We may also consider here, the form of the cutter itself, v, [fig. 1]. It is slightly conical, (more or less so according to it’s use) and of no greater diameter than the smallest width of the spaces between the teeth of the wheel. A common disk-like cutter would not produce perfect, nor even tolerable teeth on a bevil wheel. The reason of this will appear by considering that a spiral line, either on a cone or it’s base, turns more the further it is from the centre, and less the nearer it comes to it. So that a flat cutter placed at any angle, is parallel to the curve at one place only; whence the propriety of using a cutter of the kind represented in this figure. It is however true, that the first opening of the spaces may be made with a common cutter; but it should be very thin comparatively with the spaces required: and it’s cut would serve only as a sketch of such space, serving principally to permit the metal to escape while finishing the teeth with the cutter just described.
I proceed now to the examination of the plates, and the manner of adapting their length to the process of cutting spiral teeth on bevil wheels. But before entering on this subject, I would explain a kind of inadvertency into which I fell at the close of my former description of this Engine (see [page 129]). In my zeal to be candid in stating the properties of my Machines, I have suffered it to appear that I thought this an “imperfect” one:—an expression which, although modified among the errata, may still cause it to be looked upon as radically defective; than which nothing could be further from the idea I wished to convey. I intended merely to express the want of absolute connection between the two movements of the shaft—the rotatory and longitudinal motions. I meant that the process by this Machine was not theoretically certain, because dependent on the action of a weight ([Plate 16], [fig. 1 and 2]) and an unforced obedience to the direction of the plates. But this small remove from rigourous principle is in my opinion much overballanced by the facility of cutting good wheels of all diameters, by the sole change of a morsel of tin, which leaves untouched every other part of the Engine.
Entering then on this branch of the subject, I first observe that if we chuse for the teeth an inclination of 15 degrees (in imitation of the cylindrical wheels) it can only be for one point of such wheels—as observed above. This point therefore I have placed at r in the middle of the face. And supposing now that at this point the wheel O were 4 inches in diameter and the wheel S two inches, these plates would be found as before by these analogies:
(1) wr, or 2 inches : 11 inches (rad. of plate rim) ∷ 26.8 : 294.8⁄2 = 147.4 plate required.
(2) vr, or 1 inch : 11 inches (rad. of plate rim) ∷ 26.8 : 294.8⁄1 = 294.8 2d. plate required.
But it is plain that the conical face, b C, (common to both wheels) is broader than the supposed cylindrical ones b e and b d: and therefore that the above plates must be made longer (to furnish the said obliquity) in the following proportions, namely: for the wheel O in the ratio of b e to b C; and for the wheel s in that of b d to b C: that is, these plates should be lengthened as the tabular cosines of the angles B A C and D A C to radius (for b e : b C ∷ A B : A C; and b d : b C ∷ A D : A C.) Thus then,
(1) Cos. 63°27′ : radius ∷ 147.4 (present plate) : required plate x, = 147.4 r⁄Cos. 63°27′; and
(2) Cos. 26°33′ : radius ∷ 294.8 (present plate) : required plate y, = 294.8 r⁄Cos. 26°33′.
Now, by the tables, cosine 26°33′ = 894, and cosine 63°27′ (it’s complement) = 447, when radius is 1000: whence dividing the two equations by r, and substituting these values of cosines 63°27′ and 26°33′ we shall find the two quantities x and y, equal. Whence it appears that for every pair of bevil wheels, whose shafts lie at right angles, the same plate serves for both wheels: only turning it once to the right, and once to the left hand on the plate rim.
And if now we measure on a scale of equal parts, the line A r and call it 100, we shall find the line w r (near enough for practice) to be 90, and the line v r to be 45, and these numbers respectively, put for rad. for cos. 26°33′, and for cos. 63°27′, will make the first equation x = 147.4 × 100⁄45 and y = 294.8 × 100⁄90 or x = 327.55 and y = 327.55, &c. confirming the above deduction that the same plate serves for both wheels; and giving, withal, the length of the plate required.