performing the involution and evolution of numbers. 19 
gressive and rapid increase of those powers, than can be 
acquired by mere abstract reflection : and its continuation on 
the other side, shows the slow approximation to unity which 
takes place in the successive extractions of higher and higher 
roots. 
A variety of forms of construction might be given to in- 
struments operating on the principle now explained. The 
following has appeared to me, on the whole, to be the most 
convenient for practical purposes ; it is represented on a re- 
duced scale in PI. II, fig. 3. In order to preserve a sufficient mag- 
nitude of scale, I have divided the line of roots and powers into 
two parts ; placing the one above and the other below, and in- 
terposing a slider with a double scale of exponents. The slider 
of the common sliding-rule is graduated in a way that is ex- 
ceedingly well suited to this purpose, having divisions on each 
edge, and carrying two sets of numbers from 1 to 10. Adapt- 
ing a blank ruler to one of these sliders, which must be fixed 
in a proper position, I mark off, on the upper line, the series of 
numbers against their respective logarithms on the slider ; plac- 
ing 10 over the middle unit of the slider, 100 over the 2, 1000 
over the 3, and so on, proceeding towards the right from 10 to 
10000000000, the tenth power of 10, an extent which is more 
than sufficient for all useful purposes. The space to the left is 
also graduated on the same principle, from 10 to 1.259 which 
is the tenth root of 10, or io 0,1 . The upper portion of the 
rule being thus filled, I place the continuation of the same 
line on the lower portion, beginning on the right hand, and 
proceeding in a descending series of fractional powers of 10, 
corresponding with the exponents on the intermediate slider, 
which, when applied to this portion, are to be taken as only 
D 2 
