relation of the motions to the forces that produce them; and we may thence deduce an expression for the force which must animate the heavenly bodies, in order that their motions may be consistent with observation. We thus reach the principle of universal gravitation, whence we may again descend to the explanation of every astronomic phenomenon, however minute, or difficult to be separated by observation alone, from those with which it is combined.

Mechanics is a mixed science. It owes its improvement, and its extended application to the perfection of the pure mathematics, but it is founded, like all other physical sciences, upon facts deduced from observations and experience. These facts are few in number; they are in truth no more than the properties which experiment shows to be essential to the existence of matter, that substance which is the general object of physical inquiry. We find it extended in three dimensions, and thus occupying a portion of space, capable of being set in motion, and endued with the power of preventing any two of its portions from existing simultaneously in the same place.

We cannot penetrate further than this into the secrets of nature; the integral elements of matter are not to be distinguished by our obtuse senses; their combinations are alone within our reach. We may observe the effects produced by the motion of bodies, although we cannot perceive the mode in which these effects are produced, nor how the original motion is itself caused. Still, as no portion of matter contains within itself the property of moving in a given direction, rather than in any other, we infer that its motion is due to some cause acting upon it. This cause we denominate a force, but in thus naming it, it is not necessary to infer any thing in respect to its nature; no more is required than that we should determine the laws according to which it acts. If we assume a single and very small portion of matter, to which we may give the name of a material point, and suppose a force to act upon it, the point must describe a straight line. We call this line the direction of the force, and a knowledge of it, of the position of the point where it acts, and of its intensity, enables us to make it an object of mathematical investigation.

If more than a single force act upon the point at the same time, they will reciprocally modify the action of each other, and the point will move in a direction which is determined by the concurrent influence of all; it may even remain at rest in consequence of their exactly compensating each other. In the latter case, it is said to be in equilibrio, and equilibrium being the effect of counteracting forces, is equally an object of mechanical investigation with motion. Its laws are also much more readily investigated in a direct manner, but D'Alembert, by the self-evident principle that we shall have occasion to illustrate hereafter, succeeded in re

ducing the most complicated motions of systems of bodies, to the simpler questions of equilibrium. From this time the mode of proceeding in mechanics has been, first, to inquire into the conditions of equilibrium of points and masses of matter, under the action of counteracting forces, and thence to deduce the laws of their motion, when equilibrium among the forces no longer exists. Three different methods exist by which to determine the conditions of equilibrium among a set of forces acting upon a material point: the first consists in an extension and generalization of the property of the lever, discovered originally by Archimedes; the second is the principle of Virtual Velocities employed by Lagrange; the third is that which is made use of by Laplace, in the Mecanique Celeste, and which we shall proceed to explain.

When any number of forces are in equilibrio around a point, it is evident, that if all except one be supposed to be removed, and their joint action to be performed by a single force, by which the point would tend to move in the same direction, and with the same intensity as it would, had they continued to act; this assumed force would be in exact equilibrio with the single force we have supposed to be left. If then, we can determine the value of this equivalent force, when the direction and intensity of the others is given, we have the means of reaching the condition of equilibrium. Such a force, that will identically replace two, or any number of others, is called their resultant; the forces it is capable of replacing are called the components.

If the forces act in the same straight line, the resultant evidently is equal to the sum of its components; but when their directions are inclined, the investigation becomes more difficult. Newton, who, confident in his own powers, grappled directly with the most difficult questions, undertook the consideration of motion, without proceeding through the more easy, although less obvious preliminary investigation of the laws of equilibrium. Assuming that motions are in the direction of, and proportioned to the force impressed, and finding that the union of two motions, each of which would separately cause a body to describe the side of a parallelogram, would make it describe the diagonal; he inferred that the force might be represented both in magnitude and in direction by the diagonal of a parallelogram, constructed on the two forces as sides. Galileo had also been in possession of the same proposition. The same inference may however be obtained directly, and independent of the consideration of motion. This last method is much preferable to that employed by Newton, in the present state of the science, for however proper it may have been, as an introduction immediately to the consideration of motion, it is obviously unnatural and a defect in method, to obtain in the first instance the laws of equilibrium from those

* See Mecanique Celeste, Liv. I. §. 1.

of motion, and then in proceeding farther, to deduce the circumstances of motion from considerations of equilibrium. We have hence seen with some surprise that this plan is still pursued in some modern works. Among those in which it is still used, we may cite as one in which we could have hardly expected to see such a retrograde step, the masterly treatise of Venturoli. We have also to regret that it has been employed (copied we believe from Whewell) in the compilation made for Harvard University. We could have hoped that the learned collector of that course would either draw a direct demonstration of the parallelogram of forces from his own abundant stores, or copy one of of the beautiful propositions that are to be found in several authors who have treated this problem in an original manner. In this, as indeed in all other cases, we conceive that even the most elementary attainments in science should be acquired by methods analogous, if not identical with those employed in the work, which is the received standard of the highest knowledge. If to read the Mecanique Celeste be beyond the ordinary limit of a student's views, still, in acquiring knowledge of a less elevated character, he ought to be led to it by steps, such that he will not be required to renew his labours, should he wish to proceed beyond the narrow sphere of undergraduate study. And that treatise on mechanics which is founded upon Newton's laws of motion, is as great a deviation from the direct track, as those systems which teach trigonometry and the conic sections after the method of the ancient geometers.

The more remarkable of the demonstrations of this fundamental proposition of the parallelogram of forces are those of Francœur, Professor Robinson, Poisson and Laplace. The last is by far the most elegant, and, in the first part, which demonstrates that the resultant of two rectangular forces corresponds in magnitude with the diagonal of the parallelogram, is sufficiently simple in its character to adapt it for an elementary work. No farther addition than a simple figure, as given by Dr. Young in his "Elementary Illustrations," is necessary. The second part of the investigation is more complex, and although it has been freed of a part of its difficulty by that author, is still beyond the reach of ordinary students of mechanics. We conceive that

it is possible to render it intelligible to all acquainted with analytic trigonometry, and to extend it to the general case in a more easy manner than could be done by following directly the path of Laplace through the case of three rectangular forces.

By the aid of the parallelogram of forces, all questions relating to the resolution of a single given force into two others, or the composition of a single force from two given forces, become simple problems of plane trigonometry.

To determine the resultant of any number of forces, we may

proceed by finding, first the resultant of any two of them, which may be assumed as replacing them identically, and may be combined with a third force, and so on, until all have been employed. If the forces be three in number, and at right angles to each other, their resultant may be shown to be the diagonal of the rectangu lar parallelopipedon constructed on the three forces as sides; and as any force may, by the converse of this proposition, be resolved into three others at right angles to each other, and as the resultant of any number of parallel forces is equal to their sum, all forces whatsoever, acting in any manner upon a given point, may be resolved into three rectangular forces.

In this way the most abstruse investigations of mechanics may be reduced to the consideration of no more than three forces. This is called the method of rectangular co-ordinates. In it we determine the position of the point of application of a force by the perpendicular distance from it to three planes, supposed to be immoveable, and cutting each other at right angles. These perpendiculars are the co-ordinates, and the direction of the force is defined by means of the three angles it makes with these co-ordinates.

It may so happen, that, although the forces that act upon a point are not in equilibrio, it shall still remain at rest. Such is the case when the point is pressed by them against a surface. In this event, it is no longer necessary that the resultant of all the forces be equal to 0, but it is sufficient that the direction be a normal to the surface. To denote the conditions of equilibrium, we introduce an expression for the action of the surface, which is equal and contrary in direction to this resultant. If the point rest upon a given curve, we consider the curve as formed of the intersection of two surfaces, and calculate the respective effect of each, in producing the state of equilibrium.

If, by the method we have referred to, a force be resolved into three others, parallel to its projections on three co-ordinate planes, and if the several forces be multiplied by the perpendiculars let fall from the common intersection of the planes upon their respective directions, these products are called moments. They will express the effect of each force to make its point of application turn around the three intersections of the planes, which are called the axes of the co-ordinates; and for any number of forces whatsoever, the moment of the resultant is equal to the sum of the moments of its components.

As matter is incapable of setting itself in motion,† so neither can it change the motion it may have received. A material point, therefore, if once acted upon by a force, and then abantlbid, Liv. I. § 4.

* Mec. Cel. Liv I. § 3.

doned to itself, must, if it meet with no resistance, move uniformly forwards in the direction of the force. This tendency of matter to maintain its state of rest, or of motion, is called inertia, and is identical with Newton's first law of motion.

*

In uniform motions, the spaces described are proportioned to the times, but the times employed by different points, in describing the same space, differ, in consequence of the different. intensities of the moving forces. Hence arises the notion of velocity, which, in uniform motion, is the ratio between the space and the time employed in describing it. To express the space and time, we employ numbers that represent how many units of some customary lineal measure have been traversed, and how many seconds have elapsed during the motion.

Newton and his followers, assume the velocity to be directly proportioned to the moving force. Laplace,t on the other hand, has, with more judgment, considered this relation as worthy of investigation. The velocity does unquestionably depend upon the force, and may therefore be expressed mathematically as a function of the latter. Let us suppose it to be so expressed, and that a point placed upon the surface of the earth, and participating in its motion, receives a new impulse in any direction. The point will, under the action of the motion of the earth and this new force, describe their resultant, with a certain velocity which will depend on the form of the function. This, then, is to be determined, by comparing the actual motion of such a point, with what it would have under different forms of this function. Now, it is a universal result, that in all the differing circumstances of the earth's motion, at different seasons and positions in its orbit, the motion is such as it would be if the velocity were exactly proportioned to the force. Hence, it may be inferred, that it is a general law of nature, that the velocity is proportional to the force, and these two quantities may be substituted for each other, and represented by the same lines and numbers. The resultant of the forces, will also be the resultant of the velocitics. It hence results, that the relative motions of a system of bodies are independent of their common motion, and that it is impossible to judge of the absolute motion of a system, of which we ourselves form a part, from appearances alone.

If a point be acted upon by forces, which, instead of abandoning it to itself, continue to influence it during the whole period of its motion, its velocity will no longer be constant, because it receives at every instant a new impulse. § If these successive impulses are equal among each other, the accelerating force is said to be constant, and the velocity is proportioned to the time.