The Absolute and Relative Force

PROPOSITION 19. THEOREM.

150. A point can be moved along the direction AM (Fig. 16) and it is acted on, while it traverses the small distance Mm, by a force p pulling in the same direction ;the increase in the speed, that the point meanwhile acquires, is as the product of the force by the short time, in which the element of distance Mm is traversed.

DEMONSTRATION.

Let the element of time be dt, and in this time the point completes the distance Mμ, if it is not acted on by a force, but has the speed that it had at M, and it can go on moving uniformly. Truly the effect of the force is in accordance with this : as the point is drawn further forwards by μm, this extra small distance is equal to that, by which the same point initially at rest, will be drawn forwards by the same force acting for the same element of time dt, as the force is absolute, [p. 62] (111).

The increment of the speed is proportional to the time for this given distance. But if the force is the constant, then the increase of the speed is in proportion to the element of the time dt (130). [For if the force is not constant, then it will not have the same effect at different places.] Whereby when the small distance mμ, or increment of the speed shall be as the force p for a given short time interval, then the increment of the speed for any short time and for any force shall be as pdt, i. e. as the product of the force taken with the short time.

Q. E. D.

Corollary 1.

151. Let the speed of the point at M be c and the element of distance be Mm = ds, then dt  ds , since the time determined from the uniform motion in the element Mm can be c pds put in place.

Moreover since dc shall be as pdt, also dc will be as c or cdc will be as pds. Therefore the increment of the square of the speed is proportional to the product of the force by the length of the element traversed. [Thus, the time to travel the distance Mm with speed c is simply dt = Mm/c; the extra speed dc generated in this time dt is equal to the acceleration a times by the time, or a × pds dt; which is proportional to pdt, or to c as shown. Hence, cdc  pds ; and on integrating for a constant force : c 2  p.s . This would now be thought of as the conversion of work done by the constant force into kinetic energy. The reader should bear in mind that the terms work, power , strength, etc, did not have the specific physics- related meanings then that they now have;

This makes the business of translating more difficult, as these words with special meaning should not be used without qualification.

In addition, results of this nature cannot be referenced to a general principle; the work-energy principle still lay some time in the future, though there were rumblings about it in the Bernoulli camp. Euler in sect. 153 then considers the extra distance as proportional to the force by the square of the time, from which the increase in the speed is again proportional to the force times the time increment.]

Corollary 2.

152. Therefore it is apparent that not only is this theorem true, but also it is true by necessity, as thus it would involve a contradiction to put dc  p 2 dt or p3dt or another function in place of p. All of which and equally commendable are considered by the most distinguished Daniel Bernoulli in Comment. Tom. I, and I have been greatly influenced with the rigor of the demonstration of these propositions.

Scholium.

153. The demonstration of this proposition follows easily from (148), from which the element of distance mμ emerges as proportional to the [p.63] to the force p multiplied by the square of the time dt, thus so that mμ shall be as pdt 2 . But mμ divided by the time dt gives the increment of the speed, whereby the increment of the speed is as pdt, as was enunciated in the proposition.

PROPOSITION 20. THEOREM

154. The motion of the point in a direction in agreement with the direction of the force, the increment of the speed will be as the force taken with the element of time, and divided by the quantity of matter of the point is composed.

DEMONSTRATION

Let there be two points or unequal bodies A and B (Fig. 17) in motion along the line AM and BN. These are influenced by the forces p and π respectively, while they traverse the distances Mm and Nn, and the times in which these are traversed are dt and dτ. It is clear that the point B is affected by the force π in the same manner as the point A by the force A (136). Whereby by putting in place of the point B equal to the point B A, for the force π there must be substituted the force AB , and in this way we obtain the case of the preceding proposition, for which the points are put equal. Hence on account of this, the increment of the speed in traveling through the distance Mm is as the increase of the speed through pdt the distance Nn as pdt is to AB d or as A to dB (150). From which the proposition is agreed upon, that the increase of the speed is as the product of the force and the time divided by the mass or quantity of matter of the point .Q. E. D

Corollary 1.

npdt

155. If the speed of the point A were c, then dc  A , where n in all cases denotes the same number; that depends neither on the size of the force, on the element of the time, or on the quantity of matter. [There is now a constant of proportionality, enabling equations to be used rather than proportionalities. Eventually Euler sets this to 1⁄2 for convenience, as he is not required to adhere to a set or units as we now are. Later, when the need arises, he absorbs this constant into his equations to produce the correct experimental acceleration in the units chosen.

In his later works, Euler does adopt standards of mass and length and time, and moves away from ratios : see e.g. his introduction to the motion of rigid bodies presented in these translations in vol. 3]

Corollary 2.

156. The quantity of matter comes under consideration here, in as much as , in as much as it resists the tendency to be disturbed by the force i. e. as much as it agrees with the force of inertia. On this account, the increment of the speed is directly as the force acting and the of the element of time acting, and inversely as the force of inertia of the body.

Corollary 3

157. With the distance put as Mm = ds the the time dt  ds . Hence the increment in the c npds npds speed becomes dc  Ac or cdc  A . Whereby the increment of the square of the speed is proportional to the product from the force by the distance traveled divided by the mass or the mass or the force of inertia of the body.

Scholium

158. This proposition embraces all the principles expounded upon this far concerning the nature of motion to be defined and the laws of motion. On account of which if this proposition is joined with fourteen, (118), by which the effect of forces acting at angles are determined, all the principles will be put in place, from which the motion for any points can be found from any forces acting.

Corollary 4

npdt 159. Since dc  A , the distance through which the point A is lead by the force p in the element of time dt will be equal to npdt 2 . This distance is indeed the product of dc by dt. A For by saying that this distance is dz, then dc  dz (128) and thus dz  dcdt  dt npdt 2 . A