A New Century of Inventions / Being Designs & Descriptions of One Hundred Machines, Relating to Arts, Manufactures, & Domestic Life

INTRODUCTION.

In the progress of a work like the present, no competent reason could have been assigned for omitting to bring forward my System of Toothed Wheels, the Patent for which has lately expired:—a System which a few years ago, excited in this town, so much interest, aroused so much animosity, and was treated with so much illiberality:—But which, also, was fostered with so much public spirit, tried with so much candour, and adopted with so much confidence. It was I say, incumbent on me to bring the merits of this System into public view, had it only been to justify myself for proposing, and my friends for adopting it. But stronger reasons point now to the same measure. From the intimate connection the System holds with the subjects of this essay, it must be often adverted to; and I have been already obliged to speak of it in terms which can hardly have been understood by those readers who had not previously considered the general Subject. I should therefore be still in danger of filling these Pages with unintelligible assertions, did I not begin by marking out the foundations on which my statements are built; or by explaining to a certain degree, the Principles of the new System. Without then abandoning the tacit engagement I have taken with my unlearned readers—not to entangle them in too much theory, I think it indispensable to quote the Memoir I read before the Literary and Philosophical Society of Manchester, in December, 1815; which small work will form the basis of the practical remarks I shall have to make on the subject, as this work proceeds. The Memoir is thus introduced in the transactions of that learned body:

MEMOIR
ON A NEW SYSTEM OF
COG OR TOOTHED WHEELS,

By Mr. James White, Engineer.[1]

COMMUNICATED BY T. JARROLD, M. D.

(Read December 29th, 1815.)

“The subject of this paper, though merely of a mechanical nature, cannot fail to interest the Philosophical Society of a town like Manchester, so eminently distinguished for the practice of mechanical science; unless as I fear may be the case, my want of sufficient theoretic knowledge or of perspicuity in the explication, should render my communication not completely intelligible. To be convinced of the importance of the subject, we need only reflect on the vast number of toothed wheels that are daily revolving in this active and populous district, and on the share which they take in the quantity and value of its productions; and it is obvious that any invention tending to divest these instruments of their imperfections, whether it be by lessening their expence, prolonging their duration, or diminishing their friction, must have a beneficial influence on the general prosperity. Now I apprehend that all these ends will be obtained in a greater or less degree, by having wheels formed upon the new system.

I shall not content myself by proving the above theoretically, but shall present the society with wheels, the nature of which is to turn each other in perfect silence, while the friction and wear of their teeth, if any exist, are so small as to elude computation, and which communicate the greatest known velocity without shaking, and by a steady and uniform pressure.

Before I proceed to the particular description of my own wheels, I shall point out one striking defect of the system now in use, without reverting to the period when mechanical tools and operations were greatly inferior to those of modern times. Practical mechanics of late, especially in Britain, have accidentally hit upon better forms and proportions for wheels than were formerly used; whilst the theoretic mechanic, from the time of De la Hire, (about a century ago) has uniformly taught that the true form of the teeth of wheels depends upon the curve called an epicycloid, and that of teeth destined to work in a straight rack depends upon the simple cycloid. The cycloid is a curve which may be formed by the trace of a nail in the circumference of a cart wheel, during the period of one revolution of the wheel, or from the nail’s leaving the ground to its return; and the epicycloid is a curve that may be formed by the trace of a nail, in the circumference of a wheel, which wheel rolls (without sliding) along the circumference of another wheel.

Let A B (Plate 13, fig. 1.) be part of the circumference of a wheel A B F to which it is designed to adapt teeth, so formed as to produce equable motion in the wheel C, when that of the wheel A B F is also equable. Also, let the teeth so formed, act upon the indefinitely small pins r, i, t, let into the plane of the wheel C, near its circumference. To give the teeth of the wheel A B F a proper form, (according to the present prevailing system) a style or pencil may be fixed in the circumference of a circle D equal to the wheel C, and a paper may be placed behind both circles, on which by the rolling of the circle D on A B, will be traced the epicycloid d, e, f, g, s, h, of which the circle A B F is called the base, and D the generating circle. Thus then the wheel to which the teeth are to belong is the base of the curve, and the wheel to be acted upon is the generating circle; but it must be understood that those wheels are not estimated in this description at their extreme diameters, but at a distance from their circumferences sufficient to admit of the necessary penetration of the teeth; or, as M. Camus terms it, where the primitive circles of the wheels touch each other, which is in what is called in this country the pitch line.

Now it has been long demonstrated by mathematicians, that teeth constructed as above would impart equable motion to wheels, supposing the pins, r, i, t, &c. indefinitely small. This point therefore need not be farther insisted upon.

So far the theoretic view is clear; but when we come to practice, the pins r, i, t, previously conceived to be indefinitely small, must have strength, and consequently a considerable diameter, as represented at 1, 2; hence we must take away from the area of the curve a breadth as at v and n = to the semidiameter of the pins, and then equable motion will continue to be produced as before. But it is known to mathematicians that the curve so modified will no longer be strictly an epicycloid; and it was on this account that I was careful above, to say that the teeth of wheels producing equable motion, depended upon that curve; for if the curve of the teeth be a true epicycloid in the case of thick pins, the motion of the wheels will not be equable.

I purposely omit other interesting circumstances in the application of this beautiful curve to rotatory motion; a curve by which I acknowledge that equable motions can be produced, when the teeth of the ordinary geering are made in this manner. But here is the misfortune:—besides the difficulty of executing teeth in the true theoretical form, (which indeed is seldom attempted), this form cannot continue to exist; and hence it is that the best, the most silent geering becomes at last imperfect, noisy and destructive of the machinery, and especially injurious to its more delicate operations.

The cause of this progressive deterioration may be thus explained: Referring again to fig. 1, we there see the base of the curve A B divided into the equal parts a b, b c, and c d; and observing the passage of the generating circle D, from the origin of the curve at d, to the first division c on the base, we shall find no more than the small portion d e, of the curve developed, whereas a second equal step of the generating circle c b, will extend the curve forward from e to f, a greater distance than the former; while a third equal step a b, will extend the curve from f to g, a distance greater than the last; and the successive increments of the curve will be still greater, as it approaches its summit; yet all these parts correspond to equal advances of the wheel, namely, to the equal parts a b, b c and c d of the base, and to equal ones of rotation of the generating circle. Surely then the parts s g, g f, of the epicycloidal tooth will be worn out sooner than those f e, e d, which are rubbed with so much less velocity than the other, even though the pressure were the same. But the pressure is not the same. For, the line a g is the direction in which the pressure of the curve acts at the point g, and the line p q, is the length of the lever-arm on which that pressure acts, to turn the generating circle on its axis (now supposed to be fixt;) but, as the turning force or rotatory effort of the wheels, is by hypothesis uniform, the pressure at g must be inversely as p q; that is, inversely as the cosine of half the angle of rotation of the generating circle; hence it would be infinite at s, the summit of the curve, when this circle has made a semi-revolution.

Thus it appears that independently of the effects of percussion, the end of an epicycloidal tooth must wear out sooner than any part nearer its base, (and if so, much more it may be supposed of a tooth of another form;) and that when its form is thus changed, the advantage it gave must cease, since nothing in the working of the wheel can afterwards restore the form, or remedy the growing evil.

Having now shewn one great defect in the common system of wheels, I shall proceed to develope the principles of the new system, which may be understood through the medium of the three following propositions.

1. The action of a wheel of the new kind on another with which it works or geers is the same at every moment of its revolution, so that the least possible motion of the circumference of one, generates an exactly equal and similar motion in that of the other.

2. There are but two points, one in each wheel, that necessarily touch each other at the same time, and their contact will always take place indefinitely near the plane that passes through the two axes of the wheels, if the diameters of the latter, at the useful or pressing points are in the exact ratio of their number of teeth respectively; in which case there will be no sensible friction between the points in contact.

3. In consequence of the properties above-mentioned, the epicycloidal or any other form of the teeth, is no longer indispensable; but many different forms may be used, without disturbing the principle of equable motion.

With regard to the demonstration of the first proposition, I must premise an observation of M. Camus on this subject, in his Mechanics, 3d. part, page 306, viz. “if all wheels could have teeth infinitely fine, their geering, which might then be considered as a simple contact, would have the property required, [that of acting uniformly] since we have seen that a wheel and a pinion have the same tangential force, when the motion of one is communicated to the other, by an infinitely small penetration of the particles of their respective circumferences.”

Now suppose that on the cylindrical surface of a spur-wheel B c, (fig. 3) we cut oblique or rather screw-formed teeth, of which two are shewn at a c, b d, so inclined to the plane of the wheel, as that the end c of the tooth a c may not pass the plane of the axes A B c, until the end b of the other tooth b d has arrived at it, this wheel will virtually be divided into an infinite number of teeth, or at least into a number greater than that of the particles of matter, contained in a circular line of the wheel’s circumference. For suppose the surface of a similar, but longer cylinder, stripped from it and stretched on the plane A B C E (fig. 4) where the former oblique line will become the hypothenuse B C, of the right angled triangle C A B, and will represent all the teeth of the given wheel, according to the sketch E G at the bottom of the diagram. Here the lines A B and C E, are equal to the circumference of the base of the cylinder, and A C and B E to its length; and if between A and B, there exist a number, m, of particles of matter, and between A and C a number, n, the whole surfaced A B C E will contain m n particles, or the product of m and n; and the line B C, will contain a number = √m² + n², from a well known theorem; whence it appears that the line B C is necessarily longer than A B, and hence contains more particles of matter.[2]

It is besides evident, that the difference between the lines B C and A B, depends on the angle A C B; in the choice of which, there is a considerable latitude. For general use however, I have chosen an angle of obliquity of 15°, which I shall now assume as the basis of the following calculations. The tangent of 15°, per tables, is in round numbers 268 to radius 1000; and the object now is to find the number of particles in the oblique line B C, when the line A B, contains any other number, t.

By geometry, B C(x) = √r² + t² = √1000² + 268² = 1035 nearly; and this last number is to 268, as the number of particles in the oblique line B C is to the number contained in the circumference A B, of the base of the cylinder. Hence it appears, that a wheel cut into teeth of this form, contains (virtually) about four times as many teeth, as a wheel of the same diameter, but indefinitely thin, would contain. And the disproportion might be increased, by adopting a smaller angle.

Thus I apprehend it is proved, that the action of a wheel of this kind, on another with which it geers, is perfectly uniform in respect of swiftness; and hence the proof that it is likewise so, as to the force communicated.

Before I proceed to the second proposition, I ought perhaps to anticipate some objections that have been made to this system of geering, and which may have already occurred to some gentlemen present. For example, it has been supposed that the friction of these teeth, is augmented by their inclination to the plane of the wheel; but I dare presume to have already proved, that it is this very obliquity, joined to the total absence of motion in direction of the axes, that destroys the friction, instead of creating it. I acknowledge however, that the pressure on the points of contact, is greater than it would be on teeth, parallel to the axes of the wheels, and I farther concede that this pressure tends to displace the wheels in the direction of the axes, (unless this tendency is destroyed by a tooth, with two opposite inclinations.) But supposing this counteraction neglected, let us ascertain the importance of these objections. First, with regard to the increase of pressure on the point D of the line B C, (representing the oblique tooth in question,) relative to that which would be on the line B E, (which represents a tooth of common geering:) let A D be drawn perpendicular to B C. If the point D can slide freely on the line B C, (and this is the most favourable supposition for the objection,) its pressure will be exerted perpendicularly to this line; and if the point A, moves from A to B, the point D, leaving at the same moment the point A, and moving in direction A D, will only arrive at D in the same time, its motion having been slower than that of A, in the proportion of A B to A D; whence by the principle of virtual velocities, its pressure on B C is to that on A C, as the said lines A B to D A.

To convert these pressures into numbers, according to the above data; we have A C = 1000, A B = 268, B C = 1035; then from the similar triangles B A C, B D A, it will be B C : A C ∷ A B : A D = ²⁶⁸⁰⁰⁰⁄₁₀₃₅ = 259 nearly. Therefore the pressure on B C, is to that on A C, as 268 to 259, or as 1035 : 1000.

To find what part of the force tends to drive the point B, in the direction B E, (for this is what impels the wheels, in the direction of their axes,) we may consider the triangle B A C as an inclined plane, of which B C is the length, and A B the height; and the total pressure on C B, which may be represented by C B, (1035) may be resolved into two others, namely, A B and A C, which will represent the pressures on those lines respectively, (268 and 1000.) Hence the pressure on B C, is augmented only in the ratio of 1035 to 1000, or about ¹⁄₂₉ part by the obliquity; and the tendency of the wheels to move in the direction of their axes, (when this angle is used,) is the ²⁶⁸⁄₁₀₀₀ of the original stress, that is, rather more than one quarter. But since the longitudinal motion of an axis can be prevented by a point almost invisible applied to its centre, it follows that the effect of this tendency can be annulled, without any sensible loss of the active power. It may be added, that in vertical axes, those circumstances lose all their importance, since whatever force tends to depress the one and increase its friction, tends equally to elevate the other, and relieve its step of its load; a case that would be made eminently useful, by throwing a larger portion of pressure on the slow-moving axes, and taking it off from the more rapid ones.

We now proceed to the second proposition. The truth of the assertions, contained in this proposition, must, I should suppose, be evident, from the consideration of two circles touching each other, and at the point of contact, coinciding with their common tangent at that point. Let A and B be two circles, tangent to each other, (fig. 3) in e. A C is the line joining the centres, and D F the common tangent of the circles at e; which is at right angles with A C; and so are the circumferences of the two circles at the point e. For the circles and tangent coincide for the moment. Hence then I conclude, 1st that a motion (evanescently small) of the point common to the three lines, can take place without quitting the tangent D F: and 2d. that if there is an infinite number of teeth in these circles, those which are found in the line of the centres, will geer together in preference to those which are out of it, since the latter have the common tangent, and an interval of space between them.

The truth of this proposition (or an indefinite approximation to truth,) may be deduced from the supposition that the two circles do actually penetrate each other. To this end let A B a b, in fig. 5, be two equal circles, placed parallel to each other in two contiguous planes, so as for one to hide the other, in the indefinitely small curvilinear space d f e g. I say that if the arc d g is indefinitely small, the rotation of the two circles will occasion no more friction between the touching surfaces, g e f and f d g, than there would be between the two circles placed in the same plane, and touching at the point n the same common tangent.

For draw the lines D E, f d, d g, g f, g e and g D; and adverting to the known equation of the circle, let d n = x, g n = y and D g = a, the absciss, ordinate and radius of the circle; we have 2 a x - x² = y². From this equation we obtain a = (y² + x²)/2x, the denominator of this fraction (2x) being the width, d e, of the touching surfaces f d g, and f e g of the two circles. But the numerator (y² + x²) is equal to the square of the chord g d of the angle E D g, which chord I shall call z; then we have a = x²/2x from which equation we derive this proportion, a : z ∷ z : 2x = z²/a. But in very small angles, the sines are taken for the arcs without sensible error; and with greater reason may the chords; if then we suppose the arc d g, or the chord z, indefinitely small, we shall find the line d e = 2x = z²/a, indefinitely smaller; that is, of an order of infinitessimals one degree lower; for it is well known that the square of evanescent quantities are indefinitely smaller than the quantities themselves. And to apply this, if the chord z represent the circular distance of two particles of matter found in the screw-formed tooth a c, of the wheel B c, fig. 3, (referred to the circle a b, fig. 5), that distance z will be a mean proportional between the radius D g of such wheel, and the double versed sine of this inconceivably small angle.[3]

I am aware that some mathematicians maintain, that the smallest portion of a curve cannot strictly coincide with a right line; a doctrine which I am not going to impugn. But however this may be, it appears certain that there is no such mathematical curve exhibited in the material world; but only polygons of a greater or less number of sides, according to the density of the various substances, that fall under our observation. I shall therefore proceed to apply the foregoing theory, not indeed to the ultimate particles of matter, (because I do not know their dimensions,) but to those real particles which have been actually measured. Thus, experimental philosophy shews, that a cube of gold of ¹⁄₂ inch side, may be drawn upon silver to a length of 1442623 feet, and afterwards flattened to a breadth of ¹⁄₁₀₀ of an inch, the two sides of which form a breadth of ¹⁄₅₀ of an inch: so that if we divide the above length by 25, we shall have the length of a similar ribbon of metal of ¹⁄₂ an inch in breadth, namely, 57704 feet; which cut into lengths of ¹⁄₂ an inch, (or multiplied by 24, the half inches in a foot) give 1384896 such squares, which must constitute the number of laminæ of a half inch cube of gold, or 2769792 for an inch thickness. Let us suppose then a wheel of gold, of two feet in diameter, the friction of whose teeth it is proposed to determine. We must first seek what number of particles are contained in that part of the tooth or teeth, that are found in one inch of the wheel’s circumference; this we have just seen to be 2769792 thicknesses of the leaves, or diameters of the particles, such as we are now contemplating.

We shall now have this proportion, (see fig. 4) 268 (A B) : 1035 (B C) ∷ 2769792 (no. of particles in one inch of circumference of base) : x = 10696771 particles in that part of the line B C, which corresponds with that inch of the circumference. Thus each of the latter particles measured in the direction A B, is equal to the fraction ¹⁄_10696771ths of an inch. And if that fraction be taken for the arc g d, (fig. 5) then to find the length of the line d e, (on which the friction of this and all other geering depends) we must use this analogy; 12 inch (rad. of wheel) : ¹⁄_10696771 of an inch (chord g d) ∷ ¹⁄_10696771 of an inch (g d) : d e, the line required = ¹⁄_{1273050917917292} of an inch. This result is still beyond the truth, as we do not know how much smaller the ultimate molecules of gold are.

To advert now to some of the practical effects of this system, I would beg leave to present a form of the teeth, the sole working of which would be a sufficient demonstration of the truth of the foregoing theory. A, B, (fig. 6) are two wheels of which the primitive circles or pitch-lines touch each other at o. As all the homologous points of any screw-formed tooth, are at the same distance from the centres of their wheels, I am at liberty to give the teeth a rhomboidal form, o t i; and if the angle o exists all round both wheels, (of which I have attempted graphically to give an idea at D G,) in this case, those particles only which exist in the plane of the tangents f h, &c. and infinitely near that plane passing at right angles to it through the centres A and B, will touch each other; and there, as we have already proved, no sensible motion of the kind producing friction, exists between the points in actual contact. I might add, as the figure evidently indicates, that if any such motion did exist, the angles o would quit each other, and the figure of such teeth become absurd in practice; but on the other hand, if such teeth can exist and work usefully (which I assert they can, nay that all teeth have in this system a tendency to assume that form at the working points;) this circumstance is of itself a practical evidence of the truth of the foregoing theory, and of what I have said concerning it.

It must have been perceived that I have in some degree anticipated the demonstration of my third proposition, namely, that the epicycloidal or any other given form of the teeth, is not essential to this geering. It appears that teeth formed as epicycloids, will become more convex by working; since the base of the curve is the only point where they suffer no diminution by friction; whilst those of every other form, that likewise penetrate beyond the primitive circles of the wheels, will also assume a figure of the same nature, by the rounding off of their points, and the hollowing of the corresponding parts of the teeth they impel; and that operation will continue till an angle similar to that at o, but generally more obtuse, prevails around both wheels; when all sensible change of figure or loss of matter will cease, as the wheels now before you will evince.

On the right of the drawing, (fig. 6) the teeth of the wheel B are angular, (suppose square) and those of the wheel C rounded off by any curve s, within an epicycloid. All that is necessary to remark in this case is, that the teeth of the wheel B must not extend beyond its primitive circle, whilst the round parts of those of the wheel C, do more or less extend beyond its primitive circle; whence it becomes evident, that the contact of such teeth, (if infinite in number) can only take place in the plane of the common tangent at right angles to A B; also that if these teeth are sufficiently hard to withstand ordinary pressure, without indentation in these circumstances, there is no perceptible reason for a sensible change of form; since this contact only takes place where the two motions are alike, both in swiftness and direction. A fact I am going to mention may outweigh this reasoning in the minds of some, but cannot invalidate it. I caused two of these wheels made of brass, to be turned with rapidity under a considerable resistance for several weeks together, keeping them always anointed with oil and emery, one of the most destructive mixtures known for rubbing metals; but after this severe trial, the teeth of the wheels, at their primitive circles were found as entire as before the experiment. And why? Certainly for no other reason than that they worked without sensible friction.

Hitherto nothing has been said of wheels in the conical form, usually denominated mitre and bevel geer. But my models will prove, that they are both comprehended in the system. The only condition of this unity of principle is, that the axes of two wheels, instead of being parallel to each other, be always found in the same plane. With this condition, every property above-mentioned, extends to this class of wheels, which my methods of executing also include, as indeed they do every possible case of geering.

Being afraid of trespassing on the time of the society, I have suppressed a part of this paper, perhaps already too long; but I hope I may be indulged with a few remarks on the application of those wheels to practical purposes. And first, as to what I have myself seen; these wheels have been used in several important machines to which they have given much swiftness, softness or precision of motion as the case required. They have done more; they have given birth to machines of no small importance, that could not have existed without them. In rapid motions they do all that band or cord can perform, with the addition of mathematical exactness, and an important saving of power. In spinning factories these properties must be peculiarly interesting; and in calico-printing, where the various delicate operations require great precision of motion. In clock-making also, this property is of great importance in regulating the action of the weight, and thus giving full scope to the equalizing principle whatever it be. I may add, it almost annuls the cause of anomaly in these machines, since a given clock will go with less than ¹⁄₄ of the weight usually employed to move it. Another useful application may be mentioned; in flatting mills, where one roller is driven by a pinion from the other, there is a constant combat between the effort of the plate to pass equally through the rollers, and the action of the common geering, which is more or less convulsive. Whence the plate is puckered, and the resistance much increased, both which circumstances these wheels completely obviate; and many similar cases might be adduced.

I shall only add, that my ambition will be highly gratified if, through the approbation of this learned society, I may hope to contribute to the improvement and perfection of the manufactures of this county; and if the invention be found of general utility to my much loved country.”

Subsequently to the reading of the above paper, I had occasion to execute many wheels on this principle; and their appearance, and use, excited on the one hand much interest, and on the other much opposition. I had even to complain of real injury in that contest: against which I defended myself with a warmth that I thought proportionate to the attack.—But all this was local and temporary: and writing now for a more enlarged sphere, and perhaps for a more extended period, I feel inclined to lay aside every consideration, but those immediately connected with the influence of this work on the public prosperity. I shall therefore avoid all reference to the names either of my friends or my opponents. My friends will live in a grateful heart, as long as memory itself shall last; my enemies, if I have any, will be forgiven—or, at worst, forgotten; and my System is henceforward left to wind its way into public notice and usefulness, by its own intrinsic merits.

Certain Observations which I was induced to make on occasion of a re-print of the above Memoir, may assist in introducing what remains to be said on the subject. They commence thus:

The foregoing little work, which first brought this subject into public notice in this town, was not the only method employed to develope its principles, and urge its adoption. A second paper was read, at the next meeting of the society, and some time after, a third, at the Exchange Dining Room; on both which occasions new modes of reasoning were pursued, and new kinds of proof adduced. On the first, a model was exhibited of two screw-formed teeth (connected with proper centres) exactly like those represented in fig. 6; by the action of which on each other, it became manifest that teeth of this angular shape do work together without inconvenience, and therefore, that all sensible friction is, in this case, done away.

On the latter occasion (the lecture at the Exchange) two other methods were brought forward, to corroborate the principles before stated: (see Plate 14, fig. 1.) The first was a kind of transparency, in which a line of light represented the place of contact of two wheels working together; by the partial and variable obscuration of which, the successive action of every portion of the teeth was clearly shewn. The second method consisted of two pair of wheels, made from loaf sugar, the teeth of which were cut one pair in the usual form, and the other on the new principle. Here, the difference in the effects of the two methods was so great, that the common teeth were almost immediately worn or broken down, by the very same kind of impulse that the new wheels sustained without injury: and with a loss of matter almost imperceptible, since many thousand revolutions of the wheels took place without detaching so many grains of sugar!

These Observations include likewise the following remarks:

In adverting to a few of the difficulties we have encountered, it will appear curious that one of them should spring from a most useful property of the system: but the paradox is thus explained. As there is no method more effectual for giving the teeth a perfect form, than working the wheels together, (covering them with an abrasive substance) we have most frequently chosen to depend on that important property; and have therefore set the wheels at work as they came from the foundry, instead of chipping the teeth, as is usual when common wheels are expected to act well in the first instance. But our wheels being then full of asperities, their action would be of course imperfect and noisy, till time had smoothed and equalized the touching surfaces: a state of things that might well stagger the opinion of a candid observer unacquainted with the system. Happily however we can now appeal to the fact of many wheels having become silent, that were once referred to with triumph, as proofs of a radical defect in the principle. It may not be improper to add here, that if highly finished wheels were particularly desired, we would engage to cut them in metal on this principle, with all the perfection of surface given to common wheels by the first masters.

In the use of bevel wheels of this description (with singly inclined teeth) there is doubtless a tendency to approach toward or recede from each other; the extent of which (for cylindrical wheels) has been already determined. This tendency goes, so far, to give a bend to the shaft; and, if this be very weak, to create a degree of friction on the teeth as the wheels revolve. It is therefore desirable that the shafts should be rather too strong than too weak; since the principle can only exist entire, when the wheels in working, are kept in the same planes which they occupy when at rest. This is too evident to be further insisted on.

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.)

4. From the foregoing it appears that the teeth of two wheels working together tend constantly to assume a form more and more perfect: as they abrade each other while imperfect, and cannot wear themselves beyond perfection.

5. For a similar reason the division of the teeth cannot remain unequal: for those that are too far distant from a given tooth will be attacked behind, and those that are too near before; so that the division also will finally become perfect.

But it must be remembered that these recoveries of form are in their nature very slow; since the nearer the teeth come to perfection the slower is their approach to it: so that in thus dwelling on these properties, we do not advise the making of bad wheels that they may become good; but only wish to destroy an honest prejudice that has already much impeded the progress of the System; namely, that it requires great nicety to adjust them so as to work together at all: which is—(to say the least) a very great error.

In Plate 14, fig. 1, I have shewn the apparatus presented at the Exchange, as mentioned in page 110 preceding. A B is the stand; C D is a disk turning on the centre E; b a is the transparent line cut through the stand, and representing the place of contact of two wheels geering together. It is there seen, (supposing the disk to turn in the direction of the arrow) that the action of the teeth, is always progressive along the transparent line a b; whether the single or double obliquity G or F be used. In reality, the lower end of any tooth c, does not uncover the line a b, till the upper point of the succeeding tooth d has begun to cover it; whereas, observing a few of the common teeth represented at H, as directed to the centre of the disk, they would be seen to pass the line a b all at once; and thus to represent, with a certain exaggeration, the transient manner of acting of the common geering.