r 232 ] 
parallel to /3E, meeting the curve in e. Then /3£ is the 
logarithm of the ratio of /3E to £e, or of the denfity 
at B to the denfity at b. But if the greater of the 
two heights, AB and A b, bear but a. very fmail 
proportion to the femi-diameter of the earth, their 
difference B b will be very nearly equal to /2£. 
For, becaule CB : BA — CA : A/3 (by con irudtion.) 
Therefore, by converfion, CB : CA — CA : C-3 
In like manner, and by inverfion, CA : Cb “C£ : Ct . 
by equi-diftance perturbate, CB : Cb= C£: C/3, 
and converting, CB : Bb=C€ : /3b. 
by permutation, Bb : /3£=CB : C£. 
But when AB is infinitely diminifhed, CB = C 1 
ultimately. Alfo A b being infinitely diminifhed, 
C£ — CA ultimately. Therefore CB — C£ ultimately, 
and Bb —Q,& ultimately. Q. E. D 
Now AB and A b will always be fo fmail, with 
refpect to CA, if B and b be fuppofed to reprelen 
any acceffible places, that CB, C£, and B b, /3£, 
may always, in this cafe, be confidered, as in their 
ultimate proportion of equality'-. 
It is ftill therefore to be admitted as a p- maple, in 
practice, that the difference of elevation of ■ two 
places, is as the difference of the tabular 1 s.rirhms 
of the heights of the quickfilver in the barometer at 
the fame time, at both places ; that is, it is the 
logarithm of the ratio of thole heights in fome 
fyftem of logarithms. And the heights o- quick- 
filver being given, by obfervaiion, the did. ence of 
elevation will be known, if that particular fyftem 
can be determined ; that is, if the mod: dm of the 
fyftem, or the length of the fubtangent of the 
curve 
