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By means of the tangents we can draw the curve by hand through the several points found upon its contour, as
they give us the direction in which the spiral is moving.
If it is desired to imitate the spiral by means of arcs of circles joining the several points found in the above manner, the normals must be drawn perpendicular to the tangents at those points, and the point of intersection of any two consecutive normals will be the centre from which any one of those arcs must be struck.
184. To find points on the contour of the spiral in which the radius-vector varies inversely as the square root of the angle of revolution from a fixed axis.—Draw BSX (fig. 151) as a fixed axis, S being taken as the pole. Measure a length SX equal to 1000 on any scale, and from S as a centre, with SX as radius, describe a circle; from X as a centre, with a radius equal (on the same scale) to 959, draw an arc cutting the circle in K, which will be a point on the required curve. K can also be found by making the angle KSX equal to 57.3°, and measuring SK as 1000. Bisect the line KX by the line SP, taking SP equal to 1414 on the above scale; then P is a point of contrary-flexure on the curve, which now becomes convex towards the axis SX as it approaches thereto, being concave from P towards K. As the angle made by the radius-vector with the axis decreases, so the curve approaches nearer and nearer to the axis, but as the length of the radius increases at the same time, the curve never can reach the axis which is an asymptote (124) thereto.
Draw CSA perpendicular to SX, and divide each quadrant into eight equal angles, measuring the distance of each point as shown upon the figure to the same scale as was
used in measuring SK; we thus obtain a sufficient number of points to enable us to draw in the spiral by hand.
The radius of curvature (101) is of infinite length at the point of contrary-flexure, P, or the curve becomes flat at that point. From P towards A the curvature increases rapidly, or the length of the radius of curvature diminishes; thus at K the radius of curvature is not quite double the length of the radius-vector SK, at A it is less than one and a half times SA, and at D it is very little more than the length of the radius-vector SD, so that from this point the curvature of the spiral approaches very nearly to a circle.
The line SR bisects the right-angle ASX and measures 1128 on the before-mentioned scale; the line SN bisects the angle RSX and measures 1596; the line SQ bisects the angle RSN and measures 1303; the line SL bisects the angle NSX and measures 2256 ; the line SM bisects the angle NSL and measures 1840. The line S U bisects the angle ASR, and the line ST bisects the angle A SU.
185. To draw tangents at various points of the foregoing spiral.—Let $ be the angle between the tangent and radius-rector at any given point on the curve, 4 the angle (in circular measure) of rerolution made by the radius with the fixed axis SX (fig. 151), then it is found by analysis that
By help of this formula the tangent can be drawn for any given angle of revolution.
At L, take Ll, on the line SL, equal to 100 on any scale, draw lt perpendicular thereto and measuring 39 on the same scale, then tL is the tangent at L. Proceeding in
the same way, we find the tangent at P by taking Pp equal to pt, and tP is the tangent; at K, take kt equal to twice Kk, and tK is the tangent; at A, Aa is to at as 100 to 314: at E, Ee is to et as 100 to 471; at B, Bb is to bt as 100 to 628; at F, Ff is to ft as 100 to 785 ; at C, C c is to ct as 100 to 943; at D, D d is to dt as 100 to 1257. At the point D the tangent and radius-vector SD make an angle of 831, and as we follow the spiral inwards this angle gets nearer and nearer to a right-angle, or the curve approaches towards a circle in its contour.
Having drawn the tangents we can describe the curve itself by hand through the several points; or if it is desired to approximate to the curve by arcs of circles drawn from point to point, the centres of those arcs must be at the intersection of the consecutive normals or perpendiculars to the tangents at those points.
186. To apply the lituus to form the outline of a console or double volute.—Draw a horizontal axis BSX (fig. 152), take SX equal to 1000 on any scale, and from S as a centre, with SX as radius, draw the circle X.K; from X as a centre, with radius equal to 959 on the same scale, draw an arc cutting the circle in K; or draw KS, making an angle of 57.30 with the axis. Bisect the angle KS X by the line SPS', and take SP equal to 1414, then P is the point of flexure of the curve. On the same or any other scale measure PS equal to 1414, and draw B'S'X' parallel to BSX; take S'X' equal to 1000 on the same scale as PS is measured upon, and from Sas a centre, with S'X' as radius, describe a circle. Draw S'K', making with S'P an angle equal to the angle PS'X', and cutting the circle at K'. P being the point of contrary-flexure in both curves, and the two curves having a common tangent at P,