Fig. 2196.
If now we were to move the work endways upon the table, we should simply cause the moulding to be finished to shape as it passes the line a; because all the cutting is done before and up to the time that the chisel edge reaches this line; or in other words, each part of the chisel edge begins to cut as soon as it meets the moulding, and ceases to cut as soon as it reaches this line. We may now draw this circle and put on it a chisel in two positions, one at the time its lowest cutting point is crossing the line a and the other at the time the highest point on its cutting edge is passing the line, these positions being marked 1 and 2 in [Fig. 2196]; the depth of moulding to be cut being shown at s. Now, since the chisel revolves on the centre of the circle, the path of motion on its highest cutting point c will be as shown by the circle b, and that of the lowest point or end e of its cutting edge will be that of the circle d, while the depth of moulding it will cut is the distance between c and e, measured along the line a a, this depth corresponding to depth shown at s.
Clearly when the chisel has arrived at position 2, the moulding will be finished to shape, and it is therefore plain that it takes a length of cutter-edge from c to f to cut a moulding whose depth is s, or what is the same thing, c e.
But to solve the question in this way, we require for every different depth of moulding to make such a sketch, and the square bar that drives the chisel is made in various sizes, each different size again altering the length or depth of chisel edge necessary for a given depth of moulding.
Fig. 2197.
But we may carry the solution forward to the greatest simplicity for each size of square bar, and for any depth of moulding that can be dressed on that size of bar, by the following means:—In [Fig. 2197] we have the circle and the line a as before; the depth from c to e being the greatest depth of moulding to which the square bar is intended to drive the chisels; while point c is the nearest point to the square bar at which the top of the moulding must be placed. Line a represents a chisel cutting at its highest point; line b a chisel cutting the moulding to final shape at 1⁄4 inch below c, on the line a; line c a chisel cutting the moulding to final shape at a distance of 1⁄2 inch below point c and measured on the line a; lines d, e, f, g, h, and i represent similar chisel positions, the last meeting the point e, which is the lowest point at which the chisel will cut. Suppose, now, we set a pair of compasses one point at the centre a of the circle, and strike the arc j; this arc will represent the path of motion of that part of the chisel edge that would finish the moulding to shape at c; similarly arc k represents the path of motion of that part of the chisel edge that cuts the moulding to final shape on the line a, and at a distance of 1⁄4 inch below c, and so on until we come to arc r, which represents the path of motion of the end of the chisel. All these arcs are carried to meet the first chisel position c a, and from these points of intersection with this line c a we mark lines representing those on a common measuring rule. The first of these from c we mark 1⁄4, the next 1⁄2, the next 3⁄4, and so on to 2, these denoting the measurement or depth of chisel necessary to cut the corresponding depth of moulding. If, for example, we are asked to set a pair of compasses to the depth of cutting edge necessary to cut a moulding that is an inch deep, all we do is to set one leg of the compasses at c, and the other at line 1 on the line c a; or if the moulding is to be 2 inches deep, we set the compasses from c to 2 on line c a. We have here, in fact, constructed a graduated scale that is destined to be found among the tools of every workman who forms moulding cutters, and if we examine it we shall find that the line that is marked 1⁄4 inch from c is not 1⁄4 inch but about 5⁄16 inch; its distance from c being the depth of chisel edge necessary to cut a moulding that is 1⁄4 inch deep.
Again, the line marked 1 measures 13⁄16 inch from c, because it requires a chisel edge 13⁄16 deep to cut a moulding that is one inch deep. But if we measure from c to the line marked 2 we find that it is 21⁄4 inches from c, and since it represents a chisel that will cut a moulding two inches deep, we observe that the deeper the moulding is the nearer the depth of cutting edge is to the depth of moulding it will produce. This occurs because the longer the chisel the more nearly it stands parallel to the line a, at the time when its point is crossing the line a. Thus, line i is more nearly parallel to a than line a is, and our scale has taken this into account, for it has no two lines equally spaced; thus, while that marked 1⁄4 is 5⁄16 inch distant from c, that marked 1⁄2 is less than 5⁄16 inch distant from that marked 1⁄4, and this continues so that the line marked 2 is but very little more than 1⁄4 inch from that marked 13⁄4. Having constructed such a scale we may rub out the circle, the arcs, the line a, and all the lines except the line from c to a and its graduations, and we have a permanent scale for any kind of moulding that can be brought to us.
If, for example, the moulding has the four steps or members s, t, u, v, in the figure, each 1⁄2 inch deep, then we get the depth of cutter edge for the first member s on our scale, by measuring from c to the line 1⁄2 on line c a. Now the next member t extends from 1⁄2 to 1 on the moulding, and we get length of cutter edge necessary to produce it from 1⁄2 to 1 on the scale. Member u on the moulding extends from 1 to 13⁄4; that is to say, its highest point is 1 inch and its lowest 13⁄4 inch from the top of the moulding, and we get the length for this member on a scale from the 1 to the 13⁄4; and so on for any number of members.