Page:Newton's Principia (1846).djvu/319

. 1. Hence if the resistance be as the velocity, the difference of the arcs in the same medium will be as the whole arc described: and the contrary.

. 2. If the resistance be in the duplicate ratio of the velocity, that difference will be in the duplicate ratio of the whole arc: and the contrary.

. 3. And universally, if the resistance be in the triplicate or any other ratio of the velocity, the difference will be in the same ratio of the whole arc: and the contrary.

. 4. If the resistance be partly in the simple ratio of the velocity, and partly in the duplicate ratio of the same, the difference will be partly in the ratio of the whole arc, and partly in the duplicate ratio of it: and the contrary. So that the law and ratio of the resistance will be the same for the velocity as the law and ratio of that difference for the length of the arc.

. 5. And therefore if a pendulum describe successively unequal arcs, and we can find the ratio of the increment or decrement of this difference for the length of the arc described, there will be had also the ratio of the increment or decrement of the resistance for a greater or less velocity.

From these propositions we may find the resistance of mediums by pendulums oscillating therein. I found the resistance of the air by the following experiments. I suspended a wooden globe or ball weighing 57$$\scriptstyle \frac{7}{22}$$ ounces troy, its diameter 6$$\scriptstyle \frac{7}{8}$$ London inches, by a fine thread on a firm hook, so that the distance between the hook and the centre of oscillation of the globe was 10½ feet. I marked on the thread a point 10 feet and 1 inch distant from the centre of suspension; and even with that point I placed a ruler divided into inches, by the help whereof I observed the lengths of the arcs described by the pendulum. Then I numbered the oscillations in which the globe would lose $$\scriptstyle \frac{1}{8}$$ part of its motion. If the pendulum was drawn aside from the perpendicular to the distance of 2 inches, and thence let go, so that in its whole descent it described an arc of 2 inches, and in the first whole oscillation, compounded of the descent and subsequent ascent, an arc of almost 4 inches, the same in 164 oscillations lost $$\scriptstyle \frac{1}{8}$$ part of its motion, so as in its last ascent to describe an arc of 1¾ inches. If in the first descent it described an arc of 4 inches, it lost $$\scriptstyle \frac{1}{8}$$ part of its motion in 121 oscillations, so as in its last ascent to describe an arc of 3½ inches. If in the first descent it described an arc of 8, 16, 32, or 64 inches, it lost $$\scriptstyle \frac{1}{8}$$ part of its motion in 69, 35½, 18½, 9⅔ oscillations, respectively. Therefore the difference between the arcs described in the first descent and the last ascent was in the 1st, 2d, 3d, 4th, 5th, 6th cases, ¼, ½, 1, 2, 4, 8 inches respectively. Divide those differences by the number of oscillations in each case, and in one mean oscillation, wherein an arc of 3¾, 7½, 15, 30,