Turning now to [fig. 3] of this [Plate], we there see the main shaft A B, broken off at B: and the letters a b again shew the dividing plate of [figs. 1] and [2]: under this Plate is seen an alidade or moveable index, shewn by section only at c, and in elevation at d e; where it clips the plate as far as n and carries a boss between n and e, on which the dividing index e f, turns; and to which it is strongly fixed by a nut o, when the proper number to be cut is determined. Moreover, this boss forms, itself, the nut of a thumb-screw s, which, carrying a circular plate at its lower end, clothed with leather or any soft substance, connects strongly, without injuring the plate, the moveable index with any point of it, as determined by the dividing index e f. This brings us into the midst of things, as it respects the use of this Engine; for the former index c d, is furnished with a small roller, p, the motion of which all the foregoing objects must obey, when they have been fastened together by the thumb-screw s. We turn then to the [figures 1 and 2] of [Plate 16], in order to shew those parts in action: after remarking only that the form p q r of this [fig. 3], is that of the moveable index shewn before at c d; requiring only, to become complete, that the part q should be sufficiently lengthened to make the arc r q a complete semi-circle—for purposes that will shortly be explained.
In the two [figures] of [Plate 16], the Machine is shewn as placed on its bench or table, accompanied by the parts which give it a distinctive character, and in fact embody the System. In addition to the parts already described, we first remark the circular rim c d, fixed to the ends of the bar E F; and made perfectly concentric with the main shaft A B, and the dividing plate a b. This rim is shewn in section only, at v [fig. 2]. Its section resembles an L, and thus forms a basis for certain plates that will soon appear; and receives the screws by which these plates are fastened to it. This being sufficiently clear, we now proceed to describe the table and the connection of its mechanism with the foregoing.
In [Plate 16], K L is the table: to which the Engine is screwed through its feet G H. I, is a square bar of wood, sliding in a mortice through the top of the table; and connected by a joint with the lever M N—itself moving round a pin at O, and carrying a friction roller, P, which pressed by the spiral Q, as turned by the handle R, raises the bar I, and with it the main axis A B of the plate, and of course the wheel to be cut, centered as usual on this axis above B. Finally, p q r, in both figures, is the moveable index first shewn in [fig. 3] of [Plate 15]; prepared to be drawn round by a weight W, hanging to the cord x, passing over the pulley y, and tied to the right end of the arc q r, when this is to move to the left; or to its left end, when the motion is to be toward the right:—these motions depending on the right or left-handed direction of the teeth which it might be wished to cut on the Machine.
Between the two [figures 1 and 2] of this [Plate], there appears a [diagram], the base of which is nothing more than a part of the rim c d supposed straightened, and placed there that its use may be the easier understood. On the rim is seen a right angled triangle e g f, against which the roller p will lean by the action of the weight W on the cord x, and the arc q r of the moving index p q r. So THAT when, by the handle R, the spiral Q depresses the lever M N, by means of its roller P, then the bar I raises the axis A B of the Engine, and the weight W turns it at the same time, as much as the small roller p permits by rolling up the side e f of the plate e g f. And thus may a screw-formed tooth be cut in any wheel centered above B in the usual manner.
Thus then, in describing this Machine, the manner of using it has been also shewn: for the cutter, in this Machine, (to cut spur wheels) is always fixed; and all the motion is composed of the rotatory and longitudinal movements of the principal axis, which carries the wheel along with it. The cutter I say is fixed, at a proper height just above the wheel, and at an angle to the perpendicular, equal to that it is wished the teeth should form at it’s pitch line. This inclination as before observed is 15 degrees; and the tangent of 15° is in round numbers 268, when the radius is 1000. That is, in our present [figure], the basis e g of the plate e g f, occupies 268 divisions of a scale, of which the height g f contains 1000. It appears then, that to cut a tooth with 15 degrees inclination, by this Plate, the wheel receiving that tooth, must be just as large as the rim itself; for the surface of the wheel would turn more, with a given elevation, if it were larger than the rim; and would turn less, by the same elevation, if it were smaller. In a word the whole theory of this operation, is now clearly seen. The smaller the wheel to be cut, the longer, horizontally, must be the Plate; or in other words, as the diameter of the wheel is to that of the rim, (c d) so is the length e g of the Plate to the length required. Now this height f g, is always the same; all change therefore, in the plates, takes place on the horizontal length: and this length is most easily found by the foregoing RULE OF THREE. If then, instead of the triangle e f g, I had used the triangle e′ f′ g′ it would have followed at once, that to produce an inclination of 15 degrees, I must have taken a wheel of just half the diameter of the rim; for the plate e′ f′ g′ is just twice as long as that e f g. To prove this, let us suppose the diameter of a wheel wanted, to equal one half that of the rim c d: then the rule will stand thus:
1 is to 2, as 268 is to ...536, the length of the plate according to the theory; which is precisely the length it is drawn to compared with that e f g, namely twice as long. Thus the four triangles, drawn to the right and left in this diagram, represent the plates for the wheels of the following diameters respectively:
| No. | 1 | , a wheel | equal | to | the | plate rim c d; |
| 2 | do. | do. | to | 1⁄2 | do. | |
| 3 | do. | do. | to | 1⁄3 | do. | |
| 4 | do. | do. | to | 1⁄4 | do. |