iff *r. 
 
ISO 
 
HANDBOOK OF MATHEMATICS 
 FOR ENGINEERS 
 

 % Qraw~3/ill Book (n. 7ne 
 
 PUBLISHERS OF BOOKS F O R_^ 
 
 Coal Age v Electric Railway Journal 
 Electrical World * Engineering. News-Record 
 American Machinist v The Contractor 
 Engineering 8 Mining Journal ^ Power 
 Metallurgical 6 Chemical Engineering 
 Electrical Merchandising 
 
Handbook of Mathematics 
 for Engineers 
 
 BY 
 
 EDWARD V. HUNTINGTON, PH. D. 
 
 ASSOCIATE PROFESSOR OF MATHEMATICS, HARVARD UNIVERSITY 
 
 WITH TABLES OF WEIGHTS AND MEASURES BY 
 
 LOUIS A. FISCHER, B 
 
 CHIEF OF DIVISION OF WEIGHTS AND 
 
 U. 8. BUREAU OF STANDARDS 
 
 ;. S. ^rrTT 
 
 MEASURES, 4 
 
 REPRINT OF SECTIONS 1 AND 2 OF L. S. MARKS'S 
 "MECHANICAL ENGINEERS' HANDBOOK" 
 
 FIRST EDITION 
 SECOND IMPRESSION 
 
 McGRAW-HILL BOOK COMPANY, INC, 
 
 239 WEST 39TH STREET. NEW YORK 
 
 LONDON: HILL PUBLISHING CO., LTD. 
 
 6 & 8 BOUVERIE ST., E.G. 
 1918 
 
/ 
 
 c 
 
 COPYRIGHT, 1918, BY THE 
 McGRAW HILL BOOK COMPANY, INC. 
 
 COPYRIGHT, 1916, BY 
 EDWARD V. HUNTINGTON. 
 
 TH M A F L 1C X> R IB H H YOKJC PA 
 
PREFACE 
 
 This Handbook of Mathematics is designed to contain, in compact form, 
 accurate statements of those facts and formulas of pure mathematics which 
 are most likely to be useful to the worker in applied mathematics. 
 
 It is not intended to take the place of the larger compendiums of pure 
 mathematics on the one hand, or of the technical handbooks of engineering 
 on the other hand; but in its own field it is thought to be more comprehensive 
 than any other similar work in English. 
 
 Many topics of an elementary character are presented in a form which 
 permits of immediate utilization even by readers who have had no previous 
 acquaintance with the subject; for example, the practical use of logarithms 
 and logarithmic cross-section paper, and the elementary parts of the modern 
 method of nomography (alignment charts), can be learned from this book 
 without the necessity of consulting separate treatises. 
 
 Other sections of the book to which special attention may be called are 
 the chapter on the algebra of complex (or imaginary) quantities, the treat- 
 ment of the catenary (with special tables), and the brief resume of the theory 
 of vector analysis. 
 
 The mathematical tables (including several which are not ordinarily 
 found) are carried to four significant figures throughout, and no pains have 
 been spared to make them as nearly self-explanatory as possible, even to the 
 reader who makes only occasional use of such tables. 
 
 For the Tables of Weights and Measures, which add greatly to its useful- 
 ness, the book is indebted to Mr. Louis A. Fischer of the U. S. Bureau of 
 Standards. 
 
 All the matter included in the present volume was originally prepared for 
 the Mechanical Engineers' Handbook (Lionel S. Marks, Editor-in-Chief), 
 and was first printed in 1916, as Sections 1 and 2 of that Handbook. The 
 author desires to express his indebtedness to Professor Marks, not only for 
 indispensable advice as to the choice of the topics which would be most 
 useful to engineers, but also for great assistance in many details of the 
 presentation. 
 
 All the misprints that have been detected have been corrected in the plates. 
 Notification in regard to any further corrections, and any suggestions toward 
 the improvement or possible enlargement of the book, will be cordially 
 welcomed by the author or the publishers. 
 
 E. V. H. 
 
 CAMBRIDGE, MASS. 
 April 29, 1918. 
 
 M171600 
 
CONTENTS 
 
 Page 
 PREFACE v 
 
 SECTION 1. Mathematical Tables and Weights and Measures 1 
 
 (For detailed Table of Contents, see page 1.) 
 
 SECTION 2. Mathematics: 
 
 Arithmetic; Geometry and Mensuration; Algebra; Trigonometry; 
 Analytical Geometry; Differential and Integral Calculus; Graphical 
 Representation of Functions; Vector Analysis 87 
 
 (For detailed Table of Contents, see page 87.) 
 INDEX . . 187 
 
SECTION 1 
 
 MATHEMATICAL TABLES 
 
 AND 
 
 WEIGHTS AND MEASURES 
 
 BY 
 EDWARD V. HUNTINGTON, Ph. D., Associate Professor of Mathematics, 
 
 Harvard University, Fellow Am. Acad. Arts and Sciences. 
 LOUISA. FISCHER, B. S., Chief of Division of Weights and Measures, 
 
 U. S. Bureau of Standards. 
 
 CONTENTS 
 
 MATHEMATICAL TABLES 
 
 BY E. V. HUNTINGTON PAGE 
 
 Squares of Numbers 2 
 
 Cubes of Numbers 8 
 
 Square Roots of Numbers 12 
 
 Cube Roots of Numbers 16 
 
 Three-halves Powers of Numbers. . . 22 
 
 Reciprocals of Numbers 24 
 
 Circles (Areas, Segments, etc.) . 28 
 
 Spheres (Volumes, Segments, etc.).. 36 
 
 Regular Polygons 39 
 
 Binomial Coefficients 39 
 
 Common Logarithms 40 
 
 Degrees and Radians 44 
 
 Trigonometric Functions 46 
 
 Exponentials 57 
 
 Hyperbolic (Napierian) Logarithms. 58 
 
 Hyperbolic Functions 60 
 
 Multiples of 0.4343 and 2.3026 62 
 
 Residuals and Probable Errors 63 
 
 Compound Interest and Annuities. 64 
 
 Decimal Equivalents 69 
 
 WEIGHTS AND MEASURES 
 
 BY LOUIS A. FISCHER PAGE 
 U. S. Customary Weights and 
 
 Measures 70 
 
 Metric Weights and Measures 71 
 
 Systems of Units 72 
 
 Conversion Tables: 
 
 Lengths 74 
 
 Areas 76 
 
 Volumes and Capacities 76 
 
 Velocities 78 
 
 Masses (Weights) 78 
 
 Pressures 79 
 
 Energy, Work, Heat 79 
 
 Power 81 
 
 Density 81 
 
 Heat Transmission and Con- 
 duction 82 
 
 Values of Foreign Coins 82 
 
 Time - 83 
 
 Terrestrial Gravity 84 
 
 Specific Gravity and Density 84 
 
MATHEMATICAL TABLES 
 
 SQUARES OF 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 ll 
 
 1.00 
 
 1.000 
 
 1.002 
 
 1.004 
 
 .006 
 
 1.008 
 
 1.010 
 
 1.012 
 
 1.014 
 
 1.016 
 
 1.018 
 
 2 
 
 1 
 
 1.020 
 
 1.022 
 
 1.024 
 
 .026 
 
 1.028 
 
 1.030 
 
 1.032 
 
 1.034 
 
 1.036 
 
 1.038 
 
 
 2 
 
 1.040 
 
 1.042 
 
 1.044 
 
 .047 
 
 1.049 
 
 1.051 
 
 1.053 
 
 1.055 
 
 1.057 
 
 1.059 
 
 
 3 
 
 1.061 
 
 1.063 
 
 1.065 
 
 .067 
 
 1.069 
 
 1.071 
 
 1.073 
 
 1.075 
 
 1.077 
 
 1.080 
 
 
 4 
 
 1.082 
 
 1.084 
 
 1.086 
 
 .088 
 
 1.090 
 
 1.092 
 
 1.094 
 
 1.096 
 
 1.098 
 
 1.100 
 
 
 1.05 
 
 1.102 
 
 1.105 
 
 1.107 
 
 .109 
 
 1.111 
 
 .113 
 
 1.115 
 
 1.117 
 
 .119 
 
 1.121 
 
 
 6 
 
 1.124 
 
 1.126 
 
 1.128 
 
 .130 
 
 1.132 
 
 .134 
 
 1.136 
 
 1.138 
 
 .141 
 
 1.143 
 
 
 7 
 
 1.145 
 
 1.147 
 
 1.149 
 
 .151 
 
 1.153 
 
 .156 
 
 1.158 
 
 1.160 
 
 .162 
 
 1.164 
 
 
 8 
 
 1.166 
 
 1.169 
 
 1.171 
 
 .173 
 
 1.175 
 
 .177 
 
 1.179 
 
 1.182 
 
 184 
 
 1.186 
 
 
 9 
 
 1.188 
 
 1.190 
 
 1.192 
 
 .195 
 
 1.197 
 
 .199 
 
 1.201 
 
 1.203 
 
 .206 
 
 1.208 
 
 
 1.10 
 
 1210 
 
 1.212 
 
 1.214 
 
 .217 
 
 1.219 
 
 1.221 
 
 1.223 
 
 1.225 
 
 .228 
 
 1.230 
 
 
 1 
 
 1.232 
 
 1.234 
 
 1.237 
 
 .239 
 
 1.241 
 
 1.243 
 
 1.245 
 
 1.248 
 
 .250 
 
 1.252 
 
 
 2 
 
 1.254 
 
 1.257 
 
 1.259 
 
 .261 
 
 1.263 
 
 1.266 
 
 1.268 
 
 1.270 
 
 .272 
 
 1.275 
 
 
 3 
 
 1.277 
 
 1.279 
 
 1.281 
 
 1.284 
 
 1.286 
 
 1.288 
 
 1.209 
 
 1.293 
 
 .295 
 
 1.297 
 
 
 4 
 
 1.300 
 
 1.302 
 
 1.304 
 
 1.306 
 
 1.309 
 
 1.311 
 
 1.313 
 
 1.316 
 
 .318 
 
 1.320 
 
 
 1.15 
 
 1.322 
 
 1.325 
 
 1.327 
 
 1.329 
 
 1.332 
 
 1.334 
 
 1.336 
 
 1.339 
 
 .341 
 
 1.343 
 
 
 6 
 
 1.346 
 
 1.348 
 
 1.350 
 
 1353 
 
 1.355 
 
 1.357 
 
 1.360 
 
 1.362 
 
 .364 
 
 1.367 
 
 
 7 
 
 1.369 
 
 1.371 
 
 1.374 
 
 1.376 
 
 1.378 
 
 1.381 
 
 1.383 
 
 1.385 
 
 .388 
 
 1.390 
 
 
 8 
 
 1.392 
 
 1.395 
 
 1.397 
 
 1.399 
 
 1.402 
 
 1.404 
 
 1.407 
 
 1.409 
 
 .411 
 
 1.414 
 
 
 9 
 
 1.416 
 
 1.418 
 
 1.421 
 
 1.423 
 
 1.426 
 
 1.428 
 
 1.430 
 
 1.433 
 
 .435 
 
 1.438 
 
 
 1.20 
 
 1.440 
 
 1.442 
 
 1.445 
 
 1.447 
 
 1.450 
 
 1.452 
 
 1.454 
 
 1.457 
 
 .459 
 
 1.462 
 
 
 1 
 
 1.464 
 
 1.467 
 
 1.469 
 
 1.471 
 
 1.474 
 
 1.476 
 
 1.479 
 
 1.481 
 
 .484 
 
 1.486 
 
 
 2 
 
 1.488 
 
 1.491 
 
 1.493 
 
 1.496 
 
 1.498 
 
 1.501 
 
 1.503 
 
 1.506 
 
 .508 
 
 1.510 
 
 
 3 
 
 1.513 
 
 1.515 
 
 1.518 
 
 1.520 
 
 1.523 
 
 1.525 
 
 1.528 
 
 1.530 
 
 .533 
 
 1.535 
 
 
 4 
 
 1.538 
 
 1.540 
 
 1.543 
 
 1.545 
 
 1.548 
 
 1.550 
 
 1.553 
 
 1.555 
 
 1.558 
 
 1.560 
 
 
 1.25 
 
 1.562 
 
 1.565 
 
 1.568 
 
 1.570 
 
 1.573 
 
 1.575 
 
 1.578 
 
 1.580 
 
 1.583 
 
 1585 
 
 3 
 
 6 
 
 1.588 
 
 1.590 
 
 1.593 
 
 1.595 
 
 1.598 
 
 1.600 
 
 1.603 
 
 1.605 
 
 1.608 
 
 1.610 
 
 
 7 
 
 1.613 
 
 1.615 
 
 1.618 
 
 1.621 
 
 1.623 
 
 .626 
 
 1.628 
 
 1.631 
 
 1.633 
 
 1.636 
 
 
 8 
 
 1.638 
 
 1.641 
 
 1.644 
 
 1.646 
 
 1.649 
 
 1.651 
 
 1.654 
 
 1.656 
 
 1.659 
 
 1.662 
 
 
 9 
 
 1.664 
 
 1.667 
 
 1.669 
 
 1.672 
 
 1.674 
 
 1.677 
 
 1.680 
 
 1.682 
 
 1.685 
 
 1.687 
 
 
 1.30 
 
 1.690 
 
 1.693 
 
 1.695 
 
 1.698 
 
 1.700 
 
 1.703 
 
 1.706 
 
 1.708 
 
 1.711 
 
 1.713 
 
 
 1 
 
 1.716 
 
 1.719 
 
 1.721 
 
 1.724 
 
 1.727 
 
 1.729 
 
 1.732 
 
 1.734 
 
 1.737 
 
 1.740 
 
 
 2 
 
 1.742 
 
 1.745 
 
 1.748 
 
 1.750 
 
 1.753 
 
 .756 
 
 1.758 
 
 1.761 
 
 1.764 
 
 1.766 
 
 
 3 
 
 1.769 
 
 1.772 
 
 1.774 
 
 1.777 
 
 1.780 
 
 1.782 
 
 1.785 
 
 1.788 
 
 1.790 
 
 1.793 
 
 
 4 
 
 1.796 
 
 1.798 
 
 1.801 
 
 1.804 
 
 1.806 
 
 1.809 
 
 1.812 
 
 1.814 
 
 1.817 
 
 1.820 
 
 
 1.35 
 
 1.822 
 
 1.825 
 
 1.828 
 
 1.831 
 
 1.833 
 
 1.836 
 
 1.839 
 
 1.841 
 
 1.844 
 
 1.847 
 
 
 6 
 
 1.850 
 
 1.852 
 
 1.855 
 
 1.858 
 
 1.860 
 
 1.863 
 
 1.866 
 
 1.869 
 
 1.871 
 
 1.874 
 
 
 7 
 
 1.877 
 
 1.880 
 
 1.882 
 
 1.885 
 
 1.888 
 
 1.891 
 
 1.893 
 
 1.896 
 
 1.899 
 
 1.902 
 
 
 8 
 
 1.904 
 
 1.907 
 
 1.910 
 
 1.913 
 
 1.915 
 
 1.918 
 
 1.921 
 
 1.924 
 
 1.927 
 
 1.929 
 
 
 9 
 
 1.932 
 
 1.935 
 
 1.938 
 
 1.940 
 
 1.943 
 
 1.946 
 
 1.949 
 
 1.952 
 
 1.954 
 
 1.957 
 
 
 1.40 
 
 1.960 
 
 1.963 
 
 1.966 
 
 1.968 
 
 1.971 
 
 1.974 
 
 1.977 
 
 1.980 
 
 1.982 
 
 1.985 
 
 
 1 
 
 1.988 
 
 1.991 
 
 1.994 
 
 1.997 
 
 1.999 
 
 2.002 
 
 2.005 
 
 2.008 
 
 2.011 
 
 2.014 
 
 
 2 
 
 2.016 
 
 2.019 
 
 2.022 
 
 2.025 
 
 2.028 
 
 2.031 
 
 2.033 
 
 2.036 
 
 2.039 
 
 2.042 
 
 
 3 
 
 2.045 
 
 2.048 
 
 2.051 
 
 2.053 
 
 2.056 
 
 2.059 
 
 2.062 
 
 2.065 
 
 2.068 
 
 2.071 
 
 
 4 
 
 2.074 
 
 2.076 
 
 2.079 
 
 2.082 
 
 2.085 
 
 2.088 
 
 2.091 
 
 2.094 
 
 2.097 
 
 2.100 
 
 
 1.45 
 
 2.102 
 
 2.105 
 
 2.108 
 
 2.111 
 
 2.114 
 
 2.117 
 
 2.120 
 
 2.123 
 
 2.126 
 
 2.129 
 
 
 6 
 
 2.132 
 
 2.135 
 
 2.137 
 
 2.140 
 
 2.143 
 
 2.146 
 
 2.149 
 
 2.152 
 
 2.155 
 
 2.158 
 
 
 7 
 
 2.161 
 
 2.164 
 
 2.167 
 
 2.170 
 
 2.173 
 
 2.176 
 
 2.179 
 
 2.182 
 
 2.184 
 
 2.187 
 
 
 8 
 
 2.190 
 
 2.193 
 
 2.196 
 
 2.199 
 
 2.202 
 
 2.205 
 
 2.208 
 
 2.211 
 
 2.214 
 
 2.217 
 
 
 9 
 
 2.220 
 
 2.223 
 
 2.226 
 
 2.229 
 
 2.232 
 
 2.235 
 
 2.238 
 
 2.241 
 
 2.244 
 
 2.247 
 
 
 Moving the decimal point ONE place in N requires moving it TWO places in body 
 of table (see p. 6). 
 
MATHEMATICAL TABLES 
 
 SQUARES (continued) 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 ii 
 
 1.50 
 
 2.250 
 
 2.253 
 
 2.256 
 
 2.259 
 
 2.262 
 
 2.265 
 
 2.268 
 
 2.271 
 
 2.274 
 
 2.277 
 
 3 
 
 I 
 
 2.280 
 
 2.283 
 
 2.286 
 
 2.289 
 
 2.292 
 
 2.295 
 
 2.298 
 
 2.301 
 
 2.304 
 
 2.307 
 
 
 2 
 
 2.310 
 
 2.313 
 
 2.316 
 
 2.320 
 
 2.323 
 
 2.326 
 
 2.329 
 
 2.332 
 
 2.335 
 
 2.338 
 
 
 3 
 
 2.341 
 
 2.344 
 
 2.347 
 
 2.350 
 
 2.353 
 
 2.356 
 
 2.359 
 
 2.362 
 
 2.365 
 
 2.369 
 
 
 4 
 
 2.372 
 
 2.375 
 
 2.378 
 
 2.381 
 
 2.384 
 
 2.387 
 
 2.390 
 
 2.393 
 
 2.396 
 
 2.399 
 
 
 1.55 
 
 2.402 
 
 2.406 
 
 2.409 
 
 2.412 
 
 2.415 
 
 2.418 
 
 2.421 
 
 2.424 
 
 2.427 
 
 2.430 
 
 
 6 
 
 2.434 
 
 2.437 
 
 2.440 
 
 2.443 
 
 2.446 
 
 2.449 
 
 2.452 
 
 2.455 
 
 2.459 
 
 2.462 
 
 
 7 
 
 2.465 
 
 2.468 
 
 2.471 
 
 2.474 
 
 2.477 
 
 2.481 
 
 2.484 
 
 2.487 
 
 2.490 
 
 2.493 
 
 
 8 
 
 2.496 
 
 2.500 
 
 2.503 
 
 2.506 
 
 2.509 
 
 2.512 
 
 2.515 
 
 2.519 
 
 2.522 
 
 2.525 
 
 
 9 
 
 2.528 
 
 2.531 
 
 2.534 
 
 2.538 
 
 2.541 
 
 2.544 
 
 2.547 
 
 2.550 
 
 2.554 
 
 2.557 
 
 
 1.60 
 
 2.560 
 
 2.563 
 
 2.566 
 
 2.570 
 
 2.573 
 
 2.576 
 
 2.579 
 
 2.582 
 
 2.586 
 
 2.589 
 
 
 1 
 
 2.592 
 
 2.595 
 
 2.599 
 
 2.602 
 
 2.605 
 
 2.608 
 
 2.611 
 
 2.615 
 
 2.618 
 
 2.621 
 
 
 2 
 
 2.624 
 
 2.628 
 
 2.631 
 
 2.634 
 
 2.637 
 
 2.641 
 
 2.644 
 
 2.647 
 
 2.650 
 
 2.654 
 
 
 3 
 
 2.657 
 
 2.660 
 
 2.663 
 
 2.667 
 
 2.670 
 
 2.673 
 
 2.676 
 
 2.680 
 
 2.683 
 
 2.686 
 
 
 4 
 
 2.690 
 
 2.693 
 
 2.696 
 
 2.699 
 
 2.703 
 
 2.706 
 
 2.709 
 
 2.713 
 
 2.716 
 
 2.719 
 
 
 1.65 
 
 2.722 
 
 2.726 
 
 2.729 
 
 2.732 
 
 2.736 
 
 2.739 
 
 2.742 
 
 2.746 
 
 2.749 
 
 2.752 
 
 
 6 
 
 2.756 
 
 2.759 
 
 2762 
 
 2.766 
 
 2.769 
 
 2.772 
 
 2.776 
 
 2.779 
 
 2.782 
 
 2.786 
 
 
 7 
 
 2.789 
 
 2.792 
 
 2.796 
 
 2.799 
 
 2.802 
 
 2.806 
 
 2.809 
 
 2.812 
 
 2.816 
 
 2.819 
 
 
 8 
 
 2.822 
 
 2.826 
 
 2.829 
 
 2.832 
 
 2.836 
 
 2.839 
 
 2.843 
 
 2.846 
 
 2849 
 
 2.853 
 
 
 9 
 
 2.856 
 
 2.859 
 
 2.863 
 
 2.866 
 
 2.870 
 
 2.873 
 
 2.876 
 
 2.880 
 
 2.883 
 
 2.887 
 
 
 1.70 
 
 2.890 
 
 2.893 
 
 2.897 
 
 2.900 
 
 2.904 
 
 2.907 
 
 2.910 
 
 2.914 
 
 2.917 
 
 2.921 
 
 
 1 
 
 2.924 
 
 2.928 
 
 2.931 
 
 2.934 
 
 2.938 
 
 2.941 
 
 2.945 
 
 2.948 
 
 2.952 
 
 2.955 
 
 
 2 
 
 2.958 
 
 2.962 
 
 2.965 
 
 2.969 
 
 2.972 
 
 2.976 
 
 2.979 
 
 2.983 
 
 2.986 
 
 2.989 
 
 
 3 
 
 2.993 
 
 2.996 
 
 3.000 
 
 3.003 
 
 3.007 
 
 3.010 
 
 3.014 
 
 3.017 
 
 3.021 
 
 3.024 
 
 
 4 
 
 3.028 
 
 3.031 
 
 3.035 
 
 3.038 
 
 3.042 
 
 3.045 
 
 3.049 
 
 3.052 
 
 3.056 
 
 3.059 
 
 
 1.75 
 
 3.062 
 
 3.066 
 
 3.070 
 
 3.073 
 
 3.077 
 
 3.080 
 
 3.084 
 
 3.087 
 
 3.091 
 
 3.094 
 
 4 
 
 6 
 
 3.098 
 
 3.101 
 
 3.105 
 
 3.108 
 
 3.112 
 
 3.115 
 
 3.119 
 
 3.122 
 
 3.126 
 
 3.129 
 
 
 7 
 
 3.133 
 
 3.136 
 
 3.140 
 
 3.144 
 
 3.147 
 
 3.151 
 
 3.154 
 
 3.158 
 
 3.161 
 
 3.165 
 
 
 .8 
 
 3.168 
 
 3.172 
 
 3.176 
 
 3.179 
 
 3.183 
 
 3.186 
 
 3.190 
 
 3.193 
 
 3.197 
 
 3-201 
 
 
 9 
 
 3.204 
 
 3.208 
 
 3.211 
 
 3.215 
 
 3.218 
 
 3.222 
 
 3.226 
 
 3.229 
 
 3.233 
 
 3.236 
 
 
 1.80 
 
 3.240 
 
 3.244 
 
 3.247 
 
 3.251 
 
 3.254 
 
 3.258 
 
 3.262 
 
 3.265 
 
 3.269 
 
 3.272 
 
 
 1 
 
 3.276 
 
 3.280 
 
 3.283 
 
 3.287 
 
 3.291 
 
 3.294 
 
 3.298 
 
 3.301 
 
 3.305 
 
 3.309 
 
 
 2 
 
 3.312 
 
 3.316 
 
 3.320 
 
 3.323 
 
 3.327 
 
 3.331 
 
 3.334 
 
 3.338 
 
 3.342 
 
 3.345 
 
 
 3 
 
 3.349 
 
 3.353 
 
 3.356 
 
 3.360 
 
 3.364 
 
 3.367 
 
 3.371 
 
 3.375 
 
 3.378 
 
 3.382 
 
 
 4 
 
 3.386 
 
 3.389 
 
 3.393 
 
 3.397 
 
 3.400 
 
 3.404 
 
 3.408 
 
 3.411 
 
 3.415 
 
 3.419 
 
 
 1.85 
 
 3.422 
 
 3.426 
 
 3.430 
 
 3.434 
 
 3.437 
 
 3.441 
 
 3.445 
 
 3.448 
 
 3.452 
 
 3.456 
 
 
 6 
 
 3.460 
 
 3.463 
 
 3.467 
 
 3.471 
 
 3.474 
 
 3.478 
 
 3.482 
 
 3.486 
 
 3.489 
 
 3.493 
 
 
 7 
 
 3.497 
 
 3.501 
 
 3.504 
 
 3.508 
 
 3.512 
 
 3.516 
 
 3.519 
 
 3.523 
 
 3.527 
 
 3.531 
 
 
 8 
 
 3.534 
 
 3.538 
 
 3.542 
 
 3.546 
 
 3.549 
 
 3.553 
 
 3.557 
 
 3.561 
 
 3.565 
 
 3.568 
 
 
 9 
 
 3.572 
 
 3.576 
 
 3.580 
 
 3.583 
 
 3.587 
 
 3.591 
 
 3.595 
 
 3.599 
 
 3.602 
 
 3.606 
 
 
 1.90 
 
 3.610 
 
 3.614 
 
 3.618 
 
 3.621 
 
 3.625 
 
 3.629 
 
 3.633 
 
 3.637 
 
 3.640 
 
 3.644 
 
 
 1 
 
 3.648 
 
 3.652 
 
 3.656 
 
 3.660 
 
 3.663 
 
 3.667 
 
 3.671 
 
 3.675 
 
 3.679 
 
 3.683 
 
 
 2 
 
 3.686 
 
 3.690 
 
 3.694 
 
 3.698 
 
 3.702 
 
 3.706 
 
 3.709 
 
 3.713 
 
 3.717 
 
 3.721 
 
 
 3 
 
 3.725 
 
 3.729 
 
 3.733 
 
 3.736 
 
 3.740 
 
 3.744 
 
 3.748 
 
 3.752 
 
 3.756 
 
 3.760 
 
 
 4 
 
 3.764 
 
 3.767 
 
 3.771 
 
 3.775 
 
 3.779 
 
 3.783 
 
 3.787 
 
 3.791 
 
 3.795 
 
 3.799 
 
 
 1.95 
 
 3.802 
 
 3.806 
 
 3.810 
 
 3.814 
 
 3.818 
 
 3.822 
 
 3.826 
 
 3.830 
 
 3.834 
 
 3.838 
 
 
 6 
 
 3.842 
 
 3.846 
 
 3.849 
 
 3.853 
 
 3.857 
 
 3.861 
 
 3.865 
 
 3.869 
 
 3.873 
 
 3.877 
 
 
 7 
 
 3.881 
 
 3.885 
 
 3.889 
 
 3.893 
 
 3.897 
 
 3.901 
 
 3.905 
 
 3.909 
 
 3.912 
 
 3.916 
 
 
 8 
 
 3.920 
 
 3.924 
 
 3.928 
 
 3.932 
 
 3.936 
 
 3.940 
 
 3.944 
 
 3.948 
 
 3.952 
 
 3.956 
 
 
 9 
 
 3.960 
 
 3.964 
 
 3.968 
 
 3.972 
 
 3.976 
 
 3.980 
 
 3.984 
 
 3.988 
 
 3.992 
 
 3.996 
 
 
 = 9.86960 !/* = 0.101321 
 
 7.38906 
 
MATHEMATICAL TABLES 
 
 SQUARES (continued) 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 $S 
 
 2.00 
 
 4.000 
 
 4.004 
 
 4.008 
 
 4.012 
 
 4.016 
 
 4.020 
 
 4.024 
 
 4.028 
 
 4.032 
 
 4.036 
 
 4 
 
 1 
 
 4.040 
 
 4.044 
 
 4.048 
 
 4.052 
 
 4.056 
 
 4.060 
 
 4.064 
 
 4.068 
 
 4.072 
 
 4.076 
 
 
 2 
 
 4.080 
 
 4.084 
 
 4.088 
 
 4.093 
 
 4.097 
 
 4.101 
 
 4.105 
 
 4.109 
 
 4.113 
 
 4.117 
 
 
 3 
 
 4.121 
 
 4.125 
 
 4.129 
 
 4.133 
 
 4.137 
 
 4.141 
 
 4.145 
 
 4.149 
 
 4. 153 
 
 4.158 
 
 
 4 
 
 4.162 
 
 4.166 
 
 4.170 
 
 4.174 
 
 4.178 
 
 4.182 
 
 4.186 
 
 4.190 
 
 4.194 
 
 4.198 
 
 
 2.05 
 
 4.202 
 
 4.207 
 
 4.211 
 
 4.215 
 
 4.219 
 
 4.223 
 
 4.227 
 
 4.231 
 
 4.235 
 
 4.239 
 
 
 6 
 
 4.244 
 
 4.248 
 
 4.252 
 
 4.256 
 
 4.260 
 
 4.264 
 
 4.268 
 
 4.272 
 
 4.277 
 
 4.281 
 
 
 7 
 
 4.285 
 
 4.289 
 
 4.293 
 
 4.297 
 
 4.301 
 
 4.306 
 
 4.310 
 
 4.314 
 
 4.318 
 
 4.322 
 
 
 8 
 
 4.326 
 
 4.331 
 
 4.335 
 
 4.339 
 
 4.343 
 
 4.347 
 
 4.351 
 
 4.356 
 
 4.360 
 
 4.364 
 
 
 9 
 
 4.368 
 
 4.372 
 
 4.376 
 
 4.381 
 
 4.385 
 
 4.389 
 
 4.393 
 
 4.397 
 
 4.402 
 
 4.406 
 
 
 2.10 
 
 4.410 
 
 4.414 
 
 4.418 
 
 4.423 
 
 4.427 
 
 4.431 
 
 4.435 
 
 4.439 
 
 4.444 
 
 4.448 
 
 
 1 
 
 4.452 
 
 4.456 
 
 4.461 
 
 4.465 
 
 4.469 
 
 4.473 
 
 4.477 
 
 4.482 
 
 4486 
 
 4.490 
 
 
 2 
 
 4.494 
 
 4.499 
 
 4.503 
 
 4.507 
 
 4.511 
 
 4.516 
 
 4.520 
 
 4.524 
 
 4.528 
 
 4533 
 
 
 3 
 
 4.537 
 
 4.541 
 
 4.545 
 
 4.550 
 
 4.554 
 
 4.558 
 
 4.562 
 
 4.567 
 
 4.571 
 
 4.575 
 
 
 4 
 
 4.580 
 
 4.584 
 
 4.588 
 
 4.592 
 
 4.597 
 
 4.601 
 
 4.605 
 
 4.610 
 
 4.614 
 
 4.618 
 
 
 2.15 
 
 4.622 
 
 4.627 
 
 4.631 
 
 4.635 
 
 4.640 
 
 4.644 
 
 4.648 
 
 4.653 
 
 4.657 
 
 4.661 
 
 
 6 
 
 4.666 
 
 4.670 
 
 4.674 
 
 4.679 
 
 4.683 
 
 4.687 
 
 4.692 
 
 4.696 
 
 4.700 
 
 4.705 
 
 
 7 
 
 4.709 
 
 4.713 
 
 4.718 
 
 4.722 
 
 4.726 
 
 4.731 
 
 4.735 
 
 4.739 
 
 4.744 
 
 4.748 
 
 
 8 
 
 4.752 
 
 4.757 
 
 4.761 
 
 4.765 
 
 4.770 
 
 4.774 
 
 4.779 
 
 4.783 
 
 4.787 
 
 4.792 
 
 
 9 
 
 4.796 
 
 4.800 
 
 4.805 
 
 4.809 
 
 4.814 
 
 4.818 
 
 4.822 
 
 4.827 
 
 4.831 
 
 4.836 
 
 
 2.20 
 
 4.840 
 
 4.844 
 
 4.849 
 
 4.853 
 
 4.858 
 
 4.862 
 
 4.866 
 
 4.871 
 
 4.875 
 
 4.880 
 
 
 1 
 
 4.884 
 
 4.889 
 
 4.893 
 
 4.897 
 
 4.902 
 
 4.906 
 
 4.911 
 
 4.915 
 
 4.920 
 
 4.924 
 
 
 2 
 
 4.928 
 
 4.933 
 
 4.937 
 
 4.942 
 
 4.946 
 
 4.951 
 
 4.955 
 
 4.960 
 
 4.964 
 
 4.968 
 
 
 3 
 
 4.973 
 
 4.977 
 
 4.982 
 
 4.986 
 
 4.991 
 
 4.995 
 
 5.000 
 
 5.004 
 
 5.009 
 
 5.013 
 
 
 4 
 
 5.018 
 
 5.022 
 
 5.027 
 
 5.031 
 
 5.036 
 
 5.040 
 
 5.045 
 
 5.049 
 
 5.054 
 
 5.058 
 
 
 2.25 
 
 5.062 
 
 5.067 
 
 5.072 
 
 5.076 
 
 5.081 
 
 5.085 
 
 5.090 
 
 5.094 
 
 5.099 
 
 5.103 
 
 5 
 
 6 
 
 5.108 
 
 5.112 
 
 5.117 
 
 5.121 
 
 5.126 
 
 5.130 
 
 5.135 
 
 5.139 
 
 5.144 
 
 5148 
 
 
 7 
 
 5.153 
 
 5.157 
 
 5.162 
 
 5.167 
 
 5.171 
 
 5.176 
 
 5.180 
 
 5.185 
 
 5.189 
 
 5.194 
 
 
 8 
 
 5.198 
 
 5.203 
 
 5.208 
 
 5.212 
 
 5.217 
 
 5.221 
 
 5.226 
 
 5.230 
 
 5.235 
 
 5.240 
 
 
 9 
 
 5.244 
 
 5.249 
 
 5.253 
 
 5.258 
 
 5.262 
 
 5.267 
 
 5.272 
 
 5.276 
 
 5.281 
 
 5.285 
 
 
 2.30 
 
 5.290 
 
 5.295 
 
 5.299 
 
 5.304 
 
 5.308 
 
 5.313 
 
 5.318 
 
 5.322 
 
 5.327 
 
 5.331 
 
 
 1 
 
 5.336 
 
 5.341 
 
 5.345 
 
 5.350 
 
 5.355 
 
 5.359 
 
 5.364 
 
 5.368 
 
 5.373 
 
 5.378 
 
 
 2 
 
 5.382 
 
 5.387 
 
 5.392 
 
 5.396 
 
 5.401 
 
 5.406 
 
 5.410 
 
 5.415 
 
 5.420 
 
 5.424 
 
 
 3 
 
 5.429 
 
 5.434 
 
 5.438 
 
 5.443 
 
 5.448 
 
 5.452 
 
 5.457 
 
 5.462 
 
 5.466 
 
 5.471 
 
 
 4 
 
 5.476 
 
 5.480 
 
 5.485 
 
 5.490 
 
 5.494 
 
 5.499 
 
 5.504 
 
 5.508 
 
 5.513 
 
 5.518 
 
 
 2.35 
 
 5.522 
 
 5.527 
 
 5.532 
 
 5.537 
 
 5.541 
 
 5.546 
 
 5.551 
 
 5.555 
 
 5.560 
 
 5.565 
 
 
 6 
 
 5.570 
 
 5.574 
 
 5.579 
 
 5.584 
 
 5.588 
 
 5.593 
 
 5.598 
 
 5.603 
 
 5.607 
 
 5.612 
 
 
 7 
 
 5.617 
 
 5.622 
 
 5.626 
 
 5.631 
 
 5.636 
 
 5.641 
 
 5.645 
 
 5.650 
 
 5.655 
 
 5.660 
 
 
 8 
 
 5.664 
 
 5.669 
 
 5.674 
 
 5.679 
 
 5.683 
 
 5.688 
 
 5.693 
 
 5.698 
 
 5.703 
 
 5.707 
 
 
 9 
 
 5.712 
 
 5.717 
 
 5.722 
 
 5.726 
 
 5.731 
 
 5.736 
 
 5.741 
 
 5.746 
 
 5.750 
 
 5.755 
 
 
 2.40 
 
 5.760 
 
 5.765 
 
 5.770 
 
 5.774 
 
 5.779 
 
 5.784 
 
 5.789 
 
 5.794 
 
 5.798 
 
 5.803 
 
 
 1 
 
 5.808 
 
 5.813 
 
 5.818 
 
 5.823 
 
 5.827 
 
 5.832 
 
 5.837 
 
 5.842 
 
 5.847 
 
 5.852 
 
 
 2 
 
 5.856 
 
 5.861 
 
 5.866 
 
 5.871 
 
 5.876 
 
 5.881 
 
 5.885 
 
 5.890 
 
 5.895 
 
 5.900 
 
 
 3 
 
 5.905 
 
 5.910 
 
 5.915 
 
 5.919 
 
 5.924 
 
 5.929 
 
 5.934 
 
 5.939 
 
 5.944 
 
 5.949 
 
 
 4 
 
 5.954 
 
 5.958 
 
 5.963 
 
 5.968 
 
 5.973 
 
 5.978 
 
 5.983 
 
 5.988 
 
 5.993 
 
 5.998 
 
 
 2.45 
 
 6.002 
 
 6.007 
 
 6.012 
 
 6.017 
 
 6.022 
 
 6.027 
 
 6.032 
 
 6.037 
 
 6.042 
 
 6.047 
 
 
 6 
 
 6.052 
 
 6.057 
 
 6.061 
 
 6.066 
 
 6.071 
 
 6.076 
 
 6.081 
 
 6.086 
 
 6.091 
 
 6.096 
 
 
 7 
 
 6.101 
 
 6.106 
 
 6.111 
 
 6.116 
 
 6.121 
 
 6.126 
 
 6.131 
 
 6.136 
 
 6.140 
 
 6.145 
 
 
 8 
 
 6.150 
 
 6.155 
 
 6.160 
 
 6.165 
 
 6.170 
 
 6.175 
 
 6.180 
 
 6.185 
 
 6.190 
 
 6.195 
 
 
 9 
 
 6.200 
 
 6.205 
 
 6.210 , 
 
 6.215 
 
 6.220 
 
 6.225 
 
 6.230 
 
 6.235 
 
 6.240 
 
 6.245 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Moving the decimal point ONE place in N requires moving it TWO places in body 
 of table (see p. G). 
 
MATHEMATICAL TABLES 
 
 SQUARES (continued) 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 2.50 
 
 6.250 
 
 6.255 
 
 6.260 
 
 6.265 
 
 6.270 
 
 6.275 
 
 6.280 
 
 6.285 
 
 6.290 
 
 6.295 
 
 5 
 
 1 
 
 6.300 
 
 6.305 
 
 6.310 
 
 6.315 
 
 6.320 
 
 6.325 
 
 6.330 
 
 6.335 
 
 6.340 
 
 6.345 
 
 
 2 
 
 6.350 
 
 6.355 
 
 6.360 
 
 6.366 
 
 6.371 
 
 6.376 
 
 6.381 
 
 6.386 
 
 6.391 
 
 6.396 
 
 
 3 
 
 6.401 
 
 6.406 
 
 6.411 
 
 6.416 
 
 6.421 
 
 6.426 
 
 6.431 
 
 6.436 
 
 6.441 
 
 6.447 
 
 
 4 
 
 6.452 
 
 6.457 
 
 6.462 
 
 6.467 
 
 6.472 
 
 6.477 
 
 6.482 
 
 6.487 
 
 6.492 
 
 6.497 
 
 
 2.55 
 
 6.502 
 
 6.508 
 
 6.513 
 
 6.518 
 
 6.523 
 
 6.528 
 
 6.533 
 
 6.538 
 
 6.543 
 
 6.548 
 
 
 6 
 
 6.554 
 
 6.559 
 
 6.564 
 
 6.569 
 
 6.574 
 
 6.579 
 
 6.584 
 
 6.589 
 
 6.595 
 
 6.600 
 
 
 7 
 
 6.605 
 
 6.610 
 
 6.615 
 
 6.620 
 
 6.625 
 
 6.631 
 
 6.636 
 
 6.641 
 
 6.646 
 
 6.651 
 
 
 8 
 
 6.656 
 
 6.662 
 
 6.667 
 
 6.672 
 
 6.677 
 
 6.682 
 
 6.687 
 
 6.693 
 
 6.698 
 
 6.703 
 
 
 9 
 
 6.708 
 
 6.713 
 
 6.718 
 
 6.724 
 
 6.729 
 
 6.734 
 
 6.739 
 
 6.744 
 
 6.750 
 
 6.755 
 
 
 2.60 
 
 6.760 
 
 6.765 
 
 6.770 
 
 6.776 
 
 6.781 
 
 6.786 
 
 6.791 
 
 6.7% 
 
 6.802 
 
 6.807 
 
 
 1 
 
 6.812 
 
 6.817 
 
 6.823 
 
 6.828 
 
 6.833 
 
 6.838 
 
 6.843 
 
 6.849 
 
 6.854 
 
 6.859 
 
 
 2 
 
 6.864 
 
 6.870 
 
 6.875 
 
 6.880 
 
 6.885 
 
 6.891 
 
 6.896 
 
 6.901 
 
 6906 
 
 6.912 
 
 
 3 
 
 6.917 
 
 6.922 
 
 6.927 
 
 6.933 
 
 6.938 
 
 6.943 
 
 6.948 
 
 6.954 
 
 6.959 
 
 6.964 
 
 
 4 
 
 6.970 
 
 6.975 
 
 6.980 
 
 6.985 
 
 6.991 
 
 6.996 
 
 7.001 
 
 7.007 
 
 7.012 
 
 7.017 
 
 
 2.65 
 
 7.022 
 
 7.028 
 
 7.033 
 
 7.038 
 
 7.044 
 
 7.049 
 
 7.054 
 
 7.060 
 
 7.065 
 
 7.070 
 
 
 6 
 
 7.076 
 
 7.081 
 
 7.086 
 
 7.092 
 
 7.097 
 
 7.102 
 
 7.108 
 
 7.113 
 
 7.118 
 
 7.124 
 
 
 7 
 
 7.129 
 
 7.134 
 
 7.140 
 
 7.145 
 
 7.150 
 
 7.156 
 
 7.161 
 
 7.166 
 
 7.172 
 
 7.177 
 
 
 8 
 
 7.182 
 
 7.188 
 
 7.193 
 
 7.198 
 
 7.204 
 
 7.209 
 
 7.215 
 
 7.220 
 
 7.225 
 
 7.231 
 
 
 9 
 
 7.236 
 
 7.241 
 
 7.247 
 
 7.252 
 
 7.258 
 
 7.263 
 
 7.268 
 
 7.274 
 
 7.279 
 
 7.285 
 
 
 2.70 
 
 7.290 
 
 7.295 
 
 7.301 
 
 7.306 
 
 7.312 
 
 7.317 
 
 7.322 
 
 7.328 
 
 7.333 
 
 7339 
 
 
 1 
 
 7.344 
 
 7.350 
 
 7.355 
 
 7.360 
 
 7.366 
 
 7.371 
 
 7.377 
 
 7.382 
 
 7.388 
 
 7.393 
 
 
 2 
 
 7.398 
 
 7.404 
 
 7.409 
 
 7.415 
 
 7.420 
 
 7.426 
 
 7.431 
 
 7.437 
 
 7.442 
 
 7.447 
 
 
 3 
 
 7.453 
 
 7.458 
 
 7.464 
 
 7.469 
 
 7.475 
 
 7.480 
 
 7.486 
 
 7.491 
 
 7.497 
 
 7.502 
 
 
 4 
 
 7.508 
 
 7.513 
 
 7.519 
 
 7.524 
 
 7.530 
 
 7.535 
 
 7.541 
 
 7.546 
 
 7.552 
 
 7.557 
 
 
 2.75 
 
 7.562 
 
 7.568 
 
 7.574 
 
 7.579 
 
 7.585 
 
 7.590 
 
 7.596 
 
 7.601 
 
 7.607 
 
 7.612 
 
 6 
 
 6 
 
 7.618 
 
 7.623 
 
 7.629 
 
 7.634 
 
 7.640 
 
 7.645 
 
 7.651 
 
 7.656 
 
 7.662 
 
 7.667 
 
 
 7 
 
 7.673 
 
 7.678 
 
 7.684 
 
 7.690 
 
 7.695 
 
 7.701 
 
 7.706 
 
 7.712 
 
 7.717 
 
 7.723 
 
 
 8 
 
 7.728 
 
 7.734 
 
 7.740 
 
 7.745 
 
 7.751 
 
 7.756 
 
 7.762 
 
 7.767 
 
 7.773 
 
 7.779 
 
 
 9 
 
 7.784 
 
 7.790 
 
 7.795 
 
 7.801 
 
 7.806 
 
 7.812 
 
 7.818 
 
 7.823 
 
 7.829 
 
 7.834 
 
 
 2.80 
 
 7.840 
 
 7.846 
 
 7.851 
 
 7.857 
 
 7.862 
 
 7.868 
 
 7.874 
 
 7.879 
 
 7.885 
 
 7.890 
 
 
 1 
 
 7.896 
 
 7.902 
 
 7.907 
 
 7.913 
 
 7.919 
 
 7.924 
 
 7.930 
 
 7.935 
 
 7.941 
 
 7.947 
 
 
 2 
 
 7.952 
 
 7.958 
 
 7.964 
 
 7.969 
 
 7.975 
 
 7.981 
 
 7.986 
 
 7.992 
 
 7.998 
 
 8003 
 
 
 3 
 
 8.009 
 
 8.015 
 
 8.020 
 
 8.026 
 
 8.032 
 
 8.037 
 
 8.043 
 
 8.049 
 
 8.054 
 
 8.060 
 
 
 4 
 
 8.066 
 
 8.071 
 
 8.077 
 
 8.083 
 
 8.088 
 
 8.094 
 
 8.100 
 
 8.105 
 
 8.111 
 
 8.117 
 
 
 2.85 
 
 8.122 
 
 8.128 
 
 8.134 
 
 8.140 
 
 8.145 
 
 8.151 
 
 8.157 
 
 8.162 
 
 8.168 
 
 8.174 
 
 
 6 
 
 8.180 
 
 8.185 
 
 8.191 
 
 8.197 
 
 8.202 
 
 8.208 
 
 8.214 
 
 8.220 
 
 8.225 
 
 8.231 
 
 
 7 
 
 8.237 
 
 8.243 
 
 8.248 
 
 8.254 
 
 8.260 
 
 8.266 
 
 8.271 
 
 8.277 
 
 8.283 
 
 8.289 
 
 
 8 
 
 8.294 
 
 8.300 
 
 8.306 
 
 8.312 
 
 8.317 
 
 8.323 
 
 8.329 
 
 8.335 
 
 8.341 
 
 8.346 
 
 
 9 
 
 8.352 
 
 8.358 
 
 8.364 
 
 8.369 
 
 8.375 
 
 8.381 
 
 8.387 
 
 8.393 
 
 8.398 
 
 8.404 
 
 
 2.90 
 
 8.410 
 
 8.416 
 
 8.422 
 
 8.427 
 
 8.433 
 
 8.439 
 
 8.445 
 
 8.451 
 
 8.456 
 
 8.462 
 
 
 1 
 
 8.468 
 
 8.474 
 
 8.480 
 
 8.486 
 
 8.491 
 
 8.497 
 
 8.503 
 
 8.509 
 
 8.515 
 
 8.521 
 
 
 2 
 
 8526 
 
 8.532 
 
 8.538 
 
 8.544 
 
 8.550 
 
 8.556 
 
 8.561 
 
 8.567 
 
 8.573 
 
 8.579 
 
 
 3 
 
 8.585 
 
 8.591 
 
 8.597 
 
 8.602 
 
 8.608 
 
 8.614 
 
 8.620 
 
 8.626 
 
 8.632 
 
 8.638 
 
 
 4 
 
 8.644 
 
 8.649 
 
 8.655 
 
 8.661 
 
 8.667 
 
 8.673 
 
 8.679 
 
 8.685 
 
 8.691 
 
 8.697 
 
 
 2.95 
 
 8.702 
 
 8.708 
 
 8.714 
 
 8.720 
 
 8.726 
 
 8.732 
 
 8.738 
 
 8.744 
 
 8.750 
 
 8.756 
 
 
 6 
 
 8.762 
 
 8.768 
 
 8.773 
 
 8.779 
 
 8.785 
 
 8.791 
 
 8.797 
 
 8.803 
 
 8.809 
 
 8.815 
 
 
 7 
 
 8.821 
 
 8.827 
 
 8.833 
 
 8.839 
 
 8.845 
 
 8.851 
 
 8.857 
 
 8.863 
 
 8.868 
 
 8874 
 
 
 8 
 
 8.880 
 
 8.886 
 
 8.892 
 
 8.898 
 
 8.904 
 
 8.910 
 
 8.916 
 
 8.922 
 
 8.928 
 
 8.934 
 
 
 9 
 
 8.940 
 
 8.946 
 
 8.952 
 
 8.958 
 
 8.964 
 
 8.970 
 
 8.976 
 
 8.982 
 
 8.988 
 
 8.994 
 
 
 I/T = 0.101321 
 
 7.38906 
 
c 
 
 MATHEMATICAL TABLES 
 
 SQUARES (continued} 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 S?a 
 
 5* 
 
 3.00 
 
 9.000 
 
 9.006 
 
 9.012 
 
 9.018 
 
 9.024 
 
 9.030 
 
 9.036 
 
 9.042 
 
 9.048 
 
 9.054 
 
 6 
 
 1 
 
 9.060 
 
 9.066 
 
 9.072 
 
 9.078 
 
 9.084 
 
 9.090 
 
 9.096 
 
 9.102 
 
 9.108 
 
 9.114 
 
 
 2 
 
 9.120 
 
 9.126 
 
 9.132 
 
 9.139 
 
 9.145 
 
 9.151 
 
 9.157 
 
 9.163 
 
 9.169 
 
 9.175 
 
 
 3 
 
 9.181 
 
 9.187 
 
 9.193 
 
 9.199 
 
 9.205 
 
 9.211 
 
 9.217 
 
 9.223 
 
 9.229 
 
 9.236 
 
 
 4 
 
 9.242 
 
 9.248 
 
 9.254 
 
 9.260 
 
 9.266 
 
 9.272 
 
 9.278 
 
 9.284 
 
 9.290 
 
 9.296 
 
 
 3.05 
 
 9.302 
 
 9.309 
 
 9.315 
 
 9.321 
 
 9.327 
 
 9.333 
 
 9.339 
 
 9.345 
 
 9.351 
 
 9.357 
 
 
 6 
 
 9.364 
 
 9.370 
 
 9.376 
 
 9.382 
 
 9.388 
 
 9.394 
 
 9.400 
 
 9.406 
 
 9.413 
 
 9.419 
 
 
 7 
 
 9.425 
 
 9.431 
 
 9.437 
 
 9.443 
 
 9.449 
 
 9.456 
 
 9.462 
 
 9.468 
 
 9.474 
 
 9.480 
 
 
 8 
 
 9.486 
 
 9.493 
 
 9.499 
 
 9.505 
 
 9.511 
 
 9.517 
 
 9.523 
 
 9.530 
 
 9.536 
 
 9.542 
 
 
 9 
 
 9.548 
 
 9.554 
 
 9.560 
 
 9.567 
 
 9.573 
 
 9.579 
 
 9.585 
 
 9.591 
 
 9.598 
 
 9.604 
 
 
 3.10 
 
 9.610 
 
 9.616 
 
 9.622 
 
 9.629 
 
 9.635 
 
 9.641 
 
 9.647 
 
 9.653 
 
 9.660 
 
 9.666 
 
 
 1 
 
 9.672 
 
 9.678 
 
 9.685 
 
 9.691 
 
 9.697 
 
 9.703 
 
 9.709 
 
 9.716 
 
 9.722 
 
 9.728 
 
 
 2 
 
 9.734 
 
 9.741 
 
 9.747 
 
 9.753 
 
 9.759 
 
 9.766 
 
 9.772 
 
 9.778 
 
 9.784 
 
 9.791 
 
 
 3 
 
 9.797 
 
 9.803 
 
 9.809 
 
 9.816 
 
 9.822 
 
 9.828 
 
 9.834 
 
 9.841 
 
 9.847 
 
 9.853 
 
 
 4 
 
 9.860 
 
 9.866 
 
 9.872 
 
 9.878 
 
 9.885 
 
 9.891 
 
 9.897 
 
 9.904 
 
 9.910 
 
 9.916 
 
 
 3.15 
 
 9.922 
 
 9.929 
 
 9.935 
 
 9.941 
 
 9.948 
 
 9.954 
 
 9.960 
 
 9.967 
 
 9.973 
 
 9.979 
 
 
 6 
 
 9.986 
 
 9.992 
 
 9.998 
 
 10.005 
 
 
 
 
 
 
 
 6 
 
 3.1 
 
 
 
 
 
 
 
 9.99 
 
 10.05 
 
 10.11 
 
 10.18 
 
 6 
 
 2 
 
 10.24 
 
 10.30 
 
 10.37 
 
 10.43 
 
 10.50 
 
 10.56 
 
 10.63 
 
 10.69 
 
 10.76 
 
 10.82 
 
 
 3 
 
 10.89 
 
 10.96 
 
 11.02 
 
 11.09 
 
 11.16 
 
 11.22 
 
 11.29 
 
 11.36 
 
 11.42 
 
 11.49 
 
 7 
 
 4 
 
 11.56 
 
 11.63 
 
 11.70 
 
 11.76 
 
 11.83 
 
 11.90 
 
 11.97 
 
 12.04 
 
 12.11 
 
 12.18 
 
 
 3.5 
 
 12.25 
 
 12.32 
 
 12.39 
 
 12.46 
 
 12.53 
 
 12.60 
 
 12.67 
 
 12.74 
 
 12.82 
 
 12.89 
 
 
 6 
 
 12.96 
 
 13.03 
 
 13.10 
 
 13.18 
 
 13.25 
 
 13.32 
 
 13.40 
 
 13.47 
 
 13.54 
 
 13.62 
 
 
 7 
 
 13.69 
 
 13.76 
 
 13.84 
 
 13.91 
 
 13.99 
 
 14.06 
 
 14.14 
 
 14.21 
 
 14.29 
 
 14.36 
 
 8 
 
 8 
 
 14.44 
 
 14.52 
 
 14.59 
 
 14.67 
 
 14.75 
 
 14.82 
 
 14.90 
 
 14.98 
 
 15.05 
 
 15.13 
 
 
 9 
 
 15.21 
 
 15.29 
 
 15.37 
 
 15.44 
 
 15.52 
 
 15.60 
 
 15.68 
 
 15.76 
 
 15.84 
 
 15.92 
 
 
 4.0 
 
 16.00 
 
 16.08 
 
 16.16 
 
 16.24 
 
 16.32 
 
 16.40 
 
 16.48 
 
 16.56 
 
 16.65 
 
 16.73 
 
 
 1 
 
 16.81 
 
 16.89 
 
 16.97 
 
 17.06 
 
 17.14 
 
 17.22 
 
 17.31 
 
 17.39 
 
 17.47 
 
 17.56 
 
 
 2 
 
 17.64 
 
 17.72 
 
 17.81 
 
 17.89 
 
 17.98 
 
 18.06 
 
 18.15 
 
 18.23 
 
 18.32 
 
 18.40 
 
 
 3 
 
 18.49 
 
 18.58 
 
 18.66 
 
 18.75 
 
 18.84 
 
 18.92 
 
 19.01 
 
 19.10 
 
 19.18 
 
 19.27 
 
 9 
 
 4 
 
 19.36 
 
 19.45 
 
 19.54 
 
 19.62 
 
 19.71 
 
 19.80 
 
 19.89 
 
 19.98 
 
 20.07 
 
 20.16 
 
 
 4.5 
 
 20.25 
 
 20.34 
 
 20.43 
 
 20.52 
 
 20.61 
 
 20.70 
 
 20.79 
 
 20.88 
 
 20.98 
 
 21.07 
 
 
 6 
 
 21.16 
 
 21.25 
 
 21.34 
 
 21.44 
 
 21.53 
 
 21.62 
 
 21.72 
 
 21.81 
 
 21.90 
 
 22.00 
 
 
 7 
 
 22.09 
 
 22.18 
 
 22.28 
 
 22.37 
 
 22.47 
 
 22.56 
 
 22.66 
 
 22.75 
 
 22.85 
 
 22.94 
 
 10 
 
 8 
 
 23.04 
 
 23.14 
 
 23.23 
 
 23.33 
 
 23.43 
 
 23.52 
 
 23.62 
 
 23.72 
 
 23.81 
 
 23.91 
 
 
 9 
 
 24.01 
 
 24.11 
 
 24.21 
 
 24.30 
 
 24.40 
 
 24.50 
 
 24.60 
 
 24.70 
 
 24.80 
 
 24.90 
 
 
 9.86960 (x/2) 2 = 2.46740 !/ = 0.101321 
 
 Explanation of Table of Squares (pp. 2-7). 
 
 This table gives the value of N z for values of N from 1 to 10, correct to four figures. 
 (Interpolated values may be in error by 1 in the fourth figure). 
 
 To find the square of a number N outside the range from 1 to 10, note that 
 moving the decimal point one place in column N is equivalent to moving it two places 
 in the body of the table. For example: 
 
 (3.217) 2 - 10.35; (0.03217)2 = 0.001035; (3217)* = 10350000 
 
 This table can also be used inversely, to give square roots. 
 
MATHEMATICAL TABLES 
 
 SQUARES (continued) 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 & 
 
 5.0 
 
 25.00 
 
 25.10 
 
 25.20 
 
 25.30 
 
 25.40 
 
 25.50 
 
 25.60 
 
 25.70 
 
 25.81 
 
 25.91 
 
 10 
 
 1 
 
 26.01 
 
 26.11 
 
 26.21 
 
 26.32 
 
 26.42 
 
 26.52 
 
 26.63 
 
 26.73 
 
 26.83 
 
 26.94 
 
 
 2 
 
 27.04 
 
 27.14 
 
 27.25 
 
 27.35 
 
 27.46 
 
 27.56 
 
 27.67 
 
 27.77 
 
 27.88 
 
 27.98 
 
 
 3 
 
 28.09 
 
 28.20 
 
 28.30 
 
 28.41 
 
 28.52 
 
 28.62 
 
 28.73 
 
 28.84 
 
 28.94 
 
 29.05 
 
 11 
 
 4 
 
 29.16 
 
 29.27 
 
 29.38 
 
 29.48 
 
 29.59 
 
 29.70 
 
 29.81 
 
 29.92 
 
 30.03 
 
 30.14 
 
 
 5.5 
 
 30.25 
 
 30.36 
 
 30.47 
 
 30.58 
 
 30.69 
 
 30.80 
 
 30.91 
 
 31.02 
 
 31.14 
 
 31.25 
 
 
 6 
 
 31.36 
 
 31.47 
 
 31.58 
 
 31.70 
 
 31.81 
 
 31.92 
 
 32.04 
 
 32.15 
 
 32.26 
 
 32.38 
 
 
 7 
 
 32.49 
 
 32.60 
 
 32.72 
 
 32.83 
 
 32.95 
 
 33.06 
 
 33.18 
 
 33.29 
 
 33.41 
 
 33.52 
 
 
 8 
 
 33.64 
 
 33.76 
 
 33.87 
 
 33.99 
 
 34.11 
 
 34.22 
 
 34.34 
 
 34.46 
 
 34.57 
 
 34.69 
 
 12 
 
 9 
 
 34.81 
 
 34.93 
 
 35.05 
 
 35.16 
 
 35.28 
 
 35.40 
 
 35.52 
 
 35.64 
 
 35.76 
 
 35.88 
 
 
 6.0 
 
 36.00 
 
 36.12 
 
 36.24 
 
 36.36 
 
 36.48 
 
 36.60 
 
 36.72 
 
 36.84 
 
 36.97 
 
 37.09 
 
 
 
 37.21 
 
 37.33 
 
 37.45 
 
 37.58 
 
 37.70 
 
 37.82 
 
 37.95 
 
 38.07 
 
 38.19 
 
 38.32 
 
 
 2 
 
 38.44 
 
 38.56 
 
 38.69 
 
 38.81 
 
 38.94 
 
 39.06 
 
 39.19 
 
 39.31 
 
 39.44 
 
 39.56 
 
 
 3 
 
 39.69 
 
 39.82 
 
 39.94 
 
 40.07 
 
 40.20 
 
 40.32 
 
 40.45 
 
 40.58 
 
 40.70 
 
 40.83 
 
 13 
 
 4 
 
 40.96 
 
 41.09 
 
 41.22 
 
 41.34 
 
 41.47 
 
 41.60 
 
 41.73 
 
 41.86 
 
 41.99 
 
 42.12 
 
 
 6.5 
 
 42.25 
 
 42.38 
 
 42.51 
 
 42.64 
 
 42.77 
 
 42.90 
 
 43.03 
 
 43.16 
 
 43.30 
 
 43.43 
 
 
 6 
 
 43.56 
 
 43.69 
 
 43.82 
 
 43.96 
 
 44.09 
 
 44.22 
 
 44.36 
 
 44.49 
 
 44.62 
 
 44.76 
 
 
 7 
 
 44.89 
 
 45.02 
 
 45.16 
 
 45.29 
 
 45.43 
 
 45.56 
 
 45.70 
 
 45.83 
 
 45.97 
 
 46.10 
 
 
 8 
 
 46.24 
 
 46.38 
 
 46.51 
 
 46.65 
 
 46.79 
 
 46.92 
 
 47.06 
 
 47.20 
 
 47.33 
 
 47.47 
 
 14 
 
 9 
 
 47.61 
 
 47.75 
 
 47.89 
 
 48.02 
 
 48.16 
 
 48.30 
 
 48.44 
 
 48.58 
 
 48.72 
 
 48.86 
 
 
 7.0 
 
 49.00 
 
 49.14 
 
 49.28 
 
 49.42 
 
 49.56 * 
 
 49.70 
 
 49.84 
 
 49.98 
 
 50.13 
 
 50.27 
 
 
 1 
 
 50.41 
 
 50.55 
 
 50.69 
 
 50.84 
 
 50.98 
 
 51.12 
 
 51.27 
 
 51.41 
 
 51.55 
 
 51.70 
 
 
 2 
 
 51.84 
 
 51.98 
 
 52.13 
 
 52.27 
 
 52.42 
 
 52.56 
 
 52.71 
 
 52.85 
 
 53.00 
 
 53.14 
 
 
 3 
 
 53.29 
 
 53.44 
 
 53.58 
 
 53.73 
 
 53.88 
 
 54.02 
 
 54.17 
 
 54.32 
 
 54.46 
 
 54.61 
 
 15 
 
 4 
 
 54.76 
 
 54.91 
 
 55.06 
 
 55.20 
 
 55.35 
 
 55.50 
 
 55.65 
 
 55.80 
 
 55.95 
 
 56.10 
 
 
 7.5 
 
 56.25 
 
 56.40 
 
 56.55 
 
 56.70 
 
 56.85 
 
 57.00 
 
 57.15 
 
 57.30 
 
 57.46 
 
 57.61 
 
 
 6 
 
 57.76 
 
 57.91 
 
 58.06 
 
 58.22 
 
 58.37 
 
 58.52 
 
 58.68 
 
 58.83 
 
 58.98 
 
 59.14 
 
 
 7 
 
 59.29 
 
 59.44 
 
 59.60 
 
 59.75 
 
 59.91 
 
 60.06 
 
 60.22 
 
 60.37 
 
 60.53 
 
 60.68 
 
 
 8 
 
 60.84 
 
 61.00 
 
 61.15 
 
 61.31 
 
 61.47 
 
 61.62 
 
 61.78 
 
 61.94 
 
 62.09 
 
 62.25 
 
 16 
 
 9 
 
 62.41 
 
 62.57 
 
 62.73 
 
 62.88 
 
 63.04 
 
 63.20 
 
 63.36 
 
 63.52 
 
 63.68 
 
 63.84 
 
 
 8.0 
 
 64.00 
 
 64.16 
 
 64.32 
 
 64.48 
 
 64.64 
 
 64.80 
 
 64.96 
 
 65.12 
 
 65.29 
 
 65.45 
 
 
 ] 
 
 65.61 
 
 65.77 
 
 65.93 
 
 66.10 
 
 66.26 
 
 66.42 
 
 66.59 
 
 66.75 
 
 66.91 
 
 67.08 
 
 
 2 
 
 67.24 
 
 67.40 
 
 67.57 
 
 67.73 
 
 67.90 
 
 68.06 
 
 68.23 
 
 68.39 
 
 68.56 
 
 68.72 
 
 
 3 
 
 68.89 
 
 69.06 
 
 69.22 
 
 69.39 
 
 69.56 
 
 69.72 
 
 69.89 
 
 70.06 
 
 70.22 
 
 70.39 
 
 17 
 
 4 
 
 70.56 
 
 70.73 
 
 70.90 
 
 71.06 
 
 71.23 
 
 71.40 
 
 71.57 
 
 71.74 
 
 71.91 
 
 72.08 
 
 
 8.5 
 
 72.25 
 
 72.42 
 
 72.59 
 
 72.76 
 
 72.93 
 
 73.10 
 
 73.27 
 
 73.44 
 
 73.62 
 
 73.79 
 
 
 6 
 
 73.96 
 
 74.13 
 
 74.30 
 
 74.48 
 
 74.65 
 
 74.82 
 
 75.00 
 
 75.17 
 
 75.34 
 
 75.52 
 
 
 7 
 
 75.69 
 
 75.86 
 
 76.04 
 
 76.21 
 
 76.39 
 
 76.56 
 
 76.74 
 
 76.91 
 
 77.09 
 
 77.26 
 
 
 8 
 
 77.44 
 
 77.62 
 
 77.79 
 
 77.97 
 
 78.15 
 
 78.32 
 
 78.50 
 
 78.68 
 
 78.85 
 
 79.03 
 
 18 
 
 9 
 
 79.21 
 
 79.39 
 
 79.57 
 
 79.74 
 
 79.92 
 
 80.10 
 
 80.28 
 
 80.46 
 
 80.64 
 
 80.82 
 
 
 9.0 
 
 81.00 
 
 81.18 
 
 81.36 
 
 81.54 
 
 81.72 
 
 81.90 
 
 82.08 
 
 82.26 
 
 82.45 
 
 82.63 
 
 
 1 
 
 82.81 
 
 82.99 
 
 83.17 
 
 83.36 
 
 83.54 
 
 83.72 
 
 83.91 
 
 84.09 
 
 84.27 
 
 84.46 
 
 
 2 
 
 84.64 
 
 84.82 
 
 85.01 
 
 85.19 
 
 85.38 
 
 85.56 
 
 85.75 
 
 85.93 
 
 86.12 
 
 86.30 
 
 
 3 
 
 86.49 
 
 86.68 
 
 86.86 
 
 87.05 
 
 87.24 
 
 87.42 
 
 87.61 
 
 87.80 
 
 87.98 
 
 88.17 
 
 19 
 
 4 
 
 88.36 
 
 88.55 
 
 88.74 
 
 88.92 
 
 89.11 
 
 89.30 
 
 89.49 
 
 89.68 
 
 89.87 
 
 90.06 
 
 
 9.5 
 
 90.25 
 
 90.44 
 
 90.63 
 
 90.82 
 
 91.01 
 
 91.20 
 
 91.39 
 
 91.58 
 
 91.78 
 
 91.97 
 
 
 6 
 
 92.16 
 
 92.35 
 
 92.54 
 
 92.74 
 
 92.93 
 
 93.12 
 
 93.32 
 
 93.51 
 
 93.70 
 
 93.90 
 
 
 7 
 
 94.09 
 
 94.28 
 
 94.48 
 
 94.67 
 
 94.87 
 
 95.06 
 
 95.26 
 
 95.45 
 
 95.65 
 
 95.84 
 
 
 8 
 
 96.04 
 
 96.24 
 
 96.43 
 
 96.63 
 
 96.83 
 
 97.02 
 
 97.22 
 
 97.42 
 
 97.61 
 
 97.81 
 
 20 
 
 9 
 
 9801 
 
 98.21 
 
 98.41 
 
 98.60 
 
 98.80 
 
 99.00 
 
 99.20 
 
 99.40 
 
 99.60 
 
 99.80 
 
 
 10.0 
 
 100.0 
 
 
 
 
 
 
 
 
 
 
 
 Moving the decimal point ONE place in N requires moving it TWO places in body 
 of table (see p. 6). 
 
MATHEMATICAL TABLES 
 
 CUBES OP NUMBERS 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 II 
 
 1.00 
 
 1.000 
 
 1.003 
 
 1.006 
 
 1.009 
 
 1.012 
 
 1.015 
 
 1.018 
 
 1/21 
 
 1.024 
 
 1.027 
 
 3 
 
 1 
 
 1.030 
 
 1.033 
 
 1.036 
 
 1.040 
 
 1.043 
 
 1.046 
 
 1.049 
 
 1.052 
 
 1.055 
 
 1.058 
 
 
 2 
 
 1.061 
 
 1.064 
 
 1.067 
 
 1.071 
 
 1.074 
 
 1.077 
 
 1.080 
 
 1.083 
 
 1.086 
 
 1.090 
 
 
 3 
 
 1.093 
 
 1.096 
 
 1.099 
 
 1.102 
 
 1.106 
 
 1.109 
 
 1.112 
 
 1.115 
 
 1.118 
 
 1.122 
 
 
 4 
 
 1.125 
 
 1.128 
 
 1.131 
 
 1.135 
 
 1.138 
 
 1.141 
 
 1.144 
 
 1.148 
 
 1.151 
 
 1.154 
 
 
 1.05 
 
 1.158 
 
 .161 
 
 1.164 
 
 1.168 
 
 1.171 
 
 1.174 
 
 1.178 
 
 1.181 
 
 1.184 
 
 1.188 
 
 
 6 
 
 1.191 
 
 .194 
 
 1.198 
 
 1.201 
 
 1.205 
 
 1.208 
 
 1.211 
 
 1.215 
 
 1.218 
 
 1.222 
 
 
 7 
 
 1.225 
 
 .228 
 
 1.232 
 
 1.235 
 
 1.239 
 
 1.242 
 
 1.246 
 
 1.249 
 
 1.253 
 
 1.256 
 
 
 8 
 
 1.260 
 
 .263 
 
 1.267 
 
 1.270 
 
 1.274 
 
 1.277 
 
 1.281 
 
 1.284 
 
 1.288 
 
 1.291 
 
 4 
 
 9 
 
 1.295 
 
 .299 
 
 1.302 
 
 1.306 
 
 1.309 
 
 1.313 
 
 1.317 
 
 1.320 
 
 1.324 
 
 1.327 
 
 
 1.10 
 
 1.331 
 
 .335 
 
 1.338 
 
 1.342 
 
 1.346 
 
 1.349 
 
 1.353 
 
 1.357 
 
 1.360 
 
 1.364 
 
 
 1 
 
 1368 
 
 .371 
 
 1.375 
 
 1.379 
 
 1.382 
 
 1.386 
 
 1.390 
 
 1.394 
 
 1.397 
 
 1.401 
 
 
 2 
 
 1.405 
 
 1.409 
 
 1.412 
 
 1.416 
 
 1.420 
 
 1.424 
 
 1.428 
 
 1.431 
 
 1.435 
 
 1.439 
 
 
 3 
 
 1.443 
 
 1.447 
 
 1.451 
 
 1.454 
 
 1.458 
 
 1.462 
 
 1.466 
 
 1.470 
 
 1.474 
 
 1.478 
 
 
 4 
 
 1.482 
 
 1.485 
 
 1.489 
 
 1.493 
 
 1.497 
 
 1.501 
 
 1.505 
 
 1.509 
 
 1.513 
 
 1.517 
 
 
 1.15 
 
 1.521 
 
 1.525 
 
 1.529 
 
 1.533 
 
 1.537 
 
 1.541 
 
 1.545 
 
 1.549 
 
 1.553 
 
 1.557 
 
 
 6 
 
 1.561 
 
 1.565 
 
 1.569 
 
 1.573 
 
 1.577 
 
 1.581 
 
 1.585 
 
 1.589 
 
 1.593 
 
 1.598 
 
 
 7 
 
 1.602 
 
 1.606 
 
 1.610 
 
 1.614 
 
 1.618 
 
 1.622 
 
 1.626 
 
 1.631 
 
 1.635 
 
 1.639 
 
 
 8 
 
 1.643 
 
 1.647 
 
 1.651 
 
 1.656 
 
 1.660 
 
 1.664 
 
 1.668 
 
 1.672 
 
 1.677 
 
 1.681 
 
 
 9 
 
 1.685 
 
 1.689 
 
 1.694 
 
 1.698 
 
 1.702 
 
 1.706 
 
 1.711 
 
 1.715 
 
 1.719 
 
 1.724 
 
 
 1.20 
 
 1.728 
 
 1.732 
 
 1.737 
 
 1.741 
 
 1.745 
 
 1.750 
 
 1.754 
 
 1.758 
 
 1.763 
 
 1.767 
 
 
 1 
 
 1.772 
 
 1.776 
 
 1.780 
 
 1.785 
 
 1.789 
 
 1.794 
 
 1.798 
 
 1.802 
 
 1.807 
 
 1.811 
 
 
 2 
 
 1.816 
 
 1.820 
 
 1.825 
 
 1.829 
 
 1.834 
 
 1.838 
 
 1.843 
 
 1.847 
 
 1.852 
 
 1.856 
 
 
 3 
 
 1.861 
 
 1.865 
 
 1.870 
 
 1.875 
 
 1.879 
 
 1.884 
 
 1.888 
 
 1.893 
 
 1.897 
 
 1.902 
 
 
 4 
 
 1.907 
 
 1.911 
 
 1.916 
 
 1.920 
 
 1.925 
 
 1.930 
 
 1.934 
 
 1.939 
 
 1.944 
 
 1.948 
 
 5 
 
 1.25 
 
 1.953 
 
 1.958 
 
 1.963 
 
 1.967 
 
 1.972 
 
 1.977 
 
 1.981 
 
 1.986 
 
 1.991 
 
 1.996 
 
 
 6 
 
 2.000 
 
 2.005 
 
 2.010 
 
 2.015 
 
 2.019 
 
 2.024 
 
 2.029 
 
 2.034 
 
 2.039 
 
 2.044 
 
 
 7 
 
 2.048 
 
 2.053 
 
 2.058 
 
 2.063 
 
 2.068 
 
 2.073 
 
 2.078 
 
 2.082 
 
 2.087 
 
 2.092 
 
 
 8 
 
 2.097 
 
 2.102 
 
 2.107 
 
 2.112 
 
 2.117 
 
 2.122 
 
 2.127 
 
 2.132 
 
 2.137 
 
 2.142 
 
 
 9 
 
 2.147 
 
 2.152 
 
 2.157 
 
 2.162 
 
 2.167 
 
 2.172 
 
 2.177 
 
 2.182 
 
 2.187 
 
 2.192 
 
 
 1.30 
 
 2.197 
 
 2.202 
 
 Z.207 
 
 /.212 
 
 2.217 
 
 2.222 
 
 2.228 
 
 2.233 
 
 2.238 
 
 2.243 
 
 
 1 
 
 2.248 
 
 2.253 
 
 2.258 
 
 2.264 
 
 2.269 
 
 2.274 
 
 2.279 
 
 2.284 
 
 2.290 
 
 2.295 
 
 
 2 
 
 2.300 
 
 2.305 
 
 2.310 
 
 2.316 
 
 2.321 
 
 2.326 
 
 2.331 
 
 2.337 
 
 2.342 
 
 2.347 
 
 
 3 
 
 2.353 
 
 2.358 
 
 2.363 
 
 2.369 
 
 2.374 
 
 2.379 
 
 2.385 
 
 2.390 
 
 2.395 
 
 2.401 
 
 
 4 
 
 2.406 
 
 2.411 
 
 2.417 
 
 2.422 
 
 2.428 
 
 2.433 
 
 2.439 
 
 2.444 
 
 2.449 
 
 2.455 
 
 
 1.35 
 
 2.460 
 
 2.466 
 
 2.471 
 
 2.477 
 
 2.482 
 
 2.488 
 
 2.493 
 
 2.499 
 
 2.504 
 
 2.510 
 
 6 
 
 6 
 
 2.515 
 
 2.521 
 
 2.527 
 
 2.532 
 
 2.538 
 
 2.543 
 
 2.549 
 
 2.554 
 
 2.560 
 
 2.566 
 
 
 7 
 
 2,571 
 
 2.577 
 
 2.583 
 
 2.588 
 
 2.594 
 
 2.600 
 
 2.605 
 
 2.611 
 
 2.617 
 
 2.622 
 
 
 8 
 
 2.628 
 
 2.634 
 
 2.640 
 
 2.645 
 
 2.651 
 
 2.657 
 
 2.663 
 
 2.668 
 
 2.674 
 
 2.680 
 
 
 9 
 
 2.686 
 
 2.691 
 
 2.697 
 
 2.703 
 
 2.709 
 
 2.715 
 
 2.721 
 
 2.726 
 
 2.732 
 
 2.738 
 
 
 1.40 
 
 2.744 
 
 2.750 
 
 2.756 
 
 2.762 
 
 2.768 
 
 2.774 
 
 2.779 
 
 2.785 
 
 2.791 
 
 2.797 
 
 
 ] 
 
 2.803 
 
 2.809 
 
 2.815 
 
 2.821 
 
 2.827 
 
 2.833 
 
 2.839 
 
 2.845 
 
 2.851 
 
 2.857 
 
 
 2 
 
 2.863 
 
 2.869 
 
 2.875 
 
 2.881 
 
 2.888 
 
 2.894 
 
 2.900 
 
 2.906 
 
 2.912 
 
 2.918 
 
 
 3 
 
 2.924 
 
 2.930 
 
 2.936 
 
 2.943 
 
 2.949 
 
 2.955 
 
 2.961 
 
 2.967 
 
 2.974 
 
 2.980 
 
 
 4 
 
 2.986 
 
 2.992 
 
 2.998 
 
 3.005 
 
 3.011 
 
 3.017 
 
 3.023 
 
 3.030 
 
 3.036 
 
 3.042 
 
 
 1.45 
 
 3.049 
 
 3.055 
 
 3.061 
 
 3.068 
 
 3.074 
 
 3.080 
 
 3.087 
 
 3.093 
 
 3.099 
 
 3.106 
 
 
 6 
 
 3.112 
 
 3.119 
 
 3.125 
 
 3.131 
 
 3.138 
 
 3.144 
 
 3.151 
 
 3.157 
 
 3.164 
 
 3.170 
 
 
 7 
 
 3.177 
 
 3.183 
 
 3.190 
 
 3.196 
 
 3.203 
 
 3.209 
 
 3.216 
 
 3.222 
 
 3.229 
 
 3.235 
 
 
 8 
 
 3.242 
 
 3.248 
 
 3.255 
 
 3.262 
 
 3.268 
 
 3.275 
 
 3.281 
 
 3.288 
 
 3.295 
 
 3.301 
 
 7 
 
 9 
 
 3.308 
 
 3.315 
 
 3.321 
 
 3.328 
 
 3.335 
 
 3.341 
 
 3.348 
 
 3.355 
 
 3.362 
 
 3.368 
 
 
 Moving the decimal point ONE place in N requires moving it THREE places in 
 body of table (see p. 10). 
 
MATHEMATICAL TABLES 
 
 CUBES (continued) 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 & 
 
 1.50 
 
 3.375 
 
 3.382 
 
 3.389 
 
 3.395 
 
 3.402 
 
 3.409 
 
 3.416 
 
 3.422 
 
 3.429 
 
 3.436 
 
 7 
 
 1 
 
 3.443 
 
 3.450 
 
 3.457 
 
 3.464 
 
 3.470 
 
 3.477 
 
 3.484 
 
 3.491 
 
 3.498 
 
 3.505 
 
 
 2 
 
 3.512 
 
 3.519 
 
 3.526 
 
 3.533 
 
 3.540 
 
 3.547 
 
 3.554 
 
 3.561 
 
 3.568 
 
 3.575 
 
 
 3 
 
 3.582 
 
 3.589 
 
 3.596 
 
 3.603 
 
 3.610 
 
 3.617 
 
 3.624 
 
 3.631 
 
 3.638 
 
 3.645 
 
 
 4 
 
 3.652 
 
 3.659 
 
 3.667 
 
 3.674 
 
 3.681 
 
 3.688 
 
 3.695 
 
 3.702 
 
 3.709 
 
 3.717 
 
 
 1.55 
 
 3.724 
 
 3.731 
 
 3.738 
 
 3.746 
 
 3.753 
 
 3.760 
 
 3.767 
 
 3.775 
 
 3.782 
 
 3.789 
 
 
 6 
 
 3.796 
 
 3.804 
 
 3.811 
 
 3.818 
 
 3.826 
 
 3.833 
 
 3.840 
 
 3.848 
 
 3.855 
 
 3.863 
 
 
 7 
 
 3.870 
 
 3.877 
 
 3.885 
 
 3.892 
 
 3.900 
 
 3.907 
 
 3.914 
 
 3.922 
 
 3.929 
 
 3.937 
 
 
 8 
 
 3.944 
 
 3.952 
 
 3.959 
 
 3.967 
 
 3.974 
 
 3.982 
 
 3.989 
 
 3.997 
 
 4.005 
 
 4.012 
 
 8 
 
 9 
 
 4.020 
 
 4.027 
 
 4.035 
 
 4.042 
 
 4.050 
 
 4.058 
 
 4.065 
 
 4.073 
 
 4.081 
 
 4.088 
 
 
 1.60 
 
 4.096 
 
 4.104 
 
 4.111 
 
 4.119 
 
 4.127 
 
 4.135 
 
 4.142 
 
 4.150 
 
 4.158 
 
 4.166 
 
 
 
 4.173 
 
 4.181 
 
 4.189 
 
 4.197 
 
 4.204 
 
 4.212 
 
 4.220 
 
 4.228 
 
 4.236 
 
 4.244 
 
 
 2 
 
 4.252 
 
 4.259 
 
 4.267 
 
 4.275 
 
 4.283 
 
 4.291 
 
 4.299 
 
 4.307 
 
 4.315 
 
 4.323 
 
 
 3 
 
 4.331 
 
 4.339 
 
 4.347 
 
 4355 
 
 4.363 
 
 4.371 
 
 4.379 
 
 4.387 
 
 4.395 
 
 4.403 
 
 
 4 
 
 4.411 
 
 4.419 
 
 4.427 
 
 4.435 
 
 4.443 
 
 4.451 
 
 4.460 
 
 4.468 
 
 4.476 
 
 4.484 
 
 
 1.65 
 
 4.492 
 
 4.500 
 
 4.508 
 
 4.517 
 
 4.525 
 
 4.533 
 
 4.541 
 
 4.550 
 
 4.558 
 
 4.566 
 
 
 6 
 
 4.574 
 
 4.583 
 
 4.591 
 
 4.599 
 
 4.607 
 
 4.616 
 
 4.624 
 
 4.632 
 
 4.641 
 
 4.649 
 
 
 7 
 
 4.657 
 
 4.666 
 
 4.674 
 
 4.683 
 
 4.691 
 
 4.699 
 
 4.708 
 
 4.716 
 
 4.725 
 
 4.733 
 
 
 8 
 
 4.742 
 
 4.750 
 
 4.759 
 
 4.767 
 
 4.776 
 
 4.784 
 
 4.793 
 
 4.801 
 
 4.810 
 
 4.818 
 
 
 9 
 
 4.827 
 
 4.835 
 
 4.844 
 
 4.853 
 
 4.861 
 
 4.870 
 
 4.878 
 
 4.887 
 
 4.896 
 
 4.904 
 
 9 
 
 1.70 
 
 4.913 
 
 4.922 
 
 4.930 
 
 4.939 
 
 4.948 
 
 4.956 
 
 4.965 
 
 4.974 
 
 4.983 
 
 4.991 
 
 
 1 
 
 5.000 
 
 5.009 
 
 5.018 
 
 5.027 
 
 5.035 
 
 5.044 
 
 5.053 
 
 5.062 
 
 5.071 
 
 5.080 
 
 
 2 
 
 5.088 
 
 5.097 
 
 5.106 
 
 5.115 
 
 5.124 
 
 5.133 
 
 5.142 
 
 5.151 
 
 5.160 
 
 5.169 
 
 
 3 
 
 5.178 
 
 5.187 
 
 5.196 
 
 5.205 
 
 5.214 
 
 5.223 
 
 5.232 
 
 5.241 
 
 5.250 
 
 5.259 
 
 
 4 
 
 5.268 
 
 5.277 
 
 5.286 
 
 5.295 
 
 5.304 
 
 5.314 
 
 5.323 
 
 5.332 
 
 5.341 
 
 5.350 
 
 
 1.75 
 
 5.359 
 
 5.369 
 
 5.378 
 
 5.387 
 
 5.396 
 
 5.405 
 
 5.415 
 
 5.424 
 
 5.433 
 
 5.442 
 
 
 6 
 
 5.452 
 
 5.461 
 
 5.470 
 
 5.480 
 
 5.489 
 
 5.498 
 
 5.508 
 
 5.517 
 
 5.526 
 
 5.536 
 
 
 7 
 
 5.545 
 
 5.555 
 
 5.564 
 
 5.573 
 
 5.583 
 
 5.592 
 
 5.602 
 
 5.611 
 
 5.621 
 
 5.630 
 
 10 
 
 8 
 
 5.640 
 
 5.649 
 
 5.659 
 
 5.668 
 
 5.678 
 
 5.687 
 
 5.697 
 
 5.707 
 
 5,716 
 
 5.726 
 
 
 9 
 
 5.735 
 
 5.745 
 
 5.755 
 
 5.764 
 
 5.774 
 
 5.784 
 
 5.793 
 
 5.803 
 
 5.813 
 
 5.822 
 
 
 1.80 
 
 5.832 
 
 5.842 
 
 5.851 
 
 5.861 
 
 5.871 
 
 .5.881 
 
 5.891 
 
 5.900 
 
 5.910 
 
 5.920 
 
 
 1 
 
 5.930 
 
 5.940 
 
 5.949 
 
 5.959 
 
 5.969 
 
 5.979 
 
 5.989 
 
 5.999 
 
 6.009 
 
 6.019 
 
 
 2 
 
 6.029 
 
 6.039 
 
 6.048 
 
 6.058 
 
 6.068 
 
 6.078 
 
 6.088 
 
 6.098 
 
 6.108 
 
 6.118 
 
 
 3 
 
 6.128 
 
 6.139 
 
 6.149 
 
 6.159 
 
 6. 169 
 
 6.179 
 
 6.189 
 
 6.199 
 
 6.209 
 
 6.219 
 
 
 4 
 
 6.230 
 
 6.240 
 
 6.250 
 
 6.260 
 
 6.270 
 
 6.280 
 
 6.291 
 
 6.301 
 
 6.311 
 
 6.321 
 
 
 1.85 
 
 6.332 
 
 6.342 
 
 6.352 
 
 6.362 
 
 6.373 
 
 6.383 
 
 6.393 
 
 6.404 
 
 6.414 
 
 6.424 
 
 
 6 
 
 6.435 
 
 6.445 
 
 6.456 
 
 6.466 
 
 6.476 
 
 6.487 
 
 6.497 
 
 6.508 
 
 6.518 
 
 6.529 
 
 
 7 
 
 6.539 
 
 6.550 
 
 6.560 
 
 6.571 
 
 6.581 
 
 6.592 
 
 6.602 
 
 6.613 
 
 6.623 
 
 6.634 
 
 11 
 
 8 
 
 6.645 
 
 6.655 
 
 6.666 
 
 6.677 
 
 6.687 
 
 6.698 
 
 6.708 
 
 6.719 
 
 6.730 
 
 6.741 
 
 
 9 
 
 6.751 
 
 6.762 
 
 6.773 
 
 6.783 
 
 6.794 
 
 6.805 
 
 6.816 
 
 6.827 
 
 6.837 
 
 6.848 
 
 
 1.90 
 
 6.859 
 
 6.870 
 
 6.881 
 
 6.892 
 
 6.902 
 
 6.913 
 
 6.924 
 
 6.935 
 
 6.946 
 
 6.957 
 
 
 1 
 
 6.968 
 
 6.979 
 
 6.990 
 
 7.001 
 
 7.012 
 
 7.023 
 
 7.034 
 
 7.045 
 
 7.056 
 
 7.067 
 
 
 2 
 
 7.078 
 
 7.089 
 
 7.100 
 
 7.1 11 
 
 7.122 
 
 7.133 
 
 7.144 
 
 7.156 
 
 7.167 
 
 7.178 
 
 
 3 
 
 7.189 
 
 7.200 
 
 7.211 
 
 7.223 
 
 7.234 
 
 7.245 
 
 7.256 
 
 7.268 
 
 7.279 
 
 7.290 
 
 
 4 
 
 7.301 
 
 7.313 
 
 7.324 
 
 7.335 
 
 7.347 
 
 7.358 
 
 7.369 
 
 7.381 
 
 7.392 
 
 7.403 
 
 
 1.95 
 
 7.415 
 
 7.426 
 
 7.438 
 
 7.449 
 
 7.461 
 
 7.472 
 
 7.484 
 
 7.495 
 
 7.507 
 
 7.518 
 
 17 
 
 6 
 
 7.530 
 
 7.541 
 
 7.553 
 
 7.564 
 
 7.576 
 
 7.587 
 
 7.599 
 
 7.610 
 
 7.622 
 
 7.634 
 
 
 7 
 
 7.645 
 
 7.657 
 
 7.669 
 
 7.680 
 
 7.692 
 
 7.704 
 
 7.715 
 
 7.727 
 
 7.739 
 
 7.751 
 
 
 8 
 
 7.762 
 
 7.774 
 
 7.786 
 
 7.798 
 
 7.810 
 
 7.821 
 
 7.833 
 
 7.845 
 
 7.857 
 
 7.869 
 
 
 9 
 
 7.881 
 
 7.892 
 
 7.904 
 
 7.916 
 
 7.928 
 
 7.940 
 
 7.952 
 
 7.964 
 
 7.976 
 
 7.988 
 
 
 = 31.0063 I/T = 0.0322515 + 
 
10 
 
 CUBES (continued) 
 
 MATHEMATICAL TABLES 
 
 N 
 
 C 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 *S 
 
 <'-3 
 
 2.00 
 
 8.000 
 
 8.012 
 
 8.024 
 
 8.036 
 
 8.048 
 
 8.060 
 
 8.072 
 
 8.084 
 
 8.096 
 
 8.108 
 
 12 
 
 1 
 
 8.';21 
 
 8.133 
 
 8.145 
 
 8.157 
 
 8.169 
 
 8.181 
 
 8.194 
 
 8.206 
 
 8.218 
 
 8.230 
 
 
 2 
 
 8.242 
 
 8.255 
 
 8.267 
 
 8.279 
 
 8.291 
 
 8.304 
 
 8.316 
 
 8.328 
 
 8.341 
 
 8.353 
 
 
 3 
 
 8.365 
 
 8.378 
 
 8.390 
 
 8.403 
 
 8.415 
 
 8.427 
 
 8.440 
 
 8.452 
 
 8.465 
 
 8.477 
 
 
 4 
 
 8.490 
 
 8.502 
 
 8.515 
 
 8.527 
 
 8.540 
 
 8.552 
 
 8.565 
 
 8.577 
 
 8.590 
 
 8.603 
 
 
 2.05 
 
 8.615 
 
 8.628 
 
 8.640 
 
 8.653 
 
 8.666 
 
 8.678 
 
 8.691 
 
 8.704 
 
 8.716 
 
 8.729 
 
 13 
 
 6 
 
 8.742 
 
 8.755 
 
 8.767 
 
 8.780 
 
 8.793 
 
 8.806 
 
 8.818 
 
 8.831 
 
 8.844 
 
 8.857 
 
 
 7 
 
 8.870 
 
 8.883 
 
 8.895 
 
 8.908 
 
 8.921 
 
 8.934 
 
 8.947 
 
 8.960 
 
 8.973 
 
 8.986 
 
 
 8 
 
 8.999 
 
 9.012 
 
 9.025 
 
 9.038 
 
 9.051 
 
 9.064 
 
 9.077 
 
 9.090 
 
 9.103 
 
 9.116 
 
 
 9 
 
 9.129 
 
 9.142 
 
 9.156 
 
 9.169 
 
 9.182 
 
 9.195 
 
 9.208 
 
 9.221 
 
 9.235 
 
 9.248 
 
 
 2.10 
 
 9.261 
 
 9.274 
 
 9.287 
 
 9.301 
 
 9.314 
 
 9.327 
 
 9.341 
 
 9.354 
 
 9.367 
 
 9.381 
 
 
 1 
 
 9.394 
 
 9.407 
 
 9.421 
 
 9.434 
 
 9.447 
 
 9.461 
 
 9.474 
 
 9.488 
 
 9.501 
 
 9.515 
 
 
 2 
 
 9.528 
 
 9.542 
 
 9.555 
 
 9.569 
 
 9.582 
 
 9.596 
 
 9.609 
 
 9.623 
 
 9.636 
 
 9.650 
 
 14 
 
 3 
 
 9.664 
 
 9.677 
 
 9.691 
 
 9.704 
 
 9.718 
 
 9.732 
 
 9.745 
 
 9.759 
 
 9.773 
 
 9.787 
 
 
 4 
 
 9.800 
 
 9.814 
 
 9.828 
 
 9.842 
 
 9.855 
 
 9.869 
 
 9.883 
 
 9.897 
 
 9.911 
 
 9.925 
 
 
 2.15 
 
 9.938 
 
 9.952 
 
 9.966 
 
 9.980 
 
 9.994 
 
 10.008 
 
 
 
 
 
 14 
 
 2.1 
 
 
 
 
 
 
 9.94 
 
 10.08 
 
 10.22 
 
 10.36 
 
 10.50 
 
 14 
 
 2 
 
 10.65 
 
 10.79 
 
 10.94 
 
 11.09 
 
 11.24 
 
 11.39 
 
 11.54 
 
 11.70 
 
 11.85 
 
 12.01 
 
 15 
 
 3 
 
 12.17 
 
 12.33 
 
 12.49 
 
 12.65 
 
 12.81 
 
 12.98 
 
 13.14 
 
 13.31 
 
 13.48 
 
 13.65 
 
 16 
 
 4 
 
 13.82 
 
 14.00 
 
 14.17 
 
 14.35 
 
 14.53 
 
 14.71 
 
 14.89 
 
 15.07 
 
 15.25 
 
 15.44 
 
 18 
 
 2.5 
 
 15.62 
 
 15.81 
 
 16.00 
 
 16.19 
 
 16.39 
 
 16.58 
 
 16.78 
 
 16.97 
 
 17.17 
 
 17.37 
 
 20 
 
 6 
 
 17.58 
 
 17.78 
 
 17.98 
 
 18.19 
 
 18.40 
 
 18.61 
 
 18.82 
 
 19.03 
 
 19.25 
 
 19.47 
 
 21 
 
 7 
 
 19.68 
 
 19.90 
 
 20.12 
 
 20.35 
 
 20.57 
 
 20.80 
 
 21.02 
 
 21.25 
 
 21.48 
 
 21.72 
 
 23 
 
 8 
 
 21.95 
 
 22.19 
 
 22.43 
 
 22.67 
 
 22.91 
 
 23.15 
 
 23.39 
 
 23.64 
 
 23.89 
 
 24.14 
 
 24 
 
 9 
 
 24.39 
 
 24.64 
 
 24.90 
 
 25.15 
 
 25.41 
 
 25.67 
 
 25.93 
 
 26.20 
 
 26.46 
 
 26.73 
 
 26 
 
 3.0 
 
 27.00 
 
 27.27 
 
 27.54 
 
 27.82 
 
 28.09 
 
 28.37 
 
 28.65 
 
 28.93 
 
 29.22 
 
 29.50 
 
 28 
 
 1 
 
 29.79 
 
 30.08 
 
 30.37 
 
 30.66 
 
 30.96 
 
 31.26 
 
 31.55 
 
 31.86 
 
 32.16 
 
 32.46 
 
 30 
 
 2 
 
 32.77 
 
 33.08 
 
 33.39 
 
 33.70 
 
 34.01 
 
 34.33 
 
 34.65 
 
 34.97 
 
 35.29 
 
 35.61 
 
 32 
 
 3 
 
 35.94 
 
 36.26 
 
 36.59 
 
 36.93 
 
 37.26 
 
 37.60 
 
 37.93 
 
 38.27 
 
 38.61 
 
 38.96 
 
 34 
 
 4 
 
 39.30 
 
 39.65 
 
 40.00 
 
 40.35 
 
 40.71 
 
 41.06 
 
 41.42 
 
 41.78 
 
 42.14 
 
 42.51 
 
 36 
 
 3.5 
 
 42.88 
 
 43.24 
 
 43.61 
 
 43.99 
 
 44.36 
 
 44.74 
 
 45.12 
 
 45.50 
 
 45.88 
 
 46.27 
 
 39 
 
 6 
 
 46.66 
 
 47.05 
 
 47.44 
 
 47.83 
 
 48.23 
 
 48.63 
 
 49.03 
 
 49.43 
 
 49.84 
 
 50.24 
 
 40 
 
 7 
 
 50.65 
 
 51.06 
 
 51.48 
 
 51.90 
 
 52.31 
 
 52.73 
 
 53.16 
 
 53.58 
 
 54.01 
 
 54.44 
 
 42 
 
 8 
 
 54.87 
 
 55.31 
 
 55.74 
 
 56.18 
 
 56.62 
 
 57.07 
 
 57.51 
 
 57.96 
 
 58.41 
 
 58.86 
 
 44 
 
 9 
 
 59.32 
 
 59.78 
 
 60.24 
 
 60.70 
 
 61.16 
 
 61.63 
 
 62.10 
 
 62.57 
 
 63.04 
 
 63.52 
 
 47 
 
 4.0 
 
 64.00 
 
 64.48 
 
 64.96 
 
 65.45 
 
 65.94 
 
 66.43 
 
 66.92 
 
 67.42 
 
 67.92 
 
 68.42 
 
 49 
 
 1 
 
 68.92 
 
 69.43 
 
 69.93 
 
 70.44 
 
 70.96 
 
 71.47 
 
 71.99 
 
 72.51 
 
 73.03 
 
 73.56 
 
 52 
 
 2 
 
 74.09 
 
 74.62 
 
 75.15 
 
 75.69 
 
 76.23 
 
 76.77 
 
 77.31 
 
 77.85 
 
 78.40 
 
 78.95 
 
 54 
 
 3 
 
 79.51 
 
 80.06 
 
 80.62 
 
 81.18 
 
 81.75 
 
 82.31 
 
 82.88 
 
 83.45 
 
 84.03 
 
 84.60 
 
 58 
 
 4 
 
 85.18 
 
 85.77 
 
 86.35 
 
 86.94 
 
 87.53 
 
 88.12 
 
 88.72 
 
 89.31 
 
 89.92 
 
 90.52 
 
 59 
 
 4.5 
 
 91.12 
 
 91.73 
 
 92.35 
 
 92.96 
 
 93.58 
 
 94.20 
 
 94.82 
 
 95.44 
 
 96.07 
 
 96.70 
 
 62 
 
 6 
 
 97.34 
 
 97.97 
 
 98.61 
 
 99.25 
 
 99.90 
 
 100.54 
 
 
 
 
 
 64 
 
 6 
 
 
 
 
 
 
 100.5 
 
 101.2 
 
 101.8 
 
 102.5 
 
 103.2 
 
 7 
 
 7 
 
 103.8 
 
 104.5 
 
 105.2 
 
 105.8 
 
 106.5 
 
 107.2 
 
 107.9 
 
 108.5 
 
 109.2 
 
 109.9 
 
 7 
 
 8 
 
 110.6 
 
 111.3 
 
 112.0 
 
 112.7 
 
 113.4 
 
 114.1 
 
 114.8 
 
 115.5 
 
 116.2 
 
 116.9 
 
 7 
 
 9 
 
 117.6 
 
 118.4 
 
 119.1 
 
 119.8 
 
 120.6 
 
 121.3 
 
 122.0 
 
 122.8 
 
 123.5 
 
 124.3 
 
 7 
 
 Explanation of Table of Cubes (pp. 8-11). 
 
 This table gives the value of N* for values of N from 1 to 10, correct to four figures. 
 (Interpolated values may be in error by 1 in the fourth figure.) 
 
 To find the cube of a number N outside the range from 1 to 10, note that 
 moving the decimal point one place in column N is equivalent to moving it three 
 places in the body of the table. For example: 
 
 (4.852) = H4.2; (0.4852) = 0.1142; (485.2)3 = 114200000 
 
 This table may also be used inversely, to give cube roots. 
 
MATHEMATICAL TABLES 
 
 11 
 
 CUBES (continued) 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 11 
 
 5.0 
 
 125.0 
 
 125.8 
 
 126.5 
 
 127.3 
 
 128.0 
 
 128.8 
 
 129.6 
 
 130.3 
 
 131.1 
 
 131.9 
 
 8 
 
 1 
 
 132.7 
 
 133.4 
 
 134.2 
 
 135.0 
 
 135.8 
 
 136.6 
 
 137.4 
 
 138.2 
 
 139.0 
 
 139.8 
 
 
 2 
 
 140.6 
 
 141.4 
 
 142.2 
 
 143.1 
 
 143.9 
 
 144.7 
 
 145.5 
 
 146.4 
 
 147.2 
 
 148.0 
 
 
 3 
 
 148.9 
 
 149.7 
 
 150.6 
 
 151.4 
 
 152.3 
 
 153.1 
 
 154.0 
 
 154.9 
 
 155.7 
 
 156.6 
 
 9 
 
 4 
 
 157.5 
 
 158.3 
 
 159.2 
 
 160.1 
 
 161.0 
 
 161.9 
 
 162.8 
 
 163.7 
 
 164.6 
 
 165.5 
 
 
 5.5 
 
 166.4 
 
 167.3 
 
 168.2 
 
 169.1 
 
 170.0 
 
 171.0 
 
 171.9 
 
 172.8 
 
 173.7 
 
 174.7 
 
 
 6 
 
 175.6 
 
 176.6 
 
 177.5 
 
 178.5 
 
 179.4 
 
 180.4 
 
 181.3 
 
 182.3 
 
 183.3 
 
 184.2 
 
 10 
 
 7 
 
 185.2 
 
 186.2 
 
 187.1 
 
 188.1 
 
 189.1 
 
 190.1 
 
 191.1 
 
 192.1 
 
 193.1 
 
 194.1 
 
 
 8 
 
 195.1 
 
 196.1 
 
 197.1 
 
 198.2 
 
 199.2 
 
 200.2 
 
 201.2 
 
 202.3 
 
 203.3 
 
 204.3 
 
 
 9 
 
 205.4 
 
 206.4 
 
 207.5 
 
 208.5 
 
 209.6 
 
 210.6 
 
 211.7 
 
 212.8 
 
 213.8 
 
 214.9 
 
 
 6.0 
 
 216.0 
 
 217.1 
 
 218.2 
 
 219.3 
 
 220.3 
 
 221.4 
 
 222.5 
 
 223.6 
 
 224.8 
 
 225.9 
 
 II 
 
 1 
 
 227.0 
 
 228.1 
 
 229.2 
 
 230.3 
 
 231.5 
 
 232.6 
 
 233.7 
 
 234.9 
 
 236.0 
 
 237.2 
 
 
 2 
 
 238.3 
 
 239.5 
 
 240.6 
 
 241.8 
 
 243.0 
 
 244.1 
 
 245.3 
 
 246.5 
 
 247.7 
 
 248.9 
 
 12 
 
 3 
 
 250.0 
 
 251.2 
 
 252.4 
 
 253.6 
 
 254.8 
 
 256.0 
 
 257.3 
 
 258.5 
 
 259.7 
 
 260.9 
 
 
 4 
 
 262.1 
 
 263.4 
 
 264.6 
 
 265.8 
 
 267.1 
 
 268.3 
 
 269.6 
 
 270.8 
 
 272.1 
 
 273.4 
 
 
 6.5 
 
 274.6 
 
 275.9 
 
 277.2 
 
 278.4 
 
 279.7 
 
 281.0 
 
 282.3 
 
 283.6 
 
 284.9 
 
 286.2 
 
 13 
 
 6 
 
 287.5 
 
 288.8 
 
 290.1 
 
 291.4 
 
 292.8 
 
 294.1 
 
 295.4 
 
 296.7 
 
 298.1 
 
 299.4 
 
 
 7 
 
 300.8 
 
 302.1 
 
 303.5 
 
 304.8 
 
 306.2 
 
 307.5 
 
 308.9 
 
 310.3 
 
 311.7 
 
 313.0 
 
 14 
 
 8 
 
 314.4 
 
 315.8 
 
 317.2 
 
 318.6 
 
 320.0 
 
 321.4 
 
 322.8 
 
 324.2 
 
 325.7 
 
 327.1 
 
 
 9 
 
 328.5 
 
 329.9 
 
 331.4 
 
 332.8 
 
 334.3 
 
 335.7 
 
 337.2 
 
 338.6 
 
 340.1 
 
 341.5 
 
 
 7.0 
 
 343.0 
 
 344.5 
 
 345.9 
 
 347.4 
 
 348.9 
 
 350.4 
 
 351.9 
 
 353.4 
 
 354.9 
 
 356.4 
 
 15 
 
 1 
 
 357.9 
 
 359.4 
 
 360.9 
 
 362.5 
 
 364.0 
 
 365.5 
 
 367.1 
 
 368.6 
 
 370.1 
 
 371.7 
 
 
 2 
 
 373.2 
 
 374.8 
 
 376.4 
 
 377.9 
 
 379.5 
 
 381.1 
 
 382.7 
 
 384.2 
 
 385.8 
 
 387.4 
 
 16 
 
 3 
 
 389.0 
 
 390.6 
 
 392.2 
 
 393.8 
 
 395.4 
 
 397.1 
 
 398.7 
 
 400.3 
 
 401.9 
 
 403.6 
 
 
 4 
 
 405.2 
 
 406.9 
 
 408.5 
 
 410.2 
 
 411.8 
 
 413.5 
 
 415.2 
 
 416.8 
 
 418.5 
 
 420.2 
 
 17 
 
 7.5 
 
 421.9 
 
 423.6 
 
 425.3 
 
 427.0 
 
 428.7 
 
 430.4 
 
 432.1 
 
 433.8 
 
 435.5 
 
 437.2 
 
 
 6 
 
 439.0 
 
 440.7 
 
 442.5 
 
 444.2 
 
 445.9 
 
 447.7 
 
 449.5 
 
 451.2 
 
 453.0 
 
 454.8 
 
 18 
 
 7 
 
 456.5 
 
 458.3 
 
 460.1 
 
 461.9 
 
 463.7 
 
 465.5 
 
 467.3 
 
 469.1 
 
 470.9 
 
 472.7 
 
 
 8 
 
 474.6 
 
 476.4 
 
 478.2 
 
 480.0 
 
 481.9 
 
 483.7 
 
 485.6 
 
 487.4 
 
 489.3 
 
 491.2 
 
 
 9 
 
 493.0 
 
 494.9 
 
 496.8 
 
 498.7 
 
 500.6 
 
 502.5 
 
 504.4 
 
 506.3 
 
 508.2 
 
 510.1 
 
 19 
 
 8.0 
 
 512.0 
 
 513.9 
 
 515.8 
 
 517.8 
 
 519.7 
 
 521.7 
 
 523.6 
 
 525.6 
 
 527.5 
 
 529.5 
 
 
 1 
 
 531.4 
 
 533.4 
 
 535.4 
 
 537.4 
 
 539.4 
 
 541.3 
 
 543.3 
 
 545.3 
 
 547.3 
 
 549.4 
 
 20 
 
 2 
 
 551.4 
 
 553.4 
 
 555.4 
 
 557.4 
 
 559.5 
 
 561.5 
 
 563.6 
 
 565.6 
 
 567.7 
 
 569.7 
 
 
 3 
 
 571.8 
 
 573.9 
 
 575.9 
 
 578.0 
 
 580.1 
 
 582.2 
 
 584.3 
 
 586.4 
 
 588.5 
 
 590.6 
 
 21 
 
 4 
 
 592.7 
 
 594.8 
 
 596.9 
 
 599.1 
 
 601.2 
 
 603.4 
 
 605.5 
 
 607.6 
 
 609.8 
 
 612.0 
 
 
 8.5 
 
 614.1 
 
 616.3 
 
 618.5 
 
 620.7 
 
 622.8 
 
 625.0 
 
 627.2 
 
 629.4 
 
 631.6 
 
 633.8 
 
 22 
 
 6 
 
 636.1 
 
 638.3 
 
 640.5 
 
 642.7 
 
 645.0 
 
 647.2 
 
 649.5 
 
 651.7 
 
 654.0 
 
 656.2 
 
 
 7 
 
 658.5 
 
 660.8 
 
 663.1 
 
 665.3 
 
 667.6 
 
 669.9 
 
 672.2 
 
 674.5 
 
 676.8 
 
 679.2 
 
 23 
 
 8 
 
 681.5 
 
 683.8 
 
 686.1 
 
 688.5 
 
 690.8 
 
 693.2 
 
 695.5 
 
 697.9 
 
 700.2 
 
 702.6 
 
 24 
 
 9 
 
 705.0 
 
 707.3 
 
 709.7 
 
 712.1 
 
 714.5 
 
 716.9 
 
 719.3 
 
 721.7 
 
 724.2 
 
 726.6 
 
 
 9.0 
 
 729.0 
 
 731.4 
 
 733.9 
 
 736.3 
 
 738.8 
 
 741.2 
 
 743.7 
 
 746.1 
 
 748.6 
 
 751.1 
 
 25 
 
 1 
 
 753.6 
 
 756.1 
 
 758.6 
 
 761.0 
 
 763.6 
 
 766.1 
 
 768.6 
 
 771.1 
 
 773.6 
 
 776.2 
 
 
 2 
 
 778.7 
 
 781.2 
 
 783.8 
 
 786.3 
 
 788.9 
 
 791.5 
 
 794.0 
 
 796.6 
 
 799.2 
 
 801.8 
 
 26 
 
 3 
 
 804.4 
 
 807.0 
 
 809.6 
 
 812.2 
 
 814.8 
 
 817.4 
 
 820.0 
 
 822.7 
 
 825.3 
 
 827.9 
 
 
 4 
 
 830.6 
 
 833.2 
 
 835.9 
 
 838.6 
 
 841.2 
 
 843.9 
 
 846.6 
 
 849.3 
 
 852.0 
 
 854.7 
 
 27 
 
 9.5 
 
 857.4 
 
 860.1 
 
 862.8 
 
 865.5 
 
 868.3 
 
 871.0 
 
 873.7 
 
 876.5 
 
 879.2 
 
 882.0 
 
 
 6 
 
 884.7 
 
 887.5 
 
 890.3 
 
 893.1 
 
 895.8 
 
 898.6 
 
 901.4 
 
 904.2 
 
 907.0 
 
 909.9 
 
 28 
 
 7 
 
 912.7 
 
 915.5 
 
 918.3 
 
 921.2 
 
 924.0 
 
 926.9 
 
 929.7 
 
 932.6 
 
 935.4 
 
 938.3 
 
 
 8 
 
 941.2 
 
 944.1 
 
 947.0 
 
 949.9 
 
 952.8 
 
 955.7 
 
 958.6 
 
 961.5 
 
 964.4 
 
 967.4 
 
 29 
 
 9 
 
 970.3 
 
 973.2 
 
 976.2 
 
 979.1 
 
 982.1 
 
 985.1 
 
 988.0 
 
 991.0 
 
 994.0 
 
 997.0 
 
 
 10.0 
 
 1000.0 
 
 
 
 
 
 
 
 
 
 
 
 = 3 1.0063 
 
 = 0.0322515 + 
 
 Moving the decimal point ONE place in N requires moving it THREE places in body of 
 table (see p. 10). 
 
12 MATHEMATICAL TABLES 
 
 SQUARE ROOTS OF NUMBERS 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 <*)ttt 
 & 
 
 1.0 
 
 1.000 
 
 1.005 
 
 1.010 
 
 1.015 
 
 1.020 
 
 1.025 
 
 1.030 
 
 1.034 
 
 1.039 
 
 1.044 
 
 5 
 
 1 
 
 1.049 
 
 1.054 
 
 1.058 
 
 1.063 
 
 1.068 
 
 1.072 
 
 1.077 
 
 1.082 
 
 1.086 
 
 1.091 
 
 
 2 
 
 1.095 
 
 1.100 
 
 1.105 
 
 1.109 
 
 1.114 
 
 1.118 
 
 1.122 
 
 1.127 
 
 1.131 
 
 1.136 
 
 4 
 
 3 
 
 1.140 
 
 1.145 
 
 1.149 
 
 1.153 
 
 1.158 
 
 1.162 
 
 1.166 
 
 1.170 
 
 1.175 
 
 1.179 
 
 
 4 
 
 1.183 
 
 1.187 
 
 1.192 
 
 1.196 
 
 1.200 
 
 1.204 
 
 1.208 
 
 1.212 
 
 1.217 
 
 1.221 
 
 
 1.5 
 
 1.225 
 
 1.229 
 
 1.233 
 
 1.237 
 
 1.241 
 
 1.245 
 
 1.249 
 
 1.253 
 
 1.257 
 
 1.261 
 
 
 6 
 
 1.265 
 
 1.269 
 
 1.273 
 
 1.277 
 
 1.281 
 
 1.285 
 
 1.288 
 
 1.292 
 
 1.296 
 
 1.300 
 
 
 7 
 
 1.304 
 
 1.308 
 
 1.311 
 
 1.315 
 
 1.319 
 
 1.323 
 
 1.327 
 
 1.330 
 
 1.334 
 
 1.338 
 
 
 8 
 
 1.342 
 
 1.345 
 
 1.349 
 
 1.353 
 
 1.356 
 
 1.360 
 
 1.364 
 
 1.367 
 
 1.371 
 
 1.375 
 
 
 9 
 
 1.378 
 
 1.382 
 
 1.386 
 
 1.389 
 
 1.393 
 
 1.396 
 
 1.400 
 
 1.404 
 
 1.407 
 
 1.411 
 
 
 2.0 
 
 1.414 
 
 1.418 
 
 1.421 
 
 1.425 
 
 1.428 
 
 1.432 
 
 1.435 
 
 1.439 
 
 1.442 
 
 1.446 
 
 
 1 
 
 1.449 
 
 1.453 
 
 1.456 
 
 1.459 
 
 1.463 
 
 1.466 
 
 1.470 
 
 1.473 
 
 1.476 
 
 1.480 
 
 3 
 
 2 
 
 1.483 
 
 1.487 
 
 1.490 
 
 1.493 
 
 1.497 
 
 1.500 
 
 1.503 
 
 1.507 
 
 1.510 
 
 1.513 
 
 
 3 
 
 1.517 
 
 1.520 
 
 1.523 
 
 1.526 
 
 1.530 
 
 1.533 
 
 1.536 
 
 1.539 
 
 1.543 
 
 1.546 
 
 
 4 
 
 1.549 
 
 1.552 
 
 1.556 
 
 1.559 
 
 1.562 
 
 1.565 
 
 1.568 
 
 1.572 
 
 1.575 
 
 1.578 
 
 
 2.5 
 
 1.581 
 
 1.584 
 
 1.587 
 
 1.591 
 
 1.594 
 
 1.597 
 
 1.600 
 
 1.603 
 
 1.606 
 
 1.609 
 
 
 6 
 
 1.612 
 
 1.616 
 
 1.619 
 
 1.622 
 
 1.625 
 
 1.628 
 
 1.631 
 
 1.634 
 
 1.637 
 
 1.640 
 
 
 7 
 
 1.643 
 
 1.646 
 
 1.649 
 
 1.652 
 
 1.655 
 
 1.658 
 
 1.661 
 
 1.664 
 
 1.667 
 
 1.670 
 
 
 8 
 
 1.673 
 
 1.676 
 
 1.679 
 
 1.682 
 
 1.685 
 
 1.688 
 
 1.691 
 
 1.694 
 
 1.697 
 
 1.700 
 
 
 9 
 
 1.703 
 
 1.706 
 
 1.709 
 
 1.712 
 
 1.715 
 
 1.718 
 
 1.720 
 
 1.723 
 
 1.726 
 
 1.729 
 
 
 3.0 
 
 1.732 
 
 1.735 
 
 1.738 
 
 1.741 
 
 1.744 
 
 1.746 
 
 1.749 
 
 1.752 
 
 1.755 
 
 1.758 
 
 
 1 
 
 1.761 
 
 1.764 
 
 1.766 
 
 1.769 
 
 1.772 
 
 1.775 
 
 1.778 
 
 1.780 
 
 1.783 
 
 1.786 
 
 
 2 
 
 1.789 
 
 1.792 
 
 1.794 
 
 1.797 
 
 1.800 
 
 1.803 
 
 1.806 
 
 1.808 
 
 1.811 
 
 1.814 
 
 
 3 
 
 1.817 
 
 1.819 
 
 1.822 
 
 1.825 
 
 1.828 
 
 1.830 
 
 1.833 
 
 1.836 
 
 1.838 
 
 1.841 
 
 
 4 
 
 1.844 
 
 1.847 
 
 1.849 
 
 1.852 
 
 1.855 
 
 1.857 
 
 1.860 
 
 1.863 
 
 1.865 
 
 1.868 
 
 
 3.5 
 
 1.871 
 
 1.873 
 
 1.876 
 
 1.879 
 
 1.881 
 
 1.884 
 
 1.887 
 
 1.889 
 
 1.892 
 
 1.895 
 
 
 6 
 
 1.897 
 
 1.900 
 
 1.903 
 
 1.905 
 
 1.908 
 
 1.910 
 
 1.913 
 
 1.916 
 
 1.918 
 
 1.921 
 
 
 7 
 
 1.924 
 
 1.926 
 
 1.929 
 
 1.931 
 
 1.934 
 
 1.936 
 
 1.939 
 
 1.942 
 
 1.944 
 
 1.947 
 
 
 8 
 
 1.949 
 
 1.952 
 
 1.954 
 
 1.957 
 
 1.960 
 
 1.962 
 
 1.965 
 
 1.967 
 
 1.970 
 
 1.972 
 
 
 9 
 
 1.975 
 
 1.977 
 
 1.980 
 
 1.982 
 
 1.985 
 
 1.987 
 
 1.990 
 
 1.992 
 
 1.995 
 
 1.997 
 
 
 4.0 
 
 2.000 
 
 2.002 
 
 2.005 
 
 2.007 
 
 2.010 
 
 2.012 
 
 2.015 
 
 2.017 
 
 2.020 
 
 2.022 
 
 
 1 
 
 2.025 
 
 2.027 
 
 2.030 
 
 2.032 
 
 2.035 
 
 2.037 
 
 2.040 
 
 2.042 
 
 2.045 
 
 2.047 
 
 2 
 
 2 
 
 2.049 
 
 2.052 
 
 2.054 
 
 2.057 
 
 2.059 
 
 2.062 
 
 2.064 
 
 2.066 
 
 2.069 
 
 2.071 
 
 
 3 
 
 2.074 
 
 2.076 
 
 2.078 
 
 2.081 
 
 2.083 
 
 2.086 
 
 2.088 
 
 2.090 
 
 2.093 
 
 2.095 
 
 
 4 
 
 2.098 
 
 2.100 
 
 2.102 
 
 2.105 
 
 2.107 
 
 2.110 
 
 2.112 
 
 2.114 
 
 2.117 
 
 2.119 
 
 
 4.5 
 
 2.121 
 
 2.124 
 
 2.126 
 
 2.128 
 
 2.131 
 
 2.133 
 
 2.135 
 
 2.138 
 
 2.140 
 
 2.142 
 
 
 6 
 
 2.145 
 
 2.147 
 
 2.149 
 
 2.152 
 
 2.154 
 
 2.156 
 
 2.159 
 
 2.161 
 
 2.163 
 
 2.166 
 
 
 7 
 
 2.168 
 
 2.170 
 
 2.173 
 
 2.175 
 
 2.177 
 
 2.179 
 
 2.182 
 
 2.184 
 
 2.186 
 
 2.189 
 
 
 8 
 
 2.191 
 
 2.193 
 
 2.195 
 
 2.198 
 
 2.200 
 
 2.202 
 
 2.205 
 
 2.207 
 
 2.209 
 
 2.211 
 
 
 9 
 
 2.214 
 
 2.216 
 
 2.218 
 
 2.220 
 
 2.223 
 
 2.225 
 
 2.227 
 
 2.229 
 
 2.232 
 
 2.234 
 
 
 yV= 1.77245 + 
 
 = 0.56419 
 
 1.25331 
 
 1.64872 
 
 Explanation of Table of Square Roots (pp. 12-15). 
 
 This table gives the values of \/N for values of N from 1 to 100, correct to four figures. 
 (Interpolated values may be in error by 1 in the fourth figure.) 
 
 To find the square root of a number N outside the range from 1 to 100, divide 
 the digits of the number into blocks of two (beginning with the decimal point), and note 
 that moving the decimal point two places in N is equivalent to moving it one place in 
 the square root of N. For example: 
 
 X/2.718 = 1.648; A/271. 8 - 16.48; V0.0002718 = 0.01648; 
 V27.18 => 5.213; -N/2718 - 52.13; V / oTo02718 - 0.05213. 
 
MATHEMATICAL TABLES 
 SQUARE ROOTS (continued) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 
 
 
 
 <* 
 
 5.0 
 
 2.236 
 
 2.238 
 
 2.241 
 
 2.243 
 
 2.245 
 
 2.247 
 
 2.249 
 
 2.252 
 
 2.254 
 
 2.256 
 
 2 
 
 1 
 
 2.258 
 
 2.261 
 
 2.263 
 
 2.265 
 
 2.267 
 
 2.269 
 
 2.272 
 
 2.274 
 
 2.276 
 
 2.278 
 
 
 2 
 
 2.280 
 
 2.283 
 
 2.285 
 
 2.287 
 
 2.289 
 
 2.291 
 
 2.293 
 
 2.296 
 
 2.298 
 
 2.300 
 
 
 3 
 
 2.302 
 
 2.304 
 
 2.307 
 
 2.309 
 
 2.311 
 
 2.313 
 
 2.315 
 
 2.317 
 
 2.319 
 
 2.322 
 
 
 4 
 
 2.324 
 
 2.326 
 
 2.328 
 
 2.330 
 
 2.332 
 
 2.335 
 
 2.337 
 
 2.339 
 
 2.341 
 
 2.343 
 
 
 6.5 
 
 2.345 
 
 2.347 
 
 2.349 
 
 2.352 
 
 2.354 
 
 2.356 
 
 2.358 
 
 2.360 
 
 2.362 
 
 2.364 
 
 
 6 
 
 2.366 
 
 2.369 
 
 2.371 
 
 2.373 
 
 2.375 
 
 2.377 
 
 2.379 
 
 2.381 
 
 2.383 
 
 2.385 
 
 
 7 
 
 2.387 
 
 2.390 
 
 2.392 
 
 2.394 
 
 2.396 
 
 2.398 
 
 2.400 
 
 2.402 
 
 2.404 
 
 2.406 
 
 
 8 
 
 2.408 
 
 2.410 
 
 2.412 
 
 2.415 
 
 2.417 
 
 2.419 
 
 2.421 
 
 2.423 
 
 2.425 
 
 2.427 
 
 
 9 
 
 2.429 
 
 2.431 
 
 2.433 
 
 2.435 
 
 2.437 
 
 2.439 
 
 2.441 
 
 2.443 
 
 2.445 
 
 2.447 
 
 
 6.0 
 
 2.449 
 
 2.452 
 
 2.454 
 
 2.456 
 
 2.458 
 
 2.460 
 
 2.462 
 
 2.464 
 
 2.466 
 
 2.468 
 
 
 1 
 
 2.470 
 
 2.472 
 
 2.474 
 
 2.476 
 
 2.478 
 
 2.480 
 
 2.482 
 
 2.484 
 
 2.486 
 
 2.488 
 
 
 2 
 
 2.490 
 
 2.492 
 
 2.494 
 
 2.496 
 
 2.498 
 
 2.500 
 
 2.502 
 
 2.504 
 
 2.506 
 
 2.508 
 
 
 3 
 
 2.510 
 
 2.512 
 
 2.514 
 
 2.516 
 
 2.518 
 
 2.520 
 
 2.522 
 
 2.524 
 
 2.526 
 
 2.528 
 
 
 4 
 
 2.530 
 
 2.532 
 
 2.534 
 
 2.536 
 
 2.538 
 
 2.540 
 
 2.542 
 
 2.544 
 
 2.546 
 
 2.548 
 
 
 6.6 
 
 2.550 
 
 2.551 
 
 2.553 
 
 2.555 
 
 2.557 
 
 2.559 
 
 2.561 
 
 2.563 
 
 2.565 
 
 2.567 
 
 
 6 
 
 2.569 
 
 2.571 
 
 2.573 
 
 2.575 
 
 2.577 
 
 2.579 
 
 2.581 
 
 2.583 
 
 2.585 
 
 2.587 
 
 
 7 
 
 2.588 
 
 2.590 
 
 2.592 
 
 2.594 
 
 2.596 
 
 2.598 
 
 2.600 
 
 2.602 
 
 2.604 
 
 2.606 
 
 
 8 
 
 2.608 
 
 2.610 
 
 2.612 
 
 2.613 
 
 2.615 
 
 2.617 
 
 2.619 
 
 2.621 
 
 2.623 
 
 2.625 
 
 
 9 
 
 2.627 
 
 2.629 
 
 2.631 
 
 2.632 
 
 2.634 
 
 2.636 
 
 2.638 
 
 2.640 
 
 2.642 
 
 2.644 
 
 
 7.0 
 
 2.646 
 
 2.648 
 
 2.650 
 
 2.651 
 
 2.653 
 
 2.655 
 
 2.657 
 
 2.659 
 
 2.661 
 
 2.663 
 
 
 1 
 
 2.665 
 
 2.666 
 
 2.668 
 
 2.670 
 
 2.672 
 
 2.674 
 
 2.676 
 
 2.678 
 
 2.680 
 
 2.681 
 
 
 2 
 
 2.683 
 
 2.685 
 
 2.687 
 
 2.689 
 
 2.691 
 
 2.693 
 
 2.694 
 
 2.696 
 
 2.698 
 
 2.700 
 
 
 3 
 
 2.702 
 
 2.704 
 
 2.706 
 
 2.707 
 
 2.709 
 
 2.711 
 
 2.713 
 
 2.715 
 
 2.717 
 
 2.718 
 
 
 4 
 
 2.720 
 
 2.722 
 
 2.724 
 
 2.726 
 
 2.728 
 
 2.729 
 
 2.731 
 
 2.733 
 
 2.735 
 
 2.737 
 
 
 7.5 
 
 2.739 
 
 2.740 
 
 2.742 
 
 2.744 
 
 2.746 
 
 2.748 
 
 2.750 
 
 2.751 
 
 2.753 
 
 2.755 
 
 
 6 
 
 2.757 
 
 2.759 
 
 2.760 
 
 2.762 
 
 2.764 
 
 2.766 
 
 2.768 
 
 2.769 
 
 2.771 
 
 2.773 
 
 
 7 
 
 2.775 
 
 2.777 
 
 2.778 
 
 2.780 
 
 2.782 
 
 2.784 
 
 2.786 
 
 2.787 
 
 2.789 
 
 2.791 
 
 
 8 
 
 2.793 
 
 2.795 
 
 2.796 
 
 2.798 
 
 2.800 
 
 2.802 
 
 2.804 
 
 2.805 
 
 2.807 
 
 2.809 
 
 
 9 
 
 2.811 
 
 2.812 
 
 2.814 
 
 2.816 
 
 2.818 
 
 2.820 
 
 2.821 
 
 2.823 
 
 2.825 
 
 2.827 
 
 
 8.0 
 
 2.828 
 
 2.830 
 
 2.832 
 
 2.834 
 
 2.835 
 
 2.837 
 
 2.839 
 
 2.841 
 
 2.843 
 
 2.844 
 
 
 
 2.846 
 
 2.848 
 
 2.850 
 
 2.851 
 
 2.853 
 
 2.855 
 
 2.857 
 
 2.858 
 
 2.860 
 
 2.862 
 
 
 2 
 
 2.864 
 
 2.865 
 
 2.867 
 
 2.869 
 
 2.871 
 
 2.872 
 
 2.874 
 
 2.876 
 
 2.877 
 
 2.879 
 
 
 3 
 
 2.881 
 
 2.883 
 
 2.884 
 
 2.886 
 
 2.888 
 
 2.890 
 
 2.891 
 
 2.893 
 
 2.895 
 
 2.897 
 
 
 4 
 
 2.898 
 
 2.900 
 
 2.902 
 
 2.903 
 
 2.905 
 
 2.907 
 
 2.909 
 
 2.910 
 
 2.912 
 
 2.914 
 
 
 8.5 
 
 2.915 
 
 2.917 
 
 2.919 
 
 2.921 
 
 2.922 
 
 2.924 
 
 2.926 
 
 2.927 
 
 2.929 
 
 2.931 
 
 
 6 
 
 2.933 
 
 2.934 
 
 2.936 
 
 2.938 
 
 2.939 
 
 2.941 
 
 2.943 
 
 2.944 
 
 2.946 
 
 2.948 
 
 
 7 
 
 2.950 
 
 2.951 
 
 2.953 
 
 2.955 
 
 2.956 
 
 2.958 
 
 2.960 
 
 2.961 
 
 2.963 
 
 2.965 
 
 
 8 
 
 2.966 
 
 2.968 
 
 2.970 
 
 2.972 
 
 2.973 
 
 2.975 
 
 2.977 
 
 2.978 
 
 2.980 
 
 2.982 
 
 
 9 
 
 2.983 
 
 2.985 
 
 2.987 
 
 2.988 
 
 2.990 
 
 2.992 
 
 2.993 
 
 2.995 
 
 2.997 
 
 2.998 
 
 
 9.0 
 
 3.000 
 
 3.002 
 
 3.003 
 
 3.005 
 
 3.007 
 
 3.008 
 
 3.010 
 
 3.012 
 
 3.013 
 
 3.015 
 
 
 1 
 
 3.017 
 
 3.018 
 
 3.020 
 
 3.022 
 
 3.023 
 
 3.025 
 
 3.027 
 
 3.028 
 
 3.030 
 
 3.032 
 
 
 2 
 
 3.033 
 
 3.035 
 
 3.036 
 
 3.038 
 
 3.040 
 
 3.041 
 
 3.043 
 
 3.045 
 
 3.046 
 
 3.048 
 
 
 3 
 
 3.050 
 
 3.051 
 
 3.053 
 
 3.055 
 
 3.056 
 
 3.058 
 
 3.059 
 
 3.061 
 
 3.063 
 
 3.064 
 
 
 4 
 
 3.066 
 
 3.068 
 
 3.069 
 
 3.071 
 
 3.072 
 
 3.074 
 
 3.076 
 
 3.077 
 
 3.079 
 
 3.081 
 
 
 9.5 
 
 3.082 
 
 3.084 
 
 3.085 
 
 3.087 
 
 3.089 
 
 3.090 
 
 3.092 
 
 3.094 
 
 3.095 
 
 3.097 
 
 
 6 
 
 3.098 
 
 3.100 
 
 3.102 
 
 3.103 
 
 3.105 
 
 3.106 
 
 3.108 
 
 3.110 
 
 3.111 
 
 3.113 
 
 
 7 
 
 3.114 
 
 3.116 
 
 3.118 
 
 3.119 
 
 3.121 
 
 3.122 
 
 3.124 
 
 3.126 
 
 3.127 
 
 3.129 
 
 
 8 
 
 3.130 
 
 3.132 
 
 3.134 
 
 3J35 
 
 3.137 
 
 3.138 
 
 3.140 
 
 3.142 
 
 3.143 
 
 3.145 
 
 
 9 
 
 3.146 
 
 3.148 
 
 3.150 
 
 3.151 
 
 3.153 
 
 3.154 
 
 3.156 
 
 3.158 
 
 3.159 
 
 3.161 
 
 
 Moving the decimal point TWO places in N requires moving it ONE place in body 
 of table (see p. 12). 
 
14 MATHEMATICAL TABLES 
 
 SQUARE ROOTS (continued) 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 d 
 $* 
 
 10. 
 
 3.162 
 
 3.178 
 
 3.194 
 
 3.209 
 
 3.225 
 
 3.240 
 
 3.256 
 
 3.271 
 
 3.286 
 
 3.302 
 
 16 
 
 1. 
 
 3.317 
 
 3.332 
 
 3.347 
 
 3.362 
 
 3.376 
 
 3.391 
 
 3.406 
 
 3.421 
 
 3.435 
 
 3.450 
 
 15 
 
 2. 
 
 3.464 
 
 3.479 
 
 3.493 
 
 3.507 
 
 3.521 
 
 3.536 
 
 3.550 
 
 3.564 
 
 3.578 
 
 3.592 
 
 14 
 
 3. 
 
 3.606 
 
 3.619 
 
 3.633 
 
 3.647 
 
 3.661 
 
 3.674 
 
 3.688 
 
 3.701 
 
 3.715 
 
 3.728 
 
 
 4. 
 
 3.742 
 
 3.755 
 
 3.768 
 
 3.782 
 
 3.795 
 
 3.808 
 
 3.821 
 
 3.834 
 
 3.847 
 
 3.860 
 
 13 
 
 15. 
 
 3.873 
 
 3.886 
 
 3.899 
 
 3.912 
 
 3.924 
 
 3.937 
 
 3.950 
 
 3.962 
 
 3.975 
 
 3.987 
 
 
 6. 
 
 4.000 
 
 4.012 
 
 4.025 
 
 4.037 
 
 4.050 
 
 4.062 
 
 4.074 
 
 4.087 
 
 4.099 
 
 4.111 
 
 12 
 
 7. 
 
 4.123 
 
 4.135 
 
 4.147 
 
 4.159 
 
 4.171 
 
 4.183 
 
 4.195 
 
 4.207 
 
 4.219 
 
 4.231 
 
 
 8. 
 
 4.243 
 
 4.254 
 
 4.266 
 
 4.278 
 
 4.290 
 
 4.301 
 
 4.313 
 
 4.324 
 
 4.336 
 
 4.347 
 
 
 9. 
 
 4.359 
 
 4.370 
 
 4.382 
 
 4.393 
 
 4.405 
 
 4.416 
 
 4.427 
 
 4.438 
 
 4.450 
 
 4.461 
 
 11 
 
 20. 
 
 4.472 
 
 4.483 
 
 4.494 
 
 4.506 
 
 4.517 
 
 4.528 
 
 4.539 
 
 4.550 
 
 4.561 
 
 4.572 
 
 
 1. 
 
 4.583 
 
 4.593 
 
 4.604 
 
 4.615 
 
 4.626 
 
 4.637 
 
 4.648 
 
 4.658 
 
 4.669 
 
 4.680 
 
 
 2. 
 
 4.690 
 
 4.701 
 
 4.712 
 
 4.722 
 
 4.733 
 
 4.743 
 
 4.754 
 
 4.764 
 
 4.775 
 
 4.785 
 
 
 3. 
 
 4.796 
 
 4.806 
 
 4.817 
 
 4.827 
 
 4.837 
 
 4.848 
 
 4.858 
 
 4.868 
 
 4.879 
 
 4.889 
 
 10 
 
 4. 
 
 4.899 
 
 4.909 
 
 4.919 
 
 4.930 
 
 4.940 
 
 4.950 
 
 4.960 
 
 4.970 
 
 4.980 
 
 4.990 
 
 
 25. 
 
 5.000 
 
 5.010 
 
 5.020 
 
 5.030 
 
 5.040 
 
 5.050 
 
 5.060 
 
 5.070 
 
 5.079 
 
 5.089 
 
 
 6 
 
 5.099 
 
 5.109 
 
 5.119 
 
 5.128 
 
 5.138 
 
 5.148 
 
 5.158 
 
 5.167 
 
 5.177 
 
 5.187 
 
 
 7. 
 
 5.196 
 
 5.206 
 
 5.215 
 
 5.225 
 
 5.235 
 
 5.244 
 
 5.254 
 
 5.263 
 
 5.273 
 
 5.282 
 
 
 8. 
 
 5.292 
 
 5.301 
 
 5.310 
 
 5.320 
 
 5.329 
 
 5.339 
 
 5.348 
 
 5.357 
 
 5.367 
 
 5.376 
 
 9 
 
 9. 
 
 5.385 
 
 5.394 
 
 5.404 
 
 5.413 
 
 5.422 
 
 5.431 
 
 5.441 
 
 5.450 
 
 5.459 
 
 5.468 
 
 
 30. 
 
 5.477 
 
 5.486 
 
 5.495 
 
 5.505 
 
 5.514 
 
 5.523 
 
 5.532 
 
 5.541 
 
 5.550 
 
 5.559 
 
 
 
 5.568 
 
 5.577 
 
 5.586 
 
 5.595 
 
 5.604 
 
 5.612 
 
 5.621 
 
 5.630 
 
 5.639 
 
 5.648 
 
 
 2! 
 
 5.657 
 
 5.666 
 
 5.675 
 
 5.683 
 
 5.692 
 
 5.701 
 
 5.710 
 
 5.718 
 
 5.727 
 
 5.736 
 
 
 3. 
 
 5.745 
 
 5.753 
 
 5.762 
 
 5.771 
 
 5.779 
 
 5.788 
 
 5.797 
 
 5.805 
 
 5.814 
 
 5.822 
 
 
 4. 
 
 5.831 
 
 5.840 
 
 5.848 
 
 5.857 
 
 5.865 
 
 5.874 
 
 5.882 
 
 5.891 
 
 5.899 
 
 5.908 
 
 8 
 
 35. 
 
 5.916 
 
 5.925 
 
 5.933 
 
 5.941 
 
 5.950 
 
 5.958 
 
 5.967 
 
 5.975 
 
 5.983 
 
 5.992 
 
 
 6. 
 
 6.000 
 
 6.008 
 
 6.017 
 
 6.025 
 
 6.033 
 
 6.042 
 
 6.050 
 
 6.058 
 
 6.066 
 
 6.075 
 
 
 7. 
 
 6.083 
 
 6.091 
 
 6.099 
 
 6.107 
 
 6.116 
 
 6.124 
 
 6.132 
 
 6.140 
 
 6.148 
 
 6.156 
 
 
 8. 
 
 6.164 
 
 6.173 
 
 6.181 
 
 6.189 
 
 6.197 
 
 6.205 
 
 6.213 
 
 6.221 
 
 6.229 
 
 6.237 
 
 
 9. 
 
 6.245 
 
 6.253 
 
 6.261 
 
 6.269 
 
 6.277 
 
 6.285 
 
 6.293 
 
 6.301 
 
 6.309 
 
 6.317 
 
 
 40. 
 
 6.325 
 
 6.332 
 
 6.340 
 
 6.348 
 
 6.356 
 
 6.364 
 
 6.372 
 
 6.380 
 
 6.387 
 
 6.395 
 
 
 1. 
 
 6.403 
 
 6.411 
 
 6.419 
 
 6.427 
 
 6.434 
 
 6.442 
 
 6.450 
 
 6.458 
 
 6.465 
 
 6.473 
 
 
 2. 
 
 6.481 
 
 6.488 
 
 6.496 
 
 6.504 
 
 6.512 
 
 6.519 
 
 6.527 
 
 6.535 
 
 6.542 
 
 6.550 
 
 
 3. 
 
 6.557 
 
 6.565 
 
 6.573 
 
 6.580 
 
 6.588 
 
 6.595 
 
 6.603 
 
 6.611 
 
 6.618 
 
 6.626 
 
 
 4. 
 
 6.633 
 
 6.641 
 
 6.648 
 
 6.656 
 
 6.663 
 
 6.671 
 
 6.678 
 
 6.686 
 
 6.693 
 
 6.701 
 
 
 45. 
 
 6.708 
 
 6.716 
 
 6.723 
 
 6.731 
 
 6.738 
 
 6.745 
 
 6.753 
 
 6.760 
 
 6.768 
 
 6.775 
 
 7 
 
 6. 
 
 6.782 
 
 6.790 
 
 6.797 
 
 6.804 
 
 6.812 
 
 6.819 
 
 6.826 
 
 6.834 
 
 6.841 
 
 6.848 
 
 
 7. 
 
 6.856 
 
 6.863 
 
 6.870 
 
 6.877 
 
 6.885 
 
 6.892 
 
 6.899 
 
 6.907 
 
 6.914 
 
 6.921 
 
 
 8. 
 
 6.928 
 
 6.935 
 
 6.943 
 
 6.950 
 
 6.957 
 
 6.964 
 
 6.971 
 
 6.979 
 
 6.986 
 
 6.993 
 
 
 9. 
 
 7.000 
 
 7.007 
 
 7.014 
 
 7.021 
 
 7.029 
 
 7.036 
 
 7.043 
 
 7.050 
 
 7.057 
 
 7.064 
 
 
 SQUARE ROOTS OF CERTAIN FRACTIONS 
 
 N 
 
 VN 
 
 N 
 
 VN 
 
 AT 
 
 VN 
 
 N 
 
 VN 
 
 N 
 
 VN 
 
 N 
 
 VN 
 
 y* 
 
 0.7071 
 
 H 
 
 0.7746 
 
 M 
 
 0.7559 
 
 H 
 
 0.3333 
 
 Hi 
 
 0.6455 
 
 M 
 
 0.7500 
 
 y& 
 
 0.5774 
 
 % 
 
 0.8944 , 
 
 W 
 
 0.8452 
 
 % 
 
 0.4714 
 
 til 
 
 0.7638 
 
 iM 
 
 0.8292 
 
 g 
 
 0.8165 
 
 H 
 
 0.4082 
 
 M 
 
 0.9258 
 
 V* 
 
 0.6667 
 
 J M2 
 
 0.9574 
 
 13 /f 
 
 0.9014 
 
 
 0.5000 
 
 % 
 
 0.9129 
 
 M 
 
 0.3536 
 
 % 
 
 0.7454 
 
 H 
 
 0.2500 
 
 ls /fe 
 
 0.9682 
 
 94 
 
 0.8660 
 
 W 
 
 0.3780 
 
 ** 
 
 0.6124 
 
 7 ^ 
 
 0.8819 
 
 M 
 
 0.4330 
 
 to 
 
 0.1768 
 
 % 
 
 0.4472 
 0.6325 
 
 M 
 H 
 
 0.5345 
 0.6547 
 
 H 
 H 
 
 0.7906 
 0.9354 
 
 % 
 
 0.9428 
 0.2887 
 
 Me 
 Me 
 
 0.5590 
 0.6614 
 
 B 
 
 0.1250 
 0.1414 
 
MATHEMATICAL TABLES 
 SQUARE ROOTS (continued) 
 
 N 
 
 
 
 1 
 
 2 
 
 a 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 wsd 
 $* 
 
 50. 
 
 7.071 
 
 7.078 
 
 7.085 
 
 7.092 
 
 7.099 
 
 7.106 
 
 7.113 
 
 7.120 
 
 7.127 
 
 7.134 
 
 7 
 
 1. 
 
 7.141 
 
 7.148 
 
 7.155 
 
 7.162 
 
 7.169 
 
 7.176 
 
 7.183 
 
 7.190 
 
 7.197 
 
 7.204 
 
 
 2. 
 
 7.211 
 
 7.218 
 
 7.225 
 
 7.232 
 
 7.239 
 
 7.246 
 
 7.253 
 
 7.259 
 
 7.266 
 
 7.273 
 
 
 3. 
 
 7.280 
 
 7.287 
 
 7.294 
 
 7.301 
 
 7.308 
 
 7.314 
 
 7.321 
 
 7.328 
 
 7.335 
 
 7.342 
 
 
 4. 
 
 7.348 
 
 7.355 
 
 7.362 
 
 7.369 
 
 7.376 
 
 7.382 
 
 7.389 
 
 7.396 
 
 7.403 
 
 7.409 
 
 
 55. 
 
 7.416 
 
 7.423 
 
 7.430 
 
 7.436 
 
 7.443 
 
 7.450 
 
 7.457 
 
 7.463 
 
 7.470 
 
 7.477 
 
 
 6. 
 
 7.483 
 
 7.490 
 
 7.497 
 
 7.503 
 
 7.510 
 
 7.517 
 
 7.523 
 
 7.530 
 
 7.537 
 
 7.543 
 
 
 7. 
 
 7.550 
 
 7.556 
 
 7.563 
 
 7.570 
 
 7.576 
 
 7.583 
 
 7.589 
 
 7.596 
 
 7.603 
 
 7.609 
 
 
 8. 
 
 7.616 
 
 7.622 
 
 7.629 
 
 7.635 
 
 7.642 
 
 7.649 
 
 7.655 
 
 7.662 
 
 7.668 
 
 7.675 
 
 
 9. 
 
 7.681 
 
 7.688 
 
 7.694 
 
 7.701 
 
 7.707 
 
 7.714 
 
 7.720 
 
 7.727 
 
 7.733 
 
 7.740 
 
 6 
 
 60. 
 
 7.746 
 
 7.752 
 
 7.759 
 
 7.765 
 
 7.772 
 
 7.778 
 
 7.785 
 
 7.791 
 
 7.797 
 
 7.804 
 
 
 1. 
 
 7.810 
 
 7.817 
 
 7.823 
 
 7.829 
 
 7.836 
 
 7.842 
 
 7.849 
 
 7.855 
 
 7.861 
 
 7.868 
 
 
 2. 
 
 7.874 
 
 7.880 
 
 7.887 
 
 7.893 
 
 7.899 
 
 7.906 
 
 7.912 
 
 7.918 
 
 7.925 
 
 7.931 
 
 
 3. 
 
 7.937 
 
 7.944 
 
 7.950 
 
 7.956 
 
 7.962 
 
 7.969 
 
 7.975 
 
 7.981 
 
 7.987 
 
 7.994 
 
 
 4. 
 
 8.000 
 
 8.006 
 
 8.012 
 
 8.019 
 
 8.025 
 
 8.031 
 
 8.037 
 
 8.044 
 
 8.050 
 
 8.056 
 
 
 65. 
 
 8.062 
 
 8.068 
 
 8.075 
 
 8.081 
 
 8.087 
 
 8.093 
 
 8.099 
 
 8.106 
 
 8.112 
 
 8.118 
 
 
 6. 
 
 8.124 
 
 8.130 
 
 8.136 
 
 8.142 
 
 8.149 
 
 8.155 
 
 8.161 
 
 8.167 
 
 8.173 
 
 8.179 
 
 
 7. 
 
 8.185 
 
 8.191 
 
 8.198 
 
 8.204 
 
 8.210 
 
 8.216 
 
 8.222 
 
 8.228 
 
 8.234 
 
 8.240 
 
 
 8. 
 
 8.246 
 
 8.252 
 
 8.258 
 
 8.264 
 
 8.270 
 
 8.276 
 
 8.283 
 
 8.289 
 
 8.295 
 
 8.301 
 
 
 9. 
 
 8.307 
 
 8.313 
 
 8.319 
 
 8.325 
 
 8.331 
 
 8.337 
 
 8.343 
 
 8.349 
 
 8.355 
 
 8.361 
 
 
 70. 
 
 8.367 
 
 8.373 
 
 8.379 
 
 8.385 
 
 8.390 
 
 8.396 
 
 8.402 
 
 8.408 
 
 8.414 
 
 8.420 
 
 
 1. 
 
 8.426 
 
 8.432 
 
 8.438 
 
 8.444 
 
 8.450 
 
 8.456 
 
 8.462 
 
 8.468 
 
 8.473 
 
 8.479 
 
 
 2. 
 
 8.485 
 
 8.491 
 
 8.497 
 
 8.503 
 
 8.509 
 
 8.515 
 
 8.521 
 
 8.526 
 
 8.532 
 
 8.538 
 
 
 3. 
 
 8.544 
 
 8.550 
 
 8.556 
 
 8.562 
 
 8.567 
 
 8.573 
 
 8.579 
 
 8.585 
 
 8.591 
 
 8.597 
 
 
 4. 
 
 8.602 
 
 8.608 
 
 8.614 
 
 8.620 
 
 8.626 
 
 8.631 
 
 8.637 
 
 8.643 
 
 8.649 
 
 8.654 
 
 
 75. 
 
 8.660 
 
 8.666 
 
 8.672 
 
 8.678 
 
 8.683 
 
 8.689 
 
 8.695 
 
 8.701 
 
 8.706 
 
 8.712 
 
 
 6. 
 
 8.718 
 
 8.724 
 
 8.729 
 
 8.735 
 
 8.741 
 
 8.746 
 
 8.752 
 
 8.758 
 
 8.764 
 
 8.769 
 
 
 7. 
 
 8.775 
 
 8.781 
 
 8.786 
 
 8.792 
 
 8.798 
 
 8.803 
 
 8.809 
 
 8.815 
 
 8.820 
 
 8.826 
 
 
 8. 
 
 8.832 
 
 8.837 
 
 8.843 
 
 8.849 
 
 8.854 
 
 8.860 
 
 8.866 
 
 8.871 
 
 8.877 
 
 8.883 
 
 
 9. 
 
 8.888 
 
 8.894 
 
 8.899 
 
 8.905 
 
 8.911 
 
 8.916 
 
 8.922 
 
 8.927 
 
 8.933 
 
 8.939 
 
 
 80. 
 
 8.944 
 
 8.950 
 
 8.955 
 
 8.961 
 
 8.967 
 
 8.972 
 
 8.978 
 
 8.983 
 
 8.989 
 
 8.994 
 
 
 1. 
 
 9.000 
 
 9.006 
 
 9.011 
 
 9.017 
 
 9.022 
 
 9.028 
 
 9.033 
 
 9.039 
 
 9.044 
 
 9.050 
 
 
 2 
 
 9.055 
 
 9.061 
 
 9.066 
 
 9.072 
 
 9.077 
 
 9.083 
 
 9.088 
 
 9.094 
 
 9.099 
 
 9.105 
 
 5 
 
 3. 
 
 9.110 
 
 9.116 
 
 9.121 
 
 9.127 
 
 9.132 
 
 9.138 
 
 9.143 
 
 9.149 
 
 9.154 
 
 9.160 
 
 
 4. 
 
 9.165 
 
 9.171 
 
 9.176 
 
 9.182 
 
 9.187 
 
 9.192 
 
 9.198 
 
 9.203 
 
 9.209 
 
 9.214 
 
 
 85. 
 
 9.220 
 
 9.225 
 
 9.230 
 
 9.236 
 
 9.241 
 
 9.247 
 
 9.252 
 
 9.257 
 
 9.263 
 
 9.268 
 
 
 6. 
 
 9.274 
 
 9.279 
 
 9.284 
 
 9.290 
 
 9.295 
 
 9.301 
 
 9.306 
 
 9.311 
 
 9.317 
 
 9.322 
 
 
 7. 
 
 9.327 
 
 9.333 
 
 9.338 
 
 9.343 
 
 9.349 
 
 9.354 
 
 9.359 
 
 9.365 
 
 9.370 
 
 9.375 
 
 
 8. 
 
 9.381 
 
 9.386 
 
 9.391 
 
 9.397 
 
 9.402 
 
 9.407 
 
 9.413 
 
 9.418 
 
 9.423 
 
 9.429 
 
 
 9. 
 
 9.434 
 
 9.439 
 
 9.445 
 
 9.450 
 
 9.455 
 
 9.460 
 
 9.466 
 
 9.471 
 
 9.476 
 
 9.482 
 
 
 90. 
 
 9.487 
 
 9.492 
 
 9.497 
 
 9.503 
 
 9.508 
 
 9.513 
 
 9.518 
 
 9.524 
 
 9.529 
 
 9.534 
 
 
 1. 
 
 9.539 
 
 9.545 
 
 9.550 
 
 9.555 
 
 9.560 
 
 9.566 
 
 9.571 
 
 9.576 
 
 9.581 
 
 9.586 
 
 
 2. 
 
 9.592 
 
 9.597 
 
 9.602 
 
 9.607 
 
 9.612 
 
 9.618 
 
 9.623 
 
 9.628 
 
 9.633 
 
 9.638 
 
 
 3. 
 
 9.644 
 
 9.649 
 
 9.654 
 
 9.659 
 
 9.664 
 
 9.670 
 
 9.675 
 
 9.680 
 
 9.685 
 
 9.690 
 
 
 4. 
 
 9.695 
 
 9.701 
 
 9.706 
 
 9.711 
 
 9.716 
 
 9.721 
 
 9.726 
 
 9.731 
 
 9.737 
 
 9.742 
 
 
 95. 
 
 9.747 
 
 9.752 
 
 9.757 
 
 9.762 
 
 9.767 
 
 9.772 
 
 9.778 
 
 9.783 
 
 9.788 
 
 9.793 
 
 
 6. 
 
 9.798 
 
 9.803 
 
 9.808 
 
 9.813 
 
 9.818 
 
 9.823 
 
 9.829 
 
 9.834 
 
 9.839 
 
 9.844 
 
 
 7. 
 
 9.849 
 
 9.854 
 
 9.859 
 
 9.864 
 
 9.869 
 
 9.874 
 
 9.879 
 
 9.884 
 
 9.889 
 
 9.894 
 
 
 8. 
 
 9.899 
 
 9.905 
 
 9.910 
 
 9.915 
 
 9.920 
 
 9.925 
 
 9.930 
 
 9.935 
 
 9.940 
 
 9.945 
 
 
 9. 
 
 9.950 
 
 9.955 
 
 9.960 
 
 9.965 
 
 9.970 
 
 9.975 
 
 9.980 
 
 9.985 
 
 9.990 
 
 9.995 
 
 
 = 1.77245+ 
 
 0.56419 
 
 = 1.25331 
 
 = 1.64872 
 
 Moving the decimal point TWO places in N requires moving it ONE place in body of 
 table (seep. 12). 
 
16 MATHEMATICAL TABLES 
 
 CUBE ROOTS OF NUMBERS 
 
 TV 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 * 
 
 4* 
 
 1.0 
 
 1.000 
 
 1.003 
 
 1.007 
 
 1.010 
 
 1.013 
 
 1.016 
 
 1.020 
 
 1.023 
 
 1.026 
 
 1.029 
 
 3 
 
 1 
 
 1.032 
 
 1.035 
 
 1.038 
 
 1.042 
 
 1.045 
 
 1.048 
 
 1.051 
 
 1.054 
 
 1.057 
 
 1.060 
 
 
 2 
 
 1.063 
 
 1.066 
 
 1.069 
 
 1.071 
 
 1.074 
 
 1.077 
 
 1.080 
 
 1.083 
 
 1.086 
 
 1.089 
 
 
 3 
 
 1.091 
 
 .094 
 
 1.097 
 
 .100 
 
 1.102 
 
 .105 
 
 1.108 
 
 .111 
 
 1.113 
 
 1.116 
 
 
 4 
 
 1.119 
 
 .121 
 
 1.124 
 
 .127 
 
 1.129 
 
 .132 
 
 1.134 
 
 .137 
 
 1.140 
 
 1.142 
 
 
 1.5 
 
 1.145 
 
 .147 
 
 1.150 
 
 .152 
 
 1.155 
 
 .157 
 
 1.160 
 
 .162 
 
 1.165 
 
 1.167 
 
 2 
 
 6 
 
 1.170 
 
 .172 
 
 1.174 
 
 .177 
 
 1.179 
 
 .182 
 
 1.184 
 
 .186 
 
 1.189 
 
 1.191 
 
 
 7 
 
 1.193 
 
 .196 
 
 1.198 
 
 .200 
 
 1.203 
 
 .205 
 
 1.207 
 
 .210 
 
 1.212 
 
 1.214 
 
 
 8 
 
 1.216 
 
 1.219 
 
 1.221 
 
 .223 
 
 1.225 
 
 1.228 
 
 1.230 
 
 .232 
 
 1.234 
 
 1.236 
 
 
 9 
 
 1.239 
 
 1.241 
 
 1.243 
 
 1.245 
 
 1.247 
 
 1.249 
 
 1.251 
 
 .254 
 
 1.256 
 
 1.258 
 
 
 2.0 
 
 1.260 
 
 1.262 
 
 1.264 
 
 1.266 
 
 1.268 
 
 1.270 
 
 1.272 
 
 .274 
 
 1.277 
 
 1.279 
 
 
 1 
 
 1.281 
 
 1.283 
 
 1.285 
 
 1.287 
 
 1.289 
 
 1.291 
 
 1.293 
 
 .295 
 
 1.297 
 
 1.299 
 
 
 2 
 
 1.301 
 
 .303 
 
 1.305 
 
 1.306 
 
 1.308 
 
 1.310 
 
 1.312 
 
 .314 
 
 1.316 
 
 1.318 
 
 
 3 
 
 1.320 
 
 1.322 
 
 1.324 
 
 1.326 
 
 1.328 
 
 1.330 
 
 1.331 
 
 .333 
 
 1.335 
 
 1.337 
 
 
 4 
 
 1.339 
 
 1.341 
 
 1.343 
 
 1.344 
 
 1.346 
 
 1.348 
 
 1.350 
 
 .352 
 
 1.354 
 
 1.355 
 
 
 2.5 
 
 1.357 
 
 1.359 
 
 1.361 
 
 1.363 
 
 1.364 
 
 1.366 
 
 1.368 
 
 .370 
 
 1.372 
 
 1.373 
 
 
 6 
 
 1.375 
 
 1.377 
 
 1.379 
 
 1.380 
 
 1.382 
 
 1.384 
 
 1.386 
 
 .387 
 
 1.389 
 
 1.391 
 
 
 7 
 
 1.392 
 
 .394 
 
 1.396 
 
 1.398 
 
 1.399 
 
 1.401 
 
 1.403 
 
 .404 
 
 1.406 
 
 1.408 
 
 
 I 8 
 
 1.409 
 
 1.411 
 
 1.413 
 
 1.414 
 
 1.416 
 
 .418 
 
 1.419 
 
 .421 
 
 1.423 
 
 1.424 
 
 
 9 
 
 1.426 
 
 .428 
 
 1.429 
 
 1.431 
 
 1.433 
 
 1.434 
 
 1.436 
 
 1.437 
 
 1.439 
 
 1.441 
 
 
 3.0 
 
 1.442 
 
 1.444 
 
 1.445 
 
 1.447 
 
 1.449 
 
 .450 
 
 1.452 
 
 1.453 
 
 1.455 
 
 1.457 
 
 
 1 
 
 1.458 
 
 1.460 
 
 1.461 
 
 1.463 
 
 1.464 
 
 .466 
 
 1.467 
 
 1.469 
 
 1.471 
 
 1.472 
 
 
 2 
 
 1.474 
 
 .475 
 
 1.477 
 
 1.478 
 
 1.480 
 
 1.481 
 
 1.483 
 
 1.484 
 
 1.486 
 
 1.487 
 
 
 3 
 
 1.489 
 
 1.490 
 
 1.492 
 
 1.493 
 
 1.495 
 
 1.496 
 
 1.498 
 
 1.499 
 
 1.501 
 
 1.502 
 
 
 4 
 
 1.504 
 
 1.505 
 
 1.507 
 
 1.508 
 
 1.510 
 
 1.511 
 
 1.512 
 
 1.514 
 
 1.515 
 
 1.517 
 
 
 3.5 
 
 1.518 
 
 1.520 
 
 1.521 
 
 1.523 
 
 1.524 
 
 1.525 
 
 1.527 
 
 1.528 
 
 1.530 
 
 1.531 
 
 
 6 
 
 1.533 
 
 .534 
 
 1.535 
 
 1.537 
 
 1.538 
 
 1.540 
 
 1.541 
 
 1.542 
 
 1.544 
 
 1.545 
 
 1 
 
 7 
 
 1.547 
 
 1.548 
 
 1.549 
 
 1.551 
 
 1.552 
 
 1.554 
 
 1.555 
 
 1.556 
 
 1.558 
 
 1.559 
 
 
 8 
 
 1.560 
 
 1.562 
 
 1.563 
 
 1.565 
 
 1.566 
 
 .567 
 
 1.569 
 
 1.570 
 
 1.571 
 
 1.573 
 
 
 9 
 
 1.574 
 
 1.575 
 
 1.577 
 
 1.578 
 
 1.579 
 
 1.581 
 
 1.582 
 
 1.583 
 
 1.585 
 
 1.586 
 
 
 4.0 
 
 1.587 
 
 1.589 
 
 1.590 
 
 1.591 
 
 1.593 
 
 1.594 
 
 1.595 
 
 1.597 
 
 1.598 
 
 1.599 
 
 
 1 
 
 1.601 
 
 1.602 
 
 1.603 
 
 1.604 
 
 1.606 
 
 1.607 
 
 1.608 
 
 1.610 
 
 1.611 
 
 1.612 
 
 
 2 
 
 1.613 
 
 1.615 
 
 1.616 
 
 1.617 
 
 1.619 
 
 1.620 
 
 1.621 
 
 1.622 
 
 1.624 
 
 1.625 
 
 
 3 
 
 1.626 
 
 1.627 
 
 1.629 
 
 1.630 
 
 1.631 
 
 1.632 
 
 1.634 
 
 1.635 
 
 1.636 
 
 1.637 
 
 
 4 
 
 1.639 
 
 1.640 
 
 1.641 
 
 1.642 
 
 1.644 
 
 1.645 
 
 1.646 
 
 1.647 
 
 1.649 
 
 1.650 
 
 
 4.5 
 
 1.651 
 
 1.652 
 
 1.653 
 
 1.655 
 
 1.656 
 
 1.657 
 
 1.658 
 
 1.659 
 
 1.661 
 
 1.662 
 
 
 6 
 
 1.663 
 
 1.664 
 
 1.666 
 
 1.667 
 
 1.668 
 
 1.669 
 
 1.670 
 
 1.671 
 
 1.673 
 
 1.674 
 
 
 7 
 
 1.675 
 
 1.676 
 
 1.677 
 
 1.679 
 
 1.680 
 
 1.681 
 
 1.682 
 
 1.683 
 
 1.685 
 
 1.686 
 
 
 8 
 
 1.687 
 
 1.688 
 
 1.689 
 
 1.690 
 
 1.692 
 
 1.693 
 
 1.694 
 
 1.695 
 
 1.696 
 
 1.697 
 
 
 9 
 
 1.698 
 
 1.700 
 
 1.701 
 
 1.702 
 
 1.703 
 
 1.704 
 
 1.705 
 
 1.707 
 
 1.708 
 
 1.709 
 
 
 1.46459 l/V^r~= 0.682784 
 
 Explanation of Table of Cube Roots (pp. 16-21). 
 
 This table gives the values of \/TV for all values of TV from 1 to 1000, correct to four 
 figures. (Interpolated values may be in error by 1 in the fourth figure.) 
 
 To find the cube root of a number N outside the range from 1 to 1000, divide 
 the digits of the number into blocks of three (beginning with the decimal point), and 
 note that moving the decimal point three places in column N is equivalent to moving 
 it one place in the cube root of TV. For example: 
 
 y^.718 = 1.396; -y^2718 - 13.96; -^0.000002718 = 0.01396. 
 
 -^27.18 = 3.007; -^27180 = 30.07; -^0.00002718 = 0.03007. 
 
 ^271.8 = 6.477; -^271800 = 64.77; -^0.0002718 = 0.06477. 
 
MATHEMATICAL TABLES 
 CUBE ROOTS (continued) 
 
 17 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 6.0 
 
 1.710 
 
 1.711 
 
 1712 
 
 1.713 
 
 1.715 
 
 1.716 
 
 1.717 
 
 1.718 
 
 1.719 
 
 1.720 
 
 1 
 
 
 1.721 
 
 1.722 
 
 1.724 
 
 1.725 
 
 1.726 
 
 1.727 
 
 1.728 
 
 1.729 
 
 1.730 
 
 1.731 
 
 
 2 
 
 1.732 
 
 1.734 
 
 1.735 
 
 1.736 
 
 1.737 
 
 1.738 
 
 1.739 
 
 1.740 
 
 1.741 
 
 1.742 
 
 
 3 
 
 1.744 
 
 1.745 
 
 1.746 
 
 1.747 
 
 1.748 
 
 1.749 
 
 1.750 
 
 1.751 
 
 1.752 
 
 1.753 
 
 
 4 
 
 1.754 
 
 1.755 
 
 1.757 
 
 1.758 
 
 1.759 
 
 1.760 
 
 1.761 
 
 1.762 
 
 1.763 
 
 1.764 
 
 
 5.5 
 
 1.765 
 
 1.766 
 
 1.767 
 
 1.768 
 
 1.769 
 
 1.771 
 
 1.772 
 
 1.773 
 
 1.774 
 
 1.775 
 
 
 6 
 
 1.776 
 
 1.777 
 
 1.778 
 
 1.779 
 
 1.780 
 
 1.781 
 
 1.782 
 
 1.783 
 
 1.784 
 
 1.785 
 
 
 7 
 
 1.786 
 
 1.787 
 
 1.788 
 
 1.789 
 
 1.790 
 
 1.792 
 
 1.793 
 
 1.794 
 
 1.795 
 
 1.796 
 
 
 8 
 
 1.797 
 
 1.798 
 
 1.799 
 
 1.800 
 
 1.801 
 
 1.802 
 
 1.803 
 
 1.804 
 
 1.805 
 
 1.806 
 
 
 9 
 
 1.807 
 
 1.808 
 
 1.809 
 
 1.810 
 
 1.811 
 
 1.812 
 
 1.813 
 
 1.814 
 
 1.815 
 
 1.816 
 
 
 6.0 
 
 1.817 
 
 1.818 
 
 1.819 
 
 1.820 
 
 1.821 
 
 1.822 
 
 1.823 
 
 1.824 
 
 1.825 
 
 1.826 
 
 
 1 
 
 1.827 
 
 1.828 
 
 1.829 
 
 1.830 
 
 1.831 
 
 1.832 
 
 1 833 
 
 1.834 
 
 1.835 
 
 1.836 
 
 
 2 
 
 1.837 
 
 1.838 
 
 1.839 
 
 1.840 
 
 1.841 
 
 1.842 
 
 1.843 
 
 1.844 
 
 1.845 
 
 1.846 
 
 
 3 
 
 1.847 
 
 1.848 
 
 1.849 
 
 1.850 
 
 1.851 
 
 1.852 
 
 1.853 
 
 1.854 
 
 1.855 
 
 1.856 
 
 
 4 
 
 1.857 
 
 1.858 
 
 1.859 
 
 1.860 
 
 1.860 
 
 1.861 
 
 1.862 
 
 1.863 
 
 1.864 
 
 1.865 
 
 
 6.5 
 
 1.866 
 
 1.867 
 
 1.868 
 
 1.869 
 
 1.870 
 
 1.871 
 
 1.872 
 
 1.873 
 
 1.874 
 
 1.875 
 
 
 6 
 
 1.876 
 
 1.877 
 
 1.878 
 
 1.879 
 
 1.880 
 
 1.881 
 
 1.881 
 
 1.882 
 
 1.883 
 
 1.884 
 
 
 7 
 
 1.885 
 
 1.886 
 
 1.887 
 
 1.888 
 
 1.889 
 
 1.890 
 
 1.891 
 
 1.892 
 
 1.893 
 
 1.894 
 
 
 8 
 
 1.895 
 
 1.895 
 
 1.896 
 
 1.897 
 
 1.898 
 
 1.899 
 
 1.900 
 
 1.901 
 
 1.902 
 
 1.903 
 
 
 9 
 
 1.904 
 
 1.905 
 
 1.906 
 
 1.907 
 
 1.907 
 
 1.908 
 
 1.909 
 
 1.910 
 
 1.911 
 
 1.912 
 
 
 7.0 
 
 1.913 
 
 1.914 
 
 1.915 
 
 1.916 
 
 1.917 
 
 1.917 
 
 1.918 
 
 1.919 
 
 1.920 
 
 1.921 
 
 
 1 
 
 1.922 
 
 1.923 
 
 1.924 
 
 1.925 
 
 1.926 
 
 1.926 
 
 1.927 
 
 1.928 
 
 1.929 
 
 1.930 
 
 
 2 
 
 1.931 
 
 1.932 
 
 1.933 
 
 1.934 
 
 1.935 
 
 1.935 
 
 1.936 
 
 1.937 
 
 1.938 
 
 1.939 
 
 
 3 
 
 1.940 
 
 1.941 
 
 1.942 
 
 1.943 
 
 1.943 
 
 1.944 
 
 1.945 
 
 1.946 
 
 1.947 
 
 1.948 
 
 
 4 
 
 1.949 
 
 1.950 
 
 1.950 
 
 1.951 
 
 1.952 
 
 1.953 
 
 1.954 
 
 1.955 
 
 1.956 
 
 1.957 
 
 
 7.5 
 
 1.957 
 
 1.958 
 
 1.959 
 
 1.960 
 
 1.961 
 
 1.962 
 
 1.963 
 
 1.964 
 
 1.964 
 
 1.965 
 
 
 6 
 
 1.966 
 
 1.967 
 
 1.968 
 
 1.969 
 
 1.970 
 
 1.970 
 
 1.971 
 
 1.972 
 
 1.973 
 
 1.974 
 
 
 7 
 
 1.975 
 
 1.976 
 
 1.976 
 
 1.977 
 
 1.978 
 
 1.979 
 
 1.980 
 
 1.981 
 
 1.981 
 
 1.982 
 
 
 8 
 
 1 983 
 
 1.984 
 
 1.985 
 
 1.986 
 
 1.987 
 
 1.987 
 
 1.988 
 
 1.989 
 
 1.990 
 
 1.991 
 
 
 9 
 
 1.992 
 
 1.992 
 
 1.993 
 
 1.994 
 
 1.995 
 
 1.996 
 
 1.997 
 
 1.997 
 
 1.998 
 
 1.999 
 
 
 8.0 
 
 2.000 
 
 2.001 
 
 2.002 
 
 2.002 
 
 2.003 
 
 2.004 
 
 2.005 
 
 2.006 
 
 2.007 
 
 2.007 
 
 
 1 
 
 2.008 
 
 2.009 
 
 2.010 
 
 2.01 1 
 
 2.012 
 
 2.012 
 
 2.013 
 
 2.014 
 
 2.015 
 
 2.016 
 
 
 2 
 
 2.017 
 
 2.017 
 
 2.018 
 
 2.019 
 
 2.020 
 
 2.021 
 
 2.021 
 
 2.022 
 
 2.023 
 
 2.024 
 
 
 3 
 
 2.025 
 
 2.026 
 
 2.026 
 
 2.027 
 
 2.028 
 
 2.029 
 
 2.030 
 
 2.030 
 
 2.031 
 
 2.032 
 
 
 4 
 
 2.033 
 
 2.034 
 
 2.034 
 
 2.035 
 
 2.036 
 
 2.037 
 
 2.038 
 
 2.038 
 
 2.039 
 
 2.040 
 
 
 8.5 
 
 2.041 
 
 2.042 
 
 2.042 
 
 2.043 
 
 2.044 
 
 2.045 
 
 2.046 
 
 2.046 
 
 2.047 
 
 2.048 
 
 
 6 
 
 2.049 
 
 2.050 
 
 2.050 
 
 2.051 
 
 2.052 
 
 2.053 
 
 2.054 
 
 2.054 
 
 2.055 
 
 2.056 
 
 
 7 
 
 2.057 
 
 2.057 
 
 2.058 
 
 2.059 
 
 2.060 
 
 2.061 
 
 2.061 
 
 2.062 
 
 2.063 
 
 2.064 
 
 
 8 
 
 2.065 
 
 2.065 
 
 2.066 
 
 2.067 
 
 2.068 
 
 2.068 
 
 2.069 
 
 2.070 
 
 2.071 
 
 2.072 
 
 
 9 
 
 2.072 
 
 2.073 
 
 2.074 
 
 2.075 
 
 2.075 
 
 2.076 
 
 2.077 
 
 2.078 
 
 2.079 
 
 2.079 
 
 
 9.0 
 
 2.080 
 
 2.081 
 
 2.082 
 
 2.082 
 
 2.083 
 
 2.084 
 
 2.085 
 
 2.085 
 
 2.086 
 
 2.087 
 
 
 1 
 
 2.088 
 
 2.089 
 
 2.089 
 
 2.090 
 
 2.091 
 
 2.092 
 
 2.092 
 
 2.093 
 
 2.094 
 
 2.095 
 
 
 2 
 
 2.095 
 
 2.096 
 
 2.097 
 
 2.098 
 
 2.098 
 
 2.099 
 
 2.100 
 
 2.101 
 
 2.101 
 
 2.102 
 
 
 3 
 
 2.103 
 
 2.104 
 
 2.104 
 
 2.105 
 
 2.106 
 
 2.107 
 
 2.107 
 
 2.108 
 
 2.109 
 
 2.110 
 
 
 4 
 
 2.110 
 
 2.111 
 
 2.112 
 
 2.113 
 
 2.113 
 
 . 2.114 
 
 2.115 
 
 2.116 
 
 2.116 
 
 2.117 
 
 
 9.5 
 
 2.118 
 
 2.119 
 
 2.119 
 
 2.120 
 
 2.121 
 
 2.122 
 
 2.122 
 
 2.123 
 
 2.124 
 
 2.125 
 
 
 6 
 
 2.125 
 
 2.126 
 
 2.127 
 
 2.128 
 
 2.128 
 
 2.129 
 
 2.130 
 
 2.130 
 
 2.131 
 
 2.132 
 
 
 7 
 
 2.133 
 
 2.133 
 
 2.134 
 
 2.135 
 
 2.136 
 
 2.136 
 
 2.137 
 
 2.138 
 
 2.139 
 
 2.139 
 
 
 8 
 
 2.140 
 
 2.141 
 
 2.141 
 
 2.142 
 
 2.143 
 
 2.144 
 
 2.144 
 
 2.145 
 
 2.146 
 
 2.147 
 
 
 9 
 
 2.147 
 
 2.148 
 
 2.149 
 
 2.149 
 
 2.150 
 
 2.151 
 
 2.152 
 
 2.152 
 
 2.153 
 
 2.154 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 Moving the decimal point THREE places in N requires moving it ONE place in 
 body of table (see p. 16). 
 2 
 
18 
 
 MATHEMATICAL TABLES 
 
 CUBE BOOTS (continued) 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 ^ 
 <j-o 
 
 10. 
 
 2.154 
 
 2.162 
 
 2.169 
 
 2.176 
 
 2.183 
 
 2.190 
 
 2.197 
 
 2.204 
 
 2.210 
 
 2.217 
 
 7 
 
 1. 
 
 2.224 
 
 2.231 
 
 2.237 
 
 2.244 
 
 2.251 
 
 2.257 
 
 2.264 
 
 2.270 
 
 2.277 
 
 2.283 
 
 6 
 
 2. 
 
 2.289 
 
 2.296 
 
 2.302 
 
 2.308 
 
 2.315 
 
 2.321 
 
 2.327 
 
 2.333 
 
 2.339 
 
 2.345 
 
 
 3. 
 
 2.351 
 
 2.357 
 
 2.363 
 
 2.369 
 
 2.375 
 
 2.381 
 
 2.387 
 
 2.393 
 
 2.399 
 
 2.404 
 
 
 4. 
 
 2.410 
 
 2.416. 
 
 2.422 
 
 2.427 
 
 2.433 
 
 2.438 
 
 2.444 
 
 2.450 
 
 2.455 
 
 2.461 
 
 
 15. 
 
 2.466 
 
 2.472 
 
 2.477 
 
 2.483 
 
 2.488 
 
 2.493 
 
 2.499 
 
 2.504 
 
 2.509 
 
 2.515 
 
 5 
 
 6. 
 
 2.520 
 
 2.525 
 
 2.530 
 
 2.535 
 
 2.541 
 
 2.546 
 
 2.551 
 
 2.556 
 
 2.561 
 
 2.566 
 
 
 7. 
 
 2.571 
 
 2.576 
 
 2.581 
 
 2.586 
 
 2.591 
 
 2.596 
 
 2.601 
 
 2.606 
 
 2.611 
 
 2.616 
 
 
 8. 
 
 2.621 
 
 2.626 
 
 2.630 
 
 2.635 
 
 2.640 
 
 2.645 
 
 2.650 
 
 2.654 
 
 2.659 
 
 2.664 
 
 
 9. 
 
 2.668 
 
 2.673 
 
 2.678 
 
 2.682 
 
 2.687 
 
 2.692 
 
 2.696 
 
 2.701 
 
 2.705 
 
 2.710 
 
 
 20. 
 
 2.714 
 
 2.719 
 
 2.723 
 
 2.728 
 
 2.732 
 
 2.737 
 
 2.741 
 
 2.746 
 
 2.750 
 
 2.755 
 
 4 
 
 1. 
 
 2.759 
 
 2.763 
 
 2.768 
 
 2.772 
 
 2.776 
 
 2.781 
 
 2.785 
 
 2.789 
 
 2.794 
 
 2.798 
 
 
 2. 
 
 2.802 
 
 2.806 
 
 2.811 
 
 2.815 
 
 2.819 
 
 2.823 
 
 2.827 
 
 2.831 
 
 2.836 
 
 2.840 
 
 
 3. 
 
 2.844 
 
 2.848 
 
 2.852 
 
 2.856 
 
 2.860 
 
 2.864 
 
 2.868 
 
 2.872 
 
 2.876 
 
 2.880 
 
 
 4. 
 
 2.884 
 
 2.888 
 
 2.892 
 
 2.896 
 
 2.900 
 
 2.904 
 
 2.908 
 
 2.912 
 
 2.916 
 
 2.920 
 
 
 25. 
 
 2.924 
 
 2.928 
 
 2.932 
 
 2.936 
 
 2.940 
 
 2.943 
 
 2.947 
 
 2.951 
 
 2.955 
 
 2.959 
 
 
 6. 
 
 2.962 
 
 2.966 
 
 2.970 
 
 2.974 
 
 2.978 
 
 2.981 
 
 2.985 
 
 2.989 
 
 2.993 
 
 2.996 
 
 
 7. 
 
 3.000 
 
 3.004 
 
 3.007 
 
 3.011 
 
 3.015 
 
 3.018 
 
 3.022 
 
 3.026 
 
 3.029 
 
 3.033 
 
 
 8. 
 
 3.037 
 
 3.040 
 
 3.044 
 
 3.047 
 
 3.051 
 
 3.055 
 
 3.058 
 
 3.062 
 
 3.065 
 
 3.069 
 
 
 9. 
 
 3.072 
 
 3.076 
 
 3.079 
 
 3.083 
 
 3.086 
 
 3.090 
 
 3.093 
 
 3.097 
 
 3.100 
 
 3.104 
 
 
 30. 
 
 3.107 
 
 3.111 
 
 3.114 
 
 3.118 
 
 3.121 
 
 3.124 
 
 3.128 
 
 3.131 
 
 3.135 
 
 3.138 
 
 3 
 
 1. 
 
 3.141 
 
 3.145 
 
 3.148 
 
 3.151 
 
 3.155 
 
 3.158 
 
 3.162 
 
 3.165 
 
 3.168 
 
 3.171 
 
 
 2. 
 
 3.175 
 
 3.178 
 
 3.181 
 
 3.185 
 
 3.188 
 
 3.191 
 
 3.195 
 
 3.198 
 
 3.201 
 
 3.204 
 
 
 3. 
 
 3.208 
 
 3.21 1 
 
 3.214 
 
 3.217 
 
 3.220 
 
 3.224 
 
 3.227 
 
 3.230 
 
 3.233 
 
 3.236 
 
 
 4. 
 
 3.240 
 
 3.243 
 
 3.246 
 
 3.249 
 
 3.252 
 
 3.255 
 
 3.259 
 
 3.262 
 
 3.265 
 
 3.268 
 
 
 35. 
 
 3.271 
 
 3.274 
 
 3.277 
 
 3.280 
 
 3.283 
 
 3.287 
 
 3.290 
 
 3.293 
 
 3.296 
 
 3.249 
 
 
 6. 
 
 3.302 
 
 3.305 
 
 3.308 
 
 3.311 
 
 3.314 
 
 3.317 
 
 3.320 
 
 3.323 
 
 3.326 
 
 3.329 
 
 
 7. 
 
 3.332 
 
 3.335 
 
 3.338 
 
 3.341 
 
 3.344 
 
 3.347 
 
 3.350 
 
 3.353 
 
 3.356 
 
 3.359 
 
 
 8. 
 
 3.362 
 
 3.365 
 
 3.368 
 
 3.371 
 
 3.374 
 
 3.377 
 
 3.380 
 
 3.382 
 
 3.385 
 
 3.388 
 
 
 9. 
 
 3.391 
 
 3.394 
 
 3.397 
 
 3.400 
 
 3.403 
 
 3.406 
 
 3.409 
 
 3.411 
 
 3.414 
 
 3.417 
 
 
 40. 
 
 3.420 
 
 3.423 
 
 3.426 
 
 3.428 
 
 3.431 
 
 3.434 
 
 3.437 
 
 3.440 
 
 3.443 
 
 3.445 
 
 
 I. 
 
 3.448 
 
 3.451 
 
 3.454 
 
 3.457 
 
 3.459 
 
 3.462 
 
 .3.465 
 
 3.468 
 
 3.471 
 
 3.473 
 
 
 2. 
 
 3.476 
 
 3.479 
 
 3.482 
 
 3.484 
 
 3.487 
 
 3.490 
 
 3.493 
 
 3.495 
 
 3.498 
 
 3.501 
 
 
 3. 
 
 3.503 
 
 3.506 
 
 3.509 
 
 3.512 
 
 3.514 
 
 3.517 
 
 3.520 
 
 3.522 
 
 3.525 
 
 3.528 
 
 
 4. 
 
 3.530 
 
 3.533 
 
 3.536 
 
 3.538 
 
 3.541 
 
 3.544 
 
 3.546 
 
 3.549 
 
 3.552 
 
 3.554 
 
 
 45. 
 
 3.557 
 
 3.560 
 
 3.562 
 
 3.565 
 
 3.567 
 
 3.570 
 
 3.573 
 
 3.575 
 
 3.578 
 
 3.580 
 
 
 6. 
 
 3.583 
 
 3.586 
 
 3.588 
 
 3.591 
 
 3.593 
 
 3.596 
 
 3.599 
 
 3.601 
 
 3.604 
 
 3.606 
 
 
 7. 
 
 3.609 
 
 3.611 
 
 3.614 
 
 3.616 
 
 3.619 
 
 3.622 
 
 3.624 
 
 3.627 
 
 3.629 
 
 3.632 
 
 
 8. 
 
 3.634 
 
 3.637 
 
 3.639 
 
 3.642 
 
 3.644 
 
 3.647 
 
 3.649 
 
 3.652 
 
 3.654 
 
 3.657 
 
 2 
 
 9. 
 
 3.659 
 
 3.662 
 
 3.664 
 
 3.667 
 
 3.669 
 
 3.672 
 
 3.674 
 
 3.677 
 
 3.679 
 
 3.682 
 
 
 CUBE ROOTS OF CERTAIN FRACTIONS 
 
 N 
 
 y* 
 
 N 
 
 & 
 
 2V 
 
 #8 
 
 N 
 
 VN 
 
 2V 
 
 j 
 
 2V 
 
 ^2V 
 
 g 
 
 y* 
 
 H 
 
 % 
 % 
 
 .7937 
 .6934 
 .8736 
 .6300 
 .9086 
 .5848 
 .7368 
 
 % 
 % 
 H 
 % 
 M 
 M 
 . y, 
 
 .8434 
 .9283 
 .5503 
 .9410 
 .5228 
 .6586 
 .7539 
 
 t 
 
 It 
 
 tt 
 
 H 
 W 
 Vk 
 
 .8298 
 .8939 
 .9499 
 .5000 
 .7211 
 .8550 
 .9565 
 
 H 
 H 
 
 % 
 % 
 % 
 % 
 Ha 
 
 .4807 
 .6057 
 .7631 
 .8221 
 .9196 
 .9615 
 .4368 
 
 M2 
 M2 
 Hl2 
 
 He 
 ?! 6 
 Me 
 Me 
 
 .7469 
 .8355 
 .9714 
 .3969 
 .5724 
 .6786 
 .7591 
 
 We 
 
 Hi 6 
 
 "/1 6 
 
 *M6 
 & 
 
 H4 
 
 Ho 
 
 .8255 
 .8826 
 .9331 
 .9787 
 .3150 
 .2500 
 .2714 
 
MATHEMATICAL TABLES 
 CUBE ROOTS (continued) 
 
 19 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 !i 
 
 60. 
 
 3.684 
 
 3.686 
 
 3.689 
 
 3.691 
 
 3.694 
 
 3.696 
 
 3*99 
 
 3.701 
 
 3.704 
 
 3.706 
 
 2 
 
 1. 
 
 3.708 
 
 3.71 1 
 
 3.713 
 
 3.716 
 
 3.718 
 
 3.721 
 
 3723 
 
 3.725 
 
 3.728 
 
 3.730 
 
 
 2. 
 
 3.733 
 
 3.735 
 
 3.737 
 
 3.740 
 
 3.742 
 
 3.744 
 
 3.747 
 
 3.749 
 
 3.752 
 
 3.754 
 
 
 3. 
 
 3.756 
 
 3.759 
 
 3.761 
 
 3.763 
 
 2.766 
 
 3.768 
 
 3.770 
 
 3.773 
 
 3.775 
 
 3.777 
 
 
 4. 
 
 3.780 
 
 3.782 
 
 3.784 
 
 3.787 
 
 3.789 
 
 3.791 
 
 3.794 
 
 3.796 
 
 3.798 
 
 3.801 
 
 
 55. 
 
 3.803 
 
 3.805 
 
 3.808 
 
 3.810 
 
 3.812 
 
 3.814 
 
 3.817 
 
 . 3.819 
 
 3.821 
 
 3.824 
 
 
 6. 
 
 3.826 
 
 3.828 
 
 3.830 
 
 3.833 
 
 3.835 
 
 3.837 
 
 3.839 
 
 3.842 
 
 3.844 
 
 3.846 
 
 
 7. 
 
 3.849 
 
 3.851 
 
 3.853 
 
 3.855 
 
 3.857 
 
 3.860 
 
 3.862 
 
 3.864 
 
 3.866 
 
 3.869 
 
 
 8. 
 
 3.871 
 
 3.873 
 
 3.875 
 
 3.878 
 
 3.880 
 
 3.882 
 
 3.884 
 
 3.886 
 
 3.889 
 
 3.891 
 
 
 9. 
 
 3.893 
 
 3.895 
 
 3.897 
 
 3.900 
 
 3.902 
 
 3.904 
 
 3.906 
 
 3.908 
 
 3.911 
 
 3.913 
 
 
 60. 
 
 3.915 
 
 3.917 
 
 3.919 
 
 3.921 
 
 3.924 
 
 3.926 
 
 3.928 
 
 3.930 
 
 3.932 
 
 3.934 
 
 
 
 3.936 
 
 3.939 
 
 3.941 
 
 3.943 
 
 3.945 
 
 3.947 
 
 3.949 
 
 3.951 
 
 3.954 
 
 3.956 
 
 
 2! 
 
 3.958 
 
 3.960 
 
 3.962 
 
 3.964 
 
 3.966 
 
 3.968 
 
 3.971 
 
 3.973 
 
 3.975 
 
 3.977 
 
 
 3. 
 
 3.979 
 
 3.981 
 
 3.983 
 
 3.985 
 
 3.987 
 
 3.990 
 
 3.992 
 
 3.994 
 
 3.996 
 
 3.998 
 
 
 4. 
 
 4.000 
 
 4.002 
 
 4.004 
 
 4.006 
 
 4.008 
 
 4.010 
 
 4.012 
 
 4.015 
 
 4.017 
 
 4.019 
 
 
 65. 
 
 4.021 
 
 4.023 
 
 4.025 
 
 4.027 
 
 4.029 
 
 4.031 
 
 4.033 
 
 4.035 
 
 4.037 
 
 4.039 
 
 
 6. 
 
 4.041 
 
 4.043 
 
 4.045 
 
 4.047 
 
 4.049 
 
 4.051 
 
 4.053 
 
 4.055 
 
 4.058 
 
 4.060 
 
 
 7. 
 
 4.062 
 
 4.064 
 
 4.066 
 
 4.068 
 
 4.070 
 
 4.072 
 
 4.074 
 
 4.076 
 
 4.078 
 
 4.080 
 
 
 8. 
 
 4.082 
 
 4.084 
 
 4.086 
 
 4.088 
 
 4.090 
 
 4.092 
 
 4.094 
 
 4.096 
 
 4.098 
 
 4.100 
 
 
 9. 
 
 4.102 
 
 4.104 
 
 4.106 
 
 4.108 
 
 4.109 
 
 4.111 
 
 4.113 
 
 4.115 
 
 4.117 
 
 4.119 
 
 
 70. 
 
 4.121 
 
 4.123 
 
 4.125 
 
 4.127 
 
 4.129 
 
 4.131 
 
 4.133 
 
 4.135 
 
 4.137 
 
 4.139 
 
 
 1. 
 
 4.141 
 
 4.143 
 
 4.145 
 
 4.147 
 
 4.149 
 
 4.151 
 
 4.152 
 
 4.154 
 
 4.156 
 
 4.158 
 
 
 2. 
 
 4.160 
 
 4.162 
 
 4.164 
 
 4.166 
 
 4.168 
 
 4.170 
 
 4.172 
 
 4.174 
 
 4.176 
 
 4.177 
 
 
 3. 
 
 4.179 
 
 4.181 
 
 4.183 
 
 4.185 
 
 4.187 
 
 4.189 
 
 4.191 
 
 4.193 
 
 4.195 
 
 4.196 
 
 
 4. 
 
 4.198 
 
 4.200 
 
 4.202 
 
 4.204 
 
 4.206 
 
 4.208 
 
 4.210 
 
 4.212 
 
 4.213 
 
 4.215 
 
 
 75. 
 
 4.217 
 
 4.219 
 
 4.221 
 
 4.223 
 
 4.225 
 
 4.227 
 
 4.228 
 
 4.230 
 
 4.232 
 
 4.234 
 
 
 6. 
 
 4.236 
 
 4.238 
 
 4.240 
 
 4.241 
 
 4.243 
 
 4.245 
 
 4.247 
 
 4.249 
 
 4.251 
 
 4.252 
 
 
 7. 
 
 4.254 
 
 4.256 
 
 4.258 
 
 4.260 
 
 4.262 
 
 4.264 
 
 4.265 
 
 4.267 
 
 4.269 
 
 4.271 
 
 
 8. 
 
 4.273 
 
 4.274 
 
 4.276 
 
 4.278 
 
 4.280 
 
 4282 
 
 4.284 
 
 4.285 
 
 4.287 
 
 4.289 
 
 
 9. 
 
 4.291 
 
 4.293 
 
 4294 
 
 4296 
 
 4.298 
 
 4.300 
 
 4.302 
 
 4.303 
 
 4.305 
 
 4.307 
 
 
 80. 
 
 4.309 
 
 4.311 
 
 4.312 
 
 4.314 
 
 4.316 
 
 4.318 
 
 4.320 
 
 4.321 
 
 4.323 
 
 4.325 
 
 
 1. 
 
 4.327 
 
 4.329 
 
 4.330 
 
 4.332 
 
 4.334 
 
 4.336 
 
 4.337 
 
 4.339 
 
 4.341 
 
 4.343 
 
 
 2. 
 
 4.344 
 
 4.346 
 
 4.348 
 
 4.350 
 
 4.352 
 
 4.353 
 
 4.355 
 
 4.357 
 
 4.359 
 
 4.360 
 
 
 3. 
 
 4.362 
 
 4.364 
 
 4.366 
 
 4.367 
 
 4.369 
 
 4.37 1 
 
 4.373 
 
 4.374 
 
 4.376 
 
 4.378 
 
 
 4, 
 
 4.380 
 
 4.381 
 
 4.383 
 
 4.385 
 
 4.386 
 
 4.388 
 
 4.390 
 
 4.392 
 
 4.393 
 
 4.395 
 
 
 85. 
 
 4.397 
 
 4.399 
 
 4.400 
 
 4.402 
 
 4.404 
 
 4.405 
 
 4.407 
 
 4.409 
 
 4.411 
 
 4.412 
 
 
 6. 
 
 4.414 
 
 4.416 
 
 4.417 
 
 4.419 
 
 4.421 
 
 4.423 
 
 4.424 
 
 4.426 
 
 4.428 
 
 4.429 
 
 
 7. 
 
 4.431 
 
 4.433 
 
 4.434 
 
 4.436 
 
 4.438 
 
 4.440 
 
 4.441 
 
 4.443 
 
 4.445 
 
 4.446 
 
 
 8. 
 
 4.448 
 
 4.450 
 
 4.451 
 
 4.453 
 
 4.455 
 
 4.456 
 
 4.458 
 
 4.460 
 
 4.461 
 
 4.463 
 
 
 9. 
 
 4.465 
 
 4.466 
 
 4.468 
 
 4.470 
 
 4.471 
 
 4.473 
 
 4.475 
 
 4.476 
 
 4.478 
 
 4.480 
 
 
 90. 
 
 4.481 
 
 4.483 
 
 4.485 
 
 4.486 
 
 4.488 
 
 4.490 
 
 4.491 
 
 4.493 
 
 4.495 
 
 4.496 
 
 
 1. 
 
 4.498 
 
 4.500 
 
 4.501 
 
 4.503 
 
 4.505 
 
 4.506 
 
 4.508 
 
 4.509 
 
 4.511 
 
 4.513 
 
 
 2. 
 
 4.514 
 
 4.516 
 
 4.518 
 
 4.519 
 
 4.521 
 
 4.523 
 
 4.524 
 
 4.526 
 
 4.527 
 
 4.529 
 
 
 3. 
 
 4.531 
 
 4.532 
 
 4.534 
 
 4.536 
 
 4.537 
 
 4.539 
 
 4.540 
 
 4.542 
 
 4.544 
 
 4.545 
 
 
 4. 
 
 4.547 
 
 4.548 
 
 4.550 
 
 4.552 
 
 4.553 
 
 4.555 
 
 4.556 
 
 4.558 
 
 4.560 
 
 4.561 
 
 
 95. 
 
 4.563 
 
 4.565 
 
 4.566 
 
 4.568 
 
 4.569 
 
 4.571 
 
 4.572 
 
 4.574 
 
 4.576 
 
 4.577 
 
 
 6. 
 
 4.579 
 
 4.580 
 
 4.582 
 
 4.584 
 
 4.585 
 
 4.587 
 
 4.588 
 
 4.590 
 
 4.592 
 
 4.593 
 
 
 7. 
 
 4.595 
 
 4.596 
 
 4.598 
 
 4.599 
 
 4.601 
 
 4.603 
 
 4.604 
 
 4.606 
 
 4.607 
 
 4.609 
 
 
 8. 
 
 4.610 
 
 4.612 
 
 4.614 
 
 4.615 
 
 4.617 
 
 4.618 
 
 4.620 
 
 4.621 
 
 4.623 
 
 4.625 
 
 
 9. 
 
 4.626 
 
 4.628 
 
 4.629 
 
 4.631 
 
 4.632 
 
 4.634 
 
 4.635 
 
 4.637 
 
 4.638 
 
 4.640 
 
 
 Moving the decimal point THREE places in .W requires moving it ONE place in body 
 of table (seep. 16). 
 
20 
 
 MATHEMATICAL TABLES 
 
 CUBE ROOTS (continued) 
 
 N 
 
 0. 
 
 1. 
 
 2. 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 7. 
 
 8. 
 
 9. 
 
 ! 
 
 10 
 
 4642 
 
 4.657 
 
 4.672 
 
 4.688 
 
 4.703 
 
 4.718 
 
 4.733 
 
 4.747 
 
 4.762 
 
 4.777 
 
 15 
 
 1 
 
 4791 
 
 4.806 
 
 4.820 
 
 4.835 
 
 4.849 
 
 4.863 
 
 4.877 
 
 4.891 
 
 4.905 
 
 4.919 
 
 14 
 
 2 
 
 4.932 
 
 4.946 
 
 4.960 
 
 4.973 
 
 4.987 
 
 5.000 
 
 5.013 
 
 5.027 
 
 5.040 
 
 5.053 
 
 13 
 
 3 
 
 5.066 
 
 5.079 
 
 5.092 
 
 5.104 
 
 5.117 
 
 5.130 
 
 5.143 
 
 5.155 
 
 5.168 
 
 5.180 
 
 
 4 
 
 5.192 
 
 5.205 
 
 5.217 
 
 5.229 
 
 5.241 
 
 5.254 
 
 5.266 
 
 5.278 
 
 5.290 
 
 5.301 
 
 12 
 
 15 
 
 5.313 
 
 5.325 
 
 5.337 
 
 '5.348 
 
 5.360 
 
 5.372 
 
 5,383 
 
 5.395 
 
 5.406 
 
 5.418 
 
 
 6 
 
 5.429 
 
 5.440 
 
 5.451 
 
 5.463 
 
 5.474 
 
 5.485 
 
 5.496 
 
 5.507 
 
 5.518 
 
 5.529 
 
 !1 
 
 7 
 
 5.540 
 
 5.550 
 
 5.561 
 
 5.572 
 
 5.583 
 
 5.593 
 
 5.604 
 
 5.615 
 
 5.625 
 
 5.636 
 
 
 8 
 
 5.646 
 
 5.657 
 
 5.667 
 
 5.677 
 
 5.688 
 
 5.698 
 
 5.708 
 
 5.718 
 
 5.729 
 
 5.739 
 
 10 
 
 9 
 
 5.749 
 
 5.759 
 
 5.769 
 
 5.779 
 
 5.789 
 
 5.799 
 
 5.809 
 
 5.819 
 
 5.828 
 
 5.838 
 
 
 20 
 
 5.848 
 
 5.858 
 
 5.867 
 
 5.877 
 
 5.887 
 
 5.896 
 
 5.906 
 
 5.915 
 
 5.925 
 
 5.934 
 
 
 1 
 
 5.944 
 
 5.953 
 
 5.963 
 
 5.972 
 
 5.981 
 
 5.991 
 
 6.000 
 
 6.009 
 
 6.018 
 
 6.028 
 
 9 
 
 2 
 
 6.037 
 
 6.046 
 
 6.055 
 
 6.064 
 
 6.073 
 
 6.082 
 
 6.091 
 
 6.100 
 
 6.109 
 
 6.118 
 
 
 3 
 
 6.127 
 
 6.136 
 
 6.145 
 
 6.153 
 
 6.162 
 
 6.171 
 
 6.180 
 
 6.188 
 
 6.197 
 
 6.206 
 
 
 4 
 
 6.214 
 
 6.223 
 
 6.232 
 
 6.240 
 
 6.249 
 
 6.257 
 
 6.266 
 
 6.274 
 
 6.283 
 
 6.291 
 
 
 25 
 
 6.300 
 
 6.308 
 
 6.316 
 
 6.325 
 
 6.333 
 
 6.341 
 
 6.350 
 
 6.358 
 
 6.366 
 
 6.374 
 
 8 
 
 6 
 
 6.383 
 
 6.391 
 
 6.399 
 
 6.407 
 
 6.415 
 
 6.423 
 
 6.431 
 
 6.439 
 
 6.447 
 
 6.455 
 
 
 7 
 
 6.463 
 
 6.471 
 
 6.479 
 
 6.487 
 
 6.495 
 
 6.503 
 
 6.511 
 
 6.519 
 
 6.527 
 
 6.534 
 
 
 8 
 
 6.542 
 
 6.550 
 
 6.558 
 
 6.565 
 
 6.573 
 
 6.581 
 
 6.589 
 
 6.596 
 
 6.604 
 
 6.611 
 
 
 9 
 
 6.619 
 
 6.627 
 
 6.634 
 
 6.642 
 
 6.649 
 
 6.657 
 
 6.664 
 
 6.672 
 
 6.679 
 
 6.687 
 
 
 30 
 
 6.694 
 
 6.702 
 
 6.709 
 
 6.717 
 
 6.724 
 
 6.731 
 
 6.739 
 
 6.746 
 
 6.753 
 
 6.761 
 
 7 
 
 1 
 
 6.768 
 
 6.775 
 
 6.782 
 
 6.790 
 
 6.797 
 
 6.804 
 
 6.811 
 
 6.818 
 
 6.826 
 
 6.833 
 
 
 2 
 
 6.840 
 
 6.847 
 
 6.854 
 
 6.861 
 
 6.868 
 
 6.875 
 
 6.882 
 
 6.889 
 
 6.896 
 
 6.903 
 
 
 3 
 
 6.910 
 
 6.917 
 
 6.924 
 
 6.931 
 
 6.938 
 
 6.945 
 
 6.952 
 
 6.959 
 
 6.966 
 
 6.973 
 
 
 4 
 
 6.980 
 
 6.986 
 
 6.993 
 
 7.000 
 
 7.007 
 
 7.014 
 
 7.020 
 
 7.027 
 
 7.034 
 
 7.041 
 
 
 35 
 
 7.047 
 
 7.054 
 
 7.061 
 
 7.067 
 
 7.074 
 
 7.081 
 
 7.087 
 
 7.094 
 
 7.101 
 
 7.107 
 
 
 6 
 
 7.114 
 
 7.120 
 
 7.127 
 
 7.133 
 
 7.140 
 
 7.147 
 
 7.153 
 
 7.160 
 
 7.166 
 
 7.173 
 
 6 
 
 7 
 
 7.179 
 
 7.186 
 
 7.192 
 
 7.198 
 
 7.205 
 
 7.211 
 
 7.218 
 
 7.224 
 
 7.230 
 
 7.237 
 
 
 8 
 
 7.243 
 
 7.250 
 
 7.256 
 
 7.262 
 
 7.268 
 
 7.275 
 
 7.281 
 
 7.287 
 
 7.294 
 
 7.300 
 
 
 9 
 
 7.306 
 
 7.312 
 
 7.319 
 
 7.325 
 
 7.331 
 
 7.337 
 
 7.343 
 
 7.350 
 
 7.356 
 
 7.362 
 
 
 40 
 
 7.368 
 
 7.374 
 
 7.380 
 
 7.386 
 
 7.393 
 
 7.399 
 
 7.405 
 
 7.411 
 
 7.417 
 
 7.423 
 
 
 1 
 
 7.429 
 
 7.435 
 
 7.441 
 
 7.447 
 
 7.453 
 
 7.459 
 
 7.465 
 
 7.471 
 
 7.477 
 
 7.483 
 
 
 2 
 
 7.489 
 
 7.495 
 
 7.501 
 
 7.507 
 
 7.513 
 
 7.518 
 
 7.524 
 
 7.530 
 
 7.536 
 
 7.542 
 
 
 3 
 
 7.548 
 
 7.554 
 
 7.560 
 
 7.565 
 
 7.571 
 
 7.577 
 
 7.583 
 
 7.589 
 
 7.594 
 
 7.600 
 
 
 4 
 
 7.606 
 
 7.612 
 
 7.617 
 
 7.623 
 
 7.629 
 
 7.635 
 
 7.640 
 
 7.646 
 
 7.652 
 
 7.657 
 
 
 45 
 
 7.663 
 
 7.669 
 
 7.674 
 
 7.680 
 
 7.686 
 
 7.691 
 
 7.697 
 
 7.703 
 
 7.708 
 
 7.714 
 
 5 
 
 6 
 
 7.719 
 
 7.725 
 
 7.731 
 
 7.736 
 
 7.742 
 
 7.747 
 
 7.753 
 
 7.758 
 
 7.764 
 
 7.769 
 
 
 7 
 
 7.775 
 
 7.780 
 
 7.786 
 
 7.791 
 
 7.797 
 
 7.802 
 
 7.808 
 
 7.813 
 
 7.819 
 
 7.824 
 
 
 8 
 
 7.830 
 
 7.835 
 
 7.841 
 
 7.846 
 
 7.851 
 
 7.857 
 
 7.862 
 
 7.868 
 
 7.873 
 
 7.878 
 
 
 9 
 
 7.884 
 
 7.889 
 
 7.894 
 
 7.900 
 
 7.905 
 
 7.910 
 
 7.916 
 
 7.921 
 
 7.926 
 
 7.932 
 
 
 AUXILIARY TABLE OP TWO-THIRDS POWERS 
 
 AND THREE-HALVES POWERS (see pp. 22-23) 
 
 (To assist in locating the decimal point) 
 
 N 
 
 N K( = ^N^) 
 
 N**(- VF') 
 
 
 .0001 
 .001 
 .01 
 
 .1 
 1. 
 
 10. 
 100. 
 
 .002154 
 .01 
 .0464 
 . .2154 
 1. 
 4.64 
 21.54 
 
 .000001 
 00003162 
 .001 
 .03162278 
 1. 
 31.62278 
 1000. 
 
 For complete table 
 of three-halves pow- 
 ers, see pp. 22-23. 
 That table, used in- 
 versely, provides a 
 complete table of 
 two-thirds powers. 
 
 1000. 
 
 100. 
 
 31622.78 
 
 
 10000. 
 
 464.16 
 
 1000000. 
 
 
MATHEMATICAL TABLES 
 CUBE ROOTS (continued) 
 
 21 
 
 N 
 
 0. 
 
 1. 
 
 2. 
 
 a. 
 
 4. 
 
 5. 
 
 6. 
 
 7. 
 
 8. 
 
 9. 
 
 fts 
 
 <;-a 
 
 50 
 
 7.937 
 
 7.942 
 
 7.948 
 
 7.953 
 
 7.958 
 
 7.963 
 
 7.969 
 
 7.974 
 
 7.979 
 
 7.984 
 
 5 
 
 1 
 
 7.990 
 
 7.995 
 
 8.000 
 
 8.005 
 
 8.010 
 
 8.016 
 
 8.021 
 
 8.026 
 
 8.031 
 
 8.036 
 
 
 2 
 
 8.041 
 
 8.047 
 
 8.052 
 
 8.057 
 
 8.062 
 
 8.067 
 
 8.072 
 
 8.077 
 
 8.082 
 
 8.088 
 
 
 3 
 
 8.093 
 
 8.098 
 
 8.103 
 
 8.108 
 
 8.113 
 
 8.118 
 
 8.123 
 
 8.128 
 
 8.133 
 
 8.138 
 
 
 4 
 
 8.143 
 
 8.148 
 
 8.153 
 
 8.158 
 
 8.163 
 
 8.168 
 
 8.173 
 
 8.178 
 
 8.183 
 
 8.188 
 
 
 55 
 
 8.193 
 
 8.198 
 
 8.203 
 
 8.208 
 
 8.213 
 
 8.218 
 
 8.223 
 
 8.228 
 
 8.233 
 
 8.238 
 
 
 6 
 
 8.243 
 
 8.247 
 
 8.252 
 
 8.257 
 
 8.262 
 
 8.267 
 
 8.272 
 
 8.277 
 
 8.282 
 
 8.286 
 
 
 7 
 
 8.291 
 
 8.296 
 
 8.301 
 
 8.306 
 
 8.311 
 
 8.316 
 
 8.320 
 
 8.325 
 
 8.330 
 
 8.335 
 
 
 8 
 
 8.340 
 
 8.344 
 
 8.349 
 
 8.354 
 
 8.359 
 
 8.363 
 
 8.368 
 
 8.373 
 
 8.378 
 
 8.382 
 
 
 9 
 
 8.387 
 
 8.392 
 
 8.397 
 
 8.401 
 
 8.406 
 
 8.411 
 
 8.416 
 
 8.420 
 
 8.425 
 
 8.430 
 
 
 60 
 
 8.434 
 
 8.439 
 
 8.444 
 
 8.448 
 
 8.453 
 
 8.458 
 
 8.462 
 
 8.467 
 
 8.472 
 
 8.476 
 
 
 1 
 
 8.481 
 
 8.486 
 
 8.490 
 
 8.495 
 
 8.499 
 
 8.504 
 
 8.509 
 
 8.513 
 
 8.518 
 
 8.522 
 
 
 2 
 
 8.527 
 
 8.532 
 
 8.536 
 
 8.541 
 
 8.545 
 
 8.550 
 
 8.554 
 
 8.559 
 
 8.564 
 
 8.568 
 
 
 3 
 
 8.573 
 
 8.577 
 
 8.582 
 
 8.586 
 
 8.591 
 
 8.595 
 
 8.600 
 
 8.604 
 
 8.609 
 
 8.613 
 
 4 
 
 4 
 
 8.618 
 
 8.622 
 
 8.627 
 
 8.631 
 
 8.636 
 
 8.640 
 
 8.645 
 
 8.649 
 
 8.653 
 
 8.658 
 
 
 65 
 
 8.662 
 
 8.667 
 
 8.671 
 
 8.676 
 
 8.680 
 
 8.685 
 
 8.689 
 
 8.693 
 
 8.698 
 
 8.702 
 
 
 6 
 
 8.707 
 
 8.711 
 
 8.715 
 
 8.720 
 
 8.724 
 
 8.729 
 
 8.733 
 
 8.737 
 
 8.742 
 
 8.746 
 
 
 7 
 
 8.750 
 
 8.755 
 
 8.759 
 
 8.763 
 
 8.768 
 
 8.772 
 
 8.776 
 
 8.781 
 
 8.785 
 
 8.789 
 
 
 8 
 
 8.794 
 
 8.798 
 
 8.802 
 
 8.807 
 
 8.811 
 
 8.815 
 
 8.819 
 
 8.824 
 
 8.828 
 
 8.832 
 
 
 9 
 
 8.837 
 
 8.841 
 
 8.845 
 
 8.849 
 
 8.854 
 
 8.858 
 
 8.862 
 
 8.866 
 
 8.871 
 
 8.875 
 
 
 70 
 
 8.879 
 
 8.883 
 
 8.887 
 
 8.892 
 
 8.896 
 
 8.900 
 
 8.904 
 
 8.909 
 
 8.913 
 
 8.917 
 
 
 1 
 
 8.921 
 
 8.925 
 
 8.929 
 
 8.934 
 
 8.938 
 
 8.942 
 
 8.946 
 
 8.950 
 
 8.955 
 
 8.959 
 
 
 2 
 
 8.963 
 
 8.967 
 
 8.971 
 
 8.975 
 
 8.979 
 
 8.984 
 
 8.988 
 
 8.992 
 
 8.996 
 
 9.000 
 
 
 3 
 
 9.004 
 
 9.008 
 
 9.012 
 
 9.016 
 
 9.021 
 
 9.025 
 
 9.029 
 
 9.033 
 
 9.037 
 
 9.041 
 
 
 4 
 
 9.045 
 
 9.049 
 
 9.053 
 
 9.057 
 
 9.061 
 
 9.065 
 
 9.069 
 
 9.073 
 
 9.078 
 
 9.082 
 
 
 75 
 
 9.086 
 
 9.090 
 
 9.094 
 
 9.098 
 
 9.102 
 
 9.106 
 
 9.110 
 
 9.114 
 
 9.118 
 
 9.122 
 
 
 6 
 
 9.126 
 
 9.130 
 
 9.134 
 
 9.138 
 
 9.142 
 
 9.146 
 
 9.150 
 
 9.154 
 
 9.158 
 
 9.162 
 
 
 7 
 
 9.166 
 
 9.170 
 
 9.174 
 
 9.178 
 
 9.182 
 
 9.185 
 
 9.189 
 
 9.193 
 
 9.197 
 
 9.201 
 
 
 8 
 
 9.205 
 
 9.209 
 
 9.213 
 
 9.217 
 
 9.221 
 
 9.225 
 
 9.229 
 
 9.233 
 
 9.237 
 
 9.240 
 
 
 9 
 
 9.244 
 
 9.248 
 
 9.252 
 
 9.256 
 
 9.260 
 
 9.264 
 
 9.268 
 
 9.272 
 
 9.275 
 
 9.279 
 
 
 80 
 
 9.283 
 
 9.287 
 
 9.291 
 
 9.295 
 
 9.299 
 
 9.302 
 
 9.306 
 
 9.310 
 
 9.314 
 
 9.318 
 
 
 1 
 
 9.322 
 
 9.326 
 
 9.329 
 
 9.333 
 
 9.337 
 
 9.341 
 
 9.345 
 
 9.348 
 
 9.352 
 
 9.356 
 
 
 2 
 
 9.360 
 
 9.364 
 
 9.368 
 
 9.371 
 
 9.375 
 
 9.379 
 
 9.383 
 
 9.386 
 
 9.390 
 
 9.394 
 
 
 3 
 
 9.398 
 
 9.402 
 
 9.405 
 
 9.409 
 
 9.413 
 
 9.417 
 
 9.420 
 
 9.424 
 
 9.428 
 
 9.432 
 
 
 4 
 
 9.435 
 
 9.439 
 
 9.443 
 
 9.447 
 
 9.450 
 
 9.454 
 
 9.458 
 
 9.462 
 
 9.465 
 
 9.469 
 
 
 85 
 
 9.473 
 
 9.476 
 
 9.480 
 
 9.484 
 
 9.488 
 
 9.491 
 
 9.495 
 
 9.499 
 
 9.502 
 
 9.506 
 
 
 6 
 
 9.510 
 
 9.513 
 
 9.517 
 
 9.521 
 
 9.524 
 
 9.528 
 
 9.532 
 
 9.535 
 
 9.539 
 
 9.543 
 
 
 7 
 
 9.546 
 
 9.550 
 
 9.554 
 
 9.557 
 
 9.561 
 
 9.565 
 
 9.568 
 
 9.572 
 
 9.576 
 
 9.579 
 
 
 8 
 
 9.583 
 
 9.586 
 
 9.590 
 
 9.594 
 
 9.597 
 
 9.601 
 
 9.605 
 
 9.608 
 
 9.612 
 
 9.615 
 
 
 9 
 
 9.619 
 
 9.623 
 
 9.626 
 
 9.630 
 
 9.633 
 
 9.637 
 
 9.641 
 
 9.644 
 
 9.648 
 
 9.651 
 
 
 90 
 
 9.655 
 
 9.658 
 
 9.662 
 
 9.666 
 
 9.669 
 
 9.673 
 
 9.676 
 
 9.680 
 
 9.683 
 
 9.687 
 
 
 1 
 
 9.691 
 
 9.694 
 
 9.698 
 
 9.701 
 
 9.705 
 
 9.708 
 
 9.712 
 
 9.715 
 
 9.719 
 
 9.722 
 
 
 2 
 
 9.726 
 
 9.729 
 
 9.733 
 
 9.736 
 
 9.740 
 
 9.743 
 
 9.747 
 
 9.750 
 
 9.754 
 
 9.758 
 
 
 3 
 
 9.761 
 
 9.764 
 
 9.768 
 
 9.771 
 
 9.775 
 
 9.778 
 
 9.782 
 
 9.785 
 
 9.789 
 
 9.792 
 
 
 4 
 
 9.796 
 
 9.799 
 
 9.803 
 
 9.806 
 
 9.810 
 
 9.813 
 
 9.817 
 
 9.820 
 
 9.824 
 
 9.827 
 
 
 95 
 
 9.830 
 
 9.834 
 
 9.837 
 
 9.841 
 
 9.844 
 
 9.848 
 
 9.851 
 
 9.855 
 
 9.858 
 
 9.861 
 
 
 6 
 
 9.865 
 
 9.868 
 
 9.872 
 
 9.875 
 
 9.879 
 
 9.882 
 
 9.885 
 
 9.889 
 
 9.892 
 
 9.896 
 
 
 7 
 
 9.899 
 
 9.902 
 
 9.906 
 
 9.909 
 
 9.913 
 
 9.916 
 
 9.919 
 
 9.923 
 
 9.926 
 
 9.930 
 
 
 8 
 
 9.933 
 
 9.936 
 
 9.940 
 
 9.943 
 
 9.946 
 
 9.950 
 
 9.953 
 
 9.956 
 
 9.960 
 
 9.963 
 
 
 9 
 
 9.967 
 
 9.970 
 
 9.973 
 
 9.977 
 
 9.980 
 
 9.983 
 
 9.987 
 
 9.990 
 
 9.993 
 
 9.997 
 
 
 100 
 
 10.00 
 
 
 
 
 
 
 
 
 
 
 
 Moving the decimal point THREE places in N requires moving it ONE place in body 
 of table (see p. 16). 
 
22 
 
 MATHEMATICAL TABLES 
 
 THREE-HALVES POWERS OP NUMBERS (see also p. 20) 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 fl 
 
 1. 
 
 1.000 
 
 1.154 
 
 1.315 
 
 1.482 
 
 1.657 
 
 1.837 
 
 2.024 
 
 2.217 
 
 2.415 
 
 2.619 
 
 183 
 
 2. 
 
 2.828 
 
 3.043 
 
 3.263 
 
 3.488 
 
 3.718 
 
 3.953 
 
 4.192 
 
 4.437 
 
 4.685 
 
 4.939 
 
 237 
 
 3. 
 
 5.196 
 
 5.458 
 
 5.724 
 
 5.995 
 
 6.269 
 
 6.548 
 
 6.831 
 
 7.117 
 
 7.408 
 
 7.702 
 
 280 
 
 4. 
 
 8.000 
 
 8.302 
 
 8.607 
 
 8.917 
 
 9.230 
 
 9.546 
 
 9.866 
 
 10.190 
 
 
 
 313 
 
 4. 
 
 
 
 
 
 
 
 
 10.19 
 
 10.52 
 
 10.85 
 
 33 
 
 5. 
 
 11.18 
 
 11.52 
 
 11.86 
 
 12.20 
 
 12.55 
 
 12.90 
 
 13.25 
 
 13.61 
 
 13.97 
 
 14.33 
 
 35 
 
 6. 
 
 14.70 
 
 15.07 
 
 15.44 
 
 15.81 
 
 16.19 
 
 16.57 
 
 16.96 
 
 17.34 
 
 17.73 
 
 18.12 
 
 38 
 
 7. 
 
 18.52 
 
 18.92 
 
 19.32 
 
 19.72 
 
 20.13 
 
 20.54 
 
 20.95 
 
 21.37 
 
 21.78 
 
 22.20 
 
 41 
 
 8. 
 
 22.63 
 
 23.05 
 
 23.48 
 
 23.91 
 
 24.35 
 
 24.78 
 
 25.22 
 
 25.66 
 
 26.11 
 
 26.55 
 
 44 
 
 9. 
 
 27.00 
 
 27.45 
 
 27.90 
 
 28.36 
 
 28.82 
 
 29.28 
 
 29.74 
 
 30.21 
 
 30.68 
 
 31.15 
 
 46 
 
 10. 
 
 31.62 
 
 32.10 
 
 32.58 
 
 33.06 
 
 33.54 
 
 34.02 
 
 34.51 
 
 35.00 
 
 35.49 
 
 35.99 
 
 49 
 
 1. 
 
 36.48 
 
 36.98 
 
 37.48 
 
 37.99 
 
 38.49 
 
 39.00 
 
 39.51 
 
 40.02 
 
 40.53 
 
 41.05 
 
 51 
 
 2. 
 
 41.57 
 
 42.09 
 
 42.61 
 
 43.14 
 
 43.66 
 
 44.19 
 
 44.73 
 
 45.26 
 
 45.79 
 
 46.33 
 
 53 
 
 3. 
 
 46.87 
 
 47.41 
 
 47.96 
 
 48.50 
 
 49.05 
 
 49.60 
 
 50.15 
 
 50.71 
 
 51.26 
 
 51.82 
 
 55 
 
 4. 
 
 52.38 
 
 52.95 
 
 53.51 
 
 54.08 
 
 54.64 
 
 55.21 
 
 55.79 
 
 56.36 
 
 56.94 
 
 57.51 
 
 57 
 
 15. 
 
 58.09 
 
 58.68 
 
 59.26 
 
 59.85 
 
 60.43 
 
 61.02 
 
 61.62 
 
 62.21 
 
 62.80 
 
 63.40 
 
 59 
 
 6. 
 
 64.00 
 
 64.60 
 
 65.20 
 
 65.81 
 
 66.41 
 
 67.02 
 
 67.63 
 
 68.25 
 
 68.86 
 
 69.48 
 
 61 
 
 7. 
 
 70.09 
 
 70.71 
 
 71.33 
 
 71.96 
 
 72.58 
 
 73.21 
 
 73.84 
 
 74.47 
 
 75.10 
 
 75.73 
 
 63 
 
 8. 
 
 76.37 
 
 77.00 
 
 77.64 
 
 78.28 
 
 78.93 
 
 79.57 
 
 80.22 
 
 80.87 
 
 81.51 
 
 82.17 
 
 65 
 
 9. 
 
 82.82 
 
 83.47 
 
 84.13 
 
 84.79 
 
 85.45 
 
 86.11 
 
 86.77 
 
 87.44 
 
 88.10 
 
 88.77 
 
 66 
 
 20. 
 
 89.44 
 
 90.11 
 
 90.79 
 
 91.46 
 
 92.14 
 
 92.82 
 
 93.50 
 
 94.18 
 
 94.86 
 
 95.55 
 
 68 
 
 1. 
 
 96.23 
 
 96.92 
 
 97.61 
 
 98.30 
 
 99.00 
 
 99.69 
 
 100.38 
 
 
 
 
 69 
 
 I. 
 
 
 
 
 
 
 
 100.4 
 
 101.1 
 
 101.8 
 
 102.5 
 
 7 
 
 2. 
 
 103.2 
 
 103.9 
 
 104.6 
 
 105.3 
 
 106.0 
 
 106.7 
 
 107.4 
 
 108.2 
 
 108.9 
 
 109.6 
 
 7 
 
 3. 
 
 110.3 
 
 111.0 
 
 111.7 
 
 112.5 
 
 113.2 
 
 113.9 
 
 114.6 
 
 115.4 
 
 116.1 
 
 116.8 
 
 7 
 
 4. 
 
 117.6 
 
 118.3 
 
 119.0 
 
 119.8 
 
 120.5 
 
 121.3 
 
 122.0 
 
 122.8 
 
 123.5 
 
 124.3 
 
 7 
 
 25. 
 
 125.0 
 
 125.8 
 
 126.5 
 
 127.3 
 
 128.0 
 
 128.8 
 
 129.5 
 
 130.3 
 
 131.0 
 
 131.8 
 
 8 
 
 6. 
 
 132.6 
 
 133.3 
 
 134.1 
 
 134.9 
 
 135.6 
 
 136.4 
 
 137.2 
 
 138.0 
 
 138.7 
 
 139.5 
 
 8 
 
 7. 
 
 140.3 
 
 141.1 
 
 141.9 
 
 142.6 
 
 143.4 
 
 144.2 
 
 145.0 
 
 145.8 
 
 146.6 
 
 147.4 
 
 8 
 
 8. 
 
 148.2 
 
 149.0 
 
 149.8 
 
 150.5 
 
 151.3 
 
 152.1 
 
 152.9 
 
 153.8 
 
 154.6 
 
 155.4 
 
 8 
 
 9. 
 
 156.2 
 
 157.0 
 
 157.8 
 
 158.6 
 
 159.4 
 
 160.2 
 
 161.0 
 
 161.9 
 
 162.7 
 
 163.5 
 
 8 
 
 30. 
 
 164.3 
 
 " 165.1 
 
 166.0 
 
 166.8 
 
 167.6 
 
 168.4 
 
 169.3 
 
 170.1 
 
 170.9 
 
 171.8 
 
 8 
 
 I, 
 
 172.6 
 
 173 A 
 
 174.3 
 
 175.1 
 
 176.0 
 
 176.8 
 
 177.6 
 
 178.5 
 
 179.3 
 
 180.2 
 
 8 
 
 2. 
 
 181.0 
 
 181.9 
 
 182.7 
 
 183.6 
 
 184.4 
 
 185.3 
 
 186.1 
 
 187.0 
 
 187.8 
 
 188.7 
 
 9 
 
 3. 
 
 189.6 
 
 190.4 
 
 191.3 
 
 192.2 
 
 193.0 
 
 193.9 
 
 194.8 
 
 195.6 
 
 196.5 
 
 197.4 
 
 9 
 
 4. 
 
 198.3 
 
 199.1 
 
 200.0 
 
 200.9 
 
 201.8 
 
 202.6 
 
 203.5 
 
 204.4 
 
 205.3 
 
 206.2 
 
 9 
 
 35. 
 
 207.1 
 
 208.0 
 
 208.8 
 
 209.7 
 
 210.6 
 
 211.5 
 
 212.4 
 
 213.3 
 
 214.2 
 
 215.1 
 
 9 
 
 6. 
 
 216.0 
 
 216.9 
 
 217.8 
 
 218.7 
 
 219.6 
 
 220.5 
 
 221.4 
 
 222.3 
 
 223.2 
 
 224.2 
 
 9 
 
 7. 
 
 225.1 
 
 226.0 
 
 226.9 
 
 227.8 
 
 228.7 
 
 229.6 
 
 230.6 
 
 231.5 
 
 232.4 
 
 233.3 
 
 9 
 
 8. 
 
 234.2 
 
 235.2 
 
 236.1 
 
 237.0 
 
 238.0 
 
 238.9 
 
 239.8 
 
 240.8 
 
 241.7 
 
 242.6 
 
 9 
 
 9. 
 
 243.6 
 
 244.5 
 
 245.4 
 
 246.4 
 
 247.3 
 
 248.3 
 
 249.2 
 
 250.1 
 
 251.1 
 
 252.0 
 
 9 
 
 40. 
 
 253.0 
 
 253.9 
 
 254.9 
 
 255.8 
 
 256.8 
 
 257.7 
 
 258.7 
 
 259.7 
 
 260.6 
 
 261.6 
 
 10 
 
 1; 
 
 262.5 
 
 263.5 
 
 264.5 
 
 265.4 
 
 266.4 
 
 267.3 
 
 268.3 
 
 269.3 
 
 270.2 
 
 271.2 
 
 10 
 
 2. 
 
 272.2 
 
 273.2 
 
 274.1 
 
 275.1 
 
 276.1 
 
 277.1 
 
 278.0 
 
 279.0 
 
 280.0 
 
 281.0 
 
 10 
 
 3. 
 
 282.0 
 
 283.0 
 
 283.9 
 
 284.9 
 
 285.9 
 
 286.9 
 
 287.9 
 
 288.9 
 
 289.9 
 
 290.9 
 
 10 
 
 4. 
 
 291.9 
 
 292.9 
 
 293.9 
 
 294.9 
 
 295.9 
 
 296.9 
 
 297.9 
 
 298.9 
 
 299.9 
 
 300.9 
 
 10 
 
 45. 
 
 301.9 
 
 302.9 
 
 303.9 
 
 304.9 
 
 305.9 
 
 306.9 
 
 307.9 
 
 308.9 
 
 310.0 
 
 311.0 
 
 10 
 
 6. 
 
 312.0 
 
 313.0 
 
 314.0 
 
 315.0 
 
 316.1 
 
 317.1 
 
 318.1 
 
 319.1 
 
 320.2 
 
 321.2 
 
 10 
 
 7. 
 
 322.2 
 
 323.2 
 
 324.3 
 
 325.3 
 
 326.3 
 
 327.4 
 
 328.4 
 
 329.4 
 
 330.5 
 
 331.5 
 
 10 
 
 8. 
 
 332.6 
 
 333.6 
 
 334.6 
 
 335.7 
 
 336.7 
 
 337.8 
 
 338.8 
 
 339.9 
 
 340.9 
 
 342.0 
 
 10 
 
 9. 
 
 343.0 
 
 344.1 
 
 345.1 
 
 346.2 
 
 347.2 
 
 348.3 
 
 349.3 
 
 350.4 
 
 351.4 
 
 352.5 
 
 11 
 
 This table gives N? 2 from N => 1 to N = 100. Moving the decimal point TWO 
 places in N requires moving it THREE places in body of' table. Thus: 
 (7.23)^ = 19.44; (723.)^ = 19440; (0.0723)^ = 0.01944 
 (72.3)^ = 614.8; (7230.)^ = 614800; (0.723)^ = 0.6148 
 
 Used inversely, table gives M^ from M = 1 to M - 1000. Thus: (0.6148)** => 0.7230. 
 
MATHEMATICAL TABLES 
 THREE-HALVES POWERS (continued') (See also p. 20) 
 
 23 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 <H-O 
 
 50. 
 
 353.6 
 
 354.6 
 
 355.7 
 
 356.7 
 
 357.8 
 
 358.9 
 
 359.9 
 
 361.0 
 
 362.1 
 
 363.1 
 
 11 
 
 I. 
 
 364.2 
 
 365.3 
 
 366.4 
 
 367.4 
 
 368.5 
 
 369.6 
 
 370.7 
 
 371.7 
 
 372.8 
 
 373.9 
 
 11 
 
 2. 
 
 375.0 
 
 376.1 
 
 377.1 
 
 378.2 
 
 379.3 
 
 380.4 
 
 381.5 
 
 382.6 
 
 383.7 
 
 384.8 
 
 11 
 
 3 
 
 385.8 
 
 386.9 
 
 388.0 
 
 389.1 
 
 390.2 
 
 391.3 
 
 392.4 
 
 393.5 
 
 394.6 
 
 395.7 
 
 11 
 
 4. 
 
 396.8 
 
 397.9 
 
 399.0 
 
 400.1 
 
 401.2 
 
 402.3 
 
 403.4 
 
 404.6 
 
 405.7 
 
 406.8 
 
 11 
 
 55. 
 
 407.9 
 
 409.0 
 
 410.1 
 
 411.2 
 
 412.3 
 
 413.5 
 
 414.6 
 
 415.7 
 
 416.8 
 
 417.9 
 
 11 
 
 6. 
 
 419.1 
 
 420.2 
 
 421.3 
 
 422.4 
 
 423.6 
 
 424.7 
 
 425.8 
 
 426.9 
 
 428.1 
 
 429.2 
 
 11 
 
 7. 
 
 430.3 
 
 431.5 
 
 432.6 
 
 433.7 
 
 434.9 
 
 436.0 
 
 437.2 
 
 438.3 
 
 439.4 
 
 440.6 
 
 11 
 
 8. 
 
 441.7 
 
 442.9 
 
 444.0 
 
 445.1 
 
 446.3 
 
 447.4 
 
 448.6 
 
 449.7 
 
 450.9 
 
 452.0 
 
 11 
 
 9. 
 
 453.2 
 
 454.3 
 
 455.5 
 
 456.6 
 
 457.8 
 
 459.0 
 
 460.1 
 
 461.3 
 
 462.4 
 
 463.6 
 
 12 
 
 60. 
 
 464.8 
 
 465.9 
 
 467.1 
 
 468.2 
 
 469.4 
 
 470.6 
 
 471.7 
 
 472.9 
 
 474.1 
 
 475.3 
 
 12 
 
 1. 
 
 476.4 
 
 477.6 
 
 478.8 
 
 479.9 
 
 481.1 
 
 482.3 
 
 483.5 
 
 484.6 
 
 485.8 
 
 487.0 
 
 12 
 
 2. 
 
 488.2 
 
 489.4 
 
 490.6 
 
 491.7 
 
 492.9 
 
 ' 494.1 
 
 495.3 
 
 496.5 
 
 497.7 
 
 498.9 
 
 12 
 
 3. 
 
 500.0 
 
 501.2 
 
 502.4 
 
 503.6 
 
 504.8 
 
 506.0 
 
 507.2 
 
 508.4 
 
 509.6 
 
 510.8 
 
 12 
 
 4. 
 
 512.0 
 
 513.2 
 
 514.4 
 
 515.6 
 
 516.8 
 
 518.0 
 
 519.2 
 
 520.4 
 
 521.6 
 
 522.8 
 
 12 
 
 65. 
 
 524.0 
 
 525.3 
 
 526.5 
 
 527.7 
 
 528.9 
 
 530.1 
 
 531.3 
 
 532.5 
 
 533.8 
 
 535.0 
 
 12 
 
 6' 
 
 536.2 
 
 537.4 
 
 538.6 
 
 539.8 
 
 541.1 
 
 542.3 
 
 543.5 
 
 544.7 
 
 546.0 
 
 547.2 
 
 12 
 
 7. 
 
 548.4 
 
 . 549.6 
 
 550.9 
 
 552.1 
 
 553.3 
 
 554.6 
 
 555.8 
 
 5570 
 
 558.3 
 
 559.5 
 
 12 
 
 8. 
 
 560.7 
 
 562.0 
 
 563.2 
 
 564.5 
 
 565.7 
 
 566.9 
 
 568.2 
 
 569.4 
 
 570.7 
 
 571.9 
 
 12 
 
 9. 
 
 573.2 
 
 574.4 
 
 575.7 
 
 576.9 
 
 578.1 
 
 579.4 
 
 580.6 
 
 581.9 
 
 583.2 
 
 584.4 
 
 13 
 
 70. 
 
 585.7 
 
 586.9 
 
 588.2 
 
 589.4 
 
 590.7 
 
 591.9 
 
 593.2 
 
 594.5 
 
 595.7 
 
 597.0 
 
 13 
 
 1. 
 
 598.3 
 
 599.5 
 
 600.8 
 
 602.1 
 
 603.3 
 
 604.6 
 
 605.9 
 
 607.1 
 
 608.4 
 
 609.7 
 
 13 
 
 2. 
 
 610.9 
 
 612.2 
 
 613.5 
 
 614.8 
 
 616.0 
 
 617.3 
 
 618.6 
 
 619.9 
 
 621.2 
 
 622.4 
 
 13 
 
 3. 
 
 623.7 
 
 625.0 
 
 626.3 
 
 627.6 
 
 628.8 
 
 630.1 
 
 631.4 
 
 632.7 
 
 634.0 
 
 635.3 
 
 13 
 
 4. 
 
 636.6 
 
 637.9 
 
 639.2 
 
 640.4 
 
 641.7 
 
 643.0 
 
 644.3 
 
 645.6 
 
 646.9 
 
 648.2 
 
 13 
 
 75. 
 
 649.5 
 
 650.8 
 
 652.1 
 
 653.4 
 
 654.7 
 
 656.0 
 
 657.3 
 
 658.6 
 
 659.9 
 
 661.2 
 
 13 
 
 6. 
 
 662.6 
 
 663.9 
 
 665.2 
 
 666.5 
 
 667.8 
 
 669.1 
 
 670.4 
 
 671.7 
 
 673.0 
 
 674.4 
 
 13 
 
 7. 
 
 675.7 
 
 677.0 
 
 678.3 
 
 679.6 
 
 680.9 
 
 682.3 
 
 683.6 
 
 684.9 
 
 686.2 
 
 687.6 
 
 13 
 
 8. 
 
 688.9 
 
 690.2 
 
 691.5 
 
 692.9 
 
 694.2 
 
 695.5 
 
 696.8 
 
 698.2 
 
 699.5 
 
 700.8 
 
 13 
 
 9. 
 
 702.2 
 
 703.5 
 
 704.8 
 
 706.2 
 
 707.5 
 
 708.8 
 
 710.2 
 
 711.5 
 
 712.9 
 
 714.2 
 
 13 
 
 80. 
 
 715.5 
 
 716.9 
 
 718.2 
 
 719.6 
 
 720.9 
 
 722.3 
 
 723.6 
 
 725.0 
 
 726.3 
 
 727.7 
 
 13 
 
 I. 
 
 729.0 
 
 730.4 
 
 731.7 
 
 733.1 
 
 734.4 
 
 735.8 
 
 737.1 
 
 738.5 
 
 739.8 
 
 741.2 
 
 14 
 
 2. 
 
 742.5 
 
 743.9 
 
 745.3 
 
 746.6 
 
 748.0 
 
 749.3 
 
 750.7 
 
 752.1 
 
 753.4 
 
 754.8 
 
 14 
 
 3. 
 
 756.2 
 
 757.5 
 
 758.9 
 
 760.3 
 
 761.6 
 
 763.0 
 
 764.4 
 
 765.8 
 
 767.1 
 
 768.5 
 
 14 
 
 4. 
 
 769.9 
 
 771.2 
 
 772.6 
 
 774.0 
 
 775.4 
 
 776.8 
 
 778.1 
 
 779.5 
 
 780.9 
 
 782.3 
 
 14 
 
 85. 
 
 783.7 
 
 785.0 
 
 786.4 
 
 787.8 
 
 789:2 
 
 790.6 
 
 792.0 
 
 793.4 
 
 794.8 
 
 796.1 
 
 14 
 
 6. 
 
 797.5 
 
 798.9 
 
 800.3 
 
 801.7 
 
 803.1 
 
 804.5 
 
 805.9 
 
 807.3 
 
 808.7 
 
 810.1 
 
 14 
 
 ' 7. 
 
 811.5 
 
 812.9 
 
 814.3 
 
 815.7 
 
 817.1 
 
 818.5 
 
 819.9 
 
 821.3 
 
 822.7 
 
 824.1 
 
 14 
 
 8. 
 
 825.5 
 
 826.9 
 
 828.3 
 
 829.7 
 
 831.1 
 
 832.6 
 
 834.0 
 
 835.4 
 
 836.8 
 
 838.2 
 
 14 
 
 9. 
 
 839.6 
 
 841.0 
 
 842.5 
 
 843.9 
 
 845.3 
 
 846.7 
 
 848.1 
 
 849.5 
 
 851.0 
 
 852.4 
 
 14 
 
 90. 
 
 853.8 
 
 855.2 
 
 856.7 
 
 858.1 
 
 859.5 
 
 860.9 
 
 862.4 
 
 863.8 
 
 865.2 
 
 866.7 
 
 14 
 
 1. 
 
 868. J 
 
 869.5 
 
 870.9 
 
 872.4 
 
 873.8 
 
 875.2 
 
 876.7 
 
 878.1 
 
 879.6 
 
 881.0 
 
 14 
 
 2. 
 
 882.4 
 
 883.9 
 
 885.3 
 
 886.8 
 
 888.2 
 
 889.6 
 
 891.1 
 
 892.5 
 
 894.0 
 
 895.4 
 
 14 
 
 3. 
 
 896.9 
 
 898.3 
 
 899.8 
 
 901.2 
 
 902.7 
 
 904.1 
 
 905.6 
 
 907.0 
 
 908.5 
 
 909.9 
 
 15 
 
 4. 
 
 911.4 
 
 912.8 
 
 914.3 
 
 915.7 
 
 917.2 
 
 918.6 
 
 920.1 
 
 921.6 
 
 923.0 
 
 924.5 
 
 15 
 
 95. 
 
 925.9 
 
 927.4 
 
 928.9 
 
 930.3 
 
 931.8 
 
 933.3 
 
 934.7 
 
 936.2 
 
 937.7 
 
 939.1 
 
 15 
 
 6. 
 
 940.6 
 
 942.1 
 
 943.5 
 
 945.0 
 
 946.5 
 
 948.0 
 
 949.4 
 
 950.9 
 
 952.4 
 
 953.9 
 
 15 
 
 7. 
 
 955.3 
 
 956.8 
 
 958.3 
 
 959.8 
 
 961.3 
 
 962.7 
 
 964.2 
 
 965.7 
 
 967.2 
 
 968.7 
 
 15 
 
 8. 
 
 970.2 
 
 971.6 
 
 973.1 
 
 974.6 
 
 976.1 
 
 977.6 
 
 979.1 
 
 980.6 
 
 982.1 
 
 983.5 
 
 15 
 
 9. 
 
 985.0 
 
 986.5 
 
 988.0 
 
 989.5 
 
 991.0 
 
 992.5 
 
 994.0 
 
 995.5 
 
 997.0 
 
 998.5 
 
 15 
 
 100. 
 
 1000.0 
 
 
 
 
 
 
 
 
 
 
 
 Moving the decimal point TWO places in AT requires moving it THREE places in body 
 of table (see also auxiliary table on p. 20). 
 
24 MATHEMATICAL TABLES 
 
 RECIPROCALS OF NUMBERS ' 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 ii 
 
 1.00 
 
 
 .9990 
 
 .9980 
 
 .9970 
 
 .9960 
 
 .9950 
 
 .9940 
 
 .9930 
 
 .9921 
 
 .9911 
 
 -10 
 
 1 
 
 .9901 
 
 .9891 
 
 .9881 
 
 .9872 
 
 .9862 
 
 .9852 
 
 .9843 
 
 .9833 
 
 .9823 
 
 .9814 
 
 
 2 
 
 .9804 
 
 .9794 
 
 .9785 
 
 .9775 
 
 .9766 
 
 .9756 
 
 .9747 
 
 .9737 - 
 
 .9728 
 
 .9718 
 
 
 3 
 
 .9709 
 
 .9699 
 
 .9690 
 
 .9681 
 
 .9671 
 
 .9662 
 
 .9653 
 
 .9643 
 
 .9634 
 
 .9625 
 
 -9 
 
 4 
 
 .9615 
 
 .9606 
 
 .9597 
 
 .9588 
 
 .9579 
 
 .9569 
 
 .9560 
 
 .9551 
 
 .9542 
 
 .9533 
 
 
 1.05 
 
 .9524 
 
 .9515 
 
 .9506 
 
 .9497 
 
 .9488 
 
 .9479 
 
 .9470 
 
 .9461 
 
 .9452 
 
 .9443 
 
 
 6 
 
 .9434 
 
 .9425 
 
 .9416 
 
 .9407 
 
 .9398 
 
 .9390 
 
 .9381 
 
 .9372 
 
 .9363 
 
 .9355 
 
 
 7 
 
 .9346 
 
 .9337 
 
 .9328 
 
 .9320 
 
 .9311 
 
 .9302 
 
 .9294 
 
 .9285 
 
 .9276 
 
 .9268 
 
 
 8 
 
 .9259 
 
 .9251 
 
 .9242 
 
 .9234 
 
 .9225 
 
 .9217 
 
 .9208 
 
 .9200 
 
 .9191 
 
 .9183 
 
 -8 
 
 9 
 
 .9174 
 
 .9166 
 
 .9158 
 
 .9149 
 
 .9141 
 
 .9132 
 
 .9124 
 
 .9116 
 
 .9107 
 
 .9099 
 
 
 1.10 
 
 .9091 
 
 .9083 
 
 .9074 
 
 .9066 
 
 .9058 
 
 .9050 
 
 .9042 
 
 .9033 
 
 .9025 
 
 .9017 
 
 
 
 .9009 
 
 .9001 
 
 .8993 
 
 .8985 
 
 .8977 
 
 .8969 
 
 .8961 
 
 .8953 
 
 .8945 
 
 .8937 
 
 
 2 
 
 .8929 
 
 .8921 
 
 .8913 
 
 .8905 
 
 .8897 
 
 .8889 
 
 .8881 
 
 .8873 
 
 .8865 
 
 .8857 
 
 
 3 
 
 .8850 
 
 .8842 
 
 .8834 
 
 .8826 
 
 .8818 
 
 .8811 
 
 .8803 
 
 .8795 
 
 .8787 
 
 .8780 
 
 
 4 
 
 .8772 
 
 .8764 
 
 .8757 
 
 .8749 
 
 .8741 
 
 .8734 
 
 .8726 
 
 .8718 
 
 .8711 
 
 .8703 
 
 
 1.15 
 
 .8696 
 
 .8688 
 
 .8681 
 
 .8673 
 
 .8666 
 
 .8658 
 
 .8651 
 
 .8643 
 
 .8636 
 
 .8628 
 
 
 6 
 
 .8621 
 
 .8613 
 
 .8606 
 
 .8598 
 
 .8591 
 
 .8584 
 
 .8576 
 
 .8569 
 
 .8562 
 
 .8554 
 
 -7 
 
 7 
 
 .8547 
 
 .8540 
 
 .8532 
 
 .8525 
 
 .8518 
 
 .8511 
 
 .8503 
 
 .8496 
 
 .8489 
 
 .8482 
 
 
 8 
 
 .8475 
 
 .8467 
 
 .8460 
 
 .8453 
 
 .8446 
 
 .8439 
 
 .8432 
 
 .8425 
 
 .8418 
 
 .8410 
 
 
 9 
 
 .8403 
 
 .8396 
 
 .8389 
 
 .8382 
 
 .8375 
 
 .8368 
 
 .8361 
 
 .8354 
 
 .8347 
 
 .8340 
 
 
 1.20 
 
 .8333 
 
 .8326 
 
 .8319 
 
 .8313 
 
 .8306 
 
 .8299 
 
 .8292 
 
 .8285 
 
 .8278 
 
 .8271 
 
 
 1 
 
 .8264 
 
 .8258 
 
 .8251 
 
 .8244 
 
 .8237 
 
 .8230 
 
 .8224 
 
 .8217 
 
 .8210 
 
 .8203 
 
 
 2 
 
 .8197 
 
 .8190 
 
 .8183 
 
 .8177 
 
 .8170 
 
 .8163 
 
 .8157 
 
 .8150 
 
 .8143 
 
 .8137 
 
 
 3 
 
 .8130 
 
 .8123 
 
 .8117 
 
 .8110 
 
 .8104 
 
 .8097 
 
 .8091 
 
 .8084 
 
 .8078 
 
 .8071 
 
 -6 
 
 4 
 
 .8065 
 
 .8058 
 
 .8052 
 
 .8045 
 
 .8039 
 
 .8032 
 
 .8026 
 
 .8019 
 
 .8013 
 
 .8006 
 
 
 1.25 
 
 .8000 
 
 .7994 
 
 .7987 
 
 .7981 
 
 .7974 
 
 .7968 
 
 .7962 
 
 .7955 
 
 .7949 
 
 .7943 
 
 
 6 
 
 .7937 
 
 .7930 
 
 .7924 
 
 .7918 
 
 .7911 
 
 .7905 
 
 .7899 
 
 .7893 
 
 .7886 
 
 .7880 
 
 
 7 
 
 .7874 
 
 .7868 
 
 .7862 
 
 .7855 
 
 .7849 
 
 .7843 
 
 .7837 
 
 .7831 
 
 .7825 
 
 .7819 
 
 
 8 
 
 .7812 
 
 .7806 
 
 .7800 
 
 .7794 
 
 .7788 
 
 .7782 
 
 .7776 
 
 .7770 
 
 .7764 
 
 .7758 
 
 
 9 
 
 .7752 
 
 .7746 
 
 .7740 
 
 .7734 
 
 .7728 
 
 .7722 
 
 .7716 
 
 .7710 
 
 .7704 
 
 .7698 
 
 
 1.30 
 
 .7692 
 
 .7686 
 
 .7680 
 
 .7675 
 
 .7669 
 
 .7663 
 
 .7657 
 
 .7651 
 
 .7645 
 
 .7639 
 
 
 1 
 
 .7634 
 
 .7628 
 
 .7622 
 
 .7616 
 
 .7610 
 
 .7605 
 
 .7599 
 
 .7593 
 
 .7587 
 
 .7582 
 
 
 2 
 
 .7576 
 
 .7570 
 
 .7564 
 
 .7559 
 
 .7553 
 
 .7547 
 
 J54J 
 
 .7536 
 
 .7530 
 
 .7524 
 
 
 3 
 
 .7519 
 
 .7513 
 
 .7508 
 
 .7502 
 
 .7496 
 
 .7491 
 
 .7485 
 
 .7479 
 
 .7474 
 
 .7468 
 
 
 4 
 
 .7463 
 
 .7457 
 
 .7452 
 
 .7446 
 
 .7440 
 
 .7435 
 
 .7429 
 
 .7424 
 
 .7418 
 
 .7413 
 
 
 135 
 
 .7407 
 
 .7402 
 
 .7396 
 
 .7391 
 
 .7386 
 
 .7380 
 
 .7375 
 
 .7369 
 
 .7364 
 
 .7358 
 
 -5 
 
 6 
 
 .7353 
 
 .7348 
 
 .7342 
 
 .7337 
 
 .7331 
 
 .7326 
 
 .7321 
 
 .7315 
 
 .7310 
 
 .7305 
 
 
 7 
 
 .7299 
 
 .7294 
 
 .7289 
 
 .7283 
 
 .7278 
 
 .7273 
 
 .7267 
 
 .7262 
 
 .7257 
 
 .7252 
 
 
 8 
 
 .7246 
 
 .7241 
 
 .7236 
 
 .7231 
 
 .7225 
 
 .7220 
 
 .7215 
 
 .7210 
 
 .7205 
 
 .7199 
 
 
 9 
 
 .7194 
 
 .7189 
 
 .7184 
 
 .7179 
 
 .7174 
 
 .7168 
 
 .7163 
 
 .7158 
 
 .7153 
 
 .7148 
 
 
 1.40 
 
 .7143 
 
 .7138 
 
 .7133 
 
 .7128 
 
 .7123 
 
 .7117 
 
 .7112 
 
 .7107 
 
 .7102 
 
 .7097 
 
 
 1 
 
 .7092 
 
 .7087 
 
 .7082 
 
 .7077 
 
 .7072 
 
 .7067 
 
 .7062 
 
 .7057 
 
 .7052 
 
 .7047 
 
 
 2 
 
 .7042 
 
 .7037 
 
 .7032 
 
 .7027 
 
 .7022 
 
 .7018 
 
 .7013 
 
 .7008 
 
 .7003 
 
 .6998 
 
 
 3 
 
 .6993 
 
 .6988 
 
 .6983 
 
 .6978 
 
 .6974 
 
 .6969 
 
 .6964 
 
 .6959 
 
 .6954 
 
 .6949 
 
 
 4 
 
 .6944 
 
 .6940 
 
 .6935 
 
 .6930 
 
 .6925 
 
 .6920 
 
 .6916 
 
 .6911 
 
 .6906 
 
 .6901 
 
 
 1.45 
 
 .6897 
 
 .6892 
 
 .6887 
 
 .6882 
 
 .6878 
 
 .6873 
 
 .6868 
 
 .6863 
 
 .6859 
 
 .6854 
 
 
 6 
 
 .6849 
 
 .6845 
 
 .6840 
 
 .6835 
 
 .6831 
 
 .6826 
 
 .6821 
 
 .6817 
 
 .6812 
 
 .6807 
 
 
 7 
 
 .6803 
 
 .6798 
 
 .6793 
 
 .6789 
 
 .6784 
 
 .6780 
 
 .6775 
 
 .6770 
 
 .6766 
 
 .6761 
 
 
 8 
 
 .6757 
 
 .6752 
 
 .6748 
 
 .6743 
 
 .6739 
 
 .6734 
 
 .6729 
 
 .6725 
 
 .6720 
 
 .6716 
 
 
 9 
 
 .6711 
 
 .6707 
 
 .6702 
 
 .6698 
 
 .6693 
 
 .6689 
 
 .6684 
 
 .6680 
 
 .6676 
 
 .6671 
 
 
 1/ir = 0.318310 1/e = 0.367879 
 
 Moving the decimal point in either direction in N requires moving it in the OPPO- 
 SITE direction in body of table (see p. 26). 
 
MATHEMATICAL TABLES 
 RECIPROCALS (continued) 
 
 25 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 |1 
 
 1.50 
 
 .6667 
 
 .6662 
 
 .6658 
 
 .6653 
 
 .6649 
 
 .6645 
 
 .6640 
 
 .6636 
 
 .6631 
 
 .6627 
 
 -4 
 
 1 
 
 .6623 
 
 .6618 
 
 .6614 
 
 .6609 
 
 .6605 
 
 .6601 
 
 .6596 
 
 .6592 
 
 .6588 
 
 .6583 
 
 
 2 
 
 .6579 
 
 .6575 
 
 .6570 
 
 .6566 
 
 .6562 
 
 .6557 
 
 .6553 
 
 .6549 
 
 .6545 
 
 .6540 
 
 
 3 
 
 .6536 
 
 .6532 
 
 .6527 
 
 .6523 
 
 .6519 
 
 .6515 
 
 .6510 
 
 .6506 
 
 .6502 
 
 .6498 
 
 
 4 
 
 .6494 
 
 .6489 
 
 .6485 
 
 .6481 
 
 .6477 
 
 .6472 
 
 .6468 
 
 .6464 
 
 .6460 
 
 .6456 
 
 
 1.55 
 
 .6452 
 
 .6447 
 
 .6443 
 
 .6439 
 
 .6435 
 
 .6431 
 
 .6427 
 
 .6423 
 
 .6418 
 
 .6414 
 
 
 6 
 
 .6410 
 
 .6406 
 
 .6402 
 
 .6398 
 
 .6394 
 
 .6390 
 
 .6386 
 
 .6382 
 
 .6378 
 
 .6373 
 
 
 7 
 
 .6369 
 
 .6365 
 
 .6361 
 
 .6357 
 
 .6353 
 
 .6349 
 
 .6345 
 
 .6341 
 
 .6337 
 
 .6333 
 
 
 8 
 
 .6329 
 
 .6325 
 
 .6321 
 
 .6317 
 
 .6313 
 
 .6309 
 
 .6305 
 
 .6301 
 
 .6297 
 
 .6293 
 
 
 9 
 
 .6289 
 
 .6285 
 
 .6281 
 
 .6277 
 
 .6274 
 
 .6270 
 
 .6266 
 
 .6262 
 
 .6258 
 
 .6254 
 
 
 1.60 
 
 .6250 
 
 .6246 
 
 .6242 
 
 .6238 
 
 .6234 
 
 .6231 
 
 .6227 
 
 .6223 
 
 .6219 
 
 .6215 
 
 
 1 
 
 .6211 
 
 .6207 
 
 .6203 
 
 .6200 
 
 .6196 
 
 .6192 
 
 .6188 
 
 .6184 
 
 .6180 
 
 .6177 
 
 
 2 
 
 .6173 
 
 .6169 
 
 .6165 
 
 .6161 
 
 .6158 
 
 .6154 
 
 .6150. 
 
 .6146 
 
 .6143 
 
 .6139 
 
 
 3 
 
 .6135 
 
 .6131 
 
 .6127 
 
 .6124 
 
 .6120 
 
 .6116 
 
 .6112 
 
 .6109 
 
 .6105 
 
 .6101 
 
 
 4 
 
 .6098 
 
 .6094 
 
 .6090 
 
 .6086 
 
 .6083 
 
 .6079 
 
 .6075 
 
 .6072 
 
 .6068 
 
 .6064 
 
 
 1.65 
 
 .6061 
 
 .6057 
 
 .6053 
 
 .6050 
 
 .6046 
 
 .6042 
 
 .6039 
 
 .6035 
 
 .6031 
 
 .6028 
 
 
 6 
 
 .6024 
 
 .6020 
 
 .6017 
 
 .6013 
 
 .6010 
 
 .6006 
 
 .6002 
 
 .5999 
 
 .5995 
 
 .5992 
 
 
 7 
 
 .5988 
 
 .5984 
 
 .5981 
 
 .5977 
 
 .5974 
 
 .5970 
 
 .5967 
 
 .5963 
 
 .5959 
 
 .5956 
 
 
 8 
 
 .5952 
 
 .5949 
 
 .5945 
 
 .5942 
 
 .5938 
 
 .5935 
 
 .5931 
 
 .5928 
 
 .5924 
 
 .5921 
 
 
 9 
 
 .5917 
 
 .5914 
 
 .5910 
 
 .5907 
 
 .5903 
 
 .5900 
 
 .5896 
 
 .5893 
 
 .5889 
 
 .5886 
 
 
 1.70 
 
 .5882 
 
 .5879 
 
 .5875 
 
 .5872 
 
 .5869 
 
 .5865 
 
 .5862 
 
 .5858 
 
 .5855 
 
 .5851 
 
 -3 
 
 1 
 
 .5848 
 
 .5845 
 
 .5841 
 
 .5838 
 
 .5834 
 
 .5831 
 
 .5828 
 
 .5824 
 
 .5821 
 
 .5817 
 
 
 2 
 
 .5814 
 
 .5811 
 
 .5807 
 
 .5804 
 
 .5800 
 
 .5797 
 
 .5794 
 
 .5790 
 
 .5787 
 
 .5784 
 
 
 3 
 
 .5780 
 
 .5777 
 
 .5774 
 
 .5770 
 
 .5767 
 
 .5764 
 
 .5760 
 
 .5757 
 
 .5754 
 
 .5750 
 
 
 4 
 
 .5747 
 
 .5744 
 
 .5741 
 
 .5737 
 
 .5734 
 
 .5731 
 
 .5727 
 
 .5724 
 
 .5721 
 
 .5718 
 
 
 1.75 
 
 .5714 
 
 .5711 
 
 .5708 
 
 .5705 
 
 .5701 
 
 .5698 
 
 .5695 
 
 .5692 
 
 .5688 
 
 .5685 
 
 
 6 
 
 .5682 
 
 .5679 
 
 .5675 
 
 .5672 
 
 .5669 
 
 .5666 
 
 .5663 
 
 .5659 
 
 .5656 
 
 .5653 
 
 
 7 
 
 .5650 
 
 .5647 
 
 .5643 
 
 .5640 
 
 .5637 
 
 .5634 
 
 .5631 
 
 .5627 
 
 .5624 
 
 .5621 
 
 
 8 
 
 .5618 
 
 .5615 
 
 .5612 
 
 .5609 
 
 .5605 
 
 .5602 
 
 .5599 
 
 .5596 
 
 .5593 
 
 .5590 
 
 
 9 
 
 .5587 
 
 .5583 
 
 .5580 
 
 .5577 
 
 .5574 
 
 .5571 
 
 .5568 
 
 .5565 
 
 .5562 
 
 .5559 
 
 
 1.80 
 
 .5556 
 
 .5552 
 
 .5549 
 
 .5546 
 
 .5543 
 
 .5540 
 
 .5537 
 
 .5534 
 
 .5531 
 
 .5528 
 
 
 1 
 
 .5525 
 
 .5522 
 
 .5519 
 
 .5516 
 
 .5513 
 
 .5510 
 
 .5507 
 
 .5504 
 
 .5501 
 
 .5498 
 
 
 2 
 
 .5495 
 
 .5491 
 
 .5488 
 
 .5485 
 
 .5482 
 
 .M79 
 
 .5476 
 
 .5473 
 
 .5470 
 
 .5467 
 
 
 3 
 
 .5464 
 
 .5461 
 
 .5459 
 
 .5456 
 
 .5453 
 
 .5450 
 
 .5447 
 
 .5444 
 
 .5441 
 
 .5438 
 
 
 4 
 
 .5435 
 
 .5432 
 
 .5429 
 
 .5426 
 
 .5423 
 
 .5420 
 
 .5417 
 
 .5414 
 
 .5411 
 
 .5408 
 
 
 1.85 
 
 .5405 
 
 .5402 
 
 .5400 
 
 .5397 
 
 .5394 
 
 .5391 
 
 .5388 
 
 .5385 
 
 .5382 
 
 .5379 
 
 
 6 
 
 .5376 
 
 .5373 
 
 .5371 
 
 .5368 
 
 .5365 
 
 .5362 
 
 .5359 
 
 .5356 
 
 .5353 
 
 .5350 
 
 
 7 
 
 .5348 
 
 .5345 
 
 .5342 
 
 .5339 
 
 .5336 
 
 .5333 
 
 .5330 
 
 .5328 
 
 .5325 
 
 .5322 
 
 
 8 
 
 .5319 
 
 .5316 
 
 .5313 
 
 .5311 
 
 .5308 
 
 .5305 
 
 .5302 
 
 .5299 
 
 .5297 
 
 .5294 
 
 
 9 
 
 .5291 
 
 .5288 
 
 .5285 
 
 .5283 
 
 .5280 
 
 .5277 
 
 .5274 
 
 .5271 
 
 .5269 
 
 .5266 
 
 
 1.90 
 
 .5263 
 
 .5260 
 
 .5258 
 
 .5255 
 
 .5252 
 
 .5249 
 
 .5247 
 
 .5244 
 
 .5241 
 
 .5238 
 
 
 1 
 
 .5236 
 
 .5233 
 
 .5230 
 
 .5227 
 
 .5225 
 
 .5222 
 
 .5219 
 
 .5216 
 
 .5214 
 
 .5211 
 
 
 2 
 
 .5208 
 
 .5206 
 
 .5203 
 
 .5200 
 
 .5198 
 
 .5195 
 
 .5192 
 
 .5189 
 
 .5187 
 
 .5184 
 
 
 3 
 
 .5181 
 
 .5179 
 
 .5176 
 
 .5173 
 
 .5171 
 
 .5168 
 
 .5165 
 
 .5163 
 
 .5160 
 
 .5157 
 
 
 4 
 
 .5155 
 
 .5152 
 
 .5149 
 
 .5147 
 
 .5144 
 
 .5141 
 
 .5139 
 
 .5136 
 
 .5133 
 
 .5131 
 
 
 1.95 
 
 .5128 
 
 .5126 
 
 .5123 
 
 .5120 
 
 5118 
 
 .5115 
 
 .5112 
 
 .5110 
 
 .5107 
 
 .5105 
 
 
 6 
 
 .5102 
 
 .5099 
 
 .5097 
 
 .5094 
 
 .5092 
 
 .5089 
 
 .5086 
 
 .5084 
 
 .5081 
 
 .5079 
 
 
 7 
 
 .5076 
 
 .5074 
 
 .5071 
 
 .5068 
 
 .5066 
 
 .5063 
 
 .5061 
 
 .5058 
 
 .5056 
 
 .5053 
 
 -2 
 
 8 
 
 .5051 
 
 .5048 
 
 .5045 
 
 .5043 
 
 .5040 
 
 .5038 
 
 .5035 
 
 .5033 
 
 .5030 
 
 .5028 
 
 
 9 
 
 .5025 
 
 .5023 
 
 .5020 
 
 .5018 
 
 .5015 
 
 .5013 
 
 .5010 
 
 .5008 
 
 .5005 
 
 .5003 
 
 
 Moving the decimal point in either direction in N requires moving it in the OPPO- 
 SITE direction in body of table (see p. 26). 
 
26 MATHEMATICAL TABLES 
 
 RECIPROCALS (continued) 
 
 N 
 
 o 
 
 j 
 
 2 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 . 
 
 
 
 
 
 
 
 2.0 
 
 .5000 
 
 .4975 
 
 .4950 
 
 .4926 
 
 .4902 
 
 .4878 
 
 .4854 
 
 .4831 
 
 .4808 
 
 .4785 
 
 -24 
 
 I 
 
 .4762 
 
 .4739 
 
 .4717 
 
 .4695 
 
 .4673 
 
 .4651 
 
 .4630 
 
 .4608 
 
 .4587 
 
 .4566 
 
 -21 
 
 2 
 
 .4545 
 
 .4525 
 
 .4505 
 
 .4484 
 
 .4464 
 
 .4444 
 
 .4425 
 
 .4405 
 
 .4386 
 
 .4367 
 
 -20 
 
 3 
 
 .4348 
 
 .4329 
 
 .4310 
 
 .4292 
 
 .4274 
 
 .4255 
 
 .4237 
 
 .4219 
 
 .4202 
 
 .4184 
 
 - 18 
 
 4 
 
 .4167 
 
 .4149 
 
 .4132 
 
 .4115 
 
 .4098 
 
 .4082 
 
 .4065 
 
 .4049 
 
 .4032 
 
 .4016 
 
 - 17 
 
 2.5 
 
 .4000 
 
 .3984 
 
 .3968 
 
 .3953 
 
 3937 
 
 .3922 
 
 .3906 
 
 3891 
 
 3876 
 
 386 1 
 
 - 15 
 
 6 
 
 .3846 
 
 .3831 
 
 .3817 
 
 3802 
 
 3788 
 
 3774 
 
 3759 
 
 3745 
 
 3731 
 
 3717 
 
 - 14 
 
 7 
 
 .3704 
 
 .3690 
 
 3676 
 
 .3663 
 
 3650 
 
 .3636 
 
 3623 
 
 3610 
 
 3597 
 
 3584 
 
 - 13 
 
 8 
 
 .3571 
 
 .3559 
 
 .3546 
 
 3534 
 
 .3521 
 
 3509 
 
 3497 
 
 3484 
 
 .3472 
 
 3460 
 
 - 12 
 
 9 
 
 .3448 
 
 .3436 
 
 3425 
 
 3413 
 
 .3401 
 
 3390 
 
 3378 
 
 3367 
 
 .3356 
 
 .3344 
 
 -12 
 
 3.0 
 
 .3333 
 
 .3322 
 
 .3311 
 
 .3300 
 
 3289 
 
 .3279 
 
 3268 
 
 .3257 
 
 3247 
 
 3236 
 
 - 11 
 
 1 
 
 .3226 
 
 .3215 
 
 .3205 
 
 .3195 
 
 .3185 
 
 3175 
 
 .3165 
 
 .3155 
 
 .3145 
 
 3135 
 
 - 10 
 
 2 
 
 .3125 
 
 .3115 
 
 3106 
 
 .3096 
 
 3086 
 
 .3077 
 
 3067 
 
 .3058 
 
 3049 
 
 3040 
 
 - 10 
 
 3 
 
 .3030 
 
 .3021 
 
 .3012 
 
 3003 
 
 .2994 
 
 .2985 
 
 .2976 
 
 .2967 
 
 .2959 
 
 .2950 
 
 -9 
 
 4 
 
 .2941 
 
 .2933 
 
 .2924 
 
 .2915 
 
 .2907 
 
 .2899 
 
 .2890 
 
 .2882 
 
 .2874 
 
 .2865 
 
 -8 
 
 3.5 
 
 .2857 
 
 .2849 
 
 .2841 
 
 .2833 
 
 .2825 
 
 .2817 
 
 .2809 
 
 .2801 
 
 .2793 
 
 .2786 
 
 -8 
 
 6 
 
 .2778 
 
 .2770 
 
 .2762 
 
 .2755 
 
 .2747 
 
 .2740 
 
 .2732 
 
 .2725 
 
 .2717 
 
 .2710 
 
 -8 
 
 7 
 
 .2703 
 
 J2695 
 
 .2688 
 
 .2681 
 
 .2674 
 
 .2667 
 
 .2660 
 
 .2653 
 
 .2646 
 
 .2639 
 
 -7 
 
 8 
 
 .2632 
 
 .2625 
 
 .2618 
 
 .2611 
 
 .2604 
 
 .2597 
 
 .2591 
 
 .2584 
 
 .2577 
 
 .2571 
 
 -7 
 
 9 
 
 .2564 
 
 .2558 
 
 .2551 
 
 .2545 
 
 .2538 
 
 .2532 
 
 .2525 
 
 .2519 
 
 .2513 
 
 .2506 
 
 -6 
 
 4.0 
 
 .2500 
 
 .2494 
 
 .2488 
 
 J2481 
 
 .2475 
 
 .2469 
 
 .2463 
 
 .2457 
 
 .2451 
 
 .2445 
 
 -6 
 
 1 
 
 .2439 
 
 .2433 
 
 .2427 
 
 .2421 
 
 .2415 
 
 .2410 
 
 .2404 
 
 .2398 
 
 .2392 
 
 .2387 
 
 -6 
 
 2 
 
 .2381 
 
 .2375 
 
 .2370 
 
 2364 
 
 .2358 
 
 .2353 
 
 .2347 
 
 .2342 
 
 .2336 
 
 .2331 
 
 -6 
 
 3 
 
 .2326 
 
 .2320 
 
 .2315 
 
 .2309 
 
 .2304 
 
 .2299 
 
 .2294 
 
 .2288 
 
 .2283 
 
 .2278 
 
 -5 
 
 4 
 
 .2273 
 
 .2268 
 
 .2262 
 
 .2257 
 
 .2252 
 
 .2247 
 
 .2242 
 
 .2237 
 
 .2232 
 
 .2227 
 
 -5 
 
 4.5 
 
 .2222 
 
 .2217 
 
 .2212 
 
 .2208 
 
 .2203 
 
 .2198 
 
 .2193 
 
 .2188 
 
 .2183 
 
 .2179 
 
 -5 
 
 6 
 
 .2174 
 
 .2169 
 
 .2165 
 
 .2160 
 
 .2155 
 
 .2151 
 
 .2146 
 
 .2141 
 
 .2137 
 
 .2132 
 
 -5 
 
 7 
 
 .2128 
 
 .2123 
 
 .2119 
 
 2114 
 
 .2110 
 
 .2105 
 
 .2101 
 
 .2096 
 
 .2092 
 
 .2088 
 
 -4 
 
 8 
 
 .2083 
 
 .2079 
 
 .2075 
 
 .2070 
 
 .2066 
 
 .2062 
 
 .2058 
 
 .2053 
 
 .2049 
 
 .2045 
 
 -4 
 
 9 
 
 .2041 
 
 .2037 
 
 .2033 
 
 .2028 
 
 .2024 
 
 .2020 
 
 .2016 
 
 .2012 
 
 .2008 
 
 .2004 
 
 -4 
 
 I/T = 0.318310 1/e = 0.367879 
 
 Explanation of Table of Reciprocals (pp. 24-27). 
 
 This table gives the values of 1/N for values of N from 1 to 10, correct to four figures. 
 (Interpolated values may be in error by 1 in the fourth figure.) 
 
 To find the reciprocal of a number N outside the range from 1 to 10, note 
 that moving the decimal point any number of places in either direction in column N 
 is equivalent to moving it the same number of places in the opposite direction in the 
 body of the table. For example: 
 
 1 0.3108; -^r =0.0003108; nn ^ n ^ - 310.8 
 
 3.217 
 
 3217. 
 
 0.003217 
 
MATHEMATICAL TABLES 
 RECIPROCALS (continued) 
 
 27 
 
 I* 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 d 
 & 
 
 5.0 
 
 .2000 
 
 .1996 
 
 .1992 
 
 .1988 
 
 .1984 
 
 .1980 
 
 .1976 
 
 .1972 
 
 .1969 
 
 .1965 
 
 -4 
 
 .1 
 
 .1961 
 
 .1957 
 
 .1953 
 
 .1949 
 
 .1946 
 
 .1942 
 
 .1938 
 
 .1934 
 
 .1931 
 
 .1927 
 
 
 .2 
 
 .1923 
 
 .1919 
 
 .1916 
 
 .1912 
 
 .1908 
 
 .1905 
 
 .1901 
 
 .1898 
 
 .1894 
 
 .1890 
 
 
 .3 
 
 .1887 
 
 .1883 
 
 .1880 
 
 .1876 
 
 .1873 
 
 .1869 
 
 .1866 
 
 .1862 
 
 .1859 
 
 .1855 
 
 
 .4 
 
 .1852 
 
 .1848 
 
 .1845 
 
 .1842 
 
 .1838 
 
 .1835 
 
 .1832 
 
 .1828 
 
 .1825 
 
 .1821 
 
 -3 
 
 5.5 
 
 .1818 
 
 .1815 
 
 .1812 
 
 .1808 
 
 .1805 
 
 .1802 
 
 .1799 
 
 .1795 
 
 .1792 
 
 .1789 
 
 
 .6 
 
 .1786 
 
 .1783 
 
 .1779 
 
 .1776 
 
 .1773 
 
 .1770 
 
 .1767 
 
 .1764 
 
 .1761 
 
 .1757 
 
 
 .7 
 
 .1754 
 
 .1751 
 
 .1748 
 
 .1745 
 
 .1742 
 
 .1739 
 
 .1736 
 
 .1733 
 
 .1730 
 
 .1727 
 
 
 .8 
 
 .1724 
 
 .1721 
 
 .1718 
 
 .1715 
 
 .1712 
 
 .1709 
 
 .1706 
 
 .1704 
 
 .1701 
 
 .1698 
 
 
 .9 
 
 .1695 
 
 .1692 
 
 .1689 
 
 .1686 
 
 .1684 
 
 .1681 
 
 .1678 
 
 .1675 
 
 .1672 
 
 .1669 
 
 
 6.0 .1667 
 
 .1664 
 
 .1661 
 
 .1658 
 
 .1656 
 
 .1653 
 
 .1650 
 
 .1647 
 
 .1645 
 
 .1642 
 
 
 .1 
 
 .1639 
 
 .1637 
 
 .1634 
 
 .1631 
 
 .1629 
 
 .1626 
 
 .1623 
 
 .1621 
 
 .1618 
 
 .1616 
 
 
 .2 
 
 .1613 
 
 .1610 
 
 .1608 
 
 .1605 
 
 .1603 
 
 .1600 
 
 .1597 
 
 .1595 
 
 .1592 
 
 .1590 
 
 
 .3 
 
 .1587 
 
 .1585 
 
 .1582 
 
 .1580 
 
 .1577 
 
 .1575 
 
 .1572 
 
 .1570 
 
 .1567 
 
 .1565 
 
 -2 
 
 .4 
 
 .1563 
 
 .1560 
 
 .1558 
 
 .1555 
 
 .1553 
 
 .1550 
 
 .1548 
 
 .1546 
 
 .1543 
 
 .1541 
 
 
 6.5 
 
 .1538 
 
 .1536 
 
 .1534 
 
 .1531 
 
 .1529 
 
 .1527 
 
 .1524 
 
 .1522 
 
 .1520 
 
 .1517 
 
 
 .6 
 
 .1515 
 
 .1513 
 
 .1511 
 
 .1508 
 
 .1506 
 
 .1504 
 
 .1502 
 
 .1499 
 
 .1497 
 
 .1495 
 
 
 .7 
 
 .1493 
 
 .1490 
 
 .1488 
 
 .1486 
 
 .1484 
 
 .1481 
 
 .1479 
 
 .1477 
 
 .1475 
 
 .1473 
 
 
 .8 
 
 .1471 
 
 .1468 
 
 .1466 
 
 .1464 
 
 .1462 
 
 .1460 
 
 .1458 
 
 .1456 
 
 .1453 
 
 .1451 
 
 
 .9 
 
 .1449 
 
 .1447 
 
 .1445 
 
 .1443 
 
 .1441 
 
 .1439 
 
 .1437 
 
 .1435 
 
 .1433 
 
 .1431 
 
 
 7.0 
 
 .1429 
 
 .1427 
 
 .1425 
 
 .1422 
 
 .1420 
 
 .1418 
 
 .1416 
 
 .1414 
 
 .1412 
 
 .1410 
 
 
 
 .1408 
 
 .1406 
 
 .1404 
 
 .1403 
 
 .1401 
 
 .1399 
 
 .1397 
 
 .1395 
 
 .1393 
 
 .1391 
 
 
 '.2 
 
 .1389 
 
 .1387 
 
 .1385 
 
 .1383 
 
 .1381 
 
 .1379 
 
 .1377 
 
 .1376 
 
 .1374 
 
 .1372 
 
 
 .3 
 
 .1370 
 
 .1368 
 
 .1366 
 
 .1364 
 
 .1362 
 
 .1361 
 
 .1359 
 
 .1357 
 
 .1355 
 
 .1353 
 
 
 .4 
 
 .1351 
 
 .1350 
 
 .1348 
 
 .1346 
 
 .1344 
 
 .1342 
 
 .1340 
 
 .1339 
 
 .1337 
 
 .1335 
 
 
 7.5 
 
 .1333 
 
 .1332 
 
 .1330 
 
 .1328 
 
 .1326 
 
 .1325 
 
 .1323 
 
 .1321 
 
 .1319 
 
 .1318 
 
 
 .6 
 
 .1316 
 
 .1314 
 
 .1312 
 
 .1311 
 
 .1309 
 
 .1307 
 
 .1305 
 
 .1304 
 
 .1302 
 
 .1300 
 
 
 .7 
 
 .1299 
 
 .1297 
 
 .1295 
 
 .1294 
 
 .1292 
 
 .1290 
 
 .1289 
 
 .1287 
 
 .1285 
 
 .1284 
 
 
 .8 
 
 .1282 
 
 .1280 
 
 .1279 
 
 .1277 
 
 .1276 
 
 .1274 
 
 .1272 
 
 .1271 
 
 .1269 
 
 .1267 
 
 
 .9 
 
 .1266 
 
 .1264 
 
 .1263 
 
 .1261 
 
 .1259 
 
 .1258 
 
 .1256 
 
 .1255 
 
 .1253 
 
 .1252 
 
 
 8.0 
 
 .1250 
 
 .1248 
 
 .1247 
 
 .1245 
 
 .1244 
 
 .1242 
 
 .1241 
 
 .1239 
 
 .1238 
 
 .1236 
 
 
 
 .1235 
 
 .1233 
 
 .1232 
 
 .1230 
 
 .1229 
 
 .1227 
 
 .1225 
 
 .1224 
 
 .1222 
 
 .1221 
 
 
 '.2 
 
 .1220 
 
 .1218 
 
 .1217 
 
 .1215 
 
 .1214 
 
 .1212 
 
 .1211 
 
 .1209 
 
 .1208 
 
 .1206 
 
 
 .3 
 
 .1205 
 
 .1203 
 
 .1202 
 
 .1200 
 
 .1199 
 
 .1198 
 
 .1196 
 
 .1195 
 
 .1193 
 
 .1192 
 
 
 .4 
 
 .1190 
 
 .1189 
 
 .1188 
 
 .1186 
 
 .1185 
 
 .1183 
 
 .1182 
 
 .1181 
 
 .1179 
 
 .1178 
 
 -1 
 
 8.5 
 
 .1176 
 
 .1175 
 
 .1174 
 
 .1172 
 
 .1171 
 
 .1170 
 
 .1168 
 
 .1167 
 
 .1166 
 
 .1164 
 
 
 .6 
 
 .1163 
 
 .1161 
 
 .1160 
 
 .1159 
 
 .1157 
 
 .1156 
 
 .1155 
 
 .1153 
 
 .1152 
 
 .1151 
 
 
 .7 
 
 .1149 
 
 .1148 
 
 .1147 
 
 .1145 
 
 .1144 
 
 .1143 
 
 .1142 
 
 .1140 
 
 .1139 
 
 .1138 
 
 
 .8 
 
 .1136 
 
 .1135 
 
 .1134 
 
 .1133 
 
 .1131 
 
 .1130 
 
 .1129 
 
 .1127 
 
 .1126 
 
 .1125 
 
 
 .9 
 
 .1124 
 
 .1122 
 
 .1121 
 
 .1120 
 
 .1119 
 
 .1117 
 
 .1116 
 
 .1115 
 
 .1114 
 
 .1112 
 
 
 9.0 
 
 .1111 
 
 .1110 
 
 .1109 
 
 .1107 
 
 .1106 
 
 .1105 
 
 .1104 
 
 .1103 
 
 .1101 
 
 .1100 
 
 
 .1 
 
 .1099 
 
 .1098 
 
 .1096 
 
 .1095 
 
 .1094 
 
 .1093 
 
 .1092 
 
 .1091 
 
 .1089 
 
 .1088 
 
 
 .2 
 
 .1087 
 
 .1086 
 
 .1085 
 
 .1083 
 
 .1082 
 
 .1081 
 
 .1080 
 
 .1079 
 
 .1078 
 
 .1076 
 
 
 .3 
 
 .1075 
 
 .1074 
 
 .1073 
 
 .1072 
 
 .1071 
 
 .1070 
 
 .1068 
 
 .1067 
 
 .1066 
 
 .1065 
 
 
 .4 
 
 .1064 
 
 .1063 
 
 .1062 
 
 .1060 
 
 .1059 
 
 .1058 
 
 .1057 
 
 .1056 
 
 .1055 
 
 .1054 
 
 
 9.5 
 
 .1053 
 
 .1052 
 
 .1050 
 
 .1049 
 
 .1048 
 
 .1047 
 
 .1046 
 
 .1045 
 
 .1044 
 
 .1043 
 
 
 .6 
 
 .1042 
 
 .1041 
 
 .1040 
 
 .1038 
 
 .1037 
 
 .1036 
 
 .1035 
 
 .1034 
 
 .1033 
 
 .1032 
 
 
 .7 
 
 .1031 
 
 .1030 
 
 .1029 
 
 J028 
 
 .1027 
 
 .1026 
 
 .1025 
 
 .1024 
 
 .1022 
 
 .1021 
 
 
 .8 
 
 .1020 
 
 .1019 
 
 .1018 
 
 .1017 
 
 .1016 
 
 .1015 
 
 .1014 
 
 .1013 
 
 .1012 
 
 .1011 
 
 
 .9 
 
 .1010 
 
 .1009 
 
 .1008 
 
 .1007 
 
 .1006 
 
 .1005 
 
 .1004 
 
 .1003 
 
 .1002 
 
 .1001 
 
 
 Moving the decimal point in either direction in N requires moving it in the OPPOSITE 
 direction in body of table (see p. 26). 
 
28 
 
 MATHEMATICAL TABLES 
 
 CIRCUMFERENCES OF CIRCLES BY HUNDREDTHS 
 
 (For circumferences by eighths, see p. 32) 
 
 D 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 >1 
 
 <T 
 
 1.0 
 
 3.142 
 
 3.173 
 
 3.204 
 
 3.236 
 
 3.267 
 
 3.299 
 
 3.330 
 
 3.362 
 
 3.393 
 
 3.424 
 
 31 
 
 .1 
 
 3.456 
 
 3.487 
 
 3.519 
 
 3.550 
 
 3.581 
 
 3.613 
 
 3.644 
 
 3.676 
 
 3.707 
 
 3.738 
 
 
 .2 
 
 3.770 
 
 3.801 
 
 3.833 
 
 3.864 
 
 3.896 
 
 3.927 
 
 3.958 
 
 3.990 
 
 4.021 
 
 4.053 
 
 
 .3 
 
 4.084 
 
 4.115 
 
 4.147 
 
 4.178 
 
 4.210 
 
 4.241 
 
 4.273 
 
 4.304 
 
 4.335 
 
 4.367 
 
 
 .4 
 
 4.398 
 
 4.430 
 
 4.461 
 
 4.492 
 
 4.524 
 
 4.555 
 
 4.587 
 
 4.618 
 
 4.650 
 
 4.681 
 
 
 1.5 
 
 4.712 
 
 4.744 
 
 4.775 
 
 4.807 
 
 4.838 
 
 4.869 
 
 4.901 
 
 4.932 
 
 4.964 
 
 4.995 
 
 
 .6 
 
 5.027 
 
 5.058 
 
 5.089 
 
 5.121 
 
 5.152 
 
 5.184 
 
 5.215 
 
 5.246 
 
 5.278 
 
 5.309 
 
 
 .7 
 
 5.341 
 
 5.372 
 
 5.404 
 
 5.435 
 
 5.466 
 
 5.498 
 
 5.529 
 
 5.561 
 
 5.592 
 
 5.623 
 
 
 .8 
 
 5.655 
 
 5.686 
 
 5.718 
 
 5.749 
 
 5.781 
 
 5.812 
 
 5.843 
 
 5.875 
 
 5.906 
 
 5.938 
 
 
 .9 
 
 5.969 
 
 6.000 
 
 6.032 
 
 6.063 
 
 6.095 
 
 6.126 
 
 6.158 
 
 6.189 
 
 6.220 
 
 6.252 
 
 
 2.0 
 
 6.283 
 
 6.315 
 
 6.346 
 
 6.377 
 
 6.409 
 
 6.440 
 
 6.472 
 
 6.503 
 
 6.535 
 
 6.566 
 
 
 .1 
 
 6.597 
 
 6.629 
 
 6.660 
 
 6.692 
 
 6.723 
 
 6.754 
 
 6.786 
 
 6.817 
 
 6.849 
 
 6.880 
 
 
 .2 
 
 6.912 
 
 6.943 
 
 6.974 
 
 7.006 
 
 7.037 
 
 7.069 
 
 7.100 
 
 7.131 
 
 7.163 
 
 7.194 
 
 
 .3 
 
 7.226 
 
 7.257 
 
 7.288 
 
 7.320 
 
 7.351 
 
 7.383 
 
 7.414 
 
 7.446 
 
 7.477 
 
 7.508 
 
 
 .4 
 
 7.540 
 
 7.571 
 
 7.603 
 
 7.634 
 
 7.665 
 
 7.697 
 
 7.728 
 
 7.760 
 
 7.791 
 
 7.823 
 
 
 2.5 
 
 7.854 
 
 7.885 
 
 7.917 
 
 7.948 
 
 7.980 
 
 8.011 
 
 8.042 
 
 8.074 
 
 8.105 
 
 8.137 
 
 
 .6 
 
 8.168 
 
 8.200 
 
 8.231 
 
 8.262 
 
 8.294 
 
 8.325 
 
 8.357 
 
 8.388 
 
 8.419 
 
 8.451 
 
 
 .7 
 
 8.482 
 
 8.514 
 
 8.545 
 
 8.577 
 
 8.608 
 
 8.639 
 
 8.671 
 
 8.702 
 
 8.734 
 
 8.765 
 
 
 .8 
 
 8.796 
 
 8.828 
 
 8.859 
 
 8.891 
 
 8.922 
 
 8.954 
 
 8.985 
 
 9.016 
 
 9.048 
 
 9.079 
 
 
 .9 
 
 9.111 
 
 9.142 
 
 9.173 
 
 9.205 
 
 9.236 
 
 9.268 
 
 9.299 
 
 9.331 
 
 9.362 
 
 9.393 
 
 
 3.0 
 
 9.425 
 
 9.456 
 
 9.488 
 
 9.519 
 
 9.550 
 
 9.582 
 
 9.613 
 
 9.645 
 
 9.676 
 
 9.708 
 
 
 
 9.739 
 
 9.770 
 
 9.802 
 
 9.833 
 
 9.865 
 
 9.8% 
 
 9.927 
 
 9.959 
 
 9.990 
 
 10.022 
 
 31 
 
 *1 
 
 
 
 
 
 
 
 
 
 
 10.02 
 
 3 
 
 .2 
 
 10.05 
 
 10.08 
 
 10.12 
 
 10.15 
 
 10.18 
 
 10.21 
 
 10.24 
 
 10.27 
 
 10.30 
 
 10.34 
 
 
 .3 
 
 10.37 
 
 10.40 
 
 10.43 
 
 10.46 
 
 10.49 
 
 10.52 
 
 10.56 
 
 1059 
 
 10.62 
 
 10.65 
 
 
 .4 
 
 10.68 
 
 10.71 
 
 10.74 
 
 10.78 
 
 10.81 
 
 10.84 
 
 10.87 
 
 10.90 
 
 10.93 
 
 10.96 
 
 
 3.5 
 
 11.00 
 
 11.03 
 
 11.06 
 
 11.09 
 
 11.12 
 
 11.15 
 
 11.18 
 
 11.22 
 
 11.25 
 
 11.28 
 
 
 .6 
 
 11.31 
 
 11.34 
 
 11.37 
 
 11.40 
 
 11.44 
 
 11.47 
 
 11.50 
 
 11.53 
 
 11.56 
 
 11.59 
 
 
 .7 
 
 11.62 
 
 11.66 
 
 11.69 
 
 11.72 
 
 11.75 
 
 11.78 
 
 11.81 
 
 11.84 
 
 11.88 
 
 11.91 
 
 
 .8 
 
 11.94 
 
 11.97 
 
 12.00 
 
 12.03 
 
 12.06 
 
 12.10 
 
 12.13 
 
 12.16 
 
 12.19 
 
 12.22 
 
 
 .9 
 
 12.25 
 
 12.28 
 
 12.32 
 
 12.35 
 
 12.38 
 
 12.41 
 
 12.44 
 
 12.47 
 
 12.50 
 
 12.53 
 
 
 4.0 
 
 12.57 
 
 12.60 
 
 12.63 
 
 12.66 
 
 12.69 
 
 12.72 
 
 12.75 
 
 12.79 
 
 12.82 
 
 12.85 
 
 
 
 12.88 
 
 12.91 
 
 12.94 
 
 12.97 
 
 13.01 
 
 13.04 
 
 13.07 
 
 13.10 
 
 13.13 
 
 13.16 
 
 
 '.2 
 
 13.19 
 
 13.23 
 
 13.26 
 
 13.29 
 
 13.32 
 
 13.35 
 
 13.38 
 
 13.41 
 
 13.45 
 
 13.48 
 
 
 .3 
 
 13.51 
 
 13.54 
 
 13.57 
 
 13.60 
 
 13.63 
 
 13.67 
 
 13.70 
 
 13.73 
 
 13.76 
 
 13.79 
 
 
 .4 
 
 13.82 
 
 13.85 
 
 13.89 
 
 13.92 
 
 13.95 
 
 13.98 
 
 14.01 
 
 14.04 
 
 14.07 
 
 14.11 
 
 
 4.5 
 
 14.14 
 
 14.17 
 
 14.20 
 
 14.23 
 
 14.26 
 
 14.29 
 
 14.33 
 
 14.36 
 
 14.39 
 
 14.42 
 
 
 .6 
 
 14.45 
 
 14.48 
 
 14.51 
 
 14.55 
 
 14.58 
 
 14.61 
 
 14.64 
 
 14.67 
 
 14.70 
 
 14.73 
 
 
 .7 
 
 14.77 
 
 14.80 
 
 14.83 
 
 14.86 
 
 14.89 
 
 14.92 
 
 14.95 
 
 14.99 
 
 15.02 
 
 15.05 
 
 
 .8 
 
 15.08 
 
 15.11 
 
 15.14 
 
 15.17 
 
 15.21 
 
 15.24 
 
 15.27 
 
 15.30 
 
 15.33 
 
 15.36 
 
 
 .9 
 
 15.39 
 
 15.43 
 
 15.46 
 
 15.49 
 
 15.52 
 
 15.55 
 
 15.58 
 
 15.61 
 
 15.65 
 
 15.68 
 
 
 Explanation of Table of Circumferences (pp. 28-29) 
 
 This table gives the product of T times any number D from 1 to 10; that is, it is a table 
 of multiples of IT. (D diameter.) 
 
 Moving the decimal point one place in column D is equivalent to moving it one 
 place in the body of the table. 
 
 Circumference = ic X diam. = 3.141593 X diam. 
 Conversely, 
 
 Diameter = - X circumf. = 0.31831 X circumf. 
 
MATHEMATICAL TABLES 
 CIRCUMFERENCES BY HUNDREDTHS (continued) 
 
 29 
 
 D 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 9 
 
 5.0 
 
 15.71 
 
 15.74 
 
 15.77 
 
 15.80 
 
 15.83 
 
 15.87 
 
 15.90 
 
 15.93 
 
 15.% 
 
 15.99 
 
 3 
 
 .1 
 
 16.02 
 
 16.05 
 
 16.08 
 
 16.12 
 
 16.15 
 
 16.18 
 
 16.21 
 
 16.24 
 
 16.27 
 
 16.30 
 
 
 .2 
 
 16.34 
 
 16.37 
 
 16.40 
 
 16.43 
 
 16.46 
 
 16.49 
 
 16.52 
 
 16.56 
 
 16.59 
 
 16.62 
 
 
 .3 
 
 16.65 
 
 16.68 
 
 16.71 
 
 16.74 
 
 16.78 
 
 16.81 
 
 16.84 
 
 16.87 
 
 16.90 
 
 16.93 
 
 
 .4 
 
 16.96 
 
 17.00 
 
 17.03 
 
 17.06 
 
 17.09 
 
 17.12 
 
 17.15 
 
 17.18 
 
 17.22 
 
 17.25 
 
 
 5.5 
 
 17.28 
 
 17.31 
 
 17.34 
 
 17.37 
 
 17.40 
 
 17.44 
 
 17.47 
 
 17.50 
 
 17.53 
 
 17.56 
 
 
 j6 
 
 17.59 
 
 17.62 
 
 17.66 
 
 17.69 
 
 17.72 
 
 17.75 
 
 17.78 
 
 17.81 
 
 17.84 
 
 17.88 
 
 
 .7 
 
 17.91 
 
 17.94 
 
 17.97 
 
 18.00 
 
 18.03 
 
 18.06 
 
 18.10 
 
 18.13 
 
 18.16 
 
 18.19 
 
 
 .8 
 
 18.22 
 
 18.25 
 
 18.28 
 
 18.32 
 
 18.35 
 
 18.38 
 
 18.41 
 
 18.44 
 
 18.47 
 
 18.50 
 
 
 .9 
 
 18.54 
 
 18.57 
 
 18.60 
 
 18.63 
 
 18.66 
 
 18.69 
 
 18.72 
 
 18.76 
 
 18.79 
 
 18.82 
 
 
 6.0 
 
 18.85 
 
 18.88 
 
 18.91 
 
 18.94 
 
 18.98 
 
 19.01 
 
 19.04 
 
 19.07 
 
 19.10 
 
 19.13 
 
 
 .1 
 
 19.16 
 
 19.20 
 
 19.23 
 
 19.26 
 
 19.29 
 
 19.32 
 
 19.35 
 
 19.38 
 
 19.42 
 
 19.45 
 
 
 2 
 
 19.48 
 
 19.51 
 
 19.54 
 
 19.57 
 
 19.60 
 
 19.63 
 
 19.67 
 
 19.70 
 
 19.73 
 
 19.76 
 
 
 3 
 
 19.79 
 
 19.82 
 
 19.85 
 
 19.89 
 
 19.92 
 
 19.95 
 
 19.98 
 
 20.01 
 
 20.04 
 
 20.07 
 
 
 .4 
 
 20.11 
 
 20.14 
 
 20.17 
 
 20.20 
 
 20.23 
 
 20.26 
 
 20.29 
 
 20.33 
 
 20.36 
 
 20.39 
 
 
 6.6 
 
 20.42 
 
 20.45 
 
 20.48 
 
 20.51 
 
 20.55 
 
 20.58 
 
 20.61 
 
 20.64 
 
 20.67 
 
 20.70 
 
 
 .6 
 
 20.73 
 
 20.77 
 
 20.80 
 
 20.83 
 
 20.86 
 
 20.89 
 
 20.92 
 
 20.95 
 
 20.99 
 
 21.02 
 
 
 .7 
 
 21.05 
 
 21.08 
 
 21.11 
 
 21.14 
 
 21.17 
 
 21.21 
 
 21.24 
 
 21.27 
 
 21.30 
 
 21.33 
 
 
 .8 
 
 21.36 
 
 21.39 
 
 21.43 
 
 21.46 
 
 21.49 
 
 21.52 
 
 21.55 
 
 21.58 
 
 21.61 
 
 21.65 
 
 
 .9 
 
 21.68 
 
 21.71 
 
 21.74 
 
 21.77 
 
 21.80 
 
 21.83 
 
 21.87 
 
 21.90 
 
 21.93 
 
 21.96 
 
 
 7.0 
 
 21.99 
 
 22.02 
 
 22.05 
 
 22.09 
 
 22.12 
 
 22.15 
 
 22.18 
 
 22.21 
 
 22.24 
 
 22.27 
 
 
 .1 
 
 22.31 
 
 22.34 
 
 22.37 
 
 22.40 
 
 22.43 
 
 22.46 
 
 22.49 
 
 22.53 
 
 22.56 
 
 22.59 
 
 
 .2 
 
 22.62 
 
 22.65 
 
 22.68 
 
 22.71 
 
 22.75 
 
 22.78 
 
 22.81 
 
 22.84 
 
 22.87 
 
 22.90 
 
 
 .3 
 
 22.93 
 
 22.97 
 
 23.00 
 
 23.03 
 
 23.06 
 
 23.09 
 
 23.12 
 
 23.15 
 
 23.18 
 
 23.22 
 
 
 .4 
 
 23.25 
 
 23.28 
 
 23.31 
 
 23.34 
 
 23.37 
 
 23.40 
 
 23.44 
 
 23.47 
 
 23.50 
 
 23.53 
 
 
 7.5 
 
 23.56 
 
 23.59 
 
 23.62 
 
 23.66 
 
 23.69 
 
 23.72 
 
 23.75 
 
 23.78 
 
 23.81 
 
 23.84 
 
 
 .6 
 
 23.88 
 
 23.91 
 
 23.94 
 
 23.97 
 
 24.00 
 
 24.03 
 
 24.06 
 
 24.10 
 
 24.13 
 
 24.16 
 
 
 .7 
 
 24.19 
 
 24.22 
 
 24.25 
 
 24.28 
 
 24.32 
 
 24.35 
 
 24.38 
 
 24.41 
 
 24.44 
 
 24.47 
 
 
 .8 
 
 24.50 
 
 24.54 
 
 24.57 
 
 24.60 
 
 24.63 
 
 24.66 
 
 24.69 
 
 24.72 
 
 24.76 
 
 24.79 
 
 
 .9 
 
 24.82 
 
 24.85 
 
 24.88 
 
 24.91 
 
 24.94 
 
 24.98 
 
 25.01 
 
 25.04 
 
 25.07 
 
 25.10 
 
 
 8.0 
 
 25.13 
 
 25.16 
 
 25.20 
 
 25.23 
 
 25.26 
 
 25.29 
 
 25.32 
 
 25.35 
 
 25.38 
 
 25.42 
 
 
 .1 
 
 25.45 
 
 25.48 
 
 25.51 
 
 25.54 
 
 25.57 
 
 25.60 
 
 25.64 
 
 25.67 
 
 25.70 
 
 25.73 
 
 
 2 
 
 25.76 
 
 25.79 
 
 25.82 
 
 25.86 
 
 25.89 
 
 25.92 
 
 25.95 
 
 25.98 
 
 26.01 
 
 26.04 
 
 
 .3 
 
 26.08 
 
 26.11 
 
 26.14 
 
 26.17 
 
 26.20 
 
 26.23 
 
 26.26 
 
 26.30 
 
 26.33 
 
 26.36 
 
 
 .4 
 
 26.39 
 
 26.42 
 
 26.45 
 
 26.48 
 
 26.52 
 
 26.55 
 
 26.58 
 
 26.61 
 
 26.64 
 
 26.67 
 
 
 8.5 
 
 26.70 
 
 26.73 
 
 26.77 
 
 26.80 
 
 26.83 
 
 26.86 
 
 26.89 
 
 26.92 
 
 26.95 
 
 26.99 
 
 
 .6 
 
 27.02 
 
 27.05 
 
 27.08 
 
 27.11 
 
 27.14 
 
 27.17 
 
 27.21 
 
 27.24 
 
 27.27 
 
 27.30 
 
 
 7 
 
 27.33 
 
 27.36 
 
 27.39 
 
 27.43 
 
 27.46 
 
 27.49 
 
 27.52 
 
 27.55 
 
 27.58 
 
 27.61 
 
 
 .8 
 
 27.65 
 
 27.68 
 
 27.71 
 
 27.74 
 
 27.77 
 
 27.80 
 
 27.83 
 
 27.87 
 
 27.90 
 
 27.93 
 
 
 .9 
 
 27.96 
 
 27.99 
 
 28.02 
 
 28.05 
 
 28.09 
 
 28.12 
 
 28.15 
 
 28.18 
 
 28.21 
 
 28.24 
 
 
 9.0 
 
 28.27 
 
 28.31 
 
 28.34 
 
 28.37 
 
 28.40 
 
 28.43 
 
 28.46 
 
 28.49 
 
 28.53 
 
 28.56 
 
 
 .1 
 
 28.59 
 
 28.62 
 
 28.65 
 
 28.68 
 
 28.71 
 
 28.75 
 
 28.78 
 
 28.81 
 
 28.84 
 
 28.87 
 
 
 .2 
 
 28.90 
 
 28.93 
 
 28.97 
 
 29.00 
 
 29.03 
 
 29.06 
 
 29.09 
 
 29.12 
 
 29.15 
 
 29.19 
 
 
 3 
 
 29.22 
 
 29.25 
 
 29.28 
 
 29.31 
 
 29.34 
 
 29.37 
 
 29.41 
 
 29.44 
 
 29.47 
 
 29.50 
 
 
 .4 
 
 29.53 
 
 29.56 
 
 29.59 
 
 29.63 
 
 29.66 
 
 29.69 
 
 29.72 
 
 29.75 
 
 29.78 
 
 29.81 
 
 
 9.5 
 
 29.85 
 
 29.88 
 
 29.91 
 
 29.94 
 
 29.97 
 
 30.00 
 
 30.03 
 
 30.07 
 
 30.10 
 
 30.13 
 
 
 .6 
 
 30.16 
 
 30.19 
 
 30.22 
 
 30.25 
 
 30.28 
 
 30.32 
 
 30.35 
 
 30.38 
 
 30.41 
 
 30.44 
 
 
 .7 
 
 30.47 
 
 30.50 
 
 30.54 
 
 30.57 
 
 30.60 
 
 30.63 
 
 30.66 
 
 30.69 
 
 30.72 
 
 30.76 
 
 
 .8 
 
 30.79 
 
 30.82 
 
 30.85 
 
 30.88 
 
 30.91 
 
 30.94 
 
 30.98 
 
 31.01 
 
 31.04 
 
 31.07 
 
 
 .9 
 
 31.10 
 
 31.13 
 
 31.16 
 
 31.20 
 
 31.23 
 
 31.26 
 
 31.29 
 
 31.32 
 
 31.35 
 
 31.38 
 
 
 10.0 
 
 31.42 
 
 
 
 
 
 
 
 
 
 
 
 Moving the decimal point ONE place in D requires moving it ONE plaoe in body of 
 table (see p. 28). 
 
30 
 
 MATHEMATICAL TABLES 
 
 AREAS OF CIRCLES BY HUNDREDTHS 
 
 (For areas by eighths, see p. 32) 
 
 D 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 ? 
 
 <3 
 
 1.0 
 
 0.785 
 
 0.801 
 
 0.817 
 
 0.833 
 
 0.849 
 
 0.866 
 
 0.882 
 
 0.899 
 
 0.916 
 
 0.933 
 
 16 
 
 .1 
 
 0.950 
 
 0.968 
 
 0.985 
 
 1.003 
 
 1.021 
 
 1.039 
 
 1.057 
 
 1.075 
 
 1.094 
 
 1.112 
 
 18 
 
 .2 
 
 1.131 
 
 1.150 
 
 1.169 
 
 1.188 
 
 1.208 
 
 1.227 
 
 1.247 
 
 1.267 
 
 1.287 
 
 1.307 
 
 20 
 
 .3 
 
 1.327 
 
 1.348 
 
 1.368 
 
 1.389 
 
 1.410 
 
 1.431 
 
 1.453 
 
 1.474 
 
 1.496 
 
 1.517 
 
 21 
 
 A 
 
 1.539 
 
 1.561 
 
 1.584 
 
 1.606 
 
 1.629 
 
 1.651 
 
 1.674 
 
 1.697 
 
 1.720 
 
 1.744 
 
 23 
 
 1.5 
 
 1.767 
 
 1.791 
 
 1.815 
 
 1.839 
 
 1.863 
 
 1.887 
 
 1.911 
 
 1.936 
 
 1.961 
 
 1.986 
 
 24 
 
 .6 
 
 2.011 
 
 2.036 
 
 2.061 
 
 2.087 
 
 2.112 
 
 2.138 
 
 2.164 
 
 2.190 
 
 2.217 
 
 2.243 
 
 26 
 
 .7 
 
 2.270 
 
 2.297 
 
 2.324 
 
 2.351 
 
 2.378 
 
 2.405 
 
 2.433 
 
 2.461 
 
 2.488 
 
 2.516 
 
 27 
 
 .8 
 
 2.545 
 
 2.573 
 
 2.602 
 
 2.630 
 
 2.659 
 
 2.688 
 
 2.717 
 
 2.746 
 
 2.776 
 
 2.806 
 
 29 
 
 .9 
 
 2.835 
 
 2.865 
 
 2.895 
 
 2.926 
 
 2.956 
 
 2.986 
 
 3.017 
 
 3.048 
 
 3.079 
 
 3.110 
 
 31 
 
 2.0 
 
 3.142 
 
 3.173 
 
 3.205 
 
 3.237 
 
 3.269 
 
 3.301 
 
 3.333 
 
 3.365 
 
 3.398 
 
 3.431 
 
 32 
 
 .1 
 
 3.464 
 
 3.497 
 
 3.530 
 
 3.563 
 
 3.597 
 
 3.631 
 
 3.664 
 
 3.698 
 
 3.733 
 
 3.767 
 
 34 
 
 .2 
 
 3.801 
 
 3.836 
 
 3.871 
 
 3.906 
 
 3.941 
 
 3.976 
 
 4.011 
 
 4.047 
 
 4.083 
 
 4.119 
 
 35 
 
 .3 
 
 4.155 
 
 4.191 
 
 4.227 
 
 4.264 
 
 4.301 
 
 4.337 
 
 4.374 
 
 4.412 
 
 4.449 
 
 4.486 
 
 37 
 
 A 
 
 4.524 
 
 4.562 
 
 4.600 
 
 4.638 
 
 4.676 
 
 4.714 
 
 4.753 
 
 4.792 
 
 4.831 
 
 4.870 
 
 38 
 
 2.5 
 
 4.909 
 
 4.948 
 
 4.988 
 
 5.027 
 
 5.067 
 
 5.107 
 
 5.147 
 
 5.187 
 
 5.228 
 
 5.269 
 
 40 
 
 .6 
 
 5.309 
 
 5.350 
 
 5.391 
 
 5.433 
 
 5.474 
 
 5.515 
 
 5.557 
 
 5.599 
 
 5.641 
 
 5.683 
 
 42 
 
 .7 
 
 5.726 
 
 5.768 
 
 5.811 
 
 5.853 
 
 5.896 
 
 5.940 
 
 5.983 
 
 6.026 
 
 6.070 
 
 6.114 
 
 43 
 
 .8 
 
 6.158 
 
 6.202 
 
 6.246 
 
 6.290 
 
 6.335 
 
 6.379 
 
 6.424 
 
 6.469 
 
 6.514 
 
 6.560 
 
 45 
 
 .9 
 
 6.605 
 
 6.651 
 
 6.697 
 
 6.743 
 
 6.789 
 
 6.835 
 
 6.881 
 
 6.928 
 
 6.975 
 
 7.022 
 
 46 
 
 3.0 
 
 7.069 
 
 7.116 
 
 7.163 
 
 7.211 
 
 7.258 
 
 7.306 
 
 7.354 
 
 7.402 
 
 7.451 
 
 7.499 
 
 48 
 
 .1 
 
 7.548 
 
 7.596 
 
 7.645 
 
 7.694 
 
 7.744 
 
 7.793 
 
 7.843 
 
 7.892 
 
 7.942 
 
 7.992 
 
 49 
 
 .2 
 
 8.042 
 
 8.093 
 
 8.143 
 
 8.194 
 
 8.245 
 
 8.296 
 
 8.347 
 
 8.398 
 
 8.450 
 
 8.501 
 
 51 
 
 .3 
 
 8.553 
 
 8.605 
 
 8.657 
 
 8.709 
 
 8.762 
 
 8.814 
 
 8.867 
 
 8.920 
 
 8.973 
 
 9.026 
 
 53 
 
 A 
 
 9.079 
 
 9.133 
 
 9.186 
 
 9.240 
 
 9.294 
 
 9.348 
 
 9.402 
 
 9.457 
 
 9.511 
 
 9.566 
 
 54 
 
 3.5 
 
 9.621 
 
 9.676 
 
 9.731 
 
 9.787 
 
 9.842 
 
 9.898 
 
 9.954 
 
 10.010 
 
 
 
 56 
 
 .5 
 
 
 
 
 
 
 
 
 
 10.01 
 
 10.07 
 
 10.12 
 
 6 
 
 .6 
 
 10.18 
 
 10.24 
 
 10.29 
 
 10.35 
 
 10.41 
 
 10.46 
 
 10.52 
 
 10.58 
 
 10.64 
 
 10.69 
 
 6 
 
 .7 
 
 10.75 
 
 10.81 
 
 10.87 
 
 10.93 
 
 10.99 
 
 11.04 
 
 11.10 
 
 11.16 
 
 11.22 
 
 11.28 
 
 
 .8 
 
 11.34 
 
 11.40 
 
 11.46 
 
 11.52 
 
 11.58 
 
 11.64 
 
 11.70 
 
 11.76 
 
 11.82 
 
 11.88 
 
 
 .9 
 
 11.95 
 
 12.01 
 
 12.07 
 
 12.13 
 
 12.19 
 
 12.25 
 
 12.32 
 
 12.38 
 
 12.44 
 
 12.50 
 
 
 4.0 
 
 12.57 
 
 12.63 
 
 12.69 
 
 12.76 
 
 12.82 
 
 12.88 
 
 12.95 
 
 13.01 
 
 13.07 
 
 13.14 
 
 7 
 
 .1 
 
 13.20 
 
 13.27 
 
 13.33 
 
 13.40 
 
 13.46 
 
 13.53 
 
 13.59 
 
 13.66 
 
 13.72 
 
 13.79 
 
 
 .2 
 
 13.85 
 
 13.92 
 
 13.99 
 
 14.05 
 
 14.12 
 
 14.19 
 
 14.25 
 
 14.32 
 
 14.39 
 
 14.45 
 
 
 .3 
 
 14.52 
 
 14.59 
 
 14.66 
 
 14.73 
 
 14.79 
 
 14.86 
 
 14.93 
 
 15.00 
 
 15.07 
 
 15.14 
 
 
 A 
 
 15.21 
 
 15.27 
 
 15.34 
 
 15.41 
 
 15.48 
 
 15.55 
 
 15.62 
 
 15.69 
 
 15.76 
 
 15.83 
 
 
 4.5 
 
 15.90 
 
 15.98 
 
 16.05 
 
 16.12 
 
 16.19 
 
 16.26 
 
 16.33 
 
 16.40 
 
 16.47 
 
 16.55 
 
 
 .6 
 
 16.62 
 
 16.69 
 
 16.76 
 
 16.84 
 
 16.91 
 
 16.98 
 
 17.06 
 
 17.13 
 
 17.20 
 
 17.28 
 
 
 .7 
 
 17.35 
 
 17.42 
 
 17.50 
 
 17.57 
 
 17.65 
 
 17.72 
 
 17.80 
 
 17.87 
 
 17.95 
 
 18.02 
 
 
 .8 
 
 18.10 
 
 18.17 
 
 18.25 
 
 18.32 
 
 18.40 
 
 18.47 
 
 18.55 
 
 18.63 
 
 18.70 
 
 18.78 
 
 8 
 
 .9 
 
 18.86 
 
 18.93 
 
 19.01 
 
 19.09 
 
 19.17 
 
 19.24 
 
 19.32 
 
 19.40 
 
 19.48 
 
 19.56 
 
 
 Explanation of Table of Areas of Circles (pp. 30-31) 
 
 Moving the decimal point one place in column D is equivalent to moving it two 
 places in the body of the table. (D = diameter.) 
 
 Area of circle = ^ X (diam.2) = 0.785398 X (diam. 2 ) 
 Conversely, 
 
 Diam. = 
 
 = 1.128379 X 
 
MATHEMATICAL TABLES 
 AREAS OP CIRCLES BY HUNDREDTHS (continued) 
 
 31 
 
 D 
 
 
 
 1 
 
 2 
 
 a 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 $*i 
 -3 
 
 5.0 
 
 19.63 
 
 19.71 
 
 19.79 
 
 19.87 
 
 19.95 
 
 20.03 
 
 20.11 
 
 20.19 
 
 20.27 
 
 20.35 
 
 8 
 
 .1 
 
 20.43 
 
 20.51 
 
 20.59 
 
 20.67 
 
 20.75 
 
 20.83 
 
 20.91 
 
 20.99 
 
 21.07 
 
 21.16 
 
 
 .2 
 
 21.24 
 
 21.32 
 
 21.40 
 
 21.48 
 
 21.57 
 
 21.65 
 
 21.73 
 
 21.81 
 
 21.90 
 
 21.98 
 
 
 .3 
 
 22.06 
 
 22.15 
 
 22.23 
 
 22.31 
 
 22.40 
 
 22.48 
 
 22.56 
 
 22.65 
 
 22.73 
 
 22.82 
 
 
 .4 
 
 22.90 
 
 22.99 
 
 23.07 
 
 23.16 
 
 23.24 
 
 23.33 
 
 23.41 
 
 23.50 
 
 23.59 
 
 23.67 
 
 9 
 
 5.5 
 
 23.76 
 
 23.84 
 
 23.93 
 
 24.02 
 
 24.11 
 
 24.19 
 
 24.28 
 
 24.37 
 
 24.45 
 
 24.54 
 
 
 .6 
 
 24.63 
 
 24.72 
 
 24.81 
 
 24.89 
 
 24.98 
 
 25.07 
 
 25.16 
 
 25.25 
 
 25.34 
 
 25.43 
 
 
 .7 
 
 25.52 
 
 25.61 
 
 25.70 
 
 25.79 
 
 25.88 
 
 25.97 
 
 26.06 
 
 26.15 
 
 26.24 
 
 26.33 
 
 
 .8 
 
 26.42 
 
 26.51 
 
 26.60 
 
 26.69 
 
 26.79 
 
 26.88 
 
 26.97 
 
 27.06 
 
 27.15 
 
 27.25 
 
 
 .9 
 
 27.34 
 
 27.43 
 
 27.53 
 
 27.62 
 
 27.71 
 
 27.81 
 
 27.90 
 
 27.99 
 
 28.09 
 
 28.18 
 
 
 6.0 
 
 28.27 
 
 28.37 
 
 28.46 
 
 28.56 
 
 28.65 
 
 28.75 
 
 28.84 
 
 28.94 
 
 29.03 
 
 29.13 
 
 10 
 
 .1 
 
 29.22 
 
 29.32 
 
 29.42 
 
 29.51 
 
 29.61 
 
 29.71 
 
 29.80 
 
 29.90 
 
 30.00 
 
 30.09 
 
 
 .2 
 
 30.19 
 
 30.29 
 
 30.39 
 
 30.48 
 
 30.58 
 
 30.68 
 
 30.78 
 
 30.88 
 
 30.97 
 
 31.07 
 
 
 .3 
 
 31.17 
 
 31.27 
 
 31.37 
 
 31.47 
 
 31.57 
 
 31.67 
 
 31.77 
 
 31.87 
 
 31.97 
 
 32.07 
 
 
 .4 
 
 32.17 
 
 32.27 
 
 32.37 
 
 32.47 
 
 32.57 
 
 32.67 
 
 32.78 
 
 32.88 
 
 32.98 
 
 33.08 
 
 
 6.5 
 
 33.18 
 
 33.29 
 
 33.39 
 
 33.49 
 
 33.59 
 
 33.70 
 
 33.80 
 
 33.90 
 
 34.00 
 
 34.11 
 
 
 .6 
 
 34.21 
 
 34.32 
 
 34.42 
 
 34.52 
 
 34.63 
 
 34.73 
 
 34.84 
 
 34.94 
 
 35.05 
 
 35.15 
 
 
 .7 
 
 35.26 
 
 35.36 
 
 35.47 
 
 35.57 
 
 35.68 
 
 35.78 
 
 35.89 
 
 36.00 
 
 36.10 
 
 36.21 
 
 11 
 
 .8 
 
 36.32 
 
 36.42 
 
 36.53 
 
 36.64 
 
 36.75 
 
 36.85 
 
 36.96 
 
 37.07 
 
 37.18 
 
 37.28 
 
 
 .9 
 
 37.39 
 
 37.50 
 
 37.61 
 
 37.72 
 
 37.83 
 
 37.94 
 
 38.05 
 
 38.16 
 
 38.26 
 
 38.37 
 
 
 7.0 
 
 38.48 
 
 38.59 
 
 38.70 
 
 38.82 
 
 38.93 
 
 39.04 
 
 39.15 
 
 39.26 
 
 39.37 
 
 39.48 
 
 
 .1 
 
 39.59 
 
 39.70 
 
 39.82 
 
 39.93 
 
 40.04 
 
 40.15 
 
 40.26 
 
 40.38 
 
 40.49 
 
 40.60 
 
 
 .2 
 
 40.72 
 
 40.83 
 
 40.94 
 
 41.06 
 
 41.17 
 
 41.28 
 
 41.40 
 
 41.51 
 
 41.62 
 
 41.74 
 
 
 .3 
 
 41.85 
 
 41.97 
 
 42.08 
 
 42.20 
 
 42.31 
 
 42.43 
 
 42.54 
 
 42.66 
 
 42.78 
 
 42.89 
 
 12 
 
 .4 
 
 43.01 
 
 43.12 
 
 43.24 
 
 43.36 
 
 43.47 
 
 43.59 
 
 43.71 
 
 43.83 
 
 43.94 
 
 44.06 
 
 
 7.5 
 
 44.18 
 
 44.30 
 
 44.41 
 
 44.53 
 
 44.65 
 
 44.77 
 
 44.89 
 
 45.01 
 
 45.13 
 
 45.25 
 
 
 .6 
 
 45.36 
 
 45.48 
 
 45.60 
 
 45.72 
 
 45.84 
 
 45.96 
 
 46.08 
 
 46.20 
 
 46.32 
 
 46.45 
 
 
 .7 
 
 46.57 
 
 46.69 
 
 46.81 
 
 46.93 
 
 47.05 
 
 47.17 
 
 47.29 
 
 47.42 
 
 47.54 
 
 47.66 
 
 
 .8 
 
 47.78 
 
 47.91 
 
 48.03 
 
 48.15 
 
 48.27 
 
 48.40 
 
 48.52 
 
 48.65 
 
 48.77 
 
 48.89 
 
 
 .9 
 
 49.02 
 
 49.14 
 
 49.27 
 
 49.39 
 
 49.51 
 
 49.64 
 
 49.76 
 
 49.89 
 
 50.01 
 
 50.14 
 
 
 8.0 
 
 50.27 
 
 50.39 
 
 50.52 
 
 50.64 
 
 50.77 
 
 50.90 
 
 51.02 
 
 51.15 
 
 51.28 
 
 51.40 
 
 13 
 
 .1 
 
 51.53 
 
 51.66 
 
 51.78 
 
 51.91 
 
 52.04 
 
 52.17 
 
 52.30 
 
 52.42 
 
 52.55 
 
 52.68 
 
 
 .2 
 
 52.81 
 
 52.94 
 
 53.07 
 
 53.20 
 
 53.33 
 
 53.46 
 
 53.59 
 
 53.72 
 
 53.85 
 
 53.98 
 
 
 .3 
 
 54.11 
 
 54.24 
 
 54.37 
 
 54.50 
 
 54.63 
 
 54.76 
 
 54.89 
 
 55.02 
 
 55.15 
 
 55.29 
 
 
 .4 
 
 55.42 
 
 55.55 
 
 55.68 
 
 55.81 
 
 55.95 
 
 56.08 
 
 56.21 
 
 56.35 
 
 56.48 
 
 56.61 
 
 
 8.5 
 
 56.75 
 
 56.88 
 
 57.01 
 
 57.15 
 
 57.28 
 
 57.41 
 
 57.55 
 
 57.68 
 
 57.82 
 
 57.95 
 
 
 .6 
 
 58.09 
 
 58.22 
 
 58.36 
 
 58.49 
 
 58.63 
 
 58.77 
 
 58.90 
 
 59.04 
 
 59.17 
 
 59.31 
 
 14 
 
 .7 
 
 59.45 
 
 59.58 
 
 59.72 
 
 59.86 
 
 59.99 
 
 60.13 
 
 60.27 
 
 60.41 
 
 60.55 
 
 60.68 
 
 
 .8 
 
 60.82 
 
 60.96 
 
 61.10 
 
 61.24 
 
 61.38 
 
 61.51 
 
 61.65 
 
 61.79 
 
 61.93 
 
 62.07 
 
 
 .9 
 
 62.21 
 
 62.35 
 
 62.49 
 
 62.63 
 
 62.77 
 
 62.91 
 
 63.05 
 
 63.19 
 
 63.33 
 
 63.48 
 
 
 9.0 
 
 63.62 
 
 63.76 
 
 63.90 
 
 64.04 
 
 64.18 
 
 64.33 
 
 64.47 
 
 64.61 
 
 64.75 
 
 64.90 
 
 
 .1 
 
 65.04 
 
 65.18 
 
 65.33 
 
 65.47 
 
 65.61 
 
 65.76 
 
 65.90 
 
 66.04 
 
 66.19 
 
 66.33 
 
 15 
 
 .2 
 
 66.48 
 
 66.62 
 
 66.77 
 
 66.91 
 
 67.06 
 
 67.20 
 
 67.35 
 
 67.49 
 
 67.64 
 
 67.78 
 
 
 .3 
 
 67.93 
 
 68.08 
 
 68.22 
 
 68.37 
 
 68.51 
 
 68.66 
 
 68.81 
 
 68.96 
 
 69.10 
 
 69.25 
 
 
 .4 
 
 69.40 
 
 69.55 
 
 69.69 
 
 69.84 
 
 69.99 
 
 70.14 
 
 70.29 
 
 70.44 
 
 70.58 
 
 70.73 
 
 
 9.5 
 
 70.88 
 
 71.03 
 
 71.18 
 
 71.33 
 
 71.48 
 
 71.63 
 
 71.78 
 
 71.93 
 
 72.08 
 
 72.23 
 
 
 .6 
 
 72.38 
 
 72.53 
 
 72.68 
 
 72.84 
 
 72.99 
 
 73.14 
 
 73.29 
 
 73.44 
 
 73.59 
 
 73.75 
 
 
 .7 
 
 73.90 
 
 74.05 
 
 74.20 
 
 74.36 
 
 74.51 
 
 74.66 
 
 74.82 
 
 74.97 
 
 75.12 
 
 75.28 
 
 
 .8 
 
 75.43 
 
 75.58 
 
 75.74 
 
 75.89 
 
 76.05 
 
 76.20 
 
 76.36 
 
 76.51 
 
 76.67 
 
 76.82 
 
 
 .9 
 
 76.98 
 
 77.13 
 
 77.29 
 
 77.44 
 
 77.60 
 
 77.76 
 
 77.91 
 
 78.07 
 
 78.23 
 
 78.38 
 
 16 
 
 Moving the decimal point ONE place in D requires moving it TWO places in body 
 of table (see p. 30). 
 
32 
 
 MATHEMATICAL TABLES 
 
 CIRCUMFERENCES AND AREAS OF CIRCLES BY EIGHTHS, ETC. 
 
 (For tenths, see p. 28) 
 
 1 
 
 | 
 
 1 
 
 S 
 
 
 a 
 
 Circum. 
 
 1 
 
 I 
 
 Circum. 
 
 a 
 
 
 < 
 
 
 
 Q 
 
 Circum. 
 
 ot 
 
 2 
 <5 
 
 
 
 
 H 
 
 2.749 
 
 .6013 
 
 4 
 
 12.57 
 
 12.57 
 
 9 
 
 28.27 
 
 63.62 
 
 W4 
 
 .04909 
 
 .00019 
 
 6 %4 
 
 2.798 
 
 .6230 
 
 He 
 
 12.76 
 
 12.96 
 
 H 
 
 28.67 
 
 65.40 
 
 fa 
 
 .09817 
 
 .00077 
 
 *fa 
 
 2.847 
 
 .6450 
 
 H 
 
 12.96 
 
 13.36 
 
 H 
 
 29.06 
 
 67.20 
 
 fa 
 
 .1473 
 
 .00173 
 
 *%* 
 
 2.896 
 
 .6675 
 
 Me 
 
 13.16 
 
 13.77 
 
 N 
 
 29.45 
 
 69.03 
 
 Ho 
 
 .1963 
 
 .00307 
 
 'Me 
 
 2.945 
 
 .6903 
 
 H 
 
 13.35 
 
 14.19 
 
 H 
 
 29.85 
 
 70.88 
 
 % 
 
 .2454 
 
 .00479 
 
 6 ^4 
 
 2.994 
 
 .7135 
 
 Me 
 
 13.55 
 
 14.61 
 
 K^ 
 
 30.24 
 
 7276 
 
 fa 
 
 .2945 
 
 .00690 
 
 fa 
 
 3.043 
 
 .7371 
 
 
 13.74 
 
 15.03 
 
 y\ 
 
 30.63 
 
 74.66 
 
 ft*. 
 
 .3436 
 
 .00940 
 
 *%l 
 
 3.093 
 
 .7610 
 
 Mo 
 
 13.94 
 
 15.47 
 
 1& 
 
 31.02 
 
 76.59 
 
 H 
 
 .3927 
 
 .01227 
 
 1 
 
 3.142 
 
 .7854 
 
 H 
 
 14.14 
 
 15.90 
 
 10 
 
 31.42 
 
 78.54 
 
 Hi 
 
 .4418 
 
 .01553 
 
 Me 
 
 3.338 
 
 .8866 
 
 Me 
 
 14.33 
 
 16.35 
 
 H 
 
 31.81 
 
 80.52 
 
 fa 
 
 .4909 
 
 .01917 
 
 H 
 
 3.534 
 
 .9940 
 
 5 /i 
 
 14.53 
 
 16.80 
 
 H 
 
 32.20 
 
 82.52 
 
 '^4 
 
 .5400 
 
 .02320 
 
 Me 
 
 3.731 
 
 1.108 
 
 'He 
 
 14.73 
 
 17.26 
 
 H 
 
 32.59 
 
 84.54 
 
 Me 
 
 .5890 
 
 .02761 
 
 U 
 
 3.927 
 
 1.227 
 
 N 
 
 14.92 
 
 17.72 
 
 M 
 
 32.99 
 
 86.59 
 
 '%4 
 
 .6381 
 
 .03241 
 
 Me 
 
 4.123 
 
 1.353 
 
 'Me 
 
 15.12 
 
 18.19 
 
 M 
 
 33.38 
 
 88.66 
 
 fa 
 
 .6872 
 
 .03758 
 
 M 
 
 4.320 
 
 1.485 
 
 H 
 
 15.32 
 
 18.67 
 
 94 
 
 33.77 
 
 90.76 
 
 '5*4 
 
 .7363 
 
 .04314 
 
 Me 
 
 4.516 
 
 1.623 
 
 'Me 
 
 15.51 
 
 19.15 
 
 H 
 
 34.16 
 
 92.89 
 
 u 
 
 .7854 
 
 .04909 
 
 H 
 
 4.712 
 
 1.767 
 
 5 
 
 15.71 
 
 19.63 
 
 11 
 
 34.56 
 
 95.03 
 
 %4 
 
 .8345 
 
 .05542 
 
 Me 
 
 4.909 
 
 1.917 
 
 He 
 
 15.90 
 
 20.13 
 
 M 
 
 34.95 
 
 97.21 
 
 fa 
 
 .8836 
 
 .06213 
 
 5 /6 
 
 5.105 
 
 2.074 
 
 H 
 
 16.10 
 
 20.63 
 
 H 
 
 35.34 
 
 99.40 
 
 l %i 
 
 .9327 
 
 .06922 
 
 'He 
 
 5.301 
 
 2.237 
 
 Me 
 
 16.30 
 
 21.14 
 
 N 
 
 35.74 
 
 101.6 
 
 Me 
 
 .9817 
 
 .07670 
 
 N 
 
 5.498 
 
 2.405 
 
 M 
 
 16.49 
 
 21.65 
 
 H 
 
 36.13 
 
 103.9 
 
 2 ^4 
 
 1.031 
 
 .08456 
 
 'Me 
 
 5.694 
 
 2.580 
 
 Me 
 
 16.69 
 
 22.17 
 
 % 
 
 36.52 
 
 106.1 
 
 Ifa 
 
 1.080 
 
 .09281 
 
 H 
 
 5.890 
 
 2.761 
 
 N 
 
 16.89 
 
 22.69 
 
 H 
 
 36.91 
 
 108.4 
 
 2 *<J4 
 
 1.129 
 
 .1014 
 
 I?i8 
 
 6.087 
 
 2.948 
 
 Me 
 
 17.08 
 
 23.22 
 
 N 
 
 37.31 
 
 110.8 
 
 H 
 
 1.178 
 
 .1104 
 
 a 
 
 6.283 
 
 3.142 
 
 W 
 
 17.28 
 
 23.76 
 
 12 
 
 37.70 
 
 113.1 
 
 *M* 
 
 1.227 
 
 .1198 
 
 He 
 
 6.480 
 
 3.341 
 
 Me 
 
 17.48 
 
 24.30 
 
 H 
 
 38.09 
 
 115.5 
 
 l fa 
 
 1.276 
 
 .1296 
 
 H 
 
 6.676 
 
 3.547 
 
 M 
 
 17.67 
 
 24.85 
 
 H 
 
 38.48 
 
 117.9 
 
 2 ^4 
 
 1.325 
 
 .1398 
 
 Me 
 
 6.872 
 
 3.758 
 
 'He 
 
 17.87 
 
 25.41 
 
 % 
 
 38.88 
 
 120.3 
 
 Me 
 
 1.374 
 
 .1503 
 
 M 
 
 7.069 
 
 3.976 
 
 H 
 
 18.06 
 
 25.97 
 
 M 
 
 39.27 
 
 122.7 
 
 2 %4 
 
 1.424 
 
 .1613 
 
 Me 
 
 7.265 
 
 4.200 
 
 'Me 
 
 18.26 
 
 26.53 
 
 U 
 
 39.66 
 
 125.2 
 
 9$2 
 
 1.473 
 
 .1726 
 
 
 7.461 
 
 4.430 
 
 N 
 
 18.46 
 
 27.11 
 
 M- 
 
 40.06 
 
 127.7 
 
 H* 
 
 1.522 
 
 .1843 
 
 Me 
 
 7.658 
 
 4.666 
 
 'Me 
 
 18.65 
 
 27.69 
 
 ji 
 
 40.45 
 
 130.2 
 
 M 
 
 1.571 
 
 .1963 
 
 W 
 
 7.854 
 
 4.909 
 
 6 
 
 18.85 
 
 28.27 
 
 13 
 
 40.84 
 
 132.7 
 
 8 %4 
 
 1.620 
 
 .2088 
 
 Me 
 
 8.050 
 
 5.157 
 
 M 
 
 19.24 
 
 29.46 
 
 w 
 
 41.23 
 
 135.3 
 
 l fa 
 
 1.669 
 
 .2217 
 
 H 
 
 8.247 
 
 5.412 
 
 M 
 
 19.63 
 
 30.68 
 
 H 
 
 41.63 
 
 137.9 
 
 fa 
 
 1.718 
 
 .2349 
 
 H 
 
 8.443 
 
 5.673 
 
 N 
 
 20.03 
 
 31.92 
 
 N 
 
 42.02 
 
 140.5 
 
 Me 
 
 1.767 
 
 .2485 
 
 94 
 
 8.639 
 
 5.940 
 
 H- 
 
 20.42 
 
 33.18 
 
 H 
 
 42.41 
 
 143.1 
 
 3 %4 
 
 1.816 
 
 .2625 
 
 me 
 
 8.836 
 
 6.213 
 
 M 
 
 20.81 
 
 34.47 
 
 W 
 
 42.80 
 
 145.8 
 
 l fa 
 
 1.865 
 
 .2769 
 
 H 
 
 9.032 
 
 6.492 
 
 M 
 
 21.21 
 
 35.78 
 
 H 
 
 43.20 
 
 148.5 
 
 %4 
 
 1.914 
 
 .2916 
 
 'Me 
 
 9.228 
 
 6.777 
 
 N 
 
 21.60 
 
 37.12 
 
 7 /i 
 
 43.59 
 
 151.2 
 
 M 
 
 1.963 
 
 .3068 
 
 3 
 
 9.425 
 
 7.069 
 
 7 
 
 21.99 
 
 38.48 
 
 14 
 
 43.98 
 
 153.9 
 
 *H4 
 
 2.013 
 
 .3223 
 
 He 
 
 9.621 
 
 7.366 
 
 K 
 
 22.38 
 
 39.87 
 
 M 
 
 44.37 
 
 156.7 
 
 9 fa 
 
 2.062 
 
 .3382 
 
 H 
 
 9.817 
 
 7.670 
 
 M 
 
 22.78 
 
 41.28 
 
 M 
 
 44.77 
 
 159.5 
 
 *%4 
 
 2.111 
 
 .3545 
 
 Me 
 
 10.01 
 
 7.980 
 
 n 
 
 23.17 
 
 42.72 
 
 N 
 
 45.16 
 
 162.3 
 
 'Me 
 
 2.160 
 
 .3712 
 
 H 
 
 10.21 
 
 8.296 
 
 H 
 
 23.56 
 
 44.18 
 
 H 
 
 45.55 
 
 165.1 
 
 4 $B4 
 
 2.209 
 
 .3883 
 
 Me 
 
 10.41 
 
 8.618 
 
 % 
 
 23.95 
 
 45.66 
 
 H 
 
 45.95 
 
 168.0 
 
 2 fa 
 
 2.258 
 
 .4057 
 
 M 
 
 10.60 
 
 8.946 
 
 H 
 
 24.35 
 
 47.17 
 
 N 
 
 46.34 
 
 170.9 
 
 *K* 
 
 2.307 
 
 .4236 
 
 Me 
 
 10.80 
 
 9.281 
 
 X 
 
 24.74 
 
 48.71 
 
 H 
 
 46.73 
 
 173.8 
 
 H 
 
 2.356 
 
 .4418 
 
 H 
 
 11.00 
 
 9.621 
 
 8 
 
 25.13 
 
 50.27 
 
 15 
 
 47.12 
 
 176.7 
 
 *%4 
 
 2.405 
 
 .4604 
 
 Me 
 
 11.19 
 
 9.968 
 
 H 
 
 25.53 
 
 51.85 
 
 H 
 
 47.52 
 
 179.7 
 
 'fa 
 
 2.454 
 
 .4794 
 
 #* 
 
 11.39 
 
 10.32 
 
 M 
 
 25.92 
 
 53.46 
 
 H 
 
 47.91 
 
 182.7 
 
 H4 
 
 2.503 
 
 .4987 
 
 'He 
 
 11.58 
 
 10.68 
 
 H 
 
 26.31 
 
 55.09 
 
 H 
 
 48.30 
 
 185.7 
 
 'Me 
 
 2.553 
 
 .5185 
 
 N 
 
 11.78 
 
 11.04 
 
 H 
 
 26.70 
 
 56.75 
 
 M 
 
 48.69 
 
 188.7 
 
 3 %4 
 
 2.602 
 
 .5386 
 
 'Me 
 
 11.98 
 
 11.42 
 
 N 
 
 27.10 
 
 58.43 
 
 H 
 
 49.09 
 
 191.7 
 
 *fa 
 
 2.651 
 
 .5591 
 
 H 
 
 12.17 
 
 11.79 
 
 N 
 
 27.49 
 
 60.13 
 
 N 
 
 49.48 
 
 194.8 
 
 %4 
 
 2.700 
 
 .5800 
 
 is 
 
 12.37 
 
 12.18 
 
 N 
 
 27.88 
 
 61.86 
 
 K 
 
 49.87 
 
 197.9 
 
MATHEMATICAL TABLES 
 
 33 
 
 CIRCUMFERENCES AND AREAS BY EIGHTHS (continued) 
 
 s 
 
 .5 
 P 
 
 Circum. 
 
 1 
 
 P 
 
 j 
 
 a 
 
 | 
 
 Circum. 
 
 1 
 
 a 
 P 
 
 Circum. 
 
 1 
 
 16 
 
 50.27 
 
 201.1 
 
 19 H 
 
 61.26 
 
 298.6 
 
 23 
 
 7226 
 
 415.5 
 
 29 
 
 91.11 
 
 660.5 
 
 
 50.66 
 
 204.2 
 
 H 
 
 61.65 
 
 302.5 
 
 H 
 
 72.65 
 
 420.0 
 
 H 
 
 91.89 
 
 672.0 
 
 H 
 
 51.05 
 
 207.4 
 
 N 
 
 62.05 
 
 306.4 
 
 
 73.04 
 
 424.6 
 
 N 
 
 92.68 
 
 683.5 
 
 N 
 
 51.44 
 
 210.6 
 
 R 
 
 62.44 
 
 310.2 
 
 H 
 
 73.43 
 
 429.1 
 
 H 
 
 93.46 
 
 695.1 
 
 M 
 
 51.84 
 
 213.8 
 
 20 
 
 6283 
 
 314.2 
 
 H 
 
 73.83 
 
 433.7 
 
 30 
 
 94.25 
 
 706.9 
 
 S 
 
 52.23 
 
 217.1 
 
 H 
 
 6322 
 
 318.1 
 
 
 7422 
 
 438.4 
 
 H 
 
 95.03 
 
 718.7 
 
 3/ 
 
 52.62 
 
 220.4 
 
 H 
 
 63.62 
 
 322.1 
 
 ?i 
 
 74.61 
 
 443.0 
 
 H 
 
 95.82 
 
 730.6 
 
 i 
 
 53.01 
 
 223.7 
 
 N 
 
 64.01 
 
 326.1 
 
 8 
 
 75.01 
 
 447.7 
 
 N 
 
 96.60 
 
 742.6 
 
 17 
 
 53.41 
 
 227.0 
 
 M 
 
 64.40 
 
 330.1 
 
 24 
 
 75.40 
 
 452.4 
 
 31 
 
 97.39 
 
 754.8 
 
 H 
 
 53.80 
 
 230.3 
 
 N 
 
 64.80 
 
 334.1 
 
 M 
 
 76.18 
 
 461.9 
 
 H 
 
 98.17 
 
 767.0 
 
 H 
 
 54.19 
 
 233.7 
 
 
 65.19 
 
 3382 
 
 M 
 
 76.97 
 
 471.4 
 
 M 
 
 98.96 
 
 779.3 
 
 N 
 
 54.59 
 
 237.1 
 
 % 
 
 65.58 
 
 3422 
 
 N 
 
 77.75 
 
 481.1 
 
 4 
 
 99.75 
 
 791.7 
 
 M 
 
 54.98 
 
 240.5 
 
 21 
 
 65.97 
 
 346.4 
 
 25 
 
 78.54 
 
 490.9 
 
 32 
 
 100.5 
 
 804.2 
 
 &4 
 
 55.37 
 
 244.0 
 
 H 
 
 66.37 
 
 350.5 
 
 H 
 
 79.33 
 
 500.7 
 
 H 
 
 101.3 
 
 816.9 
 
 a/ 
 
 55.76 
 
 247.4 
 
 M 
 
 66.76 
 
 354.7 
 
 
 80.11 
 
 510.7 
 
 h 
 
 102.1 
 
 829.6 
 
 H 
 
 56.16 
 
 250.9 
 
 3 4 
 
 67.15 
 
 358.8 
 
 N 
 
 80.90 
 
 520.8 
 
 N 
 
 102.9 
 
 842.4 
 
 18 
 
 56.55 
 
 254.5 
 
 M 
 
 67.54 
 
 363.1 
 
 26 
 
 81.68 
 
 530.9 
 
 33 
 
 103.7 
 
 855.3 
 
 M 
 
 56.94 
 
 258.0 
 
 W 
 
 67.94 
 
 367.3 
 
 M 
 
 82.47 
 
 541.2 
 
 M 
 
 104.5 
 
 8683 
 
 H 
 
 57.33 
 
 261.6 
 
 
 68.33 
 
 371.5 
 
 
 8325 
 
 551.5 
 
 
 1052 
 
 881.4 
 
 N 
 
 57.73 
 
 265.2 
 
 j| 
 
 68.72 
 
 375.8 
 
 H 
 
 84.04 
 
 562.0 
 
 % 
 
 106.0 
 
 894.6 
 
 M 
 
 58.12 
 
 268.8 
 
 22 
 
 69.12 
 
 380.1 
 
 27 
 
 84.82 
 
 572.6 
 
 34 
 
 106.8 
 
 907.9 
 
 8 
 
 58.51 
 58.90 
 
 272.4 
 276.1 
 
 g 
 
 69.51 
 69.90 
 
 384.5 
 388.8 
 
 *5 
 
 85.61 
 86.39 
 
 583.2 
 594.0 
 
 i 
 
 107.6 
 108.4 
 
 921.3 
 934.8 
 
 J6 
 
 59.30 
 
 279.8 
 
 N 
 
 7029 
 
 393.2 
 
 N 
 
 87.18 
 
 604.8 
 
 
 109.2 
 
 948.4 
 
 19 
 
 59.69 
 
 283.5 
 
 fc 
 
 70.69 
 
 397.6 
 
 28 
 
 87.96 
 
 615.8 
 
 35 
 
 110.0 
 
 962.1 
 
 H 
 
 60.08 
 
 287.3 
 
 
 71.08 
 
 402.0 
 
 H 
 
 88.75 
 
 626.8 
 
 5-4 
 
 110.7 
 
 975.9 
 
 U 
 
 60.48 
 
 291.0 
 
 ?4 
 
 71.47 
 
 406.5 
 
 B 
 
 89.54 
 
 637.9 
 
 M 
 
 111.5 
 
 989.8 
 
 N 
 
 60.87 
 
 294.8 
 
 N 
 
 71.86 
 
 411,0 
 
 
 90.32 
 
 6492 
 
 N 
 
 112.3 
 
 1003.8 
 
 AREAS OF CIRCLES. Diameters in Feet and Inches, Areas in Square Feet 
 
 Feet 
 
 Inches 
 
 1 23456 789 10 
 
 11 
 
 
 
 1 
 2 
 3 
 4 
 
 5 
 
 6 
 7 
 8 
 ? 
 10 
 11 
 12 
 13 
 14 
 
 .0000 .0055 .0218 .0491 .0873 .1364 .1963 .2673 .3491 .4418 .5454 
 .7854 .9218 1.069 1.227 1.396 1.576 1.767 1.969 2.182 2.405 2.640 
 3.142 3.409 3.687 3.976 4.276 4.587 4.909 5.241 5.585 5.940 6.305 
 7.069 7.467 7.876 8.2% 8.727 9.168 9.621 10.08 10.56 11.04 11.54 
 12.57 13.10 13.64 14.19 14.75 15.32 15.90 16.50 17.10 1772 18.35 
 
 19.63 20.29 20.97 21.65 22.34 23.04 23.76 24,48 2522 25.97 26.73 
 28.27 29.07 29.87 30.68 31.50 32.34 33.18 34.04 34.91 35.78 36.67 
 38.48 39.41 40.34 41.28 42.24 43.20 44.18 45.17 46.16 47.17 48.19 
 50.27 51.32 52.38 53.46 54.54 55.64 56.75 57.86 58.99 60.13 61.28 
 63.62 64.80 66.00 6720 68.42 69.64 70.88 72.13 73.39 74.66 75.94 
 
 78.54 79.85 81.18 82.52 83.86 85.22 86.59 87.97 89.36 90.76 92.18 
 95.03 96.48 97.93 99.40 100.9 102.4 103.9 105.4 106.9 108.4 110.0 
 113.1 114.7 116.3 117.9 119.5 121.1 122.7 124.4 126.0 127.7 129.4 
 132.7 134.4 136.2 137.9 139.6 141.4 143.1 144.9 146.7 148.5 150.3 
 153.9 155.8 157.6 1593 161.4 1632 165.1 167.0 168.9 170.9 172.8 
 
 .6600 
 2.885 
 6.681 
 12.05 
 18.99 
 
 27.49 
 37.57 
 4922 
 62.44 
 7724 
 
 93.60 
 111.5 
 131.0 
 152.1 
 174.8 
 
 If given diameter is not found in this table, reduce diameter to feet and decimals of a 
 foot by aid of the following auxiliary table, and then find area from pp. 30-31. 
 
 From Inches and Fractions of an Inch to Decimals of a Foot 
 
 Inches 123 456 78 9 10 
 Feet .0833 .1667 .2500 .3333 .4167 .5000 .5833 .6667 .7500 .8333 
 
 11 
 .9167 
 
 Inches H H H M 5 /i 9* 7 /* 
 Feet .0104 .0208 .0313 .0417 .0521 .0625 .0729 
 Example. 5 ft. 7% in. = 5.0 + 0.5833 + 0.0313 = 5.6146 ft. 
 
 
34 
 
 MATHEMATICAL TABLES 
 
 SEGMENTS OP CIRCLES, GIVEN h/c 
 
 Given: h = height; c = chord. (For explanation of this table, see p. 38) 
 
 h 
 
 Diam. ja 
 
 Arc 
 
 Area : 
 
 Central itt 
 
 A te 
 
 c 
 
 c P 
 
 c P 
 
 h X c p 
 
 angle, Q 
 
 Diam. P 
 
 .00 
 
 
 1.000 n 
 
 .6667 n 
 
 0.00 4 , 
 
 .0000 
 
 
 25.010 174 g n 
 
 1.000 
 
 .6667 
 
 4.58 
 
 .0004 ,1 
 
 2 
 
 19 s?n i*^"u 
 
 I/.J/U *A1C7 
 
 1.001 
 
 .6669 
 
 9 .6 
 
 .00.6 ;J 
 
 3 
 
 fl 3A3 4 1 J/ 
 o.?05 *7f)7^ 
 
 1.002 i 
 
 .667. 
 
 13J3 I 5 . 7 , 
 
 .0036 ?J 
 
 4 
 
 6.290 .f073 
 
 1.004 
 
 .6675 I 
 
 18.30 Jg 
 
 .0064 g 
 
 .05 
 
 5.050 ,,0,3 
 
 1.007 
 
 .6680 , 
 
 22.84 4,3 
 
 .0099 
 
 6 
 
 4.227 |g 
 
 1.010 \ 
 
 .6686 2 
 
 27.37 
 
 .0142 
 
 7 
 
 3.641 *X7A 
 
 1.013 
 
 .6693 J 
 
 31.88 1!I 
 
 .0192 22 
 
 8 
 
 3.205 ,7 
 
 1.017 
 
 .670. 
 
 36.36 
 
 .0250 22 
 
 9 
 
 2.868 ^g 
 
 1.021 J 
 
 .6710 |0 
 
 40.82 JJ5 
 
 .0314 64 
 
 .10 
 
 2.600 ,,7 
 
 1.026 
 
 .6720 
 
 45.24 43 
 
 .0385 77 
 
 1 
 
 2.383 fl 
 
 1.032 
 
 .6731 11 
 
 49.63 1|? 
 
 .0462 o< 
 
 2 
 
 2.203 Jt2 
 
 1.038 
 
 .6743 ? 
 
 53.98 I 3 * 
 
 .0545 2^ 
 
 3 
 
 2.053 '55 
 
 1.044 2 
 
 .6756 J 
 
 58.30 1^ 
 
 .0633 
 
 4 
 
 1.926 ^ 
 
 1.051 I 
 
 .6770 {J 
 
 62.57 J27 
 
 .0727 99 
 
 .15 
 
 1.817 ,04 
 
 1.059 
 
 .6785 
 
 66.80 4,0 
 
 .0826 ,03 
 
 6 
 
 .723 ,25 
 
 1.067 
 
 .6801 2 
 
 70.98 J ? 
 
 .0929 XT' 
 
 7 
 
 1.641 JE 
 
 1.075 
 
 .6818 
 
 75.11 7A^ 
 
 .1036 VJ 
 
 8 
 
 1.569 2? 
 
 1.084 ,2 
 
 .6836 
 
 79.20 1X4 
 
 .1147 J 
 
 9 
 
 1.506 g 
 
 1.094 ^ 
 
 .6855 JJ 
 
 83.23 $5 
 
 '262 jg 
 
 .20 
 
 1.450 , n 
 
 1.103 n 
 
 .6875 71 
 
 87.21 30, 
 
 .1379 nn 
 
 1 
 
 1.400 
 
 1.114 J 
 
 .6896 ?, 7 
 
 91.13 S= 
 
 .1499 20 
 
 2 
 
 1.356 
 
 1.124 
 
 .6918 5? 
 
 95.00 W. 
 
 .1622 Jg 
 
 3 
 
 1.317 H 
 
 1.136 2 
 
 .6941 g 
 
 98.81 2J 
 
 .1746 g 
 
 4 
 
 1.282 % 
 
 1.147 
 
 .6965 24 
 
 102.56 375 
 
 .1873 Jg 
 
 .25 
 
 6 
 
 1.250 - 
 .222 28 
 
 1.159 12 
 1.171 \\ 
 
 .6989 9 c 
 .7014 ; 7 
 
 106.26 3,4 
 109.90 $52 
 
 2000 ,28 
 
 7 
 
 .196 ?6 
 
 1.184 
 
 .7041 5i 
 
 113.48 358 
 
 ^2258 !2J 
 
 8 
 
 .173 23 
 
 1.197 
 
 .7068 11 
 
 117.00 ?52 
 
 .2387 aj 
 
 9 
 
 .152 J. 
 
 1.2.1 JJ 
 
 .7096 g 
 
 120.45 345 
 
 JI517 J30 
 
 .30 
 
 1 
 2 
 3 
 
 .133 , 7 
 .116 i 
 .101 
 .088 
 
 1.225 ,4 
 1.239 7 
 1.254 \l 
 1.269 
 
 .7125 9q 
 .7154 29 
 .7185 ^ 
 .7216 3,1 
 
 123.86 334 
 127.20 334 
 130.48 325 
 133.70 322 
 
 .2647 130 
 .2777 ,o 
 .2906 HJ 
 .3034 52 
 
 4 
 
 1.075 jf 
 
 1.284 ] 
 
 .7248 55 
 
 136.86 316 
 
 3162 Jg 
 
 .35 
 
 1.064 ,n 
 
 1.300 
 
 .7280 34 
 
 139.97 3 rt , 
 
 .3289 ,-- 
 
 6 
 
 1.054 ' 
 
 1.316 
 
 .7314 5x 
 
 143.02 305 
 
 .3414 Jg 
 
 7 
 
 1.046 
 
 1.332 2 
 
 .7348 g 
 
 146.01 
 
 .3538 \ 
 
 8 
 
 1.038 2 
 
 1.349 i 
 
 .7383 ?5 
 
 148.94 ??2 
 
 .3661 g 
 
 9 
 
 1.031 J 
 
 1.366 \' 7 
 
 .7419 g 
 
 151.82 288 
 
 3783 J22 
 
 .40 
 
 1.025 
 1.020 i 
 
 1.383 ,o 
 1.401 
 
 .7455 37 
 .7492 on 
 
 154.64 -77 
 157.41 El 
 
 402 2 " 9 
 
 .W/l jj> 
 
 2 
 
 1.015 
 
 1.419 
 
 .7530 IS 
 
 160.12 %\ 
 
 .4137 JJfj 
 
 3 
 
 1.011 
 
 1.437 
 
 .7568 3*5 
 
 162.78 5?? 
 
 
 4 
 
 1.008 \ 
 
 1.455 J 
 
 .7607 ^Q 
 
 165.39 261 
 
 .4364 jj 2 
 
 .45 
 
 6 
 
 8 
 9 
 
 1.006 
 1.003 
 1.002 
 1.001 
 1.000 J 
 
 1.474 ,Q 
 1.493 
 1.512 
 1.531 ij 
 1.55. 20 
 
 .7647 40 
 .7687 TV 
 .7728 J 
 .7769 11 
 .781. 
 
 167.95 ,,, 
 170.46 gi 
 172.91 fjf 
 175.32 241 
 177.69 g7 
 
 .4475 )no 
 .4584 X^ 
 .4691 2? 
 .4796 25 
 .4899 JJJ 
 
 .50 
 
 1.000 
 
 1.571 
 
 .7854 
 
 180.00 
 
 .5000 
 
 Interpolation may be inaccurate at these points. 
 
MATHEMATICAL TABLES 
 
 .SEGMENTS OP CIRCLES, GIVEN h/D 
 
 Given: h = height; D = diameter of circle. (For explanation of this table, see p. 38) 
 
 Arc 
 
 Area 
 
 Central i 
 angle, v 
 
 Chord 
 
 Arc 
 
 Circumf. Q 
 
 Area 
 Ckde 
 
 2003 2003 
 
 3482 t1c 
 
 .4027 ,Jg 
 
 .4510 * 4 
 
 .4949 ;: 
 
 .5355 Jftn 
 5735 
 
 .'6094 'U 9 
 
 .6435 *, 
 
 .676. Jf6 
 
 .7075 JA; 
 
 .7377 JJ2 
 
 .7670 ,293 
 
 .7954 276 
 
 .8230 2;x 
 
 .8500 ^V 
 
 .8763 ;K 
 
 .9021 g* 
 
 0.9273 24R 
 
 0.9521 215 
 
 0.9764 % 
 
 1.0004 i?? 
 
 1.0239 Si 
 
 1.0472 
 1.070. 
 ..0928 
 ...152 
 1.1374 
 
 1.1593 
 1.1810 
 1.2025 
 1.2239 
 1.2451 
 
 1.2661 
 1.2870 
 1.3078 
 1.3284 
 
 1.3694 
 1.3898 
 1.4101 
 1.4303 
 1.4505 
 
 1.4706 
 1.4907 
 1.5108 
 1.5308 
 1.5508 
 
 1.5708 
 
 22g 
 
 222 
 219 
 
 217 
 
 212 
 
 
 202 
 2J2 
 
 2 
 
 .0000 
 .0013 
 .0037 
 .0069 
 .0105 
 
 .0.47 
 .0192 
 .0242 
 .0294 
 .0350 
 
 .0409 
 .0470 
 .0534 
 .0600 
 .0668 
 
 .0739 
 .0811 
 .0885 
 .0961 
 .1039 
 
 .1118 
 .1199 
 .128. 
 .1365 
 .1449 
 
 .1535 
 .1623 
 .1711 
 .1800 
 .1890 
 
 .1982 
 .2074 
 .2167 
 .2260 
 .2355 
 
 .2450 
 .2546 
 .2642 
 .2739 
 .2836 
 
 .2934 
 .3032 
 .3130 
 .3229 
 .3328 
 
 .3428 
 .3527 
 
 3727 
 3827 
 
 3927 
 
 ., 
 
 
 , 
 
 , 
 
 IUU 
 
 0.00 770 , 
 
 22.96 A 296 
 
 32.52 956 
 
 39.90 738 
 
 46.15 ,|g 
 
 51-68 .... 
 
 56.72 504 
 
 6137 * 
 
 65.72 435 
 
 69.83 ^J] 
 
 73.74 _. 
 
 77.48 *|74 
 
 81.07 359 
 
 84.54 *347 
 
 87.89 .335 
 
 91.15 ,.., 
 
 94.31 
 
 97.40 309 
 
 .00.42 302 
 
 103.37 295 
 
 106.26 
 
 109.10 284 
 
 111.89 279 
 114.63 274 
 11734 27. 
 
 266 
 
 .20.00 
 
 122.63 263 
 
 125.23 260 
 
 127.79 256 
 
 .3033 254 
 
 132.84 
 
 .35.33 249 
 
 .37.80 247 
 
 140.25 245 
 .42.67 2J2 
 
 145.08 
 
 .47.48 240 
 
 .49.86 238 
 
 152.23 237 
 
 154.58 235 
 
 156.93 , 
 
 159.26 233 
 
 161.59 233 
 
 163.90 23. 
 166.22 232 
 
 168.52 ... 
 
 .70.82 230 
 
 173.12 230 
 
 175.42 230 
 
 177.71 229 
 
 180.00 
 
 .0000 
 !2800 
 3919 
 .4359 
 
 !5I03 
 .5426 
 .5724 
 
 .6000 
 .6258 
 .6499 
 .6726 
 .6940 
 
 .7141 
 
 >513 
 .7684 
 
 .8000 
 .8146 
 .8285 
 .8417 
 
 .8660 
 .8773 
 .8879 
 .8980 
 .9075 
 
 .9165 
 .9250 
 .9330 
 .9404 
 .9474 
 
 .9539 
 .9600 
 .9656 
 .9708 
 .9755 
 
 .9798 
 .9837 
 .9871 
 .9902 
 .9928 
 
 .9950 
 .9968 
 .9982 
 .9992 
 .9998 
 
 1.0000 
 
 3.3 
 .298 
 
 241 
 J27 
 .2.4 
 
 
 .0000 
 .0638 
 .0903 
 .1108 
 .1282 
 
 .1436 
 
 !l705 
 .1826 
 .1940 
 
 .2048 
 .2.52 
 .2252 
 .2348 
 .244. 
 
 .2532 
 .2620 
 .2706 
 .2789 
 .2871 
 
 .2952 
 3031 
 3108 
 .3184 
 3259 
 
 .3333 
 .3406 
 3478 
 .3550 
 .3620 
 
 3690 
 3759 
 3828 
 3896 
 .3963 
 
 .4030 
 .4097 
 .4.63 
 .4229 
 .4294 
 
 .4359 
 .4424 
 .4489 
 .4553 
 .4617 
 
 .4681 
 .4745 
 .4809 
 .4873 
 .4936 
 
 .5000 
 
 j{j 
 
 
 
 .0000 , 7 
 
 .0017 \\ 
 
 .0048 l\ 
 
 .0087 ?? 
 
 .0134 g 
 
 .0187 , 
 
 .0245 g 
 
 .0308 g 
 
 .0375 S{ 
 
 .0446 
 
 .0520 TO 
 
 .0599 JJ 
 .0680 
 
 .0764 2; 
 
 .085. J 
 
 .0941 Q, 
 
 .1033 2 
 
 .1127 A? 
 
 .1224 II 
 
 .1323 ,99 
 
 !$ 103 
 
 .1631 !< 
 
 .1737 2$ 
 
 .1846 JJ5 
 
 .1955 m 
 
 .2066 1 
 
 .2.78 J 
 
 .2292 \\i 
 
 2407 \\l 
 
 .2523 ,, 7 
 
 2640 
 
 .2759 9 
 
 .2878 , 
 
 .2998 ]^ 
 
 .3.19 ,22 
 
 324. 22 
 
 3364 S 
 
 3487 g 
 
 .3611 j|J 
 
 3735 , 
 
 3860 g 
 
 3986 \% 
 
 .41.2 5? 
 
 .4238 Jg 
 
 .4364 . 
 
 .4491 % 
 
 .4618 ?i 
 
 .4745 g 
 
 .4873 J28 
 
 .5000 
 
 Interpolation may be inaccurate at these points. 
 
36 MATHEMATICAL TABLES 
 
 VOLUMES OF SPHERES BY HUNDREDTHS 
 
 D 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 
 
 
 
 & 
 
 1.0 
 
 .5236 
 
 .5395 
 
 .5556 
 
 .5722 
 
 .5890 
 
 .6061 
 
 .6236 
 
 .6414 
 
 .65% 
 
 .6781 
 
 173 
 
 
 .6969 
 
 .7161 
 
 .7356 
 
 .7555 
 
 .7757 
 
 .7963 
 
 .8173 
 
 .8386 
 
 .8603 
 
 .8823 
 
 208 
 
 '.2 
 
 .9048 
 
 .9276 
 
 .9508 
 
 .9743 
 
 .9983 
 
 1.0227 
 
 
 
 
 
 236 
 
 .2 
 
 
 
 
 
 
 1.023 
 
 1.047 
 
 1.073 
 
 1.098 
 
 1.124 
 
 25 
 
 .3 
 
 1.150 
 
 1.177 
 
 1.204 
 
 1.232 
 
 1.260 
 
 1.288 
 
 1.317 
 
 1.346 
 
 1.376 
 
 1.406 
 
 29 
 
 .4 
 
 1.437 
 
 1.468 
 
 1.499 
 
 1.531 
 
 1.563 
 
 1.596 
 
 1.630 
 
 1.663 
 
 1.697 
 
 1.732 
 
 33 
 
 1.5 
 
 1.767 
 
 1.803 
 
 1.839 
 
 1.875 
 
 1.912 
 
 1.950 
 
 1.988 
 
 2.026 
 
 2.065 
 
 2.105 
 
 38 
 
 .6 
 
 2.145 
 
 2.185 
 
 2.226 
 
 2.268 
 
 2.310 
 
 2.352 
 
 2.395 
 
 2.439 
 
 2.483 
 
 2.527 
 
 43 
 
 .7 
 
 2.572 
 
 2.618 
 
 2.664 
 
 2.711 
 
 2.758 
 
 2.806 
 
 2.855 
 
 2.903 
 
 2.953 
 
 3.003 
 
 48 
 
 8 
 
 3.054 
 
 3.105 
 
 3.157 
 
 3.209 
 
 3.262 
 
 3.315 
 
 3.369 
 
 3.424 
 
 3.479 
 
 3.535 
 
 54 
 
 .9 
 
 3.591 
 
 3.648 
 
 3.706 
 
 3.764 
 
 3.823 
 
 3.882 
 
 3.942 
 
 4.003 
 
 4.064 
 
 4.126 
 
 60 
 
 2.0 
 
 4.189 
 
 4.252 
 
 4.316 
 
 4.380 
 
 4.445 
 
 4.511 
 
 4.577 
 
 4.644 
 
 4.712 
 
 4.780 
 
 66 
 
 .1 
 
 4.849 
 
 4.919 
 
 4.989 
 
 5.0CO 
 
 5.131 
 
 5.204 
 
 5.277 
 
 5.350 
 
 5.425 
 
 5.500 
 
 73 
 
 .2 
 
 5.575 
 
 5.652 
 
 5.729 
 
 5.806 
 
 5.885 
 
 5.964 
 
 6.044 
 
 6.125 
 
 6.206 
 
 6.288 
 
 80 
 
 .3 
 
 6.371 
 
 6.454 
 
 6.538 
 
 6.623 
 
 6.709 
 
 6.795 
 
 6.882 
 
 6.970 
 
 7.059 
 
 7.148 
 
 87 
 
 .4 
 
 7.238 
 
 7.329 
 
 7.421 
 
 7.513 
 
 7.606 
 
 7.700 
 
 7.795 
 
 7.890 
 
 7.986 
 
 8.083 
 
 94 
 
 2.5 
 
 8.181 
 
 8.280 
 
 8.379 
 
 8.479 
 
 8.580 
 
 8.682 
 
 8.785 
 
 8.888 
 
 8.992 
 
 9.097 
 
 102 
 
 .6 
 
 9.203 
 
 9.309 
 
 9.417 
 
 9.525 
 
 9.634 
 
 9.744 
 
 9.855 
 
 9.966 
 
 10.079 
 
 
 110 
 
 .6 
 
 
 
 
 
 
 
 
 
 10.08 
 
 10.19 
 
 11 
 
 .7 
 
 10.31 
 
 10.42 
 
 10.54 
 
 10.65 
 
 10.77 
 
 10.89 
 
 11.01 
 
 11.13 
 
 11.25 
 
 11.37 
 
 12 
 
 .8 
 
 11.49 x 
 
 11.62 
 
 11.74 
 
 11.87 
 
 11.99 
 
 12.12 
 
 12.25 
 
 12.38 
 
 12.51 
 
 12.64 
 
 13 
 
 .9 
 
 12.77 
 
 12.90 
 
 13.04 
 
 13.17 
 
 13.31 
 
 13.44 
 
 13.58 
 
 13.72 
 
 13.86 
 
 14.00 
 
 14 
 
 3.0 
 
 14.14 
 
 14.28 
 
 14.42 
 
 14.57 
 
 14.71 
 
 14.86 
 
 15.00 
 
 15.15 
 
 15.30 
 
 15.45 
 
 15 
 
 .1 
 
 15.60 
 
 15.75 
 
 15.90 
 
 16.06 
 
 16.21 
 
 16.37 
 
 16.52 
 
 16.68 
 
 16.84 
 
 17.00 
 
 16 
 
 .2 
 
 17.16 
 
 17.32 
 
 17.48 
 
 17.64 
 
 17.81 
 
 17.97 
 
 18.14 
 
 18.31 
 
 18.48 
 
 18.65 
 
 17 
 
 .3 
 
 18.82 
 
 18.99 
 
 19.16 
 
 19.33 
 
 19.51 
 
 19.68 
 
 19.86 
 
 20.04 
 
 20.22 
 
 20.40 
 
 18 
 
 .4 
 
 20.58 
 
 20.76 
 
 20.94 
 
 21.13 
 
 21.31 
 
 21.50 
 
 21.69 
 
 21.88 
 
 22.07 
 
 22.26 
 
 19 
 
 3.5 
 
 22.45 
 
 22.64 
 
 22.84 
 
 23.03 
 
 23.23 
 
 23.43 
 
 23.62 
 
 23.82 
 
 24.02 
 
 24.23 
 
 20 
 
 .6 
 
 24.43 
 
 24.63 
 
 24.84 
 
 25.04 
 
 25.25 
 
 25.46 
 
 25.67 
 
 25.88 
 
 26.09 
 
 26.31 
 
 21 
 
 .7 
 
 26.52 
 
 26.74 
 
 26.95 
 
 27.17 
 
 27.39 
 
 27.61 
 
 27.83 
 
 28.06 
 
 28.28 
 
 28.50 
 
 22 
 
 .8 
 
 28.73 
 
 28.% 
 
 29.19 
 
 29.42 
 
 29.65 
 
 29.88 
 
 30.11 
 
 30.35 
 
 30.58 
 
 30.82 
 
 23 
 
 .9 
 
 31.06 
 
 31.30 
 
 31.54 
 
 31.78 
 
 32.02 
 
 32.27 
 
 32.52 
 
 32.76 
 
 33.01 
 
 33.26 
 
 25 
 
 4.0 
 
 33.51 
 
 33.76 
 
 34.02 
 
 34.27 
 
 34.53 
 
 34.78 
 
 35.04 
 
 35.30 
 
 35.56 
 
 35.82 
 
 26 
 
 .1 
 
 36.09 
 
 36.35 
 
 36.62 
 
 36.88 
 
 37.15 
 
 37.42 
 
 37.69 
 
 37.97 
 
 38.24 
 
 38.52 
 
 27 
 
 .2 
 
 38.79 
 
 39.07 
 
 39.35 
 
 39.63 
 
 39.91 
 
 40.19 
 
 40.48 
 
 40.76 
 
 41.05 
 
 41.34 
 
 28 
 
 .3 
 
 41.63 
 
 41.92 
 
 42.21 
 
 42.51 
 
 42.80 
 
 43.10 
 
 43.40 
 
 43.70 
 
 44.00 
 
 44.30 
 
 30 
 
 .4 
 
 44.60 
 
 44.91 
 
 45.21 
 
 45.52 
 
 45.83 
 
 46.14 
 
 46.45 
 
 46.77 
 
 47.08 
 
 47.40 
 
 31 
 
 4.5 
 
 47.71 
 
 48.03 
 
 48.35 
 
 48.67 
 
 49.00 
 
 49.32 
 
 49.65 
 
 49.97 
 
 50.30 
 
 50.63 
 
 33 
 
 .6 
 
 50.97 
 
 51.30 
 
 51.63 
 
 51.97 
 
 52.31 
 
 52.65 
 
 52.99 
 
 53.33 
 
 53.67 
 
 54.02 
 
 34 
 
 .7 
 
 54.36 
 
 54.71 
 
 55.06 
 
 55.41 
 
 55.76 
 
 56.12 
 
 56.47 
 
 56.83 
 
 57.19 
 
 57.54 
 
 35 
 
 .8 
 
 57.91 
 
 58.27 
 
 58.63 
 
 59.00 
 
 59.37 
 
 59.73 
 
 60.10 
 
 60.48 
 
 60.85 
 
 61.22 
 
 37 
 
 .9 
 
 61.60 
 
 61.98 
 
 62.36 
 
 62.74 
 
 63.12 
 
 63.51 
 
 63.89 
 
 64.28 
 
 64.67 
 
 65.06 
 
 38 
 
 Explanation of Table of Volumes of Spheres (pp. 36-37). 
 
 Moving the decimal point one place in column D is equivalent to moving it three 
 places in the body of the table. (D = diameter.) 
 
 Volume of sphere = 7 X (diarn.) = 0.523599 X (diam.) 
 
 Conversely, 
 
 Diam. 
 
 1.240701 X 
 
MATHEMATICAL TABLES 
 
 37 
 
 VOLUMES OF SPHERES (continued) 
 
 D 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 11 
 
 5.0 
 
 65.45 
 
 65.84 
 
 66.24 
 
 66.64 
 
 67.03 
 
 67.43 
 
 67.83 
 
 68.24 
 
 68.64 
 
 69.05 
 
 40 
 
 .1 
 
 69.46 
 
 69.87 
 
 70.28 
 
 70.69 
 
 71.10 
 
 71.52 
 
 71.94 
 
 72.36 
 
 72.78 
 
 73.20 
 
 42 
 
 .2 
 
 73.62 
 
 74.05 
 
 74.47 
 
 74.90 
 
 75.33 
 
 75.77 
 
 76.20 
 
 76.64 
 
 77.07 
 
 77.51 
 
 43 
 
 .3 
 
 77.95 
 
 78.39 
 
 78.84 
 
 79.28 
 
 79.73 
 
 80.18 
 
 80.63 
 
 81.08 
 
 81.54 
 
 81.99 
 
 45 
 
 .4 
 
 82.45 
 
 82.91 
 
 83.37 
 
 83.83 
 
 84.29 
 
 84.76 
 
 85.23 
 
 85.70 
 
 86.17 
 
 86.64 
 
 47 
 
 5.5 
 
 87.11 
 
 87.59 
 
 88.07 
 
 88.55 
 
 89.03 
 
 89.51 
 
 90.00 
 
 90.48 
 
 90.97 
 
 91.46 
 
 48 
 
 .6 
 
 91.95 
 
 92.45 
 
 92.94 
 
 93.44 
 
 93.94 
 
 94.44 
 
 94.94 
 
 95.44 
 
 95.95 
 
 96.46 
 
 50 
 
 .7 
 
 96.97 
 
 97.48 
 
 97.99 
 
 98.51 
 
 99.02 
 
 99.54 
 
 10006 
 
 
 
 
 52 
 
 .7 
 
 
 
 
 
 
 
 100.1 
 
 100.6 
 
 101.1 
 
 101.6 
 
 5 
 
 .8 
 
 102.2 
 
 102.7 
 
 103.2 
 
 103.8 
 
 104.3 
 
 104.8 
 
 105.4 
 
 105.9 
 
 106.4 
 
 107.0 
 
 5 
 
 .9 
 
 107.5 
 
 108.1 
 
 108.6 
 
 109.2 
 
 109.7 
 
 110.3 
 
 110.9 
 
 111.4 
 
 112.0 
 
 112.5 
 
 6 
 
 6.0 
 
 113.1 
 
 113.7 
 
 114.2 
 
 114.8 
 
 115.4 
 
 115.9 
 
 116.5 
 
 117.1 
 
 117.7 
 
 118.3 
 
 6 
 
 .1 
 
 118.8 
 
 119.4 
 
 120.0 
 
 120.6 
 
 121.2 
 
 121.8 
 
 122.4 
 
 123.0 
 
 123.6 
 
 124.2 
 
 
 .2 
 
 124.8 
 
 125.4 
 
 126.0 
 
 126.6 
 
 127.2 
 
 127.8 
 
 128.4 
 
 129.1 
 
 129.7 
 
 130.3 
 
 
 .3 
 
 130.9 
 
 131.5 
 
 132.2 
 
 132.8 
 
 133.4 
 
 134.1 
 
 134.7 
 
 135.3 
 
 136.0 
 
 136.6 
 
 
 .4 
 
 137.3 
 
 137.9 
 
 138.5 
 
 139.2 
 
 139.8 
 
 140.5 
 
 141.2 
 
 141.8 
 
 142.5 
 
 143.1 
 
 7 
 
 6.5 
 
 143.8 
 
 144.5 
 
 145.1 
 
 145.8 
 
 146.5 
 
 147.1 
 
 147.8 
 
 148.5 
 
 149.2 
 
 149.8 
 
 
 .6 
 
 150.5 
 
 151.2 
 
 151.9 
 
 152.6 
 
 153.3 
 
 154.0 
 
 154.7 
 
 155.4 
 
 156.1 
 
 156.8 
 
 
 .7 
 
 157.5 
 
 158.2 
 
 158.9 
 
 159.6 
 
 160.3 
 
 161.0 
 
 161.7 
 
 162.5 
 
 163.2 
 
 163.9 
 
 
 .8 
 
 164.6 
 
 165.4 
 
 166.1 
 
 166.8 
 
 167.6 
 
 168.3 
 
 169.0 
 
 169.8 
 
 170.5 
 
 171.3 
 
 
 .9 
 
 172.0 
 
 172.8 
 
 173.5 
 
 174.3 
 
 175.0 
 
 175.8 
 
 176.5 
 
 177.3 
 
 178.1 
 
 178.8 
 
 8 
 
 7.0 
 
 179.6 
 
 180.4 
 
 181.1 
 
 181.9 
 
 182.7 
 
 183.5 
 
 184.3 
 
 185.0 
 
 185.8 
 
 186.6 
 
 
 .1 
 
 187.4 
 
 188.2 
 
 189.0 
 
 189.8 
 
 190.6 
 
 191.4 
 
 192.2 
 
 193.0 
 
 193.8 
 
 194.6 
 
 
 .2 
 
 195.4 
 
 196.2 
 
 197.1 
 
 197.9 
 
 198.7 
 
 199.5 
 
 200.4 
 
 201.2 
 
 202.0 
 
 202.9 
 
 
 .3 
 
 203.7 
 
 204.5 
 
 205.4 
 
 206.2 
 
 207.1 
 
 207.9 
 
 208.8 
 
 209.6 
 
 210.5 
 
 211.3 
 
 
 .4 
 
 212.2 
 
 213.0 
 
 213.9 
 
 214.8 
 
 215.6 
 
 216.5 
 
 217.4 
 
 218.3 
 
 219.1 
 
 220.0 
 
 9 
 
 7.5 
 
 220.9 
 
 221.8 
 
 222.7 
 
 223.6 
 
 224.4 
 
 225.3 
 
 226.2 
 
 227.1 
 
 228.0 
 
 228.9 
 
 
 .6 
 
 229.8 
 
 230.8 
 
 231.7 
 
 232.6 - 
 
 233.5 
 
 234.4 
 
 235.3 
 
 236.3 
 
 237.2 
 
 238.1 
 
 
 .7 
 
 239.0 
 
 240.0 
 
 240.9 
 
 241.8 
 
 242.8 
 
 243.7 
 
 244.7 
 
 245.6 
 
 246.6 
 
 247.5 
 
 
 .8 
 
 248.5 
 
 249.4 
 
 250.4 
 
 251.4 
 
 252.3 
 
 253.3 
 
 254.3 
 
 255.2 
 
 256.2 
 
 257.2 
 
 10 
 
 .9 
 
 258.2 
 
 259.1 
 
 260.1 
 
 261.1 
 
 262.1 
 
 263.1 
 
 264.1 
 
 265.1 
 
 266.1 
 
 267.1 
 
 
 8.0 
 
 268.1 
 
 269.1 
 
 270.1 
 
 271.1 
 
 272.1 
 
 273.1 
 
 274.2 
 
 275.2 
 
 276.2 
 
 277.2 
 
 
 .1 
 
 278.3 
 
 279.3 
 
 280.3 
 
 281.4 
 
 282.4 
 
 283.4 
 
 284.5 
 
 285.5 
 
 286.6 
 
 287.6 
 
 
 .2 
 
 288.7 
 
 289.8 
 
 290.8 
 
 291.9 
 
 292.9 
 
 294.0 
 
 295.1 
 
 296.2 
 
 297.2 
 
 298.3 
 
 II 
 
 3 
 
 299.4 
 
 300.5 
 
 301.6 
 
 302.6 
 
 303.7 
 
 304.8 
 
 305.9 
 
 307.0 
 
 308.1 
 
 309.2 
 
 
 .4 
 
 310.3 
 
 311.4 
 
 312.6 
 
 313.7 
 
 314.8 
 
 315.9 
 
 317.0 
 
 318.2 
 
 319.3 
 
 320.4 
 
 
 8.5 
 
 321.6 
 
 322.7 
 
 323.8 
 
 325.0 
 
 326.1 
 
 327.3 
 
 328.4 
 
 329.6 
 
 330.7 
 
 331.9 
 
 
 .6 
 
 333.0 
 
 334.2 
 
 335.4 
 
 336.5 
 
 337.7 
 
 338.9 
 
 340.1 
 
 341.2 
 
 342.4 
 
 343.6 
 
 12 
 
 .7 
 
 344.8 
 
 346.0 
 
 347.2 
 
 348.4 
 
 349.6 
 
 350.8 
 
 352.0 
 
 353.2 
 
 354.4 
 
 355.6 
 
 
 .8 
 
 356.8 
 
 358.0 
 
 359.3 
 
 360.5 
 
 361.7 
 
 362.9 
 
 364.2 
 
 365.4 
 
 366.6 
 
 367.9 
 
 
 .9 
 
 1 
 
 369.1 
 
 370.4 
 
 371.6 
 
 372.9 
 
 374.1 
 
 375.4 
 
 376.6 
 
 377.9 
 
 379.2 
 
 380.4 
 
 13 
 
 9.0 
 
 381.7 
 
 383.0 
 
 384.3 
 
 385.5 
 
 386.8 
 
 388.1 
 
 389.4 
 
 390.7 
 
 392.0 
 
 393.3 
 
 
 .1 
 
 394.6 
 
 395.9 
 
 397.2 
 
 398.5 
 
 399.8 
 
 401.1 
 
 402.4 
 
 403.7 
 
 405.1 
 
 406.4 
 
 
 2 
 
 407.7 
 
 409.1 
 
 410.4 
 
 411.7 
 
 413.1 
 
 414.4 
 
 415.7 
 
 417.1 
 
 418.4 
 
 419.8 
 
 
 .3 
 
 421.2 
 
 422.5 
 
 423.9 
 
 425.2 
 
 426.6 
 
 428.0 
 
 429.4 
 
 430.7 
 
 432.1 
 
 433.5 
 
 14 
 
 .4 
 
 434.9 
 
 436.3 
 
 437.7 
 
 439.1 
 
 440.5 
 
 441.9 
 
 443.3 
 
 444.7 
 
 446.1 
 
 447.5 
 
 
 9.5 
 
 448.9 
 
 450.3 
 
 451.8 
 
 453.2 
 
 454.6 
 
 456.0 
 
 457.5 
 
 458.9 
 
 460.4 
 
 461.8 
 
 
 .6 
 
 463.2 
 
 464.7 
 
 466.1 
 
 467.6 
 
 469.1 
 
 470.5 
 
 472.0 
 
 473.5 
 
 474.9 
 
 476.4 
 
 15 
 
 .7 
 
 477.9 
 
 479.4 
 
 480.8 
 
 482.3 
 
 483.8 
 
 485.3 
 
 486.8 
 
 488.3 
 
 489.8 
 
 491.3 
 
 
 .8 
 
 492.8 
 
 494.3 
 
 495.8 
 
 497.3 
 
 498.9 
 
 500.4 
 
 501.9 
 
 503.4 
 
 505.0 
 
 506.5 
 
 16 
 
 9 
 
 508.0 
 
 509.6 
 
 511.1 
 
 512.7 
 
 514.2 
 
 515.8 
 
 517.3 
 
 518.9 
 
 520.5 
 
 522.0 
 
 
 10.0 
 
 523.6 
 
 
 
 
 
 
 
 
 
 
 
 Moving the decimal point ONE place in D requires moving it THREE places in body of 
 table (see p. 36). 
 
38 
 
 MATHEMATICAL TABLES 
 
 SEGMENTS OF SPHERES 
 
 (h = height of segment; D = diam. of sphere) 
 
 h 
 
 Vol. segm. 
 
 to 
 
 Q 
 
 Vol. segm. 
 
 d 
 S 
 
 Explanation of Table on this page 
 Given, h = height of segment, 
 D = diam. of sphere. 
 
 D 
 
 D 
 
 Vol. sphere 
 
 0.00 
 1 
 
 0.0000 
 0002 
 
 2 
 
 0.0000 
 0003 
 
 3 
 
 2 
 
 0.0006 
 
 4 
 
 0.0012 
 
 9 
 
 To find the volume of the segment, 
 
 3 
 
 0.0014 
 
 10 
 
 0.0026 
 
 21 
 
 form the ratio h/D and find from the 
 
 4 
 
 0.0024 
 
 14 
 
 0.0047 
 
 / 1 
 26, 
 
 table the value of (vol./D 8 ); then, by 
 
 0.05 
 
 0.0038 
 
 1 f 
 
 0.0073 
 
 of 
 
 a simple multiplication, 
 
 6 
 
 g 
 
 0.0054 
 0.0073 
 0095 
 
 lo 
 19 
 22 
 
 0.0104 
 0.0140 
 0182 
 
 31 
 
 36 
 42 
 
 vol. segment = D* X (vo\./D*) 
 The table gives also the ratio of the 
 
 9 
 
 0.0120 
 
 25 
 
 *>7 
 
 0.0228 
 
 46 
 
 11 
 
 volume of the segment to the entire 
 
 0.10 
 1 
 
 0.0147 
 0176 
 
 LI 
 
 29 
 
 0.0280 
 0336 
 
 _>/ 
 56 
 
 volume of the sphere. 
 NOTE. Area of zone = v X h X D. 
 
 2 
 
 0.0208 
 
 32 
 
 A 
 
 0.0397 
 
 61 
 
 // 
 
 (Use Table of Multiples of ir, p. 28) 
 
 3 
 
 0242 
 
 34 
 
 0463 
 
 oo 
 
 
 4 
 
 0.0279 
 
 37 
 39 
 
 0.0533 
 
 70 
 
 74 
 
 Explanation of Table on p. 34 
 
 0.15 
 6 
 
 0.0318 
 0359 
 
 41 
 
 0.0607 
 0686 
 
 79 
 
 Given, h = height of segment, 
 c = chord. 
 
 7 
 
 0.0403 
 
 44 
 
 AC 
 
 0.0769 
 
 83 
 
 QA 
 
 To find the diam. of the circle, the 
 
 8 
 9 
 
 0.0448 
 0.0495 
 
 4-> 
 
 47 
 50 
 
 0.0855 
 0.0946 
 
 OO 
 
 91 
 94 
 
 length of arc, or the area of the seg- 
 ment, form the ratio h/c, and find 
 
 0.20 
 
 0.0545 
 
 C 1 
 
 0.1040 
 
 QO 
 
 from the table the value of (diam./c), 
 
 1 
 2 
 3 
 
 0.05% 
 0.0649 
 0704 
 
 >] 
 
 53 
 55 
 
 0.1138 
 0.1239 
 1344 
 
 Vo 
 
 101 
 105 
 
 (arc/c), or (area/Ac) ; then, by a simple 
 multiplication, 
 
 4 
 
 0.0760 
 
 56 
 58 
 
 0.1452 
 
 108 
 110 
 
 diam. = c X (diam./c), 
 
 
 
 
 
 
 arc = c X (arc/c), 
 
 0.25 
 6 
 
 0.0818 
 0878 
 
 60 
 
 0.1562 
 1676 
 
 114 
 
 area = h X c X (area/Ac). 
 
 7 
 
 0.0939 
 
 61 
 
 0.1793 
 
 117 
 
 i on 
 
 The table gives also the angle sub- 
 
 8 
 
 0.1002 
 
 63 
 
 64 
 
 0.1913 
 
 120 
 
 122 
 
 tended at the center, and the ratio of 
 
 9 
 
 0.1066 
 
 65 
 
 0.2035 
 
 125 
 
 h to D. See p. 106. 
 
 0.30 
 1 
 
 0.1131 
 1198 
 
 67 
 
 0.2160 
 2287 
 
 127 
 
 Explanation of Table on p. 35 
 
 2 
 
 0.1265 
 
 67 
 
 Xrt 
 
 0.2417 
 
 130 
 
 Given, h = height of segment, 
 
 3 
 
 0.1334 
 
 ov 
 70 
 
 0.2548 
 
 134 
 
 D diam. of circle. 
 
 4 
 
 0.1404 
 
 /u 
 71 
 
 0.2682 
 
 135 
 
 To find the chord, the length of arc, 
 
 0.35 
 
 0.1475 
 
 ni 
 
 0.2817 
 
 1 Ifl 
 
 or the area of the segment, form the 
 
 6 
 7 
 8 
 
 0.1547 
 0.1620 
 0.1694 
 
 n 
 
 73 
 74 
 
 0.2955 
 0.3094 
 0.3235 
 
 1 JO 
 
 139 
 141 
 
 ratio h/D, and find from the table the 
 value of (chord/D), (arc/Z>), or 
 
 9 
 
 0.1768 
 
 74 
 75 
 
 0.3377 
 
 142 
 143 
 
 (area//) 2 ); then, by a simple multi- 
 
 
 
 MJ 
 
 
 
 plication, 
 
 0.40 
 
 0.1843 
 0.1919 
 
 76 
 
 0.3520 
 0.3665 
 
 145 
 
 chord = D X (chord/ D), 
 
 2 
 
 0.1995 
 
 76 
 
 77 
 
 0.3810 
 
 145 
 
 147 
 
 arc - D X (arc//)), 
 
 3 
 4 
 
 0.2072 
 0.2149 
 
 II 
 
 77 
 78 
 
 0.3957 
 0.4104 
 
 It/ 
 
 147 
 148 
 
 area = D* X (area/Z)"). 
 The table gives also the angle sub- 
 
 0.45 
 
 0.2227 
 
 70 
 
 0.4252 
 
 1 AQ 
 
 tended at the center, the ratio of the 
 
 6 
 7 
 8 
 
 0.2305 
 0.2383 
 2461 
 
 7o 
 78 
 78 
 
 0.4401 
 0.4551 
 4700 
 
 1^7 
 
 150 
 149 
 
 arc of the segment to the whole cir- 
 cumference, and the ratio of the area 
 
 9 
 
 0.2539 
 
 78 
 79 
 
 . 0.4850 
 
 150 
 150 
 
 of the segment to the area of the 
 
 
 
 
 
 
 whole circle. See p. 106. 
 
 0.50 
 
 0.2618 
 
 
 0.5000 
 
 
 
 NOTE. Vol. segm. - }6 * h* (3D-2h). 
 
MATHEMATICAL TABLES 
 
 39 
 
 REGULAR POLYGONS 
 
 n = number of sides; 
 
 TO = 360/n = angle subtended at the center by one side; 
 
 a = length of one side = R (2 sin |-) = r (2 tan |-) ; 
 R = radius of circumscribed circle = a ( y^ esc \ = r (sec -^-j ; 
 r = radius of inscribed circle = R(COS J = a(l$ cot s~J ; 
 Area = o*H n cot - = fl'/i n sin t> = r*n tan -. 
 
 
 
 Area 
 
 Area 
 
 Area 
 
 * 
 
 R 
 
 a 
 
 a 
 
 r 
 
 r 
 
 n 
 
 7) 
 
 a2 
 
 # 
 
 r2 
 
 
 
 r 
 
 R 
 
 r 
 
 R 
 
 a 
 
 3 
 
 120 
 
 0.4330 
 
 1.299 
 
 5:196 
 
 0.5774 
 
 2.000 
 
 1.732 
 
 3.464 
 
 0.5000 
 
 0.2887 
 
 4 
 
 90 
 
 1.000 
 
 2.000 
 
 4.000 
 
 0.7071 
 
 .414 
 
 1.414 
 
 2.000 
 
 0.7071 
 
 0.5000 
 
 5 
 
 72 
 
 1.721 
 
 2.378 
 
 3.633 
 
 0.8507 
 
 .236 
 
 1.176 
 
 1.453 
 
 0.8090 
 
 0.6882 
 
 6 . 
 
 60 
 
 2.598 
 
 2.598 
 
 3.464 
 
 1.0000 
 
 .155 
 
 1.000 
 
 1.155 
 
 0.8660 
 
 0.8660 
 
 7 
 
 5P.43 
 
 3.634 
 
 2.736 
 
 3.371 
 
 1.152 
 
 .110 
 
 0.8678 
 
 0.9631 
 
 0.9010 
 
 1.038 
 
 8 
 
 45 
 
 4.828 
 
 2.828 
 
 3.314 
 
 1.307 
 
 .082 
 
 0.7654 
 
 0.8284 
 
 0.9239 
 
 1.207 
 
 9 
 
 40 
 
 6.182 
 
 2.893 
 
 3.276 
 
 1.462 
 
 .064 
 
 0.6840 
 
 0.7279 
 
 0.9397 
 
 1.374 
 
 10 
 
 36 
 
 7.694 
 
 2.939 
 
 3.249 
 
 1.618 
 
 .052 
 
 0.6180 
 
 0.6498 
 
 0.9511 
 
 1.539 
 
 12 
 
 30 
 
 11.20 
 
 3.000 
 
 3.215 
 
 1.932 
 
 .035 
 
 0.5176 
 
 0.5359 
 
 0.9659 
 
 1.866 
 
 15 
 
 24 
 
 17.64 
 
 3.051 
 
 3.188 
 
 2.405 
 
 .022 
 
 0.4158 
 
 0.4251 
 
 0.9781 
 
 2.352 
 
 16 
 
 22. 50 
 
 20.11 
 
 3.062 
 
 3.183 
 
 2.563 
 
 .020 
 
 0.3902 
 
 0.3978 
 
 0.9808 
 
 2.514 
 
 20 
 
 18 
 
 31.57 
 
 3.090 
 
 3.168 
 
 3.1% 
 
 .013 
 
 0.3129 
 
 0.3168 
 
 0.9877 
 
 3.157 
 
 24 
 
 15 
 
 45.58 
 
 3.106 
 
 3.160 
 
 3.831 
 
 .009 
 
 0.2611 
 
 0.2633 
 
 0.9914 
 
 3.798 
 
 32 
 
 11.25 
 
 81.23 
 
 3.121 
 
 3.152 
 
 5.101 
 
 .005 
 
 0.1960 
 
 0.1970 
 
 0.9952 
 
 5.077 
 
 48 
 
 7. 50 
 
 183.1 
 
 3.133 
 
 3.146 
 
 7.645 
 
 .002 
 
 0.1308 
 
 0.1311 
 
 0.9979 
 
 7.629 
 
 64 
 
 5.625 
 
 325.7 
 
 3.137 
 
 3.144 
 
 10.19 
 
 .001 
 
 0.0981 
 
 0.0983 
 
 0.9988 
 
 10.18 
 
 BINOMIAL COEFFICIENTS 
 
 (For table giving binomial coefficients for fractional values of n, see p. 116). 
 n(n - 1) n(n - l)(n - 2) 
 
 (n)o = 1; (n)i = n; (71)2 = 
 
 (n) 
 
 1X2' 
 
 n(n - l)(n - 2) . . . (n - [r - 1]). 
 1X2X3. . . X r 
 
 1X2X3 
 Another notation: 
 
 = (n) r . 
 
 n 
 
 (n)o 
 
 ()i 
 
 (n)i 
 
 (n)i 
 
 (n)* 
 
 (n). 
 
 (n)a 
 
 ()T 
 
 (n)a 
 
 (n)a 
 
 (n)w 
 
 (n)u 
 
 (n)is 
 
 (n)i3 
 
 I 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ? 
 
 
 2 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 
 3 
 
 3 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 
 4 
 
 6 
 
 4 
 
 1 
 
 
 
 
 
 
 
 
 
 
 5 
 
 
 5 
 
 10 
 
 10 
 
 5 
 
 1 
 
 
 
 
 
 
 
 
 
 6 
 
 
 6 
 
 15 
 
 20 
 
 15 
 
 6 
 
 1 
 
 
 
 
 
 
 
 
 7 
 
 
 7 
 
 21 
 
 35 
 
 35 
 
 71 
 
 7 
 
 1 
 
 
 
 
 
 
 
 8 
 
 
 8 
 
 28 
 
 56 
 
 70 
 
 56 
 
 28 
 
 8 
 
 1 
 
 
 
 
 
 
 Q 
 
 
 9 
 
 36 
 
 84 
 
 126 
 
 126 
 
 84 
 
 36 
 
 9 
 
 1 
 
 
 
 
 
 in 
 
 
 10 
 
 45 
 
 120 
 
 210 
 
 252 
 
 210 
 
 120 
 
 45 
 
 10 
 
 | 
 
 
 
 
 11 
 
 
 11 
 
 51 
 
 165 
 
 310 
 
 462 
 
 462 
 
 330 
 
 165 
 
 55 
 
 11 
 
 1 
 
 
 
 12 
 13 
 14 
 15 
 
 
 12 
 13 
 14 
 15 
 
 66 
 78 
 91 
 105 
 
 220 
 286 
 364 
 455 
 
 495 
 715 
 1001 
 1365 
 
 792 
 1287 
 2002 
 3003 
 
 924 
 1716 
 3003 
 5005 
 
 792 
 1716 
 3432 
 6435 
 
 495 
 1287 
 3003 
 6435 
 
 220 
 715 
 2002 
 5005 
 
 66 
 286 
 1001 
 3003 
 
 12 
 78 
 364 
 1365 
 
 1 
 
 13 
 91 
 455 
 
 ..... 
 
 14 
 105 
 
 For n = 14, (n)u = 1; for n - 15, (n)u = 15, and (n) l6 - 1. 
 
40 MATHEMATICAL TABLES 
 
 COMMON LOGARITHMS (special table) 
 
 1* 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 $? 
 
 JTJ 
 
 1.00 
 
 0.0000 
 
 0004 
 
 0009 
 
 0013 
 
 0017 
 
 0022 
 
 0026 
 
 0030 
 
 0035 
 
 0039 
 
 4 
 
 1.01 
 
 0043 
 
 0048 
 
 0052 
 
 0056 
 
 0060 
 
 0065 
 
 0069 
 
 0073 
 
 0077 
 
 0082 
 
 
 1.02 
 
 0086 
 
 0090 
 
 0095 
 
 0099 
 
 0103 
 
 0107 
 
 0111 
 
 0116 
 
 0120 
 
 0124 
 
 
 1.03 
 
 0128 
 
 0133 
 
 0137 
 
 0141 
 
 0145 
 
 0149 
 
 0154 
 
 0158 
 
 0162 
 
 0166 
 
 
 1.04 
 
 0170 
 
 0175 
 
 0179 
 
 0183 
 
 0187 
 
 0191 
 
 0195 
 
 0199 
 
 0204 
 
 0208 
 
 
 1.05 
 
 0212 
 
 0216 
 
 0220 
 
 0224 
 
 0228 
 
 0233 
 
 0237 
 
 0241 
 
 0245 
 
 0249 
 
 
 1.06 
 
 0253 
 
 0257 
 
 0261 
 
 0265 
 
 0269 
 
 0273 
 
 0278 
 
 0282 
 
 0286 
 
 0290 
 
 
 1.07 
 
 0294 
 
 0298 
 
 0302 
 
 0306 
 
 0310 
 
 0314 
 
 0318 
 
 0322 
 
 0326 
 
 0330 
 
 
 .08 
 
 0334 
 
 0338 
 
 0342 
 
 0346 
 
 0350 
 
 0354 
 
 0358 
 
 0362 
 
 0366 
 
 0370 
 
 
 1.09 
 
 0374 
 
 0378 
 
 0382 
 
 0386 
 
 0390 
 
 0394 
 
 0398 
 
 0402 
 
 04C6 
 
 0410 
 
 
 .10 
 
 0.0414 
 
 0418 
 
 0422 
 
 0426 
 
 0430 
 
 0434 
 
 0438 
 
 0441 
 
 0445 
 
 0449 
 
 
 .11 
 
 0453 
 
 0457 
 
 0461 
 
 0465 
 
 0469 
 
 0473 . 
 
 0477 
 
 0481 
 
 0484 
 
 0488 
 
 
 .12 
 
 0492 
 
 0496 
 
 0500 
 
 0504 
 
 0508 
 
 0512 
 
 0515 
 
 0519 
 
 0523 
 
 0527 
 
 
 .13 
 
 0531 
 
 0535 
 
 0538 
 
 0542 
 
 0546 
 
 0550 
 
 0554 
 
 0558 
 
 0561 
 
 0565 
 
 
 .14 
 
 0569 
 
 0573 
 
 0577 
 
 0580 
 
 0584 
 
 0588 
 
 0592 
 
 0596 
 
 0599 
 
 0603 
 
 
 .15 
 
 0607 
 
 0611 
 
 0615 
 
 0618 
 
 0622 
 
 0626 
 
 0630 
 
 0633 
 
 0637 
 
 0641 
 
 
 .16 
 
 0645 
 
 0648 
 
 0652 
 
 0656 
 
 0660 
 
 0663 
 
 0667 
 
 0671 
 
 0674 
 
 0678 
 
 
 .17 
 
 0682 
 
 0686 
 
 0689 
 
 0693 
 
 0697 
 
 0700 
 
 0704 
 
 0708 
 
 0711 
 
 0715 
 
 
 .18 
 
 0719 
 
 0722 
 
 0726 
 
 0730 
 
 0734 
 
 0737 
 
 0741 
 
 0745 
 
 0748 
 
 0752 
 
 
 .19 
 
 0755 
 
 0759 
 
 0763 
 
 0766 
 
 0770 
 
 0774 
 
 0777 
 
 0781 
 
 0785 
 
 0788 
 
 x . 
 
 1.20 
 
 0.0792 
 
 0795 
 
 0799 
 
 0303 
 
 0806 
 
 0810 
 
 0813 
 
 0817 
 
 0821 
 
 0824 
 
 
 1.21 
 
 0828 
 
 0831 
 
 0835 
 
 0839 
 
 0842 
 
 0846 
 
 0849 
 
 0853 
 
 0856 
 
 0860 
 
 
 1.22 
 
 0864 
 
 0867 
 
 0871 
 
 0374 
 
 0878 
 
 0881 
 
 0885 
 
 0888 
 
 0892 
 
 0896 
 
 
 1.23 
 
 0899 
 
 0903 
 
 0906 
 
 0910 
 
 0913 
 
 0917 
 
 0920 
 
 0924 
 
 0927 
 
 0931 
 
 
 1.24 
 
 0934 
 
 0938 
 
 0941 
 
 0945 
 
 0948 
 
 0952 
 
 0955 
 
 0959 
 
 0962 
 
 0966 
 
 
 1.25 
 
 0969 
 
 0973 
 
 0976 
 
 0980 
 
 0983 
 
 0986 
 
 0990 
 
 0993 
 
 0997 
 
 1000 
 
 3 
 
 1.26 
 
 1004 
 
 1007 
 
 1011 
 
 1014 
 
 1017 
 
 1021 
 
 1024 
 
 1028 
 
 1031 
 
 1035 
 
 
 1.27 
 
 1038 
 
 1041 
 
 1045 
 
 1048 
 
 1052 
 
 1055 
 
 1059 
 
 1062 
 
 1065 
 
 1069 
 
 
 1.28 
 
 1072 
 
 1075 
 
 1079 
 
 1082 
 
 1086 
 
 1089 
 
 1092 
 
 1096 
 
 1099 
 
 1103 
 
 
 1.29 
 
 1106 
 
 1109 
 
 1113 
 
 1116 
 
 1119 
 
 1.123 
 
 1126 
 
 1129 
 
 1133 
 
 1136 
 
 
 1.30 
 
 0.1139 
 
 1143 
 
 1146 
 
 1149 
 
 1153 
 
 1156 
 
 1159 
 
 1163 
 
 1166 
 
 1169 
 
 
 1.31 
 
 1173 
 
 1176 
 
 1179 
 
 1183 
 
 1186 
 
 1189 
 
 1193 
 
 1196 
 
 1199 
 
 1202 
 
 
 1.32 
 
 1206 
 
 1209 
 
 1212 
 
 1216 
 
 1219 
 
 1222 
 
 1225 
 
 1229 
 
 1232 
 
 1235 
 
 
 1.33 
 
 1239 
 
 1242 
 
 1245 
 
 1248 
 
 1252 
 
 1255 
 
 1258 
 
 1261 
 
 1265 
 
 1268 
 
 
 1.34 
 
 1271 
 
 1274 
 
 1278 
 
 1281 
 
 1284 
 
 1287 
 
 1290 
 
 1294 
 
 1297 
 
 1300 
 
 
 1.35 
 
 1303 
 
 1307 
 
 1310 
 
 1313 
 
 1316 
 
 1319 
 
 1323 
 
 1326 
 
 1329 
 
 1332 
 
 
 1.36 
 
 1335 
 
 1339 
 
 1342 
 
 1345 
 
 1348 
 
 1351 
 
 1355 
 
 1358 
 
 1361 
 
 1364 
 
 
 1.37 
 
 1367 
 
 1370 
 
 1374 
 
 1377 
 
 1380 
 
 1383 
 
 1386 
 
 1389 
 
 1392 
 
 1396 
 
 
 1.38 
 
 1399 
 
 1402 
 
 1405 
 
 1408 
 
 1411 
 
 1414 
 
 1418 
 
 1421 
 
 1424 
 
 1427 
 
 
 1.39 
 
 1430 
 
 1433 
 
 1436 
 
 1440 
 
 1443 
 
 1446 
 
 1449 
 
 1452 
 
 1455 
 
 1458 
 
 
 1.40 
 
 0.1461 
 
 1464 
 
 1467 
 
 1471 
 
 1474 
 
 1477 
 
 1480 
 
 1483 
 
 1486 
 
 1489 
 
 
 1.41 
 
 1492 
 
 1495 
 
 1498 
 
 1501 
 
 1504 
 
 1508 
 
 1511 
 
 1514 
 
 1517 
 
 1520 
 
 
 1.42 
 
 1523 
 
 1526 
 
 1529 
 
 1532 
 
 1535 
 
 1538 
 
 1541 
 
 1544 
 
 1547 
 
 1550 
 
 
 1.43 
 
 1553 
 
 1556 
 
 1559 
 
 1562 
 
 1565 
 
 1569 
 
 1572 
 
 1575 
 
 1578 
 
 1581 
 
 
 1.44 
 
 1584 
 
 1587 
 
 1590 
 
 1593 
 
 1596 
 
 1599 
 
 1602 
 
 1605 
 
 1608 
 
 1611 
 
 
 1.45 
 
 1614 
 
 1617 
 
 1620 
 
 1623 
 
 1626 
 
 1629 
 
 1632 
 
 1635 
 
 1638 
 
 1641 
 
 
 1.46 
 
 1644 
 
 1647 
 
 1649 
 
 1652 
 
 1655 
 
 1658 
 
 1661 
 
 1664 
 
 1667 
 
 1670 
 
 
 1.47 
 
 1673 
 
 1676 
 
 1679 
 
 1682 
 
 1685 
 
 1688 
 
 1691 
 
 1694 
 
 1697 
 
 1700 
 
 
 1.48 
 
 1703 
 
 1706 
 
 1708 
 
 1711 
 
 1714 
 
 1717 
 
 1720 
 
 1723 
 
 1726 
 
 1729 
 
 
 1.49 
 
 1732 
 
 1735 
 
 1738 
 
 1741 
 
 1744 
 
 1746 
 
 1749 
 
 1752 
 
 1755 
 
 1758 
 
 
 Moving the decimal point n places to the right [or left] in the number requires adding + n 
 [or - n] ia the body of the table (see p. 42). 
 
MATHEMATICAL TABLES 
 
 41 
 
 COMMON LOGARITHMS (special table, continued) 
 
 p 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 11 
 
 1.50 
 
 0.1761 
 
 1764 
 
 1767 
 
 1770 
 
 1772 
 
 1775 . 
 
 1778 
 
 1781 
 
 1784 
 
 1787 
 
 3 
 
 1.51 
 
 1790 
 
 1793 
 
 1796 
 
 1798 
 
 1801 
 
 1804 
 
 1807 
 
 1810 
 
 1813 
 
 1816 
 
 
 1.52 
 
 1818 
 
 1821 
 
 1824 
 
 1827 
 
 1830 
 
 1833 
 
 1836 
 
 1838 
 
 1841 
 
 1844 
 
 
 1.53 
 
 1847 
 
 1850 
 
 1853 
 
 1855 
 
 1858 
 
 1861 
 
 1864 
 
 1867 
 
 1870 
 
 1872 
 
 
 1.54 
 
 1875 
 
 1878 
 
 1881 
 
 1884 
 
 1886 
 
 1889 
 
 1892 
 
 1895 
 
 1898 
 
 1901 
 
 
 1.55 
 
 1903 
 
 1906 
 
 1909 
 
 1912 
 
 19t5 
 
 1917 
 
 1920 
 
 1923 
 
 1926 
 
 1928 
 
 
 1.56 
 
 1931 
 
 1934 
 
 1937 
 
 1940 
 
 1942 
 
 1945 
 
 1948 
 
 1951 
 
 1953 
 
 1956 
 
 
 1.57 
 
 1959 
 
 1962 
 
 1965 
 
 1967 
 
 1970 
 
 1973 
 
 1976 
 
 1978 
 
 1981 
 
 1984 
 
 
 1.58 
 
 1987 
 
 1989 
 
 1992 
 
 1995 
 
 1998 
 
 2000 
 
 2003 
 
 2006 
 
 2009 
 
 2011 
 
 
 1.59 
 
 2014 
 
 2017 
 
 2019 
 
 2022 
 
 2025 
 
 2028 
 
 2030 
 
 2033 
 
 2036 
 
 2038 
 
 
 1.60 
 
 0.2041 
 
 2044 
 
 2047 
 
 2049 
 
 2052 
 
 2055 
 
 2057 
 
 2060 
 
 2063 
 
 2066 
 
 
 1.61 
 
 2068 
 
 2071 
 
 2074 
 
 2076 
 
 2079 
 
 2082 
 
 2084 
 
 2087 
 
 2090 
 
 2092 
 
 
 1.62 
 
 2095 
 
 2098 
 
 2101 
 
 2103 
 
 2106 
 
 2109 
 
 2111 
 
 2114 
 
 2117 
 
 2119 
 
 
 1.63 
 
 2122 
 
 2125 
 
 2127 
 
 2130 
 
 2133 
 
 2135 
 
 2138 
 
 2140 
 
 2143 
 
 2146 
 
 
 1.64 
 
 2148 
 
 2151 
 
 2154 
 
 2156 
 
 2159 
 
 2162 
 
 2164 
 
 2167 
 
 2170 
 
 2172 
 
 
 1.65 
 
 2175 
 
 2177 
 
 2180 
 
 2183 
 
 2185 
 
 2188 
 
 2191 
 
 2193 
 
 2196 
 
 2198 
 
 
 1.66 
 
 2201 
 
 2204 
 
 2206 
 
 2209 
 
 2212 
 
 2214 
 
 2217 
 
 2219 
 
 2222 
 
 2225 
 
 
 1.67 
 
 2227 
 
 2230 
 
 2232 
 
 2235 
 
 2238 
 
 2240 
 
 2243 
 
 2245 
 
 2248 
 
 2251 
 
 
 1.68 
 
 2253 
 
 2256 
 
 2258 
 
 2261 
 
 2263 
 
 2266 
 
 2269 
 
 2271 
 
 2274 
 
 2276 
 
 
 1.69 
 
 2279 
 
 2281 
 
 2284 
 
 2287 
 
 2289 
 
 2292 
 
 2294 
 
 2297 
 
 2299 
 
 2302 
 
 
 1.70 
 
 0.2304 
 
 2307 
 
 2310 
 
 2312 
 
 2315 
 
 2317 
 
 2320 
 
 2322 
 
 2325 
 
 2327 
 
 
 1.71 
 
 2330 
 
 2333 
 
 2335 
 
 2338 
 
 2340 
 
 2343 
 
 2345 
 
 2348 
 
 2350 
 
 2353 
 
 
 1.72 
 
 2355 
 
 2358 
 
 2360 
 
 2363 
 
 2365 
 
 2368 
 
 2370 
 
 2373 
 
 2375 
 
 2378 
 
 
 1.73 
 
 2380 
 
 2383 
 
 2385 
 
 2388 
 
 2390 
 
 2393 
 
 2395 
 
 2398 
 
 2400 
 
 2403 
 
 
 1.74 
 
 2405 
 
 2408 
 
 2410 
 
 2413 
 
 2415 
 
 2418 
 
 2420 
 
 2423 
 
 2425 
 
 2428 
 
 2 
 
 1.75 
 
 2430 
 
 2433 
 
 2435 
 
 2438 
 
 2440 
 
 2443 
 
 2445 
 
 2448 
 
 2450 
 
 2453 
 
 
 1.76 
 
 2455 
 
 2458 
 
 2460 
 
 2463 
 
 2465 
 
 2467 
 
 2470 
 
 2472 
 
 2475 
 
 2477 
 
 
 1.77 
 
 2480 
 
 2482 
 
 2485 
 
 2487 
 
 2490 
 
 2492 
 
 2494 
 
 2497 
 
 2499 
 
 2502 
 
 
 1.78- 
 
 2504 
 
 2507 
 
 2509 
 
 2512 
 
 2514 
 
 2516 
 
 2519 
 
 2521 
 
 2524 
 
 2526 
 
 
 1.79 
 
 2529 
 
 2531 
 
 2533 
 
 2536 
 
 2538 
 
 2541 
 
 2543 
 
 2545 
 
 2548 
 
 2550 
 
 
 180 
 
 0.2553 
 
 2555 
 
 2558 
 
 2560 
 
 2562 
 
 2565 
 
 2567 
 
 2570 
 
 2572 
 
 2574 
 
 
 1.81 
 
 2577 
 
 2579 
 
 2582 
 
 2584 
 
 2586 
 
 2589 
 
 2591 
 
 2594 
 
 2596 
 
 2598 
 
 
 1.82 
 
 2601 
 
 2603 
 
 2605 
 
 2608 
 
 2610 
 
 2613 
 
 2615 
 
 2617 
 
 2620 
 
 2622 
 
 
 1.83 
 
 2625 
 
 2627 
 
 2629 
 
 2632 
 
 2634 
 
 2636 
 
 2639 
 
 2641 
 
 2643 
 
 2646 
 
 
 1.84 
 
 2648 
 
 2651 
 
 2653 
 
 2655 
 
 2658 
 
 2660 
 
 2662 
 
 2665 
 
 2667 
 
 2669 
 
 
 1.85 
 
 2672 
 
 2674 
 
 2676 
 
 2679 
 
 2681 
 
 2683 
 
 2686 
 
 2688 
 
 2690 
 
 2693 
 
 
 1.86 
 
 2695 
 
 2697 
 
 2700 
 
 2702 
 
 2704 
 
 2707 
 
 2709 
 
 2711 
 
 2714 
 
 2716 
 
 
 1.87 
 
 2718 
 
 2721 
 
 2723 
 
 2725 
 
 2728 
 
 2730 
 
 2732 
 
 2735 
 
 2737 
 
 2739 
 
 
 1.88 
 
 2742 
 
 2744 
 
 2746 
 
 2749 
 
 2751 
 
 2753 
 
 2755 
 
 2758 
 
 2760 
 
 2762 
 
 
 1.89 
 
 2765 
 
 2767 
 
 2769 
 
 2772 
 
 2774 
 
 2776 
 
 2778 
 
 2781 
 
 2783 
 
 2785 
 
 
 1.90 
 
 0.2788 
 
 2790 
 
 2792 
 
 2794 
 
 2797 
 
 2799 
 
 2801 
 
 2804 
 
 2806 
 
 2808 
 
 
 1.91 
 
 2810 
 
 2813 
 
 2815 
 
 2817 
 
 2819 
 
 2822 
 
 2824 ' 
 
 2826 
 
 2828 
 
 2831 
 
 
 1.92 
 
 2833 
 
 2835 
 
 2838 
 
 2840 
 
 2842 
 
 2844 
 
 2847 
 
 2849 
 
 2851 
 
 2853 
 
 
 1.93 
 
 2856 
 
 2858 
 
 2860 
 
 2862 
 
 2865 
 
 2867 
 
 2869 
 
 2871 
 
 2874 
 
 2876 
 
 
 1.94 
 
 2878 
 
 2880 
 
 2882 
 
 2885 
 
 2887 
 
 2889 
 
 2891 
 
 2894 
 
 2896 
 
 2898 
 
 
 1.95 
 
 2900 
 
 2903 
 
 2905 
 
 2907 
 
 2909 
 
 2911 
 
 2914 
 
 2916 
 
 2918 
 
 2920 
 
 
 1.96 
 
 2923 
 
 2925 
 
 2927 
 
 2929 
 
 2931 
 
 2934 
 
 2936 
 
 2938 
 
 2940 
 
 2942 
 
 
 1.97 
 
 2945 
 
 2947 
 
 2949 
 
 2951 
 
 2953 
 
 2956 
 
 2958 
 
 2960 
 
 2962 
 
 2964 
 
 
 1.98 
 
 2967 
 
 2969 
 
 2971 
 
 2973 
 
 2975 
 
 2978 
 
 2980 
 
 2982 
 
 2984 
 
 2986 
 
 
 1.99 
 
 2989 
 
 2991 
 
 2993 
 
 2995 
 
 2997 
 
 2999 
 
 3002 
 
 3004 
 
 3006 
 
 3008 
 
 
42 
 
 MATHEMATICAL TABLES 
 
 COMMON LOGARITHMS 
 
 11 
 
 
 
 1 
 
 2 
 
 a 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Sa 
 
 <3-0 
 
 1.0 
 
 0.0000 
 
 0043 
 
 0086 
 
 0128 
 
 0170 
 
 0212 
 
 0253 
 
 0294 
 
 0334 
 
 0374 
 
 
 1.1 
 
 0414 
 
 0453 
 
 0492 
 
 0531 
 
 0569 
 
 0607 
 
 0645 
 
 0682 
 
 0719 
 
 0755 
 
 
 1.2 
 
 0792 
 
 0828 
 
 0864 
 
 0899 
 
 0934 
 
 0969 
 
 1004 
 
 1038 
 
 1072 
 
 1106 
 
 TH 
 
 1.3 
 
 1139 
 
 1173 
 
 1206 
 
 1239 
 
 1271 
 
 1303 
 
 1335 
 
 1367 
 
 1399 
 
 1430 
 
 1 
 
 1.4 
 
 1461 
 
 1492 
 
 1523 
 
 1553 
 
 1584 
 
 1614 
 
 1644 
 
 1673 
 
 1703 
 
 1732 
 
 6 
 
 ^ 
 
 13 
 
 1761 
 
 1790 
 
 1818 
 
 1847 
 
 1875 
 
 1903 
 
 1931 
 
 1959 
 
 1987 
 
 2014 
 
 1 
 
 1.6 
 
 2041 
 
 2068 
 
 2095 
 
 2122 
 
 2148 
 
 2175 
 
 2201 
 
 2227 
 
 2253 
 
 2279 
 
 03 
 
 1.7 
 
 2304 
 
 2330 
 
 2355 
 
 2380 
 
 2405 
 
 2430 
 
 2455 
 
 2480 
 
 2504 
 
 2529 
 
 ft 
 
 1.8 
 
 2553 
 
 2577 
 
 2601 
 
 2625 
 
 2648 
 
 2672 
 
 2695 
 
 2718 
 
 2742 
 
 2765 
 
 1 
 
 1.9 
 
 2788 
 
 2810 
 
 2833 
 
 2856 
 
 2878 
 
 2900 
 
 2923 
 
 2945 
 
 2967 
 
 2989 
 
 CQ 
 
 2.0 
 
 0.3010 
 
 3032 
 
 3054 
 
 3075 
 
 3096 
 
 3118 
 
 3139 
 
 3160 
 
 3181 
 
 3201 
 
 21 
 
 2.1 
 
 3222 
 
 3243 
 
 3263 
 
 3284 
 
 3304 
 
 3324 
 
 3345 
 
 3365 
 
 3385 
 
 3404 
 
 20 
 
 2.2 
 
 3424 
 
 3444 
 
 3464 
 
 3483 
 
 3502 
 
 3522 
 
 3541 
 
 3560 
 
 3579 
 
 3598 
 
 19 
 
 2.3 
 
 3617 
 
 3636 
 
 3655 
 
 3674 
 
 3692 
 
 3711 
 
 3729 
 
 3747 
 
 3766 
 
 3784 
 
 18 
 
 2.4 
 
 3802 
 
 3820 
 
 3838 
 
 3856 
 
 3874 
 
 3892 
 
 3909 
 
 3927 
 
 3945 
 
 3962 
 
 17 
 
 2.5 
 
 3979 
 
 3997 
 
 4014 
 
 4031 
 
 4048 
 
 4065 
 
 4082 
 
 4099 
 
 4116 
 
 4133 
 
 17 
 
 2.6 
 
 4150 
 
 4166 
 
 4183 
 
 4200 
 
 4216 
 
 4232 
 
 4249 
 
 4265 
 
 4281 
 
 4298 
 
 16 
 
 2.7 
 
 4314 
 
 4330 
 
 4346 
 
 4362 
 
 4378 
 
 4393 
 
 4409 
 
 4425 
 
 4440 
 
 4456 
 
 16 
 
 2.8 
 
 4472 
 
 4487 
 
 4502 
 
 4518 
 
 4533 
 
 4548 
 
 4564 
 
 4579 
 
 4594 
 
 4609 
 
 15 
 
 2.9 
 
 4624 
 
 4639 
 
 4654 
 
 4669 
 
 4683 
 
 4698 
 
 4713 
 
 4728 
 
 4742 
 
 4757 
 
 15 
 
 3.0 
 
 0.4771 
 
 4786 
 
 4800 
 
 4814 
 
 4829 
 
 4843 
 
 4857 
 
 4871 
 
 4886 
 
 4900 
 
 14 
 
 3.1 
 
 4914 
 
 4928 
 
 4942 
 
 4955 
 
 4969 
 
 4983 
 
 4997 
 
 5011 
 
 5024 
 
 5038 
 
 14 
 
 3.2 
 
 5051 
 
 5065 
 
 5079 
 
 5092 
 
 5105 
 
 5119 
 
 5132 
 
 5145 
 
 5159 
 
 5172 
 
 13 
 
 3.3 
 
 5185 
 
 5198 
 
 5211 
 
 5224 
 
 5237 
 
 5250 
 
 5263 
 
 5276 
 
 5289 
 
 5302 
 
 13 
 
 3.4 
 
 5315 
 
 5328 
 
 5340 
 
 5353 
 
 5366 
 
 5378 
 
 5391 
 
 5403 
 
 5416 
 
 5428 
 
 13 
 
 3.5 
 
 5441 
 
 5453- 
 
 5465 
 
 5478 
 
 5490 
 
 5502 
 
 5514 
 
 5527 
 
 5539 
 
 5551 
 
 12 
 
 3.6 
 
 5563 
 
 5575 
 
 5587 
 
 5599 
 
 5611 
 
 5623 
 
 5635 
 
 5647 
 
 5658 
 
 5670 
 
 12 
 
 3.7 
 
 5682 
 
 5694 
 
 5705 
 
 5717 
 
 5729 
 
 5740 
 
 5752 
 
 5763 
 
 5775 
 
 5786 
 
 12 
 
 3.8 
 
 5798 
 
 5809 
 
 5821 
 
 5832 
 
 5843 
 
 5855 
 
 5866 
 
 5877 
 
 5888 
 
 5899 
 
 11 
 
 3.9 
 
 5911 
 
 5922 
 
 5933 
 
 5944 
 
 5955 
 
 5966 
 
 5977 
 
 5988 
 
 5999 
 
 6010 
 
 11 
 
 4.0 
 
 0.6021 
 
 6031 
 
 6042 
 
 6053 
 
 6064 
 
 6075 
 
 6085 
 
 6096 
 
 6107 
 
 6117 
 
 11 
 
 4.1 
 
 6128 
 
 6138 
 
 6149 
 
 6160 
 
 6170 
 
 6180 
 
 6191 
 
 6201 
 
 6212 
 
 6222 
 
 10 
 
 4.2 
 
 6232 
 
 6243 
 
 6253 
 
 6263 
 
 6274 
 
 6284 
 
 6294 
 
 6304 
 
 6314 
 
 6325 
 
 10 
 
 4.3 
 
 6335 
 
 6345 
 
 6355 
 
 6365 
 
 6375 
 
 6385 
 
 6395 
 
 6405 
 
 6415 
 
 6425 
 
 10 
 
 4.4 
 
 6435 
 
 6444 
 
 6454 
 
 6464 
 
 6474 
 
 6484 
 
 6493 
 
 6503 
 
 6513 
 
 6522 
 
 10 
 
 4.5 
 
 6532 
 
 6542 
 
 6551 
 
 6561 
 
 6571 
 
 6580 
 
 6590 
 
 6599 
 
 6609 
 
 6618 
 
 10 
 
 4.6 
 
 6628 
 
 6637 
 
 6646 
 
 6656 
 
 6665 
 
 6675 
 
 6684 
 
 6693 
 
 6702 
 
 6712 
 
 10 
 
 4.7 
 
 6721 
 
 6730 
 
 6739 
 
 6749 
 
 6758 
 
 6767 
 
 6776 
 
 6785 
 
 6794 
 
 6803 
 
 9 
 
 4.8 
 
 6812 
 
 6821 
 
 6830 
 
 6839 
 
 6848 
 
 6857 
 
 6866 
 
 6875 
 
 6884 
 
 6893 
 
 9 
 
 4.9 
 
 6902 
 
 6911 
 
 6920 
 
 6928 
 
 6937 
 
 6946 
 
 6955 
 
 6964 
 
 6972 
 
 6981 
 
 9 
 
 log TT = 0.4971 
 log e = 0.4343 
 
 log 7T/2 = 0.1961 log 71-2 
 log (0.4343) = 0.6378 - 1 
 
 0.9943 
 
 log 
 
 0.2486 
 
 These two pages give the common logarithms of numbers between 1 and 10, correct 
 to four places. Moving the decimal point n places to the right [or left] in the number is 
 equivalent to adding n[dr-n] to the logarithm. Thus, log 0.017453 = 0.2419 - 2, 
 which may also be written 2.2419 or 8.2419 - 10. See p. 91. Graphs, p. 174. 
 log (aft) = log a + log 6 log (a N ) = N log a 
 
 log = log a - log 6 
 
 log 
 
 - log a 
 
MATHEMATICAL TABLES 
 
 43 
 
 COMMON LOGARITHMS (continued) 
 
 1* 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 !i 
 
 5.0 
 
 0.6990 
 
 6998 
 
 7007 
 
 7016 
 
 7024 
 
 7033 
 
 7042 
 
 7050 
 
 7059 
 
 7067 
 
 9 
 
 5.1 
 
 7076 
 
 7084 
 
 7093 
 
 7101 
 
 7110 
 
 7118 
 
 7126 
 
 7135 
 
 7143 
 
 7152 
 
 8 
 
 5.2 
 
 7160 
 
 7168 
 
 7177 
 
 7185 
 
 7193 
 
 7202 
 
 7210 
 
 7218 
 
 7226 
 
 7235 
 
 8 
 
 5.3 
 
 7243 
 
 7251 
 
 7259 
 
 7267 
 
 7275 
 
 7284 
 
 7292 
 
 7300 
 
 7308 
 
 7316 
 
 8 
 
 5.4 
 
 7324 
 
 , 7332 
 
 7340 
 
 7348 
 
 7356 
 
 7364 
 
 7372 
 
 7380 
 
 7388 
 
 7396 
 
 8 
 
 5.5 
 
 7404 
 
 7412 
 
 7419 
 
 7427 
 
 7435 
 
 7443 
 
 7451 
 
 7459 
 
 7466 
 
 7474 
 
 8 
 
 5.6 
 
 7482 
 
 7490 
 
 7497 
 
 7505 
 
 7513 
 
 7520 
 
 7528 
 
 7536 
 
 7543 
 
 7551 
 
 8 
 
 5.7 
 
 7559 
 
 > 7566 
 
 7574 
 
 7582 
 
 7589 
 
 7597 
 
 7604 
 
 7612 
 
 7619 
 
 7627 
 
 8 
 
 5.8 
 
 7634 
 
 7642 
 
 7649 
 
 7657 
 
 7664 
 
 7672 
 
 7679 
 
 7686 
 
 7694 
 
 7701 
 
 7 
 
 5.9 
 
 7709 
 
 7716 
 
 7723 
 
 7731 
 
 7738 
 
 7745 
 
 7752 
 
 7760 
 
 7767 
 
 7774 
 
 7 
 
 6.0 
 
 0.7782 
 
 7789 
 
 7796 
 
 7803 
 
 7810 
 
 7818 
 
 7825 
 
 7832 
 
 7839 
 
 7846 
 
 7 
 
 6.1 
 
 7853 
 
 7860 
 
 7868 
 
 7875 
 
 7882 
 
 7889 
 
 7896 
 
 7903 
 
 7910 
 
 7917 
 
 7 
 
 6.2 
 
 7924 
 
 7931 
 
 7938 
 
 7945 
 
 7952 
 
 7959 
 
 7966 
 
 7973 
 
 7980 
 
 7987 
 
 7 
 
 6.3 
 
 7993 
 
 8000 
 
 8007 
 
 8014 
 
 8021 
 
 8028 
 
 8035 
 
 8041 
 
 8048 
 
 8055 
 
 7 
 
 6.4 
 
 8062 
 
 8069 
 
 8075 
 
 8082 
 
 8089 
 
 8096 
 
 . 8102 
 
 8109 
 
 8116 
 
 8122 
 
 7 
 
 6.5 
 
 8129 
 
 8136 
 
 8142 
 
 8149 
 
 8156 
 
 8162 
 
 8169 
 
 8176 
 
 8182 
 
 8189 
 
 7 
 
 6.6 
 
 8195 
 
 8202 
 
 8209 
 
 8215 
 
 8222 
 
 8228 
 
 8235 
 
 8241 
 
 8248 
 
 8254 
 
 7 
 
 6.7 
 
 8261 
 
 8267 
 
 8274 
 
 8280 
 
 8287 
 
 8293 
 
 8299 
 
 8306 
 
 8312 
 
 8319 
 
 6 
 
 6.8 
 
 8325 
 
 8331 
 
 8338 
 
 8344 
 
 8351 
 
 8357 
 
 8363 
 
 8370 
 
 8376 
 
 8382 
 
 6 
 
 6.9 
 
 8388 
 
 8395 
 
 8401 
 
 8407 
 
 8414 
 
 8420 
 
 8426 
 
 8432 
 
 8439 
 
 8445 
 
 6 
 
 7.0 
 
 0.8451 
 
 8457 
 
 8463 
 
 8470 
 
 8476 
 
 8482 
 
 8488 
 
 8494 
 
 8500 
 
 8506 
 
 6 
 
 7.1 
 
 1 8513 
 
 8519 
 
 8525 
 
 8531 
 
 8537 
 
 8543 
 
 8549 
 
 8555 
 
 8561 
 
 8567 
 
 6 
 
 7.2 
 
 8573 
 
 8579 
 
 8585 
 
 8591 
 
 8597 
 
 8603 
 
 8609 
 
 8615 
 
 8621 
 
 8627 
 
 6 
 
 7.3 
 
 8633 
 
 8639 
 
 8645 
 
 8651 
 
 8657 
 
 8663 
 
 8669 
 
 8675 
 
 8681 
 
 8686 
 
 6 
 
 7.4 
 
 8692 
 
 8698 
 
 8704 
 
 8710 
 
 8716 
 
 8722 
 
 8727 
 
 8733 
 
 8739 
 
 8745 
 
 6 
 
 7.5 
 
 8751 
 
 8756 
 
 8762 
 
 8768 
 
 8774 
 
 8779 
 
 8785 
 
 8791 
 
 8797 
 
 8802 
 
 6 
 
 7.6 
 
 8808 
 
 8814 
 
 8820 
 
 8825 
 
 8831 
 
 8837 
 
 8842 
 
 8848 
 
 8854 
 
 8859 
 
 6 
 
 7.7 
 
 8865 
 
 8871 
 
 8876 
 
 8882 
 
 8887 
 
 8893 
 
 8899 
 
 8904 
 
 8910 
 
 8915 
 
 6 
 
 7.8 
 
 8921 
 
 8927 
 
 8932 
 
 8938 
 
 8943 
 
 8949 
 
 8954 
 
 8960 
 
 8965 
 
 8971 
 
 6 
 
 7.9 
 
 8976 
 
 8982 
 
 8987 
 
 8993 
 
 8998 
 
 9004 
 
 9009 
 
 9015 
 
 9020 
 
 9025 
 
 5 
 
 8.0 
 
 0.9031 
 
 9036 
 
 9042 
 
 9047 
 
 9053 
 
 9058 
 
 9063 
 
 9069 
 
 9074 
 
 9079 
 
 5 
 
 8.1 
 
 9085 
 
 9090 
 
 9096 
 
 9101 
 
 9106 
 
 9112 
 
 9117 
 
 9122 
 
 9128 
 
 9133 
 
 5 
 
 8.2 
 
 9138 
 
 9143 
 
 9149 
 
 9154 
 
 9159 
 
 9165 
 
 9170 
 
 9175 
 
 9180 
 
 9186 
 
 5 
 
 8.3 
 
 9191 
 
 9196 
 
 9201 
 
 9206 
 
 9212 
 
 9217 
 
 9222 
 
 9227 
 
 9232 
 
 9238 
 
 5 
 
 8.4 
 
 9243 
 
 9248 
 
 9253 
 
 9258 
 
 9263 . 
 
 9269 
 
 9274 
 
 9279 
 
 9284 
 
 9289 
 
 5 
 
 8.5 
 
 9294 
 
 9299 
 
 9304 
 
 9309 
 
 9315 
 
 9320 
 
 9325 
 
 9330 
 
 9335 
 
 9340 
 
 5 
 
 8.6 
 
 9345 
 
 9350 
 
 9355 
 
 9360 
 
 9365 
 
 9370 
 
 9375 
 
 9380 
 
 9385 
 
 9390 
 
 5 
 
 8.7 
 
 9395 
 
 9400 
 
 9405 
 
 9410 
 
 9415 
 
 9420 
 
 9425 
 
 9430 
 
 9435 
 
 9440 
 
 5 
 
 8.8 
 
 9445 
 
 9450 
 
 9455 
 
 9460 
 
 9465 
 
 9469 
 
 9474 
 
 9479 
 
 9484 
 
 9489 
 
 5 
 
 8.9 
 
 9494 
 
 . 9499 
 
 9504 
 
 9509 
 
 9513 
 
 9518 
 
 9523 
 
 9528 
 
 9533 
 
 9538 
 
 5 
 
 9.0 
 
 0.9542 
 
 9547 
 
 9552 
 
 9557 
 
 9562 
 
 9566 
 
 9571 
 
 9576 
 
 9581 
 
 9586 
 
 5 
 
 9.1 
 
 9590 
 
 9595 
 
 9600 
 
 9605 
 
 9609 
 
 9614 
 
 9619 
 
 9624 
 
 9628 
 
 9633 
 
 5 
 
 9.2 
 
 9638 
 
 9643 
 
 9647 
 
 9652 
 
 9657 
 
 9661 
 
 9666 
 
 9671 
 
 9675 
 
 9680 
 
 5 
 
 9.3 
 
 9685 
 
 9689 
 
 9694 
 
 9699 
 
 9703 
 
 9708 
 
 9713 
 
 9717 
 
 9722 
 
 9727 
 
 5 
 
 9.4 
 
 9731 
 
 9736 
 
 9741 
 
 9745 
 
 9750 , 
 
 9754 
 
 9759 
 
 9763 
 
 9768 
 
 9773 
 
 5 
 
 9.5 
 
 9777 
 
 9782 
 
 9786 
 
 9791 
 
 9795 
 
 9800 
 
 9805 
 
 9809 
 
 9814 
 
 9818 
 
 5 
 
 9.6 
 
 9823 
 
 9827 
 
 9832 
 
 9836 
 
 9841 
 
 9845 
 
 9850 
 
 9854 
 
 9859 
 
 9863 
 
 4 
 
 9.7 
 
 9868 
 
 9872 
 
 9877 
 
 9881 
 
 9886 
 
 9890 
 
 9894 
 
 9899 
 
 9903 
 
 9908 
 
 4 
 
 9.8 
 
 99 r2- 
 
 - 9917 
 
 9921 
 
 9926 
 
 9930 
 
 9934 
 
 9939 
 
 9943 
 
 9948 
 
 9952 
 
 4 
 
 9.9 
 
 9956 
 
 9961 
 
 9965 
 
 9969 
 
 9974 
 
 9978 
 
 9983 
 
 9987 
 
 9991 
 
 9996 
 
 4 
 
44 
 
 MATHEMATICAL TABLES 
 
 DEGREES AND MINUTES EXPRESSED IN RADIANS (See also p. 69) 
 
 Degrees 
 
 Hundredths 
 
 Minutes 
 
 1 
 
 .0175 
 
 61 
 
 1.0647 
 
 121 
 
 2.1118 
 
 0.01 
 
 .0002 
 
 0.51 
 
 .0089 
 
 r 
 
 .0003 
 
 2 
 
 .0349 
 
 2 
 
 1.0821 
 
 2 
 
 2.1293 
 
 2 
 
 .0003 
 
 2 
 
 .0091 
 
 2' 
 
 .0006 
 
 3 
 
 .0524 
 
 3 
 
 .0996 
 
 3 
 
 2.1468 
 
 3 
 
 .0005 
 
 3 
 
 .0093 
 
 3' 
 
 .0009 
 
 4 
 
 .0698 
 
 4 
 
 .1170 
 
 4 
 
 2.1642 
 
 4 
 
 .0007 
 
 4 
 
 .0094 
 
 4' 
 
 .0012 
 
 5 
 
 .0873 
 
 65 
 
 .1345 
 
 125 
 
 2.1817 
 
 .05 
 
 .0009 
 
 .55 
 
 .0096 
 
 5' 
 
 .0015 
 
 6 
 
 .1047 
 
 6 
 
 .1519 
 
 6 
 
 2.1991 
 
 6 
 
 .0010 
 
 6 
 
 .0098 
 
 6' 
 
 .0017 
 
 7 
 
 .1222 
 
 7 
 
 .1694 
 
 7 
 
 2.2166 
 
 7 
 
 .0012 
 
 7 
 
 .0099 
 
 7' 
 
 .0020 
 
 8 
 
 .1396 
 
 8 
 
 .1868 
 
 8 
 
 2.2340 
 
 8 
 
 .0014 
 
 8 
 
 .0101 
 
 8' 
 
 .0023 
 
 9 
 
 .1571 
 
 9 
 
 1.2043 
 
 9 
 
 2.2515 
 
 9 
 
 .0016 
 
 9 
 
 .0103 
 
 9' 
 
 .0026 
 
 10 
 
 .1745 
 
 70 
 
 1.2217 
 
 130 
 
 2.2689 
 
 0.10 
 
 .0017 
 
 0.60 
 
 .0105 
 
 10' 
 
 .0029 
 
 
 .1920 
 
 1 
 
 1.2392 
 
 1 
 
 2.2864 
 
 1 
 
 .0019 
 
 ] 
 
 .0106 
 
 11' 
 
 .0032 
 
 2 
 
 .2094 
 
 2 
 
 1.2566 
 
 2 
 
 2.3038 
 
 2 
 
 .0021 
 
 2 
 
 .0108 
 
 12' 
 
 .0035 
 
 3 
 
 .2269 
 
 3 
 
 1.2741 
 
 3 
 
 2.3213 
 
 3 
 
 .0023 
 
 3 
 
 .0110 
 
 13' 
 
 .0038 
 
 4 
 
 .2443 
 
 4 
 
 1.2915 
 
 4 
 
 2.3387 
 
 4 
 
 .0024 
 
 4 
 
 .0112 
 
 14' 
 
 .0041 
 
 15 
 
 .2618 
 
 75 
 
 1.3090 
 
 136 
 
 2.3562 
 
 .15 
 
 .0026 
 
 .65 
 
 .0113 
 
 15' 
 
 0044 
 
 6 
 
 .2793 
 
 6 
 
 1.3265 
 
 6 
 
 2.3736 
 
 6 
 
 .0028 
 
 6 
 
 .0115 
 
 16' 
 
 .0047 
 
 7 
 
 .2967 
 
 7 
 
 1.3439 
 
 7 
 
 2.3911 
 
 7 
 
 .0030 
 
 7 
 
 .0117 
 
 17' 
 
 .0049 
 
 8 
 
 .3142 
 
 8 
 
 1,3614 
 
 8 
 
 2.4086 
 
 8 
 
 .0031 
 
 8 
 
 .0119 
 
 18' 
 
 .0052 
 
 9 
 
 .3316 
 
 9 
 
 1.3788 
 
 9 
 
 2.4260 
 
 9 
 
 .0033 
 
 9 
 
 .0120 
 
 19' 
 
 .0055 
 
 20 
 
 .3491 
 
 80 
 
 1.3963 
 
 140 
 
 2.4435 
 
 0.20 
 
 .0035 
 
 0.70 
 
 .0122 
 
 20' 
 
 .0058 
 
 1 
 
 .3665 
 
 1 
 
 .4137 
 
 1 
 
 2.4609 
 
 1 
 
 .0037 
 
 1 
 
 .0124 
 
 21' 
 
 .0061 
 
 2 
 
 .3840 
 
 2 
 
 .4312 
 
 2 
 
 2.4784 
 
 2 
 
 .0038 
 
 2 
 
 .0126 
 
 22' 
 
 .0064 
 
 3 
 
 .4014 
 
 3 
 
 .4486 
 
 3 
 
 2.4958 
 
 3 
 
 .0040 
 
 3 
 
 .0127 
 
 23' 
 
 .0067 
 
 4 
 
 .4189 
 
 4 
 
 .4661 
 
 4 
 
 2.5133 
 
 4 
 
 .0042 
 
 4 
 
 .0129 
 
 24' 
 
 .0070 
 
 25 
 
 .4363 
 
 85 
 
 1.4835 
 
 145 
 
 2.5307 
 
 .25 
 
 .0044 
 
 .75 
 
 .0131 
 
 25' 
 
 0073 
 
 6 
 
 .4538 
 
 6 
 
 1.5010 
 
 6 
 
 2.5482 
 
 6 
 
 .0045 
 
 6 
 
 .0133 
 
 26' 
 
 .0076 
 
 7 
 
 .4712 
 
 7 
 
 1.5184 
 
 7 
 
 2.5656 
 
 7 
 
 .0047 
 
 7 
 
 .0134 
 
 27' 
 
 .0079 
 
 8 
 
 .4887 
 
 8 
 
 1.5359 
 
 8 
 
 2.5831 
 
 8 
 
 .0049 
 
 8 
 
 .0136 
 
 28' 
 
 .0081 
 
 9 
 
 .5061 
 
 9 
 
 1.5533 
 
 9 
 
 2.6005 
 
 9 
 
 .0051 
 
 9 
 
 .0138 
 
 29' 
 
 0084 
 
 30 
 
 .5236 
 
 90 
 
 1.5708 
 
 150 
 
 2.6180 
 
 0.30 
 
 .0052 
 
 0.80 
 
 .0140 
 
 30' 
 
 .0087 
 
 1 
 
 .5411 
 
 1 
 
 .5882 
 
 1 
 
 2.6354 
 
 1 
 
 .0054 
 
 1 
 
 .0141 
 
 31' 
 
 .0090 
 
 2 
 
 .5585 
 
 2 
 
 1.6057 
 
 2 
 
 2.6529 
 
 2 
 
 .0056 
 
 2 
 
 0143 
 
 32' 
 
 .0093 
 
 3 
 
 .5760 
 
 3 
 
 1.6232 
 
 3 
 
 2.6704 
 
 3 
 
 .0058 
 
 3 
 
 .0145 
 
 33' 
 
 .0096 
 
 4 
 
 .5934 
 
 4 
 
 1.6406 
 
 4 
 
 2.6878 
 
 4 
 
 .0059 
 
 4 
 
 .0147 
 
 34' 
 
 .0099 
 
 35 
 
 .6109 
 
 95 
 
 1.6581 
 
 155 
 
 2.7053 
 
 35 
 
 .0061 
 
 .85 
 
 .0148 
 
 35' 
 
 .0102 
 
 6 
 
 .6283 
 
 6 
 
 1.6755 
 
 6 
 
 2.7227 
 
 6 
 
 .0063 
 
 6 
 
 .0150 
 
 36' 
 
 .0105 
 
 7 
 
 .6458 
 
 7 
 
 1.6930 
 
 7 
 
 2.7402 
 
 7 
 
 .0065 
 
 7 
 
 .0152 
 
 37' 
 
 .0108 
 
 8 
 
 .6632 
 
 8 
 
 1.7104 
 
 8 
 
 2.7576 
 
 8 
 
 .0066 
 
 8 
 
 .0154 
 
 38' 
 
 .0111 
 
 9 
 
 .6807 
 
 9 
 
 1.7279 
 
 9 
 
 2.7751 
 
 9 
 
 .0068 
 
 9 
 
 .0155 
 
 39' 
 
 .0113 
 
 40 
 
 .6981 
 
 100 
 
 1.7453 
 
 160 
 
 2.7925 
 
 0.40 
 
 .0070 
 
 0.90 
 
 .0157 
 
 40' 
 
 .0116 
 
 1 
 
 .7156 
 
 1 
 
 1.7628 
 
 1 
 
 2.8100 
 
 1 
 
 .0072 
 
 1 
 
 .0159 
 
 41' 
 
 .0119 
 
 2 
 
 .7330 
 
 2 
 
 1.7802 
 
 2 
 
 2.8274 
 
 2 
 
 .0073 
 
 2 
 
 .0161 
 
 42' 
 
 .0122 
 
 3 
 
 .7505 
 
 3 
 
 1.7977 
 
 3 
 
 2.8449 
 
 3 
 
 .0075 
 
 3 
 
 .0162 
 
 43' 
 
 .0125 
 
 4 
 
 .7679 
 
 4 
 
 1.8151 
 
 4 
 
 2.8623 
 
 4 
 
 .0077 
 
 4 
 
 .0164 
 
 44' 
 
 .0128 
 
 45 
 
 .7854 
 
 105 
 
 1.8326 
 
 165 
 
 2.8798 
 
 .45 
 
 .0079 
 
 .95 
 
 .0166 
 
 45' 
 
 .0131 
 
 6 
 
 .8029 
 
 6 
 
 1.8500 
 
 6 
 
 2.8972 
 
 6 
 
 .0080 
 
 6 
 
 .0168 
 
 46' 
 
 .0134 
 
 7 
 
 .8203 
 
 7 
 
 1.8675 
 
 7 
 
 2.9147 
 
 7 
 
 .0082 
 
 7 
 
 .0169 
 
 47' 
 
 .0137 
 
 8 
 
 .8378 
 
 8 
 
 1.8850 
 
 8 
 
 2.9322 
 
 8 
 
 .0084 
 
 8 
 
 .0171 
 
 48' 
 
 .0140 
 
 9 
 
 .8552 
 
 9 
 
 1.9024 
 
 9 
 
 2.9496 
 
 9 
 
 .0086 
 
 9 
 
 .0173 
 
 49' 
 
 .0143 
 
 50 
 
 .8727 
 
 110 
 
 1.9199 
 
 170 
 
 2.9671 
 
 0.50 
 
 .0087 
 
 1.00 
 
 .0175 
 
 50' 
 
 .0145 
 
 1 
 
 .8901 
 
 1 
 
 1.9373 
 
 1 
 
 2.9845 
 
 
 
 
 
 51' 
 
 0148 
 
 2 
 
 .9076 
 
 2 
 
 1.9548 
 
 2 
 
 3.0020 
 
 
 
 
 
 52' 
 
 .0151 
 
 3 
 
 .9250 
 
 3 
 
 1.9722 
 
 3 
 
 3.0194 
 
 
 
 
 
 53' 
 
 0154 
 
 4 
 
 .9425 
 
 4 
 
 1.9897 
 
 4 
 
 3.0369 
 
 
 
 
 
 54' 
 
 .0157 
 
 55 
 
 .9599 
 
 115 
 
 2.0071 
 
 175 
 
 3.0543 
 
 
 
 
 
 55' 
 
 .0160 
 
 6 
 
 .9774 
 
 6 
 
 2.0246 
 
 6 
 
 3.0718 
 
 
 
 
 
 56' 
 
 0163 
 
 7 
 
 .9948 
 
 7 
 
 2.0420 
 
 7 
 
 3.0892 
 
 
 
 
 
 57' 
 
 .0166 
 
 8 
 
 1.0123 
 
 8 
 
 2.0595 
 
 8 
 
 3.1067 
 
 
 
 
 
 58' 
 
 0169 
 
 9 
 
 1.0297 
 
 9 
 
 2.0769 
 
 9 
 
 3.1241 
 
 
 
 
 
 59' 
 
 .0172 
 
 60 
 
 1.0472 
 
 120 
 
 2.0944 
 
 180 
 
 3.1416 
 
 
 
 
 
 60' 
 
 .0175 
 
 Arc 1 = 0.0174533 Arc 1' = 0.000290888 Arc 1" = 0.00000484814 
 1 radian - 57.295780 = 57 17'.7468 - 57 17' 44".806 
 
MATHEMATICAL TABLES 
 RADIANS EXPRESSED IN DEGREES 
 
 45 
 
 0.01 
 
 057 
 
 .64 
 
 36 67 
 
 1.27 
 
 72 77 
 
 1 90 
 
 108.86 
 
 2.53 
 
 144.% 
 
 Interpolation 
 
 2 
 
 1.15 
 
 .65 
 
 37.24 
 
 8 
 
 73.34 
 
 1 
 
 109.43 
 
 4 
 
 145.53 
 
 .0002 
 
 0.01 
 
 3 
 
 1.72 
 
 6 
 
 37.82 
 
 9 
 
 73.91 
 
 2 
 
 110.01 
 
 2.55 
 
 146MO 
 
 04 
 
 .02 
 
 4 
 
 2.29 
 
 7 
 
 38.39 
 
 1.30 
 
 74.48 
 
 3 
 
 1KP.58 
 
 6 
 
 146.68 
 
 06 
 
 .03 
 
 .05 
 
 2.86 
 
 8 
 
 38.96 
 
 1 
 
 75.06 
 
 4 
 
 111.15 
 
 7 
 
 147.25 
 
 08 
 
 .05 
 
 6 
 
 3.44 
 
 9 
 
 39.53 
 
 2 
 
 75.63 
 
 1.95 
 
 111.73 
 
 8 
 
 147 .82 
 
 .0010 
 
 0.06 
 
 7 
 
 4.01 
 
 .70 
 
 40.1 1 
 
 3 
 
 76.20 
 
 6 
 
 112.30 
 
 9 
 
 148.40 
 
 12 
 
 .07 
 
 8 
 
 4.58 
 
 1 
 
 40.68 
 
 4 
 
 76.78 
 
 7 
 
 112.87 
 
 2.60 
 
 148.97 
 
 14 
 
 .08 
 
 9 
 
 5. 16 
 
 2 
 
 41.25 
 
 1.35 
 
 77.35 
 
 8 
 
 113.45 
 
 
 149.54 
 
 16 
 
 .09 
 
 .10 
 
 5.73 
 
 3 
 
 41.83 
 
 6 
 
 77.92 
 
 9 
 
 1I4.02 
 
 2 
 
 150. 11 
 
 18 
 
 .10 
 
 1 
 
 6.30 
 
 4 
 
 42.40 
 
 7 
 
 78.50 
 
 2.00 
 
 114.59 
 
 3 
 
 I50.69 
 
 .0020 
 
 0.ll 
 
 2 
 
 6.88 
 
 .75 
 
 42.97 
 
 8 
 
 79.07 
 
 1 
 
 115M6 
 
 4 
 
 151.26 
 
 22 
 
 .13 
 
 3 
 
 7.45 
 
 6 
 
 43.54 
 
 9 
 
 79.64 
 
 2 
 
 115.74 
 
 2.65 
 
 15P.83 
 
 24 
 
 .14 
 
 4 
 
 8.02 
 
 7 
 
 44. 12 
 
 1.40 
 
 80.21 
 
 3 
 
 116.31 
 
 6 
 
 152.41 
 
 26 
 
 .15 
 
 .15 
 
 8.59 
 
 8 
 
 44.69 
 
 1 
 
 80.79 
 
 4 
 
 116.88 
 
 7 
 
 152.98 
 
 28 
 
 .16 
 
 6 
 
 9. 17 
 
 9 
 
 45.26 
 
 2 
 
 81.36 
 
 2.05 
 
 1!7.46 
 
 8 
 
 153.55 
 
 .0030 
 
 0.17 
 
 7 
 
 9.74 
 
 .80 
 
 45.84 
 
 3 
 
 81.93 
 
 6 
 
 118.03 
 
 9 
 
 154. 13 
 
 32 
 
 .18 
 
 8 
 
 10.31 
 
 1 
 
 46.41 
 
 4 
 
 82.51 
 
 7 
 
 118.60 
 
 2.70 
 
 154.70 
 
 34 
 
 .19 
 
 9 
 
 10.89 
 
 2 
 
 46.98 
 
 1.45 
 
 83.08 
 
 8 
 
 119.18 
 
 1 
 
 155.27 
 
 36 
 
 .21 
 
 .20 
 
 1I.46 
 
 3 
 
 47.56 
 
 6 
 
 83.65 
 
 9 
 
 119.75 
 
 2 
 
 155.84 
 
 38 
 
 .22 
 
 1 
 
 12.03 
 
 4 
 
 48. 13 
 
 7 
 
 84.22 
 
 2.10 
 
 120.32 
 
 3 
 
 156.42 
 
 .0040 
 
 0.23 
 
 2 
 
 12.61 
 
 .85 
 
 48.70 
 
 8 
 
 84.80 
 
 1 
 
 120.89 
 
 4 
 
 156.99 
 
 42 
 
 .24 
 
 3 
 
 13.18 
 
 6 
 
 49.27 
 
 9 
 
 85.37 
 
 2 
 
 121 .47 
 
 2.75 
 
 157 .56 
 
 44 
 
 .25 
 
 4 
 
 13.75 
 
 7 
 
 49.85 
 
 1.50 
 
 85.94 
 
 3 
 
 122.04 
 
 6 
 
 158.14 
 
 46 
 
 .26 
 
 .25 
 
 14.32 
 
 8 
 
 50.42 
 
 1 
 
 86.52 
 
 4 
 
 122.61 
 
 7 
 
 158.7I 
 
 48 
 
 .28 
 
 6 
 
 14.90 
 
 9 
 
 50.99 
 
 2 
 
 87.09 
 
 2.15 
 
 123M9 
 
 8 
 
 159.28 
 
 .0050 
 
 0.29 
 
 7 
 
 15.47 
 
 .90 
 
 5J.57 
 
 3 
 
 87.66 
 
 6 
 
 123.76 
 
 9 
 
 159.86 
 
 52 
 
 .30 
 
 8 
 
 16.04 
 
 1 
 
 52.14 
 
 4 
 
 88.24 
 
 7 
 
 124.33 
 
 2.80 
 
 160.43 
 
 54 
 
 .31 
 
 9 
 
 16.62 
 
 2 
 
 52.71 
 
 1.55 
 
 88.81 
 
 8 
 
 124.90 
 
 1 
 
 161.00 
 
 56 
 
 .32 
 
 .30 
 
 17.19 
 
 3 
 
 53.29 
 
 6 
 
 89.38 
 
 9 
 
 125.48 
 
 2 
 
 16I.57 
 
 58 
 
 .33 
 
 1 
 
 17J6 
 
 4 
 
 53.86 
 
 7 
 
 89.95 
 
 2.20 
 
 126.05 
 
 3 
 
 162. 15 
 
 .0060 
 
 0.34 
 
 2 
 
 18.33 
 
 .95 
 
 54.43 
 
 8 
 
 90.53 
 
 11 
 
 126.62 
 
 4 
 
 162 .72 
 
 62 
 
 .36 
 
 3 
 
 18.91 
 
 6 
 
 55.00 
 
 9 
 
 91.10 
 
 
 127.20 
 
 2.85 
 
 163.29 
 
 64 
 
 .37 
 
 4 
 
 19.48 
 
 7 
 
 55.58 
 
 1.60 
 
 91.67 
 
 3 
 
 127.77 
 
 6 
 
 163.87 
 
 66 
 
 .38 
 
 .35 
 
 20.05 
 
 8 
 
 56. 15 
 
 1 
 
 92.25 
 
 4 
 
 128.34 
 
 7 
 
 164.44 
 
 68 
 
 .39 
 
 6 
 
 20.63 
 
 9 
 
 56.72 
 
 2 
 
 92.82 
 
 2.25 
 
 128.92 
 
 8 
 
 165.01 
 
 .0070 
 
 0.40 
 
 7 
 
 21.20 
 
 1.00 
 
 57.30 
 
 3 
 
 93039 
 
 6 
 
 129.49 
 
 9 
 
 165.58 
 
 72 
 
 .41 
 
 8 
 
 21.77 
 
 1 
 
 57.87 
 
 4 
 
 93.97 
 
 7 
 
 130.06 
 
 2.90 
 
 166M6 
 
 74 
 
 .42 
 
 9 
 
 22.35 
 
 2 
 
 58.44 
 
 1.65 
 
 94.54 
 
 8 
 
 130.6? 
 
 1 
 
 166.73 
 
 76 
 
 .44 
 
 .40 
 
 22.92 
 
 3 
 
 59.01 
 
 6 
 
 95. 11 
 
 9 
 
 131.21 
 
 2 
 
 167.30 
 
 78 
 
 .45 
 
 1 
 
 23.49 
 
 4 
 
 59.59 
 
 7 
 
 95.68 
 
 2.30 
 
 131.78 
 
 3 
 
 167.88 
 
 .0080 
 
 0.46 
 
 2 
 
 24.06 
 
 1.05 
 
 60. 16 
 
 8 
 
 96.26 
 
 1 
 
 132.35 
 
 4 
 
 168.45 
 
 82 
 
 .47 
 
 3 
 
 24.64 
 
 6 
 
 60.73 
 
 9 
 
 96.83 
 
 2 
 
 132.93 
 
 2.95 
 
 169.02 
 
 84 
 
 .48 
 
 4 
 
 25.21 
 
 7 
 
 61.31 
 
 1.70 
 
 97.40 
 
 3 
 
 133.50 
 
 6 
 
 169.60 
 
 86 
 
 .49 
 
 .45 
 
 25.78 
 
 8 
 
 61.88 
 
 1 
 
 97.98 
 
 4 
 
 134.07 
 
 7 
 
 170. 17 
 
 88 
 
 .50 
 
 6 
 
 26.36 
 
 9 
 
 62.45 
 
 2 
 
 98.55 
 
 2.35 
 
 134.65 
 
 8 
 
 170.74 
 
 .0090 
 
 0.52 
 
 7 
 
 26.93 
 
 1.10 
 
 63.03 
 
 3 
 
 99. 12 
 
 6 
 
 135.22 
 
 9 
 
 171.31 
 
 92 
 
 .53 
 
 8 
 
 27.50 
 
 1 
 
 63.60 
 
 4 
 
 99.69 
 
 7 
 
 135.79 
 
 3.00 
 
 17P.89 
 
 94 
 
 .54 
 
 9 
 
 28.07 
 
 2 
 
 64. 17 
 
 1.75 
 
 100.27 
 
 8 
 
 136.36 
 
 
 172.46 
 
 96 
 
 .55 
 
 .60 
 
 28.65 
 
 3 
 
 64.74 
 
 6 
 
 100.84 
 
 9 
 
 136.94 
 
 1 
 
 173.03 
 
 98 
 
 .56 
 
 | 
 
 29.22 
 
 4 
 
 65.32 
 
 7 
 
 10I.41 
 
 2.40 
 
 137j51 
 
 3 
 
 173.61 
 
 
 2 
 
 29079 
 
 1.15 
 
 65.89 
 
 8 
 
 10P.99 
 
 1 
 
 138.08 
 
 4 
 
 174. 18 
 
 Multiples of v 
 
 3 
 
 30.37 
 
 6 
 
 66.46 
 
 9 
 
 102.56 
 
 2 
 
 I38.66 
 
 3.05 
 
 174J5 
 
 
 4 
 
 30.94 
 
 7 
 
 67.04 
 
 1.80 
 
 103.13 
 
 3 
 
 139.23 
 
 6 
 
 175.33 
 
 1 3.1416 
 
 180 
 
 .55 
 
 31.51 
 
 8 
 
 67.61 
 
 1 
 
 103.71 
 
 4 
 
 139.80 
 
 7 
 
 175.90 
 
 2 6.2832 
 
 360 
 
 6 
 
 32.09 
 
 9 
 
 68. 18 
 
 2 
 
 104.28 
 
 2.45 
 
 140.37 
 
 8 
 
 176.47 
 
 3 9.4248 
 
 540 
 
 7 
 
 32.66 
 
 1.20 
 
 68.75 
 
 3 
 
 104.85 
 
 6 
 
 140.95 
 
 9 
 
 177.04 
 
 4 12.5664 
 
 720 
 
 8 
 
 33.23 
 
 I 
 
 69.33 
 
 4 
 
 105.42 
 
 7 
 
 141.52 
 
 3.10 
 
 177.62 
 
 5 15.7080 
 
 900 
 
 9 
 
 33.80 
 
 2 
 
 69.90 
 
 1.85 
 
 106.00 
 
 8 
 
 142.09 
 
 1 
 
 178. 19 
 
 6 18.8496 
 
 1080 
 
 .60 
 
 34.38 
 
 3 
 
 70.47 
 
 6 
 
 106.57 
 
 9 
 
 142.67 
 
 2 
 
 178.76 
 
 7 21.9911 
 
 1260 
 
 1 
 
 34.95 
 
 4 
 
 71.05 
 
 7 
 
 107. 14 
 
 2.50 
 
 143.24 
 
 3 
 
 179.34 
 
 8 25.1327 
 
 1440 
 
 2 
 
 35.52 
 
 1.25 
 
 71.62 
 
 8 
 
 107.72 
 
 1 
 
 143.81 
 
 V 4 
 
 179.91 
 
 9 28.2743 
 
 1620 
 
 3 
 
 36. 10 
 
 6 
 
 72. 19 
 
 9 
 
 108.29 
 
 2 
 
 144.39 
 
 3.15 
 
 180.48 
 
 10 31.4159 
 
 1800 
 
46 
 
 MATHEMATICAL TABLES 
 
 NATURAL SINES AND COSINES 
 
 Natural Sines at intervals of 0.l, or 6'. (For 10' intervals, see pp. 52-56) 
 
 M 
 
 o 
 
 o 
 
 
 
 o 
 
 B 
 
 o 
 
 
 
 o 
 
 
 . 
 
 
 P 
 
 =^6o 
 
 (60 
 
 (120 (180 
 
 (240 
 
 (300 
 
 (360 
 
 (420 (480 
 
 (540 
 
 
 1* 
 
 
 
 
 
 
 
 
 
 
 0.0000 
 
 90 
 
 
 
 
 0.0000 
 
 0017 
 
 0035 0052 
 
 0070 
 
 0087 
 
 0105 
 
 0122 0140 
 
 0157 
 
 0175 
 
 89 
 
 17 
 
 1 
 
 0175 
 
 0192 
 
 0209 0227 
 
 0244 
 
 0262 
 
 0279 
 
 0297 0314 
 
 0332 
 
 0349 
 
 88 
 
 17 
 
 2 
 
 0349 
 
 0366 
 
 0384 0401 
 
 0419 
 
 0436 
 
 0454 
 
 0471 0488 
 
 0506 
 
 0523 
 
 87 
 
 17 
 
 3 
 
 0523 
 
 0541 
 
 0558 0576 
 
 0593 
 
 0610 
 
 0628 
 
 0645 0663 
 
 0680 
 
 0698 
 
 86 
 
 17 
 
 4 
 
 0698 
 
 0715 
 
 0732 0750 
 
 0767 
 
 0785 
 
 0802 
 
 0819 0837 
 
 0854 
 
 0.0872 
 
 85 
 
 17 
 
 5 
 
 0.0872 
 
 0889 
 
 0906 0924 
 
 0941 
 
 0958 
 
 0976 
 
 0993 1011 
 
 1028 
 
 1045 
 
 84 
 
 17 
 
 6 
 
 1045 
 
 1063 
 
 1080 1097 
 
 1115 
 
 1132 
 
 1149 
 
 1167 1184 
 
 1201 
 
 1219 
 
 83 
 
 17 
 
 7 
 
 1219 
 
 1236 
 
 1253 1271 
 
 1288 
 
 1305 
 
 1323 
 
 1340 1357 
 
 1374 
 
 1392 
 
 82 
 
 17 
 
 8 
 
 1392 
 
 1409 
 
 1426 1444 
 
 1461 
 
 1478 
 
 1495 
 
 1513 1530 
 
 1547 
 
 1564 
 
 81 
 
 17 
 
 9 
 
 1564 
 
 1582 
 
 1599 1616 
 
 1633 
 
 1650 
 
 1668 
 
 1685 1702 
 
 '1719 
 
 0.1736 
 
 80 
 
 17 
 
 10 
 
 0.1736 
 
 1754 
 
 1771 1788 
 
 1805 
 
 1822 
 
 1840 
 
 1857 1874 
 
 1891 
 
 1908 
 
 79 
 
 17 
 
 11 
 
 1908 
 
 1925 
 
 1942 1959 
 
 1977 
 
 1994 
 
 2011 
 
 2028 2045 
 
 2062 
 
 2079 
 
 78 
 
 17 
 
 12 
 
 2079 
 
 2096 
 
 2113 2130 
 
 2147 
 
 2164 
 
 2181 
 
 2198 2215 
 
 2233 
 
 2250 
 
 77 
 
 17 
 
 13 
 
 2250 
 
 2267 
 
 2284 2300 
 
 2317 
 
 2334 
 
 2351 
 
 2368 2385 
 
 2402 
 
 2419 
 
 76 
 
 17 
 
 14 
 
 2419 
 
 2436 
 
 2453 2470 
 
 2487 
 
 2504 
 
 2521 
 
 2538 2554 
 
 2571 
 
 0.2588 
 
 75 
 
 17 
 
 15 
 
 0.2588 
 
 2605 
 
 2622 2639 
 
 2656 
 
 2672 
 
 2689 
 
 2706 2723 
 
 2740 
 
 2756 
 
 74 
 
 17 
 
 16 
 
 2756 
 
 2773 
 
 2790 2807 
 
 2823 
 
 2840 
 
 2857 
 
 2874 2890 
 
 2907 
 
 2924 
 
 73 
 
 17 
 
 17 
 
 2924 
 
 2940 
 
 2957 2974 
 
 2990 
 
 3007 
 
 3024 
 
 3040 3057 
 
 3074 
 
 3090 
 
 72 
 
 17 
 
 18 
 
 3090 
 
 3107 
 
 3123 3140 
 
 3156 
 
 3173 
 
 3190 
 
 3206 3223 
 
 3239 
 
 3256 
 
 71 
 
 17 
 
 19 
 
 3256 
 
 3272 
 
 3289 3305 
 
 3322 
 
 3338 
 
 3355 
 
 3371 3387 
 
 3404 
 
 0.3420 
 
 70 
 
 16 
 
 20 
 
 0.3420 
 
 3437 
 
 3453 3469 
 
 3486 
 
 3502 
 
 3518 
 
 3535 3551 
 
 3567 
 
 3584 
 
 69 
 
 16 
 
 21 
 
 3584 
 
 3600 
 
 3616 3633 
 
 3649 
 
 3665 
 
 3681 
 
 3697 3714 
 
 3730 
 
 3746 
 
 68 
 
 16 
 
 22 
 
 3746 
 
 3762 
 
 3778 3795 
 
 3811 
 
 3827 
 
 3843 
 
 3859 3875 
 
 3891 
 
 3907 
 
 67 
 
 16 
 
 23 
 
 3907 
 
 3923 
 
 3939 3955 
 
 3971 
 
 3987 
 
 4003 
 
 4019 4035 
 
 4051 
 
 4067 
 
 66 
 
 16 
 
 24 
 
 4067 
 
 4083 
 
 4099 4115 
 
 4131 
 
 4147 
 
 4163 
 
 4179 4195 
 
 4210 
 
 0.4226 
 
 65 
 
 16 
 
 25 
 
 0.4226 
 
 4242 
 
 4258 4274 
 
 4289 
 
 4305 
 
 4321 
 
 4337 4352 
 
 4368 
 
 4384 
 
 64 
 
 16 
 
 26 
 
 4384 
 
 4399 
 
 4415 4431 
 
 4446 
 
 4462 
 
 4478 
 
 4493 4509 
 
 4524 
 
 4540 
 
 63 
 
 16 
 
 27 
 
 4540 
 
 4555 
 
 4571 4586 
 
 4602 
 
 4617 
 
 4633 
 
 4648 4664 
 
 4679 
 
 4695 
 
 62 
 
 16 
 
 28 
 
 4695 
 
 4710 
 
 4726 4741 
 
 4756 
 
 4772 
 
 4787 
 
 4802 4818 
 
 4833 
 
 4848 
 
 61 
 
 15 
 
 29 
 
 4848 
 
 4863 
 
 4879 4894 
 
 4909 
 
 4924 
 
 4939 
 
 4955 4970 
 
 4985 
 
 0.5000 
 
 60 
 
 15 
 
 30 
 
 0.5000 
 
 5015 
 
 5030 5045 
 
 5060 
 
 5075 
 
 5090 
 
 5105 5120 
 
 5135 
 
 5150 
 
 59 
 
 15 
 
 31 
 
 5150 
 
 5165 
 
 5180 5195 
 
 5210 
 
 5225 
 
 5240 
 
 5255 5270 
 
 5284 
 
 5299 
 
 58 
 
 15 
 
 32 
 
 5299 
 
 5314 
 
 5329 5344 
 
 5358 
 
 5373 
 
 5388 
 
 5402 5417 
 
 5432 
 
 5446 
 
 57 
 
 15 
 
 33 
 
 5446 
 
 5461 
 
 5476 5490 
 
 5505 
 
 5519 
 
 5534 
 
 5548 5563 
 
 5577 
 
 5592 
 
 56 
 
 15 
 
 34 
 
 5592 
 
 5606 
 
 5621 5635 
 
 5650 
 
 5664 
 
 5678 
 
 5693 5707 
 
 5721 
 
 0.5736 
 
 55 
 
 14 
 
 35 
 
 0.5736 
 
 5750 
 
 5764 5779 
 
 5793 
 
 5807 
 
 5821 
 
 5835 5850 
 
 5864 
 
 5878 
 
 54 
 
 14 
 
 36 
 
 5878 
 
 5892 
 
 5906 5920 
 
 5934 
 
 5948 
 
 5962 
 
 5976 5990 
 
 6004 
 
 6018 
 
 53 
 
 14 
 
 37 
 
 6018 
 
 6032 
 
 6046 6060 
 
 6074 
 
 6088 
 
 6101 
 
 6115 6129 
 
 6143 
 
 6157 
 
 52 
 
 14 
 
 38 
 
 6157 
 
 6170 
 
 6184 6198 
 
 6211 
 
 6225 
 
 6239 
 
 6252 6266 
 
 6280 
 
 6293 
 
 51 
 
 14 
 
 39 
 
 6293 
 
 6307 
 
 6320 6334 
 
 6347 
 
 6361 
 
 6374 
 
 6388 6401 
 
 6414 
 
 0.6428 
 
 50 
 
 13 
 
 40 
 
 0.6428 
 
 6441 
 
 6455 6468 
 
 6481 
 
 6494 
 
 6508 
 
 6521 6534 
 
 6547 
 
 6561 
 
 49 
 
 13 
 
 41 
 
 6561 
 
 6574 
 
 6587 6600 
 
 6613 
 
 6626 
 
 6639 
 
 6652 6665 
 
 6678 
 
 6691 
 
 48 
 
 13 
 
 42 
 
 6691 
 
 6704 
 
 6717 6730 
 
 6743 
 
 6756 
 
 6769 
 
 6782 6794 
 
 6807 
 
 6820 
 
 47 
 
 13 
 
 43 
 
 6820 
 
 6833 
 
 6845 6858 
 
 6871 
 
 6884 
 
 6896 
 
 6909 6921 
 
 6934 
 
 6947 
 
 46 
 
 13 
 
 44 
 
 6947 
 
 6959 
 
 6972 6984 
 
 6997 
 
 7009 
 
 7022 
 
 7034 7046 
 
 7059 
 
 0.7071 
 
 45 
 
 12 
 
 45 
 
 0.7071 
 
 
 
 
 
 
 
 
 
 
 
 
 =(540 
 
 (480 (420 
 
 (360 
 
 (300 
 
 (240 
 
 (180 (120 
 
 (60 
 
 (00 
 
 i 
 
 
 (For graphs, see p. 174.) 
 
 Natural Cosines 
 
MATHEMATICAL TABLES 
 
 47 
 
 NATURAL SINES AND COSINES (continued) 
 Natural Sines at intervals of 0.l, or 6'. (For 10' intervals, see pp. 52-56) 
 
 M 
 
 o 
 
 .l 
 
 2 
 
 03 
 
 4 
 
 5 
 
 6 
 
 07 03 09 
 
 
 *d 
 
 a 
 
 =(0 / 
 
 ) (60 
 
 (120 
 
 (180 
 
 (240 
 
 (300 
 
 (360 
 
 (420 (480 (540 
 
 
 <* 
 
 
 
 
 
 
 
 
 
 
 0.7071 
 
 45 
 
 
 45 
 
 0.7071 7083 
 
 7096 
 
 7108 
 
 7120 
 
 7133 
 
 7145 
 
 7157 7169 7181 
 
 7193 
 
 44 
 
 12 
 
 46 
 
 7193 7206 
 
 7218 
 
 7230 
 
 7242 
 
 7254 
 
 7266 
 
 7278 7290 7302 
 
 7314 
 
 43 
 
 12 
 
 47 
 
 731 
 
 4 7325 
 
 7337 
 
 7349 
 
 7361 
 
 7373 
 
 7385 
 
 7396 7408 7420 
 
 7431 
 
 42 
 
 12 
 
 48 
 
 743 
 
 1 7443 
 
 7455 
 
 7466 
 
 7478 
 
 7490 
 
 7501 
 
 7513 7524 7536 
 
 7547 
 
 41 
 
 12 
 
 49 
 
 7547 7559 
 
 7570 
 
 7581 
 
 7593 
 
 7604 
 
 7615 
 
 7627 7638 7649 
 
 0.7660 
 
 40 
 
 11 
 
 50 
 
 0.7660 7672 
 
 7683 
 
 7694 
 
 7705 
 
 7716 
 
 7727 
 
 7738 7749 7760 
 
 7771 
 
 39 
 
 11 
 
 51 
 
 7771 7782 
 
 7793 
 
 7804 
 
 7815 
 
 7826 
 
 7837 
 
 7848 7859 7869 
 
 7880 
 
 38 
 
 11 
 
 52 
 
 788 
 
 7891 
 
 7902 
 
 7912 
 
 7923 
 
 7934 
 
 7944 
 
 7955 7965 7976 
 
 7986 
 
 37 
 
 11 
 
 53 
 
 7986 7997 
 
 8007 
 
 8018 
 
 8028 
 
 8039 
 
 8049 
 
 8059 8070 8080 
 
 8090 
 
 36 
 
 10 
 
 54 
 
 8090 8100 
 
 8111 
 
 8121 
 
 8131 
 
 8141 
 
 8151 
 
 8161 8171 8181 
 
 0.8192 
 
 35 
 
 10 
 
 55 
 
 0.8192 8202 
 
 8211 
 
 8221 
 
 8231 
 
 8241 
 
 8251 
 
 8261 8271 8281 
 
 8290 
 
 34 
 
 10 
 
 56 
 
 829 
 
 8300 
 
 8310 
 
 8320 
 
 8329 
 
 8339 
 
 8348 
 
 8358 8368 8377 
 
 8387 
 
 33 
 
 10 
 
 57 
 
 8387 8396 
 
 8406 
 
 8415 
 
 8425 
 
 8434 
 
 8443 
 
 8453 8462 8471 
 
 8480 
 
 32 
 
 9 
 
 58 
 
 848 
 
 8490 
 
 8499 
 
 8508 
 
 8517 
 
 8526 
 
 8536 
 
 8545 8554 8563 
 
 8572 
 
 31 
 
 9 
 
 59 
 
 8572 8581 
 
 8590 
 
 8599 
 
 8607 
 
 8616 
 
 8625 
 
 8634 8643 8652 
 
 0.8660 
 
 30 
 
 9 
 
 60 
 
 0.8660 8669 
 
 8678 
 
 8686 
 
 8695 
 
 8704 
 
 8712 
 
 8721 8729 8738 
 
 8746 
 
 29 
 
 9 
 
 61 
 
 8746 8755 
 
 8763 
 
 8771 
 
 8780 
 
 8788 
 
 8796 
 
 8805 8813 8821 
 
 8829 
 
 28 
 
 8 
 
 62 
 
 882 
 
 9 8838 
 
 8846 
 
 8854 
 
 8862 
 
 8870 
 
 8878 
 
 8886 8894 8902 
 
 8910 
 
 27 
 
 8 
 
 63 
 
 8910 8918 
 
 8926 
 
 8934 
 
 8942 
 
 8949 
 
 8957 
 
 8965 8973 8980 
 
 8988 
 
 26 
 
 8 
 
 64 
 
 8988 8996 
 
 9003 
 
 9011 
 
 9018 
 
 9026 
 
 9033 
 
 9041 9048 9056 
 
 0.9063 
 
 25 
 
 7 
 
 65 
 
 0.9063 9070 
 
 9078 
 
 9085 
 
 9092 
 
 9100 
 
 9107 
 
 9114 9121 9128 
 
 9135 
 
 24 
 
 7 
 
 66 
 
 913 
 
 5 9143 
 
 9150 
 
 9157 
 
 9164 
 
 9171 
 
 9178 
 
 9184 9191 9198 
 
 9205 
 
 23 
 
 7 
 
 67 
 
 9205 9212 
 
 9219 
 
 9225 
 
 9232 
 
 9239 
 
 9245 
 
 9252 9259 9265 
 
 9272 
 
 22 
 
 7 
 
 68 
 
 927 
 
 2 9278 
 
 9285 
 
 9291 
 
 9298 
 
 9304 
 
 9311 
 
 9317 9323 9330 
 
 9336 
 
 21 
 
 6 
 
 69 
 
 9336 9342 
 
 9348 
 
 9354 
 
 9361 
 
 9367 
 
 9373 
 
 9379 9385 9391 
 
 0.9397 
 
 20 
 
 6 
 
 70 
 
 0.9397 9403 
 
 9409 
 
 9415 
 
 9421 
 
 9426 
 
 9432 
 
 9438 9444 9449 
 
 9455 
 
 19 
 
 6 
 
 71 
 
 945 
 
 5 9461 
 
 9466 
 
 9472 
 
 9478 
 
 9483 
 
 9489 
 
 9494 9500 9505 
 
 9511 
 
 18 
 
 6 
 
 72 
 
 951 
 
 1 9516 
 
 9521 
 
 9527 
 
 9532 
 
 9537 
 
 9542 
 
 9548 9553 9558 
 
 9563 
 
 17 
 
 5 
 
 73 
 
 . 9563 9568 
 
 9573 
 
 9578 
 
 9583 
 
 9588 
 
 9593 
 
 9598 9603 9608 
 
 9613 
 
 16 
 
 5 
 
 74 
 
 9613 9617 
 
 9622 
 
 9627 
 
 9632 
 
 9636 
 
 9641 
 
 9646 9650 9655 
 
 0.9659 
 
 15 
 
 5 
 
 75 
 
 0.9659 9664 
 
 9668 
 
 9673 
 
 9677 
 
 9681 
 
 9686 
 
 9690 9694 9699 
 
 9703 
 
 14 
 
 4 
 
 76 
 
 970 
 
 3 9707 
 
 9711 
 
 9715 
 
 9720 
 
 9724 
 
 9728 
 
 9732 9736 9740 
 
 9744 
 
 13 
 
 4 
 
 77 
 
 9744 9748 
 
 9751 
 
 9755 
 
 9759 
 
 9763 
 
 9767 
 
 9770 9774 9778 
 
 9781 
 
 12 
 
 4 
 
 78 
 
 978 
 
 1 9785 
 
 9789 
 
 9792 
 
 97% 
 
 9799 
 
 9803 
 
 9806 9810 9813 
 
 9816 
 
 11 
 
 3 
 
 79 
 
 9816 9820 
 
 9823 
 
 9826 
 
 9829 
 
 9833 
 
 9836 
 
 9839 9842 9845 
 
 0.9848 
 
 10 
 
 3 
 
 80 
 
 0.9848 9851 
 
 9854 
 
 9857 
 
 9860 
 
 9863 
 
 9866 
 
 9869 9871 9874 
 
 9877 
 
 9 
 
 3 
 
 81 
 
 9877 9880 
 
 9882 
 
 9885 
 
 9888 
 
 9890 
 
 9893 
 
 9895 9898 9900 
 
 9903 
 
 8 
 
 3 
 
 82 
 
 990 
 
 3 9905 
 
 9907 
 
 9910 
 
 9912 
 
 9914 
 
 9917 
 
 9919 9921 9923 
 
 9925 
 
 7 
 
 2 
 
 83 
 
 992 
 
 5 9928 
 
 9930 
 
 9932 
 
 9934 
 
 9936 
 
 9938 
 
 9940 9942 9943 
 
 9945 
 
 6 
 
 2 
 
 84 
 
 9945 9947 
 
 9949 
 
 9951 
 
 9952 
 
 9954 
 
 9956 
 
 9957 9959 9960 
 
 0.9962 
 
 5 
 
 2 
 
 85 
 
 0.9962 9963 
 
 9965 
 
 9966 
 
 9968 
 
 9969 
 
 9971 
 
 9972 9973 9974 
 
 9976 
 
 4 
 
 1 
 
 86 
 
 997 
 
 6 9977 
 
 9978 
 
 9979 
 
 9980 
 
 9981 
 
 9982 
 
 9983 9984 9985 
 
 9986 
 
 3 
 
 1 
 
 87 
 
 9986 9987 
 
 9988 
 
 9989 
 
 9990 
 
 9990 
 
 9991 
 
 9992 9993 9993 
 
 9994 
 
 2 
 
 1 
 
 88 
 
 999 
 
 4 9995 
 
 9995 
 
 9996 
 
 9996 
 
 9997 
 
 9997 
 
 9997 9998 9998 
 
 0.9998 
 
 1 
 
 
 
 89 
 
 0.9998 9999 
 
 9999 
 
 9999 
 
 9999 
 
 0000 
 
 0000 
 
 0000 0000 0000 
 
 1. 0000 
 
 
 
 
 
 90 
 
 1.0000 
 
 
 
 
 = (540 
 
 (480 
 
 (420 
 
 (360 
 
 (300 
 
 (240 
 
 (180 (120 (60 
 
 (00 
 
 $ 
 
 Q 
 
 
 Natural Cosines 
 
48 
 
 MATHEMATICAL TABLES 
 
 NATURAL TANGENTS AND COTANGENTS 
 
 Natural Tangents at intervals of 0.l, or 6'. (For 10' intervals, see pp. 52-56) 
 
 ff 
 
 .0 
 
 M 
 
 2 
 
 3 4 
 
 05 
 
 0( j 
 
 07 03 
 
 9 
 
 
 * 
 
 Q 
 
 -coo 
 
 (60 
 
 (12') 
 
 (18') (24') 
 
 (30') 
 
 (360 
 
 (420 (48') 
 
 (54') 
 
 
 IS 
 
 
 
 
 
 
 
 
 
 
 0.0000 
 
 90 
 
 
 
 
 0.0000 
 
 0017 
 
 0035 
 
 0052 0070 
 
 0087 
 
 0105 
 
 0122 0140 
 
 0157 
 
 0175 
 
 89 
 
 17 
 
 1 
 
 0175 
 
 0192 
 
 0209 
 
 0227 0244 
 
 0262 
 
 0279 
 
 0297 0314 
 
 0332 
 
 0349 
 
 88 
 
 17 
 
 2 
 
 0349 
 
 0367 
 
 0384 
 
 0402 0419 
 
 0437 
 
 0454 
 
 0472 0489 
 
 0507 
 
 0524 
 
 87 
 
 17 
 
 3 
 
 0524 
 
 0542 
 
 0559 
 
 0577 0594 
 
 0612 
 
 0629 
 
 0647 0664 
 
 0682 
 
 0699 
 
 86 
 
 18 
 
 4 
 
 0699 
 
 0717 
 
 0734 
 
 0752 0769 
 
 0787 
 
 0805 
 
 0822 0840 
 
 0857 
 
 0.0875 
 
 85 
 
 18 
 
 5 
 
 0.0875 
 
 0892 
 
 0910 
 
 0928 0945 
 
 0963 
 
 0981 
 
 0998 1016 
 
 1033 
 
 1051 
 
 84 
 
 18 
 
 6 
 
 1051 
 
 1069 
 
 1086 
 
 1104 1122 
 
 1139 
 
 1157 
 
 1175 1192 
 
 1210 
 
 1228 
 
 83 
 
 18 
 
 7 
 
 1228 
 
 1246 
 
 1263 
 
 1281 1299 
 
 1317 
 
 1334 
 
 1352 1370 
 
 1388 
 
 1405 
 
 82 
 
 18 
 
 8 
 
 1405 
 
 1423 
 
 1441 
 
 1459 1477 
 
 1495 
 
 1512 
 
 1530 1548 
 
 1566 
 
 1584 
 
 81 
 
 18 
 
 9 
 
 1584 
 
 1602 
 
 1620 
 
 1638 1655 
 
 1673 
 
 1691 
 
 1709 1727 
 
 1745 
 
 0.1763 
 
 80 
 
 18 
 
 10 
 
 0.1763 
 
 1781 
 
 1799 
 
 1817 1835 
 
 1853 
 
 1871 
 
 1890 1908 
 
 1926 
 
 1944 
 
 79 
 
 18 
 
 It 
 
 1944 
 
 1962 
 
 1980 
 
 1998 2016 
 
 2035 
 
 2053 
 
 2071 2089 
 
 2107 
 
 2126 
 
 78 
 
 18 
 
 12 
 
 2126 
 
 2144 
 
 2162 
 
 2180 2199 
 
 2217 
 
 2235 
 
 2254 2272 
 
 2290 
 
 2309 
 
 77 
 
 18 
 
 13 
 
 2309 
 
 2327 
 
 2345 
 
 2364 2382 
 
 2401 
 
 2419 
 
 2438 2456 
 
 2475 
 
 2493 
 
 76 
 
 18 
 
 14 
 
 2493 
 
 2512 
 
 2530 
 
 2549 2568 
 
 2586 
 
 2605 
 
 2623 2642 
 
 2661 
 
 0.2679 
 
 75 
 
 19 
 
 15 
 
 0.2679 
 
 2698 
 
 2717 
 
 2736 2754 
 
 2773 
 
 2792 
 
 2811 2830 
 
 2849 
 
 2867 
 
 74 
 
 19 
 
 16 
 
 2867 
 
 2886 
 
 2905 
 
 2924 2943 
 
 2962 
 
 2981 
 
 3000 3019 
 
 3038 
 
 3057 
 
 73 
 
 19 
 
 17 
 
 3057 
 
 3076 
 
 3096 
 
 3115 3134 
 
 3153 
 
 3172 
 
 3191 3211 
 
 3230 
 
 3249 
 
 72 
 
 19 
 
 18 
 
 3249 
 
 3269 
 
 3288 
 
 3307 3327 
 
 3346 
 
 3365 
 
 3385 3404 
 
 3424 
 
 3443 
 
 71 
 
 19 
 
 19 
 
 3443 
 
 3463 
 
 3482 
 
 3502 3522 
 
 3541 
 
 3561 
 
 3581 3600 
 
 3620 
 
 0.3640 
 
 70 
 
 20 
 
 20 
 
 0.3640 
 
 3659 
 
 3679 
 
 3699 3719 
 
 3739 
 
 3759 
 
 3779 3799 
 
 3819 
 
 3839 
 
 69 
 
 20 
 
 21 
 
 3839 
 
 3859 
 
 3879 
 
 3899 3919 
 
 3939 
 
 3959 
 
 3979 4000 
 
 4020 
 
 4040 
 
 68 
 
 20 
 
 22 
 
 4040 
 
 4061 
 
 4081 
 
 4101 4122 
 
 4142 
 
 4163 
 
 4183 4204 
 
 4224 
 
 4245 
 
 67 
 
 21 
 
 23 
 
 4245 
 
 4265 
 
 4286 
 
 4307 4327 
 
 4348 
 
 4369 
 
 4390 4411 
 
 4431 
 
 4452 
 
 66 
 
 21 
 
 24 
 
 4452 
 
 4473 
 
 4494 
 
 4515 4536 
 
 4557 
 
 4578 
 
 4599 4621 
 
 4642 
 
 0.4663 
 
 65 
 
 21 
 
 25 
 
 0.4663 
 
 4684 
 
 4706 
 
 4727 4748 
 
 4770 
 
 4791 
 
 4813 4834 
 
 4856 
 
 4877 
 
 64 
 
 21 
 
 26 
 
 4877 
 
 4899 
 
 4921 
 
 4942 4964 
 
 4986 
 
 5008 
 
 5029 5051 
 
 5073 
 
 5095 
 
 63 
 
 22 
 
 27 
 
 5095 
 
 5117 
 
 5139 
 
 5161 5184 
 
 5206 
 
 5228 
 
 5250 5272 
 
 5295 
 
 5317 
 
 62 
 
 22 
 
 28 
 
 5317 
 
 5340 
 
 5362 
 
 5384 5407 
 
 5430 
 
 5452 
 
 5475 5498 
 
 5520 
 
 5543 
 
 61. 
 
 23 
 
 29 
 
 5543 
 
 5566 
 
 5589 
 
 5612 5635 
 
 5658 
 
 5681 
 
 5704 5727 
 
 5750 
 
 0.5774 
 
 60 
 
 23 
 
 30 
 
 0.5774 
 
 5797 
 
 5820 
 
 5844 5867 
 
 5890 
 
 5914 
 
 5938 5961 
 
 5985 
 
 6009 
 
 59 
 
 24 
 
 31 
 
 6009 
 
 6032 
 
 6056 
 
 6080 6104 
 
 6128 
 
 6152 
 
 6170 6200 
 
 6224 
 
 6249 
 
 58 
 
 24 
 
 32 
 
 6249 
 
 6273 
 
 6297 
 
 6322 6346 
 
 6371 
 
 6395 
 
 6420 6445 
 
 6469 
 
 6494 
 
 57 
 
 25 
 
 33 
 
 6494 
 
 6519 
 
 6544 
 
 6569 6594 
 
 6619 
 
 6644 
 
 6669 6694 
 
 6720 
 
 6745 
 
 56 
 
 25 
 
 34 
 
 6745 
 
 6771 
 
 6796 
 
 6822 6847 
 
 6873 
 
 6899 
 
 6924 6950 
 
 6976 
 
 0.7002 
 
 55 
 
 26 
 
 35 
 
 0.7002 
 
 7028 
 
 7054 
 
 7080 7107 
 
 7133 
 
 7159 
 
 7186 7212 
 
 7239 
 
 7265 
 
 54 
 
 26 
 
 36 
 
 7265 
 
 7292 
 
 7319 
 
 7346 7373 
 
 7400 
 
 7427 
 
 7454 7481 
 
 7508 
 
 7536 
 
 53 
 
 27 
 
 37 
 
 7536 
 
 7563 
 
 7590 
 
 7618 7646 
 
 7673 
 
 7701 
 
 7729 7757 
 
 7785 
 
 7813 
 
 52 
 
 28 
 
 38 
 
 7813 
 
 7841 
 
 7869 
 
 7898 7926 
 
 7954 
 
 7983 
 
 8012 8040 
 
 8069 
 
 8098 
 
 51 
 
 28 
 
 39 
 
 8098 
 
 8127 
 
 8156 
 
 8185 8214 
 
 8243 
 
 8273 
 
 8302 8332 
 
 8361 
 
 0.8391 
 
 50 
 
 29 
 
 40 
 
 0.8391 
 
 8421 
 
 8451 
 
 8481 8511 
 
 8541 
 
 8571 
 
 8601 8632 
 
 8662 
 
 8693 
 
 49 
 
 30 
 
 41 
 
 8693 
 
 8724 
 
 8754 
 
 8785 8816 
 
 8847 
 
 8878 
 
 8910 8941 
 
 8972 
 
 9004 
 
 48 
 
 31 
 
 42 
 
 9004 
 
 9036 
 
 9067 
 
 9099 9131 
 
 9163 
 
 9195 
 
 9228 9260 
 
 9293 
 
 9325 
 
 47 
 
 32 
 
 43 
 
 9325 
 
 9358 
 
 9391 
 
 9424 9457 
 
 9490 
 
 9523 
 
 9556 9590 
 
 9623 
 
 0.9657 
 
 46 
 
 33 
 
 44 
 
 0.9657 
 
 9691 
 
 9725 
 
 9759 9793 
 
 9827 
 
 9861 
 
 9896 9930 
 
 9965 
 
 1.0000 
 
 45 
 
 34 
 
 45 
 
 1.0000 
 
 
 
 
 
 
 
 
 
 
 
 
 *(54 9 ') 
 
 (480 
 
 (420 (36') 
 
 (300 
 
 (240 
 
 (18') (120 
 
 (60 
 
 (00 
 
 1 
 
 
 (For graphs, see p. 174.) 
 
 Natural Cotangents 
 
MATHEMATICAL TABLES 
 
 49 
 
 NATURAL TANGENTS AND COTANGENTS (continued) 
 
 Natural Tangents at intervals of 0. 1, or 6'. (For 10' intervals, see pp. 52-56) 
 
 i 
 
 a=Y(V 
 
 ) (60 
 
 (120 
 
 (180 
 
 (240 
 
 (300 
 
 (360 (420 
 
 (480 
 
 (540 
 
 diff! 
 
 
 
 
 
 
 
 
 
 
 1.0000 
 
 45 
 
 
 45 
 
 1.0000 0035 
 
 0070 
 
 0105 
 
 0141 
 
 0176 
 
 0212 0247 
 
 0283 
 
 0319 0355 
 
 44 
 
 35 
 
 46 
 
 035 
 
 5 0392 
 
 0428 
 
 0464 
 
 0501 
 
 0538 
 
 0575 0612 
 
 0649 
 
 0686 0724 
 
 43 
 
 37 
 
 47 
 
 072 
 
 4 0761 
 
 0799 
 
 0837 
 
 0875 
 
 0913 
 
 0951 0990 
 
 1028 
 
 1067 1106 
 
 42 
 
 38 
 
 48 
 
 1106 1145 
 
 1184 
 
 1224 
 
 1263 
 
 1303 
 
 1343 1383 
 
 1423 
 
 1463 1504 
 
 41 
 
 40 
 
 49 
 
 1504 1544 
 
 1585 
 
 1626 
 
 1667 
 
 1708 
 
 1750 1792 
 
 1833 
 
 18751.1918 
 
 40 
 
 41 
 
 50 
 
 1.1918 1960 
 
 2002 
 
 2045 
 
 2088 
 
 2131 
 
 2174 2218 
 
 2261 
 
 2305 2349 
 
 39 
 
 43 
 
 51 
 
 234 
 
 9 2393 
 
 2437 
 
 2482 
 
 2527 
 
 2572 
 
 2617 2662 
 
 2708 
 
 2753 2799 
 
 38 
 
 45 
 
 52 
 
 2799 2846 
 
 2892 
 
 2938 
 
 2985 
 
 3032 
 
 3079 3127 
 
 3175 
 
 3222 3270 
 
 37 
 
 47 
 
 53 
 
 327 
 
 3319 
 
 3367 
 
 3416 
 
 3465 
 
 3514 
 
 3564 3613 
 
 3663 
 
 3713 3764 
 
 36 
 
 49 
 
 54 
 
 3764 3814 
 
 3865 
 
 3916 
 
 3968 
 
 4019 
 
 4071 4124 
 
 4176 
 
 42291.4281 
 
 35 
 
 52 
 
 55 
 
 1 .4281 4335 
 
 4388 
 
 4442 
 
 44% 
 
 4550 
 
 4605 4659 
 
 4715 
 
 4770 4826 
 
 34 
 
 55 
 
 56 
 
 4826 4882 
 
 4938 
 
 4994 
 
 5051 
 
 5108 
 
 5166 5224 
 
 5282 
 
 5340 5399 
 
 33 
 
 57 
 
 57 
 
 539 
 
 9 5458 
 
 5517 
 
 5577 
 
 5637 
 
 5697 
 
 5757 5818 
 
 5880 
 
 5941 6003 
 
 32 
 
 60 
 
 58 
 
 6003 6066 
 
 6128 
 
 6191 
 
 6255 
 
 6319 
 
 6383 6447 
 
 6512 
 
 6577 6643 
 
 31 
 
 64 
 
 59 
 
 1.6643 6709 
 
 6775 
 
 6842 
 
 6909 
 
 6977 
 
 7045 7113 
 
 7182 
 
 7251 1.7321 
 
 30 
 
 67 
 
 60 
 
 1.732 1.739 
 
 1.746 
 
 1.753 
 
 1.760 
 
 1.76} 
 
 1.775 1.782 
 
 1.789 
 
 1.797 1.804 
 
 29 
 
 7 
 
 61 
 
 1.80 
 
 4 1.811 
 
 1.819 
 
 1.827 
 
 1.834 
 
 1.842 
 
 1.849 1.857 
 
 1.865 
 
 1.873 1.881 
 
 28 
 
 8 
 
 62 
 
 1.88 
 
 1 1.889 
 
 1.897 
 
 1.905 
 
 1.913 
 
 1.921 
 
 1.929 1.937 
 
 1.946 
 
 1.954 1.963 
 
 27 
 
 8 
 
 63 
 
 1.963 1.971 
 
 1.980 
 
 1.988 
 
 1.997 
 
 2.006 
 
 2.014 2023 
 
 2.032 
 
 2.041 2.050 
 
 26 
 
 9 
 
 64 
 
 2.050 2.059 
 
 2.069 
 
 2.078 
 
 2.087 
 
 2.097 
 
 2.1062.116 
 
 2.125 
 
 2.135 2.145 
 
 25 
 
 9 
 
 65 
 
 2.145 2.154 
 
 2.164 
 
 2.174 
 
 2.184 
 
 2.194 
 
 2.204 2.215 
 
 2.225 
 
 2.236 2.246 
 
 24 
 
 10 
 
 66 
 
 2.24 
 
 6 2.257 
 
 2.267 
 
 2.278 
 
 2.289 
 
 2.300 
 
 2.311 2.322 
 
 2.333 
 
 2.344 2.356 
 
 23 
 
 11 
 
 67 
 
 2.356 2.367 
 
 2.379 
 
 2.391 
 
 2.402 
 
 2.414 
 
 2.426 2.438 
 
 2.450 
 
 2.463 2.475 
 
 22 
 
 12 
 
 68 
 
 2.47 
 
 5 2.488 
 
 2.500 
 
 2.513 
 
 2.526 
 
 2.539 
 
 2.552 2.565 
 
 2.578 
 
 2.592 2.605 
 
 21 
 
 13 
 
 69 
 
 2.605 2.619 
 
 2.633 
 
 2.646 
 
 2.660 
 
 2.675 
 
 2.689 2.703 
 
 2.718 
 
 2.733 2.747 
 
 20 
 
 14 
 
 70 
 
 2.747 2.762 
 
 2.778 
 
 2.793 
 
 2.808 
 
 2.824 
 
 2.840 2.856 
 
 2.872 
 
 2.888 2.904 
 
 19 
 
 16 
 
 71 
 
 2.904 2.921 
 
 2.937 
 
 2.954 
 
 2.971 
 
 2.989 
 
 3.006 3.024 
 
 3.042 
 
 3.060 3.078 
 
 18 
 
 17 
 
 72 
 
 3.07 
 
 8 3.096 
 
 3.115 
 
 3.133 
 
 3.152 
 
 3.172 
 
 3.191 3.211 
 
 3.230 
 
 3.251 3.271 
 
 17 
 
 19 
 
 73 
 
 3.271 3.291 
 
 3.312 
 
 3.333 
 
 3.354 
 
 3.376 
 
 3.398 3.420 
 
 3.442 
 
 3.465 3.487 
 
 16 
 
 22 
 
 74 
 
 3.487 3.511 
 
 3.534 
 
 3.558 
 
 3.582 
 
 3.606 
 
 3.630 3.655 
 
 3.681 
 
 3.706 3.732* 
 
 15 
 
 24 
 
 75 
 
 3.732 3.758 
 
 3.785 
 
 3.812 
 
 3.839 
 
 3.867 
 
 3.895 3.923 
 
 3.952 
 
 3.981 4.011 
 
 14 
 
 28 
 
 76 
 
 4.01 
 
 1 4.041 
 
 4.071 
 
 4.102 
 
 4.134 
 
 4.165 
 
 4.198 4.230 
 
 4.264 
 
 4.297 4.331 
 
 13 
 
 32 
 
 77 
 
 4.331 4.366 
 
 4.402 
 
 4.437 
 
 4.474 
 
 4.511 
 
 4.548 4.586 
 
 4.625 
 
 4.665 4.705 
 
 12 
 
 37 
 
 78 
 
 4.70 
 
 5 4.745 
 
 4.787 
 
 4.829 
 
 4.872 
 
 4.915 
 
 4.959 5.005 
 
 5.050 
 
 5.097 5.145 
 
 11 
 
 44 
 
 79 
 
 5.145 5.193 
 
 5.242 
 
 5.292 
 
 5.343 
 
 5.396 
 
 5.449 5.503 
 
 5.558 
 
 5.614 5:671 
 
 10 
 
 53 
 
 80 
 
 5,671 5.730 
 
 5.789 
 
 5.850 
 
 5.912 
 
 5.976 
 
 6.041 6.107 
 
 6.174 
 
 6.243 6.314 
 
 9 
 
 
 81 
 
 6.314 6.386 
 
 6.460 
 
 6.535 
 
 6.612 
 
 6.691 
 
 6.772 6 855 
 
 6.940 
 
 7.026 7.115 
 
 8 
 
 
 82 
 
 7.11 
 
 5 7.207 
 
 7.300 
 
 7.396 
 
 7.495 
 
 7.596 
 
 7.700 7.806 
 
 7.916 
 
 8.028 8.144 
 
 7 
 
 
 83 
 
 8.14 
 
 4 8.264 
 
 8.386 
 
 8.513 
 
 8.643 
 
 8.777 
 
 8.915 9.058 
 
 9.205 
 
 9.357 9.514 
 
 6 
 
 
 84 
 
 9.514 9.677 
 
 9.845 
 
 10.02 
 
 10.20 
 
 10.39 
 
 10.58 10.78 
 
 10.99 
 
 11.20 11.43 
 
 5 
 
 
 85 
 
 11.43 11.66 
 
 11.91 
 
 12.16 
 
 12.43 
 
 12.71 
 
 13.00 1330 
 
 13.62 
 
 13.95 14.30 
 
 4 
 
 
 86 
 
 14.3 
 
 [) 14.67 
 
 15.06 
 
 15.46 
 
 15.89 
 
 16.35 
 
 16.83 17.34 
 
 17.89 
 
 18.46 19.08 
 
 3 
 
 
 87 
 
 19.a 
 
 8 19.74 
 
 20.45 
 
 21.20 
 
 22.02 
 
 22.90 
 
 23.86 24^90 
 
 26.03 
 
 27.27 28.64 
 
 2 
 
 
 88 
 
 28.64 30.14 
 
 31.82 
 
 33.69 
 
 35.80 
 
 38.19 
 
 40.92 44.07 
 
 47.74 
 
 52.08 57.29 
 
 1 
 
 
 89 
 
 57.29 63.66 
 
 71.62 
 
 81.85 
 
 95.49 
 
 114.6 
 
 143.2 191.0 
 
 286.5 
 
 573.0 oo 
 
 
 
 
 90 
 
 00 
 
 
 
 
 
 
 
 
 
 
 
 
 =(540 
 
 (480 
 
 (420 
 
 (360 
 
 (300 
 
 (240 (180 
 
 (120 
 
 (60 (0*0 
 
 i 
 
 
 Natural Cotangents 
 
50 
 
 MATHEMATICAL TABLES 
 
 NATURAL SECANTS AND COSECANTS 
 
 Natural Secants at intervals of 0. 1, or 6'. (For 10' intervals, see pp. 52-56) 
 
 i 
 
 -(V 
 
 (60* 
 
 (120 
 
 (180 (240 
 
 (3V) 
 
 (360 
 
 (420 
 
 (480 (540 
 
 
 
 Avg. 
 diff. 
 
 
 
 
 
 
 
 
 
 
 1.0000 
 
 90 
 
 
 
 
 1.0001 
 
 3 0000 
 
 0000 
 
 0000 0000 
 
 0000 
 
 0001 
 
 0001 
 
 0001 0001 
 
 0002 
 
 89 
 
 
 
 1 
 
 000 
 
 I 0002 
 
 0002 
 
 0003 0003 
 
 0003 
 
 0004 
 
 0004 
 
 0005 0006 
 
 0006 
 
 88 
 
 
 
 2 
 
 000 
 
 i 0007 
 
 0007 
 
 0008 0009 
 
 0010 
 
 0010 
 
 0011 
 
 0012 0013 
 
 0014 
 
 87 
 
 
 3 
 
 001' 
 
 \ 0015 
 
 0016 
 
 0017 0018 
 
 0019 
 
 0020 
 
 0021 
 
 0022 0023 
 
 0024 
 
 86 
 
 1 
 
 4 
 
 002' 
 
 I 0026 
 
 0027 
 
 0028 0030 
 
 0031 
 
 0032 
 
 0034 
 
 0035 0037 
 
 1.0038 
 
 85 
 
 1 
 
 5 
 
 1.003 
 
 J 0040 
 
 0041 
 
 0043 0045 
 
 0046 
 
 0048 
 
 0050 
 
 0051 0053 
 
 0055 
 
 84 
 
 2 
 
 6 
 
 005. 
 
 > 0057 
 
 0059 
 
 0061 0063 
 
 0065 
 
 0067 
 
 0069 
 
 0071 0073 
 
 0075 
 
 83 
 
 2 
 
 7 
 
 007. 
 
 > 0077 
 
 0079 
 
 0082 0084 
 
 0086 
 
 0089 
 
 0091 
 
 0093 0096 
 
 0098 
 
 82 
 
 2 
 
 8 
 
 009* 
 
 ) 0101 
 
 0103 
 
 0106 0108 
 
 0111 
 
 0114 
 
 0116 
 
 0119 0122 
 
 0125 
 
 81 
 
 3 
 
 9 
 
 01 2f 
 
 > 0127 
 
 0130 
 
 0133 0136 
 
 0139 
 
 0142 
 
 0145 
 
 0148 0151 
 
 1.0154 
 
 80 
 
 3 
 
 10 
 
 
 \ 0157 
 
 0161 
 
 0164 0167 
 
 0170 
 
 0174 
 
 0177 
 
 0180 0184 
 
 0187 
 
 79 
 
 3 
 
 11 
 
 018} 
 
 ' 0191 
 
 0194 
 
 0198 0201 
 
 0205 
 
 0209 
 
 0212 
 
 0216 0220 
 
 0223 
 
 78 
 
 4 
 
 12 
 
 022: 
 
 I 0227 
 
 0231 
 
 0235 0239 
 
 0243 
 
 0247 
 
 0251 
 
 0255 0259 
 
 0263 
 
 77 
 
 4 
 
 13 
 
 026: 
 
 I 0267 
 
 0271 
 
 0276 0280 
 
 0284 
 
 0288 
 
 0293 
 
 0297 0302 
 
 0306 
 
 76 
 
 4 
 
 14 
 
 030* 
 
 > 0311 
 
 0315 
 
 0320 0324 
 
 0329 
 
 0334 
 
 0338 
 
 0343 0348 
 
 1.0353 
 
 75 
 
 5 
 
 15 
 
 1 .035: 
 
 I 0358 
 
 0363 
 
 0367 0372 
 
 0377 
 
 *0382 
 
 0388 
 
 0393 0398 
 
 0403 
 
 74 
 
 5 
 
 16 
 
 040: 
 
 5 0408 
 
 0413 
 
 0419 0424 
 
 0429 
 
 0435 
 
 0440 
 
 0446 0451 
 
 0457 
 
 73 
 
 5 
 
 17 
 
 0453 
 
 ' 0463 
 
 0468 
 
 0474 0480 
 
 0485 
 
 0491 
 
 0497 
 
 0503 0509 
 
 0515 
 
 72 
 
 6 
 
 18 
 
 051f 
 
 0521 
 
 0527 
 
 0533 0539 
 
 0545 
 
 0551 
 
 0557 
 
 0564 0570 
 
 0576 
 
 71 
 
 6 
 
 19 
 
 057* 
 
 0583 
 
 0589 
 
 0595 0602 
 
 0608 
 
 0615 
 
 0622 
 
 0628 0635 
 
 1.0642 
 
 70 
 
 7 
 
 20 
 
 1.064; 
 
 ' 0649 
 
 0655 
 
 0662 0669 
 
 0676 
 
 0683 
 
 0690 
 
 0697 0704 
 
 0711 
 
 69 
 
 7 
 
 21 
 
 071 
 
 0719 
 
 0726 
 
 0733 0740 
 
 0748 
 
 0755 
 
 0763 
 
 0770 0778 
 
 0785 
 
 68 
 
 7 
 
 22 
 
 078f 
 
 0793 
 
 0801 
 
 0808 0816 
 
 0824 
 
 0832 
 
 0840 
 
 0848 0856 
 
 0864 
 
 67 
 
 8 
 
 23 
 
 0864 
 
 0872 
 
 0880 
 
 0888 0896 
 
 0904 
 
 0913 
 
 0921 
 
 0929 0938 
 
 0946 
 
 66 
 
 8 
 
 24 
 
 094* 
 
 0955 
 
 0963 
 
 0972 0981 
 
 0989 
 
 0998 
 
 1007 
 
 1016 1025 
 
 1.1034 
 
 65 
 
 9 
 
 25 
 
 1.103' 
 
 f 1043 
 
 1052 
 
 1061 1070 
 
 1079 
 
 1089 
 
 1098 
 
 1107 1117 
 
 1126 
 
 64 
 
 9 
 
 26 
 
 112* 
 
 > 1136 
 
 1145 
 
 1155 1164 
 
 1174 
 
 1184 
 
 1194 
 
 1203 1213 
 
 1223 
 
 63 
 
 10 
 
 27 
 
 1222 
 
 1233 
 
 1243 
 
 1253 1264 
 
 1274 
 
 1284 
 
 1294 
 
 1305 1315 
 
 1326 
 
 62 
 
 10 
 
 28 
 
 132* 
 
 1336 
 
 1347 
 
 1357 1368 
 
 1379 
 
 1390 
 
 1401 
 
 1412 1423 
 
 1434 
 
 61 
 
 11 
 
 29 
 
 1434 
 
 1445 
 
 1456 
 
 1467 1478 
 
 1490 
 
 1501 
 
 1512 
 
 1524 1535 
 
 1.1547 
 
 60 
 
 11 
 
 30 
 
 1.1543 
 
 ' 1559 
 
 1570 
 
 1582 1594 
 
 1606 
 
 1618 
 
 1630 
 
 1642 1654 
 
 1666 
 
 59 
 
 12 
 
 31 
 
 166* 
 
 1679 
 
 1691 
 
 1703 1716 
 
 1728 
 
 1741 
 
 1753 
 
 1766 1779 
 
 1792 
 
 58 
 
 13 
 
 32 
 
 179; 
 
 1805 
 
 1818 
 
 1831 1844 
 
 . 1857 
 
 1870 
 
 1883 
 
 1897 1910 
 
 1924 
 
 57 
 
 13 
 
 33 
 
 1924 
 
 1937 
 
 1951 
 
 1964 1978 
 
 1992 
 
 2006 
 
 2020 
 
 2034 2048 
 
 2062 
 
 56 
 
 14 
 
 34 
 
 206; 
 
 2076 
 
 2091 
 
 2105 2120 
 
 2134 
 
 2149 
 
 2163 
 
 2178 2193 
 
 1.2208 
 
 55 
 
 15 
 
 35 
 
 1.220* 
 
 2223 
 
 2238 
 
 2253 2268 
 
 2283 
 
 2299 
 
 2314 
 
 2329 2345 
 
 2361 
 
 54 
 
 15 
 
 36 
 
 2361 
 
 2376 
 
 2392 
 
 2408 2424 
 
 2440 
 
 245* 
 
 2472 
 
 2489 2505 
 
 2521 
 
 53 
 
 16 
 
 37 
 
 2521 
 
 2538 
 
 2554 
 
 2571 2588 
 
 2605 
 
 2622 
 
 2639 
 
 2656 2673 
 
 2690 
 
 52 
 
 17 
 
 38 
 
 269C 
 
 2708 
 
 2725 
 
 2742 2760 
 
 2778 
 
 2796 
 
 2813 
 
 2831 2849 
 
 2868 
 
 51 
 
 18 
 
 39 
 
 2866 
 
 2886 
 
 2904 
 
 2923 2941 
 
 2960 
 
 2978 
 
 2997 
 
 3016 3035 
 
 1.3054 
 
 60 
 
 19 
 
 40 
 
 1.3054 
 
 3073 
 
 3093 
 
 3112 3131 
 
 3151 
 
 3171 
 
 3190 
 
 3210 3230 
 
 3250 
 
 49 
 
 20 
 
 41 
 
 325C 
 
 3270 
 
 3291 
 
 3311 3331 
 
 3352 
 
 3373 
 
 3393 
 
 34 M 3435 
 
 3456 
 
 48 
 
 21 
 
 42 
 
 345* 
 
 3478 
 
 3499 
 
 3520 3542 
 
 3563 
 
 3585 
 
 3607 
 
 3629 3651 
 
 3673 
 
 47 
 
 22 
 
 43 
 
 3673 
 
 3696 
 
 3718 
 
 3741 3763 
 
 3786 
 
 3809 
 
 3832 
 
 3855 3878 
 
 3902 
 
 46 
 
 23 
 
 44 
 
 3902 
 
 3925 
 
 3949 
 
 3972 3996 
 
 4020 
 
 4044 
 
 4069 
 
 4093 4118 
 
 1.4142 
 
 45 
 
 24 
 
 45 
 
 1.4142 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 t 
 
 
 
 
 
 , 
 
 
 
 
 = (540 
 
 (480 
 
 (420 (360 
 
 (300 
 
 (240 
 
 (180 
 
 (120 (60 
 
 (00 
 
 Q 
 
 
 (For graphs, see p. 174.) 
 
 Natural Cosecants 
 
MATHEMATICAL TABLES 
 
 51 
 
 NATURAL SECANTS AND COSECANTS (continued) 
 
 Natural Secants at intervals of 0.l, or 6'. (For 10' intervals, see pp. 52-56) 
 
 
 
 .o 
 
 .l 
 
 .a 
 
 .3 
 
 .4 
 
 .5 
 
 .6 .7 
 
 .8 .9 
 
 
 
 Avg. 
 
 Q 
 
 =(00 
 
 (60 
 
 (120 
 
 (180 
 
 (240 
 
 (300 
 
 (360 (420 
 
 (480 (540 
 
 
 
 diff. 
 
 
 
 
 
 
 
 
 
 
 1.4142 
 
 45 
 
 
 45 
 
 1.4142 
 
 4167 
 
 4192 
 
 4217 
 
 4242 
 
 4267 
 
 4293 4318 
 
 4344 4370 
 
 4396 
 
 44 
 
 25 
 
 46 
 
 4396 
 
 4422 
 
 4448 
 
 4474 
 
 4501 
 
 4527 
 
 4554 4581 
 
 4608 4635 
 
 4663 
 
 43 
 
 27 
 
 47 
 
 4663 
 
 4690 
 
 4718 
 
 4746 
 
 4774 
 
 4802 
 
 4830 4859 
 
 4887 4916 
 
 4945 
 
 42 
 
 28 
 
 48 
 
 4945 
 
 4974 
 
 5003 
 
 5032 
 
 5062 
 
 5092 
 
 5121 5151 
 
 5182 5212 
 
 5243 
 
 41 
 
 30 
 
 49 
 
 5243 
 
 5273 
 
 5304 
 
 5335 
 
 5366 
 
 5398 
 
 5429 5461 
 
 5493 5525 
 
 1.5557 
 
 40 
 
 31 
 
 60 
 
 1.5557 
 
 5590 
 
 5622 
 
 5655 
 
 5688 
 
 5721 
 
 5755 5788 
 
 5822 5856 
 
 5890 
 
 39 
 
 33 
 
 51 
 
 5890 
 
 5925 
 
 5959 
 
 5994 
 
 6029 
 
 6064 
 
 6099 6135 
 
 6171 6207 
 
 6243 
 
 38 
 
 35 
 
 52 
 
 6243 
 
 6279 
 
 6316 
 
 6353 
 
 6390 
 
 6427 
 
 6464 6502 
 
 6540 6578 
 
 6616 
 
 37 
 
 37 
 
 53 
 
 6616 
 
 6655 
 
 6694 
 
 6733 
 
 6772 
 
 6812 
 
 6852 6892 
 
 6932 6972 
 
 7013 
 
 36 
 
 40 
 
 54 
 
 7013 
 
 7054 
 
 7095 
 
 7137 
 
 7179 
 
 7221 
 
 7263 7305 
 
 7348 7391 
 
 1.7434 
 
 35 
 
 42 
 
 55 
 
 1.7434 
 
 7478 
 
 7522 
 
 7566 
 
 7610 
 
 7655 
 
 7700 7745 
 
 7791 7837 
 
 7883 
 
 34 
 
 45 
 
 56 
 
 7883 
 
 7929 
 
 7976 
 
 8023 
 
 8070 
 
 8118 
 
 8166 8214 
 
 8263 8312 
 
 8361 
 
 33 
 
 48 
 
 57 
 
 8361 
 
 8410 
 
 8460 
 
 8510 
 
 8561 
 
 8612 
 
 8663 8714 
 
 8766 8818 
 
 8871 
 
 32 
 
 51 
 
 58 
 
 8871 
 
 8924 
 
 8977 
 
 9031 
 
 9084 
 
 9139 
 
 9194 9249 
 
 9304 9360 
 
 1.9416 
 
 31 
 
 54 
 
 59 
 
 1.9416 
 
 9473 
 
 9530 
 
 9587 
 
 9645 
 
 9703 
 
 9762 9821 
 
 9880 9940 
 
 2.0000 
 
 30 
 
 58 
 
 60 
 
 2.00C 
 
 2.006 
 
 2.012 
 
 2.018 
 
 2.025 
 
 2.031 
 
 2.037 2.043 
 
 2.050 2.056 
 
 2.063 
 
 29 
 
 6 
 
 61 
 
 2.063 
 
 2.069 
 
 2.076 
 
 2.082 
 
 2.089 
 
 2.096 
 
 2.103 2.109 
 
 2.116 2.123 
 
 2.130 
 
 28 
 
 7 
 
 62 
 
 2.13C 
 
 2.137 
 
 2.144 
 
 2.151 
 
 2.158 
 
 2.166 
 
 2.173 2.180 
 
 2.188 2.195 
 
 2.203 
 
 27 
 
 7 
 
 63 
 
 2.203 
 
 2.210 
 
 2.218 
 
 2.226 
 
 2.233 
 
 2.241 
 
 2.249 2.257 
 
 2.265 2.273 
 
 2281 
 
 26 
 
 8 
 
 64 
 
 2.281 
 
 2.289 
 
 2.298 
 
 2.306 
 
 2.314 
 
 2.323 
 
 2.331 2.340 
 
 2349 2357 
 
 2366 
 
 25 
 
 8 
 
 65 
 
 2.36< 
 
 2375 
 
 2.384 
 
 2.393 
 
 2.402 
 
 2.411 
 
 2.421 2.430 
 
 2.439 2.449 
 
 2.459 
 
 24 
 
 9 
 
 66 
 
 2.45< 
 
 2.468 
 
 2.478 
 
 2.488 
 
 2.498 
 
 2.508 
 
 2.518 2.528 
 
 2.538 2.549 
 
 2.559 
 
 23 
 
 10 
 
 67 
 
 2.55< 
 
 2.570 
 
 2.581 
 
 2.591 
 
 2.602 
 
 2.613 
 
 2624 2.635 
 
 2.647 2.658 
 
 2.669 
 
 22 
 
 11 
 
 68 
 
 2.66' 
 
 2.681 
 
 2.693 
 
 2.705 
 
 2.716 
 
 2.729 
 
 2.741 2.753 
 
 2.765 2.778 
 
 2.790 
 
 21 
 
 12 
 
 69 
 
 2.79( 
 
 2.803 
 
 2.816 
 
 2.829 
 
 2.842 
 
 2.855 
 
 2.869 2.882 
 
 2.896 2.910 
 
 2.924 
 
 20 
 
 13 
 
 70 
 
 2.92^ 
 
 \ 2.938 
 
 2.952 
 
 2.967 
 
 2.981 
 
 2.996 
 
 3.011 3.026 
 
 3.041 3.056 
 
 3.072 
 
 19 
 
 15 
 
 71 
 
 3.07; 
 
 3.087 
 
 3.103 
 
 3.119 
 
 3.135 
 
 3.152 
 
 3.168 3.185 
 
 3.202 3.219 
 
 3.236 
 
 18 
 
 16 
 
 72 
 
 3.23( 
 
 3.254 
 
 3.271 
 
 3.289 
 
 3.307 
 
 3326 
 
 3.344 3.363 
 
 3.382 3.401 
 
 3.420 
 
 17 
 
 18 
 
 73 
 
 3.42C 
 
 3.440 
 
 3.460 
 
 3.480 
 
 3.500 
 
 3.521 
 
 3.542 3.563 
 
 3.584 3.606 
 
 3.628 
 
 16 
 
 21 
 
 74 
 
 3.62* 
 
 I 3.650 
 
 3.673 
 
 3.695 
 
 3.719 
 
 3.742 
 
 3.766 3.790 
 
 3.814 3.839 
 
 3.864 
 
 15 
 
 24 
 
 75 
 
 3.86^ 
 
 I 3.889 
 
 3.915 
 
 3.941 
 
 3.967 
 
 3.994 
 
 4.021 4.049 
 
 4.077 4.105 
 
 4.134 
 
 14 
 
 27 
 
 76 
 
 4.13^ 
 
 4.163 
 
 4.192 
 
 4.222 
 
 4.253 
 
 4.284 
 
 4.315 4.347 
 
 4379 4.412 
 
 4.445 
 
 13 
 
 31 
 
 77 
 
 4.44! 
 
 4.479 
 
 4.514 
 
 4.549 
 
 4.584 
 
 4.620 
 
 4.657 4.694 
 
 4.732 4.771 
 
 4.810 
 
 12 
 
 36 
 
 78 
 
 4.8H 
 
 1 4.850 
 
 4.890 
 
 4.931 
 
 4.973 
 
 5.016 
 
 5.059 5.103 
 
 5.148 5.194 
 
 5.241 
 
 11 
 
 43 
 
 79 
 
 5.24 
 
 5.288 
 
 5337 
 
 5.386 
 
 5.436 
 
 5.487 
 
 5.540 5.593 
 
 5.647 5.702 
 
 5.759 
 
 10 
 
 52 
 
 80 
 
 5.75 C 
 
 > 51816 
 
 5.875 
 
 5.935 
 
 5.996 
 
 6.059 
 
 6.123 6.188 
 
 6.255 6.323 
 
 6392 
 
 9 
 
 
 81 
 
 6.39; 
 
 ! 6.464 
 
 6.537 
 
 6.611 
 
 6.687 
 
 6.765 
 
 6.845 6.927 
 
 7.011 7.097 
 
 7.185 
 
 8 
 
 
 82 
 
 7.18f 
 
 7.276 
 
 7.368 
 
 7.463 
 
 7.561 
 
 7.661 
 
 7.764 7.870 
 
 7.979 8.091 
 
 8.206 
 
 7 
 
 
 83 
 
 8.2W 
 
 8.324 
 
 8.446 
 
 8.571 
 
 8.700 
 
 8.834 
 
 8.971 9.113 
 
 9.259 9.411 
 
 9.567 
 
 6 
 
 
 84 
 
 9.563 
 
 9.728 
 
 9.895 
 
 10.07 
 
 1025 
 
 10.43 
 
 10.63 10.83 
 
 11.03 11.25 
 
 11.47 
 
 5 
 
 
 85 
 
 11.42 
 
 ' 11.71 
 
 11.95 
 
 12.20 
 
 12.47 
 
 12.75 
 
 13.03 13.34 
 
 13.65 13.99 
 
 1434 
 
 4 
 
 
 86 
 
 14.3' 
 
 \ 14.70 
 
 15.09 
 
 15.50 
 
 15.93' 
 
 16.38 
 
 16.86 17.37 
 
 17.91 18.49 
 
 19.11 
 
 3 
 
 
 87 
 
 19.1 
 
 19.77 
 
 20.47 
 
 21.23 
 
 22,04 
 
 22.93 
 
 23.88 24.92 
 
 26.05 27.29 
 
 28.65 
 
 2 
 
 
 88 
 
 28.6! 
 
 30.16 
 
 31.84 
 
 33.71 
 
 35.81 
 
 38.20 
 
 40.93 44.08 
 
 47.75 52.09 
 
 57.30 
 
 
 
 89 
 
 573( 
 
 ) 63.66 
 
 71.62 
 
 81.85 
 
 95.49 
 
 114.6 
 
 143.2 191.0 
 
 286.5 573.0 
 
 00 
 
 
 
 
 90 
 
 oo 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 .9 
 
 .8 
 
 .7 
 
 .6 
 
 .5 
 
 
 
 
 M 
 
 
 
 
 =(540 
 
 (480 
 
 (420 
 
 (360 
 
 (300 
 
 (240 (180 
 
 (120 (60 
 
 (oo 
 
 a 
 
 
 Natural Cosecants 
 
52 
 
 MATHEMATICAL TABLES 
 
 TRIGONOMETRIC FUNCTIONS (at intervals of 10') 
 
 Annex -10 in columns marked *. (For O.l intervals, see pp. 46-51) 
 
 De- 
 grees 
 
 Ra- 
 dians 
 
 Sines 
 
 Cosines 
 
 Tangents 
 
 Cotangents 
 
 
 
 
 Nat. Log.* 
 
 Nat. Log.* 
 
 Nat. Log.* 
 
 Nat. Log. 
 
 
 
 (XX 
 
 0.0000 
 
 .0000 
 
 t.OOOO 0.0000 
 
 .0000 
 
 CO 00 
 
 1.5708 
 
 90 00* 
 
 10 
 
 00029 
 
 .0029 7.4637 
 
 1.0000 .0000 
 
 .0029 7.4637 
 
 343.77 2.5363 
 
 1.5679 
 
 50 
 
 20 
 
 00058 
 
 .0058 .7648 
 
 1.0000 .0000 
 
 .0058 .7648 
 
 171.89 .2352 
 
 1.5650 
 
 40 
 
 30 
 
 00087 
 
 .0087 .9408 
 
 1.0000 .0000 
 
 .0087 .9409 
 
 114.59 .0591 
 
 1.5621 
 
 30 
 
 40 
 
 0.0116 
 
 .0116 8.0658 
 
 0.9999 .0000 
 
 .0116 8.0658 
 
 85.940 1.9342 
 
 1.5592 
 
 20 
 
 50 
 
 0.0145 
 
 .0145 .1627 
 
 .9999 .0000 
 
 .0145 .1627 
 
 68.750 .8373 
 
 1.5563 
 
 10 
 
 1 00' 
 
 0.0175 
 
 .0175 8.2419 
 
 .9998 9.9999 
 
 .0175 8.2419 
 
 57.290 1.7581 
 
 1.5533 
 
 89 00' 
 
 10 
 
 0.0204 
 
 .0204 .3088 
 
 .9998 .9999 
 
 .0204 .3089 
 
 49.104 .6911 
 
 1.5504 
 
 50 
 
 20 
 
 0.0233 
 
 .0233 .3668 
 
 .9997 .9999 
 
 .0233 .3669 
 
 42.964 .6331 
 
 1.5475 
 
 40 
 
 30 
 
 0.0262 
 
 .0262 .4179 
 
 .9997 .9999 
 
 .0262 .4181 
 
 38.188 .5819 
 
 1.5446 
 
 30 
 
 40 
 
 0.0291 
 
 .0291 .4637 
 
 .9996 .9998 
 
 .0291 .4638 
 
 34.368 .5362 
 
 1.5417 
 
 20 
 
 50 
 
 0.0320 
 
 .0320 .5050 
 
 .9995 .9998 
 
 .0320 .5053 
 
 31.242 .4947 
 
 1.5388 
 
 10 
 
 2 (XX 
 
 0.0349 
 
 .0349 8.5428 
 
 .9994 9.9997 
 
 .0349 8.5431 
 
 28.636 1.4569 
 
 1.5359 
 
 88 00' 
 
 10 
 
 0.0378 
 
 .0378 .5776 
 
 .9993 .9997 
 
 .0378 .5779 
 
 26.432 .4221 
 
 1.5330 
 
 50 
 
 20 
 
 0.0407 
 
 .0407 .6097 
 
 .9992 .9996 
 
 .0407 .6101 
 
 24.542 .3899 
 
 1.5301 
 
 40 
 
 30 
 
 0.0436 
 
 .0436 .6397 
 
 .9990 .9996 
 
 .0437 .6401 
 
 22.904 .3599 
 
 1.5272 
 
 30 
 
 40 
 
 0.0465 
 
 .0465 .6677 
 
 .9989 .9995 
 
 .0466 .6682 
 
 21.470 .3318 
 
 1 .5243 
 
 20 
 
 50 
 
 0.0495 
 
 .0494 .6940 
 
 .9988 .9995 
 
 .0495 .6945 
 
 20.206 .3055 
 
 1.5213 
 
 10 
 
 3 (XX 
 
 0.0524 
 
 .0523 8.7188 
 
 .9986 9.9994 
 
 .0524 8.7194 
 
 19.081 1.2806 
 
 1.5184 
 
 87 00' 
 
 10 
 
 0.0553 
 
 .0552 .7423 
 
 .9985 .9993 
 
 .0553 .7429 
 
 18.075 .2571 
 
 1.5155 
 
 50 
 
 20 
 
 0.0582 
 
 .0581 .7645 
 
 .9983 .9993 
 
 .0582 .7652 
 
 17.169 .2348 
 
 1.5126 
 
 40 
 
 30 
 
 0.061 1 
 
 .0610 .7857 
 
 .9981 .9992 
 
 .0612 .7865 
 
 16.350 .2135 
 
 1.5097 
 
 30 
 
 40 
 
 0.0640 
 
 .0640 .8059 
 
 .9980 .9991 
 
 .0641 .8067 
 
 15.605 .1933 
 
 1.5068 
 
 20 
 
 50 
 
 0.0669 
 
 .0669 .8251 
 
 .9978 .9990 
 
 .0670 .8261 
 
 14.924 .1739 
 
 1.5039 
 
 10 
 
 4 (XX 
 
 0.0698 
 
 .0698 8.8436 
 
 .9976 9.9989 
 
 .0699 8.8446 
 
 14.301 1.1554 
 
 1.5010 
 
 86 00 
 
 10 
 
 0.0727 
 
 .0727 .8613 
 
 .9974 .9989 
 
 .0729 .8624 
 
 13.727 .1376 
 
 1.4981 
 
 50 
 
 20 
 
 0.0756 
 
 .0756 .8783 
 
 .9971 .9988 
 
 .0758 .8795 
 
 13.197 .1205 
 
 1.4952 
 
 40 
 
 30 
 
 0.0785 
 
 .0785 .8946 
 
 .9969 .9987 
 
 .0787 .8960 
 
 12.706 .1040 
 
 1.4923 
 
 30 
 
 40 
 
 0.0814 
 
 .0814 .9104 
 
 .9967 .9986 
 
 .0816 .9118 
 
 12.251 .0882 
 
 1.4893 
 
 20 
 
 50 
 
 0.0844 
 
 .0843 .9256 
 
 ,9964 .9985 
 
 .0846 .9272 
 
 11.826 .0728 
 
 1.4864 
 
 10 
 
 5 (XX 
 
 0.0873 
 
 .0872 8.9403 
 
 .9962 9.9983 
 
 .0875 8.9420 
 
 11.430 1.0580 
 
 1.4835 
 
 85 00' 
 
 10 
 
 0.0902 
 
 .0901 .9545 
 
 .9959 .9982 
 
 .0904 .9563 
 
 11.059 .0437 
 
 1.4806 
 
 50 
 
 20 
 
 0.0931 
 
 .0929 .9682 
 
 .9957 .9981 
 
 .0934 .9701 
 
 10.712 .0299 
 
 1.4777 
 
 40 
 
 30 
 
 0.0960 
 
 .0958 .9816 
 
 .9954 .9980 
 
 .0963 .9836 
 
 10.385 .0164 
 
 1.4748 
 
 30 
 
 40 
 
 0.0989 
 
 .0987 .9945 
 
 .9951 .9979 
 
 .0992 .9966 
 
 10.078 .0034 
 
 1.4719 
 
 ' 20 
 
 50 
 
 0.1018 
 
 .1016 9.0070 
 
 .9948 .9977 
 
 .1022 9.0093 
 
 9.7882 0.9907 
 
 1.4690 
 
 10 
 
 6 00' 
 
 0.1047 
 
 .1045 9.0192 
 
 .9945 9.9976 
 
 .1051 9.0216 
 
 9.5144 0.9784 
 
 1.4661 
 
 84 00' 
 
 10 
 
 0.1076 
 
 .1074 .0311 
 
 .9942 .9975 
 
 .1080 .0336 
 
 9.2553 .9664 
 
 1.4632 
 
 50 
 
 20 
 
 0.1105 
 
 .1103 .0426 
 
 .9939 .9973 
 
 .1110 .0453 
 
 9.0098 .9547 
 
 1.4603 
 
 40 
 
 30 
 
 0.1134 
 
 .1132 .0539 
 
 .9936 .9972 
 
 .1139 .0567 
 
 8.7769 .9433 
 
 1.4574 
 
 30 
 
 40 
 
 0.1164 
 
 .1161 .0648 
 
 .9932 .9971 
 
 .1169 .0678 
 
 8.5555 .9322 
 
 1.4544 
 
 20 
 
 50 
 
 0.1193 
 
 .1190 .0755 
 
 .9929 .9969 
 
 .1198 .0786 
 
 8.3450 .9214 
 
 1.4515 
 
 10 
 
 7 00' 
 
 0.1222 
 
 .1219 9.0859 
 
 .9925 9.9968 
 
 .1228 9.0891 
 
 8.1443 0.9109 
 
 1.4486 
 
 83 00' 
 
 10 
 
 0.1251 
 
 .1248 .0961 
 
 .9922 .9966 
 
 .1257 .0995 
 
 7.9530 .9005 
 
 1.4457 
 
 50 
 
 20 
 
 0.1280 
 
 .1276 .1060 
 
 .9918 .9964 
 
 .1287 .1096 
 
 7.7704 .8904 
 
 1.4428 
 
 40 
 
 30 
 
 0.1309 
 
 .1305 .1157 
 
 .9914 .9963 
 
 .1317 .1194 
 
 7.5958 .8806 
 
 1.4399 
 
 30 
 
 40 
 
 a 1338 
 
 .1334 .1252 
 
 .991 1 .9961 
 
 .1346 .1291 
 
 7.4287 .8709 
 
 1.4370 
 
 20 
 
 50 
 
 0.1367 
 
 .1363 .1345 
 
 .9907 .9959 
 
 .1376 .1385 
 
 7.2687 .8615 
 
 1.4341 
 
 10 
 
 8 00' 
 
 0.1396 
 
 .1392 9.1436 
 
 .9903 9.9958 
 
 .1405 9.1478 
 
 7.1154 0.8522 
 
 1.4312 
 
 82 00' 
 
 10 
 
 0.1425 
 
 .1421 .1525 
 
 .9899 .9956 
 
 .1435 .1569 
 
 6.9682 .8431 
 
 1.4283 
 
 50 
 
 20 
 
 0.1454 
 
 .1449 .1612 
 
 .9894 .9954 
 
 .1465 .1658 
 
 6.8269 .8342 
 
 1.4254 
 
 40 
 
 30 
 
 0.1484 
 
 .1478 .1697 
 
 .9890 .9952 
 
 .1495 .1745 
 
 6.6912 .8255 
 
 1.4224 
 
 30 
 
 40 
 
 0.1513 
 
 .1507 .1781 
 
 .9886 .9950 
 
 .1524 .1831 
 
 6.5606 .8169 
 
 1.4195 
 
 20 
 
 50 
 
 0.1542 
 
 .1536 .1863 
 
 .9881 .9948 
 
 .1554 .1915 
 
 6.4348 .8085 
 
 1.4166 
 
 10 
 
 9 (XX 
 
 0.1571 
 
 .1564 9.1943 
 
 .9877 9.9946 
 
 .1584 9.1997 
 
 6.3138 0.8003 
 
 1.4137 
 
 81 00' 
 
 
 
 Nat. Log.* 
 
 Nat. Log.* 
 
 Nat. Log.* 
 
 Nat. Log. 
 
 
 
 
 Cosines 
 
 Sines 
 
 Cotangents 
 
 Tangents 
 
 Ra- 
 dians 
 
 De- 
 grees 
 
MATHEMATICAL TABLES 
 
 53 
 
 TRIGONOMETRIC FUNCTIONS (continued) 
 
 Annex -10 in columns marked*. (For O.l intervals, see pp. 46-51) 
 
 De- 
 grees 
 
 Ra- 
 dians 
 
 Sines 
 
 Cosines 
 
 Tangents 
 
 Cotangent 
 
 
 
 
 Nat. Log.* 
 
 Nat. Log.* 
 
 Nat. Log.* 
 
 Nat. Log. 
 
 
 
 9 00' 
 
 0.1571 
 
 .1564 9.1943 
 
 .9877 9.9946 
 
 .1584 9.1997 
 
 6.3138 0.8003 
 
 1.4137 
 
 81" (XX 
 
 kio 
 
 0.1600 
 
 .1593 .2022 
 
 .9872 .9944 
 
 .1614 .2078 
 
 6.1970 .7922 
 
 1.4108 
 
 50 
 
 20 
 
 0.1629 
 
 .1622 .2100 
 
 .9868 .9942 
 
 .1644 .2158 
 
 6.0844 .7842 
 
 1.4079 
 
 40 
 
 30 
 
 0.1658 
 
 .1650 .2176 
 
 .9863 .9940 
 
 .1673 .2236 
 
 5.9758 .7764 
 
 1.4050 
 
 30 
 
 40 
 
 0.1687 
 
 .1679 .2251 
 
 .9858 .9938 
 
 .1703 .2313 
 
 5.8708 .7687 
 
 1.4021 
 
 20 
 
 50 
 
 0.1716 
 
 .1708 .2324 
 
 .9853 .9936 
 
 .1733 2389 
 
 5.7694 .7611 
 
 1.3992 
 
 10 
 
 10 00' 
 
 0.1745 
 
 .1736 9.2397 
 
 .9848 9.9934 
 
 .1763 9.2463 
 
 5.6713 0.7537 
 
 1.3963 
 
 80 (XX 
 
 10 
 
 0.1774 
 
 .1765 .2468 
 
 .9843 .9931 
 
 .1793 2536 
 
 5.5764 .7464 
 
 1.3934 
 
 50 
 
 20 
 
 0.1804 
 
 .1794 .2538 
 
 .9838 .9929 
 
 .1823 2609 
 
 5.4845 .7391 
 
 1.3904 
 
 40 
 
 30 
 
 0.1833 
 
 .1822 .2606 
 
 .9833 .9927 
 
 .1853 2680 
 
 5.3955 .7320 
 
 1.3875 
 
 30 
 
 40 
 
 0.1862 
 
 .1851 .2674 
 
 .9827 .9924 
 
 .1883 2750 
 
 5.3093 .7250 
 
 1.3846 
 
 20 
 
 50 
 
 0.1891 
 
 .1880 .2740 
 
 .9822 .9922 
 
 .1914 2819 
 
 5.2257 .7181 
 
 1.3817 
 
 10 
 
 11 00' 
 
 0.1920 
 
 .1908 9.2806 
 
 .9816 9.9919 
 
 .1944 92887 
 
 5.1446 0.7113 
 
 1.3788 
 
 790(X 
 
 10 
 
 0.1949 
 
 .1937 .2870 
 
 .9811 .9917 
 
 .1974 2953 
 
 5.0658 .7047 
 
 1.3759 
 
 50 
 
 20 
 
 0.1978 
 
 .1965 .2934 
 
 .9805 .9914 
 
 .2004 .3020 
 
 4.9894 .6980 
 
 1.3730 
 
 40 
 
 30 
 
 0.2007 
 
 .1994 .2997 
 
 .9799 .9912 
 
 2035 .3085 
 
 4.9152 .6915 
 
 1.3701 
 
 30 
 
 40 
 
 0.2036 
 
 .2022 .3058 
 
 .9793 .9909 
 
 2065 .3149 
 
 4.8430 .6851 
 
 1.3672 
 
 20 
 
 50 
 
 0.2065 
 
 2051 .3119 
 
 .9787 .9907 
 
 2095 .3212 
 
 4.7729 .6788 
 
 1.3643 
 
 10 
 
 12 00' 
 
 0.2094 
 
 .2079 9.3179 
 
 .9781 9.9904 
 
 2126 9.3275 
 
 4.7046 0.6725 
 
 1.3614 
 
 78 00' 
 
 10 
 
 0.2123 
 
 .2108 .3238 
 
 .9775 .9901 
 
 2156 3336 
 
 4.6382 .6664 
 
 1.3584 
 
 50 
 
 20 
 
 0.2153 
 
 .2136 .3296 
 
 .9769 .9899 
 
 2186 .3397 
 
 4.5736 .6603 
 
 1.3555 
 
 40 
 
 30 
 
 0.2182 
 
 .2164 .3353 
 
 .9763 .9896 
 
 2217 .3458 
 
 4.5107 .6542 
 
 1.3526 
 
 30 
 
 40 
 
 0.2211 
 
 .2193 .3410 
 
 .9757 .9893 
 
 2247 .3517 
 
 4.4494 .6483 
 
 1.3497 
 
 20 
 
 50 
 
 0.2240 
 
 .2221 .3466 
 
 .9750 .9890 
 
 2278 .3576 
 
 4.3897 .6424 
 
 1.3468 
 
 10 
 
 13 00' 
 
 0.2269 
 
 .2250 9.3521 
 
 .9744 9.9887 
 
 2309 9.3634 
 
 4.3315 0.6366 
 
 1.3439 
 
 77 (XX 
 
 10 
 
 0.2298 
 
 .2278 .3575 
 
 .9737 .9884 
 
 2339 .3691 
 
 4.2747 .6309 
 
 1.3410 
 
 50 
 
 20 
 
 0.2327 
 
 .2306 .3629 
 
 .9730 .9881 
 
 2370 .3748 
 
 4.2193 .6252 
 
 1.3381 
 
 40 
 
 30 
 
 0.2356 
 
 .2334 .3682 
 
 .9724 .9878 
 
 2401 .3804 
 
 4.1653 .6196 
 
 1.3352 
 
 30 
 
 40 
 
 0.2385 
 
 .2363 .3734 
 
 .9717 .9875 
 
 2432 .3859 
 
 4.1126 .6141 
 
 1.3323 
 
 20 
 
 50 
 
 0.2414 
 
 .2391 .3786 
 
 .9710 .9872 
 
 2462 .3914 
 
 4.0611 .6086 
 
 1.3294 
 
 10 
 
 14 00' 
 
 0.2443 
 
 .2419 9.3837 
 
 .9703 9.9869 
 
 2493 9.3968 
 
 4.0108 0.6032 
 
 1.3265 
 
 76 (XX 
 
 10 
 
 0.2473 
 
 .2447 .3887 
 
 .9696 .9866 
 
 2524 .4021 
 
 3.9617 .5979 
 
 1.3235 
 
 50 
 
 20 
 
 0.2502 
 
 .2476 .3937 
 
 .9689 .9863 
 
 2555 .4074 
 
 3.9136 .5926 
 
 1.3206 
 
 40 
 
 30 
 
 0.2531 
 
 2504 .3986 
 
 .9681 .9859 
 
 2586 .4127 
 
 3.8667 .5873 
 
 1.3177 
 
 30 
 
 40 
 
 0.2560 
 
 2532 .4035 
 
 .9674 .9856 
 
 2617 .4178 
 
 3.8208 .5822 
 
 1.3148 
 
 20 
 
 50 
 
 0.2589 
 
 2560 .4083 
 
 .9667 .9853 
 
 2648 .4230 
 
 3.7760 .5770 
 
 13119 
 
 10 
 
 15 00' 
 
 0.2618 
 
 .2588 9.4130 
 
 .9659 9.9849 
 
 2679 9.4281 
 
 3.7321 0.5719 
 
 1.3090 
 
 75 00' 
 
 10 
 
 0.2647 
 
 2616 .4177 
 
 .9652 .9846 
 
 2711 .4331 
 
 3.6891 .5669 
 
 1.3061 
 
 50 
 
 20 
 
 0.2676 
 
 2644 .4223 
 
 .9644 .9843 
 
 .2742 .4381 
 
 3.6470 .5619 
 
 1.3032 
 
 40 
 
 30 
 
 0.2705 
 
 2672 .4269 
 
 .9636 -.9839 
 
 .2773 .4430 
 
 3.6059 .5570 
 
 1.3003 
 
 30 
 
 40 
 
 0.2734 
 
 2700 .4314 
 
 .9628 .9836 
 
 2805 .4479 
 
 3.5656 .5521 
 
 1.2974 
 
 20 
 
 50 
 
 0.2763 
 
 2728 .4359 
 
 .9621 .9832 
 
 2836 -4527 
 
 3.5261 .5473 
 
 U945 
 
 10 
 
 16 00' 
 
 0.2793 
 
 2756 9.4403 
 
 .9613 9.9828 
 
 .2867 9.4575 
 
 3.4874 0.5425 
 
 12915 
 
 74 (XX 
 
 10 
 
 0.2822 
 
 2784 .4447 
 
 .9605 .9825 
 
 2899 .4622 
 
 3.4495 .5378 
 
 12886 
 
 50 
 
 20 
 
 0.2851 
 
 2812 .4491 
 
 ^9596 .9821 
 
 2931 .4669 
 
 3.4124 .5331 
 
 1.2857 
 
 40 
 
 30 
 
 0.2880 
 
 2840 .4533 
 
 .9588 .9817 
 
 2962 .4716 
 
 3.3759 .5284 
 
 1.2828 
 
 30 
 
 40 
 
 0.2909 
 
 2868 .4576 
 
 .9580 .9814 
 
 2994 .4762 
 
 3.3402 .5238 
 
 12799 
 
 20 
 
 50 
 
 0.2938 
 
 2896 .4618 
 
 .9572 .9810 
 
 .3026 .4808 
 
 3.3052 .5192 
 
 12770 
 
 10 
 
 17 (XX 
 
 0.2967 
 
 2924 9.4659 
 
 .9563 9.9806 
 
 .3057 9.4853 
 
 3.2709 0.5147 
 
 1.2741 
 
 3 (XX 
 
 10 
 
 0.2996 
 
 2952 .4700 
 
 .9555 .9802 
 
 3089 .4898 
 
 3.2371 .5102 
 
 12712 
 
 50 
 
 20 
 
 0.3025 
 
 2979 .4741 
 
 .9546 .9798 
 
 3121 .4943 
 
 3.2041 .5057 
 
 1.2683 
 
 40 
 
 30 
 
 0.3054 
 
 3007 .4781 
 
 9537 .9794 
 
 3153 .4987 
 
 3.1716 .5013 
 
 1.2654 
 
 30 
 
 40 
 
 0.3083 
 
 3035 .4821 
 
 9528 .9790 
 
 3185 .5031 
 
 3.1397 .4969 
 
 1.2625 
 
 20 
 
 50 
 
 0.3113 
 
 3062 .4861 
 
 9520 .9786 
 
 3217 .5075 
 
 3.1084 .4925 
 
 12595 
 
 10 
 
 18 (XX 
 
 0.3142 
 
 3090 9.4900 
 
 9511 9.9782 
 
 3249 9.5118 
 
 3.0777 0.4882 
 
 12566 
 
 2 (XX 
 
 
 
 Nat. Log.* 
 
 Nat. Log.* 
 
 Nat. Log.* 
 
 Nat. Log. 
 
 
 
 
 Cosines 
 
 Sines 
 
 Cotangents 
 
 Tangents 
 
 Ra- 
 dians 
 
 De- 
 grees 
 
54 
 
 MATHEMATICAL TABLES 
 
 TRIGONOMETRIC FUNCTIONS 
 
 Annex 10 in columns marked*. 
 
 (continued) 
 
 (For O.l intervals, Bee pp. 46-51) 
 
 De- 
 grees 
 
 Ra- 
 dians 
 
 Sines 
 
 Cosines 
 
 Tangents 
 
 Cotangents 
 
 
 
 
 
 Nat. Log. * 
 
 Nat. Log.* 
 
 Nat. Log.* 
 
 Nat. Log. 
 
 
 
 18 00' 
 
 0.3142 
 
 .3090 9.4900 
 
 .9511 9.9782 
 
 .3249 9.5118 
 
 3.0777 0.4882 
 
 1.2566 
 
 72 00* 
 
 10 
 
 0.3171 
 
 .3118 .4939 
 
 .9502 .9778 
 
 .3281 .5161 
 
 3.0475 .4839 
 
 1.2537 
 
 50 
 
 20 
 
 0.3200 
 
 .3145 .4977 
 
 .9492 .9774 
 
 .3314 .5203 
 
 3.0178 .4797 
 
 1.^508 
 
 40 
 
 30 
 
 0.3229 
 
 .3173 .5015 
 
 .9483 .9770 
 
 ,3346 .5245 
 
 2.9887 .4755 
 
 1.2479 
 
 30 
 
 40 
 
 0.3258 
 
 .3201 .5052 
 
 .9474 .9765 
 
 .3378 .5287 
 
 2.9600 .4713 
 
 1 .2450 
 
 20 
 
 50 
 
 0.3287 
 
 .3228 .5090 
 
 .9465 .9761 
 
 .3411 .5329 
 
 2.9319 .4671 
 
 1.2421 
 
 10 
 
 19 00' 
 
 0.3316 
 
 .3256 9.5126 
 
 .9455 9.9757 
 
 .3443 9.5370 
 
 2.9042 0.4630 
 
 1.2392 
 
 71 00' 
 
 10 
 
 0.3345 
 
 .3283 .5163 
 
 .9446 .9752 
 
 .3476 3411 
 
 2.8770 .4589 
 
 1.2363 
 
 50 
 
 20 
 
 0.3374 
 
 .3311 .5199 
 
 .9436 .9748 
 
 .3508 .5451 
 
 2.8502 .4549 
 
 1 .2334 
 
 40 
 
 30 
 
 0.3403 
 
 .3338 .5235 
 
 .9426 .9743 
 
 .3541 .5491 
 
 2.8239 .4509 
 
 1.2305 
 
 30 
 
 40 
 
 0.3432 
 
 .3365 .5270 
 
 .9417 .9739 
 
 .3574 .5531 
 
 2.7980 .4469 
 
 .2275 
 
 20 
 
 50 
 
 0.3462 
 
 .3393 .5306 
 
 .9407 .9734 
 
 .3607 .5571 
 
 2.7725 .4429 
 
 12246 
 
 10 
 
 20 00' 
 
 0.3491 
 
 .3420 9.5341 
 
 .9397 9.9730 
 
 .3640 9.5611 
 
 2.7475 0.4389 
 
 1.2217 
 
 70 00' 
 
 10 
 
 0.3520 
 
 .3448 .5375 
 
 .9387 .9725 
 
 .3673 .5650 
 
 2.7228 .4350 
 
 1.2188 
 
 50 
 
 20 
 
 0.3549 
 
 3475 .5409 
 
 .9377 .9721 
 
 .3706 .5689 
 
 2.6985 .4311 
 
 1.2159 
 
 40 
 
 30 
 
 0.3578 
 
 .3502 .5443 
 
 .9367 .9716 
 
 .3739 .5727 
 
 2.6746 .4273 
 
 1.2130 
 
 30 
 
 40 
 
 0.3607 
 
 .3529 .5477 
 
 .9356 .971 1 
 
 .3772 .5766 
 
 2.6511 .4234 
 
 1.2101 
 
 20 
 
 50 
 
 0.3636 
 
 .3557 .5510 
 
 .9346 .9706 
 
 .3805 .5804 
 
 2.6279 .4196 
 
 1.2072 
 
 10 
 
 21 00' 
 
 0.3665 
 
 .3584 9.5543 
 
 .9336 9.9702 
 
 .3839 9.5842 
 
 2.6051 0.4158 
 
 1.2043 
 
 69 0(X 
 
 10 
 
 0.3694 
 
 .361 1 .5576 
 
 .9325 .9697 
 
 .3872 .5879 
 
 2.5826 .4121 
 
 1.2014 
 
 50 
 
 20 
 
 0.3723 
 
 .3638 .5609 
 
 .9315 .9692 
 
 .3906 .5917 
 
 2.5605 .4083 
 
 .1985 
 
 40 
 
 30 
 
 0.3752 
 
 .3665 .5641 
 
 .9304 .9687 
 
 .3939 .5954 
 
 2.5386 .4046 
 
 .1956 
 
 30 
 
 40 
 
 0.3782 
 
 .3692 .5673 
 
 .9293 .9682 
 
 .3973 .5991 
 
 2.5172 .4009 
 
 .1926 
 
 20 
 
 50 
 
 0.381 1 
 
 .3719 .5704 
 
 .9283 .9677 
 
 .4006 .6028 
 
 2.4960 .3972 
 
 .1897 
 
 10 
 
 22 00' 
 
 0.3840 
 
 .3746 9.5736 
 
 .9272 9.9672 
 
 .4040 9.6064 
 
 2.4751 0.3936 
 
 .1868 
 
 68 00' 
 
 10 
 
 0.3869 
 
 .3773 .5767 
 
 .9261 .9667 
 
 .4074 .6100 
 
 2.4545 .3900 
 
 .1839 
 
 50 
 
 20 
 
 0.3898 
 
 .3800 .5798 
 
 .9250 .9661 
 
 .4108 .6136 
 
 2.4342 .3864 
 
 .1810 
 
 40 
 
 30 
 
 0.3927 
 
 .3827 .5828 
 
 .9239 .9656 
 
 .4142 .6172 
 
 2.4142 .3828 
 
 .1781 
 
 30 
 
 40 
 
 0.3956 
 
 .3854 .5859 
 
 .9228 .9651 
 
 .4176 .6208 
 
 2.3945 .3792 
 
 .1752 
 
 20 
 
 50 
 
 0.3985 
 
 .3881 .5889 
 
 .9216 .9646 
 
 .4210 .6243 
 
 2.3750 .3757 
 
 .1723 
 
 10 
 
 23 00' 
 
 0.4014 
 
 .3907 9.5919 
 
 .9205 9.9640 
 
 .4245 9.6279 
 
 2.3559 0.3721 
 
 .1694 
 
 67 00' 
 
 10 
 
 0.4043 
 
 .3934 .5948 
 
 .9194 .9635 
 
 .4279 .6314 
 
 2.3369 .3686 
 
 .1665 
 
 50 
 
 20 
 
 0.4072 
 
 .3961 .5978 
 
 .9182 .9629 
 
 .4314 .6348 
 
 2.3183 .3652 
 
 .1636 
 
 40 
 
 30 
 
 0.4102 
 
 .3987 .6007 
 
 .9171 .9624 
 
 .4348 .6383 
 
 2.2998 .3617 
 
 .1606 
 
 30 
 
 30 
 
 0.4131 
 
 .4014 .6036 
 
 .9159 .9618 
 
 .4383 .6417 
 
 2.2817 .3583 
 
 .1577 
 
 20 
 
 50 
 
 0.4160 
 
 .4041 .6065 
 
 .9147 .9613 
 
 .4417 .6452 
 
 2.2637 .3548 
 
 .1548 
 
 10 
 
 24 00' 
 
 0.4189 
 
 .4067 9.6093 
 
 .9135 9.9607 
 
 .4452 9.6486 
 
 2.2460 0.3514 
 
 .1519 
 
 66 00' 
 
 10 
 
 0.4218 
 
 .4094 .6121 
 
 .9124 .9602 
 
 .4487 .6520 
 
 2.2286 .3480 
 
 .1490 
 
 50 
 
 20 
 
 0.4247 
 
 .4120 .6149 
 
 .9112 .9596 
 
 .4522 .6553 
 
 2.2113 .3447 
 
 .1461 
 
 40 
 
 30 
 
 0.4276 
 
 .4147 .6177 
 
 .9100 .9590 
 
 .4557 .6587 
 
 2.1943 .3413 
 
 .1432 
 
 30 
 
 40 
 
 0.4305 
 
 .4173 .6205 
 
 .9088 .9584 
 
 .4592 .6620 
 
 2.1775 .3380 
 
 .1403 
 
 20 
 
 50 
 
 0.4334 
 
 .4100 .6232 
 
 .9075 .9579 
 
 .4628 .6654 
 
 2.1609 .3346 
 
 .1374 
 
 10 
 
 25 0(K 
 
 0.4363 
 
 .4226 9.6259 
 
 .9063 9.9573 
 
 .4663 9.6687 
 
 2.1445 0.3313 
 
 .1345 
 
 65 00' 
 
 10 
 
 0.4392 
 
 .4253 .6286 
 
 .9051 .9567 
 
 .4699 .6720 
 
 2.1283 .3280 
 
 .1316 
 
 50 
 
 20 
 
 0.4422 
 
 .4279 .6313 
 
 .9038 .9561 
 
 .4734 .6752 
 
 2.1123 .3248 
 
 .1286 
 
 40 
 
 30 
 
 0.4451 
 
 .4305 .6340 
 
 .9026 .9555 
 
 .4770 .6785 
 
 2.0965 .3215 
 
 .1257 
 
 30 
 
 40 
 
 0.4480 
 
 .4331 .6366 
 
 .9013 .9549 
 
 .4806 .6817 
 
 2.0809 .3183 
 
 .1228 
 
 20 
 
 50 
 
 0.4509 
 
 .4358 .6392 
 
 .9001 .9543 
 
 .4841 .6850 
 
 2.0655 .3150 
 
 .1199 
 
 10 
 
 26 00' 
 
 0.4538 
 
 .4384 9.6418 
 
 .8988 9.9537 
 
 .4877 9.6882 
 
 2.0503 0-3118 
 
 .1170 
 
 64 00' 
 
 10 
 
 0.4567 
 
 .4410 .6444 
 
 .8975 .9530 
 
 .4913 .6914 
 
 2.0353 .3086 
 
 .1141 
 
 50 
 
 20 
 
 0.4596 
 
 .4436 .6470 
 
 .8962 .9524 
 
 .4950 .6946 
 
 2.0204 .3054 
 
 .1112 
 
 40 
 
 30 
 
 0.4625 
 
 .4462 .6495 
 
 .8949 .9518 
 
 .4986 .6977 
 
 2.0057 .3023 
 
 .1083 
 
 30 
 
 40 
 
 0.4654 
 
 .4488 .6521 
 
 .8936 .9512 
 
 .5022 .7009 
 
 1.9912 .2991 
 
 .1054 
 
 20 
 
 50 
 
 0.4683 
 
 .4514 .6546 
 
 .8923 .9505 
 
 .5059 .7040 
 
 1.9768 .2960 
 
 .1025 
 
 10 
 
 27 (XX 
 
 0.4712 
 
 .4540 9.6570 
 
 .8910 9.9499 
 
 .5095 9.7072 
 
 1.9626 0.2928 
 
 1.0996 
 
 63 00' 
 
 
 
 Nat. Log.* 
 
 Nat. Log.* 
 
 Nat. Log.* 
 
 Nat. Log. 
 
 
 
 
 
 Cosines 
 
 Sines 
 
 Cotangents 
 
 Tangents 
 
 Ra- 
 dians 
 
 De- 
 grees 
 
MATHEMATICAL TABLES 
 
 55 
 
 TRIGONOMETRIC FUNCTIONS (continued) 
 
 Annex -10 in columns marked*. (For 0.l intervals, see pp. 46-51) 
 
 De- 
 grees 
 
 Ra- 
 dians 
 
 Sines 
 
 Cosines 
 
 Tangents 
 
 Cotangents 
 
 
 
 
 
 Nat. Log.* 
 
 Nat. Log.* 
 
 Nat. Log.* 
 
 Nat. Log. 
 
 
 
 27 00' 
 
 0.4712 
 
 .4540 9.6570 
 
 .8910 9.9499 
 
 .5095 9.7072 
 
 1.9626 0.2928 
 
 1.0996 
 
 63 00' 
 
 10 
 
 0.4741 
 
 .4566 .6595 
 
 .8897 .9492 
 
 .5132 .7103 
 
 1.9486 2897 
 
 1.0966 
 
 50 
 
 20 
 
 0.4771 
 
 .4592 .6620 
 
 .8884 .9486 
 
 .5169 .7134 
 
 1.9347 .2866 
 
 1.0937 
 
 40 
 
 30 
 
 0.4800 
 
 .4617 .6644 
 
 .8870 .9479 
 
 .5206 .7165 
 
 1.9210 .2835 
 
 1.0908 
 
 30 
 
 40 
 
 0.4829 
 
 .4643 .6668 
 
 .8857 .9473 
 
 .5243 .7196 
 
 1.9074 .2804 
 
 1.0879 
 
 20 
 
 50 
 
 0.4858 
 
 .4669 .6692 
 
 .8843 .9466 
 
 .5280 .7226 
 
 1.8940 .2774 
 
 1.0850 
 
 10 
 
 28 00' 
 
 0.4887 
 
 .4695 9.6716 
 
 .8829 9.9459 
 
 .5317 9.7257 
 
 1.8807 0.2743 
 
 1.0821 
 
 62 00' 
 
 10 
 
 0.4916 
 
 .4720 .6740 
 
 .8816 .9453 
 
 .5354 .7287 
 
 1.8676 .2713 
 
 1.0792 
 
 50 
 
 20 
 
 0.4945 
 
 .4746 .6763 
 
 .8802 .9446 
 
 .5392 .7317 
 
 1.8546 .2683 
 
 1.0763 
 
 40 
 
 30 
 
 0.4974 
 
 .4772 .6787 
 
 .8788 .9439 
 
 .5430 .7348 
 
 1.8418 .2652 
 
 1.0734 
 
 30 
 
 40 
 
 0.5003 
 
 .4797 .6810 
 
 .8774 .9432 
 
 .5467 .7378 
 
 1.8291 .2622 
 
 1.0705 
 
 20 
 
 50 
 
 0.5032 
 
 .4823 .6833 
 
 .8760 .9425 
 
 .5505 .7408 
 
 1.8165 .2592 
 
 1.0676 
 
 10 
 
 29 00' 
 
 0.5061 
 
 .4848 9.6856 
 
 .8746 9.9418 
 
 .5543 9.7438 
 
 1.8040 0.2562 
 
 1.0647 
 
 61 00' 
 
 10 
 
 0.5091 
 
 .4874 .6878 
 
 .8732 .941 1 
 
 .5581 .7467 
 
 1.7917 .2533 
 
 1.0617 
 
 50 
 
 20 
 
 0.5120 
 
 .4899 .690! 
 
 .8718 .9404 
 
 .5619 .7497 
 
 1.7796 .2503 
 
 1.0588 
 
 40 
 
 30 
 
 0.5149 
 
 .4924 .6923 
 
 .8704 .9397 
 
 .5658 .7526 
 
 1.7675 .2474 
 
 1.0559 
 
 30 
 
 40 
 
 0.5178 
 
 .4950 .6946 
 
 .8689 .9390 
 
 .5696 .7556 
 
 1.7556 .2444 
 
 1.0530 
 
 20 
 
 50 
 
 0.5207 
 
 .4975 .6968 
 
 .8675 .9383 
 
 .5735 .7585 
 
 1.7437 .2415 
 
 1.0501 
 
 10 
 
 30 00' 
 
 0.5236 
 
 .5000 9.6990 
 
 .8660 9.9375 
 
 .5774 9.7614 
 
 1.7321 0.2386 
 
 1.0472 
 
 60 00' 
 
 10 
 
 0.5265 
 
 .5025 .7012 
 
 .8646 .9368 
 
 .5812 .7644 
 
 1.7205 .2356 
 
 1.0443 
 
 50 
 
 20 
 
 0.5294 
 
 .5050 .7033 
 
 .8631 .9361 
 
 .5851 .7673 
 
 1.7090 .2327 
 
 1.0414 
 
 40 
 
 30 
 
 0.5323 
 
 .5075 .7055 
 
 .8616 .9353 
 
 .5890 .7701 
 
 1.6977 .2299 
 
 1.0385 
 
 30 
 
 40 
 
 0.5352 
 
 .5100 .7076 
 
 .8601 .9346 
 
 .5930 .7730 
 
 1.6864 .2270 
 
 1.0356 
 
 20 
 
 50 
 
 0.5381 
 
 .5125 .7097 
 
 .8587 .9338 
 
 .5969 .7759 
 
 1.6753 .2241 
 
 1.0327 
 
 10 
 
 31 00' 
 
 0.5411 
 
 .5150 9.7118 
 
 .8572 9.9331 
 
 .6009 9.7788 
 
 1.6643 0.2212 
 
 1.0297 
 
 59 00' 
 
 10 
 
 0.5440 
 
 .5175 .7139 
 
 .8557 .9323 
 
 .6048 .7816 
 
 1.6534 .2184 
 
 1.0268 
 
 50 
 
 20 
 
 0.5469 
 
 .5200 .7160 
 
 .8542 .9315 
 
 .6088 .7845 
 
 1.6426 .2155 
 
 1.0239 
 
 40 
 
 30 
 
 0.5498 
 
 .5225 .7181 
 
 .8526 .9308 
 
 .6128 .7873 
 
 1.6319 .2127 
 
 1.0210 
 
 30 
 
 40 
 
 0.5527 
 
 .5250 .7201 
 
 .8511 .9300 
 
 .6168 .7902 
 
 1.6212 .2098 
 
 1.0181 
 
 20 
 
 50 
 
 0.5556 
 
 .5275 .7222 
 
 .8496 .9292 
 
 .6208 .7930 
 
 1.6107 .2070 
 
 1.0152 
 
 10 
 
 32 00' 
 
 0.5585 
 
 .5299 9.7242 
 
 .8480 9.9284 
 
 .6249 9.7958 
 
 .6003 0.2042 
 
 1.0123 
 
 58 00' 
 
 10 
 
 0.5614 
 
 .5324 .7262 
 
 .8465 .9276 
 
 .6289 .7986 
 
 .5900 .2014 
 
 1.0094 
 
 50 
 
 20 
 
 0.5643 
 
 .5348 .7282 
 
 .8450 .9268 
 
 .6330 .8014 
 
 .5798 .1986 
 
 1 .0065 
 
 40 
 
 30 
 
 0.5672 
 
 .5373 .7302 
 
 .8434 .9260 
 
 .6371 .8042 
 
 .5697 .1958 
 
 1.0036 
 
 30 
 
 40 
 
 0.5701 
 
 .5398 .7322 
 
 .8418 .9252 
 
 .6412 .8070 
 
 .5597 .1930 
 
 1.0007 
 
 20 
 
 50 
 
 0.5730 
 
 5422 .7342 
 
 .8403 .9244 
 
 .6453 .8097 
 
 .5497 .1903 
 
 0.9977 
 
 10 
 
 33 (XX 
 
 0.5760 
 
 5446 9.7361 
 
 .8387 9.9236 
 
 .6494 9.8125 
 
 .5399 0.1875 
 
 0.9948 
 
 57 00' 
 
 10 
 
 0.5789 
 
 5471 .7380 
 
 .8371 .9228 
 
 .6536 .8153 
 
 .5301 .1847 
 
 0.9919 
 
 50 
 
 20 
 
 0.5818 
 
 5495 .7400 
 
 .8355 .9219 
 
 .6577 .8180 
 
 .5204 .1820 
 
 0.9890 
 
 40 
 
 30 
 
 0.5847 
 
 5519 .7419 
 
 .8339 .9211 
 
 .6619 .8208 
 
 .5108 .1792 
 
 0.9861 
 
 30 
 
 40 
 
 0.5876 
 
 5544 .7438 
 
 .8323 .9203 
 
 .6661 .8235 
 
 .5013 .1765 
 
 0.9832 
 
 20 
 
 50 
 
 0.5905 
 
 5568 .7457 
 
 .8307 .9194 
 
 .6703 .8263 
 
 .4919 .1737 
 
 0.9803 
 
 10 
 
 34 00' 
 
 0.5934 
 
 5592 9.7476 
 
 .8290 9.9186 
 
 .6745 9.8290 
 
 .4826 0.1710 
 
 09774 
 
 56 00' 
 
 10 
 
 0.5963 
 
 5616 .7494 
 
 .8274 .9177 
 
 .6787 .8317 
 
 .4733 .1683 
 
 0.9745 
 
 50 
 
 20 
 
 0.5992 
 
 5640 .7513 
 
 .8258 .9169 
 
 .6830 .8344 
 
 .4641 .1656 
 
 0.9716 
 
 40 
 
 30 
 
 0.6021 
 
 5664 .7531 
 
 .8241 .9160 
 
 .6873 .8371 
 
 .4550 .1629 
 
 0.9687 
 
 30 
 
 40 
 
 0.6050 
 
 5688 .7550 
 
 .8225 .9151 
 
 .6916 .8398 
 
 .4460 .1602 
 
 0.9657 
 
 20 
 
 50 
 
 0.6080 
 
 5712 .7568 
 
 .8208 .9142 
 
 .6959 .8425 
 
 .4370 .1575 
 
 0.9628 
 
 10 
 
 35 00' 
 
 0.6109 
 
 5736 9.7586 
 
 .8192 9.9134 
 
 .7002 9.8452 
 
 .4281 0.1548 
 
 0.9599 
 
 55 00' 
 
 10 
 
 0.6138 
 
 5760 .7604 
 
 .8175 .9125 
 
 .7046 .8479 
 
 .4193 .1521 
 
 0.9570 
 
 50 
 
 20 
 
 0.6167 
 
 5783 .7622 
 
 .8158 .9116 
 
 .7089 .8506 
 
 .4106 .1494 
 
 0.9541 
 
 40 
 
 30 
 
 0.6196 
 
 5807 .7640 
 
 .8141 .9107 
 
 .7133 .8533 
 
 .4019 .1467 
 
 0.9512 
 
 30 
 
 40 
 
 0.6225 
 
 5831 .7657 
 
 .8124 .9098 
 
 .7177 .8559 
 
 .3934 .1441 
 
 0.9483 
 
 20 
 
 50 
 
 0.6254 
 
 5854 .7675 
 
 .8107 .9089 
 
 .7221 .8586 
 
 .3848 .1414 
 
 0.9454 
 
 10 
 
 36 00' 
 
 0.6283 
 
 5878 9.7692 
 
 .8090 9.9080 
 
 .7265 9.8613 
 
 1.3764 0.1387 
 
 0.9425 
 
 54 00' 
 
 
 
 Nat. Log.* 
 
 Nat. Log.* 
 
 Nat. Log.* 
 
 Nat. Log. 
 
 
 
 
 
 Cosines 
 
 Sines 
 
 Cotangents 
 
 Tangents 
 
 Ra- 
 dians 
 
 De- 
 grees 
 
56 
 
 MATHEMATICAL TABLES 
 
 TRIGONOMETRIC FUNCTIONS (continued) 
 
 Annex -10 in columns marked*. (For 0.l intervals, see pp. 4.6-51) 
 
 De- 
 grees 
 
 Ra- 
 dians 
 
 Sines 
 
 Cosines 
 
 Tangents 
 
 Cotangents 
 
 
 
 
 
 Nat. Log.* 
 
 Nat. Log.* 
 
 Nat. Log.* 
 
 Nat. Log. 
 
 
 
 36 (XX 
 
 0.6283 
 
 3878 9.7692 
 
 .8090 9.9080 
 
 .7265 9.8613 
 
 1.3764 0.1387 
 
 0.9425 
 
 54 00' 
 
 10 
 
 0.6312 
 
 .5901 .7710 
 
 .8073 .9070 
 
 .7310 .8639 
 
 1.3680 .1361 
 
 0.9396 
 
 50 
 
 20 
 
 0.6341 
 
 .5925 .7727 
 
 .8056 .9061 
 
 .7355 .8666 
 
 1.3597 .1334 
 
 0.9367 
 
 40 ' 
 
 30 
 
 0.6370 
 
 .5948 .7744 
 
 .8039 .9052 
 
 .7400 .8692 
 
 1.3514 .1308 
 
 0.9338 
 
 30 
 
 40 
 
 0.6400 
 
 .5972 .7761 
 
 .8021 .9042 
 
 .7445 .8718 
 
 1.3432 .1282 
 
 0.9308 
 
 20 
 
 50 
 
 0.6429 
 
 .5995 .7778 
 
 .8004 .9033 
 
 .7490 .8745 
 
 1.3351 .1255 
 
 0.9279 
 
 10 
 
 37 00' 
 
 0.6458 
 
 .6018 9.7795 
 
 .7986 9.9023 
 
 .7536 9.8771 
 
 .3270 0.1229 
 
 0.9250 
 
 53 00' 
 
 10 
 
 0.6487 
 
 .6041 .781 1 
 
 .7969 .9014 
 
 .7581 .8797 
 
 .3190 .1203 
 
 0.9221 
 
 50 
 
 20 
 
 0.6516 
 
 .6065 .7828 
 
 .7951 .9004 
 
 .7627 .8824 
 
 .3111 .1176 
 
 0.9192 
 
 40 
 
 30 
 
 0.6545 
 
 .6088 .7844 
 
 .7934 .8995 
 
 .7673 .8850 
 
 .3032 .1150 
 
 0.9163 
 
 30 
 
 40 
 
 0.6574 
 
 .61 1 1 .7861 
 
 .7916 .8985 
 
 .7720 .8876 
 
 .2954 .1124 
 
 0.9134 
 
 20 
 
 50 
 
 0.6603 
 
 .6134 .7877 
 
 .7898 .8975 
 
 .7766 .8902 
 
 .2876 .1098 
 
 0.9105 
 
 10 
 
 38 00' 
 
 0.6632 
 
 .6157 9.7893 
 
 .7880 9.8965 
 
 .7813 9.8928 
 
 .2799 0.1072 
 
 0.9076 
 
 52 00' 
 
 10 
 
 0.6661 
 
 .6180 .7910 
 
 .7862 .8955 
 
 .7860 .8954 
 
 .2723 .1046 
 
 0.9047 
 
 50 
 
 20 
 
 0.6690 
 
 .6202 .7926 
 
 .7844 .8945 
 
 .7907 .8980 
 
 .2647 .1020 
 
 0.9018 
 
 40 
 
 30 
 
 0.6720 
 
 .6225 .7941 
 
 .7826 .8935 
 
 .7954 .9006 
 
 .2572 .0994 
 
 0.8988 
 
 30 
 
 40 
 
 0.6749 
 
 .6248 .7957 
 
 .7808 .8925 
 
 .8002 .9032 
 
 .2497 .0968 
 
 0.8959 
 
 20 
 
 50 
 
 0.6778 
 
 .6271 .7973 
 
 .7790 .8915 
 
 .8050 .9058 
 
 .2423 .0942 
 
 0.8930 
 
 10 
 
 39 00' 
 
 0.6807 
 
 .6293 9.7989 
 
 .7771 9.8905 
 
 .8098 9.9084 
 
 .2349 0.0916 
 
 0.8901 
 
 51 00' 
 
 10 
 
 0.6836 
 
 .6316 .8004 
 
 .7753 .8895 
 
 .8146 .9110 
 
 .2276 .0890 
 
 0.8872 
 
 50 
 
 20 
 
 0.6865 
 
 .6338 .8020 
 
 .7735 .8884 
 
 .8195 .9135 
 
 .2203 .0865 
 
 0.8843 
 
 40 
 
 30 
 
 0.6894 
 
 .6361 .8035 
 
 .7716 .8874 
 
 .8243 .9161 
 
 .2131 .0839 
 
 0.8814 
 
 30 
 
 40 
 
 0.6923 
 
 .6383 .8050 
 
 .7698 .8864 
 
 .8292 .9187 
 
 .2059 .0813 
 
 0.8785 
 
 20 
 
 50 
 
 0.6952 
 
 .6406 .8066 
 
 .7679 .8853 
 
 .8342 .9212 
 
 .1988 .0788 
 
 0.8756 
 
 10 
 
 40 00' 
 
 0.6981 
 
 .6428 9.8081 
 
 .7660 9.8843 
 
 .8391 9.9238 
 
 .1918 0.0762 
 
 0.8727 
 
 50 00' 
 
 10 
 
 0.7010 
 
 .6450 .8096 
 
 .7642 .8832 
 
 .8441 .9264 
 
 .1847 .0736 
 
 0.8698 
 
 50 
 
 20 
 
 0.7039 
 
 .6472 .81 1 1 
 
 .7623 .8821 
 
 .8491 .9289 
 
 .1778 .0711 
 
 0.8668 
 
 40 
 
 30 
 
 0.7069 
 
 .6494 .8125 
 
 .7604 .8810 
 
 .8541 .9315 
 
 .1708 .0685 
 
 0.8639 
 
 30 
 
 40 
 
 0.7098 
 
 .6517 .8140 
 
 .7585 .8800 
 
 .8591 .9341 
 
 .1640 .0659 
 
 0.8610 
 
 20 
 
 50 
 
 0.7127 
 
 .6539 .8155 
 
 .7566 .8789 
 
 .8642 .9366 
 
 .1571 .0634 
 
 0.8581 
 
 10 
 
 41 00' 
 
 0.7156 
 
 .6561 9.8169 
 
 .7547 9.8778 
 
 .8693 9.9392 
 
 .1504 0.0608 
 
 0.8552 
 
 49 00' 
 
 10 
 
 0.7185 
 
 .6583 .8184 
 
 .7528 .8767 
 
 .8744 .9417 
 
 J436 .0583 
 
 0.8523 
 
 50 
 
 20 
 
 0.7214 
 
 .6604 .8198 
 
 .7509 .8756 
 
 .8796 .9443 
 
 .1369 .0557 
 
 0.8494 
 
 40 
 
 30 
 
 0.7243 
 
 .6626 .8213 
 
 .7490 .8745 
 
 .8847 .9468 
 
 .1303 .0532 
 
 0.8465 
 
 30 
 
 40 
 
 0.7272 
 
 .6648 .8227 
 
 .7470 .8733 
 
 .8899 .9494 
 
 .1237 .0506 
 
 0.8436 
 
 20 
 
 50 
 
 0.7301 
 
 .6670 .8241 
 
 .7451 .8722 
 
 .8952 .9519 
 
 .1171 .0481 
 
 0.8407 
 
 10 
 
 42 00' 
 
 0.7330 
 
 .6691 9.8255 
 
 .7431 9.8711 
 
 .9004 9.9544 
 
 .1106 00456 
 
 0.8378 
 
 48 00' 
 
 10 
 
 0.7359 
 
 .6713 .8269 
 
 .7412 .8699 
 
 .9057 .9570 
 
 .1041 .0430 
 
 0.8348 
 
 50 
 
 20 
 
 0.7389 
 
 .6734 .8283 
 
 .7392 .8688 
 
 .9110 .9595 
 
 1 .0977 .0405 
 
 0.8319 
 
 40 
 
 30 
 
 0.7418 
 
 .6756 .8297 
 
 .7373 .8676 
 
 .9163 .9621 
 
 1.0913 .0379 
 
 0.8290 
 
 30 
 
 40 
 
 0.7447 
 
 .6777 .8311 
 
 .7353 .8665 
 
 .9217 .9646 
 
 1.0850 .0354 
 
 0.8261 
 
 20 
 
 50 
 
 0.7476 
 
 .6799 .8324 
 
 .7333 .8653 
 
 .9271 .9671 
 
 1.0786 .0329 
 
 0.8232 
 
 10 
 
 43 00' 
 
 0.7505 
 
 .6820 9.8338 
 
 .7314 9.8641 
 
 .9325 9.9697 
 
 1.0724 0.0303 
 
 0.8203 
 
 47 00' 
 
 10 
 
 0.7534 
 
 .6841 .8351 
 
 .7294 .8629 
 
 .9380 .9722 
 
 1.0661 .0278 
 
 0.8174 
 
 50 
 
 20 
 
 0.7563 
 
 .6862 .8365 
 
 .7274 .8618 
 
 .9435 .9747 
 
 1.0599 .0253 
 
 0.8145 
 
 40 
 
 30 
 
 0.7592 
 
 .6884 .8378 
 
 .7254 .8606 
 
 .9490 .9772 
 
 1.0538 .0228 
 
 0.8116 
 
 30 
 
 40 
 
 0.7621 
 
 .6905 .8391 
 
 .7234 .8594 
 
 .9545 .9798 
 
 1.0477 .0202 
 
 0.8087 
 
 20 
 
 50 
 
 0.7650 
 
 .6926 .8405 
 
 .7214 .8582 
 
 .9601 .9823 
 
 1.0416 .0177 
 
 0.8058 
 
 10 
 
 44 00' 
 
 0.7679 
 
 .6947 9.8418 
 
 .7193 9.8569 
 
 .9657 9.9848 
 
 1.0355 0.0152 
 
 0.8029 
 
 46 00' 
 
 10 
 
 0.7709 
 
 .6967 .8431 
 
 .7173 .8557 
 
 .9713 .9874 
 
 1.0295 .0126 
 
 0.7999 
 
 50 
 
 20 
 
 0.7738 
 
 .6988 .8444 
 
 .7153 .8545 
 
 .9770 .9899 
 
 1.0235 .0101 
 
 0.7970 
 
 40 
 
 30 
 
 0.7767 
 
 .7009 .8457 
 
 .7133 .8532 
 
 .9827 .9924 
 
 1.0176 .0076 
 
 0.7941 
 
 30 
 
 40 
 
 07796 
 
 .7030 .8469 
 
 .7112 .8520 
 
 .9884 .9949 
 
 1.0117 .0051 
 
 0.7912 
 
 20 
 
 50 
 
 0.7825 
 
 .7050 .8482 
 
 .7092 .8507 
 
 .9942 .9975 
 
 1.0058 .0025 
 
 0.7883 
 
 10 
 
 45 00' 
 
 0.7854 
 
 .7071 9.8495 
 
 .7071 9.8495 
 
 1.0000 0.0000 
 
 1.0000 0.0000 
 
 0.7854 
 
 45 00' 
 
 
 
 Nat. Log.* 
 
 Nat. Log.* 
 
 Nat. Log.* 
 
 Nat. Log. 
 
 
 
 
 
 Cosines 
 
 Sines 
 
 Cotangents 
 
 Tangents 
 
 Ra- 
 dians 
 
 De- 
 grees 
 
MATHEMATICAL TABLES 
 EXPONENTIALS [e and <T"] 
 
 57 
 
 n 
 
 n *H 
 
 c s 
 
 n 
 
 , g 
 
 n 
 
 e n 
 
 n 
 
 <- % 
 
 n 
 
 c~" 
 
 n 
 
 e~* 
 
 0.00 
 
 1.000 in 
 
 0.50 
 
 1.649 ,, 
 
 1.0 
 
 2.718* 
 
 0.00 
 
 1.000 ]0 
 
 0.50 
 
 .607 
 
 1.0 
 
 .368* 
 
 .01 
 
 i.oto 5 
 
 .51 
 
 1.665 S 
 
 .1 
 
 3.004 
 
 .01 
 
 0.990- J 
 
 31 
 
 .600 
 
 .1 
 
 .333 
 
 .02 
 
 J.020 2 
 
 .52 
 
 1.682 i 
 
 .2 
 
 3.320 
 
 .02 
 
 .980 ~ 2 
 
 .52 
 
 .595 
 
 2 
 
 .301 
 
 .03 
 
 1.030 9 
 
 .53 
 
 1.699 \l 
 
 3 
 
 3.669 
 
 .03 
 
 .970 -'g 
 
 .53 
 
 .589 
 
 3 
 
 .273 
 
 .04 
 
 1.041 JJ 
 
 .54 
 
 1.716 \ 7 7 
 
 .4 
 
 4.055 
 
 .04 
 
 961 I,J 
 
 .54 
 
 .583 
 
 .4 
 
 .247 
 
 0.05 
 
 1.051 .. 
 
 0.55 
 
 1.733 1R 
 
 1.5 
 
 4.482 
 
 0.05 
 
 .951 
 
 0.55 
 
 .577 
 
 13 
 
 .223 
 
 .06 
 
 .062 
 
 .56 
 
 1.751 2 
 
 .6 
 
 4.953 
 
 .06 
 
 .942 ~,J 
 
 .56 
 
 .571 
 
 . .6 
 
 202 
 
 .07 
 
 1.073 A 
 
 .57 
 
 1.768 \i 
 
 .7 
 
 5.474 
 
 .07 
 
 .932 ~' 
 
 .57 
 
 .566 
 
 .7 
 
 .183 
 
 .08 
 
 1.083 ? 
 
 .58 
 
 1.786 2 
 
 .8 
 
 6.050 
 
 .08 
 
 .923 ~ I 
 
 .58 
 
 .560 
 
 .8 
 
 .165 
 
 .09 
 
 1.094 jj 
 
 .59 
 
 1.804 ! 
 
 .9 
 
 6.686 
 
 .09 
 
 .914 I 9 9 
 
 39 
 
 .554 
 
 .9 
 
 .150 
 
 0.10 
 
 .105 ,. 
 
 0.60 
 
 1.822 ,. 
 
 2.0 
 
 7.389 
 
 0.10 
 
 .905 o 
 
 0.60 
 
 .549 
 
 2.0 
 
 .135 
 
 .11 
 
 .116 
 
 .61 
 
 1.840 
 
 .1 
 
 8.166 
 
 .11 
 
 .896 ~ A 
 
 .61 
 
 .543 
 
 .1 
 
 .122 
 
 .12 
 
 .127 I 
 
 .62 
 
 1.859 A 
 
 2 
 
 9.025 
 
 .12 
 
 .887 ~ I 
 
 .62 
 
 .538 
 
 2 
 
 .111 
 
 .13 
 
 .139 2 
 
 .63 
 
 1.878 \l 
 
 3 
 
 9.974 
 
 .13 
 
 .878 ~ J 
 
 .63 
 
 .533 
 
 3 
 
 .100 
 
 .14 
 
 .150 jj 
 
 .64 
 
 1.896 JJ 
 
 .4 
 
 11.02 
 
 .14 
 
 .869 ~ I 
 
 .64 
 
 .527 
 
 .4 
 
 .0907 
 
 0.15 
 
 .162 ,, 
 
 0.65 
 
 1.916 ]0 
 
 2.5 
 
 12.18 
 
 0.15 
 
 .861 Q 
 
 0.65 
 
 .522 
 
 23 
 
 .0821 
 
 .16 
 
 .174 2 , 
 
 .66 
 
 1.935 \l 
 
 .6 
 
 13.46 
 
 .16 
 
 .852 ~ I 
 
 .66 
 
 .517 
 
 .6 
 
 .0743 
 
 .17 
 
 .185 ' 
 
 .67 
 
 1.954 
 
 .7 
 
 14.88 
 
 .17 
 
 .844 ~ I 
 
 .67 
 
 .512 
 
 .7 
 
 .0672 
 
 .18 
 
 .197 2 
 
 .68 
 
 1.974 % 
 
 .8 
 
 16.44 
 
 .18 
 
 .835 ~ I 
 
 .68 
 
 .507 
 
 .8 
 
 .0608 
 
 .19 
 
 1.209 j| 
 
 .69 
 
 1.994 20 
 
 .9 
 
 18.17 
 
 .19 
 
 .827 I I 
 
 .69 
 
 .502 
 
 .9 
 
 .0550 
 
 0.20 
 
 1.221 
 
 0.70 
 
 2.014 7n 
 
 3.0 
 
 20.09 
 
 0.20 
 
 .819 
 
 0.70 
 
 .497 
 
 3.0 
 
 .0498 
 
 .21 
 
 1.234 \l 
 
 .71 
 
 2.034 ?,S 
 
 .1 
 
 22.20 
 
 .21 
 
 .811 ~ 
 
 .71 
 
 .492 
 
 .1 
 
 .0450 
 
 .22 
 
 1.246 2 
 
 .72 
 
 2.054 20 
 
 .2 
 
 24.53 
 
 .22 
 
 .803 - 
 
 .72 
 
 .487 
 
 .2 
 
 .0408 
 
 .23 
 
 1.259 2 
 
 .73 
 
 2.075 2, 
 
 .3 
 
 27.11 
 
 .23 
 
 .795 ~ 5 
 
 .73 
 
 .482 
 
 3 
 
 .0369 
 
 24 
 
 1.271 !| 
 
 .74 
 
 2.096 21 
 
 .4 
 
 29.% 
 
 .24 
 
 -787 I I 
 
 .74 
 
 .477 
 
 .4 
 
 .0334 
 
 0.25 
 
 1.284 n 
 
 0.75 
 
 2.117 
 
 3.5 
 
 33.12 
 
 0.25 
 
 .779 o 
 
 0.75 
 
 .472 
 
 33 
 
 .0302 
 
 .26 
 
 707 13 
 
 .^y/ | o 
 
 .76 
 
 2.138 ?! 
 
 .6 
 
 36.60 
 
 .26 
 
 .771 ~ I 
 
 .76 
 
 .468 
 
 .6 
 
 .0273 
 
 .27 
 
 1.310 ^ 
 
 .77 
 
 2.160 ?f 
 
 .7 
 
 40.45 
 
 .27 
 
 .763 ~ 5 
 
 .77 
 
 .463 
 
 .7 
 
 .0247 
 
 .28 
 
 1.323 | 
 
 .78 
 
 2.181 tl 
 
 .8 
 
 44.70 
 
 .28 
 
 .756 ~ I 
 
 .78 
 
 .458 
 
 .8 
 
 .0224 
 
 .29 
 
 1.336 JJ 
 
 .79 
 
 2.203 g 
 
 .9 
 
 49.40 
 
 .29 
 
 .748 ~ J 
 
 .79 
 
 .454 
 
 .9 
 
 .0202 
 
 O.SO 
 
 1.350 n 
 
 0.80 
 
 2.226 22 
 
 4.0 
 
 54.60 
 
 0.30 
 
 .741 
 
 0.80 
 
 .449 
 
 4.0 
 
 .0183 
 
 .31 
 
 1.363 \\ 
 
 .81 
 
 2.248 
 
 .1 
 
 60.34 
 
 .31 
 
 .733 ~ 2 
 
 .81 
 
 .445 
 
 .1 
 
 .0166 
 
 .32 
 
 1.377 
 
 .82 
 
 2.270 S 2 , 
 
 .2 
 
 66.69 
 
 .32 
 
 .726 ~ I 
 
 .82 
 
 .440 
 
 .2 
 
 .0150 
 
 .33 
 
 1.391 J 
 
 .83 
 
 2.293 g 
 
 3 
 
 73.70 
 
 .33 
 
 .719 ~ I 
 
 .83 
 
 .436 
 
 .3 
 
 .0136 
 
 .34 
 
 1.405 ]< 
 
 .84 
 
 2.3.6 g 
 
 .4 
 
 81.45 
 
 .34 
 
 .7.2 ~ ] 
 
 .84 
 
 .432 
 
 .4 
 
 .0123 
 
 0.35 
 
 1.419 14 
 
 0.85 
 
 2.340 21 
 
 4.5 
 
 90.02 
 
 0.35 
 
 .705 7 
 
 0.85 
 
 .427 
 
 4.5 
 
 .0111 
 
 .36 
 
 1.433 i 
 
 .86 
 
 2.363 5? 
 
 
 
 .36 
 
 .698 ~ i 
 
 .86 
 
 .423 
 
 
 
 .37 
 
 1.448 
 
 .87 
 
 2.387 Si 
 
 5.0 
 
 148.4 
 
 .37 
 
 .691 ~ I 
 
 .87 
 
 .419 
 
 5.0 
 
 .00674 
 
 .38 
 
 1.462 J 
 
 .88 
 
 2.411 2J 
 
 6.0 
 
 403.4 
 
 .38 
 
 .684 ~ i 
 
 .88 
 
 .415 
 
 6.0 
 
 .00248 
 
 .39 
 0.40 
 
 1.477 jf 
 
 1.492 ,e 
 
 .89 
 0.90 
 
 2.435 24 
 2.460 94 
 
 7.0 
 8.0 
 
 1097. 
 2981. 
 
 .39 
 0.40 
 
 .677 I ij 
 .670 
 
 .89 
 0.90 
 
 .411 
 .407 
 
 8.0 
 
 i. 000912 
 .000335 
 
 .41 
 
 1.507 1 
 
 .91 
 
 2.484 24 
 
 9.0 
 
 8103. 
 
 .41 
 
 .664" S 
 
 .91 
 
 .403 
 
 9.0 
 
 .000123 
 
 .42 
 
 1.522 \\ 
 
 .92 
 
 2.509 25 
 
 10.0 
 
 22026. 
 
 .42 
 
 .657 " I 
 
 .92 
 
 .399 
 
 10.0 
 
 .000045 
 
 .43 
 .44 
 
 1.537 5 
 1.553 Jf 
 
 .93 
 .94 
 
 2.535 fS 
 
 y <Mn ** 
 
 2.560 26 
 
 /2 
 fcr/2 
 
 4.810 
 23.14 
 
 .43 
 .44 
 
 .651 " S 
 644 I ? 6 
 
 .93 
 .94 
 
 .395 
 .391 
 
 IT/2 
 27T/2 
 
 .208 
 .0432 
 
 0.45 
 
 1.568 ,, 
 
 0.95 
 
 2.586 7 , 
 
 37T/2 
 
 111.3 
 
 0.45 
 
 .638 7 
 
 0.95 
 
 .387 
 
 3r/2 
 
 .00898 
 
 .46 
 
 1.584 9 
 
 .96 
 
 2.612 g 
 
 47T/2 
 
 535.5 
 
 .46 
 
 .631 " ? 
 
 .96 
 
 .383 
 
 fcr/2 
 
 .00187 
 
 .47 
 
 1.600 
 
 .97 
 
 2.638 ?9 
 
 5T/2 
 
 2576. 
 
 .47 
 
 .625 " 
 
 .97 
 
 .379 
 
 5r/2 
 
 .000388 
 
 .48 
 
 1.616 JJ 
 
 .98 
 
 2.664 $ 
 
 for/2 
 
 12392. 
 
 .48 
 
 .619 ~ 9 
 
 .98 
 
 .375 
 
 6r/2 
 
 .000081 
 
 .49 
 
 1.632 2 
 
 .99 
 
 2.691 g 
 
 lir/2 
 
 59610. 
 
 .49 
 
 .613 ~ I 
 
 .99 
 
 .372 
 
 7^/2 
 
 .000017 
 
 0.50 
 
 I/ 
 
 1.649 
 
 1.00 
 
 // 
 2.718 
 
 8^/2 
 
 286751. 
 
 0.50 
 
 0.607 
 
 1.00 
 
 368 
 
 W2 
 
 .000003 
 
 *NOTE: Do not interpolate in this column. 
 
 e = 2.71828 1/e = 0.367879 logioe = 0.4343 1/(0.4343) - 2.3026 
 
 logio(0.4343) = 1.6378 logio(e) = n(0.4343) 
 
 For table of multiples of 0.4343, see p. 62. Graphs, p. 174. 
 
58 
 
 MATHEMATICAL TABLES 
 
 HYPERBOLIC LOGARITHMS 
 
 
 n 
 
 n (2.3026) 
 
 n (0.6974-3) 
 
 These two pages give the natural (hyper- 
 bolic, or Napierian) logarithms (log) of 
 numbers between 1 and 10, correct to four 
 
 1 
 2 
 3 
 4 
 
 2.3026 
 4.6052 
 6.9078 
 9.2103 
 
 0.6974-3 
 0.3948-5 
 0.0922-7 
 0.7897-10 
 
 places. Moving the decimal point n places 
 
 5 
 
 11.5129 
 
 0.4871-12 
 
 to the right [or left] in the number is equiva- 
 
 6 
 
 13.8155 
 
 0.1845-14 
 
 lent to adding n times 2.3026 [or n times 
 3.6974] to the logarithm. Base e = 2.71828 + 
 
 ; 
 
 8 
 9 
 
 16.1181 
 18.4207 
 20.7233 
 
 08819-17 
 0.5793-19 
 0.2767-21 
 
 EU- 
 
 & 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 1.0 
 
 0.0000 
 
 0100 
 
 0198 
 
 0296 
 
 0392 
 
 0488 
 
 0583 
 
 0677 
 
 0770 
 
 0862 
 
 95 
 
 1.1 
 
 0953 
 
 1044 
 
 1133 
 
 1222 
 
 1310 
 
 1398 
 
 1484 
 
 1570 
 
 1655 
 
 1740 
 
 87 
 
 1.2 
 
 1823 
 
 1906 
 
 1989 
 
 2070 
 
 2151 
 
 2231 
 
 2311 
 
 2390 
 
 2469 
 
 2546 
 
 80 
 
 1.3 
 
 2624 
 
 2700 
 
 2776 
 
 2852 
 
 2927 
 
 3001 
 
 3075 
 
 3148 
 
 3221 
 
 3293 
 
 74 
 
 1.4 
 
 3365 
 
 3436 
 
 3507 
 
 3577 
 
 3646 
 
 3716 
 
 3784 
 
 3853 
 
 3920 
 
 3988 
 
 69 
 
 1.5 
 
 0.4055 
 
 4121 
 
 4187 
 
 4253 
 
 4318 
 
 4383 
 
 4447 
 
 4511 
 
 4574 
 
 4637 
 
 65 
 
 1.6 
 
 4700 
 
 4762 
 
 4824 
 
 4886 
 
 4947 
 
 5008 
 
 5068 
 
 5128 
 
 5188 
 
 5247 
 
 61 
 
 1.7 
 
 5306 
 
 5365 
 
 5423 
 
 5481 
 
 5539 
 
 5596 
 
 5653 
 
 5710 
 
 5766 
 
 5822 
 
 57 
 
 1.8 
 
 5878 
 
 5933 
 
 5988 
 
 6043 
 
 6098 
 
 6152 
 
 6206 
 
 6259 
 
 6313 
 
 6366 
 
 54 
 
 1.9 
 
 6419 
 
 6471 
 
 6523 
 
 6575 
 
 6627 
 
 6678 
 
 6729 
 
 6780 
 
 6831 
 
 6881 
 
 51 
 
 2.0 
 
 0.6931 
 
 6981 
 
 7031 
 
 7080 
 
 7129 
 
 7178 
 
 7227 
 
 7275 
 
 7324 
 
 7372 
 
 49 
 
 2.1 
 
 7419 
 
 7467 
 
 7514 
 
 7561 
 
 7608 
 
 7655 
 
 7701 
 
 7747 
 
 7793 
 
 7839 
 
 47 
 
 2.2 
 
 7885 
 
 7930 
 
 7975 
 
 8020 
 
 8065 
 
 8109 
 
 8154 
 
 8198 
 
 8242 
 
 8286 
 
 44 
 
 2.3 
 
 8329 
 
 8372 
 
 8416 
 
 8459 
 
 8502 
 
 8544 
 
 8587 
 
 8629 
 
 8671 
 
 8713 
 
 43 
 
 2.4 
 
 8755 
 
 8796 
 
 8838 
 
 8879 
 
 8920 
 
 8961 
 
 9002 
 
 9042 
 
 9083 
 
 9123 
 
 41 
 
 2.5 
 
 0.9163 
 
 9203 
 
 9243 
 
 9282 
 
 9322 
 
 9361 
 
 9400 
 
 9439 
 
 9478 
 
 9517 
 
 39 
 
 2.6 
 
 9555 
 
 9594 
 
 9632 
 
 9670 
 
 9708 
 
 9746 
 
 9783 
 
 9821 
 
 9858 
 
 9895 
 
 38 
 
 2.7 
 
 0.9933 
 
 9969 
 
 *0006 
 
 *0043 
 
 *0080 
 
 *0116 
 
 *0152 
 
 *OI88 
 
 *0225 
 
 *0260 
 
 36 
 
 2.8 
 
 1.0296 
 
 0332 
 
 0367 
 
 0403 
 
 0438 
 
 0473 
 
 0508 
 
 0543 
 
 0578 
 
 0613 
 
 35 
 
 2.9 
 
 0647 
 
 0682 
 
 0716 
 
 0750 
 
 0784 
 
 0818 
 
 0852 
 
 0886 
 
 0919 
 
 0953 
 
 34 
 
 3.0 
 
 1.0986 
 
 1019 
 
 1053 
 
 1086 
 
 1119 
 
 1151 
 
 1184 
 
 1217 
 
 1249 
 
 1282 
 
 33 
 
 3.1 
 
 1314 
 
 1346 
 
 1378 
 
 1410 
 
 1442 
 
 1474 
 
 1506 
 
 1537 
 
 1569 
 
 1600 
 
 32 
 
 3.2 
 
 1632 
 
 1663 
 
 1694 
 
 1725 
 
 1756 
 
 1787 
 
 1817 
 
 1848 
 
 1878 
 
 1909 
 
 31 
 
 3.3 
 
 1939 
 
 1969 
 
 2000 
 
 2030 
 
 2060 
 
 2090 
 
 2119 
 
 2149 
 
 2179 
 
 2208 
 
 30 
 
 3.4 
 
 2238 
 
 2267 
 
 2296 
 
 2326 
 
 2355 
 
 2384 
 
 2413 
 
 2442 
 
 2470 
 
 2499 
 
 29 
 
 3.5 
 
 1.2528 
 
 2556 
 
 2585 
 
 2613 
 
 2641 
 
 2669 
 
 2698 
 
 2726 
 
 2754 
 
 2782 
 
 28 
 
 3.6 
 
 2809 
 
 2837 
 
 2865 
 
 2892 
 
 2920 
 
 2947 
 
 2975 
 
 3002 
 
 3029 
 
 3056 
 
 27 
 
 3.7 
 
 3083 
 
 3110 
 
 3137 
 
 3164 
 
 3191 
 
 3218 
 
 3244 
 
 3271 
 
 3297 
 
 3324 
 
 27 
 
 3.8 
 
 3350 
 
 3376 
 
 3403 
 
 3429 
 
 3455 
 
 3481 
 
 3507 
 
 3533 
 
 3558 
 
 3584 
 
 26 
 
 3.9 
 
 3610 
 
 3635 
 
 3661 
 
 3686 
 
 3712 
 
 3737 
 
 3762 
 
 3788 
 
 3813 
 
 3838 
 
 25 
 
 4.0 
 
 1.3863 
 
 3888 
 
 3913 
 
 3938 
 
 3962 
 
 3987 
 
 4012 
 
 4036 
 
 4061 
 
 4085 
 
 25 
 
 4.1 
 
 4110 
 
 4134 
 
 4159 
 
 4183 
 
 4207 
 
 4231 
 
 4255 
 
 4279 
 
 4303 
 
 4327 
 
 24 
 
 4.2 
 
 4351 
 
 4375 
 
 4398 
 
 4422 
 
 4446 
 
 4469 
 
 4493 
 
 4516 
 
 4540 
 
 4563 
 
 23 
 
 4.3 
 
 4586 
 
 4609 
 
 4633 
 
 4656 
 
 4679 
 
 4702 
 
 4725 
 
 4748 
 
 4770 
 
 4793 
 
 23 
 
 4.4 
 
 4816 
 
 4839 
 
 4861 
 
 4884 
 
 4907 
 
 4929 
 
 4951 
 
 4974 
 
 4996 
 
 5019 
 
 22 
 
 4.5 
 
 1.5041 
 
 5063 
 
 5085 
 
 5107 
 
 5129 
 
 5151 
 
 5173 
 
 5195 
 
 5217 
 
 5239 
 
 22 
 
 4.6 
 
 5261 
 
 5282 
 
 5304 
 
 5326 
 
 5347 
 
 5369 
 
 5390 
 
 5412 
 
 5433 
 
 5454 
 
 21 
 
 4.7 
 
 5476 
 
 5497 
 
 5518 
 
 5539 
 
 5560 
 
 5581 
 
 5602 
 
 5623 
 
 5644 
 
 5665 
 
 21 
 
 4.8 
 
 5686 
 
 5707 
 
 5728 
 
 5748 
 
 5769 
 
 5790 
 
 5810 
 
 5831 
 
 5851 
 
 5872 
 
 20 
 
 4.9 
 
 5892 
 
 5913 
 
 5933 
 
 5953 
 
 5974 v 
 
 5994 
 
 6014 
 
 6034 
 
 6054 
 
 6074 
 
 20 
 
 logtX = (2.3026) logio x logioa: = (0.4343) Iog 8 x 
 
 where 2.3026 = logw and 0.4343 => logioe (see p. 62). For graph*, see p. 174. 
 
MATHEMATICAL TABLES 
 
 59 
 
 HYPERBOLIC LOGARITHMS (continued} 
 
 e^ 
 
 3 <o 
 53- 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 !i 
 
 5.0 
 
 1.6094 
 
 6114 
 
 6134 
 
 6154 
 
 6174 
 
 6194 
 
 6214 
 
 6233 
 
 6253 
 
 6273 
 
 20 
 
 5.1 
 
 6292 
 
 6312 
 
 6332 
 
 6351 
 
 6371 
 
 6390 
 
 6409 
 
 6429 
 
 6448 
 
 6467 
 
 19 
 
 5.2 
 
 6487 
 
 6506 
 
 6525 
 
 6544 
 
 6563 
 
 6582 
 
 6601 
 
 6620 
 
 6639 
 
 6658 
 
 19 
 
 5.3 
 
 6677 
 
 6696 
 
 6715 
 
 6734 
 
 6752 
 
 6771 
 
 6790 
 
 6808 
 
 6827 
 
 6845 
 
 18 
 
 5.4 
 
 6864 
 
 .6882 
 
 6901 
 
 6919 
 
 6938 
 
 6956 
 
 6974 
 
 6993 
 
 7011 
 
 7029 
 
 18 
 
 5.5 
 
 1.7047 
 
 7066 
 
 7084 
 
 7102 
 
 7120 
 
 7138 
 
 7156 
 
 7174 
 
 7192 
 
 7210 
 
 18 
 
 5.6 
 
 7228 
 
 7246 
 
 7263 
 
 7281 
 
 7299 
 
 7317 
 
 7334 
 
 7352 
 
 7370 
 
 7387 
 
 18 
 
 5.7 
 
 7405 
 
 7422 
 
 7440 
 
 7457 
 
 7475 
 
 7492 
 
 7509 
 
 7527 
 
 7544 
 
 7561 
 
 17 
 
 5.8 
 
 7579 
 
 7596 
 
 7613 
 
 7630 
 
 7647 
 
 7664 
 
 7681 
 
 7699 
 
 7716 
 
 7733 
 
 17 
 
 5.9 
 
 7750 
 
 7766 
 
 7783 
 
 7800 
 
 7817 
 
 7834 
 
 7851 
 
 7867 
 
 7884 
 
 7901 
 
 17 
 
 6.0 
 
 1.7918 
 
 7934 
 
 7951 
 
 7967 
 
 7984 
 
 8001 
 
 8017 
 
 8034 
 
 8050 
 
 8066 
 
 16 
 
 6.1 
 
 8083 
 
 8099 
 
 8116 
 
 8132 
 
 8148 
 
 8165 
 
 8181 
 
 8197 
 
 8213 
 
 8229 
 
 16 
 
 6.2 
 
 8245 
 
 8262 
 
 8278 
 
 8294 
 
 8310 
 
 8326 
 
 8342 
 
 8358 
 
 8374 
 
 8390 
 
 16 
 
 6.3 
 
 8405 
 
 8421 
 
 8437 
 
 8453 
 
 8469 
 
 8485 
 
 8500 
 
 8516 
 
 8532 
 
 8547 
 
 16 
 
 6.4 
 
 8563 
 
 8579 
 
 8594 
 
 8610 
 
 8625 
 
 8641 
 
 8656 
 
 8672 
 
 8687 
 
 8703 
 
 15 
 
 6.5 
 
 1.8718 
 
 8733 
 
 8749 
 
 8764 
 
 8779 
 
 8795 
 
 8810 
 
 8825 
 
 8840 
 
 8856 
 
 15 
 
 6.6 
 
 8871 
 
 8886 
 
 8901 
 
 8916 
 
 8931 
 
 8946 
 
 8961 
 
 8976 
 
 8991 
 
 9006 
 
 15 
 
 6.7 
 
 9021 
 
 9036 
 
 9051 
 
 9066 
 
 9081 
 
 9095 
 
 9110 
 
 9125 
 
 9140 
 
 9155 
 
 15 
 
 6.8 
 
 9169 
 
 9184 
 
 9199 
 
 9213 
 
 9228 
 
 9242 
 
 9257 
 
 9272 
 
 9286 
 
 9301 
 
 15 
 
 6.9 
 
 9315 
 
 9330 
 
 9344 
 
 9359 
 
 9373 
 
 9387 
 
 9402 
 
 9416 
 
 9430 
 
 9445 
 
 14 
 
 7.0 
 
 1.9459 
 
 9473 
 
 9488 
 
 9502 
 
 9516 
 
 9530 
 
 9544 
 
 9559 
 
 9573 
 
 9587 
 
 14 
 
 7.1 
 
 9601 
 
 9615 
 
 9629 
 
 9643 
 
 9657 
 
 9671 
 
 9685 
 
 9699 
 
 9713 
 
 9727 
 
 14 
 
 7.2 
 
 9741 
 
 9755 
 
 9769 
 
 9782 
 
 9796 
 
 9810 
 
 9824 
 
 9838 
 
 9851 
 
 9865 
 
 14 
 
 7.3 
 
 1 .9879 
 
 9892 
 
 9906 
 
 9920 
 
 9933 
 
 9947 
 
 9961 
 
 9974 
 
 9988 
 
 *OOOI 
 
 13 
 
 7.4 
 
 2.0015 
 
 0028 
 
 0042 
 
 0055 
 
 0069 
 
 0082 
 
 0096 
 
 0109 
 
 0122 
 
 0136 
 
 13 
 
 7.5 
 
 2.0149 
 
 0162 
 
 0176 
 
 0189 
 
 0202 
 
 0215 
 
 0229 
 
 0242 
 
 0255 
 
 0268 
 
 13 
 
 7.6 
 
 0281 
 
 0295 
 
 0308 
 
 0321 
 
 0334 
 
 0347 
 
 0360 
 
 0373 
 
 0386 
 
 0399 
 
 13 
 
 7.7 
 
 0412 
 
 0425 
 
 0438 
 
 0451 
 
 0464 
 
 0477 
 
 0490 
 
 0503 
 
 0516 
 
 0528 
 
 13 
 
 7.8 
 
 0541 
 
 0554 
 
 0567 
 
 0580 
 
 0592 
 
 0605 
 
 0618 
 
 0631 
 
 0643 
 
 0656 
 
 13 
 
 7.9 
 
 0669 
 
 0681 
 
 0694 
 
 0707 
 
 0719 
 
 0732 
 
 0744 
 
 0757 
 
 0769 
 
 0782 
 
 12 
 
 8.0 
 
 2.0794 
 
 . 0807 
 
 0819 
 
 0832 
 
 0844 
 
 0857 
 
 0869 
 
 0882 
 
 0894 
 
 0906 
 
 12 
 
 8.1 
 
 0919 
 
 0931 
 
 0943 
 
 0956 
 
 0968 
 
 0980 
 
 0992 
 
 1005 
 
 1017 
 
 1029 
 
 12 
 
 8.2 
 
 1041 
 
 1054 
 
 1066 
 
 1078 
 
 1090 
 
 1102 
 
 1114 
 
 1126 
 
 1138 
 
 1150 
 
 12 
 
 8.3 
 
 1163 
 
 1175 
 
 1187 
 
 1199 
 
 1211 
 
 1223 
 
 1235 
 
 1247 
 
 1258 
 
 1270 
 
 12 
 
 8.4 
 
 1282 
 
 1294 
 
 1306 
 
 1318 
 
 1330 
 
 1342 
 
 1353 
 
 1365 
 
 1377 
 
 1389 
 
 12 
 
 8.5 
 
 2.1401 
 
 1412 
 
 1424 
 
 1436 
 
 1448 
 
 1459 
 
 1471 
 
 1483 
 
 1494 
 
 1506 
 
 12 
 
 8.6 
 
 1518 
 
 1529 
 
 1541 
 
 1552 
 
 1564 
 
 1576 
 
 1587 
 
 1599 
 
 1610 
 
 1622 
 
 12 
 
 8.7 
 
 1633 
 
 1645 
 
 1656 
 
 1668 
 
 1679 
 
 1691 
 
 1702 
 
 1713 
 
 1725 
 
 1736 
 
 11 
 
 8.8 
 
 1748 
 
 1759 
 
 1770 
 
 1782 
 
 1793 
 
 1804 
 
 1815 
 
 1827 
 
 1838 
 
 1849 
 
 It 
 
 8.9 
 
 1861 
 
 1872 
 
 1883 
 
 1894 
 
 1905 
 
 1917 
 
 1928 
 
 1939 
 
 1950 
 
 1961 
 
 11 
 
 9.0 
 
 2.1972 
 
 1983 
 
 1994 
 
 2006 
 
 2017 
 
 2028 
 
 2039 
 
 2050 
 
 2061 
 
 2072 
 
 11 
 
 9.1 
 
 2083 
 
 2094 
 
 2105 
 
 2116 
 
 2127 
 
 2138 
 
 2148 
 
 2159 
 
 2170 
 
 2181 
 
 II 
 
 9.2 
 
 2192 
 
 2203 
 
 2214 
 
 2225 
 
 2235 
 
 2246 
 
 2257 
 
 2268 
 
 2279 
 
 2289 
 
 11 
 
 9.3 
 
 2300 
 
 2311 
 
 2322 
 
 2332 
 
 2343 
 
 2354 
 
 2364 
 
 2375 
 
 2386 
 
 2396 
 
 11 
 
 9.4 
 
 2407 
 
 2418 
 
 2428 
 
 2439 
 
 2450 
 
 2460 
 
 2471 
 
 2481 
 
 2492 
 
 2502 
 
 11 
 
 9.5 
 
 2.2513 
 
 2523 
 
 2534 
 
 2544 
 
 2555 
 
 2565 
 
 2576 
 
 2586 
 
 2597 
 
 2607 
 
 10 
 
 9.6 
 
 2618 
 
 2628 
 
 2638 
 
 2649 
 
 2659 
 
 2670 
 
 2680 
 
 2690 
 
 2701 
 
 2711 
 
 10 
 
 9.7 
 
 2721 
 
 2732 
 
 2742 
 
 2752 
 
 2762 
 
 2773 
 
 2783 
 
 2793 
 
 2803 
 
 2814 
 
 10 
 
 9.8 
 
 2824 
 
 2834 
 
 2844 
 
 2854 
 
 2865 
 
 2875 
 
 2885 
 
 2895 
 
 2905 
 
 2915 
 
 10 
 
 9.9 
 
 2925 
 
 2935 
 
 2946 
 
 2956 
 
 2966 
 
 2976 
 
 2986 
 
 29% 
 
 3006 
 
 3016 
 
 10 
 
 10.0 
 
 2.3026 
 
 
 
 
 
 
 
 
 
 
 
 Moving the decimal point n places to the right [or left] in the number requires adding 
 n times 2.3026 for n times (0.6974-3)] in the body of the table. See auxiliary table of 
 multiples on top of the preceding page. 
 
60 MATHEMATICAL TABLES 
 
 HYPERBOLIC SINES [sinh x = ^(e* - e~*)] 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 0.0 
 
 .0000 
 
 .0100 
 
 .0200 
 
 .0300 
 
 .0400 
 
 .0500 
 
 .0600 
 
 .0701 
 
 .0801 
 
 .0901 
 
 00 
 
 1 
 
 .1002 
 
 .1102 
 
 .1203 
 
 .1304 
 
 .1405 
 
 .1506 
 
 .1607 
 
 .1708 
 
 .1810 
 
 .1911 
 
 01 
 
 2 
 
 .2013 
 
 .2115 
 
 .2218 
 
 .2320 
 
 .2423 
 
 .2526 
 
 .2629 
 
 .2733 
 
 .2837 
 
 .2941 
 
 03 
 
 3 
 
 .3045 
 
 .3150 
 
 .3255 
 
 .3360 
 
 .3466 
 
 .3572 
 
 .3678 
 
 .3785 
 
 .3892 
 
 .4000 
 
 106 
 
 4 
 
 .4108 
 
 .4216 
 
 .4325 
 
 .4434 
 
 .4543 
 
 .4653 
 
 .4764 
 
 .4875 
 
 .4986 
 
 .5098 
 
 no 
 
 0.5 
 
 .5211 
 
 .5324 
 
 .5438 
 
 .5552 
 
 .5666 
 
 .5782 
 
 .5897 
 
 .6014 
 
 .6131 
 
 .6248 
 
 116 
 
 6 
 
 .6367 
 
 .6485 
 
 .6605 
 
 .6725 
 
 .6846 
 
 .6967 
 
 .7090 
 
 .7213 
 
 .7336 
 
 .7461 
 
 122 
 
 7 
 
 .7586 
 
 .7712 
 
 .7838 
 
 .7966 
 
 .8094 
 
 .8223 
 
 .8353 
 
 .8484 
 
 .8615 
 
 .8748 
 
 130 
 
 8 
 
 .8881 
 
 .9015 
 
 .9150 
 
 .9286 
 
 .9423 
 
 .9561 
 
 .9700 
 
 .9840 
 
 .9981 
 
 1.012 
 
 138 
 
 9 
 
 1.027 
 
 1.041 
 
 .055 
 
 1.070 
 
 1.085 
 
 1.099 
 
 1.114 
 
 1.129 
 
 1.145 
 
 1.160 
 
 15 
 
 1.0 
 
 1.175 
 
 1.191 
 
 .206 
 
 1.222 
 
 1.238 
 
 1.254 
 
 1.270 
 
 1.286 
 
 1.303 
 
 1.319 
 
 16 
 
 1 
 
 1.336 
 
 1.352 
 
 .369 
 
 1.386 
 
 1.403 
 
 1.421 
 
 1.438 
 
 1.456 
 
 1.474 
 
 1.491 
 
 17 
 
 2 
 
 1.509 
 
 1.528 
 
 .546 
 
 1.564 
 
 1.583 
 
 1.602 
 
 1.621 
 
 1.640 
 
 1.659 
 
 1.679 
 
 19 
 
 3 
 
 1.698 
 
 1.718 
 
 .738 
 
 1.758 
 
 1.779 
 
 1.799 
 
 1.820 
 
 1.841 
 
 1.862 
 
 1.883 
 
 21 
 
 4 
 
 1.904 
 
 1.926 
 
 .948 
 
 1.970 
 
 1.992 
 
 ZOI4 
 
 2.037 
 
 2.060 
 
 2.083 
 
 2.106 
 
 22 
 
 1.5 
 
 2.129 
 
 2.153 
 
 2.177 
 
 2.201 
 
 2.225 
 
 2.250 
 
 2.274 
 
 2.299 
 
 2.324 
 
 2.350 
 
 25 
 
 6 
 
 2.376 
 
 2.401 
 
 2.428 
 
 2.454 
 
 2.481 
 
 2.507 
 
 2.535 
 
 2.562 
 
 2.590 
 
 2.617 
 
 27 
 
 7 
 
 2.646 
 
 2.674 
 
 2.703 
 
 2.732 
 
 2.761 
 
 2.790 
 
 2.820 
 
 2.850 
 
 2.881 
 
 2.911 
 
 30 
 
 8 
 
 2.942 
 
 2.973 
 
 3.005 
 
 3.037 
 
 3.069 
 
 3.101 
 
 3.134 
 
 3.167 
 
 3.200 
 
 3.234 
 
 33 
 
 9 
 
 3.268 
 
 3.303 
 
 3.337 
 
 3.372 
 
 3.408 
 
 3.443 
 
 3.479 
 
 3.516 
 
 3.552 
 
 3.589 
 
 36 
 
 2.0 
 
 3.627 
 
 3.665 
 
 3.703 
 
 3.741 
 
 3.780 
 
 3.820 
 
 3.859 
 
 3.899 
 
 3.940 
 
 3.981 
 
 39 
 
 1 
 
 4.022 
 
 4.064 
 
 4.106 
 
 4.148 
 
 4.191 
 
 4.234 
 
 4.278 
 
 4.322 
 
 4.367 
 
 4.412 
 
 44 
 
 2 
 
 4.457 
 
 4.503 
 
 4.549 
 
 4.596 
 
 4.643 
 
 4.691 
 
 4.739 
 
 4.788 
 
 4.837 
 
 4.887 
 
 48 
 
 3 
 
 4.937 
 
 4.988 
 
 5.039 
 
 5.090 
 
 5.142 
 
 5.195 
 
 5.248 
 
 5.302 
 
 5.356 
 
 5.411 
 
 53 
 
 4 
 
 5.466 
 
 5.522 
 
 5.578 
 
 5.635 
 
 5.693 
 
 5.751 
 
 5.810 
 
 5.869 
 
 5.929 
 
 5.989 
 
 58 
 
 2.5 
 
 6.050 
 
 6.112 
 
 6.174 
 
 6.237 
 
 6.300 
 
 6.365 
 
 6.429 
 
 6.495 
 
 6.561 
 
 6.627 
 
 64 
 
 6 
 
 6.695 
 
 6.763 
 
 6.831 
 
 6.901 
 
 6.971 
 
 7.042 
 
 7.113 
 
 7.185 
 
 7.258 
 
 7.332 
 
 71 
 
 7 
 
 7.406 
 
 7.481 
 
 7.557 
 
 7.634 
 
 7.711 
 
 7.789 
 
 7.868 
 
 7.948 
 
 8.028 
 
 8.110 
 
 79 
 
 8 
 
 8.192 
 
 8.275 
 
 8.359 
 
 8.443 
 
 8.529 
 
 8.615 
 
 8.702 
 
 8.790 
 
 8.879 
 
 8.969 
 
 87 
 
 9 
 
 9.060 
 
 9.151 
 
 9.244 
 
 9.337 
 
 9.431 
 
 9.527 
 
 9.623 
 
 9.720 
 
 9.819 
 
 9.918 
 
 96 
 
 3.0 
 
 10.02 
 
 10.12 
 
 10.22 
 
 10.32 
 
 10.43 
 
 10.53 
 
 10.64 
 
 10.75 
 
 10.86 
 
 10.97 
 
 11 
 
 1 
 
 11.08 
 
 11.19 
 
 11.30 
 
 11.42 
 
 11.53 
 
 11.65 
 
 11.76 
 
 11.88 
 
 12.00 
 
 12.12 
 
 12 
 
 2 
 
 12.25 
 
 12.37 
 
 12.49 
 
 12.62 
 
 12.75 
 
 12.88 
 
 13.01 
 
 13.14 
 
 13.27 
 
 13.40 
 
 13 
 
 3 
 
 13.54 
 
 13.67 
 
 13.81 
 
 13.95 
 
 14.09 
 
 14.23 
 
 14.38 
 
 14.52 
 
 14.67 
 
 14.82 
 
 14 
 
 4 
 
 14.97 
 
 15.12 
 
 15.27 
 
 15.42 
 
 15.58 
 
 15.73 
 
 15.89 
 
 16.05 
 
 16.21 
 
 16.38 
 
 16 
 
 3.5 
 
 16.54 
 
 16.71 
 
 16.88 
 
 17.05 
 
 17.22 
 
 17.39 
 
 17.57 
 
 17.74 
 
 17.92 
 
 18.10 
 
 17 
 
 6 
 
 18.29 
 
 18.47 
 
 18.66 
 
 18.84 
 
 19.03 
 
 19.22 
 
 19.42 
 
 19.61 
 
 19.81 
 
 20.01 
 
 19 
 
 7 
 
 20.21 
 
 20.41 
 
 20.62 
 
 20.83 
 
 21.04 
 
 21.25 
 
 21.46 
 
 21.68 
 
 21.90 
 
 22.12 
 
 21 
 
 8 
 
 22.34 
 
 22.56 
 
 22.79 
 
 23.02 
 
 23.25 
 
 23.49 
 
 23.72 
 
 23.96 
 
 24.20 
 
 24.45 
 
 24 
 
 9 
 
 24.69 
 
 24.94 
 
 25.19 
 
 25.44 
 
 25.70 
 
 25.96 
 
 26.22 
 
 26.48 
 
 26.75 
 
 27.02 
 
 26 
 
 4.0 
 
 27.29 
 
 27.56 
 
 27.84 
 
 28.12 
 
 28.40 
 
 28.69 
 
 28.98 
 
 29.27 
 
 29.56 
 
 29.86 
 
 29 
 
 1 
 
 30.16 
 
 30.47 
 
 30.77 
 
 31.08 
 
 31.39 
 
 31.71 
 
 32.03 
 
 32.35 
 
 32.68 
 
 33.00 
 
 32 
 
 2 
 
 33.34 
 
 33.67 
 
 34.01 
 
 34.35 
 
 34.70 
 
 35.05 
 
 35.40 
 
 35.75 
 
 36.11 
 
 36.48 
 
 35 
 
 3 
 
 36.84 
 
 37.21 
 
 37.59 
 
 37.97 
 
 38.35 
 
 38.73 
 
 39.12 
 
 39.52 
 
 39.91 
 
 40.31 
 
 39 
 
 4 
 
 40.72 
 
 41.13 
 
 41.54 
 
 41.96 
 
 42.38 
 
 42.81 
 
 43.24 
 
 43.67 
 
 44.11 
 
 44.56 
 
 43 
 
 4.5 
 
 45.00 
 
 45.46 
 
 45.91 
 
 46.37 
 
 46.84 
 
 47.31 
 
 47.79 
 
 48.27 
 
 48.75 
 
 49.24 
 
 47 
 
 6 
 
 49.74 
 
 5024 
 
 50.74 
 
 51.25 
 
 51.77 
 
 52.29 
 
 52.81 
 
 53.34 
 
 53.88 
 
 54.42 
 
 52 
 
 7 
 
 54.97 
 
 55.52 
 
 56.08 
 
 56.64 
 
 57.21 
 
 57.79 
 
 58.37 
 
 58.96 
 
 59.55 
 
 60.15 
 
 58 
 
 8 
 
 60.75 
 
 61.36 
 
 61.98 
 
 62.60 
 
 63.23 
 
 63.87 
 
 64.51 
 
 65.16 
 
 65.81 
 
 66.47 
 
 64 
 
 9 
 
 67.14 
 
 67.82 
 
 68.50 
 
 69.19 
 
 69.88 
 
 70.58 
 
 71.29 
 
 72.01 
 
 72.73 
 
 73.46 
 
 71 
 
 5.0 
 
 74.20 
 
 
 
 
 
 
 
 
 
 
 
 If x > 5, sinh x = W(e*) and logio sinh x = (0.4343)z + 0.6990 1, correct to four 
 significant figures. For table of multiples of 0.4343, see p. 62. Graphs, p. 174. 
 
MATHEMATICAL TABLES 61 
 
 HYPERBOLIC COSINES [cosh x = K(e* +e~*)] 
 
 V 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 0.0 
 
 1.000 
 
 1.000 
 
 1.000 
 
 1.000 
 
 1.001 
 
 1.001 
 
 1.002 
 
 1.002 
 
 1.003 
 
 1.004 
 
 1 
 
 1 
 
 .005 
 
 1.006 
 
 1.007 
 
 1.008 
 
 1.010 
 
 1.011 
 
 1.013 
 
 1.014 
 
 1.016 
 
 1.018 
 
 2 
 
 2 
 
 .020 
 
 1.022 
 
 1.024 
 
 1.027 
 
 1.029 
 
 1.031 
 
 1.034 
 
 1.037 
 
 1.039 
 
 1.042 
 
 3 
 
 3 
 
 .045 
 
 1.048 
 
 .052 
 
 1.055 
 
 1.058 
 
 1.062 
 
 1.066 
 
 1.069 
 
 1.073 
 
 1.077 
 
 4 
 
 4 
 
 .081 
 
 1.085 
 
 .090 
 
 1.094 
 
 1.098 
 
 1.103 
 
 1.108 
 
 1.112 
 
 1.117 
 
 1.122 
 
 5 
 
 0.5 
 
 .128 
 
 1.133 
 
 .138 
 
 1.144 
 
 1.149 
 
 1.155 
 
 1.161 
 
 1.167 
 
 1.173 
 
 1.179 
 
 6 
 
 6 
 
 .185 
 
 1.192 
 
 .198 
 
 1.205 
 
 1.212 
 
 1.219 
 
 1.226 
 
 1.233 
 
 1.240 
 
 1.248 
 
 7 
 
 7 
 
 .255 
 
 1.263 
 
 .271 
 
 1.278 
 
 1.287 
 
 1.295 
 
 1.303 
 
 1.311 
 
 1.320 
 
 1.329 
 
 8 
 
 8 
 
 .337 
 
 1.346 
 
 .355 
 
 1.365 
 
 1.374 
 
 1.384 
 
 1.393 
 
 1.403 
 
 1.413 
 
 1.423 
 
 10 
 
 9 
 
 1.433 
 
 1.443 
 
 .454 
 
 1.465 
 
 1.475 
 
 1.486 
 
 1.497 
 
 1.509 
 
 1.520 
 
 1.531 
 
 11 
 
 1.0 
 
 1.543 
 
 1.555 
 
 1.567 
 
 1.579 
 
 1.591 
 
 1.604 
 
 1.616 
 
 1.629 
 
 1.642 
 
 1.655 
 
 13 
 
 1 
 
 1.669 
 
 1.682 
 
 1.696 
 
 1.709 
 
 1.723 
 
 1.737 
 
 1.752 
 
 1.766 
 
 1.781 
 
 1.796 
 
 14 
 
 2 
 
 1.811 
 
 1.826 
 
 1.841 
 
 1.857 
 
 1.872 
 
 1.888 
 
 1.905 
 
 1.921 
 
 1.937 
 
 1.954 
 
 16 
 
 3 
 
 1.971 
 
 1.988 
 
 2.005 
 
 2.023 
 
 2.040 
 
 2.058 
 
 2.076 
 
 2.095 
 
 2.113 
 
 2.132 
 
 18 
 
 4 
 
 2.151 
 
 2.170 
 
 2.189 
 
 2.209 
 
 2.229 
 
 2.249 
 
 2.269 
 
 2.290 
 
 2.310 
 
 2.331 
 
 20 
 
 1.5 
 
 2.352 
 
 2.374 
 
 2.395 
 
 2.417 
 
 2.439 
 
 2.462 
 
 2.484 
 
 2.507 
 
 2.530 
 
 2.554 
 
 23 
 
 6 
 
 2.577 
 
 2.601 
 
 2.625 
 
 2.650 
 
 2.675 
 
 2.700 
 
 2.725 
 
 2.750 
 
 2.776 
 
 2.802 
 
 25 
 
 7 
 
 2.828 
 
 2.855 
 
 2.882 
 
 2.909 
 
 2.936 
 
 2.964 
 
 2.992 
 
 3.021 
 
 3.049 
 
 3.078 
 
 28 
 
 8 
 
 3.107 
 
 3.137 
 
 3167 
 
 3.197 
 
 3.228 
 
 3.259 
 
 3.290 
 
 3.321 
 
 3.353 
 
 3.385 
 
 31 
 
 9 
 
 3.418 
 
 3.451 
 
 3.484 
 
 3.517 
 
 3.551 
 
 3.585 
 
 3.620 
 
 3.655 
 
 3.690 
 
 3.726 
 
 34 
 
 2.0 
 
 3.762 
 
 3.799 
 
 3.835 
 
 3.873 
 
 3.910 
 
 3.948 
 
 3.987 
 
 4.026 
 
 4.065 
 
 4.104 
 
 38 
 
 1 
 
 4.144 
 
 4.185 
 
 4.226 
 
 4.267 
 
 4.309 
 
 4.351 
 
 4.393 
 
 4.436 
 
 4.480 
 
 4.524 
 
 42 
 
 2 
 
 4.568 
 
 4.613 
 
 4.658 
 
 4.704 
 
 4.750 
 
 4.797 
 
 4.844 
 
 4.891 
 
 4.939 
 
 4.988 
 
 47 
 
 3 
 
 5.037 
 
 5.087 
 
 5.137 
 
 5.188 
 
 5.239 
 
 5.290 
 
 5.343 
 
 5.395 
 
 5.449 
 
 5.503 
 
 52 
 
 4 
 
 5.557 
 
 5.612 
 
 5.667 
 
 5.723 
 
 5.780 
 
 5.837 
 
 5.895 
 
 5.954 
 
 6.013 
 
 6.072 
 
 58 
 
 2.5 
 
 6.132 
 
 6.193 
 
 6.255 
 
 6.317 
 
 6.379 
 
 6.443 
 
 6.507 
 
 6.571 
 
 6.636 
 
 6.702 
 
 64 
 
 6 
 
 6.769 
 
 6.836 
 
 6.904 
 
 6.973 
 
 7.042 
 
 7.112 
 
 7.183 
 
 7.255 
 
 7.327 
 
 7.400 
 
 70 
 
 7 
 
 7.473 
 
 7.548 
 
 7.623 
 
 7.699 
 
 7.776 
 
 7.853 
 
 7.932 
 
 8.011 
 
 8.091 
 
 8.171 
 
 78 
 
 8 
 
 8.253 
 
 8.335 
 
 8.418 
 
 8.502 
 
 8.587 
 
 8.673 
 
 8.759 
 
 8.847 
 
 8.935 
 
 9.024 
 
 86 
 
 9 
 
 9.115 
 
 9.206 
 
 9.298 
 
 9.391 
 
 9.484 
 
 9.579 
 
 9.675 
 
 9.772 
 
 9.869 
 
 9.968 
 
 95 
 
 3.0 
 
 10.07 
 
 10.17 
 
 10.27 
 
 10.37 
 
 10.48 
 
 10.58 
 
 10.69 
 
 10.79 
 
 10.90 
 
 11.01 
 
 11 
 
 1 
 
 11.12 
 
 11.23 
 
 11.35 
 
 11.46 
 
 11.57 
 
 11.69 
 
 11.81 
 
 11.92 
 
 12.04 
 
 12.16 
 
 12 
 
 2 
 
 12.29 
 
 12.41 
 
 12.53 
 
 12.66 
 
 12.79 
 
 12.91 
 
 13.04 
 
 13.17 
 
 13.31 
 
 13.44 
 
 13 
 
 3 
 
 13.57 
 
 13.71 
 
 13.85 
 
 13.99 
 
 14.13 
 
 14.27 
 
 14.41 
 
 14.56 
 
 14.70 
 
 14.85 
 
 14 
 
 4 
 
 15.00 
 
 15.15 
 
 15.30 
 
 15.45 
 
 15.ftl 
 
 15.77 
 
 15.92 
 
 16.08 
 
 16.25 
 
 16.41 
 
 16 
 
 3.5 
 
 16.57 
 
 16.74 
 
 16.91 
 
 17.08 
 
 17.25 
 
 17.42 
 
 17.60 
 
 17.77 
 
 17.95 
 
 18.13 
 
 17 
 
 6 
 
 18.31 
 
 18.50 
 
 18.68 
 
 18.87 
 
 19.06 
 
 19.25 
 
 19.44 
 
 19.64 
 
 19.84 
 
 20.03 
 
 19 
 
 7 
 8 
 
 20.24 
 22.36 
 
 20.44 
 22.59 
 
 20.64 
 22.81 
 
 20.85 
 23.04 
 
 21.06 
 23.27 
 
 21.27 
 23.51 
 
 21.49 
 23.74 
 
 21.70 
 23.98 
 
 21.92 
 24.22 
 
 22.14 
 24.47 
 
 21 
 23 
 
 9 
 
 24.71 
 
 24.96 
 
 25.21 
 
 25.46 
 
 25.72 
 
 25.98 
 
 26.24 
 
 26.50 
 
 26.77 
 
 27.04 
 
 26 
 
 4.0 
 
 27.31 
 
 27.58 
 
 27.86 
 
 28.14 
 
 28.42 
 
 28.71 
 
 29.00 
 
 29.29 
 
 29.58 
 
 29.88 
 
 29 
 
 1 
 
 30.18 
 
 30.48 
 
 30.79 
 
 31.10 
 
 31.41 
 
 31.72 
 
 32.04 
 
 32.37 
 
 32.69 
 
 33.02 
 
 32 
 
 2 
 
 33.35 
 
 33.69 
 
 34.02 
 
 34.37 
 
 34.71 
 
 35.06 
 
 35.41 
 
 35.77 
 
 36.13 
 
 36.49 
 
 35 
 
 3 
 
 36.86 
 
 37.23 
 
 37.60 
 
 37.98 
 
 38.36 
 
 38.75 
 
 39.13 
 
 39.53 
 
 39.93 
 
 40.33 
 
 39 
 
 4 
 
 40.73 
 
 41.14 
 
 41.55 
 
 41.97 
 
 42.39 
 
 42.82 
 
 43.25 
 
 43.68 
 
 44.12 
 
 44.57 
 
 43 
 
 4.E 
 
 45.01 
 
 45.47 
 
 45.92 
 
 46.38 
 
 46.85 
 
 47.32 
 
 47.80 
 
 48.28 
 
 48.76 
 
 49.25 
 
 47 
 
 6 
 
 49.75 
 
 50.25 
 
 50.75 
 
 51.26 
 
 51.78 
 
 52.30 
 
 52.82 
 
 53.35 
 
 53.89 
 
 54.43 
 
 52 
 
 7 
 
 54.98 
 
 55.53 
 
 56.09 
 
 56.65 
 
 57.22 
 
 57.80 
 
 58.38 
 
 58.96 
 
 59.56 
 
 60.15 
 
 58 
 
 8 
 
 60.76 
 
 61.37 
 
 61.99 
 
 62.61 
 
 63.24 
 
 63.87 
 
 64.52 
 
 65.16 
 
 65.82 
 
 66.48 
 
 64 
 
 9 
 
 67.15 
 
 67.82 
 
 68.50 
 
 69.19 
 
 69.89 
 
 70.59 
 
 71.30 
 
 72.02 
 
 72.74 
 
 73.47 
 
 71 
 
 5.0 
 
 74.21 
 
 
 
 
 
 
 
 
 
 
 
 If x > 5, cosh x = ^i(e*) and logio cosh x = (0.4343)* + 0.6990 1, correct to four signifi- 
 cant figures. For table of multiples of 0.4343, see p. 62. Graphs, p. 174. 
 
62 MATHEMATICAL TABLES 
 
 HYPEEBOLIC TANGENTS [tanh x = (e*-e~*) /(e* +<T*) = sinh z/cosh 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 (1 
 
 8 
 
 9 
 
 |l 
 
 0.0 
 
 .0000 
 
 .0100 
 
 .0200 
 
 .0300 
 
 .0400 
 
 .0500 
 
 .0599 
 
 .0699 
 
 .0798 
 
 .0898 
 
 100 
 
 
 .0997 
 
 .1096 
 
 .1194 
 
 .1293 
 
 .1391 
 
 .1489 
 
 .1587 
 
 .1684 
 
 .1781 
 
 .1878 
 
 98 
 
 2 
 
 .1974 
 
 .2070 
 
 .2165 
 
 .2260 
 
 .2355 
 
 .2449 
 
 .2543 
 
 .2636 
 
 .2729 
 
 .2821 
 
 94 
 
 3 
 
 .2913 
 
 .3004 
 
 .3095 
 
 .3185 
 
 .3275 
 
 .3364 
 
 .3452 
 
 .3540 
 
 .3627 
 
 .3714 
 
 89 
 
 4 
 
 .3800 
 
 .3885 
 
 .3969 
 
 .4053 
 
 .4136 
 
 .4219 
 
 .4301 
 
 .4382 
 
 .4462 
 
 .4542 
 
 82 
 
 0.5 
 
 .4621 
 
 .4700 
 
 .4777 
 
 .4854 
 
 .4930 
 
 .5005 
 
 .5080 
 
 .5154 
 
 .5227 
 
 .5299 
 
 75 
 
 6 
 
 .5370 
 
 .5441 
 
 .5511 
 
 .5581 
 
 .5649 
 
 .5717 
 
 .5784 
 
 .5850 
 
 .5915 
 
 .5980 
 
 67 
 
 7 
 
 .6044 
 
 .6107 
 
 .6169 
 
 .6231 
 
 .6291 
 
 .6352 
 
 .6411 
 
 .6469 
 
 .6527 
 
 .6584 
 
 60 
 
 8 
 
 .6640 
 
 .6696 
 
 .6751 
 
 .6805 
 
 .6858 
 
 .6911 
 
 .6963 
 
 .7014 
 
 .7064 
 
 .7114 
 
 52 
 
 9 
 
 .7163 
 
 .7211 
 
 .7259 
 
 .7306 
 
 .7352 
 
 .7398 
 
 .7443 
 
 .7487 
 
 .7531 
 
 .7574 
 
 45 
 
 1.0 
 
 .7616 
 
 .7658 
 
 .7699 
 
 .7739 
 
 .7779 
 
 .7818 
 
 .7857 
 
 .7895 
 
 .7932 
 
 .7969 
 
 39 
 
 1 
 
 .8005 
 
 .8041 
 
 .8076 
 
 .8110 
 
 .8144 
 
 .8178 
 
 .8210 
 
 .8243 
 
 .8275 
 
 .8306 
 
 33 
 
 2 
 
 .8337 
 
 .8367 
 
 .8397 
 
 .8426 
 
 .8455 
 
 .8483 
 
 .8511 
 
 .8538 
 
 .8565 
 
 .8591 
 
 28 
 
 3 
 
 .8617 
 
 .8643 
 
 .8668 
 
 .8693 
 
 .8717 
 
 .8741 
 
 .8764 
 
 .8787 
 
 .8810 
 
 .8832 
 
 24 
 
 4. 
 
 .8854 
 
 .8875 
 
 .8896 
 
 .8917 
 
 .8937 
 
 .8957 
 
 .8977 
 
 .8996 
 
 .9015 
 
 .9033 
 
 20 
 
 1.5 
 
 .9052 
 
 .9069 
 
 .9087 
 
 .9104 
 
 .9121 
 
 .9138 
 
 .9154 
 
 .9170 
 
 .9186 
 
 .9202 
 
 17 
 
 6 
 
 .9217 
 
 .9232 
 
 .9246 
 
 .9261 
 
 .9275 
 
 .9289 
 
 .9302 
 
 .9316 
 
 .9329 
 
 .9342 
 
 14 
 
 7 
 
 .9354 
 
 .9367 
 
 .9379 
 
 .9391 
 
 .9402 
 
 .9414 
 
 .9425 
 
 .9436 
 
 .9447 
 
 .9458 
 
 11 
 
 8 
 
 .9468 
 
 .9478 
 
 .9488 
 
 .9498 
 
 .9508 
 
 .9518 
 
 .9527 
 
 .9536 
 
 .9545 
 
 .9554 
 
 9 
 
 9 
 
 .9562 
 
 .9571 
 
 .9579 
 
 .9587 
 
 .9595 
 
 .9603 
 
 .9611 
 
 .9619 
 
 .9626 
 
 .9633 
 
 8 
 
 2.0 
 
 .9640 
 
 .9647 
 
 .9654 
 
 .9661 
 
 .9668 
 
 .9674 
 
 .9680 
 
 .9687 
 
 .9693 
 
 .9699 
 
 6 
 
 1 
 
 .9705 
 
 .9710 
 
 .9716 
 
 .9722 
 
 .9727 
 
 .9732 
 
 .9738 
 
 .9743 
 
 .9748 
 
 .9753 
 
 5 
 
 2 
 
 .9757 
 
 .9762 
 
 .9767 
 
 .9771 
 
 .9776 
 
 .9780 
 
 .9785 
 
 .9789 
 
 .9793 
 
 .9797 
 
 4 
 
 3 
 
 .9801 
 
 .9805 
 
 .9809 
 
 .9812 
 
 .9816 
 
 .9820 
 
 .9823 
 
 .9827 
 
 .9830 
 
 .9834 
 
 4 
 
 4 
 
 .9837 
 
 .9840 
 
 .9843 
 
 .9846 
 
 .9849 
 
 .9852 
 
 .9855 
 
 .9858 
 
 .9861 
 
 .9863 
 
 3 
 
 2.5 
 
 .9866 
 
 .9869 
 
 .9871 
 
 .9874 
 
 .9876 
 
 .9879 
 
 .9881 
 
 .9884 
 
 .9886 
 
 .9888 
 
 2 
 
 6 
 
 .9890 
 
 .9892 
 
 .9895 
 
 .9897 
 
 .9899 
 
 .9901 
 
 .9903 
 
 .9905 
 
 .9906 
 
 .9908 
 
 2 
 
 7 
 
 .9910 
 
 .9912 
 
 .9914 
 
 .9915 
 
 .9917 
 
 .9919 
 
 .9920 
 
 .9922 
 
 .9923 
 
 .9925 
 
 2 
 
 8 
 
 .9926 
 
 .9928 
 
 .9929 
 
 .9931 
 
 .9932 
 
 .9933 
 
 .9935 
 
 .9936 
 
 .9937 
 
 .9938 
 
 
 2.9 
 
 .9940 
 
 .9941 
 
 .9942 
 
 .9943 
 
 .9944 
 
 .9945 
 
 .9946 
 
 .9947 
 
 .9949 
 
 .9950 
 
 1 
 
 3. 
 
 .9951 
 
 .9959 
 
 .9967 
 
 .9973 
 
 .9978 
 
 .9982 
 
 .9985 
 
 .9988 
 
 .9990 
 
 .9992 
 
 4 
 
 4. 
 
 .9993 
 
 .9995 
 
 .9996 
 
 .9996 
 
 .9997 
 
 .9998 
 
 .9998 
 
 .9998 
 
 .9999 
 
 .9999 
 
 I 
 
 5. 
 
 .9999 
 
 If x > 5, 
 
 tanh a; 
 
 = 1.0000 to four decimal places. Graphs, p 
 
 . 174. 
 
 
 
 MULTIPLES OF 0.4343 (0.43429448 = logw e) 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 0. 
 
 0.0000 
 
 0.0434 
 
 0.0869 
 
 0.1303 
 
 0.1737 
 
 0.2171 
 
 0.2606 
 
 0.3040 
 
 0.3474 
 
 0.390< 
 
 1. 
 
 0.4343 
 
 0.4777 
 
 0.5212 
 
 0.5646 
 
 0.6080 
 
 0.6514 
 
 0.6949 
 
 0.7383 
 
 0.7817 
 
 0.825: 
 
 2. 
 
 0.8686 
 
 0.9120 
 
 0.9554 
 
 0.9989 
 
 1.0423 
 
 1.0857 
 
 1.1292 
 
 1.1726 
 
 1.2160 
 
 1 .259! 
 
 3. 
 
 1 .3029 
 
 1.3463 
 
 1.3897 
 
 1.4332 
 
 1.4766 
 
 1.5200 
 
 1.5635 
 
 1.6069 
 
 1.6503 
 
 1 .6933 
 
 4. 
 
 1.7372 
 
 1.7806 
 
 1.8240 
 
 1.8675 
 
 1.9109 
 
 1.9543 
 
 1.9978 
 
 2.0412 
 
 2.0846 
 
 2.128( 
 
 5. 
 
 2.1715 
 
 2.2149 
 
 2.2583 
 
 2.3018 
 
 2.3452 
 
 2.3886 
 
 2.4320 
 
 2.4755 
 
 2.5189 
 
 2.5622 
 
 6. 
 
 2.6058 
 
 2.6492 
 
 2.6926 
 
 2.7361 
 
 2.7795 
 
 2.8229 
 
 2.8663 
 
 2.9098 
 
 2.9532 
 
 2.996* 
 
 7. 
 
 3.0401 
 
 3.0835 
 
 3.1269 
 
 3.1703 
 
 3.2138 
 
 3.2572 
 
 3.3006 
 
 3.3441 
 
 3.3875 
 
 3.43TC 
 
 8. 
 
 3.4744 
 
 3.5178 
 
 3.5612 
 
 3.6046 
 
 3.6481 
 
 3.6915 
 
 3.7349 
 
 3.7784 
 
 3.8218 
 
 3.8652 
 
 9. 
 
 3.9087 
 
 3.9521 
 
 3.9955 
 
 4.0389 
 
 4.0824 
 
 4.1258 
 
 4.1692 
 
 4.2127 
 
 4.2561 
 
 4.299* 
 
 MULTIPLES OP 
 
 2.3026 
 
 (2.3025851 = 
 
 1/0.4343) 
 
 
 
 
 
 x 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 0. 
 
 0.0000 
 
 0.2303 
 
 0.4605 
 
 0.6908 
 
 0.9210 
 
 1.1513 
 
 1.3816 
 
 1.6118 
 
 1.8421 
 
 2.0723 
 
 1. 
 
 2.3026 
 
 2.5328 
 
 2.7631 
 
 2.9934 
 
 3.2236 
 
 3.4539 
 
 3.6841 
 
 3.9144 
 
 4.1447 
 
 4.3749 
 
 2. 
 
 4.6052 
 
 4.8354 
 
 5.0657 
 
 5.2959 
 
 5.5262 
 
 5.7565 
 
 5.9867 
 
 6.2170 
 
 6.4472 
 
 6.6775 
 
 3. 
 
 6.9078 
 
 7.1380 
 
 7.3683 
 
 7.5985 
 
 7.8288 
 
 8.0590 
 
 8.2893 
 
 8.5196 
 
 8.7498 
 
 8.9801 
 
 4. 
 
 9.2103 
 
 9.4406 
 
 9.6709 
 
 9.901 1 
 
 10.131 
 
 10.362 
 
 10.592 
 
 10.822 
 
 11.052 
 
 11.283 
 
 5. 
 
 11.513 
 
 11.743 
 
 11.973 
 
 12.204 
 
 12.434 
 
 12.664 
 
 12.894 
 
 13.125 
 
 13.355 
 
 13.585 
 
 6. 
 
 13.816 
 
 14.046 
 
 14.276 
 
 14.506 
 
 14.737 
 
 14.967 
 
 15.197 
 
 15.427 
 
 15.658 
 
 15.888 
 
 7. 
 
 16.118 
 
 ' 16.348 
 
 16.579 
 
 16.809 
 
 17.039 
 
 17.269 
 
 17.500 
 
 17.730 
 
 17.960 
 
 18.190 
 
 8. 
 
 18.421 
 
 18.651 
 
 18.881 
 
 19.111 
 
 19.342 
 
 19.572 
 
 19.802 
 
 20.032 
 
 20.263 
 
 20.493 
 
 9. 
 
 20.723 
 
 20.954 
 
 21.184 
 
 21.414 
 
 21.644 
 
 21.875 
 
 22.105 
 
 22.335 
 
 22.565 
 
 22.796 
 
MATHEMATICAL TABLES 
 
 63 
 
 STANDARD 
 DISTRIBUTION OF 
 RESIDUALS (p. 121) 
 
 a = any positive quantity; 
 y = the number of residuals 
 which are numerically < a; 
 r = the probable error of a single 
 observation; 
 
 n = number of observations. 
 
 a 
 
 y 
 
 Diff. 
 
 r 
 
 n 
 
 
 0.0 
 
 .000 
 
 
 1 
 2 
 
 .054 
 .107 
 
 54 
 53 
 
 CO 
 
 3 
 4 
 
 .160 
 .213 
 
 JJ 
 
 53 
 51 
 
 0.5 
 6 
 7 
 
 .264 
 .314 
 .363 
 
 50 
 49 
 
 8 
 9 
 
 .411 
 .456 
 
 45 
 44 
 
 1.0 
 
 .500 
 
 
 1 
 
 .542 
 
 ^/ 
 
 2 
 3 
 
 .582 
 .619 
 
 40 
 37 
 
 4 
 
 .655 
 
 33 
 
 1.5 
 6 
 7 
 8 
 9 
 
 .688 
 .719 
 .748 
 .775 
 . .800 
 
 31 
 29 
 27 
 25 
 23 
 
 2.0 
 
 1 
 
 .823 
 
 .843 
 
 20 
 
 i (i 
 
 2 
 
 .862 
 
 1 7 
 
 3 
 
 4 
 
 .879 
 .895 
 
 17 
 16 
 
 
 
 13 
 
 2.5 
 
 .908 
 
 
 6 
 
 .921 
 
 
 7 
 8 
 
 .931 
 .941 
 
 10 
 10 
 
 9 
 
 .950 
 
 9 
 7 
 
 3.0 
 
 .957 
 
 
 1 
 
 .963 
 
 6 
 
 2 
 
 .969 
 
 6 
 
 3 
 
 .974 
 
 5 
 
 4 
 
 .978 
 
 4 
 
 4 
 
 3.5 
 
 .982 
 
 
 6 
 
 .985 
 
 3 
 
 7 
 
 .987 
 
 
 8 
 
 .990 
 
 3 
 
 9 
 
 .991 
 
 2 
 
 4.0 
 
 .993 
 
 6 
 
 5.0 
 
 .999 
 
 
 FACTORS FOR COMPUTING 
 PROBABLE ERROR (p. 121) 
 
 n 
 
 Bessel 
 
 Peters 
 
 0.6745 
 
 0.6745 
 
 0.8453 
 
 0.8453 
 
 V(n - 1) 
 
 Vn(n-l) 
 
 Vn(/i-l) 
 
 n\/n 1 
 
 2 
 
 .6745 
 
 .4769 
 
 .5978 
 
 .4227 
 
 3 
 
 .4769 
 
 .2754 
 
 .3451 
 
 .1993 
 
 4 
 
 .3894 
 
 .1947 
 
 .2440 
 
 .1220 
 
 5 
 
 .3372 
 
 .1508 
 
 .1890 
 
 .0845 
 
 6 
 
 .3016 
 
 .1231 
 
 .1543 
 
 .0630 
 
 7 
 
 .2754 
 
 .1041 
 
 .1304 
 
 .0493 
 
 8 
 
 .2549 
 
 .0901 
 
 .1130 
 
 .0399 
 
 9 
 
 .2385 
 
 .0795 
 
 .0996 
 
 .0332 
 
 10 
 
 .2248 
 
 .0711 
 
 .0891 
 
 .0282 
 
 11 
 
 .2133 
 
 .0643 
 
 .0806 
 
 .0243 
 
 12 
 
 .2034 
 
 .0587 
 
 .0736 
 
 .0212 
 
 13 
 
 .1947 
 
 .0540 
 
 .0677 
 
 .0188 
 
 14 
 
 .1871 
 
 .0500 
 
 .0627 
 
 .0167 
 
 15 
 
 .1803 
 
 .0465 
 
 .0583 
 
 .0151 
 
 16 
 
 .1742 
 
 .0435 
 
 .0546 
 
 .0136 
 
 17 
 
 .1686 
 
 .0409 
 
 .0513 
 
 .0124 
 
 18 
 
 .1636 
 
 .0386 
 
 .0483 
 
 .0114 
 
 19 
 
 .1590 
 
 .0365 
 
 .0457 
 
 .0105 
 
 20 
 
 .1547 
 
 .0346 
 
 .0434 
 
 .0097 
 
 21 
 
 .1508 
 
 .0329 
 
 .0412 
 
 .0090 
 
 22 
 
 .1472 
 
 .0314 
 
 .0393 
 
 .0084 
 
 23 
 
 .1438 
 
 .0300 
 
 .0376 
 
 .0078 
 
 24 
 
 .1406 
 
 .0287 
 
 .0360 
 
 .0073 
 
 25 
 
 .1377 
 
 .0275 
 
 .0345 
 
 .0069 
 
 26 
 
 .1349 
 
 .0265 
 
 .0332 
 
 .0065 
 
 27 
 
 .1323 
 
 .0255 
 
 .0319 
 
 .0061 
 
 28 
 
 .1298 
 
 .0245 
 
 .0307 
 
 .0058 
 
 29 
 
 .1275 
 
 .0237 
 
 .0297 
 
 .0055 
 
 30 
 
 .1252 
 
 .0229 
 
 .0287 
 
 .0052 
 
 31 
 
 .1231 
 
 .0221 
 
 .0277 
 
 .0050 
 
 32 
 
 .1211 
 
 . .0214 
 
 .0268 
 
 .0047 
 
 33 
 
 .1192 
 
 .0208 
 
 .0260 
 
 .0045 
 
 34 
 
 .1174 
 
 .0201 
 
 .0252 
 
 .0043 
 
 35 
 
 .1157 
 
 .0196 
 
 .0245 
 
 .0041 
 
 36 
 
 .1140 
 
 .0190 
 
 .0238 
 
 .0040 
 
 37 
 
 .1124 
 
 .0185 
 
 .0232 
 
 .0038 
 
 38 
 
 .1109 
 
 .0180 
 
 .0225 
 
 .0037 
 
 39 
 
 .1094 
 
 .0175 
 
 .0220 
 
 .0035 
 
 40 
 
 .1080 
 
 .0171 
 
 .0214 
 
 .0034 
 
 45 
 
 .1017 
 
 .0152 
 
 .0190 
 
 .0028 
 
 50 
 
 .0964 
 
 .0136 
 
 .0171 
 
 .0024 
 
 55 
 
 .0918 
 
 .0124 
 
 .0155 
 
 .0021 
 
 60 
 
 .0878 
 
 .0113 
 
 .0142 
 
 .0018 
 
 65 
 
 .0843 
 
 .0105 
 
 .0131 
 
 .0016 
 
 70 
 
 .0812 
 
 .0097 
 
 .0122 
 
 .0015 
 
 75 
 
 .0784 
 
 .0091 
 
 .0113 
 
 .0013 
 
 80 
 
 .0759 
 
 .0085 
 
 .0106 
 
 .0012 
 
 85 
 
 .0736 
 
 .0080 
 
 .0100 
 
 .0011 
 
 90 
 
 .0715 
 
 .0075 
 
 .0094 
 
 .0010 
 
 95 
 
 .06% 
 
 .0071 
 
 .0089 
 
 .0009 
 
 100 
 
 .0678 
 
 .0068 
 
 .0085 
 
 .0008 
 
64 
 
 MATHEMATICAL TABLES 
 
 COMPOUND INTEREST. AMOUNT OF A GIVEN PRINCIPAL 
 
 The amount A at the end of n years of a given principal P placed at compouni 
 interest to-day is A = P X x or A = P X y or A = P X z, according as the interes 
 (at the rate of r per cent, per annum) is compounded annually, semi-annually, o 
 quarterly; the factor x or y or z being taken from the following tables. 
 
 Values of x. (Interest compounded annually; A = P X .) 
 
 Years 
 
 r = 2 
 
 2H 
 
 3 
 
 ?M 
 
 4 
 
 4$* 
 
 5 
 
 6 
 
 7 
 
 
 i 
 
 1.0200 
 
 1.0250 
 
 1.0300 
 
 1.0350 
 
 1.0400 
 
 1.0450 
 
 1.0500 
 
 1.0600 
 
 1.0700 
 
 
 2 
 
 1.0404 
 
 1.0506 
 
 1.0609 
 
 1.0712 
 
 1.0816 
 
 1.0920 
 
 1.1025 
 
 .1236 
 
 1.1449 
 
 
 3 
 
 1.0612 
 
 1.0769 
 
 1.0927 
 
 1.1087 
 
 1.1249 
 
 1.1412 
 
 1.1576 
 
 .1910 
 
 1.2250 
 
 
 4 
 
 1.0824 
 
 1. 1038 
 
 1.1255 
 
 1.1475 
 
 1.1699 
 
 1.1925 
 
 12155 
 
 .2625 
 
 1.3108 
 
 3 
 
 5 
 
 1.1041 
 
 1.1314 
 
 1.1593 
 
 1.1877 
 
 1.2167 
 
 1.2462 
 
 1.2763 
 
 .3382 
 
 1.4026 
 
 
 6 
 
 1.1262 
 
 1.1597 
 
 1.1941 
 
 1.2293 
 
 1.2653 
 
 1.3023 
 
 1.3401 
 
 .4185 
 
 1.5007 
 
 1 
 
 7 
 
 1.1487 
 
 1.1887 
 
 1.2299 
 
 1.2723 
 
 1.3159 
 
 1.3609 
 
 1.4071 
 
 .5036 
 
 1.6058 
 
 ** 
 
 8 
 
 1.1717 
 
 1.2184 
 
 1.2668 
 
 1.3168 
 
 1.3686 
 
 1.4221 
 
 1.4775 
 
 .5938 
 
 1.7182 
 
 Ja 
 
 9 
 
 1.1951 
 
 1.2489 
 
 1.3048 
 
 1.3629 
 
 1.4233 
 
 1.4861 
 
 1.5513 
 
 1.6895 
 
 1.8385 
 
 ^ ' 
 
 10 
 
 1.2190 
 
 1.2801 
 
 1.3439 
 
 1.4106 
 
 1.4802 
 
 1.5530 
 
 1.6289 
 
 1.7908 
 
 1.9672 
 
 if 
 
 11 
 
 1.2434 
 
 1.3121 
 
 1.3842 
 
 1.4600 
 
 1.5395 
 
 1.6239 
 
 1.7103 
 
 1.8983 
 
 2.1049 
 
 t, O 
 
 12 
 
 1.2682 
 
 1.3449 
 
 1.4258 
 
 1.5111 
 
 1.6010 
 
 1.6959 
 
 1.7959 
 
 2.0122 
 
 2.2522 
 
 ^"vT 
 
 13 
 
 1.2936 
 
 1.3785 
 
 1.4685 
 
 1.5640 
 
 1.6651 
 
 1.7722 
 
 1.8856 
 
 2.1329 
 
 2.4098 
 
 v*^ 
 
 14 
 
 13195 
 
 1.4130 
 
 1.5126 
 
 1.6187 
 
 1.7317 
 
 1.3519 
 
 1.9799 
 
 2.2609 
 
 2.5785 
 
 a~^~ 
 
 15 
 
 1.3459 
 
 1.4483 
 
 1.5580 
 
 1.6753 
 
 1.8009 
 
 19353 
 
 2.0789 
 
 2.3966 
 
 2.7590 
 
 S^i 
 
 16 
 
 1.3728 
 
 1.4845 
 
 1.6047 
 
 1.7340 
 
 1.8730 
 
 2.0224 
 
 2.1829 
 
 2.5404 
 
 2.9522 
 
 2 !l 
 
 17 
 
 1.4002 
 
 1.5216 
 
 1.6528 
 
 1.7947 
 
 1.9479 
 
 2.1134 
 
 2.2920 
 
 2.6928 
 
 3.1588 
 
 .5 H 
 
 18 
 
 1.4282 
 
 1.5597 
 
 1.7024 
 
 1.8575 
 
 2.0258 
 
 2.2085 
 
 2.4066 
 
 2.8543 
 
 3.3799 
 
 
 
 19 
 
 1.4568 
 
 1.5987 
 
 1.7535 
 
 1.9225 
 
 2.1068 
 
 2.3079 
 
 2.5270 
 
 3.0256 
 
 3.6165 
 
 
 20 
 
 1.4859 
 
 1.6386 
 
 1.8061 
 
 1.9898 
 
 2.1911 
 
 2.4117 
 
 2.6^533 
 
 3.2071 
 
 3.8697 
 
 -2 
 
 25 
 
 1.6406 
 
 1.8539 
 
 2.0938 
 
 2.3632 
 
 2.6658 
 
 3.0054 
 
 3.3864 
 
 4.2919 
 
 5.4274 
 
 .2 
 
 30 
 
 1.8114 
 
 2.0976 
 
 2.4273 
 
 2.8068 
 
 3.2434 
 
 3.7453 
 
 4.3219 
 
 5.7435 
 
 7.6123 
 
 g2 
 
 40 
 
 2.2080 
 
 2.6851 
 
 3.2620 
 
 3.9593 
 
 4.8010 
 
 5.8164 
 
 7.0400 
 
 10.286 
 
 14.974 
 
 
 50 
 
 2.6916 
 
 3.4371 
 
 4.3839 
 
 5.5849 
 
 7.1067 
 
 9.0326 
 
 1 1 .467 
 
 18.420 
 
 29.457 
 
 
 60 
 
 3.2810 
 
 4.3998 
 
 5.8916 
 
 7.8781 
 
 10.520 
 
 14.027 
 
 18.679 
 
 32.988 
 
 57.946 
 
 
 Values of y. (Interest compounded semi-annually; A = P X y.) 
 
 Years 
 
 r=2 
 
 2H 
 
 3 
 
 3H 
 
 4 
 
 4H 
 
 5 
 
 6 
 
 7 
 
 
 1 
 
 1.0201 
 
 1.0252 
 
 1.0302 
 
 1.0353 
 
 1.0404 
 
 1 .0455 
 
 1.0506 
 
 1.0609 
 
 1.0712 
 
 
 2 
 
 1.0406 
 
 1.0509 
 
 1.0614 
 
 1.0719 
 
 1.0824 
 
 1.0931 
 
 1.1038 
 
 1.1255 
 
 1.1475 
 
 
 3 
 
 1.0615 
 
 1.0774 
 
 1.0934 
 
 1.1097 
 
 1.1262 
 
 1.1428 
 
 1.1597 
 
 1.1941 
 
 1 .2293 
 
 
 4 
 
 1.0829 
 
 1.1045 
 
 1.1265 
 
 1.1489 
 
 1.1717 
 
 1.1948 
 
 1.2134 
 
 1.2668 
 
 1.3168 
 
 
 5 
 
 1.1046 
 
 1.1323 
 
 1.1605 
 
 1.1894 
 
 1.2190 
 
 1.2492 
 
 1.2801 
 
 1.3439 
 
 1.4106 
 
 
 6 
 
 1.1268 
 
 1.1608 
 
 1.1956 
 
 1.2314 
 
 1.2682 
 
 1.3060 
 
 1.3449 
 
 1.4258 
 
 1.5111 
 
 
 7 
 
 1.1495 
 
 1.1900 
 
 1.2318 
 
 1.2749 
 
 1.3195 
 
 1.3655 
 
 1.4130 
 
 1.5126 
 
 1 .61 87 
 
 5 
 
 8 
 
 1.1726 
 
 1.2199 
 
 1.2690 
 
 1.3199 
 
 1.3728 
 
 1.4276 
 
 1.4845 
 
 1.6047 
 
 1.7340 
 
 g^ 
 
 9 
 
 1.1961 
 
 1.2506 
 
 1.3073 
 
 1.3665 
 
 1.4282 
 
 1.4926 
 
 1.5597 
 
 1.7024 
 
 1.8575 
 
 g 
 
 10 
 
 1.2202 
 
 1.2820 
 
 1.3469 
 
 1.4148 
 
 1.4859 
 
 1.5605 
 
 1.6386 
 
 1.8061 
 
 1.9898 
 
 ~- 
 
 11 
 
 1.2447 
 
 1.3143 
 
 1.3876 
 
 1.4647 
 
 1.5460 
 
 1.6315 
 
 1.7216 
 
 1.9161 
 
 2.1315 
 
 v ~' 
 
 12 
 
 1.2697 
 
 1.3474 
 
 1.4295 
 
 1.5164 
 
 1.6084 
 
 1.7058 
 
 1.8087 
 
 2.0328 
 
 2.2833 
 
 + 
 
 13 
 
 1.2953 
 
 1.3812 
 
 1.4727 
 
 1.5700 
 
 1.6734 
 
 1.7834 
 
 1.9003 
 
 2.1566 
 
 2.4460 
 
 H 
 
 14 
 
 1.3213 
 
 1.4160 
 
 1.5172 
 
 1.6254 
 
 1.7410 
 
 1.8645 
 
 1.9965 
 
 2.2879 
 
 2.6202 
 
 II 
 
 15 
 
 1.3478 
 
 1.4516 
 
 1.5631 
 
 1.6828 
 
 1.8114 
 
 1.9494 
 
 2.0976 
 
 2.4273 
 
 2.8068 
 
 a 
 
 16 
 
 1.3749 
 
 1.4881 
 
 1.6103 
 
 1.7422 
 
 1.8845 
 
 2.0381 
 
 2.2038 
 
 2.5751 
 
 3.0067 
 
 
 17 
 
 1 .4026 
 
 1.5256 
 
 1.6590 
 
 1.8037 
 
 1.9607 
 
 2.1308 
 
 2.3153 
 
 2.7319 
 
 3.2209 
 
 jj 
 
 18 
 
 1.4308 
 
 1.5639 
 
 1.7091 
 
 1.8674 
 
 2.0399 
 
 2.2278 
 
 2.4325 
 
 2.8983 
 
 3.4503 
 
 "9 
 
 19 
 
 1.4595 
 
 1.6033 
 
 1.7608 
 
 1.9333 
 
 2.1223 
 
 2.3292 
 
 2.5557 
 
 3.0748 
 
 3.6960 
 
 S 
 
 20 
 
 1.4889 
 
 1.6436 
 
 1.8140 
 
 2.0016 
 
 2.2080 
 
 2.4352 
 
 2.6851 
 
 3.2620 
 
 3.9593 
 
 fe 
 
 25 
 
 1.6446 
 
 1.8610 
 
 2.1052 
 
 2.3808 
 
 2.6916 
 
 3.0420 
 
 3.4371 
 
 4.3839 
 
 5.5849 
 
 
 30 
 
 1.8167 
 
 2.1072 
 
 2.4432 
 
 2.8318 
 
 3.2810 
 
 3.8001 
 
 4.3998 
 
 5.8916 
 
 7.8781 
 
 
 40 
 
 2.2167 
 
 2.7015 
 
 3.2907 
 
 4.0064 
 
 4.8754 
 
 5.9301 
 
 7.2096 
 
 10.641 
 
 15.676 
 
 
 50 
 
 2.7048 
 
 3.4634 
 
 4.4320 
 
 5.6682 
 
 7.2446 
 
 9.2540 
 
 11.814 
 
 19.219 
 
 31.191 
 
 
 60 
 
 3.3004 
 
 4.4402 
 
 5.9693 
 
 8.0192 
 
 10.765 
 
 14.441 
 
 19.358 
 
 34.711 
 
 62.064 
 
 
MATHEMATICAL TABLES 65 
 
 Values of z. (Interest compounded quarterly; A = P X 2; see opposite page) 
 
 Years 
 
 
 = 2 
 
 2H 
 
 3 
 
 3^ 
 
 4 
 
 4>i 
 
 5 
 
 6 
 
 7 
 
 
 1 
 
 
 .0202 
 
 .0252 
 
 1.0303 
 
 1.0355 
 
 1.0406 
 
 1.0458 
 
 1.0509 
 
 1.0614 
 
 1.0719 
 
 
 2 
 
 
 .0407 
 
 .0511 
 
 1.0616 
 
 1.0722 
 
 1.0829 
 
 1.0936 
 
 .1045 
 
 .1265 
 
 1.1489 
 
 
 3 
 
 
 .0617 
 
 .0776 
 
 1.0938 
 
 1.1102 
 
 1.1268 
 
 1.1437 
 
 .1608 
 
 .1956 
 
 1.2314 
 
 
 4 
 
 
 .0831 
 
 .1048 
 
 1.1270 
 
 1.1496 
 
 1.1726 
 
 1.1960 
 
 .2199 
 
 .2690 
 
 1.3199 
 
 
 5 
 
 
 .1049 
 
 .1327 
 
 1.1612 
 
 1.1903 
 
 1.2202 
 
 1.2508 
 
 .2820 
 
 .3469 
 
 1.4148 
 
 
 6 
 
 
 .1272 
 
 .1613 
 
 1.1964 
 
 1.2326 
 
 1.2697 
 
 1.3080 
 
 .3474 
 
 .4295 
 
 1.5164 
 
 _ 
 
 7 
 
 
 .1499 
 
 .1906 
 
 1.2327 
 
 1.2763 
 
 1.3213 
 
 1.3679 
 
 .4160 
 
 .5172 
 
 1.6254 
 
 X 
 
 8 
 
 
 .1730 
 
 .2206 
 
 1.2701 
 
 1.3215 
 
 1.3749 
 
 1.4305 
 
 .4881 
 
 1.6103 
 
 1.7422 
 
 o" 
 
 9 
 
 
 .1967 
 
 .2514 
 
 1.3086 
 
 1.3684 
 
 1.4308 
 
 1.4959 
 
 1.5639 
 
 1.7091 
 
 1.8674 
 
 3 
 
 10 
 
 
 .2208 
 
 .2830 
 
 1.3483 
 
 1.4169 
 
 1.4889 
 
 1.5644 
 
 1.6436 
 
 1.8140 
 
 2.0016 
 
 ^ 
 
 11 
 
 
 .2454 
 
 .3154 
 
 1.3893 
 
 1.4672 
 
 1.5493 
 
 1.6360 
 
 1.7274 
 
 1.9253 
 
 2.1454 
 
 
 12 
 
 
 .2705 
 
 .3486 
 
 1.4314 
 
 1.5192 
 
 1.6122 
 
 1.7108 
 
 1.8154 
 
 2.0435 
 
 2.2996 
 
 < 
 
 13 
 
 
 .2961 
 
 .3826 
 
 1.4748 
 
 1.5731 
 
 1.6777 
 
 1.7891 
 
 1.9078 
 
 2.1689 
 
 2.4648 
 
 ;-*. 
 
 14 
 
 
 .3222 
 
 .4175 
 
 1.5196 
 
 1.6288 
 
 1.7458 
 
 1.8710 
 
 2.0050 
 
 2.3020 
 
 2.6420 
 
 II 
 
 15 
 
 
 .3489 
 
 .4533 
 
 1.5657 
 
 1.6866 
 
 1.8167 
 
 1.9566 
 
 2.1072 
 
 2.4432 
 
 2.8318 
 
 M 
 
 16 
 
 
 .3760 
 
 .4900 
 
 1.6132 
 
 1.7464 
 
 1.8905 
 
 2.0462 
 
 2.2145 
 
 2.5931 
 
 3.0353 
 
 
 17 
 
 
 .4038 
 
 .5276 
 
 1.6621 
 
 1.8083 
 
 1.9672 
 
 2.1398 
 
 2.3274 
 
 2.7523 
 
 3.2534 
 
 J5 
 
 18 
 
 
 .4320 
 
 .5661 
 
 1.7126 
 
 1.8725 
 
 2.0471 
 
 2.2378 
 
 2.4459 
 
 2.9212 
 
 3.4872 
 
 
 19 
 
 
 .4609 
 
 .6056 
 
 ,1.7645 
 
 1.9389 
 
 2.1302 
 
 2.3402 
 
 2.5705 
 
 3.1004 
 
 3.7378 
 
 o 
 
 20 
 
 
 .4903 
 
 .6462 
 
 1.8180 
 
 2.0076 
 
 2.2167 
 
 2.4473 
 
 2.7015 
 
 3.2907 
 
 4.0064 
 
 A 
 
 25 
 
 
 .6467 
 
 .8646 
 
 2.1111 
 
 2.3898 
 
 2.7048 
 
 3.0609 
 
 3.4634 
 
 4.4320 
 
 5.6682 
 
 
 30 
 
 
 .8194 i 
 
 .1121 
 
 2.4514 
 
 2.8446 
 
 3.3004 
 
 3.8285 
 
 4.4402 
 
 5.9693 
 
 8.0192 
 
 
 40 
 
 
 !.22ii ; 
 
 '.7098 
 
 3.3053 
 
 4.0306 
 
 4.9138 
 
 5.9892 
 
 7.2980 
 
 10.828 
 
 16.051 
 
 
 50 
 
 
 5.7115 3 
 
 .4768 
 
 4.4567 
 
 5.7110 
 
 7.3160 
 
 9.3693 
 
 11.995 
 
 19.643 
 
 32.128 
 
 
 60 
 
 
 L3102 ^ 
 
 L4608 
 
 6.0092 
 
 8.0919 
 
 10.893 
 
 14.657 
 
 19.715 
 
 35.633 
 
 64.307 
 
 
 AMOUNT OP AN ANNUITY 
 
 The amount S accumulated at the end of n years by a given annual payment Y set 
 aside at the end of each year is S =* F X , where the factor v is to be taken from the 
 following table. (Interest at r per cent, per annum, compounded annually.) 
 
 Values of v 
 
 Years 
 
 r= 2 
 
 2H 
 
 3 
 
 Hi 
 
 4 
 
 ^ 
 
 5 
 
 6 
 
 7 
 
 
 1 
 
 1.0000 
 
 1.0000 
 
 1.0000 
 
 1.0000 
 
 1.0000 
 
 1.0000 
 
 1.0000 
 
 1.0000 
 
 1 .0000 
 
 
 2 
 
 2.0200 
 
 2.0250 
 
 2.0300 
 
 2.0350 
 
 2.0400 
 
 .2.0450 
 
 2.0500 
 
 2.0600 
 
 2.0700 
 
 
 3 
 
 3.0604 
 
 3.0756 
 
 3.0909 
 
 3.1062 
 
 3.1216 
 
 3.1370 
 
 3.1525 
 
 3.1836 
 
 3.2149 
 
 
 
 4 
 
 4.1216 
 
 4.1525 
 
 4.1836 
 
 4.2149 
 
 4.2465 
 
 4.2782 
 
 4.3101 
 
 4.3746 
 
 4.4399 
 
 O 
 
 5 
 
 5.2040 
 
 5.2563 
 
 5.3091 
 
 5.3625 
 
 5.4163 
 
 5.4707 
 
 5.5256 
 
 5.6371 
 
 5.7507 
 
 > 
 
 6 
 
 6.3081 
 
 6.3877 
 
 6.4684 
 
 6.5502 
 
 6.6330 
 
 6.7169 
 
 6.8019 
 
 6.9753 
 
 7.1533 
 
 .1. 
 
 7 
 
 7.4343 
 
 7.5474 
 
 7.6625 
 
 7.7794 
 
 7.8983 
 
 8.0192 
 
 8.1420 
 
 8.3938 
 
 8.6540 
 
 I 
 
 8 
 
 8.5830 
 
 8.7361 
 
 8.8923 
 
 9.0517 
 
 9.2142 
 
 9.3800 
 
 9.5491 
 
 9.8975 
 
 10.260 
 
 ^ 
 
 9 
 
 9.7546 
 
 9.9545 
 
 10.159 
 
 10.368 
 
 10.583 
 
 10.802 
 
 11.027 
 
 11.491 
 
 11.978 
 
 1 ; 
 
 10 
 
 10.950 
 
 11.203 
 
 11.464 
 
 11.731 
 
 12.006 
 
 12.288 
 
 12.578 
 
 13.181 
 
 13.816 
 
 S 
 
 11 
 
 12.169 
 
 12.483 
 
 12.808 
 
 13.142 
 
 13.486 
 
 13.841 
 
 14.207 
 
 14.972 
 
 15.784 
 
 11 o 
 
 12 
 
 13.412 
 
 13.796 
 
 14.192 
 
 14.602 
 
 15.026 
 
 15.464 
 
 15.917 
 
 16.870 
 
 17.888 
 
 !> 
 
 13 
 
 14.680 
 
 15.140 
 
 15.618 
 
 16.113 
 
 16.627 
 
 17.160 
 
 17.713 
 
 18.882 
 
 20.141 
 
 
 14 
 
 15.974 
 
 16.519 
 
 17.086 
 
 17.677 
 
 18.292 
 
 18.932 
 
 19.599 
 
 21.015 
 
 22.550 
 
 > ! 
 
 15 
 
 17.293 
 
 17.932 
 
 18.599 
 
 19.296 
 
 20.024 
 
 20.784 
 
 21.579 
 
 23.276 
 
 25.129 
 
 + 7 
 
 16 
 
 18.639 
 
 19.380 
 
 20.157 
 
 20.971 
 
 21.825 
 
 22.719 
 
 23.657 
 
 25.673 
 
 27.888 
 
 ^ 1 
 
 17 
 
 20.012 
 
 20.865 
 
 21.762 
 
 22.705 
 
 23.698 
 
 24.742 
 
 25.840 
 
 28.213 
 
 30.840 
 
 " vS 
 
 18 
 
 21.412 
 
 22.386 
 
 23.414 
 
 24.500 
 
 25.645 
 
 26.855 
 
 28.132 
 
 30.906 
 
 33.999 
 
 ^ 
 
 19 
 
 22.841 
 
 23.946 
 
 25.117 
 
 26.357 
 
 27.671 
 
 29.064 
 
 30.539 
 
 33.760 
 
 37.379 
 
 11 
 
 20 
 
 24.297 
 
 25.545 
 
 26.870 
 
 28.280 
 
 29.778 
 
 31.371 
 
 33.066 
 
 36.786 
 
 40.995 
 
 
 25 
 
 32.030 
 
 34.158 
 
 36.459 
 
 38.950 
 
 41.646 
 
 44.565 
 
 47.727 
 
 54.865 
 
 63.249 
 
 a 
 
 30 
 
 40.568 
 
 43.903 
 
 47.575 
 
 51.623 
 
 56.085 
 
 61.007 
 
 66.439 
 
 79.058 
 
 94.461 
 
 1 
 
 40 
 
 60.402 
 
 67.403 
 
 75.401 
 
 84.550 
 
 95.026 
 
 107.03 
 
 120.80 
 
 154.76 
 
 199.64 
 
 i 
 
 50 
 
 84.579 
 
 97.484 
 
 112.80 
 
 131.00 
 
 152.67 
 
 178.50 
 
 209.35 
 
 290.34 
 
 406.53 
 
 PH 
 
 60 
 
 114.05 
 
 135.99 
 
 163.05 
 
 196.52 
 
 237.99 
 
 289.50 
 
 353.58 
 
 533.13 
 
 813.52 
 
 
66 
 
 MATHEMATICAL TABLES 
 
 PRINCIPAL WHICH WILL AMOUNT TO A GIVEN SUM 
 
 The principal P, which, if placed at compound interest to-day, will amount to a giv< 
 sum A at the end of n years is P = A X x' or P = A X y' or P = A X z', according 
 the interest (at the rate of r per cent, per annum) is compounded annually, semi-annuall 
 or quarterly: the factor x' or y' or z' being taken from the following tables. 
 Values of x'. (Interest compounded annually; P = A X x') 
 
 Years 
 
 r = 2 
 
 2H 
 
 3 
 
 sw 
 
 4 
 
 4H 
 
 5 
 
 6 
 
 7 
 
 
 1 
 
 .98039 
 
 .97561 
 
 .97087 
 
 .96618 
 
 .96154 
 
 .95694 
 
 .95238 
 
 .94340 
 
 .93458 
 
 
 2 
 
 .96117 
 
 .95181 
 
 .94260 
 
 .93351 
 
 .92456 
 
 .91573 
 
 .90703 
 
 .89000 
 
 .87344 
 
 
 3 
 
 .94232 
 
 .92860 
 
 .91514 
 
 .90194 
 
 .88900 
 
 .87630 
 
 .86384 
 
 .83962 
 
 .81630 
 
 
 4 
 
 .92385 
 
 .90595 
 
 .88849 
 
 .87144 
 
 .85480 
 
 .83856 
 
 .82270 
 
 .79209 
 
 .76290 
 
 . 
 
 5 
 
 .90573 
 
 C 88385 
 
 .86261 
 
 .84197 
 
 .82193 
 
 .80245 
 
 .78353 
 
 .74726 
 
 .71299 
 
 i-H 
 
 6 
 
 .88797 
 
 .86230 
 
 .83748 
 
 .81350 
 
 .79031 
 
 .76790 
 
 .74622 
 
 .70496 
 
 .66634 
 
 
 7 
 
 .87056 
 
 .84127 
 
 .81309 
 
 .78599 
 
 .75992 
 
 .73483 
 
 .71068 
 
 .66506 
 
 .62275 
 
 
 8 
 
 .85349 
 
 .82075 
 
 .78941 
 
 .75941 
 
 .73069 
 
 .70319 
 
 .67684 
 
 .62741 
 
 .58201 
 
 1! 
 1 ^ 
 
 9 
 
 .83676 
 
 .80073 
 
 .76642 
 
 .73373 
 
 .70259 
 
 .67290 
 
 .64461 
 
 .59190 
 
 .54393 
 
 
 10 
 
 .82035 
 
 .78120 
 
 .74409 
 
 .70892 
 
 .67556 
 
 .64393 
 
 .61391 
 
 .55839 
 
 .50835 
 
 I 
 
 11 
 
 .80426 
 
 .76214 
 
 .72242 
 
 .68495 
 
 .64958 
 
 .61620 
 
 .58468 
 
 .52679 
 
 .47509 
 
 3 
 
 12 
 
 .78849 
 
 .74356 
 
 .70138 
 
 .66178 
 
 .62460 
 
 .58966 
 
 .55684 
 
 .49697 
 
 .44401 
 
 
 13 
 
 .77303 
 
 .72542 
 
 .68095 
 
 .63940 
 
 .60057 
 
 .56427 
 
 .53032 
 
 .46884 
 
 .41496 
 
 4- 
 
 14 
 
 .75788 
 
 .70773 
 
 .66112 
 
 .61778 
 
 .57748 
 
 .53997 
 
 .50507 
 
 .44230 
 
 .38783 
 
 i-H 
 
 15 
 
 .74301 
 
 .69047 
 
 .64186 
 
 .59689 
 
 .55526 
 
 .51672 
 
 .48102 
 
 .41727 
 
 .36245 
 
 D 
 
 16 
 
 .72845 
 
 .67362 
 
 .62317 
 
 .57671 
 
 .53391 
 
 .49447 
 
 .4581 1 
 
 .39365 
 
 .33873 
 
 
 17 
 
 .71416 
 
 .65720 
 
 .60502 
 
 .55720 
 
 .51337 
 
 .47318 
 
 .43630 
 
 .37136 
 
 .31657 
 
 " 
 
 18 
 
 .70016 
 
 .64117 
 
 .58739 
 
 .53836 
 
 .49363 
 
 .45280 
 
 .41552 
 
 .35034 
 
 .29586 
 
 cj 
 
 19 
 
 .68643 
 
 .62553 
 
 .57029 
 
 .52016 
 
 .47464 
 
 .43330 
 
 .39573 
 
 .33051 
 
 .27651 
 
 3 
 
 20 
 
 .67297 
 
 .61027 
 
 .55368 
 
 .50257 
 
 .45639 
 
 .41464 
 
 .37689 
 
 .31180 
 
 .25842 
 
 jjj 
 
 25 
 
 .60953 
 
 .53939 
 
 .47761 
 
 .42315 
 
 .37512 
 
 .33273 
 
 .29530 
 
 .23300 
 
 .18425 
 
 o 
 
 30 
 
 .55207 
 
 .47674 
 
 .41199 
 
 .35628 
 
 .30832 
 
 .26700 
 
 .23138 
 
 .17411 
 
 .13137 
 
 n 
 
 40 
 
 .45289 
 
 .37243 
 
 .30656 
 
 .25257 
 
 .20829 
 
 .17193 
 
 .14205 
 
 .09722 
 
 .06678 
 
 
 50 
 
 .37153 
 
 .29094 
 
 .2281 1 
 
 .17905 
 
 .14071 
 
 .11071 
 
 .08720 
 
 .05429 
 
 .03395 
 
 
 60 
 
 .30478 
 
 .22728 
 
 .16973 
 
 .12693 
 
 .09506 
 
 .07129 
 
 .05354 
 
 .03031 
 
 .01726 
 
 
 Values of y'. (Interest compounded semi-annually; P = A X y') 
 
 Years 
 
 r = 2 
 
 2V 
 
 3 
 
 W 
 
 4 
 
 4J/2 
 
 5 
 
 6 
 
 7 
 
 
 1 
 
 .98030 
 
 .97546 
 
 .97066 
 
 .96590 
 
 .96117 
 
 .95647 
 
 .95181 
 
 .94260 
 
 .93351 
 
 
 2 
 
 .96098 
 
 .95152 
 
 .94218 
 
 .93296 
 
 .92385 
 
 .91484 
 
 .90595 
 
 .88849 
 
 .87144 
 
 
 3 
 
 .94205 
 
 .92817 
 
 .91454 
 
 .90114 
 
 .88797 
 
 .87502 
 
 .86230 
 
 .83748 
 
 .81350 
 
 
 4 
 
 .92348 
 
 .90540 
 
 .88771 
 
 .87041 
 
 .85349 
 
 .83694 
 
 .82075 
 
 .78941 
 
 .75941 
 
 .' 
 
 5 
 
 .90529 
 
 .88318 
 
 .86167 
 
 .84073 
 
 .82035 
 
 .80051 
 
 .78120 
 
 .74409 
 
 .70892 
 
 \ 
 
 6 
 
 .88745 
 
 .86151 
 
 .83639 
 
 .81206 
 
 .78849 
 
 .76567 
 
 .74356 
 
 .70138 
 
 .66178 
 
 
 7 
 
 .86996 
 
 .84037 
 
 .81185 
 
 .78436 
 
 .75788 
 
 .73234 
 
 .70773 
 
 .66112 
 
 .61778 
 
 
 8 
 
 .85282 
 
 .81975 
 
 .78803 
 
 .75762 
 
 .72845 
 
 .70047 
 
 .67362 
 
 .62317 
 
 .57671 
 
 * 
 
 9 
 
 .83602 
 
 .79963 
 
 .76491 
 
 .73178 
 
 .70016 
 
 .66998 
 
 .64117 
 
 .58739 
 
 .53836 
 
 ~Z 
 
 10 
 
 .81954 
 
 .78001 
 
 .74247 
 
 .70682 
 
 .67297 
 
 .64082 
 
 .61027 
 
 .55368 
 
 .50257 
 
 
 
 11 
 
 .80340 
 
 .76087 
 
 .72069 
 
 .68272 
 
 .64684 
 
 .61292 
 
 .58086 
 
 .52189 
 
 .46915 
 
 ^ 
 
 12 
 
 .78757 
 
 .74220 
 
 .69954 
 
 .65944 
 
 .62172 
 
 .58625 
 
 .55288 
 
 .49193 
 
 .43796 
 
 , 
 
 13 
 
 .77205 
 
 .72398 
 
 .67902 
 
 .63695 
 
 .59758 
 
 .56073 
 
 .52623 
 
 .46369 
 
 .40884 
 
 4. 
 
 14 
 
 .75684 
 
 .70622 
 
 .65910 
 
 .61523 
 
 .57437 
 
 .53632 
 
 .50088 
 
 .43708 
 
 .38165 
 
 i 
 
 15 
 
 .74192 
 
 .68889 
 
 .63976 
 
 .59425 
 
 .55207 
 
 .51298 
 
 .47674 
 
 .41199 
 
 .35628 
 
 *-* 
 
 16 
 
 .72730 
 
 .67198 
 
 .62099 
 
 .57398 
 
 .53063 
 
 .49065 
 
 .45377 
 
 .38834 
 
 .33259 
 
 n 
 
 17 
 
 .71297 
 
 .65549 
 
 .60277 
 
 .55441 
 
 .51003 
 
 .46930 
 
 .43191 
 
 .36604 
 
 .31048 
 
 *a 
 
 18 
 
 .69892 
 
 .63941 
 
 .58509 
 
 .53550 
 
 .49022 
 
 .44887 
 
 .41109 
 
 .34503 
 
 .28983 
 
 
 19 
 
 .68515 
 
 .62372 
 
 .56792 
 
 .51724 
 
 .47119 
 
 .42933 
 
 .39128 
 
 .32523 
 
 .27056 
 
 J2 
 
 20 
 
 .67165 
 
 .60841 
 
 .55126 
 
 .49960 
 
 .45289 
 
 .41065 
 
 .37243 
 
 .30656 
 
 .25257 
 
 s 
 
 25 
 
 .60804 
 
 .53734 
 
 .47500 
 
 .42003 
 
 .37153 
 
 .32873 
 
 .29094 
 
 .22811 
 
 .17905 
 
 
 30 
 
 .55045 
 
 .47457 
 
 .40930 
 
 .35313 
 
 .30478 
 
 .26315 
 
 .22728 
 
 .16973 
 
 .12693 
 
 
 
 40 
 
 .45112 
 
 .37017 
 
 .30389 
 
 .24960 
 
 .20511 
 
 .16863 
 
 .13870 
 
 .09398 
 
 .06379 
 
 
 50 
 
 .36971 
 
 .28873 
 
 .22563 
 
 .17642 
 
 .13803 
 
 .10806 
 
 .08465 
 
 .05203 
 
 .03206 
 
 
 60 
 
 .30299 
 
 .22521 
 
 .16752 
 
 .12470 
 
 .09289 
 
 .06925 
 
 .05166 
 
 .02881 
 
 .01611 
 
 
MATHEMATICAL TABLES 
 
 67 
 
 Values of '. (Interest compounded quarterly; P=>A X z'', see opposite page) 
 
 Years| r = 2 
 
 2tt 
 
 3 
 
 JH 
 
 4 
 
 4J4 
 
 5 
 
 6 
 
 7 
 
 
 1 
 
 .98025 
 
 .97539 
 
 .97055 
 
 .96575 
 
 .96098 
 
 .95624 
 
 .95152 
 
 .94218 
 
 .93296 
 
 
 2 
 
 .96089 
 
 .95138 
 
 .94198 
 
 .93268 
 
 .92348 
 
 .91439 
 
 .90540 
 
 .88771 
 
 .87041 
 
 
 3 
 
 .94191 
 
 .92796 
 
 .91424 
 
 .90074 
 
 .88745 
 
 .87437 
 
 .86151 
 
 .83639 
 
 .81206 
 
 
 4 
 
 .92330 
 
 .90512 
 
 .88732 
 
 .86989 
 
 .85282 
 
 .8361 1 
 
 .81975 
 
 .78803 
 
 .75762 
 
 
 5 
 
 .90506 
 
 .88284 
 
 .86119 
 
 .84010 
 
 .81954 
 
 .79952 
 
 .78001 
 
 .74247 
 
 .70682 
 
 \ 
 1-1 
 
 6 
 
 .88719 
 
 .86111 
 
 .83583 
 
 .81132 
 
 .78757 
 
 .76453 
 
 .74220 
 
 .69954 
 
 .65944 
 
 * 
 
 7 
 
 .86966 
 
 .83991 
 
 .81122 
 
 .78354 
 
 .75684 
 
 .73107 
 
 .70622 
 
 .65910 
 
 .61523 
 
 
 8 
 
 .85248 
 
 .81924 
 
 .78733 
 
 .75670 
 
 .72730 
 
 .69908 
 
 .67198 
 
 .62099 
 
 .57390 
 
 j 
 
 9 
 
 .83564 
 
 .79908 
 
 .76415 
 
 .73079 
 
 .69892 
 
 .66849 
 
 .63941 
 
 .58509 
 
 .53550 
 
 JL* 
 
 10 
 
 .81914 
 
 .77941 
 
 .74165 
 
 .70576 
 
 .67165 
 
 .63923 
 
 .60841 
 
 .55126 
 
 .49960 
 
 o 
 
 11 
 
 .80296 
 
 .76022 
 
 .71981 
 
 .68159 
 
 .64545 
 
 .61126 
 
 .57892 
 
 .51939 
 
 .46611 
 
 V 
 
 12 
 
 .78710 
 
 .74151 
 
 .69861 
 
 .65825 
 
 .62026 
 
 .58451 
 
 .55086 
 
 .48936 
 
 .43486 
 
 ^ 
 
 13 
 
 .77155 
 
 .72326 
 
 .67804 
 
 .63570 
 
 .59606 
 
 .55893 
 
 .52415 
 
 .46107 
 
 .40570 
 
 -j- 
 
 14 
 
 .75631 
 
 .70546 
 
 .65808 
 
 .61393 
 
 .57280 
 
 .53447 
 
 .49874 
 
 .43441 
 
 .37851 
 
 
 15 
 
 .74137 
 
 .68809 
 
 .63870 
 
 .59291 
 
 .55045 
 
 .51108 
 
 .47457 
 
 .40930 
 
 .35313 
 
 ' 
 
 16 
 
 .72673 
 
 .67115 
 
 .61989 
 
 .57260 
 
 .52897 
 
 .48871 
 
 .45156 
 
 .38563 
 
 .32946 
 
 II 
 
 17 
 
 .71237 
 
 .65464 
 
 .60164 
 
 .55299 
 
 .50833 
 
 .46733 
 
 .42967 
 
 .36334 
 
 .30737 
 
 %* 
 
 18 
 
 .69830 
 
 .63852 
 
 .58392 
 
 .53405 
 
 .48850 
 
 .44687 
 
 .40884 
 
 .34233 
 
 .28676 
 
 
 19 
 
 .68451 
 
 .62281 
 
 .56673 
 
 .51576 
 
 .46944 
 
 .42732 
 
 .38903 
 
 .32254 
 
 .26754 
 
 Js 
 
 "3 
 
 20 
 
 .67099 
 
 .60748 
 
 .55004 
 
 .49810 
 
 .45112 
 
 .40862 
 
 .37017 
 
 .30389 
 
 .24960 
 
 
 25 
 
 .60729 
 
 .53630 
 
 .47369 
 
 .41845 
 
 .36971 
 
 .32670 
 
 .28873 
 
 .22563 
 
 .17642 
 
 o 
 
 30 
 
 .54963 
 
 .47347 
 
 .40794 
 
 .35154 
 
 .30299 
 
 .26120 
 
 .22521 
 
 .16752 
 
 .12470 
 
 
 
 40 
 
 .45023 
 
 .36903 
 
 .30255 
 
 .24810 
 
 .20351 
 
 .16697 
 
 .13702 
 
 .09235 
 
 .06230 
 
 
 50 
 
 .36880 
 
 .28762 
 
 .22438 
 
 .17510 
 
 .13669 
 
 .10673 
 
 .08337 
 
 .05091 
 
 .03113 
 
 
 60 
 
 .30210 
 
 .22417 
 
 .16641 
 
 .12358 
 
 .09181 
 
 .06823 
 
 .05072 
 
 .02806 
 
 .01555 
 
 
 ANNUITY WHICH WILL AMOUNT TO A GIVEN SUM (SINKING 
 FUND) 
 
 The annual payment, Y, which, if set aside at the end of each year, will amount with 
 accumulated interest to a given sum S at the end of n years is Y = S X v', where the 
 factor v' is given below. (Interest at r per cent, per annum, compounded annually.) 
 
 Values of ' 
 
 Years 
 
 r= 2 
 
 2H 
 
 3 
 
 3H 
 
 4 
 
 ^ 
 
 5 
 
 6 
 
 7 
 
 
 2 
 
 .49505 
 
 .49383 
 
 .49261 
 
 .49140 
 
 .49020 
 
 .48900 
 
 .48780 
 
 .48544 
 
 .48309 
 
 ' 
 
 3 
 
 .32675 
 
 .32514 
 
 .32353 
 
 .32193 
 
 .32035 
 
 .31877 
 
 .31721 
 
 .31411 
 
 .31105 
 
 ^^ 
 
 4 
 
 .24262 
 
 .24082 
 
 .23903 
 
 .23725 
 
 .23549 
 
 .23374 
 
 .23201 
 
 .22859 
 
 .22523 
 
 P 
 
 5 
 
 .19216 
 
 .19025 
 
 .18835 
 
 .18648 
 
 .18463 
 
 .18279 
 
 .18097 
 
 .17740 
 
 .17389 
 
 
 
 6 
 
 .15853 
 
 .15655 
 
 .15460 
 
 .15267 
 
 .15076 
 
 .14888 
 
 .14702 
 
 .14336 
 
 .13980 
 
 '" H 
 
 7 
 
 .13451 
 
 .13250 
 
 .13051 
 
 .12854 
 
 .12661 
 
 .12470 
 
 .12282 
 
 .11914 
 
 .11555 
 
 1 
 
 8 
 
 .11651 
 
 -.11447 
 
 .11246 
 
 .11048 
 
 .10853 
 
 .10661 
 
 .10472 
 
 .10104 
 
 .09747 
 
 4, 
 
 9 
 
 .10252 
 
 .10046 
 
 .09843 
 
 .09645 
 
 .09449 
 
 .09257 
 
 .09069 
 
 .08702 
 
 .08349 
 
 
 
 10 
 
 .09133 
 
 .08926 
 
 .08723 
 
 .08524 
 
 .08329 
 
 .08138 
 
 .07950 
 
 .07587 
 
 .07238 
 
 
 11 
 
 .08218 
 
 .0801 1 
 
 .07808 
 
 .07609 
 
 .07415 
 
 .07225 
 
 .07039 
 
 .06679 
 
 .06336 
 
 I* 
 
 12 
 
 .07456 
 
 .07249 
 
 .07046 
 
 .06848 
 
 .06655 
 
 .06467 
 
 .06283 
 
 .05928 
 
 .05590 
 
 
 13 
 
 .06812 
 
 .06605 
 
 .06403 
 
 .06206 
 
 .06014 
 
 .05828 
 
 .05646 
 
 .05296 
 
 .04965 
 
 
 14 
 
 .06260 
 
 .06054 
 
 .05853 
 
 .05657 
 
 .05467 
 
 .05282 
 
 .05102 
 
 .04758 
 
 .04434 
 
 j* 
 
 15 
 
 .05783 
 
 .05577 
 
 .05377 
 
 .05183 
 
 .04994 
 
 .0481 1 
 
 .04634 
 
 .04296 
 
 .03979 
 
 ! 
 
 16 
 
 .05365 
 
 .05160 
 
 .04961 
 
 .04768 
 
 .04582 
 
 .04402 
 
 .04227 
 
 .03895 
 
 .03586 
 
 
 17 
 
 .04997 
 
 .04793 
 
 .04595 
 
 .04404 
 
 .04220 
 
 .04042 
 
 .03870 
 
 .03544 
 
 .03243 
 
 
 
 18 
 
 .04670 
 
 .04467 
 
 .04271 
 
 .04082 
 
 .03899 
 
 .03724 
 
 .03555 
 
 .03236 
 
 .02941 
 
 
 19 
 
 .04378 
 
 .04176 
 
 .03981 
 
 .03794 
 
 .03614 
 
 .03441 
 
 .03275 
 
 .02962 
 
 .02675 
 
 | 
 
 20 
 
 .04116 
 
 .03915 
 
 .03722 
 
 .03536 
 
 .03358 
 
 .03188 
 
 .03024 
 
 .02718 
 
 .02439 
 
 .. ^ 
 
 25 
 
 .03122 
 
 .02928 
 
 .02743 
 
 .02567 
 
 .02401 
 
 .02244 
 
 .02095 
 
 .01823 
 
 .01581 
 
 J2 " 
 
 30 
 
 .02465 
 
 .02278 
 
 .02102 
 
 .01937 
 
 .01783 
 
 .01639 
 
 .01505 
 
 .01265 
 
 .01059 
 
 l> 
 
 40 
 
 .01656 
 
 .01484 
 
 .01326 
 
 .01183 
 
 .01052 
 
 .00934 
 
 .00828 
 
 .00646 
 
 .00467 
 
 |H 
 
 
 
 50 
 
 .01182 
 
 .01026 
 
 .00887 
 
 .00763 
 
 .00655 
 
 .00560 
 
 .00478 
 
 .00344 
 
 .00238 
 
 fe 
 
 60 .00877 
 
 .00735 
 
 .00613 
 
 .00509 
 
 .00420 
 
 .00345 
 
 .00283 
 
 .00188 
 
 .00121 
 
 
68 
 
 MATHEMATICAL TABLES 
 
 PRESENT WORTH OF AN ANNUITY 
 
 The capital C, which, if placed at interest to-day, will provide for a given annual 
 payment Y for a term of n years before it is exhausted is C = Y X w, where the factor 
 w is given below. (Interest at r per cent, per annum, compounded annually.) 
 
 Values of w 
 
 Years| r =2 
 
 2H 
 
 3 
 
 3M 
 
 4 
 
 4H 
 
 5 
 
 6 
 
 7 
 
 
 1 
 
 0.9804 
 
 0.9756 
 
 0.9709 
 
 0.9662 
 
 0.9615 
 
 0.9569 
 
 0.9524 
 
 0.9434 
 
 0.9346 
 
 
 2 
 
 1.9416 
 
 1.9274 
 
 1.9135 
 
 1.8997 
 
 1.8861 
 
 1.8727 
 
 1.8594 
 
 1.8334 
 
 1.8080 
 
 N 
 
 3 
 
 2.8839 
 
 2.8560 
 
 2.8286 
 
 2.8016 
 
 2.7751 
 
 2.7490 
 
 2.7232 
 
 2.6730 
 
 2.6243 
 
 \ 
 
 4 
 
 3.8077 
 
 3.7620 
 
 3.7171 
 
 3.6731 
 
 3.6299 
 
 3.5875 
 
 3.5460 
 
 3.4651 
 
 3.3872 
 
 1 
 
 5 
 
 4.7135 
 
 4.6458 
 
 4.5797 
 
 4.5151 
 
 4.4518 
 
 4.3900 
 
 4.3295 
 
 4.2124 
 
 4.1002 
 
 "" 
 
 6 
 
 5.6014 
 
 5.5081 
 
 5.4172 
 
 5.3286 
 
 5.2421 
 
 5.1579 
 
 5.0757 
 
 4.9173 
 
 4.7665 
 
 
 7 
 
 6.4720 
 
 6.3494 
 
 6.2303 
 
 6.1145 
 
 6.0021 
 
 5.8927 
 
 5.7864 
 
 5.5824 
 
 5.3893 
 
 \ 
 
 8 
 
 7.3255 
 
 7.1701 
 
 7.0197 
 
 6.8740 
 
 6.7327 
 
 6.5959 
 
 6.4632 
 
 6.2098 
 
 5.9713 
 
 * 
 
 9 
 
 8.1622 
 
 7.9709 
 
 7.7861 
 
 7.6077 
 
 7.4353 
 
 7.2688 
 
 7.1078 
 
 6.8017 
 
 6.5152 
 
 1- 
 
 10 
 
 8.9826 
 
 8.7521 
 
 8.5302 
 
 8.3166 
 
 8.1109 
 
 7.9127 
 
 7.7217 
 
 7.3601 
 
 7.0236 
 
 
 11 
 
 9.7868 
 
 9.5142 
 
 9.2526 
 
 9.0016 
 
 8.7605 
 
 8.5289 
 
 8.3064 
 
 7.8869 
 
 7.4987 
 
 i 
 
 12 
 
 10.575 
 
 10.258 
 
 9.9540 
 
 9.6633 
 
 9.3851 
 
 9.1186 
 
 8.8633 
 
 8.3838 
 
 7.9427 
 
 
 13 
 
 11.348 
 
 10.983 
 
 10.635 
 
 10.303 
 
 9.9856 
 
 9.6829 
 
 9.3936 
 
 8.8527 
 
 8.3577 
 
 I 
 
 14 
 
 12.106 
 
 11.691 
 
 11.296 
 
 10.921 
 
 10.563 
 
 10.223 
 
 9.8986 
 
 9.2950 
 
 8.7455 
 
 \ 
 
 15 
 
 12.849 
 
 12.381 
 
 11.938 
 
 11.517 
 
 11.118 
 
 10.740 
 
 10.380 
 
 9.7122 
 
 9.1079 
 
 ^ 
 
 16 
 
 13.578 
 
 13.055 
 
 12.561 
 
 12.094 
 
 11.652 
 
 11.234 
 
 10.838 
 
 10.106 
 
 9.4466 
 
 + 
 
 17 
 
 1 4.292 
 
 13.712 
 
 13.166 
 
 12.651 
 
 12.166 
 
 11.707 
 
 11.274 
 
 10.477 
 
 9.7632 
 
 
 18 
 
 14.992 
 
 14.353 
 
 13.754 
 
 13.190 
 
 12.659 
 
 12.160 
 
 11.690 
 
 10.828 
 
 10.059 
 
 Zl 
 
 19 
 
 15.678 
 
 14.979 
 
 14.324 
 
 13.710 
 
 13.134 
 
 12.593 
 
 12.085 
 
 11.15.8 
 
 10.336 
 
 1 
 
 20 
 
 16.351 
 
 15.589 
 
 14.877 
 
 14.212 
 
 13.590 
 
 13.008 
 
 12.462 
 
 11.470 
 
 10.594 
 
 a ^ 
 
 25 
 
 19.523 
 
 18.424 
 
 17.413 
 
 16.482 
 
 15.622 
 
 14.828 
 
 14.094 
 
 12.783 
 
 11.654 
 
 3 T 
 
 30 
 
 22.396 
 
 20.930 
 
 19.600 
 
 18.392 
 
 17.292 
 
 16.289 
 
 15.372 
 
 13.765 
 
 12.409 
 
 s " 
 
 40 
 
 27.355 
 
 25.103 
 
 23.115 
 
 21.355 
 
 19.793 
 
 18.402 
 
 17.159 
 
 15.046 
 
 13.332 
 
 o 5 
 
 50 
 
 31.424 
 
 28.362 
 
 25.730 
 
 23.456 
 
 21.482 
 
 19.762 
 
 18.256 
 
 15.762 
 
 13.801 
 
 HH 
 
 60 
 
 34.761 
 
 30.909 
 
 27.676 
 
 24.945 
 
 22.623 
 
 20.638 
 
 18.929 
 
 16.161 
 
 14.039 
 
 
 ANNUITY PROVIDED FOR BY A GIVEN CAPITAL 
 
 The annual payment Y provided for for a term of n years by a given capital C placed 
 at interest to-day is Y = C X w' . (Interest at r per cent, per annum, compounded 
 annually; the fund supposed to be exhausted at the end of the term.) 
 
 Values of w' 
 
 Years 
 
 r = 2 
 
 2% 
 
 3 
 
 3H 
 
 4 
 
 4H 
 
 5 
 
 6 
 
 7 
 
 
 2 
 
 .51505 
 
 .51883 
 
 .52261 
 
 .52640 
 
 .53020 
 
 .53400 
 
 .53780 
 
 .54544 
 
 .55309 
 
 
 3 
 
 .34675 
 
 .35014 
 
 .35353 
 
 .35693 
 
 .36035 
 
 .36377 
 
 .36721 
 
 .37411 
 
 .38105 
 
 
 
 4 
 
 .26262 
 
 .26582 
 
 .26903 
 
 .27225 
 
 .27549 
 
 .27874 
 
 .28201 
 
 .28859 
 
 .29523 
 
 c; 
 
 5 
 
 .21216 
 
 .21525 
 
 .21835 
 
 .22148 
 
 .22463 
 
 .22779 
 
 .23097 
 
 .23740 
 
 .24389 
 
 
 
 6 
 
 .17853 
 
 .18155 
 
 .18460 
 
 .18767 
 
 .19076 
 
 .19388 
 
 .19702 
 
 .20336 
 
 .20980 
 
 C^ 
 
 7 
 
 .15451 
 
 .15750 
 
 .16051 
 
 .16354 
 
 .16661 
 
 .16970 
 
 .17282 
 
 .17914 
 
 .18555 
 
 ^ 
 
 8 
 
 .13651 
 
 .13947 
 
 .14246 
 
 .14548 
 
 .14853 
 
 .15161 
 
 .15472 
 
 .16104 
 
 .16747 
 
 _i_ X. 
 
 9 
 
 .12252 
 
 .12546 
 
 .12843 
 
 .13145 
 
 .13449 
 
 .13757 
 
 .14069 
 
 .14702 
 
 .15349 
 
 ^ 
 
 10 
 
 .11133 
 
 .11426 
 
 .11723 
 
 .12024 
 
 .12329 
 
 .12638 
 
 .12950 
 
 .13587 
 
 .14238 
 
 _ TH 
 
 11 
 
 .10218 
 
 .10511 
 
 .10808 
 
 .11109 
 
 .11415 
 
 .11725 
 
 .12039 
 
 .12679 
 
 .13336 
 
 1 xb 
 
 12 
 
 .09456 
 
 .09749 
 
 .10046 
 
 .10348 
 
 .10655 
 
 .10967 
 
 .11283 
 
 .11928 
 
 .12590 
 
 ~ + 
 
 13 
 
 .08812 
 
 .09105 
 
 .09403 
 
 .09706 
 
 .10014 
 
 .10328 
 
 .10646 
 
 .11296 
 
 .11965 
 
 
 14 
 
 .08260 
 
 .08554 
 
 .08853 
 
 .09157 
 
 .09467 
 
 .09782 
 
 .10102 
 
 .10758 
 
 .11434 
 
 1' v e 
 
 15 
 
 .07783 
 
 .08077 
 
 .08377 
 
 .08683 
 
 .08994 
 
 .09311 
 
 .09634 
 
 .10296 
 
 .10979 
 
 Ii 
 
 16 
 
 .07365 
 
 .07660 
 
 .07961 
 
 .08268 
 
 .08582 
 
 .08902 
 
 .09227 
 
 .09895 
 
 .10586 
 
 
 17 
 
 .06997 
 
 .07293 
 
 .07595 
 
 .07904 
 
 .08220 
 
 .08542 
 
 .08870 
 
 .09544 
 
 .10243 
 
 ^T ^ 
 
 18 
 
 .06670 
 
 .06967 
 
 .07271 
 
 .07582 
 
 .07899 
 
 .08224 
 
 .08555 
 
 .09236 
 
 .09941 
 
 II II 
 
 19 
 
 .06378 
 
 .06676 
 
 .06981 
 
 .07294 
 
 .07614 
 
 .07941 
 
 .08275 
 
 .08962 
 
 .09675 
 
 
 20 
 
 .06116 
 
 .06415 
 
 .06722 
 
 .07036 
 
 .07358 
 
 .07688 
 
 .08024 
 
 .08718 
 
 .09439 
 
 9 
 
 25 
 
 .05122 
 
 .05428 
 
 .05743 
 
 .06067 
 
 .06401 
 
 .06744 
 
 .07095 
 
 .07823 
 
 .08581 
 
 
 
 30 
 
 .04465 
 
 .04778 
 
 .05102 
 
 .05437 
 
 .05783 
 
 .06139 
 
 .06505 
 
 .07265 
 
 .08059 
 
 "5 
 
 40 
 
 .03656 
 
 .03984 
 
 .04326 
 
 .04683 
 
 .05052 
 
 .05434 
 
 .05828 
 
 .06646 
 
 .07467 
 
 g 
 
 50 
 
 .03182 
 
 .03526 
 
 .03887 
 
 .04263 
 
 .04655 
 
 .05060 
 
 .05478 
 
 .06344 
 
 .07238 
 
 
 
 60 
 
 .02877 
 
 .03235 
 
 .03613 
 
 .04009 
 
 .04420 
 
 .04845 
 
 .05283 
 
 .06188 
 
 .07121 
 
 fc 
 
MATHEMATICAL TABLES 
 
 DECIMAL EQUIVALENTS 
 
 From minutes and 
 
 
 From decimal parts of 
 
 
 Common fractions 
 
 seconds into deci- 
 mal parts of a 
 degree 
 
 
 a degree into minutes 
 and seconds (exact 
 values) 
 
 
 8 16 32 64 
 
 ths ths nds ths 
 
 Exact 
 decimal 
 values 
 
 0' 
 
 o.oooo 
 
 0" 
 
 o.oooo 
 
 
 o.oo 
 
 0' 
 
 0.50 
 
 30' 
 
 
 1 
 
 .01 5625 
 
 1 
 
 .0167 
 
 1 
 
 .0003 
 
 
 1 
 
 0' 36" 
 
 1 
 
 30' 36" 
 
 
 1 2 
 
 .03 125 
 
 2 
 
 .0333 
 
 2 
 
 .0006 
 
 
 2 
 
 1' 12" 
 
 2 
 
 31' 12" 
 
 
 3 
 
 .04 6875 
 
 3 
 
 .05 
 
 3 
 
 .0008 
 
 
 3 
 
 V 48" 
 
 3 
 
 31' 48" 
 
 
 1 2 4 
 
 .06 25 
 
 4 
 
 .0667 
 
 4 
 
 .0011 
 
 
 4 
 
 2' 24" 
 
 4 
 
 32' 24" 
 
 
 5 
 
 .07 8125 
 
 5' 
 
 .0833 
 
 5" 
 
 .0014 
 
 
 0.05 
 
 3' 
 
 0.55 
 
 33' 
 
 
 3 6 
 
 .09 375 
 
 6 
 
 .10 
 
 6 
 
 .0017 
 
 
 6 
 
 y 36" 
 
 6 
 
 33' 36" 
 
 
 7 
 
 .10 9375 
 
 7 
 
 .1167 
 
 7 
 
 .0019 
 
 
 7 
 
 4' 12" 
 
 7 
 
 34' 12" 
 
 
 1248 
 
 .12 5 
 
 8 
 
 .1333 
 
 8 
 
 .0022 
 
 
 8 
 
 4' 48" 
 
 8 
 
 34' 48" 
 
 
 9 
 
 .14 0625 
 
 9 
 
 .15 
 
 9 
 
 .0025 
 
 
 9 
 
 5' 24" 
 
 9 
 
 35' 24" 
 
 
 5 10 
 
 .15 625 
 
 10' 
 
 0.1667 
 
 10" 
 
 0.0028 
 
 
 0.10 
 
 6' 
 
 0.60 
 
 36' 
 
 
 11 
 
 .17 1875 
 
 1 
 
 .1833 
 
 ] 
 
 .0031 
 
 
 1 
 
 6' 36" 
 
 1 
 
 36' 36" 
 
 
 3 6 12 
 
 .18 75 
 
 2 
 
 .20 
 
 2 
 
 .0033 
 
 
 2 
 
 7' 12" 
 
 2 
 
 37' 12" 
 
 
 13 
 
 .20 3125 
 
 3 
 
 .2167 
 
 3 
 
 .0036 
 
 
 3 
 
 7' 48" 
 
 3 
 
 37' 48" 
 
 
 7 14 
 
 .21 875 
 
 4 
 
 .2333 
 
 4 
 
 .0039 
 
 
 4 
 
 8' 24" 
 
 4 
 
 38' 24" 
 
 
 15 
 
 .23 4375 
 
 15' 
 
 .25 
 
 15" 
 
 .0042 
 
 
 0.15 
 
 9' 
 
 0.65 
 
 39' 
 
 
 2 4 8 16 
 
 .25 
 
 6 
 
 .2667 
 
 6 
 
 .0044 
 
 
 6 
 
 9' 36" 
 
 6 
 
 39' 36" 
 
 
 17 
 
 .26 5625 
 
 7 
 
 .2833 
 
 7 
 
 .0047 
 
 
 7 
 
 10' 12" 
 
 7 
 
 40' 12" 
 
 
 9 18 
 
 .28 125 
 
 8 
 
 .30 
 
 8 
 
 .005 
 
 
 8 
 
 10' 48" 
 
 8 
 
 40' 48" 
 
 
 19 
 
 .29 6875 
 
 9 
 
 .3167 
 
 9 
 
 .0053 
 
 
 9 
 
 11' 24" 
 
 9 
 
 41' 24" 
 
 
 5 10 20 
 
 .31 25 
 
 20' 
 
 0.3333 
 
 20" 
 
 0.0056 
 
 
 0.20 
 
 12' 
 
 0.70 
 
 42' 
 
 
 21 
 
 .32 8125 
 
 1 
 
 .35 
 
 1 
 
 .0058 
 
 
 
 12' 36" 
 
 
 42' 36" 
 
 
 11 22 
 
 .34 375 
 
 2 
 
 .3667 
 
 2 
 
 .0061 
 
 
 2 
 
 13' 12" 
 
 2 
 
 43' 12" 
 
 
 23 
 
 .35 9375 
 
 3 
 
 .3833 
 
 3 
 
 .0064 
 
 
 3 
 
 13' 48" 
 
 3 
 
 43' 48" 
 
 
 3 6 12 24 
 
 .37 5 
 
 4 
 
 .40 
 
 4 
 
 .0067 
 
 
 4 
 
 14' 24" 
 
 4 
 
 44' 24" 
 
 
 25 
 
 .39 0625 
 
 25' 
 
 .4167 
 
 25" 
 
 .0069 
 
 
 0.25 
 
 15' 
 
 0.75 
 
 45' 
 
 
 13 26 
 
 .40 625 
 
 6 
 
 .4333 
 
 6 
 
 .0072 
 
 
 6 
 
 15' 36" 
 
 6 
 
 45' 36" 
 
 
 27 
 
 .42 1875 
 
 7 
 
 .45 
 
 7 
 
 .0075 
 
 
 7 
 
 16' 12" 
 
 7 
 
 46' 12" 
 
 
 7 14 28 
 
 .43 75 
 
 8 
 
 .4667 
 
 8 
 
 .0078 
 
 
 8 
 
 16' 48" 
 
 8 
 
 46' 48" 
 
 
 29 
 
 .45 3125 
 
 9 
 
 .4833 
 
 9 
 
 .0081 
 
 
 9 
 
 17' 24" 
 
 9 
 
 47' 24" 
 
 
 15 30 
 
 .46 875 
 
 30' 
 
 0.50 
 
 30" 
 
 0.0083 
 
 
 0.30 
 
 18' 
 
 0.80 
 
 48' 
 
 
 31 
 
 .48 4375 
 
 1 
 
 .5167 
 
 1 
 
 .0086 
 
 
 1 
 
 18' 36" 
 
 1 
 
 48' 36" 
 
 
 4 8 16 32 
 
 .50 
 
 2 
 
 .5333 
 
 2 
 
 .0089 
 
 
 2 
 
 19' 12" 
 
 2 
 
 49' 12" 
 
 
 33 
 
 .51 5625 
 
 3 
 
 .55 
 
 3 
 
 .0092 
 
 
 3 
 
 19' 48" 
 
 3 
 
 49' 48" 
 
 
 17 34 
 
 .53 125 
 
 4 
 
 .5667 
 
 4 
 
 .0094 
 
 
 4 
 
 20' 24" 
 
 4 
 
 50' 24" 
 
 
 35 
 
 .54 6875 
 
 35' 
 
 .5833 
 
 35" 
 
 .0097 
 
 
 0.35 
 
 21' 
 
 0.85 
 
 51' 
 
 
 9 18 36 
 
 .56 25 
 
 6 
 
 .60 
 
 6 
 
 .01 
 
 
 6 
 
 21' 36" 
 
 6 
 
 51' 36" 
 
 
 37 
 
 .57 8125 
 
 7 
 
 .6167 
 
 7 
 
 .0103 
 
 
 7 
 
 22' 12" 
 
 7 
 
 52' 12" 
 
 
 19 38 
 
 .59 375 
 
 8 
 
 .6333 
 
 8 
 
 .0106 
 
 
 8 
 
 22' 48" 
 
 8 
 
 52' 48" 
 
 
 39 
 
 .60 9375 
 
 9 
 
 .65 
 
 9 
 
 .0108 
 
 
 9 
 
 23' 24" 
 
 9 
 
 53' 24" 
 
 
 5 10 20 40 
 
 .62 5 
 
 40' 
 
 0.6667 
 
 40" 
 
 0.01 1 1 
 
 
 0.40 
 
 24' 
 
 0.90 
 
 54' 
 
 
 41 
 
 .64 0625 
 
 1 
 
 .6833 
 
 1 
 
 .0114 
 
 
 1 
 
 24' 36" 
 
 
 54' 36" 
 
 
 21 42 
 
 .65 625 
 
 2 
 
 .70 
 
 2 
 
 .0117 
 
 
 2 
 
 25' 12" 
 
 2 
 
 55' 12" 
 
 
 43 
 
 .67 1875 
 
 3 
 
 .7167 
 
 3 
 
 .0119 
 
 
 3 
 
 25' 48" 
 
 3 
 
 55' 48" 
 
 
 11 22 44 
 
 .68 75 
 
 4 
 
 .7333 
 
 4 
 
 .0122 
 
 
 4 
 
 26' 24" 
 
 4 
 
 56' 24" 
 
 
 45 
 
 .70 3125 
 
 45' 
 
 .75 
 
 45" 
 
 .0125 
 
 
 045 
 
 27' 
 
 0.95 
 
 57' 
 
 
 23 46 
 
 .71 875 
 
 6 
 
 .7667 
 
 6 
 
 .0128 
 
 
 6 
 
 27' 36" 
 
 6 
 
 57' 36" 
 
 
 47 
 
 .73 4375 
 
 7 
 
 .7833 
 
 7 
 
 .0131 
 
 
 7 
 
 28' 12" 
 
 7 
 
 58' 12" 
 
 
 6 12 24 48 
 
 .75 
 
 8 
 
 .80 
 
 8 
 
 .0133 
 
 
 8 
 
 28' 48" 
 
 8 
 
 58' 48" 
 
 
 49 
 
 .76 5625 
 
 9 
 
 .8167 
 
 9 
 
 .0136 
 
 
 9 
 
 29' 24" 
 
 9 
 
 59' 24" 
 
 
 25 50 
 
 .78 125 
 
 50' 
 
 0.8333 
 
 50" 
 
 0.0139 
 
 
 0.50 
 
 30' 
 
 1.00 
 
 60' 
 
 
 51 
 
 .79 6875 
 
 1 
 
 .85 
 .8667 
 
 1 
 
 2 
 
 .0142 
 .0144 
 
 
 o.ooo 
 
 0".0 
 
 
 13 26 52 
 53 
 
 .81 25 
 .82 8125 
 
 3 
 
 .8833 
 
 3 
 
 .0147 
 
 
 1 
 
 3".6 
 
 
 27 54 
 
 .84 375 
 
 4 
 
 .90 
 
 4 
 
 .015 
 
 
 2 
 
 7".2 
 
 
 55 
 
 .85 9375 
 
 55' 
 
 .9167 
 
 55" 
 
 .0153 
 
 
 3 
 
 10".8 
 
 
 7 14 28 56 
 
 .87 5 
 
 6 
 
 .9333 
 
 6 
 
 .0156 
 
 
 4 
 
 14".4 
 
 
 57 
 
 .89 0625 
 
 7 
 
 .95 
 
 7 
 
 .0158 
 
 
 0.005 
 
 18" 
 
 
 29 58 
 
 .90 625 
 
 8 
 
 .9667 
 
 8 
 
 .0161 
 
 
 6 
 
 21".6 
 
 
 59 
 
 .92 1875 
 
 9 
 
 .9833 
 
 9 
 
 .0164 
 
 
 7 
 
 25".2 
 
 
 15 30 60 
 
 .93 75 
 
 60' 
 
 1.00 
 
 60" 
 
 0.0167 
 
 
 8 
 
 28".8 
 
 
 61 
 
 .95 3125 
 
 
 
 
 
 
 9 
 
 32" .4 
 
 
 31 62 
 
 .96 875 
 
 
 
 
 
 
 0.010 
 
 36" 
 
 
 63 
 
 .98 4375 
 
WEIGHTS AND MEASURES 
 
 BY 
 LOUIS A. FISCHER 
 
 In the United States the measures of weight and length commonly employed 
 are identical with the corresponding English units, but the capacity measures 
 differ from those now in use in the British Empire, the U. S. gallon being 
 defined as 231 cu. in. and the bushel as 2150.42 cu. in., whereas the corre- 
 sponding British imperial units are, respectively, 277.418 cu. in., and 2219.344 
 cu. in. (1 imp. gal. = 1.2 U. S. gal., approx.; 1 imp. bu. = 1.03 U. S. bu., 
 approx.). 
 
 The metric system of weights and measures was legalized and its use made 
 permissive in the United States by an Act of Congress, passed in 1866. 
 In 1872, by the concurrent action of the principal governments of the world, it 
 was agreed to establish an International Bureau of Weights and Measures 
 near Paris. 
 
 Prior to 1891 the British imperial yard was regarded as the real standard 
 of the United States. In 1891, the Office of Weights and Measures (now 
 Bureau of Standards) fixed the value of the United States yard in terms of 
 the international meter, according ito the ratio: one yard = 3600/3937 meters. 
 At the same time, the pound was fixed in terms of the international kilo- 
 gram, according to the relation: one pound = 453.59243 grams. 
 
 U. S. Customary Weights and Measures 
 
 Measures of Length 
 
 Measures of Area 
 
 12 inches 
 3 feet 
 5H yards = 16^ feet 
 
 40 poles = 220 yards 
 
 8 furlongs = 1760 yards 
 
 = 5280 feet 
 
 3 miles 
 
 4 inches 
 
 9 inches 
 
 = 1 foot 
 = 1 yard 
 = 1 rod, 
 
 pole or 
 
 perch 
 = 1 furlong 
 
 = 1 mile 
 
 = 1 league 
 = 1 hand 
 = 1 span 
 
 = 1 acre 
 
 640 acres = 1 square mile = 
 
 Nautical Units 
 6080.2 feet = 1 nautical mile 
 
 6 feet = 1 fathom 
 
 120 fathoms = 1 cable length 
 
 1 nautical mile per hr. = 1 knot 
 
 Surveyor's or Gunter's Measure 
 
 7.92 inches = 1 link 
 
 100 links = 66 ft. = 4 rods = 1 chain 
 80 chains = 1 mile 
 
 33H inches = 1 vara (Texas) 
 
 144 square inches = 1 square foot 
 9 square feet = 1 square yard 
 30^4 square yards = 1 square rod, pole or 
 
 perch 
 160 square rods 
 
 = 10 square chains 
 
 = 43,560sq. ft, 
 
 = 5645 sq. varas (Texas) 
 
 1 ''section" 
 of U. S. Govt. 
 surveyed 
 land 
 
 1 circular inch 
 = area of circle 1 inch 
 
 in diameter 
 1 square inch = 1.2732 cir. in. 
 1 circular mil =area of circle 0.001 in. 
 
 in diam. 
 1,000,000 cir. mils = l cir. in. 
 
 Measures of Volume 
 
 1728 cubic inches = 1 cubic foot 
 
 27 cubic feet = 1 cubic yard 
 
 1 cord of wood =128 cu. ft. 
 1 perch of masonry = 16K> to 25 cu. ft. 
 
 = 0.7854 sq. in. 
 
 70 
 
U. S. WEIGHTS AND MEASURES 
 
 71 
 
 U. S. Customary Weights and Measures (continued) 
 
 Measures of Volume 
 
 Weights 
 
 (The grain is the same in all systems) 
 
 Liquid or Fluid Measure 
 
 4 gills = 1 pint 
 
 2 pints = 1 quart 
 
 4 quarts <= 1 gallon 
 
 7.4805 gallons = 1 cubic foot 
 
 (There is no standard liquid "barrel.") 
 
 Apothecaries' Liquid Measure 
 
 60 minims = 1 liquid dram or drachm 
 
 8 drams = 1 liquid ounce 
 16 ounces = 1 pint 
 
 Water Measure 
 
 The Miner's Inch is the quantity of 
 water that will pass through an orifice 1 
 sq. in. in cross-section under a head of from 
 4 to 6^ in., as fixed by statutes, and varies 
 from Ho cu. ft. to Ho cu. ft. per sec. The 
 units now most in use are 1 cu. ft. per sec. 
 and 1 gal. per sec., the U. S. Reclamation 
 Service employing the former. See p. 260. 
 
 Dry Measure 
 
 2 pints = 1 quart 
 8 quarts = 1 peck 
 4 pecks = 1 bushel 
 
 Shipping Measure 
 1 Register ton = 100 cu. ft. 
 
 1 U. S. shipping ton = 40 cu. ft. 
 
 = f 32.14 U. S. bu. 
 m 131.14 imp. bu. 
 1 British shipping ton = 42 cu. ft. 
 
 / 32.70 imp. bu. 
 ~ \ 33.75 U. S. bu. 
 
 Board Measure 
 
 ( 144 cu. in. = volume of 
 1 board foot = { board 1 ft. sq. and 1- in. 
 
 i thick. 
 
 No. of board feet in a log = [\i(d - 4)pZ,, 
 where d = diam. of log (usually taken in- 
 side the bark at small end), in., and L = 
 length of log, ft. The 4 in. deducted are 
 an allowance for slab. This rule is vari- 
 ously known as the Doyle, Conn. River, 
 St. Croix, Thurber, Moore and Beeman, 
 and the Scribner rule. 
 
 Avoirdupois Weight 
 16 drams = 437.5 grains = 1 ounce 
 16 ounces = 7000 grains = 1 pound 
 100 pounds <= 1 cental 
 
 2000 pounds = 1 short ton 
 
 2240 pounds *= 1 long ton 
 
 Also (in Great Britain): 
 14 pounds 
 2 stone = 28 Ib. 
 4 quarters = 1 12 Ib. 
 
 20 hundredweight 
 
 = 1 stone 
 = 1 quarter 
 = 1 hundred- 
 weight (cwt.) 
 = 1 long ton 
 
 24 grains 
 
 Troy Weight 
 
 = 1 penny- 
 weight (dwt.) 
 20 pennyweights = 480 grains = 1 ounce 
 12 ounces = 5760 grains =1 pound 
 
 1 Assay Ton = 29,167 milligrams, or 
 as many milligrams as there are troy 
 ounces in a ton of 2000 Ib. avoirdupois. 
 Consequently, the number of milligrams 
 of precious metal yielded by an assay ton 
 of ore gives directly the number of troy 
 ounces that would be obtained from a ton 
 of 2000 Ib. avoirdupois. 
 
 Apothecaries' Weight 
 
 20 grains = 1 scruple 3 
 
 3 scruples = 60 grains = 1 dram 3 
 
 8 drams = 1 ounce 5 
 
 12 ounces = 5760 grains = 1 pound 
 
 Weight for Precious Stones 
 
 1 carat = 200 milligrams 
 
 (Adopted by practically all important 
 
 nations.) 
 
 Circular Measure 
 
 60 seconds = 1 minute 
 60 minutes = 1 degree 
 90 degrees = 1 quadrant 
 360 degrees = circumference 
 57.2957795 degrees =1 radian (or angle 
 ( = 57 17'44.806") having arc of length 
 equal to radius) 
 
 METRIC SYSTEM 
 
 The fundamental unit of the metric system is the meter the unit of length, 
 from which the units of volume (liter) and of mass (gram) are derived. All 
 other units are the decimal subdivisions or multiples of these. These three 
 units are simply related : one cubic decimeter equals one liter, and one liter of 
 water weighs one kilogram. The metric tables are formed by combining 
 the words "meter," "gram," and "liter" with numerical prefixes. 
 
72 
 
 WEIGHTS AND MEASURES 
 
 All lengths, areas, and cubic measures in the following conversion tables 
 are derived from the international meter. The customary weights are like- 
 wise derived from the kilogram. All capacities are based on the practical 
 equivalent: 1 cubic decimeter equals 1 liter. (The liter is defined as the 
 volume occupied by the mass of 1 kilogram of water under a pressure of 
 76 cm. of mercury and at the temperature of 4 deg. cent. According to the 
 best information, 1 liter = 1.000027 cubic decimeters.) 
 
 The customary weights derived from the international kilogram are based 
 on the value 1 avoirdupois Ib. = 453.59243 grams. The value of the troy 
 ib. is based on the same relation and also the equivalent 5760/7000 
 avoirdupois Ib. equals 1 troy Ib. 
 
 Metric Measures 
 
 Length 
 
 Area 
 
 Unit 
 
 Sym- 
 bol 
 
 Value in meters 
 
 Unit 
 
 Sym- 
 bol 
 
 Value in sq. 
 meters 
 
 Micron 
 
 M 
 mm. 
 cm. 
 dm. 
 m. 
 dkm. 
 hm. 
 km. 
 Mm. 
 
 0.000001 
 0.001 
 0.01 
 0.1 
 1.0 
 10.0 
 100.0 
 1,000.0 
 10,000.0 
 1,000,000.0 
 
 
 
 
 Millimeter.... 
 Centimeter . . 
 Decimeter. . . 
 Meter (unit). 
 Dekameter. . 
 Hectometer.. 
 Kilometer. . . 
 Myriameter.. . 
 Megameter.. . 
 
 Sq. millimeter 
 
 mm. 2 
 cm. 2 
 dm. 2 
 m. 
 a. 
 ha. 
 km." 
 
 . 000001 
 0.0001 
 0.01 
 1.0 
 100.0 
 10,000.0 
 1,000,000.0 
 
 Sq. centimeter 
 Sq. decimeter 
 
 Sq. meter (centiare) 
 Sq. dekameter (are) 
 Hectare 
 
 Sq. kilometer. . 
 
 
 
 
 
 
 
 
 Volume 
 
 Cubic measure 
 
 Unit 
 
 Symbol 
 
 Value in 
 liters 
 
 Unit 
 
 Symbol 
 
 Value in 
 cubic 
 meters 
 
 Milliliter 
 Liter (unit) 
 
 ml. or cm. 3 
 1. or dm. 8 
 kl. or m.s 
 
 cl. 
 dl. 
 dkl. 
 hi. 
 
 0.001 
 1.0 
 1,000.0 
 
 0.01 
 0.1 
 10.0 
 100.0 
 
 Cubic kilometer 
 Cubic hectometer. . . . 
 Cubic dekameter 
 
 Cubic meter 
 Cubic decimeter 
 Cubic centimeter 
 Cubic millimeter 
 Cubic micron 
 
 km. 3 
 hm. 
 dkm. s 
 
 m.8 
 dm. 3 
 cm. 3 
 mm. 3 
 M 8 
 
 10 
 10 
 10 3 
 
 1 
 
 10- 
 
 10" 
 ID' 9 
 10-18 
 
 Kiloliter 
 Also 
 Centiliter 
 
 
 Deciliter 
 
 Dekaliter 
 
 Hectoliter 
 
 
 
 
 Weight 
 
 Unit 
 
 Symbol 
 
 Value in 
 grams 
 
 Unit 
 
 Symbol 
 
 Value in 
 grams 
 
 Microgram 
 
 
 0.000001 
 
 Dekagram 
 
 dkg. 
 
 10.0 
 
 Milligram 
 
 mg. 
 
 0.001 
 
 Hectogram 
 
 fag 
 
 100 
 
 Centigram 
 Decigram 
 
 eg- 
 dg. 
 
 0.01 
 0.1 
 
 Kilogram 
 Myriagram 
 
 kg. 
 Mg 
 
 1,000.0 
 10,000 
 
 Gram (unit) 
 
 
 1 
 
 
 
 100 000 
 
 
 
 
 Ton 
 
 t. 
 
 1,000,000.0 
 
 
 
 
 
 
 
 SYSTEMS OF UNITS 
 
 The principal, units of interest to mechanical engineers can all be derived 
 from the three fundamental units of force, length, and time. These three 
 fundamental units may be chosen at pleasure; each such choice gives rise to a 
 "system" of units. The following table gives the units of the four "systems" 
 most often met with in the literature. 
 
UNITS 
 
 73 
 
 The precise definitions of the fundamental units in these systems are as follows. (In 
 these definitions the "standard pound body " and the "standard kilogram body "refer 
 to two special lumps of metal, carefully preserved at London and Paris, respectively; 
 the "standard locality" means sea level, 45 deg. latitude; or, more strictly, any locality 
 in which the acceleration due to gravity has the value 980.665 cm. per sec. 2 = 32.1740 ft. 
 per sec. 2 , which may be called the standard acceleration. 
 
 The pound (force) is the force required to support the standard pound body against 
 gravity, in vacua, in the standard locality; or, it is the force which, if applied to the stand- 
 ard pound body, supposed free to move, would give that body the "standard ac- 
 celeration." The word "pound" is used for the unit of both force and mass, and 
 consequently is ambiguous. To avoid uncertainty it is desirable to call the units 
 "pound force" and "pound mass," respectively. 
 
 The kilogram (force) is the force required to support the standard kilogram against 
 gravity, in vacua, in. the standard locality; or, it is the force which, if applied to the stand- 
 ard kilogram body, supposed free to move, would give that body the "standard accelera- 
 tion." The word "kilogram" is used for the unit of both force and mass and conse- 
 quently is ambiguous. To avoid uncertainty it is desirable to call the units "kilogram 
 force" and "kilogram mass," respectively. 
 
 The poundal is the force which, if applied to the standard pound body, would give 
 that body an acceleration of 1 ft. per sec. 2 ; that is, 1 poundal = 1/32.1740 of a pound 
 force. 
 
 The dyne is the force which, if applied to the standard gram body, would give that 
 body an acceleration of 1 cm. per sec. 2 ; that is, 1 dyne = 1/980.665 of a gram force. 
 
 Systems of Units 
 
 
 
 British 
 
 Metric 
 
 
 
 Name of 
 unit 
 
 Dimen- 
 sions of 
 units in 
 terms of 
 F,L, T 
 
 "gravita- 
 tional " sys- 
 tem, or 
 "foot-pound- 
 second" 
 
 "gravita- 
 tional " sys- 
 tem, or 
 "kilogram- 
 meter-sec- 
 
 Metric 
 "absolute" 
 system, or 
 "C. G. S." 
 system 
 
 British 
 "absolute" 
 system 
 (little used) 
 
 
 
 system 
 
 ond" system 
 
 
 
 Force 
 
 F 
 
 1 Ib. 
 
 1 kg. 
 
 1 dyne 
 
 1 poundal 
 
 Length 
 
 L 
 
 1 ft. 
 
 1 m. 
 
 1 cm. 
 
 1ft. 
 
 Time 
 
 T 
 
 1 sec. 
 
 1 sec. 
 
 1 sec. 
 
 1 sec. 
 
 
 
 
 
 
 
 Velocity 
 Acceleration . . 
 
 L/T 
 L/T* 
 
 1 ft. per sec. 
 1 ft. per sec. 2 
 
 1 m. per sec. 
 1 m. per sec. 2 
 
 1 cm. per sec. 
 1 cm. per sec. 2 
 
 1 ft. per sec. 
 1ft. per sec. 2 
 
 Pressure 
 
 F/L* 
 
 lib. per ft. 2 
 
 1 kg. per m. 2 
 
 1 dyne per cm. 2 
 
 1 pdl. per ft. 2 
 
 Impulse or 
 
 
 
 
 
 
 momentum. . 
 
 FT 
 
 1 Ib.-sec. 
 
 1 kg.-sec. 
 
 1 dyne-sec. 
 
 1 pdl.-sec. 
 
 Work or 
 
 
 
 
 
 
 energy 
 
 FL 
 
 1 ft.-lb. 
 
 1 kg.-m. 
 
 1 dyne-cm. = 
 
 1 ft. -pdl. 
 
 
 
 
 
 1 "erg." 
 
 
 Power 
 
 FL/T 
 
 1 ft.-lb. per 
 
 1 kg.-m. per 
 
 1 dyne-cm, per 
 
 1 ft.-pdl. per 
 
 
 
 sec. 
 
 sec. 
 
 sec. 
 
 sec. 
 
 Mass 
 
 F/(L/T*) 
 
 1 Ib. per (ft. 
 
 1 kg. per (m. 
 
 1 dyne per (cm. 
 
 1 pdl. per (ft. 
 
 
 
 per sec. 2 ) = 
 
 per sec. 2 ) = 
 
 per sec. 2 ) = 1 
 
 per sec. 2 ) = 
 
 
 
 1 "slug." 
 
 1 " metric 
 
 gram mass. 
 
 1 pound 
 
 
 
 
 slug." 
 
 
 mass. 
 
 NOTE. The "slug" (also called the "geepound," or the "engineer's unit of mass"), 
 the " metric slug," and the "poundal" are never used in practice. 
 
 Other common units are as follows: 
 Work: 1 joule = 10? ergs = 10,000,000 dyne-cm. 
 
 1 kilowatt-hour = 3,600,000 joules = 3600 X lO dyne-cm. 
 Power: 1 horse power = 550 ft.-lb. per sec. 
 1 poncelet = 100 kg.-m. per sec. 
 1 force de cheval = 75 kg.-m. per sec. 
 1 watt = 1 joule per sec. = 10,000,000 dyne-cm, per sec. 
 1 kilowatt = 1000 watts = 10 10 dyne-cm, per sec. 
 
 A new horse power of 550.220 ft.-lb. per sec., or 746 watts, has been proposed, but has 
 not been accepted by mechanical engineers. 
 
 The weight of a body (in a given locality) always means a force, namely, the force, re- 
 
74 
 
 WEIGHTS AND MEASURES 
 
 quired to support the body against gravity (in that locality). 'When no particular local- 
 ity is specified, the standard locality may be assumed. Thus, the "standard weight" of 
 the pound body is 1 lb.; the "standard weight" of the kilogram body is 1 kg 
 
 Dynes X 10 Kilograms Pounds Poundala 
 
 the quantity of heat required 
 to raise the temperature of 1 
 gram of water 1 deg. cent, at a 
 mean temperature of 15 deg. 
 cent., or (2) the heat required 
 to raise the temperature of 
 1 lb. of water 1 deg. fahr. The 
 former quantity is called the 
 gram-calorie (small calorie), 
 while the latter is known as the 
 British thermal unitorB.t.u. 
 The kilogram-calorie (large calorie), which ia equal to 1000 g.-cal., is largely 
 used in engineering work in metric countries. * 1 therm = 1 g.-cal. 
 CONVERSION TABLES 
 Length Equivalents 
 
 1 
 
 1.020 
 
 0.00848 
 
 2.248 ' 
 0.03518 
 
 72.33 
 1.85933 
 
 0.9807 
 1.99149 
 
 1 
 
 2.205 
 0.34334 
 
 70.93 
 
 1.85C84 
 
 0.4448 
 1.64819 
 
 0.4536 
 
 1.65667 
 
 1 
 
 32.17 
 1.50750 
 
 0.01383 
 2.14067 
 
 0.01410 
 
 2.14916 
 
 0.03108 
 
 2.49249 
 
 1 
 
 Centimeters Inches 
 
 Feet | Yards | Meters 
 
 Chains 
 
 Kilometers 
 
 Miles 
 
 1 
 
 0.3937 
 1.59517 
 
 0.03281 
 
 2.51598 
 
 0.01094 
 2.03886 
 
 0.01 
 
 2.00000 
 
 0.0 3 497I 
 4.69644 
 
 10-5 
 B". 00000 
 
 0.0 6 6214 
 6.79335 
 
 2.540 
 0.40483 
 
 1 
 
 0.0 3 8333 
 
 4.92082 
 
 0.02778 
 2.44370 
 
 0.0254 
 
 2.40483 
 
 0.0*1263 
 
 5.10127 
 
 0.0 4 254 
 
 5.40483 
 
 0.041578 
 
 5.19818 
 
 30.48 
 
 1.48402 
 
 12 
 
 1.07918 
 
 1 
 
 0.3333 
 
 T.52288 
 
 0.3048 
 
 1.48402 
 
 0.01515 
 
 2.18046 
 
 0.0 3 3098 
 4.48402 
 
 0.0s 1645 
 
 4.21608 
 
 91.14 
 
 1.96114 
 
 36 
 
 1.55630 
 
 3 
 
 0.47712 
 
 1 
 
 0.9144 
 1.96114 
 
 0.04545 
 
 2.65758 
 
 0.0s9144 
 4.96114 
 
 0.0 3 5682 
 4.75449 
 
 100 
 
 2.00000 
 
 39.37 
 
 1.59517 
 
 3.281 
 
 0.51598 
 
 1 .0936 
 
 0.03886 
 
 1 
 
 0.04971 
 
 2.69644 
 
 0.001 
 3.COOOO 
 
 0.0 3 62!4 
 
 4.79335 
 
 2012 
 3.30356 
 
 792 
 
 2.89873 
 
 66 
 
 1.81954 
 
 22 
 
 1.34242 
 
 20.12 
 1.30356 
 
 1 
 
 0.02012 
 2.30356 
 
 0.0125 
 
 2.09691 
 
 100000 
 5.00000 
 
 39370 
 4.59517 
 
 3281 
 3.51598 
 
 1093.6 
 
 3.C3886 
 
 1000 
 
 3.00000 
 
 49.71 
 
 1.69644 
 
 1 
 
 0.6214 
 1.79335 
 
 160925 
 5.20665 
 
 63360 
 
 4.80182 
 
 5280 
 3.72263 
 
 1760 
 3.24551 
 
 1609 
 3.20665 
 
 80 
 
 1.90309 
 
 1.609 
 
 0.206G5 
 
 1 
 
 The equivalents are given in the heavier type. Logarithms of the equivalents are 
 given immediately below. 
 
 Subscripts after any figure, Os, 94, etc.J mean that that figure is to be repeated the 
 indicated number of times. 
 
 Conversion of Lengths 
 
 
 Inches 
 to milli- 
 meters 
 
 Milli- 
 meters 
 to inches 
 
 Feet 
 to 
 meters 
 
 Meters 
 to 
 feet 
 
 Yards 
 to 
 
 meters 
 
 Meters 
 to 
 yards 
 
 Miles 
 to kilo- 
 meters 
 
 Kilo- 
 meters 
 to miles 
 
 1 
 
 3 
 4 
 
 6 
 7 
 
 [ 
 
 25.40 
 50.80 
 76.20 
 101 .60 
 
 127.00 
 152.40 
 177.80 
 203.20 
 228.60 
 
 0.03937 
 0.07874 
 0.1181 
 0.1575 
 
 0.1968 
 0.2362 
 0.2756 
 0.3150 
 0.3543 
 
 0.3048 
 0.6096 
 0.9144 
 1.219 
 
 1.524 
 1.829 
 2.134 
 2.438 
 2.743 
 
 3.281 
 6.562 
 9.842 
 13.12 
 
 16.40 
 19.68 
 22.97 
 26.25 
 29.53 
 
 0.9144 
 1.829 
 2.743 
 3.658 
 
 4.572 
 5.486 
 6.401 
 7.315 
 8.230 
 
 1.094 
 2.187 
 3.281 
 4.374 
 
 5.486 
 6.562 
 7.655 
 8.749 
 9.842 
 
 1.609 
 3.219 
 4.828 
 6.437 
 
 8.047 
 9.656 . 
 11.27 
 12.87 
 14.48 
 
 0.6214 
 1.243 
 1.864 
 2.485 
 
 3.107 
 3.728 
 4.350 
 4.971 
 5.592 
 
 *See Marks' MECHANICAL ENGINEERS' HANDBOOK.. 
 
CONVERSION TABLES 
 
 75 
 
 Mechanical Equivalent of Heat. See p. 311.* The value most commonly 
 accepted among American engineers as the work equivalent of 1 mean B.t.u. 
 is 777.5 ft.-lb., and the mean gram-calorie = 4.183 joules, which are the 
 values used throughout this book. The U. S. Bureau of Standards does not 
 recommend any special value; for its own purposes it takes the 59 deg. fahr. 
 B.t.u. as 778.2 ft.-lb. and the 68 deg. B.t.u. as 777.5 ft.-lb. The 15 deg. 
 calorie = 4.187 joules; 20 deg. calorie = 4.183 joules. There is an uncer- 
 tainty of about 1 part in 1000 in these values. 
 
 Conversion of Lengths : Inches and Millimeters 
 
 Common fractions of an inch to millimeters 
 (From HU to 1 in.) 
 
 64ths 
 
 Milli- 
 meters 
 
 64ths 
 
 Milli- 
 meters 
 
 64ths 
 
 Milli- 
 meters 
 
 64ths 
 
 Milli- 
 meters 
 
 64ths 
 
 Milli- 
 meters 
 
 64ths 
 
 Milli- 
 meters 
 
 1 
 
 0.397 
 
 13 
 
 5.159 
 
 25 
 
 9.922 
 
 37 
 
 14.684 
 
 49 
 
 19.447 
 
 57 
 
 22.622 
 
 2 
 
 0.794 
 
 14 
 
 5.556 
 
 26 
 
 10.319 
 
 38 
 
 15.081 
 
 50 
 
 19.844 
 
 58 
 
 23.019 
 
 3 
 
 1.191 
 
 15 
 
 5.953 
 
 27 
 
 10.716 
 
 39 
 
 15.478 
 
 51 
 
 20.241 
 
 59 
 
 23.416 
 
 4 
 
 1.588 
 
 16 
 
 6.350 
 
 28 
 
 11.113 
 
 40 
 
 15.875 
 
 52 
 
 20.638 
 
 60 
 
 23.813 
 
 5 
 
 1.984 
 
 17 
 
 6.747 
 
 29 
 
 11.509 
 
 41 
 
 16.272 
 
 53 
 
 21.034 
 
 61 
 
 24.209 
 
 6 
 
 2.381 
 
 18 
 
 7.144 
 
 30 
 
 11.906 
 
 42 
 
 16.669 
 
 54 
 
 21.431 
 
 62 
 
 24.606 
 
 7 
 
 2.778 
 
 19 
 
 7.541 
 
 31 
 
 12.303 
 
 43 
 
 17.066 
 
 55 
 
 21.828 
 
 63 
 
 25.003 
 
 8 
 
 3.175 
 
 20 
 
 7.938 
 
 32 
 
 12.700 
 
 44 
 
 17.463 
 
 56 
 
 22.225 
 
 64 
 
 25.400 
 
 9 
 
 3.572 
 
 21 
 
 8.334 
 
 33 
 
 13.097 
 
 45 
 
 17.859 
 
 
 
 
 
 10 
 
 3.969 
 
 22 
 
 8.731 
 
 34 
 
 13.494 
 
 46 
 
 18.256 
 
 
 
 
 
 11 
 
 4.366 
 
 23 
 
 9.128 
 
 35 
 
 13.891 
 
 47 
 
 18.653 
 
 
 
 
 
 12 
 
 4.763 
 
 24 
 
 9.525 
 
 36 
 
 14.288 
 
 48 
 
 19.050 
 
 
 
 
 
 Decimals of an inch to millimeters. (From 0.01 in. to 0.99 in.) 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 .0 
 
 
 0.254 
 
 0.508 
 
 0.762 
 
 1.016 
 
 1.270 
 
 1.524 
 
 1.778 
 
 2.032 
 
 2.286 
 
 .1 
 
 2.540 
 
 2.794 
 
 3.048 
 
 3.302 
 
 3.556 
 
 3.810 
 
 4.064 
 
 4.318 
 
 4.572 
 
 4.826 
 
 .2 
 
 5.080 
 
 5.334 
 
 5.588 
 
 5.842 
 
 6.096 
 
 6.350 
 
 6.604 
 
 6.858 
 
 7.112 
 
 7.366 
 
 .3 
 
 7.620 
 
 7.874 
 
 8.128 
 
 8.382 
 
 8.636 
 
 8.890 
 
 9.144 
 
 9.398 
 
 9.652 
 
 9.906 
 
 .4 
 
 10.160 
 
 10.414 
 
 10.668 
 
 10.922 
 
 11.176 
 
 11.430 
 
 11.684 
 
 11.938 
 
 12.192 
 
 12.446 
 
 .5 
 
 12.700 
 
 12.954 
 
 13.208 
 
 13.462 
 
 13.716 
 
 13.970 
 
 14.224 
 
 14.478 
 
 14.732 
 
 14 986 
 
 .6 
 
 15.240 
 
 15.494 
 
 15.748 
 
 16.002 
 
 16.256 
 
 16.510 
 
 16.764 
 
 17.018 
 
 17.272 
 
 17.526 
 
 .7 
 
 17.780 
 
 18.034 
 
 18.288 
 
 18.542 
 
 18.796 
 
 19.050 
 
 19.304 
 
 . 19.558 
 
 19.812 
 
 20.066 
 
 .8 
 
 20.320 
 
 20.574 
 
 20.828 
 
 21 .082 
 
 21.336 
 
 21.590 
 
 21 .844 
 
 22.098 
 
 22.352 
 
 22.606 
 
 .9 
 
 22.860 
 
 23.114 
 
 23.368 
 
 23.622 
 
 23.876 
 
 24.130 
 
 24.384 
 
 24.638 
 
 24.892 
 
 25.146 
 
 Millimeters to decimals of an inch. (From 1 to 99 mm.) 
 
 
 0. 
 
 1. 
 
 2. 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 7. 
 
 8. 
 
 9. 
 
 
 
 
 0.0394 
 
 0.0787 
 
 0.1181 
 
 0.1575 
 
 0.1969 
 
 0.2362 
 
 0.2756 
 
 0.3150 
 
 0.3543 
 
 1 
 
 0.3937 
 
 0.4331 
 
 0.4724 
 
 0.5118 
 
 0.5512 
 
 0.5906 
 
 0.6299 
 
 0.6693 
 
 0.7087 
 
 0.7480 
 
 2 
 
 0.7874 
 
 0.8268 
 
 0.8661 
 
 0.9055 
 
 0.9449 
 
 0.9843 
 
 1 .0236 
 
 1 .0630 
 
 1.1024 
 
 1.1417 
 
 3 
 
 1.1811 
 
 1.2205 
 
 1 .2598 
 
 1.2992 
 
 1.3386 
 
 1.3780 
 
 1.4173 
 
 1.4567 
 
 1 .4961 
 
 1.5354 
 
 4 
 
 1.5748 
 
 1.6142 
 
 1 .6535 
 
 1 .6929 
 
 1.7323 
 
 1.7717 
 
 1.8110 
 
 1.8504 
 
 1.8898 
 
 1.9291 
 
 5 
 
 1.9685 
 
 2.0079 
 
 2.0472 
 
 2.0866 
 
 2.1260 
 
 2.1654 
 
 2.2047 
 
 2.2441 
 
 2.2835 
 
 2.3228 
 
 6 
 
 2.3622 
 
 2.4016 
 
 2.4409 
 
 2.4803 
 
 2.5197 
 
 2.5591 
 
 2.5984 
 
 2.6378 
 
 2.6772 
 
 2.7165 
 
 7 
 
 2.7559 
 
 2.7953 
 
 2.8346 
 
 2.8740 
 
 2.9134 
 
 2.9528 
 
 2.9921 
 
 3.0315 
 
 3.0709 
 
 3.1102 
 
 8 
 
 3.1496 
 
 3.1890 
 
 3.2283 
 
 3.2677 
 
 3.3071 
 
 3.3465 
 
 3.3858 
 
 3.4252 
 
 3.4646 
 
 3.5039 
 
 9 
 
 3.5433 
 
 3.5827 
 
 3.6220 
 
 3.6614 
 
 3.7008 
 
 3.7402 
 
 3.7795 
 
 3.8189 
 
 3.8583 
 
 3.8976 
 
 'See Marks' MECHANICAL ENGINEERS' HANDBOOK. 
 
76 
 
 WEIGHTS AND MEASURES 
 
 Area Equivalents 
 
 (For conversion table see p. 77) 
 
 Square 
 meters 
 
 Square 
 inches 
 
 Square 
 feet 
 
 Square 
 yards 
 
 Square 
 rods 
 
 Square 
 chains 
 
 Roods 
 
 Acres 
 
 Square 
 miles or 
 sections 
 
 1 
 
 1550 
 
 3.19033 
 
 10.76 
 1.03197 
 
 1.196 
 0.07773 
 
 0.0395 
 
 2.59699 
 
 0.002471 
 
 3.39288 
 
 0.0 3 9884 
 3.99494 
 
 0.0>2471 
 
 1.39288 
 
 0.0*3861 
 
 7.58670 
 
 0.0 3 6452 
 4 80967 
 
 1 
 
 0.006944 
 3.84164 
 
 0.0011 
 
 3.88740 
 
 0.042551 
 
 5.40667 
 
 0.0*1594 
 6.20255 
 
 6 6377 
 
 7.80461 
 
 0.0e1594 
 
 7.20255 
 
 0.0 8 4910 
 
 10.39637 
 
 0.09290 
 1.96803 
 
 144 
 2.15836 
 
 1 
 
 0.1111 
 
 1.04576 
 
 0.003673 
 3.56503 
 
 0.0 3 2296 
 4.36091 
 
 0.049184 
 
 5.96297 
 
 0.0 4 2296 
 4.36091 
 
 0.0:3587 
 
 "S. 554 73 
 
 0.8361 
 
 1.92227 
 
 1296 
 3.11260 
 
 9 
 
 0.95424 
 
 1 
 
 0.03306 
 
 2.51927 
 
 0.002066 
 3.31515 
 
 0.0 3 8264 
 4.91721 
 
 0.0002066 
 4.31515 
 
 0.0 6 3228 
 
 7.50898 
 
 25.29 
 1.40300 
 
 39204 
 4.59333 
 
 272.25 
 2.43497 
 
 30.25 
 
 1.48072 
 
 1 
 
 0.0625 
 
 2.79588 
 
 0.02500 
 
 2.39794 
 
 0.00625 
 
 3.79588 
 
 0. 0*9766 
 
 6.98970 
 
 404.7 
 2.60712 
 
 627264 
 
 5.79745 
 
 4356 
 3.63909 
 
 484 
 
 2.68484 
 
 16 
 
 1.20412 
 
 1 
 
 0.4 
 
 1.60206 
 
 0.1 
 
 1.00000 
 
 0.0001562 
 
 4.19382 
 
 1012 
 
 3.00506 
 
 1568160 
 6.19539 
 
 10890 
 4.03703 
 
 1210 
 
 3.08278 
 
 40 
 
 1.60206 
 
 2.5 
 
 0.39794 
 
 1 
 
 0.25 
 
 1.39794 
 
 0.0 3 3906 
 "4.59176 
 
 4047 
 3.60712 
 
 6272640 
 6.79745 
 
 43560 
 
 4.63909 
 
 4840 
 
 3.68484 
 
 160 
 
 2.20412 
 
 10 
 
 1.00000 
 
 4 
 
 0.60206 
 
 1 
 
 0.001562 
 3.19382 
 
 2589a8 
 6.41330 
 
 
 27878400 
 
 7.44527 
 
 3097600 
 6.49102 
 
 102400 
 5.01030 
 
 6400 
 
 3.80618 
 
 2560 
 
 3.40824 
 
 640 
 
 2.80618 
 
 1 
 
 (1 hectare =100 ares = 10,000 centiares or square meters) 
 
 Volume and Capacity 'Equivalents 
 
 (For conversion table see p. 77) 
 
 Cubic 
 inches 
 
 Cubic 
 feet 
 
 Cubic 
 yards 
 
 U. S. 
 Apothe- 
 cary 
 liquid 
 ounces 
 
 U. S. quarts 
 
 U. S. gallons 
 
 Bushels 
 U. S. 
 
 Liters 
 (1) 
 
 Liquid 
 
 ry 
 
 Liquid 
 
 Dry 
 
 1 
 
 0.0 3 5787 
 
 4.76246 
 
 0.0 4 2143 
 F. 33109 
 
 0.5541 
 
 1. 74360 
 
 0.01732 
 2.23845 
 
 0.01488 
 
 2.17263 
 
 0.0 2 4329 
 3.63639 
 
 0.0 2 3720 
 
 3.57057 
 
 0.0 3 4650 
 
 4.66748 
 
 0.01639 
 2.21450 
 
 1728 
 3.23754 
 
 1 
 
 0.03704 
 
 2". 56864 
 
 957.5 
 2.98114 
 
 29.92 
 
 1.47599 
 
 25.71 
 
 1.41017 
 
 7.481 
 0.87393 
 
 6.429 
 
 0.80811 
 
 0.8036 
 
 1.90502 
 
 28.32 
 1.45205 
 
 46656 
 4.66891 
 1.805 
 0.25640 
 
 27 . 
 
 1.43136 
 001044 
 
 3.01886 
 
 1 
 
 0.0 4 3868 
 
 5.58749 
 
 25853 
 4.41251 
 
 1 
 
 807.9 
 2.90736 
 0.03125 
 
 2.49485 
 
 694.3 
 2.84153 
 0.02686 
 
 2.42903 
 
 202.0 
 2.30530 
 0.007813 
 3.89279 
 
 173.6 
 
 2.23948 
 0.006714 
 
 3.82697 
 
 21.70 
 1.33638 
 0.0 3 8392 
 
 4.92388 
 
 764.6 
 2.88341 
 0.02957 
 
 2.47091 
 
 57.75 
 
 1.76155 
 
 0.03342 
 
 2.52401 
 
 0.001238 
 3.09264 
 
 32 
 1.50515 
 
 1 
 
 0.8594 
 1.93418 
 
 0.25 
 
 1.39794 
 
 0.2148 
 1.33212 
 
 0.02686 
 2.42903 
 
 0.9464 
 1.97606 
 
 67.20 
 1.82737 
 
 0.03889 
 2.58983 
 
 0.001440 
 3.15847 
 
 37.24 
 
 1.57097 
 
 1.164 
 
 0.06582 
 
 1 
 
 2909 
 1.46376 
 
 0.25 
 
 1.39794 
 
 0.03125 
 
 2.49485 
 
 1.101 
 
 0.04188 
 
 231 
 2.36361 
 
 0.1337 
 
 1.12607 
 
 0.004951 
 3.69470 
 
 128 
 
 2.10721 
 
 4 
 0.60206 
 
 3.437 
 
 0.53624 
 
 1 
 
 0.8594 
 1.93418 
 
 0.1074 
 1.03109 
 
 3.785 
 
 0.57812 
 
 268.8 
 2.42943 
 
 0.1556 
 1.19189 
 
 0.005761 
 
 3.76053 
 
 148.9 
 2.17303 
 
 4.655 
 
 0.66788 
 
 4 
 
 0.60206! 
 
 1.164 
 
 0.06582 
 
 1 
 
 0.125 
 
 1.09691 
 
 4.405 
 0.64394 
 
 2150 
 3.33252 
 
 1.244 
 0.09498 
 
 0.04609 
 2.66362 
 
 1192 
 3.07612 
 
 37.24 
 
 1.57097 
 
 32 
 1.50515 
 
 9.309 
 
 0.96891 
 
 8 
 
 0.90309 
 
 1 
 
 35.24 
 1.54703 
 
 61.02 
 1.78550 
 
 0.03531 
 2.54795 
 
 0.001308 
 3.11659 
 
 33.81 
 
 1.52909 
 
 1.057 
 
 0.02394 
 
 0.9081 
 1.95812 
 
 0.2642 
 
 L 42188 
 
 0.2270 
 1.35606 
 
 02838 
 2.45297 
 
 1 
 
 The equivalents are given in the heavier type. Logarithms of the equivalents are 
 given immediately below. 
 
 Subscripts after any figure, Oa, 94, etc., mean that that figure ia to be repeated the 
 indicated number of timeB. 
 
CONVERSION TABLES 
 
 77 
 
 Conversion of Areas 
 
 
 Sq. in. 
 to 
 sq. cm. 
 
 Sq. cm. 
 to 
 sq. in. 
 
 Sq. ft. 
 to 
 sq. m. 
 
 Sq. m. 
 to 
 sq. ft. 
 
 Sq. yd. 
 to 
 sq. m. 
 
 Sq. m. 
 to 
 sq. yd. 
 
 Acres 
 to 
 hec- 
 tares 
 
 Hec- 
 tares to 
 acres 
 
 Sq. mi. 
 to 
 sq. km. 
 
 Sq. km. 
 to 
 sq. mi. 
 
 1 
 
 2 
 3 
 
 4 
 
 5 
 6 
 7 
 8 
 9 
 
 6.452 
 12.90 
 19.35 
 25.81 
 
 32.26 
 38.71 
 45.16 
 51.61 
 58.06 
 
 0.1550 
 0.3100 
 0.4650 
 0.6200 
 
 0.7750 
 0.9300 
 1.085 
 1.240 
 1.395 
 
 0.0929 
 0.1858 
 0.2787 
 0.3716 
 
 0.4645 
 0.5574 
 0.6503 
 0.7432 
 0.8361 
 
 10.76 
 21.53 
 32.29 
 43.06 
 
 53.82 
 64.58 
 75.35 
 86.11 
 96.87 
 
 0.8361 
 1.672 
 2.508 
 3.345 
 
 4.181 
 5.017 
 5.853 
 6.689 
 7.525 
 
 1.196 
 2.392 
 3.588 
 4.784 
 
 5.980 
 7.176 
 8.372 
 9.568 
 10.764 
 
 0.4047 
 0.8094 
 1.214 
 1.619 
 
 2.023 
 2.428 
 2.833 
 3.237 
 3.642 
 
 2.471 
 4.942 
 7.413 
 9.884 
 
 12.355 
 14.826 
 17.297 
 19.768 
 22.239 
 
 2.590 
 5.180 
 7.770 
 10.360 
 
 12.950 
 15.540 
 18.130 
 20.720 
 23.310 
 
 0.3861 
 0.7722 
 1.158 
 1.544 
 
 1.931 
 2.317 
 2,703 
 3.089 
 3.475 
 
 Conversion of Volumes or Cubic Measure 
 
 
 Cu. in. 
 to 
 cu. cm. 
 
 Cu. cm. 
 to 
 cu. in. 
 
 Cu. ft. 
 to 
 cu. m. 
 
 Cu. m. 
 to 
 cu. ft. 
 
 Cu. yd. 
 to 
 Cu. m. 
 
 Cu. m. 
 to 
 cu. yd. 
 
 Gallons 
 to 
 cu. ft. 
 
 Cu. ft. 
 to 
 gallons 
 
 2 
 3 
 4 
 
 5 
 6 
 7 
 
 8 
 9 
 
 16.39 
 32.77 
 49.16 
 65.55 
 
 81.94 
 98.32 
 114.7 
 131.1 
 147.5 
 
 0.06102 
 0.1220 
 1831 
 0.2441 
 
 0.3051 
 0.3661 
 0.4272 
 0.4882 
 0.5492 
 
 0.02832 
 0.05663 
 0.08495 
 0.1133 
 
 0.1416 
 0.1699 
 0.1982 
 0.2265 
 0.2549 
 
 35.31 
 70.63 
 105.9 
 141.3 
 
 176.6 
 211.9 
 247.2 
 282.5 
 317.8 
 
 0.7646 
 1.529 
 2.294 
 3.058 
 
 3.823 
 4.587 
 5.352 
 6.116 
 6.881 
 
 1.308 
 2.616 
 3.924 
 5.232 
 
 6.540 
 7.848 
 9.156 
 10.46 
 11.77 
 
 0.1337 
 0.2674 
 0.4011 
 0.5348 
 
 0.6685 
 0.8022 
 0.9359 
 1.070 
 1.203 
 
 7.481 
 14.96 
 22.44 
 29.92 
 
 37.41 
 44.89 
 52.36 
 59.85 
 67.33 
 
 Conversion of Volumes or Capacities 
 
 
 Liquid 
 ounces 
 to 
 cu. cm. 
 
 Cu. cm. 
 to 
 liquid 
 ounces 
 
 Pints 
 to 
 liters 
 
 Liters 
 to 
 pints 
 
 Quarts 
 to 
 liters 
 
 Liters 
 to 
 quarts 
 
 Gallons 
 to 
 liters 
 
 Liters 
 to 
 gallons 
 
 Bushels 
 to 
 hecto- 
 liters 
 
 Hecto- 
 liters 
 to 
 bushels 
 
 1 
 
 29.57 
 
 0.03381 
 
 0.4732 
 
 2.113 
 
 0.9464 
 
 1.057 
 
 3.785 
 
 0.2642 
 
 "0.3524 
 
 2.838 
 
 2 
 
 59.15 
 
 0.06763 
 
 0.9464 
 
 4.227 
 
 1.893 
 
 2.113 
 
 7.571 
 
 0.5283 
 
 0.7048 
 
 5.676 
 
 3 
 
 88.72 
 
 0.1014 
 
 1.420 
 
 6.340 
 
 2.839 
 
 3.170 
 
 11.36 
 
 0.7925 
 
 1.057 
 
 8.513 
 
 4 
 
 118.3 
 
 0.1353 
 
 1.893 
 
 8.453 
 
 3.785 
 
 4.227 
 
 15.14 
 
 1 .057 
 
 1.410 
 
 11.35 
 
 5 
 
 147.9 
 
 0.1691 
 
 2.366 
 
 10.57 
 
 4.732 
 
 5.283 
 
 18.93 
 
 1.321 
 
 1.762 
 
 14.19 
 
 6 
 
 177.4 
 
 0.2029 
 
 2.839 
 
 12.68 
 
 5.678 
 
 6.340 
 
 22.71 
 
 1.585 
 
 2.114 
 
 17.03 
 
 7 
 
 207.0 
 
 0.2367 
 
 3.312 
 
 14.79 
 
 6.625 
 
 7.397 
 
 26.50 
 
 1.849 
 
 2.467 
 
 19.86 
 
 8 
 
 236.6 
 
 0.2705 
 
 3.785 
 
 16.91 
 
 7.571 
 
 8.453 
 
 30.28 
 
 2.113 
 
 2.819 
 
 22.70 
 
 9 
 
 266.2 
 
 0.3043 
 
 4.259 
 
 19.02 
 
 8.517 
 
 9.510 
 
 34.07 
 
 2.378 
 
 3.172 
 
 25.54 
 
 Conversion of Masses 
 
 
 Grains 
 to 
 
 grams 
 
 Grams 
 to 
 grains 
 
 Ounces 
 (avoir.) 
 to 
 grams 
 
 Grams 
 to 
 ounces 
 (avoir.) 
 
 Pounds 
 (avoir.) 
 to 
 kilo- 
 grams 
 
 Kilo- 
 grams 
 to 
 pounds 
 (avoir.) 
 
 Short 
 tons 
 (2000 
 Ib.) to 
 metric 
 tons 
 
 Metric 
 tons 
 (1000 
 kg.) to 
 short 
 tons 
 
 Long 
 tons 
 (2240 
 Ib.) to 
 metric 
 tons 
 
 Metric 
 tons 
 to 
 long 
 tons 
 
 1 
 
 0.06480 
 
 15.43 
 
 28.35 
 
 0.03527 
 
 0.4536 
 
 2.205 
 
 0.907 
 
 1.102 
 
 1.016 
 
 0.984 
 
 2 
 
 0.1296 
 
 30.86 
 
 56.70 
 
 0.07055 
 
 0.9072 
 
 4.409 
 
 1.814 
 
 2.205 
 
 2.032 
 
 1.968 
 
 3 
 
 0.1944 
 
 46.30 
 
 85.05 
 
 0.1058 
 
 1.361 
 
 6.614 
 
 2.722 
 
 3.307 
 
 3.048 
 
 2.953 
 
 4 
 
 0.2592 
 
 61.73 
 
 113.40 
 
 0.1411 
 
 1.814 
 
 8.818 
 
 3.629 
 
 4.409 
 
 4.064 
 
 3.937 
 
 5 
 
 0.3240 
 
 77.16 
 
 141.75 
 
 0.1764 
 
 2.268 
 
 11.02 
 
 4.536 
 
 5.512 
 
 5.080 
 
 4.921 
 
 6 
 
 0.3888 
 
 92.59 
 
 170.10 
 
 0.2116 
 
 2.722 
 
 13.23 
 
 5.443 
 
 6.614 
 
 6.096 
 
 5.905 
 
 7 
 
 0.4536 
 
 108,03 
 
 198.45 
 
 0.2469 
 
 3.175 
 
 15.43 
 
 6.350 
 
 7.716 
 
 7.112 
 
 6.889 
 
 8 
 
 0.5184 
 
 123.46 
 
 226.80 
 
 0.2822 
 
 3.629 
 
 17.64 
 
 7.257 
 
 8.818 
 
 8.128 
 
 7.874 
 
 9 
 
 0.5832 
 
 138.89 
 
 255.15 
 
 0.3175 
 
 4.082 
 
 19.84 
 
 8.165 
 
 9.921 
 
 9.144 
 
 8.857 
 
 
78 
 
 WEIGHTS AND MEASURES 
 
 Velocity Equivalents 
 
 (For conversion table see p. 80) 
 
 Centimeters 
 per sec. 
 
 Meters 
 per sec. 
 
 Meters 
 per min. 
 
 Kilo- 
 meters 
 per hour 
 
 Feet 
 per sec. 
 
 Feet 
 per min. 
 
 Miles 
 per hour 
 
 Knots 
 
 1 
 
 0.01 
 
 0.6 
 
 1.77815 
 
 0.036 
 
 2.55630 
 
 0.03281 
 2.51598 
 
 1.9685 
 0.29414 
 
 0.02237 
 
 2.34965 
 
 0.01942 
 
 2.28825 
 
 100 
 
 2.00000 
 
 1 
 
 60 
 1.77815 
 
 3.6 
 
 0.55630 
 
 3.281 
 
 0.51598 
 
 196.85 
 
 2.29414 
 
 2.237 
 
 0.34965 
 
 1.942 
 
 0.28825 
 
 1.667 
 0.22184 
 
 0.01667 
 2.22184 
 
 1 
 
 0.06 
 
 2.77815 
 
 0.05468 
 
 2.73783 
 
 3.281 
 0.51598 
 
 0.03728 
 2.57150 
 
 0.03237 
 
 2.51018 
 
 27.78 
 1.44370 
 
 0.2778 
 1. 44370 
 
 16.67 
 
 1.22184 
 
 1 
 
 0.9113 
 
 T. 95968 
 
 54.68 
 1.73783 
 
 0.6214 
 
 1.79335 
 
 0.53960 
 
 1.73207 
 
 30.48 
 1.48402 
 
 0.3048 
 
 F. 48402 
 
 18.29 
 
 1.26217 
 
 1.097 
 
 0.04032 
 
 1 
 
 60 
 
 1.77815 
 
 0.6818 
 
 1.83367 
 
 0.59209 
 
 1.77238 
 
 0.5080 
 1.70586 
 
 0.005080 
 3". 70586 
 
 0.3048 
 
 1.48402 
 
 0.01829 
 2.26217 
 
 0.01667 
 
 2.22185 
 
 1 
 
 0.01136 
 
 2.05553 
 
 0.00987 
 3.99423 
 
 44.70 
 1.65035 
 
 0.4470 
 
 r. 65035 
 
 26.82 
 1.42850 
 
 1.609 
 
 0.20670 
 
 1.467 
 0.16633 
 
 88 
 1.94448 
 
 1 
 
 0.86839 
 1.93871 
 
 51.497 
 
 1.71178 
 
 51497 
 F. 71178 
 
 30.898 
 1.48993 
 
 1.8532 
 0.26793 
 
 1.68894 
 0.22761 
 
 101.337 
 
 2.00577 
 
 1.15155 
 
 0.06128 
 
 1 
 
 Mass Equivalents 
 
 (For conversion table see p. 77) 
 
 Kilograms 
 
 Grains 
 
 Ounces 
 
 Pounds 
 
 Tons 
 
 Troy and 
 apoth. 
 
 Avoir- 
 dupois 
 
 Troy and 
 apoth. 
 
 Avoir- 
 dupois 
 
 Short 
 
 Long 
 
 Metric 
 
 1 
 
 15432 
 4.18843 
 
 32.15 
 1.50719 
 
 35.27 
 1.54745 
 
 2.6792 
 0.42801 
 
 2.205 
 0.34333 
 
 0.0 2 1102 
 3.04230 
 
 0'0 3 9842 
 4.99309 
 
 0.001 
 3. 00000 
 
 0.046480 
 5.81157 
 
 1 
 
 0.0 2 2083 
 3.31876 
 
 0.0 2 2286 
 3.35902 
 
 0.0 3 1736 
 
 4.23958 
 
 0.0 3 1429 
 4.15490 
 
 0.0 7 7143 
 
 "8. 85387 
 
 0.0 7 6378 
 
 8.80465 
 
 0.0:6480 
 
 8.81157 
 
 0.03110 
 
 2.49281 
 
 480 
 2.68124 
 
 1 
 
 1.09714 
 0.04026 
 
 0.08333 
 
 2.92082 
 
 0.06857 
 2.83614 
 
 0.0 4 3429 
 5.53511 
 
 0.0 4 3061 
 
 5.48590 
 
 0.043110 
 
 "5.49281 
 
 0.02835 
 2.45255 
 
 437.5 
 
 2.64098 
 
 0.9115 
 
 1.95974 
 
 1 
 
 0.07595 
 
 2.88056 
 
 0.0625 
 
 2.79588 
 
 0.0 4 3125 
 
 5.49485 
 
 0.042790 
 5.44563 
 
 0.0 4 2835 
 T. 45255 
 
 0.3732 
 T.57199 
 
 5760 
 
 3.76042 
 
 12 
 
 1.07918 
 
 13.17 
 
 1.11944 
 
 1 
 
 0.8229 
 1.91532 
 
 8.0*41 H 
 
 4.61429 
 
 0.0s3673 
 
 4.56508 
 
 0.0 3 3732 
 4.57199 
 
 0.4536 
 T. 65667 
 
 7000 
 3.84510 
 
 14.58 
 1.16386 
 
 16 
 
 1.20412 
 
 1.215 
 
 0.08468 
 
 1 
 
 0.0005 
 
 T. 69897 
 
 0.0 3 4464 
 4.64975 
 
 0.0 3 4536 
 
 4.65667 
 
 907.2 
 2.95770 
 
 140e 
 
 7.14613 
 
 29167 
 4.46489 
 
 320 3 
 4.50515 
 
 2431 
 3.38571 
 
 2000 
 3.30103 
 
 1 
 
 0.8929 
 T. 95078 
 
 0.9072 
 1.95770 
 
 1016 
 3.00691 
 
 15680 4 
 7.19535 
 
 326s 
 4.51411 
 
 35840 
 4.55437 
 
 2722 
 3.43492 
 
 2240 
 3.35025 
 
 1.12 
 
 0.04922 
 
 1 
 
 1.016 
 
 0.00691 
 
 1000 
 3.00000 
 
 15432356 
 
 7.18843 
 
 32151 
 4.50719 
 
 35274 
 4.54745 
 
 2679 
 3.42801 
 
 2205 
 3.34333 
 
 1.102 
 
 0.04230 
 
 0.9842 
 1.99309 
 
 1 
 
 The equivalents are given in the heavier type. Logarithms of the equivalents are 
 given immediately below. 
 
 Subscripts after any figure, Os, 4, -etc., mean that that figure is to be repeated the 
 indicated number of times. 
 
CONVERSION TABLES 
 
 79 
 
 Pressure Equivalents 
 
 (For conversion table see p. 80) 
 
 Megabars 
 or 
 megadynes 
 per 
 sq. cm. 
 
 Kilo- 
 grams 
 per 
 sq. cm. 
 (Metric 
 atmos- 
 pheres) 
 
 Pounds 
 per 
 sq. in." 
 
 Short 
 tons 
 per 
 
 sq. ft. 
 
 Atmos- 
 pheres 
 
 Columns of 
 mercury at 
 temperature 
 0C. 
 
 Columns of water at 
 temperature 15 C. 
 
 Meters 
 
 Inches 
 
 Meters 
 
 Inches 
 
 Feet 
 
 1 
 
 1.0197 
 
 14.50 
 
 1.044 
 
 0.9S69 
 
 0.7500 
 
 29.53 
 
 10.21 
 
 401.8 
 
 33.48 
 
 
 0.00848 
 
 1.16148 
 
 0.01882 
 
 1.99427 
 
 1.87508 
 
 1.47025 
 
 1.00886 
 
 2.60402 
 
 1.52484 
 
 0.9807 
 
 1 
 
 14.22 
 
 1.024 
 
 0.9678 
 
 0.7355 
 
 28.96 
 
 10.01 
 
 394.0 
 
 32.84 
 
 1.99152 
 
 
 1.15300 
 
 0.01034 
 
 1.98579 
 
 1.86660 
 
 1.46177 
 
 1.00038 
 
 2.59555 
 
 1.51636 
 
 0.06895 - 
 
 0.07031 
 
 1 
 
 0.072 
 
 0.06804 
 
 0.05171 
 
 2.036 
 
 0.7037 
 
 27.70 
 
 2.309 
 
 2.83852 
 
 2.84700 
 
 
 2". 85733 
 
 2.83279 
 
 2.71360 
 
 0.30876 
 
 f. 84738 
 
 1.44254 
 
 0.36336 
 
 0.9576 
 
 0.9765 
 
 13.89 
 
 1 
 
 0.9450 
 
 0.7182 
 
 28.28 
 
 9.773 
 
 384.8 
 
 32.06 
 
 ,1.98119 
 
 1. 98966 
 
 1.14267 
 
 
 1.97545 
 
 1.85627 
 
 1.45143 
 
 0.99004 
 
 2.58521 
 
 1.50603 
 
 1.0133 
 
 1.0333 
 
 14.70 
 
 1.058 
 
 1 
 
 0.76 
 
 29.92 
 
 10.34 
 
 407.2 
 
 33.93 
 
 0.00573 
 
 0.01421 
 
 1.16722 
 
 0.02955 
 
 
 1.88081 
 
 1.47598 
 
 1.01459 
 
 2.60976 
 
 1.53058 
 
 1.3333 
 
 1.3596 
 
 19.34 
 
 1.392 
 
 1.316 
 
 1 
 
 39.37 
 
 13.61 
 
 535.7 
 
 44.64 
 
 0.12492 
 
 0.13340 
 
 1.28640 
 
 0.14373 
 
 0.11919 
 
 
 1.59517 
 
 1.13378 
 
 2.72894 
 
 1.64976 
 
 0.03386 
 
 0.03453 
 
 0.4912 
 
 0.03536 
 
 0.03342 
 
 0.02540 
 
 1 
 
 0.3456 
 
 13.61 
 
 1.134 
 
 2". 52975 
 
 2.53823 
 
 T. 69124 
 
 2.54857 
 
 2.52402 
 
 2.40484 
 
 
 1.53861 
 
 1.13378 
 
 0.05460 
 
 0.09798 
 
 0.09991 
 
 1.421 
 
 0.1023 
 
 0.09670 
 
 0.07349 
 
 2.893 
 
 | 
 
 39.37 
 
 3.281 
 
 2~.99114 
 
 2". 99962 
 
 0.15262 
 
 F. 00996 
 
 2.98541 
 
 2.86622 
 
 0.46139 
 
 
 1.59517 
 
 0.55198 
 
 0.002489 
 
 002538 
 
 0.03610 
 
 0.002599 
 
 0.002456 
 
 0.001867 
 
 0.07349 
 
 0.02540 
 
 1 
 
 0.08333 
 
 3.39598 
 
 3.40446 
 
 2.55746 
 
 3.41479 
 
 3.39024 
 
 3.27106 
 
 2.86622 
 
 2~.40484 
 
 
 2.92082 
 
 0.02986 
 
 0.03045 
 
 0.4332 
 
 0.03119 
 
 0.02947 
 
 0.02240 
 
 0.8819 
 
 0.3048 
 
 12 
 
 1 
 
 2.47516 
 
 2.48364 
 
 1.63664 
 
 2.49397 
 
 2.46942 
 
 2.35024 
 
 1.94540 
 
 1.48402 
 
 1.07918 
 
 
 Energy or Work Equivalents 
 
 (For conversion table see p. 80) 
 
 Joules = 
 10' ergs 
 
 Kilogram- 
 meters 
 
 Foot- 
 pounds 
 
 Kilo- 
 watt- 
 hours 
 
 Cheval- 
 vapeur- 
 hours 
 
 Horse- 
 power- 
 hours 
 
 Liter- 
 atmos- 
 pheres 
 
 Kilo- 
 gram- 
 calories 
 
 British 
 thermal 
 units 
 
 1 
 
 0.10197 
 
 1.00848 
 
 0.7376 
 
 1.86780 
 
 0.0e2778 
 7.44370 
 
 0.0o3777 
 7.57711 
 
 0.0 6 3725 
 7.57113 
 
 0.009869 
 3.99427 
 
 0.02390 
 
 T. 37848 
 
 0.0 3 9486 
 4.97709 
 
 9.80665 
 0.9915207 
 
 1 
 
 7.233 
 0.85932 
 
 0.0 5 2724 
 6.43522 
 
 0.0 8 37037 
 
 6.56863 
 
 0.053653 
 
 6.56265 
 
 0.09678 
 
 2.98579 
 
 0.002344 
 3.37000 
 
 009302 
 3.96861 
 
 1.356 
 0.13220 
 
 0.1383 
 1.14068 
 
 1 
 
 0.0 6 3766 
 7.57590 
 
 0.0 6 51206 
 
 7.70932 
 
 0.0 6 50505 
 7.70333 
 
 0.01338 
 
 2.12647 
 
 0.0 3 3241 
 
 4.51068 
 
 0.001286 
 
 3.10929 
 
 3.6X10 
 
 6.55630 
 
 3.671X105 
 
 5.56478 
 
 2.655X10" 
 6.42410 
 
 '" 
 
 1.3596 
 0.13342 
 
 1.341 
 0.12743 
 
 35528 
 4.55057 
 
 860.5 
 
 2.93478 
 
 3415 
 3.53339 
 
 2.648X10 
 6.42288 
 
 270000. 
 5.43136 
 
 1.9529X10* 
 6.29068 
 
 0.7355 
 
 1.86658 
 
 1 
 
 0.9863 
 1.99401 
 
 26131. 
 
 4.41715 
 
 632.9 
 
 2.80135 
 
 2512 
 
 3.39996 
 
 2.6845X10 8 
 
 6.42887 
 
 2. 7375X105 
 5.43735 
 
 1.98X10 
 6.29667 
 
 0.7457 
 
 1.87257 
 
 1.0139 
 0.00598 
 
 1 
 
 26494 
 4.42314 
 
 641.7 
 
 2.80735 
 
 2547 
 3.40595 
 
 101.33 
 
 2.00573 
 
 10.333 
 1.01421 ' 
 
 74.73 
 
 1.87353 
 
 0.042815 
 5.44943 
 
 0.0 4 3827 
 
 5.58284 
 
 0.0 4 3774 
 
 5.57686 
 
 1 
 
 0.02422 
 2.38425 
 
 0.09612 
 
 2.98281 
 
 4183 
 3.62153 
 
 426.6 
 2.63000 
 
 3086 
 3.48932 
 
 0.001162 
 
 3.06522 
 
 0.001580 
 
 3.19864 
 
 0.001558 
 3.19265 
 
 41.29 
 1.61579 
 
 1 
 
 3.968 
 0.59861 
 
 1054 
 
 3.02291 
 
 107.5 
 2.03139 
 
 777.52 
 2.89071 
 
 0.0 3 2928 
 
 4.46661 
 
 0.0 3 3981 
 
 4.60003 
 
 0.0 3 3927 
 
 4.59405 
 
 10.40 
 1.01719 
 
 0.25200 
 1.40139 
 
 1 
 
 The equivalents are given in the heavier type. Logarithms of the equivalents are 
 given immediately below. 
 
 Subscripts after any figure, Oi, 94, etc., mean that that figure is to be repeated the 
 indicated number of times. 
 
 V 
 
80 
 
 WEIGHTS AND MEASURES 
 
 Linear and Angular Velocity Conversion Factors 
 
 
 Cm. per 
 sec. to 
 feet per 
 min. 
 
 Feet per 
 min. to 
 cm. per 
 sec. 
 
 Cm. per 
 sec. to 
 miles 
 per hour 
 
 Miles 
 per hour 
 to cm. 
 per sec. 
 
 Feet per 
 sec. to 
 miles 
 per hour 
 
 Miles 
 per hour 
 to feet 
 per sec. 
 
 Radians 
 per sec. 
 to rev. 
 per min. 
 
 Rev. per 
 
 min. to 
 radians 
 per sec. 
 
 1 
 
 2 
 3 
 
 4 
 
 5 
 6 
 7 
 
 8 
 9 
 
 1.97 
 3.94 
 5.91 
 7.87 
 
 9.84 
 11.81 
 13.78 
 15.75 
 17.72 
 
 0.508 
 1.016 
 1.524 
 2.032 
 
 2.540 
 3.048 
 3.556 
 4.064 
 4.572 
 
 0.0224 
 0.0447 
 0.0671 
 0.0895 
 
 0.1118 
 0.1342 
 0.1566 
 0.1789 
 0.2013 
 
 44.7 
 89.4 
 134.1 
 178.8 
 
 223.5 
 268.2 
 312.9 
 357.6 
 402.3 
 
 0.682 
 1.364 
 2.046 
 2.727 
 
 3.409 
 4.091 
 4.773 
 5.455 
 6.136 
 
 1.47 
 2.93 
 4.40 
 5.87 
 
 7.33 
 8.80 
 10.27 
 11.73 
 13.20 
 
 9.55 
 19.10 
 28.65 
 38.20 
 
 47.75 
 57.30 
 66.85 
 76.39 
 85.94 
 
 0.1047 
 0.2094 
 0.3142 
 0.4189 
 
 0.5236 
 0.6283 
 0.7330 
 0.8378 
 0.9425 
 
 Conversion of Pressures 
 
 
 Pounds per 
 sq. in. to 
 kilograms 
 per sq. cm. 
 
 Kilograms 
 per sq. cm. 
 to pounds 
 per sq. in. 
 
 Atmospheres 
 to pounds 
 per sq. in. 
 
 Pounds per 
 sq. in. to 
 atmospheres 
 
 Atmospheres 
 to kilograms 
 per sq. cm. 
 
 Kilograms 
 per sq. cm. 
 to atmos- 
 pheres 
 
 1 
 
 0.0703 
 
 14.22 
 
 14.70 
 
 0.0680 
 
 1.033 
 
 0.9678 
 
 2 
 
 0.1406 
 
 28.45 
 
 29.39 
 
 0.1361 
 
 2.067 
 
 1.936 
 
 3 
 
 0.2109 
 
 42.67 
 
 44.09 
 
 0.2041 
 
 3.100 
 
 2.903 
 
 4 
 
 0.2812 
 
 56.89 
 
 58.79 
 
 0.2722 
 
 4.133 
 
 3.871 
 
 5 
 
 0.3515 
 
 71.12 
 
 73.48 
 
 0.3402 
 
 5.166 
 
 4.839 
 
 6 
 
 4218 
 
 85.34 
 
 88.18 
 
 0.4082 
 
 6.200 
 
 5.807 
 
 7 
 
 0.4922 
 
 99.56 
 
 102.9 
 
 0.4763 
 
 7.233 
 
 6.774 
 
 8 
 
 0.5624 
 
 113.8 
 
 117.6 
 
 0.5443 
 
 8.266 
 
 7.742 
 
 9 
 
 0.6328 
 
 128.0 
 
 132.3 
 
 0.6124 
 
 9.300 
 
 8.710 
 
 Conversion of Energy, Work, Heat 
 
 
 Ft.-lb. 
 to 
 kilo- 
 gram- 
 meters 
 
 Kilo- 
 gram- 
 meters 
 to 
 ft.-lb. 
 
 Ft.-lb. 
 to 
 B.t.u. 
 
 B.t.u. 
 to 
 ft.-lb. 
 
 Kilo- 
 gram- 
 meters 
 to 
 large 
 calories 
 
 Large 
 calories 
 to 
 kilo- 
 gram- 
 meters 
 
 Joules 
 to 
 small 
 calories 
 
 Small 
 calories 
 to 
 j oules 
 
 1 
 
 0.1383 
 
 7.233 
 
 0.001286 
 
 777.5 
 
 0.002344 
 
 426.6 
 
 0.2390 
 
 4.183 
 
 2 
 
 0.2765 
 
 14.47 
 
 0.002572 
 
 1555.0 
 
 0.004688 
 
 853.2 
 
 0.4780 
 
 8.367 
 
 3 
 
 0.4148 
 
 21.70 
 
 003858 
 
 2333.0 
 
 0007033 
 
 1280.0 
 
 0'.7170 
 
 12.55 
 
 4 
 
 0.5530 
 
 28.93 
 
 0.005144 
 
 3110.0 
 
 0.009377 
 
 1706.0 
 
 0.9560 
 
 16.73 
 
 5 
 
 0.6913 
 
 36.16 
 
 0.006431 
 
 3888.0 
 
 0.01172 
 
 2133.0 
 
 1.195 
 
 20.92 
 
 6 
 
 0.8295 
 
 43.40 
 
 0007717 
 
 4665.0 
 
 0.01407 
 
 2560.0 
 
 1.434 
 
 25.10 
 
 7 
 
 0.9678 
 
 50.63 
 
 0009003 
 
 5443.0 
 
 0.01641 
 
 2986.0 
 
 1.673 
 
 29.28 
 
 8 
 
 1.106 
 
 57.86 
 
 01029 
 
 6220.0 
 
 0.01875 
 
 3413.0 
 
 1.912 
 
 33.47 
 
 9 
 
 1.244 
 
 65.10 
 
 0.01157 
 
 6998.0 
 
 0.02110 
 
 3839.0 
 
 2.151 
 
 37.65 
 
 Conversion of Power 
 
 
 Horse powers 
 to kilowatts 
 
 Kilowatts to 
 horse powers 
 
 Metric 
 horse powers 
 to kilowatts 
 
 Kilowatts 
 to metric 
 horse powers 
 
 Horse powers 
 to metric 
 horse powers 
 
 Metric 
 horse powers 
 to 
 horse powers 
 
 1 
 
 0.7457 
 
 1.341 
 
 0.7354 
 
 1.360 
 
 1.014 
 
 0.9863 
 
 2 
 
 1.491 
 
 2.682 
 
 1.471 
 
 2.719 
 
 2.028 
 
 1.973 
 
 3 
 
 2.237 
 
 4.023 
 
 2.206 
 
 4.079 
 
 3.042 
 
 2.959 
 
 4 
 
 2.983 
 
 5.364 
 
 2.942 
 
 5.439 
 
 4.056 
 
 3.945 
 
 5 
 
 3.728 
 
 6.705 
 
 3.677 
 
 6.799 
 
 5.069 
 
 4.932 
 
 6 
 
 4.474 
 
 8.046 
 
 4.413 
 
 8.158 
 
 6.083 
 
 5.918 
 
 7 
 
 5.220 
 
 9.387 
 
 5.148 
 
 9.518 
 
 7.097 
 
 6.904 
 
 8 
 
 5.965 
 
 10.73 
 
 5.884 
 
 10.88 
 
 8.111 
 
 7.890 
 
 9 
 
 6.710 
 
 12.07 
 
 6.619 
 
 12.24 
 
 9.125 
 
 8.877 
 
CONVERSION TABLES 
 
 81 
 
 Power Equivalents 
 
 (For conversion table see p. 80) 
 
 Horse power 
 
 Kilo- 
 watts 
 (1000 
 joules 
 per sec.) 
 
 Cheval- 
 vapeur 
 (metric 
 h.p.) 
 
 Ponce- 
 lets 
 
 M.-kg. 
 per sec. 
 
 Ft.-lb. 
 per sec. 
 
 Kg- 
 cal. 
 per sec. 
 
 B.t.u 
 per sec. 
 
 550 stand- 
 ard ft.-lb. 
 per sec. 
 
 1 
 
 0.7457 
 
 1.014 
 
 0.7604 
 
 76.04 
 
 550 
 
 0.1783 
 
 0.7074 
 
 
 1.87256 
 
 0.00599 
 
 1.88105 
 
 1.88105 
 
 2.74036 
 
 1.25104 
 
 1.84965 
 
 1.341 
 
 1 
 
 1.360 
 
 1.020 
 
 102.0 
 
 737.6 
 
 0.2390 
 
 0.9486 
 
 0.12743 
 
 
 0.13343 
 
 0.00848 
 
 2.00848 
 
 2.86780 
 
 1.37848 
 
 1.97709 
 
 0.9863 
 
 0.7355 
 
 1 
 
 0.75 
 
 75 
 
 542.3 
 
 0.1758 
 
 0.6977 
 
 T. 99402 
 
 T. 86659 
 
 
 1.87506 
 
 1.87506 
 
 2.73438 
 
 1.24506 
 
 1.84367 
 
 1.315 
 
 0.9807 
 
 1.333 
 
 1 
 
 100 
 
 723.3 
 
 0.2344 
 
 0.9303 
 
 0.11896 
 
 1.99152 
 
 0.12493 
 
 
 2.00000 
 
 2.85932 
 
 1.37000 
 
 1.96861 
 
 0.01315 
 
 0.009807 
 
 0.01333 
 
 0.01 
 
 1 
 
 7.233 
 
 0.002344 
 
 0.009303 
 
 2.11896 
 
 3.99152 
 
 2.12493 
 
 T. 00000 
 
 
 0.85932 
 
 3.37000 
 
 2.96861 
 
 0.00182 
 
 0.001356 
 
 0.00184 
 
 0.00138 
 
 0.1383 
 
 1 
 
 0.0 3 3241 
 
 0.001286 
 
 3.25946 
 
 3.13219 
 
 3.26562 
 
 3.14067 
 
 T. 14067 
 
 
 T. 51068 
 
 T. 10929 
 
 5.610 
 
 4.183 
 
 5.688 
 
 4.266 
 
 426.6 
 
 3086 
 
 1 
 
 3.968 
 
 0.74896 
 
 0.62153 
 
 0.75494 
 
 0.63000 
 
 2.63000 
 
 3.48932 
 
 
 0.59861 
 
 1.414 
 
 1.054 
 
 1.433 
 
 1.075 
 
 107.5 
 
 777.5 
 
 0.2520 
 
 1 
 
 0.15035 
 
 0.02291 
 
 0.15632 
 
 0.03139 
 
 2.03139 
 
 2.89071 
 
 1.40138 
 
 
 The equivalents are given in the heavier type. Logarithms of the equivalents are 
 given immediately below. 
 
 Subscripts after any figure, Os, 94, etc., mean that that figure is to be repeated the 
 indicated number of times. 
 
 Density Equivalents and Conversion Factors 
 
 Equivalents 
 
 Conversion factors 
 
 Grams 
 per cu. 
 cm. 
 
 Lb. per 
 
 cu. in. 
 
 Lb. per 
 cu. ft. 
 
 Short 
 tons 
 (2000 
 lb.) per 
 cu. yd. 
 
 Lb. per 
 U. S. 
 gal. 
 
 
 Grams 
 per cu. 
 cm. to 
 lb. per 
 cu. ft. 
 
 Lb. per 
 
 cu. ft. 
 to grams 
 per cu. 
 cm. 
 
 Grams 
 per 
 cu. cm. 
 to short 
 tons per 
 cu. yd. 
 
 Short 
 tons per 
 cu. yd. 
 to grams 
 per cu. 
 cm. 
 
 1 
 
 0.03613 
 
 62.43 
 
 0.8428 
 
 8.345 
 
 1 
 
 62.43 
 
 0.01602 
 
 0.8428 
 
 1.186 
 
 
 2.55787 
 
 1.79539 
 
 1.92572 
 
 0.92143 
 
 2 
 
 124.90 
 
 0.03204 
 
 1.6860 
 
 2.373 
 
 27.68 
 
 1 
 
 1728 
 
 23.33 
 
 231 
 
 3 
 
 187.30 
 
 0.04806 
 
 2.5280 
 
 3.600 
 
 1.44217 
 
 
 3.23754 
 
 1.36792 
 
 2.36361 
 
 4 
 
 249.70 
 
 0.06407 
 
 3.3710 
 
 4.746 
 
 0.01602 
 
 0.0 3 5787 
 
 1 
 
 0.0135 
 
 0.1337 
 
 5 
 
 312.40 
 
 0.08009 
 
 4.2140 
 
 5.933 
 
 2.20466 
 
 4.76245 
 
 
 2.13033 
 
 1.12613 
 
 6 
 
 374.60 
 
 0.09611 
 
 5.0570 
 
 7.119 
 
 1.186 
 
 0.04286 
 
 74.07 
 
 1 
 
 9.902 
 
 7 
 
 437.00 
 
 0.11210 
 
 5.9000 
 
 8.306 
 
 0.07428 
 
 2.63205 
 
 1.86964 
 
 
 0.99572 
 
 8 
 
 499.40 
 
 0.12820 
 
 6.7420 
 
 9.492 
 
 0.1198 
 
 0.004329 
 
 7.481 
 
 0.1010 
 
 1 
 
 9 
 
 561.90 
 
 0.14420 
 
 7.5850 
 
 10.680 
 
 1.07855 3.63639 
 
 0.87396 
 
 1.00432 
 
 
 10 
 
 624.30 
 
 0.16020 
 
 8.4280 
 
 11.870 
 
82 
 
 WEIGHTS AND MEASURES 
 
 Conversion of Heat Transmission and Conduction 
 
 
 Small 
 
 B.t.u. 
 
 Small 
 
 B.t.u. 
 
 Small calories per 
 
 B.t.u. per hr. per 
 
 
 calories 
 
 per sq. 
 
 calories 
 
 per sq. ft. 
 
 sec. per sq. cm. 
 
 sq. ft. per 1 deg. 
 
 
 per sq. 
 cm. to 
 
 ft. to 
 small 
 
 per sq. cm. 
 per cm. to 
 
 per in. to 
 small 
 
 per 1 aeg. cent, per 
 cm. thick, to B.t.u. 
 
 fahr. per in. thick 
 to small calories 
 
 
 B.t.u. 
 
 calories 
 
 B.t.u. per 
 
 calories per 
 
 per hr. per sq. ft. 
 
 per sec. per sq. cm. 
 
 
 P 1t s q . 
 
 per sq. 
 cm. 
 
 sq. ft. 
 per in. 
 
 sq. cm. 
 per cm. 
 
 per 1 deg. fahr. 
 per in. thick 
 
 per 1 deg. cent, 
 per cm. thick 
 
 1 
 
 3.687 
 
 0.2712 
 
 1.451 
 
 0.6892 
 
 2.903X103 
 
 0.0 3 3445 
 
 2 
 
 7.374 
 
 0.5424 
 
 2.902 
 
 1.378 
 
 5.806X103 
 
 0.0 3 6890 
 
 3 
 
 11.06 
 
 0.8136 
 
 4.353 
 
 2.068 
 
 8.709X103 
 
 0.0 2 I034 
 
 4 
 
 14.75 
 
 1.085 
 
 5.804 
 
 2.757 
 
 11.61 X1Q3 
 
 0.0 2 I378 
 
 5 
 
 18.44 
 
 1.356 
 
 7.255 
 
 3.446 
 
 14.52 X10 
 
 0.0 2 1722 
 
 6 
 
 22.12 
 
 1.627 
 
 8.706 
 
 4.135 
 
 17.42 X1Q3 
 
 0.0 2 2067 
 
 7 
 
 25.81 
 
 1.898 
 
 10.16 
 
 4.824 
 
 20.32 X1Q3 
 
 0.0 2 2412 
 
 8 
 
 29.50 
 
 2.170 
 
 11.61 
 
 5.514 
 
 23.22 X103 
 
 0.0 2 2756 
 
 9 
 
 33.18 
 
 2.441 
 
 13.06 
 
 6.203 
 
 26.13 XlO 3 
 
 0.0 2 3100 
 
 NOTE. 1 gram-calorie per sq. cm. = 3.687 B.t.u. per sq. ft. 
 
 1 gram-calorie per sq. cm. per cm. = 1.451 B.t.u. per sq. ft. per in. 
 
 1 gram-calorie per sec. per sq. cm. for a temp. grad. of 1 deg. cent, per cm. 
 
 = 360 kilogram-calories per hour per sq. m. for a temp. grad. of 1 deg. cent, per m. 
 
 = 2.903 X 10 3 B.t.u. per hour per sq. ft. for a temp. grad. of 1 deg. fahr. per in. 
 
 Values of Foreign Coins 
 
 (Legal standards: (G) = gold; (S) = silver) 
 
 Country 
 
 Monetary 
 unit 
 
 Value 
 in 
 terms 
 of U. S. 
 money 
 
 Country 
 
 Monetary 
 unit 
 
 Value 
 in 
 terms 
 of U. S. 
 money 
 
 Argentina (G) 
 
 Austria-Hungary (GO 
 Belgium (G and <S) 
 
 Peso 
 
 Crown 
 Franc 
 
 CO. 9647 
 
 0.2026 
 1929 
 
 Great Britain (G) . . . . 
 
 Greece (G and S) 
 Haiti (G) 
 
 Pound ster- 
 ling. 
 Drachma. . . 
 Gourde 
 
 $4.8665 
 
 0.1929 
 9647 
 
 Bolivia (G) 
 
 
 3893 
 
 India (British) (G) 
 
 
 3244 
 
 Brazil (G) . . . 
 
 Milreis. 
 
 5463 
 
 Italy (G and <S) 
 
 Lira . . 
 
 1929 
 
 British colonies in . . 
 
 Pound ster- 
 
 
 Japan (G) 
 
 Yen 
 
 4984 
 
 Australasia and 
 
 ling. 
 
 4 8665 
 
 Liberia (G) 
 
 Dollar 
 
 1 0000 
 
 Africa (GO 
 
 
 
 Mexico (G) 
 
 Peso 
 
 4984 
 
 Canada (G) 
 
 Dollar....... 
 
 1 0000 
 
 Netherlands (G) .... 
 
 Florin 
 
 4019 
 
 Central American 
 
 
 
 Norway (G) 
 
 Crown 
 
 2679 
 
 States: 
 
 
 
 Panama (G) 
 
 Balboa 
 
 1 0000 
 
 Coeta Rica (GO 
 
 
 4653 
 
 Persia (G and <S) 
 
 Kran 
 
 Variable 
 
 British Honduras 
 (G or iS) 
 
 Dollar 
 
 1.0000 
 
 Peru (G) 
 Philippine Islands (G) 
 
 Libra 
 Peso 
 
 4 8665 
 5000 
 
 Guatemala (S).. . . 
 
 Peso 
 
 4446 
 
 Portugal (G) 
 
 Escudo 
 
 1 0805 
 
 Honduras (S) 
 
 Peso . . 
 
 4446 
 
 Roumania (G) 
 
 Leu 
 
 1929 
 
 Salvador(jS) 
 
 Peso 
 
 4446 
 
 Russia (G) 
 
 Ruble 
 
 5145 
 
 Nicaragua (S) 
 
 
 1 0000 
 
 
 Dollar 
 
 1 0000 
 
 Chile (G) ... 
 
 Peso 
 
 3649 
 
 Servia (G) 
 
 Dinar. .. 
 
 1929 
 
 China (S) 
 
 Yuan 
 
 4777 
 
 Siam (G) 
 
 Tical 
 
 3708 
 
 Colombia (G) 
 
 Pound 
 
 4.8665 
 2680 
 
 Spain (G and <S) 
 Straits Settlement (G) 
 
 Peseta 
 Dollar 
 
 0.1929 
 5677 
 
 Ecuador (G) . 
 
 Sucre 
 
 4866 
 
 Sweden (G) 
 
 Crown 
 
 2679 
 
 Egypt (G) 
 
 Pound 
 
 4 9429 
 
 Switzerland (CO 
 
 Franc 
 
 0.1929 
 
 Finland (G) 
 
 Markka 
 
 1929 
 
 Turkey (G) 
 
 Piaster 
 
 0439 
 
 France (G or S) . . . 
 German Empire (G) 
 
 Franc 
 Mark 
 
 0.1929 
 2381 
 
 Uruguay (G) 
 Venezuela (G) 
 
 Peso 
 Bolivar 
 
 1.0340 
 0.1929 
 
 
 
 
 
 
 
TIME 83 
 
 TIME 
 
 Kinds of Time. Three kinds of time are recognized by astronomers, viz., 
 sidereal, apparent solar, and mean solar time. The sidereal day is the inter- 
 val between two consecutive transits of some fixed celestial object across any 
 given meridian, or it is the interval required by the earth to make one com- 
 plete revolution on its axis. This interval is constant but it is inconvenient 
 as a time unit because the noon of the sidereal day occurs at all hours of the 
 day and night. The apparent solar day is the interval between two con- 
 secutive transits of the sun across any given meridian. On account of the 
 variable distance between the sun and earth, the variable speed of the 
 earth in its orbit, the effect of the moon, etc., this interval is not constant 
 and consequently cannot be kept by any simple mechanism, such as 
 clocks or watches. To overcome the objection noted above, the mean 
 solar day was devised. The mean solar day is the length of the average 
 apparent solar day. Like the sidereal day it is constant, and like the apparent 
 solar day its noon always occurs at approximately the same time of day. The 
 astronomical day begins at mean solar noon and the hours run from one 
 to twenty-four, while the civil day (mean solar) begins 12 hours earlier, at 
 midnight, and' the hours run from one to twelve, and then repeat from noon to 
 midnight. 
 
 The Year. There are three different kinds of year used, the sidereal, the 
 tropical, and the anomalistic. The sidereal year is the time taken by the 
 earth to complete one revolution around the sun from a given star to the same 
 star again. Its length is 365 days, 6 hours, 9 minutes, and 9 seconds. The 
 tropical year is the time included between two successive passages of the 
 vernal equinox by the sun, and since the equinox moves westward 50. "2 of 
 arc a year, the tropical year is shorter by 20'23."5 in time than the sidereal 
 year. As the seasons depend upon the earth's position with respect to the 
 equinox, the tropical year is the year of civil reckoning. The anomalistic 
 year is the interval between two successive passages of the perihelion, namely, 
 the time of the earth's nearest approach to the sun. The anomalistic year is 
 only used in special calculations in astronomy. 
 
 The Calendar. The month depended originally upon the changes of the 
 moon. The Mohammedan nations still use a lunar calendar with years of 
 twelve lunar months, which alternately contain 355 and 356 days. Accord- 
 ing to their method of reckoning the same month falls in different seasons, and 
 their calendars gain 1 year on ours every 33 years. The Julian Calendar 
 (established 45 B. C.) discards all consideration of the moon and adopts 
 36514 days as the true length of the year. It is still used in Russia and 
 generally by the Greek Church. Gregorian Calendar: The true length 
 of the tropical year is 365 days, 5 hr., 48 min., 45.5 sec., a difference of 11 
 min., 14.5 sec. by which the Julian year is too long. This amounts to a little 
 more than 3 days in 400 years. To correct for this, those century years are 
 made leap years which are divisible by 400 without remainder. 
 
 Standard Time. Prior to 1883 each city of the U. S. had its own time, 
 which was determined by the time of passage of the sun across the local merid- 
 ian. A system of standard time is used at present, according to which the 
 United States, which extends from 65 deg. to 125 deg. West longitude, is divided 
 into four sections, each of 15 deg. of longitude. The first or eastern section in- 
 cludes all territory between the Atlantic coast and an irregular line drawn from 
 Detroit, Mich., through Pittsburg to Charleston, S. C., its most southern 
 point. The time of this section is that of the 75-deg. meridian, which is 5 
 
84 
 
 WEIGHTS AND MEASURES 
 
 . 
 
 hr. slower than Greenwich time. The second (central) section includes 
 territory between the line mentioned, and an irregular line drawn from Bis- 
 marck, N. D., to the mouth of the Rio Grande. The third (mountain) sec- 
 tion includes all territory between the last-named line and a line which passes 
 through the western part of Idaho, Utah and Arizona. The fourth (Pacific) 
 section covers the rest of the country to the Pacific Ocean. Standard time 
 is uniform in each of these sections, but the time in one section differs by ex- 
 actly 1 hr. from the section next to it. In cities situated on the border line 
 of two sections, as, say, Pittsburg and Atlanta, the standard times of both sec- 
 tions are used, and in such cities when the time is given, it should be specified 
 as eastern, central, etc. The system of standard time has been adopted in 
 almost all civilized countries. All continental Europe, except Russia, uses 
 a time 1 hr. faster than that of Greenwich; in Japan and Australia the 
 time is 9 hr. faster. 
 
 TERRESTRIAL GRAVITY 
 
 By standard gravity is meant any locality where g = 980.665 cm. per 
 sec. per sec., or 32.1740 ft. per sec. per sec. This value, QO, is assumed to be 
 the value of g at sea level and latitude 45 deg. 
 
 Acceleration of Gravity 
 
 (U. S. Coast and Geodetic Survey, 1912) 
 
 Latitude, 
 
 
 
 r 
 
 
 Latitude, 
 
 
 / 
 
 
 deg. 
 
 Cm./sec. 2 
 
 Ft./sec.2 
 
 0/ffo 
 
 deg. 
 
 Cm./sec. 2 
 
 Ft./sec. 2 
 
 Q/ffo 
 
 
 10 
 20 
 30 
 40 
 
 978.0 
 978.2 
 978.6 
 979.3 
 980.2 
 
 32.088 
 32.093 
 32. 08 
 32.130 
 32.158 
 
 0.9973 
 0.9975 
 0.9979 
 0.9986 
 0.9995 
 
 50 
 60 
 70 
 80 
 90 
 
 981.1 
 981.9 
 982.6 
 .983.1 
 983.2 
 
 32.187 
 32.215 
 32.238 
 32.253 
 32.258 
 
 .0004 
 .0013 
 .0020 
 0024 
 0026 
 
 Correction for altitude above sea level: 0.3 cm. per sec. 2 for each 1000 meters; 
 - 0.003 ft. per sec. 2 for each 1000 feet. 
 
 SPECIFIC GRAVITY AND DENSITY 
 
 The specific gravity of a solid or liquid is the ratio of the mass of the 
 body to the mass of an equal volume of water at some standard temperature. 
 At the present time a temperature of 4 deg. cent. (39 deg. fahr.) is commonly 
 used by physicists, but the engineer uses 60 deg. fahr. The specific gravity 
 of gases is usually expressed in terms of hydrogen or air. 
 
 The density of a body is its mass per unit volume. If the gram is used as 
 the unit of mass and the milliliter as the unit of volume, the figures represent- 
 ing the density are the same as the specific gravity of the body referred to 
 water at 4 deg. cent, as unity. The customary unit is pounds per cu. ft. 
 
 The specific gravity of liquids is usually measured by means of an hydrom- 
 eter (see p. 254).* Special arbitrary hydrometer scales are used in various 
 trades and industries. The most common of these are the Baum6, Twaddell 
 and Beck. Twaddell's hydrometer is used for liquids heavier than water. 
 The number of degrees, N, which it indicates may be converted to specific 
 gravities, G, by the formula G = (5N + 1000) /1000. The formula for 
 the Beck hydrometer is G = 170/(170 N); for the Brix hydrometer G = 
 400/(400 N). In both of these the + sign is to be used for liquids lighter 
 than water, the sign for heavier liquids. For the salinometer (salometer), 
 see p. 1734. * The specific gravities corresponding to the indications of the 
 Baum6 hydrometer are given in the following tables. 
 See Marka' MECHANICAL ENGINEERS' HANDBOOK. 
 
SPECIFIC GRAVITY AND DENSITY 
 
 85 
 
 60 
 
 Specific Gravities at Fahr. Corresponding to Degrees Baume* 
 for Liquids Lighter than Water 
 
 60 
 Calculated from the formula, specific gravity 5 fahr. = 
 
 140 
 
 130 + Deg. 
 
 Degrees 
 Baume 
 
 Specific 
 gravity 
 
 Degrees 
 Baume 
 
 II 
 11 
 
 02 M 
 
 || 
 
 $$ 
 Qpq 
 
 Specific 
 gravity 
 
 s*> 
 
 Si 
 
 aa 
 
 Specific 
 gravity 
 
 Degrees 
 Baumfi 
 
 Specific 
 gravity 
 
 Degrees 
 Baum6 
 
 Specific 
 gravity 
 
 10 
 
 1.0000 
 
 25 
 
 0.9032 
 
 40 
 
 0.8235 
 
 55 
 
 0.7568 
 
 70 
 
 0.7000 
 
 85 
 
 0.6512 
 
 11 
 
 0.9929 
 
 26 
 
 0.8974 
 
 41 
 
 0.8187 
 
 56 
 
 0.7527 
 
 71 
 
 0.6965 
 
 86 
 
 0.6482 
 
 12 
 
 0.9859 
 
 27 
 
 0.8917 
 
 42 
 
 0.8140 
 
 57 
 
 0.7487 
 
 72 
 
 0.6931 
 
 87 
 
 0.6452 
 
 13 
 
 0.9790 
 
 28 
 
 0.8861 
 
 43 
 
 0.8092 
 
 58 
 
 0.7447 
 
 73 
 
 0.6897 
 
 88 
 
 0.6422 
 
 14 
 
 0.9722 
 
 29 
 
 0.8805 
 
 44 
 
 8046 
 
 59 
 
 0.7407 
 
 74 
 
 0.6863 
 
 89 
 
 0.6393 
 
 15 
 
 0.9655 
 
 30 
 
 0.8750 
 
 45 
 
 0.8000 
 
 60 
 
 0.7368 
 
 75 
 
 0.6829 
 
 90 
 
 0.6364 
 
 16 
 
 0.9589 
 
 31 
 
 0.8696 
 
 46 
 
 0.7955 
 
 61 
 
 0.7330 
 
 76 
 
 0.6796 
 
 91 
 
 0.6335 
 
 17 
 
 0.9524 
 
 32 
 
 0.8642 
 
 47 
 
 0.7910 
 
 62 
 
 0.7292 
 
 77 
 
 0.6763 
 
 92 
 
 0.6306 
 
 18 
 
 0.9459 
 
 33 
 
 0.8589 
 
 48 
 
 0.7865 
 
 63 
 
 0.7254 
 
 78 
 
 0.6731 
 
 93 
 
 0.6278 
 
 19 
 
 0.9396 
 
 34 
 
 0.8537 
 
 49 
 
 0.7821 
 
 64 
 
 0.7216 
 
 79 
 
 0.6699 
 
 94 
 
 0.6250 
 
 20 
 
 0.9333 
 
 " 35 
 
 0.8485 
 
 50 
 
 0.7778 
 
 65 
 
 0.7179 
 
 80 
 
 0.6667 
 
 95 
 
 0.6222 
 
 21 
 
 0.9272 
 
 36 
 
 0.8434 
 
 51 
 
 0.7735 
 
 66 
 
 0.7143 
 
 81 
 
 0.6635 
 
 96 
 
 0.6195 
 
 22 
 
 0.9211 
 
 37 
 
 0.8383 
 
 52 
 
 0.7692 
 
 67 
 
 0.7107 
 
 82 
 
 0.6604 
 
 97 
 
 0.6167 
 
 23 
 
 0.9150 
 
 38 
 
 0.8333 
 
 53 
 
 0.7650 
 
 68 
 
 0.7071 
 
 83 
 
 0.6573 
 
 98 
 
 0.6140 
 
 24 
 
 0.9091 
 
 39 
 
 0.8284 
 
 54 
 
 0.7609 
 
 69 
 
 0.7035 
 
 84 
 
 0.6542 
 
 99 
 
 0.6114 
 
 
 
 
 
 
 
 
 
 
 
 100 
 
 0.6087 
 
 60 
 
 Specific Gravities at Pahr. Corresponding to Degrees Baume 
 
 for Liquids Heavier than Water 
 
 Calculated from the formula, specific gravity fahr. = 
 
 
 
 Degrees 
 Baum6 
 
 Specific 
 gravity 
 
 Degrees 
 Baume 
 
 Specific 
 gravity 
 
 Degrees 
 Baum6 
 
 >> 
 
 11 
 
 02 00 
 
 ii 
 
 Specific 
 gravity 
 
 Degrees 
 Baume 
 
 1 
 
 Degrees 
 Baume 
 
 Specific 
 gravity 
 
 
 
 .0000 
 
 12 
 
 .0902 
 
 24 
 
 .1983 
 
 36 
 
 .3303 
 
 48 
 
 .4948 
 
 60 
 
 .7059 
 
 1 
 
 .0069 
 
 13 
 
 .0985 
 
 25 
 
 .2083 
 
 37 
 
 .3426 
 
 49 
 
 .5104 
 
 61 
 
 .7262 
 
 2 
 
 .0140 
 
 14 
 
 .1069 
 
 26 
 
 .2185 
 
 38 
 
 .3551 
 
 50 
 
 .5263 
 
 62 
 
 .7470 
 
 3 
 
 .0211 
 
 15 
 
 .1154 
 
 27 
 
 .2288 
 
 39 
 
 .3679 
 
 51 
 
 .5426 
 
 63 
 
 .7683 
 
 4 
 
 .0284 
 
 16 
 
 .1240 
 
 28 
 
 .2393 
 
 40 
 
 .3810 
 
 52 
 
 .5591 
 
 64 
 
 .7901 
 
 5 
 
 .0357 
 
 17 
 
 .1328 
 
 29 
 
 .2500 
 
 41 
 
 .3942 
 
 53 
 
 .5761 
 
 65 
 
 .8125 
 
 6 
 
 .0432 
 
 18 
 
 .1417 
 
 30 
 
 .2609 
 
 42 
 
 .4078 
 
 54 
 
 .5934 
 
 66 
 
 .8354 
 
 7 
 
 .0507 
 
 19 
 
 .1508 
 
 31 
 
 .2719 
 
 43 
 
 .4216 
 
 55 
 
 .6111 
 
 67 
 
 .8590 
 
 8 
 
 .0584 
 
 20 
 
 .1600 
 
 32 
 
 .2832 
 
 44 
 
 .4356 
 
 56 
 
 .6292 
 
 ^68 
 
 .8831 
 
 9 
 
 .0662 
 
 21 
 
 .1694 
 
 33 
 
 .2946 
 
 45 
 
 .4500 
 
 57 
 
 .6477 
 
 69 
 
 .9079 
 
 10 
 
 .0741 
 
 22 
 
 .1789 
 
 34 
 
 .3063 
 
 46 
 
 .4646 
 
 58 
 
 .6667 
 
 70 
 
 .9333 
 
 11 
 
 .0821 
 
 23 
 
 .1885 
 
 35 
 
 .3182 
 
 47 
 
 .4796 
 
 59 
 
 .6860 
 
 
 
 Mohs's Scale of Hardness 
 
 1. Talc. 2. Gypsum. 3. Calc spar. 4. Fluorspar. 5. Apatite. 
 6. Feldspar. 7. Quartz. 8. Topaz. 9. Sapphire. 10. Diamond. 
 
SECTION 2 
 MATHEMATICS 
 
 BY 
 EDWARD V. HUNTINGTON, Ph. D. 
 
 ASSOCIATE PROFESSOR OF MATHEMATICS, HARVARD UNIVERSITY, FELLOW AM. ACAD. 
 ARTS AND SCIENCES 
 
 CONTENTS 
 
 ARITHMETIC 
 
 PAGE 
 
 Numerical Computation 88 
 
 Logarithms 91 
 
 The Slide Rule 94 
 
 Computing Machines 97 
 
 Financial Arithmetic 98 
 
 GEOMETRY AND MENSURATION 
 
 Geometrical Theorems 99 
 
 Geometrical Constructions 101 
 
 Lengths and Areas of Plane Figures. 105 
 
 Surfaces and Volumes of Solids 107 
 
 ALGEBRA 
 
 Formal Algebra 112 
 
 Solution of Equations in One Un- 
 known Quantity 116 
 
 Solution of Simultaneous Equations 119 
 
 Determinants 123 
 
 Imaginary or Complex Quantities 124 
 
 TRIGONOMETRY 
 
 Formal Trigonometry 128 
 
 Solution of Plane Triangles 132 
 
 Solution of Spherical Triangles 134 
 
 Hyperbolic Functions 135 
 
 ANALYTICAL GEOMETRY 
 
 The Point and the Straight Line. ... 136 
 
 The Circle 137 
 
 PAOE 
 
 The Parabola 138 
 
 The Ellipse 140 
 
 The Hyperbola 144 
 
 The Catenary 147 
 
 Other Useful Curves 151 
 
 DIFFERENTIAL AND INTEGRAL 
 CALCULUS 
 
 Derivatives and Differentials 1 57 
 
 Maxima and Minima 159 
 
 Expansion in Series 160 
 
 Indeterminate Forms 163 
 
 Curvature 163 
 
 Table of Indefinite Integrals 164 
 
 Definite Integrals 169 
 
 Differential Equations 171 
 
 GRAPHICAL REPRESENTATION OF 
 FUNCTIONS 
 
 173 
 174 
 176 
 177 
 178 
 182 
 
 Equations Involving Two Variables 
 Equations for Empirical Curves . 
 Logarithmic Cross-section Paper . 
 
 Semi-logarithmic Paper 
 
 Equations Involving Three Variables 
 Equations Involving Four Variables 
 
 VECTOR ANALYSIS 
 
 Vector Analysis 185 
 
 COPYRIGHT, 1916, BY EDWARD V. HUNTINGTON 
 
MATHEMATICS 
 
 BY 
 EDWARD V. HUNTINGTON 
 
 ARITHMETIC 
 
 NUMERICAL COMPUTATION 
 
 Number of Significant Figures. In any engineering computation, the 
 data are ordinarily the results of measurement, and are correct o'nly to a 
 limited number of significant figures. Each of the numbers 3.840 and 
 0.003840 is said to be given "correct to four figures;" the true' value lies in 
 the first case between 3.8395 and 3.8405; in the second case, between 0.0038395 
 and 0.0038405. The absolute error is less than 0.001 in the first case, and 
 less than 0.000001 in the second; but the relative error is the same in both 
 cases, namely, an error of less than "one part in 3840." 
 
 If a number is written as 384000, the reader is left in doubt whether the number of 
 correct significant figures is 3, 4, 5, or 6. This doubt can be removed by writing the 
 number as 3.84 X 10 or 3.840 X 10 s or 3.8400 X 10* or 3.84000 X 10 5 . 
 
 In any numerical computation, the possible or desirable degree of accuracy 
 should be decided on and the computation should then be so arranged that 
 the required number of significant figures, and no more, is secured. Carry- 
 ing out the work to a larger number of places than is justified by the data, is 
 to be avoided, (1) because the form of the results leads to an erroneous impres- 
 sion of their accuracy, and (2) because time and labor are wasted in super- 
 fluous computation. The labor of working with six-place tables is nearly 
 three times as great as that with four-place tables. In computations involv- 
 ing several steps, it is desirable to retain one extra figure until just before the 
 final result is reached, in order to protect the last figure against the possible 
 cumulative effect of small tabular errors. In discarding superfluous 
 figures, if the first discarded figure is 5 or more, increase the preceding 
 figure by 1. Thus, 3.14159, written 'correct to four figures, is 3.142; correct 
 to three figures, 3.14. Again, 6.1297, correct to four figures, is 6.130. 
 
 Addition. In adding numbers, note that a doubtful final . 2056x 
 
 figure in any one number will render doubtful the whole col- 2 . 572xx 
 
 umn in which that figure lies; hence all figures to the right of 14.25xxx 
 
 that column are superfluous, and contribute nothing to the 576.1xxxx 
 
 accuracy of the result. 
 
 Subtraction. The "Austrian" or "shop" method is 593.1 
 
 recommended. The mental process is as follows, the figures here printed in 
 boldface type being the only ones written down: 
 
 [3 plus how many is 12?] 3 plus 9 is 12; 1 to carry. 14752 
 
 [7 plus how many is 15?] 7 plus 8 is 15; 1 to carry. ,J^5, 
 
 5 plus 2 is 7. 8 plus 6 is 14. 6289 
 
 88 
 
NUMERICAL COMPUTATION 89 
 
 This method is especially useful when it is desired to subtract from a given 
 number the sum of several other numbers. 
 
 7 plus 1 is 8; plus 5 is 13; plus 9 is 22; 2 to carry. 14752 
 
 5 plus is 5; plus 2 is 7; plus 8 is 15; 1 to carry. 3125~| 
 
 3 plus 1 is 4; plus 1 is 5; plus 2 is 7. 101 
 
 5 plus 3 is 8; plus 6 is 14. _5237-J 
 
 6289 
 
 The use of a wavy line to indicate subtraction is also recommended, as it 
 will minimize the danger of adding when subtraction is intended. 
 
 Multiplication. In long examples in multiplication, 4956 
 
 the arrangement of work here illustrated is recommended, 8372 
 
 since it facilitates the abbreviation of the work by the 39648 
 
 omission, in practice, of all the figures on the right of the 1486 8 
 
 vertical line. 346 92 
 
 The position of the decimal point should be determined 9 912 
 
 by reference'to the first, or left-hand, figures of the numbers, 41492|xxx 
 
 rather than by "pointing off" so-and-so many places from 
 the right-hand end. For the right-hand figures of a number are the least 
 important ones, and in many cases are entirely unknown (especially when 
 the slide rule or a computing machine is used). The mental process for 
 determining the decimal point is as follows: 
 
 (a) If the multiplier is a number like 3.1416, with only one figure preceding 
 the decimal point, think of this number as "a little over 3;" then the product 
 must be "a little over three times the number which is being multiplied;" 
 and this gives the position of the decimal point at once, by inspection. 
 
 (6) If the multiplier is a number like 3141.6 [or 0.000 003 141 6], think of 
 this number as "about 3, with the point moved three places to the right" 
 [or "about 3, with the point moved six places to the left"]; then think what 
 the answer would be if the multiplier were simply "about 3," and shift the 
 decimal point accordingly. 
 
 Multiplication Tables. Crelle's large volume (Berlin, G. Reimer) gives the product 
 of every three-figure number by every three-figure number; Peters's (Berlin, G. Reimer), 
 of every four-figure number by every two-figure number. The smaller table of H. 
 Zimmermann (Berlin, Wm. Ernst) gives the product of every three-figure number by 
 every two-figure number. 
 
 Division. In long division, where the numbers are given 23026)31416(1 
 only approximately, the work can be much abbreviated with- 23026 
 
 out loss of accuracy by "cutting off" one figure of the divisor 2303) 8390(3 
 at each step, instead of "bringing down" a doubtful zero in 6909 
 
 the dividend. Thus, 3.1416 4- 2.3026 = 1.3644. 
 
 To determine the position of the decimal point in a du; 
 problem of fractional division, shift the point (mentally) in 
 both numerator and denominator (the same number of 23) 101(4 
 places in each) until the denominator is a number in the 
 "standard form, " that is, a number with only one figure pre- 2) 9(4 
 
 ceding the decimal point. (This will not change the value 
 of the fraction.) Then estimate the approximate magnitude of the quotient 
 by inspection. Thus: 
 
 0.2718 0.000 2718 
 
 "about 0.000 09" = 0.000 08652; 
 
 3141.6 3.1416 
 
 31.416 31 416. 
 
 0.002718 2.718 
 
 "about 10 000" =11 558. 
 
90 ARITHMETIC 
 
 Reciprocals. The reciprocal of N is 1 /N. Instead of dividing by a long 
 number N, it is often better to multiply by the reciprocal of N. The table 
 of reciprocals on pp. 24-27 gives the reciprocal of any number, correct to 
 four figures. Barlow's Table (Spoil & Chamberlain, New York) gives the 
 reciprocal of every four-figure number correct to seven figures (but with- 
 out facilities for interpolation). The reciprocals of numbers having 
 more than four figures are best found by the use of a large table of 
 logarithms. 
 
 Reciprocals of I + x when x is Small. 
 1/(1 + x) = 1 - x + [error < x 2 , if x is between and 1], 
 
 = 1 x + x z [error < x 3 , if x is between and 1]. 
 1/(1 x) = 1 + x + [error < x 2 + 2z 3 , if x is between and }*], 
 
 = 1 + x + x 2 + [error < x 3 + 2z 4 , if x is between and #]. 
 
 NOTE. l/(o 6) = (l/a)[l/(l )], where x = 6/a. 
 
 Notation by Powers of 10. All questions concerning the position of the 
 decimal point are readily answered if each number is expressed in the "stand- 
 ard form," that is, as the product of two factors, one of which is a number 
 with only one figure preceding the decimal point, while the other is a positive 
 or negative power of 10. Thus, 3.1416 X 10 3 means 3.1416 with the point 
 moved three places to the right, that is, 3141.6. Again, 3.1416 X 10~ 6 means 
 3.1416 with the point moved six places to the left, that is, 0.000 003 1416. 
 This notation by powers of 10 should always be used in dealing with very 
 large or very small numbers. Among electrical engineers its use is very 
 general, even for numbers of moderate size. 
 
 Square Root, (a) If four figures of the root are sufficient, take the 
 answer directly from the table of square roots, pp. 12-15. (6) To obtain a 
 root of six or seven figures from the table, use the formula: VJV = a + 
 [(N ,a 2 )/2a] (approx.), where a is the nearest value of v~N obtainable 
 from the table, with three or four ciphers annexed. Here a 2 must be found 
 exactly, by direct multiplication, so that at least three significant figures 
 of the difference N a 2 shall be known correctly; but this done, the division 
 of N a 2 by 2a should be carried to only three figures (logarithms or slide 
 rule may be used). 
 
 NOTE. The simplest way to obtain any root of a seven-figure number correct to 
 seven figures is to use a seven-place table of logarithms, if such a table is at hand. 
 
 Square Roots of 1 x when x is Small. 
 
 (1 + *)** = 1 + \hx - [error less than ftx 2 if < x < 1] 
 = 1 + Kx - Hx z + [error < H * 3 if < x < 1] 
 
 (1 - x) W = 1 - MX - [error < #c a + Mo* 3 if < x < J4] 
 
 = 1 - )6x - ^z 2 -[error < Ho* 3 + Hex 4 if < x < ft] 
 
 NOTE. Va + b = Va (1 + x)^ t where x = 6/a. 
 
 Cube Root, (a) If four figures of the root are sufficient, take the answer 
 directly from the table of cube roots, pp. 16-21. (6) To obtain a root of 
 six or seven figures from the table, use the formula: %/N = a + [(N o 3 )/3a 2 ] 
 (approx.), where a is the nearest value of $/N obtainable from the table, with 
 three or four ciphers annexed. Here a 3 must be found correct to seven or 
 eight figures, by direct multiplication, so that at least three significant figures 
 of the difference N a 3 shall be known; but this done, the division of N a 8 
 by 3o 2 should be carried to only three or four figures (logarithms or the slide 
 rule may be used). 
 
LOGARITHMS 91 
 
 NOTE. The simplest way to obtain any root of a seven-figure number correct to 
 seven figures is to use a seven-place table of logarithms, if such a table is at hand. 
 
 Cube Roots of 1+x when x is Small. 
 
 (1 + z) H = 1 + MX - [error < H* 2 if < a: < 1], 
 
 = 1 + %x - Kx* + [error < Me* 3 if < * < 1], 
 (1 - x) % = 1 - %x - [error < %x* + Mo* 3 if < x < H], 
 
 = 1 - MX -Jte 2 - [error < Me* 3 + Ms* 4 if < x < ft]. 
 NOTE. 3/a + 6 = ^/a(l + z)**, where x = b/a. 
 
 LOGARITHMS 
 
 Tables of Logarithms. The use of a table of logarithms greatly reduces 
 the labor of multiplication, division, raising to powers, and extracting roots. 
 The table on pp. 42-43 is carried out to four significant figures, and the follow- 
 ing explanations should be sufficient to permit the use of the table readily, 
 even by one without previous experience. For algebraic theory, see p. 113. 
 
 If more than four-figure accuracy is required, recourse must be had to a larger table. 
 Five-place tables are available in great variety; the Macmillan Tables, 1913, are perhaps 
 as convenient as any. If more than five figures are required, use Bremiker's six-place 
 table, or proceed at once to a seven-place table: Schron (Vieweg und Sohn, Braun- 
 schweig); Bruhns; Vega-Bremiker. If extreme accuracy is required, use the eight-place* 
 table by Bauschinger and Peters (Engelmann, Leipzig). Logarithmic paper, see p. 176. 
 
 To Find the Logarithm of Any Given (Positive) Number. 
 
 (a) WHEN THE GIVEN NUMBER is BETWEEN 1 AND 10. 
 
 An inspection of the table on pp. 42-43 shows that as the number increases 
 from 1 to 9.99. . . the logarithm of that number increases continuously from 
 to 0.999. . . For example, log 2.97 = 0.4728; log 2.98 = 0.4742. 
 
 If the given number contains four significant figures, it is necessary to inter- 
 polate between the tabulated values, as follows: 
 
 To find log 2.973, notice that this number is fio of the way from 2.97 to 2.98; 
 hence its logarithm will be (approximately) Mo of the way from 0.4728 to 0.4742. The 
 difference here is 14 units, and iHo of this difference is 4 (to the nearest unit); hence, 
 by adding this 4 to 4728, log 2.973 = 0.4732. This process of interpolating should 
 be performed mentally; the step of finding the tabular difference will be facilitated by 
 a glance at the last column on the right, which gives, for each line of the table, the 
 average of the differences along that line. 
 
 Again, to find log 4.098: From table, log 4.09 = 0.6117; adding 9io of the difference 
 (11), or about 9, gives: log 4.098 = 0.6126. Or better, since 91o f the way forward 
 is equal to Y\Q of the way back, find in table log 4.10 = 0.6128, and subtract Y\Q of 11, 
 or 2, giving log. 4.098 = 0.6126. It should be noted that any interpolated value may 
 be in error by 1 in the last place. 
 
 If the given number contains more than four significant figures, it should 
 be cut down to four figures (see p. 88), since the later figures will not affect 
 the result in four-place computations. 
 
 (6) WHEN THE GIVEN NUMBER is LESS THAN 1 OR MORE THAN 10, it is simply 
 necessary to notice that every such number can be regarded as obtainable 
 from some number between 1 and 10 by merely shifting the decimal point 
 (see p. 90) ; and that according to the rule at the foot of the table, moving 
 the decimal point n places to the right [or left] in the number-column is 
 equivalent to adding n [or n] to the logarithm in the body of the table. 
 
 For example, to find log 2973. Here 2973 = 2.973 X 10 (i.e., 2.973 with the 
 decimal point moved 3 places to the right). From the table, log 2.973 => 0.473i. 
 Hence, log 2973 = 0.4732 + 3, which may be written as 3.4732. 
 
92 ARITHMETIC 
 
 Again, to find log 0.0002973. Here 0.0002973 = 2.973 X 10~* (i.e., 2.973 with the 
 decimal point moved 4 places to the left). From the table, log 2.973 = 0.4732. Hence, 
 log 0.0002973 = 0.4732 - 4. (This may be written as 4.4732, if desired, and is equal 
 of course, to _ 3.5268; this latter form, however, is not convenient in practice.) 
 
 It is thus evident that the logarithm of every positive number may be 
 regarded as consisting of two parts: a decimal fraction, which is always posi- 
 tive (or zero) ; and a whole number, which may be positive, negative, or zero. 
 The fractional part is called the mantissa, and is found from the table ; the 
 whole-number part is called the characteristic, and is determined by 
 inspection. 
 
 To Find the Number Corresponding to a Given Logarithm. 
 
 (a) WHEN THE GIVEN LOGARITHM is A POSITIVE DECIMAL FRACTION (CHARAC- 
 TERISTIC ZERO), simply reverse the process for finding the logarithm of a 
 number between 1 and 10. 
 
 For example, given log N = 0.4732; to find N. In the body of the table it is seen 
 that 0.4732 lies a little beyond 0.4728; hence N must lie a little beyond 2.97. By taking 
 differences it is found that 4728 is in fact #4 of the way from 0.4728 to the next 
 higher logarithm; therefore N must be y\ of the way from 2.97 to the next higher 
 number. But YU of 1 is 0.3 (to the nearest tenth), hence N = 2.973. 
 
 Again, given log N = 0.6126; to find N. Here, 0.6126 is %i of the way from 0.6117 
 to the next higher logarithm; therefore N must be JH \ of the way from 4.09 to the next 
 higher number. But %\ of 1 is 0.8 (to the nearest tenth), hence N = 4.098. 
 
 (6) WHEN THE GIVEN LOGARITHM HAS ANY GIVEN VALUE (CHARACTERISTIC 
 NOT ZERO), proceed as follows: First, be sure the given logarithm is in the 
 "standard form," that is, a positive decimal fraction (mantissa) plus a posi- 
 tive or negative whole number (characteristic). For example, if log N is 
 originally given in the form log N = 3.5268, tfiis must first be reduced to 
 the (equivalent) form log N = 0.4732 4 (or 4.4732), before entering the 
 table. Having the logarithm given in the standard form, suppose for the 
 moment that the characteristic is zero, and find in the table the number 
 corresponding to the given mantissa; then move the decimal point to the right 
 or left according as the value of the characteristic is positive or negative. 
 
 For example, given log N = 0.4732 + 3; to find N. From the table, the number 
 corresponding to 0.4732 is 2.973. The characteristic ( + 3) directs that the decimal 
 point be moved 3 places to the right; hence N = 2.973 X 10 3 = 2973. 
 
 Again, given log N = 0.4732 4; to find N. From the table, the number corre- 
 sponding to 0.4732 is 2.973. The characteristic ( - 4) indicates that the decimal 
 point is to be moved 4 places to the left; hence N = 2.973 X 10~ = 0.0002973. 
 
 The number corresponding to a given logarithm is called its antiloga- 
 rithm. Thus, if log 2973 = 0.4732 + 3, then 2973 = antilog (0.4732 + 3). 
 
 NOTE 1. In most tables of logarithms the decimal point is omitted, the tables being 
 in fact not tables of logarithms, but tables of mantissas. This omission is of no con- 
 sequence to the experienced computer, but is often perplexing to one who makes only 
 occasional use of such tables. 
 
 NOTE 2. Many computers prefer to write negative characteristics in the form of some 
 positive number minus some multiple of 10; thus, 0.4732 4 = 6.4732 10; 
 0.4732 - 13 = 7.4732 - 20; etc. 
 
 Fundamental Properties of Logarithms. The usefulness of logarithms 
 in computation depends on the following properties: 
 
 (1) log (a&) = log a + log 6; (3) log (a 71 ) = n log a; 
 
 (2) log (a/6) = log a - log 6; (4) log \/a = (1/n) log a; 
 
 (5) log 10 n = n 
 It is to be noted also that log 1 = 0, log 10 =1, and log (1/n) = log n. 
 
LOGARITHMS 93 
 
 To Multiply by Logarithms. Find from the table the log. of each factor, 
 and add; the result will be the log. of the product. Then find the product 
 itself from the table. 
 
 EXAMPLE". To find log 4.098 = 0.6126 
 
 x - (4.098) (0.0002973) (72.1). log 0.0002973 - 0.4732 - 4 
 
 Answer: x = 8.784 X lO"' lg 72.1 - 0.8579 -f 1 
 
 = 0.08784 log x = 1.9437 - 3 - 0.9437 - 2. 
 
 To Divide by Logarithms. First Method: Find from the table the 
 log. of the numerator and the log. of the denominator, and subtract the second 
 from the first; the result will be the logarithm of the quotient. Then find the 
 quotient itself from the table. 
 
 4.098 log 4.098 - 0.6126 
 
 EXAMPLE. To find x = ^^ ^ ^^ = ^732^ 
 
 Answer: x - 1.378 X 10 = 13780 log x - 0.1394 -f 4 
 
 In order to avoid negative mantissas in cases where a larger mantissa 
 would have to be subtracted from a smaller, modify the upper logarithm by 
 adding and subtracting 1. 
 
 0.0291 log 0.0291 - 0.4639 - 2 = 1.4639 - 3 
 
 EXAMPLE. To find x = _ _ 
 
 Answer: x = 4.590 X 10~ log 3 "*"- 06618^4 
 
 = 0.0004590. 
 
 But if the logarithms are written with the characteristics in front, and the "shop 
 method" of subtraction is used (see p. ] O g Q 0291 = 24639 
 
 88), then no such special device is here j og 3 4 ,_, j g021 
 
 required. Thus: T~ 
 
 log x = 4.6618 
 
 To Divide by Logarithms. Second Method: Instead of subtracting 
 the log. of a number, it is often convenient to add the cologarithm of that 
 number; the colog. of N being defined by: colog N = log (l/N) = log N. 
 
 To find the colog. of a number, write the log. of the number in the stand- 
 ard form, and subtract it from 1.0000 1, as in the following examples: 
 
 1.0000 - 1 1.0000 - 1 
 
 log 69.5 = 0.8420 + 1 log 0.0002973 = .4732_-j4 
 
 colog 69.5 = 0.1580 - 2 colog 0.0002973 = 0.5268 + 3 
 
 This subtraction should be performed mentally. Thus, to subtract the mantissa, 
 subtract each digit from 9 until the last non-zero digit is arrived at, and subtract this 
 from 10; to subtract the characteristic, follow the regular rule of algebra ("reverse the 
 sign and add"). Hence, if the logarithm itself is already written down, or can be read 
 off from the table without interpolation, the cologarithm can be written down at once, 
 by inspection. The use of cologarithms is not essential in logarithmic computation, but 
 it often facilitates a compact arrangement of the work, especially in cases where the 
 denominator of a fraction is itself the product of two or more factors. 
 
 To Find the nth Power of a Number by Logarithms. Find from the 
 table the log. of the number, and multiply it by n; the result will be the 
 logarithm of the nth power of that number. Then find the power itself from 
 the tables. 
 
 EXAMPLE 1. Find x = (0.0291) s log 0.0291 = 0.4639 - 2 
 
 Answer: x = 2.464 X 10~ 3 
 
 = 0.00002464. log x = 1.3917 - 6 - 0.3917 - 5. 
 
94 ARITHMETIC 
 
 EXAMPLE 2. Find x = (0.0291)i'i log 0.0291 = 0.4639 - 2 = - 1.5361 
 
 Answer: x = 6.825 X 10~ 8 1.41 
 
 = 0.006825 15361~ 
 
 61444 
 
 15361 
 
 logs = - 2.1659 
 = 0.8341 - 3 
 
 To Find the nth Root of a Number by Logarithms. Find from the 
 table the log. of the number, and divide it by n; the result will be the log. of 
 the nth root of that number. Then find the root itself from the table. 
 
 EXAMPLE. Find x = ^/4.098 log 4.098 = 0.6126 
 
 Answer: x = 1.600 log x = 0.2042 
 
 In order to avoid fractional characteristics, if the characteristic is 
 not divisible by n, make it so divisible by adding and subtracting a suitable 
 number before dividing. 
 
 EXAMPLE. Find x = VO-0004590. log 0.0004590 = 0.6618 - ' 
 
 Answer: x = 7.714 X 10-2 3)2.6618 - 6 
 
 = 0.07714 log x = 0^8873 - 2 
 
 But if the characteristic is positive, it is simpler to write it in front of the mantissa, 
 and then divide directly. 
 
 THE SLIDE RULE 
 
 The slide rule is an indispensable aid in all problems in multiplication, 
 division, proportion, squares, square roots, etc., in which a limited degree 
 of accuracy is sufficient. The ordinary 10-in. Mannheim rule (see below) 
 costs $3 to $4.50 and gives three significant figures correctly; the 20-in. 
 rule ($12.50) gives from three to four figures; the Fuller spiral rule ($30) 
 or the Thacher cylindrical rule ($35) gives from four to five figures. For 
 many problems the slide rule gives results more rapidly than a table of loga- 
 rithms; it requires, however, more care in placing the decimal point in the 
 answer. In all work with the slide rule, the position of the decimal point 
 should be determined by inspection (see p. 89), only the sequence of digits 
 being obtained from the instrument itself. Rapidity in the use of the in- 
 strument depends mainly on the skill with which the eye can estimate the 
 values of the various divisions on the scale; expertness in this respect comes 
 only with practice. The following explanations should be sufficient to per- 
 mit the use of the ordinary slide rule successfully without previous experience 
 and without knowledge of logarithms. 
 
 Multiplication and Division with a (Theoretical) Complete Loga- 
 rithmic Scale. Consider a complete logarithmic scale (D, Fig. 1), assumed 
 to extend indefinitely in both directions, only the main section, from 1 to 
 10, however, being usually available. Note that the divisions within the 
 several sections are indentical, except that the numeral attached to each divi- 
 sion of any one section is ten times the numeral attached to the corresponding 
 division in the preceding section. [The distances laid off from 1 are propor- 
 tional to the logarithms of the corresponding numbers, the distance from 1 to 
 10 being taken as unity.] Consider also a duplicate scale, C, numbered from 
 1 to 10, and arranged to slide along the fixed scale D as in the figures. By 
 means of such a scale D, and slide C, any two numbers between 1 and 10 
 (and hence any two numbers whatever, with proper attention to the decimal 
 point) can be multiplied or divided, as in the following examples. 
 
THE SLIDE RULE 95 
 
 To MULTIPLY 4 BY 6. In Fig. 1, starting with point 1 of the fixed scale, 
 run the eye along from 1 to 4; then set the 1 of the slide opposite this point 
 4, and run the eye forward along the slide from 1 to 6 ; the point thus reached on 
 the fixed scale is 24, which is equal to 4 X 6. This process gives the distance 
 from 1 to 4 plus the distance from 1 to 6, and is, in fact, a mechanical method 
 of adding the logarithms of these numbers; hence the result is the product 
 of the numbers. Conversely, 
 
 4 5 6 7 891 
 
 FIG. 1. 
 
 To DIVIDE 4 BY 6. In Fig. 2, starting with the point 1 of the fixed scale, 
 run the eye along from 1 to 4; then set the 6 of the slide opposite the point 4, 
 and run the eye backward along the slide from 6 to 1 ; the point thus reached on 
 the fixed scale is 0.667, which is equal to 4 -r- 6. This process gives the dis- 
 tance from 1 to 4 minus the distance from 1 to 6; and is, in fact, a mechanical 
 method of subtracting the logarithms of these numbers; hence the result is 
 their quotient. 
 
 3 4 
 
 5 6 7 8 9 l|0 _______ 20 ____ jQ_ 40 50 60 70 l-O 
 
 ____________ ^i 4 + 6 667 
 
 FIG. 2. 
 
 Multiplication and Division, Using Only a Single Section of the 
 Scale. If only the main section of scale D is available (as is usually the case 
 in practice), the result of multiplication may fall beyond the scale, as it does 
 in. Fig. 1. In such cases divide the first factor by 10 before beginning to multiply; 
 this will bring the result within the scale, without affecting the sequence of 
 digits. 
 
 For example, to multiply 4 by 6. Having found that the setting shown in Fig. 1 
 is not successful, reset the slide as in Fig. 3, with 10 instead of 1 opposite 4; run the eye 
 backward along the slide from 10 to 1, thus reaching the (unrecorded) point correspond- 
 ing to 4 -j- 10; then, continuing from "this point, run the eye forward along the slide 
 from 1 to 6, as before; the point finally reached on the main scale is 2.4, which has the 
 Bame sequence of digits as the required value 24. After a little practice, this preliminary 
 step of dividing by 10 will be performed almost intuitively. Whether or not this step 
 is necessary in any given case, can be determined only by trial. 
 
 The general rule for multiplication may be stated as follows, if pre- 
 ferred: To find the product of two factors, find one factor on the fixed scale; 
 opposite this, set (tentatively) point 1 of the slide; on the slide find the sec- 
 ond factor, and opposite this read the product on the main scale, if possible. 
 If the product falls beyond the scale, begin over again, using point 10 of the 
 slide instead of point 1. 
 
 In division also, the result may fall beyond the main section of the scale, 
 as it does in Fig. 2. In such cases, it suffices merely to multiply the result 
 by 10 in order to bring it within the scale; this will not affect the sequence of 
 digits. 
 
96 
 
 ARITHMETIC 
 
 For example, to divide 4 by 6, set the slide as in Fig. 4, and follow out mentally the 
 steps indicated by the arrows. It will be noticed that the supplementary step of multi- 
 plying by 10 is performed by simply running the eye along the slide from 1 to 10 without 
 resetting the slide; for this reason, division on the slide rule is slightly easier than 
 multiplication. 
 
 5 6 7 6 9 l|p 
 
 i 3 4 56769 l|o 
 
 -*j 4(*tO)x6'U 
 
 FIG. 3. 
 
 FIG. 4. 
 
 The Ordinary Mannheim Slide Rule has four scales, A, B, C, D, as 
 shown in Fig. 5. Scales. C and D are essentially the same as the C and D 
 scales described above, and the principle just explained shows how they are 
 used in multiplication and division. The fact that the D scale covers only the 
 main section from 1 to 10 (all decimal points being omitted) is practically no 
 restriction on the scope of the scale, as is seen in the preceding examples. 
 A runner is provided, so that intermediate positions reached in the course 
 of an extended computation may be indicated temporarily on the scale without 
 the necessity of reading off their numerical values. The best runners are 
 those which have no side frame to obscure the numerals. 
 
 f A 1 
 
 W 
 
 FIG. 5. 
 
 In problems involving successive multiplications and divisions, arrange 
 the work so that multiplication and division are performed alternately. 
 X & X c 
 
 For example, to calculate 
 
 -, divide the product a X & by d; multiply this 
 
 dX e 
 
 quotient by c; and divide this product by e. Each operation will require only one shift- 
 ing either of the slide (for multiplication) or of the runner (for division). 
 
 To multiply a number of different quantities by a constant multiplier, x, set 
 the point 1 of slide opposite x, and read, by aid of the runner, the prod- 
 ucts of x by all the quantities which do not fall beyond the scale; then reset 
 the slide, setting 10 instead of 1 opposite x, and read the products of x by all 
 the remaining quantities. 
 
 To divide a number of different quantities by a constant divisor, y, first 
 find (by the slide rule) the quotient 1 -i- y, and then use this as a constant 
 multiplier. 
 
 Scales A and B are exactly like scales C and D, except that they cover two 
 sections of the complete logarithmic scale, the graduations being only half 
 as fine. Either pair of scales may be used for multiplication and division; 
 C and D give more accurate readings, but have the disadvantage that in the 
 case of multiplication the slide must often be shifted to the other end in order 
 to keep the result on the scale an inconvenience which is not present when 
 the less accurate scales A and B are employed. 
 
 By the use of both pairs of scales, problems in squares and square roots 
 may be readily solved; for every number on A, except for the decimal point, 
 is the square of the number directly below it on D (use the runner). 
 
COMPUTING MACHINES-, FINANCIAL ARITHMETIC 97 
 
 A scale of sines, tangents, and logarithms is often printed on the back of 
 the slide. For further details concerning the use of the slide rule in various 
 problems, see the instruction books furnished with each instrument: Wm. Cox, 
 "Manual of the Mannheim Slide Rule;" F. A. Halsey, "Manual of the Slide 
 Rule;" etc. 
 
 Other Types of Slide Rules. The duplex slide rule ($5 to $18 according 
 to length) shows on one face the regular A, B, C, D scales, and on the other face the 
 scales A, B', C', D (where B' and C' are the same as B and C, only numbered in the re- 
 verse order), with a runner encircling the whole scale. This arrangement makes 
 possible the solution of more complicated problems with fewer settings of the slide, but 
 if the rule is to be used only for simple problems, the multiplicity of scales is rather con- 
 fusing. Less complicated is the polyphase rule, which is like a Mannheim rule with 
 the addition of a single inverted scale, C", printed in the middle of the slide. The log 
 log duplex slide rule (10 in., $8) is especially adapted for handling complex problems 
 involving fractional powers or roots, hyperbolic logarithms, etc. A number of circular 
 slide rules are on the market, the best of which are operated by a milled thumbnut, 
 like the stem wind of a watch. The advantage of the circular rule, aside from its com- 
 pact size (some models are scarcely larger than a watch), lies in the fact that the scale 
 is endless, so that the slide never has to be reset in order to bring the result within the 
 scale. A disadvantage is found in the necessity of reading the figures in oblique positions, 
 or else continually turning the instrument as a whole in the hand. The Fuller and 
 Thacher rules already mentioned are invaluable for problems requiring greater accuracy 
 than can be obtained with the ordinary rules. There are also many special slide rules, 
 adapted to various special types of computation, such as calculating discharge of water 
 through pipes, horse power of engines, dimensions of lumber, stadia measurements, etc. 
 One of the most recent devices of this kind is the Ross meridiograph (L. Ross, San 
 Francisco, Cal.), which is a circular slide rule for solving certain cases of spherical 
 triangles. The Eichhorn trigonometrical slide rule solves any plane triangle. 
 
 COMPUTING MACHINES 
 
 For certain purposes computing machines have ceased to be luxuries and 
 have become almost necessities; but they are expensive, and should be selected 
 with reference to the special work which is to be done. The machines may 
 be classified roughly into three groups, as follows: 
 
 Adding Machines, Non-listing. Of the machines of this kind, the most convenient 
 in the hands of a careful operator is the well-known Comptometer (Felt & Tarrant Co., 
 Chicago, 111.; $250 to $350 according to size), or the recent Burroughs non-listing 
 adding machine (Detroit, Mich., $175). To add a number, simply press a key in 
 the proper column; the result appears on the dials in front of the keyboard. Multi- 
 plication as well as addition can be performed on this machine with great rapidity, 
 and division also after a little practice. Weight, about 15 Ib. Much less rapid, but 
 less expensive and requiring somewhat less skill in operation, is the Barrett adding 
 machine (Philadelphia, Pa.) with multiplying attachment. Other key-operated 
 machines are the Mechanical Accountant (Providence, R. I.), and the Austin 
 (Baltimore, Md.). The American adding machine (American Can Co., Chicago, 
 111.; $39,50) is operated by pulling up a finger-lever for each digit. Small machines, 
 operated by the use of a stylus, are the Rapid computer (Benton Harbor, Mich., $25); 
 the Gem (Automatic Adding Machine Co., New York; $10), the Arithstyle (New 
 York, $36) and the Triumph (Brooklyn, N. Y., $35). These machines, while much 
 less rapid than the key-operated machines, are useful in simple addition. The Under- 
 wood typewriter is now supplied with a complete electrically driven adding 
 machine attached, and the Wahl adding attachment is supplied on the Rem- 
 ington and other typewriters. Ray Subtracto-Adder (Richmond, Va. f $25). 
 
 Adding and Listing Machines. The machines of this group not only add, but also 
 print the items, totals and sub-totals. The Burroughs (Detroit, Mich.), the Wales (Ad- 
 der Machine Co., Wilkes-Barre, Pa.), the Comptograph (Chicago, 111.) and the White 
 (New Haven, Conn.), resemble each other in having an 81-key keyboard; the Dalton 
 (Cincinnati, Ohio) and the Commercial (White Adding Machine Co., New Haven, 
 
98 ARITHMETIC 
 
 Conn.) have a 10-key and a 9-key keyboard respectively, admitting of operation by the 
 touch method. On all these machines, in order to add a number, first depress the proper 
 keys and then pull a handle (or, in the case of electrically driven machines, press a 
 button) to record the item. Multiplication cannot be performed conveniently, except 
 on the Dalton. Subtraction can be performed only by adding the complement, except on 
 the Commercial and on one type of the Burroughs. The prices range from $125 to 
 $600, according to size and style, new models being constantly devised for special com- 
 mercial purposes. A new and more portable machine of the 81-key type is the Barrett 
 adding and listing machine (Philadelphia, Pa., $250). A cheaper machine, with a 10- 
 key keyboard, is the Standard (St. Louis, Mo.). The new American adding and 
 listing machine (American Can Co., Chicago, 111.), operated by pulling up a finger-lever 
 for each digit, costs only $88. The Ellis (Newark, N. J.) is an elaborate adding and 
 listing machine having a complete typewriter incorporated with it. The Elliott-Fisher 
 bookkeeping machine (Harrisburg, Pa.) and the Moon-Hopkins billing ma- 
 chine (St. Louis, Mo.) are intended primarily for commercial use; the latter is a com- 
 plicated electric machine ($750) which combines many of the features of an adding and 
 listing machine with those of a calculating machine. 
 
 Calculating Machines (so-called). Machines of this third group are intended 
 primarily for multiplication and division; the types which have a keyboard can be 
 used effectively for addition and subtraction also. They are all non-listing. The 
 earliest commercially successful types were the Thomas and the Brunsviga. In both 
 these types the multiplicand is set up by moving pegs in slots, or (in the newest 
 models) by depressing keys, and the multiplication is effected by turning a handle for 
 each digit of the multiplier twice for a digit 2, three times for a digit 3, etc. ; the result 
 then appears on the dials. In the Thomas type the handle always turns in the same di- 
 rection, the change from multiplication to division being effected by a shift key. In the 
 Brunsviga type the handle is turned forward for multiplication and backward for divi- 
 sion. Among the best examples of the Thomas type now on the American market are 
 the Tim, with a single row of dials, the Unitas, with a double row of dials (both sold 
 by Oscar Miiller Co., New York City; also with keyboard and electric drive), and the 
 Reuter (Philadelphia, Pa.). Prices, $300 upward. Another machine of this type, 
 with keyboard, is the Record (U. S. Adding Machine Co., New York City). The 
 Brunsviga is represented by Carl H. Reuter, Philadelphia, Pa. ; various models. Of 
 somewhat similar type are the Triumphator (New York City; $250), and Colt's 
 calculator (Culmer Engineering Co., New York City). A new machine, on the same 
 principle, but with keyboard, is the Monroe (made in Orange, N. J.; $250). The 
 Millionaire (W. A. Morschhauser, New York City; $400), is from the mechanical point 
 of view, the only true multiplying machine on the market (except the Moon-Hopkins). 
 After the multiplicand is set up on the pegs, the digits of the multiplier are indicated 
 successively by moving a pointer, the handle being turned only once for each digit. 
 Further, the movement of the carriage is automatic. The newest models have key- 
 board and electric drive. The Ensign electric calculating machine (Boston, Mass. ; 
 $400) is a new machine with an 81-key keyboard on which it adds like an adding 
 machine, and a secondary 10-key keyboard by means of which it multiplies and 
 divides quite as rapidly as any of the calculating machines, the proper key being 
 pressed just once for each digit of the multiplier. The National calculator (New 
 York), and the Lamb calculator (Calculator Mfg. Co., New York) are less ex- 
 pensive machines devised for figuring payrolls and labor costs. A still simpler device 
 for the same purpose is the Calculacard (New York). The machine called the 
 Calculagraph (New York) is a time clock which automatically computes labor costs. 
 
 For graphical methods of computation, see pp. 106, 119, 170, 173-185. 
 
 FINANCIAL ARITHMETIC 
 
 For the facts which are commonly required in regard to compound interest, 
 sinking funds, etc., see the headings of the tables on pp. 64-68. 
 
ELEMENTARY GEOMETRY AND MENSURATION 
 
 GEOMETRICAL THEOREMS 
 
 (For geometrical constructions, see p. 101) 
 
 Right Triangles, a 2 + b 2 = c 2 . (See Fig. 1). /A-f-^=90. 
 2? 2 = mn. a 2 = me. 6 2 = nc. See also p. 105 and p. 132. 
 
 Oblique Triangles. (See also pp. 105, 134.) Sum of angles = 180. An 
 exterior angle = sum of the two opposite interior angles. (Fig. 1.) 
 
 The medians, joining each vertex with the middle point of the opposite side, 
 meet in the center of gravity G (Fig. 2), which trisects each median. 
 
 The altitudes meet in a point called the orthocenter, 0. 
 
 The perpendiculars erected at the midpoints of the sides meet in a point 
 C, the center of the circumscribed circle. [In any triangle G, O t and C lie 
 in line, and G is two-thirds of the way from O to C.] 
 
 FIG. 1. 
 
 FIG. 2. 
 
 The bisectors of the angles meet in the center of the inscribed circle (Fig. 3). 
 The largest side of a triangle is opposite the largest angle; it is less than 
 the sum of the other two sides, and greater than their difference. 
 
 FIG. 3. 
 
 FIG. 4. 
 
 Similar Figures. Any two similar figures, in a plane or in space, can be 
 placed in "perspective," that is, so that straight lines joining corresponding 
 points of the two figures will pass through a common point (Fig. 4). That is, 
 of two similar figures, one is merely an enlargement of the other. Assume 
 that each length in one figure is k times the corresponding length in the other; 
 then each area in the first figure is k 2 times the corresponding area in the second, 
 and each volume in the first figure is fc 3 times the corresponding volume 
 in the second. If two lines are cut by a set of parallel lines (or parallel planes) , 
 the corresponding segments are proportional. 
 
 The Circle. (See also pp. 106, 137.) An angle inscribed in a semicircle 
 is a right angle (Fig. 5). An angle inscribed in a circle, or an angle between 
 a chord and a tangent, is measured by half the intercepted arc (Fig. 6). An 
 angle formed by any two lines which meet a circle is measured by half the 
 sum or half the difference of the intercepted arcs, according as the point of 
 intersection of the lines lies inside (Fig. 7) or outside the circle (Fig. 8). 
 
 A tangent is perpendicular to the radius drawn to the point of contact. 
 
 If a variable line through A (Figs. 9 and 10) cuts a circle in P and Q, then 
 
100 
 
 ELEMENTARY GEOMETRY AND MENSURATION 
 
 AP X AQ is constant; in particular, if A is an external point, AP X AQ 
 - AT 2 , where AT is the tangent from A. 
 
 T 
 
 FIG. 5. 
 
 Fio. 
 
 FIG. 7. 
 
 FIG. 8. 
 
 FIG. 9. FIG. 10. 
 
 The radical axis (Fig. 11) of two circles is a straight line such that the 
 tangents drawn from any point of this line to the two circles are of equal 
 length. If the two circles intersect, the radical axis passes through their 
 points of intersection. In any case, the radical axis bisects the common 
 tangents of the two circles. The three radical axes of a set of three circles 
 meet in a common point. (For equations, see p. 137.) 
 
 FIG. 
 
 Dihedral Angles. The dihedral angle between two planes is measured 
 by a plane angle formed by two lines, one in each plane, perpendicular to the 
 edge (Fig. 12). (For solid angles, see p. 110.) 
 
 In a tetrahedron, or triangular pyramid, the four medians, joining each 
 vertex with the center of gravity of the opposite face, meet in a point, the 
 center of gravity of the tetrahedron; this point is % of the way from any 
 vertex to the center of gravity of the opposite face. The four perpendiculars 
 erected at the circumcenters of the four faces meet in a point, the center of 
 the circumscribed sphere. The four altitudes meet in a point called the 
 orthocenter of the tetrahedron. The planes bisecting the six dihedral 
 angles meet in a point, the center of the inscribed sphere. 
 
 FIG. 12. 
 
 FIG. 
 
 FIG. 14. FIG. 15. 
 
 FIG. 16. 
 
 FIG. 17. 
 
 Regular Polyhedra (see also p. 110): Regular tetrahedron (Fig. 13), 
 bounded by four equilateral triangles; cube (Fig. 14), bounded by six squares; 
 octahedron (Fig. 15), bounded by eight equilateral triangles; dodecahedron 
 (Fig. 16), bounded by twelve regular pentagons; icosahedron (Fig. 17), 
 bounded by twenty equilateral triangles. Figs. 13-17 show how these solids 
 can be made by cutting the surface out of paper and folding it together. 
 
 The Sphere. (See also p. 109.) If AB is a diameter, any plane perpen- 
 dicular to AB cuts the sphere in a circle, of which A and B are called the 
 poles. A great circle on the sphere is formed by a plane passing through 
 the center. A spherical triangle is bounded by arcs of great circles (see p. 
 
GEOMETRICAL CONSTRUCTIONS 101 
 
 134). In two polar triangles, each angle in one is tntf fcuppfeme'nt'of the 
 corresponding side in the other. In two symmetrical triangles, the sides and 
 angles of one are equal to the corresponding sides and angles of the other, 
 but arranged in the reverse order (like right-handed and left-handed gloves). 
 
 GEOMETRICAL CONSTRUCTIONS 
 
 To Bisect a Line AB (Fig. 18). (a) From A and B as centers, and with 
 equal radii, describe arcs intersecting in P and Q, and draw PQ, which will 
 bisect AB in M. 
 
 (6) Lay off AC = BD = approximately half of AB, and then bisect CD. 
 
 To Draw a Parallel to a Given Line 1 Through a Given Point A (Fig. 19). 
 With point A as center draw an arc just touching the line Z; with any point 
 O of the line as center, draw an arc BC with the same radius. Then a line 
 through A touching this arc will be the required parallel. Or, use a straight 
 edge and triangle. Or, use a sheet of celluloid with a set of lines parallel to 
 one edge and about y\ in. apart ruled upon it. 
 
 >Q 7-X (a.) (b.) 
 
 FIG. 18. FIG. 19. FIG. 20. 
 
 To Draw a Perpendicular to a Given Line from a Given Point A 
 Outside the Line (Fig. 20). (a) With A as center, describe an arc cutting 
 the line in R and S, and bisect RS in M . Then M is the foot of the perpen- 
 dicular. (&) If A is nearly opposite one end of the line, take any point B 
 of the line and bisect AB in O ; then with O as center, and OA or OB as radius, 
 draw an arc cutting the line in M . Or, (c) use a straight edge and triangle. 
 
 /?' P ^5 F*^ ~7Q P 4 B 
 
 FIG. 21. FiGT22. FIG. 23. 
 
 To Erect a Perpendicular to a Given Line at a Given Point P. 
 
 (a) Lay off PR = PS (Fig. 21), and with R and S as centers draw arcs inter- 
 secting at A. Then PA is the required perpendicular. (6) If P is near the 
 end of the line, take any convenient point O (Fig. 22) above the line as center, 
 and with radius OP draw an arc cutting the line in Q. Produce QO to meet 
 the arc in A ; then PA is the required perpendicular, (c) Lay off PB = 4 
 units of any scale (Fig. 23) ; from P and B as centers lay off PA = 3 and 
 BA = 5; then APE is a right angle. 
 
 To Divide a Line AB into n Equal Parts (Fig. 24). Through A draw 
 a line AX at any angle, and lay off n equal steps along this line. Connect 
 the last of these divisions with B, and draw parallels through the other divi- 
 
102 
 
 ELEMENTARY GEOMETRY AND MENSURATION 
 
 cdori*. These parallels wili divide the given line into n equal parts. A similar 
 method may be used to divide a line into parts which shall be proportional 
 to any given numbers. 
 
 
 / 
 
 / 
 
 ^u 
 
 \ /-r 
 
 ,' m 
 
 \ ,,--' Ix., I 
 
 A I 
 
 3 C A A fl 
 
 FIG. 25. FIG. 26. 
 
 FIG. 24. 
 
 To Construct a Mean Proportional (or Geometric Mean) Between 
 Two Lengths, m and n (Fig. 25). Lay off .AB = m and BC = n and 
 construct a semicircle on AC as diameter. Let the perpendicular erected at 
 B meet the circumference at P. Then BP = -\/mn. (See p. 115.) 
 
 To Divide a Line AB in Extreme and Mean Ratio (the "golden sec- 
 tion"). At one end, B, of the given line (Fig. 26), erect a perpendicular, BO, 
 equal to half AB, and join OA. Along OA lay off OP = OB, and along AB 
 lay off AX = AP. Then X is the required point of division; that is, AX 2 = 
 AB XBX. Numerically, AX = ^(\/5 '- 1)(AB) = 0.618(AB). 
 
 To Bisect an Angle AOB (Fig. 27). Lay off OA = OB. From A and B 
 as centers, with any convenient radius, draw arcs meeting in M ; then OM 
 is the required bisector. 
 
 To draw the bisector of an angle when the vertex of the angle is not 
 accessible (Fig. 28). Parallel to the given lines a, 6, and equidistant from 
 them, draw two lines a', &' which intersect; then bisect the angle between a' 
 and 6'. 
 
 -<* PL. 
 
 ,-'Ct' 
 
 FIG. 27. 
 
 FIG. 28. 
 
 To Draw a Line Through a Given Point A and in the Direction of 
 the Point of Intersection of Two Given Lines, when this point of inter- 
 section is inaccessible (Fig. 29). Draw any two parallel lines PQ and P'Q' 
 as in the figure; through P' draw a line parallel to PA, and through Q' draw a 
 line parallel to QA; let these lines intersect in A', and draw the line AA'. 
 This line A A' will (if produced) pass through the intersection of the two 
 given lines. 
 
 To Construct, Approximately, the Length of a Circular Arc (Rankine). 
 In Fig. 30 draw a tangent at A. Prolong the chord BA to C, making AC = 
 H AB. With C as center, and radius CB, . 
 draw arc cutting the tangent in D. Then 
 AD = arc AB, approximately (error about 4 
 min. in an arc of 60 deg.). Conversely, to 
 find an arc AB on a given circle to equal a 
 given length AD, take E one-fourth of the 
 way from A to D, and with E as center and 
 radius ED draw an arc cutting the circum- 
 ference in B. Then arc AB = AD, approxi- 
 mately. 
 
 FIG. 30. 
 
GEOMETRICAL CONSTRUCTIONS 
 
 103 
 
 To Inscribe a Hexagon in a Circle (Fig. 31). Step around the cir- 
 cumference with a chord equal to the radius. Or, use a 60-deg. triangle. 
 
 To Circumscribe a Hexagon 
 About a Circle (Fig. 32). Draw 
 a chord AB equal to the radius. 
 Bisect the arc AB in T. Draw 
 the tangent at T (parallel to AB), 
 meeting OA and OB in P and Q. 
 Then draw a circle with radius 
 OP or OQ and inscribe in it a hex- FIG. 31. FIG. 32. FIG. 33. 
 
 agon, one side being PQ. 
 
 To Inscribe an Octagon in a Square (Fig. 33). From the corners as 
 centers, and with radius equal to half the diagonal, draw four arcs, cutting 
 the sides in eight points. The points will be 
 the vertices of the octagon. 
 
 To Inscribe an Octagon in a Circle. Draw 
 two perpendicular diameters, and bisect each 
 of the quadrant arcs. 
 
 To Circumscribe an Octagon About a 
 Circle. Draw a square about the circle, and 
 draw the tangents to the circle at the points 
 where the circle is cut by the diagonals of the 
 square. 
 
 To Construct a Polygon of n Sides, One A 8 
 
 Side AB being Given (Fig. 34). With A as FIG. 34. 
 
 center and AB as radius, draw a semicircle, 
 
 and divide it into n parts, of which n 2 parts (counting from B) are to be 
 used. Draw rays from A through these points of division, and complete the 
 construction as in the figure (in which n = 7). Note 
 that the center of the polygon must lie in the perpen- 
 dicular bisector of each side. 
 
 To Draw a Tangent to a Cir- 
 cle from an external point A (Fig. 
 35). Bisect AC in M; with M as 
 center and radius MC, draw arc 
 cutting circle in P; then P is the FIG. 35. FIG. 36. 
 
 required point of tangency. 
 
 To Draw a Common Tangent to Two Given Circles (Fig. 36). Let 
 C and c be the centers and R and r the radii (R > r). From C as center, draw 
 two concentric circles with radii R + r 
 and R r; draw tangents to these 
 circles from c; then draw parallels to 
 these lines at distance r. These paral- 
 lels will be the required common tan- 
 gents. 
 
 To Draw a Circle Through Three 
 Given Points A, B, C, or to find the 
 center of a given circular arc (Fig. 37). FIG. 37. FIG. 38. 
 
 Draw the perpendicular bisectors of 
 AB and BC; these will meet in the center, 0. 
 
 To Draw a Circular Arc Through Three Given Points When the 
 Center is not Available (Fig. 38). With A and B as centers, and chord 
 
104 
 
 ELEMENTARY GEOMETRY AND MENSURATION 
 
 AB as radiua, draw arcs, cut by BC in R and by AC in S. Divide R A into 
 n equal parts, 1, 2, 3, . . . Divide BS into the same number of equal parts, 
 and continue these divisions at 1', 2', 3', ... Connect A with 1', 2', 3', . . 
 and B with 1, 2, 3, . . . 
 Then the points of intersec- 
 tion of corresponding lines 
 will be points of the re- 
 quired arc. (Construction 
 valid only when CA = CB.) 
 
 To Draw a Circle 
 Through Two Given 
 Points, A, B, and Touch- 
 ing a Given Line, 1 (Fig. FlQ> 39 FlG . 40 
 39). Let AB meet line I in 
 
 C. Draw any circle through A and B, and let CT be tangent to this circle 
 from C. Along Z, lay off CP and CQ equal toCT. Then either P or Q is the 
 required point of tangency. (Two solutions.) Note that the center of the 
 required circle lies in the perpendicular 
 bisector of AB. 
 
 To Draw a Circle Through One Given 
 Point, A, and Touching Two Given 
 Lines, 1 and m (Fig. 40). Draw the 
 bisector of the angle between I and m, and 
 let B be the reflection of A in this line. 
 Then draw a circle through A and B and 
 touching I (or m}, as in preceding con- 
 struction. (Two solutions.) 
 
 To Draw a Circle Touching Three 
 Given Lines (Fig. 41). Draw the bisec- 
 tors of the three angles; these will meet in 
 the center O. (Four solutions.) The 
 perpendiculars from O to the three lines 
 give the points of tangency. 
 
 To Draw a Circle Through Two Given Points A, B, and Touching 
 a Given Circle (Fig. 42). Draw any circle through A and B, cutting the 
 given circle in C and D. Let AB and CD meet in E, and let ET be tangent 
 from E to the circle just 
 drawn. With E as center, 
 and radius ET, draw an 
 arc cutting the given circle 
 in P and Q. Either P or 
 Q is the required point of 
 contact. (Two solutions.) 
 
 To Draw a Circle 
 Through One Given 
 Point, A, and Touching 
 Two Given Circles (Fig. 
 43). Let S be a center of 
 
 similitude for the two given circles, that is, the point of intersection of two 
 external (or internal) common tangents. Through -S draw any line cutting 
 one circle in two points, the nearer of which shall be called P, and the other 
 in two points, the more remote of which shall be called Q. Through A, P, Q 
 
 FIG. 41. 
 
 FIG. 42. 
 
 FIG. 43. 
 
LENGTHS AND AREAS OF PLANE FIGURES 
 
 105 
 
 draw a circle cutting S A in B. Then draw a circle through A and B and 
 touching one of the given circles (see preceding construction). This circle 
 will touch the other given circle also. (Four solutions.) 
 
 To Draw an Annulus Which Shall Contain a Given Number of Equal 
 Contiguous Circles (Fig. 44). (An annulus is a 
 ring-shaped area enclosed between two concentric 
 circles.) Let R + r and R r be the inner and outer 
 radii of the annulus, r being the radius of each of the 
 n circles. Then the required relation between these 
 quantities is given by r = R sin (180/n), or r = JT IG 44 
 
 (R + r)[sin (180%01/U + sin (180%*)]. 
 
 For methods of constructing ellipses and other curves, see pp. 
 139-156. 
 
 LENGTHS AND AREAS OF PLANE FIGURES 
 
 Right Triangle (Fig. 45). a 2 + 6 2 = c 2 . 
 
 Area = tf ab = Ma 2 cot A = H& 2 tan A = He 2 sin 2 A. 
 
 Equilateral Triangle (Fig. 46). Area = Ma 2 \/3 = 0.43301a 2 . 
 
 FIG. 45. 
 
 a 
 FIG. 47. 
 
 Any Triangle (Fig. 47). s = # (a + b + c), t = H(m x + m 2 
 r =-\/(s a)(s 6)(s c)/s = radius inscribed circle, 
 R = W a/ sin -A = },& /sin B = ^c/sin C = radius circumscribed circle; 
 Area = H base X altitude = \toh = tfab sin C = rs = abc/lR 
 
 = Vs(s -o)(s -&)( - c) = ^ \*(* - mi) (< - 
 
 = r 2 cot ft A cot H B cot ^ C* = 2# 2 sin A sin J5 sin C 
 
 = W { (zi2/2 22/1) + (Z2Z/3 0:32/2) + (zsl/i 12/3) } , where 
 
 (zii 2/i). (2, 1/2), (xs, 1/3) are co-ordinates of vertices. See also p. 134. 
 
 FIG. 48. 
 
 FIG. 49. 
 
 FIG. 50. 
 
 Rectangle (Fig. 48). Area = ab = &D 2 sin u. [u = angle between 
 diagonals D, D.] 
 
 Rhombus (Fig. 49). Area = a 2 sin C = tfDiDz. [C = angle between 
 two adjacent sides; DI, Dz = diagonals.] 
 
 Parallelogram (Fig. 50). Area = bh = ab sin C = MDiD* sin u. [u = 
 angle between diagonals DI andZ) 2 ;Z>i 2 + Z> 2 2 = 2(a 2 + 6 2 )]. 
 
 Trapezoid (Fig. 51). Area = H(a + b)h = WDiDt sin u. [Bases a and b 
 are parallel; u = angle between diagonals DI and D*.] 
 
106 
 
 ELEMENTARY GEOMETRY AND MENSURATION 
 
 sn u = 
 c 
 
 Quadrilateral Inscribed in a Circle (Fig. 52). Area 
 V(s - a)(s - 6)(s - c)(s - d) = tf(ac -+ bd)sin u\ s= ft (a 
 
 Any Quadrilateral (Fig. 53). Area = WDiDi sin u. 
 
 NOTE, a 2 + 6 2 + c 2 + d z = Z>i 2 + D 2 2 + 4m 2 , where m = distance between 
 midpoints of D\ and Dz. 
 
 Polygons. See table, p. 39. 
 
 FIG. 52. FIG. 53. 
 
 Circle. Area = -n-r 2 = ^O = y^Cd 
 Here r = radius, d = diam., C = circumference = 2irr = ird 
 
 Annulus (Fig. 54). Area = ir(R 2 - r 2 ) = 7r(J> 2 - d 2 )/4 
 R' = mean radius = tf(R + r), and 6 = R r. 
 
 Sector (Fig. 55). Area = Mrs = 7rr 2 (^/360) = 
 y%r z rad A, where rad A = radian measure of angle 
 A, and s = length of arc = r rad A (table, p. 44). 
 
 Segment (Fig. 56). Area = J^r 2 (rad A - sin A) 
 = yi[r(s c) + ch], where rad A = radian measure of 
 angle A (table, pp. 34-35, 44). For small arcs, 
 s = J.$(8c' c), where c' = chord of half the are. 
 (Huygens's approximation.) NOTE, c = 2\/h(d h) ; 
 c' = \/dh or d c' 2 /h, where d = diameter of circle ; 
 h =r (1 cos l /*tA), s 2r rad %A. 
 
 Ribbon bounded by two parallel curves (Fig. 57). 
 If a straight line AB moves so that it is always per- 
 pendicular to the path traced by its middle point G, 
 
 FIG. 55. 
 
 = 0.785398d 2 (table, p. 30). 
 (table, p. 28). 
 = 2-n-R'b, where 
 
 FIG. 56. 
 
 FIG. 57. 
 
 then the area of the ribbon or strip thus generated is equal to the length of 
 AB times the length of the path traced by G. (It is assumed that the radius 
 of curvature of G's path is never less than tf AB, so that successive positions 
 of the generating line will not intersect.) 
 
 Simpson's Rule (Fig. 58). Divide the 
 given area into n panels (where n is some 
 even number) by means of n + 1 parallel 
 lines, called ordinates, drawn at constant dis- 
 tance h apart; and denote the lengths of these 
 ordinates by 3/0, 2/i, 2/2, . . , 2/n. (Note that 
 2/o or y n may be zero.) Then 
 Area = ^h[(y + y n ) + 4(yi +2/3+2/5. . ) 
 + 2(2/2 + 2/4 + 2/6- . ) ]. approx. The greater 
 
 FIG. 58. 
 
 the number of divisions, the more accurate the result. Note: Taking y 
 = f(x) , where x varies from x = a to x = 6, and h = (b a) /n, then the 
 
 error = ~ ^ & ~^ f""(X), where f""(X) is the value of the fourth de- 
 
 loO 71 
 
 rivative of f(x) for some (unknown) value, x = X, between a and 6. 
 
SURFACES AND VOLUMES OF SOLIDS 
 
 107 
 
 Ellipse (Fig. 59; see also p. 140). Area of ellipse = trab. Area of shaded 
 segment = xy + ab sin" 1 (x/a). Length of perimeter of ellipse = ir(a + b)K, 
 where K = [1 + Y*m z + H*m 4 + ^sem 6 + ...], m = (a - 6)/(a + 6). 
 Form =0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 
 K= 1.002 1.010 1.023 1.040 1.064 1.092 1.127 1.168 1.216 1.273 
 
 (_.__ a _..J 
 FIG. 60. 
 Hyperbola (Fig. 60; see also p. 144). In any hyperbola, shaded area 
 
 A ab loge ( (-") In an equilateral hyperbola (a = &), area A = 
 
 \a b/ 
 
 o 8 sirih- l (y/d) = o 2 cosh- l (x/d). For tables of hyperbolic functions, see p. 60. 
 Here x and y are co-ordinates of point P. 
 
 Parabola (Fig. 61; see also p. 138). Shaded area A = ftch. In Fig. 62, 
 length of arc OP = s = tyPT + typ log e cot ^u. Here c = any chord; p = 
 semi-latus rectum; PT = tangent at P. Note: OT = OM = x. 
 
 FIG. 61. FIG. 62. 
 
 Other Curves. For lengths and areas, see pp. 147-156. 
 
 SURFACES AND VOLUMES OF SOLIDS 
 
 Regular Prism (Fig. 63). Volume = ^nrah = Bh. Lateral area = 
 nah = Ph. Here n = number of sides; B = area of base; P = perimeter of base. 
 
 Right Circular Cylinder (Fig. 64). Volume = irr 2 h = Bh. Lateral 
 area = lirrh = Ph. Here B = area of base; P = perimeter of base. 
 
 FIG. 63. 
 
 FIG. 64. 
 
 FIG. 65. 
 
 FIG. 66. 
 
 Truncated Right Circular Cylinder (Fig. 65). Volume = -n-r 2 h = Bh. 
 Lateral area = 2wrh = Ph. Here h = mean height = tf(hi + ta) ; B = area 
 of base; P = perimeter of base. 
 
108 
 
 ELEMENTARY GEOMETRY AND MENSURATION 
 
 Any Prism or Cylinder (Fig. 66). Volume = Bh = NL Lateral area 
 = Ql. Here I = length of an element or lateral edge; B = area of base; N = 
 area of normal section; Q = perimeter of normal section. 
 
 Any Truncated Prism or Cylinder (Fig. 67). Volume = Nl. Lateral 
 area = Qk. Here I = distance between centers of gravity of areas of the two 
 bases; k = distance between centers of gravity of perimeters of the two bases; 
 N = area of normal section; Q = perimeter of normal section. For a trun- 
 cated triangular prism with lateral edges a,b,c, I = k = \i(a +6 + c). Note: 
 I and k will always be parallel to the elements. 
 
 FIG. 67. 
 
 FIG. 
 
 FIG. 69. 
 
 FIG. 70. 
 
 Special Ungula of a right circular cylinder. (Fig. 68.). Volume = 
 Lateral area = 2rH. r = radius. (Upper surface is a semi-ellipse.) 
 
 Any Ungula of a right circular cylinder. (Figs. 69 and 70.) Volume = 
 H(Ha 3 c)/(r c) = H[a(r* - ^a 2 ) r 2 crad u]/(r c). Lateral area = 
 H(2ra cs)/(r c) = 2rH(a c rad u)/(r c). If base is greater (less) 
 than a semicircle, use + ( ) sign, r = radius of base; B = area of base; 
 s= arc of base; u= half the angle subtended by arc s at center; rad u = 
 radian measure of angle u (see table, p. 44). 
 
 Hollow Cylinder (right 
 and circular). Volume = 
 7rh(R* -r 2 ) =Trhb(D- b) 
 = Trhb(d + 6) = MD' = 
 irhb (R + r). Here h = 
 altitude; r,R(d,D) = inner 
 and outer radii (diameters) ; 
 b = thickness = R r; 
 D' = mean diam. =ty(d + 
 D) =D -b =d+6. 
 
 Regular Pyramid (Fig. 
 71). Volume = 1$ altitude 
 X area of base ^hran. 
 base = ^san. Here r = radius of inscribed circle; a = side (of regular 
 polygon) ; n number of sides; s = \/r* + h 2 . Vertex of pyramid directly 
 above center of base. 
 
 Right Circular Cone. Volume = H-irr z h. Lateral area = TTTS. Here 
 r =* radius of base; h = altitude; s = slant height = \/r 2 + h 2 . 
 
 Frustum of Regular Pyramid (Fig. 72). 
 
 Volume = Uhran[l + (a'/o) + (a' /)*] 
 
 Lateral area = slant height X half sum of perimeters of bases = slant 
 height X perimeter of mid-section = J /isn(r + r'). Here r,r' = radii 
 
 FIG. 71. 
 Lateral area 
 
 FIG. 72. FIG. 73. 
 
 slant height X perimeter of 
 
SURFACES AND VOLUMES OF SOLIDS 
 
 109 
 
 of inscribed circles; s = V (r r') 2 + ft 2 ; a,a r = sides of lower and upper 
 bases; n = number of sides. 
 
 Frustum of Right Circular Cone (Fig. 73). Volume = 
 
 Lateral area = irs(r + r') ;> = V(r - r') 2 + ft 2 . 
 
 Any Pyramid or Cone. Volume = ftBh. B = area of base; h = perpen- 
 dicular distance from vertex to plane in which base lies. 
 
 Any Pyramidal or Conical Frustum (Fig. 74). Volume = 
 HftCB + VBB~' + B") = XhB[l + (P'/P) + (P'/P) 2 ]- Here B, B' = areas of 
 lower and upper bases; P, P' = perimeters of lower and upper bases. 
 
 FIG. 74. 
 
 FIG. 75. 
 
 FIQ. 76. 
 
 Obelisk (Frustum of a rectangular pyramid. Fig. 75). 
 
 Volume = ^ft[(2a + ai)6 + (2ai + a) 61] = Kh[ab + (a + ai) (6 + 61) + 
 
 Wedge (Rectangular base; ai parallel to a,a and at distance ft above base. 
 Fig. 76). Volume = }6ft6(2a + ai). 
 
 Sphere. Volume = V = ^Trr 3 =. 4.188790r 3 = H^ 3 = 0.523599d 8 (table, 
 p. 36) = % volume of circumscribed cylinder. Area = A 4rnr z = four great 
 circles (table, p. 30) = ird 2 = 3.14159d 2 = lateral area of circumscribed cylinder. 
 Here r = radius; d = 2r = diameter = ^/67/7r = 1.24070 i/7 
 
 0.56419\/Z- 
 
 Hollow Sphere, or spherical shell. Volume = 
 Hir(R 3 r 3 ) = H7r(D 3 d 8 ) = 4irRiH + lint*. Here 
 -R,r = outer and inner radii; D,d = outer and inner 
 diameters; t = thickness = R r; RI = mean radius = 
 
 Spherical Segment of One Base. Zone (spher- 
 ical "cap" of Fig. 78). Volume = H7rft(3o 2 + ft 2 ) = 
 H7rft 2 (3r ft) (table, p. 38). Lateral area (of zone) = 
 27rrft= 7r(a. 2 + ft 2 ). Note: a 2 = ft(2r - ft), where r 
 = radius of sphere. 
 
 Any Spherical Segment. Zone (Fig. 77). Vol- 
 ume = ^?rft(3a 2 + 3ai 2 + ft 2 ). Lateral area (zone) 
 = 27rrft. Here r = radius of sphere. If the inscribed 
 frustum of a cone be removed from the spherical seg- 
 ment, the volume remaining is Mirhc 2 , where c = slant 
 height of frustum = \/ft 2 + (p> ~ i) 2 - 
 
 Spherical Sector (Fig. 78). Volume = W X area 
 Total area = area of cap + area of cone 
 ft(2r - ft). 
 
 
 of cap 
 2irrh + vra. Note: o 2 
 
110 
 
 ELEMENTARY GEOMETRY AND MENSURATION 
 
 (Fig. 
 area 
 
 Spherical Wedge bounded by two plane semicircles and a lune. 
 79.) Volume of wedge -r- volume of sphere = w/360 . Area of lune 
 of sphere = w/360. u = dihedral angle of the wedge. 
 
 Spherical Triangle bounded by arcs of three great circles. (Fig. 80.) 
 Area of triangle = Trr 2 E/l80 = area of octant X #/90. E = spherical 
 excess = 180 (A + B + C), where A, B, and C are angles of the triangle. 
 See also p. 134. 
 
 Solid Angles. Any portion of a spherical surface subtends what is 
 called a solid angle at the center of the sphere. If the area of the given 
 portion of spherical surface is equal to the square of 
 the radius, the subtended solid angle is called a 
 steradian, and this is commonly taken as the unit. 
 The entire solid angle about the center is called a 
 steregon, so that 4r steradians = 1 steregon. A 
 so-called "solid right angle" is the solid angle sub- 
 tended by a quadrantal (or trirectangular) spherical 
 triangle, and a "spherical degree" (now little used) 
 is a solid angle equal to ^o of a solid right angle. 
 Hence 720 spherical degrees = 1 steregon, or TT stera- 
 dians = 180 spherical degrees. If u = the angle 
 
 which an element of a cone makes with its axis, then the solid angle 'of the 
 cone contains 2?r(l cos u) steradians. 
 
 Regular Polyhedra. A = area of surface; V = volume; a = edge. 
 
 Name of solid (see p. 100) Bounded by A/a 2 V/a* 
 
 Tetrahedron ............................... 4 triangles 1.7321 0.1179 
 
 Cube ...................................... 6 squares 6.0000 1.0000 
 
 Octahedron ................................ 8 triangles 3.4641 0.4714 
 
 Dodecahedron ............................ . . 12 pentagons 20. 6457 7. 6631 
 
 Icosahedron .............................. 20 triangles 8.6603 2.1817 
 
 Ellipsoid (Fig. 81). Volume = tynrdbc, where a, b, c = semi-axes. 
 
 Spheroid (or ellipsoid of revolution). The volume of any segment made 
 by two planes perpendicular to the axis of revolution may be found ac- 
 curately by the prismoidal formula (p. 111). 
 
 FIG. 79. FIG. 80. 
 
 FIG. 81. 
 
 FIG. 82. 
 
 FIG. 83. 
 
 FIG. 84. 
 
 Paraboloid of Revolution (Fig. 82). Volume = y^rr^h = ^ volume of 
 circumscribed cylinder. 
 
 Segment of Paraboloid of Revolution (Bases perpendicular to axis, 
 Fig. 83). Volume of segment = ynr(R* + r z }h. 
 
 Barrels or Casks (Fig. 84), Volume = ],i 2 irh(2D 2 + d 2 ) approx. for cir- 
 cular staves. Volume = Wvrh(2D 2 + Dd + ^d 2 ) exactly for parabolic staves. 
 
SURFACES AND VOLUMES OF SOLIDS 
 
 111 
 
 FIG. 85. 
 
 For a standing cask, partially full, compute contents 
 by the prismoidal formula, p. 111. Roughly, the num- 
 ber of gallons, G, in a cask is given by G = 0.0034?i 2 /i, 
 where n = number of inches in the mean diameter, 
 or H(-D + d), and h = number of inches in the height. 
 
 Torus, or Anchor Ring (Fig. 85). Volume = 
 2:r 2 cr 2 . Area = 4vr 2 cr (Proof by theorems of Pappus). 
 
 Theorems of Pappus. 1. Assume that a plane figure, area A, revolves 
 about an axis in its plane but not cutting it; and let s = length of circular 
 arc traced by its center of gravity. Then volume of the solid generated by 
 A is V = As. For a complete revolution, V 2irrA, where r = distance 
 from axis to center of gravity of A. 
 
 2. Assume that a plane curve, length I, revolves about an axis in its plane 
 but not cutting it; and lets = length of circular arc traced by its center 
 of gravity. Then area of the surface generated by I is S = Is. For a 
 complete revolution, S = 2-irrl, where r = distance from axis to center of 
 gravity of I. 
 
 NOTE. If Vi or Si about any axis is known, then Vz or 2 about any 
 parallel axis can be readily computed when the distance between the axes is 
 known. 
 
 Generalized Theorems of Pappus. Consider any curved path of 
 length s. If (1) a plane figure, area A [or (2) a plane 
 curve, length I] moves so that its center of gravity 
 slides along this curved path (Fig. 86), while the 
 plane of A [or I] remains always perpendicular to the 
 path, then (1) the volume generated by A is V = As 
 [and (2) the area generated by I is S Is]. The 
 path is assumed to curve so gradually that successive positions of A [or I] 
 will not intersect. 
 
 The Prismoidal Formula (Fig. 87). Volume =Uh(A +B +.4M), 
 where h = altitude, A and B = areas of bases and M = area of a plane section 
 midway between the 
 bases. This formula is 
 exactly true for any 
 solid lying between two 
 parallel planes and such 
 that the area of a sec- 
 tion at distance x from Fm> g7 FlG> gg 
 one of these planes is 
 
 expressible as a polynomial of not higher than the third degree in x. 
 approximately true for many other solids. 
 
 Simpson's Rule may be applied to finding volumes, i'f the ordinates 
 2/i, 2/2, be interpreted as the areas of plane sections, at constant distance 
 h apart (p. 106). 
 
 Cavalieri's Theorem. Assume two solids to have their bases in the 
 same plane. If the plane section of one solid at every distance x above the 
 base is equal in area to the plane section of the other solid at the same dis- 
 tance x above the base, then the volumes of the two solids will be equal. 
 See Fig. 88. 
 
 FIG. 86. 
 
 It is 
 
ALGEBRA 
 
 FORMAL ALGEBRA 
 
 Notation. The main points of separation in a simple algebraic expres- 
 sion are the + and signs. Thus,, a + b X c d + x -)- y is to be inter- 
 preted as a + (b X c) (d -5- 3) + y. In other words, the range of opera- 
 tion of the symbols X and -i- extends only so far as the next^+ or sign. 
 As between the signs X and -f- themselves, a -f- b X c means, properly speak- 
 ing, a -r- (b X c); that is, the -5- sign is the stronger separative; but this rule 
 is not always strictly followed, and in order to avoid ambiguity it is better 
 to use the parentheses. 
 
 The range of influence of exponents and radical signs extends only over 
 the next, adjacent quantity. Thus, 2ax 3 means 2a(x 3 ), and \^2ax means 
 (\/2) (ax). Instead of \/2ax, it is safer, however, to write \/2'ax, or, bet- 
 
 ter, ox's/2- 
 
 Any expression within parentheses is to be treated as a single quantity. 
 A horizontal bar serves the same purpose as parentheses. 
 
 The notation a-b, or simply ab, means a X b; and a: b, or a/b, means a -f- b. 
 
 The symbol |a| means the "absolute value of a," regardless of sign; thus, 
 |-2| = | + 2| =2. 
 
 The symbol nl (where n is a whole number) is read: "n factorial," and 
 means the product of the natural numbers from 1 to n, inclusive. Thus 
 1! = 1; 2! = 1 X 2; 3! = 1 X 2 X 3; 4! = 1 X 2 X 3 X 4 /etc. 
 
 The symbol ^ or =f means, "not equal to"; means "plus or minus." 
 
 The symbol is som'etimes used for "approximately equal to." 
 
 Addition and Subtraction, a + b = b + a. 
 
 (a + b) + c = a + (b + c). a - ( - b) = a + b. a - a = 0. 
 
 a + (x y + z) = a+x y+z. a (x^ y + z) = a x + y z. 
 A minus sign preceding a parenthesis operates' to reverse the sign of every 
 term within, when the parentheses are removed. 
 
 Multiplication and Simple Factoring, ab = ba. (ab)c = a(bc). 
 a(b -f- c) = ab + ac. a(b c) = ab ac. Also, a X ( b) = ab, and 
 ( o) X ( b) = ab; "unlike signs give minus; like signs give plus." 
 
 (a + 6) (a - 6) = a 2 - b 2 . 
 
 (a + b) 2 ='a 2 + 2ab + b 2 , (a - 6) 2 =a 2 - 2ab + 6 2 . 
 
 (a + &) 3 = o 3 + 3o 2 & + 3a& 2 + 6 3 , (a - 6) 3 = a 3 -3a 2 &+ 3a& 2 - 6 3 ; etc. 
 (See table of binomial coefficients, p. 39; also p. 114.) 
 
 2 _ & 2 = (o - &)(o + 6), a 3 - 6 3 = (o - 6) (a 2 + ab + b 2 ). 
 
 a n - b n = (a - 6) (a 71 - 1 + a n - 2 6 + a n ~ 3 6 2 + . . . + ab n ~ 2 + 6"- 1 ). 
 
 a n + b n is factorable by a + 6 only when n is odd ; thus, 
 
 a a 4. &3 = ( O ._j_ 5)( a 2 _ ab + b 2 ), 
 
 a e + &5 = ( _|_ 6) (o 4 - a 3 b + a 2 b 2 - ab 3 + b 4 ) ; etc. 
 
 The following transformation is sometimes useful : 
 
 f / b \ 2 (Vb^^4a~c\ 2 1 
 ax* + bx+c=a[(x + -) -( ^ ) J. 
 
 ma + mb -\-rnc a+b+c 
 Fractions. If w is not zero, - - = ; that is, 
 
 mx + my x -\- y 
 
 both numerator and denominator of a fraction may be multiplied or divided 
 
 112 
 
FORMAL ALGEBRA 113 
 
 by any quantity different from zero, without altering the value of the 
 fraction. 
 
 To add two fractions, reduce each to a common denominator, and add the 
 
 a x ay bx ay + bx 
 numerators: r + ~ = ; -- H r~ = - ; - 
 
 b y by by by 
 
 a x ax a ax ax 
 
 To multiply two fractions: T X - = r~; T X z = -r X- = ~r. 
 b y by b bib 
 
 To divide one fraction by another, invert the divisor and multiply: 
 o ^5 _ ja V 2/ _ay _a _._ _ a 1 _ ^ 
 
 t ~~ T -^ ~"~ 1 > 1 X /\ 
 
 b y b . x bx b b x bx 
 
 Ratio and Proportion. The notation a:b: :c:d, which is now passing 
 out of use, is read: "a is to & as c is to d," and means simply (a/6) = (c/d), 
 or ad = be. a and d are called the "extremes," 6 and c the "means," 
 and d the "fourth proportional" to a, b, and c. The "mean proportional" 
 between two numbers is the square root of their product; also called the 
 "geometric mean" of the numbers (p. 115). If a/6 =' c/d, then (a -{-&)/& = 
 (c + d)/d, and (a - &)/& = (c d)/d; whence also, (a -\- 6) /(a - 6) = 
 (c+d)/(c-d). I*a/x=b/y=c/z = . . . = r, then 
 
 Variation. The notation x cc y is read: "a; varies directly as y," or "x 
 is directly proportional to y," and means a; = ky, where k is some constant. 
 To determine the constant k, it is sufficient to know any pair of values, as 
 x\ and 2/1, which belong together; then o?i = kyi, and hence x/xi = y/yi, or 
 = (r<i/yi)y. The expression "x varies inversely as y" or "x is inversely 
 proportional to y," means that x is proportional to 1/y, or x = A/y. 
 
 Exponents. o m+n = a w a n . a w ~ n =a m /a n . a = 1 (if a 5^ 0). o~ m = l/a m . 
 (a)* = a"" 1 , a 1 /" = AVa- Thus: a H =_ A/a", and a M = ^ou a w / n - 
 Va^. Thus:a % = Va 2 and a % = Va*. (V^) n = a. (ab) n = a n b n . 
 (a/b) n = a n /b n . ( a) n = a n if n is even. ( a) n = a n if n is odd. 
 If n is positive and increases indefinitely, a n becomes infinite if a > 1, and 
 approaches if a < 1 (a being always positive). Graphs, p. 174; series, p. 160. 
 
 Radicals. Except in the simple cases of square root and cube root, radical 
 signs should always be replaced by fractional exponents: "v/a = a n . 
 (Va) n = (a 1/n ) = a. If n is odd, \/^a = - -\/a' t but if n is even, 
 V a is imaginary. Every positive number a has two square roots, one 
 positive and the other negative; but the notation \/a always means the positive 
 root; thus, v9 = 3;_ V9 = 3. If the denominator of a fraction is of 
 the form va + v b, it is possible to "rationalize the denominator" by 
 multiplying both numerator and denominator by \/a + \/&. Thus: 
 A/a + A/6 = (Va +.Vb)(Va + A/6) _ a + b + 2A/a& 
 A/a - A/6 (A/^ - A/6)(Va + V&) ~ 0-6 
 
 Logarithms. (For the use of logarithms in numerical computation, 
 see p. 91.) The logarithm of a (positive) number N is the exponent of that 
 power to which the base (10 or e) must be raised to produce N. Thus, x 
 = logio N means that 10* = N, and x = Iog 8 N means that e x = N. Loga- 
 rithms to base 10 are called common, denary, or Briggsian logarithms. 
 For table of 4-pIace common logarithms see pp. 40-43. 
 
114 ALGEBRA 
 
 Logarithms to base e are called hyperbolic, natural, or Napierian logar- 
 ithms. Here e = 1 + 1 + 1/2! + 1 /3! + 1/4! + . . . = 2.718281828459. . . 
 For table of 4-place hyperbolic logarithms see pp. 58, 59. 
 
 If the subscript 10 or e is omitted, the base must be inferred from the 
 context, the base 10 being used in numerical computation, and the base e 
 in theoretical work. In either system, 
 
 log (o&) = log a + log b log (a n ) = n log a log = 
 
 log (0/6) = log a log b log (V / a~) = (1/w) log a log 1=0 
 
 log (1/n) = log n log (base) = 1 log co = oo 
 
 The two systems are related as follows: 
 
 M= 0.4342944819 . . .; Iog e 10 = 1/Af = 2.3025850930. . . 
 0.4343 logez; log e x = 2.3026 Iogi z. 
 
 For tables of multiples of M and 1/Af, see p. 62. For graphs of the logar- 
 ithmic and exponential functions, see p. 174; series, p. 160. 
 
 The Binomial Theorem. (For table of binomial coefficients, see p. 
 39 and p. 116.) 
 
 _ . N n(n - 1) n(n - l)(n - 2) 
 
 Let (n)l - n, (n), - - -* (), - 1X2X3 
 
 = --- 
 
 1X2X3X4 
 Then, for any value of n, provided 1 x \ < 1, 
 
 (1 + z) n = 1 + (n)ix + (n) 2 z 2 + (n) 3 * 3 + (n) 4 * 4 + . .. . 
 (If n is a positive integer, the series breaks off with the term in x n , and is 
 valid without restrictions on x, see p. 112.) 
 
 The most useful special cases are the following: 
 
 (bl < i) 
 
 (1 + o:) 3 = (1 + xY* = 1 + 
 
 128 
 
 with corresponding formulae for \/l x, etc., obtained by reversing the 
 signs of the odd powers of x. Also, provided |6| < |a|: 
 
 (a + 6) n = a n ( 1 
 
 where (n)i, (71)2, etc., have the values given above. 
 
 Arithmetical Progression. In an arithmetical progression, a; a + d\ 
 a + 2d] a + 3d; . . ., each term is obtained from the preceding term 
 by adding a constant, called the constant difference, d. If n is the number of 
 terms, the last term is I = a -j- (n l)d; the "average" term is H( + 0{ 
 
FORMAL ALGEBRA 115 
 
 and the sum of the n terms is n times the average term, or S = ftn(a + Z). 
 The arithmetical mean between a and b is (a + b)/2. 
 
 Geometrical Progression. In a geometrical progression, o; or; or 2 ; 
 or 8 ; . . . , each term is obtained from the preceding term by multiplying by a 
 constant, called the constant ratio, r. The nth term is ar n ~ l . The sum 
 of the first n terms is S = a(r n - l)/(r - 1) = a(l - r n )/(l - r). If 
 r is a positive or negative fraction, that is, if 1 < r < +1, then r n will 
 approach zero as n increases, and the sum of n terms will approach o/(l r) 
 as a limit. The geometric mean betweep a and 6 is\/~ab; also called the 
 mean proportional between a and 6 (p. 113; construction, p. 102). 
 
 The harmonic mean between a and 6 is 2ab/(a + b). 
 
 Summation of Certain Series by Second and Third Differences. 
 Let 01, a2, as, . . . On be any series of n numbers, as in the 
 first column of the adjoining scheme. By subtracting 
 each number from the next following, form the column Jo d fd 
 of "first differences," and by repeating this process, form 
 the columns of second, third, etc., differences. If the fc la 
 kth differences are all equal, so that subsequent differ- M & 
 
 ences are all zero, the original series is called an arithme- "27 37 is 
 tical series of the fcth order. In this special case the _ g 1 y -12 
 series can be summed as follows: Denote the numbers 1 i ~ 6 Q 
 which stand at the head of the successive columns of 9 1 9 6 
 differences by />',>", D"', .... Then the nth term of 8 7 ' 
 
 the series is a n , and the sum of the first n terms is S n , 
 where 
 
 (n - 
 
 1 X & 
 
 (n - l)(n -2)(n - 
 
 n(n - 1) , ' n(n-l)(n-2) 
 
 D 
 
 1X2X3 
 
 *>n 
 
 1X2X3 
 
 n(n-l)(n-2)(n-3) , 
 1X2X3X4 
 
 If the series is, for example, of the third order, each of these formulae 
 will stop with the term involving >'"; and only a few terms of the series are 
 required for the computation of the D's. (Differentials, p. 159.) 
 
 Sum of the Squares or Cubes of the First n Natural Numbers. 
 1+2+3 + . . . + (n - 1) + n = %n(n + 1). 
 
 1 2 + 22 + 3 2 + . . . + (n - I) 2 + n 2 = fcn(n + l)(2n + 1). 
 
 1 3 + 2 3 + 3 3 + . . . + (n - I) 3 + n 3 = [j*n(n + I)] 2 . 
 Formula for Interpolation by Second Differences. In any ordinary 
 
 table giving a quantity y as a function of a variable x, let it be required to 
 find the value of y corresponding to a value of x which is not given directly 
 in the table, but which lies between two tabulated values, as x\ and xz. If 
 x = xi + md, where d = xz x\ = the constant interval between two suc- 
 cessive x's, and m is some proper fraction, then the corresponding value of 
 y will be given by the formula 
 
 m(m - 1) m(m - l)(m - 2) 
 
 y = Vl + mD + 1X2 D + - : x 2<x 3 D + . . . 
 
 where D', D", D'", . . are the first, second, third, . . differences in the 
 
116 
 
 ALGEBRA 
 
 series of j/'s which begins with y\ (see above), provided the function is oi 
 such a nature that the differences of higher orders become negligibly small. 
 The coefficients of D', D", D"', ... in the formula are the binomial coeffi- 
 cients for fractional values of m (see following table). The several terms of 
 the formula (with careful attention to sign) are the successive corrections 
 which must be added to y\\ the sum of these corrections should be rounded 
 out to the nearest unit of the last significant place before adding. If D' 
 < 4, the term involving D", and later terms, can be neglected; the formula 
 then reduces to y = y\ + mD', which is the familiar formula for ordinary, 
 or "linear," interpolation. If Z>'" < 8 (or D" " < 12, or D" '" < 16), the 
 term involving D'" (or D" ", or D" '") can be neglected. 
 
 Binomial Coefficients for Fractional Values of m 
 
 m 
 
 (m). 
 
 (m), 
 
 (m) 4 
 
 (m)s 
 
 0.0 
 
 - 0.0000 
 
 0.0000 
 
 -0 0000 
 
 00000 
 
 O.I 
 
 - 0.0450 
 
 0.0285 
 
 - 0.0207 
 
 0.0161 
 
 0.2 
 
 - 0.0800 
 
 0.0480 
 
 - 0336 
 
 0255 
 
 0.3 
 
 -0.1050 
 
 0595 
 
 - 0402 
 
 0297 
 
 0.4 
 
 - 0.1200 
 
 0.0640 
 
 - 0.0416 
 
 0300 
 
 0.5 
 
 - 0.1250 
 
 0.0625 
 
 - 0.0391 
 
 0.0273 
 
 0.6 
 
 - 0.1200 
 
 0560 
 
 - 0.0336 
 
 0.0228 
 
 0.7 
 
 - 0.1050 
 
 0455 
 
 - 0.0262 
 
 0.0173 
 
 0.8 
 
 - 0.0800 
 
 0.0320 
 
 - 0.0176 
 
 0113 
 
 0.9 
 
 - 0.0450 
 
 0.0165 
 
 . - 0.0087 
 
 0.0054 
 
 - 
 Here 
 
 m(m - 
 
 , N 
 (). 
 
 w(m-l)(m-2)(m-3) 
 
 ' 6tC ' 
 
 1X2 1X2X3 1X2X3X4 
 
 Compare p. 39. 
 
 Permutations. The number of possible permutations or arrangements 
 of n different elements is 1 X 2 X 3 X . . . X n = n\ (read:"n factorial"). 
 
 If among the n elements there are p equal ones of one sort, q equal ones 
 of another sort, r equal ones of a third sort, etc., then the number of possible 
 permutations is (nl)/(p\ X q\ X rl X . . .), where p + g + r + . . . = n. 
 
 Combinations. The number of possible combinations or groups of 
 n elements taken r at a time (without repetition of any element within any 
 one group), is [n(n - l)(n - 2)(w - 3) . . . (n - r + l)]/(r!) == (n) r . 
 (See table of binomial coefficients, p. 39.) If repetitions are allowed, so 
 that a group, for example, may contain as many as r equal elements, then 
 the number of combinations of n elements taken r at a time is (m) r , where 
 m = n + r - 1. NOTE: (n)i + (n) a + . . . + (n) = 2 - 1. 
 
 SOLUTION OF EQUATIONS IN ONE UNKNOWN QUANTITY 
 
 Roots of an Equation. An equation containing a single variable x 
 will in general be true for some values of x and false for other values. Any 
 value of x for which the equation is true is called a root of the equation. 
 To "solve" an equation means to find all its roots. Any root of an equation, 
 when substituted therein for x, will "satisfy" the equation. An equation 
 which is true for all values of x, like (x + I) 2 = x z + 2x +1, is called an 
 identity [often written (x + I) 2 s x* + 2x + 1]. 
 
 Types of Equations. 
 
 (a) Algebraic Equations: 
 
 of the first degree (linear), e.gr., 2x + 6 =0 (root: x = 3); 
 of the second degree (quadratic), e.g., x z 2x 3 =0 (roots: 1, 3); 
 of the third degree (cubic), e.g., x 3 - 6x 2 + 5x + 12 =0 (roots: - 1, 3, 4). 
 
SOLUTION OF EQUATIONS IN ONE UNKNOWN QUANTITY 117 
 
 (6) Transcendental Equations: 
 
 exponential equations, e.g., 2 X = 32 (root: x = 5); 2 X = 32 (no root); 
 trigonometric equations, e.g., 10 sin x sin 3x = 4 (roots: 30, 150). 
 
 Legitimate Operations on Equations. An equation which is true for 
 a particular value of x will remain true for that value of x after any one of 
 the following operations is performed: 
 
 Adding any quantity to both sides; subtracting any quantity from both 
 sides; transposing any term from one side to the other, provided its sign 
 be changed; multiplying or dividing both sides by any quantity which is 
 not zero; changing the signs of all the terms; raising both sides to any positive 
 integral power; extracting any odd root of both sides; extracting any even 
 root of both sides, provided the sign is used; taking the logarithms 
 of both sides (both sides being positive) ; taking the sin, cos, tan, etc., of both 
 sides. 
 
 Notice, however, that the new equation obtained by some of these operations may 
 possess "additional roots" which did not belong to the original equation. This occurs 
 especially when both sides are squared; thus, x = 2 has only one root, namely, 2; 
 but z 2 =- 4, obtained by squaring, has not only the root 2 but also another root, + 2. 
 
 Equations of the First Degree (Linear Equations). Solution: Collect 
 all the terms involving x on one side of the equation, thus: ax b, where 
 a and b are known numbers. Then divide through by the coefficient of x, 
 obtaining x = b/a as the root. 
 
 Equations of the Second Degree (Quadratic Equations). Solution: 
 Throw the equation into the standard form ax 2 + bx + c = 0. Then the 
 two roots are: _ 
 
 - 6 + Vb* - 4ac -b -Vb* -4ac 
 
 Xl= ~ -2aT ** = -^a~ 
 
 The roots are real-and-distinct, coincident, or imaginary, according as 
 6 2 4ac is positive, zero, or negative. The sum of the roots is x\ + xt 
 = b/a] the product of the roots is x\xz = c/a. 
 
 GRAPHICAL SOLUTION. Write the equation in the form z 2 = px + q, and plot the 
 parabola yi = x 2 , and the straight line yi = px + q. The abscissae of the points of 
 intersection will be the roots of the equation. If the line does not cut the parabola, 
 the roots are imaginary. 
 
 Equations of the Third Degree with Term in x 2 Absent. Solution: 
 After dividing through by the coefficient of x 3 fc any equation of this type 
 can be written re 3 = Ax + B. Letp = A/3 and q = B/2. The general solu- 
 tion is as follows: 
 
 Case 1. q 2 p 3 positive. One root is real, namely 
 
 zj = %/q + Vq^ p 3 + ^/q - Vq*~^ P 3 ; 
 the other two roofs are imaginary. 
 
 Case 2. g 2 p 3 = zero. Three roots real, but two of them equal. 
 
 Case 3. g 2 p 5 negative. All three roots real and distinct. Determine 
 an angle u between and 180, such that cos u = q/(pV^p). Then 
 xi = 2Vpcos (w/3), x z = 2^/p cos (w/3 + 120), x z = 2\/pcos (w/3 + 240). 
 
 GRAPHICAL SOLUTION. Plot the curve yi = x 3 , and the straight line yt = Ax + B. 
 The abscissae of the points of intersection will be the roots of the equation. 
 
 Equations of the Third Degree (General Case). Solution: The gen- 
 eral'cubic equation, after dividing through by the coefficient of the highest 
 
118 ALGEBRA ,. 
 
 power, may be written rr 3 + ax* -f bx -f c = 0. To get rid of the term in 
 a; 2 , let x = xi a/3. The equation then becomes #i 3 = Ax\ + B, where 
 A = 3(a/3) 2 - 6, and B = - 2(a/3) 3 + 6(a/3) - c. Solve this equation 
 for xi, by the method above, and then find x itself from x = xi (a/3). 
 
 GRAPHICAL SOLUTION. Without getting rid of the term in x 2 , write the equation in 
 the form x 3 = - a[x + (&/2a)p + [a(6/2a) 2 - c], and solve by the graphical method. 
 
 General Properties of Algebraic Equations. An algebraic equation of 
 the nth degree in x is an equation of the type 
 
 aox n + aix n ~ l + oax 71 " 2 + . . . -fa n -iz+ a n = 
 
 where the a's are any given numbers (ao not zero), the expression on the 
 left being called a polynomial of the nth degree in x. Such an equation 
 will, in general, have n roots; but some of these n roots may be equal, and 
 some may be imaginary. Imaginary roots always occur in pairs. 
 
 If the equation is written in the form: (a polynomial in x) = 0, then (1) 
 if a is a root of the equation, x a is a factor of the polynomial; (2) if the 
 polynomial can be factored in the form (x p)(x q)(x r) . . . = 0, 
 each of the quantities p, q, r, . . . is a root of the equation; (3) if x is very 
 large (either positive or negative) , the higher powers of x are the most impor- 
 tant; (4) if x is very small, the higher powers may be neglected. 
 
 Short Method of Substitution in a Polynomial. To find the value of 
 4x 14z 3 + 23x 26 when x = 3, for example, first arrange the terms 
 in order of descending powers of x, and write the detached coefficients, with 
 their signs, in a row, taking care to supply 
 
 a zero coefficient for any missing term, in- 4 14 23 26 (3 
 eluding the constant term. Then, beginning 12618 15 
 
 at the left, bring down the first coefficient; - 
 
 multiply this by 3, and add to the second 4 26 511 
 
 coefficient; multiply this result by 3 again, 
 
 and add to the third coefficient; and so on. The final result, 11, is the 
 value of the polynomial when x = 3. 
 
 Short Method of Dividing a Polynomial by x a. The device just 
 explained gives not only the value of the polynomial when x = 3, but also the 
 result of dividing the polynomial by x 3. Thus, in the case illustrated, 
 the quotient is 4o; 3 2x 2 Qx + 5 and the remainder is 11. That is, 
 4z - 14z 3 + Ox 2 + 23x - 26 = (x - 3)(4z 3 - 2z 2 - 6x + 5) - 11. 
 
 Exponential Equations. To solve an equation of the form a* = 6, 
 take the logarithms of both sides: x log a = log 6, whence x = (log 6) /(log a). 
 For example, if 3* = 0.4, x = log 0.4/log 3 = (0.6021 - 1)/0.4771 = 
 0.3979/0.4771 = 0.8340. Notice that the complete logarithm must be 
 taken, not merely the mantissa. 
 
 Trigonometric Equations. (1) To solve a cos x + 6 sin x = c, where 
 a and b are positive: Find the acute angle u for which tan u = b/a, and the 
 angle v (between and 180) for which cos v = c/\/a z + 6 2 . Then xi = u + v 
 and xz = u v are roots of the equation. 
 
 (2) To solve a cos x b sin x = c, where a and 6 are positive: Find u 
 and v as above. Then x\ = (u + v) and xz = (u v) are roots of the 
 equation. 
 
 General Method of Solution by Trial and Error. This method is 
 applicable to a numerical equation of any form, and can be carried out to 
 any desired degree of approximation. It is especially useful when a first 
 approximation to a root is already known, Write the equation i^ the form 
 
SOLUTION OF SIMULTANEOUS EQUATIONS 119 
 
 f(x) = 0, where /(x) means any function of x, and plot the curve y = f(x) for 
 a sufficient number of values of x to obtain a general idea of the shape of the 
 curve. Then pick out the regions in which the curve appears to cross the 
 axis of x, and plot the curve more accurately in each of these regions. Thus, 
 <by successive approximations, plotting the important parts of the curve on a 
 larger and larger scale, determine as accurately as necessary the points where 
 the curve crosses the axis that is, the values of x which make/(z) equal to zero. 
 Thus, suppose that f(x) = 3.0 when x = 2.6 and 5.0 when x = 2.7 (see Fig. 1). 
 Then the curve must cross the axis somewhere between x = 2.6 and x = 2.7; and since 
 it will not yary greatly from a straight line between those points, it is seen that it must 
 
 -^,0.7 
 
 ;w^ 
 
 2.6 
 
 2.64 
 
 2.63 
 
 -S.O 
 
 FIG. 1. 
 
 cross near 2.64. Suppose the value of /(x) when computed for x = 2.64, is 0.2, and 
 when computed for x = 2.63 is + 0.7; then the root lies between x - 2.63 and 2.64. 
 Plotting this section on the larger scale, it is seen that the next guess should be about 
 2.638; and so on. 
 
 Instead of writing the original equation with all the terms on the left-hand side, it is 
 often better to divide the expression into two parts, say fi(x) and/Kx), writing the equa- 
 tion in the form/i(x) = MX). If then the two curves j/i = /i(x) and 3/2 = MX) be plotted 
 separately, on the same diagram, the value of x corresponding to their point of inter- 
 section will be the desired root. 
 
 SOLUTION OF SIMULTANEOUS EQUATIONS 
 
 The Meaning of a System of Simultaneous Equations. To solve a 
 system of n simultaneous equations in n unknowns, means to find all the sets 
 of values of the unknowns (if any) which, when substituted in the given 
 equations, will satisfy all the equations at the same time. If a system of 
 equations has no solution, the equations are "inconsistent;" if it has an in- 
 finite number of solutions, the equations are "not all independent." 
 
 Simultaneous Eguations of the First Degree in Two Unknowns. 
 Factors 
 
 (1) aix + b\y = c\ \ 62 
 
 (2) a z x + b 2 y = c 2 - 
 
 a z 
 
 (aibz azbi)x = bzci biCz .". x = (bzci biCz) /((lib* azbi) 
 (aibz azb\)y = a\cz azci .'. y = (a\cz azci) / (aibz azbi) 
 Here (1) is multiplied by 62, (2) by 61, and the products added so as to 
 eliminate y, again, (1) is multiplied by az, (2) by 01, and the products 
 added so as to eliminate x. (The process is most conveniently performed as 
 follows: Write the multipliers, as 6 2 and 61, at the right of the equations; 
 multiply the first term of each equation by its proper multiplier and add; 
 then multiply the second term of each equation by its proper multiplier, and 
 add; and so on. This is simpler than the common practice of multiplying 
 out each equation separately before adding.) If 0162 azbi = 0, the equa- 
 tions have no solution when c\ 7^ cz, and an infinite number of solutions when 
 
120 
 
 ALGEBRA 
 
 GI = C2. The following special solution is possible when the sum 
 difference of the two unknowns are given: 
 
 Let x + y = m (1) 
 
 and x y = n (2) 
 (1) + (2) 
 
 and 
 
 2): 2x = m + n . m .x = 
 (1) - (2): 2y = m - n .'. y = W(m - n) 
 Simultaneous Equations of the Second Degree in Two Unknowns, 
 
 (a) When the product of the unknowns, and their sum or difference, are given: 
 
 * +y = 5 
 xy = 4 
 
 Squaring (1), x* 
 From (2), 
 
 + 2xy + 2/ 2 = 
 -4*2/ 
 
 25 
 - 16 
 
 Adding, * 2 
 
 2*2/ + y 2 = 
 
 9 
 
 Hence, x 
 But * 
 
 y = 3 or 3 
 + y = 5 or 5 
 
 Therefore * 
 
 y 
 
 = 4 * = 1 
 = 1 r y = 4 
 
 
 (1) 
 
 (2) 
 
 x y = 3 
 *2/ = 4 
 
 
 (1) 
 (2) 
 
 *2 - 2*y + 2/ 2 
 4xy 
 
 = 9 
 = 16 
 
 x* + 2*2/ + 2/2 
 
 = 25 
 
 x + 2/ = 5 or - 
 x y = 3 or 
 
 - 5 
 3 
 
 x = 4 re = 
 IT- 1 r y = 
 
 - 1 
 
 - 4 
 
 (b) When the product and the sum of the squares are given: 
 
 xy = 5 (1) V(4) : x + y = 6 or 6 or 6 or 6 
 
 26 (2) 
 
 *-2/=4or-4or 4 or -4 
 
 From (1), 
 
 2*2/ = 10 (3) 
 
 .". * 
 
 = 5 
 
 I 
 
 _ i 
 
 
 - 5 
 
 (2) 
 
 + (3): *2 _|_ 2 *2/ 
 
 + 2/ 2 = 36 (4) 
 
 'y 
 
 = 1 
 
 or _ 
 o 
 
 or ^ 
 
 or 
 
 - 1 
 
 (2) 
 
 -(3): 
 
 r 2 
 
 2*2/ 
 
 + y 2 = 16 (5) 
 
 
 
 
 
 
 
 (C) 
 
 When 
 
 the sum 
 
 or difference, and the sum of 
 
 the 
 
 squares, are given: 
 
 
 
 * + 
 
 y 
 
 = 5 (1) 
 
 
 
 * - 
 
 - y 
 
 = 
 
 3 
 
 (D 
 
 
 
 *' + 
 
 2/ 2 
 
 = 17 (2) 
 
 *2 _|_ y, 
 
 17 
 
 (2) 
 
 (D s : 
 
 
 * 2 + 
 
 2*2/ 
 
 + 2/2 =25 
 
 (D 2 : 
 
 
 * 2 - 
 
 - 2xy + 2/2 = 
 
 9 
 
 
 (2) : 
 
 
 * 2 
 
 
 + 2/2 = 17 
 
 (2) : 
 
 *2 + I/ 2 = 17 
 
 (I) 2 - (2) : 
 
 2*2/ 
 
 = 8 
 
 (D 2 - 
 
 (2) : 
 
 
 - 2a-y 
 
 = 
 
 -8 
 
 
 xy =4 
 
 Then proceed as in case (a), above. 
 
 xy =4 
 
 Then proceed as in case (a), above. 
 
 (d) When one equation is of the first degree and the other of the second, as 
 ax + by = c, and Ax 2 + Bxy + Cy* + Dx + Ey + F =0: Solve the first 
 equation for y in terms of x, and substitute in the second. This will give a 
 quadratic equation in x. Solve this quadratic for the frwo values of x, and 
 for each of these values of x find the corresponding value of y by substituting 
 in the equation of the first degree. 
 
 Simultaneous Equations of the First Degree in n Unknowns. For 
 example: Factors 
 
 (a) 
 
 (c) 
 (d) 
 
 2x 
 5x 
 3x 
 4x 
 
 - y 
 
 - 4y 
 
 + 3z + 5w = 29 
 - 2z + 3; = 15 
 + 72 - w = 12 
 - 5 + 2w = 3 
 
 3 
 - 5 
 
 1 
 5 
 
 2 
 - 5 
 
 (0) 
 
 - 19* 
 
 -2ly 
 - 17y 
 
 + 192 = 
 
 + 382 
 
 + 312 = 
 
 = 12 
 = 89 
 = 43 
 
 - 2 
 
 1 
 
 - 31 
 19 
 
 
 (h) 
 
 55* 
 285* 
 
 + 5y 
 + 802/ 
 
 = 65 
 = 445 
 
 16 
 -1 
 
 
METHOD OF LEAST SQUARES 121 
 
 (;) 595* =595; . /. x = 1; 
 
 5y = 65 - 55* = 65 - 55 = 10; .'. y = 2; 
 
 192 = 12 + 19* + 13y = 12 + 19 + 26 = 57; .'.z =3; 
 
 2w = 3 - 4* - 3y + 5z = 3 - 4 - 6 + 15 = 8; .'. w = 4. 
 
 Here tu is eliminated from (a) and (6) , obtaining (e) ; from (a) and (c) , 
 obtaining (/) ; and from (a) and (d), obtaining (g). Then z is eliminated from 
 (e) and (/), obtaining (A), and from (e) and (0), obtaining (i). Then $/ is 
 eliminated from (h) and (i), obtaining 0'), which contains only the single vari- 
 able *. Hence * = 1. Now substituting this value of * in either (h) or (t), 
 y is found; substituting these values of * and y in either (e), (/), or (g), z is 
 found; and so on. (Solution by determinants, see p. 123.) 
 
 Approximate Solution of a Set of Simultaneous Equations of the 
 First Degree When the Number of Equations is Greater Than the 
 Number of Unknowns. (Method of Least Squares.) 
 
 Case 1. Single Unknown Quantity. Given n equations in one un- 
 known *; for example, n equally careful, independent measurements of some 
 physical quantity: 
 
 X Xl, X = *2, . . . * = Xn- 
 
 As the "best" value of *, take the arithmetic mean, XQ, of the several deter- 
 minations, namely, XQ = (*i + xz + . . . + *n) /n. The quantities vi = 
 *o xi, z>2 = XQ *2, . . . v n = XQ x n are called the residuals of the 
 observed values with respect to XQ, and their absolute values (that is, their 
 numerical values without regard to sign) are denoted by \vi\, \v^\, . . . |t> n |. 
 [It can be shown that the sum of the squares of the residuals with respect 
 to XQ is smaller than the sum of the squares of the residuals with respect to 
 any other value X'Q- hence the name of the method: "least squares."] 
 The quantities r and TQ, defined exactly by Bessel's formulae: 
 
 0.6745 
 
 n(n - 1) 
 or given approximately by the simpler formulae of Peters: 
 
 ' 8453 (M + H + . . + W), 
 
 are called the probable error of a single observation (r), and the probable 
 error of the mean (T-Q), for the given series of observations. Note that 
 ro = r l\/n. For tables of the coefficients, see p. 63. This quantity r (or 
 ro) is best regarded as merely a conventional means of recording the relative 
 precision of different sets of observations. If r is small, it may be inferred 
 that most errors of the " accidental" class have been eliminated; but it should 
 be especially noted that the smallness of r gives no information in regard to 
 "constant" or "systematic" errors. 
 
 A statement like ''x is equal to 2.36 with a probable error of 0.02," is 
 written: x = 2.36 + 0.02, and is usually understood to mean that the true 
 value of x, as far as can be told, is just as likely to lie inside as outside the 
 interval from 2.34 to 2.38. 
 
122 ALGEBRA 
 
 ' To test the distribution of residuals, arrange the residuals in order of 
 magnitude, without regard to sign, and count the number, y, of residuals 
 which are numerically less than some assigned value a; divide y by w, the 
 total number of observations, and divide a by r, the probable error of a single 
 observation. Do this for various values of a, and compare the results with 
 the table on p. 63, which gives the standard distribution of residuals, as 
 found from experience from a large number of different series of observations. 
 In particular, the number of residuals numerically less than r should be about 
 equal to the number numerically greater than r (if n is large). If any large 
 discrepancy appears, the series of observations should be regarded as unsatis- 
 factory. 
 
 NOTE. The "mean square error" sometimes met with is equal to the probable error 
 divided by 0.6745. 
 
 Case 2. Several Unknown Quantities. Assume that there have been 
 
 obtained by measurement or observation n different equations of the first 
 
 degree involving, say, three unknown quantities, 
 
 Given Equations x, y, z. There are then n simultaneous equations 
 
 a\x + biy + c\z = pi in three unknowns, and if n > 3 there will be, in 
 
 azx + b%y + czz = pz general, no set of values of x, y, z which will satisfy 
 
 all these n equations exactly. In such a case, 
 
 a n x -f- b n y + CnZ = Pn the "best" set of values, XQ, yo, ZQ, may be found 
 by the method of least squares as follows. (The 
 
 process usually involves a large amount of labor; the use of a computing 
 machine is advisable.) 
 
 First, arrange the n given equations in the form indicated, being careful 
 not to modify any of them by multiplication or division. (Any of the coeffi- 
 cients may of course be zero.) 
 
 Next, form the three "normal equations" as follows: (1) Multiply each 
 of the given equations by the coefficient of x in that equation, and add; the 
 
 result will be the first normal equation. 
 
 Normal Equations (2) Multiply each of the given equations 
 
 [aa]a;o + [ab]yo + [ac]zo = [ap] by the coefficient of y in that equation, and 
 [ba]xo -\- [bb]yo + [bc]zo = [bp] add; the result will be the second normal equa- 
 [ca]xo + [cb]yo + [cc]zo = [cp] tion. (3) Similarly for the third. { Nota- 
 tion: [aa] = ai 2 + a2 2 + . . . + a n 2 ; 
 [ab] = ai&i +0262 -f- . . . +a n bn', [ap] = aipi -\-aipz + . . . +a n p n : etc.} 
 
 Finally, solve the three normal equations for the three unknowns in the 
 usual way. 
 
 The quantities DI = 0,1X3 + &i2/o + C\ZQ pi, etc., are called the residuals 
 with respect to XQ, yo, ZQ. [It can be shown that the sum of the squares of the 
 residuals with respect to XQ, yo, ZQ is smaller than the corresponding quantity 
 with respect to any other set of values, x'o, y'o, z'o; this relation is taken as the 
 criterion for the "best" set of values of x, y, z.] 
 
 The probable error of a single observation is 
 
 0.6745 
 
 + vz 2 + + *>n 2 , or approximately, 
 
 (hi + M + . . . + hi), 
 
 n(n m) 
 where m = the number of unknown quantities (here m = 3). 
 
DETERMINANTS 
 
 123 
 
 DETERMINANTS 
 
 Determinants are used chiefly in formulating theoretical results; they are 
 seldom of use in numerical computation. 
 
 Evaluation of Determinants : 
 Of the second order: 
 
 |a 2 & 2 | 
 
 0162 
 
 Of the third order: 
 
 02&2C2 
 
 = Ol 
 
 & 2 c 2 
 
 &3C3 
 
 &1C1 
 
 he 
 
 &3C2) 2 (6lC3 
 
 Of the fourth order: 
 
 &2C2C?2 
 
 , *! v a ^a**a dz baCada 
 dabacada 
 
 6404^ 
 
 bzczdz 
 baCadal 
 
 etc. In general, to evaluate a determinant of the nth order, take the ele- 
 ments of the first column with signs alternately plus and minus, and form the 
 sum of the products obtained by multiplying each of these elements by its 
 corresponding minor. The minor corresponding to any element ai is the 
 determinant (of next lower order) obtained by striking out from the given 
 determinant the row and column containing 01. 
 
 Properties of Determinants. 
 
 1. The columns may be changed to rows and the rows to columns: 
 
 010203 
 
 a 2 6 2 c 2 
 
 036303 
 
 ClC 2 C3 
 
 2. Interchanging two columns changes the sign of the result. 
 
 3. If two columns are equal, the determinant is zero. 
 
 4. If the elements of one column are m times the elements of another 
 column, the determinant is zero. 
 
 5. To multiply a determinant by any number m, multiply all the elements 
 of any one column by m. 
 
 7. 
 
 Solution of Simultaneous Equations by Determinants. 
 
 If aix + biy + ciz = pi 
 
 01 + PI + 3i, bi ci 
 
 
 ai&ici 
 
 
 pibici 
 
 
 01&lCl| 
 
 a 2 + p 2 + 0.2, bz Cz 
 
 = 
 
 02&2C2 
 
 -|- 
 
 Pzbzcz 
 
 -f- 
 
 qzbzczl 
 
 03 + Pa + qa, ba ca 
 
 
 03&3C3 
 
 
 pabaca 
 
 
 qabacal 
 
 dibiCi 
 
 
 01 + mbi, &i ci 
 
 
 azbzcz 
 
 = 
 
 o 2 + mbz, bz Cz 
 
 
 aabaCa 
 
 
 03 + mba, ba Ca 
 
 
 aix + bzy + czz = p 2 
 aax + bay + caz = PS 
 then x = Di/D, 
 
 y = D 2 /Z>, where Di 
 z = Da/D, 
 
 where D 
 
 PI&IC 
 
 a 2 & 2 c 2 
 dabaca 
 
 0, 
 
 pabaca 
 
 aapac 
 
 Da = 
 
 Similarly for a larger (or smaller) number of equations. 
 
124 
 
 ALGEBRA 
 
 THE ALGEBRA OF IMAGINARY OR COMPLEX QUANTITIES 
 
 In the algebra of imaginary or complex quantities, the objects on which the 
 operations of the algebra are performed are not numbers in any ordinary 
 sense of the word, but are best thought of as points in a plane (or as vectors 
 drawn from a fixed origin to these points). The "complex plane" is de- 
 termined by three fundamental points, O, U, i, arranged as in Fig. 2 and called 
 the zero point, the unit point, and the imaginary unit point, respectively. 
 All points on the line through O and U are called real points positive if 
 on the right of O, negative if on the left. All the remain- 
 ing points in the plane are called imaginary points 
 those on the line through and i being called the pure 
 imaginary points. 
 
 The position of any point A in the plane may be de- 
 termined by the distance from the origin O, measured in 
 terms of OU as the unit length, and the angle <p which 
 OA makes with the positive direction of the axis of reals. 
 The distance r is sometimes called the modulus or ab- 
 solute value of the point; the angle <f> is sometimes called 
 the amplitude or argument of the point. The notation 
 
 FIG. 2. 
 
 A = (3, ^120) 
 
 means the point whose distance, r, is 3 times OU, and whose angle, <p, is 120. 
 The development of the algebra depends wholly on the definitions of three 
 fundamental operations denoted by A + B, A X B, and e A , as follows. 
 
 Addition and Subtraction. The sum, A + B, of two points A and B 
 is defined as the point reached by starting from A and performing a journey 
 equal in length and direction to the journey from to B. That is, the vector 
 from O to A + B is the vector sum of the vectors OA and 
 OB. In case A and B are not in line with 0, the point A + B A + B 
 
 is the fourth vertex of a parallelogram of which OA and OB 
 are the sides (Fig. 3). Conversely, if any two points A and 
 B are given, there is a definite point X such that A = B + X ; 
 this point X is called the remainder, A minus B, and is FIG. 3. 
 
 denoted by A B. The point O B is denoted for brevity 
 by B. With these definitions of A + B and A B, all the ordinary laws 
 of addition and subtraction that hold in the algebra of real numbers hold also 
 in the algebra of complex quantities. In particular, the zero point O has all 
 the formal properties of the number zero, and is denoted by 0. 
 
 [Note: If A and B are "real" points, A -f- B and A B will also be real. 
 
 Repeated Addition. Multiples and Submultiples. The point 
 A+A+A-f-. . . + A to n terms is called the nth nlultiple of A and 
 is denoted by nA. The points U, 2U, 3U, . . . are denoted, for brevity, 
 by 1, 2, 3, . . .. Conversely, if any point A, and any positive integer n 
 are given, there is a definite point X such that nX = A ; this 
 point X is called the nth submultiple of A, and is denoted by 
 A/n. The points 7/2, U/3, ... are denoted, for brevity, 
 by ^, H, . . .. 
 
 Multiplication and Division. The product, A X B, or 
 A-B, or AB, of two points A and B is defined as the point 
 whose angle is the sum of the angles of the given points, and 
 whose distance is the product of the distances. (See Fig. 4.) 
 Thus, if A = (5, ^120) and B = (2,^270), then AB = 
 (10, -^30). Conversely, if any two points A and B are given, 
 provided B is not zero, there is a definite point X such that 
 
THE ALGEBRA OF IMAGINARY OR COMPLEX QUANTITIES 125 
 
 FlO. 5. 
 
 A = BX. This point X is called the quotient, A divided by B, and is de- 
 noted by A/B (where B ^ 0). Thus, the point A/B is a point whose angle 
 is the angle of A minus the angle of B, and whose distance is the distance of 
 A divided by the distance of B. The point U/B (B 7* 0) is called the 
 reciprocal of the point.fi, and is denoted by l/B. (See Fig. 5.) With these 
 definitions of AB and A/B the elementary laws of multiplication and division 
 that hold in the algebra of real numbers hold also in the algebra of complex 
 quantities. In particular, the point U has all the formal 
 properties of the number unity, and is denoted by 1. 
 
 [Note: If A and B are real, AB and A/B will also be real.] 
 
 Repeated Multiplication. Powers and Roots. The 
 point A X A X A X . . . X A to n factors is called the 
 nth power of A and is denoted by A n (Fig. 6). Conversely, 
 if any point A (not 0) and any positive integer n are given, 
 there will be n distinct points X such that X n = A ; each of 
 these points is called an nth root of A, some one of them, 
 usually the one with the smallest positive angle, being de- 
 noted by \/A or A 1/n . Thus, the point \/A 
 is a point whose distance is the nth root of the dis- 
 tance of A, and whose angle is I/nth of the angle 
 of A. All the nth roots of A will lie on the cir- 
 cumference of a circle about O as center, and will 
 divide that circumference into n equal parts (Fig. 
 7). Every point A (not 0) has two square roots, 
 three cube roots, etc. Hence the theorem "If A n 
 = B n then A = B" does not hold in this algebra, 
 and the ordinary rules for radical signs must be 
 applied with caution. For example, if A and B 
 are positive reals, V A-V B = V ' AB and 
 not V( A)( #), which would give + \fAB. 
 
 [Note: If A is real and positive, \/A will be real and 
 positive; if A is real and negative, -\^A will be real if n 
 is odd and imaginary if n is even.] 
 
 Properties of i. The point i is the point whose dis- 
 tance is 1 and whose angle is 90 deg. It follows from 
 the definition above that multiplying any point A by 
 i has the effect of rotating the point through an 
 angle of + 90 without changing its distance from O. 
 In particular, 
 
 i 2 = - 1, i 9 - - i, i* = 1, i 5 = i, etc.; i = V-l, 
 i = V 1; where "1" denotes not the number 
 one, but the point U. 
 
 Similarly, multiplying any point A by 1 has the 
 effect of rotating the point through 180 deg. 
 
 First Standard Form for a Complex Quantity 
 (Fig. 8). Any point A can be expressed in the form 
 x + iy, where x and y are real points. For example, 
 the three cube roots of 1 are 1, # + ^i\^3, and 
 
 j-2/ 
 
 FIG. 8. 
 
 -H 
 
126 ALGEBRA 
 
 In general, (x\ + iyi) + (xz + iyd = (x\ + xz) + i(yi + 1/2) ; 
 
 (xi + iyi)(xt + ij/j) = (xix* - 2/13/2) + i(x*yi + zii/ 2 ) | 
 
 xi + iyi _ xixz + 3/13/2 _, . 333/1 xiyz 
 xz + iyz xz* + 3/ 2 2 xj + 3/ 2 2 
 
 If two complex quantities are equal, their real parts must be equal, and the 
 coefficients of their pure imaginary parts must also be equal. That is, if 
 x\ + iyi = xz + iyz, then xi = x% and y\ =* 2/2. Thus a single equation between 
 complex quantities is equivalent to two equations between real quantities. 
 
 Conjugate Imaginaries. Two points A = x + iy and B = x iy are 
 called conjugate imaginaries. Two such points are symmetrically situated 
 with regard to the axis of reals. The sum and product of two conjugate 
 imaginaries will be real. 
 
 Second Standard Form for a Complex Quantity. Since x = r cos <p and 
 y r sin <f>, any point A =* x + iy can be expressed A = r (cos <p -f- i sin <f>), 
 where r is real and positive (namely, the distance of A), and <p is real 
 (namely the angle of A). For example, the three cube roots of 1 are 1, 
 cos 120 + i sin 120, and cos 240 + i sin 240. In general, 
 [n (cos <f>i+i sin <f>i)] [r 2 (cos </>2+i sin <p 2 )] =nr 2 [(coa (<pi+<f>t) +* sin (^1+^2)]; 
 [r(cos^ + iam<p)] n = r n [coa(n<p) + ism (n<p)] (De Moivre's Theorem). 
 
 The Exponential Function, e A , or exp A, of any point A =x + iyis defined 
 aa the point whose distance is e x and whose angle (measured in radians) is y. 
 That is, .e x+iv = e*(cos y + i sin y). Here e x means the ordinary expo- 
 nential function of the real quantity x, where e = 2.718. 
 
 From this definition, the usual formal laws of exponents can be deduced: 
 
 6 A e B = e A+B f ( e A)n = g n4 > Q -A = \/ e A. e \ = e> e O = lt 
 
 The function e A is a periodic function with a pure imaginary period 2iri; 
 that is, e A k ' i ir i = e A t where k is any positive integer. 
 
 If A is made to move along a line parallel to the axis of reals [or axis of pure 
 imaginaries], the corresponding point e A will move along a straight line through 
 [or along a circle about O as center]. 
 
 Properties of e lv> . The point e l(p is a point whose distance is 1 and whose 
 angle is <f>. It follows from the definitions above that multiplying any 
 point A by e l<f> has the effect of rotating the point through an 
 angle <p, without changing its distance from O. In particular, e lir 1, 
 e~ iv = - 1; e /z = i; e~ i7r/2 = - i; e = 1. 
 
 Third Standard Form for a Complex Quantity. Any point A can be 
 expressed in the form A = re l<f> > where r is the distance and <p the angle of the 
 point. For example, the three cube roots of 1 are 1, e* 27 , e' 27r *. In general, 
 
 If x + iy = re l<f> , then r = Vz 2 + y 2 , sin * = , cos*> = , tan <? = 
 
 If two complex quantities are equal, their distances will be equal, and their 
 angles will differ at most by some multiple of 2,-ir. Thus, if ne l<f>1 = rze l<f * 
 then ri = r-i and <p\ = <pz or <p2 k2ir. Here again a single equation between 
 complex quantities is equivalent to two equations between real quantities. 
 
THE ALGEBRA OF IMAGINARY OR COMPLEX QUANTITIES 127 
 
 Definition of A B . Let A = reW; then A B = exp [(log e r + i<p)B]. 
 
 For example, i { = e-V 2 where i = \/ 1. 
 
 If a is a positive real, a x y = a x [cos (j/ log e a) + i sin (y log a)]. 
 
 Trigonometric and Hyperbolic Functions of a Complex Variable. 
 
 If A is any point, then, by definition, 
 
 e iA _ e -iA e iA + e -iA sin A 
 
 sin A = : , cos A = , tan A = (cos A /^ 0) ; 
 
 2 cos A 
 
 . e A + e A smh 
 
 smh A = - - - , cosh A = - - -- , tanh A 
 
 , , 
 
 2 2 cosh A 
 
 Hence the formulae that hold for these functions in the real case (p. 131; 
 p. 135; p. 161) hold also for the complex case. Further: 
 
 sin (a; -h iy) = sin x cosh y + i cos x sinh y, sin iy = i sinh y ; 
 
 cos (x + iV) = cos x cosh y i sin a; sinh j/, cos ty = cosh y; 
 
 sinh (aj + iy) = sinh a; cos y + i cosh a: sin j/, sinh t'y = i sin y; 
 
 cosh (x + *2/) = cosh x cos ?/ + i sinh x sin y, cosh iy = cos y; 
 where sin x, sinh x, etc., are the ordinary trigonometric and hyperbolic func- 
 tions of the real variables x and y. The functions sin A and cos A are periodic 
 with a real period 2,-tr. The functions sinh A and cosh A are periodic with a 
 pure imaginary period "2iri. 
 
 Logarithmic and Other Inverse Functions of a Complex Variable. 
 If any point A is given, there will be an infinite number of points X such 
 that e x = A; any one of these points may be called a logarithm of A, and 
 be denoted by log A. All the values of the logarithm of A may be obtained 
 from any one value by adding multiples of 2iri. 
 
 If x -f- iy = re l<f> , then log e (x + iy) = loge r + i<p + k-Zin. 
 
 If any point A is given, there will be an infinite number of points X such that 
 sin X = A; any one of these may be denoted by sin- 1 A. The functions 
 cos- 1 A, sinh- 1 A, etc., are defined in a similar way. 
 
 The elementary laws of operation which hold for these functions in the 
 algebra of reals hold also, in a general way, in the algebra of complex quanti- 
 ties; but caution must be used, on account of the ambiguity in the symbols 
 log A, sin -^A, etc., which denote many-valued functions. 
 
 Differentiation of Functions of a Complex Variable. If w =/(z), 
 _ the derivative of w with respect to z is defined as 
 
 dw/dz = lim { [f(z -f- Az) f(z)]/Az] when Az approaches 0. 
 
 It can be shown that lim {[exp Az l]/Az} = 1; hence d(e?) = e*dz, 
 d(sin z) = cos z dz, etc., so that the formulae for differentiation here are the 
 same as in the case of a real variable (p. 157). 
 
 NOTE. For the algebra of vector analysis, which differs in important respects from 
 the algebra of complex quantities, see p. 185. 
 
TRIGONOMETRY 
 
 FIG. 2. 
 
 FORMAL TRIGONOMETRY 
 
 Angles, or Rotations. An angle is generated by the rotation of a ray, 
 as Ox, about a fixed point O in the plane. Every angle has an initial line 
 (OA) from which the rotation started (Fig. 1), and a terminal line (OB) 
 where it stopped; and the counterclockwise direction of rotation is taken as 
 positive. Since the rotating ray may revolve as often as 
 desired, angles of any magnitude, positive or negative, 
 may be obtained. Two angles are congruent if they 
 may be superposed so that their initial lines coincide and 
 their terminal lines coincide. That is, two congruent 
 angles are either equal or differ by some multiple of 360 
 deg. Two angles are complementary if their sum is 90 
 deg. ; supplementary if their sum is 180 deg. (The acute 
 angles of a right-angled triangle are complementary.) If 
 the initial line is placed so that it runs horizontally to the 
 right, as in Fig. 2, then the angle is said to be an angle in 
 the 1st, 2nd, 3rd, or 4th quadrant according as the 
 terminal line lies across the region marked I, II, III, or IV. 
 The angles deg., 90 deg., 180 deg., 270 deg. are called the 
 quadrantal angles. 
 
 Units of Angular Measurement. 
 
 (1) SEXAGESIMAL MEASURE. (360 degrees = 1 revolution.) 1 degree = 
 1 = Ho of a right angle. The degree is usually divided into 60 equal parts 
 called minutes (0, and each minute into 60 equal parts called seconds ("); 
 while the second is subdivided decimally. But for many purposes it is more 
 convenient to divide the degree itself into decimal parts, thus avoiding the use 
 of minutes and seconds. (See tables, pp. 46-51.) 
 
 (2) CENTESIMAL MEASURE, used chiefly in France. (400 grades = 1 
 revolution.) 1 grade = Moo of a right angle. The grade is always divided 
 decimally, the following terms being sometimes used: 1 "centesimal minute" 
 
 = Moo of a grade; 1 "centesimal second" = Moo of a centesimal minute. In 
 reading Continental books it is important to notice carefully which systerh 
 is employed. 
 
 (3) RADIAN, OR CIRCULAR, MEASURE. (IT radians = 180 degrees.) 1 radian 
 = the angle subtended by an arc whose length is equal to the length of the 
 
 radius. The radian is constantly used in higher mathematics and in- me- 
 chanics, and is always divided decimally. Table, pp. 44^45. 
 
 1 radian = 57.30- = 57.2957795131 = 57 17' 44".806247 = 180 ,V. 
 
 1 = 0.01745 . . . radian = 0.0174532925 radian. 
 
 1' = 0.00029 08882 radian. 1" = 0.00000 48481 radian. 
 
 (For 10-place conversion tables, see the Smithsonian 
 Tables of Hyperbolic Functions, Washington, D. C.) 
 
 Definitions of the Trigonometric Functions. Let 
 x be any angle whose initial line is OA and terminal line 
 OP (see Fig. 3). Drop a perpendicular from P on OA or 
 OA produced. In the right triangle OMP, the three sides 
 are MP = "side opposite" O (positive if running upward); OM = "side 
 adjacent" to O (positive if running to the right); OP = " hypothenuse " or 
 "radius" (may always be taken as positive); and the six ratios between 
 these sides are the principal trigonometric functions of the angle x; thus: 
 
 128 
 
FORMAL TRIGONOMETRY 
 
 129 
 
 sine of = sin re = opp/hyp = MP/OP; 
 cosine of x = cosx = adj/hyp = OM/OP; 
 tangent of x = tana; = opp/adj = MP/OM; 
 cotangent of x = cot x = adj/opp = OM/MP; 
 secant of x = sec x = hyp/adj = OP/OM; 
 cosecant of x = cso x = hyp/opp = OP/MP. 
 The last three are best remembered as the reciprocals of the first three: 
 
 cot x = I/tan x] sec x = I/cos x; esc x = I/sin x. 
 
 Other functions in use are the versed sine, the coversed sine, and the ex- 
 terior secant: 
 
 vers x = 1 cos x; covers x = 1 sin x; exsec x = sec x 1. 
 For graphs, see p. 174; series, p. 161. 
 
 Signs of the Trigonometric Functions 
 
 If x is in quadrant 
 
 I 
 
 II 
 
 III 
 
 IV 
 
 sin x and esc x are . .... 
 
 -f 
 
 + 
 
 
 
 cos x and sec x are 
 
 + 
 
 
 
 
 + 
 
 tan x and cot x are 
 
 + 
 
 
 -f 
 
 
 
 
 
 
 
 vers x and covers x are always positive. 
 
 Variations in the Functions as z Varies from deg. to 360 deg. are 
 
 shown in the accompanying table. The variations in the sine and cosine are 
 
 FIG. 
 
 best remembered by noting the changes in the lines MP and OM (Fig. 4) 
 in the "unit circle" (that is, a circle with radius = OP = 1), asP moves around 
 the circumference. 
 
 X 
 
 to 90 
 
 90 to 180 
 
 180 to 270 
 
 270 to 360 
 
 Values at 
 
 30 
 
 45 
 
 60 
 
 sin x 
 
 CSC X 
 
 +0to+l 
 
 + 00 tO+1 
 
 +1to+0 
 
 +HO + OO 
 
 -Oto-l 
 
 oo tO 1 
 
 -lto-0 
 
 -ItO-oo 
 
 M 
 
 2 
 
 W2 
 V2 
 
 H\/3 
 
 ^Vs 
 
 cos x 
 
 sec x 
 
 + Ito+0 
 +1 to + oo 
 
 -Oto-l 
 
 -cotO-1 
 
 -lto-0 
 
 ItO oo 
 
 +0to+l 
 
 + ootO+l 
 
 H\/3 
 *S\/3 
 
 H\/2 
 V2 
 
 H 
 
 2 
 
 tan x 
 
 cot * 
 
 +0to+oo 
 
 + ootO+0 
 
 - oo tO -0 
 -OtO-co 
 
 +0 tO + <x> 
 + on tO +0 
 
 - oo tO -0 
 
 -Oto- 
 
 H-S/3 
 V3 
 
 1 1 
 1 
 
 V3 
 H-S/3 
 
 vers x 
 covers x 
 
 +0to+l 
 +lto+0 
 
 +lto+2 
 +0to+l 
 
 +2to+l 
 
 + lto+2 
 
 +HO+O 
 
 +2to+l 
 
 
 
 
 \/2 =1.4142; H\/2 =0.7071; A/3 =1.7321; ^VV = 0.8660; W3 =0.5774; ^A/3 =1.1547 
 
 Trigonometrical Tables. The tables on pp. 46-56 give the values of the 
 principal trigonometric functions and of their logarithms, correct to four 
 places of decimals, the angle advancing either by tenths of a degree (p. 46) 
 or by 10 min. (p. 52). These tables will be found adequate for most 
 
130 
 
 TRIGONOMETRY 
 
 computations in which an accuracy of 1 part in 1000 is sufficient. If much 
 computing is to be done, it is advisable to use a separate volume of tables, 
 containing more facilities for interpolation, and printed in larger type, 
 such as the four-place tables of E. V. Huntington (Harvard Cooperative 
 Society, Cambridge, Mass.), with convenient marginal tabs; the five-place 
 tables published by Macmillan or many others; the six-place tables of 
 Bremiker; the standard seven-place tables of Schron, Vega, or Bruhns 
 (angles advancing by 10 sec.); or the great eight-place of Bauschinger 
 and Peters (angles advancing at intervals of 1 sec. fromO deg. to 90 deg.). 
 The larger tables give only the logarithms of the functions, not the natural 
 values. 
 
 To Find Any Function of a Given Angle. (Reduction to the first 
 quadrant.) It is often required to find the functions of any angle x from a 
 table that includes only angles between deg. and 90 deg. If x is not 
 already between deg. and 360 deg., first "reduce to the first revolution" by 
 simply adding or subtracting the proper multiple of 360 deg. ; [for any func- 
 tion of (z) = the same function of (x n X 360)]. Next reduce to the 
 first quadrant as follows: 
 
 If x is between 
 
 90 and 180 
 
 180 and 270 
 
 270 and 360 
 
 Subtract 
 
 90 from x 
 
 180 from x 
 
 270 from x 
 
 Then sin x 
 
 = +cos (x-90) 
 = +sec (a; -90) 
 = -sin (x-90) 
 *=-csc (x-90) 
 = -cot (x-90) 
 = -tan (x-90) 
 
 = -sin (x-180) 
 = -csc (a: -180) 
 = -cos (x-180) 
 = -sec (x -180) 
 = +tan (x-180) 
 = +cot (s-180 ) 
 
 = -cos (x-270) 
 = -sec (x -270) 
 = +sin (x-270) 
 = +csc (x -270) 
 = - cot (x - 27C; 
 = -tan (x -270) 
 
 esc x 
 
 COS X 
 
 sec x 
 
 tan x 
 
 cot x 
 
 
 vers x 
 
 = l+sin (x -90) 
 = l-cos (a; -90) 
 
 = l+cos (a- 180) 
 = l+sin (a; -180) 
 
 = l-sin (z-27<n 
 = l + cos(a;-270 ) 
 
 covers x 
 
 The "reduced angle" (x 90, or x 180, or x 270) will in each case 
 be an angle between and 90, whose functions can then be found in the 
 table. 
 
 [NOTE. The formulae for sine and cosine are best remembered by aid of the unit circle.] 
 To Find the Angle When One of Its Functions is Given. In general, 
 there will be two angles between deg. and 360 deg. corresponding to any given 
 function. The following tabulated rules show how to find these angles. 
 
 Given 
 
 First find from the tables 
 an acute angle xo such that 
 
 Then the required angles xi 
 and xz will be 
 
 sin 3= -fa 
 cos x= -\-a 
 tan*= +o 
 cot x= +o 
 
 sin xo = a 
 cos xo = a 
 tan xo = a 
 cot xo = a 
 
 xo and 180 -xo 
 xo and [3 60 -so] 
 xo and [180+a;o] 
 xo and [180 + xo] 
 
 sin x= a 
 cos x = a 
 tan x = o 
 cot x= a 
 
 sin xo = a 
 cos xo = a 
 
 tan xo = a 
 cot xo = a 
 
 [180 + a:o]and 
 180 xo and 
 180 -xo and 
 180 -xo and 
 
 360-a:o 
 180 + xo 
 360 -xo 
 360 -xo 
 
 
 The angles enclosed in brackets lie outside the range from deg. to 180 deg., and hence 
 cannot occur as angles in a triangle. 
 
 For solution of trigonometric equations, see p. 118. 
 
FORMAL TRIGONOMETRY 
 
 131 
 
 Relations Between the Functions of a Single Angle. (See Fig. 5.) 
 sin x 1 cos x 
 
 cos x 
 
 V 1 sin 2 x 
 
 '1 + tan 2 x VI + cot 2 x 
 Functions of Negative Angles, sin ( x) = sin 
 
 cos ( x) = cos x; tan ( x) = tan x. 
 
 Functions of the Sum and Difference of Two Angles. 
 
 sin (x + y) = sin x cos y + cos sin y; 
 
 cos (x + 2/) = cos x cos y sin x sin y; 
 
 tan (x -f j/) = [tan x + tan y]/[l tan x tan j/]; 
 
 cot (x + y) = [cot x cot y l]/[cot x + cot y\\ 
 
 sin (x y) = sin x cos y cos x sin y; 
 
 cos (x y} cos x cos y + sin x sin y\ 
 
 tan (x y) = [tan x tan 2/]/[l + tan x tan j/]; 
 
 cot (x y) = [cot xcoty + l]/[cot # cot x]; 
 
 sin x + sin y = 2 sin ^(x + y) cos ^(x j/) ; 
 
 sin x sin y = 2 cos ^(x + y) sin >(x y); 
 
 cos x + cos y = 2 cos y*(x + j/) cos ^(x y}\ 
 
 cos x cos y = 2 sin W(x + #) sin \i(x y); 
 
 sin (x + y) sin (x + y) 
 
 tan & -f tan y = - ; cot x + cot j/ = 
 
 cos x cos y sin x sin |/ 
 
 FIG. 5. 
 
 sin (x y) 
 
 cot x cot y 
 
 x) 
 
 tan x tan y = 
 
 cos x cos j/ sin x sin y 
 
 sin 2 x sin 2 2/ = cos 2 y cos 2 x = sin (x + y) sin (x y); 
 
 cos 2 x sin 2 y = cos 2 |/ sin 2 x = cos (x + y) cos (x y) ; 
 
 sin (45 + x) = cos (45 - x) ; tan (45 + x) = cot (45 - x) ; 
 
 sin (45 - x) = cos (45 + x): tan (45 - x) = cot (45 + x). 
 In the following transformations, a and b are supposed to be positive, 
 c = V^ 2 + b 2 , A = the positive acute angle for which tan A = a/b, and 
 B = the positive acute angle for which tan B = b/a: 
 
 a.cos x + b sin x = c sin (A + x) = c cos (B x) ; 
 
 a cos x 6 sin x = c sin (A x) = c cos (B + x). 
 
 Functions of Multiple Angles and Half Angles. 
 
 sin 2x = 2 sin x cos x; sin x = 2 sin MX cos ^x; 
 cos 2x = cos 2 x sin 2 x = 1 2 sin 2 x = 2 cos 2 x 1 ; 
 2 tan x cot 2 x 1 
 
 tan2x 
 
 1 - tan 2 x' 
 
 cot 2x 
 
 2 cot x 
 
 3 tan x tan 3 x 
 sin 3x = 3 sin x 4sm 3 x; tan 3x = 
 
 cos 3x = 4 cos 3 x 3 cosx; 
 
 1-3 tan 2 x 
 
132 TRIGONOMETRY 
 
 sin (n*) = n sin x cos w ~ 1 x (n)a sin 3 * cos n ~ 3 * 
 
 + (n)s sin 5 a; cos n ~ 6 
 
 cos (n*) = cos n a; (n)z sin 2 a; cos n - 2 * + (n) 4 sin 4 a; cos n ~ 4 a 
 where (n) 2 , (n) 8 , . . . are the binomial coefficients (see p. 39). 
 sin M x = \A$(1 cos a;); 1 cos x =2 sin 2 MX; 
 
 cos MX = + V^(l + cos *); 1 -f- cos* = 2 cos 2 
 
 ,_ cos * sin * 1 cos * 
 
 tan . _ 
 
 cos * sin * 
 
 /* \ /I 4- sin -r 
 
 tan (a +45 C 
 
 sin * 
 
 Here the+ or sign is to be used according to the sign of the left-hand 
 side of the equation. 
 
 Relations Between Three Angles Whose Sum is 180. , 
 
 sin A + sin B + sin C = 4 cos MA cos MB cos MC; 
 
 cos A + cos B + cos C = 4 sin MA sin MB sin MC + 1 ; 
 
 sin A -\- sin B sin C = 4 sin MA sin J&B cos MC; 
 
 cos A -j- cos B cos C = 4 cos ^ A cos MB sin }C 1 ; 
 
 sin 2 A + sin 2 J5 + sin 2 C = 2 cos A cos cos C + 2; 
 
 sin 2 A + sin 2 B sin 2 C = 2 sin A sin B cos C; 
 
 tan A + tan B + tan C = tan A tan B tan (7; 
 
 cot^A+ cot MB + cot ^(7 = cot MA cot 1B cot MC; 
 
 cot A cot B + cot A cot (7 + cot B cot (7 = 1; 
 
 sin 2A + sin 2B + sin 2(7 = 4 sin A sin B sin C; 
 
 sin 2A -j- sin 2B sin 2C =4 cos A cos B sin C. 
 
 Inverse Trigonometric Functions. The notation sin" 1 * (read: anti- 
 sine of *, or inverse sine of *; sometimes written arc sin *) means the prin- 
 cipal angle whose sine is *. Similarly for cos" 1 *, tan" 1 *, etc. (The prin- 
 cipal angle means an angle between 90 and +90 in case of sin" 1 and 
 tan" 1 , and between and 180 in the case of cos" 1 .) For graphs, see p. 174. 
 
 SOLUTION OF PLANE TRIANGLES 
 
 The "parts" of a plane triangle are its three sides, a, 6, c, and its three 
 angles A, B, C (A being opposite a, B opposite b, C opposite c, and A + 
 B + (7 = 180). A triangle is, in general, determined by any three parts 
 (not all angles). To "solve" a triangle means to find the unknown parts 
 from the known. The fundamental formula? are: 
 
 Law of sines : - = . Law of cosines : c 2 = a 2 + b 2 2ab cos C. 
 b smB 
 
 Right Triangles. Use the definitions of the trigonom- 
 etric functions, selecting for each unknown part a relation 
 which connects that unknown with known quantities; then 
 solve the resulting equations. Thus, in Fig. 6, if C = 90, 
 then A + B - 90, c 2 = a 2 + 6 2 , 
 
 sin A = a/c, cos A = 6/c, tan A = a/b, cot A = b/a. 
 
 If A is very small, use tan MA = \/(c 6)/(c + 6). 
 
 Oblique Triangles. There are four cases. It is highly desirable in all 
 these cases to draw a sketch of the triangle approximately to scale before 
 commencing the computation, so that any large numerical error may be 
 readily detected. 
 
 Case 1. GIVEN Two ANGLES (provided their sum is < 180 deg.), AND ONE 
 
SOLUTION OF PLANE AND SPHERICAL TRIANGLES 
 
 133 
 
 SIDE (say o, Fig. 7). The third angle is known, since A 
 
 To find the remaining sides, use b 
 
 a sin B 
 
 a sin C 
 
 + B + C 
 
 A 
 
 = 180. 
 
 sin A sin A 
 
 Or, drop a perpendicular from either B or C on the opposite 
 side, and solve by right triangles. -p IQ 7 
 
 Check: c cos B + 6 cos C = a. 
 
 Case 2. GIVEN Two SIDES (say a and 6), AND THE INCLUDED ANGLE 
 (C) ; and suppose a > 6. Fig. 8. 
 
 First Method: Find c from c 2 = a 2 + 6 2 2ab cos C [or c 2 = (o 6) 2 + 
 2a& vers C]; then find the smaller angle, 5, from sin B = (6/c) sin C; and 
 finally, find A from A = 180 (B + C). Check: a cos B + b cos A = c. 
 
 Second Method: Find (A B) from the law of tangents: 
 
 tad #(A - J3) = [(a - 6) /(a + 6)] cot &C, 
 and tf(A + B) from #(A + B) = 90 C/2; hence A = 
 H(A + ) + #(A -5) and B = y*(A + B) -\<i(A - B). 
 Then find c from c = o sin C/sin A or c = 6 sin C/sin B. jr IQi g. 
 
 Check: o cos B + 6 cos A = c. 
 
 Third Method: Drop a perpendicular from A to the opposite side, and 
 solve by right triangles. 
 
 Case 3. GIVEN THE THREE SIDES (provided the largest is less than the 
 sum of the other two), Fig. 9. 
 
 First Method: Find the largest angle A (which may be acute or obtuse) 
 from cos A = (6 2 + c 2 - a 2 )/26c {or vers A = [a 2 - (6 - c) 2 ]/26c} and 
 then find B and C (which will always be acute) from sin B = b sin A/o and 
 sin C = c sin A/a. Check: A + B + C = 180. 
 
 Second Method: Find A, B, and C from tan \<iA = r/(s a), 
 
 tan \<iB = r/(s 6), tan tfC = r/(s c), where s = ^(a + 6 + c), and 
 
 + B +C 
 
 r =V(s - a)(s - 6)(s - c)/s. Check: A 
 Third Method: If only one angle, say A, is required, use 
 sin WA = A/(s b)(s c)/bc or 
 
 180. 
 
 cos l$A = -vs(s a) /be, 
 according as ^A is nearer or nearer 90. 
 
 Case 4. GIVEN Two SIDES (say 6 and c) AND THE ANGLE FIG. 9. 
 
 OPPOSITE ONE OF THEM (B). This is the "ambiguous 
 case" in which there may be two solutions, or one, or none (see Fig. 10). 
 
 B acute 
 
 c* c/ 
 
 'NoSoIution) (OneSolution) (TwoSolutions) 
 
 C, 
 
 (One Solution) (One Solution) 
 
 B obtuse 
 
 
 
 -v 
 
 (NoSolution) (No5olution) (NoSoMion) (No Solution) 
 
 FIG. 10. 
 
 (One Solution) 
 
 First, try to find C from sin C = c sin B/b. If sin C> 1, there is no solution. 
 If sin C = 1, C = 90 and the triangle is a right triangle. If sin C< 1, 
 this determines two angles C, namely, an acute angle Ci, and an obtuse angle 
 Cj = 180 Ci. Then Ci will yield a solution when and only when 
 
134 TRIGONOMETRY 
 
 Ci +B < 180 (see Case 1); and similarly C 2 will yield a solution when 
 
 and only when Cz + B < 180 (see Case 1). 
 
 Other Properties of Triangles. (See also p. 99 and p. 105.) 
 
 Area = tyab sin C = \/s(s a) (s b)(s c) = rs, where s = 3,i(a + b -f- c), 
 
 and r =radius of inscribed circle = \/(s a)(s 6)(s c)/s. 
 Radius of circumscribed circle = R, where 
 
 2R = a/sin A = 6 /sin B = c/sin C; r = 4R sin 4 sin ^ sin ? = ^-. 
 
 222 4/ts 
 
 The length of the bisector of the angle C is 
 
 - c) Vab[(a + b) 2 - c 2 ] 
 
 a +b a + 6 
 
 The median from C to the middle point of c is m = }S\/2(a 2 + 6 2 ) c 2 . 
 
 SOLUTION OF SPHERICAL TRIANGLES 
 
 For the occasional solution of a spherical triangle the following formulae 
 will be sufficient. For a detailed discussion, see any text-book on spherical 
 trigonometry. 
 
 Let a, b, c be the eides-of the spherical triangle, that is, portions of arcs of 
 great circles of the sphere; and let A, B, C be the angles of the triangle, that is, 
 the angles made by tangents drawn to the sides at their points of intersection 
 on the sphere. The sum of the angles will always be greater than two right 
 angles, and may be nearly six right angles. The angle E = A + B + C 
 180 is called the spherical excess of the triangle. (See also p. 100.) 
 
 sin a sin b sin 6 sin c sin c sin a 
 
 sin A sin B ' sin B sin C 1 sin C sin A 
 
 cos a = cos 6 cos c + sin b sin c cos A, 
 with similar formulae for cos b and cos c. 
 
 cos A = cos B cos C + sin B sin C cos a, 
 with similar formulae for cos B and cos C. 
 
 In the special case of a right spherical triangle, in which C = 90, 
 
 cos A cos B 
 
 cos c = cos a cos o =cotA cotB; cos a = - ; cos o = -; 
 
 sin B sin A 
 
 sin a tan b tan a 
 
 sin A = ; cos A = ; tan A = 
 
 sin c tan c sin o 
 
 The area of a spherical triangle _ spherical excess 
 area of a great circle 180. 
 
HYPERBOLIC FUNCTIONS 135 
 
 HYPERBOLIC FUNCTIONS 
 
 The hyperbolic sine, hyperbolic cosine, etc., of any number x, are 
 functions of x which are closely related to the exponential e x , and which have 
 formal properties very similar to those of the trigonometric functions, sine, 
 cosine, etc. Their definitions and fundamental properties are as follows 
 (see also p. 127; graphs, p. 175; table, p. 60; series, p. 161): 
 
 sinh x = }$(e x e~ x ); cosh a; = li(e x + e~ x '); tanh a; = sinh a; /cosh Z; 
 
 csch x = 1/sinh x; sech x = 1/cosh x; coth x = 1/tanh x; 
 
 cosh 2 x sinh 2 x =1; 1 tanh 2 x = sech 2 x; 1 coth 2 x = csch 2 x; 
 
 sinh ( x) = sinh x\ cosh ( x) = cosh x; tanh ( x) = tanh x\ 
 
 sinh {x !/) = sinh x cosh y cosh x sinh y; 
 
 cosh (x + y) = cosh x cosh y sinh x sinh y; 
 
 tanh (a; !/) = (tanh x tanh !/)/(! tanh x tanh y); 
 
 sinh 2x = 2 sinh x cosh x; cosh 2x = cosh 2 - a; + sinh 2 x ; 
 
 tanh 2x = (2 tanh x) /(I + tanh 2 x) ; 
 
 sinh^z =\As (cosh a; 1); cosh ftx = V^cosh a; + 1); 
 
 tanh MX = (cosh x l)/(sinh x) = (sinh o;)/(cosh x + 1). 
 
 The inverse hyperbolic sine of y, denoted by sinh"" 1 !/, is the number 
 whose hyperbolic sine is y; that is, the notation x = sinh" 1 !/ means 
 sinh x = y. Similarly for cosh- 1 !/, tanh- 1 !/, etc. These functions are closely 
 related to the logarithmic function, and are especially valuable in the integral 
 calculus. For graphs, see p. 175. 
 
 sinh-^/a) = log e (y + Vy + o 2 ) log e a; 
 
 cosh- l (y/a) = log e (y + VV 2 a 2 ) log e a; 
 
 y f_l-7y 7/ 1J -\- d 
 
 tanh- 1 - = ^log e -^-^; coth- 1 - = ^log, ^ ! 
 a a y a y a 
 
 The anti-gudermannian of an angle w, denoted by gd"" 1 ^, is a number 
 defined by gd~ ] w = log e tan ( l Air + tfu) = J'sec u du. When u is small, 
 gd~ l u = u + y*u* 
 
ANALYTICAL GEOMETRY 
 
 THE POINT AND THE STRAIGHT LINE 
 
 Rectangular Co-ordinates (Fig. 1). Let Pi = (xi, yi), P 2 = (22, 1/2). 
 Then, distance PiP2 = V(^i ^) 2 + (y\ 2/2)*; slope of PiP 2 = m tan u 
 = (Zft 2/i)/(2 xi); co-ordinates of mid-point are x = li(xi + xz), 
 y = Yt(yi + 2/2); co-ordinates of point (l/w)th of the way from Pi to Pa are 
 x = xi + (1AO(Z2 - si), / = 2/1 + (l/n)(2/ 2 - 3/1). 
 
 Let mi, mi be the slopes of two lines; then, if the lines are parallel, mi = m*; 
 if the lines are perpendicular to each other, mi = 1/mz. 
 
 Equations of a Straight Line. 
 
 1. Intercept Form (Fig. 2) : \- - = 1. (a, b = intercepts of the line on 
 
 a o 
 the axes.) 
 
 2. Slope Form (Fig. 3) : y = mx + b. (m = tan u = slope; 6 = inter- 
 cept on the 2/-axis; see also Fig. 2, p. 174.) 
 
 3. Normal Form (Fig. 4) : x cos v -f y sin v = p. (p = perpendicular 
 from origin to line; v = angle p makes with the x-axis.) 
 
 4. Parallel-intercept Form (Fig. 5) : - - = - (6, c = intercepts on 
 
 C ~"~ O nJ 
 
 two parallels at distance k apart.) 
 
 FIG. 1. FIG. 2. FIG. 3. 
 
 5. General Form: Ax + By + C = 0. [Here a = - C/A, b = - C/B, 
 m = A/B, cos v = A/R, sin v = B/R, p = - C/R, where R = \/4 2 + .B a 
 (sign to be so chosen that p is positive).] 
 
 6. Line Through (xi, yi) with Slope m: y yi = m(x xi). 
 
 7. Line Through (xi, yi) and (xz, yi): y yi = - - 1 (x an). 
 
 Xz zi 
 
 8. Line Parallel to a;- axis: x = a; to y-axis: y = b. 
 Angles and Distances. 
 
 If u *= angle between two lines whose slopes are mi, mz, then 
 mz mi If parallel, mi = mz. 
 
 1 + mjmi If perpendicular, mim2 = 1. 
 
 If u = angle between the lines Ax + By + C = and A'x + J5'i/ + C' = 0, 
 then 
 
 A A' + BE' If parallel, A /A' = B/B'. 
 
 COS M = - ' 
 
 + \/(A 2 + 2 ) (A /2 +5' 2 ) If perpendicular, A A' -f 5B' = 0. 
 The equations of the bisectors of the angles between the two lines just 
 mentioned are 
 
THE POINT AND THE STRAIGHT LINE; THE CIRCLE 
 
 137 
 
 The equation of a line through (xi, y\) and meeting a given line y = mx + b 
 at an angle u, is 
 
 ra + tan u 
 
 y 2/i = ; (z zO- 
 
 1 ra tan w 
 
 The 'distance from (XQ, y$) to the line Ax + By + C = is 
 
 where the vertical bars mean "the absolute value of." 
 
 The distance from (XQ, j/o) to a line which passes through (xi, y\) and makes 
 an angle u with the z-axis, is 
 
 D = (XQ xi) sin u (2/0 2/0 cos u. 
 
 Polar Co-ordinates (Fig. 6). Let (x, y) be the rec- 
 tangular and (r, 0) the polar co-ordinates of a given 
 point P. Then x = r ccys 0; y = r sin 6; x* + y z = r 2 . 
 
 Transformation of Co-ordinates. If origin is moved 
 to point (XQ, 2/0), the new axes being parallel to the old, fiQ. 6. 
 
 x XQ + x', y = yo + y'> 
 
 If axes are turned through the angle u, without change of origin, 
 
 x = x' cos u y f sin u, y = x' sin u + y' cos u. 
 
 THE CIRCLE 
 
 (See also pp. 99, 103-105, 106) 
 
 Equation of Circle with center (a,6) and radiu a r: 
 (x - a) 2 + (y - 6)2 = r 2 . 
 
 If center is at the origin, the equation becomes x* + j/ 2 = r 2 . If circle 
 goes through the origin and center is on the z-axis at point (r, 0), equation 
 becomes z 2 + y 2 = 2rx. The general equation of a circle is 
 
 x 2 + 2/ 2 + Dx + Ey +F = 0; it has center at ( -D/2, -E/2), and 
 
 radius =\/(-D/2) 2 + (E/2) 2 F (which may be real, null, or imaginary). 
 
 The equation of the radical axis of two circles, x 2 + y 2 + Dx + Ey + 
 F = and x 2 + y 2 + .D'z + JE's/ + F 1 =0, is (Z> - D')x + (E E'}y + 
 (F F') = 0: The tangents drawn to two circles from any point of their 
 radical axis are of equal length. If the circles intersect, the radical axis 
 passes through their points of intersection (see p. 100). 
 
 The equation of the tangent to x 2 + y 2 = r 2 at (xi, 2/1) is xix + y\y = r 2 . 
 The tangent tp x 2 + y 2 + Dx + Ey + F = at (xi, y\) is 
 xix + yiy + yzD(x + a*) + ftE(y +2/1) + F = 0. The line y = mx + b 
 will be tangent to the circle x 2 -}- y* =* r z if b = a\/l + m 2 . 
 
 Equations of Circle in Parametric Form. It is sometimes convenient 
 to express the co-ordinates x and y of the moving point P 
 (Fig. 7) in terms of an auxiliary variable, called a parameter. 
 Thus, if the parameter be taken as the angle u which the 
 radius OP makes with the z-axis, then the equations of the 
 circle in parametric form will be a; = acosu;y = asinu. For 
 every value of the parameter w, there corresponds a point 
 (x, y) on the circle. The ordinary equation x 2 + j/ 2 = a 2 can 
 be obtained from the parametric equations by eliminating u. 
 
 FIG. 7. 
 
138 
 
 ANALYTICAL GEOMETRY 
 
 THE PARABOLA 
 
 The parabola (see also p. 107) is the locus of a point which moves so that its 
 distance from a fixed line (called the directrix) is always equal to its distance 
 from a fixed point F (called the focus) . See Fig. 8. The point half-way from 
 iocua to directrix ia the vertex, O, The line through the focus, perpen- 
 dicular to the directrix, is the principal axis. The breadth of the curve 
 at the focus is called the latus rectum, or parameter, = 2p, where p is the 
 distance from focus to directrix. (Compare also Fig. 3, p. 174..) 
 
 H 
 
 s 
 
 
 < x *\PS 
 
 ip^ 
 
 V 
 
 
 
 I 
 
 /YlS 
 
 / 'V X 
 
 pi v 
 
 Directrix 
 
 P 
 T 
 
 \ 
 
 Fic 
 
 F M 
 p 
 
 N 
 
 u 8. 
 
 A/ 
 
 FIG. 9. 
 
 FIG. 10. 
 
 Any section of a right circular cone made by a plane parallel to a tangent plane 
 of the cone will be a parabola. 
 
 Equation of Parabola, origin at vertex (Fig. 8) : j/ 2 = 2px. 
 
 Polar Equation of Parabola, referred to F as origin and Fx as axis 
 (Fig. 9): r = p/(l - cos 6). 
 
 Equation Referred to the Tangents at the ends of the latus rectum a? 
 axes (Fig. 10) : x^ + y^ = o^, where a = p\/2. 
 
 \ 
 
 FIG. 11. 
 
 FIG. 12. 
 
 FIG. 13. 
 
 The 
 
 Equation of Tangent to y 2 = 2px at (xi,yi): y\y = p(x + xi). 
 line y = mx + b will be tangent to y* = 2px if 6 = p/(2m). 
 
 The tangent PT at any point P bisects the angle between PF and PH 
 (Fig. 8). A ray of light from F, reflected at P, will move off parallel to the 
 principal axis. The subtangent, TM , ia bisected at O. The subnormal, 
 M N, ia constant, and equal to p. The locus of the foot of the perpendicular 
 from the focus on a moving tangent ia the tangent at the vertex (Fig. 11). 
 The locus of the point of intersection of perpendicular tangents is the directrix 
 (Fig. 12). The Iocua of the mid-points of a set of parallel chords whose 
 slope is m is a straight line parallel to the principal axis at a distance 
 
THE PARABOLA 
 
 139 
 
 and is called a diameter (Fig. 13). If M is the mid-point of a chord PQ, 
 and if T is the point of intersection of the tangents atP and Q, then TM is 
 parallel to the principal axis, and is bisected by the curve (Fig. 13). 
 
 To Construct a Tangent to a Given Parabola. (1) At a given point of 
 contact, P (Fig. 14): Find T so that OT = OM, or FT = FP. Then TP is 
 the tangent at P. Or, make MAT = p = 2(OF); ihenPN is the normal atP. 
 
 (2) From a given external point, Q (Fig. 15) : With Q as center and radius 
 QF draw circle cutting the directrix in H ; draw HP parallel to principal axis; 
 then P is required point of contact. As check, note that QP is the perpen- 
 dicular bisector of FH. 
 
 M N 
 
 FIG. 14. 
 
 FIG. 15. 
 
 FIG. 16. 
 
 To Construct a Parabola. 1. GIVEN ANT Two POINTS, P AND Q, THE 
 TANGENT PT AT ONE OP THEM, AND THE DIRECTION OF THE PRINCIPAL Axis 
 OX. In Fig. 16, let K be a variable point on a line through Q parallel to 
 OX. Draw KR parallel to PT (meeting PQ in R), and draw RS parallel to 
 OX (meeting PK in S) ; then S is a point of the curve. NOTE. A line through 
 P parallel to the principal axis bisects all chords parallel to the tangent PT. 
 
 2. GIVEN THE VERTEX O AND Focus F. (a) In Fig. 17 draw Oy perpen- 
 dicular to OF, and slide the vertex of a right angle along Oy so that one side 
 always passes through F; then the other side will always be a tangent to the 
 parabola. 
 
 O , F 
 
 FIG. 17. 
 
 FIG., 18. 
 
 (6) Take a piece of paper (Fig. 18) with a straight edge, d, and mark a 
 point F near the edge. Let K be a variable point of the edge, and fold the 
 paper so that K coincides with F. The crease will be a tangent to the parabola 
 which has focus F and directrix d. 
 
 (c) In Fig. 19, let M be a variable point of the principal axis, and lay off 
 MN = 2(OF) p. WithF as center and radius FN draw a circle, cutting the 
 perpendicular at M in P. Then P is a point of the curve, and PT and PN are 
 the tangent and normal at P. 
 
 3. GIVEN Two TANGENTS AND THEIR POINTS OF CONTACT, P AND Q 
 (Fig. 20) . Divide TP and QT into any number of equal parts (here 4) . Then 
 the lines 11, 22, 33, . . . will be tangents to the parabola. This method is 
 especially advantageous in drawing rather flat arcs. 
 
140 
 
 ANALYTICAL GEOMETRY 
 
 The Radius of Curvature of y* = 2px at a point P = (x,y) is R = 
 (p + 2z)^/\/p, or R = p/sin 3 v, where v = the angle which the tangent at 
 P makes withPF (Fig. 21). At the vertex, R = p. To construct the radius 
 of curvature at any point P, lay off PR = 2(PF) parallel to the principal axis, 
 and draw RC perpendicular to the axis, meeting the normal, PN, in C. Then 
 Cis the center of curvature for the point P, and a circle about C with radius CP 
 will coincide closely with the parabola in the neighborhood of P. 
 
 FIG. 20. 
 
 FIG. 21. 
 
 THE ELLIPSE 
 
 The ellipse (see also p. 107) has two foci, F and F' (Fig. 22) , and two direc- 
 trices, DH and D'H'. If P is any point of the curve.PF +PF' is constant, =2a; 
 and PF/PH (or PF'/PH') is also constant, -e, where e is the eccentricity 
 (e<l). Either of these properties may be taken as the definition of the 
 curve. The relations between e and the semi-axes a and b are as shown in 
 Fig. 23. Thus, 6 2 = o 2 (l - e 2 ), ae = Va 2 - V, e* = 1 - (6/a) 2 . The 
 semi-latus rectum = p = a(l e 2 ) = 6 2 /o. Note that b is always less 
 than a, except in the special case of the circle, in which 6 = a and e = 0. 
 
 y 
 
 FIG. 23. 
 
 FIG. 24. 
 
 FIG. 25. 
 
 Any section of a right circular cone made by a plane which cuts all the 
 elements of one nappe of the cone will be an ellipse; if the plane is perpen- 
 dicular to the axis of the cone, the ellipse becomes a circle. 
 
 Equation of Ellipse, center as origin: 
 
 +_-=!, or y \/o 2 a; 2 . 
 a o a 
 
 If P = (x, y) ia any point of the curve, PF - a + ex, PF' = a "- ex. 
 
 Equations of the Ellipse in Parametric Form: x = a cos u, y = 
 6 sin u, where u is the eccentric angle of the point P = (z.j/). See Fig. 28. 
 
THE ELLIPSE 
 
 141 
 
 Polar Equation, focus as origin, axes as in Fig. 24: r = p/(l e cos 0). 
 
 Equation of the Tangent at (xi,yi) : b*xix + a*yiy = a 2 6 2 . 
 
 The line y = mx + k will be a tangent if k = + A/o 2 m 2 + & 2 . The normal 
 at any point P bisects the angle between PF and PF" (Fig. 25). The locus of 
 the foot of the perpendicular from the focus on a moving tangent is the circle 
 on the major axis as diameter (Fig. 26). The locus of the point of intersection 
 of perpendicular tangents is a circle with radius V a 2 + 6 2 (Fig. 27). 
 
 FIG. 26. 
 
 FIG. 27. 
 
 FIG. 28. 
 
 FIG. 29. 
 
 Ellipse as a Flattened Circle. Eccentric Angle. If the ordinates in 
 a circle are diminished in a constant ratio, the resulting points will lie on 
 an ellipse (Fig. 28). If Q traces the circle with uniform velocity, the corre- 
 sponding point P will trace the ellipse, with varying velocity. The angle u in 
 the figure is called the eccentric angle of the point P. 
 
 Conjugate Diameters are lines through the center, each of which bisects 
 all the chords parallel to the other (Fig. 29). If mi and mz are the slopes, then 
 m\mz = & 2 /o 2 . One pair of conjugate diameters are the diagonals of the 
 rectangle circumscribing the ellipse. The eccentric angles of the ends of two 
 conjugate diameters differ by 90 deg. Thus (Fig. 30), if CQ and CQ' are 
 perpendicular radii in the circle, CP and CP f will be conjugate semi-diameters 
 in the ellipse. A parallelogram formed by tangents drawn parallel to a pair of 
 conjugate diameters has a constant area, = 4a& (Fig. 31). Also, if a', b' 
 are conjugate semi-diameters, and w the angle between them, then a' 2 -f &' 2 = 
 a 2 + 6 2 and a'b' = a&/sin w. 
 
 FIG. 
 
 FIG. 31. 
 
 FIG. 32. 
 
 FIG. 33. 
 
 To Construct a Tangent to a Given Ellipse. (1) AT A GIVEN POINT OF 
 CONTACT, P. Bisect the angle between the focal radii PF and PF' (Fig. 25). 
 
 (2) FROM A GIVEN EXTERNAL POINT, R. (a) Through R draw any two 
 lines cutting the ellipse, one in A and B, the .other in C and D (Fig. 32). 
 Through the point of intersection of AD and BC and the point of intersec- 
 tion of AC and BD, draw a line cutting the ellipse in P and Q. Then P and Q 
 are the required points of contact. (&) With R as a center and radius RF, 
 draw an arc; with F' as center and radius 2a draw an arc, intersecting the 
 first in S; and let SF' meet the curve in T. Then T is the point of contact 
 (Fig. 33). 
 
142 
 
 ANALYTICAL GEOMETRY 
 
 To Construct an Ellipse, Given a and b. (1) In Fig. 34, with O as 
 center, draw circles with radii a and b (and also a third circle with radius 
 0+6). Let a variable ray through O cut these circles in J, K (and 5) ; 
 through J and K draw parallels to the axes, meeting in P. Then P is a point 
 of the ellipse (and SP is the normal atP). 
 
 (2) In Fig. 35, let P divide a line AB so that PA = a and P5 = 6. Then if 
 A and JS slide on the axes, P will describe an ellipse. 
 
 FIG. 34. 
 
 8 
 
 FIG. 35. 
 
 FIG. 36. 
 
 (3) In Fig. 36, let PBA be a straight line such that PA = a and PB = b. 
 Then if A and B slide on the axes, P will trace an ellipse. (Use a strip of 
 paper, with the points P, B, and A marked on it.) 
 
 (4) Find the foci, F and F', by striking an arc of radius a with center at B 
 (Fig. 37) . Drive pins atF, F', and B, and adjust a loop of thread around them. 
 Then remove the pin at B, and replace it by a pencil point; by moving the 
 pencil so as to keep the string taut, the complete ellipse can be drawn at one 
 sweep. Or, use a mechanical ellipsograph. 
 
 (5) and (6). Apply methods (1) and (2) of the following paragraph to 
 the special case in which OP and OQ are perpendicular semi-axes. 
 
 FIG. 37. 
 
 FIG. 38. 
 
 FIG. 39. 
 
 To Construct an Ellipse, Given a Pair of Conjugate Semi-diameters, 
 OP and OQ. (1) Complete the parallelogram, as in Fig. 38. Divide QD 
 and QO into n equal par ts, 1, 2,3, . . . and 1', 2', 3', . . . ConnectPwith 
 1, 2, 3, . . . andP' with 1', 2', 3' . ... The points of intersection of corrfe- 
 sponding lines will be points of the ellipse. 
 
 (2) Take any point K onPQ (Fig. 39). Draw EKU, and draw KV parallel 
 to OP. Then UV will be a tangent. By varying K alongPQ as many tangents 
 may be drawn as desired, thus "enveloping" the ellipse. 
 
 (3) Through P (Fig. 40), draw a perpendicular PT to OQ, and lay off PR = 
 PS = OQ. Then if the line RPT is made to slide with one end on OR and the 
 other on OQ, P will trace the ellipse. Further, the bisectors of the angle 
 ROS show the directions of the principal axes, and OR + OS = 2a and 
 
THE ELLIPSE 
 
 143 
 
 OR OS =* 26. Also, if a line through P perpendicular to RS (and there- 
 fore tangent to the ellipse atP) meets the minor axis in M, a circle with M as 
 center and MR or MS as radius will cut the major axis in the two foci. 
 
 FIG. 40. 
 
 FIG. 41. 
 
 FIG. 42. 
 
 To Construct an Ellipse Approximately by Circular Arcs. [Methods 
 (1) and (2) employ two radii, (3) and (4) employ three radii.] (1) In Fig. 41, 
 lay off OL = OA and BS = BL = a - b. Bisect SA in T, and draw THK 
 perpendicular to BA. Then H is one center, with radius HA, and K is the 
 other center, with radius KB. The junction point Q of the two arcs will fall a 
 little outside the true ellipse. 
 
 (2) In Fig. 42, lay off OU = 0V = OB = b. Draw UG perpendicular to 
 the axis and DG at 45. With G as center draw an auxiliary arc with radius 
 
 FIG. 43. 
 
 FIG. 44. 
 
 = AV = a b, and through D draw DMN just touching this arc. Then M 
 is one center (with radius M A) and N is the other center (with radius NB) . 
 The junction point P of the two arcs will be a true point of the ellipse. [E. V. 
 Huntington.] 
 
 (3) Through D (Fig. 43) draw DCiC 3 perpendicular to AB. Call CiA = n 
 and CzB = r 3 . Lay off BE = BO ( =6), and on ED as diameter draw a semi- 
 circle cutting the minor axis in W', then BW = \/o6 = r 2 . Lay off AZ = 
 
144 
 
 ANALYTICAL GEOMETRY 
 
 BW. From Ci with radius C\Z(= r z n), and from 3 with radius C&W 
 ( =r3 ra), draw arcs intersecting in Cz. Draw CzCz extended and CzCi ex- 
 tended. Then draw in the three arcs, with centers at Ci, 2, 3 and radii n, 
 rz, ra. NOTES. Since n and r$ are the radii of curvature of the ellipse at A 
 and B, this construction gives a curve which is a little too sharp at A and 
 a little too flat at B. A more accurate construction is the folio wing: 
 
 (4) In Fig. 44, lay off BE = BH BO = b. Through the mid-point X 
 of BE draw XG perpendicular to the axis, and through D draw DG at an angle 
 of 45 deg. From G as center draw an auxiliary arc with radius = DH 
 (= a b), and through D draw DCiCz just touching this arc. Take CiA 
 as n and CzB as ra. On DE as diameter draw a semi-circle cutting the minor 
 axis in W, and take BW(=\/ab) as r 2 . Lay off AZ = BW. From Ci 
 with radius CiZ( = rz n), and from 
 Cs with radius CzW( = r% rz), draw 
 arcs intersecting in Cz. Then Ci, 
 Cz, Ca are the required centers. [E. 
 V. Huntington.] 
 
 Radius of Curvature of Ellipse 
 at Any Point P = (x, y) is R = 
 o 2 6 2 (o; 2 /a 4 + y z /b*)M = p/sin 3 v, where 
 v is the angle which the tangent at P 
 makes with PF or PF'. At end of 
 major axis, R = 6 2 /a = MA; at end 
 of minor axis, R = a?/b = NB (see 
 Fig. 45). To construct the radius JT IG> 45^ JT IG 45. 
 
 of curvature at any other point P 
 
 (Fig. 46) , draw the normal at P (by bisecting the angle between PF and PF') 
 and let it meet the major axis in N. At N draw a perpendicular toPN meet- 
 ing PF in H . At H draw a perpendicular to PH meeting PN in C. Then C 
 is the center of curvature for the point P, and a circle about C with radius 
 CP will coincide closely with the ellipse in the neighborhood of P. [Note. 
 If the circle of curvature meets the ellipse in Q, then the tangent at P and 
 the line PQ are equally inclined to the axis.] 
 
 THE HYPERBOLA 
 
 The hyperbola (see also p. 107) has two foci, F and F', at distances ae 
 from the center, and two directrices, DH and D'H', at distances a/e from 
 
 FIG. 47. 
 
 FIG 
 
 the center (Fig. 47). If P is any point of the curve, \PF PF'\ is constant, 
 = 2a; &ndPF/PH (orPF'/PH') is also constant, = e (called the eccentricity), 
 where e > 1. Either of these properties may betaken as the definition of the 
 
THE HYPERBOLA 
 
 145 
 
 curve. The curve has two branches which approach more and more nearly 
 two straight lines called the asymptotes. Each asymptote makes with the 
 principal axis an angle whose tangent is b/o. The relations between e, a, and 
 b are shown in Fig. 48: b 2 = o 2 (e 2 1), ae = Va 2 + b 2 , e 2 = 1 + (b/a) 2 . 
 The semi-latus rectum, or ordinate at the focus, is p = o(e 2 1) =b 2 /a. 
 
 Any section of a right circular cone made by a plane which cuts both nappes 
 of the cone will be a hyperbola. (Compare also Fig. 3, p. 174.) 
 
 Equation of the Hyperbola, center as origin: 
 
 - r, = 1. or y 
 
 a* b* 
 
 If P = (x,y) is on the right-hand branch, PF = ex a, PF' = ex+a. 
 If P is on the left-hand branch, PF = -ex +a, PF' = -ex -a. 
 
 Equations of Hyperbola in Parametric Form. (1) x = a cosh u, 
 y b sinh u. (For tables of hyperbolic functions, see pp. 60 and 61.) Here 
 u may be interpreted as A /a 2 , where A is the area shaded in Fig. 49. 
 
 FIG. 49. 
 
 FIG. 50. 
 
 (2) x = a sec v, y = b tan v, where v is an auxiliary angle of no special 
 geometric interest. 
 
 Polar Equation, referred to focus as origin, axes as in Fig. 50: 
 
 r = p/(l e cos 6). 
 
 Equation of the Tangent at (zi,2/i): b*xix a*yiy =o 2 b 2 . 
 The line y = mx + k will be a tangent if k = VVm 2 6 2 . Thetan- 
 
 FIG. 51. 
 
 FIG. 52. 
 
 FIG. 53. \ 
 
 gent at any point P (Fig. 51) bisects the angle between PF andPF'. The locus 
 of the foot of the perpendicular from the focus on a moving tangent is the 
 circle on the principal axis as diameter (Fig. 52). The locus of the point of 
 intersection of perpendicular tangents is a circle with radius \/fl 2 b 2 , which 
 will be imaginary if 6 > a (Fig. 53). 
 10 
 
146 
 
 ANALYTICAL GEOMETRY 
 
 Properties of the Asymptotes. (Fig. 54. ) If P is any point of the curve, 
 the product of the perpendicular distances from P to the two asymptotes is con- 
 stant, = o 2 6 2 /(a 2 + 6 2 ). Also, the product of the oblique distances (the dis- 
 tance to each asymptote being measured parallel to the other) is constant, and 
 equal to }4(o 2 + b z ). If a line cuts the hyperbola and its asymptotes, the 
 parts of the line intercepted between the curve and the asymptotes are equal. 
 The part of a tangent intercepted between the asymptotes is bisected by the 
 point of contact. The triangle bounded by the asymptotes and a variable 
 tangent is of constant area, = ab. If a line through Q perpendicular to the 
 principal axis meets the asymptotes in R and S (see Fig. 54), then QR X QS 
 b*. If a line through Q parallel to the principal axis meets the asymptotes in 
 U and V, then QU X QV = a 2 . 
 
 FIG. 54. 
 
 FIG. 55. 
 
 Conjugate Hyperbolas are two hyperbolas having the same asymptotes 
 with semi-axes interchanged (Fig. 55). The equation of the hyperbola conju- 
 
 x* y* x* y 2 
 
 gate to - 2 -- 2 = 1, is - 2 -- 
 
 1. 
 
 Conjugate Diameters are lines through the center, each of which bisects 
 all the chords parallel to the other a chord which does not meet the given 
 hyperbola being understood to be terminated by the conjugate hyperbola 
 (Fig. 65). If mi and m* are the slopes, then rmmz = 6 2 /a 2 . Each asymptote, 
 regarded as a diameter, is its own conjugate. If a parallelogram is formed 
 by tangents drawn parallel to a pair of conjugate diameters, its vertices will 
 lie on the asymptotes, and its area will be constant = 4a&. If a', b' are 
 conjugate semi-diameters, and w the angle between them, then a /2 b' 2 
 = a 2 6 2 , and a'b' = ab/sin w. 
 
 Equilateral Hyperbola (a =6). Equation referred to principal axes 
 (Fig. 56) : x* y* = a 2 . NOTE, p = a. Equation referred to asymptotes 
 as axes (Fig. 57) : xy = o 2 /2. (See also Fig. 3, p. 174.) 
 
 Asymptotes are perpendicular. Eccentricity = \/2. Any diameter is equal 
 in length to its conjugate diameter. 
 
 y 
 
 Fio. 66. 
 
 FIG. 67. 
 
THE CATENARY 
 
 147 
 
 To Construct a Tangent at any given point P of a hyperbola. In Fig. 58, 
 draw PA and PS parallel to the asymptotes, and take OS = ?(OA) and OT = 
 2(OB). Then ST is the tangent at P. 
 
 FIG. 58. 
 
 To Construct a Hyperbola^ given the asymptotes and any point P. 
 
 (1) In Fig. 59 let TPT' bea variable line throughP, andlayoff T'P f = TP; 
 then P' is a point of the curve. 
 
 (2) In Fig. 60, draw PA and PB parallel to the asymptotes. Lay off 
 OA' = n(OA) and OB' = (\/ri)(OB}, where n is any number; and throughA' 
 and B f draw parallels to the axes; these will meet in a point P' of the curve. 
 
 FIG. 59. 
 
 FIG. 61. 
 
 (3) (Fig. 61.) Take any point K in the ordinate PM, and draw OK 
 meeting the line through P parallel to the z-axis in R. Draw a parallel to 
 the x-axis through K and a parallel to the y-axia through R, meeting in Q. 
 Then Q is a point of the curve. 
 
 THE CATENARY 
 
 The catenary is the curvein which aflexible chain or cord of uniform density 
 will hang when supported by the two ends. Let w = 
 weight of the chain per unit length ; T = the tension 
 at any point P; and Th,T v = the horizontal and 
 vertical components of T. The horizontal com- 
 ponent Th is the same at all points of the curve. 
 
 The length a = Th/w is called the parameter of the 
 catenary, or the distance from the lowest point O to 
 the directrix DQ (Fig. 62). When a is very large, 
 the curve is very flat. For methods of finding o in 
 any given case, see problems 1-6 below. 
 
 The rectangular equation, referred to the lowest 
 point as origin, is y a [cosh (x/a) 1]. (For 
 table of hyperbolic functions, see p. 60.) In case of 
 
 FIG. 62. 
 
148 
 
 ANALYTICAL GEOMETRY 
 
 x 2 
 very flat arcs (a large), y = - 
 
 . . . ; s = x + ^ + 
 
 approximately, 
 
 so that in such a case the catenary closely resembles a parabola. 
 
 If the perpendicular from O to the tangent at P meets the directrix in Q, 
 then DQ = arc OP = s andOQ = y + o. The radius of curvature atP is 
 R = (y +o) 2 /o, which is equal in length to the portion of the normal inter- 
 cepted between P and the directrix. 
 
 Problems on the Catenary (Fig. 62). When any two of the four 
 quantities x, y, s, T/w are known, the remaining two, and also the para- 
 meter a, can be found, as follows: 
 
 ( 1) GIVEN x AND y. Compute y/x, and find from Table 1 the value of the 
 auxiliary variable z. Then compute a = x/z, s = a sinh z, and T 
 wa cosh z. Or, having z, find s/x and wx/T by using Tables 3 and 2 inversely, 
 and hence (since x is known) compute s and T/w without the use of o. 
 
 TABLE 1. GIVING z WHEN y/x is KNOWN. THEN a = x/z 
 
 y/x 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 
 6 
 
 7 
 
 8 
 
 9 
 
 00 
 
 0.0000 
 
 0.0200 
 
 0.0400 
 
 0.0600 
 
 0.0800 
 
 0.0999 
 
 n 
 
 .1199 
 
 0.1398 
 
 0.1597 
 
 0.1795 
 
 01 
 
 0.1993 
 
 0.2191 
 
 0.2389 
 
 0.2586 
 
 0.2782 
 
 0.2978 
 
 
 
 .3173 
 
 0.3368 
 
 0.3562 
 
 0.3756 
 
 02 
 
 0.3948 
 
 0.4140 
 
 0.4332 
 
 0.4522 
 
 0.4712 
 
 0.4901 
 
 r 
 
 .5089 
 
 0.5276 
 
 0.5463 
 
 0.5648 
 
 0.3 
 
 0.5833 
 
 0.6016 
 
 0.6199 
 
 0.6381 
 
 0.6561 
 
 0.6741 
 
 n 
 
 .6919 
 
 0.7097 
 
 0.7274 
 
 0.7449 
 
 4 
 
 0.7623 
 
 0.7797 
 
 0.7969 
 
 0.8140 
 
 0.8311 
 
 0.8480 
 
 n 
 
 .8647 
 
 0.8814 
 
 0.8980 
 
 0.9145 
 
 05 
 
 0.9308 
 
 0.9471 
 
 0.9632 
 
 0.9792 
 
 0.9951 
 
 1.0109 
 
 i 
 
 .0266 
 
 1.0422 
 
 1 .0576 
 
 1.0730 
 
 0.6 
 
 1.0883 
 
 1.1034 
 
 1.1184 
 
 1.1334 
 
 1.1482 
 
 1.1629 
 
 i 
 
 .1775 
 
 1.1920 
 
 1.2064 
 
 1.2207 
 
 NOTE. 
 
 y/x - 
 
 (cosh z 
 
 DA- 
 
 
 
 
 
 
 
 
 (2) GIVEN x AND T/w. Compute wx/T, and find from Table 2 the value 
 of the auxiliary variable z. Then compute a = x/z, y = a. (cosh z l)and 
 a = a sinh z. Or, having z, find y/x and s/x by using Tables 1 and 3 inversely, 
 and hence (since x is known) compute y and s without the use of a. 
 
 TABLE 2. GIVING z WHEN wx/T is KNOWN. THEN a = x/z 
 
 wx/T 
 
 
 
 1 
 
 2 
 
 3 
 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 0.0 
 0.1 
 0.2 
 0.3 
 0.4 
 0.5 
 0.6 
 
 0.0000 
 0.1005 
 0.2042 
 0.3150 
 0.4392 
 0.5894 
 0.8053 
 
 0.0100 
 0.1107 
 0.2149 
 0.3267 
 0.4528 
 0.6068 
 0.8357 
 
 0.0200 
 0.1209 
 0.2256 
 0.3385 
 0.4666 
 0.6249 
 0.8695 
 
 0.0300 
 0.1311 
 0.2365 
 0.3505 
 0.4806 
 0.6436 
 0.9082 
 
 
 
 
 
 
 
 
 
 0400 
 
 1414 
 2474 
 3626 
 4950 
 6632 
 9541 
 
 0.0501 
 0.1517 
 0.2584 
 0.3749 
 0.5097 
 0.6836 
 1.0132 
 
 0.0601 
 0.1621 
 0.2695 
 0.3874 
 0.5248 
 0.7051 
 1.1110 
 
 0.0702 
 0.1725 
 0.2807 
 0.4000 
 0.5403 
 0.7277 
 
 0.0803 
 0.1830 
 0.2920 
 0.4129 
 0.5562 
 0.7517 
 
 00904 
 0.1936 
 0.3035 
 0.4259 
 0.5726 
 0.7775 
 
 NOTE. wx/T = z/cosh z. For every value of wx/T there are two values of z, one 
 less than 1.200 and one greater than 1.200. Only the smaller of these values is tabulated. 
 
 (3) GIVEN x AND s. Compute s/x, and find from Table 3 the value of 
 the auxiliary variable z. Then compute a = x/z, y a (cosh z l),and 
 T = wa cosh z. Or, having z, find y/x and wx/T by using Tables 1 and 2 
 inversely, and hence (since x is known) compute y and T/w without the use 
 of a. 
 
THE CATENARY 
 
 149 
 
 TABLE 3. GIVING z WHEN s/x is KNOWN. THEN a = x/z 
 
 /* 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1.000 
 
 
 0245 
 
 0.0346 
 
 0424 
 
 0.0490 
 
 0.0548 
 
 0.0600 
 
 0.0648 
 
 0.0693 
 
 0.0735 
 
 1 
 
 6 ! 0774 
 
 0.0812 
 
 0.0848 
 
 0.0883 
 
 0.0916 
 
 
 
 0948 
 
 0.0980 
 
 0.1010 
 
 0.1039 
 
 0.1067 
 
 2 
 
 0.1095 
 
 0.1122 
 
 0.1149 
 
 0.1174 
 
 0.1200 
 
 
 
 1224 
 
 0.1249 
 
 0.1272 
 
 0.1296 
 
 0.1319 
 
 3 
 
 0.1341 
 
 0.1363 
 
 0.1385 
 
 0.1407 
 
 0.1428 
 
 
 
 1448 
 
 0.1469 
 
 0.1489 
 
 0.1509 
 
 0.1529 
 
 4 
 
 0.1548 
 
 0.1567 
 
 0.1586 
 
 0.1605 
 
 0.1623 
 
 
 
 1642 
 
 0.1660 
 
 0.1678 
 
 0.1696 
 
 0.1713 
 
 1.005 
 
 0.1731 
 
 0.1748 
 
 0.1765 
 
 0.1782 
 
 0.1799 
 
 o 
 
 1815 
 
 0.1831 
 
 0.1848 
 
 0.1864 
 
 C.1880 
 
 6 
 
 0.1896 
 
 0.1911 
 
 0.1927 
 
 0.1942 
 
 0.1958 
 
 
 
 1973 
 
 0.1988 
 
 0.2003 
 
 0.2018 
 
 0.2033 
 
 7 
 
 0.2047 
 
 0.2062 
 
 0.2076 
 
 0.2091 
 
 0.2105 
 
 
 
 2119 
 
 0.2133 
 
 0.2147 
 
 0.2161 
 
 0.2175 
 
 8 
 
 0.2188 
 
 0.2202 
 
 0.2215 
 
 0.2229 
 
 0.2242 
 
 
 
 2255 
 
 0.2269 
 
 0.2282 
 
 0.2295 
 
 2308 
 
 9 
 
 0.2321 
 
 0.2334 
 
 0.2346 
 
 0.2359 
 
 0.2372 
 
 
 
 2384 
 
 0.2397 
 
 0.2409 
 
 0.2421 
 
 0.2434 
 
 1.01 
 
 0.2446 
 
 0.2565 
 
 0.2678 
 
 0.2787 
 
 0.2892 
 
 0.2993 
 
 0.3091 
 
 0.3186 
 
 0.3278 
 
 3367 
 
 2 
 
 0.3454 
 
 0.3539 
 
 0.3621 
 
 0.3702 
 
 0.3781 
 
 
 
 3859 
 
 0.3934 
 
 0.4009 
 
 0.4082 
 
 U.4153 
 
 3 
 
 0.4224 
 
 0.4293 
 
 0.4361 
 
 0.4428 
 
 0.4494 
 
 
 
 4559 
 
 0.4623 
 
 0.4686 
 
 0.4748 
 
 0.4809 
 
 4- 
 
 0.4870 
 
 0.4930 
 
 0.4989 
 
 0.5047 
 
 0.5105 
 
 
 
 5162 
 
 0.5218 
 
 0.5274 
 
 0.5329 
 
 0.5383 
 
 1.05 
 
 0.5437 
 
 0.5490 
 
 0.5543 
 
 0.5595 
 
 0.5647 
 
 
 
 5698 
 
 0.5749 
 
 0.5799 
 
 0.5849 
 
 0.5898 
 
 6 
 
 0.5947 
 
 0.59% 
 
 0.6044 
 
 0.6091 
 
 0.6139 
 
 
 
 6186 
 
 0.6232 
 
 0.6278 
 
 0.6324 
 
 0.6369 
 
 7 
 
 0.6414 
 
 0.6459 
 
 0.6504 
 
 0.6548 
 
 0.6591 
 
 
 
 6635 
 
 0.6678 
 
 0.6721 
 
 0.6763 
 
 0.6806 
 
 8 
 
 0.6848 
 
 0.6889 
 
 0.6931 
 
 0.6972 
 
 0.7013 
 
 
 
 7053 
 
 0.7094 
 
 0.7134 
 
 0.7174 
 
 0.7213 
 
 9 
 
 0.7253 
 
 0.7292 
 
 0.7331 
 
 0.7369 
 
 0.7408 
 
 
 
 7446 
 
 0.7484 
 
 0.7522 
 
 0.7559 
 
 0.7597 
 
 1.10 
 
 0.7634 
 
 
 
 
 
 
 
 NOTE: 
 
 s/x = 
 
 sinh z/z 
 
 (4) GIVEN y AND a. Then 
 
 S + l *:(?-) 
 
 tanh- 1 | -I i 
 
 (5) GIVEN y AND T/w. Then a = y, x 
 
 w 
 
 y I cosh' 
 
 T/w 
 (T/w) - y 
 
 s = V2y(T/w) - y*. Or, if y/(T/w) is small, 
 
 7 wi 
 ' 12 ~T 
 
 x \ wy 
 , - = g -y , approximately, 
 
 (6) GIVEN s AND T/w. Then a; =- A 1 - ( 
 
 Given the Length 2L of a Chain Supported at Two Points A and 
 B not in the Same Level, to find a. (See Fig. 63; b and c are supposed 
 known.) Let ( \/I/ 2 b 2 ) /c = s/x; enter Table 3 with this value of s/x, and 
 find the corresponding value of the auxiliary variable z. Then a = c/z. 
 
150 
 
 ANALYTICAL GEOMETRY 
 
 NOTE. The co-ordinates of the mid-point M of AB (see Fig. 63) are XQ = 
 a tanh" 1 (b/L), y = (L/tanh z) a, so tha't the position of the lowest point 
 is determined. 
 
 Correction for Sag in Chaining TTphill (Fig. 64). Let I = length of 
 tape (corrected for stretch and temperature), w = weight per unit length of 
 tape, A = angle between the chord AB and the horizontal. 
 
 FIG. 63. 
 
 FIG. 64. 
 
 If the tension P at the upper end is known, compute wl/P and find k from 
 Table 4. If the tension Q at the lower end is known, compute wl/Q and find 
 k from Table 5. In either case, chord AB = l kl. 
 
 TABLE 4. GIVING k 
 
 TABLE 5. GIVING k 
 
 wl 
 
 A=0 10 20 30 40 50 60 70 
 
 A =0 10 20 30 40 50 60 70 
 
 .00000 000 000 000 000 000 000 000 000 
 
 002 002 001 001 001 001 000 000 000 
 
 004 004 003 003 002 002 001 000 000 
 
 007 006 006 005 004 003 002 001 000 
 
 Oil 010 009 008 006 004 003 001 000 
 
 .00015 015 013 012 009 006 004 002 000 
 
 020 020 018 016 012 009 005 003 001 
 
 027 026 024 021 016 012 007 003 001 
 
 034 033 031 026 021 015 009 004 001 
 
 042 041 038 033 026 019 Oil 005 001 
 
 .00051 050 046 040 032 023 014 007 002 
 
 060 060 055 048 038 027 017 008 002 
 
 070 070 065 057 045 032 020 009 002 
 
 082 081 076 066 053 038 023 Oil 003 
 
 094 094 087 076 061 044 027 013 003 
 
 00107 107 100 087 070 050 031 015 004 
 
 ' 121 121 113 099 079 057 035 017 004 
 
 136 136 128 112 090 065 040 019 005 
 
 151 152 143 125 101 073 045 021 006 
 
 168 168 159 140 113 082 050 024 006 
 
 .00000 000 000 000 000 000 000 000 000 
 002 002 001 001 001 001 000 000 000 
 004 004 003 003 002 001 001 000 000 
 007 006 006 005 004 003 002 001 000 
 Oil 010 009 008 006 004 002 001 000 
 
 .00015 014 013 Oil 008 006 004 002 000 
 
 020 020 018 015 Oil 008 005 002 001 
 
 027 026 023 019 015 Oil 006 003 001 
 
 034 032 029 024 019 013 008 004 001 
 
 042 040 036 030 023 016 010 004 001 
 
 .00051 048 043 036 028 019 Oil 005 Oof 
 060 057 051 043 033 023 014 006 002 
 070 067 060 050 038 026 016 007 002 
 082 078 069 057 044 030 018 008 002 
 094 089 079 066 050 035 021 010 002 
 
 .00107 101 090 074 057 039 022 01 1 003 
 
 121 114 101 084 064 044 026 012 003 
 
 136 128 113 092 071 049 029 013 003 
 
 151 142 125 103 079 054 032 015 004 
 
 168 157 138 114 087 060 035 016 004 
 
 NOTE, k = 1 - {[l-\/l - 2m sin u + 
 given by 
 
 sin A]}, where m = wl/P and u is 
 
 [1 \/l 2m sin u + m 2 ] secw = [sinh -1 (tanw) sinh -1 (t&nu m sec w)]tan A. 
 
 Also, Q = P wl (1 k) sin A, where k is the value in Table 4 corresponding to 
 the given values of P and A. 
 
 Correction for Stretch m Chaining Uphill. Let L = unstretched length 
 of tape at working temperature, w = weight per unit length of tape, A = angle 
 
OTHER USEFUL CURVES 
 
 151 
 
 between chord AB and the horizontal, F = area of cross-section, E = Young's 
 modulus of elasticity (for steel, E = 29,000,OOQ Ib. per sq. in.), I = stretched 
 length (along curve). 
 
 If the tension P at the upper end is known, compute wL/P and find m from 
 Table 6. Then I = L + (LP/FE) (1 * m). 
 
 If the tension Q at the lower end is known, compute wL/Q and find nfrom 
 Table 7. Then I = L + (LQ/FE) (1 + n) . 
 
 TABLE 6.' GIVING m TABLE 7. GIVING n 
 
 'L 
 ?- 
 
 4 oT 10 20 30 40 50 60 70 80 90 
 
 wL 
 
 ~Q 
 
 A= 10 20 30 40 50 60 70 80 90 
 
 .00 
 .10 
 .20 
 
 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 
 .001 .010.018.026.033.039.044.047.049.050 
 .003 .021 .038 .053 .067 .078 .088 .094 .099. 100 
 
 .00 
 .10 
 .20 
 
 .000.000.000.000.000.000.000.000 000 
 .008.016.024.032.038.043.047.049.050 
 .014 .03 1 .047 .062 .075 .086 .094 .099 . 1 00 
 
 OTHER USEFUL CURVES 
 
 The Cycloid is traced by a point on the circumference of a circle which rolls 
 without slipping along a straight line. Equations of cycloid, in parametric 
 form (axes as in Fig. 65) \}x = a (rad u sin w), y = a(l cos w), where a is 
 
 FIG. 65. 
 
 FIG. 66. 
 
 the radius of the rolling circle, and rad u is the radian measure of the angle u 
 through which it has rolled. The tangent and normal at any point pass 
 through the highest and lowest points of the corresponding position of the 
 generating circle. The radius of curvature at any point P is PC = 
 4a sin(w/2) .= 2\/2ay = twice thelength of the normal, PJV. Theevolute, 
 or I ocus of centers of curvature, is an equal 
 cycloid. To construct a cycloid (Fig. 66), 
 divide the semi-circumference of the gen- 
 erating circle into n equal parts (here 4) 
 and lay off these arcs along the base (from 
 O to 4'). Describe arcs with centers at 1', 
 2', . . . and radii equal to the chords Ol, 
 O2, . . . , and sketch the cycloid as a curve 
 tangent to ail of these arcs. Or, on hori- 
 zontal lines through 1,2,. . . lay off dis- 
 tances equal to Ol', 02', etc.; the points thus reached will lie on the 
 cycloid. 
 
 The area of one arch = 37ra 2 , length of arc of one arch = 8a. Area 
 bounded by the ordinate of the point P corresponding to any value of u is 
 a 2 ($$ rad u 2 sin u + J4 sin 2w) = % ax ft y\/(2a y)y. Length of 
 arc OP = 4a (1 - cos # w) = 4a - 2V / 2a(2a - y). 
 
 FIG. 67. 
 
152 
 
 ANALYTICAL GEOMETRY 
 
 The Trochoid is a more general curve, traced by any point on a radius 
 of the rolling circle, at distance b from the center (Fig. 67). It is a prolate 
 trochoid if 6 < a, and a curtate or looped trochoid if 6 > a. The equations in 
 either case are x = a rad u b sin u, y = a b cos w. 
 
 The Epicycloid (or Hypocycloid) is a curve generated by a point on the 
 circumference of a circle of radius a which rolls without slipping on the 
 outside (or inside) of a fixed circle of radius c. For the equations, put 
 6 = a in the equations of the epi- (or hypo-) trochoid, below. The normal 
 at any point P passes through the point of contact N of the corresponding 
 position of the rolling circle. To construct the curve (Figs. 68 and 69), 
 
 Epicycloid. 
 FIG. 68. 
 
 Hypocycloid. 
 FIG. 69. 
 
 divide the semi-circumference of the rolling circle into n equal parts, by points 
 1, 2, 3 . . . , and lay off these arcs (Al, A2, A3) along the circumference of the 
 base circle, as Al', A2', A3', .... Describe circles with centers at 1', 2', 3', 
 . . . and radii equal to the chords Al, A2, A3, . . .; then the required 
 curve will be tangent to all these circles. Or, with as center, draw arcs 
 through 1, 2, 3, . . ., meeting the radius OA in 1, 2, 3, . . ..and the 
 radii Ol', O2', 03', . . . in 1", 2", 3", . . .; then from 1", 2", 3", . . . 
 lay off arcs equal to 11, 22, 33, . . . respectively; the points thus 
 reached will be points of the curve. 
 
 The area OAP = 
 
 a(c a) (c 2a) 
 ~2c 
 
 (rad u sin w) , where the upper sign 
 
 applies to the epicycloid, the lower to the hypocycloid, and rad u = the 
 radian measure of the angle u shown in Figs. 68 and 69. Arc AP = 
 (4 o/c)(c a)(l - cos J4 w); arc AD = (4a/c)(c a). [In Fig. 69, D =4".] 
 
 Radius of curvature at any point P is R = -^ '- sin Yiu\ at A, R = 0; 
 
 c 2a 
 
 at ), R 
 
 4o(c o) 
 c 2a 
 
 Special Cases. If a = ^c, the hypocycloid becomes a straight line, diam- 
 eter of the fixed circle (Fig. 70). In this case the hypo trochoid traced by any 
 
OTHER USEFUL CURVES 
 
 153 
 
 point rigidly connected with the rolling circle (not necessarily on the circum- 
 ference) will be an ellipse. If a = He, the curve generated will be the four- 
 cusped hypocycloid, or astroid, (Fig. 71), whose equation is x^ + y^ = 
 
 c^. If a = c, the epicycloid is the cardioid, whose equation in polar co- 
 ordinates (axes as in Fig. 72) is r = 2c(l + cos 0). Length of cardioid = 16c. 
 
 FIG. 70. 
 
 Astroid. 
 FIG. 71. 
 
 Cardioid. 
 FIG. 72. 
 
 The Epitrochoid (or Hypotrochoid) is a curve traced by any point rigidly 
 attached to a circle of radius a, at distance b from the center, when this circle 
 rolls without slipping on the outside (or inside) of a fixed circle of radius c. 
 
 The equations are x = (c a) cos ( ^ u J + 6 cos ( 1 ~ \ u \, 
 
 d) sin \^-u) & sin ( 1 J u .where u = the angle which the 
 
 V = (c 
 
 moving radius makes with the line of centers; take the upper sign for the epi- 
 and the lower for the hypo-trochoid. The curve is called prolate or curtate 
 according as 6 < a or 6 > a. When & = a, the special case of the epi- or hypo- 
 cycloid arises, j 
 
 The Involute of a Circle is the curve traced by the end of a taut string 
 which is unwound from the circumference of a fixed circle, of radius c. If QP 
 
 Involute of Circle. 
 FIG. 73. 
 
 Spiral of Archimedes. 
 FIG. 74. 
 
 is the free portion of the string at any instant (Fig. 73), QP will be tangent to 
 the circle at Q, and the length of QP = length of arc QA ; hence the construe- 
 
154 
 
 ANALYTICAL GEOMETRY 
 
 tion of the curve. The equations of the curve in parametric form (axes as 
 in figure) are x = c(cos u + rad u sin w), y = c (sin u rad u cos w), 
 where rad u is the radian measure of the angle u which OQ makes with the 
 x-axis. Length of arc AP = c(rad w) 2 ; radius of curvature at P is QP. 
 
 The Spiral of Archimedes (Fig. 74) is traced by a point P which, starting 
 from O, moves with uniform velocity along a ray OP, while the ray itself 
 revolves with uniform angular velocity about O. Polar equation: r = 
 k rad 0, or r = a (0/360). Here a = 2irk = the distance, measured along a 
 radius, from each coil to the next. 
 
 In order to construct the curve, draw radii Ol, O 2, O3, . . . making angles 
 
 - (360), -(360), - (360), . . with Ox, and along these radii lay 
 n n n 
 
 123 
 
 off distances equal to - a, - a, - a, . 
 n n n 
 
 the points thus reached will 
 
 lie on the spiral. The figure shows one-half of the curve, corresponding to 
 positive values of 6. 
 
 Construction for tangent and normal: Let PT and PN be the tangent 
 and normal at any point P, the line TON being perpendicular to OP. Then 
 OT = r*/k, and ON = k, where k = a/(2ir). Hence the construction. 
 
 The radius of curvature at P is R = (k 2 + r 2 )^/(2fc 2 + r 2 ). To con- 
 struct the center of curvature, C, draw NQ perpendicular to PN and PQ 
 perpendicular to OP; then OQ will meet PN in C. Length of arc OP = 
 \$k [rad 9 Vl + (rad 0) 2 + sinh -1 (rad 6)]. After many windings, arc OP = 
 , approximately. 
 
 Hyperbolic Spiral. 
 FIG. 75. 
 
 Logarithmic Spiral. 
 FIG. 76. 
 
 The Hyperbolic Spiral is the curve whose polar equation is r = o/rad 0. 
 To construct the curve, take a series of points along Ox (Fig. 75) ; through 
 each of these points, with center at O, draw an arc extending into the upper half 
 of the plane; and along each of these arcs lay off a length = a. The points 
 thus reached will lie on the curve. A line parallel to the x-axis, at distance o, 
 is an asymptote of the curve. The curve winds around and around the 
 point O without ever reaching it (asymptotic point) . The figure shows one- 
 half of the curve, corresponding to positive values of 6. HPT and PN are the 
 tangent and normal at any point P, the line TON being perpendicular to OP, 
 
OTHER USEFUL CURVES 
 
 155 
 
 then OT = a, and ON = r 2 /a. Hence a construction for the tangent and 
 normal. Radius of curvature at Pis R = r/sin 3 v, where v = angle between 
 OP and the tangent at P. Construction: At N draw a perpendicular to PN, 
 meeting PO in Q; at Q draw a perpendicular toPQ, meeting PN in C; then C is 
 the center of curvature for the point P. 
 
 The Logarithmic Spiral (Fig. 76) ,-is a curve which cuts the radii from O 
 at a constant angle v, whose cotangent is m. Polar equation: r = ae m fl . 
 Here a is the value of r when 9=0. For large negative values of 0, the curve 
 winds around O as an asymptotic point. If PT and PN are the tangent and 
 normal at P, the line TON being perpendicular to OP (not shown in fig.), 
 then ON = rm, and PN = r\A + w 2 = r/sin v. Radius of curvature at 
 P is PN. The evolute of the spiral is an equal spiral 
 whose axis makes an angle ^TT (log e m) /m with the 
 axis of th'e given spiral. Area swept out by the radius 
 r from r = (where 6 = -co) to r = r, is A = 
 rV(4ra) = half the triangle OPT. Length of arc from 
 O to P = s = r/cos v = PT. 
 
 The Tractrix, or Schiele's Anti-friction Curve 
 (Fig. 77) , is a curve such that the portion PT of the 
 tangent between the point of contact and the z-axis is 
 
 constant = o. Its equation is x = + a I cosh" 1 
 
 M T 
 Tractrix. 
 
 FIG. 77. 
 
 = a ( 
 
 or, 
 
 in 
 
 i _ 
 
 y 
 
 parametric form, x = a [t tanh t], y = a/cosh t. (For tables of hyper- 
 bolic functions, see p. 60.) The z-axis is an asymptote of the curve. 
 Length of arc BP = a log e (a/y}. The evolute (locus of centers of curvature) 
 is the catenary whose lowest point is at B, and whose directrix is Ox. 
 
 The Cissoid (Fig. 78) is the locus of a point P such that OP, laid off on a 
 variable ray from O, is equal to BD, the portion of the ray lying between a 
 fixed circle through O and a fixed tangent at the point A opposite O. If a 
 is the radius of the circle, the polar equation is r = 2a sin 2 6 /cos 6. Rec- 
 tangular equation, y*(2a x) = x 3 . 
 
 Cissoid. 
 FIG. 78. 
 
 The Lemniscate (Fig. 79) is the locus of a point P the product of whose 
 distances from two fixed points F, F r is constant, equal to Yt o 2 . The distance 
 FF f = a^/2. Polar equation is r = a A/cos 26. Angle between OP and the 
 normal at P is 29. The two branches of the curve cross at right angles at O. 
 
156 
 
 ANALYTICAL GEOMETRY 
 
 Zr H 
 
 Maximum y occurs when 9 = 30 and r = a/V^ and is equal to H a 
 Area of one loop = a 2 /2. 
 
 The Helix (Fig. 80) is the curve of a screw thread on a cylinder of radius r. 
 The curve crosses the elements of the cylinder at a con- 
 stant angle, v. The pitch, h, is the distance between two 
 coils of the helix, measured along an element of the cylinder ; 
 hence h = 2irr tan v. Length of one coil = \/(27rr) 2 + h 2 
 = 27rr/cos v. To construct the projection of a helix on 
 a plane containing the axis of the cylinder, draw a rectangle, 
 breadth 2r and height h, to represent the plane, with a 
 semicircle below it, as in the figure, to represent the base 
 of the cylinder. Divide h into equal parts (here 8), num- 
 bered from 1 to 8; think of the circumierence as also 
 divided into 8 equal parts, represented on the semicircle 
 by numbers from 1' to 4' and back again from 4' to 8'. 
 Then the point of intersection of a horizontal line through 
 1,2, . . . with a vertical line through 1', 2', ... will 
 be a point of the required projection. If the cylinder is 
 
 X, 
 
 
 ^x^ 
 
 
 
 X 
 
 
 s 
 
 
 2 
 
 
 ^s' 
 
 ^^ 
 
 
 / 
 
 
 \Jr & 
 
 -*S* 
 
 2' 
 
 Helix. 
 
 FIG. 80. 
 
 rolled out on a plane, the development of the helix will be a straight line, 
 with slope equal to tan v. 
 
DIFFERENTIAL AND INTEGRAL CALCULUS 
 
 DERIVATIVES AND DIFFERENTIALS 
 
 Derivatives and Differentials. A function of a single variable x may 
 be denoted byf(x),F(x), etc. The value of the function when x has the value 
 XQ is then denoted by f(xo),F(xo), etc. The derivative of a function y = f(x) 
 may be denoted by f'(x), or by dy/dx. The value of the derivative at a 
 given point x = XQ is the rate of change of the function at that point; 
 or, if the function is represented by a curve in the usual way (Fig. 1), the 
 value of the derivative at any point shows the slope of the curve (that is, 
 the slope of the tangent to the curve) at that point 
 (positive if the tangent points upward, and negative 
 if it points downward, moving to the right) . 
 
 The increment, Ay (read: "delta y"), in y is the 
 change produced in y by increasing x from XQ to XQ + 
 Ax; that is, Ay = f(x + Ax) - /(z ). The differ- 
 ential, dy, of y is the value which Ay would have if 
 the curve coincided with its tangent. (The differen- 
 tial, dx, of x is the same as Ax when x is the inde- 
 pendent variable.) Note that the derivative depends 
 only on the value of XQ, while Ay and dy depend not -^ IG * 
 
 only on XQ but also on the value of Ax. The ratio 
 
 Ay /Ax represents the slope of the secant, and dy/dx the slope of the tan- 
 gent (see Fig. 1). If Ax is made to approach zero, the secant approaches 
 the tangent as a limiting position, so that the derivative = f'(x) = 
 
 j A ,v. -i- rv i A i A / -i. n i A- i "-~~> My 
 
 The symbol "lim" in connection with Ax == means "the limit, as Ax 
 approaches 0, of ..." [A constant c is said to be the limit of a variable u if, 
 whenever any quantity m has been assigned, there is a stage in the variation- 
 process beyond which |c u\ is always less than m; or, briefly, c is the limit 
 of u if the difference between c and u can be made to become and remain as 
 small as we please.] 
 
 To find the derivative of a given function at a given point: (1) If the 
 function is given only by a curve, measure graphically the slope of the 
 tangent at the point in question; (2) if the function is given by a mathematical 
 expression, use the following rules for differentiation. These rules give, 
 directly, the differential, dy, in terms of dx\ to find the derivative, dy/dx, 
 divide through by dx. 
 
 Rules for Differentiation. (Here u,v,w,. . . represent any functions 
 of a variable x, or may themselves be independent variables, a is a constant 
 which does not change in value in the same discussion; e 2.71828.) 
 1. d(a + u) = du. 2. d(au) = adu. 
 
 3. d(u + v + w + . . .) = du + dv + dw + . . . . 
 
 4. d(uv) = udv + vdu. 
 
 . (du , dv dw \ 
 
 5. d(uvw . . . ) = (uvw ...)( 1 1 h . ) 
 
 \ u v w I 
 
 6 d u = vdu -udr, t 
 
 V V 2 
 
 7. d(u m ) = mu m - l du when m is not 
 
 157 
 
158 
 
 DIFFERENTIAL AND INTEGRAL CALCULUS 
 
 = 2wdu; d(w 3 ) 
 _du_ 
 
 2V 'u 
 
 = *?. 
 
 u 
 
 14. d sin M = cos udu. 
 16. d cos w = sin udu. 
 18. d tan u = sec 2 wdw. 
 
 dw 
 20. d sin- 1 w 
 
 Thus, d(w 2 ) 
 
 ,- 
 8. dVu = 
 
 10. d(e M ) = 
 12. 
 
 3w 2 dw; etc. 
 9. 
 
 11. d(a w ) = (log* a)a"dw. 
 13. d lo glo w = (logwe) 
 
 u 
 
 : (0.4343. . )^. 
 
 22. d cos- 1 u = 
 
 15. d esc M = cot u esc u du. 
 17. d sec u = tan u sec w du. 
 19. d cot w = esc 2 w du. 
 
 du 
 21. d csc- J w = - 
 
 - 1 
 
 d sec- 1 u = 
 
 dw 
 
 24. d tan- 1 w = 
 
 1 + w 2 
 
 26. d log sin u = cot w du. 
 
 25. d cot- 1 u = 
 
 27. d log tan 
 
 du 
 
 1 + w* 
 2du 
 
 sn 
 
 28. d log cos M = tan u du. 29. d log cot u = 
 
 30. d sinh w = cosh u du. 
 
 32. d cosh u = sinh w du. 
 
 34. d tanh w = sech 2 u du. 
 
 36. d sinh- 1 M = f 
 
 sin 2w 
 
 31. d csch w = csch u coth wdu. 
 33. d sech u = sech w tanh w du. 
 35. d coth w = csch 2 w du. 
 
 37. d csch- 1 M = - 
 
 + 1 
 
 38. d cosh- 1 M = 
 
 40. d tanh- 1 u 
 
 du 
 
 \/u* - I 
 du 
 
 39. d sech- 1 M = - 
 
 dw 
 
 41. d 
 
 1 - u* 
 
 1 - u* 
 
 42. d(u v ) = (u v ~ l )(u logeu dv +vdu~). 
 
 Derivatives of Higher Orders. The derivative of the derivative ia 
 called the second derivative; the derivative of this, the third derivative; and 
 BO on. Notation: if y = f(x), 
 
 /'(*) = D,y 
 
 dx' 
 
 ,; etc. 
 
 NOTE. If the notation fry/dx* is used, this must not be treated as a fraction, like dy/dx' 
 but as an inseperable symbol, made up of a symbol of operation, d 2 /dx 2 , and an operand y 
 
 The geometric meaning of the second derivative 
 is this: if the original function y = f(x) is repre- 
 sented by a curve in the usual way, then at any 
 point where f"(x) is positive, the curve is concave 
 upward, and at any point where f"(x) is negative, 
 the curve is concave downward (Fig. 2). When 
 f"(x) - 0, the curve usually has a point of 
 inflection. Fio. 2. 
 
 Differentials of Higher Orders. The differ- 
 ential of the differential is called the second differential; the differential of 
 
DERIVATIVES AND DIFFERENTIALS; MAXIMA AND MINIMA 159 
 
 this, the third differential; etc. These quantities are of little importance 
 except in the case where dx = a constant. In this case 
 
 dy=f(x-)dx; d*y =/"(*)(<**); d z y = /'"(*) -(dx) z ; . . . 
 The first, second, third, etc., differentials are close approximations to the first, 
 second, third, etc., differences (p. 115), and are therefore sometimes useful in 
 constructing tables. Thus, denoting the first, second, third, etc., differences by 
 >', D", D'", etc., and, assuming always that dx a constant, 
 
 ' =dy+Kd*y +Kd*y + Kidiy + . . . ; d*y = D>" - % D"" + . .' . 
 
 D" = d*y + d 3 y + Vi2 d*y + . . . ; d z y = D" - D"' + itfa D"" + . . . 
 
 D'" = d*y + % d*y + . . . ; dy = D' - H D" + H D'" - H D"" + . . . 
 
 Functions of Two or More Variables may be denoted by f(x, y, . . . ), 
 F(x, y, . . .),etc. The derivative of such a function u = f(x, y, . . .) formed 
 on the assumption that x is the only variable (y, . . . being regarded for the 
 moment as constants) is called the partial derivative of u with respect to 
 
 x, and is denoted by f x (x,y), or D x u, or -^-, or ^-- Similarly, the partial 
 
 dx ox 
 
 derivative of u with respect to y is f v (x,y), or D v u, or , or -r-- 
 
 dy dy 
 
 NOTE. In the third notation, d x u denotes the differential of u formed on the assump- 
 tion that x is the only variable. If the fourth notation, du/'dx, is used, this must not be 
 treated as a fraction like du/dx; the d/dx is a symbol of operation, operating on u, and 
 the " dx " must not be separated. 
 
 Partial derivatives of the second order are denoted byfxx, fx u , f vv , or by D\u, 
 
 -^ -' , .. J 
 ox z oxoy 
 
 Similarly for higher derivatives. Note that/xj, = f vx . 
 
 If increments A#, Ay, (or dx, dy) are assigned to the independent variables 
 x, y, the increment, Aw, produced in u = f(x,y) is 
 
 AM = f(x + Az, y + Ay) - f(x,y) ; 
 
 while the differential, du, that is, the value which Aw would have if the partial 
 derivatives of u with respect to x and y were constant, is given by 
 
 du = (fj-dx +V v )'dy. 
 
 Here the coefficients of dx and dy are the values of the partial derivatives of u 
 at the point in question. 
 
 If x and y are functions of a third variable t, then the equation 
 
 expresses the rate of change of u with respect to t, in terms of the separate rates 
 of change of x and y with respect to t. 
 
 For the graphical representation of u = f(x,y), see p. 178. 
 
 Implicit Functions. If f(x,y) = 0, either of the variables x and y is 
 said to be an implicit function of the other. To find dy/dx, either (1) solve 
 for y in terms of x, and then find dy/dx directly; or (2) differentiate the equa- 
 tion through as it stands, remembering that both x and y are variables, and 
 then divide by dx; or (3) use the formula dy/dx = (/*//,/), where f x and 
 f y are the partial derivatives of f(x,y~) at the point in question. 
 
 MAXIMA AND MINIMA 
 
 A Function of One Variable, as y = f(x), is said to have a maximum at 
 a point x = x<\, if at that point the slope of the curve is zero and the concavity 
 
 Dx(D u u), Dlu, or by -^ -' , .. J -^ -' the last symbols being "inseparable." 
 
 
160 
 
 DIFFERENTIAL AND INTEGRAL CALCULUS 
 
 FIG. 3. 
 
 downward (see Fig. 3) ; a sufficient condition for a maximum is f(xo) = 
 and /"(ZQ) negative. Similarly, /(z) has a minimum if the slope is zero and 
 the concavity upward; a sufficient condition for a minimum is /'(x ) = and 
 /"(XQ) positive. If /'(#o) = and f' f (xo) = 
 and f"(xo) 7* 0, the point XQ will be a 
 point of inflection. If f(x ) = and 
 f"(xo) = and f'"(xo) = 0, the point x will 
 be a maximum if /""(XQ) < 0, and a mini- 
 mum if f""(xo) > 0. It is usually sufficient, 
 however, in any practical case, to find the 
 values of x which make f(x) = 0, and then 
 decide, from a general knowledge of the 
 curve, which of these values (if any) give 
 maxima or minima, without investigating the higher derivatives. 
 
 A Function of Two Variables, as u = f(x-,y), will have a maximum 
 at a point (x ,y ) if at that point /* = 0, f y = 0, and fxx < 0, f vv < ; 
 and a minimum if at that point f x = 0, f u = 0, and f xx > 0, f yy > 0; 
 provided, in each case, (/**) (f yv ) (f xy } 2 is positive. If /* =0 and f v = 0, 
 and f x x and f yy have opposite signs, the point (#0,2/0) will be a "saddle point" 
 of the surface representing the function (p. 178). 
 
 EXPANSION IN SERIES 
 
 The range of values of x for which each of the series is convergent is stated 
 at the right of the series. 
 
 Arithmetical and Geometrical Series, and the Binomial Theorem. 
 See p. 114. 
 
 Exponential and Logarithmic Series. 
 
 a>0, 
 
 where ra = log* a = (2.3026) (logio a). 
 
 x x x 
 
 X* Or* X* X* 
 
 - 1 <x < +1. 
 
 - l<x< +1. 
 
 x< - lor + 1 <x. 
 
 < X < oo. 
 
 log.(a+*) 
 
 <a < 4- 
 -a <x < + 
 
EXPANSION IN SERIES 161 
 
 Series for the Trigonometric Functions. In the following formulae, 
 all angles must be expressed in radians. If D the number of degrees in 
 the angle, and x = its radian measure, then x = 0.017453 D. 
 
 , X 3 X 6 X 7 
 
 sin*=*-- + --^ + ...; =>< x < + oo. 
 
 X a X* X 6 X 8 
 
 C08I = 1--+---+--. . .. -.<,< + .. 
 
 x 3 2x 6 17x 7 62x 9 
 
 ' 
 
 1 x x 3 2x 6 x 1 
 cot*---------. . .; 
 
 tan- 1 y = y-- + --- + ...; - 1 y + 1. 
 
 o O / 
 
 cos" 1 y = H?r sin~' y; cot" 1 y = ^iir tan 1 y. 
 
 Series for the Hyperbolic Functions (x a pure number). 
 
 X 3 X 6 X 7 
 
 sinh x==x +^7 + 7 + 7J + ---; -eo<z<o>. 
 
 X 2 X 4 X 6 
 
 coshx = 1 + + + + . . .; -oo< x <oo. 
 
 General Formulae of Maclaurin and Taylor. If /(x) and all its deriva- 
 tives are continuous in the neighborhood of the point x = (or x = a) , then, 
 for any value of x in this neighborhood, the function /(x) may be expressed 
 as a power series arranged according to ascending powers of x (or of x o), 
 as follows: 
 
 . . . 
 
 + x"- 1 + (Pn)x". (Maclaurin.) 
 
 + (n _ (x - a)"' 1 + (Q n )(x - a)". (Taylor.) 
 
 Here (P n )x n , or (Q n )(x a) n , is called the remainder term ; the values of 
 the coefficients P n and Q n may be expressed as follows: 
 Pn = {/(")(sx)}/n! = {(1 - O"- 1 fW(tx)}/(n- 1)1 
 Qn = {/C">[a + s(x - a)]}/n! = {(1 - O w ~ l /<">[<* + t(x - a)]}/(n - 1)! 
 where s and t are certain unknown numbers between and 1 ; the s-form is 
 due to Lagrange, the /-form to Cauchy. 
 
 The error due to neglecting the remainder term is less than (P n )x n , or 
 
162 
 
 DIFFERENTIAL AND INTEGRAL CALCULUS 
 
 (Qn)( x ~ ) n where P n , or Q n , is the largest value taken on by P n , or Q n , 
 when s or t ranges from to 1. If this error, which depends on both n and 
 x, approaches as n increases (for any given value of x) , then the general - 
 expression-with-remainder becomes (for that value of x) a convergent in- 
 finite series. 
 
 The sum of the first few terms of Maclaurin's'series gives a good approxi- 
 mation to f(x) for values of x near x = 0; Taylor's series gives a similar ap- 
 proximation for values near x = a. 
 
 Fourier's Series. Let f(x) be a function which is finite in the interval 
 from x = c to x = + c and has only a finite number of discontinuities in 
 that interval (see note below), and only a finite number of maxima and 
 minima. Then, for any value of x between c and c, 
 
 f(x~) 
 
 00+ 
 
 cos + 02 cos --- \- as cos - -- \- 
 c c c 
 
 . TTX , . 2WX . 3-7TX 
 
 + &i sin +62 sin - + &3 sin -- 
 c c c 
 
 where the constant coefficients are determined as follows: 
 
 /(0COS dl, 
 
 C 
 
 /(<) sin dt. 
 c 
 
 In case the curve y = f(x) is symmetrical with respect to the origin, the 
 a's are all zero, and the series is a sine series. In case the curve is sym- 
 metrical with respect to the y-axis, the 6's are all zero, and a cosine series 
 results. (In this case, the series will be valid not only for values of x between 
 c and c, but also for x = c and x c.) A Fourier's series can be inte- 
 grated term by term; but the result of differentiating term by term will in 
 general not be a convergent series. 
 
 NOTE. If x = xo is a point of discontinuity, f(xo) is to be defined as ^[fi(xo) + /2(xo)], 
 where fi(xo) is the limit of f(x) when x approaches xo from below, and MXO) is the limit 
 of f(x) when x approaches xo from above. 
 
 ^N- 
 
 FIG. 4. 
 
 o 
 
 FIG. 5. 
 
 H*-' 
 
 /* / 
 
 / I / I * 
 
 FIG. 6. 
 
 Examples of Fourier's Series. 
 
 1. If y = f(x) is the curve in Fig. 4, 
 
 _ h _ 4A / THC 1 STI 
 V 2 ~ K* \ COS c 9 C S c 
 
 2. If i/ = /(x) is the curve in Fig. 5, 
 
 4h I . TTX 1 . Zirx 
 
 1 5irx , 
 25 C S ^- + 
 
 5irx 
 
 3. If y = /(x) is the curve in Fig. 6, 
 
 2h I . TTX 1 . 27TZ 
 
 y = ~ 8m - 
 
INDETERMINATE FORMS; CURVATURE 163 
 
 % 
 INDETERMINATE FORMS 
 
 In the following paragraphs, f(x), g(x) denote functions which approach 0; 
 F(x),G(x) functions which increase indefinitely; and U(x) a function which ap- 
 proaches 1 ; when x approaches a definite quantity a. The problem in each 
 case is to find the limit approached by certain combinations of these functions 
 when x approaches a. The symbol = is to be read "approaches." 
 
 CASE 1. "jj-" To find the limit of f(x)/g(x) when /(x) = and0(z) = 0, 
 
 use the theorem that lim = lim where f'(x) and g'(x) are the 
 
 derivatives of f(x) and g(x). This second limit may be easier to find than the 
 first. If f'(x) = and g'(x) = 0, apply the same theorem a second time: 
 
 Hm^~ " = lim ^7^; and so on. 
 g'(x) Q"(X) 
 
 CASE 2. "-^-" If F(x) = oo and(?(z) = oo, then li 
 
 precisely as in Case 1. 
 
 CASES. "O-oo." To find the limit of f(x) -F(x) when f(x) = and F(x) = oo, 
 
 write lim [f(x)-F(x)] = lim , * s , or = lim ^ ' , . ; then proceed as in 
 
 Case 1 or Case 2. 
 
 CASE 4. "0." If /(*) = and g(x) = 0, find lim [/(x)f (x) as follows: 
 let y = [f(x)}^, and take the logarithm of both sides thus: 
 
 log y = (/(x) log e /(x) ; 
 next, find lim [g(x) log e /(x)], = m, by Case 3; then lim y = e m . 
 
 CASE 5. "1." If U(x) = 1 andF(x) = oo, find lim [C7(x)]^ (x) as follows: 
 let y = [U(x)\ F ^ x \ and take the logarithm of both sides, as in Case 4. 
 
 CASE 6. " coo." if p(x) = ooand f(x) = 0, find lim [F(x)] f(x) as follows: 
 let y = [F(x)] f(x \ and take the logarithm of both sides, as in Case 4. 
 
 CASE 7. " oo- oo." If F(x) = oo and G(x) = oo, write lim [F(z) O(x)] 
 1 1 
 
 = lim -- ; then proceed as in Case 1. Sometimes it is shorter to ex- 
 
 F(x) -G(x) 
 
 pand the functions in series. It should be carefully noticed that expressions 
 like 0/0, o / 00 . etc., do not represent mathematical quantities. 
 
 CURVATURE 
 
 The radius of curvature R of a plane curve at any point P (Fig. 7) is the 
 distance, measured along the normal, on the concave side of 
 the curve, to the center of curvature, C, this point being 
 the limiting position of the point of intersection of the nor- 
 mals at P and a neighboring point Q, as Q is made to ap- 
 proach P along the curve. If the equation of the curve ia 
 V =/(*), 
 
 ds _ [1 + (yO']?* 
 ~ du ~ g" FIG. 7. 
 
154 DIFFERENTIAL AND INTEGRAL CALCULUS 
 
 where ds = -\/dx 2 + dy 2 = the differential of arc, u = tan" 1 [/'(a;)] = the 
 a,ngle which the tangent at P makes with the a>axis, and y f = f (x) and 
 y" = /" (x) are the first and second derivatives of f(x) at the point P. Note 
 that dx ds cos u and dy = ds sin u. The curvature, K, at the point P, is 
 K = 1/R = du/ds; that is, the curvature is the rate at which the angle u is 
 changing with respect to the length of arc s. If the slope of the curve is small, 
 **/<*). 
 
 If the equation of the curve in polar co-ordinates is r = /(0) , where r = radius 
 vector and 6 = polar angle, then 
 
 r 2 - rr" + 2(r') 2 
 where r' = /'(*) and r" =/"(0). 
 
 The evolute of a curve is the locus of its centers of curvature. If one curve 
 is the evolute of another, the second is called the involute of the first. 
 
 INDEFINITE INTEGRALS 
 
 An integral of f(x)dx is any function whose differential iaf(x)dx, and is 
 denoted by J"f(x)dx. All the integrals of f(x)dx are included in the ex- 
 pression y*/(30ffcc+ C, where t ff(x)dx is any particular integral, and C is an 
 arbitrary constant. The process of finding (when possible) an integral of a 
 given function consists in recognizing by inspection a function which, when 
 differentiated, will produce the given function; or in transforming the given 
 function into a form in which such recognition is easy. The most common 
 integrable forms are collected in the following brief table; for a more extended 
 list, see B. O. Peirce's "Table of Integrals" (Ginn & Co.). 
 
 GENERAL, FORMULAE 
 
 1. fadu = afdu = au + C 2. J* (u + v)dx = fudx -j- fvdx 
 3. fudv = uv -f-odu 4. fj(x)dx = f f[F(y}}F f (y)dy, x = F(y) 
 5- fdyff(x,y}dx = f dxf f(x,y}dy. 
 
 FUNDAMENTAL INTEGRALS 
 _ x n+i 
 
 6. Jx n dx = + C; when n ^ - 1 
 
 7. y = log x + C = log e ex 8. J*e x dx = e x + C 
 
 9. ^sin xdx = cos x + C 10. ^cos xdx = sin x + C 
 
 tan x + C 
 
 tan' 1 x + C = - cot- 1 x + c 
 
 RATIONAL FUNCTIONS 
 
 _L ki^n+1 
 
16. 
 17. 
 19. 
 20. 
 
 INDEFINITE INTEGRALS 
 1 
 
 dx 
 
 1-65 
 
 = loge (a + bx} 
 
 is. 
 
 (a 
 
 6(0 
 
 dx 
 
 loge 
 
 \-C = tanh" 1 * +C, 
 + C = - coth- 1 x + C, 
 
 24 r dx = - 
 
 J a + 2bx,+ ex 2 b + 
 
 when x < 1 
 
 when z > 1 
 
 when o > 0, 6 > 
 
 when 
 ac - 6 2 > ; 
 
 when 
 & - ac > 0; 
 
 dx 
 
 A+ 
 
 26. 
 
 27. 
 
 , if /(x) is a polynominal of higher than the first 
 
 degree, divide by the denominator before integrating. 
 dx 1 6 + ex 
 
 (a + 2bx 
 
 2(ac b z )(p 1) (a 
 (2p - 3)c 
 
 28 -^(^f^ 
 
 nx)dx 
 
 2(oc - 
 n 
 
 - 1) ** (o + 2bx 
 I 
 
 2c(p - 1) "(a +2bx + 
 
 me rib s* dx 
 
 ~c ^ (a + 2bx + 
 
 29. 
 
 +bx) n dx 
 
 (m 
 
 dx 
 
166 
 
 DIFFERENTIAL AND INTEGRAL CALCULUS 
 IRRATIONAL FUNCTIONS 
 
 + C 
 
 - 1 + C = - cos- 1 + c 
 a a 
 
 'a + bx, and use 21 and 22 
 
 . = log* [x + Va 2 + x 2 ] +C= sinh- 1 + c 
 = log e [x + V* z - a 2 ] + C = cosh" 1 - + c 
 
 38. y 
 
 1 
 
 + 2bx + ex 2 Vc 
 
 when c > 0; 
 
 = ^- sinh" 1 b +^ x = + C , when ac -6 2 > 0; 
 7 tc -6 2 
 
 zcosh- 1 -^^ + C, when 6 2 - ac > 0; 
 = -^Lsin' 1 b+CX + C, when c < 
 
 39. y 
 
 40. y 
 
 - V 2 + # 2 + sinh x he 
 
 a 
 
INDEFINITE INTEGRALS 
 
 167 
 
 - a 2 da; = V* 2 - a 2 - loge (x + V* 2 - a 2 ) + C 
 
 44. 
 
 TRANSCENDENTAL FUNCTIONS 
 
 45. 
 
 46. y 
 
 ax a 2 x 2 
 
 47. y loge xdx = x loge x x + C 
 
 4 8 . 
 
 49 . 
 
 71 p 
 
 _ + c 
 
 x 
 
 50. y*sin 2 xdx = \i sin 2x + \to + C = M sin x cos x + Ha; + C 
 
 51. y*cos 2 x dx = J4 sin 2x + H + C = % sin x cos x + Hx -+- C 
 
 /> . cos mx , / sin mx 
 
 52. / sm mx dx = h C 53. / cos mx dx = 1- C 
 
 m m 
 
 / . cos (m + n)x cos (m ri)x 
 
 54. / sm mx cos nx dx = f- C 
 
 2(m + n) 2(m n) 
 
 /^ . sin(m n)x sin(m + n)x , 
 
 55. / sm mx sin nx dx = 7 '- \- C 
 
 J 2(m - n) 2(m + n) 
 
 Kc p sin(m ri)x sin(m -f- ri)x , 
 
 oo. / cos mx cos nx ax = -4- C 
 
 2(m n) 2(m + n) 
 
 57. J tan xdx = log cos x + C 58. y*cot xdx = log sin x + C 
 
 59. f- - = loge tan ~ + C 60. /* = log tan ( - + - ) + < 
 
 sm x 2 *^ cos x \4 2/ 
 
 63. /'sin x cos a; da; = ^ sin 2 x + C 64. /"- 
 
 *X t-' ai 
 
 cos x 
 dx 
 
 sin x cos x 
 
 - cot - + C 
 
 = log, tan x + C 
 
 65.*y*sin n xdx = 
 
 66.*y*cos n xdx = 
 
 sin x cos" 
 n 
 
 , * -1 
 n - \ f 
 
 2 x dx 
 
 n 
 
 
 If n is an odd number, substitute cos x =- z or sin a; = z. 
 
168 
 67. 
 
 DIFFERENTIAL AND INTEGRAL CALCULUS 
 
 tan' 
 
 
 ~ z xdx 
 
 / cot n l x r> 
 68. J cot* xdx = _ J cot n ~ 2 xdx 
 
 69. r-*L_ = 
 
 J sin* -r fw. 
 
 ~ 2 
 
 (n 
 
 n ~ r <* 
 
 n I** gin"" 2 
 
 TO. y 
 
 dx 
 
 sin" x 
 
 dx 
 
 cos n x (n 1) cos""" 1 x n 1^ cos n ~ 2 x 
 sin p+1 xcos 9 ~ 1 x 
 
 k r . 
 
 71. J sm p x cos 9 xdx = 
 
 4 /*sin p xcos- J xdx 
 
 P + 3 P + <T 
 
 + 
 
 - C sin p 2 x cos' xdx 
 79 ~i~ C 
 
 sin p+1 xcos~g +1 x q - p -2 
 q - I q - I ' 
 
 74. 
 
 a + o cos a; 
 
 2 / a 
 
 5^5 -Wi 
 
 tan *** 
 
 sin p a; cos- 
 C ' when s 
 
 6 + a cos x 4- sin #v& 2 a 2 
 
 a 4 b cos x 
 
 , 2 tanh" 1 ( \/^ ^ tan \te ) 4 C, 
 &2_ a \\64a / 
 
 _ x a r> 
 
 o 4 6 cos x & & J a 4 6 cos x 
 
 when 
 a 2 < 6 s 
 
 ^ cos x dx _ x a r> dx 
 
 75 ' J n 0- h o -r = T ~ fc / /i 4- 7> nna -r "*" 
 
 76. r * 
 
 ** a 4- 
 
 ^A + 
 
 77. / ; 
 
 47 a+ 
 
 = log. (a 4 b cos x) 4 C 
 
 COS X 
 
 B cos x 4 C sin x , . / 
 
 : dx = A I 
 
 6 cos x 4 c sin x ** a 4 
 
 cos y dy 
 
 dy 
 
 pcosy 
 
 a + p cos y 
 where b = p cos w, c = p sin w and z M 
 
 a sin bx b cos & 
 78. 
 
 79. 
 
 80. 
 81. 
 82. 
 83. 
 
 sin y dy 
 a 4 pcosy' 
 
 war 1 xdx = xsin- 1 ^ + l - & + C 
 cosT 1 xdx = xcos~ l x \/l x 2 + C 
 tan" 1 xdx = x tan" 1 x - J4 log e (1 + x 2 ) 4- C 
 cot'^dx = x cot" 1 x + ^log (1 4 z a ) 4 C 
 If p or g is an odd number, substitute eos z = z or einx . 
 
DEFINITE INTEGRALS 169 
 
 84. y* sinh xdx = cosh x + C 85. y tanh xdx = log, cosh a: + C 
 
 86. y cosh x dx = sinh a; + C 87. fcoth xdx = log a sinh x + C 
 
 88. y sech xdx = 2 tan" 1 (e*) + C 89. fcach xdx = loga tanh (x/2) +C 
 
 90. A* sinh 2 xdx = ^ sinh a: cosh x MX + C 
 
 91. f cosh 2 xdx = \<i sinh at cosh x + ftx + C 
 
 92. y sech 2 xdx = tanh x + C 93. J* csch 2 xdx = - coth x + C 
 
 DEFINITE INTEGRALS 
 
 The definite integral of /(x)dx from x = o to x = 6, denoted by ^ f(x)dx, 
 is the limit (as n increases indefinitely) of a sum of n terms: . 
 
 f a b f(x~)dx = ^ [/(xi)Ax +/(x 2 )Ax +/(x 3 )Ax + . . . + /(x n )Ax], 
 
 built up as follows: Divide the interval from a to 6 into n equal parts, and call 
 each part Ax, = (b a)/n; in each of these intervals take a value of x (say 
 xi, X2, . . . x n ), find the value of the function /(x) at each of these points, 
 and multiply it by Ax, the width of the interval; then take the limit of the sum 
 of the terms thus formed, when the number of terms increases indefinitely, 
 while each individual term approaches zero. 
 
 Geometrically, J^*f(x}dx is the area bounded by the curve y = /(x), the 
 x-axis, and the ordinates x = a and x = 6 (Fig. 8) ; that 
 is, briefly, the "area under the curve, from a to b." The 
 fundamental theorem for the evaluation of a definite 
 integral is the following: 
 
 that is, the definite integral is equal to the difference be- 
 tween two values of any one of the indefinite integrals 
 of the function in question. In other words, the limit of a sum can be found 
 whenever the function can be integrated. 
 
 Properties of Definite Integrals. 
 
 THE MEAN-VALUE THEOREM FOR INTEGRALS. 
 
 f*F(.x)f(x)dx = F(X) f* f(x)dx, 
 
 provided /(x) does not change sign from x = a to x = b; here X is some (un- 
 known) value of x intermediate between a and b. 
 
 THEOREM ON CHANGE OF VARIABLE. In evaluating J* x ' a f(x)dx, f(x)dx 
 may be replaced by its value in terms of a new variable t and dt, and x a 
 and x = b by the corresponding values of t, provided that throughout the 
 interval the relation between x and t is a one-to-one correspondence (that is, 
 to each value of x there corresponds one and only one value of t, and to each 
 value of t there corresponds one and only one value of x). 
 
170 DIFFERENTIAL AND INTEGRAL CALCULUS 
 
 DIFFERENTIATION WITH RESPECT TO THE UPPER LIMIT. If b is variable, 
 then J f(x)dx is a function of 6, whose derivative is 
 
 DIFFERENTIATION WITH RESPECT TO A PARAMETER. 
 
 Functions Denned by Definite Integrals. The following definite 
 integrals have received special names, and their values have been tabulated ; 
 see, for example, B. O. Peirce's "Table of Integrals." 
 
 dx 
 
 , 
 k 2 sin 2 x 
 
 (*<!) 
 
 1. Elliptic integral of the first kind *=F(u,K) = J]" , 
 
 2. Elliptic integral of the second kind = E(u, /b) = t /^"V'l k 2 sin 2 x dx 
 (k 2 < 1) 
 
 3. 4. Complete elliptic integrals of the first and second kinds; put u = ir/2 
 in (1) and (2). 
 
 5. The Probability integral = j=-J**e~ x ' i dx 
 
 6. The Gamma function = F(n) = f^ m x n ~ 1 e~ x dx 
 
 Approximate Methods of Integration. Mechanical Quadrature. 
 
 (1) Use Simpson's rule. See p. 106. 
 
 (2) Expand the function in a power -series, and integrate term by term. 
 
 (3) Plot the area under the curve y = f(x) from x = a to x = b on squared 
 paper and measure this area roughly by "counting squares," or more accu- 
 rately, by the use of a planimeter ($14 to $35; instruction for use with each 
 instrument). 
 
 (4) Coradi's Mechanical Integraph ($240) provides a means of drawing 
 on paper the curve y = ff(x}dx, when the curve y = f(x~) is given, and can 
 be used to facilitate the solution of certain differential equations. Full 
 instructions for use with each instrument. 
 
 Double Integrals. The notation fff(x, y)dy dx 
 means J" \f f(x, y)dy}dx, the limits of integration in the 
 inner, or first, integral being functions of x (or constants). 
 
 EXAMPLE. To find the weight of a plane area whose 
 density, w, is variable, say w = f(x, y). The weight of a 
 typical element, dx dy, is f(x, y)dx dy. Keeping x and 
 dx constant, and summing these elements from, say, y jr IGt g. 
 
 Fi(x) to y = F*t(x), as determined by the shape of the 
 
 boundary, the weight of a typical strip perpendicular to the -axis is 
 fy=Fz(x) 
 
 dx I f(x,y)dy. Finally, summing these strips from, say, x a to x = b, the 
 
 Jy=Fi(x) 
 
 fx = b /y=Fz(x) 
 
 weight of the whole area is I {dx I f(x, y}dy] , or, briefly, f J*f(x, y)dydx. 
 Jx = a Jy=Fi(x) 
 
DIFFERENTIAL EQUATIONS 171 
 
 DIFFERENTIAL EQUATIONS 
 
 An ordinary differential equation is one which contains a single inde- 
 pendent variable, or argument, and a single dependent variable, or function, 
 with its derivatives of various orders. A partial differential equation is 
 
 one which contains a function of several independent variables, and its partial 
 derivatives of various orders. The order of a differential equation is the order 
 of the highest derivative which occurs in it. A solution of a differential 
 equation is any relation between the variables, which, when substituted in 
 the given equation, will satisfy it. The general solution of an ordinary 
 differential equation of the nth order will contain n arbitrary constants. 
 A differential equation is usually said to be solved when the problem is 
 reduced to a simple quadrature, that is, an integration of the form 
 y = fftx^dx. 
 
 Methods of Solving Ordinary Differential Equations 
 
 DIFFERENTIAL EQUATIONS OF THE FIRST ORDER 
 
 (1) If possible, separate the variables; that is, collect all the x's and dx on 
 one side, and all the y's and dy on the other side; then integrate both sides, 
 and add the constant of integration. 
 
 (2) If the equation is homogeneous in x and y, the value of dy/dx in terms 
 
 of x and y will be of the form = /(-). Substituting y = xt will enable 
 
 dx \x] 
 
 the variables to be separated. Solution: log e x = f~fr\ H C. 
 
 (3) The expression f(x,y)dx + F(x,y}dy is an exact differential if 
 
 = ^ ( = P, say). In this case the solution of f(x,y)dx + 
 oy ox 
 
 F(x,y)dy = is 
 
 ff(x,y)dx + f[F(x,y) - fPdx]dy = C 
 
 or fF(x,y}dy + f\f(x,y) - fPdy]dx = C 
 
 (4) Linear differential equation of the first order: -f- f(x)-y = F(x). 
 
 dx 
 
 Solution: y = e~ p { ft p F(x)dx + c} , where P = ff(x)dx. 
 
 (5) Bernoulli's equation: - |- f(x)-y =F(x)-y n . Substituting y l ~ n =v 
 
 ax 
 
 dv 
 gives + (1 ri)f(x)'V = (1 n)F(x), which is linear in v and x. 
 
 (6) Clairaut's equation: y = xp + /(p), where p = dy/dx. The solution 
 consists of the family of lines given by y = Cx + /(O , where C is any 
 constant, together with the curve obtained by eliminating p between the 
 equations y = xp + f(p) and x + f'(p) = 0, where f'(p) is the derivative of 
 
 DIFFERENTIAL EQUATIONS OF THE SECOND ORDER 
 
 (7) -, ~ n*y. Solution: y = Ci sin (nx +2) 
 ax* 
 
 or y = C z sin nx + Ci cos nx 
 
172 DIFFERENTIAL AND INTEGRAL CALCULUS 
 
 (8) = +n 2 y. Solution: y = Ci sinh (nx + C 2 ) 
 
 or y = 3 e nx + (7 4 e~ nx 
 
 (9) -- = /(y). Solution: x = f - + C 2 , where P = ff(y) dy. 
 
 VCi + 2P 
 
 (10) ^ = /(*). Solution: y = fPdx + Cix + C 2 , where P = f j(x)dx 
 
 or y = xP - J*xf(x)dx + Cix + C 2 
 
 (11) 3 = f\-r ) Putting = z, = , a; = /* - -f Ci and y = 
 ox 2 \dxj dx dx 2 dx ** /(z) 
 
 J ~7T\ ~\~ Cz't then eliminate z from these two equations. 
 
 (12> The equation for damped vibration: ~- + 26 + a 2 y = 0. 
 
 arc 2 dx 
 
 Case I. If o 2 - 6 2 > 0, let m = Va 2 - 6 2 . Solution: 
 
 y = Ci e~ bx sin (mx + ^2) or y = e~* x [Cz sin (mx) + C 4 cos (mx)] 
 Case II. If a 2 - 6 2 = 0, solution is y = e~ bx [Ci + Ctx]. 
 
 Case III. If a 2 - 6 2 < 0, let n = V& - a 2 . Solution: 
 
 y = Cie"** sinh (nx + C 2 ) or y = C 3 e~ (b+n)x + C 4 e~ (6 ~ n)a; 
 
 d*y dy c 
 
 (13) 3 + 26- h o 2 y = c. Solution: y = : 1- yi, where y\ = the solu- 
 
 ax 2 ax a 2 
 
 tion of the corresponding equation with second member zero [see (12) above]. 
 
 (14) -^ + 26-^ + a*y = c sin(fec). Solution: 
 ax* ax 
 
 y = R sin(kx - S) + yi, where R = c/V(a 2 - k 2 ) 2 + 46 2 A; 2 , 
 
 26fc 
 tan <S = 2 _ , and y\ the solution of the corresponding equation with 
 
 second member zero [see (12) above]. 
 
 d 2 y dy 
 
 (15) 3^ + 2& 3^ + aZ V = f( x )> Solution: y = y + y\, where y =-- any 
 
 particular solution of the given equation, and y\ = the general solution of the 
 corresponding equation with second member zero [see (12) above]. 
 
 If 6 2 > a 2 , y = - 
 
 b* - a 2 
 
 where m,\ = b + \/6 2 q 2 and m z = 6 
 If 6 2 <a 2 , letm=Va 2 - 6 2 ; then^/o = 
 
 e~* x | sin (mx)J*e bx cos (mx)-/(x)da; cos (mx}fe bx sin 
 If 6 2 = a 2 , 2/ e = e~ &a; ( xj*e b *f(x) dx - fx-J> x j(x) dx \ . 
 
GRAPHICAL REPRESENTATION OF FUNCTIONS 
 
 For graphical methods in statistics, etc., see W. C. Brinton's "Graphical Methods for 
 
 Presenting Facts" 
 
 EQUATIONS INVOLVING TWO VARIABLES 
 
 The Curve y = f(x). To represent graphically any function, y, of 
 a single variable, x, lay off the values of x as abscissae along a uni- 
 formly graduated horizontal axis, whose positive direc- 
 tion (as usually chosen) runs to the right, and at each 
 point on this z-axis erect a perpendicular (called an ordi- 
 nate) whose length represents the value of y at that 
 point. The unit of measurement for the y-sca\e, whose 
 
 positive. direction (as usually chosen) runs upward, need jp IG j 
 
 not be the same as the unit for the a>scale. Draw a 
 smooth curve through the extremities of the ordinates; this is the graph of 
 the given function in rectangular co-ordinates, or the curve of the function. 
 
 To measure graphically the rate of change of the function at any point P 
 (Fig. 1), draw the tangent atP; then rate of change at P = RT /PR, where 
 RT and PR are measured in units of the y-axis and x-axis, respectively. 
 This ratio, which is positive if RT runs upward, negative if RT runs down- 
 ward, is equal to the derivative of the function at the point P (see p. 157). 
 
 Graphs of Important Functions. Figs. 2-9 show the graphs (in rec- 
 tangular co-ordinates) of the most important elementary functions, namely: 
 
 The linear function, y = mx + b (Fig. 2). 
 
 The power functions, y = x n [n positive (parabolic type) ; n negative 
 (hyperbolic type)] (Fig. 3). 
 
 The exponential function, y = 10* or y =e x , and the logarithmic 
 function, y = Iog 10 x or y = log e x (Fig. 4). 
 
 The trigonometric functions (Fig. 5), and the inverse trigonometric 
 functions (Fig. 6). 
 
 The hyperbolic functions (Figs. 7 and 8) and the inverse hyperbolic 
 functions (Fig. 9). 
 
 Various special functions (Figs. 10-12). 
 
 By a slight modification, each of these diagrams may be made to represent 
 a somewhat more general function than that for which it is primarily intended. 
 For, if x is replaced by a; a in the equation, this merely requires re-number- 
 ing the x-axis so that each number is moved a units to the left; and similarly, 
 if y is replaced by y b in the equation, this merely requires re-numbering 
 the y-axis so that each number is moved b units downward. (Such a change 
 is called a translation of the curve to the right, or upward.) Further, if x is 
 replaced by x/c [or y by y/c] in the equation, it is merely necessary to multiply 
 each of the numbers written along the x-axis [or y-axis] by c, in order to 
 adapt the graph to the new equation. (Such a change is called a "stretch- 
 ing" of the curve along one of the axes.) 
 
 Empirical Curves. Any set of values of two variables x and y can be 
 represented by plotting the points (x,y) on rectangular co-ordinate paper, and 
 drawing a smooth curve through these points. The points which correspond 
 to actual data should be clearly indicated by small circles or crosses, inter- 
 mediate points being spoken of as interpolated points. While this process of 
 graphically interpolating a continuous series of points between given values is 
 usually fairly safe, the process of extrapolation that is, extending the curve 
 beyond the range of the given values, is dangerous. 
 
 173 
 
174 
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS 
 
 Linear function, y = ma; + b. 
 FIG. 2. 
 
 / o i 
 
 (Parabolic Type) (Hyperbo/fc Type) 
 
 Power function, y = x n . 
 FIG. 3. 
 
 ff 
 
 Exponential function (10* or e x ~). 
 Logarithmic function (logic x or log e x) . 
 
 FIG. 4. 
 
 Inverse trigonometric functions. 
 FIG. 6. 
 
 -4FJX 
 
 Trigonometric functions. 
 FIG. 5. 
 
 To Find a Mathematical Equation to Fit a Given Empirical Curve. 
 
 This problem is one which in general requires much patience and ingenuity. 
 Only the simplest cases can be mentioned here. 
 
 CASE 1. If the given empirical curve is a straight line, then the law con- 
 necting the given values of x and y is y = mx + 6, where ra = the slope of the 
 line, and 6 = the value of y at the point where the line crosses the 2/-axis. If 
 
EQUATIONS INVOLVING TWO VARIABLES 
 
 175 
 
 i z 3 
 
 Hyperbolic functions and inverse hyperbolic functions. 
 FIG. 7. FIG. 8. FIG. 9. 
 
 y.,< 
 
 K-or... v 
 jr.02-... s 
 
 /raaJ 
 
 FIG. 10. 
 
 /f.-/.5- /I /-f **** 
 
 FIG. 12. 
 
 the points lie only approximately on a straight line, the best position for this 
 line can usually be found by stretching a black thread among the points; or, 
 assume a law of the form y = mx -\- b, and, by substituting in this formula n 
 pairs of values of x and y, obtain n equations connecting the coefficients 
 ra and b; various pairs of these equations may then be solved for m and 6, and 
 the average of the results taken. Or, if great accuracy is required, all nof the 
 equations may be solved for m and 6 by the method of least squares (p. 121). 
 
 If any law of the formj(x,y') = m-F(x,y) + 6 is suspected, where f(x,y) and 
 F(x,y) are any expressions involving either x or y or both x and y, such a law 
 may be tested by plotting F(x,y) instead of x, andf(x,y) instead of y, on rec- 
 tangular cross-section paper, and seeing whether or not the points lie on a 
 straight line. If they do, the form of the law is verified, and the values of m 
 and b can be read from the figure as before. For example, if j/ 2 = mxy + &, 
 a straight line will be obtained by plotting y 2 against xy. Again, if xy = bx + 
 my, a straight line will be obtained by plotting y against y/x, since the equa- 
 tion may be written y = b -f m (y/x)- 
 
 CASE 2. If a law of the form y = cx n is suspected, plot the points (x,y) on 
 logarithmic paper (see below). 
 
 CASE 3. If a law of the form y =c-10 wa; [or y = c-e mx ] is suspected, plot 
 the points (x,y) on semi-logarithmic paper (see below). 
 
176 
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS 
 
 CASE 4. If the given curve resembles the logarithmic curve, y = log x, 
 interchange x and y and proceed as in Case 3. 
 
 CASE 5. If the given curve is a wavy line, resembling a sine or cosine curve, 
 try an equation of the form y = a sin bx or y = a cos bx. If the heights of the 
 waves diminish as x increases, try an equation of the form y = ae~ nx sin bx. 
 [NOTE. Any periodic function (satisfying certain simple conditions) can be 
 expressed by a Fourier's series (p. 162)]. 
 
 CASE 6. A great variety of functions can be represented approximately by 
 a polynomial of the form y = a + bx + ex 2 -f dx 3 + ex* + . . . , the first 
 three or four terms being usually sufficient. To determine the coefficients 
 a, b, c, . . . , most accurately, substitute in the formula all the given pairs of 
 values of x and y, and solve the resulting equations for a,b,c, ... by the 
 method of least squares (p. 121). 
 
 CASE 7. Many simple curves can be represented approximately by an 
 equation of the hyperbolic form, xy = c -f bx -f- ay, where a, b, and c are 
 determined by substituting the co-ordinates of three conspicuous points of the 
 curve. The lines x = a and y = b are the asymptotes of the hyperbola. 
 The equation may also be written (x a) (y b) = k, where k = ab + c. 
 
 Logarithmic Cross-section Paper. In this form of cross-section paper 
 (Fig. 13), the distance from the origin to any point on the x- or y-axis is equal 
 to the logarithm of the number written against that point. Thus, in Fig. 13 
 the distances (shown for clearness on two auxiliary scales X and Y) are the 
 logarithms of the numbers written along x and y. 
 Y v 
 
 50 100 x 
 "~~t X 
 
 FIG. 13. 
 
 3 4 56789 10 
 
 FIG. 14. 
 
 Accurately made logarithmic paper can be obtained from the principal 
 dealers in draftmen's supplies. Logarithmic paper can be easily con- 
 structed, in case of need, by copying the logarithmic scale from any ordinary 
 slide rule. The actual figures along the x- and y-axes are usually left for the 
 user to insert; in so doing, notice that the numbers . . .,0.01, 0.1, 1, 10, 
 100, . . . , or such of them as may be needed to cover any given range of 
 values, must be placed at the points of division which separate the main 
 squares. It is often convenient, however, to omit the decimal point, num- 
 
LOGARITHMIC CROSS-SECTION PAPER 177 
 
 bering each square independently from 1 to 10. The length of the side of 
 one square is called the unit or base of the logarithmic paper; the larger the 
 unit, the finer the possible subdivisions of the scale. 
 
 To plot a point (x,y) on logarithmic paper, for example, the point (3,5), 
 means to find the point of intersection of the vertical line marked x 3 and 
 the horizontal line marked y = 5. In interpolating between two lines, 
 account should be taken of the fact that the divisions are not of uniform length. 
 
 Any equation of the form y = cx n when plotted on logarithmic paper will 
 be represented by a straight line whose slope is n. For, if y\ = cx\ n and 
 yt = cx2 n , then yi/yz = (xi/xz) 11 , or (log y\ log yz) /(log x\ log xz) = n. 
 The slope must be measured by aid of an auxiliary uniform scale. 
 
 EXAMPLE. Let y = a; 3 ' 2 . When x = 1, y = 1; plot this point A on the logarithmic 
 paper, and draw the straight line AE with a slope equal to % (Fig. 13). By the aid of 
 this line, the value of y for any value of x between 1 and 100 can be read off directly; 
 for example, if x = 2.50, y = 3.95, as shown by dotted lines, so that (2.50)/z = 3.95. 
 To find the value of y for any value of x outside this range, note that moving the decimal 
 point 2 places in x is equivalent to moving it 3 places in y. The line shown in Fig. 13 
 is thus equivalent to a complete table of three-halves powers. 
 
 It will be noticed that this line crosses four squares of the logarithmic paper. By 
 superposing these four squares the whole diagram may be condensed into a single square 
 (Fig. 14), in which, however, the scales for x and y now give only the sequence of digits 
 in the answer, the position of the decimal point having to be determined by inspection. 
 
 To determine whether a given set of values, z and y, satisfies a law 
 of the form y = cx n , plot the values on logarithmic paper, and see whether 
 they lie on a straight line; if they do, then the given values satisfy a law of 
 this form ; moreover, the slope of the line gives the value of n, and the value 
 of y when x = 1 gives the value of c. 
 
 If the plotted points fail to lie exactly in line, but form a curve slightly concave up- 
 ward, try subtracting some constant b from all the y's, that is, move each point downward 
 a distance equal to b units of the y-scale at that point. If it proves possible to choose b 
 so that the resulting points lie in line, then the original values obey a law of the form 
 y b = cx n , where n is again the slope of the line, and c is the value of y b 
 when x = 1. (Conversely, if the curve is concave downward, try adding b to all the 
 y's; that is, move each point upward; if the new points lie in line, the original values 
 obey a law of the form y + b = cx n .) Another method of "straightening" the 
 curve consists of adding some constant, a, to all the values of x, which has the 
 effect of shifting all the points to the right or left (by varying amounts) ; if this 
 method succeeds, the original values obey a law of the form y = c(x + a) n . 
 
 Semi-logarithmic Cross-section Paper*. This form of paper (Fig. 15) 
 has a logarithmic scale along y and a uniform scale along x. The ''scale 
 value," k, of the paper is the number which stands, on the z-axis, at a dis- 
 tance from the origin equal to the width of one of the main horizontal strips. 
 Thus, in Fig. 15, each number shown along the auxiliary scale Y is the loga- 
 rithm of the corresponding number along y, and each number shown along 
 the auxiliary scale X is 1/fcth of the corresponding number along x (here 
 k = 5). The number k, which may be chosen at pleasure, should be taken 
 equal to some simple integer, as 1, 2, or 5, or some integral power of 10. 
 
 In preparing the paper for use it is important to notice that the numbers 
 . . .,0.01,0.1, 1, 10,100, . . . {or such of them as may be needed in any 
 given case) must be placed along the 2/-axis at the points which mark the main 
 lines of division between the horizontal strips; while the numbers . . ., 
 2fc, k, 0, + k, + 2k, . . . (or such of them as may be needed) must 
 be placed along the rr-axis at uniform intervals, each interval (from to ft, 
 from A; to 2k, etc.) being equal to the width of one of the main horizontal 
 strips. The width of one of these strips is called the unit or base of the semi- 
 
 *Made by the Educational Exhibition Co.. 26 Custom House St., Providence, R.I. 
 12 
 
178 
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS 
 
 10 x 
 
 "zx 
 
 logarithmic paper; the larger the unit, the finer the possible subdivisions of 
 the scale. 
 
 To plot a point (x,y}, as a: = 3, y 5, on semi-logarithmic paper means to 
 find the point of intersection of the vertical line marked x = 3 with the 
 horizontal line marked y = 5. 
 
 Any equation of the form y = 
 c-10 mz [or y c-e mx ] when plotted 
 on semi-logarithmic paper with scale 
 value k, will be represented by a 
 straight line whose slope is km [or 
 0.4343 few.]. By a suitable choice of 
 the scale value k, any given range of 
 values of x can be brought within 
 the size of the paper. Note that e = 
 10- 4343 . 
 
 EXAMPLE. Given y = 4-10- - 1 * [or y = 
 
 4-e- - 1 *]. In Fig. 15, when x = 0, y = 4. _! " ' " ' 
 
 By plotting this point (A) on the semi- 
 logarithmic paper, with scale value 5, and FIG. 15. 
 drawing through it a straight line with 
 
 slope equal to 0.5 [or 0.217] a graphical representation is obtained from which, for 
 any value of x, the corresponding value of y can be read off. If it is desired to condense 
 the figure, several horizontal strips may be superposed on a single strip; this of course 
 renders the decimal point in the y-scale undetermined (unless a separate y-scale is 
 provided for each section of the graph). 
 
 In order to determine whether a given set of values of x and y satisfy 
 a law of the form y = c-W mx [or y = c-e mz ], plot the values of x and y on 
 semi-logarithmic paper, with a suitable scale value k, and see whether they 
 lie on a straight line; if they do so, the law is satisfied, and the values of m 
 and c may be found as follows: m = the slope of the line divided by k [or the 
 slope of the line divided by 0.4343/c], and c = the value of y when x = 0. 
 
 If the plotted points fail to lie exactly in line, but form a curve slightly concave up- 
 ward, try subtracting some constant b from all the y's, and plot the values thus modified; 
 if & can be so chosen that the revised points lie in line, then the 
 original values obey a law of the form y b = c-10 mx [or y b = 
 c-e mx ], where m and c are to be found as before. If the curve is con- 
 cave downward, add b, instead of subtracting; and replace y bby 
 y + b in the law. 
 
 Curves in Polar Co-ordinates. Any function, r, of a single vari- 
 able, f>, can be represented by a curve in polar co-ordinates (p. 137). 
 Lay off the given values of 6 as angles, the initial line Ox running 
 toward the right, and the counterclockwise direction about the origin flQ 16 
 being taken as positive. Along the terminal side of each angle 0, 
 lay off the corresponding value of r, forward if r is positive, backward if r is negative; 
 and pass a smooth curve through the points thus determined. 
 
 The rate of change of r with respect to 6 at a given point P is represented graphically 
 as follows (Fig. 16): On the tangent at P drop a perpendicular OM from the origin; 
 then r(MP/OM) represents the rate of change, dr/de, provided is measured in 
 radians. Specially ruled polar co-ordinate paper is supplied by dealers in drafting 
 supplies. 
 
 EQUATIONS INVOLVING THREE VARIABLES 
 
 The Surface z = f(x, y). Any function, z, of two variables, x and y, 
 may be represented by a surface, as follows: Plot the given pairs of values 
 of x and y as points in a horizontal x, y plane, called the base plane; at each 
 of these points erect an ordinate, parallel to a vertical axis z, and representing 
 
EQUATIONS INVOLVING THREE VARIABLES 
 
 179 
 
 by its length the value of z at that point. Then conceive a smooth surface 
 passed through the extremities of these ordinates: this surface is said to repre- 
 sent the function. In practice, the ordinates may be made by implanting 
 stiff vertical rods in a horizontal board of soft wood which serves as the base 
 plane; the surface may then be constructed by filling in the spaces with plaster 
 of Paris. Or, more simply, pieces of cardboard may be cut out to represent 
 parallel plane sections of the surface, and then stood on edge in slots cut in 
 the board to receive them. The units employed along x, y, and z need not be 
 equal to each other. 
 
 Contour-line Charts. All the points of a surface z = f(x, y) which are 
 at any given height above the base plane form a curve on the surface, called 
 a contour line of the surface. If each of these contour lines be projected 
 on the base plane, and each labeled with the value of z to which it corresponds, 
 a complete representation of the function z = f(x, y) is obtained, all in one 
 plane. A topographical map, with contour lines showing elevations above 
 the sea, and a weather map, with contour lines showing barometric pressure, 
 are familiar' examples. If there are several values of z corresponding to any 
 given point (x, y}, there will be several contour lines whose projections pass 
 through that point. 
 
 Contour-line Charts for Simultaneous Equations [of the form z - 
 f(x,y), w = F(x,y~)]. In Fig. 17, plot the function z = f(x t y) by contour 
 lines on an x,y plane, and plot the function w = F(x,y) 
 by contour lines on the same x,y plane. Then every 
 point on the diagram (either directly or by interpola- 
 tion) is the intersection of four curves an z-curve, 
 a y-curve, a z-curve, and a w-curve. Here, by 
 "curve" is meant any line, straight or curved. By 
 the aid of such a diagram, when the values of any 
 two of these four variables are given, the values of 
 the other two can be found. The method of use 
 consists simply in entering the diagram along the two 
 given curves (or lines), tracing them to their point of 
 intersection, and then coming out again along the 
 two curves (or lines) whose values are required. The best manner of num- 
 bering the curves is indicated in the figure. 
 
 Alignment Charts for Three Variables, t, u, v. Any relation between 
 three variables, t, u, v, which can be thrown into one of the forms listed in 
 later paragraphs, can be represented graphically by a very convenient form 
 of diagram called an alignment chart. In the simplest form of an alignment 
 chart for three variables there are three scales (straight or curved), along 
 which the values of the three variables, t, u, v, are marked in such a way that 
 any three values of t, u, v which satisfy the given equation are represented 
 by three points which lie in line. Hence, if the values of any two of the vari- 
 ables are given, the corresponding value of the third can be found by simply 
 drawing a straight line through the two given points and reading the value 
 of the point where it crosses the third scale. 
 
 The most important methods of constructing alignment charts for three 
 variables are described below. Where several methods are applicable in a given 
 case, the best one must be determined largely by trial. For further informa- 
 tion see M. d'Ocagne, "Traite de Nomographie" (Gauthier-Villars, Paris); 
 Carl Runge, "Graphical Methods" (Columbia University Press) ; J. B. Peddle, 
 "Construction of Graphical Charts" (McGraw-Hill); see also page 185. 
 
180 
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS 
 
 Notation. In each of the equations which follow, U stands for any 
 function of u alone, V for any function of v alone, and.Fi(0, Fz(t) for any func- 
 tions of t alone. Any of these functions may reduce to a constant. The 
 axes of x, y, and y' which are mentioned are of merely temporary use in con- 
 structing the diagram, and the letters x, y, y' should not be written on the 
 chart. It is not necessary that the axes be at right angles, provided the x 
 of a point is always measured parallel to the z-axis, and its y parallel to the 
 y-axis. 
 
 Method 1. Given, an equation which can be thrown into the form 
 
 U-Fi() + V-Fi(t) = 1, 
 
 where, for the given range of values of u and v, the largest variations in U 
 and V are less than a certain *number m. 
 
 Draw a pair of (temporary) x,y axes (Fig. 18), and through 
 the point x = 1 draw a third axis, which may be called the axis 
 of y', parallel to the axis of y. In ordinary cases, the unit of 
 measurement along x should be nearly equal to the full width 
 of the paper. Now choose a unit for y and y' such that m 
 times this unit will about equal the height of the paper, and 
 plot, in the usual way, the points (x,y) given by 
 
 FIG. 18. 
 
 labeling each point with the value of t to which it corresponds. Connect 
 these points by a smooth curve, which gives the -scale of the diagram. [If 
 Fi(t)/Fz(t) = a constant, the -scale will prove to be a straight line parallel 
 to the j/-axis.] 
 
 Then, using the same units as above, plot along y the points given by 
 y = U, labeling each point with the corresponding value of u\ and plot along 
 y f the points given by y' = V, labeling each of these points with the corre- 
 sponding value of v. This gives the u- and v-scales of the diagram. The 
 three scales being thus constructed, the a;-axis may now be erased, and the 
 diagram is ready for use. Any three points t , u, v which lie in line correspond 
 to three values of t, u, v, which satisfy the given equation. The numbering 
 on each scale should be shown at sufficiently frequent intervals to permit of 
 easy interpolation. 
 
 200 
 100 
 
 ' 5 
 
 EO 4 
 10 
 5 
 2 
 
 U'V ' " f 
 
 FIG. 19. 
 
 u 
 
 200 - 
 
 -100- 
 
 O 
 50 J 
 
 EXAMPLE 1 (Fig. 19). Let u 1>tt = t. By taking the logarithm of both sides, and 
 dividing through by log t, reduce the equation to the form (log u) (I/log + (log t>) X 
 (1.41/log - 1. Here U = log u, V = log v, Fi(t) <*> I/log t t F*(t) - 1.41/log t, and 
 Z - 1.41/2.41 - 0.585, y - (l/2.41)log t. 
 
ALIGNMENT CHARTS 181 
 
 EXAMPLE 2 (Fig. 20). Let v = ut + 16 2 , which reduces to the form (- u/16)(l/0 
 + (/16)(l// 2 ) = 1. Here U -- u/16, V = t>/16, Fi(t) = 1/t, F 2 ) = 1/ J and 
 x - 1/(1 + 0, 2/ = V(1 + 0- 
 
 NOTE. If m = oo , values of and v which give large values of U and 7 cannot be 
 shown within the limits of the paper. In such cases, the chart may be supplemented 
 by a second chart, made according to Method 2, below. 
 
 Method 2. Given, an equation which can be thrown into the form 
 
 , _ -, 
 
 U V 
 
 where, for the given range of values of u and v, the largest variation in U is 
 less than a certain number m. and the largest variation in V is less than a 
 certain number n. 
 
 Draw.a pair of temporary x,y axes, and having chosen a unit for the z-axis 
 equal to about (l/m)th of the width of the paper, and a unit for the f/-axis 
 equal to about (l/n)th of the height, plot the points (x,y) given by 
 
 x = Fi(t), y =F 2 (0, 
 
 labeling each point of this curve with the value of t to 
 which it corresponds. Connect these points by a smooth 
 curve, which gives the <-scale of the diagram. [If 
 FiCO/ACO = a constant, the <-scale will be a straight line 
 through the origin.] 
 
 Then, using the same units as above, plot along x the 
 values of 7, labeling each point with the corresponding 
 value of u', and plot along y the values of F, labeling 
 each point with the corresponding value of v. This gives 
 the u- and v-scales of the diagram. On the chart as 
 thus completed, any three points t, u, v which lie in line 
 correspond to three values of t, u, v which satisfy the 
 given equation. 
 
 EXAMPLE (Fig. 21). Let t => (uv)/(u + ), which may be written in the form 
 t/u + t/v = 1. Here U = u, V = v, Fi(t) = t, F 2 (t) - t. 
 
 NOTE. If TO = oo and n = oo , values of u and v which give large values of U and 
 V cannot be shown within the limits of the paper. In such cases the chart may be 
 supplemented by a second chart, made according to Method 1, above. 
 
 Method 3. Given, an equation which can conveniently be thrown into 
 the form 
 
 F 2 (0 = V'Fi(t) + U, 
 
 where, for the given range of values of t, the largest variation in Fi(C) is less 
 than a certain number m, and the largest variation inFz(f) is less than a certain 
 number n. 
 
 Draw a pair of temporary x,y axes, and, having chosen a unit for x equal 
 to about (l/m)th of the width of the paper and a unit for y equal to about 
 (l/n)th of. the height, plot the points (x,y) given by 
 
 * = Fi(), y =F(0, 
 
 labeling each point of the curve with the value of I to which it corresponds. 
 Connect these points by a smooth curve, which forms the <-scale. Next, 
 using the same unit for y as above, plot along the y-axis the values of U, 
 labeling each point with the corresponding value of u. This gives the w-scale. 
 Finally, with the origin as center, and any convenient radius, draw a circle 
 cutting the z-axis in A. Along this circular arc, starting from A in the coun- 
 terclockwise direction, lay off the angles whose slopes are equal to F, 
 labeling each point of the arc with the value of v to which it corresponds. 
 
182 GRAPHICAL REPRESENTATION OF FUNCTIONS 
 
 This gives the v-scale, which in this case, however, plays a peculiar role, since, 
 in using this form of chart, two straight lines are required instead of one. 
 Thus: 
 
 In order to determine whether three values, t, u, v, 
 satisfy the given equation, lay one straight line through 
 the points t and u, and another straight line through 
 the point v and the origin; if these lines are parallel, 
 the three values of t, u, v satisfy the equation. It 
 will be noticed that the function of the r-scale here is 
 to measure, in a certain sense, the slope of the line 
 joining t and u. A chart of this type may be .called 
 " an alignment chart with a sliding scale for one of the 
 variables." , 
 
 EXAMPLE (Fig. 22). Let sin u = sin 60 sin t cos 60cos t 
 cos v, which may be put in the form 
 
 (sin 60 sin t) = cos v (cos 60 cos t) + sin u. 
 Here Fi(0 = cos 60 cos t, Fz (0 = sin 60 sin t, U = sin u, V = cos v. 
 
 Method 4. Given, an equation which can be reduced to the 
 form 
 
 C7-F(0 + V = 0, 
 
 sin u- fin 60iin t- cos 60cost cos v 
 
 FIG. 22. 
 
 where, for the given range of values of u and v, the largest varia- 
 tions in U and V are less than a certain number m. 
 
 In Fig. 23, draw temporary axes x, y, and y' , and 
 choose the units as in Method 1. To construct the 
 f-scale, which will now coincide with the a;-axis, plot 
 along x the points for which 
 
 1 
 
 ~ 1 + F(t) ' 
 
 labeling each point with the value of t to which it cor- 
 responds. The it-scale, along the axis of y, and v- 
 scale, along the axis of y' , are constructed exactly as 
 in Method 1, and the finished chart is used in the 
 same way. 
 
 FIG. 23. 
 
 15000 
 
 V 
 
 150000 
 
 10000 
 
 
 FIG. 24. 
 
 EXAMPLE (Fig. 24). Let v 0.196 thi, where u is to range from to 15,000 and v 
 from to 150,000. The equation may be written in the form (- 10 u) (0.0196**) + v 
 = 0. Here U = - 10 u, V = v, F(t) = 0.0196<. 
 
 NOTE. If m = co , values of u and v which give large values of U and V cannot be 
 shown within the limits of the paper. 
 
 EQUATIONS INVOLVING FOUR VARIABLES 
 
 [For simultaneous equations of the form z = f(x,y), w = F(x,y), see p. 179.] 
 Alignment Charts for Four Variables. The extension of the methods 
 of the alignment chart to the case x of four variables, say r, s, u, v, consists 
 essentially in replacing the f-scale of the earlier diagram by a network of two 
 scales, one for r and one for s. The point where a curve r = r\ and a curve 
 s = si intersect may be spoken of as the point (ri,si). In the following equa- 
 tions, U denotes as before any function of u alone, V any function of v alone; 
 while Fi(r,s) and Fz(r,s) represent any functions of r and s. 
 Method la. Given, an equation of the form 
 
 E7-Fi(r,) + V-F 2 (r,s) = 1. 
 
EQUATIONS INVOLVING FOUR VARIABLES 
 
 183 
 
 Draw axes x, y, and y' as in Method 1, and plot the network of curves given 
 by the equations 
 
 [To do this (Fig. 25), find the point (x,y) that corresponds to each given pair 
 of values of r and s, by direct substitution in the equations for x and y. Con- 
 nect all the points for which r = 1 by a curve, and label it r = 1; connect 
 all the points for which r = 2 by another curve, and label it r = 2 ; etc. This 
 gives the family of r-curves. Similarly, through all the points for which 
 a = 1 draw a curve labeled s = 1; through all the points for which s = 2 
 draw a curve labeled s = 2; etc. This gives the family of s-curves, intersect- 
 ing the family of r-curves. Note, however, that if it is possible to eliminate 
 s (or r-) from the equations that give x and y, the resulting equation in x, y, 
 and r (or x, y, and s) can often be plotted directly for each given value of r 
 (or of s).] 
 
 Next, construct the u- and v-scales along the axes of y and y' as in Method 1. 
 [The letters x, y, and y', and the units used in plotting along these axes, should 
 be omitted from the finished diagram, as should also the axis of x.] 
 
 In the chart, as thus completed, any three points, (r,s), u, and v which lie 
 in a straight line, correspond to values of r, s, u, v which satisfy the given 
 equation. Hence, when any three of these four values are given, the fourth 
 can be found from the chart. 
 
 FIG. 25. 
 
 O"! 2 3 i 5 
 
 FIG. 26. 
 
 Method 2a. Given, an equation of the form 
 
 + 
 
 = 1. 
 
 U V 
 
 Draw axes of x and y as in Method 2, and plot the network of curves given by 
 x =Fi(r,s), y =F a (r,s). 
 
 To do this, follow the plan outlined for a similar case under Method la. 
 labeling each curve pf the r-family (Fig. 26) with the corresponding value of r. 
 and each curve of the s-family with the corresponding value of s. Next, 
 construct the u- and v-scales along the x- and y-axes, precisely as in Method 2. 
 Then any three points, (r,s), u, and v, which lie in a straight line correspond 
 to values of r, s, u, v which satisfy the given equation. 
 
 Method 3a. Given, an equation of the form 
 F 2 (r,s) = F-Fx(r,s) + U. 
 
 Draw axes of x and y, as in Method 3, and plot the network of curves given 
 by x = Fi(r,s), y = F 2 (r,s), following the plan outlined for a similar case 
 under Method la, and labeling each curve of the r-family (or s-family) with 
 the valvte of r (or s) to which it corresponds. Next, construct the w-scale 
 
184 
 
 GRAPHICAL REPRESENTATION OF FUNCTIONS 
 
 along the y-axis, and the v-scale along a circular arc, precisely as in Method 3. 
 Then any three points, (r,s) u, and v, which are so related that the line 
 through (r,s) and u is parallel to the line joining v with the origin, will corre- 
 spond to values of r, s, u, v which satisfy the given equation. 
 
 EXAMPLE for Method 3a (Fig. 27). Let cot v = cot r cos s + esc r sin s cot u, which 
 may be written (cos r cot s) = cot v (sin r esc s) cot u. Here U = cot u, V = cot v, 
 
 x z y 2 x 2 w 2 
 
 Fi(r,s) = sin r esc s, Fz(r,s) = cos r cots, whence + =1. . , = 1, 
 
 csc 2 s cot 2 s sm 2 r cos 2 r 
 
 BO that the s-curves are ellipses and the r-curves hyperbolas. 
 
 Parallel Charts, or Proportional Charts, for Four Variables. In the 
 
 following methods of representation there are four scales, one for each of the 
 four variables, and the method of using the 
 diagram consists in connecting two pairs of 
 points by parallel lines. 
 
 Method A. Given, an equation of the form 
 
 R - S = U - V 
 
 where R, S, U, V are any functions of the 
 variables r, s, u, v, respectively. [It will be 
 noted that any proportion R/S = U/V can at 
 once be thrown into this form by taking the 
 logarithm of both sides.] 
 
 In Fig. 28, draw four vertical axes, yi, yz, y'l, 
 y'z, such that the distance between y\ and 2/1 
 (which may be zero) is equal to the distance 
 beween yz and 3/2, and so that the four zero 
 points lie in line. Along these axes, using the 
 same unit for all, plot the points given by yi =R, 
 y'l = S, yz = U, 2/2 = V, and label each point 
 with the value of r, s, u, or v to which it cor- 
 responds. (The letters y\, yz, y'l, y'z are tem- 
 porary, and should not appear on the diagram.) 
 Then if the line joining two points r and u is 
 parallel to the line joining two points s and v, 
 the four values of r, s, u, v will satisfy the given 
 equation. In this and the following methods, 
 a parallel ruler, or a pair of draf tman's triangles, 
 
 will be useful in reading the chart. A "key" stating which points are to 
 be joined with which, should be clearly given on the diagram. 
 
 EXAMPLE (Fig. 28). Let 32.2 vr = us 2 , or log r - 2 log s = log u - log (32.2 ). 
 Here R = log r, S = 2 log s, U = log u, V = log (32.2 r). 
 
 Method B. Given, an equation of the form 
 
 R, _ J7 
 S ~ V 
 
 In Fig. 29, draw a pair of axes, x,y, and parallel to them (or coinciding 
 with them) a second pair of axes, xi,yi. Using any convenient horizontal 
 unit, plot along x and xi the points given by x = R, x\ = U, and using any 
 convenient vertical unit, plot along y and y\ the points given by y = S, y\ = V. 
 Label each point with the value of r, s, u, v, to which it corresponds. (The 
 letters x, y, x\, y\ should not appear on the diagram.) Then if the line joining 
 two points r and s is parallel to the line joining two points u and v, the four 
 values r, s, u, v will satisfy the given eouation. 
 
 COt y - cot r cos s + esc r sin s cot a 
 V r , Connect^?* %%} by Parallel Una. 
 
 FIG. 27. 
 
EQUATIONS INVOLVING FOUR VARIABLES 
 
 185 
 
 Method C. Given, an equation of the form 
 
 V 
 U 
 
 R - S 
 
 In Fig. 30, take a pair of axes, x,y, and through the point x = 1 draw a 
 third axis, y', parallel to y. Also, take a second pair of axes, 2,2/2, parallel 
 to (or coinciding with) the axes of x and y. Having chosen a suitable unit 
 for x and 2, and a suitable unit for y, y', and 3/2, lay off the values of R and 
 
 500- 
 100 
 
 M /'K? 
 
 r s K'Y 
 
 40 
 
 <y 
 20 
 
 US * 
 
 FIG. 28. 
 
 v 5 Y *y 
 
 
 100- 
 
 * 
 
 -10. 
 -5. 6- 
 
 5- 
 
 4- 
 
 by Para 
 
 1 y and v\ 
 lei Lines. 
 
 50- 
 
 
 rU> 5 
 
 3- 
 
 : x \. 
 
 
 5 
 
 
 ~' 5 4 
 
 3 
 
 \ 
 
 
 N V I 1 
 
 V 54 
 
 5 6 
 
 
 
 1 ' 
 
 ' . i . . ^V* 
 
 i r* 
 
 i 
 
 
 . 1 I 3 4 5 
 
 6 
 
 U (X,) 
 
 FIG. 29. 
 
 <S along y and y f , respectively, labeling each point with the value of r or 8 to 
 which it corresponds; and lay off the values of U and V along xt and j/ 2 , label- 
 ing each point with the value of u or v to which it corresponds. Then if the 
 line joining two points r and s is parallel to the line joining two points u and v, 
 the four values r, s, u, v will satisfy the given equation. This form of chart 
 is sometimes called a "Z-chart." 
 
 For further examples, see R. C. Strachan, " Nomographic Solutions for 
 Formulas of Various Types," Trans. Am. Soc. Civil Engineers, vol. 78, 1915. 
 
 VECTOR ANALYSIS 
 
 Many problems involving directed magnitudes can be advantageously 
 treated by the methods of vector analysis. The following is a brief sum- 
 mary of the principal definitions and formulae. 
 
 A set of arrows, each arrow having a given length and pointing in a given 
 direction, is called a set of vectors, provided they combine by addition ac- 
 cording to the parallelogram law (see below). Notation: a or a for a vector; 
 a or | a | for its length. Two " free" vectors are equal if they have the same 
 length and point in the same direction; two "sliding" vectors are equal if 
 they have the same length and direction, and also lie in the same line. 
 
 A scalar is any real number, positive, negative, or zero. 
 
 Addition of vectors. If an arrow a is immediately followed, tip to tail, by 
 a second arrow b, then the arrow which runs from the beginning of a to the end 
 of b is called the sum of a and b, denoted by a + b. Conversely, if a + x = 
 b, then x = b a. The laws of operation for + and are the same as in 
 ordinary algebra (pp. 112, 124). If m is a scalar, then ma means a vector 
 having the same direction as a, and m times its length. 
 
186 VECTOR ANALYSIS 
 
 Multiplication of vectors is of two kinds, as follows: 
 
 The scalar product, or dot product, of two vectors a and b, denoted by 
 a-b or sometmes by Sab, or by (ab) in round parentheses is defined as the 
 scalar quantity ab eos 0, where is the angle between a and b. 
 
 EXAMPLE. If P is a force whose point of application moves along a vector distance x, 
 then F'X = work done by F during this displacement. 
 
 Peculiarities of scalar products: (1) Since a ; b is not a vector, expressions 
 like (a-b) -C. will not occur; (2) from a-x = a-y we cannot infer that x = y, 
 hence, quotients will not occur; (3) from a-b = 0, it follows that a is per- 
 pendicular to b (unless a or b is zero) . 
 
 On the other hand, scalar products are like ordinary products in the follow- 
 ing respects: a-b = b-a, and (a + b)-(c + d) = a-c + a-d + b-c -f- b-d; 
 also, m(a-b) = (ma-b) = a*(mb), where m is any scalar. 
 
 The vector product, or cross product, of two vectors a and b, denoted by 
 axb or sometimes by Vab, or by [ab] in square brackets is defined as the 
 vector whose length is ab sin 6, where is the angle between a and b, and whose 
 direction is perpendicular to the plane of a and b (in such a sense that a right- 
 handed screw advancing along axb would turn a toward b). 
 
 EXAMPLE. If F is a force acting on a particle whose radius vector is r, then rxF 
 = the torque of F about the origin. 
 
 Peculiarities of vector products: (1) axb = bxa, so that the order of the 
 factors is always important; (2) axa = 0; (3) it is not true that ax(bxc) = 
 (axb)xc; (4) from axx = axy it does not follow that x = y; hence, quo- 
 tients will not occur; (5) from axb = 0, it follows that a and b are parallel 
 (unless a or b is zero). 
 
 On the other hand, as in ordinary algebra 
 
 (a + b)x(c + d) = axe + axd + bxc + bxd, 
 provided the order of factors in each product is preserved; also, 
 m(axb) = (ma)xb = ax(mb), where m is any scalar. Further laws are: 
 
 a-(bxc) = b-(cxa) = c-(axb); and ax (bxc) = (a-c)b (a-b)c. 
 Vector Differentiation. If r = f (f) gives a vector r as a function of a 
 scalar t, then dr/dt = lim}[f(i + AO f(0]/A*} as At approaches zero. 
 
 d(a + b) = d& + db, d(ma) = m(da) + (dm) a, 
 d(a-b) = (da)-b + a-(db), d(axb) = (da)xb + 
 
 EXAMPLE. If r = f (<) gives the position-vector of a moving particle as a function of 
 the time t, then dr/dt = its vector velocity, v, and dv/dt = its vector acceleration, a. 
 If m and n are unit vectors in the direction of the tangent and normal to the path at the 
 time t, then v = 0m, where v = ds/dt = the (scalar) path-velocity, and dm = [(ds/R)]n, 
 where R = the (scalar) radius of curvature of the path. Then 
 
 dv dm dv 2 
 
 Here dv/dt and v 2 /R are the familiar expressions for the components of acceleration along 
 the tangent and normal. 
 
INDEX 
 
 Abscissa, 173 
 Absolute value, 112 
 Acceleration 9f gravity, 73, 84 
 Adding machines, 97 
 Addition, algebraic, 112 
 
 arithmetical, 88 
 
 of complex quantities, 124 
 
 of vectors, 185 
 Algebra, elementary, 112-123 
 
 of complex quantities, 124-127 
 
 of vectors, 18.5 
 Alignment charts, 179, 182 
 Amortization (sinking fund), 67 
 Analytical geometry, 136-156 
 Anchor ring, 111 
 Angles, bisection of, 102 
 
 complementary, 128 
 
 degrees and radians, table, 44 
 
 dihedral and solid, 110 
 
 in a circle, 99 
 
 in analytical geometry, 136 
 
 in trigonometry, 128-132 
 
 minutes and seconds, table, 69 
 
 supplementary, 128 
 
 units of, 128 
 
 Annuity tables, 65, 67, 68 
 Annulus, area of, 106 
 
 contiguous circles in, 105 
 Anti-friction curve, 155 
 Anti-gudermannian, 135 
 Anti-hyperbolic functions, 135 
 
 graphs, 175; series, 161 
 Anti-togarithms, 92 
 Anti-sines, etc., 132 
 
 graphs, 174; series, 161 
 Apothecaries' weight, 71 
 Arc, length of circular, 102, 106 
 Archimedian spiral, 154 
 Arcsin, etc., see Anti-sines, etc. 
 Area, units of, 70, 76, 77 
 Areas, approximate methods, 106, 170 
 
 of similar figures, 99 
 
 of various figures, 105 
 Arithmetic, 88-98 
 Arithmetical mean, 115 
 
 progression, 114 
 Astroid, 153 
 Asymptote, of hyperbola, 145, 146 
 
 of hyperbolic spiral, 154 
 
 of tractrix, 155 
 
 Ball-bearing (annulus), 105 
 
 Barrels, volume of, 110 
 
 Baume scale, 85 
 
 Bessel's formula, 121 
 
 Binomial coefficients, tables of, 39, 116 
 
 series, 114 
 
 theorem, 114 
 Bisection, of a line, 101 
 
 of an angle, 102 
 Bisectors (in triangle), 99, 134 
 Board measure, 71 
 
 187 
 
 Briggsian logarithms, 113 
 B.t.u., 74, 75, 82 
 Bushel, 70 
 
 Calculating machines, 98 
 Calculus, 157-172 
 
 rules for differentiation, 157 
 
 table of integrals, 164 
 Calendar, 83 
 Cardioid, 153 
 Casks, volume of, 110 
 Catenary, 147 
 Cavalieri's theorem, 111 
 Centesimal measure of angles, 128 
 Chaining up hill, 150 
 Characteristic of logarithm, 92 
 
 fractional, 94 
 Charts, alignment, 179, 182 
 
 construction of, 173 
 
 contour line, 179 
 
 parallel and proportional, 184 
 Circle, constructions for, 102-105 
 
 equation of, 137 
 
 involute of, 153 
 
 tables of areas, 30, 32 
 of circumferences, 28, 32 
 of segments, 34, 35 
 
 theorems on the, 99, 106 
 Circles, circumscribed, 99, 105, 134 
 
 great, on a sphere, 100 
 
 inscribed, 99, 105, 134 
 
 radical axis of, 100, 137 
 Circular measure of angles, 128 
 table, 44 
 
 mil, 70 
 Cissoid, 155 
 
 Coins, value of foreign, 82 
 Cologarithms, 93 
 Combinations, 116 
 Complex quantities, 124-127 
 Compound interest tables, 64, 66 
 Computation, graphical methods in, 174- 
 185 
 
 machines for, 97 
 
 numerical, 88 
 
 Cones, area and volume of, 108, 109 
 Conic sections, 138-147 
 Contour line charts, 179 
 Conversion tables, 74-82 
 Coordinates, polar, 137, 178 
 
 rectangular, 136, 173 
 Cosecant, 129; tables, 51 
 
 graph, 174 
 Cosine, 129; tables, 46, 52 
 
 graph, 174; series, 161 
 Cotangent, 129; tables, 47, 52 
 
 graph, 174; series, 161 
 Coversed sine, 129 
 Cross-section paper, 173, 178 
 logarithmic, 176 
 semi-logarithmic, 177 
 Cube In geometry), 100, 110 
 
CUBE 
 
 HYPERBOLIC LOGARITHMS 
 
 Cube roots, 90 
 of 1 x, 91 
 
 table of, 16 
 
 Cubes, summation of, 115 
 table of, 8 
 
 Cubic equation, 117 
 
 Curvature, 163 
 
 Curves, empirical, 173 
 of various functions, 174 
 in analytical geometry, 151 
 
 Cycloid, 151 
 
 Cylinder, area and volume, 107, 108 
 
 Evolutes, 164 
 
 Evolution, in algebra, 90 
 
 Expansion in series, 114, 160 
 
 Exponential equations, 118 
 function, 178 
 
 graph, 174; series, 160 
 
 in complex algebra, 126, 127 
 
 table, 57 
 
 Exponents, in algebra, 113 
 in complex algebra, 126, 127 
 
 Exsecant, 129 
 
 Extreme and mean ratio, 102 
 
 Decimal equivalents, tables, 33, 69 
 
 point, position of, 89, 90 
 Definite integrals, 169 
 Degrees, and minutes, table, 69 
 
 and radians, tables, 44, 45 
 De Moivre's theorem, 126 
 Denary logarithms, 113 
 Density, 81, 84 
 Derivatives, 157, 158 
 
 of complex quantities, 127 
 
 of definite integrals, 170 
 
 of vectors, 186 
 
 partial, 159 
 Determinants, 123 
 Diameters, conjugate, 141, 146 
 Differences, 115, 159 
 Differential calculus, 157-164 
 
 equations, 171 
 Dihedral angles, 100 
 Directrix, of catenary, 147 
 
 of ellipse, 140 
 
 of hyperbola, 144 
 
 of parabola, 138 
 Division, algebraic, 112 
 
 arithmetical, 89 
 
 by logarithms, 93 
 
 by slide rule, 95 
 
 of a line, 101, 102 
 
 of complex quantities, 124 
 Distance formula, 136 
 Dodecahedron, 100, 110 
 Dyne, 73 
 
 Eccentric angle, 141 
 Eccentricity, of ellipse, 140 
 
 of hyperbola, 144 
 Ellipse, area and perimeter of, 107 
 
 constructions for, 142 
 
 properties of, 140 
 Ellipsoid, volume of, 110 
 Elliptic integrals, 170 
 Empirical curves, 173 
 Energy, units of, 79, 80 
 Epicycloid, 152 
 Epitrochoid, 153 
 Equations, algebraic, 116 
 
 differential, 171 
 
 empirical, 174 
 
 exponential, 118 
 
 normal, 122 
 
 simultaneous, 119, 121, 179 
 
 solution of, by trial, 118 
 
 trigonometric, 118 
 
 types of, 116 
 Errors, absolute and relative, 88 
 
 mean square, 122 
 
 probable, 121 ; table, 63 
 
 Factorials, 112 
 Factoring, in algebra, 112 
 Feet and inches, table, 33 
 Figures, significant, 88 
 
 similar, 99 
 
 Financial arithmetic, 98 
 Focus, of ellipse, 140 
 
 of hyperbola, 144 
 
 of parabola, 138 
 Force, unit of, 72, 74 
 Fourier's series, 162 
 Fractions, in algebra, 112 
 Frustum of cone, 108, 109 
 Functions, defined by integrals, 170 
 
 graphs of, 174 
 
 implicit, 159 
 
 of a complex variable, 127 
 
 of two variables, 159, 160, 178 
 
 Gallon, 70 
 
 Gamma function, 170 
 Geometrical mean, 113, 115 
 construction for, 102 
 
 progression, 115 
 Geometry, analytical, 136-156 
 
 elementary, 99-111 
 Golden section, 102 
 Gram, 71 
 
 -calorie, 74, 75, 82 
 Graphs, empirical, 174 
 
 in computation, 174-185 
 
 of functions, 173 
 Gravity, acceleration of, 73, 84 
 
 specific, 84 
 Gudermannian, 135 
 
 Hardness, scale of, 85 
 Harmonic mean, 115 
 Heat units, 74, 75, 79, 80 
 Helix, 156 
 Hexagon, 103 
 Horse-power, 73 
 Huyghen's approximation, 106 
 Hyperbola, area of, 107 
 
 as type of power function, 174 
 conjugate, 146 
 constructions for, 147 
 equilateral, 146 
 properties of, 144 
 Hyperbolic logarithms, 114 
 
 table, 58 
 sines, etc., 135 
 graphs, 175 
 series, 161 
 tables, 60-62 
 of a complex variable, 127 
 
 188 
 
HYPERBOLIC LOGARITHMS 
 
 POWER FUNCTION 
 
 Hyperbolic (continued) 
 
 spiral, 154 
 Hypocycloid, 152 
 Hypotrochoid, 153 
 
 i - V^l7 125 
 Icosahedron, 100, 110 
 Identity, 116 
 Imaginary quantities, 124-127 
 
 roots of equations, 118 
 Implicit functions, 159 
 Inches, and feet, table, 33 
 
 and millimeters, table, 75 
 
 miner's, 71 
 Increment, 157 
 Indeterminate forms, 163 
 Inflection-point, 160 
 Integral calculus, 164-170 
 Integrals, approximate methods for, 
 
 definite, 169 
 
 double, 170 
 
 elliptic, 170 
 
 probability, 170 
 
 table of, 164 
 Integraph, 170 
 Interest, tables, G4, 66 
 Interpolation, 115 
 
 in logarithm tables, 91, 92 
 Intersection of lines, 102 
 Inverse sine, etc., 132 
 
 graphs, 174; series, 161 
 
 sinh, etc., 135 
 
 graphs, 175; series, 161 
 Involute, 164 
 
 of a circle, 153 
 
 Joule, 73 
 
 Mass, units of, 77, 78 
 Mathematics (contents), 87 
 
 tables (contents), 1 
 Maxima and minima, 159, 160 
 Mean, arithmetical, 115 
 
 geometrical, 102, 113, 115 
 
 harmonic, 115 
 
 proportional, 113, 115 
 construction for, 102 
 
 value theorem, 169 
 Measures, weights and, 70-85 
 Medians 9f a triangle, 99, 134 
 Mensuration, 105 
 Metric system, 72 
 Minima and maxima, 160 
 Minutes and seconds, table, 69 
 Mohs's scale, 85 
 Money, foreign, 82 
 Multiples, of IT, table, 28 
 170 of 0.4343 and 2.3026, table, 62 
 
 Multiplication, algebraic, 112 
 
 arithmetical, 89 
 
 by logarithms, 93 
 
 by slide rule, 95 
 
 of complex quantities, 124 
 
 of vectors, 186 
 
 tables (list of), 89 
 
 Napierian logarithms, 114 
 
 table, 58 
 Natural functions, tables, 46, 52 
 
 logarithms, 113; table, 58 
 Nomography, 179 
 Normal equations, 122 
 Notation, algebraic, 112 
 
 by powers of ten, 90 
 Numerical computation, 88 
 
 Kilogram, 73 
 Kilowatt, 73 
 
 Latus rectum, of ellipse, 140 
 of hyperbola, 145 
 of parabola, 138 
 Least squares, 121 
 Lemniscate, 155 
 Length, units of, 70, 74, 75 
 Line, equation of, 136 
 
 geometrical, 101 
 Linear, differential equation, 171 
 
 equation, 117 
 
 function, graph of, 174 
 Liter, 71 
 Logarithmic cross-section paper, 176 
 
 function, 173 
 
 graph, 174; series, 160 
 
 spiral, 155 
 Logarithms, tables (base e), 58 
 
 tables (common), 40 
 
 theory of, 113 
 
 use in computation, 91 
 Lune, 110 
 
 Maclaurin's theorem, 161 
 Mannheim slide rule, 96 
 Mantissa of a logarithm, 92 
 negative, 93 
 
 Obelisk, volume of, 109 
 Octagon, 103 
 Octahedron, 100, 110 
 Ordinate, 173 
 Orthocenter of a triangle, 99 
 
 Pappus, theorems of, 111 
 Parabola, area of, 107 
 
 constructions for, 139 
 
 properties of, 138 
 
 as type of power function, 174 
 Paraboloid, volume of, 110 
 Parallel charts, 184 
 
 lines, 101, 136 
 Parallelogram, area of, 105 
 Parameter, 137, 170 
 Partial derivatives, 159 
 Permutations, 116 
 Perpendicular lines, 101, 136 
 Peters's formula, 121 
 Planimeter, 170 
 Polar coordinates, 137, 178 
 
 triangles on a sphere, 101 
 Polygons, 103; table, 39 
 Polyhedra, 100, 110 
 Polynomial, 118 
 Pound and poundal, 70, 73 
 Power function, 177 
 graph, 174 
 
 units of, 73, 80, 81 
 
 189 
 
POWERS 
 
 TRUNCATED PRISM 
 
 Powers, algebraic, 113 
 
 arithmetical, 90 
 
 by logarithms, 93, 94 
 
 by slide rule, 94 
 
 in complex algebra, 125, 126, 127 
 
 of ten, notation by, 90 
 Pressure, units of, 79, 80 
 Prism, area and volume, 107, 108 
 Prismoidal formula, 111 
 Probability integral, 170 
 Probable error, 121, 122 
 
 table, 63 
 Progression, arithmetical, 114 
 
 geometrical, 115 
 Proportion, in algebra, 113 
 Proportional charts, 184 
 Pyramids, area and volume, 108, 109 
 
 Quadrant, 128, 130 
 Quadratic equations, 117 
 Quadrature, 170 
 Quadrilateral, area of, 106 
 
 Radian measure of angles, 128 
 
 tables, 44, 45 
 Radical axis, 100, 137 
 Radicals and exponents, 113 
 Ratio, in algebra, 113 
 
 extreme and mean, 102 
 Real and imaginary, 124 
 Reciprocals, 90 
 
 in complex algebra, 125 
 
 of 1 x, 90 
 
 table of, 24 
 
 Rectangle, area of, 105 
 Residuals, 121 
 
 table, 63 
 
 Rhombus, area of, 105 
 Ribbon, area of, 106 
 Roots, see Powers. 
 
 of an equation, 116 
 
 Sag, in the catenary, 150 
 Scalars, 185 
 Schiele's curve, 155 
 Secant of an angle, 129 
 
 tables, 50, 52; graph, 174 
 Sector, circular, 106 
 
 spherical, 109 
 Segments, circular, 106 
 tables, 34, 35 
 
 of paraboloid, 110 
 
 spherical, 109; table, 38 
 Semi-logarithmic paper, 177 
 Series, expansion in, 114, 160 
 
 Fourier s, 162 
 
 Maclaurin's and Taylor's, 161 
 
 summation of, 115 
 Sexagesimal measure, 128 
 Significant figures, 88 
 Similar figures, 99 
 Simpson's rule, 106, 111 
 Simultaneous equations, 119 
 by determinants, 123 
 by least squares, 121 
 contour line charts for, 179 
 Sine, 129; tables, 46, 52 
 
 graph, 174; series, 161 
 Sinking fund table, 67 
 
 Slide rule, use of, 94 
 
 types of, 97 
 Slope, 157 
 Solid angle, 110 
 Solids, areas and volumes, 107 
 Specific gravity, 84 
 Sphere, area and volume, 109 
 
 table of segments, 38 
 of volumes, 36 
 
 theorems on the, 100, 109 
 Spherical segments, 109 
 table, 38 
 
 sector, 109; wedge, 110 
 
 triangles, 134 
 area of, 110, 134 
 excess of, 134 
 Spheroid, volume, 110 
 Spiral, hyperbolic, 154 
 
 involute, 153 
 
 logarithmic, 155 
 
 of Archimedes, 154 
 Square roots, 90 
 of 1 x, 90 
 table of, 12 
 Squares, summation of, 115 
 
 table of, 2 
 
 Steradian, steregon, 110 
 Submultiples, 124 
 Subnormal, subtangent, 138 
 Subtraction, algebraic, 112 
 
 arithmetical, 88 
 
 of complex quantities, 124 
 
 of logarithms, 93 
 
 of vectors, 185 
 Summation of series, 115 
 Surface for / (x, y,} 178 
 Surfaces, areas of, 107 
 Symbols, algebraic, 112 
 Symmetrical triangles, 101 
 
 Tables, list of, 1 
 Tangent of an angle, 129 
 
 graph, 174; series, 161 
 
 tables, 48, 52 
 to circle, 99 
 
 construction of, 103 
 Tape, sag of, 150 
 Taylor's theorem, 161 
 Tens, notation by, 90 
 Tetrahedron, 100, 110 
 Therm, 74 
 
 Three-halves powers, 20, 22 
 Time, 83 
 
 Torus, area and volume, 111 
 Tractrix, 155 
 Trapezoid, area of, 105 
 Trial and error method, 118 
 Triangles, plane, 99, 105, 134 
 
 solution of, 132 
 polar, 101 
 spherical, 110, 134 
 
 solution of, 134 
 Trigonometric equations, 118 
 functions, 128 
 
 graphs, 174; series, 161 
 
 of a complex variable, 127 
 
 tables, 44, 46, 52 
 Trigonometry, 128-135 
 Trochoid, 152 
 Troy weight, 71 
 Truncated prism, 107, 108 
 
 190 
 
UNGULA ZONE 
 
 Ungula, 108 Watt, 73 
 
 Units, 72 Wedge, 109 
 
 spherical, 119 
 
 Weight, units of, 71, 77, 78 
 
 Variable, change of, 169 Weights and measures, 70-85 
 
 Vector analysis, 185 Work, units of, 73, 79, 80 
 
 Velocity, units of, 78, 80 
 
 Versed sine, 129; graph, 174 Yard, 70 
 
 Volume, units of, 70, 71, 76, 77 Year, 83 
 
 Volumes, of solids, 107 
 
 of similar figures, 99 Zone of a sphere, 109 
 
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