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Mr. Adams in England, and M. Le Verrier in France, undertook about the same time to investigate the irregularities of Uranus upon the supposition of their being produced by the action of an exterior planet, and, independently of each other, arrived at a very approximate determination of the position of the disturbing body. Upon this ground, therefore, they are severally entitled to the honour associated with the theoretical discovery of the planet Neptune. With respect to Le Verrier's researches on the limits of the orbit of the disturbing body, they have not been borne out by the results of actual observations; but this circumstance, attributable in all probability to the intricacy of the subject and the imperfect state of analysis, does not in the slightest degree impugn his claims to the great discovery just mentioned. The American astronomers and mathematicians have more especially distinguished themselves by their labours in connexion with the planet Neptune, since the epoch of its physical discovery. The results that have been deduced from Bond's observations of the satellite of Neptune and the mathematical researches of Walker and Peirce, unquestionably exhibit a degree of consistency with the actual observations of Uranus and Neptune which has not been paralleled by any similar efforts on this side of the Atlantic, while at the same time they tend to throw much interesting light on the theory of both planets. The peculiar views which Prof. Peirce was led to entertain, respecting the researches of the distinguished geometers to whom the theoretical discovery of Neptune is due, may perhaps be attributed to his having devoted his attention too exclusively to the analytical formulæ representing the action of the planets, without taking into sufficient consideration the mode in which the disturbing forces directly operate. These views were announced by Prof. Peirce in a spirit of candour and moderation highly honourable to his character as a philosopher. They are beyond all doubt erroneous, but the trifling inadvertence into which he was thus betrayed does not detract from the merit of his more substantial labours in connexion with the theory of Uranus and Neptune.

IV.

REMARKS ON THE LUNAR INEQUALITY TERMED THE EVECTION.

One of the most remarkable instances of perturbation which occurs in the solar system is the inequality in the moon's longitude termed the evection. So long as the moon was observed merely in eclipses, this inequality continued to escape the notice of astronomers. When Hipparchus, however, after having constructed the astrolabe, succeeded in determining the position of the moon in quadratures, he found that the results could not be generally reconciled with the existing theory of her motion. That great astronomer, having no similar observations of the moon anterior to his own accessible to him, was unable to arrive at a definitive conclusion respecting the anomaly; but he formally pointed out its existence, and executed a series of valuable observations with the view of aiding future astronomers in their researches on the subject. It is well known that the discovery of the law of this famous inequality is due to Ptolemy. The account which he has given of the inequality as it presented itself to his observations *, would seem to imply a law of variation Syntaxis, lib. v., cap. ii.

materially different from that suggested by the term representing the same inequality in the modern theory of the moon's motion. He states that the observed places of the moon in quadratures, whether those recorded by Hipparchus or those actually determined by himself, were found in some instances to agree very well with the computed places; in other instances to differ considerably, being sometimes in excess and at other times in defect. By attentively pursuing the inequality through its various phases, he found that it was generally insensible in sizygees. It also vanished in the quadratures when the moon was in the apogee or perigee of her epicycle (in other words, when the line of apsides was in quadratures); but it increased from those points towards the mean points of the orbit where it was greatest (in other words, it increased as the line of apsides revolved from the quadratures to the sizygees). Moreover, when the first anomaly (the equation of the centre) was subtractive, the observed place of the moon was in defect, in consequence of the new inequality; and when the first anomaly was additive, the observed place was in advance of the computed place, from the same cause.

It appeared, then, that while the inequality vanished in sizygees, its effect in quadratures was invariably to augment the equation of the centre, unless the line of apsides was in quadratures, when it vanished altogether Ptolemy, from observations of the moon in sizygees, had determined the maximum value of the equation of the centre to be 5° 1'*. In consequence of the new inequality, its value, as indicated by observations in quadratures, generally exceeded 5° 1′, increasing from that value to 7° 40′ as the line of apsides revolved from quadratures to sizygees. Hence it followed that the maximum effect of the new inequality amounted to 2° 39'.

In modern astronomy the inequality in the moon's longitude, depending on the combined effects of the equation of the centre and the evection, is represented thus:

δν

♪, = + 6° 18′ sin a + 1° 20′ sin (2 (◄ — € ) — ▲),

where a represents the mean anomaly of the moon, the mean longitude of the moon, and the mean longitude of the sun.

cases.

Nothing can at first sight appear more different than the ancient and modern modes of representing the two inequalities. With respect to the equation of the centre, its magnitude is materially different in the two The evection, however, differs not merely in absolute magnitude, but also in the law of its variation. According to Ptolemy the zero points of the inequality were fixed in position, being constantly situate in the sizygees, while its maximum value was variable. On the other hand, it is manifest, from the second term of the above equation, that the zero points of the modern inequality are variable in position relatively to the line of sizygees, but that its absolute magnitude is constant.

Ptolemy determined the ratio of the epicycle of the lunar orbit to the deferent, or, in other words, the maximum value of the equation of the centre, from three eclipses of the moon observed at Babylon, about 700 years before the Christian era, and also from three similar eclipses observed by himself. In both cases he found the ratio to be as 5 to 60, which gives 5° 1' for the equation of the centre (Syntaxis, lib. iv.). Delambre, having computed the equation of the centre by the modern analytical formulæ, found that the three Chaldean eclipses assigned 4° 59′ 16′′ as its value, and that the three eclipses of Ptolemy made it equal to 4° 59′ 42′′. The close agreement of these results affords a strong presumption, that the two sets of eclipses employed by Ptolemy in his calculations were selected on account of their mutual consistency, from a vast mass of similar observations in his possession.

Notwithstanding the striking points of dif ference referred to in the foregoing remark, the effects produced by the combination of the two constituent inequalities are identical in both cases as respects the law of variation, and are also nearly so in respect of absolute magnitude. This may be easily shewn in the following manner. Let AC represent the line of sizygees, B D the line of quadratures, E F the line of apsides, м the place of the moon in her orbit.

Let A TE, ATM = ( Hence A

ETM = 0 — 4.

2 (4

0 = 9.

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M

E

H

(0-4) = 0 + $.

Therefore ♪, = 6° 18′ sin a + 1° 20′ sin (2 (( — ©) — ^),

=

6° 18' sin (0) + 1° 20' sin (

+ $),

= 4° 58' sin (0) + 1° 20′ sin (0-4)+1° 20′ sin (§+ø), 4° 58′ sin à + 2° 40′ cos sin 0.

=

The first of these terms is manifestly the equation of the centre as deduced by Ptolemy from observations of the moon in sizygees. The second term also represents the evection as it exhibited itself to that astronomer. Thus let us suppose the moon to be in either of the sizygees. In such a case 0=0, or 180°, and consequently the second term vanishes.

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Again, if the moon be in quadratures, we have = 90°, and therefore 84° 58' sin A+ 2° 40' cos

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In this case, then, the two inequalities conspire together. The effect is obviously a maximum, when = 0, or 180°. We have then

dy = = ± 7° 38'.

These conclusions agree with Ptolemy's description, subject to a slight difference in the numerical values. Indeed the precision with which that astronomer determined the combined effects of the two inequalities in sizygees and quadratures, is one of the most astonishing circumstances connected with the ancient astronomy.

Since the evection as represented by Ptolemy has always the same sign in the quadratures as the equation of the centre, it is manifestly positive when the moon is revolving from conjunction to opposition, and negative throughout the remaining half of the orbit, or vice versa; according as the perigee is situate in the first and fourth, or in the second and third quadrants of the lunar orbit, counting from the point of conjunction in the direction of the moon's motion.

In order to determine the zero points of the evection as represented by modern astronomers, we have

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Hence it is manifest that by drawing G H, making with a c the same angle which E F makes with it, the extremities G, H, will indicate the zero points of the inequality.

It has been stated (p. 424) that Horrocks first explained the evection upon the Keplerian principles of astronomy, by supposing the eccentricity

of the lunar orbit to be variable, and attributing a libratory motion to the line of apsides. Allusion has also been made to the difficulty experienced for some time in computing, by the theory of gravitation, the motion of the lunar apogee, upon which the inequality to a great extent depends. It is worthy of remark that in the original edition of the "Principia," published in 1687, Newton states that he computed the motion of the lunar apogee in sizygees and quadratures, and also the mean motion. He asserts that he found the daily progression in sizygees to be 23', the daily regression in quadratures to be 16, and the mean annual motion to be 40°. He remarks that these results do not accord exactly with the tables, a circumstance which he thinks may be attributable to the errors of the observations. The calculations being very intricate and embarrassed with approximations, and the results not possessing all the accuracy that was desirable, he refrained from publishing the details of his researches on the subject. (Computationes autem, ut nimis perplexas et approximationibus impeditas, neque satis accuratas, apponere non lubet.)

The results which Newton obtained on this occasion cannot by any means be considered very inaccurate, when the intricacy of the subject and the imperfect state of analysis in his time are taken into account. They give 11° 21' for the monthly progression of the apogee in sizygees, and 8° 1' for the monthly regression in quadratures. The modern tables of the moon assign, in round numbers, 11° and 9° as the corresponding values of the motion of the apogee. Newton found the mean annual progression of the apogee to be 40°; the modern tables of the moon make it 40° 40′32′′. Newton appears to have been so dissatisfied with his researches on this subject, in all probability from the circumstance of the results not presenting a more complete accordance with those deducible from observation, that he suppressed all allusion to them in the second edition of the "Principia," published in 1713, under the superintendence of Cotes. Whatever may have been the method of investigation employed by him on this occasion, it was manifestly one which was capable of grappling with the main difficulties of the question. It is not improbable that a careful inspection of those manuscripts of Newton, which are still in existence, might serve to throw some light on this interesting point.

V.

NOTE RESPECTING HORROCKS.

At page 421 I have hazarded the conjecture that it was duties of a religious nature which called away Horrocks so peremptorily, while engaged in looking out for the transit of Venus on the 24th of November, 1639. This is confirmed by a note which the late Prof. Rigaud discovered in one of Hearne's Memorandum Books preserved in the Bodleian Library, Oxford, from which it appears that Horrocks was a hard-working curate at Hoole, subsisting upon a wretched pittance (Rigaud's Correspondence of Eminent Men of the Seventeenth Century, vol. ii. p. 112). It appears also, from one of Flamsteed's letters to Collins, contained in the same work, that Crabtree's death occurred in the year 1652, and not shortly after that of Horrocks, as Wallis erroneously stated in the dedicatory epistle to Lord Brouncker, inserted at the commencement of the " Opera Posthuma" of the latter.

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VI.

ACCOUNT OF SOME RECENT RESULTS OF ASTRONOMICAL OBSERVATION.

Two instances of a total eclipse of the sun have recently furnished opportunities of observing the circumstances usually attending these phenomena. The first of these eclipses happened on the 8th of August, 1850. It was visible only in the Pacific Ocean. An account of the phenomenon as observed by M. Kutczycki at Honolulu, the chief town of the Sandwich Isles, appeared in the Comptes Rendus for the 21st of April, 1851. The second eclipse happened on the 28th of July, 1851. Being visible in the northern countries of Europe, it was observed by a great number of astronomers. Two important facts were satisfactorily estab lished by the observations of these eclipses. In the first place, the reddish protuberances usually visible on such occasions, appeared in some instances to be isolated from the moon's limb. Secondly, those protuberances that were visible towards the point of immersion, were seen gradually to diminish as if concealed by the passage of the moon over the solar disk; while, on the other hand, those towards the point of emersion appeared to enlarge as if gradually disclosed to view by the same cause. Both these facts tend to support the opinion that the protuberances are solar phenomena. A serious difficulty attending the explanation of their physical cause, consists in the material difference of aspect which they exhibit to spectators distant from each other by only a very short interval.

Five more planets revolving between the orbits of Mars and Jupiter, have been discovered in addition to those referred to in the body of this work (see p. 240). Three of these bodies were discovered in the year 1850. The first (Parthenope) was discovered by De Gasparis on the 11th of May; the second (Victoria), by Hind on the 13th of September; and the third (Egeria), by De Gasparis on the 2nd of November. The remaining two planets were discovered in the course of the year 1851. The first of these (Irene) was discovered by Hind on the 19th of May, 1851. By a singular coincidence, De Gasparis also independently discovered this planet on the 23rd of the same month. The second planet (Eunomia) was discovered by De Gasparis on the 29th of July. Parthenope revolves round the sun in 1401 days, Victoria in 1303 days, Egeria in 1496 days, Irene in 1510 days, and Eunomia in 1424 days. These numbers, of course, can only be regarded as provisional. The total number of asteroids now discovered amounts to fifteen. It is not improbable that hundreds of these minute bodies may be revolving in the same region.

On the 4th of December, 1850, intelligence reached this country that on the 15th of the previous month, Mr. Bond, Director of the Observatory of Cambridge, U. S., had discovered a new ring round Saturn, interior to the bright rings already known to exist. It soon turned out that the same phenomenon had been observed in England by Mr. Dawes on the 29th of November, before he received any intimation of Mr. Bond's discovery. The most surprising circumstance, however, connected with the phenomenon is, that it was actually observed as early as the year 1838, by Dr. Galle of Berlin; although no further notice seems to have been taken of it till the announcement of its rediscovery as above mentioned. The ring now forms an interesting object of observation to astronomers armed with powerful telescopes. In brightness it is very much inferior to the outer rings. Its breadth is equal to about two-fifths of the interval included between the bright rings and the body of the planet. It would

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