But then such an arrangement would go wrong nearly three-quarters of an hour every month, and in three years would indicate new moon a day too early. In order to obtain a better train, we may compute the number of days in 2, 3, 4, 5 lunations until we get nearly a number of half-days. Now, 16 lunations consist of 472 days, 11 hours, 45 min utes, or almost exactly 945 turns of the great-wheel. This proportion can be obtained by causing a pinion of 12 teeth to lead a wheel of 81 teeth, and another pinion also of 12 teeth to lead 1 wheel of 105 teeth. This arrangement gives an error of one quarter of an hour iu 16 months, or hardlran hour in 5 years. If still greater precision be required, we must carry the multiples further: 33 lunations make 974 days, 12 hours, 134. minutes, or 1949 turns of the great-wheel of the clock; but then this number 1949 has no divisor, and it is quite impracticable to make a wheel of 1949 teeth; so that we must continuo our multiples in search of a better train. In this way, when great exactitude is desired, we often encounter an unexpected amount of labor. Forreducing this labor, the method of continued fractions is employed, and the toil is further lessened by the use of tables of divisors.
Such calculations have to be made for the construction of orreries, by which the times of the revolutions of the planets are shown; and engineers have to make them, as when a screw of a particular pitch has to be cut. If, for instance, we have to cut a screw of 200 turns to the French meter on a lathe having a leading screw of 4 turns to the English inch, the axis of the lathe must make 50 turns while the screw makes 39 and a fraction, since the meter is 39.37079 inches. By applying the method of continued fractions, we discover that, for 2,225 turns of the lathe-spindle there must be 1752 turns of the screw; and as these numbers can be reduced into products—viz., 2,225 into 5x5x89, and 1752 into 2x2x2x3x73, we can easily get trains to produce the required effect. From these illustrations, it is apparent that the computation of the trains of wheel work is inti mately connected with the doctrine of prime and composite numbers.
The general sizes of the wheels and the number of the teeth having been fixed on, the next business is to consider the shape which those teeth ought to have. Now, for the smooth and proper action of machinery, it is essential that the uniform motion of one of the wheels be accompanied by a motion also equable of the other wheel. Two curves have been known to give this quality of equable motion—viz, the epicycloid, formed by rolling one circle upon another, and the involute of the circle traced by the end of a thread, which is being wound upon a cylinder, or unwound' from it. But the general character of all curves which possess this property has been only lately examined. If it were proposed to construct two wheels which shall have their centers at the points A and B (fig. 1), and the one of which may make 5 turns while
the other makes 3, we should divide the distance AB into 8 parts, and assign 5 of these for AC, the radius of the one wheel, the remaining 3 parts for the radius BC of the other wheel. Wheels made of these sizes, and rolling upon each other, would turn equably, and if the circumferences be divided into 5 and 3 parts respectively, the points of division would come opposite to each other as the wheels turned. The circumfer ences of these circles are called the pitch-lines, and the portions of them included between two teeth is called the distance of the teeth; the distance, or arc CD, on the one wheel must be equal to the distance CE on the other wheel, in order that the motion may bring the two points D and E together. For a reason that will appear in the sequel, we cannot use wheels with so few as 3 or 5 teeth, and therefore we subdivide the dis tances CD and CE into seine number of parts, say 4, and thus obtain wheels of 20 and 12 teeth instead. Since the tooth of the one wheel must necessarily come between 2 teeth on the other, the distance between the teeth must be halved, the one half being given for tooth, and the other half for space.
Having then divided off the pitch-line of the wheelB, as in fig. 1, CD being the dis tance of the teeth, CG the half distancd, let us sketch any contour, CFGHD, for the shape of a tooth, and let us exam ine what should be the characters of this outline. In the first place, the form of this outline must he repeated for each tooth ; and in the second place, the ITne should be symmetric from the top, F, of the one, to the 'top, I, of the next tooth, in order that the wheel may be reversible face for face. These obvious conditions having been attended to, let us cut, in thin sheet brass or other convenient material, a disk having this outline, and let us pin its center at the point B. Having pre pared a blank disk on which the outline of A is to be traced, let us slip it under the edge of the previous one, and pin its center at the point A. If, now, B and A being held fast, we trace the outline of B upon A, we move each of them slightly but in the proper proportion forward, and make a new trace upon A, and so-continue as far as needed, we shall, obtain a multitude of curve lines marked upon A. The line which envelops and touches all these curves is, obviously, the proper outline for the wheel A; and thus it appears, that whatever outline, within reasonable limits, may have been assumed for the teeth of B, it is always possible by a geometrical operation, to discover the proper corresponding form for the teeth of A. These forms may be called conjugate to each other, inasmuch as, that if the disk A were now cut out and used as B has been, the identic form of B would be reproduced.