j performing the involution and evolution of numbers. 23 
Assuming, that when light is transmitted through water, one 
half of the quantity that entered is lost by passing through 
seven feet of water:* how much will be intercepted by pass- 
ing through three feet? In questions of this sort it must be 
recollected that it is the quantities of transmitted, and not of 
intercepted light, that are in geometrical progression. If 0.5 
is transmitted by seven feet, 0.5^ will be transmitted by three 
feet. As 0.5 is not contained on the rule, we must take its 
reciprocal 2, of which the -|-th power, or 1.3023, is given 
by the instrument : this number being opposite to the 3 on 
the slider, when its division 7 is placed under 2 on the rule. 
The reciprocal of 1.3023 or .9892 is the quantity transmitted; 
and therefore .0108 the quantity absorbed by three feet of 
water. 
A variety of propositions relating to the general theory of 
logarithms are illustrated by this instrument. The assump- 
tion of the number 10, as the basis of our system of loga- 
rithms is arbitrary, and is chosen only for the sake of greater 
convenience in computation. The hyperbolic system, which 
has the number 2.302585093, &c. for its basis, possesses other 
advantages, especially in the higher branches of analysis. 
The instrument may be made to exhibit at one view the se- 
ries of any particular system of logarithms, that is, of a sys- 
tem with any given basis, or any given modulus, by merely 
setting the unity of the slider against the given basis on the 
rule : or the given modulus on the slider against the number 
2.7182818, &c. on the rule. The divisions on the slider will 
then denote the logarithms of the numbers opposed to them 
on the rule. 
* Young’s Lectures on Natural Philosophy, I s 409. 
