In this way only, you will be able to meet the requirement of a great educator, who said: "I do not care to be told what a young man knows, but what he can _do_." I wish here to add my grain to the weight of instruction which you receive, line upon line, precept on precept, on this subject. The reason for these laws of centrifugal force is an extremely simple one. The first law, that this force varies directly as the weight of the body, is of course obvious. We need not refer to this law any further. The second, third, and fourth laws merely express the relative rates at which a revolving body is deflected from the tangential direction of motion, in each of the three cases described, and which cases embrace all possible conditions. These three rates of deflection are exhibited in Fig. 2. An examination of this figure will give you a clear understanding of them. Let us first suppose a body to be revolving about the point, O, as a center, in a circle of which A B C is an arc, and with a velocity which will carry it from A to B in one second of time. Then in this time the body is deflected from the tangential direction a distance equal to A D, the versed sine of the angle A O B. Now let us suppose the velocity of this body to be doubled in the same circle. In one second of time it moves from A to C, and is deflected from the tangential direction of motion a distance equal to A E, the versed sine of the angle, A O C. But A E is four times A D. Here we see in a given circle of revolution the deflection varying as the square of the speed. The slight error already pointed out in these large angles is disregarded. The following table will show, by comparison of the versed sines of very small angles, the deflection in a given circle varying as the square of the speed, when we penetrate to them, so nearly that the error is not disclosed at the fifteenth place of decimals. The versed sine of 1" is 0.000,000,000,011,752 " " " " 2" is 0.000,000,000,047,008 " " " " 3" is 0.000,000,000,105,768 " " " " 4" is 0.000,000,000,188,032 " " " " 5" is 0.000,000,000,293,805 " " " " 6" is 0.000,000,000,423,072 " " " " 7" is 0.000,000,000,575,848 " " " " 8" is 0.000,000,000,752,128 " " " " 9" is 0.000,000,000,951,912 " " " " 10" is 0.000,000,001,175,222 " " " " 100" is 0.000,000,117,522,250 You observe the deflection for 10" of arc is 100 times as great, and for 100" of arc is 10,000 times as great as it is for 1" of arc. So far as is shown by the 15th place of decimals, the versed sine varies as the square of the angle; or, in a given circle, the deflection, and so the centrifugal force, of a revolving body varies as the square of the speed.