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1891 



Cornell University Library 
arV17998 
Handbook of mathematics |w 




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HANDBOOK OF MATHEMATICS 
FOR ENGINEERS 



7^ QraW'JfillBook & Im 

PUDLISMER.S OF BOOKS F O P-^ 

Coal Age ' Electric Railway Journal 
Electrical World -^ Engineering. News-Record 
American Machinist ■» The Osntractor 
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 TJNIVEBSITT 

WITH TABLES OF WEIGHTS AND MEASURES BY 
LOUIS A. FISCHER, B. S. 

CHIEF OF DIVISION OF WEIGHTS AND MEASURES, 
17, B. BUREAU OF STANDARDS 



REPRINT OF SECTIONS 1 AND 2 OF L. S. MARKS'S 
"MECHANICAL ENGINEERS' HANDBOOK" 



First Edition 



McGRAW-HILL BOOK COMPANY, Inc. 
239 WEST 39TH STREET. NEW YORK 

LONDON: HILL PUBLISHING CO., Ltd. 

6 & 8 BOUVERIE ST„ E.G. 

1918 



Copyright, 1918, bt the 
McGbaw Hill Book Company, Inc. 



Copyright, 1916, by 
Edward V. Huntington. 



Taxi MAFIjB FKTSBfi "T O K K PX 



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 
toupd) 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. 



CONTENTS 



Preface , 



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. 
LOUIS A. FISCHER, B. S., Chief of Division of Weights and Measures, 

XJ. 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 

Compoutid Interest and Annuities. 64 

Decimal Equivalents 69 



WEIGHTS AND MEASITBE3 



BY LOUIS A. FISCHER 



Page 



U. S. Customary Weights and 

Measures 70 

Metric Weights and Measures 71 

Systema 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 



uathemat-icaij tables 



SQUARES 


OF NUMBERS 
















N 





1 


2 


3 


i 


6 


6 


7 


8 


9 




1.00 


I.OOO 


1.002 


1.004 


1.006 


1.008 


1.010 


1.012 


1.014 


1.016 


1.018 


2 


1 


1.020 


1.022 


1.024 


1.026 


1.028 


1.030 


1.032 


1.034 


1.036 


1.038 




2 


1.040 


1.042 


1.044 


1.047 


1.049 


1.051 


1.053 


1.055 


1.057 


1.059 




3 


1.061 


1.063 


1.065 


1.067 


1.069 


1.071 


1.073 


1.075 


1.077 


1.080 




4 


1.082 


1.084 


1.086 


1.088 


1.090 


1.092 


1.094 


1.096 


1.098 


1.100 




1.05 


1.102 


1.105 


1.107 


1.109 


1.111 


1.113 


1.115 


1.117 


1.119 


1.121 




6 


1.124 


1.126 


1.128 


1.130 


1.132 


1.134 


1.136 


1.138 


1.141 


1.143 




7 


1.145 


1.147 


1.149 


1.151 


1.153 


1.156 


1.158 


1.160 


1.162 


1.164 




8 


1.166 


1.169 


1.171 


1.173 


1.175 


1.177 


1.179 


1.182 


1 184 


1.186 




9 


1.188 


1.190 


1.192 


1.195 


1.197 


1.199 


1.201 


1.203 


1.206 


1.208 




LIO 


1.210 


1.212 


1.214 


1.217 


1.219 


1.221 


1.223 


1.225 


1.228 


1.230 






1.232 


1.234 


1.237 


1.239 


1.241 


1.243 


1.245 


1.248 


1.250 


1.252 




2 


1.254 


1.257 


1.259 


1.261 


\.lbl 


1.266 


1.268 


1.270 


1.272 


1.275 




3 


1.277 


1.279 


1.281 


1.284 


1.286 


1.288 


1.209 


1.293 


1.295 


1.297 




4 


1.300 


1.302 


1.304 


1.306 


1.309 


1.311 


1.313 


1.316 


1.318 


1.320 




LIS 


1.322 


1.325 


1.327 


1.329 


1.332 


1.334 


1.336 


1.339 


1.341 


1.343 




6 


1.346 


1.348 


1.350 


1353 


1.355 


1.357 


1.360 


1.362 


1.364 


1.367 




7 


1.369 


1.371 


1.374 


1.376 


1.378 


1.381 


1.383 


1.385 


1.388 


1.390 




8 


1.392 


1.395 


1.397 


1.399 


1.402 


1.404 


1.407 


1.409 


1.411 


1.414 




9 


1.416 


1.418 


1.421 


1.423 


1.426 


1.428 


1.430 


1.433 


1.435 


1.438 




i.ao 


1.440 


1.442 


1.445 


1.447 


1.450 


1.452 


1.454 


1.457 


1.459 


1.462 




I 


1.464 


1.467 


1.469 


1.471 


1.474 


1.476 


1.479 


1.481 


1.484 


1.486 




2 


1.488 


1.491 


1.493 


1.496 


1.498 


1.501 


1.503 


1.506 


1.508 


1.510 




3 


1.513 


1.515 


I.5I8 


1.520 


1.523 


1.525 


1.528 


1.530 


1.533 


1.535 




4 


1.538 


1.540 


1.543 


1.545 


1.548 


1.550 


1.553 


1.555 


1.558 


1.560 




1.86 


1.562 


1.565 


1.568 


1.570 


1.573 


1.575 


1.578 


1.580 


1.583 


1.585 


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 


1.626 


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 






1.716 


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 


1.756 


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 




1.4S 


2.102 


2.105 


2.108 


2.1 11 


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 


2.175 


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 JV requires moving it TWO places in body 
of table (see p. 6). 



MATHEMATICAL TABLES 



SQUARES (continued) 


















N 





1 


2 


3 


4 


6 


6 


7 


8 


9 




1.80 


2.250 


2.253 


2.256 


2.259 


2.262 


2.265 


2.268 


2,271 


2.274 


2.277 


3 


1 


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 


2332 


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 




1.66 


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 


2487 


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 


2.585 


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 


2686 




4 


2.690 


2.693 


2.696 


2.699 


2.703 


2,706 


2.709 


2.713 


2.716 


2.719 




1.66 


2722 


2.726 


2.729 


2.732 


2.736 


2,739 


2.742 


2.746 


2.749 


2.752 




6 


2.756 


2.759 


2 762 


2.766 


2.769 


2772 


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 


2 849 


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 




1.76 


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 




1.86 


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 




1.96 


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 



l/x' = 0.101321 e! = 7.38906 



MATHEMATICAL TABLES 



SQUARES (continued) 


















N 





1 


2 


3 


4 


S 


6 


7 


8 


9 




2.00 


4.000 


4.004 


4.C08 


4.012 


4.016 


4.020 


4.024 


4.028 


4032 


4036 


4 




4.040 


4.044 


4.048 


4.052 


4.056 


4.060 


4064 


4.068 


4072 


4.076 




2 


4.080 


4.084 


4.088 


4.093 


4.097 


4.101 


4.105 


4109 


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 




3.0S 


4.202 


4.207 


4.211 


4.215 


4.219 


4223 


4.227 


4.231 


4235 


4.239 




6 


4.244 


4.248 


4.252 


4.256 


4.260 


4.264 


4268 


4.272 


4.277 


4.281 




7 


4.285 


4.289 


4.293 


4.297 


4.301 


4.306 


4.310 


4314 


4.318 


4.322 




8 


4.326 


4.331 


4.335 


4.339 


4343 


4.347 


4.351 


4.356 


4360 


4364 




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 


4444 


4.448 




1 


4.452 


4.456 


4.461 


4.465 


4.469 


4.473 


4.477 


4.482 


4 486 


4490 




2 


4.494 


4.499 


4.503 


4.507 


4.511 


4.516 


4.520 


4.524 


4.528 


4 533 




3 


4.537 


4.541 


4.545 


4550 


4.554 


4.558 


4562 


4567 


4.571 


4.575 




4 


4.580 


4.584 


4.588 


4.592 


4.597 


4.601 


4.605 


4.610 


4614 


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 


4666 


4.670 


4.674 


4.679 


4.683 


4.687 


4.692 


4696 


4700 


4.705 




7 


4.709 


4.713 


4.718 


4.722 


4.726 


4.731 


4.735 


4.739 


4.744 


4748 




8 


4752 


4.757 


4.761 


4.765 


4770 


4.774 


4.779 


4783 


4787 


4.792 




9 


4.796 


4.800 


4.805 


4.809 


4.814 


4.818 


4.822 


4.827 


4.831 


4.836 




a.20 


4.840 


4.844 


4.849 


4.853 


4.858 


4.862 


4.866 


4.871 


4.875 


4880 




1 


4.884 


4.889 


4.893 


4.897 


4.902 


4906 


4.911 


4.915 


4920 


4924 




2 


4.928 


4.933 


4.937 


4.942 


4.946 


4951 


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.D13 




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 


5 148 




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 


5.735 


5.741 


5.746 


5.750 


5.755 




2.10 


5.760 


5.765 


5.770 


5.774 


5.779 


5.784 


5.789 


5.794 


5.798 


5.803 






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. 6). 



MATHEMATICAL TABLES 



SQUARES 


(continued) 


















N 





1 


2 


3 


4 


6 


6 


7 


8 


9 


li 


2.60 


6.250 


6.255 


6.260 


6.265 


6270 


6.275 


6280 


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 


6371 


6.376 


6.381 


6.386 


5.391 


6.395 




3 


6.401 


6.406 


6.411 


6.416 


6.421 


6.426 


6.431 


6.436 


5.441 


6.447 




4 


6.452 


6.457 


6.462 


5.467 


6.472 


6.477 


6.482 


6.487 


5.492 


5.497 




2.E6 


6.502 


6.508 


6.513 


6.518 


6.523 


6.528 


6.533 


6.538 


5.543 


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 


5.636 


6.641 


5.646 


5.551 




8 


6.656 


6.662 


6.667 


6.672 


6.677 


6.682 


6.687 


6693 


5.698 


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 


6.796 


6.802 


6.807 




1 


6.812 


6.817 


6.823 


6.828 


6.833 


6.838 


6.843 


6.849 


6.854 


5.859 




2 


6.864 


6.870 


6.875 


6.880 


6.885 


6.891 


6.896 


6.901 


6 906 


6.912 




3 


6.917 


6.922 


6.927 


6.933 


6.938 


6.943 


6948 


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 


7.155 


7J51 


7.165 


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 


7.258 


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 




I 


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 


7.425 


7.431 


7.437 


7.442 


7.447 




3 


7.453 


7.458 


7.454 


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 


7.545 


7.552 


7.557 




2.75 


7.562 


7.568 


7.574 


7.579 


7.585 


7.590 


7.595 


7.601 


7.507 


7.612 


5 


6 


7.618 


. 7.623 


7.629 


7.634 


7.640 


7.645 


7.651 


7.555 


7.562 


7.657 




7 


7.673 


7.678 


7.684 


7.690 


7.695 


7.701 


7.705 


7.712 


7.717 


7.723 




8 


7.728 


7.734 


7.740 


7.745 


7.751 


7.756 


7.752 


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 


8032 


8037 


8.043 


8.049 


8.054 


8060 




4 


8.066 


8.071 


8.077 


8083 


8.088 


8094 


8100 


8.105 


8.111 


8.117 




2.86 


8.122 


8.128 


8.134 


8.140 


8.145 


8.151 


8157 


8.162 


8158 


8174 




6 


8.180 


8.185 


8191 


8197 


8202 


8208 


8.214 


8.220 


8.225 


8.231 




7 


8.237 


8.243 


8248 


8.254 


8260 


8256 


8.271 


8.277 


8283 


8289 




8 


8.294 


8.300 


8.306 


8.312 


8317 


8.323 


8.329 


8.335 


8341 


8.345 




9 


8.352 


8.358 


8364 


8369 


8.375 


8.381 


8.387 


8.393 


8398 


8.404 




2.90 


8.410 


8.416 


8.422 


8.427 


8.433 


8.439 


8.445 


8.451 


8456 


8.452 




1 


8.468 


8.474 


8.480 


8.486 


8.491 


8497 


8.503 


8.509 


8515 


8.521 




2 


8 526 


8.532 


8.538 


8.544 


8550 


8555 


8561 


8557 


8573 


8579 




3 


8585 


8591 


8597 


8.602 


8608 


8.514 


8620 


8626 


8532 


8.538 




4 


8.644 


8.649 


8.655 


8661 


8.667 


8.573 


8.679 


8685 


8591 


8697 




2.95 


8702 


8.708 


8714 


8720 


8.726 


8732 


8738 


8744 


8.750 


8755 




6 


8.762 


8.768 


8.773 


8779 


8.785 


8791 


8.797 


8803 


8809 


8.815 




7 


8821 


8.827 


8.833 


8839 


8.845 


8851 


8857 


8.853 


8868 


8,874 




8 


8.880 


8.886 


8.892 


8898 


8904 


8.910 


8916 


8.922 


8928 


8.934 




9 


8.940 


8.946 


8.952 


8.958 


8.964 


8970 


8.975 


8.982 


8988 


8.994 





: 9.86960 l/7r2 = 0.101321 «« = 7.38906 



MATHEMATICAL TABLES 



SQUARES {continued) 


















W 





1 


2 


3 


4 


5 


6 


7 


8 


9 


ti 


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 


9.153 


9.159 


9.175 




3 


9.181 


9.187 


9.193 


9.199 


9.205 


9.211 


9.217 


9.223 


9.229 


9.235 




4 


9.242 


9.248 


9.254 


9.260 


9.265 


9.272 


9.278 


9.284 


9.290 


9J96 




S.06 


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 


9.452 


9.458 


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 




S.IO 


9.610 


9.616 


9.622 


9.629 


9.635 


9.541 


9.647 


9.553 


9.560 


9.656 




1 


9.672 


9.678 


9.685 


9.591 


9.697 


9.703 


9.709 


9.715 


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 




3.1S 


9.922 


9.929 


9.935 


9.941 


9.948 


9.954 


9.950 


9.957 


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 


10.53 


10.69 


10.75 


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 




3.6 


12.25 


12.32 


12.39 


12.45 


12.53 


12.60 


12.57 


12.74 


12.82 


12.89 




6 


12.96 


13.03 


13.10 


13.18 


13.23 


13.32 


13.40 


13.47 


13.54 


13.52 




7 


13.69 


13.76 


13.84 


,13.91 


13.99 


14.06 


14.14 


14.21 


14.29 


14.35 


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 


15.58 


15.75 


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 






16.81 


16.89 


16.97 


17.06 


17.14 


17.22 


17.31 


17.39 


17.47 


17J6 




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 


18.65 


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 




4.6 


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 


22.56 


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 


24.50 


24.70 


24.80 


24.90 





n-g = 9.86960 i.r/2y = 2.46740 lA" = 0.101321 



Explanation of Table of Squares (pp. 2-7). 

This table gives the value of N^tov values of N from 1 to 10, correct to four figures, 
(luterpolated 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)« = 10.35; (0.03217)2 = 0.001035; (3217)2 = 10350000 

This table can also be used inversely, to give square roots. 



MATHEMATICAL TABLES 



SQUARES (continued) 


















N 





1 


2 


3 


4 


6 


6 


7 


8 


9 




6.0 


25.00 


25.10 


25.20 


25.30 


25.40 


25.50 


25.60 


25.70 


25.81 


25.91 


10 




26.01 


26.11 


26.21 


26.32 


26.42 


26.52 


26.53 


26.73 


26.83 


25.94 




2 


27.04 


27.14 


27.25 


27.35 


27.46 


27.56 


27.57 


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 




5.8 


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 


32.25 


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 


34.59 


12 


9 


34.81 


34.93 


35.05 


35.16 


35.28 


35.40 


35.52 


35.54 


35.76 


35.88 




6.0 


36.00 


36.12 


36.24 


36.36 


36.48 


35.60 


35.72 


36.84 


36.97 


37.09 




1 


37.21 


37.33 


37.45 


37.58 


37.70 


37.82 


37.95 


38.07 


38.19 


38.32 




2 


38.44 


38.55 


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 


41.50 


41.73 


41.86 


41.99 


42.12 




6.S 


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 


4396 


44.09 


44.22 


44.35 


44.49 


44.62 


44.76 




7 


44.89 


45.02 


45.16 


45.29 


45.43 


45.55 


45.70 


45.83 


45.97 


45.10 




8 


46.24 


46.38 


46.51 


45.65 


45.79 


45.92 


47.05 


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 


54.45 


54.51 


15 


4 


54.76 


54.91 


55.05 


55.20 


55.35 


55.50 


55.55 


55.80 


55.95 


56.10 




7.8 


56.25 


56.40 


56.55 


55.70 


55.85 


57.00 


57.15 


57.30 


57.46 


57.61 




6 


57.76 


57.91 


58.05 


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 


60.05 


60.22 


60.37 


60.53 


60.58 




8 


60.84 


61.00 


61.15 


61.31 


51.47 


61.52 


51.78 


61.94 


62.09 


62.25 


16 


9 


62.41 


62.57 


62.73 


62.88 


63.04 


63.20 


63.36 


63.52 


63.58 


53.84 




8.0 


64.00 


64.16 


64.32 


64.48 


54.54 


54.80 


64.96 


65.12 


65.29 


65.45 




I 


65.61 


65.77 


65.93 


65.10 


66.25 


56.42 


56.59 


65.75 


66.91 


67.08 




2 


67.24 


67.40 


67.57 


67.73 


67.90 


68.06 


68.23 


58.39 


68.55 


68.72 




3 


68.89 


69.06 


69.22 


69.39 


69.56 


69.72 


59.89 


70.05 


70.22 


70.39 


17 


4 


70.56 


70.73 


70.90 


71.05 


71.23 


71.40 


71.57 


71.74 


71.91 


72.08 




8.6 


72.25 


72.42 


72.59 


72.75 


72.93 


73.10 


73.27 


73.44 


73.62 


73.79 




6 


73.95 


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 


76.55 


75.74 


75.91 


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 


80.54 


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 


84.45 




2 


84.64 


84.82 


85.01 


85.19 


85.38 


85.55 


85.75 


85.93 


85.12 


86.30 




3 


86.49 


85.68 


85.85 


87.05 


87.24 


87.42 


87.51 


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 


90.05 




9.6 


90.25 


90.44 


90.53 


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 


95.05 


95.25 


95.45 


95.65 


95.84 




8 


96.04 


96.24 


96.43 


96.53 


95.83 


97.02 


97.22 


97.42 


97.61 


97.81 


20 


9 


98.01 


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 reqiiires moving it TWO places in body 
of table (see p. 6). 



MATHEMATICAL TABLES 



CUBES OF NUMBERS 
















N 





1 


2 


3 


t 


6 


6 


7 


8 


9 


<<3 


1.00 


1.000 


1.003 


1.005 


1.009 


1.012 


1.015 


1.018 


1.021 


1.024 


1.027 


3 




1.030 


1.033 


1.036 


1.040 


1.043 


1.046 


1.049 


1.052 


1.055 


1.038 




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 


1.123 


1.128 


1.131 


1.135 


1.138 


1.141 


1.144 


1.148 


1.151 


1.134 




l.OS 


1.158 


1.161 


1.164 


1.168 


1.171 


1.174 


1.178 


1.181 


1.184 


1.188 




6 


1.191 


1.194 


1.198 


1.201 


1.203 


1.208 


1.211 


1.215 


1.218 


1.222 




7 


1.225 


1.228 


1.232 


1.235 


1.239 


1.242 


1.246 


1.249 


1.253 


1.236 




8 


1.260 


1.263 


1.267 


1.270 


1.274 


1.277 


1.281 


1.284 


1.288 


1.291 


4 


9 


1.295 


1.299 


1.302 


1.306 


1.309 


1.313 


1.317 


1.320 


1.324 


1.327 




1.10 


1.331 


1.335 


1.338 


1.342 


1.346 


1.349 


1.333 


1.357 


1.360 


1.364 




1 


1.368 


1.371 


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 


1.438 


1.462 


1.466 


1.470 


1.474 


1.478 




4 


1.482 


1.485 


1.489 


1.493 


1.497 


1.301 


1.305 


1.509 


1.513 


1.517 




1.15 


1.521 


1.525 


1.529 


1.533 


1.537 


1.341 


1.545 


1.549 


1.553 


1.557 




6 


1.561 


1.565 


1.569 


1.573 


1.577 


1.381 


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 


1.683 


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 


1.730 


1.754 


1.738 


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 


3 


1.2S 


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 


2010 


2.013 


2.019 


2.024 


2.029 


2.034 


2.039 


2044 




7 


2.048 


2.053 


2.058 


2.063 


2.068 


2.073 


2.078 


2.082 


2.087 


2092 




8 


2.097 


2.102 


2.107 


2112 


2.117 


Z122 


2.127 


2.132 


2.137 


2142 




9 


2.147 


1152 


2.157 


2.162 


2.167 


2.172 


2177 


2.182 


2.187 


2.192 




1.30 


2.197 


2.202 


Z.207 


/212 


2.217 


2.222 


2.228 


2233 


2.238 


2243 






2248 


2253 


2.258 


2.264 


2269 


2.274 


2.279 


2284 


2.290 


2295 




2 


2.300 


2305 


2.310 


2.316 


2.321 


2.326 


2331 


2.337 


2.342 


2.347 




3 


2.353 


2.358 


2363 


2.369 


2.374 


2379 


2.385 


2.390 


2.393 


2.401 




4 


Z406 


2411 


2.417 


2422 


2.428 


2.433 


2.439 


2.444 


2.449 


2.433 




1.36 


2460 


2.466 


2471 


2477 


2.482 


2488 


2493 


2.499 


2.504 


2.510 


6 


6 


2.515 


2.521 


2527 


2.532 


2538 


2343 


2549 


2.554 


2.560 


2.566 




7 


2571 


2.577 


2583 


2.588 


2.394 


2600 


2605 


2611 


2.617 


2.622 




8 


2628 


2.634 


2.640 


2.645 


2.631 


2657 


2.663 


2668 


2.674 


2.680 




9 


2.686 


2.691 


2.697 


2.703 


2.709 


2.715 


2.721 


2.726 


2.732 


Z738 




1.40 


2744 


2750 


2.756 


2.762 


2.768 


2.774 


2779 


Z785 


2.791 


2797 




1 


2.803 


2809 


2.815 


2821 


2.827 


Z833 


2839 


2.843 


2.851 


2.837 




2 


2.863 


2869 


2.875 


2881 


2888 


2.894 


2.900 


2.906 


2.912 


2918 




3 


2.924 


2930 


2.936 


2943 


2949 


2.955 


2.961 


2967 


2.974 


2980 




4 


2.986 


2.992 


2.998 


3.003 


3.011 


3.017 


3.023 


3.030 


3.036 


3.042 




1.16 


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 


3.123 


3.131 


3.138 


3.144 


3.131 


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 


3.253 


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 plaeea io 
body of table (see p. 10). 



MATHEMATICAL TABLES 



CUBES (continued) 


















N 





1 


2 


3 


4 


6 


6 


7 


8 


9 




ISO 


3.375 


3.382 


3.389 


3.395 


3.402 


3.409 


3.416 


3.422 


3.429 


3.436 


7 


I 


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 




l.SS 


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 


3.90O 


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 




1 


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 


4.355 


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 




1.66 


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 


4.955 


4.965 


4.974 


4.983 


4.991 




I 


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 




1.76 


5.359 


5.369 


5.378 


5.387 


5.396 


5.405 


5.415 


5.424 


5.433 


5.442 




6 


5.432 


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 




1.86 


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 


6518 


6.529 




7 


6.539 


6550 


6.560 


6.571 


6.581 


6.592 


6.602 


6.613 


6.623 


6634 


11 


8 


6.645 


6.655 


6.666 


6.677 


6.687 


6.698 


6708 


6.719 


6730 


6741 




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 






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 


7.111 


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 




1.96 


7.415 


7.426 


7.438 


7.449 


7.461 


7.472 


7.484 


7.495 


7.507 


7.518 


12 


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 


7.975 


7.988 





ir» = 31.0063 1A3 = 0.0322515 + 



10 



MATHEMATICAL TABLES 



CUBES {ctmlinued) 


















N 





1 


a 


3 


4 


5 


6 


7 


8 


9 




2.00 


8,000 


8.012 


8.024 


8.036 


8.048 


8.060 


8.072 


8.084 


8.096 


8.108 


12 


1 


8.121 


8.133 


8.145 


8.157 


8.169 


8.181 


8.194 


8.206 


8.218 


8.230 




2 


8242 


8.255 


8.267 


8.279 


8.291 


8.304 


8.316 


8.328 


8.341 


8.353 






8.365 


8.378 


8.390 


8.403 


8.415 


8.427 


8.440 


8.452 


8.465 


8.477 






8.490 


8.502 


8.515 


8.527 


8.540 


8.552 


8.565 


8.577 


8.590 


8.503 




2.05 


8.615 


8.628 


8.640 


8.653 


8.666 


8.678 


8.691 


8.704 


8.716 


8.729 


13 




8.742 


8.755 


8.767 


8.780 


8.793 


8.806 


8.818 


8.831 


8.844 


8.857 






8.870 


8.883 


8.895 


8.908 


8.921 


8.934 


8.947 


8.960 


8.973 


8.985 






8.999 


9.012 


9.025 


9.038 


9.051 


9.064 


9.077 


9.090 


9.103 


9ill6 






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 






9.394 


9.407 


9.421 


9.434 


9.447 


9.461 


9.474 


9.488 


9.501 


9.515 






9.528 


9.542 


9.555 


9.569 


9.582 


9.596 


9.609 


9.623 


9.635 


9650 


14 




9.664 


9.677 


9.691 


9704 


9.718 


9.732 


9745 


9.759 


9.773 


9.787 






9.800 


9.814 


9.828 


9.842 


9.855 


9.869 


9.883 


9.897 


9.911 


9.925 




2.1S 


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 




10.65 


10.79 


10.94 


11.09 


11.24 


11.39 


11.54 


11.70 


11.85 


12.01 


IS 




12.17 


12.33 


12.49 


12.65 


12.81 


12.98 


13.14 


13.31 


13.48 


13.65 


16 




13.82 


14.00 


14.17 


14.35 


14.53 


14.71 


14.89 


15.07 


15.25 


15.44 


18 


2.S 


15.62 


15.81 


16.00 


16.19 


16.39 


16.58 


16.78 


16.97 


17.17 


17.37 


20 




17.58 


17.78 


17.98 


18.19 


18.40 


18.61 


18.82 


19.03 


1925 


19.47 


21 




19.68 


19.90 


20.12 


20.35 


20.57 


20.80 


21.02 


21.25 


21.48 


21.72 


23 




21.95 


22.19 


22.43 


22.67 


22.91 


23.15 


23.39 


23.64 


23.89 


24.14 


24 




24.39 


24.54 


24.90 


25.15 


25.41 


25.67 


25.93 


26.20 


26.46 


26.73 


25 


3.0 


27.00 


27.27 


27.54 


27.82 


28.09 


28.37 


28.65 


28.93 


2922 


2950 


28 




29.79 


30.08 


30.37 


30.66 


30.96 


31.25 


31.55 


31.86 


32.16 


32.45 


30 




32.77 


33.08 


33.39 


33.70 


34.01 


34.33 


34.65 


34.97 


35.29 


35.61 


32 




35.94 


36.26 


36.59 


36.93 


37.26 


37.60 


37.93 


38.27 


38.61 


38.96 


34 




39.30 


39.65 


40.00 


40.35 


40.71 


41.05 


41.42 


41.78 


42.14 


42.51 


36 


3.6 


42.88 


43.24 


43.61 


43.99 


44.36 


44.74 


45.12 


45.50 


45.88 


45.27 


39 




46.66 


47.05 


47.44 


47.83 


4823 


48.63 


49.03 


4943 


4984 


50.24 


40 




50.65 


51.06 


51.48 


51.90 


52.31 


52.73 


53.16 


53.58 


54.01 


54.44 


42 




54.87 


55.31 


55.74 


56.18 


56.62 


57.07 


57.51 


57.96 


58.41 


58.85 


44 




59.32 


59.78 


60.24 


60.70 


61.16 


61.63 


62.10 


62.57 


63.04 


63.52 


47 


1.0 


64.00 


64.48 


64.96 


65.45 


65.94 


66.43 


66.92 


67.42 


67.92 


68.42 


49 




68.92 


69.43 


69.93 


70.44 


70.96 


71.47 


71.99 


72.51 


73.03 


73.56 


52 




74.09 


74.62 


75.15 


75.69 


76.23 


76.77 


77.31 


77.85 


78.40 


78.95 


54 




79.51 


80.06 


80.62 


81.18 


81.75 


82.31 


82.88 


83.45 


84.03 


84.60 


58 




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 


%.07 


96.70 


52 




97.34 


97.97 


98.61 


99.25 


99.90 


100.54 










54 














100.5 


101.2 


101.8 


102.5 


103.2 


7 




103.8 


104.5 


105.2 


105.8 


106.5 


107.2 


107.9 


108.5 


109.2 


1099 


7 


8 


110.6 


111.3 


112.0 


112.7 


113.4 


114.1 


114.8 


115.5 


115.2 


115.9 


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 tlie value of If> for values of iV 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 iV is equivalent to moving it three 
places in the body of the table. For example: 

(4.852)' = 114.2; (0.4852)= = 0.1142; (48S.2)» = 114200000 

This table may also be used inversely, to give cube roots. 



MATHEMATICAL TABLES 11 

CUBES (continued) 



N 





1 


2 


3 


i 


5 


6 


7 


8 


9 


|i 


S.0 


125.0 


125.8 


126.5 


127.3 


128.0 


128.8 


129.6 


130.3 


131.1 


131.9 


8 




132.7 


133.4 


134.2 


135.0 


135.8 


136.6 


137.4 


138.2 


139.0 


139.8 




2 


140.5 


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 


156.5 


9 


4 


157.5 


158.3 


159.2 


160.1 


151.0 


161.9 


162.8 


163.7 


164.5 


155.5 




5.5 


165.4 


167.3 


168.2 


169.1 


170.0 


171.0 


171.9 


172.8 


173.7 


174.7 




6 


175.5 


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 


2043 




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 


777 5 


223.6 


224.8 


225.9 


11 




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 


2609 




4 


262.1 


263.4 


264.6 


265.8 


257.1 


268.3 


269.5 


270.8 


272.1 


273.4 




6.6 


274.5 


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 


3103 


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 




357.9 


359.4 


360.9 


362.5 


364.0 


365.5 


367.1 


368.6 


3701 


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 


415.8 


418.5 


420.2 


17 


7.6 


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 




I 


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 


512.0 




8.5 


614.1 


616.3 


618.5 


620.7 


622.8 


625.0 


627.2 


629.4 


531.5 


533.8 


22 


6 


635.1 


638.3 


640.5 


642.7 


645.0 


647.2 


649.5 


651.7 


554.0 


656.2 




7 


658.5 


660.8 


663.1 


665.3 


667.5 


669.9 


672.2 


674.5 


675.8 


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 


748.5 


751.1 


25 




753.6 


756.1 


758.6 


761.0 


763.5 


756.1 


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 


830.5 


833.2 


835.9 


838.6 


841.2 


843.9 


846.5 


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 


lOOO.p 























=31.0063 l/ir» = 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 


6 


6 


7 


8 


9 




1.0 


I.OOO 


1.005 


1.010 


1.015 


1.020 


1.025 


1.030 


1.034 


1.03.9 


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 




l.S 


1.225 


1.229 


1.233 


1.237 


1.241 


1.245 


1.249 


1.253 


1.257 


1.251 




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 


1.355 


1.360 


1.364 


1.367 


I.37I 


1.375 




9 


1.378 


1.382 


1.385 


1.389 


1.393 


1.395 


1.400 


1.404 


1.407 


I.4I1 




2.0 


I.4I4 


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 


1.465 


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 




3.6 


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 


1.540 




7 


1.643 


1.646 


1.649 


1.652 


1.655 


1.658 


1.661 


1.654 


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 


1.765 


1.769 


1.772 


1.775 


1.778 


1.780 


1.783 


1.786 




2 


1.789 


1.792 


1.794 


1.797 


I.80O 


1.803 


1.805 


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.B 


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 


2.054 


2.065 


2.059 


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 


Z124 


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 


2.151 


2.163 


2.156 




7 


2.168 


2.170 


2.173 


2.175 


2.177 


2.179 


2.182 


2.184 


2.185 


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 


2 773 


2.225 


2.227 


2.229 


2.B2 


2.234 





-\/7= 1.77245 + 1/V^= 0.56419 



1.25331 Vi'= 1.64872 



Explanation of Table of Square Roots (pp. 12-15). 

This table gives the values of y/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 iV is equivalent to moving it one place in 
the square root of N. For example: 

V2.718 = 1.648; V 271.8 = 16.48; Vo.0002718 = 0.01648; 

V'27.18 =■ 5.213; V'2718 = 52.13; Vo.002718 = 0.05213. 



MATHEMATICAL TABLES 



13 



SQUARE 


ROOTS 


(continued) 
















N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


<l-3 


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 


2.295 


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.S 


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 


Z548 




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 


2.61Z 
2.636 


2.619 


2.621 


2.623 


2.625 




9 


2.627 


2.629 


2.631 


2.632 


2.634 


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 




\ 


2.665 


2.666 


2.668 


2.670 


2.672 


2.674 


2.676 


Z678 


2.680 


2.681 




2 


2.683 


2.685 


2.687 


2.689 


2.691 


2.693 


2.694 


2.695 


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 




7.6 


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 




8.6 


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 




9.6 


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 


3.135 


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 JV requires moving it ONE place in body 
of table (see p. 12). 



14 MATHEMATICAL TABLES 

SQUARE ROOTS (continued) 



N 





1 


2 


3 


4 


6 


6 


7 


8 


9 


ii 


10. 


3.162 


3.178 


3.194 


3.209 


3.225 


3.240 


3.255 


3.271 


3 285 


3.302 


15 


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 


3.535 


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 


IS. 


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 


4.335 


4.347 




9. 


4.359 


4.370 


4.382 


4.393 


4.405 


4.416 


4.427 


4.438 


4.450 


4.451 


II 


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 


4.580 




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. 




26. 


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 


^^ 


5.339 


5.348 


5.357 


5.357 


5.376 


9 


9. 


5.385 


5.394 


5.404 


5.413 


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 




1. 


5.568 


5.577 


5.586 


5.595 


5.604 


5.612 


5.621 


5.630 


5.539 


5.648 




2. 


5.657 


5.666 


5.675 


5^83 


5.692 


5.701 


5.710 


5.718 


5.727 


5.736 




3. 


5.745 


5.753 


5.762 


5771 


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 


36. 


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 


5.058 


6.056 


5.075 




7. 


6.083 


6.091 


6.099 


6.107 


6.116 


6.124 


6.132 


5.140 


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 


5.473 




2. 


6.481 


6.488 


6.496 


6.504 


6.512 


6.519 


6.527 


6.535 


6.542 


5.550 




3. 


6.557 


6.565 


6.573 


6.580 


6.588 


6.595 


6.603 


6.511 


5.618 


6.625 




4. 


6.633 


6.641 


6.648 


6.656 


6.663 


6.671 


6.578 


6.686 


6.593 


5.701 




45. 


6.708 


6.715 


6.723 


6.731 


6.738 


6.745 


6.753 


6.760 


6.768 


5.775 


7 


6. 


6.782 


6.790 


6.797 


6.804 


6.812 


6.819 


6.825 


6.834 


6.841 


5.848 




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 


5.971 


6.979 


5.986 


6.993 




9. 


7.000 


7.007 


7.014 


7.021 


7.029 


7.035 


7.043 


7.050 


7.057 


7.064 





SQUARE ROOTS OF CERTAIN FRACTIONS 



N 


Vif 


N 


Vn 


N 


Vn 


N 


Vn 


N 


y/N 


N 


y/N 


V. 


0.7071 


% 


0.7746 


Vi 


0.7559 


M 


0.3333 


Yn 


0.6455 


Mn 


0.7500 


H 


0.5774 


H 


0.8944 


>y, 


0.8452 


% 


0.4714 


M,. 


0.7638 


H<« 


0.8292 


% 


0.8165 


W 


0.4082 


w 


0.9258 


% 


0.5557 


'M2 


0.9574 


i^n 


0.9014 


H 


0.5000 


H 


0.9129 


M 


0.3536 


H 


0.7454 


Mfl 


0.2500 


'Md 


0.9682 




0.8560 


» 


0.3780 


% 


0.6124 


■% 


0.8819 


^n 


0.4330 


Mj 


0.1768 


H 


0.4472 


V, 


0.5345 


H 


0.7905 


% 


0.9428 


A4n 


0.5590 


H4 


0.1250 


% 


0.6325 


¥1 


0.6547 


H 


0.9354 


Ma 


0.2887 


Ma 


0.5514 


Ho 


0.1414 



MATHEMATICAL TABLES 



15 



SQI 


FARE 


ROOTS (continued) 
















N 





1 


2 


3 


4 


6 


6 


7 


8 


9 




SO. 


7.071 


7.078 


7.085 


7.092 


7.099 


7.105 


7.113 


7.120 


7.127 


7.134 


7 


1. 


7.141 


7.148 


7.155 


7.162 


7.159 


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 




S5. 


7.416 


7.423 


7.430 


7.435 


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 


7.553 


7.570 


7.576 


7.583 


7.589 


7.596 


7.603 


7.509 




8. 


7.616 


7.622 


7.529 


7.635 


7.M2 


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 




I. 


7.810 


7.817 


7.823 


7.829 


7.836 


7.842 


7.849 


7.855 


7.851 


7.858 




2. 


7.874 


7.880 


7.887 


7.893 


7.899 


7.905 


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 




66. 


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.I6I 


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 


8.275 


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 




I. 


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 




76. 


8.660 


8.665 


8.672 


8.678 


8.583 


8.589 


8.595 


8.701 


8.705 


8.712 




6. 


8.718 


8.724 


8.729 


8.735 


8.741 


8.745 


8.752 


8.758 


8.754 


8.769 




7. 


8.775 


8.781 


8.785 


8.792 


8.798 


8.803 


6.809 


8.815 


8.820 


8.826 




8. 


8.832 


8.837 


8.843 


8.849 


8,854 


8.860 


8.856 


8.871 


8.877 


8.883 




9. 


8.888 


8.894 


8.899 


8.905 


8.911 


8.915 


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 


9.103 


5 


3. 


9.110 


9.116 


9.121 


9.127 


9.132 


9.138 


9.143 


9.149 


9.154 


9.150 




4. 


9.165 


9.171 


9.176 


9.182 


9.187 


9.192 


9.198 


9.203 


9.209 


9.214 




86. 


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 


9.385 


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 




I. 


9.539 


9.545 


9.550 


9.555 


9.560 


9.566 


9.571 


9.576 


9.581 


9.585 




2. 


9.592 


9.597 


9.602 


9.607 


9.612 


9.618 


9.623 


9.628 


9.633 


9.538 




3. 


9.644 


9.649 


9.654 


9.659 


9.664 


9.670 


9.675 


9.680 


9.685 


9.590 




4. 


9.695 


9.701 


9.706 


9.711 


9.716 


9.721 


9.725 


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 


9.859 


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 





V^= 1.77245+ l/\/i?^ 0.56419 \/ii72= 1.25331 V7^ 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 



N 





1 


2 


3 


4 


6 


6 


7 


8 


9 


n 


1.0 


1.000 


1.003 


1. 007 


1.010 


1.013 


1.016 


1.020 


1.023 


1.026 


1.029 


3 


I 


1.032 


1.035 


1.038 


1.M2 


r.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 


I.09I 


1.094 


1.097 


1.100 


1.102 


1.105 


1.108 


1.111 


1.113 


1.115 




4 


I.M9 


I.I21 


1.124 


1.127 


1.129 


1.132 


1.134 


1.137 


1.140 


1.142 




1.5 


1.145 


1.147 


1.150 


1.152 


1.155 


I.I57 


1.160 


1.162 


I.I65 


1.167 


2 


6 


1.170 


1.172 


1.174 


1.177 


1.179 


1.182 


1.184 


1.186 


1.189 


1.191 




7 


1.193 


1. 196 


1.198 


1.200 


1.203 


1.205 


1.207 


1.210 


1.212 


1.214 




8 


1.216 


1.219 


1.221 


1.223 


1.225 


1.228 


1.230 


1.232 


1.234 


1.235 




9 


1.239 


1.241 


1.243 


U45 


1.247 


1.249 


1.251 


1.254 


1.256 


1.258 




2.0 


1.260 


1.262 


1.264 


1.266 


1.268 


1.270 


1.272 


1.274 


1.277 


1.279 




1 


1.281 


1.283 


1.285 


1.287 


1.289 


1.291 


1.293 


1.295 


1.297 


1.299 




2 


1.301 


1.303 


1.305 


1.306 


1.308 


1.310 


1.312 


1.314 


1.316 


1.318 




3 


1.320 


1.322 


1.324 


1.326 


1.328 


1.330 


1. 331 


1.333 


1.335 


1.337 




4 


1J39 


1J41 


1.343 


1.344 


1.346 


1.348 


U50 


1.352 


1.354 


1.355 




2.6 


1.357 


1.359 


1.361 


1.363 


1.364 


1.366 


1.368 


1.370 


1.372 


1.373 




6 


1.375 


1.377 


1.379 


1.380 


1.382 


1.384 


1.386 


1.387 


1.389 


1.391 




7 


1.392 


1.394 


1.396 


1.398 


1.399 


1.401 


1.403 


1.404 


1.406 


1.408 




8 


1.409 


1.411 


1.413 


1.414 


1.416 


1.418 


1.419 


1.421 


1.423 


1.424 




9 


1.426 


1.428 


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 


1.450 


1.452 


1.453 


1.455 


1.457 




1 


1.458 


1.460 


1.461 


1.463 


1.454 


1.465 


1.467 


1.469 


1.471 


1.472 




2 


1.474 


1.475 


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 


1.51 1 


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 


1.534 


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 


1J59 




8 


1.560 


1,562 


1.563 


1.565 


1.566 


1.567 


1.569 


1.570 


1.571 


1.573 




9 


1J74 


1.575 


1.577 


1.578 


1J79 


1.581 


1.582 


1.583 


1.585 


1.585 




4.0 


r.587 


r.589 


1.590 


1.591 


1.593 


1.594 


1.595 


1.597 


1.598 


1.599 




I 


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 


1.549 


1.650 




4.5 


1.651 


1.652 


1.653 


1.655 


1.655 


1.657 


1.658 


1.659 


1.661 


1.662 




6 


1.663 


1.664 


1.665 


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 


1.586 




8 


1.687 


1.688 


1.689 


1.690 


1.692 


1.693 


1.694 


1.695 


1.696 


1.597 




9 


1.698 


1.700 


1.701 


1.702 


1.703 


1.704 


1.705 


1.707 


1.708 


1.709 





\/ir= 1.46459 l/\/ir= 0.682784 



Explanation of Table of Cube Roots (pp. 16-21). 

This table gives the values of \/jV for all values of N from 1 to 1000, eorreet 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 i n the cube root of N. For example: 

- ^2.718 - 1.396; - ^2718 = 13.96; -^0.000002718 = 0.01396. 

•^27.18 = 3.007; -^27180 = 30.07; -^0.00002718 = 0.03007. 



' 6.477; -^271800 = 64.77; -^0.0002718 - 0.06477. 



MATHEMATICAL TABLES 
CUBE ROOTS (continued) 



17 



N 





1 


2 


3 


4 


6 


6 


7 


8 


9 


Mia 


S.O 


1.710 


1.711 


1712 


1.713 


1.715 


1.716 


1.717 


1.718 


1.719 


1.720 


1 


I 


1.721 


1.722 


r.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 


1.754 




S.S 


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 


1805 


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 


1833 


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 


1.855 




4 


1.857 


1.858 


1.859 


1.860 


1.860 


1.851 


1.862 


1.863 


1.864 


1.865 




6.6 


1.866 


1.867 


1.868 


1,869 


1.870 


1.871 


1,872 


1.873 


1.874 


1.875 




6 


1.875 


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 


1.695 


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 


I.9I6 


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 


r.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 


1983 


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 




SO 


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 


2.011 


2.012 


2.012 


2,013 


2.014 


2,015 


2.015 




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 




8.6 


2.041 


2.042 


2.042 


2.043 


2.044 


2,043 


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 






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 


2103 


2.104 


2.104 


2.105 


2.106 


2.107 


2.107 


2.108 


2.109 


2.110 




4 


2.1 10 


2.111 


2.112 


2.113 


2.113 


2.114 


2.115 


2.116 


2.116 


2.117 




9.6 


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 


2.145 


2.147 




9 


2.147 


2.148 


2.149 


2.149 


2.150 


2.151 


2.152 


2.152 


2.153 


2.154 





Moving the decimal point THEEB 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 




10. 


2.154 


2.162 


2.169 


2.176 


2.183 


2.190 


2.197 


2.204 


2.210 


2.217 


7 


I. 


2.224 


2.231 


2J37 


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 




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 




SO. 


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.I5I 


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 


3.211 


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 


3.299 




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 


3J97 


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 




1. 


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 


3J46 


3J49 


3J52 


3354 




.45. 


3.557 


3.560 


3.562 


3365 


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 


■^N 


N 


■^ 


N 


■^N 


N 


-^ 


N 


^N 


N 


^ 


Vi 


.7937 


% 


.8434 


Vi 


.8298 


M 


.4807 


Mj 


.7469 


Me 


.8255 


H 


.6934 


M. 


.9283 


Yf 


.8939 


H 


.6057 


M?. 


.8355 




.8826 


Vi 


.8736 


% 


.5503 


Vi 


.9499 


M, 


.7631 


iMj 


.9714 


iM« 


.9331 




.6300 


% 


.9410 


V* 


.5000 


M 


.8221 


Mn 


.3969 




.9787 


% 


.9086 


Vi 


.5228 




.7211 


'^ 


.9196 


?1(i 


.5724 


1^2 


.3150 


yi 


.5848 


■¥i 


.6586 


^ 


.8550 


% 


.9615 


Mb 


.6786 


1^4 


J500 


■a 


.7368 


■'ih 


.7539 


H 


.9565 


M2 


.4368 


Ms 


.7591 


Ho 


.2714 



MATHEMATICAL TABLES 19 

CUBE BOOTS (.continued) 



N 





1 


2 


3 


4 


6 


6 


7 


8 


9 


ii 


60. 


3.684 


3.686 


3.689 


3.691 


3.694 


3.696 


3«99 


3.701 


3.704 


3.705 


2 


1. 


3.708 


3.711 


3.713 


3.716 


3.718 


3.721 


3 723 


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 


3.766 


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 




S5. 


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 




1. 


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 


4.0O4 


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 


4.050 




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 


4.155 


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 


4J02 


4.204 


4.206 


4.208 


4.210 


4.212 


4.213 


4.215 




76. 


4.217 


4.219 


4.221 


4.223 


A225 


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 


4J93 


4.294 


4.296 


4.296 


4.300 


4.302 


4.303 


4305 


4307 




80. 


4.309 


4.311 


4.312 


4314 


4J16 


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 


4.371 


4.373 


4.374 


4.376 


4378 




4. 


4.380 


4.381 


4.383 


4.385 


4J86 - 


4.388 


4.390 


4392 


4.393 


4.395 




86. 


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 


4.445 




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 




I 


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 


4J48 


4.550 


4.552 


4.553 


4.555 


4.555 


4.558 


4.560 


4.551 




96. 


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 


4.525 




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 N requires moving it ONE place in body 
of table (seep. 16). 



20 



MATHEMATICAL TABLES 



CUBE ROOTS (continued) 



N 


0. 


1. 


2. 


3. 


4. 


B. 


6. 


7. 


8. 


9. 


M 


10 


4642 


4.657 


4.672 


4.688 


4.703 


4.718 


4.733 


4.747 


4.762 


4.777 


15 


1 


4.791 


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 


5.158 


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 


5.3(3 


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 


II 


7 


5.540 


5.550 


5.561 


5.572 


5.583 


5.593 


5.604 


5.615 


5.625 


5.635 




8 


5.646 


5.657 


5.667 


5.677 


5.588 


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 


5.018 


5.028 


9 


2 


6.037 


6.046 


6.055 


6.064 


6.073 


6.082 


6.091 


5.100 


6.109 


5.118 




3 


6.127 


6.136 


6.145 


6.153 


5.162 


6.171 


5.180 


6.188 


6.197 


5.206 




4 


6.214 


6.223 


6.232 


6.240 


6.249 


6.257 


5.265 


6.274 


6.283 


6.291 




25 


6.300 


6.308 


6.316 


6.325 


6.333 


5.341 


6.350 


5.358 


6.365 


6.374 


8 


6 


6.383 


6.391 


6.399 


6.407 


6.415 


6.423 


5.431 


6.439 


6.447 


6.455 




7 


6.463 


6.471 


5.479 


6.487 


6.495 


5.503 


5.511 


6.519 


6.527 


6.534 




8 


6.542 


6.550 


5.558 


5.565 


6.573 


5.581 


5.589 


6.595 


6.604 


6.511 




9 


6.619 


6.627 


5.534 


6.642 


6.549 


5.657 


5.664 


6.572 


6.679 


6.687 




30 


6.694 


6.702 


5.709 


6.717 


6.724 


6.731 


6.739 


6.746 


6.753 


6.761 


7 


1 


6.768 


6.775 


5.782 


6.790 


6.797 


5.804 


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 


5.931 


6.938 


6.945 


6.952 


6.959 


6.955 


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 


7.150 


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 


7.355 


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 


7.455 


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 


7.659 


7.674 


7.580 


7.585 


7.591 


7.597 


7.703 


7.708 


7.714 


S 


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 OF TWO-THIRDS POWERS 
AND THREE-HALVES POWERS (see pp. 22-23) 

(To assist in locating the decimal point) 



N 



.0001 

.001 

.01 

.1 
1. 

10. 

100. 

1000. 

10000. 



N^{: 



.002154 
.01 

.0464 
.2154 
1. 

4.54 
21 J4 
100. 
454.16 



Ar?^(= y/Jft) 



.000001 
00003152 
.001 

.03162278 
I. 
31.62278 
1000. 
31622.78 
1000000. 



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. 



MATHEMATICAL TABLES 21 

CTTBE ROOTS (continued) 



N 


0. 


1. 


2. 


3. 


i. 


6. 


6. 


7. 

• 


8. 


9. 




60 


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 




66 


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 


8.405 


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 




66 


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 


8.865 


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 




76 


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 


9.145 


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 






9.322 


9.326 


9.329 


9.333 


9.337 


9.341 


9.345 


9.348 


9.352 


9.355 




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 


9.459 




85 


9.473 


9.476 


9.480 


9.484 


9.488 


9.491 


9.495 


9.499 


9.502 


9.505 




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 


9.515 




9 


9.619 


9.6B 


9.626 


9.630 


9.633 


9.637 


9.641 


9.644 


9.648 


9.551 




90 


9.655 


9.658 


9.662 


9.666 


9.669 


9.673 


9.675 


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 




96 


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 


9.925 


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 iV requires moving it ONE place in body 
of table (see p. 16). 



22 MATHEMATICAL TABLES 

THREE-HALVES POWERS OF NUMBERS (see also p. 20) 



N 





1 

• 


2 


3 


4 


6 


6 


7 


8 


9 


^ 

H 


1. 


1.000 


1.154 


1315 


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. 


5.1% 


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 


6. 


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 


16.95 


17.34 


17.73 


18.12 


38 


7. 


18.32 


18.92. 


1932 


19.72 


20.13 


20.54 


20.95 


21.37 


21.78 


22,20 


41 


8. 


22.63 


23.05 


23.48 


23.91 


2435 


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 


3334 


34.02 


3431 


35.00 


35.49 


35.99 


49 


1. 


36.48 


36.98 


37.48 


37.99 


38.49 


39.00 


3931 


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. 


5238 


52.95 


5331 


54.08 


54;64 


55J1 


55.79 


56.36 


56.94 


5731 


57 


16. 


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.^0 


94.18 


94.85 


95.55 


68 


1. 
1. 


96.23 


96.92 


97.61 


98.30 


99.00 


99.69 


10038 
100.4 


101.1 


101.8 


1023 


69 

7 


2. 


103.2 


103.9 


104.6 


105.3 


106.0 


106.7 


107.4 


108.2 


108.9 


109.6 


7 


3. 


1103 


111.0 


111.7 


1123 


113.2 


113.9 


114.6 


115.4 


116.1 


116.8 


7 


4. 


117.6 


1183 


119.0 


119.8 


1203 


1213 


122.0 


122.8 


123.5 


1243 


7 


25. 


125.0 


125.8 


1263 


1273 


128.0 


128.8 


1293 


130.3 


131.0 


131.8 


8 


6. 


132.6 


1333 


134.1 


134.9 


135.6 


136.4 


137.2 


138.0 


138.7 


1393 


8 


7. 


1403 


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 


1503 


1513 


152.1 


152.9 


153.8 


154.5 


155.4 


8 


9. 


156J 


157.0 


157.8 


158.6 


159.4 


160.2 


161.0 


161.9 


162.7 


163.5 


8 


30. 


1643 


I65.I 


166.0 


166.8 


167.6 


168.4 


1693 


170.1 


170.9 


171.8 


8 




172.6 


173.4 


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 


1853 


186.1 


187.0 


187.8 


188.7 


9 


3. 


189.6 


190.4 


1913 


192.2 


193.0 


193.9 


194.8 


195.6 


196.5 


197.4 


9 


4. 


1983 


199.1 


200.0 


200.9 


201.8 


202.6 


2033 


204.4 


2053 


206.2 


9 


36. 


207.1 


208.0 


208.8 


209.7 


210.6 


2113 


212.4 


2133 


214.2 


215.1 


9 


6. 


216.0 


216.9 


217.8 


218.7 


219.6 


220.5 


221.4 


2223 


223.2 


224.2 


9 


7. 


225.1 


226.0 


226.9 


227.8 


228.7 


229.6 


230.6 


2313 


232.4 


2333 


9 


8. 


234J 


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 


2473 


2483 


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 


2643 


265.4 


266.4 


2673 


2683 


2693 


270.2 


2712 


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 


ID 


4. 


291.9 


292.9 


293.9 


294.9 


295.9 


296.9 


297.9 


298.9 


299.9 


300.9 


10 


46. 


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. 


322J 


323.2 


3243 


3253 


326.3 


327.4 


328.4 


329.4 


330.5 


3313 


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 


M 



This table gives JV" from iV = 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 Af*^ from ilf = ltoAf=1000. Thus: (0.6148)^ = 0.7230. 



MATHEMATICAL TABLES 
THREE-HALVES POWERS {continued) (See also p. 20) 



23 



N 





1 


2 


3 


4 


6 


6 


7 


S 


9 


ii 


60. 


353.6 


354.6 


355.7 


356.7 


357.8 


358.9 


359.9 


361.0 


362.1 


363.1 






364.2 


365.3 


366.4 


367.4 


368.5 


369.6 


370.7 


371.7 


372.8 


373.9 




i. 


375.0 


376.1 


377.1 


378.2 


379.3 


380.4 


381.5 


382.6 


383.7 


384.3 




3 


385.8 


386.9 


388.0 


3891 


3902 


391.3 


392.4 


393.5 


394.6 


395.7 




4. 


396.8 


397.9 


399.0 


400.1 


401.2 


402.3 


403.4 


404.6 


405.7 


406.8 




66. 


407.9 


409.0 


410.1 


411.2 


412.3 


413.5 


414.6 


415.7 


416.8 


417.9 




6. 


419.1 


420.2 


421.3 


422.4 


423.6 


424.7 


425.8 


426.9 


428.1 


429.2 




7. 


430.3 


431.5 


432.6 


433.7 


434.9 


436.0 


437.2 


4383 


439.4 


440.6 




8. 


441.7 


442.9 


444.0 


445.1 


446.3 


447.4 


448.6 


449.7 


450.9 


452.0 




9. 


453.2 


454.3 


455.5 


456.6 


457.8 


459.0 


460.1 


4613 


462.4 


463.6 




60. 


464.8 


465.9 


467.1 


468.2 


469.4 


470.6 


471.7 


472.9 


474.1 


4753 






476.4 


477.6 


478.8 


479.9 


481.1 


482.3 


483.5 


484.6 


485.8 


487.0 




'2. 


488.2 


489.4 


490.6 


491.7 


492.9 


494.1 


4953 


496.5 


497.7 


498.9 




3. 


500.0 


501.2 


502.4 


503.6 


504.8 


506.0 


507.2 


508.4 


509.6 


510.8 




4. 


512.0 


513.2 


514.4 


515.6 


516.8 


518.0 


519.2 


520.4 


521.6 


522.8 




66. 


524.0 


525.3 


526.5 


527.7 


528.9 


530.1 


5313 


532.5 


533.8 


535.0 




6- 


536.2 


537.4 


538.6 


539.8 


541.1 


542.3 


543.5 


544.7 


546.0 


547.2 




7. 


548.4 


549.6 


550.9 


552.1 


553.3 


554.6 


555.8 


557 


5583 


559.5 




8. 


560.7 


562.0 


563.2 


564.5 


565.7 


566.9 


568.2 


569.4 


570.7 


571.9 




9. 


573.2 


574.4 


575.7 


576.9 


578.1 


579.4 


580.6 


581.9 


583.2 


584.4 




70. 


585.7 


5S6.9 


588.2 


589.4 


590.7 


591.9 


593.2 


594.5 


595.7 


597.0 






598.3 


599.5 


600.8 


602.1 


603.3 


604.6 


605.9 


607.1 


608.4 


609.7 




2. 


610.9 


612.2 


613.5 


614.8 


616.0 


6173 


618.6 


619.9 


621.2 


622.4 




3. 


623.7 


625.0 


626.3 


627.6 


628.8 


630.1 


631.4 


632.7 


634.0 


635.3 




4. 


636.6 


637.9 


639.2 


640.4 


641.7 


643.0 


644.3 


645.6 


646.9 


648.2 




76. 


649.5 


650.8 


652.1 


653.4 


654.7 


656.0 


6573 


658.6 


659.9 


661.2 




6. 


662.6 


663.9 


665.2 


666.5 


667.8 


669.1 


6704 


671.7 


673.0 


674.4 




7. 


675.7 


677.0 


. 678.3 


679.6 


680.9 


682.3 


683.6 


684.9 


686.2 


687.6 




8. 


688.9 


690.2 


691.5 


692.9 


694.2 


695.5 


696.8 


698.2 


699.5 


700.8 




9. 


702.2 


703.5 


704.8 


706.2 


707.5 


708.8 


710.2 


711.5 


712.9 


714.2 




80. 


715.5 


716.9 


718.2 


719.6 


720.9 


722.3 


723.6 


725.0 


726.3 


727.7 




1. 


729.0 


7304 


731.7 


733.1 


734.4 


735.8 


737.1 


738.5 


739.8 


741.2 




2. 


742.5 


743.9 


745.3 


746.6 


748.0 


749.3 


7507 


752.1 


753.4 


754.8 




3. 


756.2 


757.5 


758.9 


760.3 


761.6 


763.0 


764.4 


765.8 


767.1 


768.5 




4. 


769.9 


771.2 


772.6 


774.0 


775.4 


776.8 


778.1 


779.5 


780.9 


7823 




86. 


783.7 


785.0 


786.4 


787.8 


789.2 


790.6 


792.0 


793.4 


794.8 


796.1 




6. 


797.5 


798.9 


800.3 


801.7 


803.1 


804.5 


805.9 


807.3 


808.7 


810.1 




7. 


811.5 


812.9 


814.3 


815.7 


817.1 


818.5 


819.9 


8213 


822.7 


824.1 




8. 


825.5 


826.9 


828.3 


829.7 


831.1 


832.6 


834.0 


835.4 


836.8 


838.2 




9. 


839.6 


841.0 


842.5 


843.9 


8453 


846.7 


848.1 


849.5 


851.0 


852.4 




90. 


853.8 


855.2 


856.7 


858.1 


859.5 


860.9 


862.4 


863.8 


865.2 


866.7 






868.1 


869.5 


870.9 


872.4 


873.8 


875.2 


876.7 


878.1 


879.6 


881.0 




2! 


882.4 


883.9 


885.3 


886.8 


888.2 


889.6 


891.1 


892.5 


894.0 


895.4 




3. 


896.9 


898.3 


899.8 


901.2 


902.7 


904.1 


905.6 


907.0 


908.5 


909.9 




4. 


911.4 


912.8 


914.3 


915.7 


917.2 


918.6 


920.1 


921.6 


923.0 


924.5 




96. 


925.9 


927.4 


928.9 


930.3 


931.8 


933.3 


934.7 


936.2 


937.7 


939.1 




6. 


940.6 


942.1 


943,5 


945.0 


946.5 


948.0 


949.4 


950.9 


952.4 


953.9 




7. 


955.3 


956.8 


958.3 


959.8 


961.3 


962.7 


964.2 


965.7 


967.2 


968.7 




8. 


970.2 


971.6 


973.1 


974.6 


976.1 


977.6 


979.1 


980.6 


982.1 


983.5 




9. 


985.0 


986.5 


988.0 


989.5 


991.0 


992.5 


994.0 


995.5 


997.0 


998.5 




100. 


1000.0 























Moving the decimal point TWO places in N requires moving it THREE places in body 
of table (see also auidliary table on p. 20). 



24 



MATHEMATICAL TABLES 



RECIPROCALS OF NUMBERS 














N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


P 


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 


.9543 


.9534 


.9525 


-9 


4 


.9615 


.9606 


.9597 


.9588 


.9579 


.9569 


.9560 


.9551 


.9542 


.9533 




1.05 


.9524 


.9515 


.9506 


.9497 


.9488 


.9479 


.9470 


.9451 


.9452 


.9443 




6 


.9434 


.9425 


.9415 


.9407 


.9398 


.9390 


.9381 


.9372 


.9363 


.9355 




7 


.9345 


.9337 


.9328 


.9320 


.9311 


.9302 


.9294 


.9285 


.9275 


.9258 




8 


.9259 


.9251 


.9242 


.9234 


.9225 


.9217 


.9208 


.9200 


.9191 


.9183 


-8 


9 


.9174 


.9166 


.9158 


.9149 


.9141 


.9132 


.9124 


.9115 


.9107 


.9099 




1.10 


.9091 


.9083 


.9074 


.9065 


.9058 


.9050 


.9042 


.9033 


.9025 


.9017 




1 


.9009 


.9001 


.8993 


.8985 


.8977 


.8969 


.8951 


.8953 


.8945 


.8937 




2 


.8929 


.8921 


.8913 


.8905 


.8897 


.8889 


.8881 


.8873 


.8855 


.8857 




3 


.8850 


.8842 


.8834 


.8826 


.8818 


.8811 


.8803 


.8795 


.8787 


.8780 




4 


.8772 


.8764 


.8757 


.8749 


.8741 


.8734 


.8725 


.8718 


.8711 


.8703 




1.16 


.8696 


.8688 


.8681 


.8673 


.8655 


.8558 


.8651 


.8643 


.8636 


.8628 




6 


.8621 


.8613 


.8605 


.8598 


.8591 


.8584 


.8576 


.8559 


.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 


.8395 


.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 


.8025 


.8019 


.8013 


.8005 




1.25 


.8000 


.7994 


.7987 


.7981 


.7974 


.7%8 


.7952 


.7955 


.7949 


.7943 




6 


.7937 


.7930 


.7924 


.7918 


.7911 


.7905 


.7899 


.7893 


.7886 


.7880 




7 


.7874 


.7868 


.7852 


.7855 


.7849 


.7843 


.7837 


.7831 


.7825 


.7819 




8 


.7812 


.7806 


.7800 


.7794 


.7788 


.7782 


.7775 


.7770 


.7764 


.7758 




9 


.7752 


.7746 


.7740 


.7734 


.7728 


.7722 


.7716 


.7710 


.7704 


.7698 




1.30 


.7692 


.7686 


.7680 


.7675 


.7669 


.7653 


.7557 


.7551 


.7545 


.7639 




I 


.7634 


.7628 


.7622 


.7616 


.7510 


.7605 


.7599 


.7593 


.7587 


.7582 




2 


.7575 


.7570 


.7564 


.7559 


.7553 


.7547 


.7541 


.7536 


.7530 


.7524 




3 


.7519 


.7513 


.7508 


.7502 


.7495 


.7491 


.7485 


.7479 


.7474 


.7458 




4 


.7463 


.7457 


.7452 


.7446 


.7440 


.7435 


.7429 


.7424 


.7418 


.7413 




136 


.7407 


.7402 


.7395 


.7391 


.7385 


.7380 


.7375 


.7369 


.7364 


.7358 


-5 


5 


.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 


.7158 


.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 


.5998 




3 


.6993 


.6988 


.6983 


.5978 


.6974 


.6969 


.6964 


.5959 


.6954 


.6949 




4 


.6944 


.6940 


.5935 


.6930 


.6925 


.5920 


.6915 


.6911 


.6906 


.6901 




1.45 


.5897 


.6892 


.5887 


.6882 


.6878 


.6873 


.6868 


.6863 


.5859 


.6854 




6 


.6849 


.6845 


.5840 


.6835 


.6831 


.6826 


.6821 


.6817 


.5812 


.6807 




7 


.6803 


.6798 


.6793 


.6789 


.6784 


.6780 


.6775 


.6770 


.5755 


.6761 




8 


.6757 


.6752 


.5748 


.5743 


.6739 


.6734 


.6729 


.6725 


.5720 


.6716 




9 


.6711 


.6707 


.5702 


.5598 


.5593 


.6689 


.6584 


.6680 


.5575 


.6671 





l/ir = 0.318310 1/e = 0.367879 

Moving the decimal point in either direction in JV requires moving it in the OPPO- 
SITE direction in body of table (see p. 26). 



MATHEMATICAL TABLES 



25 



RECIPROCALS (continued) 
















N 





1 


2 


3 


4 


S 


6 


7 


8 


9 


1^ 


1.60 


.6667 


.6662 


.6558 


.6653 


.5649 


.6645 


.6540 


.6635 


.5631 


.6527 


-4 


1 


.6623 


.6618 


.6514 


.6609 


.6605 


.6601 


.5595 


.6592 


.6588 


.5583 




2 


.6579 


.6575 


.6570 


.5566 


.6562 


.6557 


.6553 


.6549 


.6545 


.5540 




3 


.6536 


.6532 


.6527 


.6523 


.5519 


.5515 


.6510 


.6506 


.5502 


.5498 




4 


.6494 


.5489 


.6485 


.6481 


.5477 


.6472 


.6468 


.6464 


.6460 


.6455 




1.55 


.6452 


.5447 


.5443 


.6439 


.6435 


.6431 


.6427 


.6423 


.6418 


.6414 




6 


.6410 


.6405 


.5402 


.6398 


.6394 


.5390 


.6385 


.6382 


.5378 


.6373 




7 


.6369 


.5355 


.5351 


.6357 


.6353 


.6349 


.6345 


.5341 


.5337 


.6333 




8 


.6329 


.6325 


.6321 


.6317 


.6313 


.6309 


.5305 


.5301 


.5297 


.6293 




9 


.6289 


.6285 


.6281 


.6277 


.5274 


.6270 


.6265 


.5252 


.5258 


.6254 




1.60 


.6250 


.6246 


.6242 


.6238 


.5234 


.6231 


.5227 


.6223 


.6219 


.5215 






.6211 


.6207 


.6203 


.6200 


.5196 


.5192 


.6188 


.5184 


.6180- 


.6177 




2 


.6173 


.6169 


.6165 


.5151 


.6158 


.5154 


.6150 


.5146 


.5143 


.5139 




3 


.6135 


.5131 


.6127 


.6124 


.6120 


.5116 


.6112 


.5109 


.6105 


.5101 




4 


.6098 


.5094 


.6090 


.6086 


.5083 


.5079 


.6075 


.5072 


.5058 


.5064 




1.65 


.6061 


.6057 


.6053 


.6050 


.6045 


.6042 


.6039 


.6035 


.6031 


.6028 




6 


.6024 


.6020 


.6017 


.6013 


.6010 


.6006 


.6002 


.5999 


.5995 


.5992 




7 


.5988 


.5984 


.5981 


.5977 


.5974 


.5970 


.5957 


.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 


.5859 


.5865 


.5852 


.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 


J780 


.5777 


.5774 


.5770 


.5767 


.5764 


.5750 


.5757 


.5754 


.5750 




4 


.5747 


.5744 


.5741 


.5737 


.5734 


.5731 


.5727 


.5724 


.5721 


.5718 




1.76 


.5714 


.5711 


.5708 


.5705 


.5701 


.5698 


.5695 


.5692 


.5588 


.5685 




6 


.5682 


.5679 


.5675 


.5672 


.5669 


.5666 


.5663 


.5659 


.5655 


.5653 




7 


.5650 


.5647 


.5643 


.5640 


.5637 


.5534 


.5531 


.5627 


.5624 


3621 




8 


.5618 


.5615 


.5612 


.5609 


.5505 


.5602 


.5599 


.5595 


.5593 


.5590 




9 


.5587 


.5583 


.5580 


J577 


.5574 


.5571 


.5558 


.5555 


.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 


.5479 


.5475 


.5473 


.5470 


.5467 




3 


.5464 


.5461 


.5459 


.5456 


.5453 


.5450 


.5447 


.5444 


.5441 


.5438 




4 


J435 


J432 


.5429 


.5426 


.5423 


.5420 


.5417 


.5414 


.5411 


.5408 




1.S5 


.5405 


J402 


.5400 


.5397 


.5394 


.5391 


.5388 


.5385 


.5382 


.5379 




6 


.5376 


.5373 


.5371 


.5358 


.5365 


.5362 


.5359 


.5355 


.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 


J291 


.5288 


.5285 


.5283 


.5280 


.5277 


.5274 


.5271 


.5269 


.5266 




1.90 


.5263 


J260 


.5258 


.5255 


.5252 


.5249 


.5247 


.5244 


.5241 


.5238 




I 


.5236 


.5233 


.5230 


.5227 


.5225 


.5222 


.5219 


.5216 


.5214 


.5211 




2 


3208 


.5206 


.5203 


.5200 


.5198 


.5195 


.5192 


.5189 


.5187 


.5184 




3 


.5181 


.5179 


.5175 


.5173 


.5171 


.5168 


.5165 


.5163 


.5160 


.5157 




4 


.5155 


J152 


.5149 


.5147 


.5144 


.5141 


.5139 


.5136 


.5133 


.5131 




1.96 


.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 


.5065 


.5063 


.5061 


.5058 


.5056 


.5053 


-2 


8 


.5031 


.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 





1 


2 


3 


4 


6 


6 


7 


S 


9 


<i 


2.0 


.5000 


.4975 


.4950 


.4926 


.4902 


.4878 


.4854 


.4831 


.4808 


.4785 


-24 




.4762 


.4739 


.4717 


.4695 


.4673 


.4651 


.4630 


.4608 


.4587 


.4556 


-21 


2 


.4545 


.4525 


.4505 


.4484 


.4464 


.4444 


.4425 


.4405 


.4385 


.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 


3861 


-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 


J030 


.3021 


.3012 


3003 


.2994 


.2985 


.2976 


.2967 


.2959 


.2950 


-9 


4 


.2941 


J933 


.2924 


J915 


.2907 


.2899 


.2890 


.2882 


.2874 


.2865 


-8 


3.S 


J857 


2M9 


.2841 


.2833 


.2825 


.2817 


.2809 


.2801 


.2793 


.2786 


-8 


6 


.2778 


.2770 


.2762 


.2755 


.2747 


.2740 


.2732 


.2725 


J7\7 


.2710 


-8 


7 


.2703 


.2695 


.2688 


.2681 


.2674 


.2667 


.2660 


.2653 


.2646 


.2639 


-7 


8 


.2632 


.2623 


.2618 


.2611 


.2604 


.2597 


.2591 


.2584 


.2577 


J571 


-7 


9 


.2564 


J558 


J551 


.2545 


.2538 


.2532 


.2525 


.2519 


.2513 


.2506 


-5 


4.0 


.2500 


.2494 


.2488 


.2481 


.2475 


J469 


.2463 


.2457 


.2451 


.2445 


-6 




.2439 


.2433 


.2427 


.2421 


.2415 


.2410 


.2404 


.2398 


.2392 


.2387 


-6 


2 


.2381 


.2375 


.2370 


.2364 


.2358 


.2353 


.2347 


.2342 


.2335 


.2331 


-6 


3 


.2326 


.2320 


.2315 


.2309 


.2304 


.2299 


.2294 


.2288 


.2283 


.2278 


-5 


4 


.2273 


.2268 


.2262 


J257 


.2252 


.2247 


J242 


.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 


.2)28 


J123 


.2119 


JI14 


.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 



1/ir = 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 figureB, 
(Interpolated values may be in error by 1 in the fourth figure.) 

To find the reciprocal of a number IT 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: 

^ 0.3108; „„^,„ =0.000 3108; „„^„„,„ = 310.8 



3.217 



3217. 



0.003217 



MATHEMATICAL TABLES 27 

RECIPROCALS (.continued) 



N 





1 


2 


3 


i 


6 


6 


7 


8 


9 


^-3 


6.0 


.2000 


.1996 


.1992 


.1988 


.1984 


.1980 


.1976 


.1972 


.1969 


.1955 


-4 




.1961 


.1957 


.1953 


.1949 


.1946 


.1942 


.1938 


.1934 


.1931 


.1927 




J 


.1923 


.1919 


.1915 


.1912 


.1908 


.1905 


.1901 


.1898 


.1894 


.1890 




.3 


.1887 


.1883 


.1880 


.1876 


.1873 


.1869 


.1856 


.1862 


.1859 


.1855 




.4 


.1852 


.1848 


.1845 


.1842 


.1838 


.1835 


.1832 


.1828 


.1825 


.1821 


-3 


6.6 


.1818 


.1815 


.1812 


.1808 


.1805 


.1802 


.1799 


.1795 


.1792 


.1789 




.6 


.1785 


.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 


.1589 


.1585 


.1684 


.1681 


.1578 


.1575 


.1672 


.1659 




6.0 


.1667 


.1664 


.1651 


.1558 


.1656 


.1653 


.1550 


.1647 


.1545 


.1642 




.1 


.1639 


.1637 


.1634 


.1631 


.1629 


.1625 


.1523 


.1621 


.1518 


.1616 




.2 


.1613 


.1610 


.1508 


.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 




6.6 


.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 


.I486 


.1484 


.1481 


.1479 


.1477 


.1475 


.1473 




.8 


.1471 


.1468 


.1465 


.1454 


.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 




.1 


.1408 


.1405 


.1404 


.1403 


.1401 


.1399 


.1397 


.1395 


.1393 


.1391 




.2 


.1389 


.1387 


.1385 


.1383 


.1381 


.1379 


.1377 


.1376 


.1374 


.1372 




.3 


.1370 


.1368 


.1355 


.1354 


.1352 


.1351 


.1359 


.1357 


.1355 


.1353 




.4 


.1351 


.1350 


.1348 


.1346 


.1344 


.1342 


.1340 


.1339 


.1337 


.1335 




7.6 


.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 




,1 


.1235 


.1233 


.1232 


.1230 


.1229 


.1227 


.1225 


.1224 


.1222 


.1221 




.2 


.1220 


.1218 


.1217 


.1215 


.1214 


.1212 


.1211 


.1209 


.1208 


.1205 




.3 


.1205 


.1203 


.1202 


.1200 


.1199 


.1198 


.1196 


.1195 


.1193 


.1192 




.4 


.1190 


.1189 


.1188 


.1186 


.1185 


.1183 


.1182 


.1181 


.1179 


.1178 


-1 


8.6 


.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 


.1125 


.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 


.1058 


.1067 


.1056 


.1055 




.4 


.1064 


.1063 


.1052 


.1060 


.1059 


.1058 


.1057 


.1056 


.1055 


.1054 




9.6 


.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 


.1028 


.1027 


.1025 


.1025 


.1024 


.1022 


.1021 




.8 


.1020 


.1019 


.1018 


.1017 


.1016 


.1015 


.1014 


.1013 


.1012 


.toil 




.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 


i 


5 


6 


7 


8 


9 


Mia 


1.0 


3.142 


3.173 


3.204 


3.235 


3.267 


3.299 


3.330 


3.362 


3.393 


3.424 


31 


.1 


3.455 


3.487 


3.519 


3.550 


3.581 


3.613 


3.644 


3.675 


3.707 


3.738 




.2 


3.770 


3.801 


3.833 


3.864 


3.895 


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 


5.245 


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 


5.000 


6.032 


6.063 


6.095 


6.126 


6.158 


5.189 


6.220 


5.252 




2.0 


6.283 


6.315 


6.346 


6.377 


6.409 


6.440 


6.472 


6.503 


5.535 


6.566 




.1 


6.597 


6.629 


6.6C0 


6.692 


6.723 


6.754 


6.786 


6.817 


6.849 


6.880 




.2 


6.912 


5.943 


6.974 


7.006 


7.037 


7.069 


7.100 


7.131 


7.153 


7.194 




.3 


7.225 


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 


7.597 


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 


8.508 


8.639 


8.671 


8.702 


8.734 


8.765 




.8 


8.795 


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 


9.235 


9.268 


9.299 


9.331 


9.362 


9.393 




3.0 


9.425 


9.456 


9.488 


9.519 


9.550 


9.582 


9.513 


9.645 


9.676 


9.708 




•j 


9.739 


9.770 


9.802 


9.833 


9.865 


9.895 


9.927 


9.959 


9.990 


10.022 
10.02 


31 
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 


10.45 


10.49 


10.52 


10,55 


10 59 


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 




3.6 


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 


11.56 


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 


12.53 


12.66 


12.69 


12.72 


12.75 


12.79 


12.82 


12.85 




.1 


12.88 


12.91 


12.94 


12.97 


13.01 


13.04 


13.07 


13.10 


13.13 


13.15 




.2 


13.19 


13.23 


13.25 


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 




4.6 


14.14 


14.17 


14.20 


14.23 


14.26 


14.29 


14.33 


14.35 


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 


15.45 


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 ir times any number D from 1 to 10; that is, it is a table 
of multiples of tt. (Z) = diameter.) 

Moving the decimal point one place in column D is equivalent to moving it one 
place in the body of the table. 

Circumference = ;r X diam. = 3.141593 X diam. 
Conversely, 

Diameter = - X oiroumf. = 0.31831 X circumf. 



MATHEMATICAL TABLES 



29 



CIR 


CUMP 


EREN< 


3ES BY HUNDREDTHS 


{continued) 








D 





1 


2 


3 


4 


6 


6 


7 


8 


9 


<j-3 


6.0 


15.71 


15.74 


15.77 


15.80 


15.83 


15.87 


15.90 


15.93 


15.96 


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 


15.34 


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 


16.64 


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 




6.6 


17.28 


17.31 


17.34 


17.37 


17.40 


17.44 


17.47 


17.50 


17.53 


17.56 




.6 


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 




a 


19.48 


19.51 


19.54 


19.57 


19.60 


19.63 


19.67 


19.70 


19.73 


19.76 




2 


19.79 


19.82 


19.85 


19.89 


19.92 


19.95 


. 19.98 


20.01 


20.04 


20.07 




A 


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 




7.6 


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 


25J4 


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 




8.6 


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 




A 


29.53 


29.56 


29.59 


29.63 


29.66 


29.69 


29.72 


29.75 


29.78 


29.81 




9.S 


29.85 


29.88 


29.91 


29.94 


29.97 


30.00 


30.03 


30.07 


30.10 


30.13 




.5 


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 place in body of 
table (see p. 28}. 



30 



MATHEMATICAL TABLES 



AREAS OF CIRCLES BT HUNDREDTHS 

(For areas by eighths, see p. 32) 



D 





1 


2 


3 


i 


6 


6 


7 


S 


9 




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 


1.495 


1.517 


21 


.4 


1.539 


IJ6I 


1.584 


1.606 


1.629 


I.65I 


1.674 


1.697 


1.720 


1.744 


23 


1.6 


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 




3.464 


3.497 


3.530 


3.563 


3.597 


3.631 


3.664 


3.598 


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 


.4 


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 


5.514 


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 


S.O 


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 


J 


8.553 


8.605 


8.657 


8.709 


8.762 


8.814 


8.867 


8.920 


8.973 


9.026 


53 


.4 


9.079 


9.133 


9.186 


9.240 


9.294 


9.348 


9.402 


9.457 


9.511 


9.566 


34 


S.S 


9.621 


9.676 


9.731 


9.787 


9.842 


9.898 


9.954 


10.010 






36 


.5 
















10.01 


10.07 


10.12 


6 


.6 


10.18 


10.24 


10.29 


10.35 


10.4) 


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 




.4 


15.21 


15.27 


15.34 


15.41 


15.48 


15.55 


15.62 


15.69 


15.76 


15.83 




4.6 


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. (Z) = diameter.) 

Area of circle = t X (diam.2) = 0.785398 X (diam.^) 
Conversely, 

Diam. = V- X VSea = 1.128379 X Var^ 



MATHEMATICAL TABLES 31 

AREAS OF CIRCLES BT HUNDREDTHS (continued) 



D 





1 


2 


3 


4 


6 


6 


7 


8 


9 




6.0 


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 


27 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 J9 


23.67 


9 


6.S 


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 


25J2 


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 


27J4 


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 




6.6 


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 


36.95 


37.07 


37.18 


37.28 




.9 


3739 


37.50 


37.61 


37.72 


37.83 


37.94 


38.05 


38.15 


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 




J 


40.72 


40.83 


40.94 


41.06 


41.17 


41.28 


41.40 


41.51 


41.52 


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 




7.6 


44.18 


44.30 


44.41 


44.53 


44.65 


44.77 


44.89 


45.01 


45.13 


45.25 




.6 


45.35 


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 


47.56 




.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 


49.75 


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 




J 


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 


55.21 


56.35 


56.48 


56.51 




8.6 


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 


54.90 




.1 


65.04 


65.18 


65.33 


65.47 


65.61 


65.76 


65.90 


66.04 


65.19 


56.33 


15 


J 


66.48 


66.62 


66.77 


66.91 


67.06 


67.20 


57.35 


67.49 


57.M 


67.78 




J 


67.93 


68.08 


68.22 


68.37 


68.51 


68.65 


58.81 


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 




9.6 


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 


76.35 


76.51 


75.67 


75.82 




.9 


76.98 


77.13 


77.29 


77.44 


77.60 


77.75 


77.91 


78.07 


78.23 


7838 


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 


S 
o 


1 


e 

.3 
Q 


i 


03 

4 


i 


J 


OS 

1 


s 


D 


1 








Ji 


2.749 


.6013 


4 


12.57 


12.57 


9 


28.27 


63.62 


Hi 


.04909 


.00019 


"At 


2.798 


.6230 


Me 


12.76 


12.96 


H 


28.67 


55.40 


Hi 


.09817 


.00077 


'Hi 


2.847 


.6450 


H 


12.95 


13.35 


X\ 


29.06 


57.20 


Hi 


.1473 


.00173 


'Hi 


2.896 


.5675 


Me 


13.16 


13.77 


H 


29.45 


69.03 


Mil 


.1963 


.00307 


>M6 


2.945 


.6903 


Vi 


13.35 


14.19 


V>. 


29.85 


70.88 


^1 


.2454 


.00479 


mi 


2.994 


.7135 


Me 


13.55 


14.61 


9^ 


30.24 


72.76 


Hz 


.2945 


.00690 


'Hi 


3.043 


.7371 


H 


13.74 


15.03 


^ 


30.53 


74.66 


J«i 


.3436 


.00940 


'Hi 


3.093 


.7610 


Me 


13.94 


15.47 


"•A 


31.02 


76J9 


H 


.3927 


.01227 


1 


3.142 


.7854 


H 


14.14 


15.90 


10 


31.42 


78.54 


%i 


.4418 


.01553 


Ms 


3.338 


.8866 


Mo 


14.33 


16.35 


H 


31.81 


80.52 


hI 


.4909 


.01917 


H 


3.534 


.9940 


H 


14.53 


16.80 


Vi 


32.20 


82.52 


mi 


.5400 


.02320 


'Aa 


3.731 


1.108 


'Ms 


14.73 


17.26 


H 


32.59 


84.54 


Ms 


.5890 


.02761 


Vi 


3.927 


1.227 


M 


14.92 


17.72 


H 


32.99 


86.59 


1^4 


.6381 


.03241 


Mo 


4.123 


1.353 


'Mo 


15.12 


18.19 


M 


33.38 


88.66 


^2 


.6872 


.03758 


H 


4.320 


1.485 


H 


15.32 


18.67 




33.77 


90.76 


mi 


.7363 


.04314 


Me 


4.516 


1.623 


'Ms 


15.51 


19.15 




34.16 


92.89 


H 


.7854 


.04909 


H 


4.712 


1.767 


5 


15.71 


19.63 


11 


34.56 


95.03 


"Ai 


.8345 


.05542 


Me 


4.909 


1.917 


Ms 


15.90 


20.13 


H 


34.95 


97.21 


H2 


.8836 


.06213 


H 


5.105 


2.074 




16.10 


20.63 


Vi 


35.34 


99.40 


^%i 


.9327 


.06922 


iHe 


5.301 


2.237 


Ms 


16.30 


21.14 


^ 


35.74 


101.6 


^0 


.9817 


.07670 


M 


5.498 


2.405 


H 


16.49 


21.65 


H 


36.13 


103.9 


'Hi 


1.031 


.08456 


'Me 


5.694 


2.580 


Ms 


16.59 


22.17 




36.52 


106.1 


'Hz 


1.080 


.09281 


H 


5.890 


2.761 


H 


16.89 


22.69 




35.91 


108.4 


'Hi 


1.129 


.1014 


'Ms 


6.087 


2.948 


Ms 


17.08 


23.22 


M 


37J1 


110.8 


H 


I.I78 


.1104 


2 


6.283 


3.142 


\^ 


17.28 


23.76 


12 


37.70 


113.1 


'Hi 


1.227 


.1198 


Me 


6.480 


3.341 




17.48 


24.30 




38.09 


115.5 


^Hi 


1.276 


.1296 


H 


6.675 


3.547 


H 


17.67 


24.85 




38.48 


117.9 


'Hi 


1.325 


.1398 


Me 


6.872 


3.758 


'Ms 


17.87 


25.41 


M 


38.88 


120.3 


'Ha 


1.374 


.1503 


Vi 


7.059 


3.976 


% 


18.06 


25.97 


Vi 


39.27 


122.7 


'Hi 


1.424 


.1613 


Me 


7.265 


4.200 


'Me 


18.26 


26.53 


5i 


39.66 


125.2 


'Hi 


1.473 


.1725 


H 


7.461 


4.430 


H 


18.46 


27.11 




40.06 


127.7 


'Hi 


1.522 


.1843 


Jie 


7.658 


4.555 


'Ms 


18.65 


27.69 




40.45 


130.2 


H 


1.571 


.1963 


H 


7.854 


4.909 


6 


18.85 


28.27 


13 


40.84 


132.7 


'Hi 


1.620 


.2088 


Me 


8.050 


5.157 


H 


19.24 


29.46 


H 


41.23 


1353 


>>42 


1.669 


.2217 


H 


8.247 


5.412 


H 


19.63 


30.68 


Vi 


41.63 


137.9 


'Hi 


1.718 


.2349 


'Me 


8.443 


5.673 


H 


20.03 


31.92 


H 


42.02 


1405 


Ma 


1.767 


.2485 


M 


8.639 


5.940 


H 


20.42 


33.18 




42.41 


143.1 


'Hi 


1.816 


.2625 


'Me 


8.836 


6.213 


H 


20.81 


34.47 




42.80 


145.8 


ml 


1.865 


.2769 


H 


9.032 


6.492 




21.21 


35.78 




43.20 


148.5 


>%i 


1.914 


.2915 


'Me 


9.228 


6.777 


ji 


21.60 


37.12 


H 43J9 


15U 


W 


1.963 


.3068 


3 


9.425 


7.069 


7 


21.99 


38.48 


14 


43.98 


153.9 


*Hi 


2.013 


.3223 


Me 


9.521 


7.356 


H 


22.38 


39.87 


H 44.37 


156.7 


'Hi 


2.062 


.3382 


H 


9.817 


7.570 


M 


22.78 


41.28 


V 


44.77 


159.5 


*Hi 


2.111 


.3545 


Me 


10.01 


7.980 


H 


23.17 


42.72 


H 45.16 


1623 


•Mb 


2.160 


.3712 


H 


10.21 


8.296 


H 


23.55 


44.18 


H 45.55 


165.1 


mi 


2.209 


.3883 


Me 


10.41 


8.618 


H 


23.95 


45.55 


H 45.95 


168.0 


'Hi 


2.258 


.4057 


H 


10.50 


8.945 


% 


24.35 


47.17 


H 46.34 


170.9 


"Ai 


2.307 


.4236 


Me 


10.80 


9.281 


% 


24.74 


48.71 


H 46.73 


173.8 


H 


2.356 


.4418 


H 


11.00 


9.621 


8 


25.13 


50.27 


16 


47.12 


176.7 


*Hi 


2.405 


.4604 


Me 


11.19 


9.968 


H 


25.53 


51.85 


H 47.52 


179.7 


'Hi 


2.454 


.4794 


H 


11.39 


10.32 


Vi 


25.92 


53.45 


H 47.91 


182.7 


'Hi 


2.503 


.4987 


'Me 


11.58 


10.58 


H 


26.31 


55.09 


H 48.30 


185.7 


iM« 


2.553 


.5185 


H 


11.78 


11.04 


H 


25.70 


55.75 


H 48.69 


1M.7 


'Hi 


2.602 


.5386 


'Me 


11.98 


11.42 


H 


27.10 


58.43 


H 49.09 


191.7 




2.651 


.5591 


% 


12.17 


11.79 


H 


27.49 


60.13 


1 


i 49.48 


1H8 


mi 


2.700 


.5800 


'Me 


12.37 


12.18 


H 


27.88 


61.86 


If 


i 49.8; 


197.9 



MATHEMATICAL TABLES 



33 



CIRCUMFERENCES AND AREAS BY EIGHTHS— (con<m«ed) 


S 


6 

3 


f. 


i 




E 


a 


i 


f. 


a 


a 


<A 


(3 


5 


< 


a 


a 


<1 


Q 


6 


M 


a 


3 


< 


16 


50.27 


201.1 


19 M 


61.26 


298.6 


23 


72.25 


415.5 


29 


91.11 


660.5 


M 


50.66 


204.2 


4* 


61.65 


302.5 


H 


72.65 


420.0 


Vi 


91.89 


672.0 


H 


51.05 


207.4 


■H 


62.05 


306.4 


V4 


73.04 


424.6 




92.58 


683.5 


H 


51.44 


210.6 


'/i 


62.44 


310.2 


% 


73.43 


429.1 


Yi 


93.46 


695.1 


H 


51.84 


213.8 


20 


62 83 


314.2 


V,. 


73.83 


433.7 


30 


94.25 


706.9 




52.23 


217.1 


% 


63.22 


318.1 


^ 


74.22 


438.4 


Vi 


95.03 


718.7 


^ 


52.62 


220.4 


H 


63.62 


322.1 


H 


74.51 


443.0 


V,. 


95,82 


730.6 


H 


53.01 


223.7 


H 


64.01 


326.1 


% 


75.01 


447.7 


% 


96.60 


742.6 


17 


53.41 


227.0 


V^ 


64.40 


330.1 


21 


75.40 


452.4 


31 


97.39 


754.8 


H 


53.80 


2303 


H 


64.80 


334.1 


H 


76.18 


461.9 


Vi 


98.17 


767.0 




54.19 


233.7 


M 


65.19 


338.2 


Vr. 


75.97 


471.4 


Vi 


98.95 


779.3 


H 


54J9 


237.1 


'4 


65.58 


342.2 


M 


77.75 


481.1 


% 


99.75 


791.7 


M 


54.98 


240.5 


21 


65.97 


346.4 


26 


78.54 


490.9 


32 


100.5 


804.2 




55.37 


244.0 


H 


66.37 


350.5 


V4 


79.33 


500.7 


Vi 


101.3 


816.9 


94 


55.76 


247.4 


H 


66.76 


354.7 


Vr. 


80.11 


510.7 


w 


102.1 


829.6 


'/i 


56.16 


250.9 


H 


67.15 


358.8 


% 


80.90 


520.8 


y* 


102.9 


842.4 


18 


56.55 


254.5 


^ 


67.54 


353.1 


26 


81.58 


530.9 


33 


103.7 


855.3 


H 


55.94 


258.0 


« 


67.94 


367.3 


Vi 


82.47 


541.2 


Vi 


104.5 


8583 


M 


57.33 


261.6 


H 


68.33 


371.5 




83.25 


551.5 




105.2 


881.4 


H 


57.73 


265.2 


■'A 


68.72 


375.8 


% 


84.04 


562.0 


'A 


105.0 


894.6 


W 


58.12 


268.8 


22 


69.12 


380.1 


27 


84.82 


572.6 


31 


105.8 


907.9 


W 


58.51 


272.4 


H 


69.51 


384.5 


H 


85.51 


583.2 


Vi 


107.5 


9213 


H 


58.90 


276.1 


Vi 


69.90 


388.8 


M, 


86.39 


594.0 


H 


108.4 


934.8 


'A 


59J0 


279.8 


H 


70.29 


393.2 


Vi 


87.18 


604.8 


Vi 


109 J 


948.4 


19 


59.69 


283.5 


h 


70.69 


397.6 


28 


87.96 


515.8 


35 


110.0 


952.1 


H 


60.08 


287.3 


H 


71.08 


402.0 


Vi 


88.75 


526.8 


Vi 


110.7 


975.9 


H 


60.48 


291.0 


y* 


71.47 


405.5 


M 


89.54 


537.9 


H 


111.5 


989.8 


% 


60.87 


294.8 


-•A 


71.86 


411,0 


% 


90.32 


649.2 


Vi 


112.3 


1003.8 



AREAS OF CIRCLES. 


Diameters in Feet and Inches, Areas in Square Feet 


Feet 


Inches 


1 


23456 789 10 11 






.0000 


.0055 


.0218 


.0491 


.0873 


.1364 


.1963 


2m 


3491 


.4418 


.5454 


.5600 


1 


.7854 


.9218 


1.059 


1.227 


1.396 


1.576 


1.757 


1.969 


2.182 


2.405 


2.640 


2.885 


2 


3.142 


3.409 


3.587 


3.976 


4.276 


4.587 


4.909 


5.241 


5.585 


5.940 


6.305 


5.581 


3 


7.069 


7,467 


7.876 


«.296 


8.727 


9.158 


9.621 


10.08 


10.36 


11.04 


11.54 


12.05 


4 


12.57 


13.10 


13.64 


14.19 


14.75 


15.32 


15.90 


15.50 


17.10 


17.72 


1835 


18.99 


5 


19.63 


20.29 


20.97 


21.55 


2234 


23.04 


23.75 


24,48 


25.22 


25.97 


26.73 


27.49 


6 


28.27 


79.07 


29 87 


30.68 


31.50 


3234 


33.18 


34.04 


34.91 


35.78 


35.57 


37.57 


7 


38.48 


39.41 


40.34 


41.28 


42.24 


43.20 


44.18 


45.17 


45.16 


47.17 


48.19 


49.22 


8 


50.27 


5132 


5J38 


53.45 


54.54 


55.54 


56.75 


57.85 


58.99 


60.13 


61.28 


62.44 


9 


63.52 


64.80 


66.00 


67.20 


68.42 


69.64 


70.88 


72.13 


73.39 


74.66 


75.94 


77.24 


10 


78.54 


79 85 


81.18 


8252 


83,86 


85.22 


85.59 


87.97 


89.35 


90.75 


92.18 


93.50 




95.03 


96.48 


97.93 


99.40 


100.9 


102.4 


103.9 


105.4 


105.9 


108.4 


110.0 


111.5 


12 


113.1 


114.7 


1163 


117.9 


119.5 


I2I.I 


122.7 


124.4 


126.U 


127.7 


129.4 


131.0 


13 


132.7 


134.4 


136.2 


137.9 


139.6 


141.4 


143.1 


144.9 


146.7 


148.5 


1503 


152.1 


14 


153.9 


155.8 


157.6 


1593 


151.4 


163.2 


165.1 


157.0 


168.9 


170.9 


172.8 


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 
Feet 



1 
.0833 



2 3 

1667 .2500 



4 
3333 



4167 .5000 



7 
.5833 



8 9 10 11 

6667 .7500 .8333 .9167 



Inches H H H H ^ H ''A 

Feet .0104 .0208 .0313 .0417 .0521 .0625 .0729 

Example. 5 ft. 7H in. = 5.0 + 0.5833 + 0.0313 = 5.6146 ft. 



34 



MATHEMATICAL TABLES 



SEGMENTS OF CIRCLES, GIVEN h/c 

Given: h = height; c = chord. (For explanation of this table, see p. 38) 



Diam. 



aa 
(5 



Arc 



s 



Area 
hXc 



(3 



Central 
angle, v 



3 



Diam. 



25.010 
12.320 
8.363 
6.290 

5.050 
4.227 
3.641 
3.205 
2.868 

2.600 
2.383 
2.203 
2.053 
1.926 

1.817 
1.723 
1.641 
1.569 
1.506 

1.450 
1.400 
1.356 
1.317 
1.282 

1.250 
1.222 
1.196 
1.173 
1.152 

1.133 
1.116 
1.101 
1.088 
1.075 

1.064 
1.054 
1.046 
1.038 
1.031 

1.025 
1.020 
1.015 
1.011 
1.008 

1.006 
1.003 
1.002 
1.001 
1.000 

1.000 



12490 
•4157 
•2073 
•1240 

•823 
•586 
•436 
•337 
•268 

•217 
•180 
•150 
•127 
•109 

•94 
•82 
•72 
•63 
56 

50 
44 
39 
35 
32 

28 
26 
23 
21 
19 

17 
15 
13 
13 
II 

10 



1.000 
1.00O 
1.001 
1.002 
1.004 

1.007 
1.010 
1.013 
1.017 
1.021 

1.026 
1.032 
1.038 
1.044 
1.051 

1.059 
1.067 
1.075 
1.084 
1.094 

1.103 
1.114 
1.124 
1.136 
1.147 

1.159 
1.171 
1.184 
1.197 
1.211 

1.225 
1.239 
1.234 
1.269 
1.284 

1.300 
1.316 
1.332 
1.349 
IJ66 

1.383 
1.401 
1.419 
1.437 
1.435 

1.474 
1.493 
1.312 
1.531 
1.551 

I.57I 




1 

2 
3 

3 
3 
4 
4 
5 

6 
6 
6 
7 
8 

8 
8 
9 
10 
9 

10 
12 
II 
12 

12 
13 
13 
14 
14 

14 
15 
15 
15 
16 

16 
16 
17 
17 
17 

18 
18 
18 
18 
19 

19 
19 
19 
20 
20 



.6667 
.6667 
.6669 
.6671 
.6675 

.6680 
.6686 
.6693 
.6701 
.6710 

.6720 
.6731 
.6743 
.6756 
.6770 

.6785 
.6801 
.6818 
.6836 
.6855 

.6875 
.6896 
.6918 
.6941 
.6965 

.6989 
.7014 
.7041 
.7068 
.7096 

.7125 
.7154 
.7185 
.7216 
.7248 

.7280 
.7314 
.7348 
.7383 
.7419 

.7455 
.7492 
.7530 
.7568 
.7607 

.7647 
.7687 
.7728 
.7769 
.7811 

.7854 




2 
2 
4 
5 

6 
7 
8 
9 
10 

11 
12 
13 
14 
15 

16 
17 
18 
19 
20 

21 
22 
23 
24 
24 

25 
27 
27 
28 
29 

29 
31 
31 
32 
32 

34 
34 
33 
36 
36 

37 
38 
38 
39 
40 

40 
41 
41 
42 
43 



0.00° 
4.38 
9.16 
13.73 
18.30 

22.84° 
27.37 
31.88 
36.36 
40.82 

45.24° 
49.63 
53.98 
58.30 
62.57 

66.80° 

70.98 

73.11 

79.20 

83.23 

87.21° 
91.13 
95.00 
98.81 
102.56 

106.26° 
109.90 
113.48 
117.00 
120.43 

123.86° 
127.20 
130.48 
133.70 
136.86 

139.97° 

143.02 

146.01 

14894 

131.82 

154.64° 

157.41 

160.12 

162.78 

165.39 

167.95° 

170.46 

172.91 

175.32 

177.69 

180.00° 



458 
458 
457 
457 
434 

453 
451 
448 
446 
442 

439 
435 
432 
427 
423 

418 
413 
409 
403 
398 

392 
387 
381 
375 
370 

364 
358 
352 
345 
341 

334 
328 
322 
316 
311 

305 
299 
293 
288 
282 

277 
271 
266 
261 
256 

251 
245 
241 
237 
231 



.0000 
.0004 
.0016 
.0036 
.0064 

.0099 
.0142 
.0192 
.0250 
.0314 

.0385 
.0462 
.0545 
.0633 
.0727 

.0826 
.0929 
.1036 
.1147 
.1262 

.1379 
.1499 
.1622 
.1746 
.1873 

.2000 
.2128 
.2238 
.2387 
J317 

J647 
.2777 
.2906 
.3034 
J162 

3289 
.3414 
J538 
.3661 
J783 

.3902 
.4021 
.4137 
.4252 
.4364 

.4475 
.4584 
.4691 
.4796 
.4899 

.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, seep. 38) 



Arc 
D 



Area ia 
D' 5 



Central (g 
angle, v q 



Chord to 
D (3 



Arc 



Circumf . Q 



Area 
Circle 



0.0000 
.2003 
.2838 
.3482 
.4027 

.4310 
.4949 
J355 
.5735 
.6094 

.6435 
.6761 
.7075 
.7377 
.7670 

.7954 
.8230 
.8500 
.8763 
.9021 

0.9273 
0.9521 
0.9764 
1.0004 
1.0239 

1.0472 
1.0701 
1.0928 
1.1152 
1.1374 

1.1593 
1. 1810 
1.2025 
1.2239 
1.2451 

1.2661 
1.2870 
1.3078 
1.3284 
1.3490 

1.3694 
1.3898 
1.4101 
1.4303 
1.4505 

1.4705 
1.4907 
1.5108 
1.5308 
IJ508 

1.5708 



2003 
•835 
•644 
•545 
•483 

•439 
•406 
•380 
•359 
•341 

•326 
•314 
•302 
•293 
•284 

276 
270 
263 
258 
252 

248 
243 
240 
235 
233 

229 
227 
224 
222 
219 

217 
215 
214 
212 
210 

209 
208 
206 
206 
204 

204 
203 
202 
202 
201 

201 
201 
200 
200 
200 



.0000 ,, 

.0013 if 

.0037 iZ 

.0069 ii 

.0105 36 

.0147 45 

.0192 II 

.0242 „ 

.0294 ti 

.0350 1° 

.0409 ,, 

.0470 5! 

.0534 57 

.0600 92 

.0668 ^ 

.0739 „ 

.0811 Ij 

.0885 ii 

.0961 iS 

.1039 J5 

.1118 o, 

.1199 51 

.1281 Ji 

.1365 8^ 

.1535 oo 

.1623 S! 

.1711 II 

.1800 Z 

.1890 5^ 

.1982 ,2 

.2074 li 

.2167 li 

.2260 li 

.2355 g 

.2450 06 

.2546 2? 

.2642 2? 

.2739 l^ 

.2836 go 

.2934 98 

.3032 ll 

.3130 ll 

.3229 ^ 

332S iQQ 

.3428 go 

.3527 ,55 

.3627 gg 

.3727 gg 

.3827 jgg 

.3927 



0.00° 
22.96 
32.52 
39.90 
46.15 

51.68° 

56.72 

61.37 

65.72 

69.83 

73.74° 

77.48 

81.07 

84.54 

87.89 

91.15° 
94.31 
97.40 
100,42 
103.37 

106.26° 

109.10 

111.89 

114.63 

117.34 

120.00° 

122.63 

125.23 

127.79 

130.33 

132.84° 

135.33 

137.80 

140.25 

142.67 

145.08° 

147.48 

149.86 

152.23 

154.58 

156.93° 
159.26 
161.59 
163.90 
166.22 

168.52° 

170.82 

173.12 

175.42 

177.71 

180.00° 



2296 
•956 
•738 
•625 
•553 

•504 
•465 
•435 
•411 
•391 

•374 
•359 
•347 
•335 
•326 

316 
309 
302 
295 
289 

284 
279 
274 
271 
266 

263 
260 
256 
254 
251 

249 
247 
245 
242 
241 

240 
238 
237 
235 
235 

233 
233 
231 
232 
230 

230 
230 
230 
229 
229 



.0000 
.1990 
.2800 
.3412 
.3919 

.4359 
.4750 
.5103 
.5426 
J724 

.6000 
.6258 
.6499 
.6726 
.6940 

.7141 
.7332 
.7513 
.7684 
.7845 

.8000 
.8146 
.8285 
.8417 
.8542 

.8560 
.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 



•1990 
•810 
•612 
•507 
•440 

•391 
•353 
•323 
•298 
•276 

•258 
•241 
•227 
•214 
•201 

•191 
•181 
•171 
162 
154 

146 
139 
132 
125 
118 

113 
106 
101 
95 
90 

85 
80 
74 
70 
65 

61 
56 
52 
47 
43 

39 
34 
31 
25 
22 

18 
14 
10 
5 
2 



.a '411 

.1282 ,\ll 

'1575 •"' 

:i705 -j^o 

.1826 ti 

.1940 J>^ 

.2048 ,(w 

.2152 SJ 

.2252 'S9 

.2348 S 

.2441 I] 

.2532 M 

.2620 f? 

.2706 5» 

.2789 Si 

.2871 St 



81 



!2871 

.2952 Z 

.3031 <X 

.3108 ii 

.3184 i% 

.3259 II 

.3333 -, 

J406 „ 

.3478 ii 

.3550 II 

.3620 1° 

.3690 ,, 

.3759 ll 

.3828 ?a 

.3895 52 

J963 II 

.4030 -, 

.4097 5' 

•'"« 55 

.4229 55 

.4294 « 

.4359 ,. 

.4424 5| 

.4489 S 

.4553 ^ 

.4617 ^ 

.4581 f. 

•4745 5J 

.4809 51 

.4873 5? 

.4936 « 

.5000 



.0000 
.0017 
.0048 
.0087 
.0134 

.0187 
.0245 
.0308 
.0375 
.0446 

.0520 
.0399 
.0680 
.0764 
.0851 

.0941 
.1033 
.1127 
.1224 
.1323 



.2523 
J640 



17 
31 
39 
47 
53 

58 
63 
67 
71 
74 

79 

81 

84 
87 
90 

92 
94 
97 
99 
101 



•l527 '^ 

■l631 "M 

.low iQj 

.1953 „, 

.2066 i 

.2178 J 

.2292 i 

.2407 J!^ 



,17 
119 



122 
123 
123 



.2759 ' 

.2878 \\l 

.2998 {|y 

.3119 
.3241 

.3364 ,„ 

3487 51 

.3735 ,„ 

.3860 \ii 

.3986 f5 

.4112 25 

.4238 III 

.4364 ,„ 

.4491 g 

.4618 ™ 

.4745 ^7 

.4873 \l^ 
.5000 



• Interpolation may be inaccurate at these points. 



36 MATHEMATICAL TABLES 

VOLUMES OF SPHERES BY HUNDREDTHS 



D 





1 


2 


3 


i 


6 


6 


7 


8 


9 


li 


1.0 


J236 


J395 


.5556 


J722 


.5890 


.6061 


.6235 


.6414 


.5595 


.6781 


173 


.1 


.6969 


.7161 


.7356 


.7555 


.7757 


.7963 


.8173 


.8386 


.8603 


.8823 


208 


.2 


.9048 


.9276 


.9508 


.9743 


.9983 


1.0227 










236 


J 












1.023 


1.047 


1.073 


1.098 


1.124 


25 


J 


i.m 


1.177 


1.204 


1.232 


1.260 


1.288 


1.317 


1.345 


1.375 


1.406 


29 


.4 


1.437 


1.468 


1.499 


1.531 


1J63 


1.596 


1.630 


1.663 


1.597 


1.732 


33 


1.6 


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 


S 


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 


S.0 


4.189 


4.252 


4.316 


4.380 


4.445 


4.511 


4.577 


4.544 


4.712 


4.780 


66 


.1 


4.849 


4.919 


4.989 


5.0t0 


5.131 


5.204 


5.277 


5.350 


5.425 


5.500 


73 


.2 


5.575 


5.652 


5.729 


5.806 


5.885 


5.954 


6.044 


5.125 


6.205 


6.288 


80 


.3 


6.371 


6.454 


6.538 


6.623 


6.709 


6.795 


6.882 


5.970 


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 


S.B 


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 




no 


.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 


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 


13.85 


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 


15.58 


15.84 


17.00 


16 


J. 


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 


21J1 


21.50 


21.69 


21.88 


22.07 


22.26 


19 


S.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 


25.09 


26.31 


21 


.7 


26.52 


26.74 


26.95 


27.17 


27.39 


27.61 


27.83 


28.05 


28.28 


28.50 


22 


.8 


28.73 


28.96 


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 


35.55 


35.82 


2< 


.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 


4134 


28 


3 


41.63 


41.92 


42.21 


42.51 


42.80 


43.10 


43.40 


43.70 


44.00 


44.30 


30 


A 


44.60 


44.91 


45.21 


45.52 


45.83 


46.14 


46.45 


45.77 


47.08 


47.40 


31 


<.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 


^ 


.7 


54.36 


54.71 


55.06 


55.41 


55.76 


56.12 


56.47 


55.83 


57.19 


57.54 


35 


.8 


57.91 


58.27 


58.63 


59.00 


59.37 


59.73 


60.10 


50.48 


60.85 


61.22 


37 


.9 


61.60 


61.98 


62.36 


62.74 


63.12 


63.51 


63.89 


54.28 


64.67 


65.06 


38 



Explanation of Table of Volumes of Spheres (pp. 36-37). 

Moving the decimal point one place in column Z> is equivalent to moving it three 
places in the body of the table. (2) = diameter.) 



Conversely, 



Volume of sphere = g X (diam.") = 0.523599 X (diam.») 



Diam. 



\ 



"V' volume = 1.240701 X v^ volume 



MATHEMATICAL TABLES 37 

VOLUMES OF SPHERES (continued) 



65.45 65.84 66.24 66.64 67.03 67.43 67.83 68.24 68.64 69.05 

69.46 69.87 70.28 70.69 71.10 71.52 71.94 72.36 72.78 73.20 
73.62 74.05 74.47 74.90 75.33 75.77 76.20 76.64 77.07 77.51 
77.95 78.39 78.84 79.28 79.73 80.18 80.63 81.08 81.54 81.99 
82.45 82.91 83.37 83.83 84.29 84.76 85.23 85.70 86.17 86.64 

87.11 87.59 88.07 88.55 89.03 89.51 90.00 90.48 90.97 91.46 

91.95 92.45 92.94 93.44 93.94 94.44 94.94 95.44 95.95 96.46 

96.97 97.48 97.99 98.51 99.02 99.54 100 06 

100.1 100.6 101.1 101.6 
102.2 102.7 103.2 103.8 104.3 104.8 105.4 105.9 106.4 107.0 
107.5 108.1 108.6 109.2 109.7 110.3 110.9 111.4 112.0 112.5 

113.1 113.7 114.2 114.8 115.4 115.9 116.5 117.1 117.7 118.3 
118.8 119.4 120.0 120.6 121.2 121.8 122.4 123.0 123.6 124.2 

124.8 125.4 126.0 126.6 127.2 127.8 128.4 129.1 129.7 130.3 

130.9 131.5 132.2 132.8 133.4 134.1 134.7 135.3 136.0 136.6 
1373 137.9 138.5 139.2 139.8 140.5 141.2 141.8 142.5 143.1 

143.8 144.5 145.1 145.8 146.5 147.1 147.8 148.5 149.2 149.8 

150.5 151.2 151.9 152.6 153.3 154.0 154.7 155.4 156.1 156.8 

157.5 158.2 158.9 159.6 160.3 161.0 161.7 162.5 163.2 163.9 

164.6 165.4 166.1 166.8 167.6 168.3 169.0 169.8 170.5 1713 
172.0 172.8 173.5 1743 175.0 175.8 176.5 1773 178.1 178.8 

179.6 180.4 181.1 181.9 182.7 183.5 1843 185.0 185.8 186.6 
187.4 188.2 189.0 189.8 190.6 191.4 192.2 193.0 193.8 194.6 
195.4 196.2 197.1 197.9 198.7 199.5 200.4 201.2 202.0 202.9 

203.7 204.5 205.4 206.2 207.1 207.9 208.8 209.6 210.5 211.3 

212.2 213.0 213.9 214.8 215.6 216.5 217.4 2183 219.1 220.0 

226.2 227.1 228.0 228.9 
2353 236.3 237.2 238.1 
244.7 245.6 246.6 247.5 
2543 255.2 256.2 257.2 

264.1 265.1 266.1 267.1 

274.2 275.2 276.2 277.2 

284.5 285.5 286.6 287.6 
295.1 296.2 297.2 298.3 
305.9 307.0 308.1 309.2 

317.0 318.2 3193 320.4 

328.4 329.6 330.7 331.9 

340.1 341.2 342.4 343.6 
352.0 353.2 354.4 355.6 

364.2 365.4 366.6 367.9 

376.6 377.9 379.2 380.4 

389.4 390.7 392.0 393.3 

402.4 403.7 405.1 406.4 

415.7 417.1 418.4 419.8 

429.4 430.7 432.1 433.5 
4433 444.7 446.1 447.5 

457.5 458.9 460.4 461.8 
472.0 473.5 474.9 476.4 

486.8 4883 489.8 4913 

501.9 503.4 505.0 506.5 

517.3 518.9 520.5 522.0 



220.9 


221.8 


222.7 


223.6 


224.4 


2253 


229.8 


230.8 


231.7 


232.6 


233.5 


234.4 


239.0 


240.0 


240.9 


241.8 


242.8 


243.7 


248.5 


249.4 


250.4 


251.4 


252.3 


2533 


258.2 


259.1 


260.1 


261.1 


262.1 


263.1 


268.1 


269.1 


270.1 


271.1 


272.1 


273.1 


2783 


279.3 


2803 


281.4 


282.4 


283.4 


288.7 


289.8 


290.8 


291.9 


292.9 


294.0 


299.4 


300.5 


301.6 


302.6 


303.7 


304.8 


3103 


311.4 


312.6 


313.7 


314.8 


315.9 


321.6 


322.7 


3B.8 


325.0 


326.1 


327.3 


333.0 


334.2 


335.4 


336.5 


337.7 


338.9 


344.8 


346.0 


347.2 


348.4 


349.6 


350.8 


356.8 


358.0 


359.3 


360.5 


361.7 


362.9 


369.1 


370.4 


371.6 


372.9 


374.1 


375.4 


381.7 


383.0 


3843 


385.5 


386.8 


388.1 


394.6 


395.9 


397.2 


398.5 


399.8 


401.1 


407.7 


409.1 


410.4 


411.7 


413.1 


414.4 


421.2 


422.5 


423.9 


425.2 


426.6 


428.0 


434.9 


436.3 


437.7 


439.1 


440.5 


441.9 


448.9 


4503 


451.8 


453.2 


454.6 


456.0 


463.2 


464.7 


466.1 


467.6 


469.1 


470.5 


477.9 


479.4 


480.8 


4823 


483.8 


485.3 


492.8 


494.3 


495.8 


4973 


498.9 


500.4 


508.0 


509.6 


5II.1 


512.7 


514.2 


515.8 



523.6 



Moving the decimal point ONE place in D requires moving it THREE places in body of 
table (seep. 36). 



38 



MATHEMATICAL TABLES 



SEGMENTS OF SPHERES 

(h = height of segment; D 



diam. of sphere) 



Vol. segm. to 



0.0000 
0.0002 
0.0006 
0.0014 
0.0024 

0.0038 
0.0054 
0.0073 
0095 
0.0120 

0.0147 
0.0176 
0.0208 
0.0242 
0.0279 

0.0318 
0.0359 
0.0403 
0448 
0.0495 

0.0545 
0.0596 
0.0649 
0.0704 
0.0760 

0.0818 
0.0878 
0.0939 
0.1002 
0.1066 

0.1131 
0.1198 
0.1265 
0.1334 
0.1404 

0.1475 
0.1547 
0. 1620 
0.1694 
0.1768 

0.1843 
0.1919 
0.1995 
0.2072 
0.2149 

0.2227 
0.2305 
0.2383 
0.2461 
0.2539 

0.2618 



2 
4 
8 
10 
14 

16 
19 
22 
25 
27 

29 
32 
34 
37 
39 

41 
44 
45 
47 
50 

51 
53 
55 
56 
58 

60 
61 
63 
64 
65 

67 
67 
69 
70 
71 

72 
73 
74 
74 
75 

76 
76 
77 
77 
78 

78 
78 
78 
78 
79 



Vol. segm. 



Vol. sphere 



fa 

a 



0.0000 
0.0003 
0.0012 
0.0026 
0.0047 

0.0073 
0.0104 
0.0140 
0.0182 
0.0228 

0.0280 
0.0336 
0.0397 
0.0463 
0.0533 

0.0607 
0.0686 
0.0769 
0.0855 
0.0946 

0.1040 
0.1138 
0.1239 
0.1344 
0.1452 

0.1562 
0.1676 
0.1793 
0.1913 
0.2035 

0.2160 
0.2287 
0.2417 
0.2548 
0.2682 

0.2817 
0.2955 
0.3094 
0.3235 
0.3377 

0.3520 
0.3665 
0.3810 
0.3957 
0.4104 

0.4252 
0.4401 
0.4551 
0.4700 
0.4850 

0.5000 



3 
9 
14 
21 
26 

31 
36 
42 
46 
52 

56 
61 
66 
70 
74 

79 
83 
86 
91 
94 

98 
101 
105 



114 
117 
120 
122 
125 

127 
130 
131 
134 
135 

138 
139 
141 
142 
143 

145 
145 
147 
147 

148 

149 
150 
149 
150 
150 



Explanation of Table on this page 

Given, h = height of segment, 

D = diam. of sphere. 
To find the volume of the segment, 

form the ratio h/D and find from the 

table the value of (vol./D'); then, by 

a simple multiplication, 

vol. segment = D' X (vol./D') 
The table gives also the ratio of the 

volume of the segment to the entire 

volume of the sphere. 
Note. Area of zone = x X h. X D. 

(Use Table of Multiples of w, p. 28) 

Explanation of Table on p. 34 
Given, h = height of segment, 

c = chord. 
To find the diam. of the circle, the 
length of arc, or the area of the seg- 
ment, form the ratio h/c, and find 
from the table the value of (diam./c), 
(arc/c), or (area/Ac) ; then, by a simple 
multiplication, 

diam. = c X (diam./c), 
arc = c X (arc/c), 

area = h X c X (area/Ac). 

The table gives also the angle sub- 
tended at the center, and the ratio of 
h to D. See p. 106. 

Explanation of Table on p. 36 

Given, h — height of segment, 

D = diam. of circle. 
To find the chord, the length of arc, 
or the area of the segment, form the 
ratio h/D, and find from the table the 
value of (chord/D), (arc/2>), or 
(area/Z>2) ; then, by a simple multi- 
plication, 

chord = D X (chord/D), 
arc = D X (arc/B), 
area = D" X (area/D^). 
The table gives also the angle sub- 
tended at the center, the ratio of the 
arc of the segment to the whole cir- 
cumference, and the ratio of the area 
of the segment to the area of the 
whole circle. See p. 106. 



Note. Vol. segm. = ^i-r V (ZI>-2h). 



MATHEMATICAL TABLES 



39 



REGULAR POLYGONS 

n = number of sides; 
V = 3Q0°/n = angle subtended at the center by one side; 

a = length of one side = ^(2 sin— j = r(2tan^y, 
R = radius of circumscribed circle = a{ J-^csc — ) = r(sec — j; 
r = radius of inscribed circle = R Tcob ^ ) = a h/i cot ^ j ; 
Area = a2f}4 n cot — j = R'i()4 n sin v\ = r^fn tan — V 



n 


D 


Area 
a2 


Area 


Area 
r2 


R 

a 


B 
r 


a 
B 


a 

r 


r 
B 


r 
a 


3 
4 
5 
6 


120° 
90° 
72° 
60° 


0.4330 
1.000 
1.721 
2.598 


1.299 
2.000 
2.378 
2.598 


5.196 
4.000 
3.633 
3.464 


0.5774 
0.7071 
0.8507 
1.0000 


2.000 
1.414 
1.236 
1.155 


1.732 
1.414 
1.176 
1.000 


3.464 
2.000 
\ASi 
1.155 


0.5000 
0.7071 
0.8090 
0.8660 


0.2887 
0.5000 
0.6882 
0.8660 


7 
8 
9 
10 


51°.43 
45° 
40° 
36° 


3.634 
4.828 
6.182 
7.694 


2.736 
2.828 
2.893 
2.939 


3.371 
3.314 
3.276 
3.249 


1.152 
1.307 
1.462 
1.618 


1.110 
1.082 
1.064 
1.052 


0.8678 
0.7654 
0.6840 
0.6180 


0.%31 
0.8284 
0.7279 
0.6498 


0.9010 
0.9239 
0.9397 
0.9511 


1.038 
1.207 
1.374 
1.539 


12 
15 
16 
20 


30° 
24° 

22°. 50 
18° 


11.20 
17.64 
20.11 
31.57 


3.000 
3.051 
3.062 
3.090 


3.215 
3.188 
3.183 
3.168 


1.932 
2.405 
2.563 
3.196 


1.035 
1.022 
1.020 
1.013 


0.5176 
0.4158 
0.3902 
0.3129 


0.5359 
0.4251 
0.3978 
0.3168 


0.9659 
0.9781 
0.9808 
0.9877 


1.866 
2.352 
2.514 
3.157 


24 
32 
48 
64 


15° 

11°.25 
7°.50 
5°.625 


45.58 
81.23 
183.1 
325.7 


3.106 
3.121 
3.133 
3.137 


3.160 
3.152 
3.146 
3.144 


3.831 
5.101 
7.645 
10.19 


1.009 
1.005 
1.002 
1.001 


0.2611 
0.1960 
0.1308 
0.0981 


0.2633 
0.1970 
0.1311 
0.0983 


0.9914 
0.9952 
0.9979 
0.9988 


3.798 
5.077 
7.629 
10.18 



BINOMIAL COEFFICIENTS 

(For table giving binomial coefficients for fractional values of », see p. 1 16). 



(n)o = 1; (7j)i = n; (n)2 = -^r-sro~'' W' 



(n)r 



1 X 2 
n{n — l)(n — 2) . (n - [r ■ 



k(7i - l)(n - 2) 
1X2X3 • 



etc.; in general. 



11) 



1X2X3 



Another notation: f") =(n)r. 



n 


Wo 


Wi 


(n)2 


Wa 


in) A 


(n)5 


(n). 


M-- 


(n)8 


w. 


W.o 


W.i 


(n)i! 


W.3 


1 




2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 


























7 


1 

3 

6 

10 

15 

21 

28 

36 

45 

55 

66 

78 

91 

105 
























3 


1 

4 

10 

20 

35 

56 

84 

120 

165 

220 

286 

364 

455 






















4 


1 

5 

15 

35 

70 

126 

210 

330 

495 

715 

1001 

1365 




















5 


1 

6 

21 

56 

126 

252 

462 

792 

1287 

2002 

3003 


















6 


1 

7 

28 

84 

210 

462 

924 

1716 

3003 

5005 
















7 


1 

8 

36 

120 

330 

792 

1716 

3432 

6435 














8 


1 

9 

45 

165 

495 

1287 

3003 

6435 












1 


10 

55 

220 

715 

2002 

5005 










10 


1 

11 

66 

286 

1001 

3003 








II 


12 

78 

364 

1365 






12 
13 
14 
15 


1 

13 

91 

455 


...... 

14 
105 



For n = 14, (n)n = 1; for n = 15, (ra)u = 15, and (n)i6 = 1. 



40 



MATHEMATICAL TABLES 



COMMON LOGARITHMS {special table) 



is 





1 


2 


3 


4 


6 


6 


7 


8 


9 




1.00 


0.0000 


0004 


0009 


0013 


0017 


0022 


0026 


0030 


0035 


0039 


4 


I.OI 


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 




1.08 


0334 


0338 


0342 


0346 


0350 


0354 


0358 


0362 


0366 


0370 




1.09 


0374 


0378 


0382 


0386 


0390 


0394 


0398 


0402 


0406 


0410 




1.10 


0.0414 


0418 


0422 


0426 


0430 


0434 


0438 


0441 


0445 


0449 




1.11 


0453 


0457 


0461 


0465 


0469 


0473 


0477 


0481 


0484 


0488 




1.12 


0492 


0496 


0500 


0504 


0508 


0512 


0515 


0519 


0523 


0527. 




1.13 


0531 


0535 


0538 


0542 


0546 


0550 


0554 


0558 


0561 


0565 




1.14 


0569 


0573 


0577 


0580 


0584 


0588 


0592 


0596 


0599 


0603 




1.15 


0607 


0611 


0615 


0618 


0622 


0626 


0630 


0633 


0637 


0641 




1.16 


0645 


0648 


0652 


0656 


0660 


0663 


0667 


0671 


0674 


0678 




1.17 


0682 


0686 


0689 


0693 


0697 


0700 


0704 


0708 


0711 


0715 




1.18 


0719 


0722 


0726 


0730 


0734 


0737 


0741 


0745 


0748 


0752 




1.19 


0755 


0759 


0763 


0766 


0770 


0774 


0777 


0781 


0785 


0788 




1.20 


0.0792 


0795 


0799 


0803 


0806 


0810 


0813 


0817 


0821 


0824 




1.21 


0828 


0831 


0835 


0839 


0842 


0846 


0849 


0853 


0856 


0860 




1.22 


0364 


0867 


0871 


0874 


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 


1095 


1099 


1103 




1.29 


1106 


1109 


1113 


1116 


1119 


1123 


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] in the body of the table (see p. 42). 



'lJi.1 nJSlVlATlVALi TABLES 



41 



COMMON LOGARITHMS (^special table, continued) 








18 
1^ 





1 


2 


3 


4 


5 


6 


7 


8 


9 




1.60 


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 


1915 


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 


7777 


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 




1.80 


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 
















»l 





1 


2 


3 


4 


5 


6 


T 


8 


9 


sfa 


1-° 






















<!-a 


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 




r.2 


0792 


0828 


0864 


0899 


0934 


0969 


1004 


1038 


1072 


1106 


pH 


1.3 


1139 


1173 


1206 


1239 


1271 


1303 


1335 


1367 


1399 


1430 


I 


1.4 


1461 


1492 


1523 


1553 


1584 


1614 


1644 


1673 


1703 


1732 


1.5 


1761 


1790 


1818 


1847 


1875 


1903 


1931 


1959 


1987 


2014 


1 


1.6 


2041 


2068 


2095 


2122 


2148 


2175 


2201 


2227 


2253 


2279 


Q. 


1.7 


2304 


2330 


2355 


2380 


2405 


2430 


2455 


2480 


2504 


2529 


1.8 


2553 


2577 


2601 


2625 


2648 


2672 


2695 


2718 


2742 


2765 


J 


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 


6865 


6875 


6884 


6893 


9 


4.9 


6902 


6911 


6920 


6928 


6937 


6946 


6955 


6964 


6972 


6981 


9 



log :r = 0.4971 
log e = 0.4343 



log •7r/2 = 0.1961 log ir^ 
log (0.4343) = 0.6378 - 1 



0.9943 



log v^ = 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 « [or — 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 (flh) = log a + log 5 log (o^) = N log a 



log \~r) = log a — log 5 



log C Va) = -^logo 



MATHEMATICAL TABLES 



43 



COMMON LOGARITHMS (.continued) 












» 





1 


a 


3 


4 


6 


6 


7 


8 


9 


ti 


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 


7225 


7235 


8 


5.3 


7243 


7251 


7259 


7257 


7275 


7284 


7292 


7300 


7308 


7316 


8 


5.4 


7324 


7332 


7340 


7348 


7356 


7364 


7372 


7380 


7388 


7395 


8 


5.5 


7404 


7412 


7419 


7427 


7435 


7443 


7451 


7459 


7456 


7474 


8 


5.6 


7482 


7490 


7497 


7505 


7513 


7520 


7528 


7535 


7543 


7551 


8 


5.7 


7559 


7566 


7574 


7582 


7589 


7597 


7604 


7612 


7619 


7627 




5.8 


7634 


7642 


7649 


7557 


7654 


7672 


7579 


7686 


7694 


7701 




5.9 


7709 


7716 


7723 


7731 


7738 


7745 


7752 


7750 


7757 


7774 




6.0 


0.7782 


7789 


7796 


7803 


7810 


7818 


7825 


7832 


7839 


7845 




6.1 


7853 


7860 


7868 


7875 


7882 


7889 


7896 


7903 


7910 


7917 




6.2 


7924 


7931 


7938 


7945 


7952 


7959 


7956 


7973 


7980 


7987 




6.3 


7993 


8000 


8007 


8014 


8021 


8028 


8035 


8041 


8048 


8055 




6.4 


8062 


8069 


8075 


8082 


8089 


8096 


8102 


8109 


8116 


8122 




6.5 


8129 


8135 


8142 


8149 


8155 


8152 


8169 


8176 


8182 


8189 




6.6 


8195 


8202 


8209 


8215 


8222 


8228 


8235 


8241, 


8248 


8254 




6.7 


8261 


8267 


8274 


8280 


8287 


8293 


8299 


8305 


8312 


8319 




6.8 


8325 


8331 


8338 


8344 


8351 


8357 


8363 


8370 


8376 


8382 




6.9 


8388 


8395 


8401 


8407 


8414 


8420 


8426 


8432 


8439 


8445 




7.0 


0.8451 


8457 


8463 


8470 


8476 


8482 


8488 


8494 


8500 


8506 




7.1 


8513 


8519 


8525 


8531 


8537 


8543 


8549 


8555 


8551 


8557 




7.2 


8573 


8579 


8585 


8591 


8597 


8503 


8509 


8515 


8521 


8527 




7.3 


8633 


8639 


8645 


8651 


8657 


8653 


8659 


8675 


8581 


8685 




7.4 


8692 


8698 


8704 


8710 


8715 


8722 


8727 


8733 


8739 


8745 




7.5 


8751 


8756 


8752 


8768 


8774 


8779 


8785 


8791 


8797 


8802 




7.6 


8808 


8814 


8820 


8825 


8831 


8837 


8842 


8848 


8854 


8859 




7.7 


8865 


8871 


8875 


8882 


8887 


8893 


8899 


8904 


8910 


8915 




7.8 


8921 


8927 


8932 


8938 


8943 


8949 


8954 


8950 


8965 


8971 




7.9 


8976 


8982 


8987 


8993 


8998 


9004 


9009 


9015 


9020 


9025 




8.0 


0.9031 


9035 


9042 


9047 


9053 


9058 


9063 


9059 


9074 


9079 




8.1 


9085 


9090 


9096 


9101 


9106 


9112 


9117 


9122 


9128 


9133 




8.2 


9138 


9143 


9149 


9154 


9159 


9155 


9170 


9175 


9180 


9185 




8.3 


9191 


9196 


9201 


9205 


9212 


9217 


9222 


9227 


9232 


9238 




8.4 


9243 


9248 


9253 


9258 


9263 


9259 


9274 


9279 


9284 


9289 




8.5 


9294 


9299 


9304 


9309 


9315 


9320 


9325 


9330 


9335 


9340 




8.6 


9345 


9350 


9355 


9360 


9365 


9370 


9375 


9380 


9385 


9390 




8.7 


9395 


9400 


9405 


9410 


9415 


9420 


9425 


9430 


9435 


9440 




8.8 


9445 


9450 


9455 


9450 


9455 


9459 


9474 


9479 


9484 


9489 




8.9 


9494 


9499 


9504 


9509 


9513 


9518 


9523 


9528 


9533 


9538 




9.0 


0.9542 


9547 


9552 


9557 


9562 


9556 


9571 


9575 


9581 


9586 




9.1 


9590 


9595 


9600 


9605 


9609 


9514 


9619 


9524 


9628 


9633 




9.2 


9638 


9543 


9647 


9652 


9557 


9661 


9556 


9671 


9675 


9680 




9.3 


9685 


9589 


9694 


9599 


9703 


9708 


9713 


9717 


9722 


9727 




9.4 


9731 


9736 


9741 


9745 


9750 


9754 


9759 


9763 


9768 


9773 




9J 


9777 


9782 


9786 


9791 


9795 


9800 


9805 


9809 


9814 


9818 




9.6 


9823 


9827 


9832 


9835 


9841 


9845 


9850 


9854 


9859 


9863 




9.7 


9868 


9872 


9877 


9881 


9886 


9890 


9894 


9899 


9903 


9908 




9.8 


9912 


9917 


9921 


9925 


9930 


9934 


9939 


9943 


9948 


9952 




9.9 


9956 


9961 


9955 


9959 


9974 


9978 


9983 


9987 


9991 


9996 





44 



MATHEMATICAL TABLES 



DEGREES AND 


MINUTES EXPRESSED 


IN RADIANS (See also 


p. 69) 


Degrees | 


Hundredths | 


Minutes 


1° 


.or 75 


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 


1.0996 


3 


2.1468 


3 


.0005 


3 


.0093 


3' 


.0009 


4 


.0698 


4 


1.1170 


4 


2.1642 


4 


.0007 


4 


.0094 


4' 


.0012 


8° 


.0873 


65° 


1.1345 


126° 


2.1817 


.05 


.0009 


.55 


.0096 


5' 


.0015 


6 


.1047 


6 


1.1519 


6 


2.1991 


6 


.0010 


6 


.0098 


6' 


.0017 


7 


.1222 


7 


1.1694 


7 


2.2166 


7 


.0012 


7 


.0099 


7' 


.0020 


8 


.1396 


8 


1.1868 


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°.6O 


.0105 


10' 


.0029 


1 


.1920 


1 


1.2392 


1 


2.2864 


1 


.0019 


1 


.0106 


ir 


.0032 


2 


.2094 


2 


1.2566 


2 


2.3038 


2 


.0021 


2 


.0108 


12' 


.0035 


3 


J269 


3 


1.2741 


3 


2.3213 


3 


.0023 


3 


.0110 


13' 


.0038 


4 


J+43 


4 


1.2915 


4 


2.3387 


4 


.0024 


4 


.0112 


14' 


.0041 


15° 


.2618 


76° 


1.3090 


136° 


2.3562 


.15 


.0026 


.66 


.0113 


15' 


0044 


6 


.2793 


6 


1.3265 


6 


2.3736 


6 


.0028 


6 


.0115 


16' 


.0047 


7 


J967 


7 


1.3439 


7 


2.391 1 


7 


.0030 


7 


.0117 


17' 


.0049 


8 


.3142 


8 


1.3614 


8 


2.4086 


8 


.0031 


8 


.0119 


18' 


.0052 


9 


J3I6 


9 


1.3788 


9 


2.4260 


9 


.0033 


9 


.0120 


19' 


.0055 


ao» 


.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 


J840 


2 


1.4312 


2 


2.4784 


2 


.0038 


2 


.0126 


22' 


.0064 


3 


.4014 


3 


1.4486 


3 


2.4958 


3 


.0040 


3 


.0127 


23' 


.0067 


4 


.4189 


4 


1.4661 


4 


2J133 


4 


.0042 


4 


.0129 


24' 


.0070 


85° 


.4363 


85° 


1.4835 


146° 


2.5307 


.26 


.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 


J061 


9 


1J533 


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 


I 


1.5882 




2.6354 




.0054 


1 


.0141 


31' 


.0090 


2 


.5585 


2 


1.6057 


2 


2.6529 


2 


.0056 


2 


.0143 


32' 


.0093 


3 


J760 


3 


1.6232 


3 


2.6704 


3 


.0058 


3 


.0145 


33' 


.0096 


4 


J934 


4 


1.6406 


4 


2.6878 


4 


.0059 


4 


.0147 


34' 


.0099 


85° 


.6109 


96° 


1.6581 


166° 


2.7053 


35 


.0061 


.86 


.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 


106° 


1.8326 


166° 


2.8798 


.46 


.0079 


.96 


.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°.60 


.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 


116° 


2.0071 


176° 


3.0543 










55' 


0160 


6 


.9774 


6 


2.0246 


6 


3.0718 










56' 


!0I63 


7 


.9948 


7 


2.0420 


7 


3.0892 










57' 


!oi66 


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 = 67° 17' 44".806 



MATHEMATICAL TABLES 
RADIANS EXPRESSED IN DEGREES 



45 



0.01 


0°.57 
IMS 


.64 
.66 


35°.67 
37°.24 


1.27 

8 


72°.77 
73°.34 


1.90 

1 


108°.86 
I09°.43 


2.63 
4 


144°.% 
I45°.53 


Interpolation 


2 


.0002 


0°.01 


3 


1».72 


6 


37°.82 


9 


73°.91 


2 


110° .01 


2.66 


146°. 10 


04 


.02 


4 


2°.29 


7 


38°.39 


1.30 


74°.48 


3 


1I0».58 


6 


146°.68 


06 


.03 


.05 


2°.86 


8 


38°.95 


1 


75°.a6 


4 


111°.15 


7 


I47°.25 


08 


.05 


6 


3°.44 


9 


39''.53 


2 


75°.63 


1.96 


Ill°.73 


8 


147».82 


.0010 


0°.06 


7 


4°.01 


.70 


40°.l 1 


3 


76°.20 


6 


I12°.30 


9 


I48°.40 


12 


.07 


8 


4°38 




40°.68 


4 


76°.78 


7 


1I2°.87 


2.60 


148°.97 


14 


•22 


9 


5°.15 


2 


4r.23 


1.35 


77°.35 


8 


113°.45 


I 


I49°.54 


16 


.09 


.10 


5°.73 


3 


41°.83 


6 


77°.92 


9 


11 4° .02 


2 


150°.ll 


18 


.10 


1 


e^so 


4 


42°.40 


7 


78°.50 


2.00 


1I4°.59 


3 


15D°.69 


.0020 


OMl 


2 


6°.88 


.76 


42°.97 


8 


79°.07 


1 


115°.16 


4 


151°.26 


22 


.13 


3 


7°.45 


6 


43° .54 


9 


79°.64 


2 


115°.74 


2.65 


151°.83 


24 


.14 


4 


8'>.02 


7 


44°.12 


1.40 


80°.21 


3 


116°.31 


6 


152°.4I 


26 


.15 


.16 


8''.59 


8 


44°.69 


1 


80°.79 


4 


116°.88 


7 


I52°.98 


28 


.16 


6 


9°.17 


9 


45°.26 


2 


81°.36 


2.06 


117°.46 


8 


153°.55 


.0030 


0°.17 


7 


9°.74 


.80 


45°.84 


3 


8r.93 


6 


118°.03 


9 


154°.13 


32 


.18 


8 


ICBI 


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°.I8 


1 


155°.27 


36 


.21 


.20 


ir.46 


3 


47°.56 


6 


83°.65 


9 


119°.75 


2 


155°.84 


38 


.22 


I 


W-Oi 


4 


48°.13 


7 


84°.22 


2.10 


120°.32 


3 


156°.42 


.0040 


0°.23 


2 


12°.61 


.86 


48°.7G 


8 


84°.80 


1 


120°.89 


4 


156° .99 


42 


J4 


3 


13°.18 


6 


49°.27 


9 


85°.37 


2 


12l°.47 


2.76 


157°.56 


44 


as 


4 


130.75 


7 


49°.85 


1.50 


85».94 


3 


122°.04 


6 


158M4 


46 


.26 


.26 


I4°.32 


8 


50°.42 




86°.52 


4 


122°.61 


7 


158°.71 


48 


.28 


6 


140.90 


9 


5Q°.99 


2 


87°.a9 


2.15 


I23°.19 


8 


159°.28 


.0060 


0°.29 


7 


15°.47 


.90 


5r.57 


3 


87°.66 


6 


123°.76 


9 


159°.86 


52 


JO 


8 


16°.04 


1 


52°.14 


4 


88°.24 


7 


124°.33 


S.S0 


160°.43 


54 


3\ 


9 


16°.62 


2 


52°.7I 


1.66 


88°.81 


8 


124°.90 


1 


161°.00 


56 


21 


.30 


17M9 


3 


53°.29 


6 


89°.38 


9 


125°.48 


2 


161°.57 


58 


33 


1 


I7°.76 


4 


53°.86 


7 


89°.95 


2.20 


126°.05 


3 


162°.15 


.0060 


Q°.34 


2 


18°.33 


.95 


54°.43 


8 


90°.53 




126°.62 


4 


162 .72 


62 


.36 


3 


18°.91 


6 


55°.00 


9 


91°.10 




127°.20 


2.86 


163°.29 


64 


.37 


4 


19°.48 


7 


55°.58 


1.60 


91°.67 


3 


127°.77 


6 


I63°.87 


66 


.38 


.36 


2D°.05 


8 


56°. 15 


1 


92°.25 


4 


128°.34 


7 


164°.44 


68 


J9 


6 


20<'.63 


9 


56°.72 


2 


92°.82 


2.26 


128°.92 


8 


I65°.01 


.0070 


0°.40 


7 


21°.20 


1.00 


57°.3Q 


3 


93°.39 


6 


129°.49 


9 


I65°.58 


72 


.41 


8 


21°.77 


1 


57''.87 


4 


93°.97 


7 


130°.06 


2.90 


166°.16 


74 


.42 


9 


22''.35 


2 


58''.44 


1.66 


94°.54 


8 


130°.63 


1 


I66°.73 


76 


.44 


.40 


22°.92 


3 


59°.01 


6 


95°. 11 


9 


13I°.21 


2 


167°.30 


78 


.45 


1 


23°.49 


4 


59°.59 


7 


95°.68 


2.30 


13I°.78 


3 


I67°.88 


.0080 


0».46 


2 


24''.06 


1.06 


60°. 16 


8 


96°.26 




132°.35 


4 


I68°.45 


82 


.47 


3 


24°.64 


6 


60°.73 


9 


96°.83 


2 


132°.93 


2.95 


I69°.02 


84 


.48 


4 


25°.21 


7 


61°.31 


1.70 


97°.40 


3 


I33°.50 


6 


169°.60 


86 


.49 


.46 


25''.78 


8 


61°.88 


1 


97°.98 


4 


134°.07 


7 


17Q°.I7 


88 


JO 


6 


26°.36 


9 


62°.45 


2 


98°.55 


2.36 


134°.65 


8 


I70°.74 


.0090 


0°.52 


7 


26''.93 


1.10 


63°.03 


3 


99°.12 


6 


135°.22 


9 


171°.31 


92 


33 


8 


27''.50 


1 


63°.60 


4 


99°.69 


7 


135°.79 


3.00 


171°.89 


94 


J4 


9 


28°.07 


2 


64°.17 


1.75 


100°.27 


8 


136°J6 




172°.46 


96 


J5 


.60 


28°.65 
29°.22 


3 
4 


64°.74 
65°.32 


6 
7 


100°.84 
I01°.4I 


9 
2.40 


136°.94 
137°.51 


2 
3 


I73».03 
I73°.61 


98 


J6 


1 






2 


29°.79 


1.16 


65°.89 


8 


101°.99 


1 


138°.08 


4 


174°.18 


Multiples 


of » 


3 


30".37 
300.94 


6 
7 


66°.46 
67°.04 


9 
1.80 


102°.56 
103°.13 


2 
3 


138°.66 
139°.23 


3.05 
6 


I74°.75 
I75°.33 






4 


1 


3.1416 


180° 


.66 


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 


I76°.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 


33023 


1 


69°.33 


4 


105°.42 


7 


141°.52 


3.10 


177».62 


5 


15.7080 


900° 


9 


33°.80 


2 


69°.90 


1.86 


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° 




34°.95 


4 


71°.05 


7 


107°. 14 


2.60 


143°J4 


3 


179°.34 


8 


25.1327 


1440° 


2 


35'>.52 


1.25 


71°.62 


8 


107°.72 


1 


143°.8I 


4 


I79°.91 


9 


28.2743 


1620° 


3 


36°. 10 


6 


72°.I9 


9 


108°.29 


2 


I44°39 


3.15 


I8Q°.48 


10 


31.4159 


1800° 



46 



MATHEMATICAL TABLES 



NATURAL SINES AND COSINES 
Natural Sines at intervals of OP.l, or 6'. 



(For 10' intervals, see pp. 52-56) 



i 


".0 


°.l 


°.a 


».S 


".4 


".6 


°.6 


°.7 


".8 


».9 






ll 


a 


= ^0') 


(60 


(12') 


(18') 


(24') 


(30') 


(36') 


(42') 


(48') 


(54') 




























0.0000 


90° 




a° 


0.0000 


0017 


0035 


0052 


0070 


0087 


0105 


0122 


0140 


0157 


0175 


89 




1 


0175 


0192 


0209 


0227 


0244 


0262 


0279 


0297 


0314 


0332 


0349 


88 




2 


0349 


0366 


0384 


0401 


0419 


0436 


0454 


0471 


0488 


0506 


0523 


87 




3 


0523 


0541 


0558 


0576 


0593 


0610 


0628 


0645 


0663 


0680 


0698 


86 




4 


0698 


0715 


0732 


0750 


0767 


0785 


0802 


0819 


0837 


0854 


0.0872 


85 




5 


0.0872 


0889 


0906 


0924 


0941 


0958 


0976 


0993 


1011 


1028 


1045 


84 




6 


1045 


1063 


1080 


1097 


1115 


1132 


1149 


1167 


1184 


1201 


1219 


83 




7 


1219 


1236 


1253 


1271 


1288 


1305 


1323 


1340 


1357 


1374 


1392 


82 




8 


1392 


1409 


1426 


1444 


1461 


1478 


1495 


1513 


1530 


1547 


1564 


81 




9 


1564 


1582 


1599 


1616 


1633 


1650 


1668 


1685 


1702 


1719 


0.1736 


80° 




10° 


0.1736 


1754 


1771 


1788 


1805 


1822 


1840 


1857 


1874 


1891 


1908 


79 




II 


1908 


1925 


1942 


1959 


1977 


1994 


2011 


2028 


2045 


2052 


2079 


78 




12 


2079 


2096 


2113 


2130 


2147 


2164 


2181 


2198 


2215 


2233 


2250 


77 




13 


2250 


2267 


2284 


2300 


2317 


2334 


2351 


2368 


2385 


2402 


2419 


75 




M 


2419 


2436 


2453 


2470 


2487 


2504 


2521 


2538 


2554 


2571 


0.2588 


75 




IS 


0.2588 


2605 


2622 


2639 


2656 


2672 


2689 


2706 


2723 


2740 


2756 


74 




16 


2756 


2773 


2790 


2807 


2823 


2840 


2857 


2874 


2890 


2907 


2924 


73 




17 


2924 


2940 


2957 


2974 


2990 


3007 


3024 


3040 


3057 


3074 


3090 


72 




18 


3090 


•3107 


3123 


3140 


3156 


3173 


3190 


3206 


3223 


3239 


3256 


71 




19 


3256 


3272 


3289 


3305 


3322 


3338 


3355 


3371 


3387 


3404 


0.3420 


70° 




ao° 


0.3420 


3437 


3453 


3469 


3486 


3502 


3518 


3535 


3551 


3567 


3584 


59 




21 


3584 


3600 


3616 


3633 


3649 


3665 


3681 


3697 


3714 


3730 


3746 


58 




22 


3746 


3762 


3778 


3795 


3811 


3827 


3843 


3859 


3875 


3891 


3907 


57 




23 


3907 


3923 


3939 


3955 


3971 


3987 


4003 


4019 


4035 


4051 


4067 


56 




24 


4067 


4083 


4099 


4115 


4131 


4147 


4163 


4179 


4195 


4210 


0.4226 


55 




25 


0.4226 


4242 


4258 


4274 


4289 


4305 


4321 


4337 


4352 


4368 


4384 


54 




26 


4384 


4399 


4415 


4431 


4446 


4462 


4478 


4493 


4509 


4524 


4540 


53 


16 


27 


4540 


4555 


4571 


4586 


4602 


4617 


4633 


4648 


4664 


4679 


4695 


52 




28 


4695 


4710 


4726 


4741 


4756 


4772 


4787 


4802 


4818 


4833 


4848 


51 




29 


4848 


4863 


4879 


4894 


4909 


4924 


4939 


4955 


4970 


4985 


0.5000 


60° 




S0° 


0.5000 


5015 


5030 


5045 


5060 


5075 


5090 


5105 


5120 


5i35 


5150 


59 




31 


5150 


5165 


5180 


5195 


5210 


5225 


5240 


5255. 


5270 


5284 


5299 


58 




32 


5299 


5314 


5329 


5344 


5358 


5373 


5388 


5402 


5417 


5432 


5446 


57 




33 


5446 


5451 


5476 


5490 


5505 


5519 


5534 


5548 


5563 


5577 


5592 


55 




34 


5592 


5606 


5621 


5635 


5650 


5664 


5678 


5693 


5707 


5721 


0.5736 


55 




35 


0.5736 


5750 


5764 


5779 


5793 


5807 


5821 


5835 


5850 


5864 


5878 


54 




36 


5878 


5892 


5906 


5920 


5934 


5948 


5962 


5976 


5990 


6004 


6018 


53 




37 


6018 


6032 


6046 


6060 


6074 


6088 


6101 


6115 


6129 


6143 


6157 


52 




38 


6157 


6170 


6184 


6198 


6211 


6225 


6239 


6252 


6266 


6280 


6293 


51 




39 


6293 


6307 


6320 


6334 


6347 


6361 


6374 


6388 


6401 


6414 


0.6428 


60° 




t6* 


0.6428 


6441 


6455 


6468 


6481 


6494 


6508 


6521 


6534 


6547 


6561 


49 




41 


6561 


6574 


6587 


6600 


6613 


6626 


6639 


6652 


6665 


6678 


6691 


48 




42 


6691 


6704 


6717 


6730 


6743 


6756 


6769 


6782 


6794 


6807 


6820 


47 




43 


6820 


6833 


6845 


6858 


6871 


6884 


6896 


6909 


6921 


6934 


6947 


46 




44 


6947 


6959 


6972 


6984 


6997 


7009 


7022 


7034 


7046 


7059 


0.7071 


4B° 




45° 


0.7071 
































°.9 


°.8 


°.7 


°.6 


°.B 


°.4 


°.S 


°.2 


°.l 


°.o 


s 










=(54') (48') 


(42') 


(36') 


(30') 


(24') 


(18') 


(12') 


(6') 


(0-) 


a 





(For graphs, see p. 174.) 



Natural Cosinea 



MATHEMATICAL TABLES 



47 



NATURAL SINES AND COSINES (continued) 
Natural Sines at intervals of 0°.l, or 6'. (For lO' intervals, see pp. 52-56) 



gl 


».o 


°.l 


°.2 


°.3 


°.4 


°.5 


°.6 


°.7 


°.8 


".9 






U 





=(0 


(6') 


(12') 


(18') 


(24') 


(30') 


(35') 


(42') 


(48') 


(54') 




























0.7071 


45° 




46° 


0.7071 


7083 


7096 


7108 


7120 


7133 


7145 


7157 


7169 


7181 


7193 


44 


12 


46 


7193 


7206 


7218 


7230 


7242 


7254 


7256 


7278 


7290 


7302 


7314 


43 


12 


47 


7314 


7325 


7337 


7349 


7351 


7373 


7385 


7395 


7408 


7420 


7431 


42 


12 


48 


7431 


7443 


7455 


7466 


7478 


7490 


7501 


7513 


7524 


7535 


7547 


41 


12 


49 


7547 


7559 


7570 


7581 


7593 


7604 


7515 


7627 


7638 


7649 


0.7550 


40° 


11 


110° 


0.7660 


7672 


7683 


7694 


7705 


7715 


7727 


7738 


7749 


7760 


7771 


39 


11 


51 


7771 


7782 


7793 


7804 


7815 


7826 


7837 


7848 


7859 


7859 


7880 


38 


11 


52 


7880 


7891 


7902 


7912 


7923 


7934 


7944 


7955 


7955 


7975 


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 


8151 


8171 


8181 


0.8192 


35 


10 


55 


0.8192 


8202 


8211 


8221 


8231 


8241 


8251 


8251 


8271 


8281 


8290 


34 


10 


56 


8290 


8300 


8310 


8320 


8329 


8339 


8348 


8358 


8368 


8377 


8387 


33 


10 


57 


8387 


8396 


8406 


8415 


8425 


8434 


8443 


8453 


8452 


8471 


8480 


32 


9 


58 


8480 


8490 


8499 


8508 


8517 


8526 


8536 


8545 


8554 


8563 


8572 


31 


9 


59 


8572 


8581 


8590 


8599 


8607 


8516 


8525 


8634 


8643 


8652 


0.8550 


80° 


9 


60° 


0.8660 


8669 


8678 


8686 


8695 


8704 


8712 


8721 


8729 


8738 


8746 


29 


9 


61 


8746 


8755 


8763 


8771 


8780 


8788 


8795 


8805 


8813 


8821 


8829 


28 


8 


62 


8829 


8838 


8845 


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 


9135 


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 


9272 


9278 


9285 


9291 


9298 


9304 


9311 


9317 


9323 


9330 


9335 


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 


9455 


9461 


9466 


9472 


9478 


9483 


9489 


9494 


9500 


9505 


9511 


18 


6 


72 


9511 


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 


%32 


9636 


9541 


9546 


9650 


9655 


0.9659 


15 


5 


75 


0.9659 


9664 


9668 


9673 


9677 


9681 


9686 


9590 


9694 


9699 


9703 


14 


4 


76 


9703 


9707 


9711 


9715 


9720 


9724 


9728 


9732 


9736 


9740 


9744 


13 


4 


77 


9744 


9748 


9751 


9755 


9759 


9753 


9767 


9770 


9774 


9778 


9781 


12 


4 


78 


9781 


9785 


9789 


9792 


9795 


9799 


9803 


9806 


9810 


9813 


9816 


11 


3 


79 


9816 


9820 


9823 


9826 


9829 


9833 


9835 


9839 


9842 


9845 


0.9848 


10° 


3 


80° 


0.9848 


9851 


9854 


9857 


9850 


9863 


9866 


9869 


9871 


9874 


9877 


9 


3 


81 


9877 


9880 


9882 


9885 


9888 


9890 


9893 


9895 


9898 


9900 


9903 


8 


3 


82 


9903 


9905 


9907 


9910 


9912 


9914 


9917 


9919 


9921 


9923 


9925 


7 


2 


83 


9925 


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 


9965 


9968 


9969 


9971 


9972 


9973 


9974 


9976 


4 


1 


86 


9976 


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 


9994 


9995 


9995 


9995 


9996 


9997 


9997 


9997 


9998 


9998 


0.9998 


1 





89 


0.9998 


9999 


9999 


9999 


9999 


GOOD 


0000 


0000 


0000 


0000 


1.0000 


0° 





90° 


1.0000 
































°.9 


"".8 


°.7 


°.6 


-.5 


°.4 


°.3 


°.a 


°.l 


•.0 


S 










={54') 


(48') 


(42') 


(36') 


(30') 


(24') 


(18') 


(12') 


(6') 


(C) 


a 





Natural Cosinea 



48 



MATHEMATICAL TABLES 



NATURAL TANGENTS AND COTANGKNTS 

Natural Tangents at intervals of 0°.l, or 6'. (For lO'intervalB.Beepp. 52-56) 



i 


°.o 


°.l 


°.2 


°.3 


".4 


°.6 


°.6 


°.7 


°.S 


°.9 






^« 


a 


=(0') 


(6') 


(12') 


(18') 


(24') 


(30') 


(36') 


(42') 


(48') (54') 






^1 
























0.0000 


90° 




0° 


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 


16 


4 


0699 


0717 


0734 


0752 


0769 


0787 


0805 


0822 


0840 


0857 


0.0875 


85 


18 


5 


00875 


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 


11 


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 


M 


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 


S0° 


0.5774 


5797 


5820 


5844 


5867 


5890 


5914 


5938 


5961 


5985 


6009 


59 


24 


31 


6009 


6032 


6056 


6080 


6104 


6128 


6152 


617ft 


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 
































°.9 


°.8 


"■.7 


°.6 


°.e 


».4 


».3 


°.2 


°.l 


o.O 


s 










=(54') 


(48') 


(42') 


(36') 


(30-) 


(24') 


(18') 


(12') 


(6') 


m 


n 





(For graphs, see p. 174.) 



Natural Cotangents 



MATHEMATICAL TABLES 



49 



NATURAL TANGENTS AND COTANGENTS {continued) 

Natural Tangents at intervals of 0°.l, or 6'. (For 10' intervals, see pp. 52-56) 



» 


°.o 


°.l 


°.2 


°.S 


°.4 


°.6 


°.6 °.7 


°.8 ».9 








Avg. 


Q 


=(0') 


(60 


(12') 


(18') 


(24') 


(30') 


(36') (42') 


(48') (54') 








di£f. 




















.0000 


46° 




«5° 


1.0000 


0035 


0070 


0105 


0141 


0176 


0212 0247 


0283 0319 


0355 


44 


35 


46 


0355 


0392 


0428 


0464 


0501 


0538 


0575 0612 


0649 0686 


0724 


43 


37 


47 


0724 


0761 


0799 


0837 


0875 


0913 


0951 U99U 


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 1875I.19I8 


40° 


41 


60° 


1.1918 


I960 


2002 


2045 


2088 


2131 


2174 2218 


2261 2305 


2349 


39 


43 


31 


2349 


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 


3270 


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 


4496 


4550 


4605 4659 


4715 4770 


4825 


34 


55 


56 


4826 


4882 


4938 


4994 


5051 


5108 


5165 5224 


5282 5340 


5399 


33 


57 


57 


5399 


5458 


5517 


5577 


5637 


5697 


5757 5818 


5880 5941 


6003 


32 


60 


58 


6003 


6066 


6128 


6191 


6255 


6319 


6383 6447 


6512 6577 


5643 


31 


64 


59 


1.6643 


6709 


6775 


6842 


6909 


6977 


7045 7113 


7182 72511.7321 


30° 


57 


60° 


1.732 


1.739 


1.746 


1.753 


1.760 


1.767 


1.775 1.782 


1.789 1.797 


1.804 


29 


7 


61 


1.804 


1.811 


1.819 


1.827 


1.834 


1.842 


1.849 1.857 


1.865 1.873 


1.881 


28 


8 


62 


1.881 


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 2 023 


2.032 2.041 


2.050 


26 


9 


64 


2.050 


2.059 


2.069 


2.078 


2.087 


2.097 


2.105 2.116 


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 2.235 


2.246 


24 


10 


66 


2.246 


2.257 


2.267 


2.278 


2.289 


2.300 


2.311 2.322 


2333 2344 


2.356 


23 


11 


67 


2.356 


2.367 


2.379 


2.391 


2.402 


2.414 


2.425 2.438 


2.450 2.463 


2.475 


22 


12 


68 


2.475 


2.488 


2.500 


2.513 


2.526 


2.539 


2.552 2.555 


2.578 2.592 


2.505 


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 


2.752 


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 


3.005 3.024 


3.042 3.060 


3.078 


18 


17 


72 


3.078 


3.096 


3.115 


3.133 


3.152 


3.172 


3.191 3>2I1 


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 


3J11 


3.534 


3.558 


3J82 


3.606 


3.530 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 


4.011 


4.041 


4.071 


4.102 


4.134 


4.165 


4.198 4.230 


4.254 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 4.565 


4.705 


12 


37 


78 


4.705 


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 5.514 


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 


6314 


9 




81 


6.314 


6.386 6.460 6.535 


6.612 


6.691 


6.772 6 855 


6.940 7.025 


7.115 


8 




82 


7.115 


7.207 


7.300 


7.396 


7.495 


7.595 


7.700 7.806 


7.916 8.028 


8.144 


7 




83 


8.144 


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 


1039 


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 


14.30 


14.67 


15.06 


15.46 


15.89 


16.35 


15.83 1734 


17.89 18.45 


19.08 


3 




87 


19.08 


19.74 


20.45 


21.20 


22.02 


22.90 


23.86 24.90 


26.03 27.27 


28.54 


2 




88 


28.64 


30.14 


31.82 


33.69 


35.80 


38.19 


40.92 44.07 


47.74 52.08 


57 J9 


1 




89 
90° 


57.29 


63.66 


71.62 


81.85 


95.49 


114.6 


143.2 191.0 


286.5 573.0 


00 


0° 










°.9 


°.8 


°.7 


°.6 


°.6 


°.4 °.8 


°.2 °.l 


°.o 


M 










=(54') 


(48') 


(42') 


(36') 


(30-) 


(24') (18') 


(12') (6') 


(0') 








Natural Cotangents 



60 



MATHEMATICAL TABLES 



NATURAL SECANTS AND COSECANTS 

Natural Secants at intervals of 0°.l, or 6'. (For 10' intervals, see pp. 52-56) 



s 


°.o 


°.l. 


°.2 


°.3 


°.4 


».6 


°.6 


°.7 


°.8 


°.9 






Avg. 


Q 


=co'j 


(6') 


(12') 


(18') 


(24') 


(30') 


(36') 


(42') 


(48') 


(54') 






difl. 
























1.0000 


90° 




0° 


1.0000 0000 


0000 


0000 


0000 


uouo 


0001 


0001 


0001 


0001 


0002 


89 







0002 0002 


0002 


0003 


0003 


0003 


0004 


0004 


0005 


0006 


0006 


88 





2 


0006 0007 


0007 


0008 


0009 


0010 


0010 


0011 


0012 


0013 


0014 


87 




3 


0014 0015 


0016 


0017 


0018 


0019 


0020 


0021 


0022 


0023 


0024 


86 




4 


0024 0026 


0027 


0028 


0030 


0031 


0032 


0034 


0035 


0037 


1.0038 


85 




5 


1.0038 0040 


0041 


0043 


0045 


0046 


0048 


0050 


0031 


0053 


0055 


84 




6 


0055 0057 


0059 


0061 


0063 


0065 


0067 


0069 


0071 


0073 


0075 


83 




7 


0075 0077 


0079 


0082 


0084 


0086 


0089 


0091 


0093 


0096 


0098 


82 




8 


0098 0101 


0103 


0106 


0108 


0111 


0114 


0116 


0119 


0122 


0125 


81 




9 


0123 


0127 


0130 


0133 


0136 


0139 


0142 


0145 


0148 


0151 


1.0154 


80° 




10° 


1.0154 0157 


0161 


0164 


0167 


0170 


0174 


0177 


0180 


0184 


0187 


79 




II 


018! 


0191 


0194 


0198 


0201 


0205 


0209 


0212 


0216 


0220 


0223 


78 




12 


0223 


0227 


0231 


0235 


0239 


0243 


0247 


0251 


0255 


0259 


0263 


77 




13 


0263 


0267 


0271 


0276 


0280 


0284 


0288 


0293 


0297 


0302 


0306 


75 




M 


030( 


0311 


0315 


0320 


0324 


0329 


0334 


0338 


0343 


0348 


1.0353 


75 




15 


1.0353 


0358 


0363 


0367 


0372 


0377 


0382 


0388 


0393 


0398 


0403 


74 




16 


0403 0408 


0413 


0419 


0424 


0429 


0435 


0440 


0446 


0451 


0457 


73 




17 


045/ 


0463 


0468 


0474 


0480 


0485 


0491 


0497 


0503 


0509 


0515 


72 




18 


0515 


0521 


0527 


0533 


0539 


0545 


0551 


0557 


0564 


0570 


0576 


71 




19 


057e 


0583 


0589 


0595 


0602 


0608 


0615 


0622 


0628 


0635 


1.0642 


70° 




ao» 


1.0643 


0649 


0655 


0662 


0669 


0676 


0683 


0690 


0697 


0704 


0711 


69 




21 


0711 


0719 


0726 


0733 


0740 


0748 


0755 


0763 


0770 


0778 


0785 


68 




22 


0785 


0793 


0801 


0808 


0816 


0824 


0832 


0840 


0848 


0856 


0864 


67 




23 


0864 


0872 


0880 


0888 


0896 


0904 


0913 


0921 


0929 


0938 


0946 


66 




24 


0946 


0955 


0963 


0972 


0981 


0989 


0998 


1007 


1016 


1025 


1.1034 


65 




25 


1.1034 1043 


1052 


1061 


1070 


1079 


1089 


1098 


1107 


1117 


1126 


54 




26 


1126 


1136 


1145 


1155 


1164 


1174 


1184 


1194 


1203 


1213 


1223 


63 




27 


1223 


1233 


1243 


1253 


1264 


1274 


1284 


1294 


1305 


1315 


1325 


52 




28 


1326 


1336 


1347 


1357 


1368 


1379 


1390 


1401 


1412 


1423 


1434 


51 




29 


1434 


1445 


1456 


1467 


1478 


1490 


1501 


1512 


1524 


1535 


1.1547 


60° 




30° 


1.1543 


1559 


1570 


1582 


1594 


1606 


1618 


1630 


1642 


1654 


1666 


59 




31 


1666 


1679 


1691 


1703 


1716 


1728 


1741 


1753 


1766 


1779 


1792 


58 




32 


1792 


1805 


1818 


1831 


1844 


1857 


1870 


1883 


1897 


1910 


1924 


57 




33 


1924 


1937 


1951 


1964 


1978 


1992 


2006 


2020 


2034 


2048 


2062 


56 




34 


2062 


2076 


2091 


2105 


2120 


2134 


2149 


2163 


2178 


2193 


1.2208 


55 




35 


1.220E 


2223 


2238 


2253 


2268 


2283 


2299 


2314 


2329 


2345 


2361 


54 




36 


2361 


2376 


2392 


2408 


2424 


2440 


2456 


2472 


2489 


2505 


2521 


53 




37 


2521 


2538 


2554 


2571 


2588 


2605 


2622 


2639 


2656 


2673 


2690 


52 




38 


269C 


2708 


2725 


2742 


2760 


2778 


2795 


2813 


2831 


2849 


2868 


51 




39 


2868 


2886 


2904 


2923 


2941 


2960 


2978 


2997 


3016 


3035 


1.3054 


60° 




40° 


13054 


■ 3073 


3093 


3112 


3131 


3151 


3171 


3190 


3210 


3230 


3250 


49 


20 


41 


3250 


3270 


3291 


3311 


3331 


3352 


3373 


3393 


3414 


3435 


3456 


48 


21 


42 


3456 


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 


46° 


24 


46° 


1.4142 






























».9 


°.8 


°.7 


°.6 


°.6 


°.4 


°.S 


°.2. 


°1 


°.o 


u 








=(54') 


(480 


(42') (36') 


(30') 


(24') 


(18') 


(12') 


(6') 


(C) 


a 





(Fpr graphs, see p. 174.) 



Natural Cosecanta 



MATHEMATICAL TABLES 51 

NATURAL SECANTS AND COSECANTS {continued) 

Natural Secants at intervals of 0°.l, or 6'. (For 10' intervals, see pp. 52-56) 



s 


°.0 °.l 


°.2 


°.S 


°.4 


».5 


°.6 °.7 


°.8 


°.9 




Ave, 


a 


-(0') 


(60 


(120 


(180 


(240 


(300 


(360 (420 


(480 (540 




diff. 






















1.4142 


45° 




46° 


I.4I42 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 


SC- 


1.5557 5590 


5622 


5655 


5688 


5721 


5755 5788 


5822 


5856 


5890 


39 


33 


SI 


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 


836 


8410 


8460 


8510 


8561 


8612 


8663 8714 


8765 


8818 


8871 


32 


51 


58 


887 


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° 


2.000 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 


2.130 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 


2J73 


2.281 


26 


8 


64 


2J81 2J89 


2.298 


2306 


2314 


2323 


2331 2.340 


2349 


2357 


2366 


25 


8 


65 


2.366 2.375 


2384 


2393 


2.402 


2.411 


2.421 2.430 


2.439 


2.449 


2.459 


24 


9 


66 


2.459 2.468 


2.478 


2.488 


2.498 


2308 


2318 2.528 


2.538 


2349 


2.559 


23 


10 


67 


2359 2.570 


2.581 


2.591 


2.602 


2.613 


2 624 2.635 


2.647 


2.658 


2.669 


22 


11 


68 


2.669 2.681 


2.693 


2.705 


2.716 


2.729 


2.741 2.753 


2.765 


2.778 


2.790 


21 


12 


69 


2.790 2.803 


2.816 


2.829 


2.842 


2.855 


2.869 2.882 


2.896 


2.910 


2.924 


20° 


13 


70° 


2.924 2.938 


2.952 


2.967 


2.981 


2.996 


3.011 3.026 


3.041 


3.056 


3.072 


19 


15 


71 


3.072 3.087 


3.103 


3.119 


3.135 


3.152 


3.168 3.185 


3.202 


3.219 


3J36 


18 


16 


72 


3.236 3J54 


3.271 


3.289 


3307 


3326 


3344 3363 


3382 


3.401 


3.420 


17 


18 


73 


3.420 3.440 


3.460 


3.480 


3.500 


3321 


3342 3363 


3384 


3.606 


3.628 


16 


21 


74 


3.628 3.650 


3.673 


3.695 


3.719 


3.742 


3.766 3.790 


3.814 


3.839 


3.864 


15 


24 


75 


3.864 3.889 


3.915 


3.941 


3.967 


3.994 


4.021 4.049 


4.077 


4.105 


4.134 


14 


27 


76 


4.134 4.163 


4.192 


4.222 


4.253 


4.284 


4315 4.347 


4379 


4^412 


4.445 


13 


31 


77 


4.445 4.479 


4.514 


4.549 


4384 


4.620 


4.657 4.694 


4.732 


4.771 


4.810 


12 


36 


78 


4.810 4.850 


4.890 


4.931 


4.973 


5.016 


5.059 5.103 


5.148 


5.194 


5.241 


11 


43 


79 


5.241 5.288 


5337 


5386 


5.436 


5.487 


5340 5393 


5.647 


5.702 


5.759 


10° 


52 


80° 


5.759 5.816 5.875 


5.935 


5.996 


6.059 


6.123 6.188 


6.255 


6323 


6392 


9 




81 


6.392 6.464 


6337 


6.611 


6.687 


6.765 


6.845 6.927 


7.011 


7.097 


7.185 


8 




82 


7.185 7276 


7368 


7.463 


7.561 


7.661 


7.764 7.870 


7.979 8.091 


8.206 


7 




83 


8.206 8324 


8.446 


8.571 


8.700 


8.834 


8.971 9.113 


9.259 


9.411 


9367 


6 




84 


9.567 9.728 


9.895 


10.07 


10J5 


10.43 


10.63 10.83 


11.03 


1U5 


11.47 


5 




85 


11.47 11.71 


11.95 


12.20 


12.47 


12.75 


13.03 13.34 


13.65 


13.99 


1434 


4 




86 


1434 14.70 


15.09 


1530 


15.93 


16.38 


16.86 1737 


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 


28.65 30.16 


31.84 


33.71 


35.81 


38.20 


40.93 44.08 


47.75 


52.09 


5730 


I 




89 
90° 


5730 63.66 
oo 


71.62 


81.85 


95.49 


114.6 


U32 191.0 


2863 573.0 


09 


0° 








°.9 


°.8 


°.7 


°.6 


°.6 


°.4 °.S 


°.2 


°.l 


°.o 










=(540 


(480 


(420 


(360 


(300 


(240 (180 


(120 


(60 


(00 








Natural Cosecants 



62 



MATHEMATICAL TABLES 



TRIGONOMETRIC FUNCTIONS (at intervals of 10') 

Annex —10 in columns marked *. (For 0.°1 intervals, see pp. 46-51) 



De- 
grees 


Ra- 
dians 


Sines 


Cosines 


Tangents 


Cotangents 










Nat. 


Log.* 


Nat. Log.* 


Nat. 


Log.* 


Nat. Log. 






COO' 


0.0000 


.0000 


CQ 


1.0000 0.0000 


.0000 


CO 


CO CO 


1.5708 


90»00' 


10 


0029 


.0029 


7.4637 


1.0000 .0000 


.0029 


7.4637 


343.77 2.5363 


1.5679 


50 


20 


0058 


.0058 


.7648 


1.0000 .0000 


.0058 


.7648 


171.89 .2352 


1.5650 


40 


30 


0087 


.0087 


.9408 


1.0000 .0000 


.0087 


.9409 


114.59 .0591 


1.5621 


30 


40 


0.0116 


.0116 


8.0658 


0.9999 .0000 


.0115 


8.0658 


85.940 1.9342 


1.5592 


20 


50 


0.0145 


.0145 


.1627 


.9999 .0000 


.0145 


.1627 


68.750 .8373 


1.5553 


10 


1° 00' 


0.0175 


.0175 


8.2419 


.9998 9.9999 


.0175 


8.2419 


57.290 1.7581 


1.5533 


890 00' 


10 


0.0204 


.0204 


.3088 


.9998 .9999 


.0204 


.3089 


49.104 .6911 


1.5504 


50 


20 


0.0233 


.0233 


.3668 


.9997 .9999 


.0233 


.3569 


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»00' 


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 


1J301 


40 


30 


0.0436 


.0436 


.6397 


.9990 .9996 


.0437 


.6401 


22.904 .3599 


1.5272 


30 


40 


0.0465 


.0465 


.6677 


.9989 .9995 


.0456 


.6682 


21.470 .3318 


1.5243 


20 


50 


0.0495 


.0494 


.6940 


.9988 .9995 


.0495 


.6945 


20.206 J055 


1.5213 


10 


3° 00' 


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 J57I 


1.5155 


50 


20 


0.0582 


.0581 


.7645 


.9983 .9993 


.0582 


.7652 


17.169 J348 


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 


.0541 


.8067 


15.605 .1933 


1.5068 


20 


50 


0.0669 


.0669 


.8251 


.9978 .9990 


.0670 


.8261 


14.924 .1739 


1.5039 


10 


4° 00' 


0.0698 


.0598 


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 


.9957 .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" 00' 


0.0873 


.0872 


8.9403 


.9962 9.9983 


.0875 


B.9420 


11.430 1.0580 


1.4835 


85° 00' 


10 


0.0902 


.0901 


.9545 


.9959 .9982 


.0904 


.9563 


11.059 .0437 


1.4806 


SO 


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 .0154 


1.4748 


30 


40 


0.0989 


.0987 


.9945 


.9951 .9979 


.0992 


.9966 


10.078 .0034 


1.4719 


20 


50 


0.I0I8 


.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 .9554 


1.4632 


50 


20 


o.rro5 


.1103 


.0426 


.9939 .9973 


.1110 


.0453 


9.0098 .9547 


1.4603 


40 


30 


0.1134 


.1132 


.0539 


.9936 .9972 


.1139 


.0567 


8.7759 .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.I2SI 


.1248 


.0951 


.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 .8805 


1.4399 


30 


40 


0.1338 


.1334 


.1252 


.9911 .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 


6.8259 .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 


5.5506 .8169 


I.4I95 


20 


50 


0.1542 


.1536 


.1863 


.9881 .9948 


.1554 


.1915 


6.4348 .8085 


1.4166 


10 


9° 00' 


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 0.°1 intervals, see pp. 46-51) 



De- 
grees 


Ra- 
dians 


Sines 


Cosines 


Tangents 


Cotangents 








Nat. Log.* 


Nat. Log." 


Nat. Log.* 


Nat. Log. 






9° 00' 


0.1571 


.1564 9.1943 


.9877 9.9945 


.1584 9.1997 


6.3138 0.8003 


1.4137 


81° 00' 


ID 


0.1600 


.1593 .2022 


.9872 .9944 


.1614 .2078 


6.1970 .7922 


1.4108 


50 


20 


0.1629 


.1622 2m 


.9868 .9942 


.1544 .2158 


5.0844 .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° OC 


0.1745 


.1736 9.2397 


.9848 9.9934 


.1753 9.2453 


5.6713 0.7537 


1.3953 


80° 00' 


10 


0.1774 


.1765 .2468 


.9843 .9931 


.1793 .2535 


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 9.2887 


'5.1445 0.7113 


1.3788 


79° 00' 


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 J058 


.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 0.5725 


I.36I4 


78° 00' 


10 


0.2123 


J108 .3238 


.9775 .9901 


.2156 .3335 


4.6382 .5664 


1.3584 


50 


20 


0.2153 


.2136 .3295 


.9769 .9899 


.2186 .3397 


4.5736 .6603 


1.3555 


40 


30 


0.2182 


.2164 .3353 


.9763 .9896 


.2217 .3458 


4.5107 .6542 


1.3525 


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 .5424 


1.3468 


10 


13° 00' 


0.2269 


.2250 9.3521 


.9744 9.9887 


.2309 9.3634 


4.3315 0.5366 


1.3439 


77° 00' 


10 


0.2298 


.2278 J575 


.9737 .9884 


.2339 .3691 


4.2747 .6309 


1.3410 


50 


20 


0.2327 


.2305 .3629 


.9730 .9881 


.2370 .3748 


4.2193 .6252 


1. 3381 


40 


30 


0.2355 


.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 J786 


.9710 .9872 


.2462 .3914 


4.0611 .6086 


1.3294 


10 


14° 00* 


0.2443 


.2419 9.3837 


.9703 9.9859 


.2493 9.3958 


4.0108 0.6032 


1.3265 


75° 00* 


10 


0.2473 


.2447 J887 


.9695 .9866 


.2524 .4021 


3.9517 .5979 


1.3235 


50 


20 


0.2502 


.2476 .3937 


.9689 .9863 


.2555 .4074 


3.9136 .5925 


1.3206 


40 


30 


0.2531 


.2504 .3986 


.9681 .9859 


.2585 .4127 


3.8667 .5873 


1.3177 


30 


40 


0.2560 


.2532 .4035 


.%74 .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 J619 


1.3032 


40 


30 


0.2705 


.2672 .4269 


.9536 .9839 


.2773 .4430 


3.6059 .5570 


1.3003 


30 


40 


0.2734 


.2700 .4314 


.9628 .9835 


.2805 .4479 


3.5656 .5521 


1.2974 


20 


50 


0J763 


.2728 .4359 


.9621 .9832 


.2836 -4527 


3.5261 .5473 


1.2945 


10 


16° 00- 


0.2793 


.2756 9.4403 


.9613 9.9828 


.2867 9.4575 


3.4874 0.5425 


1.2915 


74° 00' 


10 


0.2822 


.2784 .4447 


.9605 .9825 


.2899 .4622 


3.4495 .5378 


1.2886 


50 


20 


0.2851 


.2812 .4491 


.9596 .9821 


.2931 .4659 


3.4124 .5331 


1.2857 


40 


30 


0.2880 


.2840 .4533 


.9588 .9817 


.2952 .4716 


3.3759 .5284 


1.2828 


30 


40 


0.2909 


.2868 .4576 


.9580 .9814 


.2994 .4762 


3.3402 .5238 


1.2799 


20 


50 


0.2938 


.2896 .4618 


.9572 .9810 


.3026 .4808 


3.3052 .5192 


1.2770 


10 


17° OC 


0.2967 


.2924 9.4659 


.9563 9.9806 


.3057 9.4853 


3.2709 0.5147 


1.2741 


73° 00' 


10 


0.2996 


.2952 .4700 


.9555 .9802 


.3089 .4898 


3.2371 .5102 


1.2712 


50 


20 


0.3025 


.2979 .4741 


.9546 .9798 


.3121 .4943 


3.2041 .5057 


1.2683 


40 


30 


0.3054 


J007 .4781 


.9537 .9794 


.3153 .4987 


3.1716 3013 


1.2654 


30 


40 


0.3083 


.3035 .4821 


.9528 .9790 


.3185 .5031 


3.1397 .4969 


1.2525 


20 


50 


0.3113 


.3052 .4861 


.9520 .9785 


.3217 .5075 


3.1084 .4925 


1.2595 


10 


18° 00' 


0.3142 


.3090 9.4900 


.9511 9.9782 


.3249 9.5118 


3.0777 0.4882 


1.2566 


72° 00' 






Nat. Log.' 


Nat. Log.* 


Nat. Log.* 


Nat. Log. 










Cosines 


Sines 


Cotangents 


Tangents 


Ra- 
dians 


De- 
grees 



54 



MATHEMATICAL TABLES 



TRIGONOMETRIC FUNCTIONS 


(continued) 










Annex-lOin columna marked*. 


(For 0.°1 intervals, see pp. 


46-Sl) 




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 


93118 


3.0777 0.4882 


1.2556 


72° 00- 


10 


0.3171 


3118 


.4939 


.9502 


.9778 


.3281 


3161 


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 


.9755 


.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 0.4530 


1.2392 


71° 00' 


10 


0.3345 


.3283 


.5163 


.9446 


.9752 


3475 


.5411 


2.8770 


.4589 


1.2363 


SO 


20 


0.3374 


.3311 


.5199 


.9435 


.9748 


3508 


3451 


2.8502 


.4549 


1.2334 


40 


30 


0.3403 


.3338 


3235 


.9425 


.9743 


.3541 


3491 


2.8239 


.4509 


1.2305 


30 


40 


0.3432 


.3365 


.5270 


.9417 


.9739 


3574 


3531 


2.7980 


.4469 


1.2275 


20 


50 


0.3462 


.3393 


.5306 


.9407 


.9734 


.3607 


.5571 


2.7725 


.4429 


12245 


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 


J448 


.5375 


.9387 


.9725 


3673 


3650 


2.7228 


.4350 


1.2188 


50 


20 


0.3549 


3475 


3409 


.937/ 


.9721 


.3706 


.5689 


2.6985 


.4311 


1.2159 


40 


30 


0.3578 


.3502 


3443 


.9357 


.9716 


.3739 


.5727 


2.5745 


.4273 


r.2I30 


30 


40 


0.3607 


J529 


.5477 


.9356 


.9711 


.3772 


3766 


2.5511 


.4234 


1.2101 


20 


50 


0.3635 


J557 


3510 


.9346 


.9706 


3805 


3804 


2.6279 


.4195 


1.2072 


10 


21° 00' 


0.3665 


3584 


9.5543 


.9335 


9.9702 


.3839 


93842 


2.6051 


0.4158 


1.2043 


59° 00' 


10 


0.3694 


.3611 


3575 


.9325 


.9697 


.3872 


3879 


23826 


.4121 


1.2014 


50 


20 


0.3723 


3638 


.5609 


.9315 


.9692 


3906 


.5917 


2.5605 


.4083 


1.1985 


40 


30 


0.3752 


3665 


.5641 


.9304 


.9687 


.3939 


3954 


23386 


.4046 


1.1955 


30 


40 


0.3782 


.3692 


3673 


.9293 


.9582 


3973 


.5991 


23172 


.4009 


1.1926 


20 


50 


0.381 1 


3719 


.5704 


.9283 


.9577 


.4006 


.6028 


2.4960 


.3972 


1.1897 


10 


22° 00' 


0.3840 


3746 


9.5735 


.9272 


9.9672 


.4040 


9.6054 


2.4751 


0.3936 


1.1868 


58° 00* 


10 


0.3869 


3773 


3767 


.9261 


.9567 


.4074 


.6100 


2.4545 


.3900 


1.1839 


50 


20 


0.3898 


.3800 


.5798 


.9250 


.9561 


.4108 


.6136 


2.4342 


3864 


1.1810 


40 


30 


0.3927 


.3827 


3828 


.9239 


.9555 


.4142 


.6172 


2.4142 


3828 


1.1781 


30 


40 


0.3955 


3854 


3859 


.9228 


.9651 


.4175 


.6208 


2.3945 


.3792 


1.1752 


20 


50 


0.3985 


3881 


3889 


.9216 


.9645 


.4210 


.6243 


2.3750 


.3757 


1.1723 


10 


23° 00' 


0.4014 


.3907 


9.5919 


.9205 


9.9540 


.4245 


9.6279 


2.3559 0.3721 


I.I694 


67° 00' 


10 


0.4043 


.3934 


.5948 


.9194 


.9535 


.4279 


.6314 


2.3369 


.3686 


1.1665 


50 


20 


0.4072 


.3961 


.5978 


.9182 


.9629 


.4314 


.6348 


23183 


.3652 


1.1636 


40 


30 


0.4102 


.3987 


.6007 


.9171 


.9524 


.4348 


.6383 


2.2998 


.3617 


1.1606 


30 


30 


0.4131 


.4014 


.6036 


.9159 


.9618 


.4383 


.64)7 


2.2817 


3583 


r.1577 


20 


50 


0.4160 


.4041 


.6065 


.9147 


.9613 


.4417 


.5152 


2.2637 


.3548 


1.1548 


10 


24° 00' 


0.4189 


.4067 


9.6093 


.9135 


9.9607 


.4452 


9.5486 


2.2460 0.3514 


1.I5I9 


66° 00' 


10 


0.4218 


.4094 


.6121 


.9124 


.9602 


.4487 


.6520 


2.2286 


.3480 


1.1490 


50 


20 


0.4247 


.4120 


.6149 


.9112 


.9596 


.4522 


.5553 


2.2113 


.3447 


1.1461 


40 


30 


0.4276 


.4147 


.6177 


.9100 


.9590 


.4557 


.6587 


2.1943 


.3413 


1.1432 


30 


40 


0.4305 


.4173 


.6205 


.9088 


.9584 


.4592 


.6520 


2.1775 


.3380 


1.1403 


20 


50 


0.4334 


.4100 


.6232 


.9075 


.9579 


.4628 


.6654 


2.1609 


.3346 


1.1374 


10 


25° 00' 


0.4363 


.4226 


9.6259 


.9063 


9.9573 


.4663 


9.5587 


2.1445 03313 


1.1345 


65° 00' 


10 


0.4392 


.4253 


.6286 


.9051 


.9557 


.4699 


.6720 


2.1283 


.3280 


1.1316 


50 


20 


0.4422 


.4279 


.6313 


.9038 


.9551 


.4734 


.6752 


2.1123 


.3248 


1.1286 


40 


30 


0.4451 


.4305 


.6340 


.9026 


.9555 


.4770 


.5785 


2.0965 


.3215 


1.1257 


30 


40 


0.4480 


.4331 


.6366 


.9013 


.9549 


.4806 


.6817 


2.0809 


3183 


1.1228 


20 


50 


0.4509 


.4358 


.6392 


.9001 


.9543 


.4841 


.6850 


2.0655 


3150 


1.1199 


10 


26° 00' 


0.4538 


.4384 


9.5418 


.8988 


9.9537 


.4877 


9.6882 


2.0503 0-3118 


1.1170 


64° 00* 


10 


0.4567 


.4410 


.6444 


.8975 


.9530 


.4913 


.6914 


2.0353 


.3086 


1.1141 


50 


2D 


0.4595 


.4435 


.5470 


.8962 


.9524 


.4950 


.6946 


2.0204 


.3054 


1.1112 


40 


30 


0.4625 


.4462 


.6495 


.8949 


.9518 


.4986 


.6977 


2.0057 


.3023 


1.1083 


30 


40 


0.4654 


.4488 


.5521 


.8936 


.9512 


.5022 


.7009 


1.9912 


.2991 


1.1054 


20 


50 


0.4683 


.4514 


.5546 


.8923 


.9505 


.5059 


.7040 


1.9768 


.2960 


1.1025 


10 


27° 00- 


0.4712 


.4540 


9.5570 


.8910 


9.9499 


3095 


9.7072 


1.9625 0.2928 


1.0995 


63° 00- 






Nat. 


Log.* 


Nat. 


Log.* 


Nat. 


Log.* 


Nat. 


Log. 










Cosines 


Sines 


Cotangents 


Tangents 


Ra- 
dians 


De- 
grees 



MATUMMATICAL TABLES 



55 



TRIGONOMETRIC FUNCTIONS 

Annex —10 in columns marked*. 



(fionlinuecC) 
(For O".! intervals, see pp. 46-61) 



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 


53° 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 


J280 .7226 


1.8940 .2774 


1.0850 


10 


28° OC 


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 


1.0675 


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 .6901 


.8718 .9404 


.5619 .7497 


1.7796 .2503 


1.0588 


40 


30 


0.5149 


.4924 .6923 


.8704 .9397 


.5658 .7526 


1.7575 .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 .2355 


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 


J125 .7097 


.8587 .9338 


J969 .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 


1.5426 .2155 


1.0239 


40 


30 


0.5498 


J225 .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 


J275 .7222 


.8496 .9292 


.6208 .7930 


1.6107 .2070 


1.0152 


10 


32° 00' 


0.5585 


.5299 9.7242 


.8480 9.9284 


.6249 9.7958 


1.6003 0.2042 


1.0123 


58° 00' 


10 


0.5614 


J324 .7262 


.8465 .9276 


.6289 .7986 


1.5900 .2014 


1.0094 


50 


20 


0.5643 


.5348 .7282 


.8450 .9268 


.6330 .8014 


1.5798 .1986 


1.0065 


40 


30 


0.5672 


.5373 .7302 


.8434 .9260 


.6371 .8042 


1.5697 .1958 


1.0036 


30 


40 


0.5701 


.5398 .7322 


.8418 .9252 


.6412 .8070 


1.5597 .1930 


1.0007 


20 


50 


0.5730 


.5422 .7342 


.8403 .9244 


.6453 .8097 


1.5497 .1903 


0.9977 


10 


33'' 00' 


0.5760 


.5446 9.7361 


.8387 9.9236 


.6494 9.8125 


1.5399 0.1875 


0.9948 


57° 00' 


10 


0.5789 


.5471 .7380 


.8371 .9228 


.6536 .8153 


1.5301 .1847 


0.9919 


50 


20 


0.5818 


.5495 .7400 


.8355 .9219 


.6577 .8180 


1.5204 .1820 


0.9890 


40 


30 


0.5847 


.5519 .7419 


.8339 .9211 


.6619 .8208 


1.5108 .1792 


0.9861 


30 


40 


0.5876 


.5544 .7438 


.8323 .9203 


.6661 .8235 


1.5013 .1765 


0.9832 


20 


50 


0.5905 


.5568 .7457 


.8307 .9194 


.6703 .8263 


1.4919 .1737 


0.9803 


10 


34° 00' 


0.5934 


.5592 9.7476 


.8290 9.9186 


.6745 9.8290 


1.4826 0.1710 


9774 


55° 00' 


10 


0.5963 


J616 .7494 


.8274 .9177 


.6787 .8317 


1.4733 .1683 


0.9745 


50 


20 


0.5992 


.5640 .7513 


.8258 .9169 


.6830 .8344 


1.4641 .1656 


0.9716 


40 


30 


0.6021 


.5664 .7531 


.8241 .9160 


.6873 .8371 


1.4550 .1629 


0.9687 


30 


40 


0.6050 


.5688 .7550 


.8225 .9151 


.6916 .8398 


1.4460 .1602 


0.9657 


20 


50 


0.6080 


.5712 .7568 


.8208 .9142 


.6959 .8425 


1.4370 .1575 


0.9528 


10 


35° 00' 


0.6109 


.5736 9.7586 


.8192 9.9134 


.7002 9.8452 


1.4281 0.1548 


0.9599 


55° 00' 


10 


0.6138 


.5760 .7604 


.8175 .9125 


.7046 .8479 


1.4193 .1521 


0.9570 


50 


20 


0.6167 


.5783 .7622 


.8158 .9116 


.7089 .8506 


1.4106 .1494 


0.9541 


40 


30 


0.6196 


.5807 .7640 


.8141 .9107 


.7133 .8533 


1.4019 .1467 


0.9512 


30 


40 


0.6225 


.5831 .7657 


.8124 .9098 


.7177 .8559 


1J934 .1441 


0.9483 


20 


50 


0.6254 


.5854 .7675 


.8107 .9089 


.7221 .8586 


1.3848 .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 

Annex -10 in columns marked*. 



{continued) 
(For 0°.l inteivals, see pp. 46-51) 



De- 
grees 


Ra- 
dians 


Sines 


Cosines 


Tangents 


Cotangents 










Nat. 


Log.* 


Nat. 


Log.* 


Nat. 


Log.* 


Nat. Log. 






36° OC 


0.6283 


.5878 


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 


1.3270 0.1229 


0.9250 


53° 00' 


10 


0.6487 


.6041 


.7811 


.7969 


.9014 


.7581 


.8797 


1.3190 .1203 


0.9221 


50 


20 


0.6516 


.6065 


.7828 


.7951 


.9004 


.7627 


.■8824 


1.3111 .1176 


0.9192 


40 


30 


0.6545 


.6088 


.7844 


.7934 


.8995 


.7673 


.8850 


1.3032 .1150 


0.9163 


30 


40 


0.6574 


.6111 


.7861 


.7916 


.8985 


.7720 


.8876 


1.2954 .1124 


0.9134 


20 


50 


0.6603 


.6134 


.7877 


.7898 


.8975 


.7766 


.8902 


1.2876 .1098 


0.9105 


10 


38° 00' 


0.6632 


.6157 


9.7893 


.7880 


9.8965 


.7813 


9.8928 


1.2799 0.1072 


0.9076 


52° 00' 


10 


0.6661 


.6180 


.7910 


.7862 


.8955 


.7860 


.8954 


1.2723 .1046 


0.9047 


50 


20 


0.6690 


.6202 


.7926 


.7844 


.8945 


.7907 


.8980 


1.2647 .1020 


0.9018 


40 


30 


0.6720 


.6225 


.7941 


.7826 


.8935 


.7954 


.9006 


1.2572 .0994 


0.8988 


30 


40 


0.6749 


.6248 


.7957 


.7808 


.8925 


.8002 


.9032 


1.2497 .0968 


0.8959 


20 


50 


0.6778 


.6271 


.7973 


.7790 


.8915 


.8050 


.9058 


1.2423 .0942 


0.8930 


10 


39° 00" 


0.6807 


.6293 


9.7989 


.7771 


9.8905 


.8098 


9.9084 


1.2349 0.0916 


0.8901 


51° 00' 


10 


0.6836 


.6316 


.8004 


.7753 


.8895 


.8146 


.9110 


1.2276 .0890 


0.8872 


50 


20 


0.6865 


.6338 


.8020 


.7735 


.8884 


.8195 


.9135 


1.2203 .0865 


0.8843 


40 


30 


0.6894 


.6361 


.8035 


.7716 


.8874 


.8243 


.9161 


1.2131 .0839 


0.8814 


30 


40 


0.6923 


.6383 


.8050 


.7698 


.8864 


.8292 


.9187 


1.2059 .0813 


0.8785 


20 


50 


0.6952 


.6406 


.8066 


.7679 


.8853 


.8342 


.9212 


1.1988 .0788 


0.8756 


10 


40° 00' 


0.6981 


.6428 


9.8081 


.7660 


9.8843 


.8391 


9.9238 


1.1918 0.0762 


0.8727 


50° 00' 


10 


0.7010 


.6450 


.8096 


.7642 


.8832 


.8441 


.9264 


1.1847 .0736 


0.8698 


50 


20 


0.7039 


.6472 


.8111 


.7623 


.8821 


.8491 


.9289 


1.1778 .0711 


0.8668 


40 


30 


0.7069 


.6494 


.8125 


.7604 


.8810 


.8541 


.9315 


1.1708 .0685 


0.8639 


30 


40 


0.7098 


.6517 


.8140 


.7585 


.8800 


.8591 


.9341 


1.1640 .0659 


0.8610 


20 


50 


0.7127 


.6539 


.8155 


.7566 


.8789 


.8642 


.9366 


1.1571 .0634 


0.8581 


10 


41° 00' 


0.7156 


.6561 


9.8169 


.7547 


9.8778 


.8693 


9.9392 


1.1504 0.0608 


0.8552 


49° 00' 


10 


0.7185 


.6583 


.8184 


.7528 


.8767 


.8744 


.9417 


1.1436 .0583 


0.8523 


50 


20 


0.7214 


.6604 


.8198 


.7509 


.8756 


.8796 


.9443 


1.1369 .0557 


0.8494 


40 


30 


0.7243 


.6626 


.8213 


.7490 


.8745 


.8847 


.9468 


1.1303 .0532 


0.8465 


30 


40 


0.7272 


.6648 


.8227 


.7470 


.8733 


.8899 


.9494 


1.1237 .0506 


0.8436 


20 


50 


0.7301 


.6670 


.8241 


.7451 


.8722 


.8952 


.9519 


1.1171 .0481 


0.8407 


10 


42° 00' 


0.7330 


.6691 


9.8255 


.7431 


9.8711 


.9004 


9.9544 


1.1106 00456 


0.8378 


48° 00' 


10 


0.7359 


.6713 


.8269 


.7412 


.8699 


.9057 


.9570 


1.1041 .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 


47O00' 


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 


0.7796 


.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 e""] 



57 







ta 






113 








ta 










n 


^ 


a 


n 


e» 


S 


n 


e" 


71 


e-. .- 


n 


e-" 


n 


e-n 


0.00 


1.000 


10 
10 
10 

11 

10 


50 


1.649 


16 
17 
17 
17 
17 


1.0 


2.718* 


0.00 


1.000 ,„ 

0.990- g 

.980- " 

.970-'" 

•96" -10 


0.60 


.607 


1.0 


.368* 


.01 


1.010 


.51 


1.665 




3.004 


.01 


.51 


.600 


.1 


.333 


.02 


!.020 


.52 


1.682 


2 


3.320 


.02 


.52 


.595 


2 


.301 


.03 


1.030 


.53 


1.699 


.3 


3.669 


.03 


.53 


.589 


3 


J73 


.04 


1.041 


.54 


1.716 


.4 


4.055 


.04 


.54 


.583 


.4 


.247 


0.05 


1.051 


1 1 


0.55 


1.733 


18 
17 
18 
18 
18 


1.5 


4.482 


0.05 


.951 9 
.942-, 5 
.932 '2 
.923 - I 
.914 I I 


0.55 


.577 


IJ 


.223 


.05 


1.062 


I 1 

I I 


.56 


1.751 


.6 


4.953 


.06 


.56 


.571 


.5 


.202 


.07 


1.073 


10 


J7 


1.768 


.7 


5.474 


.07 


.57 


.566 


.7 


.183 


.08 


1.083 


.58 


1.786 


.8 


6.050 


.03 


.58 


.560 


.8 


.165 


.09 


1.094 


11 

11 


.59 


1.804 


.9 


6.686 


.09 


J9 


.554 


.9 


.150 


0.10 


1.105 


1 1 


0.60 


1.822 


18 
19 
19 
18 
20 


2.0 


7.389 


0.10 


.905 o 
.896 I 
.887 - I 
.878 I 
.869 I I 


0.60 


.549 


2.0 


.135 


.11 


1.116 


1 1 


.61 


1.840 


1 


8.166 


.11 


.61 


.543 


.1 


.122 


.12 


1.127 


11 
12 


.62 


1.859 


.2 


9.025 


.12 


.62 


.538 


.2 


.111 


.13 


1.139 


.63 


1.878 


.3 


9.974 


.13 


.63 


.533 


3 


.100 


.14 


1.150 


11 
12 


.64 


1.896 


.4 


11.02 


.14 


.64 


.527 


.4 


.0907 


0.15 


1.162 


12 
11 
12 
12 
12 


0.65 


1.916 


19 
19 
20 
20 
20 


2.5 


12.18 


0.15 


.861 „ 
.852 t 
.844-5 
.835 - I 
.827 I I 


0.65 


.522 


2.5 


.0821 


.16 


1.174 


.66 


1.935 


.6 


13.46 


.16 


.66 


.517 


.6 


.0743 


.17 


1.185 


.67 


1.954 


.7 


14.88 


.17 


.67 


.512 


.7 


.0672 


.18 


1.197 


.68 


1.974 


.8 


16.44 


.18 


.68 


.507 


.8 


.0608 


.19 


U09 


.69 


1.994 


.9 


18.17 


.19 


.69 


.502 


.9 


.0550 


0.20 


(.221 


13 
12 
13 
12 
13 


0.70 


2.014 


20 
20 
21 
21 

21 


3.0 


20.09 


0.20 


.819 g 
.811 - I 
.803 - S 
.795 - 2 
.787 I \ 


0.70 


.497 


3.0 


.0498 


.21 


1.234 


.71 


2.034 


.1 


22.20 


.21 


.71 


.492 


.1 


.0450 


.22 


1.246 


.72 


2.054 


.2 


24.53 


.22 


.72 


.487 


.2 


.0408 


.23 


1.259 


.73 


2.075 


.3 


27.11 


.23 


.73 


.482 


.3 


.0369 


^■f 


1.271 


.74 


2.096 


.4 


29.96 


.24 


.74 


.477 


.4 


.0334 


0.25 


1.284 


13 
13 
13 

13 
14 


0.75 


2.117 


21 
22 
21 
22 
23 


3.5 


33.12 


0.25 


.779 „ 
.771 - S 
.763 - 2 
.756 - i 
.748 Z 7 


0.75 


.472 


3.5 


.0302 


.26 


1.297 


.76 


2.138 


.6 


36.60 


.26 


.76 


.468 


.6 


.0273 


.27 


1.310 


.77 


2.160 


.7 


40.45 


.27 


.77 


.463 


.7 


.0247 


.28 


1.323 


.78 


2.181 


.8 


44.70 


.28 


.78 


.458 


.8 


.0224 


J9 


1336 


.79 


2.203 


.9 


49.40 


J9 


.79 


.454 


.9 


.0202 


0.30 


1.350 


13 
14 
14 
14 
14 


0.80 


2.226 


22 
22 
23 
23 
24 


4.0 


54.60 


0.30 


.741 ,, 
.733 - 2 
.726 - i 
.719 - i, 
.712 Z 7 


0.80 


.449 


4.0 


.0183 


Jl 


IJ63 


.81 


2.248 


1 


60.34 


.31 


.81 


.445 


.1 


.0166 


.32 


1.377 


.82 


2.270 


.2 


66.69 


.32 


.82 


.440 


2 


•.0150 


.33 


1.391 


.83 


2.293 


3 


73.70 


.33 


.83 


.436 


3 


.0136 


.34 


1.405 


.84 


2.316 


A 


81.45 


J4 


.84 


.432 


.4 


.0123 


0.35 


1.419 


14 
15 
14 
15 
15 


0.85 


2.340 


23 
24 
24 
24 
25 


4.5 


90.02 


0.35 


.705 7 
.698 - i 
.691 - i 
.684 - i 
.677 Z 7 


0,85 


.427 


4J 


.0111 


.36 


1.433 


.86 


2.363 






.36 


.86 


.423 






.37 


1.448 


.87 


2.387 


6.0 


148.4 


.37 


.87 


.419 


6.0 


.00674 


.38 


1.462 


.88 


2.411 


6.0 


403.4 


.38 


.88 


.415 


6.0 


.00248 


J9 


1.477 


.89 


2.435 


7.0 


1097. 


.39 


.89 


.411 


7.0 


.000912 


0.40 


r.492 


15 
15 
15 
16 
15 


0.90 


2.460 


24 
25 
26 
25 
26 


8.0 


2981. 


0.40 


.670 , 
.664 - S 
.657 - { 
.651 - S 
.644; ^ 


0.90 


.407 


8.0 


.000335 


.41 


1.507 


.91 


2.484 


9.0 


8103. 


.41 


.91 


.403 


9.0 


.000123 


.42 


1.522 


.92 


2.509 


10.0 


22026. 


.42 


.92 


.399 


10.0 


.000045 


.43 
.44 


1.537 
1.553 


.93 
.94 


2.535 
2.560 


7r/2 

2t/2 


4.810 
23.14 


.43 
.44 


.93 
.94 


.395 
.391 


V2 

27r/2 


.208 
.0432 


0.45 


1.568 


16 
16 
16 
16 
17 


0.95 


2.586 


26 
26 
26 
27 
27 


3,r/2 


111.3 


0.45 


.638 , 
.631 - 1 
.625 - ? 
.619 - I 
■613 I % 
0.607 


0.95 


.387 


V2 


.00898 


.46 


1.584 


.96 


2.612 


4^/2 


535.5 


.46 


.96 


.383 


4^/2 


.00187 


.47 


1.600 


.97 


2.638 


5V2 


2576. 


.47 


.97 


.379 


5,r/2 


.000388 


.48 


1.616 


.98 


2.664 


6V2 


12392. 


.48 


.98 


.375 


6;r/2 


.000081 


.49 


1.632 


.99 


2.691 


77r/2 


59610. 


.49 


.99 


.372 


7t/2 


.000017 


0.60 


1.649 


1.00 


2.718 


8ir/2 


286751. 


0.60 


1.00 


368 


&r/2 


.000003 



* Note: Do not interpolate in this column. 

e = 2.71828 1/e = 0.367879 logioe = 0.4343 1/(0.4343) = 2.3026 

Iogio(0.4343) = 1.6378 logio(e") = n(0.4343) 

For table of multiples of 0.4343, see p. 62. Graphs, p. 174. 



58 



MATHMMATICAL TABLES 



HYPERBOLIC LOGARITHMS 





n 


n (2.3026) 


n (0.6974-3) 


These two pages give the natural (hyper- 


1 

7 


2.3026 
4 6032 


0.6974-3 
3948-5 


bolic, or Napierian) logarithms (log.) of 


1 


6.9078 


0.0922-7 


numbers between 1 and 10, correct to four 


4 


9.2103 


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- 


b 


13.8155 


0.1845-14 


lent to adding » times 2.3026 [or n times 
3.6974] to the logarithm. Base e = 2.71828H- 


y 

8 
9 


15.1181 
18.4207 
20.7233 


0,8819-17 
0.5793-19 
0.2767-21 



d 0) 





1 


2 


3 


i 


S 


6 


7 


S 


9 


ti 


1-° 






















■^■B 


1.0 


0.0000 


0100 


0198 


0296 


0392 


0488 


0583 


0677 


0770 


0852 


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 


5152 


6206 


6259 


6313 


6366 


54 


1.9 


6419 


6471 


6523 


6575 


6627 


5678 


6729 


6780 


6831 


6881 


51 


2.0 


0.6931 


5981 


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 


8529 


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 


9703 


9746 


9783 


9821 


9858 


9895 


38 


2.7 


0.9933 


9969 


•0006 


•0043 


♦0080 


•0116 


•0152 


•0188 


•0225 


•0260 


36 


2.8 


1.0295 


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 


1.0985 


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 


2295 


2326 


2355 


2384 


2413 


2442 


2470 


2499 


29 


3.5 


1.2528 


2556 


2585 


2613 


2641 


2669 


2698 


2726 


2754 


2782 


28 


3.5 


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 


3685 


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 


4655 


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 


5325 


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 


5994 


6014 


6034 


6054 


6074 


20 



log, a; ■= (2.3026) logio a; ' logioa; = (0.4343)log,a! 
where 2.3026 = log,io and 0.4343 = logioe (see p. 62). For graphs, see p. 174. 



MATHEMATICAL TABLES 



69 



HYPERBOLIC LOGARITHMS (continued) 



is 





1 


2 


3 


i 


5 


6 


7 


8 


9 


M 


1-° 






















6.0 


1.6094 


6114 


6134 


6154 


6174 


6194 


5214 


6233 


6253 


6273 


20 


5.1 


6292 


6312 


6332 


6351 


6371 


6390 


5409 


6429 


6448 


5457 


19 


5.2 


6487 


6506 


6525 


6544 


6553 


6582 


5601 


6520 


6639 


6658 


19 


5.3 


6677 


6695 


6715 


6734 


6752 


6771 


6790 


6808 


5827 


6845 


18 


5.4 


6864 


6882 


6901 


6919 


6938 


6956 


6974 


6993 


7011 


7029 


18 


5.5 


1.7047 


7065 


7084 


7102 


7120 


7138 


7156 


7174 


7192 


7210 


18 


5.6 


7228 


7245 


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 


7595 


7613 


7630 


7547 


7664 


7681 


7699 


7715 


7733 


17 


5.9 


7750 


7766 


7783 


7800 


7817 


7834 


7851 


7857 


7884 


7901 


17 


6.0 


1.7918 


7934 


7951 


7967 


7984 


8001 


8017 


8034 


8050 


8056 


16 


6.1 


8083 


8099 


8116 


8132 


8148 


8165 


8181 


8197 


8213 


8229 


16 


6.2 


8245 


8252 


8278 


8294 


8310 


8326 


8342 


8358 


8374 


8390 


16 


6.3 


8405 


8421 


8437 


8453 


8469 


8485 


8500 


8515 


8532 


8547 


16 


6.4 


8563 


8579 


8594 


8510 


8625 


8641 


8656 


8572 


8687 


8703 


15 


6.5 


1.8718 


8733 


8749 


8764 


8779 


8795 


8810 


8825 


8840 


8855 


15 


6.6 


8871 


8886 


8901 


8916 


8931 


8946 


8951 


8975 


8991 


9005 


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 


9543 


9657 


9671 


9585 


9599 


9713 


9727 


14 


7.2 


9741 


9755 


9769 


9782 


9795 


9810 


9824 


9838 


9851 


9855 


14 


7.3 


1.9879 


9892 


9906 


9920 


9933 


9947 


9951 


9974 


9988 


•0001 


13 


7.4 


2.0015 


0028 


0042 


0055 


0059 


0082 


0095 


0109 


0122 


0136 


13 


7.5 


2.0149 


0152 


0176 


0189 


0202 


0215 


0229 


0242 


0255 


0268 


13 


7.6 


0281 


0295 


0308 


0321 


0334 


0347 


0350 


0373 


0386 


0399 


13 


7.7 


0412 


0425 


0438 


0451 


0454 


0477 


0490 


0503 


0516 


0528 


13 


7.8 


0541 


0554 


0557 


0580 


0592 


0605 


0518 


0531 


0543 


0556 


13 


7.9 


0669 


0681 


0594 


0707 


0719 


0732 


0744 


0757 


0759 


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 


1065 


1078 


1090 


1102 


1114 


1125 


1138 


1150 


12 


8.3 


1153 


1175 


1187 


1199 


1211 


1223 


1235 


1247 


1258 


1270 


12 


8.4 


1282 


1294 


1305 


1318 


1330 


1342 


1353 


1365 


1377 


1389 


12 


8.5 


2.1401 


1412 


1424 


1435 


1448 


1459 


1471 


1483 


1494 


1506 


12 


8.6 


1518 


1529 


1541 


1552 


1564 


1576 


1587 


1599 


1510 


1522 


12 


8.7 


1633 


1545 


1655 


1658 


1579 


1691 


1702 


1713 


1725 


1736 


11 


8.8 


1748 


1759 


1770 


1782 


1793 


1804 


1815 


1827 


1838 


1849 


II 


8.9 


1861 


1872 


1883 


1894 


1905 


1917 


1928 


1939 


1950 


1951 


11 


9.0 


2.1972 


1983 


1994 


2005 


2017 


2028 


• 2039 


2050 


2061 


2072 


11 


9.1 


2083 


2094 


2105 


2116 


2127 


2138 


2148 


2159 


2170 


2181 


11 


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 


2450 


2471 


2481 


2492 


2502 


11 


9.5 


2.2513 


2523 


2534 


2544 


2555 


2555 


2576 


2585 


2597 


2607 


10 


9.6 


2618 


2528 


2538 


2649 


2659 


2670 


2580 


2690 


2701 


2711 


10 


9.7 


2721 


2732 


2742 


2752 


2762 


2773 


2783 


2793 


280^ 


2814 


10 


9.8 


2824 


2834 


2844 


2854 


2865 


2875 


2885 


2895 


2905 


2915 


10 


9.9 


2925 


2935 


2946 


2956 


2966 


2976 


2985 


2995 


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 (0.6974r-3)] in the body of the table. See auxihary table of 
multiples on top of the preceding page. 



60 



MATHEMATICAL TABLES 



HYPERBOLIC SINES [sinh 


X = \4(e' 


-e--)] 












X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


■Si 


0.0 


.0000 


.0100 


.0200 


.0300 


.0400 


.0500 


.0600 


.0701 


.0801 


.0901 


100 


1 


.1002 


.1102 


.1203 


.1304 


.1405 


.1506 


.1607 


.1708 


.1810 


.1911 


101 


2 


.2013 


.2115 


.2218 


.2320 


.2423 


.2526 


.2629 


.2733 


.2837 


.2941 


103 


3 


.3045 


.3150 


.3255 


.3360 


.3466 


.3572 


.3678 


.3785 


J892 


.4000 


106 


4 


.4108 


.4216 


.4325 


.4434 


.4543 


.4653 


.4764 


.4875 


.4986 


.5098 


110 


0.5 


J2II 


.5324 


.5438 


.5552 


.5666 


.5782 


.5897 


.6014 


.6131 


.6248 


115 


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 


1.055 


1.070 


1.085 


1.099 


1.114 


1.129 


1.145 


1.160 


15 


1.0 


1.175 


I.I91 


1.206 


1.222 


1.238 


1.254 


1.270 


1.286 


1.303 


1.319 


15 




1.335 


1.352 


1.369 


1.386 


1.403 


1.421 


1.438 


1.456 


1.474 


1.491 


17 


2 


1.509 


1.528 


1.546 


1.564 


1.583 


1.602 


1.621 


1.640 


1.659 


1.679 


19 


3 


1.698 


1.718 


1.738 


1.758 


1.779 


1.799 


1.820 


1.841 


1.862 


1.883 


21 


4 


1.904 


1.926 


1.948 


1.970 


1.992 


2.014 


2.037 


2.060 


2.083 


2.106 


22 


1.5 


2.129 


2.153 


2.177 


2.201 


2 775 


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 


54 


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 


10.85 


10.97 


11 




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 


25 


4.0 


27.29 


27.56 


27.84 


28.12 


28.40 


28.69 


28.98 


29.27 


29.56 


29.86 


29 




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 


4.6 


45.00 


45.46 


45.91 


46.37 


46.84 


47.31 


47.79 


48.27 


48.75 


49.24 


47 


6 


49.74 


50 24 


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 


58.95 


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 


S.O 


74.20 























If a; > S, sinh x = W,e') and logio sinh x = (0.4343)1 + 0.6990 — 1, correct to four 
significant figures. For table of multiples of 0.4343, see p. 62. Graphs, p. 174. 



MATHEMATICAL TABLES 
HYPERBOLIC COSINES [cosh x =^(e«+e-')] 



61 



X 





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 


1.020 


1.022 


1.024 


1.027 


1.029 


1.031 


1.034 


1.037 


1.039 


1.042 


3 


3 


1.045 


1.048 


1.052 


1.055 


1.058 


1.062 


1.066 


1.069 


1.073 


1.077 


4 


4 


1.081 


1.085 


1.090 


1.094 


1.098 


1.103 


1.108 


1.112 


1.117 


1.122 


5 


0.6 


1.128 


1.133 


1.138 


1.144 


1.149 


1.155 


1.161 


1.167 


1.173 


1.179 


6 


6 


1.185 


1.192 


1.198 


1.205 


1.212 


1.219 


1.226 


1.233 


1.240 


1.248 


7 


7 


1.255 


1.263 


1.271 


1.278 


1.287 


1.295 


1.303 


1.311 


1.320 


1.329 


8 


8 


1.337 


1.346 


1.355 


1.365 


1.374 


1.384 


1.393 


1.403 


1.413 


1.423 


10 


9 


1.433 


1.443 


1.454 


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 


1.765 


1.781 


1.795 


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 


2J10 


2.331 


20 


1.S 


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 


3.167 


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 


a.o 


3.762 


3.799 


3.835 


3.873 


3.910 


3.948 


3.987 


4.026 


4.065 


4.104 


38 


\ 


4.144 


4.185 


4.226 


4.267 


4.309 


4.351 


4.393 


4.435 


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 


5357 


5.612 


5.667 


5.7B 


5.780 


5.837 


5.895 


5.954 


6.013 


5.072 


58 


2.5 


6.132 


6.193 


6.255 


6.317 


6.379 


6.443 


6.507 


6.571 


6.635 


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 


7J48 


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 


9.859 


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 


12.15 


12 


2 


12.29 


12.41 


12.53 


12.65 


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 


15.61 


15.77 


15.92 


16.08 


16.25 


16.41 


16 


3.6 


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 


20.24 


20.44 


20.64 


20.85 


21.06 


21.27 


21.49 


21.70 


21.92 


22.14 


21 


8 


22.36 


22.59 


22.81 


23.04 


23.27 


23.51 


23.74 


23.98 


24.22 


24.47 


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 


4.1 


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 


59.55 


6015 


58 


8 


60.76 


61.37 


61.99 


62.61 


63.24 


63.87 


64.52 


65.16 


65.82 


56.48 


64 


9 


67.15 


67.82 


6830 


69.19 


69.89 


70.59 


71.30 


72.02 


72.74 


73.47 


71 


6.0 


74.21 























If a:> 5, cosh a; = ^(e«) and logio cosh a; = (0.4343)j; + 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 



HYPERBOLIC TANGENTS 


[tanh X 


= (e'-e- 


nne'+e-'^)' 


= sinh 


a/cosh x] 


X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


i!' 


0.0 


.0000 


.0100 


.0200 


.0300 


.0400 


.0500 


.0599 


.0699 


.0798 


.0898 


100 


1 


.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 


J080 


.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 




.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 


1 


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 


1 


5. 


.9999 


If a; > 5, 


tanh X = 


= 1.0000 to four decimal places. Graphs, p 


. 174. 







MULTIPLES OF 0.4343 


(0.43429448 


= logio e) 










X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


0. 
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 


0.0000 
0.4343 
0.8686 
1.3029 
1.7372 
2.1715 
2.6058 
3.0401 
3.4744 
3.9087 


0.0434 
0.4777 
0.9120 
1.3463 
1.7806 
2.2149 
2.6492 
3.0835 
3.5178 
3.9521 


0.0869 
0.5212 
0.9554 
1.3897 
1.8240 
2.2583 
2.6926 
3.1269 
3.5612 
3.9955 


0.1303 
0.5646 
0.9989 
1.4332 
1.8675 
2.3018 
2.7361 
3.1703 
3.6046 
4.0389 


0.1737 
0.6080 
1.0423 
1.4766 
1.9109 
2.3452 
2.7795 
3.2138 
3.6481 
4.0824 


0.2171 
0.6514 
1.0857 
1.5200 
1.9543 
2.3886 
2.8229 
3.2572 
3.6915 
4.1258 


0.2606 
0.6949 
1.1292 
1.5635 
1.9978 
2.4320 
2.8663 
3.3006 
3.7349 
4.1692 


0.3040 
0.7383 
1.1726 
1.6069 
2.0412 
2.4755 
2.9098 
3.3441 
3.7784 
4.2127 


0.3474 
0.7817 
1.2160 
1.6503 
2.0846 
2.5189 
2.9532 
3.3875 
3.8218 
4.2561 


0.3909 
0.8252 
1.2595 
1.6937 
2.1280 
2.5623 
2.9966 
3.4309 
3.8652 
4.2995 



MULTIPLES OF 2.3026 


(2.3026851 = 


1/0.4343) 








X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


0. 
1. 

8! 
9. 


0.0000 
2.3026 
4.6052 
6.9078 
9.2103 
11.513 
13.816 
16.118 
18.421 
20.723 


0.2303 
2.5328 
4.8354 
7.1380 
9.4406 
11.743 
14.046 
16.348 
18.651 
20.954 


0.4605 
2.7631 
5.0657 
7.3683 
9.6709 
11.973 
14.276 
16.579 
18.881 
21.184 


0.6908 
2.9934 
5.2959 
7.5985 
9.9011 
12.204 
14.506 
16.809 
19.111 
21.414 


0.9210 
3.2236 
5.5262 
7.8288 
10.131 
12.434 
14.737 
17.039 
19.342 
21.644 


1.1513 
3.4539 
5.7565 
8.0590 
10.362 
12.664 
14.967 
17.269 
19.572 
21.875 


1.3816 
3.6841 
5.9867 
8.2893 
10.592 
12.894 
15.197 
17.500 
19.802 
22.105 


1.6118 
3.9144 
6.2170 
8.5196 
10.822 
13.125 
15.427 
17.730 
20.032 
22.335 


1.8421 
4.1447 
6.4472 
8.7498 
11.052 
13.355 
15.658 
17 960 
20.263 
22.565 


2.0723 
4.3749 
6.6775 
8.9801 
11.283 
13.585 
15.888 
18.190 
20.493 
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: 
7t = number of observations. 



FACTORS FOR COMPUTING 
PROBABLE ERROR (p. 121) 




7i 


Bessel 


Peters 


0.6745 


0.6745 

VnCn-1) 


0.8453 


0.8453 


V(.n - 1) 


Vi(i-l) 


ns/n— 1 


2 
3 
4 


.6745 
.4769 
.3894 


.4769 
.2754 
.1947 


.5978 
.3451 
.2440 


.4227 
.1993 
.1220 


5 
6 

7 
8 
9 


.3372 
.3016 
.2754 
.2549 
.2385 


.1508 
.1231 
.1041 
.0901 
.0795 


.1890 
.1543 
.1304 
.1130 
.0996 


.0845 
.0630 
.0493 
.0399 
.0332 


10 
11 
12 
13 
14 


.2248 
.2133 
.2034 
.1947 
.1871 


.0711 
.0643 
.0587 
.0540 
.0500 


.0891 
.0806 
.0736 
.0677 
.0627 


.0282 
.0243 
.0212 
.0188 
.0167 


15 
16 
17 
18 
19 


.1803 
.1742 
.1686 
.1636 
.1590 


.0465 
.0435 
.0409 
.0386 
.0365 


.0583 
.0546 
.0513 
.0483 
.0457 


.0151 
.0136 
.0124 
.0114 
.0105 


20 
21 
22 
23 
24 


.1547 
.1508 
.1472 
.1438 
.1406 


.0346 
.0329 
.0314 
.0300 
.0287 


.0434 
.0412 
.0393 
.0376 
.0360 


.0097 
.0090 
.0084 
.0078 
.0073 


25 
26 
27 
28 
29 


.1377 
.1349 
.1323 
.1298 
.1275 


.0275 
.0265 
.0255 
.0245 
.0237 


.0345 
.0332 
.0319 
.0307 
.0297 


.0069 
.0065 
.0061 
.0058 
.0055 


30 
31 
32 
33 
34 


.1252 
.1231 
.1211 
.1192 
.1174 


.0229 
.0221 
.0214 
.0208 
.0201 


.0287 
.0277 
.0268 
.0260 
.0252 


.0052 
.0050 
.0047 
.0045 
.0043 


35 
36 
37 
38 
39 


.1157 
.1140 
.1124 
.1109 
.1094 


.0196 
.0190 
.0185 
.0180 
.0175 


.0245 
.0238 
.0232 
.0225 
.0220 


.0041 
.0040 
.0038 
.0037 
.0035 


40 
45 


.1080 
.1017 


.0171 
.0152 


.0214 
.0190 


.0034 
.0028 


50 
55 


.0964 
.0918 


.0136 
.0124 


.0171 
.0155 


.0024 
.0021 


60 
65 


.0878 
.0843 


.0113 
.0105 


.0142 
.0131 


.0018 
.0016 


70 
75 


.0812 
X784 


.0097 
.0091 


.0122 
.0113 


.0015 
.0013 


80 
85 


.0759 
.0736 


.0085 
.0080 


.0106 
.0100 


.0012 
.0011 


90 
95 


.0715 
.0696 


.0075 
.0071 


.0094 
.0089 


.0010 
.0009 


100 


.0678 


.0068 


.0085 


.0008 



64 



MATHEMATICAL TABLES 



COMPOTTITD INTEREST. AMOUNT OP A GIVEN PRINCIPAL 

The amount A at the end of n years of a given principal P placed at compound 
interest to-day 'm A = P X x ot A = P X V ot A = P X z, according as the interest 
(at the rate of r per cent, per annum) la compounded annually, semi-annually, or 
quarterly; the factor a: or y or z being taken from the following tables. 

Values of x. (Interest compounded annually; A = P X x.) 



Years 


r = 2 


2^ 


3 


m 


4 


4\i 


5 


6 


7 




1 


1.0200 


1.0250 


1.0300 


1.0350 


1.0400 


1.0450 


1.050O 


1.0600 


1.0700 




2 


1.0404 


1.0506 


1.0609 


1.0712 


1.0816 


1.0920 


1.1025 


1.1236 


1.1449 




3 


1.0612 


1.0769 


1.0927 


1.1087 


1.1249 


1.1412 


1.1576 


1.1910 


1.2250 


3 


4 


1.0824 


1.1038 


1.1255 


1.1475 


1.1699 


1.1925 


1.2155 


1.2625 


1.3108 


5 


1.1041 


1.1314 


1.1593 


1.1877 


1.2167 


1.2462 


1.2763 


1.3382 


1.4026 


a 


6 


I.I262 


1.1597 


1.1941 


1J293 


1.2653 


1.3023 


1.3401 


1.4185 


1.5007 


s 


7 


1.1487 


1.1887 


1.2299 


1.2723 


1.3159 


1.3609 


1.4071 


1.5036 


1.6058 




8 


1.1717 


1.2184 


1.2668 


1.3168 


1.3686 


1.4221 


1.4775 


1.5938 


1.7182 


■a 


9 


I.195I 


1.2489 


1.3048 


1.3629 


1.4233 


1.4861 


1.5513 


1.6895 


1.8385 


i 


10 


1.2190 


1.2801 


1.3439 


1.4106 


1.4802 


1.5530 


1.6289 


1.7908 


1.9672 


II 


1.2434 


1.3121 


1.3842 


1.4600 


1.5395 


1.6229 


1.7103 


1.8983 


2.1049 




12 


1.2682 


1.3449 


1.4258 


1.5111 


1.6010 


1.6959 


1.7959 


2.0122 


2.2522 


i^ 


13 


1.2936 


1.3785 


1.4685 


1.5640 


1.6651 


1.7722 


1.8855 


2.1329 


2.4098 


II 


14 


1.3195 


1.4130 


1.5126 


1.6187 


1.7317 


1.8519 


1.9799 


2.2609 


2.5785 


§,+ 


15 


1.3459 


1.4483 


1.5580 


1.6753 


1.8009 


1.9353 


2.0789 


2.3966 


2.7590 


r 


16 


1.3728 


1.4845 


1.6047 


1.7340 


1.8730 


2.0224 


2.1829 


2.5404 


2.9522 


°ii 


17 


1.4002 


1.5216 


1.6528 


1.7947 


1.9479 


2.1134 


2.2920 


2.6928 


3.1588 


.Sh 


18 


1.4282 


1.5597 


1.7024 


1.8575 


2.0258 


2.2085 


2.4066 


2.8543 


3.3799 


Ji 


19 


1.4568 


1.5987 


1.7535 


1.9225 


2.1068 


2.3079 


2.5270 


3.0256 


3.6165 


1 


20 


1.4859 


1.6386 


1.8061 


1.9898 


2.1911 


2.4117 


2.6533 


3.2071 


3.8697 


$ 


25 


1.6406 


1.8539 


2.0938 


2.3632 


2.6658 


3.0054 


3.3864 


4.2919 


5.4274 


■a 


30 


1.81 14 


2.0976 


2.4273 


2.8068 


3.2434 


3.7453 


4.3219 


5.7435 


7.6123 


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 


11.467 


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 =PXy.) 




Years 


r=2 


2H 


3 


3^4 


4 


m 


5 


6 


7 




1 
2 
3 
4 


1.0201 
1.0406 
1.0615 
1.0829 


1.0252 
1.0509 
1.0774 
1.1045 


1.0302 
1.0614 
1.0934 
1.1265 


1.0353 
1.0719 
1.1097 
1.1489 


1.0404 
1.0824 
1.1262 
1.1717 


1.0455 
1.0931 
1.1428 
1.1948 


1.0506 
1.1038 
1.1597 
1.2184 


1.0609 
1.1255 
1.1941 
1.2668 


1.0712 
1.1475 
1.2293 
1.3158 




5 
6 
7 
8 
9 


1.1046 
1.1268 
1.1495 
1.1726 
1.1961 


1.1323 
1.1608 
1.1900 
1.2199 
1.2506 


1.1605 
1.1956 
1.2318 
1.2690 
1.3073 


1.1894 
1.2314 
1.2749 
1.3199 
1.3655 


1.2190 
1.2682 
1.3195 
1.3728 
1.4282 


1.2492 
1.3060 
1.3655 
1.4276 
1.4926 


1.2801 
1.3449 
1.4130 
1.4845 
1.5597 


1.3439 
1.4258 
1.5126 
1.6047 
1.7024 


1.4106 
1.5111 
1.5187 
1.7340 
1.8575 


S, 


10 
11 
12 
13 
14 


1.2202 
1.2447 
1.2697 
1.2953 
1.3213 


1.2820 
1.3143 
1.3474 
1.3812 
1.4160 


1.3469 
1.3876 
1.4295 
1.4727 
1.5172 


1.4148 
1.4547 
1.5164 
1.5700 
1.5254 


1.4859 
1.5460 
1.6084 
1.6734 
1.7410 


1.5505 
1.6315 
1.7058 
1.7834 
1.8645 


1.6386 
1.7215 
1.8087 
1.9003 
1.9955 


1.8061 
1.9161 
2.0328 
2.1556 
2.2879 


1.9898 
2.1315 
2.2833 
2.4460 
2.5202 


-1- 


15 

16 
17 
18 
19 


1.3478 
1.3749 
1.4026 
1.4308 
1.4595 


1.4516 
1.4881 
1.5255 
1.5639 
1.6033 


1.5531 
1.6103 
1.6590 
1.7091 
1.7608 


1.5828 
1.7422 
1.8037 
1.8674 
1.9333 


1.8114 
1.8845 
1.9507 
2.0399 
2.1223 


1.9494 
2.0381 
2.1308 
2.2278 
2.3292 


2.0976 
2.2038 
2.3153 
2.4325 
2.5557 


2.4273 
2.5751 
2.7319 
2.8983 
3.0748 


2.8068 
3.0067 
3.2209 
3.4503 
3.6960 


1 


20 
25 
30 


1.4889 
1.6446 
1.8167 


1.6436 
1.8610 
2.1072 


1.8140 
2.1052 
2.4432 


2.0015 
2.3808 
2.8318 


2.2080 
2.5915 
3.2810 


2.4352 
3.0420 
3.8001 


2.5851 
3.4371 
4.3998 


3.2520 
4.3839 
5.8916 


3.9593 
5.5849 
7.8781 


^ 


40 
50 
60 


2.2167 
2.7048 
3.3004 


2.7015 
3.4634 
4.4402 


3.2907 
4.4320 
5.9693 


4.0064 
5.6682 
8.0192 


4.8754 
7.2446 
10.765 


5.9301 
9.2540 
14.441 


7.2096 
11.814 
19.358 


10.641 
19.219 
34.711 


15.675 
31.191 
52.064 





MATHEMATICAL TABLES 65 

Values of z. (Interest compounded quarterly; A = P X z; see opposite page) 



Years 


T =2 


2H 


3 


3H 


4 


4W 


5 


6 


7 




1 


1.0202 


1.0252 


1.0303 


1.0355 


1.0406 


1.0458 


1.0509 


1.0614 


1.0719 




2 


1.0407 


1.0511 


1.0616 


1.0722 


1.0829 


1.0936 


l.:045 


1.1265 


1.1489 




3 


1. 051 7 


1.0776 


1.0938 


1.1102 


1.1268 


1.1437 


1.1608 


1.1956 


1.2314 




4 


I.083I 


1.1048 


1.1270 


1.1496 


1.1726 


1.1950 


1.2199 


1.2590 


1.3199 




5 


I.I049 


I.I327 


1.1612 


1.1903 


1.2202 


1.2508 


1.2820 


1.3459 


1.4148 




6 


1.1272 


1.1613 


1.1964 


1.2326 


1.2597 


1.3030 


1.3474 


1.4295 


1.5164 


e* 


7 


1.1499 


1.1906 


1.2327 


1.2763 


1.3213 


1.3679 


1.4160 


1.5172 


1.6254 




8 


1.1730 


1.2206 


1.2701 


1.3215 


1.3749 


1.4305 


1.4881 


1.6103 


1.7422 


o 


9 


1.1967 


1.2514 


1.3085 


1.3684 


1.4308 


1.4959 


1.5639 


1.7091 


1.8674 


o 


10 


1.2208 


1.2830 


1.3483 


1.4169 


1.4889 


1.5544 


1.5435 


1.8140 


2.0016 


7^ 


II 


1.2454 


1.3154 


1.3893 


1.4572 


1.5493 


1.6350 


1.7274 


1.9253 


2.1454 


+ 


12 


1.2705 


1.3486 


1.4314 


1.5192 


1.5122 


1.7108 


1.8154 


2.0435 


2.2996 


13 


1. 2961 


1.3826 


1.4748 


1.5731 


1.6777 


1.7891 


1.9078 


2.1589 


2.4648 


1-t 


14 


1.3222 


1.4175 


1.5196 


1.6288 


1.7458 


1.8710 


2.0050 


2.3020 


2.6420 


II 


15 


1.3489 


1.4533 


1.5657 


1.6865 


1.8167 


1.9556 


2.1072 


2.4432 


2.8318 


«4 


16 


1.3760 


1.4900 


1.6132 


1.7464 


1.8905 


2.0462 


2.2145 


2.5931 


3.0353 


-. 


17 


1.4038 


1.5276 


1.6621 


1.8083 


1.9672 


2.1398 


2.3274 


2.7523 


3.2534 




18 


1.4320 


1.5661 


1.7125 


1.8725 


2.0471 


2.2378 


2.4459 


2.9212 


3.4872 


19 


1.4609 


1.6056 


1.7645 


1.9389 


2.1302 


2.3402 


2.5705 


3.1004 


3.7378 


20 


1.4903 


1.6462 


1.8180 


2.0076 


2.2157 


2.4473 


2.7015 


3.2907 


4.0064 


f» 


25 


1.6467 


1.8M6 


2.1111 


2.3898 


2.7048 


3.0609 


3.4634 


4.4320 


5.6682 




30 


1.8194 


2.1121 


2.4514 


2.8446 


3.3004 


3.8285 


4.4402 


5.9693 


8.0192 




40 


2.2211 


2.7098 


3.3053 


4.0306 


4.9138 


5.9892 


7.2980 


10.828 


16.051 




50 


2.7115 


3.4768 


4.4567 


5.7110 


7.3160 


9.3593 


11.995 


19.643 


32.128 




60 


3.3102 


4.4608 


5.0092 


8.0919 


10.893 


14.557 


19.715 


35.633 


64.307 





AMOUNT OF 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 = Y X v, 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 


2W 


i 


3H 


4 


4)^ 


5 


5 


7 




1 


1.0000 


1.0000 


1.0000 


1.0000 


1.0000 


1.0000 


1.000O 


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 


3.0755 


3.0909 


3.1062 


3.1216 


3.1370 


3.1525 


3.1835 


3.2149 


§ 


4 


4.1216 


4.1525 


4.1836 


4.2149 


4.2455 


4.2782 


4.3101 


4.3746 


4.4399 


5 


5 


5.2040 


5.2563 


5.3091 


5.3525 


5.4153 


5.4707 


5.5256 


5.5371 


5.7507 


6 


6.3081 


5,3877 


6.4684 


6.5502 


5.6330 


6.7169 


6.8019 


6.9753 


7.1533 


.[. 


7 


7.4343 


7.5474 


7.6625 


7.7794 


7.8983 


8.0192 


8.1420 


8.3938 


8.6540 




8 


8.5830 


8.7351 


8.8923 


9.0517 


9.2142 


9.380O 


9.5491 


9.8975 


10.250 


T-4 


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 


12.005 


12.288 


12.578 


13.181 


13.816 


= s 


11 


12.159 


12.483 


12.808 


13.142 


13.486 


13.841 


14.207 


14.972 


15.784 


■^ o 


12 


13.412 


13.795 


14.192 


14.602 


15.026 


15.464 


15.917 


16.870 


17.888 


o .^ 


13 


14.580 


15.140 


15.618 


16.113 


16.627 


17.160 


17.713 


18.882 


20.141 




14 


15.974 


15.519 


17.086 


17.577 


18.292 


18.932 


19.599 


21.015 


22.550 


> ■!• 


15 


17.293 


17.932 


18.599 


19.295 


20.024 


20.784 


21.579 


23.276 


25.129 


+ 7 


16 


18.539 


19.380 


20.157 


20.971 


21.825 


22.719 


23.657 


25.573 


27.888 


^ 1 


17 


20.012 


20.865 


21.762 


22.705 


23.698 


24.742 


25.840 


28.213 


30.840 


— .2. 


18 


21.412 


22.386 


23.414 


24.500 


25.645 


25.855 


28.132 


30.905 


33.999 


II 11 


19 


22.841 


23.946 


25.117 


26.357 


27.671 


29.054 


30.539 


33.750 


37.379 


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 


c4 

."3 


30 


40.568 


43.903 


47.575 


51.523 


55.085 


61.007 


65.439 


79.058 


94.461 


1 


40 


60.402 


67.403 


75.401 


84.550 


95.026 


107.03 


120.80 


154.75 


199.54 


50 


84.579 


97.484 


112.80 


131.00 


152.67 


178.50 


209.35 


290.34 


405.53 


f^ 


60 


114.05 


135.99 


163.05 


195.52 


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 given 
sum A at the end of n years is P = A X x' or P = A y. y' oi P = A X a', according aa 
the interest (at the rate of r per cent, per annum) is compounded annually, semi-annually, 
or quarterly; the factor x' or y' or z' being taken from the following tables. 
Values of of. (Interest compounded annually; P = AY. x') 



Years 


r = 2 


2H 


3 


m 


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 


U 


5 


.90573 


.88385 


.86261 


.84197 


.82193 


.80245 


.78353 


.74726 


.71299 


^ 


6 


.88797 


.86230 


.83748 


.81350 


.79031 


.76790 


.74622 


.70496 


.66634 




7 


.87056 


.84127 


.81309 


.78599 


.75992 


.73483 


.71068 


.66506 


.62275 


II 


8 


.85349 


.82075 


.78941 


.75941 


.73069 


.70319 


.67684 


.62741 


.58201 


i. 


9 


.83676 


.80073 


.76642 


.73373 


.70259 


.67290 


.64461 


.59190 


.54393 


o 


10 


.82035 


.78120 


.74409 


.70892 


.67556 


.64393 


.61391 


.55839 


.50835 


o 


II 


.80426 


.76214 


.72242 


.68495 


.64958 


.61620 


.58468 


.52679 


.47509 


> 


12 


.78849 


.74356 


.70138 


.66178 


.62460 


.58966 


.55684 


.49697 


.44401 




13 


.77303 


.72542 


.68095 


.63940 


.60057 


.56427 


.53032 


.46884 


.41496 


+ 


H 


.75788 


.70773 


.66112 


.61778 


.57748 


.53997 


.50507 


.44230 


J8783 




IS 


.74301 


.69047 


.64186 


.59689 


.55526 


.51672 


.48102 


.41727 


.36245 




16 


.72845 


.67362 


.62317 


.57671 


.53391 


.49447 


.4581 1 


.39365 


.33873 


■>i 


17 


.71416 


.65720 


.60502 


.55720 


.51337 


.47318 


.43630 


.37136 


.31657 


18 


.70016 


.64117 


.58739 


.53836 


.49363 


.45280 


.41552 


.35034 


.29586 


d 


19 


.68643 


.62553 


.57029 


.52016 


.47464 


.43330 


.39573 


.33051 


.27651 


■3 


20 


.67297 


.61027 


.55368 


.50257 


.45639 


.41464 


.37689 


.31180 


.25842 


a 


25 


.60953 


.53939 


.47761 


.42315 


.37512 


.33273 


.29530 


.23300 


.18425 


1 


30 


.55207 


.47674 


.41199 


J5628 


.30832 


.26700 


.23138 


.17411 


.13137 


fn 


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 


2/'. (Interest compounded semi-annually; P = 


AXv') 




Years 


r = 2 


m 


3 


3^ 


4 


4H 


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 


.901 14 


.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 


s 


6 


.88745 


.86151 


.83639 


.81206 


.78849 


.76567 


.74356 


.70138 


.66178 


II 


7 


.86996 


.84037 


.81185 


.78436 


.75788 


.73234 


.70773 


.661 12 


.61778 


8 


.85282 


.81975 


.78803 


.75762 


.72845 


.70047 


.67362 


.62317 


.57671 


S 


9 


.83602 


.79963 


.76491 


.73178 


.70016 


.66998 


.64117 


38739 


.53836 




10 


.81954 


.78001 


.74247 


.70682 


.67297 


.64082 


.61027 


.55368 


.50257 


1 




.80340 


.76087 


.72069 


.68272 


.64684 


.61292 


.58086 


.52189 


.46915 


5i 


12 


.78757 


.74220 


.69954 


.65944 


.62172 


;58625 


.55288 


.49193 


.43796 


^ 


13 


.77205 


.72398 


.67902 


.63695 


.59758 


.56073 


.52623 


.46369 


.40884 


+ 


14 


.75684 


.70622 


.65910 


.61523 


.57437 


.53632 


.50088 


.43708 


38165 


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 


.41 109 


34503 


.28983 




19 


.68515 


.62372 


.56792 


.51724 


.47119 


.42933 


39128 


.32523 


.27056 


Jj 


20 


.67165 


.60841 


.55126 


.49960 


.45289 


.41065 


.37243 


30656 


.25257 


3 


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 z'. (Interest compounded quarterly; P = A Xz'i see opposite page) 



Years 


r=2 


2H 


3 


3M 


4 


*H 


5 


6 


7 




1 


.98025 


.97539 


.97055 


.96575 


.95098 


.95624 


.95152 


.94218 


.93296 




2 


.96089 


.95138 


.94198 


.93268 


.92348 


.91439 


.90540 


.88771 


.87041 




3 


.94191 


.92796 


.91424 


.90074 


.88745 


.87437 


.86151 


.83539 


.81206 




4 


.92330 


.90512 


.88732 


.86989 


.85282 


.83611 


.81975 


.78803 


.75762 




5 


.90506 


.88284 


.86119 


.84010 


.81954 


.79952 


.78001 


.74247 


.70682 


-^ 


6 


.88719 


.86111 


.83583 


.81 132 


.78757 


.76453 


.74220 


.69954 


.65944 




7 


.86966 


.83991 


.81122 


.78354 


.75684 


.73107 


.70622 


.65910 


.61523 


n 


8 


.85248 


.81924 


.78733 


.75670 


.72730 


.69908 


.57198 


.62099 


.57390 


» 


9 


.83554 


.79908 


.76415 


.73079 


.59892 


.65849 


.63941 


.58509 


.53550 


^ 


10 


.81914 


.77941 


.74165 


.70576 


.67165 


.63923 


.60841 


.55126 


.49960 


o 

o 




.80296 


.76022 


.71981 


.68159 


.54545 


.61125 


.57892 


.51939 


.4661 1 




12 


.78710 


.74151 


.69861 


.65825 


.52026 


.58451 


.55086 


.48936 


.43485 


i^ 


13 


.77155 


.72326 


.67804 


.63570 


.59606 


.55893 


.52415 


.46107 


.40570 


+ 


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 


.45155 


.38563 


.32946 


n 


17 


.71237 


.65464 


.60164 


.55299 


.50833 


.46733 


.42957 


.36334 


.30737 


>i 


18 


.69830 


.63852 


.58392 


.53405 


.48850 


.44687 


.40884 


.34233 


.28575 




19 


.68451 


.62281 


.56673 


.51575 


.46944 


.42732 


.38903 


.32254 


.25754 


03 

a 


20 


.67099 


.60748 


.55004 


.49810 


.45112 


.40862 


.37017 


.30389 


.24960 


25 


.60729 


.53630 


.47359 


.41845 


.36971 


.32670 


.28873 


.22553 


.17642 


s 


30 


.54963 


.47347 


.40794 


.35154 


.30299 


.26120 


.22521 


.16752 


.12470 


1^ 


40 


.45023 


.36903 


.30255 


.24810 


.20351 


.15697 


.13702 


.09235 


.06230 




50 


.36880 


.28762 


.22438 


.17510 


.13569 


.10673 


.08337 


.05091 


.03113 




60 


J0210 


.22417 


.16541 


.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, wUl 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 v' 



Years 


r=2 


2H 


3 


3^ 


4 


4ii 


5 


6 


7 




2 


.49505 


.49383 


.49251 


.49140 


.49020 


.48900 


.48780 


.48544 


.48309 


ri 


3 


.32675 


J2514 


.32353 


.32193 


.32035 


.31877 


.31721 


.31411 


.31105 




4 


.24252 


.24082 


.23903 


.23725 


.23549 


.23374 


.23201 


.22859 


.22523 


n 


5 


.19216 


.19025 


.18835 


.18648 


.18463 


.18279 


.18097 


.17740 


.17389 




6 


.15853 


.15555 


.15450 


.15267 


.15076 


.14888 


.14702 


.14335 


.13980 




7 


.13451 


.13250 


.13051 


.12854 


.12561 


.12470 


.12282 


.11914 


.11555 


1 


8 


.11651 


.11447 


.11246 


.11048 


.10853 


.10561 


.10472 


.10104 


.09747 


c 


9 


.10252 


.10046 


.09843 


.09645 


.09449 


.09257 


.09069 


.08702 


.08349 


g 


10 


.09133 


.08926 


.08723 


.08524 


.08329 


.08138 


.07950 


.07587 


.07238 


O 


11 


.08218 


.0801 1 


.07808 


.07509 


.07415 


.07225 


.07039 


.06679 


.06336 


> 


12 


.07456 


.07249 


.07046 


.05848 


.06655 


.06467 


.06283 


.05928 


.05590 


+ 


13 


.05812 


.05505 


.06403 


.05206 


.06014 


.05828 


.05646 


.05296 


.04955 


14 


.05260 


.05054 


.05853 


.05557 


.05457 


.05282 


.05102 


.04758 


.04434 


^ 


15 


.05783 


.05577 


.05377 


.05183 


.04994 


.04811 


.04634 


.04296 


.03979 


•1- 


16 


.05365 


.05150 


.04951 


.04768 


.04582 


.04402 


.04227 


.03895 


.03585 




17 


.04997 


.04793 


.04595 


.04404 


.04220 


.04042 


.03870 


.03544 


.03243 


8 


18 


.04570 


.04457 


.04271 


.04082 


.03899 


.03724 


.03555 


.03235 


.02941 




19 


.04378 


.04176 


.03981 


.03794 


.03514 


.03441 


.03275 


.02962 


.02675 




20 


.04116 


.03915 


.03722 


.03536 


.03358 


.03188 


.03024 


.02718 


.02439 


I" 


25 


.03122 


.02928 


.02743 


.02567 


.02401 


.02244 


.02095 


.01823 


.01581 


30 


.02465 


.02278 


.02102 


.01937 


.01783 


.01639 


.01505 


.01265 


.01059 


40 


.01556 


.01484 


.01326 


.01183 


.01052 


.00934 


.00828 


.00646 


.00467 


S 


50 


.01182 


.01026 


.00887 


.00753 


.00555 


.00560 


.00478 


.00344 


.00238 


f^ 


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 annua] 
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 =1 


2W 


3 


3!^ 


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 


^ 


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 


n 


5 


4.7135 


4.6458 


4.5797 


4.5151 


4.4518 


4.3900 


4.3295 


4.2124 


4.1002 


o 


6 


5.6014 


5.5081 


5.4172 


5.3286 


5.2421 


5.1579 


5.0757 


4.9173 


4.7665 


o 


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 


1 


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 


o 


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 


14.292 


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 


■— ' 


19 


15.678 


14.979 


14.324 


13.710 


13.134 


12.593 


12.085 


11.158 


10.336 


1 


20 


16.351 


15.589 


14.877 


14.212 


13.590 


13.008 


12.462 


11.470 


10.594 


rt rH 


25 


19.523 


18.424 


17.413 


16.482 


15.622 


14.828 


14.094 


12.783 


11.654 


■3 7 


30 


22.396 


20.930 


19.600 


18.392 


17.292 


16.289 


15J72 


13.765 


12.409 


§ I 


40 


27.355 


25.103 


23.115 


21.355 


19.793 


18.402 


17.159 


15.046 


13.332 


o 9 


50 


31.424 


28.362 


25.730 


23.456 


21.482 


19.762 


18.256 


15.762 


13.801 


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 ia 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 


2H 


3 


3H 


4 


4^ 


5 


6 


7 




2 


.51505 


.51883 


.52261 


.52640 


.53020 


.53400 


.53780 


.54544 


.55309 




3 


.34675 


.35014 


.35353 


.35693 


.36035 


.36377 


.36721 


.37411 


.38105 


f 


4 


.26262 


J6582 


.26903 


Jim 


.27549 


.27874 


.28201 


.28859 


.29523 


|;r 


5 


.21216 


.21525 


.21835 


J2148 


.22463 


.22779 ■ 


.23097 


.23740 


.24389 


s 


6 


.17853 


.18155 


.18460 


.18767 


.19076 


.19388 


.19702 


.20336 


.20980 


1-1 


7 


.15451 


.15750 


.16051 


.16354 


.16661 


.16970 


.17282 


.17914 


.18555 


u^ 


8 


.13651 


.13947 


.14246 


.14548 


.14853 


.15161 


.15472 


.16104 


.16747 


+ s 

^ o 


9 


.12252 


.12546 


.12843 


.13145 


.13449 


.13757 


.14069 


.14702 


.15349 


10 


.11133 


.11426 


.11723 


.12024 


.12329 


.12638 


.12950 


.13587 


.14238 


"> 


11 


.10218 


.10511 


.10808 


.11109 


.11415 


.11725 


.12039 


.12679 


.13336 


1 ^ 




.09456 


.09749 


.10046 


.10348 


.10655 


.10967 


.11283 


.11928 


.12590 


ZL-\- 




.08812 


.09105 


.09403 


.09706 


.10014 


.10328 


.10646 


.11296 


.11965 


,1, ^ 




.08260 


.08554 


.08853 


.09157 


.09467 


.09782 


.10102 


.10758 


.11434 


•r E> 




.07783 


.08077 


.08377 


.08683 


.08994 


.09311 


.09634 


.10296 


.10979 


§!, 




.07365 


.07660 


.07961 


.08268 


.08582 


.08902 


.09227 


.09895 


.10586 


c-^ 




.06997 


.07293 


.07595 


.07904 


.08220 


.08542 


.08870 


.09544 


.10243 


i- 


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 


-g 


25 


.05122 


.05428 


.05743 


.06067 


.06401 


.06744 


.07095 


.07823 


.08581 


■• 


30 


.04465 


.04778 


.05102 


.05437 


.05783 


.06139 


.06505 


.07265 


.08059 


3 


40 


.03656 


.03984 


.04326 


.04683 


.05052 


.05434 


.05828 


.06646 


.07467 




50 


.03182 


.03526 


.03887 


.04263 


.04655 


.05060 


.05478 


.06344 


.07238 


g 


60 


.02877 


.03235 


.03613 


.04009 


.04420 


.04845 


.05283 


.06188 


.07121 


Em 



MATHEMATICAL TABLES 



69 



DECIMAL EQUIVALENTS 



From minutes 


and 1 


seconds into deci- | 




mal parts of 


a 




degree 




0' 


0°.0000 


0" 


O'.OOOO 




.0167 


] 


.0003 


2 


.0333 


2 


.0006 


3 


.05 


3 


.0008 


4 


.0667 


4 


.0011 


5' 


.0833 


, 5" 


.0014 


6 


.10 


6 


.0017 


7 


.1167 


7 


.0019 


8 


.1333 


8 


.0022 


9 


.15 


9 


.0025 


10' 


00.1667 


10" 


0°.0D28 


1 


.1833 




.0031 


2 


.20 


2 


.0033 


3 


.2167 


3 


.0036 


4 


.2333 


4 


.0039 


15' 


.25 


15" 


.0042 


6 


.2667 


6 


.0044 


7 


.2833 


7 


.0047 


8 


.30 


8 


.005 


9 


.3167 


9 


.0053 


20' 


0°.3333 


20" 


0°.0056 


1 


.35 


1 


.0058 


2 


.3667 


2 


.0061 


3 


.3833 


3 


.0064 


4 


.40 


4 


.0067 


25' 


.4167 


25" 


.0069 


6 


.4333 


6 


.0072 


7 


.45 


7 


.0075 


8 


.4667 


8 


.0078 


9 


.4833 


9 


.0081 


SO' 


0°.50 


30" 


0'=.Q083 


1 


.5167 


1 


.0086 


2 


.5333 


2 


.0089 


3 


.55 


3 


.0092 


4 


.5667 


4 


.0094 


35' 


.5833 


35" 


.0097 


6 


.60 


6 


.01 


7 


.6167 


7 


.0103 


8 


.6333 


8 


.0106 


9 


.65 


9 


.0108 


40' 


0°.6667 


40" 


0°.0I 1 1 


1 


.6833 


1 


.0114 


2 


.70 


2 


.0117 


3 


.7167 


3 


.0119 


4 


.7333 


4 


.0122 


45' 


.75 


45" 


.0125 


6 


.7667 


6 


.0128 


7 


.7833 


7 


.0131 


8 


.80 


8 


.0133 


9 


.8167 


9 


.0136 


60' 


0°.8333 


60" 


0°.0139 


1 


.85 


1 


.0142 


2 


.8667 


2 


.0144 


3 


.8833 


3 


.0147 


4 


.90 


4 


.015 


55' 


.9167 


55" 


.0153 


6 


.9333 


6 


.0156 


7 


.95 


7 


.0158 


8 


.9667 


8 


.0161 


9 


.9833 


9 


.0164 


60' 


1.00 


60" 


0°.0167 



From decimal parts of 


a degree into minutes 


and seconds (exact 


values) 


0°.00 


0' 


0°.50 


30' 


1 


0' 36" 


1 


30' 36" 


2 


r 12" 


2 


31' 12" 


3 


1' 48" 


3 


31' 48" 


4 


2' 24" 


4 


32' 24" 


0°.05 


3' 


0°.55 


33' 


6 


3' 36" 


6 


33' 36" 


7 


4' 12" 


7 


34' 12" 


8 


4' 48" 


8 


34' 48" 


9 


5' 24" 


9 


35' 24" 


OMO 


6' 


ceo 


36' 


1 


6' 36" 


1 


36' 36" 


2 


7' 12" 


2 


37' 12" 


3 


7' 48" 


3 


37' 48" 


4 


8' 24" 


4 


38' 24" 


0M5 


9' 


0''.65 


39' 


6 


9' 36" 


6 


39' 36" 


7 


10' 12" 


7 


40' 12" 


8 


10' 48" 


8 


40' 48" 


9 


11' 24" 


9 


41' 24" 


0°.20 


12' 


O-.TO 


42' 


1 


12' 36" 


1 


42' 36" 


2 


13' 12" 


2 


43' 12" 


3 


13' 48" 


3 


43' 48" 


4 


14' 24" 


4 


44' 24" 


0"'.25 


15' 


0°.75 


45' 


6 


15' 36" 


6 


45' 36" 


7 


16' 12" 


7 


46' 12" 


8 


16' 48" 


8 


46' 48" 


9 


17' 24" 


9 


47' 24" 


0°.30 


18' 


0°.80 


48' 


1 


18' 36" 


1 


48' 36" 


2 


19- 12" 


2 


49' 12" 


3 


19' 48" 


3 


49- 48" 


4 


20' 24" 


4 


50' 24" 


0°.35 


21' 


0''.85 


51' 


6 


21' 36" 


6 


51' 36" 


7 


22' 12" 


7 


52' 12" 


8 


22' 48" 


8 


52' 48" 


9 


23' 24" 


9 


53' 24" 


0°.40 


24' 


0°.90 


54' 


1 


24' 36" 


1 


54' 36" 


2 


25' 12" 


2 


55' 12" 


3 


25' 48" 


3 


55' 48" 


4 


26' 24" 


4 


56' 24" 


0'>.45 


27' 


0°.95 


57' 


6 


27' 36" 


6 


57' 36" 


7 


28' 12" 


7 


58' 12" 


8 


28' 48" 


8 


58' 48" 


9 


29' 24" 


9 


59' 24" 


0°.60 


30' 


1°.00 


60' 


o-.ooo 


0".0 


1 


3".6 


2 


7".2 


3 


10".8 


4 


14".4 


0°.005 


18" 


6 


21".6 


7 


25".2 


8 


28".8 


9 


32".4 


0°.010 


36" 



Common fractions 


8 16 

ths ths 


32 

nds 


64 

ths 


Exact 

decimal 

values 






1 


.01 5625 




1 


2 


.03 125 






3 


.04 6875 


1 


2 


4 


.06 25 






5 


.07 8125 




3 


6 


.09 375 






7 


.10 9375 


1 2 


4 


8 


.12 5 






9 


.14 0625 




5 


10 


.15 625 






11 


.17 1875 


3 


6 


12 


.18 75 






13 


.20 3125 




7 


14 


.21 875 






15 


.23 4375 


2 4 


8 


16 


.25 






17 


.26 5625 




9 


18 


.28 125 






19 


.29 6875 


5 


10 


20 


.31 25 






21 


.32 8125 




11 


22 


.34 375 






23 


.35 9375 


3 6 


12 


24 


.37 5 






25 


.39 0625 




13 


26 


.40 625 






27 


.42 1875 


7 


14 


28 


.43 75 






29 


.45 3125 




15 


30 


.46 875 






31 


.48 4375 


4 8 


16 


32 


.50 






33 


.51 5625 




17 


34 


.53 125 






35 


.54 6875 


9 


18 


36 


.56 25 






37 


.57 8125 




19 


38 


.59 375 






39 


.60 9375 


5 10 


20 


40 


.62 5 






41 


.64 0625 




21 


42 


.65 625 






43 


.67 1875 


11 


22 


44 


.68 75 






45 


.70 3125 




23 


46 


.71 875 






47 


.73 4375 


6 12 


24 


48 


.75 






49 


.76 5625 




25 


50 


.78 125 






51 


.79 6875 


13 


26 


52 


.81 25 






53 


.82 8125 




27 


54 


.84 375 






55 


.85 9375 


7 14 


.28 


56 


.87 5 






57 


.89 0625 




29 


58 


.90 625 






59 


.92 1875 


15 


30 


60 


.93 75 






61 


.95 3125 




31 


62 


.96 875 






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 to 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 


■ 

= 1 foot 


144 square inches =1 square toot 


3 feet 


= 1 yard 


9 square feet = 1 square yard 


5H yards = 16H feet 


= 1 rod, 


30>4 square yards = 1 square rod, pole or 




pole or 


perch 




perch 


160 square rods 




40 poles = 220 yards 


= 1 furlong 


= 10 square chfuns 




8 furlongs = 1760 yards \ , „., 


= 43,560 sq. ft. 


= 1 acre 


= S280 feet 


j ~ ^^^^° 


■= 6645 sq. varas (Texas) 




3 miles 


= 1 league 


1 "section" 


4 inches 


= 1 hand 


., ofU. S. Govt. 


9 inches 


= 1 span 


640 acres = 1 square mile = gurygyg^ 
lland 






1 circular inch 


Nautical Units 


= area of circle 1 inch = . 7854 sq. in. 


6080.2 feet 


= 1 nautical mile 


in diameter 


6 feet 


= 1 fathom 


1 square inch =1.2732 oir. in. 


120 fathoms 


= 1 cable length 


1 circiilar mil = area of circle 0.001 in. 


1 nautical mile per hr 


= ] knot 


in diam. 
1,000,000 cir. mil3 = l cir. in. 


Surveyor's or Gun 


ter's Measure 
= llink 


Measures of Volume 


7.92 inches 


1728 cubic inches = 1 cubic foot 


100 links = 66 ft. = 4 rods = 1 chain 


27 cubic feet = 1 cubic yard 


80 chains 


= 1 mile 


1 cord of wood = 128 cu. ft. 


33H inches 


= 1 vara (Texas) 


1 perch of masonry = 16>4 to 25 cu. ft. 



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 6M in-i 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 



1 British shipping too = 



40 cu. ft. 
/ 32.14 U. S. bu. 
I 31.14 imp. bu. 

42 cu. ft. 

/ 32.70 imp. bu. 

I 33.75 U. S. bu. 



Board Measure 

f 144 cu. in. = volume of 
1 board foot = < board 1 ft. sq. and 1 in. 

i thick. 
No. of board feet ina log = [M(d - 4)]2i, 
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 
2000 pounds 
2240 pounds 
Also (in Great Britain): 
14 pounds 
2 stone = 28 lb. 
4 quarters = 1 12 lb. 

20 hundredweight 



= 1 cental 
= 1 short ton 
=3 1 long ton 

= 1 stone 
= 1 quarter 
= 1 hundred- 
weight (cwt.) 
= 1 long ton 



Troy Weight 

= 1 penny- 
weight (dwt.) 
20 pennyweights = 480 grains = 1 ounce 
12 ounces = 5760 grains = 1 pound 

1 Assay Ton = 29,167 milHgrams, or 
as many milligrams as there are troy 
ounces in a ton of 2000 lb. 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 lb. avoirdupois. 



24 grains 



Apothecaries' Weight 

20 grains = 1 scruple 3 

3 scruples = 60 grains = 1 dram 5 

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. AH 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 lb. = 453.59243 grams. The value of the troy 
lb. is based on the same relation and also the equivalent 5760/7000 
avoirdupois lb. equals 1 troy lb. 

Metric Measures 



Length 


Area 


Unit 


Sym- 
bol 


Value in meters 


Unit 


Sym- 
bol 


Value in sq. 
meters 




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. . . , 


Sq. millimeter 

Sq. centimeter 

Sq. decimeter 

Sq. meter (centiare) 
Sq. dekameter (are) 


mm.' 
cm. 2 
dm .2 

m.2 
a. 

ha. 
km.' 


0.000001 
0.0001 
0.01 
1.0 
100.0 
10,000 . 


Kilometer 

Myriameter... . 


Sq. kilometer 


1,000,000.0 


















Volume 


Cubic measure 


Unit 


Symbol 


Value in 
liters 


Unit 


Symbol 


Value in 
cubic 
meters 


Milliliter 


ml. or cm. 2 
1. or dm. a 
kl. or m.* 

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 


km." 
hm.' 
dkm.' 

m.' 
dm.' 
cm.' 
mm.' 

«' 


10' ' 


Liter (unit) 


10» 


Kiloliter. . 


10' 


Also 
Centiliter 


1 






10"' 


Dekaliter 


Cubic centimeter 


10"' 




10"' 






10-18 

















Weight 



Unit 


Symbol 


Value in 
grams 


Unit 


Symbol 


Value in 
grams 






0.000001 

0.001 

0.01 

0.1 

1.0 




dkg. 

kl 
Mg. 

q. 

t. 


10 




mg. 
eg. 
dg. 




100.0 






1,000.0 






10,000.0 
100,000.0 










1,000,000.0 









SYSTEMS OP 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 mot with in the literature. 



UNITS 



73 



The preoise 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.^ = 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 vacuo, in the standard locaUty ; 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 kllo£rrani {force) is the force required to support the standard kilogram against 
gravity, in vacuo, ia 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 








Dimen- 


"gravita- 


"gravita- 


Metric 


British 
"absolute" 

system 
(little used) 


Name of 
unit 


sions of 
units in 


tional " sys- 
tem, or 


tional" sys- 
tem, or 


"absolute" 
system, or 


terms of 


"foot-pound- 


"kilogram- 


"C. G. S." 




F,L, T 


second" 


meter-sec- 


system 






system 


ond" system 






Force 


F 


1 lb. 


1kg. 


1 dyne 


1 poundal 


Length 


L 


1 ft. 


1 m. 


1 cm. 


1ft. 


Time 


T 


1 sec. 


1 sec. 


1 sec. 


1 sec. 


Velocity 


L/T 


1 ft. per sec. 


1 m. per sec. 


1 cm. per sec. 


1 ft. per sec. 


Acceleration. . 


L/T"^ 


1 ft. per sec. 2 


1 m. per sec. ^ 


1 cm. per sec.^ 


1 ft. per sec. 2 


Pressure 


F/L-^ 


lib. per ft. z 


1 kg. per m.2 


1 dyne per cm.^ 


Ipdl. perft.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 "erg." 


1 ft.-pdl. 


Power 


FLIT 


1 ft.-lb. per 


1 kg.-m. per 


1 dyne-cm. per 


1 ft.-pdl. per 






sec. 


sec. 


sec. 


sec. 


Mass 


F/iL/T-^) 


1 lb. 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 
slug." 


gram mass. 


1 pound 
mass. 



WOTE. The ■ slug l.aiso caiiea tne geepouna, or tne engineer s unit or mass J, 
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^o 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^" 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. 

Heat Units The units of Equivalents 

heat commonly used are (1) •iv.iwo c>iui>a.ioxii.a 

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 unit or B . t.u. 

The kilogram-calorie (large calorie), which is equal to 1000 g.-caJ., is largely 

used in engineering work in metric countries. * 1 therm = 1 g.-cal. 

CONVERSION TABLES 

Length Equivalents 



Dynes X 10« 


Kilograms 


Pounds 


Poundals 


1 


1.020 
0.00848 


2.248 
0.03518 


72.33 
1.85933 


0.9807 
'1.99149 


1 


2.205 
0.34334 


70.93 

1.85084 


0.4448 
i. 64819 


0.4536 
i. 65 667 


1 


32.17 

1.50760 


0.01383 

2.14067 


0.01410 
2.14916 


0.03108 

2.49249 


1 



Centimeters 


Inches 


Feet 


Yards 


Meters 


Chains 


Kilometers 


Miles 


1 


0.3937 
1.69517 


0.03281 

r.61598 


0.01094 

2.03886 , 


0.01 

2.00000 


0.0s497I 
4.69644 


lO-B 
6.00000 


0.066214 
6.79335 


2.540 

0.40483 


1 


0.038333 

4.92082 


0.02778 

2.44370 


0.0254 

2.40483 


0.0il263 

6.10127 


0.04254 

S. 40483 


0.0iI578 

6.19818 


30.48 

1.4S402 


12 

1.07918 


1 


0.3333 

1.6228S 


0.3048 

1.48402 


0.01515 

2.18046 


0.0a3098 

4.48402 


0.0,1645 
4.21608 


91.14 
1.96114 


36 
1.55630 


3 

0.47712 


1 


0.9144 
1.96114 


0.04545 

2.65768 


0.0:9144 

4.96114 


0.0.5682 

4.75449 


100 

2.00000 


39.37 
1.59517 


3.281 
0.61598 


1 .0936 
0.03886 


' 


0.04971 
2.69644 


O.OOI 
3.00000 


O.O36214 
4.79336 


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 
6.00000 


39370 
4.59517 


3281 
3.51598 


1093.6 

3.03886 


1000 
3.00000 


49.71 

1.69644 


1 


0.6214 
1.79335 


160925 

6.20665 


63360 

4.80182 


5280 
3.72263 


1760 

3.24651 


1609 
3.20665 


80 

1.90309 


1.609 
0.20665 


1 



The equivalents are given in the heavier type. Logarithms of the equivalents are 
given immediately below. 

Subscripts after any figure, O3, 94, etc., 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 Icilo- 
meters 


Kilo- 
meters 
to miles 


1 

2 
3 
4 

5 
6 
7 
8 
9 


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 Enqineeks' Handbook. 



CONVERSION TABLES 



75 



Mechanical Equivalent of Heat. Seep. 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; tor its own purposes it takes the 59 deg. tahr. 
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 fract 


ions of 


an inch to millimeterB 














(From Hi 


to 1 in.) 










64ths 


Milli- 
meters 


64th3 


MilU- 
meters 


64ths 


Milli- 
meters 


64ths 


Milli- 
meters 


64ths 


Milli- 
meters 


64thB 


Milli- 
meters 


, 


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 


5« 


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.) 









I 


2 


5 


4 


5 


6 


7 


8 


9 


.0 
.1 
.2 
.3 
.4 

.5 
.5 
.7 
.8 
.9 


2.540 
5.080 
7.620 
10.160 

12.700 
15.240 
17.780 
20.320 
22.860 


0.254 
2.794 
5.334 
7.874 
10.414 

12.954 
15.494 
18.034 
20.574 
23.114 


0.508 
3.048 
5.588 
8.128 
10.668 

13.208 
15.748 
18,288 
20.828 
23,368 


0,762 
3,302 
5,842 
8,382 
10,922 

13,462 
16,002 
18,542 
21,082 
23,622 


1,016 
3,556 
6,096 
8,636 
11,176 

13,716 
16,256 
18,796 
21,336 
23,876 


1,270 
3,810 
6,350 
8,890 
11,430 

13,970 
16,510 
19,050 
21 ,590 
24,130 


1,524 
4,064 
6,604 
9,144 
11,684 

14,224 
16,764 
19,304 
21,844 
24,384 


1,778 
4,318 
6,858 
9,398 
11.938 

14.478 
17.018 
19.558 
22.098 
24.638 


2.032 
4.572 
7.112 
9.652 
12.192 

14.732 
17.272 
19.812 
22.352 
24.892 


2.286 
4.826 
7.366 
9.906 
12.446 

14 986 
17.526 
20.066 
22.606 
25.146 



Millimeters to decimals of an inch. (From 1 to 99 mm.) 





0. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


9. 


n 




0,0394 


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 


7 


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 


A 


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 


3.8975 



'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 


Sciuare 
miles or 
sections 


1 


1550 
3.19033 


10.76 

1.03197 


1.196 
0.07773 


0.0395 

2.59699 


0.002471 
3.39288 


0.039884 
3.99494 


0.0,2471 

1.39288 


0.0.3861 
7.58670 


0.0i6452 
4 80967 


1 


0.006944 
■3.84164 


0.0011 
3.88740 


0.042551 

"6.40667 


0.0sI594 
"6.20255 


0.0.6377 

7.80461 


0.0e1594 

7.20255 


0.0)4910 

10.39637 


0.09290 

2.96803 


144 

2.15836 


1 


0.1111 

1.04576 


0.003673 

3.56503 


0.032296 
4.36091 


0.0<9184 

5.96297 


0.0<2296 

4.36091 


0.0,3587 
■B. 554 73 


0.836) 

1.92227 


1296 
3.11260 


9 

0.95424 


1 


0.03306 

2.51927 


0.002065 
3.31515 


0.038264 

4.91721 


0.0002066 
4.31515 


0.0.3228 

7.60898 


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 


0.059766 
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 


0.0.3906 
"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 


2589)8 
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 

feet 


Cubic 
yards 


U. S. 
Apothe- 
cary 
liquid 
ounces 


U. S. quarts 


U. S. gallons 


Bushels 
U. S. 




Cubic 
inches 


Liquid 


Dry 


Liquid 


Dry 


Liters 
(I) 


1 


0.035787 

4.76246 


0.0)2143 

6.33109 


0.5541 

r.74360 


0.01732 

2.23845 


0.01488 
2.17263 


0.024329 
3.63639 


0.023720 

3.57057 


0.034650 

4.66748 


0.01639 

2.21450 


1728 
3.23754 


1 


0.03704 

2.56864 


957.5 

2.98114 


29.92 
1.47699 


25.71 

1.41017 


7.481 

0.87393 


6.429 
0.80811 


0.8036 

1.90502 


28.32 
1.45205 


46656 
4.66891 


27 

1.43136 


1 


25853 
4.41251 


807.9 
2.90736 


694.3 
2.84153 


202.0 
2.30530 


173.6 

2.23948 


21.70 
1.33638 


764.6 

2.83341 


1.805 

0.25640 


0.001044 
3.01886 


0.043868 

5.58749 


1 


0.03125 
2.49485 


02686 
2.42903 


0.007813 

3.89279 


0.006714 

3". 82697 


0.038392 

4.92388 


0.02957 

2.47091 


37.75 
1.76155 


0.03342 

2.52401 


001238 

3.C9264 


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 


0.2909 
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 
r. 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.C6582 


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 

1.42188 


0.2270 

1.35606 


0.02838 

2.45297 


1 



The equivalents are given in the heavier type. Logarithms of the equivalents are 
given immediately below. 

Subscripts after any figure, O3, 94, etc., mean that that figure is to be repeated the 
indicated number of times. 



CONVERSION TABLES 



77 











Conversion of Areas 










Sq. in. 


Sq. cm. 


Sq. ft. 


Sq. m. 


Sq. yd. 


Sq. m. 


Acres 
to 
hec- 
tares 


Hec- 


Sq. mi. 


Sq. km 




to 


to 


to 


to 


to 


to 


tares to 


to 


to 




sq. cm. 


sq. m. 


sq. m. 


sq. ft. 


sq. m. 


sq. yd. 


acres 


sq. km. 


sq. mi. 


1 


6.452 


0.1550 


0.0929 


10.76 


0.8361 


1.196 


0.4047 


2.471 


2.590 


0.3861 


2 


12.90 


0.3100 


0.1858 


21.53 


1.672 


2.392 


0.8094 


4.942 


5.180 


0.7722 


3 


19.35 


0.4650 


0.2787 


32.29 


2.508 


3.588 


1.214 


7.413 


7.770 


1.158 


4 


25.81 


0.6200 


0.3716 


43.06 


3.345 


4.784 


1.619 


9.884 


10.360 


1.544 


5 


32.26 


0.7750 


0.4645 


53.82 


4.181 


5.980 


2.023 


12.355 


12.950 


1.931 


6 


38.71 


0.9300 


0.5574 


64.58 


5.017 


7.176 


2.428 


14.826 


15.540 


2.317 


7 


45.16 


1.085 


0.6503 


75.35 


5.853 


8.372 


2.833 


17.297 


18.130 


2.703 


8 


51.61 


1.240 


0.7432 


86.11 


.6.689 


9.568 


3.237 


19.768 


20.720 


3.089 


9 


58.06 


1.395 


0.8361 


96.87 


7.525 


10.764 


3.642 


22.239 


23.310 


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 


1 
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 
0.427? 
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 
lb.) to 
metric 
tons 



Metric 
tons 
(1000 

kg.) to 
short 
tons 



Long 
tons 
(2240 
lb.) to 
metric 
tons 



Metric 
tons 
to 
long 
tons 



0.06480 
0.1296 
0.1944 
0.2592 

0.3240 
0.3888 
0.4536 
0.5184 
0.5832 



15.43 
30.86 
46.30 
61.73 

77.16 
92.59 
108.03 
123.46 
138.89 



28.35 
56.70 
85.05 
113.40 

141.75 
170.10 
198.45 
226.80 
255.15 



0.03527 
0.07055 
0.1058 
0.1411 

0.1764 
0.2116 
0.2469 
0.2822 
0.3175 



0.4536 
0.9072 
1.361 
1.814 

2.268 
2.722 
3.175 
3.629 
4.082 



2.205 
4.409 
6.614 
8.818 

11.02 
13.23 
15.43 
17.64 
19.84 



0.907 
1.814 
2.722 
3.629 

4.535 
5.443 
6.350 
7.257 
8.165 



1.102 
2.205 
3.307 
4.409 

5.512 
6.614 
7.716 
8.818 
9,921 



1.016 
2.032 
3.048 
4.064 

5.080 
6.096 
7.112 
8.128 
9.144 



0.984 
1.968 
2.953 
3.937 

4.921 
5.905 
6.889 
7.874 
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 see. 


Feet 
per min. 


MUes 
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 

r. 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 

r48402 


0.01829 

2.26217 


0.01667 

2.22185 


1 


0.01136 
2.05553 


0.00987 
3.99423 


44.70 

1.65035 


0.4470 

1.65035 


26.82 
1.42850 


1.609 

0.20670 


1.467 
0.16633 


88 
1.94448 


1 


0.86839 
1.93871 


51.497 
1.71178 


0.51497 
1.71178 


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) 





Grains 


Ounces 


Pounds 


Tons 


Kilograms 


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 


O.O2IIO2 
3.04230 


0- 039842 
'4.99309 


0.001 

3.00000 


0.0i648a 
5.81157 


1 


0.0^2083 
3.31876 


0.022286 
3.35902 


0.031736 

4.23958 


O.O3I429 

4.15490 


0.0;7143 

8.85387 


O.O76378 

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)3429 
5.53511 


O.O43O6I 

5.48590 


O.O43IIO 
5.49281 


0.02835 
2.45255 


437.5 
2.64098 


0.9115 

1 .95974 


1 


0.07595 
2.88056 


0.0625 

2.79588 


0.0i3125 

5.49485 


O.O4279O 

5.44563 


0.0<2835 

5.45255 


0.3732 

T.57199 


5760 

3.76042 


12 

1.07918 


13.17 
1.11944 


1 


0.8229 
1.91532 


O.O34II4 
4.61429 


0.033673 

4.56508 


O.O33732 
4.57199 


0.4536 
1.65667 


7000 
3.84510 


14.58 

1.16386 


16 

1.20412 


1.215 

0.08468 


1 


0.0005 

4.69897 


0.034464 

4.64975 


O.O34536 
4.65667 


907.2 
2.95770 


1406 
7.14613 


29167 
4.46489 


3203 
4.50515 


2431 
3.38571 


2000 
3.30103 


1 


0.8929 
1.96078 


0.9072 
T. 95770 


1016 
3.00691 


156801 
7.19535 


3263 
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 


32I5I 
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, Oi, 94, etc., mean that that figure is to be repeated the 
indicated number of times. 



CONVERSION TABLES 



79 



Pressure Equivalents 

(For converBion table see p. 80) 



Megabars 

or 

megadynes 

per 


Kilo- 
grams 

per 
sq. cm. 
(Metric 
atmos- 
pheres) 


Pounds 

per 
sq. in. 


Short 
tons 
per 

sq. ft. 


Atmos- 
pheres 


Columns of 
mercury at 
temperature 
0°C. 


Columns of water at 
temperature 15° C. 




Meters 


Inches 


Meters 


Inches 


Feet 


1 


1.0197 


14.50 


1.044 


0.9869 


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 


1.84738 


1.44254 


0.36336 


0.9576 


0.9755 


13.89 


1 


0.9450 


0.7182 


28.28 


9.773 


384.8 


32.06 


1.93119 


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 

0.12492 


1.3596 
0.13340 


19.34 

1.28640 


1.392 
0.14373 


1.316 

0.11919 


1 


39.37 
1.69517 


13.61 

1.13378 


535.7 

2.72894 


44.64 
1.64976 


0.03386 


0.03453 


0.4912 


0.03536 


0.03342 


0.02540 


I 


0.3456 


13.61 


1.134 


2.52975 


2.53823 


1.69124 


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 


1 


39.37 


3.281 


2.99114 


2.99962 


0.15262 


1.00996 


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 


^■.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 


KUo- 
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 


0.01^778 
7.44370 


0.063777 
7.57711 


0.063725 
7.57113 


0.009869 
3.99427 


0.032390 

4.37848 


0.039486 
4.97709 


9.80665 
0.9915207 


1 


7.233 

0.85932 


0.062724 

6.43522 


0.0637037 

6.56863 


0.063653 

6.56265 


0.09678 

2.98579 


0.002344 

3.37000 


0.009302 
3.96861 


1.356 
0.13220 


0.1383 
1.14068 


1 


0.0e3766 

7.57590 


0.0651205 

7.70932 


0.0650505 
7.70333 


0.01338 

2.12647 


0.033241 

4.51068 


0.001286 
3.10929 


3.5X10« 

8.55630 


3.671X10' 

6.66478 


2.655X10« 
6.42410 


1 


1.3596 
0.13342 


1.341 

0.12743 


35528 

4.55057 


850.5 

2.93478 


3415 
3.63330 


2.648XI0" 
6.42288 


270000. 
5.43136 


1.9529X10" 
6.29068 


0.7355 

1.86658 


1 


0.9853 
1.99401 


26131. 
4.41715 


632.9 

2.80135 


2512 
3.39906 


2.6845XI0' 
6.42887 


2.7375X 10' 
5.43735 


1.98X10' 
6.29667 


0.7457 

1.87257 


1.0139 
0.00598 


I 


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 


0.0i3827 

5.58284 


0.0i3774 

5.57686 


' 


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 


0.0j2928 
4.46661 


0.033981 

4.60003 


0.0j3927 
4.59405 


10.40 
1.01719 


0.25200 
1.40139 


' 



The equivalents are given in the heavier type. Logarithms of the equivalents are 
given immediately below. 

Subscripts after any figure, Os, 9d, etc., mean that that figure is to be repeated the 
indicated number of times. 



80 



WBIQHTS AND MEASURES 





Linear and 


Angulai 


Velocity Conversion 


Factors 






Cm. per 


Feet per 


Cm. per 


Miles 


Feet per 


Miles 


Radians 


Rev. per 




sec. to 


min. to 


sec. to 


per hour 


sec. to 


per hour 


per sec. 


min. to 




feet per 


cm. per 


miles 


to cm. 


miles 


to feet 


to rev. 


radians 




mm. 


sec. 


per hour 


per sec. 


per hour 


per sec. 


per mm. 


per sec. 


1 


1.97 


0.508 


0224 


44.7 


0.682 


1.47 


9.55 


0,1047 


2 


3.94 


1.016 


00447 


89.4 


1.364 


2.93 


19.10 


0,2094 


3 


5.91 


1.524 


0.0671 


134.1 


2.046 


4.40 


28.65 


0.3142 


4 


7.87 


2.032 


0.0895 


178.8 


2.727 


5.87 


38.20 


0.4189 


5 


9.84 


2.540 


0.1118 


223.5 


3.409 


7.33 


47.75 


0.5236 


6 


11.81 


3.048 


0,1342 


268.2 


4,091 


8.80 


57.30 


0.6283 


7 


13.78 


3.555 


0.1566 


312.9 


4,773 


10.27 


66.85 


0.7330 


8 


15.75 


4.064 


0.1789 


357.6 


5,455 


11.73 


76,39 


0.8378 


9 


17.72 


4.572 


0.2013 


402.3 


6,136 


13.20 


85,94 


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 
2 
3 
4 

5 
6 
7 
8 
9 


0.0703 
0.1406 
0,2109 
0.2812 

0.3515 
4218 
0.4922 
0.5624 
0,6328 


14,22 
28.45 
42.67 
56.89 

71.12 
85.34 
99.56 
113.8 
128,0 


14.70 
29,39 
44,09 
58.79 

73.48 
88.18 
102.9 
117.6 
132.3 


0,0680 
0,1361 
0,2041 
0.2722 

0.3402 
0,4082 
0,4763 
0,5443 
0.6124 


1.033 
2.067 
3.100 
4.133 

5.166 
6.200 
7.233 
8.266 
9.300 


0.9678 
1.936 
2.903 
3.871 

4.839 
5.807 
6.774 
7.742 
8.710 







Conversion of Knergy, 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 
joules 




0.1383 


7.233 


0.001286 


777.5 


002344 


426,6 


0.2390 


4,183 




0.2765 


14,47 


0,002572 


1555.0 


00)688 


853,2 


0.4780 


8,367 




0.4148 


21,70 


0,003858 


2333.0 


0,007033 


1280,0 


0,7170 


12,55 




0.5530 


28.93 


0,005144 


3110.0 


0,009377 


1706.0 


0,9560 


16.73 




0,6913 


36.16 


006431 


3888.0 


0.01172 


2133.0 


1,195 


20.92 




0,8295 


43.40 


007717 


4665.0 


01407 


2560,0 


1,434 


25.10 




0,9678 


50,63 


009003 


5443.0 


01641 


2986,0 


1,673 


29.28 


8 


1.106 


57.86 


029 


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 


7 


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. 


per Beo. 




550 stand- 
ard ft.-lb. 
per sec. 


B.t.u 
per sec. 


1 


0.7457 

1.87256 


1.014 

0.00599 


0.7604 

1.88105 


76.04 

1.88105 


550 

2.74036 


0.1783 

1.25104 


0.7074 

1.84965 


1.341 
0.12743 
0.9863 
1.99402 


1 

0.7355 

1.86659 


1.360 
0.13343 

1 


1.020 

0.00848 

0.75 
1.87506 


102.0 
2.00848 

75 
1.87506 


737.6 

2.86780 

542.3 

2.73438 


0.2390 

1.37848 
0.1758 
T. 24506 


0.9486 
1.97709 
0.6977 

1.84367 


1.315 

0.11896 


0.9807 

1.99152 


1.333 

0.12493 


1 


100 

2.00000 


723.3 

2.85932 


0.2344 
1.37000 


0.9303 

1.96861 


0.01315 

2.11896 


0.009807 
3.99152 


0.01333 

2.12493 


0.01 

2.00000 


1 


7.233 
0.85932 


0.002344 
3.37000 


0.009303 

2.96861 


0.00182 
3.25946 


0.001356 
3.13219 


0.00184 

3.26562 


0.00138 

3.14067 


0.1383 

3.14067 


1 


0.033241 

4.51068 


0.001286 

3.10929 


5.610 

0.74896 


4.183 

0.62153 


5.688 
0.75494 


4.266 
0.63000 


426.6 
2.63000 


3086 
3.48932 


1 


3.968 
0.59861 


1.414 
0.15035 


1.054 
0.02291 


1.433 
0.15632 


1.075 
0.03139 


107.5 
2.03139 


777.5 

2.89071 


0.2520 
1.40138 


1 



The equivalents are given in the heavier type. Logarithms of the equivalents are 
given immediately below. 

Subscripts after any figure, O3, 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 grama 

per cu. 

cm. 


Grams 

per 
cu. cm. 
to short 
tons per 
cu. yd. 


Short 
tons per 

cu. yd. 

to grama 

per cu. 

cm. 


1 

27.68 
1.44217 

0.01602 
2.20466 
1.186 
0.07428 
0.1198 
r.07855 


0.03613 

2.55787 

1 

0.035787 
4.76245 
0.04286 
2.63205 
0,004329 
3.63639 


62.43 
1.79539 

1728 
3.23754 

1 

74.07 

1.86964 
7.481 
0.87396 


0.8428 
1.92572 

23.33 
1.36792 

0.0135 

2.13033 

1 

0.1010 
1.00432 


8.345 
0.92143 

231 

2.36361 

0.1337 

1.12613 
9.902 
0.99572 

1 


I 
2 

3 

4 

5 
6 
7 
8 
9 
10 


62.43 
124.90 

187.30 
249.70 

312.40 
374.60 
437.00 
499.40 
561 .90 
624.30 


0.01602 
0.03204 

0.04806 
0.06407 

0.08009 
0.09611 
0.11210 
0.12820 
0.14420 
0.16020 


0.8428 
1.6860 

2.5280 
3.3710 

4.2140 
5.0570 
5.9000 
6.7420 
7.5850 
8.4280 


1.186 
2.373 

3.600 
4.746 

5.933 
7.119 
8.305 
9.492 
10.680 
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. 
per 1 deg. cent, per 
cm. thick, to B.t.u. 


sq. ft. per 1 deg. 
fahr. per in. thick 




per sq. 


ft. to 


per sq. cm. 


per in. to 




cm. to 


small 


per cm. to 
B.t.u. per 


small 


to small calories 




B.t.u. 


calories 


calories per 


per hr. per sq. ft. 
per 1 deg. fahr. 


per sec. per sq. cm. 
per 1 deg. cent. 




per aq. 


per sq. 


sq. ft. 


sq. cm. 




ft. 


cm. 


per in. 


per cm. 


per in. thick 


per cm. thick 


I 


3.687 


0.2712 


1.451 


0.6892 


2.903X10' 


O.O33445 


2 


7.374 


0.5424 


2.902 


1.378 


5.806X103 


O.O3689O 


^ 


11.06 


0.8136 


4.353 


2.068 


8.709X10' 


0.0S1034 


4 


14.75 


1.085 


5.804 


2.757 


11.61 XIO' 


0.0:1378 


•i 


18.44 


1.356 


7.255 


3.446 


14.52 X10' 


0.0:1722 


A 


22.12 


1.627 


8.706 


4.135 


17.42 X10» 


0.0:2067 


7 


25.81 


1.898 


10.16 


4.824 


20.32 X10» 


0.0:2412 


8 


29.50 


2.170 


11.61 


5.514 


23.22X10= 


0.0:2756 


9 


33.18 


2.441 


13.05 


6.203 


26.13 X10S 


0.0:3100 



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' 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 (6) 

Au8tria-Hungary(G) 
Belgium ((? and S) . 

Bolivia (G) 

Brazil (G) 

British colonies in. . 

Australasia and 

Africa (G). 

Canada (G) 

Central American 
States: - 

Costa Rica (G)... . 

British Honduras 
(G or S). 

Guatemala (5).. . . 

Honduras (S) 

Salvador (S) 

Nicaragua (5) .... 

Chile (G) 

China (S) 

Colombia (G) 

Denmark (G) 

Ecuador (G) 

Egypt (G) 

Finland (G) 

France (G or S) . . . 
German Empire (G) 



Peso 

Crown 

Franc 

Boliviano . 
Milreis. . . . 
Pound ster- 
ling. 

Dollar..... 

Colon 

Dollar 

Peso 

Peso 

Peso 

Cordoba. . . 

Peso 

Yuan 

Pound 

Crown 

Sucre 

Pound 

Markka. . . 

Franc 

Mark 



$0.9647 

0.2026 
0.1929 
0.3893 
0.5463 

4.8665 

1.0000 



0.4653 
1.0000 

0.4446 
0.4446 
0.4446 
1.0000 
0.3649 
0.4777 
4.8665 
0.2680 
0.4866 
4.9429 
0.1929 
0.1929 
0.2381 



Great Britain (G) .... 

Greece (G and S) 

Haiti (G) 

India (British) (G) . . . 

Italy (G and S) 

Japan (G) 

Liberia (G) 

Mexico (G) 

Netherlands (G) 

Norway (G) 

Panama (G) 

Persia (G and S) 

Peru (G) 

Philippine Islands (G) 

Portugal (G) 

Roumania (G) 

Russia (G) 

Santo Domingo (G). . . 

Servia (G) 

Siam (G) 

Spain (G and S) 

Straits Settlement (G) 

Sweden (G) 

Switzerland (G) 

Turkey (G) 

Uruguay (G) 

Venezuela (G) 



Pound ster- 
ling. 
Drachma. . 

Gourde 

Rupee 

Lira 

Yen 

Dollar 

Peso 

Florin 

Crown 

Balboa. . . . 

Kran 

Libra 

Peso 

Escudo 

Leu 

Ruble 

Dollar 

Dinar 

Tioal 

Peseta 

Dollar 

Crown 

Franc 

Piaster. . . . 

Peso 

Bolivar. . . . 



S4.8665 

0.1929 
0.9647 
0.3244 
0.1929 
0.4984 
1.0000 
0.4984 
0.4019 
0.2679 
1.0000 
Variable 
4.8665 
0.5000 
1.0805 
0.1929 
0.5145 
1.0000 
0.1929 
0.3708 
0.1929 
0.5677 
0.2679 
0.1929 
0,0439 
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 
365H 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 XJ. 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 6f 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 all 
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 go = 980.665 cm. per 
sec. per sec, or 32.1740 ft. per sec. per sec. This value, go, is assumed to be 
the value of g at sea level and latitude 45 deg. 

Acceleration of Gravity 

(U. S. Coast and Geodetic Survey, 19?2) 



Latitude, 





b/Bo 


Latitude, 
deg. 


a 


g/Ot 


deg. 


Cm./sec.! 


Ft./sec' 


Cin./sec.2 


rt./seo.! 



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 
9979 
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 


1.0004 
1 0013 
1 .0020 
1 .0024 
1,0026 



Correction for altitude above sea level: — 0.3 cm. per sec. 2 for each 1000 meters; 
— 0.003 ft. per sec.^ tor 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 BaumS, Twaddell 
and Beck. Twaddell's hydrometer is used for liquids heavier than water. 
The number of degrees. A'', whieh it indicates may be converted to specific 
gravities, G, by the formula G = (5N + 1000) /lOOO. 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 
Baumfi hydrometer are given in the following tables. 
•See Marks' Mechanical Engineers' Handbook. 



SPECIFIC GRAVITY AND DENSITY 



85 



60° 

Specific Gravities at ttt Fahr. Corresponding to Degrees Baume 

dO 



for Liquids Lighter than Water 

60° 140 

Calculated from the formula, specific gravity -t^-t: fahr. = , „^ , .„ ^, 

60 130 + Deg. Be 



:] 



h 

on 


03 bo 


fin 




fin 


CO a 


C3-0 

fil 


03 M 


CDO 


•3-6 
II 


SB 
fin 


■5-e 

II 


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 


0.8046 


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 


5fl 


0.7778 


65 


0.7179 


80 


0.6667 


95 


0.6222 


21 


0.9272 


35 


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 
100 


0.6114 
0.6087 



Specific Gravities at -r^ Fahr. Corresponding to Degrees Baume 

60 



for Liquids Heavier than Water 

60° 
Calculated from the formula, specific gravity ^--^ fahr. = 



145 



145 — Deg. Baum6 



fin 




fin 


11 


Is 

bD:3 
® OS 

fin 


•s-g 

II 


q5 


•s-c 


fin 


ll 


fin 


•3 '5 

S.2 

03 bll 




1 

2 

3 

4 
5 
6 
7 

8 

9 
10 

n 


1.0000 
1.0069 
1.0140 
1.0211 

1.0284 
1.0357 
1.0432 
1 .0507 

1.0584 
1.0662 
1 .0741 
1.0821 


12 
13 
14 
15 

16 
17 
18 
19 

20 
21 
22 
23 


1.0902 
1.0985 
1.1069 
1.1154 

1 . 1240 
1 . 1328 
1.1417 
1.1508 

1.1600 
1.1694 
1.1789 
1.1885 


24 
25 
26 
27 

28 
29 
30 
31 

32 
33 
34 
35 


1 . 1983 
1.2083 
1.2185 
1.2288 

1.2393 
1.2500 
1.2609 
1.2719 

1.2832 
1.2946 
1.3063 
1.3182 


36 
37 
38 
39 

40 
41 
42 
43 

44 
45 

46 
47 


1.3303 
1.3426 
1.3551 
1.3679 

1.3810 
1.3942 
1.4078 
1.4216 

1 .4356 
1.4500 
1.4646 
1.4796 


48 
49 
50 
51 

52 
53 
54 
55 

56 
57 
58 
59 


1 .4948 
1.5104 
1.5263 
1.5426 

1.5591 
1 .5761 
1 .5934 
1.6111 

1.6292 
1.6477 
1.6667 
1.6860 


60 
61 
62 
63 

64 
65 
66 
67 

68 
69 
70 


1.7059 
1.7262 
1.7470 
1.7683 

1 .7901 
1.8125 
1.8354 
1.8590 

1.8831 
1 .9079 
1 .9333 









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 OP MATHEMATICS, HARVARD UNIVERSITY, FELLOW AM. ACAD. 
ARTS AND SCIENCES 



CONTENTS 



ARITHMETIC 



Numerical Computation. 

Logarithms 

The Slide Rule 

Computing Machines 

Financial Arithmetic 



Page 

88 
91 
94 
97 
98 



GEOMETRY AND MENSURATION 

Geometrical Theorems 99 

Geometrical Constructions 10 1 

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 136 

ANALYTICAL GEOMETRY 

The Point and the Straight Line 136 

The Circle 137 



Page 

The Parabola 138 

The Ellipse 140 

The Hyperbola 144 

The Catenary 147 

Other Useful Curves 151 

DIFFERENTIAL AND INTEGRAL 
CALCULXrS 

Derivatives and Differentials 157 

Maxima and Minima 1 59 

Expansion in Series 160 

Indeterminate Forms 163 

Curvature 163 

Table of Indefinite Integrals 164 

Definite Integrals 169 

Differential Equations 171 

GRAPHICAL REPRESENTATION OF 
FUNCTIONS 

Equations Involving Two Variables 173 

Equations for Empirical Curves. 174 

Logarithmic Cross-section Paper. 176 

Semi-logarithmic Paper 177 

Equations Involving Three Variables 1 78 

Equations Involving Four Variables 1 82 

VECTOR ANALYSIS 

Vector Analysis 185 



Copyright, 1916, by Edward V. Huntington 



87 



MATHEMATICS 

BY 
EDWARD V. HUNTINGTON 



ARITHMETIC 

NUMERICAL COMPUTATION 

Number of Significant Figures. In any engineering computation, the 
data are ordinarily tlie results of measurement, and are correct only 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* or 3.8400 X 10= or 3.84000 X 10'. 

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 0.2056x 
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" methodis 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. 14762 

[7 plus how many is 15?] 7 plus 8 is 15; 1 to carry. 8463 

S plus 2 is 7. 8 plus 6 is 14. 6289 



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. 
5 plus is 5; plus 2 is 7; plus 8 is 15; 1 to carry. 
3 plus 1 is 4; plus 1 is 5; plus 2 is 7. 
5 plus 3 is S; plus 6 is 14. 



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, °^'^ 



since it facilitates the abbreviation of the work by the 39648 

omission, in practice, of all the figures on the right of the 1486 

vertical line. 346 

The position of the decimal point should be determined ? 



92 
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: 

(o) 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. Keimer) gives the product 
of every three-figure number by every three-figure number; Peters's (Berlin, G. Reimer), 
of every four-figure nuniber 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 -H 2.3026 = 1.3644. 

To determine the position of the decimal point in a •' 

problem of fractional division, shift the point (mentally) in 
both numerator and denominator (the same number of 
places in each) until the denominator is a number in the 
"standard form, " that is, a number with only one figure pre- 
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 2V is l/Af". Instead of dividing by a long 
number N, it is often better to multiply by the reciprocal of A''. The table 
of reciprocals on pp. 24-27 gives the reciprocal of any number, correct to 
four figures. Barlow's Table (Spon & 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 1 + z when z is Small. 
1/(1 -|- a;) =1 — X -\- [error < x^, if x is between and 1], 

= 1 — X + x^ — [error < x', if x is between and 1]. 
1/(1 — X) = 1 + X ■{■ [error < x* -f- 2x', if x is between and ^], 

= 1 -|- X -(- x^ -|- [error < x' -)- 2x*, if x is between and }4]. 

Note. 1/(o ± 6) = (l/o)[l/(l ± x)\, where x = l/a. 

Notation by Powers of 10. AH 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' means 3.1416 with the point 
moved three places to the right, that is, 3141.6. Again, 3.1416 X 10~' 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— IS. (6) To obtain a 
root of six or seven figures from the table, use the formula: VJV = a + 
[(iV — o')/2o] (approx.), where a is the nearest value of V 2V obtainable 
from the table, with three or four ciphers annexed. Here o' must be found 
exactly, by direct multiplication, so that at least three significant figures 
of the difference N — a^ shall be known correctly ; but this done, the division 
ot N — a^ 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 ± z when z is Small. 

(1 -I- x)^ =\ -\-yix - [error less than Hx" if < x < 1] 
= \ +ViX — Hx^ + [error < M« a:' if < x <l] 

(1 — s)^ = 1 - ^x - [error < Hx^ + Hox' if < x < H] 

= 1 - ^x - Hx* -[error < Mox' + M«a:* if < x < J4] 

Note, y/a + b == \/o (1 + a;)", where x = b/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 + [(2V — a')/3o'] 
(approx.), where a is the nearest value of v^ obtainable from the table, with 
three or four ciphers annexed. Here o' must be found correct to seven or 
eight figures, by direct multiplication, so that at least three significant figures 
of the difference N — a^ shall be known; but this done, the division of iV — o' 
by 3a' 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 
deven 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 -I- s)*^ = 1 -I- Ha: - [error < \ix^ il < x < I], 

= I +\ix - %x' + [error < Htx^ if < x < l], 

(1 - x)^ = 1 - Mx - [error < ^x' + Hox' it < x < H], 

=: 1 - Hx -Jix^ - [error < Hex" + Htx* if < a; < «]. 
Note. \/a + b = V "(1 + ":) i where x = b/a. 

LOGARITHMS 

Tables of Loc^arithms. 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 foiu:-figure accuracy is required, recourse must be had to a larger table. 
Five-place tables are available in gteat 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 GrvEN Number is Between 1 and 10. 

An inspection of the table on pp. 42—43 shows that aa 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 ?io 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 ia 14 units, and fio 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 Ho of the difference 
(11), or about 9, gives: log 4.098 = 0.6126. Or better, since Ho of the way forward 
is equal to fio of the way back, find in table log 4.10 — 0.6128, and subtract ^o 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. 

(ft) 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 — 0.4732. 
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 lO"* (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 — 3.526S; 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. 

(o) When the gfven 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 JV = 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 ^ii of the way from 2.97 to the next higher 
number. But Yn 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 51 1 of the way from 4.09 to the next 
higher number. But Mi of 1 is 0.8 (to the nearest tenth), hence N = 4.098. 

(6) When the given logarithm has any given value (chakacteristio 
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, this 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 ( -j- 3) directs that the decimal 
point be moved 3 places to the right; hence N = 2.973 X 10= = 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 (o6) = loga -f log 6; (3) log (a") = re log o; 

(2) log (a/b) = log a - log 6; (4) log 's/a = (1/n) log a; 

(5) log 10" = n 
It is to be noted also that log 1=0, log 10 = 1, and log (1/re) = —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 

I = (4.098) (0.0002973) (72.1). log 0.0002973 = 0.4732-4 

Answer: x = 8.784 X lO-a log 72.1 = 0-8579 + 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. 



Example. 


Ta find . *-°^* 


log 4.098 = 0.6126 


0.0002973 


log 0.0002973 = 0.4732 - 4 


Answer: x 


= 1.378 X 10« = 13780 


log X = 0.1394 + 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 = 

63.4 log 63.4 = 0;8^1jfJ_= a8021J-J^ 

Answer: x = 4.590 X 10~' log x = 0,6618 - 4 

= 0,0004590. 

But if the logarithms are written with the characteristics in front, and the "shop 
method" of subtraction is used (see p. log 0.0291 = 2.4639 

88), then no such special device is here Iq„ g3_4 _ i,g021 

required. Thus: -= 

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 TV being defined by : colog iV = log (1/A'') = — logiV. 

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 69.6 = 0^20jfJ log 0.0002973 = 0^^4732^-— 4 

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 »; the result will be the 
logarithm of the rath power of that number. Then find the power itself from 
the tables. > 

Example 1. Find x = (0.0291)' log 0.0291 = 0.4639 - 2 

Answer: x = 2.464 X lO"' 5_ 

= 0.00002464. log X = 1.3917 - 6 = 0.3917 - 5. 



94 ARITHMETIC 

Example 2. Find x = (0.0291)'" log 0.0291 = 0.4639 - 2 = - 1.5361 

Answer: x = 6.825 X 10-> 1^ 

= 0.006825 15361 

61444 
15361 



log a! = -2.1859 

= 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 ra ; the result will be the log. of 
the nth root of that number. Then find the root itself from the table. 

Example. Find x = ^i.mS . log 4.098 = 0.6126 
Answer; a; = 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 = ^0.0004590. log 0.0004690 = 0.6618 - 4 

Answer: x = 7,714 X 10-« 3 )2.6618 - 6 

= 0.07714 log X = 0.8873 - 2 

But if the characteristio 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 ($36) 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 b^ 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 (Z), 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 oflf 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 Z), 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 MuLTlPLT 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, 



ijoc 



.1 .4.5 .6 .7 .8 .9 I 



II I 1 — i — I 1 — I — I— i-r-ri p 

r 8 9|[o 20 ! 30 40 50 to TO WaO^W" 



4. ^ 4x6 •i4 

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 -j- 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. 



r 1 



[Jc 



r I I > I I T T — 1 — r-T-i-r-ri _ 

5 6 7 6 9 l| 20 30 40 5 60]DtOWIiMP 



Fig. 2. 

Multiplication and Division, Usingr 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 
same 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. 

-*\ 



^ 

1 


1 2 3 456769 1|0 
1 ' 1 




1 — 1 — 1 1 F — r 1 r 1 1 

1 2l3 4 567«9l|0 


. ' . J ..,....,. 


«.__ ± --.-2 --"■"— 



FiQ. 3. 







-j 1 


? 


3 


4 5 e 7 B 9 Ik) 

-1 1 1 1 1 1 r 


I> 


2 


3 4 5 6|7 t9llo 


I... 





J 4*e('IOJ-6i7. 



Fig. 4. 



The Ordinary Mannheim Slide Rule has four scales. A, B, C, D, aa 
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. 



ja: 



j_, 



X- 



=wr 



lMi 



hH+V- 



T" 






T- 



Fig. 5. 



In problems involving successive multiplications and divisions, arrange 
the work so that multiplication and division are performed alternately. 



For example, to calculate 



o X 6 X c 



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 sUde 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 H- 2/, 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 X> (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 Kule;" F. A. Halsey, "Manual of the Slide 
Rule;" etc. 

Other Types of Slide Bules. The duplex slide rule ($5 to SIS aocording 
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., $S) 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 reqmring 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 Boss zneridlograph (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 Comptonaeter (Felt & Tarrant Co., 
Chicago, 111.; S250 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 lb. 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 Bapid conxputer (Benton Harbor, Mich., $25); 
the Gem (Automatic Adding Machine Co., New York; $10), the Arlthstyle (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. Bay Subtracto-Adder (Richmond, Va., $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, 
7 



98 ARITHMETIC 

Conn.) have a 10-key and a 9-key keyboard respectively, admitting of operation by the 
touch method. On all theee 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 Aznerican 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 bllllnff 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 TJnitas, with a double row of dials (both aold 
by Oscar Miiller Co., New York City; also with keyboard and electric drive), and the 
Renter (Philadelphia, Pa.)- Prices, $300 upward. Another machine of this type, 
with keylDoard, is the Record (U. S. Adding Machine Co., New York City). The 
Brunsviga is represented by Carl H. Renter, 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 facta 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) 

, Eight Triangles. a'+b^=c^. (See Fig. 1). ZA + ZB =90°. 
y^ = mn. a' = mc. b^ = 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, 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. 



4^^ 



Fig. 3. 




FiQ. 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 fc times the corresponding length in the other; 
then each area in the first figure is fc^ times the corresponding area in the second, 
and each volume in the first figure is k" 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 

99 



100 



ELEMENTARY GEOMETRY AND MENSURATION 



AP X AQ is constant; in particular, if A is an external point, AP X AQ 
= AT^, where AT is the tangent from A. 

J" 




(3)^ 



FiQ. 6. 



Fio. 7. 



Fig. 8. 




FiQ. 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. 11. 

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 ?4 of the way from any 
vertex to the center of gravity of the opposite face. The four perpendiculars 
erected at the ciroumcenters of the four faces meet in a point, the center of 
the circumscribed sphere. The four altitudes meet in a point called the 
orthooenter of the tetrahedron. The planes bisecting the six dihedral 
angles meet in a point, the center of the inscribed sphere. 




Fig. 12. 



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 the supplement 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). (o) 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. 

(b) 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 I; 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 tria.ngle. Or, use a sheet of celluloid with a set of lines parallel to 
one edge and about H in. apart ruled upon it. 



Xf' 



B\7C 



-^i' 



-^ 



'it 



Fig. 18. 



Fig. 19. 



M\ St 

Tt (a.) ( b.) 

FiQ. 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, (ft) 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. 



-,<4 



Fig. 21. 




Fio. 22. 



\<r. 



P 4 B 

Fig. 23. 



To Erect a Perpendicular to a Given Line at a Given Point P. 

(o) 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 (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 APB 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 



sions. These parallels will 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. 



A 



/ 
/ 



^^ 



f:-' 



\. 



Fig. 24. 



B 

Fig. 25. 



Fig. 26. 



To Construct a Mean Proportional (or Geometric Mean) Between 
Two Lengths, m and n (Fig. 25). Lay oS 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 = -ymn. (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' = 
AB XBX. Numerically, AX = His/s - 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 CM 
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'; b' which intersect; then bisect the angle between a' 
and b'. 




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 AA' will (if produced) pass through the intersection of the two 
given hues. 

To Construct, Approximately, the Lengthof 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. 




FiQ. 30. 



GEOMETRICAL CONSTRUCTIONS 



103 




Fig. 31. 



Fig. 32. 



Fig. 33. 




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- 
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 
Side AB being Given (Fig. 34). With A as 
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 re = 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 
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 7J -J- 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). 
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 




Fig. 35. 



Fig. 36. 




Fig. 37 



104 



ELEMENTARY GEOMETRY AND MENSURATION 




Fig. 40. 



AB as radius, draw arcs, out by BC in R and by AC in S. Divide RA into 
n equal parts, 1, 2, 3, . . . Divide B5 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. 
39). Let AB meet Une Hd 
C. Draw any circle through A and B, and let CT be tangent to this circle 
from C. Along I, lay off CP and CQ equal toCr. 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 5 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 




FiQ. 41. 





Fio. 42. 



Fig. 43. 



LENGTHS AND ABBAS OF PLANE FIGURES 



105 



draw a circle cutting SA 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 B — rhe 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 hy r = R sin (180° /n), or r = 
(fi + r)[sin (180° /«)]/[! + sin (180° /n)]. 

For methods of constructing ellipses and other curves, see pp. 
139-156. 




Fia. 44. 



LENGTHS AND AREAS OF PLANE FIGURES 



Right Triangle (Fig. 45). a' + b' ^ c'. 

Area = Vi ah = YiO,^ cot A = ^6" tan A = J4c' sin 2A 

Equilateral Triangle (Fig. 46). 



Area = Ho'VT = 0.43301a2. 
A 




PiQ. 45. 



FiQ. 46. 



Any T riangle (Fig. 47). g = ^ (a, +h +c), t = J^(mi -)- m2 + "is), 
r =\/(s — a)(s — b)(s — c)/s = radius inscribed circle, 
7} = )^ a/ sin A = yib/sinB = Mc/sin C = radius circumscribed circle; 
Area = yi base X altitude = Mah — yiob sin C = rs = abc/JR 

= ^s(,s -a){s -b)(s - c) = % -s/til - mi) {t - mi)(,t - ma) 

= r2 cot M A cot J^ B cot 1^ C = 2R'>- sin A sin B sin C 

= ±Vi\ (xij/2 — X2y\) + (122/3 — xsVi) + {x^Vl — iil/s) 1 , where 

(.xi, yi), (X2, 2/2), (X3, Vz) are co-ordinates of vertices. See also p. 134. 




Fig. 48. 



Fio. 49. 



£^J ^ 



FiQ. 60. 



Fig. 51. 



Rectangle (Fig. 48). Area = ah = }iD^ sin u. [u = angle between 
diagonals D, D.] 

Rhombus (Fig. 49). Area = o' sin C = IAD1D2. [C = angle between 
two adjacent sides; Di, D2 = diagonals.] 

Parallelogram (Fig. 50). Area = bh = ah sin C = ViDiD^ sin u. [u = 
angle between diagonals Di andJ[>2;I>i2 + Di^ = 2(.a' + h^)]. 

Trapezoid (Fig. 51). Area = 5^(0 + b)h = iiDiDi sin u. [Bases o andb 
are parallel ; u = angle between diagonals Di and Dt.] 



106 



ELEMENTARY GEOMETRY AND MENSURATION 



Quadrilateral Inscribed in a Circle (Fig. 52). Area = J^D^D! sin u = 
■V/(s — o)(s — 6)(s — c)(,s — d) = HCoc 4 6d)sin u; s = ii(.a + 6 + c +d). 

Any Quadrilateral (Fig. 53). Area = ^DiDj sin m. 

Note, a^ + 6^ -|. ^a ^ c;2 = ^jZ ^ 2)2=^ + 4?»2_ where m = distance between 
midpoints of Di and Dj. 

Polygons. See table, p. 39. 




Fig. 52. 



FiQ. 53. 



Fig. 64. 



<Q. 



Fig. 55. 




Circle. Area = irr^ = iiCr = YiCd = Vnrd^ = 0.785398d2 (table, p. 30). 
Here r = radius, d = diam., C = circumference = iicr — ird (table, p. 28). 

Annulus (Fig. 54). Area = vifi'^ - r') = ■n-(Z)2 - d')/i = 2TB'b, where 
R' = mean radius = ^{R + r), and b = B — r. 

Sector (Fig. 55). Area = ^rs = irrHA/SeO') = 
Hr' 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 = Hr' (rad A — sin A) 
= ^[r(s — c) + ch], where rad A = radian measure of 
angle A (table, pp. 34-35, 44). For small arcs, 
s = H(8c' — c), where c' = chord of half t he arc. 
(Huygens's approximation.) Note, c = 2'\/h{d —h) ; 
c' = ■ydh or d = c'^A, where d = diameter of circle; 
h=r {1 — cos y2A), s — 2r rad )iA. 

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, 
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 ^ 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 ri. + 1 parallel 
lines, called ordinates, drawn at constant dis- 
tance h apart; and denote the lengths of these 
ordinates by 2/0, Vi, 2/2, . . , 2/n. (Note that 
Vo or 2/n may be zero.) Then 
Area = HhKvo + Vn) + 4(yi + Vi + vs. . .) 
+ 2(j/2 + V^ + Vi. . . ) ], approx. The greater 
the number of divisions, the more accutate the result. Note: Taking y 
= f{x), where x varies from x = a to x — b, and h = (b — a)/n, then the 



Fig. 57. 




error = — 



1 (6 - a)' 
180 ' 



/""(X), where /""(X) is the value of the fourth de- 



rivative of f(,x) for some (unknown) value, i = X, between a and 6. 



auufACMS AND VOLUMES OP SOLIDS 



107 



Ellipse (Fig. 59; see also p. 140). Area of ellipse = irab. Area of shaded 
segment = xy + ab siii"^ (x/a). Length of perimeter of ellipse = ir(a + b)K, 
where K = [l + Hm'' + Htm* + ii$em^ + ...], m = (a - b)/(a + b). 
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 





Hyperbola (Fig. 60; see also p. 144). In any hyperbola, shaded area 

A == ab logs I — h r I • I'' ^1 equilateral hyperbola (o = 6) , area A = 
\o 6/ 

a' sinh— '(2//0) = a^cosh— '(x/a). 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 = %ch. In Fig. 62, 

length of arc OP = s = iiPT + Up log« cot Hu. 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^ft = Bh. Lateral 
area = 2Trrh = Ph. Here B = area of base; P = perimeter of base. 



1 

! 



ibs: 





Fig. 63. 



Fig. 64. 



Fig. 65. 



Fig. 66. 



Truncated Right Circular Cylinder (Fig. 65). Volume = Tr% = Bh. 
Lateral area = 2irrh = Ph. Here h = mean height = ii{hi + hi); 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,fe,c, I = k = H(o +& + c). Note: 
I and k will always be parallel to the elements. 




Fio. 67 



Fig. 68. 



FiQ. 70. 



Special Ungula of a right circular cylinder. (Fig. 68.). Volume = %r'fl. 
Lateral area = 2tH. r = radius. (Upper surface is a semi-ellipse.) 

Any TTngula of a right circular cylinder. (Figs. 69 and 70.) Volume = 
H(%a^ + cB)/ir + c) = H[a{r^ - Ha?) ± r'c rad u]/(r + c). Lateral area = 
H(2ra + cs)/(r + c) = 2rH(a + c rad u)/{r + e). If base is greater (less) 
than a semicircle, use -f- ( — ) sign, r = radius of base; B = area of base; 
« = 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 = 
7rA(fl2 -r') =Thb(.D- b) 
= vKb(,d + 6) = vhbD' = 
irhb {R -f r). Here h = 
altitude; r,fl(<i,D) = inner 
and outer radii (diameters) ; 
6 = thickness = iJ— r; 
D' = mean diam. =>^(d -i- 
D) =jD -6 =d-|-6. 

Regular Pyramid (Fig. 
71). Volume = H altitude 
X area of base = %hran. Lateral area = ^4 slant height X perimeter of 
base = Hsare. Here r = radius of inscribed circle; o = side (of regular 
polygon); n= number of sides; s = Vr^ -|- h^. Vertex of pyramid directly 
above center of base. 

Bight Circular Cone. Volume = Hirr%. Lateral area = vrs. Here 
r = radius of base; h = altitude; s = slant height = V r^ -f- A*. 

Frustum of Regular Pyramid (Fig. 72). 

Volume = Hhranll + (.a'/a}+ (a' /a)']. 

Lateral area = slant height X half sum of perimeters of bases = slant 
height X perimeter of mid-section = '/isn{r + r'). Here t,t' = radii 




Fig. 71. 





SURFACES AND VOLUMES OF SOLIDS 



109 



of inscribed circles; s = \/(r — r'y + h'; a,a' = sides of lower and upper 
bases; n = number of sides. 

Frustum of Right Circular Cone (Fig. 73). Volume = 
iiirr'h[l + (r'/r) + (r'/r)^] = ]^Trh{ r' + rr' + r'') = HTh[(T+ t'Y + ^{r-r'y]. 
Lateral area = ttsC?- + r'); s = "S/ {r — r'y + h^- 

Any Pyramid or Cone. Volume = \iBh. B = area of base; h = perpen- 
dicular distance from vertex to plane in which base lies. 

Any P yram idal or Conical Frustum (Fig. 74). Volume = 

HHB + VbB' + B') = HAB[1 + (.P'/P) + (P'/P)']- Here B, B' = areas of 
lower and upper bases; P,P' = perimeters of lower and upper bases. 



/ .^ 


^ 


/ 


// 


i/ii'ii 


h 


/^ 


uM 


V 




FiQ. 74. 



FiQ. 75. 



Fio. 76. 



Obelisk (Frustum of a rectangular pyramid. Fig. 75). 

Volume = MM(2a + oi)& + (2ai + a)bi] = }ih[ab + (o + ai) (b + bi) + Oibi]. 

Wedge (Rectangular base; ai parallel to a,a and at distance A above base. 
Fig. 76). Volume = i4Ab(2a + oi). 

Sphere. Volume =V = n^rr' = 4.1887907-' = Hwd' = 0.523599d' (table, 
p. 36) = % volume of circumscribed cylinder. Area = A = 4irr^ = four great 
circles (table, p. 30) = iriP = 3.14159^^ = la teral a rea of circumscribed cylinder. 
Here r = radius; d = 2r = diameter = ^/QV/tt = 1.24070 Vv = 

0.56419-\/I- 

Hollow Sphere, or spherical shell. Volume = 
Wx(iJ' - r") = HTr(X>' - d') = ^RiH + ^irtK Here 
R,r = outer and inner radii; D,d = outer and inner 
diameters; t = thickness = R — r; Ri = mean radius = 
MR + r). 

Spherical Segment of One Base. Zone (spher- 
ical "cap" of Fig. 78). Volume = HirhiSa" + h') = 
iiTrh'(,3r — h) (table, p. 38). Lateral area (of zone) = 
2TrTh= ir(.a^ + h'). Note: a' = h(,2r - h), where r 
= radius of sphere. 

Any Spherical Segment. Zone (Fig. 77). Vol- 
ume = %vh{3a'^ -{- 3ai* -|- A^). Lateral area (zone) 
= 2Trrh. Here r = radius of sphere. If the inscribed 
frustum of a cone be removed from the spherical seg- 
ment, the volume remai ning is jiThc'^, where c = slant 
height of frustum = V^^ + {a, — ai)'- 

Spherical Sector (Fig. 78). Volume = Hr X area .of 
Total area = area of cap + area, of cone = 2irrh + irra. 
h{2r - h). 




Fig 



cap = %irr'h. 
Note: o« = 



110 



ELEMENTARY GEOMETRY AND MENSURATION 



(Fig. 
area 



Spherical Wedge bounded by two plane semicircles and a lune. 
79.) Volume of wedge 4- volume of sphere = «/360°. Area of lune 
of sphere = «/360°. u = dihedral angle of the wedge. 

Spherical Triangle bounded by arcs of three great circles. (Fig. 80.) 
Area of triangle = vr'^E/lHO° = area of octant X £/90°. E = spherical 
excess = 180° — {A + -B + COi where A, B, and C are angles of the triangle. 
See also p. 134. 

Solid Angles. Any portion of a spherical surface subtends what ia 
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 4ir 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 Ho of a solid right angle. 
Hence 720 spherical degrees = 1 steregon, or t stera- 
dians = 180 spherical degrees. If w = the angle 
which an element of a cone makes with its axis, then the solid angle of the 
cone contains 2jr(l — cos u) steradians. 

Regular Polyhedra. A = area of surface; V = volume; a = edge. 

Name of solid (see p. 100) Bounded by 

Tetrahedron 4 triangles 

Cube 6 squares 

Octahedron 8 triangles 

Dodecahedron 12 pentagons 

Icoaahedron 20 triangles 

Ellipsoid (Fig. 81) . Volume = %Trahc, where a, h, 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. 



A/a' 


V/a* 


1.7321 


0.1179 


6.0000 


1.0000 


3.4641 


0.4714 


20.6457 


7.6631 


8.6603 


2.1817 





Fio. 81. 



Fig. 82. 




Fig. 83. 



|< d — H 



iT\ — 



Fia. 84. 



Paraboloid of Revolution (Fig. 82). Volume = T^irr'h = }i volume of 
circumscribed cylinder. 

Segment of Paraboloid of Revolution (Bases perpendicular to axis. 
Fig. 83). Volume of segment = ^-iriR^ + r')h. 

Barrels or Casks (Fig. 84). Volume = }iiTrh(,2D^ + d') approx. for cir- 
cular staves. Volume = i'isirh{2D' + Dd + %d^) exactly for parabolic staves. 



SURFACES AND VOLUMES OF SOLIDS 



111 




Fig. 85. 



Pot 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 = 0.003471^^, 
where n = number of inches in the mean diameter, 
01 ii(,D + d), and h = number of inches in the height. 

Torus, or Anchor Ring (Fig. 85). Volume = 
2T'crK Area = iir^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 F = 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 let s = 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 = 2Trrl, where r = distance from axis to center of 
gravity of I. 

Note. If Fi or Si about any axis is known, then Fs or Si 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 1} 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 isF = As 
[and (2) the area generated hy 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 =H'i(-A + B + 4ilf), 
where h — altitude, A and B = areas of bases and M = area of a plane section 
midway between the 
bases. This formula ia 
exactly true for any 
solid lying between two 
parallel planes and such 
that the area of a sec- 
tion at distance x from 
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, if the ordinatea 
2/1, 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 soUd at the same dis- 
tance i above the base, then the volumes of the two solids will be equal. 
See Fig. 88. 




FiQ. 86. 




Fio. 87. 



Fio. 88. 



It ia 



ALGEBRA 

FORMAL ALGEBRA 

Notation. The main points of separation in a simple algebraic expres- 
sion are the + and — signs. Thus, o+6Xc— dH-a;-|-j/ is to be inter- 
preted as a + (b X c) — (d -v- x) + y. In other words, the range of opera- 
tion of the symbols X and -s- extends only so far as the next -|- or — sign. 
As between the signs X and 4- themselves, a -i- b X c means, properly speak- 
ing, a -T- (6 X c) ; that is, the -i- 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^ means 2o(a;'), and ■V2ax means 
(-s/2) {ax) . Instead of ■V2ax, it is safer, however, to write ^/^-ax, or, bet- 
ter, ax\'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 Xb; and a: b, or a/6, means a +b. 

The symbol \a\ means the "absolute value of a," regardless of sign; thus, 
1-21 = I-I-2I =2. 

The symbol re! (where n is a whole number) is read: "n factorial," and 
means the product of the natural numbers from 1 to n, inclusive. Thua 
II = 1; 2! = 1 X 2; 3! = 1 X 2 X 3; 4! = 1 X 2 X 3 X 4; etc. 

The symbol 7^ or + means "not equal to"; + means "plus or minus." 

The symbol = is sometimes 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. (db)c = a(bc). 
a(b + c) = ab + ac. a{b — c) = 06 — ac. Also, a X ( — b) = — ab, and 
( — o) X ( — fc) = ab; "unlike signs give minus; like signs give plus." 

(o + 6) (a - 5) = a^ - b^. 

(a + 6)2 = a' + 2ab + b\ (a - 6)" =a^ - 2gs6 -|- 62. 

(o -1- 6)' = a^ + 3a% + 3ab^ + 6', (a - 6)' = o' -3a'b+ 3o6» - 6'; etc. 
(See table of binomial coefficients, p. 39; also p. 114.) 

„! _ 62 = (o - b)(a + b), a' - 6' = (a - b)(a' + ab + 52). 

o» - 6» = (o - 6)(a»-i + a"-'b + a^-^b' + . . . + o6"-2-(- ft"-'). 

a" + b" is factorable by a -f 6 only when n is odd; thus, 

a' +b^ = (a + 6)(a2 - ab + 52), 

a^ +V = (a + 5)(o* - a% + a^' - ab^ + 5<); etc. 

The following transformation is sometimes useful : 

- + - + --[('+s)"-('^^)']- ■ 

_ . . Tr . . '""^ + '"^ + "*c o -|- 6 -|- c ,, . . 

Fractions. If m 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 dififerent 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;~= • 

b y by by by 

rr. li- 1 ^ 1 J.- a X ax a ^, a ^x ax 

To multiply two fractions: — X-=;— ; -rXa;=— X-= — . 
b y by b bib 

To divide one fraction by another, invert the divisor and multiply: 

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: "o is to 6 as c is to d," and means simply (o/5) = (c/d), 
or od = 6c. o and d are called the "extremes," b and c the "means," 
and d the "fourth proportional" to o, 6, 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). lta/b= c/d, then (o +b)/b = 
(c + d)/d, and (o — 6)/6 = (c — d)/d; whence also, (o + 6)/(o — 6) = 
(c+d)/(c— d). li a/x= b/y= c/z= . . . = r, then 

(a + b+c+. . .)/{x+y+z + . . .)=T. 

Variation. The notation x « j/ is read: "x varies directly aa y," or "x 
is directly proportional to y," and means x — hy, where fc is some constant. 
To determine the constant fc, it is sufficient to know any pair of values, aa 
x\ and j/i, which belong together; then xi= ky\, and hence xlx\ = y/yi, or 
X = {xi/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 = k/y. 

Exponents, a"*''^ = a"'a'^. a"""" = a'" /a", a" = l(if o ?^ 0) . o"*" = l/a". 
(o")" = o"". aV« = -v/o. Thus: a^ = Va, and a^^ = Va'. o"/" = 

Va^- Thus: o^ = Va^ and o^ = Va'- (Va)" = a- (ofe)" = a"b\ 
(a/b)" = a" /ft". (— a)" = a" if « ia even. (— a)" = — o" if n is odd. 
If n is positive and increases indefinitely, a" becomes infinite if o > 1, and 
approaches if o < 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: va = o "■ 
( ya)" = (a"")" = o. If n is odd, v — o = — Va; but if ii, ia even, 
V — a is imaginary. Every positive number a. has two square roots, one 
positive and the other negative ; but the notation Vo always means the positive 
root; thus, VO = 3; — V 9 = — 3. If the denominator of a fraction is of 
the form y/a + V ft, it is possible to "rationalize the denominator" by 
multipljang both numerator and denominator by Vo T V fc. Thus: 
Va + Vb _ (Va + Vb) (Vo + Vb) _ g + 6 + 2 Vo5 
Va -Vb (Via - V6)(Vo + Vb) ~ a -b 

Logarithms. (For the use of logarithms in numerical computation, 
see p. 91 .) The logarithm of a (positive) number JV 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 lO"' = N, and x = log« N means that e' = iV. Loga- 
rithms to base 10 are called common, denary, or Briggsian logarithms. 
For table of 4-place common logarithms see pp. 40^3. 



114 AIGEBRA 

Logarithms to base e are called hyperbolic, natural, or Napierian logar- 
ithms. Here e = 1 + 1 + 1/2! + 1 /3!+ 1/41 + . . . = 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 iab) = log a + log 6 log (a") = n log a log = — oo 

log (a/6) = log a — log 6 log (Vo) = (1/n) log a log 1 = 

log (1/n) = — log n log (base) = 1 log oo = oo 

The two systems are related as follows: 
logioe =iW = 0.4342944819 . . .; log«10 = 1/M = 2.3025850930. . . 
logics = 0.4343 loges; logs x = 2.3026 logioa;. 

For tables of multiples of M and 1/M, 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.) 

Let (nh = n, (nh = ^^^. («)a = ix2X3 ' 

_ n(n— l){n — 2)(re — 3) 
^"''* ~ 1X2X3X4 ••■• 

Then, for any value of re, provided | a: | < 1, 

(1 -I- s)» = 1 -I- (re)ix -I- {n)2X'+ (,n)sx' + (re)4X« -|- . . . 
(If re is a positive integer, the series breaks off with the term in a", and is 
valid without restrictions on x, see p. 112.) 

The most useful special cases are the following: 



^H ^1 , 1 _ 1 2 , Jl^3_ J_ 



vT+x = (1 +x)^^ =1+2'' ~i^' + i^'''"i^^'+ • • • (I'^K^' 

i/r+~x = a +X)^ =1 +^X -^X^ +^^X' -:^X* + . . . (H<1) 

-4 (1 + x)-' == 1 - X +x^ - x' +x* - . . . (\x\ < 1) 

1 + X 

(1 +;t)-^ =1 _1 +3 , _ 5 ^3_^ 35 ^,__ _ .(i^KD 



yTTx 2 8 16 128 

,,- — = (1 + ^y^^ = ^ -h+h'-^^' + ^^'-- ■ ■ (1*1 < 1' 

i/l -I- X 3 9 81 243 



3 . 3 . 1 , . 3 , • 



Vd + x)» -(1 -l-x)'* =1 +-X +-X' - — x' +1^="'" • •(N<1) 

-7-'— -(l+x)-^=l-|x+^x^-||x3-|.f^x'-...(|x|<l) 
V(l -I- x)» 2 8 16 128 

with corresponding formulae for Vl— x, etc., obtained by reversing the 
signs of the odd powers of x. Also, provided |6| < \a\: 

(o -I- b)" = a'U + -j =0'+ (n)i a"-i6 -|- Mia'^-'b'' + (re)sa"-»b' + . .' . 

where (re)i, (re)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 re is the number of 
terms, the last term ia Z = a -|- (re — l)d; the "average" term is J4(o +0; 



FORMAL ALGEBRA 115 

and the sum of the n terms is n times the average term, or iS = yin(a + I). 
The arithmetical mean between a and 6 is (o + 6) /2. 

Geometrical Progression. In a geometrical progression, o; ar; ar'; 
ar^ ; . . ■ , each term is obtained from the preceding term by multiplying by a 
constant, called the constant ratio, /. The nth term is ar''~^. The sum 
of the first n terms is S = air" — l)/(r- - 1) = a(l - r")/(l - r). If 
r is a positive or negative fraction, that is, if — 1 < r < +1, then r" will 
approach zero as n increases, and the sum of n terms will approach o/(l — r) 
as a limit. The geometric mean between a and 6 isVobi also called the 
mean proportional between a and b (p. 113; construction, p. 102). 

The harmonic mean between a and 6 is 2ab/(a + b). 

Summation of Certain Series by Second and Third Diflerences. 
Let oi, 02, og, . . . On be any series of n numbers, as in the 
first column of the adjoining scheme. By subtracting m 
each number from the next following, form the column .S tta (ri 
of "first differences," and by repeating this process, form S ^ 'S ^ 
the columns of second, third, etc., differences. If the Z » "§ ts 
ftth differences are all equal, so that subsequent differ- "^ " cS 

ences are all zero, the original series is called an arithme- ~^i 37 _j„ 
tical series of the ftth order. In this special case the _ § ^^ _12 ^ 
series can be summed as follows: Denote the numbers — 1 i ~ 6 8 
which stand at the head of the successive columns of ^ 1 2 8 

differences by D',D", D'" Then the rath term of a ^ • 

the series is an, and the sum of the first n terms is Sn, 
where 

a„ = a. + (n - 1)D' + ^J^^^D" + 

(« - l)(n -2) (71 -3) 

1X2X3 + • • • 

®» - ""' + T3^1-^ + 1X2X3 ^ 

n(n~l)(n-2)(.n-3) „, 
"^ 1X2X3X4 -t- • • . 

If the series is, for example, of the third order, each of these formulae 
will stop with the term involving D'"; and only a few terms of the series are 
required for the computation of the D'a. (Differentials, p. 159.) 

Sum of the Squares or Cubes of the First n Natural Numbers, 

1+2+3 + . . . + (.n - 1) +n = iin(n + 1). 

12 + 22 + 32 + . . . + (re - 1)2 + 7l2 = i^„(„ + i)(2„ + 1). 

1' + 2' + 3» + . . . + (re - 1)3 + n' = [-^nin + 1)]2. 
Formula for Interpolation by Second Differences. In any ordinary 
table giving a quantity 2/ 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 Xi and xi. If 
X = xi + md, where d = X2 — xi = the constant interval between two suc- 
cessive x's, and m is some proper fraction, then the corresponding value of 
V will be given by the formula 

y y, +mu + 1X2-"+ 1X2X3 ^ + ■ ■ ■ 

where D', D", D'", . . are the first, second, third, . , . differences in the 



116 



ALGEBRA 



series of j/'s which begins with 2/1 (see above), provided the function is o) 
such a nature that the differences of higher orders become negligibly small. 
The coefficients of D',Z)", Z>"', . . . 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 yi ; 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 = yi + mD', which is the familiar formula for ordinary, 
or "linear," interpolation. If D'" < 8 (or D" " < 12, or D" '" < 16), the 
term involving D'" (or D" ", or D" '") can be neglected. 

Binomial CoefQcients for Fractional Values of m 



m 


(m)2 


(m)a 


(m)i 


(m)s 


0.0 


- 0.0000 


0.0000 


- 0.0000 


0.0000 


0.1 


- 0.0450 


0.0285 


- 0.0207 


00161 


0.2 


- 0.0800 


0.0480 


- 0335 


0255 


0.3 


-0.1050 


0.0595 


- 0402 


0297 


0.4 


- 0.1200 


0.0640 


- 0.0415 


0.0300 


0.5 


- 0.1250 


0.0625 


- 0.0391 


0.0273 


0.6 


-0.1200 


0.0560 


- 0.0335 


0228 


0.7 


- 0.1050 


0455 


- 0262 


00173 


0.8 


- 0800 


0.0320 


- 0.0175 


0113 


0.9 


- 0.0450 


0.0165 


- 0.0087 


0.0054 



m(m-l)(m-2)(m-3) 
1X2X3X4 ' 



Here (m)j = ^^^ ' ''"^' = 1X2X3 ' ''"^* " 
Compare p. 39. 

Permutations. The number of possible permutations or arrangements 
of n different elements is 1X2X3X. . .Xra=»! (read: "re factorial"). 

If among the n elements there are p equal ones of one sort, g equal ones 
of another sort, r equal ones of a third sort, etc., then the number of possible 
permutations is (re!)/(p! X 3! X r! X . . .), where 2)+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 [re(« — 1) (re — 2) (re — 3) . . . (re — r + !)]/(?•!) = (n)r. 
(See table of binomial coefBcienta, 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: (re)i + (re)2 + . . . + (re)„ = 2" - 1. 

SOLUTION OF EQUATIONS IN ONE UNKNOWN QUANTITY 

Boots 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 + 1)^ = x' + 2x + 1, is called an 
identity [often written {x + ly = x^ + 2x + 1]. 

Types of Equations. 

(a) Algebraic Equations: 
of the first degree (linear), e.ff., 2s + 6 =0 (root: x — —3); 
of the second degree (quadratic), e.g., x^ — 2x — 3 =0 (roots: — 1, 3); 
of the third degree (cubic), e.rj., x' — 6x* + Sx + 12 =0 (roots: — 1, 3,4). 



SOLUTION OP EQUATIONS IN ONE UNKNOWN QUANTITY 117 

(6) Transcendental Equations: 
exponential equations, e.g., 2" = 32 (root: a; = 5); 2=^ = — 32 (no root); 
trigonometric equations, e.g., 10 sin a; — sin 3a; = 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 X* =« 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 = 6, where 
a and 6 are known numbers. Then divide through by the coefficient of x, 
obtaining x — b/a as the roqt. 

Equations of the Second Degree (Quadratic Equations). Solution: 
Throw the equation into the standard form ax'' + 6x + c =0. Then the 
two roots are: 

- h +Vb'^ - iac -b -Vb^ - 4ac 

Xl = X2 = 

2a 2a 

The roots are real-and-distinct, coincident, or imaginary, according as 
b^ — iac is positive, zero, or negative. The sum of the roots is Xi + ij 
= — b/a; the product of the roots is xix^ = c/a. 

Gbaphicai. Solution. Write the equation in the form x2 = pa: + g, and plot the 
parabola yi = x^, and the straight line yz = 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' Absent. Solution: 
After dividing through by the coefficient of x", any equation of this type 
can be written x' = Ax + B. Letp = A/3 and g = B/2. The general solu- 
tion is as follows: 

Case 1. q' — p' positive. One root is real, namely 

X, = Vg + Vg^ - P' + V'g _ -y/ql - p3. 

the other two roots are imaginary. 

Case 2. q' — p^ = zero. Three roots real, but two of them equal. 
Xl = 2yq, Xl = — ^/q, Xa = — V3- 

Case 3. q' — p' negative. All three roots real and distinct. Determine 

an angle u between and 180°, such that cos u = q/(.p'Vp)'. Then 

Xl = 2\/poos («/3), X2 = 2\/p cos (ti/3 + 120°), ajs = 2y/peos (u/3 + 240°). 

Graphical Solution. Plot the curve yi = x', and the straight line yz = Ax + -B. 
The abscissee 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 



1J.S ALGEBRA 

power, may be written s' + ax' + fca; + c = 0. To get rid of the term in 
x', let X = xi — a/3. The equation then becomes xi' = Axi + B, where 
A = 3(a/3)2 - ft, and B = - 2(a/3)' + 6(o/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^, write the equation in 
the form x» = — alx + (,b/2a)P + la(,b/2a)' — c], and solve by the graphical method. 

Oeneral Properties of Algebraic Equations. An algebraic equation of 
the nth degree in x is an equation of , the type 

ooa;" + oix""' + 021""'' + . . . +an-ix+ On = 
where the a's are any given numbers (00 not zero), the expression on the 
left being called a polynomial of the rath 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, . . . ia a, root of the equation; (3) if a; is very 
large (either positive or negative) , the higher powers of x are the most impor- 
tant; (4) if X ia very small, the higher powers may be neglected. 

Short Method of Substitution in a Polynomial. To find the value of 
4x« — 14x' + 23x — 26 when a; = 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 
eluding the constant term. Then, beginning 
at the left, bring down the first coefficient ; - 
multiply this by 3, and add to the second 4 —2—6 6—11 

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 4x' — 2x' — 6x -(- 5 and the remainder is — 11. That is, 
4x» - 14x» -|-'0x2 -I- 23x - 26 = (x - 3)(4x' - 2x2 - 6x -|- 5) - 11. 

Exponential Equations. To solve an equation of the form o' = 5, 
take the logarithms of both sides : xloga = log 6, whence x = (log 6) /(logo). 
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 i = c, where 
a and h are positive: Find the acute angle u for which tan u = 5/o, and the 
angle » (between and 180°) for which cos v — c/vo^+S'. Then xi = u + 1 
and Xi = u — V are roots of the equation. 

(2) To solve a cos x — 6 sin x = c, where a and 6 are positive: Find « 
and !) as above. Then xi = — (w -|- «) and xa = — (m — ») 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 in the form 



SOLUTION OF SIMULTANEOUS EQUATIONS 



119 



f(x) = 0, where f(x) means any function of x, and plot the curve y = }{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 a; which make/(a;) equal to zero. 
Thus, suppose that /(i) = 3.0 when x = 2.6 and — 5.0 when a; = 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 vary greatly from a straight line between those points, it is seen that it must 



i 



"<& 



-az'--. 



-^ 



Fio. 1. 

cross near 2.64. Suppose the value oi f{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 /i(x) and/2(x), writing the equa- 
tion in the form /i(x) = /aCx). If then the two curves yi = /i(x) and 2/2 <= /zCx) 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 Equations of the First Degree in Two Unknowns. 
Factors 



(1) oix + hiy = a 

(2) OjX -I- 622/ = C2 



■ az 



(0162 — aihi)x = 62C1 — 61C2 .'. X = (62C1 — 5iC2) 7(0162 — 0261) 
(0162 — aibi)y = aid — azCi .'. y = {aid — 02Ci)/(ai62 — O261) 
Here (1) is multiplied by 62, (2) by — 61, and the products added so as to 
eliminate y; again, (1) is multiplied by — ai, (2) by oi, and the products 
added so as to eliminate x. (The process is most conveniently performed as 
follows : Write the multipliers, as 62 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 — 0261 = 0, the equa- 
tions have no solution when ci }A ct, and an infinite number of solutions when 



120 



ALQEBBA 



ci = C2. The following special solution is possible when the sum and 
difference of the two unknowns are given: 
Let X + 2/ = m (1) 
and X — y — n (2) 
(1) + (2): 2x =m +n .-. a; = H (»»+ m) 
(1) - (2): 2y = m — n ■'■ y = Vi(.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: 



X -\- y = 5 
xy = 4 



(1) 
(2) 



Squaring (1), x' + 2xy + y' = 



From (2), 

Adding, 

Hence, 

But 

Therefore 



— 4xy 



25 
- 16 



x' — 2x2/ + 2/2 = 



X -y 
X + y 


= 3 or - 3 
= 5 or 5 


1=4 
y =1 


X = 1 
°%=4 



X 


-y =3 . 
xy = i 




X2 


- 2x2/ + 2/2 
4x2/ 


= 9 
= 16 


x2 + 2X2/ + V' 


= 25 


X 
X 


+ y = 5 or 
— y = 3 or 


-5 
3 



(1) 

(2) 



= 4 
= 1 



y 



= - 1 
= -4 



(b) When the product and the sum of the squares are given: 
xy = 5 (1) \/(4): x + j/ = 6 or 6 or 

x' + y' = 26 (2 ) \/(5) : x-2/=4or— 4or 

From (1) 



6 or — 6 
4 or — 4 



. X = 5 



1 -1 -5 

5 or _ 5 o-^ _ 1 



2x2/ = 10 (3) 
(2) +(3) : x2 + 2x2/ + V' =36 (4) .■. 2/ = If 

(2) -(3): x' - 2x2/ + 2/^ = 16 (5) 
(c) When the sum or difference, and the sum of the squares, are given: 



(1)': 

(2) : 

(1)' - (2) : 



X + y = 5 (1) 

x2 + 2/2 = 17 (2) 

x' + 2x2/ +2/2 =25 
a;2 + 2/2 = 17 



2x2/ = 8 

xy =4 

Then proceed as in case (o), above. 



- 2/ = 3 (1) 

+ y' =17 (2) 

- 2x2/ + 2/2 = 9 
+ 2/2 = 17 



(1)^: 

(2) : 

(1)2 - (2) : - 2x2/ = -8 

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' + 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 two 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) 2x — y + 3z + 5w = 29 

(b) 5x + 22/ - 22 + 3u) = 15 

(c) 3x — 4y + 7z — i« = 12 

(d) 4x + 3y — 5z + 2w = 3 



(c) - 19x - 132/ + 192 = 12 
(/) 17x - 212/ + 38s = 89 
(g ) - 16x - 172/ + 3l2 = 43 



(h) 55x + 52/ = 65 
(i) 285x + 802/ = 445 



16 
-1 



31 
19 



METHOD OF LEAST SQUARES 121 

U) 595x = 595; .-. a; = 1; 

5y = 65 - 55x =65 - 55 = 10; .-. y = 2; 

19z = 12 + 19a; + 13y = 12 + 19 + 26 = 57; :.z =3; 

2to = 3 - 4a; - 32/ + 5z = 3 - 4 - 6 ■+ 15 = 8; Z. u) = 4. 

Here w is eliminated from (a) and (6), obtaining (e); from (o) and (c), 
obtaining (/) ; and from (a) and (d) , obtaining (g) . Then a is eliminated from 
(e) and (/), obtaining (A), and from (e) and (g), obtaining (i). Then j/ ia 
eliminated from (ft) and (i), obtaining 0), which contains only the single vari- 
able X. Hence a; = 1. Now substituting this value of x in either (ft) or (i), 
y is found; substituting these values of x and y in either (e), (f), or (fl), 3 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 X ; for example, n equally careful, independent measurements of some 
physical quantity: 

X = Xl, X = X2, . . . X = Xn. 

As the "best" value of x, take the arithmetic mean, xo, of the several deter- 
minations, namely, xo = (xi -f- X2 -|- . . . + Xn) /re. The quantities vi = 
Xo — Xl, S2 = Xo — X2, . . . tin = Xo — Xn are called the residuals of the 
observed values with respect to xo, and their absolute values (that is, their 
numerical values without regard to sign) are denoted by |»i|, |»2|, . . . |»»|. 
[It can be shown that the sum of the squares of the residuals with respect 
to Xo is smaller than the sum of the squares of the residuals with respect to 
any other value x'o; hence the name of the method: "least squares."] 
The quantities r and ro, defined exactly by Bessel's formulae: 

0.6745 , 



r — ■ 



Vn - 1 
0.6745 



■Wn{n — 1) 
or given approximately by the simpler formulae of Peters: 
0.8453 



Vre(re - 1) 



(|w| -I- W\ + ■ ■ + k»l). 



0.8453 ,1 I , , I , , I K 

J-o (|»i| + \A -I- ... -I- |!)„|), 

ny/n —1 

are called the probable error of a single observation (r), and the probable 
error of the mean (ro), for the given series of observations. Note that 
ro = r/^/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 hyn, 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 nutnerically 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 

aix + biy + ciz = pi in three unknowns, and if re > 3 there will be, in 
aix + biy + C22 = P2 general, no set of values of x, y, z which will satisfy 
all these n equations exactly. In such a case, 
OnX + 6nJ/ + CnZ = Pn the "best" set of values, xo, J/o, so, 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 coeflBcient 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]xo + lab]yi) + [ac]zD = [op] by the coefficient of y in that equation, and 
[bajso + [bb]ya + [bc]zo = [bp] add; theresultwillbethesecondnormalequa- 
tcalxo 4- lcb]y<i + [cc]zi) = [cp] tion. (3) Similarly for the third. ( Nota- 
tion: [aa] = oi' + oa^ -f . . . -f a„S; 
[a6] = 0161 -f 0262 -I- . . . + Onbn; [ap] = aipi + a2P2 + . . . +Onj)„;etc.) 

Finally, solve the three normal equations for the three unknowns in the 
usual way. 

The quantities vi = aixo + biyo + ci«o — pi, etc., are called the residuals 
with respect to xo, yo, zo. [It can be shown that the sum of the squares of the 
residuals with respect to xo, yo, zo is smaller than the corresponding quantity 
with respect to any other set of values, x'a, 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 . . 

*" / 'Vvi' + vi' + . . . + Vn', or approximately, 

v™ — ™ 

0.8453 „ , , , 

>■ = — r-, (Nil + H + . . . + kl), 

V»(re —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: 



Of the third order: 



ai6i 



= ai&2 — 0261 



aibici 
02&2C2 
anbacs 



= ai\ 



I&2C2 
&3C3 



„ 61C1 . 61C1 

O3C3 O2C2 



= 01(6203 — 63C2) — aiibiCa — 63C1) + 03(6102 — 62C1) 
Of the fourth order: 
aibiCid: 



026202^1 
aib3C3di 
aibiCidt 



= «i 



6202^^2 
63C3CZ3 
biCid^ 



— 02 



bicidi 
bscsds 
biC4di 



\biCidi 
+ aa 6202*^2 
IbiCidi 



■ di 



biCidi 

biCzdz 
bsCsdi 



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 ai. 

Properties of Determinants. 

1. The columns may be changed to rows and the rows to columns: 



2. 
3. 



aibici 




aia2as 


(l2b2C2 


= 


616263 


03&3C3 




C1C2C3 



Interchanging two columns changes the sign of the result. 
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. 

6. 



7. 



Solution of Simultaneous Equations by Determinants. 

If 



a, + 


Vl + Ql, 61 Ci 




016101 




j>i6ici 




gi6ici 


Oj + P2 + 52, 62 C2 


= 


026202 


+ 


P26202 


+ 


3262C2 


as + Pa + 93, 63 C3 




036303 




P363O3 




S363C3 


aibiCi 




01 + to5i, 61 01 




026202 


= 


02 + m62, 62 C2 




036303 




03 + mbn, 6 


I 03 













then 



OlX + 61V + ClZ = pi 

02a; + 622/ + C22 = P2 
03S + bay + 032 = P3 
X = Di/D, 

y = D2/D, wheie Di = 
= Da/D, 



where D = 



ai6iOi 
026202 
036303 



^ 0, 



P161C1 
P26202 
P363O3 



Di = 



OlPlOl 

02P202 
03P3C3 



Da = 



ai6iP] 
0262?! 



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 
operatiobs 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, V, 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 O f7 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 <p is sometimes called 
the amplitude or argument of the point. The notation A = (3, .^120°) 
means the point whose distance, r, is 3 times OV, and whose angle, <p, is 120°. 
The development of the algebra depends wholly on the definitions of three 
fundamental operations denoted hy A + B, A X B, and e^, 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 to A + B is the vector sum of the vectors OA and 
OB. In case A and B are not in line with O, the point 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 
denoted by A — B. The point — B is denoted for brevity 
by — B. With these definitions oi 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 has all 
the formal properties of the number zero, and is denoted by 0. 

[Note: If A and B are "real" points, A + B and A — B will also be real. 

Repeated Addition. Multiples and Submultiples. The point 
A + A + A + . . . + A to n terms is called the nth multiple 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 U/2, t//3, . . . are denoted, for brevity, 
by }4, 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 




Fig. 3. 




Fig. 



THK ALUEmiA OF IMAGINARY OR COMPLEX QUANTITIES 125 




A = BX. This point X is called the quotient, A divided by B, and is de- 
noted by A/B (where B 9^ 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 ^ 0) is called the 
reciprocal of the point B, and is denoted by 1/B. (See Fig. 5.) With these 
definitions of AB and 4 /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 V has all the formal 
properties of the number unity, and is denoted by 1. 

[Note: If A and B are real, AB and AIB will also be real.] 
Repeated Multiplication. Powers and Boots. The 
point AXAXAX. . . XAton factors is called the 
nth power of A and is denoted by .4" (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"' = 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 \/Z or A'^l". Thus, the point \/a 
is a point whose distance is the nth root of the did- ,-- 

tance of A, and whose angle is 1/nth of the angle 
of A. All the nth roots of A will lie on the cir- 
cumference of a circle about 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" 
= B" 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 

'AB and 
^AB. 



Fig. 5. 



are positive reals. 



A-V-B =■ 
which would give -\- ' 



not VJ- ^)(--B) 

[Note: If A is real and positive, -yA will be real and 
positive; if A is real and negative, v -4. 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 0. 
In particular, 

i2 = — 1, i' = — i, i* = 1, 1'S = i, etc.; i = -s/ — 1, 
— i = — ■\/~—l; where "1" denotes not the number 
one, but the point U. 

Similarly, multiplying any point Ahy — 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, —M + iii^/s, and 




Fig. 8. 



126 AWEBttA 

In general, (,xi + i|/i) + (,Xi + iyt) = (xi + xj) + i{yi + j/z) ; 

(xi + tj/i)(x2 + ij/2) = (xixii - yiVi) + i(X22/i + xij/2); 

xi + iyi _ xiX2 + i/ii/i . xai/i — xii/2 

X2 + iyi xi' + yi^ xi^ + y^ 

If two complex quantities are equal, their real parts must be equal, and the 
ooefEcients of their pure imaginary parts must also be equal. That is, if 
xi + iyi = X2 + iyi, then xi = X2 and 2/1 = 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 axe 
called conjugate imaginariea. 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 ip, any point A= x + iy can be expressed A = r (cos (p +i sin <p), 
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 <pi+i sin <pi)] [7-2(003 *;2+i sin <pi)] =nr2[(coa (ipi+<P2)+i sin (ipi+ipi)]; 
[r(co3*> + isin*!)]" = r''[cos(,n<p) + isin(n^)] (De Moivre's Theorem). 

The Exponential Function, e^, or exp A, of any point 4 =x + i2/ is defined 
as the point whose distance is e" and whose angle (measured in radians) is y. 
That is, e*"*"'" = e''(co3 y + i sin y). Here e' 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: 
e^e^ = e^+fl, (e^)" = e"'*, e~-^ = 1/e^; e' = e, e" = 1. 

The function e'^is a periodic function with a 'pure imaginary period 2jri; 
that is, e ^I'V* = e^, 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^ will move along a straight line through 
O [or along a circle about as center]. 

Properties of e'*'. The point e''° is a point whose distance is 1 and whose 
angle is <p. It follows from the definitions above that multiplying any 
point A by e*^ has the effect of rotating the point through an 
angle tp, without changing its distance from 0. In particular, e""^ = — 1, 
e-*'^ 1; e"/= = i; e-"^' i; e"^' = 1. 

Third Standard Form for a Complex Quantity. Any point A can be 

expressed in the form A = re'*'> where r is the distance and (p the angle of the 

point. For example, the three cube roots of 1 are 1, e"", e"''*. In general, 

(ne^')(r2e'*'') = (7-ir2)e''«"+*"); (re>V)" = (r»)e''''^. 

If X + ij/ = re^, then r = ■yx' + y', sin v = — , cos »> =■ — , tan y = — 

r r X 

If two complex quantities are equal, their distances will be equal, and their 

angles will differ at most by some multiple of 27r. Thus, if rie**"' = ne'*" 

thenri = n and (/>i = ^2 or ipt + fc27r. 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^. Let A = re»V; then A^ = exp [(log, r + i(p)B]. 

For example, i' = e-V^ where i = "v— 1. 

If a is a positive real, a " "'"''' = a^ [cos (y log, a) + i sin (y log, o)]. 

Trigonometric and Hyperbolic Functions of a Complex Variable. 

If A is any point, then, by definition, 

_ eiA - e-'-^ e-'* + e"*'! ^ ^ sin A , , ^ „^ 

sin A = — , cos A = , tan A = (cos A 7^ 0) ; 

2i 2 cos A 

sinh A = pr , cosh A = , tanh A — ; — - • 

2 ' 2 cosh A 

Hence the formulse that hold for these functions in the real case (p. 131; 
p. 135; p. 161) hold also for the complex case. Further: 

sin (,x + iy) = sin x cosh y + i cos x sinh y, sin iy = i sinh y ; 

cos (x + iy) = cos x cosh y — i sin x sinh y, coa iy = cosh y; 

sinh (a; + iy) = sinh a; cos y + i cosh a; sin j/, sinh iy = i sin y ; 

cosh (a; + iy) = cosh x cos y + i sinh x sin j/, cosh iy = cos j/; 

where sin x, sinh a, 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 2ir. The functions sinh A and cosh A are periodic with a 
pure imaginary period 2in. 

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^ = 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 Ziri. 

If a; + iy = re'*", then log, (a: + iy) = log,.?- +i<p + h-2in. 

It 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-iA. The functions 
C03-14, sinh-i 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 -iji, 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 {[/(« -|- Az) — f{z)]/hz} when Az approaches 0. 

It can be shown that lim | [exp Az — ll/Az] = 1; hence d(,ef) = ei'dz, 
<i(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 

FORMAL TRIGONOMETRY 

Angles, or Rotations. An angle is generated by the rotation of a ray, 
as Ox, about a fixed point 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 Fio- 2. 

quadrantal angles. 

Units of Angular Measurement. 

(1) Sbxaqesimai, Measure. (360 degrees = 1 revolution.) 1 degree = 
1° = Ho of a right angle. The degree is usually divided into 60 equal parts 
called minutes (') , 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 = Hoo of a right angle. The grade is always divided 
decimally, the following terms being sometimes used : 1 "centesimal minute" 
= Hoo of a grade; 1 "centesimal second" = Moo of a centesimal minute. In 
reading Continental books it is important to notice carefully which system 
is employed. 

(3) Radian, or Cibcuiab, Measure, {x 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 = 67°.30- = 67°.2957795131 = 57° 17' 44".806247 = 180°/5r. 

1° = 0.01745 . . . radian = 0.01745 32925 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 onOA or Fiq. 

OA produced. In the right triangle OMP, the three sides 
are MP = "side opposite" (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: 

123 




FORMAL TRIGONOMETRY 



129 



sine of X = sin s = opp/liyp = MP/OP; 
cosine of X = cos x = adj/hyp = OM/OP; 
tangent of x = tanx = 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 IMP. 
The last three are best remembered as the reciprocals of the first three: 
cot X = 1/tan x; sec x = l/cos x; esc x = l/sin x. 
Other functions in use are the versed sine, the coversed side, and the ex- 
terior secant: 

vers X = 1 — cos x; covers x = 1 — sin x; exsec i = 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 




+ 
+ 
+ 


+ 


+ 




COS X and sec x are 


+ 


tan X and cot x are 







vers X and covers x are always positive. 
Variations in the Functions as x Varies from deg. to 360 deg. are 

shown in the accompanying table. The variations in the sine and cosine are 






FiQ. 4. 

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. 





0° to 90° 


90° to 180° 


180° to 270° 


270° to 360° 


Values at 




30° 


45° 


60° 


sinz 

CSC X 


+Oto+I 
+ oo to +1 


+1 to +0 
+lto + ~ 


-Oto-I 
- CO to -1 


-Ito-0 

— 1 to — 00 


2 


HV2 
V2- 




cosz 

sec X 


+1 to +0 

+lto + - 


-Oto-1 

-ootO-1 


-lto-0 

-1 to - CO 


+Oto+1 

+ 00 tO+1 




W\/2 
V2- 


2 


tanx 

cot X 


+0tO+<x. 
+ ootO+0 


— 00 to —0 
-0 to- CO 


+0 to + 00 
+ ootO+0 


- 00 to -0 
-OtO- "o 


V3 


1 
1 


V3 


vers X 
covers X 


+Oto+1 
+1 to +0 


+lto+2 
+Oto+1 


+2to+l 
+1 to +2 


+1 to +0 
+2to+l 









■n/2 =1.4142; HV'2 =0.7071; Vs =1.7321; J^Vs =0.8660; \Wi =0.5774; %-\/z =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 



TRiaONOMETRY 



computations in which an accuracy of 1 part in 1000 is suflSoient. 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 CoSperative 
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. from deg. to 90 deg.) . 
The larger tables give only the logarithms of the functions, not the natural 
values. 

To Find Any Function of a Oiven 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 (a;) = the same function of (a; ± 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°) 
= +seo (1-90°) 
= -ain (x-90°) 
= -c8c (a!-90°) 
= -cot (x-90°) 
= -tan (1-90°) 


= -sln (x-180°) 
= — CSC Ix — 180°) 
= -cos (x-180°) 
= -sec (a- 180°) 
= -|-tan (x-180°) 
= 4- cot (1-180°) 


'--cos (x-270°) 
sec (a!-270°; 


CSC X 




= -|-sin (x-270°) 
= -fcsc (x — 270°) 
= -cot (x-270°; 
= -tan (1-270°) 


sec X 








= H-sin (1-90°) 
= l-cos (1-90°) 


= 1-1- cos (2-180°) 
= l-fsin (1-180°) 


= l-sin (1-270'? 
= H-cos(i-270°) 


covers x 



The "reduced angle" (i - 90°, or a; - 180°, or a; — 270°) will in each case 
be an angle between 0° and 90°, whose functions can then be found in the 
table. 

[NoTB. 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 X2 will be 


flin x= -\-a 
cos x= -\-a 
tana;= -f-a 
cot a;= -ha 


sin XQ = a 
cos xo = a 
tan xo = a 
cot xo = o 


xo and 180°-a;o 
xo and 360° -xo 

xo and 180°-Fxo 
xo and 180°-|-xo, 


sin X = — a 
cos x= — o 
tan x= ~a 
cot a; = — a 


sin xo = a 
cos xo = o 
tan xo = a 
cot xo = a 


[180°-Ha;o]and 
180° — xo and 
180° -xo and 
180° -xo and 


360° -xo I 
180°-|-a!o 
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. 


6.) 


• . , o , X sin a; ^ 1 cos x 




/ 


COS X tan x sin x 


/;i 


/ 






// 




cos2 X sin2 X 




, ., 




/ / 








VI H- tan2 a: VI + cot* x 


"^ 


c 


2 


1 __j. .. 


/ 


\' / 










VI + tan2 a; VI + cot^ x 




Functions of Negative Angles, sin (— x) - — einx 


'^1 




vers X 




cos ( — x) = COS x; tan ( — x) = — tan x. 




t > 


' 



Functions of the Sum and Difierence of Two Angles. 

sin (x + J/) = sin x cos y + cos x sin y ; 

COS (x + J/) = COS X COS 2/ — sin x sin j/; 

tan (x + J/) = [tan x + tan 2/]/[l — tan x tan j/]; 

cot (x + 2/) = [cot X cot J/ — l]/[cot X + cot y\\ 

Bin (x — J/) = sin x cos j/ — cos x sin j/; 

cos (x — J/) = cos X cos J/ + sin x sin y\ 

tan (x — 2/) = [tan x — tan 2/] /[l + tan x tan y] ; 

cot (x — 2/) = [cot X cot y + l]/[oot 2/ — cot x]; 

sin X + sin 2/ =2 sin }i(x + 2/) cos J4(x — j/) ; 

sin X — sin 2/ = 2 cos H(a; + 2/) sin ^(x — j/) ; 

COS X + cos 2/ = 2 cos H(a; + 2/) cos ^(x — 2/) ; 

cos X — cos 2/ = — 2 sin H(a; + 2/) sin ^(x — y) ; 



FlQ. 5. 



tan s + tan 2/ = 



tan X — tan y = 



sin (x + 2/) . 
cos X cos 2/ 
sin (x — y) 



cot X + cot 2/ = 



cot X — cot 2/ 



sin (x + y) 
sin X sin y' 
sin (2/ — x) 



cos X cos y sm X sin 2/ 

sin' X — sin' 2/ = cos' y — cos' x = sin (x + y) sin (x — 2/) ; 

cos' X — sin' 2/ = cos' y — sin' x = cos (x + y) cos (x — 2/) ; 

sin (45° + x) = cos (45° - x) ; tan (45° + x) = cot (45° - x) ; 

sin (45° - x) = cos (45° + x): tan (45° - x) = cot (45° + i). 
In the following transformations, a and 6 are supposed to be positive, 
c = y/a? + 6', A = the positive acute angle for which tan A = 0/6, aiid 
JB = the positive acute angle for which tan B = 6/0: 

a cos X + 6 sin x = c sin (4 + 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 ^x cos ^x; 

cos 2x = cos' X — sin' x = 1 — 2 sin' x = 2 cos' x — 1 ; 

2 tan X . „ cot' x — 1 

tan 2x = :; T~~'« — 't "ot 2x = • 



1 - tan' X 



2 cot X 

, . , „ 3 tan X 

Bin 3x = 3 sin x — 4 sin" x ; tan 3x = 



cos 3x = 4 cos' X — 3 cosx; 



tan' : 



1-3 tan' a 



132 TRIGONOMETRY 

sin {nx) — n sin x cos ""^ x — (jt,)t sin' x oos"~'x 

+ (»)6 sin* a; cos ""' x — 
cos (jix) = cos" X — (re)2 siu^ x cos "-^a; + (re)4sin*xcos "~* x — 
where (71)2, (n)s, . . . are the binomial coefficients (see p. 39). 
sin H x = + '\ /]A{l — cos a); 1 — cos a; = 2 sin^ ^ix; 
cos \ix = + -s/ Viil + cos x); 1 + cosx =2 cos* }Ax; 

11 — cos X sin a; 1 — cos x 

tan Hx = + \/:; — ; = ■;; — ; = : ; 

\ 1 + cos X 1 + cos X ein x 



tan (I + 45°) = ± yjl 



+ sin X 



sin X 
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 S + sin C = 4 00s HA cos HB cos aC; 

cos A + cos B + cos C = 4 sin }iA sin J4B sin }^C + 1 ; 

sin A -j- sin B — sin C = 4 sin HA sin HB cos HC; 

cos A + cos B — cos C = 4 cos HA cos HB sin yiC — 1; 

sin' A + sin'B + sin^ C = 2 cos j1 cos B cos C + 2; 

sin' A + aiu'B — sin* C = 2 sin ^ sin B cos C; 

tan A + tan B + tan C = tan A tan' B tan C; 

cot ^4+ cot iiB + cot ^C = cot iiA cot HB cot Hd 

cot A cot B + cot A cot C + cot B cot C = 1 ; 

sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C; 

sin 24 + sin 2B — sin 2C = 4 cos A cos B sin C. 
Inverse Trigonometric Functions. The notation sin~'a; (read: anti- 
sine of X, or inverse sine of x; sometimes written arc sin x) means the prin- 
cipal angle whose sine is x. Similarly for cos~'x, tan~'x, etc. (The prin- 
cipal angle means an angle between —90° and -|-90° in case of sin"' and 
tan"', and between 0° and 180° in the case of cos"'.) For graphs, see p. 174. 

SOLUTION OF PLANE TRIANGLES 

The "parts" of a plane triangle are its three sides, a, b, c, and its three 
angles A, B, C {A being opposite o, B opposite 6, C opposite c, and A ■{■ 
B + C = 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 formulse are: 

Law of sines: 7- = - — =. Law of cosines: c* = a* -f 6* — 2ab cos C. 
b sin B 

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^ = a' + b\ *'°- 

sin A = a/c, cos A = 6/c, tan A = a/5, cot A = 6/0. 

If A is very small, use tan ^lA = V(c— 6)/(c -f- 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 + B + C = 180". 



To find the remaining sides, use 6 ■■ 



a sin B 



a sin C 



FiQ. 7. 



sin A sin A 

Or, drop a perpendicular from either B or C on the opposite 
side, and solve by right triangles. 

Check: c cos B + b cos C = a. 

Case 2. Givbn Two Sides (say a and 6), and the Included Angle 
(CO ; and suppose a > 6. Fig. 8. 

First Method: Find c from c^ = o^ + 6^ — 2a6 cos C [or ti' = {a — by + 
2ab vers C]; then find the smaller angle, B, from sin B = (Jb/c) sin C; and 
finally, find A from A = 180° - (B + C). Check: a cos JS + 6 cos A = e. 

Second Method: Find yi{A — B) from the law of tangents: 
tan yi{A - B) = [(a - 6) /(a + b)] cot liC, 
and H(A + B) from >4(A. + B) = 90°— C/2; hence A = 
H(A + B) + H(A -B) and B = J^(A +B) - ^(A - B). 
Then find c from c = a sin C/sin A or c = i> sin C/sin iJ. Fiq^ g_ 

Check: o cos B + 6 cos A = a. 

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 = (b' + c^ — a'')/2bc {or vers A = [a^ — (b — cy]/2bc] and 
then find B and C (which will always be acute) from sin B = b sin A /a and 
sin C = c sin A/a. Check: A +B +C = 180°. 

Second Method: Find A, B, and C from tan ^A = r/(s — a), 

tan aB = r/(s — 6), tan ^C = r/{s — c), where s = }i(a + 6 + c), and 
r =V'(s -a)(s -b)(s - c)/s. Check: A +S + C = 180°. 

Third Method: If only one angle, say A, is required, use 
sin HA = \/ (s — b)(,s — c)/bc or 
cos HA = \/s(« — a) /be, 
according as \iA is nearer 0° or nearer 90°. 

Case 4. Given Two Sides (say b and c) and the Angle 
Opposite One op Them (B). This is the "ambiguous 
case" in which there may be two solutions, or one, or none (see Fig. 10). 




B acute 



'NaSoluthn) (OntSolution) rT^<^h,Himst ^ 



''NoSolution) (OneMutfon) fTmSalations) 



(One Solution) (One Solution) 



Bobruse ^ ^ _^ .Av^ 

(NoSolution) (NoSolution) (NoSolution) (NoSolution) 
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 (7 = 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 
Ci = 180° — Cu Then Ci will yield a solution when and only when 



134 TRIGONOMB TR Y 

Ci + B < 180° (see Case 1) ; and simUarly Cz will yield a solution when 

and only when C2 + B < 180° (see Case 1). 

Other Properties of Triangles. (See also p. 99 and p. 105.) 

Area = ^ab sin C = -Vais — a)(s — 6)(s - c) = rs, where s = }^(a +b+c), 

and r =radius of inscribed circle = V (s — a)is — 6) (s — c) /a. 
Radius of circumscribed circle = R, where 

2R = o/sin A = fc/sin B = c/ain C; r = 4K sin — sin — sin — = -=-. 

Ji Ji £t 4iV5 

The length of the bisector of the angle C is 

_ 2V'ob5(8 - c) ^ ■\/ah\(a + 6)' - c'] 
0+6 0+6 



The median from C to the middle point of c is m = }4\/2(o2 + 6") - c'. 

SOLUTION OF SPHERICAL TRIANGLES 

For the occasional solution of a spherical triangle the following formula 
will be sufficient. For a detailed discussion, see any text-book on spherical 
trigonometry. 

Let a, b, c be the sides of the spherical triangle, that is, portions of area of 
great circles of the sphere ; and let 4 , 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 b sin c sin c sin a 



sin A sin B ' sin B sin C sin C sin A 

cos a = cos 6 cos c + sin 6 sin c cos A, 

with similar formulae for cos b and cos c. 

cos A = — cos B cos C + sin B sin C cos o, 

with similar formulae for cos B and cos C. 

In the special case of a right spherical triangle, in which C = 90°, 

I. 4. A 4.T, cos A cos B 

cos c = cos a cos =cot A cotB; cos o = -; ; cos 6 = — — -; 

sm B sin A 

. sin a , tan 6 , , tan a 

sin A = -; ; cos A = ; tan A = — — ;- ■ 

sm c tan c sin b 

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*, 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 a; = H(e^ — e""); cosh a; = J^(e* + e~*); tanh a; = sinh a;/cosh a;; 

csch X = 1/sinh x\ sech x = 1/cosh x; coth X = 1/tanh x; 

cosh^ X — sinh' a; = 1; 1 — tanh' x = sech' x; 1 — coth' x = — csch' x; 

sinh ( — x) = — sinh x; cosh ( — x) = cosh x; tanh ( — x) = — tanh x; 

sinh (x + 2/) = sinh x cosh y ± cosh x sinh y; 

cosh (x + J/) = coah x cosh y ± sinh x sinh y ; 

tanh (x ±y) = (tanh x + tanh j/)/(l + tanh x tanh y)\ 

sinh 2x = 2 sinh x cosh x; cosh 2x = cosh' x + sinh' x; 

tanh 2x = (2 tanh x) /(I + ta nh' x) ; 

sinh }4x = VH(ooshx — 1) ; cosh Hx = •\/H(cosh x + 1) ; 

tanhV4x= (cosh x — l)/(sinh x) = (sinh x)/(oosh x + 1). 

The inverse hyperbolic sine of y, denoted by sinh"';/, is the number 
whose hyperbolic sine is y; that is, the notation x = sinh~'j/ means 
sinh X — y. Similarly for cosh-'j/, tanh-'j/, etc. These functions are closely 
related to the logarithmic function, and are especially valuable in the integral 
calculus. For graphs, see p. 175. 

ainh-i(j//o) = \oee(.y + V ;/' + o' ) — loge o; 
cosh-'(y/a) = log»(2/ + ■\/'y^ — a') — log« a; 

tanh-' - = }4logB ; coth-' - = Mlog« 

a a — y a y — a 

The anti-gudermannian of an angle u, denoted by gd~'«, is a number 

defined by gd~'M = logs tan (Hx + V^u) = ysec u du. When u is small, 

gd~'M = U + VtU^ + MiU^ + 6Ho4om' + . . . . 



ANALYTICAL GEOMETRY 



THE POINT AND THE STRAIGHT LINE 

Rectangular Co-ordi nates (Fig. 1). Let Pi = {xi, yi), Pa = (.xt, yi). 
Then, distance P1P2 = \/(a:i — xi)' + (2/1 — 2/2)'; slope oiPiPi = m = tan« 
= (j/2 — yi)/(.X2 — xi); co-ordinates of mid-point are x = J4(a;i -|- xi), 
y = Hiyi + 2/2); co-ordinates of point (l/78.)th of the way from Pi to P2 are 
X = XI + {l/n)(_Xi — xi), 2/ = 2/1 -f- (l/ra)(2/2 — 2/i). 

Let mi, rm be the slopes of two lines; then, if the lines are parallel, mi = mi; 
if the lines are perpendicular to each other, mi = —1/mi. 

Equations of a Straight Line. 

= 1. (a, b = intercepts of the line on 



1. Intercept Form (Fig. 2) : — h , 

a o 

the axes.) 

2. Slope Form (Fig. 3) : y = mx + b. 
cept on the 2/-axis ; see also Fig. 2, p. 174.) 

3. Normal Form (Fig. 4) : r: cos v + y sin v = p. 
from origin to line; v = angle p makes with the a-axis.) 

y — b X 

4. Parallel-intercept Form (Fig. 5): r = 7- 

c — K 

two parallels at distance k apart.) 



(to = tan u = slope; 5 = inter- 
(p = perpendicular 

(b, c = intercepts on 



? 



Fig. 1. 



Fio. 2. 



FiQ. 3. 





Fig. 4. 



Fig. 5. 



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 iS = ± y/lT+B' 
(sign to be so chosen that p is positive).] 

6. Line Through (a;i, y{) with Slope m: y — yi = m{x — xi). 

2/2 — 2/1 



7. Line Through (xi, yi) and (12, 2/2) : y — yi 



(.X - Xl). 



tan u 



X2 — Xl 

8. Line Parallel to a;-axis: x =0; to 2/-axis: y = b. 
Angles and Distances. 

If « "= angle between two lines whose slopes are mi, mi, then 
m,2 — mi If parallel, mi = mt. 

1 -(- m2mi If perpendicular, mim2 = — 1. 

If u = angle between the lines Ax + By + C = and A'x + B'y -f- C = 0, 
then 

A A' + BB' If parallel, A /A' = B/B'. 

cos u ■« , 

±V(^' +-B') (^'' +5'=) If perpendicular, XA' -|- BB' =0. 
The equations of the bisectors of the angles between the two lines just 
mentioned are 

Ax +By +C ^ A'x + B'y -f C ^ ^ 



VA.^ + B' 



VA'2 -I- B'» 
136 



THE POINT AND THE STRAJQHT LINE; THE CIRCLE 



137 



The equation of a line through (xi, j/i) and meeting a given line y = mx + 6 

at an angle u, is 

m + tan u , 

y — yi = 7 (x - xi). 

1 — m tan u 

The distance from (xo, yn) to the line Ax + By -'t C = is 

^ ^ ^Xq + gj/o + C 

where the vertical bars mean "the absolute value of." 

The distance from (xo, Vo) to a line which passes through (xi, j/i) and makes 
an angle u with the x-axis, is 

iJ = (xo — xi) sin u — (j/o — yi) cos u. 

Polar Co-ordinates (Fig. 6). Let (x, y) be the rec- 
tangular and (r, 6) the polar co-ordinates of a given 
point P. Then x = r cos 8; y = r sin 9; x^ + J/^ = '■^• 

Transformation of Co-ordinates. If origin is moved 
to point (xo, j/o). the new axes being parallel to the old, 
X = Xo + x', 2/ = j/o + y'. 



FiQ. 6. 

If axes are turned through the angle u, without change of origin, 
X = x' cos u — v' sin u. 



y = x' sin u + y' cos «. 



THE CIRCLE 

(See also pp. 99, 103-105, 106) 

Equation of Circle with center (a,b) and radius r: 
(x - a)2 + (2/ - 6)2 = tK 

If center is at the origin, the equation becomes x^ + j/* = r*. If circle 
goes through the origin and center is on the x-axis at point (r, 0), equation 
becomes x' + V' = 2rx. The general equation of a circle is 

x2 -I- y^ +Dx + Ey +F = ; it has center at ( -D/2, -E/2), and 

radius =-\/{D/2y+ (E/2y — F (which may be real, null, or imaginary). 

The equation of the radical axis of two circles, x'^ + y^ -\- Dx + Ey + 
F =0 and x' + y' + D'x + E'y +F' =0, is {D - 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^ + y^ = r" at (xi, yi) is xix -|- yiy = r'. 
The tangent to x' + y' + Dx + Ey + F = at (xi, yi) ia 
xix + yiv + \iD{x + xi) -I- yiE{y + yi) + F =0. The l ine y = mx + h 
will be tangent to the circle x^ -f 2/^ = r^ if 6 = aVl+wA 

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 x-axis, then the equations of the 
circle in parametric form will be X = acosu;y = asinu. For 
every value of the parameter u, there corresponds a point 
(i, v) on the circle. The ordinary equation x^ + j/^ = a^ can 
be obtained from the parametric equations by eliminating u. 




FiQ. 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 
iocus 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 focua to directrix. (Compare also Fig. 3, p. 174.) 



H 

■rA 






. --^py^ 





'A 




9 


p 

T 




F M N ' 
P 

N 





Fia. 8. 



FiQ. 9. 



FiQ. 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) : y^ = 2px. 

Polar Equation of Parabola, referred to F as origin and Fx as axis 
(Fig. 9): r = p/(l - cos 8). 

Equation Referred to the Tangents at the ends of the latua rectum as 



axes (Fig. 10): x 



,H 



+ J/^=a^. 



where a = p 



V2. 






Equation of Tangent to y^ = 2px at (xi,yi): yiy - pix + xi). The 
line y = mx + b will be tangent to y' = 2px if b = p/(,2m). 

The tangent PT at any point P biaecta the angle between PF and PH 
(Fig. 8). A ray of light from F, reflected at P, will move off parallel to the 
principal axia. The subtangent, TM, is bisected at 0. The subnormal, 
MN, is constant, and equal to p. The locua 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 interaeotiou of perpendicular tangenta ia the directrix 
(Fig. 12). The locua of the mid-pointa of a set of parallel chorda whose 
slope ia t» ia a straight line parallel to the principal axis at a distance p/tn. 



THE PARABOLA 



139 



and is called a diameter (Fig. 13). If M is the mid-point of a chord PQ, 
and if T ia the point of intersection of the tangents a.tP 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, oi FT = FP. Then TP is 
the tangent at P. Or, make ilf iV = p = 2(0F); then PN 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 ig the perpen- 
dicular bisector of FH. 




F 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 Dikection of the Principal Axis 
OX. In Fig. 16, let K be a variable point on a line through Q parallel to 
OX. Draw KB parallel to PT (meeting PQ in R), and draw RS parallel to 
OX (meeting Pit 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 Pr. 

2. Given the Vertex O and Focus P. (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. 





a 






r 


K 


Fig. 18. 


6 


Fig. 17. 




Fig. 19. 



(6) Take a piece of paper (Fig. 18) with a straight edge, d, and mark a 
point F near the edge. Let if be a variable point of the edge, and fold the 
paper so that K coincides with P. The crease will be a tangent to the parabola 
which has focus F and directrix d. 

(c) In Fig. 19, let JW be a variable point of the principal axis, and lay oft 
MN = 2(0P) = p. WithP as center and radiusPiV^ 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 op Contact, P and Q 
(Fig. 20). Divide rP and QT into any number of equal parts (here 4). Then 
the lines 11, 22, 33, . . . will be tangents to the parabola. This method ia 
especially advantageous in drawing rather flat arcs. 



140 



ANALYTICAL GEOMETRY 



The Radius of Curvature of y^ = 2px at a, point P = (a,j/) is fl = 
(p + 2a;)5^/Vp> or R = p/sin' v, where v = the angle which the tangent at 
P makes with Pii" (Fig. 21). At the vertex, B = p. To construct the radlua 
of curvature at any point P, lay oS 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 pointP, and a circle about C with radius CP 
will coincide closely with the parabola in the neighborhood of P. 





Fio. 20. 



FiQ. 21. 



THE ELLIPSE 

The ellipse (see also p. 107) has two foci, P and P' (Fig. 22), and two direc- 
trices, DH and D'H'. If P is any point of the curve,PP +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 5 are as shown in 
Fig. 23. Thus, 6^ = 0^(1 - e^), ae = Va^ - b\ e' = 1 - (b/a)K The 
semi-latus rectum = p = a(l - e') = bVo- Note that 6 is always less 
than a, except in the special case of the circle, in which 6 = a and e = 0. 




Fig. 23 



Fig. 24. 



Fig. 25. 



Any section of a right circular cone made by a plane which outs 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: 



— + i = 1 



or 2/ = 



+ *vV^ 



If P = (x, y) is any point of the curve, PF = a + ex, PF' = a — ex. 

Equations of the Ellipse in Parametric Form : x = a cos u, y = 
5 sin «, where u is the eccentric angle of the point P = (x.y). See Fig. 28. 



THE ELLIPSE 



141 



Polar Equation, f ocua as origin, axes as in Fig. 24: r=j)/(l— e cos S) . 

Equation of the Tangent at {xi,yi) : hhnx + a^yiy = a V. 

The line y = mx + k will be a tangent if fc = + Vo^m^ + i'. 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 Vo^+fc^ (Fig. 27) . 




Fig. 26. 



Fig. 27. 



Fig. 28. 



Fio. 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 mj are the slopes, then 
mirnt = — b^/a^. 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' 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, = 4a6 (Fig. 31). Also, if a',b' 
are conjugate semi-diameters, and w the angle between them, then a'' + b'^ = 
o* -f- 6^ and a'b' = ab/sin w. 




Fig. 30. 



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 Pi? and PP" (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. (6) With fl as a center find radius RF, 
draw an arc ; with F' as center and radius 2o draw an arc, intersecting the 
first in iS; 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 as 
center, draw circles with radii a and 6 (and also a third circle with radius 
o + 6). Let a variable ray through cut these circles in J, K (and iS); 
through / and K draw parallels to the axes, meeting in P. Then P ia a point 
of the ellipse (and SP is the normal at P) . 

(2) In Fig. 35, let P divide a line AB so that PA = a and PB = b. Then if 
A and B slide on the axes, P will describe an ellipse. 




Fig. 34. 



FiQ. 36. 



(3) In Fig. 36, let PBA be a straight line such that PA = a and PB = 6. 
Then if A and B slide on the axes, P will trace an ellipse. (Use a Btrip 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 atB 
(Fig. 37). Drive pins atii',ii",andB, 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 parts, 1, 2, 3, . . . and 1', 2', 3', . . . ConnectPwith 
1,2,3,. . . andP' withl', 2', 3' . . .. The points of intersection of corre- 
sponding lines will be points of the ellipse. 

(2) Take any point iConPQ (Fig. 39). Draw SiCt/, and draw iCV 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 oSPR =- 
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 OB ^ OS = 2o and 



THE ELLIPSE 



14d 



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. 




a 


I 




■■•^^^^ A 


y 


^^^>^l 


















\U 


/ /^ V Ji 










/ 








\ / 




'' / 


N 










o^y 





FiQ. 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 oS OU = OV = OB = b. Draw UG perpendicular to 
the axis and DG at 45°. With G as center draw an auxiliary arc with radius 




T 

\ 


-> 


H 


I 


?- 






■z>,/^ 


A 


w 

/ 
/ 








y 




'>? 


u 






■^ 


'c. 







Fig. 43. 



Fig. 44. 



= AV = a — b, and through D draw DMN just touching this arc. Then M 
is one center (with radius MA) 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 DCiCz perpendicular to AB. Call CiA = t^ 
and CsB = r^. Lay off BE = BO ( = 6) , and on ED as diameter draw a semi- 
circle cutting the minor axis in W; then BW = VoS = Ti. 'Lay off AZ ,= 



144 



ANALYTICAL GEOMETRY 



BW. From Ci with radius CiZ( = rj - ri), and from Cs with radius CtW 
{=ri — Ti), draw area intersecting in Ci. Draw CaCi extended and C'jCi ex- 
tended. Then draw in the three arcs, with centers at Ci, d, Cj and radii n, 
Ti, T}. Note. Since n and rj 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 following : 

(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 
(= o — b), and through D draw DC1C3 just touching this arc. Take CiA 
as n and C3B as ra. On DE as diameter draw a semi-circle cutting the minor 
axis in W, and take SPF(=Vob) as »-2. Lay off AZ - BW. From Ci 
with radius CiZ{ = n — n), and from 
C3 with radius CsW( = ra — n), draw 
arcs intersecting in Ci. Then Ci, 
Ci, Ca are the required centers. [E. 
V. Huntington.] 

Radius of Curvature of Ellipse 
at Any Point P = (s, j/) is iJ = 
a2fe2(a;Va* 4- 2/V&*)^ = JJ/sin'», where 
V is the angle which the tangent at P 
makes with PF or PF'. At end of 
major axis, R = b'^/a = MA ; at end 
of minor axis, R = a^/h = NB (see 
Fig, 45). To construct the radius 
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 to PN 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 linePQ 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. 46. 





Fig. 47. 

the center (Fig. 47). If P is any point of the curve, \PF — PF'\ is constant, 
= 2o; a,ndPF/PH(oTPF'/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 6/0. The relations between e, u, and 
b are shown in Fig. 48: 6^ = a\e^ — 1), ae = Vo* + b', e' = 1 + (6/0)=. 
The semi-latus rectum, or ordinate at the focus, is p = a(e^ — 1) =b^/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: 



b' 



= 1, 01 y = + — v/xs" 



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 +0, 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', where A is the area shaded in Fig- 49. 





FiQ. 49. 



FiQ. 50. 



(2) X = a aec 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. 60: 

r = p/(l — e cos 8). 
Equation of the Tangent at (11,2/1) : b'xix —a'yiy =a°6'. 
The line y = mx + k will be a tangent if fc = ± -\/a'm' — b'. Thetan- 




FiQ. 51. 





Fig. 53. 



gent at any points (Fig. 51) bisects the angle betweenPF 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 \^a' — b', which 
will be imaginary if b > a (Fig. 53) . 
10 



146 



ANALYTICAL GEOMETRY 



Properties of the Asymptotes. (Fig. 54.) IfP is any point of the curve, 
the product of the perpendicular distances from P to the two asymptotes is con- 
stant, = a%^/(a' + 6^). Also, the product of the oblique distances (the dis- 
tance to each asymptote being measured parallel to the other) is constant, and 
equal to yi{a^ -\- b^). 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 B and iS (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 XQV = aK 





FiQ. 54. 



Fig. 55. 



Conjugate Hyperbolas are two hyperbolas having the same asymptotes 
with semi-axes interchanged (Fig. 55) . The equation of the hyperbola conju- 



gate to 1 - 7^ = 1. is -; 
a' b^ a'- 



- 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. 55). If mi and mz are the slopes, then mirm = b'^ja?. 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 = 4o6. If a', b' are 
conjugate semi-diameters, and w the angle between them, then o'^ - !)'' 
= o^ — 6^, and a'V — ob/sin w. 

Equilateral Hyperbola (o =6). Equation referred to principal axes 
(Fig. 56) : i2 — j/2 = o2. Note, p = a. Equation referred to asymptotes 
as axes (Fig. 57) : xy = a^/2. (See also Fig. 3, p. 174.) 

Asymptotes are perpendicular. Eccentricity = \^. Any diameter is equal 
in length to its conjugate diameter. 




FiQ. 66. 



FiQ. 57. 



rUM CATENARY 



147 



To Construct a Tangent at any given point P of a hyperbola. In Fig. 58, 
drawP4 andPB parallel to the asymptotes, and take OS = 2(0 A) and OT = 
2 (OB). Then ST ia the tangent at P. 




To Construct a Hyperbola, given the asymptotes and any point P. 

(1) In Fig. 59 let TPT' beavariable line throughP, andlay off T'P' = 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' = (1/«)(0B), where n is any number; and throughA' 
and B' draw parallels to the axes ; these will meet in a point P' of the curve. 




FiQ. 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 s-axis in R. Draw a parallel to 
the a;-axis through K and a parallel to the j^-axia through R, meeting in Q. 
Then Q is a point of the curve. 



THE CATENARY 

The catenary ia 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,Ta = the horizontal and 
vertical components of T. The horizontal com- 
ponent Th is the same at all points of the curve. 

Thelengtho = Ta/io is called the parameter of the 
catenary, or the distance from the lowest point to 
the directrix DQ (Fig. 62). When u, is very large, 
the curve is very flat. For methods of finding a in 
any given case, see problems 1-6 below. 

The rectangular equation, referred to the lowest 
point as origin, ia y = a [cosh (x/a) — 1]. (For 
table of hyperbolic functions, see p. 60.) In case of 



y j 

,_ ;, ....J> 



Fia. 62. 



148 ANALYTICAL GEOMETRY 

x'' x' 

very flat arcs (o large) , y = - — i- . . .; a = x +ii — + . , ., 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 = arcOjP = s andOQ = y + a. The radius of curvature atP ia 
■B = (y +a,)'/a, 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, aa 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 = o sinh z, and T = 
liiacosh 2. Or, having «, find s/xand wx/r by using Tables Sand 2inversely, 
and hence (since x is known) compute s and T/w without the use of a. 





Table 1. 


GiVINQ Z WHEN y/x 


IS Known. Then a 


= x/z 




v/x 





1 


2 


3 


4 


5 


6 


7 


8 


9 


0.0 


0.0000 


0.0200 


0.0400 


0.0600 


0.0800 


0.0999 


0.1199 


0.1398 


0.1597 


0.1795 


0.1 


0. 1993 


0.2191 


0.2389 


0.2586 


0.2782 


0.2978 


0.3173 


0.3368 


0.3562 


0.3756 


0.2 


0.3948 


0.4140 


0.4332 


0.4522 


0.4712 


0.4901 


5089 


0.5276 


0.5463 


0.5648 


0.3 


5833 


0.6016 


0.6199 


0.6381 


0.6561 


0.6741 


0.6919 


0.7097 


0.7274 


0.7449 


0.4 


0.7623 


0.7797 


0.7969 


0.8140 


0.8311 


0.8480 


0.8647 


0.8814 


0.8980 


0.9145 


0.5 


0.9308 


0.9471 


0.9632 


0.9792 


0.9951 


1.0109 


1 .0266 


1 .0422 


I .0576 


1.0730 


0.6 


1 .0683 


1.1034 


I.II84 


I.r334 


1.1482 


1.1629 


1.1775 


1.1920 


1.2064 


1 .2207 



Note, y/x = (cosh z — l)/s. 



(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. 


GrviNQ 


z WHEN wx/T 


18 Known. 


Then a 


= x/z 




wx/T 


1 


2 


3 


4 


5 


6 


7 


8 


9 


00 


0.0000 0.0100 


0.0200 


0.0300 


0.0400 


0.0501 


0.0601 


0.0702 


0.0803 


00904 


0.1 


0.1005 0.1107 


0.1209 


0.1311 


0.1414 


0.1517 


0.1621 


0.1725 


0.1830 


0.1936 


02 


0.2042 0.2149 


0.2256 


0.2365 


0.2474 


0.2584 


0.2695 


0.2807 


0.2920 


0.3035 


03 


0.3150 0.3267 


0.3385 


0.3505 


0.3626 


0.3749 


0.3874 


0.4000 


0.4129 


0.4259 


0.4 


0.4392 0.4528 


0.4666 


0.4806 


0.4950 


0.5097 


0.5248 


0.5403 


0.5562 


0.5726 


0.5 


0.5894 0.6068 


0.6249 


0.6436 


0.6632 


0.6836 


0.7051 


0.7277 


0.7517 


0.7775 


0.6 


0.8053 0.8357 


0.8695 


0.9082 


0.9541 


1.0132 


1.1110 









Note. wx/T = 2/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 wxJT 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/2 




s/x 





1 


2 


3 


4 


5 


5 


7 


8 


9 


1.000 




0.0245 


0.0346 


0.0424 


0.0490 


0548 


0.0600 


0.0548 


0.0693 


0.0735 


1 


6 16774 


0.0812 


0.0848 


0.0883 


0.0916 


0.0948 


0.0980 


O.IOIO 


0.1039 


0.1067 


2 


0.1095 


0.1122 


0.1149 


0.1174 


0.1200 


0.1224 


0.1249 


0.1272 


0.1296 


0.1319 


3 


0.1341 


0.1363 


0.1385 


0.1407 


0.1428 


0.1448 


0.1469 


0.1489 


0.1509 


0.1529 


4 


0.1548 


0.1567 


0.1585 


0.1605 


0.1623 


0.1642 


0.1660 


0.1678 


0.1696 


0.1713 


1.005 


0.1731 


0.1748 


0.1765 


0.1782 


0.1799 


0.1815 


0.1831 


0.1848 


0.1854 


0.1880 


6 


0.1896 


0.1911 


0.1927 


0.1942 


0.1958 


0.1973 


0.1988 


0.2003 


0.2018 


0.2033 


7 


0.2047 


0.2062 


0.2076 


0.2091 


0.2105 


0.2119 


0.2133 


0.2147 


0.2161 


0.2175 


8 


0.2188 


0.2202 


0.2215 


0.2229 


0.2242 


0.2255 


0.2269 


0.2282 


0.2295 


0.2308 


9 


0.2321 


0.2334 


0.2346 


0.2359 


0.2372 


0.2384 


0.2397 


0.2409 


0.2421 


0.2434 


I.OI 


0.2446 


0.2565 


0.2678 


0.2787 


0.2892 


0.2993 


0.3091 


0.3186 


0.3278 


0.3367 


2 


0.3454 


0.3539 


0.3621 


0.3702 


0.3781 


0.3859 


0.3934 


0.4009 


0.4082 


0.4153 


3 


0.4224 


0.4293 


0.4361 


0.4428 


0.4494 


0.4559 


0.4623 


0.4686 


0.4748 


0.4809 


4 


0.4870 


0.4930 


0.4989 


0.5047 


0.5105 


0.5162 


0.5218 


0.5274 


0.5329 


0.5383 


1.05 


0.5437 


0.5490 


0.5543 


0.5595 


0.5647 


0.5698 


0.5749 


0.5799 


0.5849 


0.5898 


6 


0.5947 


0.5996 


0.6044 


0.6091 


0.6139 


0.6186 


0.6232 


0.6278 


0.6324 


0.5369 


7 


0.6414 


0.6459 


0.5504 


0.6548 


0.5591 


0.6635 


0.6678 


0.6721 


0.6753 


0.5805 


8 


0.6848 


0.6889 


0.5931 


0.5972 


0.7013 


0.7053 


0.7094 


0.7134 


0.7174 


0.7213 


9 


0.7253 


0.7292 


0.7331 


0.7369 


0.7408 


0.7445 


0.7484 


0.7522 


0.7559 


0.7597 


1.10 


0.7634 












Note: 


s/x = 


Binh z/z 



T s^ y 

(4) Given y and s. Then — = 77^ + -' 

w 2y 2 



»' « r^ -f / ■ n fi ^ hV ^ fA* 1 

= -■ Or, if y/s IS small, x=sl — -1-1 z\- \ — . . .• 

2y 2 ' "' L 3\s/ 15\s/ J 

T (T \ 

(5) Given 2/ AND Tlw. Then a = V, x= I y ) cosh" 

w \w / 



Tlw 
{Tlw) - 



, approximately, 



a = -\/2y(Tlw) — y'. Or, if vl(Tlw) is small, 

j2yT r 7 wy ~\ s — x _ 1 wy 

I ~ 12 "t ~ ' ' J' ~~s~~ ~3~T' 

Tlw i_ /«^\ " i_ /»^\ ' _ 1 

"L 4r 32 \r/ 128 \T/ 'J' 

(6) Given s and Tlw. Then x = — ^Jl - (~\ ' tanh-' (—\ , 

T T L /ws\ ' T f 7ws\i „ ., ,_ . „ 

«= \/l— I — li »=— \/l— I — I . Or, if wsIT la small, 

" w w\ \tJ w\ \tJ 

r l/ws\^ 11 /ws\* 1 ri/ws\ , l/w8\', 1 

^^VMy) -WoKt) -■■■}• ^='12Kt) + s[t) +■■] 
rr 1 /ws\2 1 /ws\* 1 

" = ^L'-2lW -s[-t) J- 

Given the Length 2L of a Chain Supported at Two Points A and 
B not in the Sa me Lev el, to find a. (See Fig. 63 ; b and c are supposed 
known.) Let (vl^ — b^)lc = six; enter Table 3 with this value of six, and 
find the corresponding value of the auxiliary variable z. Then a = c/2. 



150 



ANALYTICAL GEOMETRY 



Note. The co-ordinates of the mid-point M of AB (see Fig. 63) are Xo = 
a tanh~' (b/L), yo = (L/tanh z) — a, so that 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. 







y 


■^-— la 






t 


c 


^1 






"*""■ 






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 
h from Table 5. In either case, chord AB = 1 — kl. 





Table 4. Giving k 




Table 5. Giving k 


wl 

p 


A = aP 10° 20° 30° 40° 50° 60° 70° 80° 


wl 
Q 


A=0° 10° 20° 30° 40° 50° 60° 70° 80° 


.01 
.02 
.03 
.04 
.05 


.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 
007 005 006 005 004 003 002 001 000 
01 1 010 009 008 006 004 003 001 000 


.01 
.02 
.03 
.04 
.05 


.00000 000 000 000 000 000 000 000 000 
002 002 001 001 001 001 000 000 OOO 
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 


.06 
.07 
.08 
.09 
.10 


.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 


.06 
.07 
.08 
.09 
.10 


.00015 014 013 Oil 008 005 004 002 000 
020 020 018 015 Oil 008 005 002 001 
027 026 023 019 015 011 006 003 001 
034 032 029 024 019 013 008 004 001 
042 040 036 030 023 016 010 004 001 


.11 
.12 
.13 
.14 
.15 


.00051 050 046 040 032 023 014 007 002 
060 060 055 048 038 027 017 003 002 
070 070 065 057 045 032 020 009 002 
082 081 076 066 053 038 023 01 1 003 
094 094 087 076 061 044 027 013 003 


!l2 
.13 
.14 
.15 


.00051 048 043 036 028 019 Oil 005 001 
060 057 051 043 033 023 014 006 002 
070 067 060 050 038 025 016 007 002 
082 078 059 057 044 030 018 008 002 
094 089 079 066 050 035 021 010 002 


•16 
.17 
.18 
.19 
.20 


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 

168 168 159 140 113 082 050 024 005 


.16 
.17 
.18 
.19 
.20 


.00107 101 090 074 057 039 022 Oil 003 
121 114 101 084 064 044 026 012 003 
135 128 113 092 071 049 029 013 003 
151 142 125 103 079 054 032 015 004 
168 157 138 114 087 060 035 016 004 



Note. A; = 1 — {[1 — \/l — 2m sin u + m^]/[m sin. A]], where m = wl/P and m is 

given b y 

[1 — \/l — 2m sin u + wA secu = Isinh "i (tanu)' — sinh.~"i (tanu — m sec u)]tan ^4. 

Also, Q == P — wl {1 — k) ^iD. A, where k is the value in Table 4 corresponding to 
the given values of P and A. 

Correction for Stretch in 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 = 29,000,000 lb. 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 l = L + (LP/FE) (1 - m) . 

If the tension Q at the lower endi s known, compute wL/Q and find n from 
Table 7. Then l=L+ {JLQ/FE){1 + n). 





Table 6. Givinq m 




Table 7. GryiNo n 


wL 

T 


^^ 10° 20° 30° 40° 50° 60° 70° 80° 90° 


wL 
~Q 


4= 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 .0 1 6.024.032 .038 .043 .047 .049 .050 
.014 .031 .047 .062 .075 .086 .094 .099. 100 



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 «), i/ = o(l — cos u), where a is 



y 

\ 




r 

> 


V\ 


Y '' 





■■---X 


^ 


y''' 








\ 


/ 



y 


4 


zl 


f A 





/' r 3' 4' 



Fig. 65. 



FiQ. 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. Th e radius of curvature at any point P is PC = 
4osin(M/2) = 2V'2ay = twice thelength of the normal, PiV. Theevolute, 
or 1 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 !', 
2', . . . and radii equal to the chords 01, 
02, . . . , and sketch the cycloid as a curve 
tangent to all of these arcs. Or, on hori- 
zontal lines through 1, 2, . . . lay off dis- 
tances equal to 01', 02', etc.; the points thus reached will lie on the 
cycloid. 

The area of one arch = Sira', length of arc of one arch = 8a. Area 
bounded by the ordinate of the point P correspondin g to any v alue of u is 
a' (^ rad « — 2 sin u -f- H sin 2«) = ^ ax — ]^ u\^(.2a —y)y. Length of 
arc OP = 4o (1 - cos iiu) = ia - 2V'2o(2o - 2/). 




162 



ANALYTICAL GEOMETRY 



The Trochoid is a more general curve, traced by any point on a radius 
of the rolling circle, at distance h from the center (Fig. 67). It is a prolate 
trochoid if 6 < o, and a curtate or looped trochoid if 6 > o. The equations in 
either case are a; = o rad u — b sin u, y = a — b cos u. 

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 
b = 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 (A 1, A2, A3) along the circumference of the 

base circle, as AV, A2', A3', .... Describe circles with centers atl', 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 area 

through 1, 2, 3 meeting the radius OA in 1°, 2", 3" and the 

radii 01', 02', 03', ... in 1", 2", 3", . . . ; then from 1", 2", 3", . . . 

lay off arcs equal to I'l, 2°2, S'S, . . . respectively; the points thus 

reached will be points of the curve. 

aic i a)(c + 2a) 
The area OAP = — (rad u — sin it), 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 a/c)(c ± o)(l - cos Hu); arc AD = (4o/c)(c ± a). [In Fig. 69, D=i'.] 

Radius of curvature at any point P is fl = -r^ — sin Jiu; at A, B = 0; 



atD, R = 



4o(c + a) 
c + 2a 



c ± 2a 



Special Cases. If a = iic, the hypocycloid becomes a straight line, diam- 
eter of the fixed circle (Fig. 70) . In this case the hypotroohoid 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 i" + J/" = 

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 8). Length of cardioid = 16c. 






Fig. 70. 

The Epitrochoid (or Hypotrdchold) 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 e. 



The equations are a; 



y = (c ±a) sin 



in(^) 



{c ±a) cos ( — u ) + & cos 
-4.m[(l±f).] 



[(-f)-]- 



where u = the angle which the 



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 < o or 6 > o. When 6 = o, the special case of the epi- or hypo- 
cycloid arises. 

Th6 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. 
FlQ. 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(co3 u + rad u sin u), y = c (sin u — rad u cos u), 
where rad u ia the radian measure of the angle u which OQ makes with the 
X-axis. Length of arc AP — Hc(rad «)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 = 
fc rad 9, or r —a (9/360°). Here a = 2irk — the distance, measured along a 
radius, from each coil to the next. 

In order to construct the curve, draw radii 01, 02,03, . . . making angles 
12 3 

' " ' (360°), . . . with Ox, and along these radii lay 



(360°), 



(360°), 



oflE distances equal to - a, - o. 



3 

- o, 



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^/fc, and ON = k, where k = o/(2jr). Hence the construction. 

The radius of curvature at P is iJ = (k^ + r^)^^/(2k' + r'). To con- 
struct the center of curvature, C, draw NQ perpendicular to PN and PQ 
perpendicul ar to OP; th en OQ will meet PN in C. Length of arc OP = 
iik [rad 9 Vl + (rad 9)^-1- sinh~>(rad 9)]. After many windings, arc OP = 
iir'/k, approximately. 





Hyperbolic Spiral. 
Fig. 75. 



Logaritlimic Spiral. 
Fig. 76. 



The Hyperbolic Spiral is the curve whose polar equation is r = o/rad 9. 
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 without ever reaching it (asymptotic point). The figure shows one- 
half of the curve, corresponding to positive values of 6. liPT andPiV 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^/a. Hence a, construction for the tangent and 
normal. Radius of curvature atPisR = r/sin' j), where v = angle between 
OP and the tangent at P.. Construction: At iV draw a perpendicular to PiV, 
meeting PO in Q; at Q draw a perpendicular toPQ, meeting PJV in C; then C ia 
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 = oe"* ''^ '. 
Here a is the value of r when 9=0. For large negative values of 8, the curve 
winds around O as an asymptotic point. If Pr and PiV are the tangent and 
normal at P, the line TON bei ng perp endicular to OP (not shown in fig.), 
then ON = rm, and PN = rVl + m^ = 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. m) /m with the 
axis of the given spiral. Area swept out by the radius 
T from r = (where 6 = -co) to r=r, is A = 
T^/(,im) = half the triangle OPT. Length of arc from 
O to P = s = r/cos V = PT. 

The Tractrix, or Sohiele's Anti-friction Curve 
(Fig. 77) , is a curve such that the portion PT of the 
tangent between the point of contact and the K-axis ia 



/ 












a 






.y. 




h 


^^ 


■** ^ 


b 


M 


T 





Tractrix. 
FiQ. 77. 



constant = a. Its equation is a; = + a 



[...-.j-Vrrf], 



or, in 



parametric form, x = + a [t — tanh l], y = o/cosh t. (For tables of hyper- 
bolic functions, see p. 60.) The i-axia is an asymptote of the curve. 
Length of arc BP = a loge (fl/v). The evolute (locus of centers of curvature) 
ia 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 and a fixed tangent at the point A opposite O. If a 
is the radius of the circle, the polar equation is r = 2o ain' 9 /coa 9. Rec- 
tangular equation, i/2(2o — x) = x'. 






y 




^ ^ 




L. — 1 


^^ 


A. 


If 


\ 


I 'I 


O.T07la ■~-^-* 


\ 




) 



Lemniscate. 
FlQ. 79. 

The Lemniscate (Fig. 79) is the locus of a point P the product of whose 
distances from two fixed points F, F' is constant, equal to ^ o*. The distance 
FF' = oV^ Polar equation ia r = o\/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 



Maximum y occurs when = 30° and r — a/'V2, and is equal to H ay/i. 
Area of one loop = a^/2. 

The Helix (Fig. 80) ia 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 = 2irT tan v. Length of one coil = ■\/(2irT)2 + h^ 
= 27it/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 ciroumlerence 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 
rolled out on a plane, the development of the helix will be a straight line, 
with slope equal to tan v. 



K- Ji- -- H 

r ri \ [— 

A 

L. Ct 



2' 

Helix. 

FiQ. 80. 




DIFFERENTIAL AND INTEGRAL CALCULUS 

DERIVATIVES AND DIFFERENTIALS 

Derivatives and DifEerentdals. A f imction of a single variable x may 
be denoted hyf{x),F(.x), etc. The value of the function when x has the value 
Xo is then denoted by /(xo), Fixo), etc. The derivative of a function v=f(.x) 
may be denoted by f'(.x), or by dy/dx. The value of the derivative at a 
given point x = xo 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 increinent,A2/ (read: "delta y"), in yia the 
change produced in y by increasing x from xo to xi, + 
Ai; that is, Aj/ = /(xo + Ax) — /(xo). The difler- 
ential, dy, of y is the value which Aj/ 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 Xo, while Aj/ and dy depend not 
only on Xo 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 = /'(x) = 
dy lim V^v'\ lim [f(.xo + Ax) -/(xo)"l t„ \ j 

dx = Ax^o[x,j-Ax^ oL Z^ J • ^''°- ^y =/'(-) ''^- 

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 — m| 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, o is a constant 
which does not change in value in the same discussion; e = 2.71828.) 
1. d{a -\- u) = du. 2. d(flu) = adu. 

3. diu + V + w + . . .) = du + dv + dw + . . . 

4. d(uv) = udv + vdu. 

(du dv dw \ 
1 1 h . • . I 
U V w I 

„ ,u vdu — udv 
6. d- = ; - 



7. d(u'") = mu^-^du when m is not = — 1. 

157 



158 



DIFFERENTIAL AND INTEGRAL CALCULUS 



Thus, d(.u') = 2udu; d(«') = 3u'du; etc. 

\u/ u 



,- du 
8. dy/u = — ;=■ 
2Vu 



du 



10. d(e") = e'du. 

du 
12. dlogeU = — • 
u 

14. d ain u = coa u du. 

16. d cos M = — sin u<2u. 

18. d tan u = sec* udu. 

... d" 

20. d sin- 



22. d cos-' u-= 
24. d tan-' u = 



Vl - «' 
du 



11. d^a") = (log, o)a»dM. 

du 

13. dlogiou = (Iogio«) (0.4343. 

u 

15. d CSC u = — cot M cao u du. 
17. d sec u = tan u sec u du. 
19. d cot u = — csc^ u du. 

du 
21. d csc-'u = — ■ 



du 
u 



du 



--23.- d sec-' u = ■ 



uVu'' - 1 
du 



1 +u» 
26. d log« sin u = cot u du. 



uV'm'' — 1 

du 
25. d cot-' « = — 



27. d log. tan u 



1 +u» 
2du 



28. d loga COB u = — tan u du. 29, d loge cot u — 



sin 2u 
2du 



30. d sinh u = cosh u du. 
32. d cosh u = ainh u du. 
34. d tanh u = sech' u du. 

du 
36. d sinh-' u = — . 

Vu» + 1 

du 
38. d cosh-' u 



sin 2u 

31. d csch u = — csch « coth udu. 

33. d sech u = — sech u tanh u du. 

35. d coth u = — cach^ u du. 

du 
37. d csch-' u = 



40. d tanh- 



Vu' - 1 
du 



39. d sech-' u = — 



u-\/u' + 1 
du 



41. d coth-' 1* 



uVl — u' 
du 



1 - u» 



1 - u> 

42. d(u'') = (u"~') (u log, u di) +»du). 

Derivatives of Higher Orders. The derivative of the derivative is 
called the second derivative; the derivative of this, the third derivative; and 
so on. Notation: if y = f(x), 
dy 



nx) = D,y = ^; f"(x) 



ly=% r"ix) = Dly^2'- ''"■ 

Note. If the notation d^y/dx^ is used, this must not be treated as a fraction, hke dy/dx 
but as an inseperable symbol, made up of a symbol of operation, d^/dx^, and an operand y 

I The geometric meaning of the second derivative 
is this: if the original function y = fix) is repre- 
sented by a curve in the uaual way, then at any 
point where /"(s) is positive, the curve is concave 
upward, and at any point where /"(i) is negative, 
the curve is concave downward (Fig. 2). When 
/"(x) = 0, the curve usually has a point of 
inflection. "Fia. 2. 

DiSerentials of Higher Orders. The differ- 
ential of the differential is called the second differential; the differential ot 




DBRIVAriVES AND DIFFERENTIALS; MAXIMA AND MINIMA 169 

this, the third differential; etc. These quantities are of little iznportanoe 
except in the case where dx = a constant. In this case 

dy=-f'{x)dx; d'y = f"(,x)-{dx)''; d^'y = f"'(x)-{dxy; . . . 
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", D'", etc., and, assuming always that dx = a, constant, 

D' =dy+],id^y + Hd^y + l^id*y + . . . ; d'y == D"' - %D"" + . . . 

D" = dV + d'y + M2 d*y + . . .; d'^y = D" - D'" + m^ D"" + . . . 

D'" = d'y + ii d*y + . . .; dy=D'- }iD" + HD'" - HD"" + . . . 

Functions of Two or More Variables may be denoted by /(i, !/, . . .), 
F{x,y, . . .), etc. The derivative of such a function u = /(x, y, . . . ) formed 
on the assumption that x is the only variable {y, . . . being regarded for the 
moment as constants) is called the partial derivatiTO of u with respect to 

X, and is denoted by fi(x,y), or DxU, or -j— , or ^ — Similarly, the partial 

ux Ox 

dyU du 
derivative of u with respect to y is fyix,y), or DyU, or — — , or -^• 

ay ay 

Note. In the third notation, dxu denotes the differential of « formed on the aasump* 
tion that z 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 by/n, fxy, fyy, or by D*u, 

^u o ^u o ^u 
Dx(.Dyu), Dlu, or by i^r—^' -r — r— ' -r—;' the last symbols being "inseparable." 
ax' oxay ay' 

Similarly for higher derivatives. Note that /ij, =fyx- 

If increments As, Ay, (or dx, dy) are assigned to the independent variables 

X, y, the increment. Am, produced in M = f{x,y) is 

Am = /(a; + Ax, y + Ay) — f{x,y) ; 

whilethe differential, du, that is, the value which A« would have if the partial 

derivatives of u with respect to x and y were constant, is given by 

du = (Jx) ■ dx + ify) ■ 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 

du dx dy 

-dt=^^^d[ + ^^'i 

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 dj//dx = — (/i //a), where /» and 
fy are the partial derivatives of J(x,y) at the point in question. 

MAXIMA AND MINIMA 
A Function of One Variable, Bay = f{x), is said to have a maximum at 
a point X = xo, if at that point the slope of the curve is zero and the concavity 



160 



DIFFERENTIAL AND INTEGRAL CALCULUS 




Fig. 3. 



downward (see Fig. 3) ; a sufficient condition for a maximum is /'(lo) = 
and /"(so) negative. Similarly, /(x) has a minimum if the slope is zero and 
the concavity upward; a sufficient condition for a minimum is /'(lo) = and 
/"(lo) positive. If /'(xo) = and f"{xo) = 
and /'"(xo) ?^ 0, the point Xo will be a 
point of inflection. If /'(xo) = and 
/"(xo) = and/"'(xo) = 0, the point xo will 
be a maximum if /""(xo) < 0, and a mini- 
mum if /""(xo) > 0. It is usually sufficient, 
however, in any practical case, to find the 
values of x which make /'(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 = S{x,y), will have a maximum 
at a point (xo.J/o) if at that point fx = 0, fy — 0, and /ix < 0, /», <0; 
and a minimum if at that point fx = 0, /„ = 0, and fxx > 0, /„„ > 0; 
provided, in each case, (fxx) (Juy) — (/»») ^ is positive. If /» =0 and /„ = 0, 
and fxx and /j^have opposite signs, the point (xo,!/o) 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. 

X x^ x' X* 
^^=^+li + 2-!+3!+^+- • ■■• 



gx ^ gmx = 1 J. ™ ^ + ™ J.2 + 'IL a;J + 

1!2!3! 

where m = log« a = (2.3026) (logio o) . 

x^ x^ x^ x^ 
log.(l+x)=x--+---+--. 



log.(l -x) 






— CD < X < + 00, 

O>0, — CD<x<+a!, 

-1<X< +1, 
-1<X< +1. 

-I<x<+1. 

x< -lor + l<i. 

< X < ». 



{ 



0<<i< +" 
-o<x< +» 



EXPANSION IN SERIES 161 

Series for the Trigonometric Functions. In the following formuls, 
aU angles must be expressed in radians. li D = the number of degrees in 
the angle, and x = ita radian measure, then a; = 0.017453 D. 

sinx=x-|y + |^-|^ + ...; _co<x< + co. 

X^ x^ x^ x^ 
cosx=l-- + --- + --...; -co<x< + a,. 

, x' , 2x^ , 17a:' , 62x» , 
tanx=x+- + - + ^ + ^^ + . . .; -./2<x<+./2. 

1 X x' 2x'' x' 

cot X = 7, — 7~: — ST? ~ .-..g — . . . ; — jr < X < + T. 

X 3 45 945 4725 

. , . !/' 3j/' 5y' , ^ / , , 

j,3 y^ li* ^ , 

tan-> y=v-"- + f-^ + . .; -l^j/^+1. 

cos-' a = Htt — sin-' J/; cot-' y = Hit — tan-' y. 

Series for the Hyperbolic Functions (x a pure number). 

sinh ^=^+Tf + 77 + 7]+ "'' — co<a:<co. 

x^ x^ x^ 
cosh ^=1+2T'''4T"'"6^"*'" •' -<»<x<='. 

smh-',/=i/--+— --j^+ . .; _l<j,<+l. 

1/3 2/^ W'^ 

tanh-' y =y +~ +^ +Y + ■ ■ \ -l<y<+l. 

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 —a), 
as follows: 

(1) /(x) = /(O) + -j^ X + -^ x2 + — ^ x' + . . . 

+ 7— —^ «"~^ + (•?■")«"• (Maclaurin.) 
(" - 1)! 

(2) /(x) = /(o) + ^-j- (x - a) -1 ^ (x - a)2 + g, (x - o)» + . . . 

/(""') (a) 

+ -7 7T7 (x - o)"-' + (Q„)(x - o)". (Taylor.) 

(n - 1)! 

Here(Pn)x", or ((?n)(x — a)", is called the remainder term ; the values of 
the coefficients i'n and Qn may be expressed as follows: 
p„ = {/(«)(sx)l/n! = 1(1 - 0"-' fW(tx)}/{n- 1)! 

0, = l/(»)[a + s{x - a)]}/n\ = id - ()"-'/<:") [a + Kx - a)]}/{n - 1)! _ 
where s and < 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„)x", or 
11 



162 



DIFFERENTIAL AND INTEGRAL CALCULUS 



(Qn)(a; — a,)", where Pn, or Q„, is the largest value taken on by P„, or Q„, 
when « 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 tof(x) for values of x near a; = 0; Taylor's series gives a similar ap- 
proximation for values near x = a. 

Fourier's Series. Let fix) be a function which is finite in the interval 
from x = — ctox = +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, 

„ . , TTX , 2irx , Sirs , 

f(.x) = H oo -1- oi cos — +02 cos h 03 cos h . . . 

c c c 

17, ■ '^^ 1 I, • ^JTi . Sirx 

+ bi sin — +02 sin + bi sin ^r- -|- . . . 

c c c 

where the constant coefficients are determined as follows: 



1 / ' , , nirt , , 1 / 

= - I /(O cos — at, o» = - I j 

J-c J—c 



., ^ . yvKt , 

/(O sin At. 

c 



In case the curve y = f{x) is symmetrical with respect to the origin, the 
o's are all zero, and the series is a sine series. In case the curve is sym- 
metrical with respect to the 2/-axis, the &'s are all zero, and a cosine series 
results. (In this case, the series will be valid not only for values of i between 
— c and c, but also for x = — c and x = e.) 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, /(xo) is to be defined as V^t/i(xo) + /{(xo)], 
where /i(zo) is the limit of /(x) when x approaches xo from below, and /2(xo) is the Umit 
of /(x) when x approaches xo from above. 



■I. !^ t 



A 


A 


V 


A 


-2c -c 


c 


Zc 


3c 



\c 'fc 13e 




FiQ. 4. 



FiQ. 5. 



Fia. 6. 



Examples of Fourier's Series. 

1. If 1/ = /(x) is the curve in Fig. 4, 

h 
2/ = o 



i' 



4h / irx , 1 Sttx 1 

cos — + -cos — lCOS 

c 9 c 25 



2. If 2/ = /(x) is the curve in Fig. 5, 



ih I . TTX . 1 . 

1/ = — I sin 1- - 81 

IT \ c 3 



Sjtx , 1 . bvx . 

sin h - sin 1- 

c c 



Z. liy = /(x) is the curve in Fig. 6, 



V = 



2h 



.TTX 1 . 2irx , 1 . 3xx 

sin sin + - sm 

c 2 e 3 c 



5irx \ 

c ■ ■ 7 

•) 
■) 



INDETERMINATE FORMS; CURVATURE 163 

INDETEKMINATE FORMS 

In the following paragraphs, f(.x), g(x) denote functions which approach 0; 
P(x),G (a;) functions which increase indefinitely; and U(x) a function which ap- 
proaches 1 ; when x approaches a definite quantity o. 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. "-•" To find the limit of f(,x)/g{x) when f{x) — ando(x) = 0, 

use the theorem that lim — -— = lim ,, > where /'(x) and g'{x) are the 
gi,x) g'{x) 

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: 

f'(x) f"(x) 

lim — -— - = lim „, , ; and so on. 

g'(.x) g"(,x) 

Cash 2. " — " If F{x) = oo andG(s) = oo, then lim -7-^ = lim— 7-^) 

00 "(x) tr (X) 

precisely as in Case 1. 

Case 5. "Ooo." To find the limit of /(x)-2''(x) when/(x) =Oandf(x)= <x>, 

fix) P(x) 

write lim lf(,x)-F(x)] = lim , or = lim ; then proceed as in 

Case 1 or Case 2. 

Case 4. "0»." If /(x) = and g{x) = 0, find lim [/(x)]*^^) as follows: 
let y = [/(s)]* , and take the logarithm of both sides thus: 

loge y = ff(x) loge /(x) ; 
next, find lim [g(x) logi,/(x)], = m, by Case 3; then lim y = e". 

Case 5. "1"." If C7(x) = 1 andii'(x) = 00, find lim [t7(x)]^^'''as follows: 
let y = [{/(x)] "^^^'^ and take the logarithm of both sides, as in Case 4. 

Cash 6. ," ooO." If F(,x) = ooand /(x) = 0, find lim [F{x)V^''^ as foUows: 

let y = [F{x)]'^^', and take the logarithm of both sides, as in Case 4. 

Case 7. " co — »." If F(.x) = a> and (?(x) = co, write lim [F{x) — G(x)] 
_1 ]_ 

= lim — ^ ; then proceed as in Case 1. Sometimes it is shorter to ex- 



f (X) -Gix) 
pand the functions in series. It should be carefully noticed that ezpressions 
like 0/0, °° /<^, etc. , do not represent mathematical quantities. 

CURVATURE 

The radius of curvature B of a plane curve at any point P (Fig. 7) is th« 
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 is 
V = f{x), 

„ _ ds _ [1 + (y'n^ 




164 DIFFERENTIAL AND INTEGRAL CALCULUS 

where ds = y/dx'^ + dj/^ = the differential of are, u — tan"' [/'(x)] = the 
angle which the tangent at P makes with the x-axis, and y' = f\x) and 
y" = /" (s) are the first and second derivatives of /(a;) at the point P. Note 
that dx = ds cos u and dy = ds sin u. The curvature, K, at the point P, is 
K = l/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, 
K = f"{x). 

If the equation of the curve in polar co-ordinates is r = /(e) , where r = radius 
vector and S = polar angle, then 

[ r^ + (,r')^% 
r' - rr" + 2(r')^' 
where r' =/'(«) and r"=f"{e). 

Ths 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 i3f(x)dx, and is 
denoted by J'f{x)dx. All the integrals of f{x)dx are included in the ex- 
pression y/(x)dx-)- C, where S}{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 FoRMxrL.s} 

1. J adu = ajdu = om 4- C 2. J (u -f v)dx = J udx 4- Jiidx 

3. J'udv = uv —J'xdu 4. J" J{x)dx = J'f[F{y)\F'(y)dv, x =F(v) 

5. J^ dy J^ f(x,y)dx = J" dx J' S{x,y)dy. 

Fundamental Inteqbals 

a;"dx = — — + C; when nj^ -\ 
w-j-l 

7. y — = log, X -\-C = log« ex 8. J^e'dx = e' + C 

9. J sin xdx = — cos x -|- C 10. J ooa xdx — sin x -(- C 

^ dx _ ^ dx 

11. / -^— - = - cot X -I- C 12. / =tanx-)-C 

"^ sin* X "^ cos* X 

13. / — , — sin"' X -t- C = — cos"' x ■\- c 

Vl -x2 

14. y ^ ^ = tan-i X +C = - cot-' x 4- c 

Eational Functions 

15. /(a-fbx)Vx = ^^^'.fC 



INDEFINITE INTEGRALS 



165 



/» da; 1 1 

^ a +bx °^ b ^^' (a +bx) +C =- log, c(a + bx) 

17. f^dx +C 



•^ (a + 



1 



bx)' 



r+C 



19. y^r^j.= K.iog,- 

20- /; 



1 +a; 



dx 



= Hlog,^— -^ +C = 
a; + 1 



+ C = tanh-' a; + C, 

coth-i a; + C, 



b(a + bx) 
when a; < 1 

when X > 1 



-A-f^.= 7r.--'(>J^)+^ 



22. f—^^— _ 



1 , Vofc + fcx , ^ 
logs —^= h C 



1 



\/ab 



tanh' 



23. y 



dx 



.J 6 + ex 



when o > 0, 6 > 



a +2bx + ex' -y/ac - b' 
1 



tan 



■Vac — b' 



+ C 



when 
ac -b'> 0; 



■\/b' — ac — b — ex „ 

, loge , = h C 

2 Vfc* - ac Vb' -ac +b +CX 
1 



Vb' 



24 r- 



dx 



ac 
1 



26x + ex' b + ex 

^ 2c 



b + ex 

tanh-i^?^= + C, 

V 6^ - ac 

+ Ci when b' = ac 

mc — nb 



when 
6» - oc > 0; 



25- ./ ^^o., , „,. = ^ l°g« (« + 26:^ + cx») + 



26. 



•^ a + 



26x + ex' 
f{x)dx 



f„ 



dx 



a + 26x +cx' 
if /(x) is a polynominal of higher than the first 



2bx + ex' 
degree, divide by the denominator before integrating. 

27 r <^^ I ^ 6 +ex 

- ■ "^ (o + 26x + ex^)" 2(oc - b')(,p - 1) (o + 26x + cx')^^ 



+ 



(2p - 3)c 



28. /"— 
."^ {a 



(m + nx)dx 
+ 2bx + ex')" 



2(oo - 62) (p 
n 



f, 



dx 



V) ^ (o + 2&X + cx')^'^ 
1 



29. y-x-iCa +6x)»dx = 5!i;(^+M!l' 
" (m + n)b 



2c(p - 1) (o + 2fex + ex')"-! 

mc — nb p dx 

^ (o + 26x + cx^)" 



+ ■ 



(m — l)a 



x"(a + 6x)" ma 



m + ra 



. , ^,J'x'"-'(.a + bx)'' dx 
J'x'^-Ha + bx)""' dx 



m + w 



166 DIFFERENTIAL AND INTEGRAL CALCULUS 

Ibeational Functions 
30. yVcT+bxdx = ifiVa + bx)' +C 



3b' 



dx 



oc / — , = sin ' \-C = — COS ' Y c 



31. y^^i==^v^T^» + c 

Vo+6x ° 

32 /- ('" +"^^'' '' = A (3^6 _ Ian + n6i) V^T^ + C 

33 Z' ; substitute j/ = \/o + 6s, and use 21 and 22 

(m + wa;) Vo-j-fta; 

/./(r, Va+6^) J 1. i-j. i "/ , , 

34 / — , dx; substitute Vo -Yox=y 

<ia; . _, a: , „ __, s 

36. r-^= = log" [=^ + VaM^] + C = sinh-i — + c 

Vo' + x^ " 

37. r—f^= = log. [x + \/^rZ-^2] + c = cosh-i- + c 

38. f-y==^= = -^log,[6 + ex + VTVo + 26a; +cx2] + C, 

Va + 26a! + cx"^ V c 

when c > 0; 

\= sinh-i-^^^:^ + C, when ac -6^ > 0; 

Vc Vac — 6" 

= -^cosh-i ^ +''^ + C, when 6^ - ac > 0; 
Vc V 5= — oc 



-\/^ V^ 

39. / , = -VO + 26a; + ex' 

Vo + 2ba; + cx^ " 



Lsin-i ^ "^"'^ + C, when c < 



■ ac 



mc — 7i6 r* dx 



Vo + 26x + ex' 
/. x^dx x'"~'X (m — l)a y^x"'"^ dx 

40. r , ^^ ^ / — - — 

Vo + 26x + ex' »"C mc -^ A 

(2m - 1)6 y^x™-! dx ^ ^ / — — — — r-^ 

- ^^^ — I — • where X = Va + 26x + ex' 

mc ^ X 

41. J'Va'+x'dx = ^y/a' + x^ + -^ log. (x + Vo' + a;») + C 



X / a* X 

= rVo' + x' + -^ sinh-i- + 
2 2 a 



2 

c 



42. J'y/a?- x'dx = |Va= - x' + ^ sin-' - + C 



INDEFINITE INTEGRALS 167 

43. J'Vx^- a'dx = |Va:^ - a^ - ^ log, (a; + y/x^ - a') + C 

X / a^ _ X 

= -V a;* ~ "'' ■" "TT cosh '■ — \- c 

44. J'Va+ 26i + cx^dx = — — ^Va + 26s + ci« 

, ac — b^ /^ dx „ 

H / — + C 

2c " Va + 2bx + ex' 

Transcendental Functions 

45. fa^dx =r^— + C 
^ log, a 

46. / X" e"' dx = 1 h . . - . . . ± + C 

^ a »\_ ax a'x' o"»"J 

47. ^y log, xdx = X log, a; — X + C 

48. ri2Mi^d^ = _l2gl^_l+C 

^ x' XX 

49. y 02?!^ <2=^ = -4t (log. x)»+' + C 

" X re + 1 • 

50. yam' xdx = — M sin 2x + }ix + C = — H sin x cos x + Mx + C 

51. ^oos» X dx = H sin 2x + J^x + C = J^ sin x cos x + iix + C 

m /• • J cos mx .„ n , sin mx . „ 

52. / Bin mx dx = + C 53. /cos mx dx = h C 

•^ TO "^ m 

_, /» . , cos (m + n)x cos (m — n)x „ 

64. / sin mx cos nxdx = — ; \- C 

•^ 2(m + n) 2(m - n) 

._ /» . . , sin(m — n)x sinfm + n)x , _ 

65. / sin mx sin rex dx = — ; ; \- C 

^ 2(m - re) 2(to + re) 

/» , sin(m — re)x , sin(m + re)x „ 

56. / cos mx cos rex dx = — ; + C 

•^ 2(m - re) 2(m + re) 

67. J tan xdx = — log, cos x + C 58. J not xdx = log, sin x + C 

59. f-^ = log, tan I + C 60. f^^ = log, tan (^ + ^) + C 

•^ sin X 2 ^ cos X \4 2/ 

^^- fv^ tanf+C 62. y—^^ cot ^ + C 

•^ 1 + cos X 2 '^ 1 — cos X 2 

63. /sin X cos xdx = ^ sin^ x + C 64. /*-: = log, tan x + C 

" • "^ sin X cos X 



. /» . , cos X sm" 'x re — 1 /. . 

65.* / Bin" xdx = / sm" ^ xdx 

•^ rem'' 

^ /. , sin X cos""' X re — 1 /^ 
66.* / cos" xdx = / COS""' xdx 



* If n is an odd number, substitute cos x = z or sin x = z. 



168 



DIFFERENTIAL AND INTEGRAL CALCULUS 



/tan" ^ X y* 
tB.n"xdx = ; / tan""'' xdx 
n - 1 ^ 



68. J' cot" xdx 

69. f "^^ 

70. / 



cot"~i X 
n - 1 
cos X 



sm" X 
dx 



J'coV ^ xdx 

+ 



2 ^ dx 
l^ sin"- 2 : 



(re — 1) sin"-i _ 

sin a; , re — 2 /» dx 



cos" X (re — 1) COS' 



re — 2 y^ da; 
'-ix n - I-' cos"-2 



_ i /» . , sin''"'" 'x cos' 'x , g 
71. *y sin'' X cos« xdx = ^"7^1 1- 



-JsvoPxcoa^ 'xdx 



72." J'sin~''xcos''xdx= ■ 
73. *^sin''xco3-«xdx = 



P + 2 P + S" 

Bin''-'xcos'"'"^x J) — 1 /» . _„ 

; 1 ; — / sm'' ' X cos' X ox 

P + 3 P + 3 

Bin""''''"'xco3'''''x J)— g— 2 /^ . ,. 

h^^ — ^— r- / sin-''+2 X aoaixdx 

J) — 1 p — 1 ^ 

sin"""" 'xcoa"'"'' 'x . g — p — 2 /• . , 

+ ; — / sin'' X cos-«+2x dx 

g - 1 '^ 



•^ a + 



dx 



6 cos X -(/a! — 62 
1 



3 - 1 

tan"' 



(V^6*"°^'=^) 



tan Mx + C, when a' > 5> 



V6* - a> 
2 



6 + o cos X + sin xv 6' — o^ „ 

log, -T h C, 

a -'r cos x 



__ /» cos X dx 

75. / — n. 

'^ a -\-o cos X 

76. / 



/»A+Bcosx+C sin x , 
77. y r-; ; : dx 



, '- tanh-i ( \/^— -^ tan Mx) + C, 

X a /^ c 
y" b'-' o +i 



when 



dx 
b cos X 



+ C 



sin X dx 
o + b cos X 



- logs (a + b cos x) + C 



o + 6 cos X + c sin x 



^^^ 



d?/ 



pcos!/ 



cos V dy 



^ /» cos y tiy /* SIX 

+ {B cos tt + C sin v) I — ; — (B sin u — C cos «) / — — 

^ -^ o + pcosj/ '-^ a + 

where b = p cos u, c = p sin m and x — u = y. 

/• . , , o sin hx — h cos bx „ 

78. / e"* sin bxdx = ^-r-.";; e" + C 

•^ a* _|_ (j2 

»„/•„. I J acosbx +bsinbx „, , „ 

79. / e"" cos bx dx = ' , ,„ e"' + C 

^ a' + 0' 

80. r&irr^xdx = xsin^'x + •%/! — x^ + C 
8X. J' coa~^xdx = xoos-'x — \/l — x' + C 

82. y^tan"' x dx = x tan~' x — J4 log, (1 + x^) + C 

83. J'ooir^xdx = X cot"! x + Hlog, (1 + x^) + C 
* If p or g is an odd number, substitute cos x = z or sin x = z. 



sin y dy 



poos J/ 



DEFINITE INTEGRALS 



169 



84. ^y sinh xdx = ooah x -\- C 85. Ctanhxclx = loge cosh a; + C 

86. J cosh xdx = sinh x + C 87. J coth xdx = logs sinh a; + C 

88. y sech xdx = 2 tan"' (e*) + C 89. J' each xdx = log, tanh (x/2) +C 

90. ^ sinh^ xdx = \i sinh x cosh x — Yix + C 

91. J cosh? xdx = a sinh x cosh x + }4x + C 

92. J^aech' xdx = tanh x + C 93. _/*osch2 xdx = - coth x + C 

DEFINITE INTEGRALS 




The definite integral of f{x)dx from x = o to x = b, denoted by J^ f{x)dx, 
is the limit (as n increases indefinitely) of a sum of n terms: 

r''f{x)dx = '™ [/(xOAx +/(x2)Ax +/(x3)Ax + . . .+/(x„)Ax], 
•^ " yi = 00 

built up as follows: Divide the interval from o to & into n equal parts, and call 
each part Ax, = Q> — a)/n; in each of these intervals take a value of x (say 
xi, X2, . . . Xn), 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, __/ 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 yfM 
is, briefly, the "area under the curve, from a to 6." The 
fundamental theorem for the evaluation of a definite 
integral is the following: 

/;/(x)dx = [ff{x)dx ]^., - [fmdx]^^^ ; 

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. 

r = -/;= f:+f:=f:- 

The Mean-value Theorem foe Integeals. 

f'Fix) S{x)dx = F{.X) f^ Kx)dx, 

provided /(x) does not change sign from x = o to x = 6 ; here X is some (un- 
known) value of X intermediate between a and h. 

Theorem on Change of Vaeiable. In evaluating j'^ 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 = 6 by the corresponding values of (, provided that throughout the 
interval the relation between x and i 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) . 



Fig. 8. 



170 



DIFFERENTIAL AND INTEGRAL CALCULUS 



Differentiation with Respect to the XJppee Limit. If 6 is variable, 
then ^ f(x)dx is a function of 6, whose derivative is 

DiFFEBENTIATION WITH RESPECT TO A PAKAMBTEIt. 

-^ / f{x,c)dx = / — 5 ax. 

Functions Defined 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." 

r%„ dx 

1. Elliptic integral of the first kind =F(,u,k) = /„ . -^ =(fc'<l) 

V 1 — «* sin^ X 

2. Elliptic integral of the second kind = E(,u,k) = J^ " V 1 — fc' sin'x ix 
(k' < 1) 

3. 4. Complete elliptic integrals of the first and second kinds; put u = ir/2 
in (1) and (2). 

6. The Probability integral = ~~i= Jl' ^~' <^^ 

6. The Gamma function = r(m) = y^ "a;»->e-*ds 

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 = /(x) from x = ato 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 = J'f(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 S fS{x, y)dy dx 
means S\S 1^^' V)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 /(x, y)dx dy. Keeping x and 
dx constant, and summing these elements from, say, y = 
Fi(x) to y^Fiix), as determined by the shape of the 
boundary, the weight of a typical strip perpendicular to the x-axis ia 




■r,(,) 



Fig. 9. 



dx 



P' 

Jv= 



\x,y)dy. 
Fi(x) 



Finally, summing these strips from, say, x = a to x = 6, the 



weight of the whole area is 



rx^b fv 

{dx /( 
Jx=a Jy= 



= F2(x) 
Six, v)dy], or, briefly, J'J'f{x,y)dydx. 
Fi(.x) 



DIFFERENTIAL EQUATIONS 171 

DIFFERENTIAL EQUATIONS 

An ordinary diflerential 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 dlfierential 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 = fKx)dx. 

Methods of Solving Ordinary Differential Equations 

DiPFERENTIAI, EQUATIONS OF THE FiRST OhdEK 

(1) If possible, separate the variables; that is, collect all the x'a 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 -r- =/(-). Substituting y = xt will enable 

dx \x; 

/* dt ^ 

the variables to be separated. Solution: log. x = J -— + C. 

j\t) t 

(3) The expression f(x,y)dx + F(x,y)dy is an exact differential if 

— ^r-^ — = — ir- — ( = -P> say). In this case the solution of f(x,y)dx + 

ay 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 

dy 

(4) Linear differential equation of the first order: — + fix)-y = F{x). 

Solution: j/ = e"^ ( J' ePF{x)dx^- c] , where P = ff{.x)dx. 

(5) Bernoulli's equation: \- f{x)-y = F(,x)-y". Substituting j/'"" =v 

ax 

gives -7- + (1 — n)f(x)-v = (1 — n)F(,x), which is linear in v and *. 
dx 

(6) Clairaut's equation: y = xp +/(p), where p = dy/dx. The solution 
consists of the family of lines given by j/ = Cs + /(C) , 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 
/(P)- 

Differential Equations of the Second Oeder 

d^y 

(7) -T— = —n'y. Solution: y = Ci sin (ma; + C2) 
dx^ 

01 y = Cz sin nx + C4 cos nx 



172 DIFFERENTIAL AND INTEGRAL CALCULUS 

(8) -T^ = +n''y. Solution: y = Ci sinh (wx + Ci) 
ox* 

OT y = Ca e""" + d e"""^ 

(9) T^ = /(!/)• Solution: x = f ~r^- — - + Cj, where P = ff(y) dy. 



(10) -j^. = /(x). Solution: j/ = CPdx + Cix + C2, where P = C j{x)dx 
ox* 



or J/ = ^^ — J'xfix)dx + Cix + C2 

(11) _^ = /M) . Putting / = 2, V^, = -^. s = J 77T + Ci and j/ = 
dx^ Vx/ dx dx2 dx "^ /(«) 

r + C2; then eliminate z from these two equations. 

"^ /(z) 

(12) The equation for damped vibration: — + 2b- \- ahl = 0. 



Case I. If o2 - 62 > 0, let m = Va^ - b\ Solution: 

y = Ci e'^' sin (mx + Ci) or y = e~'"[Ci sin (mx) + Ci cos (mx)] 
Case II. If o2 _ 62 = 0, solution is 3/ = e-''"^[Ci + Czx]. 

Case III. If a2 - 62 < 0, let n = Vb^ - a^. Solution: 

2/ = Cie-*» sinh (nx + C2) or 3/ = (736"^''+"'^ + Cte'^''-'''"' 

(13) -r^ + 26-;^ + a'y = c. Solution: y = -z + yi, where yi = the solu- 
dx2 dx a' 

tion of the corresponding equation with second member zero [see (12) above]. 

(14) — ^ + 25-^ + o2j; = c sin(J;x). Solution: 
dx' dx 

y = R sin(J;x - <S) + yi, where R = c/Via" - k')" + ib^k\ 

2bk 

tan S = ;-, and j/i = the solution of the corresponding equation with 

a' — k2 

second member zero [see (12) above]. 

d^TJ d'u 

(15) —^ + 25— + a'^ = f{x). Solution: y = yo + yu where j/o= any 

particular solution of the given equation, and j/i = the general solution of the 
corresponding equation with second member zero [see (12) above]. 



If 52 > a\ va , ^ I e"»»: /'e-'»i^/(x)dx - e"*"^ C e''^"'' f{x)dx 

2V62 - a2 i -^ "^ 

where mi = — 6 + •\/6* — a^ and mj = — 5 — V'6'' — o*. 

If 62 <a^ letm=Vo'' -62; then yo = 
— e~** J sin (mx) J ^' cos (mx) ■f(x)dx — cos (mx) J ^' sin (mx) -/(x) dx | • 

If 62 = o2, 2/, = e-*"^ I xj'e^^'fi.x) dx - J^x-e'"f(x) dx \ ■ 




GRAPHICAL REPRESENTATION OF FUNCTIONS 

For graphical inethoda in statistics, etc., see W. C. Brinton'a "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 x-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-scaXe, whose 
positive direction (as usually chosen) runs upward, need 
not be the same as the unit for the x-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 J* (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 positive (parabolic type) ; re negative 
(hyperbolic type)] (Fig. 3). 

The exponential function, y = 10^ or y =e^, and the logarithmic 
function, y = logio x or y = loge 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 x — 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 — h in the equation, this merely requires re-numbering 
the j/-axis so that each number is moved 6 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 j/-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,j/) 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. 

ira 



174 



GRAPHICAL REPRESENTATION OF FUNCTIONS 




1 ^^^\ "' " 


-1 Tr 



Linear function, y = mx + h. 
Fig. 2. 



( Parabolic Type) (Hyperbolic Type) 

Power function, y = aj«. 
Fig. 3. 




Escponential function (10=" or e*). 
Logarithmic function (logio x or logo x) . 
Fig. 4. 




Inverse trigonometric functions. 
Fig. 6. 





y 












I' 


\ 


i 


1 




1 


/f 


^"'^ 


* 


?^ 


■M 


/zjt 




^ 


V 




/ 
/ 


/ 


'^ 






i 


-1 

-2 




^ 


1 




-f 




-1 
-I 


T 


>" 


^ 
X 


\r 


l\ 


^0* 
















1 1 


1 


-3 




! 1 


( 1 


1 



Trigonometric functions. 
Fio. 6. 



To Find a Mathematical Equation to Fit a Given Empirical Curve. 

This problem ia 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 yisy = mx + 6, where m = the slope of the 
line, and 6 = the value of y at the point where the line crosses the ^-axia. If 



EQUATIONS INVOLVING TWO VARIABLES 



175 





Hyperbolic functions and inverse hyperbolic functions. 
Fig. 8. 




Fig. 9. 



2 

»f.or.... 




K.Oi:.._ 


. 


,/r.o— ..^' 


\ 


/r.-QZ..!^" 


y\ 


K--OS-.. 


■^. 


K- -08- a 


^^^^^ 


H..,..y 


f \ t " 


K= -IS-- 


' y.e-'*Kt-" 



Fig. 10. 



Fig. 11. 



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 + 6, and, by substituting in this formula n 
pairs of values of x and y, obtain n equations connecting the coefficients 
m and 6; 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 form jix,y) = m-F{x,y) + 6 is suspected, where f(x,y) and 
F(x,y) are any expressions involving either x or 2/ or both x and y, such a law 
may be tested by plotting F(x,y) instead of x, and f(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 ft can be read from the figure as before. For example, if j/' = mxy + 6, 
a straight line will be obtained by plotting y^ 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 + m (y/x) . 

Case 2. If a law of the form y = ex" is suspected, plot the points {x,y) on 
logarithmic paper (see below). 

Case 3. If a law of the form y =c-10'"' [or y = c-e'"'] 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 = aainbx ot y = a cos bx. If the heights of the 
waves diminish as x increases, try an equation of the form y = ae~"» 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 + cx^ + dx^ + ex* + . . ., the first 
three or four terms being usually sufScient. 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 + 6x + ay, where a, b, and c are 
determined by substituting the co-ordinates of three conspicuous points of the 
curve. The lines a; = a and y ~ b are the asymptotes of the hyperbola. 
The equation may also be written {x — a)(y —b) = k, where k = o6 + 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 j/-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. 



■^1 


1 

-lonfi 




















I 




S)0 

100 
50 

10 
5 




~ 




ul I 


: 




^ 




I = 


: :; 1 










































































































/ 






? 












D 


/ 








































— t 


































































/ 






















i 








































V 


















= 






: 




^ 






::: 


















































/ 






















f 




















/ 






















/ 




















. 








5 











50 





1x 



Fio. 13. 




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 j/-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 s = 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 = ex" when plotted on logarithmic paper will 
be represented by a straight line whose slope is n. For, if j/i = cxi" and 
j/i = cXi", then yi/2/2 = (xi/xn)", or (log yi — log 2/2) /(log Xi — log xj) = n. 
The slope must be measured by aid of an auxiliary uniform scale. 

Example. Let y — x^^^. When x = 1, y = X] plot this point A on the logarithmio 
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 ir = 2.50, y = 3.95, as shown by dotted lines, so that (2.50)»'2 = 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 he determined by inspection. 

To determine whether a given set of values, z and y, satisfies a law 
of the form y = ex", plot the values on logarithmio 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'3, that is, move each point downward 
a distance equal to 6 units of the y-scale at that point. If it proves possible to choose b 
so that the resulting points he in line, then the original values obey a law of the form 
y — b = cx'*f where n is again the slope of the line, and c is the value of y — b 
when a: = 1. (Conversely, if the curve is concave downward, try adding b to all the 
2/*s: that is, move each point upward; ii the new points lie in line, the original values 
obey a law of the form y -{- b = cx^.) 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)". 

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 x-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 l/i:th of the corresponding number along x (here 
fc = 6). The number k, which may be chosen at pleasure, should be taken 
equal to some simple integer, aa 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 y-axis at the points which mark the main 
lines of division between the horizontal strips; while the numbers . . ., 
— 2k, — k, 0, + k, + 2k, ... (or such of them as may be needed) must 
be placed along the x-axis at uniform intervals, each interval (from to k, 
from k 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 




^*, 



Fio. 15. 



logarithmic paper; the larger the unit, the finer the possible subdivisions of 
the scale. 

To plot a point (.x,y), as a; = 3, j/ = 5, on semi-logarithmic paper means to 
find the point of intersection of the vertical line marked x = Z with the 
horizontal line marked y = 5. 

Any equation of the form y = 
c-lO"" [or y — ce""] when plotted 
on semi-logarithmic paper with scale 
value k, will be represented by a 
straight line whose slope is km [or 
0.4343 fcm.]. 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 = 

]^00.<343 

Example. Given y = 410-''-'" [or y = 
4.5-0. u]_ In Fig. 15, when s =■ 0, » = 4. 
By plotting this point (A) on the semi- 
logarithmic paper, with Bcale value 5, and 
drawing through it a straight line with 
slope equal to — 0.5 [or — 0.2171 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 2/-scale undetermined (unless a separate ^-scale is 
provided for each section of the graph). 

In order to determine whether a given set of values of z and y satisfy 
a law of the form y = c-lO""^ [or y = ce""'], plot the values of x and yon 
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 0.4343fc], ^nd c = the value of y when s = 0. 

If the plotted points fail to lie exactly in line, but form a curve slightly concave up- 
ward, try subtracting some constant h from all the j/'s, and plot the values thus modified; 
if h can be so chosen that the revised points lie in line, then the 
original values obey a law of the form y — b = clO*"* [or y — 6 = 
c.e«*], where m and c are to be found as before. If the curve is con- 
cave downward, add 6, instead of subtracting; and replace y — ftby 
y -|- 6 in the law. 

Curves in Polar Co-ordinates. Any function, r, of a single vari- 
able, 9j can be represented by a curve in polar co-ordinates (p. 137). 
Lay off the given values of B as angles, the initial line Ox running 
toward the right, and the counterclockwise direction about the origin 
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 d at a given point P is represented graphically 
as follows (Fig. 16): On the tangent at P drop a perpendicular Oil from the origin; 
then r{MP/OM) represents the rate of change, dr/dd, 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 




MIJUATIONS 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 out out to represent 
parallel plane sections of the surface, and then stood on edge in slots out 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 = fix, 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,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 x-curve, 
a jz-curve, a 2-curve, and a le-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, t. Any relation between 
three variables, t, u, i), 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, " Traits 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. 



AX A 



Fig. 



17. 



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(,t) , Fi{() 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 a;-axis, and its y parallel to the 
j/-axis. 

Method 1, Given, an equation which can be thrown into the form 
U-Fi(.t) + V-Fiit) = 1, 
where, for the given range of values of u and v, the largest variations in V 
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 Fro. 18. 
plot, in the usual way, the points (x,j/) given by 

^ F2(() ^ 1 

^ Flit) +F,(.t)' " Flit) +Fiit)' 
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 
Flit) /Flit) = a constant, the i-soale will prove to be a straight line parallel 
to the i/-axis.] 

Then, using the same units as above, plot along y the points given by 
y = U, labeling eaeh point with the corresponding value of M ; and plot along 
y' the points given by j/' = V, labeling each of these points with the corre- 
sponding value of V. This gives the u- and D-scales of the diagram. The 
three scales being thus constructed, the x-axis may now be erased, and the 
diagram is ready for use. Any three points t, u, T 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. 



6 

■ 5 
■4 

J,-- 



•eooti 

■ mi 

■ 50,-t 5 
4 

SO ^ 

t 



-too 



-W 




Fig. 19. 

Example 1 (Fig. 19). Let m'-" = (. By taking the logarithm of both sides, and 
dividing through by log (, reduce the equation to the form (log u) (1/log t) + (log r) X 
(1.41/log t) = 1. Here C7 = log m, F = log v, Fiit) = 1/log t.Fiit) = 1.41/Iog t, and 
X = 1.41/2.41 = 0.585, y =■ (l/2.41)log t. 



ALIGNMENT CHARTS 181 

Example 2 (Fig. 20). Let !> = «( + 16(2, which reduces to the form (- ii/16)(l/0 
+ (»/16) (!/(!) = 1. Here U = - u/16, V = V16, i'lCO = 1/'. ^2«) = lA' and 
j; = 1/(1 +t),v = tV(l + 0. 

Note. If m = co , values of u and « 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 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.j/ axes, and having chosen a unit for the s-axis 
equal to about (l/m)th of the width of the paper, and a unit for the y-axis 
equal to about (l/n)th of the height, plot the points {x,y) given by 

X =Fi(,t), V =F2it), 
labeling each point of this curve with the value of ( to 
which it corresponds. Connect these points by a smooth 
curve, which gives the i-scale of the diagram. [If 5 
Fi{t)/F2(,i) = a constant, the i-scale will be a straight line 4 
through the origin.] ^ 

Then, using the same units as above, plot along x the j 
values of U, labeling each point with the corresponding 
value of u; and plot along y the values of V, labeling 
each point with the corresponding value of v. This gives 
the u- and j)-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 = (uw)/(u + v), which may be written in the form 
t/u + t/v = 1. Here U = u, V = v, Fi(t) = t, FiW = t. 

Note. If m = 00 and n ~ co , 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 

F2(0 = V-Fi{t) + U, 
where, for the given range of values of t, the largest variation in Fi{t) is less 
than a certain number m, and the largest variation ini''2(0 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/re)th of the height, plot the points (,x,y) given by 

X =Fi(i.), y =Fi{t), 
labeling each point of the curve with the value of ( to which it corresponds. 
Connect these points by a smooth curve, which forms the i-scale. Next, 
using the same unit for y as above, plot along the j/-axis the values of V, 
labeling each point with the corresponding value of u. This gives the M-scale. 
Finally, with the origin as center, and any convenient radius, draw a circle 
cutting the a;-axi3 in A. Along this circular arc, starting from A in the coun- 
terclockwise direction, lay off the angles whose slopes are equal to V, 
labeling each point of the arc with the value of v to which it corresponds. 



/ t 


5/ 


y 


3/ 




1 2 -J 4"- S 


t'uv/iutv) 


TiQ. 21. 



182 



GRAPHICAL REPRESENTATION OF FUNCTIONS 




sinu-sfn 60 sin t- coi 60'cost cm v 

Fig. 22. 



i U, V = COB V. 

Given, an equation which can be reduced to the 



This gives the D-soale, which in this case, however, plays a peculiar rSle, 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 »-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 ( — cos 60°ooa J 
cos V, which may be put in the form 

(sin 60° sin () = cos v (ooB 60° cos t) + sin u. 
Herefi(() = cos 60° cos (, Fi (() = sin 60° sin t, U 

Method 4 
form 

U-F{t) +V = 0, 
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 
<-scale, which will now coincide with the z-axis, plot 
along X the points for which 
_ 1 
"~^+m' _ _ „„^^^,. 

labeling each point with the value of t to which it cor- 
responds. The tt-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. 

Example (Fig- 24). Let v = 0.196 (%, where u is to range from to 15,000 and • 
from to 150,000. The equation may be written in the form (— 10 m) (0.0196i>) + t 
= 0. Here U = - 10 u, Y = v, F(.t) = 0.0196(s. 

Note. If m = to , values of u and v which give large values of U and V cannot be 
shown within the limits of the paper. 




isooo- 




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 of four variables, say r, s, u, v, consists 
essentially in replacing the i-scale of the earlier diagram by a network of two 
scales, one for r and one for s. The point where a curve r = n and a curve 
s = si intersect may be spoken of as the point (n.si). In the following equa- 
tions, U denotes as before any function of u alone, V any function of v alone; 
while ifi(r,s) andii'2(r,s) represent any functions of r and s. 
Method la. Given, an equation of the form 

U-Fi(r,s) + V-Fi(.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 

^ Fi(r,s) ^ 1 

Fi(r,s) +F2(,r,s)' ^ ~ Fi(r,s) + Fa(,r,s)' 

[To do this (Fig. 25), find tfie 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 ?• = 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 
» = 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 
a (or r) from the equations that give x and y, the resulting equation in x, y, 
and T (or x, y, and s) can often be plotted directly for each given value of r 
(or of «).] 

Next, construct the u- and »-soales 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 t, 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. 



FiQ. 26. 



Method 2a. Given, an equation of the form 



Fi(r,s) 
U 



I Fi(.T,s) ^ J 



Draw axes of x and y as in Method 2, and plot the network of curves given by 
X =Fi(,r,s), y = Fi(r,a). 

To do this, follow the plan outlined for a similar case under Method \a, 
labeling each curve of the r-f amily (Fig. 26) with the corresponding value of ri 
and each curve of the s-family with the corresponding value of s. Next, 
construct the «- and s-scales along the x- and 2/-axes, precisely as in Method 2. 
Then any three points, {t,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 
Fi{r,s) = V-Fi{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 =Fi(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 value of r (or a) to which it corresponds. Next, construct the u-scale 



184 



GRAPHICAL REPRESENTATION OF FUNCTIONS 



S-ZO 



6«30' 



along the j/-axi3, and the »-acale 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 a) — cot u. Here U = — cot u, F = cot d, 

Flirts) = sin rcsc s.FzCr.a) = cos r cots, whence — i- H -t- =1, . „ — — r " 1. 

' . \ • / . cac^s cot^s sinV cos'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 > I ? Jj 

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, j/i, y2, y'l, 
y'l, such that the distance between yi andj/'i 
(which may be zero) is equal to the distance 
beween 2/2 and y'l, 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, 
V'l = iS, 3/2 = U, y'2 = V, and label each point 
with the value of r, s, u, or v to which it cor- 
responds. (The letters yi, 1/2, y'l, y'l 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, m, » 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 rrr = us', or log r - 2 log s = log u — log (32.2 »). 
Here iJ = log r, S = 2 log s, U = log u, V ■= log (32.2 t). 

Method B, Given, an equation of the form 

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 hy x = R, xi = U, and using any 
convenient vertical unit, plot along y and j/i the points given by j/ = S, j/i = V. 
Label each point with the value of r, s, u, v, to which it corresponds. (The 
letters x, y, xi, yi 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, n, u, v will satisfy the given equation. 



u=.20 




cotv'Cotr cos s *cscr sins mtu 
/rey: l<""'ect\^1"S!^") t/ n>raM Una. 

Fig. 27. 



EQUATIONS INVOLVING FOUR VARIABLES 



185 



Method C. Given, an equation of the form 

u' 



R -S = 



In Fig. 30, take a pair of axes, x,y, and through the point x — 1 draw a 
third axis, j/', parallel to y. Also, take a second pair of axes, 12,3/2, parallel 
to (or coinciding with) the axes of x and y. Having chosen a suitable unit 
for X and X2, and a suitable unit for y, y', and j/2, lay off the values of R and 



hj hi 
r 5 



K,y. 



6y rarathi Lines. 



,10 --' 

-r 

-6 



Z2.ivrmui^ 



m 



100- 
50- 






FiQ. 28. 



W 
5 



ly.) 



S. 6 



by ParaUel lines. 



VI Z 5 4- 5 6 



-uft; 



I ^ 3 4 5 ' 6 
Fia. 29 



H r W 




Fig. 30. 



S along y and j/', respectively, labeling each point with the value of r or s to 
which it corresponds; and lay off the values of TJ and V along xt and 2/2, label- 
ing each point with the value of « or » to which it corresponds. Then if the 
line joining two points r and s is parallel to the line joining two points u and «, 
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 I a I 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 cos d, where d is the angle between a and b. 

Example, If F is a force whose point of application moves along a vector distance z, 
then F'Z = work done by F during this displacement. 

PecvZiarities of scalar products: (1) Since a'b is not a vector, expressions 
like (a-b)-c will not occur; (2) from a'Z = ay we cannot infer that x = y, 
hence, quotients will not opcur; (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 -I- b)-(c + d) = a-c + a-d + b-c -f- bd; 
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 9 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 -I- b)x(c -1- d) = axe -I- 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 Diflerentiation. If r = f (t) gives a vector r as-a function of a 
scalar t, then dr/dt = lim j[f(i + At) — t(t)]/At} as At approaches zero. 
d(a +b) = da -)- db, d(ma) = m(da) -|- (dm)a, 
d(a-b) = (da)-b -t- a-(db), <i(axb) = (da)xb H- ax(db). 

Example. If r => f (0 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 = «m, where v = ds/dt = the (scalar) path-velocity, and dm = [{da/R)]n, 
where B = the (scalar) radius of curvature of the path. Then 

d(vra.) dv , dm. dv v' 

a = — ; — «= — m -h » — = — m -I- -- n. 
dt dt dt dt R 

Here dv/dt and v^/R are the familiar expressions for the components of acceleration along 
the tangent and normal. 



INDEX 



Abscissa, 173 
Absolute value, 112 
Acceleration of 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, 185 
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 sohd, 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 
Annulua, area of, 106 

contiguous circles in, 105 
Anti-friction curve, 155 
Anti-gudermannian, 135 
Anti-hyperbolic functions, 135 

graphs, 175; series, 161 
Anti-logarithme, 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 (annul us), 105 

Barrels, volume of, 110 

Baum6 scale, 85 

Bessel'a 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, S3 
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, 

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 
Diflerences, 115, 159 
Differential calculus, 157-164 

equations, 171 
Dihedral angles, 100 
Directrix, of catenary, 147 

of ellipse, 140 

of hyperbola, 144 

of parabola, 13S 
Division, algebraic, 112 

arithmetical, 89 

by logarithms, 93 

by slide rule, 95 

of a line, 101, 102 

of complex quantiti^, 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 



188 



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 



HYPERBOLIC LOGARITHMS 



POWER FUNCTION 



Hyperbolic (continued) 

spiral, 154 
Hypocycloid, 152 
Hypotrochoid, 153 



i = V - 1, 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, C4, 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 



170 



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 of 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 

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 
Polvhedra, 100, 110 
Polynomial, US' 
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 i^aper, 177 
Series, expansion in, 114, 160 

Fourier s, 162 

Maclaurin's and Taylor's, 161 

summation of, 115 
Sexagesimal measure, 128 
Significant figures, SS 
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 



DNG0LA 



ZONE 



Ungula, 108 
Units, 72 



Variable, change of, 169 
Vector analysis, 185 
Velocity, units of, 78, 80 
Versed sine, 129: graph, 174 
Volume, units of, 70, 71, 76, 77 
Volumes, of solids, 107 
of similar figures, 99 



Watt, 73 
Wedge, 109 

spherical, 119 
Weight, units of, 71, 77, 78 
Weights and measures, 70-85 
Work, units of, 73, 79, 80 

Yard, 70 
Year, 83 

Zone of a sphere, 109