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If it is desired to finish the spiral at the centre by a circular eye at any point as N, draw the normal or perpendicular to the tangent at N, and also the normal at M; then describe the eye taking the point of the intersection of these two normals for its centre.
If it is desired to draw an approximation to the spiral by means of arcs of circles joining the several points found in the foregoing manner, draw the normals at those points perpendicular to the tangents, and the point where two consecutive normals intersect will be the centre from which the arc lying between them is to be struck.
181. Table of the Squares and Cubes of Integers.
sofas e or to
6+ 12.5 210 343 512 729 1000 1331 1728 2197 2744 3:375 4096 4913 5832 0859 8000 9261 10648 12167 13:24 15425 17576 190183 21913
29791 32768 35937 39304 42875 46656 50653 54872 59319 64000 68921 74088 79507 85181 91125 97336 103823 110592 117649 125000 132651 140508 148877 157 46+ 160375 175616 185193 195112 205379 216000
Table of the Squares and Cubes of Inteyers (continued).
226981 238:28 250047 262144 274625 287496 300763 314432 328509 343000 357911 373248 389017 40522+ .421875 438976 456533 47-1552 493039 512000
531141 551368 571787 592704 614125 636056 658503 681472 704969 729000 753571 778688 804357 830584 857375 884736 912673 941232 970299 1000000
98 99 100
THE RECIPROCAL SPIRAL.
182. To find points on the spiral in which the len:yth of the radius-vector varies inversely as the angle of revolution from a fixed axis.—Let S (fig. 149) be the pole, SX the fixed axis. Draw CS A at right-angles to SX, and let the length SA represent 1000 on any scale. Draw A B parallel to SX, then A B will be an asymptote (124) to the curve. Draw SK, making an angle of 57-3° with the axis, and from Sas centre, with SA as radius, describe an arc cutting SK in K, then K is a point on the curve. Bisect the right-angle ASX by the line SI, making SI equal to 1273 on the above scale; then I is a point on the curve.
Draw SQ, bisecting the angle KSX, and equal in length to twice SK, or 2000 on the scale, then Q is a point on
the curve. Draw SR, bisecting the angle ISX, taking SR equal to twice SI, or 2546 on the scale, then R is a point on the curve, and the curve gets nearer and nearer to the asymptote AB as the angle between the radiusvector and the axis diminishes.
Draw SL, bisecting the angle . ASI, and take SL equal to one-third of SR, or to two-thirds of SI, or 819 on the scale; then L is a point on the curve.
On SA, take S M equal to half SI, or 637 on scale, and M is a point on the curve. So also SD will be half of SM, SC one-third of SM, SE one-fourth of SM, and so on.
To find other points on the spiral, divide each quadrant into four equal angles, and measure the lengths marked on the figure by the same scale as is used in the other dimensions; and in this way sixteen points on each convolution will be obtained. In figure 150 the spiral portion of the curve is drawn on an enlarged scale three times that to which figure 149 is drawn.
The portion of the curve PQR (fig. 149) is very flat, but the curvature gradually increases towards M. Thus at I the radius of curvature (101) is more than four times the length of the radius-vector SI; at K it is less than three times the radius-vector; and at M it is less than twice the radius-vector. As the curve continues in its spiral course the radius of curvature gets nearer and nearer to the radius-vector in length.
A second spiral shown by the dotted line I K'L' (fig. 149) can be drawn in the same manner as the first, by taking SA' and SK' as 1000 on some other scale than that used for SA and SK; A'B' will be the asymptote to this curve, and the other points can be found by scale as before.
Another scale of measurement can be used by dividing each right-angle, as before, into four equal angles by the lines SR, SI, SL, &c., RSX being one-fourth of a rightangle, ISX one-half, KSK three-fourths. Let SR represent unity on any scale, then S I will be half of SR, SL will be one-third of SR, SM will be one-fourth, SD will be one-eighth, SC will be one-twelfth, SE will be onesixteenth of SR, and so on round the spiral.
183. To draw tanyents at various points on the reciprocal spiral.—Let ♡ be the angle made by the radius-vector at any point on the curve with the tangent at that point, o the angle of revolution (in circular measure) of the radius from the fixed axis S X (figs. 149, 150), then it is found by analysis that
tan 0=9. The tangent Kt at K (fig. 149) makes half a right-angle with SK, so that to draw Kt we have only to take Kk and K t at right-angles, and equal to each other. To draw the tangent at M, take M m equal to 100 on any scale, draw mt a perpendicular at m, and take thereon mt equal to 157 on the same scale, then t M is the tangent at M. At D (fig. 150), take Dd equal to 100, dt equal to 314 perpendicular to D d, and t D is the tangent at D. Proceeding in the same way at other points, we have at C, Cc to ct as 10 to 47 ; at E, Ee to et as 100 to 628; at F, Ff to ft as 100 to 786 ; at G, Gg to gt as 100 to 943; at H, Hh to ht as 1 to 11; at J, Jj to jt as 100 to 1257. The tangent at any other points can be found if required in the same manner by help of the formula given above.
It will be seen that the tangent becomes more and more nearly perpendicular to the radius-vector as the spiral increases its number of convolutions.