When we have ascertained the distance of the planets from the earth, it is easy to understand how we may find their distances from each other, since, in the triangle in which each is contained, two sides are already given, and the angle to the earth can always be measured. It is only with regard to the sun and the moon that the distances to the earth are of importance. It is enough to know the distances of the planets from the sun, and of the satellites from their planets, which involve little variation. These are our means for ascertaining asti'onomical distances. As we might anticipate, our assurance is in proportion to the nearness; and great remoteness baffles us entirely. We see here again, as everywhere, that the most simple and 170 POSITIVE PHILOSOPHY.
elementary determination depends on the most delicate and complex scientific theories. This first case exhibits so much of the spirit of astronomical procedure, that we may go more rapidly through the other statical heads of celestial geometry.
T-, 1. The distances of the stars from our globe being once ascertained, we can learn what- ever we desire about their form and size by observation, if it be but precise enough. Their very distance is favour- able to this; for, while their motion or ours displays in turn all their possible aspects, our distance enables us to see at once the whole of each aspect. With regard to the most distant and the smallest, however, — to the stars outside our system, and the satellites of Uranus, and the small planets between Mars and Jupiter, — they can aj^pear to us only as points of vivid light, and their sphericity is concluded upon only through a bold induction. But, in observing the larger planets of our system, we have only to measure their apparent diameter in all directions, after allowing for refraction and parallax. It is much easier to us to learn the form and size of sun and moon than of our own globe, since we have had the aid of glasses. The only case of difiiculty is that of Saturn's rings; as it once was with the moon, whose changing aspects greatly puzzled the ancients. The most simple geometry now solves the last difficulty, and Huyghens has helped us over the first. With these exceptions, direct observation assures us that the planets are all round, with more or less flattening at the poles and bulging at the equator, in proportion to the rapidity of their rotation.
As for the size of the heavenly bodies, it is easily calcu- lated from the measurement of the apparent diameter combined with the determination of the distance; and the only reason why men were so long and so widely luistaken about the dimensions of the planets was that their real distances were unknown. No rule as yet appears which connects these results with the order of the distance of the planets from the sun. All we know is that the sun is larger than all the other bodies of the system put together, and in general that the satellites are much smaller than their planets, as the laws of celestial mechanics require.
PLANETARY ATMOSPHERES. 171 With regard to the bodies outside of our system, as we have no knowledge of their distances, we are, of course, ignorant of their dimensions.
It is by the occultation of stars, as starry p. eclipses are called, that we make observations atnio.siih(^-es. on the atmospheres of the planets, by seeing what deviation their atmospheres cause in the light of the remote stars which they eclipse. As the sun's light is pro- longed to us by the refraction of our atmosphere, the atmosphere of a planet defers (only in a much greater degree) the occultation of the star, and also shortens it; and the comparison of the apparent duration of the eclij^se with that which it would otherwise be, gives us data for the calculation of the atmosphere which causes the devia- tion. It is thus that we learn that the moon has no appreciable atmosphere. The horizontal refraction which, on our globe, would reach thirty-four minutes, does not in. the moon amount to a single second. And the inference that an atmosphere is wanting there is confirmed by M. Arago, who in a different path of inquiry, about the polarization of light reflected from liquid surfaces, has established the fact that there are not, on the surface of the moon, any great liquid masses, fitted to form an atmosphere. The next best-known case is that of Venus, which exhibits a horizontal refraction of thirty minutes, twenty-four seconds. As for the extent of the atmo- spheres, it may be roughly conjectured from the cessation of the refracting power; but such conjectures must be very loose, as the refracting power may become imper- ceptible to us, far within the limits of an atmosphere becoming attenuated towards its verge. The strangest phenomenon is that of the telescopic planets, with the ex- ception of Vesta; the atmosphere of Pallas, for instance, being more than twelve times the diameter of the planet. The usual condition, however, appears to be that shared by our globe, — of an atmosphere which is very shallow in pro- portion to the dimensions of the jilanet: and this is nearly all we know.
The remaining statical topic is that of the „.^^ ^. f form and size of the earth, which has been left j^j^^^ ^j^e to the last, on account of its special nature.
172 POSITIVE PHILOSOPHY.
-.^.,. No srlance of the eye will aid us here, nor coverv ^^J direct observation whatever. A long accumulation of indirect observations, serv- ing as a basis for complex mathematical reasonings, are our only means. The geometrical aspect of the question must be taken first, though it depends on the highest mechanical theories, and arises from a mechanical begin- ning. In the infancy of mathematical astronomy, the variations exhibited in different places by the diurnal movement fui-nished the first geometrical proof of the earth being round. It was enough to establish its evi- dently and exclusively spherical character, that the change exhibited by the height of the pole on each horizon was always in exact proportion to the length traversed accord- ing to any meridian whatever: and this remains the source of all our geometrical knowledge of the form and dimen- sions of our j^lanet. Astronomers reached their knowledge of its precise form through that of its size; for it was long before its deviation from the perfect spherical form was understood. In this, as in every case, the form of any body is appreciable only by measuring its dimensions in various directions; and here the only difficulty is in the measuring. The first principles of the discovery were given by Eratos- thenes, in the early days of the school of Alexandria; but his method was never effectually employed till the middle of the seventeenth century, when Picard undertook to measure the degree between Paris and Amiens. This was the great starting point of the measuring operations, which must have revealed, as they became more perfect, the truth that the earth is not a perfect sphere; but Newton, by his theory of gravitation, and with his one fact of the shorten- ing of the seconds pendulum at Cayenne, settled the matter, by deciding that our globe must necessarily be flattened at the poles, and bulge at the equator, in the relation of 229 to 230. The astronomers could not at once pronounce against the evidence of direct measurement, while the geometers saw the fact to be certain; and the controversy between these two orders of philosophers, for half a century, led to those scientific operations which have brought us all to one mind. The question was settled by tbe great expedition sent forth, a.bove a century ago, by PLANETARY MOTIONS. 17S the French Academy, to measure, at the equator and the pole, the two extreme degrees of latitude which must exhibit the widest variation from each other: and the comparison of these with each other, and with Picard's degree, terminated the controversy, and established, not only the truth of Newton's discovery, but the very near accuracy of his calculation. All the experiments made since, in various countries, have imited in confirming the fact of the continual lengthening of degrees in approaching the pole. It does not follow that the figure of the earth has been ascertained with absolute jjrecision. There are slight discrepancies which may either be from imperfection in our estimates, or from the earth not being precisely an ellipsoid of revolution; but whatever may be the result of future labours, we know that we are near enough to the truth for all practical purposes, unless in questions of extreme dehcacy. We have no absolute knowledge here, any more than in any other department; and we must be content to make our approximations more complex as new phenomena arise to demand it. Such is the true character of the advances that have been made in this science from the beginning. Superficial observers may call its theories arbitrary, from the incessant changes of view that have arisen: but the knowledge gained has always been positive; every scientific opinion has corresponded with the facts which gave rise to it; and such opinions remain therefore useful and sound at this day, within their own range. The science has thus always exhibited a character of stability, through all incidents of progression, from the earliest days of the Alexandrian school till now.
Such are the statical aspects of the planets of our system. We have now to look at the geometrical theory of their motions.
Like all other bodies, the planets have a.
motion composed of translation and rotation. motions ^ The connection of these two motions is so natural, that when we know of the one we look for the other. Yet they present very different degrees of difficultv, and require separate consideration.
The progression of the stars was observed long before their rotation, — the unassisted eye being enough for the 174 POSITIVE PHILOSOPHY.
first; yet the geometrical study of their rotations is easier, because the motions of the observer have no effect upon them; whereas they largely affect questions of trans- lation. And again, the question of orbits is the chief difficulty of the study of translations; and it does not enter into that of rotations. The latter nearly approaches to the character of statical questions; and. therefore it ought to be taken first in the exposition of celestial geometry, p,,. Galileo introduced the study of rotations by discovering that of the sun, which was sure to follow closely on the invention of the telescope. The method used is obvious enough, and the same in all cases; — to observe any marks that may exist on the surface of the body, their displacement and return. The more such points of observation are multiplied, the more accurate and complete will be the calculations of time, magnitude, uniformity of movement, etc., deducible from them. There is no more delicate task than this, except with regard to the sun and moon; and none that more absolutely requires a special training of the eye. It is a proof of this, that a careful and honest observer, Bianchini, suj^posed the rota- tion of Venus to be twenty-four times slower than it is. Some bodies, as Uranus, are too remote, and others, as the satellites and new planets, too small, to have their rotation established at all, though it is concluded from analogy and induction. We, as yet, know of no law determining the time of these rotatios: they are not connected with dis- tances, nor with magnitudes; and they seem only to have some general, but not invariable, connection with the degree of flattening at the poles.' But if the duration is, though regular in each case, altogether irregular as regards the different bodies, the case is much otherwise with the direction; for it is always, throughout our system, from west to east, and on planes slightly inclined to that of the solar equator: and this constitutes an important general datum in the study of our globe. rr, 1 J.- The studv of translations, much more complex, IS also much more important, it we con- ' Tlie rotations of some of the satellites are known. They all follow the law of the moon's rotation, namely, the time corresponds with the orhital periods. — J. P. N.
PLANETARY MOTION OF TRANSLATION. 175 sider the great end of astronomical pursuit — the exact prevision of the state of the heavens at some future time. Besides that the movement of the earth constitutes an important elemeut in such a study, it must make a differ- ence with regard to other stars, whether the observer is fixed or moving, as his own movement must affect his observations of other motions. We might indeed decide with certainty, without this introductory knowledge, that the sun and not the earth is the true centre of the motions of all the i:)lanets, as Tycho Brahe did when he denied our own motion; for it is enough, with this view, to establish that the distances from the planets to the sun scarcely vary at all, while their distance from us varies excessively; and again, that the solar distance between each inferior planet and the sun is less, and between each sujierior planet and the sun is greater, than our distance from the sun. But we cannot go further than this, — we cannot determine the form of the planetary orbits, or the mode in which they are traversed, without making a careful and exact allowance for the displacement of the observer. Deferring for the present the subject of the earth's motion, we will briefly notice some important data connected with the planetary motions, which may be obtained without reference to our own movement, and which are so simple as to rank among statical researches. I mean particularly the knowledge of the planes of orbits, and of the duration of the sidereal revolutions, which has nothing to do with the form of the orbits or the variable velocity of the planets. A plane being determined by three points, it is enough to observe three positions of a star to draw a geometrical conclusion about the situation of the plane of its orbit. Astronomers do not now use, in these operations, the declinations and right ascensions, which are the only co-ordinates directly observed, but, for the sake of convenience, two other spherical co- ordinates, improperly called astronomical latitude and lon- gitude, which are analogous, with regard to the ecliptic, to the others with regard to the equator. After having de- termined the latitude and longitude of the planet in the three positions, its nodes are found; that is, the points at which its orbit meets the plane of the ecliptic, and the inclination of the orbit to this plane. It is evident that 176 POSITIVE PHILOSOPHY.
confirmation may be obtained by observing otlier positions of the body, if they are chosen sufiiciently remote from each other; and thus we may obtain a far greater jjrecision than in the case of rotations. It is thus that we have learned that the planes of all the planetary orbits ^Dass through the sun; and the same with regard to the satellites of any planet; and that these planes are in general slightly inclined to the ecliptic, and more slightly still to the plane of the solar equator, except the newly-discovered planets, in whose case we find the inclination much more con- siderable.
The duration of the sidereal revolutions Sidereal revo-,. itj.ii i • ai lution may, or course, be directly observed, in the first instance, by looking for the return of the star to the same spot in relation to the centre of its motion. If we suppose its motion to be uniform, which we may for a first approximation, we can estimate its course by observ- ing the time required between any of the three positions, without waiting for the total revolution, which is sometimes very slow. The geometrical law of this motion permits us to determine, from this kind of observation, the exact time of the planetary revolution. The values of these periodic times are not irregularly divided among the bodies of our system, like the other data that we have noticed. The shorter the course, the more rapid the motion; and the dui'ation increases more rapidly than the corresponding distance; so that the mean velocity diminishes in propor- tion as the distance increases. We owe to Kepler the discovery of the harmony between these two essential elements, and it is one of the most indispensable bases of celestial mechanics.
Such is the spirit of the methods by which celestial geometry is made to yield us the elementary data which characterize the bodies of the solar system. We have still to consider those of our own planet, before we proceed to the geometrical laws of the planetary motions.
Motion of the Earth.
We are accustomed to think of the motions of transla- tion and rotation as inseparable; but, in the transition MOTION OF THE EARTH. 177 from supposing the earth to be motionless, to the present state of our knowledge, a theory existed that it whirled round its axis, but was stationary in space. Evidences of We now perceive that, in addition to the the Earth's general evidence of the double motion of the Motion, planetary bodies, we have special evidence about our owa globe, — that the annual motion could not exist without the diurnal, though we might logically suppose beforehand that it could.
As the rotation of the earth cannot be absolutely uniform in all parts of its surface, some indications of its course must exist among terrestrial phenomena. We must there- fore distinguish between the celestial and terrestrial proofs of our diurnal motion, while the annual motion admits only of the former.
this case; tor it is clear that, to our eyes (as ceptions we do not feel the rotation), it must be exactly the same thing whether we move round among the heavenly bodies, or whether they, fixed in a system, move round us in a contrary direction. There was nothing absurd in the latter supposition, in the old days when men. had no doubt of the stars being very near, and not much larger than they appear to the eye, while they exaggerated the size of the earth. They could not avoid supposing that such a mass must be immovable, while the small stars, with their little intervals, were seen moving every day. Even when the Grreek astronomers had sketched out the true geometrical theory of the movements of the planets, they treated only of the directions, and had no idea of measuring distances; and it required the whole strength of positive evidence of dimensions and distances to uproot men's strong and natural persuasion of the stability of their globe. Fi'om the moment of our obtaining an idea of the proportions of the universe, the old conception became too revolting to reason to be sustained. When it was under- stood that the earth is a mere point in the midst of pro- digious intervals, and that its dimensions are extremely small in comparison with that of the sun, and even of other bodies of our own system, it was absurd to suppose that such a universe could travel round us every day. What I. N 178 POSITIVE PHILOSOPHY.
velocities would be required to enable tlie outlying stars to complete such a daily circuit, — making allowance for their being twenty-four thousand times nearer the earth, if the •earth describes no orbit, — and how small the movement of the earth, while those prodigious masses were travelling at such speed! On mechanical grounds, the centrifugal force would be seen to be unmanageable. In every way, the supposition was perceived to be monstrous. Again, the passage of stars before each other, and in a contrary direc- tion to that of the general movement of the sky, showed that they were at different distances from each other, and not bound into an unvarying fabric. Hence arose the notion of Aristotle and Ptolemy, of a system of solid and transparent firmaments. But the existence of comets alone was enough to confute this, appearing as they f-ave wffv ^^ ^^^ regions of the sky in tui-n. As '^ ' Fontenelle said, this theory put the universe in danger of being fractured. It was, curiously enough, Tycho Brahe, the most illustrious opponent of the Coper- nican system, who provided for the overthrow of his own arguments by first presenting the true geometrical theory of comets. Long before modern precision tion ^ ^ ^*^ ''' was attained, men had been prepared by such considerations as the above to conclude uj^on the rotation of the earth. Long before Copernicus, a rough conception of the truth existed. Even Tycho Brahe felt the astronomical superiority of the true theory; but it seemed to be contradicted by what is before our eyes, — the fall of heavy bodies, etc. Copernicus himself could not remove the objections which arose out of men's ignorance of the laws of Mechanics. These objections held their ground for a century, till Galileo established the great law which we have recognized as one of the three on which Rational Mechanics is based, — that the relative motions of ■different bodies are independent of the common motion of the whole. Till this was established, the supposition of the rotation of the earth was inadmissible. It is a curious fact, •casting much light upon the action of the human mind, that the opponents of Galileo taunted him with the so-called fact that a ball let down from the top of the mast of a ship in motion would not fall at the foot of the mast, but some way PROOFS OF THE EARTIl'S ROTATION. 179 behind, — neither they nor anybody else having tried the experiment, which would have shown them that their sup- posed fact was a mistake. The followers of Copernicus did worse, — they admitted the so-called fact, but tried to reason away its bearings with fantastic subtleties. The matter was not settled even by the demonstrations of Galileo, nor till Gassendi compelled observation by a public experiment in the port of Marseilles.
That order of experiments has been carried on, and would be of high value if we could obtain perpendicular stations of sufficient height for the purpose. It is clear that a lofty tower must describe a larger circle in the same time at the top than at the base; and that any body dropped from it must share the higher rate of velocity, having a slight horizontal velocity in the direction of the earth's rotation,^ — falling therefore a little to the east of the base of the tower. Omitting the consideration of the resistance of the air, this amount is calculable in the func- tion of the height of the tower and of its latitude; but ex- periment would also be valuable; and it is to be hoped that it will be tried at the equator, where the deviation must be greater than anywhere else.
The most certain terrestrial proof of the Influence of earth's rotation is found by tracing the in- centrifugal force fluence of the centrifugal force upon the ^^Pon gravity, direction and intensity of weight. This has been done by that observation of Richer, on the shortening of the seconds- pendulum at Cayenne, which has been mentioned as having emboldened Newton to declare the true figure of the earth. The deviation from the spherical form is too small to ac- count for more than one-third of the effect observed; and the other two-thirds are precisely what would be required, at the equator, where the centrifugal force is greatest, on the supposition of the earth's rotation. Wherever the de- licate observation can be made with sufficient precision on other points of the globe's surface, the result answers to the theory. Thus, we should have sufficient assurance, in the absence of the abundant astronomical proofs that we possess of the rotation of the earth. Probably no one fact has ever, in the history of our race, produced such conse- quences as that observation of Richer's, — two-thirds of the 180 POSITIVE PHILOSOPHY.
estimated effect having completely established the rotation of our globe, and the other third having led Newton to the ascertainment of its form.