SLIDING-RULE. 



the fraftions 



— ; that is, in the former cafe, the pro- 



ve may call Q, will in like manner be found by forming 



A_Q 

 B 



duft P will ftand over B, when the i on the Aider is brought 

 under A ; and in the latter cafe, the quotient Q will Hand 

 over the i of the flider when B is brought under A." 



The fliding-rule has been varioudy modified, fo as to ferve 

 for the calculation of exchanges, the meafurement of 

 plane and folid bodies, and the computations of trigonometry ; 

 and lately by Dr. WoUalton, in his fynoptic fcale of che- 

 mical equivalents ; but notwithftanding thefe modifications, 

 its ufe is neceffarily limited to operations that are per- 

 formed by the fimple addition or fubtraftion of logarithms, 

 and to the correfponding arithmetical operations above men- 

 tioned. It is not direftly adapted to perform the involution 

 or evolution of numbers, to which the multiplication and 

 divifion of logarithms, correfpond. (See Logarithm.) 

 Neverthelefs, queftions that involve geometrical progrefllons, 

 or exponential quantities, and the computation of the terms 

 of a feries in obtainmg approximate folutions, not unfre- 

 quently occur. The common flidmij-rule affords no direft 

 method of determining even the fimple power or root of any 

 given number ; but when the exponent of the required power 

 or root is not an integral, but a fraftional number) it is Hill 

 more inadequate to the folution of the queftion. Indeed, 

 the fquares and fqnare roots are often pointed out on the 

 common rules by means of a fupplementary line graduated 

 fo that each of its divifions is double in length to thofe of the 

 other two hnes. A hue of cubes, or cube-roots, or of any 

 other given power, might, in hke manner, be fubjoined. But 

 the ufes of fuch additional lines are reilrifted to cafes where a 

 particular power is concerned ; but they afford no aflillance 

 in the cafe of any other power or root, which is not imme- 

 diately related to the power. 



Dr. Roget adopts a new mode of graduation, exhibiting 

 on fimple infpeftion all the powers and roots of any given 

 number to any given exponent, with the fame facihty, and in 

 the fame way, that produfts, quotients, and proportionals are 

 exhibited by the common fliding-rule. Accordingly it is a 

 meafure of powers, juft as the fcale of Gunter is a meafure 

 of ratios. The principle of its conftruftion will be belt il- 

 luftrated by an example. If it were required to raife the 

 number 2.123 '° ^^^ i'fth power; with the ufe of loga- 

 rithms we Ihould multiply the log. of 2.123 ('"^0.32695) 

 by 5 ; the produdl ( i .63475 ) would be found by the tables 

 to correfpond to 43.127, the fifth power of 2.1 23. If the 

 exponent, inllead of being a whole number 5, were fraftional, 

 as 4.3719, the operation may be abridged by the aid of lo- 

 garithms : thus, add the log. of 0.32695 or 9.5144813, 

 to that of 4.3719 or 0.6406702, and. the fum, or 

 0.1551525, is a logarithm anfwering to the number 1.4294, 



the produft required, -viz. 26.878, or zTiiJl '-^ " ' '. 

 In this example, the firll of the numbers added together 



is the logometnc logarithm (that is, the logarithm of the 

 logarithm) of the given root; tlie fecond is the fimple 

 logarithm of the exponent, and the fum of thefe is the 

 logometric logarithm of the power. If a table were con- 

 ftrufted having three fets of columns, the firit containing the 

 natural feries of numbers, the fecond their correfponding 

 logarithms, and the third the logarithms of thofe logarithms, 

 we (hould be able to raife any given number to any given 

 power, by the fimple addition of the num.bers in the fecond 

 and third columns ; as common multiplications are per- 

 forioed by the addition of common logarithms. Hence it 

 9 



is evident, that a line might be graduated fo that its divifiont 

 fiiould correfpond to the numbers in the third column, or re- 

 prefent the logometric logarithms of the numbers marked 

 upon them ; and if this hne were applied fo as to flide againft 

 another line logometrically divided, we might be able by it 

 to perform the operation that has been defcribed ; and we 

 fhould thus have, by infpeftion, the powers correfponding 

 to any given root and exponent. 



The inftrument propofed would therefore, in its fimpleft 

 form, confill of two graduated fcales applied to each other. 

 In PlateWW. Surveying, fig. 6. a portion of thefe fcales is 

 reprefented. The lower rule A A, called the flider, is the 

 common Gunter's double line of numbers, or a hne logometri- 

 cally divided ; the divifions of the firft half being from I to lo, 

 and repeated on the fecond half in the fame order. The up- 

 per, or fixed rule B B, is fo graduated, that each of its other 

 divifions is fet againit its relpeftive logarithm on the flider ; 

 and, confequently, all the numberson the flider will be fituated 

 immediately under thofe numbers in the upper rule, of which 

 they are the logarithins. Thus, 2 on the rule will be over 

 0.30103 of the flider ; 3 over o. 47712 ; 2 on the flider will 

 iland under lOO on the rule ; 3 under 1000, and fo on. 



As the feries of ordinary logarithms exprefs the expo- 

 nents of 10, of which the correfponding numbers are fo 

 many fucceflive powers, in this pofition of the inftrument it 

 is evident, that the upper line will exhibit the feries of the 

 powers of 10, correiponding to all the exponents marked 

 on the flider. E. gr. The 2d power of 10 is 100 ; the 3d, 

 1000, &c.; the 0.5th (or the fquare root) is 3.163 ; the 

 0.25th (or 4th root) is 1.778 ; the 0.2th (or the 5th root) 

 is 1.585, &c. 



In every other pofition of the flider, the upper rule will 

 exhibit, in like manner, the feries of powers of that num- 

 ber, under which the unit of the flider has been placed, 

 while the oppofite numbers on the flider are the exponents of 

 thofe powers. Thus, if (Plate V\\. Jig. 7.) the unit of 

 the flider be placed under the divifion 3 of the upper rule 

 (at R), the fquare of 3, or 9, will be found over the 2 of the 

 flider ; its cube, 27, ever the 3 ; its 4th power, 81, over 

 the 4, &c. Hence, in order to find a given power of any 

 number, the unit of the flider mult be fet underneath that 

 number in the upper rule ; and the number fought will then 

 be found above that number in the flider, which expreflea 

 the magnitude of the required power. The ufe of the in- 

 ftrument will be obvious in performing the contrary opera- 

 tion of finding the roots. In this cafe the root might be 

 confidered as a fraftional power ; but as this would require 

 a reduftion into decimals, the eafieft mode will be to place 

 the number expreffing the degree of the required root under 

 the given number, and the root itfelf will then be found 

 over the unit, or beginning of the fcale, in the flider. For 

 fraftional powers, the denominator of the exponent muft 

 be placed under the root, and its numerator will then point 

 out the power. By the fame mode we may difcover the ex- 

 ponent of any given power to any given root ; fince, what- 

 ever be the root over the unit of the flider, the whole feries 

 of the powers of that root, with their correfponding ex- 

 ponents, are rendered apparent. This circumftance may 

 be confidered as ah additional recommendafion to the em- 

 ployment of this inftrument ; for it affords to thofe lefs 

 verfed in the contemplation of numerical relations, an ocu- 

 lar illultration of the theory of involution. It prelents, at 

 one view, the whole feries of powers arifing from the fuc- 

 ceflive multiplication of all pollible numbers, whether entire 

 or fraftional ; and exhibits this feries in its whole continuity, 

 when the exponents are fraftional, and even incommenfurate 



with 



