In Tables 9, 10, 11 and 12, are given several sets of wheels for cutting all, and more than all, the various threads required for engine-making; but in order that any other desired step may be cut with a lathe-screw of any step, the proper modes of calculation are here given.

The general fundamental principle which is involved in the cutting of all sorts of pitches with all varieties of wheels and lathe-screws is, the relation between the pitches of two screws, which screws are the lathe-screw, and the one to be cut. As soon as these two pitches or steps are properly ascertained and stated, the relation is immediately seen ; and when this relation or ratio is known, the relation which must exist between the required wheels is also known. For example, suppose the lathe-screw with which a thread is to be cut, has one step per inch of its length, and that the desired thread is to have also one step per inch, the ratio is as one to one. Therefore, any two cog-wheels which possess this ratio to each other will cut the thread ; and any two wheels of proper sizes for the lathe can be used if their numbers of cogs are the same, whether 40 and 40, 100 and 100, 50 and 50, 134 and 134, or any other numbers.

Again, suppose that the thread to be cut is to have 7 steps per inch, and the same lathe-screw to be used. In this case the ratio is as 1 to 7, and it is only necessary to multiply each number by 20, to obtain the numbers of teeth required. Thus : - 1 x 20 = 20, and 7 x 20 = 140. 20 and 140 will therefore cut the thread. And, because the step of thread to be made is smaller than that of the lathe-screw, it is evident that the larger wheel must be put upon the lathe-screw that it may rotate slower than the screw to be cut.