Fig. 1841. Fig. 1842.
Fig. 1843. Fig. 1844.
Now, suppose the bar to have an equal flat place on its opposite side, becoming of a section shown in [Fig. 1839], upon applying the dies and pursuing a similar course of reasoning, the die with four points would reduce the bar to the size and shape shown in [Fig. 1840], or a true cylinder, while the triple-pointed cutter would produce the form shown in [Fig. 1841], which is a sort of hexagon, coinciding with the true circle in six places—a, z, b, y, d, and x—while between a and z, and opposite, between y and d, there is an elevation; also from z to b and from d to x. A flattened portion, a x, with a similar one b y, opposite, completes the profile. Suppose, now, that a bar of the form shown in [Fig. 1842], having two flat places not opposite, be taken, and the four-cutter and three-cutter dies are applied. The product of the four is shown in [Fig. 1843], and that produced by the three-cutter die in [Fig. 1844]. The section cut with four coincides with the true circle at four points, a, b, c, d, and differs from it almost imperceptibly at z, y, v, and x. There are two elevations between a and b and between b and c; also two depressions between c and d and between d and a. The section from the three-cutter die is the perfect circular form between a z, b y, and d x, with a projection from z to b and two depressions from y to d and from x to a. The four-die, applied to a section having three flats like [Fig. 1845], would produce [Fig. 1846], which does not absolutely coincide with the true circle at any point, although the difference is inconsiderable at a, z, y, c, v and x; three equidistant sections a z, y c, and v x, are elevated and the three alternate ones depressed.
