The Slide Rule

Part 1

THE SLIDE RULE: A PRACTICAL MANUAL

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.

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

Part 2

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.

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