(see that article, as likewise Gun- ning Rod, Geometry, and Logarithmic Lines, En- cyclo.) is introduced here, for the sake of a new, and (except in working direct proportions) a more com- modious method than the common, of applying the slider. This method, which is proposed by the Rev. W. Pearson of Lincoln, is as follows:

Invert the slider B on any common sliding rule, whereby the numerical figures will ascend on it, and on the fixed line A, in contrary directions; now, as the distance from unity to any multiplier, on Gunter's line, will invariably extend from any multiplicand to their product, it follows, that if any particular number on the inverted slider B be placed opposite to any other given number on A, the product of those numbers will stand on the slider B, against unity on A; for, in any position of the inverted slider, the distance from unity to the multiplier on A, instead of being carried forward on B, as when the slider is in a direct position, is brought back thereby to unity again; so that unity (or its op- posite lines where the slider is too short for the opera- tion) is invariably the index for the product of any two coincident numbers throughout the lines.

In division, by the same process, if the dividend on B be put to the index, or unity on A, the division and quotient will coincide on the two opposite lines; so that when one is given, and sought for on either line, the other is seen on its opposite line at the same time.

The next operation which offers itself here is reci- procal proportion, which can be effected by no other method than by inverting the slider, but which is ren- dered as easy by this application, as direct proportion is in the common way; for if any antecedent number on B inverted be set to its consequent on A, any other antecedent on B, in the same position, will stand against its consequent on A, so that the terms may be in a reciprocal ratio. In squaring any number, it will ap- pear, from what has been already said, that if the num- ber to be squared be placed on B, inverted against the same on A, the square will stand on B, against unity on A. Therefore, to extract the square root of any number, let that number on B stand against unity on A; and then wherever the coincident numbers are both of the same value, that point indicates the root. If two dividing lines of the same value do not exactly co- incide, the coincident point will lie at the middle of the space contained between those two which are nearest a coincidence; and as there is only one such point, there can be no mistake in readily ascertaining it. The find- ing of a mean proportional between any two numbers is extremely easy at one operation; for if one of the numbers on B inverted be set to the other on A, the coincident point of two similar numbers shows either of those to be the mean, or square root of their product, according to the preceding process. Thus have we a short and easy method of multiplying, dividing, work- ing reciprocal proportion, squaring and extracting the square root, at one position of the inverted slider, whereby the eye is directed to only one point of view for the result, after the slider is fixed; whereas, by the common method of extracting the square root by A and B direct, the slider requires to be moved back- wards and forwards by adjustment, the eye moving al- ternately to two points, till similar numbers stand, one on B against unity on A, and the other on A against the square number on B; which square number, in the case of finding a mean proportional, must be found by a previous operation. Hence, for more convenience in the extraction of roots, and measuring of solids, an ad- ditional line called D has been added to the rule, which renders it more complex, and consequently seldom un- derstood by an artificer.