BAR 



«f Falirenlieit's thermometer is performed by the aftronom 

 toyal (Phil. Tranf. vol. Ixiv. J), 162), i;i the foUowinir man- 

 ner. The French foot is to the EngHfli foot as 1.06575 to 

 I (Phil. Tranf. vol. Iviii. p. 326); and the Fahrenheit's 

 point of freezing is 52, and that of boihng water 312, hav- 

 ing an interval of iSo degrees. But M. De Luc's point of 

 boiling water or So wa<; marked when the barometer was at 

 27 French inches, that being its mean height at Geneva ; 

 but our Enghih workmen ir.ark the fame point on Fahren- 

 heit's fcale, when the barometer Hands at 30 inches, whxh 

 is equal to 28 inches 1.8 hues French meafure, or 1^8 lines 

 higher than JVI. De Luc's barometer, when he adjuiled the 

 point of boiling water on his thermometer ; and it is well 

 known, that the heat of boiling water varies with the 

 weight of the atmofphere. M. De Luc from his experi- 

 ments inferred, that an increafe of one line in the height of 

 the barometer raises the mercury of the thermometer, placed 

 in boiling water, .^j-^th part of the interval between the 

 freezing point and that of boiling water, though the rule 

 will not apply to large variations of the barometer occafion- 

 ed by vevy great heights above the earth's furface. The 

 change of the boiling pomt in Fahrenheit'.s fcale conefpond- 

 ing to a change of one line in the barometer, will be T-i-fr 

 =:o.l6; and thereforei3.S lineswiiiproduce o. 16 X I ^8 = 2. 2 

 degrees of Fahrenheit's fcale ; and a thermometer, whofe 

 point oi boiLng water was marked 212, when the barometer 

 ftcod at 30 Englifli inches =. 28 inclies 1.8 lines French 

 meafure, will, vrhen the barometer defcends to 27 French 

 inches, fink 2.2 degrees in boiling water, or to 209.8 or in 

 round numbers to 210 degrees, which is diftant only 178 

 from 32 the point of freezing. Hence it appears that an 

 extent of 80° of M. D-" Luc's thermometer correfponds to 

 an ext.-.it of 178 of our Fahrenheit's themiometer; and put- 

 ting F for the decrees of this theiinometcr, correfponding 

 to C of M. De Luc's, we (hall have C : F — 32 :: 80 : 178, 

 S^, which, fuhilitutedin M. De Luc's 



nd C = F 



^:^ 



formula, gives log. B — log. b X 1 -{- 



C -i6i _ 



= lojr. B 



215 



- log. ^ X I + 



F- 32 



i6J 



215 



=log.5— log. b 



X I 



80 



, " F - 32 - H' 16.75 = log- B - log. b 



178 ■ 2iy ° 



X I + 



3^ - il--l 



478.38 



log. B - log. ^ X I + 



. ili in French toifes. To reduce thefe to our En 



478.38 

 glifh fathoms of 6 feet each, multiply the above expreffion 

 by 1.06575, and we (hall have 



Log. B - log. b X I +1^1-11. X 1.06575 = 



Log. B - log. b X 409.11 + F X If^^li =z 



„___ 478.38 



Log. B - log. b X 2_iL._j: — or, in round numbers. 



448.S7 



= log. B - log. b xl£2+^ - log. B - log. 6X1 + 



, . 449 



^^) which expreffes the height between the two fta- 



tions in Englidi fathom?. 



In thefe expreffions B and i denote heights of the baro- 

 meter, at the lower and higher ftations, coneftcd for the 



BAR 



difference of heat between a fixed temperatuM, viz. |th of 

 the interval bctv.ten freezing and boiling water, and the 

 prefent heat, indicated by the thermometer attached to the 

 barometer at each (tation : but it will be fu.Ticimt, and 

 more co.-.venient, to comdl one barometer for the difference 

 of the heat of the two. Suppofe then the r'ppcr barometer 

 is to be correfted, to redi.ce it to the temperature of the 

 lower one, and that I fignifies the height of this baro- 

 meter, as obferved and not corredted ; the correaion, from 

 wliat has been already faid, if we call D the difterence of 

 height of the thermometer attached to the baro.meter at 

 the two ftations, e. g. at the top and bottom of the hill, will 

 , D i 



"^± J-^y 3s the thermometer ftands higheft at the lower 



or upper ftatioiT ; and the upper barometer correded, in- 



ftead of b, will he b ± — ^, which fubftituted in the for- 



mula, gives log. B — log. {b ± — =) x i 



+ F-40 

 449 



But 



the correaion, on account of ^the diffeience orfieat of the 

 baiometer at the two ftations, may be reduced to a more 

 ealy exprefTion, in which the variable quantity b, the height 

 of the upper barometer, (liaU not appear. The fluxion of a 

 logarithm is to the fluxion of its natural number as the 

 modulus of the fyftem to the natural number ; and 4^41 

 IS the modulus of the common logarithms, when the 

 four places, next the index or charaaeriftic, are 

 taken as whole numbers, inftead of decimals, which is 

 meant to be done in the ufe of the preceding formda. 



Confcquently — being very fmaU with refpeft to 



b, we (hall have variation of log, b : variation of i = 



^TTk - " ^^3+3 : ^ very nearly, and hence variation of log. 



Bb 



Hence log. ( b ± ~) = log. i^ ± 0.45 3 D ; 



+ 0.452 D. 



which, being fubftituted in the above formula, will give the 

 difference of height of the two fta tions, in En ghih fathoms, 

 in a mo re convenient e xprelTion, viz. log. B - log.b + 0.452 



^ ^- ' + —^~- ' ^'^^'"e '''= "PPer fign. — , is to be 

 ufed, when the tiitnnometer of the barometer is higheft at 

 the lovver ftation, and the lower %n, +, is to be ufed. 

 when the faid thermometer is loweft at the lower ftation! 

 ^^ hen F, the height of Fahrenheit's thermometer, is lefs 

 , g F — 40 

 than 40 , H -— -, becoming negative or fubtradive, 



muft be accordingly apphed in the calculation. In the 

 foregoing formula, B denotes the obferved altitude of the 

 barometer at the lower ftation, and b that at tiie upper 

 ftation ; log. B and log. b denote their logarithms taken 

 out of the common tables, by a(ruming the four firll figures 

 next foUowing the index, as whole numbers, and'' coni 

 fidering the three remaining figures to the right hand 

 as decimals; D figmfies the difference of height of 

 Fahrenheit's thermometer, attached to the barometer at 

 the top and bottom of the hill ; and F fignifies the mean 

 of the two heiglHs of Fahrenheit's thermometer, expofed 

 freely for a few mmutes to the open air ia the (liade at 

 the top and bottom of the hill. ' 



The formula, for the meafure of heights, may be adapted 

 4S2 to 



