expression of the laws, and to regard the numerical data or elliptic elements of the planetary orbits as not absolutely permanent, but subject to a series of extremely slow and almost imperceptible changes. These changes may be neglected when we consider only a few revolutions; but going on from century to century, and continually accumulating, they at length produce material departures in the orbits from their original state. Their explanation will form the subject of a subsequent chapter; but for the present we must lay them out of consideration, as of an order too minute to affect the general conclusions with which we are now concerned. By what means astronomers are enabled to compare the results of the elliptic theory with observation, and thus satisfy themselves of its accordance with nature, will be explained presently. (490.) It will first, however, be proper to point out what particular theoretical conclusion is involved in each of the three laws of Kepler, considered as satisfactorily established, — what indication each of them, separately, affords of the mechanical forces prevalent in our system, and the mode in which its parts are connected, — and how, when thus considered, they constitute the basis on which the Newtonian explanation of the mechanism of the heavens is mainly supported. To begin with the first law, that of the equable description of areas. — Since the planets move in curvilinear paths, they must (if they be bodies obeying the laws of dynamics) be deflected from their otherwise natural rectilinear progress by force. And from this law, taken as a matter of observed fact, it follows, that the direction of such force, at every point of the orbit of each planet, always passes through the sun. No matter from what ultimate cause the power which is called gravitation originates, — be it a virtue lodged in the sun as its receptacle, or be it pressure from without, or the resultant of many pressures or solicitations of unknown fluids, magnetic or electric ethers, or impulses, — still, when finally brought under our contemplation, and summed up into a single resultant energy— its direction is, from every point on all sides, towards the turfs center. As an abstract dynamical proposition, the reader will find it demonstrated by Newton, in the first proposition of the Principia, with an elementary simplicity to which we really could add nothing but obscurity by amplification, that any body, urged towards a certain central point by a force continually directed thereto, and thereby deflected into a curvilinear path, will describe about that center equal areas in equal times: and vice versd, that such equable description of areas is itself the essential criterion of a continual direction of the acting force towards the center to which this character belongs. The first law of Kepler, then, gives us no information as to the nature or intensity of the force urging the planets to the sun; the only conclusion it involves is, that it does so urge them. It is a property of orbitual rotation under the influence of central forces generally, and, as such, we daily see it exemplified in a thousand familiar instances. A simple experimental illustration of it is to tie a bullet to a thin string, and, having whirled it round with a moderate velocity in a vertical plane, to draw the end of the string through a small ring, or allow it to coil itself round the finger, or round a cylindrical rod held very firmly in a horizontal position. The bullet will then approach the center of motion in a spiral line; and the increase not only of its angular but of its linear velocity, and the rapid diminution of its periodic time when near the center, will express, more clearly than any words, the compensation by which its uniform description of areas is maintained under a constantly diminishing distance. If the motion be reversed, and the thread allowed to uncoil, beginning with a rapid impulse, the velocity will diminish by the same degrees as it before increased. The increasing rapidity of a dancer's pirouette, as he draws in his limbs and straightens his whole person, so as to bring every part of his frame as near as possible to the axis of his motion, is another instance where the connection of the observed effect with the central force exerted, though equally real, is much less obvious. (491.) The second law of Kepler, or that which asserts that the planets describe ellipses about the sun as their focus, involves, as a consequence, the law of solar gravitation (so be it allowed to call the force, whatever it be, which urges them towards the sun) as exerted on each individual planet, apart from all connection with the rest. A straight line, dynamically speaking, is the only path which can be pursued by a body absolutely free, and under the action of no external force. All deflection into a curve is evidence of the exertion of a force; and the greater the deflection in equal times, the more intense the force. Deflection from a straight line is only another word for curvature of path; and as a circle is characterized by the uniformity of its curvatures in all its parts—so is every other curve (as an ellipse) characterized by the particular law which regulates the increase and diminution of its curvature as we advance along its circumference. The deflecting force, then, which continually bends a moving body into a curve, may be ascertained, provided its direction, in the first place, and, secondly, the law of curvature of the curve itself, be known. Both these enter as elements into the expression of the force. A body may describe, for instance, an ellipse, under a great variety of dispositions of the acting forces: it may glide along it, for example, as a bead upon a polished wire, bent into an elliptic form; in which case the acting force is always perpendicular to the wire, and the velocity is uniform. In this case the force is directed to no fixed center, and there is no equable description of areas at all. Or it may describe it as we may see done, if we suspend a ball by a very long string, and, drawing it a little aside from the perpendicular, throw it round with a gentle impulse. In this case the acting force is directed to the center of the ellipse, about which areas are described equably, and to which a force proportional to the distance (the decomposed result of terrestrial gravity) perpetually urges it • This is at once a very easy experiment, and a very instructive one, and we shall again refer to it. In the case before us, of an ellipse described by the action of • If the suspended body be a vessel full of fine sand, hariii); a small hole at iis bottom, the elliptic trace of ita orbit will be left in a sand streak on a table placed below it. Tliii neat illustration it due, to the best of mj knowledge, to Mr. Babbage. a force directed to the focus, the steps of the investigation of the law of force are these: 1st, The law of the areas determines the actual velocity of the revolving hody at every point, or the space really run over hy it in a given minute portion of time; 2dly, The law of curvature of the ellipse determines the linear amount of deflection from the tangent in the direction of the focus, which corresponds to that space so run over; 3dly, and lastly, The laws of accelerated motion declare that the intensity of the acting force causing such deflection in its own direction, is measured by or proportional to the amount of that deflection, and may therefore be calculated in any particular position, or generally expressed by geometrical or algebraic symbols, as a law independent of particular positions, when that deflection is so calculated or expressed. We have here the spirit of the process by which Newton has resolved this interesting problem. For its geometrical detail, we must refer to the 3d section of his Principia. We know of no artificial mode of imitating this species of elliptic motion; though a rude approximation to it — enough, however, to give a conception of the alternate approach and recess of the revolving body to and from the focus, and the variation of its velocity—may be had by suspending a small steel bead to a fine and very long silk fibre, and setting it to revolve in a small orbit round the pole of a powerful cylindrical magnet, held upright, and vertically under the point of suspension. (492.) The third law of Kepler, which connects the distances and periods of the planets by a general rule, bears with it, as its theoretical interpretation, this important consequence, viz. that it is one and the same force, modified only by distance from the sun, which retains all the planets in their orbits about it. That the attraction of the sun (if such it be) is exerted upon all the bodies of our system indifferently, without regard to the peculiar materials of which they may consist, in the exact proportion of their inertiae, or quantities of matter; that it is not, therefore, of the nature of the elective attractions of chemistry or of magnetic action, which is powerless on other substances than iron and some one or two more, but is of a more universal character, and extends equally to all the material constituents of our system, and (as we shall hereafter see abundant reason to admit) to those of other systems than our own. This law, important and general as it is, results, as the simplest of corollaries, from the relations established by Newton in the section of the Principia referred to (Prop, xv.), from which proposition it results, that if the earth were taken from its actual orbit, and launched anew in space at the place, in the direction, and with the velocity of any of the other planets, it would describe the very same orbit, and in the same period, which that planet actually does, a minute correction of the period only excepted, arising from the difference between the mass of the earth and that of the planet. Small as the planets are compared to the sun, some of them are not, as the earth is, mere atoms in the comparison. The strict wording of Kepler's law, as Newton has proved in his fiftyninth proposition, is applicable only to the case of planets whose proportion to the central body is absolutely inappretiable. When this is not the case, the periodic time is shortened in the proportion of the square root of the number expressing the sun's mass or inertia;, to that of the sum of the numbers expressing the masses of the sun and planet; and in general, whatever be the masses of two bodies revolving round each other under the influence of the Newtonian law of gravity, the square of their periodic time will be expressed by a fraction whose numerator is the cube of their mean distance, i. e. the greater semi-axis of their elliptic orbit, and whose denominator is the sum of their masses. When one of the masses is incomparably greater than the other, this resolves into Kepler's law; but when this is not the cose, the proposition thus generalized stands in lieu of that law. In the system of the sun and planets, however, the numerical correction thus introduced into the results of Kepler's law is too small to be of any importance, the mass of the largest of the planets (Jupiter) being much less than a thousandth part of that of the sun. We shall presently, however, perceive all the importance of this generalization, when we come to speak of the satellites. |