Note 151 — Introductory material

(2) As regards time, no congruence is here conceivable, for to effect congruence requires always—as we saw in the case of solids—one more dimension than belongs to the magnitudes compared. No day can be brought into temporal coincidence with any other day, to show that the two exactly cover each other; we are therefore reduced to the arbitrary assumption that some motion or set of motions, given us in experience, is uniform. Fortunately, we have a large set of motions which all roughly agree; the swing of the pendulum, the rotation and revolution of the earth and the planets, etc. These do not exactly agree, but they lead us to the laws of motion, by which we are able, on our arbitrary hypothesis, to estimate their small departures from uniformity; just as the assumption of Free Mobility enabled us to measure the departures of actual bodies from rigidity. But here, as there, another possibility is mathematically open to us, and can only be excluded by its philosophic absurdity; we might have assumed that the above set of approximately agreeing motions all had velocities which varied approximately as some arbitrarily assumed function of the time, f(t) say, measured from some arbitrary origin. Such an assumption would still keep them as nearly synchronous as before, and would give an equally possible, though more complex, system of Mechanics; instead of the first law of motion, we should have the following: A particle perseveres in its state of rest, or of rectilinear motion with velocity varying as f(t), except in so far as it is compelled to alter that state by the action of external forces. Such a hypothesis is mathematically possible, but, like the similar one for space, it is excluded logically by the comparative nature of the judgment of quantity, and philosophically by the fact that it involves absolute time, as a determining agent in change, whereas time can never, philosophically, be anything but a passive form, abstracted from change. I have introduced this parallel from time, not as directly bearing on the argument, but as a simpler case which may serve to illustrate my reasoning in the more complex case of space. For since time, in mathematics, is one-dimensional, the mathematical difficulties are simpler than in Geometry; and although nothing accurately corresponds to congruence, there is a very similar mixture of mathematical and philosophical necessity, giving, finally, a thoroughly definite axiom as the basis of time-measurement, corresponding to congruence as the basis of space-measurement[159].