be a slow average apparent tendency of all the stars to the vanishing point of lines parallel to that direction, and to the region which he is leaving, however greatly individual stars might differ from such average by reason of their own peculiar proper motion. This is the necessary effect of perspective; and it is certain that it must be detected by observation, if we knew accurately the apparent proper motions of all the stars, and if we were sure that they were independent, i.e. that the whole firmament, or at least all that part which we see in our own neighbourhood, were not drifting along together, by a general set as it were, in one direction, the result of unknown processes and slow internal changes going on in the sidereal stratum to which our system belongs, as we see motes sailing in a current of air, and keeping nearly the same relative situation with respect to one another.

(854.) It was on this assumption, tacitly made indeed, but necessarily implied in every step of his reasoning, that Sir William Herschel, in 1783, on a consideration of the apparent proper motions of such stars as could at that period be considered as tolerably (though still imperfectly) ascertained, arrived at the conclusion that a relative motion of the sun, among the fixed stars in the direction of a point or parallactic apex, situated near λ Herculis, that is to say, in R. A. 17h 22m 260° 34', N. P.D. 63° 43' (1790), would account for the chief observed apparent motions, leaving, however, some still outstanding and not explicable by this cause; and in the same year Prevost, taking nearly the same view of the subject, arrived at a conclusion as to the solar apex (or point of the sphere towards which the sun relatively advances), agreeing nearly in polar distance with the foregoing, but differing from it about 27° in right ascension. Since that time methods of calculation have been improved and concinnated, our knowledge of the proper motions of the stars has been rendered more precise, and a greater number of cases of such motions have been recorded. The subject has been resumed by several eminent astronomers and mathematicians: viz. 1st, by M. Argelander, who, from the consideration of the proper motions of 21 stars exceeding

1" per annum in arc, has placed the solar apex in R. A. 256° 25', N. P.D. 51° 23'; from those of 50 stars between 05 and 10, in 255° 10′, 51° 26'; and from those of 319 stars having motions between 0"-1 and 0"-5 per annum, in 261° 11', 59° 2′: 2ndly, by M. Luhndahl, whose calculations, founded on the proper motions of 147 stars, give 252° 53', 75° 34' and 3rdly, by M. Otto Struve, whose result 261° 22', 62° 24', emerges from a very elaborate discussion of the proper motions of 392 stars. All these places are for A. D.

1790.

(855.) The most probable mean of the results obtained by these three astronomers, is (for the same epoch) R. A. =259° 9', N.P.D. 55° 23′. Their researches, however, extending only to stars visible in European observatories, it became a point of high interest to ascertain how far the stars of the southern hemisphere not so visible, treated independently on the same system of procedure, would corroborate or controvert their conclusion. The observations of Lacaille, at the Cape of Good Hope, in 1751 and 1752, compared with those of Mr. Johnson at St. Helena, in 1829-33, and of Henderson at the Cape in 1830 and 1831, have afforded the means of deciding this question. The task has very recently been. executed in a masterly manner by Mr. Galloway, in a paper published in the Philosophical Transactions for 1847 (to which we may also refer the reader for a more particular account of the history of the subject than our limits allow us to give.) On comparing the records, Mr. Galloway finds eighty-one southern stars not employed in the previous investigations above referred to, whose proper motions in the intervals elapsed appear considerable enough to assure us that they have not originated in error of the earlier observations. Subjecting these to the same process of computation he concludes for the place of the solar apex, for 1790, as follows: viz. R. A. 260° 1', N. P. D. 55° 37', a result so nearly identical with that afforded by the northern hemisphere, as to afford a full conviction of its near approach to truth, and what may fairly be considered a demonstration of the physical cause assigned.

(856.) Of the mathematical conduct of this inquiry the nature of this work precludes our giving any account; but as the philosophical principle on which it is based has been misconceived, it is necessary to say a few words in explanation of it. Almost all the greatest discoveries in astronomy have resulted from the consideration of what we have elsewhere termed RESIDUAL PHÆNOMENA*, of a quantitative or numerical kind, that is to say, of such portions of the numerical or quantitative results of observation as remain outstanding and unaccounted for after subducting and allowing for all that would result from the strict application of known principles. It was thus that the grand discovery of the precession of the equinoxes resulted as a residual phænomenon, from the imperfect explanation of the return of the seasons by the return of the sun to the same apparent place among the fixed stars. Thus, also, aberration and nutation resulted as residual phænomena from that portion of the changes of the apparent places of the fixed stars which was left unaccounted for by precession. And thus again the apparent proper motions of the stars are the observed residues of their apparent movements outstanding and unaccounted for by strict calculation of the effects of precession, nutation, and aberration. The nearest approach which human theories can make to perfection is to diminish this residue, this caput mortuum of observation, as it may be considered, as much as practicable, and, if possible, to reduce it to nothing, either by showing that something has been neglected in our estimation of known causes, or by reasoning upon it as a new fact, and on the principle of the inductive philosophy ascending from the effect to its cause or causes. On the suggestion of any new cause hitherto unresorted to for its explanation, our first object must of course be to decide whether such a cause would produce such a result in kind: the next, to assign to it such an intensity as shall account for the greatest possible amount of the residual matter in hand. The proper motion of the sun being suggested as such a cause, we have two

• Discourse on the Study of Natural Philosophy. Cab. Cyclopædia, No. 14.

things disposable-its direction and velocity, both which it is evident, if they ever became known to us at all, can only be so by the consideration of the very phænomenon in question. Our object, of course, is to account, if possible, for the whole of the observed proper motions by the proper assumption of these elements. If this be impracticable, what remains unaccounted for is a residue of a more recondite kind, but which, so long as it is unaccounted for, we must regard as purely casual, seeing that, for anything we can perceive to the contrary, it might with equal probability be one way as the other. The theory of chances, therefore, necessitates (as it does in all such cases) the application of a general mathematical process, known as "the method of least squares," which leads, as a matter of strict geometrical conclusion, to the values of the elements sought, which, under all the circumstances, are the most probable.

(857.) This is the process resorted to by all the geometers we have enumerated in the foregoing articles (art. 854,855). It gives not only the direction in space, but also the velocity of the solar motion, estimated on a scale conformable to that in which the velocity of the sidereal motions to be explained are given; i.e. in seconds of arc as subtended at the average distance of the stars concerned, by its annual motion in space. But here a consideration occurs which tends materially to complicate the problem, and to introduce into its solution an element depending on suppositions more or less arbitrary. The distance of the stars being, except in two or three instances, unknown, we are compelled either to restrict our inquiry to these, which are too few to ground any result on, or to make some supposition as to the relative distances of the several stars employed. In this we have nothing but general probability to guide us, and two courses only present themselves, either, 1st, To class the distances of the stars according to their magnitudes, or apparent brightnesses, and to institute separate and independent calculations for each class, including stars assumed to be equidistant, or nearly so: or, 2dly, To class them according to the observed amount of their apparent proper motions, on the presumption that

those which appear to move fastest are really nearest to us. The former is the course pursued by M. Otto Struve, the latter by M. Argelander. With regard to this latter principle of classification, however, two considerations interfere with its applicability, viz. 1st, that we see the real motion of the stars foreshortened by the effect of perspective; and 2dly, that that portion of the total apparent proper motion which arises from the real motion of the sun depends, not simply on the absolute distance of the star from the sun, but also on its angular apparent distance from the solar apex, being, cæteris paribus, as the sine of that angle. To execute such a classification correctly, therefore, we ought to know both these particulars for each star. The first is evidently out of our reach. We are therefore, for that very reason, compelled to regard it as casual, and to assume that on the average of a great number of stars it would be uninfluential on the result. But the second cannot be so summarily disposed of. By the aid of an approximate knowledge of the solar apex, it is true, approximate values may be found of the simply apparent portions of the proper motions, supposing all the stars equidistant, and these being subducted from the total observed motions, the residues might afford ground for the classification in question. This, however, would be a long, and to a certain extent precarious system of procedure. On the other hand, the classification by apparent brightness is open to no such difficulties, since we are fully justified in assuming that, on a general average, the brighter stars are the nearer, and that the exceptions to this rule are casual in that sense of the word which it always bears in such inquiries, expressing solely our ignorance of any ground for assuming a bias one way or other on either side of a determinate numerical rule. In Mr. Galloway's discussion of the southern stars the consideration of distance is waived altogether, which is equivalent to an admission of complete ignorance on this point, as well

• M. Argelander's classes, however, are constructed without reference to this consideration, on the sole basis of the total apparent amount of proper motion, and are, therefore, pro tanto, questionable. It is the more satisfactory then to find so considerable an agreement among his partial results as actually obtains.