263 CHAP. VIII. KEPLER'S LAW OF PERIODIC TIMES. evidently the longer the period. The order of the planets, beginning from the sun, is the same, whether we arrange them according to their distances, or to the time they occupy in completing their revolutions; and is as follows:- Mercury, Venus, Earth, Mars,- the four ultra-zodiacal planets,— Jupiter, Saturn, and Uranus. Nevertheless, when we come to examine the numbers expressing them, we find that the relation between the two series is not that of simple proportional increase. The periods increase more than in proportion to the distances. Thus, the period of Mercury is about 88 days, and that of the Earth 365-being in proportion as 1 to 4.15, while their distances are in the less proportion of 1 to 2.56; and a similar remark holds good in every instance. Still, the ratio of increase of the times is not so rapid as that of the squares of the distances. The square of 2.56 is 6.5536, which is considerably greater than 4.15. An intermediate rate of increase, between the simple proportion of the distances and that of their squares, is therefore clearly pointed out by the sequence of the numbers; but it required no ordinary penetration in the illustrious Kepler, backed by uncommon perseverance and industry, at a period when the data themselves were involved in obscurity, and when the processes of trigonometry and of numerical calculation were encumbered with difficulties, of which the more recent invention of logarithmic tables has happily left us no conception, to perceive and demonstrate the real law of their connection. This connection is expressed in the following proposition:-"The squares of the periodic times of any two planets are to each other, in the same proportion as the cubes of their mean distances from the sun." Take, for example, the earth and Mars*, whose periods are in the proportion of 3652564 to 6869796, and whose distances from the sun is that of 100000 to *The expression of this law of Kepler requires a slight modification when we come to the extreme nicety of numerical calculation, for the greater planets, due to the influence of their masses. This correction is imperceptible for the earth and Mars. 152369; and it will be found, by any one who will take the trouble to go through the calculation, that— (3652564)2: (6869796)2:: (100000)3: (152369)3. (417.) Of all the laws to which induction from pure observation has ever conducted man, this third law (as it is called) of Kepler may justly be regarded as the most remarkable, and the most pregnant with important consequences. When we contemplate the constituents of the planetary system from the point of view which this relation affords us, it is no longer mere analogy which strikes us no longer a general resemblance among them, as individuals independent of each other, and circulating about the sun, each according to its own peculiar nature, and connected with it by its own peculiar tie. The resemblance is now perceived to be a true family likeness; they are bound up in one chain-interwoven in one web of mutual relation and harmonious agreement-subjected to one pervading influence, which extends from the center to the farthest limits of that great system, of which all of them, the earth included, must henceforth be regarded as members. (418.) The laws of elliptic motion about the sun as a focus, and of the equable description of areas by lines joining the sun and planets, were originally established by Kepler, from a consideration of the observed motions of Mars; and were by him extended, analogically, to all the other planets. However precarious such an extension might then have appeared, modern astronomy has completely verified it as a matter of fact, by the general coincidence of its results with entire series of observations of the apparent places of the planets. These are found to accord satisfactorily with the assumption of a particular ellipse for each planet, whose magnitude, degree of excentricity, and situation in space, are numerically assigned in the synoptic table before, referred to. It is true, that when observations are carried to a high degree of precision, and when each planet is traced through many successive revolutions, and its history carried back, by the aid of calculations founded on these CHAP. VIII. INTERPRETATION OF KEPLER'S LAWS. 265 data, for many centuries, we learn to regard the laws of Kepler as only first approximations to the much more complicated ones which actually prevail; and that to bring remote observations into rigorous and mathematical accordance with each other, and at the same time to retain the extremely convenient nomenclature and relations of the ELLIPTIC SYSTEM, it becomes necessary to modify, to a certain extent, our verbal 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 considerable 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. (419.) 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, - No matter from what always passes through the sun. 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 sollicitations 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 sun's center. As an abstract dynamical proposition, the reader will find it demonstrated by Newton, in the 1st 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 versa, 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 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 CHAP. VIII. INTERPRETATION OF KEPLER'S LAWS. 267 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. (420.) 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 curvature 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 de |