28 Dr. Roget’s description of a new instrument , &c. 
relation will be preserved between the numbers belonging to 
any three of these perpendiculars that meet in one point ; viz. 
that the number of the perpendicular to the line of roots, 
raised to the power expressed by the perpendicular to the 
line of exponents, shall be equal to the number denoted by 
the perpendicular to the line of powers. To find, for ex- 
ample, the second power of 3 ; following the perpendicular 
from the division 3 on the line of roots, and that from the 
division 2 on the line of exponents, till they meet in the point 
e, we find among the other set of perpendiculars, the li nefg 
passing through the same point, which, followed till it meets 
the line of powers, indicates on it the number 9, which is ac- 
cordingly the second power of 3. A similar process in other 
directions will furnish the root when the power and expo- 
nent are given, or the exponent when the root and power are 
given. We may thus perceive, at a single glance, not only 
all the powers of any particular root, and all the roots of any 
particular power ; but also all the exponents of the series of 
powers belonging to the same root, as well as the similar 
powers of every possible root. 
It is, perhaps, superfluous to observe, that the same method 
is applicable to the common scale of Gunter ; and that a 
table constructed accordingly, by dividing the sides as well 
as the diagonal logometrically, and applying three sets of 
perpendiculars, would, by their intersections, exhibit in one 
view all possible products and quotients resulting from all 
possible factors or divisors. 
