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CHAP. VIII. HELIOCENTRIC PLACE OF A PLANET. 273 the perihelion, the former of which is to be determined, and the latter is one of the given elements. Lastly, the angle p S P is the heliocentric latitude of the planet, which is also required to be known.
(428.) Now, the time being given, and also the mo. ment of the planet's passing the perihelion, the interval, or the time of describing the portion A P of the orbit, is given, and the periodical time, and the whole area of the ellipse being known, the law of proportionality of areas to the times of their description gives the magni. tude of the area A SP. From this it is a problem of pure geometry to determine the corresponding angle ASP, which is called the planet's true anomaly. This problem is of the kind called transcendental, and has been resolved by a great variety of processes, some more, some less intricate. It offers, however, no peculiar dif. ficulty, and is practically resolved with great facility by the help of tables constructed for the purpose, adapted to the case of each particular planet. *
(429.) The true anomaly thus obtained, the planet's angular distance from the node, or the angle N SP, is to be found. Now, the longitudes of the perihelion and node being respectively pa and on N, which are given, their difference a N is also given, and the angle N of the spherical right-angled triangle A Na, being the inclination of the plane of the orbit to the ecliptic, is known. Hence we calculate the arc N A, or the angle NSA, which, added to A SP, gives the angle N SP required. And from this, regarded as the measure of the arc N P, forming the hypothenuse of the rightangled spherical triangle P NP, whose angle N, as before, is known, it is easy to obtain the other two sides, Np and P p. The latter, being the measure of the angle p SP, expresses the planet's heliocentric latitude ; the former measures the angle N Sp, or the planet's distance in longitude from its node, which, added to the known angle on SN, the longitude of the node, gives the heliocentric longitude. This process, however circuitous it may appear, when once well understood, may be gone through numerically, by the aid of the usual logarithmic and trigonometrical tables, in little more time than it will have taken the reader to peruse its description.
* It will readily be understood, that, except in the case of uniform cir. cular motion, an equable description of areas about any center is incompatible with an equable description of angles. The object of the problem in the text is to pass from the area, supposed known, to the angle, supposed unknown: in other words, to derive the true amount of angular motion from the perihelion, or the true anomaly from what is technically called the mean anomaly, that is, the mean angular motion which would have been performed had the motion in angle been uniform instead of the motion in area. It happens, fortunately, that this is the simplest of all problems of the transcendental kind, and can be resolved, in the most difficult case, by the rule of “ false position," or trial and error, in a very few mi. nutes. 'Nay, it may even be resolved instantly on inspection by a simple and easily constructed piece of mechanism, of which the reader may see a description in the Cambridge Philosophical Transactions, vol. iv. p. 425., by the author of this work.
(430.) The geocentric differs from the heliocentric place of a planet by reason of that parallactic change of apparent situation which arises from the earth's motion in its orbit. Were the planets' distance as vast as those of the stars, the earth's orbitual motion would be insen. sible when viewed from them, and they would always appear to us to hold the same relative situations among the fixed stars, as if viewed from the sun, i.e. they would then be seen in their heliocentric places. The difference, then, between the heliocentric and geocentric places of a planet is, in fact, the same thing with its parallax arising from the earth’s removal from the center of the system and its annual motion. It follows from this, that the first step towards a knowledge of its amount, and the consequent determination of the apparent place of each planet, as referred from the earth to the sphere of the fixed stars, must be to ascertain the proportion of its linear distances from the earth and from the sun, as compared with the earth's distance from the sun, and the angular positions of all three with respect to each other.
(431.) Suppose, therefore, S to represent the sun, E the earth, and P the planet; S r the line of equinoxes, ar E the earth's orbit, and Pp a perpendicular let fall from the planet on the ecliptic. Then will the angle
CHAP. VIII. GEOCENTRIC PLACE OF A PLANET. 275 SPE (according to the general notion of parallax conveyed in art. 69.) represent the parallax of the planet
P arising from the change of
station from S to E, E P will be the apparent direction of the planet seen from E; and if SQ be drawn parallel to
Ep, the angle or SQ will be the geocentric longitude of the planet, while p SE represents the heliocentric longitude of the earth, and or Sp that of the planet. The former of these, or SE, is given by the solar tables; the latter, a Sp is found by the process above described (art. 429.). Moreover, SP is the radius vector of the planet's orbit, and S E that of the earth's, both of which are determined from the known dimensions of their respective ellipses, and the places of the bodies in them at the assigned time. Lastly, the angle Psp is the planet's heliocentric latitude.
(432.) Our object, then, is, from all these data, to determine the angle or SQ and PE p, which is the geocentric latitude. The process, then, will stand as follows:- 1st, In the triangle S P p, right-angled at P, given S P, and the angle PSp (the planet's radius vector and heliocentric latitude), find Sp, and Pp; 2dly, In the triangle S E p, given Sp (just found), SE (the earth’s radius, vector), and the angle ESP (the difference of heliocentric longitudes of the earth and planet), find the angle SpE, and the side Ep. The former being equal to the alternate angle pSQ, is the parallactic removal of the planet in longitude, which, added to r Sp, gives its heliocentric longitude. The latter, Ep (which is called the curtate distance of the planet from the earth, gives at once the geocentric latitude, by means of the rightangled triangle PEP, of which Ep and P p are known sides, and the angle P E p is the longitude sought.
(433.) The calculations required for these purposes are nothing but the most ordinary processes of plane trigonometry; and, though somewhat tedious, are neither intricate nor difficult. When executed, however, they afford us the means of comparing the places of the planets actually observed with the elliptic theory, with the utmost exactness, and thus putting it to the severest trial; and it is upon the testimony of such computations, so brought into comparison with observed facts, that we declare that theory to be a true representation of nature.
(434.) The planets Mercury, Venus, Mars, Jupiter, and Saturn, have been known from the earliest ages in which astronomy has been cultivated. Cranus was discovered by Sir W. Herschel in 1781, March 19., in the course of a review of the heavens, in which every star visible in a telescope of a certain power was brought under close examination, when the new planet was immediately detected by its dise, under a high magnifying power. It has since been ascertained to have been observed on many previous occasions, with telescopes of insufficient power to show its disc, and even entered in catalogues as a star; and some of the observations which have been so recorded have been used to improve and extend our knowledge of its orbit. The discovery of the ultra-zodiacal planets dates from the first day of 1801, when Ceres was discovered by Piazzi, at Palermo; a discovery speedily followed by those of Juno by professor Harding, of Göttingen ; and of Pallas and Vesta, by Dr. Olbers, of Bremen. It is extremely remarkable that this important addition to our system had been in some sort surmised as a thing not unlikely, on the ground that the intervals between the planetary orbits go on doubling as we recede from the sun, or. nearly so. Thus, the interval between the orbits of the earth and Venus is nearly twice that between those of Venus and Mercury ; that between the orbits of Mars and the earth nearly twice that between the earth and Venus; and so on. The interval between the orbits of Jupiter and Mars, however, is too great, and would form an exception to this law, which is, however, again resumed in the case of the three remoter planets. It
CHAP. 'VIII. BODE'S LAW OF DISTANCES. 277 was, therefore, thrown out, by the late professor Bode of Berlin, as a possible surmise, that a planet might exist between Mars and Jupiter; and it may easily be imagined what was the astonishment of astronomers to find four, revolving in orbits tolerably well corresponding with the law in question. No account, à priori, or from theory, can be given of this singular progression, which is not, like Kepler's laws, strictly exact in its numerical veri. fication ; but the circumstances we have just mentioned lead to a strong belief that it is something beyond a mere accidental coincidence, and belongs to the essential structure of the system. It has been conjectured that the ultra-zodiacal planets are fragments of some greater planet, which formerly circulated in that interval, but has been blown to atoms by an explosion ; and that more such fragments exist, and may be hereafter discovered. This may serve as a specimen of the dreams in which astronomers, like other speculators, occasionally and harmlessly indulge.
(435.) We shall devote the rest of this chapter to an account of the physical peculiarities and probable condition of the several planets, so far as the former are known by observation, or the latter rest on probable grounds of conjecture. In this, three features principally strike us, as necessarily productive of extraordinary diversity in the provisions by which, if they be, like our earth, inhabited, animal life must be supported. There are, first, the difference in their respective supplies of light and heat from the sun ; secondly, the difference in the intensities of the gravitating forces which must subsist at their surfaces, or the different ratios which, on their several globes, the inertiæ of bodies must bear to their weights ; and, thirdly, the difference in the nature of the materials of which, from what we know of their mean density, we have every reason to believe they consist. The intensity of solar radiation is nearly seven times greater on Mercury than on the earth, and on Uranus 330 times less; the proportion between the two extremes being that of up