theoretical researches of these geometers and the actual position of the planet Neptune, must be regarded as wholly fortuitous. Let us examine some of the arguments which he adduces in support of this peculiar view of the subject.

(92.) It has been stated (p. 185) that Mr. Adams, in his final communication to the Astronomer Royal, remarked that the observations of Uranus would in all probability be satisfied best by adopting for the hypothetical planet a mean distance equal to 33.6. He was led to entertain this opinion by a comparison of the errors of his theory for the three oppositions of Uranus in 1843-44-45, resulting from the two hypotheses of the mean distance which he had already employed in his researches. It is manifest, when this inference is viewed in connexion with his previous results, that Adams had renounced all faith in even an approximation to Bode's law of the distances of the planets, and that the current of his researches was rapidly conducting him to a mean distance of the hypothetical planet agreeing with the actual mean distance of the planet Neptune. Mr. Gould seeks to depreciate the merit of this sagacious conclusion by contending that a mean distance equal to 33.6 would give erroneous results. "Le Verrier," says he, has shewn that the assumption of even 35 as the mean distance would lead to intolerable discordances. Peirce has further proved that an important change in the character of the perturbations takes place near the distance 35.3. It is therefore evident that no claims can be based upon the rough inference alluded to."

66

It has been mentioned that Peirce objected to the reasoning by which Le Verrier established the inferior limit of the mean distance of the hypothetical planet (35.04), on the ground of the continuity of the investigation being broken at the distance 35.3 by the commensurability of the mean motions of the two planets. It is plain, therefore, that the propositions announced by these two geometers are mutually incompatible; and yet Mr. Gould adduces them as confirmatory of each other! have already had occasion to remark that there do not exist grounds for supposing that either of them is entitled to any confidence.

We

(93.) Mr. Gould admits that Le Verrier may be considered the discoverer of the planet Neptune, in so far as he proved not only that it was impossible to represent the motions of Uranus without the assumption of some unknown disturbing body, but that the perturbations were of that analytical form which belongs to an exterior planet.

Now it appears to me, that the latter assertion is at direct variance with the actual state of the question. Le Verrier demonstrated, by his researches, that the perturbations were such as would be produced by the direct action of an exterior planet during the interval of time over which the observations extended; but, with respect to the analytical form of these perturbations, it depended on the elements of the disturbing planet, which were beyond the scope of investigation, and in fact turned out to be entirely different from those deduced by Le Verrier.

(94.) Mr. Gould further remarks that Le Verrier omitted the consideration of the terms depending on a near approach to commensurability; but that this, although certainly a defect, cannot be considered as an error in the theory, since within the limits where he had reason to suppose that the orbit was situated, these terms are almost uniformly negligible.

With reference to this point it may be remarked, that the irregularities in the motion of Uranus depended on the direct action of Neptune during the period of last conjunction, and not on the analytical theory of that

planet, which involves all the consequences liable to be developed in the lapse of indefinite ages. Hence it is manifest that the absence of any resemblance between the theory of the hypothetical planet of Le Verrier and that of the planet Neptune, cannot be considered as affecting in the slightest degree the merit of Le Verrier's researches, in so far as they had for their object the discovery of the disturbing body. Even in the case of Neptune, the terms to which Mr. Gould alludes do not exercise any sensible influence on the action of the planet between the

and 1845.

years 1690

(95.) With reference to the same geometer Mr. Gould makes the following statement:"His laborious and elegant researches have been crowned with brilliant success, and M. Le Verrier himself rewarded by the consciousness of having been the immediate occasion of the discovery of Neptune. And although the agreement of Neptune's direction at the time of the discovery with the direction of the theoretical planet was but accidental, it almost seems as though the heavens strove to show themselves propitious, so happy was the accident, so wonderful the coincidence." *

(96.) Leaving the above passage to the reader's own reflections, we proceed to notice one or two other statements of Mr. Gould's. Referring to the assertion of Sir John Herschel, in his "Outlines of Astronomy," that the longitude and radius vector of the hypothetical planet, whether of Adams or Le Verrier, very nearly coincided with the longitude and radius vector of Neptune during the period of its action being sensible, Mr. Gould remarks:-"But surely it cannot be considered as an analogy between the two orbits, that the perihelion of the one was so near the aphelion of the other."

The analogy between the two orbits, demanded by the question relative to the disturbing body, was confined solely to a pretty close coincidence of the paths of the hypothetical and real planets during the period of the disturbing force being sensible. Even in this case it was a near agreement of the longitudes, rather than of the distances, which was required by the conditions of the problem. With respect to the absolute identity of the two orbits, the establishment of such a condition was an object of no importance in so far as the discovery of the disturbing body was concerned, Sir John Herschel, in the work to which Mr. Gould refers-so far from attempting to demonstrate any resemblance between the elements of Neptune on the one hand, and those of the hypothetical planet of either Le Verrier or Adams on the other-on the contrary, utterly repudiates the existence of any necessary connexion between such an analogy and the question relative to the discovery of the disturbing body. But, apart from all consideration of this circumstance, it seems surprising that Mr. Gould should urge such an objection to the identity of the two orbits as that above cited, when it is borne in mind that in the one case the orbit is very eccentric, and in the other case is almost circular.

(97.) Mr. Gould concludes his Report with a remark the object of which is to reconcile the conflicting results of observation and theory. "The combined labours of Le Verrier and Peirce," says he, have incontrovertibly proved that, by reducing the limits of error assumed for the modern observations to 3", there can be but two possible solutions of the problem. There are two different mean distances of least possible error, one of which is 36, and the other 30. The one is included within the

Report, p. 51.

theory and limits of Le Verrier, and corresponds with Adams's solution; the other is the orbit of Neptune." *

With respect to the existence of two mean distances of least possible error, with an interval included between them, any mean distance corresponding to which is incapable of satisfying the observations with sufficient accuracy, it seems to be in the highest degree improbable. This will be readily seen by reference to the theoretical researches of Le Verrier and Adams. The elements of the first and second planet of Adams, and those which Le Verrier deduced from his final investigation, exhibit a successive diminution of the mean distance. Now, in each of these three cases, the mean distance was greater than the true value; but this defect was remedied by increasing the eccentricity in a corresponding degree, and placing the perihelion near the point of conjunction of Neptune and Uranus. By this means the distances of the disturbing body were rendered in each case very nearly equal to the true distances in the part of the orbit where considerable precision was indispensable; and the effect of the error in the mean distance was thrown upon the opposite portion of the orbit extending on each side of the aphelion, where it was incapable of exercising any influence. The following table will exhibit this view of the subject in a clearer light :

It appears from these results, that the perihelion distance is almost the same for each of the three planets, and that the longitude of the perihelion does not in any case differ materially from the longitude of Uranus and Neptune (273) on the occasion of their last conjunction, about the beginning of the year 1822. On the other hand, the aphelion distance varies nearly at a rate corresponding with the variation of the mean distance. Now if we suppose the mean distance of the hypothetical planet to be diminished below the value assigned by Le Verrier, so as to approach nearer the mean distance of Neptune, have we not strong reason to believe that, by similarly throwing the effect of the change mainly upon the aphelion distance, where it would be altogether uninfluential, the observations of Uranus would be satisfied with the same degree of precision as in the foregoing cases? Indeed it seems very probable that this object might be accomplished by employing any mean distance within a range extending considerably both above and below the mean distance of Neptune, the perihelion being turned towards the point of conjunction when the mean distance was greater, and the aphelion being turned towards the same point when the mean distance was less, than the true value.

(98.) We may recapitulate the conclusions suggested by the discovery of the planet Neptune in the following terms:-Two contemporary geometers, * Report, p. 55.

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.