Now let us consider the two productions ([Figs. 1840] and [1847]) that answer the requirements, the two different sections ([Figs. 1839] and [1845]) from which they were cut, and also the other two pieces ([Figs. 1841] and [1846]) that were cut from the same bars at the same time. The general shape of [Fig. 1839], is oval or four-sided, and while the four cutters operated upon it to produce perfectly circular work, the three cutters reproduced the general shape started with, only somewhat modified, as [Fig. 1841] plainly shows. Upon the blank, [Fig. 1845], the general shape of which is triangular, the very opposite is the case, for the three cutters now produce a perfect circle, while the four modify only the figure that they commenced to operate upon.
Considering that every irregular form may be approximated by a square, an equilateral triangle, or in general by either a parallelogram or a regular polygon, it will be found that from a flat, oval, or square piece of metal the four cutters will produce a true circle; from a triangular piece the three; from a heptagon neither will do so, while from a hexagon both the three and four cutters are calculated to do so. Following in the same manner, and increasing the sides, it will be found that the four cutters will produce a true circle from every parallelogram, whether all the sides are equal or not, while the three cutters will produce a true circle also from every regular polygon the number of sides of which is a multiple of three—that is, four cutters would operate correctly upon a figure having 4, 6, 8, 10, 12, &c., parallel sides, while the three would do so upon a figure having 3, 6, 9, 12, 15, &c., equal sides. Thus, for regular forms varying between these two series neither one would be adapted. Hence, if the general form of the work is represented by the first series, the four cutters are the best; if the general and average form of the material to be operated upon corresponds to the second series, then the three dies are the best adapted, so far as their two principles of action, mentioned at the outset, are concerned; hence, if it is considered that the material or bars of metal to be wrought vary from a circular form indifferently, then there is no choice between an even and an odd number merely on that account.
Placing the same dies that cut these six irregular figures upon their respective productions would not serve to correct their form; as, for instance, if the die that cut [Fig. 1846] were revolved around it—even if set up or reduced in diameter to take a cut—it would remove an equal amount all round and leave the same figure still. Similarly with, say, [Fig. 1841], cut by the three; but if the three were run over [Fig. 1846], cut by the four, it would tend to correct the errors, and likewise if the four were run over [Fig. 1841], the tendency would be to modify the discrepancies left by the three that cut it.
Fig. 1849.
Fig. 1850.