BY THE SAME AUTHOR 
 
 An Introduction to Laboratory Physics 
 
 The subject matter is similar to that of the Theory 
 of Measurements, but the treatment is adapted to 
 more mature students, and is such as will necessitate 
 more independent thought. 
 
 Cloth, 150 pp.; 5x7 X inches; $0.80. 
 
 Four-Place and Five-Place Tables 
 
 Four-place logarithms with tables of proportional 
 parts; five-place logarithms; four-place natural and 
 logarithmic circular functions ; squares ; values of 
 .67449 Vn; constants. 
 
 Cardboard, 6 pp.; 4)^x7)^ inches; $0.05. 
 
 Second Edition in Preparation : Contains also four- 
 place squares and square roots of all numbers ; exact 
 squares from I 2 to 999 2 ; inverse tangents; direct and 
 inverse circular measure; density and specific gravity 
 of water; exponentials (e n ; logio^ n ; e~ n ; e~ n *). 
 
 Cardboard, 8 pp. . . . (In press). 
 
 Theory of Measurements 
 
 Cloth, xiv+303 pp.; ... $1.25. 
 
 PUBLISHED BY 
 
 Dr. Lucius Tuttle, 
 
 Jefferson Medical College, 
 
 Philadelphia, Pa. 
 
THE THEOEY 
 
 OF 
 
 MEASUREMENTS 
 
 BY 
 
 LUCIUS TUTTLE 
 
 B.A. (YALE); M.D. (JOHNS HOPKINS) 
 
 ASSOCIATE IN PHYSICS, JEFFERSON MEDICAL COLLEGE, PHILADELPHIA 
 
 PHILADELPHIA 
 
 JEFFERSON LABORATORY OF PHYSICS 
 1916 
 
o 
 
 ' 
 
 . : 
 
 PUBLISHED BY THE AUTHOR 
 
 PRESS OF 
 
 THE NEW ERA PRINTING COMPANY 
 LANCASTER, PA. 
 
PREFACE 
 
 For the student of mathematics this book is intended 
 to furnish an introduction to some of the applications 
 of the exact sciences and their relation to the " practical" 
 sciences and useful arts, and is primarily intended to 
 give him a knowledge of facts and methods, but without 
 neglecting the accurate exercise of his reasoning powers. 
 
 For the student of physical science it is intended 
 especially to emphasize general considerations of meas- 
 urement, theory of errors, general methods of procedure, 
 quantitative accuracy, adjustment of observations, etc., 
 topics that are often merely mentioned in the intro- 
 duction or appendix of a laboratory manual, but that 
 need laboratory work and drill quite as much as the 
 measurements of the individual quantities that the stu- 
 dent will take up in his later work. Where it is impos- 
 sible to find time for a complete course of the kind 
 described here it may be helpful to use selected chapters 
 of the book as occasion arises, or the student may be 
 directed to use it as a reference book, or even to read it 
 through without performing any of the experimental 
 work. 
 
 The book is based on the mimeographed direction 
 sheets that were used in the first part of a laboratory 
 course which the writer gave at Jefferson at a time 
 when there was no elementary textbook available that 
 covered the required ground. In addition to the state- 
 ments of facts and theory e.ach of the chapters of the 
 book includes directions for actual experimental work 
 to be performed by the student, and the amount 'of this 
 work has been so planned that each lesson will require 
 
 377559 
 
vi PREFACE 
 
 about the same length of time as any of the others. 
 For the average student this will nlean about three 
 hours, but the material of the lessons can easily be 
 divided into a greater number of shorter exercises if 
 desirable. 
 
 With the more widespread use of a laboratory course 
 of this sort certain shortcomings of the author's earlier 
 " Introduction to Laboratory Physics" have been made 
 manifest. A book that demands more or less vigorous 
 mental exercise from a class of students who take a special 
 interest in the subject will naturally need more elemen- 
 tary exposition more detailed statement and less exer- 
 citational questioning if it is to be used in larger classes 
 where there is a greater likelihood of finding that some 
 of the students are lacking in interest or ability or 
 elementary mathematical training.* Accordingly, expla- 
 nations and directions have been given with considerable 
 detail, partly in order to avoid the necessity for continu- 
 ous oral assistance on the part of the instructor, and 
 partly to help the student to learn with a minimum of 
 deliberate memorizing. For the latter purpose facts 
 have occasionally been stated implicitly instead of 
 explicitly, but, in such cases, always with a later re- 
 iteration in a more expositional form. The course is 
 progressively graded in difficulty, with the object of 
 developing the student's ability as he proceeds from the 
 easier exercises to those that require more independent 
 thought. 
 
 There has been a certain demand for the "Introduction 
 to Laboratory Physics" in connection with courses of 
 
 * Such detailed directions as the instructions in regard to round 
 numbers, page 119, may seem superfluous, but they indicate faults 
 that have been found in the work of more than one of the students 
 who have taken this course. 
 
PREFACE vii 
 
 mathematics as well as for courses in physics, and for 
 this reason the requirements of the mathematician have 
 been especially kept in mind during the preparation of 
 the present book. No knowledge of trigonometry, how- 
 ever, is presupposed, and none is imposed upon the 
 reader of the book, the terms " function," " tangent," 
 " cosine," etc., that will occasionally be found being used 
 merely as convenient abbreviations for ideas that would 
 otherwise need a more cumbersome description. 
 
 In the introductory chapter the commonest mathe- 
 matical deficiencies of the student are reviewed and an 
 opportunity is given him to test his weak points. A 
 lesson on logarithms is included, which can be omitted, 
 if preferred, by a class that is familiar with the subject; 
 but there are often members of such a class who cannot 
 make practical use of logarithmic tables readily, or even 
 accurately, without additional practice, and to anyone 
 who does not need the practice it will not be at all irk- 
 some. Care has been taken to make the tables in the 
 appendix both accurate and convenient.* Experience 
 has shown that the somewhat unconventional arrange- 
 ment of the table of probable errors, page 292, is the 
 most satisfactory in actual use. The table of logarithmic 
 circular functions has been given the greatest possible 
 compactness. The columns of the table of four-place 
 logarithms are arranged especially for the convenience 
 of students who are accustomed to using scales that are 
 subdivided into tenths, and the proportional parts are 
 given in the same way as in the most carefully constructed 
 larger tables. None of the methods of arranging a five- 
 place table with proportional parts within the limits of 
 
 * All of the tables either have been verified from two independent 
 sources or have been checked by recalculation, and the proof-sheets 
 have been revised with the utmost care. 
 
viii PREFACE 
 
 two pages has ever succeeded in giving the fifth figure 
 satisfactorily, and several scientific reference books have 
 been published in which even the fourth figure of such a 
 table will often be found incorrect. Accordingly, for the 
 five-place logarithms in the present volume no attempt 
 has been made to include proportional parts, but direc- 
 tions have been given for easy interpolation with the aid 
 of three-figure logarithms. 
 
 I have replaced the perpetually misleading name for 
 the common -representative value of a set of residuals 
 by one which does not have this objectionable quality 
 and at the same time suggests the t nature of the quantity 
 in question. A few other innovations will be found here 
 and there in the text, but for the most part the book 
 follows fairly well-beaten lines. 
 
 I have found it advisable to devote the first ten or 
 fifteen minutes of the laboratory period to a rapid recita- 
 tion based on the lesson of the previous day; and have 
 allowed the students to compare many of their important 
 numerical determinations by having them record certain 
 specified results each day upon a large card (22}^" X 
 28^ /r ) that is kept on one of the laboratory tables, and 
 is ruled in separate columns headed by each student's 
 name and having separate lines for each datum. For the 
 exercises after the first chapter of this book the following 
 data may be suggested: 
 
 Weights and measures: density of a (brass-and-air) 
 weight. 
 
 Angles: largest error of the measured sines. 
 
 Significant figures: experimental value of TT. 
 
 Logarithms: calculated value of e. 
 
 Small magnitudes: results of a double weighing. 
 
 Slide rule: approximate ratio for TT, different from 22/7. 
 
 Graphic representation: temperature at 4 p. m. 
 
PREFACE ix 
 
 Curves and equations: least value of x for which 
 exp ( x 2 ) is indistinguishable from zero. 
 
 Graphic analysis : equation of the black thread experi- 
 ment. 
 
 Interpolation: population, according to second extra- 
 polation. 
 
 Coordinates in three dimensions : altitude at Sixth and 
 Market Streets. 
 
 Accuracy : measured length of the table, or its relative 
 deviation from the average. 
 
 Principle of coincidence: measured length of an inch. 
 
 Measurements: mode and extremes of measured 
 variates. 
 
 Statistics: average, median, and quartiles of variates. 
 
 Dispersion: comparison of semi-interquartile range 
 with dispersion (10 seeds). 
 
 Weights: weighted average for the density of alumi- 
 num. 
 
 Criteria of rejection: closer values for the approximate 
 ratios 10 : 12 : 15. 
 
 Least squares: comparison of black thread determina- 
 tion ( 21) with least square determination. 
 
 Indirect measurements: value of l/3 2 + 5 2 -f- 6 2 by 
 geometrical construction. 
 
 Systematic errors: direction and amount of displace- 
 ment of the second hand. 
 
 Most of the apparatus required will be found to be 
 included in that which is used in other physical experi- 
 ments; a complete list of what is needed for each group 
 of two students is given here: 
 
 2 metre sticks (graduated in tenths of an inch on the back). 
 2 30-cm. rulers (graduated in cm. and mm.). 
 1 50-cm 3 graduated cylinder. 
 1 10-cm 3 graduated pipette. 
 
x PREFACE 
 
 1 platform balance or trip scale with slide giving tenths of a gram 
 
 (and the supporting wedges used when packed for shipment). 
 1 set of brass weights (1 gm. to 500 gm.). 
 1 set of iron weights (1 oz. to 8 oz.). 
 1 pair of fine-pointed dividers. 
 1 pencil-compass. 
 1 protractor (provided with a diagonal scale). 
 
 1 brass measuring disc for the determination of TT. 
 
 2 10-inch slide rules that need not have celluloid facings, but are 
 provided with A, B, C, D, S, L, and T scales, metric equivalents, 
 and a runner. 
 
 1 hard wood block. 
 
 2 vernier calipers. 
 
 100 seeds or other variates. 
 
 1 aluminum block for density measurements. 
 
 1 set of "overflow can" and "catch-bucket" for Archimedes' 
 principle. 
 
 2 square wooden rods for the balance pans. 
 
 1 iron clamp to hold balance on cross-bar over table. 
 
 1 irregular solid (large wire nail or strip of lead that can be immersed 
 
 in the graduated cylinder). 
 1 small test-tube, 
 cardboard, 
 string, 
 fine black thread. 
 
 The student should have a watch with a second hand,* 
 a pocket-knife, and the supplies mentioned in the intro- 
 duction. A clock that beats audible seconds should 
 be available. The slide rules should have 6745 on the 
 C scale marked by making a shallow cut with a sharp 
 knife and rubbing in a little oil pigment. The note- 
 book ( 8) used at the Jefferson Laboratory of Physics 
 measures about eight by ten and a half inches and is 
 ruled both horizontally and vertically at intervals of 
 one seventh of an inch. 
 
 * A special watch for the laboratory, having a marked eccentricity 
 of the second hand, may be advisable for the use of the students who 
 have the greatest difficulty with the experiment on periodic errors. 
 
CONTENTS 
 
 PAGE 
 
 I. INTRODUCTORY 1 
 
 Object. Purpose. Continuity. Results. Fore- 
 thought. Mental Attitude. Notes. Material Equip- 
 ment. Mental Equipment. Proportionality. Vari- 
 ation. Algebraical Formulae. Mental Exercises. 
 Physical Arithmetic. Abridged Division. Abridged 
 Multiplication. Gradient. 
 
 II. WEIGHTS AND MEASURES 24 
 
 C.G.S. System. Units of Length. Units of Area 
 and Volume. Units of Mass and Density. Unit of 
 Time. Practice in Using the C.G.S. System. Rule 
 for Rounding Off a Half. The Hand as a Measure. 
 Measurement of Area. Measurement of Volume. 
 Measurement of Mass. Measurement of Density. 
 Equivalent Measures. 
 
 III. ANGLES AND CIRCULAR FUNCTIONS 37 
 
 Unit of Angle. Circular Measure. Numerical 
 Measure of an Angle. The Angle ir and the Unit 
 Angle. The Protractor. The Diagonal Scale. 
 Measures of Inclination. Use of a Table of Tan- 
 gents. Experimental Determination of Sines. Defi- 
 nition of Function. The Cosine of an Angle. 
 Circular Functions. Generalized Idea of Angle. 
 
 IV. SIGNIFICANT FIGURES 52 
 
 Estimation of Tenths. Practice. Mistakes. Value 
 of TT. Physical Measurement. Ideal Accuracy. 
 Decimal Accuracy. Significant Figures. Relative 
 Accuracy. Calculation of Relative Errors. Rule 
 for the Relative Difference of Two Measurements. 
 Accuracy of a Calculated Result. Accuracy of the 
 Abridged Methods. Standard Form. 
 
 V. LOGARITHMS 71 
 
 Definitions. Fundamental Properties. Common 
 Logarithms. Practical Tables. Use of Tables. 
 The Probability Function. 
 
xii CONTENTS 
 
 VI. SMALL MAGNITUDES 80 
 
 Approximate Values. Negligible Magnitudes. For- 
 mula for Powers. Properties of Deltas. Trans- 
 formation of Operands. Recapitulation. 
 
 VII. THE SLIDE RULE 90 
 
 Addition with Two Scales. Multiplication with 
 Logarithmic Scales. The Slide Rule. Reading 
 a Logarithmic Scale. Multiplication. Division. 
 Ratio and Proportion. Equivalent Measures. Re- 
 ciprocals. C and D Scales. Squares and Square 
 Roots. Compound Operations. Determination of 
 Circular Functions. Determination of Logarithms 
 and Antilogarithms. 
 
 VIII. GKAPHIC REPRESENTATION 102 
 
 Indication of a Point by Two Numbers. Represen- 
 tation of Two Numbers by a Point. Representation 
 of Two Variables by a Line. Graphic Diagrams. 
 Practice in Plotting Points. Orientation of a 
 Graphic Curve. Choice of Scales. General Prin- 
 ciples of Plotting. Representation of Tabular 
 Values. Smoothing of a Graphic Curve. 
 
 IX. CURVES AND EQUATIONS 115 
 
 Graphic Representation of a Natural Law. Graph 
 of an Equation. General Procedure. The Straight 
 Line. The Parabola. The Probability Curve. 
 Equation of a Graph. Change of Scales. Defini- 
 tions of Circular Functions. 
 
 X. GRAPHIC ANALYSIS 128 
 
 Interpretation of Equations. The Graph of y a -\-bx. 
 The Straight Line Law,. The "Black Thread" 
 Method. Intercept Form of a Linear Equation. 
 The Graph of y = a + bx + cx z . Law of Density- 
 Variation of Water. Typical Curves. Linear Rela- 
 tionship by Change of Variables. 
 
 XI. INTERPOLATION AND EXTRAPOLATION 142 
 
 Definitions. The Principle of Proportionate Changes. 
 Examples of Linear Interpolation. Graphic Inter- 
 polation. Interpolation along a Curve. Insuffi- 
 ciency of Data. Use of Logarithmic Paper. Semi- 
 Logarithmic Paper. Extrapolation. 
 
CONTENTS xiii 
 
 XII. COORDINATES IN THREE DIMENSIONS 158 
 
 Coordinates of a Point in Space. Convention in 
 Regard to Signs. Loci of Simple Three-Dimensional 
 Equations. Contour Lines. Use of Contour Maps. 
 Construction of a Contour Map. 
 
 XIII. ACCURACY 170 
 
 Significant Figures. Infinite Accuracy. Relative 
 Errors. Uncertain Figures. Superfluous Accuracy. 
 Finer Degrees of Accuracy. Possible Error of a 
 Measurement. Possible Error after a Calculation. 
 "Probable" Error. Accuracy Required in Special 
 
 XIV. THE PRINCIPLE OF COINCIDENCE 180 
 
 Effect of Magnitude upon Accuracy. Measurement 
 by Estimation. Measurement by Coincidence. The 
 Vernier. Use of the Vernier Caliper. Slide-Rule 
 Ratios. 
 
 XV. MEASUREMENTS AND ERRORS 188 
 
 Direct and Indirect Measurements. Independent, 
 Dependent, and Conditioned Measurements. Har- 
 mony and Disagreement of Repeated Measurements. 
 Errors of Measurements. Classification of Errors. 
 Accidental and Constant Errors. Errors and Varia- 
 tions. Measurement of Variates. 
 
 XVI. STATISTICAL METHODS 197 
 
 Frequency Distributions. Class Interval. Types of 
 Frequency Distribution. The Probability Curve. 
 Representative Magnitudes. The Average. The 
 Median. The Mode. Choice of Means. Devia- 
 tions. Average by Symmetry. Average by Parti- 
 tion. Quartiles. Semi-Interquartile Range. 
 
 XVII. DEVIATION AND DISPERSION 211 
 
 Characteristic Deviations. Total Range. Average 
 Deviation. Standard Deviation. Dispersion. Sig- 
 nificance of the Dispersion. Advantage of the Dis- 
 persion. Calculation of the Dispersion. Rule for 
 the Accuracy of the Average. Use of the Table of 
 Dispersions. Dispersions with the Slide Rule. Sigma 
 Notation. Dispersion of an Average. The State- 
 ment of a Measurement. Relative Dispersion. 
 
xiv CONTENTS 
 
 XVIII. THE WEIGHTING OF OBSERVATIONS 223 
 
 Necessity of Weights for Observations. Density by 
 Different Methods. Weights for Repeated Values. 
 The Weighted Average. Arbitrarily Assigned 
 Weights. Weight and Dispersion. Limitations of 
 w = k/d?. Exception to the Rule. 
 
 XIX. CRITERIA OF REJECTION 230 
 
 Observational Integrity. Importance of Criteria. 
 Chauvenet's Criterion. The Probable Error. 
 Graphic Approximation to Chauvenet's Criterion. 
 Irregularities of Small Groups. Justification of the 
 1 Criterion. Wright's Criterion. Comparison of 
 Characteristic Deviations. 
 
 XX. LEAST SQUARES 240 
 
 The Average as a Least-Square Magnitude. Least 
 Squares for Conditioned Measurements. Least 
 Squares and Proportionality. Least Squares for a 
 Theoretically Constant Value. Consecutive Equal 
 Intervals. Equal Intervals without Least Squares. 
 Simultaneous Indirect Measurements. 
 
 XXI. INDIRECT MEASUREMENTS 255 
 
 Importance of Indirect Measurements. Probable 
 Error of a Sum. Probable Error of a Difference. 
 Probable Error of a Multiple. Associative Law. 
 Probable Error of a Product. Probable Error of a 
 Power. Distributive Law. Recapitulation. Graphs 
 of Propagated Errors. Relative Importance of Com- 
 pounded Errors. 
 
 XXII. SYSTEMATIC AND CONSTANT ERRORS 265 
 
 Definitions. Test for Systematic Errors. Example 
 of a Systematic Error. Example of a Periodic Error. 
 Example of a Progressive Error. Constant Errors. 
 
 APPENDIX 275 
 
 Explanatory Note. Formulae. Equivalents. 
 Greek Alphabet. Size of Errors. Characteristic De- 
 viations. General Sources of Error. Density of 
 Water. Inverse Tangents and Circular Measure. 
 Squares and Square Roots. The Probability In- 
 tegral. Five-Place Logarithms. Exponentials. 
 Fifth Place of Logarithms. Four-Place Logarithms. 
 Squares. Constants. Circular Functions. 
 
 INDEX . 293 
 
I. INTRODUCTORY 
 
 1. Object. The object of a course in the theory of 
 measurements is not only to give a certain knowledge of 
 the scientific facts that are studied, but also to develop 
 the thinking and reasoning powers and to furnish the 
 special kind of mental training that results, in the first 
 place, from practice in making various kinds of measure- 
 ments with particular care for their accuracy, and, in 
 the second place, from the consideration of accuracy in 
 its quantitative aspects, from realizing that accuracy 
 itself can be made a subject of measurement, that there 
 are relative degrees of accuracy, that accuracy is im- 
 portant in one place and means only a waste of effort 
 in another, that absolute accuracy is an impossibility, 
 that a measurement by itself is of much less value thai* 
 when accompanied by a statement of its precision. 
 
 When a course of physical measurements is used as a 
 preliminary to laboratory work or practical work in some 
 subject such as astronomy, physics, psychology, or 
 surveying it is further of value in giving the student a 
 certain familiarity with apparatus and a facility in 
 handling it in such a manner that he acquires the habit 
 of utilizing it to the best advantage and of keeping it in 
 such condition that it is most fully utilizable when 
 needed. 
 
 2. Purpose. It is advisable for the student to have it 
 pointed out at the beginning of the course that his con- 
 scious purpose, throughout all of his work, should be to 
 learn, rather than to accomplish the assigned exercise. In 
 order to help him in his education and training he is 
 permitted to do certain laboratory work which will make 
 
 2 1 
 
2 l THEORY- OF .MEASUREMENTS 5 
 
 his learning easier and its effects more lasting and more 
 useful to him. He should tell himself that the work is 
 allowed, rather than that it is required, in case he has 
 any tendency to look upon his course of study as a 
 tedious job, and thus make it irksome. 
 
 3. Continuity. In any graded course of study, where 
 each exercise is an advance beyond the previous ones and 
 requires a knowledge of the earlier work, it is a decided 
 handicap for the student to miss any of the work. If 
 an absence cannot be avoided he should take particular 
 pains, for his own sake, to make up the work by outside 
 study. This is especially important for any study that 
 is at all mathematical in character. See that each topic 
 is thoroughly comprehended before going on to the next. 
 The work will usually be easier if each lesson is read over 
 before coming into class, so that it is not necessary to 
 begin the class-room work as a new and unfamiliar sub- 
 ject. 
 
 4. Results. In order to obtain good results it is 
 necessary for the student to preserve an attitude of alert 
 attention toward his own work, and especially not to omit 
 any part of it or postpone it. Be thorough without 
 being in haste; better to have half of the day's work done 
 and done well than to try to take in all of it without 
 having any of it more than half assimilated. 
 
 5. Forethought. Whenever any piece of apparatus 
 is used it must be kept in mind that it may be needed 
 again later, and it is as important to keep it in good order 
 as it is to use it efficiently. It does not take long for the 
 consequences to show in the student's work if he is 
 accustomed to pick up an instrument where he happens 
 to find it, and drop it as soon as its immediate need has 
 passed. A certain orderliness in the handling of ap- 
 paratus is a habit that is well worth cultivating, as much 
 
I INTRODUCTORY 3 
 
 for its effect on the student's individuality as for the con- 
 servation of property values. 
 
 6. Mental Attitude. In addition to keeping in mind 
 the fact that it is much more important to learn principles 
 than to "work through" each day's lesson the student 
 should adopt the motto that in all kinds of studying it is 
 better to think than to memorize. For some students it 
 seems only too easy to get into the habit of concentrating 
 upon individual items and memorizing isolated state- 
 ments of fact without ever understanding their bearings 
 or realizing their inter-relationships or acquiring a larger 
 comprehension of the body of scientific knowledge which 
 is built up on them. The best help to a broader vision 
 lies in thinking over the facts that one comes across. 
 Just as important as the question " What is true?" is the 
 further question "Why is it true?" Better than a brain 
 packed full of facts is a mind that can reason out what 
 the facts must necessarily be in particular cases. Memory 
 with little reasoning power is useless for any highly 
 organized living being; but reasoning power with little 
 memory would be perfectly practicable, as long as such 
 things as paper and pencil can be had. Furthermore, 
 the ability to think for oneself is of the greatest utility in 
 enabling the student to rely upon his own observational 
 powers. The untrained student is prone to ask "Is my 
 result right?" in circumstances where the student who 
 has learned to stand on his own feet knows that no one 
 has a better knowledge of what the "right" result is than 
 himself. 
 
 7. Notes. For any practical work, or laborator}^ work, 
 it is important that the student's notes shall be accurate 
 and that they shall be complete. Neatness is usually 
 worth while, but it is distinctly secondary in importance 
 to thoroughness and accuracy. Time spent on beauti- 
 
4 THEORY OF MEASUREMENTS 7 
 
 fying the notebook by means of elaborately shaded 
 drawings or painstaking arrangement of matter is usually 
 time wasted. All notes should be clear and intelligible, 
 and in such shape that they can be readily understood by 
 anyone who has an ordinary knowledge of the subject 
 that they deal with. Each day's work is always to be 
 dated, and it is advisable to write a heading in such 
 a way that it will catch the eye at once, and show at a 
 glance where one day's work ends and the next begins, 
 as well as indicating the nature of the following matter 
 after the manner of a title. A certain amount of " dis- 
 play," by underlining or otherwise should also be given 
 to two other things, the statement of original measure- 
 ments, especially when further calculations depend upon 
 them, and the final results of such calculations. 
 
 The matter of making on3's notes thorough needs little 
 explanation; they should be made a digest of all that the 
 student learns and does in the laboratory course, but 
 without copying or duplicating matter in his textbook 
 that he can easily turn to when it is wanted. Any 
 questions that are asked in the text should be answered 
 in the notebook. 
 
 The matter of accuracy is one that requires some care 
 and alertness. It is necessary to make it a rule that all 
 work done with a pen or pencil must be done in the note- 
 book and everything in the notebook must be put down 
 consecutively, in its natural order. If the student uses 
 the last pages of the notebook for miscellaneous calcu- 
 lations it is usually impossible to find a particular piece 
 of work when it happens to be wanted at some later 
 time. Under no circumstances is any work with pen or 
 pencil to be done on scraps of paper, or in a " temporary 
 notebook" or anywhere else except in its proper place 
 and order in the permanent notebook. The slight gain 
 
I INTRODUCTORY 5 
 
 in neatness of the student's notes, which is usually the 
 object of such procedures, is not nearly important enough 
 to counterbalance the possibility of errors in copying 
 data and the probability of being later obliged to hunt 
 in vain for statements that are not in their proper place. 
 The notebook of the scientist, like that of the accountant, 
 should be a book of " original entry," and for reasons 
 that are as important for a scientific investigation of 
 natural phenomena as they are for a legal investigation 
 of indebtedness. Furthermore, no measurement that 
 has been written down in the notebook should ever be 
 rubbed out with an eraser; if there is a reason for doubting 
 its value, or even if it is obviously wrong it may be can- 
 celed by drawing a line through it, but this should be 
 done in such a way as not to obscure what is written 
 down but to permit it to be utilized later if it is found 
 desirable to do so. 
 
 8. Material Equipment. The student's notebook 
 should be of such size and character as will be best 
 adapted to his work. If it is not furnished by the 
 Department directions will be given in regard to the kind 
 of notebook that should be used. Pen and ink will be 
 needed, for notes that are taken with a pencil are almost 
 always unsatisfactory. A fountain pen is advisable, 
 although not necessary. A piece of blotting paper should 
 be obtained which is long enough to reach across the 
 page of the notebook. A hard pencil with the point kept 
 well sharpened will also be needed. 
 
 9. Mental Equipment. The student of the theory of 
 measurement should have had a good course in algebra 
 as far as the solution of equations of the first degree; also 
 a sufficient knowledge of plane geometry to include the 
 properties of perpendiculars, equal triangles, isosceles 
 and similar triangles, the theorem of Pythagoras, and 
 
6 THEORY OF MEASUREMENTS 10 
 
 the properties of similar figures. Certain arithmetical 
 processes, such as the method of extracting square root 
 or cube root, are not needed for purposes of physical 
 measurement, but a good grasp of certain others, such 
 as proportion and variation, is almost a necessity. An 
 intelligent comprehension of principles is as important as 
 a memory of rules and formulae. 
 
 10. Proportionality. The rule that the product of the 
 means is equal to the product of the extremes is one that 
 can be used for the solution of almost any problem in pro- 
 portionality, but the important, thing for the student is to 
 understand the meaning of the combination of terms that 
 constitutes what is called a " proportion.' 7 For example, 
 the value of a commodity is proportional to its amount; 
 thus, it might be that 13 Ibs. : 26 Ibs. :: 43c. : -c. 
 The student who can handle such an example only by 
 multiplying 26 by 43 and then dividing by 13 is wasting 
 much of his time on mathematical drudgery that might 
 be much more profitably devoted to the study of other 
 subjects. A proportion is by definition an equality of 
 ratios; the ratio of 13 Ibs. to 26 Ibs. is stated to equal the 
 ratio of 43c. to some other number of cents, and a 
 glance will show that the second weight is twice as large 
 as the first, whence the second cost is twice 43c., namely 
 86c. A ratio is the same as a fraction or a quotient, and 
 13 Ibs. : 26 Ibs. means 13 lbs./26 Ibs. or 13 Ibs. -r- 26 Ibs., 
 or 1/2. It is equally correct to state that the ratio of 
 quantity to cost of a commodity is constant (cceteris 
 paribus), so that the above proportion may just as well 
 be written in the form 13 Ibs. : 43c. :: 26 Ibs. : 86c. 
 Here the first ratio is not an abstract number like 1/2, 
 but is a certain number (about .302) of pounds for a 
 cent; and the second ratio, which is stated to be equal to 
 it (:: means the same as =), must also be a number of 
 
I INTRODUCTORY 7 
 
 pounds for a cent. Some students may find it advisable 
 to write every proportion that they use in the fractional 
 form 
 
 13 Ibs. _ 26 Ibs. 13 Ibs. = 43c 
 
 ~43cT x 26 Ibs. ~~x ' 
 
 In either of these the possibility of dividing by 13 is seen 
 at once, and in this form no uncertainty can be felt in 
 regard to what should be done with proportions like 
 
 13 : 43 :: 26 : x :: 39 :y :: 6.5 : z* 
 
 Furthermore, there can be no difficulty in answering 
 questions such as the following, which sometimes make 
 trouble for the student who fails to realize the meaning 
 of proportionality: "The pressure of a gas is stated to be 
 proportional to its temperature. How can these two 
 things be proportional if it always takes four terms to 
 make a proportion?" 
 
 The equality of two or more ratios, which is the essen- 
 tial of proportionality, is often expressed by the use of 
 some symbol to denote a constant or invariable quantity. 
 Thus 13 Ibs. : 43c. = .302 lb./c.; 26 Ibs. : 84c. = .302 
 lb./c.; 6.5 Ibs. : 21. 5c. = .302 lb./c., and in general for 
 this particular commodity any weight -r- corresponding 
 value = .302 lb./c. Following the usual custom of using 
 the letter c or k to denote a constant it might be said of 
 commodities in general that w/v = k. From this it 
 follows that v/w is equal to l/k, which is another constant, 
 
 * In case any uncertainty is felt the student should attack it at 
 once, and should not be satisfied until the difficulty has been success- 
 fully overcome. It is perhaps hardly necessary to point out the 
 fact that a mathematical subject cannot usually be read as fluently 
 as a novel. To have each letter and symbol observed by the eye, 
 or even read aloud, is not enough unless the mind is given time for a 
 thorough comprehension of the meaning. 
 
8 THEORY OF MEASUREMENTS 11 
 
 say c, that is usually called " price." Since I/. 302 = 3. 31 
 the price of the substance considered above must be 3.31 
 cents per pound, which is a constant for that commodity, 
 but of course the constant c (cost divided by weight) will 
 have a value different from 3.31c./lb. if some other sub- 
 stance is considered. In technical language such a 
 " variable constant" is called a parameter. 
 
 Is there any difference between what is stated by 
 
 f Xi = cy i} 
 \ x z = cyz, 
 
 and what is stated by x\ : yi :: Xz : 2/2? Prove it. 
 
 If the absolute temperature and the pressure of a 
 given portion of gas are proportional what will happen 
 to its pressure if the gas has its absolute temperature 
 doubled or tripled? 
 
 11. Variation. If the price of a commodity remains 
 constantly 3.31 c./lb. the value is said to vary in accord- 
 ance with the weight, or, shortly, to vary as the weight. 
 or, more explicitly, to vary directly as the weight. Here 
 the weight is considered to be a variable quantity, that 
 is, we may consider any weight we please, the weight of 
 the substance may assume any numerical value for the 
 purposes of the discussion. Under these circumstances 
 the cost will also vary. Doubling the weight will double 
 the cost; cutting the weight in half will reduce the cdst 
 by 50 per cent; etc. When any change in one quantity 
 that can vary is always accompanied by an equal relative 
 chaige in a second quantity the variables are said to be 
 proportional, or to be directly proportional, and each is 
 said to vary directly as the other one. 
 
 If a portion of a gas is subjected to compression it will 
 be found that doubling the pressure exerted upon it will 
 cause its volume to decrease to only one half of its former 
 
I INTRODUCTORY 9 
 
 amount; multiplying the pressure by five will reduce the 
 volume to one fifth; etc. This kind of variation is called 
 inverse, and the pressure and volume are said to be 
 inversely proportional; the volume is said to vary in- 
 versely as the pressure. Suppose that the volume is 6 
 quarts when the pressure is one atmosphere; then if the 
 pressure is raised to 3 atmospheres the volume will be 
 reduced to 2 quarts, but if it is diminished to 1/2 at- 
 mosphere the gas will expand enough to occupy a space of 
 12 quarts. If we write an equation 
 
 v = k - 
 p 
 
 it will be evident that any increase in the size of the de- 
 nominator, p, will cause a relatively equal decrease in the 
 size of the term, v; and we have already seen that this 
 equation means the same as 
 
 vi : I/pi :: v z : l/pz ''- v 9 : I/PS 
 Clearing the equation of fractions gives 
 
 pv = k 
 
 and the ordinary way of expressing the fact that two 
 variables, such as p and v, are inversely proportional is to 
 write down an equation in which their product is stated 
 to be equal to a constant; just as direct proportionality is 
 expressed by making their quotient equal to a constant. 
 It is perhaps worth noticing here that there may easily 
 be other forms of variation, in which there is no propor- 
 tionality at all. The distance traveled by a train is not 
 usually strictly proportional to the number of hours that 
 elapse during the process, nor is an individual's wealth 
 proportional to his age. The matter of irregular vari- 
 ation will be taken up later. 
 
10 THEORY OF MEASUREMENTS 13 
 
 12. Algebraical Formulae. The following are some of 
 the facts of algebra which experience has shown that 
 students of the theory of measurement need but are not 
 in every case familiar with: 
 
 Letters are used for generalized numbers ; if (a + 6) (a b) 
 = a 2 -6 2 then (20 + 1) X(20-l) = 20 2 -1 2 , and similarly 
 for any other numbers. " Terms," between plus or 
 minus signs, are to be evaluated before performing the 
 additions or subtractions; thus 2+4X3 1+4(3 1) is 
 equal to 21, not to 31 or any other number. The product 
 a 3 X a 4 is a 7 , not a 12 ; this is obvious if it is written or con- 
 sidered as (a X a X a) X (a X a X a X a). The prod- 
 uct of a negative and a positive number is negative, but 
 of two negatives is positive; e. g., (a + b ' c)(x y) 
 = ax ay + bx by ex + cy. A negative exponent 
 indicates a reciprocal; a~ 2 means I/a 2 . A fractional 
 exponent indicates a root; a* means V a; x* = -Six* 
 = (-\/x) 3 . Fractional radicands may be simplified as 
 shown in the following example: 
 
 ~ 
 6 V 
 
 The binomial theorem is 
 (a + x) n = a n + na^x + 
 
 (a + b) 2 = a 2 + 2ab + 6 2 ; 
 (m - n) 3 = ra 3 - 3m*n + 3mn 2 - n 3 ; etc. 
 
 13. Mental Exercises. The following list of exercises 
 covers various subjects that students have occasionally 
 been found lacking in familiarity with, also many that 
 
I INTRODUCTORY 11 
 
 will be needed in different parts of the course. The 
 student should read each question and decide upon the 
 answer mentally and without hesitation. If the answer 
 is instantly apparent mark the question with a check ( \/ ) 
 and take up the next one, but if it occasions any hesitation 
 or uncertainty mark it with a plus sign (+) and if it 
 cannot be answered at all inside of a few seconds mark 
 it with a zero (0). Go through the whole list rapidly, 
 and then ask the advice of the instructor in regard to it. 
 This will often make a great difference in the ease of 
 performing the later laboratory work. Remember that 
 it is not an examination of how much it is possible to 
 recall from the depths of your memory, but a test of 
 how much mathematics you have in an immediately 
 available condition. 
 
 1. What is the square of x a? 
 
 2. What is the value of (m + x)(m x)l 
 
 3. State the value of 2 3 . 
 
 4. What is the numerical value of TT? 
 
 5. Which is the larger 37/147 or 38/148? Do you 
 know of any general method of deciding such a question? 
 
 Write the following in the form of decimal fractions: 
 
 6. 1/3. 
 
 7. 4/5. 
 
 8. 1/7. 
 
 9. 1/8. 
 
 10. 2/9. 
 
 11. 1/11. 
 
 12. Can the constant TT be called a parameter? Why? 
 
 13. Reduce 1/25 to hundredths mentally. 
 
 14. Is 6 twice as large as 4? How many times as large? 
 
 15. What is the fourth term of 1000 : 100 :: 31 : . . .? 
 
 16. State the value of 1/500 as a decimal fraction. 
 
 17. Simplify the following: (a 7 ) 5 ; a 7 + a 5 ; a 7 X a 5 . 
 
12 THEORY OF MEASUREMENTS 13 
 
 18. State the cube of a + 6. 
 
 19. (100 - 12) X (100 + 12) = - - -? 
 
 20. If the circumference of a circle is 15 cm. in what 
 way can the diameter be expressed? 
 
 21. Twenty inches on a certain map represents 2,000 
 miles. What is its scale of miles per inch numerically 
 equal to? 
 
 22. State "one out of every four" as a percentage. 
 
 23. What does "twenty percent" mean? 
 
 24. Reduce 0.375 to a percentage. 
 
 25. What is the reciprocal of 2/7? 
 
 26. What percent is the number .005 equal to? 
 
 27. What percent is .00072? 
 
 28. 2.84 X 10- 4 = ? 
 
 29. 284 X 10- 4 ?'>;? 
 
 30. Find (1 + I) 4 by the binomial theorem. 
 
 31. Solve 3 : 12 :: 16 : x. 
 
 32. How much is (- 2)(- 15)/(- 5)? 
 
 33. What is the value of 1 + J + f + J + T V + ? 
 
 34. Solve mentally 4 2 : x 2 :: 25 2 : 75 2 . 
 
 35. Is 
 
 a 
 
 true for numerical values? Give an example of it. 
 
 36. If pv is a constant how will v be affected by doubling 
 p? 
 
 37. 3 + (4 - 4 -f- 2) (7 X 4 - 3) = ? 
 
 38. l + l-2 + l-2-3 + l-2-3-4 = ? 
 
 39. State the approximate value of 8.5 -r- 10. 
 
 40. Does 5 2 X 9 2 equal 45 2 ? Use algebraical letters 
 to illustrate the general principle that is involved. 
 
 41. State the approximate square root of each of the 
 following: 2560 (ans.: about 50); 256; 25.6; 2.56; 0.256. 
 
 42. What is the approximate value of .01/2.38? 
 
INTRODUCTORY 
 
 13 
 
 43. Write "49.78 thousandths of a centimetre" in the 
 form n cm., where n is a decimal fraction. 
 
 44. Does 
 
 13.29 X 0.81 
 3826/41 
 
 have a value of about 5, or about 50, or about 500? 
 
 45. In the equation 
 
 what is the value of y if x is zero? 
 
 46. In the equation y = 2x + 4 what is the value of y 
 when x = - 3? 
 
 47. Complete the following sentence in the obvious 
 way: A cubic inch of lead weighs 165 gm., and is 150 gm. 
 heavier than 1 cubic inch of water; therefore . . . . 
 
 48. Substitute 1/ra mentally for log y in the ex- 
 pression l/( log y) and simplify. 
 
 49. Write the value_of .0011 X .00011. 
 
 50. Solve p = 2irVllg for I, and then for g. 
 
 51. If 
 
 i 
 
 21 
 
 n = . , A / 
 sr 
 
 how does n change if I becomes f of its former size? 
 t becomes f as large as before? If s is J as large? 
 52. Fill out the following table: 
 
 If 
 
 n 
 
 n* 
 
 2" -3 
 
 l/n 
 
 
 1 
 2 
 3 
 2x 
 
 
 
 
 14. Physical Arithmetic. Before performing any writ- 
 ten calculation with numbers that have been obtained 
 
14 THEORY OF MEASUREMENTS 14 
 
 from physical measurements it is advisable to make a 
 rough mental calculation of the approximate value of the 
 final result. For example, at 3.31c. per Ib. what will 
 5J Ibs. cost? One of these factors is a little more than 
 three; the other is somewhat less than six. Their 
 product, accordingly, must be in the neighborhood of 
 18. A train travels 155.8 miles in 2| hours; what is its 
 rate in miles per hour? Here the time is less than 3 
 hours, so the speed must be greater than 155.8 -f- 3; and 
 still greater than 150 -f- 3. Ans. : Somewhat faster than 
 50 miles per hour. If a still closer approximation should 
 be desired it could be obtained by noticing that the 
 actual distance is about 4 percent greater than 150, and 
 the actual time is one twelfth (say 6 percent) less than 
 the assumed time. Increasing 50 by 4 percent and then 
 by 6 percent would make it about 10 percent larger, 
 giving 55 miles per hour for a closer value of the speed. 
 The arithmetically accurate value is 56.65454545. 
 Is 
 
 4.71 X 13.8 
 9.06 
 
 equal to about 7, or about 70, or about 700? 
 
 What is the approximate value of 7.26 X .0328? 
 
 Point off 00000928800000 so as to make it equal to 
 the product of .0216 and .0043 
 
 Point off the right-hand side of the equation 2./.3 
 = 00666. 
 
 Reduce the ratios in the following expressions to ap- 
 proximate percentages, performing the calculation men- 
 tally: "8 Ibs. in every 23 Ibs. of sea water is solid salt" 
 (ans. : 8 in 24 would be 33 1 percent; 8/23 is a little greater 
 and must be 34 or 35 percent; or: 8/23 = 16/46, 16/46 
 > 15/45, /. 8/23 > 1/3); " seven inhabitants out of every 
 
I INTRODUCTORY 15 
 
 38 are voters" (ans.: 7 out of 35 would be 20 percent, 
 7/42 = 16f percent, 7/38 must have some intermediate 
 value, say 18 percent); " f ourteen-carat gold is 14/24 
 pure" (ans.: 14/24 = 28/48 = 56/96; this fraction has its 
 numerator about half as large as its denominator and so 
 will not be much changed by adding 4 to the latter if 2 
 is added to the former, 56/96 = 58/100 = 58 percent); 
 "boiling water will dissolve .000022 of its weight of 
 silver chloride" (ans.: .000022 = .0022 percent); "a 
 saturated salt solution has a strength of 5 : 13" (ans.: 
 a little less than 5 : 12J or 10 : 25 or 40 : 100, say 38 
 percent); "a steep railroad grade may have a rise of as 
 much as 180 feet per mile" (ans.: 180 ft. per 5280 ft. is 
 less than 180 per 5,000 or 360 per 10,000 or 36/1000 or 
 .036 or 3.6 percent). 
 
 Notice the different expressions that .are in common use 
 to denote the comparison between a definite fractional 
 part and the total. The same meaning is expressed by 
 each of the following phrases as by any of the other ones : 
 22 per million, 22 out of a million, 22 in 1000000, 
 22/1000000, .000022, 22 : 1000000, and .0022 per centum 
 or .0022 percent. 
 
 Observe also that the fractional notation is more con- 
 venient than the word per in naming compound units of 
 measurement, and has the same significance. Thus, a 
 
 speed of 40 miles per hour is customarily written 40 T -, 
 
 or 40 mi/hr, meaning 40 times 1 mile per hour or 40 -r-r . 
 
 That this notation is consistent will be made obvious by 
 considering that 40 X 1 mile X 1 hour would necessarily 
 be the same as 40 X 1 mile X 60 minutes, which reduces 
 to 2400 X 1 mile X 1 minute, and 40 miles per hour is 
 quite different from 2400 miles per minute; but 40 X 1 
 
16 THEORY OF MEASUREMENTS 15 
 
 mile -h 1 hour = 40 X 1 mile -r- 60 minutes = f X 1 
 mile -T- 1 minute, and f of a mile per minute is plainly 
 the same speed as 40 miles per hour. Of course there 
 are some kinds of compound units which are properly 
 expressed when their component simple units are multi- 
 plied together instead of being divided. For example, 
 one foot-pound of energy is equal to 16 foot-ounces, a 
 fact that could not be true if the unit were 1 X 1 foot 
 -r- 1 pound, but that requires it to be 1 X 1 foot X 1 
 pound. (Insulation resistance in "ohms per mile" and 
 lighting efficiency in " watts per candle" furnish illus- 
 trations of mis-named units. The longer wire has the 
 lesser insulation resistance so that 20 ohms for one mile 
 is the same as 10 for two miles and the only rational name 
 for the unit is the ohm-mile. Efficiency means light per 
 energy and its unit would properly be candle-power per 
 watt, but the illuminating engineer prefers to consider 
 inefficiency, which is properly measured by watts per 
 candle-power.) 
 
 15. Abridged Division. A number that is obtained as 
 the result of a physical measurement is frequently needed 
 for some kind of a calculation. When this is the case it 
 is a fact (as will be shown later) that the final result never 
 needs to be expressed with a greater number of figures 
 than the original data contained. Thus, it may be 
 possible to measure the width of a table so carefully as 
 to make sure that the measurement is 62 centimetres 
 + 3 millimetres + 8 tenths of a millimetre. Such a 
 quantity is preferably written as a number of centimetres, 
 and in this case is 62.38, a number consisting of four 
 figures. Suppose it is necessary to find out what one 
 third of the width will amount to. One third of 62.38 is 
 20.793333 . . . , and, as stated above, four figures of this 
 result, namely 20 79, are all that are necessary. As a 
 
I 
 
 INTRODUCTORY 
 
 17 
 
 275)1558(566545 
 1375 
 1830 
 1650 
 1800 
 1650 
 1500 
 1375 
 
 matter of fact, to keep more than four figures would be 
 decidedly objectionable. If the original measurement 
 gave the correct number of tenths and hundredths of a 
 centimetre without pretending to state any knowledge of 
 the correct number of thousandths how could any cal- 
 culation assume to give correct figures in thousandth's 
 and tens-of-thousandth's places? Similarly, in 14, 
 the " arithmetically accurate" value 
 would be wrong if the given distance 
 were even a thousandth of an inch 
 longer or if the time varied from an 
 exact 2f hours by as much as a mill- 
 ionth of a second. Here one of the 
 numbers (155.8 miles) has four figures 
 while the other can hardly be consid- 
 ered to have more than three (2.75 
 hours). In such cases it is a fact 
 that the final result will have only as 
 many trustworthy figures -as there are 
 in the shortest number from which it is 
 derived. When a number having four 
 figures is divided by a number of three 
 figures there should be only three fig- 
 ures kept in the quotient. 
 
 The principle just stated makes it 
 possible to employ the " abridged" 
 processes of multiplication and divi- 
 sion, which will automatically give 
 just the right number of figures in the 
 answer, and will also save considerable 
 labor on the part of the computer. 
 
 The first example shown in the mar- 
 gin has been worked out by ordinary 
 "long" division; in the second one 
 3 
 
 1100 
 1500 
 1375 
 125 
 
 2^)1558(567 
 
 1375 
 
 183 
 
 165 
 
 18 
 
 I 9 
 
 ABRIDGED DIVI- 
 SION. After each 
 subtraction the next 
 step is to shorten 
 the divisor instead 
 of to lengthen the 
 dividend. 
 
18 THEORY OF MEASUREMENTS 15 
 
 the abridged method has been used. The latter process 
 differs from the former in only one respect: Whenever 
 the process of " bringing down" a zero would be employed 
 the last figure of the divisor is canceled instead. In 
 order that the temporary dividend shall be larger than 
 the divisor one method stretches out the dividend by 
 affixing a cipher; the other shortens the divisor by 
 trimming off its last figure. A comparison of the two 
 examples will show that the same result is achieved in 
 each case. The beginner should work out the quotient 
 of the two numbers given above, canceling the last 
 figure of the divisor whenever he would otherwise 
 "bring down" a zero, but not referring to the illustration 
 until he was finished: After the first subtraction, when 
 the divisor, 275, is not contained in the remainder the 
 divisor is shortened by crossing off the final 5. Then, 
 27 is contained in 183 six times. The next step is to 
 multiply 27 by 6. Before saying 6 X 7 = 42 notice 
 that if the 5 had not been canceled there would have 
 been 3 to carry, resulting in 45 instead of 42. Accord- 
 ingly 5 is written down instead of 2 and the rest of 
 the multiplication proceeds as usual. After the next 
 subtraction gives a remainder of 18 the divisor is short- 
 ened to 2 instead of having the new dividend lengthened 
 to 180. Then, 2 would be contained in 18 just 9 
 times, but considering the figure that was last canceled 
 it is plain that 2.7 will not be contained much more than 
 6 times. Six times the canceled 7 would be 42 and would 
 give 4 to carry, so 6 times the 2 (plus 4) is written 16. 
 The next subtraction and cancellation puts an end to the 
 work. In the specimen given above it has been noticed 
 that the third figure of the quotient is to be the last one, 
 and it has been written down as a 7 instead of a 6 because 
 the next product, 19, comes nearer in value to the re- 
 
INTRODUCTORY 
 
 19 
 
 quired 18 than would the number 16. In other words the 
 quotient is nearer 567 than 566 and so the larger number 
 represents it more accurately than the smaller. 
 
 The quotient is to be pointed off by making a pre- 
 liminary mental calculation, as explained in 14. For 
 example if the original numbers had been 2.75 and 155.8 
 the answer would have been 56.7. 
 Similarly, 1558./.0275 = 56700; 
 1.558/27.5 = .0567; etc. 
 
 Divide 180.000 by 3.1416 
 without referring to the work 
 given in the margin until the 
 answer has been obtained. The 
 sixth figure which is given in the 
 quotient is intended to represent 
 the value of the 2/3 remaining 
 after the last subtraction. It 
 could also have been obtained 
 by continuing the regular process 
 of abridged division: cancel the 
 3, leaving 0.$ for the divisor; 
 then 2 -=- 0.3 = 7; multiplying, 
 7 X ? gives 2 to carry, 7X0 
 + 2 = 2. 
 
 Find the quotient if the divisor is 236453 and the divi- 
 dend is 6764309. The answer should come out 28.60741. 
 After the figure 4 of the quotient has been written down 
 the next step is to multiply 230^^0 by 4. Ordinarily it 
 is sufficient to take the nearest canceled figure and say 
 4 X 6 = 24, giving 2 to carry; but as the product 
 comes close to 25, which is on the boundary between 2 
 to carry and 3 to carry, it is well to investigate one more 
 canceled figure, saying 4 X = 16, giving 2 to carry 
 toward 4 X 0; the latter then becomes 26, giving 3 to 
 
 157080 
 
 22920 
 
 21991 
 
 929 
 
 628 
 
 301 
 
 283* 
 
 18 
 
 16 
 
 2 
 
 ABRIDGED DIVISION. 
 At the mark * notice that 
 3.6 comes nearer to being 
 " 4 to carry " than " 3 to 
 carry," since io is more 
 than three and a half. 
 
20 THEORY OF MEASUREMENTS 16 
 
 carry toward the written product instead of 2. Make 
 up and work out two examples in which a long number is 
 divided by a short number, and vice versa. Follow the 
 regular routine: divide, multiply, subtract, and cancel; 
 and repeat as many times as necessary. 
 
 16. Abridged Multiplication. The trustworthiness of 
 a product of two or more numbers follows the same rule 
 as that .of a quotient : no more figures of the product are 
 " significant ',' than the number of them which the shortest 
 factor contains. Thus if the diameter of a circle is 
 found to be 8 centimetres + millimetres + tenths 
 of a millimetre but nothing is stated about hundredths of 
 a millimetre, so that only three figures, 8.00, of the 
 diameter are known, it will not be possible to obtain 
 more than three figures of the circumference, even if 
 the other factor, 3.14159265359, contains a dozen figures. 
 Here too the ordinary arithmetical process can be 
 abridged so as to save time and work, and, what is more 
 important, to avoid being misled by figures that have 
 been kept when they should have been discarded. 
 
 The illustration (a) shows the ordinary process. It is 
 just as easy, although not customary, to use the figures of 
 
 (a) 65.97 (6) 65.97 (c) 60.#f 
 
 24.13 24.13 24.JL3 
 
 19791 13194 13194 
 
 6597 26388 2639 
 
 26388 6597 66 
 
 13194 19791 20 
 
 1591.8561 1591.8561 1591.9 
 
 GENESIS OF THE ABRIDGED METHOD. (a) Ordinary long multi- 
 plication. (6) The same with the figures of the multiplier used in 
 the reverse order, (c) The same as (6), but the partial products are 
 kept from stringing out to the right by progressively shortening the 
 multiplicand. 
 
1 INTRODUCTORY 21 
 
 the multiplier in the reverse order, multiplying first by 
 2, then by 4, 1, and 3, and " stepping" the partial prod- 
 ucts successively one more place to the right instead of 
 to the left. This has been done in illustration (6). 
 Examine it closely, and see that you understand just 
 how the process (6) is carried out and why it must 
 necessarily give the same result as (a). 
 
 The abridged method is shown at (c). The first 
 multiplication is by the left-hand figure 2 as in (6). 
 Then the last figure of the multiplicand, 7, is canceled; 
 and the next multiplication (by 4) is begun directly under 
 the first. The process of canceling and multiplying is 
 continued in the same way until either the multiplicand 
 is entirely canceled or the multiplier has been entirely 
 utilized, and the result is pointed off in accordance with 
 the directions in 14. The student should work out the 
 same example independently, remembering to investigate 
 how much there is "to carry" from the figure last 
 canceled. In the last partial product of (c) the 3X6 
 = 18 is increased by two units because 3 X $ gives just 
 1.5 to carry but it is evident that 3 X ffl must give a 
 number nearer to 2 than to 1. 
 
 Multiply the quotient 28.60741 given above by the 
 divisor 236453, and notice that the dividend is found cor- 
 rectly as far as six figures (more than 676430^), which is 
 all that can be expected if one factor contains only six 
 figures. 
 
 The value of 180/Tr is 57.29578. Multiply this by 
 3.141593 and see if you obtain 180.0000 correct to seven 
 figures. 
 
 Multiply any number that has five figures by some 
 number having only two figures. Repeat the multi- 
 plication, using the shorter number for multiplicand and 
 the longer one for multiplier. Which method Is pref- 
 
22 THEORY OF MEASUREMENTS 17 
 
 erable, in view of the statement at the beginning of 16? 
 
 17. Gradient. If a road that goes up-hill rises 2 feet 
 for every 5 feet of horizontal distance it is said to have a 
 slope of 2 : 5, or 2/5, or 0.4, or 2 in 5. That is, the ratio 
 of any vertical rise to the corresponding horizontal 
 distance is taken as a numerical measure of its steepness. 
 Of course the slope could also be measured in degrees; 
 in the case just mentioned the "grade angle," or angle 
 which the slant line makes with a horizontal line, would 
 be twenty-two degrees too steep to be satisfactory for a 
 road-way, but the usual custom is to state the measure 
 of a slope in terms of vertical rise per horizontal distance. 
 Another way of looking at the same thing is to consider 
 it as the amount of rise per unit of horizontal distance; 
 thus, a road that rises two feet in every five will of course 
 have a rise of 2/5 of a foot for a single foot of horizontal 
 distance. 
 
 A level road is one that has no slope. That is, its 
 slope, when measured as rise per level distance, amounts 
 to zero. If a line is made to slant more and more steeply 
 the ratio that represents its slope will obviously become 
 greater and greater without limit; thus, a slope of 1000 
 would be hardly distinguishable from a true vertical, 
 and yet between these two there must be, for example, a 
 slope of 1000000000. The diagonal of a square is inclined 
 to any of the sides at an angle of 45, and its slope is of 
 course unity. These facts may be abbreviated as follows : 
 slope of = 0; slope of 45 = 1; slope of 90 = QO. 
 
 Draw roughly an equilateral triangle that has one side 
 horizontal. Draw a vertical from its apex to the middle of 
 its base, and prove that the gradient of a sixty-degree 
 slope is equal to V 3 ; also that if the angle is half as 
 large as this the slope will be only one third as much. 
 
 It is sometimes convenient to consider the steepest 
 
I INTRODUCTORY 23 
 
 possible " slope" (a vertical line) as having 100 percent 
 of steepness. This will be the case if we measure the 
 amount of slant not by the ratio of rise to level distance 
 but by the ratio of rise to slant distance. A road which 
 has a grade angle of 22 will rise nearly 3 feet for every 
 8 feet of distance along its slanting surface, and this 
 measure of steepness may be called percent slope to 
 distinguish it from the slope, or gradient, as previously 
 defined. 
 
 Prove that the "percent slope" of 45 is H 2; of 
 is zero; of 90 is 1; of 30 is | (use the same bisected equi- 
 lateral triangle). 
 
 The steepest slopes that are generally used for road- 
 ways range from 12 percent to 15 percent. On good 
 turnpikes the grades are almost always kept below 3 
 percent. Two percent is decidedly steep for a railroad 
 grade, and in modern good railroad construction one 
 percent is about the maximum. For slight inclinations, 
 such as railroad grades, the difference between rise per 
 horizontal distance and rise per slant distance is neg- 
 ligibly small. Thus, for 1 they are respectively .017455 
 and .017452, or 92.16 feet per mile and 92.15 feet per 
 mile. 
 
II. WEIGHTS AND MEASURES 
 
 Apparatus. Ruler; metre stick; graduated cylinder; 
 graduated pipette; pair of dividers; irregular solid; 
 platform balance; set of gram weights; set of ounce 
 weights; towel; glass jar or " catch-bucket "; small test 
 tube. 
 
 18. C.G.S. System. The older units of measurement, 
 such as the length of a barleycorn, the width of a man's 
 palm, or the length of a foot or a pace, were objectionable 
 chiefly on account of their lack of uniformity. Not only 
 did different countries use Different units for measuring 
 quantities of the same kind, but even when a unit of the 
 same name was used in different localities its value was 
 not the same. In most civilized countries these older 
 units have been entirely superseded by a new system of 
 weights and measures, and in all countries this system 
 has come into universal use for every kind of scientific 
 work. It is usually called the C.G.S. System, from the 
 initial letters of the units of length (the centimetre), of 
 mass (the gram), and of time (the second). These three 
 units are called fundamental, because they have been 
 arbitrarily fixed in size, while all the other units of the 
 system have been so chosen as to make them depend 
 upon these three in as simple a manner as possible. 
 For example, the derived unit of velocity is such as will 
 denote movement through a single centimetre of distance 
 in a single second of time, thus making the measure of a 
 velocity numerically equal to the quotient of space 
 1 traversed divided by time elapsed during the process. 
 Similarly, the density of an object is defined as its mass 
 in grams divided by its volume in cubic centimetres, so 
 
 24 
 
II 
 
 WEIGHTS AND MEASURES 
 
 25 
 
 that, although no name has been given to the unit, the 
 density which is numerically equal to unity must be the 
 density of such a substance as will weigh one gram for 
 each cubic centimetre of its volume; the unit of force is 
 the force that must act for one second of time in order 
 to produce a change of one unit (one centimetre per 
 second) in the velocity of a unit mass. 
 
 19. Unit of Length. The scientific unit of length is 
 the centimetre. It is equal to about half a finger-breadth 
 and is often found on tape-measures, 
 rulers, etc. These are simply copies 
 of accurate standards belonging to 
 the manufacturer, which in turn owe 
 their accuracy to a careful comparison 
 with the standards of the government. 
 In the case of the governments that 
 subscribed to the Metric Convention, 
 including the United States, the stand- 
 ards, which are called national proto- 
 type metres, are lengths of one metre 
 (i. e., 100 centimetres) carefully laid 
 off between lines near the ends of 
 certain bars of artificially aged plat- 
 inum-iridium alloy which are 102 cen- 
 timetres long and have a cross-section 
 that somewhat resembles a letter X 
 (Fig. 1). The greater length is used 
 instead of a single centimetre because 
 it can be measured more accurately, 
 and the cross-section is for the purpose 
 of giving rigidity, and in order to allow 
 the scale to be marked on a surface 
 that would be neither stretched nor 
 compressed if the bar should be slightly bent. The 
 
 FIG. 1. TRESCA 
 CROSS-SECTION. A 
 bar of this shape has 
 great rigidity for a 
 given weight of 
 metal; and a slight 
 bending, such as 
 would stretch the 
 lower part and com- 
 press the upper 
 part, has no effect 
 on the " neutral 
 web," n, where the 
 scale is engraved. 
 The diagram shows 
 the exact size of the 
 cross-section of the 
 prototype metres. 
 
26 THEORY OF MEASUREMENTS 19 
 
 standards were constructed at Paris and distributed by 
 lot among the signatories to the Metric Convention about 
 1889, after being carefully tested and compared with one 
 another so that their relative errors and equations were 
 accurately known.* One of these, which is kept at the 
 International Bureau of Weights and Measures, near 
 Paris, is known as the international prototype metre and 
 corresponds in length with the original flat platinum bar 
 (100 cm. X 0.4 cm. X 2.5 cm.) constructed for the 
 French Government by Borda and called the metre 
 des archives. The Borda standard was intended to equal 
 one ten-millionth of the length of the meridian quadrant 
 passing through Paris from the north pole to the equator, 
 the earth itself thus furnishing the original standard. 
 The metal bar, however, is now taken as the fundamental 
 standard, not only because a microscopic measurement of 
 it can be made more easily and more accurately than a 
 geodetic survey, but also because the actual length of 
 the earth's quadrant is not constant. Its average length, 
 according to the best estimations, is about 10,002,100 
 metres. 
 
 The multiples and subdivisions of the metre that are in 
 actual use are the kilometre (1 km. is 1000 metres, or 
 about 5/8 of a mile; closer approximations are given in 
 the appendix), the centimetre (1 cm.), the millimetre 
 (0.1 cm.), and the micron (0.0001 cm.). These are more 
 convenient than a single unit in some cases, but in scien- 
 tific work it is desirable to express all lengths in terms of 
 
 * Modern processes of measurement are so accurate that a dif- 
 ference can generally be found between two standards that were 
 intended to be equal, no matter how carefully they were constructed. 
 The " equation " of a prototype metre expresses the way in which 
 its length varies when its temperature is changed. Thus at any 
 ordinary temperature, t, the length in centimetres of prototype 
 metre No. 18 is 99.9999 + .0008642^ + .000000100J 2 . 
 
II WEIGHTS AND MEASURES 27 
 
 the accepted unit, 1 cm., in order to avoid possible con- 
 fusion, or serious error in case the denomination of a 
 quantity should be accidentally omitted. Thus, the 
 length of the earth's quadrant is 1,000,210,000, but in 
 order to be doubly safe it is advisable to make it a rule 
 to write the denomination after a number in all cases 
 (1,000,210,000 cm.). 
 
 20. Units of Area and Volume The scientific unit 
 of area is the square centimetre (1 cm 2 ), the area of a 
 square each of whose sides is 1 cm. in length. A square 
 foot is about 1000 cm 2 . The unit of volume is the 
 cubic centimetre (1 cm 3 ), the volume of a cube that 
 measures 1 cm. on each edge. The dry and liquid quarts 
 are each approximately equal to 1000 cm 3 . 
 
 21. Units of Mass and Density. The scientific unit 
 of mass, or for practical purposes the unit of weight in 
 vacuo at sea level, latitude 45, is the gram (1 gm.), 
 which is divided into 1000 milligrams (mgm.) just as the 
 metre is divided into 1000 millimetres. It is derived 
 from kilogram prototype standards (1 kgm. = 1000 gm. 
 = 2.2 Ibs.) established at the same time as the standards 
 of length and was originally intended to be equal to the 
 mass of one cubic centimetre of water under standard 
 conditions. More careful measurements, however, on 
 water that has been freed from dissolved air have shown 
 that even at the temperature of its greatest density 
 (3.98 C.) a gram of water occupies a trifle more space 
 than one cubic centimetre, although the excess is only 
 one sixtieth as great as it is at ordinary room temperature. 
 In cases where the slight change of volume that is pro- 
 duced by heating or cooling can be neglected it may be 
 considered that water has a density equal to one, density 
 being defined as mass in grams divided by volume in 
 cubic centimetres. 
 
28 THEORY OF MEASUREMENTS 23 
 
 22. Unit of Time. The scientific unit of time is the 
 second, which is the 1/86400 part of the length of an 
 average day from noon to noon. As the length of the 
 solar day varies at different seasons of the year the second 
 is determined in practice as 1/86164.1 of the time of a 
 complete rotation of the earth with respect to the fixed 
 stars. This unit of time was in use before the adoption 
 of the C.G.S. System and is familiar to every one. It is 
 perhaps worth noticing that fairly accurate seconds can 
 be counted off by repeating, at ordinary conversational 
 speed, "one thousand and one, one thousand and two, 
 one thousand and three," etc. 
 
 23. Practice in Using the C.G.S. System. The scien- 
 tific system of units is so largely used that the student 
 should not be satisfied with the mere ability to translate 
 measurements from one system to the other. He 
 should practice first estimating (guessing) and then 
 measuring the dimensions of various objects that he 
 comes across, until he has acquired a certain ability to 
 "think" in centimetres, cubic centimetres, grams, etc., 
 instead of in inches, quarts, and pounds. 
 
 Across the top of one page of the notebook draw a 
 horizontal line just ten centimetres long and rule two 
 short perpendicular lines across its ends in order to 
 indicate the exact length clearly. In the same way, along 
 the right-hand edge of the page, draw a line twenty-five 
 centimetres in length. Under the first line draw five 
 others of various lengths without measuring them. After 
 they have been drawn measure each one with a scale of 
 centimetres and millimetres, and record its length to the 
 nearest millimetre. For example, if the length appears 
 to be about 152f mm. it should be recorded as 15.2 cm.; 
 if about 152f mm. it should be called 15.3 cm. Write 
 each length as a number of centimetres, not 153 mm. nor 
 
II WEIGHTS AND MEASURES 29 
 
 15 cm. 3 mm., and make it a rule to see that the denomi- 
 nation of a measurement is never omitted. 
 
 24. Rule for " Rounding Off " One Half. It may 
 happen that the measured length is so near 152J mm. 
 that it is impossible to decide between 152 and 153. A 
 fraction perceptibly less than a half should be discarded 
 and more than a half should always be considered as 
 one more unit, but when it is uncertain which figure is 
 the nearer one the universally adopted rule is to record 
 the nearest even number rather than the odd number that 
 is equally near. The reason for this procedure is that 
 in a series of several measurements of the same quantity 
 it will be as apt to make a record too large as it will to 
 make one too small, and so in the average of several such 
 values will cause but a slight error, if any. If the rule 
 were that the half should be always increased to the next 
 larger unit the errors would not balance one another and 
 the average would tend to be brought up to a larger 
 value than it should have. The same advantage would 
 of course be obtained if the nearest odd number were 
 always used, but the even number has one slight addi- 
 tional merit, namely, that in case it should have to be 
 divided by two a recurrence of the same situation would 
 be avoided. 
 
 By comparison with the lines already drawn make a 
 mental estimate of the length and width of the note- 
 book; then verify the estimate by measuring with a 
 scale. Remember to record clearly all the experimental 
 work that is done; thus, the completed notes should 
 show at a glance which number is the actual width and 
 which is the rough estimate. 
 
 25. The Hand as a Measure. Lay your hand across a 
 ruler or a metre stick and either spread the fingers slightly 
 or crowd them closer together, as may be necessary, so 
 
30 
 
 THEORY OF MEASUREMENTS 
 
 as to make a whole number of finger-breadths occupy the 
 same amount of space as a whole number of centimetres. 
 Then hold the hand in a similar manner while using it for 
 practicing approximate measurements of various objects. 
 Record also the exact measurements of the same objects 
 as they are obtained later with the graduated scale. 
 
 Separate the thumb and little finger as far as can be 
 conveniently done without special effort and measure 
 your span in centimetres and millimetres. Repeat this 
 measurement five times, being careful not to let the sight 
 of the scale under your hand influence the extent to which 
 the fingers are spread, and decide which is the most 
 satisfactory value. Measure the length and the breadth 
 of the table by means of successive spans, using hand- 
 breadths or finger-breadths for the final fraction of a 
 span, and compare the result with the actual length and 
 breadth. 
 
 26. Measurement of Area. Ask the instructor to 
 draw an irregular outline in your notebook (Fig. 2). 
 
 FIG. 2. IRREGULAR AREA. The simplest method of measuring 
 an area marked on squared paper is to count the squares that are 
 entirely within the figure as units and those that are cut by the 
 boundary as half units. The sum gives the total area almost as 
 well as if an attempt were made to estimate the fractional size of 
 each cut square. 
 
II WEIGHTS AND MEASURES 31 
 
 Count the number of the small squares of the ruled paper 
 which are entirely included within it, but do not outline 
 them on the diagram. To their total add half the number 
 of the squares that are cut by the boundary of the figure. 
 The result will be the area of the irregular outline, not 
 in square centimetres, of course, but in terms of the small 
 ruled squares, and its denomination may be written 
 " n 's." Try to obtain a more accurate value for the 
 total area by estimating as closely as possible how many 
 tenths of each cut square is included within the boundary 
 and adding these actual fractions. The first result 
 should agree quite closely with this one because any one 
 of the fractions of a square is as likely to be less than 
 one half as it is to be more than one half, and the most 
 probable value for the average of the fractions is just 
 one half. 
 
 As an alternative method, block off an equal area on 
 the same irregular figure by drawing several rectangles 
 and triangles to cover it (Fig. 3). If one corner of a 
 
 FIG. 3. AREA BY MENSURATION. An irregular figure can be 
 " blocked out " by a number of triangles and parallelograms which 
 are so drawn that an error of excess in one place is approximately 
 balanced by an error of defect in another, so as to make the com- 
 bined areas of the geometrical figures equal to the required area. 
 The geometrical areas are then evaluated by ordinary mensuration. 
 
32 
 
 THEORY OF MEASUREMENTS 
 
 28 
 
 triangle projects considerably beyond the irregular line 
 see that one of its sides is drawn so as to include less than 
 the requisite amount and try to make the two opposite 
 errors balance as nearly as possible. Then find the total 
 area of the geometrical figures by mensuration, measuring 
 their dimensions not by means of the cen- 
 timetre scale but with a scale copied from 
 the ruling of the notebook; or by transfer- 
 ring each length to the ruled page with a 
 pair of dividers, so as to obtain the area in 
 the same units as before. 
 
 27. Measurement of Volume. Examine 
 the graduated cylinder and the graduated 
 pipette and notice the volume occupied by 
 one cubic centimetre in each. Do they 
 seem larger or smaller than the space that 
 would be enclosed if an imaginary square 
 centimetre were drawn beside one linear 
 centimetre of the ruler and an imaginary 
 cube were then built up on the square? 
 Pour into a test-tube an amount of water 
 which you think will be 10 cm 3 ; then meas- 
 ure it carefully.* 
 
 Put an irregular block of metal (not one of 
 the standard weights from your set) into a 
 graduated cylinder partly filled with water 
 and determine its volume by the change in 
 the water level. In reading a cylinder or 
 a pipette the result is to be obtained by 
 noting the height of the liquid surface where it is lowest 
 in the centre (Fig. 4). 
 28. Measurement of Mass. Examine the brass 
 
 * See that a towel is at hand when water is used in any experi- 
 ment, and wipe up immediately any that is spilled on the table. 
 
 FIG. 4. 
 
 G R A DUATED 
 
 CONTAINER. 
 The scale is 
 customarily 
 arranged to 
 give the cor- 
 rect reading 
 at the lowest 
 part of the 
 meniscus, or 
 curved sur- 
 face, of the 
 liquid. 
 
V 
 
 II WEIGHTS AND MEASURES 33 
 
 weights in a set extending from one gram to 500 grams, 
 and observe especially the size of the 10-gram weight. 
 What would its volume be if its density were ten (gm. 
 per cm 3 )? If the density of brass is only 8.5 (i. e., if 
 it is less compact) will its volume be greater or less than 
 this? 
 
 Examine the platform balance and notice that there 
 are two wooden wedges that hold the pans away from 
 the beam and the beam away from its support, so as to 
 remove their weight from the accurately ground bearings 
 when the apparatus is not in use. With a fine analytical 
 balance this is such an important matter that a mechan- 
 ism is provided by means of which the user keeps the 
 scale pans and the beam " supported" while arranging 
 the weights and the object to be weighed, and only lowers 
 them upon their bearings for a few moments at the time 
 of actual weighing. Such care is not necessary with an 
 ordinary platform balance, but it is always advisable to 
 support the scale pans after one has finished using the 
 apparatus. Remove the wedges carefully and notice 
 where the moving pointer comes to rest on the arbitrary 
 scale. This need not be in the centre but may be at 
 any point on the scale, and in weighing if the weights are 
 so added as to return the pointer to this same position 
 the result will be the same as if the pointer were made to 
 take a central position in the first place. Notice the 
 counterpoise, which can be screwed toward one side or 
 the other in order to adjust the position of equilibrium, 
 but do not attempt to move it unless you are sure that 
 the apparatus is on a level part of the table and the scale 
 pans are free from dust or other adherent matter. Notice 
 whether the balance has a sliding weight that is used for 
 weighing fractions of a gram. 
 
 Find the mass, in grams, of a four-ounce avoirdupois 
 4 
 
34 THEORY OF MEASUREMENTS 30 
 
 weight, weighing it first on the left pan of the balance and 
 then on the right. Notice that when it is on the right- 
 hand pan the reading of the sliding weight must be 
 subtracted from the sum of the other weights instead of 
 being added to them. 
 
 Draw up a few cubic centimetres of water in the 
 graduated pipette, retaining it by pressing the dry finger 
 tip on the upper end of the tube. Practice letting it run 
 out slowly until you have no trouble in delivering any 
 exact amount. Then fill it again, weigh the ''catch- 
 bucket" or some other container, run just seven cm 3 of 
 water from the pipette into the container and weigh the 
 latter again. 
 
 29. Measurement of Density. Calculate the den- 
 sity of water from the data obtained in the last experi- 
 ment. 
 
 Find the volume of the 200-gram weight by measuring 
 its height and diameter as carefully as possible, but do 
 not immerse it in water. Suppose that the handle of the 
 weight were soft, like wax, and could be flattened out and 
 spread uniformly over the top of the cylindrical part, 
 and estimate as well as you can how much this would 
 add to the height of the cylinder. Then calculate the 
 density of the brass weight, remembering that multi- 
 plication and division are always to be done by the 
 abridged methods. The result may turn out to be less 
 than the usual density of brass (about 8.5 gm. per cm 3 ) 
 if the handle is a separate piece screwed into the body of 
 the weight so that an air space is left between the two 
 parts. 
 
 Find also the mass of the irregular solid whose volume 
 was determined by immersion and calculate its density. 
 
 30. Equivalents. In most English-speaking countries 
 the C.G.S. System is very little used except for scientific 
 
II WEIGHTS AND MEASURES 35 
 
 purposes, the older system still holding its own in spite of 
 such obvious disadvantages as possessing an ounce 
 (avoirdupois) that weighs about 28 gm. and another 
 ounce (Troy) that is equal to a little more than 31 gm.; 
 furthermore, the U. S. fluid ounce of water weighs more 
 than an avoirdupois ounce and less than a Troy ounce, 
 and is four percent larger than the imperial fluid ounce 
 of England. Accordingly, a knowledge of the approxi- 
 mate relationships between the units of the old system 
 and the new is almost a necessity for the scientific student 
 of today. 
 
 Turn to the tables of equivalent weights and measures 
 in the appendix of this book and use the approximate 
 equivalents for translating your own weight and height 
 into the C.G.S. System, and for answering such questions 
 as: How many kilometres is the distance from here to 
 New York (or any other city) ? What is the approximate 
 height of this room in metres? What is the C.G.S. 
 velocity of sound if it travels a mile in five seconds? 
 How many centimetres per second is your ordinary rate 
 of walking? 
 
 31. Questions and Exercises. 1. Explain why round- 
 ing off a half to the nearest even number will increase a 
 measurement, in the long run, just as often as it will 
 decrease it. 
 
 2. When you drew a line "just ten" centimetres long 
 was its length 10.0 cm.? Was it 10.00 cm.? 10.000 cm.? 
 Measure it again, and explain why it is better to use one 
 of these numbers in stating its length than to say "just" 
 ten. 
 
 3. Does the sliding weight on the platform balance add 
 its scale-indication to the right pan or to the left? How 
 can it give correct results if the zero is not at the centre 
 of its scale? 
 
36 THEORY OF MEASUREMENTS 31 
 
 4. What disadvantage is there if the finger-tip that 
 closes the top of the measuring pipette is not dry? 
 
 5. The specific gravity of a substance is denned as the 
 ratio of its mass to the mass of an equal volume of water, 
 i. e., the relative density compared with water as a stand- 
 ard. How do the specific gravity and the density of a 
 substance compare if the density of water is 1? If it is 
 a little less than one? 
 
 6. If specific volume, v, is defined for any substance as 
 the volume per gram of mass, what will the equation be 
 that shows the relationship between p (density) and u? 
 
 7. If 10 cm. = 4 inches make a rapid mental calcula- 
 tion of the C.G.S. length of 12 inches. Of 40 inches; of 
 10 inches; of 3 feet; of 7 inches. 
 
 8. State the distance of 10 kilometres as the nearest 
 whole number of miles. State 6 miles as the nearest 
 whole number of kilometres. 
 
 9. Reduce 5 pounds per square foot (pressure) to gm. 
 per cm 2 . Reduce x lbs./ft. 2 to gm./cm 2 . Write in 
 your notebook directions for reducing 1 gros (weight) per 
 square pouce (length) to grams per square centimetre, or 
 for reducing prices in roubles per pood to cents per 
 kilogram, and submit them to your instructor for ap- 
 proval. 
 
III. ANGLES AND CIRCULAR FUNCTIONS 
 
 Apparatus. A pair of dividers; a pencil compass; a 
 protractor; a ruler; a pencil with a fine point. 
 
 32. Unit of Angle. When two straight lines inter- 
 sect in such a way that each is perpendicular to the other 
 the amount of their divergence, is said to be ninety degrees 
 or one right angle. A single degree is then a compara- 
 tively small difference indirection; two lines drawn from 
 the observer toward the opposite edges of the sun's disc 
 include an angle of about half a degree. For the sake of 
 avoiding fractions at a time when decimals were not 
 used, the degree (1) was divided into sixty parts, each 
 called one minute (I'), and a sixtieth of a minute was 
 used as a still smaller unit, one second (!") These three 
 units are still in use all over the world for expressing the 
 size of an angle as a " mixed" denominate number. For 
 example the circular arc which is of the same length as 
 the radius corresponds to an angle of 57 17' 45". 
 
 33. Circular Measure. If the original right angle had 
 been divided and re-divided into tenths or hundredths 
 instead of into ninetieths and sixtieths the resultant 
 units would have been much more convenient for pur- 
 poses of calculation. The question suggests itself, 
 however, as to why the right angle should be arbitrarily 
 chosen for the quantity that is to be subdivided. Why 
 not take the whole circumference or some other amount 
 of angle? As a matter of fact, this arbitrariness is often 
 avoided for scientific purposes by measuring an angle by 
 an entirely different method: The vertex of the angle 
 is taken as a centre, around which an arc of a circle is 
 drawn extending from one side of the angle to the other 
 
 37 
 
38 THEORY OF MEASUREMENTS 34 
 
 (Fig. 5). The length of this arc will depend not only 
 upon the size of the angle but also upon the length of the 
 radius that is used, but the ratio of length of arc to length 
 
 FIG. 5. CIRCULAR MEASURE. The ratio of any arc to its radius 
 is taken as the numerical or circular measure of the angle at the 
 centre. DC/OC = BA/OA = 3/4, approximately, for the angle 
 represented here. 
 
 of radius will depend only upon the size of the angle and 
 so can be used as a measure of it. In the diagram the 
 arc AB seems to be about f as long as the radius OA, 
 the arc CD is likewise f as long as the radius OC, and the 
 circular measure of this particular angle is accordingly 
 f, or 0.75. 
 
 Draw an angle which is somewhat less than a right 
 angle. Draw its arc, and measure the curved line as well 
 as you can with an ordinary ruler. Measure the radius 
 also, and calculate the circular measure of the angle. It 
 will probably turn out to be about 1.4. Is the circular 
 measure of a right angle equal to just 1.5? State why. 
 
 34. Numerical Measure of an Angle. Since the 
 size of an angle is denned as the quotient arc divided by 
 radius it follows that this amount is not a number of 
 centimetres or of any other arbitrary units but is a pure 
 number. If the arc measures 6 centimetres and the 
 
Ill ANGLES AND CIRCULAR FUNCTIONS 39 
 
 radius is 3 centimetres the size of the angle is the abstract 
 number 2, not 2 centimetres; and if both had been 
 measured in inches the quotient would still have been 
 merely the number 2, the arc being two times the radius, 
 not two centimetres times the radius. The expression 
 numerical measure of an angle has the same meaning 
 as circular measure of an angle, and denotes the way in 
 which the size of an angle is always expressed for the- 
 oretical purposes. One of the chief advantages of this 
 method lies in the simplification which it causes. Just 
 as the foot-pound system of measures makes the density 
 of water about 62 lb./ft 3 , and hence makes specific 
 gravity approximately equal to density divided by 62, 
 instead of merely to density as in the C.G.S. System, so 
 angular velocity would be represented by 57.28 v/r 
 instead of v/r, and such higher mathematical expressions 
 as D x sin x = cos x and D x cos x = sin x would become 
 D x sin x = .01746 cos x and D cos x = TT sin x/lSO 
 if the angles were measured in degrees instead of nu- 
 merically. 
 
 A suggestion of the reason why an angle ought to be 
 expressed as an abstract number instead of in terms of 
 a unit may be obtained by imagining a length of one 
 centimeter and an angle of 45 degrees to be drawn on a 
 sheet of paper and observed through a magnifying glass. 
 The centimetre may appear to be enlarged to a length 
 of two centimetres, but the angle of 45 degrees does not 
 become an angle of 90 degrees; it remains exactly the 
 same size as before. The same thing is of course true 
 of an abstract number: with a very slight magnification 
 two objects may both be made to look larger, but no 
 amount of magnifying power will make them look like 
 three. 
 
 35. The Angle TT and the Unit Angle. Draw an angle 
 
40 THEORY OF MEASUREMENTS 36 
 
 of 180 and its arc, viz., a semicircle. Obviously its 
 numerical measure, semicircumference divided by semi- 
 diameter, is the same fraction as the ratio of the whole 
 circumference to the whole diameter, which is denoted 
 by the symbol TT and is approximately equal to 3.1416. 
 It is also clear that an angle which extends entirely 
 around a point, that is, four right angles or 360 degrees, 
 must have 2ir for its numerical measure; one right angle 
 must be equal to 7r/2, 7r/4 is the same as 45 degrees, etc. 
 If 180 = TT the numerical measure of one degree must 
 be 7T/180 and the degree-measure of an angle that is 
 numerically equal to one must be 180/Tr. 
 
 Find the number of degrees in the unit angle by writing 
 the value of TT, carried out to at least eight or ten decimal 
 places, and using the method of abridged division to 
 find out how many times it is contained in 180. The 
 latter number should be written 180.00000. . . with as 
 many ciphers as may be necessary. 
 
 Practice translating such numbers as the following into 
 degrees until it can be done without any hesitation: 
 
 ITT = ? 27T = ? JTT = ? 47T = ? 7T/3 = ? f 7T = ? 7T/4 = ? 
 
 7T/7T = ? JTT = ? If an angle is numerically equal to 
 three is it greater or less than 180? 
 
 Practice translating the following numbers of degrees 
 into circular measure until it can be done fluently: 
 90 = ? 360 = ? 45 = ? 3 right angles = ? 180 = ? 
 30 = ? 60 = ? 1 = ? 270 = ? 57.296 = ? 
 
 36. The Protractor. A protractor is a scale of angles 
 just as a graduated ruler is a scale of lengths. It consists 
 essentially of a zero line on which is a point that rep- 
 resents the vertex of the angle, and a curved scale of 
 short lines so placed that if they were sufficiently long 
 each one would pass through the vertex and make an 
 angle with the base line equal to the number of degrees 
 with which it is marked. 
 
Ill 
 
 ANGLES AND CIRCULAR FUNCTIONS 
 
 41 
 
 Examine the protractor and its scale of degrees. 
 Notice that the centre toward which the slanting lines 
 converge must be on the line joining and 180 and 
 hence must be at the top of the notch shown in the figure. 
 It is always at the corner that is made rectangular, the 
 
 FIG. 6. PROTRACTOR. The essential parts of a protractor are 
 the zero line, the central point, and the scale showing where a line 
 must pass to form the second side of any required angle. 
 
 other being obtuse or rounded. To draw a line which 
 shall make a given angle with a given line at any particu- 
 lar point the protractor is placed so that its zero coincides 
 with the given line and its centre with the given point. 
 A fine line or dot is then made opposite the required 
 number of degrees on the scale, the protractor is re- 
 moved, and a ruler is used to draw a straight line through 
 the dot and the particular point on the given line. 
 
 Use the protractor to draw a triangle of any convenient 
 size and of such shape that its three angles are 7r/2, 7r/3, 
 and 7T/6. Draw another so that its angles are ir/4, ir/4, 
 and 7T/2. 
 
 37. The Diagonal Scale. If the protractor is pro- 
 
42 THEORY OF MEASUREMENTS 38 
 
 vided with what is known as a diagonal scale notice that 
 at the top of this scale there is a horizontal line 
 which is divided into centimetres or inches and that one 
 division at the end of the scale is divided into tenths. 
 Decide for yourself how it is that any length, such as 7.4, 
 within the limits of the total length of the scale, can be 
 found already laid off as a single continuous stretch of 
 the base line, the tenths being measured from the proper 
 point to the junction of the tenths' scale and the units' 
 scale, and the units then extending onward the required 
 distance beyond the junction point. Notice that the 
 tenths' divisions are prolonged downward so as to cut diag- 
 onally across parallel horizontal lines and shift a single 
 tenth to one side while dropping ten lines downward. 
 This means that on the first level below the base line 
 their shift will be only one hundredth, and on successive 
 levels will be .02, .03, ... .10, as the student can easily 
 prove for himself by means of the principles of similar 
 triangles. Accordingly, any number of units, tenths, 
 and hundredths can be found marked off along the proper 
 level. Thus a distance of 7.43 will be found on the 
 third level between the same diagonal and vertical line 
 as mark off 7.4 on the base line. A pair of dividers should 
 be used to span an unknown length and then transfer 
 it to the diagonal scale for measurement, or to take a 
 required length from the scale for the purpose of laying 
 it off on paper. They should be held rather flat against 
 the scale, not perpendicularly, so as to avoid marring it. 
 38. Measures of Inclination. If a slanting line inter- 
 sects a level line the inclination of the former may be 
 measured either by the size of the angle between them 
 or by the rise of the inclined line per unit of level distance 
 (see 17). The second of these two amounts is said to 
 be the tangent of the first one; thus 2/5 = tan 22. This 
 
Ill 
 
 ANGLES AND CIRCULAR FUNCTIONS 
 
 43 
 
 relationship can easily be generalized so as to include 
 cases in which neither side of the angle is horizontal: 
 From any point on one side of an angle draw a straight 
 line perpendicular to the other side; this will complete a 
 right-angled triangle, and can be done in all cases where 
 the angle is acute (i. e., between and 90 in size). The 
 tangent of an acute angle of any right-angled triangle is 
 defined as the ratio of the side opposite the angle to the 
 side that is adjacent to the angle, the word "side" being 
 used here to indicate either of the right-angled sides, not 
 the hypothenuse. 
 
 In the diagram (Fig. 7) the angle G is an acute angle of 
 
 FIG. 7. ANGLES AND THEIR TANGENTS. The ratio of any vertical 
 side to the corresponding base is a number which is called the 
 tangent of the acute angle at the base. EF is the same fraction of 
 GF as AB is of GB, namely about 1/3. PQ is about twice OQ, 
 and RS is about twice OS. Accordingly tan G = 1 ,/3 and tan O = 2. 
 
 each of three different right-angled triangles. Measure 
 the height and the base of any one of these and calculate 
 the tangent of the angle G. The reason for the name 
 "tangent" may be understood by noticing that the line 
 drawn between the two sides of the angle and tangent to 
 one end of the arc will be numerically equal to the ratio 
 in question if the radius GD or GE is of unit length, for 
 then CD/GD = CD/I = CD. 
 
44 THEORY OF MEASUREMENTS 39 
 
 The inclination of two lines that form an acute angle 
 of a right-angled triangle may also be measured by the 
 ratio of the opposite side to the hypothenuse. This is 
 called the sine of the angle; for example, in Fig. 7, the 
 ratio of EF to GE, or, what amounts to the same thing, 
 EF/GD, is the sine of the angle G, often abbreviated to 
 "sin G." It is evident that these definitions of sine 
 and tangent agree with those that have been previously 
 given for the " gradient" and ''per cent slope" of an 
 angle between a sloping line and a horizontal one. 
 
 39. Use of a Table of Tangents. Close to the bottom 
 of the next unused page of your notebook draw a fine 
 horizontal line having a length of either 20, 25, or 50 of 
 the squares made by the cross-lines of the paper. Draw 
 it directly on one of the horizontally ruled lines, pref- 
 erably so that it extends nearly across the page. Call 
 its length unity (1); it may not be one footer one deci- 
 metre, but it is to be considered as having a length of 
 just one arbitrary unit, which need not be given any 
 name. Mark a figure 1 under its right-hand end, a 
 figure at the left-hand end, and the scale of numbers 
 0.1, 0.2, 0.3, etc., at intervals of two, or two and a half, 
 or five, squares, as may be required by the length of the 
 line. From the right-hand end of this base line draw a 
 fine line perpendicular to it, extending it as far as the 
 top of the page. Beginning with zero at the junction 
 with the base line lay off a similar scale along the vertical 
 line. Do not number the successive squares of the paper 
 1, 2, 3, etc., but see that this scale indicates the same 
 proportions as are given by the horizontal one. Turn to 
 the table of circular functions in the appendix; look for 
 the number 10 in the column headed DEG and notice 
 that the number opposite it in the column TAN is 1763; 
 this means that the tangent of 10 is .1763. On the 
 
Ill ANGLES AND CIRCULAR FUNCTIONS 45 
 
 vertical line just drawn make a small mark at a height 
 of .1763 above the base line according to the vertical 
 scale already made. In the same way lay off tan 20 
 on the same scale. The decimal points are omitted from 
 the table; but the tangent of a large angle is obviously 
 greater than that of a smaller angle, and the table shows 
 that for successive degrees the tangent increases gradu- 
 ally and quite regularly. Notice that the tangent of 45 
 is 1.000000. . ., and that angles and tangents larger than 
 this must be sought for above the abbreviations DEG and 
 TAN which are at the bottom of the page. If there is 
 any trouble in understanding how the table is arranged 
 use it to verify the following equations before proceeding 
 further : 
 
 Tan 1 = .0175; tan 2 = .0349; tan 6 = .1051; 
 tan 44 = .9657; tan 45 = 1.000; tan 46 = 1.036; 
 tan 84 = 9.514; tan 85 = 11.43; tan 89 = 57.29. 
 
 Lay off tan 30, tan 40, tan 50, etc., as far as the 
 length of the vertical line will allow; then draw slanting 
 lines from each of these points to the left-hand end of 
 the base line. With the protractor test the angles formed 
 in order to make sure that they are accurately 10, 20, 
 30, etc. If mistakes have been made repeat the con- 
 struction on the next page; do not correct the first 
 diagram by erasures. 
 
 40. Experimental Determination of Sines. With the 
 base line as a radius and its left-hand end as a centre draw 
 an arc on the diagram that has just been made, extending 
 it from to 90. Complete the series of angles as far 
 as 90 by laying off successive ten-degree arcs or chords 
 with a pair of dividers. Find the point where the line 
 whose slope is 10 intersects the arc, and note carefully 
 the vertical distance from the base line up to this point, 
 but do not draw a vertical line. If the diagram has been 
 
46 
 
 THEORY OF MEASUREMENTS 
 
 41 
 
 angle 
 
 sine 
 
 
 10 
 20 
 30 
 40 
 
 
 .175 - .001 
 .345 - .003 
 .500 + .000 
 .640 + .003 
 
 carefully drawn with a sharp-pointed pencil the value 
 should come out 0.174, being measured of course in terms 
 of the figured scales, not in terms of the small ruled 
 squares. Notice that this number is less than tan 10, 
 and that it corresponds to the ratio EF/GD in Fig. 7, 
 and hence is the sine of the angle 10. In the same way 
 measure sin 20, sin 30, ... sin 90, or as many of them 
 as may be directed by the instructor, and tabulate the 
 results in the first two columns of a three-column table. 
 
 Then turn to the table of cir- 
 cular functions, find the true 
 numerical values of the sines 
 from the column headed SIN, and 
 correct your measured sines by 
 adding a third column as shown 
 here; the sine of 40, for ex- 
 ample, appears to have been 
 measured as .640 and then 
 found to be actually .643, 
 whence the correction of + .003 
 in the third column. After 
 your table of sines has been 
 
 completed find the angle whose sine has the largest cor- 
 rection and divide this correction mentally by the true 
 value of the sine in order to find the relative error of the 
 measurement. Thus, if the quotient is about 3/600 
 = 1/200 = .005 your measurement has an error that 
 amounts to three parts out of a total of 600, which is the 
 same as 5 per thousand, or f percent. Call your error 
 " moderate" if it is anywhere between 0.3 percent and 
 1 percent; call it " large" if greater than 1 percent, and 
 " small" if less than 0.3 percent. 
 
 41. Definition of Function. Up to the present point it 
 has been assumed that an angle of any size from to 
 
 TABLE OF SINES. The 
 second column shows the 
 measured value; the third 
 column is the amount of 
 change that must be made 
 to obtain the true value. 
 
Ill ANGLES AND CIRCULAR FUNCTIONS 47 
 
 7T/2 or 90 will have a tangent which is a definite number 
 for each definite size of angle. A quantity which can be 
 assumed to have different sizes, whether restricted to a 
 certain range or not, is called a variable; and a second 
 quantity, which in general has a definite value for each 
 particular value of the first, is said to be a function of that 
 variable. For example x 2 3x + 2 is called a function 
 of x, because it has a definite value for any definite value 
 that may be assigned to x. Similarly sin x, tan x, V #, 
 logarithm of x, x n , a x , are all functions of x, as is also 
 any other algebraical formula which involves x. 
 
 42. The Cosine of an Angle. Another function of 
 an angle which is frequently useful is the cosine. In a 
 right triangle, such as was used in defining the tangent 
 and the sine of an angle, it is the ratio of the adjacent 
 side to the hypothenuse; the ratio of GF to GE, in Fig. 7, 
 is the cosine of the angle G, or, as it is usually abbreviated, 
 cos G = GF/GE. 
 
 43. Circular Functions. The sine, cosine, and tangent 
 are included under the general term circular functions, a 
 phrase which includes also three other functions of an 
 angle which are not so frequently used. These are the 
 cotangent, the secant, and the cosecant, and may be de- 
 fined by the equations cot x = I/tan x, sec x = I/cos x, 
 and cosec x = I/sin x. Another set of definitions, which 
 are perhaps more interesting and easier to remember, may 
 be obtained by the use of a circle diagram and extended 
 so as to include angles greater than 90 : For the sake of 
 uniformity the angle is so placed that one of its sides 
 extends out horizontally to the right. A zero angle 
 would have its second side coincident with the first, and 
 larger angles may be supposed to have been generated 
 by rotating the second side through the required angular 
 distance from the first. The usual convention is that 
 
48 
 
 THEORY OF MEASUREMENTS 
 
 43 
 
 the direction of rotation shall be counter-clockwise, i. e., 
 in the opposite direction to that in which the hands of a 
 clock turn. A circle whose radius is unity is drawn around 
 the vertex of the angle, as in Fig. 8, which shows an angle 
 
 of somewhat less than 
 45. A line (DB) is 
 drawn upward from the 
 right-hand end (D) of 
 the horizontal radius, 
 and a perpendicular 
 (AC) is dropped from 
 the end of the inclined 
 radius (A) to the base 
 line (OD). Then the 
 length of the tangent 
 to the circle (DB) is 
 the numerical tangent 
 of the angle (0), the 
 line (OB) that cuts 
 the circle is the secant 
 
 FIG. 8. CIRCULAR FUNCTIONS. 
 DB = tan DO A; AC = sin DO A; 
 OC = cos DO A; OB = sec DO A. 
 The radius is supposed to be of 
 unit length. 
 
 (Lat. secans, cutting), 
 and the perpendicular 
 (AC), which cuts off a rounded hollow of the figure, is 
 called the sine (Lat. sinus, a bay). The cosine is the 
 sine of the complementary angle (e. g., Z OAC', if x is 
 acute cosx = sin (90 x)), and the cotangent and cose- 
 cant are similarly the tangent and secant respectively 
 of the complementary angle, two angles being called 
 complementary when their sum is 7r/2 or 90. As OC 
 is considered to be a positive amount when measured to 
 the right of it is only natural to consider the cosine as a 
 negative number when C is to the left of 0. Likewise 
 CA or DB must be negative if below the base line instead 
 of above. AO and DO are to be produced if necessary. 
 
Ill 
 
 ANGLES AND CIRCULAR FUNCTIONS 
 
 49 
 
 If tan 45 = + I and sin 45 = + .707 what are the 
 values of tan (90 + 45) and sin (90 + 45)? Ans.: 
 Tan 135 = - 1; sin 
 135 = + 1. 
 
 What is the cosine of 
 135? What are the 
 tangent, sine, and cosine 
 of 180 + 45? Of 270 
 + 45? 
 
 44. Generalized Idea 
 of Angle. For some 
 purposes the term angle 
 may be defined so as to 
 include only angles less 
 than 7T/2. For other 
 purposes obtuse angles 
 (i. e., up to TT or 180) 
 may need to be in- 
 cluded. The circular 
 functions have been "al- 
 ready defined for angles 
 
 FIG. 9. FUNCTIONS OF ANY 
 ANGLES. The cosine is negative if 
 C is to the left of 0; the sine and 
 tangent are negative if A and B 
 are below the level of 0; DO and 
 AO are produced if necessary. 
 
 of all sizes up to 2ir 
 (6.28, or 360), but it is 
 obviously unnecessary to 
 stop at this figure. By 
 supposing the line OA 
 in Fig. 8 to have made more than a complete revolution 
 it can be seen that the sine, cosine, tangent, etc., of 
 360 + 40 must all have the same values as the corre- 
 sponding functions of 40. In general, any circular 
 function of any angle x is equal to the same function of 
 2ir + x, or of 4ir + x, or of 2mr + x if n is any whole 
 number. A negative angle is of course one that is gener- 
 ated by a clockwise rotation from the position of the 
 5 
 
50 THEORY OF MEASUREMENTS 45 
 
 base line; thus the angle shown in Fig. 9 may be con- 
 sidered either as + 240 or as 120, and obviously the 
 circular functions of any negative angle x have the same 
 values as those of the positive angle 2ir x. Negative 
 angles have to be considered occasionally, just as negative 
 heights or lengths need to be. Angles larger than + 2?r 
 commonly come under consideration in connection with 
 rotatory motion. Such objects as a spinning top, a 
 fly-wheel, a planet, do not commonly move through an 
 angle less than 360 and then stop, but their angular 
 motion may be of almost any amount according to the 
 extent of time occupied by it. 
 
 45. Questions and Exercises. 1. How can it be 
 proved that the ratio which gives the numerical measure 
 of an angle will have the same value whether the radius 
 is long or short? 
 
 2. Translate 135 into circular measure. Make an 
 approximate mental calculation of the number of degrees 
 in an angle whose numerical measure is 6. 
 
 3. After drawing a triangle whose angles were meant 
 to be 7T/2, 7T/3, and 7r/6 what would you do if you found 
 one of its angles inaccurate but the other two correct? 
 
 4. The tangent of 80 is given in the table as 5671. 
 Where do you place its decimal point and why? 
 
 5. The steepness of a slope is the characteristic which 
 corresponds to its sine; a larger sine means a steeper slope. 
 What characteristic can you name that will correspond 
 to its cosine? 
 
 6. Any radius of a rotating wheel describes an angle 
 which increases steadily from to (say) 407r. Explain 
 how the sine of this variable angle behaves during the 
 same time. 
 
 7. What is the approximate value of tan ( 91)? Of 
 tan (+89)? Of tan (- 89)? 
 
Ill ANGLES AND CIRCULAR FUNCTIONS 51 
 
 8. What is the approximate value of the tangent of the 
 angle A in Fig. 7? How does it compare with the value 
 of the tangent of the angle (r? What relationship is 
 there between tan T and tan 0? 
 
 9. The cotangent of any angle, A, has been denned as 
 the reciprocal of tan A, and also as the value of tan 
 (90 A). Prove that these definitions are identical 
 if A < 90 by drawing a right triangle having A for one 
 of its acute angles. 
 
 10. Draw a right triangle and prove that, if A < 90, 
 sin 2 A + cos 2 A = 1 using the theorem of Pythagoras that 
 the square of the hypothenuse is equal to the sum of the 
 squares of the other two sides.* 
 
 11. Make A an acute angle of a right triangle and 
 prove that tan A = sin A/cos A. 
 
 12. Using a circle diagram prove that in general for an 
 angle of any size, x, 
 
 (a) tan x = sin x/cos x 
 
 (6) sin 2 x + cos 2 x = I 
 
 (c) tan (90 - x) = I/tan x 
 
 (d) cos x = sin (90 - x) = sin (90 + x) 
 
 13. What synonyms have you already learned for the 
 sine and the tangent of an angle? 
 
 * Sin 2 A is the conventional abbreviation of (sin A) 2 , not of 
 sin (sin A); similarly sin 3 A, etc. Sin" 1 A, however, is always used 
 to denote the angle whose sine is A, not I/sin A. 
 
IV. SIGNIFICANT FIGURES 
 
 Apparatus. Scale of centimetres and millimetres; 
 card or strip of paper; circular brass measuring disc. 
 
 46. Estimation of Tenths. It sometimes happens 
 that a measurement requires but a slight degree of 
 accuracy, and time and trouble can be saved by making 
 it only roughly. As a general principle, however, it is 
 advantageous to make all measurements as accurately 
 as possible. Thus, measurements of length made with 
 the metre stick should be expressed not merely to the 
 nearest centimetre but to the nearest millimetre or tenth 
 of a centimetre. This is not the best that can be done, 
 however. After noticing how each centimetre of the 
 scale is divided in half by a long line and each half is 
 subdivided into fifths by four short lines, thus indicating 
 tenths of a centimetre, it is not difficult to imagine each 
 of the millimetre intervals of the scale divided in the 
 same way into tenths of a millimetre and to make a fairly 
 good estimate of just how many of these parts are in- 
 cluded in the length that is to be measured. Experienced 
 observers even attempt to make a mental subdivision 
 into hundredths of the smallest intervals on a graduated 
 scale, and find that it is only occasionally that one man's 
 estimate will differ from another's by more than one or 
 two hundredths, but for the beginner even the estimation 
 of tenths will be a rather uncertain process. To gain 
 proficiency it will be found better to begin with larger 
 subdivisions, such as a scale of centimetres that has no 
 millimetre marks. The position of a mark placed at 
 random on such a scale can be estimated mentally and 
 the accuracy of such a determination can then be tested 
 
 52 
 
IV SIGNIFICANT FIGURES ^ 53 
 
 by actual measurement with a more finely divided scale. 
 47. Practice in Estimating Tenths. Draw a short 
 line at right angles to the edge of a card or slip of paper 
 (Fig. 10) and hold this edge on a scale of centimetres and 
 millimetres in such a way 
 that the smallest gradua- 
 tions are hidden but the 
 marks indicating centime- 
 tres and half-centimetres FIG. 10. ESTIMATING TENTHS 
 are visible Notice the OF A CENTIMETRE. The card is 
 half-centimetre space in laid on the scale at random, but 
 
 in such a wav as to hide the 
 which the cross line comes gmall dividing lines The loca _ 
 
 and mentally divide it into t ion of the arrow, in millimetres, 
 five equal parts by four im- is then guessed, and afterward 
 aginary lines so as to make verified by sliding the card down- 
 
 an estimate of the location ward enou e h to ex P se the whole 
 
 ,, , scale, 
 
 of the line on the card. 
 
 For example, in the figure the arrow seems to be either 
 three fifths or four fifths of the way from the scale- 
 division 19.5 to the division 20.0, making its position 
 19.8 or 19.9, but it may be difficult to decide which of 
 these numbers is the nearer without actual measurement. 
 In practice, however, the estimate should be written 
 down and the card should then be allowed to slide care- 
 fully across the ruler until the millimetre scale is just 
 exposed. The position on the scale which the line on the 
 card occupies will then be ascertained and should be 
 written in the notebook beside the previous estimate. 
 The statement should be correct to the nearest milli- 
 metre, without any effort to decide upon fractions of a 
 millimetre (see 23 and 24). 
 
 Make ten such estimates with the card placed any- 
 where along the scale at random, and tabulate the de- 
 terminations-and the verifications in two parallel columns. 
 
54 THEORY OF MEASUREMENTS 48 
 
 Then hold the card a trifle higher, so as to hide the half- 
 centimetre graduations as well as the millimetres and 
 make twenty more determinations. These will also be 
 estimates of the nearest millimetre, but will require the 
 more difficult process of deciding upon tenths of a whole 
 centimetre instead of fifths of a half-centimetre. 
 
 If the scale that is used in the foregoing exercise is on 
 the lower edge of a metre stick there will probably be a 
 duplicate scale along its upper edge, with the help of 
 which it would be easy to make each estimate absolutely 
 correct before verifying it. This furnishes a good illus- 
 tration of the statement that the student should aim to 
 learn rather than to do (2). Finding out the correct 
 position of a line over a hidden scale is something that is 
 utterly useless to him or to anyone else ; it is the learning 
 to divide a centimetre into imagined tenths that will be 
 valuable to him later when it becomes necessary for him 
 to divide a millimetre into tenths without aid and without 
 the possibility of later verification. In making any kind 
 of a measurement care should be taken to avoid any extra- 
 neous influences, or bias or prejudice of any kind. In 
 the present case, the upper scale on the metre stick 
 should be kept covered in some way unless the student is 
 sure of his ability to disregard it completely while making 
 his estimations. 
 
 48. Mistakes in Estimating Tenths. Omitting the 
 preliminary estimate of fifths, examine your table of 
 estimated tenths closely and find out what kind of error 
 you are most apt to make. Some students find it hardest 
 to estimate 0.3 and 0.7 correctly; others have almost a 
 uniform tendency to read a position like 12.0 as either 
 11.9 or 12.1. The latter mistake is due to the fact that a 
 minute deviation from the position of a visible graduation 
 is very easily noticed and there is a tendency to consider 
 
IV SIGNIFICANT FIGURES 55 
 
 it as a single tenth. Of course if it amounts to more than 
 half of a tenth this is correct; but if it is less than half a 
 tenth it should be considered as 0.0 instead of 0.1. The 
 same bias may even cause a tendency to read 0.1 and 
 0.9 as 0.2 and 0.8. On the other hand there may be just 
 the opposite error if the graduated lines are rough or 
 coarse, unless the observer is careful to estimate from the 
 imaginary centre of such a line instead of from its margin. 
 
 If a definite kind of error is evident from a study of 
 your table see if it can be overcome when making another 
 short series of determinations. -Then draw a second line 
 on the card, place the latter in position as before, estimate 
 both points, and find the distance between them by sub- 
 traction. This is the customary method of measuring a 
 length, and is preferable to making one line coincide 
 exactly with a scale division and estimating only the 
 other one, in spite of an obvious additional source of 
 error. 
 
 49. Value of TT. An experimental determination of the 
 value of the constant, TT, can be made by rolling the brass 
 disc along a metre stick to find the length of its circum- 
 ference, then measuring its diameter and calculating 
 the ratio. Hold the disc loosely at its centre, using the 
 thumb and forefinger only. Start it with its marked 
 radius on some definite graduation of the scale and roll it 
 in a straight line until the radius again comes vertically 
 down on the scale. Read this second position, remember- 
 ing not to be satisfied with the nearest millimetre (0.1 
 cm.) but to make as good an estimate as possible of tenths 
 of a millimetre in order that the circumference may be 
 correctly measured to a hundredth of a centimetre. Use 
 the metre stick to measure the diameter of the disc with 
 the greatest possible care, avoiding the end of the stick, 
 which may be a little worn so that it does not represent 
 
56 THEORY OF MEASUREMENTS 51 
 
 precisely 0.00 cm. or 100.00 cm. Find the value of IT 
 by dividing circumference by diameter, and in this partic- 
 ular case using the unabridged method of division and 
 carrying out the result until it has two or three figures 
 that are different from the theoretical value of IT, which 
 is 3.141592653589793238462643383279502884197169399. 
 Finally round off ( 23; 24) both your result and the true 
 value to the same number of places, choosing that 
 number so that your result will show just one incorrect 
 figure; for example, 3| (or 3.1428571) will have just one 
 wrong figure if it is rounded off to 3.143, because the 
 theoretical value will round off to 3.142; while 3.14234567 
 should be rounded off to 3.1423 to compare with the 
 correct value 3.1416. 
 
 50. Physical Measurement. The operation of mak- 
 ing a measurement is merely counting; it is the deter- 
 mination of how many units of a certain kind are required 
 in order to be equal to a given quantity of the same kind. 
 But while a count such as a census of the number of 
 individuals in a town must give a perfectly definite whole 
 number it usually happens that a physical measurement 
 will not give a whole number, or even a commensurable 
 number except as the result of an error, and successive 
 repetitions of a measurement will give a number of dif- 
 ferent apparent values. (Try it; measure the circum- 
 ference of the 7r-disc a second time.) Accordingly, any 
 numerical statement of a measurement must be merely 
 an approximation to an unknown true value, and so 
 will be either indistinguishably correct or perceptibly 
 incorrect accoiding to how closely it can be examined. 
 
 51. Ideal Accuracy. The average student is liable to 
 have more or less difficulty in grasping the idea that 
 accuracy is always a relative matter and absolute precision 
 of measurement is an impossibility. This is usually 
 
IV 
 
 SIGNIFICANT FIGURES 
 
 57 
 
 because he has had very little practice in careful measure- 
 ment and at the same time his previous study of arith- 
 metic has emphasized a condition of infinite accuracy of 
 numerical values. Such a number as 12.5 has been 
 supposed not only to mean the same thing as 12.50 but 
 also to be equal to 12.500000 ... to an unlimited number 
 of decimal places. This is quite proper and satisfactory 
 as long as one realizes that he is dealing with imaginary 
 quantities, or perhaps it would be better to speak of them 
 as ideal quantities, perfections of measurement which 
 have no more reality of existence than the point, line, 
 plane, or cube, of geom- 
 etry. The smoothest sur- 
 face of a table does not 
 come as near to being a 
 plane as does the surface 
 of an " optically worked" 
 block of glass or a " Whit- 
 worth plane," and even 
 the smoothest possible sur- 
 face can be magnified so 
 as to show that it contains 
 irregularities everywhere. 
 Perhaps if it were magni- 
 fied enough we could see 
 that its shape would not 
 even remain constant, but 
 individual molecules would 
 be found swinging back 
 and forth or possibly es- 
 
 (3 
 
 FIG . 1 1 . DIAGRAMMATIC CROSS- 
 SECTION. A metal cube, greatly 
 magnified, to show that there is no 
 plane surface of contact between 
 the metal and the air above it. 
 
 caping from the surface. 
 A geometrical plane cer- 
 tainly corresponds to nothing in reality, and perfect ac- 
 curacy of number is just as much an imaginary concept. 
 
58 THEORY OF MEASUREMENTS 52 
 
 52. Decimal Accuracy. If 12.5 cm., as a measurement, 
 does not mean the ideal number 12.500000000. . . to an 
 infinite number of decimal places what does it mean? 
 As different measurements are likely to be made with 
 different degrees of accuracy the universally adopted 
 convention is merely the common-sense one that the 
 statement of a measurement must be accurate as far as it 
 goes; and it should go far enough to express the accuracy 
 of the determination. Thus "12.8 cm." means a length 
 that is nearer to precisely 12.800. . . than to precisely 
 12.7 or 12.9 cm., i. e., that its "rounded-off " value would 
 be 12.8 cm., not 12.7 or 12.9. If a length is written 
 "12.80 cm.," however, this implies that the stated meas- 
 urement is nearer to this same precise 12.8 or 12.80 or 
 12.800000. . . than it is to either 12.79 or 12.81 cm., in 
 other words, that it has been measured to hundredths 
 of a centimetre and found to be between 12.79J and 
 12.80| cm., so that it can properly be rounded off .to 
 12.80. The other description, "12.8 cm.," means 
 between 12.7| and 12. 8J; it states nothing about hun- 
 dredths of a centimetre, and can correctly represent any 
 lengths between the limits just given; for example 
 12.75, 12.76, 12.77, 12.78, 12.79, 12.80, 12.81, 12.82, 
 12.83, 12.84, or 12.85, for each one of these could be 
 rounded off to 12.8. To write such a length of 12.8 cm. 
 in the form "12.80 cm." would be to violate the rule 
 that a statement should be accurate as far as it goes, 
 for it would go as far as hundredths (stating that there 
 were eighty of them), and the chances are ten to one 
 that it would be one of the other numbers of hundredths 
 given above. On the other hand, if an observer de- 
 termined a length to be 12.80 cm., that is, if he measured 
 the length as 12 cm. + 8 tenths + hundredths if 
 he looked for hundredths and established the fact that 
 
IV SIGNIFICANT FIGURES 59 
 
 there were none of them, then to state the measurement 
 only as 12.8 cm. would not be doing justice to his own 
 accuracy, for he would imply that the correct number of 
 tenths was merely known to be nearer 8 than 7, namely 
 greater than 7.5, whereas he had already found it to be 
 nearer 8 than 7.9, namely greater than 7.95. 
 
 When a carpenter says "just 8 inches" he probably 
 means "nearer to 8f than to 8| or 7f inches," a sixteenth 
 of an inch one way or the other being unimportant. 
 When a machinist says "just 8 inches" he may mean 
 "nearer to 8-fa than to 7ff or to 8^ ," a half-sixty-fourth 
 or hundred-and-twenty-eighth of an inch being negligible 
 to him. When another person says "just 8 inches" we 
 must know what kind of materials he works with before 
 we can tell the meaning of his word "just." If decimal 
 subdivisions were everywhere used the carpenter's eight 
 inches would probably mean 8.0 while the machinist's 
 would mean 8.00; for one man "8" would mean "between 
 7.950 and 8.050" while for the other it would mean 
 "between 7.995 and 8.005." It is for the sake of avoiding 
 such ambiguities that the scientist has adopted the rule 
 that "8" means "between 7.500 and 8.500"; "8.0" 
 means "between 7.950 and 8.050"; "8.00" means 
 "between 7.995 and 8.005"; "8.000" means "between 
 7.999J and 8.000J"; etc.; in other words : no more figures 
 are to be written down than are known to be correct; and, 
 no figures that are known to be correct should be omitted. 
 This principle is simple enough when it has once been 
 properly comprehended, and after that there is not much 
 danger of the student's "rounding off" a carefully 
 obtained measurement like 2.836 gm. to 2.84 gm. merely 
 for the sake of doing some rounding off. There is a very 
 decided likelihood, however, that he will often forget to 
 write down a final significant zero; if two lengths are 
 
60 THEORY OF MEASUREMENTS 53 
 
 147 mm. and 160 mm. the tendency when writing them 
 in centimetres is to put down 14.7 and 16. If the zero 
 is as important as the seven when writing millimetres 
 the same is equally true when writing centimetres. 
 Suppose the diameter of the ?r-disc is found to be " just 8" 
 centimetres; the measurement should be stated as 8.00 
 cm. if tenths of a millimetre were estimated and none 
 were found, but it should be given as 8.0 cm. if the student 
 read the millimetres but was unable to make an estimate 
 of smaller amounts. There is nothing to show which 
 degree of accuracy was obtained if the diameter is put 
 down as 8 cm. " because it came out just even." 
 
 53. Significant Figures. In the expression 6.2 cm. 
 both the figure 6 and the figure 2 mean something or are 
 significant. In the expression 62 mm. there are likewise 
 two significant figures. When the same length is written 
 in the form .062 metre there is no difference in what is 
 signified, and although the number has three figures it is 
 still said to have only two significant figures; the zero is 
 present merely for the purpose of showing which decimal 
 places are occupied by the six and the two, or, in other 
 words, for fixing the location of the decimal point. If 
 the same length is called 62 thousands of micra or 
 62000 ju there are still only two significant figures, and 
 again the ciphers serve only to show that the six is 
 located in tens-of-thousands' place, i. e., to fix the 
 position of the decimal point. In arable notation, the 
 figures of which a number is composed, except for one or 
 more consecutive ciphers placed at its beginning or end for 
 the purpose of locating the decimal point, are called its 
 significant figures. In accordance with this definition it 
 will be clear that only two of the three figures of 0.75 
 gm. are significant, and only one of the two figures of 
 such an expression as 05c., in which a superfluous zero 
 
IV SIGNIFICANT FIGURES 61 
 
 is sometimes written. Non-significant ciphers occur 
 sometimes on the left, as in the statement that a certain 
 light-wave has a length of .00005086 cm., and sometimes 
 on the right, as when the sun is stated to be 93000000 
 miles from the earth; the first of these numbers has four 
 significant figures, the second has only two. It will be 
 noticed that a number like the last causes trouble in 
 applying the rule of making it " accurate as far as it 
 goes," for only the first two or three figures are known, 
 but eight are needed in order to place the decimal point. 
 A further source of trouble lies in the fact that the last 
 figure which is significant may happen to be a cipher 
 instead of some other digit. If this is to the right of the 
 decimal point the zero is of course written when significant 
 and omitted when not (as explained in 52), but what 
 is to be done when a similar case occurs in which the 
 significant zero occurs to the left of the decimal point? 
 If a building is said to be worth fourteen thousand 
 dollars how is any one to tell whether this means 14 
 thousands of dollars, i. e., nearer to 14 than to 13 or 15 
 of these thousands, or whether it means exactly 14000 
 dollars and no cents, or whether the number of significant 
 figures is not intended to be either two or seven but some 
 intermediate number? Suppose the number of millions 
 of miles from the earth to the sun at some particular 
 time is found to be 93.00; we need two different symbols 
 for our ciphers so that we can write 93,00o,ooo miles 
 to show that the first two ciphers are significant while 
 the last four are not. There is really no reason for 
 using ciphers at all in the last four places, except that it 
 is customary, and it would be better to use some other 
 character, such as g, and write 9300gggg. Neither of 
 these methods is ever used, however, but the same result 
 is achieved by a notation that will be explained later on. 
 
62 
 
 THEORY OF MEASUREMENTS 
 
 53 
 
 The length of an inch has been determined to be be- 
 tween 25.39977 and 25.39978 mm. To state that it is 
 25.40 mm. is correct, because this value is accurate as 
 far as it goes; but to say that 1 inch = 25.39 mm. would 
 be wrong, for the true number of hundredths is nearer to 
 40 than to 39. If it is desirable to use an approximate 
 value, so that rounding off is permissible, how many of 
 the following statements are correct and how many are 
 positively wrong? 1 in. = 25.4 mm.; 1 in. = 25.40 mm.; 
 1 in. = 25.400 mm.; 1 in. = 25.4000 mm. 
 
 Which of the following values of TT are correct, and 
 which are incorrect? 3.141592; 3.141593; 3.141600; 
 3.14160; 3.1416; 3.1415; 3.142; 3.141; 3.15; 3.1; 
 
 FIG. 12. RELATIVE AND ABSOLUTE ACCURACY. The large rec- 
 tangle comes nearer to the shape of a perfect square than the small 
 one does, although the difference in its two dimensions is precisely 
 the same as the difference between the height and the width of the 
 small rectangle. Accuracy or inaccuracy can be considered great 
 or small only relatively to the size of the quantity that is being 
 measured. 
 
 3.142857 (= 22/7); 3. (II Chronicles, chap. 4, v. 2); 
 3.16 (= V10). 
 
 At ordinary room temperature (20 C.) is the density 
 
IV SIGNIFICANT FIGURES 63 
 
 of water equal to 1? Is it 1.0? 1.00? 1.000? To what 
 point must its temperature be lowered in order to make 
 its density 1.0000? (See tables; appendix.) 
 
 54. Relative Accuracy. The diagram (Fig. 12) shows 
 two rectangles which are approximately squares. The 
 difference between the height and the width of the larger 
 one is just the same as the difference between the height 
 and the width of the smaller, yet the small rectangle is 
 obviously a less accurate approximation to the shape of a 
 perfect square than is the large one. This may serve as 
 an illustration of the important general statement that 
 accuracy is a matter of relative amount rather than of 
 absolute amount. A sixteenth of an inch has the same 
 absolute value wherever it occurs, but it is a considerable 
 part of a quarter-inch length while it is relatively insig- 
 nificant in comparison with a whole inch. 
 
 The relative accuracy of a measurement accordingly 
 depends upon two things: how much its absolute dif- 
 ference from the truth amounts to, and how large the 
 measurement itself is. If two points on the earth's 
 surface are found by careful surveying to be 10 miles 
 apart the determination of distance may easily be in 
 error by more than a foot, and even with the most ex- 
 tremely careful triangulation the error is likely to be as 
 much as four inches. An error of a quarter of an inch, 
 however, in measuring the thickness of a door could 
 hardly be made even with the clumsiest of measuring 
 apparatus. It would plainly be misleading to say that the 
 clumsy measurement should be considered more accurate 
 than the careful one on account of i inch being less than 
 4 inches. The only consistent way of looking at the 
 matter is to inquire how large a fraction of the total 
 measurement the error amounts to. Suppose the thick- 
 ness of the door is 1J inches; how large a part of this 
 
64 THEORY OF MEASUREMENTS 55 
 
 measurement is the error of J inch? Obviously it is one 
 sixth of the total or an error of more than 16 percent; 
 while four inches out of a total of ten miles is not nearly 
 a sixth, but is roughly an error of one out of a hundred 
 and fifty thousand, or about six per million, or about 
 .0006 of 1 percent. 
 
 How large an error is an eighth of an inch when half of 
 an inch is being measured (Fig. 12)? How large is a 
 sixteenth of an inch out of a total of an inch and three 
 quarters? 
 
 If a measurement is stated to be 12.8 cm. when it is 
 known to be between 12.750 and 12.850 cm. what is the 
 greatest possible error of the statement? Answer: .05 
 out of a total of 12.75. This is the same as 05. out of 
 1275., or 5 per 1275, or 1 per 255; 1/250 would be 4/1000 
 or .004, so 1/255 must be a little less than .004, i. e.j a 
 little less than 0.4 percent. 
 
 55. Calculation of Relative Errors. The relative error 
 of a measurement does not usually need to be calculated 
 with any very great care. Where numbers are as different 
 as 6 in tens-of-thousandths' place (10-mile survey, above) 
 and 16 in units' place (thickness of door) the location of 
 the decimal point is really more important than the size 
 of the significant figure that occupies either place ; to call 
 the former number "a few ten-thousandths of a per-cent" 
 and the latter "some ten or twenty percent" gives all 
 the information that is needed. This means that a 
 calculation of relative error never needs to be done on 
 paper but can always be worked out as a rough mental 
 calculation. Thus, in the illustration given above, 
 1 foot is 1/5280 of 1 mile, hence it is 1/52800 of 10 miles 
 or roughly about 1/50,000; and 4 inches, being f of 1 
 foot, is /50,000 of the whole distance, or 1/150,000. The 
 denominator of this fraction is about a sixth of a million 
 
IV 
 
 SIGNIFICANT FIGURES 
 
 65 
 
 (since 15 is about | of 100), so 1/150,000 = 6/1,000,000 
 = .000006 = .0006 percent. 
 
 Decide mentally what percent .01 is of 7.23. Ans. : 
 .0015, or .15 per cent. 
 
 What percent of 94.07 is .01? 
 
 56. " Decimal Places " versus " Significant Figures." 
 If a length is stated as 174.2 cm. the inference is that it 
 is nearer to that exact amount than to 174.1 or 174.3 
 cm., namely that its error certainly is not as much as 
 0.1 out of 174.2. This is the same as saying that it is not 
 as much as 1 out of 1742, or 1 out of nearly 2000, or 5 per 
 10000, or .0005, or .05 percent. 
 
 In the following table the left-hand column contains 
 five numbers, all of which are carried out to the same 
 number of decimal places; namely, 
 two. In the right-hand column notice 
 that the same five numbers occur, 
 but each one of them is carried out to 
 the same number (3) of significant 
 figures, no matter how many decimal 
 places there may be. If the accuracy 
 of each of these ten numbers should 
 be worked out in the way that has 
 just been explained, would the num- 
 bers in the left-hand column turn out 
 to be all of approximately equal accu- 
 racy while the right-hand column 
 showed great accuracy for one num- 
 ber and little for another, or would the 
 numbers on the right be the ones that 
 would be about equally accurate while 
 those on the left fluctuated? In other 
 words, is accuracy a matter of decimal places or of sig- 
 nificant figures? 
 6 
 
 7.23 
 
 94.07 
 
 0.52 
 
 428.00 
 
 66.67 
 
 7.23 
 94.1 
 0.522 
 
 428. 
 66.7 
 
 TABLE OF FIVE 
 NUMBERS. In the 
 left-hand column 
 each number has 
 two decimal places 
 but some have more 
 significant figures 
 than others. In the 
 right-hand column, 
 each number has 
 three significant fig- 
 ures but some have 
 more decimal places 
 than others. 
 
66 
 
 THEORY OF MEASUREMENTS 
 
 57 
 
 The table has been repeated on this page and shows the 
 answer to the question. Beside each of the ten numbers 
 its accuracy has been written down, as a percentage, and 
 it will be seen that the numbers in the right-hand column 
 all show about the same degree of accuracy, while those 
 in the left-hand column differ widely. 
 
 .1% 
 
 7.23 
 
 7.23 
 
 .1% 
 
 .01% 
 
 94.07 
 
 94.1 
 
 .1% 
 
 2. % 
 
 0.52 
 
 0.522 
 
 .2% 
 
 .002% 
 
 428.00 
 
 428. 
 
 .2% 
 
 .01% 
 
 66.67 
 
 66.7 
 
 .1% 
 
 TABLE SHOWING ACCURACY OF NUMBERS. Notice that the 
 
 number of decimal places to which a measurement is carried out 
 
 has nothing to do with its accuracy. It is the number of significant 
 figures that determines the matter. 
 
 Turn to the table in the appendix where errors are 
 classified according to their size and write the appropriate 
 word opposite each of the percentages given in the 
 right-hand column of the table on this page. Then do 
 the same way with those in the left-hand column. 
 
 57. Rule for the Relative Difference of Two Measure- 
 ments. The difference between 3.11 and TT (= 3.14) is 
 3 out of 314, and the difference between 3.17 and TT is 
 likewise 3/314, not 3/317; i. e., the numerical error is to 
 be divided by the true or theoretical value rather than 
 by the experimental or erroneous value. It is often 
 desirable, however, to compare two values which are 
 equally good, according to one's available knowledge of 
 them. When there is no standard and no reason for 
 choosing one of the measurements rather than the other 
 the accepted procedure is to divide the difference by the 
 greater value. For example, the numbers 4 and 5 would 
 be said to differ from each other by 20 percent, not 25 
 percent, for the difference divided by the greater number 
 
IV SIGNIFICANT FIGURES 67 
 
 is one fifth, not one fourth. In cases of fairly accurate 
 measurements it is unimportant whether the larger 
 number or the smaller one is taken for the divisor, but 
 for the sake of uniformity it is customary to choose the 
 larger one. 
 
 Apply this rule to your two measurements of an ir- 
 regular area ( 26), obtained by counting squares and 
 by constructing geometrical figures. How much rel- 
 ative difference is there in the results of the two methods? 
 
 58. Accuracy of a Calculated Result. Multiply 65.97 
 by 24.15, using the abridged method of multiplication. 
 Compare your product with that of example (c), 16. 
 It will be noticed that changing the fourth figure of one 
 factor has produced a change in the fourth figure of the 
 product. This means that if only three figures of the 
 factor had been known, the fourth being uncertain, no 
 calculation could have given more than three trust- 
 worthy figures of the product, because the fourth figure 
 would have depended upon the unknown fourth figure of 
 one of the factors. Likewise, in division, if only five figures 
 are known of either the divisor or the dividend there is 
 nothing to be gained by keeping more than five figures 
 in the other one; only five figures of the quotient will 
 have any meaning, and if further figures are obtained 
 by any process of calculation they will be unjustified 
 and misleading. The general rule will be obvious: the 
 result of a multiplication or division will have no greater 
 accuracy than that of the least accurate of the data 
 from which it is obtained. 
 
 59. Accuracy of the Abridged Methods. Remember- 
 ing that the accuracy with which a quantity is expressed 
 depends not upon the number of decimal places but upon 
 the number of significant figures and keeping in mind 
 the fact that the number of trustworthy figures in a 
 
68 THEORY OF MEASUREMENTS 60 
 
 product is the same as the number in its least accurate 
 factor, turn back to your notes on 15 and 16 and ob- 
 serve that the method of abridged division automatically 
 gives just the number of figures in the quotient that are 
 needed if no figures of the dividend are " brought down"; 
 and that abridged multiplication always gives at least 
 as many as are in the shortest factor. It will not give 
 any superfluous figures if the longer factor is used as the 
 multiplier, but will give as many as the longer factor 
 contains if that is used as the multiplicand. Of course 
 the best method is to round off the longer number before 
 beginning the calculation, so that it has no more figures 
 than the shorter one. 
 
 60. Standard Form. To avoid a long string of figures 
 when writing very large or very small numbers it is 
 customary to divide a number into two factors, one of 
 them being a power of ten. Thus, .00000017 and 
 632000000000 are the same as 17 X 10~ 8 and 632 X 10 9 
 respectively. -This notation also makes it possible to 
 write 93000000 unequivocally with either two significant 
 figures or four, as may be desired (see 53) , for it can be 
 put either in the form 9.3 X 10 7 or in the form 9.300 
 X 10 7 . The same value and accuracy for 9.300 X 10 7 
 would be retained just- as well by writing 93.00 X 10 6 
 or 930.0 X 10 5 , but it is customary to choose the power 
 of ten so that the other factor shall have just one sig- 
 nificant figure to the left of the decimal point. The 
 number is then said to be written in standard form. 
 
 Write the following numbers in standard form: 
 2946.3; 632 X 10 9 (ans.: 6.32 X 10 11 ); 17 X 10~ 8 ; 
 25.39978; 0.0073; .007300; 666.6; .001; .0010; 107.42; 
 186000; 2.5400; 3.1416; 9.9942; 2.54gg. Prove that 
 each result is correct by performing the indicated mul- 
 tiplication. 
 
IV SIGNIFICANT FIGURES 69 
 
 Write a definition of standard form in your own words. 
 
 61. Questions and Exercises. 1. What precautions 
 did you take in order to measure the diameter of the brass 
 disc with the greatest possible accuracy? 
 
 2. Write your measured value of ?r, carrying it out just 
 far enough to show one wrong figure. How many sig- 
 nificant figures of 3.141625 are correct? Of 3.1424? 
 
 3. State some of the possible causes that make your 
 determination of TT incorrect. Would there be any ad- 
 vantage in taking the average of several measurements of 
 the circumference? In rolling the disc through two or 
 more consecutive revolutions and measuring the total 
 distance? 
 
 4. Is there any difference in meaning between the 
 italicized statement at the beginning of 52 and the 
 one near the end of the same section? If so, what? 
 
 5. How many significant figures do you think there are 
 in the length of the earth's quadrant as given in 19? 
 
 6. Show that the last statement in 57 is, correct by 
 taking some measurement that you have made as an 
 example. 
 
 7. If one gram is equal to 15.432 grains how much is 
 five grams? If a weight of five grams is the same as 
 77.16 grains how much is one gram? (The answers 
 77.16 grains and 15.432 grains are both wrong.) 
 
 8. How is it that a metre can be measured more ac- 
 curately ( 19) than a centimetre? 
 
 9. Each side of a square measures 82.5 mm. How 
 many centimetres long is its entire periphery? (" Just 33 
 cm." is not the correct answer.) 
 
 10. Use the abridged method for multiplying 12 by 
 13, and for multiplying 13 by 12. Why does the 
 answer come out 16 each time? How should it be 
 pointed off? How many of its figures are significant? 
 
70 THEORY OF MEASUREMENTS 61 
 
 If you wanted to obtain three significant figures in the 
 product, using the abridged method, what would you do? 
 (Answer: Use three significant figures for both factors, 
 12.0 and 13.0. Try it.) 
 
 11. In exercise 9 how many significant figures did you 
 keep in the product of 82.5 X 4? How many are you 
 entitled to keep? Does the 82.5 mean the same as 82.50? 
 Does the 4 mean the same as 4.0? As 4.000? 
 
 12. Turn to the table in the appendix where the density 
 of water is given. Does water at a temperature of 4 C. 
 have a density of 1? Of 1.000? Of 1.0000? Of 
 1.00000? Correct the following statement by crossing 
 off the unjustifiable figures, but keep all that are correct: 
 "at ordinary room temperatures water has density of 
 1.00000." 
 
 13. In 35 why were you justified in adding ciphers 
 ad libitum to the number 180? 
 
V. LOGARITHMS 
 
 62. Definitions. The logarithm of a number is denned 
 as the power to which ten or some other numerical quan- 
 tity must be raised in order to give the specified number. 
 Thus, the logarithm of a thousand is 3 and the logarithm 
 of a hundred is 2, for 1000 = 10 3 and 100 = 10 2 . These 
 statements are usually abbreviated to " log 1000 = 3" 
 and "log 100 = 2." The number which is raised to 
 some power is called the base, and logarithms which 
 have 10 for a base are called common logarithms. For 
 theoretical purposes what are known as natural logarithms 
 are often used; their base is a number which is denoted 
 by the letter e (approximately 2.71828) and is equal to 
 the infinite series 1 + 1 + 1/2+1/2 -3 + 1/2 -3 -4+1/2 -3 -4 -5 
 + . . . or to the limit of [(1 + (l/n)] n when n is increased 
 indefinitely. To avoid confusion the base is often written 
 as a subscript; thus, logic 100 = 2 and log e 100 = 4.6052 
 mean exactly the same thing as 10 2 = 100 and e 4 - 6052 
 = 100. The only logarithms that will be considered here 
 are those whose base is 10. 
 
 63. Fundamental Properties of Logarithms. The table 
 on the next page gives the values of various integral 
 powers of ten; in other words it gives the numbers which 
 have integers for their logarithms. Pick out any two 
 logarithms (exponents) and add them. Then notice 
 that the sum which is thus obtained is another logarithm, 
 namely, the logarithm of the product of the two numbers 
 that correspond to the original logarithms. For example 
 the logarithm of 100 is 2, and of a thousand is 3 ; adding 
 them, 5 will be found from the table to be the logarithm, 
 not of the sum of 100 + 1000, but of the product 
 
 71 
 
72 
 
 THEORY OF MEASUREMENTS 
 
 63 
 
 100 X 1000, or 100000. One of the chief uses of loga- 
 rithms is to enable a multiplication 
 to be performed by the simpler 
 process of addition. In the par- 
 ticular case just given it is as easy 
 to multiply 100 by 1000 directly as 
 it is to add their logarithms and 
 see what number corresponds to 
 the sum, but an exercise like 6.28 
 X 17.35 is as easy as 100 X 1000 
 when worked out by logarithms 
 although it would mean much more 
 time and trouble to multiply it out, 
 even if the abridged method were 
 used. The process is simply to add 
 log 6.28 to log 17.35 and the result 
 will be the logarithm of their prod- 
 In general, 
 
 log a + log b = log (a X 6). (1) 
 
 Try numerical values for the following also, taking 
 each equation separately in turn, and extending the 
 above table if necessary: 
 
 log a - log b = log (a -I- 6), (2) 
 
 10 6 = 1000000 
 10 5 = 100000 
 10 4 = 10000 
 10 3 = 1000 
 10 2 = 100 
 10 1 = 10 
 10 = 1 
 
 io- j = .1 
 
 1C- 2 = .01 
 
 10-3 = <OQ1 
 
 TABLE OF COMMON 
 LOGARITHMS. The 
 exponent of the base, 
 10, on the left, is called 
 the logarithm of the 
 ' natural number ' on 
 the right. 
 
 uct, 6.28 X 17.35. 
 
 n X log a = log a n , 
 (log a) ^ n = log ">/a. 
 
 (3) 
 (4) 
 
 These four equations give the fundamental principles 
 involved in the use of logarithms. The student should 
 not attempt to memorize them as equations, but will need 
 to be perfectly familiar with the ideas that they express. 
 Notice that when using logarithms addition takes the 
 place of multiplication, subtraction of division, multi- 
 
LOGARITHMS 
 
 73 
 
 (from equation 3) 
 
 plication of raising to a power, and division of root ex- 
 traction. Addition and subtraction are performed on 
 logarithms; multiplication and division are also performed 
 upon logarithms but the multiplier or divisor is the 
 number itself (natural number, as it is often called to dis- 
 tinguish it from the logarithm or logarithmic number} , 
 not the logarithm of the number. The result in all 
 cases is a logarithm, and from this the required number 
 is found by consulting a table. 
 
 64. Common logarithms. The advantage of using 
 10 as a base is that 
 
 log (10 X a) = log 10 + log a (from eq. 1) 
 
 1 + log a 
 and in general 
 
 log (10 n X a) = n log 10 + log a 
 = n -h log a; 
 
 for example, log 365 = log (10 2 nat. no. 
 X 3.65) = 2 + log 3.65. Accordingly 
 tables of common logarithms are 
 made out only for natural numbers 
 between 1 and 10, the logarithms of all 
 other numbers being self-evident from 
 these. 
 
 If the logarithm of 3.65 is .562 what 
 is the logarithm of 3650? Ans. : 3.562. 
 What is log 365? Log 36.5? Log 
 .365? Ans.: - 1 + .562. Log 
 .0000365? Ans.: - 5 + .562. (Do 
 not simplify these binomial forms. 
 They are easier to use if left as they 
 are.) 
 
 The logarithms of numbers other 
 
 nat. no. 
 
 log. 
 
 1 
 
 .000 
 
 2 
 
 .301 
 
 3 
 
 .477 
 
 4 
 
 .602 
 
 5 
 
 .699 
 
 6 
 
 .778 
 
 7 
 
 .845 
 
 8 
 
 -903 
 
 9 
 
 .954 
 
 10 
 
 1.000 
 
 TABLE OF LOG- 
 ARITHMS. The 
 logarithms of the 
 natural numbers 
 from 1 to 10 are 
 given here as far as 
 the first three deci- 
 mal places. 
 
 than powers of 10 are in general incommensurable and 
 
74 THEORY OF MEASUREMENTS 65 
 
 are given only approximately in tables. Use only the 
 small table given on the last page for the exercises in 
 the following paragraph. 
 
 Find 2X3. Answer: log 2 = .301; log 3 - .477; 
 their sum is .778, and by looking in the table this is 
 found to be the logarithm that corresponds to the 
 number 6. (Six is sometimes said to be the anti-loga- 
 rithm of .778.) Using logarithms, find 2X4. (Do not 
 perform the multiplication mentally and then look for 
 the logarithm of 8 to verify the sum of log 2 and log 4, 
 but consider the product as being unknown until after 
 you have been directly led to it by following out the 
 logarithmic process.) Find 2 2 ; find 3 2 . Find 4X5; 
 V 9; 5 X 6; 50 X 6; 500 X 600. 
 
 Calculate the value of e from the infinite series given 
 above. 
 
 65. Practical Logarithm Tables. Examine the four- 
 place logarithm table in the appendix at the end of this 
 book, and notice that it contains the same succession of 
 numbers, from 1 to 9, as the small table w r hich has just 
 been used. It also contains the same succession of 
 logarithms, from .0 to .'9; but the intermediate values, 
 both of logarithms (.0000 to .9999) and of natural 
 numbers (1.00 to 9.99), are given at smaller intervals, 
 and without any decimal points. Verify each of the 
 following statements by finding the required logarithm 
 in line with the first two figures of the natural number as 
 they occur in the left-hand column, and in the column 
 that is headed by the third figure: log 3.65 = .5623; 
 log 3.66 = .5635; log 4.06 = .6085; log 7.70 = .8865; 
 log 77.0 = 1.8865 ( 64); log 7700 = 3.8865; log .00077 
 = - 4 + .8865. 
 
 Find the logarithms of 5.02; 5.01; 5.00; 50.0; 5000000. 
 
 It will have been noticed that the decimal part of 
 
V LOGARITHMS 75 
 
 a logarithm (sometimes called the mantissa) is dependent 
 only upon the arrangement of signicant figures in the 
 natural number; e. g., log 36500 = 4.5623; log .0365 
 = 2 + .5623. The integral part (sometimes called 
 the characteristic of the logarithm), however, is deter- 
 mined only by the position of the first significant figure; 
 for example, for any number beginning in tens-of- 
 thousands' place it is 4, and for any number beginning in 
 hundredths' place it is 2. In order to save space a 
 number like log .0365 is customarily written in the form 
 2.5623, the minus sign being written over the character- 
 istic to indicate that it applies only to the whole number 
 while the decimal part is always positive. 
 
 Write - 2.60 in logarithmic form. Ans.: - 2.60 = 
 - 2 - .60 = - 3 + .40 = 3.40. _ Write - 1.4377 with 
 the decimal part positive. Ans. : 2.5623. 
 
 Write the characteristic of the logarithm of each of the 
 following: 5441; 27; 79264; 264; 73; 0.73; 0.073; 0.000073. 
 Make up a rule for finding the characteristic of the 
 logarithm of any number, and write it in your notebook. 
 
 Write the logarithms of 984; 982; 981; 980; 98; 9.8; 
 .98; .098; 7; 14. Add the last two and find their sum in 
 the body of the table; see what number in the margin 
 corresponds to it, and verify the result by multiplying 
 14 by 7. 
 
 66. Use of the Table. A four-place table of loga- 
 rithms is in general satisfactory for obtaining the logarithm 
 of a number that has four significant figures; for numbers 
 of three significant figures it is not necessary to keep 
 more than three decimal places of the logarithms; for 
 five-figure accuracy a five-place table is needed; etc. A 
 four-place table is generally made more compact by 
 including only three-figure values for the natural num- 
 bers, and when the logarithm of a four-figure value is 
 
76 THEORY OF MEASUREMENTS 66 
 
 required it is found by a process called interpolation, in 
 which it is assumed that small differences between loga- 
 rithms are proportional to the corresponding differences 
 in their antilogarithms. Turn to the table and notice 
 that (log 633 log 632) is exactly one third as large as 
 (log 635 log 632) ; and of course the differences in the 
 numbers 633 632 and 635 632 are in the same ratio. 
 Suppose the logarithm of 3.142 is required: The table 
 gives log 3.14 and log 3.15. The required number 3.142 
 is larger than 3.140 but smaller than 3.150; in an orderly 
 scale of numbers it would be located just one fifth of the 
 way from 3.14 to 3.15. The assumption is, accordingly, 
 that its logarithm likewise is situated one fifth of the way 
 vfrom log 3.14 to log 3.15. Log 3.14 = .4969; log 3.15 
 = .4983. One fifth of the distance from .4969 to .4983 
 is obtained by first finding that distance subtracting; 
 4983 4969 = 14; then adding the required fraction 
 of it to the lower logarithm 1/5 of 14 = 3 (the nearest 
 whole number), and 4969 + 3 = 4972; /. log 3.142 
 = .4972. 
 
 Suppose the number is required whose logarithm is 
 .2752: Turn to the table and notice that the nearest 
 logarithms are 2742 and 2765. Their difference (called 
 the tabular difference) is 23, and the given logarithm 2752 
 is 10 larger than 2742; i. e., it lies 10/23 of the way from 
 2742 to 2765. Accordingly the required antilogarithm 
 will be 10/23 of the way from one marginal number (188) 
 to the other (189); that is, it will be 188J& r 188.4. As 
 the characteristic of the given logarithm is zero this 
 should be pointed off as 1.884. 
 
 The small multiplication tables at the side of the 
 logarithmic table enable a fraction like 10/23 to be 
 reduced to tenths mentally. Find the table headed 23 
 and notice that it gives one tenth of 23 = 2.3; .2 of 
 
V LOGARITHMS 77 
 
 23 = 4.6; etc. The number nearest to 10 is 9.2, which 
 stands opposite 4; accordingly 10 twenty-thirds comes 
 nearer to 4 tenths than to 5 tenths. Where 1/5 of 14 
 was required, above, the small tables would have been 
 used, if necessary, by finding the number which is 
 opposite 2 tenths in the fourteen table. 
 
 Find log 2.718. Ans. 0.4343. Find log 3.333; log 
 1.234; log 12.34; log 123.4; log 8888; log .4343; log 3449. 
 
 67. The Probability Function. In much of the stu- 
 dent's later work it will be important to know how e~^ 
 varies when x is given different numerical values. Be- 
 fore substituting any particular number for x it will be 
 advisable to proceed as follows: Let the function e~ x2 
 be denoted by y, then 
 
 taking logarithms of each side this becomes 
 
 log y = - x 2 log e, 
 from which it follows that 
 
 log y = x 2 log e. 
 Taking logarithms of each side again 
 
 log ( - log y) = log (z 2 ) + log (log e) 
 or 
 
 log (- log y) = log (x 2 ) + 1.6378. 
 
 In the last equation it is easy to find the value of y that 
 corresponds to any given value of x. The procedure is 
 first to calculate x 2 and find its logarithm; then add 
 T.6378. The result is stated by the equation to be the 
 logarithm of an expression enclosed in parenthesis, so the 
 numerical value of that expression is easily found by the 
 process of obtaining an antilogarithm. Then, when the 
 
78 
 
 THEORY OF MEASUREMENTS 
 
 68 
 
 value of ( log y) has been obtained it is easy to write 
 the value of (+ log y), and from the value of log y 
 there is no difficulty in finding y itself. 
 
 When such a determination is to be made for several 
 different values of x it is convenient to arrange the 
 various calculated quantities in the form of a table. 
 Using the last of the equations that was derived from 
 y = e~ x2 , and the tables of squares and of logarithms in 
 the appendix, find in succession the values of x 2 , log x 2 , 
 log x 2 + 1.6378, log (- log y), - log y, log y, and y, 
 for each one of the following values of z: 0, .2, .4, .6, .8, 
 1.0, 1.2, . . . 2.8, 3.0; 4.0; 5.0. On the next unused left- 
 hand page of your notebook tabulate the results in 
 columns headed x, x 2 , log x 2 , etc., carrying each line clear 
 across the table, as shown in the illustration, before 
 beginning the next line. 
 
 X 
 
 x z 
 
 log x 2 
 
 log (z 2 ) 
 + 1.6378 
 
 log ( - log y) 
 
 -logy 
 
 + logy 
 
 9 
 
 
 
 
 
 00 
 
 00 
 
 00 
 
 
 
 
 
 1 
 
 .2 
 
 .04 
 
 2.6021 
 
 2.2399 
 
 2.2399 
 
 .0174 
 
 1.9826 
 
 .961 
 
 .4 
 
 .16 
 
 "1.2041 
 
 2.8419 
 
 28419 
 
 .0695 
 
 1.9305 
 
 .852 
 
 .6 
 
 .8 
 
 .36 
 .64 
 
 1.5563 
 
 1.1941 
 
 1.1941 
 
 
 
 .699 
 
 
 
 
 
 
 
 
 
 TABLE CONSTRUCTED WHEN FINDING THE VALUES OF e~* 2 . It 
 is important that the table be filled out line by line, not column by 
 column. 
 
 Leave the opposite right-hand page of the notebook 
 vacant until directions for using it are reached in another 
 chapter. 
 
 68. Questions and Exercises. 1. Find the numer- 
 ical values of (1+ T V) 10 ; (1+yW 100 ; and 
 Ans.: 2.6; 2.7; 2.7. 
 
V LOGARITHMS 79 
 
 2. If log 10001 = 4.00004343 find the value of 
 
 Ans.: 2.718. 
 
 3. Find the natural logarithm of ?r. Ans.: By defi- 
 nition, e x = TT ( 62). Solving this equation will give 
 x = log 7r/log e = 1.1447. 
 
 4. Use the algebraical fact that 10*10" = IQ X+V to 
 prove equation 1 of 63, which was merely stated without 
 proof. 
 
 5. Turn back to the numbers that you have written 
 in standard form (60, If 2) and write the characteristic 
 of the logarithm of each of them. What relationship 
 can be observed? 
 
 6. Prove that the second equation of 67 must be 
 true if the first one is true. 
 
 7. Write down any number that has three significant 
 figures. Find its logarithm from the table, subtract it 
 mentally from (= log 1), and find the antilogarithm in 
 order to obtain the reciprocal of the number that was 
 written down. The mantissa is all that is needed for 
 each logarithm, as the answer can be pointed off by 
 inspection. For example, suppose the reciprocal of TT is 
 required: log 3.14 = 497; working from left to right, 
 subtract each figure of the logarithm from 9 except the 
 last one, which is to be subtracted from 10; result, 503; 
 antilog 503 = 318; pointed off, 1/3.14 = .318. 
 
 Find 1 -f- 269 by using the logarithm table, but without 
 writing down any figures. The answer must be approxi- 
 mately .04. 
 
 Find the reciprocal of a four-figure number (for ex- 
 ample, TT or e) in the same way. 
 
VI. SMALL MAGNITUDES 
 
 Apparatus. Platform balance; set of gram weights; 
 set of avoirdupois weights. 
 
 69. Approximate Values. If the sum of one hundred 
 dollars is put out at 3 percent compound interest it is 
 said to " amount " to 103 dollars after one year, about 
 106 dollars after two years, about 109 dollars after three 
 years, etc. The exact values of the latter amounts are 
 106.09 (the square of 1.03; in hundreds of dollars) and 
 109.2781 (the cube of 1.03) ; but if the principal had been 
 one single dollar the amount that could have been repaid 
 after two years or three years would necessarily have 
 been $1.06 or $1.09, respectively, on account of the true 
 amounts being less than $1.06J in the first case, and 
 nearer to $1.09 than to $1.10 in the second. The fact 
 that in United States money fractions of a cent cannot be 
 used (except for purposes of calculation) corresponds 
 exactly to the metrological fact that any measurement 
 can be carried out to some particular degree of precision 
 but no further. 
 
 70. Negligible Magnitudes. Suppose a ruler gradu- 
 ated in centimetres and millimetres is used to measure 
 the side of a square, and by estimating tenths of a milli- 
 metre the length is found to be 2.87 cm. The mathe- 
 matical square of this quantity is 8.2369 cm 2 , but it has 
 already been seen ( 58) that only three figures of this 
 area can be trusted, because nothing is known about the 
 fourth figure of the measurement from which it is de- 
 rived. Accordingly, the square is said to have an area 
 of 8.24 cm 2 . Similarly, if one side of a square measures 
 1.03 cm., the measurement being correct to tenths of a 
 
 80 
 
VI SMALL MAGNITUDES 81 
 
 millimetre but nothing being known about hundredths 
 of a millimetre, then its area will be correctly expressed 
 by the quantity 1.06 cm 3 , and the volume of a cube that 
 has this square for one of its sides will be 1.09 cm 3 . It 
 should be noticed (a) that with an ordinary ruler it is 
 impossible to measure a length of a few centimetres with 
 an accuracy greater than is expressed by three significant 
 figures; (6) that the area or volume calculated from such 
 data cannot be trusted further than its third significant 
 figure; (c) that the example just given suggests a remark- 
 ably simplified process of calculation where some quantity 
 which is to be squared or cubed is a little greater than unity, 
 namely: (one + small amount) 2 = one + twice small amt. } 
 and (one + small) 3 = 1 + 3 (small). 
 
 Decide mentally, by induction, the value of (1. .02) 2 ; 
 then prove the result in two ways: (a) expanding in 
 accordance with the binomial theorem; (b) squaring .98 
 by abridged multiplication. 
 
 The justification of the simplified process of raising a 
 number which is approximately unity to a power will 
 perhaps be made more evident by the following example : 
 
 Suppose that a metal cube has been constructed ac- 
 curately enough to measure 1.00000 cm. along each edge. 
 If it should be brought from a cold room into a warm 
 room a delicate measuring instrument might show that 
 the change of temperature had increased each dimension 
 to 1.00012 cm. and by unabridged multiplication it would 
 be easy to prove that the area of each side was 
 1.0002400144 cm 2 and that the volume had become 
 1.000360043201728 cm 3 . If the most careful measure- 
 ments make it just possible to distinguish units in the 
 fifth decimal place then tenths of those units (repre- 
 sented by the sixth decimal place) would be impossible 
 to measure, and the attempt to state not only tenths, 
 7 
 
82 THEORY OF MEASUREMENTS 71 
 
 but hundredths and thousandths of those units would 
 be absurd. By noticing that the number 1.0002400144 
 differs from the value obtained by abridged multiplication 
 (1.00024) by only a few thousandths of the smallest 
 measurable amount we can see clearly why the area of a 
 1.00012-cm. square is and must be 1.00024 cm 2 . 
 Similarly, the volume of the cube is neither more nor 
 less than 1.00036 cm 3 , and the string of figures running 
 out ten decimal places further is absolutely meaningless. 
 
 It will be noticed that the number 1.0002400144 is in 
 the same form as 1 + 2x + z 2 , the square of (1 + x), 
 where x = .00012; also that 1.000360043201728 cor- 
 responds to the cube, 1 + 3x + 3z 2 + x 3 . This is an 
 illustration of the fact that when dealing with objects 
 of the real world which is evident to our senses it may 
 happen that a measured amount is so small that its 
 higher powers, algebraically speaking, are relatively 
 minute beyond all perceptive ability. Of course this 
 must not be understood as meaning that the cube of a 
 measurable length can ever be an impalpable volume; 
 the cube of 1 + x is even a larger number, 1 + 3z; it is 
 the difference in size between this " physical " value, 
 (1 + x) s = 1 + 3z, and the true mathematical value, 
 (1 + x) 3 = 1 + 3x + 3z 2 + z 3 , that eludes perception 
 on account of x 2 being extremely small in comparison 
 with x, which is itself minute. 
 
 71. Formula for Powers. The examples that have 
 been given above suggest that, if x is small enough, 
 (1 + x) 2 = 1 + 2x, (1 + x) 3 = 1 + 3z, and in general 
 (1 + x) n = 1 + nx. The matter can be tested by mak- 
 ing use of the binomial theorem: 
 
 . n(n l) . n(nl)(n 2) , 
 lz) n = l nx+ * 2 X H ' 
 
VI 
 
 SMALL MAGNITUDES 
 
 83 
 
 This shows that (1 d= x) n is equal to 1 db nx if x is so 
 small that x 2 , x 3 , etc., are negligible, the only possible 
 
 1 X S \ S* ^ 
 
 i+S 
 
 1 x i 
 
 FIG. 13. THE SQUARE OF 1 + 5. The fact that 6 is small 
 means that 5 2 must be very small. The error produced by taking 
 the area 1 + 25 for the square of 1 + 5 is only a minute fraction of 
 the total area. 
 
 exception being in case n should be so large that it 
 could counterbalance the small size of x and prevent the 
 term 
 
 n(n 1) 
 
 from becoming negligible. In the practical use of the 
 formula, however, n is rarely larger than 2 or 3 while x 
 is at most only a few hundredths and is usually very 
 much smaller. 
 
 72. Properties of Deltas. The small quantities which 
 have been considered in this chapter are usually sym- 
 bolized by the Greek letter 8. It has been seen that 
 an important property of deltas is given by the equation 
 (1 d= 5) n = 1 nd, but the student is advised to learn 
 this in the form 
 
 (1 + 6) = 1 + n&, (1) 
 
84 THEORY OF MEASUREMENTS 72 
 
 with the understanding that 5 may have either a positive 
 or a negative value; for example the square of l + ( .03) 
 is equal to 1 + 2( .03) or 1 - .06, or 0.94. 
 
 Find the value of each of the following expressions 
 mentally: .99 2 ; .98 2 ; .98 3 ; .97 2 ; 1.00012 2 ; 1.00012 3 ; 
 (1 - .008) 2 ; .992 3 . 
 
 The ordinary process of algebraical division shows that 
 1/(1 + x) = 1 - x + x 2 - x 3 + z 4 - x 5 + - . From 
 this it follows that 
 
 Divide 1 by 0.997. Ans.: .997 = 1.-.003; !/(!-. 003) 
 = 1 + .003 = 1.003. 
 
 Find mentally the reciprocal of 1.00012. 
 
 Find 1/(1.00012) 2 mentally by using first formula (1) 
 to simplify the denominator and then formula (2) to clear 
 of fractions. 
 
 Find (1.00012)5 and complete the following formula 
 for yourself: i/l + 5 = 
 
 If a number is so small that its square is negligible it 
 will similarly be true that the product of two such 
 numbers will be negligible. For example, if a number 
 that is carried out to thousandth's place is 1.007 its 
 square will not differ from 1.014 by a single thousandth: 
 1.007 X 1.007 = 1.014049; likewise 1.007 X 1.006 will 
 not differ by a thousandth from 1.013, its " exact " 
 value being 1.013042, as the student can easily prove 
 by considering it to be a special case of (1 + x)(l + y) 
 = 1 + x + y + xy. Accordingly, 
 
 (1 + 50(1 + 5 2 ) = 1 + Si + 5 2 . (3) 
 
 This equation shows the advantage of keeping the signs 
 positive and allowing the deltas to be either positive or 
 
VI SMALL MAGNITUDES 85 
 
 negative. If the deltas were restricted to positive values 
 there would be a liability to error unless the equation 
 could be written (1 5i)(l 5 2 ) = 1 5i d z . 
 
 Find mentally 1.04 X 0.98. Ans.: .98 = 1. - .02; 
 (1 + .04) (1 - .02) = 1.02. 
 
 Find mentally 1.03 X 0.98; also 1.00012 X .99890 
 and prove the latter by abridged multiplication. 
 
 In your notebook work out neatly the value of 1.0021 
 X 1.0037 in three different ways: (a) by unabridged mul- 
 plication; (b) by abridged multiplication; (c) by the use 
 of deltas, writing the two numbers one under the other 
 and adding merely the deltas on paper in the customary 
 manner. Compare the three processes carefully and 
 draw the moral for yourself. 
 
 Remembering that a/6 is the same as a X (1/6) write 
 the formula for (1 + Si) /(I + <5 2 ). 
 
 A fact that is useful when making measurements of 
 mass is if two quantities are nearly equal to each other the 
 square root of their product (or geometrical mean) can be 
 obtained by taking their average (or arithmetical mean). 
 If the two quantities are denoted by a and a + d (to 
 indicate that their difference is a small magnitude) this 
 statement can be proved as follows: 
 
 l/a(a + 5) = i/a - o(l + d/a) 
 
 = a i/l + d/a = a(l + 5/2a), 
 since d/a is also a small magnitude. But 
 a(l + 8/2a) = a(2a + d)/2a 
 
 - (4) 
 
 If an object appears to weigh mi when placed on one 
 of the scale pans of a balance, and m,z when on the other 
 
86 THEORY OF MEASUREMENTS 72 
 
 pan it can be proved (see Fig. 14) that its true mass is 
 I/ mi 7^2. For example, a metal block weighed 15.19 and 
 15.23 gm. on the two sides of a balance. Its true weight 
 is accordingly 1/15.23 X 15.19 gm. Equation (4) shows 
 that this is the same as 15.21 gm. 
 
 1 1 + 8 
 
 m, 
 
 1+6 
 
 ~> 
 
 FIG. 14. DIAGRAM OF A BALANCE. The force multiplied by the 
 distance at which it acts is the same on each side .'. a = mi(l+5); also 
 m z =x(l +5). Eliminating (1+5) gives x/mi = m<Jx, whence x= VwiW 2 . 
 
 Weigh your whole set (16 ounces) of avoirdupois 
 weights, considered as an unknown mass on the platform 
 balance against the brass gram weights. Repeat the 
 process on the other pan of the balance, remembering 
 that the reading of the sliding weight is additive in 
 one case and subtractive in the other. Find the true 
 weight, in grams, of the avoirdupois set. 
 
 It will be almost self-evident from Fig. 15 that 
 
 tan 8 , sin 5 - ,~^ 
 
 = 1 and = 1 (5) 
 
 5 . o 
 
 if it is remembered that an angle is measured by the 
 ratio of arc to radius. 
 
 By consulting the table of circular functions find the 
 largest whole number of degrees for which tan 5 = d 
 to four decimal places. The numerical measure of 
 
VI 
 
 SMALL MAGNITUDES 
 
 87 
 
 each angle will be found in line with the number of 
 degrees, but in the column headed RAD, an abbreviation 
 of radian measure, which is another synonym for circular 
 measure. 
 
 FIG. 15. FUNCTIONS OF A SMALL ANGLE. The tangent is always 
 larger than the angle and the sine is always smaller, but the ratio 
 of either function to the angle comes nearer and nearer to unity as 
 the angle is made smaller and smaller. A formal proof of the fact 
 that lim sin x/x = lim x = lim tan x/x can easily be worked out if 
 two equal angles are juxtaposed so that their sine lines complete the 
 chord of the double arc. 
 
 If an accuracy of three decimal places is all that is 
 needed how large can d be without differing from tan 5? 
 Ans. : 6; but not 7. How large if d is to equal sin 5? 
 
 73. Transformation of Operands. It will be noticed 
 that most of the rules that have been given for operating 
 upon small quantities involve functions of 1 + 5, not of 
 d alone. This means that factoring, or some other proc- 
 ess, is often necessary in order to put an expression 
 intojthe form of 1 + <5. For example, V 50 is equal to 
 1/7 2 + 1, an expression which is not in the form of a 
 function of 1 + 5 but can be made so by dividingjthe 
 binomial by an assumed factor 49: 1/49 + 1 
 = 1/49(1 + 1/49) = 71/1+1/49 = 7(1 + 1/98). Now 
 
88 THEORY OF MEASUREMENTS 75 
 
 it will be noticed that 98 is a little less than 100, and so 
 
 02j) 
 
 -Wl + 
 
 -2/ \ r 100(1.-. I 
 
 1 + .02) } = 7(1.0102) = 7.0714. 
 
 o 
 
 The first four figures of this result are correct. Extreme 
 accuracy cannot be expected where a delta is as large as 
 one- or two percent; in most physical calculations it is 
 much smaller than this. 
 
 74. Recapitulation. The formula? for calculation 
 with small quantities are collected here for reference. 
 Notice that the second and fifth are special cases of the 
 first, and the first formula, for n equal to a whole number 
 only, is a special case of the fourth, 
 
 (1 + 5) = 1 + nd, 
 + 5) = 1 - 5, 
 
 1/1 + d = 1 + 
 
 tan 5 _ sin 3 _ 
 
 ~<T ~T 
 
 75. Questions and Exercises. 1. Find the value of 
 (Suggestion: divide both numerator and denom- 
 inator by 800 in order to convert the latter into 1 5.) 
 
 2. Find the value of .504 X .498. (Suggestion: the 
 first of these is a little more than 1/2, i. e., is equal to 
 1/2 multiplied by a little more than one; likewise, the 
 second is a little less than 1/2.) 
 
VI SMALL MAGNITUDES 89 
 
 3. Write the general formula for the value of the ex- 
 pression (1 + 5i)"(l + 5 2 ) n (l + 5 3 ) p (l + 5 4 ) 9 How 
 many of the formulae that are given in 74 does it 
 include? 
 
 4. When warmed up to ordinary room temperature 
 the brass scale of a barometer is too long; as a result its 
 reading falls short of the correct number of centimetres 
 and needs to be multiplied by 1.00037. The mercury, 
 however, expands at a different rate from the brass and 
 tends to make the reading too great; to correct this 
 error alone the reading must be divided by 1.00364. 
 The correct height of the mercury will accordingly 
 be obtained if the observed height is multiplied by 
 1.00037/1.00364. What is the percentage of difference 
 between the corrected reading and the original reading? 
 (Suggestion: call the observed height unity.) 
 
 5. In the equations of Fig. 14 eliminate x instead of d 
 
 and show that 1 + d = Vm^/mi. Then suppose that 
 m 2 = Wi + A and show that 1 + 5 = 1+ A/2w, and 
 hence that 25 = A/m. If the two weighings of your 
 avoirdupois pound (71) differed by 0.3 percent how 
 much do the two arms of the balance differ in length? 
 
 6. If V 10 = 3.16 find the value of 7r 2 . (Suggestion: 
 v = 3.14 = 3.16(1 - 2/316)). 
 
VII. THE SLIDE RULE 
 
 Apparatus. A slide rule provided with A, B, C, D, 
 L, S, and T scales, a runner, and a list of equivalent 
 measures. 
 
 76. Addition by Means of Two Scales. If two scales 
 of centimetres should be laid parallel so that the zero of 
 the second one coincided with the seventh division of the 
 first (Fig. 16) it would be evident that the fifth division 
 
 12345678 
 
 i i i i i i i 
 
 9 
 
 to 
 
 ft 
 
 12 
 
 13 
 
 
 
 1 
 
 i 
 
 2 
 
 i 
 
 3 
 
 i 
 
 4 
 
 i 
 
 5 
 
 i 
 
 6 
 
 i 
 
 7 
 
 i 
 
 FIG. 16. ADDITION WITH UNIFORM SCALES. By starting afresh 
 at 7 of the original scale and measuring 5 (or any other number, n) 
 on the new scale we reach the distance of 7 + 5 (or 7 + n, as the 
 case may be) on the original scale. It should be noticed that this 
 illustrates subtraction as well as addition: 12 5 = 7, also the 
 difference between any other pair of opposite scale numbers is equal 
 to 7. 
 
 of the second would be opposite the twelfth division of 
 the first. In other words, when a length of five has been 
 added in this manner to a length of seven the result can 
 be seen at a glance to be equal to twelve. Notice also 
 that the arithmetical difference between scale divisions 
 that lie opposite each other is everywhere the same, and 
 that it is equal to the number on the first scale which is 
 opposite the beginning of the second. 
 
 77. Multiplication with Logarithmic Scales. If the 
 same experiment is tried with two scales whose divisions 
 are not a succession of whole numbers but are the 
 logarithms of such a series the result will be different, 
 
 90 
 
VII THE SLIDE RULE 91 
 
 for adding logarithms corresponds to multiplying natural 
 numbers. Accordingly if we start at log 7 (Fig. 17) and 
 
 t5 2 3 4 5 6 7 8 9 K) 15 20 30 40 60 80 100 150 
 
 "- ! ! '''-''KVl^Mi'M. ^'.i.'iti'.iVI'.'il'V/.'.i'-v 
 
 I 1.5 2 3456 78910 15 20 25 
 
 FIG. 17. MULTIPLICATION WITH LOGABITHMIC SCALES. The 
 distances on each scale which are marked 1, 2, 3, etc., are really equal 
 to log 1, log 2, log 3, etc. Since addition of logarithms effects multi- 
 plication of their natural numbers it will be seen that the number 
 above the 5 of the lower scale is not 7 + 5 as in Fig. 16, but is equal 
 to 7 X 5, or 35. Since subtraction of logarithms corresponds to 
 division it will be evident that the opposite pairs of numbers have 
 their quotients equal to seven, instead of their differences. 
 
 measure a further distance of log 5 we shall come out with 
 log 7 + log 5, which is not equal to log (7 +' 5) but to 
 log (7X5). Notice not only that 5 on the second scale 
 comes opposite 35 on the first, but also that quotients 
 of corresponding numbers are everywhere equal to 7, 
 just as differences are in Fig. 16; and furthermore, the 
 upper scale in Fig. 17 forms a multiplication table (a 
 seven table in this case) for the numbers on the lower 
 scale just as it did an addition table in Fig. 16. 
 
 78. The Slide Rule. The apparatus called a slide rule 
 is essentially a ruler containing a groove in which is a 
 movable slide. Logarithmic scales are so marked that 
 one of them (the "A scale") can be held stationary 
 while another (the " B scale ") can be placed in any 
 required position below it (Fig. 18). 
 
 Two scales (C and D) along the lower edge of the slide 
 can be used in the same way and can be read a little more 
 accurately on account of their subdivisions being larger, 
 but the two upper scales (A and B) will be found the most 
 convenient for general use. Each one of them is really 
 
92 
 
 THEORY OF MEASUREMENTS 
 
 78 
 
 f|e 
 
 
 CQO 
 
 t 
 
 
 II 
 
 "-> M QJ 
 
 SB > 
 
 a B.-S 
 
 3 ft _. 
 
 a 
 
 
 -a 
 
 f-Jlil 
 
 !yl! ; 
 
 ^ Q O 3 
 I !i GO *3 A - ' 
 
 oq 
 
 s 
 
 ^ T3 
 
 gl 
 
 H ^ 1 ^ -3 
 
 _ +2 ^5 c-\ 
 
 OS -^ 
 
 i"^ 
 ft 2 g 
 
 o cti .2 
 
 .S 
 
VII THE SLIDE RULE 93 
 
 two complete logarithmic scales, as will readily be seen, 
 but its right-hand half is often marked 10 to 100 instead 
 of 1 to 10. In calculating with the aid of the slide rule 
 it is advisable to ignore all decimal points while using the 
 apparatus, and to point off the required answer by making 
 a rough preliminary calculation mentally, as illustrated 
 in 14. It is then possible to use either the 7, for ex- 
 ample, in the left-hand part of the A scale, or the 70 (or 
 7) in the right-hand part, whichever may happen to be 
 the more convenient, in order to represent 7000, or 700, 
 or .07, or any other number whose significant figure is 7. 
 
 By consulting Fig. 17 find the value of 70 X 20. Ans. : 
 1400. Find 3| -f- 5. Ans.: 0.7, because it is evident 
 that the quotient must be somewhat less than 1. Find 
 the approximate value of .0007 X 1.4. Ans.: nearly 
 .0010. Find 70 X 45. Ans.: as nearly as the value can 
 be read from the diagram, it seems to be 32; since 70X40 
 = 2800 the required product must be 3200, the ciphers 
 not being significant. Two-figure accuracy is the highest 
 that can be obtained from Fig. 17. 
 
 79. Reading a Logarithmic Scale. The ordinary slide 
 rule is usually constructed so carefully that three-figure 
 accuracy is always attainable, although it requires the 
 " estimation " ( 46) of halves or fifths or even tenths 
 of the graduated intervals. This is sufficient for the 
 great majority of practical calculations in every-day life; 
 if four-figure accuracy or greater is required it is usually 
 more convenient to use logarithm tables, but a slide-rule 
 makes a 3-figure logarithm table superfluous because it 
 can be used with greater rapidity. 
 
 It will be observed, both in Fig. 18 and on the slide rule 
 itself, that the distances from log 1 to log 2, log 2 to log 3, 
 log 3 to log 4, etc., grow progressively smaller and 
 smaller, so that those near the left-hand end of the scale 
 
94 THEORY OF MEASUREMENTS 79 
 
 can be more finely subdivided than those toward the 
 right-hand end. Thus it should be noticed that on the 
 A or B scale of the ordinary slide rule the third subdivision 
 to the right of 7 or 70 is 73, while on the C or D scale it 
 is 715; but the third subdivision to the right of 1 rep- 
 resents 106 on the upper scales and 103 on the lower 
 ones. To avoid mistakes in picking out the position of 
 a given number it is best to find both the next larger 
 single figure and the next smaller one; then look for the 
 long .5 graduation which is about half-way between 
 them. 
 
 Find 365 on the A scale of your slide rule by locating 
 first 3, then 4, then 3.5, then 3.6 and 3.7, and finally 365. 
 Find 235 in the same way; then by estimation set the 
 transparent runner with its vertical line as nearly as 
 possible at 238. Move the slide so that the end (1) 
 of the B scale comes under this 238, and see if 210 comes 
 just under 500. Set the transparent runner at 128; 
 then at 106, and 109; Repeat the work of this para- 
 graph using the D and C scales respectively instead of the 
 A and B scales. 
 
 In reading a given position on the scale it is likewise 
 best to read the next larger single figure and the next 
 smaller one, also the half-way point between them. 
 
 Set the left-hand 1 of the B scale at random somewhere 
 between 1 and 2 of the A scale. Then read its position 
 accurately; also the positions of 2, 3, 4, 5, 6, 7, 8, and 9. 
 On some slide rules the subdivisions between 1 and 2 of 
 the C and D scales are marked 1, 2, 3, instead of 
 1.1, 1.2, 1.3, etc. If this is the case with your slide rule 
 it will be necessary, when looking for a number such as 7, 
 to see that it belongs in the series that runs 7, 8, 9, 10, 
 out to the right-hand end of the rule, and not in the 
 series that runs merely from 1.0 to 2.0. 
 
VII THE SLIDE RULE - 95 
 
 80. Multiplication. To find the product of two 
 numbers with the slide rule either end of the B scale should 
 be set opposite one factor on the A scale, then the other 
 factor on the B scale will be opposite the required product 
 on the A scale. Try this process with small numbers by 
 setting 1 on B opposite 3 on A and observing that the 
 position of 2 on B gives the product 6 on A. 
 
 Multiply 2 X 4; 2 X 5; 2 X 6; 3 X 9; 7 X 8. Find 
 7 X 13, and 7 X 16, remembering that the two halves of 
 the B scale (and also those of the A scale) are identical, 
 so that if 16 (or 1.6) when taken on the right half of the 
 B scale falls beyond the end of the A scale the same 
 result can be read over 16 (or 1.6) on the left half. 
 
 Multiply 1.5 by 2; 1.8 X 2; 1.7 X 2.1 (estimate the 
 third figure of the product); 17.8 by 2; 1.79 by 2.53. 
 
 Find 0.6 X 183 .(caution: notl.6X 183; see end of 79) ; 
 7.3 X 1.09; 325. X 106.5; 0.073 X 0.0016. If you have 
 any difficulty in pointing off the last product master it 
 before going any further. The use of " standard form " 
 often makes the preliminary checking easier; thus in the 
 last two examples given 3 X 10 2 multiplied by 1 X 10 2 
 gives 3 X 10 4 or 3000; and (7 +) X 10~ 2 multiplied by 
 (2 -) X 10- 3 gives 14 X 10~ 5 , or .00014. 
 
 81. Division. In Fig. 16, above, notice that a length 
 of twelve may be considered as being laid off from left to 
 right, and then, beginning at the point 12, a length of 5 
 is laid off to the left, or subtracted from the original 12, 
 giving a final result of 7. Similarly, in fig. 17, log 35 
 minus log 5 equals log 7. The rule for division is ac- 
 cordingly: place the divisor on B under the dividend on 
 A and read the answer on A over either end (or the 
 middle) of the B scale. Such directions should never be 
 memorized by the student, but he should practice the 
 process until he becomes familiar with it, and in case of 
 
96 THEORY OF MEASUREMENTS 82 
 
 doubt should experiment with small numbers where the 
 answer is obvious beforehand. 
 
 Divide 35 by 5; 30 by 5; 30 by 4; 11 by 4; 11 by 7; 
 11 by 10; 11 by 11; 11 by 12; 11.8 by 99.; 114. by 3.4; 
 3.4 by 114. 
 
 82. Ratio and Proportion. In Fig. 16 it will be seen 
 that 8 1 = 9 2 = 10 3 = 7. Since subtraction 
 of logarithms corresponds to division of natural numbers 
 Fig. 17 shows that 14 -h 2 = 21/3 = 28 : 4 = 35/5, 
 etc. That is, with the slide set in a given position any two 
 opposite numbers are in the same ratio as any other two. 
 Test this on the slide rule with small numbers; for example, 
 set 6 under 2 and notice that 15 is under 5, for 6 : 2 : : 15 : 5. 
 Solve the following proportions by setting the rule so 
 that the answer is always found on the A scale.* 
 
 6 :2 : 
 
 2:6: 
 
 3 :2 ::9 :x : 
 
 15 :x, 
 15 : x, 
 12 :y :: 10 : z. 
 
 Solve 31 : 750 : : .005: x\ first notice that 750 is about 
 twenty times as large as 31. Solve 2300 : .036 :: 990 : x. 
 If 26 inches = 66 centimetres solve the following equa- 
 tion as accurately as possible : 26 : 66 : : 1 : x. What is 
 the length in centimetres of 1 inch? From the slide rule 
 find the approximate length of 4 inches. How many 
 centimetres in 41 inches? 
 
 Set 1 precisely under the special mark that indicates 
 TT (= 3.142) and notice that 7 is nearly under 22 because 
 22/7 is approximately equal to TT. Look along the scales 
 for another ratio which is equal to TT, and find one that 
 is not numerically equal to 22/7 or to 3.142857.... 
 
 * To avoid uncertainty about which scale should be read it is 
 advisable to get into the habit of setting the slide rule so that the 
 answer will always be found on one of the fixed scales, -1 or D; not 
 on either of the movable ones, B or C. 
 
VII THE SLIDE RULE 97 
 
 Then find its decimal value by unabridged division carry- 
 ing out the work until the figures become different from 
 those of the correct value of TT, as was done in 49. 
 
 83. Equivalent Measures. Turn to the back of the 
 slide rule and find a statement about centimetres and 
 inches. It will usually be given in tabular form, such as 
 cm: in. ... 26: 66, together with various other data; 
 if not, use the table given here. Find a statement of the 
 
 26 in = 66 cm 30 atmo = 31k/cm 2 
 
 292 ft = 89 m 128 lb/in 2 = 9 k/cm 2 
 
 35 yd = 32 m 500 lb/in 2 = 34 atmo 
 87 mi = 140 km 
 
 340 ft-lb - 47 kgm-m 
 
 31 in 2 = 200 cm 2 134 hph = 100 kwh 
 
 140 ft 2 = 13 m 2 67 kwh = 58000 Cal 
 
 990 HL = 61 cm 3 TT = 355/113 
 
 23/3 = 680cm 3 V TT = 39/22 
 36 in 3 = 590 cm 3 
 
 14 gal = 53000 cm 3 Zl - 57 11' 45" 
 
 - 3437'.7 
 108 gr = 7 gm = 206264 // ,8 
 
 9 % = 280 gm 1 = 1.74533 X 10~ 2 
 
 1940 av oz = 55 kgm 1' = 2.90888 X 10~ 4 
 
 22 Ib = 10 kgm I" = 4.84814 X 10~ 6 
 
 TABLE OF EQUIVALENTS FOR USE WITH THE SLIDE RULE. The 
 particular numbers are chosen so as to be accurate as far as three 
 significant figures, at least. Thus not only does 26 in. equal 66 cm., 
 but also 26.0 in. = 66.0 cm. The value of TT is not only 355/133, 
 but also 3550/1330, and 35500/13300, even to 355000/133000, as 
 may easily be verified by the process of abridged division. 
 
 relation between pounds and kilograms and calculate 
 your own weight in kilograms, remembering to set the 
 slide rule so that the answer will be found on the A 
 scale. 
 
 84. Reciprocals. Set 4 or 3 or any other small 
 
98 THEORY OF MEASUREMENTS 86 
 
 number on B under 1 on A, and notice that above 1 (or 
 10 or 100) of the B scale will be found .250 or .333 or the 
 reciprocal of whatever small number was used. Verify 
 1 -j- 7 = .142857, as accurately as the apparatus .will 
 allow. 
 
 85. C and D Scales. Experiment with any small num- 
 ber, using the C and D scales, and show where it must be 
 set in order to find its reciprocal on the (fixed) D scale. 
 Notice that if 8 on C is set over 10 on D the value of 1/8 
 will be found under 1, but if 2 is set over 1 the value of 
 1/2 will be found under the 10 instead. 
 
 Try multiplication, division, proportion, and equiva- 
 lents on the C and D scales, remembering to set the slide 
 so that the answer always comes on the stationary (D) 
 scale. If the method for any of these processes has been 
 forgotten experiment with small numbers so that you 
 know the required answer beforehand. In multiplica- 
 tion if the answer runs off the end of the rule set 10 
 instead of 1 on C opposite the first factor, and in division 
 read the result under 10 instead of under 1 when neces- 
 sary. If the fourth term of a proportion cannot be read 
 the runner must be placed over the end of the C scale and 
 the slide then moved so that the other end of the C scale is 
 under the runner. As an example, set 26 on C over 66 on 
 D and show that the number of centimetres in 5 inches 
 is 12.7. 
 
 86. Squares and Square Roots. Set the slide so that 
 the ends of the B and C scales coincide with the ends of 
 the A and D scales. Then move the runner so that its 
 vertical line falls on 9 of the C and D scales and notice 
 that it also comes on 9 2 or 81 of the A and B scales. Set 
 it at 8, 7, 6, 5, etc., on the lower scales and notice that 
 the number on the upper scales just over n on the lower 
 ones is always n 2 . For a slide rule that has no runner set 
 
VII THE SLIDE RULE 99 
 
 1 on C over n on D and 1 on B will indicate n 2 on A. 
 Read the square of 12; of 13; of 19. Find the (four- 
 figure) square of 43, knowing that the last figure must 
 be 9. 
 
 Under any number on the A (or B) scale will be found 
 its square root on the D (or C) scale, but it must be re- 
 membered that any given arrangement of significant 
 figures has two different square roots, according to how 
 it is pointed off (compare 13, ex. 41). For example, 
 V 1500 = 40-; V 150 = 12+; V 15 = 4 - ; V 1.5 
 = 1.2 +; V-15 = .4 -; V .015 = .12 +; etc. Notice 
 that under 150 of the left-hand half of the A scale one 
 of these roots, 122, is found; while the other root, 387, 
 occurs under the 150 of the right-hand half. Since one 
 of these is about three times as large as the other the 
 simple precaution of making mentally a rough pre- 
 liminary calculation of the root will avoid the possibility 
 of obtaining the wrong number. For example, is the 
 square root of .036 given by the significant figures 19 or 
 60, and how should they be pointed off? 
 
 87. Compound Operations. A quotient such as m/n 
 is found by setting n on the B scale under m on the A 
 scale and reading the value opposite the end of B. (If 
 this is not perfectly obvious try it with small numbers.) 
 If m/n is further to be multiplied by some other number, 
 say Xj notice that the slide rule does not need to be re-set 
 as it is already arranged so that the required product 
 will be found over x of the B scale. Find two sevenths 
 of thirteen in this way and notice particularly that it is 
 not necessary to read the value of the fraction 2/7; 
 merely set 7 under 2 and find the required answer over 13. 
 
 Similarly, m?n will be found on A opposite n of the 
 B scale if 1 on C is set at m on D; and Vmn will be found 
 on D under n on B if 1 on B is set to m on A. 
 
100 THEORY OF MEASUREMENTS 88 
 
 A series of fractions like a/m, b/m, c/m, d/m, - , can 
 be read off merely by setting the slide rule for 1/w and 
 looking opposite a, b, c, d, 
 
 The slide rule is usually made with two " cylinder 
 points " marked on the C scale at i/4/ir and i/40/Tr. 
 By placing either one of these opposite the diameter of a 
 cylinder the length of the cylinder on B will be found to 
 indicate its volume on A. 
 
 There is usually a special mark for TT on the left-hand 
 half of the A and B scales, and for ?r/4 (or .7854) on the 
 right-hand half. By placing the end of the B scale 
 opposite the latter the area of any circle will be found 
 on A opposite the diameter on D. 
 
 88. Determination of Circular Functions. In most 
 slide rules the back of the sliding part is provided with 
 three scales, which are named, and sometimes marked, 
 8, L, and T, from above downward. If the slide is 
 placed so that the ends of the S and T scales coincide with 
 the ends of the A and D scales respectively, the sine of 
 any angle from 1 to 90 will be found on A opposite the 
 number of degree and minutes on S; and the tangent of 
 any angle from 6 to 45 will be found on D opposite the 
 number of degrees and minutes on T. The decimal point 
 is located by recalling the facts that sin 90 = 1, tan 
 45 = 1, and if two angles differ only slightly their 
 sines (or tangents) will also be only slightly different. 
 
 By using the slide rule show that sin 70 = .940; write 
 the values of sin 50; sin 30; tan 30; sin 11 30'; tan 
 11 30'; sin 6; sin 5; sin 1. 
 
 Since tan (45 + a) and tan (45 - a) are reciprocals 
 of each other (prove by substituting 45 + a for x in the 
 equation of 45; 12, c), tangents of angles greater than 
 45 are easily obtained from the slide rule. For example, 
 to find tan 49, which is tan (45 + 4), set 41, which is 
 
VII THE SLIDE RULE 101 
 
 45 4, opposite 10 on D and read the required value, 
 1.15, on D opposite the left-hand of the T scale. 
 
 For sines less than .01 and tangents less than 0.1 differ- 
 ent types of slide rule employ different methods, usually 
 based on the formulae sin <5 = tan d = d ( 72). If no 
 special marks for angles are found on the C scale or the S 
 scale use the equations for 1, 1', and I" in 83. 
 
 89. Determination of Logarithms and Antilogarithms. 
 Set 1 on C opposite any number, n, on D, and log n 
 will be found on L opposite a special mark on the back 
 of the slide rule. Try this with small numbers whose 
 logarithms are already known; e. g., log 3 = 0.477. 
 
 90. Questions and Exercises. 1. If decimal points 
 are disregarded one square root of a given arrangement of 
 significant figures is stated ( 86) to be about three times 
 as large as the other. What is the exact value? Why? 
 
 2. Prove that the volume of a cylinder, irr 2 l, is cor- 
 rectly obtained by the process given in 87. 
 
 3. Explain how it is that the process in 88 for finding 
 tan 49 really gives the reciprocal of tan 41. 
 
 4. Set 1 on the C scale over a tentative cube root of n 
 and see whether % on B comes under n on A. Practice 
 this method of finding cube roots. In what way would 
 the marks ^10 and ^100 in Fig. 18 be helpful? 
 
 5. Find a way to use the marks " in/100, 40 20 0" 
 opposite 32 of the A scale in Fig. 18 in order to reduce 
 " American Wire Gauge " to actual diameters of the wires 
 in hundredthsof an inch. Start from the following data: 
 
 No. = .325 inch, no. 1 = .289, no. 2 = .258, no. 
 3 = .229, no. 4 = .204; no. 8 = .128, no. 12 = .081, no. 
 16 = .051, no. 20 = .032, no. 24 = .020, no. 28 = .013, 
 no. 30 = .010. Notice the marks "cm/100" that are 
 located [log] 25.4 times as far to the right, and show that 
 the diameter of no. 18 wire is very nearly 1.00 mm. 
 
VIII. GRAPHIC REPRESENTATION 
 
 Apparatus. A pencil with a sharp point for marking 
 positions accurately on the squared paper of the notebook. 
 
 91. Indication of a Point by Two Numbers. The 
 position of any point on the surface of the earth may be 
 located by two numbers. One of these, the longitude 
 of the point, expresses its distance to the east or west of 
 an arbitrary line, the meridian of reference. The other, 
 its latitude, gives its distance north or south from a 
 definite base line, the equator. 
 
 92. Representation of Two Numbers by a Point. In 
 almost all branches of science a process which is just the 
 opposite of that given above has been found to be of very 
 great value: instead of using two numbers to describe 
 the location of a point the method is reversed and any 
 two related numbers are represented diagrammatically 
 by the position of a point. For example, in order to 
 indicate that the out-door temperature on January eighth 
 was 21 F., a point could be marked on a sheet of paper 
 at a distance of 8 arbitrary units from the left-hand edge 
 of the paper and at the same time 21 units above the 
 bottom of the sheet. If the temperature had fallen to 
 17 on Jan. 9, another point could be marked, located 9 
 units from the left-hand edge and 17 units above the 
 lower edge of the paper. 
 
 93. Representation of Two Variables by a Line. By 
 continuing the process of marking down, or " plotting " 
 a new point for each successive day and its temperature 
 a series of points would be obtained; and if the tempera- 
 tures should be observed at more frequent intervals, 
 every hour or every minute, the points would come so 
 
 102 
 
VIII 
 
 GRAPHIC REPRESENTATION 
 
 103 
 
 close together that they 
 would almost make a 
 continuous line. The 
 ups and downs of the 
 irregular line would in- 
 dicate clearly to the eye 
 the fluctuations of tem- 
 perature that correspond 
 to the onward march 
 of time, which would be 
 indicated by the steady 
 progress of the line from 
 left to right. The 
 ob j ectionable feature 
 of a diagram of this 
 kind would be that no 
 temperature below zero 
 could be represented. 
 
 
 
 ordinate. 
 
 abscissa 
 FIG. 20. GRAPHIC DIAGRAM. 
 The two numbers 6 and 4 are repre- 
 sented by a point (P) whose coordi- 
 nates are the abscissa, or z-value, of 
 6 units measured horizontally, and 
 the ordinate, or ?/-value ; of 4 units 
 measured vertically. Positive values 
 are always measured to the right 
 and upward; negative ones, to the 
 left and downward. 
 
 94. Graphic Diagrams. In order to allow the repre- 
 sentation of negative values of a variable such as tem- 
 perature it is customary not to measure from the bottom 
 of the paper but from an arbitrary horizontal line ruled 
 at a sufficient height to allow space for the data that are 
 to be indicated. The other variable is not always time, 
 but may be a changing quantity which also assumes 
 negative values, so that a vertical line of reference must 
 be ruled at some distance from the left-hand edge of the 
 paper. A graphic diagram consists of these two lines of 
 reference, called axes, a numerical scale along each of 
 them, and the point or assemblage of points which cor- 
 responds to the numerical values that are to be repre- 
 sented. The vertical distance of any point from the 
 horizontal line, or x-axis, is called the ordinate of that 
 point (Fig. 20); and the horizontal distance from the 
 
104 
 
 THEORY OF MEASUREMENTS 
 
 vertical line of reference, or y-axis, to this ordinate is 
 called the abscissa of the point In the diagram the 
 point P has an abscissa of 6 and its ordinate is 4. The 
 abscissa of a point, being measured along the z-axis, is 
 sometimes called the x-value of the point; and the or- 
 dinate, or vertical distance, is called the y-value. Con- 
 sidered collectively, the two distances are called the 
 
 coordinates of the point. 
 The point whose coordi- 
 nates are both zero, viz., 
 the intersection of the 
 two axes, is called the 
 origin of the graphic dia- 
 gram. 
 
 95. Practice in Plot- 
 ting Points. Figure 21 
 shows three points rep- 
 resented by small dots 
 and three others marked 
 by crosses. The highest 
 dot has an z-value of 
 
 X 
 
 FIG. 21. -POINTS ON A GRAPHIC 
 DIAGRAM. The points shown by 
 small dots have no negative values. 
 The points shown by crosses have 
 negative values for one or both .of 
 their coordinates. 
 
 + 5 and a y-value of 
 + 3; when it is consid- 
 ered alone it may be 
 spoken of as " the point 
 
 (5, 3)," the coordinates being written, x-value first, in 
 parentheses and separated by a comma. Notice that 
 the other dots are located at (6, 0) and (7, 1). One 
 of the crosses is located at ( 2, +3). Write down the 
 location of each of the other two. 
 
 Draw two short axes in your notebook. Mark their 
 positive directions X and Y, as in Fig. 21. Lay off a short 
 scale of positive or negative numbers along each axis, 
 numbering every line, or every other line or every fifth 
 
VIII GRAPHIC REPRESENTATION 105 
 
 line, as you prefer. Make a dot for each of the following 
 points: (1, 2); (2, 1); (3, 0); (4, - 1); (5, - 2). Without 
 drawing new axes plot the following points with small 
 X 'son the same diagram: (4,0.5); (0, - 1.5); (2, - 0.5); 
 ( 1, 2); (1, 1); (3, 0). On the same diagram use 
 small +'s for the following: (- 0.5, 0); ( + 0.5, - 2); 
 (- 2, + 3); (0, - 1); (- 1.5, + 2); (- 1, + 1). Frac- 
 tional values are most conveniently plotted by drawing 
 short horizontal and vertical lines that intersect to 
 make a +. On the same diagram plot the following 
 points in this way and then connect them by a smooth 
 free-hand curve: 
 
 rr = 0.6 1.0 2.0 2.3 3.0 3.7 4.0 5.0 6.0 7.0 7.7 
 7/-3.0 3.1 3.1 3.0 2.5 2.0-1.6 1.0 0.5 0.6 1.0 
 
 Take one of the following tables, I-V, as indicated by 
 your instructor, and locate all of its points by means of 
 small +'s on a new graphic diagram. Before drawing 
 the axes find the largest and (algebraically) smallest of 
 the re-values in the table and see that the ?/-axis is not 
 drawn too far to the right or left to leave room for them. 
 Examine the ^/-values in the same way and draw the 
 re-axis as closely under your previous notes as they will 
 allow. Do not start the diagram on a new page or place 
 it so as to use up an unnecessary amount of space. Do 
 not draw lines to connect the separate points. 
 
 96. Orientation of a Graphic Curve. On the squared 
 paper of your notebook draw a rectangle that includes 
 128 of the small squares, making it 8 squares wide and 
 16 squares high. Examine the table on the next page 
 (Table A) and locate the re-axis and i/-axis so that none 
 of the points representing the tabular values shall fall 
 outside of the limits of the rectangle. Draw a second 
 
106 
 
 THEORY OF MEASUREMENTS 
 
 96 
 
 X 
 
 y x 
 
 y ^ 
 
 y x 
 
 y x 
 
 V 
 
 
 
 i - i 
 
 + 1 
 
 2 1 
 
 1 1 
 
 1 
 
 - 1 
 
 1 
 
 4 1 
 
 1 - 1 
 
 2 
 
 
 
 
 
 - 1 -2 
 
 3 
 
 - 1 2 
 
 3 - 6 
 
 
 
 1 
 
 1 
 
 5 1 
 
 - 2 - 2 
 
 5 - 8 
 
 - 2 
 
 2 
 
 1 - 1 
 
 -3 
 
 - 2 - 1 
 
 3 - 6 
 
 - 2 
 
 3 
 
 2 -3 
 
 1 -2 
 
 - 3 
 
 1 - 8 
 
 
 
 4 
 
 3 - 7 
 
 
 
 - 3 1 
 
 2 
 
 2 
 
 3 
 
 4 -6 
 
 - 1 1 
 
 - 4 - 1 
 
 2 
 
 
 
 2 
 
 3 -2 
 
 5 4 
 
 - 1 - 2 
 
 1 -3 
 
 
 
 2 
 
 - 1 - 5 
 
 2 5 
 
 -2 - 5 
 
 
 
 - 2 
 
 4 
 
 1 - 4 
 
 1 1 
 
 -5 -4 
 
 - 1 -3 
 
 2 
 
 5 
 
 
 
 5 2 
 
 -6 - 2 
 
 - 1 - 7 
 
 -3 
 
 7 
 
 -4 
 
 - 1 2 
 
 - 1 
 
 -3 - 3 
 
 3 
 
 4 
 
 -3 -5 
 
 4 4 
 
 - 3 1 
 
 - 2 - 4 
 
 - 1 
 
 3 
 
 - 4 - 4 
 
 5 - 1 
 
 3 
 
 2 - 5 
 
 1 
 
 1 
 
 - 7 -2 
 
 - 1 -2 
 
 - 1 - 3 
 
 3 - 3 ! - 1 
 
 5 
 
 - 6 - 7 
 
 - 1 
 
 -6 1|-1 -2l-4 
 
 2 
 
 -3 -5 
 
 3 1 
 
 -6 -32 -5-3 
 
 3 
 
 - 2 -3 
 
 2 - 1 
 
 - 4 1 
 
 -3 -11 
 
 4 
 
 - 1 - 1 
 
 3 1 
 
 -3 3 
 
 3 3-1 
 
 2 
 
 - 5 - 5 
 
 - 1 2 
 
 2 - 3 4 - 1 | -"3 
 
 6 
 
 - 1 -3 
 
 5 1 
 
 2 - 5 - 1 - 7 i 1 
 
 6 
 
 1 - 1 
 
 2 3 
 
 - 4 03 - 4 
 
 2 
 
 5 
 
 - 7 - 3 
 
 - 1 2 
 
 - 3 - 3 - 1 -2 
 
 2 
 
 6 
 
 - 7 - 5 
 
 1 1 
 
 - 1 - 1 -3|-5 
 
 
 
 - 7 - 1 
 
 - 1 3 
 
 -3 
 
 1 1 
 
 -3 
 
 1 
 
 -6 -3 
 
 3 1 
 
 - 1 - 1 
 
 1 - 4 
 
 - 4 
 
 3 
 
 - 7 - 5 
 
 5 
 
 1 
 
 3 - 9 
 
 - 1 
 
 4 
 
 - 5 - 1 
 
 5 
 
 - 3 
 
 5 - 2 
 
 - 1 
 
 TABLE I TABLE II TABLE III TABLE IV TABLE V 
 
 rectangle of the same size and plot the values of Table B 
 in it without going beyond its boundaries. Examine 
 Tables C and D and plot a graphic diagram for each one, 
 
 X 
 
 V 
 
 
 
 -5 
 
 1 
 
 -8 
 
 2 
 
 - 9 
 
 3 
 
 g 
 
 4 
 
 - 5 
 
 5 
 
 
 
 6 
 
 7 
 
 - 1 
 
 
 
 - 2 
 
 7 
 
 TABLE A 
 
 X 
 
 y 
 
 
 
 y 
 
 rr 
 
 y 
 
 
 
 - i 
 
 
 
 - 4 
 
 
 
 
 
 1 
 
 + 1 
 
 1 
 
 -5 
 
 0.5 
 
 0.25 
 
 2 
 
 3 
 
 2 
 
 -5.6 
 
 1 
 
 1 
 
 3 
 
 5 
 
 3 
 
 -6 
 
 2 
 
 4 
 
 4 
 
 7 
 
 5 
 
 -6.5 
 
 3 
 
 9 
 
 5 
 
 9 
 
 - 1 
 
 2 
 
 4 
 
 16 
 
 - 1 
 
 - 3 
 
 - 1.5 
 
 
 
 5 
 
 25 
 
 - 2 
 
 - 5 
 
 - 2 
 
 4 
 
 - 1 
 
 1 
 
 -3 
 
 - 7 
 
 
 
 - 2 
 
 4 
 
 
 
 
 
 - 3 
 
 9 
 
 TABLE B 
 
 TABLE C 
 
 TABLE D 
 
VIII 
 
 GRAPHIC REPRESENTATION 
 
 107 
 
 in one of the rectangles and with the axes as they have 
 already been drawn if possible, otherwise in a third rec- 
 tangle which is to be of the same size as the previous ones 
 but in which the scales of z-values and ^/-values may be 
 condensed, so that the dimensions of each square may 
 represent two, or five, or ten units, as may be most 
 convenient, or may be expanded, so that one unit may be 
 represented by two or more times the length of a single 
 square. 
 
 In the next table notice that the ^-values lie between 
 the extreme values of + 42 and + 48. In such cases 
 there is usually no objection to dispensing with the a>axis 
 
 FIG. 22. GRAPHIC DIAGRAM WITH CONDENSED SCALES. Notice 
 that the axvalues have been so condensed that the curve does not 
 extend far to the right ; and the 7/-values have been condensed to the 
 same extent, making the curve relatively flat. 
 
 FIG. 23. GRAPHIC DIAGRAM. Notice how both scales have been 
 arranged so that the same table as was illustrated by Fig. 22 now 
 has its values much better displayed. 
 
108 THEORY OF MEASUREMENTS 97 
 
 on the diagram if the scales are plainly indicated. In 
 Fig. 22 there is much wasted space between the curve 
 and the z-axis, and the curve itself is relatively flat. 
 Notice how both of these objections have been overcome 
 in Fig. 23. 
 
 x = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 
 y = 42 42-* 43 431 43 43 43| 43| 44 44 f 45| 46| 47| 48 
 
 TABLE OF VALUES REPRESENTED IN FIGS. 22 AND 23. 
 
 97. Choice of Scales. When only a few points are to 
 be plotted and the range of extreme values is not very 
 great, as in tables A, B, C, and D of 96 it is advisable 
 to use a normal scale of one unit for each square of the 
 ruled paper. All of the graphic diagrams in this chapter 
 and the next one are to be constructed in this way unless 
 special directions are given to the contrary. Condensed 
 scales are most useful for large numerical values, includ- 
 ing data given in round numbers, and for keeping a 
 diagram within the limits of a definite space. Expanded 
 scales are necessary for minute values and those which 
 need accurate representation. For example, when the 
 width of a square is only half a centimetre (one fifth of 
 an' inch) or less it is possible to divide it into tenths by 
 estimation and to show a perceptible difference in 
 position to correspond to a difference- of one tenth in 
 numerical values, but values differing only by a number 
 of hundredths or thousandths need an enlarged scale to 
 make their variations show in the graphic diagram. In 
 some cases the importance of the relation or ratio of 
 z-value to t/-value makes it necessary that both z-scale 
 and ?/-scale shall be the same, whether normal or con- 
 densed or extended. In other cases the two sets of 
 values represent measurements which are mutually in- 
 commensurable (as in the case of time and temperature, 
 
VIII GRAPHIC REPRESENTATION 109 
 
 92 and 93), so that it is theoretically of no importance 
 whether the two scales correspond or not. Where the 
 slant of the curve, or of some important part of it, is 
 fairly uniform, however, it is often more satisfactory to 
 choose the scales in such a way as to make the slope 
 approximately 45 degrees, as in Fig. 23. 
 
 98. General Principles of Plotting. When one of two 
 variable quantities changes as the result of changing the 
 other it is customary to use the horizontal scale of re- 
 values for the independent variable and the vertical 
 scale of ^/-values for the dependent or resultant variable. 
 Thus, for a graphic diagram of temperature and time, it 
 is natural to consider the temperature as being the result 
 of the time rather than the time as depending upon the 
 temperature. Before starting to draw a graphic dia- 
 gram the largest and smallest values of the variables 
 should be observed, so that the axes can be located in a 
 satisfactory position on the paper. The next step should 
 invariably be to lay off a scale along each axis before 
 plotting a single point, and to see that its zero comes at 
 the " origin " and that equal numerical intervals are always 
 represented by equal distances on any one scale, the 
 (positive) z-values always increasing from left to right, 
 and the ^/-values from below upward. This is especially 
 important when plotting a curve from a table like No. 9 
 on page 114, where the ^-values are given at larger 
 intervals in one part of. the table than in another part. 
 
 99. Representation of Tabular Values. After the 
 points corresponding to a set of tabular values have been 
 located on a graphic diagram it is customary to draw a 
 straight line from each point to the adjacent point on the 
 left or right, making a broken line for the " curve " that 
 shows the fluctuations in the value of the dependent 
 variable. If all the points lie on a smooth curve it is 
 
110 
 
 THEORY OF MEASUREMENTS 
 
 100 
 
 hour 
 
 temperature 
 
 
 A.M. 
 
 P.M. 
 
 12 
 1 
 2 
 3 
 4 
 5 
 6 
 
 36.9 C. 
 36.8 
 36.8 
 36.6 
 36.4 
 36.5 
 36.6 
 
 37.4 
 37.4 
 37.6 
 37.5 
 37.5 
 37.6 
 37.6 
 
 7 
 8 
 9 
 10 
 11 
 12 
 
 36.8 
 36.9 
 37.1 
 37.2 
 37.2 
 37.4 
 
 37.7 
 37.6 
 37.4 
 37.4 
 37.2 
 36.9 
 
 advisable to draw such a curve as evenly as possible, but 
 the experimental values given in a table are apt to show 
 
 little irregularities which make 
 it impossible to draw a smoothly 
 flowing curve through their 
 graphic points. In such cases 
 the broken line serves the pur- 
 pose of visually assembling the 
 points in a series, but is not sup- 
 posed to indicate that interme- 
 diate points, if obtainable, would 
 lie exactly along the straight 
 lines. 
 
 Draw a graphic diagram in 
 which the horizontal scale rep- 
 resenits the 24 hours of a single 
 day, beginning at midnight and 
 running through twelve o'clock, 
 noon, to the next midnight. 
 For the vertical ordinates use 
 the temperatures given in the 
 table. Connect each point with 
 the next by means of a straight 
 line, being careful not to omit any point. Notice how 
 much more striking and " graphic " the diagram is than 
 the table; how it shows at a glance facts that could be 
 gleaned from the table only with much greater effort. 
 
 100. Smoothing of a Graphic Curve. In the temper- 
 ature diagram just made it is probable that the little 
 irregularities of temperature are due to accidental causes 
 and would not be exactly repeated in taking the temper- 
 ature of another individual or even of the same individual 
 on another day. In such cases a more typical picture is 
 given by drawing a smooth, flowing curve in such a way 
 
 NORMAL TEMPERATURE 
 OF THE HUMAN BODY. 
 Use a horizontal distance 
 of one square to represent 
 one hour, and a vertical 
 distance of one square to 
 represent one tenth of a 
 degree. 
 
VIII GRAPHIC REPRESENTATION 111 
 
 that it passes through the midst of the scattering points 
 and follows their general upward and downward trend 
 without necessarily cutting most of them or even any 
 one of them. 
 
 It ought not to pass below most of the points, nor 
 above most of them, but should leave them distributed, 
 some above and some below, as evenly as possible, sub- 
 ject to the general condition that it must be a smooth 
 curve that does not show even a tendency to resemble 
 a broken line either by indicating the irregularities of 
 the table in the form of wavelets or by suggesting an 
 unduly sharp turn or " corner " at any point. 
 
 In another place on the squared paper of the notebook 
 plot merely the points . as was done for 99 ; then draw 
 a smooth curve along their general course, outlining it 
 tentatively with a light pencil mark, erasing and cor- 
 recting this until it is satisfactory, and then tracing it 
 plainly with ink.* It should show not more than one 
 downward loop and one larger upward swing. 
 
 This process, which is called smoothing a graphic dia- 
 gram, is advisable only when one can be reasonably 
 certain that the small fluctuations that are eliminated 
 by the process represent unavoidable experimental 
 errors or chance variations or else that they are due to 
 causes which are negligible in the case that is under 
 consideration. Instances have occurred in which even 
 able scientists have missed the discovery of important 
 facts on account of the " smoothing out " of what have 
 seemed to be only accidental irregularities. 
 
 101. Questions and Exercises. 1. What shape is the 
 
 * Unless special drawing apparatus is used, a curved ink line can 
 usually be drawn more neatly if it is made dotted, as in Fig. 54, 
 instead of solid. The points which it summarizes may be marked 
 rather heavily in order to aid the eye, but this must be done uni- 
 formly. 
 
112 THEORY OF MEASUREMENTS 101 
 
 curve of a graphic diagram if each of its points has a 
 2/-value that is one half as large as the corresponding 
 z-value? 
 
 2. What is the most probable value of the average 
 temperature of a healthy individual at 4 P.M.? Would 
 you prefer to decide the matter from the graphic curve 
 of 99 or from that of 100? 
 
 3. Can you see any uniformity or law in the arrange- 
 ment of the points obtained from any of the Tables A, 
 B, C, and D, of 96? Connect them with smooth, free- 
 hand curves, if this has not already been done. The 
 curve of Table A is called a parabola, B is a straight line, 
 C is a hyperbola, and D is a parabola. 
 
 4. Select any two points on the graphic diagram of 
 Table B, and find the difference in their x-values; also 
 the difference in their ^/-values. Select any two other 
 points on the same line, and treat them in the same way. 
 What relationship always exists between four such 
 differences? 
 
 5. How great is the gradient ( 17) of the line repre- 
 sented by Table B? Write an expression for the gradient 
 of any straight line drawn on a graphic diagram in terms 
 of the difference in z-values and the difference in ^/-values 
 of any two points that are located on the line. 
 
 6. 7, 8, 9, 10. Plot the points given in Tables 6, 7 (the 
 numbers that are enclosed in parentheses may be omitted), 
 8, 9, and 10, using the same scale for the z-axis as for 
 the 2/-axis, and making each curve large enough to 
 cover nearly the whole page. In each case a perfectly 
 smooth curve can be drawn which will pass through every 
 point. Number 6 is called a sinusoid or sine curve; no. 
 7 is a curve of tangents; no. 8 is & .parabola; no. 9 is a 
 logarithmic curve; no. 10 is a sinusoid. Extend the curve 
 of no. 10 in both directions as far as the limits of the 
 
VIII 
 
 GRAPHIC REPRESENTATION 
 
 113 
 
 paper will allow, using the dimensions of each small ruled 
 square to represent a distance of ir/Q along each axis. 
 
 X 
 
 y 
 
 0.0 
 
 .00 
 
 0.2 
 
 .20 
 
 0.4 
 
 .39 
 
 0.6 
 
 .56 
 
 0.8 
 
 .72 
 
 .0 
 
 .84 
 
 .2 
 
 .93 
 
 .4 
 
 .99 
 
 .6 
 
 1.00 
 
 .8 
 
 .97 
 
 2.0 
 
 .91 
 
 2.2 
 
 .81 
 
 2.4 
 
 .67 
 
 2.6 
 
 .62 
 
 2.8 
 
 .34 
 
 3.0 
 
 .14 
 
 3.2 
 
 -.06 
 
 3.4 
 
 -.26 
 
 3.6 
 
 -.44 
 
 3.8 
 
 -.61 
 
 4.0 
 
 -.76 
 
 4.2 
 
 -.87 
 
 4.4 
 
 -.95 
 
 4.6 
 
 -.99 
 
 4.8 
 
 -1.00 
 
 5.0 
 
 -.96 
 
 5.2 
 
 -.88 
 
 5.4 
 
 -.77 
 
 5.6 
 
 -.63 
 
 5.8 
 
 -.46 
 
 6.0 
 
 -.28 
 
 6.2 
 
 -.09 
 
 6.4 
 
 + .12 
 
 6.6 
 
 .31 
 
 X 
 
 y 
 
 0.0 
 
 .00 
 
 0.2 
 
 .20 
 
 0.4 
 
 .42 
 
 0.6 
 
 .68 
 
 0.8 
 
 1.03 
 
 1.0 
 
 1.56 
 
 1.2 
 
 (2.6) 
 
 1.4 
 
 (5.8) 
 
 1.6 
 
 (-34.) 
 
 1.8 
 
 (-4.) 
 
 2.0 
 
 -2.19 
 
 2.2 
 
 -1.37 
 
 2.4 
 
 - .92 
 
 2.6 
 
 - .60 
 
 2.8 
 
 - .36 
 
 3.0 
 
 - .14 
 
 3.2 
 
 + .06 
 
 3.4 
 
 .26 
 
 3.6 
 
 .49 
 
 3.8 
 
 .77 
 
 4.0 
 
 1.16 
 
 4.2 
 
 1.77 
 
 4.4 
 
 (3.1) 
 
 4.6 
 
 (8.9) 
 
 4.8 
 
 (11.) 
 
 5.6 
 
 (3.4) 
 
 5.2 
 
 -1.89 
 
 5.4 
 
 -1.22 
 
 5.6 
 
 - .81 
 
 5.8 
 
 - .52 
 
 6.0 
 
 - .29 
 
 6.2 
 
 - .09 
 
 6.4 
 
 + .12 
 
 6.6 
 
 .33 
 
 X 
 
 y 
 
 0.0 
 
 4.00 
 
 0.4 
 
 1.87 
 
 0.8 
 
 1.22 
 
 1.2 
 
 0.82 
 
 1.6 
 
 0.54 
 
 2.0 
 
 0.34 
 
 2.4 
 
 0.20 
 
 2.8 
 
 0.08 
 
 3.2 
 
 0.02 
 
 3.6 
 
 0.01 
 
 4.0 
 
 0.00 
 
 4.4 
 
 0.01 
 
 4.8 
 
 0.04 
 
 5.2 
 
 0.08 
 
 5.6 
 
 0.13 
 
 6.0 
 
 0.20 
 
 6.4 
 
 0.28 
 
 6.8 
 
 0.37 
 
 7.2 
 
 0.47 
 
 7.6 
 
 0.57 
 
 8.0 
 
 0.69 
 
 1.87 
 
 0.4 
 
 1.22 
 
 0.8 
 
 0.82 
 
 1.2 
 
 0.54 
 
 1.6 
 
 0.34 
 
 2.0 
 
 0.20 
 
 2.4 
 
 0.08 
 
 2.8 
 
 0.02 
 
 3.2 
 
 0.01 
 
 3.6 
 
 0.00 
 
 4.0 
 
 0.01 
 
 4.4 
 
 etc. 
 
 etc. 
 
 TABLE 6 TABLE 7 TABLE 8 
 
 The curves of Tables 6 and 7 should both be drawn on 
 a single diagram, and the points Tr/2 ; TT, 3ir/2, and 2ir 
 should be marked on the #-axis in addition to the usual 
 scale. The curve for Table 9 also may be drawn on the 
 same diagram; use the ^/-values as they stand. 
 9 
 
114 
 
 THEORY OF MEASUREMENTS 
 
 102 
 
 X 
 
 y 
 
 0.1 
 
 -3. + .70 
 
 0.2 
 
 -2. + .3Q 
 
 0.3 
 
 -2. + .80 
 
 0.4 
 
 -1. + .08 
 
 0.5 
 
 -1. + .31 
 
 0.6 
 
 -1. + .49 
 
 0.7 
 
 -1. + .64 
 
 0.8 
 
 -1. + .78 
 
 0.9 
 
 -1. + .89 
 
 1.0 
 
 O. + .OO 
 
 1.2 
 
 0. + .18 
 
 1.4 
 
 .34 
 
 1.6 
 
 .47 
 
 1.8 
 
 .59 
 
 2.0 
 
 .69 
 
 2.2 
 
 .79 
 
 2.4 
 
 .88 
 
 2.6 
 
 .96 
 
 2.8 
 
 1.03 
 
 3.0 
 
 1.10 
 
 3.2 
 
 1.16 
 
 3.4 
 
 1.22 
 
 3.6 
 
 1.28 
 
 3.8 
 
 1.34 
 
 4.0 
 
 1.39 
 
 X 
 
 y 
 
 
 
 7T/6 
 
 7T/3 
 7T/2 
 27T/3 
 57T/6 
 7T 
 
 
 .95X/6 
 1.657T/6 
 
 1.917T/6 
 
 1.657T/6 
 .95W6 
 
 
 7lT/6 
 87T/6 
 97T/6 
 
 10ir/6 
 
 llTT/6 
 
 2;r 
 
 - .957T/6 
 -1.657T/6 
 
 -1.917T/6 
 
 -1.657T/6 
 - .95W6 
 
 
 13W6 
 
 14ir/6 
 
 157T/6 
 167T/6 
 177T/6 
 
 .3x 
 
 + .957T/6 
 1.65W6 
 
 1.917T/6 
 
 1.657T/6 
 .957T/6 
 
 
 197T/6 
 
 etc. 
 
 - .95W6 
 etc. 
 
 -7T/6 
 
 -7T/3 
 -7T/2 
 
 etc. 
 
 - .95W6 
 -1.657T/6 
 
 -1.917T/6 
 
 etc. 
 
 TABLE 9 
 
 TABLE 10 
 
IX. CURVES AND EQUATIONS 
 
 Apparatus. A pencil with a sharp point; pencil 
 compass. 
 
 102. Graphic Representation of a Natural Law. 
 
 The graphic diagram which is obtained from a table of 
 measurements or experimental data usually shows ir- 
 regularities, which are sometimes retained by the use of 
 a broken line and are sometimes eliminated by the process 
 known as " smoothing" ( 100). Neither of these pro- 
 
 FIG. 24. PARABOLIC PATH OF A FALLING BODY. The total vertical 
 movement is proportional to the square of the horizontal movement. 
 
 cedures is necessary if the variables follow some definite 
 natural law in regard to their changes or if their relation- 
 ship can be expressed by an equation. In such cases the 
 plotted points in general will lie precisely along a smooth 
 curve without showing any irregularities whatever. For 
 example, a ball that is thrown horizontally will travel, 
 
 115 
 
116 THEORY OF MEASUREMENTS , 103 
 
 under the influence of gravity, along a curved path in 
 such a way that its progress in a horizontal direction during 
 1, 2, 3, 4, ' n seconds will be proportional to the num- 
 bers 1, 2, 3, 4, n, while its downward movement will 
 be proportional to the numbers 1, 4, 9, 16, n 2 . If 
 points are so located on a graphic diagram that the ver- 
 tical distance of each below the z-axis is proportional to 
 the square of its distance to the right of the ?/-axis 
 (Fig. 24) it will be found that a smooth curve can be 
 drawn so as to pass exactly through all of them, and the 
 relationship between the z-value and the ?/-value of each 
 and every point will of course be given by the equation 
 y = kx 2 ( 10). Such a curve is shown more or less 
 steadily by the surface of water that forms a waterfall 
 or by a jet of water that issues from a hose pipe, and is 
 known as a parabola. 
 
 It will be seen that the scales have been so chosen in 
 Fig. 24 that k = 1/5, in other words the equation 
 y = kx 2 has become y = \x 2 . Test the diagram 
 by assuming that x has the value of 3/2, finding what the 
 corresponding value of y must be by solving the equation, 
 and then locating the point (3/2, - 9/20) which has 
 these two numbers for its z-value and ?/-value. Does 
 this point lie on the same smooth curve as the points 
 that are shown? Do 'the same way with each of the 
 z-values 1/2, 5/2, and 7/2. 
 
 103. Graph of an Equation. A single equation that 
 contains two unknown quantities has an infinite number 
 of solutions, for any value whatever may be assigned to 
 one unknown quantity and the corresponding value can 
 always be calculated for the other. Any such solution 
 of an equation that involves x and y will consist of an 
 z-value and a ?/-value, and so can be represented by the 
 position of a point. All of the infinite number of solii- 
 
IX 
 
 CURVES AND EQUATIONS 
 
 117 
 
 tions of a given equation of this sort can theoretically be 
 represented by an infinite number of points on a graphic 
 diagram; thus every point that lies on the curve of Fig. 24 
 corresponds to two numbers, z-^alue and ?/-value, which 
 are a solution of the equation y = ^x 2 . In general, all 
 of the points that represent the solutions of a given equa- 
 tion will be found to lie along a smooth curved or straight 
 line, which is accordingly called the locus or "curve " of the 
 equation. If the locus of an equation extends to an 
 infinite distance, as is the case with the curve of 
 y = ^x 2 , it? distant portions are usually more or less flat 
 or straight and uninteresting, so that there is no objection 
 to omitting them from the graphic diagram. 
 
 X 
 
 y 
 
 
 
 
 
 1 
 
 
 
 2 
 
 6 
 
 3 
 
 24 
 
 4 
 
 60 
 
 5 
 
 120 
 
 -1 
 
 
 
 -2 
 
 - 6 
 
 -3 
 
 -24 
 
 TABLE OF SOLU- 
 TIONS OF THE EQUA- 
 TION y = x 3 x. 
 The ^/-values in- 
 crease so rapidly 
 beyond x = 5 that 
 the curve must be 
 nearly straight. 
 
 FIG. 25. GRAPH OF THE EQUATION y = x* x. Notice that 
 the points ( 1,0), (0, 0), and (+ 1, 0) are not close enough together 
 to determine the shape of the curve without plotting additional 
 points. 
 
 It should be carefully kept in mind that the relation- 
 ship between an equation and its " curve" or locus means 
 
118 THEORY OF MEASUREMENTS 104 
 
 that every solution of the equation gives a point which is 
 located on the curve, and that every point on the curve 
 furnishes a solution of the equation. From these two 
 statements it is evident that the coordinates of a point 
 which is not on the curve will not satisfy the equation 
 (try the point (4, 9/5) shown by the letter P in Fig. 
 24), and that an incorrect solution (try x = 2. y = + 4/5) 
 will give a point which does not lie on the curve. 
 
 It is obviously impossible to calculate the position of 
 an infinite number of points, but since an equation is 
 known to give a smooth curve it is only necessary to plot 
 enough points to prevent uncertainty as to the shape and 
 position of any part of it. 
 
 Make a graphic diagram for the equation y = x 3 x 
 by the following process : Assume that x has various small 
 integral values, positive and negative, and calculate 
 the corresponding values of y, arranging them in tabular 
 form, as shown on page 117. The z- values have not 
 been carried beyond + 5 on account of the ^/-values 
 increasing so rapidly. Negative ^-values are seen to 
 have the same numerical ^-values as positive ones but 
 with the sign changed. The next step is to choose 
 suitable axes and scales ( 98). Plot the points given 
 by the table before proceeding further. It is not prob- 
 able that the line has a straight portion between (1, 0) 
 and '( 1, 0) while the rest of it is curved, so it is neces- 
 sary to calculate at least two more points on the 
 curve, e. g., x = 1/2 and x = 1/2; and these will be 
 found sufficient to determine a smoothly flowing curved 
 outline. 
 
 104. General Procedure. In plotting the curve of an 
 equation the general procedure is to calculate the table 
 of values, substituting successive positive and negative 
 integral values of x (and fractional values if necessary) 
 
IX CURVES AND EQUATIONS 119 
 
 and solving for y, then consider the available space on 
 the paper and draw the axes in a suitable location. 
 Before locating the points that correspond to the tabular 
 values a scale of numbers should always be marked along 
 the a>axis and another along the i/-axis. The two scales 
 need not be the same ( 97), but along each axis, con- 
 sidered by itself, equal distances must always correspond 
 to equal numerical differences, and z-values must always 
 increase to the right and ?/-values increase upward. If a 
 condensed scale is used it is always advisable to let the 
 dimensions of a small square of the ruled paper represent 
 a simple round number; if the first tabular value is 
 9200 do not number the successive squares 9200, 18400, 
 27600, etc., but use the simpler round numbers 10000, 
 20000, 30000. Where the relative value of the z-unit 
 as compared with the y-umt is unimportant the scales 
 can often be advantageously chosen so as to give the 
 chief part of the curve a slope of about 45. In the rest 
 of the exercises of this chapter, however, a single square 
 of the paper is to represent a single unit, in each direc- 
 tion, except where otherwise specified. 
 
 105. The Straight Line. The ' equation y = 2x -f 3 
 represents a sloping straight line. For such a simple 
 equation it is hardly necessary to construct a table of 
 values. Locate the x-axis and ?/-axis in your notebook 
 where there is room for the y-values to extend approxi- 
 mately from + 10 to 10 and for the x- values to 
 extend at least to 5.. Plot four or five points from 
 solutions of y = 2x + 3 obtained mentally, and use a 
 ruler to draw a straight line passing through them and 
 extending a little distance beyond them in each direc- 
 tion. Label this line by writing the equation y = 2x + 3 
 alongside it, close to one end. Without making a new 
 diagram use the same axes and scales for plotting the 
 
120 THEORY OF MEASUREMENTS 105 
 
 following straight lines: y = 3 2x; y = 3 -f- Jz;* 
 y = 3 + Oz; y = 2z; y = - 1 + x; y = - 1 + Jz; 
 2/ = 1 2#; ?/ = 1 x. Label each of these in 
 the same way. Examine the lines whose equations have 
 3 for the numerical term. Do they all pass through the 
 point (0, 3)? Do the lines of the equations that have 
 1 for a numerical term all pass through the point 
 (0, 1)? Which line passes through the origin (0, 0)? 
 Does it seem probable that the equation y = a + bx 
 gives a line that passes through the point (0, a)? Test 
 the point (0, a) by ordinary algebra in order to deter- 
 mine whether it lies on the line y = a + bx. 
 
 Determine the slope ( 38 or 17) of the line which 
 you have drawn to represent the equation y = 2# -f 3, 
 and notice that if the z-value of one point on the line is 
 one unit greater than that of another point on the line 
 then the ^/-value is always two units greater; furthermore, 
 for any two points on the line the difference in ^/-values 
 is twice as great as the difference in ^-values. If the 
 difference in z-values is denoted by Ax and the corre- 
 sponding difference in ^/-values by Ay the last statement 
 can be written in the form of an equation Ay = 2Ax or 
 Ay/Ax = 2.f 
 
 In the line representing the equation y = 3 + %x 
 does a unit change in x correspond to a change of J in y? 
 In y = 3 + Ox does a unit change in x cause no change 
 in y! In y = 3 2x does a unit change in x cause an 
 increase of 2 units in y? Can the slope of this last 
 line be considered as equal to 2? Prove algebra- 
 
 * Notice that this equation, as it stands, is an explicit statement 
 of the value of y, " clearing of fractions " before starting to substitute 
 would only complicate the work. 
 
 t In this notation the symbol A is not used to denote an individual 
 quantity or factor but is a "symbol of functionality" like the "log" 
 or "cos" of the expressions log x and cos x. 
 
IX 
 
 CURVES AND EQUATIONS 
 
 121 
 
 A 
 
 ically that when x increases by one unit (e. g., when 
 
 it changes from m to m + 1) the value of y iii the 
 
 equation y = a + bx will increase 
 
 by b units; thus showing that the 
 
 equation represents a line whose 
 
 slope is numerically equal to b. 
 
 From this it is evident that Ay/ Ax 
 
 = b for the line y = a + bx, and 
 
 that the coefficient of x gives the slope 
 
 of a line if its equation is arranged in 
 
 the form y = a + bx. Turn to 
 
 your diagram and see whether the 
 
 lines y = 3 2x and y = 1 2x 
 
 are parallel (i. e., have the same 
 
 slope). The equation Ay /Ax = 2 
 
 is true of both of them and of any 
 
 other line that runs in the same 
 
 direction. Just as the line y = 3 
 
 2x represents all of the infinitely 
 
 numerous points that lie along its 
 
 locus ( 103) and has an infinite 
 
 number of solutions, so the still more 
 
 general relationship Ay/Ax = 2 
 
 represents an infinite number of 
 
 parallel lines, any one of which 
 
 may be considered as a solution of 
 
 it. 
 
 The equation y = a + bx is the general equation for 
 all straight lines except those that run parallel to the 
 i/-axis. The latter are given by the equation x = k. 
 (Plot the equation x = 2 by substituting integral values 
 of y instead of x, and solving for x instead of y.) 
 The most general equation of the straight line is 
 Ax + By + C = 0. This can be reduced to the form 
 
 A'x 
 
 FIG. 26. THE 
 STRAIGHT LINE. If 
 Pi, PZ, and PS are any 
 points on the (curved 
 or straight) line whose 
 equation is y = a + bx 
 it is easy to show alge- 
 braically that Ay/Ax 
 = A'y/A'z, and thence 
 geometrically (by simi- 
 lar triangles) to prove 
 that the "curve" is a 
 straight line. 
 
122 THEORY OF MEASUREMENTS 106 
 
 y = a-{-bx if B is different from zero, and to the form 
 x = k if B is zero. 
 
 106. The Parabola. On a single graphic diagram 
 draw and label the curves y = x 2 , y = x 2 + 3, y = x 2 , 
 and y = x 2 /lO. The first three of these need not extend 
 beyond x = =t 3, but the last one should be drawn from 
 x = 10 to x = +10. They are all similar figures and 
 are called parabolas; the curves y = x 2 and y = x 2 /lO 
 differ in size but the portion of the latter that extends 
 from 10 to + 10 is of exactly the same shape as the 
 part of the former that is included between x = 1 and 
 x = + 1. Notice that the curve y = x 2 + 3 differs 
 only in position from y = x 2 ; it appears to be the same 
 curve moved three units higher up, and the equations 
 make it clear that for each z-value the ^/-value of the 
 first curve is greater by three than that of the second. 
 
 On another graphic diagram draw and label the three 
 curves y = x 2 3x, y = x 2 6x, and y = x 2 7x. 
 Notice that they are not only of the same shape but of 
 the same size as well. 
 
 The general equation of all parabolas whose axis of 
 symmetry is vertical is y = a + bx + ex 2 . The size of 
 the curve depends only upon the value of c and it is 
 convex downward (" festoon-shaped") if c is positive, 
 convex upward (" arch-shaped") if c* is negative. A 
 change in the value of a has been shown to raise (or 
 lower) the curve through a corresponding distance, and 
 obviously it always cuts the ?/-axis at a height of a. 
 Since any curve cuts the z-axis at the points where 
 y = .it follows that the parabola y = a + bx + ex 2 
 must cross the z-axis at points whose abscissae are the 
 two roots of the quadratic equation = a + bx + ex 2 ; 
 these roots of course depend upon the values of all three 
 of the parameters a, 6, and c. 
 
IX CURVES AND EQUATIONS 123 
 
 107. The Probability Curve. In the chapter on loga- 
 rithms directions were given for calculating the value of 
 y = e~ x2 when x assumes various positive values.* 
 Turn to the table in your notebook that gives the cor- 
 responding values of x and y for this equation and plot 
 its graphic diagram on the vacant page next to it. Turn 
 the notebook so that the x-axis will lie lengthwise of the 
 page and use as large a scale as will conveniently permit 
 the x-values to extend from 0.5 to + 2.5 or 3 (say 20 
 squares = 1 unit, both horizontally and vertically). 
 For negative values of x notice that the curve is symmet- 
 rical with respect to the x-axis; that is, if one point on 
 the curve is (+ 0.5, + 0.78) or (a, b) then another one 
 will be (-0.5, +0.78) or (-a, +6). This could 
 have been inferred from the equation y = e~^, for x 
 occurs in it only as an even power and any function of 
 ( a) 2 must have the same value as the same function of 
 
 (+ a) 2 - 
 
 108. Equation of a Graph. The process of finding the 
 curve that corresponds to a given equation is not usually 
 difficult, but the reverse operation may be much harder. 
 If the given " curve" is a straight line that cuts the 7/-axis 
 its equation will be of the form y = a + bx, and the 
 values of a and b can be determined according to the 
 two italicized propositions in 105. 
 
 Draw a line through the two points (1, 3) and (2, 1), 
 and find its equation. Ans.: y = 5 2x. Draw a line 
 through (0, 2) and (3, 0), and find its equation. Find 
 the equation of a line drawn through (0, 3) and 
 
 * Although an equation is properly a sentence the student will 
 sometimes find it apparently used as a noun. In such cases one 
 side of the equation is to be considered as the substantive and all the 
 rest of it as a parenthetical statement; thus the expression in the 
 text means, "for calculating the value of y (which is the same thing 
 as e~* 2 ) when x assumes. ..." 
 
124 THEORY OF MEASUREMENTS 109 
 
 (5, 0). See if your last two equations can be transformed 
 into x/3 + y/2 = 1 and x/5 +y/ (- 3) = 1, respectively. 
 What is the apparent significance of the denominators m 
 and n in the equation x/m + y/n =1? 
 
 If the given curve is a parabola it has been seen that 
 its equation will be y = mx 2 if the origin is at the vertex 
 of the curve. Turn to your plot of y = x 2 Qx ( 106) 
 and draw a new pair of axes through the point (3, 9), 
 which is the vertex of the curve. If the coordinates of 
 points on the curve, referred to these new axes, are 
 denoted by the capital letters X and F, notice that the 
 same curve corresponds exactly to the relationship 
 F = IX 2 . From this equation it is possible to deduce 
 the original equation as follows: Let P be any point on 
 the curve; notice that its X-value is 3 less than its z-value, 
 also that its F-value is 9 more than its y-value. That 
 is, for every point on the curve the equations X = x 3 
 and F = y + 9 hold good, as well as F = IX 2 . Sub- 
 stituting in the last equation the values of X and F given 
 by the other two, y + 9 = (x 3) 2 or y = x 2 Qx. 
 
 Find the equation of the parabola that was drawn 
 for Table A, 96. 
 
 With a pencil compass draw carefully a circle whose 
 radius is 5 and whose centre is the point (4, 3) on a graphic 
 diagram. The curve will be seen to pass exactly through 
 the points (0, 0), (1, 1), (4, 2), and three corre- 
 sponding integral points in each quadrant. Mark any 
 point P on the circumference, and considering the centre 
 of the circle as a new origin draw the ordinate F of the 
 point P and from its base draw the abscissa X along the 
 new X-axis. From the theorem of Pythagoras and the 
 fact that all radii are equal the coordinates of the point 
 P must satisfy the relationship X 2 -f- F 2 = 5 2 ; and 
 since X = x 4 and F = y 3 this relationship be- 
 
IX CURVES AND EQUATIONS 125 
 
 comes (x - 4) 2 + (y - 3) 2 = 5 2 ,ory = 3 +1/25 -(x- 4) 2 , 
 which is the equation of the given circle. 
 
 109. Change of Scales. Lay off an z-scale in which 
 three squares of the ruled paper correspond to each unit 
 and a ?/-scale of two squares per unit, and plot the circle 
 x 2 + y 2 = 25 from the following twelve points (0, =b 5), 
 (=b 3, 4), ( 4, 3), (zb 5, 0), drawing as smooth a 
 curve as possible. The result is an ellipse, or a " strained" 
 (i. e. t distorted) circle. 
 
 To find the equation of such an ellipse, referred to 
 uniform scales, let each square of the ruled paper be 
 considered as equal to unity. Then each point on the 
 ellipse will be twice as high or low and three times as 
 far to the right or left as a corresponding,, point on the 
 true circle x 2 -f y' 2 = 25. That is, if the coordinates of a 
 point on the ellipse are called X and F, then X = 3x 
 and Y = 2y, in other words, J- of the X-value and J of 
 the F-value will satisfy the relationship for the circle, so 
 that (X/3) 2 + (F/2) 2 = 25. This is an illustration of 
 the general fact that re- writing an equation with xfa and 
 y/b in place of the original x and y will stretch out the 
 length and height of the diagram to a and 6 times the 
 original dimensions, respectively. 
 
 On one graphic diagram plot the loci of x + y = 1 and 
 Z/3 + 1//5 = I.' 
 
 110. Definitions of Circular Functions. The use of a 
 graphic diagram makes it possible to give very concise 
 definitions of the circular functions of an angle of any 
 size: Let a line that coincides with OX (Fig. 27) rotate 
 counterclockwise through an angle a to a new position 
 OP. If the coordinates of the point P are denoted 'by 
 x and y, and its distance from the origin by + r, then 
 sin a = y/r, cos a = x/r, tan a = y/x, cot a = x/y, 
 sec a = r/x, and esc a = r/y. These relationships hold 
 
126 
 
 THEORY OF MEASUREMENTS 
 
 111 
 
 true whether the angle is larger or smaller than 360. 
 and for negative angles when described by a clockwise 
 rotation. They furnish the easiest method of determin- 
 ing the sign of a circular function; in Fig. 28 (< 103) the 
 
 270 
 
 FIG. 27. FUNCTIONS OF AN ANGLE. For an angle of any magni- 
 tude, described counterclockwise from OX, the sine, cosine, tangent, 
 cotangent, secant, and cosecant are respectively y/r, x/r, y/x, x/y, 
 rfx, and r/y, where r is the distance from the point (x, y) on the 
 rotating line to the origin. 
 
 value of y is positive but x is negative; r is always con- 
 sidered positive, so it is immediately obvious that the 
 sine of an angle "in the second quadrant" (between 90 
 and 180) is positive while the cosine is negative. 
 
 111. Questions and Exercises. 1. If one point on the 
 curve of y = x 3 x (Fig. 25) is (a, 6) prove that another 
 one is ( a, b). 
 
 2. Knowing the shape of the graph of y = a-\-bx+cx 2 , 
 what facts can you deduce in respect to the graph of 
 x = a + by + q/ 2 ? 
 
 3. What effect will be produced if the equation of a 
 curve is re-written with x + p and y -\- q substituted 
 everywhere for the original x and yl (Try the parabola, 
 
IX CURVES AND EQUATIONS 127 
 
 y = x 2 , and p = 0, q = 3, if there is any uncertainty.) 
 
 4. What effect will be produced if the equation of a 
 curve is re-written with mx substituted everywhere for 
 the original #? If my for the original i/? 
 
 5. Make a graphic diagram of the equation y = 30/x. 
 Plot enough points to show clearly the form of both parts 
 of the cur ye. On the same diagram plot y = l/x. 
 Each of these curves is a (rectangular) hyperbola. 
 
 6. Plot the curve of y = l/x 2 carefully. How would 
 the appearance of the curve y = 30/x 2 differ? 
 
 7. Solve graphically the two simultaneous equations 
 x 2 -\- y 2 = 25 and y = x 2 3x by drawing both curves on 
 one diagram and locating their points of intersection. 
 
 8. Plot the curves of one or more of the following 
 equations: (a) the "semi-cubical" parabola y 2 = x s ; 
 (b) the finite curve y 2 = x(W x) 3 , using a condensed 
 scale along the ?/-axis; (c) the curve y = x log x] (d) 
 
 X 2 _ y 2 = Q. ( e ) X 2 + 02 , = 0. 
 
 9. Instead of being located by latitude and longitude 
 (rectilinear coordinates, referred to an z-axis and a ?/-axis), 
 the position of a point may be located by its distance and 
 direction, i. e., the length r and the angle a of 110 (its 
 polar coordinates, referred to a pole and an initial line). 
 Obviously, tan a = y/x and r 2 = x 2 + y 2 } or x = r cos a 
 and y = r sin a. The distance and angle are called polar 
 coordinates and are usually indicated by the letters r 
 (radius) and 6 (angle), or by p and 0. 
 
 Make a rough graphic diagram of the curve of the 
 equation x 2 + (y h) 2 = h 2 (a circle that passes through 
 the points (0, 0), (0, 2h), and ( h, + h); compare 108); 
 then substitute x = p cos B, y = p sin 6 and show that the 
 polar equation of this circle is p = 2h sin 6. 
 
 What must be the general shape of the curve p = 20? 
 
 Of (a) p = 2; (b) p = I/cos e or p = sec 6', (c) 6 = 1. 
 
X. GRAPHIC ANALYSIS 
 
 47 
 
 9 
 
 Apparatus. Fine black silk thread; slide rule; gradu- 
 ated ruler; pencil with a sharp point. 
 
 112. Interpretation of Equations. Turn back to the 
 diagrams that were drawn to represent the equations 
 y = 2x and y = 2x + 3 ( 105) and notice that for every 
 point of the first there is another point just three units 
 higher on the second of the two straight lines. The 
 
 equation y = 2x + 3 makes 
 the statement that the value 
 of y Is as much as the 2x of 
 the first equation with three 
 more added. Another way of 
 looking at the same equation 
 can be shown by writing the 
 terms in reversed order, y = 3 
 -f- 2x. This can be considered 
 as making the statement that, 
 for every value of x, y is equal 
 to the amount 3, to which there 
 is further added the amount 
 2x (Fig. 28). It is worth 
 while to form the habit of 
 always investigating the 
 meaning of an equation as 
 far as possible, especially for 
 the student who intends to 
 proceed further with the 
 study of any of the practical 
 applications of mathematics. For example the equa- 
 tion, s = vot + %at 2 , of uniformly accelerated motion 
 
 128 
 
 FIG. 28. GRAPH OF THE 
 EQUATION y = 3 + 2x. The 
 ordinate of any point (D) on 
 the line may be considered as 
 being either 3 more than 2x 
 (CD added on to AC} or 2x 
 more than 3 (BD added on 
 toAB). 
 
X GRAPHIC ANALYSIS 129 
 
 means that the space traversed is equal to the space, 
 vrf, that would have been traversed by an object moving 
 at an initial velocity of VQ without acceleration, plus the 
 \a& of distance that a stationary body would have been 
 made to describe if acted upon by the accelerating force 
 alone an illustration of the addition of vectors, which 
 in this case are directed along the same straight line. 
 The student of physics will find no difficulty in further 
 analyzing each of these two separate terms. 
 
 113. The Graph of y = a + bx. It has already been 
 seen ( 105) that the equation y = a + bx has a straight 
 line for its " curve," and that this line cuts the ?/-axis 
 at a height of a and has a slope that is numerically equal 
 to b. For this reason it is often spoken of as a linear 
 equation, and the law of variation which it illustrates is 
 known as the "straight line" law. It should be noticed 
 that the values of x and y are not proportional except 
 for those cases in which the straight line passes through 
 the origin. 
 
 Draw the loci of the following equations without 
 calculating the values of x and y for any point. For each 
 one rule a straight line in such a position that it inter- 
 sects the y-axis at the required point and has the re- 
 quired gradient. If there is any doubt as to the meaning 
 of a negative value for the b of the equation y = a + bx 
 a few points may be calculated from the equation (5) 
 in the usual manner. (1) y = 2 + 2x. (2) y = 2 + x. 
 (3) y = 2 + Jx. (4) y = 2 + Qx. (5) y = 2 -- \x. 
 (6) y = 4 - 2x. (7) y = 4 - x. (8) y = - J*. 
 (9) y = \x. (10) y = - 2 - x. (11) y = - 2 + x. 
 (12) y = - 2. 
 
 Lay the ruler on the squared paper at random in any 
 position and draw a straight line. Mark the axes along 
 any convenient ruled lines of the paper and determine the 
 10 
 
130 THEORY OF MEASUREMENTS 114 
 
 equation of the line that was drawn, expressing the nu- 
 merical term and the coefficient of x as decimal fractions. 
 1 14. The Straight Line Law. If a homogeneous metal 
 bar is. heated it may be expected to expand in such a way 
 that its length is always proportional to its thickness, and 
 if these two variables are denoted by x and y the relation 
 between them will be given by the equation y = kx, in 
 which k has a constant value. If we compare length and 
 temperature, however, there is no proportionality be- 
 tween them. The bar is not made twice as long by 
 heating it twice as hot, but it is not difficult to show ex- 
 perimentally that there is certain law of relationship, 
 namely, -the change in length is proportional to the 
 change in temperature. If a bar whose length is 10 at a 
 temperature of 15 C. is expanded to 12 at 20 C., then 
 its length at 30 C. will be 16.* In other words, 
 12-10 : 16-10 :: 20-15 : 30-15. Draw a graphic dia- 
 gram to illustrate this example, plotting temperature 
 horizontally ( 98) and length vertically, and notice that 
 the points (15, 10), (20, 12), and (30, 16) lie in the same 
 straight line. The variables x and y are not proportional ; 
 it is only their respective changes or differences that 
 show proportionality. Using the notation of 105, If 2, 
 we can write AT/ : Ax : : A'y : A'x, or AI//AX = k = 2/5 
 for the graph (draw these A's on your diagram), and for 
 the general physical law A(length)/A.(temp.) = k. When 
 two variables show proportional changes they are said 
 to follow a linear law, or a straight line law. 
 
 * These numbers are very greatly exaggerated as compared with 
 those for ordinary materials. Metals expand only a few hundred- 
 thousandths of any dimension when heated a few degrees. Further- 
 more the law is only approximate; careful experiments on a given 
 bar will show that its length at t C. will not be exactly expressible 
 in the form I = J (l + kt), but will need a more complicated equa- 
 tion, I = Jo(l + M.+ W) or even I = I (l + kit + 
 
GRAPHIC ANALYSIS 
 
 131 
 
 Prove (a) that proportionality is always in accordance 
 with the straight line law, but (6) that a straight line law 
 does not mean proportionality between the two variables 
 except when the straight line passes through the origin. 
 
 115. The "Black Thread" Method. If a set of ex- 
 perimental measurements, such as those of temperature 
 and length of a metal bar, are found to correspond ap- 
 proximately to a straight-line law they may be plotted 
 as the x's and y's of a graphic diagram, and their irregu- 
 larities may be eliminated (" smoothed") by drawing 
 the straight line that appears to come closest to all of 
 the points. This is called the black thread method 
 because the position of the line is decided by making use 
 of a stretched thread instead of a ruler; the thread and 
 the points can all be seen at the same time, while a ruler 
 would hide half of the points if properly placed. 
 
 Plot the values given in the table as accurately as 
 possible, marking each point by a minute dot surrounded 
 by a small circle, or by a cross com- 
 posed of a short vertical mark to in- 
 dicate the exact value of the abscissa 
 and a short horizontal line at the 
 exact height of the ordinate. 
 
 Be sure that the page of the note- 
 book rests in a perfectly flat position 
 and stretch a fine black silk thread on 
 it in such a position that it follows the 
 general direction of the points. Move 
 it a trifle toward the top or bottom of 
 the page, also rotate it slightly, both 
 clockwise and counter-clockwise. At- 
 tempt to get it into such a position that 
 it lies among the points like a smoothed 
 curve ( 100), following their general trend but not 
 
 X 
 
 y 
 
 1 
 
 9.8 
 
 2 
 
 8.5 
 
 3 
 
 8.0 
 
 4 
 
 7.2 
 
 5 
 
 6.7 
 
 6 
 
 6.5 
 
 7 6.2 
 
 8 5.5 
 
 9 
 
 5.0 
 
 10 
 
 4.1 
 
 11 
 
 3.9 
 
 12 
 
 3.2 
 
 13 
 
 2.3 
 
 EXPERIMENTAL 
 DETERMINATIONS 
 OF LINEAR VARIA- 
 TION. 
 
132 THEORY OF MEASUREMENTS 116 
 
 necessarily passing exactly through any one of them. 
 See that there are about as many points above the line 
 as below it, but if the high points are more numerous 
 toward one end of the thread and the low ones toward 
 the other end rotate the thread enough to remedy the 
 condition. When the thread is finally arranged in the 
 most satisfactory position do not attempt to draw the 
 line but notice where the thread cuts the z-axis and where 
 it cuts the y-Sixis. From these two numbers calculate 
 the slope (gradient) of the thread, noticing whether its 
 value is positive or negative. Write the equation of the 
 line that is indicated by the thread, making y equal to a 
 numerical value plus a certain number of times z; i. e., 
 write the equation in the form y = a + bx ( 105). 
 
 116. Intercept Form of a Linear Equation. The 
 equation x/m + y/n = 1 must be the equation of a 
 straight line, since it is easily reducible to the form 
 y = a + bx. Substitute zero for the value of x and notice 
 that the corresponding value of y is n. Show likewise 
 that when y is equal to zero x will be equal to m. In 
 other words the graph of x/m + y/n = 1 passes through 
 the points (0, n) and (m, 0) (compare 108, H 2). 
 
 Draw the straight line x/( 3) + y/2 = 1 by ruling 
 a line through ( 3, 0) and (0, 2); then reduce the equa- 
 tion to the form y = a + bx and see whether the coef- 
 ficients a and b verify the ^-intercept* and the gradient 
 of the ruled line. 
 
 Since m and n are the x-intercept and y-intercept of the 
 line x/m + y/n = 1 the equation of a line that cuts 
 both axes can be written immediately without any cal- 
 culation. Write the equation of your black thread de- 
 termination in this form. 
 
 * The points at which a locus cuts the z-axis and the ?/-axis are 
 called its x-intercept and y-intercept, respectively. 
 
X GRAPHIC ANALYSIS 133 
 
 117. The Graph of y = a + bx + ex 2 . Just as the 
 curve y = a + bx can be considered as having its or- 
 dinate for each value of x built up of the ordinate a 
 plus the ordinate bx ( 112, Fig. 28), so the more elabo- 
 rate equation y = a + bx + ex 2 can be considered as 
 representing a curve which is made by piling up the 
 parabola y = ex* upon the slanting line y = a -\- bx. 
 Curiously enough this also represents a parabola in 
 every case in which c is different from zero; the slant of 
 the straight line does not cause the curve to be lop-sided. 
 
 On a single graphic diagram plot both y = O.lz 2 
 and y = 3 + 0.5#. To the ordinates of the latter add 
 (or subtract, as the case may require) the ordinates of 
 the former for each integral value of x, and draw as 
 smoothly as possible the resultant curve of the equation 
 2/ = (3+0.5z)+ (O.I* 2 ). 
 
 If a curve that is obtained from experimental measure- 
 ments looks like a portion of a parabola it is possible to 
 draw an approximate tangent, find its linear equation, 
 and then determine a value of c that will make the equa- 
 tion y = a + bx -f ex 2 fit the given curve. An algebra- 
 ical procedure that accomplishes the same result is to 
 measure the coordinates of some point on the curve 
 (say (2, 3)) and substitute in the general equation (giving 
 3 = a + 6X2 + cX2 2 ); repeating this with two more 
 points gives three equations, from which the values of the 
 unknown a, 6, and c may be determined. For a graphical 
 procedure it is usually more satisfactory to complete a 
 free-hand parabola as far as its vertex, if that is not 
 already present, and then continue as in the following 
 example. 
 
 118. Law of Density- Variation for Water. The den- 
 sity of water at different temperatures is given in the 
 table. If the density is called y ( 98) and the temper- 
 
134 
 
 THEORY OF MEASUREMENTS 
 
 118 
 
 ature x, it is required to find the numerical values of the 
 coefficients of the equation y = a + bx + ex 2 that will 
 express the law of variation. 
 
 The first step is to make a careful graph. Use ex- 
 tended scales for both ^-values and ^/-values so that the 
 diagram will cover practically a whole page of your 
 notebook, noticing that the ^/-values need not include 
 
 zero but only extend 
 from .995 near the bot- 
 tom of the page through 
 .996, .997, .998, and .999 
 to 1.000 near the top. 
 The curve will be seen 
 to have the appearance 
 of an " arch-shaped " 
 parabola ( 106), so that 
 it is evident that its 
 equation will be approxi- 
 mately y = mx 2 if the 
 origin is located at the 
 vertex of the curve, 
 namely at the point (4, 
 .99997). With the origin 
 in this position the ordi- 
 nate for 4 will obviously 
 
 be zero, for a temperature 2 higher than this the ordinate 
 will be -.00003 (i. e., .99997 - .99994), for 4 higher it 
 will be .00012, etc. These ^/-values and ^-values have 
 been given in the second table, where for simplicity the 
 negative signs and the decimal points have been omitted. 
 If the values in the second table correspond to an 
 equation of the form y = kx 2 the square root of y must 
 be proportional to x itself. Using the slide rule, read 
 off the square roots of the numbers in column y and enter 
 
 temp. 
 
 dens. 
 
 
 2 
 4 
 6 
 
 8 
 10 
 
 .99984 
 .99994 
 .99997 
 .99994 
 .99985 
 .99970 
 
 12 
 
 14 
 16 
 
 18 
 20 
 
 .99950 
 .99924 
 .99894 
 .99859 
 .99820 
 
 22 
 24 
 26 
 
 28 
 30 
 
 .99777 
 .99730 
 .99678 
 .99623 
 .99564 
 
 X 
 
 y ^y 
 
 
 
 
 
 
 2 
 
 3 
 
 
 4 
 
 12 
 
 
 6 
 
 27 
 
 
 8 
 
 47 
 
 
 10 
 
 73 
 
 
 12 
 
 103 
 
 
 14 
 
 138 
 
 
 16 
 
 177 
 
 
 18 
 
 220 
 
 
 20 
 
 267 
 
 
 22 
 
 319 
 
 
 24 
 
 374 
 
 
 26 
 
 433 
 
 
 RELATIONSHIP BETWEEN TEM- 
 PERATURE AND DENSITY OF WATER. 
 
GRAPHIC ANALYSIS 
 
 135 
 
 them in the vacant column headed ^y. Since the x's 
 in the first column and the ^y'& in the third one are 
 proportional their relative magnitudes can be determined 
 by the black thread method. 
 
 Plot their values on another graphic diagram, unless 
 special directions to the contrary are given by your 
 instructor, and determine the slope with the black thread. 
 Since this is a case of proportion the thread must of 
 necessity pass through the origin ( 114, If 2), even though 
 it may appear to lie less evenly along the row of points 
 than it otherwise would. (Measurements which must 
 necessarily fulfill a certain conditional 
 relationship are called conditioned 
 measurements and will be considered 
 later. In this case notice that the 
 temperature and density of water are 
 not in themselves conditioned meas- 
 urements, but we are attempting to 
 make them satisfy a " condition," 
 namely, that they shall follow the law 
 y = a + bx + ex. 2 ) The value ob- 
 tained for the slope of the black thread 
 will probably be in the neighborhood 
 of 5/6 or 0.83. 
 
 If V y = -83x, then, it follows that 
 y = .69z 2 , y being expressed in hun- 
 dred-thousandths as in column 2 of the 
 second table and being measured downward from the 
 level of the vertex of the parabola. Instead of measuring 
 ^/-values downward from the original 99997 it will be 
 found easier to increase them by 3, as in the third column 
 of the next table, and then measure the results down- 
 ward from the level 100000, which is probably one of 
 the ruled lines of the plotting paper. 
 
 X 
 
 .69x 2 
 
 2/ + 3 
 
 
 
 
 
 3 
 
 2 
 
 3 
 
 6 
 
 4 
 
 11 
 
 14 
 
 6 
 
 25 
 
 28 
 
 8 
 
 44 
 
 47 
 
 10 
 
 69 
 
 72 
 
 12 
 
 99 
 
 102 
 
 14 
 
 135 
 
 138 
 
 16 
 
 177 
 
 180 
 
 18 
 
 224 
 
 227 
 
 20 
 
 276 
 
 279 
 
 22 
 
 334 
 
 337 
 
 24 
 
 398 
 
 401 
 
 26 
 
 467 
 
 470 
 
 TABLE OF VAL- 
 UES FOR y = .69z 2 
 AND FOR .69rc 2 + 3. 
 
136 THEORY OF MEASUREMENTS 119 
 
 On the graphic diagram already used for plotting the 
 density of water lay off the values of y + 3 (in .OOOOl's 
 of a unit of density proper) downward from the line 
 y = 1.00000, noticing particularly that x = is now at 
 the vertex of the curve (at 4, not at 0). If the work 
 has been carefully done it will be seen that this new curve 
 of the equation y = .69z 2 , or more accurately, 
 100000?/* = .69z 2 forms a fairly good approximation 
 to the unknown relationship of the empirical values of 
 density and temperature. 
 
 The last step that remains to be taken is to reduce the 
 equation 100000?/ = - .69z 2 to the original axes of 
 temperature and density. Since x is zero when t (tem- 
 perature) is 4, and 2 when t is 6, etc., it is plain that 
 x = t 4 every where; ;in the same way y = d 0.99997. 
 Substituting these values in the original equation 
 lOOOOOi/ = - .69z 2 gives 
 
 d = 0.99986 + 0.0000522J - 0.0000069J 2 
 
 which is the required law expressing the relationship be- 
 tween density and temperature. 
 
 119. Typical Curves. Any law of change (as far as 
 finite values of the variables are concerned) can be 
 expressed by an equation of the form y = a + bx 
 + ex 2 -f dx* + ex 4 + if enough terms are used, 
 but it will easily be understood that the method soon 
 becomes difficult to handle. Sometimes the appearance 
 of the curve makes it possible to guess that its equation 
 is of some particular form (Figs. 29-34), or the form 
 may be deducible from theoretical considerations (c/. 
 
 * The large coefficient is needed because the ^/-values do not 
 properly run into the hundreds of units as would appear from the 
 table but are condensed within a small fractional range (.99564 to 
 .99997) if they are to represent densities correctly. See 109, and 
 111, No. 4. 
 
GRAPHIC ANALYSIS 
 
 137 
 
 Fig. 33). For a curve that deviates only slightly from a 
 straight line it often happens that the black thread 
 method will give a linear law that is a sufficiently good 
 approximation for practical purposes. The parabola 
 can usually be fitted fairly well to a curve that shows a 
 single upward or downward sweep, and is easier to apply 
 than the exponential curve, which is so often used instead. 
 
 120. Linear Relationship by Change of Variables. 
 In the previous section the values of y were not pro- 
 portional to those of x, but a straight line was obtained 
 by plotting x and V y instead of x and y. A transforma- 
 tion of this sort can always be made when the type of 
 equation has been picked out and its numerical constants 
 are to be determined. Thus if y = e ax the transformed 
 equation log y = ax log e shows at once that x and log y 
 are proportional; if y = ax/(b -f- x) it will be found that 
 y and y/x follow the straight line law; if y = a/x either 
 of the variables will be directly proportional to the 
 reciprocal of the other; etc. 
 
 The volume of a certain mass 
 of air was found to vary, under 
 changes of pressure to which 
 it was subjected, according to 
 the numbers in the following 
 table. Assuming that the 
 pressure and volume are in- 
 versely proportional, represent 
 the relation between them by 
 an equation after plotting a 
 smoothed curve on a graphic 
 diagram with a condensed x- 
 scale (pressures) and an ex- 
 panded i/-scale (volumes) : 
 Since y is inversely proportional to # a new variable, 
 
 pressure 
 
 volume 
 
 760 mm Hg 
 
 8.1 cm 3 
 
 830 
 
 7.2 
 
 926 
 
 6.6 
 
 1022 
 
 5.8 
 
 1125 
 
 5.3 
 
 1230 
 
 4.9 
 
 1340 
 
 4.5 
 
 1410 
 
 4.1 
 
 1520 
 
 4.0 
 
 1600 
 
 3.9 
 
 EXPERIMENTAL DETER- 
 MINATION OF pv- VARIATION OF 
 A GAS. The law of varia- 
 tion for a constant mass of 
 gas is known to be of the form 
 v = k/p. 
 
138 
 
 THEORY OF MEASUREMENTS 
 
 120 
 
X GRAPHIC ANALYSIS 139 
 
 l/y, can be obtained, which will vary directly with x. 
 By plotting x and l/y and holding a black thread so as 
 to pass through the origin ( 114, If 2) find an equation 
 connecting x and l/y, and reduce it to a simple equation 
 between x and y. The result will probably be in the 
 neighborhood of v = 6000/p. Plot this equation on the 
 same graphic diagram as the smoothed curve and notice 
 how closely the two loci correspond. 
 
 FIG. 29. THE. STRAIGHT LINE. Its equation may be written 
 either y = a =fc (a/m)x, taking care that the right sign is used for 
 the gradient; or x/m + y/a = 1, if the signs of m and a are taken 
 according to the usual convention. 
 
 FIG. 30. THE PARABOLA ay = x 2 . The equation shows that it 
 cuts the line y = x at the point (a, a). While running infinitely 
 far upward the curve also extends infinitely far to the right or left, 
 but approaches verticality, and its ends subtend an angular distance 
 that, seen from the vertex, approaches zero. All parabolas are 
 similar figures. 
 
 FIG. 31. THE RECTANGULAR HYPERBOLA xy = a 2 . The curve 
 consists of two separate parts (" branches"), each of which extends 
 to an infinite distance and approaches two fixed lines, called its 
 asymptotes, without ever reaching them. All rectangular hyperbolas 
 are similar figures. 
 
 FIG. 32. THE CURVE y = a*fx 2 . I'js equation shows that it 
 passes through the points (=fc a, + a), ( 0, + QO ), and (db , +0), 
 and that it is symmetrical with respect to the ?/-axis since there is a 
 point ( m, + n) corresponding to every point (+ m, + n)> 
 
 FIG. 33. THE EXPONENTIAL CURVE y = e*. A remarkable 
 property of this curve is that its slope is everywhere equal to its 
 ordinate at the corresponding point. In the more general equation 
 y = e mx the slope is proportional to the ordinate, so that the curve 
 may be used to represent a relationship like Newton's Law of Cooling; 
 viz., the rate at which a body loses temperature is proportional to 
 the temperature itself. Curves for m = 2, m = 1 and m = 1/2 
 are shown by dotted lines. 
 
 FIG. 34. THE CURVE OF ERRORS yfb = e~ (x f a)2 . Three different 
 values of a are represented for a large value of b and one of a for a 
 smaller one of 6. 
 
140 THEORY OF MEASUREMENTS 121 
 
 121. Questions and Exercises. 1. With the black 
 thread find the best straight-line approximation for 
 v = 6000/p or for your smoothed curve. Ans. : approxi- 
 mately y = 7.7 A7x. 
 
 2. Name the kind of curve that would correspond to 
 the equation obtained by solving 
 
 8.1 = a + 6(760) + c(760) 2 , 
 5.3 = a + 6(1125) + c(1125) 2 , 
 4.0 = a + 6(1520) + c(1520) 2 , 
 
 for a, 6, and c, and substituting the values so obtained 
 in the equation y = a + bx + ex 2 . For what purpose 
 could the resultant equation be used? 
 
 3. Explain how an equation of the form y e ax 
 could be determined for the smoothed curve of the last 
 section. 
 
 4. Show that the distance between the points (3, 4) 
 and (7, 5) is equal to i/(3 - 7) 2 + (4 - 5) 2 , and write 
 a general formula for finding the distance between any 
 two points, such as (zi, yj and (x 2 , 2/2). 
 
 5. The equation of the straight line that passes through 
 the two points (xi, y\) and (z 2 , 2/2) is 
 
 2/2 - 
 
 Draw a diagram of the line and the two points, and 
 explain the significance of the subtracting, the dividing, 
 and the equating in the above formula. 
 
 6. Guess at the equation, of table 9 in 101. Test 
 the equation by substituting a few tabular values, and 
 if your first estimation was faulty, make a better one. 
 
 7. Prove that every point on the parabola of Fig. 30 
 is as far from the line y = a/4 as it is from the point 
 
X GRAPHIC ANALYSIS 141 
 
 (0, + a/4). The line is called its directrix and the point 
 its focus. 
 
 8. If e ax is always identical with (e a ) x and if 2.718 2 - 303 
 is equal to 10, turn to Fig. 33 and see how the curve 
 y = 10* would run. (The reciprocal of 2.303 is .4343.) 
 Decide how the curve x = I0 y would compare with it. 
 Have you drawn the latter curve previously? 
 
XI. INTERPOLATION AND EXTRAPOLATION 
 
 Apparatus. Black thread; slide rule; pencil with a 
 sharp point. 
 
 122. Definitions. The process of drawing the locus 
 of an equation by plotting a few isolated points and filling 
 up the intermediate positions with a smooth curved line 
 involves the tacit assumption that the values of y for the 
 intervening values of x would have been found to vary 
 in this predeterminate manner if they had been calculated. 
 That is, the value of a function that corresponds to a 
 certain magnitude of its independent variable need not 
 be obtained by calculation in all cases, but may often be 
 determined by comparing it with the values which the 
 function is known to have when the variable is larger 
 and smaller than in the particular case that is under in- 
 vestigation. When a few known values are used for the 
 purpose of determining an intermediate unknown value 
 the latter Is said to be found by interpolation. For 
 example if the population of a city were known for each 
 of the years 1850, 1860, 1870, 1880, 1890, and 1900, one 
 could guess fairly accurately what the population 
 amounted to in the year 1875, even if the given values 
 should not follow any known law or any recognizable 
 type of curve. 
 
 The process of using a certain range of values for de- 
 termining a value that lies outside of that range is called 
 extrapolation (Latin, extra, outside; inter, between; 
 polire, to make smooth) . Thus, from the data mentioned 
 above it would be possible to make some kind of an 
 estimate of the population for the year 1840, or for 
 1910, or perhaps even for 1920. 
 
 142 
 
XI INTERPOLATION AND EXTRAPOLATION 143 
 
 Turn to your diagram of the daily variations in body 
 temperature ( 99) and determine the normal temper- 
 ature of the human body at 9:30 A.M. It should be 
 about 37.07, .08, or .09. What was the temperature of 
 this individual at 9:30 A.M. on the day of the experiment. 
 Ans.: probably about 37.15 or 37.16. 
 
 Interpolation is a process that is trustworthy only 
 when the data are sufficiently numerous and are given 
 at sufficiently close intervals, and when their variation 
 is not too irregular. It would be impossible to inter- 
 polate the values y = x* x (Fig. 25) from the three 
 points (- 1,0), (0, 0), ( + 1, 0); or to fill in a free-hand 
 parabola for y = x 2 2x 3 if the only data were the 
 points (- 1, 0) and (+2, - 3). 
 
 123. The Principle of Proportionate Changes. It is 
 always necessary to make an assumption 
 of some kind when a process of interpo- 
 lation is used. In obtaining logarithms 
 from a table by interpolation it is as- 
 sumed, for example, that the logarithm ELEMENTARY 
 of 2718 is 8 tenths of the way from log STRAIGHTNESS. 
 2710 to log 2720 (66), or in general -When a circle 
 that any change in a logarithm is pro- that is drawn 
 portion al to the change in the corre- around 
 
 spending natural number. This is an n a 
 
 smooth curve is 
 instance of the linear law (114), and made smaller 
 
 may be expressed as A (log x) = k&x. and smaller it is 
 It is true only for small differences (log cut more and 
 300 - log 200 is not equal to log 200 more nearly into 
 - log 100), because thecurveof, = log 
 x is nowhere nearly a straight line unless 
 a very small stretch of it is considered by itself. 
 
 Plot a graphic diagram of y = log x from x = to 
 x = 10 using a large scale on the z-axis (5 n's = 1 unit) 
 
144 THEORY OF MEASUREMENTS 123 
 
 and a much larger one on the ?/-axis (20 n's = 1 unit). 
 Do not use any z-values except 0.1, 0.2, 0.4, 0.6, 1.0, 
 2.0, 3.0, 10.0. Draw a free-hand curve as smoothly 
 as possible through the corresponding points. Measure 
 the ^/-values for x = 3.5, 2.5, 1.4, and 0.8; then verify 
 each by referring to the tables. Notice that the points 
 on the curve for which x = 1, 2, 3, do not lie in a straight 
 line. Notice that the short stretch of curve that includes 
 the points x = 2710, 2718, 2720 has no perceptible cur- 
 vature. If the points are specified with sufficient ac- 
 curacy, however, it will be found that no three of them 
 lie in the same straight line; thus four-place logarithms 
 may be safely interpolated if the values for every three- 
 figure natural number are given, but five-place accuracy 
 for the logarithms necessitates four-figure values for the 
 natural numbers in the first part of a logarithm table, 
 that corresponds to the more sharply curved part of the 
 graph (compare tables in appendix). 
 
 The fact that a curve is " smooth" means that a very 
 short stretch of any part of it deviates very little from 
 a straight line. In other words, if a small circle is de- 
 scribed around any point of such a curve its circumference 
 will be cut by the curve at two points whose angular 
 distance approaches 180 as the circle is made smaller 
 and smaller (Fig. 35). This property of a curve is known 
 as elementary straightness, the term " element" being 
 used in the sense of "a very small portion," and is char- 
 acteristic of the curves of all equations in which y can 
 be expressed as a rational function of x. Some ." tran- 
 scendental" equations* lack elementary straightness at a 
 few points, but in general the process of linear inter- 
 
 * A transcendental equation is one that involves non-algebraical 
 functions; for example, y = a; log a:. The curve of this equation 
 cuts the elementary circle around (0, 0) at only one point, i. e., 
 comes to an abrupt end at the origin. 
 
XI 
 
 INTERPOLATION AND EXTRAPOLATION 
 
 145 
 
 polation is applicable to any tabular values that are 
 given at sufficiently small intervals. 
 
 124. Examples of Linear Interpolation. Consult the 
 table in 118 and state the density of water at 21 C. 
 
 Water boils at 100 C. when the barometric pressure 
 is equal to that of a 760-mm. column of mercury; to 
 make it boil at 101 C. the pressure must be raised to 
 788 mm. Hg. What should an accurate thermometer 
 register in boiling water when the barometer stands at 
 775? 
 
 Plot a few points for the equation y = x 2 2 and fill 
 out a free-hand curve. When x is 1 y is 1; when x 
 is 2 y is -f- 2. Accordingly, if the locus were not curved 
 it would intersect the z-axis at 
 x = 1J. Substituting x = 1.3 
 gives y = .31; substituting 
 x = 1.4 gives .04 ; substi- 
 tuting x = 1.5 gives + .25. 
 The stretch of the curve that 
 lies between 1.4 and 1.5 (Fig. 
 36) is so nearly straight that 
 the point where it intersects 
 the #-axis can be found quite 
 accurately by the law of pro- 
 portionality : Ay = .29 for 
 Ax = 0.1, so Ay should equal 
 the .04 required to bring y 
 up to zero for .04/.29 of 0.1, 
 or .014, beyond 1.4. Sub- 
 
 FIG. 36. SOLUTION BY 
 "DOUBLE POSITION." Any 
 equation can be solved ap- 
 proximately by aid of a 
 graph. Then by using two 
 z-values that are near the re- 
 quired root, x, and interpolat- 
 ing a y-value along a straight 
 line it is possible to find as 
 close an approximation, P, as 
 is desired. 
 
 stituting x = 1.414 gives 
 y = .000604; substituting x = 1.415 will be found to 
 give y = + .002225. Another application of the prin- 
 ciple of proportionality gives x = 1.414 + .000214 or 
 1.414214, and thus the calculation of the value of V2 
 11 
 
146 
 
 THEORY OF MEASUREMENTS 
 
 124 
 
 proceeds with continually increasing rapidity. This, of 
 course, is a determination of the value of one root of the 
 equation = x 2 2, and is known as the method of 
 false position, or of double position. It can be used for 
 any equation whatever containing one unknown quan- 
 tity by arranging it in the form 0(z) = 0* and drawing 
 a graphic diagram of y = <l>(x). 
 
 A delicate beam balance has a pointer which swings to 
 a position of rest at 8.0 on an arbitrary scale when an 
 unknown mass is balanced against standard weights of 
 14.837 grams but stops at 10.5 when the weights are 
 changed to 14.836. What weights would be required 
 to bring the pointer to the central position, which is at 
 10.0 on the scale? Ans.: 14.837 - f(.OOl); or 14.8362. 
 
 A telephone company charges for measured service at 
 the yearly rates given in the table. It will be plain that 
 an increase of 200 messages means an 
 additional cost of nine dollars, making 
 the rate for an increased number of 
 messages equal to $9/200 m, or 4.5c. 
 per message. At this rate 600 mes- 
 sages should cost 27 dollars ; accord- 
 ingly, it is evident that the rate for 
 any number of messages is made up 
 of two parts, a flat charge of $21 plus 
 a message rate of $9 per hundred. If 
 the number of messages is represented 
 by x and the total cost by y, then the 
 equation connecting them will be y = 21 -f .045:r, show- 
 ing straight-line variation without proportionality. 
 
 * The expression $(rc) means "any function of x," and may be 
 used as a general form to denote 3z 2 , log x, 2a x , or any other function^ 
 just as the general symbol m may be used to stand for the number 
 2, or 100, or any number whatever. As alternative notations 
 /( ), and F.( ) are often used. 
 
 no. of 
 messages 
 
 charge 
 
 600 
 
 800 
 
 1000 
 
 1200 
 
 $48 
 57 
 66 
 75 
 
 EXAMPLE OF A 
 "READ INESS-TO- 
 SERVE" CHARGE 
 INCORPORATED 
 WITH A RATE 
 CHARGE. 
 
XI INTERPOLATION AND EXTRAPOLATION 147 
 
 125. Graphic Interpolation. In general, tabular values 
 do not follow the straight-line type of variation, and 
 rather complicated formulae may be required for purposes 
 of interpolation. In case a graph can be drawn, how- 
 ever, there is usually no difficulty in constructing a 
 smooth curve (or " smoothed" curve, as occasion may 
 require) and obtaining any intermediate values simply 
 by measuring them on the graph. The process is some- 
 times uncertain or erroneous if the given values are not 
 close enough together or if their variation is too irregular. 
 It will have been noticed that the problem of finding 
 intermediate values is closely allied to the problem of 
 finding a law of variation or of finding the equation of a 
 given curve. In case a law or equation is known, unless 
 it is a complicated one it will usually be found easier to 
 substitute and calculate values than to interpolate them. 
 
 126. Graphic Tables. If the variation of two quanti- 
 ties is known to -follow the linear law it is often conveni- 
 ent to make a graphic table by plotting any two points 
 and ruling a straight line through them. 
 
 Lay off a scale of values from to 20 along the z-axis 
 and label it " inches." Lay off a scale from to 50 along 
 the ?/-axis and label it " centimetres." Rule a straight 
 line through the two points (0, 0) and (13, 33). Explain 
 how a diagram of this sort can be utilized. 
 
 According to Hooke's Law, the difference in length 
 (stretching) of a spring is proportional to the difference 
 in force applied to it. If a spring that is hung in front 
 of a scale has a length of 12 cm. when no weight is 
 attached to it, and becomes 14.85 cm. long when a 
 weight of 1 gm. is hung on it, construct a graphic table 
 which will enable you to reduce its indicated centimetres 
 to grams of weight (see 104). 
 
 127. Interpolation Along a Curve. If the change in 
 
148 THEORY OF MEASUREMENTS 128 
 
 two variables is not in accordance with a linear law it is 
 possible to use certain interpolation formulae for obtaining 
 intermediate values, but it is usually much easier to 
 make use of graphic methods. The known data are 
 plotted as a series of points, and these are either con- 
 nected by a smooth curve or are investigated with a view 
 to discovering an equation that will adequately represent 
 them. If they appear to lie along a curve that has a 
 vertical or a horizontal asymptote (Fig. 31) the hyper- 
 bola xy = k may be tried, with a suitable choice of tem- 
 porary axes and scales. If the curve has a single upward 
 or downward sweep the exponential curve y = e ax is 
 frequently used, but the parabola y = ax 2 is generally 
 easier to handle and can usually be fitted to the given 
 points just as satisfactorily. It may be turned so as to 
 have its axis horizontal, if this position seems more suit- 
 able, by interchanging the variables and writing the 
 equation ax = y 2 . 
 
 In trying to fit a parabola to the part of the curve of 
 y = log x that lies between x = 5 and x = 15 would you 
 prefer to have its axis horizontal or vertical? If vertical, 
 would its vertex be directed upward or downward? If 
 horizontal, would its vertex be directed to the left or to 
 the right? Plot a logarithmic curve rapidly, on a small 
 scale, if there is any difficulty in answering the questions; 
 compare it with the curves for y = x 2 , y = x 2 , x = y 2 , 
 and x = y 2 . 
 
 128. Insufficiency of Data. When certain tabular 
 values are given and others are to be obtained by inter- 
 polation it must always be remembered that the known 
 values are the only actual data and that nothing else can 
 be obtained without making some kind of an assump- 
 tion ( 123). If it is assumed that the points all lie 
 along a smooth curve there is always a possibility that 
 
XI INTERPOLATION AND EXTRAPOLATION 149 
 
 the assumption is incorrect. It may even happen that 
 the given points appear to be irreconcilable with a 
 smooth curve or with a uniform law, as in the following 
 case : Electricity is sold by the kilowatt-hour (abbreviated 
 KWH.) and four consumers pay the same rate to one 
 company. The first is charged $1.08 for 9 KWH.; the 
 second, $1.47 for 21 KWH.; the third, $0.99 for 11 KWH.; 
 and the fourth, $1.62 for 18 KWH. Find the rate. 
 
 Plot the four points, using 1 square along the z-axis 
 for each KWH. and 1 square along the ?/-axis for each 
 $0.10. It will be seen that it is impossible to decide 
 where a smooth curve should run. This is a case of 
 what is called a "step meter rate" and gives a broken 
 line, not a smooth curve. The rate is " 12c. per KWH. 
 if less than 10 KWH. are used; 9c. per KWH. if the con- 
 sumption is between 10 and 15 KWH.; 7c./KWH. if 
 over 15." Draw the locus for this rate on the same 
 graph and notice that it passes through the four points. 
 The objectionable feature of sometimes charging less 
 when the use of the current is greater is so apparent that 
 a rate of this kind is not often used. The following is a 
 "block meter rate" which is not so objectionable and 
 gives approximately the same income to the company: 
 "lOc. each for the first 10 KWH. used, 7c. each for the 
 next 5, 3c. each for all after the 15th." Plot this rate on 
 the same diagram; also the rate, "8c. per KWH., with a 
 minimum charge of 50c." Find a smooth curve which 
 will come fairly close to all three of these rates.* (Sug- 
 
 * In general, a "broken line" cannot be represented, by itself, 
 without using an equation containing an infinite series. This is 
 the case with the last of the above rates, which is represented by the 
 straight line y = 50 from x = to x = 6.25 only, and the straight 
 line y = Sx for the part further to the right only. 
 
 If we are not restricted to finite stretches of lines it is always 
 possible to find an equation for both of two loci each of which has 
 
150 
 
 THEORY OF MEASUREMENTS 
 
 129 
 
 0^ 
 
 gestion: consider the equation y = I2x (l/a)x 2 with 
 a suitable value for a.) 
 
 129. Use of Logarithmic Paper. A very common type 
 of variation is that in which one of the variables is pro- 
 
 !,!,,, t ... f P or tional to some power of 
 
 the other one. For ex- 
 ample, the distance trav- 
 ersed by a falling body is 
 proportional to the square 
 of the time of fall ; the 
 time of rotation of a planet 
 or satellite about a particu- 
 lar central body is propor- 
 tional to the 3/2 power of 
 its mean distance; friction 
 of water flowing through a 
 pipe varies (approximately) 
 as the 1.8 power of the ve- 
 locity. The determination, 
 for a set of experimental 
 data, of the proper value 
 of the exponent is not easily accomplished by means of 
 an ordinary graph, because the various curves y = x 2 , 
 y = x 3 , y = x*, etc., all have the same general shape 
 (Fig. 37).* If logarithms are taken, however, of both 
 sides of the equation y = x n the equivalent equation 
 log y = n log x is obtained, showing that log x and log y 
 
 its own ascertainable equation, by putting the individual equations 
 in the form </>(z, y) = and multiplying them together. Thus, 
 the two lines y = 50 and y = 8x are the locus of the single equation 
 (y - 50) X (y - 8z) = 0, or 400z - Sxy - 5Qy + y 2 = 0, as the 
 student can readily show by plotting a few points or by algebraical 
 treatment (compare 111; 8, d.). 
 
 * Experimental measurements are usually of positive quantities, 
 so that the shape of these curves to the left of the i/-axis or below 
 the x-axis is not a determining factor. 
 
 FIG. 37. Loci OF y = x n . 
 The curves are drawn for n = 
 1, 2, 3, 4, 10, 50, and 100. 
 
XI INTERPOLATION AND EXTRAPOLATION 151 
 
 10 
 
 are proportional. Accordingly, if log x and log y are 
 plotted on an ordinary graphic diagram a straight line 
 through the origin will be obtained. Now, just as a 
 
 2 3 4 5 6 7 8 9 10 
 
 FIG. 38. LOGARITHMIC PAPER. The logarithmic scales cause 
 the graph of any equation of the form y = ax n to appear like 
 y = a + nx on ordinary paper, viz., as a straight line. 
 
 slide rule is constructed by marking the numbers 1, 2, 
 3, 4, etc., at distances which are really log 1, log 2, log 3, 
 
152 THEORY OF MEASUREMENTS 129 
 
 log 4, etc., so plotting paper can be constructed with 
 logarithmic scales like those of the slide rule along each 
 axis; consequently plotting x and y according to the 
 numbered scales will really be plotting points at actual 
 distances of log x and log y from the origin; and if these 
 logarithms are known to be proportional it will be 
 evident that the graph of y = x n will be a straight line 
 through the origin. Such paper is called logarithmic paper 
 and can be obtained from dealers who handle drawing 
 materials. The lower left-hand corner of the sheet is the 
 origin and is marked 1,1, meaning of course log 1, log 1, 
 or 0, 0; and the scales extend both upward and to the 
 right from 1 to 10 as in the C and D scales of the slide 
 rule, or sometimes from 1 to 100 as in the A and B scales. 
 The latter arrangement would allow a single unbroken 
 line to be used to represent the "curve" of y = V# 
 (Fig. 38), although of course such a line could not be 
 drawn for the complete locus of y = x 3 . 
 
 If a straight line is drawn on logarithmic paper without 
 passing through the origin it must cut the y-&xis at some 
 point, such as k on- the logarithmic ?/-scale. Then 
 (log y) = (log k) + n(log x) (compare the ordinary equa- 
 tion of the straight line y = a -J- bx, 113, 105), or 
 log y = log k + log (x n ), or log y = log (k X x n ), or 
 y = kx n . In other words, a straight line on logarithmic 
 paper represents the equation y = ax n , a being the y- 
 intercept as measured by the logarithmic scale, and n 
 the true slope as measured by uniform scales. The 
 diagram (Fig. 38) shows y = V x and y = x on logarith- 
 mic paper. Any data that are suspected of following 
 the law y = ax b may be plotted directly on this ruled 
 diagram and their equation determined at once. 
 
 130. Semi-Logarithmic Paper. Consider the equation 
 of a straight line on paper that has a uniform scale 
 
XI INTERPOLATION AND EXTRAPOLATION 153 
 
 along the x-axis and a logarithmic scale along the 
 
 Instead of y = a + bx or log y = log a + & log x its 
 equation must now be log y = log a + 6z. Clearing this 
 of logarithms gives y = a X 10 & * or y = a X e bx accord- 
 
 1 2 3 4 5 6 / 9 
 
 FIG. 39. SEMI-LOGARITHMIC PAPER. The logarithmic scale 
 along the ?/-axis causes a straight line through the origin to represent 
 proportionality between x and log y, i. e., log y = kx. The exponen- 
 tial law of variation, y = ae mx , is obviously reducible to the form 
 log y = kx. 
 
 10 
 
154 
 
 THEORY OF MEASUREMENTS 
 
 130 
 
 ing to the base that is used.* Consequently, a straight 
 line on semi-logarithmic paper (Fig. 39) represents the 
 (( exponential" type of variation, y = ae mx . 
 
 Draw a straight line or hold a black thread on either 
 Fig. 38 or Fig. 39 so as to represent the area of a circle 
 (on the vertical scale) that corresponds to the radius 
 (as indicated on the horizontal scale). The formula is 
 a = irr 2 , or y = irx 2 . 
 
 Plot the locus of y = 2 x on squared paper for positive 
 integral values of # up to 6 or 7. Plot the same equation 
 up to z=10 on logarithmic paper, f and also on semi-log- 
 arithmic paper. . In which case is the locus a straight 
 line? Plot y = 2~ x on semi-logarithmic paper. 
 
 FIG. 40. EXTRAPOLATION DIAGRAM. Graph of the relation be- 
 tween date and population. 
 
 Explain how to use one of these diagrams (Figs. 38 
 and 39) to make a graphic table of the relationship 
 between the period (p) of vibration time of a pendulum 
 and the length (I) of the pendulum^ if they are related 
 according to the formula p = 2iri/l/g, g being a constant. 
 
 * These two equations are of the same form; the first can be 
 put into the form of the second merely by changing the value of 6; 
 for 10 = e 2 - 30 , so 10 te is identical with e 2 - 306 *. 
 
 f If the specially ruled paper is not on hand the work can be done 
 on thin paper laid over Fig. 38 and Fig. 39 of this book. 
 
XI INTERPOLATION AND EXTRAPOLATION 155 
 
 On Fig. 38 or Fig. 39 indicate the straight line that 
 represents the equation xy = a (suggestion: y = ax~ l ). 
 
 131. Extrapolation. The principles of extrapolation 
 are like those of interpolation, but the former process is 
 naturally more uncertain than the latter and can be 
 trusted to give good results only when 
 
 the extrapolated point is relatively year 
 near the points that correspond to 
 
 the known data. As an example of i860 
 
 the use of extrapolation the table and }|70 
 
 1850 
 
 1880 
 
 population 
 
 93,000 
 380,000 
 560,000 
 865,000 
 1,208,000 
 1,485,000 
 
 graph give the population of the state 1890 
 
 of California from 1850 to 1900. It jjjOO 
 
 is required to find the population in 
 
 1910. Continue the curve in the way 
 
 that you think it would be apt to 
 
 run, and note where it cuts the 1910 ordinate. 
 
 Reconstruct the extrapolated part of the curve, if 
 necessary, so as to make it give 2.38 X 10 6 for 1910 and 
 extrapolate again to determine its height for 1920. 
 What do you find the population will be for this date? 
 
 The following example shows that extrapolation may 
 be a very definite and decisive process if the given values 
 follow a consistent law of change and can be carried 
 close to the required value: Find the instantaneous 
 velocity of a body at a certain point of time if it is known 
 that immediately after that instant it travels 10 cm. in 
 the first second, 7.0711 cm. in the first half second, etc., 
 as given in the next table. Plot the tabular values with 
 a scale of time along the x-axis and a scale of average 
 velocity along the i/-axis and extrapolate graphically to 
 find the velocity in no interval of time.* 
 
 * In no time a moving object will of course traverse no space if 
 its velocity is not infinite, and there is strictly speaking no meaning 
 for such a phrase as "instantaneous velocity." It is convenient, 
 
156 
 
 THEORY OF MEASUREMENTS 
 
 132 
 
 time 
 (sec.) 
 
 space 
 (cm.) 
 
 velocity 
 
 (cm. /sec.) 
 
 1 
 0.5 
 0.3 
 0.2 
 0.1 
 0.05 
 
 10.000 
 7.0711 
 4.5399 
 3.0902 
 1.5643 
 0.7846 
 
 10.000 
 14.142 
 15.133 
 15.451 
 15.643 
 15.692 
 
 AVERAGE VELOCITY OF 
 A PISTON ROD THAT MOVES 
 WITH SIMPLE HARMONIC 
 MOTION. Its total path is 
 assumed to measure 20 cm. ; 
 its period, 4 seconds; the 
 required velocity is that at 
 the central point of its 
 motion. 
 
 132. Questions and Exercises. 1. Write a definition 
 of what you understand by the terms interpolation and 
 extrapolation. 
 
 2. Turn to the table of values for Chauvenet's criterion 
 ( 208) and plot I as a function 
 of n, using enough points to de- 
 termine whether the values of I 
 for integral values of n that are 
 not given in the table (e. g., 
 n = 31, n = 70) can be satisfac- 
 torily obtained by graphic in- 
 terpolation. 
 
 3 ' Turn to y ur S ra P hic dia ~ 
 S ram of V = log % and draw a 
 straight line from the point (2, 
 log 2) to the point (3, log 3), thus 
 making a chord of the curve. 
 Mark the middle point of the 
 chord. Is the ordinate of this 
 point equal to the average of log 
 2 and log 3? 
 
 What operation can be performed on any two numbers 
 by averaging their logarithms? Label your diagram so 
 as to show clearly the distance that corresponds to 
 log ^ and the distance that corresponds to log 
 [(2 X 3) 1 / 2 ]. Which of the 5-formulse for approximate 
 calculation with small magnitudes does this diagram 
 illustrate? In what way does it show the equality 
 
 however, to consider that a body which speeds up from a condition 
 of rest to a definite velocity must have passed through all inter- 
 mediate velocities successively, and as its velocity must have been 
 continually changing the concept of a definite velocity at a certain 
 point of space or time becomes almost a necessity. For purposes of 
 rigorously logical deduction this concept is defined as a limiting 
 value in the way indicated above. 
 
XI INTERPOLATION AND EXTRAPOLATION 157 
 
 expressed by the formula, and in what way does it show 
 that this equality is not exact but only approximate? 
 
 4. In finding the root of an equation by the method 
 of double position would it be satisfactory to extrapolate 
 instead of, interpolating? Explain why. 
 
 5. Calculate the roots of the equation 
 
 (3.2) (**) = TT + log (sins). 
 
 (Suggestions : Draw a rough diagram of y = sin x, say from 
 TT to + 3?r. Then add a rough outline of ?/ = log (sin x), 
 remembering that log = < , and that negative numbers 
 have no [real] logarithms. Draw also y = ir, and then 
 y = IT (# 2 ) ( * 2) . It will now be obvious that the graph of 
 y [T (# 2 ) (a;2) ] + [log (sin a;)] cannot cut the z-axis in 
 more than two points. Calculate each of them separately 
 by the method of double position. Their sum should be 
 1.368.) 
 
 6. A line is indicated on Fig. 38, page 151, which passes 
 through the points (1, 1) and (3, 8) of the logarithmic 
 paper. Determine the equation which it represents: 
 
 7. Plot the locus of y = a -f bx + k/x 2 and find the 
 asymptotes of the (oblique-angled) hyperbola which is 
 obtained. 
 
XII. COORDINATES IN THREE DIMENSIONS. 
 
 Apparatus. A pencil with a sharp point; a " quad- 
 rangle" or " topographical sheet" of the U. S. Govern- 
 ment contour map; (model of a small area of the map, 
 made by piling up contour sections sawn out of thin 
 wood). 
 
 133. Coordinates of a Point in Space. Just as the 
 position of a point in a plane (i. e., in two-dimensional 
 space) can be represented by two coordinates, an z-value 
 
 and a 2/-value, giving its 
 distance from two mutu- 
 ally perpendicular axes, 
 so the location of a point 
 in unrestricted, three- 
 dimensional space can be 
 fixed by three coordi- 
 nates, a set of three nu- 
 merical values (a;, y, z) 
 which indicate its dis- 
 tance from each of three 
 planes that intersect each 
 other at right angles. 
 
 FIG. 41. COORDINATE PLANES. 
 These planes of reference for three- 
 dimensional space correspond to the 
 base lines of reference, or coordinate 
 axes, that are used for two-dimen- 
 sional space. 
 
 Thus, the point in Fig. 
 41 is at a distance of x 
 units along the x-axis 
 from the plane YZ of the other two axes, and at a dis- 
 tance of y units, parallel to the y-axis, from the plane 
 XZ, and at a distance of z, parallel to the 2-axis, from 
 the plane XY. 
 
 134. Convention in Regard to Signs. The XY plane 
 may be thought of as being represented by the same sheet 
 
 158 
 
XII COORDINATES IN THREE DIMENSIONS 159 
 
 of paper as that on which the flat graphs of x and y have 
 previously been traced (Fig. 42). Then any point what- 
 ever must be located a certain distance above or below 
 some definite position in the plane of the paper. The 
 o>value and y-value for this position will be the x and y 
 of the point in space, and the distance from the point to 
 the plane will be the z of the point. It is customary 
 among physicists and astronomers to consider a distance 
 above the plane of the paper as a positive value of z, 
 and distance below it as negative, according to the 
 arrangement of axes shown in Figs. 41 and 42. The 
 
 X 
 
 FIG. 42. POSITIVE DIRECTIONS OF AXES. Z is considered posi- 
 tive when measured upward from the normal plane of the x-axis 
 and 7/-axis. The pin shown in this diagram would have the location 
 of its head represented approximately by the position (5, 7, 3), the 
 numbers in parenthesis being used for x, y, and z, in order. 
 
 convention among pure mathematicians is just the oppo- 
 site, viz., distances above the paper are called negative 
 and those below are positive (Fig. 43). The distinction 
 is immaterial in the greater part of the study of pure 
 mathematics, but it is important in many branches of 
 applied science; for example, the equations of the curve 
 of a right-hand screw thread in one system will represent 
 
160 
 
 THEORY OF MEASUREMENTS 
 
 134 
 
 a reversed or left-hand thread if the other system is 
 used instead. Accordingly, there is a tendency among 
 pure mathematicians to use the " physical " arrange- 
 ment for the sake of uniformity, and it is the only one 
 that will be used in this book. 
 
 That these two arrange- 
 ments exhaust the possi- 
 bilities may be seen by 
 taking any arbitrary ar- 
 rangement of axes at right 
 angles to each other and 
 rotating them as a rigid 
 figure. They can always 
 be made to coincide with 
 either Fig. 43, or Fig. 44. 
 
 
 
 FIG. 43. 
 
 44 
 
 FIG. 44. 
 
 CONVENTIONS AS TO SIGNS. 
 Fig. 43 corresponds to the ar- 
 rangement that is understood in 
 pure mathematics; Fig. 44 is that 
 of applied mathematics. 
 
 The two are essentially 
 different because neither 
 one can be rotated in 
 three-dimensional space so 
 as to coincide with the other.* 
 
 In each of the following exercises the positive direc- 
 tions of the axes are indicated. Consider each in turn 
 and state whether it is like Fig. 44 ("right") or the 
 opposite ("wrong"). 
 
 * Notice that a plane ^-diagram drawn with + Y upward but 
 with + X to the left cannot be rotated in its own two-dimensional 
 space so as to coincide with the conventional arrangement. It can 
 be turned through the third dimension, however (turning the upper 
 surface of the paper downward), and made to coincide. Similarly, 
 a solid model of Fig. 43 would need to be turned through a fourth 
 dimension of space before it could be made to coincide with a solid 
 model of Fig. 44. Since we have no appreciation of a fourth dimen- 
 sion the two figures are as essentially different to us as a capital L 
 and a Greek capital r would seem to a being whose sense perceptions 
 were limited to space of two dimensions. 
 
XII COORDINATES IN THREE DIMENSIONS 161 
 
 1. X points northward; Y, upward; Z, westward. 
 
 (Ans. : wrong.) 
 
 2. X, west; F, up; Z, north. (Ans.: right.) 
 
 3. X, down; Y, east; Z, south. 
 
 4. X, down; F, north; Z, east. 
 
 5. X, down; F, west; Z, north. 
 
 6. X, down; Y, south; Z, east. 
 
 7. X, east; F, north; Z, up. 
 
 8. X, west; F, north; Z, up. 
 
 9. X, west; F, north; Z, down. 
 
 10. X, up; F, east; Z, north. 
 
 135. Loci of Simple Three-Dimensional Equations. 
 In studying geometrical relationships in a plane we have 
 seen ( 105) that the equation y = a represents all the 
 points that are a units above the z-axis, that is, a straight 
 line parallel to the x-axis and situated at a distance a 
 above it. In the same way, the equation z = m must 
 denote all points located m units above the z?/-plane, 
 e. g., the points (2, 3, m), (0, 0, m), (0, 10, m), etc., since 
 each of these groups of values (x = 2, y = 3, z = m, for 
 example) will satisfy the equation. Similarly, x = m 
 or y = m will denote a particular plane parallel to F0Z 
 (Figs. 41 to 44) or to XOZ, and so perpendicular to the 
 x-axis or to the ?/-axis, respectively. 
 
 Consider next an equation that contains only two of 
 the three variables. On a plane surface y = x 2 is a 
 curve, a parabola. In space, any point above or below 
 any point of the curve will satisfy this equation, for it 
 has the same x and y, but a different z. Accordingly, the 
 equation represents the surface that is made up of all 
 the vertical lines that can be passed through points 
 that lie on the plane curve. 
 
 If the equation contains all three variables, any arbi- 
 trary value may be assigned to z, any value whatever to 
 
 12 
 
162 THEORY OF MEASUREMENTS 136 
 
 y, and the value of z will then be determinate. That 
 is, the points of the locus will be situated at varying 
 distances above all the points in the xy-pl&ne, and so 
 will comprise a surface. 
 
 In general, then, a single equation denotes a surface. 
 Since two surfaces intersect along some line two simul- 
 taneous equations will denote a curve in space. It has 
 been seen ( 108) that x 2 + y 2 = 5 2 must be the equation 
 of a circle; in the same way x 2 + 2/ 2 + z 2 = 5 2 is the 
 equation of a spherical surface that is everywhere 5 
 units distant from the origin. The simultaneous equa- 
 tions 
 
 * + y 2 + z 2 = 25 
 z = ?> 
 
 will have for their locus the points which satisfy the first 
 equation and at the same time satisfy the second; i. e., 
 each of the points that is located on the spherical surface 
 x 2 + y 2 + z 2 = 25 and is at the same time in the hori- 
 zontal plane z = 3. Such points must lie on the inter- 
 section of the plane and the spherical surface; and this 
 intersection is known to be a curve, namely, a circle. 
 For other values of z the circular intersection would be 
 larger or smaller; for the (tangent) plane z = 5 the 
 intersection would shrink to a point, for z = 6 there 
 would be no intersection. 
 
 136. Contour Lines. If a were given all possible 
 values in the equation z = a the resultant loci would be 
 horizontal planes at all levels, and they would intersect 
 the sphere x 2 + y 2 + z 2 = 5 2 in all the horizontal circles 
 that could be drawn on its surface. All of these inter- 
 sections, consequently, would be the spherical surface; 
 a smaller number of them would form a sort of skeleton, 
 from which the shape of the surface could be inferred 
 
XII 
 
 COORDINATES IN THREE DIMENSIONS 
 
 163 
 
 if they were sufficiently numerous. An extremely useful 
 way of representing a surface, especially an irregular 
 surface, on paper is by outlining the intersections which 
 would be formed by horizontal planes at various levels. 
 The diagram (Fig. 45) shows a portion of the surface 
 
 H 
 
 FIG. 45. THE HYPERBOLIC PARABOLOID x 2 if + z = 0. A 
 surface of this general character is sometimes spoken of as a "saddle- 
 back." 
 
 2 y 2 + z = 0, called a hyperbolic paraboloid. When 
 x is zero the equation reduces to z = y 2 ', i. e. } the inter- 
 section of the curve with the FZ-plane is the parabola 
 z = y 2 , BOE in the diagram. When y = the section 
 MOJ is a parabola z = x 2 . At any level where z = a 
 the equation of the surface becomes y 2 x 2 = a, a 
 hyperbola such as ABC and DEF or HJK and LMN; 
 for a = this degenerates into the two straight lines 
 that are common asymptotes, y = x. A series of 
 horizontal sections of the surface are shown in Fig. 46 
 as they would appear if they were all viewed from above, 
 
164 
 
 THEORY OF MEASUREMENTS 
 
 137 
 
 the algebraical signs showing whether each curve is 
 above or below the xy-plane and the smaller numbers 
 denoting the lower levels while the higher numbers indi- 
 cate the upper ones. If 
 y the numbered lines are 
 
 imagined to be raised 1, 
 2, 3, 4, 5, 6, 7, 8, and 9 
 centimetres respectively 
 above the surface of the 
 paper it will be evident 
 that a good idea of the 
 shape of the curved sur- 
 face which they outline 
 can be obtained without 
 the necessity of consult- 
 ing a perspective drawing 
 like Fig. 45. Such a sur- 
 face as this one (which 
 is convex upward along 
 the x-axis and concave 
 
 upward along the ?/-axis) is commonly called a saddle-back 
 and represents roughly the shape of the surface of the 
 earth in a mountain pass. 
 
 The irregular surface of the earth is sometimes repre- 
 sented on maps by horizontal section lines (contour lines) 
 which make it easy to find the location, height, slope, 
 etc., of any hill, valley, or other surface character by 
 inspection of the map. 
 
 137. Use of Contour Maps. The lines of horizontal 
 section usually correspond to heights taken at equidistant 
 intervals, for example, at 20, 40, 60, 80, . . . feet above 
 mean sea level, and are called contour lines. Students 
 usually find it most convenient to think of them as repre- 
 senting the new shore lines that would be formed if the 
 
 65 
 
 FIG. 46. CONTOUR LINES OF A 
 SURFACE. Horizontal sections at 
 different levels of the curved surface 
 of Fig. 45. 
 
XII 
 
 COORDINATES IN THREE DIMENSIONS 
 
 165 
 
 sea level were to rise 20 ft., 40 ft., etc., above its original 
 level. The uniform interval, twenty feet in this example, 
 is called the contour interval, and the level of reference, 
 
 FIG. 47. CONTOUR LINES OF A HILL. The upper figure is a 
 side view of a hill 80 feet high, showing in perspective the outlines 
 that would be produced if it could be cut into twenty-foot slices or 
 the new shore lines that would be produced if the surrounding 
 country could be flooded to depths of 20, 40, 60, and 80 feet. 
 
 The lower figure is a set of contour lines that represent a top view 
 of the levels shown above. The usefulness of a map is greatly 
 increased by having contour lines drawn or printed on it in some 
 distinctive color. Such maps usually have the contours in brown, 
 rivers and lakes in blue, roads, buildings, boundaries, etc., in black. 
 
 The line 20 is the locus of all points where the surface is 20 ft. 
 above the plane of reference. Notice that the ground is always 
 higher on one side of a contour line and lower on the other. 
 
166 THEORY OF MEASUREMENTS 
 
 usually mean sea level, is called the datum plane. The 
 diagrams (Fig. 47) show a vertical section of a hill 80 
 feet high and a horizontal plan of its contour lines. 
 
 Examine the contour map and notice whether the 
 datum plane and the contour interval are stated in the 
 margin. 
 
 Find a hill or other elevated area on the map, and 
 notice the arrangement of the contour lines. What is 
 the difference, as indicated on the map, between a high 
 hill and a low hill? What is the difference between a 
 steep hill-side and a more gradual slope? 
 
 The model shows the 100-foot contour lines of a hill 
 that is given on the map. See if you can identify it 
 from the shape of the horizontal sections. 
 
 Find a brook that runs down a hill. In what general 
 direction do the contour lines cross the brook? Why? 
 What is the general shape of the contour lines where 
 there is a water-course? Where a hill sends out a pro- 
 jecting buttress or ridge what is the general contour 
 form? How is a plateau formation indicated by contour 
 lines? 
 
 Why is it that contour lines where they run across a 
 road- way are never as near together as they often are in 
 other localities? Find a road (on the map) that appears 
 to have been purposely so constructed as to cut the con- 
 tour lines at considerable intervals of distance. 
 
 138. Construction of a Contour Map. Copy the fol- 
 lowing table of altitudes upon the squared paper of your 
 notebook, writing each number with ink in very small 
 figures directly on the intersection of two of the ruled 
 lines. Make all the spaces between columns of numbers 
 equally wide (say 3 or 4 squares), and space the hori- 
 zontal lines of numbers at the same distances as the 
 intervals between the columns. Omit one or two lines 
 
XII 
 
 COORDINATES IN THREE DIMENSIONS 
 
 167 
 
 at the bottom of the table or one or two columns at the 
 right-hand side rather than crowd the numbers close 
 together, if the page of your notebook is not large. 
 
 
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 CO 
 
 ALTITUDES IN CENTRAL PHILADELPHIA. The data are expressed 
 in feet above mean sea level. No datum is given here for the corner 
 of Sixth and Market streets. 
 
168 THEORY OF MEASUREMENTS 139 
 
 Copy also the dotted line that represents a 34-ft. contour, 
 noticing that it passes exactly through the points marked 
 34, runs half-way between the points 33 and 35, runs 
 between 32 and 35 twice as far from the former as from 
 the latter, etc. 
 
 Let your contour interval be 3 feet, and start a 31- 
 foot contour line somewhere near the center of the 
 diagram. Extend it carefully in both directions, remem- 
 bering that all the altitudes close to one side of it must 
 be greater than 31 and all those near the other side must 
 be less than 31. In case it is difficult or impossible to 
 extend the line further than a certain point leave it and 
 begin the construction of another contour line. After 
 the easier lines have been finished they will be found of 
 considerable aid in helping to determine the course of 
 the more difficult ones. Remember that a single level 
 may be represented by two or more lines that do not 
 join; thus, there will be a 34-foot ring around the 
 37-foot peak at Fifth and Race Streets, and this cannot 
 connect with the line shown on the diagram because of 
 the low-lying ground between. If the sea level were 
 raised 34 feet the line indicated here would be the new 
 shore line and the 34-foot ring would be the shore of a 
 separate island. 
 
 Draw new contour lines at intervals of every three 
 feet until the whole area is covered. 
 
 Estimate the elevation at the corner of Sixth and 
 Market Streets by interpolation along Market Street 
 (E. and W.), also by interpolation along Sixth Street 
 (N. and S.) and along two diagonal lines (NE. and SW., 
 and NW. and SE.). Finally decide for yourself the 
 most reasonable value for this elevation. 
 
 139. Questions and Exercises. 1. If x/a + y/b + z/c 
 = 1 is known to represent a plane surface what can 
 
XII COORDINATES IN THREE DIMENSIONS 169 
 
 you prove about the position of this plane (compare 
 116)? 
 
 2. Write the equation of the cone (curved surface) 
 produced by rotating the straight line z = 3z around the 
 2-axis (suggestion: rotation changes each point that was 
 located at a distance of x to the right of the 2-axis into a 
 horizontal circle whose radius is equal to that -value). 
 
 3. What kind of a locus corresponds in general to 
 three simultaneous equations? 
 
 4. Can two contour lines representing different levels 
 ever intersect each other on a map? What would be 
 true of the earth's surface at the point of intersection? 
 
 5. Show that the curves RCAQ and GFDP made by 
 cutting the hyperbolic paraboloid (Fig. 45) at any dis- 
 tance to the left or the right of the origin by the plane 
 y = a are " arch- shaped" parabolas. Show that the 
 curves RHKG and QLNP made by limiting the surface 
 at the front or back by the plane x = a are " festoon- 
 shaped " parabolas. 
 
XIII. ACCURACY 
 
 Apparatus. Rectangular wooden block measuring 
 about 4X8X8 cm.; centimetre and millimetre scale; 
 one scale (of centimetres and millimetres) and one steel 
 tape-measure to be used by the whole class. 
 
 140. Significant Figures. Significant figures have al- 
 ready been defined as all of those that compose a number 
 except one or more ciphers at the extreme left or right 
 which may be necessary to express the order of magni- 
 tude of the number (i. e., to locate the proper position 
 of the decimal point) but are not needed in any other way 
 for indicating its value. For example, .003803, 3.803, 
 and 3803000 have four significant figures each; provided 
 that the last one is a " round number" (i. e., is meant to 
 be accurate only to a whole number of thousands). 
 The ambiguity in the last case can best be avoided by 
 writing 3.803 X 10 6 ; and the same three numerical 
 values could all be written with seven-figure accuracy 
 by expressing them as 3.803000 X 1C)- 3 , 3.803000, and 
 3.803000 X 10 6 respectively. 
 
 It is usually understood, in any kind of careful scien- 
 tific work, that a number is never written with too many 
 significant figures, which would appear to give it an 
 unwarranted degree of accuracy; nor with too few signifi- 
 cant figures, which would mean a neglect of the full 
 extent of the accuracy that had been obtained. In 
 other words, figures are known to be correct as far as 
 they are stated and are unknown for all notational 
 "places" beyond. For example, if 1 yard is found to 
 be 83.8 cm., the best value of 1 foot that can be calcu- 
 lated from this determination is 1 ft. = 27.9 cm. and 
 
 170 
 
XIII ACCURACY 171 
 
 the ordinary arithmetical result, 1 ft. = 27.93333 cm., 
 is not only unjustified, but in this case is positively wrong 
 although the statement that 1 yard = 83.8 inches is right. 
 Which four of the following values for e are correct 
 and which six are incorrect: 2.71828182; 2.71828183; 
 2.718281; 2.7182; 2.7183; 2.7180; 2.7; 2.70; 2.71; 2.72? 
 
 141. Infinite Accuracy. A number like the ratio of 
 the diagonal of a square to its side or the ratio of circum- 
 ference to diameter for any circle, which depends only 
 upon theoretical considerations, can be calculated with 
 any desired degree of accuracy, but no number whose 
 value depends upon the measurement of a material 
 thing can be stated with more than a definite degree of 
 accuracy, on account of the limitations of the method of 
 measurement. If an imaginary circle is made large 
 enough to enclose the Milky Way its circumference, 
 measured in terms of its radius, must be equal to 
 2 X 3.1415926535897932384626433832795028841971694, 
 but if it were possible to measure a real line of corre- 
 sponding dimensions the best microscope in existence 
 would not enable us to decide whether the value of TT 
 should be as small as 3.141592653589793238462643383279 
 or as large as 3.141592653589793238462643383280.* The 
 numerical value of TT has been calculated as far as 707 deci- 
 mal places, but this of course is not even an approach 
 to perfect (i. e., infinite) accuracy. 
 
 142. Relative Errors. If the thickness of a lead pencil 
 is measured by holding it in fronq. of a scale the separate 
 
 * Such a circle might have a circumference of 100,000 "light- 
 years," or, say, 10 23 centimetres, considering the velocity of light 
 to be 30,000,000,000 cm. per second and the number of seconds in 
 a year to be over 30,000,000. If a "homogeneous immersion" 
 objective can separate two points at a distance of 1 or 2 X 10~ 5 cm. 
 it could measure with an accuracy of nearly 10~ 28 , corresponding 
 to 28 significant figures. 
 
172 THEORY OF MEASUREMENTS 142 
 
 measurements made by a class of students will be likely 
 to vary as much as 0.03 cm. This is nearly four per 
 cent of the distance measured. If the length of a six- 
 foot table top is determined with an ordinary steel 
 tape measure the results that are obtained will be apt 
 to vary two or three millimetres. This is about J^ per 
 cent, of the distance measured. In the latter case the 
 error is about ten times as great, considered as an 
 isolated length, as it is in the former. Considered in 
 relation to the thing measured, however, 0.16 per cent 
 is much smaller than 4 per cent. Since the measurement 
 of the table is obviously a more accurate process than 
 that of the lead pencil it will be evident that accuracy 
 is a matter of relative size, not of absolute size. To 
 state that a measurement is uncertain by 4 per cent 
 means something definite, but the statement that a 
 measurement is correct " within 0.03 cm. " tells us nothing 
 about its accuracy, if nothing is stated about the length 
 measured. 
 
 Measure a lead pencil and a table in the manner 
 described. The other members of the class are to meas- 
 ure the same objects with the same ruler and the same 
 tape. Report the measurements to the instructor to 
 be tabulated. Then determine their maximum dis- 
 crepancy, both in centimetres and as two percentages. 
 
 By using the most refined methods it is possible to 
 measure a distance of several miles with remarkable 
 accuracy. If an accurate steel tape is used, which is 
 stretched by a measured force when it is at a carefully 
 determined temperature, it is possible in the course of 
 a few weeks to measure a base line for surveying purposes 
 with an error of about one unit out of 500000. Make 
 a rough mental calculation (assume 5000 ft. = 1 mile) 
 of what such an error would amount to in measuring a 
 
XIII ACCURACY 173 
 
 distance of ten miles. Is it larger or smaller than the 
 0.03 cm. of the lead-pencil measurement? 
 
 Does the accuracy with which a number is stated 
 depend upon the number of decimal places to which it is 
 carried out, or upon the number of significant figures 
 which it contains? 
 
 143. Uncertain Figures. The figure that follows the 
 last trustworthy figure of a measurement may be un- 
 certain to the extent of two or three units, and yet its 
 approximate value may be definite enough to make one 
 hesitate to discard it. This would probably be the case 
 if the student attempted to estimate hundredth* when 
 using a scale of whole centimetres without millimetre 
 graduations. In such cases it is better to retain the 
 doubtful figure, keeping in mind however the fact that 
 the result obtained from it in any calculation will be 
 liable to have one uncertain figure also. 
 
 Another case in which it is sometimes desirable, to 
 keep a single uncertain figure is when a higher degree of 
 accuracy is made possible by retaining it during the 
 process of a calculation, although it is eventually dis- 
 carded in stating the final result. This is illustrated in 
 the treatment of the " figure last canceled" in the 
 'processes of abridged multiplication and division. 
 
 144. Superfluous Accuracy. An appreciation of the 
 degree of accuracy that is required in particular cases 
 often makes it possible to avoid needless trouble in 
 measuring or calculating. If the dimensions of a rec- 
 tangular block cannot be measured with more than three- 
 figure accuracy its density can be obtained by weighing 
 it merely with three-figure accuracy, and the determina- 
 tion of the fourth figure of its mass or weight will not 
 enable a better calculation of its density to be made. 
 
 Obtain the dimensions of a rectangular wooden block 
 
174 THEORY OF MEASUREMENTS 145 
 
 as accurately as possible with a scale of centimetres and 
 millimetres, estimating tenths of a millimetre. Calculate 
 its volume, using the proper number of significant figures, 
 and then find its density by weighing it and dividing 
 mass by volume. 
 
 145. Finer Degrees of Accuracy. A measurement 
 may happen to be known with an accuracy greater than 
 can be expressed by three significant figures and yet 
 not with four-figure accuracy; that is, the step from 
 any degree of accuracy to a ten-fold greater degree may 
 be too great to be suitable in all cases. For example, 
 a period of time equal to 125f seconds may have been 
 measured more accurately than to the nearest whole 
 number of seconds and yet may not have been deter- 
 mined closely enough to enable tenths of a second to be 
 stated. In such cases common fractions may be used 
 instead of decimals; and a significant zero can be em- 
 ployed in such a form as "125f seconds" without any 
 lack of intelligibility, meaning of course " between 124.9 
 and 125.1 seconds/' or " nearer to 125 seconds than to 
 124 or to 125." It is usually more satisfactory, how- 
 ever, to state a numerical value for a measurement and 
 then affix a statement of its uncertainty in the form of a 
 percentage or otherwise. 
 
 What would you understand to be the largest possible 
 error in a measurement stated to be 125 seconds? Ans.: 
 0.5 sec., or 0.4 per cent. 
 
 What would you understand to be the largest possible 
 error in a measurement stated to be 125.0 seconds? 
 What would it be for- 125f sec.? Express the answers 
 both in seconds and in percentages. 
 
 Consider carefully the stated measurement "22J cubic 
 centimetres." If it had to be written as a decimal for 
 the sake of averaging it with a set of other measurements 
 
XIII ACCURACY 175 
 
 would you prefer to write it 22.3 or 22.33, or would you 
 round it off to 22? Explain why. 
 
 146. Possible Error of a Measurement. Instead of 
 stating that a measurement may have an error in a 
 certain decimal place or significant figure although the 
 figures that precede it are correct it is usually better to 
 state the possible error in the form of a ratio or a per- 
 centage. Since the figures of a numerical statement are 
 intended to be significant and correct as far as they go 
 it is evident that the statement cannot have an error 
 larger than half of a single unit in the last decimal place, 
 or five units in the place that would follow the last one 
 that is written. Accordingly, a statement, like that of 
 56, that a length which is written 174.2 certainly 
 cannot have an error of more than 1 out of 1742, is on 
 the safe side but could be improved by making it read 
 "not more than \ out of 1742, or 1 out of 3484, or 
 .0003, or .03 per cent." 
 
 Turn to your experimental determinations of sines 
 ( 40), find the greatest possible error of each as half of a 
 single unit in the last written decimal place, and reduce 
 this possible error either to a decimal fraction (e. g., 1 
 out of 25 is the same as .04) or to a percentage (1 out of 
 25 = 4 per cent), according to which form appeals to you 
 as being the more expressive. Then mark each error 
 "small," "large," etc., as given in the table of errors 
 classified according to size (see appendix). 
 
 Notice that a gradation which is as coarse as that 
 expressed by enumerating significant figures is unsatis- 
 factory in certain cases. The number .9624 has no 
 more significant figures than 1.093 but is expressed with 
 about ten times as great accuracy as the latter. For 
 purposes of using it in a calculation along with 1.093 it 
 should be rounded off to .962. According to the rules 
 
176 THEORY OF MEASUREMENTS 147 
 
 given in the chapter on small magnitudes 1.093 X .9624 
 would equal 1. + .093 - (1 - .9624) = 1. + .093 - .0376 
 = 1.0554. Explain why this result is unjustifiable, and 
 show the fallacy in the process of obtaining it. 
 
 147. Possible Error After a Calculation. The possible 
 error of the sum of two or more measurements is equal 
 to the sum of their individual possible errors. 
 
 A table is found by measurement to be 44.3 cm. higher 
 than a bench which has a height of 42.5 cm. from the 
 floor. If the third significant figure of each measurement 
 may have a possible error of half a unit what limits can 
 be assigned for the height of the table from the floor. 
 Ans.: between 86.7 cm. and 86.9 cm. 
 
 The possible error of the difference between two measure- 
 ments is equal to the sum (not the difference) of their 
 possible errors. 
 
 A table is 51.1 cm. lower than a shelf which is between 
 137.9 cm. and 140.1 cm. above the floor. How high 
 must the table be? Ans. : from 88.75 cm. to 89.05 cm. 
 
 Would it be correct to answer the last example, "from 
 88.8 cm. to 89.0 cm."? 
 
 The possible error of the product of the two measure- 
 ments mi e\ and m-, e 2 will be found by ordinary 
 multiplication to be m\e<i + m^e\ under ordinary circum- 
 stances in which the error is small ( 70) compared with 
 the measurements themselves, so that 0162 is negligible 
 ( 72). It will be instructive, however, to re-write the 
 measurement and its possible error, m e, in the form 
 m(l d= e/m) before performing the multiplication. The 
 fraction e/m will now represent the relative error ( 54). 
 The product will be m\(\ d= ei/Wi) X w 2 (l e 2 /w 2 ) 
 = raira 2 (l d= ei/mi d= e 2 /m 2 ); * e- the possible percentage 
 error of a product is equal to the sum of the possible per- 
 centage errors of its factors. 
 
XIII ACCURACY 177 
 
 A rectangular surface measures 50 cm. X 20 cm. Cal- 
 culate (50 0.5) X (20 0.5), and show that the pos- 
 sible error of the area is 35 cm 2 ; then find the relative 
 Value of 0.5 to 50 and that of 0.5 to 20, and see whether 
 their sum is the same as the ratio of 35 cm 2 to 50 cm. 
 X 20 cm. 
 
 Similarly, the possible percentage error of a quotient is 
 equal to the sum of the possible percentage errors of the 
 divisor and dividend; for Wi(l ei/Wi) -r- w 2 (l e^/niz) 
 is equal to (1 eijm\ d= 62/^2)^1/^2. 
 
 These rules are equally applicable in cases where one 
 of the two quantities employed has no error whatever. 
 If a 50-foot tape measure has a possible error of a tenth 
 of a foot (0.2 per cent) then there may be an error of 
 three tenths of a foot in using it to measure a distance 
 of 150 feet. If the measured diameter of a circle is 
 uncertain by 3 per cent then the calculated circum- 
 ference will be liable to an error of 3 per cent. 
 
 148. "Probable" Error. The magnitude of a " pos- 
 sible error" is not actually as definite as it is usually 
 assumed to be. There may be a high degree of prob- 
 ability that a measurement has no error greater than half 
 a unit in the last decimal place that is written, but this 
 can never be so high as to amount to an absolute 
 certainty. 
 
 The most that can be said is that a certain large 
 range for the error of a measurement is very improb- 
 able, narrower limits of error are less improbable, that 
 an error is within a still smaller range may be probable 
 rather than improbable, that the error has at least a 
 certain minute magnitude may be so highly probable 
 as to be almost a certainty, etc. Without taking up a 
 quantitative discussion of the relative likelihood that 
 the magnitude of a particular error will fall within given 
 13 
 
178 THEORY OF MEASUREMENTS 150 
 
 limits we need notice here only that the most convenient 
 degree of probability that we can specify is that of a 
 half-way point between probability and improbability. 
 For a certain limited class of error it is possible to state 
 that a particular error is "just as likely as not" to be 
 within certain numerical boundaries, i. e., that the 
 error's chance of being smaller than the stated limit is 
 just equal to its chance of being larger than the limiting 
 value. Boundaries of this sort are sometimes called 
 probable errors, and will be considered in a later chapter. 
 At present all that needs to be noticed in regard to them 
 is that they are not " probable" in the sense of being 
 highly probable values nor are they even " errors" except 
 in a restricted sense of the word, so that the accepted 
 name is apt to prove a misleading one. 
 
 149. Accuracy Required in Special Cases. When only 
 a certain degree of accuracy is needed there is nothing 
 to be gained by attempting a greater degree, and there 
 is usually a considerable loss of time and labor if calcula- 
 tions are carried out to more significant figures than can 
 be utilized. For a great many scientific, engineering, 
 and commercial calculations three-figure accuracy is 
 sufficient, and in such cases the slide rule can be 
 used more readily than a table of logarithms. Four- 
 figure accuracy is attained by the use of four-place 
 logarithm tables, and is sufficient for most of the calcula- 
 tions of physics and chemistry, although five-figure 
 accuracy is needed for certain measurements, e. g., deter- 
 minations of mass, periods of time, barometric pressures, 
 etc. Six-figure accuracy is sufficient for all surveying 
 except the most precise geodetic measurements. Seven- 
 figure accuracy is also needed for certain astronomical 
 calculations. 
 
 150. Questions and Exercises. 1. An expression like 
 
XIII ACCURACY 179 
 
 " four-and-a-half-figure accuracy" is sometimes seen, 
 Write down your opinion of what this phrase means, 
 expressed as a percentage; and state your reasons for 
 choosing this number. 
 
 2. Explain why the possible error of a difference is 
 equal to the sum (instead of difference) of the errors of 
 its components. 
 
 3. If a wooden block measures 250 cm 3 , with an un- 
 certainty of 5 cm 3 , and weighs 2.00 X 10 2 gm., how 
 many gm./cm 3 does the uncertainty of its density 
 amount to? Ans. : sixteen or eighteen hundredths of a 
 unit. What can you say of the effect of the uncertainty 
 of the weight? 
 
 4. Each volume of a ten-volume encyclopedia has 2.0 
 inches thickness of leaves between two covers that are 
 each 0.2 inch thick. If the number of pages per volume 
 is liable to vary by 2 per cent and the thickness of the 
 cover by 1 per cent, what is the maximum length of 
 shelf-space that may be needed for the set? Ans.: 
 24.44 inches. 
 
 If a calculated measurement M is equal to lOwi + 20m 2 
 what will its possible error E amount to in terms of the 
 possible errors e\ and e 2 of mi and m 2 respectively? 
 Suggestion : 
 
 + where 
 
 5. The quotient in 147 should, strictly speaking, be 
 (1 =t ei/mi =F e 2 /w 2 )wi/ra 2 . Explain why the inversion of 
 the double sign is unnecessary in this connection. 
 
 6. Write in your notebook a brief resume of all of the 
 important points in chapters iv and xiii. 
 
XIV. THE PRINCIPLE OF COINCIDENCE 
 
 Apparatus. Two metre sticks; cardboard; knife; 
 rectangular wooden block; slide rule. 
 
 151. Effect of Magnitude upon Accuracy. With a 
 scale of millimetres measure a length of one inch as 
 accurately as possible, writing down the number of 
 whole centimetres, additional whole millimetres, and 
 estimated additional tenths of a millimetre which it 
 contains. Measure a length of two inches in the same 
 way and divide the result by two. Calculate one tenth 
 of the carefully measured length of an interval of ten 
 inches and compare with the two other results. Notice 
 that a large quantity can usually be measured more 
 accurately than a small one. If the method of measure- 
 ment gives the same absolute accuracy in two different 
 cases (for example, no error greater than .005 cm.) 
 the ratio of error to measurement will naturally be a 
 smaller relative error just in proportion as the measure- 
 ment itself is greater. It is for this reason that the metre 
 was chosen for a standard of length ( 19) instead of the 
 centimetre, and the kilogram for a standard of mass 
 instead of the unit, a gram. 
 
 152. Measurement by Estimation. In order to make 
 a more accurate determination of the length of an inch 
 lay one metre stick on the table with the metric gradu- 
 ations upward and place the other beside it with the 
 graduation in inches and tenths of an inch upward. 
 See that the scales are in close contact and note down 
 the number of centimetres and hundredths that happens 
 to be opposite the mark 1 inch on the other scale. Note 
 also the indication of the mark 31 inches. Move one 
 
 180 
 
XIV 
 
 THE PRINCIPLE OF COINCIDENCE 
 
 181 
 
 scale along the other at random and repeat the obser- 
 tions, unless otherwise directed, until ten sets have been 
 taken. The distance measured should always be the 
 same, 30 inches, but it should be measured at other 
 places, such as the mark 4 and the mark 34. Never use 
 the mark at the very end of the stick, as it is often 
 inaccurate on account of wear. 
 Tabulate the results in a form like the following : 
 
 MEASUREMENT OF AN INCH BY THE METHOD OF ESTIMATION. 
 
 
 1st line 
 
 2d line 
 
 Interval 
 
 Cm. scale .... 
 Inch scale .... 
 
 20.05 
 1.00 
 
 45.50 
 11.00 
 
 25.45 
 
 10.00 
 
 Cm 
 
 50.00 
 2.70 
 
 75.47 
 12.70 
 
 25.47 
 
 10.00 
 
 Inch ........ 
 
 
 Cm 
 
 10.06 
 1.50 
 
 35.46 
 11.50 
 
 25.40 
 
 10.00 
 
 Inch 
 
 
 
 
 
 
 
 
 Sum 
 
 254.02 
 25.402 
 
 100.00 
 10.000 
 
 Average 
 
 
 
 Length of an inch : 
 
 cm. 
 
 153. Measurement by Coincidence. In the method 
 of measurement by estimation one or more equal intervals 
 are measured in the customary manner, tenths of the 
 smallest scale-divisions being determined by a mental 
 estimate. For the method of measurement by coinci- 
 dence a series of equal quantities is necessary. If these 
 can be directly compared with a series of equal scale- 
 units it will usually be possible to observe that some whole 
 
182 THEORY OF MEASUREMENTS 153 
 
 number of the equal quantities is of precisely the same 
 magnitude as some other whole number of scale-divisions. 
 For example, the diagram (Fig. 48) shows in two different 
 
 l . 1|0 . 210 i C 8|0 i 
 
 I M i, i.l ,i i i i I i i .1 .1 i i i i I i i. i. i.l .1 ,i i. i, I 
 
 T-TT 
 
 i ',',', '.' i 1 u '.','.' i I'.y.'M '.','.'.' i 
 
 210 3 
 
 I, 1,1 
 
 i 'i' i M 'I"T 
 
 Ho 
 
 FIG. 48. THE METHOD OF COINCIDENCES. Each of the un- 
 known intervals is seen to be a trifle less than 1 cm., but the fact 
 that there is no distinguishable difference between 13 intervals and 
 11 cm. shows that the length of the unknown interval must be 
 11/13 of a centimetre. 
 
 locations (c) that 13 of the unknown intervals are equal 
 to 11 of the centimetre units. Accordingly, the unknown 
 interval must be 11/13 cm. 
 
 Set the two scales so that any whole number of inches 
 near one end of one metre stick is exactly opposite some 
 whole number of centimetres on the other. Hold the 
 two sticks firmly together and look again to see that the 
 coincidence is exact. Do not be satisfied unless you are 
 unable to say whether the upper mark is a little toward 
 the right or toward the left of the one below it. Without 
 allowing the two scales to slip find another place, at 
 least 30 cm. distant from this point and preferably 
 further, where there is another exact coincidence, this 
 time between any centimetre or millimetre graduation 
 and any graduation of inches or tenths. Try to decide 
 whether the coincidence is exact or whether the imaginary 
 central axis of one line lies a little beyond the other, 
 and if the coincidence is faulty choose a better one 
 elsewhere. 
 
 Record the results as in the specimen table, re-set 
 the two scales, and repeat until 10 determinations have 
 been made. Shift the position of the first coincidence, 
 as was done in the method of estimation; also, take care 
 
XIV 
 
 THE PRINCIPLE OF COINCIDENCE 
 
 183 
 
 that the interval between coincidences is not of the 
 same length each time. 
 
 MEASUREMENT OF AN INCH BY THE METHOD OF COINCIDENCES. 
 
 
 1st line 
 
 2d line 
 
 interval 
 
 cm. in 1 in. 
 
 Cm.. . 
 
 90.000 
 
 35.400 
 
 54.600 
 
 
 In 
 
 4.000 
 
 25.500 
 
 21.500 
 
 2.5395 
 
 Cm.. 
 In 
 
 15.000 
 6.000 
 
 71.400 
 28.200 
 
 56.400 
 22.200 
 
 2.5405 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Averaae . . 
 
 
 
 
 2.5401 
 
 It is not absolutely accurate to assume that the co- 
 incidences are always exact to a hundredth of the smallest 
 graduation of the scale, but the lack of perfect coincidence 
 is usually perceptible even if the discrepancy is much 
 less than a tenth of a scale-division. It will very rarely 
 exceed a hundredth of a smallest subdivision if the work 
 is carefully done. 
 
 154. The Vernier. Rule a straight line along a strip 
 of cardboard 5 or 10 cm. wide and about 20 cm. long 
 (Fig. 49). As shown in the diagram, lay off a scale of 
 
 f 
 
 i. "T 
 
 i i i i 1 i , 
 
 I 
 
 i i 
 
 
 \ 
 
 B I i i i ' | i 
 
 \ lo 5 
 
 \ <sealt 
 
 'Mill 
 lo 
 
 i i i i 
 
 15 
 
 'T 
 
 FIG. 49. MODEL OF A VERNIER CALIPER. The diagram can 
 be drawn on cardboard, cut out, and used for fairly accurate measure- 
 ments. 
 
184 THEORY OF MEASUREMENTS 154 
 
 centimetres along the lower side of the line; along the 
 upper side lay off a series of ten intervals, making each 
 one 0.9 cm. long, so that their combined length will be 
 just 9 cm. Rule a perpendicular line, AB, a little to 
 the left of the common zero point, and mark the words 
 "scale" and " vernier" as indicated. See that your 
 work corresponds in every respect to Fig. 49, except 
 that the letters A, B, and C are not marked, and the 
 scale of centimetres may be longer or shorter than shown 
 here. Observe that the mark 1 on the vernier is one 
 tenth of a centimetre to the left of the mark 1 on the 
 centimetre scale, 2 on the vernier is two tenths to the 
 left of 2 on the scale, etc. Put the card on a larger 
 piece of cardboard, to avoid marring the table, and cut 
 from C to B, but no further, also from A to B. The 
 result will be a cardboard model of a vernier caliper. 
 Notice that the zero of the vernier acts as a pointer that 
 indicates cm. when the jaws (AB) of the caliper are no 
 distance apart; consequently it must give the correct 
 reading on the centimetre scale when the jaws are slid 
 apart to any given distance. 
 
 Holding the body of the vernier caliper stationary on 
 the table move the vernier slightly to the right so that 
 its line number 1 coincides with the 1-centimetre mark; 
 then move it further until 2 and 2 coincide; then 3 and 
 3; etc. When the tenth division of the vernier has been 
 brought into coincidence with a line on the centimetre 
 scale it will be found that the zero of the vernier stands 
 at the mark 1 cm. and the jaws of the caliper are sepa- 
 rated by a distance of a single centimetre. From the 
 construction of the apparatus it will be clear that when 
 any division of the vernier (say number 3) coincides 
 with any division of the lower scale the jaws of the caliper 
 must be separated by a corresponding number (3) of 
 
XIV THE PRINCIPLE OF COINCIDENCE 185 
 
 tenths of a centimetre more than some whole number of 
 centimetres. 
 
 Separate the jaws of this model vernier caliper by a 
 distance of eight and a half centimetres, as nearly as 
 you can by estimation; then look at the vernier and 
 notice that while its zero is beyond the mark 8 of the 
 centimetre scale the division 5 of the vernier coincides 
 with some division (no matter which one) of the main 
 scale. The reading is accordingly 8.5 cm. In the same 
 way read the indicated length when the caliper is set 
 at random, and repeat the process until you are per- 
 fectly familiar with it. 
 
 155. Use of the Vernier Caliper. Hold the two parts 
 of the caliper in alignment by pinching the line BC 
 between the fore-finger and thumb of the left hand, 
 holding the lower part tightly but allowing the upper 
 part to slide; at the same time hold the same line 
 between the fore-finger and thumb of the right hand, 
 allowing them to slide on the lower part while grasping 
 the upper part firmly. This will allow the model to be 
 accurately applied to any object that is to be measured. 
 Use the apparatus to make twelve measurements of the 
 thickness of the wooden block at different positions 
 around its edge. Write down each measurement as 
 soon as it is obtained, and do not be disconcerted by 
 the fact that many of them may be identical. Find the 
 average thickness of the block, carrying the result out to 
 one more decimal place than the individual measure- 
 ments, and keep it for later reference. 
 
 If a vernier caliper is arranged to give tenths of a 
 scale-division it is possible to make a reading of half a 
 tenth, or a twentieth, in case two adjacent lines on the 
 vernier show equal and opposite deviations from co- 
 incidence with two lines of the scale. In this way an 
 
186 THEORY OF MEASUREMENTS 157 
 
 ordinary barometer scale of twentieths of an inch, pro- 
 vided with a 25-division vernier that gives 500ths, can 
 be read accurately to single thousandths of an inch. 
 
 156. Slide-Rule Ratios. It has already been seen that 
 setting a slide rule for the ratio TT and picking out a point 
 of exact coincidence will usually give a numerical value 
 for TT that has more than three-figure accuracy, i. e., 
 greater accuracy than can be obtained by the ordinary 
 process of " estimation" with the slide rule. Numbers 
 like the 26 and 66 on the back of the slide rule for the 
 ratio of inches and centimetres are so chosen as to 
 give the correct value to at least as many significant 
 figures as can be read with the apparatus used. Some 
 slide rules give the same equivalent as 50 in. = 127 cm. 
 This has the disadvantage of not being quite as easy to 
 set on the A and B scales as on C and D, but for use 
 with a very finely graduated instrument or with one 
 on which the scales are 20 inches long instead of 10 it has 
 the advantage of greater accuracy, for 66.0000 -r- 26.0000 
 = 25.3816 while 127.0000 -s- 50.00000 = 25.40000. The 
 former ratio is correct to three significant figures only, 
 while the latter is correct to five ( 53). 
 
 157. Questions and Exercises. 1. Consider carefully 
 what is meant by the expression "the principle of co- 
 incidence," and then write a definition of it in your own 
 words. See that it is a definition of a principle, not of a 
 process or of a condition. 
 
 2. What is the significance of the fact that the second 
 point of "coincidence" in Fig. 48 is not as precise as the 
 first, nor the third as good as the second? 
 
 3. Suppose the time of vibration to be just 7/8 as 
 long for one pendulum as for another. How could you 
 verify the fact by the method of coincidences. 
 
 4. If you take just seventeen steps while a man who 
 
XIV THE PRINCIPLE OF COINCIDENCE 187 
 
 is walking beside you takes just eighteen how long are 
 his steps in terms of yours, which are considered of unit 
 length? How does the frequency of his steps compare 
 with that of yours? 
 
 5. Explain why the method of coincidence is more 
 exact than the method of estimation. 
 
 6. If the numbers in the column headed "interval," 
 153, are trustworthy to five significant figures how can 
 you account for the fact that even the fourth significant 
 figure fluctuates in column of centimeters per inch? 
 
 7. If an ordinary vernier can be read to half-tenths 
 why can it not be used for other fractions of a tenth? 
 
 8. Could the upper scale in Fig. 48 be used as a 
 vernier for the centimetre scale? If so, would it indicate 
 elevenths of a centimetre, or thirteenths, or neither? 
 How would its numbers be arranged? 
 
XV. MEASUREMENTS AND ERRORS 
 
 Apparatus. A vernier caliper; 100 seeds of Phaseolus 
 (or other variates). 
 
 158. Direct and Indirect Measurements. An indirect 
 measurement is one that is obtained from another meas- 
 urement by means of a calculation. The ordinary proc- 
 esses of measuring length or weight are called direct, 
 because the unknown length is placed beside a standard 
 series of multiples and fractions of the unit of length 
 and is directly compared with it, and an unknown 
 weight is directly balanced against a series of known 
 weights until its exact equivalent is determined. An 
 example of an indirect measurement is the usual method 
 of determining density. The volume of an object and 
 its mass are determined directly, or the volume is cal- 
 culated from other measurements, and its density is 
 then obtained by making a calculation of the ratio of 
 mass to volume. 
 
 Are measurements of area usually direct or indirect? 
 Can they be made in the other way? 
 
 How could the density of a liquid be determined by a 
 direct measurement? 
 
 How could the density of a solid be determined by a 
 direct measurement? 
 
 159. Independent, Dependent, and Conditioned Meas- 
 urements. Measurements are also classified as inde- 
 pendent, dependent, and conditioned, and may belong to 
 any one of these classes whether they are direct or indirect. 
 
 Two or more measurements are said to be conditioned 
 if there is a theoretical relationship between them, 
 which must always hold good. The three angles of a 
 
 188 
 
XV MEASUREMENTS AND ERRORS 189 
 
 plane triangle furnish an illustration of this class. Their 
 sum must always be equal to w, or 180. 
 
 Measurements are said to be dependent if one of them 
 is allowed to influence or bias the observer when making 
 a later measurement of the same quantity. With accu- 
 rate determinations this effect is so hard to avoid, even 
 if the observer has the best of intentions, that it is always 
 advisable to guard against it by some such device as 
 hiding the scale of an apparatus from view until after 
 the indicating mark has been set in the position that has 
 to be read. Measurements are also dependent if any 
 essential step in making them is not repeated in succes- 
 sive determinations but is assumed to have its effect 
 remain unchanged during the series. Thus, in making 
 independent measurements by the method of coincidences 
 ( 153) the scales of length were not held in one position 
 while several coincidences were found, but were reset 
 after each determination. 
 
 It is customary to use the term independent only 
 for measurements that are at the same time neither 
 dependent nor conditioned. 
 
 If the density of an object should be determined by a 
 direct measurement would its mass, density, and volume 
 be independent, dependent, or conditioned? 
 
 160. Harmony and Disagreement of Repeated Meas- 
 urements. If the same object is measured several times 
 in succession the measurements will in general differ 
 from one another, but the differences will tend to be 
 small under either of two different circumstances: (a) if 
 the quantity is of a sharply defined character, or (6) if 
 the method of measurement is coarse or crude. Thus, 
 if a length of woolen cloth were compared with an accu- 
 rate millimetre scale it would probably be found difficult 
 to measure it the same twice in succession; but if the 
 
190 THEORY OF MEASUREMENTS 160 
 
 same length of steel rail were clamped on rigid supports 
 and measured at a constant temperature it might be 
 hard to obtain two measurements that would differ. 
 The length of one object would be a poorly defined 
 quantity; that of the other, a sharply defined quantity. 
 
 Although sharpness of definition of the quantity to 
 be investigated means that repeated measurements will 
 tend to harmonize closely, it should be carefully noted 
 that accuracy of the methods or means of measurement 
 has just the opposite result; it is the rough methods of 
 measurement that make repeated determinations iden- 
 tical with one another, and the refined methods that 
 show discrepancies. Two similar 1-lb. weights may 
 appear to have precisely the same mass on a rough 
 balance and yet differ when weighed on a more carefully 
 constructed one. If the heavier weight should then be 
 filed down just enough to make it equal to the lighter one 
 a test with a delicate chemical balance might show not 
 only that the two were still unequal but even that one 
 of them alone would not weigh the same amount twice 
 in succession. 
 
 The general statement can be made, then, that an 
 accurately defined quantity or a coarse method of meas- 
 urement will result in a series of determinations being 
 harmonious or identical, while a poorly defined quantity 
 or an accurate method of measurement will cause the 
 results to disagree or diverge. 
 
 Which measurements do you think would be most apt 
 to show variations among themselves, the measurements 
 of the wooden block which you made with the cardboard 
 caliper, or measurements of the same block made with a 
 steel vernier caliper? 
 
 It is a general truth that, no matter how sharply 
 defined a quantity may be, the use of the most precise 
 
XV MEASUREMENTS AND ERRORS 191 
 
 methods will result in successive equally careful measure- 
 ments of it differing perceptibly from one another, al- 
 though the differences may be very small. Thus, the 
 most accurate determinations of the length of a national 
 prototype meter ( 19) would be apt to differ by several 
 tenths of a micron. 
 
 If equally careful measurements of the same quantity 
 fail to agree and there is no reason for preferring one 
 rather than another it necessarily follows that the true 
 magnitude of a measured quantity is always unknown. 
 Accordingly the various approximate measures are sum- 
 marized, for actual use, by stating their average (the 
 sum of n numbers, divided by n) or some other repre- 
 sentative value, instead of by choosing one of them at 
 random. 
 
 161. Errors of Measurement. The error of a meas- 
 urement is the amount by which it differs from the true 
 value of the quantity which is measured. If the true 
 value is always unknown the error must likewise be 
 unknown. Such errors, however, can be discussed the- 
 oretically, and in this way much can be learned about 
 the best manner of dealing with them. 
 
 Errors are usually classified as constant and accidental, 
 but what are known as mistakes really belong in a sepa- 
 rate class by themselves. Constant errors affect all the 
 measurements of a series in the same manner or in the 
 same direction. Accidental errors are small errors that 
 make one measurement a trifle too large and another 
 too small, but do not tend to bias the average result. 
 Mistakes are occasional errors that are due to a lack of 
 mental alertness on the part of the observer. 
 
 CLASSIFICATION OF ERRORS. 
 
 CONSTANT: 
 
 THEORETICAL: usually calculable, such as the faulty length of 
 a linear scale, due to its expansion from heat. 
 
192 THEORY OF MEASUREMENTS 162 
 
 INSTRUMENTAL: due to faulty graduation or adjustment of 
 
 an instrument. 
 PERSONAL: some persons have a constant tendency to estimate 
 
 the instant of an occurrence a little too early; others a little 
 
 too late. 
 
 ACCIDENTAL: 
 
 INSTRUMENTAL: due to varying external influences, "play" 
 of moving parts, inconstant sensitiveness, etc. 
 
 PHYSIOLOGICAL: the senses of sight, touch, etc., have a limit 
 of sensitiveness, and this limit is not always the same. 
 
 PSYCHOLOGICAL: very likely the deductions about the outside 
 world that result from the effect of sense-impressions on the 
 mind do not correspond to the latter so closely as to be 
 absolutely free from irregular variations when dealing with 
 minute quantities. 
 
 MISTAKES: 
 
 MANIPULATIVE : doing the wrong thing. 
 
 OBSERVATIONAL: observing the wrong thing. 
 
 NUMERICAL: recording the wrong numbers. (It is especially 
 important to guard against focussing the attention so closely 
 on the minute part of a measurement, e. g., the estimation 
 of tenths of the smallest scale-division, that a mistake is 
 made in recording the figures that express the larger part of it.) 
 
 162. Accidental and Constant Errors. The difference 
 between constant errors and accidental errors can be 
 easily understood from the analogy of shots fired at 
 a target. In the diagram the average position or center 
 of clustering has a constant tendency townward and to 
 the right of the bull's-eye, while the individual shots 
 have accidental tendencies which carry them to one side 
 of this average position as much as to the other. Any 
 single shot can be considered to have a total error which 
 is the " vector sum" or geometrical combination of its 
 own accidental error plus the constant error of the whole 
 group. Physical measurements are target shots in which 
 the center of clustering can be found but in which the 
 position of the bull's-eye is unknown. 
 
XV 
 
 MEASUREMENTS AND ERRORS 
 
 193 
 
 Constant errors can be avoided or reduced to a mini- 
 mum by the help of theoretical knowledge (effect of 
 gravity in deflecting the shots downward), and by chang- 
 ing observers, methods, and conditions (repeating the 
 shots when the wind blows to the left instead of to the 
 right), and above all by judgment, 
 
 experience, care, and alert hunting t jfs$* * 
 
 for all possible sources of error. ^^\ * * 
 
 Accidental errors are much more . 9 * * * 
 easily investigated, the chief prob- 
 
 lems from the standpoint of phys- 
 ical science being the determination 
 of the center of the cluster and the 
 measurement of the degree of scat- 
 tering which takes place around it. 
 
 State one source of error that was 
 present when you performed each 
 of the experiments mentioned in 
 the following list. For example, in 
 (e), one source of error would be 
 the fact that the periphery of the 
 disc is not an exact circle. This 
 would give rise to a constant error 
 in the determination of ?r if the 
 
 diameter should be measured each time along one marked 
 radius, but would cause an accidental error if many dif- 
 ferent diameters should be measured. Mark them in 
 such a way as to show which ones gave rise to constant 
 errors, and which ones caused accidental errors. Were 
 there any mistakes? 
 
 (a) The measurement of your span. 
 
 (6) The measurement of an irregular area, first method. 
 
 (c) The measurement of the density of an irregular solid. 
 
 (d) The measurement of the sine of an angle that has a given 
 tangent. 
 
 14 
 
 FIG. 50. ILLUSTRA- 
 TION OF ERRORS OF 
 MEASUREMENT. The 
 shots (measurements) 
 aimed at the bull's-eye 
 (true value) show a 
 general drift (constant 
 error) to the right and 
 downward, and indi- 
 vidual deviations (acci- 
 dental errors) that tend 
 to extend equally in all 
 directions from the cen- 
 ter about which they 
 cluster. 
 
194 
 
 THEORY OF MEASUREMENTS 
 
 164 
 
 (e) The experimental determination of TT. 
 (/) The calculation of e~* 2 for different values of x. 
 (g) The use of the formula 1/(1 + 5) = 1 - 5. 
 (h) The "black-thread" determination. 
 
 163. Errors and Variations. The theory of accidental 
 errors runs closely parallel with the theory of variation 
 in natural objects, so that a statistical investigation 
 of the properties of several objects of the same kind, 
 or variates as they are technically called, will be found 
 to illustrate many of the facts of accidental variation 
 which could otherwise be observed only by the more 
 tedious study of the deviations of repeated measure- 
 ments of the same object. 
 
 164. Measurement of Variates. Examine the vernier 
 caliper, and if it has an extra scale that reads backwards, 
 or one scale for internal measure- 
 ments and another for external 
 ones decide which scale reads 
 the internal distance between 
 the jaws of the caliper, and 
 which point marks its zero when 
 the jaws are closed. See that 
 you understand the vernier and 
 have no trouble in reading it at 
 any setting. 
 
 Measure the length of 100 
 seeds or other variates of the 
 same kind. Make a table of 
 preliminary measurements of ten 
 variates in the order in which 
 they were measured to find their 
 general range, and then make a 
 table of lengths arranged in 
 
 cm. 
 
 1.56 
 
 1.58 
 1.59 
 1.60 
 1.61 
 1.62 
 1.63 
 1.64 
 1.65 
 1.66 
 1.67 
 1.68 
 
 frequency 
 
 EXAMPLE OF A FRE- 
 QUENCY DISTRIBUTION. 
 Notice that the expected 
 lengths are tabulated in 
 numerical order before the 
 actual measurements are 
 made. 
 
 numerical order likethe above, recording each different 
 
XV 
 
 MEASUREMENTS AND ERRORS 
 
 195 
 
 length and the number of times that it occurs. Do not 
 merely write the length when one seed is measured and 
 add a pen-stroke when another of the same length is 
 found, but see that each variate is represented by a 
 mark in the second column of your table. If an unex- 
 pectedly large or small value is found after the first col- 
 umn has been written down it may be noted anywhere 
 at the beginning or end of the table, as shown here. 
 
 Plot a graphic diagram in which the first ten variates 
 have their measurements in centimetres laid off along the 
 x-axis and the frequency of each length is represented 
 by the height of a corresponding ordinate. Do not 
 begin the scale of ^-values at zero, but allow one square 
 for each hundredth of a centimetre. The t/-scale in any 
 statistical diagram should measure one square for each 
 unit of frequency, since there are rarely enough measure- 
 ments to require a condensed scale and nothing would 
 be gained by an expanded scale where there are no frac- 
 tional values to be plotted. Connect the points of the 
 
 
 Y 
 
 
 
 
 
 
 Y 
 
 
 
 
 
 31 
 
 
 
 
 
 
 4 
 
 
 
 
 
 
 
 
 
 / 
 
 \ 
 
 
 3 
 
 
 
 
 
 
 I 
 
 / 
 
 \ 
 
 1 
 
 
 \ 
 
 2 
 
 
 
 
 
 
 
 6 
 
 
 
 
 x\ 
 
 t 
 
 
 
 
 
 
 X 
 
 
 
 
 < 
 
 ' 
 
 
 
 i 
 
 2 
 
 3 
 
 4 
 
 5 
 
 FIG. 51. FREQUENCY POLYGON AND HISTOGRAM. The ordinary 
 graphic diagram made with broken lines is called a frequency polygon 
 if the ordinates represent the frequency of occurrence of the abscissae. 
 When a length along the x-axis is used instead of a point, and a 
 vertical strip instead of a line, the diagram is called a histogram. 
 The diagrams show two slightly different ways of representing 
 
 . , tx = 0, 1, 2, 3, 4, 5. AT ., 
 
 precisely the same facts, namely, -< _ ' ' ' ' Notice 
 
 {.y u, z, i, 6, A, u. 
 
 particularly the difference in the placing of the scale numbers. 
 
196 THEORY OF MEASUREMENTS 165 
 
 diagram by a broken line and notice that the " curve" 
 is highest in the middle and slopes downward, more or less 
 uniformly, toward the ends. A graph of this sort is 
 called a frequency polygon. 
 
 Make another graphic diagram from the same table, 
 but instead of representing successive ordinates by the 
 successive vertical ruled lines of the cross-section paper 
 let them be represented by the successive white strips 
 that lie between the vertical lines (Fig. 51), and use a 
 vertical scale of one square for each measurement as 
 before. A graph constructed in this way is called a 
 histogram. Notice that the area inclosed is numerically 
 equal to the entire number of measurements or other 
 statistics that are represented. 
 
 Make a histogram from your table of the measure- 
 ments of 100 variates. 
 
 165. Questions and Exercises. 1. In addition to the 
 sources of error that you have stated for the experiments 
 listed at the end of 162, write down at least ten more 
 sources of error that were present when you performed 
 those experiments. 
 
 2. Which form of frequency diagram do you consider 
 to be the more ''graphic," the histogram or the frequency 
 polygon? 
 
XVI. STATISTICAL METHODS 
 
 Apparatus. Ruler; slide rule. 
 
 166. Frequency Distributions. A tabulation of a set 
 of measurements that shows how many times each 
 observed value occurs is said to give the frequency dis- 
 tribution of the measurements, and a graphic diagram in 
 which the ordinates give the frequencies of the measure- 
 ments that are represented by the abscissae is called a 
 frequency polygon if drawn with the usual broken line 
 or a histogram if constructed by building up rectangles 
 on successive segments of the base line. 
 
 Examine Fig. 52 and test it with a ruler held hori- 
 
 n n 
 
 156 
 
 too 
 
 1.68 
 
 f.56 
 
 1.60 
 
 164 
 
 f.68 
 
 FIG. 52. EXAMPLE OF FREQUENCY DIAGRAMS. These correspond 
 to the table in 164. 
 
 zontally in order to determine whether both frequency 
 diagrams represent the same frequency distribution. 
 Then compare, without measuring, the apparent heights 
 of the successive points on the frequency polygon with 
 the successive values given in the table in 164. Has 
 any mistake been made in attempting to represent that 
 table graphically by the construction of Fig. 52? 
 
 197 
 
198 THEORY OF MEASUREMENTS 167 
 
 167. Class Interval. When a set of variates was meas- 
 ured according to 164 the determinations were rounded 
 off to the nearest hundredth of a centimetre by the 
 inability of the apparatus to measure them more accu- 
 rately. If the five seeds, for example, which measure 
 1.62 cm. in the printed table could have been measured 
 with a much higher degree of accuracy there can be no 
 question that each of them would have a different length 
 from any of the others. 
 
 Under such circumstances the ordinary type of fre- 
 quency diagram could never be obtained, for the fre- 
 quencies would invariably be something like the series 
 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 
 and the diagram would no longer have its characteristic 
 "mound-like" shape with the ordinates high in the 
 middle of the graph and dwindling away toward each end. 
 
 The shape of a frequency diagram, accordingly, may de- 
 pend upon the numerical distance between the successive 
 abscissse that are used, the least difference between re- 
 corded measurements. In the table of 164 this amounts 
 to one hundredth of a centimetre, but if those lengths 
 had been rounded off to the nearest half millimetre 
 the frequency distribution would have become 
 
 f 1.55 1.60 1.65 1.70 centimetres, 
 [ 1 9 41 2 cases, 
 
 in which little is left of the original shape of the graph 
 except a suggestion of the general " mound-like" char- 
 acter. 
 
 The least difference in successive recorded measure- 
 ments is .05 cm. in the last illustration, .01 cm. in the 
 original table and Fig. 52, and is called the class interval. 
 
 If your frequency polygon tends to the nondescript 
 type seen when the class interval is too small the meas- 
 
XVI STATISTICAL METHODS 199 
 
 urements of the hundred variates should be regrouped. 
 Make a new table in which the values from 1.55 to 1.65 
 are all called 1.6, those from 1.65 to 1.75 are considered 
 as 1.7, etc. If several measurements have been recorded 
 as 1.65 put about half of them in the class 1.6 and the 
 other half in the class 1.7. From this table plot sepa- 
 rately both the frequency-polygon and the histogram 
 that illustrate it graphically.* Do not use an expanded 
 scale in any statistical graph either for frequency or for 
 class interval. 
 
 168. Types of Frequency Distribution. If the class 
 interval is made very small and the measurements are 
 very numerous, both forms of diagram can be considered 
 as losing their abrupt changes until they merge into 
 two identical curved lines, the notches that were orig- 
 inally present having become indefinitely small. Fre- 
 quency distributions are not invariably " mound-shaped," 
 but may be classified, according to the general shape of 
 the graphic diagram, as (a) symmetrical, (b) moderately 
 asymmetrical, (c) very asymmetrical or J-shaped, (d) 
 bilocular or U-shaped, (e) rectangular (Fig. 53). 
 
 The symmetrical type (a) is seen in physical meas- 
 urements. Notice that in it (1) the average (i. e. } 
 middle) z- value is the most frequent; (2) measurements 
 that are a given amount above or below the average are 
 less frequent than it but of equal frequency with each 
 other; and (3) extremely divergent values do not occur. 
 The asymmetrical type (6) is similar except that the 
 most frequent value does not lie in the middle of the 
 distribution and on one side of it the frequency falls 
 away more rapidly than on the other. Moderate asym- 
 metry is common in all statistical data. An example of 
 
 * If preferred, a class interval of .05 cm. may be used instead. 
 This will avoid splitting any original class into two. 
 
200 
 
 THEORY OF MEASUREMENTS 
 
 the J-shaped type (c) may be seen in the distribution of 
 wealth among any population; the frequency of indi- 
 viduals with little wealth is very great, and with in- 
 
 / b 
 
 a 
 
 FIG. 53. TYPES OF FREQUENCY DISTRIBUTION. (a) Symmet- 
 rical. (6) Positively asymmetrical (skewed toward the right), (c) 
 Extreme asymmetrical, or J-shaped. (d) Bilocular, or U-shaped. 
 (e) Rectangular. 
 
 creasing wealth the number of cases falls off until it- 
 reaches a vanishing point. The curious U-shaped type 
 (d) is seen where there are tendencies toward both 
 extremes, or the centre is a position of unstable equilib- 
 rium; and the rectangular type (e) occurs in purely 
 mathematical cases, such as the actual error of a tabu- 
 lated logarithm, which is never more than d= 0.5 of the 
 unit in the last decimal place and has all intermediate 
 values with equal frequency. 
 
 169. The Probability Curve. The measurement of 100 
 seeds will probably show a moderate degree of asym- 
 
XVI STATISTICAL METHODS 201 
 
 metry, since such objects do not grow beyond a definite 
 size but do fall short of it in many cases. This is called 
 negative asymmetry, and the curve is said to be skewed 
 toward the left. Physical measurements, however, tend to 
 be above the average just as often as they are below it, and 
 so give the symmetrical form of frequency distribution. 
 As the measurements are made more and more numerous 
 the frequency polygon approaches more and more closely 
 to the form of the so-called probability curve, y = e~**, 
 which was calculated in the lesson on logarithms and 
 plotted in the lesson on graphic representation. In 
 some cases it will appear drawn out relatively flat and 
 in others will be very high and narrow (Fig. 34), but the 
 curve is the same in all cases, except that the scales of 
 ^-values and ^/-values are condensed or spread out to 
 different degrees. 
 
 Carry out the binomial expansion that is given below 
 at least as far as n = 15, noticing that each two succes- 
 sive terms have to be added in order to obtain the term 
 below and between them. Lay off a series of equal 
 intervals on the z-axis. On these erect a series of ordi- 
 nates proportional to the successive terms of the last 
 polynomial, in order. A smooth curve through the tops 
 of the ordinates will give a very good approximation 
 to the probability curve. 
 
 (1 + D = 1 
 
 (1 + I) 1 = 1 + 1 
 
 (1 + I) 2 = 1+2 + 1 
 
 (1 + 1) 3 = 1+3+3 + 1 
 
 (1 + I) 4 = 1+4 + 6 + 4 + 1 
 (1 + I) 5 =1+5 + 10 + 10 + 5 + 1 
 (1 + 1)" = 
 
 170. Representative Magnitudes. For most scientific 
 work, the statement of a whole frequency distribution 
 
202 THEORY OF MEASUREMENTS 171 
 
 or of each one of a long series of measurements would 
 be a cumbrous process and reading such statements 
 would be a tedious task. Accordingly it is customary 
 to summarize such a set of values by stating some 
 representative value, such as the average. In special 
 cases such representative magnitudes as the geomet- 
 rical mean, M (oiO 2 - -a n ), or the harmonic mean, 
 n/(l/di + 1/02 + + I/On), or the quadratic mean, 
 V [(i 2 + o 2 2 + + o n 2 ) /n], have been employed, but 
 the most usual one is the arithmetical mean or average, 
 (01 + o 2 + H- a n )/n. The median, which is o n if 
 01, o 2 o 2tt _i are arranged in order of size, is frequently 
 useful as a representative value, as is also the mode 
 or modal value, which is simply the value that occurs 
 with the greatest frequency. The word mean is some- 
 times used in a general sense for any representative value, 
 and sometimes is restricted to the same significance as 
 the word average. 
 
 171. The Average The average is obtained by adding 
 together a set of values and dividing the sum by the 
 number of values. Thus the average of the five values 
 3, 3, 4, 5, 10, is one fifth of their sum, or 5. If certain 
 values occur repeatedly the average is obtained by divid- 
 ing the sum by the number of values, not by the number 
 of different values. Thus, the average of the measure- 
 ments given in the table at the end of the preceding 
 lesson is (2 X 1.68 + 6 X 1.67 + 3 X 1.66 - -)/(2 + 6 
 + 3 --), or 8686/53 or 1.639. 
 
 172. The Median. The median is obtained by choos- 
 ing such a value that half of the other values exceed it 
 and half are below it. // the numbers are arranged in 
 numerical order in a column the number that is half way 
 down the column is the median. Otherwise it may be 
 found by crossing off the largest number and the smallest, 
 
XVI STATISTICAL METHODS 203 
 
 and repeating the process until only one is left. If two 
 numbers are left, as will be the case if there are an even 
 number of measurements, the number half way between 
 them can be taken as the median, but in physical or 
 statistical data it usually happens that the two remaining 
 numbers are the same. 
 
 Find the median of 3, 3, 4, 5, 10. What is the median 
 of 7.4, 6.8, 7.3, 7.3, 7.2, 7.1, 7.2? Ans.: 7.2. Find the 
 median of the measurements that are tabulated in 164. 
 
 Turn to the histogram in Fig. 51, 164, and cross off 
 one of the small squares between the curve and the base 
 line at the extreme left and then one at the extreme right ; 
 repeat the process until only one (or two) squares are 
 left, and show that the median is 3. Imagine that Fig. 
 54 is similarly divided up into small squares, each square 
 representing one measurement, and satisfy yourself of 
 the truth of the fact that the ordinate that is drawn up- 
 ward from the median x-value must bisect the area of the 
 frequency curve. 
 
 173. The Mode. The mode is the most frequent 
 value. Thus in the set of values 3, 3, 4, 5, 10 the mode 
 is the number that occurs twice, namely 3. In the 
 illustration of the measurement of variates the mode is 
 1.64, the length that was found most often. 
 
 174. Choice of Means. If a frequency distribution is 
 of the symmetrical type the average, mode, and median 
 will all be the same. 
 
 Where there is a moderate degree of asymmetry the 
 median will come nearer to the mode than the aver- 
 age does, as is shown in the following diagram. The 
 mode is easily seen to be the most probable value. If 
 a seed is taken at random from the set whose measure- 
 ments were given in the table it is more likely to measure 
 1.64 cm. than any other amount. The mode has one 
 
204 
 
 THEORY OF MEASUREMENTS 
 
 174 
 
 decided disadvantage, however, in that it cannot always 
 be chosen from a given frequency distribution. For 
 example the mode cannot be obtained from the series 
 
 Qf Mo Me Av Q3 
 
 FIG. 54. TYPICAL FREQUENCY DIAGRAM. The mode has the 
 highest ordinate. The ordinate of the median bisects the area under 
 the curve. The ordinate of the average passes through the center 
 of gravity of the area. The median lies between the mode and the 
 average, and is twice as far from the former as from the latter. The 
 difference (distance) between the quartiles Q\ and Q s ( 178) is 
 called the interquartile range. 
 
 3, 4, 4, 5, 6, 6, 7, 8, except by assuming that the values 
 follow the probability law, and fitting a probability 
 curve to them as accurately as possible by a " black 
 thread" method; the position of the top of this theoret- 
 ical curve can then be determined. Where there is a 
 long series of measurements the average is sometimes 
 calculated from the formula, indicated on the above 
 diagram, that me mo = 2 (av me), or that the 
 median lies $ of the way from the average to the mode. 
 
 The median, like the mode, is easy to determine. 
 It can be used in two classes of cases where the average 
 cannot be determined. One is in case measurements 
 have been tabulated with indefinite terminal classes, 
 e. g., "less than 10 mm., 7 cases; 10 to 12 mm., 4 cases; 
 
 12 to 14, 5; 14 to 16, 3; above 16, 2 cases." Here the 
 median is the class "between 10 and 12," say 11 mm., 
 and the mode is probably the class "12 to 14," say 
 
 13 mm., but the average cannot be found on account 
 
XVI STATISTICAL METHODS 205 
 
 of nine of the numerical values not being stated; the 
 best that could be done would be to guess that the 
 average was a little less than the median. The other 
 case is where quantities can be arranged in numerical 
 order but are difficult to measure individually. Thus 
 it may be difficult or impossible to gauge the scholar- 
 ship of a student in accurate numerical terms, but if a 
 group of students can be arranged in order of scholar- 
 ship the median can be determined without difficulty. 
 Furthermore the median has a unique advantage over 
 most other representative magnitudes in that it is not 
 affected by inverting the unit of measurement. The 
 median of a number of prices will be the same whether 
 they are given as cents per dozen or as dozens per dollar. 
 Of a group of different velocities the same one will be 
 picked out by choosing the median whether the numer- 
 ical values are expressed in miles per hour or in minutes 
 for a one-mile run. It is easy to see that this will not be 
 the case if the average is used. The median, however, is 
 not quite, as good a representative value as the average, 
 in case they are different, for a series of measurements 
 that have all been made with equal care. Curiously 
 enough, however, the more extensive a series of meas- 
 urements the more likely it is to show that the individual 
 measurements are not equally trustworthy but may 
 be grouped in different classes according to their rela- 
 tive scattering. It is in such cases that the median is 
 a much better representative figure than the average, 
 for the average is influenced by an unduly large or small 
 measurement just in proportion to the aberration of 
 the latter, while as long as a measurement is above the 
 median it makes no difference how far above it may be, 
 its effect is no greater than that of any other single 
 value. (Compare the average and the median of 3, 3, 
 
206 THEORY OF MEASUREMENTS 175 
 
 4, 5, 1O, with those of 3, 3, 4, 5, 30. The very erratic 
 values ought to have the least influence instead of the 
 most.) 
 
 175. Deviations. Of almost as much importance as 
 finding the best representative value for a series of 
 measurements is the determination of how closely they 
 cluster around it or how widely they scatter from it. 
 This will help to furnish information in regard to the 
 accuracy of the measurements, and hence also in regard 
 to the accuracy of the instruments and methods em- 
 ployed in making them; it will also be useful in com- 
 paring and combining determinations made at differ- 
 ent times or by different observers. The difference, 
 m a, between any single measurement and the average 
 (or other representative magnitude) is known as the 
 deviation or variation of that measurement and is denoted 
 by the letter v. It is not the same as the error of the 
 measurement, for the error may have a constant com- 
 ponent that affects all of the measurements equally; 
 but it may be considered as the accidental error or 
 accidental component of the total error. Deviations 
 give no positive indication of any constant errors that 
 may be present. 
 
 If each one of a series of independent measurements 
 is as trustworthy as any other it can be shown mathe- 
 matically that their best representative value is their 
 arithmetical mean, or average; and accordingly the 
 average is the figure that is almost invariably used. 
 
 Copy the two following columns of figures, find the 
 average of each set, and write after each measurement 
 (m) its deviation (v) from the average of the figures 
 in the same column, marking it with a minus sign if the 
 value is less than the average, and with a plus sign if in 
 excess of the average. 
 
XVI 
 
 STATISTICAL METHODS 
 
 207 
 
 m 
 
 V 
 
 m 
 
 V 
 
 27.35 
 
 
 27.36 
 
 
 27.34 
 
 
 27.38 
 
 
 27.34 
 
 
 27.35 
 
 
 27.33 
 
 
 27.37 
 
 
 27.34 
 
 
 27.32 
 
 
 27.34 
 
 
 27.30 
 
 
 27.33 
 
 
 27.31 
 
 
 27.34 
 
 
 27.30 
 
 
 27.35 
 
 
 27.35 
 
 
 27.34 
 
 
 
 27.36 
 
 
 
 
 
 Although both averages are the same it is evident 
 that the first set of measurements must have been made 
 by a more trustworthy instrument, observer, or method 
 than the second. If the aver- 
 ages had not been of the same 
 value the first one would un- 
 doubtedly have been entitled to 
 more confidence, other things 
 being equal, than the second. 
 
 Add each column of deviations 
 and verify the fact that the alge- 
 braical sum of the deviations from 
 the average is always zero. If 
 their sum is zero their average 
 will also be zero, so that when an " average deviation" 
 is recorded the term means not the average of the alge- 
 braical deviations but the average of the values that 
 they would have if the negative signs were omitted. 
 
 176. Average by Symmetry. The foregoing property 
 suggests an easy method of finding the average in simple 
 cases: If a number can be so chosen that the individual 
 measurements are symmetrically grouped around it the 
 sum of the positive deviations will equal the sum of the 
 negative deviations and the number will be the required 
 average. 
 
 What is the average of 115 and 119? Ans.: 117, 
 because it makes the sum of the deviations (+2 and 
 2) equal to zero. 
 
 Find the average of 3, 6, 9, 12, 15 by the method of 
 symmetry. 
 
 What is the average of 16, 18, 20, 22? Of 14 and 17? 
 Of 12, 14, 17, 19? Of 126.8 and 127.4? Of 121 and 
 141? Of 161 and 191? Of 198, 199, 203? Of 8, 10J, 
 11 1? What is the value, to the nearest whole number, 
 
208 THEORY OF MEASUREMENTS 178 
 
 of the average of 117, 116, 117? (Suggestion: Is the 
 average above or below 116.5?) What is the exact 
 average of 17, 18, 18, 19, 19, 20? 
 
 177. Average by Partition. Before leaving the sub- 
 ject of the average it should be noted that there is no 
 need of adding the entire numerical values if they con- 
 tain a part in common. Thus in either of the columns 
 of figures in 175 the average is obtained by writing 
 the first three figures, which are common to all values, 
 and annexing the average of the last figure. 
 
 178. Quartiles. Just as the middle number of a series 
 arranged in ascending or descending order of magnitude 
 is called the median, or half-way value, so the middle 
 one of the numbers that lie below the median is called 
 the lower quartile, or quarter- way value; and the median 
 of the numbers that lie above the median is known as 
 the upper quartile, or three-quarter-way value. The 
 quartile abscissae are laid off on the base line of Fig. 54, 
 174, at Qi and Q 3 . It should be carefully noticed that 
 their ordinates, together with the median ordinate, divide 
 the whole area of the frequency diagram into four equal 
 parts. If there is any difficulty in understanding this 
 remember that the area of a histogram indicates the 
 total number of measurements ( 164). If half of them 
 lie below the median, by definition, and half above 
 there can be no trouble in realizing that the median is the 
 abscissa whose ordinate bisects the area of the figure. 
 Of course the same thing is not true of the average, 
 except when average and median happen to have the 
 same value. 
 
 What are the quartiles of the series 3, 4, 4, 5, 6, 8, 11? 
 
 Find the median and quartile values of the set of 
 numbers 28, 29, 31, 36, 31, 30, 35, 33, 32, 36, 29, 28, 32. 
 Notice that the numbers " below" the median need not 
 
XVI STATISTICAL METHODS 209 
 
 all be smaller numbers; the median is 31 and one of the 
 lower numbers is also 31. Notice that the lower quartile 
 is the median of the six numbers below the median, 
 not of the seven numbers that include the median itself. 
 Similarly, the upper quartile is not 33, but 34 ( 172). 
 
 179. Semi-Interquartile Range. The numerical dis- 
 tance between the lower quartile and the upper quartile 
 is called the interquartile range. In the series 7, 8, 10, 
 13, 19, 22, 27, the quartiles are 8, and 22, and their 
 difference, 14, is the range between quartiles, or inter- 
 quartile range. Half of this, or 7, is called the semi- 
 interquartile range; notice that it is the average of the 
 two distances from median to quartile. It is sometimes 
 used as a measure of the scattering or clustering of a 
 set of observations, and its theoretical importance will 
 be seen later. 
 
 Find the semi-interquartile range of each of the two 
 columns of figures in 175 after re-arranging the data 
 in numerical order. For the first column: if the five 
 smaller values are considered as lying below the median 
 and five above, that is, if the median is considered to 
 occupy no numbers in the middle of the column the 
 lower quartile will be 27.34; but if only four values are 
 considered as lying below the median, and four above, 
 that is, if the median is considered to include two numbers 
 in the middle of the column the lower quartile will be 
 27.335. As the median is actually considered to be 
 neither two numbers nor no numbers, but is one single 
 number it will be satisfactory to say that the lower quar- 
 tile is neither 27.340 nor 27.335, but is half way between 
 these two values, say 27.338. Similarly, show that the 
 upper quartile is 27.342. The second column should be 
 treated in the same manner. What do you notice about 
 the relative size of the two semi-interquartile ranges? 
 15 
 
210 THEORY OF MEASUREMENTS 180 
 
 180. Questions and Exercises. 1. Complete the fol- 
 lowing rule for finding the upper quartile of 12 measure- 
 ments, and submit it to your instructor for inspection: 
 " Arrange the measurements in order of magnitude and 
 find the median by taking the average of the sixth and 
 seventh. Then . . ." 
 
 2. When the tabulated values of 164 are re-grouped 
 in larger class intervals ( 167) there are seven measure- 
 ments of 1.65 cm., which theoretically should be sepa- 
 rated into two equal groups for inclusion with the classes 
 1.60 cm. and 1.70 cm. Would it be objectionable to add 
 exactly 3 J squares to each of the strips of the histogram, or 
 would it be better to add 3 to one and 4 to the other? 
 Explain why. 
 
 3. Why does it generally happen that the median of 
 a set of measurements fails to be exactly twice as far 
 from the mode as from the average? 
 
 4. Draw roughly a curve like Fig. 54, but having 
 negative instead of positive asymmetry. Place the 
 mode and the median where you think they should go, 
 and then locate the average. 
 
 5. Make a tabular statement of (a) the advantages 
 of the median over the average, and (6) the advantages 
 of the average over the median, as a representative value. 
 
 6. Prove that for any three numbers (say, k + di, etc.) 
 the algebraical sum of the deviations from the average 
 must be equal to zero. 
 
 7. If a train travels the first, third, fifth, seventh, 
 . . . miles at a rate of 20 miles per hour, and the second, 
 fourth, sixth, eighth, . . . miles at 30 miles per hour, 
 show that its average velocity is not the average of the 
 two rates, 25 miles per hour, but is their harmonic 
 mean ( 170) instead. 
 
 8. What would you suggest for a numerical measure of 
 asymmetry in a diagram like Fig. 54? 
 
XVII. DEVIATION AND DISPERSION 
 
 Apparatus. Vernier caliper; variates; slide rule; table 
 of logarithms. 
 
 181. Character istic Deviations. Just as the use of an 
 average or other representative magnitude obviates the 
 necessity of stating the separate measurements from 
 which it is derived, so a statement of all the deviations 
 from the average can be replaced by a single character- 
 istic deviation. A set of measurements can then be 
 summarized by two numbers, and it is customary to 
 write such a result in the form a d* where the first 
 number gives the average value of the quantity measured 
 and the second expresses the limiting distances above 
 and below the average which mark off in some way an 
 amount of deviation that is characteristic for the set 
 of measurements. 
 
 182. Total Range, The simplest way of summarizing 
 the deviations of a series of measurements is by stating 
 their total range, or the algebraical difference between 
 the smallest one and the largest one. Obviously the 
 extreme range of the measurements themselves will also 
 give the same result. Thus, in the two columns of 
 measurements in 175, show that the total range is 
 .02 for the first and .08 for the second. Unfortunately, 
 this is not only the easiest but also the worst method of 
 obtaining a characteristic deviation, for the numerical 
 value given by the total range depends upon only two 
 of the measurements of the series; and those two are 
 
 * The letter d will be used in this book to denote a particular kind 
 of characteristic deviation. A single deviation of any individual 
 measurement will be indicated by the letter v (variation). 
 
 211 
 
212 THEORY OF MEASUREMENTS 184 
 
 the least satisfactory ones, because a repetition of the 
 series of measurements would be likely to show a con- 
 siderable fluctuation in the largest and smallest values, 
 while the most closely clustered values would hardly be 
 changed at all. 
 
 What is the numerical value of the total range of the 
 data in the table of 164? In what kind of units should 
 it be expressed? 
 
 183. Average Deviation. A better index of the 
 amount of scattering is given by the average deviation. 
 This is the average of the positive arithmetical values 
 of all of the deviations. The true algebraical average 
 of the deviations cannot be used if they have been 
 calculated from the average measurement because, as 
 has been shown ( 175), its value is always zero. 
 
 The average of the positive values of the deviations 
 is always smallest when the deviations have been cal- 
 culated from the median measurement, and it is in con- 
 nection with the median that the average deviation is 
 generally used. 
 
 Find the average and the median of the numbers 3, 
 3, 4, 5, 10. Then find the deviation of each of these 
 numbers from the median, being careful not to omit one 
 of the deviations in case it happens to be zero. What is 
 the value of the average deviation when calculated from 
 the median? Find also the five deviations from the aver- 
 age, and determine the average of their (positive) arith- 
 metical values. Is the average deviation also smaller 
 when measured from the median than when measured 
 from the mode? 
 
 184. Standard Deviation. The typical deviation that 
 is most frequently used for statistical purposes is the 
 standard deviation. This is the square root of the quo- 
 tient of the sum of the squares of the deviations from 
 
XVII DEVIATION AND DISPERSION 213 
 
 the average divided by one less than the number of 
 statistical values.* If there are n quantities whose 
 deviations from their average are respectively Vi, v 2 , v 3 , 
 - v n , the standard deviation of those quantities is the 
 value of 
 
 n - 1 
 
 The standard deviation is a special case of the mean- 
 square deviation (or root-mean-square deviation), being in 
 fact the mean-square deviation from the average, or 
 approximately the quadratic mean of the deviations 
 from the average. It is also sometimes called the mean 
 deviation and the mean-square deviation, but as the term 
 mean deviation is also used for what we have defined as 
 the average deviation it is much better to avoid the use 
 of the word mean ( 170) altogether in connection with 
 a deviation. 
 
 185. Dispersion. The measure of deviation which is 
 used for physical measurements in almost all cases is that 
 particular form of characteristic deviation which is called 
 the dispersion. It is approximately two thirds of the 
 standard deviation, or, more exactly, 
 
 .674490 /*' + *'+ + 
 
 n - 1 
 
 186. Significance of the Dispersion. An important 
 characteristic of this typical value is that for a series of 
 measurements which is extensive enough to follow the 
 
 * To be strictly accurate, the standard deviation of the statis- 
 tician is V 2v 2 / V n. The reasons for using n 1 will be found in 
 the larger treatises; here it need only be noticed that the denominator 
 is diminished because the numerator is smaller than the sum of the 
 squares of the true errors (216). Of course, if n is fairly large, 
 the distinction is unnecessary. 
 
214 THEORY OF MEASUREMENTS 187 
 
 law of the probability curve the dispersion can be shown 
 mathematically to express exactly the same limits above 
 and below the average as are given with respect to the 
 median by the semi-interquartile range.* If measurements 
 follow the law of the probability curve the median and 
 the average will coincide; and the upper and lower 
 quartiles will include within the space between them 
 just half of the total number of measurements, as was 
 seen in connection with Fig. 54. In other words, the 
 right-hand half of the frequency diagram will be bisected 
 by the ordinate corresponding to the positive value of 
 the dispersion, and the corresponding abscissa will be 
 the median value of all the positive deviations. Since 
 the left-hand half of the curve is likewise bisected by 
 the negative value of the dispersion, and the graphic 
 diagram is symmetrical with respect to the ?/-axis, it 
 follows that the dispersion is the median of the absolute 
 arithmetical values of all the deviations, or that any 
 single deviation is as likely to be less than the dispersion as 
 it is to be greater. The dispersion, with its plus-and- 
 minus sign marks off a small distance above and below 
 the average, and it is just as likely as not that any single 
 measurement chosen at random will be within these 
 limits. 
 
 187. Advantage of the Dispersion. Where the num- 
 ber of measurements in a series is relatively small, say 
 not greater than 10, the dispersion obtained by calcula- 
 tion and the semi-interquartile range obtained by picking 
 out the quartile values will not usually have exactly the 
 same value, on account of the measurements not being 
 sufficiently numerous to follow the laws of probability 
 
 * Show that the median and semi-interquartile range of the 53 
 measurements given in the table of 164 are 1.64 cm. and 0.01 cm. 
 Could these two numbers be used as a rough approximation for 
 average and dispersion? 
 
XVII 
 
 DEVIATION AND DISPERSION 
 
 215 
 
 very closely. Even in this case, however, the value 
 given by the formula is preferable to half the difference 
 between the quartiles 
 because the former is 
 obtained from all the 
 measurements of the 
 series and the latter 
 from only two. Where 
 there are as many as 
 ten measurements of a 
 physical magnitude it 
 is usually found that the 
 two values will not differ 
 by more than 15 or 20 per 
 cent., and for rough pre- 
 liminary measurements 
 the semi - interquartile 
 range is almost as satis- 
 factory as the dispersion 
 and can be obtained 
 much more readily. 
 
 188. Calculation of the Dispersion. The table shows 
 the measurements (m) of ten variates taken at random 
 from the 100 that were measured before, also their 
 average (av), and their individual deviations (v) written 
 without decimal points.* The third column shows the 
 squares of the numbers in the second column, as obtained 
 from the table of squares (see appendix). According to 
 the formula in 185 the sum of these squares is to be 
 divided by 9, the square root of this quotient is then to 
 be found and multiplied by about f in order to obtain 
 the dispersion of this set of measurements. 
 
 * In the calculation of a dispersion the deviations should always 
 be written down as whole numbers. Nothing would be gained by 
 writing these v's in centimetres instead of in thousandths of -a 
 centimetre. 
 
 m 
 
 V 
 
 v 2 
 
 1.82 cm. 
 1.85 
 1.85 
 1.78 
 1.82 
 1.84 
 1.78 
 1.62 
 1.81 
 1.70 
 
 33 
 63 
 63 
 7 
 33 
 53 
 7 
 .167 
 23 
 87 
 
 1089 
 3969 
 3969 
 49 * 
 1089 
 2809 
 49 
 27889 
 529 
 7569 
 
 17.87 (10 
 1.787. .av 
 
 
 49010... 2 (v 2 ) 
 
 log 49010 = 4.6903 
 log 9 = .9542 
 
 
 2)3.7361 
 
 1.8680 
 log .6745 = 1.8290 
 
 1.6970 = log d 
 d = 49.78 
 
216 THEORY OF MEASUREMENTS 190. 
 
 If the numbers in column v are expressed in thou- 
 sandths of a centimetre the dispersion will also come 
 out in thousandths; thus, the value of d shown in the 
 illustration represents .04978 cm. This means that the 
 limits 1.737 cm. and 1.837 cm. are the boundaries between 
 which about half of the measurements should lie^see 186). 
 Verify this from the tabular values of m. 
 
 189. Rule for Accuracy of the Average. In determin- 
 ing how many figures of the average are to be retained 
 as significant it is best to follow the rule that at least 
 half of the deviations should be greater than 3. In the 
 illustrative example notice that keeping three decimal 
 places in the average has made more than half of the 
 deviations greater than 30. In such a case the average 
 could obviously be rounded off to two decimal places, 
 1.79, and at least half the deviations would still be greater 
 than 3. This will be found to simplify the calculation 
 and the result will not be essentially different. 
 
 Find the dispersion of the same ten variates by writing 
 a column of deviations from the value 1.79, and calculat- 
 ing the value of .6745 times the square root of one ninth 
 of the sum of their squares. Use logarithms but do not 
 refer to the illustrative example in 187 for each step of 
 the process; refer to the formula ( 185) instead.* 
 
 Notice that your final result agrees with the one 
 previously obtained as far as three significant figures. 
 This is ample accuracy, since two significant figures are 
 all that is usually wanted for the value of a dispersion. 
 
 190. Use of the Table of Dispersions. The root ex- 
 traction and long multiplication and division should, 
 of course, never be done by the tedious arithmetical 
 process. Even the logarithmic process used in the illus- 
 
 * Referring to the steps of a typical example is almost always the 
 easiest way of obtaining a required result; it is a very poor way of 
 earning a method of procedure. 
 
XVII DEVIATION AND DISPERSION 217 
 
 tration can be avoided as follows: 49010 -s- 9 = 5446; 
 the square root of this will be somewhat more than 70; 
 f of 70 is about 48; in the column headed (n =fc J) 2 /.6745 2 
 of the table of squares in the appendix find two numbers 
 between which 5446 lies and read the corresponding 
 number in column n. It is immediately seen to be 50, 
 which agrees with the previously obtained 49.78 as far 
 as the two significant figures which are all that is required. 
 As any arrangement of significant figures has two square 
 roots (see 13, no. 41, and 86) a place for 5446 would 
 also be found opposite 16 in the table, but the rough 
 preliminary check-calculation showed that the answer 
 should be about 48, so there could be no doubt that the 
 required answer is 50 rather than 16. 
 
 If the sum of the squares had been 490100 what would 
 be the value of the dispersion, pointed off so as to give 
 centimetres of length? Use the table of squares in the 
 manner just illustrated. 
 
 If the sum of the squares of the deviations of sixteen 
 variates is 49010 what is their dispersion? 
 
 191. Dispersions with the Slide Rule. In the rest of 
 this course the dispersions are to be calculated either with 
 the table of squares, as explained, or with the slide rule, 
 which makes the process even easier. A special line 
 is marked at 6745 on the C scale so that .6745 |/a/6 
 can be obtained with a single setting as soon as the sum 
 of the squares of the deviations is obtained. Difficulty 
 with the double square root is best avoided by setting 
 the end of the C scale to the approximate value of the 
 radical as obtained by a rough mental calculation; a 
 very slight movement of the slide is then all that is 
 needed to make an exact setting. 
 
 If the sum of the (v 2 )'s of 12 measurements is 28860 
 find the approximate value of the dispersion mentally, 
 
218 THEORY OF MEASUREMENTS 193 
 
 and then find the exact value with one setting of the 
 slide rule. 
 
 192. Sigma Notation. The capital letter sigma (S) of 
 the Greek alphabet is often used by mathematicians, 
 prefixed to an algebraical term, to denote the sum of all 
 such terms', thus the expression for the average, namely, 
 (i + a 2 + a 3 + a n )/n is abbreviated to the equivalent 
 form (2a)/w. In the same way the formula 
 
 .6745 
 
 n - 1 
 is more easily and compactly written in the form 
 
 .6745 i/2(v 2 )!(n - 1), 
 
 and the use of 2(v 2 ) will have been noticed previously 
 in the example of the calculation of a dispersion. 
 
 Rewrite the formulae for the harmonic mean and the 
 quadratic mean ( 170), using the sigma notation. Write 
 the formula for (x\ + x 2 ) z , using the same notation as 
 far as possible. 
 
 If five measurements of the quantity x give 3, 3, 4, 
 5, 10, what is the value of Zz? If the average is 5 
 what is the value of Zt>? Of 2 (mod )?* Of 2(0*)? 
 Of 2(z 2 )? 
 
 193. Dispersion of an Average. The average length 
 of 10 variates has already been calculated. If 10 more 
 were measured their average would be somewhere nearly 
 the same as the first average. If a considerable number 
 of such averages had been determined it would be a 
 simple matter to determine their dispersion, and the 
 result would naturally be a smaller number than the 
 
 * The modulus of a (real) number means its arithmetical value 
 regardless of its sign; its "absolute" value; the positive square 
 root of its square. 
 
XVII DEVIATION AND DISPERSION 219 
 
 dispersion of any of the sets of individual measurements, 
 since an average is a more trustworthy figure than a 
 single determination. In fact it can be shown mathe- 
 matically (vide infra) that if ten equally good measure- 
 ments are averaged the single measurements will show a 
 variation which is greater than the variation of such 
 averages in the proportion of 1/10 to 1, and similarly 
 that the dispersion of averages will be I/ \/n as great as 
 the dispersion of individual measurements if the latter are 
 averaged in groups of n. This means that it is not 
 necessary to calculate several averages in order to find 
 their dispersion, for it can be determined from a knowl- 
 edge of the number of measurements that go to make 
 up a single average and from the dispersion of these 
 individual measurements. Thus the dispersion of the 
 average (d av ), as it is called, of nine measurements is at 
 once seen to be one third as large as the dispersion for 
 single measurements (di), since the square root of nine 
 is three. 
 
 If a set of 16 measurements are free from constant 
 errors how much more accurate is the average than one 
 of the individual measurements? 
 
 The formula for the dispersion of an average is easily 
 written, for if 
 
 then 
 
 the second formula being I/ V n as large as the first. 
 
 194. The Statement of a Measurement. It is cus- 
 tomary to write the result of an accurate physical meas- 
 urement in the form of two numbers separated by a 
 plus-or-minus sign. The first number is the average; 
 
220 
 
 THEORY OF MEASUREMENTS 
 
 195 
 
 the second one is the dispersion of the average, not the 
 dispersion for single measurements. 
 
 Tabulate the first eleven of the twelve measurements 
 of the wooden block made with the card-board model of 
 a vernier caliper. Pick out their quartiles and find the 
 semi-interquartile range, to be used as a rough value 
 of the dispersion of the individual measurements, and 
 divide it by the square root of n. Write the thickness 
 of the block in the form av. d= d av . Divide di for your 
 measurement of 10 seeds by V 10 in order to obtain d av 
 for them, and state their measurement in the same form. 
 Make eleven more measurements of the wooden block, 
 this time with the vernier caliper that gives tenths of a 
 
 millimeter and treat them in the 
 same way as the previous eleven. 
 A typical set of results is shown 
 in the margin. Two important 
 facts are illustrated by this table : 
 
 (1) It is the accurate method 
 that shows discrepancies be- 
 tween repeated measurements 
 and the rough method that 
 shows more uniform agreement. 
 
 (2) There will b nothing falla- 
 cious about the statement that a 
 dispersion "is zero" if care is 
 taken to point off that zero. The 
 
 semi-interquartile range of the first column is .0 while 
 that of the second is .01 cm., but .0 cm. is the only correct 
 way of rounding off to one decimal place the number 
 which is expressed in the second decimal place by the 
 figures .01 cm. 
 
 195. Relative Dispersion. It has already been shown 
 that in order to see how serious an error really is it 
 
 model cat. 
 
 steel cal. 
 
 3.7 
 
 3.75 
 
 3.7 
 
 3.74 
 
 3.7 
 
 3.75 
 
 3.8 
 
 3.76 
 
 3.7 
 
 3.76 
 
 3.7 
 
 3.75 
 
 3.7 
 
 3.76 
 
 3.7 
 
 3.75 
 
 3.7 
 
 3.75 
 
 3.7 
 
 3.74 
 
 3.7 
 
 3.74 
 
 COMPARISON OF ROUGH 
 MEASUREMENTS AND PRE- 
 CISE MEASUREMENTS. 
 
XVII DEVIATION AND DISPERSION 221 
 
 should be referred to or divided by the true magnitude 
 of the quantity measured. In the same way, instead of 
 using the actual value of the dispersion, it is often found 
 more useful to obtain the proportional dispersion, or 
 relative dispersion, or fractional dispersion, as it is also 
 called; the ratio of dispersion to representative magni- 
 tude. 
 
 In the two measurements just stated, the length of 
 a seed and the thickness of the wooden block, divide 
 the dispersion of the average by the average itself in 
 order to obtain the relative dispersion of the average. 
 Express this either as a decimal or as a percentage. 
 Notice that it is smaller for the thickness of the block, 
 which is fairly uniform, than for the length of a seed, 
 which varies considerably. The absolute dispersion of 
 the average in centimetres, however, may be larger for 
 the block than for the seeds if a large dimension of the 
 block is measured while the seeds that are used are 
 small. 
 
 196. Questions and Exercises. 1. If each deviation, 
 v, is a number of thousandths explain why the mathe- 
 matical operations of finding the dispersion cause d also 
 to be given in thousandths of a unit. 
 
 2. If the results of a series of measurements are found 
 to be 8.16 cm. =b 0.033 cm. would it be advisable either 
 (a) to write 8.160 .033, or (6) to write 8.16 .03, 
 instead of using the first form? Explain why. 
 
 3. Does the number 49010 in the table of 187 mean 
 490.10 or 4.9010? Is it a number of centimetres or of 
 square centimetres? 
 
 4. Re-arrange the following table so that each col- 
 loquial statement is associated with its proper numerical 
 equivalent. 
 
222 THEORY OF MEASUREMENTS 196 
 
 "a few hundred" 20 5 
 
 "nearly a gross" 250 50 
 
 "a dozen or so" 70 10 
 
 "upward of three score" 130 10 
 
 5. Express each of the following in the form of a 
 representative magnitude and a measure of scattering: 
 
 " over a dozen " " six cr eight " 
 
 " nearly a hundred " "a few " 
 
 ' ' about a thousand " " several ' ' 
 
 " iii the neighborhood of 15 or 20 " " some " 
 
 6. If " average deviation " means the average" of the 
 (positive) arithmetical values of the deviations, what 
 would probably be meant by the expression "median 
 deviation"? Has any characteristic deviation already 
 been named which has practically the same significance? 
 
XVIII. THE WEIGHTING OF OBSERVATIONS 
 
 Apparatus. Platform balance; clamp and bar or stand 
 to support the balance 40 or 50 cm. above the table; 
 set of weights; vernier caliper; aluminum block; over- 
 flow can and catch-bucket for measuring displaced water; 
 towel; string (80 to 100 cm.) and two spreading rods; 
 fine silk thread; slide rule. 
 
 197. Necessity of Weights for Observations. A rep- 
 resentative value is often wanted for measurements 
 which are not all equally trustworthy. The accepted 
 values for such constants as the maximum density of 
 water, the mechanical equivalent of heat, the length of 
 the true ohm of mercury, the velocity of light in vacuo, 
 have all been derived from measurements by different 
 observers at various times, and in general by different 
 apparatus and methods. Any of these varying factors 
 will produce varying results, and one determination 
 can sometimes be accepted with more confidence than 
 another, and so will be entitled to greater " weight" 
 when it is necessary to decide upon a representative value. 
 
 198. Density by Different Methods. An example of 
 the effect of different methods on the determination of a 
 physical magnitude may be given by the measurement 
 of the density of a metal block. If the mass is known 
 this can be accomplished either by mensuration, or by 
 measuring displacement, or by a measurement of buoyant 
 force. According to the Principle of Archimedes the 
 apparent loss of weight of a body immersed in a fluid is 
 the same as the weight of an equal volume of the fluid. 
 If the volume of a metal block is v, its weight w, and its 
 apparent weight in water w', the density can be found 
 
 223 
 
224 THEORY OF MEASUREMENTS 198 
 
 as the ratio of the weight, w, to the loss of weight, 
 w w f , supposing that the density of water is unity; 
 or it can be determined as the ratio of the weight, w, 
 to the measured volume of water that is actually dis- 
 placed on immersion, say v', or the block can be meas- 
 ured with a caliper and the density calculated as m/v. 
 
 It will first be necessary to arrange the apparatus so 
 that the apparent weight of an object can be determined 
 while it is immersed in water. Place the platform 
 balance on the support or clamp it to the cross-bar 
 above the table in such a way that an object can be 
 weighed by suspending it under the bar with strings 
 attached to a spreading rod that is laid on one scale-pan 
 of the balance. See that the beam of the balance moves 
 freely. Use the other spreading rod as a counterpoise, 
 and make a careful allowance for the fact that they may 
 not exactly balance.* Attach the aluminum block to 
 the string by a fine thread long enough to allow it to 
 hang within the empty overflow can, and (1) weigh it as 
 accurately as possible. Fill the overflow can with water, 
 closing the spout with the finger-tip; place it in position 
 where the aluminum block is to hang, with the catch- 
 bucket under the spout; remove the finger and allow 
 the excess of water to run out of the overflow can; 
 then (2) weigh the catch-bucket with its contained water, 
 and replace it in position. Lower the aluminum block 
 carefully into the overflow can and (3) weigh it while 
 submerged; then (4) weigh the catch-bucket again in 
 order to find out how much water was displaced. 
 
 Find the density of the aluminum block (a) by com- 
 paring its weight with the weight of the overflow of 
 water actually displaced; (6) from the two values w and 
 w'; (c) by measuring the block with the vernier caliper, 
 
 * Weigh their difference; there is no need of weighing each one 
 separately. 
 
XVIII THE WEIGHTING OF OBSERVATIONS 225 
 
 computing its volume as closely as possible, and applying 
 the formula for density, d = m/v. Report your results 
 to the instructor for comparison with those of the other 
 members of the class. 
 
 199. Weights for Repeated Values. The simplest 
 case of weighting different observations is when separate 
 numerical values have each been obtained a definite 
 number of times. Suppose, for example, that the density 
 of a block of aluminum has been determined both as 2.6 
 and as 2.7, in the total of five measurements, the smaller 
 value having been found on four occasions while the 
 larger value was obtained only once. The best repre- 
 sentative figure from these data certainly would not be 
 the number 2.65, half way between 2.6 and 2.7, but 
 ought to be a number situated four times as far from the 
 least frequent measurement, 2.7, as from the most 
 frequent one, 2.6; in other words, it should be the 
 number 2.62. Moreover, this is easily seen to be the 
 same result as would be obtained by taking the average 
 of the five individual measurements. (Try it.) The 
 rule in such a case is obviously to give each numerical 
 value a weight proportional to the number of times of its 
 occurrence. 
 
 Find the weighted average of the values of a measured 
 length if it was found to be 2.345 cm. in each of six 
 trials', 2.350 cm. in twelve trials, and 2.355 in nine trials. 
 (Suggestion: it is a little easier to calculate the .value of 
 2 X 2.345 + 3 X 2.355 + 4 X 2.350.) 
 
 200. The Weighted Average. The weighted average 
 is found in any case by considering that certain values 
 have been obtained more frequently than others. In 
 the case just discussed this was a fact, in other cases it 
 is only a supposition made to fit the known or estimated 
 intrinsic value of the observations. 
 
 16 
 
226 THEORY OF MEASUREMENTS 120 
 
 If a difficult measurement had been made by an ex- 
 perienced student and found to be 0.35, while the same 
 experiment gave the value 0.41 when performed by a 
 beginner, it might be decided somewhat arbitrarily to 
 give the first number twice the weight of the second. 
 The process of finding the weighted average, (2 X 
 0.35 + 1 X 0.41)/3, would then be equivalent to sup- 
 posing that the better measurement had bden obtained 
 on two occasions but the poorer one only once. If 
 a measurement of some quantity had been found to be 
 1.36 when made under unfavorable circumstances, and 
 1.41 when made under circumstances that were more 
 favorable to experimentation it might be considered 
 best to assign the respective weights of 1 and 1.5 to 
 the two values. The weighted average would then be 
 (2 X 1.36 + 3 X 1.41) 4- 5, or 1.39, a figure which will 
 be seen to be nearer to the better value than to the poorer 
 one in exactly the ratio of 1 to 1.5. 
 
 201. Arbitrarily Assigned Weights. The objection- 
 able feature of such an arbitrary assignment of weights 
 is very obvious. The relative weights depend too much 
 upon the judgment of the individual computer; further- 
 more, it is often difficult to avoid being influenced by 
 the fact that certain determinations vary more or less 
 widely from the expected value, instead of keeping one's 
 judgment focussed on the quality of the experimental 
 work. 
 
 Which do you consider the better method of determin- 
 ing density, by buoyancy, or by displacement? Choose 
 what you consider the best ratio for their relative accu- 
 racies and find the corresponding weighted average, but 
 be careful not to give extra weight to either measure- 
 ment on account of its coming close to the third deter- 
 mination made by calculating the volume obtained by 
 mensuration (see 159, If 3). 
 
XVIII THE WEIGHTING OF OBSERVATIONS 227 
 
 202. Weight and Dispersion. Determinations of any 
 carefully measured magnitude are usually stated in the 
 form of an average and its dispersion, a d= d av . Subject 
 to the condition that the influence of constant errors 
 can be neglected, it can be shown mathematically that 
 the best value for a measurement is obtained by weighting 
 each determination of an average in inverse proportion to 
 the square of its dispersion. Thus, if one determination 
 has a dispersion of .0040 and another has a dispersion 
 of .012 the former should be given nine times as much 
 weight as the latter. This can be expressed in a general 
 formula, if d is used to denote the dispersion of an 
 average, by saying that the weighted average of 
 
 cti d= di, a% it dz, a^ d= d^, 
 is 
 
 or 
 
 w. av. = Z 
 
 but it is much better to learn the principle involved 
 than to memorize the formula. 
 
 203. Limitations of w = k/d 2 . Attention should again 
 be directed to the fact that weighting according to dis- 
 persions takes no account of the fact that constant errors 
 may be present in the given data. The dispersion sum- 
 marizes only the accidental errors, and if the constant 
 errors are greater than these the weighted average is 
 no better than the simple arithmetical average (Figs. 55, 
 56). 
 
 Tabulate the determinations, made by the various 
 members of the class, of the density of aluminum as 
 found by the effect of buoyancy. Calculate the typical 
 value in the form a d= d. 
 
228 THEORY OF MEASUREMENTS 204 
 
 Find in the same way the average and dispersion of 
 the density as determined by displacement. Use the 
 slide rule, not the tables of logarithms. 
 
 Calculate the weighted average of these two data. 
 
 55 55 
 
 / 
 
 FIGS. 55, 56. TARGET DIAGRAMS ILLUSTRATING ERRORS. In 
 Fig. 55 the two groups have nearly the same center, that with the 
 smaller dispersion naturally being the more trustworthy. In Fig. 
 56 at least one of the groups shows such a large constant error that 
 the relative difference in the two sets of accidental errors is un- 
 important. 
 
 There are several sources of constant error in each 
 of the two above methods of determining density. State 
 at least four that are common to both methods, and at 
 least one that influences one form of experiment but 
 not the other. For example, if bubbles cling to the 
 block when it is immersed (it is usually difficult to avoid 
 them altogether) the apparent weight will be lessened, 
 always causing the calculated density to be too low; 
 they will also cause too much water to overflow, again 
 making the calculated density lower than it should be. 
 Consider also the effects of such things as inequality of 
 the beam-arms of the balance, capillary attraction where 
 the thread cuts the surface of the water, etc. 
 
 204. Exception to the Rule. The method of weight- 
 ing observations in inverse proportion to their dispersions 
 is used for separate and independent data whose relative 
 accuracy is assumed to be shown by their dispersions. 
 Where two or more series of observations, however, 
 are known to have been made with equally trustworthy 
 apparatus, methods, and observers they should be 
 
XVIII THE WEIGHTING OF OBSERVATIONS ' 229 
 
 weighted merely according to the number of measure- 
 ments which each comprises, notwithstanding that their 
 dispersions might indicate a very different result.* To 
 do otherwise would be to repudiate the principle of the 
 average, which depends upon the fact that all observa- 
 tions are supposed to be equally trustworthy. On the 
 other hand, when different observations are known to 
 be unequally trustworthy, even if they occur in the 
 same series, weight may be given to the fact that some 
 are closely clustered about an apparent central position 
 while others diverge erratically. A great advantage of 
 the median, as a representative magnitude, is that it is 
 not unduly influenced by a very large or a very small 
 measurement and hence it automatically gives less weight 
 to the more aberrant measurements of a given group. 
 
 Which is to be preferred, the average or the median, 
 for a determination, like the one just made, of density 
 by buoyancy? Why? 
 
 205. Questions and Exercises. 1. Why was the 
 aluminum block hung with a thread instead of a string? 
 Would there be any advantage in using wire instead? 
 If wire should be used what kind of wire would be best? 
 
 2. Is it necessary to calculate dispersions in order to 
 weight averages in accordance with 202. What simpler 
 calculation can be used to give exactly the same result? 
 
 3. Read 186 again, and write in your note book a 
 statement of the most important fact that it contains. 
 Do not follow the form of any of the printed statements, 
 but try to write out the fact from an essentially different 
 point of view. 
 
 * Similarly, a single set of measurements known to be uniformly 
 and equally good must be simply averaged; to give more weight to 
 the individual measurements that have the smaller deviations would 
 be a procedure akin to substituting the median for the average (but , 
 see also the end of 174). 
 
XIX. CRITERIA OF REJECTION 
 
 Apparatus. Slide rule. 
 
 206. Observational Integrity. When successive re- 
 determinations of a quantity have beeri made in the 
 course of an experimental investigation it is to be sup- 
 posed that they have all been made with an equal 
 degree of care. 
 
 It is important to remember that an observation 
 should never be rejected simply because it is not in 
 satisfactory agreement with the other determinations 
 of the series. If the experimenter realizes that one of 
 his measurements was made under some kind of a handi- 
 cap or under such conditions that a faulty result would 
 be likely it is permissible to cross out the corresponding 
 value in his notes and to omit it in the final consideration 
 of the cjata, but there must be some definite and satis- 
 factory reason for discarding it other than the fact of 
 its divergence from the expected value. The tempta- 
 tion, often felt by the beginner, to omit or " re-deter- 
 mine"* a discordant result may be very perceptible, 
 but absolute freedom from prejudice (see dependent 
 measurements, 159; also 47, j[ 3) should be cultivated 
 to such a point that the experimenter is habitually 
 able to feel a certain disinterestedness in the outcome 
 of a measurement after he has first taken pains to ensure 
 its being as trustworthy as possible. His attitude should 
 be, "I have done all that can be done in the way of 
 preparations for making this measurement accurate and 
 independent; now let the result turn out as it will." 
 
 * A re-determination is not intrinsically objectionable, but it 
 should be made in addition to the other determination, not in place 
 of it. 
 
 230 
 
XIX CRITERIA OF REJECTION 231 
 
 207. Importance of Criteria. Even with all care to 
 make successive measurements equally accurate it often 
 happens that one or more of them show unduly large 
 deviations from the average. In order to prevent these 
 values from having an abnormal influence on the repre- 
 sentative value certain rules have been formulated for 
 determining whether they shall be retained or discarded, 
 for if an observer merely used his own judgment in 
 deciding the question the result would depend too much 
 upon his own individuality and temperament, and dif- 
 ferent observers would obtain different results from data 
 identically the same, just as in the case of the arbitrary 
 assignment of weights ( 201). In fact, the rejection of 
 a measurement is nothing more nor less than giving.it 
 a weight equal to zero. 
 
 208. Chauvenet's Criterion One of the easiest to 
 understand of the various devices for testing doubtful 
 observations is known as Chauvenet's criterion of rejection, 
 according to which rejectability is determined as a 
 function of deviation, dispersion, and number of meas- 
 urements. An unduly large deviation is an argument 
 for rejection, especially if the dispersion is relatively 
 small; furthermore, a deviation so large that it would 
 not be expected to occur more than once out of a hundred 
 cases might not seriously affect the average of a hundred 
 values, but if it happened to occur in a set of only ten 
 measurements it would probably exercise an altogether 
 disproportionate effect upon their average. The object 
 of a criterion of rejection is not to indicate that a certain 
 measurement is wrong, but merely to point out that it 
 is liable to be misleading it if occurs among a small 
 group.* 
 
 * Remarkably wide deviations may be expected to occur once 
 in a while if the number of measurements is extremely large; 
 
232 THEORY OF MEASUREMENTS 208 
 
 The following exercise is for the purpose of showing 
 the theory of the effect that the combined influences of 
 deviation, dispersion, and number of meas- 
 urements exert upon the determination of 
 the advisability of keeping or rejecting any 
 
 50.0 
 
 49.0 
 
 49.6 Draw a graphic diagram from the table. 
 
 Then draw the ordinates x = 10 and x = 50 
 47.2 from the base line up to the curve. The 
 
 46.0 
 
 l 49 9 single measurement of a group." 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 10 
 12 
 14 
 16 
 18 
 20 
 22 
 24 
 26 
 28 
 30 
 32 
 34 
 36 
 38 
 40 
 42, 
 44 
 46 
 48 
 50 
 52 
 54 
 56 
 58 
 60 
 
 44 ; 6 result will be the right-hand half of a nor- 
 43.3 mal frequency diagram, the ^-values corre- 
 39^9 sponding to deviations and the ^/-values to 
 36.0 the frequency of their occurrence. Remem- 
 2s!o b er that the total area of such a curve 
 24.0 corresponds to the number of deviations 
 ( 164); in the same way. the area between 
 13.8 curve and base line which is bounded on 
 the left and right by any two ordinates rep- 
 resents the number of observations whose 
 
 
 108 
 
 * 
 
 3*6 numerical deviations lie between those two 
 limits. As the dispersion is the same as the 
 median deviation ( 186) it is evident that 
 - 9 the ordinate which bisects the area (this is 
 0:5 the ordinate x = 10, for the scales used in 
 this .diagram) must have its abscissa nu- 
 O.i merically equal to the dispersion. 
 
 Suppose another ordinate is drawn at 
 o!o a position so far to the right that it 
 - includes between the ?/-axis and itself 
 
 
 
 VALUES OF n i ne tenths of the total area under the 
 
 2/ =50 e -(.o4769*) 2 curve and leaves only one tenth of the 
 
 area beyond it to the right, then the cor- 
 
 notice that there is some space between the curve y = e~* z and the 
 x-axis even at a great distance from the i/-axis ( 67,107). 
 
XIX CRITERIA OF REJECTION 233 
 
 responding abscissa would similarly have a value that 
 would be exceeded by only one tenth of the total num- 
 ber of deviations, and if any one deviation were chosen 
 at random there would be only one chance in ten that 
 it would be larger than the corresponding z-value. It 
 can be proved mathematically* that in order to include 
 nine tenths of the area the ordinate must be drawn 2.44 
 times as far to the right of the y-axis as the line which 
 bisects the area and corresponds to the dispersion. 
 
 On your diagram draw the ordinate that includes 
 nine tenths of the area and make sure that its abscissa 
 fulfills the condition stated above. If the total number 
 of measurements were ten how many would most prob- 
 ably be represented by the area to the right of ,the 
 ordinate? How many if the number of measurements 
 were 50? How many if the number were 4? How 
 many if 6? The last two questions should be answered 
 to the nearest whole number. 
 
 Since the ordinate for x = 2.44 d includes nine tenths 
 of the area and the limit 2.44 X dispersion includes 
 nine tenths of the deviations it might be said theoret- 
 ically and rather figuratively that if there were only 
 five measurements in a certain series the number 
 of measurements whose deviations were greater than 
 this limit would most probably be just one half. In 
 other words the limit would be just on such a border 
 line that if it were decreased we should expect it to 
 exclude one measurement rather than no measurements, 
 and if it were increased we should expect it to exclude 
 no measurements rather than one measurement. 
 
 * Certain statements are intended to be taken on faith by the 
 student. This one is just as true as the statement that the fre- 
 quencies of accidental errors follow the law y = e~* 2 or that fluid 
 friction in a water pipe varies as the 1.8 power of the radius; all of 
 them can be proved but none of the proofs are necessary here. 
 
234 
 
 THEORY OF MEASUREMENTS 
 
 08 
 
 It follows, then, that no one of a series of five measure- 
 ments theoretically ought to have a deviation of more 
 than 2.44 times the dispersion. This being the case it 
 is only natural to consider that one is justified in dis- 
 carding any one of the measurements of a series of five 
 if its deviation does exceed this limit. Just as the 
 ordinate for 2.44 d excludes one tenth of the area (i. e., 
 excludes half a measurement if there are five in all) so 
 the ordinate at 2.57 d excludes one twelfth of the area, 
 which would correspond to half a measurement if the 
 total number of measurements were six. Accordingly 
 no measurement in a set of six should theoretically have 
 v > 2.57 d if the set is supposed to follow the law of 
 frequency distribution for accidental errors. Chauvenefs 
 criterion is simply an extension of this delimitation 
 to other values of n as well as 5 and 6. The column, /, 
 of the following table shows the limiting z-value for 
 
 n 
 
 l 
 
 log* 
 
 | , 
 
 log i 
 
 n 
 
 i 
 
 log/ 
 
 n 
 
 i 
 
 log/ 
 
 I 
 
 2 
 3 
 4 
 5 
 
 1.00 
 1.71 
 2.05 
 2.27 
 2.44 
 
 000 
 233 
 312 
 356 
 
 387 
 
 11 
 12 
 13 
 14 
 15 
 
 2.97 
 3.02 
 3.07 
 3.11 
 3.15 
 
 473 
 480 
 487 
 493 
 498 
 
 21 
 22 
 23 
 24 
 25 
 
 3.35 
 3.38 
 3.40 
 3.43 
 3.45 
 
 525 
 529 
 532 
 535 
 538 
 
 32 
 
 34 
 36 
 
 38 
 40 
 
 3.59 
 3.62 
 3.65 
 3.68 
 3.70 
 
 554 
 556 
 561 
 566 
 570 
 
 6 
 
 7 
 8 
 9 
 10 
 
 2.57 
 2.67 
 2.76 
 
 2.84 
 2.91 
 
 410 
 427 
 441 
 453 
 464 
 
 16 
 17 
 
 18 
 19 
 20 
 
 3.19 
 3.22 
 3.26 
 3.29 
 3.32 
 
 504 
 508 
 513 
 517 
 521 
 
 26 
 
 27 
 28 
 29 
 30 
 
 3.47 
 3.49 
 3.51 
 3.53 
 3.55 
 
 540 
 543 
 546 
 549 
 551 
 
 49 
 64 
 81 
 100 
 671 
 
 3.81 
 3.95 
 4.06 
 4.16 
 5.00 
 
 581 
 597 
 608 
 619 
 699 
 
 Chauvenet's Criterion. If the most divergent measurement out 
 of a series of n determinations has a deviation more than I times 
 as great as the dispersion of the individual measurements it should 
 be rejected. 
 
 which the corresponding part of the area (see appendix) 
 included under the curve y = e~ x * is 1 l/2n and the 
 
XIX . CRITERIA OF REJECTION 235 
 
 excluded part is half of 1/w, as before, this value being 
 expressed in terms of the dispersion. 
 
 It should be carefully kept in mind when considering 
 any criterion of rejection that we are interested in the 
 individual measurements, and, accordingly, the disper- 
 sion to which the criterion applies is the dispersion of the 
 individual measurements, not the dispersion of the aver-, 
 age. Chauvenet's criterion, then, is the test of whether 
 any deviation is greater than I times the dispersion of the 
 individual values of a series of n measurements, where I 
 corresponds to n in the way shown in the table. 
 
 209. The Probable Error. By this time the student 
 ought to be thoroughly aware of the fact that the dis- 
 persion is not properly an error, but a deviation. If he 
 also realizes that deviations within its limits are no more 
 probable than improbable there can be no objection to 
 his using the term that is always employed by physicists 
 in speaking of this characteristic deviation. In this book 
 the term dispersion has been used in order to avoid 
 repeatedly informing the student that it is an error and 
 repeatedly suggesting that there is something very prob- 
 able about it. It will hereafter be spoken of as the 
 probable error, and of course it will be under- 
 stood that it is used in two forms, the prob- 'o309 
 able error of the individual measurements (di, -031 
 or pi) and the probable error of the average '9347 
 W, or p..). :o|5 
 
 In an experimental determination of the ] 38 
 specific heat of lead shot the following values -045 
 were obtained by a class of students. Test 
 them by Chauvenet's criterion to determine SPECIFIC 
 
 whether any measurement falls outside of the T E A ^ 
 
 . , i . . . - , LEAD SHOT. 
 
 theoretical limits, but if two or more such 
 
 values are found reject only the most divergent one, find a 
 
236 THEORY OF MEASUREMENTS 210 
 
 new average for those that remain, and apply the cri- 
 terion to them in turn.* Repeat the process, if necessary, 
 until no more values can be discarded, and then state 
 the best value obtainable from the figures, with its 
 " probable error." 
 
 210. Graphic Approximation to Chauvenet's Criterion. 
 Where a graphic diagram is to be used for only a 
 single series of numbers instead of for sets of values of 
 two varying quantities it is advisable to use a horizontal 
 scale and lay off the individual measurements as small 
 dots or circles (Fig. 57) unless they are sufficiently 
 numerous to allow a good histogram to be drawn. 
 
 Q1 ME Q3 
 
 
 
 1 
 
 
 50 
 
 54 
 
 58 62 
 
 66 
 
 FIG. 57. DISTRIBUTION OF A FEW MEASUREMENTS. When the 
 majority of frequencies are very small each measurement may be 
 represented by a dot instead of the formal square of the histogram. 
 The centre of gravity of such a system of equal-sized dots is more 
 readily apparent to the eye than is the position of the line that 
 would be needed to bisect the area of the frequency polygon or 
 histogram. 
 
 The following figures are the experimental values of 
 the slope of the first " black-thread" diagram as obtained 
 by a class of students: .60, ,57, .64, .53, .48, .59, .59, .61. 
 Make a graphic diagram of these values and mark the 
 median and quartiles. Do not measure the semi-inter- 
 quartile range nor multiply it by I, but mark off its 
 length on the edge of a strip of paper and apply this to 
 
 * If one large measurement and one small one are beyond the 
 limits the rejection of the more divergent one may result in shifting 
 the average far enough toward the other one to bring it within 
 bounds. Where it is obvious that this cannot happen it is un- 
 necessary to adhere to the letter of the rule. 
 
XIX 
 
 CRITERIA OF REJECTION 
 
 237 
 
 your diagram, laying it off to the right and to the left 
 of the median* as many times as may be indicated, by 
 the criterion. In this way a rough application of 
 the criterion can be made graphically and 
 the long calculation can be avoided. De- 
 termine from the diagram whether any 
 value should be rejected and then verify 
 the result by the usual form of calculation. 
 
 Use the graphic method for applying 
 Chauvenet's criterion to the following set 
 of barometer readings : 
 
 What advantage has the arithmetical 
 method over the graphic method? 
 
 Write down, in your own words, just what 
 it is that is represented by (a) the probable 
 error of a single measurement, .6745 
 
 29.986 inches 
 
 29.982 
 
 29.990 
 
 29.984 
 
 29.984 
 
 29.980 
 
 29.986 
 
 29.977 
 
 29.984 
 
 29.982 
 
 29.986 
 
 29.988 
 
 29.984 
 
 BAROMETER 
 READINGS. 
 
 (n - 1), and by 
 (6) the probable error of the average, .6745 ^2d?/n(n 1). 
 
 211. Irregularities of Small Groups. The probable 
 error, or "dispersion," cannot be considered as having 
 much meaning in cases where the total number of meas- 
 urements is less than ten, and even with ten measure- 
 ments it should be treated with a certain amount of 
 caution. A number of values less than ten will hardly 
 ever give a histogram of their frequency-distribution 
 which is recognizably similar to the graph of y = e~ x *, 
 the curve which all unbiassed measurements will be 
 found to follow if they are sufficiently numerous. 
 
 212. Justification of the Criterion. For the same 
 reason, it is hardly worth while to use a criterion of 
 rejection for less than ten measurements. The example 
 given above with only five is intended merely for an 
 illustration of the method of using the criterion, and the 
 
 * It is better to use the mid-quartile point than the median in 
 case the two are not the same. 
 
238 THEORY OF MEASUREMENTS 214 
 
 still smaller values in the table are only of theoretical 
 importance. Chauvenet's criterion is not to be con- 
 sidered as showing that any one measurement is a 
 mistake, but only as indicating that a very large deviation 
 is such a rarity that it Would have an unduly large 
 influence upon the average if it were allowed to remain 
 along with the other values of a very limited series of 
 measurements ( 208, page 231, foot-note). 
 
 213. Wright's Criterion. Another criterion of rejec- 
 tion, which is sometimes employed, is that of Wright. 
 According to this the arbitrary rejection of a single meas- 
 urement may be considered if its deviation is more than 
 five times the probable error. 
 
 Turn back to the graphic diagram of the table in 208, 
 and notice how small a part of the area of the curve lies 
 to the right of the ordinate, x = 50, which represents 
 five times the probable error. Turn to the table of 
 values for Chauvenet's criterion and note how many 
 measurements would need to be made before "half a 
 measurement" would be likely to diverge from the aver- 
 age five times as far as the probable error. 
 
 In the measurements to which you have already applied 
 Chauvenet's criterion how many would have been rejected 
 if Wright's criterion had been used instead? If a devi- 
 ation is great enough to be practically sure of rejection 
 by one of the two criteria will it ordinarily be rejected 
 by the other? If a maximum deviation is small enough 
 to avoid rejection by one criterion will it be practically 
 certain to escape rejection by the other? Explain why. 
 What are the relative advantages of the two criteria? 
 
 214. Comparison of Characteristic Deviations. Other 
 limiting values, which give practically the same result 
 as Wright's criterion, are four times the average deviation, 
 and three times the standard deviation. 
 
XIX CRITERIA OF REJECTION 239 
 
 Turn back to your notes on the use of logarithms and 
 find the graphic diagram of y = e~ x *. Mark off the 
 following values on the base line, p = .4769363, a = 
 .5641895 and s = .7071066. These represent respec- 
 tively the probable error, the average deviation, and the 
 standard deviation, and are roughly proportional to 
 10 : 12 : 15; a better approximation to their ratios than 
 is given by 10 : 12 : 15 may be found with the aid of 
 the slide rule. Draw the corresponding ordinates, and 
 notice that the last one meets the curve at the point of 
 inflection, that is, at the point where it is momentarily 
 straight as it changes from convex upward to convex 
 downward. 
 
 215. Questions and Exercises. 1. Write a definition 
 of Chauvenet's criterion in your own words. Notice 
 that the last sentence of 208 is not a satisfactory defini- 
 tion. 
 
 2. Can a set of three measurements comprise such 
 values that one of them will be rejected by Chauvenet's 
 criterion? Give an example to illustrate your answer. 
 
 3. Can a set of two measurements comprise such' 
 values that one of them will be rejected by Chauvenet's 
 criterion. Give an example to illustrate your answer. 
 
XX. LEAST SQUARES 
 
 Apparatus. Slide rule; black thread. 
 
 216. The Average as a Least-Square Magnitude. The 
 mathematical principle of least squares is that when 
 measurements are equally trustworthy their best repre- 
 sentative value is that for which the sum of the squares 
 of the deviations has the lowest numerical value.* It 
 is upon this principle that the use of the average is based, 
 for it is easy to show that the sum of the squares of the 
 deviations of any particular set of numbers will be 
 greater when measured from some other value than when 
 measured from the average. 
 
 Find the average of the numbers 3, 3, 4, 5, 10; also 
 their deviations from the average, and the sum of the 
 squares of the deviations. Find the median of 3, 3, 4, 5, 
 10: and the sum of the squares of the deviations from 
 the median. Find the sum of the squares of the devia- 
 tions from the harmonic mean (call it 4.1) or from the 
 mode, and notice that S(t; 2 ) is smaller when the deviations 
 are measured from the average than when measured 
 from any of the other numerical values. 
 
 217. Least Squares for Conditioned Measurements. 
 If we are dealing with two conditioned measurements, 
 as in the case of the ^-values and the ^/-values of the 
 black-thread experiment, the principle of least squares 
 
 * This principle can be proved from the "fact of experience" 
 that deviations follow the law of the exponential equation y = e~ xZ . 
 As an example of its application consider the marks 6, 8, and 7, 
 on a scale of ten, which one student obtained on three examination 
 questions, and the marks 7, 7, 7, which were obtained by another 
 student. Find the deviations from the theoretical mark 10, and 
 see which student has the smaller value for 
 
 240 
 
XX 
 
 LEAST SQUARES 
 
 241 
 
 shows that the line which expresses the condition or gives 
 the law of relationship between the two variables must 
 be so placed that the sum of the squares of the distances 
 from it to all of the experimental points shall have the 
 smallest possible value. 
 
 The x and y of any one of the points cannot in general 
 be substituted in the black-thread equation, y = a + bx, 
 but a + bx y will have 
 some small positive or 
 negative value instead of 
 being equal to zero. If 
 the various points are 
 considered to have the 
 definite positions de- 
 noted by (xi, 2/1) > (#2, 
 1/2) , (#3, 2/3), etc., it will 
 usually be found that 
 none of these sets of 
 values will exactly satisfy 
 the equation y = a + bx 
 or a -f bx y = 0, but 
 will give such a result 
 as a + 6x1 2/1 . di, 
 where d is some small 
 quantity whose exact 
 value need not be 
 determined ; similarly, 
 the other points will 
 give other equations, 
 
 a -f bx 2 2/2 = d 2 , a + bx 3 2/3 = d%, etc., and ac- 
 cording to the principle of least squares the sum 
 di 2 + dz 2 + d 3 2 + j must be as small as possible.* 
 
 * It can be shown mathematically that the distance from the 
 point (xi, y\) to the line y = a + bx is proportional to a + bx t y\ t 
 17 
 
 FIG. 58. THE PRINCIPLE OF 
 LEAST SQUARES. The general prin- 
 ciple is that the theoretical rela- 
 tionship is to be so arranged that 
 the sum of the squares of the dis- 
 crepancies of the actual measure- 
 ments shall be as small as possible. 
 If a straight line is to be used it 
 must be so arranged that the sum of 
 the squares of the distances to it from 
 the various points is a minimum. 
 
242 THEORY OF MEASUREMENTS 217 
 
 This means that the sum of the squares of the left-hand 
 members of the equations, or S(a + bx n y n ) 2 must 
 have its minimum value, and it can be shown by processes 
 of pure mathematics that this will be the case if 
 
 = 
 
 and 
 
 = 
 
 where x and y stand for Xi, z 2 , x 3 , and 3/1, i/ 2 , t/ 3 , 
 the #- values and y- values of the experimental points. 
 
 These equations enable one to determine the values 
 of a and b, and hence to find the position (see 105) 
 that a straight line (black thread) must have if it is to 
 be so located that the sum of the squares of the distances 
 to it from all of the experimental points shall have the 
 smallest possible value. 
 
 By similar processes a, b, and c could be found for 
 the equation of the parabola y = a + bx + ex 2 , or the 
 appropriate coefficients for curves having even more 
 complicated equations, but the processes of computation 
 become so tedious that it is better to replace the vari-. 
 ables, as explained in the lesson on graphic analysis, 
 by others that will conform to the straight line law. 
 The diagram (Fig. 58) shows some of the points that 
 correspond to the table in 118, for which the position 
 of the line through the origin was found graphically by 
 the use of a black thread. 
 
 Tabulate the values of x and y for the black-thread 
 experiment of 115, arranging them as shown in the 
 following table, and writing the proper numerical values 
 in the spaces marked Sx, 2y, 2(xy), and S(:c 2 ). Then 
 calculate the values of a and b from the formulae, arrang- 
 
XX 
 
 LEAST SQUARES 
 
 243 
 
 ing the work neatly and being careful to avoid using 
 the wrong algebraical signs or 
 confusing S(z 2 ) with (2x) 2 . 
 Keep only three significant 
 figures in the final results. 
 
 Write the equation repre- 
 senting the best position of 
 the black thread in the form 
 y = a + bx, and then in the 
 form x/m + y/n = 1. Com- 
 pare the calculated values of 
 the intercepts with your ex- 
 perimental values that were 
 obtained in the work on 
 Graphic Analysis (chap. x). 
 
 218. Least Squares and Pro- 
 portionality. If two variables, 
 x and y, are always propor- 
 tional the linear equation y = a + bx reduces to the 
 form 
 
 y = + bx 
 
 and the best value of the coefficient will be found to be 
 
 b = 
 
 X 
 
 y 
 
 xy 
 
 x 2 
 
 1 
 
 9.8 
 
 9.8 
 
 1 
 
 2 
 
 8.5 
 
 17.0 
 
 4 
 
 3 
 
 8.0 
 
 24.0 
 
 9 
 
 4 
 
 7.2 
 
 28.8 
 
 16 
 
 5 
 
 6.7 
 
 33.5 
 
 
 6 
 
 6.5 
 
 
 
 7 
 
 6.2 
 
 
 
 8 
 
 5.5 
 
 
 
 9 
 
 5.0 
 
 
 
 10 
 
 4.1 
 
 
 
 11 
 
 3.9 
 
 
 
 12 
 
 3.2 
 
 
 
 13 
 
 2.3 
 
 
 
 Sw 
 
 2(sw) 
 
 S(z 2 ) 
 
 METHOD OF LEAST SQUAEES 
 FOR A LINEAR LAW. 
 
 Let a = in the equation 
 
 then 
 
 Multiplying by nS(x), 
 
244 THEORY OF MEASUREMENTS 219 
 
 whence 
 
 But if a : b :: c : d, then a c : b d :: c : d; accord- 
 ingly, 
 
 or 
 
 In studying the density of water ( 118) the ratio of 
 V y to a: was found graphically to be about 0.83. Deter- 
 mine the accurate value of this ratio by calculating each 
 product of the variables x and ^y, summating, and 
 dividing by the summated squares of the x's. 
 
 219. Least Squares for a Theoretically Constant Value. 
 If the variation of y is supposed to be nil, i. e., if y is 
 a constant, the best representative value for the fluctuat- 
 ing experimental determinations of it can easily be found 
 by the method of least squares: 
 
 In the equation y = a-\-bxiib = Q the equation 
 
 z - nx 
 gives 
 
 S(x)S(y) = nL(xy). 
 
 Eliminating *2,(xy) between this equation and the equa- 
 tion 
 
 gives 
 
 = ( 
 
 which is obviously the same as 
 
XX LEAST SQUARES 245 
 
 In other words, the best representative value of y is 
 obtained by adding all n of its experimental values and 
 dividing by n, as already stated without proof.* 
 
 220. Consecutive Equal Intervals. If the successive 
 unknown intervals of a scale are not perceptibly different 
 from one another their value can easily be found by the 
 method of coincidences, as illustrated in Fig. 48, 153. 
 If the scale is a crude one or the method of measuring is 
 a precise one the intervals which ought to be equal will 
 be found to differ among themselves ( 160) and the 
 question arises as to the best representative value for 
 them. The principle, of course, is to find the perfectly 
 uniform scale whose intervals are of such size as to make 
 the sum of the squares of the discrepancies in the gradua- 
 tions of the irregular scale as small as possible (Fig. 59). 
 It will not do to take the average of the lengths of all 
 the consecutive intervals because the result that would 
 be obtained would depend only upon the position of 
 the first graduation and the last graduation, and the 
 data afforded by all the rest of the graduations would 
 have been neglected. 
 
 
 
 5 10 
 
 1 
 
 
 
 
 1 1 I 1 1 
 
 
 I \ '< ' * \ 
 
 Trt, 
 
 Wj 1 YV3 [ 
 
 1 111114 
 
 
 
 i , ! \ \ \ \ 
 
 
 1 | 
 
 
 II II 
 
 3 
 
 ~l A ' 
 
 FIG. 59. BEST VALUE FOR CONSECUTIVE INTERVALS. An imag- 
 inary uniform scale is to be so chosen that 2(i> 2 ) for the irregular 
 intervals is a minimum. A, actual scale; /, 7, imaginary scales. 
 
 * Remember that the principle of least squares is based on the 
 assumption Ithat all of the measurements under consideration are 
 equally trustworthy. If this condition could always be fulfilled 
 we should have little use for any representative value except the 
 average (c/. 174; 204). 
 
246 
 
 THEORY OF MEASUREMENTS 
 
 220 
 
 If the distance from the zero of the scale to the gradua- 
 tion that is marked x is called y the problem is to find the 
 best value of b in the equation y = bx (Fig. 60). The 
 formula b = 2(:n/)/S(z 2 ) of 218 cannot be used in this 
 case, for the experimental deviations cannot involve 
 
 both x and y, but must 
 be limited to the y- values 
 on account of the neces- 
 sity of keeping the differ- 
 ence between successive 
 x-values constant. It is 
 not the squares of the 
 perpendicular distances 
 in Fig. 58 whose sum 
 must be a minimum, but 
 
 the squares of the vertical 
 ,. . ** A rrn 
 
 distances in Fig. 60. The 
 
 formula that is com- 
 monly used in such cases can be obtained as follows : 
 
 If the n ^-values of the n points in Fig. 60 are Xi = 0, 
 z 2 = 1, x 3 = 2, , x n = n 1, the slope 6 which will 
 give the best value, m, of the n. 1 intervals, mi, w 2 , 
 m s , - , can be calculated as follows : 
 
 FOR EQUAL INTERVALS. 
 
 Since 
 
 +n = 
 
 and 
 
 = O 2 + I 2 + 2 2 + + n 2 = \n(n + l)(2n + 1)*, 
 
 the formula 
 
 
 * The correctness of these formulse may be taken for granted. 
 They are easily proved b}' the processes of elementary algebra. 
 
XX 
 
 becomes 
 
 LEAST SQUARES 
 
 247 
 
 n(yi + 2y 2 + 3y 3 + ---- h ny n 
 
 I) 2 - |n 2 (n 
 
 Multiplying both numerator and denominator by 12/n 
 gives 
 
 (n + l)(yi + 3/2 -J ----- h y) - 2(y t + 2y 2 H ----- h ny u ) 
 3n(n + I) 2 - 2n(n + l)(2n +"l) 
 
 , = 
 
 Simplifying the denominator and multiplying the fraction 
 by I/ 1 gives 
 
 - ( n 
 3 
 
 n 6 n 
 Collecting the coefficients of the y'a, 
 
 , a (1 n)y: + (3 
 = 
 
 Finally, collecting the coefficients of (n l), (n 3), 
 etc., gives 
 
 (n - l)(y - yQ + (n - 3)(y n _i - pp + (n - 5)(y n _ 2 - y) + 
 
 for the best value of the interval. The point Xi corre- 
 sponds, of course, to the zero of the scale and yi is 
 
 ) 
 
 i 5 i 
 
 1 
 
 FIG. 61. TREATMENT OF UNIFORM INTERVALS. The best repre- 
 sentative value, according to the formula, is 10 (yn yi) + 8(yio yz) 
 + 6(y 9 - y 3 ) + 4(y 8 - y 4 ) + 2(y 7 - y 6 ) + 0(y 6 - ye), multiplied by 
 6/n(n 2 1). Notice that an eyen number of intervals causes the 
 middle graduation to be neglected. 
 
248 
 
 THEORY OF MEASUREMENTS 
 
 221 
 
 subdivision 
 
 length 
 
 1st 
 
 160 
 
 2d 
 
 163 
 
 3d 
 
 164 
 
 4th 
 
 166 
 
 5th 
 
 165 
 
 6th 
 
 159 
 
 7th 
 
 162 
 
 8th 
 
 166 
 
 9th 
 
 165 
 
 10th 
 
 166 
 
 numerically equal to zero.* It should be noticed that 
 when there are an even number of intervals the middle 
 point does not affect the formula. Accordingly it is 
 preferable to use an odd number of intervals, unless 
 there is some reason to the contrary. 
 
 The following numbers were obtained by measuring 
 the consecutive intervals between eleven parallel lines 
 that were intended to be located 
 one tenth of a millimetre apart on 
 a glass slide used for microscopic 
 measurements. The numbers give 
 the corresponding distances in units 
 of an arbitrary scale located in the 
 eye-piece of the microscope so as to 
 appear superimposed on the image 
 of the object under examination. 
 Find the best representative value 
 for these ten subdivisions ; then note 
 which subdivision (if any) agrees in 
 length with the determination. 
 
 The same method can be used for equal changes of 
 length that are due to any uniform cause; for example, 
 a spring elongates by uniform intervals when weights 
 proportional to 1, 2, 3, 4, are hung on it. It is also 
 applicable to intervals of time, e. g., the period of a 
 pendulum or of the indicating pointer of a galvanometer 
 or analytical balance; and, indeed, to any variable that 
 is proportional to another quantity which can be varied 
 by equal intervals; thus, the vr-disc ( 49) may be rolled 
 until any given point on its circumference has touched 
 the flat scale at several equidistant points, or the area 
 
 * The student should be careful to avoid the common mistake 
 of using 2/2 instead of y\ for the first interval of the scale (Fig. 61). 
 The result would be that only the other n 2 intervals would be 
 available. 
 
 EXAMPLE OF SCALE 
 INTERVALS. 
 
XX LEAST SQUARES 249 
 
 indicated by a planimeter may be read when the tracing 
 point has been carried around the periphery once, twice, 
 thrice, etc. 
 
 221. Equal Intervals without Least Squares. The 
 black-thread method can of course be used for measure- 
 ments like those that have just been considered, but if 
 the uncertainty of judgment that is necessarily associated 
 with it is objectionable a simple calculation can be 
 performed which is free from the disadvantage of aver- 
 aging stated in the first paragraph of 220. The distance 
 from yi to 2/<+i)/2 or 2/<+2)/2 is measured, also, from y 2 to 
 2/(n+3)/2 or i/(n-i-4)/2, etc., and each of the measurements is 
 divided by (n l)/2 or n/2, the quotients being finally 
 averaged. Thus, for the scale shown in Fig. 59, one 
 fifth of the distance from the mark to the five-inch 
 mark is averaged with one fifth of the distance from 2 to 
 6, one fifth of 3 to 7, etc., so that all of the graduation 
 marks are utilized.* 
 
 222. Simultaneous Indirect Measurements. In cer- 
 tain kinds of measurement several quantities (usually 
 either two or three) have to be determined by methods 
 which will not allow each to be measured separately 
 but which furnish functional relations in which there is 
 always more than one unknown involved. For example, 
 two magnets may have the strengths of their poles 
 determined by measuring their relative intensities as a 
 ratio and their mutual attraction as a product. The 
 data x/y = a and xy = b are then sufficient to deter- 
 mine both x and y. Sometimes only one unknown 
 
 * The procedures given in 220 and 221 are the ones that are 
 ordinarily used for physical measurements. Whether their results 
 are preferable to those of the more general methods stated pre- 
 viously will be an interesting question for the student to determine 
 for himself. He should make up at least one example of an extremely 
 irregular set of scale-divisions. 
 
250 THEORY OF MEASUREMENTS 222 
 
 quantity needs to be measured, but cannot be deter- 
 mined except along with a different one. For example, 
 by the use of an ammeter to measure the flow of electric 
 current that a particular battery can drive through an 
 unknown resistance it is easy to deduce the numerical 
 value of the resistance,, but the method must be modified 
 before it can be used to measure the resistance of a 
 "ground," i. e., of the place where the current passes 
 from the negligibly resistant wire into the body of the 
 non-resistant earth, for the current must return from the 
 earth to the battery through some other resistant 
 "ground." When there are two such points, it is possible 
 to find the sum of their resistances but not the value of 
 either one alone. If three separate connections to earth 
 are available, their resistances may be called x, y, and z, 
 and after measuring in turn the resistances x + y, y -f z, 
 and x + z the value of any one of them may be found 
 by solving the three simultaneous equations that are 
 obtained. If there are four such points, however, more 
 separate and independent observations are obtainable 
 than there are unknown quantities, and the determina- 
 tions will need adjustment by the method of least 
 squares. If the resistances are called w, x, y, and z, 
 then the following data are available: 
 
 w + x = a i 
 
 x + y = b 
 
 y + z = c 
 
 w -\- y = d 
 
 x + z = e 
 
 w -f z = f 
 
 In general no exact solution will be possible, and the 
 problem is to determine such a solution that the sum of 
 the squares of the deviations of the observed values from 
 
XX 
 
 LEAST SQUARES 
 
 251 
 
 the adjusted values that are given by the assumed solution 
 shall be as small as possible. 
 
 A graphic illustration of the effect of more equations 
 than unknowns is given in Fig. 62, 
 which represents a set of experi- 
 mental determinations 
 3x - y = 3 -\ 
 
 y-x=2 k 
 2x - y = J 
 
 By plotting these three equations 
 on one diagram it will be seen 
 that the three straight lines do not 
 pass through a single common in- 
 tersection, but a point (2.4, 4.9) can 
 be found which comes fairly close 
 to satisfying the three conditions. 
 
 The general method of obtaining 
 the best representative values of 
 the unknown quantities can be 
 shown by the application of the 
 principles of the differential calcu- 
 lus to be as follows : 
 
 1. Multiply each equation by the 
 coefficient of x in that equation, 
 and add the results. The sum is 
 called a normal equation. 
 
 The equations of Fig. 62, 
 
 Y 
 
 
 
 
 / 
 
 
 
 
 
 // 
 
 7 
 
 
 
 
 / 
 
 
 
 
 1 
 
 / 
 
 9 
 
 
 
 fr 
 
 
 
 
 
 yj 
 
 1 
 
 
 3 
 
 y 
 
 LI' 
 
 9 
 
 
 / 
 
 4 
 
 W 
 
 
 
 1 
 
 # 
 
 / 
 
 
 
 
 
 / 
 
 1 
 
 'I 
 
 3 X 
 
 FIG. 62. THREE OB- 
 SERVATIONAL EQUA- 
 TIONS. Since all obser- 
 vations contain errors 
 the three equations are 
 not consistent, but the 
 best point to represent 
 the intersection of the 
 three lines can be found 
 by choosing it so that 
 the sum of the squares 
 of the discrepancies is 
 a minimum. 
 
 thus give 
 
 3x - y = 3 
 
 -x+y = 2 
 
 2x - y = 0, 
 
 x - y = - 2 
 x 2y = 
 
 I4x - 
 
 7. 
 
252 THEORY OF MEASUREMENTS 222 
 
 2. Multiply each equation by the coefficient of y in 
 that equation, and add the results in order to obtain 
 a second normal equation: 
 
 - 3z + y = - 3 
 
 - x+ y = 2 
 
 - 2x + y = 
 
 - 6z + 30 = - 1. 
 
 3. If the equations contain third, fourth, fifth, nth 
 unknown quantities, form a normal equation for each 
 one in the same way, multiplying each equation by its 
 proper coefficient of the particular unknown quantity 
 and adding, thus obtaining n normal equations in n 
 unknown quantities. 
 
 4. Solve the normal equations by any algebraical 
 process. The result will be the least-square values of 
 the unknowns. In the example under consideration this 
 gives 
 
 x = 5/2, 
 V = 14/3. 
 
 Choose any other point that you think best and show 
 that it gives a larger value for S(v 2 ) than this one does. 
 Write the equations in the form 
 
 3x y 3 = vi, 
 
 y x 2 = v Z) 
 
 2x - y = v d . 
 
 Solve the following equations by the method of least 
 squares : 
 
 x + 2.65y - 0.33^ = 6.21, 
 x + 2.370 + 0.122 = 3.18, 
 x + 2.25y - 0.292 = 5.89, 
 x - 0.870 + 3.272 = 4.28, 
 x + 3.380 + z = 4.07. 
 
XX LEAST SQUARES 253 
 
 Be careful that each product is given the right sign; 
 check them as each equation is completed. If squared 
 paper is not used be careful to keep the decimal points 
 under one another. The tedium of adding positive and 
 negative decimals can be greatly relieved by overlining 
 each negative digit* and adding en masse. 
 
 An approximate method of obtaining normal equations 
 is sometimes used in order to avoid the increased labor 
 of applying the above processes, 1, 2, 3, 4, to observation 
 equations that contain decimal fractions, but it is often 
 unsatisfactory: Make the coefficient of x positive in 
 each of the n equations by multiplying through by 1 
 where necessary; then add the resultant n equations to 
 form the first normal equation. Make the coefficients 
 of y positive in the same way, and add the n equations 
 to obtain a second normal equation. Proceed in this 
 way until n normal equations in n unknown quantities 
 have been formed; then solve. It is fairly obvious 
 that this method will be the most satisfactory in cases 
 where all the coefficients of all the observation equations 
 are of approximately the same size. 
 
 * Find the value of 2.4123 3.6418 + 5.7827 column by column 
 as follows: 
 
 2.4123 
 3.6418 
 
 5.7827 
 
 + 4.5532 
 
 Verify these also: .874 
 
 3.48 .454 
 
 7.22 _-653 
 
 6.35 2.840 
 
 3.380 
 
 1.87 
 
254 THEORY OF MEASUREMENTS 222 
 
 Apply the approximate method to the last exercise and 
 compare the ease of computation and the accuracy of 
 solution. 
 
 Apply the approximate method to the equations of 
 Fig. 62, and note one objectionable feature which it 
 may have. 
 
 To what extent are graphic methods applicable to the 
 solution of simultaneous indirect measurements? 
 
 
XXI. INDIRECT MEASUREMENTS 
 
 Apparatus. Slide rule. 
 
 223. Importance of Indirect Measurements. An in- 
 direct measurement is one that is calculated from one or 
 more direct measurements instead of being directly ob- 
 served by the experimenter. 
 
 In a certain sense almost all measurements are indirect, 
 especially those that require the highest degrees of accu- 
 racy. In making a careful determination of the weight 
 of an object the direct determinations are of the turning 
 points of the pointer that swings back and forth as the 
 beam of the balance oscillates. From these points the 
 resting position of equilibrium is calculated, and from 
 this position and a similar one obtained when the balance 
 is empty a calculation is made of the discrepancy be- 
 tween the weight of the object and that of the standard 
 weights that are balanced against it, using a previous 
 determination of the shift of resting point that is caused 
 by a very small standard weight. If the process of 
 weighing is for the purpose of obtaining mass instead of 
 weight still further calculating is necessary. 
 
 The term, indirect, however, is commonly applied to 
 measurements like that of the density obtained by 
 dividing mass by volume, one or more carefully deter- 
 mined measurements being treated by calculation in 
 such a way as to give a required " indirect" result. 
 
 In some cases values calculated from other data are a 
 necessity. The density of a solid may be obtained by 
 direct measurement if it can be tested by immersing it 
 in fluids whose densities are known, and this method is 
 found to be very convenient for testing small objects 
 
 255 
 
256 THEORY OF MEASUREMENTS 223 
 
 that do not have a very great density, such as precious 
 stones.* For denser objects, however, some indirect 
 method is necessary. The usual one is to calculate 
 d = m/v, v also being obtained indirectly from the loss 
 of weight on immersion in a fluid of known density. 
 
 In some cases much time and laborious calculation 
 can be saved by calculating the average and probable 
 error, a d av , for each quantity that is measured directly 
 and then investigating the resultant probable error of 
 the indirect measurement in accordance with theoretical 
 considerations. For example, it would take much longer 
 to calculate a density from each one of a set of twenty-five 
 determinations of mass and volume and then average 
 the results than it would to average the masses and the 
 volumes separately and perform a single division. When 
 using the latter method it is possible to find what the 
 probable error of the density will amount to by investi- 
 gating the probable errors of mass and volume. 
 
 In a case like the one just mentioned it would also 
 be possible to make twenty-five calculations of density 
 from the twenty-five sets of measurements of mass and 
 density and to compute the probable error of the 
 average density. In many cases, however, such a pro- 
 cess cannot be carried out. If there were more data at 
 hand for the mass than for the volume the excess could 
 not be utilized; and, furthermore, some of the values 
 needed in the formula may be predetermined constants, 
 such as those mentioned in chapter xviii (see 197), which 
 are given only in the form of a representative magnitude 
 and its probable error. The question then arises as to 
 
 * A solution of mercuric potassium iodide, sp. gr. = 3.11 or less, 
 is used by jewelers for this purpose. It is stated that densities as 
 high as 3.56 can be determined by using the double iodide of barium 
 and mercury. 
 
XXI INDIRECT MEASUREMENTS 257 
 
 the way in which the probable error of the indirect 
 measurement is influenced by the size of the probable 
 errors of the direct measurements on which it is based. 
 
 224. Probable Error of a Sum. The simplest case is 
 when the indirect measurement is merely the sum of the 
 two independent direct measurements. Let the direct 
 measurements be denoted by the small letters, 
 
 ai db Pi and a 2 pz, 
 
 when stated in the form of average and probable error, 
 and let the indirect measurement with its probable error 
 be represented by capitals, 
 
 AP, 
 
 the direct measurement being equal to the sum of the 
 indirect measurements, so that A = ai + a 2 ; then it can 
 be proved that 
 
 P 2 = Pi 2 + P/. 
 
 The dispersion of the sum is not as large as the sum of 
 the two direct dispersions from which it is calculated. 
 This is because positive and negative deviations will 
 counterbalance each other to a certain extent. It is 
 larger than either of the others alone, however, for the 
 extra measurement gives an extra degree of uncertainty. 
 A bench has a height of 42.50 .03 cm. above the 
 floor, and a table is 44.35 d= .04 cm. higher than the 
 bench. Write the height of the table, with its probable 
 error. 
 
 225. Probable Error of a Difference. If the indirect 
 measurement, A, is equal to the difference, a\ a 2 , 
 between two direct measurements, 0,1 and a 2 , it is perhaps 
 a natural inference that the square of its dispersion, P 2 , 
 
 18 
 
258 THEORY OF MEASUREMENTS 226 
 
 should be equal to pi 2 p 2 2 . This is not the case, how- 
 ever, but 
 
 P 2 = Pi 2 + ?2 2 
 in all cases in which 
 
 A = i CL. 
 
 The table that was measured in the last exercise is 
 51.10 .04 cm. lower than a shelf which is 137.95 .03 
 cm. above the floor. How high is the table? 
 
 Notice that the same formula applies to both of these 
 exercises. The difference of two measurements has as 
 large a probable error as their sum. Measuring first up 
 and then down does not give a more precise result than 
 measuring up and then further up. 
 
 226. Probable Error of a Multiple. If 
 
 A = citti, 
 
 where Ci is some constant not subject to error, then 
 P 2 = ci or P = 
 
 The diameter of a circular disc is found to be 7.98 .03 
 cm. What is its circumference? 
 
 Notice that the relative dispersion of the circumference 
 (3/800) is the same as the relative dispersion of the 
 diameter (vide infra). 
 
 227. Associative Law: 
 
 In general, if 
 
 A = Ciai db c 2 a 2 c 3 a 3 
 then 
 
 P2 = dV + C 2 2 P 2 2 + C 3 2 P3 2 + - (1) 
 
 where p n is the probable error of the average a n . 
 
 This formula can be used to find the probable error 
 of an algebraical expression when the probable error of 
 each of its terms is known. 
 
XXI INDIRECT MEASUREMENTS 259 
 
 A wall consists of 15 courses of bricks, each of which is 
 56.5 =t .5 mm. thick, separated by 14 layers of mortar 
 which have an average thickness of 7.5 1.5 mm. Show 
 that the probable error of the height of the wall is 22.3 
 mm., and state how much this figure would be reduced 
 if the bricks were absolutely uniform. How much if the 
 mortar was of uniform thickness instead? 
 
 228. Probable Error of a Product. If two independ- 
 ent measurements are multiplied together the probable 
 error of the product will follow the law expressed in the 
 following equation. 
 If 
 
 A = ai& 2 
 then 
 
 P 2 = piW + aiV- 
 Likewise if 
 
 A = 
 then 
 
 P 2 = 
 
 and similarly for any number of factors; but it is more 
 satisfactory in practice to make use of the relative prob- 
 able error (relative dispersion, 195) as in the following 
 form, which is easily deducible from the form just given. 
 If 
 
 A = a\ 2 3 
 then 
 
 (P/A) 2 = ( Pl /atf + (p 2 /a 2 ) 2 + (p 3 /a 3 ) 2 + - . 
 
 Prove that the last equation (as far as it goes) is 
 equivalent to 
 
 P 2 = (pic^as) 2 + (aip 2 a 3 ) 2 + (aia 2 p 3 ) 2 . 
 
 A rectangular block measures 20.00 db .04 cm. in 
 length, 10.00 =t .01 cm. in breadth, and 5.00 .01 cm. 
 in thickness. What is its volume? 
 
260 THEORY OF MEASUREMENTS 230 
 
 229. Probable Error of a Power. If 
 
 A = i n 
 then 
 
 (P/A) 2 = (rcW/arO or P y A 
 
 and, likewise, if 
 
 A = 
 then 
 
 P/A = 
 
 the constant not appearing in the formula if the relative 
 probable error is used. 
 
 In the particular case for which n = 1, notice that the 
 relative probable error of Cidi is the same as the relative 
 probable error of a\ itself (vide supra). 
 
 One edge of a cubical block is 10.00 .01 cm. What 
 is its volume? What is the area of its total surface? 
 
 Does the rectangular block (above) appear to have 
 been measured as accurately as the cubical block? How 
 do their volumes compare in respect to accuracy? What 
 is there about the data of the rectangular block which 
 make its volume determinable as closely as that of the 
 cubical block, although its measurements are less precise? 
 
 Show that the combined volume of the two blocks is 
 2000. =t 4.2 cm 3 . 
 
 230. Distributive Law : 
 
 In general, whether the exponents are positive, nega- 
 tive, or fractional, if 
 
 A = Ciai m a2 n a 3 r 
 then 
 
 (P/A) 2 = (m Pl / ai ) 2 + (np 2 /a 2 ) 2 + (rp 3 /a 3 ) 2 + - -. (2) 
 
 This formula can be used to find the probable error 
 of any single term of an algebraical expression when 
 the probable errors of its factors are known. 
 
XXI INDIRECT MEASUREMENTS 261 
 
 The formula for the volume of a cylinder is 
 v = irlr 2 . 
 
 If the measurements of I and r have respective probable 
 errors of p t and p r find the value of p v , the probable 
 error of the calculated volume. 
 
 Test the accuracy of the italicized statement (that 
 dav = di/^n) of 193 by letting p denote the probable 
 error of each of the n single measurements and finding 
 the probable error of their average. 
 
 231. Recapitulation. In the following formulae the 
 relative dispersion (P/A, or p/a) is represented by R or r. 
 
 If Then 
 
 A = o 
 
 i rb 2 =b 
 
 P 2 = Pi 2 + P2 2 + ' ' 
 
 A = c 
 
 itti 
 
 P = Cip! 
 
 A = c 
 
 lOi zt Ctttz =t ' ' 
 
 P 2 = (Ci^i) 2 + (C 2 p 2 ) 2 + ' 
 
 PROBABLE ERRORS OF INDIRECT MEASUREMENTS. In these 
 cases the probable error (p) for each average (a) is most convenient 
 for calculation. The capital letters refer to the indirect measure- 
 ment. 
 
 // 
 
 TAen 
 
 A = Oi n 
 A = CiOi n 
 
 R 2 = r! 2 - 
 R = nr 
 R = nr 
 R 2 = (mr 
 
 O 2 + (nr 2 ) 2 H 
 
 - (sr s ) 2 + - 
 
 PROBABLE ERRORS OF INDIRECT MEASUREMENTS. In these 
 cases the use of the relative probable error (r) simplifies the calcula- 
 tion. 
 
 Notice the complete formal correspondence between 
 (absolute) probable errors after adding, multiplying by a 
 coefficient, and assembling terms, and relative probable 
 errors after multiplying, raising to a power, and assem- 
 bling factors, respectively. 
 
262 THEORY OF MEASUREMENTS 232 
 
 A bowl whose interior is an exact segment of a sphere 
 is found to have a depth of 25.00 .02 centimeters 
 and a diameter across the top of 50.00 .30 centimeters. 
 Find its capacity from the formula for the volume of a 
 spherical segment, v = irhr 2 /2 + Trh s /6, where h is the 
 height or depth of the segment and r is the radius of its 
 circular base; find the probable error of the capacity 
 by applying the second general equation to each term 
 of the formula and then using the first general equation 
 to determine the final result. Notice the relative prob- 
 able error of the radius, r, is the same as that of the 
 diameter, d. Arrange the calculation systematically in 
 order to avoid numerical mistakes, and if there is any 
 trouble in making the substitution write out each step 
 of the process; for example: 
 
 ai = 25. P 2 /A 2 = 3 2 (.02) 2 /25 2 log 25 - 1.3979 
 
 pi = .02 P 2 = 3 2 (.02) 2 7r 2 25 6 /25 2 6 2 log 7r 2 0.9943 
 
 m = 3 = 7r 2 (.01) 2 25 4 4.0000 
 
 d = 7T/6 5.5916 
 
 A = 7r/* 3 /6 2.5859 
 
 logP = 
 
 232. Graphs of Propagated Errors. It has been seen 
 that the probable errors of two or more direct measure- 
 ments are propagated through any kind of a calculation 
 and give the indirect measurement a probable error 
 whose formula is of the type Vz 2 + y 2 . Since the square 
 root of the sum of two squares can always be represented 
 by the hypothenuse of a right-angled triangle a graphic 
 solution of the probable error of an indirect measurement 
 is easily effected. 
 
 From any "origin" draw a horizontal line and a 
 vertical line, making their lengths equal to the disper- 
 sions 3 and 4 of the measurements in 225. Complete 
 
XXI 
 
 INDIRECT MEASUREMENTS 
 
 263 
 
 the right-angled triangle and note that the length of the 
 hypothenuse, 5, is the " propagated" dispersion of the 
 indirect measurement, as pre- 
 viously found by calculation.* 
 The fact that pi 2 + p 2 2 + 
 p 3 2 is the same thing as 
 
 V 
 
 FIG. 63. GEOMETRIC DE- 
 TERMINATION OF PROPA- 
 GATED ERRORS. 
 
 Vpi 2 + p + Ps 2 , namely, the 
 sum of two squares, makes 
 this method extensible to any 
 number of terms (Fig. 63). 
 
 Use the geometrical con- 
 struction to find the square 
 root of the sum of the squares 
 
 of 4, 3, and 12. Afterward, verify your result by calcu- 
 lation. 
 
 233. Relative Importance of Compound Errors. The 
 fact that an indirect probable error which depends upon 
 the measurement of two or more different quantities 
 always assumes the form Vx 2 + y 2 means that it will 
 be more decidedly diminished by reducing the larger of 
 the two independent probable errors than by attempting 
 to improve the more accurate measurement. Show that 
 A/5 2 -f 2 2 is reduced by 41% if the 5 is changed to 2.5, 
 but only by 5% if the 2 is changed to 1. 
 
 Draw a right-angled triangle with one side several 
 (say 8 or 10) times as long as the other. Change the 
 long side by making it a little longer or shorter and 
 notice that the change in length of the hypothenuse 
 is almost in exact proportion. Change the short side 
 and notice that the hypothenuse is hardly affected at all. 
 
 The formula for the volume of a cylinder is v = 
 
 * These numbers refer, of course, to hundredths cf a centimetre. 
 It would be possible, but .not advisable, to perform the operation 
 A/.04 2 + .03 2 = .05 so as to obtain the answer in centimetres (188). 
 
264 THEORY OF MEASUREMENTS 234 
 
 In determining this indirect measurement which of the 
 two dimensions ought to be measured the more carefully? 
 How much more carefully? Why? 
 
 234. Questions and Exercises. 1. Devise a method 
 for determining velocities by direct measurement. 
 
 2. Each volume of a ten-volume encyclopedia has 
 5.0 cm. .1 cm. thickness of leaves between its two 
 covers. Each cover has a thickness of .35 cm. db .02 cm. 
 How much more shelf-room than 57.00 cm. is it likely 
 to need? What is the meaning of the word " likely" in 
 the last sentence? 
 
XXII. SYSTEMATIC AND CONSTANT ERRORS 
 
 Apparatus. Clock, chronometer, or time circuit, giv- 
 ing audible seconds ; watch with second hand; slide rule. 
 
 235. Definitions. It has been shown that errors may 
 be either accidental or constant. There is another class 
 of errors, often included under the term constant errors, 
 in which the error is not actually constant, nor does it 
 vary according to the law of probability. This is the 
 class of systematic errors, or errors that undergo a more 
 or less regular change during the course of making a set 
 of measurements. They may be subdivided into pro- 
 gressive errors, which show a steady increase (or decrease) 
 from one determination to the next, and periodic errors, 
 which increase for a number of measurements, then 
 decrease, and then repeat the previous cycle or period. 
 
 236. A Test for Systematic Errors. Where systematic 
 errors are absent a comparison of any measurement of a 
 series with the preceding one will tend to show an increase 
 in the numerical value about as often as a decrease; a 
 fact that can easily be tested by writing between each 
 two successive values a plus sign, a minus sign, or a zero, 
 according as the second value is respectively greater 
 than, less than, or equal to, the first, and then comparing 
 the number of the plus signs with that of the minus signs. 
 
 Where progressive errors have been greater than acci- 
 dental errors there may be all plus signs or all minus 
 signs as the result of applying the test. If the accidental 
 errors are relatively large they will probably cause several 
 of the signs to be plus and several minus, but the presence 
 of a progressive error at the same time will cause one 
 sign to appear more frequently than the other. 
 
 265 
 
266 THEORY OF MEASUREMENTS 238 
 
 If the systematic errors are periodic there will be 
 alternate groups of plus signs and minus signs, as is 
 shown in the next table. 
 
 237. Example of a Systematic Error. In an experi- 
 ment in which water in a reservoir was drawn up into 
 a tube by suction and successive readings of its height 
 were made values having the following decimals were 
 obtained in order: .76, .74, ,70, .62, .63, .61, .55, .56, 
 .51, .50, .44, .44, .39, .40, .35. .35. Are the results prob- 
 ably affected by progressive errors, or periodic errors, or 
 neither? Use a graphic diagram if the question is hard 
 to answer. What effect would you expect to result from 
 a slight leakage of air into the upper part of the tube? 
 
 238. Example of a Periodic Error. If the pivot of the 
 second hand of a watch is not exactly in the center of 
 the dial the indicated seconds will be subject to a periodic 
 error. For example, if it is located too far to the right 
 the hand may indicate 29 instead of 30 when it points 
 downward, and 1 instead of 60 when it is pointing up- 
 ward. That is, it will have a periodic error which will 
 be a maximum (positive) at 60 seconds, a minimum 
 (negative) at 30 seconds, and zero at 15 and 45 seconds 
 (Fig. 64). From the illustration it is evident that the 
 direction of displacement must be toward a point half- 
 way between the positions at which the error is greatest 
 and least. 
 
 Stand where you can hear the clock beat seconds and 
 read the time indicated by your watch. Every seven 
 seconds as indicated by the clock read the seconds and 
 estimated tenths of a second from the watch and state 
 the result to another student, who will take down the 
 values in his notebook. After three or four minutes 
 change places with him and note down the time as he 
 reads it off. Every seven-second interval should have 
 
XXII SYSTEMATIC AND CONSTANT ERRORS 
 
 267 
 
 its time by the watch noted, for a full period of seven 
 minutes. It is advisable to practice the procedure be- 
 forehand until you are sure that you can estimate tenths 
 of a second with reasonable accuracy. If your estimates 
 
 YIO error 
 
 no error 
 
 FIG. 64. ILLUSTRATION OF A PERIODIC ERROR. An eccentric 
 clock-hand will appear to be ahead of time, accurate, behind, 
 accurate, ahead, etc., in the course of its rotation. A determination 
 of the times at which the gain and the loss are most marked will 
 enable the direction and amount of displacement to be found. 
 
 are predominantly 0.5 and 0.0 the results will not be 
 satisfactory. In order to avoid losing count of the 
 (audible) seconds while you are stating the time it may 
 be advantageous to choose seven of your fingers and tap 
 on the table with each in turn at one-second intervals. 
 Concentrate your sense of sight on the watch and your 
 sense of hearing on the beats of the clock. 
 
 See that you have the complete table of sixty values 
 in your own notebook, and mark the observed tenths of 
 a second with a plus sign where they increase from one 
 observation to the next and with a minus sign where 
 they decrease, as shown below. With most watches it 
 
268 THEORY OF MEASUREMENTS 238 
 
 will be found that the second hand is not pivoted in the 
 exact centre of the graduated circle and the periodic 
 error will be shown very distinctly. 
 
 hr.min.sec. hr.min.sec. hr.min.sec. 
 
 4:42:45.2 
 52.0, 
 59.11 
 43:06.31 
 13.51 
 20.61 
 27.8 + 
 34.6 _ 
 41.3 
 48.1 _ 
 55.0 , 
 
 4:37:65.2 , 
 
 4:40:25.8 
 
 38:12.31 
 
 32.6 
 
 19.6+ 
 
 39.3 
 
 26.6 u 
 
 46.1 
 
 33.5 
 
 53.0 , 
 
 40.3 
 
 60.2 + 
 
 47.1 
 54.1 
 39:01.1 Y 
 
 08.2 + 
 
 41:07.4 + 
 14.6 + 
 
 21.7 + 
 28.7 
 
 15.4 + 
 
 35.6 
 
 22.6 + 
 
 42.2 
 
 29.6 u 
 
 49.1,. 
 
 36.4 ~ 
 
 56.1 
 
 43.1~ 
 
 42:03.2 + 
 
 50.0 ~ 
 
 57.02 
 
 10.3 + 
 17 5 
 
 64.2 + 
 
 24.7 + 
 
 40:11.4+ 
 
 31.6" 
 
 18.6 + 
 
 38.4 ~ 
 
 44:02.3 
 09.4 
 16.6 
 23.8 
 30.7 
 37.4 
 
 .^ 
 51. IX 
 
 68.12 
 
 4:45:05.2 + 
 
 APPARENT TIME OF SEVEN-SECOND INTERVALS. The tabulated 
 numbers are the times indicated by a watch at audible intervals 
 that were known to be exactly seven seconds. The presence of a 
 periodic error is shown by the tabular differences occurring in alter- 
 nate groups of positive and negative values. 
 
 Draw a graphic diagram in which the abscissae repre- 
 sent the integral part of the number of seconds in your 
 table, and the ordinates represent the corresponding 
 tenths of a second (Fig. 65). Draw a smooth curve to 
 eliminate accidental errors in the determination of time. 
 
 Determine the direction and the amount of the dis- 
 placement, and summarize the result by stating "the 
 pivot of the second hand of the watch is displaced toward 
 the figure of the dial by an amount equal to the 
 length of seconds' divisions on the graduated circle." 
 
XXII SYSTEMATIC AND CONSTANT ERRORS 269 
 
 .3 * 
 
 02 O 
 
 ll 
 
 - 
 
 11 
 
 -+-> d) 
 
 1 
 
 ^ ^ 
 
 . S C3 
 
 Eg 
 
270 
 
 THEORY OF MEASUREMENTS 
 
 239 
 
 i i 
 I 
 
 Explain how the periodic error can be eliminated in 
 case such a watch is used for determining intervals of 
 time. 
 
 239. Example of a Progressive Error. The list of fig- 
 ures given in 237 was obtained 
 t | from a determination of specific 
 
 jLJl gravity by Hare's method. If the 
 
 lower ends of two upright tubes 
 dip into two separate reservoirs 
 while their upper ends are both 
 joined to a third tube from which 
 the air can be partially exhausted 
 it can easily be proved that the 
 heights to which the fluids are 
 raised will be inversely proportional 
 to their densities ; so that if a fluid 
 whose density is unity is raised to a 
 height hi, and a heavier fluid to a 
 lesser height h%, the density or spe- 
 cific gravity of the latter must be 
 hi/hz. The complete list of deter- 
 minations of height included read- 
 ings of both columns of liquid; they 
 were made at approximately equal 
 intervals of time, and in the order 
 in which they are given in the table, 
 viz., 75.76, 73.06, 75.74, 73.04, 75.70, 73.00, 72.98, 72.95, 
 75.62, etc. 
 
 If the density is calculated by dividing 75.76 by 73.06 
 it is evident that the progressive error will make the 
 resulting figure too large, for the height of the water 
 had fallen somewhat below 75.76 when the reading of 
 the salt solution, 73.06, was taken; and if 75.74 is divided 
 by 73.06 the progressive error will make the result too 
 
 FIG. 66. HARE'S 
 METHOD OF BALANC- 
 ING COLUMNS. The 
 heights of the two liq- 
 uids are inversely pro- 
 portional to their den- 
 sities. 
 
XXII SYSTEMATIC AND CONSTANT ERRORS 
 
 271 
 
 pure 
 water 
 
 salt 
 sol lion 
 
 small, for the salt solution did not stay at 73.06 while the 
 reading 75.74 was being taken. Obviously the average 
 of 75.76 and 75.74 must be divided by 
 73.06, or 75.74 must be divided by the 
 average of 73.06 and 73.04, or some 
 other combination used in which the 
 average height of one column of liquid 
 must have occurred at the same time 
 as the average height of the other. 
 This method of eliminating progres- 
 sive errors is used in the process of 
 weighing with a delicate balance and in 
 many other processes of physical meas- 
 urement. 
 
 What set of values near the end of the 
 table can be used in the same way? 
 Make five different calculations of density 
 from successive parts of the table and see 
 whether they show any evidence of pro- 
 gressive error. 
 
 240. Constant Errors. It has already 
 been stated that constant errors are more 
 troublesome than accidental errors and 
 that the latter give very little aid in de- 
 termining the former. It is not the tar- 
 get (page 193) that is found from indi- 
 vidual measurements but only the centre 
 of clustering, and characteristic deviations 
 show only how close determinations come 
 to one another, not how close they come to 
 the truth. 
 
 Some constant errors are easily corrected with the aid 
 of theoretical considerations ; others may be very difficult 
 to eliminate. Unfortunately there is no infallible rule 
 
 75.76 
 
 75.74 
 75.70 
 
 75.62 
 
 .63 
 .61 
 
 .55 
 .56 
 
 .51 
 .50 
 
 .44 
 .44 
 
 .39 
 .40 
 
 .35 
 .35 
 
 73.06 
 73.04 
 
 73.00 
 
 72.98 
 .95 
 
 .94 
 
 .89 
 
 .88 
 
 .84 
 .84 
 
 .79 
 
 .78 
 
 .75 
 .70 
 
 .77 
 
 .77 
 
 EXAMPLE OF 
 HEIGHTS IN 
 HARE'S METHOD. 
 
272 THEORY OF MEASUREMENTS 240 
 
 for detecting them, and each experimental problem has 
 its own special sources of error. The two beam-arms 
 of a balance may be unequal, so that all weighings are 
 proportionately erroneous; the end of a metre-stick may 
 be worn, so that every setting of the zero-point is in- 
 accurate; the neutral tint of litmus may be faultily 
 judged, so that a chemical determination is biassed. 
 Consider such a simple process as the determination of 
 atmospheric pressure with a mercurial barometer. The 
 vacuum at the top is never perfect and there is often 
 capillary action, both making the reading too low. If 
 the barometer and its attached scale do not hang verti- 
 cally every apparent reading will be too high. The 
 scale itself is too long or too short except at a single 
 temperature, and the mercury may have its accepted 
 standard density only at a different temperature from 
 the one that it has when the observation is made. Even 
 if its density is standard the height of a column that will 
 give a definite pressure will depend upon the strength 
 of gravitational attraction and this varies with the lati- 
 tude and altitude of the instrument. If an aneroid 
 barometer is to be used instead of a mercurial one its 
 mechanism introduces still more sources of error. 
 
 It is evident that the amount of constant error will 
 generally be varied by changing observers, apparatus, 
 methods, and times of observation; and the more rad- 
 ically different the sets of conditions are made the better, 
 in all probability, will be the mutual neutralization of 
 constant errors when the weighted average is taken. 
 In practice, the values for most of the constants of 
 nature have been obtained under such varying condi- 
 tions. Atomic weights are obtained from various inter- 
 relations of chemical compounds obtained from different 
 sources and by different methods. The surface tension 
 
XXII SYSTEMATIC AND CONSTANT ERRORS 273 
 
 of water may be measured by the hanging drop method, 
 by the capillary wave method, by the vibrating jet 
 method, etc. The size of the molecules of a gas may 
 be calculated from the rate at which heat is conducted 
 through them, from the covolume constant, 6, of Van der 
 Waal's equation, from experimental determinations of 
 the viscosity of the gas, from measurements of the 
 maximum density obtainable by cooling and liquefying 
 or solidifying it, etc. If various determinations agree 
 closely in spite of the employment of essentially different 
 methods it becomes more probable that constant errors 
 have been satisfactorily removed, but it can never be 
 certain that all of these methods have not some common 
 source of error which would be eliminated only by using 
 some entirely different method. Constant watchfulness, 
 as stated in 162, and the exercise of good judgment are 
 of the greatest importance in guarding against constant 
 errors. If the student takes up further courses that 
 involve accurate measurement he will usually find that 
 various " sources of error" which have been found by 
 previous experimenters will be explicitly stated. Many 
 of them will be sources of constant error, and both his 
 natural ability and his progress in learning will be put 
 to the test in his management of them. 
 
 ID 
 
APPENDIX 
 
 TABLES 
 
EXPLANATORY NOTES 
 
 Formulae page 282 
 
 Equivalents page 283 
 
 The logarithm of each stated factor is given in another 
 column for convenience of computation. The table of 
 approximate equivalents is for use when no great accu- 
 racy is required. 
 
 Greek Alphabet page 284 
 
 Size of Errors page 284 
 
 The words given in this table will help to fix the 
 attention better than the numerical quantities. 
 
 Characteristic Deviations page 284 
 
 General Sources of Error " 284 
 
 Density of Water " 285 
 
 Notice that the density (mass per volume) of water 
 
 (under atmospheric pressure) is never as great as unity.* 
 
 A parallel column gives the specific gravity with reference 
 
 to water at the temperature of maximum density. 
 
 Inverse Tangents and Circular Measure .... page 285 
 
 Squares and Square Roots pages 286-287 
 
 The squares and square roots of all numbers are 
 obtained with four-figure accuracy by using this table 
 like any logarithm table. Complete five-figure and six- 
 figure squares of all three-figure number are obtained 
 
 * The terms density and specific gravity are often confused, even in 
 text books. Sometimes an arbitrary unit of volume is substituted 
 for the cubic centimetre in order that the maximum density of water 
 may appear to be unity, and a note is added to the effect that the 
 density is stated in " grams per millilitre." (A litre is defined as the 
 volume of 1000 grams of water at the temperature of maximum 
 density.) 
 
 277 
 
278 THEORY OF MEASUREMENTS 
 
 as follows: the last two figures of the required square 
 (N 2 ) are printed in italic opposite the last two figures 
 of N in the margin. These can then be used without 
 ambiguity to correct the four-figure value obtained in the 
 ordinary manner. E. g., required the square of 417. 
 The table gives the approximate value (pointed off by 
 inspection) of 173900. Opposite 17 are found the italic 
 figures 89. The square is accordingly 173889; not 173989, 
 for the latter would round off to 174000 instead of 17390Q. 
 
 The Probability Integral .page 288 
 
 The tabular value gives the area under the curve 
 y = e~ x2 between the ordinates and x in terms of the 
 total area between x = and x = oo. In the small 
 table of y and z/,47694 the value of I for Chauvenet's 
 criterion will be found in the first column opposite the 
 value of 1 l/2n in the second column. 
 
 Five-Place Logarithms page 289 
 
 This table is used like the four-place one. The first 
 three figures of the natural number (antilogarithm) are 
 found at the left, the fourth at the top, for the fifth inter- 
 polation is necessary. 
 
 If four-figure values are desired, round off a final five 
 by increasing the previous figure unless the five is fol- 
 lowed by a minus sign; e. g., log 1.055 = .0233, but log 
 1.065 = .0273, not .0274. 
 
 Fifth Place of Logarithms . . page 289 
 
 Before annexing an italicized figure the four-figure value 
 must be decreased by unity. E. g., log 839 = 92376, not 
 92386; log 274 = .43775, but log 282 = 45025. 
 
 The "tabular differences" maybe nearly as large as 
 four hundred, but are easily handled by logarithms for 
 purposes of interpolation.* E. g., to find antilog 22222: 
 
 *See Four-Place Logarithms, page 279. 
 
APPENDIX 279 
 
 antilog 22011 = 16.; antilog 22272 = 167; tab. dif.=261; 
 tabular excess of given logarithm = 211; 211/267 is about 
 0.8 ( 14) and is found by subtracting log 267 ( = 4265) 
 from log 211 ( = 3243), giving 8978 ( = log 790); /.antilog 
 22222 = 166||| = 16d79. 
 
 Exponentials page 289 
 
 A subscript number takes the place of a decimal point 
 followed by the corresponding number of ciphers. E. g., 
 4 4343 means 0.00004343, 3 1234 means .0001234, etc. 
 
 Notice that e x+y = e x e v ' } for example, e 2 - 5 is 7.3891 X 
 1.6487, and Iogi e 2 - 5 is 1.0857. 
 
 Four-Place Logarithms pages 290-291 
 
 To find the logarithm of a given number: For the integral 
 part (" characteristic") of the logarithm, count to the 
 left from units' place to the first figure (other than zero) 
 of the number, thus : 
 
 7654321:0 -1 j-2 -3 -4 
 
 93000000. or .000305 
 
 log 93000000 = 7. +.; log .000305 = - 4-. + . . .. 
 For the decimal part (" mantissa"), find the first (two) 
 figures of the given number in the left-hand column (N) 
 of the table and the third figure at the top of another 
 column. The required mantissa will be found in line 
 with the first two figures and in the column headed by 
 the third. Consider the second or third figure, if lacking, 
 to be zero. E. g., log 7 = log 70 = log 700 = .8451; 
 log .000023 = log .0000230 = - 5. + .3617. The man- 
 tissa is always kep_t positive, and a logarithm like the 
 last is abbreviated 5.3617 to save space. If the number 
 has four significant figures find the logarithms of the 
 next smaller and next larger marginal numbers and assume 
 that logarithmic differences are proportional to the corre- 
 sponding numerical differences. Thus, 1.873 would be 
 
280 THEORY OF MEASUREMENTS 
 
 located (on a scale) 3 tenths of the way from 1.87 to 1.88; 
 therefore log 1.873 is likewise '3/10 of the way from 2718 
 to 2742, namely 2725. (Three tenths of the tabular 
 difference, 24, will be found from the small marginal 
 tables to be 7, and 2718 + 7 = 2725. The approximate 
 tabular difference, D, is given for each line, so that only 
 the final digits need be subtracted.) If the given number 
 has 5 or more significant figures a table in which the 
 logarithms are stated to 5 or more places must be used. 
 
 To find the number (" antilogarithm") that corresponds 
 to a given logarithm: If the given logarithm does not 
 occur in the body of the table determine its position in 
 respect to the next higher and lower tabular logarithms 
 and use proportional parts as before. E. g., .1345 is 
 found to be 10/32 of the way from .1335 to .1367, hence 
 its antilogarithm will be 1.36J$, or 1.363. Notice that 
 10/32 can be reduced to tenths and 3/10 to twenty- 
 fourths, mentally, by using the small multiplication 
 tables (PP) in the margin. 
 
 Reciprocals are easily determined mentally by using 
 a table of logarithms. E. g., l/e = 0.3678. (Foot-note, 
 page 281.) 
 
 Squares page 292 
 
 The use of the small table of squares will be self- 
 evident. Notice that the square of a number between 
 100 and 110, say of 100 + n or 107, consists of five figures 
 which are, in order, 1, 2n, n 2 , or 1, 14, 49. The square 
 of any number between 100 and 200 can be found by the 
 same process, " carrying" mentally. Thus 
 
 112 2 = 1 173 2 = 1 
 
 24 146 
 
 144 5329 
 
 12544 29929 
 
APPENDIX 281 
 
 If either 2(v*)/n or 2(v z )/n(n - 1) is located be- 
 tween two consecutive numbers -in the third column, 
 (n -b l/2) 2 /(.67449) 2 , of the same table, then the value 
 of .67449 A/2 (v 2 ) /nor .67449 Vs(v 2 )/n(n - 1), as the case 
 may be, will be found opposite it in the first column. 
 A very rough mental calculation will prevent taking a 
 value which is VlO times too large or small. 
 
 Constants page 292 
 
 The characteristics 1 and 2 have been replaced by 
 9 and 8 respectively. 
 
 Circular Functions page 292 
 
 In the table of circular functions the " radian value," 
 natural sine, cosine, tangent, and cotangent are given for 
 every degree of the quadrant (above 45 use the lower 
 and right-hand margin), also the logarithmic sine and 
 cosine. By subtracting the two latter from each other 
 and from zero any of the six logarithmic functions may 
 be obtained from the table by inspection.* Sines and 
 
 * When two logarithms are to be added or subtracted it will be 
 found more convenient, after a little experience, to work from left 
 to right than the reverse. This is especially easy in finding recip- 
 rocals by subtracting from zero (as in 68, no. 7) : beginning at the 
 left subtract each figure from 9, except the last one, which is to be 
 subtracted from 10. For example, 
 
 log 1 = =1. 9 9 9 9 10 
 log * = 0. 4 9 7 1 5 
 
 .'. log I/TT =1. 5 2 8 5 
 
 Try working the following exercises from left to right. Before 
 each addition note whether the next pair of figures will add up to 
 more than nine and so give " one to carry." If they add up to 
 exactly nine, look a step farther to the right, and so on. 
 
 4156528436464436146 
 + 3228328943238423856 
 
 In subtracting from left to right, before setting down each partial 
 difference notice whether it will need to be decreased by unity on 
 account of the figures that follow. 
 
282 THEORY OF MEASUREMENTS 
 
 cosines of any intermediate values can safely be obtained 
 by interpolation, and tangents up to tan 70. For the 
 sine, tangent, and numerical measure of a small angle 
 the equations at the corners of the table should be used 
 as factors. E. g., sin 3' = 3 X .000290888 = .000872664. 
 For inverse " radians" and tangents see page 285. 
 
 FORMULAE 
 
 Thermometry 
 
 F = 9C/5+32 
 R = 4C/5 
 C = 5(F-32)/9 
 C = 5R/4 
 
 Logarithms 
 
 log e x = logio as/logio e = 2.3025851 logio x 
 logio x = log e z/log 10 = .4342945 log x 
 logio 2.3025851 = .3622157 
 logio .4342945 = 1.6377843 
 
 Constants 
 
 ir = 3. 141593 = 180 
 
 e = 2.718282 
 
 x 2 = 9.869604 
 
 V*= 1.772454 
 
 I/T= .3183099 
 
 V2 = 1.414214 
 
 I' 3 =1.732051 
 
 V5 = 2.236068 
 
 V7 = 2.645751 
 
 >' 10 = 3. 162278 
 
 Mensuration 
 
 triangle: base, b; altitude, a; area, ab/2. 
 parallelogram: base, b; altitude, a; area, ab. 
 circle: radius, r; circumference, 2irr; area, irr z . 
 ellipse: major axis, 2o; minor axis, 26; area, irab. 
 cylinder: radiug, r; length, 1; surface, irr z -\- 2-mrl-\- irr 2 ; volume, irrU. 
 cone: radius, r; height, h; surface, irr 2 + irrVr 2 -{-h z ; volume, irr 2 A/3. 
 pyramid: area of base, a; height, h; perimeter of base, p; slant height, s; 
 surface, ps/2; volume, ah/3. 
 
 sphere: radius, r; surface, 47ir 2 ; volume, / ltrr 3 /3. 
 
 For 5-formulse see page 88. 
 
 For American wire gauge see p. 101. 
 
APPENDIX 
 
 283 
 
 EQUIVALENTS 
 
 The best determination of the ratio of 1 metre to 1 inch is 39.37043, and this is 
 the value generally adopted in scientific work. The legal relationship, however, 
 is 1 metre = 39.37079 inches in Great Britain and 1 inch = 1 /39.37000 metre in the 
 United States, the "metre" being a number of standard inches (36ths of the 
 Imperial Standard Yard) in the former case, and the "inch" being defined as 
 certain fraction of the standard metre (International Prototype) in the latter. 
 
 The accepted ratio of 1 pound to 1 kilogram is .4535924, and the derived 
 equivalents given in the table have been calculated from these two ratios and the 
 accepted relationship 1 litre = 1000.027 cm*. 
 
 The U. S. dry measures and the Imperial measures have been calculated from 
 the assumptions that 1 U. S. bu. is equal to 2150.420 cu. in., and 1 Imperial 
 gallon (namely, the volume of 10 av. Ibs. of water at 62 F., barometer at 30 
 inches, weighed in air against brass weights) is equal to 4.545853 litres. 
 
 Unit 
 
 Equivalent 
 
 Logarithm 
 
 Centimetre 
 
 = 0.3937043 inch 
 
 5951701 
 
 Square cm . . 
 
 = 15531.64 X (Cd [red] 15 760 dry air) 
 = 0.1550030 square inch 
 
 1912173 
 1903402 
 
 Cubic cm 
 
 = 0.999973 mL 
 
 9999883 
 
 Drachm 
 U S f 3 
 
 = 16.89407 impl. HI 
 = 16.23116 U. S. TTL 
 = 3.887936 gm. 
 3 696593 cm. 3 
 
 2277342 
 2103495 
 5897191 
 5678017 
 
 Impl. f. 3 
 Foot 
 
 = 3.551543 cm. 3 
 - 30 47973 cm 
 
 5504170 
 4840111 
 
 Square ft 
 Cubic ft 
 
 = 929.0138 cm.* 
 = 28316.09 cm 3 
 
 9680222 
 4520333 
 
 Grain 
 Gram 
 
 = 64.79893 mgm. 
 = 15.43235 grains 
 
 8115678 
 1884322 
 
 Inch 
 
 = 2.539978 cm. 
 
 4048299 
 
 Square in 
 Cubic in 
 
 = 6.451487 cm.2 
 = 16.38663 cm. 3 
 
 8096598 
 2144897 
 
 Kilogram 
 
 = 2.204622 av. Ibs. 
 
 3433342 
 
 Kilometre 
 Litre (vol. of 1000 gm. 
 HiO). 
 
 Metre 
 
 = 2.679229 Troy Ibs. 
 = 0.6213767 mi. 
 = 1.056716 U. S. liquid qt. 
 = 0.9082158 U. S. dry qt. 
 = 0.8799239 impl. qt. 
 - 3 280869 ft 
 
 4280097 
 7933550 
 0239583 
 9581891 
 9444447 
 5159889 
 
 Mile 
 
 = 1.093623 yd. 
 1609 330 metres. 
 
 0388676 
 2066450 
 
 Millilitre (mL) 
 
 = 1.000027 cm. 3 
 
 0000117 
 
 Minim (m) U. S 
 Impl. 
 
 = 16.23160 U. S. TTL 
 = 16.89452 impl. TTL 
 = 0.06160990 cm. 3 
 05919238 cm 3 
 
 2103612 
 2277459 
 7896505 
 7722658 
 
 Ounce av 
 Troy 
 
 = 28.34953 gm. 
 31 10348 gm. 
 
 4525458 
 4928090 
 
 U. S. f. 5 . . 
 
 = 29.57275 mL 
 
 4708917 
 
 Impl. f. 5 
 Pound av. 
 
 1 av. oz. of water at 62 F. = 28.41234 cm. 3 
 453 5924 gm 
 
 4535070 
 6566658 
 
 Troy 
 
 = 373.2418 gm 
 
 5719903 
 
 Quart U. S. dry . 
 
 - 1101 192 cm 3 
 
 0418630 
 
 U. S. liq 
 
 9436 3280 cm 3 
 
 9760417 
 
 
 1 136599 cm 3 
 
 0556074 
 
 Ton long. . . 
 
 1016 047 kgm 
 
 0069138 
 
 Short 
 
 907 1848 kgm 
 
 9576958 
 
 Metric (1000 kgm.). 
 Yard ... 
 
 = 2204.622 av. Ib 
 91 439208 cm 
 
 3433342 
 9611324 
 
 
 
 
 APPROXIMATE EQUIVALENTS 
 
 25 mm. = 1 inch 60 mgm. = 1 gr. 15 TTL = 1 cm. 3 
 
 10 cm. = 4 in. 15 gr. = 1 gm. 30 cm. 3 = 1 fl. oz. 
 
 40 in. = 1 metre 30 gm. = 1 oz. 1 litre = 1 quart 
 
 8 km. = 5 miles 11 Ib. =5 kgm. 1000 cm. 3 = 1 litre 
 
 1000 cm. 2 = 1 square ft. 15 Ib./sq. in. = 1 atmo. = lkgm./cm. 2 
 
 For slide-rule equivalents see page 97. 
 
284 
 
 THEORY OF MEASUREMENTS 
 
 GREEK ALPHABET 
 
 Letter 
 
 Used as a symbol for 
 
 A a* 
 
 alpha 
 
 Rotation of polarized light; temperature coefficient of ex- 
 
 
 
 pansion; angle. 
 
 B & 
 
 r y 
 
 beta 
 gamma 
 
 Coefficient of expansion; angle. 
 Ratio of specific heats; angle. 
 
 A S 
 
 delta 
 
 A small quantity ; a finite difference (A) ; difference of ... 
 
 E e 
 
 epsilon 
 
 2.7183. 
 
 z f 
 
 zeta 
 
 z (in co-ordinates). 
 
 II T; 
 
 eta 
 
 Viscosity; efficiency-ratio; y (in co-ordinates). 
 
 e e 
 
 theta 
 
 Temperature; angle. 
 
 i i 
 
 iota 
 
 \ 1; intensity of electric current. 
 
 K K 
 
 kappa 
 
 Electrical conductivity; magnetic susceptibility. 
 
 A X 
 
 lambda 
 
 Wave-length; latitude. 
 
 M M 
 
 mu 
 
 Index of refraction; coeff. of friction; .0001 cm. (MM 10~ 7 
 
 N v 
 
 nu 
 
 cm.); permeability. 
 Reluctivity (= I/M). 
 
 2 
 
 xi 
 
 x (in co-ordinates). 
 
 O of 
 
 omicron 
 
 
 n IT 
 
 Pi 
 
 3.1416; product of factors such as . . . (II; cf. S). 
 
 P P 
 
 rho 
 
 Density; radius. 
 
 S rj 
 
 sigma 
 
 Sum of terms such as ... (S) ; density of air (<r) ; Poisson's 
 
 
 
 ratio (<r). 
 
 T T 
 
 tau 
 
 Time; temperature: torque. 
 
 T u 
 
 upsilon 
 
 Specific volume (= 1/p). 
 
 * 
 
 phi 
 
 Angle; function of . . . ; flux. 
 
 X x 
 
 chi 
 
 Function of 
 
 ^ l 
 
 psi 
 
 Solid angle. 
 
 CO 
 
 omega 
 
 Resistance (Q); angular velocity; dispersive power. 
 
 
 
 * Usually written with one pen-stroke, somewhat like a, to 
 avoid confusion with italic a. 
 
 
 
 t Not used as a symbol, on account of liability to confusion 
 
 
 
 with o and 0. 
 
 SIZE OP ERRORS 
 
 CHARACTERISTIC DEVIATIONS 
 
 .03% to .1% "very small 
 
 .1% to .3% "small" 
 
 .3% to 1% "moderate" 
 
 1% to 3% "large" 
 
 3% to 10% "very large' 
 
 p/m* 
 a/m 
 slm 
 
 = 
 
 .4769363 
 .5641895 
 .7071066 
 
 log p/m 
 log a/m 
 log s/m 
 
 = 
 
 6784603 
 7514250 
 8494849 
 
 pis 
 
 pja 
 a/s 
 
 = 
 
 .6744898 
 .8453475 
 .7978846 
 
 log pis 
 log p/a 
 log a/s 
 
 = 
 
 8289754 
 9270353 
 9019401 
 
 IP 
 alp 
 s/a 
 
 = 
 
 1.482602 
 1.182944 
 1.253314 
 
 log s/p 
 log alp 
 log s/a 
 
 = 
 
 1710246 
 0729647 
 0980599 
 
 * m ..." modulus ", in the equation 
 
 GENERAL SOURCES OF ERROR 
 
 Linear scales not parallel; lack of vertically; etc. 
 Faulty standardization of standards of comparison. 
 Uneven or irregular subdivision of standards. 
 Inaccurate "coincidence" or "bisection." 
 Faulty estimation of tenths or other subdivisions. 
 Parallax in reading scales or the position of a pointer. 
 Assumed zero of a scale. 
 Friction (as in a balance) and "play" or "back-lash." 
 
APPENDIX 
 
 285 
 
 DENSITY OF WATER 
 
 INVERSE TANGENTS AND CIRCULAR MEASURE 
 
 1 
 
 density 
 (gm. per 
 
 spec. grav. 
 (:H 2 O at 
 
 AO f~* \ 
 
 n 
 
 is the circular measure and 
 tangent of 
 
 n 
 
 is the circular 
 measure of 
 
 is the tangent 
 of 
 
 3 
 
 cm. 3 ) 
 
 4 C.) 
 
 nooi 
 
 00' 20". 6 = .000100 
 
 Af\ 
 
 OQO KKf f\Kff f\ 
 
 
 0C 
 
 .999841 
 
 .99987 
 
 .UUUJL 
 
 .0002 
 
 41 '.3 = '.000200 
 
 .4U 
 .41 
 
 & oo Uo .U 
 23 29 28 .6 
 
 22 17 37 !l 
 
 
 
 
 .0003 
 
 1 01 .9 = .000300 
 
 .42 
 
 24 03 51 .2 
 
 22 46 56 .7 
 
 2 
 
 .999941 
 
 .99997 
 
 .0004 
 
 1 22 .5 = .000400 
 
 .43 
 
 24 38 13 .9 
 
 23 16 03 .7 
 
 4 
 
 .999973 
 
 1.00000 
 
 .0005 
 
 1 43 .1 = .000500 
 
 .44 
 
 25 12 36 .5 
 
 23 44 58 .2 
 
 6 
 
 .999941 
 
 .99997 
 
 .0006 
 
 2 03 .8 = .000600 
 
 .45 
 
 25 46 59 .2 
 
 24 13 39 .9 
 
 8 
 
 .999849 
 
 .99988 
 
 .0007 
 
 2 24 .4 = .000700 
 
 .46 
 
 26 21 21 .8 
 
 24 42 08 .7 
 
 10 
 
 .999700 
 
 .99973 
 
 .0008 
 .0009 
 
 2 45 .0 = .000800 
 3 05 .6 = .000900 
 
 .47 
 .48 
 
 26 55 44 .5 
 27 30 07 .1 
 
 25 10 24 .7 
 25 38 27 .6 
 
 12 
 
 .999498 
 
 .99952 
 
 .001 
 
 03' 26".3 = .001000 
 
 .49 
 
 28 04 29 .7 
 
 26 06 17 .5 
 
 14 
 16 
 
 18 
 
 .999244 
 .998943 
 .998595 
 
 .99927 
 .99897 
 .99862 
 
 .002 
 .003 
 .004 
 
 6 52 .5 = .002000 
 10 18 .8 = .003000 
 13 45 .1 = .004000 
 
 .50 
 .51 
 .52 
 
 28 38' 52".4 
 29 13 15 .1 
 29 47 37 .7 
 
 26 33' 54".2 
 27 01 17 .7 
 27 28 27 .9 
 
 20 
 
 .998203 
 
 .99823 
 
 .005 
 .006 
 
 17 11 .3 = .005000 
 20 37 .6 = .006000 
 
 .53 
 .54 
 
 30 22 00 .4 
 30 56 23 .0 
 
 27 55 24 .9 
 28 22 08 .6 
 
 22 
 
 .997770 
 
 .99780 
 
 .007 
 
 24 03 .9 = .007000 
 
 .55 
 
 31 30 45 .6 
 
 28 48 38 .8 
 
 24 
 
 .997296 
 
 .99732 
 
 .008 
 
 27 30 .1 = .008000 
 
 .56 
 
 32 05 08 .3 
 
 29 14 55 .8 
 
 26 
 
 .996783 
 
 .99681 
 
 .009 
 
 30 56 .4 = .009000 
 
 .57 
 
 32 39 30 .9 
 
 29 40 59 .3 
 
 28 
 30 
 
 .996232 
 .995646 
 
 .99626 
 .99567 
 
 n 
 
 is the circular 
 measure of 
 
 is the tangent 
 of 
 
 .58 
 .59 
 
 c.r\ 
 
 33 13 53 .6 
 33 48x16 .2 
 
 QA OO' QQ" Q 
 
 30 06 49 .4 
 30 32 26 .2 
 ^fl VI' 4.Q" *! 
 
 32 
 34 
 36 
 38 
 
 .99502 
 .99437 
 .99768 
 .99296 
 
 .99505 
 .99440 
 .99371 
 .99299 
 
 .01 
 .02 
 .03 
 
 .04 
 
 34' 22".6 
 1 08 45 .3 
 1 43 07 .9 
 2 17 30 .6 
 
 34' 22".6 
 1 08 44 .7 
 1 43 06 .1 
 2 17 26 .2 
 
 ,oU 
 .61 
 .62 
 .63 
 .64 
 
 o' oo .y 
 34 57 01 .5 
 35 31 24 .2 
 36 05 46 .8 
 36 40 09 .5 
 
 OU Ot *r7 -O 
 
 31 22 59 .5 
 31 47 56 .1 
 32 12 39 .3 
 32 37 09 .3 
 
 40 
 
 .99221 
 
 .99224 
 
 .05 
 
 2 51 53 .2 
 
 2 51 44 .7 
 
 .65 
 
 37 14 32 .1 
 
 33 01 25 .7 
 
 42 
 
 .99144 
 
 .99147 
 
 .06 
 .07 
 
 3 26 15 .9 
 4 00 38 .5 
 
 3 26 01 .1 
 4 00 15 .0 
 
 .66 
 .67 
 
 37 48 54 .8 
 38 23 17 .4 
 
 33 25 29 .3 
 33 49 19 .5 
 
 44 
 
 .99063 
 
 .99066 
 
 .08 
 
 4 35 01 .2 
 
 4 34 26 .1 
 
 .68 
 
 38 57 40 .1 
 
 34 12 56 .5 
 
 46 
 
 .98979 
 
 .98982 
 
 .09 
 
 5 09 23 .8 
 
 5 08 34 .0 
 
 .69 
 
 39 32 02 .7 
 
 34 36 20 .4 
 
 48 
 
 .98893 
 
 .98896 
 
 .10 
 
 5 43' 46".5 
 
 5 42' 38". 1 
 
 .70 
 
 40 06' 25".4 
 
 34 59' 31".3 
 
 50 
 
 .98804 
 
 .98807 
 
 .11 
 
 6 18 09 .1 
 
 6 16 38 .3 
 
 .71 
 
 40 40 48 .0 
 
 35 22 29 .1 
 
 52 
 
 .98712 
 
 .98715 
 
 .12 
 
 6 52 31 .8 
 
 6 50 34 .0 
 
 .72 
 
 41 15 10 .7 
 
 35 45 14 .0 
 
 54 
 
 .98618 
 
 .98621 
 
 .13 
 
 7 26 54 .4 
 
 7 24 24 .9 
 
 .73 
 
 41 49 33 .3 
 
 36 07 46 .0 
 
 56 
 
 .98522 
 
 .98525 
 
 .14 
 
 8 01 17 1 
 
 7 58 10 .6 
 
 .74 
 
 42 23 56 .0 
 
 36 30 05 .2 
 
 58 
 
 .98422 
 
 .98425 
 
 .15 
 
 8 35 39 .7 
 
 8 31 50 .8 
 
 .75 
 
 42 58 18 .6 
 
 36 52 11 .6 
 
 60 
 
 .98321 
 
 .98324 
 
 .16 
 .17 
 
 9 10 02 .4 
 9 44 25 .0 
 
 9 05 25 .0 
 9 38 53 .0 
 
 .76 
 
 .77 
 
 43 32 41 .3 
 44 07 03 .9 
 
 37 14 05 .4 
 37 35 46 .6 
 
 62 
 
 .98217 
 
 .98220 
 
 .18 
 
 10 18 47 .7 
 
 10 12 14 .3 
 
 .78 
 
 44 41 26 .5 
 
 37 57 15 .2 
 
 64 
 
 .98110 
 
 .98113 
 
 .19 
 
 10 53 10 .3 
 
 10 45 28 .7 
 
 .79 
 
 45 15 49 .2 
 
 38 18 31 .5 
 
 66 
 
 .98002 
 
 .98005 
 
 .20 
 
 11 27' 33".0 
 
 11 18' 35". 8 
 
 .80 
 
 45 50* 11".8 
 
 38 39' 35".3 
 
 68 
 
 .97891 
 
 .97894 
 
 .21 
 
 12 01 55 .6 
 
 11 51 35 .2 
 
 .81 
 
 46 24 34 .5 
 
 39 00 26 .8 
 
 70 
 
 .97778 
 
 .97781 
 
 .22 
 
 12 36 18 .3 
 
 12 24 26 .7 
 
 .82 
 
 46 58 57 .1 
 
 39 21 06 .3 
 
 72 
 
 74 
 76 
 
 78 
 
 .97663 
 .97545 
 .97426 
 .97304 
 
 .97666 
 .97548 
 .97429 
 .97307 
 
 .23 
 .24 
 .25 
 .26 
 .27 
 
 13 10 40 .9 
 13 45 03 .5 
 14 19 26 .2 
 14 53 48 .8 
 15 28 11 .5 
 
 12 57 09 .9 
 13 29 44 .6 
 14 02 10 .5 
 14 34 27 .2 
 15 06 34 .5 
 
 .83 
 .84 
 .85 
 .86 
 
 .87 
 
 47 33 19 .8 
 48 07 47 .8 
 48 42 05 .1 
 49 16 27 .7 
 49 50 50 .4 
 
 39 41 10 .3 
 40 01 48 .9 
 40 21 52 .3 
 40 41 43 .9 
 41 01 23 .8 
 
 80 
 
 .97180 
 
 .97183 
 
 .28 
 
 16 02 34 .1 
 
 15 38 32 .1 
 
 .88 
 
 50 25 13 .0 
 
 41 20 52 .0 
 
 82 
 
 .97054 
 
 .97057 
 
 .29 
 
 16 36 56 .8 
 
 16 10 19 .8 
 
 .89 
 
 50 59 35 .7 
 
 41 40 08 .7 
 
 84 
 
 .96927 
 
 .96930 
 
 .30 
 
 17 11' 19".4 
 
 1641'57".3 
 
 .90 
 
 51 33' 58".3 
 
 41 59' 14".0 
 
 86 
 
 .96797 
 
 .96800 
 
 .31 
 
 17 45 42 .1 
 
 17 13 24 .4 
 
 .91 
 
 52 08 20 .9 
 
 42 18 07 .9 
 
 88 
 
 .96665 
 
 .96668 
 
 .32 
 
 18 20 04 .7 
 
 17 44 40 .8 
 
 .92 
 
 52 42 43 .6 
 
 42 36 50 .6 
 
 90 
 
 .96531 
 
 .96534 
 
 .33 
 .34 
 
 18 54 27 .4 
 19 28 50 .0 
 
 18 15 46 .4 
 18 46 40 .9 
 
 .93 
 .94 
 
 53 17 06 .3 
 53 51 28 .9 
 
 42 55 22 .2 
 43 13 42 .7 
 
 92 
 
 .96396 
 
 .96399 
 
 .35 
 
 20 03 12 .7 
 
 19 17 24 .2 
 
 .95 
 
 54 25 51 .6 
 
 43 31 52 .3 
 
 94 
 
 .96258 
 
 .96261 
 
 .36 
 
 20 37 35 .3 
 
 19 47 56 .0 
 
 .96 
 
 55 00 14 .2 
 
 43 49 51 .1 
 
 96 
 
 .96119 
 
 .96122 
 
 .37 
 
 21 11 58 .0 
 
 20 18 16 .1 
 
 .97 
 
 55 34 36 .9 
 
 44 07 39 .1 
 
 98 
 
 .95978 
 
 .95981 
 
 .38 
 
 21 46 20 .6 
 
 20 48 24 .4 
 
 .98 
 
 56 08 59 .5 
 
 44 25 16 .6 
 
 100 
 
 .95835 
 
 .95838 
 
 .39 
 
 22 20 43 .3 
 
 21 18 20 .8 
 
 .99 
 
 56 43 22 .2 
 
 44 42 43 .5 
 
286 
 
 THEORY OF MEASUREMENTS 
 
 COMPLETE SQUARES UP TO 999 2 4-FiG. SQUARES AND SQUARE ROOTS 
 
 N 
 
 
 
 1234 
 
 5 
 
 6789 
 
 D 
 
 PP 
 
 
 
 00 
 
 01 04 09 16 
 
 25 
 
 36 49 64 81 
 
 
 64 63 62 61 60 
 
 10 
 
 1000 
 
 1020 1040 1061 1082 
 
 1102 
 
 1124 1145 1166 1188 
 
 22 
 
 1 6.4 6.3 6.2 6.1 6.0 
 
 11 
 
 12 
 13 
 14 
 
 1210 
 1440 
 1690 
 1960 
 
 1232 1254 1277 1300 
 1464 1488 1513 1538 
 1716174217691796 
 1988 2016 2050 2074 
 
 1322 
 1562 
 1822 
 2102 
 
 1346 1369 1392 1416 
 1588 1613 1638 1664 
 1850 1877 1904 1932 
 2132 2161 2190 2220 
 
 24 
 26 
 
 28 
 30 
 
 2 12.8 12.6 12.4 12.2 12.0 
 3 19.2 18.9 18.6 18.3 18.0 
 4 25.6 25.2 24.8 24.4 24.0 
 5i 32.0 31.5 31.0 30.5 30.0 
 6 38.4 37.8 37.2 36.6 36.0 
 
 15 
 
 2250 
 
 2280 2310 2341 2372 
 
 2402 
 
 2434 2465 2496 2528 
 
 32 
 
 7 44.8 44.1 43.4 42.7 42.0 
 8 51.2 50.4 49.6 48.8 48.0 
 
 16 
 
 2560 
 
 2592 2624 2657 2690 
 
 2722 
 
 2756 2789 2822 2856 
 
 34 
 
 9 57.6 56.7 55.8 54.9 54.0 
 
 17 
 
 2890 
 
 2924 2958 2993 3028 
 
 3062 
 
 3098 3133 3168 3204 
 
 36 
 
 
 18 
 
 3340 
 
 3276 3312 3349 3386 
 
 3422 
 
 3460 3497 3533 3572 
 
 38 
 
 
 59 
 
 58 
 
 57 
 
 56 
 
 55 
 
 54 
 
 19 
 
 3610 
 
 3648 3686 3725 3764 
 
 3802 
 
 3842 3881 3920 3960 
 
 40 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 5.9 
 
 5.8 
 
 5.7 
 
 5.6 
 
 5.5 
 
 5.4 
 
 20 
 
 4000 
 
 4040 4080 4121 4162 
 
 4202 
 
 4244 4285 4326 4368 
 
 42 
 
 2 
 
 11.8 
 
 11.6 
 
 11.4 
 
 11.2 
 
 11.0 
 
 10.8 
 
 
 
 
 
 
 
 3 
 
 17.7 
 
 17.4 
 
 17.1 
 
 16.8 
 
 16.5 
 
 16.2 
 
 21 
 
 4410 
 
 4452 4494 4537 4580 
 
 4622 
 
 4666 4709 4752 4796 
 
 44 
 
 4 
 
 23.6 
 
 23.2 
 
 22.8 
 
 22.4 
 
 22.0 
 
 21.6 
 
 22 
 
 4840 
 
 4884 4928 4973 5018 
 
 5062 
 
 5108 5153 5198 5244 
 
 46 
 
 5 
 
 29.5 
 
 29.0 
 
 28.5 
 
 28.0 
 
 27.5 
 
 27.0 
 
 23 
 
 5200 
 
 5336 5382 5429 5476 
 
 5522 
 
 5570 5617 5664 5712 
 
 48 
 
 6 
 
 35.4 
 
 34.8 
 
 34.2 
 
 33.6 
 
 33.0 
 
 32.4 
 
 24 
 
 5700 
 
 5808 5856 5905 5954 
 
 6002 
 
 6052 6101 6150 6200 
 
 50 
 
 7 
 
 41 3 
 
 40.6 
 
 39.9 
 
 39.2 
 
 38.5 
 
 37.8 
 
 
 
 
 
 
 
 8 
 
 47.2 
 
 46.4 
 
 45.6 
 
 44.8 
 
 44.0 
 
 43.2 
 
 25 
 
 6,250 
 
 6300 6350 6401 6452 
 
 6502 
 
 6554 6605 6656 6708 
 
 52 
 
 9 
 
 53.1 
 
 52.2 
 
 51,3] 50.4 
 
 49.5 
 
 48.6 
 
 26 
 
 6700 
 
 6812 6864 6917 6970 
 
 7022 
 
 7076 7129 7182 7236 
 
 54 
 
 
 27 
 
 7250 
 
 7344 7398 7453 7508 
 
 7562 
 
 7618 7673 7728 7784 
 
 56 
 
 
 53 
 
 52 
 
 51 
 
 5U 
 
 tv 
 
 48 
 
 28 
 
 7S40 
 
 7896 7952 8009 8066 
 
 8122 
 
 8180 8237 8294 8352 
 
 58 
 
 
 
 
 
 
 
 
 29 
 
 84/0 
 
 8468 8526 8585 8644 
 
 8702 
 
 8762 8821 8880 8940 
 
 60 
 
 1 
 
 5.3 
 
 5.2 
 
 5.1 
 
 5.0 
 
 4.9 
 
 4.8 
 
 
 
 
 
 
 
 2 
 
 10.6 
 
 10.4 
 
 10.2 
 
 10.0 
 
 9.8 
 
 9.6 
 
 30 
 
 9000 
 
 9060 9120 9181 9242 
 
 9302 
 
 9364 9425 9486 9548 
 
 62 
 
 3 
 
 15.9 
 
 15.6 
 
 15.3 
 
 15.0 
 
 14.7 
 
 14.4 
 
 
 
 
 
 
 
 4 
 
 21.2 
 
 20.8 
 
 20.4 
 
 20.0 
 
 19.6 
 
 19.2 
 
 31 
 32 
 33 
 34 
 
 90/0 
 1024 
 1059 
 1150 
 
 9672 9734 9797 9860 
 1030 1037 1043 1050 
 1096 1102 1109 1116 
 1163 1170 1176 1183 
 
 9922 
 1056 
 1122 
 1190 
 
 9986 1005 1011 1018 
 1063 1069 1076 1082 
 1129 1136 1142 1149 
 1197 1204 1211 1218 
 
 6 4 
 
 7 
 7 
 7 
 
 5 
 
 6 
 7 
 8 
 9 
 
 26.5 
 31.8 
 37.1 
 42.4 
 47.7 
 
 26.0 
 31.2 
 36.4 
 41.6 
 46.8 
 
 '25.5 
 30.6 
 35.7 
 40.8 
 45.9 
 
 25.0 
 30.0 
 35.0 
 40.0 
 45.0 
 
 24.5 
 29.4 
 34.3 
 39.2 
 44.1 
 
 24.0 
 28.8 
 33.6 
 38.4 
 43.2 
 
 35 
 
 1225 
 
 1232 1239 1246 1253 
 
 1260 
 
 1267 1274 1282 1289 
 
 7 
 
 
 36 
 
 1250 
 
 1303 1310 1318 1325 
 
 1332 
 
 1340 1347 1354 1362 
 
 7 
 
 
 47 
 
 46 
 
 45 
 
 44 
 
 43 
 
 42 
 
 37 
 
 1305 
 
 1376 1384 1391 1399 
 
 1406 
 
 1414 1421 1429 1436 
 
 8 
 
 1 
 
 4.7 
 
 4.6 
 
 4.5 
 
 4.4 
 
 4.3 
 
 4.2 
 
 38 
 
 144-4 
 
 1452 1459 1467 1475 
 
 1482 
 
 1490 1498 1505 1513 
 
 8 
 
 2 
 
 9.4 
 
 92 
 
 9 
 
 8 8 
 
 8 6 
 
 8 4 
 
 39 
 
 1521 
 
 1529 1537 1544 1552 
 
 1560 
 
 1568 1576 1584 1592 
 
 8 
 
 :i 
 
 14.1 
 
 13'.8 
 
 13^5 
 
 13i2 
 
 12^9 
 
 12^6 
 
 40 
 
 1600 
 
 1608 1616 1624 1632 
 
 1640 
 
 1648 1656 1665 1673 
 
 8 
 
 4 
 5 
 
 18.8 
 23.5 
 
 18.4 
 23.0 
 
 18.0 
 22.5 
 
 17.6 
 22.0 
 
 17.2 
 21.5 
 
 16.8 
 21.0 
 
 41. 
 
 1681 
 
 1689 1697 1706 1714 
 
 1722 
 
 1731 1739 1747 1756 
 
 8 
 
 (i 
 
 28.2 
 
 27.6 
 
 27.0 
 
 26.4 
 
 25.8 
 
 25.2 
 
 42 
 
 1704 
 
 1772 1781 1789 1798 
 
 1806 
 
 1815 1823 1832 1840 
 
 9 
 
 7 
 
 32.9 
 
 32.2 
 
 31.5 
 
 30.8 
 
 30.1 
 
 29.4 
 
 43 
 
 1849 
 
 1858 1866 1875 1884 
 
 1892 
 
 1901 1910 1918 1927 
 
 9 
 
 8 
 
 37.6 
 
 36.8 
 
 36.0 
 
 35.2 
 
 34.4 
 
 33.6 
 
 44 
 
 1936 
 
 1945 1954 1962 1971 
 
 1980 
 
 1989 1998 2007 2016 
 
 9 
 
 9 
 
 42.3 
 
 41.4 
 
 40.5 
 
 39.6 
 
 38.7 
 
 37.8 
 
 45 
 
 2026 
 
 2034 2043 2052 2061 
 
 2070 
 
 2079 2088 2098 2107 
 
 9 
 
 
 
 
 
 
 
 
 
 41 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 46 
 
 2116 
 
 2125213421442153 
 
 2162 
 
 2172 2181 2190 2200 
 
 9 
 
 
 
 
 
 
 
 
 47 
 
 2209 
 
 2218 2228 2237 2247 
 
 2256 
 
 2266 2275 2285 2294 
 
 10 
 
 1 
 
 4.1 
 
 4.0 
 
 3.9 
 
 3.8 
 
 3.7 
 
 3.6 
 
 48 
 
 2304 
 
 2314 2323 2333 2343 
 
 2352 
 
 2362 2372 2381 2391 
 
 10 
 
 2 
 
 82 
 
 80 
 
 7 8 
 
 7 6 
 
 7.4 
 
 7 2 
 
 49 
 
 2401 
 
 2411242124302440 
 
 2450 
 
 2460 2470 2480 2490 
 
 10 
 
 3 
 
 12^3 
 
 12!o 
 
 11.7 
 
 1L4 
 
 11.1 
 
 10^8 
 
 50 
 
 2500 
 
 2510 2520 2530 2540 
 
 2550 
 
 2560 2570 2581 2591 
 
 10 
 
 4 
 5 
 
 16.4 
 20.5 
 
 16.0 
 20.0 
 
 15.6 
 19.5 
 
 15.2 
 19.0 
 
 14.8 
 18.5 
 
 14.4 
 18.0 
 
 51 
 
 260J 
 
 2611 262126322642 
 
 2652 
 
 2663 2673 2683 2694 
 
 10 
 
 
 
 24.6 
 
 24.0 
 
 23.4 
 
 22.8 
 
 22.221.6 
 
 52 
 
 2704 
 
 2714 2725 2735 2746 
 
 2756 
 
 2767 2777 2788 2798 
 
 11 
 
 7 
 
 28.7 
 
 28.0 
 
 27.3 
 
 26.6 
 
 25.9 25.2 
 
 53 
 
 2809 
 
 2820283028412852 
 
 2862 
 
 2873 2884 2894 2905 
 
 11 
 
 N 
 
 32.8 
 
 32.0 
 
 31.2 
 
 30.4 
 
 29.6 ! 28.8 
 
 54 
 
 29/0 
 
 2927 2938 2948 2959 
 
 2970 
 
 2981 2992 3003 3014 
 
 11 
 
 9 
 
 36.9 
 
 36.0 
 
 35.1 
 
 34.2 
 
 33.3 32.4 
 
APPENDIX 
 
 287 
 
 COMPLETE SQUARES UP TO 999 2 4-FiG. SQUARES AND SQUARE ROOTS 
 
 N 
 
 
 
 1234 
 
 5 
 
 6789 
 
 D 
 
 PP 
 
 55 
 
 3025 
 
 3036 3047 3058 3069 
 
 3080 
 
 3091 3102 3114 3125 
 
 11 
 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 56 
 
 3136 
 
 3147 3158 3170 3181 
 
 3192 
 
 3204 3215 3226 3238 
 
 11 
 
 1 
 
 3.5 
 
 3.4 
 
 3.3 
 
 3.2 
 
 3.1 
 
 3.0 
 
 57 
 
 3249 
 
 3260 3272 3283 3295 
 
 3306 
 
 3318 3329 3341 3352 
 
 12 
 
 2 
 
 7.0 
 
 6.8 
 
 6.6 
 
 6.4 
 
 6.2 
 
 6.0 
 
 58 
 
 3364 
 
 3376338733993411 
 
 3422 
 
 3434 3446 3457 3469 
 
 12 
 
 3 
 
 10.5 
 
 10.2 
 
 9.9 
 
 9.6 
 
 9.3 
 
 9.0 
 
 59 
 
 3481 
 
 3493 3505 3516 3528 
 
 3540 
 
 3552 3564 3576 3588 
 
 12 
 
 4 
 
 14.0 
 
 13.6 
 
 13.2 
 
 12.8 
 
 12.4 
 
 12.0 
 
 
 
 
 
 
 
 5 
 
 17.5 
 
 17.0 
 
 16.5 
 
 16.0 
 
 15.5 
 
 15.0 
 
 60 
 
 3600 
 
 3612 3624 3636 3648 
 
 3660 
 
 3672 3684 3697 3709 
 
 12 
 
 6 
 
 21.0 
 
 20.4 
 
 19.8 
 
 19.2 
 
 18.6 
 
 18.0 
 
 
 
 
 
 
 
 7 
 
 24.5 
 
 23.8 
 
 23.1 
 
 22.4 
 
 21.7 
 
 21.0 
 
 61 
 
 3721 
 
 3733 3745 3758 3770 
 
 3782 
 
 3795 3807 3819 3832 
 
 12 
 
 8 
 
 28.0 
 
 27.2 
 
 26.4 
 
 25.6 
 
 24.8 
 
 24.0 
 
 62 
 
 3844 
 
 3856 3869 3881 3894 
 
 3906 
 
 3919 3931 3944 3956 
 
 13 
 
 9 
 
 31.5 
 
 30.6 
 
 29.7 
 
 28.8 
 
 27.9 
 
 27.0 
 
 63 
 
 3969 
 
 3982 3994 4007 4020 
 
 4032 
 
 4045 4058 4070 4083 
 
 13 
 
 
 64 
 
 4096 
 
 4109 4122 4134 4147 
 
 4160 
 
 4173 4186 4199 4212 
 
 13 
 
 
 
 
 
 
 
 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 24 
 
 65 
 
 4225 
 
 4238 4251 4264 4277 
 
 4290 
 
 4303 4316 4330 4343 
 
 13 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 2.9 
 
 2.8 
 
 2.7 
 
 2.6 
 
 2.5 
 
 2.4 
 
 66 
 
 4356 
 
 4369 4382 4396 4409 
 
 4422 
 
 4436 4449 4462 4476 
 
 13 
 
 2 
 
 5.8 
 
 5.6 
 
 5.4 
 
 5.2 
 
 5.0 
 
 4.8 
 
 67 
 
 4489 
 
 4502 4516 4529 4543 
 
 4556 
 
 4570 4583 4597 4610 
 
 14 
 
 3 
 
 8.7 
 
 8.4 
 
 8.1 
 
 7.8 
 
 7.5 
 
 7.2 
 
 68 
 
 4624 
 
 4638 4651 4665 4679 
 
 4692 
 
 4706 4720 4733 4747 
 
 14 
 
 4 
 
 11.6 
 
 11.2 
 
 10.8 
 
 10.4 
 
 10.0 
 
 9.6 
 
 69 
 
 4761 
 
 4775 4789 4802 4816 
 
 4830 
 
 4844 4858 4872 4886 
 
 14 
 
 5 
 
 14.5 
 
 14.0 
 
 13.5 
 
 13.0 
 
 12.5 
 
 12.0 
 
 
 
 
 
 
 
 6 
 
 17.4 
 
 16.8 
 
 16.2 
 
 15.6 
 
 15.0 
 
 14.4 
 
 70 
 
 4900 
 
 4914 4928 4942 4956 
 
 4970 
 
 4984 4998 5013 5027 
 
 14 
 
 7 
 
 20.3 
 
 19.6 
 
 18.9 
 
 18.2 
 
 17.5 
 
 16.8 
 
 
 
 
 
 
 
 8 
 
 23.2 
 
 22.4 
 
 21.6 
 
 20.8 
 
 20.0 
 
 19.2 
 
 71 
 
 504* 
 
 5055 5069 5084 5098 
 
 5112 
 
 5127 5141 5155 5170 
 
 14 
 
 9! 26.1 
 
 25.2 
 
 24.3 
 
 23.4 
 
 22.5 
 
 21.6 
 
 72 
 
 5154 
 
 5198 5213 5227 5242 
 
 5256 
 
 5271 5285 5300 5314 
 
 15 
 
 
 73 
 
 5329 
 
 5344 5358 5373 5388 
 
 5402 
 
 5417 5432 5446 5461 
 
 15 
 
 
 
 74 
 
 5476 
 
 5491 5506 5520 5535 
 
 5550 
 
 5565 5580 5595 5610 
 
 15 
 
 
 V6 
 
 zz 
 
 zl 
 
 ZU 
 
 iy 
 
 18 
 
 75 
 
 5625 
 
 5640 5655 5670 5685 
 
 5700 
 
 5715 5730 5746 5761 
 
 15 
 
 1 
 
 2.3 
 
 2.2 
 
 2.1 
 
 2.0 
 
 1.9 
 
 1.8 
 
 
 
 
 
 
 
 2 
 
 4.6 
 
 4.4 
 
 4.2 
 
 4.0 
 
 3.8 
 
 3.6 
 
 76 
 
 5775 
 
 5791 5806 5822 5837 
 
 5852 
 
 5868 5883 5898 5914 
 
 15 
 
 3 
 
 6.9 
 
 6.6 
 
 6.3 
 
 6.0 
 
 5.7 
 
 5.4 
 
 77 
 78 
 79 
 
 5920 
 6054 
 6241 
 
 5944596059755991 
 6100 6115 6131 6147 
 6257 6273 6288 6304 
 
 6006 6022 6037 6053 6068 
 6162 6178 6194 6209 6225 
 6320 6336 6352 6368 6384 
 
 16 
 16 
 16 
 
 4 
 
 5 
 6 
 
 7 
 
 9.2 
 11.5 
 13.8 
 16.1 
 
 8.8 
 11.0 
 13.2 
 15.4 
 
 8.4 
 10.5 
 12.6 
 14.7 
 
 8.0 
 10.0 
 12.0 
 14.0 
 
 7.6 
 9.5 
 11.4 
 13.3 
 
 7.2 
 9.0 
 10.8 
 12.6 
 
 80 
 
 6400 
 
 6416 6432 6448 6464 
 
 6480 
 
 6496 6512 6529 6545 
 
 16 
 
 8 
 9 
 
 18.4 
 20.7 
 
 17.6 
 19.8 
 
 16.8 
 18.9 
 
 16.0 
 18.0 
 
 15.2 
 17.1 
 
 14.4 
 16.2 
 
 81 
 
 6561 
 
 6577 6593 6610 6626 
 
 6642 
 
 6659 6675 6691 6708 
 
 16 
 
 
 82 
 
 6724 
 
 6740 6757 6773 6790 
 
 6806 
 
 6823 6839 6856 6872 
 
 17 
 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 12 
 
 83 
 84 
 
 6889 6906 6922 6939 6956 
 7050 J7073 7090 7106 7123 
 
 6972 
 7140 
 
 6989 7006 7022 7039 
 7157 7174 7191 7208 
 
 17 
 
 17 
 
 1 
 
 1.7 
 
 1.6 
 
 1.5 
 
 1.4 
 
 1.3 
 
 1.2 
 
 
 
 
 
 
 
 2 
 
 3.4 
 
 3 2 
 
 3.0 
 
 2.8 
 
 2.6 
 
 2 4 
 
 85 
 
 7225 
 
 7242 7259 7276 7293 
 
 7310 
 
 7327 7344 7362 7379 
 
 17 
 
 3 
 
 5^1 
 
 4.8 
 
 4.5 
 
 4.2 
 
 3.9 
 
 3.6 
 
 
 
 
 
 
 
 4. 
 
 6 8 
 
 6.4 
 
 6.0 
 
 5.6 
 
 5.2 
 
 4.8 
 
 86 
 
 7396 
 
 7413 7430 7448 7465 
 
 7582 
 
 7500 7517 7534 7552 
 
 17 
 
 5 
 
 8.5 
 
 8.0 
 
 7^5 
 
 7.0 
 
 6^5 
 
 6.0 
 
 87 
 
 7569 
 
 7586 7604 7621 7639 
 
 7656 
 
 7674 7691 7709 7726 
 
 18 
 
 y 
 
 10 2 
 
 9.6 
 
 g'0 
 
 8.4 
 
 7.8 
 
 7.2 
 
 88 
 
 7744 
 
 7762 7779 7797 7815 
 
 7832 
 
 7850 7868 7885 7903 
 
 18 
 
 7 
 
 11.9 
 
 11 2 
 
 10 5 
 
 9.8 
 
 9^1 
 
 8.4 
 
 89- 
 
 7921 
 
 7939 7957 7974 7992 
 
 8010 
 
 8028 8046 8064 8082 
 
 18 
 
 8 
 
 13^6 
 
 12^8 
 
 12^0 
 
 11.2 
 
 10.4 
 
 9.6 
 
 
 
 
 
 
 
 g 
 
 is a 
 
 11 i 
 
 13 fi 
 
 12 fi 
 
 11 71 10.8 
 
 90 
 
 8100 
 
 8118813681548172 
 
 8190 
 
 8208 8226 8245 8263 
 
 18 
 
 .. 
 
 91 
 
 825* 
 
 8299 8317 8336 8354 
 
 8372 
 
 8391 8409 8427 8446 
 
 18 
 
 1 11 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 92 
 
 8464 
 
 8482 8501 8519 8538 
 
 8556 
 
 8575 8593 8612 8630 
 
 19 
 
 j 
 
 
 
 
 
 
 93 
 
 8645 
 
 8668 8686 8705 8724 
 
 8742 
 
 8761 8780 8798 8817 
 
 19 
 
 1 
 
 1.1 
 
 1.0 
 
 0.9 
 
 0.8 
 
 0.7 
 
 0.6 
 
 94 
 
 8830 
 
 8855 8874 8892 8911 
 
 8930 
 
 8949 8968 8987 9006 
 
 19 
 
 2 
 
 2.2 
 
 2.0 
 
 1.8 
 
 1.6 
 
 1.4 
 
 1.2 
 
 
 
 
 
 
 
 3 
 
 3.3 
 
 3.0 
 
 2.7 
 
 2.4 
 
 2.1 
 
 1.8 
 
 95 
 
 90,25 
 
 9044 9063 9082 9101 
 
 9120 
 
 9139915891789197 
 
 19 
 
 4 
 
 4.4 
 
 4.0 
 
 3.6 
 
 3.2 
 
 2.8 
 
 2.4 
 
 
 
 
 
 
 
 5 
 
 5.5 
 
 5.0 
 
 4.5 
 
 4.0 
 
 3.5 
 
 3.0 
 
 96 
 
 92/0 
 
 9235 9254 9274 9293 
 
 9312 
 
 9332 9351 9370 9390 
 
 19 
 
 6 
 
 6.6 
 
 6.0 
 
 5.4 
 
 4.8 
 
 4.2 
 
 3.6 
 
 97 
 
 9405 
 
 9428 9448 9467 9487 
 
 9506 
 
 9526 9545 9565 9584 
 
 20 
 
 7 
 
 7.7 
 
 7.0 
 
 6.3 
 
 5.6 
 
 4.9 
 
 4.2 
 
 98 
 
 9604 
 
 9624 9643 9663 9683 
 
 9702 
 
 9722 9742 9761 9781 
 
 20 
 
 8 
 
 8.8 
 
 8.0 
 
 7.2 
 
 6.4 
 
 5.6 
 
 4.8 
 
 99 
 
 9801 
 
 9821 9841 9860 9880 
 
 9900 
 
 9920 9940 9960 9980 
 
 20 
 
 9 
 
 9.9 
 
 9.0 
 
 8.1 
 
 7.2 
 
 6.3 
 
 5.4 
 
288 
 
 THEORY OF MEASUREMENTS 
 
 THE PROBABILITY INTEGRAL y = - / e~ xZ dx 
 
 TfJ 
 
 The tabular value gives the area under the curve y = e xZ between the ordinates and x 
 in terms of the total area between x = and x = + . For Chauvenet's criterion see p. 234. 
 
 X 
 
 
 
 1234 
 
 5 
 
 6789 
 
 D 
 
 0.0 
 
 00000 
 
 01128022560338404511 
 
 05637 
 
 06762 07886 09008 10128 
 
 1118 
 
 0.1 
 0.2 
 0.3 
 0.4 
 
 11246 
 22270 
 32863 
 42839 
 
 12362 13476 14587 15695 
 23352 24430 25502 26570 
 33891349133592836936 
 43797 44747 45689 46623 
 
 16800 
 27633 
 37938 
 47548 
 
 17901 18999 20094 21184 
 28690 29742 30788 31828 
 38933 39921 40901 41874 
 48466 49375 50275 51167 
 
 1086 
 1035 
 965 
 
 883 
 
 0.5 
 
 52050 
 
 52924 53790 54646 55494 
 
 56332 
 
 57162 57982 58792 59594 
 
 792 
 
 0.6 
 0.7 
 0.8 
 0.9 
 
 60386 
 67780 
 74210 
 79691 
 
 61168619416270563459 
 68467 69143 69810 70468 
 74800 75381 75952 76514 
 80188 80677 81156 81627 
 
 64203 
 71116 
 77067 
 82089 
 
 64938 65663 66378 67084 
 71754 72382 73001 73610 
 77610 78144 78669 79184 
 82542 82987 83423 83851 
 
 696 
 600 
 507 
 419 
 
 1.0 
 
 84270 
 
 84681 85084 85478 85865 
 
 86244 
 
 86614 86977 87333 87680 
 
 340 
 
 1.1 
 1.2 
 1.3 
 1.4 
 
 88020 
 91031 
 93401 
 95229 
 
 88353 88679 88997 89308 
 91296 91553 91805 92051 
 93606 93807 94002 94191 
 95385 95538 95686 95830 
 
 89612 
 92290 
 94376 
 95970 
 
 89910 90200 90484 90761 
 92524 92751 92973 93190 
 94556 94731 94902 95067 
 96105 96237 96365 96490 
 
 270 
 211 
 162 
 121 
 
 1.5 
 
 96611 
 
 96728 96841 96952 97059 
 
 97162 
 
 97263 97360 97455 97546 
 
 89 
 
 1.6 
 1.7 
 1.8 
 1.9 
 
 97635 
 98379, 
 98909 
 99279 
 
 97721 97804 97884 97962 
 98441 98500 98558 98613 
 98952 98994 99035 99074 
 99309 99338 99366 99392 
 
 98038 
 98667 
 99111 
 99418 
 
 98110 98181 98249 98315 
 98719 98769 98817 98864 
 99147 99182 99216 99248 
 99443 99466 99489 99511 
 
 64 
 45 
 31 
 21 
 
 2.0* 
 
 99532 
 
 *55248 57195 59063 60858 
 
 62581 
 
 64235 65822 67344 68805 
 
 1400 
 
 2.1* 
 2.2* 
 2.3* 
 2.4* 
 
 70205 
 81372 
 88568 
 93115 
 
 71548 72836 74070 75253 
 82244 83079 83878 84642 
 89124 89655 90162 90646 
 93462937939410894408 
 
 76386 
 85373 
 91107 
 94694 
 
 77472 78511 79505 80459 
 86071 86739 87377 87986 
 91548 91968 92369 92751 
 94966 95226 95472 95707 
 
 913 
 
 582 
 364 
 223 
 
 2.5* 
 
 95930 
 
 96143 96345 96537 96720 
 
 96893 
 
 97058 97215 97364 97505 
 
 135 
 
 2.6* 
 2.7* 
 2.8* 
 2.9* 
 
 97640 
 98657 
 99250 
 99589 
 
 97767 97888 98003 98112 
 98732 98802 98870 98933 
 99293 99334 99372 99409 
 99613 99636 99658 99679 
 
 98215 
 98994 
 99443 
 99698 
 
 98313 98406 98494 98578 
 99051 99105 99156 99204 
 99476 99507 99536 99563 
 99716 99733 99750 99765 
 
 79 
 46 
 26 
 14 
 
 3.0* 
 
 99779 
 
 99793 99805 99817 99829 
 
 99839 
 
 99849 99859 99867 99876 
 
 8 
 
 3.1* 
 3.2* 
 3.3* 
 3.4* 
 
 99884 
 99940 
 99969 
 99985 
 
 99891998989990499910 
 99944 99947 99951 99954 
 99971 99973 99975 99977 
 99986 99987 99988 99989 
 
 99916 
 99957 
 99978 
 99989 
 
 99921 99926 99931 99936 
 99960 99962 99965 99967 
 99980 99981 99982 99984 
 99990 99991 99991 99992 
 
 4 
 2 
 1 
 1 
 
 3.5f 
 
 99993 
 
 f99309 99358 99403 99445 
 
 99485 
 
 99521 99555 99587 99617 
 
 27 
 
 3.6f 
 
 3.71 
 3.81 
 3.91 
 
 99644 
 99833 
 99923 
 99965 
 
 99670996949971699736 
 99845 99857 99867 99877 
 99929 99934 99939 99944 
 99968 99970 99973 99975 
 
 99756 
 99886 
 99948 
 99977 
 
 99773 99790 99805 99819 
 99895 99903 99910 99917 
 99952 99956 99959 99962 
 99979 99980 99982 99983 
 
 14 
 6 
 3 
 2 
 
 4. t 
 
 99985 
 
 99993 99997 99999 00000 
 
 00000 
 
 00000 00000 00000 00000 
 
 
 . 
 
 1 
 
 
 
 
 
 x 
 .47694 
 
 y 
 
 0.0 
 
 00000 
 
 0.5 
 
 26407 
 
 1.0 
 
 50000 
 
 1.1 
 
 54188 
 
 1.2 
 
 58171 
 
 1.3 
 
 61942 
 
 1.4 
 
 65498 
 
 1.5 
 
 68833 
 
 1.6 
 
 71949 
 
 1.7 
 
 74847 
 
 1.8 
 
 77528 
 
 1.9 
 
 79999 
 
 2.0 
 
 82266 
 
 2.1 
 
 84335 
 
 2.2 
 
 86216 
 
 2.3 
 
 87918 
 
 2.4 
 
 89450 
 
 2.5 
 
 90825 
 
 2.6 
 
 92051 
 
 2.7 
 
 93141 
 
 2.8 
 
 94105 
 
 2.9 
 
 94954 
 
 3.0 
 
 95698 
 
 3.1 
 
 96346 
 
 3.2 
 
 96910 
 
 3.3 
 
 97397 
 
 3.4 
 
 97817 
 
 3.5 
 
 98176 
 
 3.6 
 
 98482 
 
 3.7 
 
 98743 
 
 3.8 
 
 98962 
 
 3.9 
 
 99147 
 
 4.0* 
 
 *30228 
 
 4.1* 
 
 43137 
 
 4.2* 
 
 53857 
 
 4.3* 
 
 62718 
 
 4.4* 
 
 69998 
 
 4.5* 
 
 75957 
 
 4.6* 
 
 80816 
 
 4.7* 
 
 84759 
 
 4.8* 
 
 87935 
 
 4.9* 
 
 90500 
 
 5.0* 
 
 92549 
 
 * Beginning with x = 2.01 and x/,47694 = 4.0 the (seven-place) y- values have their first two 
 figures (9's) omitted. 
 
 t Beginning with x = 3.51 the (nine-place) y-values have their first four figures (.9999) omitted. 
 
APPENDIX 
 
 289 
 
 FIVE-PLACE LOGARITHMS 
 
 N 
 
 
 
 1 234 
 
 5 
 
 6789 
 
 D 
 
 100 
 
 00000 
 
 00043 00087 00130 00173 
 
 00217 
 
 00260 00303 00346 00389 
 
 43 
 
 101 
 
 00432 
 
 00475 00518 00561 00604 
 
 00647 
 
 00689 00732 00775-00817 
 
 43 
 
 102 
 
 00860 
 
 00903 00945 00988 01030 
 
 01072 
 
 01115-01157 01199 01242 
 
 42 
 
 103 
 
 01284 
 
 01326 01368 01410 01452 
 
 01494 
 
 01536 01578 01620 01662 
 
 41 
 
 104 
 
 01703 
 
 01745 01787 01828 01870 
 
 01912 
 
 01953 01995-02036 02078 
 
 41 
 
 105 
 
 02119 
 
 02160 02202 02243 02284 
 
 02325 
 
 02.366 02407 02449 02490 
 
 41 
 
 106 
 
 02531 
 
 02572 02612 02653 02694 
 
 02735- 
 
 02776 02816 02857 02898 
 
 40 
 
 107 
 
 02938 
 
 02979 03019 03060 03100 
 
 03141 
 
 03181 03222 03262 03302 
 
 40 
 
 108 
 
 03342 
 
 03383 03423 03463 03503 
 
 03543 
 
 03583 03623 03663 03703 
 
 40 
 
 109 
 
 03743 
 
 03782 03822 03862 03902 
 
 03941 
 
 03981 04021 04060 04100 
 
 39 
 
 EXPONENTIALS 
 
 FIFTH PLACE OF LOGARITHMS 
 
 X 
 
 e 
 
 ogio e x 
 
 e~ x 
 
 e- 2 
 
 .0001 
 
 1.0001 
 
 044343 
 
 .9999 
 
 .0000 
 
 .0002 
 
 1.0002 
 
 Oi8686 
 
 .9998 
 
 .0000 
 
 .0003 
 
 1.0003 
 
 Osl303 
 
 .9997 
 
 .0000 
 
 .0004 
 
 1.0004 
 
 3 1737 
 
 .9996 
 
 1.0000 
 
 .0005 
 
 1.0005 
 
 Oa2171 
 
 .9995 
 
 1.0000 
 
 .0006 
 
 1.0006 
 
 3 2606 
 
 .9994 
 
 1.0000 
 
 .0007 
 
 1.0007 
 
 3 3040 
 
 .9993 
 
 1.0000 
 
 .0008 
 
 1.0008 
 
 3 3474 
 
 .9992 
 
 1.0000 
 
 .0009 
 
 1.0009 
 
 3 3909 
 
 .9991 
 
 1.0000 
 
 .001 
 
 1.0010 
 
 3 4343 
 
 .9990 
 
 1.0000 
 
 .002 
 
 1.0020 
 
 3 8686 
 
 .9980 
 
 1.0000 
 
 .003 
 
 1.0030 
 
 2 1303 
 
 .9970 
 
 1.0000 
 
 .004 
 
 1 .0040 
 
 2 1737 
 
 .9960 
 
 1.0000 
 
 .005 
 
 1.0050 
 
 0?2171 
 
 .9950 
 
 1.0000 
 
 .006 
 
 1.0060 
 
 022606 
 
 .9940 
 
 1.0000 
 
 .007 
 
 1.0070 
 
 2 3040 
 
 .9930 
 
 1.0000 
 
 .008 
 
 1.0080 
 
 2 3474 
 
 .9920 
 
 1.0000 
 
 .009 
 
 1.0090 
 
 2 3909 
 
 .9910 
 
 1.0000 
 
 .01 
 
 1.0100 
 
 024343 
 
 .9900 
 
 .9999 
 
 .02 
 
 1.0202 
 
 2 8686 
 
 .9802 
 
 .9996 
 
 .03 
 
 1.0305 
 
 Oil303 
 
 .9704 
 
 .9991 
 
 .04 
 
 1.0408 
 
 Oil737 
 
 .9608 
 
 .9984 
 
 .05 
 
 1.0513 
 
 Oi2171 
 
 .9512 
 
 .9975 
 
 .06 
 
 1.0618 
 
 Oi2606 
 
 .9418 
 
 .9964 
 
 .07 
 
 1.0725 
 
 Oi3040 
 
 .9324 
 
 .9951 
 
 .08 
 
 1.0833 
 
 Oi3474 
 
 .9231 
 
 .9936 
 
 .09 
 
 1.0942 
 
 Oi3909 
 
 .9139 
 
 .9919- 
 
 .1 
 
 1.1052 
 
 Oi4343 .9048 
 
 .9900 
 
 .2 
 
 1.2214 
 
 Oi8686 
 
 .8187 
 
 .9608 
 
 .3 
 
 1.3499 
 
 0.1303 
 
 .7408 
 
 .9139 
 
 - .4 
 
 1.4918 
 
 0.1737 
 
 .6703 
 
 .8521 
 
 .5 
 
 1.6487 
 
 0.2171 
 
 .6065 
 
 .7788 
 
 .6 
 
 1.822 
 
 0.2606 
 
 .5488 
 
 .6977 
 
 .7 
 
 2.0138 
 
 0.3040 
 
 .4966 
 
 .6126 
 
 .8 
 
 2.2255 
 
 0.3474! .4493 
 
 .5273 
 
 .9 
 
 2.459 
 
 0.3909 
 
 .4066 
 
 .4449 
 
 1. 
 
 2718 
 
 04343 
 
 .3679 
 
 .3679 
 
 2. 
 
 7.389 
 
 0.8686 
 
 .1353 
 
 i!832 
 
 3. 
 
 20.086 
 
 1.3029 
 
 .0498 
 
 3 1234 
 
 4 
 
 54.598 
 
 1.7372 
 
 .018 
 
 6 1125 
 
 5. 
 
 148.41 
 
 2.171 
 
 .006 
 
 io!389 
 
 6. 
 
 403.43 
 
 2.605 
 
 ; .002 
 
 152320 
 
 7. 
 
 1096.6 
 
 3.04011 .000 
 
 2 i5196 
 
 8. 
 
 2981.0 
 
 3.4744 .000 
 
 271604 
 
 9. 
 
 8103.1 
 
 3.908" 
 
 r .000 
 
 36 8359 
 
 10. 
 
 22029. 
 
 4.342< 
 
 ) .000 
 
 si7203 
 
 N 
 
 
 
 1234 5 
 
 6789 
 
 11 
 
 9 
 
 2250 
 
 6985 
 
 12 
 
 8 
 
 9512 1 
 
 7019 
 
 13 
 
 4 
 
 7750 3 
 
 4281 
 
 14 
 
 3 
 
 2946 7 
 
 5269 
 
 15 
 
 6 
 
 8-192 3 
 
 2060 
 
 16 
 
 2 
 
 3204 8 
 
 1219 
 
 17 
 
 5 
 
 0355 4 
 
 1725 
 
 18 
 
 7 
 
 5752 7 
 
 1466 
 
 19 
 
 6 
 
 3060 3 
 
 6775 
 
 20 
 
 3 
 
 0503 6 
 
 7765 
 
 21 
 
 2 
 
 8481 4 
 
 5664 
 
 22 
 
 2 
 
 9505 8 
 
 1334 
 
 23 
 
 3 
 
 1962- 1 
 
 1550 
 
 24 
 
 1 
 
 2219 7 
 
 4050 
 
 25 
 
 4 
 
 7023 4 
 
 t 4320 
 
 26 
 
 7 
 
 40(50 I 
 
 5135 
 
 27 
 
 6 
 
 7765 
 
 > 1540 
 
 28 
 
 6 
 
 1592 4 
 
 t 7890 
 
 29 
 
 
 
 9875 '< 
 
 \ 9627 
 
 30 
 
 2 
 
 7147 ( 
 
 ) 2456 
 
 31 
 
 fl 
 
 5543 ] 
 
 L 063.9 
 
 32 
 
 
 1505 i 
 
 ? 2570 
 
 33 
 
 1 
 
 3445 i 
 
 t 4320 
 
 34 
 
 8 
 
 5396 '< 
 
 I 8383 
 
 35 
 
 7 
 
 1470 I 
 
 } 5789 
 
 36 
 
 
 
 1110 , 
 
 9 8753 
 
 37 
 
 
 
 7417 . 
 
 J 9494 
 
 38 
 
 6 
 
 2603 t 
 
 3 0135 
 
 39 
 
 6 
 
 8990 
 
 3 0987 
 
 40 
 
 6 
 
 4318 
 
 3 3962 
 
 41 
 
 8 
 
 4050 
 
 5 9481 
 
 42 
 
 
 5147 
 
 9 1346 
 
 43 
 
 r t 
 
 8899 
 
 9 9876 
 
 44 
 
 5 
 
 4208 
 
 3 3185 
 
 45 
 
 
 8406 
 
 1 6271 
 
 46 
 
 Q 
 
 0482 
 
 5 9257 
 
 47 
 
 1 
 
 2468 
 
 9 1234 
 
 48 
 
 i 
 
 5555 
 
 4 4321 
 
 49 
 
 
 
 5753 
 
 1 8630 
 
 50 
 
 
 4073 
 
 9 5162 
 
 51 
 
 
 2726 
 
 1 5937 
 
 52 
 
 
 4703 
 
 6 9136 
 
 53 
 
 
 9134 
 
 5 6789 
 
 54 
 
 
 0000 
 
 9987 
 
 55 
 
 
 5431 
 
 9 7631 
 
 N 
 
 1234 
 
 5 
 
 6789 
 
 56 9 
 
 6415 
 
 5 
 
 551 
 
 57 7 
 
 4051 
 
 7 
 
 535 
 
 58 3 
 
 5271 
 
 6 
 
 3452 
 
 59 6 
 
 0250 
 
 2 
 
 703 
 
 60 6 
 
 7024 
 
 6 
 
 002 
 
 61 3 
 
 4567 
 
 8 
 
 000 
 
 62 {, 
 
 0005 
 
 8 
 
 765 
 
 63 4 
 
 3200 
 
 7 
 
 420 
 
 64 i 
 
 6410 
 
 6 
 
 054 
 
 65 1 
 
 5515 
 
 4 
 
 730 
 
 66 A 
 
 0617 
 
 2 
 
 353 
 
 67 1 
 
 ' 2726 
 
 
 
 037 
 
 68 ] 
 
 5526 
 
 
 
 602 
 
 69 
 
 > 5136 
 
 8 
 
 365 
 
 70 ( 
 
 ) 2467 
 
 9 
 
 0235 
 
 71 t 
 
 1 7500 
 
 1 
 
 223 
 
 72 : 
 
 5 4444 
 
 4 
 
 4333 
 
 73 '< 
 
 5 2100 
 
 
 
 5764 
 
 74 I 
 
 5 2007 
 
 6 
 
 4205 
 
 75 t 
 
 ? 4207 
 
 5 
 
 2074 
 
 76 
 
 L 5520 
 
 6 
 
 3063 
 
 77 t 
 
 t 5284 
 
 
 
 6254 
 
 78 t 
 
 t 5162 
 
 7 
 
 2735 
 
 79 ; 
 
 J 5372 
 
 7 
 
 1605 
 
 80 . 
 
 -) 3726 
 
 
 
 4715 
 
 81 , 
 
 1 2692 
 
 
 
 0255 
 
 82 
 
 L 4703 
 
 fl 
 
 5135 
 
 83 < 
 
 ? 2057 
 
 9 
 
 1346 
 
 84 
 
 3 0134 
 
 6 
 
 7501 
 
 85 
 
 2 3456 
 
 7 
 
 7500 
 
 86 
 
 3 0111 
 
 r 
 
 2222 
 
 87 
 
 2 2211 
 
 \ 
 
 0000 
 
 .88 
 
 S 5765 
 
 
 3210 
 
 89 
 
 9 8654 
 
 c 
 
 1056 
 
 90 
 
 4 2197 
 
 f 
 
 3106 
 
 91 
 
 4 2975 
 
 2 
 
 0742 
 
 92 
 
 9 6307 
 
 t 
 
 1552 
 
 93 
 
 8 5255 
 
 
 5407 
 
 94 
 
 3 0517 
 
 3 
 
 0517 
 
 95 
 
 2 5405 
 
 
 
 6172 
 
 96 
 
 7 2535 
 
 <. 
 
 5352 
 
 97 
 
 7 2716 
 
 ( 
 
 5045 
 
 98 
 
 3 715C 
 
 i 
 
 5260 
 
 99 
 
 4 715i 
 
 L 
 
 6037 
 
 10 
 
 204C 
 
 
 1523 
 
 A subscript 4 means .0000; etc. 
 
 Before annexing an italic figure subtract 1 
 from the fourth decimal place. 
 
290 
 
 THEORY OF MEASUREMENTS 
 
 FOUR-PLACE LOGARITHMS 
 
 N 
 
 
 
 1234 
 
 5 
 
 6789 
 
 D 
 
 PP 
 
 10 
 
 0000* 
 
 0043* 0086* 0128* 0170* 
 
 0212* 
 
 0253* 0294* 0334* 0374* 
 
 40 
 
 
 43 
 
 42 
 
 41 
 
 40 
 
 11 
 
 0414 
 
 0453 0492 0531 0569 
 
 0607 
 
 0645 0682 0719 0755 
 
 37 
 
 1 
 
 4.3 
 
 4.2 
 
 4.1 
 
 4.0 
 
 12 
 
 0792 
 
 0828 0864 0899 0934 
 
 0969 
 
 1004 1038 1072 1106 
 
 33 
 
 2 
 
 8.6 8.4 
 
 8.2 
 
 8.0 
 
 13 
 
 1139 
 
 1173 1206 1239 1271 
 
 1303 
 
 1335 1367 1399 1430 
 
 31 
 
 3 
 
 12.9 12.6 
 
 12.3 
 
 12.0 
 
 14 
 
 1461 
 
 1492 1523 1553 1584 
 
 1614 
 
 1644 1673 1703 1732 
 
 29 
 
 4 
 
 17.2 16.8 
 
 16.4 
 
 16.0 
 
 
 
 
 
 
 
 5 
 
 21.5! 21.0 
 
 20.5 
 
 20.0 
 
 15 
 
 1761 
 
 1790 1818 1847 1875 
 
 1903 
 
 1,931 1959 1987 2014 
 
 27 
 
 6 
 
 25.8 25.2 24.6 
 
 24.0 
 
 
 
 
 
 
 
 7 
 
 30.1 
 
 29.4 
 
 28.7 
 
 28.0 
 
 16 
 
 2041 
 
 2068 2095 2122 2148 
 
 2175 
 
 2201 2227 2253 2279 
 
 25 
 
 8 
 
 34.4 
 
 33.6 
 
 32.8 
 
 32.0 
 
 17 
 
 2304 
 
 2330 2355 2380 2405 
 
 2430 
 
 2455 2480 2504 2529 
 
 24 
 
 g 
 
 38.7 
 
 37.8 
 
 36.9 
 
 36.0 
 
 18 
 
 2553 
 
 2577 2601 2625 2648 
 
 2672 
 
 2695 2718 2742 2765 
 
 23 
 
 
 19 
 
 2788 
 
 2810 2833 2856 2878 
 
 2900 
 
 2923 2945 2967 2989 
 
 21 
 
 
 
 
 
 
 
 
 
 39 
 
 38 
 
 37 
 
 36 
 
 20 
 
 3010 
 
 3032 3054 3075 3096 
 
 3118 
 
 3139 3160 3181 3201 
 
 21 
 
 1 
 
 3.9 
 
 3.8 
 
 3.7 
 
 3.6 
 
 
 
 
 
 
 
 
 7 8 
 
 7 ft 
 
 7 4 
 
 7 2 
 
 21 
 
 3222 
 
 3243 3263 3284 3304 
 
 3324 
 
 3345 3365 3385 3404 
 
 20 
 
 Q 
 
 i .0 
 117 
 
 / .O 
 11 4 
 
 i .rt 
 11 1 
 
 10 8 
 
 22 
 23 
 
 24 
 
 3424 
 3617 
 3802 
 
 3444 3464 3483 3502 
 3636 3655 3674 3692 
 3820 3838 3856 3874 
 
 3522 
 3711 
 3892 
 
 3541 3560 3579 3598 
 3729 3747 3766 3784 
 3909 3927 3945 3962 
 
 19 
 18 
 17 
 
 2 "*! iO CD 
 
 15^6 
 19.5 
 23.4 
 
 15.2 
 19.0 
 
 22.8 
 
 14.8 
 18.5 
 22.2 
 
 14^4 
 18.0 
 21.6 
 
 25 
 
 3979 
 
 3997 4014 4031 4048 
 
 4065 
 
 4082 4099 4116 4133 
 
 17 
 
 7 
 
 27.3 
 
 26.6 
 
 25.9 
 
 25.2 
 
 
 
 
 
 
 
 8 
 
 31.2 
 
 30.4 
 
 29.6 
 
 28.8 
 
 26 
 
 4150 
 
 4166 4183 4200 4216 
 
 4232 
 
 4249 4265 4281 4298 
 
 16 
 
 9 
 
 35.1 
 
 34.2 
 
 33.3 
 
 32.4 
 
 27 
 
 4314 
 
 4330 4346 4362 4378 
 
 4393 
 
 4409 4425 4440 4456 
 
 16 
 
 
 28 
 
 4472 
 
 4487 4502 4518 4533 
 
 4548 
 
 4564 4579 4594 4609 
 
 15 
 
 
 
 29 
 
 4624 
 
 4639 4654 4669 4683 
 
 4698 
 
 4713 4728 4742 4757 
 
 14 
 
 
 30 
 
 34 
 
 33 
 
 ax 
 
 30 
 
 4771 
 
 4786 4800 4814 4829 
 
 4843 
 
 4857 4871 4886 4900 
 
 14 
 
 1 
 
 2 
 
 3.5 
 
 7.0 
 
 3.4 
 
 6.8 
 
 3.3 
 
 6.6 
 
 3.2 
 
 6.4 
 
 31 
 32 
 
 4914 
 5051 
 
 4928 4942 4955 4969 
 5065 5079 5092 5105 
 
 4983 
 5119 
 
 4997 5011 5024 5038 
 5132 5145 5159 5172 
 
 13 
 
 13 
 
 3 
 4 
 
 10.5 
 14.0 
 
 10.2 9.9 
 13.6 13.2 
 
 9.6 
 12.8 
 
 33 
 
 5185 
 
 5198 5211 5224 5237 
 
 5250 
 
 5263 5276 5289 5302 
 
 13 
 
 5 
 
 17.5 
 
 17.0 16.5 
 
 16.0 
 
 34 
 
 5315 
 
 5328 5340 5353 5366 
 
 5378 
 
 5391 5403 5416 5428 
 
 13 
 
 6 
 
 21.0 
 
 20.4 19.8 
 
 19.2 
 
 
 
 
 
 
 
 7 
 
 24.5 
 
 23.8 23.1 
 
 22.4 
 
 35 
 
 5441 
 
 5453 5465 5478 5490 
 
 5502 
 
 5514 5527 5539 5551 
 
 12 
 
 8 
 
 28.0 
 
 27.2 ' 26.4 
 
 256 
 
 36 
 
 5563 
 
 5575 5587 5599 5611 
 
 5623 
 
 5635 5647 5658 5670 
 
 12 
 
 9 
 
 31.5 
 
 30.6| 29.7 
 
 28.8 
 
 37 
 
 5682 
 
 5694 5705 5717 5729 
 
 5740 
 
 5752 5763 5775 5786 
 
 12 
 
 
 38 
 
 5798 
 
 5809 5821 5832 5843 
 
 5855 
 
 5866 5877 5888 5899 
 
 12 
 
 
 31 
 
 30 
 
 29 
 
 28 
 
 39 
 
 5911 
 
 5922 5933 5944 5955 
 
 5966 
 
 5977 5988 5999 6010 
 
 11 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 31 
 
 3.0 
 
 2.9 
 
 2.8 
 
 40 
 
 6021 
 
 6031 6042 6053 6064 
 
 6075 
 
 6085 6096 6107 6117 
 
 11 
 
 2 
 
 6.2 
 
 6.0 
 
 5.8 
 
 5.6 
 
 
 
 
 
 
 
 3 
 
 9.3 
 
 9.0 
 
 8.7 
 
 8.4 
 
 41 
 
 6128 
 
 6138 6149 6160 6170 
 
 6180 
 
 6191 6201 6212 6222 
 
 10 
 
 4 
 
 12.4 
 
 19 n 
 
 11.6 
 
 11.2 
 
 42 
 43 
 
 6232 
 6335 
 
 6243 6253 6263 6274 
 6345 6355 6365 6375 
 
 6284 
 6385 
 
 6294 6304 6314 6325 
 6395 6405 6415 6425 
 
 10 
 10 
 
 5 
 
 f, 
 
 15.5\ 15.0 
 
 18 61 1S n 
 
 14^5 
 17 4 
 
 u!o 
 
 16 8 
 
 44 
 
 6435 
 
 6444 6454 6464 6474 
 
 6484 
 
 6493 6503 6513 6522 
 
 10 
 
 7 
 
 2L7 
 
 21.0 
 
 20^3 
 
 19.6 
 
 45 
 
 6532 
 
 6542 6551 6561 6571 
 
 6580 
 
 6590 6599 6609 6618 
 
 10 
 
 8 
 9 
 
 24.8 
 27.9 
 
 24.0 
 27.0 
 
 23.2 
 26.1 
 
 22.4 
 25.2 
 
 46 
 
 6628 
 
 6637 6646 6656 6665 
 
 6675 
 
 6684 6693 6702 6712 
 
 9 
 
 
 47 
 
 6721 
 
 6730 6739 6749 6758 
 
 6767 
 
 6776 6785 6794 6803 
 
 9 
 
 
 48 
 
 6812 
 
 6821 6830 6839 6848 
 
 6857 
 
 6866 6875 6884 6893 
 
 9 
 
 
 ZY 
 
 zo 
 
 20 
 
 24 
 
 49 
 
 6902 
 
 6911 6920 6928 6937 
 
 6946 
 
 6955 6964 6972 6981 
 
 9 
 
 1 
 
 2.7 
 
 2.6 
 
 2.5 
 
 2.4 
 
 50 
 
 6990 
 
 6998 7007 7016 7024 
 
 7033 
 
 7042 7050 7059 7067 
 
 g 
 
 2 
 3 
 
 5.4 
 8.1 
 
 5.2 
 
 7.8 
 
 5.0 
 
 7.5 
 
 4.8 
 7.2 
 
 51 
 
 7076 
 
 7084 7093 7101 7110 
 
 7118 
 
 7126 7135 7143 7152 
 
 8 
 
 4 
 
 10.8 
 
 10.4 
 
 10.0 
 
 9.6 
 
 52 
 
 7160 
 
 7168 7177 7185 7193 
 
 7202 
 
 7210 7218 7226 7235 
 
 8 
 
 5 
 
 13.5 
 
 13.0 
 
 12.5 
 
 12.0 
 
 53 
 
 7243 
 
 7251 7259 7267 7275 
 
 7284 
 
 7292 7300 7308 7316 
 
 8 
 
 B 
 
 16.2 
 
 15.6 
 
 15.0 
 
 14.4 
 
 54 
 
 7324 
 
 7332 7340 7348 7356 
 
 7364 
 
 7372 7380 7388 7396 
 
 8 
 
 7 
 
 18.9 
 
 18.2 
 
 17.5 
 
 16.8 
 
 
 
 
 
 
 
 8 
 
 21.6 
 
 20.8 
 
 20.0 
 
 19.2 
 
 55 
 
 7404 
 
 7412 7419 7427 7435 
 
 7443 
 
 7451 7459 7466 7474 
 
 8 
 
 9 
 
 24.3 
 
 23.4 
 
 22.5 
 
 21.6 
 
 * Interpolated values are given on another page. 
 
APPENDIX 
 
 291 
 
 FOUR-PLACE LOGARITHMS 
 
 N 
 
 
 
 1234 
 
 5 
 
 6789 
 
 D 
 
 PP 
 
 55 
 
 7404 
 
 7412 7419 7427 7435 
 
 7443 
 
 7451 7459 7466 7474 
 
 8 
 
 
 
 
 
 
 
 
 
 23 
 
 22 
 
 21 
 
 20 
 
 56 
 
 7482 
 
 7490 7497 7505 7513 
 
 7520 
 
 7528 7536 7543 7551 
 
 
 
 
 
 
 
 57 
 
 7559 
 
 7566 7574 7582 7589 
 
 7597 
 
 7604 7612 7619 7627 
 
 
 1 
 
 2.3 
 
 2.2 
 
 2.1 
 
 2.0 
 
 58 
 
 7634 
 
 7642 7649 7657 7664 
 
 7672 
 
 7679 7686 7694 7701 
 
 
 2 
 
 4.6 
 
 4.4 
 
 4.2 
 
 4.0 
 
 59 
 
 7709 
 
 7716 7723 7731 7738 
 
 7745 
 
 7752 7760 7767 7774 
 
 
 3 
 
 6.9 
 
 6.6 
 
 6.3 
 
 6.0 
 
 
 
 
 
 
 
 4 
 
 9.2 
 
 8.8 
 
 8.4 
 
 8.0 
 
 60 
 
 7782 
 
 7789 7796 7803 7810 
 
 7818 
 
 7825 7832 7839 7846 
 
 7 
 
 5 
 
 11.5 
 
 11.0 
 
 10.5 
 
 10.0 
 
 
 
 
 
 
 
 6 
 
 13.8 
 
 13.2 
 
 12.6 
 
 12.0 
 
 61 
 
 7853 
 
 7860 7868 7875 7882 
 
 7889 
 
 7896 7903 7910 7917 
 
 
 7 
 
 16.1 
 
 15.4 
 
 14.7 
 
 14.0 
 
 62 
 
 7924 
 
 7931 7938 7945 7952 
 
 7959 
 
 7966 7973 7980 7987 
 
 
 8 
 
 18.4 
 
 17.6 
 
 16.8 
 
 16.0 
 
 63 
 
 7993 
 
 8000 8007 8014 8021 
 
 8028 
 
 8035 8041 8048 8055 
 
 
 9 
 
 20.7 
 
 19.8 
 
 18.9 
 
 18.0 
 
 64 
 
 8062 
 
 8069 8075 8082 8089 
 
 8096 
 
 8102 8109 8116 8122 
 
 
 
 65 
 
 8129 
 
 8136 8142 8149 8156 
 
 8162 
 
 8169 8176 8182 8189 
 
 6 
 
 
 19 
 
 18 
 
 17 
 
 16 
 
 
 
 
 
 
 
 1 
 
 1 n 
 
 1 8 
 
 1 7 
 
 1 6 
 
 66 
 
 8195 
 
 8202 8209 8215 8222 
 
 8228 
 
 8235 8241 8248 8254 
 
 
 i 
 2 
 
 i - - 
 3 8 
 
 l.O 
 
 3 6 
 
 1. 1 
 3 4 
 
 3.2 
 
 67 
 
 8261 
 
 8267 8274 8280 8287 
 
 8293 
 
 8299 8306 8312 8319 
 
 
 3 
 
 5^7 
 
 5.4 
 
 
 4^8 
 
 68 
 
 8325 
 
 8331 8338 8344 8351 
 
 8357 
 
 8363 8370 8376 8382 
 
 
 
 7 6 
 
 7 2 
 
 6 8 
 
 64 
 
 69 
 
 8388 
 
 8395 8401 8407 8414 
 
 8420 
 
 8426 8432 8439 8445 
 
 
 5 
 
 9^5 
 
 9^0 
 
 8.5 
 
 s'.o 
 
 
 
 
 
 
 
 g 
 
 11 4 
 
 10 8 
 
 10 2 
 
 9 6 
 
 70 
 
 8451 
 
 8457 8463 8470 8476 
 
 8482 
 
 8488 8494 8500 8506 
 
 7 
 
 7 
 
 13^3 
 
 i2.'e 
 
 1L9 
 
 112 
 
 
 
 
 
 
 
 g 
 
 15 2 
 
 14 4 
 
 13 6 
 
 12.8 
 
 71 
 
 8513 
 
 8519 8525 8531 8537 
 
 8543 
 
 8549 8555 8561 8567 
 
 
 
 17.1 
 
 ICO 
 
 i c'o 
 
 144 
 
 72 
 
 8573 
 
 8579 8585 8591 8597 
 
 8603 
 
 8609 8615 8621 8627 
 
 
 
 
 1D.Z 
 
 1O.O 
 
 i^.*t 
 
 73 
 
 8633 
 
 8639 8645 8651 8657 
 
 8663 
 
 8669 8675 8681 8686 
 
 
 
 74 
 
 8692 
 
 8698 8704 8710 8716 
 
 8722 
 
 8727 8733 8739 8745 
 
 
 
 15 
 
 14 
 
 13 
 
 12 
 
 75 
 
 8751 
 
 8756 8762 8768 8774 
 
 8779 
 
 8785 8791 8797 8802 
 
 6 
 
 1 
 
 1.5 
 
 1.4 
 
 1.3 
 
 1.2 
 
 
 
 
 
 
 
 2 
 
 3.0 
 
 2.8 
 
 2.6 
 
 2.4 
 
 76 
 
 8808 
 
 8814 8820 8825 8831 
 
 8837 
 
 8842 8848 8854 8859 
 
 
 3 
 
 4.5 
 
 4.2 
 
 3.9 
 
 3.6 
 
 77 
 
 8865 
 
 8871 8876 8882 8887 
 
 8893 
 
 8899 8904 8910 8915 
 
 
 4 
 
 6.0 
 
 5.6 
 
 5.2 
 
 4.8 
 
 78 
 
 8921 
 
 8927 8932 8938 8943 
 
 8949 
 
 8954 8960 8965 8971 
 
 
 5 
 
 7.5 
 
 7.0 
 
 6.5 
 
 6.0 
 
 79 
 
 8976 
 
 8982 8987 8993 8998 
 
 9004 
 
 9009 9015 9020 9025 
 
 
 6 
 
 9.0 
 
 8.4 
 
 7.8 
 
 7.2 
 
 
 
 
 
 
 
 7 
 
 10.5 
 
 9.8 
 
 9.1 
 
 8.4 
 
 80 
 
 9031 
 
 9036 9042 9047 9053 
 
 9058 
 
 9063 9069 9074 9079 
 
 6 
 
 8 
 
 12.0 
 
 11.2 
 
 10.4 
 
 9.6 
 
 
 
 
 
 
 
 9 
 
 13.5 
 
 12.6 
 
 11.7 
 
 10.8 
 
 81 
 
 9085 
 
 9090 9096 9101 9106 
 
 9112 
 
 9117 9122 9128 9133 
 
 
 
 82 
 
 9138 
 
 9143 9149 9154 9159 
 
 9165 
 
 9170 9175 9180 9186 
 
 
 
 83 
 
 9191 
 
 9196 9201 9206 9212 
 
 9217 
 
 9222 9227 9232 9238 
 
 
 
 11 
 
 10 
 
 9 
 
 8 
 
 84 
 
 9243 
 
 9248 9253 9258 9263 
 
 9269 
 
 9274 9279 9284 9289 
 
 
 1 
 
 1.1 
 
 1.0 
 
 0.9 
 
 0.8 
 
 85 
 
 9294 
 
 9299 9304 9309 9315 
 
 9320 
 
 9325 9330 9335 9340 
 
 5 
 
 2 
 3 
 
 2.2 
 3.3 
 
 2.0 
 3.0 
 
 1.8 
 2.7 
 
 1.6 
 2.4 
 
 86 
 
 9345 
 
 9350 9355 9360 9365 
 
 9370 
 
 9375 9380 9385 9390 
 
 
 4 
 
 4.4 
 
 4.0 
 
 3.6 
 
 3.2 
 
 87 
 
 9395 
 
 9400 9405 9410 9415 
 
 9420 
 
 9425 9430 9435 9440 
 
 
 5 
 
 5.5 
 
 5.0 
 
 4.5 
 
 4.0 
 
 88 
 
 9445 
 
 9450 9455 9460 9465 
 
 9469 
 
 9474 9479 9484 9489 
 
 
 6 
 
 6.6 
 
 6.0 
 
 5.4 
 
 4.8 
 
 89 
 
 9494 
 
 9499 9504 9509 9513 
 
 9518 
 
 9523 9528 9533 9538 
 
 
 7 
 
 7.7 
 
 7.0 
 
 6.3 
 
 5.6 
 
 90 
 
 9542 
 
 9547 9552 9557 9562 
 
 9566 
 
 9571 9576 9581 9586 
 
 4 
 
 8 
 9 
 
 8.8 
 9.9 
 
 8.0 
 9.0 
 
 7.2 
 8.1 
 
 6.4 
 
 7.2 
 
 91 
 
 9590 
 
 9595 9600 9605 9609 
 
 9614 
 
 9619 9624 9628 9633 
 
 
 
 92 
 
 9638 
 
 9643 9647 9652 9657 
 
 9661 
 
 9666 9671 9675 9680 
 
 
 
 i 
 
 
 
 
 . 
 
 93 
 
 9685 
 
 9689 9694 9699 9703 
 
 9708 
 
 9713 9717 9722 9727 
 
 
 
 
 
 
 
 94 
 
 9731 
 
 9736 9741 9745 9750 
 
 9754 
 
 9759 9763 9768 9773 
 
 
 1 
 
 0.7 
 
 0.6 
 
 0.5 
 
 0.4 
 
 95 
 
 9777 
 
 9782 9786 9791 9795 
 
 9800 
 
 9805 9809 9814 9818 
 
 5 
 
 2 
 3 
 
 1.4 
 2.1 
 
 1.2 
 1.8 
 
 1.0 
 1.5 
 
 0.8 
 1.2 
 
 96 
 
 9823 
 
 9827 9832 9836 9841 
 
 9845 
 
 9850 9854 9859 9863 
 
 
 4 
 
 2.8 
 
 2.4 
 
 2.0 
 
 1.6 
 
 97 
 
 9868 
 
 9872 9877 9881 9886 
 
 9890 
 
 9894 9899 9903 9908 
 
 
 5 
 
 3.5 
 
 3 
 
 2.5 
 
 2.0 
 
 98 
 
 9912 
 
 9917 9921 9926 9930 
 
 9934 
 
 9939 9943 9948 9952 
 
 
 6 
 
 4.2 
 
 3.6 
 
 3.0 
 
 2.4 
 
 99 
 
 9956 i 9961 9965 9969 9974 
 
 9978 
 
 9983 9987 9991 9996 
 
 
 7 
 
 4.9 
 
 4.2 
 
 3.5 
 
 2.8 
 
 
 
 
 
 
 8 
 
 5.6 
 
 4.8 
 
 4.0 
 
 3.2 
 
 100 
 
 0000 0004 0009 0013 0017 
 
 0022 
 
 0026 0030 0035 0039 
 
 4 
 
 9 
 
 6.3 
 
 5.4 
 
 4.5 
 
 3.6 
 
 For natural logarithms see page 114. 
 
292 
 
 THEORY OF MEASUREMENTS 
 
 SQUARES 
 
 CIRCULAR FUNCTIONS 
 
 n 
 
 .67449^ 
 
 21762 
 
 24234 
 
 29070 
 
 196 
 
 40061 
 
 52810 
 59843 
 J' S2 67317 
 
 lo O>i^t T^OQfi 
 
 19 361 75JdU 
 
 83583 
 
 20 4( 
 
 21 4< 
 
 90 C9Q 
 
 ? =7A 
 24 57o 
 
 26 676 
 
 97 790 
 
 27 29 
 
 12139 
 
 16623 
 
 g 17854 
 19129 
 
 91011 
 
 61 yti 
 
 32 1024 
 
 33 1089 
 
 34 1156 s 
 
 oc 199K -&blOO 
 
 ?fi Joofi 27702 
 36 129 29284 
 
 38 1444 
 
 39 1521 
 
 40 1600 
 
 41 1681 
 
 42 1764 
 
 43 1849 
 
 44 1936 
 
 45 2025 
 
 46 2116 
 
 47 2209 
 
 48 2304 
 
 49 2401 
 
 50 2500 
 
 51 2601 
 
 52 2704 
 
 53 2809 
 54 
 
 55 
 
 56 3136 
 
 57 3249 
 
 58 3364 
 
 59 3481 
 
 34296 
 
 36054 
 37857 
 
 AKCf\a 
 
 455UO 
 
 53859 
 56057 
 
 Tni cri 
 
 70 ,16?. 
 J2675 
 
 I,* 22 * 
 
 .674491 
 
 60 3600 I 78 ^ 
 
 61 3721 
 
 62 3844 
 
 co ^QftQ 
 
 6| |096 gjjf 
 
 66 |356 9 4|04 
 
 69 4761 
 
 10314 
 10617 
 
 1J237 
 
 13893 
 
 70 4900 
 
 71 fn4.1 
 
 72 5184 
 
 12200 
 
 77 5929 
 
 78 6084 
 
 79 6241 
 
 80 6400 
 
 81 6561 
 
 82 6724 
 
 83 6889 
 
 84 7056 
 
 85 7225 
 
 86 7396 
 
 89 7921 
 
 90 8100 
 
 91 8281 
 
 92 8464 
 
 93 8649 
 
 94 8836 
 
 96 9216 
 
 97 9409 
 
 98 9604 
 
 99 9801 
 
 100 10000 
 
 101 10201 
 
 102 10404 
 
 103 10609 
 
 104 10816 
 
 105 11025 
 
 106 11236 
 
 107 11449 
 
 108 11664 
 
 109 11881 
 
 17607 
 
 20469 
 20896 
 21327 
 
 21762 
 
 1^547 
 24004 
 
 CONSTANTS 
 
 RAD DEG TAN SIN g^^OS C S C T 
 
 i'=io- 
 
 2.9085 
 
 0000 
 
 0000 0000 - oo O t 1 
 
 90 * 
 
 01745* 1 
 03491 2 
 05236 3 
 06981 4 
 
 0175 0175 2419 9999 9998 5729 
 0349 0349 5428 9997 9994 2864 
 0524 0523 7188 9994 9986 1908 
 0699 0698 8436 9989 9976 1430 
 
 89 15 
 88 15 
 87 15 
 86 15 
 
 08727 5 
 
 0875 0872 9403 9983 9962 1143 
 
 85 14 
 
 1047 6 
 1222 7 
 1396 8 
 1571 9 
 
 1051 1045 0192 9976 9945 9514 
 1228 1219 0859 9968 9925 8144 
 1405 1392 1436 9958 9903 7115 
 1584 1564 1943 9946 9877 6314 
 
 84 14 
 83 14 
 82 14 
 81 14 
 
 1745 10 
 
 1763 1736 2397 9934 9848 5671 
 
 80 13 
 
 1920 11 
 2094 12 
 2269 13 
 2443 14 
 
 1944 1908 2806 9919 9816 5145 
 2126 2079 3179 9904 9781 4705 
 2309 2250 3521 9887 9744 4331 
 2493 2419 3837 9869 9703 4011 
 
 79 13 
 
 78 13 
 77 13 
 76 13 
 
 2618 15 
 
 2679 2588 4130 9849 9659 3732 
 
 75 13 
 
 2793 16 
 2967 17 
 3142 18 
 3316 19 
 
 2867 2756 4403 9828 9613 3487 
 3057 2924 4659 9806 9563 3271 
 3249 3090 4900 9782 9511 3078 
 3443 3256 5126 9757 9455 2904 
 
 74 12 
 73 12 
 72 12 
 71 12 
 
 3491 20 
 
 3640 3420 5341 9730 9397 2747 
 
 70 12 
 
 3665 21 
 3840 22 
 4014 23 
 4189 24 
 
 3839 3584 5543 9702 9336 2605 
 4040 3746 5736 9672 9272 2475 
 4245 3907 5919 9640 9205 2356 
 4452 4067 6093 9607 9135 2246 
 
 69 12 
 68 11 
 67 11 
 66 11 
 
 4363 25 
 
 4663 4226 6259 9573 9063 2145 
 
 65 11 
 
 4538 26 
 4712 27 
 4887 28 
 5061 29 
 
 4877 4384 6418 9537 8988 2050 
 5095 4540 6570 9499 8910 1963 
 5317 4695 6716 9459 8829 1881 
 5543 4848 6856 9418 8746 1804 
 
 64 11 
 63 11 
 62 10 
 61 10 
 
 5236 30 
 
 5774 5000 6990 9375 8660 1732 
 
 60 10 
 
 5411 31 
 5585 32 
 5760 33 
 5934 34 
 
 6009 5150 7118 9331 8572 1664 
 6249 5299 7242 9284 8480 1600 
 6494 5446 7361 9236 8387 1540 
 6745 5592 7476 9186 8290 1483 
 
 59 10 
 58 10 
 57 99 
 56 97 
 
 6109 35 
 
 7002 5736 7586 9134 8192 1428 
 
 55 95 
 
 6283 36 
 6458 37 
 6632 38 
 6807 39 
 
 7265 5878 7692 9080 8090 1376 
 7536 6018 7795 9023 7986 1327 
 7813 6157 7893 8965 7880 1280 
 8098 6293 7989 8905 7771 1235 
 
 54 94 
 53 92 
 52 90 
 51 89 
 
 6981 40 
 
 8391 6428 8081 8843 7660 1192 
 
 50 87 
 
 7156 41 
 7330 42 
 7505 43 
 7679 44 
 
 8693 6561 8169 8778 7547 1150 
 9004 6691 8255 8711 7431 1111 
 9325 6820 8338 8641 7314 1072 
 9657 6947 8418 8569 7193 1036 
 
 49 85 
 48 83 
 47 82 
 46 80 
 
 7854 45 
 
 1 7071 8495 8495 7071 1000 
 
 45 78 
 
 i"=io-x 
 
 4.84814 
 
 COT COS JjJjJ SIN TAN DEO RA 
 
 SYMBOL CONSTANT LOGARITHM 
 
 3.141593 0.4971499 
 
 /I 180 /ir=5717'45" 
 
 =57.29578 1.7581226 
 
 =3437'.747 3.5362739 
 
 =206264".80625 5.3144251 
 
 e 2.718282 0.4342945 
 
 M 0.4342945 9.6377843 
 
 *log T/180 = log .01745329 = 8.2418774. For sines and tangents of numerical angles see p. 1 
 
INDEX 
 
 a d, 211,220 
 Abridged division, 16 
 
 methods, 69 
 accuracy of, 67 
 
 multiplication, 20 
 Abscissa, 103 
 Abstract number, 39 
 Accidental errors, 191, 192 
 Accuracy, 170 
 
 of abridged methods, 67 
 
 of the average, 216 
 
 of calculation, 67 
 
 decimal, 58 
 
 finer degrees of, 174 
 
 ideal, 56 
 
 infinite, 171 
 
 and magnitude, 180 
 
 relative, 63 
 
 in special cases, 178 
 
 superfluous, 173 
 Addition, from left to right, 281 
 
 geometric, 90 
 Adjustment of errors, (see least 
 
 squares) 
 
 Advantage of dispersion, 214 
 Alphabet, Greek, 284 
 American wire gauge, 101 
 Analysis, graphic, 128 
 Angle(s), 37 
 
 complementary, 48 
 
 as a coordinate, 127 
 
 negative, 49 
 
 quadrant of, 126 
 
 small, 86 
 
 unit of, 37, 39 
 Angular velocity, 39 
 Antilogarithm(s), 74 
 
 with slide rule, 101 
 Appendix, tables, 275 
 Approximate calculations, 14 
 
 roots of an equation, 145 
 Arbitrary weighting, 226 
 Archimedes' principle, 223 
 Area by mensuration, 31 
 
 measurement of, 30 
 
 unit of, 27 
 
 Arithmetical mean, 202 
 Assignment of weights, 226 
 Associative law, 258 
 Assumptions, 143, 148 
 Asymmetrical frequency distribu- 
 tions, 199, 200 
 Asymmetry, 201, 210 
 Asymptote(s), 139, 148 
 Average, 191, 202, 206 
 
 accuracy of, 216 
 
 deviation, 212, 213 
 
 and other deviations, 239 
 
 as a least-square value, 240 
 
 by partition, 208 
 
 by symmetry, 207 
 
 weighted, 225 
 Axis, axes, 
 
 change of, 124 
 
 of a graph, 103 
 
 B 
 
 B. & S. wire gauge, 101 
 Balance, 33 
 
 beam arms, 89 
 
 and double weighing, 85 
 
 sensitiveness, 146 
 Balancing columns, 270 
 Base, logarithmic, 71 
 Base-line measurement, 172 
 Beam balance, 33 
 
 double weighing, 85 
 
 sensitiveness, 146 
 
 unequal arms, 89 
 Bilocular frequency distribution, 
 
 199, 200 
 Binomial expansion, 201 
 
 theorem, 10 
 Black-thread method, 131 
 
 and least squares, 240 
 
 for proportionality, 135 
 Body temperature, 110 
 Boyle's law, 137 
 Broken line, 149 
 
 293 
 
294 
 
 INDEX 
 
 C.G.S. system, 24 
 
 advantages of, 39 
 Calculation, accuracy of, 67 
 of dispersion, 215 
 possible error after, 176 
 probable error after, 257, 261 
 Caliper, vernier, 184 
 Centimetre, 25 
 cubic, 27 
 square, 27 
 
 Change of graphic axes, 124 
 proportionality of, 130, 143 
 of scales, 125 
 of variables, 124, 125, 137 
 Characteristic, 75, 279 
 
 deviations, 274 
 Chauvenet's criterion, 231 
 
 table of values, 234 
 Choice of graphic scales, 108 
 
 of means, 203 
 Circle, elementary, 143 
 equation of, 124, 125 
 Circular functions, 47 
 
 general definitions, 125 
 with slide rule, 100 
 table of, 282 
 measure, 37 
 
 inverse, 275 
 Circumference, 55 
 Class interval, 198 
 Classification of errors, 191 
 Coarse measurements, 189 
 Coefficient of expansion, 130 
 Coincidence, measurement by, 181 
 method of, 180, 182 
 principle of, 180 
 Common logarithms, 71 
 Comparison of characteristic de- 
 viations, 239 
 
 Complementary angle(s), 48 
 Compound errors, 257, 261 
 
 units, 15 
 
 Condensed graphic scales, 107 
 Condition, 135 
 Conditioned measurements, 188 
 
 and least squares, 239 
 Consecutive equal intervals, 245 
 
 249 
 
 Constant(s), 7 
 error(s), 191, 192, 271 
 
 by least squares, 244 
 table of 282 
 
 Continuity of velocity, 156 
 Contour, 162 
 interval, 165 
 lines, 162 
 
 of earth's surface, 164 
 of hyperbolic paraboloid, 164 
 map, 
 
 construction, 166 
 use, 164 
 Convention of coordinate signs, 
 
 158 
 
 Coordinates, polar, 127 
 rectilinear, 127 
 in space, 158 
 
 mathematical, 158 
 physical, 158 
 Cosecant, 47, 125 
 Cosine, 47, 125 
 Cotangent, 47, 125 
 Criterion, criteria, 
 Chauvenet's, 231 
 of rejection, 230 
 of systematic errors, 259 
 Wright's, 213 
 Crude measurements, 189 
 Cube root with slide rule, 101 
 Curve(s) (see also graphic dia- 
 grams) 
 
 drawing with ink, 111 
 and equation, 115, 117 
 of errors, 138, 139 
 exponential, 138, 139 
 interpolation along, 147 
 intersection of, 127 
 logarithmic, 114, 143 
 with new axes, 124 
 probability, 123, 200 
 of sines, 112 
 of tangents, 112 
 typical, 136, 138 
 Cylinder points, 100 
 
 D 
 
 d, 211 
 
 di, d av , 219 
 
 A, 120 
 
 Ay/Ax = k, 121, 130 
 
 Danger in smoothing graphs, 111 
 
 Datum, data, 
 
INDEX 
 
 295 
 
 insufficient, 148 
 
 plane, 166 
 Decimal (s), 
 
 accuracy, 58 
 
 negative, 253 
 
 places, 65 
 
 written as units, 215 
 Decimetre ( = 10 cm.), 44 
 Definition, sharp, 189 
 Degree, 37 
 Delta (see also difference) 
 
 5, 83 
 
 A, 120 
 
 Ay/Ax = k, 130 
 Density, by Hare's method, 270 
 
 measurement of, 34, 39, 223 
 direct, 255 
 
 unit of, 27 
 
 of water, 133, 285 
 Dependent measurements, 188, 230 
 Derived units, 24 
 Deviation(s), 206 
 
 average, 212, 213 
 
 characteristic, 211 
 compared, 238, 274 
 
 and dispersion, 211 
 
 large, 231 
 
 (root-) mean-square, 213 
 
 standard, 212 
 Diagonal scale, 41 
 Diagram, frequency, 197 
 
 graphic, 103 (see also graphic) 
 Diameters of wires, 101 
 Difference(s), in coordinate values, 
 120, 130 
 
 possible error of, 176 
 
 probable error of, 257 
 
 relative, 66 
 
 small, 85, 123 
 
 tabular, 76 
 Direct measurements, 188 
 
 of density, 255 
 Directrix of a parabola, 140 
 Disagreement of measurements, 
 
 189 
 
 Dispersion (s), 213 (see also prob- 
 able error) 
 of average (s), 218 
 and other characteristic devia- 
 tions, 239 
 and deviation, 211 
 
 fractional, 220 
 
 of individual measurements, 219, 
 235 
 
 proportional, 220 
 
 relative, 220 
 
 with slide rule, 217 
 
 table of, 216, 292 
 
 and weighting, 227 
 
 of zero value, 220 
 Distance, between two points, 140 
 
 from (xi, yi) to y = a + bx, 240 
 Distortion of a graph, 125 
 Distribution, frequency, 194, 197 
 Distributive law, 260 
 Division, abridged, 20 
 
 with slide rule, 95 
 Double position, 145 
 
 weighing, 85 
 Drawing curved lines, 111 
 
 e, 171 (see also constants') 
 Earth's quadrant, 26 
 
 surface, contours of, 164 
 Element, of a smooth curve, 143, 
 
 144 
 Elementary circle, 143 
 
 straightness, 144 
 Ellipse, 125 
 
 Equal intervals, 245, 249 
 Equation(s), (see also graph) 
 
 with changed graphic axes, 124 
 
 of a circle, 124 
 
 of an ellipse, 125 
 
 graph of, 116 
 
 of a graph, 123 
 
 interpretation of, 128 
 
 linear, 129 
 
 intercept form, 132 
 
 and locus, 115 
 
 normal, 251 
 
 approximate, 253 
 
 used as a noun, 123 
 
 of a parabola, 133 
 
 plotting of, 118 
 
 polar, 127 
 
 roots of, 122 
 
 simultaneous, 127 
 
 of a standard, 26 
 
 transcendental, 144 
 
 of two loci, 149 
 
296 
 
 INDEX 
 
 Equivalent (s) for slide rule, 97 
 
 table of, 283 
 
 weights and measures, 34, 283 
 Error(s), 56, 188 
 
 classification of, 191 
 
 constant, 191, 192, 271 
 
 curve of, 138, 139 
 
 of measurement, 191 
 
 periodic, 265, 266 
 
 possible, 175 
 
 probable, (see probable error) 
 
 progressive, 265, 270 
 
 relative, 64 
 
 size of, 284 
 
 sources of, 193, 284 
 
 systematic, 265 
 test for, 265 
 
 and variations, 194 
 Estimation, measurement by, 180 
 
 of tenths, 52 
 
 Expanded graphic scales, 107 
 Expansion, binomial, 201 
 
 by heat, 130 
 Exponential curve, 138, 139 
 
 variation, 154 
 
 Explanatory notes (tables), 277 
 Extrapolation, 142, 155 
 
 diagram, 154 
 
 F 
 
 F(x),f(x), 146 
 False position, 145 
 Figures, non-significant, 61 
 
 significant, 52, 60, 170 
 versus decimals, 65 
 
 uncertain, 173 
 
 Finer degrees of accuracy, 174 
 Finite differences, 120 
 Five-place logarithm table, 289 
 Fluid friction, 150 
 Focus of a parabola, 140 
 Form, intercept, 132 
 
 standard, 68 
 Formulae, table of, 282 
 Fourth dimension of space, 160 
 Fractional dispersion, 221 
 
 part, 15 
 
 Frequency diagram, 197, 236 
 asymmetrical, 199, 200 
 bilocular, 199, 200 
 J-shaped, 199, 200 
 
 rectangular, 199, 200 
 symmetrical, 199, 200 
 U-shaped, 199, 200 
 distribution, 194, 197 
 
 types of, 199 
 polygon, 195 
 Friction in pipes, 150 
 Function (s), 46 
 circular, 47 
 
 table of, 292 
 probability, 76 
 
 table of, 288, 292 
 of small angles, 86 
 Fundamental units, 24 
 
 G- 
 
 Gauge points, slide rule, 100 
 General sources of error, 284 
 Geometric addition, 90 
 
 mean, 202 
 
 Grade, grade angle, 22 
 Gradient, 22 (see also slope) 
 Gram, 27 
 
 Graph, (see graphic diagram) 
 Graphic analysis, 128 
 axes, change of, 124 
 determination of distance, 140 
 
 of line through two points, 140 
 diagram (s), 103 
 
 for Chauvenet's criterion, 236 
 
 equation of, 123 
 
 of an equation, 116 
 
 of a frequency distribution, 
 195 
 
 on logarithmic paper, 150, 151 
 
 of natural laws, 115 
 
 orientation .of, 105 
 
 of a product, 150 
 
 of propagated errors, 263 
 
 of proportionality, 129 
 
 shifting of, 124 
 
 of simultaneous equations, 127 
 
 slope of, 109 
 
 smoothing of, 110 
 
 stretching of, 125 
 
 of typical curves, 138 
 
 a = TIT*, 154 
 
 \'a(a +5) = a + 5/2, 156 
 
 Ax + By + C = 0, 121 
 
 ay = x\ 138, 139 
 
 ax = y*, 148 
 
INDEX 
 
 297 
 
 AT//AX = k, 120, 121, 130 
 
 log y = log a + 6 log x, 152 
 
 log y = log a + bx, 153 
 
 me mo = 2(av me), 204 
 
 p = 27r>/Z/0~, 154 
 
 0>(3) = 0, 146 
 
 ^>i(x, 2/)<p2(x, ?/) = 0, 150 
 
 p = I/cos 0, 127 
 
 p = 2h sin 0, 127 
 
 p = k, 127 
 
 s = wo* + %aP, 128 
 
 = fc, 127 
 
 x = z/ 2 , 148 
 
 x = m, 161 
 
 a; = a + by + ey*, 126 
 
 x/a + y[b +z[c = I, 168 
 
 x/m + y/n =1, 132 
 
 x 2 - I/ 2 + z = 0, 163 
 
 X 2 _|- ^2 = C 2 ? 124 
 
 a? + 2/ 2 = 0, 127 
 
 X 2 _ y2 - 0, 127 
 
 a;2 4. ^2 4 22 = 52 ? 162 
 f x 2 + j/ 2 + z 2 = 5 2 
 
 I 2 = fc, 112 
 
 x 2 + (iv - fc) 2 = /i 2 , 127 
 
 (x - a) 2 + (y - &) 2 = c 2 , 125 
 
 xy = a 2 , 138, 139 
 
 (x 2 ) (z2) = TT + log (sin x), 157 
 
 y = l/ x , y = 1/x 2 , 127 
 
 y = - l/x 2 , 127 
 
 y = 30/a;, j/ = 30/x 2 , 127 
 
 y = 3 + 23, 128 
 
 y = - l/5x 2 , 115 
 
 y = .99986 + .0000522x + 
 
 .0000069x2, 136 
 y = a + 6x, 120, 121, 129, 
 
 139, 139 
 
 y - a 4- b x -f ex 2 , 133 
 y = a + bx + ex 2 + dx 3 , 
 
 136 
 
 y = a + bx + k/x z , 157 
 y = ae mx t y = ak mx , 153 
 y = ax n , 151, 152 
 y = o 8 /3 s , 138, 139 
 j, = e *, j/ = e m:r , 138, 139 
 y = e- z2 , 123, 138, 139, 232 
 y = kx, 130 
 y = log x, 143 
 y = m, 161 
 j/ = ip(x), 146 
 j/ = <p(x, y, z), 162 
 
 y = x, 150 
 
 y = x log x, 144 
 
 y = Vx, 151, 152 
 
 y = a; 2 , 150 
 
 y = x 2 + 02, 161 
 
 2/ = z, 150, 151, 152 
 
 y = Z" - , 117, 126 
 
 2/ = x 4 , 2/ = z 10 , y = 
 
 y = x m , 150 
 y = x n , 150 
 1/2 = z(10 - z) 3 , 127 
 2/2 = x s , 127 
 2 = w, 161 
 z 2 /9 = z 2 + 3/ 2 , 169 
 
 interpolation, 147 
 
 'representation, 102 
 
 scales, 108 
 
 tables, 147 
 Gravity, specific, 36 
 
 of water, table, 285 
 Greek alphabet, 274 
 
 Half, rounding off, 29 
 Hand as a measure, 29 
 Handling algebraic decimals, 250 
 Hare's method (density), 270 
 Harmonic mean, 202, 210 
 
 motion, 156 
 
 Harmony of measurements, 189 
 Heat, specific, 235 
 Histogram, 195 
 
 modified, 236 
 Hooke's law, 147 
 Horizontal sections, 162, 163 
 Hydraulic friction, 150 
 Hyperbola, 127 
 
 rectangular, 138, 139 
 Hyperbolic paraboloid, 163 
 
 section of a surface, 163 
 
 Ideal accuracy, 56 
 
 Inclination, 22, 42 
 
 Independent measurements, 188 
 
 Indirect measurements, 188 
 and graphic errors, 262 
 and least squares, 249 
 and probable errors, 257, 261 
 
 Infinite accuracy, 171 
 
298 
 
 INDEX 
 
 Inflection, point of, 239 
 Inking curved lines, 111 
 Instantaneous velocity, 155 
 Insufficient data, 148 
 Integral, probability, 288 
 Integrity, observational, 230 
 Intercept form of equation, 132 
 International standards, 26 
 Interpolation, 142 
 
 along a curve, graphic, 147 
 
 linear, 143 
 
 logarithmic, 76 
 
 Interpretation of equations, 128 
 Interquartile range, 209 
 Intersection of two graphs, 127 
 Interval (s), class, 198 
 
 by coincidence, 186 
 
 contour, 165 
 
 and least squares, 245 
 Inverse tangents and circular 
 
 measure, 285 
 Irregular areas, 30 
 
 by mensuration, 31 
 
 surface, 163 
 Irregularities of small groups, 237 
 
 J-shaped frequency distribution, 
 199, 200 
 
 K 
 
 K.W.H., 149 
 Kilogram, 27 
 Kilometre, 26 
 Kilowatt-hour, 149 
 
 Large deviations, 231 
 Law(s), associative, 258 
 
 Boyle's, 137 
 
 of change, 115 
 
 of density of water, 133 
 
 distributive, 260 
 
 and graphs, 115 
 
 Hooke's, 147 
 
 linear, straight line, 129, 130 
 Lead shot, specific heat, 235 
 Least squares, 240 
 
 and average, 240 
 
 for black thread problem, 240 
 
 for consecutive equal intervals, 
 245 
 
 for a constant value, 242 
 
 for equal intervals, 245 
 
 for a linear law, 240 
 
 for a parabolic law, 242 
 
 for proportionality, 243 
 
 for uniform intervals, 245 
 Left-to-right addition, 281 
 Length, unit of, 25 
 Light, velocity of, 171 
 Line, broken, 149 
 
 initial, 127 
 
 law of, 130 
 
 straight, 119, 129, 138, 139 
 through origin, 131 
 through two points, 140 
 Linear equation, 129, 130 
 and black thread, 131 
 intercept form, 132 
 
 interpolation, 143 
 
 relationship, 137 
 Litre, 277 
 Locus, loci, 117 (see also graphic 
 
 diagram) 
 Logarithm (s), base of, 71 
 
 common, 71 
 
 natural, 71 
 
 properties of, 71 
 
 for reciprocals, 79 
 
 with slide rule, 101 
 
 tables, 1-place, 72 
 3-place, 73 
 4-place, 290, 291 
 5-place, 289 
 
 Logarithmic paper, 150, 151 
 Lower quartile, 208 
 
 M 
 
 Magnitude(s) and accuracy, 180 
 
 small, negligible, 80 
 Mantissa, 75 
 Mass, unit of, 27 
 
 true, 85 
 Mean(s), 202, 213 
 
 arithmetical, 202 
 
 choice of, 203 
 
 deviation, 213 
 
 geometrical, 202 
 
 harmonic, 202, 210 
 
 quadratic, 202 
 
 relation between, 204 
 Measure(s), circular, 37 
 
 equivalent, 34 
 
INDEX 
 
 299 
 
 of inclination, 42 
 
 numerical, 37 
 
 radian, 87 
 Measurement(s), 56 
 
 coarse, 189 
 
 by coincidence, 181 
 
 conditioned, 135, 188 
 and least squares, 240 
 
 crude, 189 
 
 of density, 223 
 
 dependent, 188, 230 
 
 direct, 188 
 
 disagreement of, 181 
 
 errors of, 191 
 
 by estimation, 180 
 
 harmony of, 189 
 
 independent, 188 
 
 indirect, 188 
 
 and probable errors, 257, 261 
 
 possible error of, 175 
 
 statement of, 219 
 
 true value of, 191 
 Median, 202 
 
 and average deviation, 212 
 
 and weights, 229 
 Mental approximations, 14 
 Meter rates, 149 
 Method of coincidence, 180, 182 
 Metre, 25 
 Metric convention, 25 
 
 equivalents, 283 
 Micron, 26 
 
 Microscope, resolving power, 171 
 Mid-quartile point, 237 
 Milky way, 171 
 Milligram, 27 
 Millilitre, 277 
 Millimetre, 26 
 Minimum 2 ( 2 ), 240 
 Minute of angle, 37 
 Mis-named units, 16 
 Mistakes, 191 
 
 Modal value, mode, 202, 203 
 Model vernier caliper, 183 
 Modulus, 218 
 
 Motion, simple harmonic, 156 
 Mound-like diagram, 198 
 Mountain pass, 164 
 Multiple, probable error of, 258 
 Multiplication, abridged, 20 
 with slide rule, 95 
 
 N 
 
 National prototype standards, 25 
 Natural laws and graphs, 115 
 
 logarithms, 71 
 
 number, 73 
 Negative angle, 49, 126 
 
 decimals, 253 
 
 and positive asymmetry, 201, 
 
 210 
 
 Neutral web, 25 
 Non-significant figures, 61 
 Normal equations, 251 
 approximate, 253 
 Numerator of zero, 174 
 Numerical measure, 37 
 
 O 
 
 Observations, weighting of, 223 
 Observational integrity, 230 
 Occasional large deviations, 231 
 Orbital period, 150 
 Ordinate(s), 103 
 
 summation of, 112, 133 
 Orientation of a graph, 105 
 Origin, 103 
 
 change of, 124 
 
 meaning of line through, 131 
 Overlining negative decimals, 253 
 
 7T , angle, 39 
 
 ratio, 55, 56, 171 
 7r-disc and least squares, 248 
 *>(*), 146 
 ^(x, y) = 0, 150 
 pt)-variation, 137 
 Paper, logarithmic, 150, 151 
 
 semi-logarithmic, 152, 153 
 Parabola, 122 
 
 arch-shaped, 122 
 
 festoon-shaped, 122 
 
 finding equation for, 133, 148 
 
 semi-cubical, 127 
 
 x = 2/ 2 , 148 
 Vx + Vl/ = V4, 112 
 
 y = a + bx + ex*, 122 
 
 y = - kx 2 , 116 
 
 y = mx*, 116, 122, 124 
 Parabolic section, 163 
 Paraboloid, 163 
 
300 
 
 INDEX 
 
 Parameter, 8 
 
 Partition, average by, 208 
 
 Per, 15 
 
 Percent slope, 23 
 
 Period of rotation, 150 
 
 Periodic errors, 265, 266 
 
 Physical arithmetic, 13 
 
 measurement, 56 
 Places, decimal, 65 
 Plane, datum, 166 
 Planimeter and least squares, 249 
 Plotting, principles of, 109, 118 
 
 tabular values, 109 
 Point(s), distance between two, 
 
 140 
 
 of inflection, 239 
 line through two, 140 
 mid-quartile, 237 
 Pole, 127 
 Polar coordinates, 127 
 
 equation, 127 
 Polygon, frequency, 195 
 Population and extrapolation, 55 
 Position, double, or false, 145 
 Positive and negative asymmetry, 
 
 201, 210 
 
 Possible error, 175 
 Power, probable error of, 260 
 
 resolving, 171 
 Prejudice, 54, 230 
 Principle of Archimedes, 223 
 
 of coincidence, 180 
 Probability function, calculation 
 
 of, 77 
 
 curve of, 123, 200 
 integral, table of, 288 
 Probable error(s), 117, 235 (see 
 
 also dispersion) 
 and other characteristic devia- 
 tions, 239 
 
 of indirect measurements, 261 
 associative law, 258 
 of a difference, 257 
 distributive law, 260 
 formulae for, 261 
 of a multiple, 258 
 of a power, 260 
 of a product, 259 
 of a quotient, 260 
 of a sum, 257 
 table of, 292 
 
 Product, possible error of, 176 
 
 probable error of, 259 
 
 of two equations, 150 
 Progressive errors, 265, 270 
 Propagation of errors, 255 
 Properties of deltas, 83 
 
 of logarithms, 71 
 
 Proportion with black thread, 135 
 and least squares, 243 
 
 with slide rule, 96 
 Proportional changes, 130, 143 
 
 dispersion, 221 
 Proportionality, 6 
 
 in a graph, 129 
 Prototype standards, 25, 191 
 Protractor, 40 
 
 Quadrant, 26 
 
 of an angle, 127 
 Quadratic mean, 202 
 Quantities, real and ideal, 57 
 Quartiles, upper and lower, 208 
 Quotient, possible error of, 177 
 
 probable error of, 260 
 
 R 
 
 r, P, 127 
 
 r.m.s., (see root-mean-square) 
 
 r* = z 2 + yi, 127 
 
 Radian measure, 87 
 
 Radius, 127 
 
 Range, interquartile, 209 
 
 semi-interquartile, 209 
 
 total, 211 
 
 Rates (of charge), 146, 149 
 Ratio(s), slide rule, 96, 186 
 Readiness-to-serve charge, 146 
 Reciprocals with logarithms, 79 
 
 with slide rule, 97 
 Rectangular frequency distribu- 
 tion, 199, 200 
 
 hyperbola, 127, 138, 139 
 Redetermination, 230 
 Rejection of a measurement, 230 
 Relationship, linear, 137 
 
 between means, 204 
 Relative accuracy, 63 
 
 difference, 66 
 
 dispersion, 220 
 
INDEX 
 
 301 
 
 errors, 64 
 
 Repeated values and weights, 225 
 Representative magnitude(s), 201 
 average as best, 206 
 mid-quartile point, 237 
 Resolving power of a microscope, 
 
 171 
 Root(s), cube, 101 
 
 by double position, 145 
 
 -mean-square deviation, 213 
 
 with slide rule, 98 
 
 square, table, 286, 287 
 
 of y = f(x), 145 
 Rotational period, 150 
 Rounding off, 28, 29, 59 
 Rule, slide, 90 
 
 for accuracy of average, 216 
 
 S, 218 
 
 Eva, = 0, 207 
 S(z> 2 ), 213, 240 
 Saddleback, 163, 164 
 Scale(s), diagonal, 41 
 
 graphic, change of, 125 
 choice of, 108 
 condensed, 107 
 expanded, 107 
 
 logarithmic, 90 
 
 reading, 93 
 Secant, 47, 125 
 Second, of angle, 37 
 
 of time, 28 
 Sections, plane, 162, 163 
 
 hyperbolic, 163 
 
 parabolic, 163 
 Semi-cubical parabola, 127 
 Semi-interquartile range, 209 
 
 and dispersion, 214 
 Semi-logarithmic paper, 152, 153 
 Sensitiveness and weighing, 146 
 Sharp definition, 189 
 Shifting a graphic curve, 124 
 Sigma notation, 218 
 Significant figures, 52, 60, 170 
 versus decimal places, 65 
 
 zero, 59, 61, 220 
 
 Significance of the dispersion, 213 
 Signs of three-dimensional co- 
 ordinates, 158 
 
 Simple harmonic motion, 156 
 Simultaneous equations, 127 
 
 indirect measurements, 249 
 Sine(s), 23, 44, 125 
 
 curve, 112 
 
 table, 292 
 Sinusoid, 112 
 Size of errors, 284 
 Skewedness, 201 
 Slide rule, 90 
 
 for dispersions, 217 
 ratios, 186 
 Slope, 22 
 
 of a graph, 109 
 
 of a straight line, 120 
 Small angle, 86 
 
 difference, 85, 143 
 
 groups of measurements, 237 
 
 magnitudes, 80 
 Smooth(ed) curve, 143, 144 
 Smoothing a curve, 110 
 
 with black thread, 131 
 Solution by double position, 145 
 Sources of error, 193, 284 
 Space of four dimensions, 160 
 
 of three dimensions, 158 
 Specific gravity, 36 
 of water, 285 
 
 heat, 235 
 
 volume, 36 
 
 Sphere, sections of, 162 
 Squares, least, (see least squares) 
 
 and roots, slide rule, 98 
 tables, 286, 287, 292 
 Standard(s) deviation, 212 
 and other deviations, 239 
 
 form, 68 
 
 of measurement, 25, 180 
 Statement of a measurement, 219 
 Statistics, 194, 197 
 Straight line, 119, 138, 139 
 law, 129, 130 
 slope of, 120 
 through two points, 140 
 Straightness, elementary, 144 
 Strained curve, 125 
 Stretching a graph, 125 
 Substantive use of an equation, 123 
 Substitution, 147 
 
 of x + p, y + q, 124, 126 
 
 of mx, x/a, ny, y/b, 125, 126 
 
302 
 
 INDEX 
 
 Subtraction, left to right, 281 
 
 Sum, possible error of, 176 
 probable error of, 257 
 
 Summation of ordinates, 112, 133 
 
 Superfluous accuracy, 173 
 
 Surface, irregular, 163 
 
 Surveyor's base line, accuracy of, 
 172 
 
 Symmetrical frequency distribu- 
 tion, 199, 200 
 
 Symmetry, average by, 207 
 of a curve, 123 
 
 Systematic errors, 265 
 test for, 259 
 
 6, 127 
 Tables, 275 
 
 characteristic deviations, 284 
 
 Chauvenet's criterion, 234 
 
 circular functions, 292 
 
 constants, 292 (see also ir) 
 
 6-formula3, 88 
 
 density of water, (134), 275 
 
 equivalents, 283 
 
 errors classified, 191 
 size of, 284 
 
 exponential functions, 288, 289 
 
 formulae, 282 
 
 general sources of error, 284 
 
 Greek alphabet, 284 
 
 logarithms, 1-place, 72 
 3-place, 73 
 4-place, 290, 291 
 5-place, 289 
 
 inverse tangents, 285 
 
 inverse circular measure, 285 
 
 natural logarithms, 114 
 
 probability integral, 288 
 
 probable errors (indirect), 261 
 
 sines of numerical angles, 113, 
 114 
 
 slide-rule equivalents, 97 
 
 squares, 286, 287, 292 
 
 tangents of numerical angles, 113 
 
 wire gauge, 101 
 Tabular difference, 76 
 
 values and graphs, 109 
 Tangent(s), 22, 42, 125 
 
 curve of, 112 
 
 inverse (table), 285 
 
 table of, 292 
 
 tan 6 = y/x, 127 
 Target diagram, 193 
 Temperature of the body, 110 
 Tenths, estimation of, 52 
 Test for systematic errors, 265 
 Thread method of smoothing, 131 
 Three-dimensional space, 158 
 Time, 28 
 
 intervals by coincidence, 186 
 Total range, 211 
 Transcendental equations, 144 
 Transformation to linear equa- 
 tion, 137 
 
 Tresca cross-section, 28 
 True value of a measurement, 191 
 Two points on a graph, distance, 
 
 140 
 line determined by, 140 
 
 loci, single equation of, 149 
 Typical curves, 136, 138 
 
 (see also line, parabola, etc.) 
 Types of frequency distribution, 
 
 199 
 
 U 
 U-shaped frequency distribution. 
 
 199, 200 
 
 Uncertain figures, 173 
 Uniform intervals, 245, 249 
 Unit(s), of angle, 37, 39 
 
 of area, 27 
 
 compound, 15 
 
 versus decimals, 215 
 
 of density, 27 
 
 derived, 24 
 
 fundamental, 24 
 
 of length, 25 
 
 of mass, 27 
 
 mis-named, 16 
 
 of time, 28 
 
 of volume, 27 
 
 of weight, 27 
 Upper quartile, 208 
 
 v, 206, 211 
 
 Values, physical and mathemat- 
 ical, 82 
 
 Variable(s), 8, 47 
 
 change of, 124, 125, 127 
 
INDEX 
 
 303 
 
 velocity, 155 
 Variates, 194 
 
 measurement of, 194 
 Variation (s), 8 
 
 of density (water), 133 
 
 and errors, 194 
 
 of measurements, 206 
 
 pv-, 137 
 
 Vector sum, 192 
 Velocity, angular, 39 
 
 continuity of, 156 
 
 by extrapolation, 155 
 
 of light, 171 
 
 at a point, 155 
 
 variable, 155 
 Vernier, 183 
 
 caliper, 184 
 Volume, irregular, 32 
 
 specific, 36 
 
 unit of, 27 
 
 of water, 27, 285 
 
 W 
 
 Water, density of, 133 
 and temperature, 285 
 
 mass and volume of, 27 
 
 specific gravity of, 285 
 Weighing, double, 85 
 
 and sensitiveness, 146 
 Weight(s), arbitrary, 226 
 
 and dispersion, 227 
 
 and measures, 34 
 
 for repeated values, 225 
 Weighted average, 225 
 Weighting of observations, 223 
 Wire gauge, 101 
 Wright's criterion, 238 
 
 x = k, graph of, 121 
 x-axis, 103 
 z-intercept, 132 
 a?2/-plane, zz-plane, 158 
 f x = p cos 6 
 \y = p sin6, 127 
 x 2 + 2/ 2 = c 2 , 124 
 (x - a) 2 + (y - 6) 2 = c 2 , 125 
 
 y = a + bx, 120, 121 
 \y=P sin 6 
 \x = p cos 6, 127 
 2/-axis, 103 
 ^/-intercept, 132 
 2/z-plane, 158 
 
 Z 
 
 z-axis, 158 
 
 Zero as a dispersion, 220 
 
 as a fraction, 174 
 
 numerator, 174 
 
 of a scale, 55 
 
 significant, 220 
 
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