The close connexion of these three principles, the composition of (statical) forces, the composition of (accelerating) forces with velocities, and the measure of (moving) forces by velocities, cannot be denied; yet it appears to be by no means easy to supersede the necessity of independent proofs of the last two of these principles. Both may be proved or illustrated by experiment: and the experiments which prove the one are different from those which establish the other. For example, it appears by easy calculations, that when we apply our principles to the oscillations of a pendulum, {258} the Second Law is proved by the fact, that the oscillations take place at the same rate in an east and west, and in a north and south direction: under the same circumstances, the Third Law is proved by our finding that the time of a small oscillation is proportional to the square root of the length of a pendulum; and similar differences might be pointed out in other experiments, as to their bearing upon the one law or the other. 9. _Mechanical Principles become gradually more simple and more evident._--I will again point out in general two circumstances which I have already noticed in particular cases of the laws of motion.--Truths are often at first assumed in a form which is far from being the most obvious or simple;--and truths once discovered are gradually simplified, so as to assume the appearance of self-evident truths. The former circumstance is exemplified in several of the instances which we have had to consider. The assumption, that a perpetual motion is impossible, preceded the knowledge of the first law of motion. The assumed equality of the velocities acquired down two inclined planes of the same height, was afterwards reduced to the third law of motion by Galileo himself. In the History[35\3], we have noted Huyghens's assumption of the equality of the actual descent and potential ascent of the center of gravity: this was afterwards reduced by Herman and the Bernoullis, to the statical equivalence of the solicitations of gravity and the vicarious solicitations of the effective forces which act on each point; and finally to the principle of D'Alembert, which asserts that the motions gained and lost balance each other. [Note 35\3: B. vi. c. v. sect. 2.]