PREFACE. THE present work, embracing the First Six Books of the Elements of Euclid, or in other words, the foundation of Plane Geometry, is intended as a companion volume to that issued with it on Practical Geometry; the one embracing the theory, the other the application. The text adopted is that of the distinguished mathematician Dr. Simson. In very few instances has any change been made, and that only when it seemed to be positively necessary. Several attempts have been made, by altering the text and the arrangement, to reduce the demonstrations of Euclid to what may be considered a more popular form; but these have been attended with little success; and a text like that of Simson, clear, direct, and unencumbered, like absolute truth itself, will always be adopted by scholar and student. It has been thought unnecessary to extend this volume beyond the first six books of the Elements, or to enlarge it by notes, as it will be followed by other works carrying the subject forward. A number of Exercises on each book have been given, the solutions of which will be published separately. THE ELEMENTS OF EUCLID. BOOK I. DEFINITIONS. I. A POINT is that which hath no parts, or which hath no magnitude. II. A line is length without breadth. III. The extremities of a line are points. IV. A straight line is that which lies evenly between its extreme points. V. A superficies is that which hath only length and breadth. VI. The extremities of a superficies are lines. VII. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies. VIII. A plane angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction. IX. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line. d b N.B. When several angles are at one point b, any one of them is expressed by three letters, of which the letter that is at the vertex of the angle, that is, at the point in which the straight lines that contain the angle meet one another, is put between the other two letters, and one of B |