may be to conventional ideas of rigour, have never failed to introduce perplexity and obscurity to the beginner. It may be right to remind the student that the change of place introduced by Euclid (i., prop. 4 and other propositions) has not necessarily all the concomitants of the idea of motion; geometry would not interfere to prevent the superposition from being made without the notion of the triangle, whose place is changed, passing through the intervening parts of space. It was the introduction of the idea of time which the parties who objected to the doctrine of fluxions repugned. But if we consider matter in motion, we must inquire into the external causes of motion, and the capabilities of matter with respect to motion; this we shall do in the next article [MOTION, LAWS OF], confining ourselves in the present one to the first-mentioned branch of the subject. Next to the idea of motion comes that of swiftness, rate of motion, or velocity (see also the latter word), suggested by observing different motions, or different changes of place in the same time. But here we must observe, that we are rather indebted to motion for our measure of time than to time for our measure of motion. If sentient beings, like ourselves, had lived in perpetual day, without any recurrence of periodical phenomena in nature, or any mechanical means of generating equable motion, we have no right to suppose that they would ever have learned to consider time as a measurable magnitude. They might admit that it might be more or less, as we do of industry, courage, or any other moral qualities [MAGNITUDE], but we cannot be more destitute of measures for those qualities, than they would be of means for measuring time. Since however we have obtained, though by means of equable motion, a distinct idea of successions of duration, equal in magnitude, we use this idea in the definition of motion, just as in geometry we consider the line before the surface, though we have no certainty that we ever should have a distinct notion of a line, if we had not formed lines by the intersection of surfaces. We say, though we have no certainty, but we do not forget that many philosophers are of opinion that such ideas as those of time and of a line are fundamental notions, resulting from our rational organisation, and (if we do not mistake them) anterior to observation, or, at least, not derived from it. This question is here immaterial, as we suppose all parties ready to start with a definite notion of time. Considering the motion of a simple point, which describes a line, it is called uniform when the lengths described in successive equal times are equal, whatever each time may be. It is important to remember this, since different successive motions may be uniform in some respects and not in all. Thus successive revolutions may be performed in equal times, as to whole revolutions, but equal fractions of one revolution may not be performed in equal times. In uniform motion, an arbitrary unit of time is chosen, and the length described in that time is called the velocity, which is simply the Latin for quickness. If extreme verbal correctness were required, this length should be called, not the velocity or swiftness, but the measure of the velocity. For the length described in (say) one second is not the velocity or swiftness, but something by which we judge of it. The word velocity is an abstraction from the comparison of motions; of two moving points, that one which described the greater length in a given time moved the quicker and swiftness is the absolute substantive by which we express the existence of the obvious relation, just as magnitude is that by which we express the existence of the relation of greater and less. When equal spaces are not described in equal times, we can imagine the rate of motion to change either gradually or discontinuously. Thus it can be imagined that a body which moves for some seconds uniformly at the rate of 10 feet in a second, may at once, without any intermediate state, take a velocity of 20 feet. But such a conception cannot be realised on any material body, though there may be all the appearance of it. [IMPULSE.] When the rate of motion is changing perfectly gradually, there seems to be no direct method of obtaining the rate at any one instant; for no successive equal spaces are described in equal times. This difficulty will be discussed in the article VELOCITY: for the present, it may be considered sufficient to take a length so small that the change of rate undergone in passing through it is insensible, and to consider the point as moving uniformly through that length. Let the very small portion of a foot represented by 88 be described in the small fraction of a second, represented by 8t. Let v be the number of feet which would be described in one second, at the same rate. 88 Then, v:88=1:8t.·.v= ; and 88 and 8t being taken as small as δε we please, may be taken for the velocity at any instant. 88 δι The existence of motion is detected either by a change of the distance of an object, or of its direction, or both; but it is not necessarily the object which moves. The spectator himself may be in motion unconsciously, and it is matter of common experiment that every motion of the spectator of which he is not conscious, and every rapid motion, whether he be conscious of it or not, causes surrounding objects to appear in motion. In walking, the effort necessary to maintain motion perpetually reminds us that it is ourselves who move; in a carriage, at an ordinary pace, we can always destroy the illusion of surrounding motions by a moment's thought. But if the attention drop, and we look at objects with the mind intent on other things, they soon take the motion of the carriage in a contrary direction. In the smooth motion of a boat, no effort of thought will enable the spectator to realise his own motion, and destroy that of the shore or a neighbouring vessel. We state that which we find to happen to ourselves; perhaps the experience of other persons may be different. It may also happen that the object is in motion as well as the spectator, in which case the latter motion will be transferred to the former, in the manner in which we shall describe. The whole motion of the object, compounded of that which it has of its own, and that which it appears to have from the motion of the spectator, is called the apparent or relative motion. The method of ascertaining the relative motion is as follows:-Since we only determine the positions of bodies by their distances and directions, and since we suppose the motions both of the spectator and the object to be given, let a fixed point be taken to represent the position in which the spectator imagines himself to remain, and, laying down the real distances and directions of the object at the end of successive times, set off those distances from the fixed point in the proper directions. The relative positions of the object being thus secured, the line passing through these positions will be that in which the object appears to move. For instance, let the spectator move through 123...89 while the object moves through ABC...HI, BO that when the first is at 1, the second is at A; when the first is at 2, the second is at B, and so on, the last positions being 9 and 1. Take o for a fixed point, at which the spectator fancies himself to be, and having joined 1 and A, 2 and B, &c., draw o a parallel and equal to 1 A, Ob parallel and equal to 2 B, &c., and oi parallel and equal to 9 I. Hence the spectator, fixed at o, will see the object successively at the same distances and in the same directions as a, b, &c., and i; whence the line abc...hi will be that of its apparent motion. When both motions are rectilinear and uniform, the apparent motion may be more simply obtained, as follows:-Let the spectator move uniformly from o to A, while, in the same time, the object moves from B to c. Take the following method of fixing the spectator :-As he moves forward from o to A, let the paper on which the figure is drawn move backward in the direction contrary to oA, so that by the time the spectator has reached A, the point A shall have receded to where o was. He will therefore never have changed his place, his progression on the paper having been always compensated by the retrogression of the paper itself. Take CD parallel and equal to A o, whence the point c will, by the motion of the paper, be at the end of the motion, where Fig. 2. E D B D was at its beginning. Consequently, the spectator, who imagines himself at rest, will give to в that motion which is compounded of a (CD being equal and parallel to AB), E through EF (EF being parallel to and equal to one-half of A B), G through GH, K through KL, M through M N, and let P remain at rest. Then, a spectator in a supposing himself at rest, c will appear to remain at C, E will appear to move through Ef, G through Gh, K through K, м through Mn, and P through Pp. The motion of a has been transferred in a contrary direction to each of the other vessels. When bodies are very distant their changes of distance are not soon perceived, consequently it is only by change of direction that their motion becomes visible. This is the case in all the heavenly bodies; but we shall now show what the apparent motion of a planet, superior and inferior, would be, if changes of distance, as well as direction, could be perceived and estimated. If the spectator be in motion, an object at rest appears to him to have precisely his own motion, but in a contrary direction; for if the object be o and the spectator move through ABCD, no distances would Ꭺ In the first place, it a did not move round at all, the line AG would be described; if a moved slowly round, the translation of the circle would cause an undulating curve like A H K to be described; if a moved as fast on the circle as the circle itself is moved forward, the undulation would be changed into a curve with cusps like ALG; while if a move faster on the circle than the circle is carried forward, the circle, se to speak, will not have time to get out of the way, and prevent the formation of loops, as in A M N MPQRQ.... The faster a moves, the larger and the nearer will be the loops, so that at length no one will be clear of the preceding and following, or the loops will interlace. If the circle move round another circle, the same appearances will be presented in an inverse order. Let the centre E of the circle A B C D move at all upon its circle, it would, by the motion of its circle, describe If A did not (fig. 7) be carried round the circle ET, whose centre is o. a circle (dotted) equal to ET; if a moved slowly, it would describe a succession of close loops enveloping o; if quicker, the loops would at last disengage themselves from each other; while for still more rapid motion of A the loops would become cusps, and afterwards the curve South. + 13 Fig. 4. D be changed if the spectator were fixed at o, and the object moved through ABCD, and all directions would only undergo a diametrical change. Consequently the relative motion of the object is represented by allowing it to change places with the spectator, and inverting the direction of north and south, which will have the effect of making the relative motion from west to east, if that of the spectator were from east to west, and vice versa. Let us suppose now that the earth moves round the sun in a circle, which will be near enough for our present would simply undulate. The character of these curves will be further discussed under TROCHOIDAL CURVES, and their astronomical application under PLANETARY MOTIONS. It is sufficient here to say that the apparent orbits of all the planets (or rather, the orbits as they would be if changes of distance were perceptible) are trochoidal curves of the above-described species, with loops which do not interfere with one another. The composition of motion has been virtually proved in the preceding paragraphs, combined with the account of the second law of motion. MOTION, LAWS OF.] If causes of motion act instantaneously, one of which would make a body describe a B (fig. 8) uniformly, and the other purpose; it will be immediately obvious that the direction of motion, so far as concerns the order in which constellations will be described, is the same in the relative motion of the sun round the earth as in the absolute motion of the earth round the sun. For though the absolute directions of motions are opposite, yet s, to a spectator at E, is seen towards a point of the heavens opposite to that in which E appears from s. [MOTION, DIRECTION OF.] In giving to the sun the apparent motion which answers to the real motion of the earth, the same motion must be given to the orbits in which the planets are carried round the sun. The question then is as follows-If a planet move round the sun, say with a uniform circular ▲ C, in the same time, we find in the second law of motion that the body will move so that its distance from A B at the end of any time, measured parallel to a C, is what it would have been if the cause of motion in the direction AB had never existed nor acted. Suppose, for example, that three-fifths of the whole time of motion from A to B has elapsed; take A D three-fifths of c, and the body must be then somewhere in the line DE. Again, take AF three-fifths of AB, and by the same law it follows that the body must be in the line F G,-that is, it must be at the point H, which simple geometry shows to be on the diagonal A K, and by three-fifths of that diagonal distant from A. The same may be shown for any other proportion of the whole time; consequently the body, impressed with the two motions, describes the diagonal a K uniformly, and in the same time as that in which the separate motions, from A to B, or from A to c, would have been performed. This is precisely the course the body would have taken in space, if, while it moved from A to B on the paper, the paper itself had taken the motion a 0; but the establishment of the latter assertion must not be confounded with the proof of the composition of velocities impressed on matter; the latter requires those considerations which lead to the second law of There are a great many uses of the word " motion," which are convenient, but require the introduction of arbitrary suppositions. Thus the moon never cuts the ecliptic twice running in the same place, and the intersection of her orbit with the ecliptic being called a node, it is said that the node moves, thus giving this node a sort of imaginary existence in the interval. motion. In are to be found in all works on mechanics are true; the reason for such assumption being, that if we take them for granted, and use them as the basis of a mathematical system of mechanics, all results of that system, however many the links in the chain of deduction, are found to agree with observed phenomena in species, and as nearly in magnitude as the various resistances and disturbances will allow. astronomy and optics, phenomena have been predicted with all but geometrical accuracy, by deduction from principles which would certainly be false if the received laws of motion were false. In terrestrial mechanics, the number of instances is unlimited in which these laws lead to that near approximation to prediction which is fully as much as can be expected with our imperfect knowledge of data. Many hundreds of phenomena admit, upon these laws, of an explanation which, compared with that which they could receive from any MOTION, DIRECTION OF. We have inserted this article, not for others, is as easy as the hypothesis of the motion of the earth compared the sake of rectilinear, but of circular motion, the consideration of with that of its stability. which is apt to cause some embarrassment to the beginner. In motion along a given right line there can be but two directions, in one or other of which the course must be; and these two directions are opposite to one another. But in circular or other rotatory motion, all imaginable directions are taken in the course of a revolution, and whatever linear direction the moving body takes at any one point, it has the opposite direction at the opposite point. Still, however, there are two ways of moving on a circle: the motion may either be from c to a through B, or from A to c through B. These are called, somewhat improperly, different directions of motion. MOTION, APPARENT. [MOTION.] If two bodies be moving over two circles, they are said to move in the same direction when, two radii being taken in the same direction, the linear directions of motion are the same, as BD and QR. Thus care must be taken not to compare two circular motions by positions which belong to radii in opposite directions. If, for instance, the directions of motion be A B C and q vs (which are the same), and if at the same time the two bodies be at B and s, their linear directions of motion are opposite, though according to the definition their circular motions are in the same direction. Thus, in the case of the moon, and her revolution round her own axis [Moon], the middle point of the visible moon is moving round the moon's axis in a direction opposite to the orbital motion of the moon; but the radius of that middle point is opposite in direction to the line joining the centres of the earth and moon; so that the direction of revolution of the moon's rotation is the same as that of the orbital rotation. MOTION, LAWS OF. The laws of motion mean those universal methods of receiving and losing motion which close attention to mechanical phenomena, coupled with strict inductive reasoning, has shown to be inherent in the constitution of matter. If an intelligent observer, not used to inductive reasoning, nor instructed in the results of mechanics, were required to state the views which experience had taught him of the constitution of matter, as an agent or patient in the production or reception of motion, he would perhaps reply as follows: Matter seems to have no power of moving itself, though if we judge from the fall of bodies towards the earth, the phenomena of magnetism, &c., it would appear as if matter might be the cause of motion in other matter. And it seems moreover that motion is an accident of matter which diminishes and dies out of itself, if some sustaining cause be not perpetually in action; for in all cases in which the experiment can be tried, we find that moving bodies are reduced to rest by being left to themselves. The motions of the heavenly bodies, it is true, appear to be permanent; but we have no certain assurance that there is not a constant sustaining physical cause of this permanency. There would be something of truth, and a good deal of falsehood, in the preceding conclusions, and it is not an easy thing to give that exhibition of the real constitution of matter which is placed beyond all doubt by the coincidence of its results with all the more complicated phenomena of nature. There is no question that those principles, to take two cases out of thousands, on which a ball can be projected almost unerringly to its mark from the mouth of a cannon, and the motions of the moon can be predicted within a small fraction of a second, are founded in truth; but it does not therefore follow that an à priori demonstration of them, mathematical or experimental, can be given; and in fact the method of presenting the laws of motion to a beginner is encumbered with serious difficulties. We shall begin by the assumption that those laws of motion which : So simple are the laws of motion themselves, that many have supposed them to be necessary, in the same sense as when we say it is a necessary consequence of our conception of straightness that two We shall mention this notion straight lines cannot inclose a space. again presently in the meanwhile we are in this situation, that while it is difficult, as a matter of reasoning, to disentangle the fundamental laws from the variety and complication of the phenomena in which their effects are exhibited, yet these laws themselves, when disengaged, are of that startling simplicity which disinclines the mind to receive them as the results of a train of deduction, and disposes it rather to think that it could have dictated them from its own previous conceptions. It will make some difference in our method of seeking for these laws, whether we suppose the earth to be at rest or in motion. Now the decisive proofs of the motion of the earth, as it happens, are themselves derived from certain consequences of the laws of motion. [MOTION OF THE EARTH.] We seem then to be reasoning in a vicious circle; nor do we see any mode of escape except by establishing the truth of these laws, whether the earth be at rest or in motion. And the process will be, first to detect laws for which there is a high and almost overpowering degree of probability in their favour; next to appeal to the above-mentioned uniform truth of the results deduced from the assumption of such laws for the conversion of this high state of probability into one of absolute demonstration. We will first assume the motion of the earth: every point of its surface then is in a state of revolution round the axis, while at the same time the whole is carried forward round the sun; to which we must add, the slight motion arising from the precession of the equinoxes, and the possible translation of the whole system. But this motion is very different in different parts; at the pole, for example, there is no diurnal motion, near it only a small one, and at the equator a considerable one. The points near the pole, all the motions considered, are describing a trochoidal orbit, the undulations of which are small, and the rotatory velocity small; those near the equator make larger undulations, with greater velocity of rotation. Our first idea might be, then, that at the different parts of the earth some modification of general laws would be observed, arising from the difference of the motions of the several places. It would not surprise a person wholly unacquainted with mechanics, to whom the preceding facts were stated for the first time, if he were told that some mistakes were made in the pointing of guns in our Indian battles, arising from the artillerymen having been trained by officers who had learnt their art in the latitude of Addiscombe, near Croydon, in Surrey, and had forgotten to allow for the difference in the diurnal motion of the two countries. Now the considerations which tend to establish the Second Law of Motion depend upon the fact that it never has been found necessary to take any notice of the difference of place on the earth in estimating effects of motion. It is not found necessary to write different treatises on gunnery for different latitudes, nor to alter the disposition of parts in any machine moved from one latitude to another to produce a more advantageous effect. There is, it is true, a small diminution in the weight of bodies, as they are carried toward the equator, and [CENTRIFUGAL FORCE; PENDULUM] the results of this are apparent in experiments in which the acquisition of motion depends upon weight, or rather, upon its proportion to the quantity of matter. But this very problem of the pendulum is one in which the question of the truth of the laws of motion is established by a test which would detect the smallest quantities, and furnishes an answer to those who might say that the possible effects of the difference of diurnal motions, though not distinguishable in such cases as that of a cannon-ball, might be perceptible in delicate instruments. If to the motion of the earth we superadd another, such as the motion of a carriage, the same sort of result is found. Those who move on a railroad at the rate of 30 miles an hour, or 44 feet in â second, do not find the relation in which they stand to the objects in the carriage in any degree changed by the motion. At the instant of taking the motion, or on any sudden jolt or change of motion, effects may be produced to which we shall frequently refer: but when the speed is once obtained, it is well known that a person might occupy himself in reading a work on mechanics written on terra firma (80 called), and might verify all the experimental conclusions, without coming to any result which would remind him of the difference of MOTION, LAWS OF. state between the writer and himself, as to motion. Hence we are led to the conclusion that all the relations of matter to matter remain unaltered, if the whole system be made to move, provided that the same motion be communicated to all its parts. And though we do not, absolutely speaking, know what rest is, since no point of the earth, nor of any heavenly body, can be shown to be at rest, yet since we see that the relations remain unaltered when the velocity of a whole system is changed, we are led to conclude that the same laws which prevail when all the parts of a system have the same motion, would also prevail if the whole system were at rest; the ground of our presumption being, that the laws remain unaltered under any alteration of the common motion which it is in our power to make. Let us now suppose that the carriage, instead of moving in a right line, is carried on a curved road, say a simple circle. It is no longer observed that loose objects in the carriage have a tendency to repose relatively to the carriage itself. If the motion become sufficiently rapid, or the friction of the substances on which they stand be sufficiently small, they will endeavour to move outwards, or from the centre of the circle of motion. This phenomenon can be made a consequence of the laws of motion, when the latter have obtained their simplest form; we do not at present enter into this subject further than to point out that it is only of rectilinear motions we can predicate any law as descriptive of what is inherent in matter. We have, it is true, already spoken of circular motions in taking into account those of the earth; but it must be remembered, firstly, that the circles in question are so large, that a small arc of any one is nearly a straight line; secondly, that we have been obliged to advert to this tendency outwards, which is the reason of the diminution of weight (or of much the greater part of it) detected from the oscillations of a pendulum which is carried towards the equator. This second law of motion (for such it is called, though it must be deduced first when the earth's motion is considered) may be thus stated:-If there be two or more causes of motion, taking place in two different right lines, whether inherent in the body or external to it, their effects do not interfere, nor does either diminish or augment the effect of the other. If, for instance, the body a be subject to two B D actions, one of which, being entirely in the direction AB, would bring the body to B in a given time, and the other, entirely in the direction A C, would bring it to c in the same given time; then the body will move from A to D, precisely as it would have done if, moving along A B in the manner first specified, the line A B had been translated with its extremity A moving in the second manner specified, the said line AB not changing its direction. The most simple and general method of stating this law is as follows:-The distance of a point from a straight line or plane, measured in any given direction, and as it will be at the end of a given time, is not affected by the action, during that time, of any causes of motion, provided they act in the direction of, or parallel to, that straight line or plane; or no force, in a given direction, can produce motion to or from a line in that direction. Thus if a ball were thrown up in still air, in such a manner that it would mount 50 feet in one second, no imaginable horizontal current or whirlwind, however much it might alter the actual course of the ball, would prevent its rising 50 feet in the second. The statement of the law by Newton, namely, that when a force acts upon a body in motion, the change of motion which it produces is in the direction and proportional to the magnitude of the force which acts, is perhaps rather too vague to give a distinct notion to learners. From the law just enunciated, we may learn that bodies upon the earth, moving with the earth, have the properties of bodies at rest with respect to all motions that are to be estimated relatively to the earth at least upon the supposition that the curvature of the motions of the earth is not sufficiently great to produce a sensible effect. We have then to inquire what is the natural state of matter on the earth? Can it preserve any motion of itself, or does every motion gradually slacken and die out, by the mere inability of matter to maintain it without the application of external causes? On this point we have only strong presumptions, which would be by themselves insufficient. Our first step would be to conclude, from what we actually see, that rest is the natural state of matter, and one to which it always approaches, however great a cause of motion be applied, unless that external cause, or some part of it, be maintained. On looking further however we find that terrestrial matter, immediately on its being put in motion, encounters causes of retardation. The resistance of the air, and the friction of the basis on which the substances rest, are easily shown to lessen the motion of bodies which encounter them. The more nearly these are removed, the longer does motion continue. It is certain then that these resistances contribute in a great degree to the destruction of motion; but it is not therefore to be immediately assumed that there is no other cause. with any atmosphere, would move for a long time without any sensible If we grant that a perfectly smooth ball, lying upon an indefinitely extended plane without friction, and not in contact diminution of the rate with which it was made to set out, we grant quite enough to explain all that we see, without the necessity of supposing that the motion would continue for ever. establish the first law of motion (so called), which is thus stated, that matter will retain its state of rest, or of motion, for any length of time, How then can we however great, until acted upon by some external cause? here appeal to the results of the application of this law, which have never, in any one instance, exhibited any reason to suspect that it is We must history, no one of the heavenly bodies has shown any diminution of only approximately true. Throughout the long period of astronomical its motion, or any of the consequences which would arise if the motion had a tendency to wear itself out. details of these consequences; the conclusion is, that the state which matter, independently of external bodies, has been created capable of We shall not here go into the maintaining, is not merely rest, but also uniform motion in a straight line; so that it has no more tendency of itself to part with any of its velocity, nor to move slower or faster than it was first made to move, than it has to set itself from rest into motion. most, of the mistakes which have been made by writers against the Newtonian theory of attraction, have arisen from want of proper conA great many, perhaps ception of the neutral state of matter. direction has been to them a proof of the existence of external causes maintained in action; whereas it proves nothing, but that there was at Maintenance of velocity and some time or other an external cause which acted for a longer or shorter time: the external cause steps in when the velocity changes, or the direction, or both, and not till then. matter retains, namely, either absolute rest, or any degree of velocity Properly considered, the immense number of different states which whatsoever, is as wonderful and mysterious a law as that of the attraction of matter upon matter, without any apparent intermediate agent. That matter should, without any perceptible maintainer, keep is as difficult to admit, as that the mere presence of other matter one rate of motion and one direction until acted on from without, should change that motion and that direction. What should teach blind atoms to draw straight lines in preference to circles or spirals? Have they the fundamental conceptions, according to some, or the powers of perception and inference, according to others, by which reasoning minds know or discover the simplicity of a straight line? These two consequences of observation, namely, the law of its not obliged to do so) begin from matter at rest, and establish first that The third law of motion was enunciated by Newton as follows:"Action and reaction are equal and opposite." Thus, if a body a exerts a force upon another body в by contact, tension, thrust, attraction, repulsion, or otherwise; then B exerts the very same force on A in the opposite direction. The principle thus laid down belongs as much to statics as to dynamics, and is not therefore to be considered a special law of motion. It seems indeed to establish a relation between the statical and dynamical effects of a force-the action being represented by the pressure employed, the reaction by the motion produced. But how is this motion to be measured? Newton had decided that a body's motion should be measured by its momentum, and enunciated the third law on this supposition. But taking as we now do the velocity generated in a body to be the dynamical measure of the force, that is, the reaction,-the third law of motion asserts that "the velocity generated in a body by the action of a force, whether impulsive or otherwise, is proportional to the force." Where the force continues to act during a finite interval of time, the body acquires an additional velocity in each succeeding instantan evident consequence of the second law of motion. The velocity thus generated in a unit of time (as one second) is termed by some modern writers the "rate of acceleration." And this is all that is meant by the expression "accelerating force," namely, a force that, acting continually upon a body, produces in it a rate of acceleration. Thus the force of gravity is called an accelerating force, because the motion of a body, under its influence, becomes accelerated; the rate of acceleration being about 32-2 feet per second. [ACCELERATED MOTION.] Let f and f' denote the rates of acceleration produced respectively by the action of two forces F and F'; then, by the third law of motion, ff'FF'. Also, if t be the time during which these forces act, v and v' the respective velocities acquired at the end of time t, we have v=ft and v'=f't, hence v : v=F: F'. If a body move under the action of gravitation, the rate of acceleration is g (=32.2), hence if F act upon a body whose weight is w, we have the following very useful application of the third law of motion : when v velocity acquired in time t by the action of F. The three laws of motion, then, may be thus enunciated :1. A body, if acted upon by no external force, remains at rest; or if in motion, continues to move uniformly in the same direction. 2. When any number of forces act upon a body in motion, each force produces the same effect in altering the magnitude and direction of the body's velocity, as if it acted singly on the body at rest. 3. The velocity generated in a unit of time by a force continually acting upon a body, is proportional to the force. The reader will meet with some very clear conceptions of the three laws of motion in Dr. Young's Natural Philosophy,' edited by Professor Kelland; also in O'Brien's Natural Philosophy,' published by the Society for the Promotion of Christian Knowledge. He may likewise refer to a paper by Dr. Whewell "On the Nature of the Truth of the Laws of Motion." (Camb. Phil. Trans.' vol. v. part ii.) The mistakes into which philosophers fell upon the laws of motion are uninteresting except in the applications which were made of them; and in the article MOTION OF THE EARTH will be found enough of these to give an idea of the difficulties which such fallacies placed in the way of sound knowledge. For an account of Galileo's labours, see GALILEI, in BIOG. DIV. For an account of the notions of Descartes on the same subject, see VORTICES. The first distinct enunciation of these laws appears in the Principia of Newton. Though all mechanical problems admit of solution upon the assumption of these laws, in conjunction with those which may be called the distinctive properties of the solid, fluid, and gaseous states, yet the purposes of mechanical inquiry are better served by certain general principles deduced from them, the proper conception of which can only be made by mathematicians, and are therefore referred to a purely mathematical article-VIRTUAL VELOCITIES. [PRESSURE, FORCE, ÎNERTIA, CENTRIPETAL AND CENTRIFUGAL FORCES, ACCELERATED MOTION, VELOCITY, &c.; MOVING FORCE. See particularly the article INERTIA, for the reason of the non-introduction of that word.] Among the many absurdities which have arisen out of a misapprebension of the laws of motion, is the attempt to discover what is called a perpetual motion, or a machine which of itself would never stop. The earth and planets are such machines in their rotations on their axes; and we have seen that any particle of matter, unacted on by other matter, and once in motion, is a perpetual motion. If a wheel attached to an axle could be deprived of friction at the pivots, and enclosed in a permanently air-tight and perfectly exhausted receiver, it would also, when once in motion, be a perpetual motion. But as long as any friction or resistance, however small, is perpetually retarding the motion, it is obvious that the velocity, if maintained, must be indebted to some external supply of moving power. To take the case of friction, which arises from the roughness of the supports, and which, independently of adhesion, may be considered as a rapid succession of very small jolts, by which the roughnesses of the one surface strike upon those of the other, and communicate a portion of momentum to the frame, and finally to the earth: to suppose that a wheel as above ARTS AND SCI. DIV. VOL. V. described could go on for ever, with friction, would be to suppose that there could be action without reaction. In fact, a perpetual motion, such as is intended to be made by the speculators on the subject, is nothing less than a machine which will work for ever without new moving power; it being not one bit less absurd to suppose that it would perpetually overcome friction and atmospheric resistance, than that it would continue to supply the impetus necessary to carry on the sawing of a plank or the weaving of lace. MOTION OF THE EARTH. The theory of gravitation has placed this question on a footing entirely different from that on which it was argued, whether by Ptolemaists or Copernicans. Both of the latter parties supposed the existence of a fixed central body somewhere, which the first of them would have to be the earth, and the second the sun. This centrum mundi, or centre of the universe, is exploded, and with it all the systems, whether Ptolemaic or Copernican, which preceded the discoveries of Newton. But, as noticed in COPERNICUS, in BIOG. DIV., the existing system preserves the name of that great man; the reason being, that its distinctive peculiarity is retained relatively, if not absolutely, namely, that the planets all move round the sun, or round a point near the sun. But it is added to the real Copernican system, that sun, planets, and all, may be, and probably are, in motion; the translation, as it is called, of the whole system being very nearly rectilinear, and the curvature, if any, arising from the attraction of the fixed stars. Nothing but a long course of observation can settle this last part of the question, though much has been done of late years to establish the affirmative. In approaching the old controversy on the motion of the earth, we confine ourselves rather to the arguments by which it was opposed than to those by which it was supported. For this we have two reasons: first, that the latter are well known and extensively circulated, while the former, unless preserved in historical articles, will find the oblivion from which they have no intrinsic merit to rescue them; secondly, that the controversies of the present day may be usefully illustrated by recurring to the long-decided struggle between the Copernicans and their opponents. We have now among us those who would fetter all new truths by their interpretation of the Scriptures, though they quietly acquiesce in the defeat which their own principle formerly received. The charges still brought against the cultivators of the sciences, " to the distress and disgust of every well-constituted mind," as Sir J. Herschel expresses it, should be looked at, not as the honest manifestations of an alarm newly awakened by the circumstances of the present day, but as the effects of an abiding spirit, which has always opposed investigation, and which, if it had prevailed, would have smothered all the knowledge of nature which has been acquired in the last two centuries. If some of those who have constituted themselves successors to the cardinals who forced Galileo to recant, have learnt from the past history of their own cause, and from the temper of the present age, to show the real scope of their system less openly than it appeared in the 17th century, the compliment which they thus pay to the advancing intelligence of mankind, though received with thanks and highly appreciated, should not be accepted as an equivalent for the mischiefs which must result from a successful attempt to place the great question of Revelation upon a false basis. The case of those who now endeavour to impede the progress of geology, or to limit speculation on the process of creation, is so similar in its fundamental points with that of the labourers to the same effect in the field of astronomy, that the circulation of some account of the latter will perhaps enable our readers to help themselves in forming their opinion of the former. When the work of Copernicus appeared in 1543, it seems to have been considered as a mere attempt to demonstrate (see the old use of this word in DEMONSTRATION) the motions of the heavenly bodies in a more simple way. Guarded as it was by the expressions of the preface, it was neglected as a purely speculative trial of a strange and impossible hypothesis. In 1566 Ramus (Scol. Math.') simply reproaches Copernicus with the gigantic character of his hypothesis, and says it would have been better to have taken one nearer to the truth, in a manner which implies that he thought both were agreed as to what the truth really was. Copernicus himself, as we have seen, treated his own ideas as a reproduction of those of the ancients, and in truth the existence of such a doctrine as the earth's motion was perfectly well known to all men of learning. Aristotle (in his second book on the Heavens) states that Pythagoras and his followers placed the sun in the centre, on account of the superior excellence which they attributed to the element of fire, of which they supposed the sun to be made. Different authorities give the same opinion (whether with or without the reason) to Philolaus, Anaximander, Nicetas, Seleucus, Cleanthes, Leucippus, Ecphantus, Heraclides Ponticus, and Aristarchus. The introduction of Pythagoras, as a predecessor of Copernicus, is as rational as would be the connection of the modern atomic theory with the doctrines of Epicurus; and much of the same kind is an assertion not unfrequently made, that Cardinal Cusa was a supporter of the earth's motion. This writer (De Doctà Ignorantia,' lib. ii., c. 11) certainly denies that there can be any centre of the universe; for, says he, if there were a centre, there would be a circumference, that is, a termination, to the universe; and his reasons relative to the earth's motion are of the same degree of force. He is more rational in the next chapter, where he explains that the apparent motion of other 3 F |