performing the involution and evolution of number's . 2 1 
various, and will easily present themselves. In many specu- 
lative and practical inquiries, cases occur in which geometri- 
cal progressions are concerned, and in which it becomes a 
question, the first term and the common ratio being given, to 
find the other terms ; or, knowing the first and also any other 
term, to ascertain the rate of increase. In all these cases, it 
is obvious that the first term is to be regarded as the root, or 
first power, and the unit in the slider adjusted, so as to coin- 
cide with that number in the line of powers, that is, in the 
upper and lower portions of the fixed rule. The number of 
terms will constitute the exponent of the series, and the power 
corresponding to each successive exponent 2, 3, 4, &c. will 
be the second, third, fourth, &c. term of the progression. 
The successive amounts of a sum placed at compound in- 
terest compose a geometrical progression; and accordingly 
all questions of compound interest are resolvable by this in- 
strument. The rate of interest, or the per centage per annum, 
being added to 1, gives the amount of £1. at the end of 
one year. Thus, at 5 per cent, the amount is 1.05, at 
3 per cent. 1.03, and so on. In either case this number is to 
be regarded as the first term, or root of the series. Setting 
the unit of the slider against this number on the rule, we shall 
find the amount of £ 1 . at the end of 5^ years, opposite to the 
number 5.5 on the slider, and the same of any other interval 
of time. If it be required to ascertain in what time a sum 
placed at compound interest at 3 per cent, would be doubled: 
placing the unit over 1.03, the number 2 on the rule will in- 
dicate 23.45 on the slider, as the number of years required for 
doubling the sum at that rate of compound interest. 
Questions relating to the increase of population and to the 
