junction 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.
