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MATHEMATICAL TEXTS FOR ENGINEERING STUDENTS
Edited by MANSFIELD MERRIMAN
MATllIvMATlCAL TABLES
FOR
CLASS-EOOM USE
BY
MANSFIELD MEKRIMAN
Editok-in-Chief of American Civil Engineers' Pocket Book
Member of American Mathematical Society
FIRST EDITION
FIRST THOUSAND
'
!'? !• * '
' ,* :''io,,i i' ''"'
NEW YORK
JOHN WILEY &
SONS,
Inc.
ndon: chapman &
HALL,
Limited
1915
7
coptrioht, 191.5
Bt
MANSFIELD MERRIMAN
THE SCIENTIFIC PRESS
ROBERT DRUMMOND »ND COMPANY
BROOKLYN, N. Y.
PREFACE
Computations of squares, square roots, reciprocals, areas
of circles, and other functions of numbers occur in the daily work
of every student of technology. This book gives four-place
tables by which the time spent in such computations can be much
shortened. Tables of trigonometric functions, of logarithms,
and of weights and measures are also given. These will be found
useful in cases where a high degree of precision is not required,
namely, in the great majority of problems that arise in physics
and engineering.
These tables'have been taken from the American Civil Engin-
eers' Pocket Book, and hence have a form which is approved in
the work of practical computation. It is believed that they will
be of value to students in blackboard work, in the examination
room, in the field and laboratory, and in the drawing room, as well
as in the solution of text-book problems.
Explanations, prepared especially for this volume, will render
the use of the tables easily understood, and these are accompanied
by remarks regarding the degree of precision to be expected in
interpolated tabular values. Exercises for students are given at
the end of each chapter. It is hoped that this little book may
assist both teachers and students in saving time and labor, and
thus tend to promote economy and efficiency,
3
337855
CONTENTS
Chapter 1
GENERAL EXPLANATIONS
PAGE
Art. 1. Arguments and Functions S
2. Precision of Tabular Values 8
3. Interpolation 9
4. Precision of Computed Results 10
5. Exercises for Students 10
Chapter 2
ARITHMETICAL TABLES
Art. 6. Table of Reciprocals 12
7. Table of Squares 14
8. Table of Square Roots 16
9. Table of Cubes 20
10. Table of Cube Roots 22
11. Table of Three-Halves Powers 23
12. Table of Fifth Powers and Roots 24
13. Explanations of Tables 25
14. Exercises 26
Chapter 3
TABLES OF CIRCLES AND SPHERES
Art. 15. Table of Areas of Circles (tenths) . .' 28
16. Table of Areas of Circles (eighths) 30
17. Table of Circumferences of Circles (tenths) 31
18. Table of Circumferences of Circles (eighths) .... 31
19. Table of Circular Segments 32
20. Table of Volumes of Spheres (tenths) 34
21. Table of Volumes of Spheres (eighths) 34
22. Table of Multipliers for Circular Arcs 35
23. Explanations 35
24. Exercises 36
5
6 Contents
Chapter 4
NATURAL TRIGONOMETRIC FUNCTIONS
PAGE
Art. 25. Table of Sines and Cosines 38
26. Table of Tangents and Cotangents 40
27. Table of Four-Place Trigonometric Functions. ... 42
28. Explanations 43
29. Exercises 44
Chapter 5
LOGARITHMIC TABLES
Art. 30. Table of Common Logarithms of Numbers 46
31. Table of Logarithms of Sines and Cosines 48
32. Table of Logarithms of Tangents and Cotangents 50
33. Table of Four-Place Logarithms of Trigonometric
Functions 52
34. Explanations 53
35. Exercises 54
Chapter 6
WEIGHTS AND MEASURES
Art. 36. Table of Measures of Length 56
37. Table of Measures of Area 56
38. Table of Speed and Velocity 57
39. Table of Volume and Capacity 5"
40. Table of Weight or Mass 58
41 . Table of Energy 58
42. Table of Pressure 59
43. Table of Power 59
44. Explanations 60
45. Exercises 61
Chapter 7
MISCELLANEOUS TABLES
Art. 46. Table of M.^thematical Constants 64
47. Table of Decimal Eqttivalents of Common Fractions 64
48. Table of Hyperbolic Function's 65
49. Table of Napierian Logarithms 66
50. Table of Multipliers for Transferring Logarithms 66
51 . Explanations 66
52. Exercises 67
Chapter 1
GENERAL EXPLANATIONS
8 General Explanations
1. Arguments and Functions
Arguments and Functions are the two kinds of numbers that
appear in a table, the former being the numbers which are given
and the latter those which are sought. The arguments are the
numbers for which the values of the functions have been com-
puted; thus in v^, values of 7i are the arguments and those of
Vn are the functions. An argument is at the side of the table, or
sometimes part of it is at the side and part at the top or foot;
thus when the square of 6.48 is sought from Table 7, the num-
ber 6.4 is found at the left-hand side of the table and the S at the
top or foot, then at the intersection of the horizontal row and the
vertical column is found the function or number 41.99 which is
the square of 6.48 correct to four places. In the tables of this
book the arguments are generally in bold-face typo and the func-
tions in common type. V
2. Precision of Tabular Values
The values of the functions in many of the tables of this book
are generally given only to four significant figures; thus the square
of 7.54 is given as 56.85, although its exact value is 56.8516. The
greater part of the computations in physics and engineering
require only three or four significant figures to be determined
with precision, since the given data rarely extend with accuracy
to a greater number of figures.
, Significant figures in a number are those not preceded by
ciphers after a decimal point. For example, each of the numbers
4507, 4.507, 0.04507 and 0.0004507 has four significant figures;
the number 30.2734 has six significant figures, while 0.065 and 6.5
have only two.
In any table the last figure of the argument is exact, but the
last figure of the function is liable to an error. Thus, when the
computed value of a function is 42.7854, the value given in a four-
place table is 42.79, the last figure having here an error of nearly
one-half of a unit; when the computed value is 3.7851 the value
given in a four-place table is 3.785, the last figure having here an
General Explanations 9
error of one-tenth of a unit. The maximum error in a tabulated
function is hence one-half a unit in the last figure, and the prob-
able error is one-fourth of a unit.
3. Interpolation
Interpolation is the process of finding the value of a function
when the given argument lies between two tabular arguments.
This is generall}' done by regarding the function as varying uni-
formlj' between the two adjacent tabular values. For example, if
it be required to find the reciprocal of 0.26-15 from Table G, it is
seen that the reciprocals of 0.264 and 0.265 are 3.788 and 3.774;
hence the reciprocal of 0.2645 is half-way between these, or 3.781.
Again, let it be required to find the square root of 85.04 from
Table 8; the square roots of 85.0 and 85.1 are found to be 9.220
and 9.225; the difference of these is 0.005, and 0.4X0.005 is 0.002;
then the square root of 85.04 is 9.220+0.002 = 9.222. After a little
practise interpolation in a four-place table ican be made mentally.
As another example, let it be required to find the value of sin
43° 4' from Table 25; here the sines of 43° 0' and 43° 10' are
0.68200 and 0.68412, the difference of wliich is 0.00212, thus the
difference for 1' is 0.000212 and for 4' it is 0.00085; then sin 43° 4'
is 0.68200+0.00085=0.68285. When the function decreases
as the argmnent increases, the computed difference is to be sub-
tracted from the greater value of the function; thus to find the
reciprocal of 0.5427, the reciprocals of 0.542 and 0.513 are
1.845 and 1.842; the difference of these is 0.003, and 0.003X0.7
= 0.002; hence the reciprocal of 0.5427 is 1.845 - 0.002 = 1.843.
The precision of an interpolated value is less than that of
the tabular values, because the assumption that the function
varies uniformily between the two adjacent tabular values is not
strictly correct, and because it is obtained by taking the differ-
ence of two tabular values which are not exact in the last figure.
In general the probable error in an interpolated value is at least
one-haff a unit in the last figure.
Inverse interpolation is the process of obtaining an argument
10 General Explanations
from a given function when the latter lies between two adjacent
tabular values; lliis will be explained in Arts. 28 and 34,
4. Precision of Computed Results
It is important that a computer should use the tables so as to
obtain the most precise result possible and also that he should
not attribute to the result a precision which does not exist. In
general no more tlian four significant figures can ])C' obtained from
a four-place table, and in the case of extended computations the
last figure may be liable to an error of one unit. For example,
the value of VQ.S"-+8.r- is found by the help of Tables 7 and 8
to be 10.50, which is exact, but the value of (1.25-+ 1. 45-) ^ is
found to be 13.44, which is one unit in error in the last figure.
Inexperienced computers sometimes, in making interpola-
tions, use all the figures obtained in the multiplication of differ-
ences, and thus carry the work several places further than the
tabular values warrant. This procedure not only entails additional
work and gives extra figures which are wholly inaccurate, but it
leads the computer to suppose tliat his results have a far higher
degree of precision than is actually the case, hence vitiating his
judgment and perhaps leading to the deceit of others as well as
of himself. In no case should more significant figures appear in
the final results than are given in the tables which are used.
5. Exercises for Students
1. From Table G olituin the reciprocals of 0.7G, 0.7G5, 0.766,
0.7665, 0.7668, also the reciprocals of 5.5, 5.53, 5.534, 5.536.
2. From Table 7 ohtain the t'our-iilace squares of 1.85 and 1.86;
verify the results by actual iimltiplications.
3. From Table 7 obtain the squares of 8.5, 8.53, 8.534; also of
0.85, 0.853, 0.8534.
4. Find the sine of 18° 23' from Table 25.
5. Find the value of (3.27= + 4.]82)2 by the help of Table 7; com-
pare it with the resnlt obtained by actual multiplication.
0. Find the square roots of 6.35 and 63.5 from Table 8; also the
square roots of 6.352 and 65.32.
7. Find the values of 1.42^ and 1.524^ from Table 11.
Chapter 2
ARITHMETICAL TABLES
12
Explanation on p. 38
Arithmetical Tables
6. Reciprocals
n
012345678 9
O.IO
10.00
9.901
9.804
9-709
9-615
9.524
9-434
9-344, 9-259
9-174
O.II
9.091
9.009
8.929
8.850
8.772
8.696
8.621
8.547
8.475
8.403
0.12
8-333
8.264
8.197
8.130
8.065
8.000
7-937
7-874
7-813
7752
O.I3
7.692
7-634
7-576
7-519
7 - 463
7 407
7-353
7-299
7.246
7- 194
0.14 7-143
1
7.092
7.042
6-993
6.944
6.897
6.849
6.803
6.757
6. 711
o.is 6.667
6.623
6.579
6.536
6.494
6.452
6-410
6.369
6.329
6.289
0. 16
6.250
6. 211
6.173
6.135
6.098
6.061
6.024
5.988
S-952
5-917
0.17
5-882
S-848
5-814
5-780
5-747
5-714
5.682
5-650
5.618
5-587
0.18
5-556
5-525
5 -495
5-464
5-435
5-405
5-376
5-348
5-319
5-291
0.19
S-263
5-236
5.208
5-181
5-155
5.128
5- 102
5-076
5-051
5-025
0.20
5.000
4-975
4-950
4.926
4.902
4.878
4-854
4-831
4.808
4-785
0.21
4.762
4-739
4-717
4.695
4-673
4.651
4-630
4.608
4.587
4.566
0.22 ]4.545
4-525
4-505
4.484
4.464
4-444
4-425
4.405
4.386
4.367
0.23
4 348
4-329
4.310
4.292
4-274
4-255
4-237
4.219
4. 202
4.184
0.24
4.167
4.149
4-132
4-iiS
4.098
4.082
4.065
4.049
4032
4.016
0.25
4.000
3-984
3-968
3-953
3-937
3-922
3-906
3-891
3-876
3.861
0.26
3.846
3-831
3-817
3.802
3.788
3-774
3-759
3-745
3-731
3-717
0.27
3-704
3.690
3-676
3-663
3-650
3 636
3-623
3.610
3-597
3-584
0.28
3-571
3-559
3-546
3-534
3-521
3 ■ 509
3-497
3-484
3472
3.460
0.29
3-448
3-436
3-425
3-413
3-401
3-390
3-378
3-367
3 356
3-344
0.30
3-333
3-322
3-3^^
3-300
3.289
3-279
3.268
3-257
3-247
3-236
0.31
3-226
3-215
3-205
3-195
3-185
3-175
3- "65
3-155
3-145
3-135
0.32
3-125
3-II5
3.106
3.096
3.086
3-077
3-067
3-058
3-049
3.040
0.33
3-030
3.021
3.012
3-003
2.994
2-985
2.976
2.967
2-959
2-950
0.34
2.941
2.933
2.924
2.915
2.907
2.899
2.890
2.882
2.874
2.865
0.35
2.857
2.849
2.841
2.833
2.825
2.817
2.809
2.801
2.793
2.786
0.36
2.778
2.770
2.7.62
2.755
2.747
2.740
2.732
2.725
2.717
2.710
0.37
2.703
2.695
2:688
2 . 68-1
2.674
2.667
2.660
2.653
2.646
2.639
0.38
2.632
2.625
2.618
2. 611
2.604
2-597
2.591
2.584
2.577
2.571
0.39
2.564
2.558
2.551
2-545
2.538
2.532
2.525
2.519
2.513
2. 506
0.40
2. 500
2.494
2.488
2.481
2-475
2.460
2 . 463
2.457
2.45'
2.445
0.41
2.439
2.433
2.427
2.421
2.415
2.410
2.404
2-398
2.392
2.387
0.42
2.381
2-375
2.370
2 - 364
2.358
2.353
2.347
2.342
2 . 336
2.331
0.43
2.326
2.320
2.315
2.309
2.304
2. 299
2.294
2.288
2 . 2S3
2.278
0.44
2.273
2.268
2. 262
2-257
2.252
2.247
2.242
2.237
2.232
2.227
0.45
2. 222
2.217
2. 212
2. 208
2.203
2. 198
2.193
2.188
2.183
2.179
0.46
2. 174
2. 169
2. 165
2. 160
2.15s
2. 151
2. 146
2. 141
2.137
2.132
0.47
2.128
2.123
2. 119
2. 1 14
2. no
2. 105
2. lOI
2.096
2.092
2.p88
0.48
2.083
2.079
2.075
2.070
2.066
2.062
2.058
2.053
2 049
2.045
0.49
2.041
2-037
2-033
2.028
2.024
2.020
2.0l6
2.012
2.008
2.004
0.50
2.000
1.996
r.992
1.9S8
1.984
1.980
1.976
1.972
1.969
1.965
0.51
1. 96 1
1-957
1-953
1.949
1.946
1.942
I • 938
I • 934
1.93'
1.927
0.52
1-923
1. 919
1. 916
I. 912
1.908
1 . 905
1. 901
1 . 89S
1.894
1.890
0.53
1.887
1 . 883
1.880
1.876
1-873
1.869
1.866
1.862
1 . 859
1-855
0.54
1.852
1.848
1.84s
1.842
1.838
1-835
1.832
1.828
1.825
1. 821
n
0123456789
of Numbers
Arithmetical Tables
13
n
012345678 9
0.55
I. 818
I. 815
I. 812
1.808
1.805
1.802
1.799
1.795
1.792
1.789
0.56
1.786
1.783
1-779
1.776
1-773
1.770
1.767
1.764
1.761
I. 757
0.57
r-754
1-75-
1.748
1-745
1.742
1-739
1-736
1-733
1-730
1.727
0.58
1.724
I. 721
I. 718
I. 715
I. 712
1.709
1. 706
1.704
1 . 701
1.698
0.59
1-695
1.692
1.689
1.686
1.684
1. 681
1.678
1-675
1.672
1.669
0.60
r.667
1 .664
1. 661
1.658
1.656
1-653
1.650
1.647
I ■ 645
1.642
0.61
1-639
1.637
1-634
I. 631
1.629
1.626
1.623
1.621
1.618
1. 616
0.62
1.613
1. 610
1.608
1 . 605
1.603
1 . 600
1-597
1-595
1.592
1.590
0.63
1.587
1-585
1.582
1.580
1.577
I-57S
1-572
1-570
1.567
1.565
0.64
1.562
1.560
1-558
1. 555
1.553
r-550
1-548
1-546
1-543
1. 541
0.65
1.538
1.536
1-534
1. 531
1.529
1-527
1-524
1.522
1.520
1-517
0.66
1.515
1.513
I. 511
1.508
1.506
1.504
1.502
1.499
1-497
1.495
0.67
1-493
1.490
1.488
1.486
1.484
1. 481
1-479
1.477
1-475
1.473
0.68
I. 471
1.468
1 .466
1.464
1.462
1.460
1-458
1-456
1-453
1. 451
0.69
1.449
1-447
1.445
1.443
1-441
1-439
1-437
1-435
1.433
I-431
0.70
1.429
1.427
1.425
1.422
1.420
I. 418
1 .416
1.414
1.412
1.410
0.71
1.408
1.406
1.404
1.403
1. 401
1-399
1-397
1-395
1.393
1-391
0.72
1.389
1-387
1.385
1.383
1-381
1-379
1.377
1-376
1.374
1-372
0.73
1.370
1.368
1.366
1.364
1.362
1. 361
1-359
1-357
1-355
1-353
0.74
1-351
1.350
1.348
1.346
1.344
1-342
1-340
1-339
1-337
^■ZiZ
0.75
1-333
-i-iS^
1.330
1.32S
1.326
1-325
1-323
1.321
1-319
1-318
0.76
1.316
r.314
1. 312
1. 311
1.309
1-307
I - 305
1-304
1.302
1.300
0.77
1.299
1.297
r.295
1.294
1.292
1 . 290
1.289
1.287
1.285
1.284
0.78
1.282
1.280
1.279
1.277
1.276
1.274
I. 272
I. 271
1.269
1.267
0.79
1.266
1.264
1.263
I. 261
1.259
1.258
1.256
1-255
1-253
1 . 252
0.80
1.250
1.248
1.247
1.245
1.244
r. 242
1.241
1.239
1-238
1.236
0.81
1-235
1.233
1.232
1.230
1 . 229
1.227
1.225
1.224
1 . 222
1 . 221
0.82
1.220
I. 218
I. 217
1.215
I. 214
I. 212
1.211
1. 209
1.208
1 . 206
0.83
1.205
1.203
1 . 202
1 . 200
1. 199
I. 198
1. 196
I -195
1-193
1. 192
0.84
r. 190
1. 189
1. 188
1. 186
1. 185
I. 183
1. 182
1.181
1.179
1.178
0.85
1. 176
I.I75
1. 174
1 . 172
1. 171
1. 170
1.168
1. 167
1.166
1. 164
0.86
1. 163
1. 161
1 . 160
1. 159
1-157
1. 156
I-IS5
I -153
1.152
1.151
0.87
1. 149
1. 148
1. 147
1. 145
I -144
1. 143
1. 142
1. 140
1-139
1. 138
0.88
1. 136
I.I35
1. 134
1. 133
1. 131
I. 130
1.129
1. 127
1. 126
1.125
0.89
1. 124
1 . 122
1 . 121
1 . 120
r. 119
1. 117
1. 116
1.115
1. 114
1. 112
0.90
I. Ill
I . no
1. 109
1. 107
1 . 106
r. 105
1. 104
I. 103
I. lOI
1. 100
0.91
1.099
1.098
1 . 096
1.095
1.094
1.093
1.092
1. 091
1.089
1.088
0.92
1.087
1.086
1.085
1 . 083
1.082
1. 081
1.080
1.079
1.078
1.076
0-93
1-075
1.074
1.073
1 .072
r.071
1.070
1.068
1.067
1.066
1.065
0.94
1 .064
1.063
1 .062
1 .060
1-059
1.058
1.057
1.056
1-055
1.054
0.95
1.053
1.052
1.050
1.049
1 .048
1.047
1.046
1-045
1.044
1 .043
0.96
1.042
1 .041
1 .040
1.038
1.037
1.036
1.035
1.034
1.033
1.032
0.97
1. 031
1.030
1.029
1.028
1 .027
1.026
1.025
1.024
1 .022
1 . 021
0.98
1 .020
I .oig
1. 018
1. 017
1. 016
1. 015
1 .014
1. 013
1 .012
I . on
0.99
I. 010
1.009
1.008
1.007
1 .006
1 .005
1 .004
1.003
1.002
1. 001
»
012345678 9
14
Arithmetical Tables
7. Squares of Num-
n
I.O
012345678 9
1. 000
1 .020
1 .040
1. 061
1.082
1.103
1. 124
I -145
1.166
1.188
i.i
1 . 210
1.232
1.254
1.277
1.300
1-323
1.346
1-369
1-392
1. 416
1.2
1.440
1.464
1.488
1-513
1-538
1-563
1.588
1.613
1.638
1.664
1-3
1 . 690
I. 716
1.742
1.769
1.796
1.823
1.850
1.877
1.904
1-932
1.4
1.960
1.988
2.016
2.045
2.074
2.103
2.132
2.161
2. 190
2.220
1-5
2. 250
2.280
2.310
2.341
2.372
2.403
2.434
2.465
2.496
2.528
1.6
2. 560
2.592
2. 624
2.657
2.690
2.723
2.756
2.789
2.822
2.856
1.7
2.890
2.924
2-958
2.993
3.028
3-063
3.098
3--i3,3
3-168
3-204
1.8
3-240
3-276
3-312
3-349
3-386
3-423
3.460
3-497
3-534
3-572
1-9
3.610
3-648
3.686
3-725
3-764
3 803
3-842
3-881
3.920
3.960
2.0
4.000
4.040
4.080
4. 121
4. 162
4.203
4.244
4.285
4.326
4.368
2.1
4.410
4.452
4.494
4-537
4.580
4.623
4.666
4-709
4.752
4.796
2.2
4. 840
4.884
4.928
4.973
5-018
5-063
5.108
5- 153
'5-198
5-244
2.3
5.290
5-336
5-382
5-429
5-476
5-523
5-570
5-617
5-664
5-712
2.4
5-760
5.808
5 -856
5-905
5-954
6.003
6.052
6. loi
6.150
6.200
2.5
6.250
6.300
6-350
6.401
6.452
6.503
6-554
6.605
6.656
6.708
2.6
6. 760
6.812
6.864
6.917
6.970
7-023
7.076
7.129
7.182
7-236
2.7
7.290
7-344
7-398
7-453
7-508
7-563
7.618
7-673
7.728
7-784
2.8
7.840
7.896
7-952
8.009
8.066
8.123
8.180
8.237
8.294
8-352
2.9
8.410
8.468
8.526
8.585
8.644
8.703
8.762
8.821
8.880
8.940
3-0
9.000
9.060
9. 120
9.181
9.242
9-303
9-364
9-425
9.486
9-548
3.1
9.610
9-672
9-734
9-797
9.860
9-923
9.986
10.05
10. 11
10.18
3.2
10.24
10.30
10.37
10.43
10.50
10.56
10.63
10.69
10. 76
10.82
33
10.89
10.96
11.02
11.09
11.16
11.22
11.29
11.36
11.42
11.49
3-4
11.56
11.63
II. 70
11.76
11.83
11.90
11.97
12.04
12. 11
12.18
35
12.25
12.32
12.39
12.46
12.53
12.60
12.67
12.74
12.82
12.89
3-6
12.96
13-03
13.10
13-18
13-25
13-32
13-40
13-47
13-54
13.62
3-7
13-69
13-76
13-84
13-91
13-99
14.06
14-14
14. 21
14.29
14-36
3-8
14.44
14-52
14-59
14.67
14-75
14. 82
14.90
14.98
15-05
15-13
3-9
15.21
15-29
15-37
15-44
15-52
15.60
15.68
15-76
15-84
15-92
4.0
16.00
16. oS
16.16
16. 24
16.32
16.40
16.48
16.56
16.65
16.73
4.1
16.81
16.89
16.97
17.06
17.14
17. 22
17-31
17-39
17-47
17-56
4.2
17.64
17.72
17.81
17.89
17.98
18.06
18.15
18.23
18.32
18.40
4-3
18.49
18.58
18.66
18.75
18.84
18.92
19.01
19. 10
19. 18
19.27
4-4
19-36
19-45
19-54
19.62
19.71
19.80
19.89
19.98
20.07
20. 16
4-5
20. 25
20.34
20.43
20.52
«o.6i
20. 70
20.79
20.88
20.981
21.07
4.6
21. 16
21.25
21.34
21.44
21-53
21.62
21.72
21. Si
21.90
22.00
4-7
22.09
22.18
22.28
22.37
22.47
22.56
22.66
22.75
22.8_S
22.94
4.8
23-04
23-14
23 23
23-33
23-43
23-52
23. 62
23-72
23.81
23-91
4-9
24.01
24. II
24.21
24-30
24.40
24.50
24.60
24.70
24.80
24-90
50
25.00
25.10
25.20
25-30
25-40
25-50
25.60
25.70
25.81
25-91
5-1
26.01
26. II
26.21
26.32
26.42
26. 52
26.63
26.73
26.83
26.94
5-2
27.04
27.14
27-25
27-35
27.46
27-56
27.67
27-77
27.88
27.98
53
28.09
28.20
28.30
28.41
28.52
28.62 28.73
28.84
28. 94
29.05
5-4
.?9. 16
29-27
29 -38
29.48
29-59
29.70 29.81
29.92 I30.03
30. 14
n
12345678 9
Arithmetical Tables
bers from 1.00 to 9.99
15
n
c
> I
2
3 4
5'
5 7 8
9
S-S
30
25
30
36
30
47
30.58
30.69
30
80
30
91
31.02
31- 14
31
25
5.6
31
36
31
47
31
58
31.70
31.81
31
92
32
04
32.15
32.26
32
38
5-7
32
49
32
60
32
72
32.83
32-95
33
06
33
18
33-29
33.41
33
52
5.8
33
64
33
76
33
87
33-99
34-11
34
22
34
34
34.46
34.57
34
69
5.9
34
81
34
93
35
05
35.16
35.28
35
40
35
52
35 64
35.76
35
88
6.0
36
00
36
12
36
24
36.36
36.48
36
60
36
72
36.84
36-97
31
09
6.1
37
21
37
33
37
45
37.58
37.7°
37
82
37
95
38. 07
38.19
38
32
6.2
38
44
38.56
38.69
38.81
38-94
39
06
39
19
39-31
39-44
39
56
6.3
39
69
39
82
39
94
40.07
40. 20
40
32
40
45
40.58
40.70
40
83
6.4
40
96
41
09
41
22
41.34
41.47
41
60
41
73
41.86
41.99
42
12
6.5
42
25
42
38
42
51
42.64
42-77
42
90
43
03
43.16
43.30
43
43
6.6
43
56
43
69
43
82
43.96
44.09
44
22
44
36
44.49
44.62
44
76
6.7
44
89
45
02
45
16
45-29
45-43
45
56
45
70
45.83
45-97
46
10
6.8
46
24
46.38
46
51
46.65
46.79
46
92
47
06
47.20
47.33
47
47
6.9
47
61
47
75
47
89
48.02
48.16
48
30
48
44
48.58
48.72
48
86
7.0
49
00
49
14
49
28
4942
49 •?6
49
70
49
84
49.98
50.13
50
27
7.1
50
41
SO
55
50
69
50.84
50.98
SI
12
51
27
51.41
51.55
51
70
7.2
SI
84
51
98
52
13
52.27
52-42
52
56
52
71
52.85
53-00
53
14
7.3
53
29
53
44
53
58
53.73
.53-88
54
02
54
17
S4-r32
54-46
54
61
7.4
54
76
54
91
55
06
5S-20
55-35
55
50
55
65
55-80
55-95
56
10
7.5
56
25
56
40
56
55
56.70
56.85
57
00
57
15
57-30
57.46
57
61
7.6
57
76
57
91
58
06
58-22
58.37
58
52
58
68
58.83
58.98
59
14
7.7
59
29
59
44
59
60
59.75
59.91
60
06
60
22
60.37
60.53
60
68
7.8
60
84
61
00
61
15
61.31
61.47
61
62
61
78
61.94
62.09
62
25
7.9
62
41
62
57
62
73
62.88
63.04
63
20
63
36
63.52
63.68
63
84
8.0
64
00
64
16
64
32
64.48
64. 64
64
80
64. 96
65.12
65-29
65
45
8.1
65
61
65
77
65
93
66. 10
66.26
66
42
66
59
66.75
66.91
67
08
8.2
67
24
67
40
67
57
67.73
67.90
68
06
68
23
68.39
68.56
68
72
8.3
68
89
69
06
69
22
6939
69-56
69
72
69.89
70.06
70. 22
70
39
8.4
70
56
70
73
70
90
71.06
71.23
71
40
71
57
71.74
71.91
72
08
8.5
72
25
72
42
72
59
72.76
72.93
73
10
73
27
73.44
73.62
73
79
8.6
73
96
74
13
74
30
74.48
74.65
74
82
75
00
75.17
75. '34
75-
52
8.7
75
69
75
86
76
04
76.21
76.39
76
56
76
74
76-91
77.09
77
26
8.8
77
44
77
62
77
79
77.97
78.15
78
32
78
SO
78.68
78.85
79
03
8.9
79
21
79
39
79
57
79.74
79-92
80
10
80
28
80.46
80.64
80.
82
9.0
81
00
81
18
81
36
81.54
81.72
81
90
82
08
82.26
82.45
82.
63
9.1
82
81
82
99
83
1.7
83.36
83.54
83
72
83
91
84.09
84.27
84.46]
9-2
84.64
84
82
85
01
85.19
85.38
85
56
85
75
85.93
86.12
86.
3°
9.3
86
49
86
68
86
86
87.05
87.24
87
42
87
61
87.80
87.98
88.
17
9.4
88
36
88
55
88
74
88.92
89.11
89
30
89
49
89.68
89.87
90.
06
9.5
90
25
90
44
90
63
90.82
91.01
91
20
91
39
91.58
91.78
91.
97
9.6
92
16
92
35
92
S4
92.74
92.93
93
12
93
32
93.51
93.70
93.
90
9.7
94
09
94
28
94
48
94.67
94.87
95.
06
95.
26
95.45
95.65
95-
84
9-8
96
04
96
24
96
43
96.63
96.83
97
02
97
22
97.42
97.61
97-
81
9.9
98.
01
98
21
98.
41
98.60
98.80
99-
00
99
20
99.40
99.60
99.
80
n
I
2
345678 <
J
16
Arithmetical Tables
8. Square Roots of
rt
I.O
I
2
3
4
5
6
7
8
9
1. 000
1.005
1.010
1.015
1 .020
1.025
1.030
1-034
1-039
1.044
I.l
1.049
I -054'
1.05S
1.063
1.068
1.072
1.077
1.0S2
1.086
1. 091
1.2
I.095
1. 100
1. 105
I. 109
1 . 114
1.118
1. 122
1. 127
1.131
1.136
1-3
1. 140
I -145
1.149
I. 153
1.158
1. 162
1.166
1. 170
1.175
1.179
1.4
1. 183
1.187
1. 192
1.196
1 . 200
1 . 204
1.208
1.212
1.217
1.221
1.5
1.225
1.229
1.233
1.237
1.241
1.245
1.249
1.253
1.257
1. 261
1.6
1.265
1.269
1.273
1.277
1.281
1.285
1.288
1.292
1.296
1.300
1.7
1.304
1.308
1.3"
1.315
1.319
1.323
1.327
1.330
1-334
1.338
1.8
1.342
1.345
1.349
1-353
1.356
1.360
1.364
1.367
1-371
1.37s
1.9
1.378
1.382
1-386
1.^89
1.393
1-396
1.400
1.404
1.407
1.411
2.0
1.414
1. 418
1. 421
1-425
1.428
1-432
1.435
1.439
1.442
1.446
2.1
1.449
1-453
1.456
1-459
1.463
1.466
1.470
1.473
1.476
1.480
2.2
1.483
1-487
1.490
1-493
1-497
1.500
1.503
1.507
1 . 510
1.513
2.3
1-517
1.520
1.523
1.526
1-530
1-533
1.536
1.539
1-543
1-546
2.4
1-549
1-552
1-556
1-559
1.562
1.565
1.568
1.572
1-575
1-578
2.5
I. 581
1-584
1-587
I -591
1-594
1.597
1.600
1.603
1.606
1.609
2.6
I. 612
1. 616
1 .619
1.622
1.625
1.628
1. 631
1.634
1-637
1 .640
2.7
1.643
1.646
1.649
1.652
1-655
1.658
i.66i
1.664
1.667
1.670
2.8
1-673
1.676
1.679
1.682
1.685
1.688
1.691
1.694
1-697
1.700
2.9
i-7°3
1.706
1.709
1 . 712
1-715
1.71S
I. 720
1.723
1.726
1.729
3.0
1-732
1-735
1.738
1.741
1-744
1.746
1.749
1.752
I-75S
1-758
3.1
I. 761
1-764
1.766
1.769
1.772
1-775
1.778
1.780
1.783
1.786
3-2
1.789
1.792
1-794
1.797
1. 800
1.803
I. 806
1.808
1.811
1.814
3-3
I. 817
I. 819
1.822
1.825
1.828
1.830
1-833
1.836
1.838
1.841
3.4
1.844
1.847
1.849
1.852
1.855
1.857
1.860
1.863
1.865
1.868
3-5
1. 871
1.873
1.876
1.879
1.881
1.884
1.887
1.889
1.892
1.89s
3.6
1.897
1.900
1-903
1.905
1.90S
1. 910
1. 913
1. 916
1.918
1.921
3.7
1.924
1.926
1.929
1. 931
1-934
1.936
1.939
1.942
1.944
1.947
3.8
1.949
1-952
1-954
1-957
1.960
1.962
1.965
1.967
1.970
1.972
3.9
1-975
1-977
1.980
1.982
1-985
1.987
1.990
1.992
1.995
1.997
4.0
2.000
2.002
2.005
2.007
2.010
2.012
2.015
2.017
2.020
2.022
4.1
2.025
2.027
2.030
2.032
2-035
2.037
2.040
2.042
2.045
2.047
4.2
2.049
2.052
2.054
2.057
2.059
2.062
2.064
2.066
2.069
2.071
4-3
2.074
2.076
2.078
2.081
2.083
2.086
2.088
2.090
2.093
2.09s
4.4
2.098
2. 100
2. 102
2.105
2. 107
2. 110
2. 112
2. 114
2.117
2. 119
4.5
2. 121
2. 124
2. 126
2.128
2. 131
2.133
2-135
2.138
2. 140
2. 142
4.6
2. 145
2.147
2.149
2.152
2.154
2. 156
2.159
2. 161
2.163
2. 166
4.7
2.168
2. 170
2-173
2.175
2.177
2.179
2.182
2.184
2.186
2.189
4.8
2. 191
2-193
2.195
2.198
2. 200
2.202
2. 205
2.207
2. 209
-2.211
4.9
2.214
2.216
2.218
2.220
2.223
2.225
2. 227
2.229
2.232
2.234
5.0
2.236
2.238
2.241
2 . 243
2.245
2.247
2.249
2.252
2.254
2.256
5.1
2.258
2.261
2.263
2.265
2.267
2. 269
2.272
2.274
2.276
2.278
5-2
2.280
2.283
2.2S5
2.287
2.2S9
2. 291
2.293
2.296
2.298
2.300
53
2.302
2.3°4
2.307
2.309
2. 311
2.313
2.315
2.317
2.319
2.322
5-4
11
2.324
2.326
2.328
2.330
2.332
2.33s
2.337
2.339
2.341
2.343
I
2
3
4
5
6
7
8
9
Arithmetical Tables
Numbers from 1.00 to 99.9
17
Continued on p. 18
n
01 23456789
5.5
2.345
2-347
2-349
2-352
2-354
2-356
2.358
2.360
2.362
2-364
5.6
2.366
2.369
2.371
2-373
2-375
2-377
2.379
2.381
2-383
2-385
5.7
2.387
2.390
2.392
2-394
2.396
2.398
2. 400
2.402
2.404
2.406
5.8
2 . 408
2.410
2.412
2.415
2.417
2.419
2. 421
2.423
2.425
2.427
5-9
2.429
2.431
2.433
2.435
2-437
2.439
2.441
2.443
2.445
2.447
6.0
2.449
2.452
2-454
2.456
2.458
2. 460
2. 462
2.464
2.466
2.468
6.1
2.470
2.472
2.474
2.476
2.478
2 . 4S0
2.482
2.484
2.486
2.488
6.2
2.490
2.492
2.494
2.496
2.498
2. 500
2. 502
2.504
2. 506
2 . 508
6.3
2.510
2.512
2.514
2.516
2.518
2.520
2. 522
2.524
2.526
2.528
6.4
2-530
2.532
2.534
2.536
2.538
2.540
2.542
2.544
2.546
2.548
6.5
2.550
2.551
2.553
2.555
2.557
2.559
2.561
2-563
2.565
2.567
6.6
2.569
2.571
2.573
2-575
2.577
2-579
2.581
2.583
2.585
2-587
6.7
2.588
2.590
2.592
2.594
2.596
2-598
2.600
2.602
2.604
2.606
6.8
2.608
2.610
2.612
2.613
2.615
2.617
2.619
2.621
2.623
2.625
6.9
2.627
2.629
2.631
2.632
2.634
2.636
2.638
2. 640
2.642
2.644
7.0
2. 646
2.648
2.650
2.651
2.653
2-655
2.657
2.659
2.661
2.663
7-1
2. 665
2.666
2.668
2. 670
2.672
2.674
2.676
2.678
2.680
2.681
7.2
2.683
2.685
2.687
2.689
2. 691
2.693
2.694
2. 696
2.698
2. 700
7-3
2. 702
2.704
2. 706
2.707
2. 709
2.711
2.713
2-715
2.717
2.718
7.4
2. 720
2.722
2.724
2.726
2.728
2.729
2-731
2-733
2-735
2.737
7.5
2.739
2.740
2.742
2.744
2.746
2.748
2.750
2-751
2.753
2.755
7.6
2.757
2.759
2. 760
2.762
2.764
2.766
2.768
2. 769
2.771
2-773
7-7
2-775
2.777
2.77S
2.780
2.782
2.784
2.786
2.787
2.789
2.791
7.8
2-793
2-795
2.796
2.79S
2. 800
2.802
2.804
2.805
2.807
2.809
7.9
2. 811
2.S12
2.814
2.816
2.818
2.820
2.821
2.823
2.825
2.827
8.0
2.828
2.830
2.832
2.834
2.835
2.837
2.839
2.841
2.843
2.844
8.1
2.846
2.848
2.850
2.851
2.853
2.855
2-857
2.858
2.860
2.862
8.2
2.864
2.865
2.867
2,869
2.871
2.872
2.874
2.876
2.877
2.879
8.3
2.881
2.883
2.884
2.886
2.888
2.890
2.891
2.893
2.895
2.897
8.4
2.898
2. 900
2. 902
2.903
2.905
2.907
2.909
2.910
2 . 912
2.914
8.5
2.915
2.917
2.919
2.921
2. 922
2.924
2.926
2.927
2.929
2.931
8.6
2.933
2-934
2.936
2.938
2.939
2.941'
2.943
2.944
2.946
2.948
8.7
2.950
2.951
2.953
2.955
2.956
2.958
2. 960
2.961
2.963
2.965
8.8
2. 966
2.96S
2.970
2.972
2.973
2-975
2-977
2.978
2 . 9S0
2.982
8.9
2.983
2.985
2.987
2. 988
2. 990
2. 992
2-993
2-995
2.997
2.998
9.0
3.000
3.002
3.003
3 . 005
3-007
3-008
3.010
3.012
3.013
3-oiS
9.1
3-017
3.01S
3.020
3.022
3-023
3-025
3-027
3.028
3-030
3.032
9.2
3-033
3-035
3-03S
3.038
3.040
3-041
3 -043
3 - 045
3.046
3.048
9-3
3-050
3-051
3-053
3-055
3-056
3-058
3.059
3.061
3-063
3.064
9-4
3.066
3.068
3.069
3-071
3.072
3-074
3.076
3-077
3-079
3-081
9.5
3.082
3.084
3-085
3-087
3.089
3.090
3-092
3-094
3-095
3-097
9.6
3.098
3.100
3.102
3-103
3-105
3.106
3.108
3.110
3. Ill
3-"3
9-7
3-114
3. 116
3. 118
3-119
3. 121
3.122
3-124
3- 126
3-127
3.129
9-8
3- 130
3-132
3-134
3-135
3-137
3-138
3-140
3-142
3-143
3- 145
9-9
3.146:3.148
3-150
3-151
3-153
3-154
3-156
3-158
3-159
3. 161
n
012 345678 9
18
Continued from p. 17
Arithmetical Tables
Square Roots of
n
10
.0 .1 .2 .3 -4 -5 -6 .7 -8 .9
3.162
3-178
3-194
3.209
3-225
3.240
3.256
3.271
3.286
3.302
ZI
3-317
3-332
3-347
3.362
3-376
3.391
3.406
3.421
3-435
3.450
12
3-464
3-479
3-493
3.507
3-521
3 536
3.550
3.564
3578
3-592
13
3.606
3-619
3-633
3.647
3-661
3.674
3-688
3.701
3-715
3-728
14
3-742
3-755
3-768
3.782
3.795
3.808
3.821
3.834
3-847
3-860
IS
3-873
3-886
3-899
3.912
3.924
3.937
3-950
3.962
3-975
3-987
i6
4.000
4.012
4-025
4.037
4.050
4.062
4.074
4.087
4-099
4.1H
17
4-123
4.135
4-147
4.159
4. 171
4.183
4-I9S
4.207
4.219
4-231
i8
4-243
4.254
4.266
4.278
4.290
4.301
4.313
4.324
4-336
4-347
19
4-359
4.370
4.382
4.393
4.405
4.416
4.427
4.438
4-450
4.461
20
4-472
4.483
4-494
4.506
4.517
4.528
4.539
4.550
4-561
4.572
21
4-583
4.593
4.604
4-615
4.626
4.637
4.648
4.658
4.669
4.680
22
4.690
4.701
4.712
4.722
4.733
4-743
4.754
4.764
4.775
4.785
23
4.796
4.806
4.817
4.827
4-837
4.848
4.858
4.868
4.879
4.889
24
4.899
4.909
4.919
4-930
4.940
4-950
4.960
4-970
4.980
4.990
25
5 .000
5.010
5 ■ 020
5-030
5.040
5.050
5.060
5-070
5.079
5.089
26
5-°99
5.109
5-II9
5.128
5-138
5.148
5-158
5-167
5.177
5-187
27
5-196
5.206
5-215
S-225
5-235
5-244
5-254
5-263
5.273
5.282
28
5-292
5-301
5-310
5-320
5-329
5-339
5-348
5-357
5.367
5-376
29
5.385
5-394
5-404
5.413
5 -422
5.431
5 -441
5-450
5.459
5.468
30
5-477
5-486
5-495
5.505
5-514
5.523
5-532
5-541
5.550
5-559
31
5-568
5-577
5-586
5-595
5-604
5.612
5-621
5-630
5. 639
5.648
32
5-657
5.666
5-675
5.683
5.692
5.701
5-710
S-718
5.727
5-736
33
5-745
5-753
5.762
5.771
5-779
5.788
5-797
5-805
5.814
5.822
34
5-831
5.840
5-848
5-857
5-865
5.874
5.882
5. .89 1
5.899
5.908
35
5-916
5-925
5 - 933
5-941
5-950
5.958
5-967
5-975
5.983
5-992
36
6.000
6.008
6.017
6.025
6 . 033
6.042
6.050
6.058
6.066
6075
37
5.083
6.091
6.099
6. 107
6. 116
6. 124
6.132
6. 140
6.148
6.156
38
6.164
6-173
6. 181
6.189
6.197
6.20s
6.213
6.221
6.229
6-237
39
6.245
6-253
6.261
6.269
6.277
6.28s
6.293
6.301
6.309
6-317
40
6-325
6-332
6.340
6.348
6.356
6.364
6.372
6.380
6.387
6-395
41
6.403
6. 411
6.419
6.427
6.434
6.442
6.450
6.458
6.465
6.473
42
6.481
6.488
6.496
6.504
6.512
6.519
6.527
6-535
6.542
6.550,
43
6-557
6-565
6-573
6.580
6.588
6.595
6 . 603
6. 611
6.618
6.626"
44
6-633
6.641
6.648
6.656
6.663
6.671
6.67S
6.686
6.693
6.701
45
6.708
6.716
6.723
6.731
6.738
6.745
6.753
6. 760
6.768
6-775
46
6.782
6.790
6-797
6.804
6.812
6.819
6.826
6.834
6.841
6.848
47
6.856
6.863
6.870
6.877
6.885
6.892
6.899
6.907
6.914
6.921
48
6.928
6.935
6.943
6.950
6.957
6.964
6.971
6.979
6.986
6.993
49
7.000
7.007
7.014
7.021
7.029
7.036
7.043
7.050
7.057
7.064
50
7.071
7.078
7-085
7.092
7.099
7.106
7.113
7. 120
7.127
7-134
51
7. 141
7.148
7-155
7. 162
7.169
7.176
7.183
7.190
7.197
7.204
52
7. 211
7.218
7-225
7.232
7.239
7.246
7.253
7-259
7.266
7-273
53
7.280
7.287
7-294
7 -301
7.308
7.314
7.321
7-328
7.335
7-342
54
n
7-348
7-355
7.362
7-369
7.376
7.382
7.389
7-396
7 . 403
7.409
.0 .1 .2 .3 .4 -5 -6 .7 -8 .9
Arithmetical Tables
Numbers from 1.00 to 99.9
19
n
SS
.0 .1 .2 .3 -4 -5 -6 .7 -8 .9
7.416
7-423
7-430
7-436
7-443
7-450
7-457
7-463
7-470
7-477
56
7-483
7-490
7-497
7-503
7-510
7-517
7-523
7-530
7-537
7-543
57
7-550
7-556
7-563
7-570
7-576
7-583
7-589
7-596
7.603
7.609
58
7.616
7.622
7.629
7-635
7.642
7.649
7-655
7.662
7.668
7-675
59
7.681
7.688
7-694
7.701
7-707
7-714
7.720
7-727
7-733
7-740
6o
7-746
7-752
7-759
7-765
7-772
7-778
7.785
7.791
7-797
7.804
6i
7.810
7.817
7-823
7-829
7-836
7-842
7.849
7-855
7.861
7.868
62
7.874
7.880
7-887
7-893
7-899
7.906
7.912
7.918
7-925
7-931
63
7-937
7-944
7-95°
7-956
7.962
7.969
7-975
7-981
7.987
7-994
64
8.000
8.006
8.012
8.019
8.025
8.031
8-037
8.044
8.050
8.056
65
8.062
8.068
8.075
8.081
8.087
8.093
8.099
8.106
8. 112
8. 118
66
8.124
8.130
8.136
8.142
8.149
8.155
8.161
8.167
8-173
8.179
67
8.185
8. 191
8.198
8.204
8.210
8.216
8.222
8.228
8.234
8. 240
68
8.246
8.252
8.258
8.264
8.270
8.276
8.283
8.289
8.29s
8.301
69
8.307
8-313
8-319
8.325
8.331
8.337
8.343
8.349
8-355
8.361
70
8.367
8-373
8-379
8-385
8.390
8.396
8.402
8.408
8.414
8.420
71
8.426
8.432
8.438
8.444
8.450
8.456
8.462
8.468
8.473
8.479
72
8.485
8.491
8.497
8.503
8.509
8.515
8.521
8.526
8-532
8-538
73
8.544
8-550
8.556
8.562
8.567
8.573
8.579
8-585
8.591
8.597
74
8.602
8.608
8.614
8.620
8.626
8.631
8.637
8.643
8.649
8.654
75
8.660
8.666
8.672
8.678
8.683
8.689
8.695
8.701
8.706
8.712
76
8.718
8.724
8.729
8.735
8.741
8.746
8.752
8.758
8.764
8.769
77
8.775
8.781
8.786
8.792
8.798
8.803
8.809
S.815
8.820
8.826
78
8.832
8.837
8.843
8.849
8.854
8.860
8.866
8. 871
8.877
8.883
79
8.888
8.894
8.899
8.905
8.911
8.916
8.922
8.927
8.933
8-939
80
8.944
8.950
8.955
8.961
8.967
8.972
8.978
8.983
8.989
8.994
81
9.000
9.006
9.011
9.017
9.022
9.028
9-033
9-039
9.044
9.050
82
9-055
9.061
9.066
9.072
9-077
9.083
9.088
9-094
9.099
9-105
83
9. no
9. 116
9. 121
9.127
9.132
9.138
9-143
9.149
9-154
9. 160
84
9-165
9. 171
9.176
9.182
9.187
9.192
9.198
9.203
9. 209
9.214
85
9. 220
9.225
9.230
9.236
9.241
9-247
9.252
9-257
9-263
9.268
86
9.274
9.279
9-284
9. 290
9-295
9.301
9-306
9-3II
9-317
9.322
87
9-327
9-333
9-338
9-343
9-349
9-354
9-359
9-365
9-370
9-375
88
9-381
9.386
9-391
9-397
9.402
9.407
9-413
9.418
9-423
9.429
89
9-434
9-439
9-445
9-450
9-455
9.460
9.466
9-471
9-476
9.482
90
9-487
9-492
9-497
9-503
9-508
9-513
9-518
9-524
9-529
9-534
91
9-539
9-545
9-550
9-555
9.560
9.566
9-571
9-576
9-581
9-586
92
9-592
9-597
9.602
9.607
9.612
9.618
9-623
9.628
9-633
9-638
93
9-644
9-649
9654
9-659
9.664
9.670
9-675
9.680
9.685
9.690
94
9-695
9.701
9.706
9. 711
9.716
9.721
9.726
9-731
9-737
9.742
95
9-747
9-752
9-757
9.762
9-767
9.772
9-778
9-783
9-788
9.793
96
9-798
9.803
9.808
9-813
9.818
9.823
9.829
9-834
9-839
9.844
97
9.849
9-854
9-859
9.864
9.869
9-874
9.879
9.884
9.889
9.894
98
9.899
9-905
9.910
9-915
9.920
9-925
9-930
9-935
9.940
9.945
99
n
9-950
9-955
9.960
9-965
9.970
9-975
9. 980
9-985
9-990
9-995
.0 .1 .2 .3 .4 -S .6 .7 -8 .9
20
Arithmetical Tables
9. Cuoes of Niun-
n
012345678 9
I.O
1 .000
1.030
1. 061
I 093
1. 125
1. 158
1. 191
1.225
1. 260
1.295
I.I
1-331
1.368
1.40=;
1.443
1.482
1.521
1.561
1.602
1.643
1-685
1.2
1.728
1.772
1. 8x6
1. 861
1.907
1-953
2.000
2.048
2.097
2.147
1-3
2.197
2.248
2.300
2.353
2.406
2.460
2.515
2-571
2.628
2.686
1.4
2.744
2.803
2.863
2.924
2.9S6
3-049
3. 112
3-177
3-242
3-308
i-S
3-375
3-443
3.512
3.582
3.- 652
3.724
3-796
3-870
3-944
4.020
1.6
4.096
4.173
4.252
4.331
4. 411
4.492
4-574
4.657
4.742
4.827
1-7
4.913
5.000
5.088
5-178
5.268
5-359
5-452
5.545
5.640
5-735
1.8
S-832
5-93°
6.029
6.128
6.230
6.332
6.435
6.539
6.645
6-751
1-9
6.859
6.968
7.078
7.189
7-3°i
7-415
7-530
7.645
7.762
7.881
2.0
8.000
8. 121
8.242
8.365
8.490
8.615
8.742
8.870
8-999
9.129
2.1
9. 261
9-394
9-528
9.664
9.800
9-938
10.08
10. 22
10.36
10.50
2.2
10.65
10.79
10.94
II .09
11.24
11-39
11.54
11.70
11.85
12.01
2.3
12. 17
12-33
12.49
12.65
12.81
12.98
13-14
^3-3T^
13.48
13-65
2.4
13.82
14.00
14.17
14.35
14.53
14.71
14.89
15.07
15-25
15-44
2.5
15.62
15.81
16.00
16. 19
16-39
16.58
16.78
16.97
17.17
17-37
2.6
17-58
17.78
17.98
18.19
18.40
18.61
18.82
19-03
19-25
19-47
2.7
19.68
19.90
20. 12
20.35
20.57
20.80
21.02
21.25
21. 48
21.72
2.8
21-95
22. 19
22.43
22.67
22.91
23.15
23-39
23-64
23-89
24.14
2.9
24.39
24.64
24.90
25-15
25.41
25.67
25.93
26.20
26.46
26.73
3.0
27.00
27-27
27.54
27.82
28.09
28.37
28.65
28.93
29.22
29.50
3-1
29.79
30.08
30.37
30.66
30.96
31 . 26
31.55
31.86
32.16
32.46
3-2
32-77
33-08
33-39
33-70
34-01
34.33
34.65
34.97
35.29
35.61
3-3
35-94
36.26
36.59
36.93
37.26
37.60
37.93
38.27
38.61
38.96
3.4
39-30
39-65
40.00
40.35
40.71
41.06
41.42
41.78
42.14
42.51
35
42.88
43.24
43-61
43.99
44.36
4-4.74
45-12
45.50
45.88
46.27
3.6
46.66
47-05
47-44
47.83
48.23
48.63
49-03
49.43
49.84
50.24
3.7
50-65
51.06
51-48
51-90
52.31
52.73
53-16
53.58
54.01
54.44
3.8
54.87
55-31
55-74
56.18
56.62
57-07
57.51
57-96
58.41
58.86
3-9
59-32
59-78
60.24
60.70
61.16
61.63
62. 10
62.57
63.04
63.52
4.0
64.00
64.48
64.96
65.45
65.94
66.43
66.92
67-42
67.92
68.42
4.1
68. 92
69-43
69-93
70.44
70.96
71.47
71-99
72-51
73-03
73.56
4.2
74.09
74-62
75-15
75-69
76.23
76.77
77-31
77-85
78.40
78.95
4-3
79-51
80.06
80.62
81.18
81.75
82.31
82. 88
83-45
84-03
84.60
4.4
85.18
85-77
86. 35
86.94
87.53
88.12
88.72
89-31
89-92
90.52
4-5
gi. 12
91-73
92.35
92.96
93-58
94.20
94.82
95-44
96.07
96.70
4.6
97-34
97-97
98.61
99.25
99.90
100.5
loi. 2
101.8
102. 5
1.03 . 2
4.7
103.8
104.5
105. 2
105 . 8
106. 5
107. 2
107.9
108.5
109.2
109.9
4.8
no. 6
III
3
1 12.0
112. 7
113. 4
114. 1
114.8
"5-5
116.2
116. 9
4-9
117. 6
118
4
1 19. 1
119.8
120.6
121.3
122.0
122.8
123.5
124.3
S.o
125.0
125
8
126.5
127-3
128.0
128.8
129.6
130.3
131-1
131-9
5-1
132-7
133
4
134.2
135-0
135-8
136.6
137-4
138.2
139-0
139.8
5-2
140.6
141
4
142. 2
I43-I
143-9
144.7
145-5
146.4
147-2
148.0
5-3
148.9
149
7
150.6
151-4
152-3
I53-I
154-0
154-9
155-7
156.6
5-4
I57-S
158
3
159.2
160. 1
161.
161. 9
162.8
163.7
164.6
165.5
;(
012345678 9
Arithmetical Tables
bers from 1.00 to 9.99
21
n
012345678 9
5-5
166.4
167.3
168.2
169. 1
170.0
171.
171. 9
172.8
173-7,174-7
5.6
175-6
176.6
177-5
178.5
179-4
180.4
181. 3
182.3
183-3
184.2
5-7
185.2
186.2
187. 1
188. 1
iSg.i
190. 1
191 . 1
192. 1
193- 1
194. 1
5.8
195. 1
196. 1
197. 1
198.2
199.2
200. 2
201 . 2
202.3
203.3
204.3
5-9
205.4
206.4
207-5
208.5
209.6
210.6
211 . 7
212.8
213.8
214.9
6.0
216.0
217. 1
218.2
219.3
220.3
221.4
222.5
223.6
224.8
225.9
6.1
227.0
228.1
229.2
230.3
231-5
232.6
233 -7
234-9
236.0
237-2
6.2
238.3
239-5'
240.6
241.8
243.0
244.1
245-3
246.5
247.7
248.9
6.3
250.0
251.2
252-4
253-6
254.8
2!;6.o
257-3
258.5
259-7
260.9
6.4
262. 1
263.4
264.6
265.8
267. I
268.3
269.6
270.8
272.1
273-4
6.5
274.6
275.9
277.2
278.4
279-7
281.0
282.3
283.6
284.9
286.2
6.6
287.5
288.8
290. 1
291.4
292.8
294.1
295-4
296.7
298.1
299.4
6.7
300.8
302.1
303-5
304.8
306.2
307-5
308.9
310.3
311-7
313-0
6.8
314.4
315.8
317-2
318.6
320.0
321-4
322.8
324-2
325-7
327-1
6.9
328.5
329-9
331-4
332.8
334.3
335-7
337-2
338.6
340.1
341-5
7.0
343 -o
344.5
345-9
347-4
348.9
350-4
351-9
353-4
354.9
356-4
7.1
357-9
359-4
360.9
362.5
364.0
365-5
367-1
368.6
370.1
371-7
7.2
373-2
374-8
376.4
377-9
379-5
381. 1
382.7
384-2
385-8
387-4
7.3
389.0
390.6
392.2
393-8
395-4
397-1
398.7
400.3
401.9
403.6
7.4
405.2
406.9
408.5
410.2
411-S
413-S
415-2
416.8
418.5
420.2
7.5
421.9
423.6
425-3
427.0
428.7
430-4
432-1
433-8
435.5
437-2
7.6
439-0
440.7
442.5
444-2
445-9
447-7
449-5
451-2
453-0
454-8
7.7
456-5
458.3
460. 1
461.9
463-7
465-5
467-3
469.1
470.9
472-7
7.8
474-6
476-4
478.2
4S0.0
481.9
483-7
485.6
487.4
489.3
491-2
7.9
493-0
494-9
496.8
498-7
500.6
502.5
504-4
506.3
508.2
510. 1
8.0
512.0
513-9
515.8
517-8
519-7
521-7
523-6
525-6
527-S
529-5
8.1
531.4
533-4
535-4
537-4
539-4
541-3
543-3
545-3
547-3
549-4
8.2
551-4
553-4
555-4
557-4
559-5
56I-S
563-6
565-6
567-7
569-7
8.3
571-8
573-9
575-9
578.0
580.1
582.2
584-3
586.4
588.5
590.6
8.4
592.7
594-8
596-9
599-1
601 . 2
603.4
605-5
607.6
609.8
612.0
8.5
614. 1
616.3
618.5
620.7
622.8
625.0
627.2
629.4
631.6
633-8
8.6
636.1
638.3
640.5
642.7
645-0
647.2
649-5
651.7
654.0
656.2
8.7
658.5
660. 8
663.1
665-3
667.6
669. 9
672.2
674.5
676.8
679.2
8.8
681.5
683.8
686.1
688.5
690.8
693.2
695-5
697.9
700.2
702.6
8.9
705.0
707.3
709.7
712. 1
714-5
716.9
719-3
721.7
724-2
726.6
9.0
729.0
731-4
733-9
736.3
738-8
741-2
743-7
746.1
748.6
751-1
9.1
753.6
756.1
758-6
761.0
763-6
766.1
768.6
771. 1
773-6
776.2
9.2
778.7
781.2
783.8
786.3
788.9
791-5
794.0
796.6
799.2
801.8
9-3
804.4
807.0
809.6
812.2
814.8
817-4
820.0
822.7
825.3
827-9
9.4
830.6
833.2
835-9
838.6
841.2
843-9
846.6
849-3
852.0
854-7
9-5
857-4
860.1
862.8
865.5
868.3
871.0
873-7
876.5
879-2
882.0
9.6
884.7
887.5
890-3
893-1
895-8
898.6
901.4
904.2
907.0
909-9
9-7
912.7
915-5
918.3
921 . 2
924.0
926.9
929-7
932-6
935-4
938.3
9.8
941.2
944-1
947.0
949-9
952-8
955-7
958.6
961-5
964.4
967.4
9.9
970.3
973-2
976.2
979-1
982.1
985-1
988.0
991.0
994.0
997-0
n
012345678 9
22
Arithmetical Tables
10. Cube Roots of Numbers
n
</n
V io»
11
</n
"V lo n
■y 100 n
'\Jioo n
10
2.1544
4.6416
10.000
55
3-8030
8.1932
17.652
II
2. 2240
4-7914
10.323
56
3-8259
8.2426
17-758
12
2. 2894
49324
10. 627
57
3-8485
8.2913
17.863
13
2.3513
5-0658
10. 914
S8
3-8709
8.3396
17.967
14
2.4101
5-1925
II. 187
59
3-8930
8.3872
18.070
IS
2. 4662
5-3133
11.447
60
3-9149
8.4343
18.171
i6
2.5198
5.4288
11.696
61
3-9365
8 . 4809
18.272
17
2-5713
5-5397
11-935
62
3-9579
8.5270
18.371
i8
2. 6207
5.6462
12. 164
63
3-9791
8.5726
18.469
19
2.6684
5-7489
12.386
64
4. 0000
8.6177
18.566
20
2.7144
5.8480
12-599
6S
4. 0207
8.6624
18.663
21
2.7589
5 - 9439
12.806
66
4.0412
6. 7066
18.758
22
2.8020
6. 0368
13.006
67
4.0615
8.7503
18.852
23
2.8439
6. 1269
13.200
68
4.0817
8.7937
18.945
24
2.88/J5
6.2145
13-389
69
4. 1016
8.8366
19.038
25
2.9240
6.2996
13-572
70
4.1213
8. 8790
19.129
36
2.9625
6.3825
13-751
71
4. 140S
8.9211
19.220
27
3 . 0000
6.4633
13-925
72
4. 1602
8.9628
19.310
28
3-0366
6.5421
14.095
73
4.1793
9.0041
19.399
29
3-0723
6.6191
14.260
74
4.1983
9-0450
19.487
30
3.1072
6.6943
14.422
75
4.2172
9-0856
19.574
31
3-1414
6.7679
14.581
76
4-2358
9.1258
19.661
32
3-1748
6.8399
14.736
77
4-2543
9-1657
19.747
33
3-2075
6. 9104
14.888
78
4.2727
9.2052
19.832
34
3-2396
6-9795
15-037
79
4. 2908
9-2443
19.916
35
3.2711
7-0473
15-183
80
4.3089
9.2832
20.000
36
3-3019
7.1138
15-326
81
4-3267
9-3217
20.083
37
33322
7.1791
15-467
82
4-3445
9-3599
20. 165
38
3.3620
7-2432
15-605
83
4.3621
9-3978
20.247
39
3-3912
7.3061
15-741
84
4-3795
9-4354
20.328
40
3 . 4200
7.3681
15-874
8s
4.3968
94727
20.408
41
3-4482
7.4290
16.005
86
4.4140
9-5097
20.4S8
42
3-4760
7.4889
16.134
87
4-4310
9 5464
20.567
43
3-5034
7-5478
16. 261
88
4-4480
9.5828
20.646
44
3-5303
7-6059
16.3S6
89
4-4647
9. 6190
20.724
4S
3-5569
7-6631
16.510
90
4.4814
9.6549
20, 801
46
3-5830
7-7194
16. 631
91
4.4979
9 ■ 6905
20.878
47
3.6088
7-7750
16.751
92
4-5144
9-7^59
20.954
48
3 6342
7.8297
16.869
93
4-5307
9.7610
21 .029
49
3-6593
7-8837
16.985
94
4.5468
9-7959
21. 105
50
3 ■ 6840
7-9370
17. 100
9S
4-5629
9-8305
21.179
51
3-7084
7 . 9896
17.2:3
96
4-5789
9.8648
21-253
52
3-7325
8. 0415
17-325
97
4-5947
9 . S990
21.327
S3
3 7563
8.0927
17-435
98
4 . 6104
9.9329
2 1 . 400
54
3- 7798
8-1433
17-544
99
4.6261
9.9666
21.472
Arithiietical Tables
11. Three-Halves Powers of Numbers
23
n
01 23456 789
o.o
0.000
O.OOI
0.003
0.005
0.008
O.OII
0.015
0.019
0.023
0.027
O.I
0.032
0.036
0.042
0.047
0.052
0.058
0. 064
0. 070
0.076
0.083
0.2
0.089
0.096
0.103
0. no
0.118
0. 125
0-133
0. 140
0. 148
0. 156
0.3
0. 164
0-173
0.181
0. 190
0. 198
0. 207
0. 216
0. 225
0.234
0.244
0.4
0-253
0.263
0.272
0. 282
0. 292
0.302
0.312
0.322
0-333
0.343
0.5
°-354
0.364
0-375
0.386
0.397
0.408
0.419
0.430
0.442
0-453
0.6
0.465
0.476
0.488
0.500
0.512
0.524
0.536
0.548
0.561
0-573
0.7
0.586
0.598
0.611
0.624
0.637
0.650
0.663
0.676
0.689
0.702
0.8
0.716
0.729
0.743
0.756
0.770
0.784
0.798
0.811
0.826
0.840
0.9
0.854
0.868
0.882
0.897
0.911
0.926
0.941
0-955
0.970
0-985
1.0
1. 000
1.015
1.030
1.045
1.061
1.076
1.091
1 . 107
1 . 122
1. 138
I.I
I-IS4
1. 170
1-185
1. 201
1. 217
1-233
1.249
1.266
1.282
1.298
1.2
1-315
I-33I
1-348
1.364
1.381
1-398
1.414
I -43 1
1.448
1-465
1-3
1.482
1.499
1-S17
1-534
I -551
1-569
1.586
1.604
1. 621
1-639
1.4
1.657
1.674
1.692
1 . 710
1.728
1.746
1.764
1.782
1.800
1.819
1-5
1-837
1.856
1.874
1-893
1. 911
1-930
1.948
1.967
1.986
2.005
1.6
2.024
2.043
2.062
2.081
2. 100
2. 119
2.139
2.158
2.178
2-197
1.7
2.217
2.236
2.256
2.275
2 - 295
2-315
2.335
2-355
2.37s
2-395
1.8
2.415
2-435
2-455
2.476
2.496
2.516
2.537
2-557
2-578
2-598
1.9
2.619
2.640
2.660
2.681
2. 702
2.723
2-744
2.765
2.786
2.807
2.0
2.828
2.850
2.871
2.892
2.914
2-935
2-957
2.978
3.000
3.021
2.1
3 -043
3-065
3-087
3.109
3-131
3-153
3-175
3-197
3.219
3-241
2.2
3 263
3-285
3-308
3-330
3-353
3-375
3-398
3-420
3.443
3-465
2.3
3.488
3-5"
3-534
3-557
3-580
3-602
3.626
3-649
3.672
3-695
2.4
3-718
3-741
3-765
3-788
3. 811
3-835
3-858
3-882
3 . 906
3-929
2.5
3-953
3-977
4.000
4.024
4.048
4.072
4.096
4. 120
4.144
4.168
2.6
4.192
4-217
4.241
4.263
4.289
4.314
4.338
4-363
4.387
4-412
2.7
4-437
4.461
4.486
4-511
4-536
4.560
4.585
4. 610
4-635
4.660
2.8
4.685
4.710
4.736
4-761
4.786
4.811
4-837
4.862
4.888
4-913
2.9
4-939
4.964
4.990
5-015
5-041
5.067
5-093
5-118
5-144
5-170
3-0
5-196
5.222
5-248
5-274
5-300
5-327
5-353
5.379
5-405
5-432
3-1
5-458
5-485
5-5II
5-538
5-564
5-591
5-617
5-644
5-671
5.698
3-2
5-724
5-751
5.778
5-805
5-832
5-859
5-886
5-913
5 • 940
5-968
3-3
5-995
6.022
6.049
6.077
6. 104
6.132
6-159
6.186
6.214
6.242
3-4
6. 269
6.297
6.325
6-352
6.380
6.408
6.436
6.464
6.492
6.520
3.5
6-548
6.576
6.604
6.632
6.660
6.689
6.717
6.745
6-774
6.802
3.6
6.831
6.859
6.888
6.916
6.945
6.973
7.002
7-031
7-059
7.088
3-7
7. 117
7-146
7-175
7.204
7-233
7.262
7.291
7-320
7-349
7-378
3-8
7.408
7-437
7.466
7-495
7-525
7-554
7-584
7-613
7-643
7.672
3-9
7.702
7-732
7.761
7.791
7.821
7-850
7.880
7.910
7.940
7-970
4.0
8.000
8.030
8.060
8.090
8.120
8.150
8' 181
8.211
8.241
8.272
4.1
8.302
8-332
8.363
8.393
8.424
8.454
8.485
8-515
8.546
8-577
4.2
8.607
8.638
8.669
8. 700
8-731
8.762
8.793
8.824
8-855
8.886
4-3
8.917
8.948
8-979
9.010
9.041
9-073
9.104
9-135
9.167
9.198
4.4
9.230
9.261
9-293
9-324
9-356
9-387
9-419
9-451
9-482
9-514
n
01 23456789
24 Arithmetical Tables
12. Fifth Powers and Roots; Five-Halves Powers and Roots
n
O.I
n'
»*
n^
n^
n
4.6
w"
n*
«s
n«
0.0000
0.6310
0.0032
0.3981
2059.
6 1.3569
45-383
I. 8412
0.2
. 0003
0. 7248
0.0179
0.5253
4-7
2293.
5 1.3628
47.890
1-8571
0.3
0.0024
0.7860
0.0493
0.6178
4.8
2548.
1.3685
50.47S
1.8728
0.4
0.0102
0.8326
0. 1612
0.6931
4-9
2824.
8 1-3742
53-148
1 . 8883
0.5
0.0312
0.8706
0.1768
0.7579
5-0
3125.
1.3797
55-902
1-9037
0.6
0.0778
0. 9029
0. 2789
0.8152
S.I
3450.
3 1.3852
58.739
1.9188
0.7
0.1681
0.9311
0.4100
0.8670
5.2
3802.
1.3906
61.661
1-9338
0.8
0.3277
0.9564
0.5724
0.9146
5-3
4182.
1.3959
64.668
I - 9485
0.9
0-5905
0.9791
0.7684
0.9587
5.4
4591-
7 1. 401 1
67.762
1.9632
I.O
I . 0000
I . 0000
I . 0000
1 . 0000
5-5
5032.
8 1.4063
70.943
1.9776
I.I
I. 6105
I .0192
I 2691
1.0389
5.6
5507-
3 1-4114
74.211
1-9919
1.2
2 . 4883
1.0371
1-5774
1-0757
5-7
6016.
9 I. 4164
77.569
2.0061
1.3
3-7129
1-0539
1 .9269
I. 1107
5.8
6563-
6 I. 4213
81.016
2.0201
1.4
5-3782
I . 0696
2.3191
1.1441
5-9
7149-
2 1.4262
84.553
2.0340
i.S
7-5938
1.0845
2-7557
I. 1761
6.0
7776.
3 1.4310
88.182
2.0477
1.6
10.486
1.0986
3.2382
1.2068
6.1
8446.
3 1.4357
91 . 902
2.0613
1.7
14.199
I. 1120
3.7681
1-2365
6.2
9161.
3 I -4404
95.715
2.0747
1.8
18.896
1.1247
4.3469
1.2651
6.3
9924-'
% 1-4450
99.621
2.0880
1-9
24.761
1-1370
4.9760
1.2927
6.4
10737
1.4496
103.62
2. 1012
2.0
32.000
1.14S7
5-6569
1-3195
6.5
11603
1-4541
107.72
2.1143
2.1
40. 841
1 . 1600
6.3907
1-3455
6.6
12523
1-4585
III ^ I
2. 1272
2.2
Si-536
1.1708
7.1789
1.3708
6.7
13501
I . 4629
116. 19
2.1401
2.3
64-363
1.1813
8.0227
I -3954
6.8
14539
1 . 4672
120. 58
2. 1528
2.4
79.626
1.1914
8-9234
1-4193
6.9
15640
I-471S
125.06
2.1654
2.5
97-656
I . 201 1
9.8821
1.4427
7.0
16807
1-4758
129.64
2.1779
2.6
118. 81
1 . 2106
10.900
1-4655
7-1
1S042
I 4800
134.32
2.1903
2.7
143-49
I. 2198
11.979
1.4878
7.2
19349
1.4841
139-10
2. 2026
2.8
172. 10
I. 2287
13. 119
1.5096
7-3
20731
1.4882
143-98
2.2148
2.9
205.11
1-2373
14.322
1-5309
7.4
22190
- 1-4923
148.96
2.2269
3-0
243. 00
1-2457
15-588
1-5518
7.5
23730
1-4963
154.05
2.2388
3-1
286.29
1-2539
16.920
1-5723
7.6
25355
1 . 5002
159.23
2.2507
3-2
335-54
I. 2619
18.318
1-5924
7-7
27068
1-5042
164.52
2.2625
3-3
391-35
1.2697
19-783
1.6122
7.8
28872
1.5081
169.92
2.2742
3-4
454-35
1-2773
21.316
1-6315
7.9
30771
I-5119
175.42
2.2859
3-S
525.22
1.2847
22.918
1-6505
8.0
32768
I-5157
181.02
2.2974
3.6
604 . 66
1.2920
24-590
I .6692
8.2
37074
1.5232
192.55
2.3202
3.7
693-44
I . 2991
26.333
1.6876
8.4
41821
1-5306
204.50
2.3427
3.8
7.92-35
1.3060
28. 149
1-7057
8.6
47043
1-5738
216.89
2.3648
3.9
902. 24
I. 3128
30.037
1.7236
8.8
52773
1.5449
229. 72
2.3S67
4.0
1024.0
1-3195
32.000
1.7411
9.0
59049
1-5518
243.00
2.4082
4.1
1158.6
1.3260
34-038
1-7584
9.2
65908
1-5587
256.73
2.4295
4.2
1306.9
1-3324
36.151
1-7754
9.4
73390
1 - 5654
270.91
2-4505
4-3
1470. I
1-3387
38.342
1.7922
9.6
81537
1.5720
285.55
2.4712
4.4
1649. 2
1-3449
40. 610
I . 8088
9.8
90392
1-5785
300.65
2.4917
4-5
1845-3
1-3510
42.957
1.8251
10
lOOOOC
J 1.5849
316.23
2.5119
Explanation on page 39
Arithmetical Tables 25
13. Explanations
All of the Arithmetical Tables, except 10 and 12, are four-
place tables, that is, the values of the functions are given to four
significant figures. In Tables 10 and 12 five significant figures are
given. For all these tables the probable error in the last figure
is one-fourth of a unit.
Table G gives Reciprocals of all numbers having three sig-
nificant figures by properly moving the decimal point. Thus
the reciprocals of 0.705, 7.05, 705, and 0.0705 are 1.418, 0.1418,
0.001418, and 14.18 to four significant figures.
Table 7 gives Squares of all niunbers having" three significant
figures by properly moving the decimal point. Thus the squares
of 3.94, 0.394, and 39.4 are 15.52, 0.1552, and 1552 to four sig-
nificant figures. Here the decimal point moves two places in the
function when it moves one place in the argument.
Table 8 gives Square Roots of all numbers of three significant
figures. For the numbers 5.42 and 542 the square roots 2.328
and 23.28 are found on page 16; for the numbers 54.2 and 5420 the
scjuare roots 7.302 and 73.62 are found on page 18. Here, as in
Table 7, the decimal point moves two places in the square
when it moves one place in the square root.
Table 9 gives Cubes of all numbers of three significant figures by
moving the decimal point three places in the function when it
moves one place in the argument. Thus, the cubes of 4.69 and
0.469 are 103.2 and 0.1032 to four significant figures.
Table 10 gives Cube Roots of two-place numbers. Thus, the
cube root of 35 is 3.2711, that of 350 is 7.0473, and that of 3500
is 15.183. When the given number contains a decimal point,
multiply it by 1000, take the root from the table and then divide
this by 10. In this manner the cube roots of 3.5, 0.35, and 0.0035
are found to be 1.5183, 0.7047, and 0.3271.
Table 11 gives values of ?i^ or V n' for values of n up to 4.49.
This is useful in the weir computations of hydraulics. For ex-
ample, when n is 2.59 the value of n^ is 5.432. Table 12 is also
used in hydraulic work in computations on the flow of water in
long pipes.
26 Arithmetical Tables
14. Exercises for Students
1. Find the reciprocals of 0.14, 0.145, and 0.1456; also of 1.4,
1.45, and 1.456; also of 14, 14.5, and 14.56.
2. Find the reciprocals of 0.90, 0.99, and 0.909; also of 0.695,
0.6954, and 6.954; also of 0.295, 0.2954, 29.5, and 295.4.
3. Find the squares of 3.90, 3.902, and 3.909; also of 39.0, 39.02,
and 39.09; also of 0.77, 0.777, and 0.7777; also of 0.707 and 0.7071.
4. Find the square roots of the following numbers:
2.08 2.081 2.087
2.09 2.091 2.097
9.43 9.433 9.435
20.8 20.81 20.87
20.9 20.91 20.97
94.3 94.33 94.35
0.89 0.891 0.8913
89.0 89.1 89.13
8900 8910 8913
5. Find the value of V3-+4- + 122 by the help of Tables 7 and 8.
6. Find the values of the following functions:
4.232 = 4.2312= 42.312 =
8.952= 89.52= 8952 =
0.7232= 0.07232= 7232 =
7.253= 72.5'= 725' =
7.043= 0.7043= 7043 =
0.9993 9 993 99.93 =
7. Find the value of \/¥+A^+5'^ by the help of Table 8.
8. Find the three-halves powers of 2.78 and 2.783.
9. Find the values of the following functions:
1 /32.2 =. 10/9.8 = 20/0.45
\/64r32 = V19I6 = VO 1916 =
1.35* = 1.35* = 1.35i =■ .
Chapter 3
TABLES OF CIRCLES AND SPHERES
28
Circles and Spheres
15. Areas of Circles for Diam-
d
0123456789
I.O
0.785
0.801
0.817
0.833
0. 849
0.866
0.882
0. 899
0.916
0-933
I.I
0.950
0.968
0.985
1 .003
1. 021
1-039
1-057
1-075
1.094
1. 112
1.2
1. 131
1 . 150
1. 169
1. 188
1.208
1.227
1.247
1 . 267
1.287
1-307
1.3
1.327
1-348
1-368
1-389
1 . 410
1-431
1-453
1-474
1-496
1-517
1.4
1.539
I. 561
1.584
1 . 606
1.629
1-651
1-674
1.697
1.720
1-744
1. 5
1.767
I. 791
1.815
1-839
1.863
1.8S7
1 . 91 1
1.936
1.961
1.986
1.6
2. on
2-036
2.061
2.087
2. 112
2.138
2. 164
2. 190
2.217
2.243
1.7
2.270
2-297
2-324
2-351
2-378
2.405
2-433
2.461
2.488
2-516
1.8
2.545
2-573
2.602
2-630
2-659
2.688
2.717
2.746
2.776
2.806
1.9
2.83s
2.865
2.895
2.926
2.956
2.986
3-017
3.048
3-079
3. no
2.0
3.142
3-173
3-205
3.237
3.269
3-301
Z2Z3
3.365
3-398
3-431
2.1
3.464
3-497
3530
3.563
3-597
3-631
3.664
3.698
3-733
3-767
2.2
3.801
3-836
3.871
3-906
3 941
3-976
4.012
4-047
4-083
4.119
2.3
4.155
4. 191
4.227
4-264
4-301
4.337
4-374
4-412
4-449
4-486
2.4
4-524
4.562
4.600
4.638
4-676
4.714
4-753
4-792
4-831
4-870
2.5
4.909
4-948
4.988
5.027
3-067
5.107
5-147
5-187
5.228
5.269
2.6
5-309
5-350
5-391
S-433
5-474
S-51S
5-557
5-599
5.641
5 -683
2.7
5-726
5.768
5.811
5-853
5.896
5.940
5-983
6.026
6.070
6. 114
2.8
6.158
6. 202
6.246
6. 290
6-335
6-379
6.424
6.469
6.514
6.560
2.9
6-605
6.651
6.697
6-743
6-789
6-835
6.881
6.928
6-975
7.022
3.0
7.069
7. 116
7.163
7. 211
7-258
7-306
7-354
7.402
7-451
7-499
3-1
7.548
7.596
7.64s
7.694
7-744
7-793
7-843
7-892
7-942
7.992
3-2
8.042
8.093
8.143
8.194
8-245
8.296
8.347
8-398
8-450
8.501
3-3
8.553
8.605
8.657
8.709
8. 762
8.814
8.867
8.920
8-973
9.026
3.4
9.079
9-133
9.186
9.240
9-294
9-348
9 402
9-457
9-5"
9.566
3-5
9.621
9.676
9.731
9.787
9.842
9.898
9-954
10.01
10.07
10. 12
3.6
10. 18
10. 24
10.29
10.35
10.41
10.46
10. 52
10.58
10.64
10.69
3.7
10.75
10.81
10.87
10.93
10.99
11.04
11. 10
11. 16
11.22
n.28
3.8
1 1 • 34
11.40
11.46
11.52
11-58
11.64
11.70
n.76
11.82
11.88
3-9
11.95
12.01
12.07
12.13
12. 19
12.25
12.32
12.38
12-44
12.50
4.0
12.57
12.63
12.69
12.76
12.82
12.88
12.95
13.01
13-07
13-M
4.1
13.20
13-27
13.33
13.40
13-46
13-53
13-59
13-66
13-72
13-79
4.2
13.85
13-92
13.99
14.05
14. 12
14-19
14.25
14-32
14-39
14-45
4-3
14-52
14.59
14.66
14.73
14.79
14-86
14.93
15.00
15-07
15-14
4.4
15. 21
15-27
15.34
15.41
15-48
15-55
15.62
15-69
15-76
15-83
4.5
15-90
15-98
16.05
16. 12
16. 19
16.26
16.33
16.40
16.47
i6-5S
4.6
16.62
16.69
i6.7fe
16.84
16.91
16.98
17.06
17-13
17. 20
17.28
4.7
17-35
17-42
17-50
17.57
17-65
17-72
17.80
17.87
17-95
18.02
4.8
18. 10
18.17
18.25
18.32
18.40
18.47
18-55
18.63
18.70
18.78
4-9
i8.86
18.93
19.01
19.09
19.17
19-24
19.32
19-40
19.48
19-56
S.o
19-63
19.71
19-79
19.87
19-95
20.03
20. 11
20. 19
20.27
20.35
51
20.43
20.51
20.59
20.67
20- 75
20.83
20.91
20.99
21.07
21. 16
S.2
21. 24
21.32
21.40
21.48
21-57
21.65
21-73
21. 81
21.90
21.98
5.3
22.06
22. 15
22.23
22.31
22.40
22.48
22.56
22.65
22.73
22.82
5.4
22.90
22.99
23-07
23-16
23-24
23-33
23-41
23-50
23-59
23-67
d
012345678 9
Circles and Spheres
eters in Units and Hundredths
29
d
I 2
3
4
56789
s-s
23.76
23
.84
23^93
24
. 02
24
. 1 1
24
•19
24.28
24^37
24-45
24.54
5.6
24-63
24
•72
24.81
24
.89
24
.98
25
•07
25. 16
25^25
25-34
25^43
5.7
25-52
25
.61
25.70
25
■79
25
.88
25
.97
26.06
26. 15
26. 24
26.33
5.8
26.42
26
-51
26.60
26
.69
26
•79
26
.88
26.97
27.06
27-15
27.25
5-9
27 -.34
27
•43
27^53
27
.62
27
•71
27
.81
27.90
27.99
28.09
28.18
6.0
28.27
28
•37
28.46
28
•56
28
•65
28
•75
28.84
28.94
29-03
29.13
6.1
29. 22
29
•32
29.42
29
•51
29
.61
29
•71
29.80
29.90
30.00
30.09
6.2
30-19
30
.29
30-39
30
• 48
30
•58
30
.68
30-78
30.88
30-97
31.07
6.3
31-17
31
.27
31-37
31
•47
31
•57
31
.67
31-77
31-87
31-97
32.07
6.4
32.17
32
.27
32.37
32
•47
32
•57
32
.67
32.78
32.88
32.98
33.08
6.5
33.18
Z3
■29
33-39
33,
•49
33
■59
33
.70
33-80
33.90
34.00
34-11
6.6
34.21
34
32
34.42
34
■52
34
•63
34
•73
34-84
34.94
35-05
35.15
6.7
35-26
35
36
35-47
35
•57
35
68
35
.78
35-89
36.00
36. 10
36.21
6.8
36-32
36
42
36-53
36
64
36
75
36.85
36.96
37.07
37-18
37-28
6.9
37-39
37
50
37-61
37
72
37
83
37
94
38.05
38.16
38.26
38.37
7.0
38.48
38
59
38-70
38
82
38
93
39
04
39^15
39.26
39-37
39-48
7.1
39-59
39
70
39-82
39
93
40
04
40
15
40.26
40.38
40.49
40.60
7.2
40.72
40
83
40.94
41
06
41
17
41
28
41.40
41.51
41.62
41.74
7.3
41.85
41
97
42.08
42
20
42
31
42
43
42.54
42.66
42.78
42.89
7.4
43-01
43
12
43-24
43
36
43
47
43
59
43-71
43-83
43.94
44.06
7.5
44.18
44
30
44.41
44
53
44
65
44
77
44-89
45^01
45-13
45.25
7.6
45-36
45
48
45 • 60
45
72
45
84
45
96
46.08
46.20
46.32
46.45
7.7
46.57
46.69
46.81
46
93
47
05
47
17
47-29
47^42
47-54
47.66
7.8
47-78
47
91
48.03
48
15
48
27
48
40
48.52
48.65
48.77
48.89
7.9
49.02
49
14
49.27
49
39
49
51
49
64
49-76
49.89
50.01
50-14
8.0
50.27
50.
39
50.52
50
64
50
77
50
90
51.02
51.15
51.28
51-40
8.1
51-53
51-
66
51^78
51
91
52
04
52
17
52.30
52^42
52-55
52.68
8.2
52.81
52.
94
53^07
53
20
53
33
S3
46
53-59
53^72
53-85
53^98
8.3
54.11
54-
24
54-37
54
50
54.
63
54
76
54.89
55^02
55-15
55^29
8.4
55-42
55-
55
55-68
55
81
55.
95
56.
08
56.21
56^35
56.48
56.61
8.5
56-75
56.
88
57.0I
57-
15
57-
28
57.
41
57.55
57-68
57-82
57^95
8.6
58.09
58.
22
58.36
58-
49
58-
63
58.
77
58.90
59.04
59-17
59^31
8.7
59-45
59-
58
59-72
59-
86
59-
99
60.
13
60. 27
60.41
60-55
60.68
8.8
60.82
60.
96
61. 10
61.
24
61.
38
61.
51
61.65
61.79
61-93
62.07
8.9
62.21
62.
35
62.49
62.
63
62.
77
62.
91
63-05
63-19
63-33
63.48
9.0
63.62
63-
76
63.90
64-
04
64 •
18
64.
33
64.47
64.61
64-75
64.90
9.1
65.04
65-
18
65-33
65-
47
65^
61
65.
76
65.90
66.04
66. 19
66.33
9.2
66.48
66.
62
66.77
66.
91
67.
06
67.
20
67-35
67.49
67-64
67.78
9-3
67-93
68.
08
68.22
68
37
68.
51
68.
66
68.81
68.96
69. 10
69.25
9.4
69.40
69.
55
69.69
69
84
69.
99
70.
14
70-29
70.44
70-58
70.73
9.5
70.88
71.
03
71.18
71-
2,3
71.
48
71-
63
71.78
71-93
72.08
72.23
9.6
72.38
72.
53
72.68
72
84
72.
99
73.
14
73^29
73.44
73-59
73-75
9.7
73-9°
74-
OS
74-20
74
36
74.
51
74.
66
74.82
74-97
75-12
7=;-28
9.8
75-43
75-
58
75-74
75-
89
76.
05
76.
20
76.36
76.51
76.67
76.82
9.9
76.98
77-
13
77.29
77^
44
77^
60
77^
76
77.91
78.07
78.23
78-38
d
I
2 3
456789
30
Circles akd Spheres
16. Areas of Circles
Diameters in Units
and Eiithtb
s
d
o
1/8 1/4 3/8 1/2 5/8 3/4 Vs
. 0000
0.0123
0.0491
0. 1104
0.1963
0.3068
0.4418
0.6013
I
0-7854
0. 9940
1.2272
I . 4849
1.7671
2.0739
2.4053
2.7612
2
3-1416
3-5466
3.9761
4.4301
4.9087
5.4119
5.9396
6.4918
3
7.0686
7.6699
8.2958
8.9462
9.6211
10.321
1 1 . 045
11-793
4
12.566
13-364
14.186
15.033
15-904
16.800
17.721
18.665
5
19-635
20.629
21.648
22.691
23.758
24.850
25.967
27.109
6
28.274
29465
30. 680
31-919
33.183
34472
35.785
37.122
7
38-485
39-871
41 . 282
42.718
44.179
45.664
47.173
43.707
8
50-265
51.849
53-456
55-088
56.745
58.426
60.132
61.862
9
63.617
65.397
67. 201
69.029
70.882
72.760
74.662
76-589
10
78-540
80.516
82.516
84-541
86.590
88. 664
90.763
92.886
II
95 033
97.205
99-402
lOI . 62
103.87
106. 14
108.43
110.75
13
113. 10
115.47
117.86
120. 28
122. 72
125.19
127.68
130.19
13
132-73
135.30
137-89
140.50
143 14
145.80
148.49
151.20
14
153-94
156.70
159.48
162.30
165.13
167.99
170.87
173.78
IS
176.71
179.67
182.65
185.66
188.69
191-75
194.83
197.93
i6
201.06
204. 22
207.39
210. 60
213-82
217.08
220.35
223.65
17
226.98
230-33
233.71
237.10
240.53
243-98
247.45
250.9s
i8
254-47
258.02
261.59
265.18
268.80
272.45
276. 12
279.81
19
283.53
287.27
291.04
294.83
298.65
302.49
306.35
310.24
20
314.16
318.10
322.06
326.05
330.06
334.10
338.16
342.25
21
346.36
350.50
354-66
358.84
363-05
367.28
371.54
375.83
22
380.13
384.46
388.82
393 . 20
397-61
402.04
406 . 49
410.97
23
415-48
420.00
424-56
429.13
433-74
438.36
443-01
447.69
24
452-39
457.11
461.86
466.64
471-44
476.26
481. II
485.98
25
490.87
495.79
500.74
505-71
510.71
515.72
520.77
525.84
26
530.93
536.05
541.19
546.35
551-55
556.76
562.00
567.27
27
572.56
577.87
583.21
588.57
593 . 96
599.37
604. 81
610. 27
28
615.75
621 . 26
626.80
632-36
637-94
643.55
649.18
654.84
29
660.52
666.23
671.96
677.71
683.49
689.30
695-13
700.98
30
706.86
712.76
718.69
724.64
730.62
736.62
742.64
74S.69
31
754.77
760.87
766.99
773-14
779.31
785.51
791-73
797.98
32
804. 25
810. 54
8:6.86
823.21
829.58
835-97
842.39
848. S3
33
855-30
861.79
868.31
874-85
881.41
S8S.00
894.62
901 . 26
34
907.92
914.61
921.32
928.06
934.82
941.61
948.42
955.25
35
962. II
969.00
975.91
982.84
989.80
996.78
1003.8
1010.8
36
1017.9
1025 .
1032. I
1039. 2
1046.3
1053-5
1060. 7
1068.0
37
1075.2
1082. 5
1089.8
1097-1
1104.5
1 1 1 1 . 8
1 1 19. 2
ii'26. 7
38
"34-1
II4I.6
1149.1
1 156. 6
1164.2
1171.7
1179.3
1186.9
39
1194.6
1202.3
1210.0
1217.7
1225.4
1233.2
1241.0
1 248 . 8
40
1256.6
1264.5
1272.4
1280.3
1288.2
1296. 2
1304.2
1312.2
41
1320.3
1328.3
1336.4
1344.5
1352.7
1360.8
1369-0
1377.2
42
1385.4
1393.7
1402.0
1410.3
1418.6
1427.0
1435-4
1443.8
43
1452.2
1460. 7
1469. 1
1477.6
14S6.2
1494.7
1503.3
1511.9
44
d
1520.5
1529.2
1537 -9
1546.6 1555.3
1564.0
1572-8
1581.6
1/8 1/4 3/8 1/2 6/8 3/4 7/8
Circles and Spheres
17. Circumferences of Circles
31
E
iameters
in Units
and Tenths
d
.0 .1 .2 .3 .4 .5 .6 .7 .8 .9
o. ooo
0.314
0.628
0.942
i-257li-57i 1-885 2.199
2.513
2.827
I
3-142
3-456
3-770
4.084
4.398 4.712 5.02715.341
5-655
5-969
2
6.283
6-597
6.912
7.226 7.540 7.85418.168:8.482
8.796
9. Ill
3
9425
9-739
10.05
10.37
10.68 11.00
II. 31
11.62
11.94
12. 25
4
12.57
12.88
13-19
13-51
13.82 14.14
14.45
14.77
15-08
15-39
5
15-71
16.02
16.34
16.65
16.96 17.28
17-59
17.91
18.22
18.54
6
18.85
19. 16
19.48
19-79
20. II
20.42 20.73
21.05
21.36
21.68
7
21.99
22.31
22.62
22.93
23.25
23.56 23.88
24.19
24-50
24.82
8
25-13
25-45
25-76
26.08
26.39
26.70
27.02
27-33
27-65
27.96
9
28.27
28.59
28.90
29. 22
29-53
29-85
30.16
30-47
30-79
31.10
lO
31-42
31-73
32-04
32.36
32.67
32-99
33.30
33-62
33-93
34-24
II
34-56
34.87
35-19
35-50
35-81
36-13
36.44
36.76
37-07
37.38
12
37-70
38-01
38-33
38.64
38-96
39.27 39.58
39-90
40. 21
40.53
13
40.84
41-15
41.47
41.78
42. 10
42.41
42.73
43-04
43.35
43-67
14
43-98
44 30
44.61
44.92
45-24
45-55
45-87
46.18
46.50
46.81
15
47.12
47-44
47-75
48.07
48.38
48.69
49.01
49.32
49.64
49-95
i6
50-27
50.58I 50.89
51 . 21
51-52
51-84
52.15
52.46
52.78
53-09
17
53-41
53-72
54-04
54.35 54-66
54-98
55-29
55 -61
55-92
56-23
iS
56-55
56.86
57.18
57.49I57.81
58.12
58.43 158.75
59.06
59-38
19
59-69
60.00 60.32
60.63 60.95
61.26
61. 58 61.89
62 . 20
62. 52
18. Circumferences of Circles
Diameters in Units and Eighth
s
d
1/8 1/4 3/8 1/2 5/8 3/4 7/8
0. 0000
0.3927
0.7854
1.17S1
1.5708
1-9635
2.3562
2.7489
I
3-1416
3-5343
3.9270
4-3197
4.7124
5-1051
5-4978
5-8905
2
6.2832
6.6759
7.0686
7-4613
7-8540
8.2467
8.6394
9.0321
3
9.4248
9-8175
10. 210
10.603
10. 996
11.388
II. 781
12.174
4
12.566
12-959
13-352
13-744
14-137
14-530
14.923
15-315
5
15.708
16. lOI
16.493
16.886
17.279
17.671
18.064
18-457
6
18.850
19.242
19.635
20.028
20.420
20.813
21.206
21.598
7
21 . 991
22.384
22.777
23.169
23.562
23-955
24-347
24.740
8
25-133
25 525
25.918
26.311
26. 704
27.096
27-489
27.882
9
28.274
28.667
29.060
29-452
29-S45
30-23S
30-631
31.023
10
31-416
31.809
32.201
32-594
32.987
?>3 ■ 379
33-772
34.165
II
34-558
34-950
35-343
35-736
36.128
36-521
36-914
37 306
12
37-699
38-092
38-485
38-877
39-270
39 663
40-055
40.448
13
40.841
41.233
41 . 626
42.019
42.412
42. 804
43-197
43-590
14
43-982
44-375
44.768
45. 160
45-553
45-946
46-338
46.731
IS
47.124
47-517
47-909
48.302
48-695
49.087
49.480
49-873
16
50-265
50.658
51-051
51-444
51-836
52.229
52. 622
53-014
17
53-407
53 -800
54-192
54-585
54-978
55-371
55-763
56-156
18
56-549
56-941
57-334
57-727
58.119
58-512
58-905
59-298
19
59.690
60.083
60.476
60.868
61.261
61.654
62.046
62.439
32
Circles axd Spheres
19. Circular
Central
Length
Rise
Area
Central
Length
Rise
Area
Angle
of
of
of
Angle
of
of
of
Degrees
Chord
Arc
Segment
.Degrees
Chord
Arc
Segment
I
0.0175
. 0000
0.00000
46
0.7815
0.0795
0.04176
3
0-0349
0.0002
0. 00000
47
0.7975
0.0829
0.04448
3
0.0524
. 0003
0. OOOOI
48
0.8135
0.0865
0.04731
4
0.0698
0. 0006
0.00003
49
0.8294
0.0900
0.05025
5
0.0872
0. 0010
. 00006
50
0.8452
0.0937
0.05331
6
0.1047
0.0014
O.OOOIO
51
0. 8610
0.0974
0.05649
7
0. 1221
0. 0019
0. 00015
52
0.8767
0. 1012
0.05978
8
0.139s
0.0024
0. 00023
53
0.8924
0. 1051
0.06319
9
0. 1569
0.0031
0. 00032
54
0. 9080
0. 1090
0.06673
10
0.1743
0. 0038
. 00044
55
0.9235
0.1130
0.07039
n
O.I9I7
0.0046
0.00059
56
0.9389
0. 1171
0.07417
13
0. 2091
0.0055
0.00076
57
0.9543
0. I2I2
0.07808
13
0. 2264
0. 0064
0.00097
58
0.9696
0.1254
0.08212
14
0.2437
0. 0075
0.00121
59
0.9848
0. 1296
0.08629
IS
0. 2611
0.0086
0.00149
60
I . 0000
0.1340
0.09059
i6
0.2783
0.0097
0.00181
61
1.0151
0.1384
0.09502
17
0. 2956
0. 01 10
0.00217
62
I. 0301
0. 1428
0.09958
i8
0.3129
0.0123
0.00257
63
1.0450
0.1474
0. 10428
19
0.3301
0.0137
0.00302
64
I . 0598
0. 1520
0. 10911
20
0.3473
0.0152
0.00352
65
1.0746
0. 1566
0. 1 1408
31
0.3645
0.0167
. 00408
66
1.0893
0.1613
0. 11919
32
0.3816
0. 0184
0.00468
67
I. 1039
0. 1661
0.12443
23
0.3987
0.0201
0.00535
68
I. 1 1 84
0. I7IO
0. 12982
24
0.4I5S
0. 0219
0.00607
69
I. 1328
0.1759
0.1353s
25
0.4329
0.0237
0.00686
70
I. 1472
0.1808
0. 14102
26
0.4499
0.0256
0.00771
71
I . 1614
0. 1859
0. 14683
27
0. 4669
0.0276
0. 00862
72
1. 1756
0. I9IO
•0.15279
28
0.4838
0.0297
0. 00961
73
I. 1896
0. 1961
0.15889
29
0. 5008
0.0319
0.01067
74
I. 2036
0. 2014
0. 16514
30
0. 5176
0.0341
. 1 1 80
75
I. 2175
0. 2066
0.17154
31
O.S34S
0.0364
0.01301
76
I. 2313
0. 2120
0.17808
32
0.5312
0.0387
0. 01429
77
1.2450
0.2174
0.18477
33
0. 5680
0. 0412
0.01566
78
I . 2586
0. 2229
0. 19160
34
0.5847
0.0437
0.01711
79
I . 2722
0.2284
0. 19859
35
0. 6014
0.0463
0. 01864
80
1.2856
0.2340
0.20573
36
. 6 1 80
0.0489
0.02027
81
I . 2989
0. 2396
0.-2I30I
37
0. 6346
0.0517
0.02198
82
I.3121
0.2453
0. 22045
38
0. 651 1
0.0545
0.02378
83
1.3252
0.2510
0. 22804
39
0.6676
0.0574
0.02568
84
1.3383
0. 2569
0.23578
40
0.6840
0.0603
0.02767
85
1.3512
0. 2627
0.24367
41
0. 7004
0.0633
0.02976
86
I . 3640
0. 2686
0.25171
42
0.7167
0.0664
0.03195
87
1.3767
0. 2746
0. 25990
43
0.7330
0.0696
0.03425
88
I . 3893
0. 2807
0. 26825
44
0.7492
0.0728
0.03664
89
I. 401 8
0.2867
0.2767s
45
0.7654
0. 0761
0.03915
90
1.4142
0. 2929
0. 28540
Segments
Circles and Spheres
33
Central L
ength
Rise
Area
Central L
ength
Rise
Area
Angle
of
of
Angle
of
of
of
Degrees "■
I^hord
Arc
Segment
0. 29420
Degrees C
-hord
Arc
Segment
91 I
4265
0. 2991
136 I
.8544
0. 6254
0.83949
92 I
4387
0-3053
0.30316
137 I
.8608
0.633s
0.85455
93 I
4507
0.3116
0. 31226
138 I
8672
0.6416
0.86971
94 I
4627
0.3180
0.32152
139 I
8733
0.6498
0.88497
95 I
4746
0.3244
0.33093
140 I
8794
0. 6580
0.90034
96 I
4863
0.3309
0.34050
141 I
8853
0.6662
0. 91580
97 1
4979
0.3374
0.35021
142 I
8910
0.6744
0.93135
98 I
5094
0.3439
0.36008
143 I
8966
0.6827
0. 94700
99 I
5208
0.3506
0.37009
144 I
9021
0. 6910
0. 96274
100 I
532^
°-3572
0. 38026
145 I
9074
0.6993
0.97858
lOI I
5432
0.3639
0.39058
146 I
9126
0.7076
0.99449
102 I
5543
0.3707
0.40104
147 I
9176
0. 7160
I .01050
103 I
5652
0.3775
0. 41166
148 I
9225
0.7244
1.02658
104 I
5760
0.3843
0.42242
149 I
9273
0.7328
1.04275
los I
5867
0.3912
0-43333
ISO I
9319
0.7412
1.05900
106 I
5973
0.3982
0.44439
iSi I
9363
0.7496
1-07532
107 I
6077
0.4052
0. 45560
152 I
9406
0.7581
1.09171
108 I
6180
0.4122
0. 46695
153 I
9447
0. 7666
1.10818
109 I
6282
0.4193
0.47844
154 I
9487
0.7750
I. 12472
no I
6383
0.4264
0. 49008
155 I
9526
0.7836
1.14132
in I
6483
0.4336
0. 50187
156 I
9563
0. 7921
I .15799
112 I
6581
0.4408
0.51379
157 I
9598
0.8006
I. 17472
113 I
6678
0. 4481
0.52586
158 I
9633
0. 8092
1.19151
114 I
6773
0.4SS4
0.53807
159 I
9665
0.8178
1.20835
115 I
6868
0.4627
0.55041
160 I
9696
0.8264
1.22525
116 I
6961
0.4701
0.56289
161 I
9726
0.8350
I . 24221
117 I
7053
0.4775
0.57551
162 I
9754
0.8436
I. 25921
118 I
7143
0.4850
0.58827
163 I
9780
0. 8522
I. 27626
119 I
7233
0.4925
0. 601 16
164 I
9805
0.8608
1-29335
120 I
7321
0. 5000
0. 61418
165 I
9829
0.8695
I. 31049
121 I
7407
0. 5076
0.62734
166 I
9851
0.8781
1.32766
122 I
7492
0.5152
0.64063
167 I
9871
0.8868
1.34487
123 I
7576
0. 5228
0.65404
168 I
9890
0.8955
I. 362 1 2
124 I
7659
0-5305
0.66759
169 I
9908
0. 9042
1.37940
125 I
7740
0.5383
0.68125
170 I
9924
0.9128
I. 39671
126 I
7820
0. 5460
0.69505
171 I
9938
0.9215
I. 41404
127 I
7899
0.5538
0. 70897
172 I
9951
0.9302
I. 43140
128 I
7976
0. 5616
0.72301
173 I
9963
0.9390
1.44878
129 I
8052
0.5695
0.73716
174 I
9973
0.9477
I. 46617
130 I
8126
0.5774
0.75144
175 I
9981
0.9564
1-48359
131 I
8199
0.5853
0.76584
176 1
9988
0. 9651
1.50101
132 I
8271
0-5933
0. 78034
177 I
9993
0.9738
I-51845
133 I
8341
0.6013
0.79497
178 I
9997
0.9825
1.53589
134 I
8410
0.6093
0. 80970
179 I
9999
0.9913
1-55334
135 I
8478
0.6173
0.82454
180 2
0000
I . 0000
I. 57080
34
Circles and Spheres
20. Volumes of Spheres
Diameters in Units and Tenths
d
.0 .1 .2 .3 .4 .5 .6 .7 .8 .9
0.000
O.OOl
0.004
0.014
0.034
0.065
0.113
0. 180
0.268
0.382
I
0.524
0.697
0.905
1. 150
1-437
1.767
2.145
2.572
3-054
3-591
2
4.189
4.849
5-575
6-371
7-238
8.181
9.203
10.31
11.49
12.77
3
14.14
15.60
17. 16
18.82
20.58
22.45
24.43
26.52
28.73
31.06
4
33-51
36.09
38-79
41.63
44.60
47-71
50-97
54-36
57-91
61.60
5
65-45
69.46
73.62
77-95
82.45
87.11
91-95
96.97
102.2
107-5
6
113. 1
118. 8
124.8
130.9
137-3
143-8
150-5
157-5
164.6
172.0
7
179.6
187.4
195-4
203.7
212. 2
220.9
229.8
239.0
248.5
258.2
8
268.1
278-3
288.7
299.4
310.3
321.6
333 -o
344.8
356.8
369-1
9
381-7
394-6
407-7
421 . 2
434-9
448.9
463.2
477-9
492.8
508.0
lO
523 -6
539-5
555-6
572.2
589.0
606. 1
623.6
641.4
659.6
678.1
II
696.9
716. 1
735-6
755-5
775-7
796.3
817-3
838-6
860.3
882.3
12
904.8
927.6
950.8
974-3
998.3
1023
1047
1073
1098
1124
13
1 1 50
1177
1204
1232
1260
1288
1317
1346
1376
1406
14
1437
1468
1499
1531
1563
1596
1630
1663
1697
1732
15
1767
1803
1839
187s
1912
1950
1988
2026
2065
2105
i6
2145
2185
2226
2268
2310
2352
2395
2439
2483
2527
17
2572
2618
2664
2711
2758
2806
285 s
2903
2953
3003
i8
3054
3i°S
3157
3209
3262
3315
3369
3424
3479
3535
19
3.S9I
3648
3706
3764
3823
3882
3942
4003 4064
4126
21. Volumes of Spheres
Diameters in Units and Eighths
d
1/8
1/4
3/8
1/2
5/8
3/4 7/8
0.0000
O.OOIO
0.0082
0.0276
0.0654
O.127S
0.2209
0.3508
I
0.5236
0.74S5
1.0227
1.3612
1.7671
2.2468
2 . 8062
3-451S
2
4.1888
5.0243
S-9641
7-0144
8.1812
9.4708
10.S89
12-443
3
14-137
15-979
17.974
20.129
22.449
24.942
27.612
30.466
4
33.510
36.751
40.194
43.846
47.713
51.800
56.115
60.663
5
65-450
70.482
75-766
81.308
87.114
93.189
99-541
106.17
6
113-10
120.31
127.83
135.66
143.79
152.25
161.03
170.14
7
179-59
189.39
199-53
210.03
220.89
232.12
243-73
255-71
8
268.08
280.85
294.01
307. 58
321.56
335-95
350.77
366.02
9
381.70
397-83
414.40
431.43
448.92
466.88
485-30
504.21
10
523-60
543-48
563.86
584.74
606.13
628.04
650.47
673-42
11
696.91
720.94
745-51
770.64
796.33
82 2. 58
849.40
876.80
12
904-78
933-35
962.51
992.28
1022.7
1053-6
1085.2
1117-S
13
1150.3
1183.8
1218.0
1252.8
1288.2
1324-4
1361.2
1398.6
14
1436.8
1475.6
151S-1
1555.3
1596.3
1637-9
1680.3
1723-3
15
1767.1
1811.7
1857.0
1903.0
1949.8
1997.4
204s -7
2094.8
16
2144.7
2195-3
2246.8
2299.0
2352.1
2405.9
2460.6
2516. I
17
2572.4
2629.6
2687.6
2746. s
2806.2
2866.7
2928.2
2990. 5
18
3053-6
3117-7
3182.6
3248. 5
331S-2
3382.9
3451.5
3520.9
19
3591.4
3662.7
3735-0
3808.2
38S2.4
3957.6
4033-7
4110.7
Circles and Spheres
22. Multipliers for Finding Lengths of Circular Arcs
35
I
2
3
4
5
6
7
8
9
Degrees
Minutes
Seconds
0.017453293
0.034906585
0.052359878
0.069813170
0.087266463
0.104719755
0. 122173048
0.139626340
0.157079633
0.00029088S
0.000581776
0.000872665
0.001163553
0.001454441
0.001745329
0.002036217
0.002327106
0.002617994
0.000004848
. 000009696
0.000014544
0.000019393
0.000024241
0.000029089
0.000033937
0.000038785
0.000043633
Example.
Find length of arc for a central
angle of 48° 4/ in circle of
12 ft. radius.
40° 0.698132
8° .139626
40' .011636
5' .001454
0.85085
12
Length = 10. 210 ft
23. Explanations
Table 15 gives Areas of Circles to four places for three-place
diameters. Since the area of a circle varies as the square of its
diameter, it follows that the decimal point moves two places in
the function when it moves one place in the argument. Thus,
for diameters of 4.53 and 45.3 inches the areas of the circles are
16.12 and 1612 square inches; for a diameter of 0.453 inches the
area is 0.1612 square inches.
Table 16 gives Areas of Circles when the diameters are ex-
pressed in imits and eighths-; thus for a diameter of 22| inches,
the area is 393.20 square inches. When the diameter is given to
sixteenths the area is approximately half-way between the two
nearest tabular values; thus, for a diameter of 2^6 inches the area
is 3.34 square inches.
Tables 17 and IS give Circumferences of Circles for diameters
in tenths and eighths of units. For example, circles of 7.2 and 7$
inches in diameter have circumferences of 22.62 and 22.78 inches.
Tables 17-18 can also be used for finding a diameter when
the area or circumference is given. Examples: when the areas
50.52 and 51.34 are given the corresponding diameters are 8.02
and 8.085; when the circumferences 5.027 and 5.134 are given, the
diameters are 1.600 and 1.634.
Table 19 gives properties of Segments of a Circle of radius
unity. For any other radius r the tabular lengths of chord and
36 Circles and Spheres
rise of arc are to be multiplied by r and the tabular area by r^
For example, when the radius is 20 feet and the angle at the cen-
ter of the circle is 82°, the length of the chord of the segment is
26.242 feet, the rise of the arc is 4.906 feet, and the area of the
segment is 88.18 square feet.
Tables 20 and 21 give Volumes of Spheres for diameters in
tenths and eighths. Thus, for spheres 9.1 and 9| inches in diam-
eter the volumes are 394.G and 397.8 cubic inches.
Table 22 gives Multipliers for finding lengths of Circular Arcs
of radius unity. Example: to find the length of a railroad curve
of 700 feet radius and 60° 8' central angle; here the table gives
1.0472 for C0° and 0.0023 for 8'; adding these and multiplying by
700 gives 734.65 feet for the actual length of the curve.
24. Exercises
1. Find the areas of circles whose diameters are 3.4, 3.42, and
3.421 feet; also for diameters of 340, 342, and 342.1 feet.
2. Find the area for a circle of 19.25 inches dian.cter by inter-
polation in Table 15 and comjjare the result with that given in Table
16.
3. Find circumferences of circles 20.3 and 2.03 inches diameter;
also of circles 40.6 and 4.06 feet diameter.
4. In a circle of 12 inches diameter the measured chord of a seg-
ment was 14.44 inches. What is the chord for a radius unity?
By help of Table 19 find the central angle, the rise of the arc, and the
area of the segment.
5. For a central angle of 48° 30' find the length of chord, rise of
arc, and area of segment in a circle whose radius is 60.5 centimeters.
6. What are the volumes of spheres of 0.34, 3.4, and 34 inches?
7. A cannon ball 8 inches in diameter has a specific gravity of 7.8.
If the weight of a cubic foot of water is 62.5 pounds, what is the weight
of the cannon ball?
8. Find the length of a railroad curve having a central angle of
3° 15' and a radius of 5730 feet.
Chapter 4
NATURAL TRIGONOMETRIC FUNCTIONS
38
Trigonometric Functions
25. Nahiral Sines
/'
SINE
Angl(
; 0' 10' 20' 30' 40' 50' 60'
89
o"
0.00000
0.00291
0.00582
0.00873
0. 01164
0.01454
0.01745
I
0.01745
0.02036
0.02327
0.02618
0.02908
0.03199
0.03490
88
2
0.03490
0.03781
0.04071
0.04362
0.04653
0.04943
0.05234
87
3
0.05234
0.05524
0.05814
0.06105
0.06395
0.06685
0.06976
86
4
0.06976
0.07266
0.07556
0.07846
0.08136
0.08426
0.08716
85°
S''
0.08716
0. 09005
0.0929s
0.09585
0.09874
0. 10164
0.104S3
84
6
0. 10453
0. 10742
0. 1 1 03 1
0. 11320
0. 1 1 609
0.11898
Q. I2187
83
7
0. 12187
0. 12476
0. 12764
0.13053
0.13341
0.13629
O.13917
82
8
0.13917
0. 14205
0.14493
0. 14781
0. 15069
0.15356
0.15643
81
9
0.15643
0.15931
0. 16218
0.16505
0. 16792
0. 17078
0.17365
80°
10°
0.17365
0. 17651
0.17937
0. 18224
0. 18509
0.18795
0.19081
79
II
0. I 908 I
0. 19366
0.19652
0.19937
0. 20222
0.20507
0.20791
78
12
0. 20791
0. 21076
0.21360
0.21644
0.21928
0.22212
0.2249s
77
13
0.2249s
0.22778
0.23062
0.23345
0.23627
0. 23910
0. 24192
76
14
0.24192
0.24474
o.;?4756
0.25038
0.25320
0.25601
0.25882
75°
15°
0.25882
0.26163
0.26443
0. 26724
0. 27004
0.27284
0.27564
74
i6
0.27564
0.27843
0.28123
0. 2S402
0.286S0
0.28959
0.29237
73
17
0.29237
0.29515
0.29793
0.30071
0.30348
0.30625
0.30902
72
i8
0.30902
0.31178
0.31454
0.31730
0.32006
0.32282
0.32557
71
19
0-32557
0.32832
0.33106
0.33381
0.3365s
0.33929
0.34202
70°
20°
0.34202
0.34475
0.34748
0.35041
0.35293
0.35565
0.35837
69
21
0.35837
0.36108
0.36379
0. 36650
0.36921
0.37191
0.37461
68
22
0.37461
0.37730
0.37999
0.38268
0.38537
0.38805
0.39073
67
23
0.39073
0.39341
0.39608
0.39875
0.40142
0.40408
0.40674
66
24
0.40674
0.40939
0.41204
0.41469
0.41734
0.4199S
0.42262
65°
25°
0.42262
0.42525
0.42788
0.43051
0.43313
0. 43575
0.43837
64
26
0-43837
0.44098
0.44359
0.44620
0.44880
0.45140
0.45399
63
27
0.45399
0.45658
0.45917
0.46175
0.46433
0.46690
0.46947
62
28
0.46947
0.47204
0.47460
0.47716
0.47971
0.48226
0.48481
61
29
0.48481
0.4873s
0.48989
0.49242
0.49495
0.49748
0. 50000
60°
30°
0.50000
0.50252
0.50503
0.50754
0.51004
0.51254
0.51504
59
31
0.51504
0.51753
0. 52002
0.52250
0.52498
0.52745
0.52992
58
32
0.52992
0.53238
0.53484
0.53730
0.53975
0. 54220
0.54464
57
33
0.54464
0.54708
0.54951
0.55194
0.55436
0.55678
0.55919
56
34
0.55919
0.56160
0.56401
0. 56641
0.56880
0.57119
0.57358
55°
35°
0.57358
0.57596
0.57833
0. 58070
0.58307
0.58543
0.58779
54
36
0.58779
0. 59014
0. 59248
0.59482
0.59716
0.59949
0. 60182
.53
37
0. 60182
0. 60414
0. 60645
0.60876
0. 61 107
0.61337
0. 61566
52
38
0. 61 566
0.6179s
0. 62024
0.62251
0.62479
0.62706
0.62932
51
39
0.62932
0.63158
0.633S3
0.63608
0.63832
0.64056
0.64279
50°
40°
0.64279
0.64501
0.64723
0.64945
0.65166
0.65386
0. 65606
49
41
0.65606
0.65825
0.66044
0.66262
0.66480
0.66697
0. 66913
48
42
0.66913
0.67129
0.67344
0.67559
0.67773
0.67987
0. 68200
47
43
0. 6S200
0.6S412
0. 686 24
0.6S835
0.69046
0.69256
0. 69466
46
44
0.69466
0.6967s
0.69883
0. 70091
0. 70298
0.70505
0. 70711
45
60' 50' 40' 30' 20' 10' 0' A
.ngle
COSIMB
and Cosines
Trigonometric Functions
^INE
39
Angle
0' 10' 20' 30' 40' 50' 60'
44
45"
0.70711
0.70916
0.71121
0.71325
0.71529
0.71732
0.71934
46
0.71934
0.72136
0.72337
0.72537
0.72737
0.72937
0.73135
43
47
0-73135
0.73333
0.73531
0.73728
0.73924
0. 74120
0.74314
42
48
0.74314
0.74509
0.74703
0. 74896
0. 750S8
0. 75280
0.75471
41
49
0-75471
0.75661
0.75851
0. 76041
0. 76229
0.76417
0.76604
40°
50°
0. 76604
0.76791
0.76977
0. 77162
0.77347
0.77531
0.77715
39
51
0.77715
0.77897
0. 78079
0. 78261
0. 78442
0. 78622
0. 78801
38
52
0.78801
0.78980
0.79158
0.79335
0.79512
0.79688
0. 79864
37
S3
0. 79864
0.80038
0.80212
0.80386
0.80558
0. 80730
0.80902
36
54
0.80902
0.81072
0.81242
0.81412
0.81580
0.81748
0. 81915
35°
55°
0.81915
0.82082
0.82248
0.82413
0.82577
0.82741
0.82904
34
56
0.82904
0. 83066
0.83228
0.833S9
0.83549
0.83708
0.83867
33
57
0.83867
0.84025
0.84182
0.84339
0.84495
0.84650
0.84805
32
58
0.84805
0.84959
0.85112
0.85264
0.85416
0.85567
0.85717
31
59
0.85717
0.85866
0.S6015
0.86163
0.S6310
0.86457
0.86603
30°
60°
0.86603
0.86748
0.86892
0.87036
0.87178
0.87321
0.87462
29
61
0.87462
0.87603
0.87743
0.87882
0. SS020
0.88158
0.88295
28
62
0.88295
0.8S431
0.88566
0.88701
0.88S35
0.8896S
o.Sgioi
27
63
0.89101
0.89232
0.89363
0.89493
0.S9623
0.89752
0.89879
26
64
0.89879
0.90007
0-90133
0.90259
0.90383
0.90507
0.90631
25°
65°
0.90631
0.90753
0.90875
0. 90996
0. 91116
0.91236
0.91355
24
66
0-91355
0.91472
0.91590
0. 91706
0.91822
0.91936
0.92050
23
67
0.92050
0. 92164
0.92276
0.92388
0.92499
0. 92609
0.92718
22
68
0.92718
0.92827
0.92935
0.93042
0.9314S
0.93253
0.93358
21
69
0.93358
0.93462
0.93565
0.93667
0.93769
0.93869
0.93969
20°
70°
0.93969
0.94068
0.94167
0.94264
0.94361
0. 94457
0.94552
19
71
0.94552
0.94646
0.94740
0.94832
0.94924
0.95015
0. 95106
18
72
0.95106
0.95195
0.95284
0.95372
0.95459
0.95545
0.95630
17
73
0.95630
0.95715
0.95799
0.95882
0.95964
0. 96046
0.96126
16
74
0.96126
0.96206
0.96285
0.96363
0.96440
0.96517
0.96593
15°
75°
0.96593
0.96667
0.96742
0.96815
0.96887
0.96959
0.97030
14
76
0.97030
0. 97100
0. 97169
0.97237
0.97304
0.97371
0.97437
13
77
0-97437
0.97502
0.97566
0.97630
0.97692
0.97754
0.97815
12
78
0.97815
0.97875
0.97934
0.97992
0.98050
0.98107
0.98163
II
79
0.98163
0.98218
0.98272
0.98325
0.98378
0.98430
0.98481
10°
80°
0.98481
0.98531
0.98580
0.98629
0.98676
0.98723
0.98769
9
81
0.98769
0.98814
0.98858
0.98902
0.98944
0.98986
0.99027
8
82
0. 99027
0.99067
0. 99106
0.99144
0. 99182
0.99219
0.99255
7
83
0.99255
0. 99290
0.99324
0.99357
0.99390
0. 99421
0.99452
6
84
0.99452
0. 99482
0.99511
0.99540
0.99567
0.99594
0.99619
S°
85°
0. 99619
0.99644
0.99668
0.99692
0.99714
0.99736
0.99756
4
86
0.99756
0.99776
0.99795
0.99813
0.99831
0.99847
0.99863
3
87
0.99863
0.99878
0.99892
0.99905
0.99917
0.99929
0.99939
2
88
0.99939
0.99949
0.999 58
0. 99966
0.99973
0.99979
0.99985
I
89
0.99985
0.99989
0.99993
0.99996
0.99998
I . 00000
I . 00000
0°
60' 50' 40' 30' 20' 10' 0' A
ingle
COS£N£
40
Trigonometric Functions
26.
Natural Tangents
( -.
TANGENT
Angle o' lo' 20' 30' 40' 50' 60'
89
0°
0.00000
0.00291
0.00582I0. 00873
0. 01164
0.0145s
0.01746
I
0.01746
0.02036
0.02328I
0.02619
0.02910
0.03201
0.03492
88
3
0.03492
0.03783
0.04075
0. 04366
0.04658
0.04949
0.05241
87
3
0.05241
0.05533
0.05824
0.061 16
0.06408
0.06700
0.06993
86
4
0.06993
0.07285
0.07578
0.07870
0.08163
0.08456
0.08749
85°
5°
0.08749
0.09042
0.09335
0.09629
0.09923
0. 10216
0. 10510
84
6
0. 10510
0. 10805
0. 11099
0.11394
0.11688
0.1 1983
0.12278
83
7
0. 12278
0.12574
0. 12869
0.13165
0.13461
0.13758
0.14054
82
8
0. 14054
0.1435 1
0. 14648
0.14945
0.15243
0.15540
0.15838
81
9
0.15838
0.16137
0.16435
0.16734
0.17033
0.17333
0.17633
80°
10°
0.17633
0.17933
0.18233
0.18534
0.18835
0.19136
0.19438
79
II
0. 19438
0. 19740
0. 20042
0.20345
0.20648
0.20952
0.21256
78
12
0. 2125
0.21560
0. 21864
0. 22169
0.22475
0.22781
0.23087
77
13
0.23087
0.23393
0.23700
0. 24008
0.24316
0.24624
0.24933
76
14
0.24933
0.25242
0.25552
0. 25862
0.26172
0.26483
0.26795
75°
15°
0.26795
0.27107
0.27419
0.27732
0. 28046
0.28360
0.28675
74
16
0.28675
0. 28990
0.29305
0. 29621
0.29938
0.3025s
0.30573
73
17
0.30573
0.30S91
0.31210
0.31530
0.31850
0.32171
0.32492
72
18
0.32492
0.32814
0.33136
0.33460
0.33783
0. 34108
0.34433
7X
19
0.34433
0.34758
0.35085
0.35412
0.35740
0.36068
0.36397
70°
20°
0.36397
0.36727
0.37057
0.373S8
0.37720
0.38053
0.38386
69
21
0.38386
0.38721
0.3905s
0.39391
0.39727
0. 40065
0.40403
68
22
0.40403
0.40741
0. 41081
0.41421
0.41763
0.42105
0.42447
67
23
0.42447
0.42791
0.43136
0.43481
0.43828
0.44175
0.44523
66
24
0.44523
0.44872
0.45222
0.45573
0.45924
0.46277
0.46631
65°
25°
0. 46631
0.46985
0.47341
0.47698
0.48055
0.48414
0.48773
64
26
0.48773
0.49134
0.49495
0.49858
0.50222
0.50587
0.50953
63
27
0.50953
0.51320
0.51688
0.52057
0.52427
0.52798
0.53171
62
28
0.53171
0-53545
0.53920
0. 54296
0.54673
0.55051
0.55431
61
29
0.55431
0.55812
0.56194
0.56577
0. 56962
0.57348
0.57735
60°
30°
0.5773s
0.58124
0.58513
0.58905
0.59297
0. 59691
0.60086
59
31
0. 600S6
0.60483
0.60881
0.612S0
0.61681
0.62083
0.62487
58
32
0.62487
0.62892
0.63299
0.63707
0.64117
0.64528
0.64941
57
33
0. 64941
0.65355
0.65771
0. 661 89
0.66608
0.67028
0.67451
56
34
0.67451
0.67875
0.68301
0.68728
0.69157
0.69588
0.70021
55°
35°
0. 70021
0.70455
0. 70891
0.71329
0.71769
0. 72211
0.72654
54
36
0.72654
0.73100
0.73547
0.73996
0.74447
0. 74900
0.7535s.
53
37
0.7535s
0.75812
0.76272
0.76733
0.77196
0. 77661
0. 78129
52'
38
0.78129
0.78598
0. 79070
0.79544
0.80020
0.80498
0.80978
51
39
0.80978
0.81461
0.81946
0.82434
0.82923
0.83415
0.83910
50°
40°
0.83910
0.84407
0. ■?49o6
0.85408
0.85912
0.86419
0.86929
49
41
0.86929
0.87441
0.87955
0.88473
0.88992
0.89515
0.90040
48
42
0.90040
0.90569
0.91099
0.91633
0.92170
0.92709
0.93252
47
43
0.93252
0.93797
0.94345
0. 94896
0.95451
0.96008
0.96569
46
44
0. 96569
0.97133
0.97700
0.98270
0.98843
0.99420
1 . 00000
45°
ingle
60' 50' 40' 30' 20' 10' 0' /
COTANG£NT
Trigonometric Fdnctions
41
and Cotangents
TANGENT
Angle
; 0' 10' 20' 30' 40' 50' 60'
45"!
I. 00000'
1.00583
1.01170
1.01761
1-02355
1 .02952
1-03553
44
46 11.03553
I. 04158
I .04766
1.05378
1.05994
1 .06613
1.07237
43
47 1.07237
1.07S64
I .0S496 ,
1.09131,
1.09770
1 . 10414
1 . 11061
42
48
I. iio6i
1.11713
I. 12369
1.13029
1.13694
1-14363
1-15037
41
49
1.15037
I-15715
I. 16398
I. 17085
1.17777
1.18474
1-19175
40°
50° I. "^175
I. 19882
1.20593
1 . 21310
1.22031
1.22758
1-23490
39
51 4.23490
1.24227
I . 24969
1.25717
1.26471
1.27230
1-27994
38
52 1.27994
1.28764
I. 29541
1-30323
1.31110
1.31904
1-32704
37
53 1.32704
I. 3351 1
1.34323
I-35142
1.35968
1 . 36800
1-37638
36
54
1.37638
1.38484
1.39336
I. 40195
I .41061
1.41934
I. 42815
35°
55°ii.428is
1.43703
1.44598
I. 45501
I . 46411
1-47330
1.48256
34
56
1.48256
1.49190
I. 50133
1.510S4
1.52043
1.53010
1.53987
33
57
1.53987
1.54972
1.55966
1.56969
1-57981
I . 59002
1 . 60033
32
58
I . 60033
I. 61074
I. 62125
I. 63185
1.64256
1-65337
1.66428
31
59
1.66428
1.67530
1.68643
1 . 69766
I. 70901
1.72047
1.73205
30°
60°
1.73205
1.74375
1.75556
r. 76749
1.77955
I. 79174
I . 80405
29
61
1.80405
I . 81649
1.82906
I. 84177
1.85462
1.86760
1.88073
28
62
1.8S073
I . 89400
I. 90741
1.9209S
1.93470
1.94858
1.96261
27
63
I .96261
1.97680
I .99116
2.00569
2. 02039
2.03526
2.05030
26
64
2.05030
2.06553
2.08094
2.09654
2. 11233
2. 12832
2.14451
25°
65°
2.14451
2. 16090
2.17749
2.19430
2.21132
2.22857
2. 24604
24
66
2.24604
2.26374
2. 28167
2 . 29984
2.31826
2.33693
2.35585
23
67
68
2.35585
2.47509
2.37504
2.49597_
2.39449
2.5171S
2.41421
2.53865
2.43422
2.45451
2.47509
2.60509
22
21
2. 56046
2.58261
69
2.60509
2.62791
2.65109
2.67462
2.69853
2.72281
2.74748
20°
70°
2.747.4' '2.77254
2.79802
2.82391
2.85023
2.87700
2.90421
19
71
2.90^:- J. 93189
2.96004
2.98869
3-01783
3.04749
3.0776S
18
72
3-0776'
;i . 10842
3-13972
3-17159
3.20406
3-23714
3-27085
17
73
3-2708;
3.30521
3-34023
3-37594
3-41236
3-44951
3-48741
16
74
3-48741
3.52609
3-56557
3.60588
3.64705
3.68909
3-73205
15°
75°
3-73205
3-77595
3-82083
3.86671
3-91364
3-96165
4.01078
14
76
4.01078
4.06107
4. 11256
4.16530
4-21933
4.27471
4.33148
13
77
4.33148
4.38969
4.44942
4.51071
4-57363
4.63825
4.70463
12
78
4.70463
4.77286
4.84300
4.91516
4.98940
5.06584
5-14455
II
79
5-14455
5.22566
5-30928
5-39552
5-48451
S-57638
5.67128
10°
80°
5.67128
5. 76937
5.87080
5-97576
6.08444
6.19703
6.31375
9
81
6.31375
6.43484
6.56055
6. 69116
6.82694
6.96823
7.11537
8
82
7. "537
7.26873
7.42871
7-59575
7-77035
7-95302
8.14435
7
83
84
8.14435
9-51436
8.34496
9.78817
8.55555
10.0780
8.77689
10.3854
9.00983
10.7119
9-25530
9-51436
6
5°
11.0594
11.4301
8S°
II. 4301
11.8262
12.2505
12.7062
13.1969
13.7267
14.3007
4
86
14.3007
14.9244
15.6048
16.3499
17.1693
18.0750
19.0811
3
87
19.0811
20. 2056
21.4704
22.9038
24.5418
26.4316
28.6363
2
88
28.6363 31.2416
34.3678
38.1885
42,9641
49.1039
57.2900
I
69
57.2900 OS. 7501
85.9398
,114.589
171.885
343- 774 1 ~
0°
60' 50' 40' 30' 20' 10' 0' ;
^ngle
COTANGENT
42
Trigonometric Functions
27. Natural Trigonometric Functions
Angle
Arc
Sin
Tan
Sec
Cosec
Cot
Cos
1°
0.017s
0.017s
0.0175
I . ooos
57.299
57.290
0.9998
1-5533
89
2 0.0349
0.0349
0.0349
I .0006
28. 654
28.636
0.9994
1-5359
88
3
1
0.0524
0.0523
0.0524
I . 0014
19.107
19. 081
0. 99S6
1.5184
87
4
0.0698
0.0698
0.0699
I .0024
14.336
14.301
0.9976
1.5010
86
5
0.0873
0.0872
0.0875
1.0038
11.474
11.430
0. 9962
1-4835
8s°
6°
0. 1047
0.1045
0. 1051
1.0055
9.5668
9.5144
0.9945
I. 4661
84
7
0. 1222
0. 1219
0. 1228
1.0075
8.205s
8.1443
0.9925
1 . 4486
83
8
0.1396
0.1392
0. 1405
I .0098
7.1853
7.1154
0.9903
1.4312
8a
9
0.1571
0. 1564
0. 1584
I. 0125
6.3925
6.3138
0.9877
1.4137
81
10
0.1745
0.1736
0.1763
I. 0154
5.7588
5.6713
0.9848
1-3963
80°
11°
0. 1920
0. 1908
0.1944
I. 0187
5. 2408
5.1446
0.9816
1.3788
79
12
0.2094
0.2079
0. 2126
1.0223
4.8097
4.7046
0.9781
1.3614
78
13
0. 2269
0. 2250
0.2309
1.0263
4.4454
4.331s
0.9744
1.3439
77
14
0.2443
0.2419
0.2493
1.0306
4.1336
4.0108
0.9703
1.3265
76
15
0.2618
0.2588
0.2679
1-0353
3 8637
3.7321
0.9659
1 . 3090
75°
16°
0.2793
0.2756
0.2867
I . 0403
3.6280
3.4874
0.9613
I. 2915
74
17
0. 2967
0.2924
0.3057
1.0457
3.4203
3.2709
0.9563
1.2741
73
18
0.3142
0.3090
0.3249
1. 05 1 5
3.2361
3.0777
0.9511
1.2566
72
19
0.3316
0.3256
0.3443
1.0576
3.0716
2.9042
0.9455
1.239?
71
20
0.3491
0.3420
0.3640
1.0642
2.9238
2.7475
0.9397
1. 2217
70°
21°
0.3665
0.3584
0.3839
1.0711
2.7904
2.6051
0.9336
I . 2043
69
22
0.3840
0.3746
0. 4040
1.0785
2.669s 2.4751
0.9272
1.1868
68
23
0.4014
0.3907
0.4245
1.0864
2.5593,2.3559
0.9205
1.1694
67
34
0.4189
0.4067
0.4452
1.0946
2.4586
2. 2460
0.9135
1-1519
66
25
0.4363
0.4226
0.4663
I. 1034
2.3662
2. 1445
0.9063
1-1345
65°
26°
0.4538
0.4384
0.4877
I. 1126
2.2812
2.0503
0.8988
1. 1 1 70
64
27
0.4712
0.4540
0.5095
I. 1223
2. 2027
1.9626
0. 8910
1.0996
63
28
0.4887
0.4695
0.5317
1.1326
2.1301
1.8807
0.8829
1.0821
6a
29
0. 5061
0.4848
0.5543
I. 1434
2.0627
I . 8040
0.8746
1.0647
61
30
0.5236
0. 5000
0.5774
1.1547
2 . 0000
1.7321
0.8660
1.0472
60°
31°
0.5411
0.5150
0,6009
I. 1666
I. 9416
I . 6643
0.8572
1.0297
59
32
0.5585
0.5299
0.6249
1.1792
I. 8871
I . 6003
0.8480
1:0123
58
33
0. 5760
0.5446
0.6494
1-1924
I. 8361
1.5399
0.8387
0.9948
57
34
0.5934
0.5592
0.6745
I . 2062
1.7883
1.4826
0. 8290
0.9774
56
35
0.6109
0.5736
0.7002
1.2208
1-7434
I. 4281
0.8192
0.9599
55°
36°
0.6283
O.587S
0.7265
I. 2361
1.7013
1.3764
0.8090
0.9425
54
37
0.6458
0.6018
0.7536
1.2521
I. 6616
1.3270
0.7986
0.9250
53
38
0.6632
0.6157
0.7813
I. 2690
1.6243
I. 2799
0.7880
0. 9076
52
39
0.6807
0.6293
0.8098
I . 2868
I . 5890
I . 2349
0.7771
0.S901
51
40
0.6981
0.6428
0.S391
1.3054
1-5557
I. 1918
0.7660
0.8727
50°
41°
0.7156
0.6561
0.8693
1.3250
1-5243
1.1504
0.7547
0.8552
49
42
0.7330
0.6691
0. 9004
1.3456
1.4945
1. 1106
0.7431
0.8378
48
43
0.7505
0.6820
0.9325
1.3673
1 . 4663
1.0724
0.7314
0.8203
47
44
0.7679
0.6947
0.9657
1.3902
I . 4396
1.0355
0.7193
0.8029
46
45
0.7854
0.7071
I .0000
I. 4142
1-4142
I . 0000
0.7071
0.7854
45°
Cos
Cot
Cosec
Sec
Tan
Sin
Arc
Angle
Trigonometric Functions 7"% 43
28. Explanations
Table 25 gives Natural Sines and Cosines of angles for every
10 minutes from 0° 0' to 90° 0'. When the sine is sought, the angle,
or argument, is to be looked for at the left-hand side and at the
top of the page; when the cosine is sought, the angle is to looked
for at the right-hand side and at the foot of the page. Thus the
sine of 64° 50' is 0.90507, but the cosine of 64° 50' is 0.42525.
Again, the number 0.36108 is seen to be the sine of 21° 10' or the
cosine of 68° 50'.
Table 26, which is arranged like table 25, gives Natural Tan-
gents and Cotangents of angles.
Interpolation in these tables can be made for a given angle like
13° 27' as explained in Art. 3, but the last figure of the function may
be sometimes one unit in error for the sine and cosine, and more
than one unit for a tangent of an angle greater than 60° or for a
cotangent of an angle less than 30°. For example the table gives
sin 14° 12' =0.24530 and cot 14° 12' =3.95205, the former being
in error one unit in the last place and the latter nine units.
Table 27 gives all common Trigonometric Functions to four
places. Here Arc is the length of the arc of the angle in a circle
of radius unity; thus arc 25° =0.4363, as may be otherwise found
from Table 22. The secant is the reciprocal of the cosine and the
cosecant of the sine. Interpolation need rarely be made in this
table. When the angle is less than 45° look for it at the left-
hand side of the table and for the name of the functions at the top;
for angles between 45° and 90° look for the angle at right-hand
side and for the name of the function at the foot. Thus, sin 41°
= 0.6561, cos 50° =0.6428, sec 75° =3.8637.
Inverse Interpolation is the process of finding an argument
from a given value of a function. If the sine be given as 0.70916,
the corresponding angle is seen from Table 25 to be 45° 10', and
here no interpolation is necessary. But let the sine 0.70987
be given, then the angle is seen to lie between 45° 10' and 45° 20';
the difference of the sines of these angles is 0.00205, hence the
44 Trigonometric Functions
difference for 1' is 0.000205; now the given sine is greater than the
sine of 45° 10' by 0.00071, then 71/20.7 =3.5, so that the required
angle is 45° 13. '5. It is important to note whether or not the
values of the function increase with the argument; thus, if the
cosine 0.94698 be given, the angle is seen to be less than 18° 50' and
more than 18° 40', so that the computed difference is to be sub-
tracted; here the angle will be found to be 18° 47' closely.
29. Exercises
1. Find the values of the following functions to five decimal places:
sin 25° 20'= cos 25° 20' =
sin 85° 40'= cos 85° 40' =
sin 77° 34'= cos 77° 34' =
2. Find the angle whose sine is 0.39700. Also the angle whose
tangent is 1.24312.
3. Find the values of the following functions to four decimal places:
sin 30° = cos 30° = tan 30° =
sin 60° = cos 60° = tan 60° =
sec 30° = cosec 60° = arc 60° =
4. Multiply the tangent of 11° 20' by the cotangent of the
same angle.
5. Find sin 45° and cos 45° by Table 27, and then multiply them
together.
6. Find the sine and cosine of 17°, square each by help of Table 7,
and then add these squares.
7. Find the value of arc 78° by Table 22 and also by Table 27.
8. Find the values of the following functions to five decimal places:
cos 32° 33'= cot 32° 33' =
sin 57° 27'= tan 57° 27' =
cot 40° 15'= cot 49° 45' =
0. Test the equation cos- — sin- 9 = cos 20 by assuming'a value of
e, taking the functions from Table 27, and the squares from Table 7.
10. A vertical post 3.64 foot high casts a shadow 10.0 feet long
on level ground. IIow high is the sun above the hori/on?
11. Find thefangles whose sines ^re 0.5000, 0.8660, and 0.9979;
also the angles whose tangents are 0.1, 0.3, 0.5, 0.7, and 0.9; also
the angles whose tangents are 1.0, 2.0, 3.0, and 4.0.
Chapter 5
LOGARITHMIC TABLES
46
Logarithmic Tables
30. Common Logarithms
«
lO
0123456789
00000
00432
00860
01284
01703
02 1 1^
02531
02938
03342
03743
II
04139
04532
04922
05308
05690
06070
^446
06819
07188
07555
12
07918
08279
08636
o§99i
09342
09691
10037
10380
10721
11059
13
"394
11727
12057
123S5
12710
13033
13354
13672
13988
14301
. 14
14613
14922
15229
15534
15836
16137
16435
16732
17026
17319
IS
17609
17898
18184
18469
18752
19033
19312
19590
19866
20140
i6
20412
20683
209'52
21219
214S4
21748
22011
22272
22531
22789
17
2304s
23300
23553
23805
2405s
24304
24551
24797
25042
25285
i8
25527
25768
26007
26245
26482
26717
26951
271S4
27416
27646
19
27875
28103
28330
28556
28780
29003
29226
29447
29667
29885
20
30103
30320
30535
30750
30963
31175
31387
31597
31806
32015
21
32222
32428
32634
3283S
33041
33244
33445
33646
33846
34044
22
34242
34439
3463 s
34830
35025
35218
3S4II
35603
35793
359S4
33
36173
36361
36549
36736
36922
37107
37291
37475
37658
37840
24
38021
38202
38382
38561
38739
38917
39094
39270
39445
39620
25
39794
39967
40140
40312
40483
40654
40824
40993
41162
41330
26
41497
41664
41830
41996
42160
42325
42488
42651
42813
42975
27
43136
43297
43457
43616
43775
43933
44091
44248
44404
44560
28
44716
44871
45025
45179
45332
45484
45637
45788
45939
46090
29
46240
46389
46538
46687
46S35
46982
47129
47276
47422
47567
30
47712
47857
48001
48144
48287
48430
4S572
48714
4885s
48996
31
49136
49276
49415
49554
49693
49S31
49969
50106
50243
50379
32
50515
50651
50786
50920
5105s
51188
51322
5145s
51587
51720
33
51851
51983
52114
52244
52375
52504
52634
52763
52892
53020
34
53148
53275
53403
53529
53656
53782
53908
54033
54158
54283
35
54407
54531
54654
54777
54900
55023
55145
55267
55388
55509
36
55630
55751
55871
55991
56110
56229
56348
56467
56585
56703
37
56820
56937
57054
57171
57287
57403
57519
57634
57749
57S64
38
57978
58092
58206
58320
58433
58546
58659
58771
58883
58995
39
59106
59218
59329
59439
59550
59660
59770
59879
5998S
60097
40
60206
60314
60423
60531
60638
60746
60853
60959
61066
61172
41
61278
613S4
61490
61595
61700
61805
61909
62014
62118
62221
42
62325
62428
62531
62634
62737
62839
62941
63043
63144
63246
43
63347
63448
63548
63649
63749
63849
63949
6404S
64147
64246
44
64345
64444
64542
64640
64738
64836
64933
65031
65128
65225
45
65321
65418
65514
65610
65706
65801
65S96
65992
66087
661S1
46
66276
66370
66464
66558
66652
66745
66839
66932
67025
67117
47
67210
67302
67394
67486
67578
67669
67761
67852
67943
68034
48
68124
68215
6S305
68395
68485
68574
68664
68753
68842
68931
49
69020
69108
69197
69285
693*73
69461
69548
69636
69723
69810
50
6989^
69984
70070
70157
70243
70329
70415
70501
70586
70672
51
70757
70842
70927
71012
71096
7 II 81
71265
71349
71433
71517
52
71600
71684
71767
71850
71933
72016
72099
72x81
72263
72346
53
72428
72509
72591
72673
72754
72835
72916
72997
73078
73159
54
73239
73320
73400
73480
73560
73640
73719
73799
73878
73957
0123456789
Logarithmic Tables
of Numbers from 000 to 999
47
n
01 2 34 56789J
55
74036
74115
74194
74273
743SI
74429
74507
74586
74663
74741
56
74819
74896
74974
75051
75128
75205
75282
75358
75435
755"
57
75^87
75664
7 5 740
75815
75891
75967
76042
76118
76193
76268
58
76343
7641S
76492
76567
76641
76716
76790
76S64
76938
77012
59
77085
77159
77232
77305
77379
77452
77525,
77597
77670
77743
6o
77815
77887
77960
78032
78104
78176
78247
78319
78390
78462
6i
78533
78604
78675
78746
78817
78888
78958
79029
79099
79169
62
79239
79309
79379
79449
79518
7958S
79657
79727
79796
79865
63
79934
80003
S0072
80140
80209
80277
80346
80414
80482
8oS5'ii
64
80618
S06S6
80754
80821
80889
80956
81023
81090
81158
8122J;
65
812^1
81358
81425
81491
S1558
81624
81690
81757
81823
81889
66
81954
82020
82086
82151
82217
82282
82347
82413
82478
82543
67
82607
82672
82737
82802
82866
82930
82995
83059
83123
83187
68
83251
83315
83378
83442
83506
83569
83632
83696
83759
S3822
69
83885
83948
8401 1
84073
84136
84198
84261
84323
84386
84448
70
84510
84572
84634
84696
84757
84819
84880
84942
85003
85065
71
85126
85187
85248
85309
85370
85431
85491
85552
85612
S5673
72
85733
85794
85854
85914
85974
86034
86094
86153
86213
86273
73
86332
86392
86451
86510
86570
86629
86688
86747
86806
86864
74
86923
86982
87040
87099
87157
87216
87274
87332
87390
87448
75
87506
87564
87622
87679
87737
87795
87852
87910
87967
88024
76
88081
88138
88195
88252
88309
88366
88423
88480
88536
88593
77
88649
88705
88762
88818
88874
88930
88986
89042
89098
89154
78
89209
89265
89321
89376
89432
89487
S9542
89597
89653
89708
79
89763
89818
89873
89927
89982
90037
90091
90146
90200
90255
80
90309
90363
90417
90472
90526
90580
90634
90687
90741
9079s
81
90S49
90903
90956
91009
91062
91116
91169
91222
91275
91328
82
91381
91434
91487
91540
91593
91645
91698
91751
91803
91855
83
91908
91960
92012
92065
92117
92169
92221
92273
92324
92376
84
92428
92480
92531
92583
92634
92686
92737
92788
92840
92891
85
92942
92993
93044
93095
93146
93197
93247
93298
93349
93399
86
93450
93500
93551
93601
93651
93702
93752
93802
93852
93902
87
93952
94002
94052
94101
941,51
94201
94250
94300
94349
94399
88
9444S
94498
94547
94596
94645
94694
94743
94792
94841
94S90
89
94939
94988
95036
95085
95134
95182
95231
95279
95328
95376
90
95424
95472
95521
95569
95617
95665
95713
95761
95809
95856
91
95904
95952
95999
96047
9609s
96142
96190
96237
96284
96332
92
96379
96426
96473
96520
96567
96614
96661
96708
96755
96S02
93
96848
96895
96942
96988
97035
97081
97128
97174
97220
97267
94
97313
97359
97405
97451
97497
97543
97589
97635
97681
97727
95
97772
97818
97864
97909
97955
98000
98046
98091
98137
98182
96
98227
98272
98318
98363
98408
98453
98498
98543
98588
98632
97
98677
98722
98767
98811
98856
98900
98945
98989
99034
99078
98
99123
99167
99211
99255
99300
99344
99388
99432
99476
99520
99
99564
99607
99651
99695
99739
99782
99826
99870
99913
99957
01 234 56 f 89
J
48
Logarithmic Tables
31. Common Logarithms
tOG SINE
Angle o' lo' 20'
30' 40'
50'
60'
0°
. — 00
3-46373
3-76475
3.94084
2.06578
2.16268
2. 24186
89
I
2.24186
2.30879
2.36678
2.41792
2 .46366
2.50504
2.54282
88
2
3
2.54282
2.71880
2-57757
2.74226
5.60973
2.76451
2.63968
2.78568
2 . 66769
2.80585
2 . 69400
2.82513
2.71880
87
2.84358
86
4
2.84358
2.86128
2.87829
2.89464 »2.9i040
2.92561
2.94030
85°
84
83
S
6
7
2.94030
T. 01923
T.0S589
2.95450
I. 03109
1.09606
2.96825
T. 04262
T. 10599
2.98157 -
2 • 994J0
1.00704
T. 07548
I. 01923
1.08589
1.05386
1.11570
1. 0648 1
1. 12519
I. 13447
I- 14356
82
8
9
I. 14356
1 .19433:
1-15245
1.20223
I . 16116
T. 20999
I . 16970
I . 21761
T. 17807
1.22509
I. 18628
I -19433
81
80°
1.23244
1.23967
10°
II
12
T. 23967
1.28060
T. 31788
T. 24677
1.28705
1-32378
1-25376
1.29340
T. 26063
I . 29966
X 26739
T. 27405
T. 28060
79
78
77
1.30582
'T. 34100
I. 31189
T- 34658
I. 31788
r-35209
1.32960
1-33534
13
1.35209
1-35752
1.36289
T. 36819
I. 37341
1-37858
1.38368
76
14
T. 38368
T. 38871
1.39369
I.39S60
1.40346
1.40825
1.41300
75°
15°
T. 41300
T. 41768
1.42232
I . 42690
1-43143
1-43591
1.44034
74
16
1.44034
1.44472
1.44905
1-45334
1-45758
1.4617S
1.46594
73
17
T. 46594
1.47005
1.47411
I. 47814
I. 48213
T. 48607
1 .48998
72
18
1.48998
1.49385
1.49768
I . 50148
1-50523
T. 50896
T. 51264
71
19
T. 51264
I. 51629
T.51991
1-52350
1-52705
1-53056
1-53405
70°
20°
1-53405
1.53751
1-54093
T- 54433
T- 54769
T. 55102
1-55433
69
21
1-55433
1-55761
1.56085
1.56408
1.56727
1-57044
1-57358
68
22
1-57358
1.57669
1-57978
1.582S4
T. 58588
T. 58889
I . 591S8
67
23
T. 59188
1.59484
1-59778
I . 60070
1.60359
I . 60646
T. 60931
66
24
I. 6093 I
1.61214
I. 61494
T. 61773
1.62049
1.62323
1-62595
65°
25°
1-62595
1.62865
I -63133
T. 63398
T. 63662
T. 63924
T. 64184
64
26
I. 64184
1.64442
T. 64698
1-64953
1.65205
1.65456
1-65705
63
37
1-65705
1-65952
I. 66197
I . 66441
1.66682
I .66922
1.67161
62
28
1.67161
T.6739S
1-67633
1.67866
1.68098
T. 68328
1-68557
61
29
1-68557
1.68784
I . 69010
T. 69234
1.69456
1.69677
1.69897
60°
30°
T. 69897
1.70115
1-70332
1-70547
I. 70761
T. 70973
1.71184
59
31
T. 71184
1-71393
I. 71602
I . 71809
T. 72014
T. 72218
1.72421
58
32
T. 72421
1.72622
1.72823
T. 73022
T. 73219
T. 73416
T. 73611
57
33
1.73611
1-73805
1-73997
I. 74189
1.74379
1.74568
1.74756
56
34
1-74756
I - 74943
I. 75128
1-75313
1.75496
1.75678
1.75859
55°
35°
1-75859
T. 76039
T. 76218
1-76395
T. 76572
1.76747
1.76922
54
36
1.76922
1.77095
1.77268
1-77439
1.77609
I- 77778
1-77946
53
37
T. 77946
I. 78113
I. 78280
I ■ 78445
I. 78609
1.78772
1-78934
52
38
1-78934
T. 79095
T. 79256
1-79415
1.79573
1-79731
1.79887
■51
39
1.79887
I . 80043
T. 80197
1-80351
1.80504
T. 80656
T. 80807
50°
40°
1.80807
1.80957
T. 81106
T. 81254
T. 81402
T. 81549
I . 81694
49
41
T. 81694
T. 81839
I. 81983
I . 82126
T. 82269
T. 82410
1-82551
48
42
1-82551
T. 82691
1.82830
T.8296S
T. 83106
T. 83242
1-83378
47
43
1-83378
I -83513
T. 83648
1.83781
T. 83914
1 . 84046
I. 84177
46
44
I. 84177
1.84308
1.84437
T. 84566
T. 84694
1.84822
1.84949
45°
60' 50' 4°'
30' 20'
10'
0' 1
\nglc
I.OG COSIMi:
11
90--»"
Logarithmic Tables
49
of Sines and Cosines
LOG SINE
Angl(
45"
i 0' 10' 20' 30' 40' 50' 60'
1.84949
1.85074
I . 85200
1-85324
T. 85448
1-85571
1-85693
44
46
1.85693
I. 85815
1-^5936
T. 86056
T. 86176
1.86295
1.86413
43
47
I. 86413
1.86530
T. 86647
1.86763
T. 86879
T. 86993
T.S7107
42
48
f. 87107
I. 87221
1-87334
1.87446
1-87557
1.87668
T. 87778
41
49
1.87778
1.87887
1.87996
I. 88105
I. 88212
1.88319
T. 88425
40°
50°
1.88425
T. 88531
1.88636
T. 88741
1.88844
T. 88948
I. 89050
39
51
1.89050
I. 89152
T. 89254
T- 89354
1-89455
1-S9554
1.89653
38
52
1-89653
1.89752
T. 89849
1.89947
I . 90043
1.90139
1-90235
37
53
1-90235
1-90330
1.90424
I . 9051S
I . 90611
1.90704
1.90796
36
54
1.90796
T. 90887
1.90978
I. 91069
1.91158
I. 91248
1-91336
35°
55°
1-91336
I. 91425
T . 9 1 5 1 2
I -91599
T. 91686
1.91772
1-91857
34
56
1-91857
I . 91942
I . 92027
1 . 921 1 1
T. 92194
1.92277
1-92359
33
57
1-92359
T. 92441
I . 92522
T. 92603
1.92683
1.92763
I. 92842
32
58
1.92842
T. 92921
T. 92999
1.93077
I-93154
1.93230
1-93307
31
59
1-93307
1.93382
1-93457
1-93532
1.93606
1.93680
1-93753
30°
60°
I-937S3
T. 93826
T. 93898
1.93970
I. 9404 I
I . 94112
T. 94182
29
61
I. 94182
1.94252
1-9432-1
1.94390
T. 94458
T. 94526
1-94593
28
62
1-94593
I. 94660
1.94727
1-94793
T. 94858
T. 94923
1 . 94988
27
63
T. 94988
1-95052
1.95116
1-95179
T. 95242
1-95304
T. 95366
26
64
1-95366
1-95427
T.954S8
1-95549
1.95609
T. 95668
1-95728
25°
65°
1.95728
T. 95786
1-95844
T. 95902
T. 95960
T. 96017
1.96073
24
66
1.96073
I . 96129
I. 96 1 85
T. 96240
T. 96294
1.96349
I . 96403
23
67
I . 96403
T. 96456
T. 96509
T. 96562
T. 96614
I . 96665
1,96717
22
68
T. 96717
1.96767
T. 96818
T. 96868
T. 96917
1.96966
1.97015
21
69
I-97015
1.97063
1.97111
1-97159
1.97206
T. 97252
1.97299
20 '
70°
1,97299
1-97344
1.97390
1-97435
T. 97479
1-97523
1-97567
19
71
1-97567
I. 97610
1-97653
I . 97696
1-97738
1.97779
1.97S21
18
72
I. 97821
1.97861
1.97902
T. 97942
1.97982
1.98021
1.98060
17
73
1 . 98060
1.98098
I. 98136
I. 98174
1. 982 1 1
T. 98248
T. 98284
16
74
1.982S4
T. 98320
1-98356
I -98391
1.98426
1.98460
1.98494
15°
75°
T. 98494
1.98528
I. 98561
1-98594
1.98627
T. 98659
T. 98690
14
76
T. 98690
I. 98722
1-98753
T. 98783
I. 98813
1.98843
T. 98872
13
77
T.98S72
T. 98901
1.98930
T. 98958
T. 98986
I. 99013
T. 99040
12
78
T. 99040
1 . 99067
1.99093
T.99119
1-99145
1.99170
I-99195
II
79
I-99195
I. 99219
1-99243
1.99267
1.99290
1-99313
1-99335
10°
80°
1-99335
1-99357
T-99379
1.99400
T. 99421
T. 99442
T. 99462
9
81
1.99462
1.99482
1.99501
1.99520
1-99539
1-99557
1-99575
8
82
1-99575
1-99593
I .99610
1.99627
1.99643
1.99659
1.99675
7
83
1-99675
1.99690
1-99705
1.99720
1-99734
1.99748^
1.99761
6
84
I. 99761
T-99775
1.99787
X . 99800
1.99812
1.99823
1.99834
5°
85°
1.99834
1-99845
T. 99856
T. 99866
T. 99876
T. 99885
T. 99894
4
86
1.99894
1.99903
T.99911
I. 99919
I . 99926
1-99934
1.99940
3
87
1.99940
1.99947
1-99953
1-99959
T. 99964
1.99969
1.99974
2
88
T. 99974
1.99978
1.99982
1.99985
T. 99988
1.99991
1-99993
I
89
1-99993
1-99995
1.99997
1-99998,
1-99999
. 00000
0.00000
0°
60' 50' 40' 30' 20' 10' 0' /
"ingle
r,OG coscNs:
50
Logarithmic Tables
32. Common Logarithms
LOG TANGENT
fl- i^
Ang
e 0' 10' 20' 30' 40' 50' 60'
89
0°
— CO
3-46373
3.76476
3 • 94086
1 2.06581
2.16273
2.24192
I
2. 24192
2.30888
2.366S9
2.41807
2.46385
2-50527
2.54308
88
2
2.54308
2.57788
2 . 61009
2 . 64009
2.66816
2.69453
2.71940
87
3
2.71940
2.74292
2.76525
2. 78649
2 . 80674
2.82610
2.84464
86
4
2.84464
2.86243
2-87953
2.89598
2-91185
2.92716
2-94195
85°
5°
2-94195
2.95627
2.97013
2-98358
2 .99662
T. 00930
r. 02162
84
6
I. 02162
1-03361
I .04528
1 .05666
T. 06775
1-07858
T. 08914
83
7
T. 08914
1.09947
T. 10956
1-11943
I . 12909
1-13854
T. 14780
82
8
T. 14780
I. 15688
1-16577
I- 17450
T. 18306
1. 19146
1. 19971
81
9
J0°
T.19971
1.207S2
1.21578
T. 22361
I-23130
1.23887
T. 24632
80°
T. 24632
1-25365
T. 26086
T. 26797
1-27496
T. 28186
1.28865
79
II
T. 28865
1-29535
1-30195
1.30846
1.31489
1.32122
1-32747
78
12
1-32747
1-33365
1-33974
1-34576
1-35170
I-3S757
1-36336
77
13
1-36336
1.36909
1-37476
1-38035
1-38589
1-39136
1-39677
76
14
1-39677
T. 40212
1.40742
1.41266
1-41784
1-42297
1.42805
75°
15°
T. 42805
T- 43308
T. 43806
T. 44299
1-44787
1-45271
1-45750
74
i6
1-45750
1.46224
1.46694
I. 47160
1.47622
I . 48080
1-48534
73
17
1-48534
T. 48984
T. 49430
1.49872
1-50311
T- 50746
I. 51178
72
i8
1.51178
1.51606
1.52031
1-52452
1.52870
1-53285
1-53697
71
19
1-53697
1.54106
1-54512
1-54915
1-55315
1-55712
I. 56107
70°
20°
T. 56107
T. 56498
T. 56887
1-57274
1-57658
1-58039
T. 58418
69
21
T. 58418
1-58794
1.59168
1 • 59540
1 ■ 59909
T. 60276
I. 60641
68
22
1 . 6064 1
I . 61004
T. 61364
1 . 61722
T. 62079
1-62433
1-62785
67
23
1.62785
1-63135
T. 63484
1.63S30
1-64175
T. 645 1 7
T. 64858
66
24
T.6485S
I-65197
1-65535
T. 65870
1.66204
1-66537
T. 66867
65°
25°
T. 66867
T. 67196
1.67524
T. 67850
T. 68174
1.68497
T. 68818
64
26
T. 68818
1.69138
1-69457
1-69774
T. 70089
T. 70404
T. 70717
63
27
1.70717
I. 71028
I-71339
1.71648
1-71955
1.72262
1.72567
62
28
1.72567
T. 72872
1-73175
1-73476
1-73777
1-74077
1-74375
61
29
1-74375
1.74673
T. 74969
1.75264
1-75558
1-75852
1.76144
60°
30°
I. 76144
1-76435
T. 76725
1.77015
T- 77303
1-77591
1-77877
59
31
T. 77877
T. 78163
1. 7844S
1.78732
1.79015
1-79297
1-79579
58
32
1-79579
T. 79860
T. 80140
I . 80419
T. 80697
1.80975
I. 81252
57
33
I . 81252
T. 81528
T.S1803
T. 82078
T. 82352
T. 82626
1.82899
56
34
T. 82899
T. 83171
1-83442
1-83713:
T. 83984
T. 84254
1-S4523
55°
35°
i-f'4S23
r. 84791
1-85059
T- 85327'
1-85594
1.85860
T. 86126
54
36
T. 86126
1.86392
T. 86656
1.S6921
T. 87185
T.S7448
T. 87711
53
37
T.87711
1. 87974
1.88236
T. 88498'
T. 88759
T. 89020
T. 89281
52
38
T. 89281
1.89541
T. 89801
1.90061
1.90320
T. 90578
1-90837
51
39
1.90837
T. 91095
1-91353
T. 91610
T.9186S
T. 92125
1-92381
50°
40°
1.92381
T. 92638
T. 92894
I-93150
T. 93406
T. 93661
T. 93916
49
41
I. 93916
1.94171
T. 94426
I. 9468 1
1-94935
1 95190
1-95444
48
42
I -95444
1.95698
1-95952
1-96205;
.1-96459
1.96712
T. 96966
47
43
I . 96966
T. 97219
1.97472
1-97725
1. 97978
T. 98231
T.984S4
46
44
T. 98484
1-98737
T. 98989
1.99242 1
I -99495
1.99747
0.00000
45°
60' 50' 40' 30' 20' 10' 0' /
Ingle
LOG COTA^GJiNT
5 Z, -^ . ^;.
Logarithmic Tables
of Tangents and Cotangents
L,OG TANGENT
51
Ang
e 0' 10' 20' 30' 40' 50' 60'
45°
0.00000
0.00253
0.00505
0.00758
O.OIOI I
0. 01263
0. 01516
44
46
0.01516
0.01769
0. 02022
0.02275
0.02528
0. 02781
0.03034
43
47
0.03034
0.03288
0.03541
0.03795
0.0404S
0.04302
0.04556
42
48
0.04556
0.04810
0.05065
0.05319
0.05574
0. 05829
0.06084
41
49
0.06084
0.06339
0.06594
0. 06850
0.07106
0.07362
0.07619
40°
50°
0.07619
0.07875
0.08132
0.08390
0.08647
0.08905
0.09163
39
51
0.09163
0.09422
0.09680
0.09939
0. IOI99
0. 10459
0. 10719
38
52
0. 10719
0. 10980
0. 11241
0. 11502
0. 11764
0. 12026
0. 12289
37
53
0. 12289
0. 12552
0. 12815
0.13079
0.13344
0. 13608
0.13874
36
54
0.13874
0. 14140
0. 14406
0.14673
0. 14941
0.15209
0.15477
35°
55°
0.15477
0.15746
0. 16016
0. 16287
0. 1655S
0. 16829
0. 17101
34
56
0. 17101
0.17374
0. 17648
0. 17922
0. 18197
0. 18472
0.18748
33
57
0.18748
0. 19025
0.19303
0. 19581
0. 19860
0. 20140
. 20421
32
58
0. 20421
0.20703
2. 209S5
0.21268
0.21552
0.21837
0. 22123
31
59
0.22123
0. 22409
0. 22697
0. 229S5
0.23275
0.23565
0.23856
30°
60°
0.23S56
0.24148
0.24442
0.24736
0.25031
0.25327
0.25625
29
61
0.25625
0.25923
0. 26223
0. 26524
0.26825
0.27128
0-27433
28
62
0-27433
0.27738
0. 28045
0.2S352
0.28661
0.28972
0.29283
27
63
0.292S3
0. 29596
0.29911
0. 30226
0.30543
0.30862
0. 31 182
26
64
0. 31182
0.31503
0.31826
0.32150
0.32476
0.32804
0.33133
25°
65°
o-33^33
0.33463
0.33796
0.34130
0.34465
0.34803
0.35142
24
66
0.35142
0.35483
0.35825
0.36170
0.36516
0.36865
0-37215
23
67
0.37215
0.37567
0.37921
0.38278
0.38636
0.38996
0-39359
22
68
0-39359
0.39724
0.40091
0. 40460
0.40832
0.41206
0. 41582
21
69
0.41582
0.41961
0.42342
0.42726
0.43II3
0.43502
0.43893
20°
70°
0.43893
0.44288
0.44685
0.45085
0. 45488
0.45894
0.46303
19
71
0.46303
0.46715
0.47130
0.47548
0.47969
0.48394
0.48822
18
72
0.48822
0.49254
0.49689
0.50128
0.50570
0. 51016
0.51466
17
73
0.51466
0. 51920
0.52378
0. 52840
0.53306
0.53776
0.54250
16
74
0.54250
0.54729
0.55213
0.55701
0.56194
0. 56692
0.57195
15°
75°
0.57195
0.57703
0.58216
0.58734
0.59258
0.59788
0.60323
14
76
0.60323
0. 60864
0. 61411
0. 61965
0.62524
0.63091
0.63664
13
77
0.63664
0.64243
0. 64S30
0.65424
0.66026
0.66635
0.67253
12
78
0.67253
0.67878
0.68511
0. 69154
0.69805
0.70465
0-71135
II
79
0.7113s
0. 71814
0.72504
0.73203
0.73914
0.74635
0.75368
10°
80°
0.75368
0.76113
0.76S70
0.77639
0.7S422
0. 79218
0.80029
9
81
0.80029
0.80854
0. 81694
0. S2550
0.83423
0.84312
0.85220
8
82
0.85220
0.86146
0. 87091
0.S8057
0. 89044
0.90053
0.91086
7
83
0.91086
0.92142
0.93225
0.94334
0.95472
0.96639
0.97838
6
84
0.97838
0.99070
1.00338
I .01642
1.02987
I -043 73
1.05805
5°
85°
1.05805
1.07284
1.08S15
1. 10402
1. 12047
I. 13757
1-15536
4
86'
1-15536
1.17390
I. 19326
1.21351
1-23475
1.2570S
I . 28060
3
87
I .^28060
1.30547
I. 33184
1.35991
1.38991
1 .42212
1.45692
2
88
1.45692
r. 49473
1.53615
1.58193
I. 63311
1.69112
1.75808
I
89
1.75808
1-83727
1.93419
2.05914
2.23524
2/53627
00
°°
60' 50' 40' 30' 20' 10' 0' /
Lngle
LOG COTANGENT
J l/^
52 Logarithmic Tables
33. Logarithms of Trigonometric Functions
Angle
1°
Log Arc
2 . 2419
Log Sin
Log Tan 1 Log Sec
Log Csc
Log Cot
Log Cos
2.2419
2. 2419
0. 0001
1.7581
1.7581
1-9999
0.1913 89
3
2.5429
2.5428
2.5431
0.0003
1-4572
1.4569
1-9997
0.1864
88
3
2. 7190
2.7188
2.7194
0.0006
1.2812
1.2806
1-9994
0. 1814
87
4
2.8439
2.8436
2 . 8446 ' O.OOII
1.1564
1-1554
1 . 99S9
0.1764
86
5
2 . 9408
2 . 9403
2.9420 0.0017
1-0597
1.0580
1-9983
0.1713
85°
6°
1.0200
I. 0192
T. 0216
0.0024
0.9808
0.9784
1-9976
0. 1662
84
7
1.0870
1.0859
T. 0891
0. 0032
0.9141
0. 9109
1.9968
0.1610 83
8
1.145°
T.1436
T.1478
0. 0042
0.8564
0. 8522
1.9958
0.1557
82
9
T. 1961
I. 1943
I . 1997 ' 0. 0054
0.8057
0.8003
1.9946
0.1504
81
lO
T.2419
1-2397
1.2463
0. 0066
0. 7603
0-7537
1-9934
0.1450
80°
11°
1.2833
1.2806
T.2887
0.0081
0.7194
0.7113
1-9919
0.1395
79
12
1.3211
I-3179
I . 3275 ' 0.0096
0.6821
0.6725
1.9904
0.1340
78
13
1.3558
I-3521
T.3634
0.0113
0.6479
0.6366
T.9887
0. 1284
77
14
1.3880
1-3837
1.3968
0.0131
0.6163
0. 6032
1.9869
0. 1227
76
15
I. 4180
1.4130
1.4281
0.0151
0.5870
0.5719
1-9849
0. 1169
75°
i6°
T.4460
1.4403
1.4575
0.0172
0.5S97
0.5425
T.9828
0. iiii
74
17
1.4723
1-4659
1.4853
0.0194
0.5341
0.5147
T. 9806
0. 1052
73
i8
I. 4971
T. 4900
1.5118
0.0218
0. 5100
0.4882
1.9782
0.0992
72
19
1.5206
I. 5126
1.5370
0.0243
0.4874
0.4630
1-9757
0.0931
71
20
1.5429
I -5341
1.5611
0.0270
0.4659
0.4389
1-9730
0.087a
70°
21°
1.5641
1-5543
1.5842
0.0298
0.44S7
0.4158
T.9702
0.0807
69
22
I • 5843
1-5736
1 . 6064
0.0328
0. 4264
0.3936
1-9672
0.0744I 68
23
1 . 6036
1-5919
T.6279 0.0360
0. 4081
0.3721
I . 9640
0.0G80 67
24
1 . 6221
1.6093
1.6486
0-0393
0.3907
0.3514
I . 9607
0.0614 66
25
1.6398
1.6259
T.6687
0.0427
0.3741
0-3313
1-9573
0.0548
6s°
26°
1.6569
T.6418
T.6882
0.0463
0.3582
0.3118
T-9537
0. 0481
64
27
T.6732
1.6570
1.7072
0. 0501
0.3430
0. 292S
1-9499
0. 0412
^3
28
1.6890
1.6716
1.7257
0.0541
0.3284
0.2743
1-9459
0.0343
62
29
T.7042
1.6856
1.7438
0.0582
0.3144
0. 2562
1.9418
0.0272
61
30
1.7190
I . 6990
1.7614
0.0625
0. 3010
0.2386
I -9375
0.0200
60°
31°
1.7332
T.7118
T.7788
0.0669
0.2882
0. 2212
1-9331
0. 0127
59
32
I . 7470
1.7242
1.7958
0.0716
0.2758
0.2042
1.92S4
0.0053
58
33
T. 7604
1.-361
1.8125
0.0764
0.2639
0.1875
1.9236
1-9978
57
34
1-7734
1.7476
T. 8290
0. 0814
0.2524
0. 1710
T.91S6
I .9901
56
35
1-7859
T.7586
T.8452
0.0S66
0. 2414
0.1548
1-9134
T.9822
55°
36°
1.7982
T.7692
T.8613
0. 0920
0. 2308
0.1387
1.90S0
1.9743
54
37
1.8101
1.7795
T.8771
0.0977
0. 2205
0. 1229
1-9023
I .9G62
53
38
I. 8217
1-7893
T.8928
0.1035
0. 2107
0. 1072
1-8965
I -9579
52
39
1.8329
1-7989
T.90S4
0. 1095
0.2011
0. 0916
1.8905
1-9494
51
40
I. -8439
T.8081
1.9238
0.1157
0. 1919
0.0762
1-8843
X . 9408
50°
41°
1-8547
T.8169
1-9392
0. 1222
0.1831
0.0608
1-8778
1-9321
49
42
T.8651
1-8255
1-9544
0. 1289
0.1745
0.0456
I.8711
19231
48
43
T-8753
1-8338
1.9697
0.1359
0. 1662
0.0303
T.8641
T.9140
47
44
1.8853
T.8418
T.984S
0.1431
0. 1582
0. 0152
1-8569
I . 9046
46
45
I. 895 1
1-8495
0.0000
0-1505
0.1505
0. 0000
1-8495
1-8951
45°
Log Cos
Log Cot
Log Csc
Log Sec
Log Tan
Log Sin
Log Arc
Angle
Logarithmic Tables 53
34. Explanations
Table 30 gives five-place Logarithms of three-place numbers.
The word logarithm and its abbreviation log, when used without
qualification, refer to a common logarithm which is defined by
the equation 10'°^ "=n. The table gives the decimal part, or
mantissa, of a logarithm, while the integral part, or characteristic,
is to be supplied by the follo^\ing rules: When the number is
greater than 1 , the characteristic of its log is positive and is one
less than the number of figures preceding the decimal point; thus,
log 6.54=0.81558 log 65.4 =1.81558 log 654 =2.81858 •
When the number is less than 1, the characteristic of its log is
negative and is numerically one greater than the number of ciphers
immediately following the decimal point, thus the four-place
log of 6 is 0.77S2, and
log 0.6 =T.77S2 log 0.06 = 2.7782 log 0.006 =3.7782
Here the characteristic is negative and the mantissa is positive, so
that 2.7782 is the same as —2+0.7782. When the given number
is an integral power of 10, the mantissa is zero, so that log 1000
=3, log 0.1 = - 1, log 0.01 = -2, and log 1 =0.
Multiplication and Division may be performed by the help of
logarithms and the use of the following rules:
To multiply a by 6, log a+log b =log ab
To divide a by b, log a —log b =log a/b
Here log a and log b are obtained from Table 30 and the above rules
for the characteristic; then the numbers corresponding to log cb
and log a/b are found from the Table. For example, to mAil-
tiply 68.31 by 0.2754, the sum of the logs is 1.27444 and its corre-
sponding number is 18.812, the last decim.al being in error.
Roots and Powers of numbers are m.ost conveniently computed
by logarithms and the use of the following rules:
To extract the nth root of a, -log a =log a'*
n
To raise a to the mth power, m log a =Iog a^
For example, to raise 0.8831 to the 1.53 power: 1.53X1.83448
52 Logarithmic Tables
33. Logarithms of Trigonometric Functions
Angle
1°
'Log Arc
2. 2419
Log Sin
Log Tan 1 Log Sec
Log Csc
Log Cot
Log Cos
2.2419
1
2.2419 O.OOOI
1.7581
1.7581
1.9999
0.1913' 89
2
2.5429
2.5428
2.5431 0.0003
1-4572
1.4569
1.9997 0.1864 88
3
2.7190
2.7188
2.7194 0.0006 I. 2812
1 . 2806
1-9994
0.1814 87
4
2.8439
2.8436
2.8446 o.ooii I. 1564
1.1554
1.99S9
0.1764 86
5
2 . 9408
2 . 9403
2.9420
0. 0017
1-0597
1 .05S0
1 ■ 9983
0.1713, 85°
6°
1.0200
T.0192
I. 0216
0.0024
0.9808
0.9784
1-9976
0. 1662
84
7
T.0870
1.0859
I . 0891
0.0032
0.9141
0.9109
1.996S
0.1610' 83
8
T.1450
I. 1436
I. 1478
0.0042
0.S564
0.8522
1-995S
0.1557, 82
9
I. 1961
I. 1943
I. 1997
0. 0054
0.8057
0.8003
1.9946
0.1504
81
lO
I. 2419
1.2397
I . 2463
0.0066
0.7603
0.7537
1-9934
0. 1450
80°
11°
1.2833
1.2806
T.2887 0.0081
0.7194
0.7113
1-9919
0.1395
79
12
T.3211
I. 3179
1.3275 0.0096
0.6821
0.6725
1 . 9904
0.1340
78
13
1.3558
I. 3521
T. 3634 0.0113
0.6479
0.6366
T.9887
0. 1284
77
14
T.3880
1.3837
i.3968j0.oi3i
0. 6163
0.6032
1.9869
0. 1227
76
15
T.4180
T.4130
1.4281
0.0151
0.5870
0.5719
1.9849
0.1169J 75°
i6°
T . 4460
1.4403
1.4575
0.0172
0.5597
0.5425
1.9828
0. iiii
74
17
1-4723
1.4659
1-4853
o.oi94;o.534i
0.5147
T.9806
0. 1052
73
i8
I. 4971
T.4900
1.51181 o.o2i8|o.5ioo
0.4882
1.9782
0.0992
72
19
T. 5206
I. 5126
T-5370
0.0243 0.4874
0.4630
1.9757
0.0931
71
20
1.5429
I. 5341
1.5611
0.0270 0.4659
0.4389
1.9730
0.087QJ 70°
21°
T.5641
1.5543
1.5842
0.0298I0.4457
0.4158
T.9702
0.0807
1 69
22
15843
1.5736
I . 6064
0.0328
0.4264
0.3936
T.9672
0.0744 68
23
1 . 6036
I. 5919
T. 6279 0.0360
0.4081
0.3721
1 . 9640
0.0680 67
24
T. 6221
1.6093
T.6486
0.0393
0.3907
0.3514
T.9607
0.0614 66
25
1.6398
1.6259
1.6687
0.0427
0.3741
0.3313
1-9573
0.0548 65°
26°
1.6569
T.6418
1.6S82
0.0463
0.3582
0.3118
T-9537
0.0481 64
27
1.6732
1.6570
1.7072
0.0501
0-3430
0. 2928
1.9499
0.0412 03
28
1.6890
I. 6716
1-7257
0.0541
0.3284
0.2743
1-9459
0.0343
62
29
T.7042
T.6856
I - 7438
0.0582
0.3144
0.2562
I. 9418
0.0272
61
30
I. 7190
T.6990
I. 7614
0.0625
0.3010
0.2386
1-9375
0.0200
60°
31°
1.7332
T.7118
1.7788
0.0669
0.2882
0.2212
1-9331
0. 0127
59
32
T.7470
1.7242
1.7958
0.0716
0.2758
0.2042
1.9284
0.0053
58
33
1 . 7604
I. -361
1.8125
0.0764
0.2639
0.1875
1-9236
1.9978
57
34
1-7734
1.7476
T.8290
0.0814
0.2524
0. 1710
1.9 1 86
T.9901
56
35
1.7859
T.75S6
1.8452
0.0S66
0.2414
0. 1548
1-9134
T.9S22
55°
36°
T.7982
T.7692
I. 8613
0.0920
0.2308
0.1387
T.9080
1-9743
54
37
1.8101
1-7795
T.8771
0.0977
0.2205
0. 1229
1.9023
I .9662
53
38
T.8217
1.7893
T.8928
0.1035
0. 2107
0. 1072
1-8965
1-9579
52
39
T.8329
1-7989
1.90S4
0. 1095
0. 2011
0.0916
1 - 8905
1.9494
51
40
1.8439
1.80S1
T.9238
0-1157
0. 1919
0.0762
1.8843
1.9408
50°
41°
1-8547
T.8169
1.9392
0. 1222
0.1831
0.0608
T.877S
T.9321
49
42
T.86S1
1-8255
1.9544
0. 1289
0.1745
0.0456
T.8711
1. 923 1
48
43
7-8753
1-833S
1.9697
0.1359
0. 1662
0.0303
T.8641
I. 9 140
47
44
T-S853
T.8418
T.9848
0.1431
0. 1582
0.0152
T.8569
T.9046
46
45
I. 895 1
1-8495
. 0000
0-1505
0.1505
0.0000
1-8495
1.8951
45°
Angle
L.og Cos
-og Cot
Log Csc
Log See,
Log Tan
Log Sin
Log Arc
Logarithmic Tables 53
34. Explanations
Table 30 gives five-place Logarithms of three-place numbers.
The word logarithm and its abbreviation log, when used without
qualification, refer to a common logarithm which is defined by
the equation 10'°^ "=n. The table gives the decimal part, or
mantissa, of a logarithm, while the integral part, or characteristic,
is to be supplied by the follo^ving rules: When the number is
greater than 1, the characteristic of its log is positive and is one
less than the number of figures preceding the decimal point ; thus,
log 6.54 =0.81558 log 65.4 = L81558 log 654 =2.81858
When the number is less than 1, the characteristic of its log is
negative and is numerically one greater than the number of ciphers
immediately follo^^■ing the decimal point, thus the four-place
log of 6 is 0.7782, and
log 0.6 =T.77S2 log 0.06 = 2.7782 log 0.006 =3.7782
Here the characteristic is negative and the mantissa is positive, so
that 2.7782 is the same as —2+0.7782. When the given number
is an integral power of 10, the mantissa is zero, so that log lOCO
= 3, log 0.1= -1, log 0.01 = -2, and log 1=0.
^Multiplication and Division may be performed by the help of
logarithms and the use of the following rules :
To multiply a by b, log «+log b =log ab
To divide a bj' b, log a —log b =log a/b
Here log a and log b are obtained from Table 30 and the above rules
for the characteristic; then the numbers corresponding to log ab
and log a/b are found from the Table. For example, to m.ul-
tiply 68.31 by 0.2754, the sum of the logs is 1.27444 and its corre-
sponding number is 18.812, the last decim.al being in error.
Roots and Powers of numbers are most conveniently computed
by logarithms and the use of the following rules :
To extract the nth root of a, -log a =log a"
n
To raise a to the mth. power, m log a =log a^
For example, to raise 0.G831 to the 1.53 power: 1.53X1.83448
54 Logarithmic Tables
= -1.53+1.27675 = 2.47+1.27675=1.74675, which is log of
0.55815. To find the fifth root of 0.6831: one-fifth of I.S3448
is i (-5+4.83448) =1.96690, which is log of 0.9262; or it is per-
haps better to multiply bj' 0.2 instead of diWding by 5, thus 0.2
(I.S3448) = 0.2 (- 1 + 0.83448) = - 0.2 + 0.16690 = -1 + 0.8
+0.16690=1.96690.
Tables 31 and 32 give logarithms of trigonometric functions to
five decimal places at intervals of 10', the characteristics being
given. For log sin and log tan look for the degree at the left-hand
side and for the minutes at the top; for log cos and log cot look
for the degree at the right-hand side and for the minutes at the
foot. In many books these functions arc called logarithmic sines,
logarithmic tangents, etc., while the characteristics are WTitten
8 and 9 instead of 2 and T, thus requiring some power of 10 to
be subtracted later. Here the final logarithm of a computation is
correct without such subtraction.
Table 33 gives four-place logarithms of trigonometric func-
tions, and its arrangement is the same as that of Table 29.
35. Exercises
1. Find the logarithms of 7.25, 7250, and 0.725.
2. Find the nuinbors whose logarithms are 1.64933, G.64933, 2.64933,
0.70520, 1.70520, and 0.73998.
3. Compute by logarithms the sixth powers of 3.25 and 0.325;
also the sixth roots of 3.27 and 0.327.
4. Using logarithms, multiply 32. IG by 0.01555; also divide 1825
by 0.03245.
5. Find log sinos of 44° 22' and 44° 25'; also log cosines and tan-
gents of the same angles.
0. r.ivcn a = h sin A /sin B compute the value of a when !> =973 feet,
^=24°40', and/^ = 73° 10'.
7. Compute the value of 0.375 tan 85°; also of sec 7S°Xcos7S°;
also of cot 39° lO'Xsin 39° lO'/cos 39° 10'.
8. In a right-angled triangle the hypothcnuse is 505 feet; compute
the other two sides when one of the acute angles is 53° 8'.
9. When a vortical post 3.125 feet high casts a shadow 8.275 feet
long on a level plane, what is the elevation of the svui above the horizon?
Chapter G
WEIGHTS AND MEASURES
56
Weights and Measures
36. Length
I meter = 10 decimeters =ioo centimeters = looo millimeters=io' microns =o.i deka-
meter=o.oi hectometer = o.ooi kilometer = o-oooi myriameter.
I U- S. yard = 3600/3937 meters (by definition); log = i.96ii37i.
Meters
Inches
Feet
Yards
Links
R°d^- Chains.
P°'^^'°^ Gunter's
perches
Statute
miles
U.S.
Nautical
miles
U.S.
I
39-37
1.59517*
3.2808
0.51598*
1.0936
0.03886*
4-971
0.69644*
0.1988 0.04971
1.29850* 2.69644*
0.(3)6214
4-79335*
0.(3)5396
4.73207'
0.0254
5.40483*
I
0.08333
2.920S2*
■0.02778
2.44370*
0.1263
1.10127*
0.0050S1
3-70333*
0.001263
3.10127*
0.(4)1578
SJ9818*
o.(4)i37l
S-13690
0.3048
1.48402*
12
I. 07918*
1
0.3333
1.52288*
i-SiS
0.18046*
0.06061
2.78252*
o.oisis
2. 18046*
0.(3)1894
4-27737*
o.(3)i6dS
4.21608'
0.9144
1.96114*
36
I -55630*
3
0.47712*
I
4-545
0-65758*
0.1818
1-25964*
0.0454s
2.65758*
0.(3)5682
4-75449*
o.(3)4934
4.69320*
0.2012
1.30356*
7-92
0.89S73*
0-66
I-81954*
0.22
1.34242*
I
O-04
2-60206*
O.OI
2.00000*
0.(3)1250
4.09691*
o.(3)lo86
4-03564
5.029
0.70150*
198
2. 29667*
16-5
1. 2 1 748*
5-5
0. 74036*
25
1.39794*
I
0.25
1-39794*
o.(2)3I2S
3.49485*
0.(2)2714
3-43357*
20- 12
1.30356*
792
2.89S73*
66
1.81954*
22
1.34242*
100
2-00000*
4
0-60206*
1
0.0125
2.09691*
0.010S6
2.03564*
1609.3
3.2066s*
63360
4.801S2*
5280
3.72263*
1760
3-245SI*
8000
3-90309*
320
2.50515*
80
1.90309*
1
0. 8684
1.93873
1853-25
3.26753*
72962
4.86310*
6080. 2
3.78392*
2026.73
3-30680*
9212
3-96437*
368- 5
2-56643*
92.12
1.96437*
1. 1516
0.06128*
I
I nautical mile of the British admiralty = 6080 ft. i furlong = ^3 mile = 660 feet.
1 league = 3 miles = 24 furlongs. 1 fatliom = 2 yards = 6 feet.
* Logarithm of the number immediately above.
37. Area
I hectare = 100 ares = 10 000 centares or square meters.
Square
meters
Square
inches
Square
feet
Square
yards
Square
rods
Square
chains
.\cres
Square
miles or
sections
1
1550
3-19033*
10-764
1-03197*
1. i960
0.07773*
0-03954
2.59700*
0-12)2471
3 -39 288*
0.(3)2471
4.3928S*
0. (6)3861
7- 38670'
o.(3)54S3
4.80967*
I
0-006944
3.84164*
o.C2)77i6
3.88740*
(4)2551
5-40667*
0.(0)1594
6-20255'
0.(6)1594
7.20255'
0- (9)2491
10.39637*
0.09290
2.96803*
144
2.15S36*
I
O.IllI
1.04576*
0.(2)3673
3.56503*
0.(3)2296
4-36091*
0.(4)2296
S. 3609 1*
0. (7)3587
8. 55473*
0.8361
1.92227*
1296
3- 1 1 260*
9
0.95424*
I
0-03306
2-51927*
o.(2)2o66
3-31515*
o.(3)2o66
4.31315*
0. (6)3228
7. S0898'
25.29
1.40300*
39204
4-5933*
272.25
2-43497*
30-23
1-48072*
I
0.0625
2.7958S*
0.00625
3-79588*
0. (5)9766
6.98970'
404.69
3.60712*
627264
5-79745*
4356
3-63909*
484
2.6S484*
16
1-20412*
1
O-I
I- 00000*
0. (3)IS62
4.19382*
4046.9
3.60712*
6272640
6-79745*
43560
4.63909*
4840
3.6S4S4*
160
2- 20.(I2*
10
1 . 00000*
1
0.001562
3.19382*
2589998
6. 41330*
27878400
7-44527*
3097600
6.49102*
102400
5.01030*
6400
3.80618*
640
3.80618*
I
* Logarithm of the number immediately above.
Weights and Me.^sures
57
38. Speed
and Velocity
Cm per
sec
Km per
hour
Ft per
sec
Ft per
min
Miles per
hour
Knots
I
0.036
2-53630*
0.032SI
2.51598*
1.96S5
0.29413*
0.02237
2.34965*
0.01942
2.2S825*
27-777S
1.44370*
1
0.9II34
T.9596S*
54.6806
1.737S3*
0.62137
1-79335*
0.53960
1.73207*
30.4801
1.48402*
I-0973
0.04032*
1
60
1-77S1S*
0.6S182
T.83367*
0.59209
1-77238*
0.5080
1.70586*
0.01829
2.26217*
0.01667
2.22IS5*
I
0.01136
2-05553*
0.009868
3-99423*
,44-7041
1.63035*
1-6093
0.20670*
1.46667
0.16633*
88
1.9444S*
1
0.86S39
1.93872*
SI. 4971
1.71178*
1-8332
0.26793*
1.68894
0.22761*
101.337
2.00577*
1-13155
0.06128*
I
I knot = I nautical mile per hour.
* Logarithm of the number immediately above.
39. Voliime and Capacity
I literal cubic decimeter = 1000 cubic centimeters = 10 deciliters = 100 centiliters ="
1000 milliliters = 0.1 dekaliter = 0.01 hectoliter = 0.01 kiloliter = 0.001 cubic meters or
steres.
Cubic
inches
Cubic
feet
Cubic
yards
U. S. quarts
Gallons
Bushels
U.S.
Liters
Liquid
Dry
U.S.
liquid
U.S.
dry
I
0.(3)57870
4.76246*
0. ■4^2143 0.017316
5.33109* 2.23845*
0.014881
2.17263*
0.004329
3.63639*
0.003720
3-57057*
0.(3)4650
4-66748*
0.016387
2.21430*
1728
3-23754*
I
0.037037 29.922
2.56864*11.47599*
25-714
1.41017*
7-4805
0-87393*
6.4285
0.8081 I*
0.80356
T. 90502*
28.317
1.45205*
46656
4-66891*
27
1-43136*
I
807.90 694.28
2.90736* 2.84153*
201.97
2.30330*
173-57
2.2394S*
21.696
1-33638*
764.56
2.88341*
57-75
1.76155*
0.033420
2.52401*
0.001238
3.09026*
1
0-83937
I. 93418*
0.25
1-39794*
0.21484
1.33212*
0.026855
2.42903*
0.94636
T. 97606*
67.201
1.82737*
0.038889
2.589S3*
0.001440
3-15847*
1.1637
0.06582*
1
0.29091
1.46376*
0.25
1-39794*
0.03125
2.49485*
I. 1012
0.04188*
231
2.36361*
0.13368
r. 12607*
0.004951
3-69471*
4
0.60206*1
3-4375
0.53624*
1
0-85937
T- 93418*
0.10742
T. 03109*
3-7854
0.57812*
268.80
2.42943*
0-15536
I. 19189*
0.0037C1 4.65^6
3.76o53*,o.66788*
4
0.60201*
I. 1637
0.06582*
I
0.125
1.09691*
4-4049
0.64394*
2150.4
3-33253*
1-2445
0.09498*
0.046091 37.237
2.66362* 1.57097*
32
1-50515*
9.3092
0.96891*
8
0.90309*
I
35-239
1-54703*
61.023
1.78550*
0.035313
2-54793*
0.00130S 1.0567
3.li639*|o.02394*i
0.9080S 1
r.9s8i2*j
0.26417
1.4218S*
0.22702
T. 35606*
0.028377
2-45297*
I
I U.S. liquid quart = 2 pLnts = 8 gLlls = 32 fluid ounces = 256 fluid drams = 768 fluid
scruples, i bushel = 4 piecks.
1 Imperial gallon = 1.201 U. S. gallons = 0.1605 cu ft = 4.3460 liters.
I U. S. gallon = 0.8327 Imperial gallons. 1 cubic foot = 6.229 Imperial gallons,
1 British bushel = 1.2837 cubic feet.
Shipping Measure: i register ton = 100 cu ft. i U. S. shipping ton = 40 cuft>
I British shipping ton = 42 cu ft
• Logarithm of the number immediately above.
58
Weights and Measures
40. Weight (Engineers' System) or Mass (Physicists' System)
I kilogram = iooo grams = o.ooi metric ton. i gm=io decigrams= too centigTams =
looo milligrams = o. I dekagram = o.oi hectogram = o.ooi kilogram = 0.0001 myri igrara.
1 U.S. Avoirdupois pound = 04535924277 kg = (by definition) 7000/5760 troy pounds.
Kilo-
grams
Grains
Ounces
Pounds
Tons
Avoir.
Troy and
apoth.
Troy and
apoth.
Avoir.
Short,
2000 lb
Long,
2240 lb
Metric,
1000 kg
I
15432.
4. 18843*
35-274
I-S4745*
32-151
1.50719*
2.6792
0.42801*
2 . 2046 I
0. 34333*
0.001102
3.04230*
0.(3)9842
4-99309*
O.OOI
3.00000*
0.W6480
S-81157*
I
0.(2)2286
3-3S902*
0.002083
3-31876*
0.(3)1736
4-23958*
o.(3)i42
4-15490*
0.028349
2-45255*
437-5
2.6409S*
I
0.91146
T-95974*
0-075955
2.88056*
0.0625
2.79588*
0.(4)3"S
5-49485*
0.(4)2790
S.44563*
0. (4) 283s
5-45255*
0.031103
2.49281*
480
2.68124*
I. 0971
0.04026*
I
0.083333
2.92082*
o.o6857i'o. (4)3429
2.83614* 5-53511*
o.(3)3o6i
4.56508'
0.(4)3110
S.49281*
0-37324
1.57199*
5760
3.76042*
13. 166
1.11944*
12
I. 07918*
1
0.(3)4114
4.61429*
0.(3)3673
4-56508*
0.(3)3732
4-57199*
o-4S3i59
1.65667*
7000
3.84510*
16
I. 20412*
14.583
1.16386*
I-2IS3
0.08468*
I
0.0005
4.69897*
0.(3)4464
4.6497s*
o.(3)4S36
4-65667*
907.18
2-9577°*
32000
4-S0515*
29167.
4-46489*
2430.6
3-38570*
2000
3.30103*
I
0.892S6
1.95078*
0.90718
1.95770*
1016. I
3.00691*
35840
4-55437*
32667
4.51410*
2722.2
3.43492*
2240
3-35025*
I. 12
0.04922*
I
I . I 60
0.00691*
1000
3.00000*
35274
4-54745*
32151
4.50719*
2679.2
3.42801*
2204.6
3-34333*
1.1023
0.04230*
0.98421
1.99309*
I
I quarter = 28 lb avoir, i pennyweight = 24 gr=o.o5 oz troy. 1 oz avoir. = 16 drama
avoir. =437.5 gr. 1 stone = 14 pounds. 1 centil = 100 pounds. 1 hundredweight™
T12 pounds. I apothecaries' ounce = 8 apoth. draras = 24 scruples = 48o grains.
♦ Logarithm of the number immediately above.
41. Energy or Work
Joules
= 10' erg
Meter-
kilograms
Foot-pflunds
Kilowatt-
hours
Chcval-
vapeur-
hours
Horse-
power-
hours
Rritish
thermal
units
o-(3)947S
4.97660*
I
0. 10197
1.00848*
0-73756
T.S6780*
0.(6)27778
7-4437°'
2-(6)37767
7.57711*
2-(^)3725l
7-S7II3*
9.80665
0.9915207*
I
7-2330
0.85932*
0.(5)27241
6-43522*
0.(5)37937
6.56863*
0.(5)36530
6-56265*
0.009292
3.96812*
1-3558
0.13220*
0.13826
I. 14068*
I
0.(6)37662
7-S7S90*
0.(6)51206
7.70932*
0.(6)50505
7.70333*
0.001285
3.108S0*
3.6x10"
6-55630*
3.6710 X 10'
S-56478*
2.6552X10
6.42410*
I
1-3596
0.13342*
1.3410
0.12743*
3411-
3.53290*
2.6478X10®
6.42288*
270000
S-43136*
1-9529X10®
6.29068*
0.73550
T. 86658*
I
0.98631
I.9940I*
2509.
3.39948*
2.6845X10"
6.42887*
2-7375 Xio''
5-43735*
1.98x10"
6.29667*
0-74571
T-87257*
1.0139
0.00598*
I
2544.
3.40547
to5S-
5.02340*
107.6
2.03188*
778.4
2 . 8y 1 2 0*
0.0 293a
4.46710*
o.o:>3986
4.60051*
0.033931
4-59453*
1
Logarithm of the namber immediately above.
Weights and Measures
42. Pressure
59
KUo-
grams
per sq
cm
Pounds
Short
tons,
Atmos-
Columns of
mercuryt
Columns of watert
Per sq
in
Persq
ft
persq
ft
pheres
Meters
Inches
Meters
Feet
I
14-223
1.15300*
2048.2
3-3II37*
I. 0241
0.01034*
0.96781
1.98579*
0.73553
1.86660*
28.958
I. 46177*
I . 009
1.00038*
32.837
I. 5 1636*
0.070307
2.84700*
I
144
2.15836*
0.072
2.85733*
0.06804
2.83279*
0.05 1 71 3
2.71360*
2.0359
0.30876*
0.70368
T.8473S*
2.3087
0.36336*
0.(8)4882
4.68863*
0.006944
3.84164*
I
0.0005
4.69897*
0. (3)4725
4.67442*
0.(3)3591
4-55524*
0.014138
2.15040*
0.004887
3.68901*
0.016032
2.20500*
0.97648
T. 98966*
13-889
I. 14267*
2000
3-30103*
I
0.94504
1-97545*
0.71823
1.85627*
28.277
1-45143*
9-7734
0.99004*
32.06s
1.50603*
I-0333
0.01421*
14.697
I. 16722*
2116.3
3-32558*
T.0582
0.02955*
I
O-76
T.88081*
29.921
1.47598*
10.342
1.01439*
33-929
1.53058*
1.3596
0.13340*
19-338
1.28640*
2784.6
3.44476*
1-3923
0.14373*
I. 3158
0. 11919*
I
39.37
1.59517*
13.607
1.13378*
44.644
1.64976*
0.034S33
3.53823*
0.49118
I. 69124*
70.729
I . 84960*
0.035365
2.54857*
0.033421
2.52402*
0.025400
2.40484*
I
0-34563
T. 53861*
1.1340
0.05460*
0.099913
a. 99962*
1.4211
0. 15262*
204.64
2.31099*
0.10232
T. 00996*
0.096697
2.98541*
0.073489
2.86622*
2.8933
0.46139*
I
3.2808
0.51598*
0.030453
3.48364*
0.43315
T. 63664*
62.374
1.79500*
0.031187
2-49397*
0.029473
2.46942*
0.022399
2.35024*
0.88187
T. 94540*
0.30480
T. 48402*
I
* Logarithm of the number immediately above.
t At 15° C. and5 = fo.
43. Power
I kilowatt = 1000 watts = 1000 joules per second.
1 horse-power = 550 foot-pounds per second.
I cheval-vapeur = 75 kilogram-meters per second.
Kilowatts
Cheval-
vapeur
Poncelet
Horse-
power
M-kg
per sec
Ft-lb
per sec
Kg cal
per sec
Btu
per sec
I
1 . 3600
0.13341*
1.0197
0.00848*
1. 341
0.12743*
101.97
2.00848*
737-5
2.86780*
0.2388
i. 37803*
0.9475
1.97660*
0.7355
1.86659*
I
0-75
T. 87506*
0.9863
1.99402*
75
1.87506*
542. 5
2.73438*
0.1756
T. 24456*
0.6969
1-84318
0.980665
I. 99152*
1-333
0.12493*
I
1.3151
0.11896*
100
2.00000*
723-3
2.85932*
0-2342
I. 36951*
0.9292
1-96812*
0.7457
r. 87257*
1.0139
0.00598*
0. 7604
T.S8104*
I
76.04
I. 88104*
550
2.74036*
0. 1780
T-25055*
0-7066
I- 84916*
0.009807
3-99152*
0-01333
2.12493*
O.OI
2.00000*
0.01315
2.11896*
1
7.233
0.85932*
0.002342
3-36951*
0.009292
3-96812*
0.001356
3.13220*
0.001843
3.26562*
0.00138
3.I406S*
0.001818
3-25964*
0.1383
T. 14068*
I
0.0003237
4.51016*
0.001285
3.10880*
4.188
0.62201*
5-694
0.75542*
4.271
0.63049*
5.616
0.74945*
427-1
2.63049*
3089
3.48984*
1
3-968
0.59861*
I-OS5
0.02340*
1-435
0.15682*
1.076
0.03188*
1.415
0.15084*
107.62
2.0318S*
778.4
2.89120*
0.2520
T. 40139*
I
* Logarithm of the number immediately above.
60 Weights and Measures
44. Explanations
The preceding Tables give the numerical relations between
different units of measure, all the numbers in one horizontal line
being equivalents. For example, in Table 36 the first line
shows that 1 meter is 39.37 inches, or 3.2S0S feet, or 1.0936
yards, etc.; also the seventh line from the top shows that 1 yard
is 0.9144 meters, or 36 inches, or 3 feet, or 4.545 Unks, or 0.1818
rods, etc.
When the notation (^) is seen it means that (^) is to be re-
placed by three ciphers; thus in the first line of Table 36 an
equivalent of 1 meter is O.OOOG214 statute miles and in Table 37
one square meter is 0.0000003SG1 square miles.
Below each equivalent is given its five-place logarithm marked
with a *. These are useful in converting quantities of one unit
into those of another unit. For example, to find how many
feet there are in 69.39 nautical miles: Table 36 gives 60S0.2 as
the number of feet in one nautical mile, hence the required result
is 69.39X6080.2 which may be found l)y ordinary multiplication;,
or by logarithms the log of 69.39 is taken from Table 30 while the
log of 6080.2 is found from Ta])le ,36 as 3.78392; the addition of
the two logs gives 5.62522 which is the log of 421910, where the
fifth significant figure is liable to error; hence the probable
result obtained from this table by use of the given logarithm is
that 69.39 nautical miles are equivalent to 421910 ± 2.5 feet. By
direct multiplication it is found that the number of feet is
421905.
Numbers in boldface type jare exact values, while all others in
the body of a table are liable to an error of one-fourth of a unit
in the last significant figure. The equivalents above a table and
many of those below it are also exact by definition. As a general
rule results obtained by the use of equivalents or logarithms
taken from the body of a table are liable to an error in the fifth
significant figure, except when an equivalent in the heavy type
is used directly.
Table 40 for measures of weight applies also to measures of
Weights and Measures 61
force when the engineers' system is used, since the unit of force
is the force of gravity which acts on the unit of weight at lati-
tude 45° on the surface of the earth.
Tables 41 and 43 contain some units which may not be familiar
to students who use this book, but the time will come, if they enter
on engineering work, when the equivalents of these tables may be
of great value to them. Probably all, however, know the mean-
ings of energy or work, of a horse-power and a kilowatt, and of a
British thermal unit; concerning these a few exercises are given
below.
45. Exercises
1. By Table 36 how many feet in one statute mile? how many
meters in one kilometer? how many statute miles are equivalent to
one nautical mile?
2. By Table 37 how many square feet in one acre? how many
square meters are equivalent to one square inch? how many acres
are equivalent to one square inch?
3. By Table 38 how many feet per minute are equivalent to one
mile per hour? how many statute miles per hour are equivalent to one
knot.
4. By Table 39 how many bushels are in one cubic j^ard? how many
liquid gallons are in one cubic foot? how many hters are equivalent to
1000 liquid gallons?
5. By Table 40 how many grains in one avoirdupois pound? how
many short tons in one long ton? how many kilograms in one metric
ton?
6. By Table 41 how many kilowatt-hours are equivalent to 1 foot-
pound? how many foot-pounds in one British thermal unit?
7. By Table 42 how many pounds per square inch are equivalent
to one kilogram per square centimeter? how many feet of water will
balance the pressure of one atmosphere?
8. By Table 43 how many foot-pounds per second make one
horse-power? how many kilowatts are equivalent to 100 horse-powers?
9. How many meters are equivalent to 1000 feet? how many
meters are equivalent to 300 yards? how many kilometers are
equivalent to 62.2 statute miles?
10. How many acres are equivalent to 87,120 square feet? How
many square meters are equivalent to 153,900 square inches?
62 Weights and Measures
11. How many U. S. liquid gallons are equivalent to 100,000
Imperial gallons? how many liters are equivalent to 624.3 cubic
inches?
12. How many pounds avoirdupois are 1000 kilograms? how many
long tons are 37.2 metric tons?
13. How many foot-pounds in 0.01 kilowatt-hours? How many
horse-power-hours are equivalent to 6040 foot-pounds?
14. How many atmospheres will balance a column of water 68 feet
high? how many inches of mercur)'^ will balance 100 atmospheres?
15. How many foot-pounds per second are equivalent to 100
horse-powers? how many horse-powers are equivalent to 55,000 foot-
pounds per second?
16. How many inches in one meter? how many inches in one centi-
meter? how many pounds avoirdupois in 100 long tons and how many
in one metric ton?
17. What is the definition of a horse-power? of a joule? of a British
thermal unit? of a kilowatt-hour? of a horse-power-hour?
Chapter 7
MISCELLANEOUS TABLES
64
Miscellaneous Tables
46. Mathematical Constants
Symbol
Number
Logarithm
Symbol
Number
Logarithm
2 Tt
3-I415927
6.2831853
9.4247780
0.4971499
0.7981799
0.9742711
^n
il-^n
1-7724539
0.5641896
0.2485749
1.7514251
4^
12. 5663706
1.0992099
n-^2
4.4428829
0.6476649
5^
15-7079633
I. 1961200
^2n
2.5066282
0.3990899
671
18.8495559
1.2753011
Vnl2
I. 2533141
o.o98o^99
7^
21.9911486
1.3422479
Va/-
S;r
25.1327412
1.4002399
0.7978844
I. 9019401
9;r
28.2743339
I. 4513924
£
2.7182818
0.4342945
4 7r/3
4.1S77902
0.6220886
£2
7.3890568
0.8685890
7Zl2
1-5707963
0. 1961 199
l/£
0.3678794
1-5657055
TtiA
0.7853982
T. 8950899
l/£^
0.1353353
I . 1314110
nib
0.5235988
I. 7189986
/'
0.4342945
1-6377843
~/3°
0. 1047198
T. 0200286
i/,a
2.3025S51
0.3622157
;i-/i8o
0.0174533
2.2418774
sin 1°
0.0174524
2-2418553
I/.T
0.3183099
T. 5028501
sin i'
0.0002909
4.4637261
2 In
0.6366198
T. 8038801
sin i"
0.0000048
6.6855749
i8o In
57-2957795
I. 7581226
2
2.
0. 3010300
joSoo In
3437-74677
3-5362739
V2
648000 /;:
206264.806
5-3144251
1.4142136
0.1505150
-2
9. 8696044
0.9942997
VV2
0.7071068
1.8494850
iln^
. 1 1 3 2 1 2
1.0057003
3
3-
0.4771213
n^
31.0062767
I. 4914496
^3
1.732050S
0.2385606
1/7:3
0.0322516
2.5085504
s/y-i
0-5773503
1.7614394
47. Decimal Equivalents of Common Fractions
Fract.
1/2
Decimal Logarithm
Fract
Decimal
Logarithm
Fract.
1/32
Decimal
Logarithm
0-5
1.69897
1/8
0.125
I. 09691
0.03125
2.49485
i/3
0.33333
T. 52288
3/8
0-37S
1.57403
3/32
0.09375
2.97197
■^b
0.66667
I. 82391
5/8
0.625
1.79588
5/32
0. 15625
1.19382
V4
0.25
1-39794
7/8
0.875
I. 94201
7/32
O.21S75
1.3399s
a/4
0-75
1.87506
1/12
0.08333
2 . 92082
9/32
0.28125
1.44909
V5
0.2
1-30103
5/12
0.41667
T. 61979
"/32
0.34375
1.53624
2/5
0.4
1 . 60206
7/12
0.58333
T. 76592
1-V32
0.40625
T.60S79
■•V5
0.6
1-7781S
11/12
0.91667
I. 96211
15/32
0.46S75
1.67094
^/5
Ve
0.8
0. 16667
1 . 90309
1.22185
T. 92082
Vie
0.0625
2.79588
17/32
0.53125
1.72530
5/6
0.83333
a/16
5/16
0.1875
0.3125
1.27300
1.49485
19/32
21/.-12
0.59375
0. 656''5
1.77360
I. 81707
V?
0. 14286
1.15490
7/18
0.4375
T. 64098
23/32
0.71875
T. 85658
7?
•V7
0.28571
0.42857
1-45593
1 .63202
T. 75696
1-85387
7 -9330s
0/16
0.5625
T. 75012
2,5/32
0.78125
T. 89279
■V7
6/7
0.57143
0.71429
0.85714
11/10
"/lO
15/10
0.6875
0.8125
0.9375
1.83727
1.90982
T. 97197
27/32
29/32
31/32
0.84375
0.90625
0.96S75
T. 92621
1.95725
1. 9862 1
Miscellaneous Tables
48. Natural Hyperbolic Functions
65
u
Sinh u
Cosh u
Tanh u
«
Sinh H
Cosh M
Tanh u
o.oo
0. 0000
I . 0000
. 0000
2.25
4.6912
4.7966
0.9780
0.05
0.0500
1.0013
0. 0500
2.30
4.9370
5-0372
0. 9801
O.IO
0. 1002
I .0050
0.0997
2.35
5-1951
5-2905
0.9820
0.15
0. 1506
1-0113
0. 1489
2.40
5.4662
5-5569
0.9837
0.20
0.2013
I . 0201
0.1974
2.45
5-75IO
5.8373
0.9853
0.25
0. 2526
I. 0314
0.2449
2.50
6.0502
6-1323
0.9866
0.30
0.3045
I -0453
0.2913
2.55
6-3645
6.4426
0.9879
0.35
0.3572
I .0619
0.3364
2.60
6.6947
6. 7690
0. 9890
0.40
0.4108
1.0811
0.3800
2.65
7.0417
7-1123
0. 9900
0.45
0.4653
I. 1030
0. 4219
2.70
7-4063
7-473S
0.9910
0.50
0.5211
I. 1276
0. 4621
2.75
7-7894
7-8533
0.9918
0.5S
0.5782
1-1551
0. 5005
2.80
8. 1919
8.2527
0.9q26
0.60
0.6367
1-1S55
0.5370
2.85
8.6150
8.6728
0.9933
0.65
0. 6967
I. 2188
0-5717
2.90
9-0596
9. 1146
0.9940
0,70
0.7586
1-2552
0. 6044
2.95
9-5268
9-5791
0.9945
0.7S
0.8223
1.2947
0.6352
3-00
10.018
10.068
0.9950
0.80
0.88S1
1-3374
0. 6640
3.0s
10.534
10.581
0.9955
0.85
0.9561
1-3835
0. 6911
3.10
II . 076
II . 122
0.9959
0.90
1.0265
I-4331
0.7163
3-15
11.647
11.690
0.9963
0.95
1-0995
1.4862
0.7398
3-20
12. 246
12.287
0.9967
1. 00
1-1752
I -5431
0.7616
3-25
12.876
12.915
0.9970
1.05
1-2539
1.6038
0.7818
3-30
13.538
13.575
0.9973
1. 10
1-3356
1.6685
0. 8005
3.35
14.234
14.269'
0.9976
1-15
1.4208
1-7374
0.8178
3-40
14.965
14.999
0.9978
1.20
1-5095
I. 8107
0-8337
3 -4 .=5
15.734
15.766
0.9980
1.25
I . 6019
1.8884
0.8483
3-50
16.543
16.573
0.9982
1.30
1.6984
1.9709
0.8617
3-55
17.392
17.421
0.9984
1-35
I. 7991
2.0583
0.8741
3-6o
18.285
18.313
0.9985
1.40
1-9043
2.1509
0-8854
3.65
19. 224
19.250
0.9987
1-45
2.0143
2.2488
0.8957
3-70
20. 211
20.236
0.9988
1.50
2.1293
2-3524
0. 9052
3.75
21.249
21 . 272
0. 9989
1-55
2. 2496
2.4619
0.9138
3.80
22.339
22.362
0. 9990
1.60
2-3756
2-5775
0. 9217
3.8s
23.486
23.507
0.9991
1.65
2-5075
2.6995
0. 9289
3-90
24. 691
24.711
0.9992
1.70
2.6456
2.8283
0.9354
3-95
25.958
25.977
0.9993
1.75
2.7904
2.9642
0.9414
4.0
27.290
27.308
0.9993
1.80
2.9422
3.107s
0. 9468
4.1
30.162
30.178
0.9995
1.85
3-1013
3.2585
0.9518
4.2
33-336
33.351
0.9996
1.90
3.2682
3.4177
0. 9562
4-3
36.843
36.857
0.9996
1.95
3-4432
3-5855
0.9603
4.4
40.719
40.732
0.9997
2.00
3.6269
3.7622
0.9640
4-5
45.003
45.014
0.9998
2.05
3-8196
3-9483
0.9674
4.6
49.737
49.747
0. 9998
2.10
4.0219
4- 1443
0.9705
4-7
54.969
54.978
0.9999
2.15
4.2342
4.3507
0.9732
4.8
60.751
60.759
0.9999
2.20
4-4571
4.5679
0.97S7
4-9
67.141
67.149
0.9999
Explanation on page 39
66
Miscellaneous Tables
49
. Napierian Logarithms of Numbers from i to 119
n
o. I. 2. 3. 4. 5. 6. 7. 8. g.
o
— 00
0.0000
0-6931
I . 0986
1.3863
1.6094
1.7918
I - 94S9
2-0794
2. 1972
I
2.3026
2-3979
2 - 4849
2-5649
2.6391
2.7081
2.7726
2.8332
2.8904
2.9444
2
2 9957
3 044.5
3-0910
3-1355
3-1781
3-2189
3-2581
3-2958
3-3322
3 3673
3
3.4012
3-4340
3-4657
3-4965
3- 5264
3-SSS3
3-5835
3.6109
3 6376
3-6636
4
3.6889
3-7136
3-7377
3.7612
3-7842
3-8067
3-8286
3.8501
3-8712
3.8918
5
3.9120
3-9318
3-9512
3 9703
3 • 9890
4-0073
4-0254
4-0430
4.0604
4-0775
6
4 094.?
4-I109
4-1271
4-1451
4-1589
4-1744
4. 1S97
4-2047
4.2195
4-2341
7
4.2485
4.2627
4-2767
4-2905
4.3041
4-3175
4- 3307
4- 3438
4-3567
4-3694
8
4.3820
4-3944
4-4067
4.4188
4-4308
4-4427
4-4543
4-4659
4-4773
4-4886
Q
4.4998
4-5109
4-5218
4-5326
4-5433
4-5539
4- 5643
4-5747
4- 5850
4-5951
10
4-6052
4.6151
4.6250
4- 6347
4.6444
4.6540
4-6634
4.6728
4.6821
4.6913
II
4,7005
4.7095
4-7185
4- 7274
4-7362
4-7449
4-7536
4.7622
4-7707
4-7791
50. Multipliers for Transferring Logarithms
Common to Napierian
2.302585093
4.605170186
6-907755279
9.210340372
11.512925465
13.815510558
16. 118095651
18.420680744
20.723265837
Example.
Find Nap log of 105
Com log 105 = 2.02119
2 4.605170
.02 46052
I 2303
I 230
9 207
Nap log 105 =4.65396
Napierian to Common
0.434294482
0.868588964
I . 302883446
I -737177928
2- I714724IO
2.605766891
3.040061373
3.474355855
3.908650337
E.xample.
Find number correspond
ing to Nap log 1.6078
.6
07
0.26058
304
35
+0.26397
-0.43429
Com log =
1.8297
Number = 0.6756
61. Explanations
Table 46 gives seven-place constants and their logarithms
which will often be of use in niathematical computations; tt is the
ratio of the circumference of a circle to its diameter, e is the base
of the Napierian system of logarithms, and {x is the modulus of the
common system of logarithms. In this table the shilling mark /
denotes division; thus, tt/SO is the same as ^V 't.
Table 47 gives decimal equivalents of some common frac-
tions, thus, 7/32=0.21875. The logarithms may be u.«!eful in
computations; thus to divide 0.3275 by 13/16 the log of 13/16
is subtracted from the log of 0.3275, or 1.51521 -I.909S2 = 1.60539
which is the log of 0.4031.
Miscellaneous Tables ' ' ^ ' ' ^ ' gy
Table 48 gives natural hyperbolic sines, cosines and tangents of
numbers. These are useful in engineering problems relating to
beams, to the catenary, the parabola, and other curves, also in
the theory of alternating currents. Hyperbolic functions can be
graphically represented in a rectangular hyperbola in the same
way as trigonometric functions are represented in a circle. " Sinh"
is the abbreviation for the hyperbolic sine, and " cosh " for the
hyperbolic cosine; sinh is usually pronounced shin.
Table 49 gives a few Napierian logarithms, often called hyper-
bolic logarithms. The base of this system is the number 2.71828.
Such logarithms arise in many formulas deduced by calculus,
and they are widely used in physical and engineering problems.
Table 50 shows how to obtain the Napierian logarithm of any
number from its common logarithm.
51. Exercises
I. Divide 4738 by ir, using logarithms.
2 What is the value of Air^?
3. What is the meaning of ISOVtt?
4. What is the area of a sphere whose radius is 100 feet?
5. What is the volume of a sphere whose diameter is 10 centi-
meters?
6. Find the value of 472/V|'.
7. What are the decimal equivalents of 2/7, 11/32, 80/32, 7/12
9/12, 34/64, and 35/64?
8. Given m = 1.25; square sinh u and cosh u and subtract the first
square from the second. Also do the same for another value of u.
9. Is tanh u equal to sinh t</cosh u?
10. Multiply 37 by 8, using Napierian logarithms.
II. Divide 119 by 17, using Napierian logarithms.
12. Taking the common logarithm of e from Table 47, find the
value of e^
13. If the common logarithm of a number is 4.0000, what i." its
Napierian logarithm?
14. Find the Napierian logarithm of 2.71828^
15. Find the Napierian logarithms of 3275, 3.275, 1800, and 0.18
SECOND EDITION, ENLARGED
Total Issue, Twenty Thousand
American
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2. Surveying, Geodesy, Railroad Location, by Charles B. Breed,
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3. Steam and Electric Railroads, by Walter Loring Webb, Member
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of Buildings, Borough of Manhattan, New York City.
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8. Steel Structures, by Frank P. McKibben. Professor of Civil
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formerly Professor of Civil Engineering in University of Michigan.
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It
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