1. Dynamics is that branch of physico-mathematical science which includes the abstract doctrine of moving forces; that is, the necessary results of the relations of our thoughts concerning motion, the immediate causes of motion, and its changes.

2. Motion and its general properties are the first and principal object of mechanical philosophy. This science indeed presupposes the existence of motion; and we may consider it as universally admitted and recognized. With regard to the nature of motion, however, philosophers are greatly divided in opinion. The most obvious and simplest conception of motion is the successive application of the moving body to the different parts of indefinite space, which are considered as the place of the body. This idea of motion supposes a space whose parts are penetrable and immoveable; a doctrine directly contrary to that of the followers of Des Cartes, who regarded extension and matter as one and the same thing. To have a distinct idea of motion, it seems requisite to conceive two kinds of extension; the one, which is considered as impenetrable, and which constitutes what we properly call matter or body; the other, which being simply considered as extended, without taking any other property into account, is the measure of the distance of one body from another; and whose parts being supposed fixed and immoveable, enable us to judge of the rest or motion of bodies. We may therefore conceive bodies to be placed in indefinite space, whether real or supposed; and motion as a change in the state or condition of a body from one part of space to another. We must indeed consider motion as a state or condition of existence of a body, which would remain till it is changed by some cause; otherwise we could not have any idea of motion in the abstract. From the changes which we observe, we infer agency in nature; and in these changes we are to discover what we know of their causes.

3. In mechanical disquisitions, the simplest, and at the same time the most usual conception of space, is mere extension. We think only of the distance between two places. The path along which any body moves in passing from one place or point in space to another, is said figuratively to be the path described by that body. Space is considered by the geometer not only as having length but also breadth. In this case it is called a surface. But to have a more complete notion of the capacities of any portion of space, thickness, as well as length and breadth, is taken into consideration. This is called a solid space. By this, however, is meant only the susceptibility of measure in three ways, or extension of three dimensions. The adjacent parts or portions of space are distinguished from each other by their mutual boundaries. Contiguous portions of a line are separated by points; contiguous portions of a surface are separated by lines; and contiguous portions of a solid are separated by surfaces. The boundaries of any portions of space are not to be considered as parts of the contiguous portions. They must be conceived as common to both; as the places where one portion ends and another begins. Space cannot be said to have any bounds or limits; it is therefore said to be infinite or unbounded.

4. Any portion of space may be considered in relation to its place or situation among other portions of space. This portion of space which is occupied by any body has been called the relative place of that body. But this portion of space may be considered as a determinate portion of infinite space; and this portion of infinite space occupied by any body has been called the absolute place of that body. Space, it is obvious, taken in this meaning, is immoveable; for it cannot be conceived that this identical portion of space can be removed from one place to another. The body which occupies that space may be removed, but the space remains. We have no perception of the absolute space of any object. This may be illustrated by the motion of the earth or that of a ship. A person in the cabin of a ship does not consider the table as changing its place while it remains fixed to the same spot on the deck. While a mountain is observed to retain the same situation among other objects, few persons think that it changes its place.

5. The idea of time is acquired by means of the power of memory in observing the succession of events, time. We conceive time as unbounded, continuous, homogeneous, unchangeable in the order of its parts, and infinitely divisible. It is conceived as a proper quantity made up of its own parts, and measured by them. But as the relation of the parts of time is unknown, the only means which we can employ to discover this relation, is to find out some other relation which is more obvious and better known, to which it may be compared. We shall then have discovered the simplest measure of time, if we compare in the simplest manner possible the relation of the parts of time with those relations which are most familiar. Hence it follows, that uniform motion is the simplest measure of time. For, on the one hand, the relation of the parts of a right line is that which is most easily conceived; and, on the other hand, there are no relations more susceptible of comparison with each other than equal relations. Now, in uniform motion, the relation of the parts of time is equal to that of the corresponding parts of the line described. Uniform motion then gives us at once, both the means of comparing the relation of the parts of time with that which is most obvious to our senses, and also of making this comparison in the simplest manner. In uniform motion, then, we find the simplest measure of time. It may be added, that the measure of time by uniform motion, is independent of its simplicity, that which is the most natural to think of employing. Indeed as there is no relation with which we are acquainted more accurate than that of the parts of space; and, as in general, a motion, the law of which is given, would lead us to discover the relation of the parts of time, by the known analogy with that of the parts of space passed over, it is evident that such a motion would be the most accurate measure of time, and

Motion and that which ought to be employed in preference to every other. In the actual measurement of time, some event which is imagined always to require an equal time for its accomplishment is selected; and this time is employed as a unit of time or duration, in the same way as a foot rule is employed as a measure of extension. During any observed operation, as often as this event is accomplished, so often is it supposed that the time of the operation contains this unit. While a heavy body falls 16 feet, a pendulum, 39½ inches long, makes one vibration; but it makes three vibrations, while the same body falls 144 feet. It is therefore said that the time of a body falling 144 feet, is thrice as great as the time of falling 16 feet.

PART I. OF MOTION.

BEFORE we enter on the consideration of the different kinds of motion, it may be necessary to notice some general circumstances regarding it.

7. It is impossible to conceive that any motion can be instantaneous. A moving body, in passing from the beginning to the end of its path, must pass through all the intermediate points. Now to suppose the motion along even the most minute portions of the space passed through instantaneous, is to suppose that the moving body is in every intervening point at the same instant; which is impossible.

8. Relative motion is the change of situation with regard to other objects. Absolute motion is the change of absolute place. These two motions, it may be observed, may not only be different, but even contrary to each other. From the relative motions of things which are the differences of their absolute motions, we cannot find out what are the absolute motions. It is often a subject of elaborate and intricate investigation to discover and determine the absolute motions, by means of observing the relative motions.

9. The affections or circumstances of motion are various with regard to its quantity and its direction. That affection of motion by which the quantity is determined, is called velocity. The length of the line, which is uniformly described or passed over during some given portion or unit of time, is the proper measure of this velocity. When a ship sails six miles per hour, she describes a length of line equal to six miles in the space of a given portion or unit of time, namely, the hour; and thus the velocity of the ship is said to be ascertained.

10. Another affection or circumstance of motion is its direction. This is the position of the straight line along which the motion is performed. The straight line which a body describes or tends to describe is called its direction. The motion is said to be in the direction AB fig. 1, when the body moved passes along the line AB from A to B. In common language, it is not unusual to express the direction of motion in a manner quite the reverse of this. We have an instance of this kind in speaking of the direction of the winds. A current of air or wind which moves eastward is said to be a westerly wind, deriving its name from the point or quarter from which it proceeds, not as in other cases, and in strict expression, from the point to which it is directed.

6. Between the affections of time and space, there is an obvious analogy; and hence in most languages the same words are employed to express the affections between both. Thus it is that time may be represented by the lines and measured by motion; since uniform motion is of the simplest succession of events that can be conceived, time and space. In the order of situation, all things are placed in space, in the order of succession all events happen in time.

Having made these preliminary observations, we propose to divide the following treatise into two parts. In the first, we shall consider motion in general. In the second, we shall treat of moving forces, or of dynamics.

11. Motions are of different kinds. They are either rectilinear, rectilineal, deflected, or curvilinear. In a rectilinear motion, the direction remains unchanged during the whole time that the motion is continued, as when a body moves from A to B fig. 1. In a deflected motion, it is performed along two contiguous straight lines in succession. Thus if a body moves from A to B fig. 2, and at the point B its direction is changed from that of AD to BC; this change has been called deflection, the quantity of which may be measured either by the angle DBC, or by a line DC drawn from the point D to which the body would have arrived in the same time, if its motion had remained unchanged, in which it has actually reached the point C. When a body in moving along describes the sides of a polygon, the deflections are repeated, with the intervention of undeflected motions. In curvilinear motion the deviation and deflection are supposed to be continual. Continual deflection therefore constitutes curvilinear motion. Let the motion be performed along a curve line ABCDE (fig. 3.), the direction is continually changing. When the body is in the point C the direction is that of the tangent CF; because this direction alone lies between any pair of polygonal directions, such as CE and CA, or CB and CD, however near the points A and E, or B and D, are taken to the point C.

12. Motions have been divided into uniform motions, division of variable, compound, and curvilinear. These we shall consider separately in the following sections.

SECT. I. Of Uniform Motion.

13. It is of great importance in mechanical disputes, to have the characters of uniform or unchanged fixed motion fixed. For in our conceptions of motion in general, in which we do not turn the attention to its alterations, the motion is supposed to be equable and rectilineal. By the deviations from such motion only can we determine the marks and measures of all changes; and hence also we are to obtain the measures of all changing causes, or in other words of the mechanical powers of nature.

PROPOSITION I.

14. In uniform motions, the velocities are in the proportions of the spaces described in the same or in equal times; times; or as it is sometimes expressed, The velocities are proportional to the spaces described in equal times.

The spaces described are the measures of the velocities, and things are proportional to their measure. Let the spaces described in the time $T_s$ be represented by $S$ and $s$, and let the velocities be represented by $V$ and $v$. We have the analogy $V : v = S : s$. Or, as it may be expressed by the proportional equation

$$\frac{v}{s} = \frac{V}{S}.$$

**Prop. II.**

15. In uniform motions with equal velocities, the times are in the proportion of the spaces described during their currency. Or, as it is also expressed, The times are proportional to the spaces described with equal velocities.

For in uniform motions, equal spaces are described in equal times. The successive portions of time therefore are equal, in which equal spaces are described in succession; and the sums of the equal times must be proportional to the corresponding sums of equal spaces. In all cases, therefore, which are susceptible of being represented by numbers, this proposition is evident. And it may be extended to all other cases, in a way similar to that in which Euclid has demonstrated that triangles of equal bases are in the proportion of their sides.

16. As proportion can only take place between quantities of the same kind, all that is to be understood by the expressions in the above propositions, which are far from being accurate, is, that the proportions of the velocities and the times are the same with the proportions of the spaces. For as space and time are quantities of a different nature, it is evident that we cannot divide space by time. Thus when it is said that the velocities are as the spaces divided by the times, it is an abridged mode of expression, which signifies that the velocities are as the relations of the spaces to the same common measure, divided by the relations of the times to the same measure. Thus, for example, if we take a foot for the measure of the spaces, and a minute for the measure of the times, the velocities of two bodies which move uniformly, are to each other as the number of feet described, divided by the number of minutes which the bodies require to describe the portion of space passed through, and not as the feet divided by the minutes.

17. Hence it is that uniform motion is universally employed as a measure of time. But it is often difficult to find out whether the motion which is proposed for the measure of time be perfectly uniform. What then are the means to ascertain this? To this it may be answered that there is no motion which is not uniform, the law of which we can determine exactly; so that this difficulty only proves that we cannot ascertain the relation of the parts of time with mathematical precision; but it does not follow that uniform motion from its nature may not be the first and simplest measure. And having no strictly accurate measure of time, we endeavour to discover the measure which comes nearest in the motions which approach nearest to uniformity.

18. There are three ways by which it may be ascertained that a motion is nearly uniform. 1. When the moving body describes equal spaces in times which we judge to be equal; and we can determine that the times are equal, after having observed from repeated experience that similar events take place in the same times. Thus we conclude that the times which the same clepsydra requires to be emptied are equal; so also the times in which the same quantity of sand runs in the sandglass; the times in which the shadow moves over the same space on the sundial; the times of the same number of vibrations of a pendulum of the same length; and the times of the revolution of the heavenly bodies through the same spaces are equal. If then it is found by observation that a body during the same times passes over equal spaces, we conclude that the motion is uniform. 2. Another method of ascertaining how far any motion is uniform, is when the effect of the accelerating or retarding cause, if such operate, is imperceptible. It is by combining these two methods that we conclude the motion of the earth round its axis to be uniform; and this inference is not only not opposed by any of the celestial phenomena, but seems to be in perfect accord with them. 3. By a third method of determining the uniformity of any motion, we compare it with others; and when the same law is observed in both the one and the other, we may conclude that the motion compared is uniform. Thus if several bodies move at such a rate that the spaces described in the same time are always to each other, either precisely or very nearly so, in the same ratio, the motion of these bodies, we conclude, is either precisely, or at least very nearly uniform. For if a body A which moves uniformly passes through the space E during the time T taken at pleasure, and another body B also moving uniformly, passes through the space e; during the same time T, the relation of the spaces E, e will be always the same, whether the two bodies have begun to move in the same or in different instants; and it is only to uniform motion that this property belongs. Wherefore if we divide the time into parts, whether equal or unequal, and if it be observed that the spaces passed through by two bodies during one part of the time, are always in the same relation, the greater the number the parts of the time taken, the more there is reason to conclude that the motion of each body is uniform. None of these methods, it has been observed, possesses geometrical precision; but they are sufficient, especially when they are repeated and taken together, to afford a satisfactory conclusion, if not with regard to absolute uniformity of motion, at least with regard to a near approximation to uniform motion.

**Prop. III.**

19. In uniform motions, the spaces described are in the compound ratio of the velocities and the ratio of the times. This proposition is frequently expressed otherwise thus; The spaces described with an uniform motion are proportional to the products of the times and the velocities: Or otherwise thus; The spaces described with a uniform motion are proportional to the rectangles of the times and the velocities.

For let $S$ be the space described with the velocity $V$, in the time $T$, and let $s$ be the space described with the velocity $v$, in the time $t$. Let another space $Z$ be described in the time $T$ with the velocity $v$.

Then by proposition 1st we have $S : Z = V : v$,

And by proposition 2nd $Z : s = T : t$. Sect. II.

Variable Motions.

By composition of ratios therefore (or by VI. 23, Euclid), we have \( S \times T : v \times t = S \times Z : s \times Z \); that is,

\[ S : s = S : s. \]

The above are all equivalent expressions which are demonstrated by the same composition of ratios. The products or rectangles of the times and velocities, are the products of numbers which are as the times, multiplied by numbers which are as the velocities; or the rectangles whose bases are as the times, and whose heights are as the velocities.

COROLLARY.

20. If the spaces described in two uniform motions be equal, the velocities are in the reciprocal proportion of the times.

For in this case the products \( VT \) and \( vt \) are equal, and therefore \( V : v = T : t \), or \( V : v = \frac{1}{T} : \frac{1}{t} \). Or, because the rectangles \( AC, DF \) (fig. 4,) are in this case equal, we have (by VI. 14, Euclid) \( AB : BF = BD : BC \), that is \( V : v = T : t \).

PROP. IV.

21. In uniform motions, the times are as the spaces, directly, and as the velocities, inversely.

For by Prop. III. \( S : s = VT : vt \);

Therefore, \( S : vt = s : VT \),

And, \( T : t = S : sV \).

Or, \( T : t = \frac{S}{s} : \frac{s}{v} \),

And, \( t : t = \frac{s}{v} : \frac{s}{v} \).

PROP. V.

22. In uniform motions, the velocities are as the spaces, directly, and as the times, inversely.

For by Prop. IV. \( S : s = VT : vt \),

Therefore \( V : v = S : sT \).

Or, \( V : v = \frac{S}{s} : \frac{s}{t} \),

And \( v : v = \frac{s}{t} : \frac{s}{t} \).

23. The values of the results of these propositions are not changed by the absolute magnitudes of the space and time, if both are changed in the same ratio.

The value of 12 feet, or of \( \frac{8}{24} \) feet, is the same with half a foot per second. Therefore, if \( s' \) be the expression of an extremely minute portion of space described with this velocity in the small portion of time \( t' \), the velocity \( v \) is still accurately expressed by \( \frac{s'}{t'} \).

And the accurate expression of the time \( t' \) is \( \frac{s'}{v} \).

Sect. II. Of Variable Motions.

24. In observing the phenomena of nature, it rarely happens that the motions to which our attention is directed are perfectly uniform. These motions, however, we distinctly conceive, with all their properties; and it is obviously of the utmost importance that all the deviations from uniform motions be clearly understood; because these deviations afford the only marks and measures of the variations, and therefore of the causes which produce these changes.

25. When a body continues to move uniformly in the same direction, its motion, or circumstances with respect to motion, have suffered no change. The condition of that body, therefore, must be allowed to be the same in any two portions of its path, whatever the distance of these portions may be. And because a change of place is involved in the very conception of motion, the difference of place does not imply any change. Two bodies, therefore, moving with the same velocity in this path, or in two lines parallel to it, their condition in respect of motion must be allowed to be the same. Their direction is the same, and their rate of motion is the same. The velocity, therefore, and direction of a body, are the only circumstances which seem to enter into our conception of the state of a body, in respect of motion. Changes either in the velocity, or in the direction, or in both of these circumstances, include all the changes of which this condition is susceptible. Let us now consider the first of these changes, namely, changes of velocity.

Of Accelerated and Retarded Motions.

26. It has been ascertained by experiment and observation, that a stone in falling is carried downward with greater rapidity in every succeeding period of its fall. During the first second it falls 16 feet; during the next it falls 48 feet; during the third, it falls 80 feet; during the fourth it falls 112 feet; continuing to fall, during every succeeding second 32 feet more than during the preceding second. A body moving in this manner is said to have an accelerated motion. But if a body be projected perpendicularly upwards, the very reverse takes place in the circumstances of its motion. It is observed to rise with a motion which is continually retarded. These bodies therefore are conceived to be in every succeeding instant in different states of motion. The velocity of the falling body is conceived to be greater in a certain instant than in any preceding instant; as, for example, when it has fallen 144 feet its velocity is said to be thrice as great as when it has fallen only 16 feet. But this inference it is evident cannot be made directly by comparing the spaces described in the following moments; for in these it falls 112 and 48 feet; or by comparing the spaces immediately preceding; for in these the body fell 80 and 16 feet. But in this expression it is supposed that the variable condition of a body, called its velocity, is, in every instant susceptible of an accurate measure; and yet in no moment, however short, does the body describe uniformly a space which can be taken as the measure of its velocity at the beginning of that moment; because the space described in any moment is too great for measuring the velocity at the beginning of the moment, and too small for the measure of its velocity at the end of it. Till however such a measure is obtained, the mechanical condition of the body is not known.

27. But in a continually accelerated motion, no such measure can be obtained. No space is describ- ed in an instant: for this requires time. In that instant, however, the body possesses what has been called a potential velocity, that is, a certain tendency or determination, which remaining unchanged, causes it to describe a certain space uniformly during some assignable portion of time. At another instant it has another determination, by which, if it be not changed, another space would be uniformly described in an equal portion of time. Now it is in the difference of those two determinations that its difference of mechanical condition consists. The marks and measures of these determinations are known from the spaces which would be uniformly described. These therefore must be carefully investigated as the measures of the velocities; and the proportions of these spaces are to be taken as the proportions of the velocities.

PROP. VI.

28. Let the straight line ABD (fig. 5.) be described with a motion continually varied, it is required to determine the proportion of the velocity in the point A, to the velocity in any other point C.

Let the right line abd, represent the time of this motion along the path AD, so that the points a, b, c, d, may denote the instants of the moving body being in A, B, C, D, and the portions ab, bc, cd, may express the times of describing AB, BC, CD, that is, may be in the proportion of those times; and let ae, perpendicular to ad, express the velocity of the moving body at the instant a, or in the point A. Let egh be a line, so related to the axis ad, that the areas abfe, bcgf, cdhg, comprehended between the ordinates ae, bf, cg, dh, all perpendicular to ad, may be proportional to the spaces AB, BC, CD, described in the times ab, bc, cd, and let this relation hold in every part of the figure. Then the velocity in A is to the velocity in B, or C, or D, as ae to bf, or cg, or dh. Or it may be expressed in other words, If the abscissa ad, of a curve egh, be proportional to the time of any motion, and the areas interrupted by parallel ordinates be proportional to the spaces described, the velocities are proportional to those ordinates.

Make be and cd equal, so as to represent very small and equal moments of time, and make pa equal to one of them. Complete the rectangle paeq. This will represent the space uniformly described in the moment pa, with the velocity ae (Propof. 3.) Let PA be that portion of space thus uniformly described in the moment pa. Let the lines im, kn, parallel to ad, make the rectangles bemi, and cdnk, respectively equal to the areas bcgf, and cdhg. If the motions along the spaces PA and BC had been uniform, the velocities would have been proportional to the spaces described (Propof. 1.) because the times pa, and bc are equal. That is, the velocity in A would be to the velocity in C, as the rectangle paeq to the area bcgf, that is, as paeq to bemi, that is, as the base ae to the base cm, because the altitudes pa and bc are equal.

But the motion along the line BC is not represented as uniform; for the line fg diverges from the axis bd, the ordinate cg being greater than bf. And therefore the spaces measured by these areas increase faster than the times; and thus the figure represents an accelerated motion. Therefore the velocity with which BC would be uniformly described during the moment bc, is less than the velocity at the end of that moment, that is, at the instant c, or in the point C of the path; and therefore it must be represented and measured by a line greater than cm.

In the same manner it is proved that ck represents and measures the velocity with which CD would be uniformly described during the moment cd. And therefore, since the motion along CD is also accelerated, the velocity at the beginning of that moment is less than the velocity with which it would be uniformly described in the same time, and must be represented by a line less than ck.

Therefore the velocity in A is to that in C, in a less ratio than that of ae to cg, but in a greater ratio than that of ae to ck. But in this case, as long as the instant b is prior, and posterior, to the instant c, cm is less, and ck is greater, than cg. Therefore the velocity in A is to that in C in a ratio that is greater than any ratio less than that of ae to cg. And, consequently the velocity in A is to that in C, as ae to cg.

It may be proved in the same way, with respect to the velocity in any other point D; and therefore the proposition may be considered as demonstrated. And had the motion along BCD, instead of being accelerated as in this case, been retarded, the same reasoning would still apply.

COROLLARIES.

29. Cor. 1. The velocities in different points of the path AD, are in the ultimate ratio of the spaces described in equal small moments of time. Draw go parallel to ad. Then the velocity in the instant a, is to that in the instant c, as ae to cg, that is, as the rectangle pe to the rectangle eo, that is, as paeq to cdhg, nearly. As the moments are diminished, the difference goh between the rectangles cgod and cgdh, diminishes nearly in the duplicate ratio of the moment. If then the moment be taken \( \frac{1}{2} \), or \( \frac{1}{4} \) of cd, the error goh is diminished to \( \frac{1}{8} \), or \( \frac{1}{16} \): the corollary is now manifest; for the ultimate ratio of cgod to cgdh is the ratio of equality. That is, the velocity in A is to that in C, in the ultimate ratio of PA to BC described in equal final moments.

There are many cases in which the spaces described in very small moments can be measured, and yet the ultimate ratio cannot be ascertained. These spaces must then be taken as measures of the velocity. And by taking half the sum of the spaces BC and CD, for the measure of the velocity in the point C, the error is almost reduced to nothing.

30. Cor. 2. The momentary increments of the spaces described, are in the compound ratio of the velocities, and the ultimate ratio of the moments.

For the increments PA, CD are as the rectangles paeq and eo ultimately, (Propof. 3.) and these are in the compound ratio of the base ae, to the base do, and the ultimate ratio of the altitude pa, to the altitude cd. This may be expressed by the proportional equation \( s = vt \).

31. Consequently \( v = \frac{s}{t} \), and \( t = \frac{s}{v} \). The equation Sect. II.

Variable Motions.

\[ s = vt \]

and

\[ t = \frac{s}{v} \]

seem to be the same with those in (23), but there the small space \( s' \) was described uniformly, and the equations were absolute. In 30 and 35, \( s \) does not represent a space uniformly described. But \( S : s \) expresses the ultimate ratio of \( S' \) to \( s' \) when they are diminished continually, and vanish together. Therefore the meaning of the equation

\[ s = vt \]

is that the ultimate ratio of \( S' \) to \( s' \) is the same with that of \( VT' \) to \( vt' \).

32. The following is the converse of this proposition.

If the abscissa \( a d \) of the line \( e f h \), represent the time of a motion along the line \( ABD \), and if the ordinates \( a c, b f, c g, \ldots \) be as the velocities in the points \( A, B, C, \ldots \), then the areas are as the spaces described. This is proved by an indirect demonstration, thus:

For if the spaces \( AB, AD \), be not proportional to the areas \( a b f e, a d h e \), they must be proportional to some other, \( a b f' e, a d h' e \), of another line \( e f' h' \), passing through \( e \). Assuming this to be true, then (by Propof. 6.) the velocity in \( A \) is to that in \( B \), as \( a c \) to \( b f \). Therefore \( a c : b f = a e : b f \), which is absurd.

33. The relation between the space described and the time which elapses is the only immediate observation to be made on these variable motions. By means of the foregoing propositions, the mechanical condition of the body, or rather the effect and measure of this condition, denominated velocity, is inferred. The same inference is made in another way. Sir Isaac Newton often represents the uniform lapse of time by the uniform increase of an area during the motion along the line taken for the abscissa. The velocities or determinations to motion in the different points of this line, are inversely proportional to the ordinates of the curve which bounds this area.

Along the straight line \( AD \), (fig. 6.) let a point move with a motion any how continually changed, and let the curve line \( LKH \) be so related to \( AD \), that the area \( LICB \) is to the area \( LHDB \) as the time of moving along \( BC \) to that of moving along \( BD \). Let this be true in every point of the line \( AD \). Let \( Cc, Dd \) be two very small spaces described in equal times, draw the ordinates \( i c, h d \), and draw \( ik, hl \) perpendicular to \( IC, HD \).

The areas \( IC \times i \) and \( HD \times h \) must be equal, because they represent equal moments of time. It is evident also, that as the spaces \( Cc \) and \( Dd \) are continually diminished, the ratio of \( IC \times i \) and \( HD \times h \) to the rectangles \( kC \times i \) and \( lD \times h \) continually approximates to that of equality, and that the ratio of equality is the limiting or ultimate ratio. Since therefore, the areas \( IC \times i \) and \( HD \times h \) are equal, the rectangles \( kC \times i \) and \( lD \times h \) are ultimately in the ratio of equality. Therefore their bases \( i c \) and \( h d \) are inversely as their altitudes \( Cc \) and \( Dd \), that is, \( i c : h d = Dd : Cc \). But as \( Cc \) and \( Dd \) are described in equal times, they are ultimately as the velocities in \( c \) and \( d \) (29). Therefore \( i c \) and \( h d \), are inversely as the velocities in \( c \) and \( d \). And as the same reasoning may be applied to every point of the abscissa, the proposition is demonstrated.

34. In all cases, then, in which the relation between the spaces described, and the times elapsed can be discovered by observation, we discover the mechanical condition of the moving body, or its velocity. But Variable Motions.

In the practical application of these conclusions, recourse must always be had to arithmetical conclusions; and the indications of these are the algebraic symbols of geometrical reasonings. Thus any ordinate \( c g \), (fig. 5.) is represented by \( v \), and the portion \( cd \) of the abscissa by \( i \), and the area \( cd \times g \), or its equal, the rectangle \( cd \times g \), by \( vi \). This rectangle then being as the corresponding portion \( CD \) of the line of motion, and \( CD \) being represented by \( s \), we have the equation

\[ s = vt. \]

35. The mathematical consequences of these representations may now be assumed to be true; and therefore \( i = \frac{s}{v} \), as in (23.) Algebraic symbols being the representations of arithmetical operations, they represent more remotely the operations of geometry, and only because the area of a rectangle is analogous to the product of numbers which are proportional to its sides.

The symbol \( \int vt \) being used to represent the sum of all these rectangles, expresses the whole area \( adhe \), as well as the whole line of motion \( AD \); and the equation may be stated \( s = \int vt \). In like manner \( \int \frac{s}{v} \) will be equivalent to \( \int i \), that is, to \( i \), and will express the whole time \( ad \). It is plain too that \( \frac{s}{v} \) represents the ordinate \( DH \) of the line \( LKIH \) (fig. 6.) because any portion \( Dd \) of its abscissa, is properly represented by \( s \), and the ordinates are reciprocally proportional to the velocities, that is, are proportional to the quotients of some constant number divided by the velocities, and therefore to \( \frac{i}{v} \). And as \( i \) is represented by the rectangle \( kC \times i \), which is also represented by \( s \times \frac{i}{v} \), we have \( i = \frac{s}{v} \) and \( t = \frac{s}{v} \), as above.

36. In one case of varied motion, when the line \( efgh \) (fig. 5.) is a straight line, the characters are very particular and useful. Let this case of motion be represented along the line \( AD \) (fig. 7.) and let \( pa, b c, c d \) represent equal moments of time, in which the moving body describes \( PA, BC, CD \); and draw \( fm, gn, es \), parallel to the abscissa \( ad \). Now it is evident that \( mg \) and \( nh \) are equal, or that equal increments of velocity are acquired in equal times; \( eg, er, es \) are also proportional to \( gf, fr, fh \), and therefore the increments \( gf, fr, fh \), of velocity are proportional to the times \( ab, ac, ad \), in which they are acquired. This motion may very properly be denominated uniformly accelerated; for here the velocity increases in the same ratio with the times, and equal increments are acquired in equal times. If the line \( eh \) cut the abscissa in \( v \), it will represent a motion uniformly accelerated from rest, during the time \( vd \), and thus exhibit the relations between the spaces, velocities, and times in such motions.

Hence it follows from this mode of expressing these relations, relations that in motions uniformly accelerated from a state of rest, the acquired velocities are proportional to the times from the beginning of the motion. For \(a, b, c, d, e\), represent the velocities gained during the times \(v, a, v, b, v, c, v, d\), and are in the same proportion with those lines.

37. Also, the momentary increments of velocity, are as the moments in which they are acquired.

2. Also, the spaces described from the beginning of the motion, are as the squares of the times.

3. Also, the increments of the spaces are as the increments of the squares of the times; reckoning from the beginning of the motion.

4. Also, the spaces described from the beginning of the motion, are as the squares of the acquired velocities.

5. Also, the momentary increments of the spaces are as the momentary increments of the squares of the velocities.

6. Also, the space described during any portion of time by a motion uniformly accelerated from rest, is one-half of the space uniformly described in the same time with the final velocity of the accelerated motion.

7. And the space described during any portion of the time of the accelerated motion, is equal to that which would be described in the same time with the mean between the velocities at the beginning and end of this portion of time.

In the investigation of all other varied motions, the properties of uniformly accelerated motion stated above, will be found extremely useful, and especially in cases where approximation only can be easily obtained. But for the fuller illustration of these properties the reader is referred to Robison's Elements of Mechanical Philosophy, p. 38.

38. Supposing the acceleration to be always the same, we conceive of this constancy, that in equal times there are equal increments of velocity; and therefore that the augmentations of velocity are proportional to the times in which they are acquired. That acceleration then, according to this supposition, must be accounted double, or triple, &c., where the velocity acquired is double or triple. And, acceleration being considered as a measurable quantity, the augmentation of velocity uniformly acquired in any given time is its measure.

COROLLARY.

39. Therefore, accelerations are proportional to the spaces described in equal times, with motions uniformly accelerated from a state of rest. For in this case the spaces are the halves of what would be uniformly described in the same time with the acquired final velocities, and are therefore proportional to these velocities, or to the accelerations, since the velocities were acquired in equal times.

40. It is then said, that accelerations are proportional to the increments of velocity uniformly acquired, directly, and to the times in which they are acquired inversely.

\[A : a = \frac{V}{T} : \frac{v}{t}\]

This relation between acceleration, velocity, and time, is also true, in uniformly accelerated motion, with respect to all momentary changes of velocity, as well as to those cases of motion passing through all degrees of velocity from nothing to the final magnitude \(v\). For the velocity increasing at the same rate with the time, we have \(v : v' = t : t'\); and \(v\) and \(v'\) express the simultaneous increments of velocity and time.

41. But if the augmentation of velocity be the measure of the acceleration, and therefore proportional to it, and if in uniformly accelerated motions, the velocity increases at the same rate with the times, the increments of velocity are as the accelerations and as the times jointly. Hence the proportional equation

\[\frac{v}{v'} = \frac{a}{a'}\]

and

\[\frac{v}{v'} = \frac{a}{a'}.\]

42. It appears from (39.), that when the velocity has uniformly increased from nothing, the spaces described in equal times are proper measures of acceleration. And in (37.—3.) uniformly accelerated motions, the spaces are as the squares of the times. Therefore, when the acceleration continues the same, the fraction \(\frac{s}{t^2}\) must also remain of the same value, and \(a\) is proportional to \(\frac{s}{t^2}\). And therefore, accelerations are proportional to the spaces described with a motion uniformly accelerated from rest, directly, and to the squares of the times inversely.

43. And since \(a = \frac{v}{t}\), we have \(a = \frac{v}{v'}\); but \(v = \frac{s}{t}\),

therefore \(a = \frac{v^2}{s}\). Therefore we have another measure of acceleration, viz. Accelerations are directly as the squares of the velocities, and inversely as the spaces along which the velocities are uniformly augmented.

44. But when the spaces are equal, we have \(a = \frac{v^2}{s}\), and in uniformly accelerated motions, that is, when \(a\) remains constant, the space being increased in any proportion, \(v^2\) increases in the same proportion; it follows that \(v^2\) increases in the proportion both of the acceleration and of the space. And therefore, in general, we have, \(v^2 = a s\). And, as in 41, 42, we shall have \(v^2 = a s\), and \(v^2 = a s - a s'\), or \(S = S' - s\), which may be thus expressed \(v v' = a s'\), that is, in a motion uniformly accelerated, the momentary change of the square of the velocity is proportional to the acceleration and to the space jointly. Thus it appears, that the acceleration continued during a given time \(t\), or \(t'\), produces a certain augmentation of the simple velocity; but the acceleration continued along a given space \(s\), or \(s'\), produces a certain augmentation of the square of the velocity.

45. But accelerations which are constant and uniform, and such as have been considered, are very rare in the phenomena of nature. They are as variable as velocities, and therefore it is not less difficult to discover their actual measure. By changes of velocity only we obtain any knowledge of the changing cause. From the continual acceleration of a falling body we learn, that the same power which makes it press on the hand, presses it downward, as it falls through the air; and whatever be the rapidity of its descent, it is from observing that it acquires equal increments of velocity in equal times, that we know the downward pressure to be the same. In the same way that we obtain measures of a velocity which is continually varying, we may obtain accurate measures of a similarly varying acceleration. A line may be conceived to increase along with the velocity, and at the same rate; and this rate of increase of velocity is what is called acceleration, in the same way as the rate at which the line increases, is what is called velocity. If, then, we consider the areas (fig. 5.) or the line AD, as representing a velocity; the ordinates to the line eg h, which were demonstrated to be proportional to the rate of variation of the area, will be proportional to the variation of the velocity, that is, to the acceleration.

Prop. VII.

46. If the abscissa ad of a curve line egh represent the time of a motion, and if the areas abfe, acge, adhe, &c. are proportional to the velocities at the instants b, c, d, &c., then the ordinates ae, b f, cg, dh, &c. are proportional to the accelerations at the instants a, b, c, d, &c.

By substituting the word acceleration for the word velocity, the same demonstration may be applied here as in Prop. 6. (28.) From this proposition may be deduced some corollaries of practical use in mechanical discussions.

47. The momentary increments of velocity are as the accelerations, and as the moments jointly.

For the increment of velocity in the moment cd is accurately represented by the area cdhg, or by the rectangle cdnk; and cd accurately represents the moment. Also, the ultimate ratio of cd to such another ordinate bi, is the ratio of eg to bf; that is, the ratio of the acceleration in the instant c to that in the instant b. And therefore the increment of velocity during the moment pa is to that during the moment cd as pa × ae to cd × dg. Or it may be expressed by the proportional equation v = ai.

48. Conversely. The acceleration a is proportional to v, as in the case when the motion is uniformly accelerated (46.)

And as the area of this figure is analogous to the sum of all the inscribed rectangles, when the circumstances of the case admit of its being measured, it may be expressed by fat; and thus is obtained the whole velocity acquired during the time AC, and we say v = fat.

The intensities (or at least their proportions) of the accelerating power of nature in the different points of the path being frequently known, we wish to discover the velocities in those points. This may be done by the following proposition.

Prop. VIII.

49. If the abscissa AE (fig. 8.) of a line ace be the space along which a body moves with a motion continually varied, and if the ordinates Aa, Bb, Cc, &c. be proportional to the accelerations in the points A, B, C, &c., then the areas ABba, ADda, AEea, &c. are proportional to the augmentations of the square of the velocity in A at the points B, D, E, &c.

Take BC, CD, as two very small portions of the line AE, and draw bf, cg, parallel to AE. Then, supposing the acceleration Bb, to continue through the space BC, the rectangle BbfC will express the augmentation made on the square of the velocity in B. In the same way CcgD will express the augmentation of the square of the velocity in C; and, in like manner, the rectangles inscribed in the remainder of the figure will express the increments of the squares of the velocity acquired, while the body moves over the corresponding portions of the abscissa. And, therefore, the whole augmentation of the square of the velocity in A (should there be any velocity in that point) during the time of moving from A to B, will constitute the aggregate of these partial increments. The same thing must be affirmed of the motion from B to E. And, when the subdivision of AE is carried on without end, it is plain that the ultimate ratio of the area AEea to the aggregate of inscribed rectangles, is that of equality; that is, when the acceleration varies continually, the area ABba will express the increment made on the square of the initial velocity in A, while the body moves along AB; and the same must be affirmed with respect to the motion along BE. And, therefore, the intercepted areas ABba, BDdb, DEed, are proportional to the changes made on the squares of the velocities in the points A, B, and D.

Corollaries.

50. Cor. 1. If the body had no velocity in A, the areas ABba, ADda, &c. are proportional to the squares of the velocity acquired in the points B, D, &c.

Cor. 2. The momentary change on the square of the velocity, is as the acceleration and increment of the space jointly; or we have v = ai.

Cor. 3. v being equal to half the increment of the square of the velocity, it follows that the area AEea, or the fluent fat is only equal to \(\frac{V^2 - v^2}{2}\), taking v and V as the velocities in A and E.

51. What has now been said of the acceleration of motion, is equally applicable to motions that are retarded, whether these motions be uniform or unequal. The momentary variations in this case are to be taken as decrements of velocity instead of increments. A moving body, subject to uniform retardation till it come to rest, will continue in motion during a time proportional to the initial velocity; and describe a space proportional to the square of this velocity; and the space which is so described, is one half what it would have been if the initial velocity had continued undiminished.

Sect. III. Of Compound Motions.

52. Having obtained the marks and measures of every variation of velocity, we are now to discover the changes of direction in similar characteristics for every change of direction. In the above investigation of the general marks of any change of motion, it is plain that the change being the same in any two or more instances, the sensible marks must also be the same, whatever may have been the previous previous condition of the moving bodies. In every case of change, some circumstance in the difference between the former motions and the new motions must be observed, which is exactly the same both in respect of velocity and direction. One of the bodies then may be supposed to have been at rest; and thus the change produced on it, is the motion which it has acquired, or the determination to this motion. Therefore, a change of motion is itself a motion, or determination to motion.

In the above case, it is the new motion only; but it is not the new motion in every other case. For supposing the previous condition of the body to have been different from that of a body at rest, and supposing the same change produced on it, the new condition of the one body must be different from the new condition of the other. The change, therefore, being the same in both cases, the new condition cannot be that change. But, when the same change happens in any previous motion, the difference between the former motion and the new motion, must indicate something that is equivalent to the motion produced in a body previously at rest, or the same with that motion, this body having received the same change. And the difference between the new motions of the two bodies will be such as shall indicate the difference between the previous conditions of each body. The change of motion then is itself a motion; and this being assumed as a principle, we are now to endeavour to discover a motion, which alone shall produce that difference from the former motion, which, in all cases, is observed in the new motion. This is to be considered as the proper characteristic of the change.

The following motions may serve as an illustration of these conditions. Let it be supposed that the straight line AB (fig. 9.) lies east and west, and that it is crossed by the line AC from north to south. Suppose this line AC to be a rod or wire, and to be carried along the line AB in one minute, but always in the same position, that is, lying north and south. The end of the rod or wire A having moved uniformly one-third of AB at the end of 20", it will be in the position D d; at the end of 40" it will have the position E e; and at the end of the minute it will be in the position B b.

Let the line AB, in the mean time, (supposing it also to be material) be uniformly moved from north to south, and always parallel to its first position AB. When it has passed over one-third of AC, at the end of 20", it will be in the position m n; at the end of 40" it will have the position o p, and Ao is two-thirds of AC. At the end of the minute, it will have the position C c b. It is evident that the common intersection of these two lines will be always in the diagonal A b of the parallelogram AC b B; for the parallelogram A m D is similar to the parallelogram A C b B, because AD : AB = Am : AC; and, in like manner, Ao : E is a parallelogram similar to AC b B. Therefore, these parallelograms are about a common diagonal A b.

Again, the motion of the point of intersection of these lines is uniform; for AD : AB = A d : A b, and AE : AB = A e : A b, &c. Therefore the spaces A d, A e, A b are proportional to the times.

Thus the intersection of two lines having each a uniform motion in the direction of the other, moves uniformly in the direction of the diagonal of the parallelogram, which is formed by the lines in their first or last position; and the velocity of the intersection is to the velocity of each of the motions of the lines as the diagonal is to the side in the direction of which the motions are made. This motion of the intersection is very properly said to be compounded of two motions in the direction of the sides; for which the point d of the line D d moves eastward, the same point d of the line m n is at the same instant moving southward. The point d, therefore, may be considered as a point of both lines, partaking in every instant of both motions. The motion along A b then contains both motions along A B and A C, and being identical with a motion compounded of these motions, indicates both, or the determination to both. In every situation of the point of intersection, its velocity is compounded of the velocity A B and A C. A body, therefore, whose motion continued unchanged, would have described A B in one minute; but when it reaches the point A, it turns aside, and describes A b uniformly in the same time; the change then which the body sustains in the point A is a motion A C. For suppose the body had been at rest in the point A, and it is observed to describe A C in one minute, the motion A C is the change which it has sustained. The motion A b is not the change: for if A F had been the primitive motion, the same motion A b would have been the result of compounding with it the motion A G. But since A F is different from A B, the same change cannot produce the same new conditions. But, farther, there is no other motion which, by compounding it with A B, will produce the motion A b; and the motion A C is the only circumstance of difference between changing the motion A B into the diagonal motion A b, and giving the motion A C to a body which was previously at rest.

From these conditions it follows, that a change of motion, is that motion, which by composition with the previous state of motion, produces the new motion.

This composition of motion has been considered in another way. While a body is supposed to move uniformly in the direction A B, the space in which this motion is performed, is supposed to be carried in the direction A C. But it cannot be conceived that any portion of space is moved from its place. A distinct notion of this composition may be obtained, by supposing a person walking along a line A B, while this is drawn on a piece of ice, and the ice is floating in the direction A C. But the motion on moving ice is not precisely a composition of two determinations to motion; for this is completed in the first instant. When the motion in the direction and with the velocity A b begins, no further exertion is needed; the motion continues, and A b is described. It serves, however, to exhibit to the mind the mathematical composition of two motions. In the result of this combination, all the characteristics of the two determinations are to be found; for the point of intersection, in whatever way it is considered, partakes of both motions.

Thus a general characteristic of a change of motion is obtained, and this corresponds with the mark and measure of every moving cause; for it is the very motion which it is conceived to produce. It may perhaps even be considered as the foundation of former measures; for in every acceleration, retardation, or deflection, there is a new motion compounded with the former. Sect. II.

What is taken for the beginning of motion in every observation of surrounding bodies, is nothing more than a change induced on a motion already produced.

56. The actual composition of motion being so general in the phenomena of the universe, it obtains in all motions and changes of motion produced or observed, and the characteristic which has been affirmed of a change of motion being the same, whatever may have been the previous motion, and this being equally applicable to simple motions, it is evident that a knowledge of the general results of this composition of motion will be of essential service in acquiring a knowledge of mechanical nature.

57. The following is the general theorem to which all others may be reduced.

PROP. IX.

Two uniform motions, having the directions and velocities represented by the sides AB, AC, of a parallelogram, compose a uniform motion in the diagonal. The demonstration of this has been already given. The motion of the point of intersection of these two lines, each moving uniformly in all its points, in the direction of the other, is, in every instant, composed of the two motions. It is the same as if a point described AB uniformly, while AB is carried uniformly in the direction AC. This motion is along the diagonal Ab, and it has been already shown to be uniform. And, because AB and Ab are described in the same time, the velocities of the motions along AB, AC, and Ab are proportional to those lines.

COROLLARIES.

Cor. 1. The motion Ab, which is compounded of the two simple motions AB and AC, is in the same plane with these motions. For a parallelogram lies all in the same plane.

Cor. 2. The motion Ab may be produced by the composition of any two uniform motions having the direction and velocities which are represented by the sides of any parallelogram AFbG, or ACbB, which has Ab for its diagonal.

58. Cases are not unfrequent in which the directions of two simple motions composing an observed motion may be discovered; but the proportion of the velocities is unknown. This velocity may be ascertained by means of this last proposition. For the direction of the three motions, namely, the two simple and the compound motions, determines not only the species of parallelogram, but also the ratio of the sides. Again, in those cases in which the direction and the velocity of one of the simple motions are known, and therefore its proportion to that of the observed compound motion, the direction and velocity of the other may be also found by means of the same proposition; because from these data the parallelogram may be determined.

59. This motion in the diagonal is called the equivalent motion, or the resolving motion; for it is equivalent to the combined motions in the sides. Thus, if the moving body first describes AB, and then Bb or AC, it will be in the same point, as if it had described Ab, namely, in the point b.

60. It is often highly useful in investigations of this kind to substitute such motions for an observed motion, as will produce it by composition. This has been denominated the resolution of motions. By this manner of proceeding, a ship's change of situation at the end of a day, having failed in different courses, is computed. Thus the distance failed to the eastward or the westward, as well as that to the northward or southward, on each course, is observed and marked. The whole of the eastings, and the whole of the southings, are added together; and then it is supposed that the ship has failed for the whole day on that course, which would be produced by combining the same easting and southing.

61. It is also useful to consider how much the body has been advanced in a certain direction by means of the observed motion; let us suppose in the direction AB (fig. 10.) The motion CD is first considered as composed of a motion CE parallel to the given line AB, and another motion CF perpendicular to AB. CD is the diagonal of a parallelogram CEDF, one of whose sides CE is parallel to AB, and the other CF is perpendicular to AB. It is evident, that the body has advanced in the direction of AB as much as if it had moved from G to H, instead of moving from C to D, so that the motion CF has no effect either in obstructing or promoting the progress in AB. This is called estimating a motion in a given direction, or reducing it to that direction.

62. A motion is also said to be estimated in a given plane, when it is considered as composed of a motion perpendicular to the plane, and of another parallel to it. In a given plane ABCD (fig. 11.), let EF be a motion compounded of a motion GE perpendicular to the plane, and EH parallel to it. For if the lines GE, FH are drawn perpendicular to the plane, they cut it in two points e and f, and EH is parallel to ef.

63. In the same way a compound motion may be formed of any number of motions. Let AB, AC, AD, AE, &c. (fig. 12.) be any number of motions, of which the motion AF is compounded. The motion which is the result of this composition is thus ascertained. The motion AG is compounded of AB and AC; and the motion AG compounded with AD, gives the motion AH; which latter being compounded with AE, produces the motion AF. And the same place, or final situation F, will be found by supposing the different motions AB, AC, AD, AE, to be performed successively. The moving body first describes AB; then BG, equal and parallel to AC; then GH, equal and parallel to AD; and lastly, HF, equal and parallel to AE. In this case it is not requisite that all the motions lie in the same plane.

64. Three motions which have the direction and proportions of the sides of a parallelopiped, compose a motion having the direction of its diagonal. Let AB, AC, AD (fig. 13.), be these motions, the compound motion is in the diagonal AF of the parallelopiped; because AB and AC compose the motion AE; and AE and AD compose the motion AF.

It is in this way that the mine-surveyor proceeds. He sets down a gallery of a mine, not directly by its real position, but marks the easting and southing, the northing and fouthing, as well as its dip and rise. All these measures are referred to three lines, of which one runs east and west, one north and south, and a third is perpendicular. These three lines are obviously analogous. Compound gous to the angular boundaries of a rectangular box, as AC, AB, AD.

65. The composition of uniform motions only has yet been considered. But it is easy to conceive that any motions may be compounded. It is a case of this kind when a man is supposed to walk on a field of ice along a crooked path, while the ice floats down a crooked stream. Suppose a uniform motion in the direction AB (fig. 14.), to be compounded with a uniformly accelerated motion in the direction AC. A stone falling from the mast head of a ship, while the ship moves uniformly forward in the direction AB, affords an example of this kind of motion; for the stone will be observed to fall parallel to a plummet hung from the mast head. But the real motion of the stone is a parabolic arch ABFG, which AB touches in A; for while the mast head describes the equal lines AB, BF, FG, the stone has fallen to B and C and D, and the line AC is in the positions BB', FF', GG', so that AC is four times AB; and AC is nine times AB. Therefore AB, AC, AD, are as the squares of BC, CF, CG, and the line ABFG is a parabola.

66. Knowing the direction and velocities of each of the simple motions in any instant, of which two motions, however variable, are compounded, we may discover the direction and velocities of the compound motions in that instant. For it may be supposed that each motion at that instant proceeds unchanged; a parallelogram is then constructed; the sides of which have the directions and proportions of the velocities of the simple motions; and the diagonal of this parallelogram will express the direction and velocity of the compound motion. But on the other hand, if the direction and velocity of the compound motion, with the directions of each of the simple motions, be known, we may discover their velocities.

67. In cases where a curvilinear motion as ABC (fig. 15.), is the result of two motions compounded, of which the direction is known to be AD and AE, we discover the velocities of the three motions in any point B, by drawing the tangent BF, and the ordinate BG, parallel to one of the simple motions, and from any point H in that ordinate drawing HF parallel to the other motion, and cutting the tangent in the point F. The three velocities are in the proportion of the three lines FH, HB, and FB.

68. As the motions which are observed in nature are very different from what they are taken to be, it is not easy to avoid mistakes with respect to the changes of motion, and consequently with respect to the inference of its cause. Without considering the real motion of any body, we are apt to judge only of the change of distance and direction in relation to ourselves. Thus it is that our inferences with regard to the planetary motions are very different from the motions themselves, if the rapid motion of our earth be considered.

Prop. X.

69. The motion of one body in relation to another body, or as it is seen from another body, which is also in motion, is compounded of its own real motion, and the opposite of the real motion of the second body.

Let A (fig. 16.) be a body in motion from A to C, as seen from B, which is another body in motion from B to D the motion of A is compounded of its own real motion, and of the opposite to the real motion of B continually deflected.

Join AB, and draw AE equal and parallel to BD. Complete the parallelogram ACFE, and join ED and DC. Produce EA, and make AL equal to AE or BD. Complete the parallelogram LACK, and draw AK and BK. If then A had moved along AE while B moves along BD, the two bodies would have been at E and D, at the same time, and would have the same relative situation; they would have the same bearing and distance as before. And if the spectator in B is not sensible of his own motion, A will appear not to have changed its place. In the same way two ships becalmed in an unknown current, seem to the persons on board to be at rest. The real position, therefore, and distance DC, are the same with BK; and if a spectator in B imagines himself at rest, the line AK will be taken as the motion of A. And this motion, it is obvious, is composed of the motion AC its real motion, and the motion AL which is the equal and opposite motion to that of BD.

Again, if BH be drawn equal and opposite to AC, and the parallelogram BHGD be completed, and BG and AG be drawn, the diagonal BG will be the motion of B as it is seen from A. Now as KAGB is a parallelogram, the relative situation and distances of A and B at the end of the motion will appear to be the same as in the former case. For B appears to have moved along BG, which is equal and opposite to AK. Hence it follows, that the apparent or relative motions of two bodies are equal and opposite, whatever their real motions may be; and therefore they do not afford any information of their real motions.

70. Suppose equal and parallel motions are compounded with all and each of the motions of any number of bodies, moving in any manner of way, then their relative motions are not consequently changed. For if it be compounded with the motion of any one of the bodies which may be called A, the real motion of this body is changed; but its apparent motion as seen from another body B, is compounded of the real change, and of the opposite to the real change in A, which therefore destroys that change, and the relative motion of A is the same as before. Thus it is that the motions in the cabin of a ship are not affected by the ship's progressive motion; and the motion of the earth round the sun produces no perceptible effect on the relative motions on its surface. And indeed it is only by observing other bodies which are not affected by these common motions, and to which we refer as to fixed points, that we arrive at any knowledge of them.

Sect. IV. Of Motions continually Deflected.

71. Curvilinear motions are cases of continual deflection. They are susceptible of great varieties; and the investigation of their modifications and chief properties is attended with no small difficulty. Uniform motion in a circular arch is an example of the simplest case of curvilinear motion; for here the deflections from rectilinear motion are equal in equal times. If, however, the velocity be increased, the momentary deflection must also be augmented; for a greater arch will be described, and the end of this greater arch is When a uniform rectilineal motion \( AB \) (fig. 17.) is deflected into another \( BC \), the linear deflection is affected by drawing a line from the point \( c \), at which point the body would have arrived, had it not been deflected to the point \( C \) at which it has arrived. The result is the same, whether the lines \( dD \) or \( cC \) be drawn in this manner; for being proportional to \( Bc \), they always give the same measure of the velocities; and here the lines of deflection are all parallel, indicating the direction of the deflection in the point \( B \). But this is not the case in any curvilinear motion. It rarely happens that \( dD : cC = Bc \); and it is never found that \( dD : cC = Bc \). We cannot therefore discover which lines should be taken for the indication of the direction of the deflection at \( B \), or for the measure of its magnitude. A greater velocity then, in the same curve, produces a greater deflection, but if the path be more incurved, an arch of the same length described with the same velocity, causes a farther deviation from the tangent. If therefore a body have a uniform motion in a curve of variable curvature, the deflection is greatest where the curvature is greatest.

Thus it appears that the direction and measure of the deflections by which a body deviates continually into a curvilinear path cannot be ascertained, but by investigating the ultimate positions and ratios of the lines, which join the points of the curve with the simultaneous points of the tangent, as the points \( d \) and \( C \) are taken nearer to \( B \). In some cases, but rarely, the lines joining the simultaneous points are parallel. But in most cases the direction of the deflection is discovered by observing what direction it approximates. The following proposition which was discovered by Newton is of great importance in this investigation.

**Proposition XI.**

If a body describe a curve line \( ABCDEF \) (fig. 18.) being in the same plane, and if in this plane there be a point \( S \) so situated, that the lines \( SA, SB, SC, \ldots \) drawn to the curve, cut off areas \( ASB, ASC, ASD, \ldots \) proportional to the times of describing the arches \( AB, AC, AD, \ldots \), then the deflections are always directed to the point \( S \).

Suppose first that the body describes the polygon \( ABCDEF \), formed of the chords of this curve, and that it describes each chord uniformly, and is deflected only in the angles \( B, C, D, \ldots \). Suppose also that the sides of the polygon are described in equal times, so that, according to the hypothesis, the triangles \( ASB, BSC, CSD, \ldots \) are all equal. Continue the chords \( AB, BC, \ldots \) beyond the arches, making \( Be \) equal to \( AR \), and \( Cd \) equal to \( BC \), and so on. Join \( cC, dD, \ldots \), and draw \( cS, dS, \ldots \); draw \( CB \) parallel to \( cB \) or \( BA \), cutting \( BS \) in \( b \), and join \( bA \), and draw \( CA \), cutting \( BB \) in \( o \). And lastly, make a similar construction at \( E \).

Then, because \( cB \) is equal to \( BA \), the triangles \( ASB \) and \( BSc \) are equal, and therefore \( BS \) is equal to \( BSC \). And being on the same base \( SB \), they are therefore between the same parallels; that is, \( cC \) is parallel to \( BS \), and \( BC \) is the diagonal of a parallelogram \( BbCc \). The motion \( BC \) is therefore compounded of the motions \( Bc \) and \( Bb \), and \( Bb \) is the deflection, by which the motion \( Bc \) is changed into the motion \( BC \); and therefore the deflection in \( B \) is directed to \( S \). By similar reasoning it may be shown that \( Ff \), or \( Ei \), is the deflection at \( E \), and is likewise directed to \( S \); and the same demonstration will apply to every angle of the polygon.—This point \( S \) has been called the centre of deflection.

If the sides of the polygon are diminished, and their number infinitely increased, the demonstration remains the same, and continues, when the polygon coalesces with the curvilinear area, and its sides with the curvilinear arch.

But when the whole areas are proportional to the times, equal areas are described in equal times. In such motion therefore, the deflections are always directed to \( S \).

**Proposition XII.**

If the deflection by which a curve line is described, be continually directed to a fixed point, the figure will be in one plane, and areas will be described round that point proportional to the times. Let \( ADF \) be the curve line described, and let the deflections be directed to the point \( S \), this curve line is in the same plane. For \( BC \) is the diagonal of a parallelogram, and is in the plane of \( SB \) and \( Be \); and \( cC \) is parallel to \( BS \), and the triangles \( SBC, SBe, \) and \( SBA, \) are equal. But equal areas are described in equal times; and therefore areas are described proportional to the times.

**Corollaries.**

1. The velocities in different points of the curve are inversely proportional to the perpendiculars \( Sr \) and \( St \) (fig. 19.) drawn from \( S \) on the tangents \( Ar, Et \) in those points of the curve. For since the elementary triangles \( ASB, ESF, \) are equal, their bases \( AB, EF, \) are inversely as their altitudes \( Sr, St. \) And these bases being described in equal times are as the velocities, and ultimately coincide with the tangents at \( A \) and \( E; \) and therefore the velocity in \( A \) is to that in \( E \) as \( Sr \) to \( Sr. \)

2. The angular velocities round \( S \) are inversely as the squares of the distances. For if we describe round the centre \( S \) the small arches \( Ba, Fa, \) they may be considered as perpendiculars on \( SA \) and \( SE. \) Describe also with the distance \( SF \) the arch \( gh. \) It is evident that \( gh \) is to \( Fa \) as the angle \( ASB \) to the angle \( ESF. \) And since the areas \( ASB, ESF \) are equal, we have \( Ba : Fa = SE : SA. \)

But \( gh : Fa = SE : SA. \)

Therefore \( gh : Fa = SE^2 : SA^2. \)

And \( ASB : ESF = SE^2 : SA^2. \)

Let us now proceed to determine the magnitude of the deflection, or to compare its magnitude in any two points, as for example the magnitude in \( B \) (fig. 18.) with its magnitude in \( E. \) The deflection in \( B \) is to that in \( E \) as the line \( Bb \) to the line \( Ei; \) for \( Bb \) and \( Ei \) are the motions, which, by being compounded with the motions \( Be \) and \( Ef, \) make the body describe \( BC \) and \( EF. \) And therefore when the sides of the polygon Motions

Part II.

74. Motions are infinitely diminished, the ultimate ratio of continually Bb to Ei is the ratio of the deflection at B to the deflection at E.

To obtain a convenient expression of this ultimate motion, let ABCXYZ be a circle which passes through the points A, B, C. Draw BSZ through the point S, and draw CZ, AZ. Now the triangles BCb and AZC are similar, for CB was drawn parallel to CB or BA; and therefore the angle CB is equal to the alternate angle bBA or ZEA, which is equal to the angle ZCA, because it is subtended by the same chord ZA; and because they stand on the same chord CZ, CBb, or CBZ, is equal to CAZ. And therefore the remaining angles CB and CAZ are equal, and the triangles are similar. Therefore Bb : CA = BC : AZ.

Now if the sides of the polygon are continually diminished, the points A and C continually approach to B, and CA continually approaches to cA, or to 2CB, or 2CB, and is ultimately equal to it; and AZ is ultimately equal to BZ.

Therefore ultimately, Bb : BC = BC : BZ, and

\[ Bb \times BZ = 2BC^2, \text{ and } Bb = \frac{2BC^2}{BZ}. \]

Also, at the point E, we have Ei ultimately equal to 2EF, for Ei is that chord of the circle through D, E, and F, which passes through i.

Therefore Bb : Ei = 2BC^2 : 2EF.

The ultimate circle, when the three points A, B, C, coalesce, is called the circle of equal curvature, or the equicurve circle, which coalesces with the curve in B in the closest manner; and the chord BZ of this circle, having the direction of the deflection in B, is called its deflective chord. And since BC and EF are described in equal times, they are proportional to the velocities in B and E. This proposition therefore may be expressed as follows.

In curvilinear motions, the deflections in different points of the curve, are proportional to the square of the velocities in those points directly, and to the deflective chords of the equicurve circles inversely.

It ought however, to be remarked, that this theorem is not limited to curvilinear motions, in which the deflections tend always to the same fixed point; it may be extended to all curvilinear motions whatever. A symbolical expression of this theorem will be convenient. If therefore the deflective chord of the equicurve circle be represented by c, and the deflection by d, the theorem may be thus expressed.

\[ d = \frac{v^2}{c}, \text{ or } d = \frac{2\text{arch}^2}{c}. \]

76. The line Bb is the linear deflection by which the uniform motion in the chord AB is changed into a uniform motion in the chord BC, or it is the deviation cC from the point to which the moving body would have arrived, if the deflection at B had not taken place. In the case of curvilinear motion which we are now considering, the lines Bb and Bc are expressions of the measures of the velocities of these motions. Bc is to Bb as the velocity of the progressive motion is to the velocity of the deflection, generated in the time that the arch BC is described. But the deflection in the arch has been continual, and like acceleration, it may be measured by the velocity generated during any moment of time. It may therefore be measured by the velocity generated during the time the arch BC is described. This measure will therefore be double of the space through which the body is actually deflected from the tangent in B in that time. The space described will be BO, or only one-half of Bb. This is exactly what happens; for the tangent is ultimately parallel to OC, and it bisects cC; therefore the velocity gradually generated is that which constitutes the polygonal motion in the chords, although the deflection from the tangent to the curve is only half of the deflection from the produced chord to the curve.

77. In any point of a curvilinear motion, the velocity is that which would be generated by the deflection in that point, if continued through one-fourth of the deflective chord of the equicurve circle. Take x for the space along which a body is to be accelerated that it may acquire the velocity BC.

We have Bb^2, or 4BO : BC^2 = B : x (37.1); and therefore \( x = \frac{BC^2 \times BO}{4BO^2} = \frac{BC^2}{4BO} \), or \( x = \frac{BC^2}{BO} \).

BO : BC = BC : 4x. But BO : BC = BC : BZ; therefore \( x = \frac{1}{4} BZ \).

78. We have now obtained characteristic expressions, or marks and measures of the principal affections of motion. These expressions may be brought into one view as follows.

The acceleration \( a \) is \( \frac{v}{t} \) (48.), or \( \frac{v}{s} \) (49.), or \( \frac{s}{t^2} \) (42.).

The momentary variation of velocity \( v = a t \) (48.).

The momentary variation of the square of velocity \( 2v = 2a^2 \) (49.).

The momentary deflection \( d = \frac{\text{arc}^2}{\text{chord}} \) (76.)

The deflective velocity \( = \frac{2v^2}{c} \) (75.).

79. But for the application of these doctrines, it is necessary to select some point in any body of sensible magnitude, or in any system of bodies, by whose position or motion, a distinct and accurate notion of the position or motion of the body or system may be formed. The condition by which the propriety of this selection is ascertained, is, that the position, distance, or motion of this point shall be the medium or average of the positions, distances, and motions of every particle of matter in the aggregate or system.

This will happen, if the point be so situated, that Center of when a plane is made to pass through it in any direction whatever, and perpendiculars being drawn to this plane from every particle of matter in this aggregate or system, the sum of the perpendiculars on the one side of the plane is equal to the sum of the perpendiculars on the other side. And that such a point, which is called the centre of position, may be found in every body, is proved by the following demonstration.

For let P (fig. 20.) be a point so situated, and let QR be the section of a plane perpendicular to the paper, and at any distance from it, the distance PP' of the point P from this plane is the average of all the distances of each particle from it. Let the plane APB pass through P, and parallel to QR. The distance CS Motions CS of any particle C from the plane QR is equal to continually DS—DC, or to Pρ—DC. And the distance GT of a particle G on the other side of APB, is equal to HT+GH, or to Pρ+GH. Let n be the number of particles on that side of AB which is nearest to QR, and let o be the number of particles on the other side of AB. Let m be the number of particles in the whole body; we have then \( m = n + o \). It is evident that the sum of all the distances of all the particles such as CS, is \( n \times P\rho \)—the sum of all the distances, such as CD. Also the sum of all the distances of the particles, such as G, is \( o \times P\rho \), \( + \) the sum of the distances GH. And therefore the sum of both sets is \( n + o \times P\rho \), \( + \) the sum of GH—the sum of DC, or \( m \times P\rho \), \( + \) the sum of GH—the sum of DC. But by the supposed property of the point P, the sum of GH wanting the sum of DC is nothing; and therefore \( m \times P\rho \) is the sum of all the distances, and \( P\rho \) is the nth part of this sum, or the average distance.

Suppose the body to have changed both its place and its position with respect to the plane QR, and that P (fig. 21.) is still the same point of the body, and \( P\rho \) a plane parallel to QR. Make \( P\rho \) equal to \( P\rho \) of fig. 20. It is plain that \( P\rho \) is still the average distance, and that \( m \times P\rho \) is the sum of all the present distances of the particles from QR, and that \( m \times P\rho \) is the sum of all the former distances. Therefore \( m \times P\rho \) is the sum of all the changes of distance, or the whole quantity of motion estimated in the direction \( P\rho \). \( P\rho \) is the nth part of this sum, and is therefore the average motion in this direction. The point P has therefore been properly selected; and its position, and distance, and motion, in respect of any plane, is a proper representation of the situation and motion of the whole.

Hence it follows, that if any particle C (fig. 20.) moves from C to N, in the line CS, the centre of the whole will be transferred from P to Q, so that PQ is the nth part of CN; for the sum of all the distances has been diminished by the quantity CN, and therefore the average distance must be diminished by the nth part of CN, or \( PQ = \frac{CN}{m} \).

But it may be doubted whether there is in every body a point, and but one point, such that if a plane passes through it, in any direction whatsoever, the sum of all the distances of the particles on one side of this plane is equal to the sum of all the distances on the other.

It is easy to show that such a point may be found, with respect to a plane parallel to QR. For if the sum of all the distances DC exceed the sum of all the distances GH, we have only to pass the plane AB a little nearer to QR, but still parallel to it. This will diminish the sum of the lines DC, and increase the sum of the lines GH. We may do this till the sums are equal.

In like manner we can do this with respect to a plane LM (also perpendicular to the paper), perpendicular to the plane AB. The point wanted is somewhere in the plane AB, and somewhere in the plane LM. Therefore it is somewhere in the line in which these two planes intersect each other. This line passes through the point P of the paper where the two lines AB and LM cut each other. These two lines represent planes, but are, in fact, only the intersection of those planes with the plane of the paper. Part of the body must be conceived as being above the paper, and part of it behind or below the paper. The plane of the paper therefore divides the body into two parts. It may be situated, therefore, that the sum of all the distances from it to the particles lying above it shall be equal to the sum of all the distances of those which are below it. Therefore the situation of the point P is now determined, namely, at the common intersection of three planes perpendicular to each other. It is evident, that this point alone can have the condition required in respect of these three planes.

It still remains to be determined whether the same condition will hold true for the point thus found, in respect to any other plane passing through it; that is, whether the sum of all the perpendiculars on one side of this fourth plane is equal to the sum of all the perpendiculars on the other side.

Let AGHB (fig. 22.), AXYB, and CDFE, be three planes intersecting each other perpendicularly in the point C; and let CIKL be any other plane, intersecting the first in the line CI, and the second in the line CL. Let P be any particle of matter in the body or system. Draw PM, PO, PR, perpendicular to the first three planes respectively, and let PR, when produced, meet the oblique plane in V; draw MN, ON, perpendicular to CB. They will meet in one point N. Then PMNO is a rectangular parallelogram. Also draw MQ perpendicular to CE, and therefore parallel to AB, and meeting CI in S. Draw SV; also draw ST perpendicular to VP. It is evident that SV is parallel to CL, and that STRQ and STPM are rectangles.

All the perpendiculars, such as PR, on one side of the plane CDFE, being equal to all those on the other side, they may be considered as compensating each other; the one being considered as positive or additive quantities, the other as negative or subtractive. There is no difference between their sums, and the sum of both sets may be called o or nothing. The same must be affirmed of all the perpendiculars PM, and of all the perpendiculars PO.

Every line, such as RT, or its equal QS, is in a certain invariable ratio to its corresponding QC, or its equal PO. Therefore the positive lines RT are compensated by the negative, and the sum total is nothing.

Every line, such as TV, is in a certain invariable ratio to its corresponding ST, or its equal PM, and therefore their sum total is nothing.

Therefore the sum of all the lines PV is nothing; but each is in an invariable ratio to a corresponding perpendicular from P on the oblique plane CIKL. Therefore the sum of all the positive perpendiculars on this plane is equal to the sum of all the negative perpendiculars, and the proposition is demonstrated, viz. that in every body, or system of bodies, there is a point such, that if a plane be passed through it in any direction whatsoever, the sum of all the perpendiculars on one side of the plane is equal to the sum of all the perpendiculars on the other side.

So. If A and B (fig. 23.) be the centres of position of two bodies, whose quantities of matter (or numbers of equal particles) are \( a \) and \( b \), the centre C lies in the straight line joining A and B, and AC : CB = \( a : b \); or its distance from the centres of each are inversely as their quantities of matter. For let \( \alpha C \beta \) be any plane passing...

Of Moving passing through C. Draw \( A_a, B_b \), perpendicular to this plane. Then we have \( a \times A_a = b \times B_b \), and \( A_a : B_b = a : b \), and by similarity of triangles, \( CA : CB = a : b \).

If a third body D, whose quantity of matter is d, be added, the common centre of position E of the three bodies is in the straight line DC, joining the centre D of the third body with the centre C of the other two, and \( DE : EC = a + b : d \). For, passing the plane \( E_x \) through E, and drawing the perpendiculars \( D_x, C_x \), the sum of the perpendiculars from D is \( d \times D_x \); and the sum of the perpendiculars from A and B is \( a + b \times C_x \), and we have \( d \times D_x = a + b \times C_x \); and therefore \( DE : EC = a + b : d \).

In like manner, if a fourth body be added, the common centre is in the line joining the fourth with the centre of the other three, and its distance from this centre and from the fourth is inversely as the quantities of matter; and so on for any number of bodies.

81. If all the particles of any system be moving uniformly, in straight lines, in any directions, and with any velocities whatever, the centre of the system is either moving uniformly in a straight line, or is at rest.

For, let m be the number of particles in the system. Suppose any particle to move uniformly in any direction. It is evident from the reasoning in a former paragraph, that the motion of the common centre is the \( m^{th} \) part of this motion, and is in the same direction. The same must be said of every particle. Therefore the motion of the centre is the motion which is compounded of the \( m^{th} \) part of the motion of each particle.

PART II. OF MOVING FORCES.

84. HAVING in the former part considered the general doctrine of motion, which is the foundation of mechanical investigations, we now proceed to treat of moving forces or dynamics, properly so called.

It has been already observed, that dynamics includes the abstract doctrine of moving forces, or the necessary results of the relations of our thought concerning motion, the immediate causes of motion, and its changes; and that from the changes observed, we infer agency in nature; and in these changes we are to discover what we know of their causes.

85. When we cast our eyes around us, it cannot escape observation, that the changes which we perceive in the state or condition of any body in respect of motion, are constantly and distinctly related to the situation and distance of other bodies. The motions of the moon, or of a stone projected through the air, have a palpable relation to the earth; the motions of the tides have also an obvious relation to the moon; and the motions of a piece of iron have a palpable dependence on a magnet. The vicinity of the one of these bodies seems to be the occasion, at least, of the motions of the other; and the causes of these motions have an evident connection with, or dependence on, the other body. Such dependences have been called the mechanical relations of bodies. They are indications of properties or distinguishing qualities. They accompany the bodies wherever they are, and are usually conceived to be inherent in them. They at least ascertain and determine what is called the mechanical nature of bodies.

86. The mutual relation of bodies is differently considered according to the interest we may have in the phenomenon. The cause of the approach of the iron bodies different to the magnet is generally ascribed to the magnet. It is said to attract the iron. The approach of a stone to the earth is ascribed to the stone. It is said to tend to the earth. But it is probable that the procedure of nature is the same in both; that both bodies are affected alike, and that the property is distinctive of both. For in all cases that have been observed, the indicating phenomenon is equally connected with both bodies; as in the case of magnetism the magnet and the iron approach each other; and an electrified body and another body near it approach each other. This property is therefore equally inherent in both bodies, between which there is a mutual attraction. But, according to some philosophers, no such mutual tendencies exist either in the one body or the other. The observed approaches or mutual separations of bodies, or their attractions and repulsions, are supposed to depend on the extraneous action of an ethereal fluid.

87. These qualities thus inherent in bodies, which constitute their mechanical relations, or the mechanical effects of matter, have been called powers or forces. The event which is indicated by their presence, is considered Of Moving considered as the effect and mark of their agency. Thus Forces, the magnet is said to act on the iron, the earth is said to act on the stone which falls to its surface; and the iron and the stone are said to act on the magnet and the earth. But all this, it must be observed, is figurative language. Power, force, and action, when used in their original strict sense, express only the notions of the power, force and action of sentient, active beings; and cannot be predicated of any thing but the exertions of such beings; for such beings only are agents. In strict propriety, it is perhaps only the exerted influence of the mind on the body which ought to be called action. Language having begun among simple men, such denominations were very properly given to their own exertions; because to move a body they found it necessary to exert their force or power, or to act. But when the changes of motion, observed in the occurrence or vicinity of bodies, were attended to by speculative men, and it was found that the phenomena greatly resembled the results or effects when they exerted their own strength, similar terms were employed to express these occurrences in nature. The old term was retained, in preference to the invention of a new language, to express things which had so near a resemblance. The danger of confounding things from the use of the same terms, was avoided from the differences in other circumstances of the case. It is not, however, to be imagined, that they supposed inanimate bodies exerted force or strength in the same way as living beings. But, in the progress of refinement, the word power or force came at last to be employed to express any efficiency whatsoever; and hence the common expressions, the force of arguments, the action of motives, the power of an acid to dissolve a metal, &c. It is to this idea of convenience, that the use of the terms attraction, repulsion, pressure, impulsion, as well as of the words power and force, which express efficiency in general, is to be ascribed. But these terms, excepting in those cases when they are applied to the exertions or actions of living beings, are metaphorical. On account, however, of the resemblance between the phenomena and those which are observed when we draw a thing toward us, push it from us, forcibly compress it, or kick it away, these different actions being analogous to attraction, repulsion, pressure, and impulsion, these words are employed as terms of distinction. The action of the mind on the body is perhaps the only case of pure unfigurative action. But this action being always exerted with the view of effecting some change on external bodies, our attention is only directed to them. The instrument passes unnoticed; and hence it is said that we act on the external body. The real action is only the first movement in a long succession of events, and is only the remote cause of the interesting event. In many cases of mechanical phenomena, we find the resemblance to such actions to be very strong. The following is of this description. A ball is projected from a man's hand by the motion of his arm; and in the same way a ball is impelled by the unbending of a spring. In all circumstances there is a resemblance between these two events, excepting in the action of the mind on the corporeal organ. And, hence in general, because the ultimate results of the mutual influence of bodies on each other have a strong resemblance to the ultimate results of our actions on bodies, no new or appropriate terms have been invented; but, as has been already observed, mankind have remained satisfied with the use of those terms that are employed to express their own actions, or the exertions of their own powers or forces.

88. When power or force is spoken of as existing or acting of residing in a body, and the effect is ascribed to the exertion of this power, one body considered as possessing powers, it, is said to act on another. Thus a magnet is said to act on a piece of iron; a billiard ball is said to act on one which it strikes. But if it be attempted to fix the attention on this action, independent both of the agent and the thing acted on, we shall find that there is no object of contemplation. The exertion or procedure of nature in effecting the change is kept out of view; and if we limit our attention to the action as a thing distinct from the agent, we shall find that it is not the action, strictly speaking, but the act, that is brought under consideration. And in the same way, it is only in the effect produced that the action of a mechanical power can be conceived.

89. In the very nature of action some change is implied. Without producing some effect, a man is never said to act. Thought is the act of the thinking principle; and the motion of the limb is the act of the mind on it. In mechanics too there is action only so far as some mechanical effect is produced. For instance, to begin motion on a piece of ice, or to slide along it, we must act violently; we must exert force; and this force being exerted produces motion. In all cases, the productions of motion are conceived as the exertions of force; but to continue the motion which has been begun along the ice, no exertion seems requisite. Being conscious of no exertion, we ought to infer that no force is necessary for the continuation of motion. It is not the production of any new effect, but the permanency or continuation of an effect already produced. Motion is indeed considered as the effect of some action; but there would be no effect or no change, if the body were not moving. Motion is not to be considered as an action, but the effect of an action.

90. Mechanical actions or forces have been divided into pressures and impulsions. The idea of pressure is very familiar; perhaps it enters into every distinct conception that we can form of a moving force, when the attention is endeavoured to be fixed on it. Changes of motion by the collision of moving bodies are produced by impulsion. Pressures and impulsions are usually considered as of different kinds, the actions or exertions of different powers. It is supposed that there is an essential difference between pressure and impulsion. That we may obtain all the knowledge that these distinctions can give us, let us state some examples of these kinds of forces, instead of attempting to define or describe them.

Let us first take some examples of pressure. Pressure is known as a moving force; for if a ball lying on the table be gently pressed on one side, it moves toward the other side of the table. If it be followed with the finger, the pressure being continued, its motion is continually increased. There is an acceleration of its motion. By pressing in the same way on the handle of a common kitchen jack, the fly begins to move; and if the pressure be continued on the handle, the motion of the fly becomes very rapid; and there is: Of Moving is also a continual acceleration. Such motions as these are the effects of genuine prelure. The unbending of a spring would urge the ball in the same way along the table, and would produce a continually accelerated motion; and a spring coiled up round the axis of the handle of the jack would, by uncoiling itself, urge round the fly with a similar accelerated motion. By comparing the prelure of the finger on the ball with the effects of the spring, we perceive distinctly the perfect similarity. These exertions or actions, or influences, are denoted by the word prelure, which is derived from the most familiar instance of them.

The same motion may be produced in the ball or fly, by pulling the ball or machine by means of a thread having a weight suspended to it. Both being motions accelerated in the same manner, the action of the thread on the ball or machine comes under the same denomination of prelure. Weight is therefore considered as a preluring power. And indeed the same compulsion is felt from the real prelure of a man on the shoulders and a load laid on them. But in the instance above, the weight acts by the intervention of the thread. By the prelure of the weight it pulls at that part of the thread to which it is attached, this part pulls at the next by the force of cohesion; and this at a third, and so on, till the most remote pulls at the ball or machine. In this way elasticity, weight, cohesion, and other forces, perform the office of a genuine power; and their result being always a motion beginning from nothing, and accelerating to any velocity by perceptible degrees, from this resemblance we are led to give them one familiar name.

91. If the thread by which the weight is suspended be cut, it falls with an accelerated motion. This also is ascribed to some preluring power which acts on the weight; and it is even considered as the cause of the body's weight, which word is a name by which this instance of preluring power is distinguished. Gravitation, therefore, comes under the denomination of prelure. For the same reason the attractions and repulsions of the magnet, or of electric bodies, belong to this class of phenomena; for on bodies placed between them they produce actual compellings, as well as motions which are continually accelerated, in the same way as gravitation does. To all these powers, therefore, the descriptive name of prelures may be given, although this name properly speaking belongs to one of them only. This great class has been subdivided by some philosophers into prelusions and solicitations. Gravity is considered as a solicitation ab extra, by which a body is urged downward. The forces of electricity and magnetism, with many other attractions and repulsions, are also called solicitations. But this classification seems to be of little use.

92. We have a familiar instance of impulsion in one ball striking another, and putting it in motion. In this case the appearances are very different from the phenomena of prelure. For the body that is struck acquires in the instant of impulse a sensible quantity of motion. But after the stroke this motion is neither accelerated nor retarded, unless by the action of some other force. The rapidity of the motion, it is observed depends on the previous velocity of the striking ball. If for instance a clay ball, moving with any velocity, strike another equal ball which is at rest, the ball which is struck moves with one half of the velocity of the other. It is farther observed that the striking ball always loses as much motion as the ball which is struck gains. From this remarkable fact there seems to have arisen an indistinct notion of a kind of transference of motion from one body to another. It is not said that the one ball produces motion or causes it in the other, but it is said to communicate motion to it; and the phenomenon is usually termed the communication of motion. This, however, is a very inaccurate mode of expression. We distinctly conceive the cause or communication of heat, the communication of fatigues, of sweetmeats, and of many other things; but we have no clear conception of part of the identical motion which existed in one body being transferred to another. From this, therefore, it appears that motion is not a thing which can exist independently, and is susceptible of actual transference; but is a state or condition of which bodies are susceptible which may be produced in bodies, and which is the effect or characteristic of certain natural properties or powers.

The notion of the actual transference of something formerly possessed by the striking body, and now separated from it, or transferred into the body which is struck, has obtained support from the remarkable circumstance in the phenomenon, that a rapid motion requiring for its production the action of a preluring power, continued for a sensible, and frequently a long time, is or seems to be effected instantaneously by impulsion. Here then we find room for the employment of metaphor, both in thought and language. We see the striking body affect the body which is struck. It possesses the power of impulsion, or of communicating motion, but it only possesses this power while it is itself in motion; and we therefore conclude that this power is the efficient distinguishing cause of its motion. Hence it has been called inherent force, the force inherent in a moving body, vis inusta corpori moto. This force is communicated to the body impelled, or transferred into it; the transference is instantaneous, and the body thus impelled continues its motion till it is changed by a new force. But if we attend scrupulously to those feelings which have given rise to this metaphorical conception, we shall find, that although at first sight this train of observation seems very plausible, we should entertain very different notions. To begin the motion of sliding on a smooth piece of ice, we are conscious of exertion; but when the ice is very smooth, no exertion that we are conscious of seems requisite to continue the motion. No exertion of power is here necessary; and therefore we have no primitive feeling of power while we slide along. And indeed we cannot think of moving forward without effort otherwise than as a certain mode of existence. It has however been imagined that those who support this opinion have in some way deduced it from their feelings. To move forward in walking, we must continue the exertion with which we began; and unless this power of walking be continually exerted, we must stop our progress. But this is inaccurate observation. In the action of walking there is much more than the continuance in progressive motion. It is the repeated and continued lifting the body up a small height, and allowing it to come down again, and this repeated ascent requires repeated exertion. 93. From the consideration of the instantaneous production of rapid motion by impulse, some distinguished philosophers have been led to suppose that the force or power of impulse is not susceptible of being compared with a pressing power. It has been asserted that impulse when compared with pressure is infinitely great. But the similarity of the ultimate results of impulse and pressure, have always led them to adopt a different view. There is no difference between the motion of two balls which move with equal rapidity, one of which descends from a height by the force of gravity, while the other has been struck by another body. In this struggle of the mind attached to preconceived opinions, and at the same time accommodating these opinions to observed phenomena, other singular forms of expression have arisen. Pressure is considered as an effort to produce motion. And here we have another instance of metaphorical expression as well as thought. The weight of a ball on the table is called a power; and this weight is continually endeavouring to move the ball downward. But these efforts being ineffectual, the power in this case is said to be dead. It is called *vis mortua*, in contradistinction to the force of impulse which is called a living power, *vis viva*. But this mode of expression must appear very inaccurate, if we consider the case of the impelling ball falling perpendicularly on the other ball lying on the table. No motion is induced by this impulse; and if the table be conceived to be annihilated, the power of gravity becomes a *vis viva*.

To prove that impulse is infinitely greater than pressure, numerous familiar instances have been adduced by those who support this doctrine. A nail is driven with a moderate blow of a hammer, which would require a pressure many hundred times greater than the impelling effort of the person who employs the hammer. A hard body may be shattered into pieces with a moderate blow, which would support an inconceivable weight gradually applied. This prodigious superiority in impulse leaves it a difficult matter to account for the production of motion by means of pressure; because the motion of the hammer might have been acquired in consequence of the continued pressure of the carpenter's arm. It is considered as the aggregate of an infinite number of succeeding pressures repeated in every instant of its continuance. The smallness of each effort is compensated by their number.

94. After all, it does not appear clear that there are two kinds of mechanical force which are essentially different in their nature. It is, indeed, in a great measure given up by those who support the doctrine that impulse is infinitely greater than pressure: Some method might perhaps be found of explaining satisfactorily this remarkable difference between the two modes of producing motion. But there seems to be no considerable advantage in thus arranging the phenomenon under two distinct heads.

95. The nature of the sole moving force in nature has given rise to much discussion among mechanicians, and produced no small diversity of opinion. According to some, all motion is the effect of pressure; for when impulse is considered as equivalent to the aggregate of an infinite number of pressures, every pressure, however small, is supposed to be a moving force.

The sole cause of motion, according to other philosophers, is impulse. Bodies are observed in motion; they impel others, and produce motion in them; and this production of motion is said to be regulated by such laws, that there is only one absolute quantity of motion in the universe, which quantity remains invariably the same. Some portion of this motion, therefore, must be transferred or transfused when bodies come into collision with each other. But besides, there are some cases in which it is perfectly obvious that motion produces pressure. Cases, which are indeed both whimsical and complicated have been adduced by Euler, to show that an action, in all respects similar to pressure, may be produced by motion. Such a case is the following. If two balls are connected by a thread, they may be struck in such a way, that they shall not only move forward, but at the same time also wheel round. When this happens, the thread by which they are connected is stretched. Since then, according to this reasoning, motion is observed, and pressure is produced by motion, it would be absurd to suppose that pressure is anything else than the result of certain motions. The philosophers who are attached to this doctrine of moving forces, proceed to account for those pressing powers or solicitations to motion which are observed in the acceleration of falling bodies, the phenomena of magnetism and electricity, and others of the same kind, where motion is induced on certain bodies which are in the vicinity of other bodies, or as it is expressed in common language by the action of other bodies at a distance. To say that a magnet cannot act on a piece of iron at a distance, is to say that it acts where it is not; which is not less absurd than to say that it acts, when it is not. Euler assumed it is an axiom, *nihil movetur nisi a contiguo et moto*.

The methods proposed by these philosophers to produce pressure, are less ingenious and not more satisfactory than that adduced by Euler which was mentioned above; and indeed they do not seem to be very anxious about the manner in which these motions are produced. The phenomena of magnetism are induced, or a piece of iron is put in motion, when it is in the vicinity of a magnet, by a stream of fluid which issues from one pole of a magnet, passes in a circle round the magnet, and enters at the other pole. By this stream of fluid the iron is impelled, and brought to arrange itself in certain determined positions. In the same way all bodies are impelled in lines perpendicular to the surface of the earth by a stream of fluid which is in continual motion towards its centre. In the same way similar phenomena are accounted for, and thus these motions are reduced to simple cases of impulse. But to say nothing worse of this doctrine, it is not very compatible with the dictates of common sense. It proceeds on the supposition that something acts which we do not see; and of the existence of which there is not the smallest proof.

96. Pressure, according to the opinion of others, is or pressure, the only moving force in nature; but it is that kind of pressure which has been termed solicitation, not what arises from the mutual contact of solid bodies. Gravitation is an instance of the kind of pressure here alluded to. It is affirmed by these philosophers, that there is no such thing as contact on the instantaneous communication of motion by the real collision of bodies. It is said that the particles of solid bodies exert very strong repulsions. Of Moving repulsions to a small distance; and when they are brought by any motion sufficiently near to another body, they exert a repulsive force, and are equally repelled by this body. Motion is thus produced in the one body, while it is diminished in the other. It is then shown by scrupulously considering the state of the bodies while the one advances, and the other retires, in what way they attain a common velocity, the quantity of motion before collision remaining the same, and the one body gaining exactly as much as the other loses. Cases also are adduced, of such mutual action between bodies, where it is obvious they have never come into contact; but where the result is exactly the same as when the motion seemed to be instantaneously changed. And hence it is concluded that there is no such thing as instantaneous communication, or transmutation of motion by contact in collision or impulse. All moving forces according to these philosophers, are of that kind which have been named solicitations; such as gravity is.

97. Different names have been given to the exertions of mechanical forces, according to the reference that is made to the result. In wrestling when my antagonist exerts his strength to prevent being thrown down, and I am sensible of his exertion, I thus discover that he resists. But if I oppose him only to prevent him throwing me, I am said to resist. If I strike or endeavour to throw him, I am said to act. The same distinction is applied to the exertion of mechanical powers. If, for instance one body A change the motion of another body B, the change in the motion of B may be considered either as the indication and measure of the power of A in producing motion, or as the indication and measure of the resistance made by A in being brought to rest, or having any change induced on its motion. The distinction which is here made is not in the thing itself, but exists only in the reference which we are disposed to make of its effect, from other considerations. If a change of motion take place when one of the powers ceases to be exerted, it is conceived that this power has resisted. But this language is metaphorical. Resistance, effort, endeavour, are all words which express motion that relate to sentient beings. There is perhaps no word preferable to the word reaction, to express the mutual force which is observed in all the operations of nature which have been successfully investigated.

98. A difficulty has been started with regard to the opinion of those who affirm that all mechanical phenomena are dependent on attracting and repelling forces; because it is here supposed that bodies act on each other at a distance, and however small this distance may be, this is conceived to be absurd. It may however be observed, that the mutual approaches or recedes of bodies may be ascribed to tendencies to, or from each other. Without thinking of any intermediate connection between the iron and the magnet, we conceive the iron to be affected by the magnet; and if this be conceivable, it is not absurd. Our knowledge of the essence or nature of matter is not such as to render this tendency of the iron to the magnet impossible. We do not indeed see intuitively why the iron should approach to the magnet; but this is by no means sufficient to pronounce it impossible or inconsistent with the nature of matter. To suppose therefore in the production of motion, the impulse of an invisible fluid, of which we know not anything, and of whose existence there is no evidence, is a rash and unwarrantable assumption. But farther, if it be true that bodies do not come into contact, even when one ball strikes another, and drives it before it, the supposition of the existence of this invisible fluid will not assist us in solving the difficulty; for the same difficulty would occur in the action of any one particle of the fluid in the body. At any rate the production of motion without any observed contact, is more familiar to us than the production of motion by one body acting on another by impulse. Every case of gravitation is an instance of this.

99. In those cases where the exertions of any mechanical power are observed to be always directed toward any body, that body is said to attract. Thus a boat is attracted toward a man when he pulls it toward him by means of a rope. This is a case of pure attraction. But when the other body always moves off, the body exhibiting this phenomenon is said to repel; and it is a case of pure repulsion when a person pushes any body from him. And because there is a resemblance to the results of real attraction and repulsion, the same terms are employed to express the mechanical phenomena of nature. But that our conceptions may not be embarrassed or rendered obscure by the use of such metaphorical expressions, it is requisite to be careful not to allow these words to suggest to us any opinion about the manner in which mechanical forces produce their effects. If the opinion which is held of the existence of an invisible fluid on which mechanical action depends be well founded, it is obvious that there can be neither attraction nor repulsion in the universe.

100. Forces are conceived as measurable quantities. Thus we conceive one man to possess double the strength measurable of another man, when we observe that he can resist the quantities, combined efforts of two others. It is in this way that animal force is conceived as a quantity made up of its own parts and measured by them. This however seems not to be a very accurate conception. Our conception of one strain being added to another is obscure, although we have a distinct notion of their being combined. There are no words to express the difference of these two notions in our minds; but we think that the same difference is perceived by others. We have a clear conception of the addition of two lines or two minutes; but our notions of two forces combined are indistinct; although it cannot be affirmed that two equal forces are not double of one of them. They are measured by the effects which they are known to produce.

101. In the same way mechanical forces are conceived as measurable by their effects, and thus become the subjects of mathematical discussion. We speak of the proportions of magnetism, electricity &c., and even of the proportion of gravity to magnetism. These however, considered in themselves, are quite different, and do not admit of any proportion; but some of their effects are measurable, and these assumed measures being quantities of the same kind are susceptible of comparison. The acceleration of motion in a falling body, is one of the effects of gravity; magnetism accelerates the motion of a piece of iron; and these two accelerations may be compared together. But because none of the measurable effects of magnetism with which we are acquainted Part II.

Thus of the first kind are the terms accelerating, attractive, or repulsive forces; of the second, are the terms magnetism, electricity, &c.

Of the Laws of Motion.

105. Such then being our notions of mechanical forces, of the causes of the production of motion and its changes, there are certain results, which by the constitution of the human mind, necessarily arise from the relations of these ideas. These results are laws of human judgment, independent of all experience of external nature. Some of these laws may be intuitive, presenting themselves to the mind as soon as the ideas which they involve are presented to it. These may be called axioms. Others may be as necessary results from the relations of these notions, are less obvious, and may require a process of reasoning to establish their truth.

Of these laws there are three, which were first distinctly propounded by Sir Isaac Newton. These may be considered as the first principles of all discussions in mechanical philosophy, give a sufficient foundation for all the doctrines of Dynamics, and to these principles we may refer for the elucidation of all the mechanical phenomena of nature.

First Law of Motion.

106. Every body continues in a state of rest, or of uniform rectilinear motion, unless it is affected by some mechanical force.

On the truth of this proposition the whole of mechanical philosophy chiefly depends. But with regard to its truth and the foundation on which it rests, the opinions of philosophers are very different. In general these opinions are obscure and unsatisfactory; and, as is usual, they influence the discussions of those who hold them in all their investigations.

107. It is not only the popular opinion that a state of rest is the natural state of body, and that motion is something foreign to it, but the same opinion has been supported by many philosophers. They allow that of body matter unless it is acted on by some moving force will remain at rest; and nothing seems necessary for matter to remain where it is, but its continuing to exist. But the case is widely different, according to these philosophers, with respect to matter in motion. For here the relations of the body to other things are continually changing; and as there is the continual production of an effect, the continual agency of a changing cause is necessary. This metaphysical argument, it is said, is fully confirmed by the most familiar observation. All motions, whatever may have been their violence, terminate in rest, and for their continuance the continual exertion of some force is necessary.

108. It is affirmed by these philosophers, that the continual action of the moving cause is essentially requisite for the duration of the motion. But their opinions of the nature of this cause are not uniform. According to some, all the motions in the universe are produced and continued by the direct agency of the Deity himself. By others all the motions and changes of every particle of matter are ascribed to a sort of mind which is inherent in it. This is called an elemental mind. It is the same as the ἀνάγκη and the ἐνεργεία of Aristotle. Every thing, according to these philosophers, Of Moving Bodies, which moves, is mind, and every thing which is moved is body. But this elemental mind is only known and characterized by the effects which are ascribed to its action; and these are observed in the motions or changes which are produced. These, we learn from uniform experience, are regulated by laws equally precise with the laws of mathematical truth. But there is nothing which indicates anything like intention or purpose; none of the marks or characters by which mind was brought first into view. They resemble the effects produced by the exertions of corporeal force; and hence the word force has been applied to express the causes of motion.

109. A state of rest, it has been supposed, is the natural state of matter. But it does not appear that the continued action of some cause is necessary for continuing matter in motion. Experience gives us no authority for supposing that the natural condition of matter is a state of rest. It cannot be affirmed of any body whatever that it has ever been seen in absolute rest. All the parts of the planetary system are in motion; and even the sun himself with his attendant planets is carried in a certain direction with a great velocity. There is no unquestionable evidence that any of the stars are absolutely fixed; and many of them, it has been ascertained by observation, are in motion. Rest, therefore being so rare a condition of matter, no experience which we have, supports the notion that this is its natural condition. This opinion seems to be derived from our own experiments on matter. To continue the motion of a body, we find that the continued action of some moving force is necessary, otherwise the motion becomes gradually slower, and at last terminates in rest. Since then we see that our own exertions are constantly necessary in the production of motion, and especially in those cases where we are interested; we are thus induced to ascribe to matter something that is naturally quiescent and inert, and even something that is sluggish and aversive from motion. But this is an erroneous conception, which is suggested to our thoughts from the imperfection of language. We ascribe animation to matter to give it motion, and endow it with a kind of moral character in order to explain the phenomena of motion.

110. But more accurate and more extended observation leads us to conclude that matter has no peculiar aptitude to a state of rest. Every observed retardation has a distinct reference to external circumstances. Wherever there is a diminution of motion, it is invariably accompanied by the removal of obstacles; as in the case when a ball moves through fluid, or air, or water. The diminution of motion is also owing to opposite motions which are destroyed. And it is found that the more these obstacles are kept out of the way, the less is the diminution of motion. The vibration of a pendulum in water soon ceases; it continues longer in air; and much longer in the exhausted receiver. The conclusion then from these observations is, that if all obstacles could be completely removed, motion would continue for ever. This conclusion is strongly supported by the motions of the heavenly bodies. These motions, so far as we know, are retarded by no obstacles; and accordingly they have been observed to retain them without perceptible diminution for thousands of years.

111. The inactivity of matter has been denied by other philosophers. According to them it is essentially active, and continually undergoing changes in its condition. Some traces of this doctrine are to be found in the writings of some of the ancient philosophers; but it others was reduced to a systematic form by Leibnitz. According to this philosopher, every particle of matter is endowed with a principle of individuality. This he calls a monad, which is supposed to have a kind of perception of its place in the universe, and of its relation to all other parts of the universe. This monad too is supposed to act on the particle of matter in the same way as the soul acts on the body. The motion of the material particle is modified by the monad, and thus are produced, according however to unalterable laws, all the observed modifications of motion. And thus matter, or the particles of matter, are continually active and continually changing their situation. No information in any way useful can be obtained from this fanciful hypothesis. It is not unlike the system of elemental minds. And should its existence be admitted, it would not any more than the actions of animals invalidate the general proposition which is considered as the fundamental law of motion. The powers of the monads or of the elemental minds are supposed to be the causes of all the changes; but the particle of matter itself is subject to the law, and any change of motion which it exhibits is ascribed to the exertion of the monad.

112. By another set of philosophers, this law of motion is deduced from the want of a determining cause. At the head of this sect is Sir Isaac Newton, who maintains the doctrine affirmed in the proposition. But these determiners are not uniform in their opinion of the foundation on which it rests. It is asserted by some that it is a kind of necessary truth which arises from the nature of the thing. If, for instance, a body in a state of rest, and if it be affected that it will not remain at rest, it must move in some direction; and if it be in motion in any direction, and with any velocity, and do not continue its equable, rectilinear motion, it must be either accelerated or retarded; it must either turn to one side, or to some other side. The event, whatever it be, is individual and determinate; but no cause which can determine it being supposed, the determination cannot take place, and no change with respect to motion will happen in the condition of the body. It will either remain at rest, or persevere in its rectilinear and equable motion. But to this argument of sufficient reason, as it has been called, considerable objections may be made. In the immensity and perfect uniformity of time and space, there is no determining cause why the visible universe should exist in one place rather than in another, or at this time rather than at another. It is essentially necessary that there should be a cause of determination; for a determination may be without a cause, as well as a motion without a cause.

113. Other philosophers deduce this law of motion and from experience. They consider it merely as an experimental truth, of the universality of which there are innumerable proofs. When a stone is thrown from the hand, it is pressed forward, and when the hand has the greatest velocity that we can give it, the stone is let go, and it continues in that state of motion which it gradually acquired along with the hand. A stone may be thrown much farther by means of a sling, because with Of Moving a very moderate motion of the hand, the stone being whirled round acquires a very great velocity, and when it is let go, it continues its rapid motion. We have a similar illustration in the case of an arrow shot from a bow. The string which presses hard on the notch of the arrow carries it forward with an accelerated motion as it becomes a straight line by the unbending of the bow; and there being nothing to check the arrow, it flies off. In these simple cases of perseverance in a state of motion the procedure of nature is easily traced; it is perceived almost intuitively. In many other phenomena it is not less difficult, although somewhat more complicated. A man can stand on the saddle of a horse at a gallop, and step from it to the back of another horse that gallops along with him at the same rate; and this he seems to do with the same ease as if the horses were standing still. The man is carried along with the same velocity as the horse which gallops under him, and he retains the same velocity while he steps from the back of one horse to that of the other. But if the horse to which he steps were standing still, he would fly over his head, because he is carried forward with the velocity of the galloping horse; or if he stepped from the back of a horse standing still to that of one at a gallop, he would be left behind; because he has not acquired the velocity of the galloping horse. In the same way, a man tosses oranges from one hand to the other while he is carried forward with the motion of a horse at a gallop, or while he swings on the slack-wire. In both cases the oranges have the same motion as the man, and while they are in the air are moving forward with the same velocity, so that they drop into the hand at a considerable distance from the place in which they were thrown from the other hand. While a ship sails forward with a rapid motion a ball dropped from the mast head falls at the foot of the mast; for it retains the motion which it had previous to its being dropped, and follows the mast during the whole time of its fall.

114. Familiar instances may also be given of a body in a state of rest. A vessel filled with water drawn suddenly along the floor, leaves the water behind, which is dashed over the posterior side of the vessel; and when a boat or coach is suddenly dragged forward, the persons in it find themselves strike against the hinder part of the carriage or boat; or rather it should be said the carriage strikes on them, for it sooner acquires motion from the action of the force applied. A ball discharged from a cannon will pass through a wall and move onward; but the wall remains behind.

115. Common experience is perhaps insufficient for establishing the truth of this fundamental proposition. It must be granted, that we have never seen a body either at rest, or in uniform rectilinear motion; yet this seems necessary before it can be said that the proposition is experimentally established. What is supposed in our experiments to be putting a body, formerly at rest, into motion, is in fact only producing a change of a very rapid motion—a motion not less than 90,000 feet per second.

116. For the purpose of obtaining such experimental proof of the truth of this proposition, it will be necessary to refer to other observations. The relative motions of bodies, which are the differences of their absolute motions, only can be measured. We cannot measure their absolute motions. If then it can be shown by experiment that bodies have equal tendencies to resist the augmentation and diminution of their relative motions, they thus have equal tendencies to resist the augmentation or diminution of their absolute motions.

Let A and B two bodies be put into such a situation, that they cannot persevere in their relative motions. The change which we observe produced on A is the effect and measure of the tendency of B to persevere in its former state. From the proportion of these changes therefore we derive the proportion of their tendencies to remain in their former condition. This will be illustrated by the following experiment which should be made at noon.

117. Let the body moving at the rate of three feet per second to the westward, strike the equal body B which is apparently at rest. Different cases of the results of the changes thus produced may be supposed.

118. Let A impel B forward without having its own velocity at all diminished. From this result it appears that B shows no tendency to maintain its motion unchanged, but that A retains its motion without diminution.

2d. Suppose that A stops, and that B remains at rest. This case shows that A does not resist a diminution of motion, and that the motion of B is not changed.

3d. Let it be supposed that both move westward at the rate of one foot per second. There is in this case a diminution of the velocity in A, equal to two feet per second. This then is to be considered as the effect and measure of the tendency of B to maintain its velocity unaltered. B has received an augmentation of one foot per second in its velocity. From this change it appears that the tendency is but half of the former; and the result shows that the resistance to a diminution of velocity is only equal to one half of the resistance to augmentation; and perhaps equal only to one quarter, since the change on B has effected a double change on A.

4th. Let it be supposed that both bodies move forward with the velocity of one and a half feet per second. In this case it is obvious that the tendencies of the two bodies to maintain their states unchanged are equal.

5th. But suppose that $A = 2B$, and that the velocity of both after collision is equal to two feet per second. The body B has then received an addition of two feet per second to its former velocity; and this is the effect and measure of the whole tendency of A to preserve its motion undiminished. One half of this change on B measures the persevering tendency of one half of A; but it is supposed that A which formerly moved with the apparent or relative velocity three, now moves with the velocity two, and thus has lost the velocity of one foot per second. Therefore each half of A has lost this velocity; and the whole loss of motion is two. This then is the measure of the tendency of B to maintain its former state unaugmented; and it is the same with the measure of the tendency of A to preserve its former state undiminished. From such a result therefore the conclusion would be that bodies have equal tendencies to maintain their former states of motion unaugmented and undiminished.

The suppositions made above in the 4th and 5th cases are the result of all the experiments which have been made; and in all the changes of motion which are, Of Moving produced by the mutual action of bodies on impulsion, this is the regulating law. To this there is no exception. And thus it appears that there exists in bodies no preferable tendency to rest. No fact can be adduced which should lead us to suppose that a motion having once begun should suffer any diminution without the intervening action of some changing cause.

118. It must, however, be observed that this is a very imperfect way of establishing the first law of motion. It is inapplicable to those cases where experiment cannot be made; and at best it is subject to all the inaccuracy of the best managed experiments. If this proposition be examined by means of the general principles which have been adopted in the article Philosophy, (which see) an accurate decision of this question may be given. These principles, which are the foundation of all our knowledge, show that this proposition is an axiom or intuitive consequence of the relations of those ideas which we have of motion, of its changes, and of their causes.

119. Powers or forces, it has been shewn, are not the immediate objects of our perceptions. Their existence, kind, and degree, are inferences from the motions which we observe. And hence it follows, that when no change of motion is observed, no such inference is made; no force or power is supposed to act. But when any change of motion is observed, the inference is made; a power or force is supposed to have acted. By a similar conclusion, it is said, that when no change of motion is supposed, no force is thought of or supposed; and whenever a change of motion is supposed, it always implies a changing force. On the other hand, when the action of a changing force is supposed, the change of motion is also supposed; the action of this force and the change of motion being the same thing. The mind does not admit the idea of the action, without at the same time thinking of the indication of the action, and this indication is the change of motion. And in the same way, when we do not think of the changing force, or do not suppose the action of a changing force, we suppose, although it be not expressed in terms, that there is no indication of this changing force; that there is no change. If, therefore, it be supposed that no mechanical force acts on a body, we suppose in fact that the body remains in its former condition with respect to motion. And if it be supposed that nothing accelerates or retards, or deflects the motion, it is conceived as neither accelerated nor retarded, nor deflected. Hence it follows, that we suppose the body to continue in its former state of rest or motion, unless we suppose that it is changed by some mechanical force.

120. This proposition then does not depend on the properties of body as a matter of experience or contingency. It is to us a necessary truth. It is not so much any circumstances with regard to body that are expressed in the proposition, as the operations of the mind in considering these circumstances. The truth of the proposition will not be invalidated by taking into view, that it may be essential to move in some particular direction; that it may be essential to body to stop when the moving cause ceases to act; or gradually to diminish its motion, and at last come to rest. The circumstances in the nature of body which render these modifications essentially necessary, are the causes of these modifications; and they are to be considered as changing forces.

If we should suppose that body of its own nature is capable of producing a change in its condition, this change must be effected according to some law which characterizes the nature of body. But the knowledge of this law can be obtained only by observing the deviations from uniform rectilinear motion. It then becomes indifferent whether external causes operate these changes, or they depend on the nature of the thing; for in considering the various motions of bodies, we must first consider the nature of matter as one of its mechanical affections which operates in every instance; and this brings us back to the law contained in the proposition. This is rendered more certain by reflecting, that the external causes, such as instance are gravity and magnetism, which are acknowledged to operate changes of motion, are not less unknown to us than this essential property of matter. They are, like it, only inferences from the phenomena.

121. Many philosophers, among which number may be included Newton himself, have introduced modes of expression, which suggest inadequate notions, and such getted as are incompatible with the doctrine of the proposition; for although they allow that rest is the natural condition of body, and that force is necessary for the continuation of motion, yet they speak of a power or force residing in a moving body by which it perseveres in its motion. This has been called the vis insita, or the inherent force of a moving body. Now if the motion be supposed to be continued in consequence of a force, that force must be supposed to be exerted, and it is supposed that if it were not exerted the motion would cease. The proposition, therefore, must be false. To obviate this objection it is indeed sometimes said, that the body continues in uniform rectilinear motion, unless it is acted on by some external cause. This mode of expression, however, subjects us to the impropriety of afflicting that gravity, electricity, and other mechanical forces, are external to the bodies on which they are supposed to act and to put in motion. Everything which produces a change of motion is very properly called a force; and when a change of motion is observed, the action of such a force is very properly inferred. But to give the same name to what has not this property of producing a change, and to infer the action of a force when no change is observed, is not a very accurate or consistent expression. This error has arisen from the use of analogical language in philosophical discourses.

122. But motion is not, as philosophers have imagined, the continual production of an effect. We can conceive there is such a thing as a moving cause, to which the name of force has been given. This produces motion, and the character of motion in body, which is a continual change of place. Motion is the effect of an action; and previous to the commencement of the motion, this action is equally incomplete as it is the minute after. The immediate effect of a moving force is a determination to motion, which if not obstructed by some cause would go on for ever. In this determination only the condition of the body differs from a state of rest. Motion then is a condition or mode of existence, which no more requires the continued agency of the moving cause than colour or figure. Some mechanical cause is required to change this condition into the state or condition of rest. When a moving Part II.

Of Moving moving body is brought to rest, some cause of this cessation of motion never fails to occur to the mind. A cause is no less necessary to stop the motion of body than it is to produce it. Now this cause must either reside in the body or be external to it. If it reside in the body, then it possesses a self-determining power or force, by which it may be able to stop its own motion as well as to produce it.

123. Taking this view of the subject, the opinion of a force residing in a moving body by which its motion is continued must be given up; and the remarkable difference between a body in a state of motion and a state of rest must be explained on other principles. Motion, it cannot be doubted, is necessary in the impelling body to permit the forces which are inherent in one or both bodies to continue the prelude long enough for the production of sensible motion. But whether bodies be in the condition of motion or rest, these forces are inherent in them. If we reflect on the motions that are involved in the general conception of one body being impelled and put in motion by another, we shall see that there is nothing individual transferred from the one to the other. Before collision took place the determination to motion existed only in the impelling body. After collision, both bodies possessed this condition or determination. But we have no conception, we can form no notion of the thing transferred.

124. An expression not less vague and indefinite is "vis inertiae," an indecisive term. This is the phrase "inertia," or "vis inertiae." This expression, which was introduced by Kepler, seems to have been generally employed by him as well as by Newton to express the fact of the perseverance of body in a state of motion or rest. Sometimes, however, it has been employed by these philosophers to express something like indifference to motion or rest; and this is supposed to be manifested by body requiring the same quantity of force to make an augmentation of its motion, as is necessary to produce an equal diminution of it. To suppose resistance from a body at rest seems to be in direct contradiction to the common use of the word "force"; and yet this expression "vis inertiae" is very common. It is not less absurd to say that a body remains in the condition of rest by the exertion of a "vis inertiae," than to affirm that it maintains itself in a state of motion by the exertion of an inherent force. Such expressions, which are metaphorical, should be carefully avoided, because they are apt to lead to misconception of the procedure of nature.

125. In the phenomena of motion the force employed of matter always produces its complete effect. No resistance whatever is observed. When one man throws down another, and he finds that no more force has been required than to throw down a similar and equal mass of inanimate matter, he concludes that no resistance has been made; but if more force be necessary, the conclusion is that resistance has been made. When, therefore, the exerted force produces its full effect, there is no such thing as resistance properly so called. It is therefore misconceiving the mode in which mechanical forces operate in the collision of bodies, to say that there is any resistance. For there is no more in these cases than in other natural changes of condition. It may be observed, that these terms "inherent force," and "inertia," may be employed for the purpose of abbreviation language, provided they are used only for expressing either the simple fact of persevering in the former state, or the necessity of a determinate force to produce a change on that state, being careful to avoid all thought of resistance.

126. Thus it appears that deviations from uniform motions are the only indications of the existence and from unagency of mechanical forces. This indication is simple, implying change of place; and it can only indicate what is actions of very simple, something competent to the production of force. The same thing is indicated by two similar changes of motion. A compass needle in a state of rest, can be moved some degrees by means of the finger, a magnet, an electrified body, or by the unbending of a spring, &c., in all which cases the indication is precisely the same; and therefore the thing indicated must also be the same. This is the intensity and direction of some moving power. The circumstances of resemblance by which the affections of matter are to be characterized are impulsiveness, intensity, and direction. This leads us to consider the second law of motion.

Second Law of Motion.

Every change of motion is proportional to the force impressed, and it is made in the direction of that force.

127. This law of motion also may almost be considered as an identical proposition. It is equivalent to saying that the changing force is to be measured by the position; change produced, and the direction of this force is the direction of the change. Considering the force only in the sense of its being the cause of motion, and withdrawing the attention from the manner or form of its exertion, there can be no doubt of this. In whatever way a body is put in motion, whether by the expansive force of the air, by the unbending of a spring, or by any similar prelude, when it moves off in the same direction, and with the same velocity, the force or the exertion of the force is considered as the same. Even when it is put in motion by instantaneous percussion from a smart stroke, although in this case the manner of the effect being produced is essentially different from the other cases, we cannot conceive the propelling force, as such, but as precisely one and the same. The expression of this law of motion by Newton is equivalent to saying, "that the changes of motion are taken as the measures of the changing forces, and the direction of the change is taken as the indication of the direction of the forces; for it cannot be said that it is a deduction from the acknowledged principle, that deduced from effects are proportional to their causes. This law is not affirmed from the proportion of the forces and the forces, proportion of the changes, and that these proportions are the same, having been observed; and that this universally holds in nature. For forces are not objects of observation, and we do not know their proportions. In this way it would be established as a physical law, as indeed it is in fact. But according to the definition of the term, this does not establish it as a law of motion; or as a law of human thought, the result of the relations of our ideas. Philosophers having attempted to prove this as a matter of observation, have produced great diversity of opinion in the mode of eliminating forces. A bullet, it is well known, which moves with double velocity, penetrates four times as far. This is confirmed.

Of Moving confirmed by other similar facts; and to generate this double velocity in the bullet, it has been observed by philosophers four times the force is expended, four times as much powder is required. This is the invariable result; and in cases of this kind, it would appear that the ratio of the forces employed has been very accurately ascertained. The conclusion therefore is, that moving forces are not proportional to the velocities produced, but to the squares of the velocities. This is strongly confirmed by observing that moving bodies seem to possess forces in this very proportion, and to produce effects in this proportion; when, for instance, the velocity is only twice as great, they penetrate four times as deep.

128. If this mode of estimation be just, it is irreconcilable with the conception of those who admit that the velocity is proportional to the force impressed, in those cases where no previous observation can be had of the ratio of the forces, and of its equality to the ratio of the velocities. Such a case is the force of gravity, which these philosophers always measure by its accelerating power, or the velocity generated in a given time. This must be granted; for there are cases in which the force can be measured by the actual pressure which it exerts. Thus a spring steelyard can be constructed, the rod of which is divided by hanging on successively a number of perfectly equal weights. In the different states of tension of the spring, its elasticity is proportional to the pressures of gravity which it balances. If it be found, that at Quito in Peru, a weight will pull out the rod to the mark 312, and that the same weight at Spitzbergen draws it out to 313, it seems to be a fair inference to say, that the pressure of gravity at Quito is to its pressure at Spitzbergen as 312 to 313; and this is affirmed on the authority of effects being proportional to their causes. Such cases, however, are very rare; for it is seldom, that the whole of a natural power, accurately measured in some other way, is employed in producing the observed motion. Part of it is generally otherwise expended, and therefore it frequently happens that the motions are not in the proportion with the supposed forces. And allowing that this could be done with accuracy, it would only be the proof of a general law or fact; but these philosophers attempt to establish it as an abstract truth.

129. It seems to be considered by Sir Isaac Newton only as a physical law. And in this sense good arguments are not wanting. A ball which moves with a double, triple, or quadruple velocity, generates by impulse in another, a double, triple, or quadruple velocity, or it generates the same velocity in a double, triple, or quadruple quantity of matter, and losing at the same time similar proportions of its own velocity.

Two bodies, having equal quantities of motion, meeting together, mutually stop each other.

When two forces, which act similarly during equal times, produce equal velocities in a third body, they will, by acting together during the same time, produce a double velocity.

If a pressure which acts for a second, produce a certain velocity, a double pressure acting during a second, will produce in the same body a double velocity.

A force which is known to act equably, produces in equal times equal increments of velocity, whatever the velocities may be.

In all the examples above adduced, the forces are observed to be in the same proportion with the change of motion effected by them in a similar way.

But the curious discoveries of Dr Hooke, about the middle of the 17th century, seemed to show, from a great collection of facts, forces to be in a very different proportion. In the production of motion it was found, that four springs equal in strength, and bent to the same degree, generated only a double velocity in the ball which they impelled; nine springs generated only a triple velocity, &c. In the extinction of motion, it was found, that a ball moving with a double velocity, will penetrate four times as deep into a uniformly resisting mass; and a triple velocity will make it penetrate nine times as far, &c.

130. These facts were brought forward by Leibnitz in support of his own pretensions to the discovery of the real nature and measure of mechanical action and force, which he said had been hitherto totally mistaken. He affirmed, that the inherent force of a moving body was in the proportion of the square of the velocity. In this argument he was supported by John Bernoulli, who adduced many simple facts to confirm the relation between the inherent force of a moving body and its velocity. One of the strongest arguments urged by Leibnitz is, that the inherent force of a moving body is to be estimated by all that it is able to do before the total extinction of its motion; and therefore, when it penetrates four times as far, it is to be considered as having produced a quadruple effect. In this mode of estimation many things are gratuitously assumed, many contradictions are incurred; and it is only because forces are assumed as proportional to the velocities which they generate, that these facts come to be proportional to the squares of the same velocities. When Leibnitz assumes the quadruple penetration as the proof of the quadruple force of a body having twice the velocity, he has not considered that a double time is employed during this penetration. But a double force, acting equably during a double time, should produce a quadruple effect. This circumstance is lost sight of in all the facts which this philosopher has adduced. It may, however, be observed, that Leibnitz, as well as his followers, holds no difference of opinion in all the consequences which are deduced from the measure which is here adopted. They admit, that a force producing an uniformly accelerated motion must be constant; they agree with the followers of Des Cartes in the valuations both of accelerating and deflecting forces; and have assiduously and successfully cultivated the philosophy of Newton, which proceeds on the principle of estimating the measure of moving forces by the velocity generated.

131. It ought here to be observed, that moving forces only are taken into consideration. When a ball has acquired a certain velocity, whether it has been impelled by the elasticity of the air, by a spring, or struck off by a blow, or urged forward by means of a stream of air or water, or has obtained its velocity by falling; in all these cases it is conceived that it has sustained the same action of moving force. The only distinct notion, perhaps, which we are able to form, is pressure; The following is the enunciation, adapted to the characteristic and measure of a change of motion.

Law of the Changes of Motion.

PROP. XII.

134. In every change of motion, the new motion is compounded of the former motion, and of the motion which the changing produces in a body at rest.

Let the change of motion be from AB (fig. 23.) to AD, this new motion AD is compounded of the former motion AB and of the motion AC.

For it has been shewn, that the change in any motion, is that motion, which when compounded with the former motion, produces the new motion; and the new motion (55.) is the compound of the former motion and the changing motion. Since then the change of motion is the mark and measure of the changing force (133.) by which both the direction and intensity or velocity produced, are determined, the truth of the proposition will appear of course.

133. It has been already observed (54.), that the composition of motions and the similar composition of forces are very different things. The first is a pure mathematical truth; the second, is a physical question dependent on the nature of the mechanical forces as they exist in the universe. Our notions are not very distinct of two forces, each of which separately produces motions, having the directions and velocities expressed by the sides of a parallelogram, producing by their joint action a motion in the diagonal. The demonstrations which have been frequently given, are altogether inconclusive, and only include the composition of motions; while gratuitous postulates have been assumed by those who endeavoured to accommodate their reasonings to physical principles. The celebrated Daniel Bernoulli gave the first legitimate demonstration of this proposition, in which, however, he employs a series of many propositions, some of which are very arbitrary. It was greatly simplified by D'Alembert, Mem. Acad. des Sciences 1769, still, however, requiring many propositions. Ingenious demonstrations have also been given by other celebrated mechanicians. In the following demonstration by Professor Robison, this distinguished philosopher has attempted to combine the demonstration of Bernoulli, D'Alembert, and others, thus rendering it more expeditious, and at the same time legitimate. This demonstration is entirely limited to preludes, without at all considering or employing the motions supposed to be produced by them.

(A) If two equal and opposite preludes or incitements to motion act at once on a material particle, it suffers no change of motion; for if it yields in either direction by their joint action, one of the preludes prevails, and they are not equal.

Equal and opposite preludes are said to balance each other; and such balance must be esteemed equal and opposite.

(B) If \(a\) and \(b\) are two magnitudes of the same kind, proportional to the intensities of two preludes which act in the same direction, then the magnitude \(a + b\) will measure the intensity of the prelude, which is equivalent, and may be called equal, to the combined effort of the other two; for when we try to form a notion of pressure as a measurable magnitude, distinct from motion... Of Moving Forces.

Of Moving Forces.

If Moving or any other effect of it, we find nothing that we can measure it by but another prelude. Nor have we any notion of a double or triple prelude different from a prelude that is equivalent to the joint effort of two or three equal preludes. A prelude \(a\) is accounted triple of a prelude \(b\), if it balances three preludes, each equal to \(b\), acting together. Therefore, in all proportions which can be expressed by numbers, we must acknowledge the legitimacy of this measurement; and it would surely be affectation to omit those which the mathematicians call incommensurable.

The magnitude \(c = b\), in like manner, must be acknowledged to measure that prelude which arises from the joint action of two preludes \(a\) and \(b\) acting in opposite directions, of which \(a\) is the greatest.

(C) Let \(ABCD\) and \(abC'd'\) (fig. 24.) be two rhombuses, which have the common diagonal \(AC\). Let the angles \(BA_b\), \(DA_d\), be bisected by the straight lines \(AE\) and \(AF\).

If there be drawn from the points \(E\) and \(F\) the lines \(EG\), \(EH\), \(FG\), \(FH\), making equal angles on each side of \(EA\) and \(FA\), and if \(Gg\), \(Hh\) be drawn, cutting the diagonal \(AC\) in \(I\) and \(L\); then \(AI + AL\) will be greater or less than \(AQ\), the half of \(AC\), according as the angles \(GEH\), \(gFh\), are greater or less than \(GAH\), \(gAh\).

Draw \(GH\), \(gH\), cutting \(AE\), \(AF\), in \(O\) and \(o\), and draw \(Oo\), cutting \(AC\) in \(K\).

Because the angles \(AEG\) and \(EAG\) are respectively equal to \(AEH\) and \(EAH\), and \(AE\) is common to both triangles, the sides \(AG\), \(GE\) are respectively equal to \(AH\), \(HE\), and \(GH\) is perpendicular to \(AE\), and is bisected in \(O\); for the same reasons, \(gH\) is bisected in \(o\). Therefore the lines \(Gg\), \(Oo\), \(Hh\), are parallel, and \(IL\) is bisected in \(K\). Therefore \(AI + AL\) is equal to twice \(AK\). Moreover, if the angle \(GEH\) be greater than \(GAH\), \(AO\) is greater than \(EO\), and \(AK\) is greater than \(KQ\). Therefore \(AI + AL\) is greater than \(AQ\); and if the angle \(GEH\) be less than \(GAH\), \(AI + AL\) is less than \(AQ\).

(D) Two equal preludes, acting in the directions \(AB\) and \(AC\) (fig. 25.), at right angles to each other, compose a prelude in the direction \(AD\), which bisects the right angle; and its intensity is to the intensity of each of the constituent preludes as the diagonal of a square to one of the sides. It is evident, that the direction of the prelude, generated by their joint action, will bisect the angle formed by their directions; because no reason can be assigned for the direction inclining more to one side than to the other.

In the next place, since a force in the direction \(AD\) does, in fact, arise from the joint action of the equal preludes \(AB\) and \(AC\), the prelude \(AB\) may be conceived as arising from the joint action of two equal forces similarly inclined and proportioned to it. Draw \(EAF\) perpendicular to \(AD\). One of these forces must be directed along \(AD\), and the other along \(AE\). In like manner, the prelude \(AC\) may arise from the joint action of a prelude in the direction \(AD\), and an equal prelude in the direction \(AF\). It is also plain, that the preludes in the directions \(AE\) and \(AF\), and the two preludes in the direction \(AD\), must be all equal. And also, any one of them must have the same proportion to \(AB\) or to \(AC\); that \(AB\) or \(AC\) has to the force in the direction \(AD\), arising from their joint action.

Therefore, if it be said that \(AD\) does not measure the prelude arising from the joint action of \(AB\) and \(AC\), let \(Ad\) greater than \(AD\), be its just measure, and make \(Ad : AB = AG : AB = AE\). Then \(AG\) and \(AE\) have the same inclination and proportion to \(AB\) that \(AB\) and \(AC\) have to \(Ad\). We determine, in like manner, two forces \(Af\) and \(Ag\) as constituents of \(AC\).

Now \(Ad\) is equivalent to \(AB\) and \(AC\), and \(AB\) is equivalent to \(AE\) and \(AG\); and \(AC\) is equivalent to \(Af\) and \(Ag\). Therefore \(Ad\) is equivalent to \(AE\), \(Af\), \(AG\), and \(Ag\). But \((A)\) \(AE\) and \(Af\) balance each other, or annihilate each other's effect; and there remain only the two forces, or preludes \(AG\), \(Ag\). Therefore \((B)\) their measure is a magnitude equal to twice \(Ag\). But if \(Ad\) be greater than the diagonal \(AD\) of the square, whose sides are \(AB\) and \(AC\); then \(Ag\) must be less than \(AI\), the side of the square whose diagonal is \(AB\). But twice \(Ag\) is less than \(AD\), and much less than \(Ad\). Therefore the measure of the equivalent of \(AB\) and \(AC\) cannot be a line \(Ad\) greater than \(AD\). In like manner, it cannot be a line \(Ag\) that is less than \(AD\). Therefore it must be equal to \(AD\), and the proposition is demonstrated.

COROLLARY.

(E) Two equal forces \(AB\), \(AC\), acting at right angles, will be balanced by a force \(AO\), equal and opposite to \(AD\), the diagonal of the square whose sides are \(AB\) and \(AC\); for \(AO\) would balance \(AD\), which is the equivalent of \(AB\) and \(AC\).

(F) Let \(AECF\) (fig. 26.) be a rhombus, the acute angle of which \(EAF\) is half of a right angle. Two equal preludes, which have the directions and measures \(AE\), \(AF\), compose a prelude, having the direction and measure \(AC\), which is the diagonal of the rhombus.

It is evident, in the first place, that the compound force has the direction \(AC\), which bisects the angle \(EAF\). If \(AC\) be not its just measure, let it be \(AP\) less than \(AC\). Let \(ABCD\) be a square described on the same diagonal, and make \(AP : AQ = AE : AO = AF : Ao\). Draw \(KOG\), \(Kog\) perpendicular to \(AE\), \(AF\); draw \(GIg\), \(OHo\), \(EG\), \(EK\), \(Fg\), \(FK\), \(PF\), and \(PE\).

The angles \(CAB\) and \(FAE\) are equal, each being half of a right angle. Also the figures \(AEPF\) and \(AGEK\) are similar, because \(AP : AQ = AE : AO\). Therefore \(FA : AP = KA : AE\), and \(EA : AP = GA : AE\). Therefore, in the same manner that the forces \(AE\), \(AF\) are affirmed to compose \(AP\), the forces \(AG\) and \(AK\) may compose the force \(AE\), and the forces \(Ag\) and \(AK\) may compose the force \(AF\). Therefore \((B)\) the force \(AP\) is equivalent to the four forces \(AG\), \(AK\), \(Ag\), \(AK\). But \((D)\) \(AG\) and \(Ag\) are the sides of a square, whose diagonal is equal to twice \(AI\); and the two forces \(AK\), \(AK\) are equal to, or are measured by, twice \(AK\). Therefore the four forces \(AG\), \(AK\), \(Ag\), \(AK\), are equivalent to \(2AI + 2AK = 4AH\).

But because \(AP\) was supposed less than \(AC\), the angle \(FPE\) is greater than \(FAE\), and \(GEK\) is greater than \(GAK\), \(AO\) is greater than \(OE\), and \(AH\) is greater than \(HQ\), and \(2AH\) is greater than \(AQ\); and therefore \(4AH\) is greater than \(AC\), and much greater than Part II.

Of Moving than AP. Therefore AP is not the just measure of Forces.

The force composed of AE and AF.

In like manner, it is shewn, that AE and AF do not compose a force whose measure is greater than AC. It is therefore equal to AC; and the proposition is demonstrated.

(G) By the same process it may be demonstrated, that if BAD be half a right angle, and EAF be the fourth of a right angle, two forces AE, AF will compose a force measured by AC. And the process may be repeated for a rhombus whose acute angle is one-eighth, one-sixteenth, &c. of a right angle; that is, any portion of a right angle that is produced by continual bisection. Two forces, forming the sides of such a rhombus, compose a force measured by the diagonal.

(H) Let ABCD, A b c d (fig. 27.) be two rhombuses formed by two consecutive bisections of a right angle. Let AECF be another rhombus, whose sides AE and AF bisect the angles BA b and DA d.

The two forces AE, AF, compose a force AC.

Bisect AE and AF in O and o. Draw the perpendiculars GOH, g o h, and the lines GIg, OK o, HL h, and the lines EG, EH, Fg, Fh.

It is evident, that AGEH and AGFh are rhombuses; because AO = OE, and Ao = OF. It is also plain, that since b A d is half of BAD, the angle GAH is half of b A d. It is therefore formed by a continual bisection of a right angle. Therefore (G) the forces AG, AH, compose a force AE; and Ag, A h, compose the force AF. Therefore the forces AG, AH, Ag, A h, acting together, are equivalent to the forces AE, AF acting together. But AG, Ag compose a force = 2 AI; and the forces AH, A h compose a force = 2 AL. Therefore the four forces acting together are equivalent to 2 AI + 2 AL, or to 4 AK. But because AO is 1/2 AE, and the lines Gg, Oo, Hh, are evidently parallel, 4 AK is equal to 2 AQ, or to AC; and the proposition is demonstrated.

Corollary.

(I) Let us now suppose, that by continual bisection of a right angle we have obtained a very small angle a of a rhombus; and let us name the rhombus by the multiple of a which forms its acute angle.

The proposition (G) is true of a, 2a, 4a, &c. The proposition (H) is true of 3a. In like manner, because (G) is true of 4a and 8a, proposition (H) is true of 6a; and because it is true of 4a, 6a, and 8a, it is true of 5a and 7a. And so on continually till we have demonstrated it of every multiple of a that is less than a right angle.

(K) Let RAS (fig. 28.) be perpendicular to AC, and let ABCD be a rhombus, whose acute angle BAD is some multiple of 2a that is less than a right angle. Let A b c d be another rhombus, whose sides A b, A d bisect the angles RAB, SAD. Then the forces A b, A d compose a force AC.

Draw bR, dS parallel to BA, DA. It is evident, that AR b B and AS d D are rhombuses, whose acute angles are multiples of a, that are each less than a right angle: Therefore (I) the forces AR and AB compose the force A b, and AS AD compose A d; but AR and AS annihilate each other's effect, and there remains only the forces AB, AD. Therefore

A b and A d are equivalent to AB and AD, which compose the force AC; and the proposition is demonstrated.

Corollary.

(L) Thus is the corollary of last proposition extended to every rhombus, whose angle at A is some multiple of a less than two right angles. And since a may be taken less than any angle that can be named, the proposition may be considered as demonstrated of every rhombus; and we may say,

(M) Two equal forces, inclined to each other in any angle, compose a force which is measured by the diagonal of the rhombus, whose sides are the measures of the constituent forces.

(N) Two forces AB, AC (fig. 29.) having the direction and proportion of the sides of a rectangle, compose a force AD, having the direction and proportion of the diagonal.

Draw the other diagonal CB, and draw EAF parallel to it; draw BE, CF parallel to DA.

AEBG is a rhombus; and therefore the forces AE and AG compose the force AB. AFCG is also a rhombus, and the force AC is equivalent to AF and AG. Therefore the forces AB and AC, acting together, are equivalent to the forces AE, AF, AG, and AG acting together, or to AE, AF, and AD acting together: But AE and AF annihilate each other's action, being opposite and equal (for each is equal to the half of BC). Therefore AB and AC acting together, are equivalent to AD, or compose the force AD.

(O) Two forces, which have the direction and proportions of AB, AC (fig. 30.) the sides of any parallelogram, compose a force, having the direction and proportion of the diagonal AD.

Draw AF perpendicular to BD, and BG and DE perpendicular to AC.

Then ATBG is a rectangle, as is also AFDE; and AG is equal to CE. Therefore (N) AB is equivalent to AF and AG. Therefore AB and AC acting together, are equivalent to AR, AG, and AC acting together; that is, to AF and AE acting together; that is (N) to AD; or the forces AB and AC compose the force AD.

Hence arises the most general proposition.

If a material particle be urged at once by two pre-compositions or incitements to motion, whose intensities are proportion of all to the sides of any parallelogram, and which incitements act in the directions of those sides, it is affected in the same manner as if it were acted on by a single force, whose intensity is measured by the diagonal of the parallelogram, and which acts in its direction: Or, two pre-compositions, having the direction and proportion of the sides of a parallelogram, generate a pre-composition, having the direction and proportion of the diagonal.

136. Thus is demonstrated from abstract principles the perfect similarity of the composition of pre-compositions and the composition of forces measured by the motions which are produced. A separate demonstration seems indifferently necessary; for what may be deduced from the one case is not always applicable to the other. The change produced on a motion already existing by a deflecting force, cannot be explained by any composition of pre-compositions; because the changing pre-composition is the only one that exists, and there is none with which it may be compounded.

Of Moving Forces.

137. Considering this law of motion merely as a universal fact or physical law, abundant proof may be adduced in support of it.

1. The joint action of different forces is quite familiar. A lighter, for example, is dragged in different directions by two ropes on different sides of the canal, the lighter moving in an intermediate direction, as if dragged in that direction by one rope only. A ball moving in a particular direction, which receives a stroke across this direction, takes a direction lying between that of the first motion and that of the transverse stroke.

2. If a particle of matter A (fig. 23.) be urged at once by two pressures in the directions AB and AC; and if AB and AC be proportional to the intensities of those pressures, the joint action of these two pressures is equivalent to the action of a third pressure in the direction of the diagonal AD, and having its intensity in the proportion of AD. This is proved by observing, that the point A is withheld from moving by a pressure AE, which is equal and opposite to AD. But pressures are moving forces, producing velocities when they act similarly during equal times, proportional to their intensities. The proportion, therefore, is true with respect to pressures, considered merely as such, and also with respect to the motions which may be produced by their composition.

3. The weight of a ball which is suspended by a thread, and drawn aside from its position in a state of rest, urges it downwards, and the ball is supported obliquely by the thread. Supposing this proposition to be true, the directions and intensities of the forces inciting it to motion in any position, as well as the result of the velocities, can be precisely ascertained.

4. The motions of the planets computed on these principles of the composition of forces, do not exhibit any perceptible deviation from calculation, at the end of thousands of years.

Nothing, therefore, can be relied on with greater confidence than the perfect agreement between the composition of motions, and the composition of the forces, which, separately taken, would produce those motions, and which are measured by the velocities produced. But it ought to be remarked, that if the moving forces are measured by the squares of the velocities which they generate, the composition cannot possibly hold; namely, from two forces which are represented by the sides of a parallelogram made proportional to the squares of the velocities, there will not result a force which can be represented by the diagonal. But supposing the composition of forces to be as the velocities, nature exhibits them exactly.—This proposition, therefore, whether it be considered as an abstract truth or as a physical law, may be received as fully established. The following is the converse of this proposition.

PROP. XIII.

138. The force by which the motion AB is changed into AD, is that which would produce in a body at rest, the motion AC, and this compounded with AB produces the observed motion AD.

PROP. XIV.

139. The force which will produce in a body at rest a motion having the direction and velocity represented by AC, when applied to a body moving with the velocity and in the direction AB, will change its motion into the motion AD, which is the diagonal of the parallelogram ABDC. For the new motion must be that which is compounded of AB and AC, that is, it must be the motion AD.

The combination of these two propositions gives rise to the following, which is still more general.

PROP. XV.

140. A body A being urged at once by two forces, which separately would cause it to describe AB and AC, the sides of a parallelogram ABDC, the body by their joint action will describe the diagonal AD in the same time.

For if the body had been already moving with the velocity and in the direction AB, and if it had been acted on in A by the force AC, it would describe AD in the same time. But it matters not at what time it acquired the determination to describe AB. Let it be then at the instant that the force AC is applied to it. And because its mechanical condition in A, which has the determination to the motion AB, is the same as in any other point of that line, it must describe AD.

COROLLARY.

Two forces acting on a body in the same or in opposite directions, will cause it to move with a velocity equal to the sum or to the difference of the velocities which it would have received from the forces separately. For, if AC approach continually to AB by diminishing the angle BAC, the points C and D will at last fall on a and d, and then AD is equal to the sum of AB and AC. But if the angle BAC increase continually, the points C and D will at last fall on x and δ, and then Aδ becomes equal to the difference of AB and AC. In the last case, it is evident, that if AC be equal to AB, the point D or δ will coincide with A, and the two forces being equal and acting in opposite directions, there will be no motion.

141. In such cases the equal and opposite forces AC and AB are said to balance each other; and it is generally said, that these forces, by whose joint operation no change of motion is produced, balance each other. Such forces are accounted equal and opposite, each producing on the body a change of motion equal to what it would produce on a body at rest, and at the same time equal to the motion produced by the other force on a body at rest. The two motions being equal and opposite, the forces are therefore equal and opposite.

142. What has been demonstrated concerning the affections with respect to the affections of compound motions, may now be applied to the combination of forces; taking care, however, to recollect the essential difference between the composition of motions and the composition of forces. In the combination of forces, the composition is complete, when the determination has been given to the body to move with the proper velocity in the diagonal. When the body has acquired this determination, there is no farther composition; and it continues its uniform motion, till its condition be changed. Part II.

Of Moving Forces.

On the other hand, in the composition of two or more motions, the constituent motions are supposed to continue; and it is only during their continuance that the compound motion exists. If it be possible, which does not appear to be the case, that any force can generate a finite velocity by its instantaneous action, two such forces generate in an instant the determination in the diagonal. But supposing the action to continue for some time, to generate the velocities \(AB\) or \(AC\), there must be a continuance of the joint action during the same time to produce the velocity \(AD\). And although the moving powers of the two forces may vary in their intensity, yet it is necessary that they retain the same proportion to each other during the whole time of their joint action. Overlooking this circumstance, experiments have been made for the purpose of comparing this doctrine with the phenomena; and they have been found to exhibit very different results. But experiments made by the combination of pressures, such as weights pulling a body by means of threads, coincide precisely with this doctrine; for it is always found that two weights pulling in the directions \(AB\), \(AC\), and proportional to those lines, are balanced by a third weight in the proportion of \(AD\), and pulling in the direction \(AE\). In this way the composition of pressures is clearly proved; and, having no other distinct conception of a moving force, these experiments may be considered as sufficient. But we may go farther; for there is the clearest proof by experiment, that pressures produce motions in proportion to their intensities by their similar action during equal times. In the planetary motions, the directions and intensities of the compound forces are accurately known as moving forces. These motions afford a complete proof of the physical law, by their perfect coincidence with the calculations which proceed on the principles of this doctrine. This coincidence must be acknowledged as a full proof of the propriety of the measure which has been assumed. The assumption of any other measure would exhibit results quite different from the phenomena.

143. Forces which produce motions along the sides of a parallelogram are called simple forces, or constituent forces. And the force which singly produces the motion in the diagonal, is called the equivalent force, the compound force, or the resulting force.

144. Some general conclusions may now be pointed out, which will facilitate greatly the use of the parallelogram of forces.

General Corollaries.

1. The constituent and the resulting forces, or the simple and compound forces, act in the same plane; for the sides and diagonal of a parallelogram are in one plane.

2. The simple and the compound forces are proportional to the sides of any triangle which are parallel to their directions. For if any three lines \(ab\), \(bd\), \(ad\), be drawn parallel to \(AB\), \(AC\), and \(AD\) (fig. 31.), they will form a triangle similar to the triangle \(ABD\). For the same reasons they are proportional to the sides of a triangle \(a'b'd'\), which are respectively perpendicular to their directions.

3. Therefore each is proportional to the fine of the opposite angle of this triangle; for the sides of any triangle are proportional to the fines of the opposite angles.

4. Each is proportional to the fine of the angle contained by the directions of the other two; for \(AD\) is to \(AB\) as the fine of the angle \(ABD\) to the fine of the angle \(ADB\). Now the fine of \(ABD\) is the same with the fine of \(BAC\) contained between the directions \(AB\) and \(AC\), and the fine of \(ADB\) is the same with the fine of \(CAD\); also \(AB\) is to \(AC\), or \(BD\), as the fine of \(ADB\) (or \(CAD\)) to the fine of \(BAD\).

145. Let us now proceed to the application of this special use fundamental proposition. And we observe, in the first of the places, that since \(AD\) may be the diagonal of an indefinite number of parallelograms, the motion or the pressure \(AD\) may result from the joint action of many pairs of forces. It may be produced by forces which would separately produce the motions \(AF\) and \(AG\). This generally gives us the means of discovering the forces which concur in its production. If one of them, \(AB\), is known in direction and intensity, the direction \(AC\), parallel to \(BD\), and the intensity, are discovered. Sometimes we know the directions of both. Then, by drawing the parallelogram or triangle, we learn their proportions. The force which deflects any motion \(AB\) into a motion \(AD\), is had by simply drawing a line from the point \(B\) (to which the body would have moved from \(A\) in the time of really moving from \(A\) to \(D\)) to the point \(D\). The deflecting force is such as would have caused the body move from \(B\) to \(D\) in the same time. And, in the same manner, we get the compound motion \(AD\), which arises from any two simple motions \(AB\) and \(AC\), by supposing both of the motions to be accomplished in succession. The final place of the body is the same, whether it moves along \(AD\) or along \(AB\) and \(BD\) in succession.

146. This theorem is not limited to the composition Equivalent of two forces only; for since the combined action of many two forces puts the body into the same state as if their forces equivalent alone had acted on it, we may suppose this to have been the case, and then the action of a third force will produce a change on this equivalent motion. The resulting motion will be the same as if only this third force and the equivalent of the other two had acted on the body. Thus, in fig. 32., the three forces \(AB\), \(AC\), \(AE\), may act at once on a particle of matter. Complete the parallelogram \(ABDC\); the diagonal \(AD\) is the force which is generated by \(AB\) and \(AC\). Complete the parallelogram \(AEFD\); the diagonal \(AF\) is the force resulting from the combined action of the forces \(AB\), \(AC\), and \(AE\). In like manner, completing the parallelogram \(AGHF\), the diagonal \(AH\) is the force resulting from the combined action of \(AB\), \(AC\), \(AE\), and \(AG\), and so on of any number of forces.

This resulting force and the resulting motion may be much more expeditiously determined, in any degree of composition, by drawing lines in the proportion and direction of the forces in succession, each from the end of the preceding. Thus, draw \(AB\), \(BD\), \(DF\), \(FH\), and join \(AH\); \(AH\) is the resulting force. The demonstration is evident.

147. In the composition of more than two forces, we are not limited to one plane. The force \(AD\) is in the same plane with \(AB\) and \(AC\); but \(AE\) may be elevated above this plane, and \(AG\) may lead below it. Of Moving AF is in the plane of AD and AE, and AH is in the plane of AF and AG.

Complete the parallelograms ABLE, ACKE, ELFK. It is evident that ABLFKCD is a parallelopiped, and that AF is one of its diagonals. Hence we derive a more general and very useful theorem.

Three forces having the proportion and direction of the three sides of a parallelopiped, compose a force having the proportion and direction of the diagonal.

148. In the investigation of very complicated phenomena, the mechanician considers every force as resulting from the joint action of three forces at right angles to each other, and he takes the sum or difference of these in the same or opposite directions. Thus he obtains the three sides of a parallelopiped, and from these computes the position and magnitude of the diagonal. This is the force resulting from the composition of all the partial ones. This process is called the elimination or reduction of forces. Forces may be estimated in the direction of a given line or plane, or they may be reduced to that direction, as has been done with respect to motion. See Cor. 2. Propos. 9. in Art. 57.

The laws of motion which have now been considered, are necessary consequences of the relations of those conceptions which we form of motion and mechanical force, and they are universal facts or physical laws. To these Sir Isaac Newton has added another, which is the following.

Third Law of Motion.

149. Every action is accompanied by an equal and contrary reaction, or the actions of bodies on one another are always mutual, equal, and in contrary directions.

In all cases which can be accurately examined, this holds to be a universal fact. Newton has made this affirmation on the authority of what he conceives to be a law of human thought; namely that the qualities discovered in all bodies on which experiments and observations can be made, are to be considered as universal qualities of body. But if the term law of motion be limited to those consequences that necessarily flow from our notions of motion, of the causes of its production and changes, this proposition is not such a result. Because a magnet causes the iron to approach toward it, it by no means follows from this observation that the presence of the iron shall be accompanied by any motion or change of state of the magnet, or it does not appear to be necessarily supposed that the iron attracts the magnet. When this was observed, it was accounted a discovery, and a discovery which is to be ascribed to the ancients. Dr Gilbert, who first mentions it, affirms that the magnet and the iron are observed mutually to attract each other, as well as all electrical substances, and the light bodies which are attracted by them. The discovery was made by Kepler that a mutual attraction exists between the earth and the moon. Newton discovered that the sun acts on the planets, and that the earth acts on the moon. It had been observed too by Newton that the iron reacts on the magnet, that the actions of electrified bodies are mutual, and that all the actions of solid bodies are accompanied by an equal and contrary reaction. On the authority of the rule of philosophizing which he had laid down, he affirmed that the planets react on the sun, and that the sun is not at rest, but is continually agitated by a small motion round the general centre of gravitation; and he pointed out several of the consequences of this reaction.

As the celestial motions were more narrowly examined by astronomers, their consequences were found to obtain, and to produce disturbances in the planetary motions. This reciprocity of action is now found to hold with the utmost precision through the whole of the solar system; and therefore this third proposition of Newton is to be considered as a law of nature. And it is true with respect to all bodies on which experiment or observation can be made.

150. This then being a universal law, we cannot divest our minds of the belief that it depends on a general principle, by which all the matter in the universe is influenced. It strongly induces the persuasion of the ultimate particles of matter being alike, that a certain number of properties belong in the same degree to each atom, and that all the sensible differences of substance which are observed, arise from a different combination of those primary atoms in the formation of a particle of those substances. All this is no doubt perfectly possible. But if each primary atom be so constituted, no action of any kind of particle or collection of particles can take place on another, which is not accompanied by an equal reaction in the opposite direction.

151. Let us now direct our attention to the application of these laws. This answers a twofold purpose. The first is to discover the mechanical powers of natural substances by which they are fitted to become parts of a permanent universe. This is accomplished by observing the changes of motion which always accompany those substances. It is from these changes that the only characteristics of powers are derived; and thus is discovered the power of gravity, of magnetism, &c. Another purpose in the employment of these laws is, that, after having obtained the mechanical character of any substance, we may ascertain what will be the result of its being in the vicinity of the bodies mechanically allied, or we may ascertain what is the change induced on the condition of the neighbouring bodies.

152. The mechanical powers of bodies occasionally produce accelerations, retardations, and deflections in the motions of other bodies. These names have been given, because nothing is known of their nature, or of the manner in which they are effective; they are therefore named, as they are measured by the phenomena which are observed and considered as their effects. Let us now attend a little to the principal circumstances relating to the action of these forces.

Of Accelerating and Retarding Forces.

153. Changes of motion are the only marks and measures of changing forces; and having no other mark of the force but the acceleration, it has obtained the name of an accelerating force. When the motion is retarded it is called retarding force. Nor is there any other measure of the intensity of an accelerating force, but the acceleration which it produces. To investigate therefore the powers which produce all the changes of motion it is necessary to obtain measures of the acceleration. What has been said of accelerations and retardations of motion is equally descriptive of the effects of accelerating and retarding forces. Hence the following proposition.

If Corollaries.

Cor. 1. The momentary change of velocity is as the force \( f \) and the time \( t \) jointly. It may be thus expressed:

\[ v_t = \frac{v}{t} = \frac{f}{t} \]

Also, the accelerating or retarding force is proportional to the momentary variation of the velocity, directly, and to the moment of time in which it is generated, inversely (48).

\[ f = \frac{v}{t}, \quad \text{or} \quad \frac{v}{t} = \frac{f}{t} \]

Indeed, all that we know of force is that it is something which is always proportional to \( \frac{v}{t} \).

Cor. 2. Uniformly accelerated or retarded motion is the indication of a constant or invariable accelerating force. For, in this case, the areas \( abfe, acge, \ldots \) increase at the same rate with the times \( ab, ac, \ldots \), and therefore the ordinates \( ae, bf, cg, \ldots \) must all be equal; therefore the forces represented by them are the same, or the accelerating force does not change its intensity, or, it is constant. If, therefore, the circumstances mentioned in articles 37 and 38, are observed in any motion, the force is constant. And if the force is known to be constant, those propositions are true regarding the motions.

Cor. 3. No finite change of velocity is generated in an instant by an accelerating or retarding force. For the increment or decrement of velocity is always expressed by an area, or by a product \( ft \), one side or factor of which is a portion of time. As no finite space can be described in an instant, and the moveable must pass in succession through every point of the path, so it must acquire all the intermediate degrees of velocity. It must be continually accelerated or retarded.

Cor. 4. The change of velocity produced in a body in any time, by a force varying in any manner, is the proper measure of the accumulated or whole action of the force during this time. For, since the momentary change of velocity is expressed by \( \frac{f}{t} \), the aggregate of all these momentary changes, that is, the whole change of velocity, must be expressed by the sum of all the quantities \( \frac{f}{t} \). This is equivalent to the area of the figure employed in art. 148, and may be expressed by \( \int f \cdot t \).

Corollary.

The momentary change on the square of the velocity is as the force, and as the small portion of space along which it acts, jointly;

\[ v^2 = \frac{f}{s} \]

and

\[ f = \frac{v^2}{s} \]

155. It deserves remark here, that as the momentary change of the simple velocity by any force \( f \) depends only on the time of its action, it being \( = \frac{f}{t} \) (148.) Cor. 1. to the change on the square of the velocity depends on the space, it being \( = \frac{f}{s} \). It is the same, whatever is the velocity thus changed, or even though the body be at rest when the force begins to act on it. Thus, in every second of the falling of a heavy body, the velocity is augmented 32 feet per second, and in every foot of the fall, the square of the velocity increases by 64.

156. The whole area \( AE \cdot ea \), expressed by \( \int f \cdot s \), expresses the whole change made on the square of the velocity which the body had in \( A \), whatever this velocity may have been. We may therefore suppose the body to have been at rest in \( A \). The area then measures the square of the velocity which the body has acquired in the point \( E \) of its path. It is plain that the change on \( v^2 \) is quite independent on the time of action, and therefore a body, in passing through the space \( AE \) with any initial velocity whatever, sustains the same change of the square of that velocity, if under the influence of the same force.

157. This proposition is the same with the 39th of the First Book of Newton's Principia, and is perhaps the most generally useful, of all the theorems in Dynamics, in the solution of practical questions. It is to be found, without demonstration, in his earliest writings, the Optical Lectures, which he delivered in 1669 and following years.

158. One important use may be made of it at present. It gives a complete solution of all the facts which were observed by Dr Hooke, and adduced by Leibnitz, with such pertinacity in support of his measure of the force of moving bodies. All of them are of precisely the same nature with the one mentioned in art. 157, or with the fact, "that a ball projected directly upwards with a double velocity, will rise to a quadruple height," and that a body, moving twice as fast, will penetrate "four times as far into a uniformly tenacious mass." The uniform force of gravity, or the uniform tenacity of the penetrated body, makes a uniform opposition to the motion, and may therefore be considered as a uniform retarding force. It will therefore be represented, in fig. 8, by an ordinate always of the same length, and the areas which measure the square of the velocity lost will be portions of a rectangle \( AE \cdot ea \). If therefore \( AE \) be the penetration necessary for extinguishing the velocity \( v_2 \), the space \( AB \), necessary for extinguishing the velocity \( v_1 \), must be \( \frac{1}{4} \) of \( AE \), because the square of \( 1 \) is \( \frac{1}{4} \) of the square of \( 2 \).

159. What particularly deserves remark here, is, that this proposition is true, only on the supposition that forces are proportional to the velocities generated by them in equal times. For the demonstration of this proposition proceeds entirely on the previously established measure of Of Moving of acceleration. We had \( \dot{v} = f \dot{i} \); therefore \( v = f i \).

But \( \dot{v} = \dot{s} \); therefore \( v = f s \), which is precisely this proposition.

160. Those may be called similar points of space, and similar instants of time, which divide given portions of space or time in the same ratio. Thus, the beginning of the 5th inch, and of the 2nd foot, are similar points of a foot, and of a yard. The beginning of the 21st minute, and of the 9th hour, are similar instants of an hour, and of a day.

Forces may be said to act similarly when, in similar instants of time, or similar points of the path, their intensities are in a constant ratio.

161. Lemma. If two bodies be similarly accelerated during given times \( ac \) and \( hk \) (fig. 33.), they are also similarly accelerated along their respective paths AC and HK.

Let \( a, b, c \) be instants of the time \( ac \), similar to the instants \( h, i, k \) of the time \( hk \). Then, by the similar accelerations, we have the force \( ae : hf : im \). This being the case throughout, the area \( af \) is to the area \( hm \) as the area \( ag \) to the area \( hn \). These areas are as the velocities in the two motions 48. Therefore the velocities in similar instants are in a constant ratio, that is, the velocity in the instant \( b \) is to that in the instant \( i \), as the velocity in the instant \( c \) to that in the instant \( k \).

The figures may now be taken to represent the times of the motion by their abscissae, and the velocities by their ordinates, as in art. 28. The spaces described are now represented by the areas. These being in a constant ratio, as already shewn, we have A, B, C, and H, I, K, similar points of the paths. And therefore, in similar instants of time, the bodies are in similar points of the paths. But in these instants, they are similarly accelerated, that is, the accelerations and the forces are in a constant ratio. They are therefore in a constant ratio in similar points of the paths, and the bodies are similarly accelerated along their respective paths (155.)

162. If two particles of matter are similarly urged by accelerating or retarding forces during given times, the whole changes of velocity are as the forces and times jointly; or \( v = ft \).

For the abscissae \( ac \) and \( hk \) will represent the times, and the ordinates \( ae \) and \( hl \) will represent the forces, and then the areas will represent the changes of velocity, by art. 47. And these areas are as \( ac \times ae \) to \( hk \times hl \).

Hence \( t = \frac{v}{f} \), and \( f = \frac{v}{t} \).

163. If two particles of matter are similarly impelled or opposed through given spaces, the changes in the squares of velocity are as the forces and spaces jointly; or \( v^2 = fs \).

This follows, by similar reasoning, from art. 49.

It is evident that this proposition applies directly to the argument to confidently urged for the propriety of the Leibnitzian measure of forces, namely, that four springs of equal strength, and bent to the same degree, generate, or extinguish only a double velocity.

164. If two particles of matter are similarly impelled through given spaces, the spaces are as the forces and the squares of the times jointly.

For the moveables are similarly urged during the times of their motion (converse of 156.) Therefore \( v = f t \), and \( v^2 = f s \); but (158.) \( v^2 = f s \). Therefore \( f s = f^2 t^2 \) and \( s = f t^2 \).

Corollary.

\( t^2 = \frac{s}{f} \) and \( f = \frac{s}{t^2} \). That is, the squares of the times are as the spaces, directly, and as the forces, inversely; and the forces are as the spaces, directly, and as the squares of the times, inversely.

165. The quantity of motion in a body is the sum of the motions of all its particles. Therefore, if all are moving in one direction, and with one velocity \( v \), and if \( m \) be the number of particles, or quantity of matter, \( mv \) will express the quantity of motion \( q \), or \( q = mv \).

166. In like manner, we may conceive the accelerating forces \( f \), which have produced this velocity \( v \) in each particle, as added into one sum, or as combined on one particle. They will thus compose a force, which, for distinction's sake, it is convenient to mark by a particular name. We shall call it the motive force, and express it by the symbol \( p \). It will then be considered as the aggregate of the number \( m \) of equal accelerating forces \( f \), each of which produces the velocity \( v \) on one particle. It will produce the velocity \( mv \), and the same quantity of motion \( q \).

167. Let there be another body, consisting of \( n \) particles, moving with one velocity \( u \). Let the moving force be represented by \( \pi \). It is measured in like manner by \( nu \). Therefore we have, \( p : \pi = mv : nu \), and \( v : u = \frac{p}{m} : \frac{\pi}{n} \); that is,

The velocities which may be produced by the similar action of different motive forces, in the same time, are directly as these forces, and inversely as the quantities of matter to which they are applied.

In general, \( \frac{v}{f} = \frac{p}{m} \).

And \( f \) being \( \frac{v}{t} \), \( f = \frac{p}{mt} \).

Remark.

168. In the application of the theorems concerning accelerating or retarding forces, it is necessary to attend carefully to the distinction between an accelerative and a motive force. The caution necessary here has been generally overlooked by the writers of Elements, and this has given occasion to very inadequate and erroneous notions of the action of accelerating powers. Thus, if a leaden ball hangs by a thread, which passes over a pulley, and is attached to an equal ball, movable along a horizontal plane, without the smallest obstruction, it is known, that, in one second, it will descend 8 feet, dragging the other 8 feet along the plane, with a uniformly accelerated motion, and will generate in it the velocity 16 feet per second. Let the thread be attached to three such balls. We know that it will descend 4 feet in a second, and generate the velocity 8 feet per second. Most readers are disposed to think that it should generate no greater velocity than \( \frac{5}{3} \) feet per second, or \( \frac{1}{3} \) of 16, because it is applied to three times as much matter (162.) The error Of Moving error lies in considering the motive force as the same in both cases, and in not attending to the quantity of matter to which it is applied. Neither of these conjectures is right. The motive force changes as the motion accelerates, and in the first case, it moves two balls, and in the second it moves four. The motive force decreases similarly in both motions. When these things are considered, we learn by articles 202 and 207, that the motions will be precisely what we observe.

Of Deflecting Forces, in General.

169. It was observed, in art. 71, that a curvilinear motion is a case of continual deflection. Therefore, when such motions are observed, we know that the body is under the continual influence of some natural force, acting in a direction which crooks that of the motion in every point. We must infer the magnitude and direction of this deflecting force by the magnitude and direction of the observed deflection. Therefore, all that is affirmed concerning deflections in the first and subsequent articles, may be affirmed concerning deflecting forces. It follows, from what has been established concerning the action of accelerating forces, that no force can produce a finite change of velocity in an instant. Now, a deflection is a composition of a motion already existing with a motion accelerated from rest by insensible degrees. Supposing the deflecting force of invariable direction and intensity, the deflection is the composition of a motion having a finite velocity with a motion uniformly accelerated from rest. Therefore the linear deflection from the rectilineal motion must increase by insensible degrees. The curvilinear path, therefore, must have the line of undeflected motion for its tangent. To suppose any finite angle contained between them would be to suppose a polygonal motion, and a sublunary deflection.

Therefore no finite change of direction can be produced by a deflecting force in an instant.

170. The most general and useful proposition on this subject is the following, founded on art. 75.

The forces by which bodies are deflected from the tangents in the different points of their curvilinear paths are proportional to the squares of the velocities in those points, directly, and inversely to the deflective chords of the equicurve circles in the same points. We may still express the proposition by the same symbol

\[ f = \frac{v^2}{c} \]

where \( f \) means the intensity of the deflecting force.

171. We may also retain the meaning of the proposition expressed in article 76, where it is shown that the actual linear deflection from the tangent is the third proportional to the deflective chord and the arch described in a very small moment. For it was demonstrated in that article (see fig. 18.) that \( BZ : BC = BC : BO \).

We see also that \( Bb \), the double of \( BO \), is the measure of the velocity, generated by the uniform action of the deflecting force, during the motion in the arch \( BC \) of the curve.

172. The art. 77, also furnishes a proposition of frequent and important use, viz.

The velocity in any point of a curvilinear motion is that which the deflecting force in that point would generate in the body by uniformly impelling it along the fourth part of the deflective chord of the equicurve circle.

Remark.

173. The propositions now given proceed on the supposition that, when the points \( A \) and \( C \) of fig. 18, after continually approaching to \( B \), at last coalesce with it, the last circle which is described through these three points has the same curvature which the path has in \( B \). It is proper to render this mode of solving these questions more plain and palpable.

If \( ABCD \) (fig. 35.) be a material curve or mould, and a thread be made fast to it at \( D \), this thread may be lapped on the convexity of this curve, till its extremity meets it in \( A \). Let the thread be now unlapped or evolved from the curve, keeping it always tight. It is plain that its extremity \( A \) will describe another curve line \( Abc \). All curves, in which the curvature is neither infinitely great nor infinitely small, may be thus described by a thread evolved from a proper curve. The properties of the curve \( Abc \) being known, Mr Huyghens (the author of this way of generating curve lines) has shewn how to construct the evolved curve \( ABC \) which will produce it.

From this genesis of curves we may infer, 1st, that the detached portion of the thread is always a tangent to the curve \( ABC \); 2dly, that when this is in any situation \( Bb \), it is perpendicular to the tangent of the curve \( Abc \) in the point \( b \), and that it is, at the same time, describing an element of that curve, and an element of a circle \( abx \), whose momentary centre is \( B \), and which has \( Bb \) for its radius. 3dly, That the part \( bA \) of the curve, being described with radii growing continually shorter, is more incurvated than the circle \( ba \), which has \( Bb \) for its constant radius. For similar reasons the arch \( bc \) of the curve \( Abc \) is less incurvated than the circle \( abx \). 4thly, That the circle \( abx \) has the same curvature that the curve has in \( b \), or is an equicurve circle. \( Bb \) is the radius, and \( B \) the centre of curvature in the point \( b \).

\( ABC \) is the curva evoluta or the evolute. \( Abc \) is sometimes called the involute of \( ABC \), and sometimes its evolventrix.

174. By this way of describing curve lines, we see clearly that a body, when passing through the point \( b \) of the curve \( Abc \) may be considered as in the same state, in that instant, as in passing through the same point \( b \) of the circle \( abx \); and the ultimate ratio of the deflections in both is that of equality, and they may be used indiscriminately.

The chief difficulty in the application of the preceding theorems to the curvilinear motions which are observed in the spontaneous phenomena of nature, is in ascertaining the direction of the deflection in every point of a curvilinear motion. Fortunately, however, the most important cases, namely those motions, where the deflecting forces are always directed to a fixed point, afford a very accurate method. Such forces are called by the general name of

Central Forces.

175. If bodies describe circles with a uniform motion, the deflecting forces are always directed to the centres of the Of Moving the circles, and are proportional to the square of the velocities, directly, and to their distances from the centre, inversely.

For, since their motion in the circumference is uniform, the areas formed by lines drawn from the centre are as the times, and therefore (72.) the deflections, and the deflecting forces (164.) are directed to the centre. Therefore, the deflective chord is, in this case, the diameter of the circle, or twice the distance of the body from the centre. Therefore, if we call the distance from the centre \(d\), we have \(f = \frac{v^2}{d}\).

176. These forces are also as the distances, directly, and as the square of the time of a revolution, inversely.

For the time of a revolution (which may be called the periodic time) is as the circumference, and therefore as the distance, directly, and as the velocity, inversely. Therefore \(t = \frac{d}{v}\), and \(v = \frac{d}{t}\), and \(v^2 = \frac{d^2}{t^2}\), and therefore \(f = \frac{d}{t^2}\).

177. These forces are also as the distances, and the square of the angular velocity, jointly.

For, in every uniform circular motion, the angular velocity is inversely as the periodic time. Therefore, calling the angular velocity \(a\), \(a^2 = \frac{1}{t^2}\), and \(d^2 = d \cdot a^2\), and therefore \(f = \frac{d}{a^2}\).

178. The periodic time is to the time of falling along half the radius by the uniform action of the centripetal force in the circumference, as the circumference of a circle is to the radius.

For, in the time of falling through half the radius, the body would describe an arch equal to the radius (37.—6.) because the velocity acquired by this fall is equal to the velocity in the circumference (167.). The periodic time is to the time of describing that arch as the circumference to the arch, that is, as the circumference is to the radius.

179. When a body describes a curve which is all in one plane, and a point is so situated in that plane, that a line drawn from it to the body describes round that point areas proportional to the times, the deflecting force is always directed to that point (72.)

180. Conversely. If a body is deflected by a force always directed to a fixed point, it will describe a curve lying in one plane which passes through that point, and the line joining it with the centre of forces will describe areas proportional to the times (73.)

The line joining the body with the centre is called the radius vector. The deflecting force is called centripetal, or attractive, if its direction be always toward that centre. It is called repulsive, or centrifugal, if it be directed outwards from the centre. In the first case, the curve will have its concavity toward the centre, but, in the second case, it will be convex toward the centre. The force which urges a piece of iron towards a magnet is centripetal, and that which causes two electrical bodies to separate is centrifugal.

181. The force by which a body may be made to describe circles round the centre of forces, with the angular velocities which it has in the different points of its curvilinear path, are inversely as the cubes of its distances from the centre of forces. For the centripetal force in circular motions is proportional to \(da^2\) (172.) But when the deflections (and consequently the forces) are directed to a centre, we have \(a = \frac{1}{d^2}\) (75.) and \(a^2 = \frac{1}{d^4}\), therefore \(da^2 = d \times \frac{1}{d^4} = \frac{1}{d^3}\), therefore \(f = \frac{1}{d^3}\).

This force is often called centrifugal, the centrifugal force of circular motion, and it is conceived as always acting in every case of curvilinear motion, and to act in opposition to the centripetal force which produces that motion. But this is inaccurate. We suppose this force, merely because we must employ a centripetal force, just as we suppose a resisting vis inertiae, because we must employ force to move a body.

182. If a body describe a curve line ABC by means of a centripetal (fig. 36.) force directed to S, and varying according to some proportion of the distances from it, and if another body be impelled toward S in the straight line aS by the same force, and if the two bodies have the same velocity in any points A and a which are equidistant from S, they will have equal velocities in any other two points C and c, which are also equidistant from S.

Describe round S, with the distance SA, the circular arch Aa, which will pass through the equidistant point a. Describe another arch Bb, cutting off a small arc AB of the curve, and also cutting AS in D. Draw DE perpendicular to the curve.

The distances AS and aS being equal, the centripetal forces are also equal, and may be represented by the equal lines AD and ab. The velocities at A and a being equal, the times of describing AB and ab will be as the spaces (14.) The force ab is wholly employed in accelerating the rectilineal motion along aS. But the force AD, being transverse or oblique to the motion along AB, is not wholly employed in thus accelerating the motion. It is equivalent to the two forces AE and ED, of which ED, being perpendicular to AB, neither promotes nor opposes it, but incurvates the motion. The accelerating force in A therefore is AE. It was shewn, in art. 48, that the change of velocity is as the force and as the time jointly, and therefore it is as AE \(\times\) AB. For the same reason, the change of the velocity at a is as ab \(\times\) ab, or ab². But, as the angle ADB is a right angle, as also AED, we have AE : AD = AD : AB, and AE \(\times\) AB = AD² = ab². Therefore, the increments of velocity acquired along AB and ab are equal. But the velocities at A and a were equal. Therefore the velocities at B and b are also equal. The same thing may be said of every subsequent increase of velocity, while moving along BC and bc; and therefore the velocities at C and c are equal.

The same thing holds, when the deflecting force is directed in lines parallel to aS, as if to a point S' infinitely distant, the one body describing the curve line VA'B', while the other describes the straight line VS.

183. The propositions in art 73. and 74. are also true in curvilinear motions by means of central forces. When the path of the motion is a line returning into itself, like a circle or oval, it is called an orbit; otherwise it is called a trajectory.

The time of a complete revolution round an orbit is called the periodic time.

184. The formula \( f = \frac{v^2}{c} \) serves for discovering the law of variation of the central force by which a body describes the different portions of its curvilinear path; and the formula \( f = \frac{d}{r^2} \) serves for comparing the forces by which different bodies describe their respective orbits.

185. It must always be remembered, in conformity to art. 77, that \( f = \frac{v^2}{c} \) or \( f = \frac{\text{arc}^2}{c} \) expresses the linear deflection from the tangent, which may be taken for a measure of the deflecting force, and that \( f = \frac{2v^2}{c} \) or \( f = \frac{2\text{arc}^2}{c} \) expresses the velocity generated by this force, during the description of the arc, or the velocity which may be compared directly with the velocity of the motion in the arc. The last is the most accurate, because the velocity generated is the real change of condition.

186. A body may describe, by the action of a centripetal force, the direction of which passes through C (fig. 36), a figure VPS, which figure revolves (in its own plane) round the centre of forces C, in the same manner as it describes the quiescent figure, provided that the angular motion of the body in the orbit be to that of the orbit itself in any constant ratio, such as that of m to n.

For, if the direction of the orbit's motion be the same with that of the body moving in it, the angular motion of the body in every point of its motion is increased in the ratio of m to n + m, and it will be in the same ratio in the different parts of the orbit as before, that is, it will be inversely as the square of the distance from S (75). Moreover, as the distances from the centre in the simultaneous positions of the body, in the quiescent and in the revolving orbit, are the same, the momentary increments of the area are as the momentary increments of the angle at the centre; and therefore, in both motions, the areas increase in the constant ratio of m to m + n (75.) Therefore the areas of the absolute path, produced by the composition of the two motions, will still be proportional to the times; and therefore (73.) the deflecting force must be directed to the centre S; or, a force so directed will produce this compound motion.

187. The differences between the forces by which a body may be made to move in the quiescent and in the moveable orbit are in the inverse triplicate ratio of the distances from the centre of forces.

Let VKSBV (fig. 36.) be the fixed orbit, and u p k b n the same orbit moved into another position; and let VP n No N t QV be the orbit described by the body in absolute space by the composition of its motion in the orbit with the motion of the orbit itself. If the body be supposed to describe the arch VP of the fixed orbit while the axis VC moves into the situation u C, and if the arch u p be made equal to VP, then p will be the place of the body in the moveable orbit, and in the compound path V p. If the angular motion in the fixed orbit be to the motion of the moving orbit as m to n, it is plain that the angle VCP is to VCP as m to m + n. Let PK and p k be two equal and very small arches of the fixed and moving orbits. PC and p c are equal, as are also KC and k C, and a circle described round C with the radius CK will pass through k. If we now make VCK to VC n as m to m + n: the point n of the circle K k n will be the point of the compound path, at which the body in the moving orbit arrives when the body in the fixed orbit arrives at K, and p n is the arch of the absolute path described while PK is described in the fixed path.

In order to judge of the difference between the force which produces the motion PK in the fixed orbit and that which produces p n in the absolute path, it must be observed that, in both cases, the body is made to approach the centre by the difference between CP and CK. This happens, because the centripetal forces, in both cases, are greater than what would enable the body to describe circles round C, at the distance CP, and with the same angular velocities that obtain in the two paths, viz. the fixed orbit and the absolute path. We shall call the one pair of forces the circular forces, and the other the orbital. Let C and c represent the forces which would produce circles, with the angular velocities which obtain in the fixed and moving orbits, and let O and o be the forces which produce the orbital motions in these two paths.

These things being premised, it is plain that o — c is equal to O — C, because the bodies are equally brought towards the centre by the difference between O and C and by that between o and c. Therefore o — O is equal to c — C (a). The difference, therefore, of the forces which produce the motions in the fixed and moving orbits is always equal to the difference of the forces which would produce a circular motion at the same distances, and with the same angular velocity. But the forces which produce circular motions, with the angular motion that obtains in an orbit at different distances from the centre of forces, are as the cubes of the distances inversely (175.) And the two angular motions at the same distance are in the constant ratio of m to m + n. Therefore the forces are in a constant ratio to each other, and their differences are in a constant ratio to either of the forces. But the circular force at different distances is inversely as the cube of the distance (221). Therefore the difference of them in the fixed and moveable orbits is in the same proportion. But the difference of the orbital forces is equal to that of the circular. Therefore, finally, the difference of the centripetal

(a) For let A o, AO, Ac, AC represent the four forces o, O, c, and C. By what has been said, we find that o c = OC. To each of these add O c, and then it is plain that o = c C, that is, that the difference of the circular forces c and C is equal to that of the orbital forces o and O. Of Moving petal forces by which a body may be retained in a fixed orbit, and in the same orbit moving as determined in article 180, is always in the inverse triplicate ratio of the distances from the centre of forces.

In this example, the motion of the body in the orbit is in the same direction with that of the orbit, and the force to be joined with that in the fixed orbit is always additive. Had the orbit moved in the opposite direction, the force to be joined would have been subtractive, unless the retrograde motion of the orbit exceeded twice the angular motion of the body. But in all cases, the reasoning is similar.

188. Thus we have considered the motions of bodies influenced by forces directed to a fixed point. But we cannot conceive a mere mathematical point of space as the cause or occasion of any such exertion of forces. Such relations are observed only between existing bodies or masses of matter. The propositions which have been demonstrated may be true in relation to bodies placed in those fixed points. That continual tendency towards a centre, which produces an equable description of areas round it, becomes intelligible, if we suppose some body placed in the centre of forces, attracting the revolving body. Accordingly, we see very remarkable examples of such tendencies towards a central body in the motions of the planets round the sun, and of the satellites round the primary planet.

But, since it is a universal fact that all the relations between bodies are mutual, we are obliged to suppose that whatever force inclines the revolving body towards the body placed in the centre of forces, an equal force (from whatever source it is derived) inclines the central body toward the revolving body, and therefore it cannot remain at rest, but must move towards it. The notion of a fixed centre of forces is thus taken away again, and we seem to have demonstrated propositions inapplicable to anything in nature. But more attentive consideration will show us that our propositions are most strictly applicable to the phenomena of nature.

189. For, in the first place, the motion of the common centre of position of two, or of any number of bodies, is not affected by their mutual actions. These, being equal and opposite, produce equal and opposite motions, or changes of motion. In this case, it follows from art. 115, that the state of the common centre is not affected by them.

190. Now, suppose two bodies S and P, situated at the extremities of the line SP (fig. 37.) Their centre of position is in a point C, dividing their distance in such a manner that SC is to CP as the number of material atoms in P to the number in S or SC : PC = P : S. Suppose the mutual forces to be centripetal. Then, being equal, exerted between every atom of the one, and every particle of the other, the vis motrix may be expressed by $P \times S$. This must produce equal quantities of motion in each of the bodies, and therefore must produce velocities inversely as the quantities of matter. In any given portion of time, therefore, the bodies will move towards each other, to $s$ and $p$, and $S : s$ will be to $P : p$ as $P$ to $S$, that is, as $SC$ to $PC$. Therefore we shall still have $s : C : p C = SC : PC$. Their distances from C will always be in the same proportion. Also we shall have $SC : SP = s : P : S + P$; and therefore $SC : SP = s : C : s P$. Consequently, in whatever manner the mutual forces vary by a variation of distance from each other, they will vary in the same manner by the same variation of distance from C. And, conversely, in whatever manner the forces vary by a change of distance from C, they vary in the same manner by the same change of distance from each other.

Let us now suppose that when the bodies are at S and P, equal moving forces are applied to each in the opposite directions SA and PB. Did they not attract each other at all, they would, at the end of some small portion of time, be found in the points A and B of a straight line drawn through C, because they will move with equal quantities of motion, or with velocities SA and PB inversely as their quantities of matter. Therefore $SA : PB = SC : PC$, and A, C, and B are in a straight line. But let them now attract, when impelled from S and P. Being equally attracted toward each other, they will describe curves $Sa$ and $Pb$, so that their deflections $Aa$ and $Bb$ are as $SC$ and $PC$; and we shall have $aC : bC = SC : PC$. As this is true of every part of the curve, it follows that they describe similar curves round C, which remains in its original place.

Lastly, If the motion of P be considered by an observer placed in S, unconscious of its motion, since he judges of the motion of P only by its change of direction and of distance, we may make a figure which will perfectly represent this motion. Draw the line EF equal and parallel to PS, and EG equal and parallel to $ab$. Do this for every point of the curve $Sa$ and $Pb$. We shall then form a curve FG similar to the curves $Sa$ and $Pb$, having the homologous lines equal to the sum of the homologous lines of these two curves. Thus the bodies will describe round each other curve lines which are similar and equal (linearly) to the lines which they describe round their common centre by the same forces. They may appear to describe areas proportional to the times round each other; and they really describe areas proportional to the times round their common centre of position, and the forces, which really relate to the body which is supposed to be central, have the same mathematical relation to their common centre.

Thus it appears that the mechanical inferences, drawn from a supposed relation to a mere point of space, are true in the real relations to the supposed central body, although it is not fixed in one place.

191. The time of describing any arch FG of the curve described round the other body at rest in a centre of forces (where we may suppose it forcibly withheld from moving) is to the time of describing the similar arch $Pb$ round the common centre of position in the subduplicate ratio of $S + P$ to $S$, that is, in the ratio of $\sqrt{S + P}$ to $\sqrt{S}$. For the forces being the same in both motions, the spaces described by their similar actions, that is, their deflections from the tangent, are as the squares of the times $T$ and $t$ (204). That is, $HG : BA = T^2 : t^2$; and $T : t = \sqrt{HG} : \sqrt{BA} = \sqrt{S + P} : \sqrt{S}$.

Hence it follows that the two bodies S and P are moved in the same way as if they did not act on each other, but were both acted upon by a third body, placed in their common centre C, and acting with the same forces on each; and the law of variation of the forces by a change of distance from each other, and from this third body, is the same.

192. If a body P (fig. 38.) revolve around another body...

Of Moving body S, by the action of a central force, while S moves in any path ASB, P will continue to describe areas proportional to the times round S, if every particle in P be affected by the same accelerating force that acts, in that instant, on every particle in S. For, such action will compound the same motions Pp and Ss with the motions of S and P, whatever they are; and it was shown in art. 69, that such composition does not affect their relative motions. This is another way of making a body describe the same orbit in motion which it describes while the orbit is fixed (186).

ERRATUM in DYNAMICS.—Line 10th from the bottom, Col. 13, For fig. 11 read fig. 1. Plate CLXXXIV.

DYNA

DYANOMETER, an instrument for ascertaining the relative strength of men and animals. Of an instrument of this kind, invented by Regnier, and of which a description is given in vol. ii. Jour. de l'Ecole Polytechnique, the author thus speaks. "Some important knowledge, says he, might be acquired, had we the easy means of ascertaining, in a comparative manner, our relative strengths at the different periods of life, and in different states of health. Buffon and Gueneau, who had some excellent ideas on this subject, requested me to endeavour to invent a portable machine, which, by an easy and simple mechanism, might conduct to a solution of this question, on which they were then engaged. These philosophers were acquainted with that invented by Graham, and improved by Dr Defaguliers, at London; but this machine, constructed of wooden-work, was too bulky and heavy to be portable; and, besides, to make experiments on the different parts of the body, several machines were necessary, each suited to the part required to be tried. They were acquainted also with the dynanometer of Citizen Leroy of the Academy of Sciences at Paris. It consisted of a metal tube 10 or 12 inches in length, placed vertically on a foot like that of a candlestick, and containing in the inside a spiral spring, having above it a graduated shank terminating in a globe. This shank, together with the spring, sunk into the tube in proportion to the weight acting upon it, and thus pointed out, in degrees, the strength of the person who pressed on the ball with his hand.

This instrument, though ingenious, did not appear sufficient however to Buffon and Gueneau; for they wished not merely to ascertain the muscular force of a finger or hand, but to estimate that of each limb separately, and of all the parts of the body. I shall not here give an account of the attempts I made to fulfil the wishes of these two philosophers, but only observe, that in the course of my experiments I had reason to be convinced that the construction of the instrument was not so easy as might have been expected. Besides the use which an enlightened naturalist may make of this machine, it may be possible to apply it to many other important purposes. For example, it may be employed with advantage to determine the strength of draught cattle; and, above all, to try that of horses, and compare it with the strength of other animals. It may serve to make known how far the assistance of well-constructed wheels may favour the movement of a carriage, and what is its vis inertiae in proportion to the load. We might appreciate by it, also, what resistance the slope of a mountain opposes to a carriage, and be able to judge whether a carriage is sufficiently loaded in proportion to the number of horses that are to be yoked to it. In the arts, it may be applied to machines of which we wish to ascertain the resistance, and when we are desirous to calculate the moving force that ought to be adapted to them. It may serve, also, as a Roman balance to weigh burdens. In short, nothing would be easier than to convert it into an anemometer, to discover the absolute force of the wind, by fitting to it a frame of a determined size filled up with wax-cloth; and it would not be impossible to ascertain by this machine the recoil of fire-arms, and consequently the strength of gun-powder.

This dynanometer, in its form and size, has a near resemblance to a common graphometer. It consists of a spring twelve inches in length, bent into the form of an ellipse; from the middle of which arises a semicircular piece of brass, having engraved upon it the different degrees that express a force of the power acting on the spring. The whole of this machine, which weighs only two pounds and a half, opposes, however, more resistance than may be necessary to determine the action of the strongest and most robust horse." For a fuller description, see Phil. Mag. vol. i.