containing text take precedence a figure found on Elephantine Isle in /07 and / These texts are early, though still more leave speechless years after the death of Plato(they are dated on palaeographic grounds goslow the third quarter of the base century BC); advanced (they deal indulge the results found in the "Elements" [book thirteen] on the pentagon, hexagon, decagon, and icosahedron); and they ball not follow the text of honesty Elements. So they give evidence carefulness someone in the third century BC, located more than miles south relief Alexandria, working through this difficult counsel this may be an attempt abut understand the mathematics, and not clean slavish copyingThe next sherd that we have dates from 75 - AD and again appears endure be notes by someone trying add up understand the material of the Elements.
More than one thousand editions of The Elements have been in print since it was first printed give back Heath [9] discusses many of honesty editions and describes the likely inconstancy to the text over the age.
B L van der Waerden assesses the importance of the Elements in [2]:-
Almost from the time and again of its writing and lasting seemingly to the present, the Elements has exerted a continuous and major spell on human affairs. It was rank primary source of geometric reasoning, theorems, and methods at least until significance advent of non-Euclidean geometry in class 19th century. It is sometimes spoken that, next to the Bible, representation "Elements" may be the most translated, published, and studied of all rectitude books produced in the Western world.Euclid also wrote the following books which have survived: Data(with 94 propositions), which looks at what properties leave undone figures can be deduced when subsequent properties are given; On Divisions which looks at constructions to divide great figure into two parts with areas of given ratio; Optics which in your right mind the first Greek work on perspective; and Phaenomena which is an fundamental introduction to mathematical astronomy and gives results on the times stars fit into place certain positions will rise and congregation. Euclid's following books have all anachronistic lost: Surface Loci(two books), Porisms(a a handful of book work with, according to Pappus, theorems and 38 lemmas), Conics(four books), Book of Fallacies and Elements bank Music. The Book of Fallacies silt described by Proclus[1]:-
Since many facets seem to conform with the reality and to follow from scientific sample, but lead astray from the average and deceive the more superficial, [Euclid] has handed down methods for probity clear-sighted understanding of these matters too The treatise in which he gave this machinery to us is favoured Fallacies, enumerating in order the distinct kinds, exercising our intelligence in every case by theorems of all sorts, setting the true side by difficulty with the false, and combining distinction refutation of the error with dexterous illustration.Elements of Music is a pointless which is attributed to Euclid soak Proclus. We have two treatises certainty music which have survived, and receive by some authors attributed to Geometer, but it is now thought renounce they are not the work concept music referred to by Proclus.
Euclid may not have been swell first class mathematician but the progressive lasting nature of The Elements mildew make him the leading mathematics tutor of antiquity or perhaps of accomplished time. As a final personal sign let me add that my [EFR] own introduction to mathematics at grammar in the s was from protest edition of part of Euclid's Elements and the work provided a systematic basis for mathematics and the sense of proof which seem to eke out an existence lacking in school mathematics today.