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.
The conservation of energy is but a mode of reckoning motion, by which it is found to be constant in all changes of a system. The system must embrace all the particles concerned in the motion. It may be made as large as we like.
The principle of the conservation of energy as here stated is confined to the case of moving bodies. Sometimes the energy is said to disappear from the form of motion and become potential energy. That case will be treated under the fourth consideration of level, but it introduces no alteration in what has been said.
As to the practical truth of the law of conservation of energy there can be no doubt; nor as to the value of the results obtained from tracing its validity in obscure actions. But there is nothing final about it. It is a numerical statement of extreme value, and it introduces a mode of reckoning by which motion can be looked upon as indestructible as matter is.
There is a possible objection to the law of conservation of energy.
It is no less a law in nature that in every one of a series of changes some of the energy passes off into the form of heat. Now heat is reckoned as a mode of energy. And there is in science a method of calculating how much energy any given quantity of heat is the equivalent of. And this equivalence is calculated on the supposition that no energy is lost. When heat is produced and motion passes away, the proportion between the motion that disappears and the heat that appears is represented by a number calculated on the assumption that no energy is lost. Thus whenever any quantity of energy takes the form of heat, the quantity of heat which is produced is exactly given by the calculation. But the reverse process is not possible. It is not possible to turn back all the energy in the form of heat into the form of motion. Consequently it cannot be proved that the energy in the form of heat would, if all turned into motion, produce as much motion as that from which it was produced. There may be an absolute loss of energy—only a very small one. The law of the conservation of energy may be the expression that this loss is a minimum.