independently regulated so as to go together within less than that quantity of time. In short, it would require either an infinitely long time of observation, or infinitely acute powers of measuring discrepancy, to decide positively whether two clocks were or were not in relation with each other.
A similar question actually occurs in the case of the moon’s motion. We have no record that any other portion of the moon was ever visible to men than such as we now see. This fact sufficiently proves that within the historical period the rotation of the moon on its own axis has coincided with its revolutions round the earth. Does this coincidence prove a relation of cause and effect to exist? The answer must be in the negative, because there might have been so slight a discrepancy between the motions that there has not yet been time to produce any appreciable effect. There may nevertheless be a high probability of connection.
The whole question of the relation of quantities thus resolves itself into one of probability. When we can only rudely measure a quantitative result, we can assign but slight importance to any correspondence. Because the brightness of two stars seems to vary in the same manner, there is no considerable probability that they have any relation with each other. Could it be shown that their periods of variation were the same to infinitely small quantities it would be certain, that is infinitely probable, that they were connected, however unlikely this might be on other grounds. The general mode of estimating such probabilities is identical with that applied to other inductive problems. That any two periods of variation should by chance become absolutely equal is infinitely improbable; hence if, in the case of the moon or other moving bodies, we could prove absolute coincidence we should have certainty of connection.2 With approximate measurements, which alone are within our power, we must hope for approximate certainty at the most.
The principles of inference and probability, according to which we treat causes and effects varying in amount, are exactly the same as those by which we treated simple experiments. Continuous quantity, however, affords us an infinitely more extensive sphere of observation, because every different amount of cause, however little different, ought to be followed by a different amount of effect. If we can measure temperature to the one-hundredth part of a degree centigrade, then between 0° and 100° we have 10,000 possible trials. If the precision of our measurements is increased, so that the one-thousandth part of a degree can be appreciated, our trials may be increased tenfold. The probability of connection will be proportional to the accuracy of our measurements.
When we can vary the quantity of a cause at will it is easy to discover whether a certain effect is due to that cause or not. We can then make as many irregular changes as we like, and it is quite incredible that the supposed effect should by chance go through exactly the corresponding series of changes except by dependence. If we have a bell ringing in vacuo, the sound increases as we let in the air, and it decreases again as we exhaust the air. Tyndall’s singing flames evidently obeyed the directions of his own voice; and Faraday when he discovered the relation of magnetism and light found that, by making or breaking or reversing the current of the electro-magnet, he had complete command over a ray of light, proving beyond all reasonable doubt the dependence of cause and effect. In such cases it is the perfect coincidence in time between the change in the effect and that in the cause which raises a high improbability of casual coincidence.
It is by a simple case of variation that we infer the existence of a material connection between two bodies moving with exactly equal velocity, such as the locomotive engine and the train which follows it. Elaborate observations were requisite before astronomers could all be convinced that the red hydrogen flames seen during solar eclipses belonged to the sun, and not to the moon’s atmosphere as Flamsteed assumed. As early