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tions of the axis of the atmospherical logarithmic. both suffer the same change of temperature; and Therefore, if we'multiply our common logarithms as the air may be warmed or cooled when the by 10,000, they will express the fathoms of the mercury is not, or may change its temperature axis of the atmospherical logarithmic; nothing is independent of it, still greater variations of specific more easily done. Our logarithms contain what gravity may occur. The general effect of an is called the index or characteristic, which is an augmentation of the specific gravity of the mercury integer and a number of decimal places. Let us 'must be to increase the subtangent of the atmosphejuft remove the integer-place four figures to the rical logarithmic; in which case the logarithms of right band: thus the logarithm of 60 is 1.7781513, the densities, as measured by inches of mercury, which is one integer and 7781513. Multiply in the fame proportion that the fubtangent is in

creased; or, when the air is more expanded than

$13 this by 10,000, and we obtain 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

correspond to a thicker Atratum of air. The practical application of all this reasoning is

To perfect this method, therefore, we must learn obrious and easy; observe the heights of the by experiment how much mercury expands by an mercury in the barometer at the upper and lower increase of temperature ; we must also learn how ftations in inches and decimals; take the logarithms much the air expands by the fame, or any change of these, and subtract the one from the other; the of temperature, and how much its elasticity is difference between them (accounting the four first affected by it. Both these circumstances must be decimal figures as integers) is the difference of confidered in the case of air ; for it might happen elevation of fathoms.

that the elasticity of the air is not so much affected EXAMPLL.

by heat as its bulk is. It will, therefore, be proper Merc. Height at the lower station 29,8 1.4742163 to state the experiments which have been made for upper station 29,5 1'4638930 ascertaining these two expansions.

The most accurate, and the best adapted expeDiff. of Log. X 10000

oʻ0103233 riments for ascertaining the expansion of mercury,

233 or 103 fathoms and

of a fathom, which is

are those of General Roy, published in the Philof.

Trans. vol. 67. He exposed 30 inches of mercury, 619,392 feet, or 619 feet 44 inches; differing from actually supported by the atmosphere in a barothe approximated value formerly found about į meter, in a nice apparatus, by which it could be an inch.

made of one uniform temperature through its Such is the general nature of the barometric whole length ; and he noted the expansion of it in measurement of heights first suggested by Dr decimals of an inch. These are contained in the HALLEY; and it has been verified by numberless following table; where the first column expresses comparisons of the heights calculated in this way the temperature by Fahrenheit's thermometer, the with the same height measured geometrically. It second the bulk of the mercury, and the third the was thus that the precise specific gravity of air and expansion of an inch of mercury for an increase of mercury was most accurately determined; namely, one degree in the adjoining temperatures. by observing, that when the temperature of air and

TABLE A. mercury was 32, the difference of the logarithms of the mercurial heights were precisely the fathoms

Temp. Bulk of $ ., Expan. for 1° of elevation. But it requires many corrections to adjust this method to the circumstances of the


30,5 117 0,0000763 case; and it was not till very lately that it has


0,0000787 been so far adjusted to them as to become useful.


30,4652 0,0000810 We are chiefly indebted to Mr De Luc for the

182 30,4409 0,0000833 improvements. The great elevations in Switzer.

172 30,4159 0,0000857 land enabled him to make an immense number

162 30,390% 0,0000880 of observations, in almost every variety of circum

192 30,3638

0,0000903 ftances. Sir GEORGE SHUCKBOURGH also made

142 30,3367 0,0000923 a great number with most accurate instruments in

132 30,3090 0,0000943 much greater elevations, in the same country; and

30,2807 0,0000963 he made many chamber experiments for deter

30,2518 0,0000983 mining the laws of variation in the subordinate

30,2223 0,0001003 circumstances. General Roy also made many to

92 30,1922 0,0001023 the same purpose. And to these two gentlemen

82 30,1615 0,0001043 we are chiefly obliged for the corrections which

72 30,1302 0,0001063 are now generally adopted.

62 30,0984 0,0001077 This method, however cannot apply to every

52 30,0661

0,0001093 case; it depends on the specific gravity of air and

42 30,0333 0,0001110 mercury, combined with the supposition that this

32 30,0000 0,0001127 is affected only by a change of pressure. But since


0,0001143 all bodies are expanded by heat, and perhaps not

29,9319 0,0001160 equally, a change of temperature will change the

29,8971 0,0000177 relative gravity of mercury and air, even although




I 22 II2 102


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The scale of the thermometer is constructed on products will be the corrections of the respective the supposition that the successive degrees of beat logarithms. are measured by equal increments of bulk in the There is still an easier way of applying the logamercury of the column; but that the corresponding rithmic correction. If both the mercurial temperaexpansions of this column do continually diminish, tures are the same, the differences of their logarithms General Roy attributes to the gradual detachment will be the same, although each may be a good deal of elastic matter from the mercury by heat, which above or below the standard temperature, if the expresses on the top of the column, and therefore pansion be very nearly equable. The correction will shortens it. He applied a boiling heat to the vacuum be necessary onlŷ when the temperatures at the two a-top, without producing any farther depreslion; stations are different, and will be proportional to a proof that the barometer had been carefully this difference. Therefore, if the difference of the filled. It had indeed been boiled through its mercurial temperature's be multiplied by o'0000444, whole length. He had attempted to meafure the the product will be the correction to be made on mercurial expansion in the usual way, by filling the difference of the logarithms of the mercurial 30 inches of the tube with boiled mercury, and heights. But farther, since the differences of the lo. exposing it to the heat with the open end upper- garithms of the mercurial heights are also the dif. moft. But here it is evident that the expansion of ferences of elevation in English fathoms, it follows the tube and its folid contents must be taken into that the correction is also a difference of elevation in the account. The expansion of the tube was Englith fathoms, or that the correction for one defound so exceedingly irregular, and so incapable gree of difference of mercurial temperature is 1444 of being determined with precision for the tubes of a fathom, or 32 inches, or 2 feet 8 inches. which were to be employed, that he was obliged to This correction of 2-8 for every degree of differhave recourse to the method with the real baro- ence of temperature must be subtracted from the meter. In this no regard was necessary to any cir elevation found by the general rate, when the mercumftance but the perpendicular height. There cury at the upper ftation is colder than that at the was, befides, a propriety in examining the mercury lower. For when this is the case, the mercurial in the very condition in which it was used for column at the upper ftation will appear too Mhort, measuring the pressure of the atmosphere; be the pressure of the atmosphere too small, and there. cause, whatever complication there was in the fore the elevation in the atmosphere will appear results, it was the same in the barometer in actual greater than it really is. Therefore the rule for this use.

correction will be to multiply oʻ0000444 by the de. The most obvious manner of applying these grees of difference between the mercurial tempera. experinients on the expansion of mercury to our tures at the two stations, and to add or fubtract the purpose, is to reduce the observed height of the product from the elevation found by the general mercury to what it would have been if it were rule, according as the mercury at the upper ftation of the temperature 32. Thus, fuppofe that the is hotter or colder than that at the lower. observed mercurial height is 29*2, and that the If the experiments of Gen. Roy on the expanfion temperature of the mercury is 720 make 30'5302: of the mercury in a real barometer be thought 30=29*2 : 29'0738. This will be the true meature most deserving of attention, and the expansion be ofthedensity of the air of the standard temperature. considered as variable, the logarithmic difference That we may obtain the exact temperature of the corresponding to this expansion for the mean mercury, it is proper that the observation be made temperature of the two barometers may be taken. by a thermometer attached to the barometer-frame, These logarithmic differences are contained in the so as to warm and cool along with it. Or, this following table, which is carried as far as 112°, may be done without the help of a table, and with beyond which it is not probable that any obfervasufficient accuracy, from the circumstance, that tions will be made. The number for each tem. the expansion of an inch of mercury for one degree perature is the difference between the logarithms diminishes very nearly sooth part in each fucceed- of 30 inches, of the temperature 32, and of 30 ing degree. If therefore we take from the expan- inches expanded by that temperature. fion at 32° its thousand part for each degree of any

TABLE B. range above it, we obtain a mean rate of expansion


Dec. of
Log. diff.

Ft. In. There is another way of applying this correction,

Fath. fully more expeditious and equally accurate. The difference of the logarithms of the mercurial heights


0.0000427 »427 2.7 is the measure of the ratio of those heights. In like


2.7 manner the difference of the logarithms of the ob

92 0.0000444

9444 served and corrected heights at any station is the

82 0.0000453 2453 2.9 measure of the ratio of those heights. Therefore

72 0.0000460 2460

2.9 this last difference of the logarithms is the measure

62 0.0000468 of the correction of this ratio. Now, the observed 52


)475 height is to the corrected height nearly as i to


0.0000482 ,:87 I'000102. The logarithm of this ratio, or the 32 0.0000489 1489 difference of the logarithms of rand 1'000102 is

0.0000497 9497 3.0 Oʻ0000441. This is the correction for each degree


9504 3.0 that the temperature of the mercury differs from 32. Therefore multiply oʻ0000444 by the difference It is also neceffary to attend to the temperature of the mercurial temperatures from 32, and the of the air ; and the change produced by heat in

for that range.

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its density is of much greater consequence than he wished to examine the expanfion of air twice that of the mercury. The relative gravity of the or thrice as dense, he used a column of 30 or 60 two, on which the subtangent of the logarithmic inches long; and to examine the expansion of all curve depends, and consequently the unit of our that is rarer than the external air, he placed the scale 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 vessel This adjustment is of incomparably greater diffi. containing the mixtures or water. By this policulty.than the former, and we can hardly hope tion the column of mercury was hanging in the to make it perfect. We shall relate the chief ex- tube, supported by the pressure of the atmosperiments which have been made on the expan- phere; and the elafticity of the included air was Son of air, and notice the circumstances which measured by the difference between the fufpendleave the matter still imperfec.

ed column and the common barometer. Gen. Roy compared a mercurial and an air. The following table contains the expansion of thermometer, each of which was graduated arith- 1000 parts of air, nearly of the common density, metically, that is, the units of the scales were equal by heating it from o to 212. The first column busks of mercury, and equal bulks (perhaps dif- contains the height of the barometer ; the ad conferent from the former) of air. He found their tains this height augmented by the small column progress as in the following table:

of mercury in the tube of the manometer, and TABLE C.

therefore expreffes the density of the air examin

ed; the 3d contains the total expansion of 1000 Merc.) Dift, Air. Diff.

parts: and the 4th contains the expanfion for 1°

supposing it uniform throughout. 212'o




Dengity Expanfion


Barom. of Air of 100o ptsExpanfion

examined by 212o.

by ro

29995 31'52 483.89


30'07 30°77. 482'10 2'2741

29:48 29.90 48074 262676

29'90 30'13 485.86 2.2918

29'96 30'92 489045 2.3087

29'90 30'55 47604 7*2455 It has been eftablished by many experiments


487°55 2*2998

482.80 that equal increments of heat produce equal in

2.2774 erements in the bulk of mercury. The differences


489047 2°3087 of temperature are therefore expressed by the ad

Mean column, and may be considered as equal; and

30*62 484'21 2*2840 the numbers of the 3d column must be allowed to

Hence the mean expansion of 1000 parts of air expre's the same temperatures with those of the of the density 30°62 by one degree of Fahrenheit's first. They diredly express the bulks of the air, thermometer is 2*284, or that of 1000 becomes and the numbers of the 4th column express the 1000'284. If this expansion be supposed to foilow differences of these bulks. These are evidently the same rate that was observed in the comparis unequal, and thew that common air expands most son of the mercurial and air thermometer, the ex. of all when of the temperature 62 nearly.

pansion of a thousand parts of air for one degree The next point was to determine what was the of heat at the different intermediate temperature, actwal increale of bulk by fome known increase of will be as in the following table. heat. For this purpose be took a tube, having a

TABLE E. Daltow bore, and a ball at one end. He measur.

Total Expansion ed the capacity of both the ball and the tube, and

for' 1o. divided the tube into equal spaces, which bore a determined proportion to the capacity of the ball.


2*0099 This apparatus was set in a long cylinder filled


2'0080 with frigorific mixtures, or with water, which


2*1475 could be uniformly heated up to the boiling tem


22155 peratore, and was accompanied by a nice ther


315,193 2.2840 mometer. The expansion of the air was measur.

269,513 ed by a column of mercury which rose or sunk in


7'4211 the tube. The tube being of a small bore, the


197,795 2's 124 mercury did not drop out of it; and the bore be.


172,671 2-5581 ing chosen as equal'as posible, this column re.

147,090 2'6037 mained of an uniform length, whatever part of

IZ1,053 2'5124 1:e tube it chanced to occupy. By this contri.

95,929 294211 since he was able to examine the expansibility of


2'3297. act of various denfities. When the column of

48,421 2*2383 thercury contained only a single drop or two,' the

26,038 2*1698 as was nearly the density of the external air. If

Vol. XVII. Part I.

30*60 Booo

Temp. Expansion.



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If we would have a mean expansion for any ty, being greatelt about the temperature particular ranye, as ber ween 12° and 92°; which fo that its expanfibility by heat diminit is the moft Skely to comprehend all the geodæti. 'its denfity; but he could not determin cal observations, we need only take the difference of gradation. When reduced to about of the bulks 26'038 and 222.000=195*968, and density of common air, its expansion w divide this by the interval of temperature 80°, lows: and we obtain 2-4496, or 2'45 for the mean espansion for r'. It would perhaps be better to

TABLE II. adapt the table to a mass of 1000 parts of air of


Espai the standard temperature 32° ; for in its present

Temp. Builk. form, it thews the expanfibility of air originally of the temperature c.

This will be done with fat. ficient accuracy, by saying (for 212°) 10710718:

1392.. 1134,429 12*264 1661

** 35 1484,210=1000;1:849, and so of the reft. This

1122,165 we fhall construct the tollowing table of the ex

14'150.1*070 1108015

14'151 panlion of 10,000 parts of air.

132 1093,864


071 TABLE F.



149.37 92

1064,699 Expant.

20°91T Temp. Bulk. Diff

72 for 18.


251943 1'29 52 1017,845

17.845 0891




Mean expanfion


From this very extensive and judicious

132 12272

experiments, it is evident, that the expa

of air by heat, is greatest when the air is a
443 22°1
92 11403

ordinary dentity, and that in small denfit

226 82 11177

greatly diminithed. It appears also, that 10942

235 72


of compreslion is altered'; for in this fpec
62 10704

the rare air, half of the whole expansion 1


about the temperature 99', but in air of
235 - 235

density at 103. The experiments of AMO
226 22'6

related in the Mem. of the Acad. at Pari
217 217

&c. are confiftent with these more pers
209 20.9

experiments of Gen. Roy,

After this account of the expansion of This will give for the miean expanfion of 1000 fee that the height through which we mun

! parts of air between 12° and 92=2:29. Although produce a given fall of the mercury in th it cannot happen, that in meaturing the differences meter, or the thickness of the stratum of a of elevation near the earth's surface, we shall have ponderant with a tenth of an inch of m occafion to employ air greatly exceeding the com- must increase with the expansion of air, ai mon denfity, we may insert the experiments

made if be the expanfion of one degree, w by Gen. Roy on fuch airs. They are expressed in the following table; where column firft con.

DOCO tains the densities meatured by the inches of mer. multipy the excess of the temperature of cury that they will fupport when of the tempera

above 329 by oʻ00229, and multiply the p ture 33°; column lecond is the expansion of by: 87to obtain the thickness of the ni 1000 parts of such air, by being heated from o to where the barometer stands at 30 inches: or 212 ; and column third is the mean expanlion for ever be the elevation indicated by the diff

of the baromeirical freights, upon the supp TABLE G.

that the air is of the temperature 32°, we

multiply this by o‘00229 for every degree th Expansion Expan. Density

bir is warmer, or colder than 32. The pi for 112o. for 1o.

must be added to the elevation in the first

and subtracted in the latter. 1017 451'54 2'130

Sir GEORGE SHUCKBURGH deduces c 423*23 1996

from his experiments, as the mean expanfi 8565 41209

air in the ordinary cases: and this is pro 54's 439687 2'075

nearer the truth; because Gen. Roy's experi 49*7 44324 2'o91

were made on air which was freer from Mean

than the ordinary air in the fields; and 752

minute quantity of damp increases its expa There is much more frequent occafion to ope. ty by heat in a prodigious degree.. The rate in air that is rarer than the ordinary ftate of difficulty is how to apply this correction; the fuperficial- atmosphere. Gen. Roy accor. ther, how to determine the temperature of dingly wade many experiments on such airs. He in those extensive and deep ftrata in which found is general, that their expanfibility by heat vations are measured. It seldom or neve fràs analogous to that of air in its ordinary denfi- pens, that the stratum is of the same temp $




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ruzogeout. It is commonly much colder aloft; this fate it may contain a considerable portion of

is also of different constitutions. Below it is other metals, particularly of fivér, b:(muti, anu. warm, Ioaded with vapour, and very expansible; tin, which will diminish-iis isecihe gravity. It has are it is cold, much drier, and less expansible, 'been obtained by revivification from cinnabar of bath by its dryness and its rarity. The currents 'the fpecilic gravity 14*229, and it is thought very a wind are often disposed in straia, which' long, fine if 13665. Sir George Shuckburgh found the retain their places; and as they come from diffe. quickfiver which agreed precitely with the atmofa. rat regions, are of different temperatures and pherical oblervations on which the rules are fouoddificent conftitutions. We cannot therefore de."ed, to have the ipecific gravity 13•61. It is seldom termine the expansion of the whole gratum with, obtained to heavy. It is evivent that these variapraction, and most be contented with an approxi-' tions will change the whole nesults; and that it is ratios. The best approximation that we can absolutely neceflary, to obtain precinon, that we site, is, by supposing the whole fratum of a' know the density of the mercury employed. She Dean temperature between those of its upper and subrangent of the atmospherical logarithmic; or bouc extremily, and employing the expansion cor, the height of the homogeneous anno poeie, will Teiponding to that mean temperature. This increase in the same proportion with the density kowever, is founded on a gratuitous supposition of the mercury; and the elevation coneip nu ng that the whole intermediate Aratum expands to ole tenth of an inch of barometrie beidot will ake, and that the expantion is equable in the dif. change in the fame proportion. We mat 5e canerest iatermediate temperatures; but neither of tented with the renaining imperfcctions. Fel 20$ tese are warranted by experiment. Rare air ex- purpose that can be answered by íuch incalta pards less than what is denfer; and therefore the ments of great heights, the method is lullicientiy Beteral expansion of the whole ftratum. renders exact; but it is quite inadequate to the puipote 4 denity more uniform. Dr Horsey bas point. of taking accurate levels, Tir directing the coned out some curious consequences of this in Pbil. struction of canals, aquesiucts, and other works of Traf. Vol. LXIV. There is a particular eleva. this kind, where extreme precision is absolutely 101, at which the general expanlion, inftead of neceflary. Aminilhing the denity of the air, -increases it by We shall only add a few EASY Rules for the be fuperior espansion of what is below; and we praćtice of this mode of meafürement. taos that the expansion is not equable in the in

1. M. de Luc's METHOD. mediate temperatures : but we cannot find out I. Subtract the logarithm of the barometrical apple which will give us a more accurate correc- height at the upper itation from the logarithm of 13n, thaa by taking the expansion for the mean that at the lower, and count the index and four operatore.

firit decimal figures of the remainder as tathoms, When this is done, we have carried the method the rest as a decimal fraction. Cal this the clevacf Realuriig beights by the barometer as far as it tin. can zo; and this source of remaining error makes II. Note the different temperatures of the merI needless to attend to fome other very minute cury at the two ftations, and the mean temperapisations which theory points out. Such is the ture. Multiply the logarithmic expanfion correl cinisution of the weight of the mercury by the ponding to this mean temperature in Table B.) by nge of ditance from the centre of the earth. The difference of the two temperatures, and iubThis accompanies the diminution of the weight tract the product from the elevation if the barda of the air, bat neither so as to compensate it, nor meter has been coldest at the upper ftatior, otherto go along with it pari palju. After all, there wise add it. Cait the difference or the lúm the iT ales where there is a regular deviation from approximated elevation. toe ruics, of which we cannot give any very la. lii. Note the difference of the temperatures of bufetary account. Thus, in the province of the air at the two itatioris by a detached thermo4.to la Peru, which is at a grcii elevation above ineter, and also the mean temperature and is dif

lantace of the ocean, the heights obtained by ference from 32°. Multiply this difference by the 1.4: rules fall contiderably short of the real expansion of air for ire mean temperature, and

; and at Spiubergea they considerably ex multip.ythe approximate elevation by 5 this cu them. It appears that the air in the circum- product, according as the air is above or below paar regiuns, is denter than the air of the tempe. 32°. The product is the correct elevation in faTa* dimates when of the inne beat, and imder thoms and decimals.

fie prelure; and tic contrary fienis to be EXAMPLE. Suppose that the mercury in the 10! nii the air in the torrid 201.e. It would barometer at the lower station, was at 29-4 inches, kinat the specific gravity of air to mercury is that its temperatore was soo, and the temperature # Spi:bergen about one to 19224, and in Piru of the air was 45; and let the height of the mer. about 1 10 13109. This difference is with great cury at the upper station be 25*19 inches, its tély probabii ty alcribed to the greater dryness of the perature 46, and the temperature of the air 39. Cum ar air.

Thus we have This source of error will always remain; and it 8 al Hts. Temp. $ Mean. Temp. Air. Mean, i combined with another, which thould be at 29'4


42 i tended to by all who practise this method of mea. 2519

39 twing heights, namely, a difference in the specific 1. Log. of 29.4

1°4683473 gravity of the quick-silver. It is thought fuffi Log. of 25*19

1'4012282 rently pure for a barometer when it is cleared of ilcakidable matter, so as not to fully the tube, lạ

Elevation in fathoms

671,191 II. Expanf.

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