But leaving these vague and arbitrary denominations, which neither constitute the essence of the laws of nature, nor are able to raise the simple results of the known laws of mechanics to the rank of final causes, let us return to the principle in question, which takes place in every system of bodies acting on each other in any manner whatever, whether by means of cords, inflexible lines, attractions, and also drawn by forces directed to a centre, whether the system he entirely free, or eonstrained to move about it. The sum of the products of the masses by the areas described about this centre, and projected on any plane, is always proportional to the time; so that by referring these areas to three rectangular planes.we obtain three differential equations of the first order, between the time and the co-ordinates of the curves desei ibed by the bodies; and in these equations, the nature of the principle properly exists.
The fourth principle, that of the least action, was so denominated by Maupertuis, and has since been rendered celebrated the wi kings of several illustrious authors. Analytically it is as follows: In the motion of bodies acting upon each other, the sum of the products of the masses by the velocities, and by the spaces described, is a minimum. Maupertuis has published two memoirs on this principle; one in the Transactions of the Academy of Sciences, for 1744; the other, in those of the Academy of Berlin, 1746 ; wherein he deduces from it, the laws of reflection and refraction of light, and those of the shock of bodies. It appears, however, that these applications are not only too partial fin- establish ing the truth of a general principle, but they are in them selves too vague and arbitrary ; so that the consequences attempted to be deduced become uncertain : this principle, therefl-e, deserves not to be classed with the three fitregeing. There is, however, one point of view, in which it may be considered as more general and exaet.and which alone merits the attention of geometricians. Euler first suggested the idea at the close of his Treatise on Isoperimetrieul Problems, published at Lausanne. in 1744, wherein he shows that in trajectories described by central forces, the integral of the velocity multiplied by the element of the curve is constantly either a maximutn or a minimum ; but he knew of this property only as pertaining to insulated bodies. La Grange extended it to the motion of a system of bodies acting on each "Ow. and demonstrated a new general principle, viz. That ti' sum of the products of the masses by the integrals of the velocities multiplied by the elements of the spaces described, is always a ma \imilin or a minimum.
From a combination of this latter principle with that of the outset vation of the vis viva, many difficult problems in dynamics may be solved ; as exemplified by La Grange in the Memoirs of the Academy Turin, vol. ii.
La Place, in the ifechanique Celeste. treats the doctrine of dynamics much in the same manner as La Grange. but he carries his investigatittns much farther. I le agrees with that writer in adopting the principle or D'Alembert, and in resolving every motion into two ; that which the particle had in the instant, and that which would have maintained it in cquilibrio : but he differs from him in not admitting the principle of virtual velocity to be assumed as a fundamental axiom ; which he demonstrates by a regular train of inductions.
After having established nearly the same formultr, or diti•rential equations, and deduced all the general principles in the manner just described, he introduces others in the nature of corollaries, many of which merit peculiar considera tion. From the principle of the conservation of areas, it follows, that. in the motion of a system of bodies solicited only by their mamal attraction and by forces directed to the origin of the co-ordinates, there exists a plane passing through such origin. which possesses the billowing remarkable properties: 1. That the sum of the areas traced on the plane by the projections of the radii vectores of the bodies, and multiplied by their respective masses. will be the greatest possible.
'2. That such sum is also equal 0 upon all the planes perpendicular to it.
As the principle of the vis viva, and that of areas, subsist relatively to the centre of gravity, even though the latter be supposed to have a rectilinear uniform motion, it follows, that a plane may be determined as passing through this moveable origin, on which the sum of the areas, described by the pro jections of the radii vectores, and multiplied respectively by their masses, may be the greatest possible. This plane being parallel to the one passing through the fixed origin, satisfies the same conditions ; and another plane passing through the centre of gravity. and determined according to the foregoing conditions, will remain parallel to itself during the motion of the system ; a circumstance of considerable utility and importance. To this we may add, that any plane parallel to the last-mentioned, and passing through any of the bodies, partakes of analogous properties.