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.
Figure 2 of the present Plate, represents the plan of the Machine, but
turned upside down; so that the feet _G H_ screwed _under_ the lower
plate _E F_, are wholly visible. In this figure, also, is shewn at _c
d_, the edges (without the bottom) of the horizontal slide which carries
the _stand_ for the cutter frame represented in fig. 4. This stand is
indicated by the dotted lines of this figure 2, as situated _under_ the
arm _D_ of the bar _C D_; but it is better shewn in fig. 5, where _e f_
marks the slide in which the cutter frame (fig. 4) moves up and down, by
means of the screw and handle _e f_. In general I avoid dwelling much on
these smaller parts, because they exist, probably in a more perfect
state, in most other machines. In this fig. 5, _g h_ shews the screw
that moves this stand nearer to, or further from the axis _A B_ of the
Engine, according to the diameter of the wheels: which is also a common
process in Machines of this kind, on which therefore much need not be
said. But a somewhat greater importance attaches to the _cutter frame_
represented in the 4th. figure: which is a kind of small _lathe_ whose
_spindle_ _n o_, carries the cutter _n_, _outside the frame_, for the
purpose of changing the former without displacing the latter. The cutter
(of any proper section) is placed in or near that line which is a
continuation of the centre of the fixing screw _o p_. It is _in_ that
line for wheels whose teeth can be finished with once cutting: but
_near_ it for those whose teeth must be cut at twice. In this same
figure, _i k_ represent the _ends_ of the standards that form the
vertical slide _e f_ of fig. 5; and the separate figure _p q_, shews the
_back_ of the cutter frame _l m_, the flat part of which, _p_, presses
correctly on these uprights _i k_, and thus fixes this instrument at
_any desired height_, and to _any given angle_ with the perpendicular:
the _use_ of which arrangement we shall soon have occasion to exemplify.
Turning now to fig. 3 of this Plate, we there see the main shaft _A B_,
broken off at _B_: and the letters _a b_ again shew the dividing plate
of figs. 1 and 2: under this Plate is seen an _alidade_ or moveable
index, shewn by section only at _c_, and in elevation at _d e_; where it
clips the plate as far as _n_ and carries a boss _between n and e_, on
which the dividing index _e f_, turns; and to which it is strongly fixed
by a nut _o_, when the proper number to be cut is determined. Moreover,
this boss forms, itself, _the nut_ of a thumb-screw _s_, which, carrying
a circular plate at its lower end, clothed with leather or any soft
substance, connects strongly, without injuring the plate, the moveable
index with any point of it, as determined by the dividing index _e f_.
This brings us into the midst of things, as it respects the _use_ of
this Engine; for the former index _c d_, is furnished with a small
roller, _p_, the motion of which all the foregoing objects must obey,
when they have been fastened together by the thumb-screw _s_. We turn
then to the figures 1 and 2 of Plate 16, in order to shew those parts
in action: after remarking only that the form _p q r_ of this fig. 3, is
that of the moveable index shewn before at _c d_; requiring only, to
become complete, that the part _q_ should be sufficiently lengthened to
make the arc _r q_ a complete semi-circle--for purposes that will
shortly be explained.
In the two figures of Plate 16, the Machine is shewn as placed on its
bench or table, accompanied by the parts which give it a distinctive
character, and in fact embody the System. In addition to the parts
already described, we first remark the circular rim _c d_, fixed to the
ends of the bar _E F_; and made perfectly concentric with the main shaft
_A B_, and the dividing plate _a b_. This rim is shewn in section only,
at _v_ fig. 2. Its section resembles an L, and thus forms a basis for
certain _plates_ that will soon appear; and receives the screws by which
these plates are fastened to it. This being sufficiently clear, we now
proceed to describe the table and the connection of its mechanism with
the foregoing.
In Plate 16, _K L_ is the table: to which the Engine is screwed through
its feet _G H_. _I_, is a square bar of wood, sliding in a mortice
through the top of the table; and connected by a joint with the lever _M
N_--itself moving round a pin at _O_, and carrying a friction roller,
_P_, which pressed by the spiral _Q_, as turned by the handle _R_,
raises the bar _I_, and with it the main axis _A B_ of the plate, _and
of course the wheel to be cut, centered as usual on this axis above B_.
Finally, _p q r_, in both figures, is the moveable index first shewn in
fig. 3 of Plate 15; prepared to be _drawn round_ by a weight _W_,
hanging to the cord _x_, passing over the pulley _y_, and tied to the
_right_ end of the arc _q r_, when _this_ is to move to the left; or to
its _left_ end, when the motion is to be toward the right:--these
motions depending on the right or left-handed direction of the _teeth_
which it might be wished to cut on the Machine.
Between the two figures 1 and 2 of this Plate, there appears a diagram,
the base of which is nothing more than a part of the rim _c d_ supposed
straightened, and placed there that its use may be the easier
understood. On the rim is seen a right angled triangle _e g f_, against
which the roller _p_ will lean by the action of the weight _W_ on the
cord _x_, and the arc _q r_ of the moving index _p q r_. So THAT when,
by the handle _R_, the spiral _Q_ depresses the lever _M N_, by means of
its roller _P_, _then the bar I raises the axis A B of the Engine, and
the weight W turns it at the same time_, as much as the small roller _p_
permits by rolling up the side _e f_ of the plate _e g f_. And thus may
a _screw-formed tooth_ be cut in any wheel centered above _B_ in the
usual manner.
Thus then, in describing this Machine, the manner of using it has been
also shewn: for the _cutter_, in this Machine, (to cut spur wheels) is
always _fixed_; and _all_ the motion is composed of the _rotatory and
longitudinal movements of the principal axis, which carries the wheel
along with it_. The cutter I say is fixed, at a proper height just above
the wheel, and at an angle to the perpendicular, equal to that it is
wished the teeth should form at it’s pitch line. This inclination as
before observed is 15 degrees; and the tangent of 15° is in round
numbers 268, when the radius is 1000. That is, in our present figure,
the basis _e g_ of the plate _e g f_, occupies 268 divisions of a scale,
of which the height _g f_ contains 1000. It appears then, that to cut a
tooth with 15 degrees inclination, _by this Plate_, the wheel _receiving
that tooth_, must be _just as large as the rim itself_; for the surface
of the wheel would _turn_ more, with a given elevation, if it were
_larger_ than the rim; and would turn _less_, by the same elevation, if
it were smaller. In a word the whole theory of this operation, is now
clearly seen. The smaller the wheel to be cut, the longer, horizontally,
must be the Plate; or in other words, _as the diameter of the wheel is
to that of the rim, (c d) so is the length e g of the Plate to the
length required_. Now this height _f g_, is _always the same_; all
change therefore, in the plates, takes place on the horizontal length:
and this length is most easily found by the foregoing RULE OF THREE. If
then, instead of the triangle _e f g_, I had used the triangle _e′ f′
g′_ it would have followed at once, that to produce an inclination of 15
degrees, I must have taken a wheel of just _half_ the diameter of the
rim; for the plate _e′ f′ g′_ is just _twice_ as long as that _e f g_.
To prove this, let us _suppose_ the diameter of a wheel wanted, to equal
one half that of the rim _c d_: then the _rule_ will stand thus: