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.