EUCLID OF ALEXANDRIA. 



EUCLID OF ALEXANDRIA. 



not thrir extreme precuion in the statement of what they demand ; 

 for it frequently happens that a result i* appealed to as self-evident 

 which is not to be found iu the expressed axioms. Neither does their 

 fame arise from their never as*uming what might be proved ; for in 

 the very definitions we tiud it asserted that the diameter of a circle 

 huoots the figure, which might be readily proved from the axioms. 

 Neither is it the complete freedom from redundancy, nor the perfao- 

 tion of the) arrangement; for book i., prop. 4, which is very much out 

 of place, considering that it is never wanted in the first book, is, in 

 point of fact, proved again, though not expressed, in prop. 19. Neither 

 is it thei manner in which our ideas of magnitude are rendered com- 

 plete, as well a* definite : fur instance, book iii. prop. 20 U incomplete 

 without Euclid's definition and use of the term ' angle ; ' nor with that 

 term, as used by him, can the 21st proposition of that book be fully 

 donooMrated without the help of the subsequent 22nd. In fact, the 

 1 Element*' abound in defect*, which, if we may so speak, are clearly 

 seen by the light of their excellences ; the high standard of accuracy 

 which they inculcate in general, the positive and explicit statement 

 which they make upon all real and important assumptions, the natural 

 character of the arrangement, the complete and perfect absence of 

 falsa conclusion or fallacious reasoning, and the judicious choice of the 

 demonstration*, considered with reference to the want* of the beginner, 

 arc th* oum of the universal celebrity which this book has enjoyed. 

 We shall now describe the contents of the ' Element*.' 



There are thirteen book* certainly written by Euclid, and two more 

 (the fourteenth and fifteenth) which are supposed to have been added 

 by Hypaiolee of Alexandria (about A. D. 170). 



Buok L lays down the definitions and postulates required in the 

 establishment of plane geometry, a few definitions being prefixed also 

 to books il.iii., iv., and vl It then treats of such properties of straight 

 line* and triangle* as do not require any particular consideration of 

 the properties of the circle nor of proportion. It contains the cele 

 brated proposition of Pythagoras. 



From this book it appears that Euclid lays down, as all the instru 

 mental aid permitted in geometry, the description of a right line of 

 indefinite length, the indefinite continuation of such right line, and 

 the description of a circle with a given centre, the circumference of 

 which is to pass through a given point. It is usual to say, then, that 

 the rule and compasses are the instrument* of Euclid's geometry, 

 which is not altogether correct, unless it be remembered that with 

 neither ruler nor compasses is a straight line allowed to be transferred, 

 of a given length, from on* part of space to another. It is a plain 

 ruler, whose end* are not allowed to be touched, and compasses which 

 close the moment they are taken off the paper, of which the Greek 

 geometry permit* the use. It is altogether uncertain by whom these 

 restrictive pustulates were introduced, but it must have been before 

 the time of Plato, who was contemporary with (if he did not come 

 after) the introduction of those problems whose difficulty depends 

 upon the restrictions. We may here observe that in actual construc- 

 tion the ruler might have been dispensed with. It was reserved fur 

 an Italian abbe, at th* end of the 18th century, when all who studied 

 geometry had for two thousand years admired the smallnee* of the 

 bates on which it* conclusions are built, to inquire whether, small as 

 they were, IMS would not have been sufficient. In Mscheroui' 

 'Geometria dul Compass*,' published at Pavia in 1797, it is shown 

 that all the fundamental constructions of geometry can be made with 

 out the necessity of determining any point by the intersections of 

 straight line* ; that is, by using only those of circles. This singular 

 and very original work was translated into French, and published at 

 Paris in 17W and 1888. 



Bouk ii. treat* of the squares and rectangles described upon the 

 part* into which a line ia divided. It opens the way for the applies 

 tion of geometry to arithmetic, and ends by showing how to make a 

 rectangle equal to any rectilinear figure. It also points out what 

 modification the proposition of Pythagoras undergoes in the case of a 

 triangle not right-angled. 



Book iii. treat* of the circle, establishing such properties as can be 

 deduced by mean* of the preceding books. 



Book iv. treat* of such regular figure* a* can readily be described 

 by means of th* circle only, including the pentagon, hexagon, and 

 quindeoagon. It is of no use in what immediately follows. 



Book v. treat* of proportion generally, that is, with regard to mag 

 nitud* in general. Wheth-r this most admirable theory, which 

 though abstruse ia indispensable, was the work of Euclid himself, or 

 a prediosnor, cannot now be known. Th* introduction of any 

 numerical definition of proportion i* rendered inaccurate by the 

 necessity of reasoning on quantities between which no exact numerical 

 ratio exist*. The method of Euclid avoids the error altogether, by 

 Uyiug down a definition which applies equally to commensurable, aud 

 inonmnvensurables, so that it is not even necessary to mention this 

 distinction. 



Book vi. applies the theory of proportion to geometry, and treat* o 

 similar figures, that is, of figures which differ only in .use, and not in 



: mi. 



Book Til. lays down arithmetical definitions; shows how to find 

 th* greatest common measure and least common multiple of any two 

 numbers; prove* that numbers which are the least in any ratio are 

 prime to on* another, Ac. 



Book viii. treats of continued and mean proportional*, xhowing 

 when it is possible to insert two integer mean proportionals between 

 two integers. 



Book iz. treat* of square and cubs numbers, as also of plane and 

 ulid numbers (meaning numbers of two and three factors). It also 

 continues the consideration of continued proportionals, and of prime 

 numbers, shows that there is an infinite number of prime numbers, 

 and demonstrates the method of finding what are called perftet 

 numbers. 



Book z. contains 117 propositions, and U entirely filled with the 

 nvestigation and classification of incommensurable quantities. It 

 shows how far geometry can proceed in this branch of the subject 

 without algebra ; and though of all the other books it my be said 

 [hat they remain at this time as much adapted for instruction as 

 when they were written, yet of this particular book it must be 

 asserted that it should never be read except by a student versed in 

 algebra, and then not as a part of mathematics, but of the history of 

 mathematics. The book finishes with a demonstration that the side 

 and diagonal of a square are incommensurable. From this book it 

 is nio.-t evident that the arithmetical character of geometrical magni- 

 tude had been very extensively considered ; and it teems to us 

 suilicieuUy clear that an arithmetic of a character approximating 

 closely to algebra must have been the guide, as well as that some 

 definite object was sought perhaps the attainment of the quadrature 

 of the circle. 



Book zu lays down the definitions uf solid geometry, or of geometry 

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

 proceeds to treat of the intersections of planes, and of the properties 

 of parallelepipeds, or what might be called solid rectangles. 



Book xii treats of prisms, cylinders, pyramids, and cones, estab- 

 lishing the properties which are analogous to those of triangles, etc., 

 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 their diameters, in which for the first time in Euclid, the cele- 

 brated Method of Exhaustions is employed, which, with the theory 

 of proportion, forms the most remarkable part of this most remark- 

 able work. 



Book ziii., the last of those written by Kuclid, applies some results 

 of the tenth beok to the side* of regular figures, and shows how to 

 describe the five regular bodies. 



Books ziv. and zv., attributed to Hypsicles of Alexandria, treat 

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

 their inscription iu one another. 



The writings of Kuclid continued to be the geometrical standard 

 as long as the Greek language was cultivated. The Uomans never 

 made any progress iu mathematical learning. Boethius [BoKTHltis] 

 translated, it ia said, the first book of Euclid (Cassiodorus, cited by 

 Heilbrooner), but all which has coiue down to us on the subject 

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

 contained in two books, the lir.-t of which has the enunciations and 

 figures of the principal propositions of the first four books of the 

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

 manuscripts of this writer contain an appendiz which professes to 

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

 intention of cultivating geometry throughout the Roman dominions. 

 But no such result ever arrived as lung as the Western Empire 

 lasted ; and this short account uf Roman geometry is a larger pro- 

 portion of the present article than the importance of the subject 

 warrants. 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 mado 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 

 Uclides) state* that the Orientals believe Euclid to have been a native 

 of Tyre, and also that they frequently gave his name to the science 

 which he taught. The same author gives the names of the Arabic 

 versions, one of which, that of Nasir eddin, the most celebrated of all, 

 was printed at the Medioean press at Rome in 1594. The astronomer 

 Thabet ben Korroh was one of the translators, or rather, perhaps, 

 revised thu translation of Honein ben Ishak, who died A.D. 873. There 

 is a manuscript in the Bodleian Library, purporting to be the transla- 

 tion 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.U. 1150). We havo given [CiM- 

 I'AHuaj 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 colled) of Campanus 

 (printed 1482), is in fact that of Athelard, with a commentary by 

 Campanus. 



There U a considerable number of Greek manuscripts of the Ele- 

 ment*, for which see Fabricius and Heilbroniicr. There is no account 

 of the manuscripts) which they consulted by the earlier Latin trans- 

 lators (from the Greek), nor by Gregory. It appenrs however that 

 several, if not many, of the manuscripts arc entitled EVK\CI!OI> a-roixt IWP 



