ASTRONOMY.
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nus, aiul Mercury, do in that position; but lie always shines witha full face, except near his quarlile aspect with the" sun, that is,when there is about the fourth part of the heavens between thesun and him, at which time he is observed to be gibbous like themoon three or four days before or after the opposition. That theearth is very far from being in the centre of the orbit of Mars isevident from his appearing to us almost live times larger in diame-ter when he is in opposition to the sun than when he is in conjunc-tion with him ; and consequently is almost live times nearer to usin the one position than in the other. But from observations ofhis motions as seen from the earth, it can be mathematically de-monstrated that the constantly varying distance of this planet fromthe sun arises solely from the eccentricity of his orbit and the dis-turbing force of the other planets, and therefore it is evident thatit is the sun which Mars respects as his centra. Again, since thename kind of motion in every particular is observed in Jupiter ,Saturn, and the Georgium Sidus, as in Mars , (though the dispro-portion or difference is not so great in Jupiter as in Mars , nor sogreat in Saturn as in Jupiter ; nor yet so great in the GeorgiumSidus as in Saturn , on account of their much greater distance fromthe earth than Mars is,) it is hence plain that the sun is the centreof all the motions of the superior as well as inferior planets.
(33.) Astronomers, before Kepler , supposed that the planetsmoved in circular orbits, with uniform motions, and to account forthe apparent inequality of the angular velocities, they imaginedthat the sun was placed at some distance from the centre of thecircular orbit ; let C be the center of the orbit, and S the sun,(Plate XX. Fig. 4.) it is easily shewn that while the planet, by anequable motion, passes through the equal arcs A B, BD, DM, See.the sun will apparently move among the stars, through the unequalarcs ah, bd, de, &c. " The justly celebrated Kepler would not re-ceive this hypothesis without examination, which by means of bisown, and Tvcho Brahe’s accurate observations, he accomplishedthus :
(34.) Let S, (Fig. 5,) represent the Sun, M, Mars , D, E, twoplaces of the earth when Mars is at M. When the earth wasat 1), he observed the angle M I) S, or difference of the longitudesof the Sun and Mars ; in like manner he observed the angle M ES.Mow the places 1) and E of the earth, in its orbit, being known,the distances DS, ES, and the angle DSE, will be known; hence(bv Trigonometry) MS, the distance of Mars may be found.—Kepler thus found that the greatest distance of Mars was l(St)78(>,and the least 138300, the mean distance of the earth from the sunbeing taken 100000 ; hence the mean distance of Mars was 132040,and the eccentricity of his orbit 14140. lu like manner he foundthree other distances, 147730, 163100,and 1(30235. lie next cal-culated the same three distances on the circular hypothesis, andfound them to be 148339, 103883, and 1(506 05, the errors or dif-ferences 'were 789, 783, and 350. Being persuaded the errorsproceeded not from the observations, and finding that the distancebetween the aphelion and perihelion was too great, upon the sup-position that the orbit was a circle, he concluded the orbit must bean oval, and an ellipse appearing the most simple of all ovals, besupposed the orbit to be an ellipse, and placed the sun in one of thefoci. (See his work, De Motibus Stella' Marils.) lie then calcu-lated the above observed distances together with the distances ofseveral other points of the orbit of Mars , and found that they allagreed with the distances obtained from the observations. lie thenin like manner found the same to be true for all the other planets.Hence, he ascertained, That the six primary plain ts revolve aboutthe sun in ellipses, having the sun in one of the foci.
(55.) lie had a strong passion for finding analogies in nature,and having thus discovered the relative mean distances of theplanets from the sun ; he began to seek for some relation betweenthem, and comparing t'lieir distances, he discovered the famous law,That the squares of the sidereal or periodic times of all the planets,are as the cubes of their mean distances from the sun. Hence,knowing the periodical times, the relative distances are determined.Tims, taking the distance of the earth 1, and its period 1 year, itwill he as 1, the square of 1, is to ], the cube of 1, so is the squareof the period of anv other planet to the cube of its mean distance;hence the cube root of the square of the periodic time of anyplanet being taken, we shall have its relative distance, that of theearth being 1. Again, The moon or satellite that is nearest toJupiter, is distant from Jupiter 's centre 1'51" at the mean distanceof this planet from the earth, and it performs its periodic or side-
vot. 1. —NO. 18.
real revolution in Id. 18h. 27in. 33|s. The outermost satellitedescribes its orbit in lOd. lOh. 32m. 81s. Sav, therefore, as23304192402 ", 23, the square of the sidereal revolution of the firstsatellite, is to 2079157799112 ", 25, (lie square of the periodic timeof the fourth satellite, so is 1307031” the cube of the mean distanceof the first satellite from Jupiter ’s centre, to 121704213"', 109. Ex,tract the cube root of this last number, and it will give 493”, 0,or 8' 15”, 0, for the distance of the fourth satellite from Jupiter 'scentre, at his mean distance from the sun.' Now the best obser-vations of late astronomers confirm this to be the true distance ofthis satellite from the centre of iU primary ; and the same propor-tion holds good in the other satellites, as likew ise in the satellite}of Saturn , and the Georgium Sidus.
(50.) This great astronomer also discovered from observationthat the velocity of the planets, when in their apsides, are inverselyas their distances from the sun, and that therefore in these pointsthey describe equal areas about the sun, in equal times. Hencehe thought this was true in every point of the orbit, and applyingthis hypothesis, he found that his calculations agreed with observa-tions, and hence ascertained, That the planets describe about thesun equal areas in equal times.
(57.) These laws deduced by Kepler entirely from observations,are the very same which Sir Isaac Newton lias investigated by geo-metrical reasoning, as the necessary consequence of a body revolv-ing in an ellipse by a force directed towards one of its foci, (beeCentral Forces.) \Y e are therefore warranted to conclude,That the planets are kept in their orbits by a force which is directed"to the sun, that the influence of this force extends to the utmostboundaries of the solar system, and we can assign no reason whyit may not extend to all distances whatever, observing the samelaw as the sagacity of New ton has investigated.
(58.) To illustrate this subject let AEBI1,(Plate XX. Fig. 6,)be an ellipse, to represent the orbit of. a planet, BA and HE theaxes, and S the focus in which the sun is placed. Let the planetpass over the parts of its orbit AC, CF, FP, tkc. in equal times.Join SC, SF, SP, &c. then the areas or spaces ASC, GSF, FSP.&c. described about the sun, S, in equal times are equal. Thepoint A is the aphelion, B the perihelion, and the line BA is calledthe apsides; SD is the eccentricity, and SC the mean distance ofthe planet which is equal to half the greater axis AB. Let neb h bethe orbit of another planet, ha the line of the apsides, and sc itsmean distance ; then, as the square of the periodic time, in theorbit AEBJI, is to the cube of SK, the mean distance, so is (liesquare of the periodic time in uebh to the cube of I he mean dis-tance S;’. Hence if the periodic times of the planets lie known,and the mean distance of any one of them, the distance of all (harestare found by this grand law, which is also exemplified in theorbits ol the secondary planets.
(59.) In the planetary orbits, the forms of the ellipses do notmuch differ from circles; in the orbit of ihe earth, the mean dis-tance SE being 1000, the eccentricity SI) is about 17, or nearlythe Ooth part of the mean distance. In the ditfeicnt orbits thelines of the apsides as, BA, ha, meet the heavens indifferent points,and intersect one another in the suit, at the common locus b, ofthe orbits.
(60.) 'Fhe orbits of the planets are not all in ihe same plane,but inclined to each other in small angles. Suppose the orbitaehh to turn on the line OQ, so that tiie part Hum may be alittle elevated above the plane AEB1J, (considered as the orltit ofthe earth,) and ah m a little depressed. The tingle formed bv thetwo planes is called the inclination of the orbit, the points «, m,referred to the heavens by the line n m produced, mark out in theecliptic, among the stars, the points O and D, which are called thenodes,-and O Q the line of the nodes; this line always passesthrough the sun, for the sun is in the plane of each orbit. Thatpoint where the planet ascends above the plane of the eclipticnorthward, is called the ascending node ; marked thus
(61.) Figure 1, Plate XXII. is drawn to give a just idea of theorbits of the planets, the eye being supposed to be perpendicular overthe plane of the ecliptic. The lines of the nodes and the apsidesare drawn in the figure. AC is the line of the nodes, and A theascending node of Venus ; b, the place of the sun, is the commonfocus of all the orbits; F is the other focus of Mars , /of Mercunand a of the Earth. Venus ’ orbit differs so little from a circle)that the upper focus cannot be represented in this scheme, the ( )i s Ilance between the foci being something less than -} of the distance4 A - botw'eca