the third law of motion in its most general sense; namely, that the momentum (which is proportional to the mass of the body and its velocity jointly,) may be taken, the measure of the effect; so that this momentum is as much diminished in the striking body by the resistance it experiences, as it was increased in the body struck by the impact. This was sometimes expressed by saying that the quantity of motion remains unaltered, quantity of motion being used as synonymous with momentum. Newton expressed it by saying that "action and reaction are equal and opposite," which is still one of the most familiar modes of expressing the third law of motion.

In this mode of stating the law, we see an example of a propensity which has prevailed very generally among mathematicians; namely, a disposition to present the fundamental laws of rest and motion as if they were equally manifest, and, indeed, identical. The close analogy and connexion which exists between the principles of equilibrium and of motion, often led men to confound their evidence; and this confusion introduced an ambiguity in the use of words, as we have seen in the case of momentum, force, and others. The same may be said of action and reaction, which have both a statical and a dynamical signification. And by this means, the most general statements of the laws of motion are made to coincide with the most general statical propositions. For instance, Newton deduced from his

principles the conclusion, that by the mutual action of bodies, the motion of their centre of gravity cannot be affected. Marriotte, in his Traité de la Percussion (1684), had asserted this proposition for the case of direct impact. But by the reasoners of Newton's time, the dynamical proposition, that the motion of the centre of gravity is not altered by the actual free motion and impact of bodies, was associated with the statical proposition, that when bodies are in equilibrium, the centre of gravity cannot be made to ascend or descend by the virtual motions of the bodies. This latter is a proposition which was assumed as self-evident by Torricelli; but which may more philosophically be proved from elementary statical principles.

This disposition to identify the elementary laws of equilibrium and of motion, led men to think too slightingly of the ancient solid and sufficient foundation of statics, the doctrine of the lever. When the progress of thought had opened men's minds to a more general view of the subject, it was considered as a blemish in the science to found it on the properties of one particular machine. Descartes says in his Letters, that "it is ridiculous to prove the pulley by means of the lever." And Varignon was led by similar reflections to the project of his Nouvelle Mécanique, in which the whole of statics should be founded on the composition of forces. This project was published in 1687; but the work did not appear till after the death of the author. Though the

attempt to reduce the equilibrium of all machines to the composition of forces, is philosophical and meritorious, the attempt to reduce the composition of pressures to the composition of motions, with which Varignon's work is occupied, was a retrograde step in the subject, so far as the progress of distinct mechanical ideas was concerned.

Thus, at the period at which we have now arrived, the principles of elementary mechanics were generally known and accepted; and there was in the minds of mathematicians a prevalent tendency to reduce them to the most simple and comprehensive form of which they admitted. The execution of this simplification and extension, which we term the generalisation of the laws, is so important an event, that though it forms part of the natural sequel of Galileo, we shall treat of it in a separate chapter. But we must first bring up the history of the mechanics of fluids to the corresponding point.

CHAPTER IV.

DISCOVERY OF THE MECHANICAL PRINCIPLES OF FLUIDS.

Sect. 1.-Rediscovery of the Laws of Equilibrium of

Fluids.

We have already said, that the true laws of the equilibrium of fluids were discovered by Archimedes, and rediscovered by Galileo and Stevinus; the intermediate time having been occupied by a vagueness and confusion of thought on physical subjects, which made it impossible for men to retain such clear views as Archimedes had disclosed. Stevinus must be considered as the earliest of the authors of this rediscovery; for his work was published in Dutch about 1585; and in this, his views are perfectly distinct and correct. He restates the doctrines of Archimedes, and shows that, as a consequence of them, it follows that the pressure of a fluid on the bottom of a vessel may be much greater than the weight of the fluid itself: this he proves, by imagining some of the upper portions of the vessel to be filled with fixed solid bodies, which take the place of the fluid, and yet do not alter the pressure on the base. He also shows what will be the pressure on any

portion of a base in an oblique position; and hence, by certain mathematical artifices which make an approach to the infinitesimal calculus, he finds the whole pressure on the base in such cases. This mode of treating the subject would take in a large portion of our elementary hydrostatics as it now stands. Galileo saw the properties of fluids no less clearly, and explained them very distinctly, in 1612, in his Discourse on Floating Bodies. It had been maintained by the Aristotelians, that form was the cause of bodies floating; and collaterally, that ice was condensed water; apparently from a confusion of thought as to rigidity and density. Galileo asserted, on the contrary, that ice is rarefied water, as appears by its floating; and in support of this, he proved, by various experiments, that the floating of bodies does not depend on their form. The happy genius of Galileo is the more remarkable in this case, as the controversy was a good deal perplexed by the mixture of phenomena of another kind, due to what is usually called capillary or molecular attraction. Thus it is a fact, that a ball of ebony sinks in water, while a flat slip of the same material lies on the surface; and it required considerable sagacity to separate such cases from the general rule. Galileo's opinions were attacked by various writers, as Nozzolini, Vincenzio di Grazia, Ludovico delle Colombe; and defended by his pupil Castelli, who published a reply in 1615. These opinions were generally adopted and diffused; but somewhat later,