11 



M.KK.P: SI. KKl' WALKING. 



SLIDE RULE. 



ni 



lareni by Englwh cruiser* an MtUed. [SIERRA LEOXE, in OEOO 

 Div.J Several thoiuuid* of negroes who bare eecaped from slavery 

 in the United Sute* are now settled in Canada, earning a livelihood by 

 their own industry. 



SI.KKI 1 ; SLEEP-WALKING. [8oiAiiBrusJt.] 

 sl.i, s in an instrument 



for the mechanical performance of addition and subtraction, which i 

 converted into an instrument for the mechanical performance of multi 

 plication and division by the use of logarithmic scale*, instead of 

 scales of equal parts. 



This instrument has bean greatly undervalued in our country, in 

 which it was invented, and is very little known mi the Continent ; for 

 though a French work on the rabjeot, published in lS2f>, which is fol- 

 lowed by the writer of a moiw recent m:it ln-nutic.il <lictinary in the same 

 language, assures us that in Kiiglaud the gliding rule is taught at schools 

 at the same time with tin- letters of the alphabet, we believe it would In- 

 more correct to say that nine Kngliahmen out of ten would not know 

 what the instrument was for if they saw it, and that of those who 

 even know what it is for, not one in a hundred would be able to work 

 a simple question by mean* of it. For a few shillings most persons 

 might put into their ]>ucketa some hundred times as much power of 

 calculation as they have in their heads : and the use of the instrument 

 is attainable without any knowledge of the properties of logarithms, on 

 which iu principle depend*. 



1 1 t 468789 10 



ft tl y e S f 



( h i) i 8 k B 





We have before us a logarithmic scale, of which A B, called the 

 radiui, may stand for the logarithm of 10, 100, 1000, Ac. : but if A B 

 should be, say the logarithm of 100, then A c is that of 20, A rf of 30, 

 and so on. If this scale lw repeated several times, beginning again at 

 B, and if it be also large enough to be sutxlivided to a greater extent 

 than can be shown in the diagram, any multiplication can be approxi- 

 mately performed by addition, and any division by subtraction ; which 

 may be done with a pair of compasses. That is to say, the figures of 

 the product may be found, exactly or approximately, and the meaning 

 of the figures must be settled from the known character of the result. 

 For example, to find 4 times 15 : First, let A B mean the logarithm of 

 100, then A a is that of 15; next let AB mean the logarithm of 10, 

 then A e is that of 4. Take A a on the compasses, and set it on to the 

 right of t ; it will be found that the point g is attained, directly under 

 6. But 4 times 15 must be tens ; therefore 60 is meant, or 4 x 15 = 60. 

 Next to divide 90 by 45 : from A < take A J, or set off A S from t 

 towards the left. The point c will be attained, under 2, which is the 

 quotient Next to find 7 times 5 : set off A/ from A towards the right, 

 and the point y of the scale following B will be attained, and 35 is the 

 answer. But had it been to multiply 7 by '5 or 5-tenths, this 35 would 

 have meant 3'5 or 3J. Attempts are made in works professing to ex- 

 plain the sliding-rule to give rules for the determination of the cha- 

 racter of the figures in the answer, but without any success. It is all 

 very well for a few chosen examples, but an attempt to do without the 

 book soon shows the insufficiency of rules. If, on a large scale, 653 

 should be the figures of an answer, common sense, applied to the pro- 

 blem, must say whether it is -0653, '853, 6'53, 65'3, 653, 6530, 65800, 

 Ac. which is meant. A knowledge of decimal fractions is therefore 

 indispensable. 



these additions and subtractions might be performed by a pair 



of rulers made to slide each along the other ; but whether they art- 



kept together by the hand, or whether the one niler slides in a 



: the edge of the other, matters nothing to the explana- 



Th following diagram represents the two ruler* in one relative 



-. Hero 1 on the slide is made to match 2 on the fixed ruler, 



and the instniment is now in .1 position to multiply by 2, to perform 



every division in which the quotient is 2, and to work every IP 



i uli- of three in which the ratio of the first term to either the 



or third is tint of 2 to 1, or of 1 to 2. And here let us observe, 



that much the best way of beginning to use the sliding-ruU- is not by 



working given questions, but by setting the slide at hazard, and learn- 



ing to read the questions which are thus fortuitously worked. 



3 4 S 6 7 891 



3 4 5 07B9 I 



-LU! 



i Til 



i. o 



r 



67891 

 TITTI 



In the cut before us we have the 1 of the slide placed at 2 of the fixed 

 ruler ; consequently C on the slide comes under what would bo 12 of 

 the fixed ruler if the secondary graduations were inserted. Again, 4 

 comes over 2, and 9 over 45, giving 4 : 2 : : 8 : 4'5, ' I point 



being inserted by intuition. To show the sort of results which we 

 obtain fmm such a slide of 64-inches radius (or from 1 to 1), we take 



this sort, and throw 1 of the slide at hazard bet\s 

 230 on the fixed ruler, a little farther to the right than it is on the 

 preceding diagram; guessing at th- interval, it seems ::'-:.. W. 

 detect it more exactly by looking at 5 on the slide, which is hardly 

 viable in advance of 114 on the wale. As fur then as the divisions, 



y our jiuKtuent of this interval, inform us, we have 114-=-5 



= 22-85, the truth being 22'8. Again, we find 628 (fixed} just over 

 275 (slide), the K being estimated ; hence, by the se n that 



628 275^'2-285 ; the truth U-iiiK 2".>S:W. Thirdly, we estimate 

 that 1725 (fixed) 'falls over 757 (slide), and that -J7'i ui\edi falls 

 That is, the ruler informs us that 17- 7.". 7 : : 

 J7>: : I'Jl ; the fourth terra should be l-.'l'l-.', as found l>y e-ni 

 put.itioii. We take a larger scale, having 7) inches of radius, and 



1 on the slide to ppears to fall ov. 



we hould judge to be 786 rather than 7- p >7. Now 1 725^757= 2".'7*7. 

 and 17.':.-: 7.16 = 2-2818. Both give on the s. 'nt the 



advantage is slightly in favour of the larger scale, but not so in 

 we should have expected. We now try one of -J4 im-he-i r.idi', 

 setting 1 on the slide to 2'285 on the fixed ruler, we find th 

 (fixed) falls over 754'7 (slide), the last 7 being estimation. Now 



7.". 17 ---s.'.7, which gives the advantage again (but not so 

 decidedly as might have been expected) to the large rule. The 

 that it is rather ease than proportionate accuracy which is 

 gained by the large rules: the preceding results required cai 

 close attention on the 54-inch rule ; were obtained with modem: 

 on the 74-inch; and taken off instantly from the 24-inch rule. More- 

 over, divisions on wood, made in the usual way, do not 

 racy to increase with the size : if these rulers were divid.d ,- 

 and with the precautions taken in astronomical instruments, it would 

 In- a very different thingT but after all, the wonder is that the cm 

 wooden rules should be so accurate as they are. 



The next step in the description is a* follows : It matters nothing 

 whether the second scale be really mode consecutive with the first, 

 or occupy any other part of space : provided that when 1 and 1 are 

 brought together on the first scale, 1 and 1 also come together on the 

 second, and that the first slide and its continuations slide equally. We 

 see this in the diagram before us : a 4 is one slide, anil AH ai 

 rulers on opposite sides of the groove. When <i A is pushed home. 

 A and a present coinciding scales, as do B and b : we should ratle 

 that the last is not one scale, but the end of one and the begini: 

 another; the 1 of Baud* being in the middle. The conseque: 

 that so long as 1 of the scale b is not pushed out so far as to fall out 

 of the groove (which is never necessary, since there is a whole scale on 

 B), there is always the power of reading every result of the multipli- 

 cation in hand. In the diagram, 1 on b is pushed out to 2 on i 

 on the upper scales (A and a) we see 2x2 = 4, 2x8=' 

 2x5 = 10; on the lower (B and t), 2x4 = 8, 2x5 = 10, 2x6 = 12, 

 2x7=14, 2x8 = 16, 2x9=18, 2x10 = 20. This modification was 

 invented by Mr. Silvanus Bevan ( Nicholson's 'Journal,' vol. \li\..| 

 but thirty years before this Mr. Nicholson \' Phil. Trans.', 1 7>7, p. -J IM 

 hod pointed out how to divide the whole radius into four parts, t 

 each face. 



A simple plan, and in some respects the -best, is to make a revolving 

 circle turn upon a fixed one, in which case the scale is it* own eoiitiima- 

 tion, as in the following diagram. The two circles line .1 eommon 



pivot, and the upper one turns round on the lower; the- rim of the 

 inner circle being bevelled down to the plane of the lower. -\ 

 plete logarithmic scale is marked on each I'ircumi. it will 



readily he seen that the scales are placed BO as t<> point out niuh 

 tions by 2, as in the former instances, and al-o that the rccomn 

 ment of the scale begins its continuation. Instead of two circle- 

 might be two thin cylinders, turning on a common axis, th> 

 being made on the rim. Thirty \ -it maker at 



Paris laid down logarithmic scales on the iims of tie- boi and lid of a 

 eommon circular snuff-box : one of tw o inehe-i diameter would be as 

 good an aid to calculation as tin- eommon engineer's rule. l!ut 

 calculators disliked snuff, or snnll 'takers calculation, for the scheme 

 was not found to answer, and the apparatus was broken up. 



