broken line, the parts of which make an angle; if one point be moved round another point always at an equal distance, it describes a circular line. If a point remains in its course always at the same distance from a straight line, it will describe a straight line parallel to the former. If a point in a straight line be moved in a straight course, it must either continue in the same straight line, or it must leave the direction of that line. In this latter case, the straight line which it describes will make an angle with the former straight line. This angle may be either right or oblique. II. LINES. 96. As a Line is the path described by the motion of a point, it can only have extension in length; not in breadth or thickness. All lines have in common the property of extension in length; they may differ one. from another in the quantity of the length, in their position, and in the position of their component parts. If these parts all lie in one and the same direction, the line is a straight line; if they do not all lie in the same direction, the line is either broken or curved. Three contiguous points in a curved line never lie in the same direction. Two points, let them lie as they may, can be connected by a straight line. A straight line cannot coincide with a curved line; they can only have one or more points in common. 97. A straight line is the shortest way from one point to another; every curved line between the same points is longer than the straight line. The more nearly the curved line approaches to a straight line, the shorter it will be. There may be many curved lines between the same points; there can be only one straight line, because there can be only one shortest way. In geometry, when we speak of the distance between two points, we mean the length of a straight line, and therefore of the shortest line, which can connect them. 98. Two points determine exactly the direction of a straight line. Therefore if an engineer wishes to mark out a straight line in a field, he sets a stake at each end of the line, and then sets other stakes between these two, taking care that all shall be in the direction of the first two. This he ascertains by taking sight from one of these two to the other of them. If the straight line is to be made longer, he takes sight from one stake to another, and the stakes are successively set in the line of sight. In a similar manner a row of trees is set in a straight line, or a company of soldiers drawn up. Two straight lines which have 2 points in common must have the same direction, and must coincide. MENSURATION OF STRAIGHT LINES. 99. A magnitude can be measured only by comparing it with a magnitude of the same kind; thus, length can be measured only by length, surface by surface, weight by weight. Some one known magnitude is taken as the unit, and to measure a magnitude we seek how often this unit is contained in it. Therefore, To measure a magnitude is to determine how often a known magnitude, which is taken as the unit of measure, is contained in the magnitude to be measured. There are as many different units of measure, as there are different kinds of magnitudes. In this school-room we might employ long, surface, and solid measures, since here are lengths, surfaces, and solids to be measured. 100. At present we will confine ourselves to the measure of length, or long measure. In measuring short lengths, we take as the unit of length, or linear unit, an inch, a foot, or a yard. When we actually perform the operation of measuring, we make use of a wooden or metallic rule, upon which the feet and inches are marked. If we have a line of great length to measure, we take a rod or a mile as the linear unit, and in performing the operation of measuring we make use of a wooden or metallic rod, of a tape, or of a chain. If a straight line is to be measured, we seek how often the unit of measure is contained in it. This may be done either by directly applying a rule, a rod, or a chain, or by another mode of which we shall have examples hereafter. If a curved line is to be measured, we seek how long it would be if it were extended in a straight direction. Thus a straight unit of measure is used to ascertain the length of all lines. III. OF ANGles. 1. OF ANGLES IN GENERAL. 101. An angle is the opening, or the mutual inclination of two lines meeting in a point. 102. Angles in respect to the nature of their sides. are divided into rectilineal, whose sides are straight lines; curvilineal, whose sides are curved lines; mixtilineal having one side a curved line, the other a straight line. At present we have to do with rectilineal angles only. 2. RELATIVE MAGNITUDE. 103. Two angles may be either equal or unequal. They are equal when the sides of each have a similar inclination; in such case, if one be placed upon the other, they will entirely coincide; the vertex of one will coincide with the vertex of the other, and the sides of the one with the sides of the other. Equal angles agree in all respects, and the angles which do not agree in all respects are unequal. 3. RELATIVE POSITION. 104. Two angles may lie entirely apart one from the other, so that their sides shall have no part in com mon. Or, They may have one side entirely in common, or partly in common (fig. 81.) The sides the parts of which are not in common may form one straight line, (fig. 82.) Such angles are called adjacent angles, as angles R and S. Or, Two angles may have only the vertical point in common. If such angles are equal, and the sides of one being produced coincide with the sides of the other, thus forming 2 straight lines, the angles are called vertical angles; as the angles O and X; M and N, (fig. 83.) 105. If two parallel straight lines are crossed by a third line, the angles thus made have a relative position one to another, which is expressed by particular names. Thus, (fig. 84,) C and F, D and E, &c., are alternate internal angles, because they are on opposite sides of the single line, and within the parallels. A and H, B and G, &c., are alternate external angles. D and F, C and E, &c., are interior angles upon the same side, because they are contained between the parallels, and are on the same side of the line which crosses them. A and G, B and H, &c., are exterior angles upon the same side. A and E, B and F, G and C, H and D, are external internal angles, because one is without and the other within the parallels, and both are on the same side of the single line. We have also many pairs of adjacent angles; viz., A and B, A and C, B and D, E and G, &c.; and many pairs of vertical angles, viz., A and D, B and C, E and H, &c. 4. MAGNITUDE OF ANGLES TAKEN TOgether. 106. Two adjacent angles are together equal to 2 R. A. or 180°. This follows directly from (34.) Therefore, if one of the angles R. A., then the other= 2 R. A. — — R. A. 13 R. A. The one = 30°, the other 150°. = = All the angles in the same plane about a point are together equal to 4 R. A., or 360°. Hence, when one line crosses another, since all right angles are equal, if one of the angles is a right angle, then all are R. A. 107. Two vertical angles are equal one to the other, (fig. 83.) M= N, 0=X. |