The Story of the Heavens

the sun; while on the other, it took that beautiful class of curves which had exercised the geometrical talents of the ancients, and assigned to them the dignity of defining the highways of the universe. But we have as yet only partly enunciated the first discovery of Kepler. We have seen that a planet revolves in an ellipse around the sun, and that the sun is, therefore, at some point in the interior of the ellipse—but at what point? Interesting, indeed, is the answer to this question. We have pointed out how the foci possess a geometrical significance which no other points enjoy. Kepler showed that the sun must be situated in one of the foci of the ellipse in which each planet revolves. We thus enunciate the first law of planetary motion in the following words:— Each planet revolves around the sun in an elliptic path, having the sun at one of the foci. We are now enabled to form a clear picture of the orbits of the planets, be they ever so numerous, as they revolve around the sun. In the first place, we observe that the ellipse is a plane curve; that is to say, each planet must, in the course of its long journey, confine its movements to one plane. Each planet has thus a certain plane appropriated to it. It is true that all these planes are very nearly coincident,[Pg 139] at least in so far as the great planets are concerned; but still they are distinct, and the only feature in which they all agree is that each one of them passes through the sun. All the elliptic orbits of the planets have one focus in common, and that focus lies at the centre of the sun. It is well to illustrate this remarkable law by considering the circumstances of two or three different planets. Take first the case of the earth, the path of which, though really an ellipse, is very nearly circular. In fact, if it were drawn accurately to scale on a sheet of paper, the difference between the elliptic orbit and the circle would hardly be detected without careful measurement. In the case of Venus the ellipse is still more nearly a circle, and the two foci of the ellipse are very nearly coincident with the centre of the circle. On the other hand, in the case of Mercury, we have an ellipse which departs from the circle to a very marked extent, while in the orbits of some of the minor planets the eccentricity is still greater. It is extremely remarkable that every planet, no matter how far from the sun, should be found to move in an ellipse of some shape or other. We shall presently show that necessity compels each planet to pursue an elliptic path, and that no other form of path is possible. Started on its elliptic path, the planet pursues its stately course, and after a certain duration, known as the periodic time, regains the position from which its departure was taken. Again the planet traces out anew the same elliptic path, and thus, revolution after revolution, an identical track is traversed around the sun. Let us now attempt to follow the body in its course, and observe the history of its motion during the time requisite for the completion of one of its circuits. The dimensions of a planetary orbit are so stupendous that the planet must run its course very rapidly in order to finish the journey within the allotted time. The earth, as we have already seen, has to move eighteen miles a second to accomplish one of its voyages round the sun in the lapse of 365-1⁄4 days. The question then arises as to whether the rate at which a planet moves is uniform or not. Does the earth, for instance, actually move at all times with the velocity of[Pg 140] eighteen miles a second, or does our planet sometimes move more rapidly and sometimes more slowly, so that the average of eighteen miles a second is still maintained? This is a question of very great importance, and we are able to answer it in the clearest and most emphatic manner. The velocity of a planet is not uniform, and the variations of that velocity can be explained by the adjoining figure (Fig. 38). Fig. 38.—Varying Velocity of Elliptic Motion. Let us first of all imagine the planet to be situated at that part of its path most distant from the sun towards the right of the figure. In this position the body's velocity is at its lowest; as the planet begins to approach the sun the speed gradually improves until it attains its mean value. After this point has been passed, and the planet is now rapidly hurrying on towards the sun, the velocity with which it moves becomes gradually greater and greater, until at length, as it dashes round the sun, its speed attains a maximum. After passing the sun, the distance of the planet from the luminary increases, and the velocity of the motion begins to abate; gradually it declines until the mean value is again reached, and then it falls still lower, until the body recedes to its greatest distance from the sun, by which time the velocity has abated to the value from which we supposed it to commence. We thus observe that the nearer the planet is to the sun the quicker[Pg 141] it moves. We can, however, give numerical definiteness to the principle according to which the velocity of the planet varies. The adjoining figure (Fig. 39) shows a planetary orbit, with, of course, the sun at the focus S. We have taken two portions, A B and C D, round the ellipse, and joined their extremities to the focus. Kepler's second law may be stated in these words:— "Every planet moves round the sun with such a velocity at every point, that a straight line drawn from it to the sun passes over equal areas in equal times." Fig. 39.—Equal Areas in Equal Times. For example, if the two shaded portions, A B S and D C S, are equal in area, then the times occupied by the planet in travelling over the portions of the ellipse, A B and C D, are equal. If the one area be greater than the other, then the times required are in the proportion of the areas. This law being admitted, the reason of the increase in the planet's velocity when it approaches the sun is at once apparent. To accomplish a definite area when near the sun, a larger arc is obviously necessary than at other parts of the path. At the opposite extremity, a small arc suffices for a large area, and the velocity is accordingly less. These two laws completely prescribe the motion of a planet round the sun. The first defines the path which the planet pursues; the second describes how the velocity of the body varies at different points along its path. But Kepler added to these a third law, which enables us to compare the movements[Pg 142] of two different planets revolving round the same sun. Before stating this law, it is necessary to explain exactly what is meant by the mean distance of a planet. In its elliptic path the distance from the sun to the planet is constantly changing; but it is nevertheless easy to attach a distinct meaning to that distance which is an average of all the distances. This average is called the mean distance. The simplest way of finding the mean distance is to add the greatest of these quantities to the least, and take half the sum. We have already defined the periodic time of the planet; it is the number of days which the planet requires for the completion of a journey round its path. Kepler's third law establishes a relation between the mean distances and the periodic times of the various planets. That relation is stated in the following words:— "The squares of the periodic times are proportional to the cubes of the mean distances." Kepler knew that the different planets had different periodic times; he also saw that the greater the mean distance of the planet the greater was its periodic time, and he was determined to find out the connection between the two. It was easily found that it would not be true to say that the periodic time is merely proportional to the mean distance. Were this the case, then if one planet had a distance twice as great as another, the periodic time of the former would have been double that of the latter; observation showed, however, that the periodic time of the more distant planet exceeded twice, and was indeed nearly three times, that of the other. By repeated trials, which would have exhausted the patience of one less confident in his own sagacity, and less assured of the accuracy of the observations which he sought to interpret, Kepler at length discovered the true law, and expressed it in the form we have stated. To illustrate the nature of this law, we shall take for comparison the earth and the planet Venus. If we denote the mean distance of the earth from the sun by unity then the mean distance of Venus from the sun is 0·7233. Omitting decimals beyond the first place, we can represent the periodic[Pg 143] time of the earth as 365·3 days, and the periodic time of Venus as 224·7 days. Now the law which Kepler asserts is that the square of 365·3 is to the square of 224·7 in the same proportion as unity is to the cube of 0·7233. The reader can easily verify the truth of this identity by actual multiplication. It is, however, to be remembered that, as only four figures have been retained in the expressions of the periodic times, so only four figures are to be considered significant in making the calculations. The most striking manner of making the verification will be to regard the time of the revolution of Venus as an unknown quantity, and deduce it from the known revolution of the earth and the mean distance of Venus. In this way, by assuming Kepler's law, we deduce the cube of the periodic time by a simple proportion, and the resulting value of 224·7 days can then be obtained. As a matter of fact, in the calculations of astronomy, the distances of the planets are usually ascertained from Kepler's law. The periodic time of the planet is an element which can be measured with great accuracy; and once it is known, then the square of the mean distance, and consequently the mean distance itself, is determined. Such are the three celebrated laws of Planetary Motion, which have always been associated with the name of their discoverer. The profound skill by which these laws were elicited from the mass of observations, the intrinsic beauty of the laws themselves, their widespread generality, and the bond of union which they have established between the various members of the solar system, have given them quite an exceptional position in astronomy. As established by Kepler, these planetary laws were merely the results of observation. It was found, as a matter of fact, that the planets did move in ellipses, but Kepler assigned no reason why they should adopt this curve rather than any other. Still less was he able to offer a reason why these bodies should sweep over equal areas in equal times, or why that third law was invariably obeyed. The laws as they came from Kepler's hands stood out as three[Pg 144] independent truths; thoroughly established, no doubt, but unsupported by any arguments as to why these movements rather than any others should be appropriate for the revolutions of the planets. It was the crowning triumph of the great law of universal gravitation to remove this empirical character from Kepler's laws. Newton's grand discovery bound together the three isolated laws of Kepler into one beautiful