(2) To No. 18 of Matter.
The quantity of a motion (quantitas motus, already in Descartes) is the product of the mass into the velocity.
This law is the basis not only of the doctrine of impact in mechanics, but also of that of equilibrium in statics. From the force of impact which two bodies with the same velocity exert the relation of their masses to each other may be determined. Thus of two hammers striking with the same velocity, the one which has the greater mass will drive the nail deeper into the wall or the post deeper into the earth. For example, a hammer weighing six pounds with a velocity = 6 effects as much as a hammer weighing three pounds with a velocity = 12, for in both cases the quantity of motion or the momentum = 36. Of two balls rolling at the same pace, the one which has the greater mass will impel a third ball at rest to a greater distance than the ball of less mass can. For the mass of the first multiplied by the same velocity gives a greater quantity of motion, or a greater momentum. The cannon carries further than the gun, because an equal velocity communicated to a much greater mass gives a much greater quantity of motion, which resists longer the retarding effect of gravity. For the same reason, the same arm will throw a lead bullet further than a stone one of equal magnitude, or a large stone further than quite a small one. And therefore also a case-shot does not carry so far as a ball-shot.
The same law lies at the foundation of the theory of the lever and of the balance. For here also the smaller mass, on the longer arm of the lever or beam of the balance, has a greater velocity in falling; and multiplied by this it may be equal to, or indeed exceed, the quantity of motion or the momentum of the greater mass at the shorter arm of the lever. In the state of rest brought about by equilibrium this velocity exists merely in intention or virtually, potentiâ, not actu; but it acts just as well as actu, which is very remarkable.
The following explanation will be more easily understood now that these truths have been called to mind.
The quantity of a given matter can only be estimated in general according to its force, and its force can only be known in its expression. Now when we are considering matter only as regards its quantity, not its quality, this expression can only be mechanical, i.e., it can only consist in motion which it imparts to other matter. For only in motion does the force of matter become, so to speak, alive; hence the expression vis viva for the manifestation of force of matter in motion. Accordingly the only measure of the quantity of a given matter is the quantity of its motion, or its momentum. In this, however, if it is given, the quantity of matter still appears in conjunction [pg 227] and amalgamated with its other factor, velocity. Therefore if we want to know the quantity of matter (the mass) this other factor must be eliminated. Now the velocity is known directly; for it is S/T. But the other factor, which remains when this is eliminated, can always be known only relatively in comparison with other masses, which again can only be known themselves by means of the quantity of their motion, or their momentum, thus in their combination with velocity. We must therefore compare one quantity of motion with the other, and then subtract the velocity from both, in order to see how much each of them owed to its mass. This is done by weighing the masses against each other, in which that quantity of motion is compared which, in each of the two masses, calls forth the attractive power of the earth that acts upon both only in proportion to their quantity. Therefore there are two kinds of weighing. Either we impart to the two masses to be compared equal velocity, in order to find out which of the two now communicates motion to the other, thus itself has a greater quantity of motion, which, since the velocity is the same on both sides, is to be ascribed to the other factor of the quantity of motion or the momentum, thus to the mass (common balance). Or we weigh, by investigating how much more velocity the one mass must receive than the other has, in order to be equal to the latter in quantity of motion or momentum, and therefore allow no more motion to be communicated to itself by the other; for then in proportion as its velocity must exceed that of the other, its mass, i.e., the quantity of its matter, is less than that of the other (steelyard). This estimation of masses by weighing depends upon the favourable circumstance that the moving force, in itself, acts upon both quite equally, and each of the two is in a position to communicate to the other directly its surplus quantity of motion or momentum, so that it becomes visible.
The substance of these doctrines has long ago been expressed by Newton and Kant, but through the connection and the clearness of this exposition I believe I have made it more intelligible, so that that insight is possible for all which I regarded as necessary for the justification of proposition No. 18.