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EUCLID

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Originally appearing in Volume V09, Page 880 of the 1911 Encyclopedia Britannica.
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EUCLID , See also:

Greek mathematician of the 3rd See also:century B.C.; we are ignorant not only of the See also:dates of his See also:birth and See also:death, but also of his parentage, his teachers, and the See also:residence of his See also:early years. In some of the See also:editions of his See also:works he is called Megarensis, as if he had been See also:born at See also:Megara in See also:Greece, a See also:mistake which arose from confounding him with another Euclid, a See also:disciple of See also:Socrates. See also:Proclus (A.D. 412-485), the authority for most of our See also:information regarding Euclid, states in his commentary on the first See also:book of the Elements that Euclid lived in the See also:time of See also:Ptolemy I., See also:king of See also:Egypt, who reigned from 323 to 285 B.C., that he was younger than the associates of See also:Plato, but older than Eratosthenes (276–196 B.C.) and See also:Archimedes (287–212 B.C.). Euclid is said to have founded the mathematical school of See also:Alexandria, which was at that time becoming a centre, not only of See also:commerce, but of learning and See also:research, and for this service to the cause of exact See also:science he would have deserved See also:commemoration, even if his writings had not secured him a worthier See also:title to fame. Proclus preserves a reply made by Euclid to King Ptolemy, who asked whether he could not learn See also:geometry more easily than by studying the Elements—" There is no royal road to geometry." Pappus of Alexandria, in his Mathematical Collection, says that Euclid was a See also:man of mild and inoffensive temperament, unpretending, and See also:kind to all genuine students of See also:mathematics. This being all that is known of the See also:life and See also:character of Euclid, it only remains therefore to speak of his works. Among those which have come down to us the most remarkable is the Elements (ErosxeIa) (see GEOMETRY). They consist of thirteen books; two more are frequently added, but there is See also:reason to believe that they are the See also:work of a later mathematician, Hypsicles of Alexandria. The question has often been mooted, to what extent Euclid, in his Elements, is a discoverer or a compiler. To this question no entirely satisfactory See also:answer can be given, for scarcely any of the writings of earlier geometers have come down to our times. We are mainly dependent on Pappus and Proclus for the scanty notices we have of Euclid's predecessors, and of the problems which engaged their See also:attention; for the See also:solution of problems, and not the See also:discovery of theorems, would seem to have been their See also:principal See also:object.

From these authors we learn that the See also:

property of the right-angled triangle had been found out, the principles of geometrical See also:analysis laid down, the restriction of constructions in See also:plane geometry to the straight See also:line and the circle agreed upon, the See also:doctrine of proportion, for both commensurables and incommensurables, as well as loci, plane and solid, and some of the properties of the conic sections investigated, the five See also:regular solids (often called the Platonic bodies) and the relation between the See also:volume of a See also:cone or See also:pyramid and that of its circumscribed See also:cylinder or See also:prism discovered. Elementary works had beenwritten, and the famous problem of the duplication of the See also:cube reduced to the determination of two mean proportionals between two given straight lines. Notwithstanding this amount of discovery, and all that it implied, Euclid must have made a See also:great advance beyond his predecessors (we are told that " he arranged the discoveries of See also:Eudoxus, perfected those of Theaetetus, and reduced to invincible demonstration many things that had previously been more loosely proved "), for his Elements supplanted all similar See also:treatises, and, as See also:Apollonius received the title of " the great geometer," so Euclid has come down to later ages as the elementator." For the past twenty centuries parts of the Elements, notably the first six books, have been used as an introduction to geometry. Though they are now to some extent superseded in most countries, their See also:long retention is a See also:proof that they were, at any See also:rate, not unsuitable for such a purpose. They are, speaking generally, not too difficult for novices in the science; the demonstrations are rigorous, ingenious and often elegant; the mixture of problems and theorems gives perhaps some variety, and makes their study less monotonous; and, if regard be had merely to the metrical properties of space as distinguished from the graphical, hardly any See also:cardinal geometrical truths are omitted. With these excellences are combined a See also:good many defects, some of them inevitable to a See also:system based on a very few axioms and postulates. Thus the arrangement of the propositions seems arbitrary; associated theorems and problems are not grouped together; the See also:classification, in See also:short, is imperfect. Other objections, not to mention See also:minor blemishes, are the prolixity of the See also:style, arising partly from a defective nomenclature, the treatment of See also:parallels depending on an See also:axiom which is not axiomatic, and the sparing use of superposition as a method of proof. Of the See also:thirty-three See also:ancient books subservient to geometrical analysis, Pappus enumerates first the Data (Deboje a) of Euclid. He says it contained 90 propositions, the See also:scope of which he describes; it now consists of 95. It is not easy to explain this discrepancy, unless we suppose that some of the propositions, as they existed in the time of Pappus., have since been split into two, or that what were once scholia have since been erected into propositions. The object of the Data is to show that when certain things—lines, angles, spaces, ratios, &c.—are given by See also:hypothesis, certain other things are given, that is, are determinable.

The book, as we are expressly told, and as we mdy gather from its contents, was intended for the investigation of problems; and it has been conjectured that Euclid must have extended the method of the Data to the investigation of theorems. What prompts this conjecture is the similarity between the analysis of a theorem and the method, See also:

common enough in the Elements, of reductio ad absurdum—the one setting out from the supposition that the theorem is true, the other from the supposition that it is false, thence in both cases deducing a See also:chain of consequences which ends in a conclusion previously known to be true or false. The Introduction to See also:Harmony (Elaaycayil ap/.iOV HOD, and the See also:Section of the See also:Scale (Kararo,uh Kavovos), treat of See also:music. There is good reason for believing that one at any rate, and probably both, of these books are not by Euclid. No mention is made of them by any writer previous to Ptolemy (A.D. 140), or by Ptolemy himself, and in no ancient codex are they ascribed to Euclid. The Phaenomena (cawoµeva) contains an exposition of the appearances produced by the See also:motion attributed to the See also:celestial See also:sphere. Pappus, in the few remarks prefatory to his See also:sixth book. complains of the faults, both of omission and See also:commission, of writers on See also:astronomy, and cites as an example of the former the second theorem of Euclid's Phaenomena, whence, and from the See also:interpolation of other proofs, See also:David See also:Gregory infers that this See also:treatise is corrupt. The See also:Optics and Catoptrics ('Oirruua, Ka-roirrpuKh) are ascribed to Euclid by Proclus, and by See also:Marinus in his See also:preface to the Data, but no mention is made of them by Pappus. This latter circumstance, taken in connexion with the fact that two of the propositions in the sixth book of the Mathematical Collection prove the same things as three in the Optics, is one of the reasons given by Gregory for deeming that work See also:spurious. Several other reasons will be found in Gregory's preface to his edition of Euclid's works. In some editions of Euclid's works there is given a book on the Divisions of Superficies, which consists of a few propositions, showing how a straight line may be See also:drawn to See also:divide in a given ratio triangles, quadrilaterals and pentagons.

This was supposed by See also:

John See also:Dee of See also:London, who transcribed or translated it, and entrusted it for publication to his friend Federico Commandino of See also:Urbino, to be the treatise of Euclid referred to by Proclus as Tb aepi &aip&r wv (3g3Xlov. Dee mentions that, in the copy from which he wrote, the book was ascribed to Machomet of See also:Bagdad, and adduces two or three reasons for thinking it to be Euclid's. This See also:opinion, however, he does not seem to have held very strongly, nor does it appear that it was adopted by Commandino. The book does not exist in Greek. The fragment, in Latin, De See also:levi et ponderoso, which is of no value, and was printed at the end of Gregory's edition only in See also:order that nothing might be See also:left out, is mentioned neither by Pappus nor Proclus, and occurs first in See also:Bartholomew Zamberti's edition of 1537. There is no reason for supposing it to be genuine. The following works attributed to Euclid are not now extant: 1. Three books on Porisms (Hepi Twv Tropes shrwv) are mentioned both by Pappus and Proclus, and the former gives an abstract of them, with the lemmas assumed. (See See also:PoRISM.) 2. Two books are mentioned, named T&&rwv srpbs irt0avei¢; which is rendered Locorum ad supericiem by Commandino and subsequent geometers. These books were subservient to the analysis of loci, but the four lemmas which refer to them and which occur at the end of the seventh book of the Mathematicab Collection, throw very little See also:light on their contents. R.

See also:

Simson's opinion was that they treated of curves of See also:double curvature, and he intended at one time to write a treatise on the subject. (See Trail's Life of Dr Simson). 3. Pappus says that Euclid wrote four books on the Conic Sections (f3tAi.a rkaapa KWVCK&JV), which Apollonius amplified, and to which he added four more. It is known that, in the time of Euclid, the See also:parabola was considered as the section of a right-angled cone, the See also:ellipse that of an acute-angled cone, the hyper-bola that of an obtuse-angled cone, and that Apollonius was the first who showed that the three sections could be obtained from any cone. There is good ground therefore for supposing that the first four books of Apollonius's Conics, which are still extant, resemble Euclid's Conics even less than Euclid's Elements do those of Eudoxus and Theaetetus. 4. A book on Fallacies (Hepi 1/ev&apiwv) is mentioned by Proclus, who says that Euclid wrote it for the purpose of exercising beginners in the detection of errors in reasoning. This See also:notice of Euclid would be incomplete without some See also:account of the earliest and the most important editions of his works. Passing over the commentators of the Alexandrian school, the 'first See also:European translator of any See also:part of Euclid is See also:Boetius (500), author of the De consolatione philosophiae. His Euclidis Megarensis geometriae libri duo contain nearly all the See also:definitions of the first three books of the Elements, the postulates, and most of the axioms. The enunciations, with diagrams but no proofs, are given of most of the propositions in the first, second and See also:fourth books, and a few from the third.

Some centuries afterwards, Euclid was translated into Arabic, but the only printed version in that See also:

language is the one made of the thirteen books of the Elements by Nasir AI-Din Al-Tusi (13th century), which appeared at See also:Rome in 1594. The first printed edition of Euclid was a See also:translation of the fifteen books of the Elements from the Arabic, made, it is supposed, by See also:Adelard of See also:Bath (12th century), with the comments of Campanus of See also:Novara. It appeared at See also:Venice in 1482, printed by Erhardus Ratdolt, and dedicated to the See also:doge Giovanni See also:Mocenigo. This edition represents Euclid very inadequately; the comments are often foolish, propositions are sometimes omitted, sometimes joined together, useless cases are interpolated, and now and then Euclid's order changed. The first printed translation from the Greek is that of Bartholomew Zamberti, which appeared at Venice in 1505. Its contents will be seen from the title: Euclidis megaresis philosophi platonici Mathematicaru3 disciplinaru Janitoris: Habent in hoc volumine quicugg ad mathematic¢ substantici aspir¢t: elemetorum libros xiii cu expositione Theonis insignis mathematici . Quibus . adjuneta. Deputatum scilicet Euclidi volume xiiii cu expositioe Hypsi. Alex. Itideg3 Phaeno. Specu. Perspe. cum expositione Theonis ac mirandus ille See also:liber Datorum cum expositioe Pappi Mechanici una cu See also:Marini dialectici protheoria.

See also:

Bar. Zaber. Vene. Interpte. The first printed Greek See also:text was published at See also:Basel, in 1533, with the title EUKXei8ov ITOLxE7.W1, /3113X. LE K TWV OEWVOS O•VVOVQLWV. It was edited by See also:Simon See also:Grynaeus from two See also:MSS. sent to him, the one from Venice by See also:Lazarus Bayfius, and the other from See also:Paris by John Ruellius. The four books of Proclus's commentary are given at the end from an See also:Oxford MS. supplied by John Claymundus. The See also:English edition, the only one which contains all the extant works attributed to Euclid, is that of Dr David Gregory, published at Oxford in 1703, with the title, EUKXeteov Td owj'6µeva. Euclidis quae supersunt omnia. The text is that of the Basel edition, corrected from the MSS. bequeathed by See also:Sir See also:Henry See also:Savile, and from Savile's annotations on his own copy. The Latin translation, which accompanies the Greek on the same See also:page, is for the most part that of Commandino.

The See also:

French edition has the title, See also:Les fEuvres d'Euclide, traduites en Latin et en See also:Francois, d'apres un manuscrit tres-ancien qui etait reste inconnu jusqu'd•nos jours. See also:Par F. Peyrard, Traducleur See also:des Leuvres d'Archimede. It was published at Paris in three volumes, the first of which appeared in 1814, the second in 1816 and the third in 1818. It contains the Elements and the Data, which are, says the editor, certainly the only works which remain to us of this ever-celebrated geometer. The texts of the Basel and Oxford editions were collated with 23 MSS., one of which belonged to the library of the Vatican, but had been sent to Paris by the See also:comte de Peluse (See also:Monge). The Vatican MS. was supposed to date from the 9th century; and to its readings Peyrard gave the greatest See also:weight. What may be called the See also:German edition has the title EUKXEteov Erolxeia. Euclidis Elementa ex optimis libris in usum Tironum Graece edita ab Ernesto Ferdinando See also:August. It was published at See also:Berlin in two parts, the first of which appeared in 1826 and the second in 1829. The above mentioned texts were collated with three other MSS. See also:Modern See also:standard editions are by,Dr See also:Heiberg of See also:Copenhagen, Euclidis Elementa, edidit et Latine interpretatus est J.

L. Heiberg. vols. i.-v. (Lipsiae, 1883-1888), and by T. L. See also:

Heath, The Thirteen Books of Euclid's Elements, vols. i.-iii. (See also:Cambridge, 1908). Of See also:translations of the Elements into modern See also:languages the number is very large. The first English translation, published at London in 1570, has the title, The Elements of Geometrie of the most auncient Philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, See also:Citizen of London. Whereunto are annexed certain Scholies, Annotations and Inventions, of the best Mathematiciens, both of time past and in this our See also:age. The first French translation of the whole of the Elements has the title, Les Quinze Livres des Elements d'Euclide.

Traduicts de Latin en Francois. Par D. Henrion, Mathematicien. The first edition of it was published at Paris in 1615, and a second, corrected and augmented, in 1623. See also:

Pierre Forcadel de Bezies had published at Paris in 1564 a translation of the first six books of the Elements, and in 1565 of the seventh, eighth and ninth books. An See also:Italian translation, with the title, Euclide Megarense acutissimo philosopho See also:solo introduttore delle Scientie Mathematice. Diligentemente rassettato, et ally integrity ridotto, per it degno professore di tai Scientie See also:Nicol() Tartalea Brisciano, was published at Venice in 1569, and Federico Commandino's translation appeared at Urbino in 1575; a See also:Spanish version, Los Seis Libros primeros de la geometria de Euclides. Traduzidos en legua Espanola See also:por Rodrigo Camorano, Astrologo y Mathematico, at See also:Seville in 1576; and a See also:Turkish one, translated from the edition of J. Bonnycastle by Husain Rifki, at Bulak in 1825. Dr See also:Robert Simson's editions of the first six and the See also:eleventh and twelfth books of the Elements, and of the Data.

End of Article: EUCLID

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