?2o 
MECHANICS. 
difference H—//Of—log* D—(—log. d ) Of log. D—log. d, 
orCClog. And if H=o, or D = the denfity at the 
earth’s furface, then any altitude above the furface is as 
, . , D D 
tne log. ot —. Or, generally, the log of — varies as the 
altitude of the one place above the other, whether the 
lower place be at the furface of the earth or not; and upon 
this property is founded the method of finding the heights 
of mountains, &c. by the barometer. 
Prop XLI. To determine the aElual denfity of the atmofpheric 
air at any altitude above the earth's furface. —This may be 
done with the aid of the equation in Cor. i. Prop. XL. 
but with greater facility by means of the atmofpherical lo¬ 
garithmic, thus : As the height of the homogeneous at- 
mofphere is to the modulus of Briggs’s fydem, lb is the 
given altitude in feet to a fourth number, which in the 
common tables is the logarithm of the ratio of the denfity 
of the air at the earth’s furface to its denfity exprefled by 
unity at the propofed height, on the fuppofition of equal 
gravity. But, if we attend to the variation of gravity, the 
procel's will be this: Suppofe C to be the point at which 
the denfity is required; make OC : OA :: OA : Or, or 
OC : OA :: AC : Ac; and Ac thus obtained will be the 
height to which the denfity is to be calculated by the pre¬ 
ceding analogy, on the hypothefis of equal gravity. 
Let us take for an example the height of 7 miles,and con¬ 
ceive the radius of the earth to be 4000 miles. Then fliall we 
have OC (=4007) : OA (=4000) : AC(= 7 ) : Ac— 4 °° 0>< - 
4007 
= e 6 , 98777 miles =36895! feet. Wherefore, taking 27600 
feet for the height of the homogeneous atmofphere, we dial l 
have 27600 : ’43439448 :: 36895! : ’5806957, which is the 
common logarithm of 3*80799, or 3-808 nearly. Confe- 
quently the denfity of the air at the earth’s furface is to 
its denfity at the altitude of 7 miles as to i nearly, al¬ 
lowing for the diminution of the force of gravity. This 
refult agrees nearly with experiments. Thus Mr. Cotes 
inferred from the French experiment at the Puy de Dome, 
that at the altitude of 7 miles the air was rather more than 
4 times rarer than at the furface of the earth ; but, from 
the experiments of Mr. Cafwell, at Snowden, he concluded 
that at the fame altitude of 7 miles the air was not quite 
4 times rarer than at the furface. And fir Ifaac Newton, 
in the laft edition of his Optics, dates it at 4 times rarer at 
the height of 7! miles ; which, properly reduced, gives 3’86 
for the comparative rarity at 7 miles. 
Returning to the hypothefis of equal gravity, we may 
jreadily find an equation for the altitude, which fliall in¬ 
clude the changes in temperature. Such an equation, de¬ 
duced from thedenfity of the air at the earth’s furface, has 
already been given under the article Atmosphere, vol. ii. 
p. 480. 
I11 Dr. Robifon’s method no tables are required ; it will 
be fufficiently exadt for mod purpofes, and is not diffi¬ 
cult to remember. It was deduced from thefe confidera- 
tions. 
1. The height through which we mud rife in order to 
produce any fall of the mercury in the barometer, is in- 
verfely proportional to the denfity of the air, that is, to 
the height of the mercury in the barometer. 
2. When the barometer dands at 30 inches, and the air 
and quickfilver are of the temperature 32 0 , we mud rife 
through 87 feet to produce a depreffion of ^ of an inch. 
3. But, if the air be of a different temperature, this 87 
feet mud be increafed or diminifhed by about o’2i of a 
.foot, for every degree of difference of the temperature 
from 32 0 . 
4. Every degree of difference of the temperatures of the 
mercury at the two dations makes a change of 2-833 feet, 
or 2 feet 10 inches in the elevation. 
Hence the following Rule: Firdly, Take the differ¬ 
ence of the barometric heights in tenths of an inch. 
Call this D, Secondly, Multiply the difference d be- 
tween 32 0 and the mean temperature of the air by 21, 
and take the fum or difference of this product and 87 
feet. This is the height through which we muft rife to 
caufe the barometer to fall from 30 inches to 29-9 ; and may 
be called h. Thirdly, Let m be the mean between the two 
barometric heights: then is the approximated ele¬ 
vation very nearly. Ladly, Multiply the difference 5 of 
the mercurial temperatures by 2-833 feet, and add this pro¬ 
duct to the approximated elevation if the upper barome¬ 
ter has been the warmed, otherwife fubtradt it ; then will 
the refulting fum or difference be the correfted elevation. 
Or, this Rule may be exprefled by the following for¬ 
mula ; where d is the difference between 32 0 and the mean 
temperature of the air, D is the difference of barometric 
heights in tenths of an inch, in is the mean barometric 
height, o the difference between the mercurial temperatures, 
and E is the correct elevation. F — 8 7 ~° 1 ^ ^ •*- 
. m 
^Xa-833. 
For an example, fuppofe that the mercury in the baro¬ 
meter at the lower dation was at 29-4 inches, its tempe¬ 
rature 50 0 of Fahrenheit’s thermometer, and the tempe¬ 
rature of tile air 45 0 ; the height of the mercury at the 
upper dation 25-19 inches, its temperature 46°, and the 
temperature of the air 39 0 . 
Here 0=294—251-9=42-1 
//=87-j-(icX-2i) =:8 9' r 
w=i(29-44-25-i 9 )=2 7 -295 
50 JlJiI m ■* * 
—— = approximate elevation = • • 4123-24 
Correftion for temp, of mercury 4Xa’833= n'33 
Corrected elevation in feet . . 4111*9! 
Ditto in fathoms ' • • 687^2 
Other methods are quoted under Barometer, vol. ii. 
In that ai tide alfo we have given figures of the various 
kinds Ot barometers, with the methods of making them, 
and of foretelling the weather, as well as of mea luring 
heights, by their means. And from the ufe of the molt 
accurate of thofe indruments, in the hands of the molt 
careful experimenters, has been formed the following lid 
of the altitudes of lome of the molt remarkable moun¬ 
tains See. above the ievelof the lea. 
Peak of Himmaleh, in Thibet 
Chimborazo, in South America 
Cayambe Outcou, ditto - 
Antifana - - _ 
Mountain of Potofi, in La Plata 
Volcano of Popocatipee, in Mexico 
Pichinha - 
Mont Blanc, (highed in Europe) 
Oertler Spitze, in the Tyrol 
Monte Rofa, Alps - . „ 
Mountains of Geefh, in Abyffinia 
Petcha, or Hamar, in Chinefe Tartary 
Pic of Teneriffe ... 
Mount Ophir, in Sumatra ... 
Aiguille d’Argenture, Alps 
Highed Peak of the Atlas Chain 
Pic d’Offano, Pyrennees - 
Mount St. Elias, on the N. W. coaft of America 
Mount AEtna 
Compafs Mountain, Cape of Good Hone 
City of Quito - 
Mount Lebanon and Mount Ararat 
Pic du Midi, Pyrennees ... 
Mount Cenis ... 
Mount St. Gothard - 
Canegou, Pyrennees . . 
Gondor, in Abyffinia - 
Englilh Feeff 
25000 
*9595 
* 939 * 
19290 
1 8000 
16365 
J 5670 
15662 
* 54 - 3 ® 
15084 
15050 
15000 
. 14026 
13842 
1340a 
11980 
11700 
12672 
*°954 
30000 
9977 
9500 
9300 
9212 
9075 
8 544 
8440 
Source 
