tions of the axis of the atmospherical logarithmic. Therefore, if we multiply our common logarithms by 10,000, they will exprefs the fathoms of the axis of the atmospherical logarithmic; nothing is more easily done. Our logarithms contain what is called the index or characteristic, which is an integer and a number of decimal places. Let us juft remove the integer-place four figures to the right hand: thus the logarithm of 60 is 1.7781513, 7781513 which is one integer and Multiply both suffer the same change of temperature; and as the air may be warmed or cooled when the mercury is not, or may change its temperature independent of it, ftill greater variations of fpecific gravity may occur. The general effect of an augmentation of the specific gravity of the mercury must be to increase the subtangent of the atmospherical logarithmic; in which cafe the logarithms of the denfities, as measured by inches of mercury, will express measures that are greater than fathoms in the fame proportion that the fubtangent is increased; or, when the air is more expanded than 17781,513, the mercury, it will require a greater height of homogeneous atmosphere to balance 30 inches of mercury, and a given fall of mercury will then correfpond to a thicker ftratum of air. The practical application of all this reasoning is obvious and eafy; obferve the heights of the mercury in the barometer at the upper and lower ftations in inches and decimals; take the logarithms of these, and subtract the one from the other; the difference between them (accounting the four first decimal figures as integers) is the difference of elevation of fathoms. EXAMPLE. Merc. Height at the lower station 29,8 1°4742163 upper station 29,1 1°4638930 Diff. of Log. X 10000 o'o103233 or 103 fathoms and of a fathom, which is 233 1000 619,392 feet, or 619 feet 4 inches; differing from the approximated value formerly found about an inch. Such is the general nature of the barometric measurement of heights first fuggefted by Dr HALLEY; and it has been verified by numberlefs comparisons of the heights calculated in this way with the fame height measured geometrically. It was thus that the precife specific gravity of air and mercury was most accurately determined; namely, by obferving, that when the temperature of air and mercury was 32, the difference of the logarithms of the mercurial heights were precisely the fathoms of elevation. But it requires many corrections to adjust this method to the circumftances of the cafe; and it was not till very lately that it has been fo far adjusted to them as to become ufeful. We are chiefly indebted to Mr De Luc for the improvements. The great elevations in Switzerland enabled him to make an immenfe number of observations, in almost every variety of circumftances. Sir GEORGE SHUCKBOURGH alfo made a great number with most accurate inftruments in much greater elevations, in the fame country; and he made many chamber experiments for determining the laws of variation in the subordinate circumstances. General Roy alfo made many to the fame purpose. And to these two gentlemen we are chiefly obliged for the corrections which are now generally adopted. This method, however cannot apply to every cafe; it depends on the specific gravity of air and mercury, combined with the fuppofition that this is affected only by a change of preffure. But fince all bodies are expanded by heat, and perhaps not equally, a change of temperature will change the relative gravity of mercury and air, even although To perfect this method, therefore, we must learn by experiment how much mercury expands by an increase of temperature; we must also learn how much the air expands by the fame, or any change of temperature, and how much its elasticity is affected by it. Both thefe circumftances must be confidered in the cafe of air; for it might happen that the elasticity of the air is not so much affected by heat as its bulk is. It will, therefore, be proper to ftate the experiments which have been made for afcertaining these two expansions. The most accurate, and the best adapted experiments for afcertaining the expanfion of mercury, are thofe of General Roy, published in the Philof. Tranf. vol. 67. He expofed 30 inches of mercury, actually fupported by the atmosphere in a barometer, in a nice apparatus, by which it could be made of one uniform temperature through its whole length; and he noted the expanfion of it in decimals of an inch. These are contained in the following table; where the firft column expresses the temperature by Fahrenheit's thermometer, the fecond the bulk of the mercury, and the third the expanfion of an inch of mercury for an increase of one degree in the adjoining temperatures. TABLE A. Temp. Bulk of ☀. Expan. for 1o. The scale of the thermometer is conftructed on the fuppofition that the fucceffive degrees of heat are measured by equal increments of bulk in the mercury of the column; but that the correfponding expansions of this column do continually diminish, General Roy attributes to the gradual detachment of elastic matter from the mercury by heat, which preffes on the top of the column, and therefore Thortens it. He applied a boiling heat to the vacuum a-top, without producing any farther depreffion; a proof that the barometer had been carefully filled. It had indeed been boiled through its whole length. He had attempted to measure the mercurial expanfion in the ufual way, by filling 30 inches of the tube with boiled mercury, and expofing it to the heat with the open end uppermoft. But here it is evident that the expansion of the tube and its folid contents must be taken into the account. The expansion of the tube was found fo exceedingly irregular, and fo incapable of being determined with precifion for the tubes which were to be employed, that he was obliged to have recourse to the method with the real barometer. In this no regard was neceffary to any circumftance but the perpendicular height. There was, befides, a propriety in examining the mercury in the very condition in which it was used for measuring the preffure of the atmosphere; be caufe, whatever complication there was in the results, it was the fame in the barometer in actual ufe. The most obvious manner of applying thefe experiments on the expanfion of mercury to our purpose, is to reduce the obferved height of the mercury to what it would have been if it were of the temperature 32. Thus, fuppofe that the obferved mercurial height is 292, and that the temperature of the mercury is 72° make 30°1302: 30=29°2:29°0738. This will be the true meafure of the density of the air of the standard temperature. That we may obtain the exact temperature of the mercury, it is proper that the obfervation be made by a thermometer attached to the barometer-frame, fo as to warm and cool along with it. Or, this may be done without the help of a table, and with fufficient accuracy, from the circumftance, that the expanfion of an inch of mercury for one degree diminishes very nearly 6th part in each fucceeding degree. If therefore we take from the expanfion at 32 its thousand part for each degree of any range above it, we obtain a mean rate of expansion for that range. There is another way of applying this correction, fully more expeditious and equally accurate. The difference of the logarithms of the mercurial heights is the measure of the ratio of thofe heights. In like manner the difference of the logarithms of the observed and corrected heights at any ftation is the measure of the ratio of thofe heights. Therefore this laft difference of the logarithms is the measure of the correction of this ratio. Now, the observed height is to the corrected height nearly as I to 1000102. The logarithm of this ratio, or the difference of the logarithms of 1 and 1'000102 is 00000444. This is the correction for each degree that the temperature of the mercury differs from 32. Therefore multiply o'0000444 by the difference of the mercurial temperatures from 32, and the products will be the corrections of the re logarithms. There is ftill an easier way of applying th rithmic correction. If both the mercurial te tures are the fame, the differences of their log will be the fame, although each may be a go above or below the standard temperature, if panfion be very nearly equable. The correct be neceffary only when the temperatures at t ftations are different, and will be proporti this difference. Therefore, if the difference mercurial temperatures be multiplied by oo the product will be the correction to be m the difference of the logarithms of the me heights. But farther, fince the differences of garithms of the mercurial heights are also t ferences of elevation in English fathoms, it f that the correction is alfo a difference of eleva English fathoms, or that the correction for o gree of difference of mercurial temperature is of a fathom, or 32 inches, or 2 feet 8 inche This correction of 2.8 for every degree of ence of temperature must be subtracted fro elevation found by the general rate, when the cury at the upper station is colder than that lower. For when this is the cafe, the mer column at the upper ftation will appear too the preffure of the atmosphere too small, and fore the elevation in the atmosphere will a greater than it really is. Therefore the rule fo correction will be to multiply o'o000444 by ti grees of difference between the mercurial tem tures at the two stations, and to add or fubtrac product from the elevation found by the ge rule, according as the mercury at the upper fl is hotter or colder than that at the lower. If the experiments of Gen. Roy on the expar of the mercury in a real barometer be tho moft deferving of attention, and the expanfic confidered as variable, the logarithmic differ correfponding to this expanfion for the z temperature of the two barometers may be ta Thefe logarithmic differences are contained in following table, which is carried as far as 1 beyond which it is not probable that any obfe tions will be made. The number for each t perature is the difference between the logarit of 30 inches, of the temperature 32, and of inches expanded by that temperature. TABLE B. Stcr. VII. PNEUMATICS, or thrice as denfe, he used a column of 30 or 60 its density is of much greater confequence than he wished to examine the expanfion of air twice that of the mercury. The relative gravity of the two, on which the fubtangent of the logarithmic inches long; and to examine the expanfion of all curve depends, and confequently the unit of our that is rarer than the external air, he placed the fcale of elevations, is much more affected by the tube, with the ball, uppermoft, the open end comheat of the air, than by the heat of the mercury. ing through a hole in the bottom of the veffel This adjuftment is of incomparably greater diffi- containing the mixtures or water. By this poficulty than the former, and we can hardly hope tion the column of mercury was hanging in the to make it perfect. We fhall relate the chief ex-tube, fupported by the preffure of the atmofperiments which have been made on the expanfion of air, and notice the circumstances which leave the matter still imperfect. Gen. Roy compared a mercurial and an air thermometer, each of which was graduated arithmetically, that is, the units of the fcales were equal bulks of mercury, and equal bulks (perhaps different from the former) of air. He found their progrefs as in the following table: TABLE C. 20 1762 18.8 phere; and the elasticity of the included air was The following table contains the expansion of 1000 parts of air, nearly of the common density, by heating it from o to 212. The first column contains the height of the barometer; the ad conof mercury in the tube of the manometer, and tains this height augmented by the fmall column therefore expreffes the denfity of the air examined; the 3d contains the total expansion of 1000 parts: and the 4th contains the expansion for 19, fuppofing it uniform throughout. TABLE D. Denfity Expanfion Expanfion Barom. of Air of 1000 pts Expanfion 192 194'4 18.2 by 1°. ཟça 20 157'4 132 20 138°0 19°4 112 20 1180 20'0 483.89 2.2825 20.8 92 21.6 30°07 30 77 482 10 2.2741 72 75'6 20 22.6 29°48 29'90' 480'74 2*2676 52 53'0 32 20 31°4 21.6 29'90 30'73 485.86 2.2918 12 20 11'4 20'0 29'96 30'92 489*45 2°3087 29.90 30'55 476*04 2*2455 29.95 30'60 487°55 22998 30°07 30'60 482.80 2*2774 29'48 3000 489°47 2*3087 It has been established by many experiments that equal increments of heat produce equal increments in the bulk of mercury. The differences of temperature are therefore expreffed by the 2d column, and may be confidered as equal; and the numbers of the 3d column must be allowed to exprefs the fame temperatures with thofe of the firft. They dire&ly exprefs the bulks of the air, and the numbers of the 4th column exprefs the differences of thefe bulks. Thefe are evidently unequal, and fhew that common air expands moft of all when of the temperature 62 nearly. The next point was to determine what was the actual increafe of bulk by fome known increase of heat. For this purpose he took a tube, having a narrow bore, and a ball at one end. He meatur ed the capacity of both the ball and the tube, and divided the tube into equal spaces, which bore a determined proportion to the capacity of the ball. This apparatus was fet in a long cylinder filled with frigorific mixtures, or with water, which could be uniformly heated up to the boiling temperature, and was accompanied by a nice thermometer. The expansion of the air was meafuted by a column of mercury which rose or funk in the tube. The tube being of a small bore, the mercury did not drop out of it; and the bore be ing chofen as equal as poffible, this column remained of an uniform length, whatever part of the tube it chanced to occupy. By this contri vance he was able to examine the expanfibility of air of various denfities. When the column of mercury contained only a fingle drop or two, the air was nearly the density of the external air. If VOL. XVIII. PART I. Mean 30162 484°21 2*2840 Hence the mean expansion of 1000 parts of air of the denfity 30'62 by one degree of Fahrenheit's thermometer is 2284, or that of 1000 becomes 1000 284. If this expansion be fuppofed to foilow the fame rate that was obferved in the comparipansion of a thousand parts of air for one degree fon of the mercurial and air thermometer, the exof heat at the different intermediate temperature, will be as in the following table. TABLE E. A Temp. Expanfion. TABLE II, If we would have a mean expanfion for any ty, being greatest about the temperature particular range, as between 12 and 92°; which fo that its expanfibility by heat diminis is the moft ikely to comprehend all the geodæti- its denfity; but he could not determin cal obfervations, we need only take the difference of gradation. When reduced to about of the bulks 26'038 and 222.000=195968, and denfity of common air, its expanfion w divide this by the interval of temperature 80°, lows: and we obtain 2'4496, or 2'45 for the mean expanfion for 1°. It would perhaps be better to adapt the table to a mass of 1000 parts of air of the ftandard temperature 22°; for in its prefent form, it fhews the expanfibility of air originally of the temperature c. This will be done with fatficient accuracy, by faying (for 212°) 1071718: 1484,210=1000; 1849, and fo of the reft. Thus we shall construct the following table of the expansion of 10,000 parts of air. TABLE F. Temp. Balk. Differ. ¡Espai for I 212 492. 1141,5041 7075 835 1134,429 12.264 772 1122,165 14'150 152 1108,015 147151 0*70 132 1093,864 207911 1'04 25'943 1°29 This will give for the mean expanfion of 1000 parts of air between 12° and 92—2 29. Although it cannot happen, that in meaturing the differences of elevation near the earth's furface, we fhall have occafion to employ air greatly exceeding the common density, we may infert the experiments made by Gen. Roy on fuch airs. They are expreffed in the following table; where column first contains the denfities meatured by the inches of mer cury that they will fupport when of the temperature 33°; column fecond is the expantion of 1000 parts of fuch air, by being heated from o to 212; and column third is the mean expantion for Mean expanfion From this very extensive and judicious experiments, it is evident, that the expa of air by heat, is greatest when the air is a ordinary denfity, and that in small denfit greatly diminished. It appears alfo, that of compreffion is altered; for in this fped the rare air, half of the whole expansion about the temperature 99o, but in air of denfity at 103. The experiments of Am related in the Mem. of the Acad. at Par &c. are confiftent with these more perf experiments of Gen. Roy. After this account of the expanfion of fee that the height through which we mu produce a given fall of the mercury in th meter, or the thickness of the ftratum of a ponderant with a tenth of an inch of m muft increase with the expansion of air, a if be the expanfion of one degree, w 2.29 DOCO above 32° by o'00229, and multiply the multipy the excefs of the temperature of by 87, to obtain the thickness of the where the barometer ftands at 30 inches: o ever be the elevation indicated by the dif of the barometrical heights, upon the supp that the air is of the temperature 32°, w multiply this by o'00229 for every degree ti bir is warmer or colder than 32. The p must be added to the elevation in the firi and fubtracted in the latter. Sir GEORGE SHUCKBURGH deduces from his experiments, as the mean expanf air in the ordinary cafes: and this is pro nearer the truth; becaufe Gen. Roy's exper were made on air which was freer from than the ordinary air in the fields; and a minute quantity of damp increases its expar ty by heat in a prodigious degree. The difficulty is how to apply this correction; ther, how to determine the temperature of t in thofe extenfive and deep ftrata in which th vations are measured. It feldom or never pens, that the ftratum is of the fame temper throug TABLE I If we would have a mean expanfion for any ty, being greatest about the temperature particular range, as between 12 and 92°, which fo that its expanfibility by heat diminish is the most likely to comprehend all the geodæti- its denfity; but he could not determine cal obfervations, we need only take the difference of gradation. When reduced to about of the bulks 26.038 and 222.000=195968, and denfity of common air, its expansion wa divide this by the interval of temperature 80°, lows: and we obtain 2'4496, or 2'45 for the mean expanfion for 1°. It would perhaps be better to adapt the table to a mass of 1000 parts of air of the Atandard temperature 32°; for in its prefent form, it fhews the expanfibility of air originally of the temperature c. This will be done with fatficient accuracy, by faying (for 212°) 1071*718: 1484,210=1000:1:849, and fo of the reft. Thus we than conftruct the following table of the expansion of 10,000 parts of air. TABLE F. Temp. Bulk. 212 1141,504 Differ. Expand for I 77075 0354 1134,429 12 264 6613 172 1122,165 14*1500*708 152 1108,015 14 151 0*708 0'711 112 1079,636 This will give for the mean expansion of parts of air between 12 and 922 29. Although it cannot happen, that in meafuring the differences of elevation near the earth's furface, we fhall have occafion to employ air greatly exceeding the common density, we may infert the experiments made by Gen. Roy on fuch airs. They are expreffed in the following table; where column firft contains the denfities meatured by the inches of mer. cury that they will fupport when of the temperature 33°; column, fecond is the expansion of 1000 parts of fuch air, by being heated from o to 212; and column third is the mean expantion for From this very extenfive and judicious ra experiments, it is evident, that the expan of air by heat, is greatest when the air is ab ordinary density, and that in fmall denfitie greatly diminished. It appears alfo, that ti of compreffion is altered; for in this fpeci the rare air, half of the whole expansion ha about the temperature 99o, but in air of or denfity at 105°. The experiments of AMON related in the Mem. of the Acad. at Paris &c. are confiftent with these more perfpi experiments of Gen. Roy. After this account of the expanfion of ai fee that the height through which we must produce a given fall of the mercury in the meter, or the thickness of the ftratum of air ponderant with a tenth of an inch of me: muft increase with the expanfion of air, and 2'29 if- -be the expanfion of one degree, we 1000 multipy the excefs of the temperature of th by 87% to obtain the thickness of the fir above 32° by o'00229, and multiply the pro where the barometer ftands at 30 inches: or v of the barometrical heights, upon the fuppof ever be the elevation indicated by the differ that the air is of the temperature 32°, we multiply this by o'00229 for every degree tha air is warmer, or colder than 32. The pro must be added to the elevation in the first and fubtracted in the latter. Sir GEORGE SHUCKBURGH deduces o' from his experiments, as the mean expanfio air in the ordinary cafes: and this is prob nearer the truth; because Gen. Roy's experim were made on air which was freer from da than the ordinary air in the fields; and a minute quantity of damp increafes its expanfi ty by heat in a prodigious degree. The g difficulty is how to apply this correction; or ther, how to determine the temperature of the in' thofe extenfive and deep ftrata in which the vations are measured. It feldom or never h pens, that the ftratum is of the fame temperat througho |