performing the involution and evolution of numbers. 17 
double line of numbers, or is a line logometrically divided ; 
the divisions of the first half being from 1 to 10, and being 
repeated on the second half in the same order. The upper 
or fixed rule, BB, is graduated in such a manner, that each 
of its other divisions is set against its respective logarithm on 
the slider; and, consequently, all the numbers on the slider 
will be situated immediately under those numbers in the upper 
rule, of which they are the logarithms. Thus 2 on the rule 
will be over 0.30103 of the slider ; 3 over 0.47712 : 2 on the 
slider will stand under 100 on the rule; 3 under 1000; and 
so on. 
As the series of ordinary logarithms express the exponents 
of 10, of which the corresponding numbers are so many suc- 
cessive powers, it is evident that, in this position of the in- 
strument, the upper line will exhibit the series of the powers 
of 10, corresponding to all the exponents marked on the slider. 
It will be seen, for instance, that the second power of 10 is 
100, the third, 1000, &c. : that the 0.5th (or the square root) 
is 3.163; the 0.25th (or the fourth root) is 1.778 ; the 0.2th 
(or the fifth root) is 1.585 : and so on. 
In every other position of the slider, the upper rule will 
exhibit, in like manner, the series of powers of that number 
under which the unit of the slider has been placed, while the 
opposite numbers on the slider are the exponents of those 
powers. Thus, if ( as in PI. II, fig. 2 ) the unit of the slider 
be placed under the division 3 of the upper rule (at R); the 
square of 3, or 9, will be found over the 2 of the slider: its 
cube, 27, over the 3 ; its fourth power, 81, over the 4: and 
so on for any other power. It is evident, then, that in order 
to find a given power of any number, the unit of the slider 
MDCCCXV. D 
