DYNAMICS.

BOOK II.

DEFINITIONS AND ILLUSTRATIONS.

243. DYNAMICS is that part of mechanics which has for its object the action of forces on bodies, when the result of that action is motion; and since bodies, when once put in motion continue to move for some portion of time, we introduce TIME into our investigations.

244. Forces are commonly divided into two kinds, in one the action is supposed to be instantaneous, and is called impulsive; in the other the action is continued and incessant, and is termed accelerating, or retarding, according as it acts with or opposes the bodies' motion.

From a little consideration, it will be evident that the former of these, that is impulsive force, can only take place in perfectly hard and inflexible bodies, which have no existence 14

in nature; and we may observe that in the abstract consideration of these, as well as of several other similar cases in mechanics, when metaphysical possibilities instead of the natural state of things are attended to, difficulties arise which are hardly explicable by any method of reasoning whatever: for we are certain that when finite velocity is communicated to any natural body, the time in which it is communicated must be of some finite quantity, so that, when the body acted upon begins to move from quiescence, it will, during the action of the force, possess all the intermediate degrees of velocity between nothing and the velocity ultimately communicated.

An exceeding great force, however, may be communicated, and an effect produced, in a portion of time so small as to elude the acuteness of our senses; and hence we obtain the idea of motion being derived from impulse ;-but the LAW OF CONTINUITY, by which changes are produced by imperceptible degrees, can never be violated.

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It is necessary to inform the STUDENT that the law of continuity is that by which variable quantities, passing from one magnitude to another, pass through all the intermediate magnitudes, without ever passing abruptly over any of them.

This law Boscovich proves to be universally true, in the first place, from induction. Thus the distances of two bodies can never be changed without their passing through all the intermediate distances.

We see the Planets move with different velocities in different directions, but they still observe the law of continuity. In heavy bodies projected, the velocity decreases and increases through all the intermediate velocities;---the same happens with regard to elasticity and magnetism. No body becomes more or less dense without passing through all the intermediate densities. The light of the day increases in the morning and decreases at night through all the intermediate degrees. In a word, throughout all nature we see the law of continuity takes place, if all things be rightly considered. It is true we sometimes make abrupt passages in our minds, as when we compare the length of one day with that of another immediately following it, and say that the latter is two or three minutes longer or shorter than the former, passing all at once, in our way of speaking, round the earth; but if we take all the different longitudes we shall find days of all the intermediate lengths. We likewise sometimes confound a quick motion with an instantaneous one: thus, we are apt to imagine that the ball is thrown abruptly out of the gun; But; in truth, some time is required for the gradual inflamation of the

245. Force generating velocity is called an accelerating force; generating motion it is called a moving force.

Thus, the accelerating force is measured by the velocity uniformly generated in a given time, without any regard to the quantity of matter moved; and the moving force is measured by the quantity of motion uniformly generated in a given time.

powder, for the rarefaction of the air, and for the communication of motion to the ball. In like manner, all the objections to the law of continuity may be satisfactorily solved.

Boscovich goes still farther, and maintains that a breach of this law is metaphysically impossible. This argument he draws from the nature of continuity: for it is essential to continuity that where one part of the thing continned ends, and another begins, the limit is common to both. Thus when a geometrical line is divided into two, an indivisible point is the common limit to both. Time is also continued, and where one hour ends another immediately begins, and the common limit is an indivisible instant.

Now as all variations in variable quantities are made in time, they all partake of its continuity, and hence none of them can hasten, by an abrupt passage, from one magnitude to another without passing through the intermediate magnitudes.

We cannot pass from the sixth hour to the ninth without passing through the seventh and eighth; because, if we did, there would be a common limit between the sixth hour and the ninth, which is impossible. So, likewise, we cannot go from the distance 6 to the distance 9, without passing the distances 7 and 8; because, if we did, in the instant of passage we should be both at the distance 6 and the distance 9 at the same time; which is impossible.

In like manner a body that is condensed, or rarified, cannot pass from the density 6 to the density 9, or vice versa, without passing through the densities 7 and 8; because, in the aprupt passage, there would be two densities 6 and 9 at the same instant.

The body must pass through all the intermediate densities, and this it may do either quickly or slowly, but still it must pass through them all ; the like may be said of all variable quantities, and thence we may conclude that the law of continuity is universal. But, in Creation, is there not an abrupt passage from non-existence to existence? No, there is not; for before existence a being is nothing, and therefore incapable of any state.

In creation a being does not pass from one state to another abruptly; it passes over no intermediate state; it begins to exist and to have a state, and existence is not divisible. But do we not at least, admit of an abrupt passage from fepulsive to attractive forces even in our theory itself? We do not. Our repulsive forces diminish through all the intermediate magnitudes, down to nothing; through which, as a limit, they pass to attraction. In the building of a house or ship, neither of them, is augmented abruptly; for the additions made to them are effected solely by a change of distances between the parts of which they are composed": and all the intermediate distances are gone through.

The like may be said of many other cases, and still the law of contin. uity remains firm and constant.

For more on this subject, sec vol. I, of the Supplement to the Encyclopedia Brillanica, page 105.

246. The quantity of motion or momentum of a moving body, is the product found by multiplying the quantity of matter moved into its velocity.

247. If any force act continually in the same direction upon a body in motion, its motion will be continually accelerated or retarded.

For, by art. 17, after the action of any force ceases, the velocity of the body will continue uniform; but, while the force continues to act, new velocities are generated at every instant of the body's motion. Thus, a body descending towards the earth by the force of gravity, describes unequal spaces in equal times, its motion being continually accelerated.

In the same manner, a body projected perpendicularly upwards, is opposed by the action of gravity, and its motion is continually retarded.

248, Forces are either constant or variable.

1. If, during a body's motion, equal velocities be communicated in equal successive portions of time, the force is said to be constant, and the velocity uniformly accelerated or retarded. Of this kind is the force of gravity near the earth's surface, which, in the ascent or descent of a body, creates, or destroys, a velocity of 32 feet in each second of time.

2. When unequal velocities are generated or destroyed in equal successive portions of time, the force is said to be variable. Of this kind is the force of gravity at different distances from the earth's surface, which generates or destroys unequal velocities in equal successive portions of time,

219. The moving forces which communicate the same relocities in a given time, to different bodies, will be as the quantities of matter contained in those bodies.

For if one body contain a quantity of matter ten times greater than the other, the greater may be divided into ten bodies cach equal to the less, and whatever force may be required to

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produce a certain velocity in the less body, ten such forces will be required to produce the same velocity in the ten bodies; and this will be the case whether the larger body be divided or not.

Hence, the moving force exerted by gravity upon bodies descending towards the earth, is proportional to the quantities of matter contained in them; for we know, from experience, that all bodies whatever their weights may be, descend near the earth's surface through equal spaces in equal times, abstracting from the resistance of the air.

250. The moving forces acting upon bodies, and the momenta communicated to them in a given time, are as the quantities of matter moved, and the velocities communicated jointly.

For, (art. 245.) when the velocity communicated in a given time is the same, the moving force is as the quantity of matter moved; and, by art. 18, when the quantity of matter is given, the moving force is as the velocity communicated in the same time: therefore, both the quantity of matter and the velocity communicated being different, the moving forces, and their effects, viz. the momenta produced, will be as the quantities of matter moved and the velocities communicated jointly. That is, if m denote the momentum, b the body, ƒ the force,

f

and v the velocity: then, fom xbv, and vα-.

251. The accelerating forces which communicate velocities to bodies, are as the moving forces directly and the quantities of matter moved inversely.

For, (art. 245.) the accelerating force is as the velocity generated in a given time: and the moving force is as the quantity of matter and velocity generated in the same time; therefore, the moving force is as the accelerating force and the quantity of matter moved jointly: or the accelerating force