PAGE 191. To draw the equiangular spiral when the depth and the angle . between the tangent and radius are given . . . . 157 192. To apply the equiangular spiral to the formation of volutes 193. To describe a cycloid by continuous motion . . . . 165 194. To find points upon the perimeter of the cycloid . . . 165 195. To draw the tangent and normal at any given point on the 196. Properties and applications of the cycloid . 197. To describe the curve called the companion to the cycloid . 167 198. Definition of a trochoid . . . . . . . 168 199. To find points upon a trochoid . . . . . . 168 1. DEFINITIONS.—A mathematical point is a position in space of which no part can be taken ; a mathematical line is formed by the motion of a point in space ; when the motion is always in the same direction, a right or straight line is produced. A physical line is drawn by the motion of the point of a pen or pencil over any surface. A mathematical surface is produced by the motion of a line in any direction; when the moving line is straight and the motion all in one direction, the surface is called a plane. When a portion of a surface is entirely enclosed by a line or lines, the boundary is called the perimeter or circumference. When two right lines meet together at a point they are said to form an angle with each other, which is greater or less according to the amount of turning or inclination which one line makes with the other; for example, the angle AOC (fig. 1) which 0 C makes with 0 A is greater than the angle A OB which B O makes with O A. Let a circle be drawn from 0 as a centre and with any radius, then the angle at O is measured by the arc which it subtends, the are and the angle increasing and decreasing together ; when the angle CO A is double the angle BO A, the arc A C is Fig. 1. . twice as long as the arc A B. If the line A O is produced to any point E, and the perpendicular OD is set up so as to make the angle A OD equal to the angle EOD, then each of these angles is called a right angle. The angle A OB, which is less than a right angle is called an acute angle, and the angle A OC, which is greater than a right angle, is called an obtuse angle. An angle is said to be included by the two lines forming it. The point () is called the vertex or angular point of the angles A OC, A OD, A OB. 2. To divide a right line of given length into two or more equal parts.—Let A B (fig. 2) be the given line. From Fig. 2. the points A and B as centres, and If we bisect each of the equal parts A C and BC in the manner above described, we shall have the line A B divided into four equal parts; and if each of these is again bisected the line will be divided into eight equal parts. In this manner we can divide a line into any number of equal parts, that number being a power of twó, as 4, 8, 16, 32, and so on. : When the given line is too long to allow of the arcs being struck from the two ends, its middle point can be found by measuring equal distances A G and BF from each end, and then bisecting, by the foregoing method, the line between the points F, G in the point C which will also be the middle point of the given line A B. 3. From a given point in a given right line to erect a perpendicular.—Let C (fig. 2) be the given point in the given line A B. From C as a centre and with any radius CA describe a circle cutting the line in A and B. From A and B as centres, and with any radius A F greater than AC, describe circles cutting each other in the points D and E; then the right line DCE will be perpendicular or at right angles to A B. - Another method. Let CAB (fig. 3) be the given line, A the given point. Take any length A D and measure AC equal to three times AD, and A B equal to four Fig. 3. times A D. From A as a centre, and with AC for radius, describe a circle ; from B as a centre and with B D as a radius, draw an arc cutting the circle in the point E. Then the straight line E A is perpendicular to A B. This method is used by surveyors in setting out a right angle ; let a cord or measuring tape be held tight at three points so as to form a triangle A BE, having the lengths of its sides in the proportion of 3, 4, and 5 ; then the angle B A E, which is opposite the longest side or hypotenuse BE, will be a right angle. 4. To draw a perpendicular to a given right-line from a given point outside of it.—Let A B (fig. 4) be the given line, C the given point. Take ing in C and D; draw CD cutting A B in the point H, then the line CHD is perpendicular to AB. In applying this process to surveying, it is only necessary to measure two equal lengths from C to A B, namely, CA and C B, and to bisect the line A B in H; then the line HC will be perpendicular to A B. 5. From a given point to draw a right line equal in length to a given line.—Let A B (fig. 5) be the given line, C the Fig. 5. given point. Join A C. From in E. From D as a centre, and with D E for radius, describe a circle, and produce DC to meet it in F; then CF is equal to A B, and is drawn from the given point C. |