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:
| 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