3 



The instrument contrived by Dr. Roget, is constructed to answer 

 this last purpose, with the same facility as common multiplication 

 and division are performed by the common sliding-rule. 



For in the same manner as numbers are considered as powers of 

 10, so their indices, whether integral or decimal, being, in fact, num- 

 bers, may again be regarded as powers of 10 ; and their secondary 

 indices, or logarithmic logarithms, may be laid down as logometric 

 intervals, to which other logometric intervals may be added or sub- 

 tracted mechanically, so as to present to view any multiples or ali- 

 quot part of a logarithm, and consequently any powers whatever of 

 the number to which that logarithm is index ; for when the unit of 

 Gunter's line, on the slider, is applied to any number on the scale, 

 divided into logometric logarithms, then 2 on the slider corresponds 

 to the square, 3 to the cube, &c. of the same number. 



The author enumerates various uses to which such an instrument 

 is applicable. To all cases of geometrical progression the application 

 is obvious for finding the common ratio, the number of terms, or any 

 particular term in the series. An approximate solution is thus given 

 to all questions of compound interest, to regularly progressive in- 

 crease of population, and to many calculations of chances. To cases 

 also of the reduction of temperature which a body undergoes by com- 

 munication to a surrounding medium ; and to successive stages of 

 exhaustion, by an air-pump, it may be applied with equal advantage. 



Since the scale of the instrument presents to view the proportion 

 of logarithms to each other, while the slider represents the ratios of 

 their respective numbers, it becomes a means of illustrating many 

 points relative to the general theory of logarithms, whether to ex- 

 hibit a series of logarithms formed according to any particular mo- 

 dulus, or by an inverted position of the slider to find the moduli of 

 all different systems, and accordingly in that, for instance, in which 

 the modulus is equal to the basis. 



In this inverted position of the slider it affords a ready answer to 

 various exponential equations, which do not admit of any direct so- 

 lution, as, for instance, if x* = 100. Let the unit on the slider be 

 placed opposite to 100 on the rule, then 2 will be found opposite to 

 10, which is the square root of 100; 3 will be opposite to 4'641, 

 which is the third root of 100; and by attending to the decreasing 

 numbers which correspond to increasing numbers on the slider, it will 

 be seen that 3'6 is the point at which they appear equal, showing 

 that 3'6 3 ' 6 is nearly equal to 100, and is therefore approximately the 

 root required. 



The author concludes by pointing out various forms in -which such 

 an instrument may be constructed ; since the line divided into logo- 

 metric logarithms may be turned into a spiral, or arranged according 

 to any other of the various modifications that have been given to the 

 common logometric line of Gunter. 



