method of dealing with the subject geometrically in a satis factory manner has yet been devised; yet, practically, these Propositions have fallen into disuse, and other methods of establishing the properties of proportionals have been substituted for them. The nature of the difficulty is set forth in a few introductory paragraphs which, it is hoped, will justify the method of treatment here adopted. These the student, when he has reached thus far, will easily understand. His acquaintance with Analysis may, however, be scarcely sufficient to carry him through the reasoning in the first two Propositions; on a first reading these may be omitted, and he may begin with the third Proposition. The Editors have to express their acknowledgments to Prof. EVERETT, and to the Rev. Dr. BRYCE for some valuable suggestions regarding the definitions of the First and Fifth Books; and to Mr. J. J. BLACK and Mr. JOHN THOMSON, assistants in the Mathematical Departments of the Edinburgh and Glasgow High Schools, for their careful revision of the proof sheets, and for many useful hints during the progress of the work. August, 1874 THE ELEMENTS OF EUCLID. BOOK I. DEFINITIONS. 1. A solid is that which occupies any portion of space in the three dimensions of length, breadth, and thickness. 2. A superficies or surface is that which has only length and breadth. Corollary.-The boundaries or extremities of a solid are surfaces. 3. A line is length without breadth. Cor.-The extremities or boundaries of a surface are lines; and the intersection of one surface with another is also a line. 4. A point is that which has position but not magnitude. Cor. The extremities of a line are points; and the intersection of two lines is also a point. The simplest way of conceiving of a line is of its being the edge or boundary of a surface. When, in our geometrical figures, we speak of a line, we mean either edge of the stroke upon the paper, which marks the distance between the two extreme points. 5. If two lines be such that they cannot coincide in two points without coinciding altogether, each of them is called a straight line. Cor.-Hence two straight lines cannot enclose a surface. Neither can two straight lines have a common segment. A Thus, CA cannot be a continuation of both BC and DC. 6. A plane surface or plane is that in which any two points being taken, the straight line between them lies wholly in that surface. Cor.-Hence two plane surfaces cannot enclose a space; nor can one plane surface be the continuation of two planes. 7. A plane rectilineal angle is formed by two straight lines meeting at a point; and the point of meeting is called the vertex of the angle. Scholium. An angle may be regarded as the amount of opening between any two positions of a straight line, considered as moving always in the same plane round a fixed point. Thus the opening between the two positions OA and OB of the same line moving round O as on a pivot, is the angle O, or AOB or BOA; the angle being named by a letter placed at the vertex, or by three letters, that at the vertex being put between the two others. The magnitude of an angle depends on the degree of opening or divergence of the lines, and is in no way dependent on the length of the lines. A 8. When one straight line stands on another, so as to make the two adjacent angles equal, each of them is called a right angle; and the straight line standing on the other is said to be perpendicular or at right angles to it. A B Thus, if the line OB, standing on AC, make the angle AOB equal to the angle BOC, each of these angles is called a right angle, and the line OB is said to be perpendicular, or at right angles, to AC. Schol. When the line OB, revolving round O, has reached |