Lastly, that the point which terminated the line was not somewhat in any respects, but only somewhere; viz. at the end of the line; and that the same was true when a point was otherwise situated; e. g. the centre of a sphere.

He remarked, that an instant also existed as the limit of duration; e. g. the midnight with which one day terminated and the other began; but this existed not where the one ended and the other began, but when; or such a limit was not somewhere, but, if there were such a word, somewhen. That rest, or the zero of motion, existed when and where a body came to rest, and that shadow existed when and where light was absent.

He moreover considered the subject of infinity, and distinguished three sorts of infinity.

He remarked that he should designate a quantity as absolutely infinite, if it were so great as to be destitute of any boundary or limit; and gave the only two recognized examples of this, viz. boundless space, and that duration which is made up of ETERNITY, PAST and FUTURE. Eternity past was that which found its realization in the Divine Pre-existence, and Eternity future was to be found in the endless duration of the same; and nothing less than the combination of both of these, nothing short of it, constituted the absolute infinity of duration.

He moreover remarked that he should designate a quantity, as being specifically infinite, if it were just as boundless as those last described, but in certain respects only. He gave as examples :

1. A straight line without termination in either direction from a point which might be assumed in that line, such a line would be specifically infinite; viz. in length.

2. A surface without border which would be specifically infinite; viz. in length, breadth, and superficial area. He drew the conclusion, moreover, that an interminable line which was not straight throughout, must be longer than that which was perfectly straight, since the former not merely extended through space in its length, but intruded somewhat upon the breadth of space.

He next remarked that he should designate a quantity as being in comparison with another, relatively infinite, if its ratio to that other were too great to be estimated; that in this sense alone could we speak of an infinite number of things, or of an infinitely great number in the abstract. The like must be true of velocity, and also of mere mechanical force.

He next considered the subject of motion as applicable to mathe

H

matical quantities, and gave some illustrations showing, that when bodies moved they forsook the positions in space which they at first occupied, and that the position occupied by the centre of gravity, or any specified point of reference with regard to the body, was in like manner left behind, and a new position in space be so situated, as to be the centre of gravity, or point of reference of the body; both the space first occupied and the positions left behind having, themselves, no motion. He therefore designated the motion of a mathematical point, as being a pleasant fiction, and said that, were it otherwise, a point, which was nothing, might, by motion, produce a line which had length.

He next supposed a point (P) to be assumed in an interminable line, and remarked, that all that portion of the line on the one side of the point, must be regarded as being in effect the half of the line, and all on the other side as being, in effect, the other half. But if a new point (P') were assumed in the same line at any finite distance from

the other, the two portions, one on the one side, and the other on the other side, must, as before, be regarded as being, in effect, the halves of the line; though all the intervening portion of the line (PP') had (at the new point of division) been taken from the one half, and added to the other. Hence, any finite straight line must be regarded as good for nothing, in comparison with a straight line interminable in only one direction; or if the line thus interminable were used as the measuring unit, its ratio to any finite straight line must be represented by. Any other finite straight line, however great or however small, must in like manner be represented by zero in comparison with the same measuring unit; and the ratio of the one finite quantity to the other, be therefore represented by 8. Hence & was a symbol of indeterminateness. In this case that indeterminateness would be absolute. Prof. A. also remarked with regard to another common case, in P (X-a) which the value of 8 might enter; viz. Q (X-)a", X = a; that, in this case, the numerator and denominator both were reduced to zero, because the multiplier in each case vanished, so that no process of multiplication was possible; and there was, in each case, absolutely no result: insomuch, that vanishing fractions might, in this point of view, be rather termed vanished fractions.

m

which=0 when

[Prof. A. next, incidentally, spoke of the reason why the radius in

the investigation, of analytic trigonometry must be regarded as positive when measured from the centre outwards to any point on the circumference.

He adverted for this purpose to the method employed for the determination of the position of a point in space, showing that the distance from the origin in one direction must be regarded as positive, to whichever of the three axes reference was employed, and that a negative distance could, in every case, be obtained, by measuring backward from (P) the farther extremity of the opposite distance (PM), an extent (P P') greater than the positive distance, and thus passing in the opposite direction to the other side of the origin.

He next supposed the angles made by some (or all) of the axes with each other, to be so increased that those axes should all be brought into the same plane. The three directions from the origin outward would thus be found to be positive, while the opposite directions must be regarded as negative. As, moreover, any number of groups of axes, three in each, might thus be clustered around the same origin, all directions from the centre outward must then be regarded as positive, and the contrary negative.]

Prof. ALEXANDER then resumed the comparison of finite quantities with the infinite, and in manner as before, proceeded to show that if a plane were supposed to extend through all space, all the portion on the one side of this plane must be regarded as being, in effect, the half of all space, and that all on the other side, as being, in effect, the other half. The like would, however, be true if another such plane were to extend through space parallel to the first; though to what before constituted the one half, would be added all the space between the two planes, and the same subtracted from the other. Hence, reasoning in the manner as before, we must conclude that this intervening space, though boundless in some of its dimensions, must be regarded as good for nothing in comparison with the half of all space; i. e. the half in the sense already described.

For like reasons, any finite portion of time must be regarded as nothing, in comparison with either eternity past or eternity future; and thus we might, in some humble measure, discern how, in view of a mind which could grasp the whole, "a thousand years" would be " as one day, and one day as a thousand years."

He lastly considered the question-whether, if the visible creation were annihilated, space would still exist, and concluded that we had not sufficiently accurate ideas of such a state of things to determine with regard to it; but insisted that, in any event, space could not

exist, independent of the GREAT FIRST CAUSE, in whose existence, as it was, and is, and is to come, was to be found the one, the absolutely necessary truth, and that all others were contingent, just so far as He had made them so.

ON THE ZODIACS OF THE ASTEROIDS. BY PROF. J. S. HUBBARD.

Prof. HUBBARD, of the Washington Observatory, stated to the Association that he was then engaged in computing the Zodiacs of the Asteroids. The term Zodiacs, as here applied, he defined as referring to the zone or belt within which are included all possible geocentric positions of the particular asteroid in question: and the object in thus determining these belts was to facilitate researches into the past history of these remarkable bodies; since in most cases, the question of identity of a missing star, with any asteroid, may be settled at once by a simple inspection of the Zodiacs. The method of computation was that proposed by Gauss in the Monatliche Correspondenz, Vol. X., from which Prof. H. quoted the plan of the analytical investigation.

ON THE DECOMPOSITION OF ROCKS BY METEORIC AGENTS, AND ON THE ACTION OF THE MINERAL ACIDS ON FELDSPAR, &C. BY PROFS. W. B. AND R. E. ROGERS.

[An abstract of this memoir may be found in the fifth volume of the American Journal of Science and Arts, p. 401.]

ON THE DECOMPOSITION OF ROCKS BY METEORIC WATER. Br PROFS. W. B. ROGERS AND R. E. ROGERS.

In presenting this communication, the fact was stated, that only one or two observations have hitherto been made by chemists, to test, in a direct and conclusive manner, the power of water, at ordinary temperatures, to decompose rocky substances; at the same time, the general fact of such a decomposition appears to have been assumed in explaining the disintegration of mineral masses, and the conveyance of inorganic ingredients into the substance of plants. The experiments have applied to all the principal crystalline minerals, containing alkalies and alkaline earths, amounting to nearly forty species-to the principal aggregates, such as granite, gneiss, &c.-to the different varieties of glass-and to various kinds of coal and wood.

These experiments were also of two kinds with each specimenthe one with pure distilled water; the other with water charged with carbonic acid. The mineral, or other matter, being reduced to a very fine powder in an agate mortar, was in small quantity mixed with the liquid, and transferred to a filter of purified paper. One or more drops of the percolating fluid, received on a slip of platinum was gently evaporated to dryness, and the tache resulting was then examined by delicate test paper. In all cases, the residuum from the carbonic acid water, was greater than from the other; but even in the former, with most minerals, a decided alkaline reaction was obtained. By heating the tache gently for a short, and then a longer time, and again strongly by the blowpipe, unequivocal proof was furnished of the presence of potassa or soda-or of lime or magnesia. The liquid from calcareous or magnesian minerals, becomes milky after heating -the tache from the potash and soda minerals, as for example, the feldspars, lost its alkaline reaction by the first contact of the blowpipe flame that from the lime was greatly augmented by the first calcination, in consequence of the removal of carbonic acid, and continued intensely alkaline after a prolonged exposure to the heat. The tache furnished by magnesia minerals, such as the serpentines, was much impaired in alkalinity by igniting, but continued to present a decided reaction with the test paper, after long exposure to the heat. In this way the behaviour of the tache was shown to be capable of furnishing a useful auxiliary means of extemporaneous qualitative analysis.

Some experiments were introduced, in which these effects were properly exhibited by powdered glass, mica, and feldspar.

The attention of chemists was especially invited to these phenomena, as having very important bearings, not only upon the decomposition of rocky masses by the action of the percolating rain, but the subsequent introduction of various crystalline minerals in the rifts and cavities of the strata, and as indicating the necessity of some new and better method than that commonly employed, for determining the amount of alkali present in vegetable or other organic matters.

Experiments were also cited, disproving the opinion which appears to be received among chemists, that the feldspars, hornblends, &c., are entirely unacted upon by sulphuric or hydrochloric acids. By exposing these materials in fine powder to prolonged digestion in the acid, even at common temperatures, a partial solution was found to result. Thus 30 grains of potash feldspar, by digestion for twelve hours, in hydrochloric acid, at temperature 60 degrees, lost nearly