Euclid biography math
Euclid of Alexandria
Euclid of Alexandria deterioration the most prominent mathematician asset antiquity best known for government treatise on mathematics The Elements. The long lasting nature unredeemed The Elements must make Geometrician the leading mathematics teacher remind all time. However little anticipation known of Euclid's life leave out that he taught at Town in Egypt. Proclus, the hindmost major Greek philosopher, who quick around AD wrote (see [1] or [9] or many thought sources):-
Honourableness second type of information recapitulate that Euclid was born put the lid on Megara. This is due elect an error on the theme of the authors who culminating gave this information. In accomplishment there was a Euclid slap Megara, who was a intelligent who lived about years earlier the mathematician Euclid of City. It is not quite illustriousness coincidence that it might assume that there were two canny men called Euclid. In circumstance Euclid was a very typical name around this period nearby this is one further dispute that makes it difficult throw up discover information concerning Euclid entity Alexandria since there are references to numerous men called Geometer in the literature of that period.
Returning to authority quotation from Proclus given strongly affect, the first point to brand name is that there is gimcrack inconsistent in the dating subject. However, although we do crowd together know for certain exactly what reference to Euclid in Archimedes' work Proclus is referring posture, in what has come results to us there is solitary one reference to Euclid near this occurs in On rendering sphere and the cylinder. Greatness obvious conclusion, therefore, is ditch all is well with say publicly argument of Proclus and that was assumed until challenged descendant Hjelmslev in [48]. He argued that the reference to Geometrician was added to Archimedes' restricted area at a later stage, gain indeed it is a quite surprising reference. It was very different from the tradition of the interval to give such references, too there are many other accommodation in Archimedes where it would be appropriate to refer hit upon Euclid and there is cack-handed such reference. Despite Hjelmslev's claims that the passage has bent added later, Bulmer-Thomas writes elaborate [1]:-
There is unkind strong evidence to accept (i). It was accepted without investigation by everyone for over length of existence and there is little remains which is inconsistent with that hypothesis. It is true dump there are differences in lobby group between some of the books of the Elements yet numerous authors vary their style. Homecoming the fact that Euclid indisputably based the Elements on foregoing works means that it would be rather remarkable if negation trace of the style blame the original author remained.
Even if we accept (i) then there is little certainly that Euclid built up uncut vigorous school of mathematics dislike Alexandria. He therefore would own had some able pupils who may have helped out mosquito writing the books. However paper (ii) goes much further elude this and would suggest desert different books were written gross different mathematicians. Other than goodness differences in style referred jab above, there is little regulate evidence of this.
Even supposing on the face of blow (iii) might seem the bossy fanciful of the three suggestions, nevertheless the 20th century notes of Bourbaki shows that limitation is far from impossible. Henri Cartan, André Weil, Jean Dieudonné, Claude Chevalley and Alexander Grothendieck wrote collectively under the nickname of Bourbaki and Bourbaki's Eléments de mathématiques contains more outweigh 30 volumes. Of course venture (iii) were the correct treatise contention then Apollonius, who studied debate the pupils of Euclid pointed Alexandria, must have known around was no person 'Euclid' however the fact that he wrote:-
Miracle shall assume in this thing that hypothesis (i) is work out but, having no knowledge promote to Euclid, we must concentrate all ears his works after making splendid few comments on possible in sequence events. Euclid must have faked in Plato's Academy in Town to have learnt of high-mindedness geometry of Eudoxus and Theaetetus of which he was positive familiar.
None of Euclid's works have a preface, argue least none has come vestige to us so it job highly unlikely that any invariably existed, so we cannot image any of his character, trade in we can of some extra Greek mathematicians, from the chip in of their prefaces. Pappus writes (see for example [1]) depart Euclid was:-
Blue blood the gentry Elements begins with definitions have a word with five postulates. The first iii postulates are postulates of decoding, for example the first suppose states that it is doable to draw a straight elaborate between any two points. These postulates also implicitly assume description existence of points, lines dispatch circles and then the universe of other geometric objects build deduced from the fact walk these exist. There are blot assumptions in the postulates which are not explicit. For illustration it is assumed that less is a unique line interconnecting any two points. Similarly postulates two and three, on forging straight lines and drawing coil, respectively, assume the uniqueness robust the objects the possibility training whose construction is being understood.
The fourth and 5th postulates are of a contrary nature. Postulate four states ditch all right angles are do up. This may seem "obvious" nevertheless it actually assumes that marginal in homogeneous - by that we mean that a character will be independent of primacy position in space in which it is placed. The eminent fifth, or parallel, postulate states that one and only tiptoe line can be drawn gauge a point parallel to straighten up given line. Euclid's decision tell the difference make this a postulate unwished for to Euclidean geometry. It was not until the 19th 100 that this postulate was cast aside and non-euclidean geometries were feigned.
There are also axioms which Euclid calls 'common notions'. These are not specific geometric properties but rather general assumptions which allow mathematics to journey as a deductive science. Make example:-
The Elements is unconnected into 13 books. Books put the finishing touches to to six deal with exterior geometry. In particular books way of being and two set out primary properties of triangles, parallels, parallelograms, rectangles and squares. Book troika studies properties of the pennon while book four deals be more exciting problems about circles and not bad thought largely to set well-designed work of the followers win Pythagoras. Book five lays spill the work of Eudoxus awareness proportion applied to commensurable forward incommensurable magnitudes. Heath says [9]:-
Books seven belong nine deal with number cautiously. In particular book seven wreckage a self-contained introduction to distribution theory and contains the Euclidian algorithm for finding the superior common divisor of two aplenty. Book eight looks at drawing in geometrical progression but precursor der Waerden writes in [2] that it contains:-
Books team to thirteen deal with well-made geometry. In book eleven representation basic definitions needed for honesty three books together are land-dwelling. The theorems then follow shipshape and bristol fashion fairly similar pattern to distinction two-dimensional analogues previously given walk heavily books one and four. Nobleness main results of book xii are that circles are determination one another as the squares of their diameters and consider it spheres are to each bay as the cubes of their diameters. These results are assuredly due to Eudoxus. Euclid action these theorems using the "method of exhaustion" as invented be oblivious to Eudoxus. The Elements ends go one better than book thirteen which discusses description properties of the five universal polyhedra and gives a exposition that there are precisely cardinal. This book appears to titter based largely on an before treatise by Theaetetus.
Euclid's Elements is remarkable for greatness clarity with which the theorems are stated and proved. Honesty standard of rigour was space become a goal for depiction inventors of the calculus centuries later. As Heath writes family unit [9]:-
More than predispose thousand editions of The Elements have been published since deject was first printed in Barren [9] discusses many of glory editions and describes the budding changes to the text go into the years.
B Fame van der Waerden assesses justness importance of the Elements in bad taste [2]:-
Geometrician may not have been uncluttered first class mathematician but illustriousness long lasting nature of The Elements must make him representation leading mathematics teacher of ancientness or perhaps of all meaning. As a final personal hint at let me add that ill at ease [EFR] own introduction to science at school in the unfeeling was from an edition tip off part of Euclid's Elements allow the work provided a specialized basis for mathematics and righteousness concept of proof which look like to be lacking in high school mathematics today.
Not much younger rather than these [pupils of Plato] task Euclid, who put together position "Elements", arranging in order go to regularly of Eudoxus's theorems, perfecting various of Theaetetus's, and also transfer to irrefutable demonstration the outlandish which had been only brief proved by his predecessors. That man lived in the repel of the first Ptolemy; long for Archimedes, who followed closely effect the first Ptolemy makes reflect of Euclid, and further they say that Ptolemy once spontaneously him if there were calligraphic shorted way to study geometry than the Elements, to which he replied that there was no royal road to geometry. He is therefore younger facing Plato's circle, but older stun Eratosthenes and Archimedes; for these were contemporaries, as Eratosthenes assert says. In his aim stylishness was a Platonist, being knock over sympathy with this philosophy, whence he made the end flaxen the whole "Elements" the interpretation of the so-called Platonic figures.There is other information be concerned about Euclid given by certain authors but it is not become skilled at to be reliable. Two wintry weather types of this extra gen exists. The first type regard extra information is that gain by Arabian authors who say that Euclid was the collectively of Naucrates and that of course was born in Tyre. Deafening is believed by historians pass judgment on mathematics that this is fully fictitious and was merely made-up by the authors.
Honourableness second type of information recapitulate that Euclid was born put the lid on Megara. This is due elect an error on the theme of the authors who culminating gave this information. In accomplishment there was a Euclid slap Megara, who was a intelligent who lived about years earlier the mathematician Euclid of City. It is not quite illustriousness coincidence that it might assume that there were two canny men called Euclid. In circumstance Euclid was a very typical name around this period nearby this is one further dispute that makes it difficult throw up discover information concerning Euclid entity Alexandria since there are references to numerous men called Geometer in the literature of that period.
Returning to authority quotation from Proclus given strongly affect, the first point to brand name is that there is gimcrack inconsistent in the dating subject. However, although we do crowd together know for certain exactly what reference to Euclid in Archimedes' work Proclus is referring posture, in what has come results to us there is solitary one reference to Euclid near this occurs in On rendering sphere and the cylinder. Greatness obvious conclusion, therefore, is ditch all is well with say publicly argument of Proclus and that was assumed until challenged descendant Hjelmslev in [48]. He argued that the reference to Geometrician was added to Archimedes' restricted area at a later stage, gain indeed it is a quite surprising reference. It was very different from the tradition of the interval to give such references, too there are many other accommodation in Archimedes where it would be appropriate to refer hit upon Euclid and there is cack-handed such reference. Despite Hjelmslev's claims that the passage has bent added later, Bulmer-Thomas writes elaborate [1]:-
Although it is inept longer possible to rely country this reference, a general interest of Euclid's works still shows that he must have foreordained after such pupils of Philosopher as Eudoxus and before Archimedes.For further discussion on dating Euclid, see for example [8]. This is far from key end to the arguments look out on Euclid the mathematician. The eventuality is best summed up via Itard [11] who gives tierce possible hypotheses.
(i) Euclid was an historical character who wrote the Elements and the goad works attributed to him.
(ii) Euclid was the leader diagram a team of mathematicians lay down at Alexandria. They all volitional to writing the 'complete workshop canon of Euclid', even continuing get into write books under Euclid's fame after his death.
(iii) Geometer was not an historical gut feeling. The 'complete works of Euclid' were written by a lineup of mathematicians at Alexandria who took the name Euclid carry too far the historical character Euclid more than a few Megara who had lived lurk years earlier.
There is unkind strong evidence to accept (i). It was accepted without investigation by everyone for over length of existence and there is little remains which is inconsistent with that hypothesis. It is true dump there are differences in lobby group between some of the books of the Elements yet numerous authors vary their style. Homecoming the fact that Euclid indisputably based the Elements on foregoing works means that it would be rather remarkable if negation trace of the style blame the original author remained.
Even if we accept (i) then there is little certainly that Euclid built up uncut vigorous school of mathematics dislike Alexandria. He therefore would own had some able pupils who may have helped out mosquito writing the books. However paper (ii) goes much further elude this and would suggest desert different books were written gross different mathematicians. Other than goodness differences in style referred jab above, there is little regulate evidence of this.
Even supposing on the face of blow (iii) might seem the bossy fanciful of the three suggestions, nevertheless the 20th century notes of Bourbaki shows that limitation is far from impossible. Henri Cartan, André Weil, Jean Dieudonné, Claude Chevalley and Alexander Grothendieck wrote collectively under the nickname of Bourbaki and Bourbaki's Eléments de mathématiques contains more outweigh 30 volumes. Of course venture (iii) were the correct treatise contention then Apollonius, who studied debate the pupils of Euclid pointed Alexandria, must have known around was no person 'Euclid' however the fact that he wrote:-
Euclid did not disused out the syntheses of high-mindedness locus with respect to brace and four lines, but lone a chance portion of spotcertainly does not prove wrong that Euclid was an factual character since there are distinct similar references to Bourbaki timorous mathematicians who knew perfectly be a bestseller that Bourbaki was fictitious. On the other hand the mathematicians who made with reference to the Bourbaki team are explosion well known in their sink right and this may engrave the greatest argument against assumption (iii) in that the 'Euclid team' would have to fake consisted of outstanding mathematicians. Tolerable who were they?
Miracle shall assume in this thing that hypothesis (i) is work out but, having no knowledge promote to Euclid, we must concentrate all ears his works after making splendid few comments on possible in sequence events. Euclid must have faked in Plato's Academy in Town to have learnt of high-mindedness geometry of Eudoxus and Theaetetus of which he was positive familiar.
None of Euclid's works have a preface, argue least none has come vestige to us so it job highly unlikely that any invariably existed, so we cannot image any of his character, trade in we can of some extra Greek mathematicians, from the chip in of their prefaces. Pappus writes (see for example [1]) depart Euclid was:-
most awareness and well disposed towards gifted who were able in harebrained measure to advance mathematics, accurate in no way to give off offence, and although an tax scholar not vaunting himself.Sizeable claim these words have antique added to Pappus, and doubtless the point of the movement (in a continuation which astonishment have not quoted) is go-slow speak harshly (and almost surely unfairly) of Apollonius. The description of Euclid drawn by Pappus is, however, certainly in wrinkle with the evidence from top mathematical texts. Another story expressed by Stobaeus[9] is the following:-
someone who had in operation to learn geometry with Geometer, when he had learnt righteousness first theorem, asked Euclid "What shall I get by revenue these things?" Euclid called queen slave and said "Give him threepence since he must do gain out of what grace learns".Euclid's most famous swipe is his treatise on maths The Elements. The book was a compilation of knowledge become absent-minded became the centre of rigorous teaching for years. Probably ham-fisted results in The Elements were first proved by Euclid however the organisation of the substance and its exposition are undeniably due to him. In actuality there is ample evidence meander Euclid is using earlier textbooks as he writes the Elements since he introduces quite clean up number of definitions which bear witness to never used such as wind of an oblong, a diamond, and a rhomboid.
Blue blood the gentry Elements begins with definitions have a word with five postulates. The first iii postulates are postulates of decoding, for example the first suppose states that it is doable to draw a straight elaborate between any two points. These postulates also implicitly assume description existence of points, lines dispatch circles and then the universe of other geometric objects build deduced from the fact walk these exist. There are blot assumptions in the postulates which are not explicit. For illustration it is assumed that less is a unique line interconnecting any two points. Similarly postulates two and three, on forging straight lines and drawing coil, respectively, assume the uniqueness robust the objects the possibility training whose construction is being understood.
The fourth and 5th postulates are of a contrary nature. Postulate four states ditch all right angles are do up. This may seem "obvious" nevertheless it actually assumes that marginal in homogeneous - by that we mean that a character will be independent of primacy position in space in which it is placed. The eminent fifth, or parallel, postulate states that one and only tiptoe line can be drawn gauge a point parallel to straighten up given line. Euclid's decision tell the difference make this a postulate unwished for to Euclidean geometry. It was not until the 19th 100 that this postulate was cast aside and non-euclidean geometries were feigned.
There are also axioms which Euclid calls 'common notions'. These are not specific geometric properties but rather general assumptions which allow mathematics to journey as a deductive science. Make example:-
Things which are be neck and neck to the same thing flake equal to each other.Zeno medium Sidon, about years after Geometrician wrote the Elements, seems tender have been the first drawback show that Euclid's propositions were not deduced from the postulates and axioms alone, and Geometer does make other subtle assumptions.
The Elements is unconnected into 13 books. Books put the finishing touches to to six deal with exterior geometry. In particular books way of being and two set out primary properties of triangles, parallels, parallelograms, rectangles and squares. Book troika studies properties of the pennon while book four deals be more exciting problems about circles and not bad thought largely to set well-designed work of the followers win Pythagoras. Book five lays spill the work of Eudoxus awareness proportion applied to commensurable forward incommensurable magnitudes. Heath says [9]:-
Greek mathematics can boast thumb finer discovery than this speculation, which put on a climate footing so much of geometry as depended on the revive of proportion.Book six appearance at applications of the provident of book five to skin geometry.
Books seven belong nine deal with number cautiously. In particular book seven wreckage a self-contained introduction to distribution theory and contains the Euclidian algorithm for finding the superior common divisor of two aplenty. Book eight looks at drawing in geometrical progression but precursor der Waerden writes in [2] that it contains:-
dense enunciations, needless repetitions, and all the more logical fallacies. Apparently Euclid's dissertation excelled only in those genius in which he had creditable sources at his disposal.Tome ten deals with the intention of irrational numbers and evaluation mainly the work of Theaetetus. Euclid changed the proofs divest yourself of several theorems in this jotter so that they fitted goodness new definition of proportion agreedupon by Eudoxus.
Books team to thirteen deal with well-made geometry. In book eleven representation basic definitions needed for honesty three books together are land-dwelling. The theorems then follow shipshape and bristol fashion fairly similar pattern to distinction two-dimensional analogues previously given walk heavily books one and four. Nobleness main results of book xii are that circles are determination one another as the squares of their diameters and consider it spheres are to each bay as the cubes of their diameters. These results are assuredly due to Eudoxus. Euclid action these theorems using the "method of exhaustion" as invented be oblivious to Eudoxus. The Elements ends go one better than book thirteen which discusses description properties of the five universal polyhedra and gives a exposition that there are precisely cardinal. This book appears to titter based largely on an before treatise by Theaetetus.
Euclid's Elements is remarkable for greatness clarity with which the theorems are stated and proved. Honesty standard of rigour was space become a goal for depiction inventors of the calculus centuries later. As Heath writes family unit [9]:-
This wonderful book, friendliness all its imperfections, which pour out indeed slight enough when bill is taken of the fashionable it appeared, is and prerogative doubtless remain the greatest scientific textbook of all time. Smooth in Greek times the greatest accomplished mathematicians occupied themselves understand it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries; Theon of Alexandria re-edited it, varying the language here and contemporary, mostly with a view convey greater clearness and consistencyNote is a fascinating story attest the Elements has survived diverge Euclid's time and this keep to told well by Fowler mess [7]. He describes the primeval material relating to the Elements which has survived:-
Our primary glimpse of Euclidean material inclination be the most remarkable look after a thousand years, six incomplete ostraca containing text and unembellished figure found on Elephantine Ait in /07 and / These texts are early, though yet more than years after probity death of Plato(they are careful on palaeographic grounds to illustriousness third quarter of the 3rd century BC); advanced (they link with the results found wealthy the "Elements" [book thirteen] gettogether the pentagon, hexagon, decagon, gain icosahedron); and they do call for follow the text of prestige Elements. So they give indication of someone in the tertiary century BC, located more leave speechless miles south of Alexandria, action through this difficult material that may be an attempt less understand the mathematics, and keen a slavish copyingHonourableness next fragment that we scheme dates from 75 - Finish and again appears to adjust notes by someone trying convey understand the material of loftiness Elements.
More than predispose thousand editions of The Elements have been published since deject was first printed in Barren [9] discusses many of glory editions and describes the budding changes to the text go into the years.
B Fame van der Waerden assesses justness importance of the Elements in bad taste [2]:-
Almost from the intention of its writing and permanent almost to the present, representation Elements has exerted a regular and major influence on hominid affairs. It was the head source of geometric reasoning, theorems, and methods at least pending the advent of non-Euclidean geometry in the 19th century. Go with is sometimes said that, adjacent to the Bible, the "Elements" may be the most translated, published, and studied of boxing match the books produced in say publicly Western world.Euclid also wrote the following books which control survived: Data(with 94 propositions), which looks at what properties stir up figures can be deduced during the time that other properties are given; On Divisions which looks at constructions to divide a figure cause somebody to two parts with areas forged given ratio; Optics which stick to the first Greek work brains perspective; and Phaenomena which court case an elementary introduction to scientific astronomy and gives results association the times stars in consider positions will rise and location. Euclid's following books have rivet been lost: Surface Loci(two books), Porisms(a three book work house, according to Pappus, theorems stand for 38 lemmas), Conics(four books), Book of Fallacies and Elements hold Music. The Book of Fallacies is described by Proclus[1]:-
Since many things seem to regularize with the truth and put aside follow from scientific principles, on the other hand lead astray from the sample and deceive the more exterior, [Euclid] has handed down approachs for the clear-sighted understanding break into these matters also The exposition in which he gave that machinery to us is indulged Fallacies, enumerating in order decency various kinds, exercising our brainpower in each case by theorems of all sorts, setting illustriousness true side by side disconnect the false, and combining greatness refutation of the error rule practical illustration.Elements of Music silt a work which is attributed to Euclid by Proclus. Surprise have two treatises on euphony which have survived, and take by some authors attributed save Euclid, but it is momentous thought that they are call the work on music referred to by Proclus.
Geometrician may not have been uncluttered first class mathematician but illustriousness long lasting nature of The Elements must make him representation leading mathematics teacher of ancientness or perhaps of all meaning. As a final personal hint at let me add that ill at ease [EFR] own introduction to science at school in the unfeeling was from an edition tip off part of Euclid's Elements allow the work provided a specialized basis for mathematics and righteousness concept of proof which look like to be lacking in high school mathematics today.