We have provided the definition of central forces under the word Central [1] and we would send those people to that word as well as to the division of central forces according to whether they tend to approach or to distance the body from a fixed or mobile point when one applies action to the central force . A similar word to centrifugal force means more than its ordinary use as a force by which a body moves in a circular motion and tends to continuously distance itself from the center of the circle which it describes. This force is easily sensed when a sling is in motion and we feel that the sling is stretched out as much by the stone as by the stone which is rotated with more and more speed. This tension supposes that within the stone there is an effort to distance itself from one’s hand as the center of the circle which the stone describes. In effect the stone moving circularly and continuously tends to escape tangentially by virtue of the inertial force as has been proved under the word Centrifuge. However the effort to escape tangentially tends to distance the body from the center. Therefore the stone’s effort to escape tangentially must tend towards the sling. If one wishes to understand this in a more accurate way, the body arrives at point A ( fig. 24. Mechanics ) and tends to move itself along the tangent or an infinitely small portion of the tangent A D . Due to the principle of the decomposition of forces ( See Decomposition and Composition), we can see this motion following A D as composed of two movements; one following the arc A E from the circle and the other following line E D and as we might suppose directed to the center. Of these two movements, the body preserves only the following motion A E , thus the ensuing movement E D is eliminated, and as this motion is directed from the center of the circumference, it is due to this tendency of motion that the sling is stretched.

A body which moves along on any other curve than on the circle attempts to escape tangentially at any instant; therefore we have named it centrifugal force irrespective of the curve that the body describes. In order to calculate the centrifugal force of a body located on any curve, it is necessary to know how to calculate it on a circle, since any curve can be seen as composed of an infinite number of arcs of the circle of which their centers are within the involute ( See Involutes and Points of Contact). Therefore by knowing the law of centrifugal forces in the circle, one will then understand centrifugal forces on any curve. Furthermore it is easy to calculate the centrifugal force within a circle, since by following that which we said here above, if we say that φ is the centrifugal force and dt the time taken to cover the distance AE or DE (fig. 24 Mechanics) , there will be
by seeing the circle as rigorous. However in this hypothesis we have
due to the property of the circle; thus
.

When a polygon is contained within a circle we have
; since by looking at A D as an extension of a small side of the circle, there is D E: A E; A E is to the radius AB/2 and within this hypothesis one has
; as an equation
; which is the same as the preceding one. One can then see if one has paid attention that the value of centrifugal force is found to be equivalent in both cases.

If we call u the body’s speed and if we suppose u to be equal to the speed that the body would have acquired by falling from a height of h by virtue of the weight p there will be u u = 2 p h . See Acceleration and Gravity, and what we have said hereafter on the occasion of the equation φ d e = u d u . Furthermore, there will be for the same reason
for the speed that the body would acquire by falling from the height a for a period of time θ and as this speed would travel the distance 2 a uniformly the same time θ (See Acceleration and Descent) there will be
; the
;
; and here is the proof of the theory according to Mr. Huyghens and the word Central since we will have
. One can read the consequences of this theory under the same word Central.

One may read in certain works that centrifugal force is equal to the square of the speed divided by the radii and in others that it is equal to the square of the speed divided by the diameter; this difference in expression should not be surprising since the word equal signifies proportionality as is explained in the article Equation. This signifies only that the centrifugal forces within two different circles are as the squares of the speed divided by the radii or by the diameters which is the same thing. See Equation.

In the end the reason for this apparent difference in values is that the authors of Mechanics had given to centrifugal force a result of those having been taken from the line D E to represent centrifugal force , the time d t being constant, and the others having considered D E on the curve of the polygon and the others of the rigorous curve. In the first case D E = A E² divided by the radius and in the second D E = A E² divided by the diameter. Furthermore an E here is like speed since we suppose d t constant therefore instead of A E² one can write the square of the speed. These different observations will contribute enormously to clearing up that which the different authors have written concerning central and centrifugal forces .

Since 2 p h = u u and that A B/2 is the radius of the circle, it follows that if we make the radius = r there will be
whether u and r remain constant; that is to say that the equation
or
, will occur for all the curves u being the speed and r as the radius of the involute. The centrifugal force φ is to be supposed to be directed in reference to the center of the osculating circle. This is the point where the osculating radius touches the developing curves. If one wishes to have the centrifugal or central force directed towards any other point at F this new force or k, will be the cosine of the angle that the radius follows to this point made by the osculating radius. Then by looking at force F as composed by force φ and of another directed force following the curve, one easily finds by the principle of the decomposition of forces F: φ as I: K as making I the total sine, therefore
; thus
: this is the general formula of central and centrifugal forces on any random curve.

We should allow ourselves a moment of philosophical reflection concerning the progress of the human spirit. Huyghens had discovered the law of central forces within the circle and this same geometer discovered the theory of involutes and we can see that by combining these two theories one might extract by a simple corollary the law of central forces in a random curve; however Huyghens did not take the final step which appears so simple today and that is all the more surprising since the steps that he did take were more difficult. Newton, by generalizing Huyghens’ theory, found the general theory of central forces which led him to the actual world system as he discovered differential calculus simply by generalizing Barrow’s method for tangents; a method which was, so to say, infinitely close to differential calculus. It is in this way that that the simplest corollaries of known truths, which consist of nothing more than bringing these truths together, often escape those who would seem to possess the ease and the ability to deduce them and nothing is a better example than the one we have just mentioned to confirm those reflections that we have just made concerning this point under the word Discovery.

In the formula that we have given here below for central forces , we have made an abstraction of the body mass and if one wishes to pays attention to this mass, it is evident that one will have to multiply the expression of the central forces by the body mass or whatever is simpler and that instead of seeking to make p gravity, one would look to make this quantity the body’s weight which is nothing more than the product of gravity times the mass. We are making this remark, so that we are not embarrassed when one reads the article Central, in consideration that the mass is applied into the calculations of forces.

May we add that if we are seeking another expression of centrifugal force φ than that which we have already given, we provide for these which could be useful in certain cases.

We have found
however that as the circle is supposedly described uniformly, one can instead of AE/dt , place a defined arc A divided by the time t which is used to travel the length; therefore one will obtain
. If one allows t = θ one will obtain
. Furthermore if one designates the length of a pendulum which creates a vibration in time θ that 2 π is the relation of the circumference to the radius then one will obtain π ² l = 2a. See Pendulum and Vibration. Therefore
; and if one were to suppose furthermore
which is allowed one would obtain
.

It is with these formulae that one establishes the relationship of centrifugal force to gravity at the equator. See Weight and Gravity.

Note

1. N.B. In that article, No. 12, instead of the reason for the inverse of the third involute, one should read the under-sided involute; and No 13. at the end should read sine instead of cosine.