2 6 Dr. Roget's description of a new instrument for 
the same way as the latter exhibits the series of similar 
products and quotients. 
I have also contrived another form of the instrument which 
possesses some advantages in theory, though its execution 
may perhaps be more difficult. It is evident that the whole 
scale may, like Gunter's line, be thrown into a circular form ; 
and this I have done in the way represented in PI. III. The 
circle on the outside, being logometrically divided from 1 to 
10 round the circumference, will constitute the line of expo- 
nents. The line of powers, being disposed in a spiral, will 
occupy the interior space, which may be made to revolve 
within the former, and should be provided with one or more 
threads extending from the centre to the circumference, and 
serving as radii to mark the position of all the parts of the 
spiral line with regard to the divisions of the outer circle. 
One of these threads may be fixed at the unit or beginning 
of the scale, and will serve to mark the position for the root 
of any required power. The spiral itself must be graduated 
exactly as the upper line in the first described rule : that is, 
the situation of the division 10 must be first determined upon, 
and then brought under the unit in the circle of exponents, 
that is, under the fixed thread. Every other division must 
then be marked with reference to the place of its logarithm 
on the circle, or must be made to occupy the same angular 
distance from the thread. This graduation will be most con- 
veniently made by means of the moveable leg of a sector 
revolving on the centre of the circle. The comparison of 
the divisions of the spiral with those of the circle, may be 
made, either with this moveable sector, or with the threads 
already mentioned. The numbers on the spiral will increase 
