Use of the Compass in Railroad Surveying 12 Latitude and Longitude Ranges. GEOMETRY (PART 1) PRELIMINARY DEFINITIONS NOTE. The study of Geometry is a process of systematic and orderly reasoning rather than a matter of memory. The student is advised to study the principles and propositions stated until he understands them thoroughly and sees their relation one to another, and, when a proposition is accompanied by an explanation in small type, to read over the explanation carefully one or more times, until he clearly understands the matter, following out the references to the figure when a figure is given. If he will do this he will find Geometry to be of great benefit and assistance to him in his subsequent studies. But he is not required to commit to memory the explanations or any part of the text except a few of the more important principles and propositions, such as those to which the Examination Questions relate. 1. Every material body possesses two general properties without regard to any other condition, namely: form, or shape, which is due to the relative positions of its parts; and magnitude, or size, which is due to the distance of its parts from one another. The form and magnitude of a body can be described by the relative positions of points, lines, and surfaces. 2. A point has position without magnitude. A dot is commonly used to represent a point; but a dot, no matter how small, has length, breadth, and thickness, while a theoretical point has position only. A 3. A line is the path of a point in motion; it has one dimension-length. Thus, if a point is moved from the position A, Fig. 1, to the position B, its path, or trace, is the line A B. FIG. 1 COPYRIGHTED BY INTERNATIONAL TEXTBOOK COMPANY. ENTERED AT STATIONERS' HALL, LONDON 87 FIG. 2 4. A straight line, or right line, Fig. 2, is a line that does not change its direction. 5. The distance between two points is the length of the straight line joining them. FIG. 3 FIG. 4 6. A curved line, Fig. 3, is a line that changes its direction at every point. 7. A broken line, Fig. 4, is a line that changes its direction at only certain points. It is made up wholly of different straight lines. The word line, when not qualified by any other word, is understood to mean a straight line. 8. A surface is the path of a line when moved in a direction other than its length. Thus, if a line is moved from the position AB, Fig. 5, to the position CD, the line describes the surface A B D C. B FIG. 5 9. A flat surface, plane surface, or simply a plane, is a surface such. that a straight line between any two of its points lies wholly in the surface. If a straightedge is laid on a plane surface in any direction, every point of the straightedge will touch. the surface. 10. A figure is any combination of points and lines. A figure that lies entirely in one plane is a plane figure. In referring to a figure, a point is designated by a letter placed conveniently near it; thus, in Fig. 1, the left end of the line is referred to as the point A. The entire line is referred to as "the line AB," the letters A and B designating two points, usually the ends of the line. If a line is broken or curved, as many points are named as are considered necessary to designate the line. 11. Geometry is that branch of mathematics that treats of the construction and properties of figures. 12. To produce a line is to prolong it or to increase its length. A straight line can be prolonged or produced to any extent in either direction. Thus, in Fig. 6, the straight line AB is produced to the points C and D. 13. To bisect any given magnitude is to divide it into two equal parts. Thus, the A FIG. 6 A C FIG. 7 D straight line AB, Fig. 7, is bisected at the point C if AC is equal to CB. When a given magnitude is bisected, each of the parts into which it is divided is one-half the given magnitude. STRAIGHT-LINE FIGURES ANGLES AND PERPENDICULARS 14. An angle, Fig. 8, is the opening between two straight lines that meet in a point. The two straight lines are the sides, and the point where the lines meet is the vertex, of the angle. Thus, in Fig. 8, the straight lines OA and O B form an angle at the point 0; the lines OA and OB are the sides of this angle, and the point O is its vertex. FIG. 8 B A An angle is usually referred to by naming a letter on each of its sides and a third letter at the vertex, the letter at the vertex being placed between the other two. Thus, the angle in Fig. 8 is called angle AOB or angle BOA. An angle may also be designated by a letter placed between its sides near the vertex. Thus, the two angles XCY and YCZ, Fig. 9, may be referred to as the angles. A and B, respectively. An isolated angle, that is, an angle whose vertex is not the vertex of any other angle, may be designated -X FIG. 9 by naming the letter at its vertex. For example, the angle in Fig. 8 may be called the angle O. |