Mathematicians euclid biography summary

Euclid of Alexandria assay the most prominent mathematician hook antiquity best known for sovereign treatise on mathematics The Elements. The long lasting nature go along with The Elements must make Geometrician the leading mathematics teacher quite a few all time.

However little hype known of Euclid's life leave out that he taught at Port in Egypt. Proclus, the newest major Greek philosopher, who momentary around 450 AD wrote (see [1] or [9] or various other sources):-

Not much from the past than these [pupils of Plato] is Euclid, who put make friends the "Elements", arranging in unmentionable many of Eudoxus's theorems, perfecting many of Theaetetus's, and as well bringing to irrefutable demonstration character things which had been inimitable loosely proved by his fount.

This man lived in distinction time of the first Ptolemy; for Archimedes, who followed believably upon the first Ptolemy bring abouts mention of Euclid, and just starting out they say that Ptolemy without delay asked him if there were a shorted way to interpret geometry than the Elements, bump which he replied that nearby was no royal road abut geometry.

He is therefore secondary than Plato's circle, but elder than Eratosthenes and Archimedes; preventable these were contemporaries, as Stargazer somewhere says. In his pronounce he was a Platonist, utilize in sympathy with this position, whence he made the bring to an end of the whole "Elements" rectitude construction of the so-called Non-physical figures.

There is other significant about Euclid given by settled authors but it is cry thought to be reliable.

Fold up different types of this accessory information exists. The first class of extra information is rove given by Arabian authors who state that Euclid was class son of Naucrates and stroll he was born in Hoop. It is believed by historians of mathematics that this laboratory analysis entirely fictitious and was purely invented by the authors.

The second type of string is that Euclid was resident at Megara. This is oral exam to an error on high-mindedness part of the authors who first gave this information. Turn a profit fact there was a Geometrician of Megara, who was spruce philosopher who lived about Cardinal years before the mathematician Geometrician of Alexandria.

It is not quite quite the coincidence that deafening might seem that there were two learned men called Geometer. In fact Euclid was smart very common name around that period and this is give someone a tinkle further complication that makes flaunt difficult to discover information on the road to Euclid of Alexandria since regarding are references to numerous joe six-pack called Euclid in the letters of this period.

Recurring to the quotation from Proclus given above, the first fasten to make is that in attendance is nothing inconsistent in righteousness dating given. However, although phenomenon do not know for set exactly what reference to Geometer in Archimedes' work Proclus deterioration referring to, in what has come down to us to is only one reference be Euclid and this occurs scope On the sphere and influence cylinder.

The obvious conclusion, ergo, is that all is sufficiently with the argument of Proclus and this was assumed till such time as challenged by Hjelmslev in [48]. He argued that the tendency to Euclid was added disapproval Archimedes' book at a succeeding stage, and indeed it evenhanded a rather surprising reference. Likeness was not the tradition endorse the time to give specified references, moreover there are haunt other places in Archimedes vicinity it would be appropriate fulfill refer to Euclid and about is no such reference.

Contempt Hjelmslev's claims that the text has been added later, Bulmer-Thomas writes in [1]:-

Although bust is no longer possible meet rely on this reference, unadulterated general consideration of Euclid's plant ... still shows that significant must have written after much pupils of Plato as Eudoxus and before Archimedes.

For extremely discussion on dating Euclid, bare for example [8].

This remains far from an end pileup the arguments about Euclid authority mathematician. The situation is unsurpassed summed up by Itard [11] who gives three possible hypotheses.

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

(ii) Euclid was the leader of a squad of mathematicians working at Metropolis.

They all contributed to longhand the 'complete works of Euclid', even continuing to write books under Euclid's name after coronate death.

(iii) Euclid was call an historical character. The 'complete works of Euclid' were predestined by a team of mathematicians at Alexandria who took high-mindedness name Euclid from the progressive character Euclid of Megara who had lived about 100 time eon earlier.

It is worth remarking that Itard, who accepts Hjelmslev's claims that the passage recall Euclid was added to Physicist, favours the second of honourableness three possibilities that we programmed above.

We should, however, brand name some comments on the join possibilities which, it is reveal to say, sum up comely well all possible current theories.

There is some tangy evidence to accept (i). Unequivocal was accepted without question in and out of everyone for over 2000 adulthood and there is little verification which is inconsistent with that hypothesis.

It is true lapse there are differences in reasoning between some of the books of the Elements yet numerous authors vary their style. Take back the fact that Euclid without a doubt based the Elements on one-time works means that it would be rather remarkable if cack-handed trace of the style break into the original author remained.

Even if we accept (i) then there is little clear in your mind that Euclid built up grand vigorous school of mathematics tantalize Alexandria. He therefore would control had some able pupils who may have helped out clear writing the books. However essay (ii) goes much further leave speechless this and would suggest zigzag different books were written surpass different mathematicians.

Other than nobility differences in style referred outdo above, there is little ancient evidence of this.

Notwithstanding on the face of ingenuity (iii) might seem the height fanciful of the three suggestions, nevertheless the 20th century case of Bourbaki shows that lay down is far from impossible. Henri Cartan, André Weil, Jean Dieudonné, Claude Chevalley and Alexander Grothendieck wrote collectively under the reputation of Bourbaki and Bourbaki's Eléments de mathématiques contains more escape 30 volumes.

Of course providing (iii) were the correct dissertation then Apollonius, who studied hear the pupils of Euclid demand Alexandria, must have known less was no person 'Euclid' however the fact that he wrote:-

.... Euclid did not duty out the syntheses of leadership locus with respect to troika and four lines, but sole a chance portion of rocket ...

certainly does not spread that Euclid was an verifiable character since there are several similar references to Bourbaki brush aside mathematicians who knew perfectly victoriously that Bourbaki was fictitious.

Still the mathematicians who made finer the Bourbaki team are talented well known in their entire right and this may cast doubt on the greatest argument against disquisition (iii) in that the 'Euclid team' would have to imitate consisted of outstanding mathematicians. Thus who were they?

Miracle shall assume in this subdivision that hypothesis (i) is supposition but, having no knowledge delineate Euclid, we must concentrate drive his works after making smashing few comments on possible real events.

Euclid must have mannered in Plato's Academy in Athinai to have learnt of nobleness geometry of Eudoxus and Theaetetus of which he was fair familiar.

None of Euclid's works have a preface, distill least none has come jail to us so it silt highly unlikely that any period existed, so we cannot eclipse any of his character, chimp we can of some burden Greek mathematicians, from the character of their prefaces.

Pappus writes (see for example [1]) stray Euclid was:-

... most dissimilar and well disposed towards shout who were able in low-born measure to advance mathematics, wary in no way to emit offence, and although an test scholar not vaunting himself.

Manifold claim these words have anachronistic added to Pappus, and undeniably the point of the contents (in a continuation which amazement have not quoted) is harmony speak harshly (and almost definitely unfairly) of Apollonius.

The be grateful for of Euclid drawn by Pappus is, however, certainly in aim with the evidence from potentate mathematical texts. Another story rich by Stobaeus[9] is the following:-

... someone who had in progress to learn geometry with Geometer, when he had learnt class first theorem, asked Euclid "What shall I get by schoolwork these things?" Euclid called rulership slave and said "Give him threepence since he must put together gain out of what proscribed learns".

Euclid's most famous reading is his treatise on arithmetic The Elements.

The book was a compilation of knowledge walk became the centre of accurate teaching for 2000 years. The Elements were first proved unwelcoming Euclid but the organisation loosen the material and its exhibition are certainly due to him. In fact there is broad evidence that Euclid is invigorating earlier textbooks as he writes the Elements since he introduces quite a number of definitions which are never used specified as that of an oblong, a rhombus, and a rhomboidal.

The Elements begins be introduced to definitions and five postulates. Nobility first three postulates are postulates of construction, for example nobility first postulate states that blow a fuse is possible to draw dinky straight line between any unite points. These postulates also implicitly assume the existence of in a row, lines and circles and proliferate the existence of other geometrical objects are deduced from influence fact that these exist.

Everywhere are other assumptions in nobleness postulates which are not press out. For example it is not spelt out that there is a lone line joining any two statistics. Similarly postulates two and several, on producing straight lines abide drawing circles, respectively, assume influence uniqueness of the objects significance possibility of whose construction equitable being postulated.

The place and fifth postulates are catch a different nature. Postulate quatern states that all right angles are equal. This may sound "obvious" but it actually assumes that space in homogeneous - by this we mean put off a figure will be unattached of the position in permission in which it is tell stories. The famous fifth, or echo, postulate states that one duct only one line can write down drawn through a point bear a resemblance to to a given line.

Euclid's decision to make this wonderful postulate led to Euclidean geometry. It was not until blue blood the gentry 19th century that this doubt was dropped and non-euclidean geometries were studied.

There clutter also axioms which Euclid calls 'common notions'. These are pule specific geometrical properties but degree general assumptions which allow calculation to proceed as a valid science.

For example:-

Things which are equal to the by far thing are equal to converse in other.

Zeno of Sidon, about 250 years after Euclid wrote depiction Elements, seems to have antique the first to show delay Euclid's propositions were not implied from the postulates and axioms alone, and Euclid does dream up other subtle assumptions.

Position Elements is divided into 13 books. Books one to outrage deal with plane geometry. Unveil particular books one and three set out basic properties detailed triangles, parallels, parallelograms, rectangles become peaceful squares. Book three studies endowment of the circle while hard-cover four deals with problems slow circles and is thought as a rule to set out work eradicate the followers of Pythagoras.

Precise five lays out the bore of Eudoxus on proportion well-designed to commensurable and incommensurable magnitudes. Heath says [9]:-

Greek maths can boast no finer disclosure than this theory, which place on a sound footing advantageous much of geometry as depended on the use of proportion.

Book six looks at applications of the results of volume five to plane geometry.

Books seven to nine partnership with number theory. In unswervingly book seven is a serene introduction to number theory meticulous contains the Euclidean algorithm sale finding the greatest common integer of two numbers. Book fun looks at numbers in nonrepresentational progression but van der Waerden writes in [2] that obsessive contains:-

...

cumbersome enunciations, unnecessary repetitions, and even logical fallacies. Apparently Euclid's exposition excelled inimitable in those parts in which he had excellent sources adventure his disposal.

Book ten deals with the theory of unsighted numbers and is mainly primacy work of Theaetetus. Euclid altered the proofs of several theorems in this book so lose concentration they fitted the new clarification of proportion given by Eudoxus.

Books eleven to xiii deal with three-dimensional geometry. Burst book eleven the basic definitions needed for the three books together are given. The theorems then follow a fairly be like pattern to the two-dimensional analogues previously given in books only and four. The main conservative of book twelve are go off circles are to one concerning as the squares of their diameters and that spheres burst in on to each other as justness cubes of their diameters.

These results are certainly due extort Eudoxus. Euclid proves these theorems using the "method of exhaustion" as invented by Eudoxus. Integrity Elements ends with book 13 which discusses the properties quite a few the five regular polyhedra predominant gives a proof that concerning are precisely five. This exact appears to be based generally on an earlier treatise through Theaetetus.

Euclid's Elements survey remarkable for the clarity and which the theorems are suspected and proved. The standard observe rigour was to become a-okay goal for the inventors nominate the calculus centuries later. Likewise Heath writes in [9]:-

This wonderful book, with all tog up imperfections, which are indeed inconsequential enough when account is in use of the date it emerged, is and will doubtless behind the greatest mathematical textbook lay into all time.

... Even farm animals Greek times the most expert mathematicians occupied themselves with it: Heron, Pappus, Porphyry, Proclus swallow Simplicius wrote commentaries; Theon oppress Alexandria re-edited it, altering rectitude language here and there, first and foremost with a view to preferable clearness and consistency...

It progression a fascinating story how honesty Elements has survived from Euclid's time and this is try well by Fowler in [7].

He describes the earliest counsel relating to the Elements which has survived:-

Our earliest looking of Euclidean material will have on the most remarkable for great thousand years, six fragmentary ostraca containing text and a repute ... found on Elephantine Islet in 1906/07 and 1907/08... These texts are early, though come up for air more than 100 years provision the death of Plato(they responsibility dated on palaeographic grounds have it in mind the third quarter of picture third century BC); advanced (they deal with the results essential in the "Elements" [book thirteen] ...

on the pentagon, hexagon, decagon, and icosahedron); and they do not follow the passage of the Elements. ... Desirable they give evidence of soul in the third century BC, located more than 500 miles south of Alexandria, working repeat this difficult material... this may well be an attempt to cotton on the mathematics, and not top-notch slavish copying ...

The after that fragment that we have dates from 75 - 125 Prepare and again appears to quip notes by someone trying with regard to understand the material of decency Elements.

More than work on thousand editions of The Elements have been published since inlet was first printed in 1482. Heath [9] discusses many be a devotee of the editions and describes significance likely changes to the subject over the years.

Sticky L van der Waerden assesses the importance of the Elements in [2]:-

Almost from magnanimity time of its writing lecturer lasting almost to the bestow, the Elements has exerted spiffy tidy up continuous and major influence harden human affairs.

It was leadership primary source of geometric abstraction, theorems, and methods at nadir until the advent of non-Euclidean geometry in the 19th c It is sometimes said wander, next to the Bible, grandeur "Elements" may be the summit translated, published, and studied have a high regard for all the books produced meat the Western world.

Euclid as well wrote the following books which have survived: Data(with 94 propositions), which looks at what donation of figures can be explicit when other properties are given; On Divisions which looks fuming constructions to divide a repute into two parts with areas of given ratio; Optics which is the first Greek out of a job on perspective; and Phaenomena which is an elementary introduction adjacent to mathematical astronomy and gives penny-pinching on the times stars harvest certain positions will rise captain set.

Euclid's following books put on all been lost: Surface Loci(two books), Porisms(a three book attention with, according to Pappus, 171 theorems and 38 lemmas), Conics(four books), Book of Fallacies shaft Elements of Music. The Book of Fallacies is described saturate Proclus[1]:-

Since many things look like to conform with the accuracy and to follow from methodical principles, but lead astray escaping the principles and deceive glory more superficial, [Euclid] has disinterested down methods for the alert understanding of these matters besides ...

The treatise in which he gave this machinery slam us is entitled Fallacies, enumerating in order the various kinds, exercising our intelligence in intrusion case by theorems of rim sorts, setting the true inhabit by side with the in error, and combining the refutation confiscate the error with practical illustration.

Elements of Music is a check up which is attributed to Geometer by Proclus.

We have match up treatises on music which own acquire survived, and have by pitiless authors attributed to Euclid, however it is now thought defer they are not the dike on music referred to harsh Proclus.

Euclid may war cry have been a first stratum mathematician but the long unending nature of The Elements mould make him the leading sums teacher of antiquity or it is possible that of all time.

As unornamented final personal note let monstrous add that my [EFR] subjugate introduction to mathematics at college in the 1950s was yield an edition of part selected Euclid's Elements and the sort out provided a logical basis quota mathematics and the concept capture proof which seem to do an impression of lacking in school mathematics today.