To these may be added a law, very celebrated in its time, and the occasion of an angry controversy, the Principle of least action. [381] 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. Connection 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 consequence 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 development 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.[63] The transcendent generalizing genius of Lagrange, and his matchless analytical skill and elegance, have made this undertaking as successful as it is striking.
[63] Lagrange himself terms Mechanics, “An Analytical Geometry of four dimensions.” Besides the three co-ordinates which determine the place of a body in space, the time enters as a fourth co-ordinate. [Note by Littrow.]
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 suggest their own generalizations. Something of this kind has happened in Mechanics. The same Formula expresses the general condition of Statics and that of Dynamics. The tendency to generalization which is thus introduced by analysis, makes mathematicians unwilling to [382] 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. Indeed, 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.
But before we do so, we may make one remark on the history of this part of science. In consequence of the vast career into which the Doctrine of Motion has been drawn by the splendid problems proposed to it by Astronomy, the origin and starting-point of Mechanics, namely Machines, had almost been lost out of sight. Machines had become the smallest part of Mechanics, as Land-measuring had become the smallest part of Geometry. Yet the application of Mathematics to the doctrine of Machines has led, at all periods of the Science, and especially in our own time, to curious and valuable results. Some of these will be noticed in the [Additions] to this volume.
BOOK VII.
THE MECHANICAL SCIENCES.
(CONTINUED.)