performing the involution and evolutmi of numbers. 15 
tary line graduated so that each of its divisions are double in 
length to those of the two other lines. A line of cubes, or 
cube roots, or of any other given power, might, in like man- 
ner be subjoined. But it is obvious that the uses of any such 
additional lines are confined to cases where a particular power 
is concerned : they give us no assistance in the case of any 
other power or root, which has no immediate relation with 
the former. 
A new mode of graduation has occurred to me which pos- 
sesses these requisites, and exhibits, on simple inspection, all 
the powers and roots of any given number, to any given ex- 
ponent, with the same facility, and in the same way, that 
products, quotients, and proportionals, are exhibited by the 
common sliding-rule. It is accordingly a measure of powers, 
in the same way as the scale of Gunter is a measure of ratios. 
An example will best illustrate the principle of its construc- 
tion. If it were required to raise the number 2.123 to 
fifth power : availing ourselves of logarithms, we should 
multiply the logarithm of 2.123 ( or 0-32695) by 5. The 
product ( 1.6347 5) wou ld he found by the tables to correspond 
to 43.127, which, with decimals to three places only, is the 
number required, or the fifth power of 2.123. If the expo- 
nent, instead of an entire number, as 5, were fractional, as 
4.37 19, the operation of multiplying by such a number would 
be more tedious, and might evidently again be abridged by 
having recourse to logarithms. Taking, then, the logarithm 
of - - 0.32695 or 9.5144813 
and the logarithm of - 4.3719 or 0.6406702 
and adding them, we obtain 
01351525 
