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.