atmosphere. 
burgh and General Roy, both concurring 
to shew, that such a rule for the altitudes 
and densities holds true for all heights that 
are accessible to us, when the elasticity of 
the air is corrected on account of its den- 
sity : and the result of their experiments 
shewed, that the difference of the logarithms 
of the heights of the mercury in the baro- 
meter, at two stations, when multiplied by 
10000, is equal to the altitude in English 
fathoms, of the one place above the other ; 
that is, when thq temperature of the air is 
about 31 or 32 degrees of Fahrenheit’s 
thermometer ; and a certain quantity more 
or less, according as the actual temperature 
is different from that degree. 
But it may be shewn, that the same rule 
may be deduced independent of such a 
train of experiments as those referred to, 
merely by the density of the air at the sur- 
face of the earth. Thus, let D denote the 
density of the air at one place, and d the 
density at the other ; both measured by the 
column of mercury in the barometrical 
tube: then the difference of altitude be- 
tween the two places, will be proportional 
to the log. of D — the log. of d, or to the 
log. of But as this formula expresses 
only the relation between different alti- 
tudes, and not the absolute quantity of 
them, assume some indeterminate, but con- 
stant quantity h, which multiplying the ex- 
pression log. may be equal to the real 
difference of altitude a, that is, a = A X log. 
of — . Then, to determine the value of the 
d 
general quantity h, let us take a case in 
which we know the altitude a that cor- 
responds to a known density d ; as for in- 
stance, taking a— 1 foot, or 1 inch, or some 
such small altitude : then because the den- 
sity D may be measured by the pressure 
of the whole atmosphere, or the uniform 
column of 27600 feet, when the tempera- 
ture is 55°; therefore 27600 feet will de- 
note the density D at the lower place, and 
27599 the less density d at 1 foot above it ; 
,.27600 , . , 
consequently 1 = h X log. °f— — , which 
by the nature of logarithms, is nearly — k x 
: 4 '’ 4 ~9 — or — — nearly; and hence we 
27600 63,551 
find h = 63551 feet ; which gives us this 
formula for any altitude a in general, vis. 
a = 63551 X log. of — , or a — 63551 X 
M , c M c 
log. of — feet, or 10592 X log. of — fa- 
m m 
thorns ; where M denotes the column of 
mercury in the tube at the lower place, 
and m that at the upper. This formula is 
adapted to the mean temperature of the air 
55°: but it has been found, by the experi- 
ments of Sir George Shuckburgh and Gene- 
ral Roy, that for every degree of the ther- 
mometer, different from 55°, the altitude a 
will vary by its 435th part ; hence, if we 
would change the factor h from 10592 to 
10000, because the difference 592 is the 
18tli part of the whole factor 10592, and 
because 18 is the 24th part of 435 ; there- 
fore the change of temperature, answering 
to the change of the factor h, is 24°, which 
reduces the 55° to 31°. So that, a = 10000 
M 
X log. of — fathoms, is the easiest exprea- 
m 
sion for the altitude, and answers to the 
temperature of 31°, or very nearly the 
freezing point: and for every degree above 
that, the result must be increased by so 
many times' its 435th part, and diminished 
when below it. 
From this theorem it follows, that, at 
the height of 3| miles, the density of the 
atmosphere is nearly 2 times rarer than it 
is at the surface of the earth ; at the height 
of 7 miles, 4 times rarer; and so on, ac- 
cording to the following table : 
Height in miles. 
14 
21 
28 
35 
42 
49 
56 
63 
70 
Number of times rarer. 
2 
4 
16 
64 
256 
1024 
4096 
16384 
65536 
262144 
1048576 
And, by pursuing the calculations in this 
table, it might be easily shewn, that a cu- 
bic inch of the air we breathe would be so 
much rarefied at the height of 500 miles, 
that it would fill a sphere equal in diameter 
to the orbit of Saturn. 
It has been observed above, that the at- 
mosphere has a refractive power, by which 
the rays of light are bent from the right 
lined direction, as in the case of the twilight; 
