performing the involution and evolution of numbers. 13 
numbers will therefore be the multiples of the latter by this 
constant number. Thus, by adjusting the slider, so that its 
unity shall stand under any given multiplier or divisor, the 
upper line will exhibit the series of the products of all the 
subjacent numbers by the given multiplier: and conversely, 
the slider will exhibit the series of the quotients resulting from 
the division of the numbers immediately above them by the 
given divisor.* 
* As the practical mode of using the sliding-rule is frequently not obvious even to 
those who are in possession of the principle of its construction, I shall beg leave to 
point out the following proposition, as one that leads directly to the solution of every 
case to which the instrument can be applied, and an attention to which, therefore, 
may conduce to its more ready and more general employment. In every position of 
the slider, all the fractions formed by taking the numbers on the upper line as nu- 
merators, and those immediately under them as denominators, are equal. Thus every 
corresponding numerator and denominator, having to each other the same ratio, may 
be considered as two terms of a proportion. Any two of these equivalent fractions 
will therefore furnish the four terms of a proportion ; of which any unknown term 
may be supplied, when the others are given, by moving the slider till the numbers com- 
posing the terms of the given fraction, are brought to coincide on the two lines. The 
required term will then be found occupying its proper place opposite to the other 
A C 
given term. Thus, from the proportion A : B : : C : D, we may derive — — _ . 
and adjusting the slider so that B shall stand under A, D will be found under C, when 
C is given: or C will be found over D, when D is given. A similar process would 
have furnished A or B, when one of them together with C and D, were given. Since the 
products of each numerator by the denominator of the other fraction are equal ; (that 
is, AD — BC) ; when one of the terms is unity, the question becomes one of simple 
multiplication or division. The product of A and B, which we may call P, will be 
A P 
found, as before, by placing the slider so as to express the fractions — — — . The 
quotient of A divided by B, which we may call Q, will in like manner be found by 
A Q 
forming the fractions — — : that is, in the former case, the product P will stand 
over B, when the i on the slider is brought under A ; and in the latter case, the 
quotient Qjwill stand over the i of the slider when B is brought under A. 
