613 



SLIDE RULE. 



SLIDE RULE. 



614 



The form first proposed by Oughtred (presently to be mentioned 

 was a modification of the preceding. Instead of two circles, t 

 pointing radii were attached to the centre of one circle, on which 

 number of concentric circles were drawn, each charged with 

 logarithmic scale. These pointers would either move round together 

 united by friction, or open and shut by the application of pressure 

 they were in fact a pair of compasses, laid flat on the circle, with their 

 pivot at ite centre. Calling these pointers antecedent and consequent 

 to multiply A and B the consequent arm must be brought to point tc 

 1, and the antecedent arm then made to point to A. If the pointers 

 be then moved together until the consequent arm points to B, the ante 

 ceclunt arm will point to the product of A and B. 



It will be observed that in every construction the logarithmic space: 

 are very unequal, those near the end of the scale being small when 

 compared with those at the beginning. This is not so great a disad 

 vantage as might be supposed, for it makes the liability to error increase 

 in nearly the same proportion with the result, so that the per centage 

 of error in the sliding-rule is nearly the same thing in all its parts 

 For example, the scale going from 10 to 100, the interval from 10 to 

 1 1 is to that fruin 99 to 100 as 207 to 22, nearly in the proportion o 

 10 to 1. The tendency to absolute error will be inversely as these 

 intervals, or nearly in the proportion of 1 to 10 : the tendency to 

 error is therefore about 10 times as great precisely when the resull 

 estimated becomes ten times as great. Oughtred appropriated two 

 circles to his logarithms of sines, and it would be easy in his construction 

 of the 'circles of proportion,' as he called them, to distribute the 

 scale among different circles in such a manner that the graduations 

 should be nearly equal throughout. But the mathematician will easily 

 aee that that the most perfect mode of developing this idea would be 

 to lay down the scale on a revolution of a logarithmic spiral, having 

 the pointers joined at its pole. The graduations would then be abso- 

 lutely at equal distances from each other on the arc of the spiral. 



Aii' ither modification of the principle of the slidiug-rule is as follows 

 Let the divisions be all made equal, and the numbers written upon 

 the divisions in geometrical proportion. If this were done to a 

 sufficient extent, any number might be found exactly or nearly enough 

 upon the scale ; the only difficulty being that very small divisions do 

 not give room enough to write the numbers. This modification ol 

 the principle has been applied in two very useful modes by Mr. 

 MacFarlaue. In the first, two cylinders moving on the same axis, on 

 one side and the other of a third, give the means of instantaneously 

 proposing and solving any one out of several millions of arithmetical 

 questions for the use of schools and teachers. In the second, one 

 circle revolving upon another gives the interest upon any sum, for any 

 number of days, at any rate of interest under 10 per cent. 



The rules for using the sliding-rule, in its most simple form, may be 

 symbolically expressed in the following manner : 



B 

 Alt 



B 



B -r A 



A C 



B Ox B-f- A 



Thus, if 1 on either ruler be brought opposite to A on the other, B 

 on the first ruler is brought opposite to A B on the other, lint if the 

 slide be taken out and inverted, we have the following rules : 



1 B 

 A A-f B 



A B 



1 A-r B 



A C 



B Ax B-^-C 



We now proceed to some of the addition* which are frequently made 

 to sliding-rule*, premising that we do not describe any one in particular, 

 but refer for detail to the tracts which are afterwards cited. For the 

 extraction of square or cube roots, or the formation of squares or cubes, 

 the following method is adopted : In the case of squares and square 

 roots, for instance, there is a pair of scales, one on the slide and one on 

 the fixed ruler, of different radii, the radius of one being twice as long 

 as that on the other : for cubes and cube roots the radius of one is 

 three times as long as the other. On the former scale (that of squares 

 and square roots) the rules ore now as follows : 



Longer Bad. 1 A 

 Shorter lUd. 1 AA 



VA 



A 



A B A\'B 



1 B A B 



1 B (VB):A 



A AB* B 



A C AV(C:B) 



B BC' : A" C 



The denomination of the answer, or the place of the decimal point, 

 must be determined by independent consideration, as before; but there 

 is one circumstance to be attended to in every case in which two of the 

 data are to be read on the shorter scale. For example,* suppose it is 

 required to estimate V(2: 7). By the second formula, 7 on the shorter 

 scale is placed opposite to 1 on the longer, and 2 on the shorter scale is 

 then opposite to 1693 on the longer. The answer from the scale is 

 then 'lH'J'i, to all appearance; but this is not V(2:7), but -^(2:70). 

 The place on the longer scale which should give the answer has no 

 slide opposite to it, but only empty groove. But mark where 1 on 

 the shorter scale is opposite to a part of the longer (between 119 and 

 120), and push the slide in from left to right till the first 1 on the 



* The reader will not understand this, unless with the scale in his hand. 

 The common ca^enter's rule or Sevan's rule will do, in which two consecutive 

 ridii arc on the shorter scale, and one radios of twice the length on the longer. 



shorter scale comes where the second now is ; then look under the 

 second 2 of the shorter scale, we have 534 ; and '534 is the true 

 answer so far as the scale will give it. We have taken the most 

 straightforward plan of reading the rule, and have not space for all the 

 details which are in works on the subject, particularly the method of 

 using the slide of numbers with a scale of numbers above and of square 

 roots below. The following is the general principle applicable to the 

 preceding case : 



It is well known that, whereas in common division the place of the 

 decimal point has nothing to do with the significant figures of the 

 quotient, yet in extracting the square, cube, &c., roots, the figures of 

 the root are altered by a change of the decimal point, unless it be 

 changed by an even number of places in extracting the square root, by 

 three or a multiple of three places in the cube root, and so on. In 

 extracting the square root, a number may either have two figures in its 

 first period, or one ; thus "07616 and '7616 must (in the rule for 

 extraction) be pointed 



076160 and 7616. 







Let us call numbers unidiyitul or bidigital, according as there are one 

 or two significant figures in the first period. Then the application to 

 the slidiug-rule is, that on the shorter scale numbers of the same name 

 must be read either on the same radius or with a whole radius inter- 

 vening, while numbers of different names must be read on different 

 radii. In the scale for the extraction of the cube root, numbers must 

 be distinguished into unidigital, bidigital, and tridigital ; and signifying 

 these by their initial letters, and taking the succession u B T u B T, &c., 

 there must be the same relation between the scales on which they are 

 read that there is between the places of their letters in the preceding 

 list. Thus, if u be read on one radius, T must either be read on that 

 immediately preceding, or on the next but one. Thus, in the preceding 

 question, which we first solved wrongly, we have 2 and 7 to consider ou 

 the shorter' scale, the pointing of which is 



2-0000 and 7-0000, 



and both are unidigital numbers. Bringing 7 on the shorter scale to 1 

 on the longer, we see that the next 2 is on a different radius ; it would 

 do then for 70, or '7, or "007, but not for 7- By the process we followed 

 we took not indeed a 2 on the same radius with our 7, but on the next 

 radius but one, and thus obtained the correct answer. These points, 

 and others (such as the meanings of tho lines of sines, tangents, &c., 

 annuities, &c., which are found on several rules) can only be mastered 

 by those who ore acquainted theoretically with logarithms, trigono- 

 metry, &c. ; for after all the sliding-rule will not teach the method of 

 working any question, but will only afford aid in computation in 

 common multiplication and division, to any one ; in higher rules, to 

 those who understand their principles. Oughtred, the inventor, kept 

 the instrument by him many years, out of a settled contempt for those 

 who would apply it without knowledge, having " onely the superficial! 

 scuinnie and froth of instrumentall trickes and practises ; " and wishing 

 to encourage "the way of rationall scientiallists, not of ground-creeping 

 Methodicks." A little distinction between that portion of its use 

 which is generally attainable, and that which requires mathematical 

 knowledge, would have been more reasonable. 



On the carpenter's and engineer's sliding-rule are engraved a number 

 of numerals in columns with headings, of which the following is a 

 specimen : 



SQUARX. CYLlM'ilc. GLOBE. 



FFF F1I III 

 0022 -323 3-878 



FI II 

 411 4-933 



IF I 

 0013 7-106 



These divisors (called gauge-points) are intended to convert into 

 xmnds the weight of a rectangular prism, cylinder, or globe, of cast- 

 ron ; the first, on three suppositions, namely, all dimensions in feet, 

 one in feet and two in inches, and all in inches ; thu second, on the 

 suppositions that the length is in feet and the diameter iu inches, 

 and that both are in inches ; the third, on the supposition that the 

 diameter is either in feet or in inches. We shall here content ourselves 

 with verifying one of these, say the first of those marked " cylinder," 

 vhich will show the nature of the divisor. 



The specific gravity of cast-iron is 7'207, and the content of a cylinder 

 of D iiic/iet diameter and L feel of length is -7854 x D"L divided by 141, 

 n cubic feet. A cubic foot of water weighs 62-321 pounds avoirdupois : 

 one of cast-iron, therefore, weighs 62'321 x 7'207 ; whence the weight 

 if the cylinder is, in pounds, 



62-321 x 7-207 x 7854 D 3 L + 144, or 



where o = 



144 



x 7.307 x 



or "4082 ; 



iear enough, to '411 to illustrate our object, but showing that the 

 omputer of this divisor used a specific gravity slightly differing from 

 he above. The rule in all the cases is to multiply the three dimensions 

 ogether, diameters or lengths, and to divide by the divisor given in 

 ,he table. The term gauge-point, which properly belongs to the part 



