Proposition 26

Table of Contents

Proposition 26. Theorem

The heights, by which equal small bodies fall to acquire equal speeds, vary inversely as the forces, under the hypothesis of different uniform [gravitational] forces.

DEMONSTRATION

Let the mass or weight of some small body on the surface of the earth be A, the force [on some other celestial body] some constant value g and the corresponding height [p. 83] for this speed to be acquired be v. Truly the height shall be x, by which the small body A falling under the action of the force g shall acquire a speed [c2 = ] 2 …

But 2 1=n (206), hence A gx v = or Av = gx. [On the Earth, g/A = 1 and v = x.]

Whereby, since the speeds have been produced by different forces and the bodies have been made equal, then the quantity Av is constant, and thus also gx. On account of this, x will vary inversely as g, i. e. the height, by which the body A acquires the speed v by the action of the force g, varies inversely as the force g. Q. E. D.

[No doubt by now the reader has thrown up his or her hands in horror and said : what about the conservation of mechanical energy ? The factor n = ½ has arisen from the requirement that the acceleration of gravity is one. The problem for us lies in the lack of understanding that there was at the time about kinetic and potential energy and the conservation of the sum of these for a falling body. Euler has been very clever and set up his differential equations so that they can be scaled, and when it comes down to doing a numerical example, as with the flight of the cannonball in Ch. 4, he gets the correct answers. Thus, the answers right themselves when known experimental values are put in place for the time of fall of a body. It has been convenient to put the acceleration of gravity arbitrarily as 1, from which by (209) n = ½ (for Euler does his best to get rid of constants that always appear, a tradition that has been followed by theoreticians ever since!); but if the time is measured in seconds and the distance in scruples or thousandth parts of Rhenish feet, then the acceleration of gravity is not 1, but something around 32 ft/s2 or 32000 scruples/s2 . If fact, from the equation gHV 22 = , for the speed V of a body dropped from rest from a height H is given by HHHgV 250625002 =×≈= , the scaling factor used by Euler later. Hence when experimental results are imposed, the correct value for the acceleration of gravity results, and all is well.]

Corollary 1

Newton has shown that the impressed forces on the same body put in place on the surface, and acted upon towards the centre of, the Sun, Jupiter, Saturn, or the Earth, are as 10000, 835, 525, and 400. Therefore the heights from which the body acquires equal speeds in falling on the surface of the Sun, Jupiter, Saturn, and the Earth, are between themselves as .and,, …

Corollary 2

Moreover Newton understood likewise that all bodies fall at equal rates on these surfaces, just as on the surface of the earth. Therefore there is no need to add the condition that the bodies are equal, for from the heights which are in the ratio to each other ,and,,

…

1 on the surfaces of the Sun, Jupiter, Saturn, and the Earth, any bodies dropped will acquire the same increase in their speed. [p. 84]

Scholium 1

It is understood from these that there is a two-fold effect on the body by any force : on the one hand, by which a certain force or effort is impressed on the body, and on the other, by how the body may be moved by that force. The one that is to be considered mainly in statics is the weight and how it should be measured, that the body has in an equal attempt to fall downwards, and it may be called the absolute strength of the force. In turn, truly the effect should be measured by the acceleration or the change in the speed, that is impressed on the body in a given time : this is proportional to that force divided by the mass of the body (154). This effect is called by Newton the accelerating force, and therefore the strength of the accelerating force is proportional to absolute force applied to the mass of the body, or the weight applied. On account of which, since A pds dv = (207) and A p denotes the strength of the acceleration, then dv is equal to the product of the acceleration and the element of distance travelled. Thus the absolute force of gravity is proportional to the mass of the bodies on which it acts ; for the effect of these downwards is the cause or the weight that we have shown to be in proportion to the mass. Moreover the accelerating force of gravity is equal on all bodies, since they all fall equally and they gain equal speeds in equal intervals of time.

Corollary 3

Hence the sizes of the accelerations are to each other as the absolute forces, if the bodies are of equal mass. Whereby since the strength of the acceleration due to gravity is taken as 1, as we have put in place before (205), [p. 85] then the strength of the accelerating due to gravity on the surface of the sun is equal to 24.290; the strength of the acceleration on the surface of Jupiter is 2.036; the strength of the acceleration on the surface of Saturn is 1.280. And the strength of gravity has been taken on the moon by Newton as 3 1 .

Corollary 4

Whereby if from Proposition 25 the fall of the body to the surface of the Earth is to be accounted for, then 1=A g , as we have done in (205). But truly the fall of bodies on the surface of the sun requires 290.24=A g ; or on the surface of Jupiter, 036.2=A g ; on the surface of Saturn 280.1=A g ; and for the fall of bodies on the surface of the moon, it will be 3 1=A g .

Scholium 2

Here we assume with Newton that all celestial bodies are similar to our Earth and bodies placed on the surface of these have a force pulling then to their centre, which is always similar to the force of terrestrial on bodies. Therefore, from Newton’s exposition, it is apparent that a body, the weight of which here is 1 pound [lb.], will weigh on the surface of the Sun 24.290 lb ;on the surface of Jupiter, it will be 036.2 lb; on the surface of Saturn 280.1 lb ; and on the surface of the moon one third of a pound. [p. 86]

Scholium 3

Moreover in order that the nature of the gravitational forces can be more easily compared for celestial bodies, the individual equal elements of the bodies are understood to be equally affected by gravity. From which it follows, since now it agrees with experiment, that the forces of gravity which act on any bodies, are themselves proportional to the masses or quantities of matter. Truly it has been shown previously, that if the forces are in proportion to the masses of the bodies, then their effect on moving bodies is the equal (136). On which account it is shown from these that all bodies on the surface of the earth should descent equally, and likewise for all celestial bodies.