Names of Districts and Territories. ARTICLE IV. Population of the United States of America, and Territories thereof, agreeable with an actual Enumeration made according to Law, in the Year 1810, Population of the principal Towns in the United States of America. I am indebted for the following curious speculation on this subject to John Leslie, Esq. F.R.S. E. Professor of Mathematics in the University of Edinburgh.-T. The population of America at the different periods of life may nearly represented by the differences between the complimentary spaces of a parabola. Thus, the population under ten years of age may be represented by the space A B FX, of which the boundary X F is a portion of a parabola, XZ, whose vertex is Z. From the property of the curve, DH is as D Z. Therefore D H x DZ is as D Z; and therefore any trilinear area, as DHZ, being one-third of the rectangle D H × DZ, is also as D Z3. Therefore the area AF is as A Z3 as X G H I 10 6 10 19 41 E ABCD BG If now we multiply these proportional areas by 10, and retain the nearest figure in the place of thousands, they will stand as in the second column below, while the gross return of population is as in the first. It appears, therefore, that in the first and last periods the areas are very nearly proportional to the numbers alive at those periods. But in the second, third, and fourth, periods, these areas exceed the return; which excess increases from the earlier to the more advanced ages; so that the American population is increasing more rapidly than the parabolic theory supposed. ARTICLE V. Observations on the Objections that have been made to Mr. Prevost's Explanation of the Effect produced by a Mass of Ice on a Thermometer in the Focus of a Reflecting Mirror. By Mr. Richard Davenport. SIR, (To Dr. Thomson.) BEING on a visit in Edinburgh, I took advantage of the opportunity it gave me of attending the lectures of the Professor of Chemistry, and witnessing a series of admirably conducted experiments. In a lecture on caloric an experiment was made with a conical tube of polished metal, the result of which has been considered as tending to prove the actual radiation of positive cold. On considering the phenomenon here presented, it appeared to me that it might be accounted for exactly on the principle on which an apparent radiation of cold from a body of ice in the focus of a concave metallic mirror, on a thermometer in the focus of a similar mirror set opposite to it, has been explained; which explanation I had considered as satisfactory, and I had thought was generally admitted. However, on looking into Mr. Murray's Elements of Chemistry, I found he objected to Mr. Prevost's explanation, and considered the phenomenon as not yet accounted for. I found also in the New Edinburgh Encyclopædia, Mr. Murray's objections stated and admitted, as objections unanswered, and the whole question left in an undecided state. Now as it appears to me that the phenomena both of the mirrors and the cone may be satisfactorily explained without supposing the actual existence and radiation of cold; and as the authorities I have mentioned are much looked up to; and as they leave an important question in philosophy in a state of indecision; if that explanation which I have said appears to my mind satisfactory, should on perusal appear so to you, you will perhaps not object to employ a page or two in the Annals of Philosophy by the admission of the following sketch. I am, Sir, Your obedient humble servant, It is well known that if a body be placed in the principal focus of a concave parabolic reflecting metallic disc, and a hot body in the corresponding focus of a similar reflector standing at a distance from, but duly opposite to, the first, the radiating caloric of the hot body is reflected and condensed upon the body in the focus of the first reflector; proving that radiating caloric is reflected in like manner with rays of light. But if a comparatively cold body is substituted for the hot radiating body in the focus of the second reflector, a thermometer in the focus of the first indicates a temperature below that of adjacent bodies not in that focus. This experiment was, I believe, first brought forward by Mr. Pictet, and the phenomenon at first appeared to favour the hypothesis of the actual radiation of cold; but a well-known explanation of it has been given by Mr. Prevost. I have not been able to see this explanation in Mr. Prevost's own words, but I collect it as referred to in other books. He considers that caloric is radiated from bodies at all temperatures. The thermometer, therefore, itself gives out heat by radiation. In consequence of this universal radiation, all bodies must also receive radiations. The thermometer, therefore, is receiving heat also; and while it receives an equivalent for what it gives out, no change of temperature is to be expected. But he adds, that the quantity radiated from all bodies is in proportion to the difference of their temperatures. Since, then, comparatively cold bodies emit comparatively weak radiations, when ice is placed in the focus of the second mirror the thermometer receives less than it emits, and thus loses in temperature. "But," says the writer of the article in the Encyclopædia, "this explanation is totally inconsistent with the most obvious facts attending the cooling of bodies;" for, he continues, “it is evident that on this hypothesis a hot body ought to cool more slowly when it is placed near a large body of inferior temperature than when near a small one; because in the former case it must receive more calorific emanations than in the latter." This is precisely what I deny, notwithstanding the deference I feel for the opinions of the writer of this article, as well as for those of Mr. Murray, whose objections follow. The article continues, "It is also very properly objected by Mr. Murray, that of different surfaces which at a given temperature radiate different quantities of caloric, that which radiates least must be least powerful in returning caloric to the thermometer, and must therefore have least effect in counteracting the reduction of its temperature." It then goes on to quote Mr. Murray as saying, that if Mr. Prevost's explanation were just, a blackened surface (which it is proved by Mr. Leslie's experiments radiates the largest quantity of caloric) ought to produce the least cooling effect on the thermometer, and a metallic surface (which radiates the least) ought to produce the greatest." I shall beg leave to answer this objection with the preceding one. The Encyclopædia continues, "Rejecting then this explanation of Mr. Prevost's, we shall proceed to consider that of Pictet ;" to which also they object, and therefore leave the phenomena entirely unaccounted for. The writer of the article, however, does not infer from this phenomenon the absolute existence of a frigorific principle, but only argues "that the objections against Mr. Prevost's and Mr. Pictet's explanation of the indication of decrease of temperature in the focus of the reflector, apply with equal force in cases of cooling by direct radiation and absorption;" which is certainly true. I will now describe also the experiment I have above alluded to with the metallic cone. A metallic conical tube of about 18 inches in length, whose respective apertures were one inch and five inches (the interior being polished so as to reflect powerfully and radiate little), being supported in a horizontal position, one ball of a differential thermometer was presented to the larger aperture, while a bottle of ice was presented at the smaller aperture. The liquor in the tube of the thermometer rose very little towards the bulb nearest the aperture, slightly indicating a lower temperature. The relative situations of the thermometer and the ice bottle being reversed, (i.e. the ice carried to the large end, and the bulb of the thermometer placed before the smaller end,) the liquor rose much faster towards the bulb, indicating in this position a much more rapid decrease of temperature. In this case it was argued that since the rays admitted at the wide opening were brought to a focus at the smaller one, if the thermometer at the wide end radiated caloric, its rays must be lost upon the ice in the focus, and it ought to indicate a loss of temperature; which was not the case, or only in a very small degree: but that when transposed, as this was the only focus (the experiment in that respect differing from that of the mirrors), the loss of temperature which it so rapidly indicated could arise only from a collection of actual cold rays in the focus; and that this experiment constituted another objection to Mr. Prevost's theory. This objection I shall endeavour to remove with the rest. It will be convenient first to consider in what way a focus of rays (whether of light, or of heat, or of cold,) is produced. Every point of a luminous body emits rays, proceeding in every direction in straight lines diverging from that point at all angles. Different parts of the same luminous body emit rays which cross each other at all angles; but which, after a few reflections, are absorbed, or dissipated into boundless space. But since every body possesses free caloric, more or less, and every point of these emits rays of caloric, as the points of luminous bodies do those of light, it follows that every body is surrounded on all sides by radiating and by receiving points; and may therefore be considered as surrounded by a sphere, from every point of which rays proceed and fall on it : and since the intensity is inversely as the squares of the distances, if the temperature of the parts of which the whole imaginary sphere consists is uniform, it receives equal quantities on all sides, although some of the surrounding bodies are much nearer than others; because the quantity of surface of distant bodies, intercepted by the intervening near bodies, is conversely in the same ratio; i. e. as the squares of the distances. Thus if a circular screen, whose area we will suppose equal to 36 square inches, be held at |