[ 279 ] 
I MAGINE two logarithmics, CDE, FDG (fig. 
3, 4, 5, 6.), having the right line AB, given in 
pofition, their common afymptote ; and fuppofe 
the fubtangent of each curve given in magnitude. 
Through a given point A, in the common afymptote, 
imagine the right line AC drawn, an ordinate at right 
angles with the afymptote, meeting the curves in C and 
Fj and let AC, AF be feverally given in magnitude. 
It is required to find the point where thefe curves 
interfedt. Suppofe it done, and let D be the inter- 
feftion. Draw D L perpendicular to A B. Through 
F, the point where AC meets one of the curves 
FDG, draw FM parallel to A B, meeting the other 
curve CDE in M. DrawMN perpendicular to 
A B, and take AH, AK equal to the given fub* 
tangents of the curves, CDE, FDG, relpe&ively. 
Now AL is the logarithm of the ratio of AF to 
L D, in the fyftem of the curve F D G; and (becaufe 
NM=:AF)NLis the logarithm of the fame ratio , 
in the fyftem of the curve CDE. Therefore, 
AL :LN =: AK: AH. Therefore the proportion 
of AL to LN is given j and confequently, that 
of AN to A L is given. But AN is given in mag- 
nitude. For AC and AF are given in magnitude 
(by hypothefis), Therefore, the proportion of AC to 
AF or NM is given ; and AN is the logarithm of 
that given proportion in the fyflem of the given 
curve CDE. But AN being given in magnitude, 
and the proportion of AN to A L being given, AL 
is given in magnitude. And it is given in pofition, 
and the point A is given (by hypothefis). There- 
fore the point L is given (by 27. dat.). Therefore, 
LD, 
4 
