16 On the Height of Mountains, Headlands, &fc. 



Ill) was the expression formerly given to find the altitude in 

 fathoms ; M, III being the heights of the mercury at the base 

 and summit of any eminence. 



This formula is very easily applied, and not far from the truth, 

 when an allowance is made for the increase of temperature above 

 31°, for this is the degree of temperature to which the above for- 

 mula is calculated. 



As air expands very nearly T i 3 th part of its bulk for every de- 

 gree of heat, and suffers the same contraction for every degree of 

 cold, the following rule was usually given. 



Rule. — Observe the height of the mercury at the bottom of 

 the object to be measured, and again at the top, as also the degree 

 of the thermometer at both these situations, and half the sum of 

 these two last may be accounted the mean temperature. Then 

 multiply the difference of the logarithms of the two heights of 

 the barometer by 10000, and correct the result by adding or sub- 

 trading so many times its 435th part as the degrees of the mean 

 temperature are more or less than 31° : the last number will be 

 the altitude in fathoms. 



Ex. III. If the heights of the barometer at the bottom and top 

 of a hill are 29.37 and 26.59 inches respectively, and the mean 

 temperature 26°, what is the height ? 



Log. 29.37 = 1.4679039 

 Log. 26.59 =-- 1.4247183 



0.0431856 

 And .0431856x10000=431.856. 

 Then 3l°-26°=5°= the mean temperature below 31°. 



.-.431.856 xarX5 = 431.856x T f T = 4.964, to be subtracted. 



Height, 431.856-4.964 = 426.892 fathoms. 



We shall now investigate the last formula, to give an outline 

 of the theory upon which this proposition is founded. 



Let EAR represent part of the surface of the earth, and AT a 

 column of the atmosphere. Conceive this column to be divided 

 into a number of equal and infinitely small parts, as AB, BC, CD, 

 &c, in each of which we may suppose the density to be uniform, 

 because they are infinitely small. Now since the density of the 

 air is always directly as the compressing force, therefore we have 

 the density of the air in any of the portions AB, BC, &c. as the 

 weight of the column of the atmosphere above that place ; that 

 is, if P represents generally the pressure, D the density of any 



