Part 6

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.