CHAP. VII.

MOTION OF A PROJECTILE.

233

force to put matter in motion, or to oppose and neutralize force, which gives us this internal conviction of power and causation so far as it refers to the material world, and compels us to believe that whenever we see material objects put in motion from a state of rest, or deflected from their rectilinear paths, and changed in their velocities if already in motion, it is in consequence of such an EFFORT somehow exerted, though not accompanied with our consciousness. That such an effort should be exerted with success through an interposed space, is no more difficult to conceive, than that our hand should communicate motion to a stone, with which it is demonstrably not in contact.

(371.) All bodies with which we are acquainted, when raised into the air and quietly abandoned, descend to the earth's surface in lines perpendicular to it. They are therefore urged thereto by a force or effort, the direct or indirect result of a consciousness and a will existing somewhere, though beyond our power to trace, which force we term gravity; and whose tendency or direction, as universal experience teaches, is towards the earth's center; or rather, to speak strictly, with reference to its spheroidal figure, perpendicular to the surface of still water. But if we cast a body obliquely into the air, this tendency, though not extinguished or diminished, is materially modified in its ultimate effect. The upward impetus we give the stone is, it is true, after a time destroyed, and a downward one communicated to it, which ultimately brings it to the surface, where it is opposed in its further progress, and brought to rest. But all the while it has been continually deflected or bent aside from its rectilinear progress, and made to describe a curved line concave to the earth's center; and having a highest point, vertex, or apogee, just as the moon has in its orbit, where the direction of its motion is perpendicular to the radius.

(372.) When the stone which we fling obliquely upwards meets and is stopped in its descent by the earth's surface, its motion is not towards the center, but inclined

to the earth's radius at the same angle as when it quitted our hand. As we are sure that, if not stopped by the resistance of the earth, it would continue to descend, and that obliquely, what presumption, we may ask, is there that it would ever reach the center, to which its motion, in no part of its visible course, was ever directed? What reason have we to believe that it might not rather circulate round it, as the moon does round the earth, returning again to the point it set out from, after completing an elliptic orbit of which the center occupies the lower focus? And if so, is it not reasonable to imagine that the same force of gravity may (since we know that it is exerted at all accessible heights above the surface, and even in the highest regions of the atmosphere) extend as far as 60 radii of the earth, or to the moon? and may not this be the power,for some power there must be, which deflects her at every instant from the tangent of her orbit, and keeps her in the elliptic path which experience teaches us she actually pursues?

(373.) If a stone be whirled round at the end of a string, it will stretch the string by a centrifugal force*, which, if the speed of rotation be sufficiently increased, will at length break the string, and let the stone escape. However strong the string, it may, by a sufficient rotatory velocity of the stone, be brought to the utmost tension it will bear without breaking; and if we know what weight it is capable of carrying, the velocity necessary for this purpose is easily calculated. Suppose, now, a string to connect the earth's center, with a weight at its surface, whose strength should be just sufficient to sustain that weight suspended from it. Let us, however, for a moment imagine gravity to have no existence, and that the weight is made to revolve with the limiting velocity which that string can barely counteract: then will its tension be just equal to the weight of the revolving body; and any power which should continually urge the body towards the center with a force equal to its weight would perform the office, and might supply the place of

*See Cab. Cyc. MECHANICS, chap. viii.

CH. VII.

DIMINUTION OF GRAVITY AT THE MOON. 235

the string, if divided. Divide it then, and in its place let gravity act, and the body will circulate as before; its tendency to the center, or its weight, being just balanced by its centrifugal force. Knowing the radius of the earth, we can calculate the periodical time in which a body so balanced must circulate to keep it up; and this appears to be 1h 23m 22s.

(374.) If we make the same calculation for a body at the distance of the moon, supposing its weight or gravity the same as at the earth's surface, we shall find the period required to be 10h 45m 30s. The actual period of the moon's revolution, however, is 27d 7h 43m; and hence it is clear that the moon's velocity is not nearly sufficient to sustain it against such a power, supposing it to revolve in a circle, or neglecting (for the present) the slight ellipticity of its orbit. In order that a body at the distance of the moon (or the moon itself) should be capable of keeping its distance from the earth by the outward effort of its centrifugal force, while yet its time of revolution should be what the moon's actually is, it will appear (on executing the calculation from the principles laid down in Cab. Cyc. MECHANICS) that gravity, instead of being as intense as at the surface, would require to be very nearly 3600 times less energetic; or, in other words, that its intensity is so enfeebled by the remoteness of the body on which it acts, as to be capable of producing in it, in the same time, only 3600 th part of the motion which it would impart to the same mass of matter at the earth's surface.

(375.) The distance of the moon from the earth's center is somewhat less than sixty times the distance from the center to the surface, and 3600: 1 :: 602: 12; so that the proportion in which we must admit the earth's gravity to be enfeebled at the moon's distance, if it be really the force which retains the moon in her orbit, must be (at least in this particular instance) that of the squares of the distances at which it is compared. Now, in such a diminution of energy with increase of distance, there is nothing primâ facie inadmissible. Emanations

from a center, such as light and heat, do really diminish in intensity by increase of distance, and in this identical proportion; and though we cannot certainly argue much from this analogy, yet we do see that the power of magnetic and electric attractions and repulsions is actually enfeebled by distance, and much more rapidly than in the simple proportion of the increased distances. The argument, therefore, stands thus: :- On the one hand, Gravity is a real power, of whose agency we have daily experience. We know that it extends to the greatest accessible heights, and far beyond; and we see no reason for drawing a line at any particular height, and there asserting that it must cease entirely; though we have analogies to lead us to suppose its energy may diminish rapidly as we ascend to great heights from the surface, such as that of the moon. On the other hand, we are sure the moon is urged towards the earth by some power which retains her in her orbit, and that the intensity of this power is such as would correspond to a diminished gravity, in the proportion, otherwise not

improbable, of the squares of the distances. If gravity be not that power, there must exist some other; and, besides this, gravity must cease at some inferior level, or the nature of the moon must be different from that of ponderable matter;— for if not, it would be urged by both powers, and therefore too much urged, and forced inwards from her path.

66

(376.) It is on such an argument that Newton is understood to have rested, in the first instance, and provisionally, his law of universal gravitation, which may be thus abstractly stated: Every particle of matter in the universe attracts every other particle, with a force directly proportioned to the mass of the attracting particle, and inversely to the square of the distance between them." In this abstract and general form, however, the proposition is not applicable to the case before us. The earth and moon are not mere particles, but great spherical bodies, and to such the general law does not immediately apply; and, before we can make

CHAP. VII.

ATTRACTION OF SPHERES.

237

it applicable, it becomes necessary to enquire what will be the force with which a congeries of particles, constituting a solid mass of any assigned figure, will attract another such collection of material atoms. This problem is one purely dynamical, and, in its general form, is of extreme difficulty. Fortunately, however, for human knowledge, when the attracting and attracted bodies are spheres, it admits of an easy and direct solution. Newton himself has shown (Princip. b. i. prop. 75.) that, in that case, the attraction is precisely the same as if the whole matter of each sphere were collected into its center, and the spheres were single particles there placed; so that, in this case, the general law applies in its strict wording. The effect of the trifling deviation of the earth from a spherical form is of too minute an order to need attention at present. It is, however, perceptible, and may be hereafter noticed.

(377.) The next step in the Newtonian argument is one which divests the law of gravitation of its provisional character, as derived from a loose and superficial consideration of the lunar orbit as a circle described with an average or mean velocity, and elevates it to the rank of a general and primordial relation, by proving its applicability to the state of existing nature in all its detail of circumstances. This step consists in demonstrating, as he has done * (Princip. i.17., i. 75.), that, under the influence of such an attractive force mutually urging two spherical gravitating bodies towards each other, they will each, when moving in each other's neighbourhood, be deflected into an orbit concave towards the other, and describe, one about the other regarded as fixed, or both round their common center of gravity, curves whose forms are limited to those figures known in geometry by the general name of conic sections. It

We refer for these fundamental propositions, as a point of duty, to the immortal work in which they were first propounded. It is impossible for us in this volume to go into these investigations: even did our bmits permit, it would be utterly inconsistent with our plan; a general idea, however, of their conduct will be given in the next chapter.