Page:Newton's Principia (1846).djvu/323

 greater than a third part of the whole resistance of the lesser pendulum; and thence I gathered that the resistances of the globes, when the resistance of the thread is subducted, are nearly in the duplicate ratio of their diameters. For the ratio of 7⅓ - ⅓ to 1 - ⅓, or 10½ to 1 is not very different from the duplicate ratio of the diameters 11$$\scriptstyle \frac{13}{16}$$ to 1.

Since the resistance of the thread is of less moment in greater globes, I tried the experiment also with a globe whose diameter was 18¾ inches. The length of the pendulum between the point of suspension and the centre of oscillation was 122½ inches, and between the point of suspension and the knot in the thread 109½ inches. The arc described by the knot at the first descent of the pendulum was 32 inches. The arc described by the same knot in the last ascent after five oscillations was 28 inches. The sum of the arcs, or the whole arc described in one mean oscillation, was 60 inches. The difference of the arcs 4 inches. The $$\scriptstyle \frac{1}{10}$$ part of this, or the difference between the descent and ascent in one mean oscillation, is $$\scriptstyle \frac{2}{5}$$ of an inch. Then as the radius 109½ to the radius 122½, so is the whole arc of 60 inches described by the knot in one mean oscillation to the whole arc of 67$$\scriptstyle \frac{1}{8}$$ inches described by the centre of the globe in one mean oscillation; and so is the difference $$\scriptstyle \frac{3}{5}$$ to a new difference 0,4475. If the length of the arc described were to remain, and the length of the pendulum should be augmented in the ratio of 126 to 122½, the time of the oscillation would be augmented, and the velocity of the pendulum would be diminished in the subduplicate of that ratio; so that the difference 0,4475 of the arcs described in the descent and subsequent ascent would remain. Then if the arc described be augmented in the ratio of 124$$\scriptstyle \frac{3}{31}$$ to 67$$\scriptstyle \frac{1}{8}$$, that difference 0.4475 would be augmented in the duplicate of that ratio, and so would become 1,5295. These things would be so upon the supposition that the resistance of the pendulum were in the duplicate ratio of the velocity. Therefore if the pendulum describe the whole arc of 124$$\scriptstyle \frac{3}{31}$$ inches, and its length between the point of suspension and the centre of oscillation be 126 inches, the difference of the arcs described in the descent and subsequent ascent would be 1,5295 inches. And this difference multiplied into the weight of the pendulous globe, which was 208 ounces, produces 318,136. Again; in the pendulum above-mentioned, made of a wooden globe, when its centre of oscillation, being 126 inches from the point of suspension, described the whole arc of 124$$\scriptstyle \frac{3}{31}$$inches, the difference of the arcs described in the descent and ascent was $$\scriptstyle \frac{126}{121}$$ into $$\scriptstyle \frac{8}{9\frac{2}{3}}$$. This multiplied into the weight of the globe, which was 57$$\scriptstyle \frac{7}{22}$$ ounces, produces 49,396. But I multiply these differences into the weights of the globes, in order to find their resistances. For the differences arise from the resistances, and are as the resistances directly and the weights inversely. Therefore the resistances are as the numbers 318,136 and 49,396. But that part of the resistance