BOOK I. RECTILINEAR FIGURES. INTRODUCTORY REMARKS. A ROUGH block of marble, under the stone-cutter's hammer, may be made to assume regularity of form. If a block be cut in the shape repre sented in this diagram, face. It will have six flat faces. Each face of the block is called a Sur If these surfaces be made smooth by pol ishing, so that, when a straight-edge is applied to any one of them, the straight-edge in every part will touch the surface, the surfaces are called Plane Surfaces. The sharp edge in which any two of these surfaces meet is called a Line. The place at which any three of these lines meet is called a Point. If now the block be removed, we may think of the place occupied by the block as being of precisely the same shape and size as the block itself; also, as having surfaces or boundaries which separate it from surrounding space. We may likewise think of these surfaces as having lines for their boundaries or limits; and of these lines as having points for their extremities or limits. A Solid, as the term is used in Geometry, is a limited portion of space. After we acquire a clear notion of surfaces as boundaries of solids, we can easily conceive of surfaces apart from solids, and suppose them of unlimited extent. Likewise we can conceive of lines apart from surfaces, and suppose them of unlimited length; of points apart from lines as having position, but no extent. DEFINITIONS. Magnitude 1. DEF. Space or Extension, has three Dimensions, called Length, Breadth, and Thickness. 2. DEF. A Point has position without extension. 3. DEF. A Line has only one of the dimensions of extension, namely, length. The lines which we draw are only imperfect representations of the true lines of Geometry. A line may be conceived as traced or generated by a point in motion. 4. DEF. A Surface has only two of the dimensions of extension, length and breadth. A surface may be conceived as generated by a line in motion. 5. DEF. A Solid has the three dimensions of extension, length, breadth, and thickness. Hence a solid extends in all directions. A solid may be conceived as generated by a surface in motion. Thus, in the diagram, let the upright surface ABCD move to the right to A B the position EFHK. The points. A, B, C, and D will generate the lines C F E H K AE, BF, CK, and D H respectively. The relative situation of the two points A and H involves three, and only three, independent elements. To pass from A to H it is necessary to move East (if we suppose the direction A E to be due East) a distance equal to A E, North a distance equal to E F, and down a distance equal to FH. These three dimensions we designate for convenience length, breadth, and thickness. 6. The limits (extremities) of lines are points. The limits (boundaries) of solids are surfaces. 7. DEF. Extension is also called Magnitude, f. 7 When reference is had to extent, lines, surfaces, and solids are called magnitudes. 8. DEF. A Straight line is a line which has the same direction throughout its whole extent. 9. DEF. A Curved line is a line which changes its direction at every point. 10. DEF. A Broken line is a series of con nected straight lines. When the word line is used a straight line is meant; and when the word curve is used a curved line is meant. 11. DEF. A Plane Surface, or a Plane, is a surface in which, if any two points be taken, the straight line joining these points will lie wholly in the surface. 12. DEF. A Curved Surface is a surface no part of which is plane. 13. Figure or form depends upon the relative position of points. Thus, the figure or form of a line (straight or curved) depends upon the relative position of points in that line; the figure or form of a surface depends upon the relative position of points in that surface. When reference is had to form or shape, lines, surfaces, and |