_Fifth_, that the making of apparatus to show scientific facts is more useful than making bootjacks for his father or workboxes for his mother. And, _lastly_, that a little money spent in this way will keep many a young rascal from worrying his sisters and stoning the cat; and when the inevitable time comes at which he must face the young man’s first trial, THE EXAMINER, he will often thank his stars that he learned in play the fundamental formula S = 1/2 g t², and that he knows the nature of “harmonic motion,” the two most important principles in the measurement of time. THE END. APPENDIX ON THE SHAPE OF THE TEETH OF WHEELS. [Illustration: FIG. 74.] The teeth of wheels for watches and clocks need particular care in shaping, and it may be of interest if I describe briefly the principles upon which these wheels are made. What is required is that the motion shall not be communicated by jerks as the teeth successively engage one another, but that the motion shall be perfectly smooth. The problem therefore becomes this: How are we to arrange the teeth of the wheels so that as one of them turns and drives the other round the leverage or turning power exercised by the driving wheel on the driven wheel shall always be uniform? Now if the teeth were simple spikes one can easily see that this would not be the case. For instance, as the arm _a c_ turned round, driving before it the arm _b d_, the point _c_ would scrape along, and the leverage between the two teeth would constantly alter. Evidently some other construction must be adopted. Before we can determine what it is to be, we must inquire what the leverage would be between two rods, _a c_ and _d b_, mounted on pivots at _a_ and _d_. The answer to this question is, that when a lever such as _a c_ presses with its end against another, _d b_, the power is exercised in a direction _c e_ at right angles to _d b_. Hence the leverage between the two arms is in the ratio of _a e_ to _d c_. The system is just as if we had a lever _a e_ united to a lever _d c_ by a rigid rod _e c_ at right angles to both of them. [Illustration: FIG. 75.] Whence then the ratio of the power is as _a e_ is to _d c_. [Illustration: FIG. 76.]