The Absolute and Relative Force

PROPOSITION 15. PROBLEM.

130. For a given increment of the speed, a certain force produces on the point A in the small increment of time dt. Find the increment of the speed, that the same force produces on the same point in the time increment dτ.

SOLUTION

The point A (Fig. 14) has the speed c in the same direction AB as the force has acting on it, and ao is the small distance, through which the force pulls the point A, if it were at rest, in the small increment of time dt.

If again the distance AB is taken, that the point A will traverse with speed c in the increment of time dt, then it will traverse the same distance beyond Ab with the force acting, by taking Bb = ao : and thus the distance, since it is infinitely small, is considered to be described by a uniform motion.

Therefore in the following small time dt, the body travels a distance bC = Ab with this speed, unless acted upon by a force; and with the force acting again, which it can be put to remain unchanged even through the infinitely short time it will arrive beyond C at c, by taking Cc = ao.

In the same manner in the third increment of the time dt it will traverse the distance de = dE + Ee, where again dE = Cd and Ee = ao. Indeed we have :

Ab  AB  ao; bc  AB  2ao; cd  AB  3ao; de  AB  4ao. ao is the is the increment of the speed produced by the force in the time dt ; 2dt Hence ao dt

ao increment in the small increment of the speed acquired in the time 2dt ; similarly 3dt time 3dt

Generally in the short time ndt the speed c of the point will increase by the element nao .

Put ndt  d , then n  ddt . Therefore the increment of the speed acquired dt in the time increment d will be ao.d2 . Since indeed the increment of the speed in the dt ao short time dt is dt , it will be produced in this ratio : (increment of the speed acquired in the increment of the time dt) is to (increment of the speed acquired in the time increment d ) as dt is to d . Consequently the increments of the speed are in proportion to the times in which they are produced. Q. E. I.

Corollary 1.

131. This increment of the speed does not depend on the speed c itself, but it will have the same value, however large or small a value is put for c.

From this the nature of the absolute force is better understood, since they act equally on moving bodies and on bodies at rest.

Corollary 2.

132. If c = 0, and the point A at rest is urged into moving by a force, the speeds acquired from the motion itself will be as the times : obviously twice the speed in twice the time, with the time tripled, so it will obtain three times the speed.

Corollary 3.

133. If therefore from the start, the speed of the motion acquired in a small time t is called c, and the distance traversed is s, then t = nc. But also t   ds (37). Hence the equation is c 2 2 produced nc   ds or ncdc = ds and hence s  nc2  2t n . Therefore the distances c described from the start of the motion are in the square ratio of the times or of the speeds acquired in that distance. [p. 53; we see that the constant n is the inverse of the acceleration.]

Scholium 1

134. This proposition says that the increments of the speed are in proportion with the increments of time in which they are generated.

This also agrees with finite quantities as long as the force acting on the point:

• stays the same
• always retains the same direction as the motion of the point itself.

For infinitely small times it is not necessary to have this restriction; for no force of any variation greater than the smallest is to be considered. Moreover we are soon to show what the effect of the different forces may be, and also how points are influenced by forces of different kinds that we put in place, as another larger or smaller force can be put in the given ratio.

This does not adversely affect extremely small points, but indeed not only points that we understand to be mathematical but also the physical points that arise from the composition of bodies. For two or more points can be conceived to be merged into one, since the points still remain of infinitely small magnitude, and nevertheless it is better with single points.

Scholium 2.

135. GALILEO was the first person to use [this principle] in the investigation of a falling weight, for the solution of the problem found for this theorem.

He did not give a demonstration of this, but yet on account of the conspicuous nature of this principle from many similar phenomena, he did not wish to be doubted. Indeed he had refuted other opinions regarding this matter, from which he had greatly confirmed his own views.

Others indeed were of the opinion that increments not with time but with distance traversed were in proportion; truly the absurdity of this had been established by Galileo, and then by many philosophers. Moreover it appears, if the actions of the forces follow this law, that no body is able to lead in the motion at any time.

It might be that dc , that one body might evade the other is = nds and c = ns, truly the time t, that is  ds c   1n l c  1n l ns + const., which constant ought to be   1n l 0n. Clearly equal to 1n ds c the times described by the logarithm of the distance divided by the zero distance should be in proportion, but is thus infinite. Therefore no body can ever lead in the motion from rest.

Thus correctly Galileo responded to his opponents arguments, that instantaneous finite motion would have to arise from that [initial] position, otherwise forwards motion would not be possible.

Although indeed in the beginning the speed at the point is made infinitely small, yet that speed from an imaginary force of this kind will never be able to be finite. Now from the given solution of the problem it is understood that it is necessary to find a law, so that neither can any other force with contradictions arise in the beginning.