disclosed to astronomers any traces of a libratory motion of this character; whence we may conclude that the motions of rotation and revolution did not originally differ by a sensible quantity *.

Lagrange next considered the libration of the moon in latitude. In this memoir he did not succeed in explaining the singular fact discovered by Cassini relative to the coincidence of the nodes of the lunar orbit and equator. When he assumed this coincidence in the outset of his researches, he found that the lunar equator, instead of being fixed with respect to the ecliptic, continually approached towards that plane, while observation on the other hand went to prove that it was inclined to it at nearly an invariable angle. This portion of his researches being imperfect, he resumed the subject fifteen years afterwards, and in the volume of the Berlin Academy for 1780, he published an admirable memoir, in which he completed the theory of the moon's motion about her centre of gravity. On this occasion he was conducted to the remarkable conclusion, that if the mean nodes of the lunar equator and orbit be supposed to have originally coincided, the action of the earth upon the lunar ellipsoid would constantly maintain this coincidence. He also determined the laws of the small oscillations, by which the true node of the lunar equator deviates from the mean node.

The only question connected with this subject which still remained to be examined was the effect which the secular inequalities in the mean motion of the moon might produce upon the appearance of the lunar surface. These inequalities will one day derange the mean place of the moon to the extent of several circumferences of the circle, and if the rotatory motion of the moon remained constant during the whole period of their developement, the inevitable consequence would be, that the moon would present the whole of her surface in gradual succession towards the

Several writers on astronomy, when describing the various librations of the moon, affirm that the fourth, or physical libration, was discovered by Lagrange. If this refers to the libratory motion mentioned in the text, it cannot be called a discovery, since its actual existence has not yet been established by astronomers. The only real libration which observation has detected is that depending on the lunar inequalities in longitude (chiefly the annual equation; see Chapter XI.), and this phenomenon was first remarked as a theoretical truth by the great founder of Physical Astronomy, who unfolded the whole mechanism of the planetary system, and by his unrivalled sagacity anticipated those results which his successors, by the aid of a refined analysis, have been enabled only to confirm and extend. Laplace is surprised that Newton should have failed to notice that, in order to assure the constant equality of the motions of rotation and revolution, it was not absolutely necessary that at the origin they should have been exactly equal. This, however, might be considered as a natural corollary to the remark of Newton, that any disturbance of the elongated axis of the moon would merely result in an oscillatory motion on each side of its mean place; for the possibility of allowing the arbitrary constants of any system to vary a little on each side of a mean state, without occasioning any permanent derangement of the system, is a manifest attribute of the condition of stable equilibrium, and such a condition is clearly implied in Newton's words:-"Unde ad hunc situm semper oscillando redibit."-Princip., lib. iii. prop. xxxviii. If the motions of rotation and revolution had differed a little at the origin, as Laplace conceived they might, it is clear that the elongated axis would not have coincided exactly with the line joining the earth and moon; and hence, according to Newton's statement, it would oscillate continually on each side of that line. Newton, however, evidently refers to the difference in the two motions occasioned by the inequalities in the moon's longitude. It is natural enough, indeed, to suppose that the illustrious author of the Principia did not feel any anxiety to repudiate the original equality of the motions of rotation and revolution-a relation which, although perhaps difficult to explain by the doctrine of chances, becomes very interesting and suggestive when it is considered as the result of Supreme Intelligence.

earth. Laplace investigated this interesting question, and arrived at the conclusion that such a condition was inconsistent with the theory of gra vitation. He found, in fact, that the terrestrial attraction would always draw the moon's axis into coincidence with the line joining the earth and moon, so that the rotatory motion will participate in the secular acceleration of the motion in longitude, and consequently the lunar hemisphere, which is turned away from us, will remain for ever concealed from view, with the exception of the small portion disclosed by the periodic inequalities.

Laplace has considered the circumstances which determine the stability of the singular mechanism with which the planet Saturn is furnished. He considers that the rotatory motion of the rings may be accounted for by supposing the particles composing them to be homogeneous, and to move freely among each other like the particles of a fluid. Under such conditions, he shews that they would be maintained in equilibrium by the action of the planet and the centrifugal force generated by their own rotatory motion, the exterior surfaces assumed by both rings being such, that all sections perpendicular to them, and passing through the centre of the planet, would be ellipses, whose major axes when produced would pass through that point. Laplace hence concluded that the period of the rotation of the rings is equal to that of a satellite revolving at the distance of the centre of the generating ellipse. This period he found to be equal to 10h. 33'.36". It is remarkable that Herschel inferred, from certain periodic changes in the appearance of the rings, that they accomplished a revolution round the planet in 10h. 32'.15′′*.

If the rings were uniform and circular, and were not exposed to the action of any extraneous force, it would still be possible for them to revolve constantly round the planet; but it is clear that the least disturbance, as the action of a satellite or comet, would affect their stability, and ultimately precipitate them upon the body of the planet. In order, therefore, to assure the permanence of the rings, Laplace conceived that it was necessary to suppose their figures to be irregular, so that any disturbance either of them might suffer would be rapidly checked in course of rotation by the unequal distribution of the mass.

CHAPTER VII.

Jupiter's Satellites.-Galileo.-Simon Marius.-Hodierna.-Borelli.-Cassini.-His first Tables. He is invited to France. He publishes his Second Tables.-His Rejection of the Equation of Light.-Researches of Maraldi I.-He discovers that the Inclination of the second Satellite is variable.-Bradley's Discoveries.-Maraldi II.-His Discoveries relative to the third and fourth Satellites. He adopts the Equation of Light. -Wargentin. He discovers the Inequalities in Longitude of the first and second Satellites. He remarks that the third Satellite has two Equations of the Centre.Motion of the Nodes of the fourth Satellite.-Inclination of the third Satellite.Libratory Motion of the Nodes.-Inclination of the fourth Satellite.

THE discovery of Jupiter's satellites is one of the most interesting events in the history of astronomy. Even in any age it would have been deemed

Phil. Trans. 1790.

an important contribution to science; but in the beginning of the seventeenth century, when men's minds were wavering between the ancient and modern ideas of the system of the world, it exercised an influence of which it is impossible to form an adequate conception in the present day. The existence of four bodies revolving round one of the principal planets of the solar system, exhibited a beautiful illustration of the moon's motion round the earth, and furnished an argument of overwhelming force in favour of the Copernican theory. The announcement of this fact pointed out also the long vista of similar discoveries which have continued from time to time down to the present day to enrich the solar system, and to shed a lustre on the science of astronomy. In more recent times the physical theory of Jupiter and his attendants has supplied evidence of the most varied and satisfactory character in favour of the principle of Universal Gravitation. All the irregularities which arise from the mutual action of the larger bodies of the system are here exhibited in miniature. Their study also offers peculiar advantages to the mathematician, for, as they generally pass through all their values in short periods, their real character is readily appreciable, and on this account they are eminently favourable for testing the conclusions of his theory. Nor is it merely in its relation to speculative science that the discovery of Jupiter's satellites is to be regarded as of capital importance. The eclipses of these bodies. soon suggested a new solution of the great problem of the longitude. Their theory thus came to be associated with one of those questions which most deeply affect the progress of civilization-the promotion of mutual intercourse between the various nations of mankind,-and a more earnest and more generally diffused interest was naturally felt in the researches connected with its improvement.

When Galileo first turned his telescope to the planets, he was delighted to perceive that they exhibited a round appearance like the sun or moon. Jupiter presented a disc of considerable magnitude, but in no other respect was he distinguishable from the rest of the superior planets. Having, however, examined him with a new telescope of superior power on the 7th January, 1610, his attention was soon drawn to three small but very bright stars that appeared in his vicinity, two on the east side and one on the west side of him. He imagined them to be three fixed stars, and still there was something in their appearance which excited his admiration. They were all disposed in a right line parallel to the plane of the ecliptic, and were brighter than other stars of the same magnitude.

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This did not, however, induce him to alter his opinion that they were fixed stars, and therefore he paid no attention to their distances from each other, or from the planet. Happening, by mere accident, to examine Jupiter again on the 8th January, he was surprised to find that the stars were now arranged quite differently from what they were when he

"Cum autem die octava, nescio quo fato ductus, ad inspectionem eandem reversus essem."- Sidereus Nuncius, p. 20.

first saw them. They were all now on the west side of the planet, and

were nearer to each other than they had been on the previous evening; they were also disposed at equal distances from each other. The strange fact of the mutual approach of the stars did not yet strike his attention, but it excited his astonishment, that Jupiter should be seen to the east of them all, when only the preceding night he had been seen to the west of two of them. He was induced, on this account, to suspect that the motion of the planet might be direct, contrary to the calculations of astronomers, and that he had got in advance of the stars by means of his proper motion. He therefore waited for the following night with great anxiety, but his hopes were disappointed, for the heavens were on all sides enveloped in clouds. On the 10th he saw only two stars, and they were both on the east side of Jupiter. He suspected that the third

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might be concealed behind the disc of the planet. They appeared as before in the same right line with him, and lay in the direction of the zodiac. Unable to account for such changes by the motion of the planet, and being at the same time fully assured that he always observed the same stars, his doubts now resolved themselves into admiration, and he found that the apparent motions should be referred to the stars themselves and not to the planet. He therefore deemed it an object of paramount importance to watch them with increased attention.

On the 11th he again saw only two stars, and they were also both on the east

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side of Jupiter. The more eastern one appeared nearly twice as large as the other, although on the previous evening he had found them almost equal. This fact, when considered in connexion with the constant change of the relative positions of the stars and the total disappearance of one of them, left no doubt on his mind of their real character. He therefore came to the conclusion, that there are in the heavens three stars revolving round Jupiter in the same manner as Venus and Mercury revolve round the sun. On the 12th he saw three stars; two on the east side of

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Jupiter, and one on the west side. The third began to appear about three o'clock in the morning, emerging from the eastern limb of the

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planet; it was then exceedingly small, and was discernible only with great difficulty. On the 13th he finally saw four stars. Three of them

were on the west side of the planet, and the remaining one on the east side. They were all arranged in a line parallel to the ecliptic, with the exception of the central star of the three western ones, which declined a little towards the north. They appeared of the same magnitude, and, though small, were very brilliant, shining with a much greater lustre than fixed stars of the same magnitude *.

The future observations of Galileo established beyond all doubt that Jupiter was attended by four satellites. He continued to examine them until the latter end of March, noting their configurations, and recording the stars which appeared in the same field of view with them.

Soon after Galileo's famous discovery, he perceived the utility of the satellites for finding the longitude, and he continued for many years to make observations on them, with the view of constructing a theory of their motions. Much has been said about his tables of the satellites, which were to have been published by his friend and pupil Rimieri, but which, by some unaccountable accident, disappeared at the death of that person, and could nowhere be found, until they were finally discovered a few years ago in a private library at Rome. We know that Galileo himself was very sanguine of their practical utility, but his opinion of their merits does not seem to be borne out by the actual examination of them consequent on their rediscovery. Indeed, when we reflect on the many painful efforts which it cost his successors to arrive at even a tolerable knowledge of the elements of the satellites, we might very reasonably conclude, à priori, that his tables can only be regarded in the present day as an object of scientific curiosity. An interesting fragment of his early researches on the satellites is to be found in one of his letters to Welser, the person through whom he carried on the controversy with Schener the Jesuit, relative to the discovery of the solar spots. At the end of a letter dated December 1st, 1612, he gives a sketch in rough drawings of the configurations of the satellites from 1st March till 7th May of the follow ing year.

Simon Mayer, the German astronomer, who contended for the independent discovery of the satellites, resolved to strengthen his claims by the construction of tables of their motions. The crude labours of this impudent pretender were, however, no sooner given to the world than they fell into deserved oblivion. Hodierna, a Sicilian astronomer, is the next person who is mentioned as having devoted his attention to this subject. In 1656 he published his observations on the satellites, accompanied with remarks on the theory of their motions. He is the first astronomer who pointed out the superior importance of eclipses of the satellites as compared with other phenomena. He also calculated tables of their motions, but they are said to have been so very inaccurate, that in a few years they even ceased to represent the configurations of the different bodies. In 1666 Borelli attempted to establish a theory of the satellites,

The preceding configurations are derived from those given by the illustrious discoverer in the Sidereus Nuncius.