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diff --git a/75904-0.txt b/75904-0.txt new file mode 100644 index 0000000..59bb40f --- /dev/null +++ b/75904-0.txt @@ -0,0 +1,6495 @@ + +*** START OF THE PROJECT GUTENBERG EBOOK 75904 *** + + + + + + THE + SLIDE RULE: + A PRACTICAL MANUAL + + + BY + + CHARLES N. PICKWORTH + + WHITWORTH SCHOLAR; EDITOR OF “THE MECHANICAL WORLD”; AUTHOR OF + “LOGARITHMS FOR BEGINNERS”; “THE INDICATOR: ITS CONSTRUCTION AND + APPLICATION”; “THE INDICATOR DIAGRAM: ITS ANALYSIS AND CALCULATION,” + ETC. + + _SEVENTEENTH EDITION_ + + MANCHESTER: + EMMOTT AND CO., LIMITED, + 65 KING STREET; + + NEW YORK: + D. VAN NOSTRAND CO., + 8 WARREN STREET. + + LONDON: + EMMOTT AND CO., LIMITED, + 20 BEDFORD STREET, W.C. + + AND + PITMAN AND SONS, LIMITED, + PARKER ST., KINGSWAY, W.C. 2. + + [_Three Shillings and Sixpence net_] + + + + + _All rights reserved._ + + + + + PREFACE TO THE FIFTEENTH EDITION. + + +Several new slide rules for special calculations are described in this +edition, and the contents further extended to include a section dealing +with screw-cutting gear calculations by the slide rule—an application of +the instrument to which attention has been given recently. + +Mention should be made of the fact that some of the special slide rules +described in previous editions are no longer obtainable. As, however, +the descriptive notes may be of service to those possessing the +instruments, and are, in some measure, of general interest, they have +been allowed to remain in the present issue. + +The author tenders his thanks to the many who have evinced their +appreciation of his efforts to popularise the subject; also for the many +kind hints and suggestions which he has received from time to time, and +with a continuance of which he trusts to be favoured in the future. + + C. N. P. + +WITHINGTON, MANCHESTER, _November 1917_. + + + PREFACE TO THE SEVENTEENTH EDITION. + +The sustained demand for this very successful work having resulted in +the early call for a new edition, the opportunity has been taken to +introduce descriptions of new slide rules and to effect some slight +revisions. + + C. N. P. + +WITHINGTON, MANCHESTER, _December 1920_. + + + + + CONTENTS. + + + PAGE + Introductory 5 + The Mathematical Principle of the Slide Rule 6 + Notation by Powers of 10 8 + The Mechanical Principle of the Slide Rule 9 + The Primitive Slide Rule 10 + The Modern Slide Rule 12 + The Notation of the Slide Rule 14 + The Cursor or Runner 17 + Multiplication 19 + Division 24 + The Use of the Upper Scales for Multiplication and Division 26 + Reciprocals 27 + Continued Multiplication and Division 28 + Multiplication and Division with the Slide Inverted 30 + Proportion 31 + General Hints on the Elementary Uses of the Slide Rule 36 + Squares and Square Roots 37 + Cubes and Cube Roots 40 + Miscellaneous Powers and Roots 45 + Power and Roots by Logarithms 45 + Other Methods of Obtaining Powers and Roots 47 + Combined Operations 49 + Hints on Evaluating Expressions 52 + Gauge Points 53 + Examples in Technical Calculations 56 + Trigonometrical Application 74 + Slide Rules with Log-log Scales 84 + Special Types of Slide Rules 92 + Long-Scale Slide Rules 96 + Circular Calculators 101 + Slide Rules for Special Calculations 109 + Construction Improvements in Slide Rules 110 + The Accuracy of Slide Rule Results 111 + Appendix:— + New Slide Rules 113 + The Solution of Algebraic Equations 122 + Screw-Cutting Gear Calculations 124 + Gauge Points and Signs on Slide Rules 126 + Tables and Data 128 + Slide Rule Data Slips 133 + + + + + THE SLIDE RULE. + + + + + INTRODUCTORY. + + +The slide rule may be defined as an instrument for mechanically +effecting calculations by logarithms. Those familiar with logarithms and +their use will recognise that the slide rule provides what is in effect +a concisely arranged table of logarithms, together with a simple and +convenient means for adding and subtracting any selected values. Those, +however, who have no acquaintance with logarithms will find that only an +elementary knowledge of the subject is necessary to enable them to make +full use of the slide rule. It is true that for simple slide-rule +operations, as multiplication and division, a knowledge of logarithms is +unnecessary; indeed, many who have no conscious understanding of +logarithms make good use of the instrument. But this involves a blind +reliance upon rules without an appreciation of their origin or +limitations, and this, in turn, engenders a want of confidence in the +results of any but the simplest operations, and prevents the fullest use +being made of the instrument. For this reason a brief, but probably +sufficient _résumé_ of the principles of logarithmic calculation will be +given. Those desiring a more detailed explanation are referred to the +writer’s “Logarithms for Beginners.” + +The slide rule enables various arithmetical, algebraical and +trigonometrical processes to be performed with ease and rapidity, and +with sufficient accuracy for most practical purposes. A grasp of the +simple fundamental principles which underlie its operation, together +with a little patient practice, are all that are necessary to acquire +facility in using the instrument, and few who have become proficient in +this system of calculating would willingly revert to the laborious +arithmetical processes. + + + + + THE MATHEMATICAL PRINCIPLE OF THE SLIDE RULE. + + +Logarithms may be defined as a series of numbers in _arithmetical_ +progression, as 0, 1, 2, 3, 4, etc., which bear a definite relationship +to another series of numbers in _geometrical_ progression, as 1, 2, 4, +8, 16, etc. A more precise definition is:—The logarithm of a number to +any base, is the _index of the power_ to which the base must be raised +to equal the given number. In the logarithms in general use, known as +_common logarithms_, and with which we are alone concerned, 10 is the +base selected. The general definition may therefore be stated in the +following modified form:—_The common logarithm of a number is the index +of the power to which 10 must be raised to equal the given number._ +Applying this rule to a simple case, as 100 = 10^2, we see that the base +10 must be squared (_i.e._, raised to the 2nd power) in order to equal +100, the number selected. Therefore, as 2 is the index of the power to +which 10 must be raised to equal 100, it follows from our definition +that 2 is the common logarithm of 100. Similarly the common logarithm of +1000 will be 3, while proceeding in the opposite direction the common +log. of 10 must equal 1. Tabulating these results and extending, we +have:— + + Numbers 10,000 1000 100 10 1 + Logarithms 4 3 2 1 0 + +It will now be evident that for numbers + + between 1 and 10 the logs. will be between 0 and 1 + „ 10 „ 100 „ „ 1 „ 2 + „ 100 „ 1000 „ „ 2 „ 3 + „ 1000 „ 10,000 „ „ 3 „ 4 + +In other words, the logarithms of numbers between 1 and 10 will be +wholly fractional (_i.e._, decimal); the logs. of numbers between 10 and +100 will be 1 _followed by a decimal quantity_; the logs. of numbers +between 100 and 1000 will be 2 followed by a decimal quantity, and so +on. These decimal quantities for numbers from 1 to 10 (which are the +logarithms of this particular series) are as follows:— + + Numbers 1 2 3 4 5 6 7 8 9 10 + Logarithms 0 0·301 0·477 0·602 0·699 0·778 0·845 0·903 0·954 1·000 + +Combining the two tables, we can complete the logarithms. Thus for 3 +multiplied successively by 10, we have:— + + Numbers 3 30 300 3000 30,000 etc. + Logarithms 0·477 1·477 2·477 3·477 4·477 „ + +We see from this that for numbers having the _same significant figure_ +(or figures), 3 in this case, the decimal part or _mantissa_ of the +logarithm is the same, but that the integral part or _characteristic_ is +always _one less than the number of figures before the decimal point_. + +For numbers less than 1 the same plan is followed. Thus extending our +first table downwards, we have:— + + Numbers 1 0·1 0·01 0·001 0·0001 etc. + Logarithms 0 −1 −2 −3 −4 „ + +so that for 3 divided successively by 10, we have:— + + Numbers 3 0·3 0·03 0·003 0·0003 etc. + Logarithms 0·477 ̅1·477 ̅2·477 ̅3·477 ̅4·477 „ + +Here again we see that with the same significant figures in the numbers, +the mantissa of the logarithm has always the same (_positive_) value, +but the characteristic is _one more_ than the _number of 0’s immediately +following the decimal point_, and is _negative_, as indicated by the +minus sign written over it. Only the decimal parts of the logarithms of +numbers between 1 and 10 are given in the usual tables, for, as shown +above, the logarithms of all tenfold multiples or submultiples of a +number can be obtained at once by modifying the characteristic in +accordance with the rules given. + +An examination of the two rows of figures giving the logarithms of +numbers from 1 to 10 will reveal some striking peculiarities, and at the +same time serve to illustrate the principle of logarithmic calculation. +First, it will be noticed that the addition of any two of the logarithms +gives the logarithm of the _product_ of these two numbers. Thus, the +addition of log. 2 and log. 4 = 0·301 + 0·602 = 0·903, and this is seen +to be the logarithm of 8, that is, of 2 × 4. Conversely, the difference +of the logarithms of two numbers gives the logarithm of the _quotient_ +resulting from the division of these two numbers. Thus, log. 8 − log. 2 += 0·903 − 0·301 = 0·602, which is the log. of 4, or of 8 ÷ 2. + +One other important point is to be noted. If the logarithm of any number +is _multiplied_ by 2, 3, or any other quantity, whole or fractional, the +result is the logarithm of the original number, raised to the 2nd, 3rd, +or other power respectively. Thus, multiplying the log. of 3 by 2, we +obtain 0·477 × 2 = 0·954, and this is seen to be the log. of 9, that is, +of 3 raised to the 2nd power, or 3 _squared_. Again, log. 2 multiplied +by 3 = 0·903—that is, the log. of 8, or of 2 raised to the 3rd power, or +2 _cubed_. Conversely, dividing the logarithm of any original number by +any number _n_, we obtain the logarithm of the _n_th root of the +original number. Thus, log. 8 ÷ 3 = 0·903 ÷ 3 = 0·301, and is therefore +equal to log. 2 or to the log. of the _cube root_ of 8. + +Only simple logs. have been taken in these examples, but the student +will understand that the same reasoning applies, whatever the number. +Thus for 20^3 we prefix the characteristic (1 in this case) to log. 2, +giving 1·301. Multiplying by 3, we have 3·903 as the resulting +logarithm, and as its characteristic is 3, we know that it corresponds +to the number 8000. Hence 20^3 = 8000. + +In this brief explanation is included all that need now be said with +regard to the properties of logarithms. The main facts to be borne +clearly in mind are:—(1.) That to find the _product_ of two numbers, the +logarithms of the numbers are to be _added_ together, the result being +the logarithm of the product required, the value of which can then be +determined. (2.) That in finding the _quotient_ resulting from the +division of one number by another, _the difference_ of the logarithms of +the numbers gives the logarithm of the quotient, from which the value of +the latter can be ascertained. (3.) That to find the result of _raising +a number to the nth power_, we _multiply_ the logarithm of the number by +_n_, thus obtaining the logarithm, and hence the value, of the desired +result. And (4.) That to find the n_th root of a number_, we _divide_ +the logarithm of the number by _n_, this giving the logarithm of the +result, from which its value may be determined. + + + + + NOTATION BY POWERS OF 10. + + +A convenient method of representing an arithmetical quantity is to split +it up into two factors, of which the first is the original number, with +the decimal point moved so as to immediately follow the first +significant figure, and the second, 10^{_n_} where _n_ is the number of +places the decimal point has been moved, this index being _positive_ for +numbers greater than 1, and _negative_ for numbers less than 1.[1] In +this system, therefore, we regard 3,610,000 as 3·61 × 1,000,000, and +write it as 3·61 × 10^6. Similarly 361 = 3·61 x 10^2; 0·0361 (= +(3·61)/(100)) = 3·61 × 10^{−2}; 0·0000361 = 3·61 × 10^{−5}, etc. To +restore a number to its original form, we have only to move the decimal +point through the number of places indicated by the index, moving to the +right if the index is positive and to the left (prefixing 0’s) if +negative. This method, which should be cultivated for ordinary +arithmetical work, is substantially that followed in calculating by the +slide rule. Thus with the slide rule the multiplication of 63,200 by +0·0035 virtually resolves itself into 6·32 × 10^4 × 3·5 × 10^{−3} or +6·32 × 3·5 × 10^{4–3} = 22·12 x 10^1 = 221·2. It will be seen later, +however, that the result can be arrived at by a more direct, if less +systematic, method of working. + + + + + THE MECHANICAL PRINCIPLE OF THE SLIDE RULE. + + +[Illustration: FIG. 1.] + +The mechanical principle involved in the slide rule is of a very simple +character. In Fig. 1, A and B represent two rules divided into 10 equal +parts, the division lines being numbered consecutively as shown. If the +rule B is moved to the right until 0 on B is opposite 3 on A, it is seen +that any number on A is equal to the coinciding number on B, plus 3. +Thus opposite 4 on B is 7 on A. The reason is obvious. By moving B to +the right, we add to a length 0·3, another length 0·4, the result read +off on A being 7. Evidently, the same result would have been obtained if +a length 0·4 had been added, by means of a pair of dividers, to the +length 0·3 on the scale A. By means of the slide B, however, the +addition is more readily effected, and, what is of much greater +importance, the result of adding 3 to _any one of the numbers_ within +range, on the lower scale, is _immediately_ seen by reading the adjacent +number on A. + +Of course, subtraction can be quite as readily performed. Thus, to +subtract 4 from 7, we require to deduct from 0·7 on the A scale, a +length 0·4 on B. We do this by placing 4 on B under 7 on A, when over 0 +on B we find 3, on A. It is here evident that the _difference_ of any +pair of coinciding numbers on the scales is constantly equal to 3. + +[Illustration: FIG. 2.] + +An important modification results if the slide-scale B is inverted as in +Fig. 2. In this case, to find the sum of 4 and 3 we require to place the +4 of the A scale to 3 on the B scale, and the result is read on A over 0 +on B. Here it will be noted, the _sum_ of any pair of coinciding numbers +on the scales is constant and equal to 7. This case, therefore, +resembles that of the immediately preceding one, except that the _sum_, +instead of the _difference_, of any pair of coinciding numbers is +constant. + +To find the difference of two factors, the converse operation is +necessary. Thus, to subtract 4 from 7, 0 on B is placed opposite 7 on A, +and over 4 on B is found 3 on A. + +From these examples it will be seen that with the slide _inverted_ the +methods of operation are the reverse of those used when the slide is in +its normal position. + +It will be understood that although we have only considered the primary +divisions of the scales, the remarks apply equally to any subdivisions +into which the primary spaces of the scales might be divided. Further, +we note that the length of scale taken to represent a unit is quite +arbitrary. + + + + + THE PRIMITIVE SLIDE RULE. + + +The application of the foregoing principles to the slide rule can be +shown most conveniently by describing the construction of a simple form +of slide rule:—Take a strip of card about 11 in. long and 2 in. wide; +draw a line down the centre of its width, and mark off two points, 10 +in. apart. Draw cross lines at these points and figure them 1 and 10 on +each side, as in Fig. 3. Next mark off lengths of 3·01, 4·77, 6·02, +6·99, 7·78, 8·45, 9·03 and 9·54 inches, from the line marked 1. Draw +cross lines as before, and figure these lines, 2, 3, 4, 5, 6, 7, 8 and +9. To fill in the intermediate divisions of the scale, take the logs, of +1·1, 1·2, 1·3, etc. (from a table), multiply each by 10, and thus obtain +the distances from 1, at which the several subdivisions are to be +placed. Mark these 1·2, 1·3, 1·4, etc., and complete the scale, making +the interpolated division marks shorter to facilitate reading, as with +an ordinary measuring rule. Cutting the card cleanly down the centre +line, we have the essentials of the slide rule. + +[Illustration: FIG. 3.] + +The fundamental principle of the slide rule is now evident:—Each scale +is graduated in such a manner that the _distance of any number from 1 is +proportional to the logarithm of that number_. + +[Illustration: FIG. 4.] + +“We know that to find the product of 2 × 3 by logarithms, we add 0·301, +or log. 2, to 0·477, the log. of 3, obtaining 0·778, or log. 6. With our +primitive slide rule we place 1 on the lower scale to 3·01 in. (which we +have marked 2) on the upper scale (Fig. 4). Then over 4·77 in. on the +lower scale (which we marked 3), we have 7·78 in. (which we marked 6) on +the upper scale. Conversely, to divide 6 by 3, we place 3 on the lower +scale in agreement with 6 on the upper, and over 1 on the lower scale +read 2 on the upper scale. This method of adding and subtracting scale +lengths will be seen to be identical with that used in the simple case +shown in Fig. 1. + + + + + THE MODERN SLIDE RULE. + + +The modern form of slide rule, variously styled the Gravêt, the +Tavernier-Gravêt, and the Mannheim rule, is frequently made of boxwood, +but all the leading instrument makers now supply rules made of boxwood +or mahogany, and faced with celluloid, the white surface of which brings +out the graduations much more distinctly than lines engraved on a +boxwood surface. The celluloid facings should not be polished, as a dull +surface is much less fatiguing to the eyes. The most generally used, and +on the whole the most convenient size of rule, is about 10½in. long, +1¼in. wide, and about ⅜in. thick; but 5 in., 8 in., 15 in., 20 in., 24 +in. and 40 in. rules are also made. In the centre of the stock of the +rule a movable slip is fitted, which constitutes the slide, and +corresponds to the lower of the two rules of our rudimentary examples. + +[Illustration: FIG. 5.] + +From Fig. 5, which is a representation of the face of a Gravêt or +Mannheim slide rule, it will be seen that four series of logarithmic +graduations or scale-lines are employed, the upper and lower being +engraved on the stock or body of the rule, while the other two are +engraved upon the slide. The two upper sets of graduations are exactly +alike in every particular, and the lower sets are also similar. It is +usual to identify the two upper scale-lines by the letters A and B, and +the two lower by the letters C and D, as indicated in the figure at the +left-hand extremities of the scales. + +Referring to the scales C and D, these will each be seen to be a +development of the elementary scales of Fig. 3, but in this case each +principal space is subdivided, more or less minutely. The principle, +however, is exactly the same, so that by moving the slide (carrying +scale C), multiplication and division can be mechanically performed in +the manner described. + +The upper scale-line A consists of two exactly similar scales, placed +end to end, the first lying between IL and IC, and the second between IC +and IR. The first of these scales will be designated the _left-hand A +scale_, and the second the _right-hand A scale_. Similarly the +coinciding scales on the slide are the _left-hand B scale_ and the +_right-hand B scale_. Each of these four scales is divided (as finely as +convenient) as in the case of the C and D scales, but, of course, they +are exactly one half the length of the latter. + +The two end graduations of both the C and D scales are known as the +_left-_ and _right-hand indices_ of these scales. Sometimes they are +figured 1 and 10 respectively; sometimes both are marked 1. Similarly IL +and IR are the left- and right-hand indices of the A and B lines, while +IC is the centre index of these scales. Other division lines usually +found on the face of the rule are one on the left-hand A and B scales, +indicating the ratio of the circumference of a circle to its diameter, π += 3·1416; and a line on the right-hand B scale marking the position of +(π)/(4) = 0·7854, used in calculating the areas of circles. Reference +will be made hereafter to the scales on the under-side of the slide, and +we need now only add that one of the edges of the rule, usually +bevelled, is generally graduated in millimetres, while the other edge +has engraved on it a scale of inches divided into eighths or tenths. On +the bottom face inside the groove of the rule either one or the other of +these scales is continued in such a manner that by drawing the slide out +to the right and using the scale inside the rule, in conjunction with +the corresponding scale on the edge, it is possible to measure 20 inches +in the one case, or nearly 500 millimetres in the other. On the back of +the rule there is usually a collection of data, for which the slips +given at the end of this work may often be substituted with advantage. + + + + + THE NOTATION OF THE SLIDE RULE. + + +Hitherto our attention has been confined to a consideration of the +primary divisions of the scales. The same principle of graduation is, +however, used throughout; and after what has been said, this part of the +subject need not be further enlarged upon. Some explanation of the +method of reading the scales is necessary, as facility in using the +instrument depends in a very great measure upon the dexterity of the +operator in assigning the correct value to each division on the rule. By +reference to Fig. 5, it will be seen that each of the primary spacings +in the several scales is invariably subdivided into ten; but since the +lengths of the successive primary divisions rapidly diminish, it is +impossible to subdivide each main space into the same number of parts +that the space 1–2 can be subdivided. This variable spacing of the +scales is at first confusing to the student, but with a little practice +the difficulty is soon overcome. + +With the C or D scale, it will be noticed that the length of the +interval 1–2 is sufficient to allow each of the 10 subdivisions to be +again divided into 10 parts, so that the whole interval 1–2 is divided +into 100. The shorter main space 2–3, and the still shorter one 3–4, +only allow of the 10 subdivisions of each being divided into five parts. +Each of these main spaces is therefore divided into 50 parts. For the +remainder of the scale each of the 10 subdivisions of each main space is +divided into two parts only; so that from the main division 4 to the end +of the scale the primary spaces are divided into 20 parts only. + +In the upper scales A or B, it will be found that—as the space 1–2 is of +only half the length of the corresponding space on C or D—the 10 +subdivisions of this interval are divided into five parts only. +Similarly each of the 10 subdivisions of the intervals 2–3, 3–4, and 4–5 +are further divided into two parts only, while for the remainder of the +scale only the 10 subdivisions are possible, owing to the rapidly +diminishing lengths of the primary spacings. + +The values actually given on the rule run from 1 to 10 on the lower +scales and from 1 to 100 on the upper scales, and, as explained on page +9, all factors are brought within these ranges of values by multiplying +or dividing them by powers of 10. By following this plan, we virtually +regard each factor as merely a series of significant figures, and make +the necessary modification due to the “powers of 10” when fixing the +position of the decimal point in the answer. + +Many, however, find it convenient in practice to regard the values on +the rule as multiplied or divided by such powers of 10 as may be +necessary to suit the factors entering into the calculation. If this +plan is adopted, the values given to each graduation of the scales will +depend on that given to the left index figure (1) of the lower scales, +this being any multiple or submultiple of 10. Thus IL on the D scale may +be regarded as 1, 10, 100, 1000, etc., or as 0·1, 0·01, 0·001, 0·0001, +etc.; but once the initial value is assigned to the index, the ratio of +value must be maintained throughout the whole scale. For example, if 1 +on C is taken to represent 10, the main divisions 2, 3, 4, etc., will be +read as 20, 30, 40, etc. On the other hand, if the fourth main division +is read as 0·004, then the left index figure of the scale will be read +as 0·001. The figured subdivisions of the main space 1–2 are to be read +as 11, 12, 13, 14, 15, 16, 17, 18 and 19—if the index represents 10,—and +as corresponding multiples for any other value of the index. + +Independently considered, these remarks apply equally to the A or B +scale, but in this case the notation is continued through the second +half of the scale, the figures of which are to be read as tenfold values +of the corresponding figures in the first half of the scale. + +The reading of the intermediate divisions will, of course, be determined +by the values assigned to the main divisions. Thus, if IL on D is read +as 1, then each of the smallest subdivisions of the space 1–2 will be +read as 0·01, and each of the smallest subdivisions of the spaces 2–3 or +3–4 as 0·02, while for the remainder of the scale the smallest +subdivisions are read as 0·05. In the A or B scale the subdivisions of +the space 1–2 of the first half of the scale are (if IL = 1) read as +0·02, 0·04, etc.; for the divisions 2–3, 3–4, and 4–5, the smallest +intervals are read as 0·05 of the primary spaces, and from 5 to the +centre index of the scale the divisions represent 0·1 of each main +interval. Passing the centre index, which is, now read as 10, the +smallest subdivisions immediately following are read 10·2, 10·4, etc., +until 20·0 is reached; then we read 20·5, 21·0, 21·5 22·0, etc., until +the figured main division 5 is reached. The remainder of the scale is +read 51, 52, 53, etc., up to 100, the right-hand index. + +Further subdivision of any of the spaces of the rule can be effected by +the eye, and after a little practice the operator will become quite +expert in estimating any intermediate value. It affords good practice to +set 1 on C to 1·04, 1·09, etc. on D, and to read the values on D, under +4, 6, 8, etc. on C. As the exact results are easily calculated mentally, +the student, by this means, will receive better instruction in +estimating intermediate results than can be given by any diagram. + +Some rules will be found figured as shown in Fig. 5; in others, the +right-hand upper scales are marked 10, 20, 30, etc. Again, others are +marked decimally, the lower scales and the left-hand upper scales being +figured 1, 1·1, 1·2, 1·3 ... 2·5, etc. The latter form has advantages +from the point of view of the beginner. + +The method of reading the A and B scales, just given, applies only when +these scales are regarded as altogether independent of the lower pair of +scales C and D. Some operators prefer to use the A and B scales, and +some the C and D scales, for the ordinary operations of proportion, +multiplication, and division. Each method has its advantages, as will be +shown, but in the more complex calculations, as involution and +evolution, etc., the relation of the upper scales to the lower scales +becomes a very important factor. + +The distance 1–10 on the upper scales is one-half of the distance 1–10 +on the lower scales. Hence any distance from 1, taken on the upper +scales, represents _twice the logarithm_ which the same distance +represents on the lower scales. In other words, the length which +represents log. N on D, would represent 2 log. N on A; and, conversely, +the length which represents log. N on A, would represent (log. N)/(2) on +D. + +Now we have seen (page 8) that multiplying the log. of a number by 2 +gives the log. of the square of the number. Hence, above any number on D +we find its _square_ on A, or, conversely, below any number on A, we +find its _square root_ on D. Thus, above 2 we find 4; under 49, we find +7 and so on. Obviously the same relation exists between the B and C +scales. + + + + + THE CURSOR OR RUNNER. + + +All modern slide rules are now fitted with a _cursor_ or _runner_, which +usually consists of a light metal frame moving under spring control in +grooves in the edges of the stock of the rule. This frame carries a +piece of glass, mica or transparent celluloid, about 1 in. square, +across the centre of which a fine reference line is drawn exactly at +right angles to the line of scales. To “set the cursor” to any value on +the scales of the rule, the frame is taken between the thumb and +forefinger and adjusted in position until the line falls exactly upon +the graduation, or upon an estimated value, between a pair of +graduations, as the case may be. Having fixed one number in this way, +another value on either of the scales on the slide may be similarly +adjusted in reference to the cursor line. The cursor will be found very +convenient in making such settings, especially when either or both of +the numbers are located by eye estimation. It also finds a very +important use in referring the readings of the upper scale to those of +the lower, or _vice versa_, while as an aid in continued multiplication +and division and complex calculations generally, its value is +inestimable. + +_Multiple Line Cursors._—Cursors can be obtained with _two_ lines, the +distance between them being that between 7·854 and 10 on the A scale. +The use of this cursor is explained on page 57. Another multiple line +cursor has short lines engraved on it, corresponding to the main +graduations from 95 to 105 on the respective scales. This is useful for +adding or deducting small percentages. + +_The Broken Line Cursor._—To facilitate setting, broken line cursors are +made, in which the hair-line is not continued across the scales, but has +two gaps, as shown in Fig. 6. + +_The Pointed Cursor_ has an index or pointer, extending over the +bevelled edge of the rule, on which is a scale of inches. It is useful +for summing the lengths of the ordinates of indicator diagrams, and also +for plotting lengths representing the logarithms of numbers, sometimes +required in graphic calculations. + +_The Goulding Cursor._—It has been pointed out that in order to obtain +the third or fourth figure of a reading on the 10 in. slide rule, it is +frequently necessary to depend upon the operator’s ability to mentally +subdivide the space within which the reading falls. This subdivision can +be mechanically effected by the aid of the Goulding Cursor (Fig. 7), +which consists of a frame fitting into the usual grooves in the rule, +and carrying a metal plate faced with celluloid, upon which is engraved +a triangular scale A B C. The portion carrying the chisel edges E is not +fixed to the cursor proper, but slides on the latter, so that the index +marks on the projecting prongs can be moved slightly along the scales of +the rule, this movement being effected by the short end of the bent +lever F working in the slot as shown. D is a pointer which can be moved +along F under spring control. As illustrating the method of use, we will +assume that 1 on C is placed to 155 on D, and that we require to read +the value on D under 27 on C. This is seen to lie between 4150 and 4200, +so setting the pointer D to the line B C—always the first operation—we +move the whole along the rule until the index line on the lower prong +agrees with 4200. We then move F across the scale until the index line +agrees with 4100, set the pointer D to the line A C, and move the lever +back until the index line agrees with 27 on the slide. It will then be +found that the pointer D gives 85 on A B as the value of the +supplementary figures, and hence the complete reading is 4185. + +[Illustration: FIG. 6.] + +[Illustration: FIG. 8.] + +[Illustration: FIG. 7.] + +[Illustration: FIG. 9.] + +_Magnifying Cursors_ are of assistance in reading the scales, and in a +good and direct light are very helpful. In one form an ordinary lens is +carried by two light arms hinged to the upper and lower edges of the +cursor, so that it can be folded down to the face of the rule when not +in use. A more compact form, shown in Fig. 8, consists of a strip of +plano-convex glass, on the under-side of which is the hair-line. In a +cursor made by Nestler of Lahr, the plano-convex strip is fixed on the +ordinary cursor. The magnifying power is about 2, so that a 5 in. rule, +having the same number of graduations as a 10 in. rule, can be read with +equal facility, by the aid of this cursor. + +The Digit-registering Cursor, supplied by Mr. A. W. Faber, London, and +shown in Fig. 9, has a semicircular scale running from 0 at the centre +upward to −6 and downward to +6. A small finger enables the operator to +register the number of digits to be added or subtracted at the end of a +lengthy operation, as explained at page 28. + + + + + MULTIPLICATION. + + +In the preliminary notes it was shown that by mechanically adding two +lengths representing the logarithms of two numbers, we can obtain the +_product_ of these numbers; while by subtracting one log. length from +another, the number represented by the latter is divided by the number +represented by the former. Hence, using the C and D scales, we have the + +RULE FOR MULTIPLICATION.—_Set the index of the C scale to one of the +factors on D, and under the other factor on C, find the product on D._ + +[Illustration: FIG. 10.] + +Thus, to find the product of 2 × 4, the slide is moved to the right +until the left index (1) of C is brought over 2 on D, when under the +other factor (4) on C, is found the required product (8) on D. Following +along the slide, to the right, we find that beyond 5 on C (giving 10 on +D), we have no scale below the projecting slide (Fig. 10). If we imagine +the D scale prolonged to the right, we should have a repetition of the +earlier portion, but, as with the two parts of the A scales, the +repeated portion would be of tenfold value, and 10 on C would agree with +20 on the prolonged D scale. We turn this fact to account by moving the +slide to the left until 10 on C agrees with 2 on D, and we can then read +off such results as 2 × 6 = 12; 2 × 8 = 16, etc., remembering that as +the scale is now of tenfold value, there will be two figures in the +result. Hence, for those who prefer rules, we have the + +RULE FOR THE NUMBER OF DIGITS IN A PRODUCT.—_If the product is read with +the slide projecting to the_ LEFT, ADD THE NUMBER OF THE DIGITS IN THE +TWO FACTORS; _if read with the slide to the_ RIGHT, _deduct 1 from this +sum_. + + + EX.—25 × 70 = 1750. + + The product is found with the slide projecting to the _left_, so the + number of digits in the product = 2 + 2 = 4. + + EX.—3·6 × 25 = 90. + + The slide projects to the _right_, and the number of digits in the + product is therefore 1 + 2 − 1 = 2. + + EX.—0·025 × 0·7 = 0·0175. + + The product is obtained with the slide projecting to the _left_, and + the number of digits is therefore −1 + 0 = −1. + + EX.—0·000184 × 0·005 = 0·00000092. + + The sum of the number of digits in the two factors = −3 + (−2) = −5, + but as the slide projects to the _right_, the number of digits will be + −5 − 1 = −6. + + +From the last two examples it will be seen that when the first +significant figure of a decimal factor does not immediately follow the +decimal point, the minus sign is to be prefixed to the number of digits, +counting as many digits _minus_ as there are 0’s following the decimal +point. Thus, 0·03 has −1 digit, 0·0035 has −2 digits, and so on. Some +little care is necessary to ensure these minus values being correctly +taken into account in determining the number of digits in the answer. +For this reason many prefer to treat decimal factors as whole numbers, +and to locate the decimal point according to the usual rules for the +multiplication of decimals. Thus, in the last example we take 184 × 5 = +920, but as by the usual rule the product must contain 6 + 3 = 9 decimal +places, we prefix six cyphers, obtaining 0·00000092. When both factors +consist of integers as well as decimals, the number of digits in the +product, and therefore the position of the decimal point, will be +determined by the usual rule for whole numbers. + +Another method of determining the number of digits in a product deserves +mention, which, not being dependent upon the position of the slide, is +applicable to all calculating instruments. + +GENERAL RULE FOR NUMBER OF DIGITS IN A PRODUCT.—_When the first +significant figure in the product is smaller than in_ EITHER _of the +factors, the number of digits in the product is equal to the_ SUM _of +the digits in the two factors. When the contrary is the case, the number +of digits is 1_ LESS _than the sum of the digits in the two factors. +When the first figures are the same, those following must be compared._ + +_Estimation of the Figures in a Product._—We have given rules for those +who prefer to decide the number of figures by this means, but experience +will show that to make the best use of the instrument, the result, as +read on the rule, should be regarded merely as the _significant figures +of the answer_, the position of the decimal point, if not obvious, being +decided by a very rough mental calculation. In very many instances, the +magnitude of the result will be evident from the conditions of the +problem—_e.g._, whether the answer should be 0·3 in., 3 in., or 30 in.; +or 10 tons, 0·1 ton, 100 tons, etc. In those cases where the magnitude +of the answer cannot be estimated, and the factors contain many figures, +or have a number of 0’s following the decimal point, the use of notation +by powers of 10 (page 8) is of considerable assistance; but more usually +it will be found, that a very rough calculation will settle the point +with comparatively little trouble. Considerable practice is needed to +work rapidly and with certainty, when using rules. Moreover, the +experience thus acquired is confined to slide-rule work. The same time +spent in practising the “rough approximation” method will enable +reliable results to be obtained rapidly, with the advantage that the +method is applicable to calculations generally. However, the choice of +methods is a matter of personal preference. Both methods will be given, +but whichever plan is followed, the student is strongly advised to +cultivate the habit of forming an idea of the magnitude of the result. + + + EX.—33·6 × 236 = 7930. + + Setting 1 on C to 33·6 on D, we read under 236 on D and find 793 on + D, as the significant figures of the answer. A rough calculation, as + 30 × 200 = 6000, indicates that the result will consist of 4 + figures, and is therefore to be read as 7930. + + EX.—17,300 × 3780 = 65,400,000. + + By factorising with powers of 10 + + 1·73 × 10^4 × 3·78 × 10^3 = 1·73 × 3·78 × 10^7. + + Setting 1 on C to 1·73 on D, we read, under 3·78 on C, the result of + the simple multiplication, as 6·54. Multiplying by 10^7 moves the + decimal point 7 places to the right, and the answer is 65,400,000. + + +If it is required to find a series of products of which one of the +factors is _constant_, set 1 on C to the constant factor on D and read +the several products on D, under the respective variable factors. + +If the factors are required which will give a constant _product_ (really +a case of division), set the cursor to the constant product on D. Then +obviously, as the slide is moved along, any pair of factors found +simultaneously under the cursor line on C, and on D under index of C, +will give the product. A better method of working will be explained when +we deal with the inversion of the slide. + +It is sometimes useful to remember that although we usually set the +slide to the rule, we can obtain the result equally well by setting the +rule to the slide. Thus, bringing 1 (or 10) on D to 2 on C, we find on +C, _over_ any other factor, _n_ on D, the product of 2 × _n_. But note +that the slide and rule have now changed places, and if we use rules for +the number of digits in the result, we must now deduct 1 from the sum of +the digits in the factors, when the _rule projects_ to the _right of the +slide_. + +With the ordinary 10 in. rule it will be found in general that the +extent to which the C and D scales are subdivided is such as to enable +not more than three figures in either factor being dealt with. For the +same reason it is impossible to directly read more than the first three +figures of any product, although it is often possible—by mentally +dividing the smallest space involved in the reading—to correctly +determine the fourth figure of a product. Necessarily this method is +only reliable when used in the earlier parts of the C and D scales. +However, the last numeral of a three-figure, and in some cases the last +of a four-figure, product can be readily ascertained by an inspection of +the factors. + +EX.—19 × 27 = 513. Placing the L.H. index of C to 19 on D, we find +opposite 27 on C, the product, which lies between 510 and 515. A glance +at the factors, however, is sufficient to decide that the third figure +must be 3, since the product of 9 and 7 is 63, and the last figure of +this product must be the last figure in the answer. + +EX.—79 × 91 = 7189. + +In this case the division line 91 on C indicates on D that the answer +lies between 7180 and 7190. As the last figure must be 9, it is at once +inferred that the last two figures are 89. + +When there are more than three figures in either or both of the factors, +the fourth and following figures to the right must be neglected. It is +well to note, however, that if the first neglected figure is 5, or +greater than 5, it will generally be advisable to increase by 1 the +third figure of the factor employed. Generally it will suffice to make +this increase in one of the two factors only, but it is obvious that in +some cases greater accuracy will be obtained by increasing both factors +in this way. + +CONTINUED MULTIPLICATION.—To find the product of more than two factors, +we make use of the cursor to mark the position of successive products +(the value of which does not concern us) as the several factors are +taken into the calculation. Setting the index of C to the 1st factor on +D, we bring the line of the cursor to the 2nd factor on C, then the +index of C to the cursor, the cursor to the 3rd factor, index of C to +cursor, and so on, reading the final product on D under the last factor +on C. (Note that the 1st factor and the result are read on D; all +intermediate readings are taken on C.) + +If the rule for the number of digits in a product is used, it is +necessary to note the number of times multiplication is effected with +the slide projecting to the right. This number, deducted from the sum of +the digits of the several factors, gives the number of digits in the +product. Ingenious devices have been adopted to record the number of +times the slide projects to the right, but some of these are very +inconvenient. The author’s method is to record each time the slide so +projects, by a minus mark, thus −. These can be noted down in any +convenient manner, and the sum of the marks so obtained deducted from +the sum of the digits in the several factors, gives the number of digits +in the product as before explained. + +EX.—42 × 71 × 1·5 × 0·32 × 121 = 173,200. + +The product given, which is that read on the rule, is obtained as +follows:—Set R.H. index of C to 42 on D, and bring the cursor to 71 on +C. Next bring the L.H. index of C to the cursor, and the latter to 1·5 +on C. This multiplication is effected with the slide to the right, and a +memorandum of this fact is kept by making a mark −. Bring the R.H. index +of C to the cursor and the latter to 0·32 on C. Then set the L.H. index +of C to the cursor and read the result, 1732, on D under 121 on C, while +as a slide again projects to the right, a second − memo-mark is +recorded. There are 2 + 2 + 1 + 0 + 3 = 8 digits in the factors, and as +there were 2 − marks recorded during the operation, there will be 8 − 2 += 6 digits in the product, which will therefore read 173,200 +(173,194·56). + +For a very rough evaluation of the result, we note that 1·5 × 0·3 is +about 0·5; hence, as a clue to the number of figures we have + + 40 × 70 × 60 = 3000 × 60 = 180,000. + + + + + DIVISION. + + +The instructions for multiplication having been given in some detail, a +full discussion of the inverse process of division will be unnecessary. + +RULE FOR DIVISION.—_Place the divisor on C, opposite the dividend on D, +and read the quotient on D under the index of C._ + +EX.—225 ÷ 18 = 12·5. + +Bringing 18 on C to 225 on D, we find 12·5 under the L.H. index of C. + +As in multiplication, the factors are treated as whole numbers, and the +position of the decimal point afterwards decided according to the +following rule, which, as will be seen, is the reverse of that for +multiplication:— + +RULE FOR THE NUMBER OF DIGITS IN A QUOTIENT.—_If the quotient is read +with the slide projecting to the_ LEFT, _subtract the number of digits +in the divisor from those in the dividend; but if read with the slide to +the_ RIGHT, ADD _1 to this difference_.[2] + +In the above example the quotient is read off with the slide to the +right, so the number of digits in the answer = 3 − 2 + 1 = 2. + +EX.—0·000221 ÷ 0·017 = 0·013. + +Here the number of digits in the dividend is −3, and in the divisor −1. +The difference is −2; but as the result is obtained with the slide to +the right, this result must be increased by 1, so that the number of +digits in the quotient is −2 + 1 = −1, giving the answer as 0·013. + +If preferred, the result can be obtained in the manner referred to when +considering the multiplication of decimals. Thus, treating the above as +whole numbers, we find that the result of dividing 221 by 17 = 13, since +the difference in the number of digits in the factors, which is 1, is, +owing to the position of the slide, increased by 1, giving 2 as the +number of digits in the answer. Then by the rules for the division of +decimals we know that the number of decimal places in the quotient is +equal to 6 − 3 = 3, showing that a cypher is to be prefixed to the +result read on the rule. + +As in multiplication, so in division, we have a + +GENERAL RULE FOR NUMBER OF DIGITS IN A QUOTIENT.—_When the first +significant figure in the_ DIVISOR _is greater than that in the_ +DIVIDEND_, the number of digits in the quotient is found by subtracting +the digits in the divisor from those in the dividend. When the contrary +is the case, 1_ IS TO BE ADDED _to this difference. When the first +figures are the same, those following must be compared._ + +ESTIMATION OF THE FIGURES IN A QUOTIENT.—The method of roughly +estimating the number of figures in a quotient needs little explanation. + +EX.—3·95 ÷ 5340 = 0·00074. + + + Setting 534 on C to 3·95 on D we read under the (R.H.) index of C, the + significant figures on D, which are 74. Then 3·9 ÷ 5 is about 0·8 and + 0·8 ÷ 1000 gives 0·0008 as a rough estimate. + + +EX.—0·00000285 ÷ 0·000197 = 0·01446. + + + Regarding this as 2·85 × 10^{−6} ÷ 1·97 × 10^{−4} we divide 2·85 by + 1·97 and obtain 1·446. Dividing the powers of 10 we have 10^{−6} ÷ + 10^{−4} = 10^{−2}, so the decimal point is to be moved two places to + the left and the answer is read as 0·01446. + + +Another method of dividing deserves mention as of special service when +dividing a number of quantities by a _constant divisor_:—Set the index +of C to the divisor on D and over any dividend on D, read the quotient +on C. + +For the division of a _constant dividend_ by a variable divisor, set the +cursor to the dividend on D and bring the divisor on C successively to +the cursor, reading the corresponding quotients on D under the index of +C. Another method which avoids moving the slide is explained in the +section on “Multiplication and Division with the Slide Inverted.” + +CONTINUED DIVISION, if we can so call such an expression as + + (3·14)/(785 × 0·00021 × 4·3 × 64·4) = 0·0688, + +may be worked by repeating as follows:—Set 7·85 on C to 3·14 on D, bring +cursor to index of C, 2·1 on C to cursor, cursor to index, 4·3 to +cursor, cursor to index, 6·44 to cursor, and under index of C read 688 +on D as the significant figures of the answer. + +For the number of figures in the result, we deduct the sum of the number +of digits in the several factors and add 1 for each time the slide +projects to the right, which in this case occurs once. There are 3 + +(−3) + 1 + 2 = 3 denominator digits, 1 numerator digit, and 1 is to be +added to the difference. Therefore there are 1 − 3 + 1 = −1 digits in +the answer, which is therefore 0·0688. The foregoing method of working +may confuse the beginner, who is apt to fall into the process of +continued multiplication. For this reason, until familiarity with +combined methods has been acquired, the product of the several +denominators should be first found by the continued multiplication +process, and the figures in this product determined. Then divide the +numerator by this product to obtain the result. + +As the denominator product will be read on D, we may avoid resetting the +slide by bringing the numerator on C to this product and reading the +result on C _over_ the index of D. The slide and rule have here changed +places; hence if rules are followed for the number of figures in the +result, 1 must be added to the difference of digits, when the _rule +projects_ to the _right of the slide_. + +The author’s method of recording the number of times division is +performed with the slide to the right is by vertical memorandum marks, +thus |. The full significance of these memo-marks will appear in the +following section. + +For a rough calculation to fix the decimal point, in this example we +move the decimal points in the factors, obtaining + + (3)/(0·8 × 2 × 4 × 6) = (3)/(40) = 0·075. + + + + + THE USE OF THE UPPER SCALES FOR MULTIPLICATION AND DIVISION. + + +Many prefer to use the upper scales A and B, in preference to C and D. +The disadvantage is that as the scales are only one-half the length of C +or D, the graduation does not permit of the same degree of accuracy +being obtained as when working with the lower scales. But the result can +always be read directly from the rule without ever having to change the +position of the slide after it has been initially set. Hence, it +obviates the uncertainty as to the direction in which the slide is to be +moved in making a setting. + +When the A and B scales are employed, it is understood that the +left-hand pair of scales are to be used in the same manner as C and D, +and so far the rules relating to the latter are entirely applicable. But +in this case the slide is always moved to the right, so that in +multiplication the product is found either upon the left or right scales +of A. If it is found on the left A scale, the rule for the number of +digits in the product is found as for the C and D scales, and is equal +to the _sum of the digits in the two factors, minus 1_; but if found on +the right-hand A scale, the number of digits in the product is equal to +the sum of the digits in the two factors. + +In division, similar modifications are necessary. If when moving the +slide to the right the division can be completely effected by using the +L.H. scale of A, the quotient (read on A above the L.H. of index B) has +a number of digits equal to the number in the dividend, less the number +in the divisor, _plus 1_. But if the division necessitates the use of +both the A scales, the number of digits in the quotient equals the +number in the dividend, less the number in the divisor. + + + + + RECIPROCALS. + + +A special case of division to be considered is the determination of the +_reciprocal_ of a number _n_, or (1)/(_n_). Following the ordinary rule +for division, it is evident that setting _n_ on C to 1 on D, gives +(1)/(_n_) on D under 1 on C. It is more important to observe that by +inverting the operation—setting 1 (or 10) on C to _n_ on D—we can read +(1)/(_n_) on C over 1 (or 10) on D. Hence whenever a result is read on D +under an index of C, we can also read its reciprocal on C over whichever +index of D is available. + +_The Number of Digits in a Reciprocal_ is obvious when _n_ = 10, 100, or +any power (_p_) of 10. Thus (1)/(10) = 0·1; (1)/(100) = 0·01; +(1)/(10^{_p_}) = 1 preceded by _p_ − 1 cyphers. For all other cases we +have the rule:—_Subtract from 1 the number of digits in the number._ + +EX.—(1)/(339) = 0·00295. + +There are 3 digits in the number; hence, there are 1 − 3 = −2 digits in +the answer. + +EX.—(1)/(0·0000156) = 64,100. + +There are −4 digits in the number; hence, there are 1 − (−4) = 5 digits +in the result. + + + + + CONTINUED MULTIPLICATION AND DIVISION. + + +By combining the rules for multiplication and division, we can readily +evaluate expressions of the form (_a_)/(_b_) × (_c_)/(_d_) × (_e_)/(_f_) +× (_g_)/(_h_) = _x_. The simplest case, (_a_ × _c_)/(_b_) can be solved +by one setting of the slide.[3] Take as an example, (14·45 × 60)/(8·5) = +102. Setting 8·5 on C to 14·45 on D, we can, if desired, read 1·7 on D +under 1 on C, as the quotient. However, we are not concerned with this, +but require its multiplication by 60, and the slide being already set +for this operation, we at once read under 60 on C the result, 102, on D. +The figures in the answer are obvious. + +When there are more factors to take into account, we place the cursor +over 102 on D, bring the next divisor on C to the cursor, move the +cursor to the next multiplier on C, bring the next divisor on C to the +cursor, and so on, until all the factors have been dealt with. Note that +only the first factor and the result are read on D; also _that the +cursor is moved for multiplying and the slide for dividing_. + +_Number of Digits in Result in Combined Multiplication and +Division._—For those who use rules the author’s method of determining +the decimal point in combined multiplication and division may be used. +Each time _multiplication_ is performed with the slide projecting to the +_right_, make a − mark; each time _division_ is effected with the slide +to the right, make a | mark; _but allow the_ | _marks to cancel the_ − +_marks as far as they will_. Subtract the sum of the digits in the +denominator from the sum of digits in the numerator, and to this +difference _add_ any uncancelled memo-marks, if of | character, or +_subtract_ them if of − character. + +EX.—(43·5 × 29·4 × 51 × 32)/(27 × 3·83 × 10·5 × 1·31) = 1468. + +[Sidenote: ⵜ + ⵜ + ⵏ + ⵏ] + +Set 27 on C to 43·5 on D, and as with this _division_ the slide is to +the right, make the first ⵏ mark. Bring cursor to 29·4 on C, and as in +this _multiplication_ the slide is to the right, make the first − mark, +cancelling as shown. Setting 3·83 on C to the cursor, requires the +second ⵏ mark, which, however, is cancelled in turn by the +multiplication by 51. The division by 10·5 requires the third ⵏ mark, +and after multiplying by 32 (requiring no mark) the final division by +1·31 requires the fourth ⵏ mark. Then, as there are 8 numerator digits, +6 denominator, and 2 uncancelled memo-marks (which, being 1, are +additive) we have + + Number of digits in result = 8 − 6 + 2 = 4. + +Had the uncancelled marks been − in character, the number of digits +would have been 8 − 6 − 2 = 0. + +For quantities less than 0·1 the digit place numbers will be _negative_. +The troublesome addition of these may be avoided by transferring them to +the opposite side and treating them as positive. + + _2_ _4_ + 0·00356 × 27·1 × 0·08375 + Thus:— ───────────────────────── = 288 + 0·1426 × 9·85 × 0·00002 + _2_ _1_ _1_ + +The first numerator, 0·00356, has −2 digits. Note this by placing 2 +_below the lower line_ as shown. 27·1 has 2 digits; place 2 over it. +0·08375 has −1 digit; hence place 1 _below the lower line_. The first +denominator has no digits; the second, 9·85, has 1 digit; hence place 1 +under it. 0·00002 has −4 digits; place 4 _above the upper line_. The sum +of the top series is 2 + 4 = 6; of the bottom series 2 + 1 + 1 = 4. +Subtracting the bottom from the top, we have 6 − 4 = 2 digits, to which +1 has to be added for an uncancelled memo-mark, and the result is read +as 288. + +Moving the decimal point often facilitates matters. Thus, (32·4 × 0·98 × +432 × 0·0217)/(4·71 × 0·175 × 0·00000621 × 412000) is much more +conveniently dealt with when re-arranged as (32·4 × 9·8 × 432 × +2·17)/(4·71 × 17·5 × 6·21 × 4·12) = 141. + +To determine the number of figures in the result by rough cancelling and +mental calculation, we note that 4·71 enters 432 about 100 times; 9·8 +enters 17·5 about 2; 6·21 into 32·4 about 5; and 2·17 into 4·12 about 2. +This gives (500)/(4) = 125, showing that the result contains 3 digits. +From the slide rule we read 141, which is therefore the result sought. + +The occasional traversing of the slide through the rule, to interchange +the indices—a contingency which the use of the C and D scales always +involves—may often be avoided by a very simple expedient. Such an +example as (6·19 × 31·2 × 422)/(1120 × 8·86 × 2.09) = 3·93 is sometimes +cited as a particularly difficult case. Working through the expression +as given, two traversings of the slide are necessary; but by taking the +factors in the slightly different order, (6·19 × 31·2 × 422)/(8·86 × +2·09 × 1120), _so that the significant figures of each pair are more +nearly alike_, we not only avoid any traversing the slide, but we also +reduce the extent to which the slide is moved to effect the several +divisions. + +Such cases as (_a_ × _b_)/(_c_ × _d_ × _e_ × _f_ × _g_) or (_a_ × _b_ × +_c_ × _d_ × _e_)/(_f_ × _g_) really resolve themselves into (_a_ × _b_ × +1 × 1 × 1)/(_c_ × _d_ × _e_ × _f_ × _g_) and (_a_ × _b_ × _c_ × _d_ × +_e_)/(_f_ × _g_ × 1 × 1 × 1), but, of course, if rules are used to +locate the decimal point, the 1’s so (mentally) introduced are not to be +counted as additional figures in the factors. + + + + + MULTIPLICATION AND DIVISION WITH THE SLIDE INVERTED. + + +If the slide be inverted in the rule but with the same face uppermost, +so that the Ɔ scale lies adjacent to the A scale, and the right and left +indices of the slide and rule are placed in coincidence, we find the +product of any number on D by the coincident number on Ɔ (readily +referred to each other by the cursor) is always 10. Hence, by reading +the numbers on Ɔ as decimals, we have over any unit number on D, its +_reciprocal_ on Ɔ. Thus 2 on D is found opposite 0·5 on Ɔ; 3 on D +opposite to 0·333; while opposite 8 on Ɔ is 0·125 on D, etc. The reason +of this is that the sum of the lengths of the slide and rule +corresponding to the factors, is always equal to the length +corresponding to the product—in this case, 10. + +It will be seen that if we attempt to apply the ordinary rule for +multiplication, with the slide inverted, we shall actually be +multiplying the one factor taken on D by the _reciprocal_ of the other +taken on Ɔ. But multiplying by the _reciprocal of a number_ is +equivalent to _dividing_ by that number, and _dividing_ a factor by the +_reciprocal_ of a number is equivalent to _multiplying_ by that number. +It follows that with the slide inverted the operations of multiplication +and division are reversed, as are also the rules for the number of +digits in the product and the position of the decimal point. Hence, in +multiplying with the slide inverted, we place (by the aid of the cursor) +one factor on Ɔ opposite the other factor on D, and read the result on D +under either index of Ɔ. It follows that with the slide thus set, any +pair of coinciding factors on Ɔ and D will give the same constant +product found on D under the index of Ɔ. One useful application of this +fact is found in selecting the scantlings of rectangular sections of +given areas or in deciding upon the dimensions of rectangular sheets, +plates, cisterns, etc. Thus by placing the index of Ɔ to 72 on D, it is +readily seen that a plate having an area of 72 sq. ft. may have sides 8 +by 9 ft., 6 by 12, 5 by 14·4, 4 by 18, 3 by 24, 2 by 36, with +innumerable intermediate values. Many other useful applications of a +similar character will suggest themselves. + + + + + PROPORTION. + + +With the slide in the ordinary position and with the indices of the C +and D scales in exact agreement, the _ratio_ of the corresponding +divisions of these scales is 1. If the slide is moved so that 1 on C +agrees with 2 on D, we know that under any number _n_ on C is _n_ × 2 on +D, so that if we read numerators on C and denominators on D we have + + C 1 1·5 2 3 4 + ───────────────────────────────────────── + D1 2 3 4 6 8. + +In other words, the numbers on D bear to the coinciding numbers on C a +ratio of 2 to 1. Obviously the same condition will obtain no matter in +what position the slide may be placed. The rule for proportion, which is +apparent from the foregoing, may be expressed as follows:— + +RULE FOR PROPORTION.—_Set the first term of a proportion on the C scale +to the second term on the D scale, and opposite the third term on the C +scale read the fourth term on the D scale._ + + + EX.—Find the 4th term in the proportion of 20 ∶ 27 ∷ 70 ∶ _x_. Set 20 + on C to 27 on D, and opposite 70 on C read 94·5 on D. Thus + + C 20 70 + ───────────────── + D 27 94·5. + + +It will be evident that this is merely a case of combined multiplication +and division of the form, (20 × 70)/(27) = 94·5. Hence, given any three +terms of a proportion, we set the 1st to the 2nd, or the 3rd to the 4th, +as the case may be, and opposite the other given term read the term +required.[4] + +Thus, in reducing vulgar fractions to decimals, the decimal equivalent +of (3)/(16) is determined by placing 3 on C to 16 on D, when over the +index or 1 of D we read 0·1875 on C. In this case the terms are +3 ∶ 16 ∷ _x_ ∶ 1. For the inverse operation—to find a vulgar fraction +equivalent to a given decimal—the given decimal fraction on C is set to +the index of D, and then opposite any denominator on D is the +corresponding numerator of the fraction on C. + +If the index of C be placed to agree with 3·1416 on D, it will be clear +from what has been said that this ratio exists throughout between the +numbers of the two scales. Therefore, against any _diameter_ of a circle +on C will be found the corresponding _circumference_ on D. In the same +way, by setting 1 on C to the appropriate conversion factor on D, we can +convert a series of values in one denomination to their equivalents in +another denomination. In this connection the following table of +conversion factors will be found of service. If the A and B scales are +used instead of the C and D scales, a complete set of conversions will +be at once obtained. In this case, however, the left-hand A and B scales +should be used for the initial setting, any values read on the +right-hand A or B scales being read as of tenfold value. With the C and +D scales a portion of the one scale will project beyond the other. To +read this portion of the scale, the cursor or runner is brought to +whichever index of the C scale falls within the rule, and the slide +moved until the other index of the C scale coincides with the cursor, +when the remainder of the equivalent values can then be read off. It +must be remembered that if the slide is moved in the direction of +notation (to the _right_), the values read thereon have a tenfold +_greater_ value; if the slide is moved to the _left_, the readings +thereon are _decreased_ in a tenfold degree. Although preferred by many, +in the form given, the case is obviously one of multiplication, and is +so treated in the Data Slips at the end of the book. + + TABLE OF CONVERSION FACTORS. + ─────────────────────────────────────────────────────────────── + GEOMETRICAL EQUIVALENTS. + ──────────────────────────┬──────────────────────────┬───────── + SCALE C. │ SCALE D. │If C = 1, + │ │ D = + ──────────────────────────┼──────────────────────────┼───────── + Diameter of circle │Circumference of circle │3·1416 + „ „ │Side of inscribed square │0·707 + „ „ │„ equal square │0·886 + „ „ │„ „ equilateral │ + │ triangle │1·346 + Circum. of circle │„ inscribed square │0·225 + „ „ │„ equal square │0·282 + Side of square │Diagonal of square │1·414 + Square inch │Circular inch │1·273 + Area of circle │Area of inscribed square │0·636 + ──────────────────────────┴──────────────────────────┴───────── + MEASURES OF LENGTH. + ──────────────────────────┬──────────────────────────┬───────── + Inches │Millimetres │25·40 + „ │Centimetres │2·54 + 8ths of an inch │Millimetres │3·175 + 16ths „ „ │„ │1·587 + 32nds „ „ │„ │0·794 + 64ths „ „ │„ │0·397 + Feet │Metres │0·3048 + Yards │„ │0·9144 + Chains │„ │20·116 + Miles │Kilometres │1·609 + ──────────────────────────┴──────────────────────────┴───────── + MEASURES OF AREA. + ──────────────────────────┬──────────────────────────┬───────── + Square inches │Square centimetres │6·46 + Circular „ │„ „ │5·067 + Square feet │„ metres │0·0929 + „ yards │„ „ │0·836 + „ miles │„ kilometres │2·59 + „ „ │Hectares │259·00 + Acres │„ │0·4046 + ──────────────────────────┴──────────────────────────┴───────── + MEASURES OF CAPACITY. + ──────────────────────────┬──────────────────────────┬───────── + Cubic inches │Cubic centimetres │16·38 + „ „ │Imperial gallons │0·00360 + „ „ │U.S. gallons │0·00432 + „ „ │Litres │0·01638 + Cubic feet │Cubic metres │0·0283 + „ „ │Imperial gallons │6·23 + „ „ │U.S. gallons │7·48 + „ „ │Litres │28·37 + „ yards │Cubic metres │0·764 + Imperial gallons │Litres │4·54 + „ „ │U.S. gallons │1·200 + Bushels │Cubic metres │0·0363 + „ │„ feet │1·283 + ──────────────────────────┴──────────────────────────┴───────── + MEASURES OF WEIGHT. + ──────────────────────────┬──────────────────────────┬───────── + Grains │Grammes │0·0648 + Ounces (Troy) │„ │31·103 + „ (Avoird.) │„ │28·35 + „ „ │Kilogrammes │0·02835 + Pounds (Troy) │„ │0·3732 + „ (Avoird.) │„ │0·4536 + Hundredweights │„ │50·802 + Tons │„ │1016·4 + „ │Metric tonnes │1·016 + ──────────────────────────┴──────────────────────────┴───────── + COMPOUND FACTORS—VELOCITIES. + ──────────────────────────┬──────────────────────────┬───────── + Feet per second │Metres per second │0·3048 + „ „ │„ minute │18·288 + „ „ │Miles per hour │0.682 + „ minute │Meters per second │0·00508 + „ „ │„ minute │0·3048 + „ „ │Miles per hour │0·01136 + Yards per „ │„ „ │0·0341 + Miles per hour │Metres per minute │26·82 + Knots │„ „ │30·88 + „ │Miles per hour │1·151 + ──────────────────────────┴──────────────────────────┴───────── + COMPOUND FACTORS—PRESSURES. + ──────────────────────────┬──────────────────────────┬───────── + Pounds per sq. inch │Grammes per sq. mm. │0·7031 + „ „ │Kilos. per sq. centimetre │0·0703 + „ „ │Atmospheres │0·068 + „ „ │Head of water in inches │27·71 + „ „ │„ „ feet │2·309 + „ „ │„ „ metres │0·757 + „ „ │Inches of Mercury │2·04 + Inches of water │Pounds per square inch │0·0361 + „ „ │Inches of mercury │0·0714 + „ „ │Pounds per square foot │5·20 + Inches of mercury │Atmospheres │0·0333 + Atmospheres │Metres of water │10·34 + „ │Kilos. per sq. cm. │1·033 + Feet of water │Pounds per square foot │62·35 + „ „ │Atmospheres │0·0294 + „ „ │Inches of mercury │0·883 + Pounds per sq. foot │„ „ │0·01417 + „ „ │Kilos. per square metre │4·883 + „ „ │Atmospheres │0·000472 + Pounds per sq. yard │Kilos. per square metre │0·5425 + Tons per sq. inch │„ square mm. │1·575 + „ sq. foot │Tonnes per square metre │10·936 + ──────────────────────────┴──────────────────────────┴───────── + COMPOUND FACTORS—WEIGHTS, CAPACITIES, ETC. + ──────────────────────────┬──────────────────────────┬───────── + Pounds per lineal ft. │Kilos. per lineal metre │1·488 + „ per lineal yd. │„ „ „ │0·496 + „ per lineal mile │Kilos. per kilometre │0·2818 + Tons „ „ │Tonnes „ │0·6313 + Feet „ „ │Metres „ │1·894 + Pounds per cubic in. │Grammes per cubic cm. │27·68 + „ per cubic ft. │Kilos. per cubic metre │16·02 + „ per cubic yd. │„ „ „ │0·593 + Tons per cubic yard │Tonnes „ „ │1·329 + Cubic yds. per pound │Cubic metres per kilo. │1·685 + „ per ton │„ „ per tonne │0·7525 + Cubic inch of water │Weight in pounds │0·03608 + Cubic feet of water │„ „ │62·35 + „ „ │„ kilos │28·23 + „ „ │Imperial gallons │6·235 + „ „ │U.S. gallons │7·48 + Litre of water │Cubic inches │61·025 + Gallons of water │Weight in kilos │4·54 + Pounds of fresh water │Pounds of sea water │1·026 + Grains per gallon │Grammes per litre │0·01426 + Pounds per gallon │Kilos. per litre │0·0998 + „ per U.S. gal. │„ „ │0·115 + ──────────────────────────┴──────────────────────────┴───────── + COMPOUND FACTORS—POWER UNITS, ETC. + ──────────────────────────┬──────────────────────────┬───────── + British Ther. Units. │Kilogrammetres. │108 + „ „ │Joules │1058 + „ „ │Calories (Fr. Ther. units)│0·252 + „ „ per sq. ft. │„ per square metre │2·713 + „ „ per pound │„ per kilogramme │0·555 + Pounds per sq. ft. │Dynes, per sq. cm. │479 + Foot-pounds │Kilogrammetres │0·1382 + „ „ │Joules │1·356 + „ „ │Thermal Units │0·00129 + „ „ │Calorie │0·000324 + Foot-tons │Tonne-metres │0·333 + Horse-power │Force decheval (Fr.H.P.) │1·014 + „ „ │Kilowatts │0·746 + Pounds per H.P. │Kilos. per cheval │0·447 + Square feet per H. P. │Square metres per cheval │0·0196 + Cubic „ „ │Cubic „ „ │0·0279 + Watts │Ther. Units per hour │3·44 + „ │Foot-pounds per second │0·73 + „ │„ per minute │44·24 + Watt-hours │Kilogrammetres │367 + „ „ │Joules │3600 + Kilogrammetres │„ │9·806 + ──────────────────────────┴──────────────────────────┴───────── + +_Inverse Proportion._—If “more” requires “less,” or “less” requires +“more,” the case is one of _inverse_ proportion, and although it will be +seen that this form of proportion is quite readily dealt with by the +preceding method, the working is simplified to some extent by inverting +the slide so that the C scale is adjacent to the A scale. By the aid of +the cursor, the values on the inverted C (or Ɔ) scale, and on the D +scale, can be then read off. These will now constitute a series of +inverse ratios. For example, in the proportion + + ─────────── + Ɔ 8 4 + ─────────── + D 1·5 3 + +the 4 on the Ɔ scale is brought opposite 3 on D, when under 8 on Ɔ is +found 1·5 on D.[5] + + + GENERAL HINTS ON THE ELEMENTARY USES OF THE SLIDE RULE. + +Before the more complex operations of involution, evolution, etc., are +considered, a few general hints on the use of the slide rule for +elementary operations may be of service, especially as these will serve +to enforce some of the more important points brought out in the +preceding sections. + +Always use the slide rule in as _direct_ a light as possible. + +Study the manner in which the scales are divided. Follow the graduations +of the C and D scales from 1 to 10, noting the values given by each +successive graduation and how these values change as we follow along to +the right. Do the same with the two halves of the A and B scales and +note the difference in the value of the subdivisions, due to the shorter +scale-lengths. + +Practise reading values by setting 1 on C to some value on D and reading +under 2, 3, 4, etc., on C, checking the readings by mental arithmetic. +To the same end, find squares, square roots, etc., comparing the results +with the actual values as given in tables. Practise setting both slide +and cursor to values taken at random. Aim at accuracy; speed will come +with practice. + +When in doubt as to any method of working, verify by making a simple +calculation of the same form. + +Follow the orthodox methods of working until entirely confident in the +use of the instrument, and even then do not readily make a change. If +any altered procedure is adopted, first work a simple case and guard +carefully against unconsciously lapsing into the usual method during the +operation. + +Unless the calculation is of a straightforward character, time taken in +considering how best to attack it (rearranging the expression if +desirable) is generally time well spent. + +In setting two values together, set the cursor to one of them on the +rule, and bring the other, on the slide, to the cursor line. + +In multiplying factors, as 57 × 0·1256, take the fractional value first. +It is easier to set 1 on C to 1256 on D and read under 57 on C, than to +reverse the procedure. When both values are eye-estimated, set the +cursor to the second factor on C and read the result on D, under the +cursor line. + +In continuous operations avoid moving the slide further than necessary, +by taking the factors in that order which will keep the scale readings +as close together as possible. + + + SQUARES AND SQUARE ROOTS. + +We have seen that the relation which the upper scales bear to the lower +set is such that over any number on D is its square on A, and, +conversely, under any number on A is its square root on D, the same +remarks applying to the C and B scales on the slide. Taking the values +engraved on the rule, we have on D, numbers lying between 1 and 10, and +on A the corresponding squares extending from 1 to 100. Hence the +squares of numbers between 1 and 10, or the roots of numbers between 1 +and 100, can be read off on the rule by the aid of the cursor. All other +cases are brought within these ranges of values by factorising with +powers of 10, as before explained. + +The more practical rule is the following:— + +_To Find the Square of a Number_, set the cursor to the number on D and +read the required square on A under the cursor. The rule for + +_The Number of Digits in a Square_ is easily deducible from the rule for +multiplication. If the square is read on the _left_ scale of A, it will +contain _twice_ the number of digits in the original number _less_ 1; if +it is read on the _right_ scale of A, it will contain _twice_ the number +of digits in the original number. + + + EX.—Find the square of 114. + + Placing the cursor to 114 on D, it is seen that the coinciding number + on A is 13. As the result is read off on the _left_ scale of A, the + number of digits will be (3 × 2) − 1 = 5, and the answer is read as + 13,000. The true result is 12,996. + + EX.—Find the square of 0·0093. + + The cursor being placed to 93 on D, the number on A is found to be + 865. The result is read on the _right_ scale of A, so the number of + digits = −2 × 2 = −4, and the answer is read as 0·0000865 + [0·00008649]. + + +_Square Root._—The foregoing rules suggest the method of procedure in +the inverse operation of extracting the square root of a given number, +which will be found on the D scale opposite the number on the A scale. +It is necessary to observe, however, that if the number consists of an +_odd_ number of digits, it is to be taken on the _left-hand_ portion of +the A scale, and the number of digits in the root = (N + 1)/(2), N being +the number of digits in the original number. When there is an even +number of digits in the number, it is to be taken on the _right-hand_ +portion of the A scale, and the root contains _one-half_ the number of +digits in the original number. + + + EX.—Find the square root of 36,500. + + As there is an _odd_ number of digits, placing the cursor to 365 on + the L.H. A scale gives 191 on D. By the rule there are (N + 1)/(2) = + (5 + 1)/(2) = 3 digits in the required root, which is therefore read + as 191 [191·05]. + + EX.—Find √(0·0098.) + + Placing the cursor to 98 on the right-hand scale of A (since −2 is an + _even_ number of digits), it is seen that the coinciding number on D + is 99. As the number of digits in the number is −2, the number of + digits in the root will be (−2)/(2) = −1. It will therefore be read as + 0·099 [0·09899+]. + + EX.—Find √(0·098). + + The number of digits is −1, so under 98 on the left scale of A, we + find 313 on D. By the rule the number in the root will be (−1 +1)/(2) + = 0, and the root is therefore read as 0·313 [0·313049+]. + + EX.—Find √(0·149.) + + As the number of digits (0) is _even_, the cursor is set to 149 on the + right-hand scale of A, giving 386 on D. By the rule, the number of + digits in the root will be (0)/(2) = 0, and the root will be read as + 0·386 [0·38605+]. + + +Another method of extracting the square root, by which more accurate +readings may generally be obtained, is by using the C and D scales only, +with the slide inverted. If there is an _odd_ number of digits in the +number, the _right_ index, or if an even number of digits the _left_ +index, of the inverted scale Ɔ is placed so as to coincide with the +number on D of which the root is sought. Then with the cursor, the +number is found on D which coincides with the same number on Ɔ, which +number is the root sought. + + + EX.—Find √(22·2.) + + Placing the left index of Ɔ to 222 on D, the two equal coinciding + numbers on Ɔ and D are found to be 4·71. + + +Note that under the cursor line we have the original number, 22·2, on A, +and from this the number of digits in the root is determined as before. + +The plan of finding the square of a number by ordinary multiplication is +often very convenient. The inverse process of finding a square root by +trial division is not to be recommended. + +To obtain a close value of a root or to verify one found in the usual +way, the author has, on occasion, adopted the following plan:—Set 1 (or +10) on B to the number on the A scale (L.H. or R.H. as the case may +require), and bring the cursor to the number on D. If the root found is +correct, the readings on C under the cursor and on D under the index of +C, will be in exact agreement. + +If 1 on B is placed to a number _n_ on the L.H. A scale, the student +will note that while root _n_ is read on D under 1 on C, the root of 10 +_n_ is read on D under 10 on B. Hence, if preferred, the number can be +taken always on the first scale of A and the root read under 1 or 10 on +B, according to whether there is an odd or even number of digits in the +number. Obviously the second root is the first multiplied by √(10). + + + CUBES AND CUBE ROOTS. + +In raising a number to the third power, a combination of the preceding +method and ordinary multiplication is employed. + +TO FIND THE CUBE OF A NUMBER.—_Set the_ L.H. _or_ R.H. _index of C to +the number on D, and opposite the number_ ON THE LEFT-HAND _scale of B +read the cube on the_ L.H. _or_ R.H. _scale of A_. + +By this rule four scales are brought into requisition. Of these, the D +scale and the L.H. B scale are _always_ employed, and are to be read as +of equal denomination. The values assigned to the L.H. and R.H. scales +of A will be apparent from the following considerations. + +Commencing with the indices of C and D coinciding, and moving the slide +to the right, it will be seen that, working in accordance with the above +rule, the cubes of numbers from 1 to 2·154 (= ∛(10)) will be found on +the first or L.H. scale of A. Moving the slide still farther to the +right, we obtain _on the_ R.H. _A scale_ cubes of numbers from 2·154 to +4·641 (or ∛(10) to ∛(100)). Had we a _third_ repetition of the L.H. A +scale, the L.H. index of C could be still further traversed to the +right, and the cubes of numbers from 4·641 to 10 read off on this +prolongation of A. But the same end can be attained by making use of the +R.H. index of C, when, traversing the slide to the right as before, the +cubes of numbers from 4·641 to 10 on D can be read off _on the_ L.H. _A +scale_ over the corresponding numbers on the L.H. B scale. Hence, using +the L.H. index of C, the readings on the L.H. A scale may be regarded +comparatively as units, those on the R.H. A scale as tens; while for the +hundreds we again make use of the L.H. A scale in conjunction with the +_right-hand_ index of C. + +By keeping these points in view, the number of digits in the cube (N) of +a given number (_n_) are readily deduced. Thus, if the units scale is +used, N = 3_n_ − 2; if the tens scale, N = 3_n_ − 1; while if the +hundreds scale be used, N = 3_n_. Placed in the form of rules:— + +N = 3_n_ − 2 when the product is read on the L.H. scale of A with the +slide to the _right_ (units scale). + +N = 3_n_ − 1 when the product is read on the R.H. scale of A; slide to +the _right_ (tens scale). + +N = 3_n_ when the product is read on the L.H. scale of A with the slide +to the _left_ (hundreds scale). + +With decimals the same rule applies, but, as before, the number of +digits must be read as −1, −2, etc., when one, two, etc., cyphers follow +immediately after the decimal point. + +EX.—Find the value of 1·4^3. + +Placing the L.H. index of C to 1·4 on D, the reading on A opposite 1·4 +on the L.H. scale of B is found to be about 2·745 [2·744]. + +EX.—Find the value of 26·4^3. + +Placing the L.H. index of C to 26·4 on D, the reading on A opposite 26·4 +on the L.H. scale of B is found to be about 18,400 [18,399·744]. + +EX.—Find the value of 7·3^3. + +In this case it becomes necessary to use the R.H. index of C, which is +set to 7·3 on D, when opposite 7·3 on the L.H. scale of B is read 389 +[389·017] on A. + +EX.—Find the value of 0·073^3. + +From the setting as before it is seen that the number of digits in the +number must be multiplied by 3. Hence, as there is −1 digit in 0·073, +there will be −3 in the cube, which is therefore read 0·000389. + +The last two examples serve to illustrate the principle of factorising +with powers of 10. Thus + + 0·073 = 7·3 × 10^{−2}; 0·073^3 = 7·3^3 × (10^{−2})^3 = 389 × 10^{−6} = + 0·000389. + +_Cube Root_ (_Direct Method_).—One method of extracting the cube root of +a number is by an inversion of the foregoing operation. Using the same +scales, _the slide is moved either to the right or left until under the +given number on A is found a number on the_ L.H. _B scale, identical +with the number simultaneously found on D under the right or left index +of C_. This number is the required cube root. + +From what has already been said regarding the combined use of these +scales in cubing, it will be evident that in extracting the cube root of +a number, it is necessary, in order to decide which scales are to be +used, to know the number of figures to be dealt with. We therefore (as +in the arithmetical method of extraction) point off the given number +into sections of three figures each, commencing at the decimal point, +and proceeding to the left for numbers greater than unity, and to the +right for numbers less than unity. Then if the first section of figures +on the left consists of— + +1 figure, the number will evidently require to be taken on what we have +called the “units” scale—_i.e._, on the L.H. scale of A, using the L.H. +index of C. + +If of 2 figures, the number will be taken on the “tens” scale—_i.e._, on +the R.H. scale of A, using the L.H. index of C. + +If of 3 figures, the number will be taken on the “hundreds” +scale—_i.e._, on the L.H. scale of A, using the R.H. index of C. + +To determine the number of digits in cube roots it is only necessary to +note that when the number is pointed off into sections as directed, +there will be one figure in the root for every section into which the +number is so divided, whether the _first_ section consists of 1, 2, or 3 +digits. + +Of numbers wholly decimal, the cube roots will be decimal, and for every +group of _three_ 0s immediately following the decimal point, _one_ 0 +will follow the decimal point in the root. If necessary, 0s must be +added so as to make up complete multiples of 3 figures before proceeding +to extract the root. Thus 0·8 is to be regarded as 0·800, and 0·00008 as +0·000080 in extracting cube roots. + +EX.—Find ∛(14,000.) + +Pointing the number off in the manner described, it is seen that there +are _two_ figures in the first section—viz., 14. Setting the cursor to +14 on the R.H. scale of A, the slide is moved to the right until it is +seen that 241 on the L.H. scale of B falls under the cursor, when 241 on +D is under the L.H. index of C. Pointing 14,000 off into sections we +have 14 000—that is, _two_ sections. Therefore, there are two digits in +the root, which in consequence will be read 24·1 [24·1014+]. + +EX.—Find ∛(0·162.) + +As the divisional section consists of _three_ figures, we use the +“hundreds” scale. Setting the cursor to 0·162 on the L.H. A scale, and +using the R.H. index of C, we move the slide to the left until under the +cursor 0·545 is found on the L.H. B scale, while the R.H. index of C +points to 0·545 on D, which is therefore the cube root of 0·162. + +EX.—Find ∛(0·0002.) + +To make even multiples of 3 figures requires the addition of 00; we have +then 200, the cube root of which is found to be about 5·85. Then, since +the first divisional group consists of 0s, one 0 will follow the decimal +point, giving ∛(0·0002) = 0·0585 [0·05848]. + +_Cube Root (Inverted Slide Method)._—Another method of extracting the +cube root involves the use of the inverted slide. Several methods are +used, but the following is to be preferred:—_Set the_ L.H. _or_ R.H. +_index of the slide to the number on A, and the number on ᗺ (i.e., B +inverted), which coincides with the same number on D, is the required +root._ + +Setting the slide as directed, and using first the L.H. index of the +slide and then the R.H. index, it is always possible to find _three_ +pairs of coincident values. To determine which of the three is the +required result is best shown by an example. + + + EX.—Find ∛(5,) ∛(50,) and ∛(500.) + + Setting the R.H. index of the slide to 5 on A, it is seen that 1·71 on + D coincides with 1·71 on ᗺ. Then setting the L.H. index to 5 on A, + further coincidences are found at 3·68 and at 7·93, the three values + thus found being the required roots. Note that the first root was + found on that portion of the D scale lying under 1 to 5 on A; the + second root on that portion lying under 5 to 50 on A; and the third + root on that portion of D lying under 50 to 100 on A. In this + connection, therefore, scale A may always be considered to be divided + into three sections—viz., 1 to _n_, _n_ to 10_n_, and 10_n_ to 100. + For all numbers consisting of 1, 1 + 3, 1 + 6, 1 + 9—_i.e._, of 1, 4, + 7, 10, or −2, −5, etc., figures—the coincidence under the first + section is the one required. If the number has 2, 5, 8, or −1, −4, −7, + etc., figures, the coincidence under the second section is correct, + while if the number has 3, 6, 9, or 0, −3, etc., figures, the + coincidence under the last section is that required. The number of + digits in the root is determined by marking off the number into + sections, as already explained. + + +_Cube Root (Pickworth’s Method)._—One of the principal objections to the +two methods described is the difficulty of recollecting which scales are +to be employed and with which index of the slide they are to be used. +With the direct method another objection is that the readings to be +compared are often some distance apart, the maximum distance intervening +being _two-thirds_ of the length of the rule. To carry the eye from one +to another is troublesome and time-taking. With the inverted scale +method the reading of a scale reversed in direction and with the figures +inverted is also objectionable. + +With the author’s method these objections are entirely obviated. The +_same scales and index are always used_, and are read in their normal +position. The three roots of _n_, 10_n_ and 100_n_ (_n_ being less than +10 and not less than 1) are given with one setting and appear in their +natural sequence, no traversing of the slide being needed. The readings +to be compared are always close together, the maximum distance between +them being _one-sixth_ of the length of the rule. The setting is always +made in the earlier part of the scales where closer readings can be +obtained, and finally, if desired, the result may be readily verified on +the lower scales by successive multiplication. + +For this method two gauge points are required on C. To conveniently +locate these, set 53 on C to 246 on D; join 1 on D to 1 on A with a +straight-edge and with a needle point draw a short fine line on C. Set +246 on C to 53 on D, and repeat the process at the other end of the +rule. The gauge points thus obtained (dividing C into three equal parts) +will be at 2·154 and 4·641, and should be marked ∛(10) and ∛(100) +respectively.[6] + + + EX.—Find ∛(2·86,) ∛(28·6) and ∛(286). + + Set cursor to 2·86 on A and drawing the slide to the right find 1·42 + under 1 on C, when 1·42 on B is under the cursor. Then reading under + 1, ∛(10) and ∛(100,) we have + + ∛(2·86) = 1·42; ∛(28·6) = 3·06 and ∛(286) = 6·59. + + +It will be seen that factorising with powers of 10, we multiply the +initial root by ∛(10) and ∛(100). Obviously the three roots will always +be found on D, in their natural order and at intervals of one-third the +length of the rule. The number of digits in the roots of numbers which +do not lie between 1 and 1000, is found as before explained. + +In any method of extracting cube roots in which the slide has to be +adjusted to give equal readings on B and D, the author has found it of +advantage to adopt the following plan:—The cursor being set to, say, 4·8 +on A, bring a near _main_ division line on B, as 1·7, to the cursor; +then 1 on C is at 1·68 on D. The difference in the readings is two small +divisions on D, and moving the slide forward by _one-third the space +representing this difference_, we obtain 1·687 as the root required. +With a little practice it is possible to obtain more accurate results by +this method than by comparing the reading on D with that on the less +finely-graded B scale. + + + MISCELLANEOUS POWERS AND ROOTS. + +In addition to squares and cubes, certain other powers and roots may be +readily obtained with the slide rule. + +_Two-thirds Power._—The value of N^⅔ is found on A over ∛̅N on D. The +number of digits is decided by the rule for squares, working from the +number of digits in the cube root. It will often be found preferable to +treat N^⅔ as N ÷ ∛̅N, as in this way the magnitude of the result is much +more readily appreciated. + +_Three-two Power._—N^{³⁄₂} can be obtained by cubing the square root, +deciding the number of digits in each process. For the reason just +given, it is preferable to regard N^{³⁄₂} as N × √̅N. + +_Fourth Power._—For N^4 set the index of C to N on D and over N on C +read N^4 on A; or find the square of the square of N, deciding the +number of digits at each step. + +_Fourth Root._—Similarly for ∜̅N, take the square root of the square +root. + +_Four-third Power._—N^{⁴⁄₃} = N^{1·33} (useful in gas-engine diagram +calculations) is best treated as N × ∛̅N. + +Other powers can be found by repeated multiplication. Thus setting 1 on +B to N on A, we have on A, N^2 over N; N^3 over N^2; N^4 over N^3; N^5 +over N^4, etc. In the same way, setting N on B to N on D, we can read +such values as N^¾, N^⅞, etc. + + + POWERS AND ROOTS BY LOGARITHMS. + +For powers or roots other than those of the simple forms already +discussed, it is necessary to employ the usual logarithmic process. Thus +to find _a^n_ = _x_, we multiply the logarithm of _a_ by _n_, and find +the number _x_ corresponding to the logarithm so obtained. Similarly, to +find _ⁿ√̅a_ = _x_ we divide the logarithm of _a_ by _n_, and find the +number _x_ corresponding to the resulting logarithm. + +_The Scale of Logarithms._—Upon the back of the slide of the Gravêt and +similar slide rules there will be found three scales. One of +these—usually the centre one—is divided equally throughout its entire +length, and figured from right to left. It is sometimes marked L, +indicating that it is a scale giving logarithms. The whole scale is +divided primarily into ten equal parts, and each of these subdivided +into 50 equal parts. In the recess or notch in the right-hand end of the +rule is a reference mark, to which any of the divisions of this +evenly-divided scale can be set. + +As this decimally-divided scale is equal in length to the logarithmic +scale D, and is figured in the reverse direction, it results that when +the slide is drawn to the right so that the L.H. index of C coincides +with any number on D, the reading on the equally-divided scale will give +the decimal part of the logarithm of the number taken on D. Thus if the +L.H. index of C is placed to agree with 2 on D, the reading of the back +scale, taken at the reference mark, will be found to be 0·301, the +logarithm of 2. It must be distinctly borne in mind that the number so +obtained is the _decimal part_ or _mantissa_ of the logarithm of the +number, and that to this the characteristic must be prefixed in +accordance with the usual rule—viz., _The integral part, or +characteristic of a logarithm is equal to the number of digits in the +number, minus 1. If the number is wholly decimal, the characteristic is +equal to the number of cyphers following the decimal point, plus 1._ In +the latter case the characteristic is negative, and is so indicated by +having the minus sign written _over_ it. + +To obtain any given power or root of a number, the operation is as +follows:—Set the L.H. index of C to the given number on D, and turning +the rule over, read opposite the mark in the notch at the right-hand end +of the rule, the decimal part of the logarithm of the number. Add the +characteristic according to the above rule, and multiply by the exponent +of the power, or divide by the exponent of the root. Place the _decimal +part_ of the resultant reading, taken on the scale of equal parts, +opposite the mark in the aperture of the rule, and read the answer on D +under the L.H. index of C, pointing off the number of digits in the +answer in accordance with the number of the characteristic of the +resultant. + + + EX.—Evaluate 36^{1·414}. + + Set 1 on C to 36 on D and read the decimal part of log. 36 on the + scale of logarithms on the back of the slide. This value is found to + be 0·556. As there are two digits in the number, the characteristic + will be 1; hence log. 36 = 1·556. Multiply by 1·414, using the C and D + scales, and obtain 2·2 as the log. of the result. Set the decimal + part, 0·2, on the log. scale to the mark in the notch at the end of + the rule and read 1585 on D under 1 on C. Since the log. of the result + has a characteristic 2, there will be 3 digits in the result, which is + therefore read as 158·5. + + +This example will suffice to show the method of obtaining the nth power +or the _n_th root of _any_ number. + + + + + OTHER METHODS OF OBTAINING POWERS AND ROOTS. + + +A simple method of obtaining powers and roots, which may serve on +occasion, is by scaling off proportional lengths on the D scale (or the +A scale) of the ordinary rule. Thus, to determine the value of +1·25^{1·67} we take the actual length 1–1·25 on D scale, and increase it +by any convenient means in the proportion of 1 ∶ 1·67. Then with a pair +of dividers we set off this new length from 1, and obtain 1·44 as the +result. One convenient method of obtaining the desired ratio is by a +pair of proportional compasses. Thus to obtain 1·52^{¹⁷⁄₁₆}, the +compasses would be set in the ratio of 16 to 17, and the smaller end +opened out to include 1–1·52 on the D scale; the opening in the large +end of the compasses will then be such that setting it off from 1 we +obtain 1·56 on D as the result sought. + +[Illustration: FIG. 11.] + +The converse procedure for obtaining the _n_th root of a number N will +obviously resolve itself into obtaining (1)/(_n_)th of the scale length +1-N, and need not be further considered. + +Simple geometrical constructions are also used for obtaining scale +lengths in the required ratio. A series of parallel lines ruled on +transparent celluloid or stout tracing paper may be placed in an +inclined position on the face of the rule and adjusted so as to divide +the scale as desired. When much work is to be done which requires values +to be raised to some constant but comparatively low power, _n_, the +author has found the following device of assistance:—On a piece of thin +transparent celluloid a line OC is drawn (Fig. 11) and in this a point B +is taken such that (OC)/(OB) is the desired ratio. It is convenient to +make OB = 1–10 on the A scale, so that assuming we require a series of +values of _v_^{1·35}, OB would be 12·5 cm. and OC, 16·875 cm. On these +lines semi-circles are drawn as shown, both passing through the point O. + +Applying this cursor to the upper scales so that the point O is on 1 and +the semi-circle O M B passes through _v_ on A, the larger semi-circle +will give on A the value of _v^n_. Thus for _p_ _v^n_ = 39·5 × +4·9^{1·35}, set 1 on B to 39·5 on A (Fig. 12) and apply the cursor to +the working edge of B, so that O agrees with 1 and O M B passes through +4·9 on B. The larger semi-circle then cuts the edge of the slide on a +point, giving 337 on A as the result required. + +Of course any number of semi-circles may be drawn, giving different +ratios. If a number of evenly-spaced divisions are used as bases, the +device affords a simple means of obtaining a succession of small powers +or roots, while it also finds a use in determining a number of geometric +means between two values as is required in arranging the speed gears of +machine tools, etc. The converse operation of finding roots will be +evident as will also many other uses for which the device is of service. + +[Illustration: FIG. 12.] + +The lines should be drawn in Indian ink with a very sharp pen and on the +_under_ side of the celluloid so that the lines lie in close contact +with the face of the rule. + +_The Radial Cursor_, another device for the same purpose, is always used +in conjunction with the upper scales. As will be seen from Fig. 13, the +body of the cursor P carries a graduated bar S which can be removed in a +direction transverse to the rule, and adjusted to any desired position. +Pivoted to the lower end of S is a radial arm R of transparent celluloid +on which a centre line is engraved. + +A reference to the illustration will show that the principle involved is +that of similar triangles, the width of the slide being used as one of +the elements. Thus, to take a simple case, if 2 on S is set to the index +on P, and 1 on B is brought to N on A, then by swinging the radial arm +until its centre line agrees with 1 on C, we can read N^2 on A. +Evidently, since in the two similar triangles A O N^2 and N _t_ N^2 the +length of A O is twice that of N _t_, it results that A N^2 = 2 A N. In +general, then, to find the _n_th power of a number, we set the cursor to +1 or 10 on A, bring _n_ on the cross bar S to the index on the cursor, +and 1 on B to N on A. Then to 1 on C we set the line on the radial arm, +and under the latter read N^{_n_} on A. The inverse proceeding for +finding the _n_th root will be obvious. + +[Illustration: FIG. 13.] + +An advantage offered by this and analogous methods of obtaining powers +and roots is that the result is obtained on the ordinary scale of the +rule, and hence it can be taken directly into any further calculation +which may be necessary. + + + COMBINED OPERATIONS. + +Thus far the various operations have been separately considered, and we +now pass on to a consideration of the methods of working for solving the +various formulæ met with in technical calculations. We propose to +explain the methods of dealing with a few of the more generally used +expressions, as this will suffice to suggest the procedure in dealing +with other and more intricate calculations. In solving the following +problems, both the upper and lower scales are used, and the relative +value of the several scales must be observed throughout. Thus, in +solving such an expression as √((74·5)/(15·8)) = 6·86, the division is +first effected by setting 15·8 on B to 745 on A. From the relation of +the two parts of the upper scales (page 37) we know that such values as +7·45, 745, etc., will be taken on the _left-hand_ A and B scales, while +values as 15·8, 1580, etc., will be taken on the _right-hand_ A and B +scales. Hence, 15·8 on the R.H. B scale is set to 745 on the L.H. A +scale, and the result read on D under the index of C. Had both values +been taken on the L.H. A and B scales, or both on the R.H. A and B +scales, the results would have corresponded to _x_ = √((7·45)/(1·58)) = +2·17, or to _x_ =√((74·5)/(15·8)) = 2·17, _i.e_., to (6·86)/(√(10)). +Hence if a wrong choice of scales has been made, we can correct the +result by multiplying or dividing by √(10) as the case may require. If +the result is read on D, set to it the centre index (10) of B and read +the corrected result under the index of C. + +To solve _a_ × _b_^2 = _x_. Set the index of C to _b_ on D, and over _a_ +on B read _x_ on A. + +To solve (_a_^2)/(_b_) = _x_. Set _b_ on B to _a_ on D by using the +cursor, and over index of B read _x_ on A. + +To solve (_b_)/(_a_^2) = _x_. Set _a_ on C to _b_ on A, and over 1 on B +read _x_ on A. + +To solve (_a_ × _b_^2)/(_c_) = _x_. Set _c_ on B to _b_ on D, and over +_a_ on B read _x_ on A. + +To solve (_a_ × _b_)^2 = _x_. Set 1 on C to _a_ on D, and over _b_ on C +read _x_ on A. + +To solve ((_a_)/(_b_))^2 = _x_. Set _b_ on C to _a_ on D, and over 1 on +C read _x_ on A. + +To solve √(_a_ × _b_) = _x_. Set 1 on B to _a_ on A, and under _b_ on B +read _x_ on D. + +To solve √((_a_)/(_b_)) = _x_. Set _b_ on B to _a_ on A, and under 1 on +C read _x_ on D. + +To solve _a_ (_b_)/(_c_^2) = _x_. Set _b_ on C to _c_ on D and over _a_ +on B read _x_ on A. + +To solve _c_√((_a_)/(_b_)) = _x_. Set _b_ on B to _a_ on A, and under +_c_ on C read _x_ on D. + +To solve (√_̅a_)/(_b_) = _x_. Set _b_ on C to _a_ on A, and under 1 on C +read _x_ on D. + +To solve (_a_)/(√_̅b_) = _x_. Set _b_ on B to _a_ on D, and under 1 on C +read _x_ on D. + +To solve _b_√_̅a_ = _x_. Set 1 on C to _b_ on D, and under _a_ on B read +_x_ on D. + +To solve √(_a_^3) = _x_. Treat as _a_√_̅a_. + +To solve _a_√(_b_^3) = _x_. Treat as _a_√_̅b_ × _b_. + +To solve (√_̅a_^3)/(_b_) = _x_. Treat as (√_̅a_ × _a_)/(_b_). + +To solve √((_a_^3)/(_b_)) = _x_. Treat as (√_̅a_ × _a_)/(√_̅b_) = +√((_a_)/(_b_)) × _a_. + +To solve √((_a_ × _b_)/(_c_)) = _x_. Set _c_ on B to _a_ on A, and under +_b_ on B read _x_ on D. + +To solve (_a_ × _b_)/(√_̅c_) = _x_. Set _c_ on B to _b_ an D, and under +_a_ on C read _x_ on D. + +To solve √((_a_^2 × _b_)/(_c_)) = _x_. Set _c_ on B to _a_ on D, and +under _b_ on B read _x_ on D. + +To solve (_a_^2 × _b_^2)/(_c_) = _x_. Set _c_ on B to _a_ on D, and over +_b_ on C read _x_ on A. + +To solve (_a_√_̅b_)/(_c_) = _x_. Set _c_ on C to _b_ on A, and under _a_ +on C read _x_ on D. + +To solve ((_a_ × √_̅b_)/(_c_))^2 = _x_. Set _c_ on C to _a_ on D, +and over _b_ on B read _x_ on A. + + + HINTS ON EVALUATING EXPRESSIONS. + +As a general rule, the use of cubes and higher powers should be avoided +whenever possible. Thus, in the foregoing section, we recommend treating +an expression of the form _a_√(_b_^3) as _a_ × _b_ × √_̅b_; the +magnitudes of the values thus met with are more easily appreciated by +the beginner, and mistakes in estimating the large numbers involved in +cubing are avoided. + + + EX.—7·3 × √(57^3) = 3140. + + Set 1 on C to 57 on D; bring cursor to 57 on B (R.H., since 57 has an + _even_ number of digits); bring 1 on C to cursor, and under 7·3 on C + read 3140 on D. As a rough estimate we have √(57), about 8; 8 × 57, + about 400; 400 × 7, gives 2800, showing the result consists of 4 + figures. + + +An expression of the form _a_∛(_b_^2), or _a_ _b_^⅔, is better dealt +with by rearranging as _a_ × (_b_)/(∛_b_). + + + EX.—3·64∛(4·32^2) = 9·65. + + Set cursor to 4·32 on A, and move the slide until 1·63 is found + simultaneously under the cursor on B and on D under 1 on C; bring + cursor to 1 on C; 4·32 on C to cursor, and _over_ 3·64 on D read 9·65 + on C. (Note that in this case it is convenient to read the answer on + the _slide_; see page 22). From the slide rule we know ∛(4·32) = about + 1·6; this into 4·32 is roughly 3; 3·64 × 3 is about 10, showing the + answer to be 9·65. + + +Similarly products of the form _a_ × _b_^{⁴⁄₃} are best dealt with as +_a_ × _b_ × ∛_b_. + +Factorising expressions sometimes simplifies matters, as, for instance, +in _x_^4 − _y_^4 = (_x_^2 + _y_^2)(_x_^2 − _y_^2). Here, working with +the fourth powers involves large numbers and the troublesome +determination of the number of digits in each factor; but squares are +read on the rule at once, the number of digits is obvious, and, in +general, the method should give a more accurate result. Take the +expression, D_{1} = ∛((D^4 − _d_^4)/(D)) giving the diameter D_{1} of a +solid shaft equal in torsional strength to a hollow shaft whose external +and internal diameters are D and _d_ respectively. Rearranging as D_{1} += ∛(((D^2 + _d_^2)(D^2 − _d_^2))/(D)) and taking, as an example, D = 15 +in. and _d_ = 7 in., we have D^2 + _d_^2 = 274 and D^2 − _d_^2 = 176; +hence D_1 = ∛((274 × 176)/(15)) = ∛(3210) = 14·75 in. + +_Reversed Scale Notation._—With expressions of the form 1 − _x_, or 100 +− _x_, it is often convenient to regard the scales as having their +notation reversed, _i.e._, to read the scale backwards. When this is +done the D scale is read as shown on the lower line— + + Direct Notation 1 2 3 4 5 6 7 8 9 10 + D Scale + Reversed Notation 9 8 7 6 5 4 3 2 1 0 + +The new reading can be found by subtracting the ordinary reading from 1, +10, 100, etc., according to the value assigned to the R.H. index, but +actually it is unnecessary to make this calculation, as with a little +practice it is quite an easy matter to read both the main and +subdivisions in the reversed order. Applications are found in plotting +curves, trigonometrical formulæ, etc. + + + EX.—Find the per cent. of slip of a screw propeller from + + 100 − S = (10133V)/(PR) + + taking the speed, V, as 15 knots, the pitch of the propeller, P, as 27 + ft. 6 in., and the revolutions per minute, R, as 60. + + Set 27·5 on B to 10133 on A (N.B.—Take the setting near the _centre_ + index of A); bring the cursor to 15 on B and 60 on B to cursor. + Reading the L.H. A scale backwards, the slip, S, = 8 per cent. is + found on A over 10 on B. + + +_Percentage Calculations._—To increase a quantity by _x_ per cent. we +multiply by 100 + _x_; to diminish a quantity by _x_ per cent. we +multiply by 100 − _x_. Hence, to add _x_ per cent., set 100 + _x_ on C +to 1 on D and read new values on D under original values on C. To deduct +_x_ per cent. read the D scale backwards from 10 and set R.H. index of C +to _x_ per cent. so read. Then read as before. + + + GAUGE POINTS. + +Special graduations, marking the position of constant factors which +frequently enter into engineering calculations, are found on most slide +rules. Usually the values of π = 3·1416 and (π)/(4) = 0·7854—the “gauge +points” for calculating the circumference and area of a circle—are +marked on the upper scales. The first should be given on the lower +scales also. Marks _c_ and _c_^1 are sometimes found on the lower scales +at 1·128 = √((4)/(π)) and at 3·568 = √((40)/(π)). These are useful in +calculating the contents of cylinders and are thus derived:—Cubic +contents of cylinder of diameter _d_ and length _l_ = (π)/(4)_d_^2_l_; +substituting for (π)/(4) its reciprocal (4)/(π), the formula becomes +(_d_^2)/(1·273 × _l_), and by taking the square root of the fractional +part we have (_d_)/(1·128)^2 × _l_. This is now in a very convenient +form, since by setting the gauge point _c_ on C to _d_ on D, we can read +over _l_ on B the cubic contents on A. This example indicates the +principle to be followed in arranging gauge points. Successive +multiplication is avoided by substituting the reciprocal of the +constant, thus bringing the expression into the form (_a_ × _b_)/(_c_), +which, as we know, can be resolved by one setting of the slide. The +advantage of dividing _d_ before squaring is also evident. The mark +_c_^1 = _c_ × √(10) is used if it is necessary to draw the slide more +than one-half its length to the right. + +A gauge point, M, at 31·83 = (100)/(π) is found on the upper scales of +some rules. Setting this point on B to the diameter of a cylinder on A, +the circumference is read over 1 or 100 on B or the area of the curved +surface over the length on B. + +As another example of establishing a gauge point, we will take the +formula for the theoretical delivery of pumps. If _d_ is the diameter of +the plunger in inches, _l_ the length of stroke in feet, and Q the +delivery in gallons, we have + +Q = _d_^2 × (π)/(4) × _l_ × (12)/(277). (N.B.—277 cubic inches = 1 +gallon.) + +Multiplying out the constant quantities and taking its reciprocal, we +readily transform the statement into Q = (_d_^2_l_)/(29·4) or +((_d_)/(5·42))^2 × _l_. Hence set gauge point 5·42 on C to _d_ on +D and over length of stroke in feet on B, read delivery in gallons per +stroke on A; or over piston speed in feet per minute on B, read +theoretical delivery in gallons per minute on A. + +Several examples of gauge points will be found in the section on +calculating the weights of metal (see pages 59 and 60). In most cases +their derivation will be evident from what has been said above. In the +case of the weight of spheres, we have Vol. = 0·5236_d_^3, and this +multiplied by the weight of 1 cubic inch of the material will give the +weight W in lb. Hence for cast-iron, W = 0·5236 × _d_^3 × 0·26, which is +conveniently transformed into W = (_d_ × _d_^2)/(7·35) as in the example +on page 60. + +With these examples no difficulty should be experienced in establishing +gauge points for any calculation in which constant factors recur. + +_Marking Gauge Points._—The practice of marking gauge points by lines +extending to the working edge of the scale is not to be recommended, as +it confuses the ordinary reading of the scales. Generally speaking, +gauge points are only required occasionally, and if they are placed +clear of the scale to which they pertain, but near enough to show the +connection, they can be brought readily into a calculation by means of +the cursor. Usually there is sufficient margin above the A scale and +below the D scale for various gauge points to be marked. Another plan +consists in cutting two nicks in the upper and lower edges of the cursor +near the centre and about ⅛ in. apart. These centre pieces, when bent +out, form a tongue, which are in line with the cursor line and run +nearly in contact with the square and bevelled edges of the rule +respectively. A fine line in the tongue can then be set to gauge points +marked on these two edge strips, the ordinary measuring graduations +being removed, if desired, by a piece of fine sand-paper. + +For gauge points marked on the face of the rule, the author prefers two +fine lines drawn at 45°—thus, ✕—and crossing in the exact point which it +is required to indicate. With the “cross” gauge point the meeting lines +facilitate the placing of the cursor, and an exact setting is readily +made.[7] All lines should be drawn in Indian ink with a very sharp +drawing pen. For a more permanent marking the Indian ink may be rubbed +up in glacial acetic acid or the special ink for celluloid may be used. +If any difficulty is found in writing the distinguishing signs against +the gauge point, the inscription may be formed by a succession of small +dots made with a sharp pricker. + + + EXAMPLES IN TECHNICAL CALCULATIONS. + +In order to illustrate the practical value of the slide rule, we now +give a number of examples which will doubtless be sufficient to suggest +the methods of working with other formulæ. A few of the rules give +results which are approximate only, but in all cases the degree of +accuracy obtained is well within the possible reading of the scales. In +many cases the rules given may be modified, if desired, by varying the +constants. In most of the examples the particular formula employed will +be evident from the solution, but in a few of the more complicated cases +a separate statement has been given. + + + MENSURATION, ETC. + +Given the chord _c_ of a circular arc, and the vertical height _h_, to +find the diameter _d_ of the circle. + +Set the height _h_ on B to half the chord on D, and over 1 on B read _x_ +on A. Then _x_ + _h_ = _d_. + + + EX.—_c_ = 6; _h_ = 2; find _d_. Set 2 on B to 3 on D, and over 1 on B + read 4·5 on A. Then 4·5 + 2 = 6·5 = _d_. + + +Given the radius of a circle _r_, and the number of degrees _n_ in an +arc, to find the length _l_ of the arc. + +Set _r_ on C to 57·3 on D, and over any number of degrees _n_ on D read +the (approximate) length of the arc on C. + + + EX.—_r_ = 24; _n_ = 30; find _l_. + + Set 24 on C to 57·3 on D, and over 30 on D read 12·56 = _l_ on C. + + +Given the diameter _d_ of a circle in _inches_, to find the +circumference _c_ in _feet_. + +Set 191 on C to 50 on D, and under any diameter in inches on C read +circumference _c_, in feet on D. + + + EX.—Find the circumference in feet of a pulley 17 in. in diameter. Set + 191 on C to 50 on D, and under 17 on C read 4·45 ft. on D. + + +Given the diameter of a circle, to find its area. + +Set 0·7854 on B to 10 (centre index) on A and over any diameter on D +read area on B. + +When the rule has a special graduation line = 0·7854, on the right-hand +scale of B, set this line to the R.H. index of A and read off as above. +If only π is marked, set this special graduation on B to 4 on A. + +On the C and D scales of some rules a gauge point marked _c_ will be +found indicating √((4)/(π)) = 1·1286. In this case, therefore, set 1 on +C to gauge point _c_ on D, and read area on A as above. If the gauge +point _c_′ is used, divide the result by 10. Or set _c_ on C, to +diameter on D, and over index of B read area on A. Cursors are supplied, +having _two_ lines ruled on the glass, the interval between them being +equal to (4)/(π) = 1·273 on the A scale. In this case, if the right hand +of the two cursor lines be set to the diameter on D, the _area_ will be +read on A under the _left_-hand cursor line. For diameters less than +1·11 it is necessary to set the middle index of B to the L.H. index of +A, reading the areas on the L.H. B scale. The confusion which in general +work is sometimes caused by the use of two cursor lines might be +obviated by making the left-hand line in two short lengths, each only +just covering the scales. + +Given diameter of circle _d_ in _inches_, to find area _a_ in square +_feet_. + +Set 6 on B to 11 on A, and over diameter in inches on D read area in +square feet on B. + +To find the surface in square feet of boiler flues, condenser tubes, +heating pipes, etc., having given the diameter in inches and length in +feet. + +Find the circumference in feet as above and multiply by the length in +feet. + + + EX.—Find the heating surface afforded by 160 locomotive boiler tubes + 1¾in. in diameter and 12 ft. long. + + Set 191 on C to 50 on D; bring cursor 1·75 on C, L.H. index of C to + cursor; cursor to 12 on C; 1 on C to cursor; and under 160 on C read + 880 sq. ft. of heating surface on D. + + +If the dimensions are in the same denomination and the rule has a gauge +point M at 31·83 (= (100)/(π)), set this mark on B to diameter of +cylinder on A, and read cylindrical surface on A over length on B. + +To find the side _s_ of a square, equal in area to a given rectangle of +length _l_ and breadth _b_. + +Set R.H. or L.H. index of B to _l_ on A, and under _b_ on B read _s_ on +D. + + + EX.—Find the side of a square equal in area to a rectangle in which + _l_ = 31 ft. and _b_ = 5 ft. + + Set the (R.H.) index of B to 31 on A, and under 5 on B read 12·45 ft. + on D. + + +To find various lengths _l_ and breadths _b_ of a rectangle, to give a +constant area _a_. + +Invert the slide and set the index of Ɔ to the given area on D. Then +opposite any length _l_ on Ɔ find the corresponding breadth _b_ on D. + + + EX.—Find the corresponding breadths of rectangular sheets, 16, 18, 24, + 36, and 60 ft. long, to give a constant area of 72 sq. ft. + + Set the R.H. index of Ɔ to 72 on D, and opposite 16, 18, 24, 36, and + 60 on Ɔ read 4·5, 4, 3, 2, and 1·2 ft., the corresponding breadths on + D. + + +To find the contents in cubic feet of a cylinder of diameter _d_ in +inches and length _l_ in feet. + +Find area in feet as before, and multiply by the length. + +If dimensions are all in inches or feet, set the mark _c_ (= 1·128) on C +to diameter on D and over length on B, read cubic contents on A. + +To find the area of an ellipse. + +Set 205 on C to 161 on D; bring cursor to length of major axis on C, 1 +on C to cursor, and under length of minor axis on C read area on D. + + + EX.—Find the area of an ellipse the major and minor axes of which are + 16 in. and 12 in. in length respectively. + + Set 205 on C to 161 on D; bring cursor to 16 on C, 1 on C to cursor, + and under 12 on C read 150·8 in. on D. + + +To find the surface of spheres. + +Set 3·1416 on B to R.H. or L.H. index of A, and over diameter on D read +by the aid of the cursor, the convex surface on B. + +To find the cubic contents of spheres. + +Set 1·91 on B to diameter on A, and over diameter on C read cubic +contents on A. + + + WEIGHTS OF METALS. + +To find the weight in lb. per lineal foot of square bars of metal. + +Set index of B to weight of 12 cubic inches of the metal (_i.e._, one +lineal foot, 1 square inch in section) on A, and over the side of the +square in inches on C read weight in lb. on A. + + + EX.—Find the weight per foot length of 4½in. square wrought-iron bars. + + Set middle index of B to 3·33 on A, and over 4½ on C read 67·5 lb. on + A. + + +(N.B.—For other metals use the corresponding constant in column (2), +below). + +To find the weight in lb. per lineal foot of round bars. + +Set R.H. or L.H. index of B to weight of 12 cylindrical inches of the +metal on A (column (4), below), and opposite the diameter of the bar in +inches on C, read weight in lb. per lineal foot on A. + + + EX.—Find the weight of 1 lineal foot of 2 in. round cast steel. + + Set L.H. index of B to 2·68 on A, and over 2 on C read 10·7 lb. on A. + + +To find the weight of flat bars in lb. per lineal foot. + +Set the breadth in inches on C to (1)/(weight of 12 cub. in.) of the +metal (column (3), below) on D, and above the thickness on D read weight +in lb. per lineal foot on C. + + + EX.—Find the weight per lineal foot of bar steel, 4½in. wide and ⅝in. + thick. + + Set 4·5 on C to 0·294 on D, and over 0·625 on D read 9·56 lb. per + lineal foot on C. + + +To find the weight per square foot of sheet metal, set the weight per +cubic foot of the metal (col. 1) on C to 12 on D, and + + ────────────┬──────────┬──────────┬──────────┬─────────── + │ (1) │(2) │ (3) │ (4) + │Weight in │Weight of │ (1)/(Wt. │ Weight of + Metals. │ lb. per │12 cubic │of 12 cub.│ 12 + │cubic ft. │in. │ in.) │cylindrical + │ │ │ │ in. + ────────────┼──────────┼──────────┼──────────┼─────────── + Wrought iron│480 │3·33 │0·300 │2·62 + Cast iron │450 │3·125 │0·320 │2·45 + Cast steel │490 │3·40 │0·294 │2·68 + Copper │550 │3·82 │0·262 │3·00 + Aluminium │168 │1·166 │0·085 │0·915 + Brass │520 │3·61 │0·277 │2·83 + Lead │710 │4·93 │0·203 │3·87 + Tin │462 │3·21 │0·312 │2·52 + Zinc (cast) │430 │2·98 │0·335 │2·34 + „ (sheet) │450 │3·125 │0·320 │2·45 + ────────────┴──────────┴──────────┴──────────┴─────────── + +above the thickness of the plate in inches on D read weight in lb. per +square foot on C. + + + EX.—Find the weight in lb. per square foot of aluminium sheet ⅜in. + thick. + + Set 168 on C to 12 on D, and over 0·375 on D read 5·25 lb. on C. + + +To find the weight of pipes in lb. per lineal foot. + +Set mean diameter of the pipe in inches (_i.e._, internal diameter +_plus_ the thickness, or external diameter _minus_ the thickness) on C +to the constant given below on D, and over the thickness on D read +weight in lb. per lineal foot on C. + + ┌────────────┬─────────────────────┬─────────────────────┐ + │ Metals. │ Constant for Pipes. │Constant for Spheres.│ + ├────────────┼─────────────────────┼─────────────────────┤ + │Wrought iron│ 0·0955 │ 6·87 │ + │Cast iron │ 0·1020 │ 7·35 │ + │Steel │ 0·0936 │ 6·73 │ + │Brass │ 0·0882 │ 6·35 │ + │Copper │ 0·0834 │ 6·00 │ + │Lead │ 0·0646 │ 4·65 │ + └────────────┴─────────────────────┴─────────────────────┘ + + + EX.—Find the weight per foot of cast-iron piping 4 in. internal + diameter and ½in. thick. + + Set 4·5 on C to 0·102 on D, and over 0·5 on D read 22·1 lb. on C, the + required weight. + + +To find the weight in lb. of spheres or balls, given the diameter in +inches. (W = 0·5236_d_^3 × wt. of 1 cub. in. of material). + +Set the constant for spheres (given above) on B to diameter in inches on +A, and over diameter on C read weight in lb. on A. + + + EX.—Find the weight of a cast-iron ball 7½in. in diameter. + + Set 7·35 on B to 7·5 on A, and over 7·5 on C read 57·7 lb. on A. + + +To find diameter in inches of a sphere of given weight. + +Set the cursor to the given weight in lb. on A, and move the slide until +the same number is found on C under the cursor that is simultaneously +found on A over the constant for the sphere on B. + + + EX.—Find diameter in inches of a sphere of cast-iron to weigh 7½lb. + + Setting the cursor to 7·5 on A, and moving the slide, it is found that + when 3·8 on C falls under the cursor, 3·8 on A is simultaneously found + over 7·35 on B. The required diameter is therefore 3·8 in. + + +The rules for cubes and cube roots (page 40) should be kept in view in +solving the last two examples. + + + FALLING BODIES. + +To find velocity in feet per second of a falling body, given the time of +fall in seconds. + +Set index on C to time of fall on D, and under 32·2 on C read velocity +in feet per second on D. + +To find velocity in feet per second, given distance fallen through in +feet. + +Set 1 on C to distance fallen through on A, and under 64·4 on B read +velocity in feet per second on D. + + + EX.—Find velocity acquired by falling through 14 ft. + + Set (R.H.) index of C to 14 on A, and under 64·4 on B read 30 ft. per + second on D. + + +To find distance fallen through in feet in a given time. + +Set index of C to time in seconds on D, and over 16·1 on B read distance +fallen through in feet on A. + + + CENTRIFUGAL FORCE. + +To find the centrifugal force of a revolving mass in lb. + +Set 2940 on B to revolutions per minute on D; bring cursor to weight in +lb. on B; index of B to cursor, and over radius in feet on B read +centrifugal force in lb. on A. + +To find the centrifugal stress in lb. per square inch, in rims of +revolving wheels of cast iron. + +Set 61·3 on C to the mean diameter of the wheel in feet on D, and over +revolutions per minute on C read stress per square inch on A. + + + EX.—Find the stress per square inch in a cast-iron fly-wheel rim 8 ft. + in diameter and running at 120 revolutions per minute. + + Set 61·3 on C to 8 on D, and over 120 on C read 245 lb. per square + inch on A. + + + THE STEAM ENGINE. + +Given the stroke and number of revolutions per minute, to find the +piston speed. + +Set stroke in inches on C to 6 on D, and over number of revolutions on D +read piston speed in feet per minute on C. + +To find cubic feet of steam in a cylinder at cut-off, given diameter of +cylinder and period of admission in inches. + +Set 2200 on B to cylinder diameter on D, and over period of admission on +B read cubic feet of steam on A. + + + EX.—Cylinder diameter 26 in., stroke 40 in., cut-off at ⅝ of stroke. + Find cubic feet of steam used (theoretically) per stroke. + + Set 2200 on B to 26 on D, and over 40 × ⅝ or 25 in. on B, read 7·68 + cub. ft. on A, as the number of cubic feet of steam used per stroke. + + +Given the diameter of a cylinder in inches, and the pressure in lb. per +square inch, to find the load on the piston in tons. + +Set pressure in lb. per square inch on B to 2852 on A, and over cylinder +diameter in inches on D read load on piston in tons on B. + + + EX.—Steam pressure 180 lb. per square inch; cylinder diameter, 42 in. + Find load in tons on piston. + + Set 180 on B to 2852 on A, and over 42 on D read 111 tons, the gross + load, on B. + + +Given admission period and absolute initial pressure of steam in a +cylinder, to find the pressure at various points in the expansion period +(isothermal expansion). + +Invert the slide and set the admission period, in inches, on Ɔ to the +initial pressure on D; then under any point in the expansion stroke on Ɔ +find the corresponding pressure on D. + + + EX.—Admission period 12 in., stroke 42 in., initial pressure 80 lb. + per square inch. Find pressure at successive fifths of the expansion + period. + + Set 12 on Ɔ to 80 on D, and opposite 18, 24, 30, 36 and 42 in. of the + whole stroke on Ɔ find the corresponding pressures on D:—53·3, 40, 32, + 26·6 and 22·8 lb. per square inch. + + +To find the mean pressure constant for isothermally expanding steam, +given the cut-off as a fraction of the stroke. + +Find the logarithm of the ratio of the expansion _r_, by the method +previously explained (page 46). Prefix the characteristic and to the +number thus obtained, on D, set 1 on C. Then under 2·302 on C read _x_ +on D. To _x_ + 1 on D set _r_ on C, and under index of C read mean +pressure constant on D. The latter, multiplied by the initial pressure, +gives the mean forward pressure throughout the stroke. (N.B.—Common log. +× 2·302 = hyperbolic log.) + + + EX.—Find the mean pressure constant for a cut-off of ¼th, or a ratio + of expansion of 4. + + Set (L.H.) index of C to 4 on D, and on the reverse side of the slide + read 0·602 on the logarithmic scale. The characteristic = 0; hence to + 0·602 on D set (R.H.) index of C, and under 2·302 on C read 1·384 on + D. Add 1, and to 2·384 thus obtained on D set _r_ (= 4) on C, and + under 1 on C read 0·596, the mean pressure constant required. + + +Mean pressure constants for the most usual degrees of cut-off are given +below:— + + Cut-off in fractions of stroke Mean pressure constant + ¾ 0·968 + ⁷⁄₁₀ 0·952 + ⅔ 0·934 + ⅝ 0·919 + ⅗ 0·913 + ½ 0·846 + ⅖ 0·766 + ⅜ 0·750 + ⅓ 0·699 + ³⁄₁₀ 0·664 + ¼ 0·596 + ⅕ 0·522 + ⅙ 0·465 + ⅐ 0·421 + ⅛ 0·385 + ⅑ 0·355 + ⅒ 0·330 + ¹⁄₁₁ 0·309 + ¹⁄₁₂ 0·290 + ¹⁄₁₃ 0·274 + ¹⁄₁₄ 0·260 + ¹⁄₁₅ 0·247 + ¹⁄₁₆ 0·236 + +To find mean pressure:—Set 1 on C to constant on D, and under initial +pressure on C read mean pressure on D. + +Given the absolute initial pressure, length of stroke, and admission +period, to find the absolute pressure at any point in the expansion +period, it being assumed that the steam expands adiabatically. (P_{2} = +(P_{1})/(R^{¹⁰⁄₉}) in which P_{1} = initial pressure and P_{2} the +pressure corresponding to a ratio of expansion R.) + +Set L.H. index of C to ratio of expansion on D, and read on the back of +the slide the decimal of the logarithm. Add the characteristic, and to +the number thus obtained on D set 9 on C, and read off the value found +on D under the index of C. Set this number on the logarithmic scale to +the index mark, in the opening on the back of the rule, and under L.H. +index of C read the value of R^{¹⁰⁄₉} on D. The initial pressure divided +by this value gives the corresponding pressure due to the expansion. + + + EX.—Absolute initial pressure 120 lb. per square inch; stroke, 4 ft.; + cut-off ¼. Find the respective pressures when ½ and ¾ths of the stroke + have been completed. + + In the first case R = 2. Therefore setting the L.H. index of C to 2 on + D, we find the decimal of the logarithm on the back of the slide to be + 0·301. The characteristic is 0, so placing 9 on C to 0·301 on D, we + read 0·334 as the value under the R.H. index of C. (N.B.—In locating + the decimal point it is to be observed that the log. of R has been + multiplied by 10, in accordance with the terms of the above + expression.) Setting this number on the logarithmic scale to the back + index, the value of R^{¹⁰⁄₉} is found on D, under the L.H. index of C, + to be 2·16. Setting 120 on C to this value, it is found that the + pressure at ½ stroke, read on C over the R.H. index of D, is 55·5 lb. + per square inch. In a similar manner, the pressure when ¾ths of the + stroke is completed is found to be 35·4 lb. per square inch. + + +For other conditions of expanding steam, or for gas or air, the method +of procedure is similar to the above. + +To find the horse-power of an engine, having given the mean _effective_ +pressure, the cylinder diameter, stroke, and number of revolutions per +minute. + +To cylinder diameter on D set 145 on C; bring cursor to stroke in feet +on B, 1 on B to cursor, cursor to number of revolutions on B, 1 on B to +cursor, and over mean effective pressure on B find horse-power on A. + +(N.B.—If stroke is in inches, use 502 in place of 145 given above.) + + + EX.—Find the indicated horse-power, given cylinder diameter 27 in., + mean effective pressure 38 lb. per square inch, stroke 32 in., + revolutions 57 per minute. + + Set 502 on C to 27 on D, bring cursor to 32 on B, 1 on B to cursor, + cursor to 57 on B, 1 on B to cursor, and over 38 on B read 200 I.H.P. + on A. + + +To determine the horse-power of a compound engine, invert the slide and +set the diameter of the _high_-pressure cylinder on Ɔ to the cut-off in +that cylinder on A. Use the number then found on A over the diameter of +the _low_-pressure cylinder on Ɔ as the cut-off in that cylinder, +working with the same pressure and piston speed, and calculate the +horse-power as for a single cylinder. + +To find the cylinder ratio in compound engines, invert the slide and set +index of Ɔ to diameter of the low-pressure cylinder on D. Then over the +diameter of the high-pressure cylinder on C, read cylinder ratio on A. + + + EX.—Diameter of high-pressure cylinder 7¾in., low-pressure 15 in. Find + cylinder ratio. + + Set index on Ɔ to 15 on D, and over 7·75 on Ɔ read 3·75, the required + ratio, on A. + + +The cylinder ratios of triple or quadruple-expansion engines may be +similarly determined. + + + EX.—In a quadruple-expansion engine, the cylinders are 18, 26, 37, and + 54 inches in diameter. Find the respective ratios of the high, first + intermediate, and second intermediate cylinders to the low-pressure. + + Set (R.H.) index of Ɔ to 54 on D, and over 18, 26, and 37 on Ɔ read 9, + 4·31, and 2·13, the required ratios, on A. + + +Given the mean effective pressures in lb. per square inch in each of the +three cylinders of a triple-expansion engine, the I.H.P. to be developed +in each cylinder, and the piston speed, to find the respective cylinder +diameters. + +Set 42,000 on B to piston speed on A; bring cursor to mean effective +pressure in low-pressure cylinder on B, index of B to cursor, and under +I.H.P. on A read low-pressure cylinder diameter on C. To find the +diameters of the high-pressure and intermediate-pressure cylinders, +invert the slide and place the mean pressure in the low-pressure +cylinder on ᗺ to the diameter of that cylinder on D. Then under the +respective mean pressures on ᗺ read corresponding cylinder diameters on +D. + + + EX.—The mean effective pressures in the cylinders of a + triple-expansion engine are:—L.P., 10·32; I.M.P., 27·5; and H.P., 77·5 + lb. per square inch. The piston speed is 650 ft. per minute, and the + I.H.P. developed in each cylinder, 750. Find the cylinder diameters. + + Set 42,000 on B to 650 on A, and bring cursor to 10·32 on B. Bring + index of B to cursor, and under 750 on A read 68·5 in. on C, the L.P. + cylinder diameter. Invert the slide, and placing 10·32 on ᗺ to 68·5 on + D, read, under 27·5 on ᗺ, the I.M.P. cylinder diameter = 42 in., on D; + also under 77·5 on ᗺ read the H.P. cylinder diameter = 25 in., on D. + + +To compute brake or dynamometrical horse-power. + +Set 525 on C to the total weight in lb. acting at the end of the lever +(or pull of spring balance in lb.) on D; set cursor to length of lever +in feet on C, bring 1 on C to cursor, and under number of revolutions +per minute on C find brake horse-power on D. + +Given cylinder diameter and piston speed in feet per minute, to find +diameter of steam pipe, assuming the maximum velocity of the steam to be +6000 ft. per minute. + +Set 6000 on B to cylinder diameter on D, and under piston speed on B +read steam pipe diameter on D. + +Given the number of revolutions per minute of a Watt governor, to find +the vertical height in inches, from the plane of revolution of the balls +to the point of suspension. + +Set revolutions per minute on C to 35,200 on A, and over index of B read +height on A. + +Given the weight in lb. of the rim of a cast-iron fly-wheel, to find the +sectional area of the rim in square inches. + +Set the mean diameter of the wheel in feet on C to 0·102 on D, and under +weight of rim on C find area on D. + +Given the consumption of coal in tons per week of 56 hours, and the +I.H.P., to find the coal consumed per I.H.P. per hour. + +Set I.H.P. on C to 40 on D, and under weekly consumption on C read lb. +of coal per I.H.P., per hour on D. + + + EX.—Find coal used per I.H.P. per hour, when 24 tons is the weekly + consumption for 300 I.H.P. + + Set 300 on C to 40 on D, and under 24 on C read 3·2 lb. per I.H.P. per + hour on D. + + +(N.B.—For any other number of working hours per week divide 2240 by the +number of working hours, and use the quotient in place of 40 as above.) + +To find the tractive force of a locomotive. + +Set diameter of driving wheel in inches on B to diameter of cylinder in +inches on D, and over the stroke in inches on B read on A, tractive +force in lb. for each lb. of effective pressure on the piston. + + + STEAM BOILERS. + +To find the bursting pressure of a cylindrical boiler shell, having +given the diameter of shell and the thickness and ultimate strength of +the material. + +Set the diameter of the shell in inches on C to twice the thickness of +the plate on D, and under strength of material per square inch on C read +bursting pressure in lb. per square inch on D. + + + EX.—Find the bursting pressure of a cylindrical boiler shell 7 ft. 6 + in. in diameter, with plates ½in. thick, assuming an ultimate strength + of 50,000 lb. per square inch. + + Set 90 on C to 1·0 on D, and under 50,000 on C find 555 lb. on D. + + +To find working pressure for Fox’s corrugated furnaces by Board of Trade +rule. + +Set the least outside diameter in inches on C to 14,000 on D, and under +thickness in inches on C read working pressure on D in lb. per square +inch. + +To find diameter _d_ in inches, of round steel for safety valve springs +by Board of Trade rule. + +Set 8000 on C to load on spring in lb. on D, and under the mean diameter +of the spring in inches on C read _d_^3 on D. Then extract the cube root +as per rule. + + + SPEED RATIOS OF PULLEYS, ETC. + +Given the diameter of a pulley and its number of revolutions per minute, +to find the circumferential velocity of the pulley or the speed of +ropes, belts, etc., driven thereby. + +Set diameter of pulley in inches on C to 3·82 on D, and over revolutions +per minute on D read speed in feet per minute on C. + + + EX.—Find the speed of a belt driven by a pulley 53 in. in diameter and + running at 180 revolutions per minute. + + Set 53 on C to 3·82 on D, and over 180 on D read 2500 ft. per minute + on C. + + EX.—Find the speed of the pitch line of a spur wheel 3 ft. 6 in. in + diameter running at 60 revolutions per minute. + + Set 42 in. on C to 3·82 on D, and over 60 on D read 660 ft. per minute + on C. + + +Given diameter and number of revolutions per minute of a driving pulley, +and the diameter of the driven pulley, to find the number of revolutions +of the latter. + +Invert the slide and set diameter of driving pulley on Ɔ to given number +of its revolutions on D; then opposite diameter of any driven pulley on +Ɔ read its number of revolutions on D. + + + EX.—Diameter of driving pulley 10 ft.; revolutions per minute 55; + diameter of driven pulley 2 ft. 9 in. Find number of revolutions per + minute of latter. + + Set 10 on Ɔ to 55 on D, and opposite 2·75 on Ɔ read 200 revolutions on + D. + + + BELTS AND ROPES. + +To find the ratio of tensions in the two sides of a belt, given the +coefficient of friction between belt and pulley μ and the number of +degrees θ in the arc of contact (log. R = (μθ)/(132)). + +Set 132 on C to the coefficient of friction on D, and read off the value +found on D under the number of degrees in the arc of contact on C. Place +this value on the scale of equal parts on the back of the slide, to the +index mark in the aperture, and read the required ratio on D under the +L.H. index of C. + + + EX.—Find the tension ratio in a belt, assuming a coefficient of + friction of 0·3 and an arc of contact of 120 degrees. + + Set 132 on C to 0·3 on D, and under 120 on C read 0·273. Place this on + the scale to the index on the back of the rule, and under the L.H. + index C read 1·875 on D, the required ratio. + + +Given belt velocity and horse-power to be transmitted, to find the +requisite width of belt, taking the effective tension at 50 lb. per inch +of width. + +Set 660 on C to velocity in feet per minute on D, and opposite +horse-power on D find width of belt in inches on C. + +Given velocity and width of belt, to find horse-power transmitted. + +Set 660 on C to velocity on D, and under width on C find horse-power +transmitted on D. + +(N.B.—For any other effective tension, instead of 660 use as a gauge +point:—33,000 ÷ tension.) + +Given speed and diameter of a cotton driving rope, to find power +transmitted, disregarding centrifugal action, and assuming an effective +working tension of 200 lb. per square inch of rope. + +Set 210 on B to 1·75 on D, and over speed in feet per minute on B read +horse-power on A. + + + EX.—Find the power transmitted by a 1¾in. rope running at 4000 ft. per + minute. + + Set 210 on B to 1·75 on D, and over 4000 on B read 58·3 horse-power + on A. + + +Find the “centrifugal tension” in the previous example, taking the +weight per foot of the rope as = 0·27_d_^2. + +Set 655 on C to the diameter, 1·75 in., on D, and over the speed, 4000 +ft. on C, read centrifugal tension = 114 lb. on A. + + + SPUR WHEELS. + +Given diameter and pitch of a spur wheel, to find number of teeth. + +Set pitch on C to π (3·1416) on D, and under any diameter on C read +number of teeth on D. + +Given diameter and number of teeth in a spur wheel, to find the pitch. + +Set diameter on C to number of teeth on D, and read pitch on C opposite +3·1416 on D. + +Given the distance between the centres of a pair of spur wheels and the +number of revolutions of each, to determine their diameters. + +To twice the distance between the centres on D, set the sum of the +number of revolutions on C, and under the revolutions of each wheel on C +find the respective wheel diameters on D. + + + EX.—The distance between the centres of two spur wheels is 37·5 in., + and they are required to make 21 and 24 revolutions in the same time. + Find their respective diameters. + + Set 21 + 24 = 45 on C to 75 (or 37·5 × 2) on D, and under 21 and 24 + on C find 35 and 40 in. on D as the respective diameters. + + +To find the power transmitted by toothed wheels, given the pitch +diameter _d_ in inches, the number of revolutions per minute _n_, and +the pitch _p_ in inches, by the rule, H.P. = (_n_ _d_ _p_^2)/(400). + +Set 400 on B to pitch in inches on D; set cursor to d on B, 1 on B to +cursor, and over any number of revolutions n on B read power transmitted +on A. + + + EX.—Find the horse-power capable of being transmitted by a spur wheel + 7 ft. in diameter, 3 in. pitch, and running at 90 revolutions per + minute. + + Set 400 on B to 3 on D; bring cursor to 84 in. on B, 1 on B to cursor, + and over 90 revolutions on B read 170, the horse-power transmitted, on + A. + + + SCREW-CUTTING. + +Given the number of threads per inch in the guide screw, to find the +wheels to cut a screw of given pitch. + +Set threads per inch in guide screw on C, to the number of threads per +inch to be cut on D. Then opposite any number of teeth in the wheel on +the mandrel on C, is the number of teeth in the wheel to be placed on +the guide screw on D. + + + STRENGTH OF SHAFTING. + +Given the diameter _d_ of a steel shaft, and the number of revolutions +per minute _n_, to find the horse-power from:— + +H.P. = _d_^3 × _n_ × 0·02. + +Set 1 on C to _d_ on D, and bring cursor to _d_ on B. Bring 50 on B to +cursor, and over number of revolutions on B read H.P. on A. + + + EX.—Find horse-power transmitted by a 3 in. steel shaft at 110 + revolutions per minute. + + Set 1 on C to 3 on D, and bring cursor to 3 on B. Bring 50 on B to + cursor, and over 110 on B read 59·4 horse-power on A. + + +Given the horse-power to be transmitted and the number of revolutions of +a steel shaft, to find the diameter. + +Set revolutions on B to horse-power on A, and bring cursor to 50 on B. +Then move the slide until the same number is found on B under the cursor +that is simultaneously found on D under the index of C. This number is +the diameter required. + +To find the deflection _k_ in inches, of a round steel shaft of diameter +_d_, under a uniformly distributed load in lb. _w_, and supported by +bearings, the centres of which are _l_ feet apart (_k_ = (_w_ +_l_^3)/(78,000_d_^4)). + +Modifying the form of this expression slightly, we proceed as +follows:—Set _d_ on C to _l_ on D, and bring the cursor to the same +number on B that is found on D under the index of C. Bring _d_ on B to +cursor, cursor to _w_ on B, 78,800 on B to cursor, and read deflection +on A over index of B. + + + EX.—Find the deflection in inches of a round steel shaft 3½in. + diameter, carrying a uniformly distributed load of 3200 lb., the + distance apart of the centres of support being 9 ft. + + Set 3·5 on C to 9 on D, and read 2·57 on D, under the L.H. index of + C. Set cursor to 2·57 on B, and bring 3·5 on B to cursor, cursor to + 3200 on B, 78,000 on B to cursor, and over L.H. index of B read + 0·199 in., the required deflection on A. + + +To find the diameter of a shaft subject to twisting only, given the +twisting moment in inch-lb. and the allowable stress in lb. per square +inch. + +Set the stress in lb. per square inch on B to the twisting moment in +inch-lb. on A, and bring cursor to 5·1 on B. Then move the slide until +the same number is found on B under the cursor that is simultaneously +found on D under the index of C. + + + EX.—A steel shaft is subjected to a twisting moment of 2,700,000 + inch-lb. Determine the diameter if the allowable stress is taken at + 9000 lb. per square inch. + + Set 9000 on B to 2,700,000 on A, and bring the cursor to 5·1 on B. + Moving the slide to the left, it is found that when 11·51 on the + R.H. scale of B is under the cursor, the L.H. index of C is opposite + 11·51 on D. This, then, is the required diameter of the shaft. + + +(N.B.—The rules for the scales to be used in finding the cube root (page +42) must be carefully observed in working these examples.) + + + MOMENTS OF INERTIA. + +To find the moment of inertia of a square section about an axis formed +by one of its diagonals (I = (_s_^4)/(12)). + +Set index of C to the length of the side of square _s_ on D; bring +cursor to _s_ on C, 12 on B to cursor, and over index of B read moment +of inertia on A. + +To find the moment of inertia of a rectangular section about an axis +parallel to one side and perpendicular to the plane of bending. + +Set index of C to the height or depth _h_ of the section, and bring +cursor to _h_ on B. Set 12 on B to cursor, and over breadth _b_ of the +section on B read moment of inertia on A. + + + EX.—Find the moment of inertia of a rectangular section of which _h_ = + 14 in. and _b_ = 7 in. + + Set index of C to 14 on D, and cursor to 14 on B. Bring 12 on B to + cursor, and over 7 on B read 1600 on A. + + + DISCHARGE FROM PUMPS, PIPES, ETC. + +To find the theoretical delivery of pumps, in gallons per stroke. + +Set 29·4 on B to the diameter of the plunger in inches on D, and over +length of stroke in feet on B read theoretical delivery in gallons per +stroke on A. + +(N.B.—A deduction of from 20 to 40 per cent. should be made to allow for +slip.) + +To find loss of head of water in feet due to friction in pipes (Prony’s +rule). + +Set diameter of pipe in feet on B to velocity of water in feet per +second on D and bring cursor to 2·25 on B; bring 1 on B to cursor, and +over length of pipe in miles on B, read loss of head of water in feet, +on A. + +To find velocity in feet per second, of water in pipes (Blackwell’s +rule). + +Set 2·3 on B to diameter of pipe in feet on A, and under inclination of +pipe in feet per mile on B read velocity in feet per second on D. + +To find the discharge over weirs in cubic feet per minute and per foot +of width. (Discharge = 214√(_h_^3)) + +Set 0·00467 on C to the head in feet _h_ on D, and under _h_ on B read +discharge on D. + +To find the theoretical velocity of water flowing under a given head in +feet. + +Set index of B to head in feet on A, and under 64·4 on B read +theoretical velocity in feet per second on D. + + + HORSE-POWER OF WATER WHEELS. + +To find the effective horse-power of a Poncelet water wheel. + +Set 880 on C to cubic feet of flow of water per minute on D, and under +height of fall in feet on C, read effective horse-power on D. + +For breast water wheels use 960, and for overshot wheels 775, in place +of 880 as above. + + + ELECTRICAL ENGINEERING. + +To find the resistance per mile, in ohms, of copper wire of high +conductivity, at 60° F. the diameter being given in mils. (1 mil. = +0·001 in.). + +Set diameter of wire in mils. on C to 54,900 on A, and over R.H. or L.H. +index of B read resistance in ohms on A. + + + EX.—Find the resistance per mile of a copper wire 64 mils. in + diameter. + + Set 64 on C to 54,900 on A, and over R.H. index of B read 13·4 ohms + on A. + + +To find the weight of copper wire in lb. per mile. + +Set 7·91 on C to diameter of wire in mils. on D, and over index of B +read weight per mile on A. + +Given electromotive force and current, to find electrical horse-power. + +Set 746 on C to electromotive force in volts on D, and under current in +ampères on C read electrical horse-power on D. + +Given the resistance of a circuit in ohms and current in ampères, to +find the energy absorbed in horse-power. + +Set 746 on B to current on D, and over resistance on B read energy +absorbed in H.P. on A. + + + EX.—Find the H.P. expended in sending a current of 15 ampères through + a circuit of 220 ohms resistance. + + Set 746 on B to 15 on D, and over 220 on B read 66·3 H.P. on A. + + + COMMERCIAL. + +To add on percentages. + +Set 100 on C to 100 + given percentage on D, and under original number +on C read result on D. + +To deduct percentages. + +Set R.H. index of C to 100 − the given percentage on D, and under +original number on C read result on D. + + + EX.—From £16 deduct 7½ per cent. + + Set 10 on C to 92·5 on D and under 16 on C, read 14·8 = £14, 16s. on + D. + + +To calculate simple interest. + +Set 1 on C to rate per cent. on D; bring cursor to period on C and 1 on +C to cursor. Then opposite any sum on C find simple interest on D. + +For interest per annum. + +Set R.H. index on C to rate on D, and opposite principal on C read +interest on D. + + + EX.—Find the amount with simple interest of £250 at 8 per cent., and + for a period of 1 year and 9 months. + + Set 1 on C to 8 on D; bring cursor to 1·75 on C, and 1 on C to + cursor; then opposite 250 on C read £35, the interest, on D. Then + 250 + 35 = £285 = the amount. + + +To calculate compound interest. + +Set the L.H. index of C to the amount of £1 at the given rate of +interest on D, and find the logarithm of this by reading on the reverse +side of the rule, as explained on page 46. Multiply the logarithm, so +found, by the period, and set the result, on the scale of equal parts, +to the index on the under-side of the rule; then opposite any sum on C +read the amount (including compound interest) on D. + + + EX.—Find the amount of £500 at 5 per cent. for 6 years, with compound + interest. + + Set L.H. index of C to £1·05 on D, and read at the index on the + scale of equal parts on the under-side of rule, 0·0212. Multiply by + 6, we obtain 0·1272, which, on the scale of equal parts, is placed + to the index in the notch at the end of the rule. Then opposite 500 + on C read £670 on D, the amount required, including compound + interest. + + + MISCELLANEOUS CALCULATIONS. + +To calculate percentages of compositions. + +Set weight (or volume) of sample on C, to weight (or volume) of +substance considered, on D; then under index of C read required +percentage on D. + + + EX.—A sample of coal weighing 1·25 grms. contains 0·04425 grm. of ash. + Find the percentage of ash. + + Set 1·25 on C to 0·04425 on D, and under index on C read 3·54, the + required percentage of ash on D. + + +Given the steam pressure P and the diameter _d_ in millimetres, of the +throat of an injector, to find the weight W, of water delivered in lb. +per hour from W = (_d_^2√̅P)/(0·505). + +Set 0·505 on C to P on A; bring cursor to _d_ on C and index of C to +cursor. Then under _d_ on C read delivery of water on D. + +To find the pressure of wind per square foot, due to a given velocity in +miles per hour. + +Set 1 on B to 2 on A, and over the velocity in miles per hour on D read +pressure in lb. per square foot on B. + +To find the kinetic energy of a moving body. + +Set 64·4 on B to velocity in feet per second on D, and over weight of +body in lb. on B read kinetic energy or accumulated work in foot-lb. on +A. + + + + + TRIGONOMETRICAL APPLICATIONS + + +_Scales._—Not the least important feature of the modern slide rule is +the provision of the special scales on the under-side of the slide, and +by the use of which, in conjunction with the ordinary scales on the +rule, a large variety of trigonometrical computations may be readily +performed. + +Three scales will be found on the reverse or under-side of the slide of +the ordinary Gravêt or Mannheim rule. One of these is the evenly-divided +scale or scale of equal parts referred to in previous sections, and by +which, as explained, the decimal parts or mantissæ of logarithms of +numbers may be obtained. Usually this scale is the centre one of the +three, but in some rules it will be found occupying the lowest position, +in which case some little modification of the following instructions +will be necessary. The requisite transpositions will, however, be +evident when the purposes of the scales are understood. The upper of the +three scales, usually distinguished by the letter S, is a scale giving +the logarithms of the sines of angles, and is used to determine the +natural sines of angles of from 35 minutes to 90 degrees. The notation +of this scale will be evident on inspection. The main divisions 1, 2, 3, +etc., represent the degrees of angles; but the values of the +subdivisions differ according to their position on the scale. Thus, if +any primary space is subdivided into 12 parts, each of the latter will +be read as 5 minutes (5′), since 1° = 60′. + +_Sines of Angles._—To find the sine of an angle the slide is placed in +the groove, with the under-side uppermost, and the end division lines or +indices on the slide, coinciding with the right and left indices of the +A scale. Then over the given angle on S is read the value of the sine of +the angle on A. If the result is found on the left scale of A (1 to 10), +the logarithmic characteristic is −2; if it is found on the right-hand +side (10 to 100), it is −1. In other words, results on the right-hand +scale are prefixed by the decimal point only, while those on the +left-hand scale are to be preceded by a cypher also. Thus:— + + Sine 2° 40′ = 0·0465; sine 15° 40′ = 0·270. + +Multiplication and division of the sines of angles are performed in the +same manner as ordinary calculations, excepting that the slide has its +under-face placed uppermost, as just explained. Thus to multiply sine +15° 40′ by 15, the R.H. index of S is brought to 15 on A, and opposite +15° 40′ on S is found 4·05 on A. Again, to divide 142 by sine 16° 30′, +we place 16° 30′ on S to 142 on A, and over R.H. index of S read 500 on +A. + +The rules for the number of integers in the results are thus determined: +Let N be the number of integers in the multiplier M or in the dividend +D. Then the number of integers P, in the product or Q, in the quotient +are as follows:— + + When the result is found to the right of M or D, │P = N − 2│Q = N + and in the same scale │ │ + When the result is found to the right of M or D, │P = N − 1│Q = N + 1 + and in the other scale │ │ + When the result is found to the left of M or D, and│P = N − 1│Q = N + 1 + in the other scale │ │ + When the result is found to the left of M or D, and│P = N │Q = N + 2 + in the same scale │ │ + +If the division is of the form (20° 30′)/(50), the result cannot be read +off directly on the face of the rule. Thus, if in the above example 20° +30′ on S, is placed to agree with 50 on the right-hand scale of A, the +result found on S under the R.H. index of A is 44° 30′. The required +numerical value can then be found: (1) By placing the slide with all +indices coincident when opposite 44° 30′ on S will be found 0·007 on A; +or (2) In the ordinary form of rule, by reading off on the scale B +opposite the index mark in the opening on the under-side of the rule. +The above rules for the number of integers in the quotient do not apply +in this case. + +If it is required to find the sine of an angle simply, this may be done +with the slide in its ordinary position, with scale B under A. The given +angle on scale S is then set to the index on the under-side of the rule, +and the value of the sine is read off on B under the right index of A. + +Owing to the rapidly diminishing differences of the values of the sines +as the upper end of the scale is approached, the sines of angles between +60° and 90° cannot be accurately determined in the foregoing manner. It +is therefore advisable to calculate the value of the sine by means of +the formula: + +Sine θ = 1 − 2 sin^2 (90 − θ)/(2). + +To determine the value of sin^2 (90 − θ)/(2). With the slide in the +normal position, set the value of (90 − θ)/(2). on S to the index on the +under-side of the rule, and read off the value _x_ on B under the R.H. +index of A. Without moving the slide find _x_ on A, and read under it on +B the value required. + + + EX.—Find value of sine 79° 40′. + + Sine 79° 40′ = 1 − 2sin^2 5° 10′. + + But sine 5° 10′ = 0·0900, and under this value on A is 0·0081 on B. + Therefore sine 79° 40′ = 1 − 0·0162 = 0·9838. + + +The sines of very small angles, being very nearly proportional to the +angles themselves, are found by direct reading. To facilitate this, some +rules are provided with two marks, one of which, a single accent (′), +corresponds to the logarithm of (1)/(sine 1′) and is found at the number +3438. The other mark—a double accent (″)—corresponds to the logarithm of +(1)/(sine 1″) and is found at the number 206,265. In some rules these +marks are found on either the A or the B scales; sometimes they are on +both. In either case the angle on the one scale is placed so as to +coincide with the significant mark on the other, and the result read off +on the first-named scale opposite the index of the second. + +In sines of angles under 3″, the number of integers in the result is −5; +while it is −4 for angles from 3″ to 21″; −3 from 21″ to 3′ 27″; and −2 +from 3′ 27″ to 34′ 23″. + + + EX.—Find sine 6′. + + Placing the significant mark for minutes coincident with 6, the value + opposite the index is found to be 175, and by the rule above this is + to be read 0·00175. For angles in seconds the other significant mark + is used; while angles expressed in minutes and seconds are to be first + reduced to seconds. Thus, 3′ 10″ = 190″. + + +_Tangents of Angles._—There remains to be considered the third scale +found on the back of the slide, and usually distinguished from the +others by being lettered T. In most of the more recent forms of rule +this scale is placed near the lower edge of the slide, but in some +arrangements it is found to be the centre scale of the three. Again, in +some rules this scale is figured in the same direction as the scale of +sines—viz., from left to right,—while in others the T scale is reversed. +In both cases there is now usually an aperture formed in the back of the +left extremity of the rule, with an index mark similar to that already +referred to in connection with the scale of sines. Considering what has +been referred to as the more general arrangement, the method of +determining the tangents of angles may be thus explained:— + +The tangent scale will be found to commence, in some rules, at about +34′, or, precisely, at the angle whose tangent is 0·01. More usually, +however, the scale will be found to commence at about 5° 43′, or at the +angle whose tangent is 0·1. The other extremity of the scale corresponds +in all cases to 45°, or the angle whose tangent is 1. This explanation +will suggest the method of using the scale, however it may be arranged. +If the graduations commence with 34′, the T scale is to be used in +conjunction with the right and left scales of A; while if they commence +with 5° 43′ it is to be used in conjunction with the D scale. + +In the former case the slide is to be placed in the rule so that the T +scale is adjacent to the A scales, and, with the right and left indices +coinciding, when opposite any angle on T will be found its tangent on A. +From what has been said above, it follows that the tangents read on the +L.H. scale of A have values extending from 0·01 to 0·1; while those read +on the R.H. scale of A have values from 0·1 to 1·0. Otherwise expressed, +to the values of any tangent read on the L.H. scale of A a cypher is to +be prefixed; while if found on the R.H. scale, it is read directly as a +decimal. + + + EX.—Find tan. 3° 50′. + + Placing the slide as directed, the reading on A opposite 3° 50′ on T + is found to be 67. As this is found on the L.H. scale of A, it is to + be read as 0·067. + + + EX.—Find tan. 17° 45′. + + Here the reading on A opposite 17° 45′ on T is 32, and as it is found + on the R.H. scale of A it is read as 0·32. + + +As in the case of the scale of sines, the tangents may be found without +reversing the slide, when a fixed index is provided in the back of the +rule for the T scale. + +We revert now to a consideration of those rules in which a single +tangent scale is provided. It will be understood that in this case the +slide is placed so that the scale T is adjacent to the D scale, and that +when the indices of both are placed in agreement, the value of the +tangent of any angle on T (from 5° 43′ to 45°) may be read off on D, the +result so found being read as wholly decimal. Thus tan. 13° 20′ is read +0·237. + +If a back index is provided, the slide is used in its normal position, +when, setting the angle on the tangent scale to this index, the result +can be read on C over the L.H. index of D. + +The tangents of angles above 45° are obtained by the formula: Tan. θ = +(1)/(tan. (90 − θ)). For all angles from 45° to (90° − 5° 43′) we +proceed as follows:—Place (90 − θ) on T to the R.H. index of D, and read +tan. θ on D under the L.H. index of T. The first figure in the value +thus obtained is to be read as an integer. Thus, to find tan. 71° 20′ we +place 90° − 71° 20′ = 18° 40′ on T, to the R.H. index of D, and under +the L.H. index of T read 2·96, the required tangent. + +The tangents of angles less than 40′ are sensibly proportional to the +angles themselves, and as they may therefore be considered as sines, +their value is determined by the aid of the single and double accent +marks on the sine scale, as previously explained. The rules for the +number of integers are the same as for the sines. + +Multiplication and division of tangents may be quite readily effected. + + + EX.—Tan. 21° 50′ × 15 = 6. + + Set L.H. index of T to 15 on D, and under 21° 50′ on T read 6 on D. + + + EX.—Tan. 72° 40′ × 117 = 375. + + Set (90° − 72° 40′) = 17° 20′ on T to 117 on D, and under R.H. index + of T read 375 on D. + + +_Cosines of Angles._—The cosines of angles may be determined by placing +the scale S with its indices coinciding with those of A, and when +opposite (90 − θ) on S is read cos. θ on A. If the result is read on the +L.H. scale of A, a cypher is to be prefixed to the value read; while if +it is read on the R.H. scale of A, the value is read directly as a +decimal. Thus, to determine cos. 86° 30′ we find opposite (90° − 86° +30′) = 3° 30′ on S, 61° on A, and as this is on the L.H. scale the +result is read 0·061. Again, to find cos. 59° 20′ we read opposite (90° +− 59° 20′) or 30° 40′ on S, 51 on A, and as this is found on the R.H. +scale of A, it is read 0·51. + +In finding the cosines of small angles it will be seen that direct +reading on the rule becomes impossible for angles of less than 20°. It +is advisable in such cases to adopt the method described for determining +the _sines_ of the _large_ angles of which the complements are sought. + +_Cotangents of Angles._—From the methods of finding the tangents of +angles previously described, it will be apparent that the cotangents of +angles may also be obtained with equal facility. For angles between 5° +45′ and 45°, the procedure is the same as that for finding tangents of +angles greater than 45°. Thus, the angle on scale T is brought to the +R.H. index of D, and the cotangent read off on D under the L.H. index of +T. The first figure of the result so found is to be read as an integer. + +If the angle (θ) lies between 45° and 84° 15′, the slide is placed so +that the indices of T coincide with those of D, and the result is then +read off on D opposite (90 − θ) on T. In this case the value is wholly +decimal. + +_Secants of Angles._—The secants of angles are readily found by bringing +(90 − θ) on S to the R.H. index of A and reading the result on A over +the L.H. index of S. If the value is found on the L.H. scale of A, the +first figure is to be read as an integer; while if the result is read on +the R.H. scale of A, the first _two_ figures are to be regarded as +integers. + +_Cosecants of Angles._—The cosecants of angles are found by placing the +angle on S to the R.H. index of A, and reading the value found on A over +the L.H. index of S. If the result is read on the L.H. scale of A, the +first figure is to be read as an integer; while if the result is found +on the R.H. scale of A, the first _two_ figures are to be read as +integers. + +It will be noted that some of the rules here given for determining the +several trigonometrical functions of angles apply only to those forms of +rules in which a single scale of tangents T is used, reading from left +to right. For the other arrangements of the scale, previously referred +to, some slight modification of the method of procedure in finding the +tangents and cotangents of angles will be necessary; but as in each case +the nature and extent of this modification is evident, no further +directions are required. + + + + + THE SOLUTION OF RIGHT-ANGLED TRIANGLES. + + +From the foregoing explanation of the manner of determining the +trigonometrical functions of angles, the methods of solving right-angled +triangles will be readily perceived, and only a few examples need +therefore be given. + +Let _a_ and _b_ represent the sides and _c_ the hypothenuse of a +right-angled triangle, and _a_° and _b_° the angles opposite to the +sides. Then of the possible cases we will take + +(1.) Given _c_ and _a_°, to find _a_, _b_, and _b_°. + +The angle _b_° = 90 − _a_°, while _a_ = _c_ sin _a_° and _b_ = _c_ sin +_b_°. To find _a_, therefore, the index of S is set to _c_ on A, and the +value of _a_ read on A opposite _a_° on S. In the same manner the value +of _b_ is obtained. + + + EX.—Given in a right-angled triangle _c_ = 9 ft. and _a_° = 30°. Find + _a_, _b_, and _b_°. + + The angle _b_° = 90 − 30 = 60°. To find _a_, set R.H. index of S to 9 + on A, and over 30° on S read _a_ = 4·5 ft. on A. Also, with the slide + in the same position, read _b_ = 7·8 ft. [7·794] on A over 60° on S. + + +(2.) Given _a_ and _c_, to determine _a_°, _b_°, and _b_. + +In this case advantage is taken of the fact that in every triangle the +sides are proportional to the sines of the opposite angles. Therefore, +as in this case the hypothenuse c subtends a right angle, of which the +sine = 1, the R.H. index (or 90°) on S is set to the length of _c_ on A, +when under _a_ on A is found _a_° on S. Hence _b_° and _b_ may be +determined. + +(3.) Given _a_ and _a_°, to find _b_, _c_, and _b_°. + +Here _b_° = (90 − _a_°), and the solution is similar to the foregoing. + +(4.) Given _a_ and _b_, to find _a_°, _b_°, and _c_. + +To find _a_°, we have tan. _a_° = _a_/_b_, which in the above example +will be (4·5)/(7·8) = 0·577. Therefore, placing the slide so that the +indices of T coincide with those of D, we read opposite 0·577 on D the +value of _a_° = 30°. The hypothenuse _c_ is readily obtained from _c_ = +_a_/(sin _a_°). + + + + + THE SOLUTION OF OBLIQUE-ANGLED TRIANGLES. + + +Using the same letters as before to designate the three sides and the +subtending angles of oblique-angled triangles, we have the following +cases:— + +(1.) Given one side and two angles, as _a_, _a_°, and _b_°, to find _b_, +_c_, and _c_°. + +In the first place, _c_° = 180° − (_a_° + _b_°); also we note that, as +the sides are proportional to the sines of the opposite angles, _b_ = +(_a_ sine _b_°)/(sine _a_°) and _c_ = (_a_ sine _c_°)/(sine _a_°). + +Taking as an example, _a_ = 45, _a_° = 57°, and _b_° = 63°, we have _c_° += 180 − (57 + 63) = 60°. To find _b_ and _c_, set _a_° on S to _a_ on A, +and read off on A above 63° and 60° the values of _b_ (= 47·8) and _c_ +(= 46·4) respectively. + +(2.) Given _a_, _b_, and _a_°, to find _b_°, _c_°, and _c_. + +In this case the angle _a_° on S is placed under the length of side _a_ +on A and under _b_ on A is found the angle _b_° on S. The angle _c_° = +180 − (_a_° + _b_°), whence the length _c_ can be read off on A over +_c_° on S. + +(3.) Given the sides and the included angle, to find the other side and +the remaining angles. + +If, for example, there are given _a_ = 65, _b_ = 42, and the included +angle _c_° = 55°, we have (_a_ + _b_) ∶ (_a_ − _b_) = tan. (_a_° + +_b_°)/(2) ∶ tan. (_a_° − _b_°)/(2). Then, since _a_° + _b_° = 180° − 55° += 125°, it follows that (_a_° + _b_°)/(2) = (125°)/(2) = 62° 30′. + +By the rule for tangents of angles greater than 45°, we find tan. 62° +30′ = 1·92. Inserting in the above proportion the values thus found, we +have 107 ∶ 23 = 1·92 ∶ tan. (_a_° − _b_°)/(2). From this it is found +that the value of the tangent is 0·412, and placing the slide with all +indices coinciding, it is seen that this value on D corresponds to an +angle of 22° 25′. Therefore, since (_a_° + _b_°)/(2) = 62° 30′, and +(_a_° − _b_°)/(2) = 22° 25′, it follows that _a_° = 84° 55′, and _b_° = +40° 5′. Finally, to determine the side _c_, we have _c_ = (_a_ sin +_c_°)/(sin _a_°) as before. + + + PRACTICAL TRIGONOMETRICAL APPLICATIONS. + +A few examples illustrative of the application of the methods of +determining the functions of angles, etc., described in the preceding +section, will now be given. + +To find the chord of an arc, having given the included angle and the +radius. + +With the slide placed in the rule with the C and D scales outward, bring +one-half of the given angle on S to the index mark in the back of the +rule, and read the chord on B under twice the radius on A. + + + EX.—Required the chord of an arc of 15°, the radius being 23 in. + + Set 7° 30′ on S to the index mark in the back of the rule, and under + 46 on A read 6 in., the required length of chord on B. + + +To find the area of a triangle, given two sides and the included angle. + +Set the angle on S to the index mark on the back of the rule, and bring +cursor to 2 on B. Then bring the length of one side on B to cursor, +cursor to 1 on B, the length of the other side on B to cursor, and read +area on B under index of A. + + + EX.—The sides of a triangle are 5 and 6 ft. in length respectively, + and they include an angle of 20°. Find the area. + + Set 20 on S to index mark, bring cursor to 2 on B, 5 on B to cursor, + cursor to 1 on B, 6 on B to cursor, and under 1 on A read the area = + 5·13 sq. ft. on B. + + +To find the number of degrees in a gradient, given the rise per cent. + +Place the slide with the indices of T coincident with those of D, and +over the rate per cent. on D read number of degrees in the slope on T. + +As the arrangement of rule we have chiefly considered has only a single +T scale, it will be seen that only solutions of the above problem +involving slopes between 10 and 100 per cent. can be directly read off. +For smaller angles, one of the formulæ for the determination of the +tangents of submultiple angles must be used. + +In rules having a double T scale (which is used with the A scale) the +value in degrees of any slope from 1 to 100 per cent. can be directly +read off on A. + +To find the number of degrees, when the gradient is expressed as 1 in +_x_. + +Place the index of T to _x_ on D, and over index of D read the required +angle in degrees on T. + + + EX.—Find the number of degrees in a gradient of 1 in 3·8. + + Set 1 on T to 3·8 on D, and over R.H. index of D read 14° 45′ on T. + + +Given the lap, the lead and the travel of an engine slide valve, to find +the angle of advance. + +Set (lap + lead) on B to half the travel of the valve on A, and read the +angle of advance on S at the index mark on the back of the rule. + + + EX.—Valve travel 4½in., lap 1 in., lead ⁵⁄₁₆in. Find angle of advance. + + Set 1⁵⁄₁₆ = 1·312 on B to 2·25 on A, and read 35° 40′ on S opposite + the index on the back of the rule. + + +Given the angular advance θ, the lap and the travel of a slide valve, to +find the cut-off in percentage of the stroke. + +Place the lap on B to half the travel of valve on A, and read on S the +angle (the supplement of the _angle of the eccentric_) found opposite +the index in the back of the rule. To this angle, add the angle of +advance and deduct the sum from 180°, thus obtaining the _angle of the +crank_ at the point of cut-off. To the cosine of the supplement of this +angle, add 1 and multiply the result by 50, obtaining the percentage of +stroke completed when cut-off occurs. + + + EX.—Given the angular advance = 35° 40′, the valve travel = 4½in., and + the lap = 1 in., find the angle of the crank at cut-off and the + admission period expressed as a percentage of the stroke. + + Set 1 on B to 2·25 on A, and read off on S opposite the index, the + supplement of the angle of the eccentric = 26° 20′. Then 180° − (35° + 40′ + 26° 20′) = 118° = the crank angle at the point of cut-off. + Further, cos. 118° = cos. 62° = sin (90° − 62°) = sin 28°, and placing + 28° on S to the back index, the cosine, read on B under R.H. index of + A, is found to be 0·469. Adding 1 and placing the L.H. index of C to + the result, 1·469, on D, we read off under 50 on C, the required + period of admission = 73·4 per cent. on D. + + +The trigonometrical scales are useful for evaluating certain formulæ. +Thus in the following expressions, if we find the angle _a_ such that +sin. _a_ = _k_, we can write:— + + (_k_)/(√1 − _k^2_) = tan. _a_; (√1 − _k^2_)/(_k_) = cot. _a_; √(1 − + _k^2_) = cos. _a_; etc. + +In the first expression, take _k_ = 0·298. Place the slide with the sine +scale outward and with its indices agreeing with the indices of the +rule. Set the cursor to 0·298 on the (R.H.) scale of A, and read 17° 20′ +on the sine scale as the angle required. Then under 17° 20′ on the +tangent scale, read 0·312 on D as the result. + + + SLIDE RULES WITH LOG.-LOG. SCALES. + +For occasional requirements, the method described on page 45 of +determining powers and roots other than the square and cube, is quite +satisfactory. When, however, a number of such calculations are to be +made, the process may be simplified considerably by the use of what are +known as _log.-log._, _logo-log._, or _logometric_ scales, in +conjunction with the ordinary scales of the rule. The principle involved +will be understood from a consideration of those rules for logarithmic +computation (page 8) which refer to powers and roots. From these it is +seen that while for the multiplication and division of numbers we _add_ +their logarithms, for involution and evolution we require to _multiply_ +or _divide_ the logarithms of the numbers by the exponent of the power +or root as the case may be. Thus to find 3^{2.3}, we have (log. 3) × 2·3 += log. _x_, and by the ordinary method described on page 45 we should +determine log. 3 by the aid of the scale L on the back of the slide, +multiply this by 2·3 by using the C and D scales in the usual manner, +transfer the result to scale L, and read the value of _x_ on D under 1 +on C. By the simpler method, first proposed by Dr. P. M. Roget,[8] the +multiplication of log. 3 by 2·3 is effected in the same way as with any +two ordinary factors—_i.e._, by adding their logarithms and finding the +number corresponding to the resulting logarithm. In this case we have +log. (log. 3) + log. 2·3 = log. (log. _x_). The first of the three terms +is obviously the _logarithm of the logarithm_ of 3, the second is the +simple logarithm of 2·3, and the third the _logarithm of the logarithm +of_ the answer. Hence, if we have a scale so graduated that the +distances from the point of origin represent the logarithms of the +logarithms (the log.-logs.) of the numbers engraved upon it, then by +using this in conjunction with the ordinary scale of logarithms, we can +effect the required multiplication in a manner which is both expeditious +and convenient. Slightly varying arrangements of the log.-log. scale, +sometimes referred to as the “P line,” have been introduced from time to +time, but latterly the increasing use of exponential formulæ in +thermodynamic, electrical, and physical calculations has led to a +revival of interest in Dr. Roget’s invention, and various arrangements +of rules with log.-log. scales are now available. + +_The Davis Log.-Log. Rule._—In the rule introduced by Messrs. John Davis +& Son Limited, Derby, the log.-log. scales are placed upon a separate +slide—a plan which has the advantage of leaving the rule intact for all +ordinary purposes, while providing a length of 40 in. for the log.-log. +scales. + +In the 10 in. Davis rule one face of the slide, marked E, has two +log.-log. scales for numbers greater than unity, the lower extending +from 1·07 to 2, and the upper continuing the graduations from 2 to 1000. +On the reverse face of the slide, marked -E, are two log.-log. scales +for numbers less than unity, the upper extending from 0·001 to 0·5, and +the lower continuing the graduations from 0·5 to 0·933. Both sets of +scales are used in conjunction _with the lower or D scale of the rule_, +which is to be primarily regarded as running from 1 to 10, and +constitutes a scale of exponents. In the 20 in. rule the log.-log. +scales are more extensive, and are used in conjunction with the upper or +A scale of the rule (1 to 100); in what follows, however, the 10 in. +rule is more particularly referred to. + +It has been explained that on the log.-log. scale the distance of any +numbered graduation from the point of origin represents the log.-log. of +the number. The point of origin will obviously be that graduation whose +log.-log. = 0. This is seen to be 10, since log. (log. 10) = log. 1 = 0. +Hence, confining attention to the E scale, to locate the graduation 20, +we have log. (log. 20) = log. 1·301 = 0·11397, so that if the scale D is +25 cm. long, the distance between 10 and 20 on the corresponding +log.-log. scale would be 113·97 ÷ 4 = 28·49 mm. For numbers less than 10 +the resulting log.-logs. will be negative, and the distances will be +spaced off from the point of origin in a negative direction—_i.e._, from +right to left. Thus, to locate the graduation 5, we have + + log. (log. 5) = log. 0·699 = ̅1·844; _i.e._, −1 + 0·844 or −0·156; + +so that the graduation marked 5 would be placed 156 ÷ 4 = 39 mm. distant +from 10 in a _negative_ direction, and proceeding in a similar manner, +the scale may be extended in either direction. In the -E scale, the +notation runs in the reverse direction to that of the E scale, but in +all other respects it is precisely analogous, the distance from the +point of origin (0·1 in this case) to any graduation _x_ representing +log. [-log. _x_.]. It follows that of the similarly situated graduations +on the two scales, those on the -E scale are the _reciprocals_ of those +on the E scale. This may be readily verified by setting, say, 10 on E to +(R.H.) 1 on D, when turning to the back of the rule we find 0·1 on -E +agreeing with the index mark in the aperture at the right-hand extremity +of the rule. + +In using the log.-log. scales it is important to observe (1) that the +values engraved on the scale are definite and unalterable (_e.g._, 1·2 +can only be read as 1·2 and not as 120, 0·0012, etc., as with the +ordinary scales); (2) that the upper portion of each scale should be +regarded as forming a prolongation to the right of the lower portion; +and (3) that immediately above any value on the lower portion of the +scale is found the 10th power of that value on the upper portion of the +scale. Keeping these points in view, if we set 1·1 on E to 1 on D we +find over 2 on D the value of 1·1^2 = 1·21 on E. Similarly, over 3 we +find 1·1^3 = 1·331, and so on. Then, reading across the slide, we have, +over 2, the value of 1·1^{2 × 10} = 1·1^{20} = 6·73, and over 3 we have +1·1^{3 × 10} = 1·1^{30} = 17·4. Hence the rule:—_To find the value of +x^n, set x on E to 1 on D, and over n on D read x^n on E._ + +With the slide set as above, the 8th, 9th, etc., powers of 1·1 cannot be +read off; but it is seen that, according to (2) in the foregoing, the +missing portion of the E scale is that part of the upper scale (2 to +about 2·6) which is outside the rule to the left. Hence placing 1·1 to +10 on D, the 8th, 9th, etc., powers of 1·1 will be read off _on the +upper part_ of the E scale. In general, then, + +If _x_ on the _lower_ line is set to 1 on D, then _x^n_ is read directly +on that line and _x_^{10_n_} on the upper line. + +If _x_ on the _upper_ line is set to 1 on D, then _x^n_ is read directly +on that line and _x_^{_ⁿ⁄₁₀_} on the lower line. + +If _x_ on the _lower_ line is set to 10 on D, then _x_^{_ⁿ⁄₁₀_} is read +directly on that line and _x^n_ on the upper line. + +If _x_ on the _upper_ line is set to 10 on D, then _x_^{_ⁿ⁄₁₀_} is read +directly on that line and _x_^{_ⁿ⁄₁₀₀_} on the lower line. + +These rules are conveniently exhibited in the accompanying diagram (Fig. +14). They are equally applicable to both the E and -E scales of the 10 +in. rule, and include practically all the instruction required for +determining the _n_th power or the _n_th root of a number. They do not +apply directly to the 20 in. rule, however, for here the relation of the +lower and upper scales will be _x^n_ and _x_^{100_n_}. + + + EX.—Find 1·167^{2·56}. + + Set 1·167 on E to 1 on D, and over 2·56 on D read 1·485 on E. + + + EX.—Find 4·6^{1·61}. + + Set 4·6 on upper E scale to 1 on D, and over 1·61 on D read 11·7 + (11·67) on E. + + + EX.—Find 1·4^{0·27} and 1·4^{2·7}. + + Set 1·4 on E to 10 on D, and over 2·7 on D read 1·095 = 1·4^{0·27} on + lower E scale and 2·48 = 1·4^{2·7} on upper E scale. + + +[Illustration: FIG. 14.] + + + EX.—Find 46^{0·0184} and 46^{0·184}. + + Set 46 on upper E scale to 10 on D, and over 1·84 on D read 1·073 on + lower E scale and 2·022 (2·0228) on upper E scale. + + + EX.—Find 0·074^{1·15}. + + Using the -E scale, set 0·074 to 1 on D, and over 1·15 on D read 0·05 + on -E. + + +The method of determining the root of a number will be obvious from the +preceding examples. + + + EX.—Find ^{1.4}√(17) and ^{14}√(17). + + Set 17 on E to 1·4 on D, and over 1 on D read 7·56 on upper E scale + and 1·224 on lower E scale. + + EX.—Find ^{0·031}√(0·914). + + Set 0·914 on -E to 3·1 on D, and over 10 on D read 0·055 on upper -E + scale. + + +When the exponent _n_ is fractional, it is often possible to obtain the +result directly with one setting of the slide. Thus to determine +1·135^{¹⁷⁄₁₆} by the first method we find ¹⁷⁄₁₆ = 1·0625, and placing +1·135 on E to 1 on D, read 1·144 on E over 1·0625 on D. By the direct +method we place 1·135 on the E scale on 1·6 on D, and over 1·7 on D read +1·144 on E. It will be seen that since the scale D is assumed to run +from 1 to 10 we are unable to read 16 and 17 on this scale; but it is +obvious that the _ratios_ (1·7)/(1·6) and (17)/(16) are identical, and +it is with the ratio only that we are, in effect, concerned. + +Since an expression of the form _x_^{-_n_} = (1)/(_x^n_) or +((1)/(_x_))^{_n_}, the required value may be obtained by first +determining the reciprocal of _x_ and proceeding as before. By using +both the direct and reciprocal log.-log. scales (E and -E) in +conjunction however, the required value can be read directly from the +rule, and the preliminary calculation entirely avoided. In the Davis +form of rule, the result can be read on the -E scale, used in +conjunction with the D scale of the rule, _x_ on E being set to the +index mark in the aperture in the back of the rule. + + + EX.—Find the value of 1·195^{−1·65}. + + Set 1·195 on E to the index in the left aperture in the back of the + rule, and over 1·65 on D read 0·745 on the -E scale. + + +It may be noted in passing that the log.-log. scale affords a simple +means for determining the logarithm or anti-logarithm of a number to any +base. For this purpose it is necessary to set the base of the given +system on E to 1 on D, when _under_ any number on E will be found its +logarithm on D. Thus, for common logs., we set the base 10 on E to 1 on +D, and under 100 we find 2, the required log. Similarly we read log. 20 += 1·301; log. 55 = 1·74; log. 550 = 2·74, etc. Reading reversely, over +1·38 on D we find its antilog. 24 on E; also antilog. 1·58 = 38; +antilog. 1·19 = 15·5, etc. + +For logs. of numbers under 10 we set the base 10 to 10 on D; hence the +readings on D will be read as one-tenth their apparent value. Thus log. +3 = 0·477; log. 5·25 = 0·72; antilog. 0·415 = 2·6; antilog. 0·525 = +3.·35, etc. + +The logs. of the numbers on the lower half of the E scale will also be +found on the D scale; but a consideration of Fig. 14 will show that this +will be read as _one-tenth_ its face value if the base is set to 1 on D, +and as _one-hundredth_ if the base is set to 10. + +For natural, hyperbolic, or Napierian logarithms, the base is 2·718. A +special line marked ε or _e_ serves to locate the exact position of this +value on the E scale, and placing this to 1 on D we read log._{_e_} 4·35 += 1·47; log._{_e_} 7·4 = 2·0; antilog._{_e_} × 2·89 = 18, etc. The other +parts of the scale are read as already described for common logs. +Calculations involving powers of _e_ are frequently met with, and these +are facilitated by using the special graduation line referred to, as +will be readily understood. + +If it is required to determine the power or root of a number which does +not appear on either of the log.-log. scales, we may break up the number +into factors. Usually it is convenient to make one of the factors a +power of 10. + + + EX.—3950^{1·97} = 3·95^{1·97} × 10^{3 × 1·97} = 3·95^{1·97} × + 10^{5·91}. + + Then 3·95^{1·97} = 15, and 10^{5·91} (or antilog.) 5·91 = 812,000. + Hence, 15 × 812,000 = 12,180,000 is the result sought. + + +Numbers which are to be found in the higher part of the log.-log. scale +may often be factorised in this way, and greater accuracy obtained than +by direct reading. + +The form of log.-log. rule which has been mainly dealt with in the +foregoing gives a scale of comparatively long range, and the only +objection to the arrangement adopted is the use of a separate slide. + +_The Jackson-Davis Double Slide Rule._—In this instrument a pair of +aluminium clips enable the log.-log. slide to be temporarily attached to +the lower edge of the ordinary rule, and used, by means of a special +cursor, in conjunction with the C scale of the ordinary slide. In this +way both the log.-log. and ordinary scales are available without the +trouble of replacing one slide by the other. Since the scale of +exponents is now on the slide, the value of _x^n_ will be obtained by +setting 1 on C to _x_ on E and reading the result on E under _n_ on C. + +By using a pair of log.-log. slides, one in the rule and one clamped to +the edge by the clips, we have an arrangement which is very useful in +deducing empirical formulæ of the type _y_ = _x^n_. + +_The Yokota Slide Rule._—In this instrument the log.-log. scales are +placed on the face of the rule, each set comprising three lines. These, +for numbers greater than 1, are found above the A scale while the three +reciprocal log.-log. lines are below the D scale. Both sets are used in +conjunction with the C scale on the slide. Other features of this rule +are:—The ordinary scales are 10 in. long instead of 25 cm. as hitherto +usual; hence the logarithms of numbers can be read on the ordinary scale +of inches on the edge of the rule. There is a scale of cubes in the +centre of the slide and on the back of the slide there is a scale of +secants in addition to the sine and tangent scales. + +[Illustration: FIG. 15.] + +_The Faber Log.-log. Rule._—In this instrument shown in Fig. 15, the two +log.-log. scales are placed on the face of the rule. One section, +extending from 1·1 to 2·9, is placed above the A scale, and the other +section, extending from 2·9 to 100,000, is placed below the D scale. +These scales are used in conjunction with the C scale of the slide in +the manner previously described. The width of the rule is increased +slightly, but the arrangement is more convenient than that formerly +employed, wherein the log.-log. scales were placed on the bevelled edge +of the rule and read by a tongue projecting from the cursor. + +[Illustration: FIG. 16.] + +Another novel feature of this rule is the provision of two special +scales at the bottom of the groove, to which a bevelled metal index or +marker on the left end of the slide can be set. The upper of these +scales is for determining the efficiency of dynamos and electric motors; +the lower for determining the loss of potential in an electric circuit. + +_The Perry Log.-log. Rule._—In this rule, introduced by Messrs. A. G. +Thornton, Limited, Manchester, the log.-log. scales are arranged as in +Fig. 16, the E scale, running from 1·1 to 10,000, being placed above the +A scale of the rule, and the -E or E^{−1} scale running from 0·93 to +0·0001, below the D scale of the rule. These scales are read in +conjunction with the B scales on the slide by the aid of the cursor. + +The following tabular statement embodies all the instructions required +for using this form of log.-log. slide rule:— + + When _x_ is greater than 1. + + _x^n_ Set 1 on B to _x_ on E; over _n_ on B read _x^n_ on E + _x_^{-_n_} Set 1 on B to _x_ on E; under _n_ on B read _x_^{-_n_} on + E^{−1} + _x_^{_ⁱ⁄ₙ_} Set _n_ on B to _x_ on E; over 1 on B read _x_^{_ⁱ⁄ₙ_} on + E + _x_^{_⁻ⁱ⁄ₙ_} Set _n_ on B to _x_ on E; under 1 on B read _x_^{_⁻ⁱ⁄ₙ_} + on E^{−1} + + When _x_ is less than 1. + + _x^n_ Set 1 on B to _x_ on E^{−1}; under _n_ on B read _x^n_ on + E^{−1} + _x_^{-_n_} Set 1 on B to _x_ on E^{−1}; over _n_ on B read _x_^{-_n_} + on E + _x_^{_ⁱ⁄ₙ_} Set _n_ on B to _x_ on E^{−1}; under 1 on B read + _x_^{_ⁱ⁄ₙ_} on E^{−1} + _x_^{_⁻ⁱ⁄ₙ_} Set _n_ on B to _x_ on E^{−1}; over 1 on B read + _x_^{_⁻ⁱ⁄ₙ_} on E + +If 10 on B is used in place of 1 on B, read _x_^{_ⁿ⁄₁₀_} in place of +_x^n_ on E, and _x_^{-_ⁿ⁄₁₀_} in place of _x_^{-_n_} on E^{−1}. If 100 +on B is used, these readings are to be taken as _x_^{_ⁿ⁄₁₀₀_} and +_x_^{-_ⁿ⁄₁₀₀_} respectively. + +In rules with no -E scale the value of _x_^{-_n_} is obtained by the +usual rules for reciprocals. We may either determine _x^n_ and find its +reciprocal or, first find the reciprocal of _x_ and raise it to the +_n_th power. The first method should be followed when the number _x_ is +found on the E scale. + + + EX.—3·45^{−1·82} = 0·105. + + Set 1 on C to 3·45 on E, and under 1·82 on C read 9·51 on C. Then set + 1 on B to 9·5 on A, and under index of A read 0·105 on B. + + +When _x_ is less than 1 the second method is more suitable. + + + EX.—0·23^{−1·77} = ((1)/(0·23))^{1·77} = 4·35^{1·77} = 13·5 + + Set 1 on B to 0·23 on A, and under index of A read (1)/(0·23) = 4·35 + on B. + + Set 1 on C to 4·35 on E, and under 1·77 on C read 13·5 on E. + + +As with the Davis rule, the exponent scale C will be read as ⅒th its +face value if its R.H. index (10) is used in place of 1. + + + SPECIAL TYPES OF SLIDE RULES. + +In addition, to the new forms of log.-log. slide rules previously +described, several other arrangements have been recently introduced, +notably a series by Mr. A. Nestler, of Lahr (London: A. Fastlinger, Snow +Hill). These comprise the “Rietz,” the “Precision,” the “Universal,” and +the “Fix” slide rules. + +THE RIETZ RULE.—In this rule the usual scales A, B, C, and D, are +provided, while at the upper edge is a scale, which, being three times +the range of the D scale, enables cubes and cube roots to be directly +evaluated and also _n_^{³⁄₂} and _n_^⅔. + +A scale at the lower edge of the rule gives the mantissa of the +logarithms of the numbers on D. + +THE PRECISION SLIDE RULE.—In this rule the scales are so arranged that +the accuracy of a 20 in. rule is obtainable in a length of 10 in. This +is effected by dividing a 20 in. (50 cm.) scale length into two parts +and placing these on the working edges of the rule and slide. On the +upper and lower margins of the face of the rule are the two parts of +what corresponds to the A scale in the ordinary rule; while in the +centre of the slide is the scale of logarithms which, used in +conjunction with the 50 cm. scales on the slide, is virtually twice the +length of that ordinarily obtainable in a 10 in. rule. The same remark +applies to the trigonometrical scales on the under face of the slide. +Both the sine and tangent scales are in two adjacent lengths, while on +the edge of the stock of the rule, below the cursor groove, is a scale +of sines of small angles from 1° 49′ to 5° 44′. This is referred to the +50 cm. scales by an index projection on the cursor. + +If C and C′ are the two parts of the scale on the slide and D and D′ the +corresponding scales on the rule, it is clear that in multiplying two +factors 1 on C can only be set directly to the upper scale D; while 10 +on C′ can only be set directly to the lower scale D′. Hence if the first +factor is greater than about 3·2, the cursor must be used to bring 1 on +C to the first factor on D′. Similarly, in division, numerators and +denominators which occur on C and D′ or on C′ and D cannot be placed in +direct coincidence but must be set by the aid of the cursor. + +Any uncertainty in reading the result can be avoided by observing the +following rule: _If in setting the index_ (1 _or_ 10) _in +multiplication, or in setting the numerator to the denominator in +division, it is necessary to cross the slide, then it will also be +necessary to cross the slide to read the product or quotient._ + +THE UNIVERSAL SLIDE RULE.—In this instrument the stock carries two +similar scales running from 1 to 10, to which the slide can be set. +Above the upper one is the logarithm scale and under the lower one the +scale of squares 1 to 100. On the edge of the stock of the rule, under +the cursor groove, is a scale running from 1 to 1000. An index +projecting from the cursor enables this scale to be used with the scales +on the face of the rule, giving cubes, cube roots, etc. + +On the slide, the lower scale is an ordinary scale, 1 to 10. The centre +scale is the first part of a scale giving the values of sin _n_ cos _n_, +this scale being continued along the upper edge of the slide (marked +“sin-cos”) up to the graduation 50. On the remainder of this line is a +scale running from right to left (0 to 50) and giving the value of +cos^2_n_. In surveying, these scales greatly facilitate the calculations +for the horizontal distance between the observer’s station and any +point, and the difference in height of these two points. + +On the back of the slide are scales for the sines and tangents of +angles. The values of the sines and tangents of angles from 34′ to 5° +44′ differ little from one another, and the one centre scale suffices +for both functions of these small angles. + +THE FIX SLIDE RULE.—This is a standard rule in all respects, except that +the A scale is displaced by a distance (π)/(4) so that over 1 on D is +found 0·7854 on A. This enables calculations relating to the area and +cubic contents of cylinders to be determined very readily. + +THE BEGHIN SLIDE RULE.—We have seen that a disadvantage attending the +use of the ordinary C and D scales, is that it is occasionally necessary +to traverse the slide through its own length in order to change the +indices or to bring other parts of the slide into a readable position +with regard to the stock. To obviate this disadvantage, Tserepachinsky +devised an ingenious arrangement which has since been used in various +rules, notably in the Beghin slide rule made by Messrs. Tavernier-Gravêt +of Paris. In this rule the C and D scales are used as in the standard +rule, but in place of the A and B scales, we have another pair of C and +D scales, displaced by one-half the length of the rule. The lower pair +of scales may therefore be regarded as running from 10^{_n_} to 10^{_n_ ++ 1}, and the upper pair as running from √(10) × 10^{_n_} to √(10) × +10^{_n_ + 1}. With this arrangement, _without moving the slide more than +half its length_, to the left or right, it is always possible to compare +_all values between_ 1 _and_ 10 _on the two scales_. This is a great +advantage especially in continuous working. + +Another commendable feature of the Beghin rule is the presence of a +reversed C scale in the centre of the slide, thus enabling such +calculations as _a_ × _b_ × _c_ to be made with one setting of the +slide. On the back of the slide are three scales, the lowest of which, +used with the D scale, is a scale of squares (corresponding to the +ordinary B scale), while on the upper edge is a scale of sines from 5° +44′ to 90°, and in the centre, a scale of tangents from 5° 43′ to 45°. +On the square edge of the stock, under the cursor groove, is the +logarithm scale, while on the same edge, above the cursor groove, are a +series of gauge points. All these values are referred to the face scales +by index marks on the cursor. + +THE ANDERSON SLIDE RULE.—The principle of dividing a long scale into +sections as in the Precision rule, has been extended in the Anderson +slide rule made by Messrs. Casella & Co., London, and shown in Fig. 17. +In this the slide carries a scale in four sections, used in conjunction +with an exactly similar set of scale-lines in the upper part of the +stock. On the lower part of the stock is a scale in eight sections +giving the square roots of the upper values. In order to set the index +of the slide to values in the stock, two indices of transparent +celluloid are fixed to the slide extending over the face of the rule as +shown in the illustration. As each scale section is 30 cm. in length, +the upper lines correspond to a single scale of nearly 4 ft., and the +lower set to one of nearly 8 ft. in length, giving a correspondingly +large increase in the number of subdivisions of these scales, and +consequently much greater accuracy. + +In order to decide upon which line a result is to be found, sets of +“line numbers” are marked at each end of the rule and slide and also on +the metal frame of the cursor. In multiplication, the line number of the +product is the sum of the line numbers of the factors if the left index +is used, or 1 more than this sum if the right index is used. The +illustration shows the multiplication of 2 by 4. The left index is set +to 2 (line number, 1), and the cursor set to 4 on the slide (line +number, 2); hence, as the left index is used, the result is found on +line No. 3. Similar rules are readily established for division. The +column of line numbers headed 0 is used for units, that headed 4 for +tens, and so on; one column is given for tenths, headed −4. The square +root scale bears similar line numbers, so that the square root of any +value on the upper scales is found on the correspondingly figured line +below. + +[Illustration: FIG. 17.] + +THE MULTIPLEX SLIDE RULE differs from the ordinary form of rule in the +arrangement of the B scale. The right-hand section of this scale runs +from left to right as ordinarily arranged, but the left-hand section +runs in the reverse direction, and so furnishes a reciprocal scale. At +the bottom of the groove, under the slide, there is a scale running from +1 to 1000, which is used in conjunction with the D scale, readings being +referred thereto by a metal index on the end of the slide. By this means +cubes, cube roots, etc., can be read off directly. Messrs. Eugene +Dietzgen & Co., New York, are the makers. + +THE “LONG” SLIDE RULE has one scale in two sections along the upper and +lower parts of the stock, as in the “Precision” rule. The scale on the +slide is similarly divided, but the graduations run in the reverse +direction, corresponding to an inverted slide. Hence the rules for +multiplication and division are the reverse of those usually followed +(page 30). On the back of the slide is a single scale 1–10, and a scale +1–1000, giving cubes of this single scale. By using the first in +conjunction with the scales on the stock, squares may be read, while in +conjunction with the cube scale, various expressions involving squares, +cubes and their roots may be evaluated. + +HALL’S NAUTICAL SLIDE RULE consists of two slides fitting in grooves in +the stock, and provided with eight scales, two on each slide, and one on +each edge of each groove. While fulfilling the purposes of an ordinary +slide rule, it is of especial service to the practical navigator in +connection with such problems as the “reduction of an ex-meridian sight” +and the “correction of chronometer sights for error in latitude.” The +rule, which has many other applications of a similar character, is made +by Mr. J. H. Steward, Strand, London. + + + LONG-SCALE SLIDE RULES + +It has been shown that the degree of accuracy attainable in slide-rule +calculations depends upon the length of scale employed. Considerations +of general convenience, however, render simple straight-scale rules of +more than 20 in. in length inadmissible, so that inventors of long-scale +slide rules, in order to obtain a high degree of precision, combined +with convenience in operation, have been compelled to modify the +arrangement of scales usually employed. The principal methods adopted +may be classed under three varieties: (1) The use of a long scale in +sectional lengths, as in Hannyngton’s Extended Slide Rule and Thacher’s +Calculating Instrument; (2) the employment of a long scale laid in +spiral form upon a disc, as in Fearnley’s Universal Calculator and +Schuerman’s Calculating Instrument; and (3) the adoption of a long scale +wound helically upon a cylinder, of which Fuller’s and the “R.H.S.” +Calculating Rules are examples. + +FULLER’S CALCULATING RULE.—This instrument, which is shown in Fig. 18, +consists of a cylinder _d_ capable of being moved up and down and around +the cylindrical stock _f_, which is held by the handle. The logarithmic +scale-line is arranged in the form of a helix upon the surface of the +cylinder _d_, and as it is equivalent to a straight scale of 500 inches, +or 41 ft. 8 in., it is possible to obtain four, and frequently five, +figures in a result. + +Upon reference to the figure it will be seen that three indices are +employed. Of these, that lettered _b_ is fixed to the handle; while two +others, _c_ and _a_ (whose distance apart is equal to the axial length +of the complete helix), are fixed to the innermost cylinder _g_. This +latter cylinder slides telescopically in the stock _f_, enabling the +indices to be placed in any required position relatively to _d_. Two +other scales are provided, one (_m_) at the upper end of the cylinder +_d_, and the other (_n_) on the movable index. + +[Illustration: FIG. 18.] + +In using the instrument a given number on _d_ is set to the fixed index +_b_, and either _a_ or _c_ is brought to another number on the scale. +This establishes a ratio, and if the cylinder is now moved so as to +bring any number to _b_, the fourth term of the proportion will be found +under _a_ or _c_. Of course, in multiplication, one factor is brought to +_b_, and _a_ or _c_ brought to 100. The other factor is then brought to +_a_ or _c_, and the result read off under _b_. Problems involving +continuous multiplication, or combined multiplication and division, are +very readily dealt with. Thus, calling the fixed index F, the upper +movable index A, and the lower movable index B, we have for _a_ × _b_ × +_c_:—Bring _a_ to F; A to 100; _b_ to A or B; A to 100; _c_ to A or B +and read the product at F. + +The maximum number of figures in a product is the sum of the number of +figures in the factors and this results when all the factors except the +first have to be brought to B. Each time a factor is brought to A, 1 is +to be deducted from that sum. + +For division, as _a_/(_m_ × _n_), bring _a_ to F; A or B to _m_; 100 to +A; A or B to _a_; 100 to A and read the quotient at F. + +[Illustration: FIG. 19.] + +The maximum number of figures in the quotient is the difference between +the sum of the number of figures in the numerator factors and those of +the denominator factors, _plus_ 1 for each factor of the denominator and +this results when A has to be set to all the factors of the denominator +and all the factors of the numerator except the first brought to B. Each +time B is set to a denominator factor or a numerator factor is brought +to A, 1 is to be deducted. + +Logarithms of numbers are obtained by using the scales _m_ and _n_ and +hence powers and roots of any magnitude may be obtained by the procedure +already fully explained. The instrument illustrated is made by Messrs. +W. F. Stanley & Co., Limited, London. + +THE “R.H.S.” CALCULATOR.—In this calculator, designed by Prof. R. H. +Smith, the scale-line, which is 50 in. long, is also arranged in a +spiral form (Fig. 19), but in this case it is wrapped around the central +portion of a tube which is about ¾in. in diameter and 9½in. long. A +slotted holder, capable of sliding upon the plain portions of this tube, +is provided with four horns, these being formed at the ends of the two +wide openings through which the scale is read. An outer ring carrying +two horns completes the arrangement. + +One of the horns of the holder being placed in agreement with the first +factor, and one of the horns of the ring with the second factor, the +holder is moved until the third factor falls under the same horn of the +ring, when the resulting fourth term will be found under the same (right +or left) horn of the holder, at either end of the slot. In +multiplication, 100 or 1000 is taken for the second factor in the above +proportion, as already explained in connection with Fuller’s rule; +indeed, generally, the mode of operation is essentially similar to that +followed with the former instrument. + +The scale shown on one edge of the opening in the holder, together with +the circular scale at the top of the spiral, enables the mantissæ of +logarithms of numbers to be obtained, and thus problems involving powers +and roots may be dealt with quite readily. This instrument is supplied +by Mr. J. H. Steward, London. + +THACHER’S CALCULATING INSTRUMENT, shown in Fig. 20, consists of a +cylinder 4 in. in diameter and 18 in. long, which can be given both a +rotary and a longitudinal movement within an open framework composed of +twenty triangular bars. These bars are connected to rings at their ends, +which can be rotated in standards fixed to the baseboard. The scale on +the cylinder consists of forty sectional lengths, but of each scale-line +that part which appears on the right-hand half of the cylinder is +repeated on the left-hand half, one line in advance. Hence each half of +the cylinder virtually contains two complete scales following round in +regular order. On the lower lines of the triangular bars are scales +exactly corresponding to those on the cylinder, while upon the upper +lines of the bars and not in contact with the slide is a scale of square +roots. + +[Illustration: FIG. 20.] + +By rotating the slide any line on it may be brought opposite any line in +frame and by a longitudinal movement any graduation on these lines may +be brought into agreement. The whole can be rotated in the supporting +standards in order to bring any reading into view. As shown in the +illustration, a magnifier is provided, this being conveniently mounted +on a bar, along which it can be moved as required. + +SECTIONAL LENGTH OR GRIDIRON SLIDE RULES.—The idea of breaking up a long +scale into sectional lengths is due to Dr. J. D. Everett, who described +such a gridiron type of slide rule in 1866. Hannyngton’s Extended Slide +Rule is on the same principle. Both instruments have the lower scale +repeated. H. Cherry (1880) appears to have been the first to show that +such duplication could be avoided by providing two fixed index points in +addition to the natural indices of the scale. These additional indices +are shown at 10′ and 100′ in Fig. 21, which represents the lower sheet +of Cherry’s Calculator on a reduced scale. The upper member of the +calculator consists of a transparent sheet ruled with parallel lines, +which coincide with the lines of the lower scale when the indices of +both are placed in agreement. To multiply one number by another, one of +the indices on the upper sheet is placed to one of the factors, and the +position of whichever index falls under the transparent sheet is noted +on the latter. Bringing the latter point to the other factor, the result +is found under whichever index lies on the card. In other arrangements +the inventor used transparent scales, the graduations running in a +reverse direction to those of the lower scale. In this case, a factor on +the upper scale is set to the other factor on the lower, and the result +read at the available index. + +[Illustration: FIG. 21.] + +PROELL’S POCKET CALCULATOR is an application of the last-named +principle. It comprises a lower card arranged as Fig. 21, with an upper +sheet of transparent celluloid on which is a similar scale running in +the reverse direction. For continued multiplication and division, a +needle (supplied with the instrument) is used as a substitute for a +cursor, to fix the position of the intermediate results. A series of +index points on the lower card enable square and cube roots to be +extracted very readily. This calculator is supplied by Messrs. John J. +Griffin & Sons, Ltd., London. + + + + + CIRCULAR CALCULATORS. + + +Although the 10 in. slide rule is probably the most serviceable form of +calculating instrument for general purposes, many prefer the more +portable circular calculator, of which many varieties have been +introduced during recent years. The advantages of this type are: It is +more compact and conveniently carried in the waistcoat pocket. The +scales are continuous, so that no traversing of the slide from 1 to 10 +is required. The dial can be set quickly to any value; there is no +trouble with tight or ill-fitting slides. The disadvantages of most +forms are: Many problems involve more operations than a straight rule. +The results being read under fingers or pointers, an error due to +parallax is introduced, so that the results generally are not so +accurate as with a straight rule. The inner scales are short, and +therefore are read with less accuracy. Special scale circles are needed +for cubes and cube roots. The slide cannot be reversed or inverted. + +[Illustration: FIG. 22.] + +[Illustration: FIG. 23.] + +THE BOUCHER CALCULATOR.—This circular calculator resembles a +stem-winding watch, being about 2 in. in diameter and ⁹⁄₁₆in. in +thickness. The instrument has two dials, the back one being fixed, while +the front one, Fig. 22 (showing the form made by Messrs. W. F. Stanley, +London), turns upon the large centre arbor shown. This movement is +effected by turning the milled head of the stem-winder. The small centre +axis, which is turned by rotating the milled head at the side of the +case, carries two fine needle pointers, one moving over each dial, and +so fixed on the axis that one pointer always lies evenly over the other. +A fine index or pointer fixed to the case in line with the axis of the +winding stem, extends over the four scales of the movable dial as shown. +Of these scales, the second from the outer is the ordinary logarithmic +scale, which in this instrument corresponds to a straight scale of about +4¾in. in length. The two inner circles give the square roots of the +numbers on the primary logarithmic scale, the smaller circle containing +the square roots of values between 1 and 3·162 (= √(10)), while the +other section corresponds to values between 3·162 and 10. The outer +circle is a scale of logarithms of sines of angles, the corresponding +sines of which can be read off on the ordinary scale. + +On the fixed or back dial there are also four scales, these being +arranged as in Fig. 23. The outer of these is a scale of equal parts, +while the three inner scales are separate sections of a scale giving the +cube roots of the numbers taken on the ordinary logarithmic scale and +referred thereto by means of the pointers. In dividing this cube-root +scale into sections, the same method is adopted as in the case of the +square-root scale. Thus, the smallest circle contains the cube roots of +numbers between 1 and 10, and is therefore graduated from 1 to 2·154; +the second circle contains the cube roots of numbers between 10 and 100, +being graduated from 2·154 to 4·657; while the third section, in which +are found the cube roots of numbers between 100 and 1000, carries the +graduations from 4·657 to 10. + +What has been said in an earlier section regarding the notation of the +slide rule may in general be taken to apply to the scales of the Boucher +calculator. The manner of using the instrument is, however, not quite so +evident, although from what follows it will be seen that the operative +principle—that of variously combining lengths of a logarithmic scale—is +essentially similar. In this case, however, it is seen that in place of +the straight scale-lengths shown in Fig. 4, we require to add or +subtract arc-lengths of the circular scales, while, further, it is +evident that in the absence of a fixed scale (corresponding to the stock +of the slide rule) these operations cannot be directly performed as in +the ordinary form of instrument. However, by the aid of the fixed index +and the movable pointer, we can effect the desired combination of the +scale-lengths in the following manner. Assuming it is desired to +multiply 2 by 3, the dial is turned in a backward direction until 2 on +the ordinary scale lies under the fixed index, after which the movable +pointer is set to 1 on the scale. As now set, it is clear that the +arc-length 1–2 is spaced off between the fixed index and the movable +pointer, and it now only remains to add to this definite arc-length a +further length of 1–3. To do this we turn the dial still further +backward until the arc 1–3 has passed under the movable pointer, when +the result, 6, is read under the fixed index. A little consideration +will show that any other scale length may be added to that included +between the fixed and movable pointers, or, in other words, any number +on the scale may be multiplied by 2 by bringing the number to the +movable pointer and reading the result under the fixed index. The rule +for multiplication is now evident. + +_Rule for Multiplication._—_Set one factor to the fixed index and bring +the pointer to 1 on the scale; set the other factor to the pointer and +read the result under the fixed index._ + +With the explanation just given, the process of division needs little +explanation. It is clear that to divide 6 by 3, an arc-length 1–3 is to +be taken from a length 1–6. To this end we set 6 to the index +(corresponding in effect to passing a length 1–6 to the left of that +reference point) and set the pointer to the divisor 3. As now set, the +arc 1–6 is included between 1 on the scale and the index, while the arc +1–3 is included between 1 on the scale and the pointer. Obviously if the +dial is now turned forward until 1 on the scale agrees with the pointer, +an arc 1–3 will have been deducted from the larger arc 1–6, and the +remainder, representing the result of this operation, will be read under +the index as 2. + +_Rule for Division._—_Set the dividend to the fixed index, and the +pointer to the divisor; turn the dial until 1 on the scale agrees with +the pointer, and read the result under the fixed index._ + +The foregoing method being an inversion of the rule for multiplication, +is easily remembered and is generally advised. Another plan is, however, +preferable when a series of divisions are to be effected with a constant +divisor—_i.e._, when _b_ in (_a_)/(_b_) = _x_ is constant. In this case +1 on the scale is set to the index and the pointer set to _b_; then if +any value of a is brought to the pointer, the quotient _x_ will be found +under the index. + +_Combined Multiplication and Division_, as (_a_ × _b_ × _c_)/(_m_ × _n_) += _x_, can be readily performed, while cases of continued multiplication +evidently come under the same category, since _a_ × _b_ × _c_ = (_a_ × +_b_ × _c_)/(1 × 1) = _x_. Such cases as _a_/(_m_ × _n_ × _r_) = _x_ are +regarded as (_a_ × 1 × 1 × 1)/(_m_ × _n_ × _r_) = _x_; while (_a_ × _b_ +× _c_)/(_m_) = _x_ is similarly modified, taking the form (_a_ × _b_ × +_c_)/(_m_ × 1) = _x_. In all cases the expression must be arranged so +that there is _one more factor in the numerator_ than _in the +denominator_, _1’s being introduced as often as required_. The simple +operations of multiplication and division involve a similar disposition +of factors, since from the rules given it is evident that _m_ × _n_ is +actually regarded as (_m_ × _n_)/(1), while (_m_)/(_n_) becomes in +effect (_m_ × 1)/(_n_). It is important to note the general +applicability of this arrangement-rule, as it will be found of great +assistance in solving more complicated expressions. + +As with the ordinary form of slide rule, the factors in such an +expression as (_a_ × _b_ × _c_)/(_m_ × _n_) = _x_ are taken in the +order:—1st factor of numerator; 1st factor of denominator; 2nd factor of +numerator; 2nd factor of denominator, and so on; the 1st factor as _a_ +being set to the index, and the result _x_ being finally read at the +same point of reference. + + + EX.—(39 × 14·2 × 6·3)/(1·37 × 19) = 134. + + Commence by setting 39 to the index, and the pointer to 1·37; bring + 14·2 to the pointer; pointer to 19; 6·3 to the pointer, and read the + result 134 at the index. + + +It should be noted that after the first factor is set to the fixed +index, the _pointer_ is set to each of the _dividing_ factors as they +enter into the calculation, while the _dial_ is moved for each of the +_multiplying_ factors. Thus the dial is first moved (setting the first +factor to the index), then the pointer, then the dial, and so on. + +_Number of Digits in the Result._—If rules are preferred to the plan of +roughly estimating the result, the general rules given on pages 21 and +25 should be employed for simple cases of multiplication and division. +For combined multiplication and division, modify the expression, if +necessary, by introducing 1’s, as already explained, and subtract the +sum of the denominator digits from the sum of numerator digits. Then +proceed by the author’s rule, as follows:— + +_Always turn dial to the_ LEFT; _i.e._, _against the hands of a watch_. + +_Note dial movements only; ignore those of the pointer._ + +_Each time 1 on dial agrees with or passes fixed index_, ADD _1 to the +above difference of digits_. + +_Each time 1 on dial agrees with or passes pointer_, DEDUCT _1 from the +above difference of digits_. + +Treat continued multiplication in the same way, counting the 1’s used as +denominator digits as one less than the number of multiplied factors. + + + EX.—(8·6 × 0·73 × 1·02)/(3·5 × 0·23) = 7·95 [7·95473+]. + + Set 8·6 to index and pointer to 3·5. Bring 0·73 to pointer (noting + that 1 on the scale passes the index) and set pointer to 0·23. Set + 1·02 to pointer (noting that 1 on the scale passes the pointer) and + read under index 7·95. There are 1 + 0 + 1 = 2 numerator digits and 1 + + 0 = 1 denominator digit; while 1 is to be added and 1 deducted as + per rule. But as the latter cancel, the digits in the result will be 2 + − 1 = 1. + + +When moving the dial to the left will cause 1 on the dial to pass _both_ +index and pointer (thus cancelling), the dial may be turned back to make +the setting. + +It will be understood that when 1 is the _first_ numerator, and 1 on the +dial is therefore set to the index, no digit addition will be made for +this, as the actual operation of calculating has not been commenced. + +In the Stanley-Boucher calculator (Fig. 23) a small centre scale is +added, on which a finger indicates automatically the number of digits to +be added or deducted; the method of calculating, however, differs from +the foregoing. To avoid turning back to 0 at the commencement of each +calculation, a circle is ground on the glass face, so that a pencil mark +can be made thereon to show the position of the finger when commencing a +calculation. + +_To Find the Square of a Number._— Set the number, on one or other of +the square root scales, to the index, and read the required square on +the ordinary scale. + +_To Find the Square Root of a Number._—Set the number to the index, and +if there is an _odd_ number of digits in the number, read the root on +the inner circle; if an even number, on the second circle. + +_To Find the Cube of a Number._—Set 1 on the ordinary scale to the +index, and the pointer (on the back dial) to the number on one of the +three cube-root scales. Then under the pointer read the cube on the +ordinary scale. + +_To Find the Cube Root of a Number._—Set 1 to index, and pointer to +number. Then read the cube root under the pointer on one of the three +inner circles on the back dial. If the number has + + 1, 4, 7, 10 or −2, −5, etc., digits, use the inner circle. + 2, 5, 8, 11 or −1, −4, etc., „ „ second circle. + 3, 6, 9, 12 or −0, −3, etc., „ „ third circle. + +_For Powers or Roots of Higher Denomination._—Set 1 to index, the +pointer to the number on the ordinary scale, and read on the outer +circle on the back dial the mantissa of the logarithm. Add the +characteristic (see p. 46), multiply by the power or divide by the root, +and set the pointer to the mantissa of the result on this outer circle. +Under the pointer on the ordinary scale read the number, obtaining the +number of figures from the characteristic. + +_To Find the Sines of Angles._—Set 1 to index, pointer to the angle on +the outer circle, and read under the pointer the _natural sine_ on the +ordinary scale; also under the pointer on the outer circle of the back +dial read the _logarithmic sine_. + +THE HALDEN CALCULEX.—After the introduction of the Boucher calculator in +1876, circular instruments, such as the Charpentier calculator, were +introduced, in which a disc turned within a fixed ring, so that scales +on the faces of both could be set together and ratios established as on +the slide rule. Cultriss’s Calculating Disc is another instrument on the +same principle. The Halden Calculex, of which half-size illustrations +are given in Figs. 24 and 25, represents a considerable improvement upon +these early instruments. It consists of an outer metal ring carrying a +fixed-scale ring, within which is a dial. On each side of this dial are +flat milled heads, so that by holding these between the thumb and +forefinger the dial can be set quickly and conveniently. The protecting +glass discs, which are not fixed in the metal ring but are arranged to +turn therein, carry fine cursor lines, and as these are on the side next +to the scales a very close setting can be made quite free from the +effects of parallax. This construction not only avoids the use of +mechanism, with its risk of derangement, but reduces the bulk of the +instrument very considerably, the thickness being about ¼in. + +On the front face, Fig. 24, the fixed ring carries an outer +evenly-divided scale, giving logarithms, and an ordinary scale, 1–10, +which works in conjunction with a similar scale on the edge of the dial. +The two inner circles give the square roots of values on the main scales +as in the Boucher calculator. On the back face, Fig. 25, the ring bears +an outer scale, giving sines of angles from 6° to 90° and an ordinary +scale, 1–10, as on the front face. The scales on the dial are all +reversed in direction (running from right to left), the outer one +consisting of an ordinary (but inverse) scale, 1–10, while the three +inner circles give the cube roots of values on this inverse scale. As +the fine cursor lines extend over all the scales, a variety of +calculations can be effected very readily and accurately. + +[Illustration: FIG. 24.] + +[Illustration: FIG. 25.] + +SPERRY’S POCKET CALCULATOR, made by the Keuffel and Esser Company, New +York (Fig. 26), has two rotating dials, each with its own pointer and +fixed index. The S dial has an outer scale of equal parts, an ordinary +logarithmic scale, and a square-root scale. The L dial has a single +logarithmic scale arranged spirally, in three sections, giving a scale +length of 12½in. The pointers are turned by the small milled head, which +is concentric with the milled thumb-nut by which the two dials are +rotated. The gearing is such that both the L dial and its pointer rotate +three times as fast as the S dial and pointer. All the usual +calculations can be made with the spiral scale, as with the Boucher +calculator, and the result read off on one or other of the three +scale-sections. Frequently the point at which to read the result is +obvious, but otherwise a reference to the single scale on the S dial +will show on which of the three spirals the result is to be found. + +[Illustration: FIG. 26.] + +_The K and E Calculator_, also made by the Keuffel and Esser Company, is +shown in Figs. 27 and 28. It has two dials, of which only one revolves. +This, as shown in Fig. 27, has an ordinary logarithmic scale and a scale +of squares. There is an index line engraved on the glass of the +instrument. The fixed dial has a scale of tangents, a scale of equal +parts and a scale of sines, the latter being on a two-turn spiral. The +pointers, which move together, are turned by a milled nut and the +movable dial by a thumb-nut, as in Sperry’s Calculator, Fig. 26. + +[Illustration: FIG. 27.] + +[Illustration: FIG. 28.] + + + SLIDE RULES FOR SPECIAL CALCULATIONS. + +ENGINE POWER COMPUTER.—A typical example of special slide rules is shown +in Fig. 29, which represents, on a scale of about half full size, the +author’s Power Computer for Steam, Gas, and Oil Engines. This, as will +be seen, consists of a stock, on the lower portion of which is a scale +of cylinder diameters, while the upper portion carries a scale of +horse-powers. In the groove between these scales are two slides, also +carrying scales, and capable of sliding in edge contact with the stock +and with each other. + +This instrument gives directly the brake horse-power of any steam, gas, +or oil engine; the indicated horse-power, the dimensions of an engine to +develop a given power, and the mechanical efficiency of an engine. The +calculation of piston speed, velocity ratios of pulleys and gear wheels, +the circumferential speed of pulleys, and the velocity of belts and +ropes driven thereby, are among the other principal purposes for which +the computer may be employed. + +[Illustration: FIG. 29.] + +THE SMITH-DAVIS PIECEWORK BALANCE CALCULATOR has two scales, 11 feet +long, having a range from 1d. to £20, and marked so that they can be +used either for money or time calculations. The scales are placed on the +rims of two similar wheels and so arranged that the divided edges come +together. The wheels are mounted on a spindle carried at each end in the +bearings of a supporting stand. The wheels are pressed together by a +spring, and move as one. + +To set the scales one to the other, a treadle gear is arranged to take +the pressure of the spring so that when the fixed wheel is held by the +left hand the free wheel can be rotated by the right hand in either +direction. When the amount of the balance has been set to the combined +weekly wage the treadle is released locking the two wheels together, +when the whole can be turned and the amounts respectively due to each +man read off opposite his weekly wage. The Smith-Davis Premium +Calculator is on the same principle but the scales are about 4 feet 6 +inches long and the wheels spring-controlled. Both instruments are +supplied by Messrs. John Davis & Son, Ltd., Derby. + +THE BAINES SLIDE RULE.—In this rule, invented by Mr. H. M. Baines, +Lahore, four slides carrying scales are arranged to move, each in edge +contact with the next. The slides are kept in contact and given the +desired relative movement one to the other, by being attached (at the +back), to a jointed parallelogram. On this principle which is of general +application, the inventor has made a rule for the solution of problems +covered by Flamant’s formula for the flow of water in cast-iron pipes:—V += 76·28_d_^{⁵⁄₇}_s_^{⁴⁄₇}, in which _s_ is the sine of the inclination +or loss of head; _d_ the diameter of the pipe in inches and V the +velocity in feet per second. The formula Q = AV is also included in the +scope of the rule, Q being the discharge in cubic feet per second and A +the cross sectional area of the pipe in square inches. + +FARMAR’S PROFIT-CALCULATING RULE.—The application of the slide rule to +commercial calculations has been often attempted, but the degree of +accuracy required necessitates the use of a long scale, and generally +this results in a cumbersome instrument. In Farmar’s Profit-calculating +Rule the money scale is arranged in ten sections, these being mounted in +parallel form on a roller which takes the place of the upper scale of an +ordinary rule. The roller, which is ¾in. in diameter, is carried in +brackets secured to each end of the stock, so that by rotating the +roller any section of the money scale can be brought into reading with +the scale on the upper edge of the slide and with which the roller is in +contact. This scale gives percentages, and enables calculations to be +made showing profit on turnover, profit on cost, and discount. The lower +scale on the slide, and that on the stock adjacent to it, are similar to +the A and B scales of an ordinary rule. The instrument is supplied by +Messrs. J. Casartelli & Son, Manchester. + + + CONSTRUCTIONAL IMPROVEMENTS IN SLIDE RULES. + +The attention of instrument makers is now being given to the devising of +means for ensuring the smooth and even working of the slide in the stock +of the rule. In some cases very good results are obtained by slitting +the back of the stock to give more elasticity. + +In the rules made by Messrs. John Davis & Son, a metal strip, slightly +curved in cross section as shown at A (Fig. 30), runs for the full +length of the stock to which it is fastened at intervals. Near each end +of the rule, openings about 1 in. long are made in the metal backing +through which the scales on the back of the slide can be read. To +prevent warping under varying climatic conditions both the stock of the +rule and the slide are of composite construction. The base of the stock +is of mahogany, while the grooved sides, firmly secured to the base, are +of boxwood. Similarly the centre portion of the slide is of mahogany and +the tongued sides of boxwood. Celluloid also enters into the +construction, a strip of this material being laid along the bottom of +the groove in the stock. A fine groove runs along the centre of this +strip in order to give elasticity and to allow the sides of the stock to +be pressed together slightly to adjust the fitting of the slide. As a +further means of adjustment the makers fit metal clips at each end of +the rule, so that by tightening two small screws the stock can be closed +on the slide when necessary. + +[Illustration: FIG. 30.] + +[Illustration: FIG. 32.] + +[Illustration: FIG. 31.] + +In the rule made by the Keuffel and Esser Company of New York, one strip +is made adjustable (Fig. 32). + + + THE ACCURACY OF SLIDE RULE RESULTS. + +The degree of accuracy obtainable with the slide rule depends primarily +upon the length of the scale employed, but the accuracy of the +graduations, the eyesight of the operator, and, in particular, his +ability to estimate interpolated values, are all factors which affect +the result. Using the lower scales and working carefully the error +should not greatly exceed 0·15 per cent. with short calculations. With +successive settings, the discrepancy need not necessarily be greater, as +the errors may be neutralised; but with rapid working the percentage +error may be doubled. However, much depends upon the graduation of the +scales. Rules in which one or more of the indices have been thickened to +conceal some slight inaccuracy should be avoided. The line on the cursor +should be sharp and fine and both slide and cursor should move smoothly +or good work cannot be done. Occasionally a little vaseline or clean +tallow should be applied to the edges of the slide and cursor. + +That the percentage error is constant throughout the scale is seen by +setting 1 on C to 1·01 on D, when under 2 is 2·02; under 3, 3·03; under +5, 5·05, etc., the several readings showing a uniform error of 1 per +cent. + +A method of obtaining a closer reading of a first setting or of a result +on D has been suggested to the author by Mr. M. Ainslie, B.Sc. If any +graduation, as 4 on C, is set to 3 on D, it is seen that 4 main +divisions on C (40–44) are equal in scale length to 3 main divisions on +D (30–33). Hence, very approximately, 1 division on C is equal to 0·75 +of a division on D, this ratio being shown, of course, on D under 10 on +C. Suppose √(4·3) to be required. Setting the cursor to 4·3 on A, it is +seen that the root is something more than 2·06. Move the slide until a +main division is found on C, which exactly corresponds to the interval +between 2 and the cursor line, on D. The division 27–28 just fits, +giving a reading under 10 on C, of 74. Hence the root is read as 2·074. +For the higher parts of the scale, the subdivisions, 1–1·1, etc., are +used in place of main divisions. The method is probably more interesting +than useful, since in most operations the inaccuracies introduced in +making settings will impose a limit on the reliable figures of the +result. + +For the majority of engineering calculations, the slide rule will give +an accuracy consistent with the accuracy of the data usually available. +For some purposes, however, _logarithmic section paper_ (the use of +which the author has advocated for the last twenty years) will be found +especially useful, more particularly in calculations involving +exponential formulæ. + + + + + APPENDIX. + + + NEW SLIDE RULES—FIFTH ROOTS, ETC.—THE SOLUTION OF ALGEBRAIC + EQUATIONS—GAUGE POINTS AND SIGNS ON SLIDE RULES—TABLES AND DATA—SLIDE + RULE DATA SLIPS. + +THE PICKWORTH SLIDE RULE.—In this rule, made by Mr. A. W. Faber, the +novel feature is the provision of a scale of cubes (F) in the stock or +body of the rule. From Fig. 33 it will be seen that the scale is fixed +on the bevelled side of a slotted recess in the back of the rule. The +slide carries an index mark, which is seen through the slot and can be +set to any graduation of the scale; in its normal position it agrees +with 1 on the scale. The C scale on the face of the rule is divided into +three equal parts by two special division lines, marked II. and III., +which, together with the initial graduation 1 of the scale, serve for +setting or reading off values on the D scale. Similar division lines are +marked on the D scale. + +[Illustration: FIG. 33.] + +In using the rule for cubes or cube roots the slide is drawn to the +right, this movement never exceeding one-third of the length of the D +scale. With this limited movement, and with a single setting of the +slide, the values of ∛_̅a_, ∛(_a_ × 10), and ∛(_a_ × 100)) (_a_ being +less than 10 and not less than 1) are given simultaneously and without +any uncertainty as to the scales to use or the values to be read off. + +_To Find the Cube of a Number._—The marks II. and III. on D divide that +scale into three equal sections. If the number to be cubed is in the +first section, I. on C is set to it; if in the second section, II. on C +is set to it; if in the third section, III. on C is set to it. Then, +under the index mark on the back of the slide will be found the +significant figures of the cube on the scale F. If I. on C was used for +the setting, the cube contains 1 digit; if II. was used, 2 digits; if +III. was used, 3 digits. If the first figure of the number to be cubed +is not in the units place, the decimal point is moved through _n_ places +so as to bring the first significant figure into the units place, the +cube found as above, and the decimal point moved in the _reverse +direction_ through 3_n_ places. + +_To Find the Cube Root of a Number._—The index mark is set to the +significant figures of the number on scale F, and the cube root is read +on D under I., II. or III. on C, according as the number has 1, 2 or 3 +digits preceding the decimal point. Numbers which have 1, 2 or 3 figures +preceding the decimal point are dealt with directly. Numbers of any +other form are brought to one of the above forms by moving the decimal +point 3 places (or such multiple of 3 places as may be required), the +root found and its decimal point moved 1 place for each 3-place +movement, but in the _reverse direction_. + +THE “ELECTRO” SLIDE RULE.—In this special rule for electrical +calculations, made by Mr. A. Nestler, the upper scales run from 0·1 to +1000, and are marked “Amp.” and “sq. mm.” respectively. The lower scale +on the slide running from 1 to 10,000 is marked M (metres), while the +lower scale on the rule (0·1 to 100) is marked “Volt.” The latter scale +is so displaced that 10 on M agrees with 0·173 on the Volt scale. The +four factors involved are the current strength (in Amp.); the area of a +conductor (in sq. mm.); the length of the conductor (in metres); and the +permissible loss of potential (in volts). Having given any three of +these, the fourth can be found very readily. On the back of the slide +are a scale of squares, a scale of cubes and a single scale +corresponding to the D scale of an ordinary rule. Hence, by reversing +the slide, it is possible to obtain the 2nd, 3rd and 4th powers and +roots of numbers. In another form of the rule, the scale of metres is +replaced by one of yards, while instead of the area of the conductor in +sq. mm., the corresponding “gauge” sizes of wires are given. + +THE “POLYPHASE” SLIDE RULE.—This instrument, made by the Keuffel & Esser +Company, New York, has, in addition to the usual scales, a scale of +cubes on the vertical edge of the stock of the rule, while in the centre +of the slide there is a reversed C scale; _i.e._, a scale exactly +similar to an ordinary C scale but with the graduations running from +right to left. The rule is specially useful for the solution of problems +containing combinations of three factors and problems involving squares, +square roots, cubes, cube roots and many of the higher powers and roots. +It is specially adapted for electrical and hydraulic work. + +THE LOG-LOG DUPLEX SLIDE RULE.—The same makers have introduced a log-log +duplex slide rule, in which the log-log scale is in three sections, +placed one above the other, these occupying the position usually taken +up by the A scale. These scales are used in the manner already described +(page 86), but some advantage is obtained by the manner in which the +complete log-log scale is divided, the limits being _e_^{¹⁄₁₀₀} to _e_^⅒ +(on Scale L.L. 1); _e_^⅒ to _e_ (on Scale L.L. 2); and _e_ to _e_^{10} +(on Scale L.L. 3), _e_ being the base of natural or hyperbolic +logarithms (2·71828). In this way a total log-log range of from 1·01 to +22,000 is provided, meeting all practical requirements. These log-log +scales are read in conjunction with a C scale placed at the upper edge +of the slide. A similar C scale, but reversed in direction, is placed at +the lower edge of the slide, this having red figures to distinguish it +readily. The adjacent scale on the body of the rule is an ordinary D +scale, and under this is an equally-divided scale giving the common +logarithms of values on D. In the centre of the slide is a scale of +tangents. + +It will be understood that a “duplex” rule consists of two side strips +securely clamped together at the two ends, forming the body of the rule, +the slide moving between them; hence both front and back faces of the +rule and slide are available, graduations on the one side being referred +to those on the other by the cursor which extends around the whole. In +this instrument, the scales on the back face are the ordinary scales of +the standard rule with the addition of a scale of sines which is placed +in the centre of the slide. It will be evident that this instrument is +capable of dealing with a very wide range of problems involving +exponential and trigonometrical formulæ. + +SMALL SLIDE RULES WITH MAGNIFYING CURSORS.—Several makers now supply 5 +in. rules having the full graduations of a 10 in. rule, and fitted with +a magnifying cursor (Fig. 34). This forms a compact instrument for the +pocket, but owing to the closeness of the graduations it is not usually +possible to make a setting of the slide without using the cursor. This, +of course, involves more movements than with the ordinary instrument. It +is also very necessary to use the magnifying cursor in a _direct_ light, +if accurate readings are to be obtained. If these slight inconveniences +are to be tolerated, the principle could be extended, a 10 in. rule +being marked as fully as a 20 in., and fitted with a magnifying cursor. +The author has endeavoured, but without success, to induce makers to +introduce such a rule. + +The magnifying cursor, supplied by Messrs. A. G. Thornton, Limited, has +a lens which fills the entire cursor. It has a powerful magnifying +effect, and the change from the natural to the magnified reading is less +abrupt than with the semicircular lens. + +[Illustration: FIG. 34.] + +THE CHEMIST’S SLIDE RULE.—A slide rule, specially adapted for chemical +calculations, has been introduced recently by Mr. A. Nestler. In this +instrument the C and D scales are as usually arranged; but, in place of +the A and B scales, there are a number of gauge points or marks denoting +the atomic and molecular weights of the most important elements and +combinations. The scales on the back of the slide are similarly +arranged, so that by reversing the slide the operations can be extended +very considerably. The rule finds its chief use in the calculation of +analyses. Thus, to find the percentage of chlorine if _s_ grammes of a +substance have been used and the precipitate of Ag.Cl. weighs _a_ +grammes, we have the equation, _x_ = (Cl.)/(Ag.Cl.) × (_a_)/(_s_). +Hence, the mark Ag.Cl. on the upper scale of the slide is set to the +mark Cl. on the upper scale of the rule, when under _a_ on the C scale +is found the quantity of chlorine on D. By setting the cursor to this +value and bringing _s_ on C to the cursor, the percentage required can +be read on C over 10 on D. + +The rule is also adapted to the solution of various other chemical and +electro-chemical calculations. + +THE STELFOX SLIDE RULE.—This rule, shown in Fig. 35, has a stock 5 in. +long, fitted with a 10 in. slide jointed in the middle of its length by +means of long dowels. By separating the parts the compactness of a 5 in. +rule is obtained. The upper scales on the rule and slide resemble the +usual A and B scales. The D scale on the lower part of the stock is in +two sections, the second portion being placed below the first, as shown +in the illustration. The centre scale on the slide corresponds to the +usual C scale, while on the lower edge of the slide is a similar scale, +but with the index (1) in the middle of its length. The arrangement +avoids the necessity of resetting the slide, as is sometimes necessary +with the ordinary rule, and in general it combines the accuracy of a 10 +in. rule with the compactness of a 5 in. rule; but a more frequent use +of the cursor is necessary. This rule is made by Messrs. John Davis & +Son, Limited, Derby. + +[Illustration: FIG. 35.] + +ELECTRICAL SLIDE RULE.—Another rule by the same makers, specially useful +for electrical engineers, has the usual scales on the working edges of +the rule and slide, while in the middle of the slide is placed a scale +of cubes. A log-log scale in two sections is provided; the power +portion, running from 1·07 to 2, is found on the lower part of the +stock, and the upper portion, running from 2 to 10^3, on the upper part +of the stock. The uppermost scale on the stock is in two parts, of which +that to the left, running from 20 to 100 and marked “Dynamo,” gives the +efficiencies of dynamos; that on the right, running from 20 to 100 and +marked “Motor,” gives the efficiencies of electric motors. The lowest +scale on the stock, marked “Volt,” gives the loss of potential in copper +conductors. The ordinary upper scale on the stock is marked L (length of +lead) at the left, and KW (kilowatts) at the right; the ordinary upper +scale on the slide is marked A (ampères) and mm^2 (sectional area) at +the left, and HP (horse-power) at the right. Additional lines on the +cursor enable the electrical calculations to be made either in British +or metric units. + +THE PICOLET CIRCULAR SLIDE RULE.—A simple form of circular calculator, +made by Mr. L. E. Picolet of Philadelphia, is shown in Fig. 36. It +consists of a base disc of stout celluloid on which turns a smaller disc +of thin celluloid. A cursor formed of transparent celluloid is folded +over the discs, and is attached so that the friction between the cursor +and the inner disc enables the latter to be turned by moving the former. +By holding both discs the cursor can be adjusted as required. The +adjacent scales run in opposite directions, so that multiplication and +division are performed as with the inverted slide in an ordinary rule. +The outer scale, which is two-thirds the length of the main scale, +enables cube roots to be found. Square roots are readily determined and +continuous multiplication and division conveniently effected. Modified +forms of this neatly made little instrument are also available. + +[Illustration: FIG. 36.] + +OTHER RECENT SLIDE RULES.—Among other special types of slide rule, +mention should be made of the _Jakin_ 10 in. rule for surveyors, made by +Messrs. John Davis & Son, Limited, Derby. By the provision of a series +of short subsidiary scales, the multiplication of a sine or tangent of +an angle by a number can be obtained to an accuracy of 1 in 10,000. The +_Davis-Lee-Bottomley_ slide rule, by the same makers, has special scales +provided for circle spacing. The division of a circle into a number of +equal parts, often required in spacing rivets, bolts, etc., and in +setting out the teeth of gearwheels, is readily effected by the aid of +this instrument. The _Cuntz_ slide rule is a very comprehensive +instrument, having a stock about 2¼ in. wide, with the slide near the +lower edge. Above the slide are eleven scales, referable to the main +scales by the cursor. These scales enable squares and square roots, +cubes and cube roots, and areas and circumferences of circles to be +obtained by direct reading. A much more compact instrument could be +obtained by removing one-half the scales to the back of the rule and +using a double cursor. + +[Illustration: FIG. 37.] + +In one form of 10 in. rule, supplied by Mr. W. H. Harling, London, the +body of the rule is made of well-seasoned cane, with the usual celluloid +facings. The rule has a metal back, enabling the fit of the slide to be +regulated. This backing extends the full length of the rule, openings +about 1 in. long being provided at each end, enabling the scales on the +back of the slide to be set with greater facility than is possible with +the notched recesses usually adopted. The author has long endeavoured, +but without success, to induce makers to fit windows of glass or +celluloid in place of the notched recesses. This would allow the +graduation of the S and T scales to be set more accurately, and enable +both to be used at each end of the rule—an advantage in certain +trigonometrical calculations. It would have the further advantage of +permitting each alternate graduation of the evenly-divided or logarithm +scale to be placed at opposite sides of one central line, enabling the +reading to be made more accurately and conveniently. + +Many special slide rules have lately been devised for determining the +time necessary to perform various machine-tool operations and for +analogous purposes, while attention has again been given to rules for +calculating the weights of iron and steel bars, plates, etc. + +THE DAVIS-STOKES FIELD GUNNERY SLIDE RULE.—This rule, which is adapted +for calculations involved in “encounter” and “entrenched” field gunnery, +is designed for the 18 pr. quick-firing gun. The upper and lower +portions of the boxwood stock are united by a flexible centre of +celluloid, thus providing grooves front and rear to receive boxwood +slides. Each of the nineteen scales is marked with its name, and +corresponding scales are coloured red or black. The front edge is +bevelled and carries a scale of 1 in 20,000. The rule solves +displacement problems, map angles of sight, changes of corrector and +range corrections for changes in temperature, wind and barometer, etc. A +special feature for displacement calculations is the provision of a 50 +yd. sub-base angle scale, by which the apex angle is read at one +setting. + +THE DAVIS-MARTIN WIRELESS SLIDE RULE.—In wireless telegraphy it is +frequently necessary to determine wave-length, capacity or +self-induction when one or other of the factors of the equation, λ = +59·6√(LC) is unknown. The Davis-Martin wireless rule is designed to +simplify such calculations. The upper scale in the stock (inductance) +runs from 10,000 to 1,000,000; the adjacent scale on the slide +(capacity) runs from 0·0001 to 0·01 but in the reverse direction. The +lower scale on the stock (wave-length) runs from 100 to 1000, giving +square roots of the upper scale; while on the lower edge of the scale +are several arrows to suit the various denominations in which the +wave-length and capacity may be expressed. + +IMPROVED CURSORS.—In some slide-rule operations, notably in those +involved in solving quadratic and cubic equations, it not infrequently +happens that readings are obscured by the frame of the cursor. Frameless +cursors have been introduced to obviate this defect. A piece of thick +transparent celluloid is sometimes employed, but this is liable to +become scratched in use. Fig. 37 shows a recent form of frameless glass +cursor made by the Keuffel & Esser Company, Hoboken, N.J., which is +satisfactory in every way. + +Cursors having three hair lines are now fitted to some rules, the +distance apart of the lines being equal to the interval 0·7854–1 on the +A scale. + +THE DAVIS-PLETTS SLIDE RULE.—In this rule a single log.-log. scale and +its reciprocal scale are arranged opposite the ordinary upper log. +scale. Thus, common logarithms can be read directly, while by taking +advantage of the properties of characteristics and mantissas of common +logarithms, the scale can be extended indefinitely. As 10 is the highest +number on the log.-log. scale, it is carried down to within 0·025 of +unity. The reading of log.-log. values above 10 is effected in a very +simple manner. There is also a scale in the centre of the slide which, +used in conjunction with the upper log. scale enables the natural +logarithm of any number between 0·0001 and 10,000 to be read direct, +while any number on the upper log. scale can be multiplied or divided by +_e^x_ if the latter is between these limits. On the back of the slide +are scales for all circular and hyperbolic functions, these being used +in conjunction with the upper log. scales. + +THE CROMPTON-GALLAGHER BOILER EFFICIENCY CALCULATOR has a stock in the +thickness of which is a slot admitting a chart which can be moved at +right angles to the two separate slides. On the bevelled edge of one +slide, the graduations are continued so as to read against curves on the +chart, through an opening in the stock. + +THE DAVIS-GRINSTED COMPLEX CALCULATOR.—This slide rule is of +considerable service in connection with calculations involving the +conversion of complex quantities from the form _a_ + _j_ _b_ to the form +R∠θ, and _vice versa_. The usual process of conversion necessitates +repeated reference to trigonometrical tables, and is both tedious and +time-taking. The Complex Calculator enables the conversions to be +effected without reference to tables and with the minimum expenditure of +time and labour. + +The rule, which is about 16 in. long, has five scales. The upper one (A) +is an ordinary logarithmic scale thrice-repeated. The adjacent scales on +the slide comprise (1) a logarithmic scale of tangents (B) ranging from +0·1° to 45°, and (2) a logarithmic scale of secants (C) from 0° to 45°. +The lower scales D and E are identical with the A scale, and are +provided to enable multiplication, etc., to be performed without the +need for a separate slide rule. Readings can be transferred from A to +the lower scales by means of the cursor. + +In using the rule to convert _a_ + _j_ _b_ to R∠θ, the index (45°) of +the B scale is set to the larger component and the cursor to the smaller +component, on scale A. Then θ (or its complement if _b_ is greater than +_a_) is read on B under the cursor. The cursor is then set to θ on the C +scale, and R is read on A under the cursor. The rule is made by Messrs. +John Davis & Son, Limited, Derby. + + + THE SOLUTION OF ALGEBRAIC EQUATIONS. + +The slide rule finds an interesting application in the solution of +equations of the second and third degree; and although the process is +essentially one of trial and error, it may often serve as an efficient +substitute for the more laborious algebraic methods, particularly when +the conditions of the problem or the operator’s knowledge of the theory +of equations enables some idea to be obtained as to the character of the +result sought. The principle may be thus briefly explained:—If 1 on C is +set to _x_ on D (Fig. 38), we find _x_(_x_) = _x_^2 on D under _x_ on C. +If, however, with the slide set as before, instead of reading under _x_, +we read under _x_ + _m_ on C, the result on D will now be _x_(_x_ + _m_) += _x_^2 + _mx_ = _q_. Hence to solve the equation _x_^2 + _mx_ − _q_ = +0, we reverse the above process, and setting the cursor to _q_ on D, we +move the slide until the number on C under the cursor, and that on D +under 1 on C, _differ by m_. It is obvious from the setting that the +_product_ of these numbers = _q_, and as their difference = _m_, they +are seen to be the roots of the equation as required. For the equation +_x_^2 − _mx_ + _q_ = 0, we require _m_ to equal the _sum_ of the roots. +Hence, setting the cursor as before to _q_ on D, we move the slide until +the number on C under the cursor, and that on D under 1 on C, are +_together equal to_ _m_, these numbers being the roots sought. The +alternative equations _x_^2 − _mx_ − _q_ = 0, and _x_^2 + _mx_ + _q_ = 0 +are deducible from the others by changing the signs of the roots, and +need not be further considered. + +[Illustration: FIG. 38.] + + + EX.—Find the roots of _x_^2 − 8_x_ + 9 = 0. + + Set the cursor to 9 on D, and move the slide to the right until when + 6·64 is found under the cursor, 1·355 on D is under 1 on C. These + numbers are the roots required. + + +The upper scales can of course be used; indeed, in general they are to +be preferred. + + + EX.—Find the roots of _x_^2 + 12·8_x_ + 39·4 = 0. + + Set the cursor to 39·4 on A, and move the slide to the right until we + read 7·65 on B under the cursor, and 5·15 on A over 1 on B. The roots + are therefore −7·65 and −5.15. + + +With a little consideration of the relative value of the upper and lower +scales, the student interested will readily perceive how equations of +the third degree may be similarly resolved. The subject is not of +sufficient general importance to warrant a detailed examination being +made of the several expressions which can be dealt with in the manner +suggested; but the author gives the following example as affording some +indication of the adaptability of the method to practical calculations. + + + EX.—A hollow copper ball, 7·5 in. in diameter and 2 lb. in weight, + floats in water. To what depth will it sink? + + The water displaced = 27·7 × 2 = 55·4 cub. in. The cubic contents of + the immersed segment will be (π)/(3)(3_r_ _x_^2 − _x_^3), _r_ being + the radius and _x_ the depth of immersion. Hence (π)/(3)(3_r_ _x_^2 − + _x_^3) = 55·4, and 11·25_x_^2 − _x_^3 = 52·9. + + To solve this equation we place the cursor to 52·9 on A, and move the + slide until the reading on D under 1 and that on B under the cursor + together amount to 11·25. In this way find 2·45 on D under 1, with 8·8 + on B under the cursor _c_, _c_, as a pair of values of which the sum + is 11·25. Hence we conclude that _x_ = 2·45 in. is the result sought. + + With the rule thus set (Fig. 39) the student will note that the slide + is displaced to the right by an amount which represents _x_ on D, and + therefore _x_^2 on A; while the length on B from 1 to the cursor line + represents 11·25 − _x_. Hence the upper scale setting gives + _x_^2(11·25 − _x_) = 11·25_x_^2 − _x_^3 = 52·9 as required. + + +[Illustration: FIG. 39.] + +When in doubt as to the method to be pursued in any given case, the +student should work synthetically, building up a simple example of an +analogous character to that under consideration, and so deducing the +plan to be followed in the reverse process. + + + SCREW-CUTTING GEAR CALCULATIONS. + +The slide rule has long found a useful application in connection with +the gear calculations necessary in screw-cutting, helical gear-cutting, +and spiral gear work. + +SINGLE GEARS.—For simple cases of screw-cutting in the lathe it is only +necessary to set the threads per inch to be cut to the threads per inch +in the guide screw (or the pitch in inches in each case, if more +convenient). Then any pair of coinciding values on the two scales will +give possible pairs of wheels. + + + EX.—Find wheels to cut a screw of 1⅝ threads per inch with a guide + screw of 2 threads per inch. + + Setting 1·625 on C to 2 on D, it is seen that 80 (driver) and 65 + (driven) are possible wheels. + + +COMPOUND GEARS.—When wheels so found are of inconvenient size, a +compound train is used, consisting (usually) of two drivers and two +driven wheels, the product of the two former and the product of the two +latter being in the same ratio as the simple wheels. Thus with 60 and 40 +as drivers, and 65 and 30 as driven, we have, (60 × 40)/(65 × 30) = +(2400)/(1950) = (2)/(1·625) as before. + +With the slide set as above, values convenient for splitting up into +suitable wheels are readily obtainable. Thus, (1600)/(1300); +(2400)/(1950); (4000)/(3250); (4800)/(3900) are a few suggestive values +which may be readily factorised. + +SLIDE RULES FOR SCREW-CUTTING CALCULATIONS.—Special circular and +straight slide rules for screw-cutting gear calculations have long been +employed. For compound gears these usually entail the use of six scales, +two on each of the two slides and two on the stock. The upper scale on +the stock may be a scale of threads per inch to be cut, the adjacent +scale (on the upper slide) a scale of threads per inch in the guide +screw. Setting the guide screw-graduation to the threads to be cut, the +lower slide is adjusted until a convenient pair of drivers is found in +coincidence on the central pair of scales, while a pair of driven wheels +are in coincidence on the two lower scales. + +Some years ago, a slide rule was introduced by which compound gears +could be obtained with a single slide. Assuming the set of wheels +usually provided—20 to 120 teeth advancing by 5 teeth—the products of 20 +× 25, 20 × 30, etc., up to 115 × 120 were calculated. These products +were laid out along each of the two lower scales. The upper scales were +a scale of threads per inch to be cut and a scale of the threads per +inch of various guide screws. Setting the guide screw-graduation to the +threads to be cut, any coinciding graduations on the lower scales gave +the required pairs of drivers and driven wheels. + +FRACTIONAL PITCH CALCULATIONS.—The author has long advocated the use of +the slide rule for determining the wheels necessary for cutting +fractional pitch threads, and it is gratifying to find its value in this +connection is now being appreciated. For the best results a good 20 in. +rule is desirable, but with care very close approximations can be found +with an accurate 10 in. rule. In any case a magnifying cursor or a hand +reading-glass is of great assistance. + + + EX.—Find wheels to cut a thread of 0·70909 in. pitch; guide screw, 2 + threads per inch. + + To 0·70909 on D, set 0·5 (guide screw pitch in inches) on C. To make + this setting as accurately as possible, the method described on page + 112 may be used. Set 10 on C to about 91 on D, and note that the + interval 77–78 on C represents 0·91 of the interval 70–71 on D. Set + the cursor to 78 on C and bring 5 to the cursor. The slide is then set + so that 5 on C agrees with 7·091 on D. + + +Inspection of the two scales shows various coinciding factors in the +ratio required. The most accurate is seen to be (55 on C)/(78 on D). +These values may be split up into (55 × 50)/(65 × 60) to form a suitable +compound train of gears. + + + GAUGE POINTS AND SIGNS ON SLIDE RULES. + +Many slide rules have the sign (Prod.)/(−1) at the right-hand end of the +D scale, while on the left is (Quot.)/(+1.) It is somewhat unfortunate +that these signs refer to rules for determining the number of digits in +products and quotients, which are used to a considerable extent on the +Continent, and conflict with those used in this country. By the +Continental method the number of digits in a product is equal to the sum +of the digits in the two factors, if the result is obtained on the LEFT +_of the first factor_; but if the result is found on the RIGHT of the +first factor, it is equal to this sum − 1. The sign (Prod.)/(−1) the +_right_-hand end of the D scale provides a visible reminder of this +rule. + +Similarly for division:—The number of digits in a quotient is equal to +the number of the digits in the dividend, minus those in the divisor, if +the quotient appears on the RIGHT _of the dividend_, and to this +difference + 1, if the quotient appears on the LEFT of the dividend. The +sign (Quot.)/(+1) at the _left_-hand end of the D scale provides a +visible reminder of this rule. + +The sign + + +ⵏ– + ⟵ⵏ⟶ + –ⵏ+ + +found at both ends of the A scale is of general application but of +questionable utility. It is assumed to represent a fraction, the +vertical line indicating the position of the decimal point. If the +number 455 is to be dealt with in a multiplication on the lower scales, +we may suppose the decimal point moved two places to the left, giving +4·55, a value which can be actually found on the scale. If we use this +value, then to the number of digits in this result, as many must be +added as the number of places (two in this case) by which the decimal +point was moved. If the point is moved to the right, the number of +places must be subtracted. Similarly, in division, if the decimal point +in the divisor is moved _n_ places to the left, then _n_ places must be +subtracted at the end of the operation; while if the point is moved +through _n_ places to the right, then _n_ places must be added. The sign +referred to, which, of course, applies to all scales, completely +indicates these processes and is submitted as a reminder of the +procedure to be followed by those using the method described. + +The signs π, _c_, _c′_, and M are explained in the Section on “Gauge +Points,” p. 53. + +On some rules additional signs are found on the D scale. One, locating +the value (180 × 60)/(π) = 3437·74 and hence giving the number of +minutes in a radian, is marked ρ′. Another, representing the value (180 +× 60 × 60)/(π) = 206265, and hence giving the number of seconds in a +radian is marked ρ″. A third point, marked ρ_{˶}, placed at the value +(200 × 100 × 100)/(π) = 636620, is used when the newer graduation of the +circle is employed. + +These gauge points are useful when converting angles into circular +measure, or _vice versa_, and also for determining the functions of +small angles. + +A gauge point is sometimes marked at 1146 on the A and B scales. This is +known as the “Gunner’s Mark,” and is used in artillery calculations +involving angles of less than 20°, when, for the purpose in view, the +tangent and circular measure of the angle may be regarded as equal. For +this constant, the angle is taken in minutes, the auxiliary base in +feet, and the base in yards. The auxiliary base in feet on B is set to +the angle in minutes on A when over 1146 on B is the base in yards on A. +The value (1)/(1146) = (π × 3)/(180 × 60). + + + TABLES AND DATA. + + + MENSURATION FORMULAE. + + + Area of a parallelogram = base × height. + + Area of rhombus = ½ product of the diagonals. + + Area of a triangle = ½ base × perpendicular height. + + Area of equilateral triangle = square of side × 0·433. + + Area of trapezium = ½ sum of two parallel sides × height. + + Area of any right-lined figure of four or more unequal sides is found + by dividing it into triangles, finding area of each and adding + together. + + Area of regular polygon = (1) length of one side × number of sides × + radius of inscribed circle; or (2) the sum of the triangular areas + into which the figures may be divided. + + Circumference of a circle = diameter × 3·1416. + + Circumference of circle circumscribing a square = side × 4·443. + + Circumference of circle = side of equal square × 3·545. + + Length of arc of circle = radius × degrees in arc × 0·01745. + + Area of a circle = square of diameter × 0·7854. + + Area of sector of a circle = length of arc × ½ radius. + + Area of segment of a circle = area of sector − area of triangle. + + Side of square of area equal to a circle = diameter × 0·8862. + + Diameter of circle equal in area to square = side of square × 1·1284. + + Side of square inscribed in circle = diameter of circle × 0·707. + + Diameter of circle circumscribing a square = side of square × 1·414. + + Area of square = area of inscribed circle × 1·2732. + + Area of circle circumscribing square = square of side × 1·5708. + + Area of square = area of circumscribing circle × 0·6366. + + Area of a parabola = base x ⅔ height. + + Area of an ellipse = major axis × minor axis × 0·7854. + + Surface of prism or cylinder = (area of two ends) + (length × + perimeter). + + Volume of prism or cylinder = area of base × height. + + Surface of pyramid or cone = ½(slant height × perimeter of base) + + area of base. + + Volume of pyramid or cone = (⅓)(area of base × perpendicular height). + + Surface of sphere = square of diameter × 3·1416. + + Volume of sphere = cube of diameter × 0·5236. + + Volume of hexagonal prism = square of side × 2·598 × height. + + Volume of paraboloid = ½ volume of circumscribing cylinder. + + Volume of ring (circular section) = mean diameter of ring × 2·47 × + square of diameter of section. + + + SPECIFIC GRAVITY AND WEIGHT OF MATERIALS. + + METALS. + ─────────────────────┬───────────────┬───────────────┬─────────────── + METAL. │ Specific │ Weight of 1 │ Weight of 1 + │ Gravity. │Cub. Ft. (Lb.).│Cub. In. (Lb.). + ─────────────────────┼───────────────┼───────────────┼─────────────── + Aluminium, Cast │ 2·56│ 160│ 0·0927 + Aluminium, Bronze │ 7·68│ 475│ 0·275 + Antimony │ 6·71│ 418│ 0·242 + Bismuth │ 9·90│ 617│ 0·357 + Brass, Cast │ 8·10│ 505│ 0·293 + „ Wire │ 8·548│ 533│ 0·309 + Copper, Sheet │ 8·805│ 549│ 0·318 + „ Wire │ 8·880│ 554│ 0·321 + Gold │ 19·245│ 1200│ 0·695 + Gun metal │ 8·56│ 534│ 0·310 + Iron, Wrought (mean) │ 7·698│ 480│ 0·278 + „ Cast (mean) │ 7·217│ 450│ 0·261 + Lead, Milled Sheet │ 11·418│ 712│ 0·412 + Manganese │ 8·012│ 499│ 0·289 + Mercury │ 13·596│ 849│ 0·491 + Nickel, Cast │ 8·28│ 516│ 0·300 + Phosphor Bronze, Cast│ 8·60│ 536·8│ 0·310 + Platinum │ 21·522│ 1342│ 0·778 + Silver │ 10·505│ 655│ 0·380 + Steel (mean) │ 7·852│ 489·6│ 0·283 + Tin │ 7·409│ 462│ 0·268 + Zinc, Sheet │ 7·20│ 449│ 0·260 + „ Cast │ 6·86│ 428│ 0·248 + ─────────────────────┴───────────────┴───────────────┴─────────────── + + MISCELLANEOUS SUBSTANCES. + ────────────┬──────────┬────────── + SUBSTANCE. │ Specific │Weight of + │ Gravity. │1 Cub. In. + │ │ (Lb.). + ────────────┼──────────┼────────── + Asbestos │ 2·1–2·80 │·076-·101 + Brick │ 1·90 │ ·069 + Cement │2·72–3·05 │·0984-·109 + Clay │ 2·0 │ ·072 + Coal │ 1·37 │ ·0495 + Coke │ 0·5 │ ·0181 + Concrete │ 2·0 │ ·072 + Fire-brick │ 2·30 │ ·083 + Granite │ 2·5–2·75 │·051-·100 + Graphite │ 1·8–2·35 │·065-·085 + Sand-stone │ 2·3 │ ·083 + Slate │ 2·8 │ ·102 + Wood— │ │ + Beech │ 0·75 │ ·0271 + Cork │ 0·24 │ ·0087 + Elm │ 0·58 │ ·021 + Fir │ 0·56 │ ·0203 + Oak │ ·62-·85 │·025-·031 + Pine │ 0·47 │ ·017 + Teak │ 0·80 │ ·029 + ────────────┴──────────┴────────── + + ULTIMATE STRENGTH OE MATERIALS. + ──────────────────┬────────────┬────────────┬────────────┬──────────── + MATERIAL. │ Tension in │Compression │Shearing in │ Modulus of + │lb. per sq. │ in lb. per │lb. per sq. │ Elasticity + │ in. │ sq. in. │ in. │ in lb. per + │ │ │ │ sq. in. + ──────────────────┼────────────┼────────────┼────────────┼──────────── + Cast Iron │ 11,000 to│ 50,000 to│ │ 14,000,000 + │ 30,000│ 130,000│ │ to + │ │ │ │ 23,000,000 + „ aver.│ 16,000│ 95,000│ 11,000│ + Wrought Iron │ 40,000 to│ │ │ 26,000,000 + │ 70,000│ │ │ to + │ │ │ │ 31,000,000 + „ aver.│ 50,000│ 50,000│ 40,000│ + Soft Steel │ 60,000 to│ │ │ 30,000,000 + │ 100,000│ │ │ to + │ │ │ │ 36,000,000 + Soft Steel aver.│ 80,000│ 70,000│ 55,000│ + Cast Steel aver.│ 120,000│ │ │ 15,000,000 + │ │ │ │ to + │ │ │ │ 17,000,000 + Copper, Cast │ 19,000│ 58,000│ │ + „ Wrought │ 34,000│ │ │ 16,000,000 + Brass, Cast │ 18,000│ 10,500│ │ 9,170,000 + Gun Metal │ 34,000│ │ │ 11,500,000 + Phosphor Bronze │ 58,000│ │ 43,000│ 13,500,000 + Wood, Ash │ 17,000│ 9,300│ 1,400│ + „ Beech │ 16,000│ 8,500│ │ + „ Pine │ 11,000│ 6,000│ 650│ 1,400,000 + „ Oak │ 15,000│ 10,000│ 2,300│ 1,500,000 + Leather │ 4,200│ │ │ 25,000 + ──────────────────┴────────────┴────────────┴────────────┴──────────── + + POWERS, ROOTS, ETC., OF USEFUL FACTORS. + _n_ │(1)/(_n_)│ _n_^2 │ _n_^3 │ √_̅n_ │ (1)/(√_̅n_) │ ∛_̅n_ │ (1)/(∛_̅n_) + ────────────────┼─────────┼───────┼──────────┼──────┬┴───────────┬─┴────┬──┴───────── + π = 3·142 │ 0·318│ 9·870│ 31·006│ 1·772│ 0·564│ 1·465│ 0·683 + 2π= 6·283 │ 0·159│ 39·478│ 248·050│ 2·507│ 0·399│ 1·845│ 0·542 + (π)/(2) = 1·571 │ 0·637│ 2·467│ 3·878│ 1·253│ 0·798│ 1·162│ 0·860 + (π)/(3) = 1·047 │ 0·955│ 1·097│ 1·148│ 1·023│ 0·977│ 1·016│ 0·985 + (4)/(3)π = 4·189│ 0·239│ 17·546│ 73·496│ 2·047│ 0·489│ 1·612│ 0·622 + (π)/(4) = 0·785 │ 1·274│ 0·617│ 0·484│ 0·886│ 1·128│ 0·923│ 1·084 + (π)/(6) = 0·524 │ 1·910│ 0·274│ 0·144│ 0·724│ 1·382│ 0·806│ 1·241 + π^2 = 9·870 │ 0·101│ 97·409│ 961·390│ 3·142│ 0·318│ 2·145│ 0·466 + π^3 = 31·006 │ 0·032│961·390│29,809·910│ 5·568│ 1·796│ 3·142│ 0·318 + (π)/(32) = 0·098│ 10·186│ 0·0095│ 0·001│ 0·313│ 3·192│ 0·461│ 2·168 + _g_ = 32·2 │ 0·031│1036·84│ 33,386·24│ 5·674│ 0·176│ 3·181│ 0·314 + 2_g_ = 64·4 │ 0·015│4147·36│ 267,090│ 8·025│ 0·125│ 4·007│ 0·249 + ────────────────┴─────────┴───────┴──────────┴──────┴────────────┴──────┴──────────── + + + HYDRAULIC EQUIVALENTS. + + 1 foot head = 0·434 lb. per square inch. + 1 lb. per square inch = 2·31 ft. head. + 1 imperial gallon = 277·274 cubic inches. + 1 imperial gallon = 0·16045 cubic foot. + 1 imperial gallon = 10 lb. + 1 cubic foot of water = 62·32 lb. = 6·232 imperial gallons. + 1 cubic foot of sea water = 64·00 lb. + 1 cubic inch of water = 0·03616 lb. + 1 cubic inch of sea water = 0·037037 lb. + 1 cylindrical foot of water = 48·96 lb. + 1 cylindrical inch of water = 0·0284 lb. + A column of water 12 in. long 1 in. square = 0·434 lb. + A column of water 12 in. long 1 in. diameter = 0·340 lb. + Capacity of a 12 in. cube = 6·232 gallons. + Capacity of a 1 in. square 1 ft. long = 0·0434 gallon. + Capacity of a 1 ft. diameter 1 ft. long = 4·896 gallons. + Capacity of a cylinder 1 in. diameter 1 ft. long = 0·034 gallon. + Capacity of a cylindrical inch = 0·002832 gallon. + Capacity of a cubic inch = 0·003606 gallon. + Capacity of a sphere 12 in. diameter = 3·263 gallons. + Capacity of a sphere 1 in. diameter = 0·00188 gallon. + 1 imperial gallon = 1·2 United States gallon. + 1 imperial gallon = 4·543 litres of water. + 1 United States gallon = 231·0 cubic inches. + 1 United States gallon = 0·83 imperial gallon. + 1 United States gallon = 3·8 litres of water. + 1 cubic foot of water = 7·476 United States gallons. + 1 cubic foot of water = 28·375 litres of water. + 1 litre of water = 0·22 imperial gallon. + 1 litre of water = 0·264 United States gallon. + 1 litre of water = 61·0 cubic inches. + 1 litre of water = 0·0353 cubic foot. + + ───────────────────────────────────────────────────────────────────────── + EQUIVALENTS OF POUNDS AVOIRDUPOIS. + ─┬───────┬────────────┬────────────────┬────────────────┬──────────────── + │ 10 │ 100 │ 1000 │ 10,000 │ 100,000 + ─┼───────┼────────────┼────────────────┼────────────────┼──────────────── + │qr. lb.│cwt. qr. lb.│ton cwt. qr. lb.│ton cwt. qr. lb.│ton cwt. qr. lb. + 1│ 0 10│ 0 3 16│ 0 8 3 20│ 4 9 1 4│ 44 12 3 12 + 2│ 0 20│ 1 3 4│ 0 17 3 12│ 8 18 2 8│ 89 5 2 24 + 3│ 1 2│ 2 2 20│ 1 6 3 4│ 13 7 3 12│133 18 2 8 + 4│ 1 12│ 3 2 8│ 1 15 2 24│ 17 17 0 16│178 11 1 20 + 5│ 1 22│ 4 1 24│ 2 4 2 16│ 22 6 1 20│223 4 1 4 + 6│ 2 4│ 5 1 12│ 2 13 2 8│ 26 15 2 24│267 17 0 16 + 7│ 2 14│ 6 1 0│ 3 2 2 0│ 31 5 0 0│312 10 0 0 + 8│ 2 24│ 7 0 16│ 3 11 1 20│ 35 14 1 4│357 2 3 12 + 9│ 3 6│ 8 0 4│ 4 0 1 12│ 40 3 2 8│401 15 2 24 + ─┴───────┴────────────┴────────────────┴────────────────┴──────────────── + + + TRIGONOMETRICAL FUNCTIONS. + + + RIGHT-ANGLED TRIANGLES. + +[Illustration: [Right-angled Triangle]] + +Sin. A = (_a_)/(_b_) Sec. A = (_b_)/(_c_) Tan. A = (_a_)/(_c_) + +Cos. A = (_c_)/(_b_) Cosec. A = (_b_)/(_a_) Cotan. A = (_c_)/(_a_) + +Versin. A = (_b_ − _c_)/(_b_). Coversin. A = (_b_ − _a_)/(_b_). + + ───────┬─────────┬───────────────────────────────────────────────────── + Given. │Required.│ Formulæ. + ───────┼─────────┼───────────────────────────────────────────────────── + _a_,_b_│ A,C,_c_ │Sin. A = (_a_)/(_b_) Cos. C = (_a_)/(_b_) _c_ = + │ │ √((_b + a_)(_b − a_)) + │ │ + _a_,_c_│ A,C,_b_ │Tan. A = (_a_)/(_c_) Cotan. B = (_a_)/(_c_) _b_ = + │ │ √(_a_^2 + _c_^2) + │ │ + A,_a_ │C,_c_,_b_│ C = 90° − A _c_ = _a_ × Cotan. A _b_ = + │ │ (_a_)/(Sin. A) + │ │ + A,_b_ │C,_a_,_c_│C = 90° − A _a_ = _b_ × Sin. A _c_ = _b_ × Cos. + │ │ A + │ │ + A,_c_ │C,_a_,_b_│ C = 90° − A _a_ = _c_ × Tan. A _b_ = + │ │ (_c_)/(Cos. A) + │ │ + ───────┴─────────┴───────────────────────────────────────────────────── + + + OBLIQUE-ANGLED TRIANGLES. + +_s_ = ½(_a + b + c_) + +[Illustration: [Oblique-angled Triangle]] + + ───────────┬─────────┬───────────────────────────────────────────────── + Given. │ │ Formulæ. + ───────────┼─────────┼───────────────────────────────────────────────── + A,B,C,_a_ │ Area= │(_a_^2 × Sin. B × Sin. C) ÷ 2 Sin. A + A,_b_,_c_ │ „ │½(_c_ × _b_ × Sin. A) + _a_,_b_,_c_│ „ │√(_s_(_s_ − _a_)(_s_ − _b_)(_s_ − _c_)) + ───────────┼─────────┼───────────────────────────────────────────────── + Given. │Required.│ Formulæ. + ───────────┼─────────┼───────────────────────────────────────────────── + A,C,_a_ │ _c_ │ _c_ = _a_(Sin. C)/(Sin. A) + ───────────┼─────────┼───────────────────────────────────────────────── + A,_a_,_c_ │ C │ Sin. C = (_c_ Sin. A)/(_a_) + ───────────┼─────────┼───────────────────────────────────────────────── + _a_,_c_,B │ A │ Tan. A = (_a_ Sin. B)/(_c_ − _a_ Cos. B) + ───────────┼─────────┼───────────────────────────────────────────────── + _a_,_b_,_c_│ A │Sin. ½A = √(((_s_ − _b_)(_s_ − _c_))/(_b_ × _c_)) + „ │ „ │ Cos. ½A = √((_s_(_s_ − _a_))/(_b_ × _c_)); + „ │ „ │ Tan. ½A = √(((_s_ − _b_)(_s_ − _c_))/(_s_(_s_ − + │ │ _a_))) + ───────────┴─────────┴───────────────────────────────────────────────── + + + COMPOUND ANGLES. + + Sin. (A + B) = Sin. A Cos. B + Cos. A Sin. B. + Sin. (A − B) = Sin. A Cos. B − Cos. A Sin. B. + Cos. (A + B) = Cos. A Cos. B − Sin. A Sin. B. + Cos. (A − B) = Cos. A Cos. B + Sin. A Sin. B. + +Tan. (A + B) = (Tan. A + Tan. B)/(1 − Tan. A Tan. B). + +Tan. (A − B) = (Tan. A − Tan. B)/(1 + Tan. A Tan. B). + + + SLIDE RULE DATA SLIPS, COMPILED BY C. N. PICKWORTH, WH.SC. + + (_It is suggested that this page be removed by cutting through the above + line, and selected portions of the Sectional Data Slips attached to the + back of the Slide Rule._) + + ¹⁄₃₂ │0·03125 + ¹⁄₁₆ │0·0625 + ³⁄₃₂ │0·09375 + ⅛ │0·125 + ⁵⁄₃₂ │0·15625 + ³⁄₁₆ │0·1875 + ⁷⁄₃₂ │0·21875 + ¼ │0·25 + ⁹⁄₃₂ │0·28125 + ⁵⁄₁₆ │0·3125 + ¹¹⁄₃₂ │0·34375 + ⅜ │0·375 + ¹³⁄₃₂ │0·40625 + ⁷⁄₁₆ │0·4375 + ¹⁵⁄₃₂ │0·46875 + ¹⁷⁄₃₂ │0·53125 + ⁹⁄₁₆ │0·5625 + ¹⁹⁄₃₂ │0·59375 + ⅝ │0·625 + ²¹⁄₃₂ │0·65625 + ¹¹⁄₁₆ │0·6875 + ²³⁄₃₂ │0·71875 + ¾ │0·75 + ²⁵⁄₃₂ │0·78125 + ¹³⁄₁₆ │0·8125 + ²⁷⁄₃₂ │0·84375 + ⅞ │0·875 + ²⁹⁄₃₂ │0·90625 + ¹⁵⁄₁₆ │0·9375 + ³¹⁄₃₂ │0·96875 + +Circ. of circle = 3·1416 _d_. + +Area „ „ = 0·7854 _d_^2. + +Sq. eq. area to cir., _s_ = 0·886 _d_. + +Circle eq. to sq., _d_ = 1·128 _s_. + +Sq. inscbd. in circ., _s_ = 0·707 _d_. + +Circsb. circ. of sq., _d_ = 1·414 _s_. + +Area of ellipse = 0.7854 _a_ × _b_. + +Surface of sphere = 3·1416 _d_^2. + +Volume „ „ = 0·5236 _d_^3. + + „ „ cone = 0·2618 _d_^2 _h_. + +Radian = (180°)/(π) = 57·29 deg. + +Base of nat. or hyp. log. = e = 2·7183. + +Nat. or hyp. log. = com. log. × 2·3026. + +g (at London) 32·18 ft. per sec., per sec. + +Abs. temp. = deg. F. + 461° = deg. C. + 274°. + +C.° = (5)/(9)(F.° − 32°); F.° = (9)/(5)C.° + 32°. + +Cal. pr.—Ther. units per lb.: Coal, 14,300; + + petrol’m, 20,000; coal gas per cu. ft., 700. + +Sp. heat:—Wt. iron, 0·1138; C.I., 0·1298; + + copper, brass, 0·095; lead, 0·0314. + +Inch = 25·4 mil’metres; mil’metre = 0·03937 in. + +Foot = 0·3048 metres; metre = 3·2809 feet. + +Yard = 0·91438 metre; metre = 1·0936 yards. + +Mile = 1·6093 kilomtrs.; kilomtr. = 0·6213 mile. + +Sq. in. = 6·4513 sq. cm.; sq. cm. = 0·155 sq. in. + +Sq. ft. = 9·29 sq. decmtr.; sq. decmtr. = 0·1076 sq. ft. + +Sq. yd. = 0·836 sq. metre; sq. metre = 1·196 sq. yds. + +Sq. ml. = 258·9 hectares; hectare = 0·00386 sq. ml. + +Cu. in. = 16·386 c. cm.; c. cm. = 0·06102 cu. in. + +Cu. ft. = 0·0283 c. metre; c. metre = 35·316 cu. ft. + +Grain = 0·0648 gramme; gram. = 15·43 grs. + +Ounce = 28·35 grams.; „ = 0·03527 oz. + +Pound = 0·4536 kilogm.; kilogm. = 2·204 lb. + +Ton = 1·016 tonnes; tonne = 0·9842 ton. + +Mile per hr. = 1·466 ft., or 44·7 cm., per sec. + +Lb. per cu. in. = 0·0276 kilogram per cu. cm. + +Kilogram per cu. cm. = 36·125 lb. per cu. in. + +Lb. per cu. ft. = 16·019 kilogm. per cu. mtre. + +Grain per gall. = 0·01426 gramme per litre. + +Gramme per litre = 70·116 grains per gall. + + Ultimate Strength│Lb. per Sq. in. + „ │Tens’n.│Comp’n. + ─────────────────┼───────┼─────── + Wt. iron │ 50,000│ 50,000 + Cast „ │ 16,000│ 95,000 + Steel │ 80,000│ 70,000 + Copper │ 21,000│ 50,000 + Brass │ 18,000│ 10,500 + Lead │ 2,500│ 7,000 + Pine │ 11,000│ 6,000 + Oak │ 15,000│ 10,000 + + Weight of Metals.│ Cub. In. │ Cub. Ft. │12 Cu. In. + ─────────────────┼──────────┼──────────┼────────── + Wt. iron │ 0·277│ 480│ 3·33 + Cast „ │ 0·260│ 450│ 3·12 + Steel │ 0·283│ 490│ 3·40 + Copper │ 0·318│ 550│ 3·82 + Brass │ 0·300│ 520│ 3·61 + Zinc │ 0·248│ 430│ 2·98 + Alumin’m │ 0.096│ 168│ 1·16 + Lead │ 0.411│ 710│ 4·93 + + Lb. per sq. in. = 2·31 ft. water = 2·04 in. mercury = 0·0703 kilo. per + sq. cm. + Atmosphere = 14·7 lb. per sq. in. = 33·94 ft. water = 1·0335 „ „ + Ft. hd. water = 0·433 lb. per sq. in. = 62·35 lb. per sq. ft. = 0·0304 „ + „ + Cub. ft. of water = 62·35 lb. = 0·0278 ton = 28·315 litres = 7·48 U.S. + galls. + Gall. (Imp.) = 277·27 cu. in. = 0·1604 cu. ft. = 10 lb. water = 4·544 + litres. + Litre = 1·76 pints = 0·22 gall. = 61 cu. in. = 0·0353 cu. ft. = 0·264 + U.S. gall. + Horse-power = 33,000 ft.-lb. per min. = 0·746 kilowatt = 42·4 heat units + per min. + Heat unit = 778 ft.-lb. = 1055 watt-sec. = 107·5 kilogrammetres = 0·252 + calorie. + Foot-pound = 0·00129 heat unit = 1·36 joules = 0·1383 kilogrammetres. + Kilowatt = 1·34 H.P. = 44,240 ft.-lb. per min. = 3412 heat units per + hour. + +----- + +Footnote 1: + + It will be recognised that n is the characteristic of the logarithm of + the original number. + +Footnote 2: + + The special case in which the numerator is 1, 10, or any power of 10 + must be treated by the rule for reciprocals (page 27). + +Footnote 3: + + The possible need for traversing the slide, to change the indices, + when using the C and D scales, is not considered as a setting. + +Footnote 4: + + The reader may be reminded that cross-multiplication of the factors in + any such slide rule setting will give a constant product, _e.g._, 20 × + 94·5 = 27 × 70. + +Footnote 5: + + In this case cross _dividing_ gives a constant quotient, _e.g._, 8 ÷ 3 + = 4 ÷ 1·5. Since the upper scale is now a scale of reciprocals, the + ratio is really + + O ⅛ ¼ + ─────────── + D 1·5 3 + +Footnote 6: + + These lines should not be brought to the working edge of the scale but + should terminate in the horizontal line which forms the border of the + finer graduations, their value being read into the calculation by + means of the cursor (see page 55). + +Footnote 7: + + The same principle may be applied to the cursor. + +Footnote 8: + + Philosophical Transactions of the Royal Society, 1815. + +------------------------------------------------------------------------ + + + + + _BY THE SAME AUTHOR._ + + + LOGARITHMS FOR BEGINNERS. + +“An extremely useful and much-needed little work, giving a complete +explanation of the theory and use of logarithms, by a teacher of great +clearness and good style.”—_The Mining Journal._ + + 1s. 8d. Post Free. + + + THE INDICATOR HANDBOOK. + +Comprising “The Indicator: Its Construction and Application” and “The +Indicator Diagram: Its Analysis and Calculation.” Complete in One +Volume. + + 7s. 10d. Post Free. + +“Mr. Pickworth’s judgment is always sound, and is evidently derived from +a personal acquaintance with indicator work.”—_The Engineer._ + + + POWER COMPUTER FOR STEAM, GAS AND OIL ENGINES, Etc. + +“Accurate, expeditious and thoroughly practical.... We can confidently +recommend it, and engineers will find it a great boon in undertaking +tests, etc.”—_The Electrician._ + + 7s. 6d. Post Free. + +------------------------------------------------------------------------ + + + + + ADVERTISEMENTS. + + + LOGARITHMS FOR BEGINNERS + +For a full and intelligent appreciation of the Slide Rule and its +various applications an elementary knowledge of logarithms is necessary. +All that is required will be found in this little work, which gives a +simple, detailed and practical explanation of logarithms and their uses, +particular care having been taken to elucidate all difficult points by +the aid of a number of worked examples. + + Seventh Edition, 1s. 8d. Post Free. + + + POWER COMPUTER + for + STEAM, GAS, AND OIL ENGINES, Etc. + +Gives The Indicated Horse-Power of Steam, Gas, and Oil Engines—The Brake +Horse-Power of Steam, Gas, and Oil Engines—The Size of Engine Necessary +to Develop any Given Power—The Mechanical Efficiency of an Engine—The +Ratio of Compound Engine Cylinders—The Piston Speed of an Engine—The +Delivery of Pumps with any Efficiency—The Horse-Power of Belting—The Rim +Speeds of Wheels, Speeds of Ropes, Belts, etc.—Speed Ratios of Pulleys, +Gearing, etc. + + Pocket size, in neat case, with instructions and examples. + + Post Free, 7s. 6d. net. + + + C. N. PICKWORTH, Withington, Manchester + + ┌─────────────────────────────────────────────────────────────────────┐ + │ W. P. THOMPSON, G. C. DYMOND, │ + │ F.C.S., M.I.Mech.E., F.I.C.P.A. M.I.Mech.E., F.I.C.P.A. │ + │ │ + │ W. P. Thompson & Co., │ + │ 12 CHURCH STREET, LIVERPOOL, │ + │ CHARTERED PATENT AGENTS. │ + │ │ + │ H. E. POTTS, J. V. ARMSTRONG, │ + │ M.Sc., Hon. Chem., F.I.C.P.A. M.Text.I., F.I.C.P.A. │ + │ │ + │ W. H. BEESTON, R.P.A. │ + └─────────────────────────────────────────────────────────────────────┘ + +------------------------------------------------------------------------ + + + + + BRITISH + SLIDE RULES + + for all + ARTS and + INDUSTRIES + + including + +[Illustration: [Slide Rule]] + + _LOG-O-LOG + DR. YOKOTA’S + SURVEYORS’ + WIRELESS + GUNNERY + ELECTRICAL RULES, Etc._ + + SEND FOR LIST 55 + +MADE BY— + + JOHN DAVIS & SON (Derby), Ltd. + ALL SAINTS’ WORKS, DERBY + +------------------------------------------------------------------------ + + + + + K & E Slide Rules + +are constantly growing in popularity, and they can now be obtained from +the leading houses in our line throughout the United Kingdom. + +[Illustration: [Slide Rule]] + + We manufacture a complete line of ENGINE-DIVIDED SLIDE RULES, and call +special attention to our Patent Adjustment, ensuring smooth working of +the Slide; also to our new “Frameless” Indicator, which hides no figures +on the Rule. + +[Illustration: [Thacher’s Calculating Instrument]] + +THACHER’S CALCULATING INSTRUMENT, for solving problems in +multiplication, division, or combinations of the two; has upwards of +33,000 divisions. Results can be obtained to the fourth and usually to +the fifth place of figures with a surprising degree of accuracy. + + We also make + + ALL METAL, CIRCULAR, STADIA, CHEMISTS’, ELECTRICAL, and OTHER SPECIAL + SLIDE RULES + _DESCRIPTIVE CIRCULARS ON REQUEST_ + + KEUFFEL & ESSER CO. + 127 Fulton St., NEW YORK General Office and Factories, HOBOKEN, + N.J. + CHICAGO − ST. LOUIS − SAN FRANCISCO − MONTREAL + + _DRAWING MATERIALS_ + _MATHEMATICAL and SURVEYING INSTRUMENTS_ + _MEASURING TAPES_ + +------------------------------------------------------------------------ + + + + +[Illustration: 6 in. Standard with magnifying Cursor complete in pocket +case, 5/-] + + NORTON + & + GREGORY + LTD. + + + Head Office + + CASTLE LANE, WESTMINSTER, LONDON, S.W. 1 + + Branches + + 71 QUEEN STREET, GLASGOW. + PHOENIX HOUSE, QUEEN STREET and SANDHILL, NEWCASTLE-ON-TYNE. + + + SLIDE RULES in Stock, from 17/6 to 27/6 + + Special Quotations to the Trade for Quantities + + + For particulars of Surveying, Measuring and Mathematical Instruments, + Appliances and Material of all kinds for the Drawing Office, write + to the Head Office. + +[Illustration: NORTON & GREGORY LTD] + +------------------------------------------------------------------------ + + + + + NORTON & GREGORY, LTD., + London. + + + “DIAMOND” + DRAWING INSTRUMENTS + Manufactured at our London Works. + + CENTRE SCREW SPRING BOW HALF SET. + +[Illustration: [Centre Screw Spring Bow Half Set]] + +4 inch Spring Bow Half Set centre screw adjustment, with interchangeable +needle, pen, and pencil points Price 17/6 + +The Centre Screw Spring Bow Half Set of Compasses, as illustrated, +possesses the advantage of COMBINING IN ONE INSTRUMENT THE SET OF THREE +SEPARATE SPRING BOWS hitherto in use, while the centre screw makes for +ease and accuracy of manipulation, at the same time providing a radius +of over 2 inches, or double that of the old pattern. + +This instrument is less expensive than the set of 3 bows, while +considerably stronger in construction. + +The fixed needle point is shouldered. + +This illustration is given as an indication of the various Drawing +Instruments manufactured by us. + +Illustrated Booklet giving full particulars and prices of other +Instruments and Cases of Instruments sent on application. + + + Specially arranged Sets of Instruments made for Colleges, Schools, + Technical Institutes + + Estimates submitted on Application. + + + _Write to our Head Office_: + CASTLE LANE, WESTMINSTER, LONDON, S.W. 1. + +------------------------------------------------------------------------ + + + DRAWING AND SURVEYING INSTRUMENTS + + A. G. THORNTON Ltd. + SLIDE RULES FOR Paragon Works ACCURATE SECTIONAL + ENGINEERS 2 King St. West PAPERS AND CLOTHS + MANCHESTER + + D 1916 Illustrated Catalogue, just published, in two editions; Drawing + Office (448 pages); Draughtsman’s (160 pages): the most complete + Catalogues in the trade. + + _CONTRACTORS TO H.M. WAR OFFICE AND ADMIRALTY_ + _Manufacturers also of Drawing Materials and Drawing Office Stationary._ + + (ALSO AT MINERVA WORKS AND ALBERT MILLS, MANCHESTER.) + + + MATHEMATICAL INSTRUMENTS + SURVEYING INSTRUMENTS + SLIDE RULES + For Students and Engineers + + MANNHEIM, POLYPHASE, DUPLEX, ELECTRICAL, LOG-LOG, AND CALCULEX + + J. H. STEWARD LTD. + Scientific Instrument Makers + + 406 STRAND, and 457 WEST STRAND + LONDON, W.C. 2 + +------------------------------------------------------------------------ + + + + + _A True Friend and Trusty Guide_ + + + THE + ‘HALDEN CALCULEX’ + +[Illustration: [Halden Calculex]] + + ACTUAL SIZE| | | BRITISH MADE + + The handiest and most perfect form of Slide Rule. + Does all that can be done with a straight rule. + Complete in Case, with book of instructions, + 27/6 post free. + + J. HALDEN & CO., LTD., 8 ALBERT SQUARE MANCHESTER + + _Depots_—London, Newcastle-on-Tyne, Birmingham, Glasgow, and Leeds + +------------------------------------------------------------------------ + + +[Illustration: [Rope]] + + ENGINEERING, + SURVEYING + AND + MATHEMATICAL + INSTRUMENTS, + ETC. + + + SLIDE RULES. + + JOSEPH CASARTELLI & SON, + 43 MARKET STREET, MANCHESTER. + Tel. No. 2958 City.| | | Established 1790. + +ROPE DRIVING + + Is the most EFFICIENT and most ECONOMICAL METHOD of Power + Transmission. + +The LAMBETH Cotton Driving Rope + + Is the most EFFICIENT and most ECONOMICAL ROPE for Power Transmission. + +[Illustration: Made 4 Strand or 3 Strand.] + +SPECIAL FEATURES: + + LESS STRETCH THAN ANY OTHER ROPE. MORE PLIABLE THAN ANY OTHER ROPE. + GREATER DRIVING POWER THAN ANY OTHER ROPE. + + THOMAS HART LTD., Lambeth Works, BLACKBURN. + +------------------------------------------------------------------------ + + + + + TRANSCRIBER’S NOTES + + + Page Changed from Changed to + + 24 the right, so the number of the right, so the number of + digits in the answer = 3 − 2 × 1 digits in the answer = 3 − 2 + 1 + = 2 = 2 + + 116 grammes, we have the equation, grammes, we have the equation, + _x_ × (Cl.)/(Ag.Cl.) × _x_ = (Cl.)/(Ag.Cl.) × + (_a_)/(_s_). Hence, the mark (_a_)/(_s_). Hence, the mark + + ● Typos fixed; non-standard spelling and dialect retained. + ● Used numbers for footnotes, placing them all at the end of the last + chapter. + ● Enclosed italics font in _underscores_. + ● Enclosed bold font in =equals=. + ● The caret (^) serves as a superscript indicator, applicable to + individual characters (like 2^d) and even entire phrases (like + 1^{st}). + ● Subscripts are shown using an underscore (_) with curly braces { }, + as in H_{2}O. + + + +*** END OF THE PROJECT GUTENBERG EBOOK 75904 *** |