Book ii. treats of the squares and rectangles described upon the parts into which a line is 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 whin modification the proposition of Pythagoras undergoes in the case of s triangle not right-angled.
Book ill. treats of the circle, establishing such properties as can be deduced by means of the preceding books.
Book iv. treats of such regular figures as can readily be described by means of the circle only, including the pentagon, hexagon, and quindecagon. It is of no use in what immediately follows.
Book v. treats of proportion generally, that is, with regard to meg nitude in general. Whether this most admirable theory, whist: though abstruse is was the work of Euclid himself, or • predecessor, cannot now be known. The introduction of and numerical definition of proportion is rendered inaccurate by tbe necessity of reasoning on quantities between which no exact numerics ratio exists. The method of Euclid avoids the error altogether, 1).1 laying down a definition which applies equally to commenaurables and incommemurables, so that it is not even necessary to mention this distinction.
Book vi. applies the theory of proportion to geometry, and treats o similar figures, that is, of figures which differ only in size, and not ii form.
Book vii. lays down arithmetical definitions; shows how to fins the greatest common measure and least common multiple of any tea numbers; proves that numbers which are the least in any ratio an prime to one another, &c.
Book viii. treats of coutinued and mean proportionals, showing then it is possible to insert two integer mean proportional, between two integers.
Book Ix. treats of square and cube numbers, as also of plane and ruled numbers (meaning numbers of two and three factors). It also iontiuues the consideration of continued proportional% and of prime Dumber', shows that there is an infinite number of prime numbers, end demonstrate the method of finding what are called perfect 'lumbers.
Book x. contains 117 propositions, and is entirely filled with the Investigation 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 may be said that 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 tho side and diagonal of a square are inoommensurable. From this book it
is most evident that the arithmetical character of geometrical magni tude had been very extensively considered ; and it seems to us sufficiently clear that an arithmetic of a character approximating closely to algebra must have been tho guide, as well as that some definite object was sought—perhaps the attainment of the quadraturo of the circle.
Book xi. 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, and of the properties of parallelopipeds, or what might be called solid rectangles.
Book zits treats of prisms, cylinders, pyramids, and cones, estab lishing 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 oubes on their diameters, in which for the first time in Euclid, the cele brated Method of Exhauations is employed, which, with the theory of proportion, forms the most remarkable part of this most remark able work.
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.
Books xiv. and iv., attributed to ITypsicles of Alexandria, treat entirely of the relative proportions of the five regular solids, and of their inscription to one another.
The writinga of Euclid continued to be the geometrical etandard as long as the Greek language was cultivated. The Roinaus never made any progress in mathematical learning. Boethius [Boeenitis) translated, it is said, the first book of Euclid (Cassiodorus, cited by Heilbrouner), 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 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 appendix which professes to give an account of a letter of Julius Caesar, in which he expresses his intention of cultivating geometry throughout the Rennin dominions. But no such result ever arrived as lung as the Western Empire tented ; and this abort account of Roman geometry is a larger pro portion of the present article than the importance of the subject warrants. These books of Boethius continued to bo tho standard text books until Euclid was brought in again from the Arabs.