24 Dr. Roget’s description of a new instrument for 
Let it be required to determine the particular system of 
logarithms, in which the modulus shall be equal to the basis. 
Take out the slider, and introduce it in an inverted position, 
so that the numbers on it shall increase from right to left : 
and place the number .4343, &c. ( the modulus of the common 
system) under 10 (its corresponding basis) on the rule, as 
represented in PI. II, fig. 4. We shall find that in this position, 
all the other numbers on the slider will be the moduli corre- 
sponding to the respective bases of each different system, on 
the rule. Thus, the 1 on the slider, or the modulus of the 
hyperbolic system, is opposite to 2.718, the basis of that sys- 
tem. On the other hand, the division 2 on the rule is oppo- 
site to 1.4427, which is the modulus of the system having for 
its basis the number 2. Carrying the eye still more to the 
left, and observing the point where similar divisions appear 
both on the rule and the slider, we shall find it to be at the 
number 1.76315, which therefore expresses the modulus and 
the basis in that particular system in which they are both 
equal. The reason of the above process will readily appear 
when it is considered, that the modulus of every system is the 
reciprocal of the hyperbolic logarithm of its basis. 
This inverted condition of the slider will also afford an easy 
method of solving exponential equations, for which there 
exists no direct analytical method. The following may serve 
as an example. Let the root of the equation x x = 100 be 
required. Set the unit of the inverted slider under 100 on the 
rule, and observe, as before, the point where similar divisions 
coincide ; this will be at 3.6, which is a near approximation 
to the required root: and accordingly 3.o 3 ' 6 == 100. 
The principle of the ‘instrument above described might be 
