“To mark the pitch line we proceed as follows:—
Fig. 194.
“In [Fig. 194], a a and b b are centre lines passing through the major and minor axes of the ellipse, of which a is the axis or centre, b c is the major and a e half of the minor axis. Draw the rectangle b f g c, and then the diagonal line b e; at a right angle to b e draw line f h cutting b b at i. With radius a e and from a as a centre draw the dotted arc e j, giving the point j on the line b b. From centre k, which is on line b b, and central between b and j, draw the semicircle b m j, cutting a a at l. Draw the radius of the semicircle b m j cutting f g at n. With radius m n mark on a a, at and from a as a centre, the point o. With radius h o and from centre h draw the arc p o q. With radius a l and from b and c as centres draw arcs cutting p o q at the points p q. Draw the lines h p r and h q s, and also the lines p i t and q v w. From h as centre draw that part of the ellipse lying between r and s. With radius p r and from p as a centre draw that part of the ellipse lying between r and t. With radius q s and from q draw the ellipse from s to w. With radius i t and from i as a centre draw the ellipse from t to b. With radius v w and from v as a centre draw the ellipse from w to c, and one half the ellipse will be drawn. It will be seen that the whole construction has been performed to find the centres h p q i and v, and that while v and i may be used to carry the curve around the other side or half of the ellipse, new centres must be provided for h p and q; these new centres correspond in position to h p q.
“If it were possible to subdivide the ellipse into equal parts it would be unnecessary to resort to these processes of approximately representing the two curves by arcs of circles; but unless this be done, the spacing of the teeth can only be effected by the laborious process of stepping off the perimeter into such small subdivisions that the chords may be regarded as equal to the arcs, which after all is but an approximation; unless, indeed, we adopt the mechanical expedient of cutting out the ellipse in metal or other substance, measuring and subdividing it with a strip of paper or a steel tape, and wrapping back the divided measure in order to find the points of division on the curve.
Fig. 195.
“But these circular arcs may be rectified and subdivided with great facility and accuracy by a very simple process, which we take from Prof. Rankine’s “Machinery and Mill Work,” and is illustrated in [Fig. 195]. Let o b be tangent at o to the arc o d, of which c is the centre. Draw the chord d o, bisect it in e, and produce it to a, making o a = o e; with centre a and radius a d describe an arc cutting the tangent in b; then o b will be very nearly equal in length to the arc o d, which, however, should not exceed about 60°; if it be 60°, the error is theoretically about 1⁄900 of the length of the arc, o b being so much too short; but this error varies with the fourth power of the angle subtended by the arc, so that for 30° it is reduced to 1⁄16 of that amount, that is, to 1⁄14400. Conversely, let o b be a tangent of given length; make o f = 1⁄4 o b; then with centre f and radius f b describe an arc cutting the circle o d g (tangent to o b at o) in the point d; then o d will be approximately equal to o b, the error being the same as in the other construction and following the same law.