A B, Plate 7, fig. 2 and 3, is a ring or wheel fixed to the frame C D; and having all round it’s inside, teeth directed to the centre. F is a wheel of half the diameter, and exactly half the number of teeth of the wheel A B. It turns on a Crank-arm, E F, whose radius is equal to one quarter of the diameter of the fixed wheel A B—in the centre of which the axis of this Crank finds it’s due position. The latter, therefore, so conveys the wheel F round the inside of the fixed wheel A B, that the teeth of both are constantly geering to a proper depth: and a stud being fixed on the face of the wheel F, opposite the middle of any tooth, a, directly over the centre of the Crank E, this stud describes the perpendicular diameter of the large wheel: and will either receive motion from the rod R of a Steam Engine Piston, so as to give the fly I K, a rotatory motion; or communicate to a Pump-piston a reciprocating motion, drawn from the rotatory one of the fly, when that is the effect desired to be produced.

This Invention will be remembered, as having procured me a remunerating Medal from the late Napoleon Bonaparte, then first Consul of the French Republic. That period, however, (1801) was not the real date of this production, although then first made public. I have proof, on the contrary, of its existence with me several years before; and it is generally ascribed to me by the publicists. I might quote in particular Doctor Gregory: who likewise mentions its having been executed by Messrs. Murray and Wood, of Leeds, subsequently to it’s exhibition at Paris. The Doctor commits, however, a small error in calling me an Anglo-American; but this is accounted for by my then living in a country where to be an Englishman was itself a crime! and where some kind friends, wishing to hide me from the relentless decrees of the day, felt justified in using this sort of pious fraud in my favour: a resource from which, though I did not authorize it, I reaped no small advantage; and still think of with gratitude, though not with unmixed approbation.

I think it a duty more imperious than agreeable, to expostulate a little with Messrs. Lanz & Betancourt, on their apparent partiality in giving an account of this Machine. In their work on the construction of machines, art. 97, page 37, they make M. de la Hire the inventor of it, by the terms in which they introduce his treatise on Epicycloids: and they leave me the thread-bare merit of having “presented a model of this movement at the last exposition but one,” &c. Now, although I do not attach great importance to this kind of misrepresentation, I cannot but observe, that neither my Machine or their description of it can be called a Theorem! nor especially a theorem relating solely to the Epicycloid, as M. de la Hire’s was. These Gentlemen knew that he insisted principally on the application of this curve to the teeth of wheels, with which my Invention has nothing to do. On the contrary, my Machine is a combination of two curves at least, on which de la Hire says absolutely nothing. Is this then inadvertency? or is it uncandid nationality? I hope, the former.

A further remark on the utility of this System as a first motion, may be of use in this place. It respects the geering of the fixed and moveable wheels A B, and F, on the perfection of which depends the truth of the statement, that the stud, a, describes a diameter of the large wheel. Now, perfection is too much to be expected from common teeth when of the necessary strength; so that my Patent Geering is an indispensable complement to this Invention: as by its use, the principle is made practically true; this line becoming really straight, and this motion, under proper circumstances, being unattended with noise or commotion. In a word, I cannot move a step in this mechanical field, without meeting with instances where the new System shews its superiority to the old: whence it becomes a duty for me to commence the consideration of this subject in the very next part of this publication.

OF
A SYSTEM OF CONCENTRIC PULLEYS,
Already known as White’s Patent Pulleys.

A very considerable improvement in the construction of pulleys has been made by Mr. James White, who obtained a Patent for his Invention, of which he gives the following description: “Fig. 4, Plate 7, of this work, shews the Machine, consisting of two pullies, Q and R; the former fixed, the other moveable. Each of these has six concentric grooves, capable of having a line put round them, and thus of acting like as many different pulleys having diameters equal to those of the grooves. Supposing then, each groove to be a distinct pulley, and that all these diameters were equal, it is evident, that if the weight 144 were to be raised by pulling at S, till the pulleys touched each other, the first pulley must receive the length of line as many times as there are parts of the line hanging between it and the lower pulley. In the present case there are 12 lines, b, d, f, &c. hanging between the two pulleys, formed by its revolution about the six upper and six lower grooves. Hence as much line must pass over the uppermost pulley as is equal to 12 times the distance of the two. But, from an inspection of the figure, it is plain that the second pulley R S, cannot receive the full quantity of line by as much as is equal to the distance betwixt it and the first. In like manner, the third pulley receives less than the first, by as much as is equal to the distance between the first and the third; and so on to the last which receives only ¹⁄₁₂ of the whole: for this receives it’s share of line n, from a fixed point in the upper frame which gives it nothing: while all the others in the same frame receive the line partly by moving to meet it, and partly by the line coming to meet them.”

“Supposing now these pulleys to be equal in size, and to move freely as the line determines them, it appears from the nature of the system, that the number of their revolutions, and consequently their velocities, must be in proportion to the number of suspending parts, that are between the fixed point above-mentioned, (n) and each pulley respectively. Thus the outermost pulley would go twelve times round in the time that the pulley under which the part n of the line passes, (if equal to it) would revolve only once; and the intermediate times and velocities would be a series of arithmetical proportionals of which, if the first term were l, the last would always be equal to the whole number of terms. Since then, the revolutions of equal and distinct pulleys are measured by their velocities, and that it is possible to find any proportion of velocity on a single body running on a centre, viz. by finding proportional distances from that centre; it follows, that if the diameters of certain grooves in the same body be exactly adapted to the above series, (the line itself being supposed inelastic and of no magnitude) the necessity of using several pulleys in each frame will be obviated, and with that some of the inconveniences to which the use of the common pulley is liable.”

“In the figure referred to the coils of rope, by which the weight is supported, are represented by the lines a, b, c, &c. a is the line of traction commonly called the fall, which passes over and under the proper grooves, until it is fastened to the upper frame just above n. In practice, however, the grooves are not arithmetical proportionals; nor can they be so, for the diameter of the rope employed must be deducted from each term, without which, the small grooves to which the said diameter bears a greater proportion than to the larger ones, will tend to rise and fall faster than the latter, and thus introduce worse defects than those which they were intended to obviate.”

“The principal advantage of this kind of pulley is, that it destroys lateral friction, and that kind of shaking motion which are so inconvenient in the common pulley; and lest, says Mr. White, (I quote Dr. Gregory) this circumstance (of a long pin) should give the idea of weakness, I would observe, that to have pins for pulleys to run upon, is not the only, nor perhaps the best method: but that I sometimes use centres fixed in the pulleys, and revolving on a short bearing in the side of the frame, by which strength is increased, and friction much diminished: for to the last moment of duration, the motion of the pulley is circular, and this very circumstance is the cause of it’s not wearing out in the centre as soon as it would, assisted by the ever increasing irregularities of a gullied bearing.—These pullies when well executed, apply to Jacks and other Machines of that nature with great advantage: both as to the time of their going and their own durability: and it is possible to produce a System of pulleys of this kind, composed of six or eight parts only, and adapted to the pocket, which by means of a skain of sewing silk, would raise more than a hundred weight.”

There are several real and solid advantages attending the use of this pulley; some of which are only hinted at in this description. I have thought, therefore, it might be useful to introduce here an account of some trials which the System underwent a few years ago at Portsmouth,—at the request of an Officer of the Navy, who had re-invented it with some ingenious additions to my ideas. Not being at present in correspondence with that Gentleman, I hardly think myself at liberty to mention his name; but fully so to give an extract from the report which followed these experiments—in which the superiority of the System in respect of power, is made evident, although some less favourable circumstances prevented its adoption on that occasion.

“With a view to comparison, it was settled with Lieutenant S. that his blocks should be made to correspond with the treble and double 16 inch blocks of a 24 gun ship, which carry a 4¹⁄₂ inch rope. The sheeves in the new blocks are fixed upon the pin, revolving therewith, and are of different diameters proportioned to the velocity of the parts of the rope that pass over them; they are also reeved with a double rope so that there are two grooves of each size, the diameter of the smallest groove in this tackle being 2⁸⁄₁₂, and of the largest 15 inches. The diameter of the sheeves of the common blocks would have been (as usually made) 9¹⁄₈ to the bottom of the grooves, but were reduced at the request of Lieutenant S. in the treble block to 8¹⁄₈, and in the double block to 8⁷⁄₈, in order that the sum of the diameters of the sheeves in each tackle should be the same. The Lieutenant intending in the first instance, to have used a roller under the pin, for the purpose of diminishing friction, but afterwards laying aside this idea on account of it’s complication, was the reason that he had not made his sheeves in the same proportion with the common blocks: the weight and length of the respective blocks are as follows:

“Lieutenant S.’s blocks were reeved with a 2¹⁄₂ inch double rope, and the common block with a 4¹⁄₂ inch single rope, and both tackles suspended from a beam, and their respective falls let over the single blocks, so as to keep the weight applied as a power, just clear of the weight to be lifted, thus forming a power of six to one; the following experiments were made:

“After reeving the common blocks with a 3¹⁄₂ inch rope in lieu of a 4¹⁄₂ inch rope, it was as follows: 5376 1101 1232.

“It must be observed, that the double 2¹⁄₂ inch rope in Lieutenant S.’s blocks, is not of equal strength with the single 4¹⁄₂ inch rope first used in the common blocks; and that his blocks had an undue advantage in the first experiment over the common blocks, in respect to the pliability of the rope. The rope should therefore, be taken larger in the one or smaller in the other case, on this account: The common blocks were reeved in the last experiment, with a 3¹⁄₂ inch rope, which is as near as may be of the same strength as the double 2¹⁄₂ inch rope.

“In these experiments it was observable, that the tar was much more squeezed out of the parts of the rope that passed over the smallest sheeves in Lieutenant S.’s blocks, than out of those passing over the larger sheeves, or out of those passing over the sheeves of the common blocks; by which, as well as by the nature of the thing, we judge that with blocks requiring such small sheeves, the ropes would be more crippled and broken than by the common blocks, especially if any constant strain or weight in motion, as on ship board, should be held by them. In regard to our opinion of the merits of the blocks proposed by Lieutenant S. compared with common blocks, we beg leave to submit, that the mechanical principle of them is very inviting, and it is not to be wondered that an ingenious person should pursue the idea; yet allowing there would be a saving of power, which is attained in so great a degree with the common blocks, but considering the greater complication, weight, and expence of these blocks, and their greater disposition to cripple the ropes, we do not perceive any application of them on ship board, for which we could recommend them in preference to common blocks; neither do we perceive any purposes on shore, for the services of the dock yards in which to recommend their application in preference to the other powers in use.”

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:

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:

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 (y² + x²) is equal to the square of the chord g d of the angle E D g, which chord I call c. This gives a = c²/(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:

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.