Fourth.—­With a given actual velocity, or speed in feet per minute, inversely as the radius of the circle.

[Footnote 1:  A revolving body is said to have the same angular velocity, when it sweeps through equal angles in equal times.  Its actual velocity varies directly as the radius of the circle in which it is revolving.]

Of course there is a reason for these laws.  You are not to learn them by rote, or to accept them on any authority.  You are taught not to accept any rule or formula on authority, but to demand the reason for it—­to give yourselves no rest until you know the why and wherefore, and comprehend these fully.  This is education, not cramming the mind with mere facts and rules to be memorized, but drawing out the mental powers into activity, strengthening them by use and exercise, and forming the habit, and at the same time developing the power, of penetrating to the reason of things.

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.