BAR 



As the heiglits of the mercury in the barometei- in all 

 acceffible elevations indicate the denfities of the air at thcfe 

 elevations, the method of taking heights by this inftrument 

 may be illuftrated in the following familiar manner. 



It has already been obfervcd, that if the mercni-y in the 

 barometer fland at 30 inches, and the air and mercurj be of 

 the fame temperature of 32° Fahrenheit, a column of air 

 87 feet thick has the fame v/eight with a ccliimn of mercury 

 yV of an inch thick : and therefore if in afcending the mer- 

 cury (inks^to 29.9 inches, the interval of aftent is 87 feel. 

 Suppofe the mercury at a h'gher elevation to ftand at 29.8 

 inches, and it be required to know the height to v.-hich the 

 barometer has been carried. The ftratum through which 

 it has been raifed, as the air is lefs comprefTrd and rarer, 

 muil of courfe be thicker. The denfity of the firil ftra- 

 tum may be called 300, eflimating the dcnfity by the num- 

 ber of tenths of an inch of mercury which its elafticity 

 proportional to its denfity enables it to fupport. In the 

 fame manner the denfity of the fecond ftratum mull be 299. 

 But when the weights are equal, the bulks ?rc inverfely as 

 the denf:ties ; and when the bafes of the ftrata are equal, 

 the bulks are as the thickneffes. In order therefore to ob- 

 tain the thickncfs of the fecond ftratum, fay 299 : 300 :: 

 87 : 07. 29, which denotes the thicknefs of the fecond ftra- 

 tum ; and therefore the whole interval of the elevation of 

 the barometer has been 174.29 feet. When the barome- 

 ter at a higher elevation, fhews the denfity to be 

 298, fay 29S : 300 :: 87 : 87.584 the ttiicknefs of the third 

 ftratum, and 261.875 will be the whole alcent. By this 

 method m.ay be computed the following table, in which the 

 iirft column is the height of the mercuiy in the barometer, 

 the fecond column is the thicknefs of the ftratum, or the 

 elevation above the preceding ftation, and the third column 

 is the whole elevation above the firft; ftation. 



In order to meafure any elevation within the limits of this 

 table, obferve the barometer at the lower and at the upper 

 ftations, and write down the correfponding elevations ; 

 fubtrafl. the one from the other, and the remainder is the 

 height required. E. G. Suppofe that at the lower ftation 

 the mercurial height was 29.8, and that at the upper ftation 

 it was 29. 1. 



29.1 - - - - 793.644 



29.8 - - - - 174.291 



6 1 9.353 the elevation required. 



Without the aid of the table, let in reprcfent the medium 

 of the mercurial heights, and d their difference in tenths of 

 an inch ; then fay, as m is to 300, fo is %'dx.o the height 



Vol. III. 



BAR 



required /. ; or /. = Z222iM=2^S2i . Thus in the 



m m 



preceding example, m is 29.45, ^nd ^ = .7 ; and therefore, 



, .7 26100 18270.0 , ,.- . , 



*= = ■ = 620.4, Qinenn^ only ens 



29.45 29.45 



foot from the former value. The whole error of the eleva- 

 tion 883 feet 4 inches, the extent of the table, cftima- 

 ted in either of thefc methods, is only |"" of an inch. It it 

 needlcfs however to recur to approximations, when the 

 fcientiiic and more accurate method fiift praftifed by Dr. 

 Halley is equally eafy. Upon the fuppofition of equal gra- 

 vity, as we have already (hewn, the dcnfities of the air arc 

 as the ordinates of a logarithmic curve whofe axis is the 

 hne of elevations. It has been alfo iliewn, that, in the true 

 theory of gravity, if the diftances from the centre of 

 the earth increafe in an hai-monic progrellion, the denfitiet 

 will decreafe in an arithmetical progreftion ; but if the 

 greateft elevation above the furface be but a few miles, thii 

 harmonic progreffion will fcarcely differ from an arithmeti- 

 cal one. Thus if yf^, ^c, Ad, arc 1, 2, and 3 miles, the 

 correfponding elevations AB, AC, AD, will be fenfibly ia 

 arithmetical progreffion alfo ; for the earth's radius AC, is 

 nearly 4000 miles. Hence it follows that BC — AB it 



= . of a mile, or t4^ of an inch, 



400c X 400 1 1 6004000 



which is a quantity altogether infignificant. We may 

 therefore affume, that in all acccffible places, the elevations 

 increafe in an arithmetical progreffion, while the dcnfities de- 

 creafe in a geometrical progreffon. Confequently the ordi- 

 nates are proportional to the numbers which are taken to 

 meafure the denfities, and the portions of the axis are pro- 

 portional to the logarithms of thefe numbers. Hence it 

 follows, that we may take fuch a fcale for meafuring the 

 denfities, that the logarithms of the numbers of this fcale 

 fhall be the portions of the axis, that is, of the vertical line 

 in feet, yards, fathoms, or any other meafure ; and we mav, 

 on the other hand, chufe fuch a fcale for meafuring our 

 elevations that the logarithms of our fcale of denfities (hall 

 be parts of this fcale of elevations ; and either of thefe fcalcs 

 may be found fcientifically. For it is a known property of 

 the logarithmic curves, that when the ordinates are the fame, 

 the intercepted portions of the abfciffae are proportional to 

 their fubtangents. But the fubtangent of the atm.ofpheri- 

 cal logarithmic is known ; it is the height of the homogeneous 

 atmofphere in any meafure we pleafe, e. g. fathoms ; and 

 we find this height by comparing the gravities of air and 

 mercury, when both are of fome determined dtnfiiv. 

 Thus in the temperature of 32° of Fahrenheit, when the 

 barometer ftands at 30 inches, it is known, as the refult of 

 many experiments, that mercury is 10423.068 times 

 heavier than air; therefore the height of the counter-ba- 

 lancing column of homogeneous air will be 10423.068 tim'.i 

 30 inches, that is, 4342.945 Enghih fathoms. It is a!fo 

 known that the fubtangent of our common logarithmic ta- 

 bles, where i is the logarithm of the number 10, is 

 0.4342945. Confequently the number 0.4342945 is to 

 the difference D of the logarithms of any two barometric 

 heights as 4342.945 fathoms are to the fathoms F contain- 

 ed in the portion of the axis of the atmofpherical logarith- 

 mic, which is intercepted between the ordinates equal to thefc 

 barometrical heights ; or that 0.4342945 ; D :: 4342.945 : 

 F, and 0.4342945 : 4342.945 :: D : F; but 0.4342945 

 is the ten thoufandth part of 434?. 945, and therefore JtJ 

 is the ten thoufandth part of F. 



Thus it accidentally happens, that the logarithms of the 

 4 S denliiietj 



