scribed in and circumscribed about another rectilineal figure, lie has given no proposition showing how in any case such inscription or circumscription may be effected. The equilateral triangle, the square, the regular pentagon, and such regular polygons as can be derived from these, were the only regular figures known to be inscriptible in a circle by means of elementary geometry, till Gauss discovered, in 1796, that the circumference of a circle could be divided into 17 equal parts. In his Disquisitiones Arithmeticæ, published in 1801, it is proved that there can be inscribed in a circle any regular polygon, the number of whose sides is prime, and is denoted by 2n + 1. Euclid's second book presupposes, that is, depends to some extent upon, the first; the third presupposes both the first and second; the fourth presupposes the first three; arid all four are largely concerned with the discussion of the absolute equality or inequality of certain magnitudes. The fifth book stands alone, depending upon none of the preceding books, and contains the theory of proportion, with respect not merely to geometrical magnitudes, such as lines, angles, areas, solids, but to any magnitudes of which multiples can be formed. The diagrams consist of straight lines, probably for convenience of construction, but the enunciations of the propositions and the reasoning are perfectly general. With the exception of his treatment of parallels, Euclid's doctrine of proportion has been the subject of more discussion than any other part of the Elements. The foundation of the doctrine is the criterion of proportionality laid down in the famous fifth definition. The necessity or the appropriate ness of such a criterion can be seen only when the distinction between number and magnitude has been clearly apprehended, or, what amounts to the same thing, when an adequate conception has been formed of incommensurables. The ordinary arithmetical test of proportionality will then be found to suit only certain cases which occur those, namely, where the magnitudes considered are commensurable, and if the theory of proportion is to be rigorous and complete (as Euclid's is), it must be extended to incommensurables by the notions of continuity and limits. The difficulty therefore which Is felt by readers of the fifth book in grasping Euclid's doctrine is due mainly to the nature of the subject, and no very material simplification of the full treatment of proportion is possible. The sixth book contains the application of the theory of proportion, mostly to rectilineal figures. In the last proposition, the second part of which is due to Theon, it is noteworthy that the restricted definition of an angle, given in the first book, and adhered to throughout, is tacitly abandoned. The seventh, eighth, and ninth books are arithmetical, that is, treat of the properties of numbers. The definitions relating to them occur at the beginning of the seventh book, and some of these show perhaps the tendency of the Greeks, natural enough to a scientific people with a defective numerical notation, to consider quantity from a geometrical point of view. A. number composed of two factors was called a plane number, one composed of three a solid number, and the factors themselves were called sides. The test by which numbers are recognized to be proportionals is different from that given in the fifth book, for here it requires to be ap plied only to quantities which are commensurable, namely, integers. The last proposition of the ninth book shows how to construct a perfect number, that is, a number which is equal to the sum of all its divisors; for example, 6 = 1 + 2 + 3, 28 = 1 + 2 + 4 + 7 + 14, &c. The tenth book is the longest of the Elements. It is occupied with the consideration of commensurable and incommensurable magnitudes, and ends with the proposition that the diagonal and the side of a square are incommensurable. With regard to straight lines, Euclid distinguishes between those which are commensurable or incommensurable in length, and those which are so in power, the latter being defined to be straight lines the squares on which have or have not a common measure. There are three sets of definitions to this book, the second set inserted before the forty-ninth pro position, and the third before the eighty-sixth. The eleventh, twelfth, and thirteenth books treat mainly of solid geometry. In the eleventh are given the definitions which serve for the three books, the principal properties of straight lines and planes, of solid angles, and of parallelepipeds. The twelfth book begins with two theorems of plane geometry, and then discusses chiefly the properties of pyramids, cones, and cylinders. The last two propositions relate to spheres, the last being to prove that spheres have to one another the triplicate ratio of their diameters. In this book is exemplified the method of Exhaustions, which is founded on the principle that by taking away from a magnitude more than its half, from the remainder more than its half, and so on, a remainder is at length reached which is less than any assignable magnitude (book x. prop. 1 ). Other applications of this method, the nearest approach made by the ancients to the differential calculus, are to be found in the works of Archimedes (see his Measurement of the Circle, Conoids and Spheroids, Sphere and Cylinder). The thirteenth book treats of lines divided in extreme and mean ratio, of some regular figures inscribed in circles, and of the five regular solids, the last proposition being to exhibit the edges of these five solids, and to compare them with one another.
The question has often been mooted, to what extent Euclid, in his Elements, is a discoverer or a compiler. To this question no entirely satisfactory answer can be given, for scarcely any of the writings of earlier geometers have come down to our times. We are dependent on Pappua and Proclus for the scanty notices we have of Euclid's predecessors, and of the problems which engaged their attention; for the solution of problems, and not the discovery of theorems, would seem to have been their principal object. From these authors we learn that the property of the right-angled triangle had been found out, the principles of geometrical analysis laid down, the restriction of constructions in plane geometry to the straight line and the circle agreed upon, the doctrine of proportion, as well as loci, plane and solid, and some of the properties of the conic sections investigated, the five regular solids (often called the Platonic bodies) and the relation between the volume of a cone or pyramid and that of its circumscribed cylinder or prism discovered. Elementary works had been written, and the famous problem of the duplication of the cube reduced to the determination of two mean proportionals between two given straight lines. Notwithstanding this amount of discovery, and all that it implied, Euclid must have made a great advance beyond his predecessors (we are told that "he arranged the discoveries of Eudoxus, perfected those of Theaetetus, and reduced to invincible demonstration many things that had previously been more loosely proved"), for his Elements supplanted all similar treatises, and, as Apollonius received the title of "the great geometer." so Euclid has come down to later ages as " the elementator,"
The first six and, less frequently, the eleventh and twelfth books are the only parts of the Elements which are now read in the schools or universities of the United Kingdom; and, within recent years, strenuous endeavours have been made by the Association for the Improvement of Geometrical Teaching to supersede even these. On the Continent, Euclid has for many years been abandoned, and his place supplied by numerous treatises, certainly not models of geometrical rigour and arrangement. The fact that for twenty centuries the Elements, or parts of them, have held their ground as an introduction to geometry is a
