1. If A produces B, then 2 A will produce 2 B.
This is the blank form of causal relation quantitatively considered, when the causes and effects are simple—that is, are unimpeded by other causes and uncomplicated by other effects; and whenever two or more causes co-operate, there is no possibility of determining the relation between the compound cause and the compound effect except by assuming that between each co-operating cause and its separate effect there exists this same quantitative relation.
2. This truth holds whatever the natures of the simple causes and simple effects; and is an a priori assumption made in conducting every experiment and in reasoning from it.
Every process of weighing, every chemical analysis, every physical investigation, proceeds on this truth without assigning warrant for it; and in allowing for the effect of any minor cause that interferes with the major cause, this same truth is assumed.
3. When A is an impressed force and B the produced motion, then the general truth that if A produces B, 2 A will produce 2 B, becomes the more special truth called the Second Law of Motion.
Newton’s amplified statement of this Law is:—“If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively.” And his further clause, asserting that this law holds whether the directions of the forces are or are not the same, asserts a proportionality between each force and its produced motion, such as we have seen to be invariably assumed between each cause and its separate effect, when there are co-operating causes.
4. This Law may be affirmed, without specification of the modes in which the impressed force and the resulting motion are to be estimated.
Newton’s statement is abstract. Taking for granted right modes of measurement, it asserts that the alteration of motion (rightly measured) is proportional to the impressed force (rightly measured).
5. No a posteriori proof of the general ultimate physical truth (or of this more special truth it includes) is possible; because every supposed process of verification assumes it.
These, cleared from entanglements, are the theses held by me, and defended in the following pages. {298}