obtained by ascending gradually from particulars to truths more and more general, respecting the motions of fluids; but were caught at once, by a perception that the parts of fluids are included in that range of generality which we are entitled to give to the supreme laws of motion of solids. Thus, solid and fluid dynamics resemble two edifices which have their highest apartment in common, and though we can explore every part of the former building, we have not yet succeeded in traversing the staircase of the latter, either from the top or from the bottom. If we had lived in a world in which there were no solid bodies, we should probably not yet have discovered the laws of motion; if we had lived in a world in which there were no fluids, we should have no idea how insufficient a complete possession of the laws of motion may be, to give us a true knowledge of particular results.

14. Various General Mechanical Principles.-The generalised laws of motion, the points to which I have endeavoured to conduct my history, include in them all other laws by which the motions of bodies can be regulated; and among such, several laws which had been discovered before the highest point of generalisation was reached, and which thus served as stepping-stones to the ultimate principles. Such were, as we have seen, the principles of the conservation of vis viva and of the motion of the centre of gravity, and the like. These principles may, of course, be deduced from our elementary laws, and were finally

established by mathematicians on that footing. There are other principles which may be similarly demonstrated; among the rest, I may mention the principle of the conservation of areas, which extends to any number of bodies a law analogous to that which Kepler had observed respecting the areas described by each planet round the sun. I may mention also, the principle of the immobility of the plane of maximum areas, a plane which is not disturbed by any mutual action of the parts of any system. The former of these principles was published about the same time by Euler, D. Bernoulli, and Darcy, under different forms, in 1746 and 1747; the latter by Laplace.

To these may be added a law, very celebrated in its time, and the occasion of an angry controversy, the principle of least action. Maupertuis conceived that he could establish à priori, by theological arguments, that all mechanical changes must take place in the world so as to occasion the least possible quantity of action. In asserting this, it was proposed to measure the action by the product of velocity and space; and this measure being adopted, the mathematicians, though they did not generally assent to Maupertuis' reasonings, found that his principle expressed a remarkable and useful truth, which might be established on known mechanical grounds.

15. Analytical Generality. Connexion of Statics and Dynamics.-Before I quit this subject, it is important to remark the peculiar character which the science of mechanics has now assumed, in con

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sequence of the extreme analytical generality which has been given it. Symbols, and operations upon symbols, include the whole of the reasoner's task; and though the relations of space are the leading subjects in the science, the great analytical treatises upon it do not contain a single diagram. The Mécanique Analytique" of Lagrange, of which the first edition appeared in 1788, is by far the most consummate example of this analytical generality. "The plan of this work," says the author, "is entirely new. I have proposed to myself to reduce the whole theory of this science, and the art of resolving the problems which it includes, to general formulæ, of which the simple developement gives all the equations necessary for the solution of the problem."-" The reader will find no figures in the work. The methods which I deliver do not require either constructions, or geometrical or mechanical reasonings; but only algebraical operations, subject to a regular and uniform rule of proceeding." Thus this writer makes mechanics a branch of analysis; instead of making, as had previously been done, analysis an implement of mechanics. The transcendent generalising genius of Lagrange, and his matchless analytical skill and elegance, have made this undertaking as successful as it is striking.

The mathematical reader is aware that the language of mathematical symbols is, in its nature, more general than the language of words; and that in this way truths, translated into symbols, often

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suggest their own generalisations. this kind has happened in mechanics. The same formula expresses the general condition of statics and that of dynamics. The tendency to generalisation which is thus introduced by analysis, makes mathematicians unwilling to acknowledge a plurality of mechanical principles; and in the most recent analytical treatises on the subject, all the doctrines are deduced from the single law of inertia. deed, if we identify forces with the velocities which produce them, and allow the composition of forces to be applicable to force so understood, it is easy to see that we can reduce the laws of motion to the principles of statics; and this conjunction, though it may not be considered as philosophically just, is verbally correct. If we thus multiply or extend the meanings of the term force, we make our elementary principles simpler and fewer than before; and those persons, therefore, who are willing to assent to such a use of words, can thus obtain an additional generalisation of dynamical principles; and this, as I have stated, has been adopted in several recent treatises. I shall not further discuss here how far this is a real advance in science.

Having thus rapidly gone through the history of force and attraction in the abstract, we return to the attempt to interpret the phenomena of the universe by the aid of these abstractions thus established.

NOTE ON LEONARDO DA VINCI.

WHEN the preceding pages were prepared for the press, I had not seen Venturi's "Essai sur les Ouvrages Physico-Mathématiques de Léonard da Vinci, avec des Fragmens tirés de ses Manuscrits apportés d'Italie. Paris, 1797." Leonardo was born in 1452, and died in 1519; and was an eminent mathematician and engineer, as well as a painter, sculptor, and architect. It is proper to examine, therefore, whether he claims a place in the history of astronomy and mechanics. The following statements will show that he is no inconsiderable figure in the prelude to the great discoveries in both these sciences; if, indeed, we do not put him in the stead of Stevinus, as the first person who clearly understood the oblique action of pressure.

Leonardo da Vinci, about 1510, explained how a body, by describing a kind of spiral, might descend towards a revolving globe, so that its apparent motion, relatively to a point in the surface, might be in a straight line tending to the centre. He thus showed that he had entertained in his thoughts the hypothesis of the earth's rotation, and was employed in removing the difficulties which accompanied it, by the consideration of the composition of motions.

He also, as early as 1499, gave a perfectly correct ștatement of the proportion of the forces exerted by a cord which acts obliquely and supports a weight on a lever. He distinguishes between the real lever, and the potential levers, that is, the perpendiculars drawn from the centre upon the directions of the forces. Nothing can be more entirely sound and satisfactory than this. It is quite as good as the proof of Stevinus. These views must, in all probability, have been sufficiently promulgated by the time of Galileo, to influence his reasonings concerning the lever; which, indeed, much resemble those of Leonardo.

Da Vinci also anticipated Galileo in asserting that the time of descent of a body down an inclined plane is to the time of descent down its vertical height, in the proportion of the length