education. Commencing with the simplest ideas of number, the successive volumes conduct the student by easy gradation through all the essential principles of Arithmetic, Algebra, Geometry, Trigonometry, Analytics, and the Calculus. The entire course is treated as a unit, each part being adapted to all the others, and all the parts being so arranged as to form a symmetrical whole. No effort has been spared to make the definitions clear, and the illustrations full, and the treatment of the subject throughout has been intended to afford the best results with the least expenditure of time and labor-in a word, to make the series concise in demonstration, consecutive in arrangement, complete in detail. GREENWICH, CONN., July 4, 1876. MANUAL OF GEOMETRY. INTRODUCTION. FUNDAMENTAL CONCEPTIONS. 1. A portion of space, limited in all directions, is called a volume; that which separates a volume from the rest of space is called a surface. If two surfaces intersect, or cut each other, that which is common to both is called a line; if two lines intersect, that which is common to both is called a point. General Definitions. 2. A magnitude is anything that can be measured, that is, whose value can be expressed in terms of some thing of the same kind, taken as a unit. 3. A point is that which has position, but no magnitude. 4. A line is that which has length, without breadth or thickness. 5. A surface is that which has length and breadth without thickness. 6. A volume is that which has length, breadth, and thickness. Straight Lines and Curved Lines. 7. A straight line is a line whose direction does not change at any point; thus, AC is a straight line. A It is assumed that one straight line, and only one, can be drawn through any two points. 8. A broken line is a line made up of straight lines lying in different directions; thus, ACDE is a broken line. 9. A curved line, or a curve, is a line whose direction changes at every point; thus, ACD is a curved line. When the term line is used by itself, a straight line is always meant. Planes and Curved Surfaces. 10. A plane is a surface such that a straight line through any two of its points lies wholly in the surface. 11. A curved surface is a surface of which no part is plane. Generation of Lines and Surfaces. 12. If a point moves through space in accordance with a fixed law it is said to generate, or trace out, a line; the moving point is called the generatrix, and the law that governs its motion is called the law of generation. The law of generation determines the kind of lines generated. 13. If a line moves through space in accordance with a fixed law it is said to generate a surface; the moving line is the generatrix, and the law which governs its motion is the law of generation. Angles. 14. An angle is the inclination of two lines that meet at a common point; the two lines are called sides and their common point is the vertex: thus, the inclination of OA and OC is an angle, whose sides are OA and OC, and whose vertex is 0. |