GEOMETHY. 



GEOMETRY. 



are 



evident that the arithmetical character of geometrical magnitude 

 bad been very extensively conaidered; and it seems to us sufficiently 

 clear that an arithmetic of a character approximating closely to 

 algebra must hare been the guide, as well as that some definite 

 object was sought perhaps the attainment of the quadrature of the 

 circle. 



Book zi. lays down the definitions of solid geometry, or of geometry 

 which considers lines in different planes and solid figures. It then 

 proceeds to treat of the intersections of planes, ami of the proiwrtiet) 

 of paraUelopipedd, or what might be called solid rectangle* or right 

 solids. 



Book zii. treats of prisms, cylinders, pyramids, and cones, establishing 



the properties which are analogous to those of triangles, &c., in the 

 first and sixth books. It also shows that circles are to one another as 

 the squares on their diameters, and spheres as the cubes on tm-ir 

 diameters, in which, for the first time in Euclid, the celebrated method 

 of EXHAUSTIONS is employed, which, with the theory of proportion, 

 forms the most remarkable part of this most remarkable work. In the 

 article just cited we have referred to the present one for some account 

 of this method, which we now give. 



The only method of reasoning upon the length, area, or solidity, of 

 curve lines or surfaces, is by observing the properties of inscribed 

 polygons or polyhedrons, which may, by sufficiently increasing the 

 number of their sides or faces, be made to approach as near as we 

 pleasu to continuous curviliuearity. But since the rigour of geometry 

 is not content with proving that a proposition may be considered as 

 nearly true as we please, and will not infer that one line is equal 

 to another because it can be shown that their difference is (no 

 matter how) small ; Euclid (or some of his predecessors, but most 

 probably Euclid, if we may judge by the character of his dis- 

 coveries given by Proclus) invented this method of exhaustions, 

 which may be considered as contained in the following two pro- 

 positions. 



I. If from A more than its half be taken, and from the remainder 

 more than its half, and so on, the remainder will at last become less 

 than 1), where B is any magnitude named at the outset (and of the 

 came kind as A), however small. This proposition may be easily 

 proved, and is equally true if the fixed proportion abstracted each time 

 be half or less than half. 



II. Let there, be two magnitudes, F and q, both of the same kind; 

 and let a succession of other magnitudes, called x,, x,, X, ... be 

 each nearer and nearer to r, so that any one, x ., shall differ from P 

 less than half as much as its predecessor differed. Let Y,, T-, Y, . . . be 

 a succession of quantities similarly related to Q ; and let the ratios of 

 X, to T,, of x, to v,, and so on, be all the same with each other, and 

 the same with that of A to B. Then it must be that r is to Q as A to B. 

 (It is obvious, from the conditions, that if x, be greater than P, v, is 

 greater than q, *c.,&c.) Suppose x,, x,, &c., less than r, and therefore 

 T,, T,, &c., less than q. Then, if A be nut to B as P to Q, A in to B as 

 P to some other quantity 8 greater or less than q : say less than Q. 

 Then (by hypothesis and I.), we can find some one of the series Y,, 

 T, . . . (say v . ) which is nearer to q than 8 is to q ; and which is 

 therefore greater than s. Then, since x. is to Y. as A to B, or as p to 

 8, we have x. is to Y. as P to s, or x. to r as Y to s : from which, 

 since x. is lees than p, T. is less than s. But Y. is also greater than s, 

 which is absurd ; therefore A is not to B as p to less than q. Neither 

 is A to B as P to more than q (which call s), for in that case s is to p as 

 BtoA:letsbetopasqtoT, then sistoqasPtoT; from which, 

 8 being greater than q, P is greater than T. But BiatoAosstop, 

 that is, as q to less than p, which is proved to be impossible by the 

 reasoning of the last case. Consequently, A is not to B as P to 

 more than q, or to less than q ; that is, A is to B as p to q. Which 

 was to be shown. Let P and q be two circles, A and u the squares on 

 their diameters, x, and Y, inscribed squares, x, and Y, inscribed 

 regular octagons, x, and Y, inscribed regular figures of sixteen sides, 

 ic : the preceding process gives the proof that circles are to one 

 another as the squares on their diameters. 



Book xiii., the last of those written by Euclid, applies some results 

 of the tenth book to the sides of regular figures, and shows how to 

 describe the five regular bodies. [Soucs, REGULAR.] 



Books xiv. and xv., attributed to Hypnicles of Alexandria, treat 

 entirely of the relative proportions of the five regular solids, and of 

 their inscription in one another. 



The writings of Euclid continued to be the geometrical standard as 

 long as the Greek language was cultivated. The Romans never made 

 any progress in mathematical learning. Boethius [Bo ETUI us, in Bioo. 

 Div.J translated, it is said, the first book of Euclid (Casciodorus, cited 

 by Heilhronner) ; but all which has come down to us on the subject 

 from this writer (who lived at the beginning of the 6th century) is 

 contained in two books, the first of which has the enunciations and 

 figures of the principal prujKisitions of the first four books of the 

 Elements, and the second of which is arithmetical. Some of the 

 manuscript* of this writer contain an appendix which professes to give 

 an account of a letter of Julius Cscsar, in which he expresses his 

 intention of cultivating geometry throughout the Roman dominions. 

 But no such result ever arrived so long as the Western Empire lasted; 

 and this short account of Koman geometry is a larger proportion ..i tin- 

 present article than the importance of the subject warrant,". These 



books of Boethius continued to be the standard text-books until Euclid 

 was brought in again from the Arabs. 



Among the last-mentioned race geometry made no actual progress, 

 though many of the works of the Greek writers were translated, and 

 Euclid among the rest. There are several Arabic versions, the most 

 perfect of which is that of Othman of Damascus, who augmented the 

 usual imperfect translations by means of a Greek manuscript which he 

 saw at Rome. D'Herbelot (at the words Aklides and Oclides) states 

 that the Orientals believe Euclid to have been a native of Tyre, and 

 al> that they frequently gave his name to the science which he taught. 

 The same author gives the name of the Arabic versions, one of which, 

 that of Nasir-eddin, the most celebrated of all, was printed at the 

 Medicean press at Rome in 1594. The astronomer Thabet ben Korrah 

 [ASTRONOMY] was one of the translators, or rather, perhaps, revised the 

 translation of Honein ben Ishak, who died A.D. 873. There is a manu- 

 script in the Bodleian Library, purporting to be the translation of the 

 latter edited by the former. 



The first translation of Euclid into Latin, of which the date can be 

 tolerably well fixed, is that of Athelard, or Adelard, a monk of Bath, who 

 lived under Henry I. (about A.D. 1150). We have given [CAMPA 

 Bioo. Di v.] a summary of authorities to show that Campanus, supposed 

 to be another translator of Euclid, lived after this period ; but we are 

 inclined to believe that this translation (so called) of Campanux (printed 

 A.D. 1482), is in fact that of Athelard, with a Commentary by Cam- 

 panus. For Campanus is not expressly described in the book as a 

 translator (see article cited), but as a 'commentator; add to which, 

 that there is in the Bodleian Library a manuscript entitled ' Euclidis, 

 &c.. ex versione Adelardi de Arabico, una cum commento Magistri 

 Campani Novarieusis.' Scheibel (cited by Camerer and Hauber, in 

 the preface of then- edition, presently noticed) states that in his copy 

 of Campanus the fact of the translation being that of Adelard was 

 noted in a handwriting apparently as old as the edition itself. The 

 point might be settled by a comparison of the printed Campanus and 

 the manuscript in the Bodleian. With regard to this version, it U 

 stated (in the preface just cited) that it differs from the one of Nasir- 

 eddin. With the precedent just cited, we may be allowed to state 

 that in a copy of Campanus which we have examined, some ancient 

 handwriting, completely obliterated, is attached to Ratdolt's preface. 

 Chemical means have succeeded in recovering a few unconnected words 

 only, among which are " ben Honein " and " Tebit ben Corra," ex- 

 pressing perhaps the opinion of the writer that the version chosen by 

 Adelard or Campanus was that of the two Eastern editors who have 

 been previously mentioned. 



There is a considerable number of Greek manuscripts of the Elements, 

 for which see Fabricius, Heilbronner, and the preface of Peyrard. There 

 is no account of the manuscripts which they consulted by the earlier 

 Latin translators (from the Greek), nor by Gregory. It appears how- 

 ever that several, if not many, of the manuscripts are entitled EuxAtiSoi/ 

 tTToix'iw /3i/3Aio if in ruf Siayos avrovauan, from which it was in: 

 that the compilation of the elements was the work of Theon, from the 

 materials left by Euclid. It is certain that Theon, in his commentary 

 On the Almagest, speaks of his edition (Moan) of Euclid, and mentions 

 that the part of the last proposition which relates to the sectors was 

 added by himself. On looking at that proposition, it is found that the 

 demonstration relative to the sectors comes after the " iwtp ISti 8<t<u," 

 with which Euclid usually ends his propositions. And Alexander, the 

 commentator on Aristotle, who lived before Theou, calls that the fourth 

 prohibition of the tenth book which is the fifth in all the manuscripts. 

 \\ r i in tlifii oi.-tinctly trace the hand of Theon as a commentator, 

 anil may suspect that he performed the duty of a revising editor to 

 the work of Euclid as it now appears ; but there is not the smallest 

 reason to suppose that Theon actually digested the work into tin turn 

 which it now has. These remarks relative to the claims of Theou 

 were first made by Sir Henry Savile, who opened the chair of geometry 

 which he founded at Oxford by thirteen lectures on the fundamental 

 parts of the first book of Eucid, which were delivered in 1620, and 

 published in 1621. 



We now give a short summary of the early editions of Euclid, which 

 have appeared in Greek or Latin. It is unnecessary to .-p. . ii\ tlie 

 common editions of Siuisou, Playfair, Ac., &c., which confine tln-m- 

 selves to the first six books, and the eleventh and twelfth, and arc 



grin-rally know 11. 



I. Editions of the whole of Euclid's works. An imperfect Latin 

 edition, by Bartholomew Zamberti, Venice, 1505. But the principal 

 edition of all the works of Euclid is that published by the Oxford 

 press in 1703, under the care of David Gregory, then Savilian pro 



|, .,,,!-. 



II. Greek editions of the Elements only. (1.) An edition curt 

 Simonis Urymci, Basel, 1533, with the commentary of Proclus. (2.) 

 The Paris edition by Peyrard, 1814-18, in three volumes quail. ., rn- 

 taining the Elements and Data, Greek and French. It in the tn.-f. 

 edition which has readings from various manuscripts. (3.) The 1'., i hn 

 edition, liy August, 1826, 8vo., containing the thirteen books of the 

 Element*, ill Greek, with a nelection of readings from Peyrard and 

 iioni other manuscripts. 



HI. Latin editions of the Elements only. (1.) That of Cainpouux, 



the first Eurlid printed, Ratdolt, Venice, 1482. (2.) A reprint of the 



.;ig, marked ' YinucuUu:, auiio nalutis, 14U1.' (3.) Au edition 



