A path is a line whose beginning and end are distinguished. Two paths are considered to be equivalent, which, beginning at the same point, lead to the same point. Thus the two paths, A B C D E and A F G H E (Fig. 3), are equivalent. Paths which do not begin at the same point are considered to be equivalent, provided that, on moving either of them without turning it, but keeping it always parallel to its original position, [so that] when its beginning coincides with that of the other path, the ends also coincide. Paths are considered as geometrically added together, when one begins where the other ends; thus the path A E is conceived to be a sum of A B, B C, C D, and D E. In the parallelogram of Fig. 4 the diagonal A C is the sum of A B and B C; or, since A D is geometrically equivalent to B C, A C is the geometrical sum of A B and A D.
Figure 3.
Figure 4.
All this is purely conventional. It simply amounts to this: that we choose to call paths having the relations I have described equal or added. But, though it is a convention, it is a convention with a good reason. The rule for geometrical addition may be applied not only to paths, but to any other things which can be represented by paths. Now, as a path is determined by the varying direction and distance of the point which moves over it from the starting-point, it follows that anything which from its beginning to its end is determined by a varying direction and a varying magnitude is capable of being represented by a line. Accordingly, velocities may be represented by lines, for they have only directions and rates. The same thing is true of accelerations, or changes of velocities. This is evident enough in the case of velocities; and it becomes evident for accelerations if we consider that precisely what velocities are to positions—namely, states of change of them—that accelerations are to velocities.
Figure 5.
The so-called “parallelogram of forces” is simply a rule for compounding accelerations. The rule is, to represent the accelerations by paths, and then to geometrically add the paths. The geometers, however, not only use the “parallelogram of forces” to compound different accelerations, but also to resolve one acceleration into a sum of several. Let A B (Fig. 5) be the path which represents a certain acceleration—say, such a change in the motion of a body that at the end of one second the body will, under the influence of that change, be in a position different from what it would have had if its motion had continued unchanged, such that a path equivalent to A B would lead from the latter position to the former. This acceleration may be considered as the sum of the accelerations represented by A C and C B. It may also be considered as the sum of the very different accelerations represented by A D and D B, where A D is almost the opposite of A C. And it is clear that there is an immense variety of ways in which A B might be resolved into the sum of two accelerations.
After this tedious explanation, which I hope, in view of the extraordinary interest of the conception of force, may not have exhausted the reader’s patience, we are prepared at last to state the grand fact which this conception embodies. This fact is that if the actual changes of motion which the different particles of bodies experience are each resolved in its appropriate way, each component acceleration is precisely such as is prescribed by a certain law of Nature, according to which bodies in the relative positions which the bodies in question actually have at the moment,[[32]] always receive certain accelerations, which, being compounded by geometrical addition, give the acceleration which the body actually experiences.