out any reasonable hope of ever arriving at anything ultimate and fundamental.

But though this principle cannot be looked upon as a great discovery of Galileo, it is a highly useful rule; and the varied forms under which he and his successors urged it, tended much to dissipate the vague wonder with which the effects of machines had been looked upon; and thus to diffuse sounder and clearer notions on such subjects.

This principle of virtual velocities also affected the progress of mechanical science in another way; it suggested some of the analogies by the aid of which the third law of motion was made out; leading to the adoption of the notion of momentum as the arithmetical product of weight and velocity. Since on a machine on which a weight of two pounds at one part balances three pounds at another part, the former weight would move through three inches while the latter would move through two inches; we see (since three multiplied into two is equal to two multiplied into three,) that the product of the weight and the velocity is the same for the two balancing weights: and if we call this product momentum, the law of equilibrium is, that when two weights balance on a machine, the momentum of each would be the same, if they were put in motion.

The notion of momentum was here employed in connexion with virtual velocities: but it also came under consideration in treating of actual velocities, as we shall soon see.

Sect. 5.-Attempts at the Third Law of Motion.Notion of Momentum.

IN the questions we have hitherto had to consider respecting motion, no regard is had to the size of the body moved, but only to the velocity and direction of the motion. We must now trace the progress of knowledge respecting the mode in which the mass of the body influences the effect of force. This is a more difficult and complex branch of the subject; but it is one which requires to be noticed, as obviously as the former. Questions belonging to this department of mechanics, as well as to the others, occur in Aristotle's Mechanical Problems. "Why," says he, "is it, that neither very small nor very large bodies go far when we throw them; but, in order that this may happen, the thing thrown must have a certain proportion to the agent which throws it? Is it that what is thrown or pushed must react13 against that which pushes it; and that a body so large as not to yield at all, or so small as to yield entirely, and not to react, produces no throw or push?" The same confusion of ideas prevailed after his time: and mechanical questions were in vain discussed by means of general and abstract terms, employed with no distinct and steady meaning; such as impetus, From power, momentum, virtue, energy, and the like.

13 ἀντερείδειν.

some of these speculations we may judge how thorough the confusion in men's heads had become. Cardan perplexes himself with the difficulty, already mentioned, of the comparison of the forces of bodies at rest and in motion. If the force of a body depends on its velocity, as it appears to do, how is it that a body at rest has any force at all, and how can it resist the slightest effort, or exert any pressure? He flatters himself that he solves this question, by asserting that bodies at rest have an occult motion. "Corpus movetur occulto motu quiescendo."Another puzzle, with which he appears to distress himself rather more wantonly, is this: "If one man can draw half of a certain weight, and another man also one half; when the two act together, these proportions should be compounded; so that they ought to be able to draw one half of one half, or one quarter only." The talent which ingenious men had for getting into such perplexities, was certainly at one time very great. Arriaga, who wrote

in 1639, is troubled to discover how several flat weights, lying one upon another on a board, should produce a greater pressure together than the lowest one alone produces, since that alone touches the board. Among other solutions, he suggests that the board affects the upper weight, which it does not touch, by determining its ubication, or whereness.

Aristotle's doctrine, that a body ten times as heavy

11 Rod. de Arriaga, Cursus Philosophicus, Paris, 1639.

as another, will fall ten times as fast, is another instance of the confusion of statical and dynamical forces: the force of the greater body, while at rest, is ten times as great as that of the other; but the force, as measured by the velocity produced, is equal in the two cases. The two bodies would fall downwards with the same rapidity, except so far as they are affected by accidental causes. The merit of proving this by experiment, and thus refuting the Aristotelian dogma, is usually ascribed to Galileo, who made his experiment from the famous leaning tower of Pisa, about 1590. But others about the same time had not overlooked so obvious a fact.F. Piccolomini, in his Liber Scientiæ de Natura, published at Padua, in 1597, says, "On the subject of the motion of heavy and light bodies, Aristotle has put forth various opinions, which are contrary to sense and experience, and has delivered rules concerning the proportion of quickness and slowness, which are palpably false. For a stone twice as great does not move twice as fast." And Stevinus, in the Appendix to his Statics, describes his having made the experiment, and speaks with great correctness of the apparent deviations from the rule, arising from the resistance of the air. Indeed, the result followed by very obvious reasoning; for ten bricks, in contact with each other, would obviously fall in the same time as one; and these might be conceived to form a body ten times as large as one of them. Accordingly, Benedetti, in 1585, reasons in this manner

with regard to bodies of different size, though he retains Aristotle's error as to the different velocity of bodies of different density.

The next step in this subject is more clearly due to Galileo; he discovered the true proportion which the accelerating force of a body falling down an inclined plane bears to the accelerating force of the same body falling freely. This was at first a happy conjecture; it was then confirmed by experiments, and, finally, after some hesitation, it was referred to its true principle, the third law of motion, with proper elementary simplicity. The principle here spoken of is this; that for the same body, the dynamical effect of force is as the statical effect; that is, the velocity which any force generates in a given time when it puts the body in motion, is proportional to the pressure which the same force produces in a body at rest. The principle, so stated, appears very simple and obvious; yet this was not the form in which it suggested itself either to Galileo or to other persons who sought to prove it. Galileo, in his Dialogues on Motion, assumes, as his fundamental proposition on this subject, one much less evident than that we have quoted, but one in which that is involved. His postulate is, that when the same body falls down different planes of the same height, the velocities acquired are equal. He confirms and illustrates this by a very ingenious experiment on a