performing the involution and evolution of numbers. 25 
/ 
applied in a variety of different forms to these several 
purposes : and 1 shall beg leave to notice one or two that 
offer some peculiarities. If to the upper scale, which we may 
suppose to be fixed, and to be graduated logometrically, con- 
stituting, as we have already seen, the line of exponents, a 
slider be adjusted, graduated on both edges, according to the 
logometric logarithms ; and the line below, which like the 
upper one is supposed to be fixed, be graduated in the same 
manner as the slider, the instrument will possess the following 
property. When the division 10 on the slider is set against 
any particular number, or exponent, in the upper line, all the 
numbers on the lower line will be the powers, to the same 
degree, of the numbers opposite to them on the slider : the 
degree of the power being marked by the exponent on the 
upper line which is above the 10 on the slider. The lower 
line, therefore, will exhibit the whole series of similar powers 
belonging to all possible roots ; and conversely, the slider 
will exhibit all the roots of the same dimension, with regard 
to all possible numbers. Thus, if the 10 on the slider be 
under 3 in the line of exponents, it will itself be above 1000 
(which is its cube) in the lower line; all the other numbers in 
that line will be the cubes of their opposites on the slider; 
and, conversely, the former will every where be the cube 
roots of the latter. This will be sufficiently apparent, when 
it is recollected that the addition or subtraction of logometric 
logarithms answer to the multiplication or division of simple 
logarithms, and therefore to the involution and evolution of 
numbers. The rule in this form, therefore, bears a closer 
analogy to the common sliding-rule ; since in every position 
it exhibits the series of similar powers and roots, exactly in 
mdcccxv. E 
