A D V E N T U R E S in C Y B E R S O U N DEuclid (alt: Euklid, Eucleides) : 365 - 300 BC
Euclid's The Optics is the earliest surviving work on geometrical optics, and is generally found in Greek manuscripts along with elementary works on spherical astronomy. There were a number of medieval Latin translations, which became of new importance in the fifteenth century for the theory of linear perspective.
Euclid, Greek Eucleides (fl. c. 300 BC, Alexandria), the most prominent mathematician of Greco-Roman antiquity, best known for his treatise on geometry, the Elements.
Of Euclid's life it is known only that he taught at and founded a school at Alexandria in the time of Ptolemy I Soter, who reigned from 323 to 285/283 BC. Medieval translators and editors often confused him with the philosopher Eucleides of Megara, a contemporary of Plato about a century before, and therefore called him Megarensis. Writing in the 5th century AD, the Greek philosopher Proclus told the story of Euclid's reply to Ptolemy, who asked whether there was any shorter way in geometry than that of the Elements - "There is no royal road to geometry." Another anecdote relates that a student, probably in Alexandria, after learning the very first proposition in geometry, wanted to know what he would get by learning these things, whereupon Euclid called his slave and said, "Give him threepence since he must needs make gain by what he learns."
Euclid compiled his Elements from a number of works of earlier men. Among these are Hippocrates of Chios (5th century BC), not to be confused with the physician Hippocrates of Cos (flourished 400 BC). The latest compiler before Euclid was Theudius, whose textbook was used in the Academy and was probably the one used by Aristotle. The older elements were at once superseded by Euclid's and then forgotten. For his subject matter Euclid doubtless drew upon all his predecessors, but it is clear that the whole design of his work was his own. He evidently altered the arrangement of the books, redistributed propositions among them and invented new proofs if the new order made the earlier proofs inapplicable. Thus, while Book X was mainly the work of the Pythagorean Theaetetus (flourished 369 BC), the proofs of several theorems in this book had to be changed in order to adapt them to the new definition of proportion developed by Eudoxus (q.v.). According to Proclus, Euclid incorporated into his work many discoveries of Eudoxus and Theaetetus. Most probably Books V and XII are the work of Eudoxus, X and XIII of Theaetetus. Book V expounds the very influential theory of proportion that is applicable to commensurable and incommensurable magnitudes alike (those whose ratios can be expressed as the quotient of two integers and those that cannot). The main theorems of Book XII state that circles are to one another as the squares of their diameters and that spheres are to each other as the cubes of their diameters. These theorems are certainly the work of Eudoxus, who proved them with his "method of exhaustion," by which he continuously subdivided a known magnitude until it approached the properties of an unknown. Book X deals with irrationals of different classes. Apart from some new proofs and additions, the contents of Book X are the work of Theaetetus; so is most of Book XIII, in which are described the five regular solids, earlier identified by the Pythagoreans. Euclid seems to have incorporated a finished treatise of Theaetetus on the regular solids into his Elements. Book VII, dealing with the foundations of arithmetic, is a self-consistent treatise, written most probably before 400 BC. Other books of the Elements are not on this high mathematical level. In Book VIII, the second of the three arithmetical books, are found cumbersome enunciations, needless repetitions, and even logical fallacies. Apparently Euclid's exposition excelled only in those parts in which he had excellent sources at his disposal.
In ancient times, Hero and Pappus of Alexandria and Proclus and Simplicius all wrote commentaries. Theon of Alexandria (4th century AD) brought out a new revision of the work with textual changes and some additions; his version was the basis of all published Greek texts and translations until, early in the 19th century, an important Greek manuscript containing an ante-Theonine text was discovered in the Vatican. Three Arabic translations were made in the middle ages: by al-Hajjaj ibn Yusuf ibn Matar, first for the Abbassid caliph Harun ar-Rashid (ruled 786-809) and again for the caliph al-Ma`Mun (ruled 813-833); by Hunayn ibn Ishaq (ruled 808-873), in Baghdad, whose translation was revised by Thabit ibn Qurrah (died 901); and by Nasir ad-Din at-Tusi in the 13th century. Euclid was first made known in the West through Latin translations of these Arabic versions. The first extant Latin translation of the Elements was made about 1120 by Adelard of Bath, who obtained a copy of an Arabic version in Spain, where he travelled while disguised as a Muslim student. Adelard also composed an abridged version and an edition with commentary. Hermann of Carinthia translated Books I-XII from the same Arabic version. Gerard of Cremona (c. 1114-87) translated the 15 books of Euclid from the Ishaq-Thabit version. The first Latin translation to be printed was by Johannes Campanus in the 13th century. The first direct translation from the Greek without the Arabic intermediary was made by Bartolomeo Zamberti and published in Vienna in Latin in 1505; and the editio princeps of the Greek text was published at Basel in 1533 by Simon Grynaeus. But the most important Latin translation of this period was by Federico Commandino in 1572. The first edition of the complete works of Euclid was the Oxford edition of 1703, in Greek and Latin, by David Gregory. All texts are now superseded by Euclidis Opera Omnia (8 vol. and a supplement, 1883-1916), edited by J.L. Heiberg and H. Menge. The first English translation of the Elements was by Sir Henry Billingsley. The many later editions include Robert Simson's in Latin and English, containing Books I-VI, XI, XII, and the Data, in 1756; and the definitive The Thirteen Books of Euclid's Elements by T.L. Heath, with introduction and commentary (3 vol., 1908; 2nd ed., 1926).
Other extant works of Euclid include two belonging to elementary geometry: the Data, containing 94 propositions, which demonstrates that, if certain elements in a figure are given, then other things are given, i.e., can be determined; and a book On Divisions (of figures), discovered in both Arabic and Latin versions and restored and edited in 1915, which deals with problems of dividing a given figure by one or more straight lines into parts that are equal or that have given ratios to one another or to other given areas. The Optics of Euclid is extant in Greek in two forms, one being Euclid's own treatise, and the other a critical revision by the Greek writer Theon. The Catoptrica ("Reflections") is not by Euclid but is, rather, a later compilation from ancient works on the subject. The Phaenomena, extant in Greek, is a treatise on the geometry of the sphere for use in astronomy and is similar in content to the work, by Autolycus of Pitane, Moving Sphere. The Elements of Music is attributed to Euclid by Proclus and Marinus, the latter being another Greek commentator. Included in this work are two treatises that probably are not by Euclid: the Sectio canonis ("Division of the Scale"), which gives the Pythagorean theory of music with some later additions; and the Introductio harmonica ("Introduction to Harmony"), written by Cleonides, a student of Aristoxenus, in which an identifiable tone separates notes on the scale. Four lost works in geometry are described in Greek sources and attributed to Euclid. The purpose of the Pseudaria ("Fallacies"), it is said, was to distinguish and to warn beginning students against different types of fallacies to which they might be susceptible in geometrical reasoning. The Porisms, in three books, was an advanced work of which Pappus gave a summary account. Although the word means "corollaries," Euclid apparently meant a statement that was intermediate in significance between a problem and a theorem. The Conics, made up of four books on conic sections, corresponded in content to the first four books of Apollonius' Conics, although Apollonius added new theorems to his own treatment. Euclid called the conics by their previous designations, sections of a right-angled cone, an obtuse-angled cone, and an acute-angled cone, respectively; it was Apollonius who first gave them the names parabola, hyperbola, and ellipse, the descriptions of which he derived. Pappus also mentioned the Surface-loci, which is made up of two books that probably dealt with loci on surfaces, perhaps also loci which are surfaces, and with conic sections. A fragment in Latin, De levi et ponderoso, which is included in Gregory's edition of Euclid, contains a statement of the principles of Aristotle's dynamics but is not by Euclid.
Almost from the time of its writing and lasting almost to the present, the Elements has exerted a continuous and major influence on human affairs. It was the primary source of geometric reasoning, theorems, and methods at least until the advent of non-Euclidean geometry in the 19th century. It is sometimes said that, next to the Bible, the Elements may be the most translated, published, and studied of all the books produced in the Western world. Euclid may not have been a first-class mathematician. He certainly was, however, a first-class teacher of mathematics, inasmuch as his textbook has remained in use practically unchanged for more than 2,000 years.
Euclid's extant works are collected in Euclidis Opera Omnia, ed. by J.L. Heiberg and H. Menge, 8 vol. and a supplement (1916), containing... The Elements, Books I-XIII (The Thirteen Books of Euclid's Elements, trans. by Sir Thomas Heath, 1952) Data (Euclid's Data, restored to their true and genuine order, trans. by R. Jack, 1756) Euclid's Book on Divisions of Figures, ed. by R.C. Archibald (1915) The Optics The Phaenomena
Sarton, George, Introduction to the History of Science, vol. 1, pp. 153-156 (1927, reprinted 1968) Heath, Thomas L., A History of Greek Mathematics, vol. 1, pp. 354-446 (1921, reprinted 1965) Lejeune, Albert, Euclide et Ptolémée, deux stades de l'optique géométrique grecque (1948) Clagett, Marshall, The Medieval Latin Translations from the Arabic of The Elements of Euclid, with Special Emphasis on the Versions of Abelard of Bath, Isis, 44:16-42 (1953) Morrow, Glenn R., (ed. and trans.), Proclus' Commentary on the First Book of Euclid's Elements (1970)
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