sustaining a small weight; this weight will stretch the line in the direction in which gravity acts; hence this line gives the direction of gravity. This direction is always the same at the same place, being invariably perpendicular to the surface of still water; and for all places near each other the directions are sensibly parallel, for a small extent of the surface of still water may be considered as a plane, and consequently the perpendiculars to it will be all parallel. But at places at a considerable distance from each other, as at London and Paris, the directions of gravity are inclined to each other. For the surface of still water at these places being portions of parallel spherical surfaces, or very nearly so, the perpendiculars to these surfaces at those places would, if produced, meet near the centre of the earth; and at places more distant still, as, for instance, at Paris and the Cape of Good Hope, the directions of gravity will be nearly at right angles to each other, and at our antipodes the directions are on the same line, but opposite to each other, both being towards the centre. It is generally said that the direction of gravity is perpendicular to the surface of the earth; now by this we do not mean the actual surface, such as it generally exists with its numerous inequalities, but an ideal surface, such as the earth would present were it uniformly curved like a large tract of still water, or a calm sea.

41. Falling Bodies. In observing falling bodies the first thing which strikes us is the different rates of their descent. A piece of lead, or any other metal, falls very rapidly, the leaf of a tree falls very slowly; this cannot arise from the difference of the weight of two substances, for since the greater weight implies a greater number of particles, and gravity acts equally on all particles, there ought to be no difference in the descent of the two bodies. But the real cause of this apparent difference is the resistance of the air, which acts much more effectively on the extended surface of the leaf than on a smaller and heavier substance; and that this is the true

explanation the well-known guinea and feather experi ment renders evident; for in this experiment a guinea and a feather reach the bottom of an exhausted receiver at the same instant. Having shewn then that all bodies fall with the same velocity, we must proceed to inquire what is this one velocity, or law, according to which the descent of every species of matter takes place; or, in other words, what relation subsists between the space which a heavy body passes over, and the time which it employs in passing over it. This relation is the law of the motion which

gravity impresses on matter. The readiest apparent method of solving this question is to procure a long glass tube, exhaust the air, and allowing a body to fall from the top, to mark the point which it passes, after one second, two seconds, and so on. But we should observe here, as we may observe in the descent of bodies in the air, that at the commencement of their fall they move slowly, but their velocity augments so rapidly, that in a few seconds it is impossible to note, at all accurately, the point which the body passes at a given instant. A fall so accelerated precludes all direct observation, and recourse must be had to indirect methods, either to the inclined plane of Galileo, or to the machine' of Atwood (Art. 43). We cannot, however, establish a law by direct experiment, since the errors of observation can never be altogether prevented, but we can verify an assumed law, and this is precisely what is done in treating of gravity. It is proved by observation that gravity satisfies the defined laws of a uniformly accelerated motion, as we shall proceed to shew.

42. Uniformly accelerated Motion.-We have just seen that the velocity of a falling body increases rapidly during its descent. Now the motion of a body is said to be uniformly accelerated when the velocity added is equal for equal times; that is, the body must receive the same addition of velocity in the 4th or 5th second as it receives in the 2nd or 3rd; when this is the case, the body is said to be acted on by a uniform accelerating force. We must con

sider in all cases of accelerated motion, that were the action of the force to cease at any instant, the body would go on moving, according to the first law of motion, with the velocity which it then had; but, that as this force does not cease at any instant, velocity is added continuously to that already existing; the velocity added in any time being measured (Art. 26) by the space which would be passed over by the body, moving with that velocity, in the given time.

Uniformly accelerating force is measured by the velocity added in a given time, as, for instance, one second. Thus, if it should appear from observation that the velocity of a falling body receives the same addition every second, then will the force of gravity be an uniformly accelerating force, and will be measured by this quantity.

From these principles there will result the following laws:-1o. The velocity generated in any time by an uniformly accelerating force is proportional to the time.' 2o. The space described from the beginning of the motion is as the square of the time.' 3°. The space described by a body uniformly accelerated from rest, is half the space described in the same time with the last velocity; or, in other words, the velocity generated in one second will in the next second carry the body over double the space described in the first.'

The preceding are the laws of uniformly accelerated motion, and it remains to shew that the motion produced by gravity is subject to these laws; whence we may conclude that gravity is an uniformly accelerating force.*

43. Atwood's Machine.-The velocity produced by the undiminished force of gravity being much too great to be conveniently subjected to experimental examination, it

The exact laws expressing the relation betwixt the velocity, space, time, and force, are v=ƒt, and s=}ƒ t2. In these equations ƒ is the accelerating force, that is, the velocity added or generated in each second, and the whole velocity generated in t seconds; also s is the space described in the same time.-See Whewell's Elementary Mechanics, Art. 161-164.

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occurred to Atwood that all the phenomena of falling bodies might be experimentally exhibited and accurately observed, if a force of the same kind as gravity, but of much less intensity, were used; so that the motion would still be governed by the same law of gravity, while the intensity of the agent being diminished, and consequently the quantity of motion, the velocity might, even after several seconds, not have become too rapid to admit of accurate measurement. With this view Atwood suspended two equal weights, one at each end of a string, passing over a wheel, turning very easily on its

axle. The equal weights are representedred at A and B, and the wheel at c.

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weights balance each other perfectly, andoai $450 no motion can take place. To one of the bom weights, as to B, a slight addition is made;ose ed this was generally done by placing a small impo-2 bar upon в; the bar is represented at G, 0998-8 and is too long to pass the ring F. ring can be attached to any part graduated scale represented on D E.

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pose that в is exactly opposite the zero of the scale, and that the additional bar is laid upon it. The loaded weight will begin to descend, drawing up the unloaded weight A. The descent of the loaded weight is a motion of exactly the same kind, as of a heavy body falling freely; hence the laws of its motion will be the same; the absolute amount of velocity will be very different in the two cases, but the rate of its increase will be the same, and this is what we wish to discover.

Let us see now whether the result obtained will accord with the laws of the preceding articles. For this purpose, let the body B, being loaded, begin to descend from opposite the zero division of the scale, and let it pass by the first division in one second; then, according to the law that the space is proportional to the square of the

time, it ought after two seconds to be opposite division 4, after three seconds to be opposite division 9, and so on; and the experiments with this machine shew that this is the case. We will now examine what will take place when the additional weight & acts only for a short time. Let the ring F be placed at division one of the scale, then the weight G will be taken off at the end of the first second, and в will continue to descend with its acquired velocity. According to law 30 (Art. 42), it ought to pass by two divisions in the next and every subsequent second; and this is found to be the case; after 2" it will be opposite division 3, after 3" opposite 5, and so on.

The same results may be obtained by making bodies descend down inclined planes of different heights and lengths. It was assumed by Galileo, and may be shewn to be true, for the descent of bodies down inclined planes, that the velocity acquired down all planes, whose perpendicular heights are equal, is the same; and equal to the velocity acquired by falling down the perpendicular height.' The time of descent down an inclined plane varies with its length, hence the motions may be made as slow as we please, and the preceding laws of accelerated motion may be verified in the case of gravity.

From these, and other far more accurate experiments, which the pendulum will give us, it appears that the laws of accelerated motion, by gravity, satisfy those of uniformly accelerated motion; hence we may conclude, that gravity at the earth's surface is an uniformly accelerating force.

44. Law of Gravity at the Earth's surface.-From the experiments detailed in the preceding article it appears that gravity at the earth's surface is an uniform force; and this conclusion was arrived at by shewing that gravity is in its nature such as satisfies the definitions of that kind of force. But it must be observed, that nothing has hitherto been determined respecting the intensity of this force; the experiments with Atwood's machine shew the rate by which a given quantity increases, but not the amount of that