To find what part of the force tends to drive the point B, in the direction B E, (for this is what impels the wheels, in the direction of their axes,) we may consider the triangle B A C as an inclined plane, of which B C is the length, and A B the height; and the total pressure on C B, which may be represented by C B, (1035) may be resolved into two others, namely, A B and A C, which will represent the pressures on those lines respectively, (268 and 1000.) Hence the pressure on B C, is augmented only in the ratio of 1035 to 1000, or about 1⁄29 part by the obliquity; and the tendency of the wheels to move in the direction of their axes, (when this angle is used,) is the 268⁄1000 of the original stress, that is, rather more than one quarter. But since the longitudinal motion of an axis can be prevented by a point almost invisible applied to its centre, it follows that the effect of this tendency can be annulled, without any sensible loss of the active power. It may be added, that in vertical axes, those circumstances lose all their importance, since whatever force tends to depress the one and increase its friction, tends equally to elevate the other, and relieve its step of its load; a case that would be made eminently useful, by throwing a larger portion of pressure on the slow-moving axes, and taking it off from the more rapid ones.
We now proceed to the second proposition. The truth of the assertions, contained in this proposition, must, I should suppose, be evident, from the consideration of two circles touching each other, and at the point of contact, coinciding with their common tangent at that point. Let A and B be two circles, tangent to each other, ([fig. 3]) in e. A C is the line joining the centres, and D F the common tangent of the circles at e; which is at right angles with A C; and so are the circumferences of the two circles at the point e. For the circles and tangent coincide for the moment. Hence then I conclude, 1st that a motion (evanescently small) of the point common to the three lines, can take place without quitting the tangent D F: and 2d. that if there is an infinite number of teeth in these circles, those which are found in the line of the centres, will geer together in preference to those which are out of it, since the latter have the common tangent, and an interval of space between them.
The truth of this proposition (or an indefinite approximation to truth,) may be deduced from the supposition that the two circles do actually penetrate each other. To this end let A B a b, in [fig. 5], be two equal circles, placed parallel to each other in two contiguous planes, so as for one to hide the other, in the indefinitely small curvilinear space d f e g. I say that if the arc d g is indefinitely small, the rotation of the two circles will occasion no more friction between the touching surfaces, g e f and f d g, than there would be between the two circles placed in the same plane, and touching at the point n the same common tangent.
For draw the lines D E, f d, d g, g f, g e and g D; and adverting to the known equation of the circle, let d n = x, g n = y and D g = a, the absciss, ordinate and radius of the circle; we have 2 a x - x² = y². From this equation we obtain a = (y² + x²)/2x, the denominator of this fraction (2x) being the width, d e, of the touching surfaces f d g, and f e g of the two circles. But the numerator (y² + x²) is equal to the square of the chord g d of the angle E D g, which chord I shall call z; then we have a = x²/2x from which equation we derive this proportion, a : z ∷ z : 2x = z²/a. But in very small angles, the sines are taken for the arcs without sensible error; and with greater reason may the chords; if then we suppose the arc d g, or the chord z, indefinitely small, we shall find the line d e = 2x = z²/a, indefinitely smaller; that is, of an order of infinitessimals one degree lower; for it is well known that the square of evanescent quantities are indefinitely smaller than the quantities themselves. And to apply this, if the chord z represent the circular distance of two particles of matter found in the screw-formed tooth a c, of the wheel B c, [fig. 3], (referred to the circle a b, [fig. 5]), that distance z will be a mean proportional between the radius D g of such wheel, and the double versed sine of this inconceivably small angle.[3]
[3] I ought perhaps to have introduced this reasoning on the [5th. figure] by observing, that every projection of every part of a screw, on a plane at right angles with the axis of such screw, is a circle; and that therefore the chord z, or the line g d, is the true projection of a proportionate part of any line, B C, [fig. 4], when wrapped round a cylinder of equal diameter with the circle a b, [fig. 5].
I am aware that some mathematicians maintain, that the smallest portion of a curve cannot strictly coincide with a right line; a doctrine which I am not going to impugn. But however this may be, it appears certain that there is no such mathematical curve exhibited in the material world; but only polygons of a greater or less number of sides, according to the density of the various substances, that fall under our observation. I shall therefore proceed to apply the foregoing theory, not indeed to the ultimate particles of matter, (because I do not know their dimensions,) but to those real particles which have been actually measured. Thus, experimental philosophy shews, that a cube of gold of 1⁄2 inch side, may be drawn upon silver to a length of 1442623 feet, and afterwards flattened to a breadth of 1⁄100 of an inch, the two sides of which form a breadth of 1⁄50 of an inch: so that if we divide the above length by 25, we shall have the length of a similar ribbon of metal of 1⁄2 an inch in breadth, namely, 57704 feet; which cut into lengths of 1⁄2 an inch, (or multiplied by 24, the half inches in a foot) give 1384896 such squares, which must constitute the number of laminæ of a half inch cube of gold, or 2769792 for an inch thickness. Let us suppose then a wheel of gold, of two feet in diameter, the friction of whose teeth it is proposed to determine. We must first seek what number of particles are contained in that part of the tooth or teeth, that are found in one inch of the wheel’s circumference; this we have just seen to be 2769792 thicknesses of the leaves, or diameters of the particles, such as we are now contemplating.