SLIDING-RULE. 



with the MOt itfelf. The produftion of the upper line in 

 one direftion, conveys a more accurate notion of the pro- 

 greflive and rapid incrcafe of thofe powers, than can be ac- 

 quired by mere abftradl refledlion ; and its continuation on 

 the other fide (hews the flow approximation to unity which 

 takes place in the fucceflive extraftions of higher and higher 

 roots. Among the variety of forms of con Itruftion of which 

 inllruments that operate on the principle now explained are 

 capable, Dr. Roget conceives the fjUowirg to be, upon the 

 whole, the moll convenient for practical purpofes. Its re- 

 prefentation, on a reduced fcale, appears in P/ateV II. Jig- 8. 

 With a view of preferving a fufficient extent of fcale, the 

 line of root? and powers is divided into two parts ; one being 

 placed above and the other below, and between them a 

 Aider with a double fcale of exponents. The Aider of the 

 common fliding-rule is graduated in a way that is exceedingly 

 well fuited to this purpofe, having divifions on each edge, 

 and carrying two fets of numbers from i to lo. Adapting 

 a blank ruler to one of thefe Aiders, which mult be fixed in 

 a proper pofition. Dr. Roget marks off, on the upper line, 

 the feries of numbers againll their refpeftive logarithms on 

 the Aiders; placing lo over the middle unit of the Aider, 

 loo over the 2, 1000 over the 3, and fo on, proceeding 

 towards the right, from 10 to loooooooooo, the loth 

 power of 10 ; an extent which is more than fufRcient for all 

 ufeful purpofes. The fpace to the left is alfo graduated on 

 the fame principle, from 10 to 1.259, which is the loth 

 root of 10, orTo'"-'. The upper portion of the rule 

 being thus filled, Dr. R. places the continuation of the fame 

 line on the lower portion, beginning on the right hand, and 

 proceeding in a defcending feries of fraftional powers of 10, 

 correfponding with the exponents on the intermediate Aider, 

 which, when applied to this portion, are to be taken as only 

 -r^Tfth of their value when applied to the upper portion. While 

 1.259 therefore is marked on the right, 1.0233 (= io\°"°') 

 will occupy the middle, and 1.002305 (= io)"'°"') the 

 left end of the lower line. The graduation, it is plain, might 

 be thus continued indefinitely in both direftions. But for 

 all praftical purpofes, the limits thus obtained will be found 

 amply fufficient. In proportion as numbers in a defcending 

 feries approach very near to unity, their logarithms bear 

 more and moreexaftly a conftant ratio to the excefs of thofe 

 numbers above unity, viz. ihe ratio expreflcd by the mo- 

 dulus of the fyltcm, or i to .4342944819. As we defceiid 

 in the fcale, therefore, the decimal part of the exponents 

 becoming fmaller and fmaller, the correfponding logarithms 

 will approximate fo nearly to the multiple of that decimal 

 part by this modulus, that no fenfiblc error will refult from 

 an"um!iig them to be the fame. Thus the log. of 1.05 is 

 .02 1 1 89 ; that of 1 .005 is .002 1 66 1 ; and that of i .0005 is 

 .00021709, which differs from the produdl of the moduliif 

 by .0005 (or .0002 17 15) by a quantity affedliiig only the 4th 

 fignificant figure. The roots 1.0005, 1.00005, 1.000005, 

 &c. may, therefore, without fenfible error, be conlidered 

 as coinciding with the <?vifion 217 on the Aider. Heiice, 

 the divifions to the left ot the lower portion of the rule may 

 be taken as fnffitiently accurate reprefcntations of the divi- 

 fions which would occur in the fucceeding portions of the 

 line, if it were prolonged indefinitely in that direction. 



This inftriiment may, from the principle of its conltruc- 

 tion, be applied to the folutioii ot various problems in which 

 geometrical progrcffions are concerned. E. ^. As tlie fuc- 

 ccffive amounts of a fum placed at compound interell com- 

 pofe a geometrical progreAion, all fjuellions of compound 

 interelt are refolvable by this inllrument. The rate of in- 



tereft, or the^^r eentage per annum, being added to I, gives 

 the amount of i/. at the end of one year. Thus, at 5 per 

 cent, the amount is 1.05, at 3 per cent. 1. 03, &c. In either 

 cafe this number is to be regarded as the firft term, or 

 root of the feries. Setting the unit of the Aider againft this 

 number on the rule, we (hail find the amount of i/. at the 

 end of 5^ years oppofite to the number 5.5 on the Aider, 

 and the fame of any other interval of time. If it be required 

 to afcertain in what time a fum placed at compound interell 

 at 3 per cent, will be doubled ; placing the unit over 1.03, 

 the number 2 on the rule will indicate 23.45 on the Aider, as 

 the number of years required for doubling the fum at that 

 rate of compound interell. The interpolation of a given 

 number of mean proportionals between two given numbers, 

 is fometimes required for the lolution of a problem ; and it 

 is eafily effeitled by the rule above defcribed. Thus, in di- 

 viding the mufical oftave into 12 equal femitones, the follow- 

 ing feries of numbers mull be calculated, oiiz. 2iV» 2tV» 2tV» 

 2xV> 2tV> 2-iV> i-frt 2rV> 2 ,Vt 2 14', 2.i: this ma^ rea- 

 dily be done in one pofition of the Aider ; fur when the 12 

 marked on it is placed under 2 on the rule, the i of the 

 Aider will point to 1.0595 — 2-rV) the 2 of the Aider will 

 indicate 1.1225 = 2t't> the 3, 1.1892 = 2^, &c. 



A variety of queftions relating to the general theory of 

 logarithms may be illuftrated bv this inllrument. The af- 

 fumption of the number 10, as the bafis of our fyllem of lo- 

 garithms, is arbitrary, and chofen merely for greater conve- 

 nience in computation. The hyperbolic fyllem, whofe 

 bafis is the number 2.302585093, &c. has its pecuhar 

 advantages, efpecially in the higher branches of analyfis. 

 The inilrumeiit may be made to exhibit at one view the fe- 

 ries of any particular fyftem of logarithms, that is, of a fyf- 

 tem with any given bafis, or any given msdulus, by merely 

 fetting the unity of the Aider againll the given bafis on the 

 rule, or the given modulus on the Aider again!! the number 

 2.7182818, &c. on the rule. The divifions on the Aider will 

 then denote the logarithms of the numbers oppofed to them 

 on the rule. 



Let it be required to determine the particular fyftem 

 of logarithms, in which the modulus (hall be equal to the 

 bafis. Take out the Aider, and introduce it in an inverted 

 pofition, fo that the numbers on it Aiall iiicreafc from right 

 to left ; and place the numbers -4343, &c. (the modulus of 

 the common fyftem) under 10 (its correfponding bafis) on 

 the rule, as reprefented in PLile VII. Jg. 9. We (hall find 

 that, in thi.s pofition, all the other numbers oa the Aider will 

 be the moduli correfponding to the relpcilive bafcs of each 

 different fyftem on the rule. Thus, the I on the Aider, or 

 the modulus of the hyperbolic fyftem, is oppofite to 2.718, 

 the bafis of that fyitem. On the other hand, the divifioa 

 2 on the rule is oppofite to 1.4427, which is the modulus 

 of the lyllem having for its balls the number 2. Carrying 

 the eye Hill more to the left, and obfervlng the point where 

 fimilar divifions appear, both on the rule and the Aider, we 

 fhall find it to be at the number 1.76315, which therefore 

 expred'es the modulus and the bafis in tliat particular fyftem 

 in which they are both equal. The reafon of the above 

 procefs will readily appear, when it if confidercd, that the 

 modulus of every fyllem is the reciprocal of the hyperbolic 

 l>ig:>rlthm of its balls. 



This inverted condition of the Aider will alfo afford an eafy 

 method of (olving cxoonential equations, for which there 

 exifts no diieil analy'.ical method. Eg. Let the root of 

 the equation .v' — 100 be required. Set the unit of the in- 

 verted Aider under 100 on the rule, and oblcrve, as before, 

 the point wiicre fimilar divifions coincide : this will be at 



3-6. 



