Though it is admitted by all writers on this subject, that the most abstruse propositions may be deduced from a few simple principles, yet few arc found who entirely agree in their choice of such principles. The an )st, ad vanta(reous. and indeed the most natural method, seems to be that wherein the relation between various forces in a state of equilibrium is first investigated, and then the consideration extended to a body in motion. if a body remain in equilibrium, at the same time that it is solicited by several forces, each free is supposed to produce only a tendency to motion, which is measured by the motion it would produce were it not checked by the power of the others: therefitre. after expressing the ()Get of any one of the forces by unity, the relative force of the others may likewise be expressed by 'words or numbers.
La Place merely assumes the two foregoing principles, and speaks of them as experimental facts; while 1)r. Young does not scruple to declare them capable of demonstration. (See his Lectures.) But this difference of opinion is of little fin putt:ince, since the principles themselves ark.: universally admitted.
La Grange has founded the whole doctrine of the equi librium of forces on the well-known principle of the lever, the composition or motion, and the principle of virtual velocity ; each („If which we shall here notice.
The principle y* the lever may be derived from the com position of forces, or even from much less complicated con siderations.
Archimedes, the earliest author on record, who attempted to demonstrate the property of the lever, assumes the equi librium of equal weights at equal distances from the fulcrum, as a mechanical axiom ; and he reduces to this simple and primitive case that of unequal weights, by supposing them, when commensurable, to be divided into equal parts, placed at equal distances on different points of the lever, which may thus be loaded with a number of small equal weights, at equal distances lomn the fulcrum.
The principle of the straight and horizontal lever being admitted, the law of equilibrium in other machines may be deduced from it. Though it is not without difficulty that the inclined plate is referred to this principle ; the laws to which have been lint lately known.
Stevinus, mathematician to Prince Naurice of Nassau, first demonstrated the principle of the inclined plane by a very indirect, though curious mode of reasoning. lie con siders the case of a solid triangle resting on its horizontal base, sides then become two inclined planes : over these he supposes a chain to lie thrown, consisting of small equal weights threaded together ; the upper part of such chain resting on the two inclined planes, and the ends hanging at liberty below the Coot of the base, llis reason :1g is. that if the chain be not in equilibria, it %sill begin to
slide along the plane, and would continue so to do, the same cause still existing, rut ever ; thus producing a perpetual motion. Butt as this implies a contradiction, we roust con clude the chain to be in cquilibrio; in which case. as the efforts of all the weights appded to one side would be an exact counterpoise to those applied to the other, and the number of w mild be in the same ratio as the lengths of the planes ; he concludes that the weights will be in equi libria on the inclined planes when they are to each other as the lengths of the 'Janes ; but that when the plane is vertical, the power is equal to the weight ; and that therefore, in every inclined plane, the power is to the weight as the height of the plane to its length.
Virtual velocity is that which a body in equilibrium is disposed to receive whenever the equilibrium is disturbed ; in other words, it is what it body actually receives in the first moment of its !notion.
The principle of virtual velocity. in its roost general form, is as t'ollows: suppose a system in equilibrium composed of a of points. drawl in any direction. by m hatever to he so put in motion, as that every point shall de scribe an infinitely small space, indicative of its virtual veloc ity; the stun of the forces being each multiplied by the space described by the point to wli ch it is applied, in the direction of the force, will equal 0; the small spaces described in the direction of the forces being estimated as positive, and those in a contrary direction as negative. Galileo, in his Treatise on ..11echanicnI Science. and in his Dialogues., proposes this principle as a general prorrty in the equilibrinm of machines; lie appears to have been the first writer on mechanics, who was aelnainted with it. His disciple Tot ricelli was the author of another principle, which seems to be but a neces sar• consequence. of Galileo's. lie supposes two weights to be so connected, that however placed, their el.ntre of gravity shall neither rise nor ; in every situation. there.