those obtained by Le Verrier from the perturbations of Herschel, do not amount to one-tenth of the several elements, he has reached as great a degree of accuracy as the data at his command would admit of. Now an error of such an amount, nay, even far greater, is nothing new in the history of astronomy, and of this we will cite a very familiar instance.

The distance between the sun and earth, and thence the dimensions of the solar system, were inferred, up to the middle of the last century, from observations of the parallax of the planet Mars. By this method it was concluded that this distance was about eighty millions of miles. It is unnecessary to enter into a description of the methods employed in determining this parallax. We need only state that the possible errors of observation bore so large a proportion to the whole parallax, as to render the inferred distance doubtful, to those who really understood the subject. A better method was suggested in the determination of the parallax of Venus, from observations of its transit over the sun's disk. Such transits are rare phenomena. One of them, however, occurred in 1761, and another in 1769. Both were observed. In the observations of the first date, some of the observers committed the error of recording the minute wrong, while the seconds and fractions were right. By this error of record the distance of the sun and earth, when calculated, was still eighty millions of miles. The greater part of the observers, who had correctly written the minute within which their observation occurred, were induced, by the correspondence of the new and old results, to fancy and admit that the error had been committed by them. It was not, therefore, until the observations of 1769 had been calculated, that it was admitted that the distance between the sun and earth amounted to ninety-six millions of miles. Here, then, we have an error in the first approximation amounting to one-fifth of the whole.

The data, whence alone Le Verrier could possibly have deduced his theory of the new planet, are, in the first place, the observed positions of Herschel; and, in the second, the positions predicted from the best tables. Even with the best modern instruments, the uncertainty in the observed positions may amount to 2". The tables are calculated from the general laws of planetary motion in an ellipse whose dimensions are given, and the places thus obtained are corrected for the disturbances produced by the action of other bodies. Of the bodies which act upon Herschel, Saturn has the greatest influence, and his action depends upon the relation of his mass to that of Herschel, and their ever-varying distance. The latter is well known; but in respect to the mass of Herschel,

the two methods which have been employed for determining it give results which differ from each other in the proportion of 75%. Herschel has been carefully observed for more than sixty years, and while this uncertainty remains as to his mass, we must not be surprised that the attempts which have been made to determine the mass of Le Verrier's planet should make it no more than 5%, or even of that predicted by him. Herschel may also be acted upon by a planet still unknown, and superior to Neptune, and the two united may leave an uncertainty in the predicted place of the former, which may amount to 10" or 12". That no question may remain in relation to Le Verrier's claims, he had himself pointed out the possible errors of his data, in papers read before the actual discovery of the planet, and had explained the method by which he proposed to limit the uncertainty thereby caused in his theory. He had, also, before the publication of the calculated orbit of Walker, inserted a complete memoir, in which these limits were pointed out, in the Connaissance des Temps. It is sufficient for our conviction that Neptune, as discovered by Galle, is the planet whose existence was inferred by Le Verrier, that we should find all the calculated elements of Walker within the limits assigned in these papers. Every one of them is in truth thus included, and no doubt can possibly remain of the soundness of his claims, and still more of the extraordinary sagacity he had exhibited in selecting that position within his possible limits, in which the planet was so exactly found. It is not to be forgotten, that after pointing out this more probable position as 5° from the star we have named above, he had stated that it might possibly be necessary to extend the exploration to a distance of 5° on each side, and that the research should not be considered as having totally failed until it had been made to cover 18°. The elements of the first orbit computed by Walker, if written side by side with those predicted by Le Verrier, are as follows:

Walker, however, some months later, re-calculated the orbit from a longer series of observations. Although the results differ in some respects from those we have cited, they are, upon the whole, even farther removed from the predictions of Le Verrier. We might FOURTH SERIES, VOL. I-31

have compared the theoretic orbit with either of them without changing our conclusions.

The table we have given would seem to present enormous differences. But when we compare them within the limits to which his data restricted Le Verrier, the discrepancy becomes as small as could be possibly hoped.

1. The mean distance is by Walker, in terms of the mean distance of the earth, 30-25; while by Le Verrier it was predicted to be 36·15. The difference is one-fifth of the former. But the actual distance of the planet from the sun at the epoch of the discovery was, by the first, 30-026; while by the second, it was no more than 32.8,-the difference falling to one-fifteenth, or far within the possible limits of uncertainty.

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2. The mass was predicted by Le Verrier to be probably 300. By Struve's calculation it is no more than 1. But, besides the possibility of error in the data for an investigation of so much delicacy, it is to be seen that the difference bears an obvious relation to the difference in the distances. If Le Verrier's distance turn out to be correct, the mass will become larger than that given by Struve. 3. The position is the element in relation to which the investigation promises the greatest degree of accuracy; for when the direction of an attractive body is known, its action may remain the same if its distance is increased, provided the mass be augmented correspondingly. The discovery of the planet in conformity with Le Verrier's directions shows how closely he had approached the truth; and here the comparison is not with a calculated orbit, but with the actual position in the heavens. If, however, we have recourse to Walker's orbit, we find that in 1812 the difference of the planet in longitude in Le Verrier's own orbit from that calculated by Walker is only 307; while in 1842 it is less than one-fifth of a degree. It so happens, that the action of this planet upon Herschel only began to be sensible in 1812, and ceased to be sensible in 1842. In 1812, when the disturbances were only beginning, the error, although far from large, is the greatest, and diminishes as the action becomes more sensible; while in 1842, when Le Verrier was enabled to bring all the disturbances during thirty years to bear upon the question, the assigned position is so near that calculated by Walker, that the approximation appears marvellous. We may now, therefore, infer,— 1. That the planet discovered by Galle is the very body whose action on Herschel was investigated by Le Verrier, and whose existence was inferred from the disturbances it produced. 2. The direction which Le Verrier assigned to the planet, is nearer the truth than could possibly have been hoped for. 3. That the error in the

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distance from the sun in Le Verrier's orbit does not differ enormously from that calculated by Le Verrier, and was at the time of the discovery far from being great. 4. That if Struve's calculation of the mass be admitted to be correct, the difference between his result and the prediction of Le Verrier is little greater than the doubt which still exists in relation to the mass of Herschel.

We have already expressed our doubts whether the orbit of Walker is yet founded on sufficient data. We now say, further, that this hesitation to receive his results is not founded upon any denial of the accuracy of his observations, and far less upon any want of skill on his part in making the calculations. The observations at Washington are not only satisfactory in themselves, but are thoroughly confirmed at many other observatories; while his skill and accuracy as a calculator are perhaps unrivalled. We think, however, that if his calculations were translated into ordinary language, it would present an argument that would be not only metaphorically, but literally, reasoning in a circle. Searching the probable path of the observed planet, a star is found wanting, which, upon the hypothesis of uniform motion, of course in a circle, is considered to represent a former place of the body. From that place, and those whence the rate of the motion is derived, an orbit is calculated which differs but little from a circle. Can it be doubted for a moment, that the original proposition has been again reached in the course of the argument? From this orbit, the periodic time, the mean distance, and the elliptical elements are no more than inferences. That an arc as great as that between the star of La Lande, and the observed places of the planet, will not give the true orbit of a planet, may be shown by an instance which happens to lie before us. If the planet Mars had been seen for the first time on the 23d of July, and again on the 16th of December, 1807, he would have been found to have described in the interval one-fourth part of his path among the stars. In the absence of other data, it might have been inferred that his periodic time was 580 days, with a corresponding mean distance. Mars, however, would not have described half his apparent path until the 21st of May, 1808, increasing the resulting periodic time to 604 days. Again, Mars did not return to the place in which we have supposed him to have been first observed, until the 8th of June, 1809; and now the periodic time can be determined to be actually within a few hours of 687 days. The relative difference between the first and last of these numbers is only less than that between the periodic times given by Le Verrier and Walker, in the ratio of 19:189.

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In conclusion, we remain of opinion, that, as far as any direct

proof has yet been given, the orbit of Le Verrier may be as near the truth as the orbit of Walker. Nothing can decide between them but observations continued for five or six years to come. If these should confirm the orbit of Walker, Le Verrier will lose nothing except the reputation of having made a good choice of the observations to be combined in the calculation of the most probable elements, for the orbit of Walker falls within his previously announced limits. But he must forever retain the glory of having inferred the most probable direction of the disturbing body, and of having, if only in this one point, solved a problem believed to be impracticable.

ART. IX.-CHEEVER'S LECTURES ON "THE PILGRIM'S

PROGRESS."

Lectures on the Pilgrim's Progress, and on the Life and Times of John Bunyan. By Rev. GEORGE B. CHEEVER, D. D. Seventh Edition. New-York: Edward Walker, 114 Fulton-street. 1847.

JUST after reading Southey's Life of Bunyan, we sat down to the perusal of the work which forms the subject of this article. The transition was peculiarly gratifying. It was as if the fog of a dull morning had suddenly cleared away, revealing a beautiful landscape. Southey's work is an ignorant misrepresentation of Bunyan's character, and a burlesque on true religion. It is sadly deficient in moral principle, and is, in fact, an apology for intolerance and persecution. Dr. Cheever, on the contrary, has done Bunyan's religious experience and feelings ample justice, especially in his frequent comparisons of Bunyan's autobiography, in his "Grace Abounding to the Chief of Sinners," with his "Pilgrim's Progress."

"As you read the Grace Abounding you are ready to say, at every step, Here is the future author of the Pilgrim's Progress. It is as if you stood beside some great sculptor, and watched every movement with his chisel, having had his design described to you beforehand, so that at every blow some new trait of beauty in the future statue comes clearly into view. In the Grace Abounding you see at every step the work of the Divine Artist on one of the most precious living stones that ever his wisdom and mercy selected in this world to shine in the glory of his living temple. Nay, to lay aside every figure but that employed by the Holy Spirit, you see the refiner's fire, and the crucible, and the gold in it, and the heavenly Refiner himself sitting by it, and bending over it, and carefully removing the dross, and tempering the heat, and watching and waiting for his own perfect image. How beautiful, how sacred, how solemn, how interesting, how thrilling the process!" Pp. 13, 14.