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half of SD, and on SG take SC, equal to three-fourths of SD. So also for the four angles into which the first quadrant is divided, the length of the radius-vector drawn from S is one-fourth of S A at the first quarter of the rightangle, one half S A at the second quarter, and three-fourths SA for the third quarter. In the other quadrants the length of each radius-vector increases in the same way by one-fourth of SA for every quarter of a right-angle.

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If we bisect each of the 16 angles, the increase of the radius-vector at each successive angle will be one-eighth of SA.

The distance apart, D H, of any two convolutions of this spiral, as measured along the radius-vector, is everywhere the same, being equal to the length S D.

175. To draw the tanyent at various points of the foregoing spiral.—Let o be the angle between the tangent and radius at any point of the curve after the radius-vector has passed over an amount of revolution from the axis which is represented in magnitude by 6 (taken in circular measure); then it is found by the differential calculus that

. The tan =4. From this formula it appears that the angle which the tangent makes with the radius-vector is very acute for any point near the pole S, and increases rapidly as its distance therefrom is increased. Thus at A, the tangent makes an angle TAS of 57.52° with the radius-vector ; at B the angle TBS is 72:67°; at C the angle TCS is 78° ; at D the angle TDS is 81°; at E the angle TES is 82.75° ; at F the angle TFS is 84°; at G the angle TGS is 84.8°; and at H the angle is 85.45°. And as the number of convolutions increases, so the curvature of the spiral gets nearer and nearer to that of a circle having S for its centre.

To draw the tangent at A, make A a perpendicular to SA, and equal on any scale to 157 ; draw a T at right angles to A a, and equal to 100 on the same scale, then TA is the tangent at A. Proceeding in the same way at the other points, at B make B b equal to 314, 6 T to 100; then TB is the tangent at B. At C, CT the tangent is drawn by taking C c to c T as 471 to 100. At D, take Dd to dT as 628 to 100. At E, take Ee to e T as 785 to 100. At F, take Ff to Tf as 942 to 100. At G, take Gg to Tg, as 11 to 1.

176. To find points on the spiral whose distance from the pole is proportional to the square of the angle made by the radius-vector with the fixed axis.Let S (figs. 143, 144) be the pole, SX the fixed axis. Draw KSM (fig. 1.13) per

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pendicular to SX, and divide each quadrant into four equal angles. Measure along these lines from S the squares of

Fig. 143

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successive integers beginning with unity, as given in the Table (181), on any convenient scale ; then sixteen points will be obtained on each convolution of the spiral. To

save reference to the Table, the numbers are marked on the figures themselves, indicating the distance of the various

Fig. 144.

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points from the pole. The spiral can be drawn by hand through the points thus found.

177. To draw the tangents at various points of the fore. going spiral.-If we call the angle which the tangent at

any point of the curve makes with the radius-vector drawn to that point, and 6 the amount of revolution described by the radius-vector from the axis (taken in circular measure), it is found by the differential calculus that

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From this formula it appears that the angle between the tangent and radius is very acute for any point near the pole S, but increases rapidly as the number of convolutions is increased.

Let S P (fig. 144) make half a right angle with S X, its length being 4; draw Pp perpendicular to PS, and measure Pp, equal to 39 on any scale, make p T at right-angles to p P and equal to 100 on the same scale, then TP is the tangent at P. Proceed in the same way at the other points, taking A a to T a as 78 to 100, and T A is the tangent at A. At Q, take Qq to Tq as 118 to 100, and T Q is the tangent; at B, take B b to Tb as 157 to 100, and T B is the tangent; at R, take Rr to Tr as 196 to 100, and TR is the tangent. At C (fig. 143), take C c to Tc as 236 to 100, and TC is the tangent; at D, take D d to Td as 314 to 100; at E, take Ee to Te as 393 to 100; at F, take Ff to Tf as 471 to 100. At G, take Gg to Tg as 55 to 10, and TG is the tangent; at H, take H h to T h as 63 to 10 ; at I, take I i to Ti as 71 to 10; at J, take J; to T; as 79 to 10; at K, take K k to Tk as 86 to 10; at L, take Ll to Tl as 94 to 10; at M, take M m to Tm as 102 to 10, and T M is the tangent.

Before drawing any spiral by hand, it is essential to draw as many tangents as possible, since they give the direction in which the curve is moving at the several points.

178. To find points on the spiral whose distance from the pole is proportional to the cube of the angle made by the

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