Chapter VIII: Introduction (1)

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._)

[1] N. B. A Patent was taken out for the Invention some years ago.

“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 = √(_m_² + _n_²), 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]

[2] It need hardly be observed, that whatever is true of the whole triangle C A B, (fig. 4) is true of every similar part of it, be it ever so small: and in fact, when the hypothenuse B C, is folded again round the cylinder, from which we have supposed it stripped, the acting part will be very small indeed; but it will still act in the way here described, and give tendencies to the wheel it acts on, and to its axis, precisely proportionate to the quantities here mentioned.

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)_ = √(_r_² + _t_²) = √(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_ - _x_² = _y_². From this equation we obtain _a_ = (_y_² + _x_²)/2_x_, the denominator of this fraction (2_x_) being the width, _d e_, of the touching surfaces _f d g_, and _f e g_ of the two circles. But the numerator (_y_² + _x_²) 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_ = _x_²/2_x_ from which equation we derive this proportion, _a_ : _z_ ∷ _z_ : 2_x_ = _z_²/_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_ = 2_x_ = _z_²/_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]

[3] I ought perhaps to have introduced this reasoning on the 5th. figure by observing, that every projection of every part of a screw, on a plane at right angles with the axis of such screw, is a circle; and that therefore the chord _z_, or the line _g d_, is the true projection of a proportionate part of any line, _B C_, fig. 4, when wrapped round a cylinder of equal diameter with the circle _a b_, fig. 5.

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.

Some knowledge of the nature of this geering may be gathered from its very appearance: see fig. 5 Plate 14. To represent these teeth properly, no light must appear between them. The tops of the teeth offer a continued circular line, similar to what it would be if there were no teeth at all: and the latter are distinguished only by a different shading of their front and lateral surfaces. The reason (as has been already observed) is, that they are necessarily so placed, as that the _last_ end of any tooth shall not quit the plane of the centres, until the _first_ end of the succeeding tooth arrives at it; which principle precludes the possibility of any space remaining between the teeth, that an eye directed _parallelly_ to the axes could penetrate. Such a space indeed would introduce a portion of the properties of the old geering, which it is the object of this System to avoid. As this wheel then _appears_ in fig. 5, _so it acts_: that is equally and perpetually.

It were well also to observe the appearance of these wheels on their edges; or in the planes which, as wheels they occupy. The 4th. figure of this Plate is outlined with some care, in order to shew the varying, and seemingly anomalous form which the teeth assume as they approach the boundaries of the figure. Although cut as obliquely to the axis there, as any where else, the receding cylindrical surface, thus seen, appears to take this obliquity away; and the _very_ outward teeth seem nearly parallel to the axis of the wheel. But this is only appearance: and we give here _one_ example of it, that we may not be obliged to lose much time hereafter, in drawing correctly, wheels on this principle--a process indeed which in many cases, would be found very difficult, if not impossible.

We have already adverted to the oblique tendencies of these wheels, when used with a single inclination of the teeth; from which, among other things, it follows that, in the act of urging the shafts endwise, they tend also to bend these shafts: for which reason the shafts require to be stronger than those of common wheels--that is, when the effort bears any proportion to their stiffness--a circumstance which, in light rapid movements, is of small moment. And in heavier works, when it is desirable to get rid of these tendencies altogether, we have peremptory means of avoiding the very appearance of this evil.

Suppose then (fig. 2 and 3, plate 14) _a b_ to be a straight rack on this principle; driven by the wheel or pinion _c_. The motion, backward, of the pinion, tends, clearly, to urge the pinion endwise towards _d_, and the rack sideways towards _a b_. But either of these motions is prevented by fixing to the pinion, or the rack, a _cheek_ _e f_, to support them against this lateral pressure. But then, exclaims a doubting friend, you introduce _friction_: and it is true: there is now a real rubbing of the _ends_ of the teeth against this cheek; but the pressure there being only one quarter of what it would be on the front of straight teeth, we avoid (on a rough estimate) three quarters of the friction; while preserving _all_ the constancy and smoothness of motion which the system gives; and which after all, is the most important part of the business.

This idea then applies among other things to the racks of slide-lathes; giving a regular motion to the _rest and cutting tool_, thereby adding to the perfection of the turning process: and many other cases might be adduced.

But instead of using a rack and pinion, as thus described, _two wheels_, of any desired proportions might have been thus treated, and the result would have been the same. _They_ would have worked with perfect smoothness, under about one quarter of the friction attendant upon common wheels in similar circumstances. There are cases therefore, in which it would be expedient thus to employ the System. I cannot but observe likewise, that this method of using _cheeks_ to prevent any side motion in spur wheels, might also be applied to bevel-wheels, to prevent the _angular_ tendency which the obliquity of their teeth gives them: and that I prefer such a method of obviating this evil (where it is one) to any attempt at using teeth in the V form, on bevel wheels. Still however, as before observed, this counteraction of the oblique tendencies is not always necessary. It may be dispensed with in all light and rapid movements; especially in the use of perpendicular shafts; and where the _driven_ wheels are small and distributed round a central wheel in positions nearly opposite each other: of all which cases we shall see examples in the spinning machinery to be described hereafter.

OF
THE CUTTING ENGINE,
_To form Spur-wheels, on my late Patent principle_.

The figures of this Engine (see Plates 15 and 16,) are drawn to a scale, from the Machine itself, now before me. The scale of the objects on Plate 15, is one inch and three quarters to the foot; and that of the objects on Plate 16, one inch and one third. These were convenient proportions for introducing this object into the present work; but the size itself of the Machine is arbitrary. I did not make it according to my ideas of the _best_ dimensions: but bought it as a common cutting Engine, and gave it those other properties that my System required.

The first remarkable deviation from the usual form is in the shaft or axis of the dividing plate. See fig. 1 and 2 of the Plates 15 and 16. The dividing plate _a b_, is concentric with, and fixed to an axis _A B_ made as perfectly cylindrical as possible, so as _both to slide and turn_ in the bars _C D_, and _E F_ composing the frame. These bars are _bushed_, to fit the axis _A B_, either with a contracting ring of brass, as usual in some mathematical instruments; or with _type metal_, cast around the axis into _rough_ holes in those bars:--which metal, closing upon the axis makes a good centre; and will last a long time. My Engine is made in this manner; and has been renewed in this part only twice in several years. This frame _C D E F_ of the Engine, is strongly connected with the feet _G G H H_, by means of the nuts _E F_ in the plan: and by these feet it is fixed to its bench or table, as will be seen in Plate 16.