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 = /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 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 = (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. *SJ 0) CO CM t 2 ?, OJ Ol 3 CM CM CO CO CO ts CM Ol CO Ol s 3 CM Ol 4 co 8 > o CO 0) O ^ CM CM CO Ol 6 10 CM to CM s 5 OJ CO 8 C^ 0* CO CM CO CO CM 01 a o fO CO CM 6 CM CO CO 5J % CO CM a s CM CO CO CO 7 CO 8 X#'' ! CM eu IS (0 CM CM s n 1 o 1 OJ G* en CM o <0 to 8 CO CO or, , CO CM CM CO CO PJ co 9 to f) 8 00 O CO * CO CO o CO g s in CO in CO 10 co CO r-t CO s & (a CM -^ co ^> 4--- CO 01 CO 11 CO CO CM co co \" co CO .^ x< CO CO 12 8 TO co O 5 3 co m '' CO \ s co CO o CO *"~N 13 CO 51 n CO * 5 %> CO C- BD s CO CO 2 ^ 5 CM * o> CO >J CO 15 $ CO t-- CM CO CO ; s 5 co fe '"co TO '"in CO 16 8 5 m co CO CD ^ CO s co' CO co , in 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? 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 ( 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/)(*), 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 THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $1.OO ON THE SEVENTH DAY OVERDUE. - ;5 IQ 1938 SEP 523 1938 Ofif IHW \- pjp ! NOV 6 |94i 2 ; J jj % S J ^ *. N. , u ff 2? i ,-f *--* 4 j ^ CO < w c 5 u rj < ? tt ~> N l 1 " 2 '2 LD 21-95m-7,'37 I I 377559 .-.o UNIVERSITY OF CALIFORNIA LIBRARY