Mr. Nicholson’s Method Sc c. 247 
4. If the differences of the logarithms of numbers he laid 
in order upon an arrangement of equi-difbmt parallel right 
lines, in fuch a manner as that a right line, drawn acrofs the 
whole, {hall interfecl it at divifions which denote numbers in 
geometrical progreffion ; then, from the condition of the ar- 
rangement and the property of this logarithmic line, it fol- 
lows, firft, that every right line, fo drawn, will, by its inter- 
fedtions, indicate a geometrical feri.es of numbers*; fecondlv, 
* Let AB, CD, EF (Tab. X. fig. 5 .) be portions of the logarithmic line, arranged 
according to the condition : let GFI be a right line drawn acrofs, fo as to pafs 
through points of divifion e y c , <j, denoting numbers in geometrical progreffion : 
then will any other line IK, drawn acrofs the arrangement, alfo pafs through 
points f y dy by denoting numbers in geometrical progreffion. 
Demonftration. From one of the extreme points of interfe£lion f in the laft 
named line IK draw the right line fg , parallel to GII, and interfering the 
■arrangement in the points h ; and the ratios of the numbers e : fy c : and 
a : by will be equal, becaufe the intervals on the logarithmic line, or differences of 
the logarithms of thofe numbers, are equal : 
e f c i 
Or - =4 and - “ — . 
ci ah 
e c 
But - — - by the condition. 
c a 
f t 
Therefore 4 = ~ ; or the numbers /", i, h , are in the fame continued ratio as 
i h 
the numbers e, c, a. 
Again, the pointy, the line /V, and the line hby are in arithmetical progreffion, 
and denote the differences of the logarithms of the numbersy and y, i and d % 
h and b. 
The quotients of the numbers themfelves are therefore in geometrical pro- 
oreffion, that is, 
y d . b i _dh 
f : 7 : h’ or h~bi 
Or - — — , by fubftituting — for its equal — • 
d bf f t 
Whence — = -7 or f : d : b. Q. E. D. 
d b 
O O 2 
that 
