Accelerating forces. A dead force can only be understood when it is transformed into an accelerating or delayed force that still possesses the full freedom to exercise itself due to its continued action in which motion either accelerates or is delayed if it travels in the opposite direction. See Acceleration. However this way of considering accelerating forces appears to be subject to great difficulties. In effect, we could we ask, if motion is produced by an accelerating force such as gravity which begins at zero speed and if suspended by a string, why does it experience any resistance from that which supports it? It would be exactly the same for a body placed on a horizontal plane and attached to a string which is also horizontal at the end of which is placed a force . This force would not have to make any effort to retain the body since the body is at rest or that which is the same since the speed at which it moves is zero. Furthermore, if the first speed with which a body tends to move is also equal to zero as it is supposed, then why is the effort that must be used to retain it not also zero? During its descent this body will undoubtedly assume a finite speed at the end of some indeterminate time; as long as the effort that is made to support it, does not act against the speed that it will become; if it acts against that which tends to place it in motion then that would be against zero speed. In one word, a weighted body suspended by a string tends to move horizontally and vertically at zero speed, from where does the effort arise that prevents it from moving vertically and is none required to prevent it from moving horizontally? There are only two ways in which to answer these questions, neither one nor the other is adequate.

One can say in the first instance that it is wrong to suppose that the initial speed at which a body drops is absolute zero, that this speed is finite even though very small and as little as one might wish to suppose it to be it appears difficult to conceive how a speed which has started at absolute zero would eventually become real; how can a force which has no motion produce real motion by the passage of time and that gravity is a force of the same type as centrifugal force which we will find out following this article. That this latter force is portrayed in nature not as an infinitely small force but a finite force which is very small, the bodies of which follow a curve but do not in reality describe rigorous curves but polygonal curves composed of finite and very large quantity of small, straight contiguous lines set against very obtuse angles. This is the first answer.

After which I add, first: that it is difficult and perhaps impossible to understand how a force , which begins to produce in a body with no speed, can by successive bodies which are reiterated to infinity produce within the body a finite speed; it is no easier to understand how a solid is formed through the motion of a surface without depth; how a series of indivisible points can form space; how a series of indivisible instants make time even how points and indivisible instants succeed one another; how an atom at rest at any randomly selected point can be transported elsewhere; finally how the ordinate of a curve which is zero at its summit becomes real through the passage of this ordinate along the abscissa; all of these difficulties and similarly others, contain an unknown essential and one that is always incomprehensible to motion, space and time. Therefore as they do not prohibit us from recognizing the reality of space, time and motion, the difficulty encountered against the passage from zero speed to finite speed must not be assumed to be decisive. Second: without doubt centrifugal force , either in rigorous curves or in curves considered as infinite polygons, is comparable in its outcome to gravity. But how do we arrive at the fact that no portion of a curve described by a body in nature is not rigorous and that all others are polygons with a finite number of sides but very large? These would be sides of finite number, very small and would be perfectly straight lines. Why do we find it more difficult to suppose that there are very small perfectly straight lines than very small perfectly curved lines? I cannot seem to understand the reason for this preference since its absolute correctness is just as difficult to determine in a portion of space as the curve should be absolute. Third: It is here that the difficulty emerges in regards to the first answer; that if the nature of the accelerating force is to produce at the first instant a very small speed, then this force acting at each instant during a finite time, would by the end of this time, produce an infinite speed which is definitely contrary to the experiment.

One might say that the nature of gravity is not to act at every instant, but rather to provide for small finite pushes which succeed one another like as would jerks in the intervals of finite time, even though they are very small; however it is felt that this supposition is purely arbitrary and why would gravity act in a jerking fashion and not by a continuous uninterrupted effort? One would be unable to admit this hypothesis except in cases where one would see weight similarly as the effect of impulsion of a fluid. And we are aware how doubtful it is that weight should come from such an impulse, since until now the phenomena of weight has not been able to be established or even appears to be of a contrary notion. See Weight, Gravity and Gravitation. We have seen by all of these thoughts that the first answer to the difficultly that we proposed on the nature of accelerating forces is itself subject to considerable difficulties.

One might add as a second response to this difficulty, that a body containing weight or any other body moved by any accelerating force must begin its motion at zero speed, but that this body is no less disposed to move vertically as long as not impedes it in doing so since then is no more reason for it to move horizontally; and that there is consequently a tendency for this body to move vertically called a nisus, which is not so for a horizontal motion and that it is this nisus, this tendency which is supported in the first case and that does not need to be supported in the second case and that it can only have a counter-effect by a nisus, which is a similar tendency. That the effort that is needed to support a weight is the same as the weight itself, and that this effort will produce in all facts at the first moment of an infinitely small speed which is very different from zero effort, since zero effort would produce no movement whatsoever and that the effort to which we refer would produce a finite motion at the end of finite time. This second response is no more satisfying than the other; for what is a nisus in motion which does not produce finite speed in the first instance? What are we to make of such an effort? Furthermore why the effort to support a heavy is weight necessarily so much greater than what is needed to stop a billiard ball which moves at a finite speed? It appears that the latter would have to be greater, if in effect the force of the weight was zero in relation to that of the impact.

It so happens that of everything that we have said the proposed difficulty deserves the attention of all Physicists and Geometers. We invite them to seek out ways in which to solve with greater success that which we have just done, assuming that it is possible.

Laws of accelerating forces and the methods to compare them. Irrespective of what has been said on these thoughts on the nature of accelerating forces, at least there is a certainty in the sense that we have explained the word Accelerator; that if we say that φ is the acceleration force of a body, dt the time variable, du that of speed, we will have φ dt=du; if the force is delayed instead of accelerating one would have φ dt = -du since when t increases u diminishes; kindly refer to my treatise on Dynamics, articles 19 and 20. Then by designating e as distance travelled one has u= de / dt (see Speed); then the equation φ dt = ±du also provides the following φ dt² = ±dde, that is to say that the small spaces that an accelerating or delaying force allows to be travelled are as time².

This equation φ dt² = ±dde or the equation φ dt = ±du which comes to the same is not a principle of mechanics as other authors have thought but a simple definition. The accelerating force can only be known to us through its effect. This effect is nothing more than the speed that it produces within a certain time and when we say for example that the acceleration of a body is reciprocally proportional to the square of the distance, one only wishes to say that du / dt is reciprocally proportional to this square; thus φ is nothing else but the abbreviated expression of du / dt and the second part of the equation which expresses the value of du / dt. See Accelerator and my treatise on Dynamics already quoted.

The equation dde / dt² = φ allows us to see that during an instant the effect of all of any accelerating force is similar to time² since the variable quantity φ is presumed to be a constant during the instant dde / dt².

Consequently dde is equal to dt. Therefore during any instant that the small spaces of an accelerating force travel between themselves are similar time² or similar to their corresponding instants.

All accelerating effects act instantaneously in the same way and following the same laws as gravity acts in a defined time, since the distance that gravity covers is as time². See Acceleration and Descent. Therefore if we assign a space that weigh p which will cover some random time θ ,
consequently
we will obtain the general formula to compare weight p and an accelerating force φ .

However there is an important statement to make concerning this; it will not occur unless one notices the rigorous curve which would have t as its time and e for the abscissa for the spaces as the ordinates or that which comes to the same which would be represented by the equation between the coordinates e and t . See Equation. For if one looks at this curve as a polygon, then dde is seen as an ordinary differential equation would have a doubled value which is the one which is contained within the rigorous curve and consequently it would be necessary to suppose
in order to maintain the same value for φ . See the following words Polygonal curve and Differential, page 988. col 1. It was due to this lack of detail that the famous Mr. Newton had been mistaken on the mesure of central forces in the first edition of the Principia . Mr. Bernoulli proved this in the 1711 Academy of Science mémoires at the time that Mr. Newton’s new edition was being prepared in England for which this great man corrected himself without responding and allowing for a better understanding with a simple example how this distinction between the two equations is necessary, I will suppose that φ is constant and equal to p ;
which is obtained by the first equation by integration
.

Then if t = θ , one will obtain e = a / 2 which does not comply with the hypothesis, since we have designated a = to the space described in time ς and that consequently if t = θ one will obtain e = a ; contrary by making
we will find as we should e = a . This statement is extremely important since it will allow us to avoid paradoxes.

The equation φ dt = du , provides φ de = u du since dt = de/u ; therefore uu=2 s φ d e ; another equation between speed and distance for accelerating forces . Therefore by example φ is constant, there will be u u =2 φ e and this is the equation for distance and speeds within the motion of bodies that weight animates.