rote with very different success, no man having been more praised or more criticized than he was: his literary paradoxes, his singular systems, in all branches of polite learning, and above all, his judgment upon the ancients, which, like those of Petronius, were thought disrespectful and detracting, raised him up formidable adversaries. Racine, Boileau, Rousseau, and Madam Dacier, were among the number of those who made it their business to avenge antiquity on a man who, with more wit than genius or learning, assumed a kind of dictatorial authority in the province of belles lettres. He became blind in the latter years of his life, and died in 1731: a complete edition of all his works was published in 11 vols., 8vo. in 1754; though, as has been said of our Swift, his reputation had been better consulted by reducing them to three or four.is defined to be the continued and successive change of place.

There are three general laws of motion.

1. That a body always perseveres in its state of rest, or of uniform motion in a right line, till by some external force it be made to change its state: for as body is passive in receiving its motion, and the direction of its motion; so it retains them, or perseveres in them without any change, till it be acted on by something external. From this law it appears why we inquire not, in philosophy, concerning the cause of the continuation of the motion or rest in bodies, which can be no other than their inertia; but if a motion begin, or if a motion already produced is either accelerated or retarded, or if the direction of the motion is altered, an inquiry into the power or cause that produces this change is a proper subject of philosophy.

2. The second general law of motion is, The change of motion is proportional to the force impressed, and is produced in the right line in which that force acts. When a fluid acts upon a body, as water or air upon the vanes of a mill, or wind upon the sails of a ship, the acceleration of the motion is not proportional to the whole force of those fluids, but to that part only which is impressed upon the vanes or sails, which depends upon the excess of the velocity of the fluid above the velocity which the vane or sail has already acquired; for if the velocity of the fluid be only equal to that of the vane or sail, it just keeps up with it, but has no effect either to advance or retard its motion. Regard must always be had to the direction in which the force is impressed, in order to determine the change of motion produced by it: thus, when the wind acts obliquely with respect to the direction of a ship, the change of her motion is first to be estimated in the direction of the force impressed; and thence, by a proper application of mechanical and geometrical principles, the change of the motion of the ship in her own direction is to be deduced.

3. The third general law of motion is, That action and reaction are equal, with opposite directions, and are to be estimated always in the same right line. Body not only never changes its state of itself, but resists by its inertia every action that produces a change in its motion: hence when two bodies meet, each endeavours to persevere in its state, and resists any change; the one acquires no new motion, but what the other loses in the same direction; nor does this last lose any force, but what the other acquires: and hence, though, by their collision, motion passes from the one to the other; yet the sum of their motions, estimated in a given direction, is preserved the same, and is unalterable by their mutual actions upon each other.

All motion may be considered absolutely or relatively. Absolute or real motion, says Mr Maclaurin, is when a body changes its place in absolute space; and relative motion, is when a body changes its place only with relation to other bodies.

From the observation of nature, every one knows that there is motion; that a body in motion perseveres in that state, till by the action of some power it is necessitated to change it; that it is not in relative or apparent motion in which it perseveres in consequence of its inertia, but in real or absolute motion. Thus the apparent diurnal motion of the sun and stars would cease Motion cease, without the least power or force acting upon them, if the motion of the earth was stopt; and if the apparent motion of any star was destroyed by a contrary motion impressed upon it, the other celestial bodies would still appear to persevere in their course.

To make this matter still plainer, Mr Martin observes, that space is nothing but an absolute and infinite void, and that the place of a body is that part of the immense void which it takes up or possesses; and this place may be considered absolutely, or in itself, in which case it is called the absolute place of the body; or else with regard to the place of some other body, and then it is called the relative or apparent place of the body.

Now, as a motion is only the change of place in bodies, it is evident that it will come under the same distinction of absolute and relative, or apparent. All motion is in itself absolute, or the change of absolute space; but when the motions of bodies are considered and compared with each other, then are they relative and apparent only: they are relative, as they are compared to each other; and they are apparent only, insomuch that not their true or absolute motion, but the sum or difference of the motions only is perceivable to us.

In comparing the motions of bodies, we may consider them as moving both the same way, or towards contrary parts: in the first case, the difference of motion is only perceived by us; in the latter, the sum of the motions. Thus, for example; suppose two ships, A and B, set sail from the same port upon the same rhumb, and that A sails at the rate of five miles per hour, and B at the rate of three; here the difference of the velocity (viz. two miles per hour) is that by which the ship A will appear to go from the ship B forwards, or the ship B will appear at A to go with the same velocity backwards, to a spectator in either respectively.

If the two ships, A and B, move with the same degree of velocity, then will the difference be nothing, and so neither ship will appear to the other to move at all. Hence it is, that though the earth is continually revolving about its axis; yet as all objects on its surface partake of the same common motion, they appear not to move at all, but are relatively at rest.

If two ships, A and B, with the degrees of velocity as above, meet each other, the one will appear to the other to move with the sum of both velocities, viz. at the rate of eight miles per hour; so that in this case the apparent motion exceeds the true, as in the other it fell short of it. Hence the reason why a person, riding against the wind, finds the force of it much greater than it really is; whereas if he rides with it, he finds it less.

The reason of all these phenomena of motion will be evident, if we consider that we must be absolutely at rest, if we would discern the true or real motion of bodies about us. Thus a person on the strand will observe the ships sailing with their real velocity. A person standing still will experience the true strength and velocity of the wind; and a person placed in the regions between the planets will view all their true motions, which he cannot otherwise do, because in all other cases the spectator's own motion must be added to or subtracted from that of the moving body.

Motion is either equable or accelerated.

Equable motion is that by which a body passes over equal spaces in equal times.

Accelerated motion is that which is continually augmented or increased, as retarded motion is that which continually decreases; and if the increase or decrease of motion be equal in equal times, the motion is then said to be equally accelerated or retarded.

Equable motion is generated by a single impetus or stroke; thus the motion of a ball from a cannon is produced by the single action of the powder in the first moment; and therefore the velocity it first sets out with would always continue the same, were it void of gravity, and to move in an unresisting medium; which therefore would be always equable, or such as would carry it through the same length of space in every equal part of time.

Hence we may determine the theorems for the expressions of the time (T) the velocity (V) and the space (S) passed over in equable or uniform motion very easily thus:

If the time be given, or the same, the space passed over will be as the velocity, viz. \( S : V \); that is, with twice the velocity, twice the space; with three times the velocity, three times the space, will be passed over in the same time; and so on.

If the velocity be given, or remain the same, then the space passed over will be as the time, viz. \( S : T \); that is, it will be greater or less, as the time is so.

But if neither the time nor velocity be given or known, then will the space be in the compound ratio of both, viz. \( S : TV \). Hence, in general, since \( S : TV \), we have \( V = \frac{S}{T} \); that is, the velocity is always directly as the space, and inversely as the time. And also \( T : \frac{S}{V} \); that is, the time is as the space directly, and as the velocity inversely; or, in other words, it increases with the space, and decreases with the velocity.

If, therefore, in any rectangle, one side represent the time, and the other side the velocity, it is evident that the area of the said rectangle will represent the space passed over by an uniform motion in that time, and with that velocity.

Accelerated motion is produced by a constant impulse or power, which keeps continually acting upon the body, as that of gravity which produces the motion of falling bodies; which sort of motion is constantly accelerated, because gravity every moment adds a new impulse, which generates a new degree of velocity; and the velocity thus increasing, the motion must be quickened each moment, or the body must fall faster and faster the lower it falls.

In like manner a body thrown perpendicularly upward, as a ball from a cannon, will have its motion continually retarded, because gravity acts constantly upon it in a direction contrary to that given it by the powder; so that its velocity upwards must be continually diminished, and so its motion as continually retarded, till at last it be all destroyed. The body has then attained its utmost height, and is for a moment motionless; after which it begins to descend with a velocity in the same manner accelerated, till it comes to the earth's surface.

Since the momentum (M) of a body is compounded of the quantity of matter (Q), and the velocity (V), we have this general expression \( M = QV \), for the force force of any body A; and suppose the force of another body B be represented by the same letters in Italics, viz. \( M = QV \).

Let the two bodies A and B in motion impinge on each other directly; if they tend both the same way, the sum of their motions towards the same part will be \( QV + QV \). But if they tend towards contrary parts, or meet, then the sum of their motions towards the same part will be \( QV - QV \); for since the motion of one of the bodies is contrary to what it was before, it must be connected by a contrary sign. Or thus; because, when the motion of B confides with that of A, it is added to it; so, when it is contrary, it is subtracted from it, and the sum or difference of the absolute motions is the whole relative motion, or that which is made towards the same part. Again, this total motion towards the same parts, is the same both before and after the stroke, in case the two bodies A and B impinge on each other; because, whatever change of motion is made in one of those bodies by the stroke, the same is produced in the other body towards the same part; that is, as much as the motion of B is increased or decreased towards the same part by the action of A, just so much is the motion of A diminished or augmented towards the same part by the equal reaction of B, by the third law of motion.

In bodies not elastic, let \( x \) be the velocity of the bodies after the stroke (for, since we suppose them not elastic, there can be nothing to separate them after collision; they must therefore both go on together, or with the same celerity). Then the sum of the motions after collision will be \( Qx + Qx \); whence, if the bodies tend the same way, we have \( QV + QV = Qx + Qx \), or if they meet, \( QV - QV = Qx + Qx \); and accordingly \( \frac{QV + QV}{Q + Q} = x \), or \( \frac{QV - QV}{Q + Q} = x \).

If the body (B) be at rest, then \( V = 0 \), and the velocities of the bodies after the stroke will be \( \frac{QV}{Q + Q} = x \).

Thus if the bodies be equal (viz. \( Q = Q \)) and A with 10 degrees of velocity impinge on B at rest; then \( \frac{QV}{Q + Q} = \frac{10}{2} = 5 \times \). If \( Q = Q \), and \( V : V :: 10 : 6 \) we have \( \frac{QV + QV}{Q + Q} = \frac{16}{2} = 8 \times \), the velocity after the stroke.

If the bodies are both in motion, and tend the contrary way; then when \( Q = Q \) and \( V = V \), it is plain \( \frac{QV - QV}{Q + Q} = 2 \times \); that is, the bodies which meet with equal bulks and velocities will destroy each other's motion after the stroke, and remain at rest. If \( Q = Q \) but \( V : V :: 6 : 14 \), then \( \frac{QV - QV}{Q + Q} = \frac{-8}{2} = -4 \times \); which shews that equal bodies meeting with unequal velocities, they will, after meeting the stroke, both go on the same way which the most prevalent body moved before.

If the velocity \( \frac{QV + QV}{Q + Q} \) be multiplied by the quantities of matter \( Q \) and \( Q \), we shall have \( \frac{Q^2V + Q^2V}{Q + Q} = \) the momentum of A after the stroke;

and \( \frac{QV + QV}{Q + Q} = \) the momentum B; therefore \( QV \)

\( \frac{Q^2V + Q^2V}{Q + Q} = \) the quantity of the motion lost in A after the stroke, and consequently is equal to what is gained in B, as may be shewn in the same manner.

But since a part of this expression (viz. \( \frac{Q^2}{Q + Q} \)) is constant, the loss of motion will ever be proportional to the other part \( V = W \). But this loss or change of motion in either body is the whole effect, and so measures the magnitude or energy of the stroke. Wherefore any two bodies, not elastic, strike each other with a stroke always proportionable to the sum of their velocities (\( V + V \)) if they meet, or to the difference of their velocities (\( V - V \)) if they tend the same way.

Hence, if one body (B) be at rest before the stroke, then \( V = 0 \); and the magnitude of the stroke will be as \( V \); that is, as the velocity of the moving body A; and not as the square of its velocity, as many philosophers (viz. the Dutch and Italians) maintain.

In bodies perfectly elastic, the restituent power or spring by which the parts displaced by the stroke restore themselves to their first situation, is equal to the force impressed, because it produces an equal effect; therefore, in this sort of bodies, there is a power of action twice as great as in the former non-elastic bodies; for these bodies not only strike each other by impulse, but likewise by repulse, they always repelling each other after the stroke. But we have shewn, that the force with which non-elastic bodies strike each other is as \( V = V \); therefore the reaction of elastic bodies is the same; that is, the velocity with which elastic bodies recede from each other after the stroke, is equal to the velocity with which they approached each other before the stroke. Whence if \( x \) and \( y \) be the velocities of two bodies A and B, tending the same way after the stroke, since \( V = V = y - x \), we have \( x + V = V = y - x \); whence the motion of A after the stroke will be \( Qx \), and that of B will be \( Qx + QV - QV \); and the sum of these motions will be equal to the sum of the motions before the stroke, viz. \( Qx + Qx + QV - QV = QV + QV \). Whence, by reducing the equation, it will be \( Qx + Qx = QV - QV + 2QV \); and \( x = \frac{QV - QV + 2QV}{Q + Q} \) = the velocity of the body A.

Again, the velocity of B is \( x + V = V = \frac{QV - QV + 2QV}{Q + Q} + V - V = \frac{2QV - QV + 2QV}{Q + Q} \). Here we suppose the bodies tend the same way before the stroke; and it is evident from the equation above, that so long as \( QV + 2QV \) is greater than \( QV \), the velocity (\( x \)) of A after the stroke will be affirmative, or the body A will move the same way after the stroke as before; but when \( QV \) is greater than \( QV + 2QV \), the velocity (\( x \)) will be negative, or the body A will be reflected back.

If the body B be at rest, then \( V = 0 \); and \( x = \frac{QV - QV}{Q + Q} \), which shews the body A will go forwards or backwards, as \( QV \) is greater or lesser than \( QV \), or A greater or lesser than B. If \( Q = 3 \), \( Q = V = 10 \), and \( V = 0 \); then after the stroke the velocity of A will be \( x = \frac{QV - QV}{Q + Q} = \frac{30 - 20}{5} = 2 \), and the velocity of B will be \( y = \frac{2QV}{Q + Q} = 12 \).

If the bodies are both in motion, and \( V = 5 \), the rest is the same as before; then \( \frac{QV - QV + 2QV}{Q + Q} = 6 \) velocity of A after the stroke, and \( \frac{2QV - QV + QV}{Q + Q} = 11 \) velocity of B after the stroke.

If the bodies A and B move towards contrary parts, or meet each other, then will the relative velocity, to which the force of the stroke is proportional, be \( V + V \); and so the velocities of A and B after the stroke will be \( x \) and \( x + V + V \); and so the motion of A will be \( Qx + Qx + QV + QV \); the sum of these motions in \( Qx + Qx + QV + QV = QV - QV \) motion towards the same part before the stroke. Whence we have \( x = \frac{QV - QV - 2QV}{Q + Q} \), and therefore the velocity of B will be \( \frac{QV - QV - 2QV}{Q + Q} + V + V = \frac{2QV + QV - QV}{Q + Q} \).

If \( QV + 2QV \) be greater than \( QV \), the motion of the body A will be backwards; otherwise it will go forwards as before.

If \( Q = 3 \), \( Q = 2 \), \( V = 10 \), and \( V = 5 \); then will the velocity of A be \( \frac{QV - QV - 2QV}{Q + Q} = \frac{10}{5} = 2 \); and so the body A will go back with two degrees of velocity. The velocity of B, after the stroke, will be \( \frac{2QV + QV - QV}{Q + Q} = 13 \).

If the bodies are equal, that is, if \( Q = Q \), then \( x = \frac{-2QV}{2N} = -V \); which shows, that when equal bodies meet each other, they are reflected back with interchanged velocities; for in that case also the velocity of B becomes \( \frac{2QV}{Q} = V \).

If the bodies are equal, and one of them at rest, as B, then since \( Q = Q \), and \( V = 0 \), we have the velocity of A after the stroke \( x = 0 \); or the body A will abide at rest, and the velocity of B will be \( = V \), the velocity of A before the impulse, as appears by the example in the figure referred to.

If several bodies B, C, D, E, F, are contiguous in a right line, and another equal body A strike B with any given velocity, it shall lose all its motion, or be quiescent after the stroke; the body B which receives it will communicate it to C, and C to D, and D to E, and E to F; and because action and re-action between the bodies B, C, D, E, are equal, as they were quiescent before, they must continue so; but the body F, having no other body to re-act upon it, has nothing to obstruct its motion; it will therefore move on with the same velocity which A had at first, because it has all the motion of A, and the same quantity of matter by hypothesis.

Let there be three bodies A, B, C, and let A strike B at rest; the velocity generated in B by the stroke will be \( y = \frac{2QV}{Q + Q} \), and so the momentum of B will be \( \frac{2QV}{Q + Q} = Qy \). With this momentum B will strike C at rest and contiguous to it; the velocity generated in C will be \( \frac{2Qy}{Q + Q} \); and its momentum will be \( \frac{2Qy}{Q + Q} \).

If now we suppose B a variable quantity, while A and C remain the same, we shall find what proportion it must have to each of them, in order that the momentum of C may be a maximum, or the greatest possible, by putting the fluxion thereof equal to nothing; that is,

\[ 4Q^2C^2VQ - 4Q^2C^2Q = 0; \quad \text{whence we get} \quad QC - QC + Q + Q = 0, \]

and so \( QC = QC \); consequently \( Q : Q : Q : C \), or \( A : B : C : C \); that is, the body B is a geometrical mean between A and C. Hence if there be any number (n) of bodies in a geometrical ratio (r) to each other, and the first be A, the second will be \( rA \), the third \( r^2A \), and so on to the last, which will be \( r^{n-1}A \).

Also, the velocity of the first being \( V \), that of the second will be \( \frac{2V}{1 + r} \) (for \( \frac{2QV}{Q + Q} \) is here \( \frac{2AV}{A + rA} \)), that of the third \( \frac{4V}{1 + r^2} \), that of the fourth \( \frac{8V}{1 + r^3} \), and so on to the last, which will be \( \frac{2}{1 + r^n}V \).

The momentum of the first will be \( AV \), that of the second \( \frac{2rAV}{1 + r} \), that of the third \( \frac{4r^2AV}{1 + r^2} \), that of the fourth \( \frac{8r^3AV}{1 + r^3} \), and so on to the last, which will be \( \frac{2}{1 + r^n}AV \).

To give an example of this theorem; if \( n = 100 \), and \( r = 2 \), then will the first body A be to the last \( \frac{2}{1 + r^n} = \frac{2}{1 + 2^{100}} \approx 1 \times 10^{-7} \); nearly; and its velocity to that of the last nearly as \( 2^{100} \approx 1 \times 10^{30} \) to 1; lastly, the momentum of the first to that of the last will be nearly as 1 to \( 2^{33848} \approx 1 \times 10^{99} \).

If the number (n) of bodies be required, and the ratio of the momenta of the first and last be given as 1 to M, and the ratio of the series r given also; then, putting \( \frac{2r}{1 + r} = R \), we have the momentum of the last body expressed by \( \frac{2}{1 + r^n} = M = R^{n-1} \); therefore the logarithm of \( M \) (\( l_M \)) is equal to the logarithm of \( R \) (\( l_R \)) multiplied by the power \( n-1 \); that is, \( l_M = n-1 \times l_R \); consequently \( \frac{l_M}{l_R} + 1 = n \), the number of bodies required.