I i AM ACCOUNT OF THE ASTRONOMICAL DISCOVERIES or KEPLER. ACCOUNT OF THB ASTRONOMICAL DISCOVERIES Of KEPLER; INCLUDING AN HISTORICAL REVIEW OF THE SYSTEMS WHICH HAD SUCCESSIVELY PREVAILED BEFORE HIS TIME. BY ROBERT SMALL, D. D. FELLOW OF THE ROYAL SOCIETY OF EDINBURGH. LONDON: PRINTED FOR. J. MAWMAN, NO. 22 fOULTRY, 1804. By T GUUt. Salis TO THE RIGHT HONOURABLE THE EARL OF LAUDERDALE, THE FOLLOWING ACCOUNT OF THE ASTRONOMICAL DISCOVERIES . OF KEPLER, IS GRATEFULLY INSCRIBED, BY HIS LORDSHIP'S MOST OBEDIENT AND DEVOTED SERVANT, R. SMALL. CONTENTS. Page Introduction, ...., 1 CHAP. I. Of the principal Motions and Inequalities of the Celestial Bodies 2 CHAP. II. Of the more Ancient Theories and Planetary Systems, and especially of the Ptolemaic System 32 CHAP. III. Of the Copernican System 8 i CHAP. IV. Of the System of Tycho BraU . . - l 126 ^ d r CHAP. V. Of the Preparations to Kepler s Discoveries, and of his Original Intentions 145 CHAP. VI. Of Kepler's Planetary Theoy, founded on ap- parent Oppositions, and of its total Failure. . 1 63 \. CHAP. \Tlil CONTENTS. Page CHAP. VII. Of Kepler s Solar Theory, that s, Us Theory of the Second Inequalities ..., IQ7 CHAP. VIII. Of the Theory of Mars resumed, and the Appli- cation to this Planet of the physical Method of Equations ; together with its important Consequences 227 NOTES , 307 t!> - AN ACCOUNT OF THB ASTRONOMICAL DISCOVERIES OF KEPLER, &c. AS the discoveries of Kepler have contributed more than all other causes to raise the science of astronomy to its present state of improvement, they not only deserve full and particular explica- tion, but also all the circumstances which led to them, and even the mistakes committed in their prosecution, become interesting objects of curiosity. It is a just observation of his, that we not only pardon Columbus and the Portuguese navigators, for relating their errors ; the former in the disco- very of America, and the latter in the circumnavi- gation of Africa; but should be deprived of much instruction and satisfaction if those errors were omitted. My principal intention, therefore, in the present publication, is to give a more full and par- ticular account of Kepler's discoveries, than any to be found in the usual systems, or the general his- tories of Astronomy ; and to extract the account from his own investigations. These are chiefly con- tained in his Commentary on the Motions of Mars; and I have often regretted that a work, containing B such 1 ASTRONOMICAL DISCOVERIES such invaluable discoveries, should not be more generally and distinctly known. This work claims attention for another reason, that it exhibited, even prior to the publication of Bacon's Novum Qrga- num, a more perfect example,, than perhaps ever was given, of legitimate connection between theory and experiment ; of experiments suggested by theory, and of theory submitted without preju- dice to the test and decision of experiments. But, in order to form a just estimate of those disco- veries, nay, perhaps, a distinct conception of the investigations by which they were produced, it seemed absolutely necessary to prefix an account of the more ancient astronomical theories, and of the principal phenomena which they were con- tPived and supposed to explain. CHAP. I. Of tire principal Motions and Inequalities of the Celestial Bodies. 1. In the contemplation of the heavens, the at- tention of mankind must have been first attracted by the splendor and configurations of the celestial bo- dies, and the constant diurnal revolution in whicii they scern in general to partake.. They all appear to be placed in the vast concave surface of the ethereal expanse, at equal distances from the spec- tator's eye, and indeed from the earth itself; for no change af the spectator's situation, upon the surface of the earth, is found to produce any gene- ral and sensible ehauge on their relative positions, It was concluded, therefore, that this expanse was a sphere, o which the earth is the centre ; and of such OP KEPLER, 3 such immense magnitude, that the earth, in com- First mo- parison with it, was to be considered only as a sin- tion,ordi- gle point : and, by the constant rotation of this ""^ expanse upon its axis, the whole phenomena of the diurnal revolution, called the first motion, were explained. 2. But in attending to the circumstances of the diurnal revolution, it was discovered that seven of the celestial bodies, to wit, the Sun, the Moon, Mercury, Venus, Mars, Jupiter, and Saturn, though constantly carried round the earth in the course of the diurnal revolution, neither described their circles with the same accuracy as the rest, nor in the same precise portion of time. On mark- ing the points of the horizon, where on any parti- cular day they rose and set, they were always found after some interval to rise and set at different points, more to the north, or to the south : and, in the same manner, the points where they crossed the meridian were also observed to vary, though within certain limits ; and their transits over it to be made at unequal distances, both from the zenith and from either pole. Those diurnal circles were seldom great circles of the sphere : they were ra- Peculiar ther a perpetual spiral than a series of circles, and m^ons of resembled a thread regularly wound about a cylin- by means of Meton's cycle, or perhaps some other still less perfect, the mean time of both, and consequently the mean motion of the moon in longitude, for any particular portion of time, was in some manner, however im- perfectly, obtained. 8. This determination, however, of the moon's mean motion in longitude, though imperfect, was sufficient to demonstrate, that her real motion was unequal : for it was found that, if her mean and true places had at any time coincided, in about 7 dnys after they generally differed 5 or 6 degrees ; in about 14 days they again coincided ; in 21 days they again differed 5 or 6 degrees in a contrary di- rection ; and, in 2/ days, and something more than an half, they again coincided as at first. It appeared therefore, that there were two opposite Inequality points of the lunar orbit, afterwards called apsides of the xi. i i -^ i i i-rr- i moon's mo- asm the solar orbit, in which the differences be- t ion in ion. tween the moon's mean and real places disappeared ; and ie ASTRONOMICAL DISCOVERIES and two others, at equal distances from either of these, in which they came to their greatest amount ; so that, in the one semicircle, the velocity was in- creasing from its least to its mean, and then to it$ greatest rate ; and, in the other, decreasing in a reversed order, though in the same proportion. To ascertain the period of those inequalities, the an- cients employed lunar eclipses, the observations of them being the only ones in which the place of the moon was not deranged by parallax : and, if in any two eclipses the motions of the moon had been found eqwal and of the same kind, either increas- ing ie velocity, or decreasing, they considered this as a proof, that in both she occupied the same point of her orbit, or was in the same degree of anomaly* Consequently, they were enabled, by their records of her retardations and accelerations, to compute how many repetitions of those inequa- lities had been made in the interval, or how many revolutions the moon had made with respect to the apsides of her* orbit: and their conclusions were confirmed, if two other eclipses could be found, whose interval was the same, where the moon's motions were also equal, but in the oppo- site simicircle of anomaly, with the velocity de- creasing, ^s in the former it had been increasing. Hipparchus accordingly found two such pairs of eclipses, at the interval of 126,007 days, and 1 hour, during which he computed the moon to have performed 426; synodical revolutions, and 46l2 tropical revolutions, wanting 7 3O' ; while the re- turns of the inequality now mentioned, that is, the revolutions in the orbit were 4573. It be- came evident, therefore, that the returns of this inequality had neither any connection with the synodical revolution, (because in 4267 lunations^ the moon had returned to the same degree of ano- maly OF KEPLER. 11 Vnuly 4573 times) ; nor with the tropical revolu- tion, (because during those 4573 revolutions of anomaly, she had been nearly 46 12 times in the same degree of longitude). The lunar apsides, therefore, were variable, and not fixed points in the zodiac, and appeared to advance, during every lunation, about 3 degrees in longitude, so as in () years to complete the circuit of the heavens : and if the greatest difference between the mean, and true places of the moon had happened, that is, if the greatest equation of 5 or 6 degrees had been applied, in one lunation, at any particular point of longitude, it was not applicable in the next lunation at the same point, but at one about 3 more advanced. By this investigation also of liipparchus, a new cycle for calculating the lunar motions was discovered, namely, that in 251 luna- tions the moon performs 269 revolutions of ano- maly : for the ratio of 4267 to 4573, is the same with that of 251 to 269. Q. It was likewise discovered, chiefly by the observation of lunar eclipses, that, as the ecliptic intersects the equator, so the lunar orbit inter- sects the ecliptic in 'two opposite points, called nodes ; and that these opposite points are, like the apsides, variable, though in a contrary di- rection, in their longitude. For it was only in the nodes that the moon was seen to be without latitude ; in all the other points of her orbit she was found to be cither on the north, or on the south of the ecliptic; till, at the distance of 00 from the nodes, her latitude, or deviation from the ecliptic, amounted to 5 degrees: and, since in lunar eclipses she must always be very near one or other of the nodes, it appeared that she was not involved in the shadow of the earth, at the same constant l ASTRONOMICAL DISCOVERIES of constant points of the zodiac. The position of l!be the moon's no des, therefore, was variable ; and their motion became also evident from her occultations of stars situated in the ecliptic. When an occultation of such stars, as for example, of Regulus, happens, the moon is certainly in one of the nodes : but, as in something less than 5 years, she is seen to pass 5 degrees to th-e north or the south of Regulus; and as this is the greatest latitude at which she is known to arrive, she Hiust have departed from the node to the distance of QO degrees. To determine the rate and direction of the motion of the nodes, Hipparchus compared all the lunar eclipses which fead been observed from the times of the Chaldean astronomers to his own, and found that, in 5458 lunations, the moon had been in the same node 5Q23 times. By the same comparison another cycle was discovered of great importance in the calculatiorrof lunar eclipses : for, in 223 Itinations, that is, in 18 years and 10 days, the moon had made 242 revolutions to the same node, 241 and 1O 46' more to the same degree of longitude, and 23Q to the same degree of anomaly : so that while the anornalistical revolution was longer than the tropical, and consequently the apsides were con- tinually advancing in the ecliptic, the revolution rn latitude, or to the nodes, was shorter ; and the. motion of these points therefore retrograde. 1O. Modern astronomers, possessing many ad- vantages above the ancients, for determining the mean times of the various lunar -revolutions, have accordingly applied themselves to this subject with the greatest care. D. Cassini, for example, com- pared a lunar eclipse, recorded by Ptolemy to have been observed at Babylon, 720 years before the Christian sera, with an eclipse observed -by himself m OP KEPLER. 13 in the year of that aera 1/17 The Chaldean eclipse, he finds, must have happened at Paris on the igth of March, at 6h. 48', O. S ; and his own was observed there, on the Qth of September, at 6 h. 2'. The moon's longitude in the former was 5s. 21 27', and in the latter 11 s. 27 34'; and the interval consists of 890,288 days, wanting 46 minutes. In this interval he calculated,, that the moon had performed 32,585 revolutions to the same point of longitude, and 6s. 6 7' more ; and consequently, dividing the interval by the number of revolutions, that the mean time of every one was 27 d. 7 h. 43' 5". But, as there might be some uncertainty about the number of the revolu- tions, and the omission of one in so long a period would not produce a difference of above one minute in the length of each, he also made comparisons between less distant eclipses, and by these estab- lished his conclusion. . The moon's mean diurnal Meantro- motion, therefore, in longitude, according to D. Cassini, is 13lO / 35 // ; and multiplying this by the 36o|, he found her annual motion ; and this pro- duct again by 10O, her secular motion ; that is, the remainders of both, after subtracting the com- plete revolutions. By still later investigations of the same kind, the tropical revolution, at the begin- ning of the eighteenth century, is found to be per- formed in 27 d. 7 h. 43' 4 /y .648 ; and consequently the mean diurnal motion in longitude, to be 13 JO' 35 // .02847, and the mean secular motion lOs. 7 53' 35". From the time of a tropical revolution thus Sidereal re* ascertained, that of a sidereal one is easily inferred ; volution - for, in 27 d. 7 h. 43' 4 /; .648, the equinoctial points move in precession through 4 ;/ of longitude ; and as the moon's mean motion in 1 hour is through 32' Sf) 7 '. 458, she will employ 7" of time in moving through 14 ASTRONOMICAL DISCOVERIES through 4^. The time, therefore, of a mean real revolution of the moon, in the 18th century, lS27d. 7h. 43' Il /; .648. From the same investigation, the time will also easily follow of a mean synodical revolution. For the times of the mean tropical revolutions of the sun and moon are given, and the arches they de- scribe are inversely as the times. Let / be the time of a tropical revolution of the moon, and T that of the sun ; and let the arch described by the sun, in the time of a synodical revolution, be called x, and revolution. conse q U ently that described by the moon in the same time 36o + # Since, therefore, T : / :: 360 + x : x 9 we shall have T tit:: 360 : x ; and this arch therefore will be found : and, since the time of the moon*s tropical revolution, or of describing 360, is given, the whole time of de- scribing 360 + x will be found to consist of 2Qd. 12 h. 44' 2".8291. The time, again, of an anornalistical revolution is, in the same manner, determined by comparisons between the places occupied by the apsides, in ages distant from one another. By observing the points where the moon's mean and true places coincide, the longitude of the apsis is given ; for it is, in these circumstances, the same with the longitude of the moon:' and, from the nature of the lunar motion about the time of the observation, it may always be distinguished, whether the apsis under consideration be the apogee, or the perigee. Or *^ e pl acc of either apsis may be more accurately found, by observing the points of longitude where the greatest equations of contrary kinds, or, in ge- neral, where any two equal equations of contrary kinds take plnce^ because the apsides are situated precisely in the middle between them. To these means of discovering the position of the apsides, th KEPLER: -15 the moderns add observations of the lunar diame- ter ; for, as its apparent magnitude varies between the two extremes of 29' 30", and 33' 30 ;/ , the moon is evidently in the apogee in the former case, and in the perigee in the latter. Accordingly, by the comparison of such places, the apsides were found to perform their revolution with respect to either of the equinoctial points, in 323 I'd. 8h. 24' 57 /7 .6; and with respect to any particular star, in 3232 d. 11 h. 14' 31" : so that their mean diurnal motion towards the former is 6' 41 ".0698 15, and towards the latter 6' 40^.931992 ; and the time of the ano- malistical revolution is 27 d. 13 h. 18' 34 // .022. Finally, the places of the nodes were determined Revolution by observations of the moon's latitude, especially of the in eclipses ; and, by like comparisons between n< them, they are found to perform their retrograde revolution, with respect to the equinoctial points, in 6798 d. 4h. 52' 52 /y ; so that the time of the moon's revolution, with respect to the nodes, con- sists of 27 d. 5 h, 5' 49". 1709. Thus-, no less than five kinds of lunar revolutions came to be distinguished, the tropical, which brings back the. moon to the same longitude ; the sidereaf, which brings her back to the longitude of some particular star ; the synodical, in which she goes through all her varieties of phasis, and returns to Jher conjunction with the sun ; the anomalistical, in which she returns to the apogee of her orbit, and during which, the inequality above mentioned performs its course ; and her revolution to either of the nodes, which brings back her whole varie- ties of latitude. 1 1. The greatest amount of the inequality hi- therto spoken of, and 'called the equation of the orbit, the first inequality, and the mtequalitas so- luta, J$ ASTRONOMICAL DISCOVERIES luta, was fixed by Hipparchus and Ptolemy, at 5 l' : and, while the attention of astronomers was principally turned to lunar eclipses, no other had been distinguished, at least accurately ascertained. But when armill a d of Venus, 470 48 ^ to tne westward of the sun, she not only overtakes him in the superior conjunction, but also gets beyond the place he occupies, and attains to a like elongation to the eastward: in the firft of which circumstances she is called the f morning star, and in the last, the evening star. She then becomes stationary for about 36 hours, when, her retrograde motion begins ; and it continues till, after passing the inferior conjunction, she becomes again stationary at her greatest western elongation. As this planet is likewise retained as an attendant upon the sun, it is obvious that her mean returns to the same point of longitude must be also made in the time of a solar revolution ; and that h.er re- volution through the zodiac must be performed sooner than the return of all the vicissitudes now described. 17. A synodical revolution of the planet Mars h performed in about 780 days, and during 707 of these OF KEPLER. 21 these bis motion is progressive, and in general more rapid than that of the sun. From an elon- gation of about 134 to the weft ward of the sun, he not only comes up with the sun in the con- junction, but passes beyond the conjunction, and attains to a similar eastern elongation. In this configuration he is, as it were, arrested in his course, becomes for two days stationary, and then seems to be drawn backward to the opposition. This retrograde motion continues for about 73 days, and through an arch from 10 to 18 degrees ; till after the opposition the planet is found to the westward of the sun, and before the renewal of its progressive course besomcs stationary a second time. The period necessary for the return of all these vicissitudes is longer than the period of the planet's revolution in the zodiac : for this is perform- ed in 6s6 days. The apparent motions and irregu- larities of Jupiter and Saturn are similar to those and of the r TV it til i r lupcnor of Mars, excepting only that the completion of their vicissitudes takes un less time than their re- volution in the zodiac. The mean sy nodical re- volution of the former employs 398 days, in 12O ot which the planet's motion is retrograde, through an arch of about 1C degrees; his stations continue about 4 days each, and they happen when his elon- gations from the sun arise to 115. The sy nodical revolution again of Saturn employs 378 days, in 138 of which his motion is retrograde, through an arch of 6 or 7 degrees, and each of his stations continues about 8 clays. To these it may be added, that the synodical revolutions of the planet disco- vered by Herschel are performed in about 37 O days, that its retrograde motion continues about 142 days, through an arch of 4, and that each of its stations continues about 1 1 days. C 3 18. But 22 ASTRONOMICAL DISCOVERIES 18. But it was not enough to know that all the vicissitudes now described return and complete their course, in the time of a synodical revolution; for it was equally necessary to obtain the motion of the planets in the zodiac separate from all those sy nodical derangements: and, that this might be accomplished, the ancients endeavoured to compare their motions with the sun's mean progress on the ecliptic, as a standard which was not affected by any inequality. In making the comparison with respect to Mars, Jupiter, and Saturn, called the superior planets, they turned their attention chiefly to their oppositions; as being not only the points in which they were moft observable, but in which also they were thought to occupy the same places in the zodiac, as if they had not been subjected to any synodical vicissitudes. By a comparison 'of such oppositions it was discovered that these vicis- situdes were noc the only irregularities to which the superior planets were subjected ; but that their progressive motions in the zodiac were also, like those of the sun and moon, inequable in themselves. The mean returns of those planets to their opposi- tions were made, as has been said, in the following intervals, of Mars in 779 d. 22 h. 28' 26 /y , of Ju- piter in 398 d. 21 h. 15' 44 ;/ .6, and of Saturn in 3/8 d. 2 h. 8' 7 /; .8. Bui the real oppositions to the mean place of the sun seldom, or never, re- turned in these intervals, and very seldom after any other equal intervals: and the points where they took place were very seldom at equal distances in the zodiac. The oppositions, for example, of Ju- piter, as they return almofl every year, and this planet performs its revolution round the zodiac in nearly 12 years, ought to happen at the distance of about one sign from one another. But when Jupiter is OP KEPLER. 23 is in the neighbourhood of Libra, they happen at a First and less distance; and, in the neighbourhood of Aries, JJShS" at a greater. His motion therefore, abstracting ot the from all consideration of his occasional accelera- P lanets - tions and retrogradations, is evidently subject to another inequality, and of a different kind: and his progress in one part of his orbit is slower than in another. This, although probably the latest in being discovered, was called the first inequality, f and, as it had no dependence upon the synodical revolution, the mtfqualitas soluta. The other, more striking one, whose returns always keep pace with the synodical revolution, was called the second ine- quality, and the insequalitas alligaia. ]<}. In order then to discover the course and law of the first inequality, in any planetary orbit, it was necessary to find the mean motion of the planet, or the time which it employs in performing its revolution in the zodiac: for, in the same manner as in the cases of the sun and moon, the comparison of the real place, observed at any time when it was divested of the second inequality, with the calculated mean place, would give the corre- spondent first inequality, or (hew the points where it disappeared. The molt effectual method for this purpose was, in the orbits of the iuperior planets, to employ the same oppositions by which this inequality was at first discovered: only taking care to choose the most distant ones which might lie found. This was necessary upon two accounts; first, of the errors which might be supposed to have been committed in observation; and secondly, of the still more important errors which might arise from the observations having been made in different* or opposite, degrees of anomaly. For these errors, even the last of them, unless where C 4 the 24 ASTRONOMICAL DISCOVEBIES the inequality was very great, by being thus distri- buted among a very great number of revolutions, would disappear. But a previously necessary step was to approximate to the time of a revolution, by a comparison of less distant oppositions: for, till this was nearly found, it could not be known how many revolutions had been performed in the in- terval. D. Cassini's procedure, in determining the mean motions of Saturn, will serve as an example. First, to approximate to the time of a revolution, he compared together the three oppositions, of the l6th of September 1701, at 2h. 0'; of the 10th of September 1730, at 12 h. 27 '; and of the 23d of September 1731, at 15 h. 31'. The observed lon- gitude of the planet, in the first of these opposi- tions, was 11 s. 23 21' }&'-, in the second, 11s. 17 53' 57"; and in the third, Os 30' 50 /7 . The interval between the first and second consists of 29 years, of which seven are bissextile, wanting 5 d. 13 h. 33'; and between the second and third, pf 378 d. 3 h. 24': and the difference of the two first longitudes is 5 27' 1Q /7 , and of the two last nosa 12 36/ 53// * Tlleref re t0 find the time in Which turn " Saturn will describe the 5 27' IQ 11 necessary to complete his circle ; that is, to bring him to the same point of longitude which he occupied at the first opposition; he had this analogy, 12 36' 53 ;/ : 5 27' 19 ;/ : : 3/8 d. 3 h. 24': l63 d. 12 h. 41'. By adding this quantity therefore to the first interval, he had, for a first approximation to the time of a tropical revolution of Saturn, 2Q y. 104 d. 23 h. S': whence the mean annual motion of the planet is found to be 12 13' 23". 8'; and the mean diurnal motion 2' o".466. Next, to deiermine the time of a revolution more accurately, he employed an ancient observa- tion OF KEPLER. 5 tion made at Babylon, on the 24th of the month Tybi, in the olyth year of the aera of Nabonassar, and uhich time he found to be the 1st of March, in the 228th year before the Christian aera, accord- ing to the Julian calendar. In this observation, Saturn, when in opposition to the sun, was in con- junction with the star marked y in the south shoulder of Virgo; and Cassini computes that this conjunction took place at h. on the meridian of Babylon, or at 3 h. on the meridian of Paris. Hence he concludes, by taking into the account the precession of the equinoxes, that, in the 228th year before the Christian sera, Saturn was in oppo- sition on the2d of March O S. at 1 h. on the me- ridian of Paris, in 5 s. 8 25' of iongi'ude. With this ancient opposition he compares that of Fe- bruary ]5, 1714, at 8 h. 15', in 5 s. 7 b6' 4' of longitude; and that of February 28, 17 it), at 16 h. 55' in 5s. 21 3' 14". The interval between the first and second oppositions consists of ilMSy. 105 d. 7 h. 15'; and in the second the planet wanted 28' 14" to complete its circle. The inrerval again between the two last is 378 d. 8 h. 4O', and the difference of their longitudes is 18 6' -IS 11 . Therefore 13 6' 28' 7 : 28' 14*: : 3/8 H. 8 h. 4O': 13 d. 14 h.; and consequently 13 d. 14 h. added to the first interval, gives 1Q43 y. Il8d. 21 h. la' for the period which brings Saturn to the point or lon- gitude which he occupied in the first opposition. Dividing this period by 2Q y. 164 d. 23 h. 8', the approximate time before found of a tropical revolu- tion, it appeared that the planet had performed GO such revolutions in 1Q43). 118 d. 21 h. ]f>':so that the mean time of each was 'jQy 162 d. 4 n. '27', and the planet's mean annual motion in longitude 12 13' 35' .233. The same astronomer endeavoured, by a like pro- ccuure, 25 ASTRONOMICAL DISCOVERIES cedure, to determine the mean motion of Jupiter, ff Jupiter, To approximate to this he first employed the mo^ dern oppositions of 1699, 171O, 17 11, which gave for the time of a revolution lly. 313d. ldh.34'; secondly, those of 1672, 1731, 1732, which gave 11 y. 315 d. 7 h. 13'; and thirdly, those of 1(J2O, and 1703, which gave 11 y. 315 d. 11 h. 25': a mean of which is 11 y. 314 d. igh. 47' Then, for a more accurate determination, he compared Ptolemy's oppositions, first of the year 133 with those of 1698 and 1699; secondly, of the year 136 with that of 1713; and thirdly that of 137 with the opposition of 1714. The results for the time of a tropical revolution of Jupiter were 11 y. 315 d. 11 h. 25'; 11 y. 315 d. 17 h. 6'; and 11 y. 315 d. 16 h. 32 ; . A mean of these is lly. 315d. 14 h. 36 / ~4430 d. 14 h. 36': and consequently the mean annual motion of Jupiter in longitude is 3O 20' 3l t '.S3 ) and his mean diurnal motion 4'59". By similar researches, first from modern obser- vations, and secondly from a comparison of these with the ancient oppositions of Ptolemy, in the years 130, 135, and 139, he found the time and of of a tropical revolution of Mars to consist of Mars. (js6d. 22 h. IS'; his mean annual motion in lon- gitude to be 6 s. 11 17' 9 ;/ .5; and his mean di- urnal motion to be 31' 26 ;/ : and care was taken, in all these comparisons, to associate the opposi- tions, where it was found, by a previous knowledge of the theories of those planets, that the planet was nearly in the same degree of anomaly : and spe- cial regard was had to such as were at a conside- rable distance from the apsides. Surreal ^o. The times of the mean sidereal revolutions revoiu- r . , . lions. of the planets are, on account of the precession of the OP KEPLER, 27 the equinoxes, longer than those of the tropical re- volutions. According to the determinations of D. Cassini and Halley, they are, for Saturn, 29 y. J7 d. 6 h. 36' 2(j"zz 10759 d. <5h. 30' 26"; for Jupiter, 4332 d. 12 h. 20' 25"; and for Mars, 686 d. 23 h. 27 ' 30". 21. When, by researches of this kind, the an- cients had determined the mean motions of those three planets, they were enabled to compare toge- ther the mean \vith the observed places, and conse- quently to find their differences. By following the course of such differences through all parts of the zodiac, they perceived that, in the same manner as in the orbits of the sun and moon, there were two opposite points, or apsides, of every planetary orbit, in which the differences entirely disappeared, and two other opposite points, in the middle be- tween the former, where they came to their greatest amount, though in different directions: for, in the one semicircle, the mean place always preceded the true, and in the opposite semicircle, it was al- ways left as far behind. The motions therefore of Equations the superior planets, as viewed from the earth, were of the evidently unequal, independently of their annual centie - vicissitudes; and equations were necessary in their orbits as well as in those of the sun and moon. These were called the equations of the centre; and, in the orbit of Saturn, they were found to arise to 6^ 30'; in that of Jupiter to 5 30'; and of Mars to 10 D 39'. 22. The motions of Venus and Mercury, called' the inferior planets, differed in two essential cir- cumstances from those of the superior. For, as has been seen, the general progress of the superior planets in their orbits, and the times of their revo- lutions, 28 ASTRONOMICL DISCOVERIES hitions, bad no dependence upon the revolutions of the sun; and it was only their stations, accele- rations, and retrogradations, that were connected with his positions, and completed their course in the times of his synodical revolutions. But it was, on the contrary, the apparent revolutions of Venus and Mercury in their orbits which were connected with the sun, for they were performed in the precise time of the sun's tropical revolution ; and their stations and other vicissitudes of second inequality had no dependence on any of his periods. The mean motions therefore of Venus and Mercury, or their mean progress in their orbits, were the same with the mean motion of the sun: and the determina- tion of it did not require from the ancients such laborious researches as have been now exemplified. But, judging their real progress to be unequal, they found as great difficulty in determining it, as in de- termining the real progress of the superior planets. For, as the unequal distances of the times and points of the oppositions shewed the digressions of the superior planets fronvthe point opposite to the Fit-it inc. sun t o b e unequal, so the visible unequal digres- 2?vJmM sions of Venus and Mercury from the sun himself, and Mer- were supposed to evidence like inequalities in the ewy. progress of these inferior planets in their orbits : and, the prejudices of the ancients in favour of the perfect regularity of the celestial motions, not per- mitting them to believe that these apparent un- equal digressions were really unequal, no hypo- thesis was left to account for their differences, ex- cept what was founded on the difference of the dis- tances from the earth at which the digressions were viewed. Ascribing therefore to the inferior planets inequalities to which they were not subject, they supposed the orBits of Venus and Mercury to have, each of them, an apogee, in which the arches OF KEPLER. 2$ arches of digression, subtending the smallest angle at the earth, would appear smallest ; a perigee, in which they appeared greatest; and two other points in the middle between these apsides, where the apparent magnitude of those arches would be a mean between both extremes. The first inequa- lity therefore of each inferior planet followed as a necessary consequence ; for its apparent velocity, independent of all secondary vicissitudes, would be always inversely at its distance : and, as in the progress of the planet from its apogee to its return, all its first inequalities completed their course, so the variations of the second inequalities, or greatest digressions, depended upon the degree of anomaly which the planet occupied at the time of their taking place. But though the variations of the greatest digressions of Venus seemed to agree with these suppositions, they were not fufricient to ex- plain the variations of the digressions of Mercury: for though his greatest digression subtended the smallest angle at the apogee, the perigee was not the point of the orbit where it subtended the greatest angle ; and the angle which it subtended was found, or at least believed, to be the greatest, at two points on either side of the perigee, and dO distant from it, 23. As the stations and other vicissitudes of second inequality of the inferior planets had no connection with any of the solar periods, it was a' business of equal importance, to determine their laws and progress, as to determine the motions in the orbits. The inferior planets, as has been said, arc never in opposition ; their conjunctions, til! telescopes were invented, were all invisible ; their stations were not instantaneous ; and, from their nearness, especially that of Merc urv, to the sun, all 3O ASTRONOMICAL DISCOVERIES all the observations of them were made near the horizon, and were therefore precarious and doubt-' fill. The only method, therefore, which remained to the ancients, was to find two observations of either of those planets, as distant as possible from each other, in which the planet was both in the same degree of longitude in the zodiac, and at the same distance in the same direction from the mean place of the sun : for its returns to the same point of longitude shewed that it had performed round the earth a certain number of revolutions in the zodiac, and its returns to the same distance from the sun shewed that it had also completed a ccr- Secondin- tain number of revolutions of second inequality: anc | t j ie num b er O f foQth being supposed to have been, either observed during the interval, or known by other means, the times in which each was per- formed were consequently given. Accordingly, it was found in this manner by Hipparchus, that in 8 years, wanting 2 d. 6 h. 83, there had been five returns of all the vicissitudes of the second in- equality of Venus ; so that the mean time of each consisted of 583 d. 22 h.; and that in 46 y. 1 d. ()h. 15', the second inequalities of Mercury had returned 145 times, so that every one of his mean synodical revolutions employed about 115 days. 25. The ancients also found that the sarne equ- tions, between the mean and true places of the Motion of planets, did not continue to be applicable in the the planet, same points of longitude, and that therefore their ide*. P ~ apsides were not fixed points in the zodiac, but moveable,like those of the sun and moon, according to the order of the signs. But they considered the motions of all as equal, and the same in quantity with the precession of the equinoxes. 25. As OF KEPLER. 31 25. As all the great circles, in which the planets performed their peculiar motions, wore early known to be distinct from the equator, so it was also soon discovered that neither did they coincide with the solar orbit, or ecliptic ; but that they were all inclined to it in different angles, and intersected it in opposite points, called, as in the lunar orbit, nodes. It was only at these nodes that any planet Nodes of was ever seen in the ecliptic: and at all the other jjl eor - points of their orbits they were observed to have latitude, north or south, according to the passage of the planet through the ascending or descending node. The latitudes were also seen to be extremely variable, yet not so as to exceed the limits of the zodiac. These intersections of the planetary orbits with the ecliptic, afforded another method of deter- mining the times of their revolutions, and conse- quently of finding their mean motions. For, if a planet has been observed in two successive passages through the same node, the interval, if the node has no motion, is evidently the time of its tropical revolution: and, if the node has any motion, it must be added to the motion of the planet, or sub- tracted from it. CHAP. 32 ASTRONOMICAL DISCOVERIES CHAP. II. Of the more Ancient Theories and Planetary Systems, and especially of the Ptolemaic System. $6. r | 1 HE purpose intended in all systems was Jl_ to give a just representation of the dif- fered phenomena of the celestial bodies, and espe- cially to explain the causes and laws of those inequ- lities in their motions, which observation from time to time discovered. When these motions were so perplexed, and apparently irregular, as has been seen in the foregoing account, it is evident that the explication of them was an attempt of the greatest difficulty ; and that, instead of its being surprising that the ancients were unsuccessful in it, the inge- nuity which was capable of reducing them to any settled rules whatever, is entitled to our higheil praise. The natural difficulty of accounting for so* many complicated appearances was also increased by physical prejudices; and by this in particular, that all the celestial motions, notwithstanding their apparent irregularity, were perfectly circular and uniform. A circular motion was considered as not only the most perfect of all motions, and therefore the only one suited to the nature of such divine and perfect beings as the stars were conceived to be; but also as the only motion which they could perform without extrinsic violence. The great object there- fore of the ancients in all their systems was, as has been formerly mentioned, to delineate the circles, andcompositionsof circles, by means of which all the apparently unequal motions of the celestial bodies might OP KELLER. 33 might be resolved into such as were regular and uniform. 27. The most celebrated of the ancient systems, appears to have been invented by Apollonius, not long after Aristotle ; and, though it takes its name and received many improvements from Ptolemy, to have been carried to its perfection chiefly by Hip- parchus. In this system the earth is supposed to be immoveably fixed in the centre of the starry sphere, and round the earth all the motions, both diurnal and periodical, of the celestial bodies are represented to be performed. The order, in which those bodies, or rather the spheres in which they move, are arranged in it, is the following. The Ptolemaic lowest of all the spheres, or the least distant from s > tcm - the earth, was that of the moon : and this place was justly assigned to it, both on account of the occul* tations by the moon of all the celestial bodies which she encounters in her progress through the zodiac, find because she is more than any other deranged by the diurnal parallax. The sphere of Mercury was placed next to the moon. Then followed those of Venus,- the Sun, Mars, Jupiter, and Sa- turn, in the order now enumerated; and beyond Saturn the sphere of the fixed stars was placed, at a distance so immense, that no change of the ob- server's situation upon the earth made any change on the relative positions of the bodies situated in it, and to its semidiameter the semidiameter of the earth had no sensible proportion. This sphere Eighth was called the eighth, or the frimum mobile\ it '^^^ uniformly revolved round the earth, from east to mobile, west, in 24 sidereal, or in 23 h. 56" 4 ;/ .098 mean Solar hours; and, in its revolution, all the inferior orbits were supposed to partake, with degrees of Telocity modified by their respective periodical re- D volutions 34 ASTRONOMICAL DISCOVERIES volutions in directions nearly opposite. But this system was not implicitly received by all the an- cients, nor does it appear upon what principles all its parts were formed. Many objections were made, especially to the arrangement of Venus and Mer- cury: and Geber of Seville, in particular, an Ara- bian astronomer, remarks that in this part Ptolemy had directly contradicted his own doctrines and observations: for, though he supposed the diurnal parallaxes of Venus and Mercury to be insensible, and the sun to have a parallax, of 2' 51 y/ , he not- withstanding placed them below the sun; and so far below him, that the parallax of Venus ought sometimes to arise to 3 7 , and of Mercury to no less than J f . Geber also adduces, as an argument against the arrangement of Ptolemy, a circumstance in which that arrangement was not really in fault, and which the modern invention of telescopes would have turned into a presumption in its favour. For he demonstrates, from the latitudes of Venus and Mercury, that these two planets ought to be frequently seen passing over the solar disk; and as no such phenomenon had ever been observed, he infers that there was strong reason to suspect the theory to be false. It was probably on account of these and similar difficulties, that another sys- tem was formed, which received its denomination from Plato, and seems never to have been destitute of illustrious adherents. In it the sun was placed next to the moon, after him Venus, and next Mer- cury; in other respects it agreed with the Ptole- maic. Much regard was also at all times paid to the ancient Egyptian system; in which Venus and Mercury were supposed to be satellites, or attend- ants upon the sun, and to revolve round him con- tinually, while he performed his annual revolution round the earth. Among the disciples of this system OF KEPLER. 35 system were Vitruvius, Martian us Capella, Macro- bius, Bede, &c. if not Plato himself. 28. Though some philosophers, especially Anax- agoras and Democritus, taught that the celestial bodies moved in spaces void of all resistance, their doctrine, as it assigned no physical causes of the celestial motions, was never prevalent among astro- nomers. The current opinion was, that those bo- dies were fixed in transparent solid spheres, to which motions were communicated, from some original power or cause, by mechanical operations r- and, it being likewise an inviolable maxim that all Riseofth* the celestial motions were perfectly regular and cir- cular, these two principles seem to have given rise to the first and most ancient method of accounting for the observed inequalities, and to which the name of the concentric theory was given. For when the same sphere, or the same great circle of it, in which any moveable celestial body was plac- ed, could not admit of all the motions peculiar to that body, another subordinate one, like a smaller wheel moving upon the circumference of a greater, with different and even contrary motions, was supposed to be adapted to it: and the body was conceived to be placed in the circumference of this subordinate sphere or circle, and to be carried round by its motion on its axis. A subordinate sphere or circle, which thus moved in the circumference of a large one, was called an epicycle; and the larger one, on which it moved, was called its defe- rent. By means of an epicycle, one motion ap- parently irregular, was resolved into two that were circular and uniform: and when the observed mo- tion was so irregular and complicated, as not to be represented by one epicycle, there was no other expedient but to super-add, more. In short, the D 2 whole 35 ASTRONOMICAL DISCOVERIES whole celestial physics of the ancients hinged on two principles, supposed to be unquestionable} first) that all the celestial motions were perfectly circular; and secondly, that they were really uni- form, when referred to their proper centres: and when an ancient astronomer had investigated the circles and epicycles which seemed best to agree with his observations, he rested satisfied that he had divined the true system of nature. '2Q. The representation made by this Concentric theory of the solar inequalities in longitude, was as follows. Let C (fig. J.) be the centre both of the earth and of the circle FBD; and let HGK be ^a smaller circle : called an epicycle, whose cente B moves uniformly in the circumference FBD, from west to east, or in consequentia, w.hile the sun moves also uniformly, and with the same velocity in the circumference of the epicycle, in antecedent la in the Concentric upper part, but in cvnsequentia in the lower. If the solar u >- p j nt Q O f t h e e pj c y c ] e? called its apogee, as being most distant from the earth, be supposed, at the beginning of the anomalistical revolution, to be placed in the point A of CF produced ; and if, when it comes to G, the arch GH be taken similar to FB, the point H will be the place of the sun when the centre of the epicycle has moved from F to B. If then in CF, to which BH is parallel, we take CErrBH, and on E as a centre with the distance EAzrCF describe the circle A HP, tha sun would be seen from E to move in this circle equably; for the angle AEH is equal to the angle FCB: but seen from C, the centre of the earth, he will appear to move in it inequably; for tha angle ACH, in the first semicircle of anomaly, that is, in the passage of the sun from A to P, is always less than AEH or FCB; and his true place OP KEPLER. 37 place H will be less advanced in longitude than his mean place B. When, again, the centre of the epicycle, or the mean place of the sun, having do- scribed a semicircle, shall have come to D, the sun having also described a semicircle of the epicycle," \viil be found in P, the perigee of the orbit A HP, Apogee and his mean and true places B and H will be seen *" l e pcrl " from C to coincide, as they did in A, the apogee. But in the sun's passage from P to A, that is in the second semicircle of anomaly, his true place H, as seen from C, will be always more advanced in lon- gitude than his mean place B: for, in this semi- circle, the angle PCH is always greater than PEH, or DOB. The angle EHC, or BCH, which is the difference between the mean and true places of the sun, is called the equation of the orbit: as Equation being that quantity which added to the true mo- bttheorblt - tiori ACH of the planet in its orbit AIIP in the first semicircle of anomaly, and subtracted from it in the opposite semicircle, will render it equal to the mean motion AEH, or FCB: and it is evident that this equation, or difference, will be greatest in N or M, where the centre B of the epicycle is (,0 distant from either apsis. Any lines drawn from Deferent. E, the centre of the orbit AH P., to the true place of the sun in H, and from C the centre of the earth and of the deferent FBD to the mean place of the sun in B, are equally called the line of the mean motion of the sun; because these lines are always parallel, and mark the same point in the zodiac: and any line drawn from C to the true place of the sun is the line of the sun's true mo- tion. 30. It was thus that the ancients originally pro- ceeded in their representation of the solar inequa- lities, and the representation seemed to be sufH- D 3 ciertly 38 ASTRONOMICAL DISCOVEHIES ciently justified by observation: at least, till the days of T. Brahe, no observations had been made with sufficient accuracy to subject it to suspicion. Their success also, while no lunar inequality ex- cept the simple anomalistical one was discovered, was equal in the application of the same concentric theory to the motions of the moon: and having, in two cases, thus successfully, by means of one subordinate sphere or epicycle, reconciled apparent inequality of motion with real uniformity, it was natural to suppose that other inequalities, though more various and complicated, might be explained in a similar manner, and required only the addi- tion of other epicycles. The same method of pro- cedure therefore was continued, and every new inequality which observation discovered, was ac- counted for by a new sphere or epicycle producing it, till the whole number employed in the system amounted to 34. Aristotle, on narrower exami- nation, found these insufficient, and added to them 2: but ftill they were deemed insufficient ; and the number was at last increased to 72. But though it was not till long after the days of Aris- totle, that the theory was carried to such a degree of extravagance, the multiplication of epicycles rendered it, even in his time, almost as intricate and complex as the appearances which it was in- tended to explain. Some examples of this kind will occur on the revival of it by Copernicus and T. Brahe: and when Hipparchus and Ptolemy in- roduced eccentric orbits, and by means of them somewhat diminished the multiplicity of the spheres employed by their predecessors, they were thought to do a signal service to astronomy. 3 1 . The manner in which Hipparchus explained the solar inequalities, in his eccentric theory, was to OF KEPLER. 39 to this purpose. Let O (fig. 2.) be the centre of the earth and of the starry sphere. Let BCDE be the ecliplic, or great circle in the primum mobile in which the sun seems to perform bib annual revolu- tions; and, in the same plane, but on a different centre Z, let \he circle ALP be described. This is the circle, or orbit, in which the sun is supposed actually to move, and to describe round its centre equal arches or angles in equal times: or rather, he is supposed to be carried round by the equable motion of the circle itself; which, because its centre is not occupied by the earth, is called an cxcentric circle. It is evident, in this representa- Excentric tion, that if the earth were placed in Z, a spectator solar lht * on it would perceive the sun, since he is supposed to move uniformly in the excentric, to move also uniformly in the ecliptic. But the earth is placed in O, at the distance OZ from the centre of the excentric; and therefore, when his motions are referred to the ecliptic by a spectator in O, they must appear unequal. When, for example, he de- parts from A the apogee of the excentric, and comes to K, he would be seen from Z in the point R of the ecliptic; but from O, the centre of the earth, he is seen in C, a point less advanced in lon- gitude. On the contrary, when he departs from P, the perigee of the excentric, and comes to N, his place in the ecliptic, as seen from Z r is the point V; but seen from O it is the point F, more advanced in longitude than V. Any line, as ZK, drawn from the centre of the excentic to the sun, or any parallel to it drawn from O, is called the line of mean motion, and determines the mean anomaly AZK; and any line, as OK, drawn from the sun to the centre of the earth, is called the line of true motion, and determines the true anomaly AOK; and the angle OKZ, which is the difference D 4 between 40 ASTRONOMICAL DISCOVERIES between the mean and true anomalies, is the equa- tion of the orbit. In the apogee and perigee this equation vanishes, in the same manner as in the coneentric theory, because there the lines of mean and true motion coincide; and at the points L, L, \vhcre a perpendicular to the line of apsides, drawn through O, meets the excentric, it comes to its greatest amount. Thus, by the single supposition that the solar orbit was excentric to the earth, Hip- parchus supplied the place of the epicycle added to the concentric : nor is it difficult to perceive that the representations, given by both theories of the solar inequalities, were in their effects precisely the same. 32. In both these theories, it is evident that the; inequalities of the sun were considered as purely optical: and what was principally required was to find the point O, in the line AP of the apsides, in which the earth must be situated, in order to give Jo the solar motions the just inequality which ob- servation required; and to determine the longitude A of the solar apogee, that' is the point B of the ecliptic to which it is referred from O. Without finding the just excentricity OZ, the calculated differences, or equations, between the sun's mean and true places, would not correspond with the observed differences: and without discovering the position of the apogee, the calculated equations, however accurate, would not be applied in their proper places. In these investigations the proce- dure of Hipparchus was as follows. Let B (fig. 3.) be the place of the sun at the vernal equinox, BDan arch of the excentric equal to his mean motion for 94 J days, that is, from the vernal equinox to the summer solstice (5.), DF an arch equal to his mean motion for Q2| days, or from OF KEPLER. 4] from that solstice to the autumnal equinox: let the chord BF be drawn, and from D another chord DEG perpendicular to BF. The point E of the intersection of these chords is evidently the point where the earth must be situated: for it is the only point from which B, D, F, G, can appear at the distance of QO from one another, and as they ap- pear actually in the heavens. It was required therefore to determine the ex- centricity EC, or the distance of the point E from C, the centre of the solar orbit. Since the arch BDF of the mean motion, and which the ancients supposed to be the only real motion of the sun, from B the vernal to F the autumnal equinox, is given by means of the annual revolution, and con- sists of 184 ? 2O', its half BH will consist of gi 10'. Through H draw the diameter HC; and, through C the centre of the orbit, the diameter CK perpen- dicular to HC, and meeting DE in L; and join CE. If from BD, the mean motion for Q4f days, and Q3 Q', we subtract BHz=92 10', the re- mainder DH will be foundnzog': 'and if from BH we subtract the quadrant KH, the remainder BK will be ~2 1O'. Therefore CLand EL, the sines of the arches DH, BK, will be given in parts of CDzrlO,OOO; the former to witzz!7'2, and the latter ~378. Therefore the excentricity EC, being the hypothenuse of the right-angled triangle ELC, will be found in the same parts: for EC'zrCL 2 -f- EL 2 , and therefore EC~415. Bv producing EC to meet the excentric in A and P, we shall have the line AP of the apsides: and the longitude of A the apogee will be found Longitude from the same triangle ELC; for CE:EL::R: f sin. ECL KCA BEA 6530'. SinceBthere- fore represents the point of Aries, the place A of the apogee tell short of the solstitial point D,243O', in 42 ASTRONOMICAL DISCOVERIES in the days of Hipparchus. It is now between 9* and 10 more advanced. 33. When the excentricity and the longitude of the apsides were thus determined, the equations of the orbit, or the differences between the mean and true places of the sun, were obtained by a very simple trigonometrical computation. For in the excentric theory, (fig. 2) when this equation comes to its greatest magnitude, in the points L, L, or M, M, where perpendiculars to the line of apsides ir drawn through O or Z meet the excentric, the ex- ! " Centricity OZ becomes the tangent of the angle HOBS. ZMO to the radius of the orbit ZM: and at any other point K, with the excentricity OZ, the radius ZK, and the angle KZO the supplement of the mean anomaly AZK, the equation ZKO will be found by the analogy KZ-f-ZO:KZ~ZO : : tan. iAZK: tan, {(ZOK ZKO). The calculation will be precisely the same in the concentric theory if EC betaken in CA, (fig 1.) equal to the radius of the epicycle, that is to the excentricity OZ, (fig. 2.) and will produce the same results: and the excentric theory will be found to differ from the concentric only in simplicity. It is probably need^- less to observe that the parallels OL and ZM will mark the same point in the zodiac. 34. As the first inequality of the moon (8) was .the only one which for a long time was known to the ancients, it was, like the solar inequality, re- presented by either of the theories now described ; and Hipparchus appears to have used both promis- cuously. But as the greatest equations of the moon did not take place in the same fixed points of the zodiac, but were found to advance about 3 a eastward in every tropical revolution, this peculia- rity OF KEPLER. 43 rity was accounted for, in the excentnc theory, by Excentric an- orbit AGPH (fig. 4.) not fixed and unmovcable Jj^S^rt like the solar orbit ; but moveable upon the point T firs- inc. where the centre of the earth is placed, and de- 1^"*' scribing with its centre C, round that point, a small circle Cc. If therefore, in any one revolution, that orbit had been in the position AGPH, with the greatest equations EK and FL, it would in the next tropical revolution come into the position a g p h, and the greatest equations would be the arches c k and fl, equal indeed to the former, but more advanced in longitude. The effect thus re- presented was produced by the different durations of the tropical and anomalistical revolutions: for as the latter are about five hours longer than the former, (lO) it must happen that, when the moon returns to the longitude of the point B, she has not yet returned to the apogee of her orbit; nor will she return to it, till after advancing about 3 far- ther in the zodiac, 35. In the concentric theory of the moon there Concentric was no difference from the solar concentric theory, theol 'y of except that in the latter the revolutions of the sun i in the circumference of the epicycle, and of the tie$ - centre of the epicycle in the deferent, were made in times precisely equal; whereas in the former they were made in unequal times; the moon's re- volution in the epicycle requiring 27 d. 13 h. &c. and the revolution of the centre of the epicycle requiring only 27 d. 7 h. &c. Consequently when the centre of the epicycle, after a tropical revolu- tion, returns to its first situation in A, (fig. 5.) the moon has not yet returned to'E. the apogee of the epicyle; nor will return to it, till the centre has ad- vaurcd in longitude 3 beyond A. The indepen- dency therefore of the anomalistical revolution upon 44 ASTRONOMICAL DISCOVERIES upon the tropical, and the advanced position of the greatest equations in the zodiac, became as evident on this supposition, as on the former. 36. What was principally required in these theories was to ascertain, in the first, the ratio of the excentricity, and in the second, that of the sc- midiameter of the epicycle, to the semidiamcter of the orbit; and to investigate the moon's equa- tions. The procedure in the concentric theory xvas to this purpose. Ptolemy had observed three lunar eclipses; the first in the year 133 of the Christian sera, when the moon was seen in 7s. 13 35'; the second in 134, in Os. 25 10'; and the third in 136, in 5s. 14 5': and Copernicus found that, in time reckoned on the meridian of Cracow, the first had happened on the 6th of May, at J 1 h, 15'; and that the interval between the first and Second consisted of 531 d, 13 h. 29'; and, between the second and third, of 5O2 d. 5 h. 3O'. The ap* Tofind,in parent motion, therefore, of the moon in longitude, theccncen- in passing from 7s. J3 15' to Os. '25 10', after ttetttS? neglecting the complete revolutions, was 161 55'; oftheepi- and, in passing from s. 25 10' to 5s. 14 5', it eycJc was 138 55'. But the mean motion of the moon, that is, of the centre of the epicycle, known by the time of her tropical revolutions, was in the first in- terval, 169 37', and in the second, 13/ 34': and her anomalistical motion in the epicycle, being the difference between the mean motion of the moon and that of her apogee, was also, in the first interval 1 JO 21'; and, in the second, 81 36'. Let then ABC (fig. 6.) be the lunar epicycle, and E its centre revolving in comequentla round T, the centre of the earth: and let it be supposed that the first eclipse was observed in the point A of the epicycle, a place somewhat more advanced in longitude than the .OF KEPLER. 45 the mean place E or D. Since the first eclipse was observed in A, it is evident, that, if the moon's apparent motion in longitude had been, in the first interval, equal to her mean motion, the second eclipse would have also happened at the point A; the angle ATE continuing in that case invariable. But since the moon's apparent motion, in the first interval, is 7 42' less than the mean, the second eclipse will not happen in the point A, but in a point B, less advanced in longitude by 7 42': and the anomalistical arch of 11O 21', which she also described in this interval, she must have described vi antecedentia, and therefore in the most distant part of her epicycle, and its position must have been such as to subtend, at the centre of the earth, the angle ATB of 7 42'. In like manner, since the apparent motion of the moon in longitude is, in the second interval, greater by J 21 ' than the mean motion, the third eclipse will not happen at the same point B with the second, but at a point C more advanced in longitude than B by 1 21': and the moon, when describing in the same interval the anomalistical arch BC of 81 36'^ must have been descending to the lower part of the epicycle; .and even moving in that lower part; because the angle BTC of 1 21' is described in conse- quentia. Since then the arch AB is towards the apogee of the epicycle, the line T A, if it be not a tangent to the epicycle, will meet it again in some point F towards the perigee. Join BF, CF, BC. In the triangle BFT all the angles are given: for AFB standing on the arch ABzzl 1O 21' 5 and consequently =55 10' 30", is the supplement of BFT, andBTFz=/42'; and therefore the side FT, in parts of the scmid. of the circumscribing eircle, will be found =73698, and BF=133Qy. The * ASTRONOMICAL DISCOVERIES The angles also of the triangle CFT are given 5 forCTFzzBTF BTCzz6 4l'; and AFC the supplement of CFT stands on the arch ABCzzlpt 57' , and is therefore 95 58' 3d'. Therefore FT, in parts of the semid. of a circle described about CFT~999SO, and CFzr 11060; and therefore, in parts of BF found formerly, CF = 8151. But in the triangle BFC, the angle BFC, standing on the arch BC = 81 36', is equal to 40 48', and therefore with the sides BF=* 13399 and CF = 8151, the remaining side BC will be found = 8980. But BC being the chord of 81 36', will be =* 130684, in parts of the semid. of the epicycle == 100000; and therefore, in like parts, FT ^ 107'2684, and CF=1 18637; and consequently the arch CKF of which it is the chord ** 72 46' 10*. Consequently, FHA = CFA CKF * 95 16' 5O", and its chord FA = 14778(); and therefore TFA =*TF + FA = 1220470. Suppose E the centre of the epicycle found, and the line TED of apsides drawn: and then, since AT. TF *= DT. TK, and DT. TK -f KE 2 =TE 2 , this square will be given; and therefore also TE = 1148558, in parts of KE= 10000O. Therefore in parts of TE = 100000, we shall have KE 8?06. By the same method of investigation, Copernicus found, from three lunar eclipses observed by himself, that TE: KE:s 100000:860-1. Thesemidiameter of the epicycle being deter- andthee- mined, the equations \Vere thus investigated. Ff0m thc cenlre E draw EG perpendicular to AF, and meeting the epicycle in H. Since TE, TF, and FA are given, in parts of KE = 1000OO, GFwill be found, in the same parts = 73893, arid GT = 1146577. Therefore, in the right-angled triangle TGE, the angle TEG, which is measured by thc arch K H, will be found OF KEPLER. 47 found = 86 38' 30 V ; and consequently its sup- plement HED, or the arch HD=g321 / 3O": and since HA = |FA= 47 38' 25", the arch AD of the moon's distance from the apogee, in the first eclipse, will be = HD HA = 45 43' ; BD the anomaly, in the second eclipse, will be = BA AD = (J4 38'; and CBD the anomaly, in the third eclipse, will be = CB + BD = 146 14'. Since also TEG = 86 38' 30", andTGE a right angle, the angle GTE or ATD, the equa- tion at the first eclipse, will be = 3 22'; at the second eclipse the equation will be BTD = ATB ATD = 4 20'; and at the third eclipse the equation will be CTD = BTD ETC = 2 59'; and in the same manner the equations in all other degrees of anomaly might be derived. 37. In the excentric theory the procedure was to this purpose. Let D (fig. 7.) be the excentric place of the earth 1n the circle ABC, in the cir- cumference of which the moon performs her ano- malistical revolution ; and A, B, C, the three eclipses observed from D. Let E be the centre of the circle; join AD, BD, CD, AE, BE, CE; produce any of the lines AD. drawn from the moon to the earth, to meet the circle again in G; and join GB, CC. Since all the lunar motions are To find the supposed to be uniform round the centre E, and ** their periods given, the anomalistical angles AEB, the\cen- BEC will be given, for they are equal to the dif- '"ic theory, ierences between the mean motion of the moon, and the motion of the apogee, in the intervals of the eclipses. As also the apparent motion of the moon in these intervals is known by her observed longitude in each eclipse, the angles ADB, BDC are likewise given; for they are equal to the difV ferences 48 ASTRONOMICAL DISCOVERIES ferences between the motion of the apogee and the moon's apparent motion. Thus Moon's mean mot. 1st interval 169 37'; Motion of the apogee 5Q 16 j Therefore AEB 11O 21 ; Moon's apparent motion l6l 55' Motion of the apogee 59 1.6 andADB 102 3c)> Moon's mean mot. 2d interval 137 34'; Motion of the apogee 55 58; Therefore BE C 81 3(5 Moon's apparent motion 138 55' Motion of the apogee 55 58 and BDC 82 57 Then 1st, in the triangle DBG, with the given angles BDG the supplement of ADB, and BGD the half of AEB, and BD assumed of any definite length, suppose ==200000, DG may be found. 2dly, In the triangle DCG, with DG and the given angles DGC = AEC, and CDGthe sup- ple, of ADC, the side DC may be found- 3dly, Jn the triangle BDC, with the slides BD, DC, arid the included angle BDC we may find DBC and BC. 4thly, In the isosceles triangle BEC with BC and BEC we may find BE. 5thly, In the triangle EBD, with the sides BD, BE, and the included angle EBD - EBC DBC, we may find the excentricity DE in parts of BD assumed, and the angle BDE, which is the distance of the moon from the apogee in the second eclipse. 6thly, Since OF KEPLER. 4g Since BE is found in parts of BD assumed, if we make BE = IOOOOO, the excentricity-ED will be found in the same parts, and the result will be the same as in the concentric theory. 38. Though both these theories were equivalent in their effects, and therefore equally fitted to re- present the first lunar inequality, yet whsn the cvection, or second inequality, came to be consi- dered, they were insufficient to represent both combined; and Ptolemy found it necessary to in- vent another much more complicated. In this the moon was still supposed to perform her anp- malistical revolution in the circumference of an epicycle, but the deferent on which the centre of that epicyle moved was no longer concentrical, but excentrical, to the earth; and the epicycle re- volved on this deferent, in such a manner, as in all mean conjunctions and oppositions to be always in its apogee, and, in all quadratures, in its perigee. f This effect was produced by two supposed con- trary motions performed round the centre of the earth ; to wit, of the epicycle in conseqtienlia, and of the apsides together with the centre of the de- ferent inantecedentia\ and the rates of motion were Pfolemy's such, that the angular distances on both sides of theor y ot ., P, t , ,,,, theevec* the mean place of the sun were always equal, ihe tiou. representation of it was thus effected. Let E. (fig. 8.) be the centre of the earth, O the centre of the deferent AFP supposed to revolve round E, and to describe with its apogee A the circle ADCB in the course of one lunation; and let this circle be divided into four quadrants by the perpendicular diameters AC, BD. If at the time of the mean conjunction in the line EA the apogee of the deferent be conceived to coincide tf ith the centre of the epicycle, and, if, for the E sake 50 ASTRONOMICAL DISCOVERIES sake of simplicity, we suppose the sun to continue in this line during the whole lunation, the conse- quenses will be these: C will be the point of the full moon or opposition, B and D the points of quadrature, and the apogee A moving in antece- dentia from the line EA will against the first quadrature come to D, having described the arch AD, or angle AED of 90, while the centre of the epicycle GH, by a like motion in consequentia^ will describe the equal angle AEK. Though there- fore their angular distance from the sun in EA is only Q0, they will be diametrically opposite to each other; that is, the centre of the epicycle will be in the perigee of the deferent, and the epicycle, which represents the lunar inequality, being now at the least distance from the earth, will be seen under the greatest possible angle, to wit, LEK or NEB much greater than MEA; and the equa- tion NEB will now rise^to 7 4o', whereas MEA, the greatest equation which can take place at the conjunction, is only 5 I 1 ' Thus the evection or second lunar inequality was by this theory repre- sented to be merely an optical amplification of the first, and wholly produced by the approach of the lunar epicycle, especially at the quadratures, to the earth: and the theory also explained the perplex- ing circumstance of its not coming to its full amount in every quadrature ; for, if the moon should then happen to be in the apogee or the perigee of her epicycle, as the first inequality could not take place in such quadrature, so neither could the second: or if the moon should be then found in some point Q of the epicyle not far dis- tant from either of its apsides, the inequality would indeed take place, but in a much less degree. It is true that the sun does not continue in the line E A during the whole lunation, and consequently neither cuoa Of- KEPLER. 5 I ^either will the points B and D be the precise points of quadrature, nor will the angles A ED, AEK, made by the line of apsides and the line of the moon's mean motion, with the line of the sun's mean motion, be eqtial, for the sun advances in con- sequentia at the mean daily rate of 59' 8 ;/ : in order therefore to render these angles always equal, the daily motion ascribed to the centre of the epicycle was 13 ll', and to the lunar apogee only 11 2', and the equality of the angles AED, AEK, was in all cases thus preserved. 3Q. In this representation, however, the prin- ob j t , ciple of uniform motion in perfect circles was not to this in all respects inviolably maintained. For, if by theor >'- means of the supposed contrary revolutions of the epicycle and the apogee of the deferent, the arch AR of the zodiac intercepted, (fie. 9.) at the first octant for example, between the lines EKR and E A, or the angle AER which is subtended at the centre of the earth, was = 90, the arch AK of the deferent which the centre of the epicycle really de- scribed, or the angle AOK at the centre of the orbit which AK subtended, was of necessity greater than 90; and it evidently followed that the motion of the centre of the epicycle, though apparently equal in the zodiac, was really unequal in its own proper orbit. 4O. The merits of the ancient astronomers, in discovering and separating between two inequali- ties, whose returns were so irregular, and their connection so intricate, were certainly very great: and when Ptolemy formed a theory which fnight subject them to any settled rules, he was justly en- titled to the credit he obtained. But, when the ex- centricity necessary for representing the second in- E 2 equality 52 ASTRONOMICAL DISCOVERIES equality came to be afterwards investigated ; and the distance, at which it placed the moon in qua- drature from the earth, came to be compared with the distances given by her diurnal parallax, an ob- jection altogether unanswerable was presented to his theory. For, let RS (fig. 9, n. 2) be the ex- centric orbit, in the pofilion which Ptolemy as- signed to it in the quadratures, M its centre, and Tthe centre of the earth ; and let RT& ==5 1', be the angle under which the semi-diameter of the epicycle would appear from T, if its centre were in R, and ST// = 7 42', the angle under which it appears when the centre is in S. It is evident, that the distances RT and ST, are inversely as the tangents of these angles ; for, taking in TR the line Ty = TS, and drawing^ parallel to R, and meeting Tk in z, we have Ty : TR ::yz : lifc ; that is TS : TR :: tan. /Is : (R = Sh ==> tan. Another ST// ; or TS : TR :: tan. RT : tan. STA. There- more dcci- fore , tan> ST ^ + tan> RT ^ . i (jtan> STA _ tan> RT/') :: i (TR + TS) : | (TR TS) ; that is, 11MQ35 : 237117 :: 1000000 : 213164 == TM. Hence TR, the distance of the lunar apogee from the centre of the earth, will be 1 2-13164, The distance, according to Ptolemy and the other ancients, is found, by their observations of parallax, to be equivalent tp 60 serpi-diameters of the earth; and consequently, by subtracting from it a poth part, we shall have 1192945 for the distance of the apogee of the excentric from the earth's surface. The semi-diameter of the epicycle, found in like parts by the analogy R : tan. RT7; :: TR : R, will be = 306304 ; and this added to 1192945, will give 1299449 for the distance of the moon from the earth's surface, when the centre of the epicycle is io the apogee of the excentric, and the moon at the same time in the apogee of the epicycle. But .when OP KEPLER. 53 when this centre comes in the quadratures to the perigee of the excentric, and the moon happens at this time to be also in the perigee of the epicycle, her distance from the surface of the earth will be little more than the half of this quantity. For, if to double the excentricity TM, we add the semi-dia- incter of the epicycle, and subtract the sum = 532832 from lie)2Q45, the remainder, which is the moon's least distance' from the earth's surface, will not exceed 660113. The apparent diameter, therefore, and the diurnal parallaxes of the moon ought, as Copernicus justly observed, to be found, in some quadratures, nearly twice as great as in some oppositions. But by all the ancient observa- tions, the difference was found so inconsiderable as, hardly to deserve attention. 41. Ptolemy accounted more happily for the la- ' titudes of the moon, ascribing them to their proper cause, the intersection of the plane of the lunar orbit in the plane of the ecliptic ; but, as he sup- posed the inclination of these planes invariable, it is probable that many of the inequalities in latitude had escaped his notice. The inclination, he agreed Latitudes with Hipparchus in fixing at 5; and, as the lati- of fhe tudes did not return regularly with any of the other lunar revolutions, this variation of them was also rightly ascribed to the revolution of the nodes round the ecliptic in a retrograde direction. This revolution of the nodes was supposed to be per- formed in 38y. 223d. 12 h. 6'; after which pe- riod the varieties of latitude returned regularly in their order. 42. The inequalities of the planets were so vari- ous and intricate,, that the explications of them were for a long time extremely imperfect, and so E 3 partial, moon, 54 AST110NOMICL DISCOVERIES partial, that no Grecian astronomer before Ptolemy had supposed it practicable to give a compleat theory of all. In the more ancient times the ex- plications of them appear to have been made by orbits concentric to the earth, and charged with epicycles : but, as Ptolemy had found no method of representing the second inequalities, except by means of epicycles, so, to avoid the perplexity oc- J casioned by the multiplication of them, he gave the preference to an excentric orbit for the representa- tion of the_first; and, by the su peri ^simplicity of the represerifaTion, the authority of the excentric theory was for many centuries established. With respect to those first inequalities, at least of the su- perior planets, it appears to have been originally supposed, that they might be sufficiently accounted for by the more simple solar hypothesis (3l). For, if the planet, in consequence of its second inequa- lities, be represented as moving in the circumfe- rence ale (fig. 2) of an epicycle, the centre A, L, P, Q, of this epicycle, will represent the places which it would occupy, if it were divested of all second inequality :-and it was thought asufficienc explication of the first inequality, to suppose that this centre moved equally round Z, the centre of the orbit, and consequently inequably round O, the centre of the earth. But when, according to Keplers conjectures on this subject, they endea- voured to account for the inequality of the lati- tudes in opposition, and of the digressions of the planets, especially their greatest digressions, from the point opposite to the sun, by variations of their distances from O, it appeared that the point Z, round which the planet moved equably, could not be the same with the centre of the orbit ; for, both the latitudes, and also the angles aOc, dQf, which the epicycle subtended at the centre of the earth, were OF KEPLER. 55 \Vere found to be greater at the apogee, and less Theory of at the perigee than the limits of the excentric A L th PQ permitted ; and that, consequently the centre or of the orbit occupied a place nearer than Z to the centre of the earth. It was therefore a matter of much greater difficulty to form an hypothcsU for the motions of the planets, than for those of the sun : for, if their first inequalities required one determined excentricity, or distance between the centre of uniform motion and the centre of the earth, the variations of their latitudes, and of their second inequalities, showed that this was not the excentricity of the orbit in which the epicycle moved, and that this orbit evidently required an- other. In what proportion the distance ZO be- tween the centre of uniform motion and the centre of the earth, ought to be divided by the centre of the orbit, appears to have been for a long time a matter of much uncertainty. But Ptolemy tells us that, on applying himself to investigate the measure of the approach of the centre of the epicycle, within the circle ALPO at the apogee, and its conse- quent withdrawing from it beyond the perigee, he found, by multiplied observations, that the centre of the orbit lay precisely in the middle, between Z the centre of uniform motion, and O the centre of the earth. This is the famous principle, known by the name of the bisection of the excentricity : and. Bisection as Ptolemy gives no account of the means by which of thcex - j r i r i * centricity. it was discovered, nor or the observations from which it was inferred, his assuming it has justly excited the wonder^of all astronomers. The greater part believed him to have assumed it merely from con- jecture, and riot to have derived it, as Kepler more generously supposed, from any observations ; and there seems to be spine reason for thinking, E 4 that 56 ASTRONOMICAL DISCOVERIES that it came to him by tradition, from the more ancient astronomy of the east. 43. In consequence of adopting this principle of the bisection, Ptolemy's explication of the first ine- qualities of Mars, Jupiter, and Saturn, was to this effect. Let ABC (fig. 10.) be the orbit of any su- perior planet, and D its centre : let Tbe the earth at the distance DT from D the centre of the orbit, A DP' the line of apsides, and B the centre of the epicycle in the circumference of which the planet is supposed to move. This centre B, does not, like the sun, move equably round D, the centre of Deferent ABC, called the deferent of the epicycle ; but round. the centre E of another circle FGL, called the equant ; and the point E is so taken between D and the apogee A, that its distance ED from D, the centre of the deferent, may be =; TD. The first inequality therefore of tne planet was supposed to be produced by the earth's place in T, at the distance TE from the centre of the equant, call- ed the excentricity of the equant: for, on this equ y am,and account the lines TB of the true motion of the the orbit, centre of the epicycle, and EB, or its parallel TM, of the mean motion, could never coincide, except at the apsides A and P ; and every where else they would make an angle with each other, as EBT, or its equal JBTM, subtractive from the mean place of the centre in the first six signs of anomaly, in order to find the true, and additive in the six last. This angle is called the equation, Equation * m & T of the cen- f the centre, to wit, of the epicycle; ana it is ma- to, nifest that it must be much greater, than if the centre had moved equably in the circumference ABC, and its only excentricity had been DT, call- ed the excentricity qf the orbit, 44. TQ OP KEPLEH. , 57 , 44. To explain again the second inequality, or the change of a superiour planet's motion from di- rect to retrograde, and from retrograde to direct, and their stations between these two opposite changes, the ancients supposed that, while the cerh- tre B of the epicycle moved inequably round the earth in T, though equably round the point E, the planet itself performed various revolutions in the circumference of the epicycle ; not in a similar direction indeed to those of the sun and moon in their epicycles, but in a contrary direction, to wit, in consequently, or from K to H and O in the upper part of it, and in antecedents, in the lower. These motions also were supposed to be made uniformly round the centre of the epicycle ; and, as i the middle, between the two opposite directions of the motions, there are two small arches of the epicycle sensibly coincident with the tangents TO, TQ drawn to it from the earth, they supposed, that in describing these the planet would seem to be sta- tionary. Hence therefore arises another equation, the ciitference, to wit, between the place of the centre of the epicycle, and the place of the planet in its circumference, as referred to the zodiac from T the centre of the earth. For example, suppos- ing the planet in O, this equation to the angle OTB, called the equation of the argument, and of Equation the orbit : and as, when the centre of the epicycle ofthear - r r,i gument. or was supposed to move in the circumference of the rbit. cquant, the calculated amount of this angle was always less towards the apogee, and greater to* wards the perigee than the observed amount, this was one of the reasons by which Kepler conceived Ptolemy to be induced, when he placed the centie of the circle in which the epicycle actually moved nearer than the centre of the equant to the earth, The circumstance, concerning the revolutions of the 5S ASTRONOMICAL DISCOVERIES the superior planets in their epicycles, which was most surprizing to the ancients, was their close con* nection with the motion of the sun, and their being always made in the precise time of his synodical revolutions to the same planets ; so that, if the mean place of the sun was found in B, the planet in conjunction with the, sun, or in the line TB, was always in the true apogee K of its epicycle, and, producing TB to V, the planet in opposition to the mean place of the sun in V, was always in the true perigee R of its epicycle ; and in all other situations of the planet, as at Z, the line BZ of the planet's true motion in the epicycle, was always parallel to TX, the line of the sun's mean motion round the earth. The point H of the epicycle, found by drawing through its centre B a line EB, from the centre of the equant, is called the mean apogee ; and it is on the distance of the planet, as HBZ, from this mean apogee, that the amount of the equation ZTB partly depends ; and this dis- tance HBZ is part of the argument of that equa- tion. In forming this part of the argument, all that is necessary is to subtract the mean motion of the centre B of the epicycle, from the mean motion of the sun ; for we have seen, that HBZ is always equal to MTX, or EEa : and it is manifest that, the more slowly the centre B revolves in the defe- rent ABC, the sun's returns to the planet, and con- sequently the revolutions of all its second inequa- lities will be the more frequent 45. To find the excentricity TE, and the place of the apogee, which corresponded to the first in- equality of any superior planet, its observed oppo- sitions were employed, because in these it was not affected by any second inequality : and Ptolerny supposed, for a first approximation^ that the centre of OP KEPLER. 59 of the deferent coincided with the centre of the equant. In this approximation, therefore, his pro-? cedure was the same" as in deducing from lunar eclipses the same elements in the excentric theory of the moon (37). Only here, the motion of the apsides of the planets being considered as the same with that of the fixed stars in precession, there was no necessity, in a short period, for distinguishing between the anomalistical and the tropical revolu- tions. But as the deferent did not coincide with the equant, this approximation was by no means sufficient. For, let ABC (fig. 1 1), be the equant described on the centre D, and MNO the planet's apparent path in the zodiac described on E, the centre of the earth : let ED, the excentricity of To find the the equant, be bisected, as Ptolemy directed, in F, and on this point as a centre, with the semi- din meter FG =: DA, or EM, describe the defer- ent, or orbit GKL. The places of the centre of the epicycle, or of the planet itself in opposition, at the limes represented by A, B, C, will be mark- ed in this orbit by lines drawn to those points from D the centre of the equant, and cutting the orbit in G, K, L : and, if lines be drawn from G, K, L to E, the centre of the earth, and meeting the ecliptic in P, Q, R, they will represent the longi- tudes in which the three oppositions were observed, But these known arches PQ, GUI, of the ecliptic, are not subtended by the known arches AB, BC, of the equant, but the arches subtended by AB, BC, are the unknown onesMN, NO ; and, till tha differences PM, QN, OR, between thcie and PQ, QR, be found, the excentricity and the position of the line HS of apsides cannot be accurately deter- mined. The method therefore of Ptolemy, to ob- tain those arches, proceeded thus : first, in any one &{ the triangles, as BDE, which have for their common 60 ASTRONOMICAL DISCOVERIES common base the excentricity DE of the equant ; and for their sides lines drawn to given points, as B, of the same circle, from its centre D, and from E the centre of the earth, he determined, with BD the semidiameter = 100000, and with DE the excentricity, and the included angle BDE, to both and Ion i wmcn ne na< ^ approximated, the angle DBE. tude of the Secondly, in the triangle KDF, with the same angle at D, and the sides DF = iDE, and FK s= BD, he determined the side DK. Lastly, in the triangle KDE, with the same angle at D and the including sides, he determined the angle EKD. By these means he found the angle BEK = EKD EBD, and consequently the arch QN ; and in the same manner the other arches OR and PM. Having added these to, or subtracted them from, the given arches PQ,, Q.R, according as their po- sitions required, he proceeded anew, and with the arches MN, NO, instead of PQ, QR, deduced another excentricity DE, and another place of the apogee (37), that with these he might determine three other small arches, by which MN, NO, might be corrected : and he repeated the same ap- proximations, till at last the corrections of MN, NO became insensible. By investigations of this kind the excentricity of the equant of Saturn was fixed at 11388, of Jupiter at 9167, and of Mars at 2000O, in parts of which the semi-diameter of each orbit contained 1 00000. / 46. The situation of the centre of uniform mo- tion, in a point different from the centre of the orbit, rendered two operations necessary in calcu- lating the equations of the centre, that is, the an- gles TBE, crBTM; (fig. 10), and they could not, as in the solar theory, be obtained by a single operation. First, since the longitude of A, the apogee, OF KEPLER. 6l apogee, is found, and the mean motion of the cen- tre of the epicycle given, (18), the angle AEB of mean anomaly is also given : and therefore, in theTocalcu- triangle DBE, with DB the semi-diameter of the Jjj t ^ g e ; deferent, and DE = JTE the excentricity of the the centre, equant, the angle DBE may be found. Next, in the triangle DBT, with the sides DB, DT, and the included angle BDT = BED + DBE, the angle TBD may be found ; and, ;if required, the radius vector, or distance TB between the centres of the epicycle and the earth. The two angles TBD, BDE, make up the whole equation of the centre, or of the first inequality ; and the first of these was by the later disciples of Ptolemy, called the optical equation, as arising merely from the situation of the observer : the other was called the physical equation, and, according to this theory, though perhaps without the knowledge of its au- thor, was a real variation of the planet's velocity, and an exception from the supposed universal law of uniform motion. 47. In order to calculate the second inequalities, it was necessary to determine the semi-diameter of the epicycle, that is, its ratio to the semi-diameter of the orbit. This was obtained by observing the planet out of opposition, and comparing its place with the' calculated place of the centre of the epi- cycle ; that is, the place which it was found to occupy when the equation TBE was applied. If the planet was observed in Z, the difference would Semi-dia- be the angle BTZ. But the angle HBZ, the tance of the planet from the mean apogee H of epicycle is given, it being always equal to or MTX, the distance of the mean place of the sun from the mean place of the planet; (43), and adding to this, or subtracting from it, HBK ==- TBE, 62 ASTRONOMICAL DISCOVERIES TBE, the result wilt be KBZ, the planet's dis- tance from the true apogee. Therefore, with TBZ, the suppl. of KBZ, the side TB, and the angle BTZ, it was easy to determine, in the triangle BTZ, the side BZ. In this manner, supposing the semi-diameter of each orbit = 100000, the semi-diametar of the epicycle of Saturn was found to contain 10833 of such parts, that of Jupiter 19166, and of Mars 65833. But, if the Ptolemaic theory thus determined the ratios of the epicycles to their orbits, it contained no principle whereby to determine the ratios of the orbits to one another. 48. When the semi-diameter of the epicycle was To caku- determined, it was easy to calculate, for any given late the e- time, the equation of the orbit; or the derange- ?hTorbit f ment ma de by the second inequality on the planet's place in the zodiac. The argument of this equa- tion, called also the equation of the argument, is the distance KBZ, from the true apogee of the epicycle, now described; and with this, and the sides BZ, BT of the triangle BTZ, the angle, or equation, BTZ may be always found. The greatest equation of this kind will evidently take place, when a line drawn from the centre of the earth to the planet is a tangent to the epicycle. But the equations of the orbit do not continue equal, in all degrees of anomaly, even when the planet is found in the same point of the epicycle ; for, near the apogee of the excentric, its semi-diameter must appear under a less angle than towards the peri- gee ; and, in all such cases, an operation, equiva- lent to a new determination of it, was necessary. As the planet also, towards the apogee, described its epicycle in less time than towards the perigee, another deviation from the law of uniform motion evidently committed. According OP KEPLER. 63 According to this theory, the greatest equations of the centre do not take place, as in the solar or 1 - bit, at the points C, N, where a perpendicular to the line of apsides, drawn through the centre of the earth, meets the orbit ; but in two other points , , found by bisecting TD in Y, and drawing through Y a line parallel to CN. The points c, n, are those also in which the epicycle is at its mean distances from the earth ; for cT = cD. 4Q. Though the ancients employed epicycles to explain the whole second inequalities of the pla- nets, it did not follow, that either their stations or their retrogradations would be always visible ; and no such vicissitudes had ever been observed, not- withstanding their epicycles, in the motions of the sun and moon. Apollonius therefore demonstrated that these phenomena depended on a certain ratio, between the velocity of the centre of the epicycle, and the velocity of the planet in its circumference ; and, that if this were not less than the ratio of the semi-diameter of the epicycle to its distance from the earth, the planet never could appear retro- grade. His demonstration depends upon the following lemma : Let DGE (fig. 13) be a triangle whose base DE is so divided in C, that DC is not less than the adjacent side DG, the ratio of DC to CE will be greater than that of the angle DEG to the angle EDG. Let DC be equal to DG, join GC, to which draw from D the parallel DA, to meet EG produced ; and, from G to DA, the line GB parallel to DC.. Thus DC will be = GB, and GC = DB; and a circle described on G with the semirdiameter GB a will pass through D, because DC = GB = DG. The 64 ASTRONOMICAL DISCOVERIES The ratio therefore of the triangle ABG to the tri-* angle DGB, will be greater than that of the sector" FGB to the triangle DGB ; because the sector FGB is only a part of the triangle ABG. But the ratio of the sector FBG to the triangle DGB, is greater than that of the sector FBG to the sector DGB ; and therefore the ratio of the triangle AGB to the triangle DGB, is much greater than that of the sector FBG to the sector DGB. But the triangle AGB is to the triangle DGB :: AB : BD :: AG : GE :: DC : CE, and the sector FGB is to the sector DGB, as the angle FGB to the an- gle DGB. Therefore the ratio of DC to CE is much greater than that of FGB to DGB, that is of the angle DEG to the angle EDG. If DC were greater than DG, the truth of the lemma would be still more evident. The proposition then, that, if the ratio of the velocity of the centre D, to the velocity of the pla- net in G, be not less than that of DC to CE, when a (fig. 12), the planet never can appear retrograde, P lan * n \vas thus demonstrated. If. this should be supposed possible, it must be in some arch GC, near the perigee of the epicycle, where the motion is actually retrograde. Join DG, which will be equal to DC. Since the base DE, of the triangle DGE, is -so divided in C, that DC is not less than the adjacent side DG, the ratio of the angle DEG, to the angle EDG, will be less than that of DC to CE. But the ratio of the velocity of the centre D, to the;* velocity of the planet in G, was supposed not to }>e less than that of DC to CE ; and therefore ^nust be much greater than that of DEG to EDG. But the velocity of the planet in G, is as the angle CDG, or EDG ; and therefore the velocity of the centre D. must be as an angle LED, greater than GED. ' The OP KEPLER. 65 The truth of the proposition is therefore mani- fest. If D were at rest while the planet described the angle GDC, the planet would, no doubt, ap- pear to be retrograde, because the arch GC sub- tends the visual angle GED ; but, since D is not when a at rest, but has described in the same time the an- planet will gle DEL, greater than DEG in consequential , the planet must also have appeared to move /;/ conse- quentia, through an arch PL, or to have described an angle PEL = DEL DEG. It is equally manifest that, if the ratio of the ve- locity of D to the velocity of the planet in G were less than that of DC to CE, the planet would ap- pear to be retrograde in describing an arch GC near the perigee ; for then DEL would be less than DEG, and DEL DEG would be negative, that is, PEL would lie on the other side of PE. Finally, if the ratios were equal, the planet must appear for some time to be stationary ; because then DEL DEG = 0. The stations therefore of a planet will happen when any small retrograde arch, which it describes of the epicycle, and the contemporary direct arch described by the centre of the epicycle, shall subtend the same visual .an- gle ; or when the velocity of the centre shall be to the velocity of the planet, as the distance of the planet from the centre to the distance of the planet from the earth. The reason is hence also evident why the sun and moon, though supposed to move in epicycles, are never seen to be retrograde, or so much as stationary, even while describing the retrograde arches of those epicycles. For the velocity of the sun, and the centre of his epicycle are precisely equal, and of the moon and centre of her epicycle nearly so : whereas the distances are, in the former case, as '24 to f, and in the latter nearly as 10 to l, F 66 ASTRONOMICAL DISCOVERIES If the line ED be produced to meet the epicycle again in A, and other lines be also drawn from E, on both sides of EA, to meet the epicycle, as in G, B, and in H, K ; and if the intercepted parts BG, KH> be bisected in P, M, by perpendiculars from D, the ratio of PG to GE, or of MH to HE, will be less than that of DC to CE : and, as the lines diverge from DE, this ratio will continually diminish, till at last, when LE, NE, become tan- gents to the epicycle, it entirely vanishes. The planet therefore may be found in a point G, or H, of its epicycle such, that the velocity of D shall be to the velocity of G, as PG to GE, or as MH to HE : so that its first station will be in G, and its second in H. The distance GC, or HC, of either of these points from C the perigee of the epicycle, and con- sequently the length of the whole retrograde arch GH, was thus investigated. The ratio of PG to GE, being the same with that of the velocity of D to the velocity of G, is given, and therefore that of BG to GE, and of BE to GE. But the ratio of DC, in parts of the semi-diameter of the planet's orbit zz 100000, to CE is given ; and consequently that of AE to CE ; and, since AE.CE = BE.GE, the rectangle BE.GE, will be also given in parts of CE. If therefore we make BE : GE :: BE.GE : GE Z , the side GE of this square will also be found in parts of CE; and consequently BR, BG and PG. Therefore, in the right angled triangle PDG, with PG and DG, we may find the angle PDG, and the side DP ; and, in the right-angled triangle EPD, with DP and PE, we may find EDP. If, therefore, we subtract PDG from EDP, the remainder GDC will be measured by the re- quired arch GC, the distance of the planet at the first point of station from the perigee C. In the same OP KEPLER. 67 same manner HC may be found ; and the retro- grade arch GH is = GC + HC. 50. The varying latitudes of the superior planets were accounted for by supposing the planes of their orbits were, not only like that of the moon, inclined to the ecliptic; but that the planes of the epicycles were also inclined to the orbits. Let AB (fig. 14), be the plane of the ecliptic, and CD that of any superior planet's orbit, intersecting each other in a line passing through the 1 centre of the earth, or rather of the ecliptic, and called, as in the lunar orbit, the line of nodes. The incli- nation of these planes was supposed to be invari- able, and stated at 2 30' for the orbit of Saturn, at 1 3O' for Jupiter, and for Mars at 1. On this single supposition there could be no varieties of latitude, except those produced by its change from Latitudes north to south, and from south to north, according Je^fST- to the passage of the centre of the epicycle, from nets, the one side of the plane of the ecliptic to the other, and arising from O' in the nodes, to the greatest amount in the limits ; and those occasion- ed by the position of the earth in a point different from the centre of the orbit. But if the plane EF, or GH, of the epicycle, shall so cross the plane of the orbit, as to be parallel .to AB the plane of the ecliptic, the planet, when viewed from the earth, in E, or H, the apogee of the epicycle, will have a less distance from AB ; and, when viewed in the psrigee F, or G, a greater distance from AB, than if the plane of the epicycle had co-incided with the plane of the orbit : and, as the revolutions of the planets in the epicycles, had no connection with the revolutions of the epicycles in the orbits, the varieties of the observed latitudes would evi- dently be very great. But these suppositions were F 2 found 68 ASTRONOMICAL DISCOVERIES found insufficient, and there was added to them a libration of the plane of the epicycle on a diameter perpendicular to the line of apsides. The inclina- tion therefore of the epicycle was conceived to be variable; and the variation, in Saturn's orbit, in- creased to 4 30'; in that of Jupiter, to 2 30'; and in that of Mars, to 2 15'. It does not however appear, that this variation of inclination was ad- mitted by Ptolemy himself; and Kepler was of, opinion that, on the contrary, the varieties of la- titude were partly the cause, which suggested to him the bisection of the excentricity ; and that this was the principle, from which he endeavoured to deduce them, and not from any supposed oscil- lation of the epicycle. 51. Though the motions of the inferior planets differed from those of the superior in this essential circumstance, ttrat their mean revolutions in their orbits were alone connected with the solar revolu- tion, and their revolutions in their epicycles totally unconnected with it, the Ptolemaic theory, at least of Venus, did not essentially differ from that of the superior planets. Let T (fig. 15), be the cen- Theoryof tre of the earth, or of the ecliptic ABCD ; let E Venus. b e the centre of the solar orbit GHKL, and O the centre of the orbit of Venus MNP. Join TO pro- ducing it to meet the orbit in M and Q, and in OM, taking OF zz OT, the point F will be the centre of the cquant of Venus, and about which, and not about O, her mean motion, that is, the revolution of the centre of her epicycle was per- formed ; and as this was always equal to the mean motion of the sun, it followed that the lines of both were always parallel, as FM to EL, FN to ER, FQ to ES, &c. As F is thus the centre of the OF KEPLER. 69 the uniform motion of Venus, and T of her appa- rent motion, the first inequalities were represented by drawing from the centre of her epicycle, in N or P for example, lines to T and F ; and the equa- tions of this centre were the angles TNF, TPF. As Ptolemy supposed that the excentricity TF of the equant of Venus .had the same ratio to its v . . ... , i i First m semi-diameter, which the solar excentricity had to quality, the semi-diameter of the solar orbit, the equations were the same as those of the sun ; but, as he also supposed that D was the longitude of the apogee of Venus, and A that of the solar apogee, the points differed where the same equations were ap- plied ; for example, the greatest equation of Ve- nus is at the point N or P, but of the sun in H or V. But some of Ptolemy's disciples, particu- larly the framers of the Alphonsine tables, supposed that the line of the apsides of Venus coincided with the line AET of the solar apsides, and con- sequently that her orbit was in the position XYZ; and others, that her orbit, and that of the sun, were in all respects co-incident ; and that, as in the solar theory, the centres of the orbit and the equant were the same. 52. The second inequalities of the inferior pla- second i nets were likewise represented in the same manner equality, as those of the superior planets, by means of an epicycle. But, though the planet was always in conjunction with the sun in the apogee of its epi- cycle, those conjunctions, as the periods of the revolutions in the epicycle and the orbit were dif- ferent, happened in all points of the orbit indif- ferently ; and, neither had the stations any con- nection with the revolution of the centre of the epicycle, that is, of the sun, or with its position. As also the greatest digressions of Venus never F 3 exceeded 70 ASTRONOMICAL DISCOVERIES exceeded 48, and those of Mercury did not amount to 28, it was evident that the limits of their epicycles did not permit them to be in oppo- sition to the sun, like the superior planets, when they came to the perigees of their epicycles ; and that there also they would again be found in con- junction with him. But both kinds of conjunc- tion were alike invisible to the ancients; and, their observations of the inferior plants being therefore limited to their greatest digressions, they endea- voured to account for the variations of these by the excentricities of their orbits ; which made the semi-diameters of the epicycles appear, at different points of anomaly, under different angles. Yet the digressions of Mercury were found to be so extremely various as not to admit of this explica- tion ; and it was found necessary to have recourse also to variations of the excentricity, and oscilla- tions of the apogee ; and by these the theory of this planet was rendered much more perplexed and intricate than that of Venus. Theor of ^ 3 * ^ n ^ e *h eorv f Mercury the excentricity of the equant was not, as in the orbits of the superior planets, double of the excentricity of the orbit; but, on the contrary, the centre of the equant bisected the mean distance between the centres of the orbit and the earth. Let TA (fig. 16), be the mean ex- centricity of the orbit, let B be the point in which it is bisected by the centre of the equant CDEFG, and let BC, BD, BE, &c. be the line of the mean motion of the centre of the epicycle, always pa- rallel, as in the case of Venus, to TC the line of the mean motion of the sun, and performing its revolution in consequent'ia, from C to D, E, &c. On the centre A, with the semi-diam. AB zz BT, describe the small circle KLMBNO : and, to re- present OF KEPLER. 71 present the supposed variations of excentricity and oscillations of the apogee, let the centre of the orbit be conceived to revolve in a direction con- trary to that of BC 5 though with the same velocity, in the circumference of this small circle, from K, to L, M, B, &c. The orbit, in these different positions of its centre, is marked by the dotted lines of the figure ; and without a complicated scheme of this kind, it was thought that the whole varieties of the greatest digressions of Mercury, be- tween the extremes of 28 and 16, could not be justly represented. For, if the line PQ of the ap- sides should remain immoveable, as the first in- equality would disappear at these apsides, and be- come greatest at points equally distant from them in the zodiac, so, with respect to the second in- equality, the greatest digressions of the planet from the sun would take place in the perigee, be- cause there the epicycle approaching nearest to the earth, would be seen under the greatest angle; the least digressions would, on the contrary, hap- pen in the apogee; and the mean digressions about the points of the zodiac, or of the orbit equally dis- tant from them. They would not, indeed, invari- ably return, like the first inequalities, with the re- turn of the centre of the epicycle to these parti- cular points of the orbit ; because the periods of the revolutions in the epicycle and in the orbit are different ; and, where a digression from the sun had taken place, there might, in a following revo- lution, be a conjunction with him. But, if any digressions did take place, they would observe these constant rules, and the apparent magnitude of the epicycle would always be the same in the same de- gree of anomaly reckoned from the immoveable line PQ. But, as the greatest digressions observed no such constant rules, it became necessary, to F 4 suppose 72 ASTRONOMICAL DISCOVERIES suppose that this line PQ of the apsides was move- able upon the point T, and in such a manner, that when the line BC of mean motion had revolved through two signs, and had come to the situation BD, the centre of the excentric should also move through two signs in the opposite direction, in the circumference of the small circle, from K to L ; and carry along with it the whole excentric orbit, and consequently the line of apsides PTQ into a new situation RTS. By these means the centre of the epicycle was brought to a point of the line BD, much nearer both to the earth and to the perigee, than if the situation of the orbit had been immoveable: for, on this supposition, it would Jiave been found in Y, with the anomaly PKY, whereas, on the supposition of a moveable orbit, it is found in Z, with a much greater anomaly RLZ. In like manner, when the line of mean motion had pro- ceeded through two signs more to BE, and the centre of the excentric through as many, in ante- cedentia to M, the line of apsides would come into the position VTX, and the centre of the epicycle to the point a, very near the perigee X ; and conse- quently, the apparent magnitude of the semi-dia- meter of the epicycle, that is, the greatest digres- sion from the sun, would at this time be very great. Indeed, in no other point could the digres- sions appear greater ; for, as the line TP of the apsides only oscillates, and does not revolve, about the point T, it would happen that, when the line of mean motion proceeded farther to the situation BF in the perigee of the equant, the line of ap- sides would be in the position CTF, having re ? turned from TV to TP. But, as the orbit now coincided with the equant CDEF, the centre of the epicycle would be found in F, the common perigee of both, and at a greater distance from the. OP KEPLER. 73 the earth than when it was in a ; for TF is evi- dently greater than Ta. The appearances in the other half of the revolution are perfectly similar. Two points a and I were thus assigned, in every revolution of the centre of the epicycle, where the greatest digressions of Mercury came to their maximum, and each of them at the distance of 60 from the perigee of the equant ; whereas, in the immoveable orbit they could not have come to their maximum, except at the perigee. By this theory also, the first inequalities were represented as more variable than in an immoveable orbit, and the path, in which the planet actually moved, as an oval, and not a circle. (B). 54. It was impossible to determine the excentri- cities of the inferior planets, and the longitudes of their apogees, in the same manner as those of the superior planets, because in all their conjunctions they were in visible; and as their greatest digressions were extremely variable, and of necessity, especi- ally in the case of Mercury, observed too near the horizon, not to be greatly affecled by derange- ments from refraction, the conclusions from them with respect to these elements, were neither made so early, nor with such certainty. The greatest digression of an inferior planet is the difference between the place of the planet, in the point of its greatest elongation from the sun, and the mean place of the sun determined by cal- culation ; for the lines of the mean motion of the sun and of the planet, being always either parallel or coincident, (50) terminate in the same points of the zodiac. This difference is the angle LTM, or FTO, (fig. 18), or the arch of the zodiac which measures it : and, as it is the angle which is sub- tended by the semi-diameter of the epicycle, it follows. 74 ASTRONOMICAL DISCOVERIES Longitude follows, that the greater the distance of the epi- of theapo- C y C l e from the eye in T, the less will be the angle inferior*" which it subtends. If then, among a number of planet. such observed angles, two equal ones, as LTM, FTO, shall be found in opposite directions from the sun; that is, an evening digression LTM, equal to the morning digression FTO ; it is evi- dent, that the line of apsides ATH will, according to the ancient principles, bisect the angle MTO, formed at the centre of the earth by the lines of the true motion of the planet, or the angle DTB, formed by the lines of the mean motion of the sun. If also two other greatest digressions KTP, GTQ, of contrary kinds, shall be found, differing in magnitude from the former, but equal between themselves, it will be known whether the apsis A, whose longitude was determined, were the apogee or the perigee ; for, if the two last digressions were the greatest, it must have been the apogee. Nor is it an objection to this procedure, that Pto- lemy introduced into his theory oscillations of the line of apsides, for what is required is, its longi- tude in its mean position. 55. When the longitude of the apogee was de- - termined, the excentricity CT of the orbit was found in this manner. Let the greatest digressions ETV, and HTY, be observed, when the first, be- ing in the apogee, is a minimum, and the last, at the perigee, a maximum. Or, though the position of the apsides has not been determined, if these greatest digressions were certainly of the least and greatest kinds, the mean place of the sun was of necessity, at the time of the former, in the apogee, and, of the latter, in the perigee: and,, in the theory of Venus, these two points are in the same straight line ; and, in that of Mercury, supposed in OF KEPLER. 75 in it. Now, since the epicycles, though appearing in the heavens, and represented in the figure, as unequal, were notwithstanding, according to the theory, really equal ; their distances ET and HT will be reciprocally proportional to the tangents of the angles ETS, HTX; (4O) 'that is, TEL: TE :: tan. ETS : tan. HTX. These distances therefore will be given, in parts of the semi-diameter of the orbit ~ 10000O ; and consequently, in the same parts, the exccntricity TC =r { (ET - HT). The Excemri- semi-diameter EV, or HY, also of the epicycle k given in the same parts; for R : sin. HTY :: TH : HY. By these means Ptolemy determined that the mean excentricity of the orbit of Mercury was 9498, and the semi-diameter of his epicycle 35738; Semidiam and the excentricity of the orbit of Venus 233p, f c e ! and the serni-diameter of her epicyle 72094 ; in parts of which the semi-diameter of each orbit contained IOOOOO. 56. But though these were the excentricities of the orbits, and their centres, as C, the points from which the epicycles, in the apsides, would appear of the same magnitude, it remained to enquire, whether an epicycle would continue to appear at C of the same magnitude in all the other points of the orbit; that is, if equants were not also neces- sary to the inferior planets, and at what distances from the centre of the earth the centres of these equants should be placed : and we find, that though Ptolemy gives no account of his procedure in this enquiry in the case of the superior planets, he thus describes it with respect to Mercury and Venus. Let AB, (fig. 19), be the line of the ap- sides of Venus, T the centre of the earth, E the centre of the equant, and let AED, the angular distance of the sun, or centre of the epicycle, from - 76 ASTRONOMICAL DISCOVERIES from A the apogee of Venus, be, for example, == Q0. In this situation of the centre of the epi- cycle, let DTG be an observed greatest evening digression of the planet, and at some other time, when the sun occupies the same point of the equant, let DTF be an observed greatest morn- ing digression. These digressions, though unequal when viewed from T, will, notwithstanding, be equal when viewed from E, and a mean of them will be the angle DTF. Therefore, the other acute angle FDT, in the right-angled triangle DTF, is given ; and, since the side DF, the se- mi-diameter of the epicycle, is given in parts of the semi-diameter ED of the equant r: 100000, (54) the secant TD of the angle FDT to the radius DF will be given in the same parts. Therefore, in the right-angled triangle DET, with TD and ED, Excentri- the required excentricity TE will readily be found. city of the In this manner Ptolemy discovered that the excen- tricity of the equant of Venus, as in the case of the superior planets, was double the excentricity of the orbit ; whereas that of the equant of Mer- cury was only one-half of the excentricity of the orbit. 57. When the excentricities of the orbit and equant, and the longitude of the apogee were de- termined, the place of an inferiour planet, at any given time in the zodiac, was to be found, in the same manner as of the superior planets, by calcu^ Jating the equations of the centre andot the orbit. (45. 47). But as the excentricity of Venus was inconsiderable, and her orbit and equant seemed almost to coincide, and the centre of the equant of Mercury bisected the excentricity of the orbit, Equations instead of the double operation for the equation oi the cen- c .\_ . 11,11 tte. or the centre, it was calculated by one operation, as OP KEPLER. 77 as in the solar orbit. (33), Yet, when great pre- cision was studied, it was previously necessary, in the orbit of Mercury, and even of Venus, to in- vestigate the variations of the excentricity. Thus, let L (fig. 16) be the place of the centre of the de- ferent, at the distance, suppose of ()0* from K, and join AL. Then, in the triangle TAL, the varied excentricity TL was investigated from TA and AL, and the angle TAL given. Or, accord- ing to a more approved hypothesis, which consi- dered the line of apsides as immoveable, and the motion of the centre of the deferent as only ima- ginary in the small circle KLM, while its real mo- tion was a libration in the diameter KB, the varied excentricity was found by drawing to KB from L the perpendicular L?//; and was rz TA -f- Am, and, universally, zz TA =t cos. KAL. The equation also of the orbit was calculated in the same manner as for the superior planets, (47), excepting that the argument of this equation was not the difference between the mean motion of the sun and the mean motion of the planet in its orbit, but between the former and the mean mo- tion of the planet in its epicycle ; and that it was previously necessary to investigate the variations of the apparent magnitude of the semi-diameter of the epicycle. For this last purpose the procedure was as follows. Let AB, (fig. 20), be the semi-diameter of the epicycle of Venus at its mean distance from the earth, and zr 718QO, in parts of the semi-diameter of the orbit, and let BC = 2500 be the difference between its greatest and least apparent semi-dia- meters, of which the first takes place in the peri- gee, and the second in the apogee. To find then the semi-diameter in any other point of the orbit, describe on B, with the distance BC, the circle CEP; in *vhich let CE be equal to the given ano- malv. 7 ASTRONOMICAL DISCOVERIES maly. The required variation of the semi-diameter AB will be BF, the cosine of CE; subtractive from AB near the apogee, that is, in the first and fourth quadrants of anomaly, and additive to AB in the second and third. , . t . 58. The various latitudes of the inferior planets Latitudes . . , . ,, . . -n^ ,/ N of the in- were explained in this manner. Let ABC (fig. 21) feriorpla- b e the plane of an orbit, BD the line of apsides, B the apogee, and AC the intersection of this plane with the ecliptic. It was supposed, first, that the plane of the orbit oscillated on the line AC; coin- ciding with the ecliptic, when the centre of the epicycle was in A or C ; receding from it, on the departure of this centre from A, till it came to B the apogee ; and again approaching to the ecliptic, in the progress of the epicycle from B to C : and this oscillatory motion of the plane of the orbit Deviation, was called the deviation. Secondly, it was sup- inclination posed, that the plane of the epicycle was inclined to tne pl ane of the orbit, intersecting it in the line EF; and that it oscillated on this line accord- ing to a law the contrary of the deviation; the in- clination disappearing at the apsides, and coming in the nodes to its greatest amount. Finally, it was supposed that another oscillation of the epi- cycle, called the reflexion, took place upon aline GH, perpendicular to EF, and observed the law of the deviation. The nodes were placed at the distance of QO from the apsides, and both were supposed t& move according to the apparent motion given by the precession of the equinoxes to the fixed stars. Though the latitudes of the moon and the planets required that reductions should be made from their peculiar orbits to the ecliptic, Pto- lemy docs not appear to have perceived their ne- cessity ; nor, till the later times, wer# they it- tended to by his followers. OP KEPLER. 79 50. Not with standing the labour employed, and the ingenuity displayed in the formation of this system, its imperfections were so great and nu- merous, that it was impossible it should always continue to maintain its credit. Though the ori- ginal introduction of excentric orbits by Hippar- chus banished some of the ancient epicycles, and promised greater simplicity than had been attained in the ancient concentric theory, the more exten- sive application, which Ptolemy and his followers made of the excentric theory, rendered it at last almost as perplexed and intricate as the theory which it had displaced. Its suppositions were often rmperfec _ dissimilar, and even inconsistent with one another; tions of the and instead of answering to the title of a system, it was an assemblage of parts connected by no gene- ral principle of union, and between which there subsisted no known mutual relations : for, though the ratio of an epicycle to its peculiar orbit might be determined, those of the orbits to one another were entirely arbitrary. In many respects it was destitute of the authority even of sense, to which it principally appealed; and, notwithstanding the pre- tensions of reducing the celestial phenomena to the rules of geometrical calculation, it was not un- common to find differences of hours and days, and even of months, between the calculation and the fact ; nay, it frequently happened, that the predic- tions, made according to its rules, entirely failed. But the objection, which seems to have struck at the credit of the Ptolemaic theory more than all its inaccuracy in representing the phenomena, was its contradiction to the supposed inviolable law of circular and uniform motion, which it was the principal object of all systems to establish and con- firm. Not only did its oscillations and librations produce perpetual deviations, both from the plane and 80 ASTRONOMICAL DISCOVERIES and the circumference of the circle ; but also the uniformity aimed at by the cquant itself was purely imaginary ; for it took place in an orbit where the celestial body was hardly ever found ; and, by the introduction of it, a real inequality of velocity was acknowledged in the orbit, which the body actually described. The position also of the centre of the equant was regulated by no general law : for, in the theories of Venus and the superior planets, its distance from the centre of the earth was bisected by the centre of the orbit ; in the theory of Mer- cury, on the contrary, the mean excentricity of the orbit was bisected by it ; in the lunar theory, it continually varied its position ; and, in the solar theory, it coincided with the centre of the orbit. Finally, by the complicated motions ascribed to some of the circles, especially in the theories of the moon and Mercury, those bodies were brought, m some parts of their orbits, so near to the earth, that their position was wholly inconsistent with all ob- servations of diurnal parallax. CHAP. KEPLER. 81 CHAP. III. Of tbe Copernlcan System. &). fT'HOUGH the imperfections of the Pto- JL lemaic system were not immediately per- ceived, especially during the confusion w"bich at- tended the decline and destruction of the Roman empire, their effects did not fail, in process of time, to become fully evident. In the ninth cen- tury, on the revival of science in the east, under the encouragement of the caliphs, surnamed Abassides, Ptolemy's astronomical tables were found to deviate so widely from the actual situa- tions of the celestial bodies, as to be no longer useful in calculations t and it became necessary for the Saracen "astronomers at Bagdat to form tables entirely new. The Saracens carried their astrono- mical knowledge with them into Spain ; and, in the thirteenth century again, the new tables were found unfit to represent the celestial motions ; and, to supply their place, the tables, called Al- phonsine, were constructed, by the direction of Alphonso the 10th, king of Castile. The errors even of the Alphonsine tables became, in the fif* teenth century, equally sensible with the former. But though it was hence evident, that the revolu- tions of the celestial bodies were not precisely what the excentric circles and epicycles of the ancients represented, such was the veneration of astrono- mers for Hipparchus and Ptolemy, from whom they derived all that was valuable in their science ; and, indeed, so great was the merit of subjecting motions, so intricate as the apparent celestial re- G volutions, cus 82 ASTRONOMICAL DISCOVERIES volutions, to any settled rules ; that no suspicions concerning the principles of the theory appear to have been entertained. All that, was proposed even by eminent astronomers, such as Purbach and Muller, surnamed Regiomontanus, was only to correct the theory by more accurate observa- tions : and none of the amendments, which they introduced, was inconsistent with its principles. It Coperm. wag ^g ce i e b ra t ec l Copernicus, a native of Thorn in Polish Prussia, who first called in question the principles themselves, and to whom the exclu- sive honour belongs, of substituting for the Ptole- maic a new and more beautiful system, represent- ing the celestial motions with much more simpli- city, and, after its principles were fully understood, with incomparably greater accuracy. 6l. It does not appear that Copernicus origi- nally meditated such a total revolt from the autho* rity of Ptolemy, as that to which, in the course of forming his theory, he was eventually led. Though equally sensible with others of the deviation of the Ptolemaic tables from the actual state of the hea- vens, the chief cause of his dissatisfaction with Ptolemy's theory related to his explications of the first planetary inequalities, and was his departure in these explications from the principle of uniform motion in perfect circles, which alj astronomers considered as sacred and inviolable. When Hip- parchus introduced an excentric orbit into the so- lar theory, no trespass against this principle vva3 committed ; because the sun was supposed to move uniformly round its centre : but when Ptolemy extended the application of excentrics to the pla- netary orbits, and supposed every epicycle to move uniformly round a point, not in the centre of the deferent, the principle was undeniably abandoned: for OP KEPLER. 83 for the uniformity attained was in a foreign orbit, called an equant ; and there was a real and evi- dent inequality in its own* It was this part of the theory of Ptolemy which, Copernicus tells us, he chiefly disapproved ; and, as he. found the explica- tion of the first inequality, by means of amexcen- tric with an equant, irrecohcil cable with his fa- t i i i i r " 1 l " vounte principle, he seems to have had at first no na i design, higher purpose in view, than to substitute an ex- plication of a different kind, and more consonant to that principle. 62. The theory which he proposed to substitute, for explaining the first inequalities in a manner more consistent with the principle of uniformity, was the ancient concentric one : and it was in re- storing this from its neglected state, that the dis- tinguishing and essential part of his system, in which he ventured to depart, not only from the authority of Ptolemy, but from all the established opinions of mankind, seems first to have presented itself to his thoughts. When both inequalities were represented by means of concentric circles with epicycles, the necessary multiplicity of epi- cycles confounded the imagination ; and the chief recommendation of the excentric theory, and even the origina^ caUvSe of framing it, was the banishing .several of the epicycles by which the imagination had been perplexed. The same desire therefore of simplicity, which led Ptolemy to substitute an ex- centric orbit, in the explication of the first inequa- lity, for the concentric with its epicycle, seems to have had equal influence, in suggesting to Coper- Probable nicus a like substitution for the epicycles used in motlves the explication of the second. He discovered in the annual orbit of the sun, or earth, an universal epicycle, which explained the second inequalities G 2 more 84 ASTRONOMICAL DISCOVERIES more advantageously than all the various separate ones with which the planetary orbits had been en- cumbered : and in fact the solar orbit had been already employed in this explication, at least in some degree ; for while the ancients formed the argument of the equation of the second inequa- lity, by taking the difference between the places of the sun and the planet (47), they were actually converting the sun's orbit into an epicycle. These seeili to have been the motives by which Coperni- cus was led to conceive the bold design of attri- buting motion to the earth ; and, by the applica- tion which he made of her annual orbit, he found the simplicity which he sought ; not like Ptolemy, at the exp'ence of the sacred principle of unifor- mity, but in some sense perfectly consistent with it. This attachment indeed to the doctrines of uniform circular motion, which made him reject the excentric of Ptolemy, was merely a prejudice connected with the imperfect state of physical knowledge ; for the motions of the planets are in reality neither circular, nor uniform : but, in the present instance, it produced the happiest and most important effects, and proved the introduc- tion of all that is true and valuable in astronomy. 63. When the design was thus conceived of ascribing motion to the earth, and displacing her from the centre of the planetary system, Coperni- cus found that, bold as it was, it was not destitute of support from many powerful arguments, and even from several striking astronomical pheno- mena. In particular, he found that an acknow- ledgment of its propriety was made, however un- designedly, in the whole theory of the epicycles, which, by the annual orbit of the earth, he pro- posed to abolish. The ratio of the semi<-diameter of OP KEPLER* 85 of the epicycle of Mars, to the semi-diameter of Confirms- his orbit, exceeded that of 6 to 10 ; and, in the theory of Venus it was still greater ; for it exceed- ed that of 7 to 10; and therefore, when the dis- tance of the former planet from the earth varied from 4 to l6, and of the latter from 3 to 17 , it was undeniably absurd to consider the earth as the centre of their motions : and the absurdity was the same, though not so evident, of supposing the earth to be the centre of the motions of the other planets. Copernicus found also that the Ptolemaic arrangement of the inferior planets had not always been generally received ; for Plato, and his fol- lowers, placed them beyond the sun ; and he* saw that the reasons for inverting this order, and in- cluding their orbits between those of the sun and moon, were unsatisfactory and inconclusive. The distance of the sun from the earth, Ptolemy reck- oned at 1 1 60 of the earth's semi-diameters, and that of the moon in apogee, at 64. The interja- cent space between these two orbits he filled up, first, with the epicycle of Mercury of 177 in dia- meter, and next with that of Venus pf Q10 ; thus bringing Mercury's epicycle almost in contact with the lunar orbit, and the epicycle of Venus al- most in contact wij;h the sojar orbit : and the prin- pipal reason which he assigns for crowding them in these positions, is the improbability of supposing a space so vast to have been left wholly empty ; for- getting that, by this very arrangement, he had allotted, to the single epicycle of Venus, a space more than four times, or according to a more pro-- bable deduction, more than six times as extensive in breadth, as the space allotted to the earth, the moon, and Mercury, alt together. The other rea- son alleged for this arrangement, viz, the propriety of the sun's holding the middle station, between G 3 the 86 ASTRONOMICAL DISCOVERIES the planets whose digressions permitted them to come into opposition with him, and those whose digressions were more limited, was both false and frivolous : for the moon's digressions were urili^ mited, though her orbit was arranged on the same side of the sun with those of Venus and Mercury, No rea'son also appeared from this arrangement, why the digressions of Venus and Mercury should be limited, why their revolutions should be so in^ timately connected with those of the sun, or why any one of all the planets, the superior ones not excepted, should be placed nearer to the earth than any other. On these accounts Copernicus could not fail to consider the theory of the inferior planets, attributed to the ancient Egyptians, and held by several Latin astronomers, and particujarly by Martian us Capella in the fifth century, as much more worthy of attention than the Ptolemaic, and as giving a much more consistent explication of Egyptian their phenomena. In this, the sun was the centre about which Venus and Mercury performed their revolutions: and, as the earth was not included within their orbits, it was impossible that they should be seen from the earth, to make any greater digressions from the sun than the limits of these orbits would allow : and the reason was also ma- nifest, both of the relative positions assigned them with respect to the sun, and of the intimate con- nection of their apparent annual circuit round the earth with the apparent solar revolution. This theory Copernicus applied to the superior planets, and found the application attended with like suc- cess ; for, though their oppositions shewed that the earth was included within their orbits, their near approaches to the earth in their oppositions, and the vast distances to which they removed in their conjunctions, made it impossible that the earth OF KEPLER. 87 earth could be the centre of their motions. This Extension variation of distance was especially remarkable in of il the planet Mars ; who, in oppositions, appears equal in size to Jupiter, but towards his conjunc- tions no larger than a star of the second or third magnitude ; and afforded an unquestionable proof that none of the superior orbits approaches so near the earth. It was from this extension of the Egyp- tian theory concerning the inferior planets to the superior, and making the sun the centre of all the planetary orbits, that the transition seems to have Transition been more immediately made to the doctrine of h m ^ the motion, or revolution, of the earth, like any system, other planet, round the sun. For, as the varia- tions of distance shewed the earth to be nearest to the orbits of Mars and Venus, and as she was evidently within the former, and without the lat- ter, no reason appeared why she should not par- take of the revolutions round the sun, in which so many other bodies, and some of them thought to be of greater magnitude, on both sides of her, were supposed to be involved : nay, on the contrary, strong probability appeared that she did partake in those revolutions ; for, on this principle, all the varieties of the distances pf >the planets, .and all the circumstances of their second inequalities were at once and easily explained : and they could not be explained otherwise, without the improbable supposition of the annual revolution of the whole planetary orbits, round her centre. The only ce- lestial body, which could not be subjected to the general law of describing an orbit round the sun as a centre, was the moon : for, though her oppo- sitions, like those of Mars, proved that the earth was included within her orbit, her parallaxes shewed her distances to be much less than the least dis- tance of Venus ; and the variations of her distance G 4 were 89 ASTRONOMICAL DISCOVERIES were so inconsiderable as not to require, or even to admit, of any other centre of her motions than the earth. In the interjacent space, therefore, be- tween the orbits of Mars and Venus, and where the former system made the sun to move, Coper- nicus placed the orbit of the moon with the earth in its centre ; and supposed both together, like some great planet, to revolve round the sun, in the precise time of an apparent solar revolution. The sun was now considered as the only immoveable body in the system : for Copernicus was not in- fluenced by the objection of his apparent progress through the zodiac; being convinced, from innu- merable examples, how unavoidably we ascribe to surrounding objects all the real motions of which we are not sensible : and, by the immoveable po. sition of the sun in the centre, and the continual revolutions of the earth arid planets round it, at different distances, and in different times, not only were the second inequalities explained in general, without the embarrassment of epicycles, but the causes also of the different times of the revolutions in these imaginary epicycles, and of their different magnitudes, became fully evident. In particular, the cause became evident, why both the direct and retrograde arches of Jupiter were greater, and re* quired longer time than those of Saturn, and less than those of Mars ; and the like arches of Venus greater than those of Mercury : and why those vicissitudes returned more frequently in Saturn than in Jupiter and Mars ; and more frequently in Mercury than in Mars and Venus. It is true that no reciprocations of this kind had been observed in any of the fixed stars : but this Copernicus boldly, though justly ascribed to their immense distance, in comparison with which the diameter of the whole terrestrial orbit, though bearing a sensible ratio to the OF KEPLER. 8 net, GH a portion of the zodiac, E the planet supposed at rest ; and let the earth be supposed to move, according to the order of the signs, in that part of her orbit which is between the planet and the sun. Here it is equally obvious, as in the former cafe, that during the progress of the earth from A to B, where the observer finds the planet in opposition, and from thence to C ; that is, while the earth describes the arch ABC comprehended between two tangents drawn to her orbit from E, the planet must appear retrograde. For, when the earth is in A, the planet's place in the zodiac will be G, marked by the line AEG, drawn from the centre of the earth through E ; when the earth and of the advances to I, the place of the planet in the zodiac superior. w m b e j . anc j j t W JH a pp ea r in M, when the earth proceeds to L ; and in H, when the earth arrives at C. But in that part of the earth's orbit which is most remote from the planet, or in the whole progress of the earth from C to N, and thence to the conjunction in P, and from P through O, till her return to A, the motions of the planet will appear direct : for NM is more advanced in the zodiac than CH, and OK than NM, and AG more advanced than all the three. It is true here also, that, during this progress of the earth through all the signs, the planet has not continued \m* moveable, but has advanced through some portion of its own orbit: but, as its progress is much slower than that of tfye earth, the only difference pro- duced upon the appearances will relate, as before, to the extent of the retrograde and direct arches, the duration of their description, and the points of longitude in the zodiac by which they are ter- minated. The reason is also equally evident, why a superior planet, in describing its retrograde arch, will OP KEPLER. Q5 will seem to move with the difference only, and, in describing its direct arch, with the sum, of both velocities. 69. Between the direct and retrograde motions of the planet, there must be a time when the one changes into the other; and when, of consequence, the planet must appear stationary, and seem to partake of no motion except the diurnal revolution. If the earth continued at rest in the point T of her orbit, (fig. 24), the planet Venus, for example, would always appear stationary when she was found in any of the two tangents, as TK, which might be drawn from T to the planetary orbit : only the change of her motion from direct to retrograde would not be instantaneous, because there is a small portion of the orbit, towards K, which is to sense confounded with the tangent. But as the earth docs not continue fixed, but moves in her orbit from T to D, Venus, though found in the tangent TK, will not immediately appear stationary, but seem to continue the course she held before she arrived at K, because the tangent itself alters its direction. In a short time, however, the visual stations of ray MN, drawn from the earth to Venus, will the P la - cease to be a tangent to the orbit, and will meet it in another point O : yet, as long as the contem- porary arches, TM and KN, described by the earth and the planet, shall be so situated with respect to each other, as to render the lines MN and TK^ which include them, parallel, the planet will appear stationary, and be referred to the same point in the starry heavens. For the distance between all pa- rallel straight lines, which can be drawn from the earth, in any of all her different positions, bears no sensible ratio to the immense distance of the celes- tial sphere. It is obvious that the duration of the stations nets, $ ASTRONOMICAL DISCOVERIES stations will be longer, than if the earth had coil* tinued immoveable. (C.) 70. It was in this simple and convincing man- ner that Copernicus accounted for the second in- equalities of the planets, by substituting the orbit of the earth for the three epicycles of the superior planets, and the two deferents of the inferior. By this alone the causes of all the circumstances and varieties of these inequalities were, in the most sa- tisfactory manner, exhibited to view : whereas the ancient system shewed no cause, either of the phe- nomenon itself, or of the magnitude of the epi- cycle by which it was represented, or of the ratio between the number of the revolutions in the epi- cycle and the number of the revolutions in the real or supposed orbit. of In determining the times and points in which " t ^ ie P^ anets became stationary, Copernicus, not- withstanding his new explication of their second inequalities, continued to employ the ancient me* thod suggested by Apollonius (4$), in which the orbits are considered as circular and concentrical to the sun : and it is evident that this method was alike applicable to his system as to the Ptolemaic. Indeed, the orbits are still considered as circular and concentrical to the sun in the greatest part of the modern solutions of this problem ; and solu- tions, on such conditions, are supposed to be suf- ficiently exact for any useful purpose to which they can be applied. (D.) 71. An equal advantage of the system of Coper- nicus was, that it determined the mutual ratios of the distances of the planets, from the common centre of their motions ; and consequently, it was no longer an aggregate of parts disproportioned and OP KEPLER* 97 arbitrarily assumed. When a celestial body is referred to one point of the starry sphere from the centre of the earth, and to another from any station on its surface ; that is, when it is subject to a diurnal parallax ; it was known that this pa- rallax was an indication of its distance from the centre of the earth being less than the semi-dia- meter of that sphere ; and presented the means of . comparing the distance with the semi-diameter of the earth. But the comparison was in general in- accurate, because the ratio of the semi-diameter of the earth to the distances of all the celestial boi dies, except the moon, was so inconsiderable, that it was incapable of being ascertained by the ancient observations. But, by the annual motion of the earth, two stations were found whose distance was not less than the semi-diameter, or even the dia- meter of the whole terrestrial orbit ; and there was no planet in the system whose place in the starry sphere did not appear to be changed, by viewing it from stations so remote. Let (fig. 27), be Annual the sun, or rather the centre of the terrestrial orbit P aiallax > BAT ; let T, M, J, S, be the Earth, Mars, Jupi- ter, and Saturn, in their several orbits ; and let CDEF be a portion of the starry sphere. The dif- ferences of the places of these planets, called their annual parallaxes, as seen from T the centre of the earth, and from the centre of the sun, are the arches CF, CE, CD, of the zodiac ; or, since the parallels TC, 0G, when produced, mark the same point of the zodiac, the arches GF, GE, GD, which measure the angles G0F, G0E, G0D, or their equals 0MT, JT, 0ST, under which the semi-diameter of the terrestrial orbit is viewed from every planet. It is indeed supposed, that one of the stations is in the centre of the orbit ; but, from the theory of the first inequalities formed on op* H positions, Q8 ASTRONOMICAL DISCOVERIES positions, the place of the planet, as viewed from the centre of the orbit, may at any time be calcu* lated : nor* Whether this centre be occupied by the earth, or by the sun, will there be any difference on the result of the calculation* The difference between the place of the planet, observed from the earth, and its place thus calculated for the time of the observation, is the required parallax : and, with the difference between either of these and the place of the sun, the ratio between the semi-dia* meters of the planetary and terrestrial orbits was given. 72. Copernicus, for example, determined ii\ this manner the ratio between the distances of Sa-r turn and the earth, from the sun. He observed the longitude of Saturn, on the 24th of February, 1514, at 17 h. in the line T^C, and found it =s 20Q, Then, either by his own theory of the first inequalities, or by that of Ptolemy (46), (for, ex- cept as to the supposed positions of the earth and sun, these theories chiefly differed in form,) he calculated for the same instant the longitude as seen from the sun, in the line 0SD, or which would have been seen from the earth, in the point L of her orbit, and found it to be 203 1(5'; that is, the heliocentric longitude was 5 44' less than the geocentric. The two visual rays, therefore, or lines TC and 0D, proceeding from the earth and sun, intersected each other at the centre of Saturn, in an angle CSD, or TS0, of 5 44': and this an- s gle, which in the Ptolemaic system was called the equation of the orbit, or of the argument, is the Angles of parallax. Having also the place of the sun in K, comumta- * vvn ^ cn tne place of the earth, as seen from K or tion,ande- 0, is always opposite, the difference between this anc j ^ helioqcntric longitude of Saturn, gave him the OF KEPLER. 9Q the angle S0T, called the angle of commutation; and which, in the present case, was = 67 35'. Consequently, he had also, in the triangle S0T, the elongation, or angle ST0, zr 106 43'; and therefore the ratio of the distance T0, of the earth from the sun, to the distance S0, of Saturn from the sun, was given : for sin. TS0 : sin. ST0 :: 0T : 0S; and this ratio, supposing 0T = 1, is that of 1 to 9.6. The procedure with respect to every other planet was the same: for the parallaxes 0JT> <3>MT of Jupiter and Mars, as well as those of the infe- rior planets, are in like manner the differences be- tween the observed, or geocentric, longitudes TJ, TM, and the calculated, or heliocentric, longi- tudes 0J, M, obtained from the theory of the first inequalities ; and the commutations J0T, M0T, the differences between the heliocentric longitudes of the planet and the earth, as well as the elongations JT0, MT0, the differences be- tween the geocentric longitudes of the planet and the sun, are given. On these principles, the dis- To find the tances of the planets in their order from the sun [^.bil, were found to be, to one another, as the numbers to one an- 4, 7, 1O, 15, 52, 95 ; and though the investigation other - was not made with all the advantages afforded by the Copernican system, because the true form of the orbits was not yet discovered, and no consi- deration was had of their situation in different planes, it became the foundation of one of Kep- ler's most celebrated laws, that the squares of the periodical times of the planets are to one another as the cubes of their distances from the sun. The procedure indeed is not materially different from the procedure of Ptolemy, in finding the ratio be- tween the semi-diameter of any single orbit and the semi-diameter of its epicycle. For the paral- H2 lax 100 ASTRONOMICAL DISCOVERIES lax 0ST is the same, and found in the same mat)* ner, with the equation of the orbit S0H; and the commutation S0T, which is its argument, is the same with 0SH, the supplement of DSH, the pla- net's mean motion from the true apogee D of the epicycle ; which was the argument of the equation of the orbit (48), and, to the astonishment of the ancients, found always equal to D0K, the sun's mean motion from the same apogee. But from Ptolemy's procedure, nothing more could be found than the ratio between a single orbit and its par- ticular epicycle : whereas, in the Copernican sys- tem, the orbit of the earth being made the gene* ral epicycle, the mutual ratios of the orbits easily followed, and the reasons became manifest of the order in which they were arranged. It is obvious that all the circumstances of the second inequali- ties depend upon the annual parallaxes. The less the parallax, when the angles of commutation are equal, the less will always be the extent of the di* rect and retrograde arches of the planet, and the greater their number, during the course of every revolution in their orbits : and, as the distance of every planet from the sun is variable, upon account of the excentricity of their orbits, the magnitude of these arches will not be constant, nor the times pf describing them the same. 73. Another no less signal advantage of the Copernican system was, that it led to a simple and satisfactory explication of the latitudes of the pla- nets, and pointed out the reason of their being found very different, even in the same distances from the nodes. Ptolemy, to explain the latitudes of the superior planets, supposed a double inclina^ tion, first of the planes of the orbits to the plane of the ecliptic ; and, secondly, of the epicycles to the 01? KEfcLEfc. 101 the orbits ; and, for the inferior planets, besides the same double inclination, no less than three os- cillatory motions. But, in the system of Coper- nicus, there was no necessity for such improbable suppositions. For, let ABCD (fig. 28), be the or- bit of the earth, in the plane of the ecliptic EFG, and let KFL be the orbit of the planet Mars, for example, inclined to the ecliptic in the angle KFE, and intersecting it in the line of nodes FH, pass- ing through the centre S of ABCD ; and, through the same centre, draw ESG perpendicular to FH> in the plane' of the ecliptic ; and also KSL likewise perpendicular to FH in the plane of the orbit. A perpendicular KE drawn from the point K of the orbit to E, or from L to SG, will mark its greatest elevation above, or depression below, the plane of the ecliptic : and the angle under which this perpendicular is seen from S will be the pla- net's heliocentric latitude, (because S is also sup- posed to be the centre of the sun,) and will be tncand equal to the angle KFE of the inclination : and the only variations to which this heliocentric lati- tude is subject, arise from its gradual increase, as the planet moves from the ascending node H, where it is = O, to the limit K, where it comes to its greatest amount; and from its gradual decrease, in the descent of the planet to the opposite node F. But the angle KTE is the geocentric latitude, and this, according to the Copernican system, is vari- able, not Only on account of the variations of the heliocentric latitudes, between the limits and the nodes; but much more on account of the varia- tions of distance between the planet and earth, produced by the revolution of the latter in her or- bit. Even at the same point of the orbit N, the same heliocentric latitude MN is seen from T, under the angle MTN, and from A under a dif- H 3 ferent 102 ASTRONOMICAL DISCOVERIES ferent angle MAN ; and these angles would be still more various at B, C, V, &c. 74. To be enabled at any time to calculate the latitudes, whether heliocentric or geocentric, it was necessary to determine the inclinations of the planetary orbits to the ecliptic : and Copernicus, for this purpose, employed the following method. He determined, by some theory of the first in- equalities, the excentricity of the planet's orbit, and the place of its aphelion ; whence, as he sup- posed every orbit to be circular, the distance of the planet from the sun, in the centre of the ecliptic, could be easily calculated for any given point of anomaly ; and having also the ratio between this distance and the semi-diameter of the terrestrial orbit (72), together with the observed latitudes ; all the conditions necessary for the solution were given. But his success in this part of his under- taking did not correspond to the justness of his principles; for his observations of latitude were chiefly taken from Ptolemy, and his deference for this ancient astronomer prevented him from per- ceiving that they were not always to be depended upon. Ptolemy had, for example, transmitted an observation of Mars, in opposition at his perihe- lion, in which his latitude, observed from the earth, was (5 50 7 south ; and, to deduce hence the incli- nation of the orbit, Copernicus thus proceeded. To deter- Let A (fig. 29), be the centre of the ecliptic, that inclination * S ' * ne G % n place of the sun, B the place of the of ihe or- earth, C Mars in opposition, and CBD the ob- bltl> served latitude. Then, since AC the distance of Mars from the centre of the ecliptic was given, by the theory of the first inequalities, in parts of the x semi-diameter of the planet's own orbit; and, con- sequently, from the determined ratio of the orbits, was OP KEtLER. 103 was found rr 1370 in parts of AB = 100O, and the angle CBD was given ; he had this ana- logy for the angle ACB, AC AB :: sin. CBD : sin. ACB : 4 5(/; and therefore the inclination CAB = CBD ACB was found to be = 1 51', r?nd, as is now ascertained, very nearly accurate. Copernicus also found this result to he in some de- gree confirmed by another observation transmitted by Ptolemy of the planets north latitude i for pro- ducing the line of apsides CA to F, where FA is ==. iftjO in parts of AE = 1000, and, with these lines and the angle EAF of inclination now de- termined, calculating the angle EFA, he found it = 24O'; whence the geocentric latitude, when the planet, in aphelion, is in opposition to the earth in E, ought to be 4 31 ; = FEG : now Pto- lemy tells us, that he observed this to be = 4 20'. But, by another observation of Ptolemy's, near the conjunction in the perihelion, from which he de- duced that the south latitude CED, if observed in the conjunction, would be no more than 5', it was found, by resolving the triangle CAE, that the inclination CAD was no greater than Q x ; and by a like procedure, founded on Ptolemy's ob- servations, the results for the inclinations of Saturn and Jupiter, deduced from oppositions in the li- mits, were 2 44', and ] 42'; but, from conjunc- tions in the limits, only 2 16', and 1 18'. These results Copernicus considered as sufficient evi- dences that the inclinations of the orbits to the plane of the ecliptic were variable ; and, as no va- riation could be ascribed to the plane of the eclip- tic, he supposed the planes of the orbits to oscillate backwards and forwards, each on its line of nodes; and the oscillation of the orbit of Mars was sup- posed to be through an arch, or angle, of 14J', of Jupiter through 24', and of Saturn through 28'; H 4 and 104 ASTRONOMICAL DISCOVERIES and in such a manner, that the inclination always increased in proportion to the increase of anomaly. Nor was his deference for Ptolemy the only cause of concealing from him the full value and impor- tant consequences of his principles ; but the two unhappy preconceived opinions, that every orbit was perfectly circular, and that the planes of the orbits intersected the plane of the ecliptic in lines passing through its centre, contributed equally to the same effect : for by these means all the dis- tances, as AF and AE, whether of the planet or of the earth from the sun, which entered into his investigation, were false. It is evident, that the system of Copernicus afforded an equally simple and satisfactory explication of the various latitudes of the inferior planets : but the same causes ope- rated here also, and prevented the just use and application of its principles. His theory for the latitudes of Venus and Mercury, was even more complicated than for the latitudes of the superior planets ; and, except in name and form, was very little different from that of Ptolemy. Nor was he more successful in his deductions of the longitudes and motions of the nodes ; for, in the orbits of the inferior planets, he considered the line of nodes as coincident with the line of apsides ; and in the or- bit of Mars it was perpendicular to the line of ap- sides ; and, in all orbits, he gave no different mo- tion, either to their apsides or to their nodes, ex- cept the apparent. motion of the fixed stars, on ac- count of the regression of the equinoxes. 75, Copernicus seems, like Ptolemy, to have thought the inclinations of the superior planets so inconsiderable, that reductions from the planes of their orbits to the plane of the ecliptic might be neglected as unnecessary ; and, accordingly, in his investigation OF KEPLER. 105 investigation of the distance of Saturn from the sun, (72), no reduction was employed ; but he saw the necessity of them in the orbits of the moon R e d uc tioij and the inferior planets, for that otherwise their to the calculated and observed places would not in ge- ecllpl - c * neral coincide. His methods, with respect to Ve- nus and Mercury, were of necessity equally com? plicated \viih his theory of their latitudes ; and, as they were not required by his system, need not be repeated ; but, with respect to the moon, they were of the same simple kind which are now em- ployed. Let ABC (fig. 30), be the orbit of the earth, and aEP the orbit of the moon, or of any planet, intersecting the plane of the earth's orbit, that is, of the ecliptic, in the line of nodes BD. If a line PC be drawn from the planet in P per- pendicular to the plane of the ecliptic, it will re- present the latitude of the planet, geocentric if seen from the earth, and heliocentric if seen from the sun in S, supposed to be the centre of the ecliptic; for it is the sine of the angle PSC, or of the arch of a circle of latitude by which PSC is measured ; and the inclination of the orbits will be the angle CBP. The arch BP of the orbit, computed in consequentia from the ascending node B, is called the argument of latitude ; because, Argument when the angle CBP of inclination is given, it is of latitude, the condition, or circumstance, on which the magr nitude of the angle PSC of heliocentric latitude depends. When the inclination CBP, and this argument of latitude BP are given, the arch BC, or the distance of the planet from the node reckon- ed upon the ecliptic, may be found by this analogy K : cos. CBP :: tan. BP : tan. BC ; and the dif- ference between BP and BC is the required reduc- tion> subtractive from BP in the first and third quadrants 166 ASfROttdMldAt quadrants of anomaly; and additive to it, in the Second and fourth. (E.) 70- The vicissitudes' of the seasons are generally ttides ef the explained, in the Copernican system, by supposing - t k atj while the inclination of the axis of the equa- tor to the plane of the ecliptic is the same as in the Ptolemaic system, its positions in the annual trans- lation of the earth through all the signs of the ecliptic continue sensibly parallel to one another; by which means all the circles parallel to the etjiiator, are 1 daily intersected in variable propor- tions, by the great circle on the earth's surface which separates light and darkness ; and several of them are generally found wholly within, or wholly Without its limits. But the effects produced by this invariable direction of the axis of the equator. Were explained by Copernicus himself in a less Simple manner; viz* by ascribing to the earth an fittnual rotation upon the axis of the ecliptic; or tnore strictly speaking, upon an axis of its own perpendicular to the ecliptic. For he supposed the annual revolution of the earth round the snn to resemble the menstrual revolution of the moon roUrid the earth, or that of a ball fixed at one extremity of a lever, the other extremity of which turns round Upon a pivot; and seems to have con- sidered this as the only method in which a revolu- tion about a centre could be performed. Conse^ quently it became necessary for him, in explain- ing the vicissitudes of the seasons* to ascribe also to the earth the annual rotation now spoken of, &ntl performed in a contrary direction to the an- nual revolution : and otherwise, if we abstract from the diurnal rotation, only one hemisphere of the earth would have been presented to the sun> and OP KEPLER. 107 and it would have been presented to him invari- ably ; while the opposite hemisphere was involved in perpetual darkness. 77. But while the merits of Copernicus were so great, in these essential and distinguishing parts of his system, which related to the second inequali- ties of the planets, all their first inequalities, and all the inequalities in general of the sun and moon, remained in their former obscure and imperfect state ; and all the explications of them, which he attempted to give, differed from those of Ptolemy chiefly in outward form. A minute detail of his whole doctrines on these subjects would be unne- cessary, and it will be sufficient to produce some of the most remarkable. 7S. Copernicus was dissatisfied with Ptolemy's lunar theory, chiefly on two accounts ; the first arising merely from the ancient prejudice in favour of uniform motion in perfect circles, with which the explication of the second lunar inequality was inconsistent, (3Q); but the other unanswerable and decisive, that the variations of the moon's distance, which this explication rendered necessary, were irreconcileable with all observations, both of her diurnal parallax, and her apparent diameter, (40). His own lunar theory, therefore, was to this pur- pose. Let A (fig. 35), be the centre of the earth, and of the lunar deferent BCD ; let C be tbecen- tre of the epicycle EFGH, carried about in the circumference of the deferent ; let E be the centre of a smaller epicycle KLM, carried about in the circumference of the greater, and let AC be the line of the mean motion of the sun, or earth. Co- pernicus supposed, that C the centre of the greater epicycle moved in consequential from C to D, with a velocity 16ft Velocity equal to that of the moon's mean motion in the zodiac; from which, subtracting the ap-> parent mean motion of the sun, the remainder CD was the mean motion of the moon from the sun ; &o that her return to the line AC was made in the time of a synodical revolution < But the centre B of the smaller epicycle was supposed to be carried from E, the apogee of the greater, towards F, G^ Hj with a motion equal to that of the moon in ftiesn anomaly $ while the moon was carried in the Circumference of the smaller, from K its perigee^ towards L, Mj with a velocity double of her rate of mean motion from the sun. By these means he supposed that the first and second lunar inequa- lities would be justly represented : for, supposing both to disappear at an opposition to the sun in C > arid that then the centre C of the greater epicycle, the centre E of the smaller^ and K the centre of the fnoon, were in the same fight line AC, it Would happen that when the centre of the greater epicycle came to the quadrature in D, and the centre of the smaller one to F, the moon herself having a motion double of CD would be found in M, having described the whole semicircle KLM ; find) joining MA> the equation, or angle MAD, Would be the greatest possible, and amount to 7 40'i But in other situations of the centre E of the smaller epicycle at the opposition, and which Would necessarily take place, on account of the difference between the synodical and anomalistical revolutionsj the place both of it and of the moon flt the quadrature might be less remote from D, than E and M in the present figure^ and the equa- tion would be liable to all possible varietieSj from ? 4(/ to O 0'* To represent the first inequality separately from the second, the smaller epicycle Was supposed to describe with its perigee K the circle OP KEPLER. 109 circle KN ; to this a tangent AN was drawn from A, and the angle KAN, which never exceeded 5, was the first inequality ; while CAO, formed by 3 tangent drawn from A to the circle MO described by the apogee of the smaller epicycle, represented both inequalities, and amounted to 7 40'. It will however be understood, that though both are re^ presented at the opposition in C, they could not meet together ami be accumulated, except at the quadratures in B and D, Hence it was easy to rind the semi-diameters of the epicycles, at least their ratio to the semi-diameter of the lunar orbit ; for, in the right angled triangles CNA, COA, CN is the sine of 5 and CO the sine of 7 4O' to the same radius CA rr 10000O, and consequently CN ZT 8716, and CO =s 13341, Their difference NO is = 4625 ; so that FN, the semi- diameter of the smaller epicycle, is = 2312, and CF, the semi- diameter of its deferent, = 1 1O28, This was the principal part of the lunar theory of Copernicus, nearly as complicated as that of Ptolemy, and liable also, like his, to various objections. One in particular was made by Kepler, that it rendered the distance MA of the moon, in quadrature the earth, sometimes greater than the se ter of the orbit, though all observations concurred to make it Jess, 79- Nor was Copfcrnicus more successful in bfc explications of the first inequalities of the planets. In ascertaining the mean motions and revolutions of the superior planets, his procedure was the same with the ancient one ; for, whether the cen* tre of the system were occupied by the sun or by the earth, the place of a planet in oppositioa viewed from either was the same. But, as Coper* nicus considered Ptolemy's representation of the first lift ASTRONOMICAL DISCOVERIES first inequalities, by an excentric with an equani, (o be a trespass against the law of uniform' circular motion, he returned, as was mentioned, to the more ancient concentric theory with its epicycles, which Ptolemy had abandoned ; and one of his . explications was to this purpose. Let A (fig. 36), be the sun, or centre of the ecliptic, BCD the or- thefirstln- bit of a superior planet, and BD the line of ap- equities sides. On B, with a semi-diameter BF equal to | erioi e $ la- f Ptolemy's excentricity of the equant, describe nets. the greater epicycle FGHK; and on F, its apogee, with the semi-diameter FN equal to the remain- ing |, describe the smaller epicycle LMNO. Co- pernicus supposed, nearly as in his lunar theory, that B the centre of the greater epicycle, was car- ried round the orbit in consequentia, from B, to C, D ; that F, the centre of the smaller, was, in the Same precise time, carried round the circumference FGHK, from F to G, H, K ; and that the planet irioved with a double velocity in the circumference LMNO, from L to M, N, O ; and it was also sup- posed that, when the centre F was in the apogee t>f the greater epicycle, the place of the planet was in L, the perigee of the smaller. By these means, \vhen the centre B advanced to C Q0 distant, the centre of the smaller epicycle was found in G ; but the planet in N, having described the whole semi-circle LMN ; and the equation NAC, which at Q0 of anomaly is subtractive from the mean longitude AC, would come to its greatest amount. When again the centre B, or C, had described another quadrant, and was brought to D, and the centre of the smaller epicycle, having also describ- ed another quadrant, was found in H the perigee of the greater ; the planet had moved through its other semicircle NOL to L the apogee of the Snoaller epicycle, and therefore the equation would disappear. OF KEPLER. Jll disappear. This theory, as has been said, hardly differs from that of Ptolemy, except in outward form. No new excentricity is introduced, and th$ mutual ratio of the epicycles is equivalent to the ancient bisection ; for, if from L the perigee of the smaller epicyie we take in LA the part LP ~ AB, the part AP will be precisely equal to half the excentricity of Ptolemy's equant ; and if on P f with the semi-diameter PL, we describe a circle, this will be the planet's path in the heavens ac cording to ihe cxccntric theory, and nearly its path according to the present theory of Copernicus. Indeed it was the object of Copernicus rather tQ confirm Ptolemy's explication of the first inequa^ lities, than to depart from it ; and to remove from it the supposed absurdity of assigning to the pla* nets a motion which was uniform only in a fictjtjr ous orbit. But his explication was much more complicated than that of Ptolemy ; and, though the motions in all his three circles were uniform, the real path of the planet was no longer circular, but extended beyond the circle in all points es*- cept the apsides; for example, towards 90 of ano maly, the line PN, drawn from P the centre of the Ptolemaic orbit, to the planet, is the secant of th angle LPN, The uniformity attained was there*- fore merely imaginary, 80. Another explication given by Copernicus of the first inequalities of the superior planets was less complicated, and in form more resembling thg Ptolemaic. It was by means of an excentric orbit FCDE (fig. 37), the distance AB of whose centre, from A the centre of the ecliptic, was equal to the semi-diameter of the former greater epicycle ; 3n4 he only retained the smaller epicycle, in the cir? umference of which the planet was supposed tp Ill ASTRONOMICAL blSCOVERlES move, with half only of the former velocity, but in the same direction. Two uniform equal mo- tions were, by this theory, substituted for the sin* gle unequable motion of Ptolemy; the one of the centre of the epicycle round the centre of the or- bit, and the other of the planet round the centre of the epicycle : but it is evident that this theory Another was in all respects equivalent to the former. In theory. neither of them, did Copernicus so much attempt to investigate the exccntricity and place of the aphelion, as to examine the conclusions of Ptolemy respecting these elements ; and to make trial how far the equations deduced from them agreed with observation. In this examination his procedure was to this purpose. Having three mean opposi- tions, that is, three observations of a planet when, the centre of the epicycle was in the points D, E, F, and the planet in the points G, H, K, he either adopted Ptolemy's determinations of the excentri- city AB, and the longitude ABC of the aphelion ; or he approximated to them in the manner of Pto- lemy, by supposing CDEF to be the orbit described by the planet, and not by the centre of the epi- cycle (45). With these elements, thus rudely found, he had first, in the triangle FBA, the ex- Centricity AB in parts of FB = 100000, and* the angle FBA, known by the motion of the centre of the epicycle from C, the supposed place of the aphelion ; and could therefore find the base FA, and the angles AFB, FAB ; and secondly, in the triangle FAK, he had the angle KFN == FBC, and therefore KFA = KFN + AFB, and conse- quently, with the sides KF, FA, the angle FAK; which, subtracted from FAB, gave the distance 'KAC of the planet from the aphelion ; and afford- ed one part of the evidence by which he was to judge how far his conclusion, or assumption, for the OP KEPLER. 113 the place of the aphelion erred from the truth, Excentri- In the same manner, by means of the triangles C j^ e a n f d the DBA, DAG, he calculated the distance G A C aphelion. of the planet from the supposed aphelion, at the time of the opposition in D ; and, by means of the triangles- EBA, EAH, the distance HAG of the planet from the aphelion, at the time of the opposition in E. After these operations, by adding together KAC, and GAC, and subtracting GAG from HAG, he obtained the angles KAG, GAH; and, these being found equal to the observed dif- ferences of longitude in Ptolemy's three opposi- tions, he concluded that, as far as the observations could be depended upon, Ptolemy's determinations of the excentricity and place of the aphelion were just. But, on applying the same elements to some oppositions observed by himself, he found that the calculated differences of longitude did not agree so perfectly with his observed differences ; and he was therefore obliged to make some alterations of the Ptolemaic suppositions : these alterations were especially found necessary in the theory of Mars; where, instead of fixing AB at 15OO, in parts of BC =z IOOOO, that is. at f of Ptolemy's excentri- city of the equant, he fixed it at 1460, while the semi-diameter of the epicycle continued at 500 ; so that, in his calculations for this planet, he did not precisely adopt the principle of the bisection. In the former theory, the manner of the calcu- The ktion would have been different. Let E (fig. 36) in tlie be the centre of the greater epicycle in any one of theol > the oppositions, and F its apogee : and let Q be the centre of the smaller epicycle. Then, on ac- count of the equal motion of both centres, FEQ ~ BAE ; and, on account of the double velocity ofR the planet, EQR will be either = 2FEQ, or the suppl. of 2FEQ to 180; and therefore, I adopting same 114 ASTRONOMICAL DISCOVEBIES adopting Ptolemy's elements, varied to suit the concentric form, the points Q and R are given. Join AQ, AR, RQ. Then, in the triangle AEQ, with the sides AE, EQ, and the included angle AEQ, the angles EQA, EAQ, and the base AQ may be found : and, in the same manner, in the triangle ARQ, with AQ, QR, and AQR = EQR > EQA, the angle QAR. Consequently the whole angle EAR is given ; and, subtracted from EAB, gives the required distance RAB, from the aphelion B. t 81. The Copernican system produced an alte- ration in the manner of considering the mean mo- tions of the inferior planets; for it represented their only motions in longitude to be performed in what the ancients called their epicycles. The mean time therefore of a revolution in the ancient orbit was not now required, it being the same with that of a solar revolution, but only the time of a revolution in the epicycle. But though the manner of con- sidering the motions of the inferior planets was simplified, the investigation of them continued ta be attended with the same difficulties as in the an- cient system; and, till telescopes were invented, the greatest digressions continued to be the only means of determining the times of their revolu- tions. According to the principles of the Co- pernican system, a right line CE, (fig. 38), drawn to the planet from the earth in C, at the time of a greatest digression, is a tangent to the orbit: and, as it is also perpendicular to a right line BE drawn to the planet from the sun in B, the parallax BEC Mean ino- 1Tlust at this ^ me be ~ 3 signs, or QO. If then tions of the the planet's motion were circular and uniform, an ^ ^ e sun * n ^ ie centre ? tne circular orbit ; and if also the instant of the greatest digression could OP KEPlER. 115 fcould be exactly observed, the planet seen from C, in two successive greatest digressions of the same kind, and in the same point of geocentric longi- tude, would be precisely in the same point of its heliocentric orbit ; and the interval would be the precise time of its revolution round the sun. Even, if the observations of two such digressions had been made in distant ages, from the same point of the earth's orbit, the time of any one revolution might be ascertained, by the knowledge of the whole number of revolutions performed in the interval. But none of these required conditions takes place : for neither is the planet's motion uniform, nor its orbit a circle whose centre is occupied by the sun ; nor, -since the tangent and the orbit are, for some time about the greatest digression, sensibly coinci- dent, can the instant of any one be observed with precision. Yet, by employing the observations of distant ages, and thus distributing the errors aris- ing from those causes among a great number of particular revolutions, the method has been ap- . plied with considerable success to the motions of Venus, ami the time of her heliocentric revolution determined with considerable accuracy. Cassini's procedure in this investigation was as follows* He compared an observation of a greatest digression, made by Ptolemy in the year 136, and which he found must have taken place at Paris on the 25th of December, at 4 h., with another made by Julius Byrgius, in 15Q4, and which also took place at Paris on the 1/th of December, at 4h. SO'. The geocentric longitude of the planet^ in the first of these observations, was J s. 20 13' 45 /; , and in the second Is. 23 V 3& , differing 2 47' 5l". As in both observations the right lines drawn from the earth and the sun to the planet, are supposed to be perpendicular to each other, the same must have I 2 been 116 ASTRONOMICAL DISCOVERIES been the difference between their heliocentric Ion* gitudes : and from a ruder determination of the motions of Venus, deduced from less distant obser*- vations, Cassini found that the time, in the year 1594, when the heliocentric longitude was the same as on the 25th day of December 136, at 4 h* must have been, December the 15th, at 10 h. 36'. In this interval he knew from his ruder determina- tion, that Venus had made 2370 revolutions in her orbit : and, therefore, dividing the interval by 237O, he found that the time of each consisted of 224 d. 16 h. 39' 4"; and consequently, that the mean annual motion of the planet in longitude was 193. 1447 / 45 // , that is, 7s. 14 47' 45" more than a compleat revolution ; and the mean diurnal motion 1 36' 8 /; . But the application of this me- thod to Mercury was, on account of the greater excentricity and the difficulty of observing this planet, attended \vith less success; indeed hardly with any, till Kepler, by remarking that the time of a greatest digression must be also the time when a telescope shews the disk of the planet accurately dichotomized, and, happily distinguishing such instants, attained to a determination of the time of a revolution much more precise than he had pre- sumed to hope. The more accurate method of modern astronomers of determining those mean motions, is .by observations of conjunctions, and especially of the transits of the inferior planets over the solar disk. By these methods the time of a tropical revolution of Venus is now fixed at 224 d. 16 h. 40' 30".6, and of Mercury, at 87 d. 23 h, 14' 34".4. 82. In determining the place of the aphelion for an inferior planet, Copernicus did nothing more than adopt the methods used by Ptolemy to the forms OF KEPLER. 117 forms of his own system. Let A be the sun in the centre of the terrestrial orbit CD, B the centre of the orbit of Venus, and AB its line of apsides pro- duced to meet the earth's orbit in C and D. From the same reasonings which Ptolemy employed, Co- pernicus also supposed that, if two greatest digres- sions ECB, GCB, in opposite directions, were ob- served from C to be precisely equal, the line of ap- sides must be BC bisecting the intercepted arch GE of the orbit, and passing through the centre of the earth in C ; and if from the opposite point D of the earth's orbit two other greatest digres- sions BDF, BDH should also be observed precisely equal, but different from the former, the line of apsides would again bisect the intercepted arch FH, and pass through the centre of the earth in D. If therefore the latter digressions were greater than the former, the point of bisection L was the pla- ^ he ,^ of net's aphelion, where it was most distant from the the inferior sun, and nearest to the earth ; and if they were P lancls ' less, the point K of bisection must have been the perihelion. If, again, the opposite equal digres- sions should be of their mean magnitude, the earth must have been in a line perpendicular to the line of apsides ; and in both cases the longitude of the apsides was given by means of the longitude of the earth. In this investigation Copernicus em- ployed not only the principles, but the observations of Ptolemy, and concluded as he had done, that the longitude of the aphelion of Venus was Is. 25, and of Mercury 6s. 1O. Hence the excentricity of the orbit easily follow- ed ; for, if the digressions towards the opposite ap- tieToTthe 1 " sides were of the greatest and least kinds, he drew orbits, from these apsides to the orbit the tangents CE, DF perpendicular to the semi- diameters BE, BF, and consequently the lines BC, BD were the se- I 3 cants 118 ASTRONOMICAL DISCOVERIES cants of the angles BCE, BDF to the radius BE. Half their sum. is AC, the semi-diameter of the or- bit of the earth : and assuming this zz 100OOO, BE, the semi-diameter of the orbit of Venus was, zr 7193, and AB, its excentricity, -= 208 ; and the semi-diameter of the orbit of Mercury zz 3773, and its excentricity -r Q48. 83. Copernicus also employed the ancient me- thod of investigating the position of the centre of uniform motion in an inferior orbit ; and of deter- mining if it was different from the centre of the or- bk, which he had now found. Let AB (fig. 3Q), be the line of the apsides of Mercury, F the centre of his orbit, and E the earth in a line EG perpen- dicular to AB ; though no more than in Ptolemy's investigation was this absolutely necessary, it being sufficient if the angle AGE was given : it is re- quired to find the point G, in which the centre of the planet's orbit ought to be situated, in order to nlake the digressions on both sides of it appear equal to a spectator at E ; that is, the distance GF of this point from F the real centre of the orbit, and if it differed from the excentricity AB found by the former investigation. Here Copernicus employed two of Ptolemy's? greatest digressions, the one CEG = 2O 15', and and of the the other DEG = 26 J5 X . Their sum gave the P!" tof whole angle CED = 46 30', and bisecting this xiniform , . .. & ^^ , . , it. /* ? motion, by the line EF, which meets the line of apsides in the real centre F, the angle CEF was found to be = 23 15'. Therefore GEF = CEF CEG = 3, and consequently GF the tangent of this angle to the radius EG = 100000, was found zz 524. This was little more than the half of Q48, the excentricity of the orbit : so that in Ptolemy's language this excentricity was bisected by the centre OP KEPLER. lig centre of the equant ; and, contrary to what hap* pened in the case of the superior planets, the or- bit at the apsides extended beyond the equant. On the contrary, the distance of the centre of the uniform motion of Venus from the centre of her orbit was found, by a similar procedure, to be =z 41 6 parts of EG ; that is, to be double of 208, the excentricity of the orbit : and consequently the same bisection took place in her theory, which had been introduced into the theories of the superior planets. 84. The substitutions which Copernicus made for the Ptolemaic equant, in the theories of the inferior planets, were different from what he em- ployed in those of the superior. Instead of suppos- ing the inferior planet to move in an epicycle car- ried round in the circumference of an excentric orbit, (80), he supposed the centre of the orbit to revolve in what he called an hypocycle. Let A, (fig*. 40), be the centre of the terrestrial orbit BCDE, let the diameter BAD coincide with the Jine of the apsides of Venus, in its mean situation, and let the diameter CAE be perpendicular' to BAD. Let also AF = 2l6 be the mean excen^ tricity of the orbit of Venus, and AG = 41 6, what Ptolemy called the excentricity of her equant. If FG be bisected in H, and on H, with the semi- diameter HF, a circle FLGK be described, this will be the hypocycle in which the centre of the orbit is supposed to revolve. The law of the revo- lution is, that when the earth is in B, the longi- tude of the aphelion of Venus, the centre of the orbit is F the perihelion of the hypocycle ; when cl ^ p< the earth has revolved to C, the centre of the orbit having a double velocity is found in G, the aphe- lion of the hypocycle, and has described the whole I 4 semi- 120 ASTRONOMICAL DISCOVERIES semi-circle FLG; and when the earth has proceed- ed to D, the centre is again found in F, having described the other semi-circle GKF. Thus the physical part of the first inequality, which Ptolemy made to arise from the distance between the cen- tres of the orbit and equant, was explained as ef- fectually in the case of the inferior planets by means of this hypocycle, as in the case of the su- perior by means of the epicycle LGO (fig. 37) ; for, whether the planet shall describe such an epi- cycle in the circumference of its orbit, or the cen- tre of the orbit shall describe the hypocycle FLGK, the appearances to a spectator from the earth will evidently be the same : and thus also the second inequalities were much more diversified than if this hypocycle had not been employed ; for, when, the earth was in D, the longitude of the planet's perihelion, the greatest digression was MDA of the least kind ; when the earth was in C, 90 from D or B, it was NCA much greater ; and the greatest of all the digressions OBA took place when the earth was in B the longitude of the pla^ net's aphelion, that is, of its perigee. The calculations were much the same with those formerly exemplified. Let the earth be in the given point Q, and the position of P the centre of the planet's orbit will be also given, for FHP = 2DAQ. Join QA, QP, AP ; and draw RS pa- rallel to QA. 1. In the triangle AHP, with the given sides AH, HP, and the angle AHP, the ex- centricity AP, and the angle PAH, are given. 2. In the triangle PAQ, with the sides AP, AQ, and the an 3 JG QAP = GAH ~ PAH ' the side QP 5 and the angle PQA = QPR are given : and QPR is in Ptolemy's language the equation of the centre, and the point R the mean perigee of the epicycle (47). 3. The arch RT, or angle RPT, is OP KEPLER. 121 is the motion of the planet from the mean perigee of its orbit, or epicycle, and consequently given: and, if with this and the angle QPR, we form the angle QPT, we may thence, having the sides PQ^ FT, find the equation of the orbit PQT. 85. The same principle, with some variations,, \vas employed in the orbit of Mercury. In this AE, (fig. 4l) = 948 is the cxcentricity of the or- bit, and AF what Ptolemy called the excentricity of the equant, EF is bisected in G, and on this centre, with the semi-diameter GE = 2CJ2, the. hypocycle is described. In this the centre of the orbit was supposed to move in consequent ia, with a velocity double that of the earth, and in such a manner that, when the earth is in B the longitude Theory of of the aphelion, the centre of the orbit is in E, and Mercur y when the earth has described the quadrant EC, the centre of the orbit is found in F. But this re- presentation was not sufficient to display all the irregularities of Mercury ; and therefore Coperni- cus added an epicycle NM nearly equal to EF, and supposed the planet, not indeed to move in its circumferences, but to librate' in its diameter, ac- cording to this law, that when the earth was in B, the planet should be in M, the perihelion of the epicycle ; when the earth had described the quad- rant BC, the planet should be in N the aphelion ; and that it should return to M, against the time that the earth was diametrically opposite to B. By these means the orbit was not only displaced, ac- cording to the different positions of the earth, but even its magnitude was varied : for, when the earth was in the line of apsides, as at B, the semi- diameter of the planet's path was EM less than EH, and its digressions especially of the earth were at the point opposite to B, the least possible. When 122 ASTRONOMICAL DISCOVERIES When again the earth was in C, the semi-4iamete? of the planet's path was FN, greater than FH = EH, and therefore the digression NCA greater than the former, both on this account, and also because the earth is at a less distance from the or- bit. By this representation Copernicus also ex- plained the phenomenon which, according to Pto- lemy, was so singular in the orbit of Mercury ; that the greatest of all the digressions should take place, not when the earth is in B the longitude of the planet's aphelion, nor yet at C where the semi- diameter FN of the planet's path is greatest ; but when the earth is in the point K, or L, 60 re- moved from B. For, as the motion of the centre of the orbit is double to that of the earth, its place at that time in the hypocycle will be O, found by taking the angle EGO = 2BAK; and the planet's, place in the diameter of the epicycle will be P, found by taking MHR = EGO, and drawing from R to MN the perpendicular RP ; so that the digression becomes PKA, greater than NCA, though FN be greater than OP, and even greater than MBA at the planet's perigee. 86. To avoid the prolixity produced by this tlouble epicycle, Copernicus afterwards employed a different representation. Let A, (fig. 42), be the centre of the circle BCD, divided into quadrants by the diameters BD, CC, and on the same centre with a semi-diameter double of GF in the former figure, describe the hypocycle GQF, meeting BD in F and G. Also on F, with the semi-diameter FH = AB, describe the circle HEK, and on G the equal circle MNO. Suppose that these two last circles move in consequential round the centre A of BCD, with a velocity equal to the difference between the velocity of Mercury and the earth ; that OP KEPLER. 123 that Mercury moves from H in consequents ', with a velocity equal to that of the earth ; and that the centre of HEK is, in the time of an annual revo- lution, first carried from F upwards to G, and thence back again to F. By these means, when the centre is in F, and Mercury in H, the planet, since its orbit revolves on the centre A, will appear as if it described the smaller circle HLO, and its di- gressions will therefore be of the least kind ; and they will also be of the least kind, when the planet is in O, and the centre of the orbit in G. But when the planet comes to C, QO from H or B, and the centre of the orbit HEK to A, it will ap- pear to describe the circle BCD, and its digressions will be the greatest possible. The planet however, though at the distance of Q0 from H, is not there- fore 90 distant from the apsides of its orbit, be- cause these apsides revolve also ; and the difference between their velocity and that of the planet, will make the greatest digressions take place, as the observations were believed to require, when the planet is at the distance of 60 from the aphelion. 87- The principal cause which engaged Coper- nicus in all this waste of labour and ingenuity, was the unhappy prejudices so often mentioned in favour of uniform and circular motion, joined to his excessive deference for Ptolemy. This deference was indeed the sole cause of that most vexatious part of his labour, which regarded the motions of Mercury : for his situation on the low and foggy banks of the Vistula, rendered it impossible for him to make observations himself on that excentric planet ; and he had not presumed, like Kepler, to suspect that Ptolemy, instead of deducing his theory from observations, had sometimes corrected the observations to suit the theory. When he introduced J24 ASTRONOMICAL DISCOVERIES introduced therefore such a complication of circles, and perplexed compositions of their motions, to account for the singular phenomenon of Mercury's greatest digressions taking place at 60 on each side of his least distance from the earth, it was because he had no opportunity of discovering that the phenomenon was imaginary ; and the obser- vations, from which it was inferred, either inac- curately made, or unfaithfully transmitted. of Copernicus was not receivcc? T - on its appearance, with any degree of that appro- . ^ a tion which it deserved, and which it now uni- versally obtains. Its cold reception, indeed, fully justified the hesitation and tardiness of its author, to communicate it to the world. Yet, his want of success> in explaining the latitudes and first in- equalities of the planets in fongitucle, and the in- tricacy of his theories on these subjects, were not the principal causes of rejecting his opinions. On the contrary, those were the parts of his labours which, on their first publication, were chiefly va- lued : and his theory of Mercury, especially, not- withstanding its being encumbered with more epi- cycles than his explication of the second inequali- ties had banished, excited the admiration of many eminent astronomers. But his system was chiefly opposed, on account of all in it that was valuable and distinguishing : and the substitution of the diurnal and annual motions of the earth, for the; apparent diurnal revolution of the heavens, and the annual motion of the sun, was such a violent contradiction, both of the philosophical principles of the age, and the immediate evidence of sense, that all its advantages were undervalued, and proved insufficient to procure to it general credit. The conception of Copernicus, which represented the OF KEPLER. 125 the distance of the fixed stars from the sun to be so immense, that in comparison with it, the whole diameter of the terrestrial orbit shrunk into an imperceptible point, was too great to be adopted suddenly by men accustomed to refer all magni- tudes to the earth, and to consider the earth as the principal object in the universe. Instead of being reckoned an answer to the objection against the annual revolution of the earth, that her axis was not found directed to different stars, it was rather considered as the subterfuge of one who had in- vented, and therefore tried to vindicate, an absur- dity : and, when in answer to another equally powerful objection, that no varieties of phase were seen in the planets, especially in Venus and Mer- cury, Copernicus could only express his hopes that such varieties would be discovered in future times, his reply, though it now raises admiration, could not in his own times make the least impression on those who opposed his system. The earth was uni- versally supposed to be so immense and ponderous as to be incapable of any kind of motion : and the diurnal rotation, in particular, was thought to be decisively confuted by the consideration of centrifu- gal force; which would throw offall bodies, animate and inanimate, from its surface. These objections, and many others of no force in themselves, but in that age deemed irresistible, by reason of the low state of human knowledge, prevented the Copcr- nican system from being generally considered in any other light than as a mere hypothesis, and were the principal causes of the celebrity for some time sjaintained by the system of T. Brahe. CHAP. 126 ASTRONOMICAL DISCOVERIES CHAP. IV. Of tie System of Tycho Brake. gg. fTT'HIS celebrated astronomer was a Dane* JL of a noble family in Schonen, a province now subject to Sweden. His strong and early pro- pensity for astronomical studies, the opposition given him by his family in indulging it, his perse- verance in these studies so as to devote to them his whole time and fortune, and the truly royal en- couragement giveri him by Frederick, the first king of this name in Denmark, are very generally known. This enlightened prince conferred on him the island of Wien, or Huen, at the entrance of the Baltic, as a proper and undisturbed retreat for his observations ; built for him a castle distinguished by the name of Uraniburg, erected an observatory, and contributed to the expence of his astronomical instruments and assistants. In this sequestered place he employed himself for fifteen years, from 1582 to 15Q8, in those assiduous observations of the celestial bodies, which have been the principal foundations both of his own fame, and of the whole of modern astronomy. But, after the death of his patron, his labours suffered the most mortifying in- terruption ; and the persecutions he experienced, not only from envy, but from bigotry, to which science, in all its branches, is a constant object of jealousy, obliged him to abandon his native coun- try. He found, however, a new patron in the Emperor of Germany, Rhodolph the second, and an asylum at Prague under his protection : and, assisted by Rhodolph's liberality, he resumed his studies OF KEPLER. 127 studies with his former ardour. But a final period was soon after put to them, by his death, in the year l6oi, of an acute distemper, at the age of 55. QO. It was impossible for T. Brahe, with his abi- lities and advantages, not to perceive the incom- parable superiority of the Copernican system to the Ptolemaic, both in .'he arrangement of the planets, and in the simple and satisfactory explication which it gave of their second inequalities. What Coper- nicus assumed from more doubtful evidence, and perhaps partly from theory, T. Brahe found to be ascertained by indubitable observations, that Venus and Mercury were not always less distant than the sun from the centre of the earth, as in the system of Ptolemy, but that they were frequently much farther distant ; and thai Venus sometimes ap- proached much nearer to the earth than Mercury. The places therefore assigned to them by Ptolemy could not possiby be just; nor, however it might be contended that the earth was the centre of the solar revolutions, could it with any propriety be reckoned the centre of the revolutions of the infe- rior planets: and the representation of these, given in the ancient Egyptian system, was found to be unquestionable, that as they constantly attended the sun, and frequently crossed his apparent orbit, the sun was the centre about which their motions were performed. But T. Brahe's observations also System of demonstrated, that the planet Mars sometimes ap- T * Biahe " preached nearer than the sun to the earth, and was found within the solar orbit, and sometimes re- moved to a much greater distance; and induced him to conclude, that the sun, and not the earth, was the centre also of the motions of Mars: and, since the distances of Jupiter and Saturn from the earth were likewise extremely variable, he con- cluded 128 ASTRONOMICAL DISCOVERIES eluded in general, that the sun was the centra of the whole planetary system. His approbation how- ever of the principles of Copernicus went no far~ ther. The motion of the earth in any sense, either annual or diurnal, he conceived to be impossible; and, as the principal objections against it had never received any satisfactory confutation, he considered them as unanswerable. He applied himself there- fore to form another hypothesis, which might be more consistent with celestial observations than the Ptolemaic, and less- contradictory to first appear- ances and established principles than the Coperni* can. Accordingly, in this hypothesis, the earth continued, as before, immoveable in the centre of the universe : and, notwithstanding that his obser- vations on comets had demolished the whole an- cient fabrick of solid spheres, he thought it less repugnant to reason, that the immense sphere of the. stars, and together with it those of the moon, the sun, and the five planets, though connected by no conceivable principle, should daily revolve round the earth, than that the earth should be subjected to a daily rotation upon an axis. But, while the sun, the moon, and the planets, were thus carried round the immoveable earth, by the diurnal revolution of the starry sphere, from east to west, he explained the periodical revolutions, which arc made in a contrary direction from west to east, by supposing the earth to be the centre of the motions only of the moon and sun : while the lat- ter, as his observations required, was the centre of the revolutions of the five planets; with which, as satellites at certain assignable distances, he was al- ways accompanied in revolving round the earth e The invention indeed of this system is not solely to be attributed to T. Brahe, for it had originally occurred to Copernicus, before his prejudices against OP KEPLER. against the motion of the earth were fully con- quered : nor was it adopted in its full extent by all T. Brahe's followers ; for Lojigomontanus, his assistant and disciple, did not admit the doctrine of diurnal revolution of the heavens. 91. T. Brabe's explication of the second inequa- H . je ^ lities of the inferior planets was perfectly the same cation of with that given in the supposed system of the an- f he * cient Egyptians ; and, as to its effects, the same "i e e 8 q . ua with that given by Copernicus. For, whether the earth revolve round the sun, or the sun round the earth, carrying along with him those orbits of which he is the centre, the appearances, to a spec- tator upon the earth, of the motions performed in them will be perfectly the same. The explication - indeed, given in this system of the second inequa- lities of the inferior planets, did not differ in effect from the theory of Ptolemy. For, though Ptolemy did not suppose-them to describe their obits, or, in his language, their epicycles, about the sun as their centre, but round centres nearer to the earth; these centres were always in the line of the mean motion of the sun ; and the ratio between their distances and that of the sun, from the earth, did not enter into his calculations. 92. As it may be more difficult to conceive the manner in which the second inequalities of the superior planets are produced, in this system, than that of those of the inferior, the following is a de- scription of it. Let T (fig. 43) be the earth, ABCD the solar orbit, HK the orbit of Jupiter, and let Jupiter in H be in conjunction with the sun in A. While the sun describes his annual revolution m consequentia, Jupiter also moves in consequentia K through ISO ASTRONOMICAL DISCOVERIES through nearly 30 degrees of his orbit ; and there- fore his next conjunction with the sun will not he in the point E of the zodiac, but in a point G 30 more to the east. In the fourth part therefore of the interval between the conjunctions, the sun, having described somewhat more than a fourth part of His annual circle, will be found in B, and the situation of the line HK in Jupiter's orbit, will be LM. The planet therefore having advanced about 7 30' in that orbit, will be found in N ; and, by ajine TNO, drawn from the earth, will be re- ferred to the point O of the zodiac. When again the sun has proceeded from B to C, the situation of HK will be PQ ; Jupiter, having departed about 15 from P, will be found in R the point opposite to the sun ; and by a line TRF, drawn from the earth, will be referred to F. Further, when Jupiter has described 7 3O' more of his or- bi f , and the sun has come to D, the situation of HK will be SV; and the planet, being found in X, will be referred by the line TXY to Y. Finally, at the next conjunction, the orbit will be restored to its first position ; and the sun, having proceeded to a, and Jupiter having advanced about 30 from H, he will be found in Z, and referred from the earth, by a line TZG to G. Thus, in his progress in the zodiac, from E to G, his motion will first appear to be direct, between E and O; then retro- grade, between O and Y ; then a second time di- rect, between Y and G ; and about O and Y, he will seem to be stationary. While Jupiter there- fore seems to perform these motions in the zodiac, his real path, according to this system, is the looped curve HNRXZ ; and, it is evident that, the greater the distance of the planet from the earth, and the slower its motion, if a superior planet ; x but the quicker if inferior; the greater will be the number of OP KEPLER. 13J of such reduplications; in the course of its own periodical revolution, in the former case ; and in the course of the solar revolution, in the latter: and, if the distance of the mean place of the same planet from the earth, and its velocity, should be variable, so will likewise be the extent of its re- trograde arches in the zodiac, and the time of their description. By this representation, the intricacy and perplexity of T. Brahe's system, in comparison with the Copernican, may clearly appear : for, in the latter, nothing more is necessary to bring back the whole reciprocations of a planet, than the dif- ference between its velocity and the velocity of the earth; whereas, in the former, the whole or- bit must be displaced, and the difference, between the velocities of the planet and the sun, is not Sufficient. Q3. This system had the same advantages with the Copernican, in explaining the various latitudes of the planets, and in determining their various distances from the sun ; that is, the ratios of the semi-diameters of their orbits to the serni-diame- ter of the orbit of the earth. But the imperfec- tions of these determinations continued to be very great : and neither did T. Brahe, more than Co- pernicus, avail himself of all the advantages afford- ed by his hypothesis. The greut cause of the imperfection of T. Brahe's Imperfect investigations on these subjects, was the perplexity "auction! of his system, which prevented him from accurately distinguishing between the real and apparent orbit of a planet. The orbit of the earth, according to the Copernican system, is NH (fig. 44), and of a superior planet MK ; and, if their planes intersect each other in a line MG, passing through G the place of the sun, the place of the node in the zodiac K 2 : will 132 ASTRONOMICAL DISCOVERIES will be A ; and the orbit of the earth will be re- ferred from G to the great circle AEB, and of the planet to the great circle AFD. But if K be the place of the planet in opposition, and H that of the earth, the planet will be referred from H by the line HKC, to the point C of the zodiac, CHE, or its sine CFE will be its apparent latitude, and AC its apparent orbit. Now T. Brahe, by placing the earth in the centre of the system, was led to consider CFE as the real latitude : and, having compared the observed latitudes in opposition, con- cluded the greatest of them to be the latitude in the limits, and consequently the measure of the inclination CAE, which angle he supposed to con- tinue constant. With any other observation, there- fore, of latitude in opposition, he calculated in the right-angled triangle CAE, from the angle CAE and the side CE given, the sides AE and AC, and the difference between them was his required re- duction between the ecliptic and the orbit. But, as Kepler first and justly remarked, the heliocen- tric place of the planet in the zodiac is in the line GKF drawn from G through the planet ; and, since GKF and HKC are in the plane of the same circle CFE of latitude, and the angle CHE of ob- served latitude in oppositions is always greater than that of heliocentric latitude, the line GKF will meet the arch CFE in a point F, between C and E ; and the reductions from AC to AK will be all too great. Neither is the circle AC the same cir- cle with that which is described on the same cen- tre H to pass through the node and the planet in either limit, but another of the smaller circles of the sphere : and, if with this reduction applied to AE, the observed distance from the node, so as to make it zz AC, and the line AC thus determined, the angle CAE should be calculated, it would,. contrary OF KEPLER. 133 contrary to the supposition, be found a variable angle. All the calculations therefore of latitude or distance founded on these suppositions were unavoidably deranged, and it was impossible for T. Brahe and his disciples to reject the Copernican or Ptolemaic oscillations of the orbit. One phe- nomenon especially rendered them necessary, name- ly, that the difference of the observed latitudes in the opposite limits was very* great : for, in the northern limit they did not exceed 4 34' 3 whereas in the southern limit they arose to 6 26'; and, without supposing the orbit to oscillate, it must have been believed to be infracted in the line where its plane intersects the plane of the ecliptic. 94. In explaining the first inequalities of the First Inc. planets, T. Brahe adopted the concentrical orbit qualitie ** of Copernicus with its two epicycles, and for the same reasons; namely, that by Ptolemy's introduc- tion of an equant, a real, and not merely an opti- cal, inequality took place in his excentric orbit, (79) ; and that the distances of the planet, given by this excentric theory, from the centre of the ecliptic were often inconsistent with the observa- tions of parallax. But the magnitudes of his epi- cycles were not precisely the same with those of the epicycles of Copernicus : that is, if his scheme had been reduced to the Ptolemaic form, the cen- tre of his orbit would not have bisected the excen- tricity of the equant, but have divided it in a dif- ferent ratio. 95. The principal merit of T. Brahe, and in Great me. which he far excelled all the preceding astrono- " mers of whom we have any knowledge, was that of a zealous, indefatigable, and most ingenious, observer of the heavens ; and indeed, in the time K 3 in 134 ASTRONOMICAL DISCOVERIES in which he lived, this was the chief and most im- portant distinction which an astronomer could at- tain, Me properly considered observations as the only foundation of a just astronomy ; and finding astronomy as it then stood, in a great measure destitute of this foundation, he extended them to the greatest part of the celestial phenomena. Though he had to contrive and form the greatest part of his instruments, he determined, without any assistance fi;om the pendulum, and by the la- borious method of distances, the positions of no less than 777 fixed stars ; the parallaxes, refrac- tions, diameters, and whole peculiarities, of the sun, rnoon, planets, and even the comets which then appeared, were subjected to his examination ; and, by the uncommon magnitude of his instru- ments, and the ingenuity of their construction, he pot only attained to an accuracy before unknown, but also made several perfectly new discoveries, still allowed to be of the most delicate and subtle kind, and most apt to elude observation. 96. Before we can properly understand his ex- plications of the lunar inequalities which he disco- vered, a short account is necessary of his lunar theory. T. Brahe's Though, for the same reasons with Copernicus, lunar the- he rejected the Ptolemaic hypothesis, he did not imitate him in representing the first lunar inequa- lity by a single epicycle ; but he divided its semi- diameter, which consisted of 8696 parts of the semi-diameter of the orbit = 100000, into two portions; one BF (fig. 45) = 57Q7.3, and the other FM == 28Q8.6 : and : having on A the cen- tre of the earth described the lunar orbit BCD, he described on B, with the semi-diameter BF, the epicycle FGH ; and on F its apogee, with the semi- OF KEPLER. 135 semi-diameter FM, the epicycle KLM. He sap- posed the centre of the greater of these to move in ihe circumference of the orbit, in consequents, or from B to C, and the centre of the smaller in the circumference of the greater, in antecedentia, or from F to G, H ; and both to complete their cir- cuits in 27 d. 13 h. 18' 35", the time of an ano- malistical revolution. But he supposed the moon to move in the circumference of the smaller epi- cycle, in consequents, with a velocity double of the former ; and in such a manner that, if at the con- junction the line of apsides had happened to co- incide with the line of syzigy, her centre should be in K ; but towards the quadrature, when the cen- tre of the greater epicycle had described the qua- drant BC, and the centre of the smaller the qua- drant FG, the moon's centre should be found in M, having described the semi-circle KLM. The F; rst ; ne . first inequality was therefore represented by the quality, angle CAM ; and when, as in the present case, the conjunction happened at the apsis, the inequa- lity arose to its greatest amount at C the quadra- ture. In other cases, when the moon at the con- junction was not found in K, so neither at the quadrature would she be found in M ; and the equation would be less, or even wholly disappear. To represent the evection, or second inequality, second i- be took in AB (fig. 46), the line AN = 2174 of equaiit y ,or the parts now mentioned, and with the semi-dia- meter AN described on Nthe hypocycle AOP ; in the circumference of which the centre of the orbit was supposed to perform a revolution in conse-. quentia, in half the time of the synodical revolu- tion, and in such a manner, that in the conjunc- tion it should be found in A, and consequently at the quadrature in P, having described the semi- circle AOP. The orbit therefore would be dis- K 4 placed ; 136 ASTRONOMICAL DISCOVERIES placed ; and, if the conjunctions had happened near its apsides, both equations would be accumu- lated at the quadrature : and the whole equation would not, as before, be the angle HPM of nearly 5,, but a greater angle CAM, amounting to Third ine- 97. .-With these preparations, the variation, or ^ e al ^ a _ or third inequality, and which T. Brahe first disco- tion. vered, was thus explained. He supposed another small circle to be described, at the conjunction or opposition, on B the centre of the greater epi- cycle, (fig. 4/), with a semi-diameter BQ = Il63, and that its diameter QR continued always per- pendicular to a line BA, or SO, drawn from its centre to the centre of the orbit. In this diameter he supposed the centre of the greater epicycle, carrying with it the smaller one ML, to librate ; and that the libration, from S or B to R, from R to Q, and from Q back to S, was also performed in half the time of the synoclical revolution. In the conjunction, therefore, the centre of the epi- cycle FG coincided with S, the centre of QR ; but, against the time of the first octant, when the centre of the orbit was in O, at the distance of 90 from A, the centre of FG was not found in S, but at the extremity R of the diameter QR : and the mean place of the moon had advanced beyond the point S, where it would have been, if not affected by this libration, and required an equation ROS of 40 X 30 7/ . But, against the first quadrature, the centre of the epicycle FG would librate back to the centre S of QR, and there would be no such variation. Fourth, or gg. As to the annual equation, which depends quation!" u P on * ne anomaly of the sun, and in virtue of which OP KEPLER. 137 which the moon moves with greatest velocity when the sun is in the apogee of his orbit, no represen- tation of it was given. It was sufficient to have discovered the existence of this inequality, and the law which it observed ; and T. Brahe applied its equation, converted into time, as a correction of the time for which he had made any calculation of the moon's longitude. To correct the time, or to correct the longitude corresponding to it, are operations perfectly equivalent ; and in this prac- tice T. Brahe was followed both by Kepler and Horrox. 99. The last discovery of T. Brahe, relating to Variable the variable inclination of the lunar orbit, and the inclination variable motion of the nodes, he explained by b it> n< ascribing to each of the poles of the orbit a revo- lution, in a small circle, round the point which it would have always occupied, if the inclination had never varied from its mean amount. According to his observations, the inclination, in conjunc- tions and oppositions, was 4 58' 3O Y ; and it rose to 5 17' 30" in quadratures. Since the mean of these is 5 8', he laid off from A the pole of the ecliptic (fig. 48), in the great circle AD, an arch AB = 5 8', and supposed B to be the mean place of the pole of the lunar orbit ; so that, if the inclination had been constant, it would have been always measured by AB ; and the point E, at the- distance of 90 both from A and B, would have been the constant position of the node. To repre- sent therefore the variations of both, he described on the centre B, with the semi-diameter BM = 9' 30 Y the circle MPLN ; and supposed the pole of the lunar orbit to revolve in this circle, with double the velocity of the moon in her synodical revolution ; and in such a manner that, if NP be perpen- 138 ASTRONOMICAL DISCOVERIES perpendicular to AB or ML, the pole at every sy- zigy was found in M, at every quadrature in L, at the first and third octants in P, and at the se- cond and fourth in N. According to this repre- sentation, the least inclination AM at the syzigies was found, by subtracting M'B g' 30" from AB 5 8', to be no greater than 4 58' 3O"; and the greatest inclination AL at the quadratures, by adding LB to AB, to arise to 5 17' 30". At the second and fourth octants, it was AN, measuring the angle AGN, formed by perpendiculars from each pole to the arches EC and HK of the orbits of the earth and moon, and equal to EGH their inclination ; and, by resolving the right angled spherical triangle ABN, found to be ~ 5 8' g". Variable ] n other positions of the pole of the lunar orbit, as the nodes. at Q> when the moon's distance from the quadra- ture is = 22 30', and consequently the angle QBL = 45, the inclination is measured by the arch AQ; and, by resolving the triangle ABQ, it is found to be = 5 14' 18 ;/ . The unequal mo- tion of the nodes was a consequence of these sup- positions, and each of them appeared to librate continually to and from the point E, through an arch double of EG ; for, if the distance AB of the two poles were invariable, the node would always be in the point E, where perpendiculars from them to both orbits intersect each other ; and this would also be its position, when the lunar pole was in L or M : but at an octant, as when this pole is in N, and the lunar orbit in the position HGK, such perpendiculars will intersect each other in the point G of the ecliptic; and the variation of the longi- tude of the node will be EG, measuring the angle BAG = DAO, that is BAN. This equation of the node BAN, is to be found by resolving the tri- angle ABN, where, with the former data BAN, its greatest OP KEPLER. 139 greatest amount will come out = 1 46' Q"; and in other cases, as when the pole is in Q, and it is then = 1 13' 28 // .4, by resolving the triangle AEQ. These variations of the inclination and places of Variationt the nodes must obviously produce correspondent of latitude, variations of the moon's latitudes, and they will not continue to be the same even in the same degrees of longitude : and variations must also follow on the reductions to the ecliptic. The methods how- ever of calculating these need not be explained. It is here impossible to forbear remarking, both the accuracy of T. Brahe's observations on a sub- ject so delicate, and the ingenuity of his explica- tions. The only correction made on his conclu- sions is, that the mean inclination is now stated at 5 8' 52 7/ , instead of 5 8', and consequently BN the semi-diameter of the circle in which the lunar pole revolves at &' 4Q lf : and, though Sir I. New- ton assigned the cause of the phenomenon, no alteration was made, either by him or others, on the manner of deducing the various latitudes, till T. Mayer, in his lunar theory, found a method of calculating them by a single operation. 100. Great however as the ingenuity and merits T. Brahms of T. Brahe appear to be, we can hardly fail to per- system pre. ceive the inferiority of his system to the Coperni- I can, in perspicuity, simplicity, and symmetry of mean, parts. The periodical motions which he ascribed to the planets were double, and performed round two different centres ; for, considering the orbits of the planets as epicycles, a planet had first to move in the circumference of its epicycle round the sun, and then the sun, carrying along with him all the planetary epicycles, had to move round the centre of the ecliptic : and, if we also take into consideration 140 ASTRONOMICAL DISCOVERIES consideration the diurnal revolution, the system of T. Brabe will be found more complex than that of Ptolemy, who gave one common centre to all the revolutions, both periodical and diurnal. But, not- withstanding the perplexed nature of this system, and that it gave no account why the sun, moon,, and planets, should obey the diurnal revolution of a sphere in which they did not move ; or why the planets should obey the annual revolution of the solar sphere, from which the distance of some was immense; or why the earth, though placed between the spheres of Mars and Venus, should alone re- sist the influence, whatever it might be, which carried these bodies and so many others on each side~of it round the sun ; such was the difficulty of conceiving and admitting the motion of the earth, that this intricate and incoherent system was preferred to the simple and beautiful system of Copernicus, by all the vulgar ; and for a long time rivalled, and even surpassed it, in reputation, among the learned. 101. The first steps which tended to abate the general prejudice against the Copernican system, appear to have been taken by its author's friends and disciples, Reinholdus and Rheticus, men of eminence, and joint professors of the mathemati- cal sciences at Wittemberg. The astronomical tables, called the Prutenic, which Reinholdus cal- p r0 g r e SS of culated on the principles of Copernicus, were the latter found to correspond more accurately with the state reputation ^ ^ e heavens than any before in use ; and many astronomers were thereby induced to think favour- ably of the system on which they were founded. But a circumstance of greater importance towards its reception, was the spirit of philosophical specu- lation, and particularly the ardour in astronomical researches, OF KEPLER; 141 researches, which the Copernican system contri- buted to excite. The important doctrine, for ex- ample, of the composition of motion, taught by the celebrated and unfortunate Gallileo, and found- ed on reason and experiment, decisively confuted the grand objection against tjie rotation of the earth, drawn from the motion of projectiles, and especially of falling bodies on its surface ; and this decisive confutation, of an objection thought to be so powerful, weakened the influence of others, and produced suspicions of their being alike founded on ignorance and mistake. 102. But what principally contributed to remove causes of the general prejudice against the Copernican sys- it tem, was the invention of telescopes, and the ap- plication of them, especially by Gallileo, to the observation of celestial objects. His discoveries, by means of the telescopes, were so many and striking as to attract the attention even of the vulgar, and to render familiar and acceptable those parts of the theory, which originally appeared most improbable and paradoxical. By his observations, for example, Telescopic of the moon, and similar, though later observa- disco y eri tions made by others on the planets, these bodies by P Galli- appeared to be of the same nature with the earth, k. and some of them far to exceed the earth in mag- nitude : and when such bodies were seen to per- form revolutions through the heavens, the annual revolution of the earth ceased to be thought such a violent and improbable supposition, as it had ap- peared, while the evidences presented of their mag- nitude were of a less familiar kind. Of the same tendency were his discoveries of the satellites of Jupiter, and the later discoveries, when telescopes were more improved, of like satellites to Saturn. When 142 ASTRONOMICAL DISCOVERIES When Copernicus had displaced the earth front the irnmoveable station which Ptolemy supposed her to occupy, and made the sun the common centre, both of her orbit and of the planetary re- volutions, he had been obliged to except the moon from this general rule, and to acknowledge that no other centre than the earth could be given to the lunar orbit. This therefore seemed to be an incoherent and contradictory part of his hypo- thesis, and he appeared here to depart from its boasted simplicity, and to return to the complex motion in epicycles, which it was one of the prin- cipal merits of his system to destroy. But, after those examples, the revolution of the moon, in an epicycle round the earth, could no longer be viewed as a singular and solitary fact ; and the attendance of the moon upon the earth in her annual orbit, in the same manner as so many va- rious satellites of other planets were seen to do, could no longer be deemed contradictory to the common course of nature. Of the same kind also, though less decisive in its tendency, for the argu- ment applied equally to the system of T. Brahe, was the discovery made by Gallileo and others, of the phases of Mars, Mercury, and Venus. It was objected to the Copernican system by the disci- ples of Ptolemy, that, on the supposition of its truth, these planets ought to be seen from the earth under varieties of form, similar to those of the moon : and we have seen the acknowledge- ment of Copernicus, that such variations of form ought undoubtedly to take place ; and his predic- tion, that they would become evident in future times. His prediction was fulfilled by the telescope ; the phases of Venus were recognized almost imme- diately after the date of that invention, and the discovery OF KEPLER. 143 discovery of those of Mercury and Mars soon fol- lowed it ; and if like variations of form are not also observable in Jupiter and Saturn, the prin- ciples of the system sufficiently explain the rea- son. The telescope also furnished strong proba- bilities in favour of the diurnal rotation of the earth ; for it discovered the rotation of the sun about an axis, and, besides those of some other planets, the extremely rapid rotation of Jupiter. On account of this, while the whole heavens ap- pear from the earth to revolve in 24 hours, they must appear from Jupiter to revolve in Q hours 56' : and the reasons for concluding both revo- lutions to be merely apparent, are the same. By these discoveries, the original prejudices against the system of Copernicus were in a great degree removed, and it rose in the esteem of astrono- mers, not only above the system of Ptolemy, but above the more coherent one of T. Brahe. Even the diurnal rotation of the heavens became more questionable than the authority of sense had re- presented it : and, though no annual parallax of the fixed stars had been discovered, the irregular path, which the annual revolution of the plane- tary orbits represented the planets as describing through the heavens, seemed to be a more impro- bable and violent supposition, than that the dia- meter of the terrestrial orbit should be considered as evanescent, when compared with the immense distance of the fixed stars. But perhaps no cause finally contributed so much to promote the credit of the Copernican system, as the authority and discoveries of Kepler ; and by him such laws were demonstrated to be established, among the celestial bodies of which it is composed, as were wholly inconsistent with every other. But 144 ASTRONOMICAL DISCOVERIES But the value of Kepler's discoveries rises far beyond that of being adminicles to any system : and it was chiefly to exhibit that value in a just and convincing light, that the foregoing detail, of sys- tems and of the previous state of astronomy, has been given. CHAP. OF KEPLER. 145 CHAP. V. Of the Preparations to Kepler s Discoveries, and of his Original Intentions. 103. IT" EPLER was born in the year 1571, in iJ^ the dutchy of Wirtemberg, of a family ranked among ihenobkfsc, or gentry; but, in con- sequence of military service, the only business thought reputable by the noblesse of many coun- tries, reduced to indigence. By the favour of his prince he was educated at Tubingen, a seat of learning then so eminent, that masters of science were frequently selected from among its scholars, to supply the universities of other countries. Though he prosecuted his studies with success, Occasion^ and was a disciple of Maestlinus, an astronomer of ^ ^^J* eminence, and of the Copernican school, he in- toastrono- forms us, that he had no peculiar predilection for m ?' astronomy. His passion was rather for studies more flattering to the ambition of a youthful mind; and when his prince selected him, in 15Q1, to fill the vacant astronomical chair at Gratz, in Stiria, it was purely from deference to his authority, and the persuasions of Maestlinus, who had high ex- pectations from his talents, that he reluctantly ac- cepted of the office. He appears to have thought it unsuitable to his pretensions ; and the state of astronomy was besides so low, uncertain, and in many respects visionary, that he had no hope of attaining to eminence in it. But what he under- took with reluctance, and as a temporary provi- sion conferred on a dependant by his prince, soon L engaged 14-6 ASTRONOMICAL DISCOVERIES engaged his ardour, and engrossed almost bis whole attention. His first 1O4. The first fruits of his application to astro- astronomi- nomical studies appeared in his Mysterium Cosmo- cal pubh- 7 , . , ' ' , , r i cations, graphicum, published about two years aner his set- tlement in Gratz. Though he had adopted the Copernican arrangement of the planets, the prin- ciples of connection subsisting between its parts appeared to him to be in a great degree unknown. No causes especially were assigned for the differ- ent positions of the centre of uniform motion in the different orbits, nor for the irregular distances of the planets from the sun ; and no proportion was discovered between these distances and the times of the planetary revolutions. He considered the arrangement as deficient and unsatisfactory, till all those causes should be discovered: and in I he work now mentioned he supposed that he had dis- covered them, at least he traced out several simi- lar analogies, in the Pythagorean, or Platonic doc- trines concerning numbers, in the proportions of the regular solids in geometry, and in the divisions of the musical scale : and these analogies seemed to assign the reason why the primary planets should be only six in number. Hasty and juvenile as this production was, it displayed so many marks of genius, and such indefatigable patience in the toil of calculation, that, on presenting it to T. Brahe, it procured him the esteem of this illustrious astro- nomer, and even excited his anxiety for the proper direction of talents so uncommon. Accordingly, not contented with exhorting Kepler to prefer the road of obvservation to the more uncertain one of theory, T. Brahe added a generous and unsolicited invitation to live with him at Uraniburg ; where his whole observations would be open to Kepler's perusal, OP KEPLER. perusal, and those advantages found for making others, which his situation at Gratz denied. The opinion of other astronomers, concerning this pro- duction, was no less favourable : and, were there no other evidence of his just and general concept tions, the remark, which afterward led to such important consequences, that the cause of the equant, whatever it might, be, ought to operate universally, is a sufficient attestation of them; and a proof that, even in this early period of his studies, the possibility at least of deducing the equations of a planet, from the relation between its different degrees of velocity and distances from the sun, had presented itself to his thoughts. 105. Notwithstanding the ardent desire of Kep- His i ler to be admitted to the perusal of T. Brahe's c& servations, it was probable that the distance of the place, and the difficulty of the journey, to one in Kepler's situation, would have for ever prevented the gratification of it. But the persecutions notf arose which drove T. Brahe from his native coon- try, and from which he at last found a refuge in Prague, the capital of Bohemia. Thither, accord- ingly, Kepler repaired in the year 160O; and, that he might be under no necessity of returning to Gratz, he obtained, by T. Brahe's interest with the Emperor, his patron, the appointment of im- perial mathematician. On his arrival at Prague, another circumstance occurred, equally important to his success and fame, and which he piously ascribes to the kind direction of Providence. T. Brahe, with his assistant Longomontarius, wa3 employed about his theory of Mars ; no doubt, in- duced by the favourable opportunity of verifying it, by observations of this planet in its approaching opposition in the sign of Leo. KeplerV attention 1 2 xv a* 148 ASTRONOMICAL DISCOVERIES was therefore, of course, directed to the same pla- net ; where the great degree of excentricity ren- ders its inequalities peculiarly remarkable, and leads with proportional advantage to the discovery of their laws ; and it would have been directed to observations much less instructive, had those as- tronomers been engaged about any other planet. 106. The former planetary theories were in ge- neral unlike, and founded upon different princi- ples. The ancients, as Ptolemy and his followers, considered every planet separately, and supposed that their several motions and inequalities arose from causes peculiar to every orbit, and with which the other orbits had no connection. The moderns, again, considered the orbits as connected by a com- mon principle; and, remarking all that was similar in their motions, endeavoured to derive it from a common cause. But they disagreed about this principle ; for the cause of the second inequalities of the planets, according to Copernicus, was the annual revolution of the earth ; while T. Brahe and his disciples ascribed them to the attendance of the planets upon the sun, in his annual revolu- tion. But, notwithstanding these and other less important differences, the effects of all the theories, in representing the motions of the planets, and on the calculations of their places, were very nearly equal, and Kepler found them almost equally erro- neous : for the longitudes of Mars, calculated for August 1598, and August 1608, even from the Prutenic tables of Reinholdus, fell short of the ob- served longitudes, the first nearly 4, and the last no less than nearly 5; and errors of this magni- tude were not peculiar to the Prutenic tables, for similar and even greater errors were found in all the others. These errors, so demonstrative of the imperfect OF KEl'tER. 14() imperfect state of the science to which they ad- hered, Kepler chiefly imputed to the practice uni- versally followed by astronomers, of observing the mean oppositions only of a planet, and forming their theories for the first inequalities upon these. It was not at all surprizing that this practice should have been introduced ; for the points of the zo- diac, which a planet occupies in oppositions, are not, as in lunar eclipses, distinguished by any sen- sible marks, but must be determined by calcula- tion : and to determine when the mean place of the sun was opposite to a planet, required only the knowledge of the mean solar motions; whereas, to determine the same thing of the true place of the sun, required an accurate solar theory ; of which, in the ruder ages of their science, astronomers were destitute. As the authority of the ancients who had introduced this practice, continued to be followed, so Copernicus, for example, when he ob- served a planet about the time of an expected op- position, calculated the mean place of the sun known by his mean motion ; and if this differed precisely six signs, or 1 80', from the observed place of the planet, the instant of the observation was the required instant of the opposition : but, if the difference was not 18O precisely, he endeavoured, from a comparison of the motions of the planet viewed for several nights successively, to find out the instant when this precise difference of ISO 3 took place. (G.) Thus he supposed the eye to be placed in the centre A of the earth's orbit, round which the mean motions of the sun or earth were believed to be performed, (fig. 4Q); and the obser- vations of the planet to be made in the points J, H, K, of the planetary orbit, marked by lines pass- ing from A through the earth in O, M, N, and not from the centre of the sun in B. Consequently, L 3 when 150 ASTRONOMICAL DISCOVERIES when he had obtained as many longitudes of the planet in such points, and their correspondent times, as he thought necessary, and proceeded with these to investigate the elements of his theory, the elements investigated were, in Ptolemy's lan- guage, the distance AE of the centre of the pla- net's equant from the point A, and the longitude marked by the line AEF, which he reckoned th line of apsides, in the zodiac. Now this was the practice against which Kepler chiefly objected, and to which he chiefly ascribed the imperfection of all the former theories : and, without immediately de- ciding on their respective merits, the principal im- provement, which he originally had in view, was to introduce the practice of employing apparent oppositions, which supposed the observations to be made from the centre of the sun in B, and to form his projected theory on these. Consequently, in his theory, the line of apsides would be BEG, drawn from the sun through the centre of the equal 1 !*, and not AEF from the centre of the eclip- tic : and the elements to be investigated for the first inequalities, would be the longitude marked in the zodiac by this lincBEC, and the excentri- city BE ; or the distance of the centre of uniform juotion from the sun, and not AE its distance from the centre of the ecliptic, or earth's orbit OMN. tThis was Kepler's principal, at least his original, design ; and the immediate reason of his anxious desire, to obtain a perusal of T. Brahe's observa- tions, was to ascertain the propriety of the altera- tion. (H.) 107. The necessity of this substitution was part- ly suggested to Kepler by the principles implied in the Copernican system ; and, with the strong propensity by which he was distinguished, of en- quiring OF KEPLER. 151 quiring into the physical causes, especially of astro- nomical phenomena, the design of introducing it was what he could hardly fail to entertain. After T. Brahe's observations on comets had exploded the doctrine of solid spheres, it seemed to the dis- ciples of Copernicus more likely than ever, that the force which controuled and gave motion to the planets proceeded from the sun : and Kepler tells us, that he could conceive no reason, why, in particular, they should he retarded at the points of their orbits most distant from the sun, and ac- celerated in the parts least distant, except that this force became more languid, in proportion to the increase of the distance ; and grew more vigorous in proportion as the distance was diminished. But, if the line of apsides should pass, not through B the sun, but through the centre A of the earth's orbit, the planet's motion will not be slowest in C the point of greatest distance from the sun, and quickest at the point D of least distance, but at the points Fand G of greatest and least distance from the centre A, where no force could be con- ccived to reside. Or, if to evade this argument, the acceleration and retardation of a planet should be ascribed to some innate affection or propensity of the planet, or of the intelligence or power by which it might be supposed to be animated ; it was equally inconceivable how this intelligence should be induced to overlook the sun, an object of so great magnitude and splendour, and with respect to which it might regulate its variations of velocity according to geometrical principles; and Reasons in preference to the sun to fix its attention on an inducing imaginary point, distinguished by no mark, .and him * placed at no greater distance from the sun than two or three solar semi-diameters : especially, when Copernicus had acknowledged this point to be also L 4 variable ASTKQNOMICAI, DISCOVERIES* variable in position, and had taught that the ex^ ccntricity BA of the earth's orbit was changed, though the position of the sun in B remained the same. But there was also a practical reason which urged Kepler to his projected innovation, namely, his perceiving that the ancient practice in a great degree defeated the design for which oppositions were observed, and involved the planets more or less in those second inequalities, which it was the sole purpose in observing their oppositions to avoid. Indeed, in the rare case of an observation of this kind, in the line ABH of the solar apsides, the lines AH and BH of mean and true oppositions were coincident ; and the planet, considered with respect to either, equally divested of every degree of second inequality. But in the observations on either side of ABH, and especially at the distance of Q0 from it, the apparent opposition will happen in the line BQP, for example, and at the time marked by the line AQ ; and the mean opposition will not happen till afterwards, when the planet is found in the line AN, and at the time which this line marks. Now, at this time, when the earth has proceeded to N, the planet will not be seen to have advanced in its orbit ; a line drawn to it from the earth in N, will not even be parallel to PQ, and make it seem stationary in P; but, as its motion is slower than that of the earth, that mo- tion will seem to be retrograde; and the planet's geocentric place will be p, a point less advanced than P; or, which is the same, in k less advanced than K; that is, it will be involved in the second inequality, and,, instead of being seen in the same point from t'ne centres of the earth and sun, lines drawn from the earth to the sun and the planet will make with each other the angle BN/'. In the OF KEPLER. 153 the opposite semi-circle, again, of the earth's ano- maly, the mean oppositions will be more early than the apparent, and the planet will be more advanced in longitude : and these differences may arise, in the case of Mars, to 5 li/ ; of Jupiter, to 3O'; and of Saturn, to 15': that is, to inter- vals of many days. Corrections, therefore, on all the mean oppositions, except those observed in the line of the solar apsides, seemed to Kepler to be absolutely necessary: on those, to wit, observed in the first semi-circle of the earth's anomaly, by additions to the planet's longitude and subtrac- tions from the time ; and, on those observed in the opposite semi-circle, by additions to the time and subtractions from the longitude : and he be- lieved, that without such corrections it was vain to expect more perfection from any theory, however sound all the other principles might be which should be employed in forming it. The conse- quent difference, therefore, made on the ancient theories, would be, that the planet's excentricity should be measured from the centre of the sun in B, and that the line of apsides should be trans- lated from its former |osition AEF. to a new po- sition EEC ; and the greatest accelerations would take place, as they ought, at the planet's least dis* tances, and the greatest retardations at its greatest distances from the sun in B, and not from the imaginary point A. By these means, not only would the equations for the first inequality be va- ried, both in magnitude and position ; but much greater variations, since the centre of the orbit was also brought from L toT, might be expected upon the equations of second inequality and the calcu- lated latitudes. In short, Kepler proposed to demonstrate, that the planets moved in orbits, whose planes inter- sected 154 ASTRONOMICAL DISCOVERIES ?ected one another, in lines passing through (he centre of the sun, and not through the centre of the ecliptic ; and this was the great innovation which he originally intended, and was chiefly de- sirous to introduce into astronomy. In fact, ex- cepting only the system of Copernicus, it was an improvement more important, and of greater con- sequence, to simplify the science, than any which had been introduced in all the preceding ages ; an/i his successful and decisive establishment of its truth and propriety, may be justly ranked among his greatest discoveries ; and equally deserves our attention with those which have been more gene- rally celebrated. 1O8\ On Kepler's coming to Prague to live with T. Brahe, he found greater necessity for his pro- jected innovations than he had formerly believed. T. Brahe and Longomontanus had made a cata- logue of all the mean oppositions of Mars observed from the year 158O, and formed a theory repre- senting them, as they said, so nearly, that its errors in longitude never exceeded two minutes. In this theory, the place *f the apogee of Mars, for the year 1585, was in 4s. 23 4o' ; the whole cxcentricity was 20l6o, or nearly the same with that of Ptolemy's equant ; and of this they had Confirms a ^ otte ^ 16380 for the semi-diameter of the greater tions of epicycle. Consequently, to express their theory them. j n Ptolemy's language, they did not bisect the ex- centricity of the equant by the centre of the orbit, but divided it in the ratio of 1260O to 756O. Sa- tisfied with the accuracy of these elements, they had calculated the equations of the centre, that is, of the first inequality, for every degree of ano- mal) ; and, making an addition of 1' 40 11 to the mean motion of ttie Prutenic tables, had formed a new OF KEPLER. 155 new table of the mean motions of the planet, of its apogee, and of its nodes, for 400 years. But the ancient difficulties about the second inequalities, or the equations of the orbit, and the latitudes even in opposition, remained unconquered ; and not only abated their confidence in their theory, but put a stop to their farther procedure in com- pleting it. Now these were the very imperfections to which Kepler had supposed that the principles on which they formed their theory would lead. For, by employing mean opposHons. and making IJA the exccntricity of their equant, they had dis* placed the planetary orbit, describing it on the centre L, a point of the line EA : arid the error produced on the equations of second inequality could not fail, in some configurations of the earth with the planet and the sun, by thus displacing the orbit from its proper centre T, to be consider- able. For example, if the earth should be in the point I, and the planet falsely represented in S, while its real place was in K, the error of the equa* tion of the orbit would be the angle RIS. If also the planes of the orbits should intersect each other in a line passing through the sun in B, their sup- position, of this intersection taking place in the centre A of the ecliptic, could not fail to produce correspondent errors on the calculated latitudes. 10p. But, convincing as these reasons appeared to Kepler, and at last, as we are informed, to T. Brahe himself, the first -proposal of the altera- tion gave rise to much debate between those astro- nomers. The n asons certainly were not obvious why T. Brahe's hypothesis should not be supposed as perfect as the intended hypothesis, which was to rest on apparent oppositions ; and it seemed to be impossible that a theory, which was affirmed to 15(3 ASTRONOMICAL DISCOVERIES to repfesent the mean oppositions so exactly, in all points of the orbit without distinction, should be false. This debate, and the desire of giving satis- faction to T. Brahe about the subject of it, was the occasion of engaging Kepler in all the labon* ous and unpleasant investigations contained in the pose of first part of his commentary ; in which he shews, first b v demonstrations adapted, not only to the three pi-'s different astronomical systems, but even to both men* j ne kinds of theory, concentric as well as excentric, that though a false position were given to the or- bit, the longitudes of a planet might be so re- presented, by a proper position of the centre of the equant, as never to err, in oppositions, above 5 minutes from those given by observation ; but, that by the same false position of the orbit, and though no other error of theory should be com- mitted, the second inequalities and the latitudes would be much more considerably, and indeed very greatly, deranged : and he demonstrates, in par- ticular, that when the planet is in Q s. 8 33' &' of mean longitude, and the earth in Qs. 16 O' l& f > the error produced on the calculated longitude, or the angle RIS, would arise to 1 3' 32 7/ . These investigations, therefore, however unpleasant and laborious, served to confirm Kepler in his purpose; and that part of them, which related to the simi* Jarity of the excentric and concentric theories in their effects, served also another important pur- pose ; for it demonstrated, that the labour and in- genuity employed in tie formation of those theo- ries, and in the substitution of the one for the other, had been in a great measure expended in vain, and were of little importance to the real im- provement of astronomy. (I.) 110. After OP KEPLER. 157 1 10. After thus examining the principles of the Strucfur>f ancient theories, and pointing out their defects, T.Brah|' his noxt step was to examine the structure of T. Brahe's theory in particular, in order to discover, if in all parts' of it the principles had been pre- served, and if its success in the representation even of the first inequalities, were as complcat as had been supposed. By this examination, errors, some- times very considerable, were discovered in those observations of mean opposition, from which its elements had been deduced. In mean oppositions, the computed mean place of the sun, and the ob- served place of the planet, should be precisely six. signs distant. But in none of all the ten con- tained in T. Brahe's table, from 1580 to j6oO, and employed in the formation of his theory, did this essential condition accurately take place ; and in the greater part the mean place of the sun was at a considerable distance from the point opposite to the planet. For in that table the oppositions, and their circumstances, stood as follows ; Times 158 ASTRONOMICAL DISCOVERIES Times of opp. Obs. long, in ecllp. M. long. d. it t & o x a $ f! 1580. Nov. 17 9 40 2 6 46 8 6 48 32 1582. Dec. 28 12 16 3 16 46 10 9 16 50 58 1585. Jan. 31 19 35 4 21 10 26 10 21 1O 13 1587- Mar. 7 17 22 5 25 10 20 11 '25 5 57 1589- Apr. 15 13 34 7 3 58 10 1 3 53 32 1591. June 8 16 25 8 26 42 2 26 45 24 1593. Aug. 24 2 13 1 1 12 43 45 5 12 34 36 1595. Oct. 29 21 22 1 17 56 15 7 17 56 17 1597- Doc. 13 13 35 3 2 28 9 2 28 51 i6oo. Jan. 19 9 40 4 8 18 10 8 18 43 Difference. Latitudes. M. long. S long. Apogee / // / n 5. o / it s. o / // 2 22 . 1 40 N 1 27 2Q 46 3 25 21 4O 4 48 4 6 N 3 11 34 56 22 17 13 + 4 32 10 N 4 22 37 46 55 4 23 + 3 38 12 N 6 3 27 46 23 32 4 38 + 1 6 45 N 7 16 53 7 24 10 13 24 .. _, 39 S 9 7 47 30 48 9 9 + 6 3 S 11 JO 53 50 25 26 O 2 + o o 15 N l 8 26 47 27 35 51 _ 33 N 2 24 55 47 29 5 O 43 4 30 50 N 4 6 46 16 30 6 Precession. Reduction. Ob*, long, orbit. Calculate^ o f n 27 58 50 410 + / // 5O 10 50 4O 23 O 38 5 20 + 51 3O 51 26 2 25 036 9 so 941 4 10 5 10 5 1O 4 50 5 55 3 35 54 35 54 33 7 47 10 20 + 42 40 23 040 8 45 35 O 34 36 11 27 12 + 56 5 57 14 13 20 6 + 34 O 32 2O 15 5 045 + 18 45 19 57 The UP KEPLER. 159 error in some of these opposition* appears from this table to have been very great. Particu- larly, on the 8th of June 15Q1. the sun was ad- vanced no less than 13' '24 lf beyond the point op- posite ro the planet. No restraint,' therefore, was laid on Kepler, in the introduction of his theory, from the accuracy observed in the structure of the former; for this error was about three times greater than any difference which the proposed transposi- tion of the orbit could produce. 111. These errors arose from the inaccurate conceptions entertained by T. Brahe and his assist- ant, concerning oppositions ; their supposing, to wit, that they took place, not when the sun was distant 180 from the point of the ecliptic in which the planet was observed, but from the point of the ecliptic in which it would be found after applying the reduction. It was with design, therefore, that they observed the planet in the points marked in the table ; because they thought them the only points, where the application of the proper reduc- tions would produce the effect they wished. But, even if this principle had been just, their rules of reduction were not constant. The nodes of the Cause of orbit of Mars were in 1 s. 17, and 7s. 17, and consequently the limits in 4s. 17, and IDs, 17 and, though therefore the reductions ought to have been greatest at 45 from either, yet at the obser- vation of 1582, of the planet in as. \6 *& 10", that is, at the distance of 30 17' 5Q" from the limit, their reduction was 4'48*j and at the obser- vation of 1597, in 3s. 2 28' o", at the greater distance of 44 32' from the limit, it was only 51 ;/ , With the inclination which they gave to the pla- net's orbit, the just reductions were those which are marked in a subsequent part of the table : and rors. 160 ASTRONOMICAL DISCOVERIES and upon the application of them, the places 6f the planet and the sun were not found precisely opposite, but stood as follows : Long. $ reduced to the orbit. M. long. . Difference*. 50' 10" 48' 32" 1' 38 ;/ -p 51 30 5O 58 1 28 + 9 50 1O 13 O 23 5 1O 5 57 O 47 54 35 53 32 13 + 42 O 45 24 3 24 ~ 35 O 34 36 O 24 + 56 5 6 17 O 12 34 O 28 51 5 g + 18 45 18 43 O2 + No errors of. any consequence were found to have been committed about the planet's mean lon- gitude ; nor, when the theory was formed, and the true longitudes calculated according to its princi- ples, did the calculated places differ, any where, above 2' from the places given by observation, ex- cept at the opposition of *5Ql. In this the helio- centric longitude, which, as deduced from the ob- servation, was 8s 20 42 ; O ;/ , came out, by T. Brahe's calculation, from his theory, 8 s. 26 4O / 23'', as in the table But, by a more accurate cal- culation made by Kepler, on the same principles, it was found to be 8 s. 26 34' 43 /; ; from which, subtracting 1O X 3O" for the reduction to the eclip- tic, it becomes 8s. 2(3 24' J3 7/ ; and. therefore, as the sun's mean longitude was at this time 2s. 26 45 7 24", the error of the theory, instead of 2^ arises to no less than 21' J l". 1 12. While the planet's apparent orbit was con- foundedj by T. Brahe and his disciples, in the same OF KEPLEK, l6l same manner as by the ancients, with its real orbit; they justly thought that, if they should continue, like the ancients, to reckon the opposition from the point of the ecliptic where the planet was ob- served, it would not be perfectly divested of the second inequality : for, the supposed inclination being far greater than the truth, the planet in the point E of observation (fig. 44), was brought too near the node. But by taking from the node A an arch AB of the ecliptic equal to AC, that is, by reckoning the opposition from the point B, in which the planet was found after the application of the reduction, Kepler remarks, that it is as truly involved in the second inequality, as if no such correction had been made, and the ancient prac- tice had been continued. For, let AFD be the Demon- real or heliocentric orbit; which, as the heliocen* *j j r e ation of trie latitudes in opposition are less than the geo- centric, must intersect the circle of latitude CE, in a point F between C and E ; and let CB be joined, cutting AF in D. The arch AD there- fore is shorter than either of the sides AC, AB, of the isosceles triangle ABC, and the point B more remote from A than the point D, which the planet, according to this practice, would oc- cupy in its real orbit ; that is, the sun, when op- posite to B, has passed the point of his opposition to the planet in D. But, though the observations were made errone- ous, by this inaccuracy in computing the opposi- ' tions, they were rendered much more erroneous by the excessive reductions, even if they had been regularly applied, which the mistake of the appa- rent for the real orbit produced. T. Brahe's re- duction was BE, for example, the difference be- tween AE the observed distance from the node, and AC the distance in the supposed orbit calcu- M lated 102 ASTRONOMICAL DISCOVERIES lated from CE and CAE given (93). Bat AC is an arch only of the apparent orbit ; and, unless the planet at an opposition be actually in the node, must always be greater than AF the distance in the real orbit. The inclination EAF of the real orbit, as will afterwards appear, does not much exceed 1 50', and consequently the reduction from it; even when AE is rr 45, will not exceed l': but T. Brahe considered CAE as the inclina- tion of the orbit ; and, as this angle was valued at 4 58' in the northern semi-circle, and at 6 2(3' in the southern, the reductions rose to 8', and even to 1C/. Though therefore the error committed, if AC had been the real orbit, by making AB = AC, and counting B the point of opposition, would have been inconsiderable, the effects of it, when AC is an arch only of the apparent orbit, on the observations, may sometimes arise to '. On these accounts it was evident to Kepler, that a theory founded on such erroneous principles, could not be depended upon. But he still con- tinued his examination, extending it to the man- ner in which T. Brahe had deduced, from those principles, the times and points of his oppositions. Here, however, no considerable errors were found; and the greatest, which took place at the opposi- tion of 15Q7, appeared to arise from inaccurate observations unavoidably produced by his persecu- tions. (J.) CHAP. OF KEPLER. 163 CHAP. VI. Of Kefler s Theory, founded on apparent Oppositions, ami of its total Failure. 1 13. r TT 1 HE principal reason which induced Kep- X ler to substitute apparent for mean op- positions, in the formation of his planetary theories, was, that a planet could not be found perfectly di- vested of its second inequalities, except in the points of apparent opposition. But since the point of the orbit, which a planet occupies in an appa- rent opposition, cannot be accurately cle r ermined without just and legitimate reductions from the ecliptic, a necessary previous step to the introduc- tion of his theory, was to investigate the inclina- tion of the orbit on which the reductions depend. The determination again, not only of the inclina- tion, but also of the place of the nodes, depended on the diurnal parallax ; and it was not yet known, with certainty how far any observation might be affected by it* ] 14. The diurnal parallax of Mars was a sub- ject to which T. Brahe had paid great attention ; and especially about the opposition of the planet in Cancer, in the year J582, he had made a great number of observations for the express purpose of discovering it. By these he supposed it to be ascertained, that towards oppositions this parallax was remarkably greater than that of^the sun ; and concluded that, from 1'Q 77 , its amount at the point of the orbit most distant from the earth, it arose to no less than 8' 35^ at the point least distant. M 2 But, ]64 ASTRONOMICAL DISCOVERIES But, on examining the observations, it was found that they were not made with all the ingenuity which a subject so delicate required ; for the fixed stars, with which he had compared the planet, were generally at a great distance from it, and also near the ecliptic : and,, therefore, the star with which the comparison was made in the morning was commonly set, or so near the horizon as to be deranged by refraction, before the evening obser- vation ; and it became necessary to supply its place Inaccuracy w ^ n another. Kepler also found reason to sup- in the in- pose, that in ascribing so great a parallax to Mars, onfofthe T - Brah ^ had been misled b y the calculations of parallax of his assistants, who, in making them, had entirely Mars. mistaken the intention of their master. He had, at different times of the same days, taken vari- ous distances of the planet from various zodiacal stars ; and had committed to his assistants the care of calculating, without regard to any theory, by a comparison of those distances, what parallax they would produce. But, to Kepler's surprize, they had applied themselves to calculate what place the planet would occupy, according to the forms and proportions of the concentric theory of Coperni- cus ; and from the difference between this 'and any of the observed places, endeavoured to deduce the horizontal parallax. It became necessary, therefore, that Kepler should have recourse to the original observations of 15S2, as much a if they had not been subjected to any previous exami- nation. 115. T. Brahe had endeavoured to investigate the horizontal parallax of Mars from the sum of two parallaxes in longitude, found by observations of the planet made on each side of the nonagesi- mal ; that is, by the excess of the apparent motion in OF KEPLER. 105 in longitude above the real motion : and this real motion he carefully ascertained by the observa- tions of successive days. The most remarkable results which Kepler found from the examination of this procedure, were the following : 1. From one pair of such observations made on the 26th of December 1582, at an interval of JOh. 46', the sum of the parallaxes in longitude was 1' 10 /7 .0 2. From a pair made on the l6th of January 1583, at an interval of gh. SO' 0' 22 /7 .5 3. From a pair on the 17th, at an in- terval of 9 h. 40', it was 1" 6 /7 .6 4. From another pair on the 37th, at an interval of gh. 18', 2' ig /7 .2 5. From a pair on the 18th, at an in- terval of 7 h. 53', it was 1 ' 46". 2 The only conclusion which he thought himself Para] j axof entitled to draw from these results, so far below Mars iTn- the common opinions of the age, was, that the certain determination of the horizontal parallax of Mars was a matter of the greatest delicacy, and almost an unconquerable difficulty : and this conclusion was confirmed by other observations in different years, all of which were found equally indecisive. In the examination of the foregoing observa- tions, Kepler employed also the converse method; and, supposing the planet's horizontal parallax, on the J7th of January, 1583, for example, to be 4', he calculated the sum which this supposition would give for the parallaxes of longitude ; in the above- mentioned interval of gh. 4O X , beginning at 5h. 2O' and ending at 15h, the times of T. Brahe's obser- vations. But, instead of 1 ' 6 H .(j given by the ob- servations, the result was 4' 8 /7 . M 3 Nor l66 ASTRONOMICAL DISCOVERIES Nor were the conclusions more decisive which he attempted to draw from some observations of his own. The method of deducing the horizontal parallax from motions in longitude having proved so unsuccessful, he attempted to employ observa- tions of latitude. These he made in February, 1604, at. a time peculiarly favourable, when Mars was stationary, and when, at the times of observa- tion, a -circle of latitude passing through the pla- net, nearly coincided with a vertical : and the in- struments used were an iron sextant of 2-| feet ra- dius, and a brass. quadrant of 3j. But his obser- vations_, after all the attention he could bestow, appeared to him inferior to those of T. Brahe : and on the whole he was obliged to rest in this general conclusion, that the horizontal parallax of Mars certainly did not exceed 4', even in his nearest approaches to the earth, and that, very probably, it was a great deal less. (K.) Il6, When Kepler had thus satisfied himself that no considerable errors could take place in re- gard to the observations on which he was to found his theory, by adopting the common opinions con- cerning the parallax of Mars ; his next object was Investlga- to investigate the longitudes of the nodes. T. t ion of the g ra h' s procedure in this business was founded on a false principle ; for it supposed AC (fig. 44), the path of the planet seen from the earth in H, to be its real orbit, and that the angle CAE was con- stant : and, therefore, unless the latitude EC, from which, and the angle CAE, he calculated the dis- tance AE, was very small, his conclusion for the place of the node A was not to be depended upon. But Kepler s procedure consisted in observing the planet when actually in the node : /md conse- quently, when the observed longitude was cleared of OP KEPLER. of refraction and parallax, the true longitude of the node was found. Four observations were accordingly found, in T. Bra he's register, of the planet in the ascending, and two in the descending node. By the first four it was represented to have been in the former, on the 7th of March 1'5QO, the 23d of January 15Q2, the 10th of December 15g3, and the 28th of Oc- tober 15Q5 : and by the two last, in the latter, on the gth of May 158Q, and the 2Qth of December 15Q4. The longitudes of the nodes were therefore found by calculating, for these times, the longi- tudes of the planet. This was done by T. Brahe's theory. For example ; the mean longitude of Mars, on the 2Qth of December 1 5Q4, was 7s. 27 14' 30"; and on the 28th of October 15Q5, it was ] s. 5 31' O 7 ; and the equations corresponding to these were 1 1 30' 30" subtractive, and 1 1 17' o" additive. The heliocentric longitude therefore of the descending node was 7s. 15 44', and of the ascending node Is. 16 48'. The same must have also been their longitude very nearly in the other observations : for, though their position is not ab- solutely invariable, the variation in the course of a few years is almost insensible. (L.) 117. After determining, at least nearly the lon- gitude of the nodes, the remaining and most im- portant business was, to investigate the inclination i nc i; nat ; on of the orbit to the ecliptic. For this purpose Kep- of thcor- ler employed three methods. One, which is a p.- blt * plicable to all the planets except Mercury, requires observations of latitude made in the limits, when the distance of the planet from the earth and the sun are equal': and it supposes that the mutual ratio of the orbiis and the heliocentric distance of M 4 the J08 ASTEONOMICAL DISCOVERIES the planet may be nearly, at least, determined by some of the former theories. Thus, since the node G (fig. 52), is in Is. 17, and the other F in 7s. 17, one of the limits C will be in 4s. 17, and the other in IDs. 17. On C then, with the distance CB, describe a circle ABH, cutting the terrestrial orbit in A and H ; so that, by joining AB, AC, there will be formed, upon the base AB, the 'isosceles triangle CAB. When the planet is in C, in 4s. l/, its heliocentric distance CB, cal- culated from T. Brahe's theory, is 1 666. 6 in parts of AB = 1000 : and it is required to find, in this case, what the relative positions of the sun, the planet, and the earth, will be, when AC becomes First me- GC l u ^ to B(U. For this purpose, draw from C the thod tode- line CD, bisecting AB in D. Then, since AC = . j 666.6, and AD = 50O, the angle CAD will be found = 72 23', and thence ACB = 2ACD = 34 54'. Therefore, since Mars, when seen from the sun in 4s. 17 is in the line BC, and ACB is n: 34 s. 54', he will, at the time when AC is =2 BC, be seen from the earth in 5s. 21 54': and, since CAB = 72 23', the sun will a,t the same time be in 8s. 4 27'. Or, if the earth should be in the point H of her orbit, the planet would be seen from the earth in 3s. 12 6', and the sun in Os. 28 33'. When C represents the opposite limit, the con- figuration will be somewhat different. For, then CB, according also to T. Brahe's theory, is z^ }375: and CAB will therefore be found = 68 4 1 ', and ACB = 42 28'. Consequently, in this li- mit, the planet will be seen from the earth in gs. 4 22', and the sun in 6s. 25 5 i ', when AC becomes equal to BC ; or, in lls. 29 28', while the sun is in 2s. 6 9'. As the invariable inclination of the planetary orbits OF KEPLER, orbits to the ecliptic, which was established by this investigation, may justly be ranked among Kep- ler's great discoveries, the methods of investigation merit a particular detail. Five observations were found, in T. Brahe's re- gister, of the planet near to 5s. 21 24', while the sun was also nearly in 8s. 4 27'. On the th of November 1588, at 18h. 3O', the planet was seen in 5s. 25 31% with 1 36' 45" of north latitude, while the sun was in 7s. 28. Since, therefore, it was neither precisely in the limit, nor the sun precisely in 8s. 4 27', the angle CAB will be only 02 2C)', instead of 72 23', and, AC being greater than BC, the observed latitude must be less than the inclination. On the 4th of the following December, Mars was observed in 6s. Q 1Q' 24", with 1 53' 50" of north latitude, and the sun was in 8s. 23. The elongation CAB was therefore 73 40' 36"; and, consequently AC being less than BC, the observed latitude was greater than the inclination. But the planet was more than 18 C beyond the limit, and the inclination at the point of heliocentric longi- tude which it then occupied, was not the greatest inclination. On balancing the effects of those op- posite causes, the greatest inclination was stated at 1 50'. On the 21st of October 1586, at 18h. Mars was seen in 5s. O 7', with 136'.6 of latitude, and the sun was in 7s. 8; so that the elongation was only 67 33', and the planet had not yet reached the limit. On both these accounts the latitude must have been considerably less than the incli- nation. On the 1st of November following, at l6h.4O', Mars was seen in 5s. 5 52', with 1 47' of lati- tude, and the longitude of the sun was 7s. 19 40'. Consequently 1.70 ASTRONOMICAL DISCOVERIES Consequently the elongation was 73 48', and the planet 16 short of the limit. Kepler considers this observation as very accurate, and in the limit it wouhl have given a latitude =r 1 50'. On the 3Oth of the same month, at iph. 30', Mars appeared in 5s. 20 4/, that is, almost in the limit, with 2 13' 30 V of north" latitude. But, the sun's longitude being 8 18", the elongation was 87 56' instead of 72 23'. The distance AC therefore, was much less than BC, and the ob- served latitude considerably greater than the incli- nation. Two observations were also found of the planet, near the same limit, from the earth towards the point H. On the gth of March 1506, at 8 h. the planet's observed longitude was 2s. 15 14 7 , with 1 4Q X 50 V of north latitude, and the sun's longitude Os. 0O / . The elongation therefore was too great, but the planet was 26 short of the limit ; and, balancing the effects, Kepler states the inclination at J 50'- On the 22d of April 1583, at Qh. 45', the pla- net's geocentric longitude was 4s. 1 17', with 1 5O' 40 /x of latitude, and the sun's longitude j s. 11. The elongation therefore was too great by about 8 ; but the planet had passed IQ beyond the limit, and the inclination in the limit is again stated at 1 50 X . ]Mo more than two observations of the planet were found in the opposite limit, and in both the earth was on the same side of it. The first was on the 5th of September 158p, at 7h. 35 7 , when the planet's geocentric longitude was 8s. 16 45' 40", with l ^}' 40" of south la- titude, and the sun's longitude 6s. 2. The elongation therefore was 74 45' 40 y/ instead of 68 41', and the planet fell 17 36' short of the limit. OF KEPLER. 171 limit. Balancing the opposite effects of these causes, the inclination at this limit also appears to be r* 50 X . The other was on the 1st of the following No- vember, at 6h. K)'; when the observed longitude was ps. 2O 50', with 1 36' of latitude,' and the sun's longitude 7s 1C) . The elongation therefore was only 6l 50', and the planet also 16 beyond the limit. These causes combined, must render the latitude less than the inclination, which again appears to be about 1 5O'. In th'e same investigation Kepler employed an- other method, equally of his own invention, and which required, neither any pre-conceived opinion concerning the ratio of the orbits, nor the aid of calculation to distinguish the observations proper for the purpose. It depends on this principle, Second that when two planes ABCD, and ABEF (fig. 53), method, intersect each other, two perpendiculars CB, EB, drawn in both planes to the point B of the line of intersection AB, will contain an angle CBE equal to the angle DAF contained by two other perpen- diculars drawn in the same planes to any other point A of the line AB, and also equal to the in- clination of the planes. If, therefore, there be any observations of a planet, when both the sun and the earth are in the line of its nodes, and the geocentric place of the planet QO from the nodes, the observed latitude will at once give the incli- nation. Though observations with such conditions are rare, four were found in T. Brahe's collection, in circumstances not much different. One was, that ot April 22d, 1583, already employed. The sun was in 1 s. 1 1, that is, 5 or 6 degrees below the node, and consequently the earth as ranch above the opposite one. But, though this tended to in- crease t J/2 AStROKOMICAL DISCOVERIES crease the apparent latitude, the elongation, which was only 80 17', instead of C)0, tended rather more to diminish it; and Kepler concluded, ^Yhat it might be considered as very nearly equal to the inclination. Again, on the 12th of November 1584, at 13 h. r 20', the sun was in 8s. 1, that is, 14 or 1 5 Q below the line of nodes, and Mars was seen in 4s. 23 1 4', with 2 12' 24" of north latitude. The planet therefore had passed the limit, and was not in the point of greatest inclination. But the elongation was g746 / , instead of 90, and the latitude, on account of the^ nearness of the earth, much more increased than by the other cause di- minished 5 and on balancing both, the inclination appeared to be 1 5O'. On the 26th of April 1585, at Qh. 42', Mars was observed in 4s 21 6', with 1 49' 45 ff of la- titude, while the sun was in Is. l6, that is, almost in the node. The elongation was too great, being Q5 6'; but the planet was beyond the limit, and therefore the latitude somewhat less than the in-*, di nation. Also, on the l6th of October 15Q1, at 6h. 30', Mars, towards the opposite limit, was observed in 10s. 27 2O X , with 2 iO' 3O" of south latitude de- creasing, while the sun was in 7s, 2 30', that is, 15 above the node. The observed latitude there- fore, as the earth was too near the limit, was greater than the inclination. The latitude had decreased 28', from the 2d to the l6th of October; and, if it had continued to decrease at the same rate for the next fourteen days, about which time the sun would have occupied the node, it would have been reduced to 145 / . But, on the with- drawing of the earth from the planet, the rate of decrease varies, and the conclusion of 1 50' for the inclination is confirmed. OF KEPLER. 173 This method of determining the inclination may Extenslon be made general, for any geocentric position of the O t it. planet ; for any two lines BG, BK, drawn in both planes to any point B of the line of nodes, will include an angle GBK, equal to the angle HAL, included between two other lines AH, AL, re- spectively parallel to BG, BK, and also drawn in both planes to any other point A of the line of nodes. Thus, in the observation of April 26, 1555, when the sun's longitude was Is. 1(3, the planet was seen from B in 4s. 21 6' 9 with a latitude GBK = 149'45". This therefore is equal to the heliocentric latitude HAL of the planet, in 4s. 21 6' of heliocentric longitude; and as this point is distant 4 fi' from the limit, or 85 54' from the node, the inclination, or heliocentric la* titude in the limit, will be found by this analogy, sin. 85 54' : sin. 149'45" :: sin. 90 : sin. 1 5O'. To confirm this conclusion from so many ob- servations, Kepler also employed the method ori- ginally used by Copernicus, though it supposes a pre-conceived opinion concerning the ratio of the orbits. Let A (fig. 29), be the sun, B the earth, Third me- C the planet Mars in his orbit, and D the point of the ecliptic to which he is referred by a circle of latitude. Let CBD, the observed latitude, at the opposition, for example, of the 24th of Au- gust 1 5Q3, be 6 3' south; and let it be previ- ously known, that in this point of the orbit AB : : AC :: 10OO : 138Q. Then, since AC >. AB :: sin. ABC : sin. ACB, this angle will be found = 4 21 y 6", so that, subtracting it from CBD, the heliocentric latitude CAD will be found = 1 41' 54": and, since the longitude of Mars in this op- position was lls. 12 30' 3 and consequently his distance 174 ASTRONOMICAL DISCOVERIES distance from the node rr 6/1 30', the analogy, sin. 64 30' : sin. gO : sin. 1 4)' 54" : sin. 1 5O' 2O", will give this last angle for the required in- clination. 1 J8. It was by such careful and multiplied in- vestigations, and from so majiy .different observa-* tions, in the most various configurations of the planet with the sun and the earth, at 60 or 70 a of elongation, as in the first method, at both quadratures, reckoned from the nodes as in the second, and at any opposition, taken at pleasure^ as in the third, that Kepler established this most Important important conclusion, and which, though funda- mental to every just planetary theory, had never ^ een before established, that the inclination of the was inva- orbits is invariable and constant; and that, in the . . * * orbit of Mars in particular, instead of amounting to 4 33' in the northern limit, and 6 26' in the southern, as T. Brahe had supposed^ it was very little more than 1 50'. By this discovery the oscillations, which, in consequence of the variety of observed latitudes, Ptolemy had ascribed to the epicycle, and Copernicus, on the contrary, to the orbit of the planet, and T. Rrahe's equally im- probable infraction of the orbit, were demonstrated to have no existence. Ptolemy appears to have been led to his improbable doctrine, in the same manner with T. Brahe, by the intricacy and per- plexity of his hypothesis, which prevented him from perceiving, that the whole varieties of lati- tude might have been justly represented by the single supposition of the parallelism of the plane of the epicycle to the plane of the ecliptic : and besides, his observations of latitude were not suf- ficiently numerous, nor did he pay equal attention to OP KEPLER. 175 to all ; but, misled by this theory, frequently re- jected those which disagreed remarkably with the greater number. Copernicus again was misled by bis reverence for Ptolemy, and his not forming a just estimation of the value* of his own theory, * nor placing in it sufficient confidence. Though he beheld, as we are told, with great satisfaction, ru s. its explication of the increase of the latitudes as the earth approached the planets, and of their de- crease on the withdrawing of the earth ; when he found that increase and decrease not entirely con- sonant to Ptolemy's account of his observations, he introduced oscillations similar to Ptolemy's ; and, as it was impossible that the ecliptic should be subjected to them, he was obliged to transfer them to the planetary orbits. This doctrine, how- ever, of an inclination which varied, not accord- . ing to the situation of the planet in the oscil- lating orbit, but of the earth in an orbit which did not oscillate, was what Kepler tells us he al- ways thought incredible, even before he had the good fortune of perusing T. Brahe's observations; and his opinion was not more judiciously pre-con- ccived, than we have seen it to be decisively con- firmed, by experiments most numerous and most fairly conducted. HQ. After having thus determined the longi- tiide of the nodes, and the inclination of the orbit n . atlon of . , i i 11 . , the opposi te the ecliptic, he next proceeded to ascertain the u points of the ecliptic where the apparent opposi- tions on which he was to form his theory took, place; and he was now enabled to reduce them justly and legitimately to the points of the orbit "which the planet really occupied. His procedure in deducing, from observations near tha opposi- tion, us. 176 ASTRONOMICAL DISCOVERIES tiori, the time and point of the ecliptic at which the opposition actually happened, was the same in principle with those since employed by the mo- dern astronomers (G) ; and the same laborious at- tention and accuracy were displayed in it which distinguished all his investigations. The opposi- tions which he thus determined were twelve in number, consisting of the ten employed by T. Brahe, and two observed by himself; and he had the advantage of verifying some of them by means of the observations made by David Fabricius, in East Friesland. The mean longitudes of Mars were calculated from T. Brahe's tables, and the whole catalogue is as follows. (M.) OF KEPLER. 177 ~~co m TT o 10 o m co . "*> "^Xl^COCOiOrH-iioOiO fc Z co en z 55 2 5 =5O COO^CCOOO ^ rp iO O p on~', E, F, G, will be given : and, since AC is the common base of the four triangles CFA, CEA, CDA, CGA, if it be denoted by any known number, the four lines AF, AE, AD, AG, will be also given in parts of AC. Since also in the other four triangles FAE, EAD, DAG, GAF, the sides with the in- cluded angles are given, the angles AFE, AFG, ADE, ADG, will be given ; that is, the two an- gles EFG, EDG. But these, by the supposition, are angles at the circumference of a circle, and the opposite angles of the quadrilateral figure DEFG inscribed in it : therefore their sum should N3 be 182 ASTRONOMICAL DISCOVERIES^ be equal to two right angles : and, if it be either greater or less, one of the assumptions for the longitudes of AH and CF 5 and perhaps both, are false. ^Formation Retain! DP*, therefore, the same mean longitudes, of Kepler s . . , r i IT TT i i i theory, let the longitudes or the aphelion H he changed ; by which means, though the angles FAH, FCFI, of anomaly are varied, the former equations AFC, AEC, &c. will continue: and let, the operation be repeated, till the sum of the angles at the cir- cumference be found a second time. If the dif- ference between it and two right angles be greater than before, the change upon the position of ACH has been made for the worse; and, if its longi- tude by the change was increased, it must be now diminished ; or, if diminished, it must be now in- creased. But, if the result comes nearer than the former to the sum of two right angles, it shews that you are in the way of coming at the truth : and a comparison of the present error with the former, will assist you in judging how much the longitude of AH should be again diminished, or increased. The result, however, upon the angles at D and F, "will not be in the simple ratio of "the change made on the position of AH : but the ope- ration must be, probably many times, repeated, before the sum of these angles be found accurately, or even but nearly, equal to two right angles. When you have at last obtained a sum of the two angles EFG, EDG, equal, or at least very nearly equal, to the sum of two right angles, and thus found that D, E, F, G, are point.s in the cir- cumference of the same circle, it is next to be enquired, if B the centre of it lies in the line AC. For otherwise the motion will not be slowest at the greatest distance from the earth, or the sun, in A ; as Ptolemy supposed that his observations required, OF KEPLER. 183 required, and Kepler, that also physical causes rendered necessary.- In prosecuting this enquiry, from the sides AE, AG of the triangle AEG, which were before found, and the given angle GAE zz GAD + DAK, the side GE, and the angle AGE, arc first to he determined Next, in the isosceles triangle BEG, where the angle GBE is given, being double of the angle GFE at the circumfer- ence, and consequently the angle at the base BGE, the semi-diameter BG is to be determined in parts of AC. Finally, in the triangle AGB, \vith the given sides AG, BG, and the included angle AGB = BGE AGE, the angle BAG is to be determined : and, if this shall differ from CAG, or HAG, before assumed, the centre B does not fall in the line AC ; and the suppositions for the longitudes- of FC and ACH must, one, or perhaps both, be false. But no new supposition can be made for the position of AH, unless one be also made for FC, and the other mean longitudes, because we have already investigated the only longitude of AH, which, with the assumed mean longitude, will per- mit the four points D, E, F, G, to be in a circle. In varying therefore the position of AH, we must also vary the position of FC, and the other mean longitudes ; and then the position of AH may be changed five or six times, if necessary, and the first operation as often repeated, till, with the new assumptions for FC, EC, and the rest, the sum of the angles at F and D be again found equal to two right angles.' When this shall be accom- plished, the second part of the process, namely, to find the angle BAG, is to be repeated : and, when this angle BAG is compared with the new as- sumption for CAG, it will be discovered, whether you are approaching to your purpose, or depart- N 4 ing 184 ASTRONOMICAL DISCOVERIES ing from it: and you will have directions in what manner other assumptions are to be made, with which all the former operations are to be repeat- ed, till at last BAG, and CAG, come out equal, while D, , F, G, are in the circumference of the circle described on the centre B. When this is at last accomplished, if the semi- diameter BG of this circle be denoted by any number of known parts, suppose J 00000, BA its excentricity, and CA the excentricity of the equant, will easily be found in the same parts ; and the suppositions for the place of the aphelion, and the mean longitude of the planet, at any one of the oppositions, made immediately before the last re- petition of the operations, will be established. (N.) 123. This then was the method in which Kep- ler investigated the elements of his first theory of Mars, which he afterward distinguished by the name of the vicarious theory ; and, when the de- tail of it is so fatiguing, he appeals to his readers if he, who had gone over all its steps no less than seventy times, was not justly entitled to their sympathy ; and if it was at all surprizing that four years had been consumed in the forma- tion of it. If we consider that logarithms were not then invented, the justice of his appeal will be readily admitted. The elements of the theory deduced from this painful investigation were, that, in the opposition of 1587, on the 6th of March, at 7h. 23', the longitude of the aphelion of Mars was 4s. 28 48' 55 /7 ; that the planet's mean longitude was 6s. 51' 35", that is, 3' 55" more advanced, than according to T. Brahe's tables ; that the excen- tricity of the equant was 18564, in parts of the semi-diameter of the orbit =: 100000, and that the excentricity OF KEPLER. 185 cxcentricity of the orbit was 1 1332. The excen- tricity therefore of the equant was not bisected, as in the Ptolemaic theory, but divided by the centre of the orbit in the ratio of 1 1332 to 7232: and, according to the forms of Copernicus and T. Brahe, 1'JpSS was allotted for the semi-diameter of the greater, and 3628 for the semi-diameter of the smaller epicycle. The improvement therefore of astronomy, which Kepler originally meditated, was now completed. In it he agreed with preceding astronomers, in supposing the orbit to be circular, and that there was a fixed point within it, about which the planet described angles proportional to the times : but he differed from them with respect to its relative position to the eye ; for drawing his line of apsides through the centre of the sun, he rendered the real motions slowest, as the phy- sical causes seemed to require, at the points most distant from the sun ; and avoided the absurdity of assigning such points to the greatest aud least velocities as seemed inconsistent with their appa- rent cause. 124. It only remained that this vicarious theory should be verified, by the exactness with which it represented the heliocentric longitudes, in all the observed oppositions. But, that the verification might be legitimate, it was previously necessary to determine the motion of the aphelion. This determination, however, could not be precisely ac- curate ; for it depended entirely on the observa- tions transmitted by Ptolemy, and these were not placed entirely beyond suspicion. Let A, (fig. 49), be the centre of the terrestrial orbit MN ; let E be the centre of the equant of IVJnrs, and B the centre of the sun. According to Ptolemy's accounts, the line AB of the solar apsides 1S6 ASTRONOMICAL DISCOVERIES apsides was in 2s. 5 3G X , and AEF the line of the apsides of Mars, in 3 25' 30 /; ; so that the angle BAE was = oO. According also to Pto- lemy, the solar excentricity AB was 4153, in purls of the semi-cjiameter AM z= 100000 : and AE, the excentricity of the cquant of Mars, in the same parts = 30380, so that by resolving the triangle AEB, the angle ABE comes out 123 27'. The longitude therefore of BEC, the new line of apsides, according to Kepler's theory, was 4s. 2 3' in the days of Ptolemy : and Kepler was disposed to consider it as no small presumption in favour of this conclusion, that BE the excen- tricity of the cqnantj given by the solution of this triangle, is, in parts of the semi-diameter TC = 1000OO, very nearly the same with what had been determined by his own theory. As Ptolemy's account of the precession was 5r- phe. reconcfleable with every other, it was impossible st f r Kepler to deduce from it the longitude of the apogee in 1587, and he was obliged to compare the place of it with that of a fixed star. By an observation of Ptolemy, the longitude of Cor leonis, in the year 140, was 4s. 2 3O' ; and, as Kepler found the apogee then in 4s. 2 3', its distance from that star, in antecedenlia^ must have been 27'. But T. Brahe found the same star, in 1587, to be in 4 s. '24 5'; and, according to the vicarious theory, -the apogee was also in 4s 28 4Q', and distant from Cor leonis 4 44', in consequential. Its whole motion, therefore, with respect to this star, in 1447 years, that is, from 140 to 1587, was 5 1 l'; and consequently its annual motion nearly 13 H , or in 30 years 6 2Q ;/ . If we add to this the motion of the fixed stars in precession, and which, according to T. Brahe, amounts to 25 X 30 ;/ in 3O years, the motion of the aphelion of Mars, with respect OF KEPLER. 1S7 respect to the equinoctial points, will, in 3O years, be 31' 5Q" in consequntia, and its annual motion = l ; 4 y/ . , Nor does this determination differ much from the most accurate determination of later times; and in .particular, in De la Lunde's tables, it is reckoned at 1' 7 7/ . As Kepler purposed also to verify his theory by means of the latitudes of the planet, so he endea- voured on this occasion to determine the motion of the nodes. He collected from Ptolemy's ac- counts, that the distance of the northern limit from Cor Iconis was, about the year MO, 3 2Q', or 3 30', in anlecedentia. But, in 1587, its distance from the same star, also in antccedentia, was 7 1? The difference is 4 15', and this therefore was the re- trograde motion of the limit, or of the nodes, from Cor Iconis in 1447 years. The annual motion, therefore, of the nodes, with respect to the fixed stars, is lO^.sO, and for 30 years 5' 17 7/ , in untece- dcntla ; and, subtracting this from 25' 30", the precession for 3O years, the remainder '2O' 13 /7 , shews that the motion of the nodes, with respect to the equinoctial points, is in consequential. De la Lande makes it 1Q' 54' v . 125. After thus determining the motion of the aphelion, Kepler proceeded to the verification of his theory, by calculating from it the longhudes of the planet in all the observed oppositions, and comparing them with the observed longitudes. He sets before us every step of his twelve calcu- lations ; but an example of one of them will be here sufficient. Suppose that it is required to calculate the planet's longitude at the opposition of la 188 ASTRONOMICAL DISCOVERIES In 1587, the longiUide of the s . , ^ t i on ' f the aphelion now found was 4 28 48 53 theory, by Its motion, in consequentm, for . "early 35 years, is 1 5 56 Longitude of the aphelion, at the -- opposition of 1 602 - 4 2Q 4 5 1 Mean longitude of the planet, from T. Brabe's tables + 3' 55" 515 3 32 Mean anomaly, or angle FCH - (fig. 54) 35 58 41 Since BC is zz 7232 in parts of BF = 10000O, we have, in the triangle BCF, sin. BFC = BC. sin. FCH = 7232.37528 = 199 1. Therefore the angle BFC, or the physical equation is = 1 8' 26". Since alsoFBH = FCH BFC = 14 50' 15", and BA = 1 1332, we have, in the triangle AFB, tan. 4 (BAF AFB) = tan. iFBH = 79643.1302] = 10370. Whence |(BAF AFB) = 5 55' 14", and the optical equation BKA = 1 29 / 53 // .5. The whole equation there- fore is 2 38' 19 /x -5 ; and the true . , anomaly FAH == - O 13 20 21^ Adding to this the longitude of the aphelion 4 2Q 451 The calculated long, of Mars, at -- the opposition of 1603, is 5 12 25 12.5 The observed longitude was - 5 12 27 Difference, though one of the --- greatest, no more than 1 4/.5 The verification therefore of the theory, as far as depended on longitudes in opposition, was com- pleat. As it was superior to all that preceded it, in its principle, in the accuracy of its structure, and in the legitimate manner of determining all the OF KEPLER. 189 the conditions by which it could be affected ; so likewise it was superior in the exactness with which it represented all the longitudes in opposition. Its errors never exceeded 2'; that is, they never ex- ceeded the supposed apparent magnitude of the planet : and if we consider how far it surpassed in accuracy the theory of T. Brahe, it will not be surprizing that Kepler indulged the most sanguine expectations from it. 126. Contrary however to all expectation and Refutation probability, and notwithstanding the present veri- of lts fication by oppositions, this theory, on* which so much time and labour had been expended, was found to be false, and soon received the most de- cisive confutation. The. principle especially of dividing the excentricity in a different ratio from bisection, and which Kepler had supposed to be of the greatest importance, was found altogether inadmissible, not merely in his own theory, but even in the ancient ones : and it seemed probable to him, that, when Ptolemy originally introduced the bisecti9n, when Copernicus was so scrupulous in departing from it, and when T. Brahe hesitated and left his theory unfinished, they had been in- duced by the reasons which convinced himself of the impropriety of rejecting it. The first refutation of the vicarious theory arose First t> y from the latitudes in opposition, and especially j from those observed near to the apsides and thetion, limits. Let DE (fig. 55), be a line in the plane of the orbit of Mars, and HL a line in the plane of the ecliptic, both passing through A the centre of the sun. Let D be a point towards the northern limit, and E towards the southern : and, let AD, AH, be-in the plane of the same circle of latitude, as also AE, AL. Let the earth, sit the opposition of 1QO ASTRONOMICAL DISCOVERIES of 1585, be in B, and at the opposition of in C. Since the planet's longitude, in the former, was 4s. 21, and, in the latter, 10s. 12; and, since the longitude of the earth's aphelion is 9s. 5, it is evident that the distance AC of the earth in 10s. 12 from the sun, must be greater than AB, its distance in 4 s. '21. On T. Brahe's supposi- tion, that the excentricity cf the earth's orbit is 3584, in parts of the semi- diameter 10000O, AC will be = 10J40O, and AB = 97500. If (he point B were precisely in the limit, the angle BAD of heliocentric latitude would be = 1 5O' (118), but, as it is 4 or 5 degrees from the limit, BAD at this point will be only 1 4$' 3O /; . But the observed latitude HBD, was 4 32' 10", and therefore BDA =z 2 42' 40 /; ; and consequently AD = AB ^|^ = 163000. Or, if according to the theories founded on mean oppositions, A were the centre, not of the sun, but of the earth's orbit, and consequently AB zz 100000 ; AD zs A ^ sin. HBD TII/- /^ AB - . , would be found zi 107000. sin. BDA' In the same manner the angle CAE of helio- centric latitude, at the opposition of 15Q3, is found to be 1 39': for the points C and E are 64 distant from the node, and sin. 90 : sin. 1 5O' :: sin. 64 : sin. 1 39'. But the geocentric lati- tude ECL was 6 3', and consequently AEC zi 4 24'. Therefore AE = AC ^ ~ = 13930O. sm. ALC Or, if AC were = 100000, then AE = 13738O. If the excentricity of the earth's orbit should not amount to 3584, but, as Kepler suspected, to little more than one-half of it, the results for AD and AE would be different ; greater, to wit, than the first, and less than the second. From OF KEPLER. From these lengths of AD and AE, the former in 4s. 21, and the latter in lOs. 12 of longitude, their lengths at the apsides of Mars in 4s. 28, and 10s. 28, may at least nearly be deduced ; for AD, at the distance of 8 degrees from the aphe- lion, would in the aphelion be 1 5O parts longer ; and AE, at the distance of 16 degrees from the perihelion, would in the perihelion be 300 parts shorter. When, therefore, DA and AE coincide with the line of apsides, AD will certainly be between 163150 & 167350 ; and AC between 13QOOO & 137080 : so that DE between 3O21 5O 8c 3044 3O ; and DK = JDE between 151O75 & 152215 ; and the excentricity AK = AD DK, certainly between 120/5 & l'5135. Or, in parts of the semi-diame- ter of the planet's orbit = 1000OO, between 8000 & Q913. But, by the theory now investigated, AK was found to be = J 1332. Some of the principles assumed in forming it must therefore be false. One of these was, that the orbit of the planet is a perfect circle ; and the other, that there is a fixed, point in the line of apsides, about which the pla- net describes equal angles in equal times. Since then the observations were accurate, one of these principles, or perhaps both, must be false. The same conclusion holds equally in the theories formed on mean oppositions : for the excentricity, instead of arising to 12352, as T. Brahe had de- termined, is by these observations of latitude re- duced to QQ43. 127. It was on account of such observations of latitude, that Kepler supposed Ptolemy to have introduced 1Q2 ASTKONOMTCAL DISCOVERIES Probable introduced the doctrine of the bisection of the ex- origin of centricity. For, as the excentricity of the equant don/* m Kepler's theory is 18564, the half of this, to wit, 9282, differs only 31O parts from $971, the mean between 8000 and 9943, the t\vo extreme results from the present observations of latitude : and if, conversely, we apply the principle of bi- section to the calculation of the longitudes in op- position, the differences from the observed longi- tudes will be found, at least in 90 of distance from the apsides, to be very inconsiderable. Even in the octants, they were not of such consequence as to attract the attention of Ptolemy, whose me- thods of observation appear to have been very im- perfect, in comparison with those of later times. For example, if at the opposition of 1503, where the planet's mean anomaly was 6s. 11 3' l6 /y , we substitute in the analogy BD : BC :: sin. DCK : sin. BDC for the physical equation, (fig. 54), Q.282 for 7232, the amount of BC, according to the vicarious theory, this equation will become J 1 ' 12"; and by substituting also 9282 for 1 1332 in the analogy for the optical equation, it will be- come 1 13' 26". CDA, therefore, the sum of both, will be 2 1*4' 38 ;/ ; and, consequently, the longitude = Us. 12 13' 37", differing only 3' 5" in defect from that given by the vicarious theory. If the bisection be in like manner employed, at the opposition of 1 582, when the planet's distance from the aphelion was about 41, the calculated longitude will become 3s. 174 / 45 // ; greater by I 1 41" than according to the vicarious theory, and by 9' 15" than according to the observation. When Ptolemy therefore was induced, as Kepler supposed, by the latitudes in opposition, to adopt the principle of the bisection, it was impossible that these errors produced by it upon the longi- tudes OP KEPLER. 3Q3 tudcs should divert him from his purpose : for, as he did not pretend to greater accuracy of observa- tion than within 10', and those errors never ex- ceeded 8' or 9', and in many parts of the orbit were much more inconsiderable, they were all within the limits of his acknowledged errors of observation. But the advantage which Kepler en- joyed, of perusing such observations as those made by T. Brahe, no longer permitted him to ascribe errors of theory 'to inaccuracy of observation: and he was perfectly convinced by his present deduc- tions, that, if the orbit were a perfect circle,, there was no fixed point within it, about which the pla- net could describe equal angles in equal times. From these deductions it undeniably appeared that, if at 90 and 270 of anomaly the bisection might with propriety be introduced, yet, near the apsides, the distance BC of the centres of the or- bit and cquant was greater, and, near the octants, less, than 9282, the half of 18564. If the orbit, Cause of therefore, was to be accounted circular, and its "' excentricity BA, as the latitudes required = 9282, no method seemed to be left for extinguishing the differences between the observed and calculated longitudes, except the libration of the centre of theequant; and this was a supposition, both impro- bable, and apparently incapable of being derived from any natural cause. These eight minutes there- fore of difference, produced by the bisection, be- tween the observed and calculated longitudes, Kepler informs us, were the sole cause of his far- ther researches in astronomy, and of the total re- formation of it, which he eventually introduced. If, like Ptolemy, he could have neglected these 8 7 , and ascribed them to errors of observation, he would have rested content with the principle of O the 1Q4 ASTRONOMICAL DISCOVERIES the bisection, and accounted it a sufficient cor* rection of his former theory. Second re- 128. The second refutation of the vicarious lutation, theory, notwithstanding the accuracy with which it represented the longitudes in opposition, was derived from longitudes observed at points not in opposition, when the planet was near the apsides. For example, on the 5th of March, 1600, at 12b. Mars was observed in 3s. 2Q 12' 30", with 3 23' of north latitude. His mean longitude was 4s. 2Q b 14' 58", and the longitude of his aphelion 4s. 29 2' 45 /x . The mean anomaly therefore was 12' 13", and, by the vicarious theory, the equa- tion was 2 7 subtractive. The true heliocentric longitude therefore was 4s. 29 12' 58"; while at the same time the sun's longitude was 11s. 25 45 / 51 // . Let A (%. 56), be the sun, B Mars, and C the earth. Since CB is in 3s. 29 12' 3O", and AB in 4s. 29 12' 58", the angle CBA of annual pa- rallax will be zz 3O 0' 28"; and, since CA is in 31s. 25 45' 51", the angle BCA of elongation . will be = 123 26' 39". Since also CA the dis- tance of the earth from the sun, in this part of the orbit, calculated for T. Brahe's excentricity =; ini *T r*\ * sin. .BC-A. 3584, is 99302, we shall have AB = CA sin CB ^ = 165680. Or, by the theory of mean opposi- tions, where CA is the semi-diameter of the earth's orbit and = 10OOOO, AB will be = 166846. by longi- Again, on the 30th of July, 1593, at 13 h. 45', tudes out Mars was observed in 1 1 s. 17 39' 3O" of longi- of opposi- tud ^ with g g/ 15// of SQuth j atitnc j e- Ris mean longitude was 10s. 26 16' 38"; and, as the aphe- lion was in 4s. 28 55' 43", the mean anomaly was 5s. 27 20 7 55^ and the equation 32 ; sub- tractive. OP KEPLER. 195 tractive. The true heliocentric longitude therefore was lOs. 25 44' 38'', and the sun's longitude was 4s. 17 3'. Let then D be Mars, and E the earth. The parallax EDA will now be 21 44' 5'i", and the elongation AED = l/ip c 23' 30": and since, ac- cording to the theory of apparent oppositions, AE is the distance of the earth from the centre of the sun, and in the present case =. 102689, we shall have AD = AE J^jJ = 140080. Or, by the theory of mean oppositions, where AE is =s looooo, AD = AE ' = 136409. As the distance of AB from the aphelion was only 12' 11", any correction for its length in the aphelion would be insensible ; but, as AD was 3 11' 7" distant from the perihelion, its length in the perihelion would be less by 15, and conse- quently == 140005, or 136394. The' heliocentric latitude FAB, at the point B of the orbit, is = J 48'; and, therefore AF in the plane of the orbit 82 parts longer than AB ; and the heliocentric latitude GAD is rz 1 36', and therefore AG 72 parts longer than AD. Therefore AF is between 165762 and 166928 AG between 140137 and 136466 Consequently FG - 305899 303394 and HG = FG 152949.5 151697 and the excentricity AH = AF -r- HF = 12812.5 15231; and in parts of HF or HG = 100000, it is between 8377 and 10106, A mean of these is 9241.5, differing very little from 9'282, found by bisecting 18564, the whole excentricity of the vicarious theory. O 2 Since, 1Q0 ASTRONOMICAL DISCOVERIES Since, therefore, the excentricity deduced from latitudes, and from longitudes out of opposition, differed so remarkably from that given by the lon- gitudes in opposition, and investigated in forming the vicarious theory, one at least of the principles of that theory must be false. If the orbit of the planet were a perfect circle, it undeniably appear- ed, there could be no fixed point within it, about which the planet moved uniformly; or, if the centre of uniform motion were a fixed point in the line of apsides, the orbit of the planet could not be a perfect circle. CHAP. OP KEPLBR. 1Q7 CHAP. VIL Of Kepler's Solar Theory, that is, hh Theory of the Second Inequalities. 12Q. A S Kepler's labours had been attended Cause of y\ with so little success, while, in imitation beginning of former astronomers, he began his researches S earSe from the first inequalities of the planets, he resolved from the to reverse this procedure, and to begin from their solarthe - second inequalities; that is, he resolved to inves- tigate the whole circumstances of the annual mo- tion of the earth, by which these second inequali- ties were produced. The theory, which he had formed from observations of longitude in opposi- tion, had been directly contradicted by observa- tions equally authentic of other kinds; and the principles of it partly subverted, and partly ren- dered doubtful ; and he expected by this alteration of procedure, to determine which of those doubt- ful principles was to be retained, or whether they ought not both to be rejected. Among the rea- sons which led him to this alteration of procedure, the investigations immediately preceding were the chief ; for, as in these, the" distances of the earth from the sun were necessary conditions, it was evi- dent that, if the excentricity of the terrestrial or- bit were not justly determined, all the distances would be false, and consequently all the investiga- tions founded on them. The knowledge, there*- fore, of all the circumstances of the second in- equality, that is, an accurate and authentic solar theory, was considered by Kepler as the key to all O3 the J08 AStRONOMICAL DISCOVERIES the mysteries of astronomy ; and in fact, by his use of it, the greatest part of them has been hap- pily disclosed. Suspicion 130. One of the propensities, by which Kepler of the ne- \vas remarkably distinguished from other men, was, ah equant an unconquerable desire of discovering the causes to the of natural phenomena, and of tracing them up to earth. general analogies and laws. Accordingly, when, in his most early and juvenile production, he en- deavoured to assign a cause for the planetary equants, or second epicycles ; that is, a reason why the accelerations and retardations of the pla- nets were not wholly optical, but partly real ; it occurred to him that* if this cause were just, its operation ought to be general, and to produce similar effects on all the moveable bodies in the system. But as no equant, or second epicycle, had been found necessary to the earth, or sun, an objection to his conclusions arose from this cir- cumstance, and his speculations were left unfi- nished. From that period, however, he enter- tained a suspicion, that, notwithstanding the con* , trary opinion of all preceding astronomers,, the earth, or sun, required an equant, in the same manner with every one of the other planets ; and, if the necessity of it should be demonstrated, that is, if the centre of the earth's uniform motion should be found different from the centre of the orbit ; it evidently followed that, not only would the excentricity, and the whole radii vector es in that orbit be different from those employed by T. Brahe and his predecessors, but the excentrici- ties, and indeed the whole elements of the plane- tary orbits, would also differ from the ancient de- terminations. This suspicion was strongly con- firmed by his Correspondence with T. Brahe ; who wrote OF KEPLER. 1Q9 wrote to him, while he was yet in Stiria, that the orbit ascribed by Copernicus to the earth, did not always appear to be of the same magnitude, but sensibly varied its ratio to the three superior orbits; and especially, that in some particular situations of Mars, the angles, which its semi-diameter sub- tended at that planet, sometimes differed by no less than 1 4o'. Expressions, which Kepler reckoned of the same import, occurred also in the letters which T. Brahe published ; and particularly where he mentions, that some inequality, which seemed to arise from variations of the solar excen- tricity, affected the longitudes of this planet in particular situations. 131. On receiving this intelligence, Kepler im- mediately perceived that the error, which he had always suspected in the solar theory, of confound- ing together the orbit and the equant, gave a suf- ficient explication of it ; and, that there was no necessity for having recourse to the improbable supposition of a variation of the excentricity, or of the magnitude of the semi-diameter. Let DE (fig. 57), be the line of the earth's apsides, A the sun, and B the centre of the orbit. When the planet is in T, a point of DE produced, and ob- served from the earth in the opposite quadratures Q and R, it was evident, that whatever were the position of the centre C of uniform motion in DE, and whether it coincided with B or not, the angles of parallax QTC, RT(3, under which the supposed semi-diameter of the orbit appeared from T, would be always equal; because they were the same with the angles QTB, RTB. But, if the planet were in F, a point of the line FC, perpendicular to DE in the centre of uniform motion C, and observed from the earth in the opposite apsides D and E, O4 the 200 ASTRONOMICAL DISCOVERIES the angles DFC, EFC, of parallax could not be* equal, unless C should coincide with B ; and they who believed these points to coincide, and there- fore considered DC and EC as the semi-diameters of the orbit, would of necessity be led to conclude' it variable in magnitude. Accordingly, when Kepler came to examine T. Brahe's observations, the variations of parallax were found to be pre- cisely what he had supposed * None took place, in the first configuration, when the planet in T \vas observed from the earth in Q and R ; nor in* deed from any other opposite points as X, Y, of any line perpendicular to the line of apsides : But, in the second configuration, when the planet in F was observed from the earth at D and E, or from L and K, making with FC the equal angles FCL, FCK, the variations of parallax were consider- able ; and, as this was the law which they con- stantly observed, he concluded it to be impossible that the centres of the equant and orbit could co* incide. First proof No pairs of observations indeed of Mars in of this nc- quadrature, from the earth precisely in the apsides, were to be found in T. Brahe's whole collection ; but two of a less perfect kind were found from the earth at L and K, whose distance LCD, KCE, from the apsides were nearly equal ; and where, if Kepler's suspicions were just, the parallaxes would also be affected, as he supposed, though, no doubt, in a less degree. On the 3Oth of May, 1585, at 5h. when the heliocentric longitude of Mars was 6s. 13 Q.S 1 !&', and the earth at K in 8s. 17 51' 46 /; , the planet was observed in 5s. 6 37' ; and, on the 2Oth of January, 15Q1, at Oh., when the heliocentric longitude was also 6s. 13 28' ]6", and the earth at L, in 4s. Q 4 X 46", the planet's geocentric longitude was ?s. 2134 / . Now, at these OP KEPLER* these times, the angles of commutation LCF, KCF, being the differences between the heliocen- tric longitudes of the planet and the earth, were equal, for each was = 64 23'' 2O' 7 : but the pa- rallaxes or differences between the heliocentric and geocentric longitudes of the planet were unequal ; for KFC was only = 36 5l', whereas LFC was = 38 6'; that is, greater than KFC by no less than 1 15'. The consequence was, that CL must be greater than CK, and that the point C, about which the equal angles LCF, KCF, were described in equal times, was not the centre of the orbit, but a point in the line of apsides more distant from the sun. On applying himself to investigate the distance CB of 'this point from the centre of the orbit, he found it to be 1837,, in parts of BD = 100000; that is, only 45 of such parts greater than 17C)2, the half of 3584, which was the whole excentricity of T. Brahe. As it was therefore only in the circumstances here described, that the dif- ferences now mentioned took place, the centre of the earth's orbit, in the same manner with those of the orbits of the superior planets, appeared to bisect the excentricity of the equant ; and the co- incidence of the orbit and equant could no longer be admitted. (O.) 132. As Kepler thus found means to account for the inequalities of parallax, without having re- course to the improbable supposition of variations in the magnitude of the terrestrial orbit, and merely by displacing the point of uniform motion, from its centre ; his next experiment, to ascertain the propriety of the new position given to that point, was of a less timid kind ; and consisted, not like the former, of a few observations subject to rare conditions, but of any number of observations taken r 202 ASTRONOMICAL DISCOVERIES taken promiscuously, and subject only to this sin- gle condition, that the planet in all should occupy the same point of heliocentric longitude. Accordingly, the observations of Mars, em- ployed in this experiment, were those of March 5th, I5g0, at 7 h. 10'; January 2 1st, 1502, at 6 h. 41'; December 8th, 15Q3, at 6h. 12 X ; and Octo- ber 26th, 1595, at 5h. 44'. At the first of these times, the true heliocentric longitude of the planet, calculated by T. Brand's theory, was Is. 15 53' 45", and increased, at the others, by the proper additions for the precession. Second Let C (fig. 58), be the centre of the earth's uni- moie deci- form motion, and A the sun, in the line of ap- l ' sides AC; let D be the place assumed for the aphelion, and to be ascertained by the result; and let the earth, at the time of the first observation, be in E, with 5s. 22 58' 46" of mean longitude ; at the time of the second, in Fwith 4s. 10 5' 57"; of the third, in G with 2s. 27 13' 12"; and of the fourth, in H with is. 14 20' 25". It is required to determine the lines CE, CF, CG, CH ; for, if they all come out equal, Kepler observes, that his opinions concerning the position of the centre of the equant must, notwithstanding the former ex- periment, be false ; but, if unequal, his cause is decisively gained. In 15QO, the planet, at the point M of the line CM, was in is. 15 53' 45 /x of heliocentric longi- tude, and the geocentric longitude observed from E was Os. 25 6'. Consequently, the parallax EMC = 20 47 ' 45", and, from the commutation ECM = 127 5' J", it appears that the elongation CEM was = 32 7 ' 14". Therefore, OP KEPLER. 203 CE = CM ^ = 66774; & in like manner sin. = CM. = 67467; = CM, All these are in parts of CM = 100000; and it therefore clearly appears, that when the earth was in G, with 2s. 27 J3' I2 V of longitude, and con- sequently near the perihelion P in 3s. 5 30', the distance CG was the greatest; CE, at 77 from the perihelion, was the least ; and CF, and CH, nearly equal, and of an intermediate length be- tween both extremes. CH, indeed, as being more remote from P, ought to be less than CF ; but the error may be justly ascribed to the minuteness of the angles at C and M, in the triangle CHM. The consequence was what Kepler had always believed. The point C, about which the planet describes the equal angles ECF, FCG, GCH, in equal times, (for each of them is determined by the time of the revolution of Mars), is not the centre of the earth's orbit EFGH. The centre of this orbit is a point B nearer to A the sun ; and, on investigating the distance BC of the point C from the centre of the orbit, by means of the dis- tances CE, CF, CG, and the given angles EAF, FAG, he found it to be = 1023, in parts of CM = 100000; or = 1530, in parts of BE = 1OOOOO; that is, somewhat less than the half of T. Brahe's eccentricity. It clearly appeared, therefore, that, even in this ruder form of investigation, where both the mean motions and the equations were taken from T. Brahe's tables, constructed on the ancient principles, an equant was necessary for the earth 504 ASTRONOMICAL DISCOVERIES earth as much as for the planets; and that her motions are accelerated in approaching the sun, and retarded in withdrawing from him ; not in ap- pearance only, but in reality. (P.) 133. The next step of Kepler's procedure was, to investigate what might be the distance AB in the opposite direction, between the centre of the orbit and the sun ; for, if this also should be but one-half of T. Brahe's excentricity, the evidence was compleat, that the equant and orbit were cir- cles wholly different. In this investigation the procedure was more accurate than in the former ; for, though the same observations were employed, the reductions of them to the proper times were made with greater care ; the heliocentric longitude of Mars was calculated from the principles of the vicarious theory ; and true longitudes of the earth substituted for mean longitudes. The distances of the earth from the sun, calculated on these con- ditions, were as follows. Long, of the earth. Distances from the $no. E in 5s.l4 (X 25" (fig. 5Q) AE = 67467 F in 4 JO 17 8 AF = 66652 G in 2 25 53 24 AG = 66420 H in 1 U 41 34 AH = 67220; all in parts of AM = IOOOOO. From these distances it appeared, even without the aid of calculation, that the centre of the orbit very nearly bisected the excentricity of the equant. For, according to T. Brahe's- tables, the longitude of P, the earth's perihelion, was 3s, 5 3O' ; and, consequently, the distance AG, at 9 from this apsis, between the earth and the sun, could not be much greater than the perihelion distance ; and AE, at 79 from it, could not be much less than the mean distance* Therefore OP KEPLER. 2O5 Therefore their difference 1038, in parts of Ai\l = 1OOOOO, or 1539, in parts of BE = 1OOOOO* though less than the just excentricity, shews that this cannot exceed 1800, the half of the excen- tricity deduced from the solar equations. On more particular investigation, this excentricity of 180O was established beyond suspicion ; and, in further confirmation of it, Kepler reversed the procedure; and with this excentricity, and the distance AM of Mars from the sun = 14770O, that is, nearly equal to 147443, found by the direct method ( 128) and on the supposition that the planet between the apsides retired within the circle, calculated the geocentric longitudes, and found them hardly to differ from those given by observation. (Q.) 134. As a doctrine so important required vari- Fourt!l ous and multiplied evidence, and as the proof of pnwf. it now given had employed longitudes deduced by the vicarious theory, which, as well as the mean longitudes to which they were supposed to correspond, might be suspected of inaccuracy ; so, in the next proof which he produced, nothing is supposed to be ascertained concerning the planet, except the time of its periodical revolution. A supposition is, indeed, made for its heliocentric longitude ; but this is merely for the purpose of establishing or correcting it by the methods of false position. It is obvious that the accuracy of the parallaxes EMA, FMA, &c. (fig. 60), and, consequently, of the distances EA, FA, &c. of the earth from the sun, depends on the justness of the position given to the line AM. By assuming it therefore, and calculating accordingly the lines EA, FA, &c., Kepler proceeded to deduce, from these and the given angles at A, which thev in- cluded, the angles EOF, and EHF. If these came 206 ASTRONOMICAL DISCOVERIES came out equal, as they ought, since the earth's orbit was supposed to be circular, the position as- sumed for AM was just ; and, if not, new suppo- sitions for it were to be made. When this, after repeated trials, was at last effected, and the just heliocentric longitude of Mars, at the time of his first observation, found to be 6s. 5 IQ' 42", he proceeded, in his usual manner, to investigate the excentricity AB, aad the longitude of the earth's aphelion. The result for the first was 1653, and for the latter Qs. 10 1Q ; ; and he considers it as evident, that, with a more just place of the aphelion, the excentricity would rise to 1800. It also appeared that AM, the distance of Mars from the sun, in the point which he now occupied of his orbit, was = 16285J, in parts of BE = 3000OO. -Here likewise, reversing the procedure, and di- minishing the heliocentric longitude of Mars V 30 ;/ , and supposing, as before, the orbit of the earth to retire within the circle, he calculated, with the excentricity 1800, and the place of the aphelion in gs, 5 3O', the geocentric longitudes. These were found to be Calculated 4s. 2655 A 0"; 5 s. 8 11 ' 4O"; ) Observed 4 26 54 30 ; 5 8 12 O;> Calculated 7s. 8 49' ;/ ; 7s 9 44' 20"; Observed 7 8 48 ]5 ; 7 9 47 10: and he also tells us that, on calculating, with the same suppositions, the geocentric longitude of Mars, for the 9th of February, l()04, at the time when the planet was observed by himself in the meri- dian in 6s. 26 18' 48", he found it to be 6s. 26 Q 17' 3O". With these suppositions also the distance AM was, OF KEPLER. 207 was, in this last investigation, found to be some- what less than 1 63 100. (R.) 135. The last evidence which Kepler produced, that an equant which should not coincide with the orbit was necessary to the earth, consisted, in in- vestigating with any number of longitudes of the earth, and their distances from the sun, at the times when Mars was in the same point of his orbit, and with as many geocentric longitudes of the planet, whether any different combinations of these data would give the same results for the heliocentric longitude and distance of the planet from the sun ; when the earth's distances were calculated for a circular orbit, whose excen- tricity was zz 1800. When the periodical time of Mars is known, it Fifth is evident that, though his heliocentric longitudes p roof - were unknown, we may, by reckoning from any particular instant as an epoch, find any number of others, in all of which his longitude must be the same. At five such times observations of the geo- centric longitude were made from the earth in E, F, G, H, K, and the apparent longitudes of the sun were given. The object, therefore, of the in- vestigation was, to find what results any pairs of these observations would give for the length and position of AM. With the data at G and F, the triangle GAF was resolved to find its base GF, and the angles at the base AGF, AFG ; and by subtracting these from the given angles of elongation AGM, AFM, the angles at the base of the triangle MFG were found, and consequently its side FM. With this therefore, and FA, and the given angle AFM, in the triangle AFM, the side AM and the angle FAM were both found ; so that the length of AM was 208 ASTRONOMICAL DISCOVERIES was by this investigation = 1(56208, and, by sub- tracting FAM = 21 26 X 32" from the longitude of the point F, its position was in 5s. 8 14' 32". A like investigation from E and H gave 166179 for the length of AM, and 5s 8 13' 8" for ils position ; and on employing the observation in K, with AM = 1662O8, the position found was 5s. 8 15' 4". Nothing therefore could appear with fuller evi- dence, than that the distances AE, AF, AG, AH, AK, of the earth from the sun, calculated for a circular orbit, whose excentricity was 1800, werfc at least nearly just ; and that, therefore, this was the true excentricity of the terrestrial orbit. Wherefore T. Brahe's excentricity of 3600, the results for the length and position of AM, inves- tigated in the same manner, would have been wholly irreconcileable. (S.) Other evi- 136. These demonstrations of the necessity of ences. an e q uant to the earth, and of the bisection of its excentricity by the centre of the orbit, received farther, and indeed intuitive, confirmation from the variations of the sun's apparent diameter ; which were not above half of what would have been produced by an excentricity of 3600. When the distances also of the earth from the sun were calculated on the principle of bisection, and the equations deduced, it was found that the differences between them and the ancient equa- tions, calculated on the supposition of the coinci- dence of the equant and orbit, were altogether in- considerable. Even at the octants, where the greatest differences might have been expected, the new equations were 1 27' 24", at 45 of anomaly, and 127 X 28'', at J35; the ancient equations, at both these points, being l 27 '31". (T.) 137. But OP KEPLER. 137. But in all this attention and labour, be-Newme- stowed by Kepler upon the solar theory, his object thud of / r i equations* was not merely to render the representations or the solar motions more correct, nor even to furnish him with the means of perfecting the planetary theories, but it was also to introduce a method of deriving equations, from a more just and natural principle than the imaginary equant, which all former astronomers had employed. This principle was the relation between the velocity of a planet in any point of its orbit, and the distance of that point from the sun ; or, in other words, the acce- leration of the planet, in approaching the sun, and its retardation in retiring from him. Such a varia- tion of velocity had been, at least virtually, ac- knowledged by Ptolemy in assigning equants to the planets, and by Copernicus and T. Brahe, ia ascribing to the superior planets epicycles, and to the inferior hypocycles ; but especially by Ptolemy in his doctrine of the bisection. Kepler's design of applying, to the important purpose of deducing the equations of the planets, a principle thus uni- versally acknowledged to regulate their motions, appears to have been conceived from the most early period of his astronomical studies : but as no equant was then thought necessary to the earth, and the principle had not been demonstrated to be general in its operation, his speculations concern- ing it were interrupted. But, now that T. Brahe's observations had enabled him to demonstrate ia the most convincing manner, that its operation affected the earth as well as the other planets, arid that the retardation of every planet in retiring from the sun, and its acceleration in approaching him, was the general law of the whole system, he was encouraged to resume a design so happily con- P ceived, 210 ASTRONOMICAL DISCOVERIES ceived, and so constantly entertained ; and to en- deavour to, carry it into execution. . 138. Though the consequences of endeavouring to derive the equations of a planet, from the rela- tion between its velocity and its distance from the sun, were as important as the ingenuity of the con- ception was admirable, the evidences, drawn from fact and experiment, on which the law was founded, were more satisfactory, than the geometrical demon- stration which he attempted. This demonstration was not general, for it is applicable only to circular orbits,and even to the very small arches only of those which are next to the apsides; nor are the principles rigorously exact, for it depends upon a compensa tion or balancing of errors by one another ; and though the proportion, between the times of de- scribing these small arches and their distances from the sun, be strictly true, it is deduced only in an approximate manner. But, as every step which Kepler took towards his invaluable discovery I that the planets in revolving round the sun describe areas proportional to the times, must be deemed of the greatest importance, the omission of ,any one would be unpardonable. The proposition to be demonstrated is, that on the Ptolemaic hypothesis of an equant, whose ex- centricity is bisected by the centre of the orbit, the times of describing very small -arches, at the aphe- lion and perihelion, are proportional to the dis- tances of these apsides from the sun ; and the de- monstration is to this purpose. Let A (fig. 62), be the sun, B the centre of the orbit, C the centre of the equant ; let AC be bisected by the centr^B, and on B and C let the equal circles DFEG and HLKM be described. Let FD be a very small portion of the orbit at the ; aphelion, OF KEPLER. 211 aphelion, and draw CFL to meet the equant in L; and, in the same manner, let EG be a very small portion of the orbit at the perihelion, and draw CMG to meet the equant in M : so that, accord- ing to the received principles of Ptolemy, the arch HL of the equant will measure the time of de- scribing DF ; and the arch KM, the time of de- scribing EG. 1. Because HL, DF, do not sensibly differ from straight lines, and the two triangles CHL, CDF, may be considered as right-lined similar triangles, we shall have CH : CD :: HL : DF. But, on ac- count of the bisection of AC in B, the lines CD, CH (zr BD), and AD are arithmetically propor- tional ; and an arithmetical mean between two terms, whose difference is inconsiderable in com- parison with either term, is but insensibly greater than a geometrical mean. Consequently, CH : CD :: AD : BD; only the farmer ratio is insensibly greater than the "latter. Therefore HL : DF :: AD : BD (= BE). 2. It follows also, from like reasons, that CE : CK :: EG : KM. But CE, CK (= BE), and AE are, in the same roinner, arithmetically pro- portional ; and, therefore CE : CK :: BE : AE ; only the former ratio is insensibly less than the latter : and therefore EG : KM :: BE : AE. 3. Since then the ratio of AD to BD is but in- sensibly less than that of BE to AE, because AD, BD, or BE, and AE are also, in the same man- ner, arithmetically proportional ; and it has been proved, that HL : DF is but insensibly greater than AD : BD the least ; and, that EG : KM, is but insensibly less than BD : AE the greatest ; Kepler supposes it to follow, that the ratio of HL to DF is accurately the same with that of EG to KM. That is, HL : DF :: EG : KM. P2 4. If, 212 ASTRONOMICAL DISCOVERIES 4. If, therefore, the two small arches DF and EG be taken equal, either of them will be a mean proportional between HL and KM. Therefore HL : KM :: (HL 2 : DF 2 ::) AD 2 : BD 2 ; or HL : KM :: (EG 2 : KM 2 : :) BE 2 : AE 2 ; of which ra- tios AD 2 : BD 2 is the least, and BE 2 : AE 2 the greatest. 5. But, as has been seen, AD : BD :: BE (== BD) : AE; and, consequently AD : AE :: AD 2 : BD' :: BE 2 : AE 2 . Therefore HL : KM :: AD : AE. That is, the time of describing a very small arch DF, at the aphelion, is to the time of de- scribing the equal arch EG, at the perihelion, as the aphelion distance AD to the perihelion distance AE ; and, in general, Kepler concluded, though from reasonings less geometrical, that, in all other points of the orbit, as N, O, P, Q, marked by lines drawn from A, the time of describing any small arch, at N for example, is to the time of describing an equal arch, at any other point P, as AN to AP. Of this demonstration the three last steps seern to be wholly unnecessary. For, since HL : DF :: AD : BD; and EG : KM :: (BE =) BD : AE; it follows, when DF is taken = EG, that HL : KM :: AD : AE; and, indeed, it is remarked by Kepler himself, that this conclusion does not follow from the two proportions HL : DF :: EG : KM, and AD : BD :: BD : AE, except by means of a compensation of errors ; the ratio of HL to DF being as much less than the ratio of EG to KM, as the ratio of AD to BD is less than that of BD to AE. An easier, and less exceptionable, demonstra- tion, for the case of small arches at the apsides, will be as follows. Since the two triangles CDF, CHL, may be considered as right-lined and similar, CD : CH :: DF OP KEPLER. DF : HL. Consequently, CD . HL = CH . DF : or, because CD = AE, and CH = BD, we have AE.HL ^ BD.DF. Since also the two triangles CEG, CKM, may be considered as right-lined and similar, CE : CK :: EG : KM. Consequently, CE . KM = CK . EG; or, because CE = AD, and CK = BD, we have AD. KM = BD.EG. If, then, the two small arches DF and EG be taken equal, BD . DF will be = BD . EG ; and consequently AE . HL = AD . KM. Therefore HL : KM :: AD : AE. From this proposition Kepler appears to have de- rived another, which he employs afterwards (148). It is this, that the diurnal motions, at the apsides, are in the inverse duplicate proportion of the dis- tances from the sun. The derivation may be made in this manner. it has been seen that AE . HL = BD . DF, and BD . EG = AD . KM ; and therefore BD : HL :: AE : DF, and BD : KM :: AD : EG. Suppose, then, that the very small arches DF and EG are described in equal times, and the arches HL and KM, which are the measures of these times, will be equal. Consequently, BD : HL :: BD : KM; and, therefore AE : DF :: AD : EG ; and, hence AE . EG = AD . DF. But, since ADF and AEG may be considered as right-angled plane triangles, it is manifest, that DF = AD. tan. DAF, and EG = AE. tan. EAG; and therefore, by substituting these values, AE 2 . tan. EAG = AD 1 , tan. DAF. Therefore AE Z : AD Z :: tan. DAF : tan. EAG. But, since the angles DAF and EAG are very small, the ratio of their tangents will not sensibly differ from the ratio of the angles themselves; and therefore AE* : AD* :: DAF : EAG. P3 It 214 ASTRONOMICAL DISCOVERIES It may justly excite surprize, that Kepler does not appear to have been fully sensible of the im- portant uses which he might have made of this proposition, in many of his future disquisitions. 13Q. Since, then, it appeared, that the times employed by any planet, in describing very small and equal arches of its orbit, were always propor- tional to the distances of these arches from the sun ; or, in other words, that the greater the dis- tance of the planet, the more feebly and slowly is it always impelled forwards, and the more forcibly and rapidly the less the distance ; it was impossible that Kepler should refrain from a variety of specu- lations, concerning the power or virtue which pro- duced such effects, and the source from which it flowed. The whole of these it would be unne- cessary to repeat, and they are only of importance, as they assisted or retarded his progress in his fu- ture investigations. In these speculations, one of the two things con- tained in this alternative seemed evident, that the force, which produced the planetary revolutions, must reside, cither in the sun, or in the planet ; and, as no planet, without some animal activity, could be supposed to change its place ; nor, even though endowed with animation, without some mechanical means or bodily organs, by which the translation might be made ; and since, after T. Brahe's observations on comets had exploded the ancient transparent solid spheres, no probability appeared of its possessing such means or organs ; nothing seemed to be left, except to ascribe these revolutions to some force proceeding from the sun. In confirmation of this opinion, the sun was also seen to be the source of light ; and as light pro- ceeded from him in straight lines, and in all direc- tions, OP KEPLER. 215 tions, and illuminated all bodies, not equally, but according to various circumstances by which it was modified, a probability seemed to arise, that the present force, in like manner, proceeded from the sun ; not producing equal velocities in every planet, nor even uniform velocity in the same planet, but modified also by various circumstances; such as the distance, the resistance of the medium, and the supposed disposition or proneness of the planet to rest, and which he termed its vis inertite. But the example, which Kepler thought most ap- plicable to his subject, was that of a magnet, to which W. Gilbert, in England, had compared the earth, in her relation to the moon ; and by supposing the sun to possess a similar virtue, and to have a constant rotation upon the axis of the ecliptic, a cause seemed to be presented from which the periodical motions of the planets might be derived. For, as a magnet seems not to at- Force tract iron in all its parts, but in those which are njo ... r >? i 1. pla at equal distances from its poles, gives only to the their or- iron a direction parallel to its own fibres ; so bils Kepler conceived, that neither does the sun attract planets placed in the zodiac, but, sending forth magnetical rays or fibres, and whirling continually upon his axis, impels forwards, with these extended fibres, the planets from west to east in the direc- tion of his own rotation. According to this con- ception, the planets in their revolution were not active, but merely passive : nor was it an objec- tion to the theory, but rather a confirmation of it, that the planets were not all carried with the same velocity ; for, as the rotation of the sun, with his magnetical fibres extending in the direction of the zodiac, was the common cause of their revolutions, and the impulse of those fibres could not commu- nicate equal velocity to every planet, but only in P4 the nets m 216 ASTRONOMICAL DISCOVERIES the inverse ratio of the resistance compounded with the distance from the source of power ; like the impulse of a feeble lever, which is the more easily bent, the greater that the distance is of its extremity from ilic fulcrum; so those 'planets, whose mass was greatest, and which from their greater distance were impelled by more feeble and dis- persed rays, would be impelled with least celerity ; and none of them, not even Mercury, though im- pelled most forcibly, would attain in its revolution the same celerity with the rotation of the sun. So far, indeed, did Kepler carry his speculations on this part of the subject, as to attempt to deduce the time of a solar rotation from the period of a planet's revolution ; and, having observed that the ratio of the semi-diameter of the sun to the semi- diameter of the orbit of Mercury, was nearly the same with that of the semi diameter of the earth to the semi-diameter of the lunar orbit, he sup- posed that the ratio of 'he revolutions to the rota- tions might also be the same : and he thence de- duced this consequence, that the time of a solar rotation consisted of nearly three days. 14O. But, though the impulse of magnetical -fibres, extended from the sun, and carried round by his continual rotation, was considered as -the cause of the periodical revolutions of the planets, no explication could be derived from it of the va- riations of their velocities and distances from the, sun ; because the rotation was supposed to be per- formed with perfect equability. It was therefore necessary to suppose also, that the planets were endowed with some intrinsic force, and peculiar energy of their own, by which they could vary their distances from the sun, and consequently their velocities : but, while the prejudice in favour of OP KEPLER. 217 of circular orbits continued, it was hardly possi- ble to form any probable conceptions, either about the nature of this force, or the manner of its ope- ration. By supposing the centre H, (fig. 63), of the Causeof . epicycle GD, to revolve in the circle KH, and in a^o^o? the direction KH, round A its centre, which is velocity, also the centre of the sun, and the planet to re- volve in the circumference of this epicycle in the contrary direction from G to D, a representation may be made of an excentric orbit perfectly cir- cular. For, as was formerly explained (2u), if the * arch KH of the circle described by the centre of the epicycle should be always similar to the arch GD of the epicycle described by the planet, the path of the planet through the heavens would be the excentric circle CDP, whose excentricity AB is equal to HD. But, whatever might be the nature of the in- trinsic force of the planet, no means could be conceived, by which this effect should be pro- duced, which did not involve great absurdities. For, ], it was necessary to suppose that the planet had some propensity, or was somehow excited, to endeavour to extricate itself from the force of the impelling ray, or fibre, AG ; and to ascribe to it a power of moving in a contrary direction, from G to D : and of this no planet, even though en- dowed with animation and intelligence, seemed, without bodily and mechanical organs, to be ca- pable. 2. It this should be supposed possible, and the arches GD and KH should be so described as to be always .similar ; and therefore AH, the ray from the sun, always parallel to BD, the line drawn from the planet to the centre of the orbit; the con- sequence would be, that AH would not be car- ried round A with the invariable and constant velocity 218 ASTRONOMICAL DISCOVERIES velocity of the sun's rotation, but with the incon- stant velocity of the line BD. For it is to be ob- served, that the epicycle GD represents only the optical inequality of the planet, the real inequality must be represented by the variable motion of BD ; and therefore AH would not only be vari- able in its motion, though the rotation of the suri be invariable ; but variable according to the dis- tance AD of the point D from the sun, a point entirely foreign and unconnected with it. Other suppositions were found to involve equal absurdities ; and the last which Kepler made, be- fore his prejudice in favour of circular orbits was removed ; a prejudice which he feelingly laments, as involving him in many vexatious and fruitless * Nocens labours, and robbing him of much precious* time; temporis waS) ft^t tne p] ane t did not at all move in the circumference of the epicycle, but only librated backwards arid forwards in its diameter GHP, (No. 2.) : but, though it did not move in the cir^ cumference, the libration GO, or PO, was deter- mined as if it had moved in it in the manner now described. But besides that no explication could be found of the cause which rendered the motion of H variable, it was evident, that though astro- nomers might calculate the distance AD, by means of the imaginary arch GD of a supposed epicycle, no rule, or mark, could be conceived by which the planet, even if endowed with intelligence and memory, might be enabled to preserve its distance AD, such as the limits of the excentric required ; except the variations which it might perceive and recollect on the apparent solar diameter. 141. Unsatisfactory, however, as all the attempts of Kepler were, to account for the causes by which a planet is determined, and the means by which it is OP KEPLER. is enabled to describe a circular orbit excentric to the sun, since it was undeniable, that the times in which it described very small and equal arches of it, were always as its distances from the sun, he resolved to carry into execution his long conceived design of applying this .principle to the deduction of equations. As his procedure for this purpose was the original of the important and celebrated discovery, that all bodies revolving round the sun describe areas proportional to the times, it de- serves a full and particular explication. His first attempt, for deriving the equations of F a planet from this natural principle, was to divide the excentric orbit into 360 degrees, or equal equations, parts, as if these had been the least possible ; and, supposing that the distance, during the description of any one of these, suffered no variation, he pro- ceeded to calculate for the orbit of the sun or earth, with which alone he was at present con- cerned, the distance of every particular degree or part. (T ) He next summed all these distances, and thence had this analogy, as the sum of the 36o distances of the earth from the sun is to the time of the annual revolution, that is, to 365 d. (5 h. 5 so is the sum of any given number of distances, reckoning them in their order in consequentia from the aphelion, to the time in which the degrees correspondent to this given number of distances are described. As the periodical time 365 d. 6h. did not consist of an integral number of days, so, to facilitate his calculation, he designed it, in the usual manner of astronomers, by the integral num- ber of 3(30 of mean anomaly ; and then his ana- logy was thus expressed, as 36000OOO (supposing the sum of the 30'0 distances to be equal to the sum of 360 semi-diameters) is to 360, so is 3053Q6, the sum of the distances from A of the 3 degrees, for 22O ASTRONOMICAL DISCOVERIES for example, of the excentric DH (fig. 6l), which are next to the aphelion, to 3 I 7 7 /y , the time, or First at- Tnean anomaly, corresponding to these 3 decrees tempt. of the excentric- orbit. The difference 3' 7 /; is the physical equation CHB, produced by the real re- tardation of the earth in this most distant part of the excentric ; and adding to it the optical equa- tion BHA, which may be easily calculated, espe- cially when the distance HA is given, or, in orbits so little excentric as that of the earth, may be taken equal to the physical part, the result for the whole equation CHA will be 6' iV. 142. But, as this method was tedious and me- chanical, and the physical equation could not be obtained for any particular degree of the excentric, without previously calculating and summing the distances of all the preceding degrees of anomaly, it was necessary to think of other methods. Ac- cordingly, it occurred to Kepler, that the whole plane of the excentric might be considered as com- posed of the sum of the distances from A of all the minute equal parts into which its circumference might be divided, and which are here supposed to be 360 ; and he recollected that Archimedes, in attempting the quadrature of the circle, had di- vided it, by lines drawn from the centre, into an indefinite number of triangles. In his next at- tempt therefore, besides dividing the circumference of the excentric into 360 equal parts, he divided its plane or area into 360 parts, by drawing lines not only from the centre, in imitation of Archi- niedes, but also from the excentric point A, to all the points of equal division in the circumference. Second at- Let ABC (fig. 04), be the line of apsides, A the tempi. sun, B the centre of the orbit, and let the semi- circle CFK be divided into any indefinite number of OP KEPLER. 221 of equal parts, in the points D, E, F, &c. and to all these points let straight lines from A and B be drawn. It is evident that the sectors CBD, DBE, EBF, &e. are all equal ; and that the areas CBD, CBE, CBF, &c. will represent the excentric ano- malies as accurately, as the arches CD, CE, CF, &c. or the angles at the centre which these arches of the excentric measure. But as the area of the semi-circle CFK may be considered as composed of the whole sum of the lines drawn from B to the indefinite number of points D, E, F, &c. in its circumference ; and this sum may be also considered as equivalent to the sum of the indefinite number of sectors, into which it is by these lines divided ; so the same things may be observed concerning the lines which are drawn from A. They may likewise be consi- dered as composing the area of the semi-circle CFK, and as equivalent to the sum of the inde- finite number of small areas CAD, DAE, EAF, &c. into which the semi-circle is divided by lines from A. If, therefore, the areas CAD, CAE, CAF, &c. could be exactly calculated, Kepler supposed that they would be equivalent to the required sum of the distances from A of the points of equal di- vision in the arches CD, CE, CF, &c. and con- sequently represent the times in which these arches were described ; and that the areas CAD, CAE, CAF, &c. would, equally with the actual sums of their distances, become the mean anomalies to the excentric anomalies CD, CE, CF, &c. or to the sectors CBD, CBE, CBF, &c. by which they are represented. That is, he concluded this to be a just analogy, " As the semi- circular area CFK is * c to half the periodical time, or 180, so are the " particular areas CAD, CAE, CAF, &c. to the " times in which the arches CD, CE, CF, &c. " described, 222 ASTRONOMICAL DISCOVERIES " described, or to their correspondent mean ano- " malies." Since, then, the sector CBD is the excentric anomaly of the earth in the point D, and the area CAD the mean anomaly; their difference, the area of the triangle DAB, will be the physical equation at this point : and, as the angle ADB is the optical equation, both parts will be obtained by the resolution of the same triangle. 143. The area of the triangle ABD, or the phy- sical equation, is easily determined. When the earth is in F, and the excentric anomaly CBF = QO, this equation becomes the area of the triangle AT-T> U' 1 t T^D A D 100000.1800 AFB; which is = FB.AB = = . 90,000,000. This area will be reduced to parts of mean anomaly by this analogy, as the whole area of the circle, that is, 31,415,926,536, is to 360, so is 90,000,000 to 3713" = 1 1' 53 /y . The mean anomaly therefore, when the earth is in this point of the excentric, will be zr 91 I' 53"; and if the optical equation be calculated by means of its tangent AB, it will be found = 1 1' 52 ;/ . The whole equation therefore corresponding to 91 I 1 53 /; of mean anomaly is 2 1' 45 /; , and the cor- respondent true anomaly, that is, the angle CAF is = 88 58" 8 /; . Or, by merely doubling the phy- sical part, the whole equation will be 2 3' 4& 9 and the true anomaly = 88 58' 7 /; . Calcula-" When the area AFB is found and expressed in tionofthe parts of mean anomaly, the physical equation, at by^ts'prin- an Y ther point D of the excentric, will be deter- ciples. mined by means of the well known proposition, that all triangles which have the same base ore as their altitudes. Through D draw to BE, for ex- ample, the line DM parallel to ABC, the line of apsides; OP KEPLEJR. 223 apsides; and from D, E, M, the lines DN, EO, MP, perpendicular to it, and join MA. The tri- angles DAB, MAB, are equal, and the triangles EAB, MAB, are to each other as their altitudes EO, MP, or DN. That is, EAB : DAB :: sin. EBC : sin DBG. Let the earth therefore be in E, and the excen- tric anomaly EBC = 45. The physical equa- tion, or area BAE, will be found by the analogy, sin. FBC : sin. EBC :: triang. FBA : triang. EBA; that is, sin. QO C : sin. 45 :: 3713 /; : 2(J25" =. EBA = 43' 45". The mean anomaly, therefore, or the area CAE zz CBE -f- BAE zz 4543'45 // ; and calculating the optical equation, or the angle BEA, by means of the formula, tan. 1(BAE BEA) zz tan. |CBE (||-=lA?), it will be found = 43' 14 /; . The whole equation therefore in 45 43' 45" of mean anomaly, is>l 26 X 5g" y and the true anomaly zz 44 16' 46". This then was Kepler's new method for calcu- lating the equations of a planet in an excentric circular orbit ; in which he substituted, for the planet's supposed uniform progress in an imaginary equant, a principle which took place in nature, that the times of describing any given arches of that orbit, are as the sums of the distances of their indefinitely small divisions from the sun ; or, as the areas bounded by these arches, and in- cluded between two lines drawn to the sun from the extremities of each : and this was the original draught of the celebrated law by which the pla- nets are found to be regulated, that in performing their revolutions they describe about the sun areas proportional to the times. It was afterwards found that this law was universal, and that it not only regulated the planets and the other celestial bodies which 224 ASTRONOMICAL DISCOVERIES which revolve about the sun, but also the moon in her revolutions about the earth, and all other secondary planets in their revolutions about their primary ones. As the equations for the solar orbit, calculated on this principle, suited the phenomena with equal, or even greater accuracy, than those derived from the ancient principles, the law, as far as this orbit was concerned, seemed to be suf- ficiently established. tiX>nhis 144 ' But, notwithstanding the justness of the method, principle from which this method of equations was derived, and the accuracy with which these corresponded with the phenomena ; Kepler was well aware that his reasonings in forming it had not been unexceptionable. Besides that the planetary orbits, as he afterwards discovered, were not pre- cisely circular, he saw that the planes, or areas, which he employed, were not accurate measures of those distances, or sums of distances, which he had found to be equivalent to the times or mean anomalies. For, when the area of a circle is di- vided into 360 equal parts, by ISO diameters, since each of these diameters is denoted by 200,000, the sum of the whole must be 36,000,000 ; but, when lines, as AF, AQ; AE, AR; AD, AS ; are drawn from the excentric point A to the extremi- ties of these diameters, they will form with all of them, except the diameter which passes through A, such triangles as AFQ, AER, ADS; and, as the sum of any two sides of a triangle is greater than the third, the whole sum of the lines drawn from A must be greater than the sum of the 18O diameters passing through B; that is, the sum of the distances from A is greater than the area of the circle employed as 'their measure ; and, con- sequently, any particular sum of the distances, as of OP KEPLER. 225 of those belonging to the arches CD, CE, CF, &c. of the excentric, must 'be greater than the area CDA, CEA, CFA, &c. In fact, the areas em- ployed in this method, instead of being equivalent to the sums of the distances from A, were equiva- lent only to the sums of the distances from such points as T and V, where perpendiculars from A meet the various diameters. But, what Kepler counted very remarkable (miraculi loco), it after- wards appeared, when the truth came' to be dis* covered, that the errors committed by this pro- cedure exactly balanced one another : for if, on the one hand, the area CAD, for example, was too small to measure the distances from A of all the points of equal division in the circular arch CD ; that is, to represent the time employed by the earth in describing this circular arch ; those dis- tances, on the other hand, were uniformly greater than the real distances of the earth from the sun ; the real orbit not being a circle, but an el- lipse. (U.) 145. By the foregoing investigations, however, the solar theory, that is, the theory of the annual motion of the earth, which Copernicus justly con- sidered as the cause of the second inequalities of the planets, was carried to a degree of perfection never before attained : and Kepler, having de- monstrated that the bisection of the excentricity took place in the earth's orbit, in the same manner as the latitudes and longitudes, except those at op- position, required it in the orbit of Mars \ having made the corrections, which this principle rendered necessary, on the former erroneous distances be- tween the earth and sun ; having found that the inequalities of the motion of the earth were to be Q, ascribed 226 ASTRONOMICAL DISCOVERIES ascribed to the same common cause which pro- duced the first inequalities of the planets ; and having, from the consideration of this cause, de- rived a new and accurate method of investigating the earth's equations he was enabled to proceed to the theory of the planets, with greater advan- tage than any of his predecessors ; and to view their motions more perfectly separated from every mixture with the solar motions, and unaffected by the errors of the solar theory. CHAP. OF KEPLER. 227 CHAP. VIII. Of the Theory of Mars resumed, and the Application to this Planet of the physical Method of Equations ; together with its important Consequences. 146. TN Kepler's former attempt to investigate a JL theory for Mars, he had endeavoured to deduce all its elements from observed oppositions, after the manner of the ancients; and the only dif- ference was, that he employed real instead of mean oppositions. But though the equations, calculated by the theory then investigated, gave the planet's longitudes with considerable accuracy, the excen- tricity and distances of the planet from the sun were totally irreconcileable with the variations, both of latitude and longitude, produced by the annual motion of the earth. (126.128.) That these distances, therefore, might be determined with greater certainty, he had reversed the ancient or- der of procedure, and had begun his researches with investigating the demerits of the solar theory; that is, the whole peculiarities of the annual mo- tion of the earth : for, while these were unknown, or assumed erroneously, it was impossible that the theory of Mars, which was in a great degree de- pendent on them, could be accurate. As this was now therefore accomplished, it was to be expected that no farther difficulties about the theory of this planet should remain ; and certainly, had its orbit been a precise circle, the expectation would have been fully gratified. For, when the distances of the earth from the sun were correctly given, the Q 2 distance 228 ASTRONOMICAL DISCOVERIES distance of Mars from the sun was also given, by means of the angles of parallax and elongation (72); and, in a circular orbit excentric to the sun, from any three such distances of a planet given, and the angles at the sun given which are included by lines drawn from the planet, the magnitude and position of the orbit with respect to the point oc- cupied by the sun, that is, the excentricity and longitude of the aphelion, may be always readily determined. (P.) Deduction 147. Accordingly, in the investigation of these oftheeie- e l erne nts, three distances of Mars carefully de- Mars, with duced from his parallaxes and elongation, and the thecor- earth's distances calculated by the improved solar reeled dis- . /rr , x i i //~ \^c \ tances of theory (T), were now employed, (fig. 66.) from the Helioc. long. Mars. Dist. of Mars from the sun. 1st. is. 14 16' 52" 147750 = AE. 2d. 5 8 19 4 166225 = AF. 3d. 6 5 24 21 163100 = AG. Let A be the sun, and from A let the lines AE, AF, AG, be drawn of the given lengths, and so as to make with each other the given angles EAF zz 114 2' 12", and FAG = 25 5' 17 /; . It is re- quired to determine the magnitude and position of the circle passing through E, F, G. In the triangle EAF, with AE, AF, and the in- cluded angle, the angle AEF will be found = 35 10' 17"; and from the like data, in the triangle EAG, the angle AEG will be found = 2O 26' 13 /x . Therefore GEF = 14 44' 4", and its double GBF = 2Q 28 ; 8"; and its chord GF = 50868, in parts of the semi-diameter BG 100OOO. Again, in the triangle GAF, the angle AGF will, from like data, be found == 78 44' l", and GF = 77J87, in parts of AG = 1 63 100. There- fore sun. OP KEPLER. 22Q fore AG will be zz 1O7486, in parts of BG = 100000 ; and, since GBF = 29 28' 8", we shall have EGA = 3 28' 5" (= AGF BGF.) Therefore, in the triangle EGA, with the sides AG, BG, and the angle BGA, the angle GAB will be found zz 38 15' 4o y/ ; and, since the po- sition of AG is in 6s. 5 24' 21", that of ABC will be in 4s. 2/ 8' 3&' ; and the excentricity AB will be also found = 9768, in parts of BG zz 100OOO, and BG = 151740, in parts of AG = 163JOO. But how false these conclusions are, will appear by repeating the same operation with a combina- tion of any other three distances of Mars, or even from one other distance combined with any two of the present three ; for the results both for the ex- centricity and the longitude of the aphelion would be found entirely different. As it was thus found impossible to determine these elements from three given longitudes, and three distances of the planet in those points of longitude from the sun, though deduced from observation with the greatest care ; and as these data would have been quite sufficient, if the orbit had been circular; strong cause of sus- picion was given that it was not circular, and that the ancient principle on this subject was less sacred and inviolable than had been supposed. 148. A different course was therefore to be pur- sued, and Kepler, laying aside all preconceived opinions concerning the form of the orbit, now resolved to investigate the aphelion and perihelion distances of the planet from the sun, purely by means of observations. As it was therefore previously known, by the vicarious theory, that the aphelion of Mars was, very nearly at least, in 4s. 28 53', he endea- voured to collect from T. Brahe's register all the Q 3 observations 13O ASTRONOMICAL DISCOVERIES observations of the planet made near to this point of heliocentric longitude : and from one observa- tion in 1585., three about the end of 1586, two in 1588, one in 1590, and two in 1600, he found, that when the planet was in 4s. 29 18' 36" of heliocentric longitude, its geocentric longitudes, the heliocentric longitudes of the earth, and the distances of the earth from the sun, calculated according to his improved solar theory, were these '(fig. 67.) Times. Geoc. long. Mars. d. h. , s. o / // 1585. Feb. 17 10 o 4 15 12 30 1587. Jan. 5 Q31 6 2 8 3O 1588. Nov. 22 Q 11| 6 2 35 40 15QO. Oct. 10 8 35 5 2O 13 30 ISOO. Mar. 6 6 17 3 29 18 3O Long, of the Earth. Distances Earth. s. o / // "5 g 22 37 3 25 21 16 2 10 55 8 26 58 46 .5 26 31 36 991/0 = AD. 98300 = AE. 98355 = AF. 993OO =; AG. 99667 = AH. Let, then, A be the sun, B the centre of the orbit of Mars, C the aphelion, in or near to which the planet is seen from the earth in D, E, F, G, H. The procedure of Kepler to determine AC, was first to assume it zz 166700 ; and with this, and the given angles of elongation CDA, CEA, CFA, CGA, CHA, and the given distances of the earth from the sun, to calculate the parallaxes DCA, EGA, FCA, GCA, HCA : for, if these concurred to place the planet in the same point of heliocen- tric longitude, the assumption for AC was un- doubtedly OP KEPLER. 231 doubtedly just; and if not, new suppositions for Newand A. * i ,1 i i . r 1 more sue- it were to be made, and the calculations of paral- cessful me- lax repeated, till this effect should at last be pro- duced. At present it happened that no new sup- positions for AC were necessary ; since the results for the planet's heliocentric longitude, after cor- rections for the precession, were, from the obser- vation of 15S5atD 1587 at E 1588 at F 4S. 20 18' IQ" 4S. 2Q J9' 21" 4s. 29 20' 40" 1590 at G 1600 at H 4S. 29 20' 30" 4S. 29 28' 44". By the vicarious theory they should have been 4s. 29 I/ 7 O" 4s. 29 18'36" 4s. 2920 X 12" 4s. 29 21 x 48" 4s. 29 29' 51". It follows, therefore, that the assumption for AC \vasjust ; and after the reduction to the orbit was added, the heliocentric latitude at this point being 1 48', it became 166780. The observations of the planet towards the peri- helion P were less numerous : but from one of 1589, one f 159 1 * a d two of 1593, it appeared that the necessary conditions of the investigation were, Times. Geoc. long. Mars. 1589. Nov. Id. l6h. 10' 9s. 20 59' 15" J591. Sept. 39 d. 5h. 42' 0s. 14 20' 30" Io93. Aug. 6d. 5h. 14' lls. id 56' O" Long, of the Earth. Distances. is. 19 13' 56" 98730 = AK. Os. 5 47' 5" 99946 = AL. 10s. 23 26' 13" 101 183 = AM. Q 4 The 232 ASTRONOMICAL DISCOVERIES The assumption for AP was less fortunate than for AC, and it required repeated trials before it was ascertained, that the only supposition which would give nearly the same results for the helio- centric longitude, was 138,430, in the plane of the ecliptic, or 138,500, in the plane of the orbit. These results, after corrections for the precession, were, 158Q at K ISQl at L 15Q3atM 10S. 29 54' 53" 10s. 2Q 56' 30" lOs. 2Q 58 X 6" By the vicarious theory they should have been 10s. 29 52' 56" 10s. 29 54' 31" 10s. 29 56' 1" From these results for the planet's heliocentric longitude, the longitude of the apsides was thus investigated. The heliocentric longitude of Mars, near the perihelion, on the 1st of November, at 6h. 10', was 10s. 20' 54' 53"; and near the aphelion, on the 22d of November, 1588, at 9h. 1 1 1 30", it was 4s. 29 20' 12". The interval was 343d. 21 h. 52' 30", and the time of half a revolution of the pla- net is 343d" J 1 h. 46'. The interval therefore was lOh. 6' 30" greater than the time of half a revo- lution. The arch described in this interval was 180 34' 41", that is, 180 33' 53", after subtract- ing 48" for the precession. If, therefore, this excess of 33' 53' above 18O had been described, after the planet's being in the perihelion, the longitude of the aphelion would have been precisely 4 s. 29 20' 12". But it was demonstrated (138), that the diurnal motions of a planet at the apsides, are very nearly in the inverse duplicate ratio of the distances. That is, the diurnal arch described at the aphelion, is to the diurnal arch described at the perihelion as OF KEPLER. 233 as AP Z to AC% Since, then, the mean diurnal motion 31' 27 " is given, the diurnal motion at the aphelion will be 26' 13", and at the perihelion 3-8' 2". Therefore, since Mars, if he were supposed to depart from the precise point of the aphelion, would, at the end of half his periodical time, be found precisely in the perihelion, it follows that, if we begin to reckon this time 24 hours after his leaving the aphelion, the arch described during the course of it, will begin at a point more ad- vanced in longitude than the aphelion by 26' 13", and terminate at a point more advanced than the perihelion by 38' 2 7/ ; and that it will consist of 1 1 ' 49" more than 180 degrees. But the present interval is longer than half the periodical time by lOh. & 30"; and consequently, whatever may have been the point at which the arch described in it began, it must upon this ac- count exceed 180. If, therefore, we suppose, that half the excess of the interval, that is, 5h. 3' 15", elapsed before the planet reached the aphe- lion, and half after it passed the perihelion ; since the motion in 5h. 3' 15" is 5' l6" about the aphe- lion, and 8' 1" towards the perihelion, the point where the motion begins must be carried forward 5' l6" beyond the place which the planet occu- pied at the beginning of the interval, and will be found in 4s. 29 25' 28 /x ; and the point where it ends must be brought back 8 1 1": That is, the in- tercepted arch must be diminished J3' 17"; and, as it amounted to 180 33' 53" in half the perio- dical time, and lOh. 6' 3O" more ; so now, when this excess of time is taken away, it will consist of 180 20 7 36". Since then it appeared, that when the arch de- scribed in half the periodical time, consisted of 180 234 ASTRONOMICAL DISCOVERIES ton ilucle > of the reckoned 24 hours after the planet had passed the aphelion, aphelion ; it follows, that when the arch consists, as at present, of 180 2O' 36", the time of de- scribing it must begin to be reckoned 41 hours and 54 minutes after the planet was in the aphelion. As therefore the planet, when near the aphelion, describes an arch of 45' 42" in 4lh. 54 X , the aphe- lion must be brought back 45' 42" from 4s. 2Q 25' 28", the point whence the motion began to be reckoned, and will fall in 4s. 28 3Q' 46". By the vicarious theory the longitude of the aphelion, in November 1588, was 4s. 28 5O' 44", differing from the present determination no more than 10' 58". Correction By thus altering the longitude of the aphelion, '" 1588 ' fr m 4S> 28 50/ AA " t0 4S * 28 3 & 46// ' the planet's mean longitude at that epoch also must be altered. For, if at the time when Mars was supposed to be in the aphelion, without any equation, he was in fact 10 X 58" beyond it, an ad- dition of 4 X must be made to his true place, in order to find the mean. When the planet, there- fore, was in 4s. 28 50' 44" of true longitude, the mean longitude was 4s. 28 54 7 44". Excentri- To determine the excentricity of the orbit, cor- cityofthe rec tions of the distances now found would be ne- cessary, if the planet were in any considerable de- gree remote from the apsides; but at present such corrections would be insensible. Since then The aphelion dist. AC is found to be zz 166780, and the perihelion distance AP = 1385OO, a mean of them, that is, the semi-dia- -- meter BC will be = 152640, and the excentricity AB = BC AP = 14140, er in parts of BC = 100000, AB will be = 9264. But OF KEPLER. 235 But the exccntricity of the,equant given by the vicarious theory, was 18564 ; and its half Q282, differs only 18 from the present result. The bi- section therefore takes place in this orbit, in the same manner as in the orbit of the earth ; and, if it were really circular, the equations might be derived from the same natural principles. To confirm this conclusion for the length and position of AC, they were calculated by the method of art. 135, from the observations at D and F, and AC was found ~ 166725, and its longitude in 1587 was 4s. 2Q 19' 4Q", differing only 28" from the result by the present investigation. Nor did the combination of any one of these, with the observation at G, produce any sensible difference. 149. As the planet Mars thus appeared to be governed by the same law which regulated the motions of the earth or sun, and the bisection taking place in this orbit demonstrated, that the times of describing any arches of it, were as the sums of their distances from the sun (138), it was to be expected, that the equations might be derived from this principle with the same ac- curacy as in the solar orbit. It was, indeed, vain to make the attempt, with the uncertain and con- tradictory elements investigated by the common methods (147); but, now that these elements had been determined with so much accuracy by another method, no reason appeared why Kepler should dread any disappointment. Accordingly he proceeded to the calculation of the equations, and in the same manner as in the solar theory considered the circular areas CAD, DAE, (fig. 64), included between two lines drawn to the sun in A, from the extremities of the arches CD, DE, &c. as equivalent to the sum of the distances 136 ASTRONOMICAL DISCOVERIES distances from the sun of all the minute equal parts into which those arches were divided, and consequently to the times in which they were de- scribed. Nor was it possible that the equations of Mars thus calculated, should have been less accurate, if the orbit had been truly circular, than those which he had formerly deduced from the same principle for the sun. Calcula- The excentricity AB, being now found to be tionofthe Q2<)4, is, in Q0 of excentric anomaly CBF, the ofMars 18 ' tan g eilt f tne optical equation AFB to the radius by the new FB = 100000, and this optical equation is there- method. f ore __ 5 o 17' 34". an( | the physical equation is the area AFB = |AB . FB = 463,200*000 ; to which the whole area of the circle bears the same ratio that 36o bears to 5 J8' 28". The mean anomaly therefore is Q5 18' 18", and subtracting from it the whole equation = 1O 36' 'I", the true anomaly, 'or angle CAF, will be . = 84 42' 26". But by the vicarious theory, which is sufficiently accurate as to longitudes, the true anomaly ought to be 84 42 X 2" 5 so that' an error is in this case committed of 24" in excess. Its failure When the excentric anomaly is 45. or 135, the error will be more considerable. For, since R : sin. CBE :: ar. ABF : ar. ABE, this last area will be found zz 3 45' 12"; so that the mean anoma- lies will be 48 45' 12", and 138 45' 12"; and, by resolving the triangles ABE, ABG, the optical equation at the first octant will be found = 3 31' 6", and at the second 4 O x 35". The true ano- malies therefore will be CAE = 41 28' 54", and CAG = 130 59' 25". By the vicarious theory they are 4J 20 ; 33 ;/ , and 131 ?'26 // ; so that there is an error in the first of 8' 21 7/ in excess, and in the second of 8 7 1" in defect; and the planet is represented as moving too rapidly about the OP KEPLER. 237 the apsides, and too slowly towards the mean dis- tances. But, notwithstanding these errors, Kepler was not immediately induced to believe that the an- cient doctrine of precisely circular orbits was false, and merely founded in prejudice ; but was rather inclined to suspect the falsehood of his new me- thod of equations. For, in calculating the equa- tions in the ancient manner, by means of an equant, on {he principle of the bisection, the errors com- mitted, (which were indeed as great, but of a con- trary kind), were considered as evidences of the falseness of Ptolemy's principles, and the present errors might equally be considered as evidences of the falseness of his own. It first occurred to him, that his errors might be ascribed to the substitution of areas for the sums of the distances, and his supposing that these areas were adapted equally with the distances to repre- sent the times. But, on examination, this was found to be impossible. For, after calculating notow - in with the greatest care the excess of the sum of the to its own whole distances in the semi-circle above its area ; that is, the excess of the sum of the distances from the excentric point A, above the sum of the dis- tances from the centre B ; it amounted to no more than 7000 parts of the semi-circular area which contained 18000000; and if this area be equiva- lent to 180, the sum of the excesses must be equivalent to little more than 4' 30"; and these 4 X 3O" are not accumulated at any particular point of the excentric, but diffused over the whole semi- circle. Besides the effect of the substitution of areas for the sums of distances, would chiefly be to represent them as too short in the intermediate parts of the excentric ; as at the point F Q0 from the apsides, where BF = 100OOO, would be sub- * stituted 38 ASTRONOMICAL DISCOVERIES stituted for AF, the secant of the greatest optical equation BF&, and = 100432. The times there- fore would, in consequence of this imperfection, be rendered too short in the intermediate parts, and the planet's motions represented as too rapid : whereas, the effects of the present errors are to make the motions too rapid about the apsides, and too slow in the intermediate parts. It was more evidently impossible, that his errors could arise from his rejection of the double epi- cycle of Copernicus and T. Brahe, (fig. 36,) which represents the planet's path as an oval running out beyond the circle at Q0 of excentric anomaly 246 parts of 1OOOOO ; and the effect of the method of areas, substituted for the distances in this theory, would have been to increase the rapidity much more. (V.) Suspicion 150. But though Kepler was inclined to ascribe bit "as not ^ e errors f m ' s equations to the imperfection of circular, the method of areas from which they were de- duced, the considerations now mentioned rendered this no longer possible, for it was undeniable that this method would have led to errors directly con- trary. No conceivable cause therefore remained to which they could be imputed, except what he had begun to suspect, the form which he and all preceding astronomers had falsely ascribed to the planetary orbits ; and, on examination, convincing evidence was found of the justice of his suspicions; and that the false opinion of orbits precisely cir- cular, notwithstanding all'the authority on which it had been established, was the real and only cause of all his past vexatious labours and disap- pointment. His first proof of the falseness of this opinion, was the circumstance which first gave rise to his suspicions, OP KEPLER. suspicions, that when, on the supposition of its Proof of truth, the longitude of the aphelion, the excentri- ^ e city of the orbit, and its ratio to the solar orbit, j^ were calculated, though by the most legitimate deduction, and from the mo3t unquestionable and accurate distances, they were totally irreconcileable with the determination of the same elements drawn from direct and immediate observation ; and that the results for them, given by any one combina- tion of distances, were generally irreconcileable with those given by any other. Thus the distances AE (fig. 66), in longitude ] s. 14 l6'42"; AF in longitude 5s. 8 19' 4' 7 , and AG in 6s. 5 24' 21", had been carefully deduced from accurate obser- vations; (133, 134, 135.) but when they were sup- posed to be the distances of the planet in the points now mentioned of a circular orbit, the aphelion distance AC, calculated from them, came out = 166562, the perihelion distance AD ~ 136918, the mean distance BD = 15174O, and the excentricity AB = 14822 (147). Whereas, by another investigation (148), from multiplied and equally accurate observations, and independent of all theory about the form of the orbit, they were AC = 166780, AD =: 138500, BD = 152640, and AB = 14140. A proof still more evident, that the orbit could not be circular, and which even gave some indi- cations of its real form, arose from a comparison of the distances AE, AF, AG, deduced from ob- servation, with the results for them calculated in an orbit precisely circular. The method of cal- culation was the same as in the solar theory ; for, in October 15QO, the longitude of the aphelion was 4s. 28 41' 40", and from the heliocentric lon- gitudes of the planet in E, F, G, the angles CAE, CAF, CAG, were therefore given ; and with BE 240 ASTRONOMICAL DISCOVERIES = 1 5264O, and AB = 14140, the distances re- quired for the circular orbit were found, by the resolution of the triangles AFB, AGB, AEB, to be AF = 166605, AG = 163883, AE = 148539; whereas by actual observation they were AF = 166225, AG = 163100, AE = 147750; shorter than the former by 350, 783, 78Q; o that the path of the planet, in moving between the apsides, clearly appears to come within the circle. At F in 5s. 8, where the planet is within 10 of the aphelion, the deficiency is least : but at E, where the distance from the aphelion is 104, and at G, where it is 37, the deficiency is much greater ; and indeed too great, especially at G, no doubt from some inaccuracy, especially of observa- tion. The conclusion therefore was undeniable, that the orbit of the planet is not a circle, but a curve, which coincides with the circle at the ap- sides, and then retires more and more within it till, at Q0 and 270 of excentric anomaly, it comes to its greatest deviation from the circle. Another argument to the same purpose arose from the errors committed in attempting to de- duce the equations for an excentric circular orbit, by the method of areas (149). In that method the area of the circle, though somewhat less than the sum of the distances of the indefinite number of equal parts of its circumference from the excen- tric point which the sun occupied, was supposed to be equivalent to the whole time of the planet's revolution ; and any particular area of it, included by an arch of the circle, and two lines drawn to the same excentric point from its extremities, to be equivalent also to the time in which the planet described OP KEPLER. 241 described that particular arch. Bat if the real path of the planet should not be a circle, but a curve retiring within the circle at all points except the apsides, the errors committed, by employing circular areas to measure the times in which the arches of this curve are described, would be un- avoidable. For, every particular area, bounded in the manner now mentioned, would, every where, except about the apsides, be greater than the sum of the distances of the correspondent arch; and especially towards the points in QO or 270 of ex- centric anomaly ; and therefore it would represent the time as too long, and the motion of the planet while it describes the arch, as too slow. But no such errors can be committed by employing the areas of the curve which the planet really describes; for, as its area is less than that of the circle, and yet measures the same periodical time, the parti- cular portions of this time will, on account of its form, be more properly distributed. The particular areas of the curve, nearly coinciding at the apsides with the circular areas, will there represent a longer time; and, at the distance of Q0 from the apsides, retiring more within the circle, will there represent a shorter time. The errors therefore committed by the circular areas will be remedied, and the planet's motions rendered, as they ought to be, slower at the apsides, and quicker in the interme- diate parts, than by the former calculations. It was not however by this argument that Kep- ler was first led to believe that the orbit was not circular ; nor was it from any hope of escaping the errors produced by employing circular areas, that he was induced to forsake the ancient opinion. On the contrary, so far was he from imagining this to be an evidence of its falsity, that, after a long time spent in vain and perplexing attempts to R reconcile 242 ASTRONOMICAL DISCOVERIES reconcile with a circular orbit the equations given by circular areas, he abandoned altogether his me- thod of areas, and gave up every hope of being able to apply the natural principle on which it was founded to any real use. Nor did he resume this method till he was convinced by other arguments, that the circular form ascribed to the orbit 'was the great cause of his failure ; and especially till he had found the distances in a circular orbit to be totally inconsistent with those deduced from obser- vation. 151. But after Kepler was thus convinced that the ancient circular form ascribed to the orbit was merely a prejudice, a theory which he precipitately formed about the cause which produced its devia- tion from the circle, involved him in equal diffi- culties, and occasioned equal vexation and loss of time. Some account was given (140) of his at- tempts to assign a natural cause of the excentricity of a planet's orbit, and to explain the means by which it was enabled to describe a perfect circle round the sun, though not situated in its centre. One of these consisted in supposing the planet to be possessed of some intrinsic force, by which it endeavoured to extricate itself from the influence of the solar fibres, and to describe an epicycle in a contrary direction ; and the supposed law of the description was, that the line DH, (fig. 63), drawn from the planet to the centre of the epicycle, should be equal to the excentricity AB, and always parallel to the line of apsides ABC; and from thence it followed, that the line DB, drawn from the planet to the centre of the orbit, would be also parallel to AH, a line drawn from the sun to the centre of the epicycle. But though the orbit DC thus described, was evidently a perfect circle, the explication OF KEPLER. 243 explication was rejected ; for, no representation having been made of the physical equation, it was necessary that the motion of AH should be vari- able, even while the distance of H from the sun continued invariable ; and consequently the pla- net's motion in the epicycle would be also variable, and regulated by the laws of the extraneous force impelling AH, which it had no conceivable means, even if it were endowed with intelligence, to know. This theory, however, Kepler now, in some degree, resumed ; and overlooking the first diffi- culty about the inequable motion of the centre H ccrningthe of the epicycle, that is, of the line AH, supposed oval orbit ' the planet to move always equably in the circum- ference of the epicycle, at the rate of the mean velocity of H. By these means he expected that the errors of his equations would be remedied, and the planet brought within the excentric as far as the observations required. For, when it revolved in the epicycle from G towards D, L, always at the mean rate of H ; and the line AH again in passing, for example, from the aphelion C to G, in the first quadrant of anomaly, moved slower than its mean rate ; the planet's place, when AC coincided with AG, would not be in the line HD parallel to ABC, but in a line HL inclined to it, and entering within the circle CD ; and, as the angles GHL and CBD were no longer equal, the arches GL and CD would be no longer similar. Consequently, the distance AL of the planet from the sun, when the centre of the epicycle was in H, would be less than AD, which would have been its distance if GL had been similar to CD ; the greater distance AD would take place in a position of the centre of the epicycle nearer to the aphelion C, and in a less degree of anomaly; and the physical equations, consisting of an accumula- R 2 tion 244 ASTRONOMICAL DISCOVERIES tion of distances, would become more accurate ; for this accumulation would consist both of greater and of more numerous distances towards the aphe- lion, and there measure longer time ; and of more dispersed and shorter distances towards Q0 from the aphelion, and there measure shorter time, and represent a more rapid motion than formerly. In this theory, therefore, the cause which brought the planet within the circle in all the parts between the apsides, was, that in the first semicircle of ano- maly, the intrinsic virtue of the planet antici- pated the solar virtue impelling it forwards ; and made it attain to the distances correspondent to given arches of the excentric, before the solar virtue had impelled the centre of the epicycle to describe these arches ; and that, in the second semicircle, the latter virtue anticipated the former, and impelled the centre of the epicycle to describe the arches, before the intrinsic virtue Tiad made the planet attain to their proper distances. Accord- ingly Kepler tells us, that confiding in this theory, which seemed to promise effects so consonant to observation, and supposing that he could not fail to find a method of reducing it to calculation, he began again to entertain the most sanguine hopes of conquering all the difficulties which the motions of Mars had so long presented ; and was induced to reject the more just hypothesis, to which he had once given the preference ; that the planet did not at all revolve in the epicycle, but only librated upwards and downwards in its diameter. Some account of the vexatious investigations, in which this precipitancy engaged him, will be deemed not unworthy of attention. 152. His first fruitless labour, in the prosecu- tion of his theory, was to describe the curve in which OP KEPLER. 245 which it supposed the planet to revolve. For this purpose he transformed the concentric KHH, with its epicycle GDL, into an excentric CDP, whose excentricity AB was equal to the semi-diameter DH of the epicycle ; and it was well known, and had indeed been formerly demonstrated, that these two forms were perfectly equivalent. According to this representation, it is evident that, if the planet were always impelled forward by equal degrees of the solar virtue, it would move as equably in this excentric, as it is supposed to do by its own intrinsic virtue in the epicycle; and would not only describe round its centre the equal arches CD, DE, EF, &c. (fig. 68), or angles, or even sectors, CBD, DBE, EBF, &c. in equal times ; but also its distances DA, EA, FA, from the sun, at the end of those times, would be the same, both as to magnitude and position, as they would have been after describing, with the same uniformity, similar arches, or angles, or sectors, of the epicycle. But, since it is only the planetary virtue which acts equably, and its effects, in the passage of the planet from the aphelion C to the perihelion K, anticipate the effects of the solar virtue, the planet in the time CBD, for example, will attain to the distance DA before the solar virtue has impelled it forwards through the arch CD ; and since, in passing from the perihelion K the effects of the planetary virtue are anticipated by the solar, the planet will, on the contrary, be impelled forwards through an arch K// = CD, before it attains to the distance Z;A ; and, when it is actually found in the lines DA, ^A, its distances from the sun will be less than the distances of the points D, //. At all the points, therefore, D, E, F, &c. of this excentric, the path of the planet will retire within R 3 it, 246 ASTRONOMICAL DISCOVERIES it, and its distances, at the end of the equal times CD, DE, EF, &c., though the same in magni- tude with the lines DA, EA, FA, &c. will not be the same in position. Attempts As it is therefore only the magnitude of those the oval" e lines, as DA, which is determined by the sup- posed equal action of the planetary virtue, which carries the planet through the arch CD in the time represented by the sector CBD ; and the position of DA depends upon the inequable action of the solar virtue, which requires the time represented by the area CAD to carry the planet through the arch CD; so Kepler first thought of determining its position by this analogy, as 'the area CAD is to the sector CBD, so is the arch CD described by the solar virtue in the time CAD, to the arch CL, which will be described by the same virtue in the time CBD ; that is, since arches of a circle are as their correspondent sectors, by the analogy CAD : CBD :: CBD : CBL. Thus the sector CBD was considered as a mean proportional between the areas CAD and CBL ; and, by converting the area CAD into circular parts, and dividing the an- gle CBD by the line BL, in the ratio of the num- ber of the circular parts in CAD, to the number of circular parts in CBD, he imagined that the arch CL, or the position of the planet when its distance LA was = DA, would be found. But the imperfections of this attempt were many. For, 1. the area CAD is not precisely equal to the sum of the distances from A (149), and therefore does not denote the time of describing CD with perfect accuracy : 2. Though any single distance from A be to the indefinitely small arch corresponding to it, inversely, as the distance from B to its cor- respondent indefinitely small arch, it does not fol- low that this will continue to be the ratio of ac- cumulations OP KEPLER. 247 cumulations of these distances and arches; for if, for example, BD were = 10, and the minute arch corresponding to it = 10', while AD was = 12, the arch which is inversely to the given arch 1O' as AD to BD, would be found by the analogy 32 : JO :: 1O' : 8^, and if AD were =11, the analogy would be 1 1 : 10:: \0 f : QVr ; but 1'2 + 1 1 = 23, is not to 2O as 20' to 8' T + QVr = 17 if, for 23 : 20 :: 20 : 17 T V 3diy. Though CBD were an accurate mean proportional between CAD and CBL, the angle CBL could not be de- termined geometrically, because there was no geo- metrical method of dividing the angle CBD in a given ratio. 4thly. Though the angle CBL were accurately determined, and consequently the cir- cular arch CL, the arch of the curve which the planet really describes will not thereby be deter- mined ; and therefore the position of LA = DA will remain unknown. A second attempt, therefore, was to this pur- pose. Since the sectors CBD, CBE, &c. represent the times in which the planet, moving equably round B, would attain to the distances DA, EA ; and the areas CA A, CA m, of the real orbit which the planet, moving unequably by the impulse of the solar virtue, describes before it attains to the distances A A, m A, respectively equal to DA, EA, also represent the same times ; let the lines LA, MA, be so drawn from the given point A, as to cut off from the plane of the excentric the areas CAL, CAM, respectively equal to CBD, CBE ; or, in other words, so as lo render the space /DL, which is cut off from CBD, equal to /AB, the space added to it : and on A as a centre with the distances DA, EA, describe arches cutting the lines LA, MA, in the points A, m. These will nearly be the points of the orbit to which the planet R 4 will 24S ASTRONOMICAL DISCOVERIES will come in the times CBD, CBE, and where it attains to the required distances. But the defects in point of geometrical accuracy were similar to those of the former method. The areas were not precisely equal to the sums of the distances. There was no known method of dividing a semicircular area in a given ratio by a line drawn from a given excentric point in the diameter : and it was uncer- tain if the differences of the circular areas CAL, CAM, from the areas CAx, CAw, of the orbit, were every where proportional. As no geometrical method, therefore, could be found of describing the curve in which this theory supposed the planet to move, Kepler endeavoured to supply the want of it by the assistance of his vicarious theory. Let A (fig. 69), be the sun, ABE the line of ap- sides, AD the excentricity of the equant = 18564, and AC = 11332 the excentricity of the orbit LGM. In this theory the angle LDG will repre- sent the mean anomaly, and the angle LAG the true, when the planet is in the point G. But the distance AG will be false, because AC, the excen- tricity of the orbit, is false, and its centre C does not bisect the line AD. Let AD, therefore, be bisected in B, and on B with the semi- diameter BE = CL, describe the excentric EHF, in which the planet would, by its own intrinsic virtue, move equably round B, and make the angle EBH ~ LDG: so that, after the time EBH or LDG, it will attain to its just distance AH. But the positron of the planet is not in the line AH, but in some point of the line AG ; because the vicarious theory is, with respect to longitudes, sufficiently exact. If, therefore, on the centre A, with the semi-diameter AH, an arch, HK, be de- scribed meeting AG in K, the line KA will be the required OF KEPLER. 24Q required distance of the planet from the sun, and both its magnitude and position will be sufficiently accurate. These were all the attempts that could be made for describing the curve required by the present theory: and Kepler observes, that in all of them it is an oval, and not an ellipse ; and that the seg- ment, or lunula, CFKP (fig. 68), cut off by it from the plane of the excentric, is broadest to- wards the perihelion. 153. As it was impossible, even after attaining to and to find the description of this oval, to deduce its equations lts( l uadra - by the method of areas, without attaining also to its quadrature, so the quadrature of it was another labour in which Kepler found himself, in conse- quence of his precipitancy, engaged. But this la- bour was equally fruitless with the former ; and the whole result of it was to find, that if the oval should be supposed sensibly equal to an ellipse of the same excentricity, and described upon the same greater axis, the lurmla cut off by it from the area of the semicircle would be but insensibly greater than half the area of a circle, whose dia- meter is equal to the excentricity AB : or, sup- posing the oval to pass through the point X, that BT.TX = AB 2 . Consequently, since BT : AB :: AB : TX, the extreme breadth of the lunula, and AB = Q264, we shall have TX = 858 : and since also BT : TX :: BT 2 : AB 2 ; that is, as the area of the circle described upon BT to the area of the circle described upon AB ; this last area will be found = 2(k)500000, and subtracting it from 3141590000O, the area of the circle described on BT, the remainder will be the area of the oval, and =: 31146400000. (W.) But, though the accurate quadrature of the oval should 250 , ASTRONOMICAL DISCOVERIES should be supposed to be thus attained, it was still impossible to deduce its equations by the method of areas, unless means should be also found of di- viding it in any giv,cn ratio by lines drawn from A. When the orbit was considered as a perfect circle, and the planet in any given point F of excentric anomaly FBC, it was easy, with the given excen- tricity AB, to determine the area FAB, which, added to the given sector FBC, makes up the area FAC, or represents the time in which the planet would describe the arch CF : and it was also easy with the same data to determine the true anomaly, or angle CAF. But because the planet, in v de- scending from C, retires within the circle, it was necessary to determine not only the angle CAF, but also the portion of the oval area intercepted between the lines CA and FA, or at least the por- tion of the lunula intercepted between them, that by subtracting it from CAF, we may find the oval area. This, however, was found impracticable by any geometrical methods. 154. Finding thus no assistance from geometry, it was necessary for him to have recourse to other methods. In one of these, he was freed from the necessity of attempting the particular quadrature of the oval ; for, by continuing to suppose it sen- sibly coincident with a perfect ellipse, he was enabled to derive its areas from the known ratio between it and a circle described on its greater axis as a diameter. For let CPK be an ellipse de- scribed on CK, the diameter of the circle CFK as its greater axis ; let A be the focus of it, which is occupied by the sun ; let BP be the conjugate axis meeting the circle in F, and o m n an ordinate to the greater axis; and let n A, m A, be joined ; it was known that the areas CPK and CFK, or Cm A OF KEPLER. 151 C m A and C n A, were always to each other in the constant ratio of BP to BF; and if therefore the quadrature of the circle was supposed to be at- tained, that of the ellipse was attained also. In this method, therefore, the circular area C n A was found by the usual analogy; as the whole pe- riodical time, or 360, is to the whole area of the excentric, so is the time given since the passage of the planet from the aphelion to the area C n A. But to find the angle CA n of this area at the sun, it was necessary to employ the methods of false position ; and to assume the excentric anomaly CB n such, that when the triangle n AB was added to the sector CBn, their sum should be 'precisely equal to the area C n A. When this was effectu- ated, it was easy to find the true anomaly, or angle CA m\ for, with the angle ABw, #nd the sides n B,BA, the angle CA n is given ; and the tan- gents of the angles CA n and CAtf* to the radius o A, are to each other in the constant ratio of BF toBP. For example, let the mean anomaly be = Q5 c Anattempt IS' IS 1 '. The equivalent area CAF will be found JjJ/"^ by this analogy, 360 : 3141592653d :: Q5 i8'tionTby" 28" : 8317172671 = CAF. Then to find the meansof angle CAF, let CBF be assumed n Q(P, so tbat^w^ the sector CBF will be = J of the circular area, * h e ellipse that is = 7853Q81634, and BF = 100OOO. There- fore the area of the triangle FAB = f AB.BF = 4632 X IOOOOO = 463200OOO ; and this added to the sector CBF, gives, for the area CAF, the sum 8317181634, insensibly greater than the for- mer. It appears, therefore, that the assumption for CBF was nearly accurate ; and since its sine BF is = 100000, and PF the breadth of the lu- nula at this point = 858, the semi-axis minor BP of the ellipse will be = QQ142. Wherefore, since in 252 ASTRONOMICAL DISCOVERIES in the triangle PAB, we have PB : BA :: R : tan. APB, this angle will be found = 5 2O' 18", and consequently the true anomaly CAP = 84 39' 42". By the vicarious theory it is = 84 42' 2 /; . Again, let the mean anomaly be = 48 45' 12"; atid since the whole circular area may be repre- sented by 360, as well as by its superficial amount, the area of the triangle FAB, represented in the same manner, will be = 1C) 108". To find then the angle CAw, let CEn be assumed = 45, and consequently its sine n o =r 70711. The area, therefore, of the triangle n AB, which is to FAB as 7071 J to 100OOO will be = igios" X 70711 = 13512" = 3 45' 12". This, added to the sector CB n = 45, will give the area CA n = 48 45' 12", that is, equal to the given mean anomaly, and shews that the assumption for CB n was just. But BF : BP :: on : om = 7O1O4 : and A o = cos. CB n + AB = 707 1 1 + 9264 = 79975. Therefore, since A o : om :: R : tan. m A o, this angle, that is the true anom. CA m, \vill be found = 41 14' 9". The vicarious theory gives it = 4120 / 33". A comparison of the effects of the different the- ories is as follows : Mean OP KEPLER. 253 o 00 Oco o - CO CO CO CM tO t>. CO CO to c* Oi CO CO 5; 00 o O) 00 00 CO CO CO "f ^ ^f CO "f 00 00 00 00 Tji cS r-t 10 O o - o - CO CO CO 00 CO CO I O i cr 1 v s . o a. 3 W3 p o B 'S i ^- +- C 'o cfl O C "3 ti '- y 0-13 ^o u u 4 ^ o S ^u- ^ o I I I I p, f^ o " 1 . s ' E e^2 3 ! S 2 i *0 1 p c .g ^ ^ 0) O "C c S 2 C3 C C J C ^ 5 g| &C U, S -^ o ~ ?S CO i S o g a. (U CJ ^ > s |J8_ *j *- jz> *- 4: rs >.^ 5^5^ c R t> 5 c 1> Q 3 Q 3 3 ja H H w ca j^ oa *e H ^-S o 0*0 o = o c c ' c a c g> C3 C3 CC c3 +- ^ll^l 1 ! HH H H From 254 ASTRONOMICAL DISCOVERIES From this comparison it appears, that, if the vi- carious theory be considered as the standard, the calculation by the elliptical areas of the present method approaches nearer to the truth than the calculation by circular areas : and, what may per- haps seem surprizing, that with a very small in- crease of excentricity, it would be nearly equiva- lent to the Ptolemaic method. As this, therefore, was found to be incorrect and imperfect, because it made the motions too slow about the apsides, and too quick about the mean distances, it followed that the present method was equally imperfect ; and that it produced errors of a contrary kind to those produced by areas in a circular orbit : though it was yet uncertain whether the errors arose from the theory itself, or from the imperfect manner in which its principles had been expressed in the cal- culations. It likewise appeared, that a mean of the results, from the circular and elliptical areas, approached very nearly to the truth, that is, to the results from the vicarious theory : and, if this cir- cumstance did not first suggest to Kepler the idea of another ellipse, which should every where bisect the space cut off from the circular area by the present ellipse, it certainly at least confirmed him in his intention of introducing it. 155. As the calculations hitherto made, on the principles of the oval theory, had been in many respects deficient as to geometrical accuracy, so that it might be certainly determined whether the errors produced arose from the theory, or from the methods of calculation, Kepler next applied him- self to deduce the equations immediately from the distances in the oval, and to employ the sums of these in place of the elliptical areas. The calcula- tion and summation of these distances was indeed a most OP KEPLER. 255 a most laborious business; but the errors, what- ever they might be, which arose from the areas not being precise measures of the corresponding dis- tances ; from the ratio of any sum of distances to the sum of the corresponding arches not continu- ing the same with the ratio of any particular dis- tance to its particular arch ; from the impossibility of obtaining a geometrical quadrature of the areas; and from the differences between circular and oval sectors; would/ by means of this labour, be avoided (152). In calculating the distances, he employed the An attempt excentric circle CHK (fig. 70), in which the pla- b ^ actual summation net, moving equably, as the theory supposed it to of thcdis- do in the epicycle, would describe equal arches tanc - CD, or angles CBD, about the centre B in equal times : and dividing the circumference into 360 equal part?, as being the least possible, he calcu- lated for the excentricity AB = Ql65, the distance of every one of these 360 divisions from the sun in A : and, lest any error should arise from any vari- ation of distance in the time of describing a single division, he took a mean of the two distances by which every particular division was terminated. Thus the distances, as DA, of the planet from the sun, at all the particular points of equal division in the excentric, or at all the particular times marked by these points, were accurately determined as to length ; and they were also the distances of the planet, at the same particular times, in the various points of its oval orbit from the sun : but their po- sitions were not just ; for the planet does not in- fact describe equal angles, CBD, in equal times about the centre B, but, in consequence of the solar force acting inversely as the distance, de- scribes in equal times unequal angles about this centre. Those distances, however, were the real distances 256 ASTRONOMICAL DISCOVERIES distances of the planet, and their sum, therefore, was equivalent to the time of describing the whole oval, and found to be = 36075562. Since, then, the arches described by the planet are inversely as the distances, to find the length of the oval arch corresponding to every distance, Kepler had this analogy; as the whole oval circumference is to the whole sum of distances, so is the distance of every particular arch to the arch itself. Thus supposing CD to be an arch of the excentric = 1, the dis- tance of the planet from the sun at the end of the time CD will be DA ; and as the whole length of the oval circumference is to 36075562, so is DA to the oval arch CE : and if, on the centre A, with the distance DA, an arch DE were described, and on the centre C, with the distance CE considered as a straight line, another arch were described cut- ting the former in E, this would be nearly the point of the orbit in which the planet would be found after the time CBD. In the very same manner the length of the next oval arch, advanc- ing from the point E, may be also found ; arid the position of its distance with respect to the distance AE. But to determine the position of AE in a more accurate manner, that is the angle CAE of true anomaly, it was necessary to find the angle CBE, at the centre of the orbit; and this is not mea- sured by the known oval arch CE, but by the un- known arch CG of the excentric, greater than CE. But, since CG and CE, when viewed from B, ap- pear to be equal, let them, for a first approxima- tion, be supposed actually eqaal ; and also, as if the very small arch GF were insensible, let each of them be supposed = CF. On these suppositions, the angle FBC, measured by CF = CE, is given : and therefore, with its supplement FBA ; and the sides OP KEPLER. 257 sides FB, BA, the base AF of the triangle ABF may be found; and therefore EF zr: AF AC. But EF is almost accurately equal to EG, the quantity by which the oval arch CE falls wit hi a the excentric at its extremity E. By bisecting, therefore, the line EG, the quantity will be also found by which the whole arch CE would fall within the cxcentric, if the approach to B of all its parts were uniform. The visible magnitude* there- fore, of CE, viewed from B, will b? given, and the angle CB, which was assumed less than the truth, and zr CBF, will be corrected. With this corrected angle, therefore, and the sides AE, A3, the true anomaly CAE was finally determined. By this method of procedure, no equation, except for the first degree of anomaly, could be separately calculated, and all the rest, to the ISOth degree, required the equation immediately preceding to be found. But this was not its only defect; for the foundation of the whole was wanting; that is, the length of the whole oval circumference was un- known : nor could Kepler find any means to de- termine it, except the indirect methods of false position, in which it was assumed, and the assump- tion verified by adding together, after the 180th operation, all its particular arches CE, or rather angles CAE, in the order in which they had been calculated : for if the sum came out greater or less than ISO , the assumption was evidently liilse. In fact, it was not till after many most laborious and obstinate calculations of this kind, that the whole length of the semi-oval circumference was deter- mined to be = 179 14' 15"; that is, 45' 45 7/ shorter than the circumference CHK. In this method, laborious as it was, Kepler cal- culated the equations for every point of the excen- tric, that is, according to the present representa- S lion, 158 ASTRONOMICAL DISCOVERIES tion, in which the motion was supposed to be equable round the centre B, for every degree of mean anomaly, no less than three several times. First, with an excentricity = 9165, and conse- quently too small ; and with the length of the semi-oval assumed too great: for it was supposed to exceed 180, and consequently the sum of the angles CAE came out also greater than 180. Se- condly, with the same excentricity, and the semi- oval assumed = 179 14' J5": and as the equa- tions in the first quadrant, and especially at 90 of mean anomaly, came out too small, it appeared that the excentricity was too small. Thirdly, with the same supposition for the length of the semi- oval, and the excentricity == 9230: and, after the calculation of 180 true anomalies, the results were Mean anom. True anom. By the vicarious theory. Differences. 45 90 235 38 /( l 24 79 26 49 126 56 25 38 4 54 79 27 41 126 52 / // 2 30 ; O 52 . 4 25 +. The planet, therefore, still moved too slowly about the apsides, and too rapidly about the mean distances: and any incrense of excentricity would have increased the errors. But, as the equations approached nearer to those of the vicarious theory than any given by the former calculations, Kepler considered them as confirmations of his present hypothesis; and began to flatter himself again with the expectation, that, the more exactly its prin- ciples were expressed, the effects would always ap- proach so much nearer to the truth. (X.) 156. This method, however, was so vexatious, and its foundations so unsettled, that it was im- possible OP KEPLER. 259 possible for Kepler to continue satisfied with it. The argumentation employed in it was of that vi- ciou* kind, which is called reasoning in a circle: for the length cf the oval which the planet was supposed to describe, could not be ascertained without determining the deviation of all its points from the excentric, and the measures of this devia- tion could not be determined without ascertaining the length of the oval circumference. It seemed therefore to follow that, either a more legitimate method of calculation should be found, or that the theory should be abandoned, as involving an ab- surdity unexampled in any other law of nature. There was also reason to suspect, that even in this method, the theory had not been justly represent- ed. The oval which the planet, in consequence of the solar force, and its own intrinsic force, was supposed to describe, had been divided into un- equal portions, each of which was inversely as the time, or distance from the sun. But, according to the theory, it is only the solar force which acts in the inverse ratio of the distances; and the pla- net's intrinsic force is invariable in every distance. There seemed, therefore, to be some reason to doubt, whether the portions of the oval orbit, which is the joint production of both forces, were accurately measured. Accordingly Kepler, laying wholly aside the oval orbit produced by the com- position of two forces, and all considerations of its quadrature, resolved to return to the original principles of his theorv, and to deduce his equa- tions immediately from these. Let A (fig. 03), be the centre of the sun, and from it, with the distance AK, describe the circle KH, in which the centre of the t picycle is sup- posed to revolve, and with the distance AC an- other circle OG, in which its aphelion revolves. S 2 According -ASTRONOMICAL DISCOVERIES According to the principles of the theory, while the centre of the epicycle passes from K to H, or its aphelion from C to G, the planet moves equably in its circumference, from G to L, describing angles, GHL, proportional to the times; and the line AH or AG moves uncquably, describing an- gles, KAH, which are inversely as the distances AL of the planet from the sun. It is therefore required to find these distances AL, and angles KAH, and thence the true anomalies CAL. Pednction Since the planet moves equably in the epicycle, of the the mean anomalies may be represented by the S"heo"L an S lcS GHL > r the arcbCB GL f ltS circumfe - ginai form rencc, divided into 3f)0 degrees, as its smallest equal of the divisions; and, therefore, from those angles, with, the semi-diameter AH m 10UOOO, and the semi- diameter HL of ihe epicycle = tj r 2(34, the distance* AL, and angles HAL, corresponding to every complete degree of mean anomaly, may easily be found. When these whole distances are calculated, their sum will be nearly equal to 3(3075562, as in the former method, and the 36oth part of this sum is 10O21O. Every particular angle KAH, therefore, will be found by this analogy, as every particular distance AL, in its order, is to the mean distance 100210, so -is l c or 60', to the arch KH, or angle Kx\H, corresponding to the given dis- tance AL. Thus all the different angles KAH will be found, which are described by the centre of the epicycle ; and if, beginning with the first, we add to it the second, and to their sum a third, and a fourth, till the whole 180 are summed, we shall have a table of all the mean anomalies, as GHL, accompanied with all their ^corresponding distances AL, and angles HAL and KAH ; and, therefore, by subtracting from every angle KAH, its correspondent angle HAL, we shall also have the OP KEPLER. 261 the true anomaly KAL. Thus, if GIIL were = 45, this table would shew that the angle KAH corresponding to it was = 41 27' O", and HAL = 3 30' 17"; so that KAL = KAH HAL = 37 56" 43". The effects of this method in the following mean anomalies were Mer.n anom. True anora. Vicarious theory. Differences. 45 37 56' 43" 38 4' 54" 8' 1 l ;/ . 90 79 2(5 35 79 17 O O 25 . J20 l!0 28 8 110 J8 3O 38 +. 15O 1-14 16 4() 144 8 O 8 4Q +. It clearly appeared, therefore, that though the principles of the theory were now fully introduced and fairly expressed in the calculation, the errors became greater than ever; and the planet rendered still slower about the apsides, and quicker than before, about the mean distances. The failure therefore of the calculation, by the method of art. J54, was not so much owing to its imperfec- tions in point of geometrical accuracy, or to mis- representation of the theory, as to the falsity of its principles. (Y.) 157- But a more decisive evidence of the im- perfection, or indeed the falsity, of the theorv, arose from :>n accurate investigation of the planet"* - actual distances from the sun, as given by obser- vation in all the various degrees of anomaly, and a comparison of them with the distances cal- culated for the present oval orbit. The investi- gation of the distances obtained by observation was made for both semicircles of anomaly, and for the corresponding points of both : and as by a like investigation full evidence was given that the orbit was not circular (150), equal evidence was now given that neither did the planet so move in 3 3 i %2 ASTRONOMICAL DISCOVERIES epicycle, as to describe the oval orbit supposed to be produced, by the combination of its own uni- form motion with the unequable motion arising from the solar impulse. As the consequence of this investigation was so important, some examples of it are justly entitled to our attention. Distances To find the distance, for example, in the &7th given hy ^ ec r ree o f mean anomaly, he employed two obser- obstrva- v . r i ^ i / / " t uon. vations; one of the Oth of May 1580, at 11 h. 2O , when the planet's observed longitude was = 6s. 27 T 2O", the sun's apparent longitude = Is. 25 48' 40", and his distance from the earth =: 101365. In this observation, also, the planet's Hiean heliocentric longitude was 7s. 26 0' 36", and therefore its true heliocentric longitude 7s. 15 32' 33", by the vicarious theory. But, as this theory did not, in J58Q, represent the longitude in the opposition of the preceding April, with greater certainty than within 1' 12" (125), he added to this observation another, made on the 27th of December 15CM, at IQh. 15', when the mean heliocentric longitude was 7s. 20 13' 3y /x , the observed longitude 8s. 8 46' 3O", the sun's longitude gs. 16 4/ y JO'', and his distance from thcTeanh 08232. Let A (fig. 72), be the sun, B the place of the cjfrth, at the time of the first observation in 7s. 25 48' 40", and D its place at the second in 3s. 16 47 X JO"; and though the planet is not in both observations at the same precise point of he- liocentric longitude, let C represent its position in both. Since, then, in the iirst observation, the elongation ABC is = 151 18 X 40", and AB = 1()I3()5. if we shall assume AC = J54'2O0 3 the parallax ACB will be found = 1 8 23 y 43" ; and consequently AC in 7s. 15 3J ; 3 X/ of heliocentric longitude. But, if in this first observation AC be assumed OP KEPLER. 263 assumed = 15420O, it must in the second be assumed 32 parts shorter ; for in this part of the orbit a difference of one degree in longitude varies the distance 24O parts, and the present difference is 13', or, after subtracting the precession, 8'. Therefore, if with AC = 154168, AD ~ 98232, and the elongation ADC = 38 D'-IC/, we calcu- late the parallax -ACD, it will be found = 23 6 21"; and, consequently, AC in 7s. 154O'9" of heliocentric longitude ; that is, its position at the second observation, will differ 9' fi" from its posi- tion in the first. But the difference ought to be somewhat greater ; for the difference of the mean longitudes is ]3'3", including the precession, and therefore the true longitude ought to differ 12'37 /; ; and consequently AC, in the second observation, ought to have been in 7s. 15 43' 40". This Kepler found would be effected by rseuming AC = 154400 in the first observation, and == Io4368 in the second ; and these therefore he concluded to be their lengths, the first in 8/ 9' 24'' of mean anomaly, and the second in 87 }& 30 7 . No re- duction of these distances to the plane of the orbit was thought necessary, because the latitude in the first observation was only 6' 4O" north, and in the second 3' 31" also north. Corrections for refrac- tion had been applied both to the observed longi- tudes and latitudes, but the diurnal parallaxes were considered as insensible. For the distance AF, towards the correspondent point of the opposite semi-circle, he employed an observation of the 17th of December J 595, at 7h. 6', when the planet was seen from E the earth to be in 1 s. 1 1 31' 27 ;/ , with 1 4O 7 44" of north latitude. The mean heliocentric longitude was 2s. 2 4' 22 v , and the aphelion being in 4s. 23 58' 10", and consequently the mean anomaly S4 =3 264 ASTRONOMICAL DISCOVERIES = 273 6' 12"; the true heliocentric longitude, by the vicarious theory, was '2s. 12 32' 1 22 / . The fun's longitude was Qs 53C) / 3", and his distance AE = 98200. The parallax AFE' therefore was zr 3iO' 55", and the elongation AEF = 125 52' 2*1"; and from these angles, and the distance AE, the planet's distance AF in the plane of the ecliptic came out = 154432. From this 60 parts arc to be subtracted, because the planet was. ID' 36 /x nearer the aphelion than in the observation of 158p ; but, on the contrary, 15.5 parts are to be added as the reduction to the plane of the orbit, because the latitude was considerable. The dis- tance therefoie of the planet from the sun, when the complement of the mean anomaly is 87 9' 24" was found 'to be = 154387, differing only 13 from AC = 15440O. Such also was Kepler's attention, that though this observation might have been safely confided in, because the he'iocentric longitudes of Mars, given by the vicarious theory for the year 1595, were very nearly accurate (125), he chose to employ another, made on the 29th of October, 1597, at I/ h. from which the distance AF in ihe same point of the orbit was found to be = 154272, or more probably, supposing an error of 3' to have been committed in the observed longitude, = 154387, that is, precisely equal to the former. This error of observation may be supposed to be very probable, for T. Brahe, after his expulsion from Denmark, had not yet obtained a fixed place of residence. In the same manner, at 70 55' ;/ of mean anomaly, the distance of the planet from the sun was by one observation found to be 156090, and by another 1 5SH 1 ; and in the correspondent point of the opposite semi-circle, to wit, at 28p Q 5 Q" of mean anomaly, it was 158217. At OP KEPLER. 265 At 43 13' Si' 7 of mean anomaly, it was found, by a single observation, to be, 163100; and at 3 16 36' '29", it was by one observation = 1 63051, and by another = 162996. At 11 37' O", it was by one observation = 166 180, and by another, = 165208 ; and at 348 23' O", it was by one observation = 1 66230. In the same manner also, towards the perihelion, it was found to be = 147820, or = 14770O, in 1 13 of mean anomaly, and = 147443 or 14/75O in 247. Finally, in 162 and 198 of mean anomaly, it was between 138Q54 and I3QOOO. (Z.) A comparative view of these observed distances, and the distances calculated for the same degrees of anomaly, in an oval orbit, where the mean dis- tance was 152350, and the escentricity = 14115, is as follows : Mean anom. Observed dist. 11 37 O 166194 ^ 43 23 31 16310O 7O 55 15810O Calculated* 87 9 24 1 13 O 162 o o 154387 147/6O 138954 166179 162895 15/53O Compl. mean anom. 11 37 x Observed dist. 16623O 153697 146928 43 23 31 163028 138991 70 55 158217 87 9 1* 154400 J 13 O 147591 162 o o 1 3QOOO J It seemed, therefore, to be clearly established, that the orbit of the planet, in all points except the apsides, went beyond the oval assigned to it by the present 266 ASTRONOMICAL DISCOVERIES present theory and that, towards the mean dis- tances, the difference exceeded 600 parts. ] 58< Before the Distances of Mars from the sun, bits ppss found hy this laborious investigation, were employ- thec!t e< ^ ^7 Kepler for his present purpose of shewing f the sun. that the planet's real orbit extended beyond the oval, in all points except the apsides, he applies it to demonstrate the propriety of the innovation which he had originally intended to introduce into astronomy (}O(J). He had conceived, that the plane of every planetary orbit, and the line of its apsides, ought to pass through the centre of the sun, and not through the centre of the ecliptic: and he was persuaded that, till this substitution should be adopted, it was vain to expect any im- portant improvement of the science. The argu- ments hitherto adduced in support of this opinion were imperfect, and not fully conclusive ; but the equality of the planet's distances from the sun, which he had now so laboriously established, in so many corresponding points of both semi-circles of anomaly, supplied him with a strict demonstration of it, and enables us to rank it among his greatest and most important discoveries. Let A (fig. 73,*) be the centre of the sun, AG the line of the apsides of Mars, DGE his excentric described on the centre C 3 G its aphelion in 4s. 29, AC the excentricity = 14 J 40 in parts of which the semi-diameter of the earth's orbit con- tains 100000, F the centre of the equant, GFE the mean anomaly when the planet is in E, equal to GFD, the complement of the mean anomaly when the planet is in D. Suppose each of those angles = 87 g' 24"; so that, if the lines AE, AD, be drawn, each of them, as has been now investi- * In fig. 73, F FD ought to be joined. gated, OP KEPLER. 2()7 gated, will be = 1 54400. Let also B be the centre of the earth's orbit, and H of her eqnant; OABN her line of apsides, where the aphelion N is in C)s 5 3O', and AB the excentricity = 180O. Join also BE. BD. Since N is in ()s 5 Q 30', and E in 7s l.o 30' (J57), the angle EAB will be = 50'--, and AEB being a very small angle, the angle EBA will be obtuse; and consequently EA longer than E13. In like manner, since D is in 2s. 12 3O' (157), a"d O the earth's perihelion in 3s. 5 3O', the angle GAD will be = 23, and its supplement DAB = 157, anc l consequently BD longer than AD, or AE. Therefore BD is much longer than BE; that is, the lines drawn from the centre of the earth's orbit to any two correspondent points of the opposite semi-circles are unequal. It is even in general manifest, that there are no points within the orbit, except those which lie ia the line GA passing through the centre of the sun, from which equal straight lines can possibly be drawn to any two such correspondent points. It is true that by joining BC, and producing it to L, we may suppose this to be a new line of ap- sides, and it may be said, that since D is Jess re- mote than E from its aphelion L, it is not to be wondered at that BD should be longer than BE. But whatever lines of this kind may be drawn, it has been proved by the present investigation^ 1 57), that the lines AE, AD, must .ilvvays continue equal. Now it was before proved (see note on Article 3 06), that no new line of apsides, drawn through the centre of the orbit described on C, could, equally with AG, suit the observations in oppo- sition: and if, to suit these, a third line HKF of apsides should he drawn through H and F, the centres of uniform motion, and the centre of the orbit should be thereby displaced from C to K, the ASTRONOMICAL DISCOVERIES the orbit would also be displaced, and come into the situation MPQ. ; and consequently the lines AEj AD, terminating in it, could no longer con- tinue equal, as the observations out of opposition have been now found to require. It is also to be considered, that the orbit of Mars was proved (150) to retire within the excen- tric in all points between the apsides ; and it has now been proved (157), that any two points equally remote from either apsis fall equally within it. The orbit, therefore, must be described on GFCAR as its greater axis, and not on MFKHP : and if the distances from the point H should be calculated, they would be found altogether irregular, and in- capable of being bounded by any known curve. To complete the evidence of a fact so impor- tant, Kepler afterwards added two other proofs equally decisive. 1. That the places of the nodes were found to lie precisely in a line passing thro' the centre of the sun ; whereas, if they had been situated in a line passing through the centre of the ecliptic, the longitude of the ascending node of Mars must have been 1 ]' 33" less, and that of the descending node 1 l' 33 ff greater than those given by observation: and 2, That the in- clination of the orbits could not otherwise be in- variable, as had been before demonstrated (117); for had their intersection taken place in the centre of the ecliptic, the heliocentric latitude at thd northern limit must have been two minutes less, and at the southern two minutes greater than it was actually found. It was thus undeniably ascertained, that the motion of every planet is in one invariable plane passing through the centre of the sun: and the discovery was of greater importance to the im- provement OF KEPLER. provement and simplification of astronomy than any which had ever before been introduced, ex- cept, perhaps, the Copernican system alone. 159. The undoubted conclusion from the in- vestigations of Article 157 was, that the real path of the planet went beyond the oval which it was erroneously supposed to describe : but Kepler justly thought that a conclusion so important ought not to rest on any single method of proof, and he therefore employed another equally deci- sive. It was drawn from observations of the pla- net before and after its oppositions, when the dis- tances of the earth from the line of syzigy were equal. Let A (fig. 7-1) be the sun. B the earth before o the opposition, and ABD the angle of elongation ; and C the earth after the opposition, and A CD gated by the angle of elongation ; and let AD be the line ano( >[ of syzigy. The planet therefore, at the first ob- servation, will be seen in some point G of the line BD ; and at the second, in some point as H of the line CD ; and from the given interval of the observations, the angle GAH, described by the planet, will be found with sufficient accuracy by means of the vicarious theory. The distances also of the planet from the sun, in the degrees of ano- maly given by the vicarious theory, will be also found by the usual methods of calculation, so as not to err widely from the truth ; at least, if the interval be not too great, and especially, if the op- position has happened, either near the apsides, or at the points Q0 remote from them, the differ- cnces of the distances will be nearly found ; nor will the form of the orbit, whether circular or oval, make, any considerable alteration on them. ASTRONOMICAL DISCOVERIES If, therefore, with these distances AG, AH, we resolve the triangles ABG, ACH, in which the angles of elongation are given by observation, and the sides AB, AC, by the solar theory, in order to find the nngles of commutation BAG, CAH, and thence the angle G\H ; this angle, if AG, AH, be too long, will come out less than the amount of it given by the interval of the observations ; and, if AG, AH, be too short, it will come out greater ; that is, if for AG, AH, we have used AE, AF, the angle will be EAF instead of GAH; and, if we -have used AK, AL, it will be KAL. We shall also discover, by resolving the triangles ABG, ACH, with the given angles BGA, CHA, of parallax, instead of the sides AG, AH, whether any error has been committed by the vicarious theory, as to the positions of AG, AH ; for, if AG has been erroneously advanced to AE, and AH to.AL, we shall find for AG a distance AE too long, and for AH a distance AL too short. Accordingly, the observations before and after the opposition on the 28th of December, 1582, were these OP KEPLER. 271 (M x- 2 2 ^ O & co OiO^cOfo GO m H co co V O co rf3 CO ^ C< O Q ^ o * ^^^ * ^-4 a CO . O x CO x O CO CO O O O co c^ ^ ^H CM r-i O CO Oi f O -51 O m c^J co 'x 1 (M O co co co Oi ^O ^O CO f-i CO x 7; D vl 9 T* CO 3 O CO s W3 i 3 O 3 o o V^ .. G g a ? c c ^^J^.2 fcJOC3c3^G 8 8js'i3'S S .a jS ** OO^SSQC ASTRONOMICAL DISCOVERIES The difference of the two distances AE, AF, in the second and third observations, is = 424O, and AF, though nearest to the aphelion, is the shortest. From the previous calculation this dif- ference ought to have been only 336. Taking therefore a mean of both, we shall have AE = 322054 536 -L' , . 32204 + 336 - - - = 100859 * and AF = -- ^ - == l6l!Q5 ; and with these resolving the triangles ABE, ACF, we shall find AE in 3s. 16 5', and AF in 3 ?. l/ 55', each less advanced by about }' 3O' than according to the vicarious theory. They seem therefore to be" nearly the just dis- tances ; though, on account of the smallness of the angles at E and F, they are not to be depend- ed upon. Greater confidence may be placed on the results from the, first and fourth observations. The dif- ference between AG and AH is = 5236. But by the previous calculation it ought to be about 55/O. Therefore AG = M** 5 *- 55 _ 155790; and resolving the triangles ABG, ACH, AG will be found m 3s 41', and AH in 4s. 8 X 30 ;/ ; each less advanced also by about 3 'than according to the vicarious theory. The results therefore for AG and All seem to be almost accurate, espe- cially as, by the observations of four successive days about the opposition, the planet's longitude in it given by the vicarious theory appeared to be I 7 30' too great ; and they confirm the results from the two other observations, because the angles at G and H are so considerable, that an error of I' in observing could not produce an error of above 50 parts on either distance* ' Kepler OF KEPLfifi. $73 Kepler investigated, in the same manner, no less than 23 distances, in different points of the orbit ; and for the verification of (hem, he em- ployed them conversely in deducing the geocentric longitudes ; and the differences found between the longitude thus calculated, and the longitude ob- served, were in general very inconsiderable. Only \vhcn the planet was in Cancer, they fell 4 minutes short of the observed longitudes, and in the oppo- site part of the orbit as much exceeded them. But it was impossible that those errors should have proceeded from false distances; for the effects pro- duecd by this cause could not have been the same on both sides of an opposition, but would have been contrary ; and they were more probably sup- po-ed to arise from an error in the position of the aphelion. (Z.) 160. By these two different methods were the distances of Mars from the sun investigated in no less than forty different points of the orbit : but before they should be compared with the distances given by the oval theory, Kepler thought it neces- sary to employ them in a new investigation of the ratio of the orbits. This he had formerly at- tempted by observations near the apsides ; but he now proposed to add the testimony of some of the present distances. It appeared from five observations, that when Newtnvet- Mars was in 11 37' of mean anomaly, or more ^ra^tf accurately in 1 J 52', according to a correction of the orbits. 15' found necessary on the vicarious theory, his distance from the sun in the plane of the ecliptic was J66j80, or 160208 (157); and therefore, as this point was 23 distant from the northern limit, the distance in the plane of the orbit must bs ld(J25O, or 166273. It also appeared, that when T 274 ASTKONOMICAL DISCOVERIES the complement of the mean anomaly was 1 ()' 4l", or more correctly, Q 54' 4 1/ 7 , the distance, was 166311, or subtracting 1' 3O /; from thejongi- tude given by the vicarious theory, 1662O8 ; and, consequently, by an addition of 1 57' ig" to the complement of mean anomaly, so as to make it 11 52', the distance will be diminished, and be- come =. 166113, or in the plane of the orbit = 166193. It was likewise found, that when the complement of the mean anomaly was 8 2' 2i /A , or more accurately 7 47 x 5 l", the distance in the plane, of the ecliptic was 166396; and therefore at a point 4 4' g" more remote from the aphelion, it will be = 166204, and in the plane of the or- bit 166284. A mean of the results then, when the complement of mean anomaly was 1 i 52', 13 166238; and a mean of the results in art, 13o> probably still more accurate, is 166260. Since, then, at 11 52 A of mean anomaly, or of its complement, the distance is 166260, it is evident that, whatever be the form of the orbit, the excess of the aphelion distance above l6626o>. cannot be greater than 250 ; and, if the orbit be circular, it will be less. The aphelion distance therefore, by the present investigation, is 16651Q. In the investigation of art. 148, it was 166780,, but from less accurate observations. The perihelion .distance again was found in art. 148, to be = 13S500. But, in art. 157, it was found that the distance, when the complement of mean anomaly was 160 45' 30 ^ or more cor- rectly, l6o 30' 3O /X , came out = 138984 in the plane of the ecliptic, or 139000. Suppose it then = 139000, when the planet is in 11s. 21 of he- liocentric longitude; and consequently, since this point is 35 beyond the limit, the distance in the jpiane of the orbit will be =? 1 39019. The excess of OP KEPLER. 2/5 of this above the perihelion distance cannot be greater than 876, and if the orbit were circular it would be less. The perihelion distance therefore will be ^ 138173, less than 1385OO by 327. - Since, then, the aphelion distance now found is = - 16651O, and the perihelion distance = 138173, half their sum, or the semi-diameter of the orbit, will be = ' 152342 ; and the excentricity = 14169, or. in 'parts of the semi-diameter of thq earth's orbit = 10000O, the excentricity will be = - 9301. But, as an excentricity so great did not agree with the other observations ; and as Kepler had .found, by the greatest variety of trials, that Q2(35, the excentricity of art. 143, was the most probable, and gave the most accurate physical equation?, he adapted this to the. present aphelion distance, in order to discover another perihelion distance, by means of the analogy 100000 + 9260 : 1OOOOO C)265 :: 16651O : 138274 ; whence the serni- diameter of the orbit came out = 1524OO. But, as too much confidence might thus seem to be placed on the aphelion distance = 1660 10, and too little on the perihelion distance = 138173, the conclusions which he finally adopted were Aphelion dist. Perihelion dist. Semi-d. of the orbit. Excentricity* 160465 138234 15235O 14115; whereas before they were 166780 138500 152640 14140; and it was on the elements now investigated, that the comparison of art. 157 was made; and bv which it was so clearly established, that the oval retired as far within the real orbit as the circle went beyond it. T2 161. I 1J ASTRONOMICAL DISCOVERIES Decisive |Q| 4 Kepjer appears to have made a much more confutation . . r ' , cttheoval mm lite comparison than that now mentioned; and theory. f O nave submitted to the labour of deducing the distances for his oval orbit from the present ele- ments, in all the 4O points of mean anomaly which he had employed ; and finding, that all of them which lay between the apsides, fell generally as much short of the observed distances, as the calculated circular distances had exceeded them ; the evidence was rendered undeniable, and corn- pleat, that neither is the orbit of the planet a circle, nor does it retire within the circle so far as the oval of his present theory reouired, but passes nearly in the middle between them. Nor was he satisfied with a comparison thus la- borious and minute ; for he also conversely calcu- lated what geocentric longitudes would be given by the distances of his oval theory ; and found that in all the seven oppositions employed in his second method of investigation (J5()), these lon- gitudes, as must have been expected from distances erring in defect, were always too far advanced be- fore the opposition, and too little advanced after it. This especially was evident at the oppositions of 1589 anc ^ 15Q1 in the descending, and of '1582 and I 505 in the ascending, semi-circle of anomaly ; in all which the distance from the apsides was con- siderable ; and as the oval in these points retired too far within the orbit, and the distances which it gave erred sometimes 660 parts in defect, the errors produced on the geocentric longitudes rose sometimes to '20'. Kepler informs us also, that this was observed by D. Fabricius, in East Fries- land, as well as by himself; and so nearly was he anticipated by this astronomer in the discovery of the real form of the orbit, that, on communicating to him the oval theory, his answer was, that all its distances OF KEPLER. 277 - distances towards the intermediate parts were too short. An evidence to the same purpose was also given by the errors of the equations of art. 156, though calculated according to the original and strictest principles of the theory. For they concurred in representing the motions of the planet as by far too rapid in the intermediate parts of the orbit, and consequently shewed the times or distances in those parts to be shortened a great deal too much. The conclusion, therefore, was, that the oval theory, in endeavouring to avoid the errors of the circle in excess, commits others equally conspicu- ous in defect ; and the physical causes which Kepler supposed to produce it, to use his own language, fly off in smoke. 162. When it thus appeared that the breadth Accidental of the lunula, cut off by the real orbit of ^*$%L from the exccntric, was but half of that cut off law of the by the oval, and that even at QO from the a p- d istanccs - sides, where it was greatest, it did not exceed d()0 parts of a semi-diameter = 152350 ; Kepler tells us, that the vexation which he suffered was very great ; not merely from the mortification of find- ing a theory confuted on which he had founded the most sanguine expectations, but principally, from anxious and fruitless speculations concerning the causes of his disappointment. But while his mind was thus occupied, accident, as he candidly informs us, was more friendly to him than all the exertions of his ingenuity. The 660 parts of a semi-diameter = l5235Owere equivalent to 432 parts of a semi-diameter ~ 10OOOO, that is, nearly to 429 the half of 858, which he had found to be in the same parts, the extreme; breadth of the lunula cut off in the oval theory ; and, happening T 3 to ASTRONOMICAL DISCOVERIES to tarn his attention to the greatest optical equa- tion of Mars, which is between 5 18 X and 5 1Q', he perceived that 42Q was also the excess of the secant of 5 18' above the radius 10OOOO. By the accidental discovery of this coincidence, he tells us he was suddenly roused as from sleep, and a new light seemed to dart in upon him. His reasonings on the discovery were to this purpose. It is at the points F and Q (i% 64), of the excen- tric, which are equally remote from either apsis, that the greatest optical equation AFB takes place ; and these are also the points where the orbit retires farthest within the circle, and the ex- treme breadth of the lunula, instead of amount- ing to 858, arises only to 42Q. If, therefore, the semi-diameter BF or BQ, each = 10000O, should be employed instead of the secants AF and AQ, each of which is = 100i'2(), as the distances of the planet in these points F and Q, of excentric anomaly from the sun, they would be the very distances required by tire observations; and, ex- tending this reasoning to all the other divisions of the excentric, he concluded in general, that if, at the opposite points of it, E and R, D and S, &c., he should substitute, for AE, AR, and AD, AS, &c. the lines VE, VR, TD, TS, &c., deter- mined by perpendiculars AV, AT, &c. drawn from A to the diameters ER, DS, See. ; the distances would be every where of the just amount required by the observations, ?md, would neither err in ex- cess, as those given by the circular theory, nor as those given by the oval theory in defect. . But though the propriety of substituting BF for AF, at 90 of excentric anomaly, was evident, it remained to be proved, whether the like substi- tution of VE for.AR, and of TD for AD, &c., was equally admissible in the other points of the orbit. 4DF KEPLER. orbit. Accordingly, to all the true anomalies of art. 157, which from the longitude of the aphe- lion and the heliocentric longitude of the planet were given, he calculated the excentric anomalies CBD, CBE, and the optical equations ADB, AEB. Then, with the distances AD, AE, or AS, AR, given in the circular theory, he calculated, by the analogy sec. ADB : R :: AD : DT, all the lines TD, VE, or TS, VR, and found their coincidence with the observed distance in these points almost perfect ; at least, the differences were altogether insignificant. The following is a comparative view of both. In the descending semi-circle. Observed distances. Calculated. Differences. 166194 16622S 34 +. 163022 163160 138 + 358101 158074 27 . 154100 154338 62 . 147760 147913 158 -f . 13900O 139093 93 +. In the ascending semi-circle. Observed distances. Calculated. Differences. 166348 166228 120 . 163100 163160 60 +. 158217 1580/4 143 . 154278 154338 60 +. 147871 147918 47 +. 138984 139093 109 +. Where two conclusions for the observed dis- tance had been made, a mean of them was taken; and it is evident, that as the differences are so irregular, and at the same time so inconsiderable, T 4 they 260 ASTRONOMICAL DISCOVERIES they are principally to be ascribed to errors of ob r servation. It is evident that the substitution now described was the same in effect, as if, in the original form of the theory (fig. 63, No. l), AM were substi- tuted for AD ; or if, in the separate epicycle, (fig. 63, No. 2), AO were substituted for AM or AN, and AS for AQ, or AR ; and, that by it the planet is not at all supposed to move in the cir- cumference GNR of its epicycle, but only to Jibrate upwards and downwards in its diameter ; and it therefore became unnecessary to institute a like comparison with the distances giv 7 en by the second method of investigation (I5(j), because in it the librations of the planet, or the variations of its distance from the sun, had been the only sub- jects of consideration. Diametral It appeared therefore, from the most numerous cumfercn- observations made in all points of the orbit pro- tiaidis- miscuously, that the lines TD, TS, or VE, VR, tapccs. which Kepler called diametral distances, were the exact measures of the real distances of the planet from the sun, in those points of the orbit which it occupied, when, according to the circular theory, it would have been found in the points D, S, or E, R, with the distances AD, AS, or AE, AR ; which he called circumferential ; as supposing the planet to revolve in the circumference of its epi- cycle, and not to librate in its diameter. (Aa.) 163. It was now to be expected, that when Kepler had discovered a method of calculating the just distances cf the planet in all points of its orbit from the sun, that no farther difficulty would oc- cur in deducing from them just equations. But, notwithstanding this important discovery, he was uot immediately successful ; and, by a false appli- cation OP KEPLER. 281 ' cation of his distances, though in themselves just and accurate, his fears of a new disappointment were again revived. Accordingly, in one of those lively sallies of fancy, by which he was as much distinguished as by sagacity of perception, he tells us that, though the last and most secret retreat of the truth was discovered, the reluctant inhabitant did not immediately deliver hcrselt up to his pos- session, and applies to her the verses of Virgil ; Malo me Galatea petit, lasciva puclla ; Et fugit ad salius, et se cupit ante videii. The cause of this new alarm he thus describes. On the points A, B, (fig. 77), let the equal circles HKS, GDT, be described, and let AB be the exccntricity of the orbit. Let the equal arches HK, GD, represent the same exccntric anomaly, and join DK, which will be parallel and equal to AB. On K describe, with the distance DK, the epicycle LD, and draw AK producing it, if ne- cessary, to meet the epicycle again in L; by which means LD will be an arch of the epicycle similar to GD. Join BD, and from D draw to AL, AG, the perpendiculars DE, DC. It is evident, from what has been now proved, that the accurate dis- tance of the planet from the sun, at the time \vhen, if it moved in the circle ADT, it would be found in the point D, is the line AE ; and it is required to know what time it has employed in attaining this exccntric anomaly GBD. Now, be- cause GC, the versed sine of GD, that is, alter reduction to the dimensions of the epicycle, the libration LE, subtracted from AL = AG, gives the accurate length of AE, Kepler tells us, he supposed that the position of its extremity E was to be sought for in the semi diameter BD of the cxcentric, and not in the perpendicular DC; and to find it, he described on A as a centre, with the distance 282 ASTRONOMICAL DISCOVERIES distance AE, the arch EZF, meeting BD in Z. Falseposi- According to this supposition' AZ would have been Sianutral 6 ^ e accurate distance of the planet, in position as distances, well as magnitude, when GBD was its excentric anomaly ; and the angle GAZ would have been the true anomaly ; though, as was afterward dis- covered, the true anomaly corresponding to the time, or area DAG, ought to have been the angle GAF, determined by the line AF, drawn to A from the point F, where the arch ZF meets the perpendicular DC. The mistake was first discovered by experiment. On calculating the mean anomaly, or circular area DAG, corresponding to the excentric anomaly GBD, he found that, whether the calculation was made by taking the sum of the diametral distances from A of all the points of equal division in GD, or by adding the triangle DAB to the sector GBD, the substitution of the angle GAZ for GAF, al- ways rendered the true anomaly greater by 5' 30 /; , about the first and fourth octants, than the result from the vicarious theory ; and, about the second and third octants, nearly as much less. As the equations therefore, even when deduced in this manner from librations, differed from the truth, he ascribed his errors to the librations ; and, not- withstanding the just ^distances which they were found to give, he for some time abandoned them. The ellipse therefore seemed now to be his only resource for applying to any use, the natural prin- ciple of the motions being inversely as the times; and he supposed that, in having recourse to the ellipse, he was returning to a theory perfectly dif- ferent from his theory of librations. His argumentation in returning to the ellipse, was to this purpose : " The circle gave true ano- malies erring in excess, and the oval substituted fOF OF KEPLER.^ 283 for it, and which had been considered in his cal- culations as coincident with an ellipse, gave them erring in defect ; and, since the errors were of contrary kinds and equal, another ellipse must therefore be the only 'curve which can annihilate them. But, if the path of the planet be an ellipse, it must follow, that the point Z cannot be substi- tuted for F, because this would render the plane of the orbit too broad towards the aphelion, and transform the ellipse into an oval. For, let K, , be two positions of the centre of the epicycle Cfjuallv remote from the opposite apsides. Since the angles EDB, fd'B, are right, and the sines DE, de, of the excentric anomalies equal, it is evident that the arch EZ of less curvature will cut off a smaller portion DZ of the semi-diameter DB, than that which the arch ez of greater cur- vature cuts off from the semi-diameter vill be a point in the circumference of the same ellipse, and FC an ordinate to the axis GBT. Cor. If GT be also the transverse axis of the ellipse GKT (fig. 78), the distance AF of the point F, where the ellipse is cut by the perpendi- cular DC, drawn to GT from any point D of equal division in the circumference GMT, will be equal to the diametral distance XD of that point D; and the sum of the distances of such points F from A, the focus of the ellipse, whether in the whole el- liptical circumference, or in the particular arch GF, will be equal to the correspondent sum of the diametral distances^ either in the whole cir- cular $95 ASTRONOMICAL DISCOVERIES civlar circumference GMT, or the particular arch GD. For, if the diameter DJ pass through the two opposite points of division D, d, and there be drawn through A the line Ee parallel to DJ, and from D and d^ the perpendiculars DE, da to E meets that semi- diameter, that he was doubtful about the truth of his reasoning ; for the ellipse was divided by such ordinates into unequal arches, and the effect produced was to determine the times of de- scribing these ; whereas the end which seemed to be proposed, was to estimate the times of de- scribing equal arches. It was therefore by trial and experience, that his conviction/ of the truth of his OP KEPLER. 280 his reasoning was principally established ; and, in- deed, the fact was ascertained before the reason- ings were employed. His principal experiment was to this purpose. He calculated the whole diametral distances for every degree of excentric anomaly, and adding them together in their order, not only found the whole sum, which was zz 3(3000000 ; but also the particular sum corresponding to the degrees of the same anomaly in any given arch GD. Then, drawing from the given point D the ordinate DC, cutting the ellipse in F, he joined AD, AF ; and supposing the whole sum 36000000, to be equi- valent to the whole elliptical area, and the parti- cular sum corresponding to the degrees in GD to be equivalent to the particular elliptical area GFA, he had this analogy, as 36000000 is to 3]415g26536, the whole area of the circle, so is the sum corresponding to the divisions of GD, or the equivalent area GFA to the circular area GDA. The result for the area GDA left no doubt about the justice of the supposition ; for, when calcu- lated by this analogy, it came out perfectly the same as if he had calculated it by finding the amount of the triangular area ABD, and adding this area to the sector GBD (Dd.) He found also that, by supposing the anomalis- tical arch to terminate in the semi-diameter BD, and not in the ordinate DC, the same error en- sued which had formerly made him abandon the theory of librations. For, an area GZA, and not GFA, became the mean anomaly corresponding to the excentric arch GD; and the angle GAZ, the true anomaly : and this angle, in the same man- ner as formerly, was found to exceed the truth in the first and fourth quadrants, and to fall short of it in the second and third. But the angle GAF U was 2QO ASTRONOMICAL DISCOVERIES was always found to be the same with the true anomaly of the vicarious theory ; that is, it en- tirely agreed with the observations. 167. In this manner it was that Kepler dis- covered and established the celebrated law, by which all the planets are regelated in describing their various orbits ; namely, that every such orbit is an ellipse, in one of the foci of which the sun is situated ; and that, in revolving round this com- mon focus, the planet describes areas, as GFA, of its ellipse proportional to the times in which it moves through the elliptical arches, as GF, by which those areas are subtended. It only remained therefore, to explain or investigate the methods of deducing the equations of a planet from the prin- ciples now established. From the The first problem proposed for that purpose was, anomll^to from the given excentricity and excentric anomaly, find the to find the correspondent mean anomaly. The, excentric anomaly is the angle or sector GBD, or the arch GD which measures them : or it is the elliptical arch GF, because this arch is divided by ordinates to the axis into the same number of parts with GD. The required mean anomaly, again is the circular area GDA, or the elliptical area GFA, because each of them has a constant and determined ratio to the sum of the diametral or elliptical distances, that is, to the time ; and therefore will justly measure it. The circular area, GDA, consists of two parts. The first is the sector GBD ; which, because the arch GD is given, will be also given in parts of which the whole area of the circle contains 360. For example, if the arch GD, or the angle GBD, were = 10, the sector, or area, GBD, would be also = 10. The OF KEPLER. 2Q1 The other part of the area GDA, is the trian- gular space ABD. To find this in the most ex- peditious manner, Kepler previously calculated the area of the triangle MAB, when the excen- tric anomaly GBM is == Q0. This area is = M_B .AB _ R whole circular area is to 36o, so is this area 463200000 to JQ108" = 5 18' 28". Now, MAB : DAB :: MB : DC :: R : sin. GD ; and therefore, if the given excentric anomaly GD were = 10, the triangular space DAB would be = MAB. sin. GD = 33 IS 77 .; = 55' \s".7. Therefore GDA = GBD + DAB = 10 55 ' IB 1 '. 7. With equal ease he solved a second problem From the from the given excentric anomaly, to find the true^omah% anomaly, that is, the angle GAP. For the co- tofindtl * sine BC of the given arch GD, is given in parts of BD = 1000OO; and, when GD is = 10, will be = Q848I. Since also AB, in the same part?, is = $264, and R : cos. ABX (= GBD) :: AB : BX, the line BX will be = 0124. Therefore DB + BX = AE = AF 109124 ; and, since AC = AB + BC = 107746, we shall have m the right-angled triangle FAC, AF : AC :: R : cos. GAF = Q 4' 56 /x . The whole equation therefore, or difference between the true and mean anoma- lies, will be = 1 50' 22". 7. In the second and third quadrants of excentric anomaly, A/ will be = dE ca BX, and Ac = AB C/D BC. l63. It was a business of greater difficulty to From the find the excentric anomaly, from the cxcentricity lfadbl' and the true anomaly given. Two solutions, how- excentric ever, of this problem were found, and the princi- ancmaiy * pies of the first were these. U 2 i . The 2Q2 ASTRONOMICAL DISCOVERIES solution j. The parts GD, of the ordinates to the greater axis KP, which are intercepted between the ellipse KHP (fig. 80), and the excentric KEP described upon KP as its diameter, increase in the ratio of the sines of excentric anomaly. For GD : HE :: CD : FE. 2. The tangent MD of the angles DBG, by which the intercepted parts GD of the ordinates are subtended at B, the centre of the orbit, would increase in the ratio of the cosines of excentric anomaly, if GD were supposed to continue in- variable. If BG be produced to meet the tangent in M, we shall have DM : GD :: sin. DGM : sin.DMG; whence DM. sin. DMG = GD. sin. DGM. But the difference between DMG and a right angle is almost insensible ; for its complement DBG, un- der which GD is seen from B, does not even in the orbit of Mars, and at its maximum, exceed 8 minutes. Therefore the tangent DM will be as GD. sin. DGM = GD. sin. BGC ; or, since GD is supposed to continue invariable, as sin. BGC. But BGC is very nearly equal to BDC, the com- plement of the excentric anomaly KBD, and cannot at its maximum exceed BDC above 8 mi- nutes. Therefore DM will always be very nearly as sin. BDC, that is, as cos. KBD ~ BC. This tangent therefore, if GD were invariable, would, at the points K, and P, subtend the greatest angle DBM ; for there it coincides with GD, and at these points the cosine BC is greatest ; and, at L, and the opposite point, Q0 remote from K and P, it coincides with BL, and the angle which it sub- tends vanishes with BC. 3. The tangent of the angle DBG, or, since very small angles are as their tangents, the angle DBG OP KEPLEH. 2Q3 DBG increases nearly in the compound ratio of the sine and cosine of the excentric anomaly. The magnitude of the angle DBG depends, 1st, on the magnitude of the intercepted part DG of the ordinate DC, which increases from nothing in K to its greatest amount in L, according to the ratio of the sines; and, 2dly, on the apparent magnitude of the tangent DM, to the eye placed in B ; and this apparent magnitude has been seen to decrease from K to L, in the ratio of the co- sines. On the first account the angle DBG is = 0, in K ; because there the sine of excentric anomaly is = : and on the second account DBG is also = O, in L; because there the co- sine of excentric anomaly is = 0. But this angle, at 45 from K and L, has risen to more than half its greatest amount ; for the sine of 45 is zr 7O711, that is, greater than half the radius, and the cosine of 45 is equal to its sine. The rect- angle, therefore, under the sine and co-sine of ex- centric anomaly, is greatest at 45, and becomes a square equal to half the square of the radius. 4. Let KBD be the excentric anomaly, from the focus A draw to the excentric the line AE pa- rallel to BD, from D and E draw ta the axis KP the ordinates DC, EF, meeting the ellipse in G and H, and let BG, AH, be joined ; the angle DBG will be equal to the angle EAH. For CD : DG :: FE : EH, and CD : DB :: FE : EA. Therefore DG : DB :: EH : EA ; and, since the angles CDB, FEA, are equal, the angle DBG is also equal to the angle EAH. 5. The angle EAH is accurately as sin. KAE. cos. KAH ; (for its equal DBG has been proved to be as sin. CBD. cos. KBG), and therefore nearly as sin. KAE. cos. KAE. This angle EAH, at its greatest amount in 45 Ua of 294 ASTRONOMICAL DISCOVERIES of excentric anomaly, may be found in this man- ner. When KBL is == 90, the intercepted part LN of BL is = 429, or 432 (162); and, since BL : LN :: CD : DG, or 10OOOO : 432 :: 70711 : DG, this intercepted part will, in 45 of excen- tric anomaly, be found = 315 ; and consequently CG = 70396. Since, then, CB = 70/11, the angle CBG will be = 44 52' 19", and DBG '== CBD CBG = 7' 41" == EAH. Hence the angle EAH at any other point of ex. centric anomaly, suppose at 30" 1 , will be readily found. For, since sin. 45. cos. 45 zz 500OOOOOOO, and DBG at 45 of excentric anomaly = 7' 4l ;/ , the angle EAH at 30 of the same anomaly, will be found by this analogy, sin. 45. cos. 45 : 7 X 41" :: sin. 30. cos. 30 : -^~. C S> ^ . 46]" == sin. 45. cos. 45 4330150000 fi 86603^ , ,, _ _ fi/ 5000000000' 4 ' 100000 ' ? = EAH. Suppose then, -that it were required to find the excentric anomaly KBE, corresponding to the given true anomaly KAH. The angle EAH, as has now been seen, is zz & 38", and consequently KAE zz 30 6 X 38". Therefore we shall have in I . , T-> A T TTT* A -A. 13. Sill. J_)Ahi the triangle EAB, sin. BEA = ^ == , and BEA zz 2 3Q' 4 9". 7. Whence the required excentric anomaly KBE will be zz 32 46' 27".7 ; and from KBE thus found, the mean anomaly, or area KEA, in parts of the circular area ; or KHA, in parts of the ellip- tical area, will be found as in the former article. A practical rule, arising from the above investi- gation, for finding the anglfc EAH at any point of true anomaly, was this : Multiply the sine of the given true anomaly into the cosine, double the product, OP KEPLER. 2Q5 product, cut off the five last figures, and multiply the result into 46l /; . The importance of this ab- breviation, while logarithms were unknown, is obvious. 169. The other solution was by means of an solution algebraical analysis. Let the given true anomaly second. GAP (fig. 78), be = Rj> rr^i f R? R + V R : x = . Therefore = ^~ cos. # : a /z K and by the solution of this simple equation, the value of BX = j, and consequently of BC = x, the cosine of the required excentric anomaly will be found. In general ~ = ~ y cos. p co ^. For example, let GAP =

* _ s j n . . Whence V I (r sin. **) = * (r 2 fl2 ' s ' **), and VVJ 2 t>V 2 = V^ 2 sin. 2 * *> 2 a 2 sin. 2 *. Whence and sn. * = =fc v V^a a z a 2 ' W a a* Since in the planetary system V : v :: V 6 : V a we shall have 2 . sin. *x b : :: /- 2 -- - : r 2 sin. ^ ; whence sin. 2 ^ = - ; or making r = a = 1, o^ a 3 # ^ b* = - - - = ',-. ; a na sin. a- = - If V were = T, we should have sin. x = ; and, conse- quently, the points of station would be in the conjunctions and oppositions. For, since AD : BD :: V : v, it is impossible that AD can be equal to BD, except when both are parallel, and therefore infinite. If v were greater than V, and at the same AS = a, so related to BS = <5, that V 2 /> 2 were greater than tra 2 , the planet never would appear stationary. E. The argument of latitude is formed by subtracting the lon- gitude of the node from the longitude of the planet, reckoning both on the planet's orbit. Thus, if A (fig. 30), be the place of the point of Aries in the ecliptic, an arch B// = BA is to be computed in the orbit, and this arch BA will be the longitude of the node, while PB^ is the longitude of the planet, and the argument of latitude is BP = PB# B#. When the lon- gitude of the node is greater than the longitude of the planet, 360 are to be added to the last. Thus, if the planet were in p with the longitude pa = 20, while Ba the longitude of the node was = 60, the argument would be 360 -f pa B0 = 380 60 = 320 = ^DPB ; and the nearest distance from the node would be p& = 40. Though the rule thus expressed is general, it may perhaps be NOTES. 311 fee thought mote easy in practice to find ^B directly, by sub- tracting ap from B ; for /;B would thus also be =s 60 20 3 = W. The chief calculations relating to the second inequalities of Calcula- of a planet, according to the principles of the Copernican tionsrelat- system, with the addition of the reductions now employed, ^f^'j^ will be easily understood by the following description. Let q ua iities- S (fig. 31, 32), be the sun in the centre of the ecliptic, T the place of the earth in the orbit CDT, and P the place of a planet in its orbit APB. By drawing from the planet in P the perpendicular PQ to the plane of the ecliptic, the angle PSQ will be its heliocentric latitude, and PTQ its geocentric latitude ; SP and TP will he its real distances in its orbit from the sun, and from the earth ; and SQ, TQ its distances in the plane of the ecliptic, called the reduced or shortened distances. The heliocentric longitude in the orbit would be marked by the line SP, and the geocentric longitude by the line TP ; but these longitudes reduced to the ecliptic are L and O, marked by the lines SQ, TQ. The angle SQT, therefore, which is the difference of these reduced heliocen- tric and geocentric longitudes, is the annual parallax ; the difference TSQ between the heliocentric longitudes of the planet and the earth is the commutation, and the difference STQ between the geocentric longitudes of the sun and the planet is the elongation. In the triangle STQ, therefore, \vith any three of its six parts, (that is, its sides and angles) given, the other three may always readily be found. Suppose, for example, that the distance of a planet in its particuiar- orbit from the sun, and its heliocentric longitude, also inty or the . the orbit, together with the longitude of the earth, were found for any particular time, by means of some theory of their first inequalities, and that it were required to calculate for the same time its geocentric longitude. Since the incli- nation CBP (fig. 30), of the orbit is given, and invariable, and the argument PB of latitude may be formed from the given longitude of the node, the reduction to the ecliptic will easily be found, and ther.ce also the longitude CBA. of the planet reckoned on the ecliptic ; and therefore in the triangle SQT, the commutation TSQ, which is the difference be- tween this longitude and that of the earth, will be found. From the inclination CBP and the argument of latitude, the heliocentric latitude PSQ, or CSP (fig. 30), will also be found by the analogy R ': sin. CBP :: sin. BP : sin. PSC, that is, sin. PSQ = sin. inclination, sin. argument latitude ; and, therefore, in the ri<^ht-angled plane triangle PQS, the reduced distance SQ will be = SP. cos. PSQ^; for R : cos. PSQ :: SP : SQ. If, then, we assume the semi-diameter ST X4 of 312 NOTES. of the earth's orbit = 1 ; we shall with SQ, in like parts, and the included angle TSQ, the difference of the longitudes, find the angle of elongation STQ, by t$ie usual analogy " ST -f- SQ r^STco SQ::(SQT + S FQ) : J(SQT c/; STQ); and this angle added to, or subtracted from, the longitude of the sun TS, which is always six signs greater than the lon- gitude of the earth, will give the required geocentric longi- tude TQ. This is the longitude in the ecliptic, and which observations give, while the ger. centric longitude in the or- bit is marked by TP ; and ft is evident that the operation now exemplified, is the same with the calculation of the equation of tiie orbit, in the Ptolemaic theory, excepting only in what relates to the reductions. In forming the angle of commuta- tion, the longitude of a superior planet is always to be sub- tracted from the longitude of the earth ; and the longitude of the earth is always to be subtracted from that of an inferior planet, because this angle always increases by the excess of the greater velocity above the least. and lati- The geocentric latitude of a planet PTQ, for any given rude, time, may be calculated by this analogy sin. TSQ : sin. STQ :: tan. PSQ : tan. PTQ ; that is, sin. commutation : sin. elongation :: tan. heliocentric latitude ; tan. geocentric lati- tude. For, in the triangle PTQ, TQ : QP :: R : tan. PTQ ; and in the triangle PSQ QP : SQ :: tan. PSQ : R ; and therefore TQ : SO :: tan. PSQ : tan. PTQ. But in the tria. SQT t TQ : SQ :: sin. TSQ : sin. STQ ; and therefore sin. TSQ : sin. STQ :: tan. PSQ : tan. PTQ. Distance At oppositions of the superior planets, and conjunctions of found by the inferior, their distances from the sun, if the inclinations SOIM"? 1 ^ t ^ le or bi ts ai *d the longitudes of the nodes are given, may titude in " ^ e ^ oun ^ ky observations of their geocentric latitudes. Let opposi. and the earth be in B (fig. 29), and C the planet observed at its conjunc. opposition, with the latitude CBD. Therefore, its place, in the ecliptic, is the point D of the line DBA, passing through the centres of the earth and sun ; and if the inclination of the orbit, and the argument cf latitude are given, the angle CAB of heliocentric latitude will be found ; and with the two angles CAB, CBA, of the triangle CAB, and the side AB, which is the distance of the earth from the sun, given by the solar theory, we may easily find also the side AC re- quired. F. The usual method of representing the vicissitudes of the seasons, according to the Copernican principles, is as follows., Let NOTES. 313 Let ABCD (fig. 34), be the orbit of the earth, S the sun, Vicissi- PE the axis ofdie earth, JEQ the equator, and TR, tr t the tu tropics of the terrestrial globe. It is manifest that, if the se axis PE were perpendicular to the plane of the ecliptic, this great circle would coincide with the equator ; and, as the. rays of the sun must always illuminate one hemisphere of the earth, that, by their direction in the plane of the equator, they would constantly shine from pole to pole, and there could be no variation, either of the seasons, or of the length of day and night. But, siace the planes of the equator and the ecliptic intersect each other in an angle of about 23 3(Y 9 the axis PE will be inclined to the latter plane in an angle of about 66 30' ; and, because this angle continues constant during the whole annual revolution, and the direction of the axis invariable, the rays of the sun will be very seldom in the direction of the plane of the equator, and the effects will be very different. When the earth is in the position D, suppose about the 21st of June, with the axis so inclined to the ecliptic that its north pole P is nearest the sun, the solar rays will be perpendicular to the earth's surface at the tropic of Cancer T ; and, in the course of the diurnal rotation, every place under this tropic will have the sun in its zenith. In this po- sition also, as the solar rays illuminate a whole hemisphere of the earth, the light will extend as far beyond the north pole P as the distance between the tropic and the equator, and will fall as far short of the other pole ; the whole region within the arctic circle, consisting of 23 3(/ round P, will enjoy light, while a region of equal extent round the opposite pole E will be involved in darkness ; and the greater portion of all the parallels between the equator and the arctic circle will be illuminated, while the greater portion of all between the equator and the antarctic circle will be darkened. Again, in the position A, 90 distant from D, because the axis PE invariably retains the same direction, both its poles will be equally distant from the sun, and his light will extend to both ; a ray drawn from him to the earth will be perpendi- cular to its surface at the intersection of the equator with the ecliptic ; all ,the points of the equator will successively, in the course of the diurnal rotation, have the sun in their zenith; and all its parallels being equally divided by the circle which terminates the light, the days in all places of the earth will be equal to the nights. Farther, in the position B, diame- trically opposite to D, about the 21st of December, the whole appearances will be contrary to those of the 21st of June ; tor, since the direction of the earth's axis remains invariable, the pole P, which was formerly inclined to the sun, will be now as much withdrawn from him ; his rays will fall as much short 314 NOTES. short of this pole, as, in the position D, they went beyond it ; the places under fr, the tropic of Capricorn, will have the sun in their zenith ; and to all the countries on the north of the equator it will be winter. Finally, in the position C, opposite to A, where both poles are again equally distant from the sun, the appearances at A will return ; a ray from the sun perpendicular to the earth's surface, will be in the planes of both the equator and the ecliptic ; -and those inha* bitants of the earth, whose nearest pole is about to incline to- wards the sun, will have spring ; while it will be autumn to those whose nearest pole is about to be^ withdrawn from him. G. Method of As the oppositions of the superior planets, and the con- observing junctions of the inferior, are the conditions of principal im- oppositions. portance in the formation of their theories, it will not be improper to give in this place some account of the procedure of astronomers in observing them. One of -D. Cassini's methods, for example, consists in determining the right as- cension and declination of the planet, by means of a com- parison with some remarkable fixed star, and thence calcu- lating its longitude in the ecliptic. If the difference between this and the real place'of the sun be precisely ISO , the time and point, in which the planet was observed, are the time and point of the opposition ; but, if it be greater or less, the ex- cess or defect is to be noted ; and, as it arises from the mo- tion of the sun or earth in consequenfia, and the motion of the planet, which, toward the oppositions, is always in antecc- dcntia, or retrograde, it will consist of the sum of both mo- tions. Consequently, by comparing it with the sum of their motions in 24- hours, in the parts of their orbits which they then describe, the interval between the time of'the observation and the opposition will be discovered, and the point and time of the last determined. It ought however to be observed that, in order to obtain proper accuracy, the precession, aber- ration, nutation, refraction, and diurnal parallax, are all to be taken into the account. Example At the opposition of Saturn in 1773, the zenith distances of this planet, and of the star /3 in Cancer, were taken at the observatory in Milan, by M. de Caesaris, with a sextant of six feet radius ; while, at the same instants, M. Reggio observed their transits over the meridian, with a transit instrument of five feet radius, whose plane had been accurately placed in the meridian by means of a great num- ber of correspondent altitudes. The position of the fixed star had been previously determined by Mr. Bradley and de la Caille. NOTES. 315 Time by the Clock. On the 27th of Feb. 1773, the star crossed the meridian at - * 9h. 6' 4-1" -. the centre of Saturn crossed at 11 50 19.8 Apparent zenith distance of the star - 35 31' 2". 3 , of Saturn's centre - 35 31 40.3 On the 28th of Feb. the transit of the star was made at - 9h. #54".? . and of "Saturn's centre at 11 46 15.6 Apparent zenith distance of the star - 35 31 ' 2". 3 . of Saturn's centre - 35 29 50.5 Apparent noon on the 27th at llh. 47' 20" . and onthe23hat - 11 47 18.2 Interval according to the clock - 23 59 58.2 This interval in seconds - - 86398".2 Interval between apparent noon and the tiansitof Saturn, on the 27th - 43379".8 Diurnal motion of the sun in long, at the part of his orbit whicli he then occupied, by calcul. 361 0".8 Diurnal motion of Saturn, concluded from his transits, on the 26th, 27th, and 23th - 2S5".2 Sun's longitude on the 27th at noon, cal- culated for the meridian of Milan, from De la Caille, - - 11s. 9 15' 30'' His motion therefore in longitude, during the interval between his transit and that of Saturn on the 27th, was found by this ana- logy, 86398".2 : 43^79".8 :: 3610".8 : 1813" = 30* 13" Whence the sun's longitude at the time of Saturn's transit on the 27th, was 11s. 9 45' 45" Supposing then the right ascension and declination of /3 in Cancer to have been accurately determined for 1750, by applying the corrections for precession, aberration, and nutation, the right ascension of this star, at the time of the observations, will be - - 121 3' 25".5 And its north declination, corrected for re- fraction . . . 9 51' 57".S But the observed interval, by the clock, be- tween the star's transits, on the 27th and 28th, is - - 23h. 56' 13".4 And between the transits of the star and Saturn, on the 27th, - 2 43 38.8 Therefore, as 23 h. 56' 13". 4 : 2h. 43' 38".8, so is a compleat revolution of the star, or 360, to 41 1' 3".2,the difference between Sl6 KOTES. between the right ascension of the star and Saturn, at the" time of Saturn's transit ; so that the right ascension of Sa- turn, at his transit on the 27th, was 162 4' 28".7 In like manner, his right ascension on the 28th, was 162 (X 2".5 As the differences between the zenith distances of the star and Saturn, at the times of their transits, did not amount to 2', and the correct declination of the star had been deter- mined, the declination of Saturn maybe safely deduced from it, without particularly calculating his refraction. Saturn's declination therefore, on the 27th, was North - - 9.'1'19".8 And on the 28th, also North 9 52 9.6 Therefore his apparent longitude, at his transit on the 27th, was - 5s. 9 43' 26". and on the 28th, - 5 9 38 41.7 and the difference - ' - 4' 44".4 This is Saturn's change of apparent longitude in the time of his own revolution to the meridian ; but, as his motion is retrograde, and the time of his revolution 4' shorter than that of the apparent soiar revolution, in the present circum- stances, 0".8 is to be added to this difference ; so that Sa- turn's variation of longitude at present, and in the time of the present solar apparent revolution, is 4' 45".2 Since, then, the apparent longitude of the sun, at the time of Saturn's transit on the 27th, was 11s. 9 45' 43", and that of Saturn 5s. 9 43' 26". 1; and the difference, instead of amounting to six signs, or 180, precisely, falls 2' 16".9 short; and since, also, as the earth's motion is direct, and that of the planet retrograde, the arch 2' 16".9 must continue to in- crease ; it follows that the opposition was now past. Some previous instant therefore must be calculated, in which the sun's longitude was less, and that of Saturn greater ; and such an instant that, by correcting the present longitudes of both, in the proper ratio of the motions of each, two longi- tudes may be found, whose difference shall be 180 precisely. To find this instant there is evidently given the following analogy; as the sum of the motions of the sun and planet, during the time of the sun's present apparent revolution, is to the time of the sun's apparent revolution, so is the sum of the motions of both bodies beyond the point of opposition, to the time elapsed since the opposition, or from the opposition to the observation ; that is, 60' 10". 8 + 4' 45".2 : 23h. 59' *8".2 :: 2' 16".9 ; to the time required j or 3896" : 86398".2 NOTES. 317 :: I36".9 : 3035".9 = 50' 35". 9 ; and this subtracted from llh. 50' 19".8, the time of Saturn's transit on the 27th, gives lOh. 59' 43".9 for the time of the opposition ; and, since the sun's motion in 50' 35".9, according to its present rate, is 2' 6".88, and Saturn's motion 10".02, also at its present rate, the longitude of the sun at this time will be found = Us. 9 43' 36". 12, and that of Saturn 5s. 9 -13' 36". 12, differing 180 precisely. The time however, which is now found, is only time reckoned by the clock ; and the clock, on the 27th of Febr. at noon, was 12' 40" slower than the sun, and 12' 41".8 slower than the sun on the 2Sth at noon. The interval there- fore between apparent noon, on the 27th, and the instant of opposition reckoned by the clock, is lOh. 59' 45".9 + 12' 40" = llh. 12' 23".9; and, since the clock fell back 1'.8 in the time of the whole revolution of the sun to the meridian, it must, in this interval, have fallen back 0".8. The apparent time, therefore, of the opposition, reckoned on the meridian of Milan, was llh. 12' 24/',7 of the 27th of February 1773. It ought however to be noticed, that the place in which the planet appears, is not that which it really occupies ; for, in the present example, the very small equations cf aberration and nutation were not applied to the planet ; and, if[ they had, its place would have been found a very small degree more advanced. It is obvious that the same method is equally applicable to conjunctions. By means of the comparison cf very distant oppositions, Mean or conjunctions, astronomers have principally determined the motions, mean motions of the planets in longitude, or the times of their tropical revolutions. From these, again, the times of their sidereal revolutions are deduced ; for, if the arch of precession be given, and called />, and the sidereal revolution be denoted by 360, the arch described by the planet in the time / of its tropical revolution will be = 360 / ; and 360 p : / : : p : ^-~ ; and this portion of time added to / will give the time of a sidereal revolution. The procedure is the same with respect to anomaltstical and synodical revo- lutions ; and the mean diurnal, annual, and secular motions ef a planet likewise follow from the time of the tropical re- volution. For this time is to 360 as the Julian year, or 365 d. 6h. to the mean annual motion; and this again, di- vided by 365 d. Gh. gives the diurnal motion; and multi- plied by 100, the secular ; that is, the arch which may be found to remain after the compleat revolutions are rejected. The SIS NOTES. The following table of the most important of these motions- is given by M. de la Lande. The secular motions of Saturn, Jupiter, and the Moon, are those only which take place in the present century ; for the revolutions of the two last are found to be performed in less time, and of Saturn in more, than they required in former ages ; and both the secular and diurnal motions respect the equinoctial points. Mercury Venus Sun Mars Jupiter Saturn - Moon Mercury Venus Sun Mars Jupiter Saturn Moon Tropical Revolutions. d. h. , u 87 23 14 25,9 224 16 41 30.4 - 365 5 4-8 45.5 686 22 18 $7.3 4330 8 58 27.3 10749 7 21 50 27 7 43 4.6 Secular Motions. s% o i u 2 14 12 10 6 19 12 12 46 10 2 1 42 Sidereal Revolutions, d. h. , H 87 23 15 37.0 224 16 49 12.7 365 6 9 11.2 686 23 30*43.3 4382 8 51 25.6 10761 14 36 42.5 27 7 43 11 5 4 10 10 6 27 30 23 14 30 7 53 35 Diurnal Motions. I g 32.570376 1 ,30 7.806488 59 8.330458 31 26.656536 4 59.281314 2 0,565914 13 10 35.02847 The tropical revolution of the newly discovered planet em- ploys 83y. 52d. 4h. the sidereal 83 y. 150 d. 18 h. The se- cular motion is 2s. 13 16' 55"> and the mean diurnal motion (X 42".678026. H. In order to demonstrate that the great error in all the former theories, whether excentric or concentric, was, their being founded on observations, not of real, but of mean op- positions, it was necessary for him previously to shew, that, as far as the first inequalities were concerned, they were per- fectly, or very nearly, equivalent in their effects. In those cases, however, where the first inequalities were believed to be purely optical, as in the orbits of the sun and moon, their coincidence has been already pointed out (31), and the re- petition of K'epler's reasoning becomes unnecessary. But, even in the planetary orbits, where the first inequalities were thought to be partly real, the difference in the effect of the theories is very inconsiderable. For, if, instead of bisecting, like NOTES. 319 like Ptolemy, the excentricity of Mars, we should introduce E *. into the excentric theory the correction made by T. Brahe, lency . of and make DE (fig. 10) = 7560, and DT = 12GOO, and theories. thence calculate the equation fcBT for 45 of anomaly ; we shall -find EBD = 1 = 3 4' 52", and from JJi> the analogy DB + DT : DB DT :: tan.|ADB : tan.|(DTB DBTJ" that DBT is = 4- 24' 4"; and therefore the whole equation EBT = 7 26' 56". Or, if in the concentric theory, (fig. 36) where EQ is = 16380, and QR = 3780, and where, since BAF = 45, the angle EQR is a right angle, we calculate the equation RAE ; we shall find by drawing QS, RT, perpendicular to AC, that ES, the sine of 45, will in the same parts be = 70711, RT t= EQ + ES == 87091, and AT = AS OR = 66931 ; so that BAR = R TA TRA = -~- = 37 32' 37", and the equation RAE = BAE BAR = 7 27' 23"; that is, only 1' 23" less than the former. When the mean anomaly is 90, the equation by the former theory will be f 1 25' 58", and by the latter 11 23' 53"; still differing only 1' 55". Every proof therefore of inaccuracy of principle, is appli* cable to both theories ; and their difference of form has no effect to remove or diminish the errors. I. If the first inequalities of the planets could be considered as purely optical, and the centre of uniform motion as coin- cident with the centre of the orbit, as in the simple solar hypothesis, it would be of no consequence whether their theory were founded on mean, or apparent oppositions. For if, in the concentric theory,- ^we describe on the centre A (fig. 50), the arch BD of the orbit, and on B, with the dis* tance BC, determined by Copernicus, describe an epicycle, and on D, with the same distance, another epicycle ; and if the places of the planets in the epicycles be C and F, such that DF may be always parallel to ABC, here supposed to be the line of apsides ; this scheme will represent the theory, founded on mean oppositions : and the planet's path through the heavens will be the circular arch CF, similar and equal to BD, and described upon the centre E, found in AB, by taking AE = BC. The theory, again, founded on apparent oppositions, will be represented by removing the line of up- side* 320 NOTES. sides from the position ABC to the position AHK, and de- scribing on the points B, H, D, of the orbit a different epicycle, in which, however, the planet so continues to move, that BL, DM, are always parallel to AHK. But though the path of the planet through the heavens is thus removed from CF to LM, it is still a circular arch LKM, similar and equal to CF or BD : and its centre N is found in AH, by taking AN = HK. In the excentric form of Ptolemy, on the centre 763 in parts of BD = 100000, and AG the solar excentricity, in the same parts = 2346. Con- sequently by resolving die triangle AGC, the apogee D of the new orbit was found to be in 4s. 29 0' 3", and the ex- centricity AC = 1S034-; and dividing this in B in the ratio of GL = l'>35* to LC = 74-11, AB was found = 11271, and BC = 1)76:5 : and B was the centre of the new orbit DEF. The point of this orbit where the greatest differences upon the appearances from G take place, will be found by draw- ing through C, the centre of the equant, the line CVY at right angles to CG, and cutting both orbits- in VY. For if ia LV and BY, and draw to CV the line BZ, parallel Y td 322 NOTES. to GC, we shall have in the right-angled triangle LCV, with LV and LC, the line CV = 99725, and in the right-angled triangle BZC, with ZBC = ACG, and BC, the line BZ = 67-"2, and CZ = 644-. Consequently, in the triangle BZY, also right-angled, with BY and BZ, we shall have ZY = 99733; and therefore CY .= 100417 ; and VY = CY ~CV = 692. On the first The line VY measures the quantity by which the planet at inequali- the point Y would be displaced by the substitution of the new ties, orbit DEF, and at the instant of time marked by CVY ; and VGY will be the difference produced by the substitution on the former equation of first inequality seen from G. To find this difference, we have in the right-angled triangle GCV, from GC and CV given, the angle CGV = 7S 47' 30", and in the right-angled triangle GCY, from GC and CY, the angle CGY = % 78 51' 54" ; and therefore VGY = 4' 24". This is the greatest difference on the old equations of first inequality that the new theory could produce ; for, though VY is not the greatest distance of the orbits, that distance is least obliquely viewed from the point G. and on the But the effects which the proposed theory would produce second. on the ancient equations of second inequality, were* by r*o means thus inconsiderable. Through the centres TL, of both orbits (fig. 49), draw the line TLS, and from E, the centre of the equant, draw towards the parts nearest to the earth's orbit, the line FRS, cutting the new orbit in R, and the old in S ; and marking the same instant of time on both. When the earth is in the point a, where this line cuts the ter- restrial orbit, the planet will be seen in the same point of the zodiac, according to either theory. But, when the earth is on either side of ERS, the intercepted part RS, which is very nearly equal to TL, will subtend at the eye a greater er a less angle, according to the circumstances in which it is viewed ; and this angle will evidently be greatest at the point I, where a circle passing through R, S, touches the earth's orbit. To find therefore the point of contact I, it is necessary to determine RS, both in magnitude and position, and to draw from A the line AX perpendicular to ES. In the triangle ELS, with EL = 7411, and ALS = 47 59' 16", the side ES will be found = 105123, and the angle LES = 44 59' 10". But LET was found = 5 27' 47", and therefore TER = 50 26' 57"; and with this angle, and ET = 6763, ER will be found, in the triangle ETK = 104170; and therefore RS = ES ER = 953, and ESL = 3 (7 6'". Again, in the right-angled triangle AXE, with AEX =- 44 59' 10", and AE = 19763, AX will be found = 13971, and EX = 13978. NOTES. 323 1397S. The semi-diameter AI also of the earth's orbit will be found = 656565, in parts of TR, or LS = 100000, from AB = 2346. Since the circle whiclr passes through R, S, must touch the earth's orbit in I, its centre will be in the semi-diameter AI produced, and at the point Y of its intersection with ZY, a perpendicular to RS bisecting it in Z. Join RY, and from A draw A

Then r = A I* 4- 2AI . IY 4- IV s . But IY = RY = RZ\4- * 2 Therefore f-. AI 2 4- 2Af. v'RZ^ -f X* 4- RZ 2 4- x". Again f = A

2 . Or AI 1 -f RZ 2 4- 2AI. vRZ= A? 2rZ<, 4- Z^. By resolving this equation Kepler finds AT = ZY = 25772. His solution is subjoined, as an example of his patience in calculation. By the substitution of known values, the first equation,^* = AI 2 4- RZ 2 4- * 2 4- 2AI. V RZ^ + ^ be- comes f = 4.310.974.527 4- 2 (97.876.383.536.363. 4- 4.310.747. 475. 52)2 4. ft- ; and, by a like substitution, the second equa- tion y r = A<$> 2 4- J 2 2/Z<^ 4- Z- becomes /- = 8.415.875.065 27942* 4- * 2 . Therefore 4.310.974.527 4- 2(97.876.383 536.363 4-4.310.747.475.^)^ 8.415.875.065 2794-2 .r. Therefore 2(97.876.383.536.363 4- 4.310.747.475 *)* , = 4.104.900.538 ~ 27942^. Or (97.876.383.536.363 4- 4.310. 747.475 .r 2 )* = 2.052.450.269 13971 x. And, by involving both sides of the equation 4.310.747.475.J 1 4- 97.896.383.556.363= 19.503.841.*' 57-349.565 4 16.398.* 4- 421.252.106.718.172.361. Whence 4. 11 5.558.634. r* 4- 57.349-565.416.398.* = 42J. 156.228.334.645.988. \Yhence again x* 4- 13934.* = 1.023.329.6CO; and x = 25772 = ZY. 324 NOTES. Havfng thus found the semi-diameter ZY of the reqrired circle, the positions of 1 and S, and the angle RIS, will be easily determined. For Z<}> co ZY = Y = 11801 ; and, since A<$> =. ZX =r 90665.5, and A^Y a right-angle, the an- gle 4>AY will be found = 7 6 30' 10". Since also ESL s= 3 0' 6", and SL, or ABM, the line of the solar apsides, is 3 s. 5 30' of longitude, SE or >A will be in 3 s. 8 30' 6" of longitude; and, "adding AY = 7 30' 10", YA will be in 3s. 16 0' 16" ; that is, 1 the position of the earth, when the greatest difference of second inequality takes place, is in 9s. 16 0' 16". The difference itself RIS, is easily found, with the sides RZ, ZY, of the right-angled triangle RZY ; for from these RYZ = RIS, comes out = 1 3' 32". The mean longitude also of the planet in R is easily found : for AE, according to T. Brahe, is in 4s. 23 32' 16" ; and since SEL = 44 59' 1C", its supplement SEF is = 135 Q' 50" = 4s. 15 0' 50"; and therefore ER in 9s. 8 33' 6". The true longitude, after applying the equation of first inequality, would be 8s. 27. The solution of this problem is perfectly the same with that given long afterwards by Sir Isaac Newton, in his Aritlmttica Uhiccisal.s, Problem 45. Yet these laborious investigations are not the only evidences of the scrupulous exactness, with which Kepler examined every part of the ancient theory, before he adventured to in- troduce his own. For, out of deference to T. Brahe, he mi- nutely traced all the possible effects of the proposed substitu- tion, in every one of the three systems ; and the trouble given, him by the complicated form of T. Brahe's system and theory was very great. But Kepler had given to T. Brahe, in his dying moments, a promise to this purpose, and not only the present investigations, but the two following parts of his commentary, are proofs how sacredly he fulfilled it. The present investigations also are not only given in the excentric forms, but even in the concentric. i. To communicate just notions of Kepler's scrupulous atten- tion and candour, it would be necessary to transcribe his whole book. In the present examination these three examples must suffice. On the 13th of April 1589, at 12h. 5', the longitude of Mars, deduced by T. Brahe with great attention from an ob- servation, 'was 7s. 3" 58' 31 ", and, after a correction for pa- rallax, stated at 7s. 3 57' 11". The time was Ih. 30 / before the opposition j and, as the diurnal motion found by the ob- servations NOTES. 325 serrations of some successive days was 22', the longitude in, opposition ou^ht to have been 7s. 3 55' 4-9". The table makes it 7s. 3 58' 10". On the 10th of November 1597, at 8h. SO', the observed longitude of Mars was doubly marked, first in 3 s. 3 30', and next in 5s. 4 1'. A mean of both is 3s. 3 45' 3(X'. The time assigned for the opposition is 3d. 5h. 5' later, in which interval the motion /';/ /://f/\Wr/i//V/, taken from Magini's tables, is 1 ].*/. The longitude in opposition ought therefore to have been 3 s. 2 M 30". In the table it US s. 2 2S'. The cause of the uncertainty, marked with respect to this obser- vation, is evident from -its date. T. Brahe had been obliged to withdraw from Huenna ; and, though his other instru- ments were left with his assistants, had found means to carry off with him (his radius) one of the most valuable. The loss to astronomy, by his removal at this time, was great. The opposition of 1597 was of the utmost consequence for detect- ing the parnl!ax cf Mars, and very few so favourable occur in the coarse cf a whole life-time. On the 13th of January, at Prague, the right ascension of Examina- Mars, determined by distances, at'llh. 50', was tion of th meanoppo- 1st. from the bright star in the first of Geiriini 134 23 39 which T 2d. from Cor leonis, or Regulus - - 134 27 37 Brahe , 3d. from Pollux - - - - 134 23 18 f " nned hi 4th. from the third star in the wing of Virgo 134 29 43 theor ^ Consequently, a mean of all the four determina- tionsis ..... 13421-33 ^Whence the longitude of Mars, at 1 1 h. 40' of mean time, re- duced to the meridian of Uraniburg, was 4s. 10 3 more than 3 f 43".8 in the interval 5h. 30' ; so that all which is left for the sum of the parallaxes is only 1' 4-6".2. Finding the results from these observations, when examined in a direct manner, so inconsiderable, and so far below the common opinion, Kepler proceeded to examine them by a converse method ; and, since the solar parallax was then esti- mated at 3', he supposed the horizontal parallax of Mars in his present position to be 4', and proceeded to investigate what sum this would give for the parallax in longitude in the former interval of 9h. 40' on the 17th of January. Tire sun's longitude at 5h. 20' was 10s. 7 22', and 10s. 7 31' at 15h. and therefore with the obliquity of the ecliptic given, His right ascensions were found to be 309 47 & f>09 56 To which, adding the apparent time con- verted into degrees - - 79 225 The right ascension of the meridian was 2845 17456 Longitude of the culminating point of the ecliptic - - - 3056 174-29 Its declination - - . 1150 212 Its altitude - - - 45 55 36 1 8 Angle of the ecliptic with the meridian 69 2.9 66 31 Altitude of the Nonagesimal - - % 49 12 42 20 Distance between the culminating and ori- ent points of the ecliptic 108 45 61 31 Distance between the culminatine point and the Nonagesimal - - 18 45 28 29 Longitude of the Nonagesimal - - 49 42 202 53 Longitude of Mars - - - 90 90 10 Distance of Mars from the Nonagesimal 40 19 112 43 Correspondent parallaxes of longitude 157.5 237.8 Y 4 The 328 NOTES. The sum of these parallaxes is 4' 35".3. As therefore the real motion at this time was 6', it must have been increased to 10' 35".3 by the sum of the parallaxes ; and as it was in- creased no farther than to 7' it is impossible that the hori- zontal parallax should be so great as 4'. Other observations made by T. Brahe for the same purpose, in the years 1585, 1587* &c. were examined with equal caie, and the results were found to be equally unsatisfactory. 8orre. gave a parallax so inconsiderable as was thought far below the truth, some gave none at all, and other results were di- rectly repugnant to the possibility of its existence. Kepler next details his own obs rvations on this subject, and nothing can exceed the candour and fairness with which they are related, or the ardour and patience with which they were made. In his investigation he employed the method of latitudes; and, as the longitudes xof Mars and Arcturus \vere at that time nearly the same, his observations were chiefly of distances from this fixed star. On the 17th and 19th of February 1601, his observations cliiefry served to prove that Mars was stationary, aad that his real distance from Arcturus was 29 18'. But on the 22d, when Os leonis was in the meridian, it was 29 15', and when Cor Scorpii was in the meridian, it was 29 19'; so that a variation of about 4' 15" on the latitude had taken place in the interval ; which, as Mars was still stationary, argued an horizontal parallax of no less than 9'. But, when Os leo, nis was in the meridian, the altitude of Mars was only ! 2 30'; and, if we apply to the planet T. Brahe's refractions of the fixed stars, the variation of latitude must be diminished 2 1 ' 18'\ so that it will become 1' 57"> and give an horizontal parallax of 4^ II'. But if we apply his solar refractions, the varia- tion of latitude must be diminished 4' 36", that is, the result \vould be contradictory to every degree of parallax. The observations of the 26th were also unfavourable to parallax, and the variation of distance from Arcturus amount- ed at the utmost to no more than 1'. L. The observations of the planet in the node, or near to it, were these. 1. On the 4th of March 1590, at 7h. 10', he found from the right ascension of Mars = 22 35' 10", and his declination = 9 26' north, that the longitude was Os 24 22' 56", and the latitude 3' 12" south. As the effects of pa- rallaj: and refraction seemed here to be equal, they were ne- glected. 2. On the 23d of 'January 1592, at lOh. 15', Mars found in Os. li 9 34' 30" pf longitude, with 2' of south Utitude j NOTTS. 329 latitude ; and here also the refraction was neglectecj, the al- p ctcrm ; na . titude being 25 in the meridian. But as the elongation from tion of the " the sun was 60, and therefore the horizontal parallax the longitude same with that of the sun, \vhich was reckoned at 3', a change of the WHS thereby supposed to be produced upon the latitude of nodes, nearly 2; for the circle of latitude nearly coincided with the meridian, and the planet was concluded to be actually in the node. 3. On the 10th of December 1593, Mars was also in the node ; for, after corrections for parallax and refraction, the latitude was no greater than 45'' north. 4. On the,27th of October 1595, at 12h. the latitude of Mars, cleared of paral- lax and refraction, was 2 20" south ; and on the 28th, after like corrections, it was 25" north ; and Kepler concluded that the planet had been in the node at Oh. on the 28th. These observations were mutually confirmed by reckoning 637 days, the time of a tropical revolution, backwards from Oh. on the 2Sth of October 1595; x by which means we are brought to the 10th of December 1593, the date of the third observation, and thence another perjod of 687 days will bring us to the 23d of January 1592, -when the planet was actually in the node ; and a third period cf (587 days will terminate at noon, on the 7th of March 15^0, when the arch of 3' 12' south latitude is found from the diurnal motion in latitude to have actually been described. In like manner, on the 3d of June 1595, at 19 h. 1(X, when the altitude of Mars was 8, his longitude was 7s. 13 36'4-CX', and latitude 3' 16" nonh. The parallax was insensible, be- cause the distance was greater than that of the sun; but the refraction was great, and chiefly affected the latitude, so that instead of 3' 16" north, it may be reckoned at c / 3", or even more minutes south. Also, on the 15th of April 1589, the observed latitude, at the time of the least distance of the pla- net from the earth, was 1 7' 0" north, and after 21 days it decreased to 6' K>' north. The rate of decrease was not indeed uniform, bat became gradually slower; the error hov.ever will be insensible, though we should calculate the time when the planet was in the ecliptic by the common ana- logy 6' : 2 8". The planet therefore was in the node on the 9th of May, or near the end of the 8th. Hence, reckoning three times 687 days backward, we are brought to the 29th of December 1594-, towards the end of that day, when the planet ought to be again found in the node. Accordingly, on the 3d of January 1595, that is, five cays after, the observed latitude at 19h. 10' was found to be *>' 16" south. Nor is any error committed by now giving five days to the motion of 3' 16" in latitude, and in 15S9 giving only 2J. 8h. to die motion of 6' 40" in latitude, for in 1594 the 330 NOTES. the planet was near the conjunction, and in 1589 near the opposition. M. The attention of Kepler in calculating the times and points of his apparent oppositions, was the same which we have seen employed in his examination of T. Brahc's mean oppositions. Apparent To prevent the possibility of suspecting that he had suited oppositions, the observations to his theory, he gives a particular account of his treatment of every one, noticing their minutest circum- stances, and especially pointing out those which might intro- duce the least ambiguity or uncertainty ; whether they arose from the imperfection of the observation, or from the inac- curate doctrines taught concerning parallax and refraction. In treating his own observations, he even describes the whole procedure of making them ; and his attention and fairness, in the conclusions deduced from all, certainly was never ex- ceeded. N. Tt is impossible to convey a just notion of the difficulties which attended the formation of Kepler's theory, or of his indefatigable patience in it, except by a repetition of his pro- cedure. This perhaps will be acceptable as an object of curiosity, and an example of the management of calculations, before the discovery of logarithms. The oppositions he employed were those of the years 1587; 1591, 1593, and 1595 ; in which the observed longitudes were 5s. 25 43' 0"; 8s. 26 43' 0''; lls. 12 16' 0"; and 1 s. 17 31' 40"; and the mean longitude, calculated from T. Brahe's tables, 6s, 4?' 40"; 9s. 5 43' 55" ; lls. 9 55' 4" ; and Is. 7 14' 9". To clear these of every degree of foreign mo- tion, they were reduced to the year 1587, by corrections taken from the same tables for the precession ; and these corrections were, for the first interval from 15b7 to 1591, & 37"; for the second, from 1587 to 1593, 5' 30": and 7' 18" for the third interval from 1587 to 1595. They therefore became App. long. AF = 5* 25 43 ; AE = 8 26 39 43 ; Mean long. CF = 6 47 40 ; CE = 9 5 40 18 ; App. long. AD = 11 12 10 30 ; AG = 1 17 24 22 Mean long. CD = 11 9 49 34; CG = 1 7 6 51. In NOTES. 331 .J 5 1 J H "^ 2 i < 6 c 2 ! J> 2 3 S = T. K O rt b ** j ^ w 's S 8 S 1o* a^a 1^5i"5l*2 P 1-7-T ^ 5jQ rrf W O ~ ^ c X"B ^ ^ ^rt-5^^0 =coo K" "* f ^ CJ . r-s S 5J QJ f"i r< ~ wj ^|o^J . CJU ^ if 11*1 - s ! OU X g O o ** 3 uo 5 O oo oc; CM CO ^ -^ CO 010 '0 S 01 oo Q, ,o ,0 .5 c -- .2 H J w *SD b C I s - . r , ^ S8B O il O O .So Brj'Sa nj ^*^ gj ^ .5 "3 ^ C C*. Q ^ O, .. < 0-S'So 1 l i ~< o> M 8 < fa/3 ^ rt .s ;*; s fcJO J c o g-a 332 NOTES. ^H GO 01 ^ Ofa PO O O ,.0*0 "CO - Itt v-1 488 5 US & 00 T* CO 2 '0 -H O1 CO CO 00 t i i co ~ t- CO o 02 j^ CM Ol CO 1-- -* t- 1:0 s ^ 2 < II II GO << , s 2O O1 <:o . t>- 01 oo GO 00 ^Jg-fe^^O g II II i , WO ^ s II II < rt t h x-s ^ P C' \_j hH > ( tfPOC S b ii ^ - ta -g v, -r G a > a .5 " ~ | 2 y -Ifg g| c .5: S 15 s II II << S << + I ^o H II WO QQ + S O X) | S sgrf C tf-lficJ -S5S"?<= I O *- - II "^ -g rt ^ H rt*-* J r^'S > (J M? 5 a-a w P ^ rtl CN O S 2 3 513 GO ^> ^ CO 334 NOT2S. at D being now greater by 1' 48" than the defect at F, it fol- lows, that the points D, E, F, G, though more nearly than before, are not yet precisely in a circle. The former excess at D was less by 29 1 5" than the defect at F, and the present excess at D is 1' 48" greater ; and there- fore the addition to the longitude of the aphelion H has been too great. But, instead of again repeating the whole tedious process, it will be sufficient to employ the common rule of proportion. When the error was 29' 15", the sum of all the four half differences was 12 1' 43"; and now when the error is 1'48", their -sum is 12 19 46. The diiference is 18' 3"; and the difference of the errors, as they are in contrary di- rections, is 31' 3". As therefore 31' 3", the difference of the errors, to IS/ 3", the excess of the present sum abpve the former false one, so will be the present error 1' 48", to the ex- cess of the present sum above its just amount. This will be found = 1' 2" ; and subtracted from 12 Iff 46" gives 12 18' 44" for the just sum required ; so that the excess at I), and the defect at F, ought each to be = 6 9> 22". Supposing then that, in consequence of this correction, the points D, K, F, G, are in the circumference of the same cir- cle, it is next to be enquired if its centre B be situated in AC. Here 1. Since the \ suppl. of FAE is = 44 31' 48", and of FAG = 25 50' 41", their sum will be = 70 22' 29" ; whence, subtracting 6 9 22'', which ought to be the defect at F, the remainder will be 64 13' 1" = EFG ; and, conse- quently EBG \vill be = 128 26/ 14". 2. Since EAG = (EAD + DAG =) 140 44/59", its \ suppl. will be = 19 3?' 40". But, in resolving the triangle EAG, in order to find EG and AEG, a correction must be made upon the sides AE, AG, in consequence of the change on the angles at D and F. This will be found by consider- ing, that since the increase of the sum of all the four half differences, from 12 1' 43" to 12 I*/ 46", was too great, the variations of AE and AG must have been feoo great also ; and, as the first was increased by 37' and the second dimi- nished by 21', that now, since 12 ]& 46 ;/ is by a correction of 1'2" reduced to 12 18' 44*, AE and AG must be again varied, by shortening the first and lengthening the second. As therefore 18' 8", the first variation of the sum of the half differences, is to 58', the sum of the changes produced upon AE and AG, so is V 2", the second variation of the sum of he half differences, to 3 the sum of the changes now to be made upon AE and AG ; and dividing this in the ratio of 37 to 21, AE will become = 50738.7, and AG = 52282.3. Therefore tan. (AEG AGE) = tan. suppl. EAG (AG NOTES. 335 and AEG = 19 5^51"; as also EG = sin. AEG 52282.6:3271 _ 34088 3. Since the base EG of the isosceles triangle EBG, and the vertical angle EBG are thus found, the angle BEG at the base is given, and = 25 46' 53" ; and, therefore, BE = EG. sin. BEG 97041,43494 7832 4. In the triangle BE A, the angle BE A is given, for it is = BEG AEG = 5 51' 2", and also its \ suppl. = 87" 4' 27". Therefore tan. \ (BAE ABE) = tan. * suppl. 17' 8"; so that BAE =117 21 '37". But, since in the second operation the aphelion H was found to be too far advanced in longitude, let it now, in con- sequence of the last correction, be considered as advanced no more than 3' 8", instead of 3' 20" beyond the longitude first assumed. Then, since AH is in 4s. 28 47' 8", and AE in 8s. 26 39' 23", CAR or HAE will be = 117 52' 5"; that is, greater by 3(X 28" than BAE ; and B is not situated in AC, but on side of it towards E. The suppositions there- -fore for FAH and FCH, must, one of them, or perhaps both, be false. But these angles of anomaly cannot be varied by the mere variation of the assramed longitude of the aphelion ; because no other position of it will permit the points D, K, F, G, to be situated in the circumference of the sam.e circle ; and be- fore it can be farther varied, the mean longitudes, or the po- sition of the lines FC, FE, &c. must be varied. This, there- fore, was the next step of Kepler's procedure ; and he tells us, it was not till after a great variety of unsuccessful trials, that he found his purpose would be nearly accomplished by the addition of 2' more to the longitude of the aphelion, and of 30" at the same time to the mean longitudes. By these additions the mean anomalies FCH, ECH, &c. are all dimi- nished V 30" each ; and we have FCH = 32 3' 36"; KCE = 53 7' 2"; KCD = 11 0' 44"; and KCG = 68 18' J". The angles again of equation will become AFC = 5 8' 26"; A EC = 9 4' 41"; ADC = 2 17' 1C"; and AGC = 10 13' 45"; being increased 30" in the first semi- circle of anomaly, and as much diminished in the second : consequently, the * lines 336 NOTES. o t? S a fe 60 II W l~ 1 rt 1 II r 6 ii s "rt < S I r-i I G S O O 'S .S .g | 'o 8 O S "5 00 CO ^*5o c ^^ .o C ^ L O ulr* O pC _^ fi JJ ^ S ,-sl 3.1s n "*S ^ I s - ca t* %+ & ^^ Ol I I 30 6) -3 g cc ^o co^ I! W ^ GO I! ii O ^^ n-g aS o w o W y w S e 766 ." <} J i .- .s ^ nn o I ^ ^s. > l *S 'Zl G S I c D . sbts rt a + .2 OD

, and no two observations from the earth in these opposite points, made of the planet, either in 6s % 5 30', or Os. 5 3 a, could be found in T. Brahe's whole collection. It became necessary, therefore, to investigate the times in which, while Mars w r as near to either of these points of quadrature, the angles of commu- tation KCF, LCF, though not right angles, were equal to each other. When these times were found, T. Brahe's col- lection was next to be .examined, to discover if in them any observations of the planet had been made ; and if his ardour had not been indefatigable, no pair with such concurring conditions could have been found. As the heliocentric places of Mars were required to be in 6 s. 5 3C' and Os. 5 30', and the apogee of his orbit, accord- ing to T. Brahe, was in 4s. 23 30', the required true ano- maly was 42, or 318; and T. Brahe's equation, at either of these points, was 8 15' 16". The requisite mean anomaly therefore was 50 15' 56", or 309 44' 4" ; and Mars was twelve times in the former of these during the course of T. Brahe's observations. It was therefore to be enquired if any of these twelve times, in which Mars was in 50 15' 16" of mean anomaly, and con- sequently in a line perpendicular to the line of the earth's ap- sides, the earth was to be found in two points G, H, or K, L, of her orbit, making with the planet the equal angles of commutation FCG, FCH, or FCK, FCL. A revolution of Mars is performed in 687 days, and two revolutions NOTES. 339 revolutions of the earth in 730* days. The difference of these periods is 4.S* ; during which time the earth moves through 42 54' 23". If, then, in the space of two years, two places of the earth be required, which shall make equal angles, in equal times, with a perpendicular from Mars to the line of apsides, on each side of it, each place must make an angle with it of 21 27'. In four years they must form similarly situated angles of 42 />4 ; ; in six years, of 64- 22'; and in eight years, of 85 4*8'. No such pair at the distance of eight years were to be found in T. Brahe's register ; but a pair, at the distance of six years, was found as proper for the purpose as could reasonably be expected. These were the observations of the 30th of May 1585, at 5h. and of January 20th, 1.391, at Oh. ; in both which the mean heliocentric lon- gitude of Mars was 6 s. 22 43' 8", and therefore the true he- liocentric longitude, after applying T. Brahe's equation, 6 s. 13 28' 16"; and the earth in the former being at K, in 8s. 17 51' 46", and in the latter at L, in 4s. 9 4' 46", the angles of commutation KCF, LCF, were each = 64 23' 30". The observations, indeed, were not actually made at the precise instants now mentioned, and only reduced to them from May 18th and January 22d, by means of the diurnal motions in Magini's tables ; because the commutations would not other- wise have been equal. In the former observation, indeed, the distance of the earth from the aphelion, in 9 s. 5 30' was only 17 38', in antecedenlia ; and in the latter, the distance from the perihelion was no less In conscquentia than 33 35'; but this was an imperfection which could not be avoided. The investigation of the distance BC, between the centres of the orbit and the equant, was to this effect. Join BK, BL, KL ; from L draw LP perpendicular to KC produced ; draw, from B and C, the perpendiculars BM and CN to KL ; and from N draw to BM die line NO parallel to the line of apsides DE. Assume FC = 100000. Since KFC = 36 51', and KCF = 64 23' 30", we shall haye the angle FKC = 78 45' 30", and therefore KC = 6114& ; and, in the same manner, since LFC = 38 6', LC will be found = 63186. Again, since KCL = 128 47' 19", PCL will be = 51 12' 41", and PLC = 38 47' 19"; and therefore, since LC = 63186, PL will be found = 49251, and PC = 39583, PL Therefore PK = 100731 ; and since tan. LKP = - , thij PK angle will be found = 26 3' 21", and its secant KL '= 112126, and therefore LM = KL = 56063. But CLK = 180 (KCL + LKP) = 25 9' 21", and therefore CN = CL, sin. CLK = 26859, and LN = CL. cos CLK = 57193. Z 2 Therefore 340 NOTES. Therefore MN = IN LM = 1130. Since also, KCE =* 17 38', and LKC = 26 3' 21", we shall have ONM = LKC KCE = 8 25' 21". Therefore MO = MN. tan. ONM = 167 ; and NO = BC = 11 43. But BM == CN + MO = 27025; and KB = RMf.Of BM* ** 62237 ; and therefore BC, in parts of KB = 1 00000, will be found == 1837. Two conditions are indeed assumed in this investigation, viz. that the earth's orbit is a perfect circle, and that the place of its aphelion, as given by T. Brahe, is just. But though neither of these be accurately true, the certainty of the general conclusion will not thereby be affected. p. In the above experiment, the mean longitude of Mars, on the 5th of March 1590, at 7h. 10', was Is. 4 38' 50", ac- cording to the Rhudolphine tables, and the equation 11 14' 55" additive ; so that the true heliocentric longitude was 1 s. 15 53' 45". To the mean longitude 1' 36" was added for the precession during every interval, and the remaining true lon r gitudes became Is. 15 55' 23"; ls.U5 56' 56"; Is. 15 58' 30"; on the application of the proper equations. The mean places of the earth were, E in 5s. 22 58' 46"; F in 4s. 10 5' 57"; G in 2s. 27 13' 12"; H in Is. 14 20' 25". The angles of commutation therefore were ECM = 127 5' 1"; FCM=: 84 10' 34"; GCM = 41 16' 16"; and HCM = 1 38' 5". The observed longitudes of Mars were, from E, Os 25 6' 0"; from F, Os. 19 19' 0"; from G, Os. 3 35' 30"; and from H, Is. 19 21' 35". Consequently the parallaxes were EMC = 20 47' 45' ; FMC = 35 46' 23"; GMC = 42 21' 26"; and HMC = 3 23' 5". Whence the elongations came out CEM = 32 7' 14"; CFM = 60 3' 3"; CGM == 96 22' 18"; and CHM 174 58' 50". When, with the.se data, the distances CE, CF, CQ-, CH ? were found, the distance BC of C from the centre of the or- bit was to be investigated. Join BE, BF, .BG, and EF, EG, FG. As all the angles about C are equal, and each of them, as ECF consists of 42 52' 47", we shall ha've in the triangle ECF, with this angle, and the sides CE, CF, the base EF 49169, and CEF 69 18' 46": and in the same manner, in the triangles FCQ, ECG, the angle CGF will be found = 68 12' 6"; and CGE = 46 39' 10" " therefore EGF = CGF CGE =* 21 33' 16", and thence NOTES. 341 thence EBF at the centre = 43 d 6' 30", and BEF = 68 26' 44' ; and from these, with EF also found, the semi-diameter BF = 6692-3. Therefore, finally, in the triangle BEC, with the angle BEC = CEF BEF = 52' 2", and the sides Bt, CE, the base BC will be found = 1023, that is, = 1530 in parts of BE = 100000; and die angle ECB =97 50' 3; f ;. When 3.17 When the true anomaly DAH is = 45*, the distance AH will be found by drawing from the centre B, the line BN, perpendicular to AH, and joining BH. Then, AN = AB. cos. 45" = 1272 = BN. Whence BHN == 4:r 45", and its co-sine HN = 99992, and therefore AH = HN -j- AN = 101264. Whence also, in the opposite degree 225 of true anomaly, by producing AH to meet the orbit again in K, we shall have AK = HN AN = 98720. In the former hypothesis these distances are AQ = 102514, and AV = 97422, found by drawing from C the line CX -perpendicular to OAV. It likewise appears that the mean distances, though the same as in the ancient hypothesis, do not take place at 90 ' and 270 3 of true anomaly ; but, as Reinholdus had observed with respect to the planetary orbits, at two points where a parallel to FAG, bisecting AB, meets the orbit DHFE ; and that, as Ptolemy had also observed, in all cases where the bisection takes place, the greatest equations do not hap- pen at these points of mean distance, but at F and G, in 90* and 270 of true anomaly. But, though Kepler expected that these new distances of the earth from the sun, would produce considerable altera- tions on the elements of the planetary theories, he shews by experiment, that their effect on the ancient solar equations would be altogether inconsiderable. At the mean distances towards F and G, the effect is alto- gether insensible. For, whether CA be considered as the sine of the angle CRA to the radius CR, or" as the tangent of the angle CFA to the radius AF, the equations will be almost perfectly the same. In other points, as at the octants, though the equations must be calculated by different methods, their difference is still very inconsiderable. In the ancient theory, the equation at 45 is found by a single operation, to wit, sin. AQC = AC 1 *r 4'" ' = ! 27' 31"; and this is also its amount at 135 of true anomaly. In the new theory, again, it is found by AB. sin. 45 a double operation; for 1st. sm. AHB = ^-r-. '- 43 13 ri 46"; and, 2dly, tan. \ (BCH ~ BHC) = tun. suppU T TT "PO CBH (-oTr TTTT) ; whence the physical equation at 45^ of anomaly, will be BHC = 43' 38"; and at 135', it will be = 43 42 : and consequently the whole equation AHC, at the Srst octant/will be == P 27' 24", and at the second 1 27' 28''. As 313 NOTES. As the differences between the physical and optical equa- tions, in an orbit so little excentrical, were in most cases in- sensible, Kepler thought it sufficient to calculate the optical equation only, and by doubling it to obtain the whole. So that in both octants it was P 27' 32". The disposition of the distances in Kepler's table of them was peculiar. They were calculated for compleat degrees of true anomaly, but the corresponding true anomalies are not given. The anomalies, which are marked in the table, on each side of every distance, are, the one less than the mean anomaly by the real part of the sol v ar equation, and the other less than the true by the optical part of it. Con- sequently, as Kepler considered these two- parts of the equa- tion to be sensibly equal, the true anomaly is the mean be- tween the anomalies marked in the table. The disposition was made in this manner. Mean anomalies. Distances* True anomalies. Mean of them, " o i ti o i ii o 45 43 45 101265 44 16 15 45 46 44 30 101242 45 15 30 46 47 45 15 101219 46 14 45 4? Thus the anomalies, entitled mean, are not the angles de- scribed about C the centre of the equant, but the angles form- ed at B the centre of the orbit, when the optical equations r as BHA, are added to the angles formed at A : and the ano- malies, entitled true, are net the angles described about the sun in A, but angles less than these, and found by subtract- ing equations, equal to the optical ones, from the angles at A. It is true that when, in consequence of this arrangement, you apply to the table with a given true anomaly, as 44 16' 15", the corresponding distance 101265 is not the radius vec- tor of the earth in 44 16' 15" of true anomaly, but in 45; and it is therefore too short : or, if you apply to the table with a given mean anomaly, as 45 43' 45", the distance 101265 is too short for this mean anomaly, for it really be- longs to the mean anomaly 46 2/' 30". But this was done purposely, when Kepler discovered that the orbit between the apsides retired within the circle, th^t the distances might fall as much short of the circular distances as the limits of the oval required. v U.. 1. In calculating equations, according to this natural prin- ciple, Kepler generally abridged his labour, by reckoning the physical and optical parts of the equation equal. Indeed, ia NOTES. 349 in orbits so little excentrical as that of the earth, the differ- ence was almost insensible. For, if on the centre fcl (fig. 64-), with the distance EB we describe the arch BX, to meat EA in X, the sector BEX will represent the optical equation as accurately as the angle BE A. Now this sector dilfers from the area BEA, that is, from the physical equation, only by the very small area, or triangle, BXA ; and, in the solar or- bit, this difference, even at the octants, does not exceed 33 seconds. But in more excentrical orbits it is by no means to be neglected. 2. As Kepler observed that, when a circle is divided into 3t30, or any indefinite number of equal parts by diameters, the sum of the distances of the points of division, from any excentric point A, is greater than the sum of the distances of the same points from the centre B ; that is, greater than the area of the circle : so he likewise observed, that if as many lines, making equal angles with one another, be drawn through A to the circumference, the sum of the distances of their ex- tremities from A, will be less than the sum of the distances BD, BY, of the same extremities from the centre B, that is, less than the area of the circle ; this area being in both cases considered as equivalent to the sum of the indefinite semi- diameters. 3. To shew the real excess of the sums of the distances from A, in the arches CD, CE, CF, &c. above the areas CD A, CEA, CFA, &c. employed to measure them, he sup- posed a straight line cdtfghk (fig. 65), to have been found equal to the circumference CDEFGHK, and its equal parts pd, de, ef, &c. equal to the arches CD, DE, EF, &c. From c, d, <-, &c. he drew lines cl- 9 db^ r-6, Sec. each equal to the semi-diameter BC, and perpendicular to and by joining the points <, 6, 6 9 Tiad a parallelogram bckb, double of the tri- angle by which Archimedes attempted to measure the area of the semi-circle; and small parallelograms c6 t db, &c. double of every particular sector CD B, DkB, &c. He took also in c6, db> eb> &c. produced when necessary, the distances ca, da, ea, c. respectively equal to AC, AD, AE, &c. ; and joining the points a, a, a, bf the curve line aaa, had the figure acka, double of the sum of the distances AC, AD, AE, &c. ; and in which the lines ca, da, ckb ; and consequently double of the semi-circular area : so that the lunula comprehended between the two curves was double the excess of the sum of the distances AC, AD, AE, &c. above the sum of the areas CD A, DEA, &c. that is, above the area of the semi-circle As Kepler could find no accurate quadrature of these areas, he calls upon all the geometricians of the age for their assistance. He observes also, that the breadth or the lunula is greatest towards the points p and q\ and that it seems to be bisected, not by the line af, but by bbb. This however he also recommends to all geometricians, as a subject worthy of $heir investigation. V. In shewing how inconsiderable the difference is between the area of a semi-circle, that is, the sum of all the distances of the indefinite number of equal parts of its circumference from the centre, and the sum of the distances of the same parts from the excentric point A (fig. 64), he observes, that this difference is the space anmopyaa comprehended between the curves of fig. 65. The greatest breadth of this space is at the point o 9 where the line fo = />, is equal to FB the semi-diameter of the orbit ; and /?; is equal to FA =1004-32, the secant of the"greatest optical equation BFA. From their difference an 432, the sum of the other excesses na, ma % pa, qa, may nearly be inferred. For, if all the particular excesses, on each side of the greatest no, were to ao, as the sum of all the sines of excen- tric anomaly on each side of the semi-diameter, or sinus totvs Y FB, is to FB, then as FB = 100000 to 11458869, the sum f>f the sines, so would the excess ao = 432 be to 49934, the srm of the excesses of the secants of the optical equations above their correspondent rad.-i.- But the excesses na, ma, on> pa, qa, are not to one another as the sines of excentric anomaly, but nearly in the duplicate ratio of those sines. The sine of 90, for example, is double the sine of 30. But the optical equation at 90 of excentric Anomaly, is 5 !&', and the excess of its secant above -the ra- dius 482. The optical equation again at 30 of excentric anomaly, like the sine of 3u, is very nearly the half of 5 19'. But the excess of the secant of this optical equation above the radius is only 107> that is but a fourth part of 432. After- wards tkese excesses become almost insensible ; and, by ac- tually computing them, the sum amounts to no more than 7000, which is not fully the fotiitli part of 499^4. The NOTES. 351 W. The demonstration that the lunula cut ofF by the ellipse of the oval theorv from the plane of the excentric, is but in- sensibly greater than the area of the semi-circle AZB, is to this purpose- JLet AB (fig. 68), be bisected in Q, draw QR perpendi- cular to AB, and meeting the excentric in R ; join RA, RB, and from S, the centre of the equant, draw to the same cir- cumference the line ST parallel to RB ; join BT, AT, and from A as a centre, with the distance AR, describe an arch cutting AT in V, and BT in X, and join AX. Since R is at an equal distance from both A and B, AR will be the mean distance of the planet from A the sun. But because S T is parallel to BR, and AX ^ AR, the point X is the position of the planet in its oval orbit, at the time CBR or CST, and when in the excentric it would be found in R ; and TX, that part of BT which is intercepted between the arch RX and the excentric, will nearly measure the extreme breadth of the lunula CXKRT. It would be accurately measured by a line from X parallel to QR. From B draw to AR the perpendicular BZ. Then will XT be double of AZ. For, join RT, and from R draw to BT the perpendicular RN; and from X to AR the perpen- dicular XO. Then, since ABR = BST, and BT = BR = AR, and BS = AB, the triangle TSB will be similar and equal to RBA. Therefore AB will be parallel and equal to TR ; and, since RN is parallel and equal to BZ, AZ will be = TN. Since, also, RN = OX, and BR = AR = AX, and the angles at O and N right, the triangles BRN, AXO, will be similar and equal; and therefore AC = BN; and, consequently AZ = NX. Therefore TN = NX, and TX = 2AZ. Since, then, a perpendicular RN is drawn from a point R in the circumference of a circle to its diameter J3T, we shall have RN* = (BT + BN) TN. But TR> = AB* = RN 2 4- TN 2 ; and therefore AB = (BT +. BN) TN -f TN 2 = 2BT. TN = BT. TX. But, according to Archi- medes, BT 2 is to BT. TX nearly as the area of the circle to the area of the ellipse ; and therefore BT a BT. TX, or BT 2 AB 2 will be nearly the difference between these areas, that is, nearly equal to twice the lunula CXKRT. The point X is a point of the orbit described by the assist- ance of the vicarious theory : and as it falls so far within the ellipse CNK, whose excentricity is BA, and its greater axis. CK, the errors of the oval theory, and the propriety of the improvement afterwards made upon it, become evident al- most by inspection. 352 NOTES. But it was not only necessary to find the quadrature of the lunula, but also to divide it in any given ratio by a line drawn from the excentric point A ; and one of Kepler's attempts for this purpose was as follows. Let the se.ru-circle CFK (fig. 64*) be, as formerly, extended into the straight line tfk (fig. 65), which is divided into the same number of equal parts, by per- pendiculars ca 9 rfa> &c. representing the distances from A. Let the quadrant CF be = as this also is to the semi-diameter ; and so on continually, till all the lines of this kind be determined ; so as to represent the breadth of the lunula at all the correspond- ing points of the semi-circle: and let all the points thus found be joined by the curve ctwxk* Since, then, the whole space included between rfk and the curve aoa, is double of the area of the semi-circle, it seemed probable that the space included between r/fr, and the curve ci-k, was also double of the lunula ;- and every portion of it, as ctdy double of the correspondent portion of the lunula. But not finding any geometrical demonstration of these things, be again implores the assistance of geometers. X, In his attempts for the rectification of the oval circumfer- ence, Kepler endeavoured to derive some assistance from geometry. For, let BC : BA :: BA : CN (fig. 70), so that CN shall be equal to the extreme breadth of the lunula ; and from B, towards N, take BL = CN. On the centre L, with the distance LC, describe the circle CQM, touching the ex- centric in C ; and on the centre B, with the distance BN, describe the circle NOM, which consequently will touch the oval CEK in O, the extremity of its conjugate axis BO, and the circle CQM in M, a point of the line of apsides. It is evident that the circle NOM is less than CQM, and CQM less than CHK ; and, since the semi-diameters CB, CL, and BN, of tkese circles are in arithmetical proportion, the cir- cumfcience CQM will be an arithmetical mean between CHK and NOM. Since, therefore, the oval circumference COK touches CHK in C and K, and NOM in O, and the point opposite to O, it seems not to differ sensibly in length from the cir* cumference COM, It appears however to be somewhat, though very inconsi- derably, longer. For tjie area of a circle, whose semi-diameter NOtES. 353 is a geometrical mean between the two semi-axes of ah ellipse, is eqital to the area of the ellipse ; and supposing the present oval sensibly equal to an ellipse described on the same axes, its circumference will therefore be longer than that of the circle now mentioned, and probably therefore longer than the: circumference CQM. For the semi-diameter of CQM is an arithmetical mean between CB and BN ; and therefore longer than a geometrical mean, though, in the present circum- stances, very inconsiderably. Since, then, CN = 858, BL = 4-29, and BC = 100000, we shall have LC == 99571 ; and, since 100000 : 99571 :: 3(>a> : ;*58 J 27' 20 ', this will be the length of the whole oval circumference, and therefore its half CO K = 1~9 13' 40''. It was not however fn-m this investigation, but from the la- borious calculations above mentioned, that Kepler concluded it to be = 179 14-' 15". As the decurtation of the oval circumference is nearly equal to the optical amplification of it, Kepler also observes, that the laborious operation, of first shortening and after- wards enlarging every particular arch of the oval, might perhaps be thought unnecessary ; and no doubt it would have been unnecessary, if the decurtation and amplification had in all points of the orbit been the same. But though the de- curtation be in all points the same, the optical amplification is variable ; and, for example, the defect of the whole oval semi-circumference being 45' 4-5", the decurtation of every decree of it must be = 15"; whereas, at the aphelion, th optical amplification of an arch of 1 does not exceed l". Y. Though the attempts to be now mentioned for deducing the equations of Mars, in the oval theory, were equally un- successful with the rest, it seems impossible to form a just conception, either of the character of Kepler, or of his pro- gress in the prosecution of his discoveries, without consider- ing them. 1. He calculated for the excentricity 9165, all the dis- tances of the planet from the sun, which corresponded to com- pleat decrees of that kind of anomaly, which is a mean be- tween the true and mean anomalies ; and which, as being only useful in calculating distances, he called the distantiarj anomaly. This is the angle LAH (fig. 69), for LBH in that figure represents the mean anomaly. The sum of these dis- tances he found = 35921-252, and less than 36000000, the sum of 360 semi-dia meters ; for it is the sum only of 180 lines drawn through A to meet the excentric* He then had A a this 354 % NOTES. tins analogy, a $ 35924252 to 360, or the whole periodical time, so is the sum of any given number of distances to the time required ; and with a view of obtaining accuracy, each distance employed was a mean of the two by which every arch was terminated. This procedure however led to an absurdity ; for, let B (fig. 71), be the centre of the excentric DCL, used for com- puting the distances, and where DBC represents the .mean anomaly, and let A be the sun, and on A , with the distance AK t= BD, describe the circle KHP. Then, if the distan^ twry anomaly DAC be = 45, the sum of the distances of all its 4* degrees from A will be = 4867852 ; and from the analogy 35924252 : 360 :: 4867852 : 48 46' 51", this last will represent the time of describing the arch DC ; that is, it will be equivalent to the area DAC. But this is only a very little greater than the sector DBC ; for if, with the given angle DAC, and the sides BC, BA, we calculate the angle or sector DBC, it will be found = 48 42' 5)5". In contra- diction therefore to the hypothesis, the planet wa ; represented as moving round A with almost perfect equability. The cause of the error soon appeared. The distances cor- responded to equal divisions of the angle CAD, that is, of the arch KH ; and consequently to unequal divisions of the arch DC : and, the divisions of DC being longest where the distances from A were greatest, the distances were too few in number ; and therefore the area CAD, supposed to be equivalent to their sum, too small. That his labour, how- ever, might not be wholly lost, he subtracted from the angle CAD = 45 of distantiary anomaly, the difference 3 46' 51", between it and the mean anomaly CBD, and thus obtained the true anomaly EAD = 41 13' 9". Then, supposing that the planet, in the time CBD, described about B an angle EBD = CAD, he thus collected together as many distances in the smaller arch ED, or OD, and distributed them among equal parts of it, as had by mistake been distributed among unequal parts of CD ; and thus from the distantiary anomaly CAD and BC, BA, given, he was enabled to calculate every distance CA = EA, and every physical equation 33EA = C -VE. But, though the lines EB and CA were here falsely- supposed parallel, the equations differed very little from those Deduced in the original form of the theory (156). For to the Mean anomalies. The true anom. were Vicarious theory. Differences* 48* 42' 59" 41 13' 9" 41 21' 0" 7' 51" 95 15 31 84 44 18 84 39 18 5 -f- 138 42 59 131 20 24 131 4 7 Ifr 17 -f and the planet's motions at t{;e apsides continued too slow, and at the mean distances too rapid. 2, He NOTES. 355 2. He calculated with the same excentricity a set of third proportionals, every one of which was to every one of the" former distances, as that distance to the semi-diameter of die orbit. These third proportionals were also measures of the times in which the planet describes very small arches of its orbit. For, since the times of describing such arches were as the squares of their distances, that is, as AH- to AC 2 (138), and AH : AC :: AC : AF, these times will also be as AH to AF. This procedure however supposed, that the distance AC was both of a just magnitude, and in a ju-t position ; or that the orbit, contrary to the theory, was a perfect circle. The effects therefore were what might have been expected. The distances between the apsides being too long, the third proportionals were also too long ; and consequently the pla- net's motions were in these parts too slow, and at the apsides too rapid. For to the True anom. The mean anom. were Vicarious theory. Differences. 45 (X 0" 52 39' 40" 52 53' 0" 13'20:' 90 00 100 29 12 100 34? 30 5 18 135 142 10 47 142 90 1 47 4- On adding together the third proportionals, Kepler found, with some surprize, that their sum was precisely = 36000000; and he proposes the fact in this manner for the investigation of geometers. Let two equal circles CD, KH, be described on the centres B and A ; join AB, producing it to meet CD in D, L, and KH in K, P ; divide the circumference KHP into 360 equal arches, draw through all the points of division lines AK, AH, AP, &c. cutting the circumference DCL in D, C, L, &c. and make AK : AD :: AD : AG ; and AH : AC :: AC : AF ; and AP : AL :: AL : AM, &c. It is re^ quired to demonstrate, that AG + AF + AM, &c. = 86000000. 3. As, by calculating the distances and third proportionals corresponding to equal divisions of the angle CAD, this angle no longer represented the distantiary anomaly, but instead of it the true, and rendered the orbit in the second attempt a circle ; so, for a third attempt he reduced, by the common rule of analogy, his distances from unequal divisions of the excentric, to become the distances for equal divisions of it ; and then, with the excentric anomaly CD, or CBD, consist- ing of any given number, suppose 45, of these divisions, he calculated the angle CAD, and added together the 45 dis- tances of the divisions in CD, in order to give him the time in which CAD was described. But the same mistake was by these means repeated ; for CAD being the true, and not the distantiary anomaly, the distances were represented as being A a 2 just, NOTES. just, in position a's well as magnitude, and the orbit agtfitt became a perfect circle, in which the motions, as before, were .too- rapid at the apsider,, and too slow about the mean dis- tances. For the effects were these : Mean anom. True anom. Vicarious theory. Differences. 48*38' 31" 41 31' 0" 41 17' 6" 13' 54" + 95 13 58 84 45 50 84 37 45 85 + 138 45 41 131 1 52 131 7 13 5 24. 4. He next calculated a set of third proportionals to the distances now described, which corresponded to compleat degrees of excentric anomaly CD. Their sum was found to be =,36384621. But by these the errors were increased. For the sum of the distances was 36075562 ; and if the first 45 of these gave too long a time for describing the arch CD, the third proportionals belonging to them must have given a time still longer. 5. His next recourse therefore was to the table of true ano- malies, calculated on the principles of the vicarious theory, and to the former combination of it with the oval theory, m order to obtain distances, accurate both in position and mag- nitude (152). Since, then, the distances AH (fig. 69), cor- responding to compleat degrees of mean anomaly LBH, OP LDG, corresponded of consequence to fractional parts of true anomaly LAK, he reduced them by the common rule of proportion to compleat degrees of true anomaly, and found that the sum of such distances, now corresponding to all the equal angles about the point A, was 35770014. But the pro- cedure continued to be unsuccessful. For, though the sum was less than 360 semi-diameters, the divisions of the angle LAG being equal, those of the oval EK were rendered un- equal, and too few distances were allotted to it in the parts towards the aphelion ; insomuch, that when their sum was added together to make up the area EKA, it did not come out equal to the assumed mean anomaly EBH. For example, if EBH be equal to 48 44', the angle EAH of distantiary anomaly will, from BH and BA also given, be found = 45, and the distance AH = 105784 ; but this distance is, by the vicarious hypothesis, carried upwards to AK; and the true anomaly given by that hypothesis is EAK = 41 22'. When this angle EAK, therefore, is divided into equal parts, it contains only 41 distances, and something more than a third ; and the sum of these will not give an area EKA equal to the sector or mean anomaly EBH, but an area mueh less. 6. A sixth attempt therefore was made, by calculating a set of third proportionals to the semi-diameters, and every one NOTES. 357 er.e of the distances now described; and these therefore were measures of the times in which the planet describes those arches of its orbit, which subtend equal and indefinitely small angles of true anomaly. Their sum was 356924-08 ; and, as this was equivalents the whole periodical time, or 360*, so the time, or mean anomaly corresponding to any given num- ber of them, was found by the common rule of analogy. The results were these : True anom. IVIean acorn. Vicarious theory. Differences. 41 0' 0" 48 24.' 3" 48 12' 9" 5' 1" + 81 91 30 39 91 34 8 3 29 91 101 28 10 101 34. 7 5 27 131 138 28 5 138 39 28 11 23 It is true, that by increasing the excentricity, the errors vottld be diminished ; but the planet would still be too slow about the apsides, and too rapid in the intermediate parts. This method was equivalent to that of art. 156, and in prac- tice more convenient ; and it neglected the supposed revolu- tion of the planet in an epicycle, and only considered the variation of its distances from the sun ; that is, its librations. But though, on this account, it approached a step nearer to the truth, the errors shewed the librations to be too great : and they clearly appeared to arise from the principles of the theory, and not from the imperfect manner of expressing them in the calculation. z. That elongation of the earth, from a line joining the centres of the planet and the sun, which will most sensibly shew any error committed on the planet's distance from the sun, may be thus investigated. Let A (fig. 75), be the sun, B the planet, CD the orbit of the earth. From B draw the straight line BE perpendicular to BA, and find in BE a point E, such that a circle de- scribed upon it with the semi-diameter BE, shall touch the orbit of the earth. Through the point of contact C, draw ACE, and the line CF perpendicular to AB. The sine CF of the required angle CAB, will be to the distance CA of the earth from the sun, as the excess BF of the distance of the planet from the sun, above the co-sine of CAB, is to the planet's distance CA. For EC (= EB) : EA :: CF : CA; and EC : EA :: BF : BA ; and, therefore CF : CA :: BF : BA ; and, since BF differs but little from BD, CF : CA :: BD : BA. Aa3 If 358 KOTES, < II M5 ssi 00 CO co P> O3 o 22^ c^ o o o Ol ^ CO 1^ ^ o *- ?o -< >c * co -* O * O CO O CO CD C7^ O O O CO CO CO Tf CO CO O >O 01 - 01 CO -< CX rH NOTES. '359 ill! a 2 2 C5 W X X ci rf ^r en *~ X CO Cl 2.SSJ 00 X X 00 O ..''"* -r ox So t- as as o fc i- r- t- i- t~ 0000 J Tf CC WJ 0-1 C-l c 71 s ^ -r 71 o 01 g CM o o o * 'o o - - o f 71 C X V5 Q t U) U T -00^ 5 o M + I S "rt -000 .5 . ^ 2220 -3 ~ ^ ^ g*^ o "S "^ ft* Ml O X C?) x\ P-i *** o o -^ *s e-3 Ci ;0 01 rS 2.} ^ 2 o -S j* ^ c^^o. J3 fi S ."2 M O C7 C5 c-i c; i- r^ ?1 X I- *0 r- x T} x- ;^ '"= - 1 ci Cl C5 Ci *-* *T3 ^ b -s s - 71 01 X ti U - - C 3 rt tail - 2 Aa-t 2 rt ^ T3 C^ U C 2 w ^C P .2 rt br. C T3 < 3 ' o c -s '- ' - 82-1*1 ' (U w ^ r-< 3 - o bo - S - &55 360 NOTES. extremity G, will be less than PL, at the nearest extremity P, though the arches GN, PR of the epicycle are equal. But, since it is now found that the orbit can no longer be consi- dered as circular, and that the planet's distances from the sun, corresponding to these equal arches of the epicycle, are AO, AS, and not AK, AL ; the more distant libration GO is rendered precisely equal to the least distant one PS. It is true, that still the middle part OS, is greater than either of the extremes GO, PS, though the arches GN, NR, PR, are equal ; but Kepler supposed that he could assign a natural and sufficient cause of this exception. As he had ascribed the revolution of a planet to the im- pulse of the sun's magnetical fibres, every way extended in straight lines from his centre, and continually carried about by his rotation on an axis ; he endeavours also to shew, that the librations of a planet, through the versed sines of the arches of its epicycle, may be ascribed to a like mechanical cause ; and that there is no necessity for supposing the planet endowed with animation or intelligence. His theory on this subject principally consists in applying to the planets Gilbert's doctrine cdncerning the earth, and which he had also applied to the sun, that every one of them may be considered as a great magnet ; and he endeavours to account for the librations now demonstrated to take place, and which were afterwards found to produce an elliptical orbit, by the constant parallelism of that axis-, which joins the planet's magnetical poles, to a. line perpendicular to the line of apsides. Let EGH (fig. 76), be a great magnet representing the planet Mars, EF the magnetical axis which joins the attrac- tive pole E, and the repulsive pole F, and whose positions, in all the various points C, D, K, T, of the orbit, are always in a line perpendicular to ABC ; and let the axis EF be bi- sected by the perpendicular diameter GH. It is evident that, when, in the course of the planet's constant revolution, it is found in the apsis C, or T, it will have no tendency either to approach the sun, or to withdraw from him ; because the perpendicular GH, coinciding with the line of apsides, passes through the centre of the sun ; and the angles ACE, ACF, being equal, the attractive and repulsive forces are in equi- libria But when, in the descending semi-circle, the planet comes to the point K, and the magnetical axis is in the line KA, passing through the centre of tjie sun, the attractive force, because the attractive pole E is nearest the sun, will be at its maximum ; and the portion of the diameter through which it librates (as OS in fig. 63), will be greatest ; and in the opposite point the repulsive force will in like manner be at NOTES. 36l at its maximum ; and in either case such force may be repre- sented by the whole line EF. It may hence also appear why the extreme librations of the planet, (as GO, PS, in fig. 63), though equal in length, are made in unequal times ; both, because magnets attract each other more forcibly at a less than at a greater distance, and because the planet, near the perihelion, is more rapidly impelled forwards by the solar fibres. In all cases the attractive force will be as the sine of the true anomaly. Let the planet, for example, be in the point D, where the true anomaly is the angle CAD = RDH, and the attractive pole E be inclined, the angle EDA, or EDR, to the line DR, passing through the centre of the sun: and through R draw to Er the perpendicular RL. According to the principles of the balance, the attractive force will, in this case, be represented by the line FL, and the repulsive force by EL ; and, since the whole line FE represents either, when at their maximum, if we cut off from FL the part FQ = EL, the line QL will represent the whole attractive force which acts upon the planet in D ; or if DE, the half of FE, be supposed to measure this force when at its maximum, DL, the half of QL, will measure it when the planet is in D. But DL is = RM, the sine of the true anomaly RUH ; and -therefore the force by which the planet is attracted to the sun, in the descending semi-circle, and by which it is in like manner repelled from him in the ascending semi-circle of ex- centric anomaiy, is always as the sine of the true anomaly. But all observations testify that the libration itself, pro- duced by either of these forces, is not as the sine of the true anomaly, but as the versed sine of the excentric anomaly ; and therefore, if this should be also found to be the effect produced by magnetical force, th-.re would be a probable method for accounting for the planetary librations, without having recourse to intelligence or animation. Certainly this seems to be very nearly the case ; for, since the sine of every arch RH of true anomaly is the measure of the attractive force in that degree ef it, the sum of these sines will be nearly the measure of all the attractions or impressions made by it, in all degrees of anomaly ; and the libration is the effect of all these impressions joined together. But the, sum of the sines of the arch RH, answering to all the degrees, or indefinitely small equal parts, into which it is divided, is to the sum of the sines of the quadrant ERH, divided into like parts, nearly as HM, the versed sine of RH, to the semi- diameter HD. For example, if RH be = 15, since the sum of the sines of 15 is = 208166, and the sum of the sines of 90 = 57S9431, we shall have 57S9131 : 203166 :: 100000 : 3591 ; 362 NOTES. 3594- ; now the versed sine of 15 to the same radius, is s= 3047. In like manner the sum of 30 sines is = 792598; and .57894.31 : 79-'598 ;: 1 00000 : 13691 ; and the versed sine of 30 is = 13397 : and still in like manner the sum of 60 sines is = 2908017 ; and 57894-31 : 2908017 :: 100000 : 50000, which is also the versed of 60. These ratios, indeed, differ considerably at the beginning of the quadrant ; for 174-5 is the sine of 1 ; and 5789431": 174<5 :: 100000 : 30 ; whereas the versed sine of 1 is only 15. But it ought to be consi- dered, that as the libratkms increase, the difference of the ratios becomes insensible. This reasoning, indeed, only shews that the librations are nearly as the versed sines of the true anomalies, whereas the observations testify that they are as the versed sines of the ex- centric anomalies HJ3N = CBD ; which in the first and fourth quadrants are always greater than the true. But it is .evident that, by the substitution of the versed sines of excen- tric anomaly, the reasoning' would become less exception- able ; for, as they are greater than the versed sines of true anomaly, so likewise are the last terms in ail the above ana- logies. It appears, then, from the example of a 'magnet, that those Jibrations of a planet which produce an elliptical orbit, may with some probability be ascribed to a mechanical cause ; without having recourse to the supposition of the planet's being endowed with intelligence ; and Kepler also endea- vours to prove, that this supposition would render the expli- cation much more difficult, if not impossible. In this explication the magnetical axis is supposed to be perfectly different from that of the planet's diurnal rotation ; ai\d the magnet to be an interior globe, within the exterior surface of the planet, and unaffected by its rotation. Bb. That the excentric area GDA increases in the same ratio with the sum of the correspondent diametral distances, may easily appear from the principles employed in the fluxionary calculu^s ; and these are the principles from which Kepler evidently derived the proposition. Let GB (fig. 79), be the diameter of the circle GED, whose circumference is divided into any number (suppose 360), of indefinitely small equal parts GE, EF, EG, GH, HD, &c. so that the arches GE, GF, GG, GJH, GD, of ex- centric anomaly increase uniformly by the common difference EG ; the sine DC of the arch GD, will be equal to EG (cos. GE -f cos. GF + cos. GG + cos. GH -f cos. GD). Let NOTES. 363 Let EP, FQ, GR, HS, DC, be the sines, and BP, BQ, BR, BS, BC, the co-sines of the arches GE, GF, GG, GH, GD ; from E, F, G, H, draw to the sines the perpendiculars EV, FX, GY, HZ, and join BE, BF, BG, BH, BD. It is evident that DC = EP + FV + GX + HY + DZ. But BE : BP :: GE : EP = GE.BP = GE. cos. GE, (for GE is sensibly = tan. GE) and BF : BQ :: EF : FV = EF.BQ = GE. cos. GF, BG : BR :: FG : GX = FG.BR = GE. cos. GG, BH : BS :: GH : HY = GH.BS = GE. cos. GH, BD : BC :: HD : DZ = AD.BC = GE. cos. GD. Therefore DC = GE (cos. GE + cos. GF + cos. GG 4. cos. GH + c s - GD). The sines therefore of excentric ano- maly increase in the ratio of the sum of as many co-sines of excentric anomaly, as there are points of equal division in the arches of which they are the sines. But the circular area GD A consists of two parts ; the sec- tor GBD, and the triangular area ABD. And the sum of the diametral distances consists also of two parts ; the sum of the semi-diameters drawn from B, to the points of equal division in GD, and the sum of the lines BK, BL, BM, BN, BO. Now the sector GBD increases precisely as the number of the semi-diameters. And the triangle ABD increases as the sine DC ; for it is .equal to DC. -~ > and it has now been proved that tlje 2i sine DC increases as the sum of the co-sines of GD. Therefore the triangle ABD increases as the sum of the co-sines of GD. But the lines BK, BL, BM, BN, BO, are the co-sines cor- responding to the same divisions of GD, multiplied into AB, and therefore their sum increases in the same ratio with the sum of the co-sines of GD ; that is, in the same ratio with the triangle ABD. Therefore the whole area GDA increases in the same ratio with the sum of the diametral distances corresponding to the equal divisions of GD. The reason, therefore, was evident why the method of areas was so unsuccessful, while the circumferential distances were considered as the just distances. For the sum of any pair of them drawn from A to opposite points of the excen- tric near the extremities of the diameter GAT (fig. 78), is indeed nearly equal to the sum of two semi-diameters ; but, the sum of AD, Ad, drawn to the points D, PROPAGATION OF HEAT. BY J. LESLIE, ESQ. ^i'st's.ssrsssiBs! . sssssss _ a "^mediate recall. AUG281999 U.C.BERKELEY LIBRARIES