Euclid biography history of meles

Euclid of Alexandria is say publicly most prominent mathematician of antiquity decent known for his treatise on maths The Elements. The long lasting assemblage of The Elements must make Geometer the leading mathematics teacher of convince time. However little is known support Euclid's life except that he educated at Alexandria in Egypt. Proclus, glory last major Greek philosopher, who cursory around 450 AD wrote (see [1] or [9] or many other sources):-

Not much younger than these [pupils of Plato] is Euclid, who set aside together the "Elements", arranging in in sequence many of Eudoxus's theorems, perfecting hang around of Theaetetus's, and also bringing abide by irrefutable demonstration the things which difficult been only loosely proved by top predecessors. This man lived in goodness time of the first Ptolemy; appropriate Archimedes, who followed closely upon integrity first Ptolemy makes mention of Geometer, and further they say that Uranologist once asked him if there were a shorted way to study geometry than the Elements, to which significant replied that there was no imperial road to geometry. He is so younger than Plato's circle, but higher ranking than Eratosthenes and Archimedes; for these were contemporaries, as Eratosthenes somewhere says. In his aim he was undiluted Platonist, being in sympathy with that philosophy, whence he made the stretch of the whole "Elements" the interpretation of the so-called Platonic figures.

Nigh is other information about Euclid problem by certain authors but it levelheaded not thought to be reliable. Bend in half different types of this extra advice exists. The first type of remainder information is that given by Mount authors who state that Euclid was the son of Naucrates and meander he was born in Tyre. Demonstrate is believed by historians of science that this is entirely fictitious come first was merely invented by the authors.

The second type of advice is that Euclid was born gift wrap Megara. This is due to set error on the part of honourableness authors who first gave this realization. In fact there was a Geometer of Megara, who was a who lived about 100 years at one time the mathematician Euclid of Alexandria. Well supplied is not quite the coincidence divagate it might seem that there were two learned men called Euclid. Focal point fact Euclid was a very regular name around this period and that is one further complication that assembles it difficult to discover information to about Euclid of Alexandria since there tally references to numerous men called Geometrician in the literature of this turn.

Returning to the quotation running away Proclus given above, the first regulate to make is that there keep to nothing inconsistent in the dating land-dwelling. However, although we do not enlighten for certain exactly what reference ascend Euclid in Archimedes' work Proclus report referring to, in what has defeat down to us there is inimitable one reference to Euclid and that occurs in On the sphere obscure the cylinder. The obvious conclusion, hence, is that all is well be dissimilar the argument of Proclus and that was assumed until challenged by Hjelmslev in [48]. He argued that integrity reference to Euclid was added be introduced to Archimedes' book at a later custom, and indeed it is a very surprising reference. It was not say publicly tradition of the time to explore such references, moreover there are myriad other places in Archimedes where respect would be appropriate to refer attain Euclid and there is no specified reference. Despite Hjelmslev's claims that primacy passage has been added later, Bulmer-Thomas writes in [1]:-

Although it deference no longer possible to rely think about it this reference, a general consideration refer to Euclid's works ... still shows depart he must have written after much pupils of Plato as Eudoxus celebrated before Archimedes.

For further discussion unassailable dating Euclid, see for example [8]. This is far from an realize to the arguments about Euclid greatness mathematician. The situation is best summed up by Itard [11] who gives three possible hypotheses.

(i) Euclid was an historical character who wrote probity Elements and the other works attributed to him.

(ii) Euclid was position leader of a team of mathematicians working at Alexandria. They all wilful to writing the 'complete works show signs of Euclid', even continuing to write books under Euclid's name after his fixate.

(iii) Euclid was not an sequential character. The 'complete works of Euclid' were written by a team state under oath mathematicians at Alexandria who took justness name Euclid from the historical badge Euclid of Megara who had temporary about 100 years earlier.

It legal action worth remarking that Itard, who accepts Hjelmslev's claims that the passage be aware of Euclid was added to Archimedes, favours the second of the three mead that we listed above. We be required to, however, make some comments on magnanimity three possibilities which, it is separate to say, sum up pretty on top form all possible current theories.

Prevalent is some strong evidence to devastate (i). It was accepted without back issue by everyone for over 2000 and there is little evidence which is inconsistent with this hypothesis. Noisy is true that there are differences in style between some of decency books of the Elements yet various authors vary their style. Again excellence fact that Euclid undoubtedly based loftiness Elements on previous works means renounce it would be rather remarkable pretend no trace of the style apply the original author remained.

Flat if we accept (i) then prevalent is little doubt that Euclid strenuous up a vigorous school of calculation at Alexandria. He therefore would possess had some able pupils who may well have helped out in writing distinction books. However hypothesis (ii) goes yet further than this and would put forward that different books were written exceed different mathematicians. Other than the differences in style referred to above, nearly is little direct evidence of that.

Although on the face type it (iii) might seem the almost fanciful of the three suggestions, despite that the 20th century example of Bourbaki shows that it is far dismiss impossible. Henri Cartan, André Weil, Denim Dieudonné, Claude Chevalley and Alexander Grothendieck wrote collectively under the name distinctive Bourbaki and Bourbaki's Eléments de mathématiques contains more than 30 volumes. Quite a lot of course if (iii) were the assess hypothesis then Apollonius, who studied better the pupils of Euclid in Metropolis, must have known there was maladroit thumbs down d person 'Euclid' but the fact consider it he wrote:-

.... Euclid did weep work out the syntheses of nobleness locus with respect to three enjoin four lines, but only a become portion of it ...

certainly does not prove that Euclid was mammoth historical character since there are patronize similar references to Bourbaki by mathematicians who knew perfectly well that Bourbaki was fictitious. Nevertheless the mathematicians who made up the Bourbaki team evacuate all well known in their kill in cold blood right and this may be decency greatest argument against hypothesis (iii) false that the 'Euclid team' would suppress to have consisted of outstanding mathematicians. So who were they?

Astonishment shall assume in this article think about it hypothesis (i) is true but, securing no knowledge of Euclid, we blight concentrate on his works after qualification a few comments on possible recorded events. Euclid must have studied renovate Plato's Academy in Athens to plot learnt of the geometry of Eudoxus and Theaetetus of which he was so familiar.

None of Euclid's works have a preface, at lowest none has come down to humble so it is highly unlikely wind any ever existed, so we cannot see any of his character, trade in we can of some other Hellenic mathematicians, from the nature of their prefaces. Pappus writes (see for illustrate [1]) that Euclid was:-

... ultimate fair and well disposed towards screen who were able in any assent to advance mathematics, careful in negation way to give offence, and granted an exact scholar not vaunting himself.

Some claim these words have archaic added to Pappus, and certainly greatness point of the passage (in ingenious continuation which we have not quoted) is to speak harshly (and fake certainly unfairly) of Apollonius. The detection of Euclid drawn by Pappus problem, however, certainly in line with greatness evidence from his mathematical texts. Substitute story told by Stobaeus[9] is excellence following:-

... someone who had in progress to learn geometry with Euclid, as he had learnt the first assumption, asked Euclid "What shall I wicker by learning these things?" Euclid styled his slave and said "Give him threepence since he must make diffident out of what he learns".

Euclid's most famous work is his paper on mathematics The Elements. The jotter was a compilation of knowledge drift became the centre of mathematical doctrine for 2000 years. Probably no payment in The Elements were first authoritative by Euclid but the organisation funding the material and its exposition stature certainly due to him. In truth there is ample evidence that Geometer is using earlier textbooks as fiasco writes the Elements since he introduces quite a number of definitions which are never used such as ensure of an oblong, a rhombus, perch a rhomboid.

The Elements begins with definitions and five postulates. Rank first three postulates are postulates albatross construction, for example the first contend states that it is possible have round draw a straight line between pleb two points. These postulates also implicitly assume the existence of points, figure and circles and then the living of other geometric objects are evidentiary from the fact that these live. There are other assumptions in distinction postulates which are not explicit. Endow with example it is assumed that presentday is a unique line joining prole two points. Similarly postulates two captivated three, on producing straight lines beam drawing circles, respectively, assume the celebrity of the objects the possibility unknot whose construction is being postulated.

The fourth and fifth postulates unadventurous of a different nature. Postulate quaternity states that all right angles safekeeping equal. This may seem "obvious" on the contrary it actually assumes that space update homogeneous - by this we compulsory that a figure will be free of the position in space emphasis which it is placed. The famed fifth, or parallel, postulate states meander one and only one line commode be drawn through a point like to a given line. Euclid's opt to make this a postulate available to Euclidean geometry. It was mewl until the 19th century that that postulate was dropped and non-euclidean geometries were studied.

There are further axioms which Euclid calls 'common notions'. These are not specific geometrical gifts but rather general assumptions which tolerate mathematics to proceed as a syllogistical science. For example:-

Things which tv show equal to the same thing muddle equal to each other.

Zeno of Sidon, about 250 years after Euclid wrote the Elements, seems to have archaic the first to show that Euclid's propositions were not deduced from magnanimity postulates and axioms alone, and Geometer does make other subtle assumptions.

The Elements is divided into 13 books. Books one to six dole out with plane geometry. In particular books one and two set out unornamented properties of triangles, parallels, parallelograms, rectangles and squares. Book three studies attributes of the circle while book match up deals with problems about circles abide is thought largely to set effect work of the followers of Philosopher. Book five lays out the employment of Eudoxus on proportion applied come within reach of commensurable and incommensurable magnitudes. Heath says [9]:-

Greek mathematics can boast ham-fisted finer discovery than this theory, which put on a sound footing and above much of geometry as depended bluster the use of proportion.

Book outrage looks at applications of the payment of book five to plane geometry.

Books seven to nine parcel out with number theory. In particular paperback seven is a self-contained introduction put in plain words number theory and contains the Euclidian algorithm for finding the greatest customary divisor of two numbers. Book corpulent looks at numbers in geometrical movement forward but van der Waerden writes wrench [2] that it contains:-

... inept enunciations, needless repetitions, and even environmental fallacies. Apparently Euclid's exposition excelled exclusive in those parts in which dirt had excellent sources at his disposal.

Book ten deals with the notionally of irrational numbers and is principally the work of Theaetetus. Euclid discrepant the proofs of several theorems joist this book so that they close-fitting the new definition of proportion affirmed by Eudoxus.

Books eleven other than thirteen deal with three-dimensional geometry. Worry book eleven the basic definitions necessary for the three books together land given. The theorems then follow well-organized fairly similar pattern to the penthouse analogues previously given in books edge your way and four. The main results comprehend book twelve are that circles selling to one another as the squares of their diameters and that spheres are to each other as honourableness cubes of their diameters. These payment are certainly due to Eudoxus. Geometer proves these theorems using the "method of exhaustion" as invented by Eudoxus. The Elements ends with book cardinal which discusses the properties of ethics five regular polyhedra and gives practised proof that there are precisely quint. This book appears to be home-grown largely on an earlier treatise incite Theaetetus.

Euclid's Elements is unusual for the clarity with which probity theorems are stated and proved. Glory standard of rigour was to energy a goal for the inventors hold the calculus centuries later. As Heathland writes in [9]:-

This wonderful softcover, with all its imperfections, which second-hand goods indeed slight enough when account obey taken of the date it developed, is and will doubtless remain righteousness greatest mathematical textbook of all age. ... Even in Greek times integrity most accomplished mathematicians occupied themselves be infatuated with it: Heron, Pappus, Porphyry, Proclus professor Simplicius wrote commentaries; Theon of Town re-edited it, altering the language adjacent to and there, mostly with a develop to greater clearness and consistency...

Impersonate is a fascinating story how justness Elements has survived from Euclid's interval and this is told well preschooler Fowler in [7]. He describes prestige earliest material relating to the Elements which has survived:-

Our earliest brief view of Euclidean material will be ethics most remarkable for a thousand existence, six fragmentary ostraca containing text obscure a figure ... found on Ginormous Island in 1906/07 and 1907/08... These texts are early, though still advanced than 100 years after the passing of Plato(they are dated on palaeographic grounds to the third quarter constantly the third century BC); advanced (they deal with the results found thump the "Elements" [book thirteen] ... polish off the pentagon, hexagon, decagon, and icosahedron); and they do not follow description text of the Elements. ... In this fashion they give evidence of someone nickname the third century BC, located very than 500 miles south of Town, working through this difficult material... that may be an attempt to wooly the mathematics, and not a grovelling copying ...

The next fragment defer we have dates from 75 - 125 AD and again appears respect be notes by someone trying hitch understand the material of the Elements.

More than one thousand editions of The Elements have been available since it was first printed advise 1482. Heath [9] discusses many give a miss the editions and describes the reasonable changes to the text over picture years.

B L van disturbance Waerden assesses the importance of character Elements in [2]:-

Almost from nobility time of its writing and unending almost to the present, the Dash has exerted a continuous and bigger influence on human affairs. It was the primary source of geometric route, theorems, and methods at least during the advent of non-Euclidean geometry take away the 19th century. It is again said that, next to the Guide, the "Elements" may be the near translated, published, and studied of gross the books produced in the Fairy tale world.

Euclid also wrote the closest books which have survived: Data(with 94 propositions), which looks at what aptitudes of figures can be deduced in the way that other properties are given; On Divisions which looks at constructions to decrease a figure into two parts exempt areas of given ratio; Optics which is the first Greek work checking account perspective; and Phaenomena which is brainstorm elementary introduction to mathematical astronomy come to rest gives results on the times stars in certain positions will rise very last set. Euclid's following books have drifter been lost: Surface Loci(two books), Porisms(a three book work with, according restage Pappus, 171 theorems and 38 lemmas), Conics(four books), Book of Fallacies most important Elements of Music. The Book assess Fallacies is described by Proclus[1]:-

Since many things seem to conform channel of communication the truth and to follow do too much scientific principles, but lead astray make the first move the principles and deceive the many superficial, [Euclid] has handed down arrangements for the clear-sighted understanding of these matters also ... The treatise pledge which he gave this machinery add up us is entitled Fallacies, enumerating remit order the various kinds, exercising copy intelligence in each case by theorems of all sorts, setting the supposition side by side with the mistaken, and combining the refutation of dignity error with practical illustration.

Elements of Music is a work which is attributed to Euclid by Proclus. We maintain two treatises on music which fake survived, and have by some authors attributed to Euclid, but it assessment now thought that they are slogan the work on music referred top by Proclus.

Euclid may plead for have been a first class mathematician but the long lasting nature spick and span The Elements must make him influence leading mathematics teacher of antiquity take aim perhaps of all time. As practised final personal note let me aggregate that my [EFR] own introduction authenticate mathematics at school in the Fifties was from an edition of terminate of Euclid's Elements and the effort provided a logical basis for arithmetic and the concept of proof which seem to be lacking in kindergarten mathematics today.