motion, and at the same time bears them all together in its orbit round the sun, which perhaps in its turn carries the earth and the whole system of planets toward some distant constellation. Relative rest, therefore, is really the only kind of rest which can actually take place among the objects to which our attention is directed. It is at least all that we can ever be certain of observing.

16. We are hence led to make a similar distinction with respect to motion, and to separate the absolute motions of bodies, considered with reference to immoveable space, from the relative changes of position which may happen among them. These last therefore may be called relative motions, whether that body of the system to which they are referred, be itself in motion or at rest. The changes of place, for example, among the heavenly bodies, which we observe from the surface of the earth, are not absolute but relative motions, because the earth to which we refer them, as a fixed centre, has actually a motion of rotation on its axis and a progressive motion about the sun. Even when by calculation we have inferred from these observations the actual motions of the heavenly bodies as they would appear, if seen from the sun, we cannot affirm positively that these are absolute motions, since it may be that the sun and the whole planetary system have a common motion in space.

17. According to the idea of inertia which we derive from experience, we must regard the state of motion and that of rest, as simple accidents of matter, which it is incapable of imparting to itself, and which it can only receive from without, and which once received, it cannot alter. When therefore we see a body passing from one of these states to the other, we must regard this change as produced and determined by the action of external causes. These causes, whatever they may be, are denominated forces. Nature presents us with an infinite number of them which are at least in appearance of different kinds. Such are the forces produced by the muscles and organs of living animals, the exercise of which, for the most part, depends solely on the will. Such are also the forces of physical agents, as the expansion of bodies by heat, and their contraction by cold, &c. There are moreover others which scem to be inherent in certain bodies, as the attraction of the magnet for iron, and that which is manifested among electrified bodies.

From the very nature of matter as thus presented to our consideration, it will be seen that a body once put into a state of motion or rest, by any cause whatever, must continue in that state forever, if no new cause is made to act upon it. If it cannot give itself motion when at rest, it cannot stop itself when in motion, for this would be equivalent to giving itself motion in the opposite direction; neither can it change its velocity or direction, for this would equally imply a new force. Thus, motion is naturally equal or uniform and rectilinear.

18. When several forces are applied at the same time to a body, they are mutually modified by the connexion which exists among the different parts of the body, and which prevents each from taking the motion which the force exerted upon it tends to produce. If these forces happen entirely to destroy each other, so that the body remains at rest, we say that the forces are in equilibrium, or that the body is in equilibrium, under the action of these forces.

19. Mechanics is the science which treats of the equilibrium and motion of bodies. That part, the object of which is to discover the conditions of equilibrium, is called statics.* We give the name of dynamicst to the other part which has for its object to determine the motion which a body takes, when the forces applied to it are not in equilibrium. The general laws of statics and dynamics are applicable to fluids; but on account of the peculiar difficulty attending the consideration of this class of bodies we are accustomed to treat them separately. That part of the mechanics of fluids which relates to their eqilibrium is called hydrostatics, and that which comprehends their motions, hydrodynamics.§

20. In our inquiries on these subjects, we first proceed upon the supposition that there are no other bodies, and no other forces, in nature, except those under consideration. Thus all bodies. are supposed to be destitute of weight, and free from friction, resistance, and obstructions of every kind. Regard is afterwards had to these causes; but to estimate their effects, it is necessary to begin by investigating each point separately.

* From 1, I stand.

From dwp, water, and 1oryui.

† From δυναμις, power.

From up and durars.

STATICS.

Of Uniform Motion.

21. A body is said to have a uniform motion when it passes continually over the same space in the same time.

In order to compare the motions of two bodies which move uniformly, it is necessary to consider the space which each describes in the same determinate time, as one minute, one second, &c. This space is what is called the velocity of the body.

22. The volocity of a body therefore is, properly speaking, only the space which this body is capable of describing uniformly in the interval of time which we take for unity.

Thus in the uniform motion of two bodies, the time being reckoned in seconds, if one passes over five feet in a second, and the other six feet in a second, we say that the velocity of the first is five feet, and that of the second six feet.

23. But if, the second being always taken as the unit of time, I am told that a body passes over 100 feet in 5 seconds, 100 feet does not express the velocity, since this space is not that which answers to the unit of time, a second; but it will be perceived, that in each second it would pass over a fifth part of this 100 feet, or 20 feet; that is, in order to find the volocity, I divide the number 100, the parts of the space passed over, by 5, the number of units in the elapsed time. Hence universally, the velocity is equal to the space divided by the time; for it is clear, that if we divide the whole space into as many equal parts, as there are units in the time elapsed, each part will be the space described during this unit of time, and will consequently be the velocity according to our definition. Thus calling the velocity and st

the space passed over in any portion of time denoted by t, we shall have

2=

this is one of the fundamental principles of mechanics.

24. The equation v = gives not only the measure of the velocity, but also that of the time and space. Indeed if we considered t and s as unknown quantities successively, we shall have, by the common rules of algebra,

Thus, to find the time we divide the space by the velocity; and, to find the space we multiply the velocity by the time.

If, for example, it is asked what time is required to describe 200 feet, when the body in question has a uniform velocity of 5 feet in a second; it is evident that it would require as many seconds as there are 5 feet in 200 feet; that is, we should have the time sought, or the number of seconds, by dividing the space 200 by the velocity 5; we shall find for the answer 40 seconds; or, in other words, a number of seconds equal to the quotient arising from dividing the space by the time.

In like manner, if it is asked what space would be described in 20 seconds by a body moving with a constant velocity of 5 feet in a second; it is manifest that it would describe 20 times 5 feet; that is, it is necessary in this case to multiply the velocity by the time.

Thus, although we have here employed algebraic characters, it is not because they are necessary to the investigation of these fundamental truths, but because, by means of them, the propositions, and their dependence, the one upon the other, are more concisely expressed, and more easily remembered. Indeed it will be seen by the above example, that the first principle, expressed algebraically, being once fixed in the mind, the two others are readily deduced from it by the most familiar rules.

25. It will be casy now to compare the uniform motions of two, or of a greater number of bodies. If it is asked, for exam

ple, what is the ratio of the velocities of two bodies which describe the known spaces s, s', in the times t, respectively; by calling v, v', the velocities of these two bodies respectively, we shall have

that is, the velocities are as the spaces divided by the times.

In a word, if it is proposed to compare the velocities, the spaces, or the times, the principle above laid down, will give the expression for each of these particulars with respect to each body; we have therefore only to compare together these expressions. For example, if we would compare the spaces, the fundamental proposition = gives svt; we have in like manner for the

that is, the spaces are as the velocities multiplied by the times.

26. Of these three things, namely, the space, time, and velocity, if we would compare two together, when the third is the same for each body, we have only to deduce from the same fundamental theorem, the expression for this third particular, with respect to each body, and to put these two expressions equal to each other. If, for example, we would know the ratio of the spaces when the velocities are the same, we should have

ť

whence, since by supposition = ', we have, and accordingly

s: s': :t: ť'

that is, the velocities being equal, the spaces are as the times. It will be found in like manner that, the times being equal, the spaces are as the velocities; and that the spaces being equal, the velocities must be

23.