20 
Dr. Roget’s description of a new instrument for 
one hundredth of their value when applied to the upper portion. 
While 1.259 therefore is marked on the right, 1.0233 == 
io°‘° s ) will occupy the middle, and 1.002305 (= io°‘ 001 ) the 
left end of the lower line. It is evident that the graduation 
might thus be continued indefinitely in both directions. But 
for all practical purposes the limits thus obtained will be found 
amply sufficient: for the well known property of the log- 
arithms of roots in a descending series, enables us to dispense 
with all farther continuation of the scale in that direction. In 
proportion as numbers in a descending series approach very 
near to unity, their logarithms bear more and more exactly 
a constant ratio to the excess of those numbers above unity, 
namely, the ratio expressed by the modulus of the system, or 
1 to .434,294,4819, See. As we descend in the scale, therefore, 
the decimal part of the exponents becoming smaller and 
smaller, the corresponding logarithms will approximate so 
nearly to the multiple of that decimal part by this modulus, 
that no sensible error will result from assuming them to be 
the same.* The divisions to the left of the lower portion of 
the rule may therefore be taken as sufficiently accurate re- 
presentations of the divisions which would occur in the suc- 
ceeding portions of the line, if it were prolonged indefinitely 
in that direction. 
The applications of which this instrument is susceptible are 
* Thus the logarithm of 1.05 is .021189 
that of 1.005 * s .0021661 
and of 1.0005 is .00021709 
which differs from the product of the modulus by .0005 (or .00021715) by a quan- 
tity affecting only the fourth significant figure. The roots 1.C005, 1.00005, 1.000005, 
&c. may, therefore, without sensible error, be considered as coinciding with the divi- 
sion 217 on the slider. 
