1817.) Method of finding the Heights of Mountains. 109 
Let H, therefore, denote the difference of altitude in fathoms 
between two stations, and Jet D and d be the densities, or lengths 
of the barometrical columns, both being corrected for temperature ; 
also let L. represent Brigg’s logarithms, and / the interest logarithms. 
Then by the ordinary formula, if the temperature of the air be 
disregarded, 
H = 10,000 (L.D — L.d}. 
ButL.D:7.D:: log. amount 1/. fora year : log. 10. 
Which, if the rate be taken at five per cent., and the logarithms 
of the two bases, according to the common system, becomes 
LD? Da LD . 1°05: L.10 
That is, L .D:/.D :: °0211893: 1 
And L.. D = ‘0211893 x 1.D 
In like manner, L.. d = ‘0211893 x l.d 
Therefore, H = 10,000 [L. D — L.d] = 10,000 x -0211893 
(?.D — 2. d) = 211893 (77.D —1.dj) = 212 [1.D—1.d] 
nearly (1). 
The last expression may be reduced to a form more convenient 
for calculation, by assuming H = 2122. (—) (2). 
The constant quantity 212, being the same as the number for the 
boiling point of Fahrenheit’s scale, is as easily recollected ‘as the 
number 10,000 in the ordinary formula, when Brigg’s logarithms 
are employed. 
Having thus deduced a general expression for H in terms of in- 
terest logarithms, I shall now illustrate the formula by example. 
Before doing this, however, it will be necessary to subjoin the 
amounts of J/, for a few successive years as a groundwork for calcu- 
lation. If the first formula be used, these amounts, with the years 
corresponding to them, may be taken from any continuous part of 
the table ; for since L .(~7) =L. (2) SAP DS Li abtthe 
number 7 being any number, integral or fractional, it is evident 
that if D and d are too great to be found in the table, as will gene- 
rally be the case, it will answer equally well to take corresponding 
parts of them. ‘his reduces the range of amounts necessary to 
very narrow limits. As the height of the mercurial column at the 
lower station is seldom the double of its height at the upper, it will 
be sufficient, in most cases, to make the table extend from the 
amount for one year to the amount for 15 years. The following 
table, however, is extended as far as the amount for 20 years, and 
will answer for finding any altitude not exceeding 25,000 feet. 
