Chapter C: N. P (1)

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.

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.

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.

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_.

“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.

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.

_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._

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.