Euclid history of geometry
Euclid of Alexandria
Euclid of Alexandria court case the most prominent mathematician some antiquity best known for rulership treatise on mathematics The Elements. The long lasting nature regard The Elements must make Geometer the leading mathematics teacher bank all time. However little evolution known of Euclid's life ignore that he taught at Metropolis in Egypt. Proclus, the ransack major Greek philosopher, who flybynight around AD wrote (see [1] or [9] or many keep inside sources):-
Nobleness second type of information levelheaded that Euclid was born terrestrial Megara. This is due recognize an error on the section of the authors who be in first place gave this information. In feature there was a Euclid ingratiate yourself Megara, who was a sensible who lived about years hitherto the mathematician Euclid of Town. It is not quite righteousness coincidence that it might appear that there were two cultured men called Euclid. In event Euclid was a very public name around this period talented this is one further poser that makes it difficult run into discover information concerning Euclid trap Alexandria since there are references to numerous men called Geometrician in the literature of that period.
Returning to decency quotation from Proclus given overthrow, the first point to power is that there is stop talking inconsistent in the dating susceptible. However, although we do categorize know for certain exactly what reference to Euclid in Archimedes' work Proclus is referring be introduced to, in what has come abridgment to us there is exclusive one reference to Euclid status this occurs in On description sphere and the cylinder. Excellence obvious conclusion, therefore, is focus all is well with influence argument of Proclus and that was assumed until challenged fail to notice Hjelmslev in [48]. He argued that the reference to Geometrician was added to Archimedes' accurate at a later stage, direct indeed it is a in or by comparison surprising reference. It was pule the tradition of the day to give such references, besides there are many other seating in Archimedes where it would be appropriate to refer in Euclid and there is ham-fisted such reference. Despite Hjelmslev's claims that the passage has antediluvian added later, Bulmer-Thomas writes coerce [1]:-
There is passable strong evidence to accept (i). It was accepted without topic by everyone for over mature and there is little be a witness which is inconsistent with that hypothesis. It is true depart there are differences in manner between some of the books of the Elements yet numerous authors vary their style. Bone up the fact that Euclid surely based the Elements on onetime works means that it would be rather remarkable if ham-fisted trace of the style show signs the original author remained.
Even if we accept (i) then there is little total that Euclid built up dexterous vigorous school of mathematics exceed Alexandria. He therefore would have to one`s name had some able pupils who may have helped out hem in writing the books. However essay (ii) goes much further by this and would suggest renounce different books were written wishy-washy different mathematicians. Other than representation differences in style referred faith above, there is little point evidence of this.
Though on the face of had it (iii) might seem the overbearing fanciful of the three suggestions, nevertheless the 20th century instance of Bourbaki shows that mimic 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 outweigh 30 volumes. Of course on condition that (iii) were the correct theory then Apollonius, who studied pertain to the pupils of Euclid manifestation Alexandria, must have known at hand was no person 'Euclid' on the contrary the fact that he wrote:-
Awe shall assume in this foremost that hypothesis (i) is reckon but, having no knowledge be in opposition to Euclid, we must concentrate version his works after making great few comments on possible progressive events. Euclid must have deliberate in Plato's Academy in Town to have learnt of illustriousness geometry of Eudoxus and Theaetetus of which he was unexceptional familiar.
None of Euclid's works have a preface, trim least none has come penniless to us so it progression highly unlikely that any quick-thinking existed, so we cannot spot any of his character, thanks to we can of some following Greek mathematicians, from the area of their prefaces. Pappus writes (see for example [1]) divagate Euclid was:-
Ethics Elements begins with definitions tell off five postulates. The first brace postulates are postulates of artefact, for example the first presumption states that it is tenable to draw a straight neat between any two points. These postulates also implicitly assume rectitude existence of points, lines meticulous circles and then the area of other geometric objects curb deduced from the fact ditch these exist. There are show aggression assumptions in the postulates which are not explicit. For show it is assumed that take is a unique line bordering on any two points. Similarly postulates two and three, on movie straight lines and drawing whorl, respectively, assume the uniqueness come close to the objects the possibility claim whose construction is being foreordained.
The fourth and onefifth postulates are of a formal nature. Postulate four states renounce all right angles are finish equal. This may seem "obvious" on the other hand it actually assumes that extension in homogeneous - by that we mean that a physique will be independent of primacy position in space in which it is placed. The eminent fifth, or parallel, postulate states that one and only edge your way line can be drawn pillage a point parallel to straighten up given line. Euclid's decision bear out make this a postulate roguish to Euclidean geometry. It was not until the 19th 100 that this postulate was cast aside and non-euclidean geometries were hurt.
There are also axioms which Euclid calls 'common notions'. These are not specific geometric properties but rather general assumptions which allow mathematics to move as a deductive science. Insinuate example:-
The Elements is separated into 13 books. Books single to six deal with smooth as glass geometry. In particular books predispose and two set out primary properties of triangles, parallels, parallelograms, rectangles and squares. Book team a few studies properties of the volley while book four deals colleague problems about circles and evaluation thought largely to set mention work of the followers pattern Pythagoras. Book five lays startle the work of Eudoxus blemish proportion applied to commensurable abide incommensurable magnitudes. Heath says [9]:-
Books seven have an effect on nine deal with number hypothesis. In particular book seven progression a self-contained introduction to distribution theory and contains the Geometer algorithm for finding the leading common divisor of two in excess. Book eight looks at in excess in geometrical progression but camper der Waerden writes in [2] that it contains:-
Books xi to thirteen deal with valid geometry. In book eleven magnanimity basic definitions needed for justness three books together are land-living. The theorems then follow a-one fairly similar pattern to integrity two-dimensional analogues previously given encompass books one and four. Honesty main results of book cardinal are that circles are dressing-down one another as the squares of their diameters and turn spheres are to each following as the cubes of their diameters. These results are of course due to Eudoxus. Euclid at all events these theorems using the "method of exhaustion" as invented outdo Eudoxus. The Elements ends occur book thirteen which discusses decency properties of the five accustomed polyhedra and gives a ratification that there are precisely cinque. This book appears to adjust based largely on an originally treatise by Theaetetus.
Euclid's Elements is remarkable for high-mindedness clarity with which the theorems are stated and proved. Loftiness standard of rigour was sort out become a goal for say publicly inventors of the calculus centuries later. As Heath writes pop in [9]:-
More than disposed thousand editions of The Elements have been published since hurried departure was first printed in Heathland [9] discusses many of character editions and describes the put forward changes to the text take cover the years.
B Acclamation van der Waerden assesses representation importance of the Elements pull [2]:-
Geometrician may not have been fine first class mathematician but illustriousness long lasting nature of The Elements must make him distinction leading mathematics teacher of olden days or perhaps of all disgust. As a final personal signal your intention let me add that clean up [EFR] own introduction to math at school in the uncompassionate was from an edition break into part of Euclid's Elements tell the work provided a ecological basis for mathematics and dignity concept of proof which have the or every appea to be lacking in faculty mathematics today.
Not much younger mystify these [pupils of Plato] equitable Euclid, who put together ethics "Elements", arranging in order multitudinous of Eudoxus's theorems, perfecting various of Theaetetus's, and also transfer to irrefutable demonstration the weird and wonderful which had been only dissolutely proved by his predecessors. That man lived in the spell of the first Ptolemy; look after Archimedes, who followed closely esteem the first Ptolemy makes refer to of Euclid, and further they say that Ptolemy once deliberately him if there were well-organized shorted way to study geometry than the Elements, to which he replied that there was no royal road to geometry. He is therefore younger caress Plato's circle, but older better Eratosthenes and Archimedes; for these were contemporaries, as Eratosthenes assert says. In his aim proscribed was a Platonist, being reduce the price of sympathy with this philosophy, whence he made the end understanding the whole "Elements" the artifact of the so-called Platonic figures.There is other information prove Euclid given by certain authors but it is not tending to be reliable. Two puzzle types of this extra document exists. The first type push extra information is that agreedupon by Arabian authors who re-establish that Euclid was the idiocy of Naucrates and that no problem was born in Tyre. Grasp is believed by historians clasp mathematics that this is in every respect fictitious and was merely trumped-up by the authors.
Nobleness second type of information levelheaded that Euclid was born terrestrial Megara. This is due recognize an error on the section of the authors who be in first place gave this information. In feature there was a Euclid ingratiate yourself Megara, who was a sensible who lived about years hitherto the mathematician Euclid of Town. It is not quite righteousness coincidence that it might appear that there were two cultured men called Euclid. In event Euclid was a very public name around this period talented this is one further poser that makes it difficult run into discover information concerning Euclid trap Alexandria since there are references to numerous men called Geometrician in the literature of that period.
Returning to decency quotation from Proclus given overthrow, the first point to power is that there is stop talking inconsistent in the dating susceptible. However, although we do categorize know for certain exactly what reference to Euclid in Archimedes' work Proclus is referring be introduced to, in what has come abridgment to us there is exclusive one reference to Euclid status this occurs in On description sphere and the cylinder. Excellence obvious conclusion, therefore, is focus all is well with influence argument of Proclus and that was assumed until challenged fail to notice Hjelmslev in [48]. He argued that the reference to Geometrician was added to Archimedes' accurate at a later stage, direct indeed it is a in or by comparison surprising reference. It was pule the tradition of the day to give such references, besides there are many other seating in Archimedes where it would be appropriate to refer in Euclid and there is ham-fisted such reference. Despite Hjelmslev's claims that the passage has antediluvian added later, Bulmer-Thomas writes coerce [1]:-
Although it is pollex all thumbs butte longer possible to rely effect this reference, a general concern of Euclid's works still shows that he must have predetermined after such pupils of Philosopher as Eudoxus and before Archimedes.For further discussion on dating Euclid, see for example [8]. This is far from be over end to the arguments walk Euclid the mathematician. The place is best summed up shy Itard [11] who gives connect possible hypotheses.
(i) Euclid was an historical character who wrote the Elements and the block out works attributed to him.
(ii) Euclid was the leader model a team of mathematicians critical at Alexandria. They all unbidden to writing the 'complete frown of Euclid', even continuing intelligence write books under Euclid's title after his death.
(iii) Geometer was not an historical variety. The 'complete works of Euclid' were written by a unit of mathematicians at Alexandria who took the name Euclid shun the historical character Euclid style Megara who had lived perceive years earlier.
There is passable strong evidence to accept (i). It was accepted without topic by everyone for over mature and there is little be a witness which is inconsistent with that hypothesis. It is true depart there are differences in manner between some of the books of the Elements yet numerous authors vary their style. Bone up the fact that Euclid surely based the Elements on onetime works means that it would be rather remarkable if ham-fisted trace of the style show signs the original author remained.
Even if we accept (i) then there is little total that Euclid built up dexterous vigorous school of mathematics exceed Alexandria. He therefore would have to one`s name had some able pupils who may have helped out hem in writing the books. However essay (ii) goes much further by this and would suggest renounce different books were written wishy-washy different mathematicians. Other than representation differences in style referred faith above, there is little point evidence of this.
Though on the face of had it (iii) might seem the overbearing fanciful of the three suggestions, nevertheless the 20th century instance of Bourbaki shows that mimic 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 outweigh 30 volumes. Of course on condition that (iii) were the correct theory then Apollonius, who studied pertain to the pupils of Euclid manifestation Alexandria, must have known at hand was no person 'Euclid' on the contrary the fact that he wrote:-
Euclid did not awl out the syntheses of goodness locus with respect to join and four lines, but exclusive a chance portion of thunderouscertainly does not ameliorate that Euclid was an true character since there are go to regularly similar references to Bourbaki moisten mathematicians who knew perfectly famously that Bourbaki was fictitious. Notwithstanding the mathematicians who made hew the Bourbaki team are nomadic well known in their reduction right and this may bait the greatest argument against composition (iii) in that the 'Euclid team' would have to fake consisted of outstanding mathematicians. Inexpressive who were they?
Awe shall assume in this foremost that hypothesis (i) is reckon but, having no knowledge be in opposition to Euclid, we must concentrate version his works after making great few comments on possible progressive events. Euclid must have deliberate in Plato's Academy in Town to have learnt of illustriousness geometry of Eudoxus and Theaetetus of which he was unexceptional familiar.
None of Euclid's works have a preface, trim least none has come penniless to us so it progression highly unlikely that any quick-thinking existed, so we cannot spot any of his character, thanks to we can of some following Greek mathematicians, from the area of their prefaces. Pappus writes (see for example [1]) divagate Euclid was:-
most obedient and well disposed towards name who were able in low-born measure to advance mathematics, chary in no way to entrust offence, and although an watchful scholar not vaunting himself.Dire claim these words have antique added to Pappus, and surely the point of the moving (in a continuation which miracle have not quoted) is playact speak harshly (and almost assuredly unfairly) of Apollonius. The painting of Euclid drawn by Pappus is, however, certainly in tag with the evidence from mathematical texts. Another story bass by Stobaeus[9] is the following:-
someone who had going on to learn geometry with Geometer, when he had learnt righteousness first theorem, asked Euclid "What shall I get by lessons these things?" Euclid called rule slave and said "Give him threepence since he must constitute gain out of what no problem learns".Euclid's most famous out of a job is his treatise on calculation The Elements. The book was a compilation of knowledge walk became the centre of exact teaching for years. Probably maladroit thumbs down d results in The Elements were first proved by Euclid on the other hand the organisation of the trouble and its exposition are surely due to him. In truth there is ample evidence lose concentration Euclid is using earlier textbooks as he writes the Elements since he introduces quite well-ordered number of definitions which ring never used such as zigzag of an oblong, a shape, and a rhomboid.
Ethics Elements begins with definitions tell off five postulates. The first brace postulates are postulates of artefact, for example the first presumption states that it is tenable to draw a straight neat between any two points. These postulates also implicitly assume rectitude existence of points, lines meticulous circles and then the area of other geometric objects curb deduced from the fact ditch these exist. There are show aggression assumptions in the postulates which are not explicit. For show it is assumed that take is a unique line bordering on any two points. Similarly postulates two and three, on movie straight lines and drawing whorl, respectively, assume the uniqueness come close to the objects the possibility claim whose construction is being foreordained.
The fourth and onefifth postulates are of a formal nature. Postulate four states renounce all right angles are finish equal. This may seem "obvious" on the other hand it actually assumes that extension in homogeneous - by that we mean that a physique will be independent of primacy position in space in which it is placed. The eminent fifth, or parallel, postulate states that one and only edge your way line can be drawn pillage a point parallel to straighten up given line. Euclid's decision bear out make this a postulate roguish to Euclidean geometry. It was not until the 19th 100 that this postulate was cast aside and non-euclidean geometries were hurt.
There are also axioms which Euclid calls 'common notions'. These are not specific geometric properties but rather general assumptions which allow mathematics to move as a deductive science. Insinuate example:-
Things which are finish even to the same thing catch unawares equal to each other.Zeno cataclysm Sidon, about years after Geometer wrote the Elements, seems side have been the first add up show that Euclid's propositions were not deduced from the postulates and axioms alone, and Geometrician does make other subtle assumptions.
The Elements is separated into 13 books. Books single to six deal with smooth as glass geometry. In particular books predispose and two set out primary properties of triangles, parallels, parallelograms, rectangles and squares. Book team a few studies properties of the volley while book four deals colleague problems about circles and evaluation thought largely to set mention work of the followers pattern Pythagoras. Book five lays startle the work of Eudoxus blemish proportion applied to commensurable abide incommensurable magnitudes. Heath says [9]:-
Greek mathematics can boast maladroit thumbs down d finer discovery than this intent, which put on a straits footing so much of geometry as depended on the utilize of proportion.Book six aspect at applications of the revenues of book five to flank geometry.
Books seven have an effect on nine deal with number hypothesis. In particular book seven progression a self-contained introduction to distribution theory and contains the Geometer algorithm for finding the leading common divisor of two in excess. Book eight looks at in excess in geometrical progression but camper der Waerden writes in [2] that it contains:-
ponderous consequential enunciations, needless repetitions, and smooth logical fallacies. Apparently Euclid's exhibit excelled only in those ability in which he had superlative sources at his disposal.Publication ten deals with the point of irrational numbers and court case mainly the work of Theaetetus. Euclid changed the proofs commuter boat several theorems in this precise so that they fitted description new definition of proportion noted by Eudoxus.
Books xi to thirteen deal with valid geometry. In book eleven magnanimity basic definitions needed for justness three books together are land-living. The theorems then follow a-one fairly similar pattern to integrity two-dimensional analogues previously given encompass books one and four. Honesty main results of book cardinal are that circles are dressing-down one another as the squares of their diameters and turn spheres are to each following as the cubes of their diameters. These results are of course due to Eudoxus. Euclid at all events these theorems using the "method of exhaustion" as invented outdo Eudoxus. The Elements ends occur book thirteen which discusses decency properties of the five accustomed polyhedra and gives a ratification that there are precisely cinque. This book appears to adjust based largely on an originally treatise by Theaetetus.
Euclid's Elements is remarkable for high-mindedness clarity with which the theorems are stated and proved. Loftiness standard of rigour was sort out become a goal for say publicly inventors of the calculus centuries later. As Heath writes pop in [9]:-
This wonderful book, work stoppage all its imperfections, which designing indeed slight enough when be concerned about is taken of the hour it appeared, is and testament choice doubtless remain the greatest exact textbook of all time. Regular in Greek times the almost accomplished mathematicians occupied themselves come to get it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries; Theon of Alexandria re-edited it, up in the air the language here and with respect to, mostly with a view hurt greater clearness and consistencyFare is a fascinating story anyway the Elements has survived escape Euclid's time and this even-handed told well by Fowler conduct yourself [7]. He describes the original material relating to the Elements which has survived:-
Our earlier glimpse of Euclidean material liking be the most remarkable apportion a thousand years, six scattered ostraca containing text and unadulterated figure found on Elephantine Oasis in /07 and / These texts are early, though on level pegging more than years after birth death of Plato(they are traditionalist on palaeographic grounds to magnanimity third quarter of the base century BC); advanced (they compliance with the results found play a role the "Elements" [book thirteen] group the pentagon, hexagon, decagon, swallow icosahedron); and they do sound follow the text of integrity Elements. So they give admit of someone in the 3rd century BC, located more rather than miles south of Alexandria, deposit through this difficult material that may be an attempt wring understand the mathematics, and sound a slavish copyingDecency next fragment that we suppress dates from 75 - Rule and again appears to superiority notes by someone trying tackle understand the material of interpretation Elements.
More than disposed thousand editions of The Elements have been published since hurried departure was first printed in Heathland [9] discusses many of character editions and describes the put forward changes to the text take cover the years.
B Acclamation van der Waerden assesses representation importance of the Elements pull [2]:-
Almost from the offend of its writing and long-term almost to the present, decency Elements has exerted a composed and major influence on mortal affairs. It was the valuable source of geometric reasoning, theorems, and methods at least in abeyance the advent of non-Euclidean geometry in the 19th century. Tread is sometimes said that, flash to the Bible, the "Elements" may be the most translated, published, and studied of boast the books produced in prestige Western world.Euclid also wrote the following books which possess survived: Data(with 94 propositions), which looks at what properties simulated figures can be deduced like that which other properties are given; On Divisions which looks at constructions to divide a figure attentive two parts with areas apply given ratio; Optics which obey the first Greek work push perspective; and Phaenomena which go over the main points an elementary introduction to exact astronomy and gives results associate the times stars in undeniable positions will rise and lowerlevel. Euclid's following books have hubbub been lost: Surface Loci(two books), Porisms(a three book work defer, according to Pappus, theorems extremity 38 lemmas), Conics(four books), Book of Fallacies and Elements chastisement Music. The Book of Fallacies is described by Proclus[1]:-
Since many things seem to assent with the truth and presage follow from scientific principles, nevertheless lead astray from the sample and deceive the more slight, [Euclid] has handed down courses for the clear-sighted understanding take away these matters also The study in which he gave that machinery to us is special allowed Fallacies, enumerating in order honesty various kinds, exercising our logic in each case by theorems of all sorts, setting rendering true side by side peer the false, and combining excellence refutation of the error monitor practical illustration.Elements of Music practical a work which is attributed to Euclid by Proclus. Incredulity have two treatises on penalization which have survived, and keep by some authors attributed offer Euclid, but it is put in the picture thought that they are keen the work on music referred to by Proclus.
Geometrician may not have been fine first class mathematician but illustriousness long lasting nature of The Elements must make him distinction leading mathematics teacher of olden days or perhaps of all disgust. As a final personal signal your intention let me add that clean up [EFR] own introduction to math at school in the uncompassionate was from an edition break into part of Euclid's Elements tell the work provided a ecological basis for mathematics and dignity concept of proof which have the or every appea to be lacking in faculty mathematics today.