SLIDING-RULE. 



then againft 42 on the girt-line is 233 feet for tlic content 

 fought ; whereas, by the connmon way, there arife only 

 184 feet. In efFedl, the common meafure is only to the 

 true meafure, as 1 1 to 14. See Timber. 



3. To meafure a cube. Suppofe the fides to be 6 feet 

 each: fet 12 on the girt-line D, to 6 on C ; then againit 

 72 inches (the inches in 6 feet) on the girt-line are 216 

 feet on C, which give the content required. 



4. To meafure unequally fquared timber ; that is, where 

 the breadth and depth are not equal. Meafure the length 

 of the piece, and the breadth and depth (at the end) in 

 inches ; then find a mean proportional between the breadth 

 and depth of the piece. This mean proportional is the fide 

 of a fquare equal to the end of the piece : which found, 

 the piece may be meafured as fquare timber. 



For an inftance : let the length of the piece of timber be 

 13 feet, the breadth 23 inches, and the depth 13 inches: 

 fet 23 on the girt-line D, to 23 on C ; then againll 13 on 

 C is 17.35 on the girt-line D for the mean proportional. 

 Again, letting 12 on the girt-line D to 13 feet, the length 

 on the line C, againft I7-3S> on the girt-line is 27 feet, the 

 content. 



5. To meafare taper timber. The length being mea- 

 fured in feet, note one-third of it ; which is found thus : 

 fet 3 on the line A, to the length on the hne B ; then 

 againft i on A is the third part on B : then if the folid be 

 round, meafure the diameter at each end in inches, and 

 fubtraft the lefs diameter from the greater ; add half the 

 difference to the lefs diameter ; the fum is the diameter in 

 the middle of the piece. Then fet 13.54 on the girt-line 

 D to the length on the line C, and againft the diam.eter in 

 the middle, on the girt-line, is a fourth number on the 

 line C. Again, fet 13.54 on the girt-line to the third part 

 of the length on the line C ; then againft half tlie difference 

 on the girt-line, is another fourth number on the line C : 

 thefe two fourth numbers, added together, give the con- 

 tent. For an inftance : let the length be 27 feet (one- 

 third of which is 9), the greater diameter 22 inches, and 

 tlie lefs 18; the fum of the two will be 40, their differ- 

 ence 4, and half the difference 2, which, added to the lefs 

 diameter, gives 20 inches for the diameter in the middle of 

 the pjece. Now, fet 13.54 on the girt-line, to 27 on the 

 line C, and againft 20 on D is 5%. 9 feet. Again, fet 13.54 

 on the girt. line to 9 on the line C ; and againft 2 on the 

 girt-line (reprefented by 20) is .196 parts; therefore, by 

 adding 58.9 feet to .196 feet, the fum is 59.096 feet, the 

 content. 



If the timber be fquare, and has the fame dimenfions ; 

 that is, the length 27 feet, the fide of the greater end 22 

 inches, and that of the lefl'er 18 inches; to find the con- 

 tent : fet 12 on the girt-line to 27, the length on the line 

 C ; and againft 20 inches, the fide of the mean fquare on 

 the girt-line, is 75.4 feet. Again, fet 12 on the girt-hnc 

 to 9 feet, one-tliird of the lenp^th, on the line C ; and againft 

 2 inches, lialf the difference of the lides of the fquares of 

 the ends on the girt-line, is .25 parts of a foot : both to- 

 gether make 75.65 feet, the content of the folid. 



The girt or circumference of a tree, or round piece of 

 timber, given ; to find the fide of the fquare within, or the 

 number of inches of a fide wlicn the round timber is fquared. 

 Set 10 on A to 9 on B ; then againft the girt on A are 

 the inches for tlie fide of the fquare on the line B. See 

 Dendkometeh. 



Dr. Roget has lately communicated to the Royal Society 

 the " Defcription of a new InUrumeiit for performing the 

 Involution and Evolution of Npmbers," by which he has very 



much improved the conftruftion, and extended the applica- 

 tion of the " fliding-rule." 



In his ingenious paper on this fubjeft, he has ftated and 

 elucidated the general principles on which inltruments of this 

 kind are conftrufted, and the ordinary purpofes to which they 

 are applicable. It is well known, that the Gunter's line 

 and common fiiding-rule are derived from the properties of 

 logarithms, and that their primary ufe confills in facihtating 

 the mulliplicalion and d'tvifwn of numbers. (See Gi;nter'x 

 Line and Gunter'j Scale.) Our author's inilrument is 

 founded on a particular mode of employing logarithms, and 

 is defigned for immediate application to the involution and 

 evolution of numbers. The common Hiding-rule, it is ob- 

 vious, ferves for comparing the intervals between the num- 

 bers on Gunter's fcale ; and for this purpofe it confifts of a 

 fcale, called the Aider, exaftly of the fame length with the 

 former, and bearing the fame divifions, which, by being move- 

 able along the fide of the other, allows of the application of 

 any part of the one to any part of the other. If the two 

 fcales originally coincide, and the Aiding fcale is the under- 

 moft, then, by advancing the Aider through any given dif- 

 tance, each of its divifions will be brought under thofe of 

 the fixed fcale, which before were refpeftively fituated far- 

 ther forwards by an interval equal to that given diftance. 

 Every number in the upper fcale will therefore have to the 

 number ftanding under it on the Aider the fame conftant ratio ; 

 a ratio indicated by the number under which the unity, or 

 commencement of the fcale of the Aider has been placed. 

 The former numbers will therefore be the multiples of the 

 latter by this conftant number. Thus, by adjufting the 

 Aider lo that its unity ffiall Hand under any given multiplier 

 or divifor, the upper line will exhibit the feries of the pro- 

 dufts of all the fubjacent numbers by the given multiplier ; 

 and converfely, the Aider will exhibit the feries of the quo- 

 tients refulting from thedivifion of the numbers immediately 

 above them by the given divifor. In order to faciUtate the 

 general ufe of the Aiding-rule, Dr. Roget has pointed out the 

 following propofition, as leading diredly to the folution of 

 every cafe to which the inftrument can be applied. " ,In 

 every pofition of the Aider, all the fraftions formed by taking 

 the numbers on the upper line as numerators, and thofe im- 

 mediately under them as denominators, are equal. Thus, 

 every correfponding numerator and denominator, having to 

 each other the fame ratio, may be confidercd as two terms 

 of a proportion. Any two of thefe equivalent fradions 

 will therefore furniih the four terms of a proportion ; of 

 which any unknown term may be fupplied when the others 

 are given, by moving the Aider till the numbers corapofing 

 the terms of the given fraftion are brought to coincide on 

 the two lines. The required term will then be found occu- 

 pying its proper place oppofite to the other given term. 

 Thus, from the proportion A : B :: C : D, wo may derive 



A C 



— = — ; and adjufting the Aider fo that B fhall ftand under 

 B D J 6 



A, D will be found under C, when C is given ; or C will 

 be found over D, when D is given. A fimilar procefs would 

 have furnifiied A or B, when one of them, together with 

 C and D, were given. Since the produAs of each numerator 

 by the denominator of the other fraftion are equal (that is, 

 AD = B C) ; when one of the terms is unity, the queftion 

 becomes one of fimple multiplication or divifion. The pro- 

 dud of A and B, which we may call P, will be found as 

 before, by placing the Aider fo as to exprels the frac- 



A P 



tions - - = TT- The quotient of A divided by D, which 



T . 



we 



