k sin. !=-0.000219 sin. ( - ) -Q R011 (* -0.006699 sin. 3( - -00000 es -- +0.000001 sin. 11 ( - ) +0.000001 sin. 19 ( 1) k', cos. 0,= -0.000002 sin. 3 ( -)-0.000001 cos. 3 (* ) -0.000009 sin. 5(x - $) +0.000027 cos, 6 ( - ) +-0.000003 cos. 11 (W --) Mr. Pierce remarked that his original views were unchanged in regard to the importance to be attached to the vast discrepancies between the predicted and observed orbits of the planet which disturbs the motions of Uranus. "Neptune is not the planet designated by geometry, although it is a perfect solution of the problem which analysis had undertaken to investigate, and had really solved, but in a form radically different from the actual solution of nature. This is not a personal question ; it is certainly not one in which the reputations of Adams and Leverrier are concerned. The accuracy of their investigations is not assailed; but it is expressly admitted that they announced the correct results of most profound analytical researches. “ The fair consideration of this question cannot be made without recalling the true office and position of geometry in science, which alone entitles it to the appellation of the key to the physical world. Mathematics is the science of exact measurement; accuracy is its sole aim and object, and it is this which places it in harmony with a creation, which is subject to perfect law and undeviating order. An inaccurate result cannot be a geometrical one ; a result, inaccurate beyond certain well-defined limits, does not belong to the exact science; an inconsistency, which exceeds a certain amount, may not be neglected by him who deals with nothing but more or less, without disturbing the very foundations of his faith. “The geometrical statement was distinctly made, that the planet which disturbed Uranus could not be at a less mean distance from the sun than thirty-five times the earth's mean distance from the sun; that is, that no planet which was within this distance could cause the observed irregularities in the motions of Uranus. Neptune's mean distance from the sun is only thirty times the earth's mean distance, and yet Neptune does account for the perturbations of Uranus. It is five hundred millions of miles nearer the sun than it was distinctly stated by geometry that it possibly could be, in order to be capable of producing the effect which it actually does produce. The spirit of mathematical accuracy cannot be supposed to be sufficiently elastic to embrace so great an inconsistency, amounting to one sixth part of Neptune's distance from the sun, and to one half of the distance of his orbit from that of Uranus. “Whence comes this enormous difference between the theoretical and observed planets? Had it been quite small, it might have been regarded as an excusable numerical error. Had it even amounted to once or twice the radius of the earth's orbit, it might have been deemed an error, although it would then have been a grievous one, and would have seriously marred the beauty of the result. But as it is, it cannot be assumed to be a mere error, without admitting that such an one radically vitiates the whole theory. Whoever adopts this opinion, be it the author of the theory himself, is bound to show where the error is, and how far it extends. Such an opinion has never been advanced by me, and I am not responsible for it. I admit, however, that I have not fully investigated this point, but maintain that the profound geometry of M. Leverrier is not to be set aside without proof, or even argument. M. Leverrier found that the planet which would best account for the disturbed motion of Uranus was at the mean distance 36 from the sun ; and that, by increasing or decreasing the mean distance of the hypothetical disturber, the want of coincidence between the observed and computed motions of Uranus increased until, at the mean distances of 38 on the increase and 35 on the decrease, the residual differences between theory and observation became so great as to be wholly inadmissible. He therefore came to the natural conclusion, from such a result, that the mean distance of the required planet from the sun could not be less than 35, or more than 38; and he contented himself with this conclusion, without extending his inquiries to still smaller mean distances; and any facts in regard to these inner distances which are at variance with this result are certainly not to be included under his theory. I have confined my remarks to M. Leverrier's researches, but nothing in Mr. Adams's less comprehensive investigations, in which there is no attempt to ascertain the limits, is opposed to these conclusions. “It has been intimated, that too rigorous an agreement with obser. vation was insisted upon in the original inquiries, and that the limits might have been extended to include Neptune, by a more liberal concession to other unknown planets, or to an error in the mass of Saturn. The inspection of the preceding table completely refutes such a sug. gestion, for it now appears that Neptune satisfies the observations of Uranus more perfectly than the best planet of previous theory. If Leverrier was, as I have supposed, correct in his former computations, he must have found by extending them, that, although the action of his hypothetical planet agreed less perfectly with observation by the contraction of the radius of its orbit from 36 to 35, and that this disagreement would have still farther increased by a still farther contraction, there was a distance at which the disagreement ceased to increase, and would, on the contrary, begin to diminish, until at the distance 30 it would have vanished, and the disturbed motions of Uranus would have been wholly explained. But this singular change in the character of the disturbing force, if it really occurs, - and the only doubt in regard to it is derived from a supposed but unproved inaccuracy in Leverrier's investigations, — was excluded from the range of this geometer's investigations, and now that observation has led to its discovery, geometry cannot claim it as one of its predictions. The defect of the theory must be as frankly admitted as the more serious charge of error is boldly repelled. “ From some indistinct remarks which have been thrown out in regard to the mass of Neptune, which is not too small to be excluded from the limits of the theory, there seems to be an indisposition to confess this defect. But on turning to the original formulæ, it will be found that, although this small mass is not positively excluded, its adoption does not contribute to advance the claim of geometry upon the planet. It shows, on the contrary, most decisively, that the orbits of theory are all of them fundamentally different from those of Neptune. For the mean distance which corresponds to this mass in the theory is about 351, and the eccentricity very much greater than in the best hypothetical orbit, while the discrepancy between the theoretical and observed action on Uranus is increased beyond the admitted limits. “ The case might safely rest there, but I desire to dwell upon the essential and radical difference between Neptune's action upon Uranus and that of the planets of theory. For this purpose, I will read an extract from a report made by me last September to the honorable committee of the Overseers of Harvard University who visited the Observatory. “ The differences are not accidental, but inherent in the very nature of the case, while the points of resemblance are purely accidental. The solutions of Adams and Leverrier are perfectly correct for the assumption to which they are limited, and must be classed with the boldest and most brilliant attempts at analytical investigation, richly entitling their authors to all the éclat which has been lavished upon them, on account of the singular success with which they are thought to have been crowned. But their investigations are nevertheless wholly inap |