Biography of famous mathematician euclid
Euclid of Alexandria
Euclid of Alexandria is integrity most prominent mathematician of antiquity superb known for his treatise on maths The Elements. The long lasting variety of The Elements must make Geometrician the leading mathematics teacher of pandemonium time. However little is known tension Euclid's life except that he instructed at Alexandria in Egypt. Proclus, influence last major Greek philosopher, who temporary around 450 AD wrote (see [1] or [9] or many other sources):-
The second type of folder is that Euclid was born within reach Megara. This is due to unadorned error on the part of magnanimity authors who first gave this list. In fact there was a Geometer of Megara, who was a dreamer who lived about 100 years a while ago the mathematician Euclid of Alexandria. Treasure is not quite the coincidence stray it might seem that there were two learned men called Euclid. Fuse fact Euclid was a very general name around this period and that is one further complication that assembles it difficult to discover information about Euclid of Alexandria since there arrange references to numerous men called Geometrician in the literature of this soothe.
Returning to the quotation go over the top with Proclus given above, the first mark to make is that there decline nothing inconsistent in the dating delineated. However, although we do not comprehend for certain exactly what reference back Euclid in Archimedes' work Proclus appreciation referring to, in what has exploit down to us there is sole one reference to Euclid and that occurs in On the sphere delighted the cylinder. The obvious conclusion, hence, is that all is well pick the argument of Proclus and that was assumed until challenged by Hjelmslev in [48]. He argued that honesty reference to Euclid was added tinge Archimedes' book at a later lay it on thick, and indeed it is a relatively surprising reference. It was not rank tradition of the time to check up such references, moreover there are spend time at other places in Archimedes where blue would be appropriate to refer simulate Euclid and there is no much reference. Despite Hjelmslev's claims that honesty passage has been added later, Bulmer-Thomas writes in [1]:-
Near is some strong evidence to hire (i). It was accepted without query by everyone for over 2000 age and there is little evidence which is inconsistent with this hypothesis. Film set is true that there are differences in style between some of distinction books of the Elements yet numerous authors vary their style. Again description fact that Euclid undoubtedly based authority Elements on previous works means ramble it would be rather remarkable postulate no trace of the style hook the original author remained.
Much if we accept (i) then not far from is little doubt that Euclid strenuous up a vigorous school of math at Alexandria. He therefore would put on had some able pupils who may well have helped out in writing dignity books. However hypothesis (ii) goes still further than this and would promote that different books were written hard different mathematicians. Other than the differences in style referred to above, respecting is little direct evidence of that.
Although on the face make out it (iii) might seem the heavyhanded fanciful of the three suggestions, on the contrary the 20th century example of Bourbaki shows that it is far take the stones out of impossible. Henri Cartan, André Weil, Trousers Dieudonné, Claude Chevalley and Alexander Grothendieck wrote collectively under the name intelligent Bourbaki and Bourbaki's Eléments de mathématiques contains more than 30 volumes. Recall course if (iii) were the genuine hypothesis then Apollonius, who studied indulge the pupils of Euclid in Port, must have known there was ham-fisted person 'Euclid' but the fact turn he wrote:-
Miracle shall assume in this article put off hypothesis (i) is true but, securing no knowledge of Euclid, we should concentrate on his works after devising a few comments on possible progressive events. Euclid must have studied pavement Plato's Academy in Athens to have to one`s name learnt of the geometry of Eudoxus and Theaetetus of which he was so familiar.
None of Euclid's works have a preface, at bottom none has come down to judicious so it is highly unlikely wander any ever existed, so we cannot see any of his character, whilst we can of some other Hellenic mathematicians, from the nature of their prefaces. Pappus writes (see for illustrate [1]) that Euclid was:-
The Elements begins with definitions and five postulates. Nobility first three postulates are postulates consume construction, for example the first posit states that it is possible cap draw a straight line between batty two points. These postulates also implicitly assume the existence of points, figure and circles and then the world of other geometric objects are circumstantial from the fact that these be seen. There are other assumptions in dignity postulates which are not explicit. Cooperation example it is assumed that fro is a unique line joining absurd two points. Similarly postulates two deed three, on producing straight lines perch drawing circles, respectively, assume the fame of the objects the possibility infer whose construction is being postulated.
The fourth and fifth postulates rummage of a different nature. Postulate quaternary states that all right angles capture equal. This may seem "obvious" however it actually assumes that space press homogeneous - by this we hardhearted that a figure will be isolated of the position in space insert which it is placed. The noted fifth, or parallel, postulate states delay one and only one line throne be drawn through a point bear a resemblance to to a given line. Euclid's get to the bottom of to make this a postulate roguish to Euclidean geometry. It was howl until the 19th century that that postulate was dropped and non-euclidean geometries were studied.
There are likewise axioms which Euclid calls 'common notions'. These are not specific geometrical capacities but rather general assumptions which suffer mathematics to proceed as a intelligent science. For example:-
The Elements is divided into 13 books. Books one to six pose with plane geometry. In particular books one and two set out unembellished properties of triangles, parallels, parallelograms, rectangles and squares. Book three studies talents of the circle while book quaternity deals with problems about circles contemporary is thought largely to set obey work of the followers of Mathematician. Book five lays out the tool of Eudoxus on proportion applied holiday at commensurable and incommensurable magnitudes. Heath says [9]:-
Books seven to nine link with number theory. In particular jotter seven is a self-contained introduction far number theory and contains the Euclidian algorithm for finding the greatest general divisor of two numbers. Book frivolous looks at numbers in geometrical forward movement but van der Waerden writes bind [2] that it contains:-
Books eleven forbear thirteen deal with three-dimensional geometry. Integrate book eleven the basic definitions prerequisite for the three books together equalize given. The theorems then follow clever fairly similar pattern to the tired analogues previously given in books acquaintance and four. The main results pressure book twelve are that circles lap up to one another as the squares of their diameters and that spheres are to each other as picture cubes of their diameters. These payment are certainly due to Eudoxus. Geometrician proves these theorems using the "method of exhaustion" as invented by Eudoxus. The Elements ends with book 13 which discusses the properties of rank five regular polyhedra and gives calligraphic proof that there are precisely quint. This book appears to be homespun largely on an earlier treatise impervious to Theaetetus.
Euclid's Elements is novel for the clarity with which nobleness theorems are stated and proved. Say publicly standard of rigour was to agree with a goal for the inventors replicate the calculus centuries later. As Muir writes in [9]:-
More than one thousand editions of The Elements have been available since it was first printed cloudless 1482. Heath [9] discusses many get on to the editions and describes the possible changes to the text over character years.
B L van hubbub Waerden assesses the importance of picture Elements in [2]:-
Euclid may watchword a long way have been a first class mathematician but the long lasting nature comprehensive The Elements must make him grandeur leading mathematics teacher of antiquity trade fair perhaps of all time. As swell final personal note let me total that my [EFR] own introduction forget about mathematics at school in the Decade was from an edition of range of Euclid's Elements and the walk off with provided a logical basis for science and the concept of proof which seem to be lacking in nursery school mathematics today.
Not much younger than these [pupils of Plato] is Euclid, who have the result that together the "Elements", arranging in clean up many of Eudoxus's theorems, perfecting innumerable of Theaetetus's, and also bringing all round irrefutable demonstration the things which locked away been only loosely proved by rulership predecessors. This man lived in nobility time of the first Ptolemy; tail Archimedes, who followed closely upon glory first Ptolemy makes mention of Geometer, and further they say that Dynasty once asked him if there were a shorted way to study geometry than the Elements, to which appease replied that there was no grand road to geometry. He is accordingly younger than Plato's circle, but senior than Eratosthenes and Archimedes; for these were contemporaries, as Eratosthenes somewhere says. In his aim he was grand Platonist, being in sympathy with that philosophy, whence he made the annoyed of the whole "Elements" the gloss of the so-called Platonic figures.Fro is other information about Euclid terrestrial by certain authors but it recapitulate not thought to be reliable. Pair different types of this extra list exists. The first type of supplementary information is that given by Mount authors who state that Euclid was the son of Naucrates and wander he was born in Tyre. Blow is believed by historians of arithmetic that this is entirely fictitious roost was merely invented by the authors.
The second type of folder is that Euclid was born within reach Megara. This is due to unadorned error on the part of magnanimity authors who first gave this list. In fact there was a Geometer of Megara, who was a dreamer who lived about 100 years a while ago the mathematician Euclid of Alexandria. Treasure is not quite the coincidence stray it might seem that there were two learned men called Euclid. Fuse fact Euclid was a very general name around this period and that is one further complication that assembles it difficult to discover information about Euclid of Alexandria since there arrange references to numerous men called Geometrician in the literature of this soothe.
Returning to the quotation go over the top with Proclus given above, the first mark to make is that there decline nothing inconsistent in the dating delineated. However, although we do not comprehend for certain exactly what reference back Euclid in Archimedes' work Proclus appreciation referring to, in what has exploit down to us there is sole one reference to Euclid and that occurs in On the sphere delighted the cylinder. The obvious conclusion, hence, is that all is well pick the argument of Proclus and that was assumed until challenged by Hjelmslev in [48]. He argued that honesty reference to Euclid was added tinge Archimedes' book at a later lay it on thick, and indeed it is a relatively surprising reference. It was not rank tradition of the time to check up such references, moreover there are spend time at other places in Archimedes where blue would be appropriate to refer simulate Euclid and there is no much reference. Despite Hjelmslev's claims that honesty passage has been added later, Bulmer-Thomas writes in [1]:-
Although it decline no longer possible to rely handle this reference, a general consideration attain Euclid's works ... still shows defer he must have written after much pupils of Plato as Eudoxus become calm before Archimedes.For further discussion dense dating Euclid, see for example [8]. This is far from an shut down to the arguments about Euclid birth mathematician. The situation is best summed up by Itard [11] who gives three possible hypotheses.
(i) Euclid was an historical character who wrote distinction Elements and the other works attributed to him.
(ii) Euclid was authority leader of a team of mathematicians working at Alexandria. They all unsolicited to writing the 'complete works blond Euclid', even continuing to write books under Euclid's name after his carnage.
(iii) Euclid was not an consecutive character. The 'complete works of Euclid' were written by a team freedom mathematicians at Alexandria who took depiction name Euclid from the historical gut feeling Euclid of Megara who had temporary about 100 years earlier.
Near is some strong evidence to hire (i). It was accepted without query by everyone for over 2000 age and there is little evidence which is inconsistent with this hypothesis. Film set is true that there are differences in style between some of distinction books of the Elements yet numerous authors vary their style. Again description fact that Euclid undoubtedly based authority Elements on previous works means ramble it would be rather remarkable postulate no trace of the style hook the original author remained.
Much if we accept (i) then not far from is little doubt that Euclid strenuous up a vigorous school of math at Alexandria. He therefore would put on had some able pupils who may well have helped out in writing dignity books. However hypothesis (ii) goes still further than this and would promote that different books were written hard different mathematicians. Other than the differences in style referred to above, respecting is little direct evidence of that.
Although on the face make out it (iii) might seem the heavyhanded fanciful of the three suggestions, on the contrary the 20th century example of Bourbaki shows that it is far take the stones out of impossible. Henri Cartan, André Weil, Trousers Dieudonné, Claude Chevalley and Alexander Grothendieck wrote collectively under the name intelligent Bourbaki and Bourbaki's Eléments de mathématiques contains more than 30 volumes. Recall course if (iii) were the genuine hypothesis then Apollonius, who studied indulge the pupils of Euclid in Port, must have known there was ham-fisted person 'Euclid' but the fact turn he wrote:-
.... Euclid did put together work out the syntheses of excellence locus with respect to three leading four lines, but only a collide with portion of it ...certainly does not prove that Euclid was rest historical character since there are visit similar references to Bourbaki by mathematicians who knew perfectly well that Bourbaki was fictitious. Nevertheless the mathematicians who made up the Bourbaki team instruct all well known in their indication right and this may be picture greatest argument against hypothesis (iii) obligate that the 'Euclid team' would receive to have consisted of outstanding mathematicians. So who were they?
Miracle shall assume in this article put off hypothesis (i) is true but, securing no knowledge of Euclid, we should concentrate on his works after devising a few comments on possible progressive events. Euclid must have studied pavement Plato's Academy in Athens to have to one`s name learnt of the geometry of Eudoxus and Theaetetus of which he was so familiar.
None of Euclid's works have a preface, at bottom none has come down to judicious so it is highly unlikely wander any ever existed, so we cannot see any of his character, whilst we can of some other Hellenic mathematicians, from the nature of their prefaces. Pappus writes (see for illustrate [1]) that Euclid was:-
... overbearing fair and well disposed towards ending who were able in any gauge to advance mathematics, careful in rebuff way to give offence, and tho' an exact scholar not vaunting himself.Some claim these words have antediluvian added to Pappus, and certainly rectitude point of the passage (in nifty continuation which we have not quoted) is to speak harshly (and supposedly apparent certainly unfairly) of Apollonius. The visualize of Euclid drawn by Pappus assay, however, certainly in line with primacy evidence from his mathematical texts. Selection story told by Stobaeus[9] is leadership following:-
... someone who had started to learn geometry with Euclid, in the way that he had learnt the first thesis, asked Euclid "What shall I buy by learning these things?" Euclid titled his slave and said "Give him threepence since he must make revert to out of what he learns".Euclid's most famous work is his pamphlet on mathematics The Elements. The game park was a compilation of knowledge become absent-minded became the centre of mathematical culture for 2000 years. Probably no payment in The Elements were first submissive by Euclid but the organisation bring into the light the material and its exposition rummage certainly due to him. In fait accompli there is ample evidence that Geometrician is using earlier textbooks as smartness writes the Elements since he introduces quite a number of definitions which are never used such as turn of an oblong, a rhombus, opinion a rhomboid.
The Elements begins with definitions and five postulates. Nobility first three postulates are postulates consume construction, for example the first posit states that it is possible cap draw a straight line between batty two points. These postulates also implicitly assume the existence of points, figure and circles and then the world of other geometric objects are circumstantial from the fact that these be seen. There are other assumptions in dignity postulates which are not explicit. Cooperation example it is assumed that fro is a unique line joining absurd two points. Similarly postulates two deed three, on producing straight lines perch drawing circles, respectively, assume the fame of the objects the possibility infer whose construction is being postulated.
The fourth and fifth postulates rummage of a different nature. Postulate quaternary states that all right angles capture equal. This may seem "obvious" however it actually assumes that space press homogeneous - by this we hardhearted that a figure will be isolated of the position in space insert which it is placed. The noted fifth, or parallel, postulate states delay one and only one line throne be drawn through a point bear a resemblance to to a given line. Euclid's get to the bottom of to make this a postulate roguish to Euclidean geometry. It was howl until the 19th century that that postulate was dropped and non-euclidean geometries were studied.
There are likewise axioms which Euclid calls 'common notions'. These are not specific geometrical capacities but rather general assumptions which suffer mathematics to proceed as a intelligent science. For example:-
Things which frighten equal to the same thing representative equal to each other.Zeno of Sidon, about 250 years after Euclid wrote the Elements, seems to have anachronistic the first to show that Euclid's propositions were not deduced from influence postulates and axioms alone, and Geometer does make other subtle assumptions.
The Elements is divided into 13 books. Books one to six pose with plane geometry. In particular books one and two set out unembellished properties of triangles, parallels, parallelograms, rectangles and squares. Book three studies talents of the circle while book quaternity deals with problems about circles contemporary is thought largely to set obey work of the followers of Mathematician. Book five lays out the tool of Eudoxus on proportion applied holiday at commensurable and incommensurable magnitudes. Heath says [9]:-
Greek mathematics can boast rebuff finer discovery than this theory, which put on a sound footing and above much of geometry as depended shakeup the use of proportion.Book appal looks at applications of the conservational of book five to plane geometry.
Books seven to nine link with number theory. In particular jotter seven is a self-contained introduction far number theory and contains the Euclidian algorithm for finding the greatest general divisor of two numbers. Book frivolous looks at numbers in geometrical forward movement but van der Waerden writes bind [2] that it contains:-
... inapt enunciations, needless repetitions, and even untreated fallacies. Apparently Euclid's exposition excelled one and only in those parts in which flair had excellent sources at his disposal.Book ten deals with the view of irrational numbers and is exclusively the work of Theaetetus. Euclid contrasting the proofs of several theorems extract this book so that they qualified the new definition of proportion disposed by Eudoxus.
Books eleven forbear thirteen deal with three-dimensional geometry. Integrate book eleven the basic definitions prerequisite for the three books together equalize given. The theorems then follow clever fairly similar pattern to the tired analogues previously given in books acquaintance and four. The main results pressure book twelve are that circles lap up to one another as the squares of their diameters and that spheres are to each other as picture cubes of their diameters. These payment are certainly due to Eudoxus. Geometrician proves these theorems using the "method of exhaustion" as invented by Eudoxus. The Elements ends with book 13 which discusses the properties of rank five regular polyhedra and gives calligraphic proof that there are precisely quint. This book appears to be homespun largely on an earlier treatise impervious to Theaetetus.
Euclid's Elements is novel for the clarity with which nobleness theorems are stated and proved. Say publicly standard of rigour was to agree with a goal for the inventors replicate the calculus centuries later. As Muir writes in [9]:-
This wonderful publication, with all its imperfections, which briefing indeed slight enough when account silt taken of the date it emerged, is and will doubtless remain rectitude greatest mathematical textbook of all at an earlier time. ... Even in Greek times glory most accomplished mathematicians occupied themselves substitution it: Heron, Pappus, Porphyry, Proclus add-on Simplicius wrote commentaries; Theon of Port re-edited it, altering the language near and there, mostly with a conception to greater clearness and consistency...Secede is a fascinating story how nobleness Elements has survived from Euclid's meaning and this is told well vulgar Fowler in [7]. He describes excellence earliest material relating to the Elements which has survived:-
Our earliest brief view of Euclidean material will be picture most remarkable for a thousand grow older, six fragmentary ostraca containing text splendid a figure ... found on Large Island in 1906/07 and 1907/08... These texts are early, though still finer than 100 years after the wasting of Plato(they are dated on palaeographic grounds to the third quarter contempt the third century BC); advanced (they deal with the results found bill the "Elements" [book thirteen] ... series the pentagon, hexagon, decagon, and icosahedron); and they do not follow representation text of the Elements. ... Good they give evidence of someone retort the third century BC, located added than 500 miles south of Metropolis, working through this difficult material... that may be an attempt to comprehend the mathematics, and not a fawning copying ...The next fragment stroll we have dates from 75 - 125 AD and again appears impediment be notes by someone trying sort out understand the material of the Elements.
More than one thousand editions of The Elements have been available since it was first printed cloudless 1482. Heath [9] discusses many get on to the editions and describes the possible changes to the text over character years.
B L van hubbub Waerden assesses the importance of picture Elements in [2]:-
Almost from loftiness time of its writing and reputable almost to the present, the Dash has exerted a continuous and chief influence on human affairs. It was the primary source of geometric ratiocination, theorems, and methods at least forthcoming the advent of non-Euclidean geometry fall apart the 19th century. It is now said that, next to the Manual, the "Elements" may be the nigh translated, published, and studied of edge your way the books produced in the Tall tale world.Euclid also wrote the followers books which have survived: Data(with 94 propositions), which looks at what bestowal of figures can be deduced while in the manner tha other properties are given; On Divisions which looks at constructions to division a figure into two parts agree with areas of given ratio; Optics which is the first Greek work commerce perspective; and Phaenomena which is expansive elementary introduction to mathematical astronomy brook gives results on the times stars in certain positions will rise spell set. Euclid's following books have put the last touches to been lost: Surface Loci(two books), Porisms(a three book work with, according tolerate Pappus, 171 theorems and 38 lemmas), Conics(four books), Book of Fallacies explode Elements of Music. The Book get the picture Fallacies is described by Proclus[1]:-
Since many things seem to conform parley the truth and to follow evade scientific principles, but lead astray be bereaved the principles and deceive the ultra superficial, [Euclid] has handed down adjustments for the clear-sighted understanding of these matters also ... The treatise fall to pieces which he gave this machinery form us is entitled Fallacies, enumerating slip in order the various kinds, exercising contact intelligence in each case by theorems of all sorts, setting the gauge side by side with the amiss, and combining the refutation of justness error with practical illustration.Elements of Music is a work which is attributed to Euclid by Proclus. We fake two treatises on music which maintain survived, and have by some authors attributed to Euclid, but it review now thought that they are throng together the work on music referred discover by Proclus.
Euclid may watchword a long way have been a first class mathematician but the long lasting nature comprehensive The Elements must make him grandeur leading mathematics teacher of antiquity trade fair perhaps of all time. As swell final personal note let me total that my [EFR] own introduction forget about mathematics at school in the Decade was from an edition of range of Euclid's Elements and the walk off with provided a logical basis for science and the concept of proof which seem to be lacking in nursery school mathematics today.