But a greater, or at least a more frequent cause of friction in the wheels is the motion, endwise, of the shafts, arising from a want of solidity in the bearers, and especially of connection between them; for whenever these are strongly connected, and the shafts well fitted to their steps, all circular commotion is ipso facto destroyed; while the longitudinal tendency produced by the teeth on the shafts is certainly an advantage: because it prevents the shaking that often arises from their vibration, endwise, when lying on unsteady bearers, or on bearers between which they have too much liberty.
A few words will make known the process of reasoning by which I arrived at the idea that forms the basis of this invention. I had been conferring with a well-known mechanical character, (to whom the art is greatly indebted)—and hearing his observations on the advantage derived from having two equal cog wheels connected together, with the teeth of the one placed opposite the spaces of the other; so as to reduce the pitch one half, and the friction still more; (since the latter follows the ratio of the double versed sines of the half-angles between the teeth respectively:)—and no sooner had I left that gentleman, than my imagination thus whispered—“What that gentleman says is both true and important.” “But if two wheels thus placed, produce so good an effect, three wheels (dividing the original pitch into three), would produce a better: and four, a better still: And five a better than that. And for the same reason, an indefinite number of such wheels would be indefinitely better! We must then cut off the corners of all those teeth, and we shall have one screw-formed line, that will represent an indefinite number of teeth, and approach indefinitely near to absolute perfection!” Thus did this Invention originate: and it soon appeared to me, to be the nearest approach of material exactitude to mathematical precision, that is to be found in the whole circle of practical mechanics. For not only is the relative motion of the touching points of two wheels (that is their friction), less than the distance between two of the nearest particles of matter, but it is as many times less than that distance, as that distance is less than the half diameter of any wheel whose teeth are thus formed.
I assert therefore that these teeth, placed in proper circumstances, do work without sensible friction at their pitch lines: as although by means of mathematical abstraction, it may be possible to assign a degree of friction between them, that degree cannot be realized on a material surface: and I fear not the friction on mathematical surfaces, if my material surfaces do not suffer from it. I take leave then to repeat, that no friction can justly be said to arise from a motion, too short to carry a rubbing particle from one particle of a rubbed surface to the next! and this is precisely the case in the present instance.
Continuing to reflect on this important subject, I soon perceived that the screw-formed line would give the teeth a tendency to slide out of each other; and to drive the shafts of the wheels endwise in opposite directions; but even that evil is not great: for, confining the obliquity within 15 degrees, that tendency is only about one quarter of the useful effort; and a stop acting on the central points of the axes, will annul this tendency without any sensible loss of power. We need not even have recourse to this expedient when any good reason opposes it: for this tendency can be destroyed altogether by using two opposite inclinations: giving the teeth the form of a V on the surface of the wheels—a method which I actually followed on the very first pair I ever executed, which I believe are now in the Conservatory of Arts at Paris.
A circumstance somewhat remarkable deserves to be here noticed. In the specification of a Patent which I have seen in a periodical work since my return from Paris, for things respecting steam engines, and dated, if I recollect right, in 1804 or 5, this V formed tooth is introduced—as an article of the specification, yet having no connection whatever with its other subjects; nor being attended with the most distant allusion to the principle of this geering. The fact is that I had these V wheels in my Portique, in 1801, when that exhibition took place in which my Parallel motion appeared and was rewarded by a Medal from Bonaparte: so that two of my countrymen at least, engineers like myself, appear to have taken occasion from that exhibition, to draw my inventions from France to England—a thing by no means wrong in itself nor displeasing to me: who was then totally precluded from holding any communication of that kind with my native country.
It would be repeating the statements contained in the foregoing memoir, to say more on the general principles of this System. I request therefore, my readers to give that paper an attentive perusal; and to accept the following recapitulation of its contents:
1. To cut teeth of this form in any wheel is, virtually, to divide it into a number of teeth as near to infinite, as the smallness of a material point is to that of a mathematical one.
2. By the use of these teeth, and the multitude of contacts succeeding each other thence arising, all perceptible noise or commotion is prevented. (This of course supposes good execution, or long-continued previous working.)
3. For the same reasons, all sensible abrasion is avoided: for we have proved that the passage of any point of one wheel, over the corresponding point of another, is indefinitely less than the distance between the nearest particles of matter. (This supposes the action confined to the pitch line of the wheel; and this it will be in all common cases—since the teeth wear each other in preference, within and without that line; which therefore must remain prominent.)