BY J O H N C A S E Y, LL. D., F. R. S., FELLOW OF THE ROYAL UNIVERSITY OF IRELAND; MEMBER OF COUNCIL, ROYAL IRISH ACADEMY; MEMBER OF THE MATHEMATICAL SOCIETIES OF LONDON AND FRANCE; AND PROFESSOR OF THE HIGHER MATHEMATICS AND OF MATHEMATICAL PHYSICS IN THE CATHOLIC UNIVERSITY OF IRELAND. PIC THIRD EDITION, REVISED AND ENLARGED. DUBLIN: HODGES, FIGGIS, & CO., GRAFTON-ST. LONDON: LONGMANS, GREEN, & CO. 1885. DUBLIN PRINTED AT THE UNIVERSITY PRESS, BY PONSONBY AND WELDRICK ________.

This edition of the Elements of Euclid, undertaken at the request of the principals of some of the leading Colleges and Schools of Ireland, is intended to supply a want much felt by teachers at the present day—the production of a work which, while giving the unrivalled original in all its integrity, would also contain the modern conceptions and developments of the portion of Geometry over which the Elements extend. A cursory examination of the work will show that the Editor has gone much further

Geometry is the Science of figured Space. Figured Space is of one, two, or three dimensions, according as it consists of lines, surfaces, or solids. The boundaries of solids are surfaces; of surfaces, lines; and of lines, points. Thus it is the province of Geometry to investigate the properties of solids, of surfaces, and of the figures described on surfaces. The simplest of all surfaces is the plane, and that department of Geometry which is occupied with the lines and curves drawn on a plane is

T h e P o i n t . i . A point is that which has position but not dimensions. A geometrical magnitude which has three dimensions, that is, length, breadth, and thickness, is a solid; that which has two dimensions, such as length and breadth, is a surface; and that which has but one dimension is a line. But a point is neither a solid, nor a surface, nor a line; hence it has no dimensions—that is, it has neither length, breadth, nor thickness. T h e L i n e . i i . A line is length without breadth.

Every Proposition in the Second Book has either a square or a rectangle in its enunciation. Before commencing it the student should read the following preliminary explanations: by their assistance it will be seen that this Book, which is usually considered difficult, will be rendered not only easy, but almost intuitively evident. 1. As the linear unit is that by which we express all linear measures, so the square unit is that to which all superficial measures are referred. Again, as there are di

________________ DEFINITIONS. i . Equal circles are those whose radii are equal. This is a theorem, and not a definition. For if two circles have equal radii, they are evidently congruent figures, and therefore equal. From this way of proving this theorem Props. x x v i .– x x i x . follow as immediate inferences. i i . A chord of a circle is the line joining two points in its circumference. If the chord be produced both ways, the whole line is called a secant, and each of the parts into which a

i . If two rectilineal figures be so related that the angular points of one lie on the sides of the other—1, the former is said to be inscribed in the latter; 2, the latter is said to be described about the former. i i . A rectilineal figure is said to be inscribed in a circle when its angular points are on the circumference. Reciprocally , a rectilineal figure is said to be circumscribed to a circle when each side touches the circle. i i i . A circle is said to be inscribed in a rectilineal fig

Introduction. —Every proposition in the theories of ratio and proportion is true for all descriptions of magnitude. Hence it follows that the proper treatment is the Algebraic. It is, at all events, the easiest and the most satisfactory. Euclid’s proofs of the propositions, in the Theory of Proportion , possess at present none but a historical interest, as no student reads them now. But although his demonstrations are abandoned, his propositions are quoted by every writer, and his nomenclature i

i . Similar Rectilineal Figures are those whose several angles are equal, each to each, and whose sides about the equal angles are proportional. Similar figures agree in shape; if they agree also in size, they are congruent. 1. When the shape of a figure is given, it is said to be given in species . Thus a triangle whose angles are given is given in species. Hence similar figures are of the same species. 2. When the size of a figure is given, it is said to be given in magnitude ; for instance, a

i . When two or more lines are in one plane they are said to be coplanar . i i . The angle which one plane makes with another is called a dihedral angle . i i i . A solid angle is that which is made by more than two plane angles, in different planes, meeting in a point. i v . The point is called the vertex of the solid angle. v . If a solid angle be composed of three plane angles it is called a trihedral angle; if of four, a tetrahedral angle; and if of more than four, a polyhedral angle . PROP.