the first growth of trees, in this region, must have taken root on the surface of new-made ground, deposited from the Mississippi, and that ground, or deposite, of the depth of 1056 feet, with a continued tendency to settle into a more compact condition by incumbent pressure, molecular affinity, and arrangement of particles, as we found it to be the case with the settling of our sediment. Every year's sinking down, in this case, would be fully compensated by every year's deposite, for the lower it at any time sunk the longer it would be subject to inundation by the subsequent overflow, so that this sinking and compensatory process would go on together, and continue about equal, which well accounts for the circumstances of former forests being now found so far below the level of the sea. Should this subsidiary process be now, in any measure, incomplete in that region where the City of New Orleans now stands, and which we indeed very much question, in that case the compensatory equivalent being cut off by embankments, or the levying out of the waters, the time may, by possibility, yet come, when the ground on which the City of New Orleans now stands, may sink to a level with the ocean, or even below it; at least, should there be any further settlement at all, it must, in the same ratio, approximate that level; for the final adjustment of particles in a deposite of 1056 feet deep, that will preclude any further settling, may be expected to take an immensity of time. ANDREW BROWN, M. W. DICKESON, Committee. Table of Calculations made use of in the foregoing Report. 1. Quantity of water discharged by the Mississippi river, annually, 14,883,360,636,880 cubic feet. 2. Quantity of sediment discharged by the Mississippi river, annually, 28,188,083,892 cubic feet. 3. Area of the delta of the Mississippi, according to Mr. Lyell, 13,600 square miles. 4. Depth of the delta, according to Prof. Riddell, 1056 feet. 5. The delta, therefore, according to 3 and 4, contains 400,378,429, 440,000 cubic feet, or 2720 cubic miles. 6. According to 2, it would require, for the formation of one cubic mile of delta, 5 years 81 days. 7. For the formation of one square mile, of the depth of 1056 feet, 1 year 16 days. 8. For the formation of the delta, according to 2, 3, 4, 14,203 years. 9. The Valley of the Mississippi, from Cape Girardeau to the delta, is estimated to contain 16,000 square miles, of 150 feet deep; it therefore contains 66,908,160,000,000 cubic feet, or 454 cubic miles. Prof. GERMAIN gave some views of his connected with this subject. On motion of Prof. W. R. JOHNSON, it was recommended to the Association that thanks be presented to the Committee for the very able manner in which they have discharged their duty in this matter, and that the report be published in full as early as practicable. Dr. R. COATES moved that this Section adjourn to meet to-morrow at 10 A. M., which was agreed to. R. W. GIBBES, Sec'ry. September 22. SECTION OF GENERAL PHYSICS, &c. Second Meeting. The Section met at 9. A. M. The following papers were read: ON THE FUNDAMENTAL PRINCIPLES OF MATHEMATICS. BY PROF. STEPHEN ALEXANDER. Prof. ALEXANDER remarked, the object of all scientific research was truth; a term too valuable to be misunderstood, and yet too general to admit of a ready definition. He proceeded, however, to characterize it, in some of its various aspects, observing that while each is applicable to its own object, that is true in mathematics, which, under the existing system of things, is supposable; that is true in physics, which, under the existing system of things, has been permitted to exist; that is true in matters of taste which is consistent with the laws of beauty, deduced from the observation of things actual; and that is true in morals, (in the highest and best sense in which it is good,) which is consistent with what is found in the GREAT SOURCE OF ALL GOOD. He next proceeded to state that mathematics had not to do with things, but the relations of things, and it was sufficient that those relations should be supposable; and that the certainty of mathematical reasoning rested upon the fact, that those relations could be more readily understood and completely defined, than the properties of the things themselves. He stated, moreover, that these were constituted relations, and not mere figments of the human mind; the things which we had to deal with, being made, in certain respects, not merely what they were, but as they were. Thus, that two bodies did not occupy the same space, and that it was true that one day of the week must follow another, were not true because his audience and himself might think so, but because the Creator had made them so. He next commented upon the general term which was used to designate that to which mathematical reasoning was applicable; viz. quantity; and said, that in so far as mathematics had to do with it, it was that which admitted of the distinction of greater or less. Moreover, quantities were of the same species when each in itself exceeded its less in the self same respect in which the other in itself exceeded its less; whatever might be true of the boundaries or limits of either. Thus, a straight line and a curve were of the same species, since each exceeded its less in length; so, also, an hour and a minute were of the same species, since each exceeded its less in duration. He remarked, that the only point of resemblance between quantities of different species, was to be found in the fact, that the distinction of greater and less was admissible in the case of every species; and hence it was possible to compare the ratio of two quantities of one species, with that of two quantities of another species. He proceeded to the more special consideration of the two great relations of things, time, and space; remarking, that space might be described, as that wherein there was room for the separate existence of material substances; and duration, as that wherein there was room in another sense for the separate, and therefore successive, occurrence of events. He next commented upon the nature of zero; showing that it implied the absence of the species of quantity which happened to be the subject of investigation, and not the absence of every other. That, thus, the surface which bounded a solid quantity, was not somewhat in the same sense in which the solid was somewhat, viz. in the property of occupying space, but only somewhere, viz. where the solid terminated, and space met it; the space without met it, though that surface was still somewhat in superficial extent. That the line which bordered the surface was not somewhat in this last respect, but only somewhere; though still somewhat in length. Lastly, that the point which terminated the line was not somewhat in any respect, 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 recognised 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 (P P') 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 . 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 which the value of might enter; viz. P(X-a)" which Q(X-a)", m = & when 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. [Prof. A. next, incidentally, spoke of the reason why the radius in |