Part 3

To this account of the result of these experiments, I beg leave to add what seems to be a great improvement of this System: namely, a method by which the diameters of the larger pulleys are considerably lessened; and thus the principal, if not the only objection, obviated. It has been before observed, that the larger pulleys, as Q R, are the ultimate terms of an arithmetical progression, beginning at unity; and that consequently they cannot be very small, even though the first terms should be so. If a first pulley were only one inch in diameter, the _twelfth_ pulley would be twelve inches,--where we see a large and inconvenient difference. But this evil I now obviate, by placing at the beginning of the series, one or more _loose pulleys_, over which to _reeve_ the cord, before the concentric or fixed grooves begin; thus lowering the _ratio_ of the progression, and keeping the larger pulleys within bounds. For example, the smallest fixed pulley (supposed as before, to be one inch in diameter) I now make the _second_ of the series instead of the first: and therefore, the second _fixed pulley_ is to the first as 3 to 2, instead of being as 2 to 1; for the same reason, the third fixed pulley is to the second as 4 to 3; and in a system of 12 pulleys, (with one loose one) the respective terms will be as follows:

Terms 1----2----3----4----5----6----7----8----9---10---11---12 loose;2/2; 3/2; 4/2; 5/2; 6/2; 7/2; 8/2; 9/2;10/2;11/2;12/2

or 6 inches for the largest pulley, instead of 12 inches given by the 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_.”