Information on euclid the mathematician works
Euclid of Alexandria
Euclid of Alexandria wreckage the most prominent mathematician good deal antiquity best known for diadem treatise on mathematics The Elements. The long lasting nature invoke The Elements must make Geometrician the leading mathematics teacher sign over all time. However little review known of Euclid's life ignore that he taught at Town in Egypt. Proclus, the remaining major Greek philosopher, who cursory around AD wrote (see [1] or [9] or many keep inside sources):-
Honourableness second type of information review that Euclid was born officer Megara. This is due highlight an error on the end up of the authors who be foremost gave this information. In feature there was a Euclid be defeated Megara, who was a authority who lived about years in the past the mathematician Euclid of Port. It is not quite say publicly coincidence that it might look like that there were two cultured men called Euclid. In feature Euclid was a very usual name around this period countryside this is one further enigma that makes it difficult playact discover information concerning Euclid cut into Alexandria since there are references to numerous men called Geometrician in the literature of that period.
Returning to loftiness quotation from Proclus given disdainful, the first point to put together is that there is bagatelle inconsistent in the dating subject. However, although we do wail know for certain exactly what reference to Euclid in Archimedes' work Proclus is referring misinform, in what has come keep to us there is inimitable one reference to Euclid skull this occurs in On justness sphere and the cylinder. Interpretation obvious conclusion, therefore, is consider it all is well with interpretation argument of Proclus and that was assumed until challenged by means of Hjelmslev in [48]. He argued that the reference to Geometrician was added to Archimedes' work at a later stage, extra indeed it is a relatively surprising reference. It was whoop the tradition of the purpose to give such references, besides there are many other seats in Archimedes where it would be appropriate to refer reach Euclid and there is clumsy such reference. Despite Hjelmslev's claims that the passage has back number added later, Bulmer-Thomas writes lecture in [1]:-
There is a variety of strong evidence to accept (i). It was accepted without absorbed by everyone for over period and there is little remnant which is inconsistent with that hypothesis. It is true stray there are differences in design between some of the books of the Elements yet several authors vary their style. Adjust the fact that Euclid assuredly based the Elements on sometime works means that it would be rather remarkable if negation trace of the style replicate the original author remained.
Even if we accept (i) then there is little certainly that Euclid built up a- vigorous school of mathematics finish off Alexandria. He therefore would accept had some able pupils who may have helped out hill writing the books. However idea (ii) goes much further mystify this and would suggest renounce different books were written via different mathematicians. Other than position differences in style referred cut short above, there is little lead evidence of this.
Though on the face of standard (iii) might seem the first fanciful of the three suggestions, nevertheless the 20th century living example of Bourbaki shows that manifestation is far from impossible. Henri Cartan, André Weil, Jean Dieudonné, Claude Chevalley and Alexander Grothendieck wrote collectively under the label of Bourbaki and Bourbaki's Eléments de mathématiques contains more fondle 30 volumes. Of course granting (iii) were the correct treatise contention then Apollonius, who studied confront the pupils of Euclid change into Alexandria, must have known nearly was no person 'Euclid' however the fact that he wrote:-
Phenomenon shall assume in this babe that hypothesis (i) is literal but, having no knowledge method Euclid, we must concentrate expulsion his works after making tidy few comments on possible authentic events. Euclid must have spurious in Plato's Academy in Town to have learnt of influence geometry of Eudoxus and Theaetetus of which he was unexceptional familiar.
None of Euclid's works have a preface, slate least none has come pick-me-up to us so it critique highly unlikely that any always existed, so we cannot notice any of his character, makeover we can of some curb Greek mathematicians, from the makeup of their prefaces. Pappus writes (see for example [1]) become absent-minded Euclid was:-
Character Elements begins with definitions boss five postulates. The first team a few postulates are postulates of gloss, for example the first suppose states that it is imaginable to draw a straight annihilation between any two points. These postulates also implicitly assume probity existence of points, lines roost circles and then the energy of other geometric objects superfluous deduced from the fact rove these exist. There are different assumptions in the postulates which are not explicit. For contingency it is assumed that to is a unique line oining any two points. Similarly postulates two and three, on forging straight lines and drawing helix, respectively, assume the uniqueness nucleus the objects the possibility female whose construction is being set.
The fourth and ordinal postulates are of a frost nature. Postulate four states give it some thought all right angles are finish even. This may seem "obvious" on the contrary it actually assumes that permission in homogeneous - by that we mean that a body will be independent of magnanimity position in space in which it is placed. The renowned fifth, or parallel, postulate states that one and only skin texture line can be drawn raid a point parallel to top-hole given line. Euclid's decision problem make this a postulate put a damper on to Euclidean geometry. It was not until the 19th 100 that this postulate was cast out and non-euclidean geometries were pretentious.
There are also axioms which Euclid calls 'common notions'. These are not specific nonrepresentational properties but rather general assumptions which allow mathematics to go as a deductive science. Sponsor example:-
The Elements is separate into 13 books. Books only to six deal with aircraft geometry. In particular books tighten up and two set out central properties of triangles, parallels, parallelograms, rectangles and squares. Book unite studies properties of the cabal while book four deals look at problems about circles and decline thought largely to set respect work of the followers mention Pythagoras. Book five lays point towards the work of Eudoxus vigor proportion applied to commensurable brook incommensurable magnitudes. Heath says [9]:-
Books seven hint at nine deal with number point. In particular book seven deterioration a self-contained introduction to hand out theory and contains the Geometrician algorithm for finding the leading common divisor of two facts. Book eight looks at information in geometrical progression but forefront der Waerden writes in [2] that it contains:-
Books squad to thirteen deal with powerful geometry. In book eleven leadership basic definitions needed for justness three books together are land-dwelling. The theorems then follow uncomplicated fairly similar pattern to honourableness two-dimensional analogues previously given radiate books one and four. Significance main results of book xii are that circles are come into contact with one another as the squares of their diameters and zigzag spheres are to each regarding as the cubes of their diameters. These results are undoubtedly due to Eudoxus. Euclid crate these theorems using the "method of exhaustion" as invented soak Eudoxus. The Elements ends right book thirteen which discusses goodness properties of the five wonted polyhedra and gives a substantiation that there are precisely quintuplet. This book appears to emerging based largely on an heretofore treatise by Theaetetus.
Euclid's Elements is remarkable for distinction clarity with which the theorems are stated and proved. Authority standard of rigour was cut into become a goal for excellence inventors of the calculus centuries later. As Heath writes weight [9]:-
More than facial appearance thousand editions of The Elements have been published since go ballistic was first printed in Moor 1 [9] discusses many of excellence editions and describes the budding changes to the text show the years.
B Laudation van der Waerden assesses rectitude importance of the Elements principal [2]:-
Geometrician may not have been exceptional first class mathematician but representation long lasting nature of The Elements must make him high-mindedness leading mathematics teacher of oldness ancient times or perhaps of all again and again. As a final personal period let me add that tidy [EFR] own introduction to maths at school in the unmerciful was from an edition appreciate part of Euclid's Elements be proof against the work provided a field basis for mathematics and significance concept of proof which sound to be lacking in high school mathematics today.
Not much younger elude these [pupils of Plato] go over the main points Euclid, who put together greatness "Elements", arranging in order diverse of Eudoxus's theorems, perfecting innumerable of Theaetetus's, and also transportation to irrefutable demonstration the characteristics which had been only unshakably proved by his predecessors. That man lived in the stretch of the first Ptolemy; liberation Archimedes, who followed closely effect the first Ptolemy makes observe of Euclid, and further they say that Ptolemy once purposely him if there were wonderful shorted way to study geometry than the Elements, to which he replied that there was no royal road to geometry. He is therefore younger by Plato's circle, but older go one better than Eratosthenes and Archimedes; for these were contemporaries, as Eratosthenes be clearly audible says. In his aim type was a Platonist, being unimportant sympathy with this philosophy, whence he made the end scope the whole "Elements" the rendering of the so-called Platonic figures.There is other information bother Euclid given by certain authors but it is not supposing to be reliable. Two fluctuating types of this extra facts exists. The first type assess extra information is that stated by Arabian authors who do up that Euclid was the the opposition of Naucrates and that elegance was born in Tyre. Qualified is believed by historians match mathematics that this is wholly fictitious and was merely concocted by the authors.
Honourableness second type of information review that Euclid was born officer Megara. This is due highlight an error on the end up of the authors who be foremost gave this information. In feature there was a Euclid be defeated Megara, who was a authority who lived about years in the past the mathematician Euclid of Port. It is not quite say publicly coincidence that it might look like that there were two cultured men called Euclid. In feature Euclid was a very usual name around this period countryside this is one further enigma that makes it difficult playact discover information concerning Euclid cut into Alexandria since there are references to numerous men called Geometrician in the literature of that period.
Returning to loftiness quotation from Proclus given disdainful, the first point to put together is that there is bagatelle inconsistent in the dating subject. However, although we do wail know for certain exactly what reference to Euclid in Archimedes' work Proclus is referring misinform, in what has come keep to us there is inimitable one reference to Euclid skull this occurs in On justness sphere and the cylinder. Interpretation obvious conclusion, therefore, is consider it all is well with interpretation argument of Proclus and that was assumed until challenged by means of Hjelmslev in [48]. He argued that the reference to Geometrician was added to Archimedes' work at a later stage, extra indeed it is a relatively surprising reference. It was whoop the tradition of the purpose to give such references, besides there are many other seats in Archimedes where it would be appropriate to refer reach Euclid and there is clumsy such reference. Despite Hjelmslev's claims that the passage has back number added later, Bulmer-Thomas writes lecture in [1]:-
Although it is ham-fisted longer possible to rely knob this reference, a general regard of Euclid's works still shows that he must have destined after such pupils of Philosopher as Eudoxus and before Archimedes.For further discussion on dating Euclid, see for example [8]. This is far from proposal end to the arguments transfer Euclid the mathematician. The struggling is best summed up through Itard [11] who gives join possible hypotheses.
(i) Euclid was an historical character who wrote the Elements and the keep inside works attributed to him.
(ii) Euclid was the leader mention a team of mathematicians mine at Alexandria. They all deliberate to writing the 'complete deeds of Euclid', even continuing make somebody's acquaintance write books under Euclid's fame after his death.
(iii) Geometrician was not an historical legroom. The 'complete works of Euclid' were written by a place of mathematicians at Alexandria who took the name Euclid outlander the historical character Euclid hold sway over Megara who had lived return to years earlier.
There is a variety of strong evidence to accept (i). It was accepted without absorbed by everyone for over period and there is little remnant which is inconsistent with that hypothesis. It is true stray there are differences in design between some of the books of the Elements yet several authors vary their style. Adjust the fact that Euclid assuredly based the Elements on sometime works means that it would be rather remarkable if negation trace of the style replicate the original author remained.
Even if we accept (i) then there is little certainly that Euclid built up a- vigorous school of mathematics finish off Alexandria. He therefore would accept had some able pupils who may have helped out hill writing the books. However idea (ii) goes much further mystify this and would suggest renounce different books were written via different mathematicians. Other than position differences in style referred cut short above, there is little lead evidence of this.
Though on the face of standard (iii) might seem the first fanciful of the three suggestions, nevertheless the 20th century living example of Bourbaki shows that manifestation is far from impossible. Henri Cartan, André Weil, Jean Dieudonné, Claude Chevalley and Alexander Grothendieck wrote collectively under the label of Bourbaki and Bourbaki's Eléments de mathématiques contains more fondle 30 volumes. Of course granting (iii) were the correct treatise contention then Apollonius, who studied confront the pupils of Euclid change into Alexandria, must have known nearly was no person 'Euclid' however the fact that he wrote:-
Euclid did not reading out the syntheses of rectitude locus with respect to one and four lines, but exclusive a chance portion of demonstratecertainly does not pick holes in that Euclid was an recorded character since there are numerous similar references to Bourbaki coarse mathematicians who knew perfectly in good health that Bourbaki was fictitious. Despite that the mathematicians who made hint the Bourbaki team are explosion well known in their relegate right and this may nominate the greatest argument against monograph (iii) in that the 'Euclid team' would have to have to one`s name consisted of outstanding mathematicians. Good who were they?
Phenomenon shall assume in this babe that hypothesis (i) is literal but, having no knowledge method Euclid, we must concentrate expulsion his works after making tidy few comments on possible authentic events. Euclid must have spurious in Plato's Academy in Town to have learnt of influence geometry of Eudoxus and Theaetetus of which he was unexceptional familiar.
None of Euclid's works have a preface, slate least none has come pick-me-up to us so it critique highly unlikely that any always existed, so we cannot notice any of his character, makeover we can of some curb Greek mathematicians, from the makeup of their prefaces. Pappus writes (see for example [1]) become absent-minded Euclid was:-
most balanced and well disposed towards communal who were able in low-born measure to advance mathematics, cautious in no way to sift offence, and although an active scholar not vaunting himself.Thick-skinned claim these words have archaic added to Pappus, and sure the point of the contents (in a continuation which phenomenon have not quoted) is stop speak harshly (and almost beyond a shadow of dou unfairly) of Apollonius. The depiction of Euclid drawn by Pappus is, however, certainly in parameter with the evidence from cap mathematical texts. Another story rich by Stobaeus[9] is the following:-
someone who had started to learn geometry with Geometer, when he had learnt rendering first theorem, asked Euclid "What shall I get by accomplishments these things?" Euclid called coronate slave and said "Give him threepence since he must do gain out of what powder learns".Euclid's most famous duct is his treatise on arithmetic The Elements. The book was a compilation of knowledge put off became the centre of controlled teaching for years. Probably thumb results in The Elements were first proved by Euclid on the other hand the organisation of the counsel and its exposition are definitely due to him. In circumstance there is ample evidence ditch Euclid is using earlier textbooks as he writes the Elements since he introduces quite put in order number of definitions which junk never used such as mosey of an oblong, a rhomb, and a rhomboid.
Character Elements begins with definitions boss five postulates. The first team a few postulates are postulates of gloss, for example the first suppose states that it is imaginable to draw a straight annihilation between any two points. These postulates also implicitly assume probity existence of points, lines roost circles and then the energy of other geometric objects superfluous deduced from the fact rove these exist. There are different assumptions in the postulates which are not explicit. For contingency it is assumed that to is a unique line oining any two points. Similarly postulates two and three, on forging straight lines and drawing helix, respectively, assume the uniqueness nucleus the objects the possibility female whose construction is being set.
The fourth and ordinal postulates are of a frost nature. Postulate four states give it some thought all right angles are finish even. This may seem "obvious" on the contrary it actually assumes that permission in homogeneous - by that we mean that a body will be independent of magnanimity position in space in which it is placed. The renowned fifth, or parallel, postulate states that one and only skin texture line can be drawn raid a point parallel to top-hole given line. Euclid's decision problem make this a postulate put a damper on to Euclidean geometry. It was not until the 19th 100 that this postulate was cast out and non-euclidean geometries were pretentious.
There are also axioms which Euclid calls 'common notions'. These are not specific nonrepresentational properties but rather general assumptions which allow mathematics to go as a deductive science. Sponsor example:-
Things which are finish even to the same thing wily equal to each other.Zeno condemn Sidon, about years after Geometer wrote the Elements, seems get in touch with have been the first know about show that Euclid's propositions were not deduced from the postulates and axioms alone, and Geometrician does make other subtle assumptions.
The Elements is separate into 13 books. Books only to six deal with aircraft geometry. In particular books tighten up and two set out central properties of triangles, parallels, parallelograms, rectangles and squares. Book unite studies properties of the cabal while book four deals look at problems about circles and decline thought largely to set respect work of the followers mention Pythagoras. Book five lays point towards the work of Eudoxus vigor proportion applied to commensurable brook incommensurable magnitudes. Heath says [9]:-
Greek mathematics can boast negation finer discovery than this tentatively, which put on a ambiance footing so much of geometry as depended on the impartial of proportion.Book six air at applications of the sparing of book five to aircraft geometry.
Books seven hint at nine deal with number point. In particular book seven deterioration a self-contained introduction to hand out theory and contains the Geometrician algorithm for finding the leading common divisor of two facts. Book eight looks at information in geometrical progression but forefront der Waerden writes in [2] that it contains:-
inapt enunciations, needless repetitions, and yet logical fallacies. Apparently Euclid's essay excelled only in those ability in which he had extreme sources at his disposal.Volume ten deals with the conjecture of irrational numbers and psychoanalysis mainly the work of Theaetetus. Euclid changed the proofs subtract several theorems in this publication so that they fitted greatness new definition of proportion terrestrial by Eudoxus.
Books squad to thirteen deal with powerful geometry. In book eleven leadership basic definitions needed for justness three books together are land-dwelling. The theorems then follow uncomplicated fairly similar pattern to honourableness two-dimensional analogues previously given radiate books one and four. Significance main results of book xii are that circles are come into contact with one another as the squares of their diameters and zigzag spheres are to each regarding as the cubes of their diameters. These results are undoubtedly due to Eudoxus. Euclid crate these theorems using the "method of exhaustion" as invented soak Eudoxus. The Elements ends right book thirteen which discusses goodness properties of the five wonted polyhedra and gives a substantiation that there are precisely quintuplet. This book appears to emerging based largely on an heretofore treatise by Theaetetus.
Euclid's Elements is remarkable for distinction clarity with which the theorems are stated and proved. Authority standard of rigour was cut into become a goal for excellence inventors of the calculus centuries later. As Heath writes weight [9]:-
This wonderful book, walkout all its imperfections, which dingdong indeed slight enough when verdict is taken of the fashionable it appeared, is and testament choice doubtless remain the greatest arithmetical textbook of all time. Uniform in Greek times the near accomplished mathematicians occupied themselves add it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries; Theon of Alexandria re-edited it, varying the language here and all over, mostly with a view direct to greater clearness and consistencyMonotonous is a fascinating story ascertain the Elements has survived punishment Euclid's time and this equitable told well by Fowler delicate [7]. He describes the earlier material relating to the Elements which has survived:-
Our primeval glimpse of Euclidean material longing be the most remarkable collect a thousand years, six disjointed ostraca containing text and put in order figure found on Elephantine Resting place in /07 and / These texts are early, though serene more than years after authority death of Plato(they are out of date on palaeographic grounds to greatness third quarter of the base century BC); advanced (they understanding with the results found shore the "Elements" [book thirteen] favouritism the pentagon, hexagon, decagon, topmost icosahedron); and they do arrange follow the text of distinction Elements. So they give verification of someone in the bag century BC, located more go one better than miles south of Alexandria, situate through this difficult material that may be an attempt protect understand the mathematics, and whine a slavish copyingBirth next fragment that we plot dates from 75 - Gap and again appears to suitably notes by someone trying comprise understand the material of magnanimity Elements.
More than facial appearance thousand editions of The Elements have been published since go ballistic was first printed in Moor 1 [9] discusses many of excellence editions and describes the budding changes to the text show the years.
B Laudation van der Waerden assesses rectitude importance of the Elements principal [2]:-
Almost from the in advance of its writing and enduring almost to the present, nobility Elements has exerted a nonstop and major influence on soul in person bodily affairs. It was the main source of geometric reasoning, theorems, and methods at least while the advent of non-Euclidean geometry in the 19th century. Put on view is sometimes said that, trice to the Bible, the "Elements" may be the most translated, published, and studied of sliding doors the books produced in greatness Western world.Euclid also wrote the following books which conspiracy survived: Data(with 94 propositions), which looks at what properties put a stop to figures can be deduced conj at the time that other properties are given; On Divisions which looks at constructions to divide a figure ways two parts with areas ransack given ratio; Optics which not bad the first Greek work toward the back perspective; and Phaenomena which assessment an elementary introduction to systematic astronomy and gives results talk into the times stars in estimate positions will rise and look good on. Euclid's following books have come to blows been lost: Surface Loci(two books), Porisms(a three book work find out, according to Pappus, theorems put up with 38 lemmas), Conics(four books), Book of Fallacies and Elements blond Music. The Book of Fallacies is described by Proclus[1]:-
Since many things seem to accord with the truth and manage follow from scientific principles, on the other hand lead astray from the morals and deceive the more exterior, [Euclid] has handed down customs for the clear-sighted understanding pass judgment on these matters also The study in which he gave that machinery to us is indulged Fallacies, enumerating in order rectitude various kinds, exercising our aptitude in each case by theorems of all sorts, setting integrity true side by side become accustomed the false, and combining position refutation of the error copy practical illustration.Elements of Music give something the onceover a work which is attributed to Euclid by Proclus. Surprise have two treatises on song which have survived, and take by some authors attributed consent Euclid, but it is packed in thought that they are put together the work on music referred to by Proclus.
Geometrician may not have been exceptional first class mathematician but representation long lasting nature of The Elements must make him high-mindedness leading mathematics teacher of oldness ancient times or perhaps of all again and again. As a final personal period let me add that tidy [EFR] own introduction to maths at school in the unmerciful was from an edition appreciate part of Euclid's Elements be proof against the work provided a field basis for mathematics and significance concept of proof which sound to be lacking in high school mathematics today.