CHAPTER V. SPIRALS. 171. DEFINITIONS.—-When a point is made to describe a series of convolutions about a fixed point called the pole, so that its distance therefrom is continually varying, the curve which it marks out is called a spiral. Any line drawn from the pole to the revolving point is called a radius-vector to the curve or the radius which the moving point carries round with it. It is usual to consider the radius-vector as starting from some straight line or axis, which passes through the pole, and to which it returns after every successive convolution. The simplest form of spiral is that which is obtained when a thread is unwound from a circular cylinder, the spiral marked out by the end of the thread being called the Involute of the circle (fig. 141), in which the tangents of the circle are normals to the spiral. If a rod is made to 'revolve round a pole from a fixed axis, and a point starting from the pole moves along the rod so that its distance from the pole is always proportional to the angle which the rod has revolved over from the axis, the curve which the point will mark out is called the spiral of Archimedes (fig. 142). This spiral is one of a large family of spirals in which the length of the radiusvector varies according to some power of the angle it has revolved over from the fixed axis. Thus we can have a spiral in which the length of the radius-vector is proportional to the square or second-power of the angle of revolution (figs. 143, 144); or one in which it is proportional to the cube or third-power of that angle (figs. 145, 146, 147, 148). When the length of the radius-vector, starting from a fixed axis, diminishes in proportion as its angular distance from that axis increases, the curve marked out by the moving-point is called the reciprocal-spiral (figs. 149, 150), the radius-vector in this case being said to vary inversely as the angle of revolution. If the length of the radius-vector varies inversely as the square-root of its angular distance from the axis, the curve marked out is called the lituus (fig. 151); and a variety of this spiral can be obtained by making the radiusvector vary inversely as the cube-root of the angle revolved through (fig. 153). If the length of the radius-vector is increased in geometrical ratio, while the angle of revolution increases in arithmetical ratio, the spiral obtained is called the equiangular or logarithmic spiral (figs. 154, 156, 158). It is called equiangular because the tangent at every point in the same spiral makes everywhere the same angle with the radius-vector; and it is termed logarithmic because the angle of revolution is proportional to the logarithm of the length of the radius-vector. This spiral can be drawn in infinite variety, and has geometrically neither beginning nor ending, as it consists of an endless series of convolutions, all of which are exactly similar in the same curve, and only differing in actual size or scale. THE INVOLUTE OF THE CIRCLE. 172. To draw the involute by continuous motion. Let S be the pole, S A the axis (fig. 141) of the proposed spiral. Fig. 141. Take a circular cylinder of radius SA, S being its centre, and wrap a thread round it. Attach a pencil to the outer end, A, of the thread, and unwind the thread from the cylinder, keeping it tight by the pencil ; then the spiral AECD will be marked out. If we take AC perpendicular to A S, when the pencil arrives at C the quantity of thread unwound will be equal to the circumference of the cylinder; and since A C is the tangent to the circle at A, the tangent to the spiral at C is parallel to A S, being always at rightangles to the tangent of the circle ; and A C is the normal to the spiral at C. Let it be required to draw a spiral of a given size and given number of convolutions ; if A C is given for the first convolution, we must find the diameter A F of the generating circle or cylinder by taking A F to A C in the ratio of 100 to 314, or of 7 to 22. If A C is given for two convolutions, then we must take A F to AC as 100 to 628, or twice 314; if A C is given for three convolutions, take A F to A C as 100 to three times 314, or 942 ; and so on, according to the number of convolutions required. 173. To find points upon the involute.-- Divide the generating-circle A BF (fig. 141) into four quadrants, and each quadrant into four equal angles. Draw the tangents to the circle (48) at each of the 16 points into which its circumference is divided. Let the diameter of the circle A F represent 100 on any scale ; then to find the corresponding points on the spiral, measure along the tangents to the circle by the same scale the distances shown by the numbers on the figure from the points where the tangents touch the circle, beginning from the point A, where the spiral first starts from the circle. In this way 16 points will be found on the first convolution from A to C. To find points on the second convolution, produce the abovenamed tangents indefinitely, and measure a distance of 314 on the same scale as before, from each point on the first convolution ; these will give 16 points on the second convolution. To find points on the third convolution, measure 314 from each point on the second, and so on for any number that may be required. A larger number of points can be found upon the spiral if desired. Suppose it is required to find 32 points on each convolution ; bisect each of the 16 angles into which the circle was divided above, and draw the tangents at the extremities of the radii. The length of the tangent at the end of the first radius after leaving A will be 314 divided by 32 ; that of the second, twice that of the first; of the third, three times that of the first, and so on for all the thirty-two points round the circle. For the points on the second convolution we have only to proceed as before, by adding 314 to the length of each normal to the first convolution, and in the same way for the third convolution. When a cogged-rack is driven by a pinion-wheel, the outline of the cogs is the involute of the circle which forms the 'pitch line of the driver.' THE SPIRAL OF ARCHIMEDES. 174. To draw a spiral whose distance from the pole is directly proportional to the angle made by the radius-vector with the fixed axis.—Let FSH (fig. 142) be the fixed axis, S the fixed point, or pole. The spiral can be drawn by continuous motion if a rod is made to revolve about S, and at the same time a pencil moves along the rod at a rate proportional to the velocity of revolution. To describe the spiral by finding points on its contour, draw ESG at right-angles to SH, and divide each rightangle into four equal angles. Let D be the point where the spiral is to cut the axis after one convolution ; on S E take SA, equal to one-fourth of SD, on SF take SB, equal to |