When writers who understand the subject talk about the centripetal and centrifugal forces being different names for the same force, and about equal action and reaction, and employ other confusing expressions, just remember that all they really mean is to express the universal relation between force and resistance. The expression "centrifugal force" is itself so misleading, that it becomes especially important that the real nature of this so-called force, or the sense in which the term "force" is used in this expression, should be fully explained.[1] This force is now seen to be merely the tendency of a revolving body to move in a straight line, and the resistance which it opposes to being drawn aside from that line. Simple enough! But when we come to consider this action carefully, it is wonderful how much we find to be contained in what appears so simple. Let us see.

[Footnote 1: I was led to study this subject in looking to see what had become of my first permanent investment, a small venture, made about thirty-five years ago, in the "Sawyer and Gwynne static pressure engine." This was the high-sounding name of the Keely motor of that day, an imposition made possible by the confused ideas prevalent on this very subject of centrifugal force.]


I have called your attention to the fact that the direction in which the revolving body is deflected from the tangential line of motion is toward the center, on the radial line, which forms a right angle with the tangent on which the body is moving. The first question that presents itself is this: What is the measure or amount of this deflection? The answer is, this measure or amount is the versed sine of the angle through which the body moves.

Now, I suspect that some of you--some of those whom I am directly addressing--may not know what the versed sine of an angle is; so I must tell you. We will refer again to Fig. 1. In this figure, O A is one radius of the circle in which the body A is revolving. O C is another radius of this circle. These two radii include between them the angle A O C. This angle is subtended by the arc A C. If from the point O we let fall the line C E perpendicular to the radius O A, this line will divide the radius O A into two parts, O E and E A. Now we have the three interior lines, or the three lines within the circle, which are fundamental in trigonometry. C E is the sine, O E is the cosine, and E A is the versed sine of the angle A O C. Respecting these three lines there are many things to be observed. I will call your attention to the following only:


Their length is always less than the radius. The radius is expressed by 1, or unity. So, these lines being less than unity, their length is always expressed by decimals, which mean equal to such a proportion of the radius.


The cosine and the versed sine are together equal to the radius, so that the versed sine is always 1, less the cosine.


If I diminish the angle A O C, by moving the radius O C toward O A, the sine C E diminishes rapidly, and the versed sine E A also diminishes, but more slowly, while the cosine O E increases. This you will see represented in the smaller angles shown in Fig. 2. If, finally, I make O C to coincide with O A, the angle is obliterated, the sine and the versed sine have both disappeared, and the cosine has become the radius.


If, on the contrary, I enlarge the angle A O C by moving the radius O C toward O B, then the sine and the versed sine both increase, and the cosine diminishes; and if, finally, I make O C coincide with O B, then the cosine has disappeared, the sine has become the radius O B, and the versed sine has become the radius O A, thus forming the two sides inclosing the right angle A O B. The study of this explanation will make you familiar with these important lines. The sine and the cosine I shall have occasion to employ in the latter part of my lecture. Now you know what the versed sine of an angle is, and are able to observe in Fig. 1 that the versed sine A E, of the angle A O C, represents in a general way the distance that the body A will be deflected from the tangent A D toward the center O while describing the arc A C.

The same law of deflection is shown, in smaller angles, in Fig. 2. In this figure, also, you observe in each of the angles A O B and A O C that the deflection, from the tangential direction toward the center, of a body moving in the arc A C is represented by the versed sine of the angle. The tangent to the arc at A, from which this deflection is measured, is omitted in this figure to avoid confusion. It is shown sufficiently in Fig. 1. The angles in Fig. 2 are still pretty large angles, being 12° and 24° respectively. These large angles are used for convenience of illustration; but it should be explained that this law does not really hold in them, as is evident, because the arc is longer than the tangent to which it would be connected by a line parallel with the versed sine. The law is absolutely true only when the tangent and arc coincide, and approximately so for exceedingly small angles.

Fourth 531 4a

Fig. 2

In reality, however, we have only to do with the case in which the arc and the tangent do coincide, and in which the law that the deflection is equal to the versed sine of the angle is absolutely true. Here, in observing this most familiar thing, we are, at a single step, taken to that which is utterly beyond our comprehension. The angles we have to consider disappear, not only from our sight, but even from our conception. As in every other case when we push a physical investigation to its limit, so here also, we find our power of thought transcended, and ourselves in the presence of the infinite.

We can discuss very small angles. We talk familiarly about the angle which is subtended by 1" of arc. On Fig. 2, a short line is drawn near to the radius O A'. The distance between O A' and this short line is 1° of the arc A' B'. If we divide this distance by 3,600, we get 1" of arc. The upper line of the Table of versed sines given below is the versed sine of 1" of arc. It takes 1,296,000 of these angles to fill a circular space. These are a great many angles, but they do not make a circle. They make a polygon. If the radius of the circumscribed circle of this polygon is 1,296,000 feet, which is nearly 213 geographical miles, each one of its sides will be a straight line, 6.283 feet long. On the surface of the earth, at the equator, each side of this polygon would be one-sixtieth of a geographical mile, or 101.46 feet. On the orbit of the moon, at its mean distance from the earth, each of these straight sides would be about 6,000 feet long.