The inhabitants of the valley would not have been right in saying that one act of a routine caused another: But they were right in saying that the amount of sensation was constant, and that some of it passed off in a form in which they could not feel it.
And so let us not say that one action causes another. Let us not say, for example, that the downward swing of a pendulum is the cause of its upward swing. But let us simply say that the one follows the other; that the amount of energy present is the same except for the small portion that passes off into the form of heat.
Suppose certain sets of numbers were being presented to us one after the other, and amongst these three consecutive sets were the following. First set: 3, 5, 6. Second set: 8, -2, -1, 1. Third set: 7, 4, 2, -1.
A little consideration will show us that there is a certain uniformity in these sets.
Take the square of each of the numbers in the first set and add them together, the result is 70. Thus 3² + 5² + 6² = 9 + 25 + 36 = 70.
The sums of the squares of the numbers in the second set come to the same. 8² + (-2)² + (-1)² + 1² = 64 + 4 + 1 + 1 = 70. Also in the third, 7² + 4² + 2² + (-1)² = 49 + 16 + 4 + 1 = 70, and so on.
Having noticed this we should regard it as a purely formal law, having nothing to do with why the numbers were presented to us. But we should consider it likely that it would characterize all the numbers that were presented to us. And if this expectation were found to be realized, we should after a time feel a certain assurance that the next set of numbers presented would satisfy the same law. If this assurance was indefinitely satisfied we should get to regard the satisfying this law as an invariable condition of the numbers presented. But we should never regard this purely formal law—that is, a law about the particular characteristics of the numbers—we should never regard this formal law as the cause of the next set of numbers appearing after the first had gone.
When, however, we talk about the conservation of energy we are apt to think of it as more than a merely formal law, more than a statement about numbers which has been found to hold true.
Yet it is no more. The law of the conservation of energy asserts that in any system in motion the sum of the squares of the velocities of the particles at any one moment is equal to the sum of the squares of the velocities of the particles at the next moment.