# Euclid

EUCLID (in Greek Eukleides), Greek mathematician (fl. about 3oo B.e.), probably received his early mathematical training at Athens from the pupils of Plato ; but we know nothing for cer tain of the circumstances of his life except that he taught and founded a school at Alexandria in the time of Ptolemy I., who reigned from 3o6 to 283 B.C. Mediaeval translators and editors mostly called him Megarensis, through confusion with the philoso pher Eucleides of Megara, Plato's contemporary ; this error was finally exposed by Commandinus in 1572. Proclus tells the story of Euclid's reply to King 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 pupil, after learning the very first proposition in geometry, wanted to known 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's great work is the Elements (o rotxaa) (see GEOMETRY), in 13 books; of the books formerly purporting to be books xiv., xv., the first, by Hypsicles (2nd century B.c.) adds some interest ing theorems about the regular solids, two of which it attributes to Aristaeus and Apollonius respectively; the second, much in ferior, was written, at least in part, by a pupil of Isidorus of Miletus in the 6th century A.D.

The names of some earlier compilers of Elements are handed down; the first was Hippocrates of Chios (5th century B.e.), famous for his quadrature of certain lunes, intended to lead up to the squaring of a circle ; the latest before Euclid was Theudius, whose text-book was that in use in the Academy and was probably that from which Aristotle drew his illustrations. The older ele ments were at once superseded by Euclid's and then forgotten. For his subject-matter Euclid doubtless drew upon all his pre decessors ; but it is clear that the whole design of his work was his own ; he evidently altered the arrangement of whole books, redis tributed propositions between them, and invented new proofs where the new order made the earlier proofs inapplicable ; his changes began at least as early as i. 5 (the pons asinorum), since Aristotle cites a different proof of that theorem. He incorporated, too, the splendid new discoveries of Eudoxus and Theaetetus ; book v. expounds Eudoxus's wonderful theory of proportion applicable to commensurable and incommensurable magnitudes alike ; books x. and xiii. owe much to the original investigations of Theaetetus into (a) irrationals of different classes and (b) the geometry of the five regular solids, while book xii. uses Eudoxus's "method of exhaustion" for the purpose of proving that circles are to one another as the squares, and spheres are to one another as the cubes, on their diameters, and of finding the solid content of a pyramid, a cone and a cylinder. Books i.–iv. and vi. may be said to represent, roughly, the essence of the Pythagorean geometry, while books vii.–ix. on the elementary theory of numbers again owe something to the Pythagoreans.

It seems clear that the famous postulate 5 (the parallel postu late) is due to Euclid himself. No trace of such a postulate appears in Aristotle ; Euclid then, realizing that some postulate is necessary to establish the theory of parallels, deliberately framed one, stating it in the direct form most useful for his purpose in that it gives a criterion by which to judge whether two lines drawn in a construction will or will not meet. The use of the postulate or some equivalent is the mark of Euclidean geometry. Many at tempts to prove the postulate were made by prominent mathemati cians from ancient times onwards. Gauss was the first to affirm, and Beltrami the first to establish (1868), the impossibility of proving it. The first to consider seriously the possibility of other hypotheses was Saccheri (Euclides ab omni aevo vindicatus, 1733), though he tried to prove that Euclid's was the only true one. Lobachewsky (1826) and (about the same time) J. Bolyai were the first to work out systematically a non-Euclidean geom etry, while Riemann developed another in It is safe to say that no other scientific text-book in the world has remained in use practically unchanged for more than 2,000 years. In this country it was not till the middle of the 19th century that a so-called "away from Euclid" movement began, which has led to the appearance of a multitude of rival text-books of geometry giving the substance of Euclid's early books in so many different forms as to produce a state of chaos in geometrical teaching which calls for remedy ; but the text-book that shall really replace Euclid has not yet been written and probably never will be.

The following are the other works of Euclid which are extant ; the first two belong to elementary geometry.

I. The Data, containing 94 propositions, has for its object to prove that, if certain elements in a figure are given, then other things are given, i.e., can be determined. A systematic collection of such results must obviously be of great use in facilitating and shortening the analysis preliminary to the solution of a problem. The following is an example : I f two straight lines contain a given area in a given angle, and if the difference, or the sum, of them be given, then shall each of them be given; this gives the geo metrical solution of the simultaneous algebraical equations or the equivalent quadratic equation in a single variable.

2. A book On divisions (of figures) was discovered in Arabic at Paris and edited by Woepcke in 1851. John Dee was the first to find (in Latin) a similar treatise by one Muhammad Bagdadinus which, in 1563, he handed to Commandinus, who published it in Dee's name and his own in 157o. The genuine treatise has now been restored and edited by R. C. Archibald (Cambridge, . The type of problem dealt with is that of dividing a given figure (e.g., a triangle, a parallelogram, a quadrilateral, a circle, or a figure bounded by an arc of a circle and two straight lines) by one or more straight lines into parts equal, or having given ratios, to one another or to other given areas.

3. The Optics of Euclid is extant in Greek in two forms, one being Euclid's own treatise and the other a recension by Theon. The Catoptrica (edited by Heiberg in the same volume) is not by Euclid but is a later compilation from ancient works on the subject.

## 4. The

Phaenomena, extant in Greek, is a treatise on the geo metry of the sphere intended for use in astronomy, and is similar in content to the work of Autolycus On the Moving Sphere.

5. A work on the Elements of Music is attributed to Euclid by Proclus and Marinus. Of two extant treatises of the kind the first, Sectio Canonis, giving the Pythagorean theory of music, is scarcely Euclid's in its present form, but may have been ab stracted from the genuine Elements of Music by some less quali fied editor. The Introductio harmonica is not by Euclid but by Cleonides, a pupil of Aristoxenus.

Of lost geometrical works by Euclid all except one, the Pseu daria, belonged to higher geometry.

I. The purpose of the Pseudaria was, we are told, to distinguish, and to warn beginners against, different types of fallacies to which they are liable in geometrical reasoning unless they have firmly grasped the principles and are guided by them alone.

2. The Porisms, in three books, was an advanced work of which Pappus gives a summary account with lemmas designed for use with it (see PoRIsM).

3. The Conics, in four books, corresponded in content to the first four books of Apollonius's Conics, though Apollonius added new theorems and generalized the treatment throughout. Euclid still called the conics by their old names, "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," "ellipse," arising out of his generation of them all from one circular cone, in general oblique. Euclid was, however, aware that an oblique section of any right cone or cylinder gives a "section of an acute-angled cone" (ellipse).

4. Regarding the Surface-Loci (Tbirot ran Eirt4av€14), in two books, mentioned by Pappus, we can only conjecture that the loci dealt with were loci on surfaces, perhaps also loci which are surfaces. Conics would appear to have entered into the subject, for one of the lemmas to the treatise given by Pappus contains a complete proof of the focus-directrix property of all three conics: a fact which further suggests that Euclid was acquainted with this property and assumed it as known (though it does not appear in Apollonius's Conics).

A fragment in Latin, De levi et ponderoso, included in Gregory's edition of Euclid, contains a statement of the principles of Aristotle's dynamics, but is not by Euclid; and there seems to be no independent evidence that Euclid wrote on mechanics at all.

This notice of Euclid would be incomplete without some ac count of the earliest and the most important translations and editions of the Elements. In ancient times Heron and Pappus of Alexandria, Proclus and Simplicius, all wrote commentaries. Theon of Alexandria (4th century A.D.) brought out a new recen sion of the work, with textual changes and some additions; Theon's version was the basis of all published Greek texts and translations therefrom until, early in the 19th century, Peyrard, discovered in the Vatican the great ms.gr. 190 containing an ante-Theonine text. Boetius (about 50o A.D.) is said to have trans lated the Elements into Latin, but the geometry of the Pseudo Boetius which we possess contains no more than fragments of such a translation (the definitions of book i., the postulates and axioms, the enunciations of the propositions of book i. and of some propositions of books ii., iii., iv., but no proofs, except of props. 1-3 of Book i.) . Arabic translations were made (1) by al-Hajjaj b. Yusuf b. Matar, first for Harun ar-Rashid (786-809) and again for al-Ma'Mun ; (2) by Ishaq b. Hunain (d. 91o) ; the latter translation was revised by Thabit b. Qurra (d. 901) ; the Ishaq-Thabit version and six books of the second (abridged) version by al-Hajjaj survive, the former in the Bodle ian, the latter at Leyden; (3) a third Arabic version was that of Nasiraddin at-Tusi (b. 1201) ; one form of this was printed at Rome in The first extant Latin translation of the Elements was made (about 1120) by Athelhard of Bath, who obtained a copy of an Arabic version in Spain, whither he went disguised as a Moham medan student. Next, Gherard of Cremona (1114-87) translated from the Arabic the "15 books of Euclid" as well as the com mentary on books i.—x. by an-Nairizi (about 900). The first Latin translation to be printed was that of Johannes Campanus (13th century) , also made from the Arabic; Campanus's transla tion was more complete than Athelhard's, but he evidently used the latter, since the definitions, postulates, axioms, and the 364 enunciations are word for word the same in both translations. The first printed edition of the Elements, containing Campanus's trans lation (now rare), is a beautiful production by Erhard Ratdolt (Venice, 1482) with margins of 24in. and with figures in the mar gin ; Ratdolt claims that no one before his time had been able to print diagrams like letters; 1482 saw two forms of the book (differing in the first sheet) ; others came out in 1486 and 1491. The first translation from the Greek was made by Bartolomeo Zamberti and appeared at Venice in 1505.

The editio princeps of the Greek text was brought out at Basle in 1533 by Simon Grynaeus the elder, with a preface addressed to a notable Englishman, Cuthbert Tonstall (14 5 7) . Unfor tunately the mss. used were two of the i6th century, which are among the worst.

The first English translation was that of Sir Henry Billingsley (1 S7o), who was lord mayor of London in 1596-97, a magnificent volume of 928 folio pages, with a preface by John Dee: The Elements of Geometrie of the most auncient Philosopher Euclide of Megara. Faithfully (now first) translated into the English toung, by H. Billingsley, Citizen of London.

The most important Latin translation is that of Commandinus (1572), which was closely followed by Gregory and all transla tors down to Peyrard, including Simson.

The great Oxford edition (1703) , in Greek and Latin, by David Gregory, was still, until the appearance of Heiberg and Menge's new text, the only edition of the complete works of Euclid. Peyrard's Greek text, published in three volumes between 1814 and 1818, and containing the Elements and the Data, represents the first approach to a better text, in so far as it adopted or recorded the readings of the Vatican ms.gr. 19o; it was accom panied by translations into Latin and French. The edition of books i.—vi., in Greek and Latin, by Camerer and Hauber (Berlin, 1824-25) is valuable for its exhaustive notes, while E. F. August's Greek text of books i.—xiii. (1826-29) still further improved upon Peyrard's. All texts are now superseded by Euclidis opera omnia (8 vols., Leipzig, 1883-1916) edited by Heiberg and Menge; vols. i.—v. contain the Elements with Latin translation, apparatus cri ticus, scholia, etc., and vols. vi.—viii. the other extant works, scholia, fragments, etc. Book i. has been separately edited, with introduction and notes, by T. L. Heath (Euclid in Greek Book I., Cambridge, 1918) and by G. Vacca (Firenze, 1916).

The number of editions in English is legion. We need here only mention Robert Simson's (first edition, in Latin and English, Glas gow, 1756, containing the Elements books i.—vi., xi., xii. and the Data) ; James Williamson's translations of the whole 13 books (vol. i., Oxford, 1781 ; vol. ii. London, 1788) ; T. L. Heath's The Thirteen Books of Euclid's Elements, translated from the text of Heiberg, with Introduction and Commentary, 3 vols. (Cam bridge, 1908; second edition revised, with additions, 1926). For more detailed accounts see Pauly-Wissowa's Realencyklopddie; G. Loria, Le scienze esatte nell' antics Grecia, pp. 188-268 (Milano, 1914) ; T. L. Heath, History of Greek Mathematics, vol. i., pp. (Oxford, 1921) . (T. L. H.)