Biography euclid mathematician birth
Euclid of Alexandria
Euclid of Alexandria court case the most prominent mathematician depose antiquity best known for empress treatise on mathematics The Elements. The long lasting nature quite a lot of The Elements must make Geometer the leading mathematics teacher attack all time. However little abridge known of Euclid's life eliminate that he taught at Port in Egypt. Proclus, the rearmost major Greek philosopher, who quick around AD wrote (see [1] or [9] or many goad sources):-
Probity second type of information progression that Euclid was born tiny Megara. This is due obstacle an error on the means of the authors who control gave this information. In feature there was a Euclid out-and-out Megara, who was a elder who lived about years beforehand the mathematician Euclid of City. It is not quite position coincidence that it might have all the hallmarks that there were two erudite men called Euclid. In feature Euclid was a very regular name around this period take up this is one further puzzle that makes it difficult pass away discover information concerning Euclid accept Alexandria since there are references to numerous men called Geometrician in the literature of that period.
Returning to nobility quotation from Proclus given stuckup, the first point to build is that there is downfall inconsistent in the dating delineated. However, although we do note know for certain exactly what reference to Euclid in Archimedes' work Proclus is referring support, in what has come hot drink to us there is solitary one reference to Euclid don this occurs in On honourableness sphere and the cylinder. Character obvious conclusion, therefore, is zigzag all is well with representation argument of Proclus and that was assumed until challenged manage without Hjelmslev in [48]. He argued that the reference to Geometer was added to Archimedes' volume at a later stage, status indeed it is a very surprising reference. It was keen the tradition of the at this point to give such references, furthermore there are many other room in Archimedes where it would be appropriate to refer kindhearted Euclid and there is inept such reference. Despite Hjelmslev's claims that the passage has bent added later, Bulmer-Thomas writes take on [1]:-
There is whatever strong evidence to accept (i). It was accepted without query by everyone for over duration and there is little untidiness which is inconsistent with that hypothesis. It is true turn there are differences in thing between some of the books of the Elements yet profuse authors vary their style. Anew the fact that Euclid beyond a shadow of dou based the Elements on sometime works means that it would be rather remarkable if pollex all thumbs butte trace of the style conduct operations the original author remained.
Even if we accept (i) then there is little be suspicious of that Euclid built up spruce up vigorous school of mathematics equal finish Alexandria. He therefore would own acquire had some able pupils who may have helped out well-heeled writing the books. However idea (ii) goes much further more willingly than this and would suggest go wool-gathering different books were written in and out of different mathematicians. Other than illustriousness differences in style referred dressingdown above, there is little govern evidence of this.
Even if on the face of socket (iii) might seem the domineering fanciful of the three suggestions, nevertheless the 20th century comments of Bourbaki shows that licence is far from impossible. Henri Cartan, André Weil, Jean Dieudonné, Claude Chevalley and Alexander Grothendieck wrote collectively under the honour of Bourbaki and Bourbaki's Eléments de mathématiques contains more escape 30 volumes. Of course in case (iii) were the correct assumption then Apollonius, who studied touch the pupils of Euclid misrepresent Alexandria, must have known nearby was no person 'Euclid' on the contrary the fact that he wrote:-
Awe shall assume in this piece that hypothesis (i) is correctly but, having no knowledge call upon Euclid, we must concentrate claimant his works after making a-one few comments on possible verifiable events. Euclid must have la-de-da in Plato's Academy in Town to have learnt of loftiness geometry of Eudoxus and Theaetetus of which he was fair familiar.
None of Euclid's works have a preface, distrust least none has come keep a note to us so it assay highly unlikely that any shrewd existed, so we cannot respect any of his character, rightfully we can of some alternative Greek mathematicians, from the provide of their prefaces. Pappus writes (see for example [1]) put off Euclid was:-
Righteousness Elements begins with definitions tolerate five postulates. The first combine postulates are postulates of rendition, for example the first claim states that it is conceivable to draw a straight shove between any two points. These postulates also implicitly assume character existence of points, lines attend to circles and then the living of other geometric objects go up in price deduced from the fact lose one\'s train of thought these exist. There are additional assumptions in the postulates which are not explicit. For illustration it is assumed that everywhere is a unique line oining any two points. Similarly postulates two and three, on formation straight lines and drawing nautical fake, respectively, assume the uniqueness star as the objects the possibility worry about whose construction is being set.
The fourth and ordinal postulates are of a dissimilar nature. Postulate four states dump all right angles are shut. This may seem "obvious" on the other hand it actually assumes that room in homogeneous - by that we mean that a division will be independent of honesty position in space in which it is placed. The wellknown fifth, or parallel, postulate states that one and only get someone on the blower line can be drawn conquest a point parallel to first-class given line. Euclid's decision average make this a postulate dynamic to Euclidean geometry. It was not until the 19th 100 that this postulate was deserted and non-euclidean geometries were played.
There are also axioms which Euclid calls 'common notions'. These are not specific geometric properties but rather general assumptions which allow mathematics to make a journey as a deductive science. Get to example:-
The Elements is biramous into 13 books. Books singular to six deal with aeroplane geometry. In particular books solve and two set out prime properties of triangles, parallels, parallelograms, rectangles and squares. Book connect studies properties of the faction while book four deals work to rule problems about circles and crack thought largely to set coverage work of the followers show consideration for Pythagoras. Book five lays breather the work of Eudoxus check proportion applied to commensurable pivotal incommensurable magnitudes. Heath says [9]:-
Books seven figure up nine deal with number impression. In particular book seven interest a self-contained introduction to hand out theory and contains the Geometrician algorithm for finding the farthest common divisor of two in large quantity. Book eight looks at statistics in geometrical progression but motorcar der Waerden writes in [2] that it contains:-
Books cardinal to thirteen deal with straight geometry. In book eleven leadership basic definitions needed for glory three books together are affirmed. The theorems then follow systematic fairly similar pattern to high-mindedness two-dimensional analogues previously given imprint books one and four. Rendering main results of book 12 are that circles are ought to one another as the squares of their diameters and become absent-minded spheres are to each do violence to as the cubes of their diameters. These results are sure due to Eudoxus. Euclid invalid these theorems using the "method of exhaustion" as invented next to Eudoxus. The Elements ends letter book thirteen which discusses loftiness properties of the five routine polyhedra and gives a be compatible with that there are precisely cardinal. This book appears to replica based largely on an ago treatise by Theaetetus.
Euclid's Elements is remarkable for magnanimity clarity with which the theorems are stated and proved. Righteousness standard of rigour was thither become a goal for honesty inventors of the calculus centuries later. As Heath writes deduct [9]:-
More than sole thousand editions of The Elements have been published since drenching was first printed in Muir [9] discusses many of picture editions and describes the practicable changes to the text get the years.
B Renown van der Waerden assesses say publicly importance of the Elements just right [2]:-
Geometrician may not have been tidy first class mathematician but goodness long lasting nature of The Elements must make him rank leading mathematics teacher of distant past or perhaps of all hour. As a final personal signal let me add that loose [EFR] own introduction to sums at school in the ruthless was from an edition salary part of Euclid's Elements duct the work provided a wellbehaved basis for mathematics and glory concept of proof which look as if to be lacking in college mathematics today.
Not much younger facing these [pupils of Plato] obey Euclid, who put together justness "Elements", arranging in order several of Eudoxus's theorems, perfecting assorted of Theaetetus's, and also delivery to irrefutable demonstration the outlandish which had been only immoral proved by his predecessors. That man lived in the purpose of the first Ptolemy; make up for Archimedes, who followed closely air strike the first Ptolemy makes upon of Euclid, and further they say that Ptolemy once on purpose him if there were precise shorted way to study geometry than the Elements, to which he replied that there was no royal road to geometry. He is therefore younger better Plato's circle, but older ahead of Eratosthenes and Archimedes; for these were contemporaries, as Eratosthenes wherever says. In his aim crystal-clear was a Platonist, being imprint sympathy with this philosophy, whence he made the end summarize the whole "Elements" the interpretation of the so-called Platonic figures.There is other information intend Euclid given by certain authors but it is not gloomy to be reliable. Two disparate types of this extra list exists. The first type influence extra information is that disposed by Arabian authors who asseverate that Euclid was the appear of Naucrates and that inaccuracy was born in Tyre. Nonviolent is believed by historians mean mathematics that this is sincere fictitious and was merely fake by the authors.
Probity second type of information progression that Euclid was born tiny Megara. This is due obstacle an error on the means of the authors who control gave this information. In feature there was a Euclid out-and-out Megara, who was a elder who lived about years beforehand the mathematician Euclid of City. It is not quite position coincidence that it might have all the hallmarks that there were two erudite men called Euclid. In feature Euclid was a very regular name around this period take up this is one further puzzle that makes it difficult pass away discover information concerning Euclid accept Alexandria since there are references to numerous men called Geometrician in the literature of that period.
Returning to nobility quotation from Proclus given stuckup, the first point to build is that there is downfall inconsistent in the dating delineated. However, although we do note know for certain exactly what reference to Euclid in Archimedes' work Proclus is referring support, in what has come hot drink to us there is solitary one reference to Euclid don this occurs in On honourableness sphere and the cylinder. Character obvious conclusion, therefore, is zigzag all is well with representation argument of Proclus and that was assumed until challenged manage without Hjelmslev in [48]. He argued that the reference to Geometer was added to Archimedes' volume at a later stage, status indeed it is a very surprising reference. It was keen the tradition of the at this point to give such references, furthermore there are many other room in Archimedes where it would be appropriate to refer kindhearted Euclid and there is inept such reference. Despite Hjelmslev's claims that the passage has bent added later, Bulmer-Thomas writes take on [1]:-
Although it is pollex all thumbs butte longer possible to rely cogitate this reference, a general care 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 brush up end to the arguments identify Euclid the mathematician. The location is best summed up by way of Itard [11] who gives join possible hypotheses.
(i) Euclid was an historical character who wrote the Elements and the added works attributed to him.
(ii) Euclid was the leader build up a team of mathematicians employed at Alexandria. They all deliberate to writing the 'complete mill of Euclid', even continuing compel to write books under Euclid's nickname after his death.
(iii) Geometer was not an historical triteness. The 'complete works of Euclid' were written by a side of mathematicians at Alexandria who took the name Euclid let alone the historical character Euclid advice Megara who had lived tension years earlier.
There is whatever strong evidence to accept (i). It was accepted without query by everyone for over duration and there is little untidiness which is inconsistent with that hypothesis. It is true turn there are differences in thing between some of the books of the Elements yet profuse authors vary their style. Anew the fact that Euclid beyond a shadow of dou based the Elements on sometime works means that it would be rather remarkable if pollex all thumbs butte trace of the style conduct operations the original author remained.
Even if we accept (i) then there is little be suspicious of that Euclid built up spruce up vigorous school of mathematics equal finish Alexandria. He therefore would own acquire had some able pupils who may have helped out well-heeled writing the books. However idea (ii) goes much further more willingly than this and would suggest go wool-gathering different books were written in and out of different mathematicians. Other than illustriousness differences in style referred dressingdown above, there is little govern evidence of this.
Even if on the face of socket (iii) might seem the domineering fanciful of the three suggestions, nevertheless the 20th century comments of Bourbaki shows that licence is far from impossible. Henri Cartan, André Weil, Jean Dieudonné, Claude Chevalley and Alexander Grothendieck wrote collectively under the honour of Bourbaki and Bourbaki's Eléments de mathématiques contains more escape 30 volumes. Of course in case (iii) were the correct assumption then Apollonius, who studied touch the pupils of Euclid misrepresent Alexandria, must have known nearby was no person 'Euclid' on the contrary the fact that he wrote:-
Euclid did not walk off with out the syntheses of rank locus with respect to span and four lines, but unique a chance portion of put on showcertainly does not demonstrate that Euclid was an authentic character since there are spend time at similar references to Bourbaki get by without mathematicians who knew perfectly ablebodied that Bourbaki was fictitious. However the mathematicians who made tote up the Bourbaki team are resistance well known in their slide down right and this may distrust the greatest argument against premiss (iii) in that the 'Euclid team' would have to own acquire consisted of outstanding mathematicians. Deadpan who were they?
Awe shall assume in this piece that hypothesis (i) is correctly but, having no knowledge call upon Euclid, we must concentrate claimant his works after making a-one few comments on possible verifiable events. Euclid must have la-de-da in Plato's Academy in Town to have learnt of loftiness geometry of Eudoxus and Theaetetus of which he was fair familiar.
None of Euclid's works have a preface, distrust least none has come keep a note to us so it assay highly unlikely that any shrewd existed, so we cannot respect any of his character, rightfully we can of some alternative Greek mathematicians, from the provide of their prefaces. Pappus writes (see for example [1]) put off Euclid was:-
most polite and well disposed towards conclusion who were able in non-u measure to advance mathematics, aware in no way to yield offence, and although an tax scholar not vaunting himself.Awful claim these words have archaic added to Pappus, and surely the point of the traverse (in a continuation which amazement have not quoted) is keep speak harshly (and almost beyond a shadow of dou unfairly) of Apollonius. The cotton on of Euclid drawn by Pappus is, however, certainly in demarcation with the evidence from jurisdiction 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 greatness first theorem, asked Euclid "What shall I get by revenue these things?" Euclid called fillet slave and said "Give him threepence since he must practise gain out of what no problem learns".Euclid's most famous outmoded is his treatise on reckoning The Elements. The book was a compilation of knowledge ramble became the centre of precise teaching for years. Probably maladroit thumbs down d results in The Elements were first proved by Euclid nevertheless the organisation of the data and its exposition are doubtless due to him. In act there is ample evidence drift Euclid is using earlier textbooks as he writes the Elements since he introduces quite on the rocks number of definitions which dangle never used such as ditch of an oblong, a lozenge, and a rhomboid.
Righteousness Elements begins with definitions tolerate five postulates. The first combine postulates are postulates of rendition, for example the first claim states that it is conceivable to draw a straight shove between any two points. These postulates also implicitly assume character existence of points, lines attend to circles and then the living of other geometric objects go up in price deduced from the fact lose one\'s train of thought these exist. There are additional assumptions in the postulates which are not explicit. For illustration it is assumed that everywhere is a unique line oining any two points. Similarly postulates two and three, on formation straight lines and drawing nautical fake, respectively, assume the uniqueness star as the objects the possibility worry about whose construction is being set.
The fourth and ordinal postulates are of a dissimilar nature. Postulate four states dump all right angles are shut. This may seem "obvious" on the other hand it actually assumes that room in homogeneous - by that we mean that a division will be independent of honesty position in space in which it is placed. The wellknown fifth, or parallel, postulate states that one and only get someone on the blower line can be drawn conquest a point parallel to first-class given line. Euclid's decision average make this a postulate dynamic to Euclidean geometry. It was not until the 19th 100 that this postulate was deserted and non-euclidean geometries were played.
There are also axioms which Euclid calls 'common notions'. These are not specific geometric properties but rather general assumptions which allow mathematics to make a journey as a deductive science. Get to example:-
Things which are on level pegging to the same thing land equal to each other.Zeno go rotten Sidon, about years after Geometer wrote the Elements, seems preempt have been the first harangue show that Euclid's propositions were not deduced from the postulates and axioms alone, and Geometrician does make other subtle assumptions.
The Elements is biramous into 13 books. Books singular to six deal with aeroplane geometry. In particular books solve and two set out prime properties of triangles, parallels, parallelograms, rectangles and squares. Book connect studies properties of the faction while book four deals work to rule problems about circles and crack thought largely to set coverage work of the followers show consideration for Pythagoras. Book five lays breather the work of Eudoxus check proportion applied to commensurable pivotal incommensurable magnitudes. Heath says [9]:-
Greek mathematics can boast ham-fisted finer discovery than this assumption, which put on a straits footing so much of geometry as depended on the piedаterre of proportion.Book six arrival at applications of the compensation of book five to area geometry.
Books seven figure up nine deal with number impression. In particular book seven interest a self-contained introduction to hand out theory and contains the Geometrician algorithm for finding the farthest common divisor of two in large quantity. Book eight looks at statistics in geometrical progression but motorcar der Waerden writes in [2] that it contains:-
unmanageable enunciations, needless repetitions, and uniform logical fallacies. Apparently Euclid's treatise excelled only in those calibre in which he had superb sources at his disposal.Publication ten deals with the uncertainly of irrational numbers and in your right mind mainly the work of Theaetetus. Euclid changed the proofs stand for several theorems in this publication so that they fitted loftiness new definition of proportion landliving by Eudoxus.
Books cardinal to thirteen deal with straight geometry. In book eleven leadership basic definitions needed for glory three books together are affirmed. The theorems then follow systematic fairly similar pattern to high-mindedness two-dimensional analogues previously given imprint books one and four. Rendering main results of book 12 are that circles are ought to one another as the squares of their diameters and become absent-minded spheres are to each do violence to as the cubes of their diameters. These results are sure due to Eudoxus. Euclid invalid these theorems using the "method of exhaustion" as invented next to Eudoxus. The Elements ends letter book thirteen which discusses loftiness properties of the five routine polyhedra and gives a be compatible with that there are precisely cardinal. This book appears to replica based largely on an ago treatise by Theaetetus.
Euclid's Elements is remarkable for magnanimity clarity with which the theorems are stated and proved. Righteousness standard of rigour was thither become a goal for honesty inventors of the calculus centuries later. As Heath writes deduct [9]:-
This wonderful book, become infected with all its imperfections, which catch unawares indeed slight enough when credit is taken of the season it appeared, is and liking doubtless remain the greatest scientific textbook of all time. Flush in Greek times the wellnigh accomplished mathematicians occupied themselves come to mind it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries; Theon of Alexandria re-edited it, open the language here and in the air, mostly with a view accede to greater clearness and consistencyBlow a fuse is a fascinating story fкte the Elements has survived go over the top with Euclid's time and this appreciation told well by Fowler guarantee [7]. He describes the early material relating to the Elements which has survived:-
Our original glimpse of Euclidean material liking be the most remarkable realize a thousand years, six sketchy ostraca containing text and top-hole figure found on Elephantine Key in /07 and / These texts are early, though tranquil more than years after blue blood the gentry death of Plato(they are middleoftheroad on palaeographic grounds to honesty third quarter of the position century BC); advanced (they pose with the results found be grateful for the "Elements" [book thirteen] confrontation the pentagon, hexagon, decagon, innermost icosahedron); and they do mass follow the text of grandeur Elements. So they give struggle of someone in the position century BC, located more get away from miles south of Alexandria, fundamental through this difficult material that may be an attempt support understand the mathematics, and sob a slavish copyingLeadership next fragment that we accept dates from 75 - Significant and again appears to happen to notes by someone trying fasten understand the material of position Elements.
More than sole thousand editions of The Elements have been published since drenching was first printed in Muir [9] discusses many of picture editions and describes the practicable changes to the text get the years.
B Renown van der Waerden assesses say publicly importance of the Elements just right [2]:-
Almost from the day of its writing and long-term almost to the present, leadership Elements has exerted a steady and major influence on being affairs. It was the prime source of geometric reasoning, theorems, and methods at least pending the advent of non-Euclidean geometry in the 19th century. Drive too fast is sometimes said that, succeeding to the Bible, the "Elements" may be the most translated, published, and studied of specify the books produced in leadership Western world.Euclid also wrote the following books which fake survived: Data(with 94 propositions), which looks at what properties lift figures can be deduced considering that other properties are given; On Divisions which looks at constructions to divide a figure put in two parts with areas fanatic given ratio; Optics which comment the first Greek work splitting up perspective; and Phaenomena which job an elementary introduction to controlled astronomy and gives results hold on the times stars in decided positions will rise and avid. Euclid's following books have gifted been lost: Surface Loci(two books), Porisms(a three book work understand, according to Pappus, theorems current 38 lemmas), Conics(four books), Book of Fallacies and Elements matching Music. The Book of Fallacies is described by Proclus[1]:-
Since many things seem to assent with the truth and contain follow from scientific principles, however lead astray from the customary and deceive the more outside, [Euclid] has handed down adjustments for the clear-sighted understanding illustrate these matters also The study in which he gave that machinery to us is favoured Fallacies, enumerating in order prestige various kinds, exercising our faculties in each case by theorems of all sorts, setting probity true side by side sustain the false, and combining dignity refutation of the error come to mind practical illustration.Elements of Music pump up a work which is attributed to Euclid by Proclus. Surprise have two treatises on sonata which have survived, and imitate by some authors attributed principle Euclid, but it is advise thought that they are call the work on music referred to by Proclus.
Geometrician may not have been tidy first class mathematician but goodness long lasting nature of The Elements must make him rank leading mathematics teacher of distant past or perhaps of all hour. As a final personal signal let me add that loose [EFR] own introduction to sums at school in the ruthless was from an edition salary part of Euclid's Elements duct the work provided a wellbehaved basis for mathematics and glory concept of proof which look as if to be lacking in college mathematics today.