stands. An angle is also frequently designated by putting a letter or figure in it and near the vertex. 23. The Size of an Angle depends upon the rapidity with which its sides separate, and not upon their length. ILL.-The angles BAC and MON, Fig. 10, are equal, since the sides separate at the same rate, although the sides of the latter are more prolonged than those of the former. The sides DF and DE separate faster than AB and AC, hence the angle EDF is greater than the angle BAC. 24. Adjacent Angles are angles so situated as to have a common vertex and one common side lying between them. ILL.-In Fig. 10, angles 4 and 5 are adjacent, since they have the common vertex L, and the common side LR. Angles 9 and 10 are also adjacent, as are also 8 and 9. 25. Angles are distinguished as Right Angles and Oblique Angles. Oblique angles are either Acute or Obtuse. 26. A Right Angle is an angle included between two straight lines which meet each other in such a manner as to make the adjacent angles equal. An Acute Angle is an angle which is less than a right angle, i. e., one whose sides separate less rapidly. An Obtuse Angle is an angle which is greater than a right angle, i.e., one whose sides separate more rapidly. B ILL.-As in common language an angle is called a corner, so a right angle is called a square corner ; an acute, a sharp corner; and an obtuse angle might be called a blunt corner. In Fig. 11, BAC and DAB are right angles. In Fig. 10, 1, 2, 3, 5, 8, 9, and 10 are acute angles, 4 and 6 are obtuse, and 7 is a right angle. A A SOLID. 27. A Solid is a limited portion of space. It may also be conceived as the path of a surface in motion. ILL-Suppose you have a block of wood like that represented in Fig. 12, with all its corners (angles) square corners (right angles). Hold it still in your fingers a moment, and fix your mind fort you can think of it just as well as of B the wood. This space is an example of what we call a Solid in Geometry. In Ak fact, the solids of Geometry are not solids FIG. 12. at all in the common sense of solids; they are only places of certain shapes. Again, hold your ball still a moment in your fingers and then let it drop, and think of the place it filled when you had it in your fingers. It is this place, shaped just like your ball, that we think about, and talk about as a solid, in Geometry. In order to see how a solid may be conceived as the path of a surface, suppose you cut out a piece of paper of just the same size as the end of the block represented in Fig. 12. Let ABCD represent this piece of paper. Now, holding the paper in a perpendicular position, as ABCD is represented in the figure, move it along to the right, so that its angles shall trace the lines AG, BH, DE, and CF. When the paper has moved to the position CHFE, its path will be just the same space as the block of wood occupied. This path, or the space through which the surface represented by the piece of paper moved, is the solid. Ex. 1. If a semicircle is conceived as revolved around its diameter, what is the path through which it moves? See Fig. 7. Ex. 2. If the surface OMNP, Fig. 9, is conceived as revolved around OP, what is the path through which it moves? CAUTION.-The student needs to be careful and distinguish between the surface traced by the line MN, and the solid traced by the surface OMNP. Ex. 3. If the surface represented by ABC be conceived as revolved about its side CA, what kind of a solid is its path? [NOTE. As has been said before, the student is not necessarily expected to name these solids, but rather to show, in his own language, that he has the conception.] C Ex. 4. As you fill a vessel with water, what is the A solid traced by the surface of the water? Ans. The same as the space within the vessel. FIG. 13. Ex. 5. If a circle is conceived as lying horizontally, and then moved directly up, what will be the solid described, i. e., its path? Do not confound the surface described with the solid. What describes the surface? What the solid? EXTENSION AND FORM. 28. Extension means a stretching or reaching out. Hence, a Point has no extension. It has only position (place). A Line stretches or reaches out, but only in length, as it has no width. Hence, a line is said to have One Dimension, viz., length. A Surface extends not only in length, but also in breadth; and hence has Two Dimensions, viz., length and breadth. A Solid has Three Dimensions, viz., length, breadth, and thickness. ILL.-Suppose we think of a point as capable of stretching out (extending) in one direction. It would become a line. Now suppose the line to stretch out (extend) in another direction-to widen. It would become a surface. Finally, suppose the surface capable of thickening, that is, extending in another direction. It would become a solid. 29. The Limits (extremities) of a line are points. The Limits (boundaries) of a surface are lines. The Limits (boundaries) of a solid are surfaces. 30. Magnitude (size) is the result of extension. Lines, surfaces, and solids are the geometrical magnitudes. A point is not a magnitude, since it has no size. The magnitude of a line is its length; of a surface, its area; of a solid, its volume. 31. Figure or Form (shape) is the result of position of points. The form of a line (as straight or curved) depends upon the relative position of the points in the line. The form of a surface (as plane or curved) depends upon the relative position of the points in it. The form of a solid depends upon the relative position of the points in its surface. Lines, surfaces, and solids are the geometrical figures.* ILL.-In Fig. 14, it is easy to conceive the form of the lines by knowing the position of points in the lines. By taking a quantity of common pins of different lengths, sticking them upright in a board, and conceiving the heads to represent points in a surface, we can readily see how the position of the points in a surface determine its form. FIG. 14. O Ex. 1. Suppose a line to begin to con * Lines, surfaces, and solids are called magnitudes when reference is had to their extent, and figures when reference is had to their form. tract in length, and continue the operation till it can contract no longer, what does it become? That is, what is the minor limit of a line? Ex. 2. If a surface contracts in one dimension, as width, till it reaches its limit, what does it become? If it contracts to its limit in both dimensions, what does it become? Ex. 3. If a solid contracts to its limit in one dimension, what does it pass into? If in two dimensions? If in three dimensions? Ex. 4. What kind of a surface is that, every point in which is equally distant from a given point? 32. Geometry treats of magnitude and form as the result of extension and position. The Geometrical Concepts are points, lines, surfaces (including plane and spherical angles), and solids (including solid angles). The Object of the science is the measurement and comparison of these concepts. Plane Geometry treats of figures all of whose parts are confined to one plane. Solid Geometry, called also Geometry of Space, and Geometry of Three Dimensions, treats of figures whose parts lie in different planes. The division of Part II. into two chapters is founded upon this distinction. In the Higher or General Geometry these divisions are marked by the terms "Of Loci in a Plane," and "Of Loci in Space." PART I. A FEW OF THE MORE IMPORTANT FACTS OF THE SCIENCE. SECTION I. ABOUT STRAIGHT LINES. 33. Prob.-To measure a straight line with the dividers and scale. A SOLUTION.-Let AB, Fig. 15, be the line to be measured. Take the dividers, Fig. 1 (frontispiece), and placing the sharp point A firmly upon the end A of the line AB, open the dividers till the other point B (the pencil point) just reaches the other end of the line B. Then letting the dividers reK main open just this amount, place the point A on the lower end of the left hand scale, as at o, Fig. 1, and notice where the point B reaches. In this case it reaches 3 spaces beyond the figure 1. Now, as this scale is inches and tenths of inches,* the line AB is 1.3 inches long. Ex. 1. What is the length of CD? Ex. 2. What is the length of EF? Ex. 3. What is the length of GH? Ex. 4. What is the length of IK? Ex. 8. Draw a line .85 of an inch long. Ans. .15 of a foot. Ans. 75 of an inch. Ans. 14 inches. Ans. .18 of a foot. * The next scale to the right is divided into 10ths and 100ths of a foot. Thus from p to 10 is 1 tenth of a foot, and the smaller divisions are hundredths. |