last progression.
So likewise, if we take _two_ loose pulleys, (which will not add much to
the complication of the Machine) and make the third term 1 inch, the
fourth will become 4/3, shewing the _ratio_ of the progression to be
1/3, so that the series of 12 terms will stand thus:
Terms, 1-----2----3---4----5----6----7----8----9---10---11---12
loose;loose; 1; 4/3; 5/3; 6/3; 7/3; 8/3; 9/3;10/3;11/3;12/3; or,
four inches for the largest groove in the concentric part of the System.
Now we saw before, that the first and last pulley were in diameter to
each other, as 1 to 12; whereas, here, with only two loose pulleys,
these extremes are but as 1 to 4: dimensions much more convenient and
manageable. The 5th. figure of the Plate 7, is intended to shew
graphically, the effect of this modification of the principle. In that
figure, if the line a, be the diameter of the _first_ pulley, that of
the sixth pulley will be shewn by the line b c; but if the same line a
be made the _second_ pulley, the diameter of the sixth will be shewn by
the line e d; only 2/3 of the former. And in fine, if the same a, be the
third pulley, the sixth will have it’s diameter reduced to the line f
g, only one half of what it was in the first case. In a word, the more
loose pulleys are put before the fixed ones begin, the nearer to
cylindrical will the general form become; and the more conveniently may
pulleys be used for general purposes. I might even assert, that if
_one_, or at most two loose pulleys had been used in the above-mentioned
experiments, the result would have been as favourable to the System,
with respect to the _weight of the tackle and stress on the ropes_, as
it was in respect of _power_; where it’s advantages were important and
undeniable.
OF
A POWER-WHEEL,
_Turned by heated Air, Gas, &c._
This Wheel (see Plate 8, fig. 1,) is technically called a Bucket-wheel.
It is plunged almost entirely in water, oil, mercury (or other heavy
fluid) contained in the vessel A B. It’s axis carries a _waved_ wheel a
b, on which rolls a friction-pulley p, running on a pin in the mortice
of the bar c d. This bar works the pump f; which by the descent of it’s
_loaded_ Piston, drives _cold_ air (or gas) into the tube g,
communicating with _several_ collateral ones placed _across_ the vessel,
so as to convey the air to h, below and beyond the centre of the wheel.
A fire being made at F under this vessel, the water (or other fluid) is
brought to a proper heat; and if then the pump f, be made to give a
stroke or two, air will be forced from the tubes at h, which having been
heated in the passage, will bubble up into the buckets h, i, k, &c. and
turn the wheel so as to perpetuate it’s own supplies from the Pump, and
furnish a surplus of _power_ for other purposes. This results from the
fact, that air (for example) in rising to the temperature of boiling
water, expands, under the pressure of the atmosphere, to about three
times the volume it occupied at the mean temperature: so that it resists
the entrance into the vessel as _unity_, and acts (when heated) as 3:
leaving a power of _two_, in the form of a rotatory motion.
It will occur to many readers, that azotic gas or nitrogen, might be
used with advantage to turn this wheel: only adding to the Machine a
_long_ returning tube, leading from the top of the vessel, through air
or water, to the _suction valve_ of the pump f; and _that_ in order to
bring down the temperature of the gas from the heat it had acquired in
the vessel, to the mean temperature; at which this gas is said to occupy
only 1/7 of the space it fills when at the heat of boiling water.
I have now to observe that this invention was _executed_ in 1794, of
which abundant proof remains. Since then, it has been proposed by other
persons, and is I think, patentized either in France or England: but a
different method is employed of introducing the cold _air_, namely an
inverted screw of Archimedes, whose manner of working I do not entirely
recollect. What I here wish to observe is, that this concurrence of idea
between others and myself, gives me no pain; since it would be more
strange if it did not happen, while so many active minds are ransacking
nature for the very purpose of unveiling her secrets. Only I think it
incumbent upon me to use every method, consistent with truth and honour,
to avoid being thought unjust enough to purloin other people’s ideas,
and call them my own.
OF
AN EQUABLE PUMP,
_Or Machine for raising Water without interruption or concussion_.
This Machine is represented in Plate 8, fig. 2 and 3. It is composed of
two barrels A B, both of them forming part of the column of water to be
raised; connected together by a crooked tube C, of equal diameter, out
of which the lower Piston-rod passes through a stuffing box into the
air: as does the upper Piston-rod at D, where the column leaves the Pump
to pass upward. The two Pistons fixed to the rods E and F, are of the
bucket kind; made as thin and light as possible; their valves opening
upwards and their motions being such, generally, that when one of them
is drawn up, the water rises through the other, _then descending_: But
here lies both the novelty and utility of this Machine; these upward and
downward motions are _not_ reciprocal: Both Pistons fall faster than
they rise, and thus leave an interval of time _when they both rise
together_; during which their valves, respectively, close by their own
weight _before_ the column of water falls upon them. In such manner,
indeed, that the column never _falls_ at all. By this important
arrangement, the work is constantly going on, and _no commotion_ occurs
to absorb _Power_ uselessly, or to destroy, prematurely, the Machine;
circumstances which _constantly_ attend every Pump Machine acting by
merely reciprocal motion.
This non-reciprocity then, I produce by several methods; one of which
(perhaps the most easily understood) is that shewn in fig. 2: There, A B
are two friction-rollers, made as large as possible, rolling on the
curves C X, the ascending and descending parts of which are essentially
_unequal_. For example, the rising part of the curve occupies 2/3 of the
whole circumference; and the falling part 1/3 only; so that both curves
recede from the centre at the same time, during 1/6 of a revolution, at
the two opposite positions, A C and X Y. Applying then, these curves and
levers to the Pump-barrels represented in fig. 3, we obtain that
_continuity of uniform motion_, which is necessary to doing the greatest
quantity of work with the least power; and to securing the greatest
durability of the Machine. Having hinted at a _minimum_ of power, I must
add here that this Machine appears to promise that result, much more
credibly than any reciprocating pump whatever; especially if to this
continuity of motion we add a certain _largeness_ of dimension that
shall produce the required quantity of water, with the slowest possible
motion of each particle; and even here this _continuative_ principle
helps us much; since pistons and valves of the largest dimensions may be
used without introducing any convulsive, or (what is synonymous) any
destructive effects.
One particular remains to be noticed in fig. 2. It relates to the means
by which the _perpendicularity_ of the motion in the Piston-rods is
secured. The arcs M are portions of cylinders having the bolts Z, for
their centres, and which, _rolling_ up and down against the
perpendicular plane O N, secure a similar motion to the bolts. The
_tenons_ P, are cycloidal, on their upper and lower surfaces; and work
in square or oblong holes in the plane N O, being kept _in_ their holes
by the action of the two springs on a pin let through these tenons: and
thus is the motion of the point Z of the levers M B, a perpendicular
one; and that of the friction rollers A B, very nearly so.
My object in this work, is to make known the principles, and _some_ of
the forms of these Inventions, but my limits will not permit their being
dilated on; else I could give several more useful forms of this Machine:
but, to make room for other subjects, I must hasten forward--reserving
to some future period, many hints respecting the adaptation of those
ideas to particular cases. Those of my readers who love to speculate on
the doctrine of _permutations_, will anticipate how much may be done by
the _combination of a hundred Machines_ with each other: and they will
give me credit for detached items of knowledge--useful in themselves,
though too minute to be severally brought forward. Should, however, the
degree of patronage I have already experienced, be proportionably
extended as the work advances, _I can and will_ follow it up with many
useful hints, tending to shew the extent of some of my present subjects,
and the amplitude of the sphere in which they roll.
It should be observed, in concluding this article, that the present
Machine was executed in France, in 1793, and also proposed to the
Government, as a substitute for the celebrated Machine of Marly. In the
report then published, it was preferred to the whole multitude of former
projects; but left _in equilibrio_ with _one_ modern Machine,--a
competition which prevented it’s adoption for the moment--and indeed
till I was _glad to escape the notice_, instead of courting the favour
of the then rapidly succeeding governments.
OF
A SIMPLE MACHINE,
_For Protracting the Motions of Weight-Machinery_.
Let A, Fig. 4 Plate 8, be the barrel-wheel of a Clock, or other Machine,
already in use, and driven by a weight; and let the _similar_ barrel B
be added to the former; the motion of both being connected by the
_unequal_ wheels C D. The rope or chain E F, is then led from the barrel
A under the pulley P to the barrel B: By which arrangement, when the
weight has occasioned _one_ revolution of the barrel and wheel A C,
_those_ B D, will have made a lesser portion of a revolution in the
ratio of the wheel C and D; (namely as 22 to 24,) and that motion will
have _taken up_ 11/12 of the line which the barrel A has _given off_. By
these means, the motion of the whole may be prolonged almost
indefinitely. This System may appear to some persons open to the
objection that the friction of the wheels C D, will absorb so much of
the power, as to leave the rotatory tendency too feeble for it’s
intended purpose. But I again take refuge in the well proved property of
my patent geering,--of not impeding (sensibly) the motion of any Machine
in which it is used.
Should it further be suggested, that this is only an awkward parody on
the _differential wheel and axle_, ascribed by Dr. Gregory (in the
introduction to his work, page 4,) to the celebrated George Eckhardt: I
would answer, that I made _that invention also_; though doubtless
_after_ Mr. Eckhardt; and especially after the date of the figure given
by the Doctor, as coming from China, “among some drawings of nearly a
century old;” Of course then, I do not pretend to priority of invention:
but _truth herself_ authorises me to say, that I did invent this Machine
also, _in the night between the 17th. and 18th. of January, 1788, and
drew it in bed by moonlight, that it might not escape me!_ It was the
result of a previous _fit_ of close thinking: and of the conclusion I
_then_ drew, that in whatever way, _slowness_ of motion is obtained by
the connection of two movements, _power_ is invariably gained for the
same reason, and in the same proportion. The fact is, that all my ideas
respecting differential motions, have flowed from this source; as will
be evident to the attentive reader of these pages.
OF
AN INSTRUMENT
_For drawing Portions of Circles, and finding their Centres by
inspection_.
It is a known property of _an angle_ such as g d f (plate 9 fig. 1) when
touching two fixed points g f, and gliding from one of these points to
the other, to describe a portion of a circle g d f. My object in this
instrument is to determine, by inspection, the radius of such circle in
all cases.
To do this, I connect with the jointed rule m d n, another rule like
itself but shorter g e f, so as that the figure g d e f shall be a
perfect parallelogram: and I then say that knowing the distance of the
points d and e, (the distance d f being given) I know the radius of the
circle of which g d f is a portion. To prove this, a little calculation
is necessary: In the circles A B and a b (fig. 6) draw the lines E D; _f
d_, _d g_, _g f_, _g e_, and _g D_; and bearing in mind the known
equation of the circle, let _d n_ = _x_, _g n_ = _y_; and g D = a, the
absciss, ordinate, and radius respectively. The equation is 2ax - x² =
y²: from which we get _a_ = (y² + x²)/(2x) the denominator of this
fraction being the line _d e_. But further its numerator (_y_² + _x_²)
is equal to the square of the chord g d of the angle E D g, which chord
I call _c_. This gives _a_ = _c_²/(line _d e_); from which equation we
derive this proportion _a_ : _c_ ∷ _c_ : line _d e_; Putting then the
chord _c_ = 1 (one foot for instance) this proportion becomes _a_ : 1 ∷
1 : 1/_a_; whence we draw this useful conclusion, that, whatever portion
of a foot is contained in the line _d e_, (expressed by a fraction
having _unity_ for its numerator) the radius of the circle will be
expressed _in feet_ by the denominator of that fraction. Thus if the
line _d e_, be 1 inch or 1/12 of a foot (and the line _g d_ or _d f_ be
1 foot) the radius of the circle will be 12 feet; and so for every other
fraction. Now in the instrument itself the two points _d_ and _e_, _are
connected by a micrometer-screw_ (not here drawn) of the kind described
in a subsequent article, and by which an inch is divided in 40,000
parts, each of which therefore is the 1/3333.33, &c. part of a foot: so
that if the distance _d e_, were only _one_ of these parts, we should
produce a portion g d f of a circle of 3333.33, &c. feet radius--being
more than half a mile.
I had omitted to observe, that the _points_ or studs, against which the
rulers m n slide, to trace the curve (_by a style in the joint d_,) that
these studs I say are fixed to a detached ruler o p, laid _under_ the
parallelogram on the paper, and having two _stump points_ to hold it
steady: _one_ of the studs being moveable in a slide, in order that it
may adapt the distance f g, to _any_ required distance of the points _d
e_: We note also that the dotted curve g d f is _not_ the very circle
drawn, but one parallel to it and distant one half the width of the
rulers. In fact the mortices of these rulers are properly the acting
lines, and _not their edges_. I expect, for several reasons, to resume
the subject of this instrument before the work closes.
OF
AN INCLINED HORSE WHEEL,
_Intended to save room and gain speed_.
My principal inducements for giving this Wheel the form represented, by
a section, in fig. 3, (see Plate 9) were to save _horizontal room_; and
to gain speed by _a Wheel_ smaller than a common horse-walk,--and _yet_
requiring less obliquity of effort on the part of the horse. With this
intention, the horse is placed _in a conical_ Wheel A B, more or less
inclined, and not much higher than himself: where, nevertheless, his
head is _seen_ to be at perfect liberty out of the cone as at C. The
horse then walks _in_ the cone, and is harnessed to a fixed bar
introduced from the open side where, by a proper adjustment of the
traces, he is made to act partly by his weight, so as to exert his
strength in a favourable manner. This Machine applies with advantage
where a horse’s power is wanted, _in a boat or other confined place_:
and it is evident, by the relative diameters of the wheel and pinion A B
and D, (as well as by the small diameter of the wheel) that a
considerable velocity will be obtained at the source of power,--whence,
of course, the subsequent _geering_ to obtain the swifter motions, will
be proportionately diminished.
OF
A DIFFERENTIAL COMBINATION OF WHEELS,
_To count very high numbers, or gain immense power_.
In fig. 2, of Plate 9, (which offers an horizontal section of the
Machine), A B is an axis, to the cylindrical part of which the wheels C
D are fitted, so as to turn with ease in either direction. Each of these
wheels, C and D, has two rims of teeth, _a b_, and _c d_; and between
those _b d_ are placed an intermediate pinion W, connected by it’s
centre with the arm _x_, which forms a part of the axis A B. There is
likewise a fourth wheel or pinion Z, working in the outer rims _a c_ of
the wheels C and D. It appears from the figure itself, that the action
of this Machine depends on the greater or lesser _difference_ between
the motion _forward_ of the wheel C, and the motion _backward_ of the
wheel D; for if these opposite motions were exactly alike, the wheels
would indeed all turn, but produce no effect on the arm _x_, or the axis
A B: whereas _this_ motion is the very thing required. Since then the
motion of the bar _x_, and finger _g_ depends on the difference of
action of the wheels C and D on the intermediate pinion W, we now
observe, that in the present state of things, the rims _a_, _b_, _c_,
_d_, have respectively 99, 100, 100, and 101 teeth: and that when _one
revolution_ has been given to the wheel C, the rim _b_ of this wheel
has acted, by 100 of its teeth, on those of the intermediate pinion W;
insomuch that if the opposite wheel D had been immoveable, the arm _x_
would have been carried round the common centre a portion equal to 50
teeth, or one half of it’s circumference (which effect takes place
because the pinion W _rolls_ against the wheels C and D, it’s centre
progressing only half as fast as it’s circumference.) But instead of the
wheel D standing still, it has moved in a direction opposite to the
former, a space equal to 99/100 of a revolution, and brought into the
teeth of the pinion W, 99/100 of 101 teeth; that is, 99 teeth, and 99
hundredths of one tooth: so that the _account_ between the two motions
stands thus:
The forward motion by the wheel C, is equal to 100,00 teeth.
And the backward motion by the wheel D, is 99,99 „
------
And the difference in favour of the forward motion is 00,01 of 1
tooth.
Or, dividing the whole circumference into 101 parts (each one equal to a
tooth of the rim _d_,) this difference becomes 1/100 part of 1/101 =
1/10100 of a revolution of the axis A B, for each revolution of the
wheel C. But we have observed, that the arm _x_ progresses only _half_
as much, on account of the _rolling_ motion: whence it appears that the
wheel C, must make 20200 turns to produce _one_ turn of this axis A B.
And if, with 20 teeth in the pinion Z, we suppose the movement to be
given by the handle _y_, this handle must make _more_ than 20200
revolutions, in the proportion of 99 (the teeth in the wheel) to 20, the
teeth in the pinion Z. Thus the said 20200 turns must be multiplied by
the fraction 99/20 which gives 99990 turns of the handle, for one of the
axis A B. And finally, if instead of turning this Machine by the handle
and pinion _y_ Z, we turned it by an endless screw, taking into the rim
_c_, of 100 teeth; the handle of such screw must revolve 2020000 times
to produce one single revolution of the axis A B; or to carry the finger
_g_, once round the common centre.
The above calculations are founded on the very numbers of a Machine of
this kind I made in Paris: and of which I handed a model to a public man
nearly thirty years ago. I need not add that this kind of movement
admits of an almost endless variety: since it depends both on the
numbers of the wheels and their differences; nay, on the differences of
their differences. I might have gone to some length in these
calculations had I not conceived it more important to bring other
objects into view, than to touch at present the extensive discussions
_this subject_ invites and will doubtless suggest to many. Suffice it
now to say, that here is a simple Machine which gains power (or
occasions slowness), in the ratio of two millions and twenty thousand to
one; giving, (if executed in proper dimensions) to a man of ordinary
strength, the power _of raising, singly, from three to four hundred
millions of pounds_. It may be useful to observe that using this Machine
for an opposite purpose, that of _gaining speed_, _extreme rapidity_ may
be caused by a power acting very slowly on the axis A B; only in that
case, the _difference_ must be enlarged, and the diameters and numbers
of the wheels be calculated _on the principles of perfect
geering_--which is as easy in this Machine as in any other.
OF
A CRANE,
_Which combines_ VARIABLE POWERS _with speed and safety_.
Doctor Gregory (in his Mechanics 2d. volume page 157,) thus introduces
the description of this Crane, and the observations with which he tags
that description.
“The several Cranes described in this article, as preferable to the
common walking Crane, while they are free from the dangers attending
that Machine, lose at the same time one of it’s advantages, that is,
they do not avail themselves of that addition to the moving power which
the weight of the men employed may furnish: yet this advantage has been
long since insured by the mechanists on the continent: who cause the
labourers to walk upon an inclined plane, turning upon an axis, after
the manner shewn in the figure referred to under the article
_foot-mill_,--where we have described a contrivance of that kind, well
known in Germany nearly 150 years ago. The same principle has been
lately brought into notice (probably without knowing it had been adopted
before) by Mr. Whyte, (White) of Chevening in Kent: His Crane is
exhibited,--fig. 2 and 4, Plate 10, _as it was described in the
Transactions of the Society for the Encouragement of Arts_.”
“A, Plate 9, fig. 4, (of this Work) is a circular inclined plane, moving
on a pivot under it, and carrying round with it the axis E. A person
walking on this plane at A, and pressing against a lever, throws off a
gripe or brake, and thus permits the plane to move freely, and raise the
weight G by the coiling of the rope F, round the axis E. To shew more
clearly the construction and action of the lever and gripe, _a plan_ of
the plane connected with them, is added in fig. 5, where B represents
the lever, and D the gripe: where it is seen that when the lever B is in
the situation in which it now appears, the brake or gripe D, _presses
against the periphery of the plane_; but when the lever B is driven out
to the dotted line H, the gripe D is detached, and the whole Machine
left at liberty to move: a rope or cord of a proper length, being
fastened to B, and to one of the uprights in the frame, to prevent this
lever from being pushed too far towards H, by the man working at the
Crane.”
“The _supposed properties_ of this Crane, (says Dr. Gregory) for which
the premium of forty guineas was adjudged by the society to the
Inventor, are as follows:”
“‘1. It is simple, consisting merely of a wheel and axle:
“‘2. It has comparatively little friction, as is obvious from the bare
inspection of the figure:
“‘3. It is durable from the two properties above mentioned:
“‘4. It is safe: for it cannot move but during the pleasure of the man,
and while he is actually pressing on the gripe lever:
“‘5. This Crane admits of an almost infinite variety of different
powers; and this variation is obtained without the least alteration of
any part of the Machine. If in unloading a vessel, there should be found
goods of every weight, from a few hundreds to a ton and upwards, the
workman will be able so to adapt his strength to each, as to raise it in
a space of time, (inversely) proportionate to it’s weight, he walking
always with the same velocity as nature and his greatest ease may teach
him.’”