engineering handbook

Chapter 1

Mathematical Tables and Units and Systems of Weights and Measures Philip Franklin, Massachusetts Inst. of Technology*

L.E. Barbrow. U.S. Nat]. Bureau of Standards

Contents

‘

Mathematical Tables By Philip Franklin

Numbers

Table 1 .l-Squares I-2 Table I.2-Gibes I-7 Table 1.3-Square Roots .i-ii Table 1.4-Cube Roots l-14 Table 1.5-Three-Halves Pow& : : : : : : : : : : : : : : : : l-19 Table 1.6--Reciprocals 1-21

Circles

Table 1.7-Circumferences by Hundredths l-24 Table 1.8-Areas by Hundredths .I-26 Table 1.9-Circumferences and Areas by Eighths l-28 Table 1 .lO-Areas in Square Feet I-30 Table 1 .l l-Segments, ‘Given h/c ............... l-31 Table 1.12-Segments. Given h/D ................ 1-32

Spheres

Table 1.13-Segments ......................... 1-33 Table 1,14-Volumes by Hundredths .............. 1-34 Table 1 .I5-Regular Polygons .................. l-36 Table 1.16--Binomial Coefficients ............... 1-37 Table 1,17-Common Logarithms (1 .OO to 2.00) 1-38 Table I .18-Common Loqarithms I-40 Table 1.19-Degrees and-Minutes in Radians ....... l-42 Table 1.20--Radians in Degrees ...... ......... 1-43 Table 1.21-Natural Sines and Cosines l-44 Table 1.22-Natural Tangents and Cotangents l-46 Table 1.23-Natural Secants and Cosecants l-48 Table 1.24-Trigonometric Functions ........... I-50 Table 1.25-Exponentials ........... ........... 1-55 Table 1.26-Natural Logarithms I-56 Table 1.27-Hyperbolic Sines

Cosines : : : : : : : : : : : : : 1-58

Table 1.28-Hyperbolic 1-59 Table 1.29-Hyperbolic Tangents I-60 Table 1.30-Multiples of 0.4343 l-60 Table 1.31-Multiples of 2.3026 Table 1.32-Standard Distribution of Residuals :

I-60 I-61

Table 1.33-Factors for Computing Probable Error 1-61

This chapter I” the 1962 edMn was written by Ph,l,p Franklm and Laws ”
Judson (both deceased)
Compound Interest

Table 1.34-Amount of a Given Principal Table 1.35-Amount of an Annuity Table 1.36-Principal Amounting to a Given Sum

Annuities

Table 1.37-Amountino to a Given Sum (Sinking Fund) y.,

Table 1.38-Present Worth Table 1.39-Provided for by a Given Capital Table 1.40-Decimal Equivalents

Units and Systems of Weights and Measures by L.E. Barbrow

Conversion Tables

Table 1.41-Length Equivalents Table 1.42-Conversion of Lengths : : : : : : Table 1.43-Common Fractions of an Inch to Millimeters

Table 1.44-Decimals of an Inch to Millimeters Table 1.4%Millimeters to Decimals of an Inch Table 1.46-Area Equivalents Table 1.47-Volume and Capacity Equivalents Table1.48-Areas Table 1.49-Volumes or Cubic Meters Table 1.50-Volumes or Capacities : : : Table 1.51-Mass Equivalents Table 1.52-Masses .‘,‘,’ ‘.’ ‘.’ Table 1.53-Velocity Equivalents Table 1.54-Linear and Angular Velocities : : : Table 1.55--Pressures Table 1.56-Pressure Eauivalents ‘.‘. Table 1.57-Energy or Work Equivalents Table 1.58-Energy, Work, Heat Table 1.59-Pow& Equivalents Table 1.60-Power . . ‘, Table 1.61--Density Equivalents Table 1.62-Thermal Conductivity Table 1.63-Thermal Conductance : : Table 1.64-Heat Flow Table 1.65-Relative Densities Correspondtng to

OAPI and Weights per U.S. Gallon

I-62 l-63 l-64

l-65 l-66 1-66 l-67

1-71 I-71

I-72 l-72 I-72 l-73 I-73 I-74 l-74 I-74 l-75 I-75 l-76 I-76 l-76 l-77 l-77 I-78 I-78 l-78 l-79 l-79 I-79 I-79

l-80

l-2 PETROLEUM ENGINEERING HANDBOOK

TABLE 1 .I -SQUARES OF NUMBERS

1 2 3 4 5 6 7 8 9 Average Difference

1.000 1.002 1.004 1.020 1.022 1.024 1.040 1.042 1.044 1.081 1.083 1.065 1.082 1.084 1.086

1.006 1.026 I.047

1.012 1.014 1.018 1.018 2

1.05 1.08 1.07 1.08 1.09

1.102 1.124 1.145 1.166 1.188

1.210 1.232 1.254 1.277 1.300

1.105 1.128 1.147 1.189 1.190

1.107

1.087 1.088

1.109

1.008 1.028 1.049 1.089 1.090

1.010 1.030 1.051 1.071 1.092

1.032 1.034 1.036 1.053 1.055 1.057 1.073 1.094

1.111 1.128 1.130 1.132 1.149 1.151 1.153

1.113 1.134 1.156 1.177 1.199

1.221 1.243 1.266 1.288 1.311

1.334 1.357 1.381 1.404 1.428

1.452 1.476 1.501 1.525 1.550

1.575 1.6OC 1.626 1.651 1.677

1.703 1.729 1.756

1.115 1.138 1.158 1.179 1.201

1.223 1.245 1.268 1.290 1.313

1.075 1.096

1.117

1.077 1.098

1.119 1.141 1.162 1.184 1.206

1.228 1.250 1.272 1.295 1.318

1.038 1.059 1.080 1.100

1.121

1.10 1.11 1.12 1.13 1.14

1.212 1.234 1.257 1.279 1.302

1.171 1.192

1.214 1.237 1.259 1.281 1.304

1.175 1.197

1.219 1.241 1.263 1.286 1.309

1.138 1.160 1.182 1.203

1.225 1.248 1.270 1.293 1.316

1.143 1.184 1.186 1.208

1.230 1.252 1.275 1.297 1.320

1.15 1.16 1.17 1.18 1.19

1.322 1.325 1.327 1.346 1.348 1.350 1.369 1.371 1.374 1.392 1.395 1.397 1.416 1.418 1.421

1.336 1.339 1.360 1.362 1.383 1.385 1.407 1.409 1.430 1.433

1.341 1.343 1.364 1.387 1.388 1.390 1.411 1.414 1.435 1.438

1.20 1.440 1.442 1.445 1.21 1.484 1.467 1.469 1.22 1.488 1.491 1.493 1.23 1.513 1.515 1.518 1.24 1.538 1.540 1.543

1.25 1.26 1.27 1.28 1.29

1.173 1.195

1.217 1.239 1.281 1.284 1.306

1.329 1.353 1.376 1.399 1.423

1.447 1.471 1.496 1.520 1.545

1.570 1.595

1.30 1.31 1.32 1.33 1.34

1.582 1.585 1.568 1.588 1.590 1.593 1.813 1.815 1.618 1.638 1.641 1.644 1.684 1.667 1.669

1.690 1.893 1.695 1.716 1.719 1.721 1.742 1.745 1.748 1.769 1.772 1.774 1.796 1.798 1.801

1.821 1.648 1.672

1.898 1.724 1.750 1.777 1.804

1.332 1.355 1.378 1.402 1.426

1.450 1.474 1.498 1.523 1.548

1.573 1.596 1.823 1.849 1.874

1.700 1.727 1.753

1.454 1.457 1.459 1.462 1.479 1.481 1.484 1.486 1.503 1.506 1.508 1.510 1.528 1.530 1.533 1.535 1.553 1.555 1.558 1.560

1.583 1.585 3 1.606 1.610 1.833 1.838 1.659 1.862 1.685 1.887

1.780 1.782 1.806 1.809

1.578 1.580 1.803 1.805 1.628 1.831 1.654 1.658 1.680 1.682

1.708 1.708 1.732 1.734 1.758 1.761 1.785 1.788 1.812 1.814

1.711 1.713 1.737 1.74c 1.764 1.766 1.790 1.793 1.617 1.820

1.35 1.822 1.825 1.828 1.831 1.38 1.850 1.852 1.855 1.858 1.37 1.877 1.880 1.882 1.885 1.38 1.904 1.907 1.910 1.913 1.39 1.932 1.935 1.938 1.940

1.839 1.841 1.844 1.847 1.886 1.889 1.871 1.874 1.893 1.896 1.899 1.902 1.921 1.924 1.927 1.929 1.949 1.952 1.954 1.957

1.40 1.960 1.963 1.966 1.968 1.41 1.988 1.991 1.994 1.997 1.42 2.016 2.019 2.022 2.025 1.43 2.045 2.048 2.051 2.053 1.44 2.074 2.076 2.079 2.082

1.977 1.980 1.982 1.985 2.005 2.008 2.011 2.014 2.033 2.036 2.039 2.042 2.062 2.065 2.068 2.071 2.091 2.094 2.097 2.100

1.45 2.102 2.105 2.108 2.111 1.48 2.132 2.135 2.137 2.140 1.47 2.181 2.164 2.187 2.170 1.48 2.190 2.193 2.198 2.199 1.49 2.220 2.223 2.226 2.229

2.120 2.123 2.126 2.129 2.149 2.152 2.15" 2.158 2.179 2.182 2.184 2.187 2.208 2.211 2.214 2.217 2.238 2.241 2.244 2.247

1.50 1.51 1.52 1.53 1.54

2.250 2.253 2.258 2.259 2.280 2.283 2.286 2.289 2.310 2.313 2.316 2.320

1.833 1.880 1.888 1.915 1.943

1.971 1.999 2.028 2.056 2.085

2.114 2.143 2.173 2.202 2.232

2.282 2.292 2.323

1.836 1.863 1.891 1.918 1.946

1.974 2.002 2.031 2.059 2.088

2.117 2.146 2.176 2.205 2.235

2.285 2.295 2.326

2.268 2.271 2.298 2.301 2.329 2.332 2.359 2.362 2.390 2.393

2.274 2.277 2.304 2.307 2.335 2.338

2.341 2.344 2.347 2.350 2.353 2.356 2.372 2.375 2.378 2.381 2.384 2.387

2.385 2.389 2.396 2.399

1.55 2.402 2.406 2.409 2.412 2.415 2.418 2.421 2.424 2.427 2.430 1.56 2.434 2.437 2.440 2.443 2.448 2.449 2.452 2.455 2.459 2.462 1.57 2.465 2.488 2.471 2.474 2.477 2.481 2.484 2.487 2.490 2.493 1.58 2.496 2.500 2.503 2.506 2.509 2.512 2.515 2.519 2.522 2.525 1.59 2.528 2.531 2.534 2.538 2.541 2.544 2.547 2.550 2.554 2.557

Exolanation of Table of Souares

This table givesthe value of N2 for values of N from 1 to 10, correct to four figures. (Interpolated values may be in error by 1 in the fourth figure.) To find the square of a number N outside the range from 1 to 10, note that moving the decimal point one place in

Column N is equivalent to moving it two places in the body of the table. For example, (3.217)'=10.35, (0.03217)'=0.001035, and (3,217)' =10,035,000. This table also can be used inversely to give square roots.

MATHEMATICAL TABLES & UNITS & SYSTEMS OF WEIGHTS & MEASURES 1-3

TABLE l.l-SQUARES OF NUMBERS(continued)

7 8

2.582 2.586 2.815 2.818 2.647 2.650 2.680 2.883 2.713 2.716

Average 9 Difference

2.589 3 2.621 2.654 2.688 2.719

2.748 2.749 2.752 2.779 2.782 2.788 2.812 2.816 2.819 2.848 2.849 2.853 2.880 2.883 2.887

2.914 2.948

2.962 2.965 2.989 2.972 2.978 2.979 2.983 2.996 3.000 3.003 3.007 3.010 3.014 3.017

3.045 3.049 3.052

3.080 3.084 3.087 3.115 3.119 3.122 3.151 3.154 3.158

2.917 2.921 2.952 2.955 2.986 2.989 3.021 3.024 3.058 3.059

N 0 1 2 3

2.570

4 5

1.60 1.61 1.62 1.83 1.84

2.560 2.592 2.824 2.657 2.690

2.583 2.595 2.828 2.680 2.693

2.566 2.599 2.602 2.631 2.634 2.683 2.667 2.696 2.899

2.729 2.732 2.762 2.768 2.796 2.799 2.829 2.832 2.863 2.866

2.897 2.900 2.931 2.934

2.573 2.805 2.637 2.670 2.703

2.576

6

2.579 2.608 2.811 2.641 2.644

1.65 2.722 2.726 1.88 2.758 2.759 1.87 2.789 2.792 1.88 2.822 2.828 1.89 2.856 2.859

2.738 2.789 2.802 2.836 2.870

1.70 2.890 1.71 2.924 1.72 2.958 1.73 2.993 1.74 3.028

2.893 2.928

2.904 2.938

2.673 2.676 2.706 2.709

2.739 2.742 2.772 2.776 2.806 2.809 2.839 2.843 2.873 2.876

2.907 2.910 2.941 2.945

1.75 3.082 1.78 3.098 1.77 3.133 1.78 3.188 1.79 3.204

3.031

3.066 3.101 3.138 3.172 3.208

3.035

3.070 3.105 3.140 3.176 3.211

3.247

3.038

3.073 3.108 3.144 3.179 3.215

3.042

3.077 3.112 3.147 3.183 3.218

3.254

3.091 3.126 3.161

3.186 3.190 3.193 3.197 3.222 3.226 3.229 3.233

1.80 1.81 1.82 1.83 1.84

3.240 3.244 3.251 3.258 3.262 3.285 3.289 3.276 3.280 3.283 3.287 3.291 3.294 3.298 3.301 3.305 3.312 3.316 3.320 3.323 3.327 3.331 3.334 3.338 3.342 3.349 3.353 3.356 3.360 3.364 3.367 3.371 3.375 3.378 3.386 3.389 3.393 3.397 3.400 3.404 3.408 3.411 3.415

1.85 3.422 3.426 3.430 3.434 3.437 3.441 3.445 3.448 3.452 1.88 3.480 3.463 3.467 3.471 3.474 3.478 3.482 3.486 3.489 1.87 3.497 3.501 3.504 3.508 3.512 3.516 3.519 3.523 3.527 1.88 3.534 3.538 3.542 3.546 3.549 3.553 3.557 3.561 3.565 1.89 3.572 3.578 3.580 3.583 3.587 3.591 3.595 3.599 3.602

1.90 1.91 1.92 1.93 1.94

3.610 3.614 3.618 3.821 3.825 3.829 3.833 3.837 3.648 3.652 3.656 3.680 3.663 3.667 3.671 3.675 3.888 3.725 3.784

3.713 3.752 3.791

3.640 3.679 3.717 3.758 3.795

1.95 3.802 1.98 3.842 1.97 3.881 1.98 3.920 1.99 3.960

3.890 3.694 3.698 3.702 3.706 3.709 3.729 3.733 3.738 3.740 3.744 3.748 3.787 3.771 3.775 3.779 3.783 3.787

3.806 3.810 3.814 3.818 3.822 3.826 3.846 3.849 3.853 3.857 3.861 3.865 3.885 3.889 3.893 3.897 3.901 3.905 3.924 3.928 3.932 3.964 3.968 3.972

3.944 3.984

4.024 4.064 4.105 4.145 4.188

3.830 3.834 3.869 3.873 3.909 3.912 3.948 3.952 3.988 3.992

2.00 4.000 4.004 4.008 4.012 2.01 4.040 4.044 4.048 4.052 2.02 4.080 4.084 4.088 4.093 2.03 4.121 4.125 4.129 4.133 2.04 4.162 4.186 4.170 4.174

3.936 3.976

4.016 4.056

3.940 3.960

4.097 4.137 4.178

4.020 4.060 4.101 4.141 4.182

4.028 4.032 4.036 4.068 4.072 4.076 4.109 4.113 4.117 4.149 4.153 4.158 4.190 4.194 4.198

2.05 4.202 2.08 4.244 2.07 4.285 2.08 4.328 2.09 4.388

4.207 4.211 4.215 4.219 4.223 4.227 4.231 4.235 4.248 4.252 4.258 4.280 4.284 4.268 4.272 4.277 4.289 4.293 4.297 4.301 4.331 4.335 4.339 4.343 4.372 4.378 4.381 4.385

4.314 4.318 4.358 4.360 4.397 4.402

2.10 4.410 4.414 2.11 4.452 4.458 2.12 4.494 4.499 2.13 4.537 4.541 2.14 4.580 4.584

4.418 4.423 4.427 4.461 4.465 4.469 4.503 4.507 4.511

4.306 4.310 4.347 4.351 4.389 4.393

4.431 4.435 4.473 4.477 4.516 4.520

4.545 4.550 4.554 4.558 4.562 4.588 4.592 4.597 4.801 4.605

4.439 4.444 4.448 4.482 4.486 4.490 4.524 4.528 4.533 4.587 4.571 4.575 4.610 4.814 4.618

2.15 4.622 4.627 4.831 4.635 4.640 4.844 4.648 4.653 4.857 2.18 4.886 4.670 4.674 4.679 4.683 4.887 4.692 4.696 4.700 2.17 4.709 4.713 4.716 4.722 4.728 4.731 4.735 4.739 4.744 2.18 4.752 4.757 4.761 4.765 4.770 4.774 4.779 4.783 4.787 2.19 4.796 4.800 4.805 4.809 4.814 4.818 4.822 4.827 4.831

3.094 4 3.129 3.165 3.201 3.236

3.272 3.309 3.345 3.382 3.419

3.458 3.493 3.531 3.566 3.606

3.844 3.683 3.721 3.760 3.799

3.838 3.877 3.916 3.956 3.996

4.239 4.281 4.322 4.364 4.406

4.661 4.705 4.748 4.792 4.836

r =9.66960; l/r2 2

=0.101321, e2 =7.38906. x2 =9.669w; (T/z)* =2.46740; l/n2 =0.101321.

(continued on next page)

I-4 PETROLEUM ENGINEERING HANDBOOK

TABLEl.l-SQUARES OF NUMBERS(continued)

4 Average

9 Difference N 0 1 2 - - ~ 5 8 7

4.840 4.844 4.849 4.884 4.928 4.973 5.018

4.889 4.893 4.933 4.937 4.977 4.982 5.022 5.027

3

4.853 4.897 4.942 4.986 5.031

4.858 4.882 4.888 4.871 4.902 4.906 4.911 4.915 4.948 4.951 4.955 4.960 4.991 4.995 5.000 5.004 5.036 5.040 5.045 5.049

5.062 5.067 5.072 5.076 5.081 5.085 5.090 5.094 5.108 5.112 5.117 5.121 5.128 5.130 5.135 5.139 5.153 5.157 5.162 5.167 5.171 5.176 5.180 5.185 5.198 5.203 5.208 5.212 5.217 5.221 5.226 5.230 5.244 5.249 5.253 5.258 5.262 5.267 5.272 5.276

5.290 5.295 5.338 5.341 5.382 5.387

5.299 5.304 5.308 5.313 5.318 5.355 5.359 5.364 5.401 5.408 5.410 5.448 5.452 5.457 5.494 5.499 5.504

5.322 5.327 5.388 5.373 5.415 5.420

8

4.875 2.20 2.21 2.22 2.23 2.24

2.25 2.26 2.27 2.28 2.29

2.30 2.31 2.32 2.33 2.34

2.35 2.38 2.37 2.38 2.39

2.40 2.41 2.42 2.43 2.44

5.429 5.434 5.478 5.480

5.345 5.350 5.392 5.396

5.522 5.527 5.570 5.574 5.871 5.822 5.884 5.889 5.712 5.717

5.438 5.443 5.485 5.490

5.532 5.537 5.579 5.584 5.626 5.631 5.674 5.879 5.722 5.726

5.482 5.488 5.508 5.513

5.541 5.548 5.551 5.555 5.580 5.565 5.588 5.593 5.598 5.603 5.807 5.612 5.636 5.641 5.645 5.650 5.855 5.660 5.883 5.688 5.693 5.698 5.703 5.707 5.731 5.738 5.741 5.746 5.750 5.755

5.780 5.808 5.856 5.905 5.954

5.785 5.770 5.774 5.779 5.784 5.813 5.818 5.823 5.827 5.832 5.861 5.866 5.871 5.876 5.881 5.910 5.915 5.919 5.924 5.929 5.958 5.963 5.988 5.973 5.978

5.794 5.842 5.890 5.939 5.988

2.45 2.48 2.47 2.48 2.49

8.002 6.007 6.012 6.017 6.022 6.027 8.502 6.057 6.061 6.088 8.071 6.078 8.101 6.108 6.111 6.116 6.121 6.126

5.789 5.837 5.885 5.934 5.983

6.032 8.081 6.131

6.037 6.086 6.136

6.150 6.155 6.180 6.185 8.170 8.175 6.180 6.185 6.200 6.205 6.210 6.215 6.220 6.225 6.230 6.235

2.50 8.250 6.255 6.260 8.265 6.270 6.275 6.280 6.285 2.51 8.300 6.305 6.310 8.315 6.320 6.325 6.330 6.335 2.52 6.350 6.355 6.380 8.366 8.371 6.378 8.381 6.388 2.53 6.401 6.406 6.411 8.416 8.421 6.428 8.431 8.438 2.54 6.452 6.457 6.462 8.467 6.472 6.477 8.482 6.487

2.55 8.502 6.508 6.513 8.518 8.523 6.528 8.533 8.538 2.56 6.554 8.559 6.564 6.569 8.574 6.579 8.584 6.589 2.57 6.605 8.810 6.815 8.620 8.625 6.831 6.636 8.641 2.58 6.656 6.862 6.667 8.672 8.677 6.882 6.687 8.693 2.59 6.708 8.713 6.718 6.724 8.729 6.734 6.739 8.744

2.60 2.61 2.62 2.63 2.64

6.780 8.765 6.812 8.817 6.884 8.870 6.917 6.922 6.970 8.975

6.770 6.323 6.375

8.776 8.781 8.833 8.885 6.938 8.991

6.788 6.791 8.798 6.849 6.901 6.954 7.007

2.65 7.022 7.028 2.68 7.076 7.081 2.87 7.129 7.134 2.68 7.182 7.188 2.89 7.238 7.241

2.70 7.290 7.295 2.71 7.344 7.350 2.72 7.398 7.404 2.73 7.453 7.458 2.74 7.508 7.513

6.927 f5.980

7.033

8.828 8.880 6.933 8.985

6.838 6.843 6.891 6.896 6.943 6.948 6.996 7.001

7.086 i.140

7.038 7.044 7.049 7.092 7.097 7.102 7.145 7.150 7.158 7.198 7.204 7.209 7.252 7.258 7.263

7.306 7.312 7.317 7.360 7.366 7.371 7.415 7.420 7.426 7.469 7.475 7.480 7.524 7.530 7.535

2.75 2.76 2.77 2.78 2.79

7.562 7.618 7.673 7.728 7.784

7.588 7.623 7.678 7.734 7.790

7.193 7.247

7.301 7.355 7409 7.484 7.519

7.574 7.629 7.884 7.740 7.795

7.579 7.585 7.590 7.634 7.840 7.645 7.690 7.895 7.701 7.745 7.751 7.756 7.801 7.806 7.812

7.054 7.060 7.108 7.113 7.161 7.166 7.215 7.220 7.288 7.274

7.322 7.328 7.377 7.382 7.431 7.437 7.486 7.491 7.541 7.546

7.596 7.801 7.851 7.656 7.708 7.712 7.782 7.767 7.818 7.823

4.880 4 4.924 4.988 5.013 5.058

4.920 4.964 5.009 5.054

5.099 5.103 5 5.144 5.148 5.189 5.235 5.281

5.194 5.240 5.285

5.331 5.378 5.424 5.471 5.518

5.798 5.803 5.847 5.852 5.895 5.900 5.944 5.949 5.993

6.042 6.091 6.140 6.190 6.240

6.290 8.340 6.391 6.441 8.492

8.543 6.595

5.998

6.047 6.096 6.145 8.195 6.245

6.295 5 6.345 6.396 6.447 6.497

6.548 6.600

8.848 8.651 8.698 8.703 8.750

8.802 8.854 8.906 6.959 7.012

7.085 7.118 7.172 7.225 7.279

7.333 7.388 7.442 7.497 7.552

7.607 7.662 7.717 7.773 7.829

6.755

8.807 8.859 6.912 6.964 7.017

7.070 7.124 7.177 7.231 7.285

7.339 7.393 7.447 7.502 7.557

7.812 8 7.867 7.723 7.779 7.834

r’=966960. -In2 =0.101321, e2 =7389X r2=9.86960, (n/2)* =2.46740. l/n2 =0 101321

MATHEMATICAL TABLES & UNITS & SYSTEMS OF WEIGHTS & MEASURES l-5

ce
N 0
2.80 2.81 2.82 2.83 2.84

7.840 7.896 7.952 8.009 8.066

2.85 8.122 2.86 8.180 2.87 8.237 2.88 8.294 2.89 8.352

2.90 8.410 2.91 8.468 2.92 8.526 2.93 8.585 2.94 8.644

2.95 2.96 2.97 2.98 2.99

8.702 8.762 8.821 8.880 8.940

3.00 9.000 3.01 9.060 3.02 9.120 3.03 9.181 3.04 9.242

3.05 3.06 3.07 3.08 3.09

9.302 9.364 9.425 9.486 9.548

3.10 3.11 3.12 3.13 3.14

9.610 9.672 9.734 9.797 9.860

3.15 9.922 3.16 9.986

3.1 3.2 3.3 3.4

10.24 10.89 11.56

3.5 12.25 3.6 1296 3.7 13.69 3.8 1444 3.9 15.21

4.0 16.00 4.1 16.81 4.2 17.64 4.3 18.49 4.4 19.36

4.5 20.25 4.6 21.16 4.7 22.09 4.6 23.04 4.9 24.01

5.0 25.00 5.1 26.01 5.2 27.04 5.3 28.09 5.4 29.16

1 2 3 4 5 6 7 8

7.646 7.851 7.857 7.862 7.868 7.874 7.879 7.885 7.902 7.907 7.913 7.919 7.924 7.930 7.935 7.941 7.958 7.964 7.969 7.975 7.981 7.986 7.992 7.998 8.015 8.020 8.026 8.032 8.037 8.043 8.049 8.054 8.071 8.077 8.083 8.088 8.094 8.100 8.105 8.111

8.128 8.134 8.140 8.145 8.151 8.157 6.162 8.168 8.185 8.191 8.197 8.202 8.208 8.214 8.220 8.225 6.243 8.248 8.254 8.260 8.266 8.271 8.277 8.283 8.300 8.306 8.312 8.317 8.323 8.329 8.335 8.341 8.358 8.384 8.369 8.375 8.381 8.387 8.393 8.398

8.416 8.422 6.427 8.433 8.439 8.445 8.451 8.456 8.474 8.480 8.486 8.491 8.497 8.503 8.509 8.515 8.532 8.538 8.544 8.550 8.556 8.561 8.567 8.573 8.591 8.597 8.602 8.606 8.614 8.820 8.626 8.632 8.649 8.655 8.661 8.667 8.673 8.679 8.885 8.691

8.708 8.714 8.720 8.726 8.732 8.738 8.744 8.750 8.768 8.773 8.779 8.785 8.791 8.797 8.803 8.809 8.827 8.833 8.839 8.845 8.851 8.857 8.863 8.868 8.886 8.892 8.898 8.904 8.910 8.916 8.922 8.928 8.946 8.952 8.958 8.964 8.970 8.976 8.982 8.988

9.006 9.012 9.018 9.024 9.030 9.036 9.042 9.048 9.066 9.072 9.078 9.084 9.090 9.096 9.102 9.108 9.126 9.132 9.139 9.145 9.151 9.157 9.163 9.169 9.187 9.193 9.199 9.205 9.211 9.217 9.223 9.229 9.248 9.254 9.260 9.266 9.272 9.278 9.284 9.290

9.309 9.315 9.321 9.327 9.333 9.339 9.345 9.351 9.370 9.376 9.382 9.388 9.394 9.400 9.406 9.413 9.431 9.437 9.443 9.449 9.456 9.462 9.468 9.474 9.493 9.499 9.505 9.511 9.517 9.523 9.530 9.536 9.554 9.560 9.567 9.573 9.579 9.585 9.591 9.598

9.616 9.622 9.629 9.635 9.641 9.647 9.653 9.660 9.678 9.685 9.691 9.697 9.703 9.709 9.716 9.722 9.741 9.747 9.753 9.759 9.766 9.772 9.778 9.784 9.803 9.809 9.816 9.822 9.828 9.834 9.841 9.847 9.866 9.872 9.878 9.885 9.891 9.897 9.904 9.910

9.929 9.935 9.941 9.948 9.954 9.960 9.967 9.973 9.992 9.998 10.005

9.99 10.05 10.11 10.30 10.37 10.43 10.50 10.56 10.63 10.69 10.76 10.96 11.02 11.09 11.16 11.22 11.29 11.36 11.42 11.63 11.70 11.76 11.83 11.90 11.97 12.04 12.11

12.32 12.39 12.46 12.53 12.60 12.87 12.74 12.82 13.03 13.10 13.18 13.25 13.32 13.40 13.47 13.54 13.76 13.84 13.91 13.99 14.06 14.14 14.21 14.29 14.52 14.59 14.67 14.75 14.82 14.90 14.98 15.05 15.29 15.37 15.44 15.52 15.60 15.68 15.76 15.84

16.08 16.16 16.24 16.32 16.40 16.48 16.56 16.65 16.89 16.97 17.06 17.14 17.22 17.31 17.39 17.47 17.72 17.81 17.89 17.98 18.06 18.15 18.23 18.32 18.58 18.66 18.75 18.84 18.92 19.01 19.10 19.18 19.45 19.54 19.62 19.71 19.80 19.89 19.98 20.07

20.34 20.43 20.52 20.61 20.70 20.79 20.88 20.98 21.25 21.34 21.44 21.53 21.62 21.72 21.81 21.90 22.18 22.28 22.37 22.47 22.56 22.66 22.75 22.85 23.14 23.23 23.33 23.43 23.52 23.62 23.72 23.81 24.11 24.21 24.30 24.40 24.50 24.60 24.70 24.80

25.10 25.20 25.30 25.40 25.50 25.60 25.70 2581 26.11 26.21 26.32 26.42 26.52 26.63 26.73 26.83 27.14 27.25 27.35 27.46 27.56 27.67 27.77 27.88 28.20 28.30 28.41 28.52 28.62 28.73 28.84 28.94 29.27 29.38 29.48 29.59 29.70 29.81 29.92 30.03

TABLE l.l-SQUARES OF NUMBERS (continued)

r2 =986960. l/r2 =0101321.e2=736906 r2 =9.66960. (T/Z)~ =2.46740. l/n2 =o 101321

9

7.890 7.947 6.003 8.060 8.117

8.174 8.231 8.289 8.346 8.404

8.462 8.521 8.579 8.638 8.697

8.756 8.815 8.874 8.934 8.994

9.054 9.114 9.175 9.236 9.296

9.357 9.419 9.480 9.542 9.604

9.666 9.728 9.791 9.853 9.916

9.979

10.18 10.82 11.49 12.18

12.89 13.62 14.36 15.13 15.92

16.73 17.56 18.40 19.27 20.16

21.07 22.00 22.94 23.91 24.90

25.91 26.94 27.98 29.05 30.14

Average Differen

6

8

9

10

11

(contmued on next page)

l-6 PETROLEUM ENGINEERINGHANDBOOK

e

TABLE l.l-SQUARES OF NUMBERS (continued)

Average DifferencN 0 1 2

5.5 30.25 30.36 30.47 5.6 31.36 31.47 31.58 5.7 32.49 32.60 32.72 5.8 33.64 33.76 33.87 5.9 34.81 34.93 35.05

6.0 36.00 6.1 37.21 6.2 38.44 6.3 39.69 6.4 40.96

36.12 37.33 38.56 39.82 41.09

36.24 36.36 37.45 37.58 38.69 38.81 39.94 40.07 41.22 41.34

6.5 42.25 6.6 43.56 6.7 44.89 6.8 46.24 6.9 47.61

42.38 42.51 43.69 43.82 45.02 45.16 46.38 46.51 47.75

7.0 49.00 49.14 7.1 50.41 50.55 7.2 51.84 51.98 7.3 53.29 53.44 7.4 54.76 54.91

47.89

49.28 50.69 52.13 53.58 55.06

7.5 7.6 7.7 7.8 7.9

8.0 8.1 8.2 8.3 8.4

8.5 8.6 8.7 8.8 8.9

56.25 57.76 59.29 60.84 62.41

64.00 65.61 67.24 68.89 70.56

72.25 73.96 75.69 77.44 79.21

56.40 56.55 56.70 56.85 57.00 57.15 57.30 57.46 57.61 57.91 58.06 58.22 58.37 58.52 58.68 58.83 58.98 59.14 59.44 59.60 59.75 59.91 60.06 60.22 60.37 60.53 60.68 61.00 61.15 61.31 61.47 61.62 61.78 61.94 82.09 62.25 6257 62.73 62.88 63.04 63.20 63.36 63.52 83.68 63.84

64.16 64.32 64.48 64.64 64.80 64.96 65.12 65.29 65.45 65.77 65.93 66.10 66.26 66.42 66.59 66.75 66.91 67.08 67.40 67.57 67.73 67.90 68.06 68.23 68.39 68.56 68.72 69.06 69.22 69.39 69.56 69.72 69.89 70.06 70.22 70.39 70.73 70.90 71.06 71.23 71.40 71.57 71.74 71.91 72.08

72.42 72.59 72.76 74.13 74.30 74.48 75.86 76.04 76.21 77.62 77.79 77.97 79.39 79.57 79.74

9.0 81.00 81.18 81.36 9.1 82.81 82.99 83.17 9.2 84.64 84.82 85.01 9.3 86.49 86.68 86.86 9.4 88.36 88.55 88.74

9.5 9.6 9.7 9.8 9.9

90.25 so.44 90.63 92.16 92.35 92.54 94.09 94.28 94.48 94.67 96.04 96.24 96.43 96.63 98.01

100.0

98.21 98.41 98.60

10.0

3 4 5 6 7

30.80 30.91 31.02

8 9

31.14 32.26 33.41

31.25 32.38 33.52 34.69 35.88

32.15 33.29

31.92 32.04 33.06 33.18 34.22 34.34 35.40 35.52

34.46 34.57 35.64 35.76

36.60 36.72 36.84 36.97 37.09 37.82 37.95 38.07 38.19 38.32 39.06 39.19 39.31 39.44 39.56 40.32 40.45 40.58 40.70 40.83 41.60 41.73 41.86 41.99 42.12

30.58 30.69 11

12

13

14

15

16

17

18

19

20

31.70 32.83 33.99 35.16

31.81 32.95 34.11 35.28

36.48 37.70 38.94 40.20 41.47

42.77 44.09 45.43 46.79 48.16

49.56

42.64 43.96

42.90 43.03 43.16 43.30 43.43 44.22 44.36 44.49 44.62 44.76

45.29 46.65 48.02

49.42

45.56 45.70 45.83 45.97 46.92 47.06 47.20 47.33 48.30 48.44 48.58 48.72

49.70 49.84 49.98 50.13

46.10 47.47 48.86

50.27 51.70 53.14

50.84 50.98 51.12 52.27 52.42 52.56 53.73 53.88 54.02 55.20 55.35 55.50

51.27 52.71

51.41 51.55 52.85 53.00 54.32 54.46 55.80 55.95

54.17 55.65

54.61 56.10

72.93 73.'10 73.27 73.44 73.62 73.79 74.65 74.82 75.00 75.17 75.34 75.52 76.39 76.56 76.74 76.91 77.09 77.26

81.54 83.36 85.19 87.05 88.92

90.82 92.74

78.15 78.32 78.50 78.68 78.85 79.03 79.92 80.10 80.28 80.46 80.64 80.82

81.72 81.90 82.08 82.26 82.45 82.63 83.54 83.72 83.91 84.09 84.27 84.46 85.38 85.56 85.75 85.93 86.12 86.30 87.24 87.42 87.61 87.80 87.98 88.17 89.11 89.30 69.49 89.68 89.87 90.06

91.01 91.20 91.39 91.58 91.78 91.97 92.93 93.12 93.32 93.51 93.70 93.90 94.87 95.06 95.28 95.45 95.65 95.84 96.83 97.02 97.22 97.42 97.61 97.81 98.80 99.00 99.20 99.40 99.60 99.80

r =986960 lh2 =o 101321. e2 =736906 2 r2=966960.(r12)2=246740. l/r2 =o 101321


E

Tb

N(

TABLE1.2-CUBESOFNUMBERS

2 3 4 5 6 7 8 Average

9 Difference N 0

1.00 1.01 1.02 1.03 1.04

1.000

1

1.003 1.006 1.009 1.012 1.030 1.033 1.061 1.064 1.093 1.125

1.036 1.067 1.099 1.131

1.05 1.06 1.07 1.08 1.09

1.10 1.11 1.12 1.13 1.14

1.096 1.128

1.161 1.194 1.228 1.263 1.299

1.335 1.371

1.338 1.375 1.412 1.451 1.489

1.15 1.16 1.17 1.18 1.19

1.20 1.21 1.22 1.23 1.24

1.158 1.191 1.225 1.260 1.295

1.331 1.368 1.405 1.443 1.482

1.521 1.561 1.602 1.643 1.685

1.728 1.772 1.816 1.861 1.907

1.164 1.198 1.232 1.267 1.302

1.409 1.447 1.485

1.525 1.565 1.606

1.529 1.569 1.610

1.647 1.651 1.656 1.660 1.689 1.694 1.698 1.702

1.732 1.737 1.741 1.745 1.750 1.776 1.780 1.785 1.789 1.794 1.820 1.825 1.829 1.834 1.838 1.865 1.870 1.075 1.879 1.884 1.911 1.916 1.920 1.925 1.930

1.25 1.26 1.27 1.28 1.29

1.30 1.31 1.32 1.33 1.34

1.953 1.958 1.983 1.967 1.972 1.977 2.000 2.005 2.010 2.015 2.019 2.024 2.048 2.097 2.147

2.197

2.053 2.102 2.152

2.248 2.300 2.353 2.406

2.202 2.253 2.305 2.358 2.411

2.058 2.107 2.157

2.207 2.258 2.310 2.363 2.417

1.35 2.460 2.466 2.471 1.36 2.515 2.521 2.527 1.37 2.571 2.577 2.583 1.38 2.628 2.634 2.640 1.39 2.686 2.691 2.697

1.40 1.41 1.42 1.43 1.44

2.744 2.750 2.756 2.803 2.809 2.815 2.863 2.869 2.924 2.930 2.986 2.992

1.45 3.049 3.055 1.46 3.112 3.119 1.47 3.177 3.183 1.48 3.242 3.248 1.49 3.308 3.315

1.50 1.51 1.52 1.53 1.54

3.375 3.382 3.443 3.450 3.512 3.519

2.875 2.936 2.998

3.081 3.125 3.190 3.255 3.321

3.389 3.457 3.528

3.582 3.589 3.596 3.652 3.659 3.667

1.55 3.724 3.731 3.738 1.56 3.796 3.804 3.811 1.57 3.870 3.877 3.885 1.58 3.944 3.952 3.959 1.59 4.020 4.027 4.035

1.015 1.046 1.077

1.018 1.021 1.024 1.027 3 1.040 1.043 1.071 1.074

1.049 1.080 1.112 1.144

1.052 1.083

1.055 1.086 1.118 1.151

1.058 1.090

1.102 1.106 1.135 1.138

1.168 1.171 1.201 1.205 1.235 1.239 1.270 1.274 1.306 1.309

1.342 1.346 1.379 1.382 1.416 1.420 1.454 1.458 1.493 1.497

1.533 1.537 1.573 1.577 1.614 1.618

1.109 1.141

1.115 1.148

1.181 1.215 1.249 1.284 1.320

1.357 1.394 1.431 1.470 1.509

1.549 1.589 1.631

1.122 1.154

1.188 1.222 1.256 1.2Sl 4 1.327

1.364 1.401

1.174 1.178 1.208 1.211 1.242 1.246 1.277 1.281 1.313 1.317

1.349 1.353 1.386 1.390 1.424 1.428 1.462 1.466 1.501 1.505

1.541 1.545 1.585 1.626 1.668 1.711

1.754 1.798 1.843 1.888 1.934

1.981 2.029 2.078 2.127 2.177

2.228

1.184 1.218 1.253 1.288 1.324

1.360 1.397 1.435 1.439 1.474 1.478

1.672 1.715

1.513

1.553 1.593 1.635 1.677 1.719

1.758 1.763 1.802 1.807 1.847 1.852 1.893 1.897 1.939 1.944

1.517

1.557 1.598 1.639

1.581 1.622 1.664 1.706

1.681 1.724

1.986 1.991 2.034 2.039 2.082 2.087 2.132 2.137 2.182 2.187

2.233 2.238 2.274 2.279 2.284 2.290 2.326 2.331 2.337 2.342

2.395 2.449

1.767 1.811 1.856 1.902 1.948 5

1.996 2.044 2.092 2.142 2.192

2.243 2.295 2.347 2.401 2.455

2.504 2.510 6 2.560 2.566 2.617 2.622 2.674 2.680 2.732 2.738

2.785 2.791 2.845 2.851 2.906 2.912 2.967 2.974 3.030 3.036

3.093 3.099 3.157 3.164 3.222 3.229 3.288 3.295 3.355 3.362

3.422 3.429

2.797 2.857 2.918 2.980 3.042

3.106 3.170 3.235 3.301 7 3.368

3.438

2.063 2.068 2.112 2.117 2.162 2.167

2.212 2.217 2.264 2.269 2.316 2.321 2.369 2.374 2.422 2.428

2.477 2.482 2.532 2.538 2.588 2.594 2.645 2.651 2.703 2.709

2.762 2.768 2.821 2.827 2.881 2.888 2.943 2.949 3.005 3.011

3.068 3.074 3.131 3.138 3.196 3.203 3.262 3.268 3.328 3.335

3.395 3.402

2.073 2.122 2.172

2.222

2.379 2.385 2.390 2.433 2.439 2.444

2.480 2.493 2.499 2.543 2.549 2.554 2.600 2.605 2.611 2.657 2.663 2.668 2.715 2.721

2.774 2.779 2.833 2.839 2.894 2.900 2.955 2.961 3.017 3.023

3.080 3.087 3.144 3.151 3.209 3.216 3.275 3.281 3.341 3.348

3.409 3.416

2.726

3.464 3.470 3.477 3.484 3.491 3.498 3.505 3.533 3.540 3.547 3.554 3.561 3.568 3.575 3.603 3.610 3.674 3.681

3.746 3.753 3.818 3.826 3.892 3.900 3.967 3.974 4.042 4.050

3.617 3.624 3.688 3.695

3.760 3.767 3.833 3.840 3.907 3.914 3.982 3.989 4.058 4.065

3.631 3.638 3.645 3.702 3.709 3.717

3.775 3.782 3.789 3.848 3.855 3.863 3.922 3.929 3.937 3.997 4.005 4.012 a 4.073 4.081 4.088

xplanation of Table of Cubes

his table gives the value of N3 for values of Nfrom 1 to 10, correct to four figures. (Interpolated values may be in error y 1 in the fourth figure.) To find the cube of a number Noutsidethe range from 1 to 10, notethatmovingthe decimal point one place in Column is equivalent to moving it three places in the body ofthelable. For example,(4.852)3 = 114.2,(0.4852)3 =0.1142, and 485.2)3 =114,200.000. This table also can be used inversely to give cube roots.


1-B PETROLEUM ENGINEERINGHANDBOOK

TABLE 1.2-CUBES OF NUMBERS (continued)

Average Difference

8

9

IO

11

12

13

14

14

14 15 16 18

9 N 0 1 2 - - 3 4 5

4.119 4.127 4.135

6

4.197 4.204 4.212 4.275 4.283 4.291 4.355 4.363 4.371 4.435 4.443 4.451

4.142 4.220 4.299 4.379 4.460

4.533 4.541 4.616 4.624 4.699 4.708 4.784 4.793 4.870 4.878

4.956 4.965 5.044 5.053 5.133 5.142 5.223 5.232 5.314 5.323

7 8

4.150 4.158 4.228 4.236 4.307 4.315 4.387 4.395 4.468 4.476

4.550 4.558 4.632 4.641 4.716 4.725 4.801 4.810 4.887 4.896

4.974 4.983 5.062 5.071 5.151 5.160 5.241 5.250 5.332 5.341

5.424 5.433 5.517 5.526 5.611 5.621 5.707 5.716 5.803 5.813

5.900 5.910

1.60 1.61 1.62 1.63 1.64

4.096 4.104 4.111 4.189 4.267

4.173 4.181 4.252 4.259 4.331 4.339 4.411 4.419

4.347 4.427

1.65 4.492 4.500 4.508 1.66 4.574 4.583 4.591 1.67 4.657 4.666 4.674 1.68 4.742 4.750 4.759 1.69 4.827 4.835 4.844

1.70 4.913 4.922 4.930 1.71 5.000 5.009 5.018 1.72 5.088 5.097 5.106 1.73 5.178 5.187 5.196 1.74 5.268 5.277 5.286

1.75 1.76 1.77 1.78 1.79

1.80 1.81 1.82 1.83 1.84

5.359 5.369 5.378 5.452 5.461 5.470 5.545 5.640 5.735

5.832 5.930 6.029 6.128 6.230

5.555 5.649 5.745

5.842 5.940 6.039 6.139 6.240

6.342 6.445 6.550 6.655 6.762

6.870 6.979 7.089

5.564 5.659 5.755

5.851 5.949 6.048 6.149 6.250

1.85 6.332 1.86 6.435 1.87 6.539 1.88 6.645 1.89 6.751

6.352 6.456 6.560 6.666 6.773

1.90 1.91 1.92 1.93 1.94

6.859 6.968 7.078 7.189 7.200 7.301 7.313

6.881 6.990 7.100 7.211 7.324

1.95 7.415 7.426 7.438 1.96 7.530 7.541 7.553 1.97 7.645 7.657 7.669 1.98 7.762 7.774 7.786 1.99 7.881 7.892 7.904

2.00 8.000 8.012 2.01 8.121 8.133 2.02 8.242 8.255 2.03 8.365 8.378 2.04 8.490 8.502

8.024 8.145 8.267 8.279 8.291 8.390 8.403 8.415

2.05 2.06 2.07 2.08 2.09

2.10 2.11 2.12 2.13 2.14

8.615 8.628 8.742 8.755 8.870 8.883 8.999 9.012 9.129 9.142

9.261 9.274 9.394 9.528

8.515

8.640 8.767 8.895 9.025 9.156

9.287 9.407 9.542

9.421 9.555

2.15

2.1 2.2 2.3 2.4

9.664 9.677 9.691 9.800 9.814 9.828

9.938 9.952 9.966

10.65 10.79 10.94 11.09 11.24 12.17 12.33 12.49 12.65 12.81 13.82 14.00 14.17 14.35 14.53

4.166 4.244 4.323 4.403 4.484

4.566 4.649 4.733 4.818 4.904

4.991 5.080 5.169 5.259 5.350

5.442 5.536 5.630 5.726 5.822

5.920 6.019 6.118

4.517 4.525 4.599 4.607 4.683 4.691 4.767 4.776 4.853 4.861

4.939 4.948 5.027 5.035 5.115 5.124 5.205 5.214 5.295 5.304

5.387 5.396 5.480 5.489 5.573 5.583 5.668 5.678 5.764 5.774

5.861 5.871 5.959 5.969 6.058 6.068 6.159 6.169 6.260 6.270

6.362 6.373 6.466 6.476

5.405 5.498

5.415 5.508

5.592 5.602 5.687 5.697 5.784 5.793

5.881 5.891 5.979 5.989 6.078 6.088

5.999 6.009 6.098 6.108 6.199 6.209 6.301 6.311

6.179 6.280

6.383 6.487

6.189 6.291

6.393 6.497 6.602 6.708 6.816

6.219 6.321

6.404 6.414 6.508 6.518 6.613 6.623 6.719 6.730 6.827 6.837

6.935 6.946 7.045 7.056 7.156 7.167 7.268 7.279 7.381 7.392

7.495 7.507 7.610 7.622 7.727 7.739 7.845 7.857 7.964 7.976

8.084 8.096 8.206 8.218 8.328 8.341 8.452 8.465 8.577 8.590

8.704 8.716 8.831 8.844

6.424 6.529 6.634 6.741 6.848

6.957 7.067

6.571 6.581 6.592 6.677 6.687 6.698

6.805 6.783 6.794

6.892 6.902 7.001 7.012 7.111 7.122 7.223 7.234 7.335 7.347

7.449 7.461

6.913 6.924 7.023 7.034 7.133 7.245 7.358

7.472

7.798 7.810 7.916 7.928

8.036 8.048 8.157 8.169

7.144 7.256 7.369

7.484 7.599 7.715

7.564 7.576 7.587 7.680 7.692 7.704

7.821 7.833 7.940 7.952

7.178 7.290 7.403

7.518 7.634 7.751 7.869 7.988

8.108 8.060 8.072 8.181 8.194 8.304 8.316 8.427 8.440 8.552 8.565

8.653 8.666 8.678 8.780 8.793 8.806

8.934 9.064 9.195

8.691 8.818 8.947 9.077 9.208

9.327 9.341 9.461 9.474 9.596 9.609

8.230 8.353 a.477 8.603

8.729 8.857 8.986 9.116 9.248

8.527 8.540

8.908 8.921 9.038 9.051 9.169 9.182

9.301 9.314 9.434 9.447 9.569 9.582 9.704 9.718 9.842 9.855

9.980 9.994

8.960 8.973 9.090 9.103 9.221 9.235

9.354 9.367 9.488 9.501 9.623 9.636 9.759 9.773 9.897 9.911

9.381 9.515 9.650 9.787 9.925

9.732 9.745 9.869 9.883

10.008

9.94 11.39 12.98 14.71

10.08 10.22 10.36 10.50 11.54 11.70 11.85 12.01 13.14 13.31 13.48 13.65 14.89 15.07 15.25 15.44

r3= 31.0063 and l/n3 =0.0322515+

MATHEMATICALTABLES& UNITS& SYSTEMSOF WEIGHTS& MEASURES 1-9

TABLE1.2-CUEIESOF NUMBERS(continued)

Average Difference N 0 3 4 -- 5 6 8 9 2 1

15.81 17.78 19.90 22.19 24.64

27.27 30.08 33.08 36.26 39.65

43.24

2.5 2.6 2.7 2.8 2.9

3.0 3.1 3.2 3.3 3.4

3.5 3.6 3.7 3.8 3.9

4.0 4.1 4.2 4.3 4.4

4.5 4.6 4.6 4.7 4.8 4.9

5.0 5.1 5.2 5.3 5.4

5.5 5.6 5.7 5.8 5.9

6.0 6.1 6.2 6.3 6.4

6.5 6.6 6.7 6.8 6.9

7.0 7.1 7.2 7.3 7.4

7.5 7.6 7.7 7.8 7.9

8.0 8.1 8.2 8.3 8.4

15.62 17.58 19.68 21.95 24.39

27.00 29.79 32.77 35.94 39.30

42.88

16.00 17.98 20.12 22.43 24.90

27.54 30.37 33.39 36.59 40.00

43.61

16.19 16.39 16.58 16.78 16.97 17.17 18.19 18.40 18.61 18.82 19.03 19.25

17.37 19.47 21.72 24.14 26.73

20 21 23 24 26

28 30 32 34 36

39 40 42 44 47

49 52 54 58 59

62 64 7 7 7 7

8

9

IO

11

12

13

14

15

18

17

18

19

20

21

20.35 20.57 20.80 21.02 22.67 22.91 23.15 23.39 25.15 25.41 25.67 25.93

27.82 28.09 28.37 28.65 30.66 30.96 31.26 31.55 33.70 34.01 34.33 34.65 36.93 37.26 37.60 37.93 40.35 40.71 41.06 41.42

43.99 44.36 44.74 45.12

21.25 21.48 23.64 23.89 26.20 26.46

28.93 29.22 31.86 32.16 34.97 35.29 38.27 38.61 41.78 42.14

45.50 49.43 53.58 57.96 62.57

67.42 72.51 77.85 83.45 89.31

95.44

45.88 49.84 54.01 58.41 58.86 63.04 63.52

67.92 68.42 73.03 73.56 78.40 78.95 84.03 84.60 89.92 90.52

96.07 96.70

101.8 102.5 108.5 109.2 115.5 116.2 122.8 123.5

130.3 131.1 138.2 139.0

29.50 32.46 35.61 38.96 42.51

46.27 50.24 54.44

46.66 47.05 47.44 47.83 48.23 48.63 49.03 50.65 51.06 51.48 51.90 52.31 52.73 53.16 54.87 59.32

64.00 68.92 74.09 79.51 85.18

91.12 97.34

103.8 110.6 117.6

55.31 59.78

64.48 69.43

55.74 60.24

64.96 69.93 75.15 80.62 86.35

56.18 56.62 60.70 61.16

65.45 65.94 70.44 70.96 75.69 76.23 81.18 81.75 86.94 87.53

92.96 93.58 99.25 99.90

105.8 106.5 112.7 113.4 119.8 120.6

57.07 61.63

66.43 71.47 76.77 82.31 88.12

94.20 100.54 100.5 107.2 114.1 121.3

128.8 136.6

57.51 62.10

66.92 71.99 77.31 82.88 88.72

94.82

101.2 107.9 114.8 122.0

103.2 109.9 116.9 124.3

146.4 147.2 154.9 155.7 163.7 164.6

131.9 139.8 148.0 156.6 165.5

172.8 173.7 174.7 182.3 183.3 184.2 192.1 193.1 194.1 202.3 203.3 204.3 212.8 213.8 214.9

223.6 224.8 225.9

74.62 80.08 85.77

91.73 97.97

104.5 111.3 118.4

125.8 133.4

92.35 98.61

105.2 112.0 119.1

126.5 134.2 142.2 150.6 159.2

168.2 177.5 187.1 197.1 207.5

218.2 229.2 240.6 252.4 264.6

277.2 290.1

125.0 132.7 140.6 148.9 157.5

166.4 175.6 185.2 195.1 205.4

216.0 227.0 238.3 250.0 262.1

274.6 287.5

127.3 128.0 135.0 135.8 143.1 143.9 151.4 152.3

129.6 137.4 145.5 154.0

141.4 149.7

144.7 153.1

158.3

167.3 176.6 186.2 196.1 206.4

217.1 228.1 239.5 251.2 263.4

160.1 161.0

169.1 170.0 178.5 179.4 188.1 189.1 198.2 199.2 208.5 209.6

219.3 220.3

161.9

171.0 180.4 190.1 200.2 210.6

221.4 232.6 244.1 256.0 268.3

281.0 294.1

162.8

171.9 181.3 191.1 201.2 211.7

222.5 230.3 231.5 241.8 243.0

233.7 234.9 236.0 237.2 245.3 246.5 247.7 248.9

253.6 254.8 265.8 267.1

257.3 258.5 259.7 269.6 270.8 272.1

307.5 321.4

282.3 283.6 295.4 298.7 308.9 310.3 322.8 324.2

284.9 298.1

335.7

350.4 365.5 381.1 397.1 413.5

337.2

351.9 367.1 382.7 398.7 415.2

432.1 449.5 467.3

338.6

353.4 368.6 384.2 400.3 416.8

311.7 325.7 340.1

354.9 370.1 385.8 401.9 418.5

260.9 273.4

286.2 299.4 313.0 327.1 341.5

356.4 371.7 387.4 403.6 420.2

430.4 447.7 465.5 483.7 502.5

435.5 453.0 470.9

485.6 504.4

433.8 451.2 469.1 487.4 506.3

489.3 508.2

437.2 454.8 472.7 491.2 510.1

521.7 523.6 525.6 527.5 529.5 541.3 543.3 545.3 547.3 549.4 561.5 583.6 565.6 567.7 569.7 582.2 584.3 586.4 588.5 590.6 603.4 605.5 607.6 609.8 612.0

275.9 288.8 302.1 315.8

278.4 279.7 291.4 292.8 304.8 306.2 318.6 320.0

300.8 314.4 328.5

343.0 357.9 373.2 389.0 405.2

329.9 331.4

344.5 345.9 359.4 360.9 374.8 376.4 390.6 392.2 406.9 408.5

421.9 423.8 439.0 440.7 456.5 458.3

303.5 317.2

474.6 476.4 493.0 494.9

425.3 442.5 460.1 478.2 496.8

512.0 513.9 515.8 531.4 533.4 535.4 551.4 553.4 555.4 571.8 573.9 575.9 592.7 594.8 596.9

332.8 334.3

347.4 348.9 362.5 364.0 377.9 379.5 393.8 395.4 410.2 411.8

427.0 428.7 444.2 445.9 461.9 463.7 480.0 481.9 498.7 500.6

517.8 519.7 535.4 539.4 557.4 559.5 578.0 580.1 599.1 601.2

r =310063and 1h3=O0322515+ 3



N

8.5 614.1 8.6 636.1 8.7 658.5 8.8 681.5 8.9 705.0

9.0 9.1 9.2 9.3 9.4

729.0 753.6 778.7 804.4 807.0 830.6 833.2

9.5 9.6 9.7 9.8 9.9

10.0

857.4 860.1 862.8 884.7 887.5 890.3 912.7 915.5 918.3 941.2 944.1 947.0 970.3 973.2 976.2

1000.0

0

616.3 638.3 660.8 683.8 707.3

731.4 756.1 781.2

2

TABLE 1.2-CUBES OF NUMBERS (continued)

618.5 640.5 663.1 686.1 709.7

733.9 758.6 783.8 809.6 835.9

3 4

620.7 622.8 642.7 645.0 665.3 667.6 688.5 690.8 712.1 714.5

736.3 738.8 761 .O 763.6 786.3 788.9 812.2 814.8 838.6 841.2

865.5 868.3 893.1 895.8 921.2 924.0 949.9 952.8 979.1 982.1

5

625.0 647.2 669.9 693.2 716.9

741.2 766.1 791.5 817.4 043.9

627.2 649.5 672.2 695.5 719.3

743.7 768.6 794.0 820.0 846.6

746.1 771.1 796.6 822.7 849.3

871 .O 873.7 876.5 898.6 901.4 904.2 926.9 929.7 932.6 955.7 958.6 961.5 985.1 988.0 991 .o

6 7

629.4 651.7 674.5 697.9 721.7

8 9 Average Difference

631.6 633.8 654.0 656.2 678.8 679.2 700.2 702.6 724.2 726.6

748.6 773.6 799.2 825.3 852.0

879.2 907.0 935.4 964.4 994.0

751.1 776.2 801.8 827.9 854.7

882.0 909.9 938.3 967.4 997.0

22

23 24

25

26

27

28

29

T =31 0063 and 1/x3 =00322515+ 3

MATHEMATICALTABLES &UNITS &SYSTEMSOFWElGHTS&MEASURES 1-11

TABLE 1.3-SQUARE ROOTS OF NUMBERS


1.044 5 1.091 1.136 4 1.179 1.221

N

1.0 1.1 1.2 1.3 1.4

0

1.000

1.5 1.6 1.7 1.8 1.9

1.049 1.095 1.140 1.183

1.255 1.265

2.0 2.1 2.2 2.3 2.4

2.5 2.6 2.7 2.8 2.9

3.0 3.1 3.2 3.3 3.4

1.789

1.304

1.378

1.414

1.342

1.449

1.817

1.483 1.517 1.549

1.581 1.612 1.643 1.673 1.703

1.732 1.761

3.5 3.6 3.7 3.8 3.9

4.0 4.1 4.2 4.3 4.4

4.5 4.6 4.7 4.8 4.9

1.844

1.871 1.897 1.924 1.949 1.975

2.000 2.025 2.049 2.074 2.098

2.121 2.145 2.168 2.191 2.214

5.0 2.236 5.1 2.258 5.2 2.280 5.3 2.302 5.4 2.324

5.5 2.345 5.6 2.366 5.7 2.387 5.8 2.408 5.9 2.429

6.0 6.1 6.2 6.3 6.4

6.5 6.6 6.7 6.8 6.9

2.449 2.452 2.454 2.456 2.458 2.470 2.472 2.474 2.476 2.478 2.490 2.492 2.494 2.496 2.498 2.510 2.512 2.514 2.516 2.518 2.530 2.532 2.534 2.536 2.538

2.550 2.569 2.588 2.608 2.627

2 3 4

1.010 1.015 1.020

5

1.025 1.072 1.118 1.162 1.204

6

1.030 1.077 1.122 1.166 1.208

1.249 1.288

7

1.034 1.082 1.127 1.170 1.212

8

1.039 1.088

1

1.005 1.054 1.100 1.145 1.187

1.229 1.269

1.058 1.083 1.068 1.105 1.109 1.114 1.149 1.153 1.158 1.192 1.196 1.200

1.131 1.175

1.233 1.273 1.311 1.349

1.237 1.277

1.241 1.281 1.319 1.356 1.393

1.428 1.463 1.497 1.530 1.562

1.245 1.285

1.217

1.253 1.257 1.261 1.292 1.296 1.300

1.315 1.353 1.389

1.323 1.360 1.396

1.432 1.466 1.500 1.533 1.565

1.327 1.364

1.386

1.421 1.425 1.456 1.459 1.490 1.493 1.523 1.526 1.556 1.559

1.400

1.435 1.470 1.503 1.536 1.568

1.330 1.334 1.338 1.367 1.371 1.375 1.404 1.407 1.411

1.439 1.442 1.446 1.473 1.476 1.480 3 1.507 1.510 1.513 1.539 1.543 1.546 1.572 1.575 1.578

1.587 1.591 1.594 1.597 1.600 1.603 1.606 1.609 1.619 1.622 1.625 1.628 1.631 1.634 1.637 1.640 1.649 1.652 1.655 1.658 1.661 1.664 1.667 1.670 1.679 1.682 1.685 1.688 1.691 1.694 1.697 1.700 1.709 1.712 1.715 1.718 1.720 1.723 1.726 1.729

1.382

1.308

1.418 1.453

1.345

1.487 1.520 1.552

1.584 1.616 1.646 1.676 1.706

1.735 1.764

1.738 1.741 1.744 1.746 1.766 1.769 1.772 1.775

1.803 1.830 1.857

1.749 1.778 1.806 1.833 1.860

1.752

1.808 1.836 1.863

1.780

1.889 1.916 1.942 1.967 1.992

2.017 2.042 2.066 2.090 2.114

2.138 2.161

1.811

1.755

1.838 1.885

1.783

1.884 1.887 1.910 1.913 1.936 1.939 1.962 1.965 1.987 1.990

1.892 1.918 1.944 1.970 1.995

2.012 2.015 2.037 2.040 2.062 2.064 2.086 2.088 2.110 2.112

2.020 2.045 2.069 2.093 2.117

2.133 2.156 2.179 2.202 2.225

2.247 2.269 2.291 2.313 2.335

2.356 2.377 2.398

2.140 2.163

2.135 2.159 2.182 2.184 2.186 2.205 2.207 2.209

1.814 1.841

1.758

1.868

1.895 1.921 1.947

1.786

1.972 1.997

2.022

1.847

1.792

1.873 1.900 1.926

1.819

1.952 1.977

2.002 2.027 2.052

1.794 1.822 1.849

1.876 1.903 1.929 1.954 1.980

2.005

1.797 1.800 1.825 1.828 1.852 1.855

1.879 1.881 1905 1.908 1.931 1.934 1.957 1.960 1.982 1.985

2.007 2.010 2.030 2.032 2.054 2.057 2.078 2.081 2.102 2.105

2.035 2.059 2.083 2.107

2.131 2.154 2.177 2.200 2.223

2.245 2.267 2.289 2.311 2.332

2.354

2.047 2 2.071

2.076 2.100

2.124 2.147 2.170 2.193 2.216

2.238 2.261 2.283 2.304 2.326

2.347

2.095 2.119

2.142 2.166 2.189 2.211 2.234

2.256 2.278 2.300 2.322 2.343

2.364

2.126 2.128 2.149 2.152 2.173 2.175 2.195 2.198 2.218 2.220 2.227 2.229 2.232

2.249 2.252 2.254 2.272 2.274 2.276 2.293 2.296 2.298 2.315 2.317 2.319 2.337 2.339 2.341

2.358 2.360 2.382

2.241 2.243 2.263 2.265 2.285 2.287 2.307 2.309 2.328 2.330

2.349 2.352 2.369 2.371 2.390 2.392

2.373 2.375 2.394 2.398 2.415 2.417 2.435 2.437

2.379 2.381 2.383 2.385 2.400 2.402 2.404 2.406 2.421 2.423 2.425 2.427 2.441 2.443 2.445 2.447

2.410 2.412 2.431 2.433

2.419 2.439

2.460 2.462 2.464 2.466 2.468 2.480 2.482 2.484 2.486 2.488 2.500 2.502 2.504 2.506 2.508 2.520 2.522 2.524 2.526 2.528 2.540 2.542 2.544 2.546 2.548

2.557 2.559 2.561 2.563 2.565 2.567 2.577 2.579 2.581 2.583 2.585 2.587 2.596 2.598 2.600 2.602 2.604 2.606 2.615 2.617 2.619 2.621 2.623 2.625 2.634 2.636 2.638 2.640 2.642 2.644

2.551 2.553 2.555 2.571 2.573 2.575 2.590 2.592 2.594 2.610 2.612 2.613 2.629 2.631 2.632

Exolanation of Table of Sauare Roots

This table gives the value of fiforvaluesof Nfrom 1 to lOO,correcttofourfigures.(lnterpolated values maybe inerror 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

oftwo(beginning with thedecimalpoint),and note that movingthedecimal lace in the square root of N. For example, &%-8=1.648,

ointtwo placesin N is equivalent to moving it one +=

J--Y= 00002718 001648, and Jo7oo2718=0.05213.

2718 5213, &%%=18.48, ~@%3=52.13,



TABLE 1.3-SQUARE ROOTS OF NUMBERS(continued)

7 8 9

2.659 2.661 2.663

2 3 4 5 6

2.650 2.651 2.653 2.655 2.657 2.668 2.670 2.672 2.674 2.676 2.678 2.687 2.689 2.691 2.693 2.694 2.696 2.706 2.707 2.709 2.711 2.713 2.715 2.724 2.728 2.728 2.729 2.731 2.733

2.742 2.744 2.746 2.748 2.750 2.751 2.760 2.762 2.764 2.766 2.768 2.769 2.778 2.780 2.782 2.784 2.786 2.787 2.796 2.798 2.800 2.802 2.804 2.805 2.814 2.816 2.818 2.820 2.821 2.823

2.832 2.834 2.835 2.837 2.839 2.841 2.850 2.851 2.853 2.855 2.857 2.858 2.867 2.869 2.871 2.872 2.874 2.876 2.884 2.886 2.888 2.890 2.891 2.893 2.902 2.903 2.905 2.907 2.909 2.910

2.919 2.921 2.922 2.924 2.926 2.927 2.936 2.938 2.939 2.941 2.943 2.944 2.953 2.955 2.956 2.958 2.960 2.961 2.970 2.972 2.973 2.975 2.977 2.978 2.987 2.988 2.990 2.992 2.993 2.995

3.003 3.005 3.007 3.008 3.010 3.012 3.020 3.022 3.023 3.025 3.027 3.028 3.036 3.038 3.040 3.041 3.043 3.045 3.053 3.055 3.056 3.058 3.059 3.061 3.069 3.071 3.072 3.074 3.076 3.077

3.085 3.087 3.089 3.090 3.092 3.094 3.102 3.103 3.105 3.106 3.108 3.110 3.118 3.119 3.121 3.122 3.124 3.126 3.134 3.135 3.137 3.138 3.140 3.142 3.150 3.151 3.153 3.154 3.156 3.158

3.194 3.209 3.225 3.240 3.256 3.271 3.347 3.362 3.376 3.391 3.406 3.421 3.493 3.507 3.521 3.536 3.550 3.564 3.633 3.647 3.661 3.674 3.688 3.701 3.768 3.782 3.795 3.808 3.821 3.834

3.899 3.912 3.924 3.937 3.950 3.962 4.025 4.037 4.050 4.062 4.074 4.087 4.147 4.159 4.171 4.183 4.195 4.207 4.266 4.278 4.290 4.301 4.313 4.324 4.382 4.393 4.405 4.416 4.427 4.438

4.494 4.506 4.517 4.528 4.539 4.550 4.604 4.615 4.626 4.637 4.648 4.658 4.712 4.722 4.733 4.743 4.754 4.764 4.817 4.827 4.837 4.848 4.858 4.868 4.919 4.930 4.940 4.950 4.960 4.970

5.020 5.030 5.040 5.050 5.060 5.070 5.119 5.128 5.138 5.148 5.158 5.167 5.215 5.225 5.235 5.244 5.254 5.263 5.310 5.320 5.329 5.339 5.348 5.357 5.404 5.413 5.422 5.431 5.441 5.450

5.495 5.505 5.514 5.523 5.532 5.541 5.586 5.595 5.604 5.612 5.621 5.630 5.675 5.683 5.692 5.701 5.710 5.718 5.762 5.771 5.779 5.788 5.797 5.805 5.848 5.857 5.865 5.874 5.882 5.891

5.933 5.941 5.950 5.958 5.967 5.975 6.017 6.025 6.033 6.042 6.050 6.058 6.099 6.107 6.116 6.124 6.132 6.140 6.181 6.189 6.197 6.205 6.213 6.221 6.281 6.269 6.277 6.285 6.293 6.301

2.680 2.681 2.698 2.700 2.717 2.718 2.735 2.737

2.753 2.771 2.789 2.807 2.825

2.843

2.755 2.773 2.791 2.809 2.827

2.844 2.860 2.877 2.895 2.912

2.862 2.879 2.897 2.914

2.929 2.931 2.946 2.948 2.963 2.965 2.980 2.982 2.997 2.998

3.013 3.015 3.030 3.032 3.046 3.048 3.063 3.064 3.079 3.081

3.095 3.097 3.111 3.113 3.127 3.129 3.143 3.145 3.159 3.161

3.286 3.302 3.435 3.450 3.578 3.592 3.715 3.728 3.847 3.860

3.975 4.099 4.219 4.336 4.450

3.987 4.111 4.231 4.347 4.461

4.561 4.669 4.775 4.879 4.980

4.572 4.680 4.785 4.889 4.990

5.079 5.089 5.177 5.187 5.273 5.282 5.367 5.376 5.459 5.468

5.550 5.559 5.639 5.648 5.727 5.736 5.814 5.822 5.899 5.908

5.983 5.992 6.066 6.075 6.148 6.156 6.229 6.237 6.309 6.317

Vr=i.77245+. l/V'*=O.56419, V,*/Z=l 25331, and ~e=,.64872

Average Difference

2

16 15 14

13

12

11

10

9

8

N

7.0 7.1 7.2 7.3 7.4

0

2.646 2.665 2.683 2.702 2.720

1

2.648 2.666 2.685 2.704

7.5 7.6 7.7 7.8 7.9

2.739 2.757

8.0 8.1 8.2 8.3 8.4

2.775 2.793 2.811

2.828

2.722

2.740 2.759 2.777 2.795 2.812

2.830 2.846 2.848 2.864 2.865 2.881 2.883 2.898 2.900

8.5 2.915 2.917 8.6 2.933 2.934 8.7 2.950 2.951 8.6 2.966 2.968 8.9 2.983 2.985

9.0 9.1 9.2 9.3 9.4

3.000 3.002 3.017 3.018

9.5 9.6 9.7 9.8 9.9

3.033 3.050 3.066

3.082 3.098 3.114 3.130 3.146

10.0 11.0 12.0 13.0 14.0

3.035 3.051 3.068

3.084 3.100 3.116 3.132 3.148

3.178 3.162 3.317 3.332 3.464 3.479 3.606 3.742

15.0 3.873 16.0 4.000 17.0 4.123 18.0 4.243 19.0 4.359

3.619 3.755

3.886 4.012 4.135 4.254 4.370

20.0 4.472 21.0 4.583 22.0 4.690 23.0 4.796 24.0 4.899

4.483 4.593

25.0 5.000 26.0 5.099 27.0 5.196 28.0 5.292 29.0 5.385

30.0 31.0 32.0 33.0 34.0

5.477

4.701 4.806 4.909

5.010 5.109 5.206 5.301 5.394

5.486 5.568 5.577 5.657 5.666

35.0 36.0 37.0 38.0 39.0

5.745 5.753 5.831 5.840

5.916 5.925 6.000 6.008 6.083 6.091 6.164 6.173 6.245 6.253

MATHEMATICALTABLES& UNITS& SYSTEMSOFWElGHTS& MEASURES 1-13

TABLE 1.3-SQUARE ROOTS OF NUMBERS(continued)

N 0 1 2 3 4 5 6 7 8 9 Average Difference

40.0 41 0 42.0 43.0 44.0

6.325 6.332 6.340 6.348 6.356 6.364 6.372 6.380 6.387 6.395 8 6.411 6.419 6.427 6.434 6.442 6.450 6.458 6.465 6.473 6.488 6.496 6.504 6.512 6.519 6.527 6.535 6.542 6.550 6.565 6.573 6.580 6.588 6.595 6.603 6.611 6.618 6.626 6.641 6.648 6.656 6.663 6.671 6.678 6.686 6.693 6.701

45.0 46.0 47.0 48.0 49.0

6.403

6.856

6.481 6.557

6.928

6.633

6.708 6.782

6.716 6.790

6.731 6.738 6.745 6.753 6.804 6.812 6.819 6.826

50.0 51.0 52.0 53.0 54.0

7.029

6.885

7.099 7.169 7.239 7.308

6.957

7.376

7.036

7.106

6.892

7.176 7.246 7.314 7.382

6.964

55.0 56.0 57.0 56.0 59.0

7.000

7.071 7.141 7.211 7.280 7.348

7.416 7.483 7.550 7.616 7.681

7.443 7.450 7.510 7.517 7.576 7.583 7.642 7.649 7.707 7.714

6.760 6.834 6.907 6.979 7.050

7.120 7.190 7.259 7.328 7.396

7.463 7.530 7.596 7.662 7.727

60.0 7.746 61.0 7.810 62.0 7.874 63.0 7.937 64.0 8.000

7.772 7.778 7.836 7.842 7.899 7.906 7.962 7.969 8.025 8.031

65.0 66.0 67.0 68.0 69.0

8.062 8.124 8.185 8.246 8.307

7.007

6.863

7.078 7.148

6.935

7.218 7.287 7.355

7.423 7.490 7.556 7.622 7.688

7.752 7.817 7.880 7.944 8.006

8.068 8.130 8.191 8.252 8.313

6.723 6.797 6.870 6.943 7.014

7.085 7.155 7.225 7.294 7.362

7.430 7.497 7.563 7.629 7.694

7.759 7.823 7.887 7.950 8.012

8.075 8.136 8.198 8.258 8.319

8.087 8.093 8.149 8.155 8.210 8.216 8.270 8.276 8.331 8.337

70.0 8.367 8.373 8.379 71.0 8.426 8.432 8.438 72.0 8.485 8.491 8.497 73.0 8.544 8.550 8.556 74.0 8.602 8.608 8.614

6.877

7.021

7.092 7.162 7.232

6.950

7.301 7.369

7.436 7.503 7.570 7.635 7.701

7.765 7.829 7.893 7.956 8.019

8.081 8.142 8.204 8.264 8.325

8.385 8.390 8.396

7.043

7.113 7.183

6.899

7.253 7.321 7.389

7.457

6.971

7.523 7.589 7.655 7.720

7.785 7.849 7.912 7.975 8.037

8.099 8.161 8.222 8.283 8.343

8.402

7.791 7.855 7.918 7.981 8.044

8.106 8.167 8.228 8.289 8.349

8.408 8.414 8.444 8.450 8.456 8.462 8.468 8.473 8.479 8.503 8.509 8.515 8.521 8.526 8.532 8.538 8.562 8.567 8.573 8.579 8.585 8.620 8.626 8.631 8.637 8.643

75.0 8.660 8.666 8.672 8.678 8.683 8.669 8.695 8.701 76.0 8.718 8.724 8.729 8.735 8.741 8.746 8.752 8.758 77.0 8.775 8.781 8.786 8.792 8.798 8.803 8.809 8.815 78.0 8.832 8.837 8.843 8.849 8.854 8.860 8.866 8.871 79.0 8.888 8.894 8.899 8.905 8.911 8.916 8.922 8.927

80.0 8.944 8.950 8.955 81.0 9.000 9.006 9.011 82.0 9.055 9.061 9.066 83.0 9.110 9.116 9.121 84.0 9.165 9.171 9.176

85.0 86.0 67.0 88.0 89.0

9.220 9.225 9.230 9.274 9.279 9.284 9.327 9.333 9.338 9.381 9.386 9.391 9.434 9.439 9.445

8.961 9.017 9.072 9.127 9.182

9.236 9.290 9.343 9.397 9.450

90.0 91.0 92.0 93.0 94.0

9.487 9.492 9.497 9.503 9.539 9.545 9.550 9.555 9.592 9.597 9.602 9.607

9.077

8.967

9.132

9.022

9.187

9.241 9.295 9.349 9.402 9.455

9.508 9.560 9.612 9.664 9.716

9.083

8.972

9.138 9.192

9.028

9.247 9.301 9.354 9.407 9.460

9.513

8.978

9.198

9.252

9.033

9.306 9.359

9.088

9.413 9.466

9.143

9.518

9.203

8.983

9.257 9.311

9.039

9.365 9.418

9.094

9.471

9.524

9.149

9.644 9.649 9.654 9.659 9.695 9.701 9.706 9.711

9.566 9.571 9.576 9.581 9.586 9.618 9.623 9.628 9.633 9.638 9.670 9.675 9.680 9.721 9.726 9.731

95.0 9.747 9.752 9.757 9.762 9.767 9.772 9.778 9.783 96.0 9.798 9.803 9.808 9.813 9.818 9.823 9.829 9.834 97.0 9.849 9.854 9.859 9.864 9.869 9.874 9.879 9.884 98.0 9.899 9.905 9.910 9.915 9.920 9.925 9.930 9.935 99.0 9.950 9.955 9.960 9.965 9.970 9.975 9.980 9.985

6.768 6.841 6.914 6.986 7.057

7.127 7.197 7.266 7.335 7.403

7.470 7.537 7.603 7.668 7.733

7.797 7.861 7.925 7.987 8.050

8.112 8.173 8.234 8.295 8.355

6.775 7 6.848 6.921 6.993 7.064

7.134 7 7.204 7.273 7.342 7.409

7.477 7.543 7.609 7.675 7.740 6

7.804 7.868 7.931 7.994 8.056

8.118 8.179 8.240 8.301 8.361

8.420

8.591 8.597 8.649 8.654

8.706 8.712 8.764 8.769 8.820 8.826 8.877 8.883 8.933 8.939

8.989 8.994 9.044 9.050 9.099 9.105 5 9.154 9.160 9.209

9.263 9.317 9.370 9.423 9.476

9.529

9.214

9.268 9.322 9.375 9.429 9.482

9.534

9.685 9.690 9.737 9.742

9.788 9.793 9.839 9.844 9.889 9.894 9.940 9.945 9.990 9.995

SQUAREROOTSOFCERTAlNFRACTlONS

N &i N Jii N v% N J- N fi N -J%

0.7071- % - __ - '/2 0.7746 % 0.7559 '/9 0.3333 -0.64559/rs- s/1* 0.7500 '/a 0.5774 a/5 0.8944 % 0.8452 2/ 0.4714 y,p 0.7638 "A6 0.8292 v3 0.8165 '/6 0.4082 % 0.9258 % 0.6667 1%~ 0.9574 '3/G 0.9014 '/4 0.5000 5/s 0.9129 '/A 0.3536 =/g 0.7454 'h6 0.2500 '5/e 0.9662 v4 0.8660 '/$ 0.3780 % 0.6124 '/g 0.8819 3/ls 0.4330 '/!32 0.1768 '/s 0.4472 2/7 0.5345 % 0.7906 8/s 0.9428 %s 0.5590 '/6* 0.1250 75 0.6325 3/7 0.6547 '/a 0.9354 '/'2 0.2887 ‘/,s 0.6614 'ho 0.1414


TABLEl.4-CUBEROOTSOFNUMBERS


1.029 3

4 5 6 7 N 0 1 2 - - 3

1.010

8

1.026 1.0 1.1 1.2 1.3 1.4

1.000 1.003 1.007 1.013 1.032 1.035 1.038 1.042 1.045 1.063 1.066 1.069 1.071 1.074

1.020 1.023

1.094 1.097 1.100 1.102 1.121 1.124 1.127 I.129

1.016 1.048 1.077 1.105 1.132

1.051 1.054 1.080 1.083 1.108 1.111 1.134 1.137

1.5 1.6 1.7 1.8 1.9

1.147 1.172 1.196 1.219 1.241

1.155 1.157 1.160 1.162 1.165 1.167 2 1.179 1.182 1.184 1.186 1.189 1.191 1.203 1.205 1.207 1.210 1.212 1.214 1.225 1.228 1.230 1.232 1.234 1.236 1.247 1.249 1.251 1.254 1.256 1.258

2.0 2.1 2.2 2.3 2.4

2.5 2.6 2.7 2.8 2.9

1.091 1.119

1.145 1.170 1.193 1.216 1.239

1.260 1.281 1.301 1.320

1.262 1.283 1.303 1.322

1.150 I.174 1.198 1.221 1.243

1.264 1.285 1.305 1.324

1.339

1.357 1.375 1.392 1.409 1.426

1.341 1.343

1.359 1.361 1.377 1.379 1.394 1.396 1.411 1.413 1.428 1.429

1.152 I.177 1.200 1.223 1.245

1.266 1.287 1.306 1.326 1.344

1.363 1.380 1.398 1.414 1.431

1268 1.270 1.272 1.289 1.291 1.293 1.308 1.310 1.312 1.328 1.330 1.331 1.346 1.348 1.350

1.364 1.366 1.368 1.382 1.384 1.386 1.399 1.401 1.403 1.416 1.418 1.419 1.433 1.434 1.436

3.0 1.442 1.444 1.445 1.447 1.449 1.450 1.452 3.1 1.458 1.460 1.461 1.463 1.464 1.466 1.467 3.2 1.474 1.475 1.477 1.478 1.480 1.481 1.483 3.3 1.489 1.490 1.492 1.493 1.495 1.496 1.498 3.4 1.504 1.505 1.507 1.508 1.510 1.511 1.512

3.5 1.518 1.520 1.521 3.6 1.533 1.534 1.535 3.7 1.547 1.548 1.549 3.8 1.560 1562 1.563 3.9 1.574 1.575 1.577

1.524 1.538 1.552 1.566 1.579

4.0 1.567 1.589 1.590 4.1 1.601 1.602 1.603 4.2 1.613 1.615 1.616 4.3 1.626 1627 1.629 4.4 1.639 1.640 1.641

1.523 1.537 1.551 1.565 1.578

1.591 1.604 1.617 1.630 1.642

1.593 1.606 1.619 1.631 1.644

1.525 1.527 1.540 1.541 1.554 1.555 1.567 1.569 1.581 1.582

1.594 1.595 1.607 1.608 1.820 1.621 1.632 1.634 1.645 1.646

4.5 4.6 4.7 4.8 4.9

1.651 1.652 1.653 1.655 1.656 1.658 1.663 1.664 1.675 1.676 1.687 1.688 1.698 1.700

1.666 1.677 1.689 1.701

1.657 1.669 1.681

1.711 1.712 1.722 1.724 1.734 1.735 1.745 1.746 1.755 1.757

1.766 1.767 1.777 1.778

1.667 1.668 1.679 1.680 1.690 1.692 1.702 1.703

1.693 1.704

1.670 1.682 1.694 1.705

5.0 1.710 5.1 1.721 5.2 1.732 5.3 1.744 5.4 1.754

1.713 1.715 1.716 1.717 1.718 1.719 1.720 1.725 1.726 1.727 1.728 1.729 1.730 1.731 1.736 1.737 1.738 1.739 I.740 1.741 1.742 1.747 1.748 1.749 1.750 1.751 1.752 1.753 1.758 1.759 1.760 1.761 1.762 1.763 1.764

5.5 5.6 5.7 5.8 5.9

6.0 6.1 6.2 6.3 6.4

6.5 6.6 6.7 6.8 6.9

1.765 1.776

1.768 1.769 I.771 1.779 1.780 1.781

1.786 1.787 1.788 1.789 1.790 1.792 1.797 1.798 1.799 1.800 1.801 1.802 1.807 1.808 1.809 1.810

1.817 1.818 1.819 1.820 1.827 1.828 1.829 1.830 1.837 1.838 1.839 1.840

1.811

1.847 1.848 1.849 1.850 1.857 1.858 1.859 1.860

1.821 1.831 1.841 1.851 1.860

1.812

1.822 1.832 1.842

1.772 1.782 1.793 1.803 1.813

1.823 1.833 1.843

1.852 1.853 1.861 1.862

1.866 1.867 1.868 1.869 1.870 1.871 1.872 1.873 1.874 1.875 1.876 1.877 1.878 1.879 1.880 1.881 1.881 1.882 1.883 1.884 1.885 1.886 1.887 1.888 1.889 1.890 1.891 1.892 1.893 1.894 1.895 1.895 1.896 1.897 1.898 1.899 1.900 1.901 1.902 1.903 1904 1.905 1.906 1.907 1.907 1.908 1.909 1.910 1.911 1.912

1.057 1.060 1.086 1.089 1.113 1.116 1.140 1.142

1.274 1.295 1.314 1.333 1.352

1.370 1.387 1.404 1.421 1.437

1.453

1.277 1.279 1.297 1.299 1.316 1.318 1.335 1.337 1.354 1.355

1.372 1.373 1.389 1.391 1.406 1.408 1.423 1.424 1.439 1.441

1.455 1.457 I.471 1.472 1.486 1.487 1.501 1.502 1.515 1.517

1.530 1.531 1.544 1.545 1

1.469 1.484 1.499 1.514

1.528 I.542 1.556 1.570 1.583

1.597 1.610 1.622 1.635 1.647

1.659 1.671 1.683

1.558 I .559 1.571 1.573 1.585

1.598 1.811 1.624 1.636 1.649

1.681 1.673 1.685

1.586

1.599 1.612 1.625 1.637 1.650

1.662 1.674 1.686 1.697 1.709

1.695 1.696 1.707 1.708

1.773 1.783 1.794 1.804 1.814

1.824 1.834 1.844 1.854 1.883

1.774 1.784 1.795 1.805 1.815

1.825 1.835 1.845

1.775 1.785 1.796 1.806 1.816

1.826 1.836 1.846

1.855 1.856 1.864 1.865

Explanation of Table of Cube Roots

This table gives the value of fi for values of N from 1 to 1,000, 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 1,000, divide the digits of the number into blocks

of three (beginning with the decimal point), and note that moving the decimal point fhree laces in column N is equivalent to moving it one 6.477 tb-?% =13.96

lace in the cube root of N. For example, %%%? = 1.396, &iii6 =30.07, m =64.77,

Kit = 3.007 ?/zix=

and m =0.06477. ?0.000002718 =0.01396, w =0.03007,

MATHEMATlCALTABLES&UNlTS & SYSTEMSOFWElGHTS& MEASURES 1-15

TABLE 1.4-CUBE ROOTS OF NUMBERS (continued)

8

1.920 1.929 1.938 1.947 1.956


1.921 1 1.930 1.939 1.948 1.957

1.964 1.965 1.973 1.974 1.981 1.962 1.990 1.991 1.998 1.999

N 0 2 6 7

1.918 1.919 1.927 1.928

4 5

1.917 1.917 1.926 1.926 1.935 1.935 1.943 1.944 1.952 1.953

3

1.916 1.925

1

1.914 1.923 1.932 1.941 1.950

7.0 1.913 7.1 1.922 7.2 1.931 7.3 1.940 7.4 1.949

1.915 1.924 1.933 1.942 1.950

1.934 1.943 1.951

1.936 1.937 1.945 1.946

1.955

7.5 1.957 7.6 1.966 7.7 1.975 7.8 1.983 7.9 1.992

1.958 1.959 1.960 1.961 1.962 1.967 1.968 1.969 1.970 1.970

1.964 1.972

1.976 1.976 1.977 1.978 1.979 1.984 1.985 1.986 1.987 1.987 1.992 1.993 1.994 1.995 1.996

1.954

1.963 1.971 1.980 1.988 1.997

2.005 2.013 2.021

1.981 1.989 1.997

8.0 2.000 2.001 2.002 2.002 8.1 2.008 2.009 2.010 2.011 8.2 2.017 2.017 2.018 2.019 8.3 2.025 2.026 2.026 2027 8.4 2.033 2.034 2.034 2.035

2.003 2.004 2.012 2.012 2.020 2.021

2.006 2.014 2.022

2.028 2.029 2.030 2.030 2.036 2.037 2.038 2.038

2.007 2.007 2.015 2.016 2.023 2.024 2.031 2.032 2.039 2.040

8.5 2.041 2.042 2.042 2.043 2.044 2.045 2.046 2.046 2.047 2.046 8.6 2.049 2.050 2.050 2.051 2.052 2.053 2.054 2.054 2.055 2.056 8.7 2.057 2.057 2.058 2.059 2.060 2.061 2.061 2.062 2.063 2.064 8.8 2.065 2.065 2.066 2.067 2.068 2.068 2.069 2.070 2.071 2.072 8.9 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 9.1 2.088 2.089 2.089 2.090 9.2 2.095 2.096 2.097 2.098 9.3 2.103 2.104 2.104 2.105 9.4 2.110 2.111 2.112 2.113

2.083 2.084 2.085 2.091 2.092 2.092 2.098 2.099 2.100 2.106 2.107 2.107

2.085 2.086 2.087 2.093 2.094 2.095 2.101 2.101 2.102 2.108 2.109 2.110 2.116 2.116 2.117

9.5 9.6 9.7 9.8 9.9

10.0 11.0 12.0 13.0 14.0

15.0 16.0 17.0 18.0 19.0

2.118 2.125 2.133 2.140 2.147

2.154 2.224 2.289 2.351 2.410

2.466

2.119 2.119 2.120 2.126 2.127 2.128 2.133 2.134 2.135 2.141 2.141 2.142 2.148 2.149 2.149

2.113 2.114 2.115

2.121 2.122 2.122 2.128 2.129 2.130 2.136 2.136 2.137 2.143 2.144 2.144 2.150 2.151 2.152

2.183 2.190 2.197 2.251 2.257 2.264 2.315 2.321 2.327

2.123 2.124 2.125 2.130 2.131 2.132 2.138 2.139 2.139 2.145 2.146 2.147 2.152 2.153 2.154

2.162 2.231 2.296 2.357 2.416

2.169 2.176 2.237 2.244 2.302 2.308

2.204 2.210 2.270 2.277 2.333 2.339

2.363 2.369 2.375 2.381 2.387 2.393 2.399 2.422 2.427 2.433 2.438 2.444 2.450 2.455

2.217 7 2.263 6 2.345 2.404 2.461

2.515 5 2.520 2.571 2.621 2.668

2.472 2.525 2.576 2.626 2.673

2.477 2.530 2.581 2.630 2.678

2.483 2.488 2.535 2.541 2.586 2.591 2.635 2.640 2.682 2.687

2.728 2.732 2.772 2.776 2.815 2.819 2.856 2.860 2.896 2.900

2.493 2.546 2.596 2.645 2.692

2.499 2.551 2.601 2.650 2.696

2.741 2.785 2.827 2.868 2.908

2.947 2.985 3.022 3.058 3.093

3.128 3.162 3.195

2.504 2.556 2.606 2.654 2.701

2.509 2.561 2.566 2.611 2.616 2.659 2.664 2.705 2.710

20.0 21.0 22.0 23.0 24.0

2.714 2.719 2.723 2.759 2.763 2.768 2.802 2.806 2.811 2.844 2.648 2.852 2.884 2.888 2.892

2.924 2.928 2.932 2.962 2.966 2.970

2.737 2.781 2.823 2.864 2.904

2.746 2.750 2.755 4 2.789 2.794 2.798 2.831 2.836 2.840 2.872 2.876 2.880 2.912 2.916 2.920

25.0 26.0 27.0 28.0 29.0

2.936 2.974

3.004 3.040

3.000 3.037 3.072

3.107 3.141 3.175 3.208 3.240

3.271 3.302 3.332

3.007 3.044

3.011 3.047

2.951 2.989 3.026 3.062

2.955 2.959 2.993 2.996

3.076 3.079 3.083

2.940 2.978 3.015 3.051 3.086

3.121 3.155 3.188 3.220 3.252

2.943 2.981 3.018 3.055 3.090 3.097

3.131 3.165 3.198

3.029 3.033 3.065 3.069 3.100 3.104

30.0 31.0 32.0 33.0 34.0

3.111 3.114 3.118 3.145 3.148 3.151 3.178 3.181 3.185 3.211 3.214 3.217 3.243 3.246 3.249

3.124 3.158 3.191 3.224 3.255

3.135 3.138 3 3.168 3.171 3.201 3.204

3.227 3.230 3.233 3.236 3.259 3.262 3.265 3.268

35.0 36.0 37.0 38.0 39.0

3.362 3.391

3.274 3.277 3.280 3.283 3.287 3.290 3.293 3.296 3.299 3.305 3.308 3.311 3.314 3.317 3.320 3.323 3.326 3.329 3.335 3.338 3.341 3.344 3.347 3.350 3.353 3.356 3.359 3.365 3.368 3.371 3.374 3.377 3.380 3.382 3.385 3.388 3.394 3.397 3.400 3.403 3.406 3.409 3.411 3.414 3.417

-=14&%59aI nd I/ 3- x=0 592794


1-16 PETROLEUM ENGINEERING HANDBOOK


3

3.428 3.457

5 - - 3.434 3.462

6 8 Average

9 Difference N 0 1 2 - - 7

3.440 3.468

40.0 41.0 42.0 43.0 44.0

3.420 3.423 3.426 3.448 3.451 3.454

45.0 46.0 47.0 48.0 49.0

3.479 3.506 3.533

3.560 3.586 3.611

3.482 3.484 3.509 3.512 3.536 3.538

3.637 3.662

3.562 3.566 3.614 3.639 3.664

50.0 51.0 52.0 53.0 54.0

3.476 3.503 3.530

3.557 3.583 3.609 3.634 3.659

3.684 3.708 3.733 3.756 3.780

3.686 3.689 3.691 3.694 3.696 3.711 3.713 3.716 3.718 3.721 3.735 3.737 3.740 3.742 3.744 3.759 3.761 3.763 3.766 3.768 3.782 3.784 3.787 3.789 3.791

55.0 56.0 57.0 58.0 59.0

60.0 61.0 62.0 63.0 64.0

3.803 3.805 3.826 3.828 3.849 3.851 3.871 3.873 3.893 3.895

3.808 3.810 3.812 3.814 3.817 3.819 3.630 3.833 3.835 3.837 3.839 3.842

3.915 3.917

3.853 3.855 3.857 3.875 3.878 3.880 3.897 3.900 3.902

3.919 3.921 3.924 3.936 3.939 3.941 3.943 3.945 3.958 3.960 3.962 3.964 3.966

65.0 66.0 67.0 68.0 69.0

3.979 4.000

4.021 4.041 4.062 4.082 4.102

4.121 4.141

3.981 4.002

4.023 4.043 4.064 4.084 4.104

70.0 71.0 72.0 73.0 74.0

4.160 4.179 4.198

4.217 4.236 4.254 4.273 4.291

4.309

4.123 4.143 4.162 4.181 4.200

3.983 3.985 3.987 4.004 4.006 4.008

4.025 4.027 4.029 4.045 4.047 4.049 4.066 4.068 4.070 4.086 4.088 4.090 4.106 4.108 4.109

4.125 4.127 4.129 4.145 4.147 4.149 4.164 4.166 4.168 4.170 4.183 4.185 4.187 4.189

75.0 76.0 77.0 78.0 79.0

4.219 4.238 4.256 4.274 4.293

80.0 81.0 82.0 83.0 84.0

4.311

4.202 4.204 4.206 4.208 4.210 4.212 4.213 4.215

4.221 4.223 4.225 4.227 4.228 4.230 4.232 4.234 4.240 4.241 4.243 4.245 4.247 4.249 4.251 4.252 4.258 4.260 4.262 4.264 4.265 4.267 4.269 4.271 4.278 4.278 4.280 4.282 4.284 4.285 4.287 4.289 4.294 4.296 4.298 4.300 4.302 4.303 4.305 4.307

4.312 4.314 4.316 4.318 4.320 4.321 4.323 4.325 4.327 4.329 4.330 4.344 4.346 4.348 4.362 4.364 4.366 4.380 4.381 4.383

85.0 4.397 4.399 4.400 86.0 4.414 4.416 4.417 87.0 4.431 4.433 4.434 88.0 4.448 4.450 4.451 89.0 4.465 4.466 4.468

90.0 4.481 4.483 4.485 91.0 4.498 4.500 4.501 92.0 4.514 4.516 4.518 93.0 4.531 4.532 4.534 94.0 4.547 4.548 4.550

95.0 4.563 4.565 4.566 96.0 4.579 4.580 4.582 97.0 4.595 4.596 4.598 98.0 4.610 4.612 4.614 99.0 4.626 4.628 4.629

3.431 3.459

3.437 3.465

3.443 3.471

3.495 3.498 3.522 3.525 3.549 3.552

3.445 3 3.473 3.501 3.528 3.554

3.575 3.578 3.580 3.601 3.604 3.606 3.627 3.629 3.632

3.487 3.514 3.541

3.567 3.593 3.619

3.565 3.591 3.616 3.642 3.644 3.867 3.669

3.490 3.517 3.544

3.570

3.493 3.520 3.546

3.573 3.599 3.624

3.596 3.622 3.647 3.672

3.649 3.652 3.654 3.657 2 3.674 3.677 3.679 3.682

3.699 3.723 3.747

3.701 3.725 3.749 3.773 3.796

3.704 3.706 3.728 3.730 3.752 3.754

3.770 3.794

3.775 3.798

3.821 3.844 3.866 3.889 3.911

3.932

3.777 3.801

3.824 3.846 3.869 3.891 3.913

3.934

3.860 3.882 3.904

3.926 3.947 3.968 3.990 4.010

4.031 4.051 4.072 4.092 4.111

4.131 4.151

3.862 3.864 3.884 3.886 3.906 3.908

3.928 3.930 3.949 3.951 3.954 3.956 3.971 3.973 3.975 3.977 3.992 4.012

4.033 4.053 4.074 4.094 4.113

3.994 3.996 3.998 4.015 4.017 4.019

4.035 4.037 4.039 4.055 4.058 4.060 4.076 4.078 4.080 4.096 4.098 4.100 4.115 4.117 4.119

4.135 4.137 4.139 4.154 4.156 4.158 4.174 4.176 4.177 4.193 4.195 4.196

4.133 4.152 4.172 4.191

4.332 4.334 4.336 4.337 4.339 4341 4.350 4.352 4.353 4.355 4.357 4.359

4.343 4.360 4.370 4.395

4.367 4.369 4.371 4.373 4.365 4.366 4.388 4.390

4.402 4.404 4.419 4.421 4.436 4.438 4.453 4.455 4.470 4.471

4.486 4.488 4.503 4.505

4.405 4.423 4.440 4.456 4.473

4.490 4.506

4.407 4.424 4.441 4.458 4.475

4.491 4.508

4.374 4.392

4.409 4.426 4.443 4.460 4.476

4.493 4.509

4.376 4.393

4.411 4.412 4.428 4.429 4.445 4.446 4.461 4.463 4.478 4.480

4.519 4.521 4.523 4.524 4.526 4.536 4.537 4.539 4.540 4.542

4.555 4.556 4.558

4.495 4.496 4.511 4.513 4.527 4.529 4.544 4.545 4.560 4.561

4.571 4.572 4.574 4.576 4.577 4.587 4.588 4.590 4.592 4.593 4.603 4.604 4.608 4.607 4.609 4.618 4.620 4.621 4.623 4.625 4.634 4.635 4.637 4.638 4.640

4.552 4.553

4.568 4.569 4.584 4.585 4.599 4.601 4.615 4.617 4.631 4.632

xc 1 46459 and II a- T = 0 682784



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

Average Difference

15 14 13

12

11

10

9

8

7

6

N

10 11 12 13 14

15 16 17 18 19

20 21 22 23 24

Cube roots of numbers from 100.0 to 499.0 4.642 4.657 4.672 4.688 4.703 4.718 4.733 4.747 4.762 4.777 4.791 4.806 4.820 4.932 4.946 4.960 5.066 5.079 5.092 5.192 5.205 5.217

5.313 5.325 5.337 5.429 5.440 5.451

4.835 4.849 4.863 4.877 4.891 4.905 4.919 4.973 4.987 5.000 5.013 5.027 5.040 5.053 5.104 5.117 5.130 5.143 5.155 5.168 5.180 5.229 5.241 5.254 5.266 5.278 5.290 5.301

5.540 5.550 5.561 5.646 5.657 5.667

5.346 5.463

6.162

5.360 5.474 5.583

6.249

5.888 5.789

5.887 5.981 6.073

5.372 5.363 5.395 5.485 5.496 5.507

5.406 5.418 5.518 5.529 5.625 5.636 5.729 5.739

5.749 5.759

5.848 5.858 5.944 5.953 6.037 6.046 6.127 6.136 6.214 6.223

5.769

5.867 5.963 8.055

5.572 5.677 5.779

6.145 8.232

5.877 5.972 6.064 6.153 6.240

5.593

6.171

5.604 5.698

6.180

5.708 5.799

8.257

5.809

6.266

5.896 5.906 5.991 6.000 6.082 6.091

5.815 5.718 5.819

5.915 6.009 6.100 6.188 6.274

25 26

;i 29

30 31 32 33 34

35 36 37 38 39

40 41 42 43 44

45 46 47 48 49

6.300 6.308 6.316

5.82(

5.92! 6.Oli 6.109 6.197 6.283

6.383 6.391 6.399 6.463 6.471 6.479 6.542

7.884

6.619

6.694 6.768 6.840 6.901 6.980

7.047 7.114 7.179 7.243 7.306

7.368 7.429 7.489 7.548 7.606

7.663 7.719 7.775 7.830

6.550 6.627

6.702 6.775

6.558 6.634

6.325 6.407 6.487 6.565 6.642

6.847 6.917 6.986

7.054 7.120 7.186

6.709 6.717 6.782 6.790 6.854 6.924 6.993

7.061 7.127 7.192

6.333 6.415 6.495 6.573 6.649

6.724 6.797 6.868 6.938 7.007

7.074 7.140 7.205

6.341 6.423 6.503 6.581 6.657

6.731 6.804 6.875 6.945 7.014

7.081

6.350 8.431 6.511 8.589 8.664

6.739 6.811 8.882 6.952 7.020

7.087

6.358 6.366 6.439 6.447 6.519 6.527 6.596 6.604 6.672 6.679

5.838

5.934 8.028 6.118 6.206 6.291

6.374 6.455 6.534 6.611 6.667

6.861 6.931 7.000

7.067 7.133 7.198 7.262 7.325

6.746 6.753 6.761 6.818 6.826 6.833

7.147 7.153 7.211 7.218

6.889 6.896 8.903 6.959 6.966 6.973 7.027 7.034 7.041

7.094 7.101 7.107

7.250 7.256 7.312 7.319

7.268 7.331

7.166 7.230

7.374 7.380 7.386 7.393 7.435 7.441 7.447 7.453 7.495 7.501 7.507 7.513 7.554 7.560 7.565 7.571 7.612 7.617 7.623 7.629

7.275 7.281 7.337 7.343

7.399 7.405 7.459 7.465 7.518 7.524 7.577 7.563 7.635 7.640

7.691 7.697 7.747 7.753 7.802 7.808 7.857 7.862 7.910 7.916

7.160 7.224 7.287 7.350

7.411 7.471 7.530 7.589 7.646

7.703 7.758 7.813 7.868 7.921

7.294 7.356

7.417 7.477 7.536 7.594 7.652

7.708 7.764 7.819 7.873 7.926

7.173 7.237 7.300 7.362

7.669

7.889

7.674 7.725

7.894

7.731 7.780 7.786 7.835 7.841

7.680 7.686 7.736 7.742

7.423 7.463 7.542 7.600 7.657

7.791 7.797 7.846 7.851 7.900 7.905

Cube roots of numbers from 500.0 to 1.000.0

7.714 7.769 7.824 7.878 7.932

50 7.937 7.942 7.948 51 7.990 7.995 8.000 52 8.041 8.047 8.052 53 8.093 8.098 8.103 54 8.143 8.148 8.153

55 8.193 8.198 8.203 56 8.243 8.247 8.252 57 8.291 8.296 8.301 58 8.340 8.344 a.349 59 8.387 8.392 8.397

7.953 7.958 7.963 7.969 8.005 8.010 8.016 8.021 8.057 8.062 8.067 8.072 8.108 8.113 8.118 8.123 8.158 8.163 8.168 8.173

8.208 8.213 8.218 8.223 8.257 8.262 8.267 8.272

7.974 8.026 8.077 8.128 8.178

7.979 8.031 8.082 8.133 8.183

7.984 8.036 8.088 8.138 8.188

8.228 8.277

8.233 8.282 8.330 8.378

8.238 8.286

8.306 8.311 8.316 8.320 8.325 8.354 8.359 8.363 8.368 8.373

8.335 8.382

60 61 62

6":

8.434 8.439 8.444 8.481 8.486 8.490 8.527 8.532 8.536

8.401 8.406 8.411 8.416 8.420 8.425 8.430

8.448 8.453 8.458 8.462 8.487 8.472 8.476 8.495 6.499 8.504 8.509 8.513 6.518 8.522 8.541 8.545 8.550 8.554 8.559 8.564 8.568

8.573 6.577 8.582 8.586 8.591 8.595 8.600 8.604 8.609 8.613 8.618 8.622 8.627 8.631 8.636 8.640 8.645 8.649 8.853 8.658

65 8.662 8.667 8.671 8.676 8.680 8.685 8.689 8.693 8.698 8.702 66 8.707 8.711 6.715 8.720 8.724 6.729 8.733 8.737 8.742 8.746 67 8.750 8.755 8.759 8.763 8.768 8.772 8.776 8.781 8.785 8.789 68 8.794 8.798 8.802 8.607 8.811 8.815 8.819 8.824 8.828 8.632 69 8.837 8.841 8.845 8.849 8.854 8.858 8.862 8.866 8.871 8.875

?-i= 1 46459 and l/x= 0.662764

(contmed on next page)



N a 1 2 3 4 5 6 7 6 Average

9 Difference

70 8.679 8.683 71 6.921 6.925 72 8.963 8.967 73 9.004 9.008 74 9.045 9.049

9.066 9.090 9.126 9.130 9.166 9.170

6.867 6.929 8.971 9.012 9.053

9.094 9.134 9.174

8.892 8.896 8.900 8.904 8.909 6.913 8.917 4 8.934 6.938 8.924 8.946 8.950 6.955 8.959 6.975 6.979 8.984 8.968 8.992 6.996 9.000 9.016 9.021 9.025 9.029 9.033 9.037 9.041 9.057 9.061 9.065 9.069 9.073 9.078 9.062

75 76 77 76 79

9.205 9.209 9.213 9.244 9.248 9.252

9.098 9.102 9.106 9.110 9.114 9.118 9.138 9.142 9.146 9.150 9.154 9.158 9.178 9.182 9.185 9.189 9.193 9.197 9.217 9.221 9.225 9.229 9.233 9.237 9.256 9.260 9.264 9.268 9.272 9.275

9.122 9.162 9.201 9.240 9.279

BO 9.283 9.267 9.291 9.295 9.299 9.302 9.306 9.310 9.314 9.318 81 9.322 9.326 9.329 9.333 9.337 9.341 9.345 9.348 9.352 9.356 82 9.360 9.364 9.366 9.371 9.375 9.379 9.383 9.366 9.390 9.394 83 9.396 9.402 9.405 9.409 9.413 9.417 9.420 9.424 9.428 9.432 64 9.435 9.439 9.443 9.447 9.450 9.454 9.456 9.462 9.465 9.469

85 86 87 88 69

9.473 9.476 9.510 9.513

90 91 92 93 94

9.546 9.550 9.583 9.586 9.619 9.623

9.655 9.656

9.480 9.484 9.486 9.491 9.495 9.499 9.502 9.517 9.521 9.524 9.528 9.532 9.535 9.539 9.554 9.557 9.561 9.565 9.568 9.572 9.576 9.590 9.594 9.597 9.601 9.605 9.608 9.612 9.626 9.630 9.633 9.637 9.641 9.644 9.646

9.506 9.543

9.662

9.761 9.796

9.630 9.865 9.899 9.933 9.967

10.00

9.666 9.701 9.736 9.771 9.806

9.669 9.673 9.676 9.683 9.691 9.694 9.698 9.726 9.729 9.733

9.705 9.740

9.706 9.743

9.712 9.747

9.680 9.715 9.750 9.765 9.620

9.579 9.615 9.651

9.687 9.719 9.754

9.722 9.758

9.764 9.766 9.799 9.803

9.775 9.776 9.762 9.810 9.813 9.617

9.789 9.792 9.824 9.627

95 96 97 96 99

9.634 9.637 9.841 9.844 9.846 9.651 9.655 9.858 9.861 9.868 9.872 9.875 9.879 9.882 9.885 9.669 9.892 9.696 9.902 9.906 9.909 9.913 9.916 9.919 9.923 9.926 9.930 9.936 9.940 9.943 9.946 9.950 9.953 9.956 9.960 9.963 9.970 9.973 9.977 9.980 9.983 9.987 9.990 9.993 9.997

100

CUBE ROOTS OF CERTAIN FRACTIONS

%N% N% N% N -- - - - - ~ % N % - - 0.7469 %6 0.8255 0.7937 3/5 0.8434 % 0.8296 '/9 0.4807 %2

0.6934 % 0.9263 % 0.6939 2/9 0.6057 %? 0.6355 "/lj 0.6826 0.9714 '3/16 0.9331 0.3969 's6 0.9767 0.5724 '/a2 0.3150 0.6786 '164 0.2500 0.7591 %o 0.2714

.'- 0.8736 '/, 0.5503 % 0.9499 4/ 0.7631 "A2 0.6300 y6 0.9410 '/6 0.5000 79 0.6221 %6

0.9086 '/, 0.5226 %6 0.7211 x3 0.9196 %6

0.5648 +$ 0.6566 6/S 0.8550 Y9 0.9615 %6

0.7368 % 0.7539 X8 0.9565 'A2 0.4368 %6

K= 146459 and li r ~=0.682764


Explanation of Table

N 0 1 2 3 4 5 6 7 6 9

1.0 2.0 3.0 4.0 4.0

1.000 1.154 1.315 1.482 1.657 1.837 2.024 2.626 3.043 3.263 3.466 3.718 3.953 4.192 5.196 5.458 5.724 5.995 6.269 6.546 6.831 8.000 8.302 8.607 8.917 9.230 9.546 9.866

2.217 4.437 7.117

10.190 10.19

2.415 2.619 4.685 4.939 7.408 7.702

10.85

5.0 11.18 6.0 14.70 7.0 18.52 8.0 22.63 9.0 27.00

11.86 15.44 19.32 23.46 27.90

12.90 16.57 20.54 24.76 29.28

10.0 31.62 11.0 36.48 12.0 41.57 13.0 46.87 14.0 52.38

32.56 37.48 42.61 47.96 53.51

34.02 39.00 44.19 49.60 55.21

14.33 16.12 22.20 26.55 31.15

35.99 41.05 46.33 51.82 57.51

15.0 16.0 17.0 18.0 19.0

20.0 21.0 21.0 22.0 23.0 24.0

58.09 64.00 70.09 76.37 82.82

89.44 96.23

11.52 15.07 16.92 23.05 27.45

32.10 36.98 42.09 47.41 52.95

58.68 64.60 70.71 77.00 83.47

90.11 96.92

59.26 65.20 71.33 77.64 84.13

90.79 97.61

12.20 12.55 15.61 16.19 19.72 20.13 23.91 24.35 26.36 28.82

33.06 33.54 37.99 38.49 43.14 43.66 48.50 49.05 54.08 54.64

59.85 60.43 65.81 66.41 71.96 72.58 78.26 76.93 84.79 85.45

91.46 92.14 98.30 99.00

61.02 67.02 73.21 79.57 86.11

92.82 99.69

13.61 17.34 21.37 25.66 30.21

35.00 40.02 45.26 50.71 56.36

62.21 68.25 74.47 60.87 87.44

94.18

10.52

13.97 17.73 21.78 26.11 30.68

35.49 40.53 45.79 51.26 56.94

62.80 66.66 75.10 81.51 88.10

94.86

63.40 69.48 75.73 82.17 68.77

95.55

103.2 103.9 104.6 105.3 106.0 106.7 110.3 111.0 111.7 112.5 113.2 113.9 117.6 118.3 119.0 119.8 120.5 121.3

102.5 109.6 116.8 124.3

25.0 26.0 27.0 28.0 29.0

30.0 31.0 32.0 33.0 34.0

35.0 36.0 37.0 38.0 39.0

40.0 41.0 42.0 43.0 44.0

45.0 46.0 47.0 48.0 49.0

125.0 125.8 126.5 127.3 128.0 128.8 132.6 133.3 134.1 134.9 135.6 136.4 140.3 141.1 141.9 142.6 143.4 144.2 148.2 149.0 149.8 150.5 151.3 152.1 156.2 157.0 157.8 158.6 159.4 160.2

101.8 108.9 116.1 123.5

131.0 138.7 146.6 154.6 162.7

164.3 165.1 172.6 173.4 181.0 181.9 189.6 190.4 198.3 199.1

207.1 216.0 225.1 234.2 243.6

253.0 262.5 272.2 282.0 291.9

301.9 312.0 322.2 332.6 343.0

208.0 216.9 226.0 235.2 244.5

166.0 174.3 182.7 191.3 200.0

208.8 217.8 226.9 236.1 245.4

166.8 167.6 168.4 175.1 176.0 176.8 183.6 184.4 185.3 192.2 193.0 193.9 200.9 201.8 202.6

209.7 210.6 211.5 218.7 219.6 220.5 227.6 226.7 229.6 237.0 238.0 238.9 246.4 247.3 246.3

255.6 256.8 257.7 265.4 266.4 267.3 275.1 276.1 277.1 264.9 285.9 286.9 294.9 295.9 296.9

304.9 305.9 306.9 315.0 316.1 317.1 325.3 326.3 327.4 335.7 336.7 337.8 346.2 347.2 348.3

13.25 16.96 20.95 25.22 29.74

34.51 39.51 44.73 50.15 55.79

61.62 67.63 73.84 60.22 66.77

93.50 100.38 100.4 107.4 114.6 122.0

129.5 137.2 145.0 153.0 161.9

169.3 177.6 186.1 194.8 203.5

212.4 221.4 230.6 239.8 249.2

258.7 266.3 278.0 267.9 297.9

307.9 318.1 326.4 336.6 349.3

359.9 370.7 381.5 392.4 403.4

101.1 108.2 115.4 122.8

130.3 138.0 145.8 153.8 161.9

170.1 178.5 187.0 195.6 204.4

213.3 222.3 231.5 240.8 250.1

259.7 269.3 279.0 288.9 298.9

308.9 319.1 329.4 339.9 350.4

170.9 179.3 187.9 196.5 205.3

214.2 223.2 232.4 241.7 251.1

131.8 139.5 147.4 155.4 163.5

171.8 180.2 188.7 197.4 206.2

215.1 224.2 233.3 242.6 252.0

253.9 254.9 263.5 264.5 273.2 274.1 283.0 283.9 292.9 293.9

302.9 303.9 313.0 314.0 323.2 324.3 333.6 334.6 344.0 345.1

260.6 261.6 270.2 271.2 280.0 281.0 269.9 290.9 299.9 300.9

310.0 311 .o 320.2 321.2 330.5 331.5 340.9 342.0 351.4 352.5

50.0 353.6 354.6 355.7 356.7 357.8 358.9 51.0 364.2 365.3 366.4 367.4 366.5 369.6 52.0 375.0 376.1 377.1 376.2 379.3 380.4 53.0 365.6 386.9 388.0 389.1 390.2 391.3 54.0 396.8 397.9 399.0 400.1 401.2 402.3

361.0 362.1 363.1 371.7 372.6 373.9 362.6 383.7 384.6 393.5 394.6 395.7 404.6 405.7 406.8

55.0 407.9 409.0 410.1 411.2 412.3 413.5 414.6 415.7 416.8 418.0 56.0 419.1 420.2 421.3 422.4 423.6 424.7 425.8 426.9 428.1 429.2 57.0 430.3 431.5 432.6 433.7 434.9 436.0 437.1 436.3 439.4 440.6 58.0 441.7 442.9 444.0 445.1 446.3 447.4 446.6 449.7 450.9 452.0 59.0 453.2 454.3 455.5 456.6 457.8 459.0 460.1 461.3 462.4 463.6

TABLEl.5-THREE-HALVES POWERS OFNUMBERS

Average Difference

183 237 280 313 33

35 38 41 44 46

49 51 53 55 57

59

:A 65 66

66 69 7 7 7 7

6 0 9 9 9

9 9 9

i

10 IO 10 10 10

IO 10 10 10 11

11 11 11 11 11

11 11 11 11 12

This table gives N3” from N= 1 to N= 100. Moving the decimal point two places in N requires moving it three places in body of table. For example, (7.23)3’” = 19.44, (723.)3” = 19,440, (0.0723)3’2 =0.01944, (72.3)3’2 =614.8, (7,230)3’ =614,800, and (0.723)3'2 =0.6148.

Used inversely, this table gwes M2’3 from M= 1 to M=l,OOO. For example, (0 6148)z3 =0.7230.

(conhnued on next page)


TABLE 1.5-THREE-HALVES POWERS OF NUMEERS(continued)

N 0 -- 66.0 464.8 61.0 476.4 62.0 488.2 63.0 500.0 64.0 512.0

65.0 524.0 66.0 536.2 67.0 548.4 68.0 560.7 69.0 573.2

70.0 585.7 71.0 598.3 72.0 610.9 73.0 623.7 74.0 636.6

75.0 649.5 76.0 662.6 77.0 6757 78.0 688.9 79.0 702.2

60.0 715.5 81.0 729.0 62.0 742.5 83.0 756.2 84.0 769.9

85.0 783.7 86.0 797.5 87.0 811.5 88.0 825.5 89.0 839.6

90.0 853.8 91.0 868.1 92.0 882.4 93.0 896.9 94.0 911.4

95.0 925.9 96.0 940.6 97.0 955.3 98.0 970.2 99.0 985.0

100.0 l,ooo.o

1 2 3 4 5 6 7 8 9

465.9 467.1 477.6 478.7 489.4 490.6 501.2 502.4 513.2 514.4

472.9 474.1 475.3 484.6 485.8 487.0 496.5 497.7 498.9 508.4 569.6 510.8 520.4 521.6 522.8

525.3 537.4 549.6 562.0 574.4

586.9 599.5 612.2 625.0 637.9

650.8 663.9 677.0 690.2 703.5

526.5 538.6 550.9 563.2 575.7

588.2 600.8 613.5 626.3 639.2

652.1 665.2 678.3 691.5 704.8

468.2 469.4 470.6 471.7 479.9 481.1 482.3 483.5 491.7 492.9 494.1 495.3 503.6 504.8 506.0 507.2 515.6 516.8 518.0 519.2

527.7 528.9 530.1 531.3 539.8 541.1 542.3 543.5 552.1 553.3 554.6 555.8 564.5 565.7 566.9 568.2 576.9 578.1 579.4 580.8

589.4 590.7 591.9 593.2 602.1 603.3 604.6 605.9 614.8 616.0 617.3 618.6 627.6 628.8 630.1 631.4 640.4 641.7 643.0 644.3

653.4 654.7 656.0 657.3 686.5 867.8 669.1 670.4 679.6 680.9 682.3 683.6 692.9 694.2 695.5 696.8 706.2 707.5 708.8 710.2

719.6 720.9 722.3 723.6 733.1 734.4 735.8 737.1 746.6 748.0 749.3 750.7 760.3 761.6 763.0 764.4 774.0 775.4 776.8 778.1

787.8 789.2 790.6 792.0 801.7 803.1 804.5 805.9 815.7 817.1 818.5 819.9 829.7 831.1 832.6 834.0 843.9 845.3 848.7 848.1

858.1 859.5 860.9 862.4 872.4 873.8 875.2 876.7 886.8 888.2 889.6 891.1 901.2 902.7 904.1 905.6 915.7 917.2 918.6 920.1

930.3 931.8 933.3 934.7 945.0 946.5 948.0 949.4 959.8 961.3 962.7 964.2 974.6 976.1 977.6 979.1 989.5 991.0 992.5 994.0

532.5 544.7 557.0 569.4 581.9

533.8 546.0 558.3 570.7 583.2

595.7 608.4 621.2 634.0 646.9

659.9 673.0 686.2 699.5 712.9

535.0 547.2 559.5 571.9 584.4

594.5 607.1 619.9 632.7 645.6

658.6 671.7 684.9 898.2 711.5

597.0 609.7 622.4 635.3 648.2

661.2 674.4 687.6 700.8 714.2

716.9 718.2 730.4 731.7 743.9 745.3 757.5 758.9 771.2 772.6

726.3 739.8 753.4 767.1 780.9

785.0 786.4 798.9 800.3 812.9 814.3 826.9 828.3 841.0 842.5

725.0 738.5 752.1 765.8 779.5

793.4 807.3 821.3 835.4 849.5

794.8 808.7 822.7 836.8 851.0

855.2 856.7 869.5 870.9 883.9 885.3 898.3 899.8 912.8 914.3

863.8 865.2 878.1 879.6 892.5 894.0 907.0 908.5 921.6 923.0

727.7 741.2 754.8 768.5 782.3

796.1 810.1 824.1 838.2 852.4

886.7 881.0 895.4 909.9 924.5

927.4 928.9 942.1 943.5 956.8 958.3 971.6 973.1 986.5 988.0

936.2 937.7 939.1 950.9 952.4 953.9 985.7 967.2 968.7 980.6 982.1 983.5 995.5 997.0 998.5

AUXILIARY TABLE OF TWO-THIRDS POWERS AND THREE-HALVES POWERS

(to assist in locating the decimal point)

For complete table of three-halves powers, see preceding. That table, used inversely, provides a completetableoftwc-thirds powers.

N /‘J”( = G) N”(= d@)

0.0001 0.001 0.01 0.1 1.0

10.0 100.0

l,ooo.o 10,000.0

0.002154 0.000001 0.01 0.00003162 0.0464 0.001 0.2154 0.03162278 1.0 1.0 4.64 31.62278

21.54 l,ooo.o 100.0 31,622.78 464.16 1,000,000.0

Average Difference

12 12 12 12 12

12 12 12 12 13

13 13 13 13 13

13 13 13 13 13

13 14 14 14 14

14 14 14 14 14

14 14 14 15 15

E 15 15 15

MATHEMATICAL TABLES & UNITS 8 SYSTEMS OF WEIGHTS & MEASURES l-21

Te

o

1 ‘ABLE 1.6 --RECIPR IOCALS 0 IF NUMBE :RS

N” 0 1 2 3 4 5 6 7 8 9 Average Difference

1.00 1 .Ol 1.02 1.03 1.04

0.9901 0.9804 0.9709 0.9615

0.9990 0.9980 0.9970 0.9960 0.9950 0.9940 0.9930 0.9921 0.9891 0.9881 0.9872 0.9862 0.9852 0.9643 0.9833 0.9823

0.9747 0.9653 0.9560

1.05 1.06 1.07 1.08 1.09

1.10 1.11 1.12 1.13 1.14

0.9524 0.9434 0.9346 0.9259 0.9174

0.9794 0.9785 0.9775 0.9786 0.9756 0.9699 0.9690 0.9681 0.9671 0.9662 0.9606 0.9597 0.9588 0.9579 0.9569

0.9515 0.9506 0.9497 0.9488 0.9479 0.9425 0.9416 0.9407 0.9398 0.9390 0.9337 0.9328 0.9320 0.9311 0.9302 0.9251 0.9242 0.9234 0.9225 0.9217 0.9186 0.9158 0.9149 0.9141 0.9132

0.9470 0.9381 0.9294 0.9208 0.9124

0.9911 0.9814 0.9718 0.9625 0.9533

0.9443 0.9355 0.9268 0.9183 0.9099

0.9091 0.9083 0.9074 0.9066 0.9058 0.9050 0.9042 0.9009 0.9001 0.8993 0.8985 0.8977 0.8969 0.8961 0.8929 0.8921 0.8913 0.8905 0.8897 0.8889 0.8881 0.8850 0.8842 0.8834 0.8826 0.8818 0.8811 0.8803 0.8772 0.8784 0.8757 0.8749 0.8741 0.8734 0.8726

0.9737

0.8953

0.9728

0.8945

0.9643

0.8873

0.9634 0.9551

0.8865

0.9542

0.9461 0.9452 0.9372 0.9363 0.9285 0.9276 0.9200 0.9191 0.9116 0.9107

0.9033 0.9025 0.8937 0.8857

0.9017

0.8795 0.8787 0.8780 0.8718 0.8711 0.8703

1.15 0.8696 0.8688 1.16 0.8621 0.8613 1.17 0.8547 0.8540 1.18 0.8475 0.8487 1.19 0.8403 0.8398

0.8681 0.8673 0.8606 0.8598 0.8532 0.8525 0.8460 0.8453 0.8389 0.8382

0.8319 0.8313

0.8666 0.8591 0.8518 0.8446 0.8375

0.8658 0.8584 0.8511 0.8439 0.8388

0.8299

0.8651 0.8643 0.8636 0.8628 0.8576 0.8569 0.8562 0.8554 0.8503 0.8496 0.8489 0.6482 0.8432 0.8425 0.8418 0.8410 0.8361 0.8354 0.8347 0.8340

1.20 1.21 1.22 1.23 1.24

0.8333 0.8328 0.8258 0.8190 0.8123 0.8058

0.8306 0.8292 0.8264 0.8197

0.8251 0.8183

0.8237 0.8230 0.8170 0.8163

0.8130 0.8065

0.8000 0.7937 0.7874 0.7812 0.7752

0.7692 0.7634 0.7576 0.7519 0.7463

0.8117 0.8052

0.8244 0.8177 0.81 IO 0.8045

0.8285 0.8278 0.8271 0.8217 0.8210 0.8203 0.8150 0.8143 0.8137

0.8104 0.8097 0.8039 0.8032

1.25 1.26 1.27 1.28 1.29

1.30 1.31 1.32 1.33 1.34

0.7994 0.7987 0.7981 0.7974 0.7968 0.7930 0.7924 0.7918 0.7911 0.7905 0.7868 0.7862 0.7855 0.7849 0.7843 0.7806 0.7806 0.7794 0.7788 0.7782 0.7748 0.7740 0.7734 0.7728 0.7722

0.8224 0.8157 0.8091 0.8026

0.8084 0.8078 0.8071 0.8019 0.8013 0.8006

0.7686 0.7680 0.7675 0.7669 0.7683 0.7628 0.7822 0.7616 0.7610 0.7605 0.7570 0.7564 0.7559 0.7553 0.7547 0.7513 0.7508 0.7502 0.7496 0.7491 0.7457 0.7452 0.7446 0.7440 0.7435

0.7962 0.7955 0.7949 0.7899 0.7893 0.7886 0.7837 0.7831 0.7825 0.7776 0.7770 0.7764 0.7716 0.7710 0.7704

0.7657 0.7651 0.7645 0.7599 0.7593 0.7587 0.7541 0.7536 0.7530 0.7485 0.7479 0.7474 0.7429 0.7424 0.7418

0.7943 0.7880 0.7819 0.7758 0.7898

0.7639 0.7582 0.7524 0.7468 0.7413

1.36 0.7407 0.7402 0.7396 0.7391 0.7386 0.7380 0.7375 0.7369 0.7364 0.7358 1.36 0.7353 0.7348 0.7342 0.7337 0.7331 0.7326 0.7321 0.7315 0.7310 0.7305 1.37 0.7299 0.7294 0.7289 0.7283 0.7278 0.7273 0.7267 0.7262 0.7257 0.7252 1.38 0.7246 0.7241 0.7236 0.7231 0.7225 0.7220 0.7215 0.7210 0.7205 0.7199 1.39 0.7194 0.7189 0.7184 0.7179 0.7174 0.7168 0.7163 0.7158 0.7153 0.7148

1.40 1.41 1.42 1.43 1.44

0.7143 0.7092 0.7042 0.6993 0.6944

0.6897 0.6849 0.6803 0.6757 0.6711

0.6667 0.6623

0.7138 0.7087 0.7037 0.6988 0.6940

0.6892 0.6845 0.6798 0.6752 0.6707

0.6662 0.6618

0.7133 0.7082 0.7032 0.6983 0.6935

0.7107 0.7102 0.7057 0.7052 0.7008 0.7003 0.6959 0.6911

1.45 1.48 1.47 1.48 1.49

1 so 1.51 1.52 1.53 1.54

0.6887 0.8840 0.6793 0.6748 0.6702

0.6658 0.6614

0.7128 0.7123 0.7077 0.7072 0.7027 0.7022 0.6978 0.6974 0.6930 0.6925

0.6882 0.6878 0.8835 0.8831 0.6789 0.6784 0.6743 0.8739 0.6698 0.6693

0.6653 0.6649 0.6609 0.6805

0.7117 0.7067 0.7018 0.6969 0.6920

0.6873 0.6826 0.6780 0.8734 0.6689

0.6645 0.8601 0.6557 0.6515 0.6472

0.7112 0.7062 0.7013 0.6964 0.6916

0.6868 0.6821 0.6775 0.6729 0.6684

0.6640 0.8596

0.6863 0.6817 0.6770 0.6725 0.6680

0.6636 0.6592

0.7097 0.7047 0.6998 0.6949 0.6901

0.6854 0.6807 0.6761 0.6718 0.6671

0.6627 0.6583

0.6579 0.6575 0.6570 0.6566 0.6562 0.6536 0.6532 0.6527 0.6523 0.6519

0.6954 0.6906

0.6859 0.6812 0.6766 0.8720 0.6676

0.6631 0.8588

0.6494 0.6489 0.6485 0.6481 0.6477

0.6553 0.6549 0.6545 0.6540 0.6510 0.6506 0.6502 0.6498 0.6468 0.6464 0.6460 0.6456

- 10

-9

-8

-7

-6

-5

-4

Explanation of Table of Reciprocals

his table gives the values of i/N for values of N from 1 to 10, correct to four figures. (Interpolated values may be in rror by 1 in the fourth figure.) To find the reciprocal of a number N outside the range from 1 to 10, note that moving the decimal point any number f 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, l/3.217=0.3109, 1/3,217=0.0003109, and 1/0.003217=310.9.



TABLE 1.6--RECIPROCALS OF NUMBERS(continued)

0 1 Average

2 3 4 5 6 7 8 9 Difference -- N

1.55 0.6452 1.56 0.6410 1.57 0.8369 1.58 0.6329 1.59 0.6289

0.6447 0.6406 0.6365 0.6325

0.6443 0.6402 0.6361 0.6321 0.6281

0.6439 0.6435 0.6431 0.6427 0.6423 0.6396 0.6394 0.6390 0.6366 0.6362 0.6357 0.6353 0.6349 0.6345 0.6341 0.6317 0.6313 0.6309 0.6305 0.6301

0.6285 0.6277 0.6274 0.6270

0.6234 0.6231 0.6196 0.6192 0.6158 0.6154 0.6120 0.6116 0.6083 0.6079

0.6046 0.6042 0.6010 0.6006 0.5974 0.5970 0.5938 0.5935 0.5903 0.5900

0.5669 0.5865

0.6266 0.6262

0.6223 0.6164 0.6146 0.6109 0.6072

0.6035 0.5999

0.6418 0.6414 0.6376 0.6373 0.6337 0.6333 0.6297 0.6293 0.6258 0.6254

1.60 0.6250 0.6246 0.6242 0.6238 1.61 0.6211 0.6207 0.6203 0.6200 1.62 0.6173 0.6169 0.6165 0.6161 1.63 0.6135 0.6131 0.6127 0.6124 1.64 0.6098 0.6094 0.6090 0.6086

0.6227 0.6168 0.6150 0.6112 0.6075

0.6219 0.6215 0.6180 0.6177 0.6143 0.6139 0.6105 0.6101 0.6068 0.6064

1.65 1.66 1.67 1.66 1 .a9

0.6061 0.6024 0.5988 0.5952 0.5917

0.5882 0.5848 0.5814

0.6057 0.6020 0.5984 0.5949 0.5914

0.5879 0.5645 0.5811

0.6053 0.6050 0.6017 0.6013 0.5981 0.5977 0.5945 0.5942 0.5910 0.5907

1.70 1.71 1.72 1.73 1.74

0.5875 0.5872 0.5841 0.5638 0.5807 0.5804

0.5634 0.5831 0.5800 0.5797 0.5767 0.5764 0.5734 0.5731

0.6039 0.6002 0.5967 0.5931 0.5896

0.5862 0.5628 0.5794

0.6031 0.6028 0.5995 0.5992

0.5963 0.5959 0.5956 0.5926 0.5924 0.5921

0.5780 0.5747

0.5777 0.5744

0.5774 0.5741

0.5770 0.5737

0.5893 0.5889

0.5858 0.5855 0.5824 0.5621 0.5790 0.5787

0.5760 0.5757 0.5754 0.5727 0.5724 0.5721

1.75 0.5714 0.5711 0.5708 0.5705 0.5701 0.5698 0.5695 0.5692 0.5688 1.76 0.5682 0.5679 0.5675 0.5672 0.5669 0.5666 0.5663 0.5659 0.5656 1.77 0.5650 0.5647 0.5643 0.5640 0.5637 0.5634 0.5631 0.5627 0.5624 1.78 0.5618 0.5615 0.5612 0.5609 0.5605 0.5602 0.5599 0.5596 0.5593 1.79 0.5587 05583 0.5580 0.5577 0.5574 05571 0.5568 0.5565 0.5562

1.80 1.81 1.82 1.83 1.84

0.5556 0.5552 0.5549 0.5546 0.5543 0.5540 0.5537 0.5525 0.5522 0.5519 0.5516 0.5513 0.5510 0.5507 0.5495 0.5491 0.5488 0.5485 0.5482 0.5479 0.5464 0.5461 0.5459 0.5456 0.5453 0.5450 0.5435 0.5432 0.5429 0.5426 0.5423 0.5420

0.5534 0.5504 0.5473 0.5444 0.5414

0.5531 0.5501 0.5470 0.5441 0.5411

1.85 1.86 1.87 1.88 1.89

0.5405 0.5402 0.5400 0.5376 0.5373 0.5371 0.5348 0.5345 0.5342 0.5319 0.5316 0.5313 0.5291 0.5288 0.5285

1.90 0.5263 0.5260 0.5256 1.91 0.5236 0.5233 0.5230 1.92 0.5206 0.5206 0.5203 1.93 0.5181 0.5179 0.5176 1.94 0.5155 0.5152 0.5149

0.5397 0.5368 0.5339 0.5311 0.5263

0.5255 0.5227 0.5200 0.5173 0.5147

0.5394 0.5391 0.5365 0.5362 0.5336 0.5333 0.5308 0.5305 0.5280 0.5277

0.5385 0.5382 0.5356 0.5353 0.5326 0.5325 0.5299 0.5297 0.5271 0.5269

0.5252 0.5225 0.5198 0.5171 0.5144

0.5249 0.5222 0.5195 0.5166 0.5141

0.5476 0.5447 0.5417

0.5388 0.5359 0.5330 0.5302 0.5274

0.5247 0.5219 0.5192 0.5165 0.5139

0.5244 0.5241 0.5216 0.5214 0.5169 0.5167 0.5163 0.5160 0.5157 0.5136 0.5133 0.5131

1.95 0.5128 0.5126 0.5123 0.5120 1.98 0.5102 0.5099 0.5097 0.5094 1.97 0.5076 0.5074 0.5071 0.5068 1.98 0.5051 0.5048 0.5045 0.5043 1.99 0.5025 0.5023 0.5020 0.5018

0.5118 0.5092

0.5115

0.5066 0.5040

0.4975 0.4950 0.4739 0.4717 0.4525 0.4505 0.4329 0.4310

0.5089 0.5063 0.5038 0.5013 0.5015

0.5112 0.5110 0.5107 0.5105 0.5086 0.5084 0.5081 0.5079 0.5061 0.5058 0.5056 0.5053 0.5035 0.5033 0.5030 0.5028 0.5010 0.5006 0.5005 0.5003

2.0 2.1 2.2 2.3 2.4

0.5000 0.4762 0.4545 0.4348

0.4926 0.4902 0.4878 0.4854 0.4831 0.4808 0.4785 0.4695 0.4673 0.4651 0.4630 0.4808 0.4587 0.4566

2.5 2.6 2.7 2.8 2.9

0.4167 0.4149

0.4000 0.3984 0.3846 0.3831 0.3704 0.3690 0.3571 0.3559 0.3448 0.3438

0.3333 0.3322

0.4132

0.3966 0.3817 0.3676

0.4484 0.4464 0.4292 0.4274 0.4115 0.4098

0.3953 0.3937 0.3802 0.3788 0.3663 0.3650

0.4444 0.4425 0.4405 0.4386 0.4367 0.4255 0.4237 0.4219 0.4202 0.4184 0.4082 0.4065 0.4049 0.4032 0.4016

0.3922 0.3906 0.3891 0.3676 0.3861 0.3774 0.3759 0.3745 0.3731 0.3636 0.3623 0.3610 0.3597

0.3546 0.3534 0.3521 0.3509 0.3497 0.3484 0.3472 0.3425 0.3413 0.3401 0.3390 0.3378 0.3367 0.3356

3.0 3.1 3.2 3.3 3.4

0.3226 0.3215 0.3125 0.3115

0.3311 0.3300 0.3289 0.3279 0.3268 0.3257 0.3247 0.3236 0.3205 0.3195 0.3185 0.3175 0.3165 0.3155 0.3145 0.3135 0.3106 0.3096 0.3086 0.3077 0.3067 0.3058 0.3049 0.3040 0.3012 0.3003 0.2994 0.2985 0.2976 0.2967 0.2959 0.2950 0.2924 0.2915 0.2907 0.2899 0.2890 0.2882 0.2974 0.2865

0.3030 0.3021 0.2941 0.2933

-4

-3

-2

-24 -21 -20 -16 -17

-15 -14 -13 -12 -12

-11 -10 -10 -9 -8

0.5886

0.5851 0.5617 0.5784 0.5750 0.5718

0.5685 0.5653 0.5621 0.5590 0.5559

0.5528 0.5496 0.5467 0.5438 0.5408

0.5379 0.5350 0.5322 0.5294 0.5266

0.5238 0.5211 0.5164

0.3717 0.3564 0.3460 0.3344

l,*=O 316310 and lk?=0.367679

MATHEMATlCALTABLES&UNlTS & SYSTEMS OF WEIGHTS & MEASURES 1-23

TABLE 1.6-REC

3

:IPROCAI 3 OF NI UMBERS(< :ontinue 4

Average Difference N 9 0 1 2 4 5 6 a

3.5 0.2857 0.2849 0.2841 0.2833 0.2825 0.2817 0.2809 0.2801 3.6 0.2778 0.2770 0.2762 0.2755 0.2747 0.2740 0.2732 0.2725 3.7 0.2703 0.2695 0.2688 0.2681 0.2674 0.2667 0.2660 0.2653 3.6 0.2632 0.2625 0.2618 0.2611 0.2604 0.2597 0.2591 0.2584 3.9 0.2564 0.2558 0.2551 0.2545 0.2538 0.2532 0.2525 0.2519

4.0 0.2500 4.1 0.2439 4.2 0.2381 4.3 0.2326 4.4 0.2273

0.2494 0.2433 0.2375 0.2320

0.2488 0.2461 0.2427 0.2421 0.2370 0.2364 0.2315 0.2309 0.2262 0.2257

0.2475 0.2469 0.2463 0.2457 0.2415 0.2410 0.2404 0.2398

0.2268

0.2217 0.2169 0.2123 0.2079 0.2037

0.1996

0.2358 0.2353 0.2347 0.2342 0.2304 0.2299 0.2294 0.2288

0.2247 0.2242 0.2237 0.2252

0.2203 0.2155 0.2110 0.2066 0.2024

0.1964

0.2793 0.2717 0.2646 0.2577 0.2513

0.2451 0.2392 0.2336 0.2283 0.2232

4.5 4.6 4.7 4.8 4.9

0.2222 0.2174

5.0 5.1 5.2 5.3 5.4

0.2128 0.2083 0.2041

0.2000

0.2212 0.2208 0.2165 0.2160 0.2119 0.2114 0.2075 0.2070 0.2033 0.2026

0.1961 0.1957 0.1923 0.1919 0.1887 0.1883 0.1652 0.1848

0.1992 0.1953 0.1916 0.1880 0.1645

5.5 0.1818 0.1815 a.1812 5.6 0.1786 0.1783 0.1779 5.7 0.1754 0.1751 0.1748 5.8 0.1724 0.1721 0.1718 5.9 0.1695 0.1692 0.1669

6.0 6.1 6.2 6.3 8.4

0.1667 0.1664 0.1661 0.1656 0.1656 0.1653 0.1650 0.1647 0.1645 0.1639 0.1637 0.1634 0.1631 0.1629 0.1626 0.1623 0.1621 0.1618

0.1610 0.1608 0.1605 0.1603 0.1585 0.1582 0.1580 0.1577 0.1560 0.1558 0.1555 0.1553

6.5 6.6 6.7 6.8 6.9

0.1536 0.1534 0.1531 0.1529 0.1513 0.1511 0.1508 0.1506 0.1490 0.1488 0.1486 0.1484 0.1468 0.1466 0.1464 0.1462 0.1447 0.1445 0.1443 0.1441

7.0 7.1 7.2 7.3 7.4

0.1613 0.1587 0.1563

0.1538 0.1515 0.1493 0.1471 0.1449

0.1429 0.1408 0.1389 0.1370 0.1351

0.1427 0.1425 0.1422 0.1420 0.1406 0.1404 0.1403 0.1401 0.1387 0.1385 0.1383 o.t38i 0.1368 0.1366 0.1364 0.1362 0.1350 0.1348 0.1346 0.1344

7.5 0.1333 7.6 0.1316 7.7 0.1299 7.6 0.1262 7.9 0.1266

0.1330 0.1312 0.1295 0.1279 0.1263

8.0 6.1 a.2 6.3 8.4

9.5 6.6 8.7 8.8 8.9

0.1250 0.1235

0.1332 0.1314 0.1297 0.1260 0.1264

0.1248 0.1233 0.1216 0.1203 0.1189

0.1175 0.1161 0.1148 0.1135 0.1122

9.0 9.1 9.2 9.3 9.4

0.1220 0.1205 0.1190

0.1176 0.1163 0.1149 0.1136 0.1124

0.1111 0.1099 0.1087 0.1075 0.1064

0.1053 0.1042 0.1031 0.1020 0.1010

0.1110 0.1098 0.1086 0.1074 0.1063

0.1247 0.1232 0.1217 0.1202 0.1188

0.1174 0.1160 0.1147 0.1134 0.1121

0.1109 0.1096 0.1085 0.1073 0.1062

9.5 9.6 9.7 9.8 9.9

0.1052 0.1041 0.1030 0.1019 0.1009

0.1050 0.1049 0.1048 0.1047 0.1046 0.1040 0.1038 0.1037 0.1036 0.1035 0.1029 0.1028 0.1027 0.1026 0.1025 0.1018 0.1017 0.1016 0.1015 0.1014 0.1008 0.1007 0.1006 0.1005 0.1004

0.2786 0.2710 0.2639 0.2571 0.2506

0.2445 0.2387 0.2331 0.2278 0.2227

-8 -6 -7 -7 -6

-6 -6 -6 -5 -5

-5 -5 -4 -4 -4

-4

-3

-2

-1

0.1986

0.2198 0.2151 0.2105 0.2062 0.2020

0.2193 0.2188 0.2183 0.2179 0.2146 0.2141 0.2137 0.2132

0.1980

0.2101 0.2058 0.2016

0.1976 0.1938 0.1901 0.1866 0.1632

0.2096 0.2092 0.2088 0.2053 0.2049 0.2045 0.2012 0.2008 0.2004

0.1972 0.1969 0.1965 0.1934 0.1931 0.1696 0.1894 0.1882 0.1859 0.1626 0.1825

0.1927 0.1890 0.1855 0.1621

0.1799 0.1795 0.1792 0.1769 0.1767 0.1764 0.1761 0.1757 0.1736 0.1733 0.1730 0.1727 0.1706 0.1704 0.1701 0.1698 0.1676 0.1675 0.1672 0.1669

0.1949 0.1946 0.1942 0.1912 0.1906 0.1905 0.1876 0.1873 0.1642 0.1638

0.1808 0.1805 0.1776 0.1773 0.1745 0.1742 0.1715 0.1712 0.1686 0.1684

0.1869 0.1835

0.1802 0.1770

0.1592 0.1567 0.1543

0.1642 0.1616 0.1590 0.1565 0.1541

0.1520 0.1517 0.1497 0.1495 0.1475 0.1473 0.1453 0.1451 0.1433 0.1431

0.1412 0.1410 0.1393 0.1391 0.1374 0.1372

0.1739 0.1709 0.1681

0.1600 0.1597 0.1575 0.1572 0.1550 0.1548

0.1527 0.1524 0.1504 0.1502 0.1481 0.1479 0.1460 0.1458 0.1439 0.1437

0.1418 0.1416

0.1595 0.1570 0.1546

0.1522 0.1499 0.1477 0.1456 0.1435

0.1414 0.1399 0.1379

0.1397 0.1395 0.1377 0.1376 0.1359 0.1357 0.1340 0.1339

0.1361 0.1342

0.1325 0.1307 0.1290 0.1274 0.1258

0.1242 0.1227 0.1212 0.1196 0.1183

0.1170 0.1156 Q.1143

0.1355 0.1337

0.1353 0.1335

0.1328 0.1326 0.1311 0.1309 0.1294 0.1292 0.1277 0.1276 0.1261 0.1259

0.1245 0.1244 0.1230 0.1229 0.1215 0.1214 0.1200 0.1199 0.1186 0.1185

0.1172 0.1171 0.1159 0.1157 0.1145 0.1144

0.1323 0.1321 0.1305 0.1304 0.1289 0.1287 0.1272 0.1271 0.1256 0.1255

0.1241 0.1239 0.1225 0.1224 0.1211 0.1209 0.1196 0.1195 0.1182 0.1181

0.1168 0.1167 0.1155 0.1153 0.1142 0.1140

0.1319 0.1318 0.1302 0.1300 0.1285 0.1284 0.1269 0.1267 0.1253 0.1252

0.1238 0.1236 0.1222 0.1221 0.1208 0.1206 0.1193 0.1192 0.1179

0.1133 0.1131 0.1130 0.1129 0.1127 0.1120 0.1119 0.1117 0.1116 0.1115

0.1166 0.1152 0.1139 0.1126 0.1114

0.1107 0.1106 0.1105 0.1104 0.1103 0.1101 0.1095 0.1094 0.1093 0.1092 0.1091 0.1089 0.1083 0.1082 0.1081 0.1080 0.1079 0.1078 0.1072 0.1071 0.1070 0.1068 0.1067 0.1066 0.1060 0.1059 0.1058 0.1057 0.1056 0.1055

0.1045 0.1044 0.1034 0.1033 0.1024 0.1022 0.1013 0.1012 0.1003 0.1002

0.1176

0.1164 0.1151 0.1136 0.1125 0.1112

0.1100 0.1088 0.1076 0.1065 0.1054

0.1043 0.1032 0.1021 0.1011 0.1001

r=031a310 an* lie=0 367879

1-24 PETROLEUM ENGINEERINGHANDBOOK

TABLEl.7-CIRCUMFERENCESOFCIRCLESSY HUNDREDTHS

3 4 6 7 8 Average

9 Difference

3.424 31 3.738 4.053 4.367 4.681

4.995 5.309 5.623 5.938 6.252

d

1.0 1.1 1.2 1.3 1.4

0 1

3.173 3.487 3.801

2

3.142 3.456 3.770 4.084 4.398

4.115 4.430

3.204 3.236 3.519 3.550 3.633 3.864 4.147 4.178 4.461 4.492

5

3.299 3.613 3.927

3.362 3.393 3.676 3.707

4.241 4.555

1.5 4.712 1.6 5.027 1.7 5.341 1.8 5.655 1.9 5.969

4.744 5.058 5.372 5.686 6.000

6.315 6.629 6.943 7.257 7.571

7.685 8.200 8.514 8.828 9.142

9.456 9.770

4.775 4.807 5.069 5.121 5.484 5.435 5.718 5.749 6.032 6.063

4.869 5.184 5.498 5.812 6.126

6.440 6.754

3.990 4.021 4.304 4.335 4.618 4.650

4.932 4.964 5.246 5.278 5.561 5.592

5.906 6.220

2.0 2.1 2.2 2.3 2.4

6.283 6.597 6.912 7.226 7.540

6.346 6.377 6.680 6.692 6.974 7.006 7.288 7.320 7.683 7.634

7.069 7.383

6.535 6.566 6.849 6.660 7.163 7.194 7.477 7.508 7.791 7.823

2.5 7.854 2.6 8.168 2.7 6.462 2.6 6.796 2.9 9.111

7.917 7.948 6.231 8.262 8.545 6.577 8.859 8.891 9.173 9.205

8.105 8.137 8.419 8.451 6.734 8.765 9.048 9.079 9.362 9.393

3.0 3.1 3.1 3.2 3.3 3.4

9.425 9.739

9.466 9.519 9.602 9.833

3.267 3.581 3.696 4.210 4.524

4.838 5.152 5.466 5.781 6.095

6.409 6.723 7.037 7.351 7.665

7.980 6.294 8.606 8.922 9.236

9.550 9.865

7.697

8.011 8.325 6.639 8.954 9.266

9.562 9.896

3.330 3.644 3.958 4.273 4.587

4.901 5.215 5.529 5.843 6.158

6.472 6.786 7.100 7.414 7.728

8.042 8.357 8.671 8.985 9.299

9.613 9.927

5.875 6.189

6.503 6.817 7.131 7.446 7.760

8.074 8.388 8.702 9.016 9.331

9.645 9.959

9.676 9.990

9.708 10.022 31 10.02 3

10.05 10.37 10.68

10.08 10.40 10.71

11.03

10.12 10.15 lo.18 10.21 10.43 10.46 10.49 10.52 10.74 10.78 10.81 10.84

10.24 10.56 10.87

11.18 11.50 11.81 12.13 12.44

10.27 10.30 10.59 10.62 10.90 10.93

3.5 11.00 3.6 11.31 3.7 11.62 3.8 11.94 3.9 12.25

11.34 11.66 11.97 12.28

11.06 11.09 11.12 11.15 11.37 11.40 11.44 11.47 11.69 11.72 11.75 11.78 12.00 12.03 12.06 12.10 12.32 12.35 12.38 12.41

11.22 11.25 11.53 11.56 11.84 11.88 12.16 12.19 12.47 12.50

4.0 12.57 12.60 12.63 12.66 12.69 12.72 12.75 12.79 12.82 4.1 12.86 12.91 12.94 12.97 13.01 13.04 13.07 13.10 13.13 4.2 13.19 13.23 13.26 13.29 13.32 13.35 13.38 13.41 13.45 4.3 13.51 13.54 13.57 13.60 13.63 13.67 13.70 13.73 13.76 4.4 13.82 13.85 13.89 13.92 13.95 13.98 14.01 14.04 14.07

10.34 10.65 10.96

11.28 11.59 11.91 12.22 12.53

12.85 13.16 13.46 13.79 14.11

4.5 14.14 14.17 14.20 14.23 14.26 14.29 14.33 14.36 14.39 14.42 4.6 14.45 14.48 14.51 14.55 14.58 14.61 14.64 14.67 14.70 14.73 4.7 14.77 14.80 14.83 14.86 14.89 14.92 14.95 14.99 15.02 15.05 4.8 15.08 15.11 15.14 15.17 15.21 15.24 15.27 15.30 15.33 15.36 4.9 15.39 15.43 15.46 15.49 15.52 15.55 15.58 15.61 15.65 15.68

5.0 15.71 15.74 15.77 15.80 15.83 5.1 16.02 16.05 16.06 16.12 16.15 5.2 16.34 16.37 16.40 16.43 16.46 5.3 16.65 16.68 16.71 16.74 16.78 5.4 16.96 17.00 17.03 17.06 17.09

15.87 16.18 16.49 16.81 17.12

15.90 16.21 16.52 16.84 17.15

15.93 16.24 16.56 16.87 17.18

15.96 16.27 16.59 16.90 17.22

15.99 3 16.30 16.62 16.93 17.25

Exotanation of Table of Circumferences

This table gives the product of % times any number d (diameter) from 1 to 10; i.e., it is a table of multiples 01 r. Moving the decimal point one place in Column d is equivalent to moving it one place in the body of the table

Circumference=ax d= 3.141593xd. Conversely, d= 1/rrxcircumference=0.31831xcircumference.

MATHEMATICALTABLES& UNlTS&SYSTEMSOFWElGHTS&MEASURES 1-25

TABLE1.7-CIRCUMFERENCESOFCIRCLES BY HUNDREDTHS(continued)

Average 0 1 2 3 4 5 6 7 6 9 Difference

-17.31 17.28 17.34 17.37 17.40 17.44- 17.47 17.50- 17.53 17.56 3 17.59 17.62 17.66 17.69 17.72 17.75 17.78 17.81 17.84 17.88

d

5.5 5.6 5.7 5.8 5.9

17.91 17.94 17.97 18.00 18.03 18.22 18.25 18.28 18.32 18.35 18.54 18.57 16.60 18.63 18.66

18.13 18.16 18.44 18.47 18.76 18.79

6.0 18.85 18.88 6.1 lg.16 19.20 6.2 19.48 19.51 6.3 19.79 19.82 6.4 20.11 20.14

6.5 20.42 20.45 6.6 20.73 20.77 6.7 21.05 21.08 6.8 21.36 21.39 6.9 21.68 21.71

18.91 18.94 18.98 19.23 19.26 19.29 19.54 19.57 19.60 19.85 19.89 19.92 20.17 20.20 20.23

20.48 20.51 20.55 20.80 20.83 20.86 21.11 21.14 21.17 21.43 21.46 21.49 21.74 21.77 21.80

18.06 18.10 18.38 18.41 18.69 18.72

19.01 19.04 19.32 19.35 19.63 19.67 19.95 19.98 20.26 20.29

20.50 20.61 20.89 20.92 21.21 21.24 21.52 21.55 21.83 21.87

19.07 19.38 19.70 20.01 20.33

20.64 20.95 21.27 21.58 21.90

19.10 19.42 19.73 20.04 20.36

20.67 20.99 21.30

21.99 22.02 22.05 22.09 22.12 22.15 22.18 22.31 22.34 22.37 22.40 22.43 22.46 22.49

21.61 21.93

7.0 7.1 7.2 7.3 7.4

22.62 22.93 23.25

22.65 22.68 22.71 22.75 22.78 22.81 22.97 23.00 23.03 23.06 23.09 23.12 23.28 23.31 23.34 23.37 23.40 23.44

23.59 23.62 23.66 23.69 23.72 23.75

22.21 22.24 22.53 22.56 22.84 22.87 23.15 23.18

7.5 7.6 7.7 7.8 7.9

23.56 23.88 24.19 24.50 24.82

23.91 24.22 24.54 24.85

23.94 24.25 24.57 24.88

23.97 24.00 24.03 24.06 24.28 24.32 24.35 24.38 24.60 24.63 24.66 24.69 24.91 24.94 24.98 25.01

23.47 23.50

23.78 23.81 24.10 24.13 24.41 24.44 24.72 24.76 25.04 25.07

0.0 25.13 25.16 25.20 25.23 25.26 25.29 25.32 25.35 25.38 8.1 25.45 25.48 25.51 25.54 25.57 25.60 25.64 25.67 25.70 8.2 25.76 25.79 25.82 25.86 25.89 25.92 25.95 25.98 26.01 8.3 26.08 26.11 26.14 26.17 26.20 26.23 26.26 26.30 26.33 8.4 26.39 26.42 26.45 26.48 26.52 26.55 26.58 26.61 26.64

a.5 8.6 8.7 8.8 8.9

26.70 26.73 26.77 26.80 26.83 26.86 26.89 26.92 27.02 27.05 27.08 27.11 27.14 27.17 27.21 27.24 27.33 27.65 27.96

9.0 28.27 9.1 28.59 9.2 28.90 9.3 29.22 9.4 29.53

27.36 27.39 27.43 27.46 27.49 27.68 27.71 27.74 27.77 27.80 27.99 28.02 28.05 28.09 28.12

28.31 28.34 28.37 28.40 28.43 28.62 28.65 28.68 28.71 28.75 28.93 28.97 29.00 29.03 29.06

27.55 27.87 28.18

28.49 28.81 29.12 29.44 29.75

30.07 30.38 30.69 31.01 31.32

26.95 27.27 27.58 27.90 28.21 28.24

29.25 29.28 29.31 29.34 29.37 29.56 29.59 29.63 29.66 29.69

27.52 27.83 28.15

28.46 28.78 29.09 29.41 29.72

28.53 28.56 28.84 28.87 29.15 29.19 29.47 29.50 29.78 29.81

9.5 29.85 29.88 29.91 29.94 29.97 30.00 30.03 9.6 30.16 30.19 30.22 30.25 30.28 30.32 30.35 9.7 30.47 30.50 30.54 30.57 30.60 30.63 30.66 9.8 30.79 30.82 30.85 30.88 30.91 30.94 30.98 9.9 31.10 31.13 31.16 31.20 31.23 31.26 31.29

30.10 30.13 30.41 30.44 30.72 30.76 31.04 31.07 31.35 31.38

10.0 31.42

18.19 18.50 18.82

19.13 19.45 19.76 20.07 20.39

20.70 21.02 21.33 21.65 21.96

22.27 22.59 22.90 23.22 23.53

23.84 24.16 24.47 24.79 25.10

25.42 25.73 26.04 26.36 26.67

26.99 27.30 27.61 27.93


TABLEl.6-AREASOFCIRCLESBY HUNDREDTHS

Average Difference d

1.0 1.1 1.2 1.3 1.4

1.5 1.6 1.7 1.8 1.9

2.0 2.1 2.2 23 2.4

2.5 26 2.7 2.8 2.9

3.0 3.1 3.2 3.3 3.4

3.5

3.6 3.7 3.8 3.9

4.0 4.1 4.2 4.3 4.4

4.5 4.6 4.7 4.8 4.9

5.0 5.1 5.2 5.3 5.4

0

0.785 0.950 1.131 1.327 1.539

1.767 2.011

2 3

0.817 0.833

4 5

0.866

6

0.985 1.003 1.169 1.188 1.368 1.389 1.584 1.606

0.849 1.021 1.208 1.410 1.629

1.039 1.227 1.431 1.651

0.882 1.057 1.247 1.453 1.674

1.791 1.815 1.839 2.036 2.061 2.087

1.887 2.138

2.324 2.602 2.895

3.205 3.530 3.871 4.227 4.600

4.988 5.391 5.811 6.246 6.697

7.163 7.645 8.143 8.657 9.188

9.731

2.351 2.630 2.926

1.863 2.112 2.378 2.659 2.956

2.405 2.688 2.986

3.301 3.631 3.976 4.337 4.714

1.911 2.164 2.433 2.717 3.017

3.237 3.269 3.563 3.597 3.906 3.941 4.264 4.301 4.638 4.676

3.333 3.664 4.011 4.374 4.753

5.027 5.067 5.433 5.474 5.853 5.896 6.290 6.335 6.743 6.789

7.211 7.258 7.694 7.744 8.194 8.245 8.709 8.762 9.240 9.294

9.787 9.842

5.107 5.515 5.940 6.379 6.835

7.306 7.793 8.298 8.814 9.348

5.147 5.557 5.983 6.424 6.881

9.898

7.354 7.843 8.347 8.867 9.402

9.954

10.29 10.35 10.41 10.46 10.52 10.87 10.93 10.99 11.04 11.10 11.46 11.52 11.58 11.64 11.70 12.07 12.13 12.19 12.25 12.32

12.69 13.33 13.99 14.66 15.34

15.98 16.05

12.76 12.82 13.40 13.46 14.05 14.12 14.73 14.79 15.41 15.48

16.12 16.19

12.88 13.53 14.19 14.86 15.55

16.26 16.98 17.72 18.47 19.24

20.03 20.83 21.65 22.48

12.95 13.59 14.25 14.93 15.62

16.33 17.06 17.80 18.55 19.32

20.11 20.91 21.73 22.56

7 8 9 - - 0.899 0.916 0.933

1

0.801 0.968 1.150 1.348 1.561

1.112 1.307 1.517 1.744

16 18 20 21 23

2.270 2.545 2.835

3.142 3.464 3.801 4.155 4.524

4.909 5.309 5.726 6.158 6.605

7.069 7.548 8.042 8.553 9.079

9.621

2.297 2.573 2.865

3.173 3.497 3.836 4.191 4.562

4.948 5.350 5.768 6.202 6.651

1.986 2.243 2.516 2.808 3.110

3.431 3.767 4.119 4.486 4.870

5.269 5.683 6.114 6.560 7.022

24 26 27 29 31

32 34 35 37 38

40 42 43 45 46

7.116 7.596 8.093 8.605 9.133

9.676

7.499 7.992 8.501 9.026 9.566

10.18 10.24 10.81 11.40 12.01

12.63 13.27 13.92 14.59 15.27

1.075 1.094 1.267 1.287 1.474 1.496 1.697 1.720

1.936 1.961 2.190 2.217 2.461 2.488 2.746 2.776 3.048 3.079

3.365 3.398 3.698 3.733 4.047 4.083 4.412 4.449 4.792 4.831

5.187 5.228 5.599 5.641 6.026 6.070 6.469 6.514 6.928 6.975

7.402 7.451 7.892 7.942 8.398 8.450 8.920 8.973 9.457 9.511

10.010 10.01 10.07 10.58 10.64

10.12 10.69 11.28 11.88

48 49 51 53 54

56

: 10.75 11.34 11.95

12.57 13.20 13.85 14.52 15.21

15.90

7

16.62 16.69 16.76 16.84 16.91 17.35 17.42 17.50 17.57 17.65

11.16 11.22 11.76 11.82 12.38 12.44

13.01 13.07 13.66 13.72 14.32 14.39 15.00 15.07 15.89 15.76

16.40 16.47

12.50

13.14 1379 14.45 15.14 15.83

16.55

18.10 18.17 18.25 18.32 18.86 18.93 19.01 19.09

19.63 19.71 19.79 19.87 20.43 20.51 20.59 20.67 21.24 21.32 21.40 21.48 22.06 22.15 22.23 22.31

18.40 19.17

19.95 20.75 21.57 22.40

17.13 17.20 17.28 17.87 17.95 18.02

18.78 19.56

18.63 18.70 19.40 19.48

20.19 20.27 20.99 21.07 21.81 21.90 22.65 22.73

8

20.35 21.16 21.98 22.82

22.90 22.99 23.07 23.16 23.24 23.33 23.41 23.50 23.59 23.67 9

Explanation of Table of Areas of Circles

Moving the decimal point one place in Column d (diameter) IS equivalent to moving it two places in the body of the table. For example, area of circle= R/~XC? =0,785398x(d'). Conversely, diameter of circle=&x &=1.128379x 6.

MATHEMATICAL TABLES & UNITS 8 SYSTEMS OF WEIGHTS & MEASURES l-27

TABLE 1.8-AREAS OF CIRCLES BY HUNDREDTHS (continued)

Average Difference

9

10

11

12

13

14

15

16

d 0

5.5 5.6 5.7 5.8 5.9

6.0 6.1 6.2 6.3 6.4

23.76 23.84 24.63 24.72 25.52 25.61 25.70 25.79 25.88 25.97 26.42 26.51 26.60 26.69 26.79 28.88 27.34 27.43

28.27 28.37 29.22 29.32 30.19 30.29 31.17 31.27 32.17

6.5 6.6 8.7 6.8 6.9

33.18 34.21 35.26 36.32 37.39

7.0 38.48 7.1 39.59 7.2 40.72 7.3 41.85 7.4 43.01

7.5 44.18 7.6 45.36 7.7 46.57 7.8 47.78 7.9 49.02

8.0 50.27 8.1 51.53 8.2 52.81 8.3 54.11 8.4 55.42

8.5 56.75 8.6 58.09 8.7 59.45 8.8 60.82 8.9 62.21

9.0 9.1 9.2 9.3 9.4

9.5 9.6 9.7 9.8 9.9

63.62 65.04 66.48 67.93 69.40

70.88 72.38 73.90 75.43 76.98

4 5 6 7 8 9 2 3

23.93 24.02 24.81 24.89

24.45 24.54 25.34 25.43 26.24 26.33 27.15 27.25 28.09 28.18

29.03 29.13 30.00 30.09 30.97 31.07 31.97 32.07 32.98 33.08

1

24.11 24.19 24.98 25.07

24.28 24.37 25.16 25.25 26.06 26.15 26.97 27.06 27.90 27.99

28.84 28.94 29.80 29.90 30.78 30.88 31.77 31.07 32.78 32.88

27.53 27.62 27.71 27.81

28.46 28.56 28.65 28.75 29.42 29.51 29.61 29.71 30.39 30.48 30.58 30.68 31.37 31.47 31.57 31.67

32.27

33.29 34.32 35.36 36.42 37.50

38.59 39.70 40.83 41.97 43.12

44.30 45.48 46.69 47.91 49.14

50.39 51.66 52.94 54.24 55.55

56.88 58.22 59.58

32.37 32.47 32.57 32.67

33.39 33.49 33.59 33.70 33.80 33.90 34.00 34.11 34.42 34.52 34.63 34.73 34.84 34.94 35.05 34.15 35.47 36.53 37.61

38.70 39.82 40.94 42.08 43.24

44.41 45.60 46.81 48.03 49.27

50.52 51.78 53.07 54.37 55.68

57.01 58.36 59.72

35.57 36.64 37.72

38.82 39.93 41.06 42.20 43.36

44.53 45.72 46.93 48.15 49.39

50.64 51.91 53.20 54.50 55.81

35.68 36.75 37.83

38.93 40.04 41.17 42.31 43.47

35.78 35.89 36.00 36.85 36.96 37.07 37.94 38.05 38.16

39.04 39.15 39.26 40.15 40.26 40.38 41.28 41.40 41.51 42.43 42.54 42.66 43.59 43.71 43.83

44.65 45.84 47.05 48.27 49.51

44.77 45.96 47.17 48.40 49.64

50.90 52.17 53.46 54.76 56.08

44.89 45.01 46.08 46.20 47.29 47.42 48.52 48.65 49.76 49.89

36.10 37.18 38.26

39.37 40.49 41.62 42.78 43.94

45.13 46.32 47.54 48.77 50.01

50.77 52.04 53.33 54.63 55.95

51.02 51.15 51.28 51.40 52.30 52.42 52.55 52.68 53.59 53.72 53.85 53.98 54.89 55.02 55.15 55.29 56.21 56.35 56.48 56.61

57.15 57.28 57.41 57.55 57.68 57.82 58.49 58.63 58.77 58.90 59.04 59.17 59.86 59.99 60.13 60.27 60.41 60.55

36.21 37.28 38.37

39.48 40.60 41.74 42.89 44.06

45.25 46.45 47.66 48.89 50.14

57.95 59.31 60.68 62.07 63.48

64.90 66.33 67.78

60.96 61.10 61.24 61.38 61.51 61.65 61.79 61.93 62.35 62.49 62.63 62.77 82.91 63.05 63.19 63.33

63.76 63.90 64.04 64.18 64.33 64.47 65.18 65.33 65.47 65.61 65.76 65.90 68.62 66.77 66.91 67.06 67.20 67.35 68.08 68.22 68.37 68.51 68.66 68.81 69.55 69.69 69.84 69.99 70.14 70.29

71.03 72.53 74.05 75.58 77.13

71.18 72.68

71.33 71.48 71.63 72.84 72.99 73.14 74.36 74.51 74.66 75.89 76.05 76.20 77.44 77.60 77.76

71.78 73.29

74.20 75.74 77.29

74.82 74.97 75.12 76.36 76.51 76.67 77.91 78.07 78.23

64.61 66.04 67.49 88.96 70.44

71.93 73.44

64.75 66.19 67.64 69.10 70.58

72.08 73.59

69.25 70.73

72.23 73.75 75.28 76.82 78.38


TABLE I.S-CIRCUMFERENCES AND AREAS OF CIRCLES BY EIGHTHS

Circumference Area

0.04909 0.00019 0.09817 0.00077 0.1473 0.00173 0.1963 0.00307 0.2454 0.00479 0.2945 0.00690 0.3436 0.00940 0.3927 0.01227 0.4418 0.01553 0.4909 0.01917 0.5400 0.02320 0.5890 0.02761 0.6381 0.03241 0.6872 0.03758 0.7363 0.04314 0.7854 0.04909 0.8345 0.05542 0.8836 0.06213 0.9327 0.06922 0.9817 0.07670 1.031 0.08456 1.080 0.09281 1.129 0.1014 1.178 0.1104 1.227 0.1198 1.276 0.1296 1.325 0.1398 1.374 0.1503 1.424 0.1613 1.473 0.1726 1.522 0.1843 1.571 0.1963 1.620 0.2088 1.669 0.2217 1.718 0.2349 1.767 0.2485 1.816 0.2625 1.865 0.2769 1.914 0.2916 1.963 0.3068 2.013 0.3223 2.062 0.3382 2.111 0.3545 2.160 0.3712 2.209 0.3883 2.258 0.4057 2.307 0.4236 2.356 0.4418 2.405 0.4604 2.454 0.4794 2.503 0.4987 2.553 0.5185 2.602 0.5386 2.651 0.5591 2.700 0.5800

Circumference Area

2.749 0.6013 2.798 0.6230 2.847 0.6450 2.896 0.6675 2.945 0.8903 2.994 0.7135 3.043 0.7371 3.093 0.7610 3.142 0.7854 3.338 0.8866 3.534 0.9940 3.731 1.108 3.927 1.227 4.123 1.353 4.320 1.485 4.516 1.623 4.712 1.767 4.909 1.917 5.105 2.074 5.301 2.237 5.498 2.405 5.694 2.580 5.890 2.761 6.087 2.948 6.283 3.142 6.480 3.341 6.676 3.547 6.872 3.758 7.069 3.976 7.265 4.200 7.461 4.430 7.658 4.666 7.854 4.909 8.050 5.157 8.247 5.412 8.443 5.673 6.639 5.940 8.836 6.213 9.032 6.492 9.228 6.777 9.425 7.069 9.621 7.366 9.817 7.870 10.01 7.980 10.21 8.296 10.41 8.618 10.60 8.946 10.80 9.281 11.00 9.621 11.19 9.968 11.39 10.32 11.58 10.68 il.78 11.04 11.98 11.42 12.17 11.79 12.37 12.18

Circumference Area

12.57 12.57 12.76 12.96 12.96 13.36 13.16 13.77 13.35 14.19 13.55 14.61 13.74 15.03 13.94 15.47 14.14 15.90 14.33 16.35 14.53 16.80 14.73 17.26 14.92 17.72 15.12 18.19 15.32 18.67 15.51 19.15 15.71 19.63 15.90 20.13 16.10 20.63 16.30 21.14 16.49 21.65 16.69 22.17 16.89 22.69 17.08 23.22 17.28 23.76 17.48 24.30 17.67 24.85 17.87 25.41 18.06 25.97 18.26 26.53 18.46 27.11 18.65 27.69 18.85 28.27 19.24 29.46 19.63 30.68 20.03 31.92 20.42 33.18 20.81 34.47 21.21 35.78 21.60 37.12 21.99 38.48 22.38 39.87 22.78 41.28 23.17 42.72 23.56 44.18 23.95 45.66 24.35 47.17 24.74 48.71 25.13 50.27 25.53 51.85 25.92 53.46 26.31 55.09 26.70 56.75 27.10 58.43 27.49 60.13 27.88 61.86

MATHEMATlCALTABLES& UNITS& SYSTEMS OF WEIGHTS & MEASURES 1-29

TABLE1.9-CIRCUMFERENCES AND AREAS OFCIRCLES BY ElGHTHS(continued)

Circumference

28.27 63.62 28.67 65.40 29.06 67.20 29.45 69.03 29.85 70.88 30.24 72.76 30.63 74.66 31.02 76.59 31.42 78.54 31.81 80.52 32.20 82.52 32.59 84.54 32.99 86.59 33.38 88.66 33.77 90.76 34.16 92.89 34.56 95.03 34.95 97.21 35.34 99.40 35.74 101.6 36.13 103.9 36.52 106.1 36.91 108.4 37.31 110.8 37.70 113.1 38.09 115.5 38.48 117.9 38.88 120.3 39.27 122.7 39.66 125.2 40.06 127.7 40.45 130.2 40.84 132.7 41.23 135.3 41.63 137.9 42.02 140.5 42.41 143.1 42.80 145.8 43.20 148.5 43.59 151.2 43.98 153.9 44.37 156.7 44.77 159.5 45.16 162.3 45.55 165.1 45.95 168.0 46.34 170.9 46.73 173.8 47.12 176.7 47.52 179.7 47.91 182.7 48.30 185.7 48.69 188.7 49.09 191.7 49.48 194.8 49.87 197.9

Circumference

50.27 201.1 23 50.66 204.2 '/a 51.05 207.4 '/4 51.44 210.6 '/4 51.84 213.8 ‘12

52.23 217.1 78 52.62 220.4 %l 53.01 223.7 % 53.41 227.0 24 53.80 230.3 '/4 54.19 233.7 '12 54.59 237.1 VI 54.98 240.5 25 55.37 244.0 '/4 55.76 247.4 '12 56.16 250.9 %l 56.55 254.5 26 56.94 258.0 '12 57.33 261.6 ‘/2

57.73 285.2 3/i

58.12 268.8 27 58.51 272.4 'A 58.90 276.1 ‘12

59.30 279.8 3/k

59.69 283.5 28 60.08 287.3 'h 60.48 291.0 '/2 60.87 294.8 v4 61.26 298.6 29 61.65 302.5 '/a 62.05 306.4 '12 62.44 310.2 Y4 62.83 314.2 30 63.22 318.1 l/4 63.62 322.1 '/2 64.01 326.1 3/b 64.40 330.1 31 64.80 334.1 '/4 65.19 338.2 'I2 65.58 342.2 v4 65.97 346.4 32 66.37 350.5 '/4 66.76 354.7 ‘12

67.15 358.8 %I 67.54 363.1 33 67.94 367.3 '/4 68.33 371.5 '12 68.72 375.8 =/4 69.12 380.1 34 69.51 384.5 'I4 69.90 388.8 % 70.29 393.2 v4 70.69 397.6 35 71.08 402.0 '/4 71.47 406.5 '12 71.86 411.0 3/i

Circumference Area

72.26 415.5 72.65 420.0 73.04 424.6 73.43 429.1 7383 433.7 74.22 438.4 74.61 443.0 75.01 447.7 75 40 452.4 76.18 461.9 7697 471.4 77.75 481.1 78.54 490.9 79.33 500.7 80.11 510.7 80.90 520.8 81.68 530.9 82.47 541.2 83.25 551.5 84.04 562.0 84.82 572.6 85.61 583.2 86.39 594.0 87.18 604.8 87.96 615.8 88.75 626.8 89.54 637.9 90.32 649.2 91.11 660.5 91.89 672.0 92.68 683.5 93.46 695.1 94.25 706.9 95.03 718.7 95.82 730.6 96.60 742.6 97.39 754.8 98.17 767.0 98.96 779.3 99.75 791.7 100.5 804.2 101.3 816.9 102.1 829.6 102.9 842.4 103.7 855.3 104.5 868.3 105.2 881.4 106.0 894.6 106.8 907.9 107.6 921.3 108.4 934.8 109.2 948.4 110.0 962.1 110.7 975.9 111.5 989.8 112.3 1003.8


TABLE l.lO-AREAS OF CIRCLES-DIAMETERS IN FEET AND INCHES, AREAS IN SQUARE FEET

Inches

Feet

0

0 1 2 3 4

0.0000 0.0055 0.0216 0.0491 0.0673

07854 09216 1.069 1.227 1.396 3142 3.409 3687 3976 4276

7.069 7 467 7676 8.296 8.727 1257 13.10 13.64 14.19 14.75 19.63 20.29 20.97 21.65 2234 2627 29.07 29 67 30.66 31 50 3048 39.41 40.34 41.26 42 24 50 27 51 32 5236 5346 54 54

6362 6460 66 00 67.20 6842 78 54 7985 81.16 82.52 83.66 95.03 96.48 97 93 99.40 100.9

113.1 1147 1163 1179 119.5 1327 134.4 1362 1379 139.6 153.9 1556 1576 159.5 161.4

5 6

0.1364 0.1963 1.576 1.767 4587 4909

9.168 9.621 1532 15.90 2304 23.76 32 34 33.18 43.20 44.18 55.64 56.75 69.64 70.88 85.22 8659 1024 103.9 121 1 122.7 1414 143.1

163.2 165.1

7 8 0.2673

1.969 5.241

10.08 16.50 24.48 34.04 45.17

57.86 72.13 87.97 105.4 124.4 144.9 167.0

0.3491

2.182 5.585

10.56 17.10 25.22 3491

46.16 58.99 73.39 89.36

106.9 1260 146.7

168.9

9 0.4416

2405 2.640 2885 5.940 6.305 6681

11.04 11.54 1205 17.72 18.35 1899 25.97 26.73 2749 35.78 36.67 3757 47.17 48.19 49 22 60.13 61.28 6244 74.66 75.94 77 24 90.76 92.18 9360

108.4 110.0 111 5 127.7 129.4 131 0 148.5 150.3 1521 170.9 172.8 1748

10 11

0.5454 06600

From Inches and Fractions of an Inch to Decimals of a Foot” Inches 1 2 3 4 5 6 7 6 9 10 11 Feet 0.0833 0.1667 0.2500 03333 0.4167 0.5000 0.5833 0.6667 0.7500 0.8333 09167

Inches '/a % % '12 % v4 % Feet 0.0104 0.0208 0.0313 00417 00521 0.0625 0 0729

MATHEMATICALTABLES& UNlTS&SYSTEMSOFWElGHTS&MEASURES 1-31

hlc

0.00

0.01 0.02 0.03 0.04

d/c Difference

25.010 12.520

8.363 6.290

0.05 0.06 0.07 0.08 0.09

0.10 0.11 0.12 0.13 0.14

0.15 0.16 0.17 0.18 0.19

5.050 4.227

3.641 3.205 2.868

2.600 2.383 2.203 2.053 1.926

1.617 1.723 1.641 1.569 1.506

0.20 1.450 0.21 1.400 0.22 1.356

0.23 1.317 0.24 1.282

0.25 0.26 0.27 0.28 0.29

0.30 0.31 0.32 0.33 0.34

0.35 0.36 0.37

0.38 0.39

1.250 1.222 1.196 1.173 1152

1.133 1.116 1.101 1.088 1.075

1.064 1.054 1.046 1.038 1.031

0.40 1.025 0.41 1.020

0.42 1.015

0.43 1.011 0.44 1.008

0.45 1.006 0.46 1.003

0.47 1.002 0.48 1.001

0.49 1.000

0.50 1.000

12,490 4.157' 2,073' 1,240'

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 11

10 8 8 7 6

5 5 4

3 2

3 1 1 1

0

Arc/c

1.000 1.000 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 1.254 1.269 1.284

1.300 1.316 1.332

1.349 1.366

1.383 1.401

1.419 1.437 1.455

1.474 1.493 1.512 1.531 1.551

1.571

TABLE 1 .l l-SEGMENTS OF CIRCLES, GIVEN h/c

Difference

3 3 4 4

5

6

6 6 7

8

8 8 9 10 9

11 10 12 11 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

Arealhxc

0.6667 0.6667 0.6669 0.6671 0.6675

0.6680 0.6686 0.6693 0.6701 0.6710

0.6720 0.6731 0.6743 0.6756 0.6770

0.6785 0.6801 0.6818 0.6836 0.6855

0.6875 0.6896 0.6918 0.6941 0.6965

0.6989 0.7014 0.7041 0.7088 0.7096

0.7125 0.7154 0.7185 0.7216 0.7248

0.7280 0.7314 0.7348

0.7383 0.7419

0.7455 0.7492 0.7530

0.7568 0.7607

0.7647 0.7687 0.7728 0.7769 0.7811

0.7854

Explanation of Table of Segments of Circles, Given h/c

Difference

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 35

36 36

37 38 38 39 40

40 41 41 42

43

Central Angle, CY (degrees)

0.00 4.58 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 75.11 79.20 83.23

87.21 91.13 95.00 98.81 102.56

106.26 109.90 113.48 117.00 120.45

123.86 127.20 130.48 133.70 136.86

139.97 143.02 146.01

148.94 151.82

154.64 157.41 160.12 162.78 165.39

167.95 170.46 172.91 175.32

177.69

180.00

Difference

458 458 457 457 454

453 451

448 446 442

439 435 432 427 423

418 413 409 403 399

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

hld

0.0000

0.0004 0.0016

0.0036 0.0064

0.0099 0.0142

0.0192 0.0250 0.0314

0.0385 0.0462 0.0545 0.0633 0.0727

0.0826 0.0929 0.1036 0.1147 0.1263

0.1379 0.1499 0.1622 0.1746 0.1873

0.2000 0.2128

0.2258 0.2387 0.2517

0.2647 0.2777 0.2906 0.3034 0.3162

0.3289 0.3414 0.3538 0.3661 0.3783

0.3902 0.4021 0.4137 0.4252 0.4364

0.4475 0.4584 0.4691 0.4796 0.4899

0.5000

Difference

4 12 20 28 35

43 50

58 64 71

77

83 88 94 99

103 107 111 116 116

120 123 124 127 127

128 130 129 130

130

130 129

128 128 127

125 124 123 122 119

119

116 115 112 111

109 107 105 103

101

Given: h= height of segment and c=chord, to find the drameter, d, of the circle, the length of the arc, or the area of the segment, form the ratio h/c, and find from the table the value of d/c, arc/c, or arealhc; then, by a simple multiplication, d= cx (d/c), arc=c x (arc/c), and area= h x c x (arealhc). The table also gives the angle subtended at the center, and the ratio of h to d.


TABLE 1.12-SEGMENTS OF CIRCLES, GIVEN h/d

Cf?ntL4 Angle, (v Arc Area

hid Arc/d Difference Area/d' D!fference (degrees) Difference c/d Difference Circumference Drfference Circle Difference

0.00 0.000 0.01 0.2003

0.02 0.2638

0.03 0.3482

0.04 0.4027

0.05 0.4510

0.06 0.4949

0.07 0.5355

0.08 0.5735

0.09 0.6094

0.10 0.6435

0.11 0.6761

0.12 0.7075

0.13 0.7377

0.14 0.7670

0.15 0.7954

016 0.8230

0.17 0.8500

0.18 0.8763

0.19 0.9021

0.20 0.9273

0.21 0.9521

0.22 0.9764

0.23 1.0004

024 1.0239

0.25 1.0472

0.26 1.0701

0.27 1.0928

0.26 1.1152

0.29 1.1374

0.30 1.1593

0.31 1.1810

0.32 1.2025

0.33 1.2239

0.34 1.2451

0.35 1.2661

036 12870

0.37 1.3078

038 1.3284

0.39 1.3490

0.40 1.3694

0.41 13898

0.42 14101

043 1.4303

044 1.4505

0.45 14706

046 1.4907

047 1.5106

0 48 1.5308

049 1.5508

0.50 15708

2003

835'

644'

545'

483'

439'

406'

380'

359'

341'

326'

314'

302'

293'

284'

276

270

263

256

252

248

243

240

235

233

229

277

224

222

219

217

215

214

212

210

209

208

206

206

204

204

203

202

202

201

201

201

200

200

200

0.0000 0.0013

0.0037

0.0069

0.0105

0.0147

0.0192

0.0242

0.0294

0.0350

0 0409

00470

00534

00600

00666

0 0739

00811

00885

00961

01039

01118

0 1199

01281

01365

0 1449

0 1535

01623

01711

01800

0.1890

0.1982

0.2074

02167

0.2260

0.2355

02450

0.2546

0.2642

0 2739

02836

0 2934

03032

03130

0 3229

0 3328

0.3428

03527

03627

03727

0.3627

0.3927

13

24

32

36

42

45

50

52

56

59

61

64

66

68

71

72

74

76

76

79

81

82

84

64

86

88

88

89

90

92

92

93

93

95

95

96

96

97

97

98

98

98

99

99

100

99

100

100

100

100

0.00 22.96 32.52

39.90

46.15

51.68

56.72

61.37

65.72

69.83

7374

77 48

81.07

8454

87 89

91.15

94.31

9740

10042

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

13780

140.25

14267

14506

147.46

14986

15223

15458

15693

15926

161 59

16390

16622

16652

17082

17312

17541

17771

18000

Explanatcon of Table of Segments of Circles. Given h/d

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

229

230

229

0.1990

0.2800

0.3412

0.3919

0.4359

0.4750

0.5103

05426

0.5724

0 6000

0.6258

0.6499

0.6726

0.6940

,0.7141

0.7332

0.7513

07684

0.7846

0.8000

0.8146

0.8285

0.8417

0.8542

0.8660

0.8773

0.8879

0 8980

0.9075

0.9165

0.9250

0.9330

0.9404

0.9474

0.9539

0 9600

0 9656

0.9708

0.9755

0.9798

0.9837

0.9871

0 9902

0 9928

0.9950

0 9968

0.9982

0 9992

0 9998

10000

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

58

52

47

43

39

34

31

26

22

18

14

10

6

2

0 0000 0.0638

0.0903

0.1108

0.1282

0.1436

0.1575

0.1705

0.1826

0.1940

0.2048

0.2152

0.2252

0.2348

0.2441

0.2532

0.2820

0.2706

0 2789

0.2871

0.2952

0.3031

0.3108

0.3184

0.3259

0.3333

0.3406

0.3478

0.3550

0.3820

0.3690

0.3759

0.3828

0.3898

0.3963

0.4030

0.4097

0.4163

0.4229

0.4294

0.4359

0.4424

0 4489

0.4553

0.4617

0.4681

04745

0.4809

04873

0.4936

0.5000

638‘

265'

205'

174'

154'

139.

130'

121

114

108

104

100

96

93

91

88

86

83

82

81

79

77

76

75

74

73

72

72

70

70

69

69

68

67

67

67

66

66

65

65

65

65

64

64

64

64

64

64

63

64

0.0000 0.0017

0.0046

0.0087

0.0134

0.0187

0.0245

0.0308

0.0375

0 0446

0.0520

0.0598

0.0680

0.0764

0.0851

0.0941

0.1033

0.1127

0.1224

0 1323

0.1424

0.1527

0.1631

0.1738

01846

01955

02066

02178

0 2292

02407

0.2523

0.2640

0.2759

0.2878

0 2998

0.3119

03241

0.3364

0.3487

0.3611

0.3735

0.3860

0.3986

0.4112

0.4236

0.4364

04491

0.4618

04745

0.4873

0.5000

17

31

39

47

53

58

63

67

71

74

78

82

84

87

90

92

94

97

99

101

103

104

107

108

109

111

112

114

115

116

117

119

119

120

121

122

123

123

124

124

125

126

126

126

126

127

127

127

128

127

Gwen: h=helght of segment and d= dtameter of circle, to find the chord, the length of arc, or the area of the segment, form the ratlo h/d and find from the table the value of (c/d), (arc/d), or (area/d*), then by a simple multipllcatlon. c=dx(cld), arc=dx(arc/d), and

area=d'x (area/d') The table also gives the angle subtended at the center. the ratlo of the arc of the segment to the whole circumference. and the ratlo of

the area 01 the segment to the area of the whole cvcle


h/d

0.00

0.01 0.02

0.03 0.04

Volume segment

d3 Difference

0.0000

0.05 0.06 0.07 0.08 0.09

0.10 0.11 0.12

0.13 0.14

0.0002 0.0006 0.0014 0.0024

0.0038 0.0054 0.0073 0.0095 0.0120

0.0147 0.0176 0.0208 0.0242 0.0279

0.15 0.0318 0.16 0.0359

0.17 0.0403 0.18 0.0448

0.19 0.0495

0.20 0.21 0.22 0.23 0.24

0.25 0.26 0.27

0.28 0.29

0.30 0.31 0.32 0.33 0.34

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

0.35 0.1475

0.36 0.1547

0.37 0.1620 0.38 0.1694

0.39 0.1768

0.40 0.1843

0.41 0.1919

0.42 0.1995

0.43 0.2072

0.44 0.2149

0.45 0.2227 0.46 0.2305 0.47 0.2383

0.48 0.2461

0.49 0.2539

0.50 0.2618

TABLE 1.13~SEGMENTS OF SPHERES

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

Volume segment

Volume sphere

0.0000 0.0003 0.0012 0.0026

0.0047

0.0073 0.0104 0.0140 0.0182 0.0228

0.0260 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

‘Volume segment= 1/6rh2(3c-2h)

Explanation of Table of Segments of Spheres

Difference

3 9

14 21

26

31 36 42

46 52

56 61

66 70 74

79 83 86 91 94

98 101 105

108 110

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

Given: h = height of segment and d = diameter of sphere, to find the volume of the segment, form the ratio h/d and find from the table the value of (volume/d3): then. by a simple multiplication, volume segment= d3 x (volume/d3). The table also gives the ratio of the volume of the segment to the entire

volume of the sphere. Note that the area of zone= ?TX hx d.


d 0

-0.5236 1.0 1.1 0.6969 1.2 0.9048 1.2 1.3 1.150 1.4 1.437

1 2 3 4 5 6 7 8 9 Difference

0.5395 0.5556 0.7161 0.7356 0.9276 0.9508

0.5722 0.7555

0.6236 0.6414 0.6596 0.6781 0.8173 0.8386 0.8603 0.8823

0.9743

0.5890 0.7757 0.9983

1.177 1.204 1.232 1.260 1.468 1.499 1.531 1.563

0.6061 0.7963 1.0227 1.023 1.288 1.596

1.047 1.073 1.098 1.124 1.317 1.346 1.376 1.406 1.630 1.663 1.697 1.732

173 208 236 25 29 33

1.5 1.767 1.803 1.839 1.875 1.912 1.950 1.988 2.026 2.065 2.105 1.6 2.145 2.185 2.226 2.268 2.310 2.352 2.395 2.439 2.483 2.527 1.7 2.572 2.618 2.664 2.711 2.758 2.806 2.855 2.903 2.953 3.003 1.8 3.054 3.105 3.157 3.209 3.262 3.315 3.369 3.424 3.479 3.535 1.9 3.591 3.648 3.706 3.764 3.823 3.882 3.942 4.003 4.064 4.126

38 43 48

:;

2.0 4.189 4.252 4.316 4.380 4.445 4.511 4.577 4.644 4.712 4.780 66 2.1 4.849 4.919 4.989 5.060 5.131 5.204 5.277 5.350 5.425 5.500 73 2.2 5.575 5.652 5.729 5.806 5.885 5.964 6.044 6.125 6.206 6.288 80 2.3 6.371 6.454 6.538 6.623 6.709 6.795 6.882 6.970 7.059 7.148 87 2.4 7.238 7.329 7.421 7.513 7.606 7.700 7.795 7.890 7.986 8.083 94

2.5 8.181 2.6 9.203 2.6 2.7 10.31 2.8 11.49 2.9 12.77

8.280 8.379 8.478 8.580 8.682 8.785 8.888 9.309 9.417 9.525 9.634 9.744 9.855 9.966

9.097

10.42 10.54 10.65 10.77 10.89 11.01 11.13 11.62 11.74 11.87 11.99 12.12 12.25 12.38 12.90 13.04 13.17 13.31 13.44 13.58 13.72

8.992 10.079 10.08 il.25 12.51 13.86

10.19 11.37 12.64 14.00

102 110 11 12 13 14

3.0 14.14 14.28 14.42 14.57 14.71 14.86 15.00 15.15 15.30 15.45 15 3.1 15.60 15.75 15.90 16.06 16.21 16.37 16.52 16.68 16.84 17.00 16 3.2 17.16 17.32 17.48 17.64 17.81 17.97 18.14 18.31 18.48 18.65 17 3.3 18.82 18.99 19.16 19.33 19.51 19.68 19.86 20.04 20.22 20.40 18 3.4 20.58 20.76 20.94 21.13 21.31 21.50 21.69 21.88 22.07 22.26 19

3.5 22.45 22.64 22.84 23.03 23.23 23.43 23.62 23.82 24.02 2423 20 3.6 24.43 24.63 24.84 25.04 25.25 25.46 25.67 25.88 26.09 26.31 21 3.7 26.52 26.74 26.95 27.17 27.39 27.61 27.83 28.06 28.28 2850 22 3.8 28.73 28.96 29.19 29.42 29.65 29.88 30.11 30.35 30.58 30.82 23 3.9 31.06 31.30 31.54 31.78 32.02 32.27 32.52 32.76 33.01 33.26 25

4.0 33.51 33.76 34.02 34.27 34.53 34.78 35.04 35.30 35.56 35.82 26 4.1 36.09 36.35 36.62 36.88 37.15 37.42 37.69 37.97 38.24 38.52 27 4.2 38.79 39.07 39.35 39.63 39.91 40.19 40.48 40.76 41.05 41.34 28 4.3 41.63 41.92 42.21 42.51 42.80 43.10 43.40 43.70 44.00 44.30 30 4.4 44.60 44.91 45.21 45.52 45.83 46.14 46.45 46.77 47.08 47.40 31

4.5 47.71 48.03 48.35 48.67 49.00 49.32 49.65 49.97 50.30 50.63 33 4.6 50.97 51.30 51.63 51.97 52.31 52.65 52.99 53.33 53.67 54.02 34 4.7 54.36 54.71 55.06 55.41 55.76 56.12 56.47 56.83 57.19 57.54 35 4.8 57.91 58.27 58.63 59.00 59.37 59.73 60.10 60.48 60.85 61.22 37 4.9 61.60 61.98 62.36 62.74 63.12 63.51 63.89 64.28 64.67 65.06 38

5.0 65.45 65.64 66.24 66.64 67.03 67.43 67.83 68.24 68.64 69.05 40 5.1 69.46 69.87 70.28 70.69 71.10 71.52 71.94 72.36 72.78 73.20 42 5.2 73.62 74.05 74.47 74.90 75.33 75.77 76.20 76.64 77.07 77.51 43 5.3 77.95 78.39 78.84 79.28 79.73 80.18 80.63 81.08 81.54 81.99 45 5.4 82.45 82.91 83.37 83.83 84.29 84.76 85.23 85.70 86.17 86.64 47

TABLE 1.14-VOLUMES OF SPHERES BY HUNDREDTHS

Explanation of Table of Volumes of Spheres

Moving the decimal point one place in Column d (diameter) is equivalent to moving it three places in the body of the table.

For example, volume of sphere= ~/6x (d3)=0.523599x(d3). Conversely, d= ?/%x $k%%= 1.240701 x %%r%e.

1-35 MATHEMATICALTABLES & UNITS& SYSTEMSOFWEIGHTS & MEASURES

TABLE1.14-VOLUMESOF SPHERES BY HUNDREDTHS(continued)

d 0 1 2 3 4 5 6 7 8 9 Difference -~ 5.5 87.11 5.6 91.95 5.7 96.97

5.7 5.8 102.2 5.9 107.5

87.59 88.07 88.55 89.03 89.51 92.45 92.94 93.44 93.94 94.44 97.48 97.99 98.51 99.02 99.54

90.00 94.94 100.06 100.1 105.4 110.9

90.48 90.97 91.49 95.44 95.95 96.46

102.7 103.2 103.8 104.3 104.8 108.1 108.6 109.2 109.7 110.3

100.6 101.1 101.6 105.9 106.4 107.0 111.4 112.0 112.5

6.0 113.1 113.7 114.2 114.8 115.4 115.9 116.5 117.7 117.1 118.3 6.1 118.8 119.4 120.0 120.6 121.2 121.8 122.4 123.6 123.0 124.2

6.2 124.8 125.4 126.0 126.6 127.2 127.8 128.4 129.1 129.7 130.3 6.3 130.9 131.5 132.2 132.8 133.4 134.1 134.7 135.3 136.0 136.6

6.4 137.3 137.9 138.5 139.2 139.8 140.5 141.2 141.8 142.5 143.1

6.5 143.8 144.5 145.1 145.8 146.5 147.1 147.8 148.5 149.2 149.6 6.6 150.5 151.2 151.9 152.6 153.3 154.0 154.7 155.4 156.1 156.8 8.7 157.5 158.2 158.9 159.6 160.3 161.0 161.7 162.5 163.2 163.9 6.8 164.6 165.4 166.1 166.8 167.6 168.3 169.0 169.8 170.5 171.3 6.9 172.0 172.8 173.5 174.3 175.0 175.8 176.5 177.3 178.1 178.8

7.0 179.6 7.1 187.4 7.2 195.4 7.3 203.7 7.4 212.2

7.5 220.9 7.6 229.8 7.7 239.0 7.8 248.5 7.9 258.2

8.0 268.1 8.1 278.3 8.2 288.7 8.3 299.4 8.4 310.3

180.4 181.1 181.9 182.7 183.5 184.3 185.0 185.8 186.6 188.2 189.0 189.8 190.6 191.4 192.2 193.0 193.8 194.6 196.2 197.1 197.9 198.7 199.5 200.4 201.2 202.0 202.9 204.5 205.4 206.2 207.1 207.9 208.8 209.6 210.5 211.3 213.0 213.9 214.8 215.6 216.5 217.4 218.3 219.1 220.0

221.8 222.7 223.6 224.4 225.3 226.2 227.1 228.0 228.9 230.8 231.7 232.6 233.5 234.4 235.3 236.3 237.2 238.1 240.0 240.9 241.8 242.8 243.7 244.7 245.6 246.6 247.5 249.4 250.4 251.4 252.3 253.3 254.3 255.2 256.2 257.2 259.1 260.1 261.1 262.1 263.1 264.1 265.1 266.1 267.1

269.1 270.1 271.1 272.1 273.1 274.2 275.2 276.2 277.2 279.3 280.3 281.4 282.4 283.4 284.5 285.5 286.6 287.6 289.8 290.8 291.9 292.9 294.0 295.1 296.2 2972 298.3 300.5 301.6 302.6 303.7 304.8 305.9 307.0 308.1 309.2 311.4 312.6 313.7 314.8 315.9 317.0 318.2 319.3 320.4

8.5 321.6 8.6 333.0 8.7 344.8 8.6 356.8 8.9 369.1

9.0 381.7 9.1 394.6 9.2 407.7 9.3 421.2 9.4 434.9

9.5 448.9 9.6 463.2 9.7 477.9 9.8 492.8 9.9 508.0

10.0 523.6

322.7 323.8 325.0 326.1 327.3 328.4 329.6 330.7 331.9 334.2 335.4 336.5 337.7 338.9 340.1 341.2 342.4 343.6 346.0 347.2 348.4 349.6 350.8 352.0 353.2 354.4 355.6 358.0 359.3 360.5 361.7 362.9 364.2 365.4 366.6 367.9 370.4 371.6 372.9 374.1 375.4 376.6 377.9 379.2 380.4

383.0 384.3 385.5 386.8 388.1 389.4 390.7 392.0 393.3 395.9 397.2 398.5 399.8 401.1 402.4 403.7 405.1 406.4 409.1 410.4 411.7 413.1 414.4 415.7 417.1 418.4 419.8 422.5 423.9 425.2 426.6 428.0 429.4 430.7 432.1 433.5 436.3 437.7 439.1 440.5 441.9 443.3 444.7 446.1 447.5

450.3 451.8 453.2 454.6 456.0 457.5 458.9 460.4 461.8 464.7 466.1 467.6 469.1 470.5 472.0 473.5 474.9 476.4 479.4 480.8 482.3 483.8 485.3 486.8 488.3 489.8 491.3 494.3 495.8 497.3 498.9 500.4 501.9 503.4 505.0 506.5 509.6 511.1 512.7 514.2 515.8 517.3 518.9 520.5 522.0

48 50 52 5 5 6

6

7

8

9

10

11

12

13

14

15

16


n

3 4 5 6 7 a 9 10 12 15 16 20 24 32 48 64

DI

1200 900 72O 600

510.43 450 400 36" 300 24'

220.50 la0 150

110.25 70.50 5O.625

Area Area Area

L2 R2 r2

R R L

L r R

0.4330 1.299 5196 0.5774 2000 1.732 1.000 2.000 4.000 0.7071 1.414 1.414 1.721 2.378 3633 0.8507 1.236 1.176 2.598 2.598 3464 1.000 1.155 1.000 3.634 2.736 3.371 1.152 1.110 0.8678 4.828 2.828 3314 1307 1.082 0.7654 6.182 2 a93 3276 1.462 1.064 0.6840 7.694 2.939 3249 I.618 1.052 0.61 a0

11.20 3.000 3215 1.932 1.035 0.5176 17.64 3.051 3 188 2.405 1.022 0.4158 20.11 3.062 3 ia3 2.563 1.020 0.3902 31.57 3.090 3.168 3196 1.013 0.3129 45.58 3.106 3160 3.831 1.009 0.2611 al.23 3.121 3152 5.101 1.005 0.1960 183.1 3.133 3146 7.645 1.002 0.1308 325.7 3.137 3144 10.19 1.001 0.0981

TABLE 1.15-REGULAR POLYGONS

L r r

r R L

3.464 0.5000 0.2887 2.000 0.7071 0.5000 1 453 0.8090 0.6882 1.155 0.8660 0.8660 0.9631 0.9010 1.038 0.8284 0.9239 1.207 0.7279 0.9397 1.374 0.6498 0.9511 1.539 0.5359 0.9659 1.866 0.4251 0.9781 2.352 0.3978 0.9808 2.514 0.3168 0.9877 3.157 0.2633 0.9914 3.798 0.1970 0.9952 5.077 0.1311 0.9979 7.629 0.0983 0.9968 10.18

n = number of sides, N = 3600/n = angle subtended at the center by one side, 1 = length of one side = R (2 sin (~12) = r(2 tan a/2), R = radius of circumscribed circle = L(% csc u/2) =r(sec a/2).

r = radius of inscribed circle = R (cos a/2) = L(% cot ~12). and

area = L* (l/4 n cot (u/2) = R2 (% n sin (u) = r2 (n tan 0(/2).


TABLE 1.16-BINOMIAL COEFFICIENTS

(n), =I. (n), =R. (n)? =[n(n l)]/(l x 2). (r~)~ =[n(n l)(n 2)]/(1 x 2 x 3). etc., m general, (n), = n(n - l)(n - 2) [n - (r- l)]/(l x 2 x 3 xr) Other notations nCr=(nir) = (nJr

” cm, (n). m, w,

1 I 1

2 1 2 3 1 3 ; 1

4 1 4 6 4

5 1 5 10 10

6 1 6 15 20

7 1 7 21 35

8 1 a 28 56 9 1 9 36 a4

10 1 10 45 120

11 1 11 55 165

12 1 12 66 220

13 1 13 78 286 14 1 14 91 364

I5 1 15 105 455

bid

5 1 15 6

35 21

70 56

126 126

210 252

330 462

495 792

715 1,287 1.001 2,002

1.365 3.003

ml,

1

7

28 64

210

462

924

1.716 3,003

5.005

Hi m 8

21 1

36 9

120 45 330 165

792 495 1.716 I ,287 3,432 3.003

6,435 6,435

11 1

2:: 12 1

78 13 1 1 001 364 91 14

3.003 1.365 455 105

l-38 PETRC ILEUM E :NGINEERING HANDBOOK

TABLE 1.17-COMMON LOGARITHMS (1 .OO TO 2.00)


0.0039 4 0.0082 0.0124 0.0166 0.0208

Number 0 1 2 3 4 5 6

0.0009 0.0013 o.0017 o.0022 0.0026 1.00 1.01 1.02 1.03 1.04

0.0000 0.0043 0.0086 0.0128 00170

0.0004 0.0048 0.0090 0.0133 00175

0.0030 0.0052 0.0056 0.0060 0.0065 0.0069 0.0095 a0099 0.0103 0.0107 0.0111

8

0.0035

0.0137 6.0141 0.0145 0.0179 0.0183 0.0187

1.05 0.0212 0.0216 0.0220 0.0224 0.0228 1 06 0.0253 0.0257 0.0261 0.0265 0.0269 1.07 0.0294 0.0298 0.0302 0.0306 0.0310 1.08 0.0334 0.0338 0.0342 0.0346 0.0350 109 0.0374 0.0378 0.0382 0.0386 0.0390

0.0149 0.0191

0.0233 0.0273 0.0314 0.0354 0.0394

0.0154 0.0195

0.0073 0.0077 0.0116 00120 0.0158 0.0162 0.0199 0.0204

0.0237 0.0241 0.0278 0.0282 0.0318 0.0322 0.0358 0.0362 0.0398 0.0402

00245 0.0249 0.0286 0.0326 0.0366 00406

0.0290 0.0330 0.0370 0.0410

1.10 0.0414 0.0418 0.0422 00426 00430 0.0434 0.0438 0.0441 00445 0.0449 1.11 0.0453 0.0457 0.0461 00465 0 0469 0.0473 0.0477 0.0481 00484 0.0488 1.12 0.0492 0 0496 0 0500 00504 00508 0.0512 0.0515 0.0519 00523 0.0527 1.13 0.0531 0.0535 0.0538 0.0542 00546 0.0550 0.0554 0.0558 00561 0.0565 1 14 0.0569 00573 0.0577 00580 00584 00588 0.0592 0.0596 0 0599 0.0603

1.15 1.16 1.17 1.18 1.19

1.20 1 21 1.22 1.23 1.24

0.0607 0.0645 00682 0.0719 0.0755

0.0792 0.0828 0.0864 0.0899 0.0934

1.25 0.0969 1.26 0.1004 1.27 0.1038 1.28 0.1072 1.29 01106

1.30 1.31 1.32 1.33 1.34

0.1139 0.1173 0.1206 0.1239 0.1271

1.35 1.36 1.37 1.38 1.39

0.1303 0.1307 0.1310 01313 01316 0.1319 0.1323 0.1326 0.1335 0.1339 0.1342 0.1345 0 1348 0.1351 0.1355 0.1358

1.40 1.41 1.42 1.43 1.44

1.45 1.46 1.47 1.48 1.49

0.1367 0.1399 0.1430

0.1461 01492 0.1523 0.1553 0.1584

0.1614 0 1644 0.1673 0.1703 0.1732

0.0633 0.0671 0.0708 0.0745 0.0781

0.0817 0.0853 0.0888 0.0924 0.0959

0.0993

00637 00674 00711 0.0748 00785

0.0821 00856 0 0892 0 0927 0 0962

0.0997

0.0641 0.0678 0.0715 0.0752 0.0788

0.0611 0.0615 0.0618 0.0622 0.0626 0.0630 0.0648 0.0652 00656 0.0660 0.0663 0.0667 0.0686 0.0689 0.0693 0 0697 0.0700 0.0704 0.0722 0.0726 0.0730 00734 0.0737 0.0741 0.0759 0.0763 0.0766 00770 0.0774 0.0777

0 0795 0.0799 0.0803 00806 0.0810 0.0813 0.0831 0.0835 0.0839 00842 0.0846 0.0849 0.0867 0.0871 0.0874 0.0878 0.0881 0.0885 0.0903 0.0906 0.0910 0.0913 0.0917 0.0920 0.0938 0.0941 0.0945 0.0948 0.0952 0.0955

0.0973 0.0976 0.0980 0 0983 0.0986 0.0990 0.1011 0.1014 01017 0.1021 0.1024 0.1045 0.1048 0.1052 0.1055 0.1059 0.1079 0.1082 01086 0.1089 0.1092 0.1113 0.1116 01119 0.1123 0.1126

0.1146 0.1149 01153 0.1156 0.1159 0.1179 0.1183 01186 0.1189 0.1193

0.0824 0.0860 0.0896 0.0931 0.0966

0.1000 3 0.1007 0 1041

0.1028 01031 0.1035 0.1062 01065 0.1069

01075 0.1109

0.1143 01176 0.1209 0.1242 0.1274

0.1096 0.1129

0.1163 0.1196 0.1229 0.1261 0.1294

0.1099 0.1133

01166 0.1199 01232 01265 01297

01329 01361

0.1103 0.1136

0.1169 0.1202 0.1235 0.1268 0.1300

0.1332 0.1364 0.1396 0.1427 0.1458

0.1489 0.1520 0.1550 01581 0.1611

0.1641 0 1670 01700 0.1729 0.1758

0.1212 0.1216 0.1219 0.1222 0.1225 0.1245 0.1248 0 1252 0.1255 0.1258 0.1278 0.1281 01284 0.1287 0.1290

0.1370 0.1402 0.1433

0.1464 0.1495 0.1526 0.1556 0.1587

0.1617 0.1647 0.1676 0.1706 0.1735

0.1374 0.1377 0 1405 0.1408 0.1436 0.1440

0.1467 0.1471 0.1498 01501 0.1529 01532 01559 0.1562 0.1590 0.1593

0.1620 0.1623

0.1380 0.1383 01411 0.1414 0 1443 0.1446

01474 0.1477 01504 01508 01535 0.1538 0.1565 0.1569 01596 0.1599

0.1626 0.1629

0.1386 0.1389 0.1418 0.1421 0.1449 0.1452

0.1480 0.1483 0.1511 0.1514 0.1541 0.1544 0.1572 0.1575 0.1602 0.1605

0.1632 0.1635

01392 01424 01455

01486 01517 01547 01578 01608

01638 0.1649 0.1652 0 1655 0.1658 0.1661 0.1679 0.1682 0.1685 0.1688 0.1691

0.1664 01667 0.1694 0.1697 0.1723 01726 0.1752 01755

01708 01711 01714 01717 0.1720 0.1738 0.1741 0.1744 0.1746 0.1749

Moving the decimal polntn places to the right (or left) In the number requires adding +n (or-n) in the body of the table See Table 1.18.

MATHEMATICALTABLES & UNITS &SYSTEMSOFWElGHTS& MEASURES I-39

Number

1 so 1.51 1.52 1.53 1.54

Average 0 1 2 3 4 5 6 7 8 9 Difference

0.1761 0.1764 0.1767 0.1770 0.1772 0.1775 0.1778 0.1781 01784 0.1787 3 0.1790 0.1793 0.1796 0.1798 0 1801 0.1804 0.1807 0.1810 0 1813 0.1816 0.1818 0.1821 0.1824 0.1827 0.1830 0.1833 0.1836 0.1838 0.1841 0.1844 0.1847 0.1850 0.1853 0.1855 0.1858 0.1861 0.1864 0.1867 0 1870 0 1872 0.1875 0.1878 0.1881 0.1884 0.1886 0.1889 0.1892 0.1895 0.1898 0.1901

1.55 0.1903 0.1906 0.1909 0.1912 0.1915 0.1917 0.1920 0.1923 01926 0.1928 1.56 01931 0.1934 0.1937 0.1940 0.1942 0.1945 0.1948 0.1951 01953 0.1956 1.57 0.1959 0.1962 0.1965 0.1967 0.1970 0.1973 0.1976 0.1978 0.1981 0.1984 1.58 0.1987 0.1989 0.1992 0.1995 0.1998 0.2000 0.2003 0.2006 0.2009 0.2011 1.59 02014 0.2017 0.2019 0.2022 0.2025 0.2028 0.2030 0.2033 0.2036 0.2038

1.60 02041 0.2044 1.61 0.2068 0.2071 1 62 0.2095 0.2098 1 63 02122 0.2125 1.64 02148 0.2151

0.2052 0.2055 0.2057 0.2060 0.2063 0.2066 0.2079 0.2082 0.2084 0.2087 0.2090 0.2092 0.2106 0.2109 0.2111 0.2114 0.2117 0.2119 0.2133 0.2135 0.2138 0.2140 0.2143 0.2146 0.2159 0.2162 0.2164 0.2167 0.2170 0.2172

1.65 1.66 167 1.68 1.69

1.70 1.71 1 72 1 73 1.74

02175 0.2177 0.2201 0.2204 02227 0.2230 02253 0.2256 0 2279 0.2281

0.2047 0.2049 0.2074 0.2076 0.2101 0.2103 0.2127 0.2130 0.2154 0.2156

0.2180 0.2183 0.2206 0.2209 0.2232 0.2235 0.2258 0.2261 0.2284 0.2287

0.2185 0.2188 0.2191 0.2193 0.2196 0.2198 0.2212 0.2214 0.2217 0.2219 0.2222 0.2225 0.2238 0.2240 0.2243 0.2245 02248 0.2251 0.2263 0.2266 0.2269 0.2271 02274 0.2276 0.2289 0.2292 0.2294 0.2297 0.2299 0.2302

1.75 176 1.77 1.78 1.79

02304 0.2307 0.2310 0.2312 0.2315 0.2317 0.2320 0.2322 02325 0.2327 02330 0.2333 0.2335 0.2338 0.2340 0.2343 0.2345 0.2348 0.2350 0.2353 0.2355 0.2358 0.2360 0.2363 0.2365 0.2368 0.2370 0.2373 02375 0.2378 0.2380 0.2383 0.2385 0.2388 0.2390 0.2393 0.2395 0.2398 02400 0.2403 0.2405 0.2408 0.2410 0.2413 0.2415 0.2418 0.2420 0.2423 0.2425 0.2428 2

02430 0.2433 0.2435 0.2438 0.2440 0.2443 0.2445 0.2448 0.2450 0.2453 0.2455 0.2458 0.2460 0.2463 0.2465 0.2467 0.2470 0.2472 0.2475 0.2477 0.2480 0.2482 0.2485 02487 0.2490 0.2492 0.2494 0.2497 0 2499 0.2502 0.2504 0.2507 0.2509 0.2512 0.2514 0.2516 0.2519 0.2521 02524 0.2526 0.2529 0.2531 0.2533 0.2536 0.2538 0.2541 0.2543 0.2545 0.2548 0.2550

1.80 0.2553 0.2555 0.2558 0.2560 0.2562 0.2565 0.2567 0.2570 0.2572 0.2574 1.81 0 2577 0.2579 0.2582 0.2584 0.2586 0.2589 0.2591 0.2594 0.2596 0.2598 1.82 0.2601 0.2603 0.2605 02608 0.2610 0.2613 0.2615 0.2617 0.2620 0.2622 1.83 0.2625 0.2627 0.2629 0.2632 0.2634 0.2636 0.2639 0.2641 02643 0.2646 1.84 0.2648 0.2651 0.2653 02655 0.2658 0.2660 0.2662 0.2665 0.2667 0.2669

1.85 1.86 1.87 1.88 1 89

0.2672 0.2674 0.2676 02679 0.2681 0.2683 0.2686 0.2688 0.2690 0.2693 0 2695 0.2697 0.2700 0.2702 0.2704 0.2707 0.2709 0.2711 0 2714 0.2716 0.2718 0.2721 0.2723 0.2725 0.2728 0.2730 0.2732 0.2735 0.2737 0.2739 0.2742 0.2744 0.2746 0 2749 0.2751 0.2753 0.2755 0.2758 0.2760 0.2762

1.90 1.91 1 92 1 93 194

195 196 197 1 98 1 99

0.2765

0.2788 0.2810 0.2833 0.2856 0.2878

0.2900 0.2923 0.2945 0.2967 0.2989

0.2767

0.2790 0.2813 0.2835 0.2858 0.2880

0.2903 0.2925 0.2947 0.2969 0.2991

0.2769

0.2792 0.2815 0.2838 0.2860 0.2882

0.2772

0.2794 0.2817 02840 02862 02885

0.2905 0.2927 0.2949 0.2971 0.2993

0 2907 0 2929 0.2951 0.2973 0.2995

0.2774

0.2797 0.2819 0.2842 0.2865 0.2887

0.2909 0.2931 0.2953 0.2975 0.2997

0.2776 0.2781 0.2783 0.2785

0.2799 0.2822 0.2844 0.2867 0.2889

0.2778

0.2801 0.2824 0.2847 0.2869 0.2891

0.2914 0.2936 0.2958 0.2980 0.3002

0.2804 0.2826 0.2849 0.2871 0.2894

0.2911 0.2934 0.2956 0.2978 0.2999

0.2916 0.2938 0.2960 0.2982 0.3004

02806 0.2808 0.2828 0 2831 0.2851 0.2853 0.2874 0.2876 0.2896 0.2898

0.2918 0.2920 0.2940 0.2942 0.2962 0.2964 0.2984 0.2986 0.3006 0.3008

TABLE 1.17-COMMON LOGARITHMS (1.00 TO 2.00) (continued)

Mowng the decimal point n places 10 the right [or left)in the number requires adding +n (or-r?) in the body of the table See Table 1.18


TABLEl.lB-COMMON LOGARITHMS

Average 0 1 2 3 4 5 6 7 8 9 Difference __ ~ ~

0.0000 0.0043 0.0086 0.0128 0 0170 0.0212 0.0253 0.0294 0.0334 0.0374 See Table 1.17

Number

1.0 1.1 1.2 1.3 1.4

1.5 1.6 1.7 1.8 1.9

0.0414 0.0453 0.0792 0.0828 0.1139 0.1173 0.1461 0.1492

0.1761 0.1790 0.2041 0.2068 0.2304 0.2330 0.2553 0.2577 0.2788 0.2810

2.0 0.3010 0.3032 2.1 0.3222 0.3243 2.2 0.3424 0.3444 2.3 03617 0.3636 2.4 0.3802 0.3820

2.5 0.3979 0.3997 2.6 0.4150 0.4166 2.7 04314 0.4330 2.8 04472 04487 2.9 0.4624 0.4639

3.0 31 3.2 3.3 3.4

0.4771 0.4786 0 4914 0.4928 0.5051 0.5065 0.5185 0.5198 0.5315 0.5328

3.5 3.6 3.7 3.8 3.9

4.0 4.1 4.2 4.3 4.4

0.5441 0.5453 0.5563 0.5575 0.5682 0.5694 0.5798 0.5809 0.5911 0.5922

0.6021 0.6031 0.6128 0.6138 0.6232 0.6243 0.6335 0.6345 0.6435 0.6444

4.5 4.6 4.7 4.8 4.9

5.0 5.1 52 5.3 5.4

0.6532 0.6542 0.6551 0.6628 0.6637 0.6646 0.6721 0.6730 0.6739 0.6812 0.6812 0.6830 0.6902 0.6911 0.6920

0.6990 0.6998 0.7007 0.7076 0.7084 0.7093 0.7160 0.7168 0.7177 0.7243 0.7251 0.7324 0.7332

5.5 0.7404 0.7412 5.6 0.7482 0 7490 5.7 0.7559 0.7566 5.8 0.7634 0.7642 59 0.7709 0.7716

0.0492 0.0864 0.1206 0.1523

0.1818 0.2095

0.0531 0.0899 0.1239 0.1553

0.1847 0.2122

0.0569 0.0607 0.0645 0.0682 0.0719 0 0934 0.0969 0.1004 0.1038 0.1072 0.1271 0.1303 0.1335 0.1367 0.1399 0.1584 0.1614 0.1644 0.1673 0.1703

01875 0.1903 0.1931 0.1959 0.1987 0.2148 0.2175 0.2201 0.2227 0.2253

0.0755 0.1106 0.1430 0.1732

0.2014 0.2279

0.2355 0.2380 0.2405 0.2430 0.2455 0.2480 0.2504 0.2529 0.2601 0.2625 0.2648 0.2672 0.2695 0.2718 0.2742 0.2765 0.2833 0.2856

0.3483 0.3674 0.3856

0.4031 04200 0.4362

0.2878

0.3096 0.3304 0.3502 0.3692 0.3874

0.4048 04216 0.4378

0.2900

0.3054 0.3075 0.3263 0.3284 0.3464 0.3655 0.3838

0.4014 0.4183 0.4346 0.4502 0.4654

0.3118 0.3324 0.3522 0.3711 0.3892

0.4065

0.2923

0.3139 0.3345 0.3541 0.3729 0.3909

0.4082

0.2945 0.2967 0.2989

See Table 1.17 See Table 1.17 See Table 1.17 See Table 1.17

See Table 1.17 See Table 1.17 See Table 1.17 See Table 1.17 See Table 1.17

0.3160 0.3365 0.3560 0.3747 0.3927

0.4232 0.4249 0.4393 0.4409 0.4548 0.4564 0.4698 0.4713

0.4099 0.4265 0.4425 0.4579 0.4728

0.3181 0.3385 0.3579 0.3766 0.3945

0.4116 04281 0.4440

0.3201 0.3404 0.3598 0.3784 0.3962

0.4133 0.4298 0.4456 0.4609 0.4757

21 20 19 18 17

04518 0.4533 0.4669 0.4683

0.4594 0.4742

17 16 16 15 15

0.4800 0.4814 0.4829 0.4843 0.4857 0.4871 0.4886 0.4900 14 0.4942 0.4955 0.4969 0.4983 0.4997 0.5011 0.5024 0.5038 14 0.5079 0.5092 0.5105 0.5119 0.5132 0.5145 0.5159 0.5172 13 0.5211 0.5224 0.5237 0.5250 0.5263 0.5276 0.5289 0.5302 13 0.5340 0.5353 0.5366 0.5378 0.5391 0.5403 0.5416 0.5428 13

0.5465 0.5478 0.5587 0.5599 0.5705 0.5717 0.5821 0.5832 0.5933 0.5944

0.5490 0.5502 0.5611 0.5623 0.5729 0.5740 0.5843 0.5855

0.5527 0.5539 0.5551 0.5647 0.5658 0.5670

0.5955 0.5966

0.5514 0.5635 0.5752 0.5866 0.5977

0.5763 0.5775 0.5786 0.5877 0.5888 0.5899

12 12 12 11 11

0.6042 0.6149 0.6253 0.6355 0.6454

0.6053 0.6064 0.6075 0.6160 0.6170 0.6180

0.6274 0.6284 0.6375 0.6385 0.6474 0.6484

06571 0.6580

0.6263 0.6365 0.6464

0.6561 0.6656 0.6749 0.6839 0 6928

0.7016 0.7101 0.7185 07267 07348

07427 0.7505 07582 07657 07731

0.6085 0.6191 0.6294 0.6395 0.6493

0.5988 0.5999 0.6010

0.6096 0.6107 0.6117 0.6201 0.6212 0.6222 0.6304 0.6314 0.6325 0.6405 0.6415 0.6425 0.6503 0.6513 0.6522

0.6599 0.6609 0.6618

11 10 10 10 10

0.6590 0.6665 0.6675 0.6684 0.6693 0.6702 0.6712 0.6758 0.6767 0.6776 0.6785 0.6794 0.6803 0.6848 0.6857 0.6866 0.6875 0.6884 0.6893 0.6937 0.6946 0.6955 0.6964 0.6972 0.6981

10 10 9 9 9

0.7259 0.7340

0.7419 0.7497 0.7574 0.7649 0.7723

0.7024 0.7033 0.7042 0.7050 0.7059 0.7067 07110 0.7118 0.7126 0.7135 0.7143 0.7152 0.7193 0.7202 0.7210 0.7218 0.7226 0.7235 0.7275 0.7284 0.7292 0.7300 0.7308 0.7316 07356 0.7364 0.7372 0.7380 0.7388 0.7396

0.7435 0.7443 0.7451 0.7459 0.7466 0.7474 0.7513 0.7520 0.7528 0.7536 0.7543 0.7551 0.7589 0.7597 0.7604 0.7612 0.7619 0.7627 0.7664 0.7672 0.7679 0.7686 0.7694 0.7701 0.7738 0.7745 0.7752 0.7760 07767 0.7774

9 8' 8 8 8

8 8 8 7 7

This table givesthecommon logarithms of numbers between 1 and 10,correclto four places. Moving the decimalpointn places to the right (or left) in the number is equivalent to adding n (or -n) to the logarithm. Thus, log 0.017453=0.2419-2, which may also be written 2.2419 or 8.2419- 10.

For example, log(ab)=log a+log b, log (aN)=N log a, log(alb)=log a-log b, and log(?'a)= l/N log a.

MATHEMATlCALTABLES& UNlTS&SYSTEMSOFWElGHTS& MEASURES l-41

TABLE 1.18- -COMMON LOGARITHMS (continued)

Average Difference Number

6.0 6.1 6.2 6.3 6.4

0 1 2 3

0.7782 0.7789 07796 0.7803 0.7853 0.7924 0.7993 0.8062

0.7860 07868 0.7875 0.7931 0.7938 0.7945 0.8000 0.8007 0.8014 0.8069 0.8075 0.8082

6.5 0.8129 0.8136 6.6 0.8195 0.8202 6.7 0.8261 0.8267 6.8 0.8325 0.8331 6.9 0.8388 0.8395

7.0 0.8451 0.8457 7.1 0.8513 0.8519 7.2 0.8573 0.8579 7.3 0.8633 0.8639 7.4 0.8692 0.8698

7.5 76 7.7 7.8 7.9

0.8751 0.8808 0.8865

0.8756 0.8814 0.8871 0.8927 0.8982

8.0 81 82 8.3 8.4

0.8921 0.8976

0.9031 0.9085 09138 0.9191 0 9243

0.9036 0.9042 0.9047 0 9053 0.9058 0.9063 0.9069 0.9074 0.9090 0.9096 0.9101 0.9106 0.9112 0.9117 0.9122 0.9128 09143 0.9149 0.9154 09159 0.9165 0.9170 0.9175 09180 0.9196 0.9201 0.9206 0.9212 0.9217 0.9222 0.9227 0.9232 0 9248 0.9253 0.9258 0.9263 0 9269 0.9274 0.9279 0.9284

85 8.6 8.7 8.8 8.9

0 9294 0.9345 0.9395 0.9445 0.9494

9.0 9.1 9.2 9.3 9.4

0.9542 0.9590 0.9638 0.9685 0.9731

9.5 0.9777 9.6 0.9823 9.7 0.9868 9.8 0.9912 9.9 0.9956

0.9299 0.9350 0.9400 0.9450 0.9499

0.9547 0.9595 0.9643 0.9689 0.9736

0.9782 0.9827 0.9872 0.9917 0.9961

4 5 6 7 8 9

0.7810 0.7818 0.7825 0.7832 0.7839 0.7846 0.7882 0.7889 0.7896 0.7903 0.7910 0.7917 0.7952 0.7959 0.7966 0.7973 0.7980 0.7987 0.8021 0.8028 0.8035 0.8041 0.8048 0.8055 0.8089 0.8096 0.8102 0.8109 0.8116 0.8122

0.8189 0.8254 0.8319 0.8382 0.8445

0.8506 0.8567 0.8627 0.8686 0.8745

0.8802 0.8859 0.8915 0.8971 0.9025

0.9079 0.9133 0.9186 0.9238 0.9289

0.9340 0.9390 0.9440 0.9489

0.8142 0.8149 0.8156 0.8162 0.8169 0.8176 0.8182 0.8209 0.8215 0.8222 0.8228 0.8235 0.8241 0.8248 0.8274 0.8280 0.8287 0.8293 0.8299 0.8306 0.8312 0.8338 0.8344 0.8351 0.8357 0.8363 0.8370 0.8376 0.8401 0.8407 0.8414 0.8420 0.8426 0.8432 0.8439

0.8463 0.8470 0.8525 0.8531 0.8585 0.8591 0.8645 0.8651 0.8704 0.8710

0.8762 0.8768 0.8820 0.8825 0.8876 0.8882 0.8932 0.8938 0.8987 0.8993

0.8476 0.8482 0.8488 0.8494 0.8500 0.8537 0.8543 0.8549 0.8555 0.8561 0.8597 0.8603 0.8609 0.8615 0.8621 0.8657 0.8663 0.8669 0.8675 0.8681 0.8716 0.8722 0.8727 0.8733 0.8739

0.8774 0.8779 0.8785 0.8791 0.8797 0.8831 0.8837 0.8842 0.8848 0.8854 0.8887 0.8893 0.8899 0.8904 0.8910 0.8943 0.8949 0.8954 0.8960 0.8965 0.8998 0.9004 0.9009 0.9015 0.9020

6 6 6

6 6 6 6 6

6 6 6 6 5

5 5 5 5 5

5 5 5 5 5

5 5 5 5 5

5 4 4 4 4

0.9304 0.9309 0.9315 0.9320 0.9325 0.9330 0.9335 0.9355 0.9360 0.9365 0.9370 0.9375 0.9380 0.9385 0.9405 0.9410

0.9832 0.9836 0.9877 0.9881 0.9921 0.9926 0.9965 0.9969

0.9455 0.9460 0.9504 0.9509

0.9552 0.9557 0.9600 0.9605 0.9647 0.9652 0.9694 0.9699 0.9741 0.9745

0.9786 0.9791 0.9841 0.9845 0.9850 0.9854 0.9859

0.9415

0.9886 0.9890 0.9894 0.9899 0.9903

0.9420 0.9425 0.9430 0.9465 0.9469 0.9474 0.9479 0.9513 0.9518 0.9523 0.9528

0.9562 0.9566 0.9571 0.9576 0.9609 0.9614 0.9619 0.9624 0.9657 0.9661 0.9666 0.9671 0.9703 0.9708 0.9713 0.9717 0.9750 0.9754 0.9759 0.9763

0.9795 0.9800 0.9805 0.9809

0.9435 0.9484 0.9533

0.9581 0.9628 0.9675 0.9722 0.9768

0.9814

0.9538

0.9586 0.9633 0.9680 0.9727 0.9773

0.9818 0.9863 0.9908 0.9952 0.9996

0.9930 0.9934 0.9939 0.9943 0.9948 0.9974 0.9978 0.9983 0.9987 0.9991

109 v=o 4971 log &2=0 1961, log $t2=09943 log \,<=02466. log e-04343, and log (04343)=06378- 1


TABLE 1.19-DEGREES AND MINUTES EXPRESSED IN RADIANS’

1 00175 2 0.0349 3 0.0524 4 0.0698 5 0.0873 6 0.1047

L 0.1222 0 1396

9 01571 10 0.1745 11 0.1920 12 0.2094 13 0.2269 14 02443 15 02618 16 0 2793 17 0.2967 18 0.3142 19 0.3316 20 0.3491 21 0.3665 22 0.3840 23 0.4014 24 0.4189 25 0.4363 26 0.4538 27 0.4712 28 0.4887 29 0.5061 30 0.5236 31 0.5411 32 0.5585 33 0.5760 34 0.5934 35 0.6109 36 0.6283 37 0.6458 38 0.6632 39 0.6807 40 0.6981 41 0.7156 42 0.7330 43 0.7505 44 0.7679 45 0.7854 46 0.8029 47 0.8203 48 0.8378 49 0.8552 50 0.8727 51 0 8901 52 0.9076 53 0.9250 54 0.9425 55 0 9599 56 0.9774 57 0 9948 58 1.0123 59 1.0297 60 1.0472

Degrees

61 1.0647 62 1.0821 63 1.0996 64 1.1170 65 1.1345 66 1.1519 67 1.1694 68 1.1868 69 1.2043 70 1.2217 71 1.2392 72 1.2566 73 1.2741 74 12915 75 1.3090 76 1.3265 77 1.3439 78 1.3614 79 1.3788 80 1.3963 81 1.4137 82 1.4312 83 1.4486 84 1.4661 a5 1.4835 86 1.5010 87 1.5184 88 1.5359 89 1.5533 90 1.5708 91 1.5882 92 1.6057 93 1.6232 94 1.6406 95 1.6581 96 1.6755 97 1.6930 98 1 7104 99 1.7279

100 1.7453 101 1.7628 102 1.7802 103 1.7977 104 1.8151 105 18326 106 18500 107 1.8675 108 1.8850 109 1.9024 110 1.9199 111 1.9373 112 1.9548 113 1.9722 114 1.9897 115 20071 116 2.0246 117 2.0420 118 2.0595 119 2.0769 120 2.0944

121 2.1118 122 2.1293 123 2.1468 124 21642 125 2.1817 126 21991 127 2.2166 128 22340 129 2.2515 130 22689 131 2.2864 132 23038 133 2.3213 134 2.3387 135 23562 136 2.3736 137 2.3911 138 24086 139 24260 140 2.4435 141 2.4609 142 2.4784 143 2.4958 144 2.5133 145 25307 146 25482 147 2.5656 148 2.5831 149 2.6005 150 2.6180 151 2.6354 152 2.6529 153 2.6704 154 2.6878 155 27053 156 27227 157 2.7402 158 2.7576 159 2.7751 160 2.7925 161 2.8100 162 2.8274 163 28449 164 2.8623 165 28798 166 2.8972 167 29147 168 2.9322 169 2.9496 170 2.9671 171 2.9845 172 3.0020 173 3.0194 174 3.0369 175 3.0543 176 3.0718 177 3.0892 178 3.1067 179 3.1241 180 3.1416

O”.Ol 00002 OO.51 00.02 00003 OO.52 00 03 00.04 00.05 OO.06 00.07 O" 08 00.09 00.10 00.11 00 12 00.13 00.14 00.15 OO.16 00.17 OO.18 00.19 00.20 00.21 00.22 OO.23 OO.24 OO.25 0".26 00.27 OO.28 00 29 0".30 00.31 0".32 00.33 00.34 00.35 OO.36 00.37 0".38 00.39 OO.40 00.41 OO.42 0043 00.44 00.45 OO.46 00 47 OO.48 00.49 00.50

Hundredths Minutes

0.0005 00.53 00007 00 54 0.0009 00.55 0.0010 OO.56 00012 00.57 00014 OO.58 0.0016 00.59 00017 OO.60 0.0019 OO.61 00021 OO.62 0.0023 OO.63 00024 OO.64 00026 0".65 00028 00.66 00030 OO.67 0.0031 00.68 00033 OO.69 00035 00.70 00037 00.71 0.0038 OO.72 0.0040 00.73 0.0042 00.74 00044 00.75 00045 OO.76 00047 00.77 0.0049 00.78 0.0051 00.79 0.0052 00.80 0.0054 OO.81 00056 0".82 0.0058 OO.83 0.0059 0".84 0.0061 OO.85 0.0063 00.86 0.0065 OO.87 0.0066 00.88 0.0068 OO.89 0.0070 00.90 0.0072 00.91 0.0073 00.92 0.0161 0.0075 00.93 0.0162 0.0077 00.94 0.0164 0.0079 00.95 0.0166 0.0080 OO.96 0.0168 00082 00.97 0.0169 0.0084 0".98 0.0171 0.0086 00.99 0.0173 0.0087 10.00 0.0175

0.0089 0.0091 0.0093 0.0094 0.0096 0.0098 0.0099 0.0101 0.0103 0.0105 0.0106 0.0108 0.0110 0.0112 0.0113 0.0115 0.0117 0.0119 0.0120 0.0122 0.0124 0.0126 0.0127 0.0129 0.0131 0.0133 0.0134 0.0136 0.0138 0.0140 0.0141 0.0143 0.0145 0.0147 0.0148 0.0150 0.0152 0.0154 0.0155 0.0157 0.0159

1 0.0003 2 0.0006 3 00009 4 0.0012 5 0.0015 6 0.0017 7 0.0020 8 00023 9 0.0026 10 0.0029 11 0.0032 12 0.0035 13 0.0038 14 0.0041 15 0.0044 16 00047 17 0.0049 18 00052 19 0.0055 20 0.0058 21 0.0061 22 0.0064 23 0.0067 24 0.0070 25 0.0073 26 0.0076 27 0.0079 28 0.0081 29 00084 30 0.0087 31 0.0090 32 0.0093 33 0.0096 34 0.0099 35 0.0102 36 0.0105 37 00108 38 0.0111 39 0.0113 40 0.0116 41 0.0119 42 0.0122 43 0.0125 44 0.0128 45 0.0131 46 00134 47 0.0137 48 0.0140 49 00143 50 0.0145 51 00148 52 0.0151 53 00154 54 00157 55 00160 56 0.0163 57 0.0166 58 0.0169 59 0.0172 60 0.0175

Arc ,“=00,74533. Arc ,‘=0000290888,Arc 1~=000000484814 1 radnn = 57”0295780=57°17’07468=57017’44”0806 ‘See aim Table 1 40

MATHEMATICALTABLE !S & UI VlTS& SYSTEMSOFWEIGHTS & MEASURES 1-43

TABLEl.PO-RADIANSEXPRESSED

36O 67 1.27 72O.77 1.90 108O.86 37".24 1.28 730.34 1.91 1090.43

IN DEGREES

2.53 144O 96 2.54 1450 53

lnterpolatlon

0.0002 00.01 0.01 00.57 0.02 10.15

0 64 0.65 066 0 67 0.68 0 69

0.03 I" 72 0.04 20.29 0.05 20.86 0.06 30.44 0.07 40.01 0.08 4O.58

37O.82 1.29 73" 91 1.92 110~.01 2.55 146O IO 0.0004 00.02 38O.39 1.30 74' 48 1.93 IlOO. 2.56 146O.68 0.0006 00.03 38O .96

5OO.42

1.31

1.51

39".53 1.32 400.11

500.99

1.33 4OO.68

1.52

1.34 41O.25 1.35 41".83

510.57

1.36

1.53

42O.40 1.37 42O.97

52O.14

1.38 430.54

1.54

1.39 44O.12 1.40 44O.69 1.41 45O.26 1.42 45O.84 1.43 46O.41 1.44 46O.98 1.45 47O.56 1.46 480.13 1.47 48O.70 1.48 49O.27 1.49 49O .85 1.50

75O.06 1.94

86O.52

1110.15

2.14

2.57

122O.61

147O 25 75'63 1.95

87O.09

1110.73

2.15

2.58

123O.19

147'82 76O.20 1.96

87O.66

112O.30

2.16

2.59

123O.76

148O.40 76".78 1.97

88O.24

112O.87

2.17

2.60

124O.33

148O.97 770.35 1.98 1130.45 2.61 1490.54 770.92 1.99 114O.02 2.62 150~11 780.50 2.00 1140.59 2.63 150°69 790.07 2.01 115".16 2.64 151° 26 79O.64 2.02 1150.74 2.65 151O.83 8OO.21 2.03 116O.31 2.66 152O.41 8OO.79 2.04 1160.88 2.67 152O.98 81O.36 2.05 117O.46 2.68 1530.55 81O.93 2.06 118°.03 2.69 1540.13 82O.51 2.07 1180.60 2.70 154O.70 83O.08 2.08 119O.18 2.71 155O.27 83O.65 2.09 1190.75 2.72 155O.84 84O.22 2.10 12OO.32 2.73 156O.42 84O.80 2.11 12OO.89 2.74 156O .99 85".37 2.12 121O.47 2.75 157O.56 85O.94 2.13 1220.04 2.76 158O.14

2.77 158O.71 2.78 159O.28 2.79 1590.86 2.80 16OO.43 2.81 161'.00 2.82 161O.57 2.83 162O.15 2.84 162O.72 2.85 163O.29 2.86 163O87 2.87 164O.44 2.88 165O 01 2.89 165O.58 2.90 166O16 2.91 166O.73

0.0008 00 05 0.0010 0".06 0.0012 00.07 0.0014 0".08 0.0016 OO.09 0.0018 00.10 0.0020 00.11 0.0022 00.13 0.0024 OO.14 0.0026 00.15 0.0028 OO.16 0.0030 00.17 0.0032 OO.18

0.70 0.71

0.09 5O.16 0.10 50.73

0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80

0.11 6O.30 0.12 60.88 0.13 70.45 0.14 8O.02 0.15 8O.59 0.16 90.17 0.17 90.74 0.18 100.31 081 0.19 lo".89 0.82

0.0034 00.19 0.0036 00.21

0.20 II='.46 0.83 0.21 12O.03 0.84 0.22 12O.61 0.85 0.23 13O.18 0.86 0.24 130.75 0.87 0.25 14O.32 0.88 0.26 140.90 0.89 0.27 150.47 0.90 0.28 16O.04 0.91 0.29 16O.62 0.92 0.30 170.19 0.93 0.31 17".76 0.94 0.32 18O.33 0.95 0.33 18O.91 0.96 0.34 IgO. 0.97 0.35 2OO.05 0.98 0.36 2OO.63 0.99 0.37 210.20 1.00 0.38 210.77 1.01 0.39 22O.35 1.02 0.40 220.92 1.03 0.41 230.49 1.04 042 24O.06 1.05 0.43 24O.64 1.06

0.0038 00.22 0.0040 0".23 0.0042 OO.24 0.0044 OO.25 0.0046 OO.26 0.0048 00.28 0.0050 00.29 0.0052 00.30 0.0054 00.31

52O.71 1.55 530.29 1.56 53O.86 1.57 540.43 1.58 550.00 1.59 55O.58 1.60 56O.15 1.61 56O.72 1.62 570.30 1.63 57O.87 1.64 58".44 1.65 590.01 1.66 590.59 1.67

880.81 2.18 124O.90 89O.38 2.19 125O.48 89".95 2.20 126O.05 900.53 2.21 126O.62 910.10 2.22 127O.20 91° 67 2.23 127O.77 920.25 2.24 128O.34 92O 82 2.25 128O.92 930.39 2.26 1290.49 930 97 2.27 130a.06 940.54 2.28 13OO.63 950.11 2.29 131O.21 95O.68 2.30 131O.78

0.0056 OO.32 0.0058 00.33 0.0060 00.34 0.0062 OO.36 0.0064 00.37 0.0066 OO.38 0.0068 00.39 00070 00.40 0.0072 00.41 0.0074 OO.42 0.0076 00.44 0.0078 OO.45 0.0080 On.46

2.92 167O.30 2.93 167O.88 2.94 168O.45 2.95 169O.02

600.16 1.68 96O.26 2.31 132O.35 6OO.73 1.69 96O.83 2.32 132O.93

0.0082 00.47 0.0084 OO.48

0.44 25O.21 0.45 25".78 0.46 26O.36 0.47 26O.93

1.07 1.08

61O.31 1.70 97O.40 2.33 133O.50 2.96 169O.60 610.88 171 97O.98 2.34 134O.07 2.97 170~.17 62O.45 172 98O.55 2.35 134O.65 2.98 1700.74 63'=.03 173 99O.12 2.36 135O.22 2.99 1710.31

0.0086 00.49 0.0088 00.50 0.0090 OO.52 0.0092 00.53 0.0094 00.54 0.0096 OO.55 0.0098 OO.56

Multiples of r

1.09 1.10

0.48 27O.50 1.11 63O.60 1.74 99O.69 2.37 1350.79 3.00 71O.89 0.49 28".07 1.12 64O.17 1.75 lOOO.27 2.38 136O.36 3.01 72O.46 0.50 28O.65 1.13 64O.74 1.76 loo='.84 2.39 136O.94 3.02 730.03 0.51 290.22 1.14 65O.32 1.77 1010.41 2.40 1370.51 3.03 73O.61 0.52 290.79 1.15 65O.89 1.78 1010.99 2.41 138O.08 3.04 74O.18 0.53 300.37 1.16 66'=.46 1.79 102O.56 2.42 138O.66 3.05 740.75 0.54 30".94 1.17 67O.04 1.80 1030.13 2.43 1390.23 3.06 750.33 0.55 310.51 1.18 67O.61 1.81 1030.71 2.44 139O.80 3.07 750.90 0.56 320.09 1.19 680.18 1.82 104O.28 2.45 1400.37 3.08 76O.47 0.57 32O.66 1.20 68O.75 1.83 104O.85 2.46 140".95 3.09 1770.04 0.58 33O.23 1.21 69".33 1.84 105O.42 2.47 141O.52 3.10 177O.62 0.59 33O.80 1.22 69".90 1.85 106°.00 2.48 142O.09 3.11 178O.19 0.60 34O.38 1.23 7OO.47 1.86 106O.57 2.49 142O.67 3.12 178O.76 0.61 34O.95 1.24 71O.05 1.87 107O.14 2.50 143O.24 3.13 1790.34 0.62 35O.52 1.25 71O.62 1.88 107O.72 2.51 144O.81 3.14 1790.91

1 3.1416 180° 2 6.2832 360° 3 9.4248 5400 4 12.5664 720° 5 15.7080 9000 6 18.8496 1,080° 7 21.9911 1,260" 8 25.1327 1,4400 9 28.2743 1,620"

0.63 36O.10 1.26 72O.19 1.89 108O.29 2.52 144O.39 3.15 18OO.48 IO 31.4159 1.800°


TABLE 1 .Pl-NATURAL SINES AND COSINES

Natural Stnes at Intervals of O0 01 or 6"

00070 0.0244

00419 0 0593

00767

0.0941 01115

00087 0.0105 0.0122 00140 00157 00262 0 0279 0 0297 00314 00332

00436 00454 0.0471 0.0488 00506 00610 0 0628 0.0645 00663 0.0680

00785 0 0802 00819 00837 00854

0 0958 01132

01 0

(60')

0 0000

0.0175 0 0349

0.0523 0.0698 0.0872

17 17

17 17

17

01305 01478

01650

0 0976 0 0993 0.1011 0.1028 0.1045 01149 01167 0 1184 01201 0.1219 01323 0 1340 0.1357 01374 0.1392

01495 01513 0 1530 01547 0.1564

01668 01685 0 1702 01719 0.1736

17 17 17

17

17

01822 0.1840 0.1994 0.2011 02164 0 2181 0.2334 02351 0.2504 0 2521

0.1857 02028 0 2198

0.2368 0 2538

0.1874 0.1891 0.1908 0.2045 0.2062 0.2079 0.2215 0.2233 0.2250 0.2385 0.2402 0.2419 0.2554 0.2571 0.2588

Degrees

90 89

E 86

a5

84 83 82 81

80

79 78 77

76 75

17 17 17

17 17

0.2672 0.2689 0.2706 02723 0.2740 0.2756 74 17 0.2840 02857 02074 0.2890 0.2907 0.2924 73 17 0.3007 0.3024 0.3040 0 3057 0.3074 0.3090 72 17

0.3173 03190 0.3206 0.3223 0.3239 0.3256 71 17 0.3338 03355 0.3371 0.3387 0.3404 0.3420 70 16

0.9 154'1

0.0000 0.0017 00035 00052 0.0175 00192 00209 00227

0.0349 00366 00384 0.0401 0.0523 00541 0.0558 00576

0.0698 0.0715 00732 00750

(498') Degrees

Average

Difference

5 6 7 8

9

10 11 12

13 14

0.0872 0.0889 00906 0.0924 0.1045 01063 0.1080 0.1097

0.1219 01236 0.1253 01392 01409 0.1426

0.1564 01582 0.1599

0.1736 0.1754 0.1771 01908 0.1925 01942 02079 02096 02113

0 2250 0.2267 0.2284

02419 02436 02453

0.2588 02605 0.2622 02756 0.2773 0.2790 0 2924 0.2940 0 2957

03090 03107 0.3123 0.3256 03272 0.3289

0.1271 0.1444

0.1616

01288 01461

0.1788 01959 02130

0.2300 02470

15 16 17

18

19

20 21 22

23 24

0.2639 0.2807 0 2974 0.3140

03305

01633

01805 01977 02147

02317 02407

0.2656 02823 0 2990

03156 03322

0.3420 0.3437 0.3453 0 3469 0 3584 0.3600 0.3616 0.3633 03746 0.3762 0.3778 0 3795

0.3907 0.3923 0.3939 0.4067 04083 04099

0.3955 0.4115

03486 0.3649 03811 0.3971 04131

03502 0.3665 0.3827

0.3987 0.4147

0.3518 0.3535 03551 0.3567 0.3584 69 16 0.3681 0 3697 0.3714 0.3730 03746 68 16 03843 0.3859 0.3875 0.3891 0.3907 67 16 0.4003 04019 0 4035 0.4051 0.4067 66 16 04163 04179 04195 04210 0.4226 65 16

25 0.4226 04242 04258 04274 0 4289 04305 0.4321 04337 04352 0.4368 0.4384 64 16 26 04384 0.4399 0.4415 0.4431 0.4446 0.4462 04478 0.4493 0.4509 0.4524 0.4540 63 16 27 0 4540 04555 0.4571 0.4586 04602 0.4617 04633 04648 0.4664 0.4679 0 4695 62 16

28 04695 04710 0.4726 04741 04756 0.4772 04787 0.4802 0.4818 0.4833 0.4848 61 15 29 0 4848 0.4863 0.4879 0.4894 0.4909 0.4924 0 4939 0 4955 0.4970 0.4985 0.5000 60 15

30 05000 0.5015 0.5030 0.5045 0.5060 0.5075 31 0.5150 0.5165 0.5180 0.5195 05210 0.5225

32 05299 0.5314 0.5329 0.5344 05358 0.5373

33 0 5446 0.5461 0.5476 0.5490 0.5505 0.5519

34 05592 05606 05621 05635 05650 0.5664

0 5090 05105 0.5120 0.5135 0.5240 0 5255 0.5270 0.5284

0.5388

0.5534 0.5678

0.5402 0.5417

0 5548 0.5563

0 5693 0.5707

0.5432 0.5577 0.5721

59 58

57 56

55

35 05736 05750 05764 36 0.5878 0.5892 0.5906

37 06018 06032 0.6046 38 06157 06170 06184

39 06293 06307 0.6320

0 5779 0.5920 0.6060 06198

0 5793 0.5807 05821 0 5934 0.5948 0.5962

06074 06088 0.6101 06211 06225 0 6239

06347 0.6361 0.6374

0.5835 0.5976

0.6115 0.6252

0.6388

0.5850 0.5864 0.5990 0.6004

0.6129 0.6143 0.6266 0.6280

54 53

52 51

50

40 06428 06441 0.6455 41 0.6561 0.6574 0.6587

42 0.6691 0.6704 06717

43 0 6820 0.6833 06845 44 0.6947 0.6959 06972

0.6334

0.6468 0.6600 0.6730

06858 0 6984

0.6401

06481 0.6494 0.6508 0.6521 0.6534 06613 06626 0.6639 0.8652 06665 06743 06756 0.6769 0.6782 0.6794 06871 06884 0.6896 0.6909 06921 0 6997 0 7009 0.7022 0.7034 0.7046

0.5150 0.5299 0.5446 0 5592

05736

0.5878 06018

0.6157 0.6293

0.6428

0.6561 06691 0.6820 0.6947

07071

49 48 47

46

45

45 0 7071 07083 07096 07108 0 7120 0.7133 0.7145 0.7157 0.7169

46 07193 07206 07218 07230 07242 07254 0.7266 0 7278 0 7290

47 07314 0.7325 07337 0 7349 07361 07373 07385 0 7396 0 7408

48 0 7431 0.7443 07455 07466 07478 0 7490 0 7501 0.7513 0 7524

49 07547 0.7559 07570 07581 0 7593 07604 0.7615 0.7627 0 7638

0.6414

0.6547 0 6678 0 6807 0 6934

0.7059

0 7181

0 7302 0 7420

0 7536

0 7649

07193 44 07314 43 07431 42 07547 41

0 7660 40

15 15

15 15 14

14 14

14 14

13

13 13 13 13 12

12 12 12

12 11

0.0

(0')

Comes

01 =;5: )

“8 (46') (6’)

Natural

'For IO' mtervals. see Table 1 24


TABLE 1.21-NATURAL SINES AND COSINES (continued)

Natural Smes at Intervals 01 O" 01 or 6"

(i3) 04

(24’)

0 7694 0.7705 07716 07804 0 7615 0 7826

07912 0 7923 0 7934 0 8018 0 8028 0 8039 0.8121 0.8131 0 8141

08221 0.8231 0 8241 0.8320 0 8329 08339

0.8415 0.8425 08434

0.8508 08517 0 8526 0.8599 0.8607 08616

0.7727 07837

0 7944

0 8049 08151

08251 08348

08443

08536 08625

08712 0 8796 0.8678 0 8957 0 9033

07738 0 7749 0.7760 07771 07848 0 7859 0.7869 0.7880 0 7955 0 7965 0.7976 0.7986 0 8059 08070 0.8080 0.8090 08161 08171 0.8181 0.8192

08261 0.8358

0.8453

08545 0.8634

08686 0 8695 0 8704 0.8771 0.8780 0.8788 0.8854 0.8862 0.8670 0 8934 0 8942 0 8949 0.9011 09018 0 9026

08721 0.8805

0.8886 08965 0.9041

0 9085 0 9092 09100 0.9107 09114 0.9157 0.9164 0.9171 0.9178 0.9184 0.9225 0.9232 a.9239 0.9245 0 9252 0.9291 0.9298 0.9304 0.9311 0.9317 0.9354 0.9361 0 9367 0 9373 0.9379

0.9415 0.9472 0.9527 0.9578

0.9627

0.9673 09715 0.9755 0.9792

0.9826

0.9421 0.9426 0.9432 0.9438 0.9478 0.9483 0 9489 0.9494 0.9532 0.9537 0.9542 0 9548 0.9583 0.9588 0.9593 0.9598

0.9632 0.9636 09641 0 9646

0.9677 0 9681 0 9686 0.9728 09767

0.9803 0 9836

0 9866 0.9893 0.9917

0.9938

0.9956

0.9971 0 9982 0.9991

0 9997 10000

0 9690

"6

(36') (2) (i88') 01.0

Wl Degrees Average Difference

0.8281 0.8290 08377 0.8387 08471

08563 0.8652

0.8480 0.8572 0.8660

08738 0.8746 0.8821 0.8829 0.8902 0.8910 0.8980 0.8988 0.9056 0.9063

39 38

2 35

34 33

32 31

30

29 28 27 26 25

11 11

11

10 10

10 10

9 9 9

09128 0.9135 24 7 0.9198 0.9205 23 7 0.9265 0.9272 22 7 0.9330 0.9336 21 6 0.9391 0.9397 20 6

0.9449 0.9505

0.9558 0.9608

0.9655

0.9699 0.9740 0.9778 0.9813

0.9845

0.9455 0.9511

0.9563 0.9613

0.9659

19 6 18 6 17 5 16 5

15 5

0.9703 14 0.9744 13 0.9781 12

0.9816 11

0.9848 10

0.9874 0.9900 0.9923

0.9943

0.9960

0.9974 0 9985 0.9993

0.9998 10000

0.9877 0 9903 0.9925

0.9945 0 9962

0.9976 4 0 9986 3 0 9994 2

0.9998 1 10000 0

4

4 4

3 3

3 3 2

2 2

1 1

:, 0

Degrees

50 51

52

53 54

55 56

57

58 59

60 61

62 63 64

65 66 67 68

69

70 71

72 73 74

75 76 77

78

79

80 81 82

83

84

85 86 87

88 89

90

0 7672 0.7683 0 7782 07793

0 7891 0.7902

0.8007 0 7997 08100

0 8202 08300

0.8396

0 8490 0.8581

08111

0.8211 0.8310

0.8406

0.8499 0.8590

08678 08763

0.8846 08926 0.9003

0.9078 0.9150

0.9219 0.9285 0.9348

07660 07771

07880

0.9732 09770

0.9806 0.9839

0.9869 0.9895 0.9919

0.9940

0.9957

0 8271 08368

08462

08554 08643

0 8729 08813 0 8894 0 8973 0 9048

09121 09191 0 9259 09323

0 9385

09444 0.9500 0 9553 09603

0 9650

0 9694

0.9736 0.9774

09810

0.9842

09871 0.9898 09921

0.9942 0.9959

0.9972 0.9973 0.9983 0 9984 0.9992 0 9993 0.9997 0.9998 10000 10000

0 7986 0 8090

0 8192 08290

0.8387

08480 08572

08660 0.8746

0.8829 08910

0 8988

0 9063 0.9135

0.9205 0.9272

0 9336

0.9397 0.9455

0.9511 0.9563 0.9613

0.9659 0.9703 0.9744

0.9781

0.9816

0.9848 0 9877 0.9903

0 9925 0.9945

0 8669 0.8755

0.8830 08918

0.8996

0.9070 0.9143 0.9212

0.9278 0.9342

0.9403 0.9409 09461 0.9466

09516 0.9521 0.9568 0.9573

0.9617 0.9622

0.9664 0.9668 0 9707 0.9711 0 9748 09751

0 9785 0.9789

0.9820 0 9823

0.9851 0.9854 0.9857 0 9880 0 9882 0 9885 0 9905 0 9907 0.9910

0 9928 0 9930 0 9932

0 9947 0 9949 09951

0 9720

0.9796

0.9829

0.9759 0.9799

0 9724

0.9833

0.9863

0.9763

0 9890 09914

0 9936

0.9954

0.9969 09981 0 9990

0 9997 10000

0.9860 0.9888 09912

0 9934 0.9952

0.9968 0 9980 0 9990

0 9996 0 9999

0.9962 0 9976 0 9986

0.9994 0 9998

1.0000

0 9963 0 9965 0.9966 0 9977 0 9978 0 9979

0 9987 0 9988 0.9989

0 9995 0 9995 0 9996 0 9999 0 9999 0 9999

03

(18') ,,"%,

01

(6 ) Natural

09 =(54 )

'For IO intervals see Table 1 24


TABLE 1.22-NATURAL TANGENTS AND COTANGENTS

Natural Tangents at Intervals of On.1 or 6"

0.0 ".6 0.7 0.9 "1 0 Degrees =(O') (;8') (36’1 (42') (54') 630 1

Average Degrees Difference

0 0000 0.0017 00035 00175 0.0192 0.0209

0 0349 00367 0.0384 00524 0.0542 0 0559

0 0699 0.0717 0 0734

0.0052 0.0227

0.0402 0.0577

0.0752

0.0070 0.0087 00105 0.0122 0 0140 0.0157 00244 00262 0.0279 0.0297 0.0314 0.0332 0.0419 0.0437 00454 0.0472 0.0489 0.0507 0.0594 00612 00629 0.0647 0.0664 0.0682 0.0769 00787 00805 0.0822 0 0840 00857

0.0000 90 00175 89 0 0349 88 0.0524 07 0 0699 86 0.0675 65

17 17

17 18 18

5 00875 0.0892 00910 0.0928 0.0945 00963 00981 0.0998 0.1016

6 0 1051 01069 0 1086 0.1104 0.1122 0.1139 0.1157 0.1175 0.1192 7 0 1228 0.1246 01263 0.1281 0.1299 0.1317 01334 0.1352 0.1370 a 01405 0.1423 01441 01459 0.1477 01495 01512 0.1530 01548 9 0 1584 0.1602 0.1620 01638 0.1655 01673 0 1691 0.1709 0.1727

0.1033

0.1210 0.1388 0 1566 01745

01051 64 18 01228 63 18 0.1405 82 18 0.1584 81 18 01763 80 18

10 01763 0.1761 01799 01817 0.1835 01853 0.1871

11 01944 0.1962 01980 0.1996 0.2016 02035 02053 12 02126 02144 02162 0.2180 0.2199 0.2217 02235

13 0 2309 02327 02345 0.2364 0.2382 02401 02419

14 0 2493 02512 0.2530 0 2549 02566 02586 0.2605

01690 0.2071 0.2254 0.2438 02623

0.1908 0.1926 01944 79 18 0.2089 0.2107 0.2126 78 18 0.2272 0.2290 02309 77 18 0.2456 0.2475 0 2493 76 19 0.2642 0.2661 0 2679 75 19

15 16 17 18 19

0 2679 02867

0 2698 0.2717 02736 0.2754 02773

0.2886 0 2905 0 2924 0.2943 0 2962 0.3076 0.3096 03115 0.3134 03153 0.3269 03268 0 3307 03327 03346 0.3463 0.3462 0 3502 03522 03541

0 2792

03057 0 3249 03443

0.2981 03172

0.3365 0.3561

0.2811 0.2630 0.2849 0.2867 74 19 0 3000 0.3019 0.3038 03057 73 19 0 3191 0.3211 0.3230 03249 72 19 03385 0.3404 0.3424 0.3443 71 19 0.3561 03600 03620 0.3640 70 20

20 03640 0 3659 0 3679 0 3699 0.3719 03739 0.3759 0 3779 0.3799 0.3819 0 3839 69 20 21 0 3839 0.3859 0 3879 0 3099 0.3919 0 3939 0.3959 0.3979 0.4000 0.4020 0.4040 68 20 22 04040 0.4061 04061 04101 0.4122 04142 0.4163 0.4163 0.4204 0.4224 04245 67 21 23 04245 04265 0.4266 04307 04327 04348 0.4369 0.4390 0.4411 0.4431 04452 66 21 24 04452 04473 0.4494 04515 04536 04557 0.4578 0.4599 0.4621 0.4642 04663 65 21

25 04663 04664 04706 0.4727 0.4746 04770 04791

26 04877 0 4899 04921 0 4942 0 4964 0 4986 05008 27 0 5095 05117 0.5139 0.5161 05184 05206 0.5228

28 0.5317 05340 0.5362 0 5364 05407 05430 05452 29 0.5543 05566 0.5589 0 5612 05635 05658 05681

0.4813

0.5029 0.5250 0.5475

0.5704

30 0.5774 05797 0.5820 0.5844 0.5067 0 5890 05914 0.5938 31 0.6009 06032 06056 0 6080 06104 06128 06152 0.6176 32 0.6249 06273 0.6297 0 6322 06346 06371 0.6395 0.6420 33 0.6494 0.6519 06544 0 6569 0 6594 06619 0 6644 0.6669 34 0.6745 06771 0.6796 0.6822 0.6847 06873 0 6899 0.6924

0.4834 04856 0.4877 64 21 0.5051 05073 0 5095 63 22 0.5272 0.5295 05317 62 22

0.5498 05520 05543 61 23 0.5727 05750 05774 60 23

0.5961 0.6200 0.6445 0.6694 0.6950

05985 06009 59 24 06224 06249 58 24

0 6469 0.6494 57 25 06720 06745 56 25 06976 07002 55 26

35 0.7002 07028 07054 07080 36 0.7265 07292 07319 0 7346

37 0.7536 07563 0 7590 07618 38 0.7813 07841 0 7869 0.7898 39 0 8098 08127 08156 0.8185

0.7107 0.7373

0.7646 0 7926 08214

07133 0 7400

07673 0 7954 08243

07159 0.7427 07701 0.7983 08273

0.7186 0.7212 0.7239 07265 54 26 0.7454 0.7481 0.7508 07536 53 27 0.7729 0.7757 07785 07813 52 28 08012 0.8040 0.8069 0 8098 51 28 0.8302 0.8332 08361 06391 50 29

40 06391 08421 08451 0.8481 08511 08541 0.8571 0.8601 41 0.6693 08724 08754 08785 08816 08847 0.8678 0 8910 42 0 9004 0 9036 0.9067 0.9099 09131 09163 0.9195 0.9228

43 0 9325 0.9358 09391 0.9424 0 9457 0.9490 09523 0 9556 44 0 9657 09691 0 9725 0.9759 0 9793 0 9827 0.9661 0.9896

08632 0.8941 0.9260 0.9590 0.9930

0.8662 0.8693 49 30 08972 0 9004 46 31 0.9293 0 9325 47 32

0 9623 0 9657 46 33 0 9965 10000 45 34

".8 07 '.6

(48') 142') 136’) (2, 0.4 0.3

(24') (18’) (6’) Natural Cotangents

'For 10 intervals see Table I 24

MATHEMATICAL TABLES & UNITS & SYSTEMS OF WEIGHTS & MEASURES I-47

TABLE 1.22-NATURAL TANGENTS AND COTANGENTS (continued)

Natural Tanaenls al Intervals of 0" 1 0' 6 '

02 00 =(O 1 16 I I12 /

45 10000 10035 1.0070 46 10355 10392 1.0428 47 10724 10761 1.0799

40 11106 1 1145 1.1184 49 11504 1 1544 1.1585

1.0141

"3

(18 1 10105 10464 10837 1 1224

11626

07

(42 I

1.0501 10875

10176 10538 10913

1 1263 1 1303 11667 1 1708

10212 1.0247 10575 10612 10951 1.0990 1 1343 1.1383 11750 1.1792

“8 “9 01 0 (48 ) (54 ) (60 )

1.0283 1.0319 1.0355 10649 10686 1.0724 11028 1 1067 1.1106

1 1423 I 1463 1 1504 11833 1 1875 1.1918

Average Degrees Difference

44 43 42 41

40

35 37 38

40 41

50 11918 1.1960 1.2002 12045 1.208R 1.2131 12174 1.2218 1.2261 12305 1.2349 39 43 51 12349 12393 12437 12482 1.2527 12572 12617 1.2662 12708 12753 12799 38 45 52 1.2799 1.2846 1.2892 1.2938 12985 13032 13079 1.3127 13175 1.3222 13270 37 47 53 1.3270 1.3319 1.3367 13416 13465 1.3514 13564 1.3613 13663 13713 1.3764 36 49 54 1.3764 13814 1.3865 13916 13968 14019 14071 1.4124 14176 14229 1.4281 35 52

55 56 57 58 59

14281 14826 1.5399 1.6003 1.6643

14335 1.4388 14442 1.4496 14550 14882 14938 14994 1.5051 1.5108 15458 1.5517 1.5577 15637 15697 16066 1.6128 16191 16255 16319 16709 1.6775 16842 1.6909 16977

1.4605 1.4659 1.5166 15224 1.5757 15818 1.6383 1.6447 1.7045 1.7113

1775 1.782 1.849 1.857 1.929 1937 2.014 2023 2.106 2.116

14715 14770 1.4826 34 55 15282 1.5340 15399 33 57 15880 15941 16003 32 60 16512 1.6577 16643 31 64 17182 17251 17321 30 67

1789 1797 1804 29 7 1865 1873 1.881 28 a 1946 1.954 1.963 27 8 2 032 2.041 2.050 26 9 2 125 2.135 2.145 25 9

2204 2.215 2225 2236 2.311 2.322 2 333 2.344

2450 2.463 2578 2592 2 718 2 733

2.246 24 10 2356 23 11 2475 22 12 2605 21 13 2747 20 14

2.872 2.888 2.904 19 16 3 042 3.060 3.078 16 17 3 230 3.251 3 271 17 19 3 442 3465 3487 16 22 3 681 3 706 3.732 15 24

3952 3.981 4.011 14 28 4 264 4.297 4.331 13 32 4 625 4.665 4.705 12 37 5050 5.097 5.145 11 44 5 558 5.614 5 671 10 53

6174 6.243 6.314 9 6 940 7.026 7115 a 7916 8.028 8.144 7 9205 9.357 9.514 6

1099 11.20 11.43 5

1362 1789 2603 4774

2865

13.95 1430

18.46 1908 2727

52.08 573.0

2864

5729 m

60 1732 1739 1746 1753 1.760 1767 61 1804 1811 1819 1827 1.834 1842 62 1881 1889 1.897 1905 1.913 1.921 63 1 963 1 971 1980 1988 1.997 2006 64 2.050 2.059 2069 2078 2.087 2097

65 66 ti7

68 69

2.145 2.154 2 164 2174 2 246 2.257 2267 2278 2.356 2.367 2379 2391 2 475 2.488 2.500 2513 2 605 2.619 2 633 2646

2.184 2.289 2.402 2.526 2.660

2194 2.300 2414

70 2747 2 762 71 2.904 2921 72 3078 3.096 73 3271 3.291 74 3.487 3.511

2 778

2937

2.539 2.675

3.115 3312

3534

2793 2.954 3133 3333 3558

2808 2.971 3152 3354 3582

75 3.732 3.758 3.785 3.812 3839 76 4.011 4.041 4.071 4102 4 134 77 4331 4.366 4402 4.437 4.474 78 4705 4.745 4.787 4.829 4.872 79 5.145 5.193 5.242 5292 5.343

80 5 671 5.730 5 789 5.850 81 6.314 6.386 6.460 6.535 82 7.115 7.207 7300 7396 83 8.144 8.264 8386 a513 84 9 514 9.677 9 845 1002

5 912

2.824

2.989 3.172 3.376 3606

3.867

4 165 4.511 4915

5396

5 976 6 691 7 596 8777

10.39

12 71

1635

6612 7495 8.643

10.20

85 11.43 11.66 11.91 12.16 1243 86 1430 14.67 1506 15.46 15.90 87 19 0.3 19.74 2045 21 20 22 02 88 2864 30.14 31 82 33.69 3580 89 57.29 63.66 71 62 81 a5 95 49

2290 38.19

1146

90 oc

2.426 2.438 2.552 2.565 2.689 2.703

2.840 2.856 3.006 3.024 3.191 3.211 3 398 3420 3.630 3.655

3.895 3.923 4.198 4.230 4.548 4586 4959 5.005 5.449 5.503

6.041 6.107 6.772 6.855 7.700 7806 8.915 9.058

10.58 1078

13.00 13.30 16.83 17.34 2386 2490 40.92 4407

143.2 191.0

0.2 01 0.0 0.9 0.8

=(54') (48') (2, (12’) (6’) (0’) Natural Cotanoents


TABLE 1.23-NATURAL SECANTS AND COSECANTS

Degrees

10 11 12 13 14

15 16 17 18 19

20 21 22 23 24

25 26 27 28 29

30 31 32 33 34

35 36 37 38 39

40 41

42 43 44

0.0

=(O’) (61,

0.2

(12’)

10000 1 0000 10000

10002 10002 1.0002

10006 10007 10007 10014 1.0015 1.0016 10024 10026 10027

10038 1.0040 10041 1.0055 10057 10059

1.0075 10077 10079 1 0098 10101 10103 1.0125 10127 10130

1.0154 1.0157 1.0161 1.0187 1.0191 1.0194 1.0223 1.0227 1.0231 1.0263 1.0267 10271 10306 1.0311 10315

10353 10358 10363 10403 1.0408 10413 10457 1.0463 1.0468 10515 10521 10527 10576 1.0583 10589

10642 1.0649 10655 10711 10719 1 0726 t 0785 10793 10801 10864 1.0872 1 0880 10946 1.0955 1.0963

1 1034 1.1043 1.1052 1 1126 11136 1.1145 11223 11233 1.1243 1 1326 1 1336 1.1347 1 1434 1 1445 1.1456

11547 11559 1.1570 1.1666 1 1679 1.1691 1.1792 11805 1.1818 1.1924 1 1937 1.1951 1.2062 12076 1.2091

1.2208 12223 12238 1.2361 1.2376 12392 12521 1.2538 12554 12690 12708 1.2725 1.2868 1.2886 1.2904

1.3054 1.3073 1.3093 13250 13270 13291

13456 1.3478 1.3499 13673 13696 1.3718 1 3902 13925 13949

10000 10000 10000 10001 1.0001 10001

1.0003 1.0003 1.0003 10004 10004 1.0005

1.0008 10009 10010 10010 10011 1.0012 1.0017 1.0018 10019 10020 10021 1.0022 1.0028 10030 10031 10032 10034 10035

1.0043 10045 10046 10048 10050 10051 1.0061 10063 10065 10067 10069 1.0071

1.0082 10084 10086 10089 1.0091 10093 1.0106 1.0108 10111 10114 1.0116 1.0119 10133 1.0136 10139 10142 1.0145 1.0148

1.0164 10167 1.0170 10174 10177 10180 1.0198 10201 1.0205 10209 1.0212 10216 1.0235 10239 1.0243 10247 10251 10255 10276 1.0280 1.0284 10288 10293 1.0297 10320 10324 1.0329 10334 1.0338 10343

1.0367 10372 10377 10382 1.0388 1.0393 1.0419 10424 1.0429 1.0435 1.0440 1.0446 1.0474 10480 1.0485 10491 1.0497 1.0503 1.0533 10539 1.0545 10551 1.0557 1.0564 1.0595 10602 10608 10615 1.0622 1.0628

1.0662 10669 10676 10683 1.0690 1.0697 1.0733 10740 1.0748 1.0755 1.0763 1.0770 10808 10816 10824 10832 10840 1.0848 1.0888 10896 10904 10913 1.0921 1.0929 10972 10981 1.0989 10998 1.1007 1.1016

1.1061 1 1070 1.1079 1 1089 1.1098 1.1107 1.1155 1 1164 1.1174 11184 1 1194 .1203 1.1253 11264 1 1274 1 1284 1.1294 .I305 1.1357 1.1368 11379 1 1390 1.1401 .1412 1 1467 1.1478 1 1490 1 1501 1.1512 1524

1.1582 11594 11606 1 1618 1.1630 1 1642 11703 1.1716 1 1728 1.1741 1.1753 1766 11831 1.1844 1 1857 1 1870 1.1883 1897 1 1964 1.1978 1 1992 12006 12020 .2034 12105 1.2120 12134 12149 1.2163 2178

1.2253 12268 1.2283 12299 1.2314 1.2329 1.2408 12424 12440 12456 12472 12489 12571 12588 12605 12622 1.2639 1.2656 1.2742 12760 12778 12796 1.2813 1.2831 12923 1.2941 1.2960 1.2978 12997 13016

13112 1.3131 1.3151 1.3171 1.3190 1.3210 13311 13331 1.3352 13373 13393 1.3414 13520 13542 1.3563 1.3585 13607 13629 13741 13763 1.3786 13809 13832 13855 13972 13996 14020 14044 1.4069 1.4093

0.9 a.0 07 06 a.5 04 03 02 =(54 ) (48') (42’) (36 I (30') (24') (18') (12’)

".6 0.7 ".8

(36’1 (42') (48')

“9 01 0

(54') (60 I 10000 10002 10006

Average

Degrees Difference

10001 1.0006 1.0013 10023 10037

10014 10024

10038

90 89 88 87 86

85

10053 10055 04 10073 10075 83 10096 10098 82 10122 10125 81 10151 10154 80

10184 10187 79 10220 10223 78 1.0259 10263 77 10302 10306 76

10348 10353 75

10398 10403 74 5 1.0451 10457 73 5 10509 10515 72 6 1.0570 10576 71 6 1.0635 10642 70 7

1.0704 10711 69 7 1.0778 10785 68 7 1.0856 10864 67 8 1.0938 10946 66 8 1.1025 1 1034 65 9

1.1117 1 .1213 1 .I315 1 .1423 1 1535

1 1126 64 1 1223 63 1 1326 62 1 1434 61 11547 60

9 10 10 11

11

1 1654 1 1666 59 1 1779 1 1792 58 1 .1910 1 1924 57 1 2048 12062 56 1 2193 12208 55

12 13 13 14

15

12345 12361 54 15 12505 12521 53 16 12673 1 2690 52 17 1.2849 12868 51 18 13035 13054 50 19

1.3230 13250 49 20 13435 13456 48 21 13651 1.3673 47 22 13878 13902 46 23 14118 14142 45 24

00

(0')

Natural Cotanoents

I-49 MATHEMATICAL TABLES & UNITS & SYSTEMS OF WEIGHTS & MEASURES

TABLE 1.23-NATURAL SECANTS AND COSECANTS (continued)

Natural Secants at Intervals of OO.1 or 6"

"0 0 1 02 "3 04 05 =10 ) (6 ) I12 1 (18 1 (24 I (30 I

“6 Deqrees (36 J

07 00 09 "10 Average

(42 ) (48 1 (54 I (60 1 Deqrees Difference

45 14142 1.4167 14192 14217 14242 14267 1.4293 1.4318 14344 1.4370 1 4396 44 25 46 1.4396 1.4422 1.4440 1.4474 14501 14527 1.4554 1.4561 1.4608 14635 1.4663 43 27 47 14663 1.4690 1.4718 1.4746 14774 1.4802 1.4830 14659 1.4867 14916 1.4945 42 28 48 14945 1.4974 15003 1.5032 15062 15092 15121 1.5151 1.5182 1.5212 1.5243 41 30 49 15243 1.5273 1.5304 1.5335 15366 15398 1.5429 1.5461 15493 1.5525 15557 40 31

50 15557 1.5590 1.5622 1.5655 1.5686 15721

51 15890 1.5925 1.5959 1.5994 1.6029 16064 52 16243 16279 16316 16353 16390 16427

53 16616 16655 16694 1.6733 1.6772 1.6812

54 17013 17054 1 7095 1.7137 1.7179 17221

1.5755 1.6099 1.6464 16852 17263

1.5788 1.5822 15856 1.5890 39 33 1.6135 1.6171 1.6207 1.6243 38 35 1.6502 1.6540 1.6578 16616 37 37 16892 1.6932 1.6972 1.7013 36 40 17305 1.7348 1.7391 1.7434 35 42

55 17434 1.7476 1.7522 1.7566 17610 17655 1.7700 1.7745 17791

56 17883 1.7929 1.7976 1.8023 18070 18118 18166 1.6214 18263

57 18361 18410 1.9460 1.8510 1.8561 1.8612 18663 18714 18766

58 18871 1.8924 18977 1.9031 19064 19139 1.9194 19249 19304

59 19416 1.9473 1 9530 19587 19645 19703 1.9762 1.9821 19880

34 45 33 48 32 51 31 54 30 58

60 2 000 2006 2.012 2016 2025 2031 2037 2.043 2.050

61 2063 2069 2.076 2082 2089 2096 2.103 2 109 2.116

62 2 130 2.137 2.144 2.151 2 158 2166 2.173 2.180 2.188

63 2 203 2.210 2.218 2.226 2233 2.241 2249 2.257 2.265

64 2.261 2.289 2.298 2306 2314 2323 2331 2.340 2.349

2366 2375 2459 2468 2 559 2570

1.7637 17863 1.8312 1.6361 18618 18871 1.9360 1.9416 1.9940 20000

2.056 2.063 2 123 2.130 2 195 2.203 2 273 2.261 2.357 2.366

29 6 28 7 27 7 26 8 25 8

65 66 67 68 69

2.449 2.459 2 549 2.559

2669 2661 2 790 2.603

2.364 2.478 2.581 2693 2.816

2 393 2402 2488 2498 2591 2602 2 705 2716 2.829 2842

2.411 2.421 2.430 2439

2.508 2 518 2 528 2.536 2.613 2.624 2.635 2.647

2.729 2.741 2 753 2.765 2.855 2669 2.682 2.896

2.658 2.669 2.778 2.790 2.910 2.924

24 9 23 10 22 11 21 12

20 13

3.041 3.056 3.072 19 15 3.202 3.219 3.236 18 16 3.382 3.401 3.420 17 16 3.564 3.606 3.628 16 21 3.814 3.839 3.864 15 24

4.077 4.379 4.732

5.148 5647

14 27 13 31 12 36 11 43 10 52

5996 6.059 6.123 6.188 6255

6687 6765 6.845 6.927 7.011

7561 7661 7 764 7670 7.979

8 700 8.834 8.971 9.113 9 259

1025 1043 10.63 10.83 11 03

4.105 '4.134 4.412 4.445 4.771 4.610 5.194 5.241 5.702 5.759

6.323 6.392 7.097 7.185 8091 8.206 9.411 9.567

11.25 1147

1365 1399 1434 1791 18.49 19.11 26.05 2729 28.65 47.75 5209 57.30

2865 573.0 m

2.952 3.103

70 2924 2938 71 3072 3087 72 3 236 3.254 73 3420 3.440 74 3 628 3650

3.271

3.460 3 673

2967 3 119 3.289 3.480 3 695

2 981 3135 3307

2.996 3152 3.326 3521 3742

3.011 3.168

3500 3 719

3.344 3542 3.766

75 3.864 3.889 3.915 3.941 3967 3994 4.021

76 4134 4163 4 192 4222 4253 4284 4315

77 4445 4.479 4.514 4549 4584 4.620 4.657

78 4 810 4.850 4 890 4931 4973 5016 5059

79 5 241 5.288 5337 5386 5436 5487 5540

80

81

:: 84

5759 5.816 5875 6 392 6.464 6.537 7 185 7.276 7 368

8 206 a.324 8 446 9567 9.728 9 895

5935 6611 7.463 8571

1007

85 66 a7 88

89

11.47 14.34 1911 2865

5730

11.71 11 95 12.20 1247 1275 14.70 15.09 1550 1593 1636 19.77 20 47 21 23 22 04 2293 30.16 31 84 33 71 35 81 3820

6366 71 62 81 85 95 49 114 6

13.03 16.86

2368 4093

143.2

90 m

3.026 3.185 3.363 3.563 3.790

4.049 4347

4.694 5.103 5.593

13.34 17.37

24.92 4408

191.0

04

124'1

02

112'1

Natural Cotangents

09

=154')

'For 10' intervals see Table 1 24


TABLE 1.24-TRIGONOMETRIC FUNCTIONS(ATINTERVALS OF 10')'

Sines Cosines Tangents Cotangents

Degrees Radians Natural Log" Natural Log'* Natural Log" Natural Log'* ~_____ 00 00' 0.0000 0.0000 03 1 .oooo 0.0000

1.0000 0.0000 00 10' 0.0029 0.0029 7.4637 00 20' 0.0058 0.0058 7.7648 00 30' 0.0087 0.0087 7.9408

0.0000 0.0029 7.400637

1 .oooo 0.0000 0.0058 7.7648 1.0000 0.0000 0.0087 7.9409

343po77 2.5:63 171.89 2.2352 114.59 2.0591

1.5708 900 00’ 1.5679 900 50’ 1.5650 900 40' 1.5621 90" 30' 1.5592 900 20' 1.5563 90" 10'

1.5533 1.5504 1.5475 1.5446 1.5417 1.5388

1.5359 1.5330 1.5301

880 30' 880 20'

89" 00' 89O 50' 89" 40' 890 30' 890 20' 89O 10'

880 00' 880 50' 88O 40'

00 40' 00 50'

10 00' 10 10' 10 20' 10 30' 1" 40' IO 50'

20 00' 20 10' 20 20'

0.0116 0.0116 8.0658 0.9999 0.0000 0.0145 0.0145 8.1627 0.9999 0.0000

0.0175 0.0175 8.2419 0.0204 0.0204 8.3088 0.0233 0.0233 8.3668 0.0262 0.0262 8.4179 0.0291 0.0291 8.4637 0.0320 0.0320 8.5050

0.0349 0.0349 8.5428 0.0378 0.0378 8.5776 0.0407 0.0407 8.6097

0.9998 9.9999 0.9998 9.9999 0.9997 9.9999 0.9997 9.9999 0.9996 9.9998 0.9995 9.9998

0.9994 9.9997 0.9993 9.9997 0.0378 8.5779 0.9992 9.9996 0.0407 8.6101

0.0116 8.0658 0.0145 8.1627

0.0175 8.2419 0.0204 8.3089 0.0233 8.3669 0.0262 8.4181 0.0291 8.4638 0.0320 8.5053

0.0349 8.5431

85.940 1.9342 68.750 1.8373

57.290 1.7581 49.104 1.6911 42.964 1.6331 38.188 1.5819 34.368 1.5362 31.242 1.4947

28.636 1.4569 26.432 1.4221 24.542 13899 22.904 1.3599 21.470 1.3318

29 30' 0.0436 0.0436 8.6397 20 40' 0.0465 0.0465 8.6677

0.9990 9.9996 0.9989 9.9995 0.9988 9.9995

0.9986 9.9994 0.9985 9 9993 0.9983 9.9993

0.0437 8.6401 0.0466 8.6682 0.0495 8.6945

0.0524 8.7194 0.0553 8.7429 0.0582 8.7652

1.5272

1.5213 1.5243

2" 50' 0.0495 0.0494 8.6940

3" 00' 0.0524 0.0523 8.7188 30 10' 0.0553 0.0552 8.7423 3" 20' 0.0582 0.0581 8.7645

20.206 1.3055

19.081 1.2806 18075 12571 17.169 1.2348 16350 12135 15.605 1.1933 14.924 1.1739

1.5184 1.5155 1.5126 1.5097 1.5068 1.5039

1.5010 1.4981 1.4952 1.4923 1.4893 1.4864

1.4835 1.4806 1.4777 1.4748 1.4719 1.4690

1.4661 1.4632 1.4603 1.4574 1.4544 1.4515

1.4486 1.4457 1.4428 1.4399 1.4370 1.4341

880 IO'

87" 00' 870 50' 870 40' 870 30' 870 20' 87O 10'

860 00' 86" 50' 860 40' 86O 30' 860 20' 860 10'

850 00' 85O 50' 850 40' 85' 30' 85O 20' 850 10'

840 00' 84O 50' 84" 40' 840 30' 840 20' 84' IO'

83O 00' 83O 50' 830 40' 83" 30' 83O 20' 830 10'

30 30' 0.0611 0.0610 8.7857 09981 9 9992 0.0612 8.7865 3" 40' 0.0640 0.0640 8.8059 0.9980 9.9991 0.0641 8.8067 3" 50' 0.0669 0.0669 8.8251 0.9978 9.9990 0.0670 8.8261

0.9976 9.9989 0.0699 8.8446 0.9974 9.9989 0.0729 8.8624 0.9971 9.9988 0.0758 8.8795 0.9969 9.9987 0.0787 8.8960 0.9967 9.9986 0.0816 8.9118 0.9964 9.9985 0.0846 8.9272

0.9962 9.9983 0.0875 8.9420 0.9959 9.9982 0.0904 8.9563 0.9957 9.9981 0.0934 8.9701 0.9954 9.9980 0.0963 8.9836

40 00' 0.0698 0.0698 8.8436 40 10' 0.0727 0.0727 8.8613

14.301 1.1554 13.727 1.1376

40 20' 40 30' 40 40' 40 50'

50 00' 5" 10' 50 20' 50 30' 50 40' 50 50'

6O 00' 6O IO' 6" 20' 6' 30' 6" 40' 6O 50'

70 00' 70 10' 70 20'

0.0756 0.0756 8.8783 0.0785 0.0785 8.8946 0.0814 0.0814 8.9104 0.0844 0.0843 8.9256

0.0873 0.0902 0.0931 0.0960 0.0989 0.1018

0.1047 0.1076 0.1105 0.1134 0.1164 0.1193

0.1222 0.1251 0.1280

0.0872 8.9403 0.0901 8.9545 0.0929 8.9682 0.0958 8.9816 0.0987 8.9945 0.1016 9.0070

0.1045 9.0192 0.1074 9.0311 0.1103 9.0426 0.1132 9.0539 0.1161 9.0648 0.1190 9.0755

0.1219 9.0859 0.1248 9.0961 0.1276 9.1060

13.197 1.1205 12.706 1.1040 12251 1.0882 11.826 1.0728

11.430 1.0580 11.059 1.0437 10.712 1.0299 10.385 1.0164 10078 1.0034 9.7882 0.9907

9.5144 0.9784 9.2553 0.9664 9.0098 0.9547 8.7769 0 9433 8.5555 0.9322 8.3450 0.9214

8.1443 0.9109 7.9530 0.9005 7.7704 0.8904 7.5958 0.8806 7.4287 08709 7.2687 0.8615

0.9951 9.9979 0.0992 8.9966 0.9948 9.9977 0.1022 9.0093

0.9945 9.9976 0.1051 9.0216 0.9942 9.9975 0.1080 9.0336 0.9939 9.9973 0.1110 9.0453 0.9936 9.9972 0.1139 9.0567 0.9932 9.9971 0.1169 9.0678 0.9929 9.9969 0.1198 9.0786

0.9925 9.9968 0.1228 9.0891 0.9922 9.9966 0.1257 9.0995 0.9918 9.9964 0.1287 9.1096 0.9914 9.9963 0.1317 9.1194 0.9911 9.9961 0.1346 9.1291 0.9907 9.9959 0.1376 9.1385

0 9903 9.9958 0.1405 9.1478 7.1154 0.8522 0.9899 9.9956 0.1435 9.1569 6.9682 08431 0.9894 9.9954 0.1465 9.1658 6.8269 08342 0.9890 9.9952 0.1495 9.1745 66912 08255 0.9886 9.9950 0.1524 9.1831 6.5606 0.8169 09881 9 9948 0.1554 9.1915 64348 08085

0 9877 9.9946 0.1584 9.1997 0.9872 9.9944 0.1614 9.2078

63138 08003 6.1970 0.7922 60844 07842 5.9758 0.7764

70 30' 0.1309 0.1305 9.1157 70 40' 0.1338 0.1334 9.1252 70 50' 0.1367 0.1363 9 1345

8" 00' 0.1396 0.1392 9.1436 8O IO' 0.1425 0.1421 9.1525 8" 20' 0.1454 0.1449 9.1612 8O 30' 01484 0.1478 9.1697 80 40' 0.1513 0.1507 9.1781 8O 50' 0.1542 0.1536 91863

90 00' 01571 0.1564 91943 9" 10' 0.1600 0.1593 9.2022

1.4312 1.4283 1.4254 1.4224 1.4195 1.4166

820 00' 82O 50' 82" 40' 82O 30' 820 20' 820 10'

810 00' 81' 50' 810 40' 810 30' 81° 20' 810 10'

14137 1.4108 14079 1.4050 1.4021 1.3992

9'1 20' 0.1629 0.1622 9.2100 90 30' 0.1658 0.1650 92176 90 40' 0.1687 0.1679 9.2251 90 50' 0.1716 0.1708 9.2324

0.9868 9.9942 0 9863 9.9940

0.1644 9.2158 0.1673 9.2236 0.1703 9.2313 0.1733 9.2389

0.9858 9.9938 0.9853 9.9936

5.8708 0.7687 5.7694 07611

Natural Loo** Natural Loa" Natural Log" Natural Log"

Cosines Sines Cotangents Tangents Radians Degrees

'Far OD 1 mtervals, see Tables 1 21. 1 22. and 1 23 'Add 10 I" ,h?se columns

MATHEMATICALTABLES& UNlTS&SYSTEMSOFWElGHTS &MEASURES I-51

TABLE 1.24-TRIGONOMETRIC FUNCTIONS (AT INTERVALS OF 10’)’ (continued)

nes Sil

Natural

Tange

Natural

nts Cotangents

Natural Log" ~~ 5.6713 0.7537 5.5764 0.7464 5.4845 0.7391 5.3955 0.7320 5.3093 0.7250 5.2257 0.7181

Log"

9.2463 92536 9.2609 92680 9.2750 92819

0.1763 0.1793 0.1823 0.1853 0.1883 0.1914

Degrees Radians Loo" Natural

100 00' 100 10'

0.1745 0.1774

0.1736 9.2397 0.9848 0.1765 92468 0.9843 0.1794 9.2538 0.9838 0.1822 92606 0.9833 0.1851 9.2674 0.9827 0.1880 92740 0.9822

1.3963 80" 00' 1.3934 80° 50' 1.3904 80' 40' 1.3875 80° 30' 1.3846 80' 20' 1.3817 80° 10'

IO" 20' 100 30' 10" 40' 100 50'

11~00' 110 IO' 110 20' 110 30' 11" 40' 110 50'

120 00'

0.1804 0.1833 0.1862 0.1891

0.1920 0.1908 0.1949 0.1937

92806 92870 9.2934 9.2997 93058 93119

09816 0.9811

99919 0.1944 9.2887 5.1446 0.7113 1.3788 9.9917 0.1974 9.2953 5.0658 0.7047 1.3759 9.9914 0.2004 9.3020 4.9894 0.6980 1.3730 9.9912 0.2035 9.3085 4.9152 0.6915 1.3701

790 00’ 790 50'

0.1978 0.2007 0.2036 0.2065

0.2094 02123 0.2153 0.2182 0.2211 0.2240

0.2269 0.2298 0.2327

0.1965 0.1994 0.2022 0.2051

0.9805 0.9799 0.9793 0.9787

0.2079 9.3179 0.9781 0.2108 9.3238 0.9775 0.2136 0 3296 0.9769 0.2164 93353 0.9763 0.2193 9.3410 0.9757 0.2221 9.3466 0.9750

9.9909 9.9907

9.9904 9.9901 9.9899 9.9896 9.9893 9.9890

0.2250 0.2278 0.2306 0.2334 0.2363 0.2391

93521 0.9744 9.9887

0.2065 9.3149 4.8430 0.6851 1.3672 0.2095 9.3212 4.7729 0.6788 1.3643

790 40' 790 30' 790 20' 790 IO'

0.2126 9.3275 4.7046 0.6725 1.3614 78O 00' 0.2156 9.3336 4.6382 0.6664 1.3584 78' 50' 0.2186 9.3397 4.5736 0.6603 1.3555 78O 40' 0.2217 9.3458 4.5107 0.6542 1.3526 7%" 30' 0.2247 9.3517 4.4494 0.6483 1.3497 78O 20' 0.2278 9.3576 4.3897 0.6424 1.3468 78" 10'

0.2309 9.3634 4.3315 0.6366 1.3439 770 00' 0.2339 9.3691 4.2747 0.6309 1.3410 770 50' 0.2370 9.3748 4.2193 0.6252 1.3381 77" 40' 0.2401 9.3804 4.1653 0.6196 1.3352 770 30' 0.2432 9.3859 4.1126 0.6141 1.3323 770 20' 0.2462 9.3914 4.0611 0.6086 1.3294 770 IO'

0.2419 9.3837 0.9703 9.9869 0.2493 9.3968 0.2447 9.3887 0.9696 9.9866 0.2524 9.4021 0.2476 9.3937 0.9689 9.9863 0.2555 9.4074

4.0108 0.6032 3.9617 0.5979 3.9136 0.5926 3.8667 0.5873 3.8208 0.5822 3.7760 0.5770

1.3265 1.3235 1.3206

0.2531 0.2504 9.3986 0.9681 9.9859 0.2586 9.4127 0.2560 0.2532 94035 0.9674 9.9856 0.2617 9.4178

0.9667 9.9853 0.2648 9.4230

0.9659 9.9849 0.2679 9.4281 0.9652 9.9846 0.2711 9.4331

3.7321 0.5719 3.6891 0.5669 3.6470 0.5619 3.6059 0.5570 3.5656 0.5521 3.5261 0.5473

1.3177 1.3148 1.3119

1.3090 1.3061

76O 00' 76" 50' 76O 40' 76' 30' 76O 20' 76" IO'

75" 00' 750 50'

3.4874 0.5425 3.4495 0.5378 3.4124 0.5331 3.3759 0.5284 3.3402 0.5238 3.3052 0.5192

3.2709 0.5147 3.2371 0.5102 3.2041 0.5057 3.1716 0.5013 3.1397 0.4969 3.1084 0.4925

3.0777 0.4882 3.0475 0.4839 3.0178 0.4797 2.9887 0.4755 2.9600 0.4713 2.9319 0.4671

2.9042 0.4630 2.8770 0.4589 2.8502 0.4549 2.8239 0.4509 2.7980 0.4469 2.7725 0.4429

120 10' 120 20' 120 30' 120 40' 120 50'

130 00' 130 IO' 130 20'

93575 0.9737 9.9884 9.3629 0.9730 9.9881

130 30'

130 50'

140

130 40'

00’ 140 10' 140 20' 140 30' 140 40' 140 50'

150 00' 150 10' 150 20' 150 30' 150 40' 150 50'

16O 00' 16" IO' 16O 20' 16' 30' 16O 40' 16O 50'

170 00' 170 IO' 170 20' 170 30' 170 40' 170 50'

18O 00' 18' 10' 18O 20' 18' 30' la0 40' 18O 50'

190 00' 190 IO' 190 20' 19" 30' 190 40' 19" 50'

0.2414

0.2356

0.2443 0.2473 0.2502

0.2385 9.3682 0.9724 9.9878 9.3734 0.9717 9.9875 93786 0.9710 9.9872

0.2589 0.2560 9.4083

0.2618 0.2588 9.4130 0.2647 0.2616 9.4177 0.2676 0.2644 9.4223 0.2705 0.2672 9 4269

0.9644 9.9843 0.2742 0.9636 9.9839 0.2773 0.9628 9.9836 0.2805 0.9621 9.9832 0.2836

9.4381 9.4430

1.3032 75" 40' 1.3003 750 30'

0.2734 0.2700 9.4314 0.2763 0.2728 9 4359

9.4479 9.4527

9.4575 9.4622 9.4669 9.4716 9.4762 9.4808

9.4853

1.2974 75" 20' 1.2945 750 IO'

1.2915 740 00' 1.2886 740 50' 1.2857 740 40' 1.2828 740 30' 1.2799 740 20' 1.2770 74" IO'

1.2741 730 00' 1.2712 730 50' 1.2683 730 40' 1.2654 730 30' 1.2625 730 20'

0.2793 0.2822 0.2851 0.2880 0.2909 0.2938

0.2967 0.2996 0.3025 0.3054 0.3083 0.3113

0.2756 0.2784 0.2812 0.2840 0.2868 0.2896

0.2924

94403 0.9613 9.4447 0.9605 94491 0.9596 94533 0.9588 94576 0.9580 9.4618 0.9572

9.4659 9.4700 9.4741 9.4781 9.4821 9.4861

0.9563 0.9555 0.9546 0.9537 0.9528

9.4900 9.4939

0.9520

0.9511 0.9502 0.9492 0.9483 0.9474 0.9465

0.9455 0.9446

9.9828 9.9825 9.9821 9.9817 9.9814 9.9810

9.9806

0.2867 0.2899 0 2931 0.2962 0 2994 0.3026

0.3057 0.2952 0.2979

9.9802 9.9798

0.3089 9.4898 0.3121 9.4943 0.3153 9.4987 0.3185 9.5031

0.3007 0.3035 0.3062

9.9794 9.9790 9.9786

9.9782 9.9778 9.9774 9.9770 9.9765 9.9761

9.9757 9.9752

0.3217 9.5075

0.3249 9.5118 0.3281 9.5161 0.3314 9.5203 0.3346 9.5245 0.3378 9.5287 03411 9.5329

0.3443 9.5370 0.3476 9.5411

1.2595

1.2566 1.2537 1.2508 1.2479 1.2450 1.2421

1.2392 1.2363

730 IO'

72' 00' 72' 50' 72O 40' 72O 30' 72O 20' 72O IO'

710 00' 710 50' 710 40' 710 30' 710 20' 710 10'

0.3142 0.3090 0.3171 0.3118 0.3200 0.3145 0.3229 0.3173

9.4977 9.5015

0.3258 0.3201 9.5052 0.3287 0.3228 9.5090

0.3316 0.3256 9.5126 0.3345 0.3283 9.5163 0.3374 0.3311 9.5199 0.9436 9.9748 0.3508 9.5451 0.3403 0.3338 9.5235 0.9426 9.9743 0.3541 9.5491

1.2334 1.2305 1.2275 1.2246

0.3432 0.3365 9.5270 0.9417 9.9739 0.3574 9.5531 0.3462 0.3393 9.5306 0.9407 9.9734 0.3607 9.5571

Natural Log" Natural Log" Natural Log'* Natural Log**


'For 0' 1 ~nlervals. see Tables 1 21. 1 22. and 1 23 "Add 10 I" these columns



TABLE 1.24--TRIGONOMETRIC FUNCTIONS (AT INTERVALS OF 10’)’ (continued)

CotangE

Natural

2.7475 2.7228 2.6985 2.6746 2.6511 2.6279

Cow ies

Natural Log'*

0.9397 9.9730 0.9387 9.9725

Tang{

Natural

0.3640 0.3673 0.3706 0.3739 0.3772 0.3805

Sines

Natural Log" ~ ~

fnts

Log"

9.5611 9.5650

!rlts

Log"

0.4389 0.4350

Dearees Radians

200 00' 0.3491 0.3420 200 10' 0.3520 0.3448 200 20' 0.3549 0.3475 20" 30' 0.3578 0.3502 200 40' 0.3607 0.3529 200 50' 0.3636 0.3557

210 00' 210 10' 21" 20' 210 30' 210 40' 21" 50'

0.3665 0.3694 0.3723 0.3752 0.3782 0.3811

0.3840 0.3869 0.3898 0.3927 0.3956 0.3985

0.3584 0.3611

0.3665 0.3692

0.3638

0.3719

220 00' 220 IO' 220 20' 220 30' 220 40' 220 50'

0.3746 0.3773 0.3800 0.3827 0.3854 0.3881

23O 00' 0.4014 23O IO' 0.4043 23O 20' 0.4072 23' 30' 0.4102 23O 40' 0.4131 23" 50' 0.4160

24" 00' 0.4189 24O 10' 0.4218 24O 20' 0.4247 24' 30' 0.4276 24" 40' 0.4305 24' 50' 0.4334

250 00' 25' 10' 25O 20' 25' 30' 25O 40' 25' 50'

26' 00' 26O IO' 26' 20' 26O 30' 260 40' 26O 50'

270 00' 27' 10' 270 20' 27" 30' 27O 40' 27O 50'

0.4363 0.4392 0.4422 0.4451 0.4480 0.4509

0.4538 04567 0.4596 04625 0.4654 04683

0.4712 0.4741 0.4771 0.4800 0.4829 0.4858

280 00' 280 10' 28" 20' 28O 30' 28O 40' 28O 50'

290 00' 290 10' 290 20' 290 30' 290 40' 290 50'

04887 0 4695 0.4916 04720 0.4945 04746 0.4974 0.4772 0.5003 0 4797 05032 0.4823

05061 04848 0.5091 04874 0.5120 0.4899 05149 0.4924 0.5178 0.4950 0.5207 0.4975

9.5341 9.5375 9.5409 9.5443 9.5477 9.5510

9.5543

0.4311 0.4273 0.4234 0.4196

1.2217 1.2188 12159 1.2130 12101 1.2072

0.4158 1.2043

700 00' 70" 50' 700 40' 7o" 30' 700 20' 700 10'

0.9377 0.9367

9.9721 9.9716 9.9711 9.9706

9.5689 9.5727 9.5766 9.5804

0.9356 0.9346

0.9336 12014 1.1985 1.1956 1 1926 1.1897

1.1868 1.1839 1.1810 1 1781 1 1752 1.1723

69" 00' 69O 50' 69O 40' 69O 30' 69O 20' 69O 10'

680 00' 680 50' 68O 40' 680 30' 680 20' 680 10'

1.1694 67O 00' 1.1665 67" 50' 1 1636 67O 40' 1 1606 67O 30' 1.1577 67'= 20' 1.1548 67O 10'

1.1519 1 1490 1 1461 1 1432 1.1403 1.1374

1.1345 1 1316 1.1286 1.1257 1.1228 1.1199

1.1170 1.1141 1.1112 1.1083 1.1054 1.1025

660 00' 660 50' 66" 40' 66" 30' 66" 20' 660 IO'

9.9702 0.3839 9.5842 2.6051 9.9697 0.3872 9.5879 2.5826 9.9692 0.3906 9.5917 2.5605

9.5641

9.5576

9.5673 9.5704

9.5609

9.5736 9.5767 9.5798 9.5828 9.5859 9.5889

0.9325 0.9315 0.9304 0.9293

0.4121 0.4083 0.4046 0.4009

0.4173 0.4200

0.3907 0.3934 0.3961 0.3987 0.4014 0.4041

0.4067 0.4094 0.4120 0.4147

0.9283 9.9677 0.4006 9.6028 2.4960 0.3972

0.9272 9.9672 0.4040 9.6064 2.4751 0.3936 0.9261 9.9667 0.4074 9.6100 2.4545 0.3900 0.9250 9.9661 0.4108 9.6136 2.4342 0.3864 0.9239 9.9656 0.4142 9.6172 2.4142 0.3828 0.9228 9.9651 0.4176 9.6208 2.3945 0.3792 0.9216 9.9646 0.4210 9.6243 2.3750 0.3757

9.5919 0.9205 9.5948 0.9194 9.5978 0.9182 9.6007 0.9171 9.6036 0.9159 9.6065 0.9147

9.6205 9.6232

9.6093 9.6121 9.6149 9.6177

0.9135 9.9607 0.4452 9.6486 0.9124 9.9602 0.4487 9.6520 0.9112 9.9596 0.4522 0.9100 9.9590 0.4557 0.9088 9.9584 0.4592 0.9075 9.9579 0.4628

9.9687 0.3939 9.5954 2.5386 9.9682 0.3973 9.5991 2.5172

2.3559 2.3369 2.3183 2.2998

0.3721 0.3686 0.3652 0.3617

2.2817 0.3583 2.2637 0.3548

2.2460 0.3514 2.2286 0.3480 2.2113 0.3447 2.1943 0.3413 2.1775 0.3380 2.1609 0.3346

2.1445 2.1283 2.1123 2.0965 2.0809 2.0655

2.0503 2.0353 2.0204 2.0057 1.9912

0.3313 0.3280 0.3248 0.3215 0.3183 0.3150

0.3118 0.3086 0.3054 0.3023 0.2991

1.9768 0.2960

65O 00' 65O 50' 65O 40' 65O 30' 65O 20' 65O IO'

64O 00' 64O 50' 64" 40' 64O 30' 64O 20' 64O 10'

1.9626 0.2928 1.0996 63O 00' 1.9486 0.2897 1.0966 63O 50' 1.9347 0.2866 1.0937 63O 40' 1.9210 0.2835 1.0908 63O 30' 1.9074 0.2804 1.0879 63'= 20' 1.8940 0.2774 1.0850 63O IO'

1.8807 1.8676 1.8546 1.8418 1.8291 1.8165

1.8040 1.7917 1.7796 1.7675 1.7556 1.7437

0.2743 1.0821 62O 00' 0.2713 1.0792 62'= 50' 0.2683 1.0763 62O 40' 0.2652 1.0734 62O 30' 0.2622 1.0705 62O 20' 0.2592 1.0676 62O 10'

0.2562 1.0647 61° 00' 61° 50' 61' 40' 61° 30' 61'= 20' 61° 10'

9.6279 9.6314 9.6348 9.6383 9.6417 9.6452

9.9640 0.4245 9.9635 0.4279 9.9629 0.4314 9.9624 0.4348 9.9618 0.4383 9.9613 0.4417

9.6553 9.6587 9.6620 9.6654

9.6687 9.6720 9.6752 9.6785 9.6817 9.6850

9.6882

0.4226 9.6259 0.4253 9.6286 0.4279 9.6313 04305 9.6340 0.4331 9.6366 0.4358 9.6392

0.4384 9.6418 04410 9.6444 0.4436 9.6470

0.9063 9.9573 0.9051 9.9567 0.9038 9.9561 0.9026 9.9555 0.9013 9.9549 0.9001 9.9543

0.8988 9.9537

0.4663 0.4699 0.4734 0.4770 04806 04841

04877 0.8975 9.9530 0.8962 9.9524 0 8949 9.9518 0.8936 9.9512

0.4913 9.6914 0.4950 9.6946

04462 9.6495 04488 9.6521

0.8923 9.9505

0.8910 0.9499 0.8897 9.9492 0.8884 9.9486 0.8870 9.9479 0.8857 9.9473 0.8843 9.9466

0 4986 9.6977 0.5022 9.7009 0 5059 9.7040

0 5095 9.7072 0.5132 9.7103 05169 9.7134 05206 9.7165 0.5243 9.7196 0.5280 9.7226

9.9459 0.5317 9.7257 9.9453 0.5354 9.7287 9.9446 0.5392 9.7317 9.9439 0 5430 9.7348 9.9432 0.5467 9.7378 9 9425 0.5505 9.7408

9.9418 0.5543 9.7438 9.9411 0.5581 9.7467 9.9404 0.5619 9.7497 9.9397 0.5658 9.7526 9.9390 0.5696 9.7556 9.9383 0.5735 9.7585

04514

04540 04566 0 4592 0.4617 0.4643 0 4669

9.6546

9.6570 9.6595

0.8829 0.8816 0.8802 0.8788 0.8774 0.8760

0.8746 0 8732 0.8718 0.8704 0.8689 0.8675

9.6620 9.6644 9.6668 9.6692

9.6716 9.6740 9.6763 9.6787 9.6810 9.6833

9.6856 9.6878 9.6901 9.6923 9.6946 9.6968

0.2533 1.0617 0.2503 1.0588 0.2474 1.0559 0.2444 1.0530 0.2415 1.0501

Natural Log" Natural Log'* Natural Log* * Natural Log"


‘For 0” 1 intervals. see Tables 1 21 1.22. and I 23 “Add 10 I” these columns

MATHEMATlCALTABLES& UNlTS&SYSTEMSOFWElGHTS & MEASURES 1-53

TABLE 1.24-TRIGONOMETRIC FUNCTIONS (AT INTERVALS OF 10')' (continued)

Sines Comes Tangents Cotangents

Degrees Radians Natural Log" Natural Log" Natural Log*' Natural Log"

300 00' 0.5236 0.5000 9.6990 0.8660 9.9375 0.5774 9.7614 1.7321 0.2386 1.0472 60a 00' 30" 10' 0.5265 0.5025 9.7012 0.8646 9.9368 0.5812 9.7644 1.7205 0.2356 1.0443 60° 50' 300 20' 0.5294 0.5050 9.7033 0.8631 9.9361 0.5851 9.7673 1.7090 0.2327 1.0414 60° 40' 30" 30' 0.5323 0.5075 9.7055 0.8616 9.9353 0.5890 9.7701 1.6977 0.2299 1.0385 60° 30' 300 40' 0.5352 0.5100 9.7076 0.8601 9.9346 0.5930 9.7730 1.6864 0.2270 1.0356 60° 20' 300 50' 0.5381 0.5125 9.7097 0.8587 9.9338 0.5969 9.7759 1.6753 0.2241 1.0327 60' 10'

310 00' 0.5411 0.5150 9.7118 08572 9.9331 0.6009 9.7788 1.6643 0.2212 1.0297 590 00' 310 IO' 0.5440 0.5175 9.7139 0.8557 9.9323 0.6048 9.7816 1.6534 0.2184 1.0268 590 50' 31" 20' 0.5469 0.5200 9.7160 0.8542 9.9315 0.6088 9.7845 1.6426 0.2155 1.0239 59" 40' 31'1 30' 0.5498 0.5225 9.7181 0.8526 9.9308 0.6128 9.7873 1.6319 0.2127 1.0210 590 30' 31" 40' 0.5527 0.5250 9.7201 0.8511 9.9300 0.6168 9.7902 1.6212 0.2098 1.0181 590 20' 310 50' 0.5556 0.5275 9.7222 0.8496 9.9292 0.6208 9.7930 1.6107 0.2070 1.0152 590 10'

32O 00' 0.5585 0.5299 9.7742 0.8480 9.9284 0.6249 9.7958 1.6003 0.2042 1.0123 58O 00' 32" 10' 0.5614 0.5324 9.7262 0.8465 9.9276 0.6289 9.7986 1.5900 0.2014 .I.0094 58" 50' 32O 20' 0.5643 0.5348 9.7282 0 8450 9.9268 0.6330 9.8014 1.5798 0.1986 1.0065 58O 40' 32" 30' 0.5672 0.5373 9.7302 0.8434 9.9260 0.6371 9.8042 1.5697 0.1958 1.0036 58' 30' 32O 40' 0.5701 0.5398 9.7322 0 8418 9.9252 0.6412 9.8070 1.5597 0.1930 1.0007 58O 20' 32' 50' 0.5730 0.5422 9.7342 0.8403 9.9244 0.6453 9.8097 1.5497 0.1903 0.9977 58' IO'

330 00' 330 10' 33" 20' 339 30' 330 40' 330 50'

340 00' 34" 10' 340 20'

0.5760 0.5446 9.7361 0.8387 9.9236 0.6494 9.8125 1.5399 0.5789 0.5471 9.7380 0.8371 9.9228 0.6536 9.8153 1.5301 0.5818 0.5495 9.7400 0.8355 9.9219 0.6577 9.8180 1.5204 0.5847 0.5519 9.7419 0.8339 9.9211 0.6619 9.8208 1.5108 0.5876 0.5544 9.7438 0.8323 9.9203 0.6661 9.8235 1.5013 0.5905 0.5568 9.7457 0.8307 9.9194 0.6703 9.8263 1.4919

0.5934 0.5592 9.7476 0.8290 9.9186 0.6745 9.8290 1.4826 0.5963 0.5616 9.7494 0.8274 9.9177 0.6787 9.8317 1.4733 0.5992 0.5640 9.7513 0.8258 9.9169 0.6830 9.8344 1.4641

0.1875 0.1847 0.1820 0.1792 0.1765 0.1737

0.1710 0.1683 0.1656

0.9948 0.9919 0.9890 0.9861 0.9832 0.9803

0.9774 0.9745 0.9716

57" 00' 570 50' 570 40' 570 30' 570 20' 570 IO'

56O 00' 56O 50' 56'= 40'

340 30' 0.6021 0.5664 9.7531 0.8241 9.9160 0.6873 9.8371 1.4550 0.1629 0.9687 56O 30' 340 40' 0.6050 0.5688 9.7550 0.8225 9.9151 0.6916 9.8398 1.4460 0.1602 0.9657 56" 20' 340 50' 0.6080 0.5712 9.7568 0.8208 9.9142 0.6959 9.8425 1.4370 0.1575 0.9628 56O IO'

350 00' 0.6109 0.5736 9.7586 0.8192 9.9134 0.7002 9.8452 1.4281 0.1548 0.9599 559 00' 350 IO' 0.6138 0.5760 9.7604 0.8175 9.9125 0.7046 9.8479 1.4193 0.1521 0.9570 550 50' 350 20' 0.6167 0.5783 9.7622 0.8158 9.9116 0.7089 9.8506 1.4106 0.1494 0.9541 550 40' 350 30' 0.6196 0.5807 9.7640 0.8141 9.9107 0.7133 0.8533 1.4019 0.1467 0.9512 550 30' 350 40' 0.6225 0.5831 9.7657 0.8124 9.9098 0.7177 9.8559 1.3934 0.1441 0.9483 550 20' 35" 50' 0.6254 0.5854 9.7675 0.8107 9.9089 0.7221 9.8586 1.3848 0.1414 0.9454 550 10'

36" 00' 0.6283 0.5878 9.7692 0.8090 9.9080 0.7265 9.8613 1.3764 0.1387 0.9425 540 00' 36O IO' 0.6312 0.5901 9.7710 0.8073 9.9070 0.7310 9.8639 1.3680 0.1361 0.9396 540 50' 36' 20' 0.6341 0.5925 9.7727 0.8056 9.9061 0.7355 9.8666 1.3597 0.1334 0.9367 540 40' 36O 30' 0.6370 0.5948 9.7744 0.8039 9.9052 0.7400 9.8692 1.3514 0.1308 0.9338 540 30' 36" 40' 0.6400 0.5972 9.7761 0.8021 9.9042 0.7445 9.8718 1.3432 0.1282 0.9308 54" 20' 36O 50' 0.6429 0.5995 9.7778 0.8004 9.9033 0.7490 9.8745 1.3351 0.1255 0.9279 540 10'

370 00' 0.6458 0.6018 9.7795 0.7986 9.9023 0.7536 9.8771 1.3270 0.1229 0.9250 530 00' 37" 10' 0.6487 0.6041 9.7811 0.7969 9.9014 0.7581 9.8797 1.3190 0.1203 0.9221 530 50' 370 20' 0.6516 0.6065 9.7828 0.7951 9.9004 0.7627 9.8824 1.3111 0.1176 0.9192 530 40' 370 30' 0.6545 0.6088 9.7844 0.7934 9.8995 0.7673 9.8850 1.3032 0.1150 0.9163 53" 30' 370 40' 0.6574 0.6111 9.7861 0.7916 9.8985 0.7720 9.8876 1.2954 0.1124 0.9134 530 20' 370 50' 0.6603 0.6134 9.7877 0.7898 9.8975 0.7766 9.8902 1.2876 0.1098 0.9105 53" 10'

38O 00' 0.6632 0.6157 9.7893 0.7880 9.8965 0.7813 9.8928 1.2799 0.1072 0.9076 52O 00' 38O IO' 0.6661 0.6180 9.7910 0.7862 9.8955 0.7860 9.8954 1.2723 0.1046 0.9047 52O 50' 38O 20' 0.6690 0.6202 9.7926 0.7844 9.8945 0.7907 9.8980 1.2647 0.1020 0.9018 52' 40' 38O 30' 0.6720 0.6225 9.7941 0.7826 9.8935 0.7954 9.9006 1.2572 0.0994 0.8988 52O 30' 38O 40' 0.6749 0.6248 9.7957 0.7808 9.8925 0.8002 9.9032 1.2497 0.0968 0.8959 52" 20' 38O 50' 0.6778 0.6271 9.7973 0.7790 9.8915 0.8050 9.9058 1.2423 0.0942 0.8930 52O 10'

390 00’ 0.6807 0.6293 9.7989 0.7771 9.8905 0.8098 9.9084 1.2349 0.0916 0.8901 510 00' 390 10' 0.6836 0.6316 9.8004 0.7753 9.8895 0.8146 9.9110 1.2276 0.0890 0.8872 51" 50' 39" 20' 0.6865 0.6338 9.8020 0.7735 9.8884 0.8195 9.9135 1.2203 0.0865 0.8843 510 40' 390 30' 0.6894 0.6361 9.8035 0.7716 9.8874 0.8243 9.9161 1.2131 0.0839 0.8814 510 30' 390 40' 0.6923 0.6383 9.8050 0.7698 9.8864 0.8292 9.9187 1.2059 0.0813 0.8785 51" 20' 390 50' 0.6952 0.6406 9.8066 0.7679 9.8853 0.8342 9.9212 1.1988 0.0788 0.8756 510 IO'

Natural Log*' Natural Log** Natural Log"' Natural Log"'

Cosines Smes Cotangents Tangents Radians Degrees

(contmued on next page)


Degrees Radtans

400 00' 400 10' 400 20' 400 30' 400 40' 400 50'

410 00' 410 10' 41" 20' 410 30' 410 40' 410 50'

42O 00' 42' IO' 42O 20' 42O 30' 42O 40' 420 50'

430 00' 430 10' 430 20' 430 30' 430 40' 43" 50'

0.6981 0.7010 0.7039 0.7069 0.7098 0.7127

0.7156 0.7185 0.7214 0.7243 0.7272 0.7301

0.7330 0.7359 0.7389 0.7418 0.7447 0.7476

0.7505 0.7534 0.7563 0.7592 0.7621 0.7650

44" DO' 440 IO' 44" 20' 440 30' 44" 40' 440 50'

450 00'

0.7679 0.7709 0.7738 0.7767 0.7796 0.7825

0.7854

TABLE 1.24-TRIGONOMETRIC FUNCTIONS(AT INTERVALS OF 10')' (continued)

Sines

Natural Log** Natural Log"

Tangents

Natural Log** Natural

0.6428 9.8081 0.7660 9.8843 0.8391 9.9238 1.1918 0.6450 9.8096 07642 9.8832 0.8441 9.9264 1.1847 0.6472 9.8111 0.7623 9.8821 0.8491 9.9289 1.1778 0.6494 9.8125 0.7604 9.8810 0.8541 9.9315 1.1708 0.6517 9.8140 0.7585 9.8800 0.8591 9.9341 1.1640 0.6539 9.8155 0.7566 9.8789 0.8642 9.9366 1.1571

Log*'

0.0762 0.0736 00711 0.0685 0.0659 0.0634

0.8727 500 00' 0.8698 50" 50' 0.8668 500 40' 0.8639 50" 30' 0.8610 500 20' 0.8581 500 10'

0.6561 9.8169 0.7547 9.8778 0.8693 9.9392 1.1504 0.0608 0.8552 490 00' 0.6583 9.8184 07528 9.8767 0.8744 9.9417 1.1436 0.0583 0.8523 490 50' 0.6604 9.8198 0.7509 9.8756 0.8796 9.9443 1.1369 0.0557 0.8494 49" 40' 0.6626 9.8213 0 7490 9.8745 0.8847 9.9468 1.1303 0.0532 0.8465 490 30' 0.6648 9.8227 0.7470 9.8733 0.8899 9.9494 1.1237 0.0506 0.8436 49" 20' 0.6670 9.8241 074.51 9.8722 0 8952 9.9519 1.1171 0.0481 0.8407 490 10'

0.6691 9.8255 0 7431 9.8711 0.9004 9.9544 1.1106 00456 0.8378 48O 00' 0.6713 9.8269 07412 9.8699 0.9057 9.9570 1.1041 0.0430 0.8348 48" 50' 0.6734 9.8283 0.7392 9.8688 0.9110 9.9595 1.0977 00405 0.8319 48O 40' 0.6756 9.8297 0.7373 9.8676 0.9163 9.9621 1.0913 0.0379 0.8290 480 30' 0.6777 9.8311 0.7353 9.8665 0.9217 9.9646 1.0850 0.0354 0.8261 48O 20' 0.6799 9.8324 0.7333 9.8653 0.9271 9.9671 1.0786 0.0329 0.8232 48" IO'

0.6820 9.8338 0.7314 9.8641 0.9325 9.9697 1.0724 0.0303 0.8203 0.6841 9.8351 0.7294 9.8629 0.9380 9.9722 1.0661 0.0278 0.8174 0.6862 9.8365 0.7274 9.8618 0.9435 9.9747 1.0599 0.0253 0.8145 0.6884 9.8378 0.7254 9.8606 0.9490 9.9772 1.0538 0.0228 0.8116 0.6905 9.8391 0.7234 9.8594 0.9545 9.9798 1.0477 0.0202 0.8087 0.6926 9.8405 07214 9.8582 0.9601 9.9823 1.0416 0.0177 0.8058

0.6947 9.8418 0.7193 9.8569 0.6967 9.8431 0.7173 9.8557 0.6988 9.8444 0.7153 9.8545 0.7009 9.8457 0.7133 9.8532 0.7030 9.8469 0.7112 9.8520 0.7050 9.8482 0.7092 9.8507

1.0355 0.0152 0.8029 1.0295 0.0126 0.7999 1.0235 0.0101 0.7970 1.0176 0.0076 0.7941 1.0117 0.0051 0.7912 1.0058 0.0025 0.7883

0.7071 9.8495 0.7071 9.8495

Natural Loq" Natural LOCI"

0.9657 9.9848 0.9713 9.9874 0.9770 9.9899 0.9827 9.9924 0.9884 9.9949 0.9942 9.9975

1.0000 0.0000

Natural Log'*

Cotangents

47" 00' 470 50' 470 40' 470 30' 470 20' 470 IO'

46O 00' 46O 50' 46' 40' 46O 30' 46' 20' 46O IO'

1 .oooo 0.0000

Natural Log"

0.7854 450 00'

Sines

Cotangents

Tangents Radians Degrees

MATHEMATICAL TABLES & UNITS & SYSTEMS OF WEIGHTS & MEASURES I-55

TABLE 1.25-EXPONENTIALS (e” and em”)’

” e

0.00 1000

Dlllerence

10

10 10

11 10

11 11

10

11 11

11

11

12 11

12

12

11

12

12

12

13 12

13 12 13

13

13

13

13 14

13 14

14 14 14

14

15 14

15 14

15

15 15 16

15

16

16

16

16

17

n e’ Otfferenc~ D~iierence r> e n e

_n 0.50 0 607 1.0 0368"

”

1.0 11 12

13

14

e

2 718'

3 004 3 320

0 01 0 990 002 0980

3669 003 0970 4055 0 04 0 961

15 4 482

16 4 953 1 7 5 474

18 6050

19 6 686

005 0951

0 06 0.942

007 0932 0 08 0 923

009 0914

2.0 7389 0.10 0 905 21 8 166 0 11 0 896 22 9025 012 0887

23 9 974 013 0878 24 11 02 014 0869

25 1218 015 0861 26 1346 016 0852 27 1488 017 0844 28 1644 018 0835

29 1817 019 0827

3.0

31 32

33

34

20 09 2220

24 53 27 11

2996

0.20 0 819 0 21 0 811 022 0803

023 0795

0 24 0 787

35 3312 025 0779 36 3660 026 0771

37 4045 027 0763

38 44 70 028 0756 39 4940 029 0748

4.0 41

42

43 44

5460 0.30 0 741 60 34 0 31 0 733

6669 032 0726

7370 033 0719 81 45 034 0712

45 9002 035 0705 036 0698

5.0 1484

60 4034 70 1097

80 2.981

90 8 103

10.0 22 026

;I2

2si2

3r/2 4ri2

5ii2

6ri2

7~12 7112

4 810

23 14 111 3

5355

2.576 0

037 0691

038 0684

0 39 0 677

0.40 0 670

0 41 0 664

042 0657 043 0651

044 0644

12.392 0

59.610 0

286.751 0

045 0638 046 0631

047 0625

048 0619

049 0613

0.50 0 607

” e’

* 0.00 1000

0 01 1010 0.02 1020 003 1030

0 04 1041

005 1051

0 06 1062 007 1073

0 08 1083

0 09 1094

0.10 1 105

0 11 1116

0 12 1127

0 13 1139

014 1150

0 15 1 162 0 16 1 174

017 i 185 0 18 1197

0 19 1209

0.20 1221

0 21 1234

0 22 1 246

023 1259

0 24 1.271

025 1284

0 26 1 297

027 1310

028 1323 0 29 1 336

0.30 1.350

0 31 1.363

032 1377

0 33 1391

034 1405

035 1419

0 36 1433

0 37 i 448

0.38 1 462

0 39 1477

0.40 1492

0 41 1 507

042 1522 043 1537

044 1553

0 45 1 568 0 46 i 584

047 1600

048 1616

049 1632

0.50 1649

0.50 1649

051 1 665 0 52 1682

0 53 1699

0 54 1716

0 55 1733

0 56 1751

0 57 1 768 0 58 1786

0 59 1804

0.60 1822

061 1840

0 62 1859

0 63 i a78

064 1896

0 65 1916 066 1935

067 1954

0 68 1974

0 69 1994

0.70 2 014

0 71 2034

0 72 2054

0 73 2 075 0 74 2 096

0 75 2 117 0 76 2 138

0 77 2 160

0 78 2181 0 79 2203

0.80 2226

081 2248

082 2 270 0 83 2 293 0 64 2 316

085 2340 0 86 2363

087 2 387

088 2411

0 89 2 435

0.90 2460

091 2484

0 92 2 509 0 93 2535

0 94 2560

0 95 2 586 0 96 2612

0 97 2638

0 98 2664

0 99 2691

1.00 2718

16

17 17

17 17

18 17

18

18 18

18

19

19 18

20

19

19

20

20 20

20 20

21 21

21

21

22

21

22

23

22 22

23

23 24

23

24 24

24 25

24

25

26 25

26

26

26 26

27

27

- I” -10 051 0600 11 0333 -10 052 0595 12 0301

9 053 0 589 13 0213

-10 054 0583 14 0247

0 055 0577 15 0223

J

-10 0 56 0 571 16 0202

n 057 0566 17 0183 I ; 058 0560 18 0165

059 0554 19 0150 - 9

0.60 0 549 2.0 0135 - 9

n 061 0543 21 0122 - 3

9 062 0538 22 0111 -

9 063 0533 23 0100

-

- 8 064 0 527 24 00907

25 00821 26 00743

27

28

29

3.0

31 32

33

34

35

36

37

38 39

00672 00608

00550

0 0498

00450

00408

0 0369

00334

00302 00273

00247

00224 00202

00183

00166 00150

00136 00123

00111

000674

000248 0000912

0000335

0.000123

-8 0.80 0 449 4.0

- 7 081 0445 4 1

- 7 082 0440 42

- 7 083 0436 4.3 -7 084 0432 44

-7 085 0427 45

7 086 0423

-7 087 0419 5.0

- 7 088 0415 60

- 7 089 0411 70

- 6 0.90 0 407 8 0

_ 091 0403 90 I

6 092 0399 10.0 0000045

- 7

093 0 395 -

6 094 0391 r;/2 0 208

2712 00432

- 7 095 0387 3~12 000898

6 096 0383 4~12 000187

- ^ 097 0379 5~12 0000388

1; 098 0375 6ni2 0000081

^ 099 0372 7~12 0000017 b

l.oo 0368 8~12 0000003

*e=pjy828, I,~=o.s~~Ts, lag,,e=04343. l/(04343)=2.3026. log,,(O 4343)=1.6378. log,,e" =n(O4343). For tables of multiples of 0.4343, see Table 1 30 "Do not ~nterpcdate m this column


RALLOGARITHMS'

n(0.6974-3)

TABLE 1.26-NATUI

" n(2 3026)

1 2.3026 2 4.6052 3 6.9078 4 9.2103 5 11.5129 6 13.8155 7 16.1181 8 18.4207 9 20.7233

0 6974-3 0 3948-5 0.0922 -7 0.7897-10 0.4871-12 0.1845-14 0.8819-17 0.5793-19 0 2767-21

8 9

0.0770 0.1655 0.2469 0.3221 0.3920

0.0862 0.1740 0.2546 0.3293 0.3988

Average Difference

95 87 80 74 69

0.4574 0.4637 65 0.5188 0.5247 61 0.5766 0.5822 57 0.6313 0.6366 54 0.6831 0.6881 51

0.7324 0.7372 0.7793 0 7839 0.8242

0.9083

1.3029

0.9478

0.8671

0.9858

1.3297

1.0225 1.0578 1.0919

1 .I249 1.1569 1.1878 1.2179 1.2470

1.2754

0.9123

09517

0.8286

0 9895

0.8713

1.0260 10613 1.0953

1.1282 1 1600 1.1909 1.2208 1.2499

1.2782

49 47 44 43 41

39 38 36 35 34

33 32 31 30 29

1.3558 1.3813

28 27 27 26 25

1.4061 1.4303 1.4540 1.4770 1.4996

1.3584

1.3056

1.3838

1.4085

1.3324

1.4327 14563 1.4793 15019

15239 1.5454 1.5665 1.5872 1.6074

1.6273 1.6467 1.6658 1.6845 1.7029

25 24 23 23 22

1.5217 1.5433 1.5644 1.5851 1.6054

1.6253 1.6448 1.6639 1.6827 1.7011

22

;1 20 20

20 19 19 18 18

Number 0 2 3 4 5 6 7

0.0198 0.0296 0.0392 0.0488 0.0583 0.0677

1

0.0100 01044 0.1906 02700 0.3436

0.4121 0.4762

1.0 1.1 1.2 1.3 1.4

1.5 1.6 1.7 1.8 1.9

0.0000 0.0953 0.1823 0.2624 0.3365

0.4055 0.4700 0.5306 0.5878 0.6419

0.6931 0.7419 0.7885 0.8329 0.8755

0.9163

0.1133 01222 01310 0.1989 0.2070 0.2151

0.6471

0.6981 0.7467 0.7930 0.8372 0.8796

0.2776 0.2852 0.2927 0.3507 0.3577 0.3646

0.4187 0.4253 0.4318 0.4824 0.4886 0.4947 0.5423 0.5481 0.5539 0.5988 0.6043 0.6098 0.6523 06575 0.6627

0.5365 0.5933

2.0 2.1 2.2 2.3 2.4

0.7031 07080 0.7129 0.7514 0.7561 0.7608

2.5 2.6 2.7 2.8 2.9

0.9203

0.7975 0.8020 0.8065 0.8416 0.8459 0.8502 0.8838 0.8879 0.8920

0.9243 0.9282 0.9322 0.9555 0.9594 0.9632 0.9670 0.9708 0.9933 0.9969 1.0006 1.0043 1.0080 1.0296 1.0332 1.0367 1.0403 1.0438 1.0647 1.0682 1.0716 1.0750 1.0784

3.0 1.0986 3.1 1.1314 3.2 1.1632 3.3 1.1939 3.4 1.2238

1.1019 1.1346 1.1663 1 .I969 1.2267

1.1053 1.1086 1.1119 1.1378 1.1410 1.1442 1.1694 1.1725 1.1756 1.2000 1.2030 1.2060 1.2296 1.2326 1.2355

3.5 1.2528 1.2556 1.2585 1.2613 1.2641 3.6 1.2809 1.2837 1.2865 1.2892 1.2920 3.7 1.3083 13110 1.3137 13164 1.3191 3.8 1.3350 1.3376 1.3403 1.3429 1.3455 3.9 13610 13635 1.3661 13686 1.3712

4.0 1.3863 1.3888 1.3913 13938 1.3962 4.1 1.4110 1.4134 1.4159 1.4183 1.4207 4.2 1.4351 1.4375 1.4398 1.4422 1.4446 4.3 1.4586 1.4609 1.4633 1.4656 1.4679 4.4 1.4816 1.4839 1.4861 1.4884 1.4907

4.5 4.6 4.7 4.8 4.9

5.0 5.1 5.2 5.3 5.4

1.5041 1.5063 1.5085 1.5107 1.5129 1.5151 1.5173 1.5195 1.5261 1.5282 1.5304 1.5326 1.5347 1.5369 1.5390 1.5412 1.5476 1.5497 1.5518 1.5539 1.5560 1.5686 1.5707 1.5728 1.5748 1.5769 1.5892 1.5913 1.5933 15953 1.5974

1.6094 1.6114 1.6134 1.6154 1.6174 1.6292 1.6312 1.6332 1.6351 1.6371 1.6487 1.6506 1.6525 1.6544 1.6563 1.6677 1.6696 1.6715 1.6734 1.6752 1.6771 1.6790 1.6808 1.6864 1.6882 1.6901 1.6919 1.6938 1.6956 1.6974 1.6993

01398 0.1484 0.1570 0.2231 0.2311 0.2390 0.3001 0.3075 0.3148 0.3716 0.3784 0.3853

0.4383 0.4447 0.5008 0.5068 0.5596 0.5653 0.6152 0.6206 0.6678 0.6729

0.7178 0.7227 07655 0.7701 0.8109 0.8154 0.8544 0.8587 0.8961 0.9002

0.4511 0.5128 0.5710 0.6259 0.6780

0.7275 0.7747 0.8198 0.8629 0.9042

0.9361 0.9400 0.9439 0.9746 0.9783 0.9821 1.0116 1.0152 1.0188 1.0473 1.0508 1.0543 1.0818 1.0852 1.0886

1.1151 1.1184 1.1217 1.1474 1.1506 1.1537 1.1787 1.1817 1.1848 1.2090 1.2119 1.2149 1.2384 1.2413 1.2442

1.2669 1.2698 1.2726 1.2947 1.2975 1.3002 13218 1.3244 1.3271 1.3481 1.3507 1.3533 13737 1.3762 1.3788

13987 1.4012 1.4036 1.4231 1.4255 1.4279 1.4469 1.4493 1.4516 1.4702 1.4725 1.4748 1.4929 1.4951 1.4974

1.5581 1.5602 1.5623 1.5790 1.5810 1.5831 1.5994 1.6014 1.6034

1.6194 1.6214 1.6233 1.6390 1.6409 1.6429 1.6582 1.6601 1.6620

This table gives the natural or Napierian logarithms (In) of numbers between 1 and 10, correct to four places. Moving the decimal pointn places to the right (or left) in the number is equivalent to adding R times 2 3026 (or n times 3.6974) to the logarithm. Base e=2.71828+.

‘,nw=(2.3026,,og,,x, ,og,,x=(O4343),nx, whete23026=ln 10 andO4343=log,:e

MATHEMATlCALTABLES&UNlTS& SYSTEMSOFWElGHTS&MEASURES 1-57

TABLE 1.26 -NATURAL LOGARITHMS' (continued)

n nf2.30261 n(0.6974-31

2.3026 0.6974-3 4.6052 0.3948-5 6.9078 0.0922- 7 9.2103 0.7897-10 11.5129 0.4871-12 13.8155 0.1845-14 16.1181 0.8819-17 18.4207 0.5793-19 20.7233 0.2767-21

Number

5.5 5.6 5.7 5.8 5.9

6.0 6.1 6.2 6.3 6.4

6.5 6.6 6.7 6.8 6.9

7.0 7.1 7.2 7.3 7.4

7.5 7.6 77 78 79

8.0 8.1 8.2 8.3 8.4

8.5 8.6 8.7 8.8 8.9

9.0 9.1 9.2 9.3 9.4

9.5 9.6 9.7 9.8 9.9

10.0

0 1 2 3 4 5 6 7 8 9 ____ 1.7047 1.7066 1.7084 1.7102 1.7120 1.7138 1.7156 1.7174 1.7192 1.7210 1.7228 1.7246 1.7263 1.7281 1.7299 1.7317 1.7334 1.7352 1.7370 1.7387 1.7405 1.7422 1.7440 1.7457 1.7475 1.7579 1.7596 1.7613 1.7630 1.7647 1.7750 1.7766 1.7783 1.7800 1.7817

1.7918 1.7934 1.7951 1.7967 1.7984 1.8083 1.8099 1.8116 1.8132 1.8148 1.8245 1.8262 1.8278 1.8294 1.8310 1.8405 1.8421 1.8437 1.8453 1.8469 1.8563 1.8579 1.8594 1.8610 1.8625

1.8718 1.8733 1.8749 1.8764 1.8779

1.7492 1.7509 1.7664 1.7681 1.7834 1.7851

1.8001 1.8017 1.8165 1.8181 1.8326 1.8342 1.8485 1.8500 1.8641 1.8656

1.8795 1.8810

1.7527 1.7544 1.7561 1.7699 1.7716 1.7733 1.7867 1.7884 1.7901

1.8034 1.8050 1.8066 1.8197 1.8213 1.8229 1.8358 1.8374 1.8390 1.8516 1.8532 1.8547 1.8672 1.8687 1.8703

1.8825 1.8840 1.8856 1.8871 1.8886 1.8901 1.8916 1.8931 1.8946 1.8961 1.8976 1.8991 1.9006 1.9021 1.9036 1.9051 1.9066 1.9081 1.9095 1.9110 1.9125 1.9140 1.9155 1.9169 1.9315

1.9459 1.9601 1.9741 1.9879 2.0015

2.0149 2.0281 2.0412 2.0541 2.0669

1.9184 1.9330

1.9473 1.9615 1.9755 1.9892 2.0028

1.9199 1.9344

1.9488 1.9629 1.9769 1.9906 2.0042

2.0162 2.0176 2.0295 2.0308 2.0425 2.0438 2.0554 2.0567 2.0681 2.0694

1.9213 1.9228 1.9242 1.9257 1.9272 1.9286 1.9301 1.9359 1.9373 1.9387 1.9402 1.9416 1.9430 1.9445

1.9502 1.9643 1.9782 1.9920 20055

2.0189 2.0321 2.0451 2.0580 2.0707

1.9516 1.9530 1.9544 1.9559 1.9657 1.9671 1.9685 1.9699 1.9796 1.9810 1.9824 19838 1.9933 1.9947 1.9961 1.9974 2.0069 2.0082 2.0096 20109

2.0202 2.0215 2.0229 20242 2.0334 2.0347 2.0360 2.0373 2.0464 2.0477 2.0490 2.0503 2.0592 2.0605 2.0618 2.0631 2.0719 2.0732 2.0744 2.0757

1.9573 1.9713 1.9851 1.9988 2.0122

2.0255 2.0386 2.0516 2.0643 2.0769

1.9587 1.9727 1 9865 2.0001 2.0136

2.0268 2.0399 2.0528 2.0656 2.0782

2.0794 2.0807 2.0819 2.0832 2.0844 2.0857 2.0869 2.0882 2.0894 2.0906 2.0919 2.0931 2.0943 2.0956 2.0966 2.0980 2.0992 2.1005 2.1017 2.1029 2.1041 2.1163 2.1282

2.1401 2.1518 2.1633 2.1748 2.1861

2.1972

2.1054 2.1066 2.1078 2.1090 2.1102 2.1175 2.1187 2.1199 2.1211 2.1223 2.1294 2.1306 2.1318 2.1330 2.1342

2.1412 2.1424 2.1436 2.1448 2.1459 2.1529 2.1541 2.1552 2.1564 2.1576 2.1645 2.1656 2.1668 2.1679 2.1691 2.1759 2.1770 2.1782 2.1793 2.1804 2.1872 2.1883 2.1894 2.1905 2.1917

2.1983 2.1994 2.2006 2.2017 2.2028

2.1114 2.1235 2.1353

2.1471 2.1587 2.1702 2.1815 2.1928

2.2039

2.1126 2.1138 2.1150 2.1247 2.1258 2.1270 2.1365 2.1377 2.1389

2.1483 2.1494 2.1506 2.1599 2.1610 2.1622 2.1713 2.1725 2.1736 2.1827 2.1838 2.1849 2.1939 2.1950 2.1961

2.2050 2.2061 2.2072 2.2083 2.2094 2.2105 2.2116 2.2127 2.2138 2.2148 2.2159 2.2170 2.2181 2.2192 2.2203 2.2214 2.2225 2.2235 2.2246 2.2257 2.2268 2.2279 2.2289 2.2300 2.2311 2.2322 2.2332 2.2343 2.2354 2.2364 2.2375 2.2386 2.2396 2.2407 2.2418 2.2428 2.2439 2.2450 2.2460 2.2471 2.2481 2.2492 2.2502

2.2513 2.2523 2.2534 2.2544 2.2555 2.2565 2.2576 2.2586 2.2597 2.2607 2.2618 2.2628 2.2638 2.2649 2.2659 2.2670 2.2680 2.2690 2.2701 2.2711 2.2721 2.2732 2.2742 2.2752 2.2762 2.2773 2.2783 2.2793 2.2803 2.2814 2.2824 2.2834 2.2844 2.2854 2.2865 2.2875 2.2885 2.2895 2.2905 2.2915 2.2925 2.2935 2.2946 2.2956 2.2966 2.2976 2.2986 2.2996 2.3006 2.3016

2.3026

Average Difference

18 18 17 17 17

16 16 16 16 15

15 15 15 15 14

14 14 14 13 13

13 13 13 13 12

12 12 12 12 12

12 12 11 11 11

II 11 11 11 11

IO IO 10 IO 10

Thts table gives the natural or Naprerian logarithms (In) of numbers between 1 and 10, correct to four places. Moving the decimal pornt n places to the right (or left) In the number is equivalent to adding n times 2.3026 (or n times 3.6974) to the logarithm. Base e=2 71828+

-In x=/2 3026) log .>x bg ,,x=(O 4343)ln x. where 2 3026=In 10 and 0 4343=log,,e


TABLE 1.27-HYPERBOLIC SINES [sinh x = Xte” -e -“)I’

x 0

0.0 0.0000 0.1 0 1002 0.2 0.2013 0.3 0.3045 04 0.4108

0.5 0.5211 0.6 0.6367 0.7 0.7586 0.8 0.8881 0.9 1.027

1 .o 1.175 1.1 1.336 1.2 1.509 1.3 1.698 1.4 1.904

1.5 2.129 1.6 2.376 1.7 2 646 1.8 2 942 1.9 3 268

2.0 3 627 2.1 4.022 2.2 4457 2.3 4.937 2.4 5.466

2.5 6.050 2.6 6.695 2.7 7.406 2.8 8.192 2.9 9.060

3.0 10.02 3.1 11.08 3.2 12.25 3.3 13.54 3 4 14.97

1 2 3 4 5 6 7 8 9

0.0100 0.0200 0.1102 0.1203 0.2115 0.2218 0.3150 0.3255 0.4216 0.4325

0.0400 0.0500 0.0600 0.0701 0.0801 0.0901 100 0.1405 0.1506 0.1607 0.1708 0.1810 0.1911 101 0.2423 0.2526 0.2629 0.2733 0.2837 0.2941 103 0.3466 0.3572 0.3678 0.3785 0.3892 0.4000 106 0.4543 0.4653 0.4764 0.4875 0.4986 0.5098 110

0.5324 05438 0.6485 0.6605 0.7712 0.7838 0.9015 0.9150 1.041 1055

0.0300 0.1304 0.2320 0.3360 0.4434

0.5552 0.6725 0.7966 0.9286 1.070

1.222

0.5666 0.5782 0.5897 0.6014 0.6131 0.6248 116 0.6846 0.6967 0.7090 0.7213 0.7336 0.7461 122 0.8094 0.8223 0.8353 0.8484 0.8615 0.8748 130 0.9423 0.9561 0.9700 0.9840 0.9981 1.012 138 1.085 1.099 1.114 1.129 1.145 1.160 15

1.191 1206 1.352 1369 1.528 1546 1.718 1.738 1.926 1948

1.238 1.386 1.403 1.564 1.583 1.758 1.970

1.779 1992

1.254 1.270 1.286 1.303 1.319 16 1.421 1.438 1.456 1.474 1.491 17 1.602 1.621 1.640 1.659 1.679 19 1.799 1.820 1.841 1.862 1.883 21 2.014 2.037 2.060 2.083 2.106 22

2.153 2 177 2.201 2.225 2.250 2.274 2.299 2401 2428 2.454 2.481 2.507 2.535 2.562 2.674 2 703 2 732 2 761 2.790 2820 2.850 2973 3.005 3.037 3069 3.101 3.134 3.167 3303 3 337 3372 3.408 3.443 3479 3.516

3665 3 703 3741 3.780 4.064 4.106 4.148 4 191 4.503 4549 4.596 4643 4.988 5.039 5.090 5.142 5.522 5 578 5.635 5.693

6.112 6.174 6.237 6.300 6.763 6.831 6.901 6.971 7.481 7.557 7.634 7.711 8.275 8.359 8.443 8.529 9.151 9.244 9.337 9 431

3.820 4.234 4.691 5.195 5.751

6.365 7.042 7.789 8.615 9.527

3.859 3.899 4.278 4.322 4739 4.780 5.248 5.302 5810 5.869

6.429 6.495 7.113 7.185 7.868 7.940 8.702 8.790 9623 9.720

10.12 10.22 11.19 11.30 12.37 12.49 13.67 13 81 15.12 15.27

10.32 10.43 11.42 11.53 12.62 12.75 13.95 14.09 15.42 15.58

10.53 1064 10.75 10.86 10.97 11 1165 11 76 11.88 12.00 12.12 12 12.88 13.01 13.14 13.27 13.40 13 14.23 14.38 14.52 14.67 14.82 14 15.73 15.89 16.05 16.21 1638 16

3.5 16.54 16.71 16.88 17.05 17.22 17.39 17.57 17.74 3.6 18.29 18.47 18.66 18.84 19.03 19.22 19.42 19.61 3.7 20.21 20.41 20.62 20.83 21.04 21 25 21 46 21.68 3 8 22.34 22.56 22.79 2302 23.25 23.49 2372 23.96 3.9 24.69 24.94 2519 25.44 25.70 25.96 26.22 26.48

4.0 27.29 27.56 4.1 30.16 30.47 42 33.34 33.67 43 36.84 37.21 4.4 40.72 41 13

27.64 30.77 34.01 3759 41.54

45.91 50.74 5608 61 98 68.50

28.12 31.08

28.40 31.39 34.70 38.35 42 30

28.69 31.71

34.35 37.97

35.05 38.73

41.96 42.81

28.98 32.03 3540 39.12 43.24

4.5 45.00 45.46 4 6 49.74 50.24 4.7 54.97 55.52 4.8 60.75 61.36 4.9 67.14 67.82

4637 51.25 56.64 62.60 69.19

46.84 47.31 51.77 52.29 57.21 57.79 63.23 63.87 69.88 70.58

47.79 52.81 58.37 64 51 71 29

29.27 32.35 35.75 39.52 43.67

48.27 53.34 58.96 65.16 72.01

5.0 74.20

Average Difference

2.324 2.590 2.881 3.200 3.552

3.940 4.367 4.837 5.356 5.929

6.561 7.258 8.028 8.879 9.819

2.350 25 2.617 27 2.911 30 3.234 33 3.589 36

3.981 4.412 4.887 5.411 5.989

6.627 7.332 8.110 8.969 9.918

39 44 48 53 58

64 71 79 87 96

17.92 19.81 21.90 24.20 26.75

29.56 32.68 36.11 39.91 44.11

18.10 17 20.01 19 22.12 21 24.45 24 27.02 26

29.86 29 33.00 32 36.48 35 4031 39 44.56 43

40.75 49.24 47 53.88 54.42 52 59.55 60.15 58 65.81 67.47 64 72.73 73.46 71

MATHEMATIC :AL TAB1 -ES&UNlTS&SYSTEMSOFWElGHTS& MEASURES I-59

0

TABLE 1.28-HYPERBOLIC COSINES [cash x ,=M(e* +e-“)I’

Average Difference 9 x 1 2 3 4 5 6

1.000 1 .ooo 1.000 1.001- 1.001 1.002

7 8

1.002 1.003 1.014 1.016 1.037 1.039 1.069 1.073 1.112 1.117

1.167 1.173 1.233 1.240 1.311 1.320 1.403 1.413 1.509 1.520

1.629 1.642 1.766 1.781

0.0 01 0.2 0.3 0.4

0.5

1.000 1.004 1.018 1.042 1.077 1.122

1.179 1.248 1.329 1.423 1.531

1.655 1.796

6

; 10 11

1.921 1.937 1.954 2.095 2.113 2.132

2.331

13 14 16 18 20

2.554 2.802 3.078 3.385 3.726

23 25 28 31 34

4.104 4.524 4.988 5.503 6.072

38 42 47 52 58

6.702 64 7.400 70 a.171 78 9.024 86 9.968 95

11.01 11 12.16 12 13.44 13 14.85 14 16.41 16

la.13 17 20.03 19 22.14 21 24.47 23 27.04 26

29.88 29 33.02 32 36.49 35 40.33 39 44.57 43

49.25 47 54.43 52 60.15 58 66.48 64 73.47 71

1.005 1.006 1.007 1.008 1.010 1.011 1.013 1.020 1.022 1.024 1.027 1.029 1.031 1.034 1.045 1.081

1.128 0.6 1.185 0.7 1.255

1.052 1.090

1.138

1.055 1.058 1.062 1.066 1 094 1.098 1.103 1.108

1.198 1.271

0.8 0.9

1.337 1.433

1.048 1.085

1.133 1.192 1.263 1.346 1.443

1.355 1.454

1.144 1.149 1.155 1.161 1.205 1.212 1.219 1.226 1.278 1.287 1.295 1.303 1.365 1.374 1.384 1.393 1.465 1.475 1.486 1.497

1 .o 1.543 1.555 1.567 1.579 1.591 1.604 1.616 1.1 1.669 1.682 1.696 1.709 1.723 1.737 1.752 1.2 1.811 1.826 1.841 1.857 1.872 1.888 1.905 1.3 1.971 1.988 2.005 2.023 2.040 2.058 2.076 1.4 2.151 2.170 2.189 2.209 2.229 2.249 2.269

1.5 2.352 1.6 2.577 1.7 2.828 1.8 3.107 1.9 3.418

2.0 3.762 2.1 4.144 2.2 4.568 2.3 5.037 2.4 5.557

2.5 2.6 2.7 2.8 2.9

6.132 6.769 7.473 8.253 9.115

10.07 11.12 12.29 13.57 15.00

16.57 18.31

2.290 2.310

2.507 2.530 2.750 2.776 3.021 3.049 3.321 3.353 3.655 3.690

4.026 4.065 4.436 4.480 4.891 4.939 5.395 5.449 5.954 6.013

6.571 6.636 7.255 7.327 8.011 a.091 8.847 8.935 9.772 9.869

10.79 10.90 11.92 12.04 13.17 13.31 14.56 14.70 16.08 16.25

17.77 17.95 19.64 19.84 21.70 21.92 23.98 24.22 26.50 26.77

29.29 29.58 32.37 32.69 35.77 36.13 39.53 39.93 43.68 44.12

48.28 48.76

2 374 2.395 2.417 2.439 2.462 2.484 2.601 2.625 2.650 2.675 2.700 2.725 2.855 2.882 2.909 3.137 3.167 3.197 3.451 3.484 3.517

3.799 3.835 3.873 4.185 4.226 4.267 4.613 4.658 4.704 5.087 5.137 5.188 5.612 5.667 5.723

2.936 2.964 2.992 3.228 3.259 3.290 3.551 3.585 3.620

3.910 3.948 3.987 4.309 4.351 4.393 4.750 4.797 4.844 5.239 5.290 5.343 5.780 5.837 5.895

6.193 6.255 6.317 6.379 6.443 6.507 6.836 6.904 6.973 7.042 7.112 7.183 7.548 7.623 7.699 7.776 7.853 7.932 8.335 8.418 8.502 8.587 8.673 8.759 9.206 9.298 9.391 9.484 9.579 9.675

10.17 10.27 10.37 10.48 10.58 10.69 11.23 11.35 11.46 11.57 11.69 11.81 12.41 12.53 12.66 12.79 12.91 13.04 13.71 13.85 13.99 14.13 14.27 14.41 15.15 15.30 15.45 15.61 15.77 15.92

16.74 16.91 17.08 17.25 17.42 17.60 18.50 18.68 18.87 19.06 19.25 19.44 20.44 2064 20.85 21.06 21.27 21.49 22.59 22.81 23.04 23.27 23.51 23.74 24.96 2521 25.46 25.72 25.98 26.24

27.58 30.48 33.69 37.23 41.14

45.47 50.25 55.53 61 37 67.82

27.86 30.79 3402 3760 41 55

4592 5075 56.09 61 99 68.50

28.14 28.42 28.71 31.10 31 41 31.72 34.37 34 71 35.06 3798 3836 38.75 41.97 4239 42.82

46.38 46.85 47.32 51.26 51 78 52.30 56.65 57.22 57.80 62.61 63.24 63.87 69.19 69 a9 70.59

29.00 3204 3541 39.13 4325

4780 5282 58.38 64.52 71.30

3.0 3.1 3.2 3.3 3.4

3.5 3.6 3.7 3.8 3.9

20.24 22.36 24.71

4.0 4.1 4.2 4.3 4.4

4.5 4.6 4.7 4.8 4.9

27.31 30.18 33.35 36.86 40.73

45.01 49.75 54.98 60 76 67.15

5.0 74.21

57.35 57.89 58.96 59.56 65.16 65.82 72.02 72.74


TABLE 1.29-HYPERBOLIC TANGENTS [tanh x = (e” - e -“)/(e” + e -“) = sinh xkosh x]

x 0

0.0 0.0000 0.1 0.0997 0 2 0.1974 0.3 0.2913 0.4 0.3800

0.5 0.4621 0.6 0.5370

2 3

0.0200 0.0300

4

0.0400 0.1391 0.2355 03275 04137

0 4930 0.5649

5

0.0500

6

0.0599 0.1587 0.2543 0.3452 0.4301

0.5080 0.5784

a 9

0.0898 0.1878 0.2821 0.3714 0.4542

Average Difference

0.0699 0.0798 0.1684 01781 0.2636 0.2729 0.3540 0.3827 0.4382 0.4462

100 98 94 89 82

0.5154 05227 0.5299 75 0.5850 0.5915 0.5980 67 0.6469 0.6527 0.6584 60 0.7014 0.7064 0.7114 52 0.7487 0.7531 0.7574 45

39 33 28 24 20

17 14 11 9 8

0.0100 0.1096 0.2070 0.3004 0.3885

0.4700 0.5441

0.1194 0.2165 0.3095 0.3969

0.4777 0.5511

0.1293 0.2260 0.3185 0.4053

0.4854 0.5581 0.6231 0.6805 0.7306

0.1489 0.2449 0.3364 0.4219

0.5005 0.5717 0.6352 0.6911 0.7398

0.7 0.6044 0.6107 0.6169 0.8 0.6640 0.6696 0.6751

0.6291 0.6858

0.6411 0.6963

0 9 0.7163

1.0 07616 1.1 0.8005 1.2 0.8337 1.3 0.8617 1.4 0.8854

1.5 0.9052 1.6 0.9217 1.7 0.9354 1.8 0.9468 1.9 09562

2.0 0.9640 2.1 0.9705 2.2 0 9757 2.3 0 9801 2.4 0 9837

0.7211 0 7259

0.7658 0.8041 0.8367 0.8643 0.8875

0.9069 0.9232 0.9367 0.9478 0.9571

0.7699 0.8076 0.8397 0.8668 0.8896

0.9087 0.9246 0.9379 0.9488 0.9579

0.9647 0.9654 0.9710 0.9716 0.9762 0.9767 0.9805 0.9809 0.9840 0.9843

2.5 0 9866 0.9869 2.6 0 9890 0.9892 2.7 0 9910 0.9912 28 09926 0.9928

0.9871 0.9895 0.9914 0.9929

0.7352 0.7443

0.7739 0.7779 0.8110 0.8144

0.7818 0.8178 0.8483 0.8741 0.8957

0.7857 0.7895 0.8210 0.8243

0.7932 0.8275 0.8565 0.8810 0.9015

0.9186 0.9329 0.9447 0.9545 0.9626

0.7969 0.8306 0.8591 0.8832 0.9033

0.9202 0.9342 0.9458 0.9554 0.9633

0.9699 0.9753 0.9797 0.9834 0.9863

0.9888 0.9908 0.9925 0.9938 0.9950

0.9992 0.9999

0.8426 0.8455 0.8693 0.8717 0.8917 0.8937

0.9104 0.9121 0.9261 0 9275 0.9391 0.9402 0.9498 0.9508 0.9587 0.9595

0.9661 0 9668 0.9674 0.9680 0.9727 0.9732 0.9738 0.9776 0.9780 0.9785 0.9816 0.9820 0.9823 0 9849 0.9852 0.9855

0.8511 0.8764 0.8977

0.9138 0.9154 0.9289 0.9302 0.9414 0.9425 0.9518 0.9527 0.9603 0.9611

09876 0.9879 0.9881 09899 0.9901 0.9903

0.8538 0.8787 0.8996

0.9170 0.9316 0.9436 0.9536 0.9619

0.9687 0.9743 0.9789 0.9827 0.9858

0.9884 0.9905 0.9922 0.9936 0.9947

0.9722 0.9771 0.9812 0.9846

0.9874 0.9897 0.9915 0.9931

0.9693 0.9748 0.9793 0.9830 0.9861

0.9917 0.9932

0.9919 0.9933

0.9920 0.9935

0.9942 0.9943 0.9944 0.9945 0.9946

0.9886 0.9906 0.9923 0.9937 0.9949

0.9967 0.9973 0.9978 0.9982 0.9985 0.9988 0.9990 0.9996 0.9996 0.9997 0.9998 0.9998 0.9998 0.9999

2.9 0.9940

3.0 0.9951 4.0 0.9993 5.0 0.9999

0.9941

0.9959 0.9995

TABLE 1 JO-MULTIPLES OF 0.4343 (0.43429448 = 109 roe)

2

0.0869

3 4 6

0.0

5

0.2171

8 9

0.3474 0.3909 1 .o 2.0

0.8252 1.2595

3.0 4.0

0.0000 0.0434 0.4343 0.4777 0.8686 0.9120 1.3029 1.3463 1.7372 1.7806

0.5212 0.9554 1.3897 1.8240

0.1303 0.1737 0.5646 0.6080 0.9989 1.0423 14332 1.4766 1.8675 1.9109

0.6514 1.0857 1.5200 1.9543

0.2606 0.3040 0.6949 0.7383 1.1292 1.1726 1.5635 1.6069 1.9978 2.0412

0.7817 1.2160 1.6503 2.0846

1.6937 2.1280

5.0 2.1715 2.2149 2.2583 2.3018 2.3452 2.3886 2.4320 2.4755 2.51 a9 2.5623 6.0 2.6058 2.6492 2.6926 2.7361 2.7795 2.8229 2.8663 2.9098 2.9532 2.9966 70 3.0401 3.0835 3.1269 3.1703 3.2138 3.2572 3.3006 33441 3.3875 3.4309 80 3.4744 3.5178 3.5612 3.6046 3.6481 3.6915 3.7349 3.7784 3.8218 3.8652 9.0 3.9087 3.9521 3.9955 40389 4.0824 4.1258 4.1692 4.2127 4.2561 4.2995

TABLE 1.31-MULTIPLES OF 2.3026 (2.3025851 =In,, = 110.4343)

1 2 3 4 5 6 7 8

0.2303 0.4605 0.6908 0.9210 1.1513 1.3816 1.6118 1.8421 2.5328 2.7631 2.9934 3.2236 3.4539 3.6841 3.9144 4.1447 4.8354 5.0657 5.2959 5.5262 5.7565 59867 6.2170 6.4472 7.1380 7.3683 75985 7.8288 8.0590 8.2893 8.5196 8.7498 9.4406 9.6709 9 9011 10.131 10.362 10.592 10.822 11.052

11.743 11.973 12.204 12.434 12.664 12.894 13.125 13.355 14046 14.276 14.506 14.737 14967 15.197 15.427 15.658 16.348 16.579 16 809 17039 17269 17.500 17.730 17.960 18651 la.881 19111 19342 19572 19.802 20.032 20.263 20.954 21.184 21 414 21.644 21 875 22.105 22.335 22.565

9 Y 0

0.0000 0.0 2.0723 4.3749 6.6775 8.9801 11 283

1.0 2.0 3.0 4.0

23026 4.6052 6.9078 9.2103

5.0 11 513 6.0 13.816 7.0 16.118 8.0 18421 90 20.723

13585 15.888 18.190 20.493 22.796

MATHEMATlCALTABLES& UNlTS&SYSTEMSOFWElGHTS& MEASURES I-61

TABLE 1.32-STANDARD DISTRIBUTION OFRESIDUALS

a/r yin Difference

0.0 0.000 0.1 0.054 0.2 0.107 0.3 0.160 0.4 0.213

0.5 0264 0.6 0.314 0.7 0 363 0.8 0.411 0.9 0 456

1.0 0.500 1.1 0.542 1.2 0.582 1.3 0.619 1.4 0.655

1.5 0.688 1.6 0.719 1.7 0748 1.8 0.775 1.9 0.800

2.0 0823 2.1 0.843 2.2 0 862 2.3 0.879 2.4 0.895

2.5 0.908 2.6 0 921 2.7 0.931 2.8 0.941 2.9 0.950

3.0 0.957 3.1 0.963 3.2 0 969 3.3 0.974 3.4 0 978

3.5 0 982 36 0985 3.7 0 987 38 0990 3.9 0991

4.0 0993

5.0 0.999

54 53 53 53 51

50 49 48 45 44

42 40 37 36 33

31 29 27 25 23

20 19 17 16 13

13 10 IO 9 7

6

a = any positive quantity, y = number of residuals that are

numerically smaller than 1, r = probable error of a single

observation. and n = number of observations.

TABLE1.33-FACTORS FORCOMPUTING PROBABLEERROR

Bessel Peters

0.6745

n/2 W) - 2 0.6745 3 0.4769 4 0.3894

5 0.3372 6 0.3016 7 0.2754 a 0.2549 9 0.2385

IO 0.2248 11 0.2133 12 0.2034 13 0.1947 14 o.la7l

15 0.1803 16 0.1742 17 0.1686 ia 0.1636 19 0.1590

20 0.1547 21 0.1508 22 0.1472 23 0.1438 24 0.1406

25 0.1377 26 0.1349 27 0.1323 28 0.1298 29 0.1275

30 0.1252 31 0.1231 32 0.1211

z: 0.1192 0.1174

35 01157 36 0.1140 37 0.1124 38 0.1109 39 0.1094

40 0.1080 45 0.1017

50 0.0964 55 0.0918

60 0.0878 65 0.0843

70 0.0812 75 0.0784

a0 0.0759 a5 0.0736

90 0.0715 95 0.0696

100 0.0678

0.6745

\l;r(n-1)

0.4769 0.2754 0.1947

0.1508 0.1231 0.1041 0.0901 0.0795

0.0711 0.0643 0.0587 0.0540 0.0500

0.0465 0.0435 0.0409 0.0386 0.0365

0.0346 0.0329 0.0314 0.0300 0.0287

0.0275 0.0265 0.0255 0.0245 0.0237

0.0229 0.0221 0.0214 0.0208 0.0201

0.0196 0.0190 0.0185 0.0180 0.0175

0.0171 0.0152

0.0136 0.0124

0.0113 0.0105

0.0097 0.0091

0.0085 0.0080

0.0075 0.0071

0.0068

0.8453 0.8453

~~)

0.5978 0.3451 0.2440

0.1890 0.1543 0.1304 0.1130 0.0996

0.0891 0.0806 0.0736 0.0677 0.0627

0.0583 0.0546 0.0513 0.0483 0.0457

0.0434 0.0412 0.0393 0.0376 0.0360

0.0345 0.0332 0.0319 0.0307 0.0297

0.0287 0.0277 0.0268 0.0260 0.0252

0.0245 0.0238 0.0232 0.0225 0.0220

0.0214 0.0190

0.0171 0.0155

0.0142 0.0131

0.0122 0.0113

0.0106 0.0100

0.0094 0.0089

0.0085

nm 04227 01993 0.1220

0.0845 00630 0 0493 0 0399 00332

00282 00243 0.0212 00188 00167

00151 00136 00124 00114 00105

0 0097 0 0090 00084 00078 0.0073

0 0069 00065 00061 00058 00055

00052 00050 00047 00045 0.0043

00041 00040 00038 00037 00035

00034 00028

0.0024 0.0021

00018 0.0016

00015 0.0013

0.0012 0.0011

0.0010 0.0009

0.0008


TABLE1.34-COMPOUND INTEREST-AMOUNTOFAGIVEN PRINCIPAL

The amount A at the end of n years of a given principal P placed at compound Interest today isA = Px xor A = Px yorA = P xz,according asiheinterest(atthe rateofipercentperannum) iscompounded annually, semtannually, or quarterly; the factorx or yorz being taken from the following tables.

Values of x-Interest Compounded Annually: A=Pxx’

Years 2'/2 3

1.0300 1.0609 1.0927 11255

1.1593 1.1941 1.2299 12668 13048

13439 1.3842

3'12

1.0350 1.0712 1.1087 1.1475

4 4'12 5 6

1.0400 10450 1.0500 1.0600 1.0816 1.0920 1.1025 1.1236 1.1249 1.1412 1.1576 1.1910 1.1699 1.1925 1.2155 1.2625

1.1877 1.2167 1.2462 1.2763 1.3382 1.2293 1.2653 1.3023 1.3401 1.4185 1.2723 1.3159 1.3609 1.4071 1.5036 1.3168 1.3686 1.4221 1.4775 1.5938 1.3629 1.4233 14861 1.5513 1.6895

7

1.0700 1.1449 12250 1.3108

1.4026 1.5007 1.6058 1.7182 1.8385

1.9672 2.1049

1.0200 1.0250 1.0404 1.0506 1.0612 1.0824

1.1041 1.1262 1.1487 1.1717 11951

1.2190 1.2434

1.0769 1.1038

8 9

10 11

1.1314 1.1597 1.1887 1.2184 1.2489

12 1.2682 13 1.2936 14 1.3195

1.3459 1.3728 1.4002 1.4282 1.4568

1.2801 1.3121 1.3449 1.3785 1.4130

15 16 17 18 19

1.4483 1.4845 1.5216 1.5597 1.5987

20 1.4859 1.6386 25 1.6406 1.8539 30 1.8114 2.0976

40 2.2080 2.6851 50 2.6916 3.4371 60 3.2810 4.3998

1.4106 1.4802 1.5530 1.6289 1.7908 1.4600 1.5395 1.6229 1.7103 1.8983

14258 14685

1.5111 1.5640 1.6187

22522 2.4098 2.5785

1.6753 1.7340 1.7947 18575 1.9225

1.6010 1.6959 1.7959 1.6651 1.7722 1.8856 1.7317 1.8519 1.9799

1.8009 1.9353 2.0789 1.8730 2.0224 2.1829 1.9479 2.1134 2.2920 2.0258 2.2085 2.4066 2.1068 2.3079 2.5270

2.0122 2.1329 2.2609

2.3966 2.7590 2.5404 2.9522 2.6928 3.1588 2.8543 3.3799 3.0256 3.6165

1.9898 2.1911 2.4117 2.6533 3.2071 3.8697 2.3632 2.6658 3.0054 3.3864 4.2919 5.4274 2.8068 3.2434 3.7453 4.3219 5.7435 7.6123

3.9593 4.8010 5.8164 7.0400 10.286 14.974 5.5849 7.1067 9.0326 11.467 18.420 29.457 7.8781 10.520 14.027 18.679 32.988 57.946

15126

15580 1.6047 1.6528 1.7024 1.7535

1.8061 20938 2.4273

32620 43839 5.8916

‘This table IS computed from the formula Y= 1 +(!/lClO)”

Values of y-interest Compounded Semiannually: A = Pxy’

Years i=2 2'/2 3 3% 4 4% ~ 1.0201 1.0252 1.0302 1.0353 1 1.0404 1.0455

2 1.0406 1.0509 1.0614 1.0719 1.0824 1.0931 3 1.0615 1.0774 1.0934 1.1097 1.1282 1.1428 4 1.0829 1.1045 1.1265 1.1489 1.1717 1.1948

5 1.1046 1.1323 1.1605 1.1894 1.2190 1.2492 6 1.1268 1.1608 1.1956 1.2314 1.2682 1.3060 7 1.1495 1.1900 1.2318 1.2749 1.3195 1.3655 8 1.1728 1.2199 1.2690 1.3199 1.3728 1.4276 9 1.1961 1.2506 1.3073 1.3665 1.4282 1.4926

IO 1.2202 1.2820 1.3469 1.4148 1.4859 1.5605 11 1.2447 1.3143 1.3876 1.4647 1.5480 1.6315 12 2.2697 1.3474 1.4295 1.5164 1.6084 1.7058 13 1.2953 1.3812 1.4727 1.5700 1.6734 1.7834 14 1.3213 1.4160 1.5172 1.6254 1.7410 1.8645

15 1.3478 1.4516 1.5631 1.6828 1.8114 1.9494 16 1.3749 1.4881 1.6103 1.7422 1.8845 2.0381 17 1.4026 1.5256 1.6590 1.8037 1.9607 2.1308 18 1.4308 1.5639 1.7091 1.8674 2.0399 2.2278 19 1.4595 1.6033 1.7608 1.9333 2.1223 2.3292

20 1.4889 1.6436 1.8140 2.0016 2.2080 2.4352 25 1.6446 1.8610 2.1052 2.3808 2.6916 3.0420 30 1.8167 2.1072 2.4432 2.8318 3.2810 3.8001

40 2.2167 2.7015 3.2907 4.0064 4.8754 5.9301 50 2.7048 3.4634 4.4320 5.6682 7.2446 9.2540 60 3.3004 4.4402 5.9693 8.0192 10.765 14.441

‘This table IS computed from the formula y = t + (W’OO)“’

6 7 5

1.0506 1.1038 1.1597 12184

1.2801 1.3449 1.4130 1.4845 1.5597

16386 1.7216

1.0609 I.1255 1.1941 1.2668

1.0712 1.1475 1.2293 1.3168

1.3439 1.4106 1.4258 1.5111 1.5126 1.6187 1.6047 1.7340 17024 1.8575

1.8061 1.9161 2.0328 2.1566 2.2879

2.4273 2.5751 2.7319 2.8983 3.0748

1.9898 2.1315 2.2833 2.4460 2.6202

2.8068 3.0067 3.2209 3.4503 3.6960

3.2620 4.3839 5.8916

10.641 19.219 34.711

3.9593 5.5849 7.8781

15.676 31.191 62.064

18087 1.9003 1.9965

2.0976 2.2038 2.3153 2.4325 2.5557

2.6851 3.4371 4.3998

7.2096 11.814 19.358

MATHEMATlCALTABLES8, UNITS& SYSTEMSOF WEIGHTS & MEASURES 1-63

V

TABLE 1.34-COMPOUND INTEREST-AMOUNT OF A GIVEN PRINCIPAL (continued)

Values of z--Interest Compounded Quarterly: A=Pxz'

Years 1=2

1.0202 1.0407 1.0617 1.0831

1.1049 1.1272 1.1499 1 1730 1.1967

1.2208 1.2454 1.2705

2112

1.0252 1.0511 1.0776 1.1048

1.1327 1 1613 1.1906 12206 12514

1.2830 13154 1.3486 1.3826 14175

1.4533

3

10303 1.0616 10938 1 1270

3% 4

1.0406

4% 5

1.0355 1.0722 1.1102 1.1496

1.0458 1.0509 1.0829 1.0936 1.1045 1.1268 1.1437 1.1608 1.1726

1.1612 1.1903 12202 11964 12326 1.2697 1.2327 1.2763 1.3213 12701 1.3215 1.3749 1.3086 1.3684 1.4308

1.1960 1.2199

1.2508 1.2820 1.3080 1.3474 1.3679 1.4160 1.5172 14305 1.4881 1.6103 1.4959 1.5639

1.3483 1.4169 1.4889 1.5644 1.6436 13893 14672 1.5493 1.6360 1.7274 14314 1.5192 1.6122 1.7108 1.8154 14748 1.5731 16777 1.7891 1.9078 1.5196 1.6288 1.7458 1.8710 2.0050

1.5657 1.6866 1.8167 1.7464 1.8905 1.8063 1.9672

1.9566 2.1072 2.4432 2.8318 2.0462 22145 2.5931 3.0353 2.1398 2.3274 2.7523 3.2534 22378 2.4459 2.9212 3.4872 2.3402 2.5705 3.1004 3.7378

6

1.0614 1.1265 1.1956 1.2690

1.3469 1.4295

7

1.0719 1.1489 1.2314 1.3199

1.4146 1.5164 1.6254 1.7422 1.8674

2.0016 2.1454 2.2996 2.4648

2

4

5 6

8 9

10 11 12 13 14

15

1.7091

1.8140 1.9253 2.0435 2.1689 1.2961

1.3222 2.3020 2.6420

1.3489 16 1.3760 1.4900 1.6132 17 1.4038 1.5276 1.6621 18 1.4320 1.5661 17126 1.8725 2.0471 19 1.4609 1.6056 17645 1.9389 2.1302

20 1.4903 1.6462 1 8180 2.0076 2.2167 2.4473 2.7015 3.2907 4.0064 25 1.6467 1.8646 2.1111 23898 2.7048 3.0609 3 4634 4.4320 5.6682 30 18194 2.1121 2.4514 28446 3.3004 3.8285 4.4402 5.9693 8.0192

40 2.2211 2.7098 3.3053 4.0306 49138 5.9892 7.2980 10.828 16051 50 27115 3.4768 4.4567 5.7110 7.3160 9.3693 11.995 19.643 32.128 60 3.3102 44608 6.0092 8.0919 10893 14.657 19.715 35.633 64.307

'Th,s table IS compuled from ,he formula z= 1 +(,,400)111

TABLE 1.35-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= Yx v. where the factor \I is taken from the following table. (Interest at i percent per annum, compounded annually)

alues of V*

Years

10 11 12 13 14

15 16 17 18 19

20 25 30

40 50 60

2% 3 A 7. 3%

1.0000 2.0350 3 1062 4.2149

53625 65502 7.7794 90517 10368

11 731 13.142 14602 16113 17677

4 4'h 5

10000 10000 1 0000 2 0400 20450 20500 3.1216 3.1370 3.1525 4.2465 4.2782 4.3101

5.4163 5.4707 55256 66330 6.7169 6.8019 7.8983 8.0192 8.1420 9.2142 9.3800 9.5491 10.583 10.802 11.027

12006 12.288 12.578 13.486 13.841 14.207 15026 15.464 15.917 16627 17.160 17.713 18 292 18.932 19.599

20024 20.784 21 579 21.825 22.719 23.657

r=2

10000 20200 30604 41216

52040 63081 74343 85830 9.7546

10.950 12169 13.412 14.680 15974

17293 18639

10000 1.0000 20250 2.0300

1.0000 1.0000 2.0600 2.0700

3.0756 4.1525

5.2563

3.0909 41836

5.3091 64684 7.6625 8.8923 10.159

11.464 12808 14192 15.618 17086

18599 20157

3.1836 3.2149 4.3746 4.4399

5.6371 5.7507 6.9753 7.1533 8.3938 8.6540 9 8975 10260 11.491 11.978

13.181 13.816 14972 15784 16870 17888 18882 20.141 21 015 22550

23276 25129 25673 27.668

6.3877 7.5474 87361 9 9545

11.203 12.483 13.796 15.140 16519

17.932 19.380

19296 20971 22705 24500

20012 20.865 21 762 21 412 22386 23.414

23 698

27.671

24.742

29.064

25840

30.539

29.778 31 371

25.645

33.066 41.646

26.855

44.565

28.132

47.727 56.085 61.007 66.439

95.026 107.03 120.80 152.67 17850 209.35 237.99 289 50 353.58

33 760 37.379

28213 30840 30906 33.999

22.841 23.946 25.117 26357

26 870 28280 36.459 38950 47.575 51 623

75.401 84.550 112.80 131 00 163.05 19652

24297 25545 32030 34.158

36786 40.995 54865 63249

40568 43.903

60.402 67403 84.579 97484 11405 135.99

79058 94461

154.76 19964 290.34 406.53 533.13 81352


TABLE 1.36-PRINCIPAL AMOUNTING TO A GIVEN SUM

The pnncipal P whrch. tf placed at compound interest today, will amount to a given sum A at the end ofn years IS P=A xx'or P=A xy'or P= A xz', according as the interest (at the rate of i percent per annum)is compounded annually, semiannually,orquarterly; thefactorx'or y'orz'beingtaken from the following tables.

Values of x,-Interest Compounded Annually: P=A xx”

Years

9

10

i=2

0.98039 0.96117 0 94232 0.92385

0 90573 0.88797 0.87056 0.85349 0.83676

0.82035

2%

0.97561 0.95181 0.92860 0.90595

3

0 97087 0 94260 091514 0 88849

3%

0.96618 0.93351 0.90194 0.87144

0.88385 0.86261 0.84197 0.86230 083748 0.81350 0.84127 0 81309 0.78599 0.82075 0.78941 0.75941 0.80073 0.76642 0.73373

0.78120 0.74409 0.70892

4 4%

0.96154 0.95694 0.92456 0.91573 0.88900 0.87630 0.85480 0.83856

0.82193 0.80245 0.79031 0.76790 0.75992 0.73483 0.73069 0.70319 0.70259 0.67290

0.67556 0.64393

5 6 7

0.95238 0.94340 0.93458 0.90703 0.89000 0.87344 0.86384 0.83962 0.81630 0.82270 0.79209 0.76290

0.78353 0.74726 0.71299 074622 0.70496 0.66634 0.71068 0.66506 0.62275 067684 0.62741 0.58201 0.64461 0.59190 0.54393

0.61391 0.55839 0.50835 11 0.80426 076214 0.72242 068495 0.64958 0.61620 0.58468 0.52679 0.47509 12 0.78849 0.74356 0.70138 0.66178 0.62460 0.58966 0.55684 0.49697 0.44401 13 0.77303 0.72542 0.68095 0.63940 0.60057 0.56427 0.53032 0.46884 0.41496 14 0.75788 0.70773 0.66112 0.61778 0.57748 0.53997 0.50507 0.44230 0.38783

15 0.74301 0.69047 0.64186 0.59689 0.55526 0.51672 0.48102 0.41727 0.36245 16 0.72845 0.67362 0.62317 0.57671 0.53391 0.49447 0.45811 0.39365 0.33873 17 0.71416 0.65720 0.60502 0.55720 0.51337 0.47318 0.43630 0.37136 0.31657 18 0.70016 0.64117 0.58739 0.53839 0.49363 0.45280 0.41552 0.35034 0.29586 19 0.68643 0.62553 0.57029 0.52016 0.47464 0.43330 0.39573 0.33051 0.27651

20 0.67297 0 61027 0.55368 0.50257 0.45639 0.41464 0.37689 0.31180 0.25842 25 0.60953 0.53939 0.47761 0.42315 0.37512 0.33273 0 29530 0.23300 0.18425 30 0.55207 0.47674 0.41199 0.35628 0.30832 0.26700 0.23138 0.17411 0.13137

40 0.45289 0.37243 0.30656 0.25257 0.20829 0.17193 0.14205 0.09722 0.06678 50 0.37153 0.29094 0.22811 0.17905 0.14071 0.11071 0.08720 0.05429 0.03395 60 0 30478 0.22728 0.16973 0.12693 0.09506 0.07129 0.05354 0.03031 0.01726

Values of y'-Interest Compounded Semiannually: P=Axy’

Years i=2 2% 3 3% 4 4% 5 6 7

1 0.98030 0.97546 097066 0.96590 0.96117 0.95647 0.95181 0.94260 0.93351 2 0.96098 0.95152 094218 0.93296 0.92385 0.91484 0.90595 0.88849 0.87144 3 0.94205 0.92817 0.91454 0 90114 0.88797 0.87502 0.86230 0.83748 0.81350 4 0.92348 0.90540 0.88771 0 87041 0.85349 0.83694 0.82075 0.78941 0.75941

5 0.90529 0.88318 0.86167 0 84073 0.82035 0.80051 0.78120 0.74409 0.70892 6 0.88745 0.86151 0.83639 0 81206 0.77849 0.76567 0.74356 0.70138 0.66178 7 0.86996 0.84037 0.81185 0 78436 0.75788 0.73234 0.70773 0.66112 0.61778 8 0.85282 0.81975 0.78803 0.75762 0.72845 0.70047 0.67362 062317 0.57671 9

10 11 12 13 14

15 16 17

0.83602 0.79963 0.76491 0 73178 0.70016 0.66998 0.64117 0 58739 0.53836

0.81954 0.78001 0.74247 0.70682 0.67297 0.64082 0.61027 055368 0.50257 0.80340 0.76087 0.72069 0.68272 0.64684 0.61292 0.58086 052189 0.46915 0.78757 0.74220 0.69954 0.65944 0.62172 0.58625 0.55288 049193 0.43796 077205 0.72398 0.67902 0 63695 0.59758 0.56073 0.52623 0 46369 0.40884 0.75684 0.70622 0.65910 061523 0.57437 0.53632 0.50088 043708 0.38165

0.74192 0.68889 0.63976 0 59425 0.55207 0.51298 0.47674 041199 0.35628 0.72730 0.67198 0.62099 0 57398 0.53063 0.49065 045377 038834 0 33259 0.71297 0.65549 0.60277 0 55441 0.51003 0.46930 0.43191 0.36604 0.31048

18 069892 0.63941 058509 0.53550 0.49022 0.44887 0.41109 0.34503 0.28983 19 0.68515 0.62372 056792 051724 0.47119 0.42933 0.39128 0.32523 0.27056

20 0.67165 0.60841 0 55126 0.49960 0.45289 0.41065 0.37243 0.30656 0.25257 25 060804 0.53734 047500 042003 0.37153 032873 0.29094 0.22811 0.17905 30 055045 0.47457 040930 035313 030478 0.26315 0.22728 0.16973 0.12693

40 0.45112 0.37017 0 30389 0.24960 0.20511 0.16863 0.13870 0.09398 0.06379 50 0.36971 0.28873 0 22563 0.17642 0.13803 0.10806 0.08465 0.05203 0.03206 60 0.30299 0.22521 016752 0.12470 0.09289 0.06925 005166 002881 001611

MATHEMATICALTABLES & UNITS &SYSTEMS OF WEIGHTS & MEASURES 1-65

TABLE 1.36-PRINCIPAL AMOUNTING TO A GIVEN SUM (continued)

Values of z’-Interest Comoounded Quarterlv: P=A xz’*

Years i=2 2% 3 3% 4 4% 5 6 7

1 0.98025 0.97539 0.97055 0.96575 0.96098 0.95624 0.95152 0.94218 0.93296 2 0.96089 0.95138 0.94198 0.93266 0.92348 0.91439 0.90540 0.88771 0.87041 3 0.94191 0.92796 0.91424 0.90074 0.88745 0.87437 0.86151 0.83639 0.81206 4 0.92330 0.90512 0.88732 0.86989 0.85282 0.83611 0.81975 0.78803 0.75762

5 0.90506 0.88284 086119 0.84010 0.81954 0.79952 0.78001 074247 0.70682 6 0.88719 0.86111 0.83583 0.81132 0.78757 0.78453 0.74220 0.69954 0.65944 7 0.86966 0.83991 0.81122 0 78354 0.75684 0.73107 0.70622 0.65910 0.61523 8 0.85248 0.81924 0.78733 0 75670 0.72730 0.89908 0.87198 0.62099 0.57390 9 0.83564 0.79908 0.76415 0 73079 0.69892 0.66849 0.63941 0.58509 0.53550

IO 0.81914 0.77941 11 0.80296 0.76022 12 0.78710 0.74151 13 0.77155 0.72326 14 0.75631 0.70546

0.60841 0.55126 0.49960 0.57892 0.51939 0.46611 0.55086 0 48936 0.43486 0.52415 046107 0.40570 0.49874 0.43441 0.37851

15 16 17 18 19

20 25 30

40 50 60

0.74137 0.68809 0.72673 0.67115 0.71237 0.65464 0.69830 0.63852 0.68451 0.62281

0.74165 070576 0.67165 0.63923 0.71981 068159 0.64545 0.61126 0.69861 065825 0.62026 0.58451 0.67804 0.63570 0.59606 0.55893 0.65808 0.61393 0.57280 0.53447

0.63870 0.59291 0.55045 0.51108 0.61989 0.57260 0.52897 0.48871 0.60164 0 55299 0.50833 0.46733 0.58392 0.53405 0.48850 0.44687 0.56673 0.51576 0.46944 0.42732

0.47457 0.45156 0.42967 0.40884 0.38903

0.67099 0.60748 0.60729 0.53630 0.54963 0.47347

0.45023 0.36903 0.36880 0.28762 0.30210 0.22417

0.55004 0.49810 0.45112 0.40862 0.37017 0.47369 041845 0.36971 0.32670 0.28873 0.40794 035154 0.30299 0.26120 0.22521

0.30255 0.24810 0.20351 0.16697 0.13702 0.22438 0.17510 0.13669 0.10673 0.08337 0.16641 012358 0.09181 0.06823 0.05072

0.40930 0.35313 0.38563 0.32946 0.36334 0.30737 0.34233 0.28676 032254 0.26754

0.30389 0.24960 0.22563 0.17642 0.16752 0.12470

0.09235 0.06230 0.05091 0.03113 0.02806 0.01555

'The formula used for this lable IS L'= [l +(1/100)] = l/z

TABLE 1.37-ANNUITY AMOUNTING TO A GIVEN SUM (SINKING FUND)

The annual payment, V, which, if set aside at the end of each year, will amount with accumulated interesttoagivensum Sattheendofnyearsis Y=Sxv', wherethefactorv'isgiven below. Interest is i percent per annum, compounded annually.)

Values of v"

Years

2 3 4

5 6

8 9

i=2

0.49505 0.32675 024262

019216 015853 013451 0.11651 010252

2% 3 3% 4 4% 5 6 7

0.49383 0.49261 0.49140 0.49020 0.48900 0.48780 0.48544 0 48309 0.32514 0.32353 0.32193 0.32035 0.31877 0.31721 0.31411 031105 0.24082 0.23903 0.23725 0.23549 0.23374 0.23201 0.22859 022523

0.19025 0.18835 0.18648 0 18463 0.18279 0.18097 0.17740 017389 0.15655 0.15460 0.15267 0.15076 0.14888 0.14702 0.14336 0 13980 013250 013051 0.12854 012661 0.12470 0.12282 011914 011555 0.11447 0.11246 0.11048 0.10853 0.10661 0.10472 0.10104 0 09747 0.10046 0.09843 0.09645 0.09449 0.09257 0.09069 0.08702 0 08349

10 009133 0.08926 0.08723 0.08524 008329 0.08138 0.07950 0.07587 007238 11 0.08218 0.08011 0.07808 0.07609 0.07415 0.07225 0.07039 0.06679 0.06336 12 0 07456 0.07249 0.07046 0.06848 0.06655 0.06467 0.06283 0.05928 0.05590 13 006812 0.06605 0.06403 0.06206 0.06014 0.05828 0.05646 0.05296 004965 14

15 16 17 18 19

0.06260

0.05783 0.05365 0 04997 0.04670 0.04378

20 0.04116 25 003122 30 0.02465

40 0.01656 50 0.01182 60 0.00877

0.06054

0.05577 0.05160 0.04793 0.04467 0.04176

0.03915 0.02928 0.02278

0.01484 0.01026 0.00735

0.05853

0.05377 0.04961 0.04595 0.04271 0.03981

0.03722 0.02743 0.02102

0.01326 0.00887 0.00613

0.05657 005467 0.05282

0.05183 0.04994 0.04811 0.04768 0.04582 0.04402 0.04404 0.04220 0.04042 0.04082 0.03899 0.03724 0.03794 003614 0.03441

0.03536 003358 0.03188 0.02567 002401 0.02244 0.01937 001783 0.01639

0.01183 0.01052 0.00934 0.00763 000655 0.00560 0.00509 0.00420 0.00345

0.05102

0.04634 0.04227 0.03870 0.03555 0.03275

0.03024 0.02095 0.01505

0.00828 0.00478 0.00283

0.04758

0.04296 0.03895 0.03544 0.03236 0.02962

0.02718 0.01823 0.01265

0.00646 0.00344 0.00188

0.04434

0.03979 0.03586 003243 0.02941 0.02675

0.02439 0.01581 0.01059

0.00467 000238 0.00121

'The fctmula used for this table IS v'=(1/100111~[1 +(r/IOOJ] - 1 I = II"


TABLE 1.38-PRESENT WORTHOFAN ANNUITY

Thecaprtal C, which. tf placed al interesttoday,wrllprovide for a given annual payment Yfor a term of "years before it IS exhausted is C= YX w,wherethefactor w IS given below (Interestatipercent per annum, compounded annually.)

Values of w*

Years /=2 2% 3 3% 4 4% 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 3 2.8839 4 3.8077

5 4.7135 6 5.6014

i 6.4720 7.3255

9 8.1622

10 8.9826 11 9.7868

1.9274 2.8560 3.7620

4.6458 5.5081 6.3494 7.1701 7.9709

1.9135 1.8997 1.8861 1.8727 1.8594 1.8334 2.8286 2.8016 2.7751 2.7490 2.7232 2.6730 3.7171 3.6731 3.6299 3.5875 3.5460 3.4651

4.5797 4.5151 4.4518 4.3900 4.3295 4.2124 5.4172 5.3286 5.2421 5.1579 5.0757 4.9173 6.2303 6.1145 6.0021 5.8927 5.7864 5.5824 7.0197 6.8740 6.7327 6.5959 6.4632 6.2098 7.7861 7.6077 7.4353 7.2688 7.1078 6.8017

1.8080 2.6243 3.3872

4.1002 4.7665 5.3893 5.9713 6.5152

8.7521 8.5302 8.3166 8.1109 7.9127 7.7217 7.3601 7.0236 9.5142 9.2526 9.0016 8.7605 8.5289 8.3064 7.8869 7.4987

12 10.575 10.258 9.9540 9 6633 9 3851 9.1186 8.8633 8.383% 7.9427 13 11.348 10983 10.635 10303 9.9856 9.6829 9.3936 8.8527 8.3577 14 12.106 11.691 11.296 10921 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 il.274 10.477 9.7632 18 14992 14.353 13.754 13190 12.659 12.160 11.690 10.828 10.059 19 15.678 14.979 14.324 13.710 13.134 12.593 12065 11.158 10.336

20 16.351 15.589 14.877 14.212 13.590 13.008 12.462 11.470 10.594 25 19.523 18.424 17.413 16.482 1562'2 14.828 14093 12.783 11.654 30 22.396 20.930 19.600 18.392 17.292 16.289 15.372 13.765 12.409

40 27.355 25.103 23.115 21 355 19.793 18.402 17.159 15.046 13.332 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

‘The formula used lor thls table IS w = : 1 -[1 + (l/l OO)] I /(r/loo, = "ix

TABLEl.39-ANNUITY PROVIDED FOR BYAGIVENCAPITAL

The annual payment Yprovided for for a term ofn years by agiven capitalcplaced atinteresttoday is Y=Cx w'. Interest at i percent per annum, compounded annually; the fund is supposed lo be exhausted at the end of the term.

Values of w"

Years i=2 2% 3 3% 4 4% 5 6 7

~ 2 0.51505 0.51883 0.52261 0.52640 0.53020 0.53400 0.53780 0.54544 0.55309 3 0.34675 0.35014 0.35353 035693 0.36035 0.36377 0.36721 0.37411 0.38105 4 0.26262 0.26582 0.26903 0.27225 0.27549 0.27874 0.28201 0.28859 0.29523

E 0.21216 0.17853 0.21525 0.18155 021835 018460 0.22148 0.18767 0.22463 0.19076 0.22779 0.19388 0.23097 019702 0.23740 0.20336 0.24389 0.20980 7 0.15451 0.15750 0 16051 0.16354 0.16661 0.16970 0.17282 0.17914 0.18555 8 0.13651 0.13947 0.14246 0.14548 0.14853 0.15161 0 15472 0.16104 0.16747 9 0.12252 0.12546 0.12843 0.13145 0.13449 0.13757 0.14069 0.14702 0.15349

10 0.11133 0.11426 0.11723 0.12024 0.12329 0.12638 0.12950 0.13587 0.14238 11 0.10218 0.10511 0.10808 0.11109 0.11415 0.11725 0.12039 0.12679 0.13336 12 0.09456 0.09749 0.10046 0.10348 0.10655 0.10967 0.11283 0.11928 0.12590 13 0.08812 0.09105 0.09403 0.09706 0.10014 0.10328 0.10646 0.11296 0.11965 14 0.08260 0.08554 0.08853 0.09157 0.09467 0.09782 0.10102 0.10758 0.11434

15 0.07783 16 0.07365 17 0.06997 18 0.06670 19 0.06378

20 0.06116 25 0.05122 30 0.04465

40 0.03656 50 0.03182 60 0.02877

0.08077 0.07660 0.07293 0.06967 0.06676

0.06415 0.05428 0.04778

0.03984 0.03526 0.03235

0.08377 0.08683 0.08994 0.09311 0.09634 0.10296 0.10979 0.07961 0.08268 0.08582 0.08902 0.09227 0.09895 0.10586 0.07595 0.07904 0.08220 0.08542 008870 0.09544 0.10243 0.07271 0.07582 0.07899 0.08224 0.08555 0.09236 0.09941 0.06981 0.07294 0.07614 0 07941 0.08275 0.08962 0.09675

0.06722 0.07036 0.07358 0.07688 0.05743 0.06067 0.06401 0.06744 0.05102 0.05437 0.05783 0.06139

0.04326 0.04683 0.05052 0.05434 0.03887 0.04263 0.04655 0.05060 0.03613 0.04009 0.04420 0.04845

0.08024 0.08718 0.07095 0.07823 0.06505 0.07265

0.05828 0.06646 0.05478 0.06344 0.05283 0.06188

0.09439 0.08581 0.08059

0.07467 0.07238 0.07121

MATHEMATlCALTABLES&UNlTS&SYSTEMSOFWElGHTS& MEASURES 1-67

0’

2' 3' 4' 5' 6' 7' 8' 9' IO' 11' 12' 13' 14' 15' 16' 17' 18' 19' 20' 21' 22' 23' 24' 25' 26' 27' 20' 29' 30' 31' 32' 33' 34' 35' 36' 37' 30' 39' 40' 41' 42' 43' 44' 45' 46' 47' 40' 49' 50' 51' 52' 53' 54' 55' 56' 57' 58' 59' 60'

From Mrnutes and Seconds into Decimal Parts of a Degree

0~.0000 0' 00 .oooo 0~0167 1 " 0?0003 00.0333 2" 0°.0006 00.05 3" 0" 0008 00.0667 4" 0~.0011 00.0833 5" 0~.0014 00.10 6" 0~.0017 O".1167 7" 0".0019 00.1333 8" 000022 0".15 9" 0".0025 00 1667 10" 0" 0028 OO.1833 11" 0~.0031 00 20 12" 000033 OO.2167 13" On.0036 OO.2333 14" 00.0039 OO.25 1 5" 0'.0042 OO.2667 16" 00.0044 O" 2833 17" 00 0047 00.30 1 8" 00.005 00.3167 19" 00.0053 00.3333 20" 0'.0056 00.35 21" O".0058 0".3667 22" 0".0061 OO.3833 23" 0?0064 00.40 24" 0".0067 0°.4167 25" 0~.0069 OO.4333 26" 0".0072 00.45 27" 00.0075 OO.4667 28" 0'.0078 OO.4833 29" 0°0081 00.50 30" O".0083 OO.5167 31" OO.0086 OO.5333 32" 0" 0089 00.55 33" 00 0092 OO.5667 34" 0" 0094 OO.5833 35" 0?0097 OO.60 36" O".Ol OO.6167 37" 0?0103 OO.6333 38" 0".0106 OO.65 39" O".0108 OO.6667 40" O?Olll OO.6833 41" 0~.0114 00.70 42 " 00.0117 OO.7167 43" 0?0119 00.7333 44" 00.0122 00.75 45 I' OO.0125 OO.7667 46" OO.0128 O" 7833 47" 0?0131 O".l30 48" 00.0133 0'=.8167 49 " 0?0136 OO.8333 50" 0".0139 OO.85 51 " 0°.0142 OO.8667 52" oa.0144 OO.8833 53" 000147 0".90 54 VI 0".015 OO.9167 55 'I 00.0153 00 9333 56" OO.0156 00 95 57" 0'.0158 OO.9667 58" O".0161 OO.9833 59" OYO164 10.00 60" 00.0167

TABLE 1.40-DECIMAL EQUIVALENTS

Common Fractrons

From Decrmal Parts of a Degree into Minutes

and Seconds (exact valuesi

30.00 IO.01 IO.02 IO.03 IO.04 IO.05 IO.06 30.07 IO.08 Y.09 30.10 IO.11 30.12 Y.13 IO.14 IO.15 IO.16 IO.17 IO.18 IO.19 IO.20 10 21 IO.22 IO.23 I" 24 IO.25 IO.26 IO.27 IO.28 10 29 IO.30 IO.31 I" 32 IO.83 IO.34 IO.35 IO.36 IO.37 IO.38 IO.39 30.40 IF.41 IO.42 25' 12" 00.92 55' 12" IO.43 25' 48" 00 93 55' 48" IO.44 IO.45 IO.46 IO.47 IO.48 30 49 I".50

0’ 0' 36" 1' 12" 1' 48" 2' 24" 3' 3' 36" 4' 12" 4' 48" 5' 24" 6' 6' 36" 7' 12" 7' 48" 8' 24" 9' 9' 36" IO' 12" 10' 48" 11' 24" 12' 12' 36" 13' 12" 13' 48" 14' 24" 15' 15' 36" 16' 12" 16' 48" 17' 24" 18' 18' 36" 19' 12" 19' 48" 20' 24" 21' 21' 36" 22' 12" 22' 48" 23' 24" 24' 24' 36"

26' 24" 27' 27' 36" 28' 12" 28' 48" 29' 24" 30'

00.50 00 51 OO.52 00.53 00.54 00.55 O" 56 00.57 OO.58 00.59 OO.60 OO.61 00 62 OO.63 OO.64 0".65 00 66 OO.67 00.68 O" 69 OO.70 00.71 O" 72 00 73 0" 74 00.75 0" 76 00 77 0".78 00.79 00.80 OO.81 0".82 OO.83 0".84 OO.85 00.86 0' 87 00 88 O" 89 00.9c 00 91

00.94 56' 24" 00 95 57' 0".96 57' 36" 00.97 58' 12" 0".98 58' 48" 00.99 59' 24" I".00 60'

30 30' 36" 31' 12" 31' 48" 32' 24" 33' 33' 36" 34' 12" 34' 48" 35' 24" 36' 36' 36" 37' 12" 37' 48" 38' 24" 39' 39' 36" 40' 12" 40' 48" 41' 24" 42' 42' 36" 43' 12" 43' 48" 44' 24" 45' 45' 36" 46' 12" 46' 48" 47' 24" 48' 48' 36" 49' 12" 49' 48" 50' 24" 51' 51' 36" 52' 12" 52' 48" 53' 24" 54' 54' 36"

0”.000 0” .o O%OOl 3" 6 0".002 7" 2 00.003 10" .8 00.004 14" .4 0?005 1 8" O".006 21" .6 00.007 25" .2 0“.008 28" 8 00.009 32" 4

16ths 32nds

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

Exact Decrmal

64ths Values

1 0.01 5625 2 0.03 125 3 0.04 6875 4 0.06 25 5 0.07 8125 6 009 375 7 0.10 9375 8 0.12 5 9 0.14 0625 10 0.15 625 11 017 1875 12 0.18 75 13 0.20 3125 14 0.21 875 15 0.23 4375 16 0.25 17 026 5625 18 0.28 125 19 0.29 6875 20 031 25 21 0.32 8125 22 0.34 375 23 035 9375 24 0.37 5 25 0.39 0625 26 0.40 625 27 0.42 1875 28 0.43 75 29 0.45 3125 30 0.46 875 31 0.48 4375 32 0.50 33 051 5625 34 0.53 125 35 0.54 6875 36 0.56 25 37 0.57 8125 38 059 375 39 060 9375 40 0.62 5 41 0.64 0625 42 0.65 625 43 0.67 1875 44 068 75 45 0.70 3125 46 0.71 a75 47 0.73 4375 48 0.75 49 0.76 5625 50 0.78 125 51 0.79 6875 52 0.81 25 53 0.82 8125 54 0.84 375 55 0.85 9375 56 0.87 5 57 0.89 0625 58 0.90 625 59 0.92 1875 60 0.93 75 61 0.95 3125 62 0.96 875

O".OIO 36" 63 0.98 4375

Units and Systems of Weights and Measures L.E. Barbrow. U.S. N,itl Bureau of Standards

The U.S. Natl. Bureau of Standards was established by

act of congress in 1901 to serve as a national scientific

laboratory in the physical sciences and to provide fun-

damental measurement standards for science and indus-

try. In carrying out these related functions the bureau

conducts research and development in many fields of

physics, mathemstics, chemistry, and engineering. At the

time of its founding, the bureau had custody of two

primary standards-the meter bar for length and the kilo-

gram cylinder for mass (or weight). With the phenome-

nal growth of science and technology over the past half

century, the bureau has become a major research institu-

tion concerned not only with everyday weights and measures but also with hundreds of other scientific and

engineering standards that have become necessary to the

industrial progress of the U.S. Nevertheless, the U.S.

still looks to the bureau for information on the units of

weights and measures. particularly their definitions and

equivalents.

The subject of weights and measures can be treated from

several different standpoints. Scientists and engineers are

interested in the methods by which precision meas-

urements are made. general knowledge of weights and

measures, the present status of units and standards, and

miscellaneous facts that will be useful in their everyday

professional life.

The expression “weights and measures” is used here

in its basic sense of referring to measurements of length,

mass, and capacity, thus excluding such topics as electri-

cal and time measurements and thermometry. This sec-

tion on units and systems of weights and measures presents

some fundamental information to clarify thinking on this

subject and to eliminate erroneous and misleading use of terms.

Unit and Standard Definitions It is essential that there be established and kept in mind

the distinction between the terms “units” and “standards”

of weights and measures.

A unit is a value, quantity, or magnitude in terms of

which other values, quantities, or magnitudes are ex-

pressed. In general, a unit is fixed by definition and is

independent of such physical conditions as temperature.

Examples include the yard. the pound. the gallon, the

meter, the liter, the gram. A .stumlurd is a physical embodiment of a unit. In gener-

al it is not independent of physical conditions, and it is

a true embodiment of the unit only under specified con-

ditions. For example, a yard standard has a length of one yard when at some definite temperature and supported in

a certain manner. If supported in a different manner, it

might have to be at a different temperature in order to

have a length of I yard.

The Metric System Definition, Origin, and Development

The metric system* is the international system of weights

and measures based on the meter and the kilogram. The

essential features of the system were embodied in a report

made to the French Natl. Assembly by the Paris Acade-

my of Sciences in 1791. The definitive action taken in 1791 was the outgrowth of recommendations along simi-

lar lines dating back to 1670. The adoption of the system

in France was slow, but its desirability as an internation-

al system was recognized by geodesists and others. On

May 20, 1875. an international treaty known as the Intl,

Metric Convention was signed providing for an Intl.

Bureau of Weights and Measures, thus ensuring “the in-

ternational unification and improvement of the metric sys-

tem.” The metric system is now either obligatory or

permissible throughout the world.

Although the metric system is a decimal system, the

words ‘*metric” and “decimal” are not synonymous, and

care should be taken not to confuse the two terms.

‘See also Chap 58


Units and Standards of the Metric System

In the metric system, the fundumcntai units of length and

mass arc the tneter and the kilogram. The other units of

length and mass. as well as all units ofarea. volume. and

compound units such as density. are derived from thcxc

two fundamental units.

From I960 to 1983. the meter was defined as the length

equal to I 650 763.73 vvavelengths in vacuum of the radi-

ation corresponding to the transition between the levels

21?tn and 5d5 of the krypton 86 atom. Since 1983. the

meter has been defined as the length of the path travelled

by light in a vacuum during a time interval of

I1299 792 458 of a second. The kilogram previously dc-

fined as the mass of I cubic decimeter of water at the tem- pcraturc of maximum density was known as the Kilogram

of the Archives. It was replaced after the Intl. Metric Con-

vention in 1875 by the Intl. Prototype Kilogram. which

became the unit of mass vvithout reference to the mass

of a cubic decimeter of water or to the Kilogram of the

Archives.

The liter is a unit of capacity. In 1964 the 12th General

Conference on Weights and Measures redefined the liter

as being I cubic decimeter. By its previous definition-

the volume occupied. under standard conditions. by a

quantity of pure water having a mass of I kilogram-the

liter was larger than the cubic decimeter by 28 parts in

I 000 000; except for determinations of high precision,

this difference is so small as to be of no consequence.

The tnodernized metric system includes “base” units

such as units of temperature and time. as well as many

“derived” units such as units of force and work. For de-

tails, see Chap. 58.

The Intl. Bureau of Weights and Measures

The lntl. Bureau of Weights and Measures was established

at Scvres, a suburb of Paris, in accordance with the Intl.

Metric Convention of May 20. 1875. At the bureau are

kept the Intl. Prototype kilogram, many secondary stan-

dards of all sorts. and equipment for comparing standards

and making precision measurements. The Bureau, main-

tained by assessed contributions of the signatory govcrn-

merits, is truly international.

In recent years the scope of the work at the Intl. Bureau

has been considerably broadened. It now carries on re-

starch in the fields of electricity and photometry in addi-

tion to its former work in weights and measures, which included such allied fields as thermometry and the meas-

urement of barometric pressures.

Present Status of the Metric System in the U.S.

The use of the metric system in this country was legal-

ized by act of congress in 1866. but was not made ob-

ligatory.

The speed of light in a vacuum and U.S. Prototype Kilo-

gram No. 20 are recognized as the primary standards of

length and mass for both the metric and the inch-pound

(customary) systems of measurement in this country be-

cause these standards are the most precise and reliable

standards available. Since I893 in the U.S. the yard has

been defined in terms of the meter and the pound in tertns

of the kilogram. There is in the U.S. no primary stan-

dard either of length or mass in the inch-pound system.

From 1893 until 1959. the yard was defined as being

equal exactly to 360013937 111. In 1959 a small change

was tnade in the definition of the yard to resolve discrepan-

cies both in the U.S. and abroad. Since 1959. the yard

is defined as being equal exactly to 0.9144 m; the new

yard is shorter than the old vard by exactly two parts in

a million. At the same time,*it was decided that anv data

expressed in feet derived from geodetic survey within

the U.S. would continue to bear the relationship as dc-

fined in 1893 (I ft equals 1200/3937 m). This foot is called

the U.S. survey foot, while the foot defined in 1959 is

called the international foot. Measurements expressed in

statute miles, survey feet, rods, chains, links. or the

squares thereof, and also acres should therefore be con-

vcrted to the corresponding metric values by using

pre-1959 conversion factors whet-c more than tivc-

significant-figure accuracy is involved.

In 1971 the Natl. Bureau of Standards completed a

3-year study of the impact of increasing worldwide met-

ric use on the U.S. The study ended with a report to the

Congress entitled A MmYc Auwrictr-A Dwi.sio~t W/MN Ti/irc Hus Cornc. In the last few years metric USC has been

increasing rapidly in the U.S., principally in the tnanu- facturing and educational sectors. Public Law 93-380.

enacted Aug. 2 I, 1974, states that It IS the policy of the

U.S. to encourage educational agencies and institutions

to prepare students to use the tnetric system of mcasure- tnent with ease and facility as a part ofthe regular educa-

tion program. On Dec. 23. 1975. President Gerald Ford

signed Public Law 94-168. the Metric Conversion Act

of 1975. This act declares a national policy of coordinat-

ing the increasing use of the metric system in the U.S.

It established a U.S. Metric Board. whose functions as

of Oct. I. 1982 were transferred to the Dept. of Con-

mercc, Office of Metric Progratns, to coordinate the

voluntary conversion to the tnetric system.

British and U.S. Systems The implication is sometimes made that the inch/pound

system of weights and measures in the British Common-

wealth countries and the U.S. system are identical. It is

true that the U.S. and the British inch are defined identi-

cally for scientific work. that they are practically identi-

cal in commercial usage, that a similar situation exists for

the U.S. and the British pound, and that many tables. such

as 12 in. = I ft, 3 ft= I yd, and 1,760 yd= I international

mile, arc the same in both countries; but there are some

very important differences.

First. the U.S. bushel. the U.S. gallon. and their sub-

divisions differ from the corresponding British units. Also.

the British ton is 2,240 Ibm. whereas the ton generally

used in the U.S. is the short ton of 2,000 Ibm. The Ameri-

can colonists adopted the English wine gallon of 23 I cu

in. The English of that period used this wine gallon. but

they also had another gallon. the ale gallon of 282 cu in.

In 1824 these two gallons were abandoned by the British

when they adopted the British imperial gallon. which is

defined as the volume of IO Ibtn of water. at a tempera-

ture of 62°F. which, by calculation, is equivalent to

277.42 cu in. At the same time, the bushel was rcdcfined

as 8 gal, In the British system. the units of dry mcasurc

are the sdtne as those of liquid muasurc. In the U.S. these

two are not the same. the gallon and its subdivisions being


used in the measurement of liquids, while the hushcl, with

its subdivisions. is used in the measurement of certain dry

commodities. The U.S. gallon is divided into 4 liquid

quarts and the U.S. bushel into 32 dry quarts. All the units

of capacity mentioned thus far are larger in the British

system than in the U.S. system. But the British fluid ounce

is smaller than the U.S. fluid ounce. because the British

quart is divided into 40 fluid ounces. whereas the the U.S.

quart is dlvlded into 32 fluid ounces.

From the foregoing it is seen that in the British system

an avoirdupois ounce of water at 62°F has a volume of

I fluid oz. because 10 Ibm is equivalent to I60 avoirdupois oz. and I gal is equivalent to 4 quarts, or 160 fluid oz.

This convenient relation does not exist in the U.S. sys-

tem because a U.S. gallon of water at 62°F weighs about

8% Ibm, or 133% avoirdupois oz, and the U.S. gallon

is cquivalcnt to 4 X 32, or 128 fluid oz. In comparison.

I U.S. fluid oz=1.041 British fluid oz; I British fluid

oz=O.961 U.S. fluid oz; 1 U.S. gal=0.833 British in-

pcrial gal: and I British imperial gal= I.201 U.S. gal.

Subdivision of Units In general, units are subdivided by one of three systems:

(I) decimal, i.e., into tenths; (2) duodecimal, into

twelfths; or (3) binary, into halves. Usually the subdivi-

sion is continued by use of the same system. Each method

has advantages for certain purposes, and it cannot prop-

erly be said that any one method is “best” unless the use

to which the unit and its subdivisions are to be put is

known.

For example, if we are concerned only with measure-

ments of length to moderate precision, it is convenient

to measure and to express these lengths in feet, inches,

and binary fractions of an inch, thus 9 ft 4% in. If, how-

ever. these measured lengths are to be used subsequently

in calculations of area or volume, that method of subdi-

vision at once becomes extremely inconvenient. For that

reason, engineers, instead of dividing the foot into inches

and binary subdivisions of the inch, divide it decimally:

that is. into tenths, hundredths, and thousandths of a foot. The method of subdivision of a unit is thus made largely

on the basis of convenience to the user. The fact that units

have commonly been subdivided into certain subunits for

centuries does not preclude their also having another mode

of subdivision in some frequently used cases where con-

venience indicates the value of such other method. Thus

the gallon is usually subdivided into quarts and pints, but

the majority of gasoline-measuring pumps of the price-

computing type are graduated to show tenths of a gallon.

Although the mile has for centuries been divided into rods,

yards, feet, and inches. the odometer part of an automo-

bile speedometer indicates tenths of a mile. Although our

dollar is divided into 100 parts, we habitually use and

speak of halves and quarters.

Standards Length

A specified spectral line emitted by krypton 86 is the

international standard on which all length measurements

are based. To obtain a constant and uniform wavelength,

krypton lamps are operated at the temperature of the triple

point of nitrogen.

One yard is defined as 0.9144 m. The inch is thcrcfore

exactly equal to 25.4 mm.

Mass

The primary standard of mass for this country is U.S.

Prototype Kilogram 20, which is a platinum-iridium cyl-

inder kept at the Natl. Bureau of Standards. The value

of this mass standard is known in terms of the Intl.

Prototype Kilogram, a platinum-iridium standard, which

is kept at the Intl. Bureau of Weights and Measures.

The avoirdupois pound is defined in terms of the kilo-

gram by the relation I avoirdupois lbm=0.453 592 37 kg.

Mass vs. Weight. The mass of a body is a measure of

its inertial property. The ,t~i#r of a body has in the past

been used at times to designate its mass and at other times

to designate a force that is related to gravitational attrac-

tion. Because these two concepts of weight are incorn-

patible, and have therefore resulted in confusion, the term

“weight” should be avoided in technical practice except

under circumstances in which its meaning is completely

clear. When the term is used, it is important to know

whether mx~ orfbrw is intended and to use SI units prop-

erly by using kilograms for mass and newtons for force.

See also Chapter 58.

As weighing and measuring are important factors in our

everyday lives, it is quite natural that questions arise about

the use of various units and terms and about the magni-

tude of quantities involved.

The ton is used as a unit of measure in two distinct

senses: (I) as a unit of weight. and (2) as a unit of capac-

ity or volume.

In the weight sense the term means: (I) the shorr. or

rzet ton of 2,000 Ibm; (2) the long, ~KLKY, or .shipprr’.r ton of 2240 Ibm; (3) the n?~vic ton of 1000 kg, or 2204.6

Ibm. In the capacity sense “ton ” is restricted to uses re-

lated to ships.

In the U.S. and Canada the ton (weight) most commonly

used is the short ton, in Great Britain it is the /orzg ton,

and in countries using the metric system it is the metric,

ton.

Effect of Air Buoyancy. Another point to consider in the

calibration and use of standards of mass is the buoyancy

or lifting effect of the air. A body immersed in any fluid

is buoyed up by a force equal to the force of gravity on

the displaced fluid. Two bodies of equal mass. if placed

one on each pan of an equal-arm balance. will balance

each other in a vacuum. A comparison in a vacuum against

a known mass standard gives “true mass.” If compared

in air, however, they will not balance each other unless

they are of equal volume. If of unequal volume, the larg-

er body will displace the greater volume of air and will

be buoyed up by a greater force than will than will the

smaller body, and the larger body will appear to be of

less mass than the smaller body. The greater the differ-

ence in volume, and the greater the density of the air in

which the comparison weighing is made, the greater will

be the apparent difference in mass. For that reason. in

assigning a precise numerical value of mass to a standard,

it is necessary to base this value on definite values for

the air density and the density of the mass standard of

reference.


TABLE 1.41-LENGTH EGUIVALENTS

Inches

39.37 i.59517

Feet

3.261 0.51598

Yards

1.0936 0.03886

Chains

0.04971 2.69644

Meters 1

Centimeters

100 2.00000

Kilometers Miles

0.001 3.00000

0.0,6214 6.79335

0.01 2.00000

1 0.3937 0.03281 0.01094 1.59517 2.51598 2.03886

0.0,4971 10-5 4.69644

0.0,6214 5.00000 4.79335

0.0254 2.540 2.40483 0.40483

1 0.08333 0.02770 0.001263 0.0,254 5.40483

0.0,1578 2.92082 2.44370 3.10127 5.19818

0.3048 30.46 12 (1.3333 6.01515 1.48402 1.48402 1.07918 1.52288 2.18046

Q.Q,3048 4.48402

0.0,1894 4.27736

0.9144 91.44 36 3 1.96114 1.96114 1.55630 0.47712

1 0.04545 2.65758

0.0,9144 4.96114

0.0,5682 4.75449

20.12 2012 792 66 22 0.02012 0.0125 1.30356 3.30356 2.89873 1.81954 1.34242 2.30356 2.09691

1000 100 000 39,370 3,281 1,093.6 49.71 3.00000 5.00000 4.59517 3.51598 3.03886 1.69644

1

1609 160 935 63,360 5,280 1,760 80 1.609 3.20665 5.20665 4.80182 3.72263 3.24551 1.90309 0.20665

0.6214 i.79335

The equivalents are given in the bold-faced type. Logarithms of the equivalents are given immediately below with the characteristic of the logarithm (i.e., 1.59517 = 0.59517). In some cases, theequivalents have been rounded off, whtle the logarithm corresponds to the equivalent carried to a greater number of decimal places. Subscripts after any figure-O,, 9,-mean that that figure IS to be repeated the indicated number of times.

TABLE 1.42-CONVERSION OF LENGTHS’

Inches to

Millimeters

1 25.40 2 50.80 3 76.20 4 101.60

Millimeters to

Inches

0.03937 0.07874 0.1181 0.1575

Feet to

Meters

0.3048 0.6096 0.9144 1.219

Meters to

Feet

3.281 6.562 9.842 13.12

Yards to

Meters

0.9144 1.829 2.743 3.658

Meters to

Yards

1.094 2.187 3.281 4.374

Miles to

Kilometers

1.609 3.219 4.828 6.437

5 127.00 0.1968 1.524 16.40 4.572 5.468 8.047 6 152.40 0.2362 1.829 19.68 5.486 6.562 9.656 7 177.80 0.2756 2.134 22.97 6.401 7.655 11.27 8 203.20 0.3150 2.438 26.25 7.315 8.749 12.87

Kilometers to

Miles

0.6214 1.243 1.864 2.485

3.107 3.728 4.350 4.971

9 228.60 0.3543 2.743 29.53 8.230 9.842 14.48 5.592

‘For ewample, 1 I” = 25 40 mm

The corrections furnished by the U.S. Natl. Bureau of

Standards (NBS) for the more precise mass standards are

given on two bases: comparison in vacuum and compari-

son against normal brass standards in air under standard

conditions. with no correction applied for the buoyant ef-

fect of the air. By definition, brass standards have a

density of 8400 kg per m3 at 0°C and a coefficient of

cubical thermal expansion of 0.000 054 per “C. Standard conditions are defined as air of I .2 kg per m3 and tem-

perature of 20°C. The corrections to be used with pre-

cise analytical weights are ordinarily given only in terms

of apparent mass against normal brass standards.

A full discussion of this topic is given in NBS Mono-

graph 133, Mass and Mass Values, by Paul E. Pontius

(Nat]. Technical Information Service, 528.5 Port Royal

Road, Springfield, VA 22161; COM 7450309).

Capacity

Units of capacity. being derived units, are defined in the

U.S. in terms of linear units and are not represented by

fundamental standards. Laboratory standards have been

constructed and are maintained at NBS. These have va-

lidity only by calibration with reference either directly

or indirectly to the linear standards.


TABLE 1.43-COMMON FRACTIONS OF AN INCH TO MILLIMETERS (FROM ye4 to 1 in.)

64ths Millimeters 64ths Millimeters 64ths Millimeters 64ths Millimeters 64ths Millimeters 64ths

1 0.397 13 5.159 25 9.922 37 14.684 49 19.447 57 2 0.794 14 5.556 26 10.319 38 15.081 50 19.844 58 3 1.191 15 5.953 27 10.716 39 15.478 51 20.241 59 4 1.588 16 6.350 28 11.113 40 i 5.875 52 20.638 60

5 I ,984 6 2.381 7 2.770 a 3.175

9 3.572 10 3.969 11 4.366 12 4.763

17 6.747 29 11.509 41 18 7.144 30 11.906 42 19 7.541 31 12.303 43 20 7.938 32 12.700 44

21 8.334 33 13.097 45 22 8.731 34 13.494 46 23 9. I 28 35 13.891 47 24 9.525 36 14.288 48

16.272 16.669 17.066 17.463

17.859 18.256 18.653 19.050

53 21.034 61 54 21.431 62 55 21.828 63 56 22.225 64

Millimeters

22.622 23.019 23.416 23.813

24.209 24.606 25.003 25.400

0.0 0.1 0.2 0.3 0.4

0.5 0.6 0.7 0.8 0.9

TABLE 1.44-DECIMALS OF AN INCH TO MILLIMETERS (FROM 0.01 in. TO 0.99 in.)

0 1 2 3 4 5 6 7 8 9

0.254 0.508 0.762 1.016 1.270 1.524 i ,778 2.032 2.286 2.794 3.048 3.302 3.556 3.810 4.064 4.318 4.572 4.626 5.334 5.588 5.842 6.096 6.350 6.604 6.858 7.112 7.366 7.874 8.128 8.382 8.636 a.890 9.144 9.398 9.652 9.906

10.414 10.668 10.922 11.176 11.430 11.684 ii ,938 12.192 12.446

12.954 13.208 13.462 13.716 13.970 14.224 14.478 14.732 14.986

2.540 5.080 7.620 10.160

12.700 15.240 15.494 15.748 16.002 16.256 16.510 16.764 17.018 17.272 17.526 17.780 18.034 18.288 18.542 18.796 19.050 19.304 19.558 19.812 20.066 20.320 20.574 20.828 21.082 21.336 21.590 21 .a44 22.098 22.352 22.606 22.860 23.114 23.368 23.622 23.876 24.130 24.384 24.638 24.892 25.146

TABLE 1.4% -MILLIMETERS TO DECIMALS OF AN INCH (FROM 1 TO 99 mm)

0. 1.

0 0.0394 1 0.3937 0.4331 2 0.7874 0.8268 3 l.tatt 1.2205 4 1.5748 1.6142

5 1.9685 2.0079 6 2.3622 24016 7 2.7559 2.7953

2. 3. 4. 5. 6. 7. 8.

0.0787 0.1181 0.1575 0.1968 0.2362 0.2756 0.3150 0.4724 0.5118 0.5512 0.5906 0.6299 0.6693 0.7087 0.8661 0.9055 0.9449 0.9842 1.0236 1.0630 1.1024 1.2598 1.2992 1.3386 1.3780 1.4173 1.4567 1.4961 1.6535 1.6929 1.7323 1.7716 1.8110 1.8504 1.8898

2.0472 2.0866 2.1260 2.1654 2.2047 2.2441 2.2835 2.4409 2.4803 2.5197 2.5590 2.5984 2.6378 2.6772 2.8346 2.8740 2.9134 2.9528 2.9921 3.0315 3.0709

9.

0.3543 0.7480 1.1417 1.5354 1.9291

2.3228 2.7165 3.1102

a 3.1496 3 la90 3.2283 3.2677 3.3071 3.3464 3.3858 3.4252 3.4646 3.5039 9 3.5433 3 5827 3.6220 3.6614 3.7008 3.7402 3.7795 3.8189 3.8583 3.8976


TABLE1.46-AREAEQUIVALENTS'

Square Square Square Square Square Square Square Mrles or Meters Inches Feet Yards Rods Chains Roods Acres Sections

1 1,550 10.76 1.196 0.0395 0.002471 0.0,9884 0.0,2471 4.39288

0.0,3861 3.19033 1.03197 0.07773 2.59699 3.39288 4.99494 7.58670

0.0,6452 1 0.006944 0.0,7716 0.0,2551 0.0,1594 0.0,6377 0.0,1594 0.0,2491 4.80967 3.84164 4.88740 5.40667 6.20255 7.80461 7.20255 10.39637

g.09290 144 1 0.1111 0.003673 0.0,2296 0:0,9184 0.0,2296 0.0,3587 2.96803 2.15836 1.04576 3.56503 4.36091 5.96297 5.36091 8.55473

0.8361 1,296 9 1 0.03306 0.002066 0.0,8264 0.0002066 0.0,3228 1.92227 3.11260 0.95424 2.51927 3.31515 4.91721 4.31515 7.50898

25.29 39,204 272.25 30.25 1 0.0625 g.02500 0.00625 0.0,9766 1.40300 4.59333 2.43497 1.48072 2.79588 2.39794 3.79588 6.98970

404.7 627,264 4,356 484 16 1 0.4 0.1 0.0001562 2.60712 5.79745 3.63909 2.68484 1.20412 1.60206 1 .ooooo 4.19382

1012 1,568,160 10,890 1,210 40 2.5 1 0.25 0.0,3906 3.00506 6.19539 4.03703 3.08278 1.60206 0.39794 1.39794 4.59176

4047 6,272,640 43,560 4,840 160 IO 4 1 0.001562 3.60712 6.79745 4.63909 3.68484 2.20412 1 .ooooo 0.60206 3.19382

2 589 998 27,878,400 3,097,600 102,400 6,400 2,560 640 1 6.41330 7.44527 6.49102 5.01030 3.80618 3.40824 2.80618

The equivalents are given in the bold-faced type Logarithms of the equivalents are given immediately below with the characteristic of the logarithm (i.e., 2.59699 = 0.059699). In some cases, the equivalents have been rounded off, whrfe the logarithm corresponds to the equivalent carried to a greater number of decimal places. Subscripts after any figure-O,, 9,, etc.-mean that the figure is to be repeated the Indicated number of times.

'1 heclare= 100 acres=10 000 mp

Liters

1

Cubic Cubic Cubic Inches Feet Yards

61.03 0.03532 0.001308 1.78551 2.54796 3.11659

U.S. Apothecary

Fluid Ounces Drv

33.81 1.52909

Liquid

1.057 0.02394

0.9081 i .95812

Gallons Bushels

0.2642 0.02838 1.42188 2.45297

0.01639 2.21450

0.0,5787 0.0,2143 0.5541 0.01732 0.01488 4.76246

0.0 2 4329 0.0,4650 5.33109 1.74360 2.23845 2.17263 3.63639 4.66746

28.32 1728 1 0.03704 957.5 29.92 25.71 7.481 0.8036 1.45205 3.23754 2.56864 2.98114 1.47599 1.41017 0.87393 1.90502

764.5 46656 27 1 25853 807.9 694.3 202.2 21.70 2.88340 4.66891 1.43136 4.41251 2.90736 2.84153 2.30530 1.33638

0.02957 1.805 2.47091 0.25640

0.001044 0.0,3868 3.01886 5.58749

0.03342 0.001238 2.52401 3.09264

0.03889 0.001440 2.58983 3.15847

0.1337 0.004951 1.12607 3.69470

1.244 0.04609 0.09498 2.66362

1 0.03125 2.49485

6.02686 2.42903

0.007812 0.0,8392 3.89279 4.92388

0.9463 67.75 1.97604 1.76155

32 1.50515

1 0.8594 0.25 0.02686 i .9341a 1.39794 2.42903

1.101 67.20 0.04188 1.82737

37.24 1.164 0.2909 0.03125 1.57097 0.06582 1.46376 2.49485

3.785 231 0.57812 2.36361

128 4 3.437 0.1074 2.10721 0.60206 0.53624 1.03109

35.24 2150 1.54703 3.33252

1192 37.24 32 9.309 3.07612 1.57097 1.50515 0.96891

TABLEl.47-VOLUME AND CAPACITY EQUIVALENTS

U.S. Quarts U.S. u.s

The equivalents are given in the bold-faced type. Logarithms of the equivalents are given immediately below with the characteristic of the logarithm (i.e., 2.54796 =0.054796). In some cases, the equivalents have been rounded off, while the logarithm corresponds to the equivalent carried to a greater number of decimal places, Subscripts after any figure-O,, 9,, etc.-mean that the figure is to be repeated the indicated number of times.


‘F

TABLE 1.48-CONVERSION OF AREAS*

sq In. t0

cm2

cm’ t0

so tn.

6.452 0.1550 0.0929 10.76 0.8361 12.90 0.3100 0.1858 21.53 1.672 19.35 0.4650 0.2787 32.29 2.508 25.81 0.6200 0.3716 43.06 3.345 32.26 0.7750 0.4645 53.82 4 181 38.71 0.9300 0.5574 64.58 5.017 45.16 1.085 0.6503 75.35 5.853 51.61 1.240 0.7432 86.11 6 689 58.06 1.395 0.8361 96.87 7 525

sq ft

j.yz

mL to

sa ft

sq Yd

3

m’ to

sq yd

1.196 2.392 3.588 4.784 5.980 7.176 8.372 9.568 10.764

acres to

hectares

0.4047 0.8094 1.214 1.619 2.023 2.428 2.833 3.237 3.642

hectares to

acres

2.471 4.942 7.413 9.884

12.355 14.826 17.297 19.768 22.239

sq miles to

km’

2.590 5.180 7.770

10.360 12.950 15.540 18.130 20.720 23.310

or example. 1 sq in =6 452 cm2

km’ to

sq miles

0.3861 0.7722 1.158 1.544 1.931 2.317 2.703 3.089 3.475

TABLE 1.49-CONVERSION OF VOLUMES OR CUBIC MEASURES

C” I”. to

cm3

cm3 to

cu tn.

cu ft

;a

m3 to

cu ft

cu yd

;3

m3 gal to to

cu vd cu ft

1 16.39 0.06102 0.02832 35 31 0.7646 1.308 0.1337 2 32.77 0.1220 0.05663 70.63 1.529 2.616 0 2674 3 49.16 0.1831 0.08495 105.9 2.294 3.924 0.4010 4 65.55 0.2441 0.1133 141.3 3.058 5.232 0 5347 5 81.94 0.3051 0.1416 176.6 3.823 6.540 0.6684 6 98.32 0.3661 0 1699 211.9 4.587 7.848 0.8021 7 114.7 0.4272 0 1982 247.2 5.352 9.156 0.9358 8 131.1 0.4882 0 2265 282.5 6.116 10.46 1.069 9 147.5 0.5492 0.2549 317.8 6.881 11.77 1.203

cu ft to

gal

7.481 14.96 22.44 29.92 37.40 44 88 52.36 59.84 67.32
Fluid oz to
cm3

29.57 59.15 88.72 118.3 147.9 177.4 207.0 236.6 266.2

cm3 to

fluid oz

0.03381 0.06763 0.1014 0.1353 0.1691 0.2092 0.2367 0.2705 0.3043

TABLE 1.50-CONVERSION OF VOLUMES OR CAPACITIES

liquid pints to

liters

0.4732 0.9463 1.420 1.893 2.366 2.839 3.312 3.765 4.259

liters lrqurd to quarts

liqurd to pints liters

2.113 0 9463 4.227 1.893 6.340 2.839 8.454 3.785 10.57 4.732 12.68 5.678 14.79 6.624 16.91 7.571 19.02 8.517

liters to

liquid gal to

quarts liters

1.057- 3.785 2.113 7.571 3.170 11.36 4.227 15.14 5.284 la93 6.340 22.71 7.397 26.50 8.454 30.28 9.510 34.07

liters bushels hectolitersto to to

gal hectoliters bushels

0.2642 0.3524 2.838 0.5284 0.7048 5.676 0.7925 1.057 8.513 1.057 1.410 11.35 1.321 1.762 14.19 1.585 2.114 17.03 i .a49 2.467 19.86 2.113 2.819 22.70 2.378 3.171 25.54

MATHEMATICAL TABLES 8 UNITS 8 SYSTEMS OF WEIGHTS & MEASURES I-75

TABLE 1.51- -MASS EQUIVALENTS

Ibm tons

0.0,1102 3.04230

long

0.0,9842 4.99309

metric

0.001 3.00000

0.0,7143 0.0,6378 0.0,6480 a.85387 8.80465 8.81157

0.0,3429 0.0,3061 0.0,3110 5.53511 5.48590 5.49281

OZ

troy and apothecary avoirdupois

32.15 35.27 1.50719 1.54745

0.0,2083 0.0,2286 3.31876 3.35902

1 1.09714 0.04026

troy and apothecary avoirdupots

2.6792 2.205 0.42801 0.34333

0.0,1736 0.0,1429 4.23958 4.15490

0.08333 0.06857 2.92082 2.83614

0.07595 0.0625 2.88056 2.79588

ka grains

15,432 4.18843

1

0.0,6480 581157

0.03110 480 2.49281 2.68124

0.02835 437.5 0.9115 1 2.45255 2.64098 1.95974

0.0,3125 0.0,2790 0.0,2835 5.49485 5 44563 5.45255

0.3732 5,760 12 13.17 1 0.8229 0.0,4114 0.0,3673 0.0,3732 1.57199 3.76042 1 07918 1.11944 1.91532 4.61429 4.56508 4.57199

0.4536 7,000 14.58 16 1.215 1 0.0005 0.0,4464 0.0,4536 1.65667 3.84510 1 .I6386 1.20412 0.08468 4.69897 4.64975 4.65667

907.2 140, 2.95770 7.14613

1016 15680, 3.00691 7.19535

29,167 320, 2431 2,000 4.46489 4.50515 3.38571 3.30103

32,667 35,840 2722 2,240 4.51411 4.55437 3.43492 3.35025

32,151 35,274 2679 2,205 4.50719 4.54745 3.42801 3.34333

1 0.8929 0.9072 1.95078 1.95770

1.12 1.016 0.04922 0.00691

1000 15,432,356 3.00000 7.18843

1.102 0.9842 0.04230 1.99309

1

The equivalents are given in the bold-faced type. Logarithms of the equivalents are given immediately below with the characteristic of the logarithm (i.e., 3.04230 =0.0004230). In some cases, the equivalents have been rounded off, while the logarithm corresponds to the equivalent carried to a greater number of decimal places. Subscripts after any figure-O,, 9,, etc.-mean that the figure is to be repeated the indicated number of times.

TABLE 1.52-CONVERSION OF MASSES

short metric ionq tons tons tons

(2.000 lbn 1) (1000 kg) (2 240lbm) to t0 to

metric short metric

tons tons tons long

tons

0 907 1 102 1 016 0 984 1 a14 2 205 2032 1968 2 722 3 307 3 048 2 953 3629 4409 4 064 3937 4536 5 512 5 080 4921 5443 6614 6 096 5 905 6350 7 716 7 112 6889 7257 8 818 8 128 7874

8 165 9921 9 144 8858

Ibm kg (avordupots) 10

to lbm

kg (avo~rdupo!sj

04536 2 205

0 9072 4409

1361 6614 1814 8818

2 268 11 02 2 722 1323 3 175 1533

3 629 1764

a 082 19 84

oz grams

lavorrdupois) IO

to oz

grams (avoirdupois)

28 35 003527 5670 007055

a5 05 0 1058 11340 01411 141 75 0 1764 17010 02116 19845 0 2469 22680 0 2822 255 15 0 3175

gmns grams

10 to

grams gram

0 06480 1543 0 1296 3086

0 1944 4630 0 2592 61 73 0 3240 77 16 0 3888 92 59 04536 10803

0 5184 12346

0 5832 13889


TABLEl.53-VELOCITY EQUIVALENTS

60 3.6 100 3.281 196.85 2.237 1.77815 0.55630 2.00000 0.51598 2.29414 0.34965

0.01667 2.22104

I 0.06 1.667 0.05468 3.281 0.03728 0.03238 2.77815 0.22184 2.73783 0.51598 2.57150 2.51022

0.2778 16.67 1.44370 1.22184

1 27.78 0.9113 54.68 0.6214 0.53959 1.44370 I .95908 1.73783 1.79335 1.73205

0.01 0.6 0.036 1.77815 2.55630

1 0.03281 1.9685 0.02237 0.01943 2.51598 0.29414 2.34965 2.28836

0.3048 18.29 1.097 30.48 1.46402 1.26217 0.04032 1.48402

I 60 1.77815

0.005080 0.3048 0.01629 0.5080 0.01887 3.70586 1.48402 2.26217 1.70586 2.22185

1

0.8818 1.63367

0.01136 2.05553

0.4470 26.82 1.609 44.70 1.467 88 1.65035 1.42850 0.20670 1.65035 0.16633 1.94446

1

0.51479 30.887 1.8532 51.479 1.68894 101.337 1.15155 1.71163 1.48978 0.26793 1.71163 0.22761 2.00577 0.06128

mimin kmlhr cm/set ft/sec ftlmin mph knots

1.943 0.28836

0.59209 1.77238

0.00987 3.99423

0.86839 1.93871

1

TABLE1.54-CONVERSIONOFLlNEARANDANGULARVELOClTlES

revimin to

radslsec

0.1047 0.2094 0.3142 0.4189 0.5236 0.6283 0.7330 0.8378 0.9425

rads/sec t0

revlmin

9.55 19.10 28.65 38.20 47.75 57.30 66.84 76.39 85.94

cm/s 10

ftlmin

1.97 3.94 5.91 7.87 9.84 II.81 13.78 15.75 17.72

ftlmin to

cm/s

0.508 1.016 1.524 2.032 2.540 3.048 3.556 4.064 4.572

cmls mph to t0

mph cm/s

0.0224 44.70 0.0447 89.41 0.0671 134.1 0.0895 178.8 0.1118 223.5 0.1342 268.2 0.1566 312.9 0.1790 357.6 0.2013 402.3

ftls to

mph

0.682 1.364 2.045 2.727 3.409 4.091 4.773 5.455 6.136

mph to ft/s

1.47 2.93 4.40 5.87 7.33 8.80

10.27 11.73 13.20

Ibm per sq in to

kilopascals

1 6.8948 2 13789 3 20684 4 27.579 5 34.474 6 41.369 7 48.263 8 55.158 9 62.053

TABLEl.55-CONVERSION OF PRESSURES'

kilopascals atmospheres to Ibm to Ibm

oer sa in. oer sa in.

0.1450 14.70 0.2901 29.39 0.4351 44.09 0.5802 58.78 0.7252 73.48 0.8702 88.18 1.0153 102.9 1.1603 117.6 1.3053 132.3

Ibm per atmospheres kilopascals sq in. to to IO

atmospheres kilopascals atmospheres

0.0680 101.3 0.00987 0.1361 202.6 0.01974 0.2041 304.0 0.02961 0.2722 405.3 0.03947 0.3402 506.6 0.04935 0.4083 608.0 0.05922 0.4763 709.3 0.06908 0.5444 810.6 0.07895 0.6124 911.9 0.08882


TABLE1.56-PRESSURE EQUIVALENTS

columns of mercury at

temperature 0% columns of water at temperature 15%

and and g=9.80665 m/s2 g = 9.80665 m/s*

kilopascals

1

kg/cm’ (metric

atmospheres)

0.010197 2.00848

1

short tons per

lbmlsq in. sq ft

0.14504 0.010443 1.16148 2.01882

14.22 1.024 1.15300 0.01034

1 0.072 2.85733

atmospheres

0.009869 3.87510

98.066 1.99152

0.9678 1.98579

6.8948 0.07031 0.83852 2.84700

95.760 0.9765 1.98119 1.98966

0.06805 2.83280

OOb:'sOl o(lndl3 (m)

(in.)

i.99427 1.47025 . 1.00886 0.1021 0.60402 4.018

0.7356 28.96 10.01 394.1 1.86662 1.46177 1.00038 2.59556

0.05171 2.036 0.7037 27.70 2.71360 0.30876 1.84738 1.44254

13.89 1 0.9451 0.7183 28.28 9.774 384.8 1.14267 1.97547 1.85628 1.45143 0.99006 2.58521

101.325 1.0332 14.70 1.058 2.00573 0.01420 1.16722 0.02453

133.322 1.3595 19.34 1.392 2.12490 0.13338 1.28640 0.14373

1 0.76 29.92 10.34 407.1 1.8808 1 1.47598 1.01459 2.60975

1.316 39.37 13.61 535.7 0.11919 1.59517 113378 2 72894

(fU 0.3349 1.52485

32.84 1.51636

2.309 0.36336

32.07 1.50604

3.386 0.03453 0.52975 2.53823

9.798 0.09991 0.99114 2.99962

0.2489 0.002538 1.39598 3.40446

2.926 0.03045 0.47516 2.48364

0.4912 0.03536 0.03342 0.02540 0.3456 13.61 1.69124 2.51857 2.52402 2.40484 1.53861 1.13378

1.421 0.1023 0.15262 1.00996

0.03609 0.002599 2.55745 3.41479

0.4331 0.03119 1.63663 2.49397

0.09670 0.07349 2.893 39.37 2.98541 2.86622 0.46139 1.59517

0.002456 0.001867 0.07349 0.02540 3.39024 3.27106 2.86622 2.40484

0.02947 0.02240 0.8819 0.3048 12 2.46942 2.35024 1.94540 1.48402 1.07918

33.93 1.53058

44.64 1.64976

1.134 0.05460

3.281 0.51598

0.08333 2.92082

1

TABLE 1.57-ENERGY OR WORK EQUIVALENTS

J

1

kg-m ft-lbf

0.10197 0.7376 1.00848 1.86760

kW hours

0.0,2778 7.44370

metric hp

hours

0.0,3777 7.57711

hp hours

0.0,3725 7.57113

L-atm

0.009869 3.99427

kilocalories Btu's

0.0,2388 0.0,9478 4.37809 4.97670

9.80665 0.9915207

1 7.233 0.0,2724 0.0,37037 0.0,3653 0.09678 0.002342 0.009295 0.85932 6.43521 6.56863 6.56265 2.98579 3.36961 3.96825

1.356 0.1383 0.13220 1.14068

1 0.0,3766 0.0,51208 0.0,50505 0.01338 0.0,3238 0.001285 7.57590 7.70932 7.70333 2.12647 4.51029 3.10890

3.600~10~ 3.671 x lo5 2.655~10' 1.3598 1.341 35 526 859.9 3,412 6.55630 5.56478 6.42410 0.13342 0.12743 4.55057 2.93443 3.53303

2.648x lo6 270 000 1.9529x10~ 0.7355 0.9863 26 131 632.4 2,510 6.42288 5.43136 6.29068 1.86658 1.99401 4.41715 2.80098 3.39961

2.8845~ lo6 2.7375~10~ 1.98x106 0.7457 1.0139 6.42887 5.43735 6.29667 1 87356 0.00598

1 26 493 641.2 2,544 4.42314 2.80699 3.40557

101.33 10.333 74.74 0.0,2815 0.0,3827 0.0,3775 2.00573 1 .01421 1.87353 5.44952 5.58284 5.57686

1

4187 426.9 3,088 0.001163 0.001581 0.001560 41.32 3 62191 2.63036 3.48971 3.06558 3.19902 3.19304 1.61618

0.02420 2.38382

1

0.09604 2.98246

3.988 0.59861

1055 107.6 778.2 0.0,2931 0.0,3985 0.0,3930 10.41 0.25200 3.02360 2.03178 2.89110 446697 4.60042 4.59444 1.01757 1.40139

1

The equivalents are given in bold-faced type. Logarrthms of the equivalents are given immediately below with the characteristrc of the logarithm (1.e , 1.00848 = 0.00848). In some cases, the equivalents have been rounded off, although the logarithm corresponds to the equivalent carried to a greater number of dectmal places. Subscripts after any figure-O IS to be repeated the Indicated number of times.

3, 9,, etc.-mean that the figure


TABLE 1.58-CONVERSION OF ENERGY, WORK, HEAT

calories to

ft-lbf kg-m to IO

kg-m ft-lbf - ~ 0.1383 7.233 0.2765 14.47 0.4148 21.70 0.5530 26.93 06913 36.16 0.8295 43.40 09678 50.63 1.106 57.86 1.244 65.10

ft-lbf Btu to to Btu ft-lbf

0.001285 778.2 0.002570 1,556. 0.003855 2,334. 0.005140 3,113. 0.006425 3,891. 0.007710 4,669. 0.008995 5,447. 0.01028 6,225. 0.01156 7,003.

kg-m to

kilocalories

0.002342 0.004685 0.007027 0.009369 0.01172 0.01405 0.01640 0.01874 0.02108

kilocalorles IO

kg-m

4269 8539

1,281. 1,708. 2,135. 2,562. 2,989. 3,415. 3,642.

to calories

0.2388 0.4777 0.7165 0.9554 1.194 1.433 1.672 1.911 2.150

: 4.187 6.374

3 12.56 4 16.75 5 20.93 6 25.12 7 29.31 8 33.49 9 37.68

TABLE 1.59-POWER EQUIVALENTS’

metric hp poncelets kg-m/s ft-lbflsec kilocalories/s Btulsec

1.360 1.020 102.0 737.6 0.2388 0.9478 0.13343 0.00848 2.00848 2.86780 1.37813 1.97673

kW hD

1 1.341 0.12743

0.7457 1.87256

1 1.014 0.7604 76.04 550 0.1781 0.7068 0.00599 1.88105 188105 2.74036 1.25066 1.84936

0.7355 0.9863 1.86658 1.99402

1 0.75 75 542.5 0.1757 0.6971 1.87506 1.87506 2.73438 1.24467 1.84328

0.9807 1.315 1.333 100 723.3 0.2342 0.9295 1.99152 0.11896 0.12493 2.00000 2.85932 1.36961 1.96825

0.009807 0.01315 0.01333 0.01 7.233 0.002342 0.009295 3.99152 2.11896 2.12493 2.00000 0.85932 3.36961 3.96825

0.001356 0.00182 0.00164 0.00136 0.1383 0.0,3238 0.001285 3.13220 3.25946 3.26562 3.14067 1.14067 4.51029 3.10890

4.187 5.615 5.692 4.269 426.9 3,088 3.968 0.62187 0.74934 0.75530 0.63036 2.63036 3.48971 0.59861

1.055 1.415 1.434 1.076 107.6 778.2 0.2520 0.02320 0.15074 0.15668 0.03178 2.03178 2.89110 1.40138

1

‘For example 1 hp=07457 k\ni

TABLE 1.60-CONVERSION OF POWER

kW hp kW metric hp to to to to metw metric hp kW b hp

1.341 0.7355 1360 1.014 2.682 1.471 2.719 2.028 4.023 2.206 4.079 3.042

hp to kW

metric hp to hp

0.9863 1.973 2.959 3.945 4.932 5.918 6.904 7.891 8.877

1.491 2.237

4 2.983 5 3.729 6 4.474 7 5.220 8 5.966 9 6711

5.384 2.942 5.438 4.055 6.705 3.677 6.798 5.069 8.046 4.412 8.158 6.083 9.387 5.147 9.520 7.097 10.73 5.883 10.88 8.111 12.07 6.618 12.24 9.125


TABLE 1.61-DENSITY EQUIVALENTS AND CONVERSION FACTORS’

Equivalents Conversion Factors

kg/m 3

1

g/cm3

0.001

lbmlcu in.

0.0,3613 5.55787

1 0.03613 2.55787

27.60 1 1.44217

1000

27680 4.44217

16.019 1.20466

119.83 2.07855

‘For example, 1 kg/m’ = 0 0624 Ibmlcu ft.

0.01602 0.0,5787 2.20466 4.76245

0.1198 0.004329 1.07055 3.63639

lbmlcu ft IbmilJS. gal

0.0624 0.00835 2.79539 3.92143

62.43 8.345 3 1.79539 0.92143 4

1,728 231 5 3.23754 2.36361 6

1

7.481 0.87396

0.1337 7 1.12613 8

1 9 10

2

Ibm/cu ft kg/m 3 to to

kg/m 3 Ibm/cu 11

16.019 0.0624 32.037 0.1249

48.055 0.1873 64.074 0.2497

80.092 0.3121 96.111 0.3746

112.129 0.4370 128.148 0.4994

144.166 0.5619 160.185 0.6243

g/cm 3 kg/m 3 to to

kg/m 3 g/cm 3

1000 0.001 2000 0.002

3000 4000

0.003 0.004

5000 0.005 6000 0.006

7000 0.007 8000 0.008

9000 10 000

0.009 0.010

TABLE 1.62-THERMAL CONDUCTIVITY

W/m.% Cal/[s(cm’/cm)°C] Wl[(cm’lcm)“C] Call[h(cm2/cm)aC] Btu/[hr(sq ft/ft)OF]

1 0.002388 0 0100 8.598 0.5778 418.7 1 4.187 3600. 241.9 100.0 0.2388 1 860 57.78

0 1163 0.0002778 0001163 1 0 0672 1.731 0.004134 0.01731 14.88 1

TABLE 1.63-THERMAL CONDUCTANCE

W/m’.% Cal/(s.cm*~” C) W/cm’.% Cal/(h.cm2.%) Btu(hr-sq ft-OF)

1 0.0,2388 0.0001 0.08598 0.1761 41868 1 4.187 3600 7373 10000 0.2388 1 860 1761

11.63 0.0,2778 0.001163 1 2.048 5.678 0.0,1355 0.0,5678 0.4882 1

TABLE 1.64-HEAT FLOW

Wlm’ Cal/s.cm* Wlcm 2 Cal/h cm ’ Btulhr-sq ft

1 0.0,2388 0.0001 0.08598 0.3170 41868 1 4 187 3600 13272 10000 0.2388 1 860 3170

11.63 0.0,2778 0 001163 1 3.687 3.155 0.0,7535 0.0,3154 0.2712 1


TABLE 1.65~-RELATIVE DENSITIES CORRESPONDING TO OAPI AND WEIGHTS PER U.S. GALLON

Relative Densitv

Relative Relative Density

1.0000 0.9930 0.9861 0.9792 0.9725 0.9659 0.9593 0.9529 0.9465 0.9402 0.9340 0.9279 0.9218 0.9159 0.9100 0.9042 0 8984 0.8927 0.8871 0.8816 0.8762 0.8708 0.8654

Relative Density

0.7547 0.7507 0.7467 0.7428 0.7389 0.7351 0.7313 0.7275 0.7238 0.7201 0.7165 0.7128 0.7093 0.7057 0.7022 0.6988 0.6953 0.6919 0.6886 0.6852 0.6819 0.6787 0.6754

OAPI

10 11 12 13 14 15 16 17 18 19 20

Ibm1U.S. gal OAPI

8.328 33 8.270 34 8.212 35 8.155 36 8.099 37 8.044 38 7.989 39

Ibm/U.S. gal OAPI

7.163 56 7119 57 7076 58 7.034 59 6.993 60 6.951 61 6.910 62

Ibm0J.S. gal

6.283 6.249 6.216 6.184 6.151 6.119 6.087 6.056 6.025 5.994 5.964

='APl Density

79 0.6722 80 0.6690

1bmlU.S. gal

5.595 5.568 5.542 5.516 5.491 5.465 5.440 5.415 5.390 5.365 5.341 5.316 5.293 5.269 5.246 5.222 5.199 5.176 5.154 5.131 5.109 5.086

0.8602 0.8550 0.8498 0.8448 0.8398 0.8348 0.8299 0.8251 0.8203 0.8155 0.8109 0.8063 0.8017 0.7972 0.7927 0.7883 0.7839 0.7796 0.7753 0.7711 0.7669 0.7628 0.7587

81 0.6659 82 0.6628 83 0.6597 84 0.6566 85 0.6536 86 0.6506 87 0.6476 88 0.6446 89 0.6417 90 0.6388 91 06360 92 0.6331 93 0.6303

7.935 40 7.882 41 7.830 42 7.778 43

6.870 63 6.830 64 6.790 65 6752 66

21 22

7.727 44 7.676 45 7.627 46 7.578 47 7.529 48 7.481 49 7.434 50

6.713 67 6.675 68 6.637 69 6 600 70 6.563 71 6526 72 6.490 73 6.455 74 6.420 75 6.385 76 6.350 77 6.316 78

5.934 5.904

23 24 25 26 27 28 29 30 31 32

5.874 5.845 5.817 5.788 5.759 5.731 5.703 5.676 5.649 5.622

94 0.6275 95 0.6247 96 0.6220 97 0.6193 98 0.6166 99 0.6139 100 0.6112

7.387 51 7.341 52 7.296 53 7.251 54 7.206 55

*Calculated from the formula relatw densely= (141 5)/(131 5+ -, hp, ) The wghls I” this table are wghts I” air at 60°F wth humldtty 50% and pressure 760 mm fig

Relative Density and Density The relative density 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 tem-

perature of4”C [39”F] is commonly used by physicists,

but the engineer uses 60°F. The relative density 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 millimeter

as the unit of volume, or if the kilogram is used as the

unit of mass and the liter as the unit of volume, the figures

representing the density are the same as the relative den-

sity of the body. The inch-pound unit is pounds per cubic

foot.

The relative density of liquids is usually measured by

a hydrometer, and different special arbitrary hydrometer

scales are used in various trades and industries. The most

common of these are the API and BaumC. The API scale

is approved by the American Petroleum Inst., the ASTM,

the U.S. Bureau of Mines, and the Natl. Bureau of Stan-

dards and is recommended for exclusive use in the U.S.

petroleum industry, superseding the Baume scale for liq-

uids lighter than water. The relation between API degrees

and relative density is expressed by the followmg

equation.

The relative densities corresponding to “API and

weights per U.S. gallon are given in Table 1.65.

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