i8 Dr. Roget’s description of a new instrument for 
must be set underneath that number in the upper rule ; and 
that the number sought will then be found above that number 
in the slider which expresses the magnitude of the required 
power. 
Such being the mode of its application to the finding of 
powers, its use will be obvious in performing the contrary 
operation of finding roots. The root might, for this purpose, 
be considered as a fractional power: but as this would require 
a reduction to decimals, the easiest mode will be to place the 
number expressing the degree of the required root under the 
given number, and the root itself will then be found over the 
unit, or beginning of the scale, in the slider. For fractional 
powers, the denominator of the exponent must be placed 
under the root, and its numerator will then point out the 
power. 
It is hardly necessary to add, that by the same mode we 
may discover the exponent of any given power to any given 
root : since, whatever be the root over the unit of the slider, 
the whole series of the powers of that root, with their 
corresponding exponents, are rendered apparent. This cir- 
cumstance may indeed be considered as an additional re- 
commendation to the employment of this instrument : for it 
affords to those less versed in the contemplation of numerical 
relations an ocular illustration of the theory of involution. It 
presents, at one view, the whole series of powers arising from 
the successive multiplication of all possible numbers, whether 
entire or fractional ; and exhibits this series in all its conti- 
nuity when the exponents are fractional, and even incommen- 
surate with the root itself. The production of the upper line 
in one direction conveys a more accurate notion of the pro- 
