Fig. 157.
Having found the point k, we mark (on the outside of the circle, so as to keep the marks distinct from those first marked) the division b, c, d, [Fig. 156], &c., up to g, the number of divisions between b and g being one quarter of those in the whole circle. Then, beginning at k, we mark off also one quarter of the number of divisions arriving at m in the figure and producing the point 3. By a similar operation on the other side of the circle, we get the true position of point No. 4. If, in obtaining points 3 and 4, the compasses are not found to be set dead true, the necessary adjustment must be made; and it will be seen that, so far, we have obtained four true positions, and the process of obtaining each of them has served as a justification of the distance of the compass points. From these four points we may proceed in like manner to mark off the holes or points between them; and the whole will be as true as it is practicable to mark them off upon that size of circle. In cases, however, where mathematical precision is required upon flat and not circumferential surfaces, the marking off may be performed upon a circle of larger diameter, as shown in [Fig. 157]. If it is required to mark off the circle a, [Fig. 157], into any even number of equidistant points, and if, in consequence of the closeness together of the points, it becomes difficult to mark them (as described) with the compasses, we mark a circle b b of larger diameter, and perform our marking upon it, carrying the marks across the smaller circle with a straightedge placed to intersect the centres of the circles and the points marked on each side of the diameter. Thus, in [Fig. 157], the lines 1 and 2 on the smaller circle would be obtained from a line struck through 1 and 4 on the outer circle; and supposing the larger circle to be three times the size of the smaller, the deviation from truth in the latter will be only 1⁄3 of whatever it is in the former.
In this example we have supposed the number of divisions to be an even one, hence the point k, [Fig. 152], falls diametrically opposite to a, whereas in an odd number of points of division this would not be the case, and we must proceed by either of the two following methods:—
Fig. 158.
In [Fig. 158] is shown a circle requiring to be divided by 17 equidistant points. Starting from point 1 we mark on the outside of the circumference points 2, 3, 4, &c., up to point 9. Starting again from point 1 we mark points 10, 11, &c., up to 17. If, then, we try the compasses to 17 and 9 we shall find they come too close together, hence we take another pair of compasses (so as not to disturb the set of our first pair) and find the centre between 9 and 17 as shown by the point a. We then correct the set of our first pair of compasses, as near as the judgment dictates, and from point a, we mark with the second compasses (set to one half the new space of the first compasses) the points b, c. With the first pair of compasses, starting from b, we mark d, e, &c., to g; and from i, we mark divisions h, i, &c., to k, and if the compasses were set true, k and g would meet at the circle. We may, however, mark a point midway between k and g, as at 5. Starting again from points c and i, we mark the other side of the circle in a similar manner, producing the lines p and q, midway between which (the compasses not being set quite correct as yet) is the true point for another division. After again correcting the compasses, we start from b and 5 respectively, and mark point 7, again correcting the compasses. Then from c and the point between p and q, we may mark an intermediate point, and so on until all the points of division are made. This method is correct enough for most practical purposes, but the method shown in [Fig. 159] is more correct for an odd number of points of division. Suppose that we have commenced at the point marked i, we mark off half the required number of holes on one side and arrive at the point 2; and then, commencing at the point i again, we mark off the other half of the required number of holes, arriving at the point 3. We then apply our compasses to the distance between the points 2 and 3; and if that distance is not exactly the same to which the compasses are set, we make the necessary adjustment, and try again and again until correct adjustment is secured.
Fig. 159.