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NA TURE 



\Aitgust 2 1, 1 1 



well as working drawings for machinery and plans for 

 building, all depend for their usefulness on the repre- 

 sentation of magnitudes by lines of proportional length. 

 In the widest application of plain scales we may say that 

 the relations between material things are represented by 

 relations between magnitudes in space, and they have in 

 this way been of the utmost service in scientific discovery, 

 presenting to the eye the general nature of the relation 

 between two associated quantities, and suggesting to the 

 mind the probable law of their connection in cases when 

 the law is unknown. 



A convenient form in which to use ordinary scales is 

 to have a foot-rule divided into inches, and into the half, 

 quarter, and eighth of an inch, like plotting scales, and 

 then subdivided on the one side decimally and on the other 

 side duodecimally, the edges of the rule being bevelled 

 off so as to enable distances to be immediately pricked 

 off from the scale on to the drawing. Frequently scales 

 are required different from those which are usually made, 

 and it is then necessary to make a scale of the required 

 size on the drawing. This is also required when measure- 

 ment has to be made on the drawing itself. The scale 

 must then be put on the paper at the same time that the 

 drawing is made, so that if the paper should alter its 

 dimensions, the scale will alter in the same proportion. 

 A valuable adjunct to a scale of this kind is a vernier 

 scale, which enables us to take off small distances 

 with far more accuracy than the ordinary diagonal 

 scales. 



Of the scales in the construction of which numbers 

 found by calculation are used, the commonest are those 

 found on the ordinary sector, which contains a scale of 

 chords by means of which an angle may be more accu- 

 rately set out than by the ordinary plain scale protractor. 

 By it we are enabled also to set oft" lines proportional to 

 the trigonometrical functions, to solve all questions in 

 proportion, to reduce or enlarge drawings in a required 

 ratio, to describe a polygon of a given number of sides, 

 and to perform calculations by means of the logarithmic 

 line. This last is a line numbered from I to ioo, the 

 distance from I to any number being made proportional 

 to the logarithm of that number. Thus, since the loga- 

 rithm of 10 is I and of ioo is 2, the scale consists of two 

 parts, the part from 10 to ioo being a repetition of that 

 from I to 10, since the logarithm of a number between 10 

 and ioo, say 40, is equal to the logarithm of 10 added to 

 the logarithm of the same number divided by 10, such as 

 4. Thus by the compasses alone we are able to perform, 

 with a certain degree of accuracy, the operations of multi- 

 plication, division, finding a third or fourth proportional, 

 and evolution and involution. For instance, to multiply 

 35 by 27 we should first multiply 3'5 by 27, or of 35 by 27, 

 in order that the product might be less than 100, and 

 afterwards multiply the result by 10. Taking in the com- 

 passes the distance on the scale from 1 to 3'5 we should 

 set that interval beyond the 27 on the scale. We should 

 then find the leg of the compass furthest from the be- 

 ginning of the scale pointing to 94'5, so that the product 

 required would be 945. A similar process obviously 

 enables us to perform division. There is, however, some 

 inconvenience in using the compasses, and this may be 

 avoided by the use of the slide-rule. This rule consists of 

 two parts, one fixed, which we shall call A, the other 

 sliding, which we shall call B ; on each of these parts a 

 logarithmic line of numbers is placed. Hence by the 

 sliding of the rule we can perform the same operations 

 which would otherwise require the use of compasses. For 

 instance, to divide xhyy, place the number on B denoting 

 y against the number on A denoting x, then the number 

 on A which is opposite to the beginning of B will give 

 the quotient required. Similarly the square root of a 

 number may be extracted by so sliding B that the number 

 on A opposite to I on B may be the same as the number 

 on B which is opposite the number on A, the square root 



of which is required. The rule may be arranged in other 

 ways so as to give at once the squares of numbers, the 

 lengths of the spaces being made proportional to the 

 logarithms of the squares of the numbers indicated. 

 This is used, for instance, in finding the content of 

 timber. 



A slide-rule which has lately been devised by Major 

 General Hannyngton, whilst remaining very compact in 

 size, is capable of much greater accuracy. Here instead 

 of one very long rule, the rule is divided into a number of 

 parts which are placed under each other, each part being 

 a continuation of the part above it ; the slide also consists 

 of a number of parts arranged under each other, consisting 

 of a set of bars with spaces between, which are united at 

 the extremities by cross pieces. The bars on the slide 

 fit into grooves in the fixed part or stock, and are so 

 arranged that the numbers on the stock and on the slide 

 both begin together, although the former is longer than 

 the latter, in order that in every position the slide may 

 have a part of the stock opposite to it. The use of this 

 " extended slide-rule " is the same as that of the ordinary 

 rule, but in the case of the largest which is made, it is as 

 exact as a rule ten feet long, whilst it is compactly arranged, 

 so as to be only one foot long. By this rule all the opera- 

 tions performed by the ordinary slide-rule may be effected, 

 but with much greater accuracy. On account of this it 

 would seem as if this rule ought to become very popular 

 when its merits become known. 



The graduating of a scale so that the distances from 

 the end of it may be proportional to the logarithms of the 

 numbers which are marked on it, which is the principle of 

 the slide-rule, is evidently capable of a greatly extended 

 application, and different scales may be devised intended 

 for different purposes. Thus a set of three scales has 

 been devised by Mr. Lala Ganga Ram, intended for the 

 use of engineers, architects, and builders. The first of 

 these is intended to show at a glance the scantlings of 

 timber in beams and joists, and to obtain the stresses in 

 trusses. The principle employed is correct, and the 

 results obtained are very approximate. The depth and 

 breadth of a beam sufficient either for strength or stiff- 

 ness, can be found by the same rule. It has on the 

 reverse side a scale, by means of which the stresses in 

 the principal rafter and the beam of a king post truss 

 may be found, and then the same quantities may be de- 

 termined for trusses of different form by multiplying by 

 a certain coefficient marked on the edge of the scale. 

 This gives without any difficulty the maximum stresses 

 coming on the principal rafter and tie-beam, and is all 

 that is usually required, since the scantlings or minor com- 

 ponents of a truss are generally determined from practical 

 rather than from theoretical considerations. The second 

 scale is designed to give the thickness of retaining walls. 

 By means of information contained on the back of it the 

 thickness may be found for various forms of wall and 

 kinds of loading. Here again the method of using 

 the scales could not be simpler, and the results are 

 such as agree with calculation. The third of the set 

 enables us to find the stresses (or, as they are called by 

 the inventor, in accordance with ancient custom, strains) 

 on girders. When we state that this scale enables us to 

 ascertain the stress on the flanges at any point of a beam 

 up to 200 feet span, and also the shearing stress at any 

 point of beams with different systems of bracing under 

 both uniform and travelling loads, and that this is effected 

 by merely sliding the scale, it is evident that we have here 

 a means of obtaining at sight results which would other- 

 wise require a considerable amount of calculation. The 

 results are such as, for all practical purposes, seem to be 

 abundantly accurate. 



The principle of the slide-rule is thus one which is 

 susceptible of almost indefinite application. It may be 

 used in all cases when the results we wish to attain depend 

 on calculations for which logarithms are ordinarily used 



