OUR laCURE
MATHEMATIOAL TABLES
OTTOMLEY
IN MEMORIAM
FLORIAN CAJO
» • • I U. > U • A,^ B - 11-
^
:l
FOUR FIGURE
MATHEMATICAL TABLES.
f
FOUR FIGURE
MATHEMATICAL TABLES
COMPRISING LOGARITHMIC AND TRIGONOMETRICAL TABLES, AND
TABLES OF SQUARES, SQUARE ROOTS, AND RECIPROCALS.
BY
J. T. BOTTOMLEY, M.A, LL.D., D.Sc,
F.R.S., F.R.S.E., F.C.S.,
LATE LECTURER IN NATURAL PHILOSOPHY IN THE UNIVERSITY OF GLASGOW.
MACMILLAN AND CO., Limited
NEW YORK : THE MACMILLAN COMPANY
1905
A/i rights reserved
First Edition 18S7.
Reprinted 1890, 1893, 1894, 1S96, i?97, 1899 (twice), 1900, 1901,
1902, 1903 (twice). With additions 1904, 1905 (twice).
CAJORI
GLASGOW : PRINTED AT THE UNIVERSITY PRESS
BY ROBERT MACLEHOSE AND CO. LTD.
■' > > ' J 1 ) J * ' »
«i> » » • JO
■» > ) > .J
EXPLANATIONS AND RULES
FOR THE
USE OF THE ACCOMPANYING TABLES.
The logarithm of a number consists in general of two parts, an
integer part and a decimal. The integer part is called the Index or
Characteristic ; the decimal part is called the Mantissa.
RULE I, The Index of the logarithm of a number greater than
unity is the number which is less by one than the number of digits in
the integral part of the given number.
Thus, the index of the logarithm of 47320 is 4.
473-2 is 2.
4732 is o.
RULE n. The Index of the logarithm of a number less than unity,
and reduced to the form of a decimal fraction, is negative, and is a
higher number by one than the number of zeros that follow the decimal
point of the given number.
Thus, the index of the logarithm of '4732 is - i
•004732 is - 3
To denote that the index is negative the sign minus is often
written above it ; thus T, 3.
RULE III. To find the mantissa of the logarithm of a given number
consisting of four figures. — Find the Jirst two figures in the left hand
column of the table. Pass along the corresponding horizontal line and
take the number in the vertical column headed by the third figure.
To this number add the number found in the difference columns under
the fourth figure of the given number. The sum with a decimal point
prefixed is the required mantissa.
911380
6 MATHEMATICAL TABLES.
Example. Find the mantissas corresponding to the sequences of
figures 4732 and 6985.
473 : ; -': - - - - 6749
; '2.from dif. col. - - 2
.CC C,f«CC I •
• » •
•6751
698 8439
5 from dif. coL - - 3
•8442
RULE IV. To find the logarithm of a given number consisting of
four figures.— Find the mantissa corresponding to the given four figures,
and to it prefix the proper index. The number thus obtained is the
required logarithm.
Exa//:p/es.
log 47320 - - - is 4-6751
log 47-32 - - - is 1-6751
log 6-985 - - - is 0-8442
log 0-006985 - - - is 3 8442
A^o^e. — A logarithm whose index is negative really consists of a
positive mantissa with a negative index algebraically added to it.
Thus:— 7-8442 = +0-8442- I. It is important to bear this in mind
in numerical operations on logarithms. For example, in taking the
square root of 0-6985, the logarithm of that number is divided by 2,
and in taking the cube by 3. The simplest way of doing this is
as follows : —
|(-8442- i) = 1(1-8442 -2) = -9221 -I
IC8442- 0 = ^2-8442 -3; = -9481 - I
RULE V. To find the anti-logarithm of a given logarithm, i.e., the
number corresponding to the given logarithm. — Find in the table of
anti-logarithms, proceeding as in Rule III., the sequence of figures cor-
responding to the man/issa of the given logarithm. To these figures
place a decimal point, in the position indicated by the index of the
given logarithm, prefixing or affixing zeros, if necessary. (See Rules I.
and II.) The number thus obtained is that required.
Examples. Given the logarithm 27834 find the anti-logarithm.
783 6067
4 from dif col. - - 6
6073
Hence the number whose logarithm is 2-7834 is 607-3.
EXPLANATIONS AND RULES. f
The number corresponding to the logarithm 67834 is 6073000 ; that
corresponding to 47834 is "0006073.
A'otc. — The use of Rules I. and II., which are commonly given for
the purposes of finding the index and of placing the decimal point
in an anti-logarithm, may be dispensed with altogether if the prin-
ciple on which these rules are founded is kept in view ; and in
reality the principle is more simple than the rules and easier to
remember. The logarithm, to the base 10, of any number greater than
I and less than 10 is a positive proper fraction, and is given in the
tables as a decimal without whole number. On the other hand the
anti-logarithm of a decimal without whole number is a number greater
than I and less than 10.
Thus log 7"32 = o"8645 ; and the logarithm o'693i corresponds to the
number 4'9j3.
Any number such as 7320, or '000732 is derived from 7*32 by
multiplying or dividing by a power of 10 ; and the corresponding
change in the logarithm is made by adding or subtracting the index of
that power of 10.
Thus 7320 = 7*32 X 10^ ; log. 7320= '8645 -1-3
■000732 = 7"32 X I o"* ; log. '000732 = "8645 -4
In the same way since '6931 as a logarithm corresponds to 4"933, it
follows that 2'693i, or '6931-1-2, corresponds to 4*933 xio^; and
3'693i, or "6931 —3, corresponds to 4*933 x io~^.
RULE VI. Given any angle less than 90° to find its natural sine,
cosine, tangent, etc., or its value in radian measure.* — Find the
degrees in the left hand column of the proper table. Pass along the
corresponding horizontal line, and take out the number in the vertical
column headed by the number of minutes lower than, and nearest to,
the given number of minutes. Take the difference between the number
of minutes given and the number of minutes just found, and from the
difference columns find in the same horizontal line the corresponding
correction. This correction is additive in the cases of the sine, tangent,
secant, and radian measure. In the cases of cosine, cotangent, and
cosecant it is subtractive.
Note. — It will be observed that the main division of the degree in
the trigonometrical tables is into parts of 6' each. This corresponds
to decimals of the degree. Thus, I2°i8'=i2°'3.
Note. — In the tables of natural sines and cosines the decimal points
are omitted. In the other tables the decimal points and the whole
numbers which precede them are omitted in all the columns except
* Formerly called " circular measure."
8
MATHEMATICAL TABLES.
that headed o' ; and excepting also the case of a few numbers at the
extremities of the tables, where the variation of the trigonometrical
function is extremely rapid. At the extremities of some of the tables
differences are not given, as the variation of the function is so rapid as
to make the differences unserviceable.
Examples. Find the sine and cosine of i8°27', and the tangent and
secant of 58°44'.
From table of sines l8°24' . - - '3156
3' - ■ - 8
sin i8''27'
From table of cosines i8°24'
3'
cos i8°27'
From table of tangents s8°42'
2'
tan 58°44'
From table of secants 58°42'
2'
sec 58°44'
RULE VII. To find the logarithmic sine, tangen
•3164
•9489
3
•94S6 ■
1-6447
21
1-6468
1-9249
18
I -9267
, cosine, cotangent,
secant, or cosecant of an angle less than 90°, Proceed as in Rule VI.,
using the proper table.
iV(?/^.-— The sines of all angles, and the tangents of angles less than
45°, being less than unity, the logarithms of these sines and tangents
are preceded by a negative index. In order to avoid the writing of
these negative indices the number 10 is added to the nvz/ value of the
log. sin. log. tan. etc., and the number so found is entered in the
tables. In all calculations this must be borne in mind, and allowance
must be made.
RULE VIII. To find the angle in degrees and minutes, or in
degrees and decimals of a degree, corresponding to any given natural
or logarithmic sine, cosine, tangent, etc. Find in the proper table the
number nearest to that given, interpolating, if necessary, by means of
the difference columns ; and by reversing the process of Rules VI. and
VII. obtain the corresponding number of degrees and minutes, or
degrees and decimals of a degree.
The preceding explanations are easily applicable to the remaining
EXPLANATIONS AND RULES. 9
tables of squares, square roots, and reciprocals. With regard to the
tables of squares and square roots, it is to be noticed that while the
square of such a number as 528 is found from the square of 5'28 simply
by multiplying by a power of 10, a similar relation does not hold
always in the case of the square root. It is necessary, therefore, to have
two tables of square roots — one extending from i or 100 to 9.99 or 999,
and the other from 10 or 1000 to 99.99 or 9999.*
RULE IX. To find the Neperian or hyperbolic logarithm of a
number. — If the number be greater than i and less than 10 its Neperian
logarithm is found directly from the proper table in the manner
explained in Rule 111. If the number is greater than 10 or less than
I, it may always be expressed as the product of two factors, of which
one is a power of 10, and the other a n'jmber greater than i and less
than 10 ; the latter being simply the original series of figures with the
decimal point suitably moved. The sum of the Neperian logarithms of
these two factors is the Neperian logarithm of the given number. A
table of Neperian logarithms of powers of 10 is given on pp. 54, 55.
Exaviplcs.
Find the Neperian logarithms of 3'24i, 324T, and •0003241.
(I)
log 3-241 - - is ri759
(2)
log^324-i=log^3-24ixio2
log^3-24i - - 1-1759
loggio- - - 4-6052
logg324'i - - 5781 1
(3) iog^-ooo324r =log^3'24i x 10-*
log^3-24i - - I-I759
log IQ-* - - 10-7897
loge-ooo324r - - 9'9656
In calculating the value of a fraction, of which the numerator and
denominator each consists of two or more factors, it is often of advan-
tage, instead of siibtractiui^ the logarithms Qf the denominator factors,
to add in the logarithms of their reciprocals — the complemental
logarithms or co-logs as they are sometimes called.
• That which causes the necessity for two s\ich tables gives rise also to the necessity foi
watchfulness on the part of the calculator. Probably the 'jest preventive against niisiakea
is the habit, excellent in all calculations, of making a mental estimate of the number to be
expected as the result of taking the square root. Mistakes may also be avoided easily
and with little loss of time by comparing with the table of squares.
lo MATHEMATICAL TABLES.
RULE X. To find the mantissa of the logarithm of a reciprocal. —
Write down the difference between the mantissa of the logarithm of
the given number and I'oooo ; or simply, commencing at the left hand,
write down the series of numbers which will make each figure of the
mantissa of the logarithm of the number up to be equal to 9, except
the last significant figure, which must be made up to 10.
RULE XL Otherwise : To find the mantissa of the logarithm of a
reciprocal. — Proceed as in Rule II L, using the Table of Logarithms of
Reciprocals.
RULE XIL To find the index for the logarithm of a reciprocal. —
If the given number consist of a whole number and a decimal,
the index is ecjual to the number of the digits which constitute the
whole number, and is negative. If the given number is a decimal
without a whole number the index is equal to the number of zeros
which follow the decimal point and is positive.
Examples. Find
log — ^
237-4
Mantissa of co-log from table, p. 16. - - '6246.
Index —3
log ="6246-3 or r6246
237'4
log = '6246 + 2 or 2 '6246
-002374
Re7nark. — In finding the logarithm of the reciprocal of a trigono-
metrical function it is only necessary to subtract the tabular logarithm
from 10. This will readily be seen from an example.
Since (see Note to Rule VII.),
log sin 36° = Tab. log sin 36°- 10
we have
log i— —^ j= -log sin 36°= 10- Tab. log sin 36°.
The subtraction from 10 is most easily performed by writing down
the numbers which make up the figures of the tabular logarithm to 9,
as in Rule X., except in the case of the last significant figure, for
which write the number which, if added to it, would make it up to
be 10.
An example of calculation is given here in order to show a con-
venient way of writing down the given numbers and their logarithms.
It is scarcely possible to overestimate the importance of strict ad-
herence to method ; for instance, in physical calculations. In the
EXPLANATIONS AND RULES. n
first place errors are thus most easily avoided or detected ; and it is
also frequently useful to be able to return on the arithmetical steps in
order to make an alteration of form, or, if improved data are forth-
coming, to obtain a result true to a closer degree of approximation.
Example. Calculate the value of
2734XOT325 X sin 29°
I4"23 X •00176 X tan 34°"
Numbers. Logs.
27'34 -4368 + 1
0-1325 (see table, p. 12) - '1222-1
sin 29° 9-6856-10
1/14-23 -8468-2
1/-00176 7545 + 2
i/tan34° 0-1710 + 0
-0169 + 2
Result - - 1-040x10^.
In bringing out a Second Edition of this Rook of Tables, I desire
to acknowledge the kind assistance of friends ; and in particular the
valuable criticisms and suggestions which I have received from Prof.
Sir G. Gabriel Stokes, Bart., Pres. R. S., and from Prof. G. H.
Darwin, F.R.S.
J. T. B.
February 18, 1890. •
12
LOGARITHMS 1000 to 1409
.
0
1
2
3
4
5
6
7
8
9
100
oooo
0004
0009
0013
0017
0022
0026
0030
0035
0039
101
102
103
0043
0086
012S
0048
0090
0133
0052
0095
0137
0056
0099
0141
0060
0103
0145
0065
0107
0149
0069
OIII
0154
0073
0116
015S
0077
0120
0162
0082
0124
0166
104
105
106
0170
0212
0253
0175
0216
0257
0179
0220
0261
0183
0224
0265
01S7
0228
0269
0191
0233
0273
0195
0237
0278
0199
0241
0282
0204
0245
0286
020S
0249
0290
107
108
109
0294
0334
0374
0298
0338
0378
0302
0342
0382
0306
0346
0386
0310
03 5&
0390
0314
0354
0394
0318
0358
0398
0322
0362
0402
0326
0366
0406
0330
0370
0410
110
0414
0418
0422
0426
0430
0434
043S
0441
0445
0449
111
112
113
0453
0492
0531
0457
0496
0535
0461
0500
0538
0465
0504
0542
0469
0508
054&
0473
0512
0550
0477
0515
0554
0481
0519
0558
0484
0523
0561
0488
0527
0565
114
115
116
117
118
119
120
121
122
123
124
125
126
0569
0607
0645
0573
06 1 1
. 064S
0577
0615
0652
0580
061S
0656
0584
0622
0660
058S
0626
0663
0592
0630
0667
0596
0633
0671
0599
0637
0674
0603
0641
0678
06S2
0719
0755
0686
0722
0759
^689
0726
0763
0693
0730
0766
0697
0734
0770
0700
0737
0774
0704
0741
0777
0708
0745
0781
07 1 1
0748
0785
0715
0752
0758
0792
0795
0799
0803
0S06
0810
0813
0817
0821
0S24
0828
0864
0S99
0831
0867
0903
0S35
0871
0906
0S39
0S74
0910
0842
0878
0913
0846
0881
0917
0849
0885
0920
0853
0858
0924
0856
0892
0927
0S60
0896
0931
0934
0969
1004
0938
0973
1007
0941
0976
lOII
0945
09S0
1014
0948
0983
1017
0952
0986
1021
0955
0990
1024
0959
0993
1028
0962
0997
103 1
0966
1000
1035
127
128.
129
103S
1072
1 106
1041
1075
1 109
1045
1079
1113
1048
1082
1116
1052
1086
1119
1055
1089
1123
1059
1092
1126
1062
1096
1 1 29
1065
1099
"33
1069
1 103
1136
130
"39
1 143
1 146
1 149
"53
1156
1159
1 163
1 166
1 1 69
131
132
133
1173
1206
1239
1176
1209
1242
1179
1212
1245
11S3
1216
1248
1 186
1219
1252
1 189
1222
1255
"93
1225
1258
1 196
1229
1 26 1
"99
1232
1265
1202
1235
1268
134
135
136
1271
1303
1335
1274
1307
1339
1278
1310
1342
1281
1313
1345
12S4
1316
134S
1287
1319
1351
1290
1323
1355
1294
1326
1358
1297
1329
1361
1300
1332
1364
137
133
139
1367
1399
1430
1370
1402
1433
1374
1405
1436
1377
140S
1440
13S0
1411
1443
1383
1414
1446
1386
1418
1449
13S9
1421
1452
1392
1424
1455
1396
1427
1458
140
1461
1464
1467
1471
1474
1477
14S0
14S3
i486
1489
LOGARITHMS OF RECIPROCALS looo to 1409.
13
0
1
2
3
4
6
6
7
8
9
100
9996
9991
99S7
9983
9978
9974
9970
9965
9961
101
102
103
9957
9914
9872
9952
9910
^ 9867
9948
9905
9863
9944
9901
9859
9940
9897
9855
9935
9893
9851
9931
9^)89
9846
9927
9884
9842
9923
9880
9838
9918
9S76
9834
104
105
106
9830
9788
9747
9S25
9784
9743
9821
9780
9739
9817
9776
9735
9813
9772
9731
9809
9767
9727
9805
9763
9722
9801
9759
9718
9796
9755
9714
9792
9751
9710
107
108
109
9706
9666
9626
9702
9662
9622
9698
0658
9618
9694
9654
9614
9690
9650
9610
96S6
9646
9606
9682
9642
9602
9678
9638
9598
9674
9634
9594
9670
9630
9590
110
9586
9582
9578
9574
9570
9566
9562
9559
9555
9551
111
112
113
9547
9508
9469
9543
9504
9465
9539
9500
9462
9535
9496
9458
9531
9492
9454
9527
9488
9450
9523
9485
9446
9519
9481
9442
9516
9477
9439
9512
9473
9435
114
115
116
9431
9393
9355
9427
9389
9352
9423
93S5
9348
9420
9382
9344
9416
9378
9340
9412
9374
9337
9408
9370
9333
9404
9367
9329
9401
9363
9326
9397
9359
9322
117
118
119
9318
9281
9245
9314
9278
9241
93"
9274
9237
9307
9270
9234
9303
9266
9230
9300
9263
9226
9296
9259
9223
9292
9255
9219
9289
9252
9215
9285
9248
9212
120
9208
9205
9201
9197
9194
9190
9187
9183
9179
9176
121
122
123
9172
9136
9101
9169
9133
9097
9165
9129
9094
9161
9126
9090
915S
9122
9087
9154
9119
9083
9151
9115
9080
9147
9112
9076
9144
910S
9073
9140
9104
9069
124
125
126
9066
9031
8996
9062
9027
8993
9059
9024
89S9
9055
9020
89S6
9052
9017
8983
9048
9014
8979
9045
9010
8976
9041
9007
8972
9038
9003
8969
9034
9000
8965
127
128
129
8962
8928
8894
8959
8925
8891
8955
8921
8887
8952
8918
8884
S948
8914
8881
8945
891 1
8877
8941
8908
8874
S938
8904
8871
8935
8901
8867
8931
8897
8S64
130
8861
8857
8854
8851
8847
8844
8841
8837
8834
8831
131
132
133
8827
8794
8761
8824
8791
8758
8821
8788
8755
8817
8784
8752
8814
8781
8748
881 1
8778
874s
8807
8775
8742
8804
8771
8739
8801
8768
8735
8798
8765
8732
134
135
136
8729
8697
8665
8726
8693
8661
8722
8690
8658
8719
8687
8655
8716
8684
8652
8713
8681
8649
8710
8677
8645
8706
8674
S642
8703
8671
8639
8700
8668
8636
137
138
139
8633
8601
8570
8630
8598
8567
8626
8595
8564
8623
8592
8560
8620
8589
8557
8617
8586
8554
8614
8582
8551
861 1
8579
8548
8608
8576
8545
8604
8573
8542
140
8539
8536
8533
8529
8526
S523
8520
8517
8514
8511
14
LOGARITHMS.
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
0
oooo
1
2
3
4
5
6
7
8
9
12 3
4 5 6
7 8 9
0043
0086
0128
0170
0212
0253
0294
0334
0374
4 8 12
17 21 25
29 33 37
0414
0792
1 139
0453
0828
111?,
0492
0864
1206
0531
0899
1239
0569
0934
1271
0607
0969
1303
0645
1004
1335
0682
1038
1367
0719
1072
1399
0755
1 106
1430
4 8 II
3 7 10
3 6 10
15 19 23
14 17 21
13 16 19
26 30 34
24 28 31
23 26 29
146 1
1761
2041
1492
1790
2068
1523
1818
2095
1553
1847
2122
1584
1875
2148
1614
1903
2175
1644
1931
2201
1673
1959
2227
1703
1987
2253
1732
2014
2279
369
3 6 8
3 5 8
12 15 18
II 14 17
II 13 16
21 24 27
20 22 25
iS 21 24
2304
2553
2788
2330
2577
2810
2355
2601
2833
2380
2625
2856
2405
2648
2878
2430
2672
2900
2455
2695
2923
2480
2718
2945
2504
2742
2967
2529
2765
2989
2 5 7
257
247
10 12 15
9 12 14
9 II 13
17 20 22
16 19 21
16 18 20
3010
3032
3054
3075
3096
3118
3139
3160
31S1
3201
246
8 II 13
15 17 19
3222
3424
3617
3243
3444
3636
3263
3464
3655
3284
3483
3674
3304
3502
3692
3324
3522
3711
3345
354^^
3729
3365
3560
3747
3385
3579
3766
3404
3598
3784
246
246
246
8 10 12
8 10 12
7 9 II
14 16 18
14 15 17
13 15 17
3802
3979
4150
3820
3997
4166
3838
4014
41S3
3856
4031
4200
3874
4048
4216
3892
4065
4232
3909
4082
4249
3927
4099
4265
3945
4116
4281
3962
4133
4298
2 4 5
2 3, 5
2 3 5
7 9 II
7 9 10
7 8 10
12 14 16
12 14 15
II 13 15
4314
4472
4624
4330
4487
4639
4346
4502
4654
4362
4518
4669
4378
4533
4683
4393
45,48
469S
4409
4564
4713
4425
4579
4728
4440
4594
4742
4456
4609
4757
2 3 5
2 3 5
I 3 4
6 8. 9
.6 8 9
679
II 13 14
II 12 14
10 12 13
4771
4786
4800
4814
4829
4843
4857
4871
4886
4900
I 3 4
679
10 II 13
4914
5051
51S5
4928
5065
5198
4942
5079
-52 II
4955
5092
5224
4969
5105
5237
4983
5119
5250
4997
5132
5263
501 1
5145
5276
5024
5159
5289
5038
5172
5302
I 3 4
I 3 4
I 3 4
678
5 7 8
568
10 II 12
9 11 12
9 10 12
5315
5441
5563
5328
5453
5575
5340
5465
5587
5353
5478
5599
5366
5490
56 1 1
537S
5502
5623
5391
5514
5635
5403
5527
5647
5416
5539
5658
5428
5551
5670
I 3 4
I 2 4
I 2 4
568
567
5 6 7
9 10 II
9 10 II
8 10 II
56S2
5798
59"
5694
5809
5922
5705
5821
5933
5717
5832
5944
5729
5843
5955
5740
5855
5966
5752
5866
5977
5763
5877
5988
5775
58SS
5999
57S6
5899
6010
I 2 3
I 2 3
123
5 6 7
5 6 7
4 5 7
8 9 10
8 9 10
8 9 10
6021
6128
6232
6335
6031
6042
6053
6064
6075
6085
6096
6107
6117
I 2 3
4 5 6
8 9 10
6138
6243
6345
6149
6253
6355
6i6o
6263
6365
6170
6274
6375
6180
6284
6385
6191
6294
6395
6201
6304
6405
6212
6314
6415
6222
6325
6425
I 2 3
I 2 3
I 2 3
456
456
4 5 6
7 8 9
7 8 9
7 8 9
6435
6532
6628
6444
6542
6637
6454
6551
6646
6464
6561
6656
6474
6571
6665
6484
6580
6675
6493
6590
6684
6503
6599
6693
6513
6609
6702
6522
6618
6712
X 2 3
I 2 3
I 2 3
456
4 5 6
456
7 8 9
7 8 9
7 7 8
6721
6812
6902
6730
6S21
691 1
6739
6830
6920
6749
6839
6928
6758
6848
6937
6767
6857
6946
6776
6866
6955
6785
6875
6964
6794
6S84
6972
6803
6893
6981
I 2 3
I 2 3
I 2 3
4 S 5
4 4 5
4 4 5
6 7 8
678
678
6990
6998
7007
7016
7024
7033
7042
7050
7059
7067
I 2 3
3 4 5
678
7076
7160
7243
7084
7168
7251
7093
7177
7259
7101
7185
7267
7110
7193
7275
7118
7202
7284
7126
7210
7292
7135
7218
7300
7143
7226
7308
7152
7235
7316
I 2 3
I 2 2
I 2 2
3 4 5
3 4 5
3 4 5
678
6 7 7
667
7324
7332
7340
7348
7356
7364
7372
7380
73S8
7396
I 2 2
3 4 5
667
LOGARITH^IS.
15
65
66
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76.
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
0
1 2 3
4 5
6
7
i "
9 12 2
4 5 e
1
.780
7404
7412 7419
7427
7435
7443 7451
7459
7466
7474
I 2 2
3 4 f
5 6 7
7482
7559
7^54
7490
7566
7042
7497
7574
7649
7505
7582
7657
7513
7589
7664
7520
7597
7672
752S
7604
7679
7536
7612
76S6
7543
7619
7694
7551
7627
7701
122
122
I I 2
3 4 S
3 4 5
3 4 4
5 6 7
5 6 7
5 6 7
7709
7782
7^53
7716
77^^9
7860
7723
7796
7S0S
7731
7803
7875
7738
7810
78S2
7745
7S18
7SS9
7752
7825
7896
7760
7832
7903
7767
7839
791C
7774
7S46
7917
I I 2
I I 2
I I 2
3 4 4
3 4 4
M 4 4
5 6 7
566
5 6 6
7924
7993
8062
7931
8000
8069
7938
8007
8075
7945
8014
8082
7952
8021
8089
7959
8028
8096
7966
8035
8102
7973
8041
8109
798c
S04S
8116
7987
8055
8122
I I 2 3 3 4
I I 2 3 3 4
I I 2! 3 3 4
566
5 5 6
5 5 6
8129
8136
8142
8149
8156
8162
S169
8176
8182
8189
I I 2
3 3 4
5 5 6
8195
8261
S325
S202
8267
8331
8209
8274
8338
8215
8280
8344
8222
8287
8351
8228
8293
8357
8235
8299
83&3
8241
8306
8370
8248
8312
8376
8254
8319
8382
112
I I 2
I I 2
3 3 4
3 3 4
3 3 4
5 5 6
5 5 6
4 5 6
83SS
8451
8513
8395
8457
8519
8401
8463
8525
8407
8470
8531
8414
8476
8537
8420
84S2
8543
8426
84S8
8549
8432
8494
8555
8439
8500
8561
8445
8506
8567
I I 2
I I 2
I I 2
234
234
234
456
456
4 5 5
S573
8633
8692
8579
8639
8698
8585
8645
8704
8591
8651
8710
8597
8657
8716
8603
8663
8722
S609
8669
8727
8615
8675
8733
8621
8681
8739
8627
8686
8745
I I 2
I I 2
I I 2
234
234
,234
4 5 5
4 5 5
4 5 5
8751
8756
8762
876S
8774
8779
8785
8791
8797
8802
112
233
4 5 5
8808
8865
8921
8814
8S71
8927
8820
8876
8932
8825
88S2
8938
8831
8887
8943
8837
8S93
8949
8S42
8S99
8954
8848
8904
8960
8S54
8910
8965
8859
S915
8971
112
112
I I 2
233
233
233
4 5 5
4 4 5
4 4 5
8976
9031
9085
8982
9036
9090
89S7
9042
9096
8993
9047
9101
8998
9053
9106
9004
9058
9112
9009
9063
9117
9015
9069
9122
9020
9074
9128
9025
9079
9133
I I 2
112
I I 2
2 J J
233
233
4 4 5
4 4 5
4 4 5
9138
9191
9243
9143
9196
9248
9149
9201
9253
9154
9206
9258
9159
9212
9263
9165
9217
9269
9170
9222
9274
9175
9227
9279
9180
9232
9284
9186
9238
9289
112
112
I I 2
233
233
233
4 4 5
4 4 5
4 4 5
9294
9299
9304
9309
9315
9320
9325
9330
9335
9340
112
233
4 4 5
9345
9395
9445
9350
9400
9450
9355
9405
9455
9360
9410
9460
9365
9415
9465
9370
9420
9469
9375
9425
9474
9380
9430
9479
9385
9435
9484
9390
9440
94S9
I I 2
0 I I
01 I
2 3 3
223
223
4 4 5
3 4 4
3 4 4
9494
9542
9590
9499
9547
9595
9504
9552
9600
9509
9557
9605
9513
9562
9609
9518
9566
9614
9523
9571
9619
9528
9576
9624
9533
9581
9628
9538
9586
9633
0 I I
0 I I
0 I I
223
223
223
3 4 4
3 4 4
3 4 4
9638
9685
9731
9643
9689
9736
9647
9694
9741
9652
9699
9745
9657
9703
9750
9661
9708
9754
9666
9713
9759
9671
9717
9763
9675
9722
9768
9680
9727
9773
0 I I
0 1 1
0 I I
223
223
223
3 4 4
3 4 4
3 4 4
95
96
97
98
99
9777
9782
9786
9791
9795
9800
9S05
9809
9814
9818
0 I I
223
3 4 4
9823
9868
9912
9827
9872
9917
9832
9827
9921
9836
9881
9926
9841
9S86
9930
9845
9S90
9934
9S50
9S94
9039
9854
9899
9943
9859
9903
9948
9863
9908
9952
0 I I
0 I I
0 I I
223
223
2. 2 3
3 4 4
3 4 4
3 4 4
9956
9961
99659969
9974 9978
99S3
9987
9991
9996
0 I I
223
3 3 4
i6
LOGARITHMS OF RECIPROCALS.
10
n
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
0
1
2
3
4
5
6
7
8
9
12 3
4 5 6
7 8 9
9957
9914
9872
9S30
97S8
9747
9706
9666
9626
4 8 12
17 21 25
29 21 2,7
9586
9208
8861
9547
9172
8827
9508
9136
8794
9469
9101
8761
9431
9066
8729
9393
9031
8697
9355
8996
8665
9318
8962
8633
9281
8928
8601
9245
8S94
8570
4 8 II
3 7 10
3 6 10
15 19 23
14 17 21
13 16 19
26 30 34
24 28 31
23 26 29
8539
8239
7959
7696
7447
7212
S508
8210
7932
8477
8182
7905
8447
8153
7878
8416
8125
7852
8386
8097
7825
8356
8069
7799
8327
8041
7773
8297
8013
7747
8268
7986
7721
369
3 6 8
3 5 8
12 15 18
11 14 17
II 13 16
21 24 27
20 22 25
18 21 24
7670
7423
7190
7645
7399
7167
7620
7375
7144
7595
7352
7122
7570
7328
7100
7545
7305
7077
7520
7282
7055
7496
7258
7033
7471
7235
701 1
2 5 7
2 5 7
2 4 7
10 12 15
9 12 14
9 II 13
17 20 22
16 19 21
16 18 20
6990
6968
6946
6925
6904
6882
6861
6840
6819
6799
246
8 u 13
15 17 19
6778
6576
6383
6757
6556
6364
6737
6536
6345
6716
6517
6326
6696
6497
6308
6676
6478
6289
6655
6459
6271
6635
6440
6253
6615
6421
6234
6596
6402
6216
246
246
2 4 6
8 10 12
8 10 12
7 9 II
14 16 18
14 15 17
13 15 17
6198
6021
5850
6180
6003
5834
6162
5986
5817
6144
5969
5800
6126
5952
578^
6108
5935
5768
6091
5918
5751
6073
5901
5735
6055
5884
5719
6038
5867
5702
245
235
2 3 5
7, 9 II
7 9 10
7 8 10
12 14 x6
12 14 IS
II 13 15
5686
S52S
5376
5670
5513
5361
5654
5498
5346
5638
5482
5331
5622
5467
5317
5607
5452
5302
5591
5436
5287
5575
5421
5272
5560
5406
5258
5544
5391
5243
2 3 5
235
I 3 4
689
689
679
II 13 14
II 12 14
10 12 13
5229
5214
5200
5186
5171
5157
5143
5129
5114
5100
I 3 4
679
10 II 13
31
32
33
34
35
36
37
33
39
40
«1
42
43
44
45
46
47
48
49
50
51
52
53
54
50S6
4948
4815
5072
4935
4802
5058
4921
4789
5045
4908
4776
5031
4895
4763
5017
48S1
4750
5003
486S
4737
4989
4855
4724
4976
4841
4711
4962
4828
469S
I 3 4
I 3 4
I 3 4
678
5 7 8
568
10 II 12
9 II 12
9 10 12
4685
4559
4437
4672
4547
4425
4660
4535
4413
4647
4522
4401
4634
4510
4389
4622
4498
4377
4609
4486
4365
4597
4473
4353
4584
4461
4342
4572
4449
4330
I 3 4
I 2 4
I 2 4
568
5 6 7
5 6 7
9 10 II
9 10 II
9 10 II
4318
4202
4089
4306
4191
4078
4295
4179
4067
4283
4168
4056
4271
4157
4045
4260
4145
4034
4248
4134
4023
4237
4123
4012
4225
4112
4001
4214
4101
3990
I 2 3
I 2 3
I 2 3
5 i 7
567
467
8 9 10
8 9 10
8 9 10
3979
39^9
3958
3947
3936
3925
3915
3904
3893
3883
I 2 3
456
7 8 -9
3872
376S
3665
3862
3757
3655
3851
3747
3645
3840
3737
3635
3830
3726
3625
3820
3716
3615
3809
3706
3605
3799
3696
3595
3788
3686
3585
377S
3675
3575
I 2 3
I 2 3
I 2 3
456
456
456
7 8 9
7 8 9
7 8 9
3565
3468
3372
3556
3458
3363
3546
3449
3354
3536
3439
3344
3526
3429
3335
3516
3420
3325
3507
3410
3316
3497
3401
3307
3487
3391
3298
3478
3382
328S
I 2 3
I 2 3
I 2 3
456
456
4 5 6
7 8 9
7 8 9
7 7 8
3279
318S
3098
3270
3179
3089
3261
3170
3080
3251
3161
3071
3242
3152
3063
3233
3143
3054
3224
3134
3045
3215
3125
3036
3206
3116
3028
3197
3107
3019
I 2 3
I 2 3
I 2 3
4 5 5
4 4 5
4 4 5
678
678
678
3010
3002
2993
2984
2976
2967
2958
2950
2941
2933
I 2 3
3 4 5
678
2924
2840
2757
2916
2832
2719
2907
2823
2741
2899
2815
2733
2890
2807
2725
2882
2798
2716
2874
2790
2708
2865
2782
2700
2857
2774
2692
2848
2765
2684
I 2 3
I 2 2
I 2 2
3 4 5
3 4 5
3 4 5
678
6 7 7
667
2676
266S
2660
2652
2644
2636
2628
2620
2612
2604
I 2 2
3 4 5
667
H.'SL — If umbers tn Difference Columns to be Subtracted, not Added.
LOGARITHMS
OF
RECIPROCALS.
17
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80,
81
82
83
84
85
86
87
88
89
1 °
1
2
3
4
5
6
7
8
9
12 3
4 6 6 7 8 9 1
2596
2518
2588
2510
2581
2503
2573
2495
2565
24S7
2557
2480
2549
2472
2541
2464
2534 2526
2457 2449
I 2 2
I 2 2
3 4 5
3 4 5
5 6 7
5 6 7
2441
2366
2291
2434
2358
2284
2426
2351
2277
2418
2343
2269
2411
2336
2202
2403
2328
2255
2396
2321
224S
238S
2314
2240
2381
2306
2233
2373
2299
2226
I 2 2
I I 2
112
3 4 5
3 4 4
3 4 4
5 6 7
5 6 7
5 6 7
2218
2211
2204
2197
2190' 2182 2175
2168
2161
2154
I I 2
3 4 4
566
2147
2076
2007
2140
2069
2000
2I32'2I25
2062 2055
1993! 1986
2Il8,21IIJ2I04
2048 2041 j 2034
1979; I9721 1965
2097
2027
1959
2090 2083
2020 2013
19521 1945
I I 2
I I 2
112
3 4 4
3 3 4
3 3 4
5 6 6
566
5 5 6
1938
1871
1805
193'
1864
i79«
I925J I918
IS58 1S5I
I79I 1785
I9II
1844
1778
1904
1838
1772
1898
1S31
1765
1891
1824
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1580
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1630
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1186
1296
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1290
1232
1175
1284
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223 344
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0453
0448
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0438
0434
0429
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0419
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0 I I
0 I I
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223344
91
92
93
94
95
96
97
98
99
0410
0362
03 1-5
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0357
0311
0400
0353
0306
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0301
0391
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0376
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0372
0325
0278
0367
0320
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223
223
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3 4 4
3 4 4
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0246
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0155
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0195
0150
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0191
0146
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0 I I
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223
3 4 4
3 4 4
3 4 4
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0044
0128
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0079
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0119
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01 14
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0026
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0066
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0106
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0097
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0092
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oil
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3 4 4
3 4 4
3 3 4
M.& — Nnaibers in IHfferenee Columns to be Subtracted, not Added.
6
18
ANTILOGARITHMS.
•00
■01
•02
•03
•04
•05
•06
•07
•08
■09
•10
0
1
2
3
4
5
6
7
IOI6
8
9
12 3
4 5 6789
.
lOOO
1002
1 00s
1007
1009
1012
1014
1019
1021
001
I 11 222
1023
10^7
1072
1026
1050
1074
1028
1052
1076
1030
1054
1079
1033
1057
1081
1035
1059
1084
1038
1002
1086
1040
1064
1089
1042
1067
1091
1045
1069
1094
001
001
001
III
III
III
222
2 2,2
222
1096
II22
1 148
1099
1125
115^
1 102
1127
"53
1 104
1130
1156
1107
1132
1159
1 109
"35
1161
1112
1138
1164
III4
1 140
II67
1117
"43
1169
1119
1 146
1172
oil
oil
oil
112
112
112
'2 2 2
2 2 2
2 2 2
"75
1202
1230
1 178
1205
1233
1180
1208
1236
1183
I3II
1239
1186
1213
1242
1189
1216
1245
1191
1219
1247
II94
1222
1250
"97
1225
1253
"99
1227
1256
oil
oil
oil
112
112
112
2 2 2
223
223
1259
1262
1265
1268
1271
1274
1276
1279
1282
1285
oil
112
223
•11
•12
•13
•14
•15
•16
•17
•18
•19
•20
•21
•22
•23
■24
•25
•26
•27
■28
■29
•30
■31
•32
•33
•34
•35
■36
•37
■38
■39
■40
•41
■42
•43
1288
1318
1349
1291
1321
1352
1294
1324
1355
1297
1327
1358
1300
1330
1361
1303
1334
1365
1306
1337
1368
1309
1340
I37I
1312
1343
1374
1315
1346
^377
oil
oil
oil
122
122
122
223
223
233
1380
1413
1445
1384
1416
1449
1387
1419
1452
1390
1422
1455
1393
1426
1459
1396
1429
1462
1400
1432
1466
1403
1435
1469
1406
1439
1472
1409
1442
1476
oil
oil
oil
122
122
122
23.3
233
233
1479
1514
1549
1483
1517
1552
i486
1521
1556
1489
1524
1560
1493
1528
1563
1496
1531
1507
1500
1535
1570
1503
1538
1574
1507
1542
1578
1510
1545
1581
oil
oil
oil
I 22
122
122
233
233
3 3 3
1585
1589
1592
1596
1600
1603
1607
161I
1614
1618
oil
122
3 3 3
1622
1660
1698
1626
1663
1702
1629
1667
1706
1633
1671
1710
1637
1675
1714
1641
1679
1718
1644
1683
1722
1648
1687
1726
1652
1690
J 730
1656
1694
1734
oil
oil
oil
222
222
222
3 3 3
3 3 3
3 3 4
1738
1778
1820
1742
1782
1824
1746
1786
1828
1750
1791
1832
1754
1795
1837
175^
1799
1841
1762
1803
1845
1766
1807
1849
1770
1811
1854
1774
1816
1858
oil
oil
oil
222
2 22
223
3 3 4
3 3 4
3 3 4
1
1862
1905
1950
1866
1910
1954
1871
1914
1959
1875
1919
1963
2009
1879
1923
1968
1884
1928
1972
1S88
1932
1977
1892
1936
1982
1897
1941
1986
1 901
1945
1991
oil
oil
oil
223
223
223
3 3 4
3 4 4
3 4 4
1995
2000
2004
2014
2018
2023
2028
2032
2037
oil
223
3 4 4
2042
2089
2138
2046
2094
2143
2051
2099
2148
2056
2104
2153
2061
2109
2158
2065
2113
2163
2070
2Il8
2168
2075
2123
2173
2080
2128
2178
2084
2133
2183
oil
oil
oil
223
223
223
3 4 4
3 4 4
3 4 4
2188
2239
2291
2193
2244
2296
2198
2249
2301
2203
2254
2307
2208
2259
2312
2213
2265
2317
2218
2270
2323
2223
2275
2328
2228
2280
2333
2234
2286
2339
112
112
112
233
233
233
4 4 5
4 4 5
4 4 5
2344
2399
2455
2350
2404
2460
2355
2410
2466
2360
2415
2472
2366
2421
2477
2371
2427
2483
2377
2432
2489
2382
2438
2495
2388
2443
2500
2393
2449
2506
112
112
112
233
233
233
4 4 5
4 4 5
4 5 5
2512
2518
2523
2529
2535
2541
2547
2553
2559
2564
112
234
4 5 5
2570
2630
2692
2576
2636
2698
2582
2642
2704
2588
2649
2710
2594
2655
2716
2600
2661
2723
2606
2667
2729
2612
2673
2735
2618
2679
2742
2624
2685
2748
112
112
112
234
234
3 3 4
4 5 5
456
456
•44
•45
•46
■47
■48
■49
2754
2818
2884
2761
2825
2891
2767
2831
2897
2773
2838
2904
2780
2844
291 1
2786
2851
2917
2793
2858
2924
2799
2864
2931
2805
2871
2938
2812
2877
2944
112
112
112
3 3 4
3 3 4
3 3 4
456
5 5 6
5 5 6
2951
3020
3090
2958
3027
3097
2965
3034
3105
2972
3041
3112
2979
3048
3"9
2985
3055
3126
2992
3062
3133
2999
3069
3006
3076
3148
3013
3083
3155
I 12
112
112
3 3 4
3 4 -i
3 4 4
5 5 6
566
566
ANTILOGARITHMS.
19
•50
0
1
2
3
4
5
6
7
8
9
1234 5 67 8 9
3162
3170
3177
31S4
3192
3199
3206
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3221
322fa
112
3 4 4
567
•51
•52
•53
•54
•55
•56
•57
•58
•59
•60
•61
•62
•63
•64
•65
•66
•67
•68
•69
•70
•71
•72
•73
•74
•75
•76
•77
•78
•79
•80
•81
•82
•83
•84
•85
•86
•87
•88
•89
•90
•91
•92
•93
•94
•95
•96
•97
•98
•99
3^30
3311
3388
3243
3319
3396
3251
3327
3404
3258
3334
3412
3266
3342
3420
3273
335c
3428
3281
3357
3436
328c
336=
3443
329^
3372
3451
3304
3381
3459
122
122
122
3 4 5
3 4 5
3 4 5
567
667
3467
3548
3631
3475
3556
3t'39
3483
3565
3648
3491
3573
3(356
3499
3581
3064
3508
3589
3673
3516
3597
3681
3524
3606
3690
3532
3614
3698
3540
3622
3707
122
122
123
3 4 5
3 4 5
3 4 5
667
677
678
3715
3802
3890
3724
3811
3899
3733
3819
3908
3741
3828
3917
3750
3837
3926
3758
3846
3936
37<i7
3855
3945
3776
3864
3954
3784
3873
3963
3793
3882
3972
123
123
I 2 3
3 4 5
4 4 5
4 5 5
678
678
678
3981
3990
3999
4009
4018
4027
4036
4046
4055
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4074
4109
4260
4083
4178
4276
4093
4188
4285
4102
4198
4295
4111
4207
4305
4121
4217
4315
4130
4227
4325
4140
4236
4335
4150
4246
4345
4159
4256
4355
I 2 3
123
123
456789
456789
456789
4365
44(^7
4571
4375
4477
4581
4385
4487
4592
4395
4498
4603
4406
4508
4613
4416
4519
4624
4426
4529
4634
4436
4539
4645
4446
4550
4656
4457
4560
4667
I 2 3
123
I 2 3
4 5 6
456
4 5 6
789
789
7 9 10
4677
4786
4890
4688
4797
4909
4699
4808
4920
4710
4819
4932
4721
4831
4943
4732
4842
4955
4742
4853
4966
4753
4864
4977
4764
4875
4989
4775
4887
5000
123
I 2 3
I 2 3
4 5 7
467
567
8 9 10
8 9 10
8 9 10
5012
5023
5035
5047
5058
5070
5082
5093
5105
5T-T^7
124
567
8 9 II
5129
5248
5370
5140
5260
5383
S152
5272
5395
5164
5284
5408
5176
5297
5420
5188
5309
5433
5200
5321
5445
5212
5333
5458
5224
5346
5470
5236
5358
5483
I 2 4
I 2 4
I 3 4
S 6 7
5 6 7
568
8 10 II
9 10 II
9 10 II
5495
5^23
5754
5508
5636
5768
5521
5649
5781
5534
5662
5794
5546
5675
5808
5559
5689
5821
5572
5702
5834
5585
5715
5848
5598
5728
5861
5610
5741
5875
I 3 4
I 3 4
I 3 4
568
5 7 8
5 7 8
9 10 12
9 10 12
9 II 12
5800
6026
6166
5902
6039
6180
5916
6053
6194
5929
6067
6209
5943
6081
6223
5957
609s
6237
5970
6109
6252
5984
6124
6266
5998
6138
6281
6012
6152
6295
I 3 4
I 3 4
I 3 4
5 7 8
678
679
10 II 12
10 II 13
10 II 13
6310
6324
6339
6353
6368
6383
6397
6412
6427
6442
r 3 4
679
10 13 13
6457
6607
6761
6471
6622
6776
6486
6637
6792
6501
6653
6808
6516
6668
6823
6531
6683
6839
6546
6699
6855
6561
6714
6871
6577
6730
6S87
6592
6745
6992
235
235
235
689
689
689
II 12 14
II 13 14
II 13 14
69x8
7079
7244
6934
7096
7261
6950
7112
7278
6966
7129
7295
6982
7145
731 1
6998
7161
7328
7015
7178
7345
7031
7194
7362
7047
72 1 1
7379
7063
7228
7396
235
235
235
6 8 10
7 8 10
7 8 10
11 13 15
12 13 15
12 13 15
7413
7586
7762
7430
7603
7780
7447
7621
7798
7464
7638
7816
7482
7656
7834
7499
7674
7852
7516
7691
7870
7534
7709
7889
7551
7727
7907
7568
7745
7925
235
245
245
7 9 10
7 9 II
7 9 II
12 14 16
12 14 16
13 14 16
7943
7962
7980
7998
8017
8035
8054
8072
8091
8110
246
7 9 11
13 15 17
8128
8318
8511
8147
8337
8531
8166
8356
8551
8185
8375
8570
8204
8395
8590
8222
8414
8610
8241
8433
8630
8260
8453
8650
8279
8472
8670
8299
8492
8690
246
246
246
8 9 II
8 10 12
8 10 12
13 IS 17
14 IS 17
14 16 18
8710
8913
gi2o
8730
8933
9141
8750
8954
9162
8770
8974
9183
8790
8995
9204
8810
9016
9226,
8831
9036
9247
8851
9057
9268
8872
9078
9290
8892
9099
93"
246
246
246
8 10 12
8 10 12
8 II 13
14 16 18
15 17 19
15 17 19
9333
9550
9772
9354
9572
9795
9376
9594
9817
9397
9616
9840
9419
9638
9863
9441
9661
9886
9462
9683
9908
9484
9705
9931
9506
9727
9954
9528
9750
9977
247
247
2 5 7
9 II 13
9 II 13
9 II 14
15 17 20
16 18 20
16 18 20
20
LOGARITHMIC SINES.
0°
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
~27^
28
29
30
0'
6'
12'
18'
24'
30'
36'
42'
48'
54'
12 3
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Inf. Neg.
7-2419
5429
7190
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1450
1961
21 41 62
82 103
8-2419
8-5428
8-71S8
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7330
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7468
3558
6035
7602
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6220
7731
4179
6397
7857
4459
6567
7979
4723
6731
8098
4971
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0334
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9655
0403
8849
9736
0472
8946
9816
0539
9042
9894
0605
9135
9970
0670
9226
0046
0734
9315
0120
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16 32 48
13 26 39
II 22 33
64 80
52 65
44 55
9-0859
9'i436
9-1943
0920
14S9
1991
09S1
1542
2038
1040
1594
2085
1099
1646
2131
"57
1697
2176
1214
1747
2221
1271
1797
2266
1326
1847
2310
1381
1895
2353
10 19 29
8 17 25
8 15 23
38 48
34 42
30 38
9-2397
2439
24S2
2524
2565
2606
2647
26S7
2727
2767
7 14 20
27 34
9-2806
9-3179
9-3521
2845
3214
3554
2SS3
3250
3586
2921
3284
3618
2959
3319
3650
2997
0 ■> r T
oj5j
3682
3034
3387
3713
3070
3421
3745
3107
3455
3775
3'43
3488
3806
6 12 19
6 II 17
5 II 16
25 31
23 28
21 26
9-3S37
9-4130
9-4403
3867
4158
4430
3S97
4186
4456
3927
4214
4482
3957
4242
4508
39S6
4269
4533
4015
4296
4559
4044
4323
4584
4073
4350
4609
4102
4377
4634
5 10 15
5 9 14
4 9 13
20 24
18 23
17 21
9-4659
9-4900
9-5126
46S4
4923
5148
4709
4946
5170
4733
4969
5192
4757
4992
5213
4781
5015
5235
4805
5037
5256
4829
5060
5278
4853
5082
5299
4876
5104
5320
4 8 12
4 8 II
4 7 II
16 20
15 19
14 18
9-5341
5361
5382
5402
5423
5443
5463
5484
5504
5523
3 7 10
14 17
9-5543
9-5736
9-5919
5563
5754
5937
5583
5773
5954
5602
5792
5972
5621
5810
5990
5641
5828
6007
5660
5847
6024
5679
5865
6042
5698
5883
6059
5717
5901
6076
3 6 10
369
369
13 16
12 15
12 15
9-6093
9-6259
9-6418
6110
6276
6434
6127
6292
6449
6144
6308
6465
6161
6324
64S0
6177
6340
6495
6194
6356
6510
6210
6371
6526
6227
6387
6541
6243
6403
6556
368
3 5 8
3 5 8
II 14
II 13
10 13
9-6570
9-6716
9-6856
65S5
6730
6S69
6600
6744
6S83
6615
6759
6896
6629
6773
6910
6644
6787
6923
6659
6801
6937
6673
6814
6950
668 7
6S28
6963
6702
6S42
6977
2 5 7
2 5 7
247
10 12
9 12
9 II
9-6990
7003
7016
7029
7042
7055
7068
7080
7093
7106
246
9 II
31
32
33
9-7118
9-7242
9-7361
7131
7254
7373
7144
7266
7384
7156
7278
7396
7168
7290
7407
7181
7302
7419
7193
7314
7430
7205
7326
7442
7218
7338
7453
7230
7349
7464
246
246
246
8 10
8 10
8 ic
34
35
36
9-7476
9-75S6
9-7692
74S7
7597
7703
7498
7607
7713
7509
7018
7723
7520
7629
7734
7531
7640
7744
7542
7650
7754
7553
7661
7764
7564
7671
7774
7575
7682
7785
246
2 4 5
2 3 5
7 9
7 9
7 9
37
38
39
9-7795
9-7893
9-79S9
7805
7903
7998
7815
7913
8007
7S25
7922
8017
7835
7932
8026
7844
7941
8c35
7854
7951
8044
7864
7960
8053
7874
7970
8063
7884
7979
8072
235
2 3 5
2 3 5
7 8
6 8
6 8
40
9-8081
Sogo
8099
8108
8117
8125
8134
8143
8152
8161
I 3 4
6 7
41
42
43
9-8169
9-S255
9-8338
8178
8264
8346
8187
8272
8354
8195
8280
8362
8204
8289
8370
8213
8297
8378
8221
830 s
83S6
8230
8313
8394
8238
8322
8402
8247
8330
8410
I 3 4
I 3 4
I 3 4
6 7
6 7
5 7
44
9-8418
8426
8433
8441
8449
8457
8464
8472
84S0
8487
I 3 4
5 6
. LOGARITHMIC SINES. ^^^ /^^^^ 0 4.^1,^/
45^
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
£3
84
£5
86
87
88
89
0'
6'
12'
18'
24'
30
36'
42'
48'
54'
12 3
4 5
9-8495
8502
8510
8517
8525
8532
8540
8547
8555
8562
I 2 4
5 6
9-8569
9-8641
9-8711
8577
8648
871S
8584
8655
8724
8591
8662
8731
8598
8669
8 738
8606
8676
8745
8613
868 3
875''
8620
8690
8758
8627
8697
8765
•8634
8704
8771
I 2 4
I 2 3
I 2 3
5 6
5 6
4 6
9-8778
9-8843
9-8905
8784
8849
891 1
8791
8855
S917
8797
8862
8923
8S04
8868
8929
8810
8874
8935
8817
8880
8941
8S23
S8S7
8947
SS30
8S93
8953
8836
8S99
8959
I 2 3
I 2 3
I 2 3
4 5
4 5
4 5
9-8965
9-9023
9-90S0
8971
9029
9085
8977
9035
9091
8983
9041
9096
S989
9046
9101
8995
9052
9107
9000
9057
9112
9006
9063
9118
9012
9069
9123
9018
9074
9128
I 2 3
I 2 3
I 2 3
4 5
4 5
4 5
9-9134
9139
9144
9149
9155
9160
9165
9170
9175
9181
I 2 3
3 4
9-91S6
9-9236
9-9284
9191
9241
9289
9196
9246
9294
9201
9251
9298
9206
9255
9303
9211
9260
9308
9216
9265
9312
9221
9270
9317
9226
9275
9322
9231
9279
9326
I 2 3
122
I 2 2
3 4
3 4
3 4
9-9331
9-9375
9-9418
9335
9380
9422
9340
9384
9427
9344
9388
9431
9349
9393
9435
9353
9397
9439
9358
9401
9443
9362
9406
9447
9367
9410
9451
9371
9414
9455
I I 2
I I 2
1 I 2
3 4
3 4
3 3
9-9459
9-9499
9-9537
9463
9503
9540
9467
9507
9544
9471
9510
954S
9475
9514
9551
9479
9518
9555
94S3
9522
9558
9487
9525
9562
9491
9529
9566
9495
9533
9569
I I 2
I I 2
I I 2
0 3
3 3
2 3
9-9573
9576
9580
95S3
9587
9590
9594
9597
9601
9604
I I 2
2 3
9-9607
99640
9-9672
961 1
9643
9675
9614
9647
9678
9617
9650
96S1
9621
9653
9684
9624
9656
9687
9627
9659
9690
9631
9662
9693
9634
9666
9696
9637
9669
9699
I I 2
I I 2
0 I I
2 3
2 3
2 2
9-9702
99730
9-9757
9704
9733
9759
9707
9735
9762
9710
9738
9764
9713
9741
9767
97:6
9743
9770
9719
9746
9772
9722
9749
9775
9724
9751
9777
9727
9754
9780
oil
0 I I
0 I I
2 2
2 2
2 2
9-9782
9-9S06
9-982S
9785
9S0S
9S31
9787
9811
9833
9789
98 1 3
9835
9792
9815
9837
9794
9S17
9839
9797
9820
9841
9799
9822
9S43
9801
9824
9845
9804
9826
9847
0 I I
0 I I
0 I I
2 2
2 2
I 2
9-9849
9S51
9S53
9855
9S57
9859
9S61
9863
9865
9867
oil
I 2
9-9869
9-9887
9-9904
9871
9889
9906
9S73
9891
9907
9875
9892
9909
9S76
9894
9910
9878
9896
9912
9S80
9897
9913
98S2
9899
9915
9S84
9901
9916
9885
9902
9918
0 I I
0 I I
oil
I 2
9-9919
9-9934
9-9946
9921
9935
9947
9922
9936
9949
9924
9937
9950
9925
9939
9951
9927
9940
9952
9928
9941
9953
9929
9943
9954
9931
9944
9955
9932
9945
9956
001
0 0 I
001
9 -99 58
9-9968
9-9976
9959
9968
9977
9960
9969
9978
9961
9970
997S
9962
9971
9979
9963
9972
9980
9964
9973
9981
9965
9974
9981
9966
9975
9982
9967
9975
9983
0 0 I
000
000
0 I
9-9983
99S4
9985
9985
9986
9987
9987
9988
9988
9989
000
0 0
9-9989
9-9994
9-9997
9990
9994
9998
9990
9995
9998
9991
9995
9998
9991
9996
9998
9992
9996
9999
9992
9996
9999
9993
9996
9999
9993
9997
9999
9994
9997
9999
000
000
000
0 0
0 0
0 0
9-9999
9999
0000
cooo
0000
0000
0000
0000
cooo
-
cooo
000
0 0
22
LOGARITHMIC
COSINES.
0°
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
0'
6'
12'
18'
24'
30' 36'
42'
48'
54'
12 3
4 5
lO'OOOO
0000
0000
0000
000c
0000
0000
0000
0000
9-9999
000
0 0
9-9999
9-9997
9-9994
9999
9997
9994
9999
9997
9993
9999
9996
9993
9999
9996
9992
9999
9996
9992
9998
9996
9991
9998
9995
9991
9998
9995
9990
9998
9994
9990
000
000
000
0 0
0 0
0 0
9-9989
9-9983
9-9976
9989
9983
9975
99SS
9982
9975
9988
99S1
9974
9987
9981
9973
9987
9980
9972
9986
9979
9971
9985
9978
9970
9985
9978
9969
9984
9977
9968
000
000
000
0 0
0 I
1 I
9-9968
9-9957
9-9946
9967
9956
9945
9966
9955
9944
9965
9954
9943
9964
9953
9941
9963
9952
9940
9962
9951
9939
9961
9950
9937
9960
9949
9936
9959
9947
9935
0 0 I
0 0 I
0 0 I
9 '9934
9932
9931
9929
9928
9927
9925
9924
9922
9921
0 0 I
9-9919
9-9904
9-9887
9918
9902
9885
9916
9901
98S4
9915
9899
98S2
9913
9897
9880
9912
9896
9878
9910
9894
9876
9909
9892
9875
9907
9891
9S73
9906
9S89
9871
oil
0 I J
0 I I
I 2
9-9869
9-9849
99828
9867
9847
9S26
9865
9S45
9824
9S63
9843
9822
9861
9841
9820
9859
9839
9S17
9857
9837
9815
9S55
9835
9813
9853
9S33
9811
9851
9S31
9808
0 I I
0 I I
0 I I
I 2
1 2
2 2
9-9806
9-9782
9'9757
9804
97 So
9754
9801
9777
9751
9799
9775
9749
9797
9772
9746
9794
9770
9743
9792
9767
9741
9789
9764
9738
9787
9762
9735
9785
9759
0 I I
0 I I
0 I I
2 2
2 2
2 2
9-9730
9727
9724
9722
9719
9716
9713
9710
9707
9704
0 I I
2 2
9-9702
9-9672
9-9640
9699
9669
9637
9696
9666
9634
9693
9662
9631
9690
9659
9627
9687
9656
9624
9684
9653
9621
9681
9650
9617
9678
9647
9614
9675
9643
961 1
oil
I I 2
I I 2
2 2
2 3
2 3
9-9607
9*9573
9'9537
9604
9569
9533
9601
9566
9529
9597
9562
9525
9594
9558
9522
9590
9555
9518
9587
9551
9514
9583
9548
9510
9580
9544
9507
9576
9540
9503
112
I I 2
I I 2
2 3
2 3
-t ^
9-9499
9-9459
9-9418
9495
9455
9414
9491
9451
9410
9487
9447
9406
94S3
9443
9401
9479
9439
9397
9475
9435
9393
9471
9431
938S
9467
9427
93S4
9463
9422
9380
I I 2
I I 2
I I 2
3 3
3 3
3 4
9-9375
9371
9367
9362
9358
9353
9349
9344
9340
9335
I I 2
3 4
9-9331
9-9284
9-9236
9326
9279
9231
9322
9275
9226
9317
9270
9221
9312
9265
9216
9308
9260
9211
9303
9255
9206
9298
9251
9201
9294
9246
9196
9289
9241
9191
I 2 2
122
I 2 3
3 4
3 4
3 4
9-9186
9'9i34
9-9080
9181
9128
9074
9175
9123
9069
9170
911S
9063
9165
9112
9057
9160
9107
9052
9155
9101
9046
9149
9096
9041
9144
9091
9035
9139
90S5
9029
I 2 3
I 2 3
I 2 3
3 4
4 5
4 5
9-9023
9-8965
9-8905
9018
8959
8899
9012
8953
8893
9006
S947
88S7
9000
8941
8880
S995
8935
SS74
8989
8929
8868
8983
8923
8862
8977
8917
S855
8971
8911
8S49
I 2 3
I 2 3
I 2 3
4 5
4 5
4 5
9-8843
8836
8830
8823
8817
8810
8804
S797
8791
8784
•123
4 5
9-8778
9-8711
9-S641
8771
8704
8634
8765
8697
8627
8758
8690
S620
8751
8683
S613
8745
8676
S606
8738
8669
8598
8731
8662
8591
8724
8655
8584
8718
8648
8577
I 2 3
I 2 3
I 2 4
5 6
5 6
5 6
9-8569
8562
8555
8547 8540
8532
8525 8517
8510
8502
I 2 4
5 6
N.B. — Numbers in difference columns to be subtracted, not added. — See Rules.
LOGARITHMIC COSINES.
23
0'
6'
12'
18'
24' 1 30'
36
42'
48'
54'
12 3 4 5
45°
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
83
9-8495
8487
8480 8472
8464
8457
8449
8441
8433
8426
I 3 4
5 6
9-8418
9-8338
9-8255
8410
8330
8247
8402
S322
8238
8394
8313
8230
8386
8305
8221
8378
8297
. S213
870
82S9
8204
8362
82S0
8195
8354
8272
81S7
S346
S264
8178
I 3 4
I 3 4
I 3 4
5 7
6 7
6 7
9"8i69
9-SoSi
9-7989
8161
8072
7979
8152
8063
7970
8143
S053
7960
8134
8044
7951
8125
8035
7941
8117
8026
7932
8108
8017
7922
8099
8007
7913
8090
7998
7903
1 3 4
235
2 3 5
6 7
6 8
6 8
9-7893
9-7795
97692
7S84
77S5
76S2
7874
7774
7671
7S64
7764
7661
7S54
7754
7650
7844
7744
7640
7835
7734
7629
7S25
7723
7618
7815
7713
7607
7S05
7703
7597
2 3 5
2 3 5
2 4 5
7 8
7 9
7 9
97586
7575
7564
7553
7542
7531
7520
7509
7498
74S7
246
7 9
97476
97361
97242
7464
7349
7230
7453
7338
7218
7442
7326
7205
7430
7314
7193
7419
7302
7181
7407
7290
7168
7396
7278
7156
7384
7266
7144
7373
7254
7131
246
246
246
8 10
8 10
8 10
9711S
9-6990
9-6856
7106
6977
6S42
7093
6963
6828
70S0
6950
6814
706S
6937
6801
7055
6923
6787
7042
6910
6773
7029
6896
6759
7or6
6S83
6744
7003
6869
6730
246
247
2 5 7
9 II
9 II
9 12
9-6716
9-6570
9-6418
6702
6556
6403
66S7
6541
63S7
6673
6520
6371
6659
6510
6356
6644
6495
6340
6629
64S0
6324
6615
6465
6308
6600
6449
6292
6585
6434
6276
257
3 5 8
3 5 8
10 12
10 13
11 13
9-6259
6243
6227
6210
6194
6177
6161
6144
6127
6110
3 6 8
II 14
9-6093
9-5919
9-5736
6076
5901
5717
6059
5S83
5698
6042
5S65
5679
6024
5847
5660
6007
5S28
5641'
5990
5810
5621
5972
5792
5602
5954
5773
5583
5937
5754
5563
369
369
3 6 10
12 15
12 15
13 16
9-5543
9-5341
9-5126
5523
5320
5104
5504
5299
5082
54S4
5278
5060
5463
5256
5037
5443
5235
5015
5423
5213
4992
5402
5192
4969
5382
5170
4946
5361
514S
4923
3 7 10
4 7 II
4 8 II
14 17
14 18
15 19
9-4900
9-4659
9-4403
4876
4634
4377
4853
4609
4350
4829
45S4
4323
4805
4559
4296
4781
4533
4269
4757
450S
4242
4733
4482
4214
4709
4456
4186
46S4
4430
4158
4 8 12
4 9 13
5 9 14
16 20
17 21
18 23
9-4130
4102
4073
4044
4015
3986
3957
3927
3897
3867
5 10 15
20 24
9-3837
9-3521
9-3179
3S06
34S8
3143
3775
3455
3107
3745
3421
3070
3713
33S7
3034
3682
3353
2997
3650
3319
2959
3618
3284
2921
3586
3250
2SS3
3554
3214
2845
5 II 16
6 II 17
6 12 19
21 26
23 28
25 31
9-2806
9-2397
9-1943
2767
2353
1895
2727
2310
1S47
26S7
2266
1797
2647
2221
1747
2606
2176
1697
2565
2131
1646
2524
20S5
1594
2482
2038
1542
2439
1 99 1
1489
7 14 20
8 15 23
8 17 25
27 34
30 38
34 42
9-1436
9-0S59
90192
1381
0797
0120
1326
; 07^4
0046
1271
0670
9970
1214
0605
9S94
"57
0539
9S16
1099
0472
9736
1040
0403
9655
0981
0334
9573
0920
0264
9489
10 19 29
11 22 33
13 26 39
38 48
44 55
52 65
8-9403
9315
9226
9135
9042
8946
8849
8749
8647
8543
16 32 48
64 So
8-8436
S-7188
8-5428
8326
7041
5206
8213
6889
4971
8098
6731
4723
7979
6567
4459
7S57
6397
4179
773^
6220
3SS0
7602
6035
3558
7468
5S42
3210
7330
5640
2S32
21 41 62
82 103
8-2419
1961
1450
0870
0200
7-9408
8439
7190
5429
2419
N.B. — Numbers in difference columns to be subtracted, not added. — See Utiles
24
LOGARITHMIC TANGENTS.
0'
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
0'
6'
12'
18'
24'
30' 36'
42'
48'
54' 12 3
4 5
Inf. Neg.
7-2419
5429
7190
S439
9409
0200
0870
^450
1962
29 58 87
21 41 62
116 145
83 303
8-2419
8-5431
8-7194
2833
5643
7337
3211
5845
7475
3559
6038
7609
38S1
6223
7739
41S1
6401
7S65
4461
6571
79S8
4725
6736
8107
4973
6894
8223
5208
7046
8336
8-8446
8-9420
9'02i6
8554
9506
02S9
8659
9591
0360
S762
9674
0430
8862
9756
0499
8960
9836
0567
9056
9915
0633
9150
9992
0699
9241
006S
0764
9331
0143
0S28
16 32 48
13 26 40
II 22 34
64 8i
53 66
45 56
9-0891
9-1478
9-1997
0954
1533
2046
1015
1587
2094
1076
1640
2142
1135
1693
2189
1194
1745
2236
1252
1797
22S2
1310
1848
2328
1367
1898
2374
1423
1948
2419
10 20 29
9 17 26
8 16 23
39 49
35 43
31 39
9-2463
2507
2551
2594
2637
2680
2722
2764
2805
2846
7 14 21
28 35
9-2887
9-3275
9-3634
2927
3312
3668
2967
3349
3702
3006
3385
3736
3046
3422
3770
3085
3458
3804
3123
3493
3837
3162
3529
3870
3200
3564
3903
3237
3599
3935
6 13 19
6 12 iS
6 II 17
26 32
24 30
22 28
9-3968
9-4281
9-4575
4000
4311
4603
4032
4341
4632
4064
4371
4660
4095
4400
4688
4127
4430
4716
4158
4459
4744
41S9
4488
4771
4220
4517
4799
4250
4546
4826
5 10 16
5 10 15
5 9 14
21 26
20 25
19 23
9-4853
9-5118
9-5370
48S0
5143
5394
4907
5169
5419
4934
5195
5443
4961
5220
5467
49S7
5245
5491
5014
5270
5516
5040
5295
5539
5066
5320
5563
5092I
5345
5587,
4 9 13
4 8 13
4 8 12
18 22
17 21
16 20
9-5611
5634
5658
5681
5704
5727
5750
5773
5796
5819
4 8 12
IS 19
9-5842
9 6064
96279
5864
6086
6300
5S87
6108
6321
5909
6129
6341
5932
6151
6362
5954
6172
6383
5976
6194
6404
5998
6215
6424
6020
6236
6445
6042;
6257
6465I
4 7 II
4 7 II
3 7 10
15 19
14 18
14 17
9-6486
9-6687
9-6882
6506
6706
6901
6527
6726
6920
6547
6746
6939
6567
6765
6958
65S7
6785
6977
6607
6S04
6996
6627
6824
7015
6647
6S43
7034
6667;
6S63
7053
3 7 10
3 7 10
369
13 17
13 16
13 16
9-7072
9-7257
9-7438
7090
7275
7455
7109
7293
7473
7128
7491
7146
7330
7509
7165
734S
7526
7183
7366
7544
7202
7384
7562
7220
7402
7579
7238,
7420,
7597,
369
369
369
12 15
12 15
12 15
9-7614
7632
7649
7667
7684
7701
7719
7736
7753
7771!
369
12 14
9-7788
9-7958
9-8125
7805
7975
8142
7822
7992
8158
7839
S008
8175
7S56
8025
8191
7873
8042
820S
7S90
8059
8224
7907
8075
8241
7924
8092
8257
7941;
8109'
8274I
369
368
3 5 8
II 14
II 14
II 14
9-8290
9-8452
9-8613
8306
8468
8629
8323
8484
8644
8339
8501
8660
8355
8517
8676
8371
8533
8692
8388
8549
8 70S
8404
8565
8724
8420
8581
8740
8436
8597
S755
3 5 8
3 5 8
3 5 8
II 14
II 13
II 13
9-8771
98928
9-9084
8787
8944
9099
8803
8959
9115
8818
8975
9130
8834
8990
9146
8S50
9006
9161
8865
9022
9176
8881
9037
9192
8897
9053
9207
8912
9068
9223
3 5 8
3 5 8
3 5 8
10 13
10 13
10 13
9-9238
9254
9269
9284
9300
9315
9330
9346
9361
9376
3 5 8
10 13
9-9392
9-9544
9-9697
9407
9560
9712
9422
9575
9727
943S
9590
9742
9453
9605
9757
9468
9621
9773
9483
963C
978S
9499
9651
9803
9514
9666
9818
9529
9681
9833
3 5 8
3 5 8
3 5 8
10 13
10 13
10 13
9-984S
9864
9S79
9894
9909
9924
9939
9955
9970
9985
3 5 8
10 13
LOGARITHMIC TANGENTS.
25
45°
O'
6'
12'
18'
24'
30'
36'
42'
48'
54'
12 3
4 5
lO'OOOO
0015
0030
0045
0061
0076
0091
0106
0121
0136
3 5 8 10 13
46
47
48
10-0152
10 0303
io*0456
0167
0319
0471
01S2
0334
0486
0197
0349
0501
0212
0364
0517
0228
0379
0532
0243
0395
0547
0258
0410
0562
0273
0425
0578
0288
0440
0593
3 5 J
3 5 ^
3 5 ^
i 10 13
10 13
10 13
43
50
51
52
53
54
55
10 "0608
10-0762
10-0916
0624
0777
0932
0639
0793
0947
0654
0S08
0963
0670
0S24
0978
0685
0839
0994
0700
0S54
1010
0716
0870
1025
0731
0885
1041
0746
or 01
1056
3 5 S
3 5 8
3 5 8
10 13
10 13
10 13
10-1072
10-1229
10-13S7
1088
1245
1403
1 103
1260
1419
1119
1276
1435
1135
1292
145 1
1150
1308
1467
1166
1324
14S3
1182
1340
1499
1197
1356
1516
1213
1371
1532
3 5 8 10 13
3 5 8; 11 13
3 5 8 II 13
10-1548
1564
15S0
1596
1612
1629
1645
1661
1677
1694
3 5 8 II 14
56
57
58
59
60
61
62
63
64
10-1710
10-1S75
I c -2042
1726
1891
2059
1743
1908
2076
1759
1925
2093
1776
1941
2110
1792
1958
2127
1809
1975
2144
1825
1992
2161
1842
2008
2178
1858
2025
2195
3 5 8
368
369
11 14
II 14
11 14
10-2212
10-23S6
10-2562
2229
2403
25S0
2247
2421
2598
2264
243S
2616
2281
2456
2634
2299
2474
2652
2316
2491
2670
2333
2509
2689
2351
2527
2707
2368
2545
2725
369
369
369
12 14
12 15
12 15
10-2743
10-2928
10-3118
2762
2947
3137
2780
2966
3157
2798
2985
3176
2817
3004
3196
2S35
3023
321S
2854
3042
3235
2S72
3061
3254
2S91
3080
3274
2910
3099
3294
369
369
3 6 10
12 15
13 16
13 16
65
66
67
68
10-3313
3333
3353
3373
3393
3413
3433
3453
3473
3494
3 7 10
13 17
10-3514
10-3721
10-3936
3535
3743
3958
3555
3764
3980
3576
3785
4002
3596
3806
4024
3617
3S28
4046
3638
3S49
4068
3659
3871
4091
3679
3892
4113
3700
3914
4136
3 7 10
4 7 II
4 7 II
14 17
14 18
15 19
69
70
71
10-4158
10-43S9
10-4630
4181
4413
4655
4204
4437
4680
4227
4461
4705
4250
4484
4730
4273
4509
4755
4296
4533
4780
4319
4557
4805
4342
45S1
4831
4366
4606
4S57
4 8 12
4 8 12
4 8 13
15 19
16 20
17 21
72
73
74
75
76
77
78
10-4882
10-5147
10-5425
4908
5174
5454
4934
5201
5483
4960
5229
5512
4986
5256
5541
5013
52S4
5570
5039
5312
5600
5066
5340
5629
5093
5368
5659
5120
5397
56S9
4 9 13
5 9 14
5 10 15
18 22
19 23
20 25
10-5719
5750
5780
5811
5842
5S73
5905
5936
5968
6000
5 10 16
21 26
10-6032
10-6366
10-6725
6065
6401
6763
6097
6436
6800
6130
6471
6838
6163
6507
6S77
6196
6542
6915
6230
6578
6954
6264
6615
6994
6298
6651
7033
6332
66SS
7073
6 II 17
6 12 18
6 13 19
22 28
24 30
26 32
79
80
81
10-7113
10-7537
10-8003
7154
7581
8052
7195
7626
8102
7236
7672
8152
7278
7718
8203
7320
7764
8255
7363
7811
8307
7406
7858
8360
7449
7906
8413
7493
7954
8467
7 14 21
8 16 23
9 17 26
28 35
31 39
35 43
82
83
84
85
86
87
88
89
10-8522
10-9109
10-9784
8577
9172
9857
8633
9236
9932
8690
9301
0008
874S
9367
0085
8S06
9433
6164
8S65
9501
0244
8924
9570
0326
8985
9640
0409
9046
9711
6494
10 20 29
11 22 34
13 26 40
39 49
45 56
53 66
1 1 -0580
0669
0759
0850
0944
1040
1138
1238
1341
1446
i6 32 48
64 81
11-1554
1 1 -2806
11-4569
1664
2954
4792
1777
3106
5027
1893
3264
5275
2012
3429
5539
2135
3599
5819
2261
3777
6119
2391
3962
6441
2525
4155
6789
2663
4357
7167
20 41 62
29 58 87
83 103
116 144
11-7581
8038
8550
9130
9800
0591
1561
2810
4571
7581
26
LOGARITHMIC
COTANGENTS.
0°
1
2
3
4
5
6
7
8
9
0'
6'
12'
18'
24'
30'
36'
42'
48'
54'
12 3
4 6
Inf.
12-76
4571
2810
1561
0591
9S00
9130
8550
8038
29 59 88
21 42 63
iiS 147
83 104
11-7581
11-4569
1 1 -2S06
7167
4357
2663
6789
4155
2525
6441
3962
2391
6119
3777
2261
5S19
3599
2135
5539
3429
2012
5275
3264
1893
5027
3106
1777
4792
2954
1664
"•I554
11-0580
10-9784
1446
0494
9711
1341
0409
9640
1238
0326
9570
1138
0244
9501
1040
0164
9433
0944
0085
9367
0850
0008
9301
0759
9932
9236
0669
9857
9172
16 32 49
13 26 40
II 23 34
65 Si
53 66
45 55
lo-gioQ
10-8522
10-8003
9046
8467
7954
8985
8413
7906
8924
8360
7858
8865
8307
7811
8806
8255
7764
8748
820^,
771S
8690
S152
7672
8633
8102
7626
8577
8052
7581
10 20 29
9 17 26
8 16 23
39 49
35 43
31 39
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
10-7537
7493
7449
7406
7363
7320
7278
7236
7195
7154
7 14 21
28 35
10-7113
10-6725
10-6366
7073
6688
6332
7033
6651
629S
6994
6615
6264
6954
6578
6230
6915
6542
6196
6877
6507
6163
6S38
6471
6130
6800
6436
6097
6763
6401
6065
6 13 19
6 12 18
6 II 17
26 32
24 30
22 28
10-6032
10-5719
10-5425
6000
56S9
5397
596S
5659
5368
5936
5629
5340
5905
5600
5312
5S73
5570
5284
5842
5541
5256
5811
5512
5229
5780
5483
5201
5750
5454
5174
5 10 15
5 10 15
5 9 14
21 26
20 25
19 23
10-5147
10-4882
10-4630
5120
4857
4606
5093
4S31
45S1
5066
4805
4557
5039
4780
4533
5013
4755
4509
49S6
4730
4484
4960
4705
4461
4934
4680
4437
4908
4655
4413
4 9 13
4 9 13
4 8 12
18 22
17 21
16 20
10-4389
4366
4342
4319
4296
4273
4250
4227
4204
41S1
4 8 12
15 19
10-4158
10-3936
10-3721
4136
3914
3700
4113
3S92
3679
4091
3871
3659
406S
3S49
3638
4046
3828
3617
4024
3806
3596
4002
37S5
3576
3980
3764
3555
3958
3743
3535
4 7 II
4 7 II
3 7 10
15 19
14 18
14 17
10-3514
10-3313
10-3118
3494
3294
3099
3473
3274
30S0
3453
3254
3061
3433
3235
3042
3413
3215
3023
3393
3196
3004
0 -t 'T ->
oj7j
3176
2985
3353
3157
2966
3333
3137
2947
3 7 10
3 6 10
369
13 17
13 16
13 16
10-2928
10-2743
10-2562
2910
2725
2545
2891
2707
2527
2S72
2689
2509
2854
2670
2491
2835
2652
2474
2S17
2634
2456
2798
2616
2438
2780
259S
2421
2762
2580
2403
369
369
369
12 15
12 15
12 15
10-2386
2368
2351
2333
2316
2299
22S1
2264
2247
2229
369
12 14
10-2212
10-2042
10-1875
2195
2025
1S58
217S
200S
1S42
2161
1992
1S25
2144
1975
1809
2127
1958
1792
2110
1941
1776
2003
1925
1759
2076
1908
1743
2059
1891
1726
369
3 5 8
3 5 8
II 14
II 14
II 14
10-1710
10-1548
10-13S7
1694
1532
1371
1677
1516
1356
1661
1499
1340
1645
1483
1324
1629
1467
1308
1612
1451
1292
1596
1435
1276
1580
1419
1260
1564
1403
1245
3 5 8
3 5 8
3 5 8
11 14
II 13
II 13
10-1229
10-1072
10-0916
1213
1056
0901
1 197
1041
0S85
1 182
1025
0S70
1 166
lOIO
0854
1 1 50
0994
0839
"35
0978
0824
1119
0963
080S
1 103
0947
0793
1088
0932
0777
3 5 8
3 5 8
3 5 8
10 13
10 13
10 13
10-0762
0746
0731
0716
0700
06S5
0670
0654
0639
0624
3 5 8
10 13
10-0608
10-0456
10-0303
0593
0440
0288
057s
0425
0273
0562
0410
0258
0547
0395
0243
0532
0379
022S
0517
0364
0212
0501
0349
0197
0486
0334
0182
0471
0319
0167
3 5 8
3 5 8
3 5 8
10 13
10 13
10 13
10-0152
0136
0121
0106
0091
0076
0061
0045
0030
0015
3 5 8
10 13
N.B. — Numbers in difference columns to be subtracted, not added.— .5"^^? Kiiks.
LOGARITHMIC
COTANGENTS.
27
45°
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
0'
6'
12'
18'
24'
30'
36'
42'
48'
54'
12 3
4 5
lo-oooo
99S5
9970
9955
9939
9924
9909
9894
9879
i864
3 5 8
10 13
9-9848
99697
9'9544
9S33
9681
9529
9S18
9666
9514
9803
9651
9499
97S8
9636
9483
9773
9621
9468
9757
9605
9453
9742
9590
943S
9727
9575
9422
9712
9560
9407
^ 5 8
3 5 8
3 5 8
10 13
10 13
10 13
99392
9-9238
9-90S4
9376
9223
9068
9361
9207
9053
9346
9192
9037
9330
9176
9022
9315
9161
9006
9300
9146
8990
9284
9130
8975
9269
9115
8959
9254
9099
8944
3 5 8
3 5 8
3 5 8
10 13
10 13
10 13
9-8928
9-8771
9-8613
8912
8755
S597
8S97
S740
85S1
888 1
8724
S565
8865
S708
8549
8850
S692
S533
8S34
8676
8517
8818
S660
8501
8S03
8644
8484
8787
8629
8468
3 5 8
3 5 8
3 5 8
10 13
" 13
" 13
9-8452
S436
8420
8404
8388
8371
8355
8339
8323
8306
3 5 8
II 14
9-8290
9-8125
9-7958
8274
8109
7941
8257
8092
7924
8241
8075
7907
8224
8059
7890
8208
S042
7S73
8191
8025
7S56
8175
8008
7S39
815S
7992
7822
8142
7975
7S05
3 5 8
368
369
II 14
II 14
II 14
9-7788
97614
9-7438
7771
7597
7420
7753
7579
7402
7736
7562
7384
7719
7544
7366
7701
7526
7348
7684
7509
7330
7667
7491
7311
7649
7473
7293
7632
7455
7275
369
369
369
12 14
12 15
12 15
97257
9-7072
9-6882
723S
7053
6863
7220
7034
6843
7202
7015
6824
7183
6996
6804
7165
6977
6785
7146
6958
6765
7128
6939
6746
7109
6920
6726
7090
6901
6706
369
369
3 7 10
12 15
13 16
13 16
9-6687
6667
6647
6627
6607
6587
6567
6547
6527
6506
3 7 10
13 17
9-6486
9-6279
9-6064
6465
6257
6042
6445
6236
6020
6424
6215
5998
6404
6194
5976
6383
6172
5954
6362
6151
5932
6341
6129
5909
6321
610S
5S87
6300
60S6
5864
3 7 10
4 7 II
4 7 II
14 17
14 18
15 19
9-5842
9-561 1
9-5370
5819
55S7
5345
5796
5563
5320
5773
5539
5295
5750
5516
5270
5727
5491
5245
5704
5467
5220
5681
5443
5195
5658
5419
5169
5634
5394
5143
4 8 12
4 8 12
4 8 13
15 19
16 20
17 21
9-5118
9-4853
9-4575
5092
4S26
4546
5066
4799
4517
5040
4771
448S
5014
4744
4459
49S7
4716
4430
4961
4688
4400
4934
4660
4371
4907
4632
4341
4880
4603
43"
4 9 13
5 9 14
5 10 15
18 22
19 23
20 25
9-4281
4250
4220
4189
4158
4127
4095
4064
4032
4000
5 10 16
21 26
9-3968
9-3634
9-3275
3935
3599
3237
3903
3564
3200
3S70
3529
3162
3837
3493
3123
3804
3458
3085
3770
3422
3046
3736
3385
3006
3702
3349
2967
3668
3312
2927
6 II 17
6 12 iS
6 13 19
22 28
24 30
26 32
9-2887
9-2463
9-1997
2846
2419
1948
2805
2374
189S
2764
232S
1848
2722
2282
1797
2680
2236
1745
2637
2189
1693
2594
2142
1640
2551
2094
1587
2507
2046
1533
7 14 21
8 16 23
9 17 26
28 35
31 39
35 43
9-1478
9-0891
9-0216
1423
0828
0143
1367
0764
0068
1310
0699
9992
1252
0633
9915
1 194
0567
9836
8960
"35
0499
9756
1076
0430
9674
1015
0360
9591
0954
0289
9506
10 20 29
" 23 34
13 27 40
39 49
45 56
53 66
8-9420
9331
9241
9150
9056
8862
8762
8659
8554
16 32 49
65 81
8-8446
8-7194
8-5431
8336
7046
5208
8223
6894
4973
8107
6736
4725
7988
6571
4461
7865
6401
4181
7739
6223
3881
7609
6038
3559
7475
5845
3211
7337
5643
2833
21 42 63
29 59 88
83 104
118 147
8-2419
1962
1450
0870
0200
9409
8439
7190
5429
2419
iV^i)'.— Numbers in difTerence columns to be subtracted, not added.- Sec- Knlcs.
28
LOGARITHMIC SECANTS.
0'
6'
12'
18'
24'
30'
36'
42'
48'
54'
12 3
4 5
0°
lO'OOOO
0000
OOOC
OOOC
0000
0000
0000
0000
0000
0001
000
0 0
1
2
3
lO'OOOI
I o -0003
lo'ooob
0001
0003
0006
0001
0003
0007
0001
0004
0007
0001
0004
0008
0001
0004
0008
0002
0004
0009
0002
0005
0009
0002
0005
0010
0002
0006
0010
000
000
000
0 0
0 0
0 0
4
5
6
lO'OOII
lo'ooi;
10 '0024
001 1
0017
0025
0012
0018
0025
0012
0019
0026
0013
0019
0027
0013
0020
0028
0014
0021
0029
0015
0022
0030
0015
0022
0031
0016
0023
0032
000
000
000
0 0
0 I
7
8
9
IO"0032
1 0-0042
io'oo54
0033
0044
0055
0034
0045
0056
0035
004 b
0057
0036
0047
0059
0037
0048
0060
0038
0049
0061
0039
0050
0063
0040
0051
0064
0041
0053
0065
000
0 0 I
0 0 I
10
10-0066
0068
C069
0071
0072
0073
0075
0076
0078
0079
0 0 I
11
12
13
lo'ooSi
10 '0096
10-0113
0082
0098
0115
0084
0099
0II6
0085
0101
0118
0087
0103
0120
0088
0104
0122
0090
0106
0124
0091
0108
0125
0093
0109
0127
0094
OIII
0129
0 I I
0 I I
0 I I
I 2
14
15
16
10-0131
10-0151
10-0172
0133
0153
0174
0135
0155
0176
0137
0157
0178
0139
0159
0180
0141
0161
0x83
0143
0163
0185
0145
0165
0187
0147
0167
01S9
0149
0169
0192
0 I I
0 I I
0 I I
I 2
I 2
I 2
17
18
19
10-0194
10-0218
10-0243
0196
0220
0246
0199
0223
0249
0201
0225
0251
0203
0228
0254
0206
0230
0257
0208
0233
0259
02 1 1
0236
0262
0213
0238
0265
0215
0241
0267
0 I I
0 I I
0 I I
2 2
2 2
2 2
20
10-0270
0273
0276
0278
0281
0284
0287
0290
0293
0296
0 I I
2 2
21
22
23
10-0298
10-0328
10-0360
0301
0331
0363
0304
0334
0366
0307
0338
0369
0310
0341
0373
0313
0344
0376
0316
0347
0379
0319
0350
0383
0322
0353
0386
0325
0357
03S9
0 I I
1 I 2
I I 2
2 2
2 3
2 3
24
25
26
27
28
29
10-0393
10-0427
10-0463
0396
0431
0467
0399
0434
0471
0403
0438
0475
0406
0442
047S
0410
0445
0482
0413
0449
0486
0417
0452
0490
0420
0456
0494
0424
0460
0497
I I 2
I I 2
I I 2
2 3
2 3
100501
10-0541
1005S2
0505
0545
0586
0509
0549
0590
0513
0553
0594
0517
0557
0599
0521
0561
0603
0525
0565
0607
0529
0569
C6l2
0533
0573
0616
0537
0578
0620
I I 2
I I 2
I I 2
3 3
3 3
3 4
30
10-0625
0629
0633
0638
0642
0647
0651
0656
0660
0665
I I 2
3 4
31
32
33
10-0669
10-0716
10*0764
0674
0721
0769
0678
0725
0774
0683
0730
0779
0688
0735
0784
0692
0740
0789
0697
0745
0794
0702
0749
0799
0706
0754
0804
07 1 1
0759
0S09
122
122
122
3 4
3 4
3 4
34
35
36
io-o8i4
10 -0866
10-0920
0819
0872
0926
0825
0877
0931
0830
0882
0937
0835
0S88
0943
0S40
0893
094S
0845
0S99
0954
0851
0904
0959
0856
0909
0965
0861
0915
0971
I 2 3
I 2 3
I 2 3
3 4
4 5
4 5
37
38
39
10-0977
10-1035
10-1095
0982
1041
IIOI
09S8
1047
1 107
0994
1053
III3
1000
1059
1120
1005
1065
1126
lOII
1071
1132
1017
1077
1138
1023
1083
"45
1029
1089
1151
I 2 3
I 2 3
I 2 3
4 5
4 5
4 5
40
10-1157
1 164
1170
II77
1183
1190
1196
1203
1209
1216
I 2 3
4 5
41
42
43
IO-I222
IO-I2S9
10-1359
1229
1296
1366
1235
1303
1373
1242
1310
1380
1249
1317
13S7
1255
1324
1394
1262
1 33 1
1402
1269
I33S
1409
1276
1345
1416
1282
1352
1423
I 2 3
I 2 3
I 2 4
4 6
5 6
5 6
44
10-1431
1438
1445
1453
1460
1468
1475
I4S3
1490
1498
I 2 4
-1
LOGARITHMIC SECANTS.
29
0'
6'
12'
18'
24'
30'
36'
42'
48'
54' 1 2 3
4 5
45°
10-1505
1513
1520
1528
1536
1543
1551
1559
1567
1574
I 3 4
5 6
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
io"i5S2
10-1662
10-1745
1590
1670
1753
1598
167S
1762
1606
1687
1770
1614
1695
1779
1622
1703
17S7
1630
1711
1796
1638
1720
1805
1646
1728
1S13
1654
1736
1S22
I 3 4
I 3 4
I 3 4
5 7
6 7
6 7
10-1831
10-1919
10-201 1
1839
1928
2021
1S4S
1937
2030
1857
1947
2040
1866
1956
2049
1875
1965
2059
1883
1974
2068
1892
1983
2078
1901
1993
20S7
1910
2002
2097
1 3 4
2 3 5
2 3 5
6 7
6 8
6 8
10-2107
10-2205
10-2308
2116
2215
2318
2126
2226
2329
2136
2236
2339
2146
2246
2350
2156
2256
2360
2165
2266
2371
2175
2277
23S2
2185
22S7
2393
2195
2297
2403
2 3 5
2 3 5
245
7 8
7 9
7 9
10-2414
2425
2436
2447
2458
2469
2480
2491
2502
2513
245
7 9
10-2524
10-2639
10-2758
2536
2651
2770
2547
2662
2782
2558
2674
2795
2570
2686
2S07
2581
2698
2S19
2593
2710
2S32
2604
2722
2844
2616
2734
2856
2627
2746
2869
246
246
246
8 10
8 10
8 10
10-2S82
10-3010
10-3144
2894
3023
3158
2907
3037
3172
2920
3050
31S6
2932
3063
3199
2945
3077
3213
2958
3090
3227
2971
3104
3241
2984
3117
3256
2997
3131
3270
246
247
2 5 7
8 II
9 II
9 12
10-3284
10-3430
10-3582
3298
3444
3597
3313
3459
3613
3327
3474
3629
3341
3490
3644
3356
3505
3660
3371
3520
3676
3385
3535
3692
3400
3551
3708
3415
3566
3724
2 5 7
3 5 8
3 5 8
10 12
10 13
11 13
10-3741
3757
3773
3790
3S06
3823
3839
3856
3873
3890
3 6 8
11 14
10-3907
10-4081
10-4264
3924
4099
42S3
3941
4117
4302
3958
4135
4321
3976
4153
4340
3993
4172
4359
4010
4190
4379
4028
4208
4398
4046
4227
4417
4063
4246
4437
369
369
3 6 10
12 14
12 15
13 16
10-4457
10-4659
10-4874
4477
4680
4S96
4496
4701
4918
4516
4722
4940
4537
4744
4963
4557
4765
49S5
4577
4787
5008
4598
4808
5031
4618
4830
5054
4639
4852
5077
3 7 10
4 7 II
4 8 II
13 17
14 18
15 19
10-5100
10-5341
10-5597
5124
5366
5623
5147
5391
5650
5171
5416
5677
5195
5441
5704
5219
5467
5731
5243
5492
5758
5267
5518
57S6
5291
5544
5814
5316
5570
5842
4 8 12
4 9 13
5 9 14
16 20
17 21
18 23
10-5870
5898
5927
5956
5985
6014
6043
6073
6103
6133
5 10 15
20 24
10-6163
106479
10-6821
6194
6512
6S57
6225
6545
6893
6255
6579
6930
6287
6613
6966
6318
6647
7003
6350
6681
7041
6382
6716
7079
6414
6750
7117
6446
67S6
7155
5 II 16
6 II 17
6 12 19
21 26
23 28
25 31
10-7194
10-7603
10-8057
7233
7647
8105
7273
7690
S153
7313
7734
8203
7353
7779
8253
7394
7S24
8303
7435
7869
8354
7476
7915
8406
7518
7962
8458
7561
8009
S511
7 14 20
8 15 23
S 17 25
27 34
30 38
34 42
10-8564
10-9141
10*9808
8619
9203
9880
8674
9266
9954
8729
9330
0030
8786
9395
0106
8843
9461
0184
8901
9528
0264
8960
9597
0345
9019
9666
0427
90S0
9736
0511
10 19 29
11 22 33
13 26 39
38 48
44 56
53 66
11-0597
0685
0774
0865
0958
1054
1151
1251
1353
1457
16 32 48
64 81
11-1564
11-2812
11-4572
1674
2959
4794
1787
31 II
5029
1902
3269
5277
2021
3433
5541
2143
3603
5821
2269
37S0
6120
2398
3965
6442
2532
4158
6790
2670
4360
7168
11-7581
8039
8550
9130
9800
0592
1561
2810
4"57i
7581
3°
LOGARITHMIC COSECANTS.
0°
0'
6'
12'
18'
24'
30'
36'
42'
48'
54'
12 3
4 5
Inf.
12-76
4571
2810
1561
0592
980c
913c
855c
803c
,
1
2
3
4
5
6
II-7581
11-4572
II'28l2
7168
4360
2670
679c
415S
2532
6442
3965
2398
6l2C
378c
2269
5S21
3603
2143
5541
3433
2021
5277
3269
1902
5025
3111
1787
4794
2959
1674
11-1564
11-0597
10-9808
1457
05 1 1
973(>
1353
0427
9666
1251
0345
9597
II5I
0264
9528
1054
0184
9461
095S
0106
9395
0865
0030
9330
0774
9954
9266
0685
9880
9203
16 32 48
13 26 39
II 22 33
64 81
53 66
44 56
7
8
9
IO-9I4I
IO-S564
10-8057
90S0
8511
8009
9019
8458
7962
S960
8406
7915
S901
8354
7S69
8843
8303
7824
8786
S253
7779
8729
8203
7734
8674
8153
7690
86ig
8105
7647
10 19 29
8 17 25
8 15 23
38 48
34 42
30 38
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
10-7603
7561
7518
7476
7435
7394
7353
7313
7273
7233 7 14 20
27 34
10-7194
10-6821
10-6479
7155
6786
6446
7117
6750
6414
7079
6716
6382
7041
6681
6350
7003
6647
6318
6966
6613
6287
6930
6579
6255
6S93
6545
6225
6S57
6512
6194
6 12 19
6 II 17
5 II 16
25 31
23 28
21 26
10-6163
IO-5S7O
10-5597
6133
5S42
5570
6103
5S14
5544
6073
57S6
55i«
6043
575S
5492
6014
5731
5467
5985
5704
5441
5956
5677
5416
5927
5650
5391
5898
5623
5366
5 10 15
5 9 14
4 9 13
20 24
18 23
17 21
10-5341
10-5100
IO-4S74
5316
5077
4852
5291
5054
4830
5267
5031
4808
5243
5008
4787
5219
4985
4765
5195
4963
4744
5171
4940
4722
5147
491S
4701
5124
4896
4680
4 8 12
4 8 II
4 7 II
16 20
15 19
14 18
10-4659
4639
4618
4598
4577
4557
4537
4516
4496
4477
3 7 10
13 17
10-4457
10-4264
10-4081
4437
4246
4063
4417
4227
4046
4398
4208
4028
4379
4190
4010
4359
4172
3993
4340
4153
3976
4321
4135
3958
4302
4117
3941
4283 3 6 10
4099 369
3924 369
13 16
12 15
12 14
10-3907
10-3741
10-3582
3890
3724
3566
3S73
3708
3551
3856
3692
3535
3S39
3676
3520
3823
3660
3505
3806
3644
3490
3790
3629
3474
3773
3613
3459
3757
3597
3444
^ 6 8
3 5 8
3 5 8
II 14
II 13
10 13
10-3430
10-3284
10-3144
3415
3270
3131
3400
3256
3117
3385
3241
3104
3371
3227
3090
3356
3213
3077
3341
3199
3063
3327
31S6
3050
3172
3037
3298
3158
3023
2 5 7
2 5 7
247
IQ 12
9 12
9 II
10-3010
2997
2984
2971
2958
2945
2932
2920
2907
2894
246
8 II
31
32
33
34
35
36
37
38
39
IO-28S2
10-2758
10-2639
2869
2746
2627
2856
2734
2616
2S44
2722
2604
2S32
2710
2593
2819
2698
2581
2807
26S6
2570
2795
2674
2558
2782
2662
2547
2770
2651
2536
246
246
246
8 10
8 10
8 10
10-2524
10-2414
10-2308
2513
2403
2297
2502
2393
2287
2491
2382
2277
2480
2371
2266
2469
2360
2256
245S
2350
2246
2447
2339
2236
2436
2329
2226
2425
2318
2215
2 4 5
245
2 3 5
7 9
7 9
7 9
10-2205
10-2107
10-201 1
2195
2097
2002
2185
2087
1993
2175
207S
1983
2165
2068
1974
2156
2050
1965
2146
2049
1956
2136
2040
1947
2126
2030
1937
2116
2021
1928
2 3 5
2 3 5
2 3 5
7 8
6 8
6 8 :
40
41
42
43
44
10*1919
1910
1901
1892
1883
1875
1866
1857
I84S
1839
I 3 4
6 7
10-1831
IO-I745
10-1662
1822
1736
1654
1813
1728
1646
1805
1720
1638
1796
1711
1630
1787
1703
1622
1779
1695
1614
1770
1687
1606
1762
1678
1598
1753
1670
1590
I 3 4
I 3 4
I 3 4
6 7
6 7 .
5 7
10-1582
1574
1567
1559I1551
1543
1536
1528
1520
1513
I 3 4
5 6
J\f,£, — Numbers in difference columns to be subtracted, not added. — Stfe Kuics.
LOGARITHMIC COSECANTS.
31
0'
6'
12'
18'
24'
30' 36'
42'
48'
54'
12 3
4 5
45°
10-1505
149S
1490
1483
1475
1468
1460
1453
1445 1438
1 2 4
5 6
46
47
48
10-1431
10-1359
10-1289
1423
1352
1282
1416
1345
1276
1409
1338
1269
1402
1331
1262
1394
1324
1255
13S7
1317
1249
1380
1310
1242
1373 1366
1303 1296
12351229
1 2 4
1 2 3
1 2 3
5 <3
5 6
4 6
49
60
61
10-1222
IO-II57
10-1095
1216
1151
10S9
1209
"45
1083
1203
1138
1077
1196
1132
1071
1190
1126
1065
1183
1120
1059
1177
1113
1053
1170
1 107
1047
'1164
not
1041
1 2 3
1 2 3
I 2 3
4 5
4 5
4 5
62
53
64
10-1035
10-0977
10-0920
1029
0971
0915
1023
0965
0909
1017
0959
0904
ion
0954
0899
1005
0948
0S93
1000
0943
0888
0994
0937
0882
0988
0931
0877
09S2
0926
0872
I 2 3
I 2 3
1 2 3
4 5
4 5
4 S
55
10-0S66
0S61
0856
0851
0S45
0840
0835J0S30
0825
0S19
I 2 3
3 4
56
67
58
10-0S14
10-0764
10-0716
0809
0759
0711
0S04
0754
0706
0799
0749
0702
0794
0745
0697
0789
0740
0692
0784
0735
0688
0779
0730
0683
0774
0725
0678
0769
0721
0674
122
122
122
3 4
3 4
3 4
69
60
61
10-0669
10-0625
10-05S2
0665
0620
057S
0660
0616
0573
0656
0612
0569
0651
0607
0565
0647
0603
0561
0642
0599
0557
0638
0594
0553
0633
0590
0549
0629
0586
0545
112
112
1 1 2
3 4
3 4
3 3
62
63
64
10-0541
io'05oi
10-0463
0537
0497
0460
0533
0494
0456
0529
0490
0452
0525
04S6
0449
0521
0482
0445
0517
0478
0442
0513
0475
0438
0509
0471
0434
0505
0467
0431
112
I 1 2
1 1 2
3 3
3 3
2 3
65
10-0427
0424
0420
0417
0413
0410
0406
0403
0399
0396
1 I 2
2 3
66
67
68
10-0393
10-0360
10-0328
0389
0357
0325
0386
0353
0322
0383
0350
0319
0379
0347
0316
0376
0344
0313
0373
0341
0310
0369
0338
0307
0366
0334
0304
0363
0331
0301
I 1 2
1 I 2
0 1 I
2 3
2 3
2 2
69
70
71
10-0298
10-0270
10-0243
0296
0267
0241
0293
0265
0238
0290
0262
0236
0287
02S9
0233
0284
0257
0230
02S1
0254
0228
0278
0251
0225
0276
0249
0223
0273
0246
0220
0 1 1
0 1 1
0 I I
2 2
2 2
2 2
72
73
74
10-0218
10-0194
10-0172
0215
0192
0169
0213
0189
0167
02 II
0187
0165
0208
0185
oi6j
0206
0183
0161
0203
oiSo
0159
0201
0178
0157
0199
0176
0155
0196
0174
0153
oil
0 1 1
oil
2 2
I 2
I 2
76
10-0151
0149
0147
0145
0143
0141
0139
0137
0135
0133
0 1 1
1 2
76
77
78
10-0131
10-0113
10-0096
0129
on I
0094
0127
0109
0093
0125
0108
0091
0124
0106
0090
0122
0104
00S8
0120
0103
0087
0118
OIOI
0085
0116
0099
0084
0115
0098
0082
0 1 1
0 I 1
0 I I
1 2
79
80
81
10-0081
10-0066
10-0054
0079
0065
0053
0078
0064
0051
0076
0063
0050
0075
0061
0049
0073
0060
0048
0072
0059
0047
0071
0057
0046
0069
0056
0045
0068
0055
0044
001
0 0 I
0 0 I
82
83
84
10-0042
10-0032
10-0024
0041
0032
0023
0040
0031
0022
0039
0030
0022
003S
0029
0021
0037
0028
0020
0036
0027
0019
0035
0026
0019
0034
0025
0018
0033
0025
0017
000
000
000
0 1
85
10-0017
0016
0015
0015
0014
0013
0013
0012
0012
0011
000
0 0
86
87
88
10-001 1
10-0006
10-0003
0010
0006
0002
0010
0005
0002
0009
0005
0C02
0009
0004
0002
oooS
0004'
0001
0008
0004
0001
0007
0004
0001
0007
0003
0001
0006
0003
0001
000
000
000
0 0
0 0
0 0
89
lo-oooi 0001
0000
0000
0000
0000 0000 1 0000
0000
0000
000
0 0 1
32
NATURAL SINES.
0°
O'
6'
12'
18
24'
30'
36'
42'
48'
54'
12 3
4 5
oooo
C017
0035
0052
0070
0087
0105
0122
0140
0157
369
12 15
1
2
3
OI75
0349
0523
0192
0366
0541
0209
03S4
0558
0227
0401
0576
0244
0419
0593
0262
0436
0610
0279
0454
0628
0297
0471
0645
0314
048S
0663
0332
0506
06S0
369
369
369
12 15
12 15
12 15
4
5
6
7
8
9
069S
0S72
1045
0715
0889
1063
0732
0906
1080
0750
0924
1097
0767
0941
i"5
0785
095 8
1132
0S02
0976
1 149
0819
0993
1167
0837
lOII
11S4
0S54
1028
1201
369
369
369
12 15
12 14
12 14
1219
1392
1564
1236
1409
1582
1253
1426
1599
1271
1444
1616
1288
1461
1633
1305
1478
1650
1323
1495
1668
1340
1513
1685
1357
1530
1702
1374
1547
1719
369
369
369
12 14
12 14
12 14
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
1736
1754
1771
1788
1805
1822
1S40
1S57
1S74
1891
369
12 14
1908
2079
2250
1925
2096
2267
1942
2113
2284
1959
2130
2300
1977
2147
2317
1994
2164
2334
201 1
2181
2351
2028
2198
2368
2045
2215
23S5
2062
2232
2402
369
369
368
II 14
II 14
II 14
2419
25 S8
2756
2436
2605
2773
2453
2622
2790
2470
2639
2S07
2487
2656
2823
2504
2672
2840
2521
26S9
2857
2538
2706
2874
2554
2723
2890!
2571
£740.
368
3 6 8
6 6 8
II 14
II 14
II 14
2924
3090
3256
2940
3107
3272
2957
3123
3289
2974
3140
3305
2990
3156
3322
3007
3173
3024
3190
3355
3040
3206
3371
3057
3223
3387
3074
3239
3404
368
3 6 8
3 5 8
II 14
II 14
II 14
3420
3437
3453
3469
3486
3502
351S
3535
3551
3567
3 5 8
II 14
3584
3746
3907
3600
3762
3923
3616
3778
3939
3633
3795
3955
3649
3811
3971
3665
3827
39S7
36S1
3843
4003
3697
3S59
4019
3714
3875
4035
3730
3891
4051
3 5 8
3 5 8
3 5 8
II 14
4067
4226
4384
4083
4242
4399
4099
4258
4415
4115
4274
4431
4131
4289
4446
4147
4305
4462
4163
4321
4478
4179
4337
4493
4195
4352
4509
4210
4368
4524
3 5 8
3 5 8
3 5 8
II i3
II 13
10 13
4540
469s
4848
4555
4710
4863
4571
4726
4879
4586
474c
4894
4602
4756
4909
4617
4772
4924
4633
47S7
4939
4648
4802
4955
4664
4S18
4970
4679
4833
49S5
3 5 8
3 5 8
3 5 8
10 13
10 13
10 13
5000
5015
5030
5045
5060
5075
5090
5105
5120
5135
3 5 8
10 13
5150
5299
5446
5165
5314
5461
5180
5329
5476
5195
5344
5490
5210
5358
5505
5225
5373
5519
5240
5388
5534
5255
5402
554S
5270
5417
5563
5284
5432
5577
2 5 7
2 5 7
2 5 7
10 12
10 12
10 12
5592
5736
5S78
5606
5750
5892
5621
5764
5906
5635
5779
5920
5650
5793
5934
5664
5S07
5948
5678
5S21
5962
5693
5S35
5976
5707
5850
5990
5721
5864
6004
2 5 7
2 5 7
2 5 7
10 12
10 12
9 12
6018
6157
6293
6032
6170
6307
6046
6184
6320
6060
6198
6334
6074
6211
6347
6088
6225
6361
6101
6239
6374
6115
62^2
63S8
6129
6266
6401
6143
6280
6414
2 5 7
2 5 7
247
9 12
9 II
9 II
6428
6441
6455
6468
6481
6494
6508
6521
6534
6547
247
9 II
6561
6691
6820
6574
6704
6S33
6587
6717
6845
6600
6730
6858
6613
6743
6871
6626
6756
6SS4
6639
6769
6896
6652
67S2
6909
6665
6794
6921
667S
6807
6934
2 4 7
246
246
9 II
9 II
8 II
6947
6959
6972
69S4
6997
7009
7022 7034
7046
7059 246
8 10
NATURAL SINES
z^
■^1
3-
^-•,^-
45°
46.
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
67
68
69
70
71
~72~
73
74
75
76
77
7d
79
80
81
82
83
84
85
86
87
88
89
0'
6'
12'
18'
24'
30'
36'
42'
48'
54'
12 3
4 5
7071
70S3
7096
7108
7120
7133
7145
7157
7169
7181
246
8 10
7193
7314
7431
7206
7325
7443
7218
7337
7455
7230
7349
7466
7242
7361
7478
7254
7373
7490
7266
7385
7501
7278
7396
75^3
7290
7408
7524
7302
7420
7536
246
246
246
8 10
8 10
8 10
7547
7660
7771
7558
7672
77S2
7570
7683
7793
75S1
7094
7804
7593
7705
7S15
7604
7716
7826
7615
7727
7837
7627
7738
7848
7638
7749
7859
7649
7760
7869
246
246
2 4 5
8 9
7 9
7 9
7880
79S6
8090
7891
7997
Sioo
7902
S007
8111
7912
S018
8121
7923
8028
8131
7934
8039
8141
7944
8049
8151
7955
8059
8161
7965
8070
8171
7976
80S0
8181
2 4 5
2 3 5
2 3 5
7 9
7 9
7 8
S192
S202
82 1 1
8221
8231
8241
8251
8261
8271
8281
2 3 5
7 8
8290
S3S7
84S0
8300
S396
8'49o
8310
8406
8499
8320
8415
8508
8329
8425
8517
8339
S434
8526
8348
8443
8536
8358
8453
S545
8368
8462
8554
8377
8471
8563
2 3 5
2 3 5
2 3 5
6 8
6 8
6 8
8572
S660
S746
8581
8669
875s
8590
m
8599
pi686
^771
8607
S095
8780
S616
S704
87S8
8625
8712
8796
8634
8721
8S05
8643
8729
8813
8652
873S
8S21
I 3 4
I 3 4
I 3 4
6 7
6 7
6 7
SS29
S910
S9S8
SS38
89 1 8
8996
8S46
8926
9003
8854
8934
901 1
8862
8942
9018
8870
8949
9026
8S78
S957
9033
8S86
8965
9041
8894
8973
9048
S902
89S0
9056
I 3 4
I 3 4
I 3 4
5 7
5 6
5 6
9063
9070
9078
9085
9092
9100
9107
9114
9121
9128
I 2 4
5 6
9135
9205
9272
9143
9212
9278
9150
9219
92S5
9157
9225
9291
9164
9232
929S
9171
9239
9304
9178
9245
93"
9184
9252
9317
9191
9259
9323
9198
9265
9330
123
I 2 3
I 2 3
5 6
4 6
4 5
9336
9397
9455
9342
9403
9461
9348
9409
9466
9354
9415
9472
9361
9421
9478
9367
9426
9483
9373
9432
9489
9379
943S
9494
938s
9444
9500
9391
9449
9505
I 2 3
I 2 3
I 2 3
4 5
4 5
4 5
9511
9563
9613
9516
9568
9617
9521
9573
9622
9527
9578
9627
9532
9583
9632
9537
9588
9636
9542
9593
9641
9548
9598
9646
9553
9603
9650
9558
9608
9655
I 2 3
122
122
4 4
3 4
3 4
9659
9664
9668
9673
9677
9681
96S6
9690
9694
9699
I I 2
3 4
9703
9744
9781
9707
9748
9785
9711
9751
9789
9715
9755
9792
9720
9759
9796
9724
9763
9799
9728
9767
9803
9732
9770
9806
9736
9774
9810
9740
9778
9813
112
I I 2
I I 2
3 3
3 3
2 3
9816
9848
9S77
9820
9S51
9880
9823
9854
9882
9826
9S57
9885
9829
9S60
9888
9S33
9863
9890
9S36
9866
9S93
9839
9809
9S95
9842
9S71
9S98
9S45
9S74
9900
I I 2
0 I I
0 I I
2 3
2 2
2 2
9903
9925
9945
9905
9928
9947
9907
9930
9949
9910
9932
9951
9912
9934
9952
9914
9936
9954
9917
9938
9956
9919
9940
9957
9921
9942
9959
9923
9943
9960
0 I I
0 I I
0 I 1
2 2
1 2
I I
9962
9963
9965
9966
9968
9969
9971
9972
9973
9974
0 0 I
I I
9976
99S6
99^14
9977
99S7
9005
9978
99SS
9995
9979
9989
9996
9980
9990
9996
9981
9990
9097
9982
9991
9997
9983
9992
9997
9984
9993
9998
99S5
9993
9998
0 0 I
000
000
I I
I I
0 0
9998
9999
9999
9999
9999
I'OOO
nearly.
I'OOOjI'OOO
nearly, jiie.irly.
rooo
iiearh'.
rooo
itearly.
000
0 0
34
NATURAL
COSINES.
0°
0'
6'
12' 18'
24'
30' 36'
42'
48'
54'
12 3
4 5
'
I'OOO
I '000
nearly.
I'OOO
nearly.
rooo
nearly.
rooo
nearly.
9999
9999
9999
9999
9999
000
0 0
<
1
2
3
4
5
6
9998
9994
99^6
9998
9993
9985
9998
9993
99S4
9997
9992
99S3
9997
9991
99S2
9997
9990
9981
9996
9990
99S0
9996
99S9
9979
9995
9988
9978
9995
9987
9977
000
000
GDI
0 0
1 I
I I
9976
9962
9945
9974
9960
9943
9973
9959
9942
9972
9957
9940
9971
9956
9938
9969
9954
9936
9968
9952
9934
9966
9951
9932
9965
9949
9930
9963
9947
9928
0 0 I
0 I I
0 I I
I I
I 2
I 2
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
9925
9903
9S77
9923
9900
9S74
9921
9S98
9871
9919
9895
9S69
9917
9893
9S66
9914
9890
9863
9912
98S8
9860
9910
9SS5
9857
9907
9S82
9S54
9905
9S80
9S51
0 I I
0 I I
0 I I
2 2
2 2
2 2
9S48
9S45
9S42
9S39
9836
9833
9S29
9826
9S23
9S20
I I 2
2 3
9S16
97S1
9744
9813
9778
9740
9S10
9774
9736
9S06
9770
9732
9S03
9767
972S
9799
9763
9724
9796
9759
9720
9792
9755
9715
97S9
9751
97 1 1
97S5
9748
9707
I I 2
I I 2
I I 2
2 3
3 3
3 3
9703
9659
9613
9699
9655
9608
9694
9650
9603
9690
9646
9598
96S6
9641
9593
96S1
9636
95S8
9677
9632
9583
9673
9b27
9578
9668
9622
9573
9664
9617
9568
I I 2
I 2 2
I 2 2
3 4
3 4
3 4
9563
9511
9455
9558
9505
9449
9553
9500
9444
9548
9494
943S
9542
94S9
9432
9537
9483
9426
9532
9478
9421
9527
9472
9415
9521
9466
9409
9516
9461
9403
I 2 3
I 2 3
I 2 3
4 4
4 5
4 5
9397
9391
9385
9379
9373
9367
9361
9354
9348
9342
I 2 3
4 5
9336
9272
9205
9330
9265
9198
9323
9259
9191
9317
9252
9184
93"
9245
9178
9304
9239
9171
9298
9232
9164
9291
9225
9157
92S5
9219
9150
927S
9212
9143
I 2 3
I 2 3
I 2 3
4 5
4 6
5 6
9135
9063
S9S8
912S
9056
89S0
9121
9048
8973
9114
9041
8965
9107
9033
8957
9100
9026
8949
9092
9018
8942
90S5
9011
8934
9078
9003
8926
9070
8996
8918
I 2 4
I 3 4
I 3 4
5 6
5 6
5 6
8910
SS29
8746
8902
8S21
8738
8894
8813
8729
8SS6
8S05
8721
8S78
8796
S712
8S70
87S8
8704
8S62
S7S0
8695
8S54
8771
8686
8S46
8763
8678
8858
S755
8669
I 3 4
I 3 4
I 3 4
5 7
6 7
6 7
8660
8652
8643
8634
8625
S616
8607
8599
85 90
8581
I 3 4
6 7
8572
8480
83S7
8563
8471
8377
8554
8462
8368
8545
S453
8358
8536
8443
8348
8526
8434
8339
8517
8425
8329
8508
8415
8320
8499
8406
8310
8490
8396
8300
2 3 5
2 3 5
2 3 5
6 8
6 8
6 8
'
8290
8192
8090
8281
8181
80S0
8271
8171
8070
8261
8i6[
8059
8251
8151
8049
8241
8141
8039
8231
8131
S028
8221
8121
S018
82 1 1
8111
8007
8202
8100
7997
2 3 5
2 3 5
2 3 5
7 8
7 8
7 9
7986
7880
7771
7976
7869
7760
7965
7S59
7749
7955
7S48
7738
7944
7837
7727
7934
7S26
7716
7923
7S15
7705
7912
7804
7694
7902
7793
7683
7891
7782
7672
2 4 5
2 4 5
246
7 9
7 9
7 9
'<
7660
7649
7638
7627
7615
•7604
7593
75S1
7570
7559
246
8 9
7547
7431
7314
7536
7420
7302
7524
7408
7290
7513
7396
7278
7501
73S5
726a
7490
7373
7254
7478
7361
7242
7466
7349
7230
7455
7337
7218
7443
7325
7206
246
246
246
8 10
8 10
8 10
7193
71S1
7169
7157
7145
7i.>j
7120
710S 7096
70S3
246
8 ID
N.B. — Numbers in difference columns to be subtracted, not added; — See Riiies.
NATURAL COSINES.
35
O'
6'
12'
18'
24'
30'
36'
42'
48'
54'
12 3
4 5
45°
46
47
48
49
50
51
52
53
54
7071
7059
7046
7034
7022
7009
6997
69S4
6972
6959
246
8 10
6947
6S20
6691
6934
6S07
6678
6921
6794
6065
6909
6782
6652
6S96
6769
6639
6884
6756
6626
6871
6743
6613
6858
6730
6600
6S45
6717
6587
6833
6704
6574
246
246
247
8 II
9 II
9 11
6561
642S
6293
6157
601S
5S78
6547
6414
62S0
6534
6401
6266
6521
6388
6252
6508
6374
6239
6494
6361
6225
6481
6347
621 1
6468
6334
6198
6455
6320
6184
6441
6307
6170
247
247
257
9 II
9 II
9 II
6143
6004
5S64
6129
5990
5850
6115
5976
5835
6101
5962
5821
60S8
5948
5807
6074
5934
5793
6060
5920
5779
6046
5906
5764
6032
5892
5750
2 5 7
2 5 7
257
9 12
9 12
9 12
55
5736
5721
5707
5093
5678
5664
5650
5635
5621
5606
257
10 12
56
57
58
59
60
61
5592
5446
5299
5577
5432
5284
5563
5417
5270
5548
5402
5255
5534
5388
5240
5519
5373
5225
5505
5358
5210
5490
5 544
5195
5476
5329
5 1 So
5461
5314
5165
2 5 7
2 5 7
2 5 7
10 12
10 12
10 12
5150
5000
4S48
5135
49S5
4833
5120
4970
4818
5105
4955
4802
5090
4939
4787
5075
4924
4772
50C0
4909
4756
5045
4894
4741
5030
4S79
4726
5015
4S63
4710
3 5 8
3 5 8
3 5 8
10 13
10 13
10 13
62
63
64
65
66
67
68
69
70
71
72
73
74
75
4695
4540
43S4
4226
4679
4524
4368
4664
4509
4352
4648
4493
4337
4633
4478
4321
4617
4462
4305
.4602
4446
4289
4586
4431
4274
4571
4415
4258
4555
4399
4242
3 5 8
3 5 8
3 5 8
10 13
10 13
11 13
4210
4195
4179
4163
4147
4131
4115
4099
40S3
3 5 8
II 13
4067
3907
3746
4051
3S91
3730
4035
3875
3714
4019
3S59
3697
4003
3843
3681
3087
3665
3971
3811
3649
3955
3795
3633
3939
3778
3616
3923
3762
3600
3 5 8
3 5 8
3 5 8
II 14
II 14
II 14
3584
3420
3256
3567
3404
3239
3551
3387
3223
3535
3371
3206
3518
3355
3190
3502
3173
34S6
3322
3156
3469
3305
3140
3453
3289
3123
3437
3272
3107
3 5 8
3 5 8
3 6 S
II 14
II 14
II 14
3090
2924
2756
3074
2907
2740
3057
2890
2723
3040
2S74
.2706
3024
2857
2689
3007
2840
2672
2990
2823
2656
2974
2807
2639
2957
2790
2622
2940
2773
2605
368
3 6 8
3 6 8
II 14
II 14
II 14
25S8
2419
2250
2079
1908
1736
1564
2571
2554
2538
2521
2504
2487
2470
2453
2436
3 6" 8
II 14
76
77
78
2402
2233
2062
2385
2215
2045
2368
2198
2028
2351
21S1
201 1
2334
2164
1994
2317
2147
1977
2300
2130
1959
2284
21 13
1942
2267
2096
1925
368
369
369
II 14
I.I 14
II 14
79
80
81
82
83
84
85
1S91
1719
1547
1874
1702
1530
1857
16S5
1513
1840
1668
1495
1822
1650
1478
1805
1633
1461
17S8
1616
1444
1771
1599
1426
1754
1582
1409
369
369
369
12 14
12 14
12 14
1392
1219
1045
1374
1201
1028
1357
1 1 84
lOII
1340
1167
0993
1323
1 149
0976
1305
II32
0958
1288
iiiS
0941
1271
1097
0924
1253
loSo
0906
1236
1063
08S9
369
369
369
12 14
12 14
12 14
0S72
0S54
0837
0S19
0S02
0785
0767
0750
0732
0715
369
12 15
86
87
88
0698
0523
0349
06S0
0506
0332
0663
04S8
0314
0645
0471
0297
0628
0454
0279
0610
0436
0262
0593
0419
0244
0576
0401
0227
0558
0384
02C9
0541
0366
0192
369
369
369
12 15
12 15
12 15
89
0175
0157
0140
0122
0105
0087
0070
0052
0035
0017
3 6 9
12 15
I/.B. — Numbers in differeuce columns tg be subtracted, not added —See RuUs.
36
NATURAL
TANGENTS.
0°
1
2
3
4
5
6
O'
6'
12'
18'
24'
30'
36'
42'
48'
54'
12 3
4 5
•oooo
0017
0035
0052
0070
0087
0105
0122
0140
0157
369
12 14-
•OI7S
•0349
•0524
0192
0367
0542
0209
0384
0559
0227
0402
0577
0244
0419
0594
0262
0437
0612
0279
0454
0629
0297
0472
0647
0314
0489
0664
0332
0507
0682
369
369
369
12. 15
12 '15
12 15
•0699
•0875
•1051
0717
0892
1069
0734
0910
1086
0752
0928
1 104
0769
0945
1122
0787
0963
"39
0805
0981
1157
0822
0998
"75
0S40
1016
1 192
0857
1033
1210
369
369
369
12 15
12 15
12 15
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32-
33
34
35
36
37
38
39
•1228
•1405
■1584
1246
1423
1602
1263
1441
1620
12S1
1459
1638
1299
1477
1655
1317
1495
1673
1334
1512
1691
1352
1530
1709
1370
1548
1727
1388
1566
1745
369
369
369
12 15
12 15
12 15
•1763
1781
1799
1817
1835
1853
1871
1890
1908
1926
369
12 IS
•1944
•2126
•2309
1962
2144
2327
1980
2162
2345
1998
2180
2364
2016
2199
2382
2035,
2217'
2401
2053
2235
2419
2071
2254
2438
20S9
2272
2456
2107
2290
2475
369
369
369
12 15
12 15
12 15
•2493
•2679
•2867
2512
2698
2886
2530
2717
2905
2549
2736
2924
2568
2754
2943
25S6
2773
2962
2605
2792
2981
2623
2811
3000
2642
2830
3019
2661
2849
3038
369
369
369
12 16
13 16
13 16
•3057
•3249
•3443
3076
3269
3463
3096
3288
3482
3115
3307
3502
3134
3327
3522
3153
3346
3541
3172
3365
3561
3191
3385
3581
3211
3404
3600
3230
3424
3620
3 6 10
3 6 10
3 6 10
13 16
13 16
13 17
•3640
3659
3679
3699
3719
3739
3759
3779
3799
3819
3 7 10
13 17
•3839
•4040
•4245
3859
4061
4265
3879
4081
4286
3S99
4101
4307
3919
4122
4327
3939
4142
4348
3959
4163
4369
3979
4183
4390
4000
4204
44"
4020
4224
4431
3 7 10
3 7 10
3 7 10
13 17
14 17
14 17
■4452
•4663
•4877
4473
4684
4899
4494
4706
4921
4515
4727
4942
4536
4748
4964
4557
4770
49S6
4578
4791
5008
4599
4813
5029
4621
4834
5051
4642
4856
5073
4 7 10
4 7 II
4 7"
14 18
14 18
15 18
•5095
•5317
•5543
5"7
5340
5566
5'39
5362.
5589
5161
► 5384
5612
5184
5407
5635
5206
5430
5658
5228
5452
5681
5250
5475
5704
5272
5498
5727
5295
5520
5750
4 7 II
4 8 II
4 8 12
15 18
15 19
15 19
•5774
5797
5820
5844
5867
5890
5914
5938
5961
5985
4 8 12
16 20
•6009
•6249
■6494
6032
6273
6519
6056
6297
6544
6080
6322
6569
6104
6346
6594
6128
6371
6619
6152
6395
6644
6176
6420
6669
6200
6445
6694
6224
6469
6720
4 8 12
4 8 12
4 8 13
16 20
16 20
17 21
•6745
•7002
•7265
6771
7028
7292
6796
7054
7319
6822
7080
7346
6847
7107
7373
6873
7133
7400
6899
7159
7427
6924
71S6
7454
6950
7212
7481
6976
7239
7508
4 9 13
4 9 13
5 9 14
17 21
18 22
18 23
■7536
■7813
•8098
7563
7841
8127
7590
7S69
8156
7618
7898
8185
7646
7926
8214
7673
7954
8243
7701
7983
8273
7729
8012
8302
7757
8040
8332
7785
8069
8361
5 9 14
5 10 14
5 10 15
18 23
19 24
20 24
40
41
42
43
•8391
8421
8451
84S1
8511
8541
8571
8601
8632
8662
5 10 15
20 25
•S693
•9004
•9325
8724
9036
9358
8754
9067
9391
8785
9099
9424
8816
9131
9457
8847
9163
9490
8878
9195
9523
8910
9228
9556
8941
9260
9590
8972
9293
9623
5 10 16
5 " 16
6 II 17
21 26
21 27
22. 28
44
•9657
9691
9725
9759
9793
9827
9861
9896
9930
9965
6 II 17
23. 29
NATURAL TANGENTS.-
^^••37
45°
46
47
48
49
50
51
52
53
54
55
56
57
58
69
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
0'
&
12'
18'
24'
30'
36'
42'
48'
54'
12 3
4 5
i-oooo
0035! 0070
0105
0141
0176
0212
0247
0283
0319
6 12 18
24 30
I -0355
1-0724
i*iio6
0392
0761
1 145
0428
0799
11S4
0464
0837
1224
0501
0875
1263
0538
0913
1303
0575
0951
1343
0612
0990
1383
0649
1028
1423
0686
1067
1463
6 12 18
6 13 19
7 13 20
25 31
25 32
26 33
1-1504
1-191S
1-2349
1544
i960
2393
15S5
2002
2437
1626
2045
2482
1667
2088
2527
1708
2131
2572
1750
2174
2617
1792
22 1 8
2662
1833
2261
2708
1875
2305
2753
7 14 21
7 14 22
8 15 23
28 34
29 36
30 38
1-2799
1-3270
1-3764
2846
3319
3814
2S92
3367
3865
2938
3416
3916
2985
3465
3968
3032
3514
4019
3079
3564
4071
3127
3613
4124
3175
3663
4176
3222
3713
4229
8 16 23
8 16 25
9 17 26
31 39
33 41
34 43
I -428 1
4335
4388
4442
4496
4550
4605
4659
4715
4770
9 18 27
36 45
I -4S26
1-5399
I -6003
48S2
5458
6066
4938
5517
6128
4994
5577
6191
5051
5637
6255
5108
5697
6319
5166
5757
6383
5224
5818
6447
5282
58S0
6512
5340
5941
6577
lo 19 29
10 20 30
11 21 32
38 48
40 50
43 53
1-6643
1-7321
I -8040
6709
7391
8115
6775
7461
8190
6842
7532
8265
6909
7603
8341
6977
7675
8418
7045
7747
8495
7113
7820
8572
7182
7893
8650
7251
7966
8728
n 23 34
12 24 36
13 26 38
45 56
48 60
51 64
I -8S07
I -9626
2-0503
SSS7
9711
0594
8967
9797
0686
9047
9883
0778
9128
9970
0872
9210
0057
0965
9292
0145
1060
9375
0233
"55
9458
0323
1251
9542
0413
1348
14 27 41
15 29 44
16 31 47
55 68
58 73
63 78
2-1445
1543
1642
1742
1842
1943
2045
2148
2251
2355
17 34 51
68 85
2-2460
2-3559
2-4751
2566
3673
4876
2673
3789
5002
2781
3906
5129
2889
4023
5257
2998
4142
5386
3109
4262
5517
3220
4383
5649
3332
4504
5782
3445
4627
5916
18 37 55
20 40 60
22 43 65
74 92
79 99
87 108
2-6051
27475
2-9042
6187
7625
9208
6325
7776
9375
6464
7929
9544
6605
8083
9714
6746
8239
9887
6S89
8397
0061
7034
8556
0237
7179
8716
0415
7326
8878
0595
24 47 71
26 52 78
29 58 87
95 "8
104 130
115 144
3-0777
3-2709
3-4874
0961
2914
5105
1 146
3122
5339
1334
3332
5576
1524
3544
5816
1716
3759
6059
1910
3977
6305
2106
4197
6554
2305
4420
6806
2506
4646
7062
32 64 96
36 72 108
41 82 122
129 161
144 iSo
162 203
3-7321
7583
7848
811S
8391
8667
8947
9232
9520
9S12
46 94 139
186 232
4-0108
4-3315
4-7046
0408
3662
7453
0713
4015
7S67
1022
4374
828S
1335
4737
8716
1653
5107
9152
1976
5483
9594
2303
5864
0045
2635
6252
0504
2972
6646
0970
53 107 160
62 124 186
73 146 219
214 267
248 310
292 365
5-1446
5-6713
6-3138
1929
7297
3859
2422
7894
4596
2924
8502
5350
3435
9124
6122
3955
9758
6912
4486
0405
7720
5026
1066
8548
5578
1742
9395
6140
2432
0264
87 175 262
350 437
Difference- columns
cease to be useful,
owing to the rapidity
with which the value
of the tangent changes.
7-1154
8-1443
9-5144
2066
2636
9-677
3002
3863
9-845
3962
5126
I0-02
4947
6427
10-20
5958
7769
10-39
6996
9152
10-58
8062
0579
10-78
9158
2052
10-99
0285
3572
u-20
11-43
11-66
1 1 -91
12-16
12-43
12-71
13-00
13-30
13-62
13-95
86
87
88
89
14-30
19-08
28-64
14-67
19-74
30-14
15-06
20-45
31-82
15-46
21-20
,33-69
1 5 -89
22 -02
35-80
16-35
22-90
38-19
16-83
23-86
I40-92
17-34
24-90
44-07
17-89
26-03
47-74
18-46
27-27
52-oS
57-29
63-66
71-6281-85
95-49]ii4-6|i43-2,i9i-o
286-5
573-0
1
38
NATURAL COTANGENTS.
0'
6'
12'
18'
24'
30'
36'
42'
48'
54'
Difference- columns
not useful here, owing
to the rapidity with
which the value of the
cotangent changes.
0°
Inf.
573-0
286-5
191-0
143-2
114-6
95-49
81-85
71-62
63-66
1
2
3
4
6
6
57-29
28-64
19-08
52-08
27-27
18-46
47 -74 44-07
26-0324-90
17-8917-34
40-92
23-86
16-83
38-19
22-90
16-35
35-80
22 -02
15-89
33-69
21-20
15-46
31-82
20-45
15-06
30-14
19-74
14-67
14-30
1 1 '43
9'5i44
13-95
11 -20
3572
13-62
10-99
2052
13-30
10-78
0579
13-00
10-58
9152
12-71
10-39
7769
12-43
IO-20
6427
12-16
10 -02
5126
1 1 -91
9-845
3S63
11-66
9-677
2636
7
8
9
8-1443
7-1154
6-3138
0285
0264
2432
9158
9395
1742
8062
8548
106C
6996
7720
0405
595S
6912
9758
4947
6122
9124
3962
5350
8502
3002
4596
7894
2066
3859
7297
12 3
4 5
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
5-6713
6140
5578
5026
44S6
3955
3435
2924
2422
1929
5 '1446
4-7046
4-3315
0970
6646
2972
0504
6252
2635
0045
5864
2303
9594
5483
1976
9152
5 107
1653
8716
4737
1335
8288
4374
1022
7867
4015
0713
7453
3662
0408
74 148 222
63 125 188
53 107 160
296 370
252 314
214 267
4-0108
3-7321
3-4874
9812
7062
4646
9520
6806
4420
9232
6554
4197
8947
6305
3977
8667
6059
3759
8391
5816
3544
8x18
5576
3332
7848
5339
3122
7583
5105
2914
46 93 139
41 82 122
^6 72 loS
1S6 232
163 204
144 180
3-2709
3-0777
2 -9042
2506
0595
8S78
2305
0415
8716
2106
0237
8556
1910
0061
8397
1716
9887
8239
1524
9714
8083
1334
9544
7929
1 146
9375
7776
0961
9208
7625
32 64 96
29 58 87
26 52 78
129 161
115 144
104 130
2-7475
7326
7179
7034
6889
6746
6605
6464
6325
61S7
24 47 71
95 "8
2-6051
2-4751
2-3559
5916
4627
3445
5782
4504
3332
5649
43S3
3220
5517
4262
3109
5386
4142
2998
5257
4023
2889
5129
3906
27S1
5002
37S9
2673
4876
3673
2566
22 43 65
20 40 60
18 37 55
87 108
79 99
74 92
2-2460
2-1445
2-0503
2355
1348
0413
2251
1251
0323
2148
"55
0233
2045
1060
0145
1943
0965
0057
1842
0872
9970
1742
0778
9S83
1642
06S6
9797
1543
0594
9711
17 34 51
16 31 47
15 29 44
68 85
63 78
58 73
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
1 44
1 -9626
1-8807
1 -8040
9542
8728
7966
9458
8650
7893
9375
8572
7820
9292
8495
7747
9210
8418
7675
9128
8341
7603
9047
S265
7532
8967
8190
7461
SS87
S115
7391
14 27 41
13 26 38
12 24 36
55 68
51 64
48 60
1-7321
7251
7182
7113
7045
6977
6909
6S42
6775
6709
II 23 34
45 56
1 -6643
1 -6003
1-5399
6577
5941
5340
6512
5880
5282
6447
5818
5224
6383
5757
5166
6319
5697
5108
6255
5637
5051
6191
5577
4994
6128
5517
4938
6066
5458
4S82
II 21 32
10 20 30
10 19 29
43 53
40 50
38 48
1-4826
I -4281
I -3764
4770
4229
3713
4715
4176
3663
4659
4124
3613
4605
4071
3564
4550
4019
3514
4496
3968
3465
4442
3916
3416
4388
3S65
3367
4335
3814
3319
9 18 27
9 17 26
8 16 25
36 45
34 43
33 41
1-3270
1-2799
1-2349
3222
2753
2305
3175
2708
2261
3127
2662
2218
3079
2617
2174
3032
2572
2131
2985
2527
2088
2938
24S2
2045
2S92
2437
2002
2846
2393
i960
8 16 23
8 15 23
7 14 22
31 39
30 38
29 36
1-1918
1S75
1833
1792
1750
1708
1667
1626
1585
1544
7 14 21
28 34
1-1504
1-1 106
1-0724
1463
1067
0686
1423
1028
0649
1383
0990
0612
1343
0951
0575
1303
0913
053S
1263
0875
0501
1224
0S37
0464
1184
0799
042S
"45
0761
0392
7 13 20
6 13 19
6 12 18,
26 33
25 32
25 31
1-0355
0319
0283
0247
0212
0176
0141
0105
0070
0035
6 12 iS 24 30 1
/v'./>. — Numbers in difference columns to be subtracted, not added. — See Kjilcs
NATURAL COTANGENTS.
39
O'
6' 12'
18'
24'
30'
36'
42'
48'
54'
12 3
4 5
45°
I'O
0-9965
0-9930
) 0-9896
0-986 J
0-9827
0-979;
o'975S
) 0-972C
0-969]
6 II 17
23 29
46
47
48
•9657
•93^5
•9004
9623
9293
S972
9590
9260
8941
9556
9228
8910
9523
9195
8878
9490
9163
8847
9457
9131
8S16
9424
9099
8785
9391
9067
8754
9358
9036
8724
6 II 17
5 II 16
5 10 16
22 28
21 27
21 26
49
50
51
■S693
•Sj9i
•S098
S662
8361
S069
8632
S332
8040
S601
S302
S012
S571
8273
7983
S541
8243
7954
8511
8214
7926
84S1
S1S5
7898
8451
8156
7869
8421
8127
7S41
5 10 15
5 10 15
5 10 14
20 25
20 24
19 24
52
53
54
7S13
7265
77S5
750S
7239
7757
7481
7212
7729
7454
7186
7701
7427
7159
7673
7400
7133
7646
7373
7107
7618
7346
7080
7590
7319
7054
7563
7292
702S
5 9 14
5 9 14
4 9 13
18 23
18 23
18 22
55
56
57
58
59
60
61
7002
6976
6950
6924
6S99
6873
6S47
6822
6796
6771
4 9 13
17 21
•6745
•6494
•6249
6720
6469
6224
6694
6445
6200
6669
6420
6176
6644
6395
6152
6619
6371
6128
6594
6346
6104
6569
6322
6080
6544
6297
6056
6519
6273
6032
4 8 13
4 8 12
4 8 12
17 21
16 20
16 20
•6009
•5774
•5543
59S5
5750
5520
5961
5727
5498
5938
5704
5475
5914
5681
5452
5S90
5658
5430
5S67
5635
5407
5S44
5612
53S4
5820
5589
5362
5797
5566
5340
4 8 12
4 8 12
4 8 II
16 20
15 19
15 19
62
63
64
65
66
67
68
69
70
71
72
73
74
■5317
•5095
•4877
5295
5073
4856
5272
5051
4834
5250
5029
4813
5228
5008
4791
5206
49S6
4770
5184
4964
4748
5161
4942
4727
5139
4921
4706
5117
4S99
4684
4 7 II
4 7 II
4 7 II
15 18
15 18
14 18
•4663
4642
4621
4599
4578
4557
4536
4515
4494
4473
4 7 10
14 18
•4452
•4245
•4040
4431
4224
4020
441 1
4204
4000
4390
4183
3979
4369
4163
3959
4348
4142
3939
4327
4122
3919
4307
4101
3899
4286
4081
3879
4265
4061
3859
3 7 10
3 7 10
3 7 10
14 17
14 17
13 17
•3S39
•3640
•3443
3S19
3620
3424
3799
3600
3404
3779
3581
3385
3759
3561
3365
3739
3541
3346
3719
3522
3327
3699
3502
3307
3679
34S2
3288
3659
3463
3269
3 7 10
3 6 10
3 6 10
13 17
13 17
13 16
•3249
•3057
•2S67
3230
3038
2849
3211
3019
2S30
3191
3000
281 1
3172
2981
2792
3153
2962
2773
3134
2943
2754
3115
2924
2736
3096
2905
2717
3076
2SS6
269S
3 6 10
369
369
13 16
13 16
13 16
75
•2679
2661
2642
2623
2605
25S6
2568
2549
2530
2512
369
12 16
76
77
78
•2493
•2309
•2126
2475
2290
2107
2456
2272
20S9
2438
2254
2071
2419
223s
2053
2401
2217
2035
2382
2199
2016
2364
21 So
1998
2345
2162
1980
2327
2144
1962
369
369
369
12 15
12 15
12 15
79
80
81
82
83
84
85
86
87
88
•1944
•1763
•15S4
1926
1745
1566
190S
1727
1548
1890
1709
1530
1S71
1691
1512
1S53
1673
1495
1835
1655
1477
1S17
163S
1459
1799
1620
1441
1781
1602
1423
369
369
369
12 15
12 15
12 15
•1405
•122S
•1051
13S8
1210
1033
1370
1 1 92
1016
1352
1175
0998
1334
1157
0 81
1317
1139
0963
1299
1122
0945
1281
1 104
0928
1263
1086
0910
1246
1069
0S92
369
369
369
12 15
12 15
12 15
•CS75
0S57
0S40
0822
0S05
07S7
0769
0752
0734
0717
369
12 15
•0699
•0524
■0349
06S2
0507
0332
0664
04S9
0314
0647
0472
0297
0629
0454
0279
0612
0437
0262
0594
0419
0244
0577
0402
0227
0559
0384
0209
0542
0367
0192
369
369
369
12 15
12 15
12 15
89
•0175
0157
0140
0122
0105
00S7 0070
0052
0035
0017
369
12 14
N.B. — Numbers in difference columns to be subtracted, not added. — See Rules.
40
NATURAL SECANTS.
0'
6'
12'
18'
24'
30'
36'
42'
48'
54'
12 3
4 5
0°
1
2
3
4
5
6
7
8
9
I "OOOO
0000
0000
0000
0000
0000
OOOI
OOOI
OOOI
OOOI
000
0 0
I -0002
I -0006
1-0014
0002
0007
0015
0002
0007
0016
0003
0008
0017
0003
0009
0018
0003
0010
C019
0004
0010
0020
0004
001 1
0021
0005
0012
0022
0006
0013
0023
000
000
0 0 I
0 0
0 0
1 I
I "0024
I -0038
1-0055
0026
0040
0057
0027
0041
0059
002 8
0043
0061
0030
0045
0063
0031
0046
0065
0032
004S
0067
0034
0050
0069
0035
0051
0071
0037
0053
0073
0 0 I
oil
0 I I
I I
I I
I 2
I -0075
I -0098
l'OI25
0077
OIOI
0127
0079
0103
0130
0082
0106
0133
0084
0108
0136
0086
OIII
0139
0089
0II4
0142
0091
0II6
0145
0093
0II9
0148
0096
0122
0I5I
0 I I
0 I I
0 I I
2 2
2 2
2 2
10
11
12
13
1-0154
0157
0161
0164
0167
0170
0174
0177
0180
0184
I I 2
2 3
1-0187
1-0223
I -0263
0191
0227
0267
0194
0231
0271
0198
0235
0276
0201
0239
0280
0205
0243
0284
0209
0247
0288
0212
0251
0293
0216
0255
0297
0220
0259
0302
I I 2
112
I I 2
3 3
3 4
14
15
16
I -0306
I -0353
I -0403
0311
0358
0408
0315
0363
0413
0320
0367
0419
0324
0372
0424
0329
0377
0429
0334
03 i' 2
0435
0338
0388
0440
0343
0393
0446
0348
0398
0451
122
I 2 3
I 2 3
3 4
3 4
4 5
17
18
19
I -0457
I-05I5
1-0576
0463
0521
0583
0468
0527
0589
0474
0533
0595
04S0
0539
0602
0485
0545
0608
0491
0551
0615
0497
0557
0622
0503
0564
0628
0509
0570
0635
I 2 3
I 2 3
I 2 3
4 5
4 5
4 5
20
I -0642
0649
0655
0662
0669
0676
0683
0690
0697
0704
I 2 3
5 6
21
22
23
I -07 1 1
1-0785
I -0864
0719
0793
0872
0726
0801
0S80
0733
0S08
088S
0740
0816
0896
0748
0824
0904
0755
0832
0913
0763
0S40
0921
0770
0S48
0929
077S
0S56
093S
I 2 4
I 3 4
I 3 4
5 6
5 6
6 7
24
25
26
I -0946
I -1034
I-II26
0955
1043
1 1 36
0963
1052
1145
0972
1061
"55
0981
1070
1 164
0989
1079
1174
0998
1089
1 184
1007
1098
1 194
IOI6
1 107
1203
1025
III7
I2I3
1 3 4
2 3 5
2 3 5
6 7
6 8
6 8
27
28
29
1-1223
1-1326
I '1434
1233
1336
1445
1243
1347
1456
1253
1357
1467
1264
1368
1478
1274
1379
1490
1284
1390
1 501
1294
I40I
I5I2
1305
I4I2
1524
I3I5
1423
1535
2 3 5
2 4 5
246
7 9
7 9-
8 9
30
31
32
33
34
35
36
I-I547
1559
1570
1582
1594
1606
I6I8
1630
1642
1654
246
8 10
I -1666
1-1792
1-1924
1679
1805
1937
1691
181S
1951
1703
1831
1964
1716
1844
1978
1728
1857
1992
I74I
1870
2006
1753
1883
2020
1766
1897
2034
1779
I9I0
2048
246
2 4 7
2 5 7
8 10
9 II
9 12
I -2062
I -2208
1-2361
2076
2223
2376
2091
223S
2392
2105
2253
240S
2120
2268
2424
2134
2283
2440
2149
2299
2456
2163
2314
2472
2178
2329
2489
2193
2345
2505
2 5 7
3 5 8
3 5 8
10 12
10 13
11 13
37
38
39
I -2521
I -2690
I -2868
2538
2708
2886
2554
2725
2904
2571
2742
2923
2588
2760
2941
2605
2778
2960
2622
2796
2978
2639
2813
2997
2656
2831
3016
2673
2849
3035
368
369
369
11 14
12 15
12 16
40
41
42
43
44
1-3054
3073
3093
3112
3131
3151
3I7I
3190
3210
3230
3 7 10
13 16
1-3250
1-3456
1-3673
327c
3478
3696
3291
3499
3718
33"
3520
3741
3331
3542
3763
3352
3563
3786
3373
3585
.3809
3393
3607
3832
3414
3629
3855
3435
3651
3878
3 7 10
4 7 II
4 8 II
14 17
14 18
15 19
I -3902
3925
3949
3972
3996
4020
4044
4069
4093
4118
4 8 12
16 20
NATURAL SECANTS.
41
45"
46
47
48
49
50
51
O'
6'
12'
18'
24'
30'
36'
42' 48'
54'
12 3
4 5
1-4142
4167
4192
4217
4242
.4267
4293
4318
4344
4370
4 8 13
17 21
I '4396
I '4663
1 4945
4422
4690
4974
4448
4718
5003
4474
4746
5032
4501
4774
5062
4527
4802
5092
4554
4830
5121
4581
4859
5151
4608
4S87
5182
4635
4916
5212
4 9 13
5 9 14
5 10 15
18 22
19 23
20 25
1-5243
1-5557
1-5890
5273
5590
5925
5304
5622
5959
5335
5655
5994
5366
56SS
6029
5398
5721
6064
5429
5755
6099
5461
5788
6135
5493
5822
6171
5525
5856
6207
5 10 16
6 II 17
6 12 18
21 26
22 28
24 29
52
53
54
55
56
57
58
59
60
61
1-6243
I -6616
1-7013
6279
6655
7054
6316
6694
7095
6353
6733
7137
6390
6772
7179
6427
6812
7221
6464
6852
7263
6502
6892
7305
6540
6932
7348
6578
6972
7391
6 12 19
7 13 20
7 14 21
25 31
26 33
28 35
1-7434
7478
7522
7566
7610
7655
7700
7745
7791
7837
7 15 22
30 37
1-7SS3
1-8361
1-8871
7929
8410
8924
7976
8460
8977
8023
8510
9031
8070
8561
9084
8118
S612
9139
8166
8663
9194
8214
8714
9249
8263
8766
9304
8312
8818
9360
8 16 24
9 17 26
9 18 27
32 40
34 43
36 45
I -9416
2 -0000
2-0627
9473
0061
0692
9530
0122
0757
95S7
0183
0S24
9645
0245
0890
9703
0308
0957
9762
0371
1025
9821
0434
1093
9880
0498
1162
9940
0562
1231
10 19 29
10 21 31
11 22 34
39 49
42 52
45 56
62
63
64
65
"66~
67
68
69
70
71
2-1301
2-2027
2-2S12
1371
2103
2894
1441
2179
2976
1513
2256
3060
1584
2333
3144
1657
2412
3228
1730
2490
3314
1803
2570
3400
1877
2650
3486
1952
2730
3574
12 24 36
13 26 39
14 28 43
48 61
52 66
57 71
2 -3662
3751
3841
3931
4022
4114
4207
4300
4395
4490
15 31 46
62 77
2-4586
2-5593
2-6695
4683
5699
6S11
47S0
5S05
6927
4S79
5913
7046
4978
6022
7165
5078
6131
7285
5180
6242
7407
5282
6354
7529
5384
6466
7653
54S8
6580
7778
17 34 50
18 37 55
20 40 60
67 84
73 92
81 lOI
2-7904
2-923S
30716
8032
9379
0872
8161
9521
1030
8291
9665
1 190
8422
9811
1352
8555
9957
1515
8688
6106
1681
8824
0256
1848
8960
0407
2017
9099
0561
2188
22 44 67
25 49 74
27 55 82
89 III
99 123
no 137
72
73
74
3-2361
3-4203
3-6280
2535
4399
6502
2712
4598
6727
2891
4799
6955
3072
5003
71S6
3255
5209
7420
3440
541S
7657
3628
5629
7897
3S17
5843
8140
4009
6060
8387
31 61 92
35 69 104
39 79 iiS
123 154
138 173
157 196
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
3-S637
8890
1627
4793
8496
9147
9408
9672
9939
021 1
0486
0765
1048
45 90 135
180 225
4-1336
44454
4-8097
1923
5137
8901
2223
5486
9313
2527
5841
9732
2837
6202
0159
3150
6569
0593
3469
6942
1034
3792
7321
1484
4121
7706
1942
52 104 156
61 122 1S2
72 144 2l6
208 260
243 304
287 359
5-240S
5-7588
6-3925
28S3
8164
4637
3367
8751
5366
3860
9351
6111
4362
9963
6874
4S74
0589
7655
5396
1227
8454
592S
1880
9273
6470
2546
0112
7023
3228
0972
86 173 259
345 432
7-1853
8-2055
9-5668
11-47
2757
3238
7283
11-71
3684
4457
8955
4635
57"
0685
561 1
7004
2477
6613
8337
4334
7642
9711
6261
8700
1129
8260
97S7
2593
11-03
0905
4105
11-25
11-95
12-20
12-4712-7513-03
13-34
13-65
13-99
14-34
19-11
28-65
14-70
19-77
30-16
15-09
20-47
31-84
15-50
21-23
33-71
15-9316-3816-86
22-04 22-93 23-8S
35-81 38-2040-93
17-37
24-92
44-oS
17-91
26-0=;
47-75
18-49
27-29
52-09
57-30
63-66
71-62
81-85
95-49114-6,143-2
191-0
2S6-5
573-0
42
NATURAL COSECANTS,
0°
1
2
3
4
5
6
7
8
9
10
0'
&
12'
18'
24'
30'
36'
42'
48'
54'
12 3
4 5
Inf.
573-0
2S6-5
191-0
143-2 114-6
95-49
81-85
71-62
63-66
61 122 182
52 104 156
243 304
20S 260
57-30
28-65
19-11
52-09
27-29
18-49
47-75
26-05
17-91
44-08
24-92
17-37
40-9338-20
23-8822-93
16-8616-38
35-81
22-04
15-93
33-71
21-23
15-50
31-84
20-47
15-09
30-16
19-77
14-70
14-34
11-47
9-5668
13-99
11-25
4105
13-65
11-03
2593
13-34
10-83
1 1 29
13-03
10-63
9711
.-75
10-43
8337
12-47
10-25
7004
12-20
10-07
57"
11-95
9-895
4457
11-71
9-728
3238
8-2055
7-1853
6-3925
0905
0972
322S
9787
0112
2546
8700
9273
1880
7642
8454
1227
6613
7655
0589
561 1
6874
9963
4635
6111
9351
3684
5366
8751
2757
4637
8164
5-7588
7023
6470
5928
5396
4874
4362
3860
3367
2883
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
5 -240S
4-8097
4 "44 54
1942
7706
4121
1484
7321
3792
1034
6942
3469
0593
6569
3150
0159
6202
2S37
9732
5841
2527
9313
5486
2223
S901
5137
1923
8496
4793
1627
4-1336
3-8637
3-6280
1048
83S7
6060
0765
8140
5843
0486
7897
5629
02 1 1
7657
5418
9939
7420
5209
9672
71S6
5003
940S
6955
4799
9147
6727
4598
8S90
6502
4399
45 90 135
39 79 118
35 69 104
180 225
157 196
138 173
3-4203
3-2361
3-0716
4009
218S
0561
3817
2017
0407
3628
1848
0256
3440
1681
0106
3255
1515
9957
3072
1352
981 1
2891
1 190
9665
2712
1030
9521
2535
0872
9379
31 61 92
27 55 82
25 49 74
123 154
no 137
99 123
2-9238
9099
8960
S824
8688
8555
8422
8291
8161
8032
22 44 67
89 III
2-7904
2-6695
2-5593
7778
65S0
54SS
7653
6466
53S4
7529
6354
5282
7407
6242
51S0
7285
6131
5078
7165
6022
497S
7046
5913
4879
6927
5805
47S0
681 1
5699
4683,
20 40 60
18 37 55
17 34 50
81 lOI
73 92
67 84
2-4586
2-3662
2-2812
4490
3574
2730
4395
34S6
2650
4300
3400
2570
4207
3314
2490
4114
3228
2412
4022
3144
2333
3931
3060
2256
3S41
2976
2179
3751
2S94
2103
15 31 46
14 28 43
13 26 39
62 77
57 71
52 65
2-2027
2-1301
2-0627
1952
1231
0562
1877
1 162
049S
1803
1093
0434
1730
1025
0371
1657
0957
0308
15S4
0S90
0245
1513
0S24
0183
1441
0757
0122
1371
0692
0061
12 24 36
II 22 34
10 21 31
48 60
45 56
42 52
2 -0000
9940
9S80
9821
9762
9703
9645
95S7
9530
9473
10 19 29
39 49
I -9416
1-SS71
1-8361
9360
8818
8312
9304
8766
8263
9249
8714
8214
9194
8663
8166
9139
8612
8118
90S4
8561
8070
9031
8510
8023
8977
8460
7976
8924
S410
7929
9 18 27
8 17 25
8 16 24
36 45
34 42
32 40
1-7883
1-7434
1-7013
7837
7391
6972
7791
7348
6932
7745
7305
6892
7700
7263
6852
7655
7221
6812
7610
7179
6772
7566
7137
6733
7522
7095
6694
7478
7054
6655
7 15 22
7 14 21
7 13 20
30 37
28 35
26 33
I -6616
1-6243
1-5890
6578
6207
5856
6540
6171
5822
6502
6135
5788
6464
6099
5755
6427
6064
5721
6390
6029
5688
6353
5994
5655
6316
5959
5622
6279
5925
5590
6 12 19
6 12 18
6 II 17
25 31
23 29
22 28
1-5557
5525
5493
5461
5429
5398
5366
5335
5304
5273
5 10 16
21 26
1-5243
1-4945
I -4663
5212
4916
4635
5182
4887
4608
5151
4S59
4581
5121
4830
4554
5092
4S02
4527
5062
4774
4501
5032
4746
4474
5003
4718
4448
4974
4690
4422
5 10 15
5 9 14
4 9 13
20 25
19 23
18 22
I -4396
4370
4344
431S
4293
4267
4242
4217
4192
4167
4 8 13
17 21
J^'.B. — Numbers in difference columns to be subtracted, not added. — See Rii/es.
NATURAL COSECANTS.
43
1
o
6'
12'
18'
24'
30'
36' 1
42'
48'
54'
12 3
4 5
45'
1-4142
411S
4093
4069
4044
4020
3996
3972
3949
3925
4 8 12
16 20
46
47
48
49
50
61
I '3902
1-3456
3S7S
3651
3435
3230
3035
2849
3S55
3629
3414
3S32
3607
3393
3S09
35S5
37^6
3563
3352
3763
3542
3331
3741
3520
33"
3718
3499
3291
3696
347S
3270
4 8 II
4 7 II
3 7 10
15 19
14 iS
14 17
I -3250
1-3054
I -2868
3210
3016
2S31
3190
2997
2813
3I7I
2978
2796
3151
2960
2778
3131
2941
2760
3112
2923
2742
3093
2904
2725
3073
2886
2708
3 7 10
369
369
13 16
12 15
12 15
62
53
54 J
I -2690
1-2521
1-2361
2673
2505
2345
2656
2489
2329
2639
2472
2314
2622
2456
2299
2605
2440
2283
25S8
2424
2268
2571
2408
2253
2554
2392
2238
2538
2376
2223
368
3 5 8
3 5 8
II 14
II 13
10 13
55
I -2208
2193
2178
2163
2149
2134
2120
2105
2091
2076
257
10 12
56
57
58
1-2062
1-1924
1-1792
2048
1910
1779
2034
1897
1766
2020
1SS3
1753
2006
1870
I74I
1992
1857
1728
1978
1844
1716
1964
1831
1703
1951
1818
1691
1937
1S05
1679
257
2 4 7
246
9 12
9 II
8 10
59
60
61
1-1666
I -1547
I '1434
1654
1535
1423
1642
1524
1412
1630
1512
1401
I618
I5OI
1390
1606
1490
1379
1594
147S
1368
15S2
1467
1357
1570
1456
1347
1559
1445
1336
246
2 4 6
2 4 5
8 10
8 9
7 9
62
63
64
66
1-1326
1-1223
1-1126
1315
1213
1117
1305
1203
1 107
1294
1 194
IC98
1284
I I 84
IOS9
1274
1174
1079
1264
1 1 64
1070
1253
1155
1061
1243
1145
1052
1233
1136
1043
2 3 5
2 3 5
2 3 5
7 9
6 8
6 8
I -1034
1025
1016
1007
0998
0989
09S1
0972
0963
0955
I 3 4
6 7
66
67
68
I -0946
I -0S64
1-0785
0938
0856
0778
0929
0S48
0770
0921
0S40
0763
C9I3
0832
0755
0904
0S24
0748
0896
0816
0740
0S88
0S08
0733
oSSo
oSoi
0726
0872
0793
0719
I 3 4
I 3 4
I 2 4
6 7
5 7
5 6
69
70
71
72
73
74
I -07 1 1
I -0642
1-0576
0704
0635
0570
0697
0628
0564
0690
0622
0557
O6S3
0615
0551
0676
0608
0545
0669
c6o2
0539
0662
0595
0533
0655
0589
0527
0649
05S3
0521
I 2 3
I 2 3
I 2 3
5 6
4 5
4 5
1-0515
I -0457
I -0403
0509
0451
0398
0503
0446
0393
0497
0440
0388
0491
0435
O3S2
0485
0429
0377
0480
0424
0372
0474
0419
0367
0468
0413
0363
0463
0408
0358
I 2 3
I 2 3
122
4 5
4 4
3 4
75 ; I -0353
034S
0343
0338
0334
0329
0324
0320
0315
031 1
I 2 2
3 4
76
77
78
79
80
81
1-0306
I -0263
1-0223
0302
0259
0220
0297
0255
0216
0293
0251
0212
0288
0247
0209
0284
0243
0205
0280
0239
0201
0276
0235
0198
0271
0231
0194
0267
0227
0191
I I 2
I I 2
I I 2
3 4
3 3
3 3
1-0187
I -01 54
I -0125
01S4
0151
0122
0180
0148
0119
0177
0145
0116
0174
0142
OII4
0170
0139
OIII
0167
0136
oioS
0164
0133
0106
0161
0130
0103
0157
0127
OIOI
I I 2
0 I I
0 I I
2 3
2 2
2 2
82
83
84
85
86
87
88
I -0098
1-0075
1-0055
0096
0073
0053
0093
0071
0051
0091
0069
0050
0089
0067
0048
0086
0065
0046
0084
0063
0045
0082
0061
0043
0079
0059
0041
0077
0057
0040
oil
0 I I
0 I I
2 2
I 2
I I
I -0038
0037
0035
0034
0032
0031
0030
0028
0027
0026
0 0 I
I I
I -0024
1-0014
I -0006
0023
0013
0006
0022
0012
jOoo5
0021
001 1
0004
0020 0019
0010 0010
0004 0003
0018
0009
0003
0017
0008
0003
0016
0007
0002
0015
0007
0002
000
000
000
I I
I I
0 0
89
I -0002
0001 0001
0001
i
0001 1 0000
0000
0000
0000
0000
000
0 0
iV.^'. — Numbers in difference columns to be subtracted, not added. — Si;e Kiilcs.
44
RADIAN MEASURE.
0°
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
0'
6'
12'
18'
24'
30'
36'
42'
48'
54'
12 3
4 5
O'OOOO
0017
0035
0052
0070
0087
0105
0122
0140
0157
369
12 15
0-OI75
0-0349
0-0524
0192
0367
0541
0209
0384
0559
0227
0401
0576
0244
0419
0593
0262
0436
061 1
0279
0454
0628
0297
0471
0646
0314
04S9
0663
0332
0506
0681
369
369
369
12 15
12 15
12 15
0-0698
0-0873
0-1047
0716
0890
1065
0733
0908
1082
0750
0925
1 100
076S
0942
1117
0785
0960
"34
0803
0977
1 1 52
0820
0995
1 169
0838
1012
1187
0855
1030
1204
369
369
369
12 15
12 15
12 15
0-I222
0-1396
0-1571
1239
1414
1588
1257
1431
1606
1274
1449
1623
1292
1466
1641
1309
1484
1658
1326
1501
1676
1344
1518
1693
1361
1536
1710
1379
1553
1728
369
369
369
12 15
12 15
12 15
0-I745
1763
1780
1798
1815
1833
1S50
1868
1885
1902
369
12 15
0-1920
0-2094
0-2269
1937
2112
2286
1955
2129
2304
1972
2147
2321
1990
2164
2339
2007
2182
2356
2025
2199
2374
2042
2217
2391
2059
2234
2409
2077
2251
2426
369
369
369
12 15
12 15
12 15
0-2443
0-2618
0-2793
2461
2635
2810
2478
2653
2827
2496
2670
2845
2513
26S8
2862
2531
2705
2880
2548
2723
2897
2566
2740
2915
2583
2758
2932
2601
2775
2950
369
369
369
12 15
12 15
12 15
1 0-2967
0-3142
I 0-3316
2985
3159
3334
3002
3176
3351
3019
3194
3368
3037
32 1 1
33S6
3054
3229
3403
3072
3246
3421
3089
3264
3438
3107
3281
3456
3124
3299
3473
369
369
369
12 15
12 15
12 15
0-3491
3508
3526
3543
3560
3578
3595
3613
3630
3648
369
12 15
0-3665
0-3840
0-4014
3683
3857
4032
3700
3875
4049
3718
3892
4067
3735
3910
4084
3752
3927
4102
3770
3944
4119
3787
3962
4136
3805
3979
4154
3822
3997
4171
369
369
369
12 15
12 15
12 15
0-4189
0-4363
0-4538
4206
4381
4555
4224
4398
4573
4241
4416
4590
4259
4433
4608
4276
4451
4625
4294
4468
4643
4311
4485
4660
4328
4503
4677
4346
4520
4695
369
369
369
12 15
12 15
12 15
0-4712
0-4S87
0-5061
4730
4904
5079
4747
4922
5096
4765
4939
5114
4782
4957
5131
4800
4974
5149
4817
4992
5166
4835
5009
5184
4852
5027
5201
4869
5044
5219
369
369
369
12 15
12 15
12 15
0-5236
5253
5271
52S8
5306
5323
5341
5358
5376
5393
369
12 15
0-5411
0-5585
0-5760
5428
5603
5777
5445
5620
5794
5463
5637
5812
5480
5655
5839
5498
5672
5847
5515
5690
5864
5533
5707
5882
5550
5725
5899
5568
5742
5917
369
369
369
12 15
12 15
12 15
0-5934
0-6109
0-62S3
5952
6126
6301
5969
6144
6318
59S6
6161
6336
6004
6178
6353
6021
6196
6370
6039
6213
6388
6056
6231
6405
6074
6248
6423
6091
6266
6440
369
369
369
12 15
12 15
12 15
0-6458
0-6632
0-6S07
6475
6650
6S24
6493
6667
6842
6510
6685
6S59
6528
6702
6877
6545
6720
6S94
6562
6737
6912
6580
6754
6929
6597
6772
6946
6615
6789
6964
369
369
369
12 15
12 15
12 15
0-6981
6999
7016
7034
7051
7069
70S6
7103
7121
7138
369
12 15
0-7156
0-7330
0-7505
7173
734S
7522
7191
7365
7540
720S
7383
7557
7226
7400
7575
7243
7418
7592
7261
7435
7610
7278
7453
7627
7295
7470
7645
7313
7487
7662
369
3-69
369
12 15
12 15
12 15
0-7679
7697
7714 7732
7749
7767
7784
7S02
7819
7837
3t6 9
12 15
RADIAN MEASURE.
45
45'
46
47
48
49
60
51
52
53
64
55
66
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
O'
6'
12'
18'
24'
30'
36'
42'
48'
54'
12 3
4 5
07S54
7871
7SS9
7906
7924
7941
7959
7976
7994
Sou
369
12 15
o-So29
o'S203
0-8378
S046
8221
S395
8063
S238
8412
8081
8255
8430
8098
8273
S447
8116
8290
8465
8133
8308
8482
8151
8325
8500
8168
8343
S517
8186
8360
8535
369
369
369
12 15
12 15
12 15
0-8552
0-S727
0-S901
8570
8744
S919
8587
8762
S936
8604
8779
8954
8622
8796
S971
8639
8814
8988
S657
8831
9006
8674
8849
9023
8692
8866
9041
8709
8884
9058
369
369
369
12 15
12 15
12 15
0-9076
0-9250
0-9425
9093
9268
9442
9111
9285
9460
9 1 28
9303
9477
9146
9320
9495
9163
9338
9512
9 1 So
9355
9529
9198
9372
9547
9215
9390
9564
9233
9407
9582
369
369
369
12 15
12 15
12 15
0-9599
9617
9634
9652
9669
9687
9704
9721
9739
9756
369
12 15
0-9774
0-9948
1-0123
9791
9966
0140
9809
9983
0158
9826
0001
0175
9S44
6018
0193
9S61
0036
0210
9S79
0053
0228
9896
0071
0245
9913
0088
0263
9931
6105
02S0
369
369
369
12 15
12 15
12 15
I -0297
1 -0472
1-0647
0315
0489
0664
0332
0507
0681
0350
0524
0699
0367
0542
0716
0385
0559
0734
0402
0577
0751
0420
0594
0769
0437
0612
0786
0455
0629
0S04
369
369
369
12 15
12 15
12 15
1-0821
I -0996
1-1170
0S38
1013
11S8
0856
1030
1205
0873
104S
1222
0S91
1065
1240
090S
10S3
1257
0926
1 100
1275
0943
1118
1292
0961
1135
1310
0978
"53
1327
369
369
369
12 15
12 15
12 15
I*i345
1362
1380
1397
1414
1432
1449
1467
1484
1502
369
12 15
1-1519
I -1694
I -1868
1537
1711
1886
1554
1729
1903
1572
1746
1921
15S9
1764
1938
1606
1781
1956
1624
1798
1973
1641
1816
1990
1659
1833
2008
1676
1851
2025
369
369
369
12 15
12 15
12 15
1 -2043
1-2217
1-2392
2060
2235
2409
2078
2252
2427
2095
2270
2444
2113
2287
2462
2130
2305
2479
2147
2322
2497
2165
2339
2514
21S2
2357
2531
2200
2374
2549
369
369
369
12 15
12 15
12 IS
1-2566
1-2741
1-2915
2584
2758
2933
2601
2776
2950
2619
2793
2968
2636
2811
29S5
2654
2828
3003
2671
2846
3020
2689
2863
3038
2706
2881
3055
2723
2898
3073
369
369
369
12 15
12 15
12 15
1 -3090
3107
3125
3142
3160
3177
3195
3212
3230
3247
369
12 15
1 -3265
1-3439
1-3614
3282
3456
3631
3299
3474
3648
3317
3491
3666
3334
3509
3683
3352
3526
3701
3369
3544
3718
3387
3561
3736
3404
3579
3753
3422
3596
3771
369
369
369
12 15
12 15
12 15
1-37SS
I -3963
I '4137
3S06
3980
4155
3823
3998
4172
3840
4015
4190
3858
4032
4207
3875
4050
4224
3S93
4067
4242
3910
4085
4259
3928
4102
4277
3945
4120
4294
3 6 .9
369
369
12 15
12 15
12 15
I -4312
I -4486
I -4661
4329
4504
4678
4347
4521
4696
4364
4539
4713
43S2
4556
4731
4399
4573
4748
4416
4591
4765
4434
4608
4783
4451
4626
4800
4469
4643
4818
369
369
369
12 15
12 15
12 15
1-4835
4853
4870
4888
4905
4923
4940
4957
4975
4992
369
12 15
I -5010
I -5184
1-5359
5027
5202
5376
5045
5219
5394
5062
5237
541 1
5080
5254
5429
5097
5272
5446
5115
5289
5464
5132
5307
5481
5149
5324
5499
5167
5341
5516
3 69
369
369
12 15
12 15
12 15
1-5533
5551
5568
55S6
5603
5621
5638
5656
5673
5691
3 6' 9
12 15
46
SQUARES.
1-0
1-1
1-2
1-3
1-4
1-5
1-6
1-7
1-8
1-9
2-0
2-1
2-2
2-3
2-4
2-5
2-6
0
1
2
3
4
5
6
7
8
.9
12 3
4 5 6
7 8 9
I'OOO
I -020
1-040
I -061
1-082
1-103
I -124
1*145
I -166
I -188
246
8 10 13
15 17 19
I-2I0
1-440
I "690
1-232
1-464
1-716
1-254
1-488
1-742
1-277
1-513
1-769
1-300
1-53S
1796
1-323
1-563
1-823
1-346
1-588
1-850
1-3^9
I -613
1-877
1-392
1*638
1-904
1-416
1-664
1-932
2 5 7
257
3 5 8
9 11 14
10 12 15
11 13 16
16 18 2;
17 20 22
19 22 24
I '960
2-250
2-560
1-988
2-280
2-592
2-016
2-310
2-624
2-045
2-341
2*657
2-074
2-372
2-690
2-103
2-403
2-723
2-132
2-434
2-756
2-i6i
2-465
2-789
2-190
2-496
2-822
2 -220
2-528
2-856
369
369
3 7 10
12 14 17
12 15 19
13 16 20
20 23 26
22 25 28
23 26 30
2-890
3-240
3-610
2-924
3 •■^76
3-648
2-958
3-3*2
3-686
2-993
3*349
3-725
3-028
3-386
3-764
3-063
3-423
3-S03
3-098
3-460
3-842
3-497
3-881
3*168
3*534
3*920
3-204
3-572
3-960
3 7 10
4 7 11
4 ,8 12
14 17 21
15 18 22
16 19 23
24 28 31
26 30 33
27 31 35
4-000
4-040
4-080
4-I2I
4-162
4-203
4-244
4*285
4-326
4-368
4 8 12
16 20 25
29 33 37
4-410
4-840
5-290
4*452
4-S84
5-336
4*494
4-928
5-382
4-537
4-973
5-429
4-580
5-018
5-476
4-623
5-063
5-523
4-666
5-108
5*570
4-709
5-153
5-617
4-752
5-19S
5-664
4-796
5-244
5-712
4 9 13
4 9 13
5 9 14
17 21 26
18 22 27
19 23 28
30 34 39
31 36 40
33 38 42
5-760
6-250
6-760
5-808
6-300
6-812
5-856
6-350
6-864
5-905
6-401
6-917
5-954
6-452
6-970
6-003
6-503
7-023
6-052
6*554
7-076
6-IOI
6-605
7-129
6-150
6-656
7-182
6 -200
6-708
7-236
5 10 15
5 10 15
5 11 16
20 24 29
20 25 31
21 26 32
34 39 44
36 41 46
37 42 48
2-7
2-8
2-9
3-0
31
31
3-2
3-3
3-4
3-5
3-6
3-7
3-8
3-9
4-0
4-1
4-2
4-3
4-4
4-5
4-6
4-7
4-8
4-9
5-0
6-1
5-2
5-3
5-4
7-290
7-840
8-410
7-344
7-896
8-468
7-398
7-952
8-526
7*453
8-009
8-585
7-508
8-066
8-644
7-563
8-123
8-703
7-618
8-i8o
8762
7-673
8-237
8-821
7-728
8-294
8-S80
7-784
8-352
8-940
5 11 16
6 II 17
6 12 18
22 27 11
23 28 34
24 29 35
38 44 49
40 46 51
41 47 53
9-000
9-060
9*120
9-181
9-242
9*303
9-364
9*4259-486
9-548
6 12 18
24 30 37
43 49 55
9-610
10-24
10-89
9-672
10-30
10-96
9-734
10-37
11-02
9-797
10-43
11-09
9-860
10-50
11-16
9-923
jo-56
11-22
9-986
10-63
11-29
10-05
10-69
11-36
lO-II
10-76
11-42
10-18
10-S2
11-49
6 13 19
112
I I 2
I I 2
25 31 38
3 3 4
3 3 4
3 3 4
44 50 57
5 5 6
5 5 6
5 5 6
11-56
12-25
12-96
11-63
12-32
13-03
11-70
12*39
13-10
n-76
12-46
13-18
11-83
12-53
13-25
11-90
1 2 -60
13-32
11-97
12-67
13-40
12-04
12-74
13-47
12-11
12-82
13-54
12-18
12-89
13-62
I I 2
112
I I 2
3 3 4
3 4 4
3 4 4
566
566
5 6 7
13-69
14-44
15-21
13-76
14-52
15-29
13-84
14-59
15-37
13-91
14-67
15-44
13-99
14-75
15-52
14-06
14-82
15-60
14-14
14-90
15-68
14-21
14-98
15-76
14-29
15-05
15-84
14-36
15-13
15-92
I' 2 2
I 2 2
I 2 2
3 4 5
3 4 5
3 4 5
5 6 7
5 6 7
667
16-00
16-08
16-16
1 6 -24
16-32
16-40
16-48
16-56
16-65
16-73
I 2 2
3 4 5
667
16-81
17-64
18-49
16-89
1772
18-58
16-97
17-81
18-66
17-06
17-89
1875
17-14
17-98
18-84
17-22
1 8 -06
18592
17-31
18-15
19-01
17-39
18-23
19-10
17-47
18-32
19-18
17-56
18-40
19-27
I 2 2
1 2 3
I 2 3
3 4 5
3 4 5
3 4 5
6 7 7
678
678
19-36
20-25
21-16
19-45
20-34
21-25
19-54
20-43
21-34
19-62
20-52
21-44
19-71
20-61
21-53
19-80
20-70
21-62
19-89
20-79
21-72
19-98
20-8S
21-81
20-07
20-98
21-90
20-l6
21-07
22-00
I 2 3
I 2 3
I 2 3
4 5 5
4 5 5
4 5 6
678
6 7 8
7 7 8
22-09
23-04
24-01
22-iS
23-14
24-11
22 -28
23-23
24-21
22-37
23-33
24-30
22-47
23-43
24-40
22-56
23-52
24-50
22-66
23-62
24-60
22-75
23-72
24-70
22-85
23-81
24-80
22-94
23-91
24-90
I 2 3
I 2 3
I 2 3
4 5 6
456
456
7 8 9
7 8 9
7 8 9
25-00
25-10
25-20
25-30
25-40
25-50
25-60
25-7025-8125-91
I 2 3
456
7 8 9
26-01
27-04
28-09
26-11
27-14
28-20
26-21
27-25
28-30
26-32
27-35
28-41
26-42
27-46
28-52
26-52
27-56
28-62
26-63
27-67
28-73
26-7326-8326-94
27-7727-8827-98
2S-S42S-94 29-05
I 2 3
I 2 3
I 2 3
4 5 6
4 5 6
456
7 8 9
7 8 9
7 9 10
29-16
29-27
29-38
29-48
29-59
29-70
29-81
29-9230-0330-14
I 2 3
4 6 7
8 9 10
SQUARES.
47
5-5
1
5-6
5-7
5-8
5.-9
6-0
6-l!
0
1
2
3
4
5
6
7
8
9
12 3
4 5 6
7 8 9
30-25
30-36
30-47
30-5S
30-69
30 -So
30-91
31-0231-14
31-25
I 2 3
4 6 7
8 9 10
31-36
32-49
33-64
31 '47
32-60
33-76
31-58
32-72
33-87
31-70
32-83
33 '99
31-81 31-92
32-9533-06
34-1134-22
32-04
33-18
34-34
32-15
33-29
34-46
32-26
33-41
34-57
32-38
34-69
I 2 3
I 2 3
I 2 4
5 6 7
5 6 7
5 6 7
8 9 10
8 9 10
8 9 II
34-81
30-00
37-21
34-93
36-12
J/ OJ
35-05
36-24
37-45
35-16
36-36
37-58
35-28
36-48
37-70
35-40
36-60
37-82
35-52
36-72
37-95
35-64
36-84
38-07
35-76
36-97
38-19
35-88
37-09
38-32
I 2 4
I 2 4
I 2 4
5 6 7
5 6 7
5 6 7
8 10 II
9 10 II
9 10 II
6-2
6-3
6-4
38 -44
39-69
40-96
38-56
39-82
41-09
38-69 38-81
39-9440-07
4i-22|4i-34
38-94
40-20
41-47
39-06
40-32
41-60
39-19
40-45
41-73
39-31
40-58
41-86
39-44
40-70
41-99
39-56
40-83
42-12
I 3 4
I 3 4
I 3 4
568
5 6 8
568
9 10 II
9 10 II
9 10 12
6-5'
6-6i
6-7,
6-8
6-9'
7-0
7-1]
7-2
7-3
7-4
7-5
7-6
7-7
7 -81
7-9
8-0
8-1
8-2
8-3
8-4
8-5
8-6
8-7
8-8
8-9
90
9-1
9-2
9-3
9-4
9-5!
9-6
9-7
9-8
9-9|
42-25
42-38
42-51
42-64
42-77
42-90
43-03
43-16
43-30
43-43
I 3 4
5 7 8
9 10 12
43-56
44-89
46-24
43-69
45-02
46-38
43-82
45-16
46-51
43-96
45-29
46-65
44-09
45-43
46-79
44-22
45-56
46-92
44-36
45-70
47-06
44-49
45-83
47-20
44-62
45-97
47-33
44-76
46-10
47-47
I 3 4
I 3 4
I 3 4
5 7 8
5 7 8
5 7 8
9 II 12
9 II 12
10 II 12
47-61
49-00
50-41
4775
49-14
50-55
47-89
49-28
50-69
48-02
49-42
50-84
48-16
49-56
50 -98
48-30
49-70
51-12
48-44
49-84
51-27
48-58
49-98
51-41
48-72
50-13
51-55
48-86
50-27
51-70
I 3 4
I 3 4
I 3 v4J
678
678
679
10 II 13
10 II- 13
10 II 13
51-84
53-29
5476
51-98
53*44
54-91
52-13
53-58
55-06
52-27
53-73
55-20
52-42
53-88
55-35
52-56
54-02
55-50
52-71
54-17
55-65
52-85
54-32
55-80
53-00
54-46
55-95
53-14
54-61
56-10
I 3 4
I 3 4
I 3 4
679
679
679
10 12 13
10 12 13
10 12 13
56-25
56-40
56-55
56-70
56-85
57-00
57-15
57-30
57-46
57-61
235
689
II 12 14
57-76
59-29
60-84
57-91
59-44
61-00
58-06
59-60
61-15
58-22
59-75
61-31
58-37
59-91
61-47
5S-52
60-06
61-62
58-68
60-22
61 -78
58-83
60-37
61-94
58-98
60-53
62-09
59-14
60 -68
62-25
2 3 5
2 3 S
2 3 5
689
689
689
II 12 14
II 12 14
II 13 14
62-41
64-00
65-61
62-57
64-16
65-77
62-73
64-32
65-93
62-8S
64-48
66-10
63-04
64-64
66-26
63-20
64-80
66-42
63-36
64-96
66-59
63-52
65-12
66-75
63-68
65-29
66-91
63-84
65-45
67-08
2 3 5
235
2 3 5
6 8 10
6 8 10
7 8 10
II 13 14
II 13 14
II 13 15
67-24
68-89
70-56
67-40
69-06
7073
67-57
69-22
70-90
67-73
69-39
71-06
67-90
69-56
71-23
68 -06
69-72
71-40
68-23
69-89
71-57
68-39
70-06
71-74
68-56
70-22
71-91
68-72
70-39
72 -oS
2 3 5
235
2 3 5
7 8 10
7 8 10
7 8 10
12 13 15
12 13 15
12 14 15
72-25
72-42
72-59
72-76
72-93
73-10
73-27
73-44
73-62
7379
2 3 5
7 9 10
12 14 15
73-96
75-69
77-44
74-13
75-86
77-62
74-30
76-04
77-79
74-48
76-21
77-97
74-65
76-39
78-15
74-82
76-56
78-32
75-00
7674
7S-50
75-17
76-91
78-68
75-34
77-09
78-85
75-52
77-26
79-03
235
2 4 5
245
7 9 10
7 9 II
7 9 II
12 14 16
12 14 16
12 14 16
79-21
81-00
82-81
79-39
81-1S
82-99
79-57
81-36
83-17
79-74
81-54
83-36
79-92
81-72
83-54
80-10
81-90
83-72
80-28
82-08
83-91
80-46
82-26
84-09
So -64
82-45
84-27
So-82
S2-63
84-46
2 4 5
2 4 5
2 4 5
7 9 II
7 9 II
7 9 II
13 14 16
13 14 16
13 15 16
84 -64
86-49
88-36
84-82
86-68
88-55
85-01
86-86
88-74
85-19
87-05
88-92
85-38
87-24
89-11
85-56
87-42
89-30
8575
87-61
89-49
85 -93
87-80
89-68
86-12
87 -98
89 -87
86-30
88-17
90-06
246
246
246
7 9 II
7 9 II
8 9 II
13 15 17
13 15 17
13 15 17
90-25
90-44
90-63
90-82
91-01
91-20
91-39
91-58
91-78
91-97
246
8 10 II
13 15 17
92-16
94-09
96-04
92-35
94-2S
96-24
92-54
94-48
96-43
92-74
94-67
96-63
92-93
94-87
96-83
93-12
95-06
97-02
93-32
95-26
97-22
93-51
95-45
97-42
93-70
95-65
97-61
93-90
95-84
97-81
246
214 6
2 '• 4 6
8 10 12
8 10 12
8 10 12
14 15 17
14 16 18
14 16 18
98-01
98 -21
98-41
98 -60
98 -80
99-00
99-20
99-40
99-60
99 -So
246
8 10 12
14 16 18
C5-0-IN-
48
SQUARE ROOTS
FROM 100 TO 999-9.
10
n
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
33
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
0
1
2
3
4
5
6
7
8
9
•1 -2 -3
•4 -5 -6
•7 -8 -9
lo-oo
10-05
10-10
10-15
10-20
10-25
10-30
10-34
10-39
10-44
oil
223
3 4 4
IO-49
IO-95
11-40
10-54
11-00
11-45
10-58
11-05
11-49
10:63
1 1 -09
"•53
10-68
11-14
11-58
10-72
II-18
11-62
10-77
11-22
11-66
10-82
11-27
11-70
10-86
11 -31
"-75
10-91
11-36
11-79
oil
oil
oil
223
223
223
3 4 4
3 4 4
3 3 4
11-83
12-25
12-65
11-87
12-29
12-69
11*92
12-33
12-73
11-96
12-37
12-77
I2-00
12-41
I2-8I
12-04
12-45
12-85
1 2 -08
12-49
12-88
12-12
12-53
12-92
1217
12-57
12-96
12-21
12-61
13-00
oil
0 I I
0 I 1
2 2 2
2 2 2
2 2 2
3 3 4
3 3 4
3 3 4
13-04
13-42
1378
13-08
13-45
13-82
13-11
13-49
13-86
13-15
13-53
13-89
13-19
13-56
13-93
13-23
13-60
13-96
13-27
13-64
14-00
13-30
13-67
14-04
13-34
13-71
14-07
13-38
13-75
14-11
oil
0 I I
oil
2 2 2
I 2 2
12 2
3 3 3
3 3 3
3 3 3
14-14
14-18
14-21
14-25
14-28
14-32
14-35
14-39
14-42
14-46
oil
I 2 2
233
14-49
14-83
15-17
14-53
14-87
15-20
14-56
14-90
15-23
14-59
14-93
15-26
14-63
14-97
15-30
14-66
15-00
15-33
14-70
15-03
15-36
14-73
15-07
15-39
14-76
1510
15-43
14-80
15-13
15-46
0 I I
oil
0 I I
12 2
I 2 2
12 2
233
233
233
15-49
15-81
l6-I2
15-52
15-84
16-16
15-56
15-87
16-19
15-59
15-91
16-22
15-62
15-94
16-25
15-65
15-97
16-28
15-68
16-00
16-31
15-72
16-03
16-34
1575
1 6 -06
16-37
15-78
16-09
16-40
0 I I
oil
0 I I
12 2
I 2 2
1 2 2
233
233
223
16-43
16-7^
17-03
16-46
16-76
17-06
16-49
16-79
17-09
16-52
16-82
17-12
16-55
16-85
17-15
16-58
16-88
17-18
16-61
16-91
17-20
16-64
16-94
17-23
16-67
16-97
17-26
16-70
17-00
17-29
oil
oil
0 I I
1 2 2
112
I I 2
223
223
223
17-32
17-35
17-38
17-41
17-44
17-46
17-49
17-52
17-55
17-58
0 1 I
I I 2
223
i7'6i
17-89
18-17
17-64
17-92
18-19
17-66
17-94
18-22
17-69
17-97
18-25
17-72
1800
18-28
1775
18-03
18-30
17-78
18 -06
18-33
17-80
18-08
18-36
17-83
18-11
18-38
17-86
18-14
18-41
oil
oil
0 I I
1 I 2
I I 2
I I 2
223
222
222
18-44
18-71
18-97
18-47
1873
19-00
18-49
iS-76
19-03
18-52
18-79
19-05
18-55
i8-8i
19-08
18-57
18-84
19-10
1 8 -60
18-87
19-13
18-63
18-89
19-16
18-65
18-92
19-18
18-68
18-95
19-21
0 I I
oil
0 I I
112
I I 2
112
222
222
222
19-24
19-49
1975
1926
19-52
19-77
19-29
19-54
19-80
19-31
19-57
1982
19-34
19-50
19-85
19-36
19-62
19-87
19-39
19-65
19-90
19-42
19-67
19-92
19-44
19-70
19-95
19-47
19-72
1997
oil
oil
0 I I
112
I I 2
1 I 2
222
222
222
20-00
20-02
20-05
20-07
20-10
20-12
20-15
20-17
20-20
20-22
001
I I I
222
20-25
20-49
20-74
20-27
20-52
20-76
20-30
20-54
20-78
20-32
20-57
20 -S I
20-35
20-59
20-83
20-37
20-62
20-S6
20-40
20-64
20-8S
20-42
20-66
20-90
20-45
20-69
20-93
20-47
20-71
20-95
0 0 I
001
0 0 1
I I I
III
I I I
222
222
222
20-98
21-21
21-45
2I*00
21-24
21-47
21 -02
21-26
21-49
21-05
21 -28
21-52
21-07
21-31
21-54
21-10
21-33
21-56
21-12
21-35
21-59
21-14
21-38
21-61
21-17
21-40
21-63
21-19
21-42
21-66
0 0 I
001
0 0 I
III
I I I
I I I
222
222
222
21-68
21-91
22-14
21-70
21-93
22-l6
21-73
21-95
22-18
21-75
21-98
22-20
21-77
22-00
22-23
21-79
22-02
22-25
21-82
22-05
22-27
21-84
22-07
22-29
21-86
22-09
22-32
21-89
22-11
22-34
0 0 1
0 0 1
0 0 I
I I 1
1 1 I
I I I
222
222
222
22-36
22-38
22-41
22-43
22-45
22-47
22-49
22-52
22-54
22-56
001
I I 1
222
22-58
22 -So
23-02
22-61
22-83
23-04
22-63
22-85
23-07
22-65
22-87
23-09
22-67
22-89
23-11
22 69
22-91
23-13
22-72
22-93
23-15
22-74
22 96
23-17
22-76
22-98
23-19
22-78
23-00
23-22
0 0 I
0 0 I
001
I I I
III
1 I I
222
222
222
23-24
23-26
23-28
23-30
23-32
23-35
23-37
23-39
23-41
23-43
0 0 I
I I I
I 2 2
rAcL
SQUARE
ROOTS
FROM 100 TO 999*9.
49
66
56
67
68
69
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
0
1
2
3
4
6
6
7
8
9
-1 -2 -3
•4 -5 -6
•7 -8 9
23-45
23 06
23-4723-4923-52
23-6923-7123-73
23-54
2375
23-5623-58
23-7723-79
23-60
23-81
23-62,23-64
23-832385
0 0 I
0 0 I
122
122
23-87
24-08
24-29
23-9023-92
24-1024-12
24-31 24-33
23-94
24-15
24-35
23-96 23-98'24-oo
24-1724-1924-21
24-37 24-39 24-41
24-02
24-23
24-43
24 -04 24 -06
24-2524-27
24-45|24-47
0 0 I
0 0 I
0 0 1
1 2 2
I 2 2
122
24-49
24-5224-54
24-56
24-58
24-60
24-62
24-64
24-6624-68
0 0 I
122
24-70
24-90
251C
24-7224-74
24-92:24-94
25-1225-14
24-76
24-96
25-16
24-78
24-98
25-18
24-80 24 82
25-0025-02
25-2025-22
24-84
25-04
25-24
24-86^24 88
25 -06 25 08
25-2625-28
001
0 0 I
0 0 I
1 2 2
1 2 2
1 2 2
25-30
25-50
25-09
25-32
25-51
25-71
25-3425-36
25-5325-55
25-732575
25-38
25-57
25-77
25-40
25-59
25-79
25-42
25-61
25-81
25-44
25-63
25-83
25-46
25-65
25-85
25.48
25-07
25-87
0 0 1
0 0 I
001
1 2 2
I 2 2
I 2 2
25-88
2608
26-27
25-90
26-10
26 29
25-92
26-12
26-31
25-94
26-13
26-32
25-96
26-15
26-34
25-98
26-17
26-36
26-00
26-19
26-38
26-02
26-21
26-40
26-04
26-23
26-42
26-06
26-25
26-44
001
001
0 0 1
I 2 2
122
I 2 2
26-46
26-48
26-50
26-51
26-53
26-55
26-57
26-59
26-61
26-63
0 0 I
I 2 2
26-65
26-83
27-02
26-66
26-85
2704
26-6S
26-87
27-06
26-70
26-89
27-07
26-72
26-91
27-09
26-74
26-93
27-11
26-76
26-94
27-13
26-78
26-96
27-15
26-80
26-98
27-17
26-81
27 00
27-18
001
0 0 I
0 0 I
1 I 2
1 1 2
1 1 2
27-20
27-39
27-57
27-22
27-40
27-59
27-24
27-42
27 60
27-26
27-44
27-62
27-28
27-46
27-64
27-29
27-48
27-66
27-31
27-50
27-68
27-33
2751
27-69
27-35
27-53
27-71
27-37
27-55
27-73
001
0 0 1
0 0 1
112
112
112
27 75
27-93
28-11
27-77
27-95
28-12
27-78
27-96
2814
27 So
27-98
28-16
2782
28-00
28-18
27-84
28-02
28-20
27 -86
28-04
28-21
27-87
2805
28-23
27 -89
28-07
28-25
27-91
28-09
28-27
0 0 I
0 0 I
0 0 I
1 1 2
1 1 2
1 I 2
28-28
28-30
28-32
28-34
28-35
28-37
28-39
28-41
28-43
28-44
0 0 I
1 I 2
28-46
28-64
28-81
28-48
28-65
28-S3
28-50
28-67
28-84
28-51
28-69
28-86
2S-53
28-71
28-88
28-55
28-72
28-90
28-57
28-74
28-91
28-58
2S-76
28-93
28-60
28-77
2S-95
28-62
28-79
28-97
001
0 0 I
001
1 I 2
1 I 2
112
28-98
I29-15
29-33
29 00
29-17
29-34
29-02
29-19
29-36
29-03
29-21
29-38
29-05
29-22
29-39
29-07
29-24
29-41
29-09
29-26
29-43
29-10
29-27
29-44
29-12
29-29
29-46
29-14
2931
29-48
001
0 0 I
001
1 1 2
1 1 2
1 I 2
29-50
2966
29-83
29-51
2968
29-85
29-53
29-70
29-87
29-55
29-72
29-88
29-56,29-58
297329-75
29-9029-92
29-60
29-77
29-93
29-61
29-78
29-95
29-63
29-80
29-97
29-65
29-82
29-98
0 0 1
001
0 0 1
112
112
I I 2
30-00
30-02
30-03
30-05
30-07 30-08
30-10
30-12
30-13
30-15
000
III
30-17
30-33
30-50
30-18
30-35
30-51
30-20
30-36
30-53
30-22
30-38
30-55
30-23
30-40
30-56
30*25
3a-4i
30-58
30-27
30-43
30-59
30-28
30-45
30-61
30-30
30-46
30-63
30-32
30-48
30-64
000
000
000
I I I
I 1 I
1 I I
94
95
96
30-66
3082
30-98
30-68
30-84
31-00
30-6930-71
30-S530-S7
31-0231-03
30-72
3089
31-05
30-74
30-90
31-06
30-76
30-92
31-08
30-7730-79
30-94 30-95
31-1031-11
30-81
30-97
31-13
000
000
000
I I 1
I I 1
I 1 1
97
98
99
31-14
31-30
31-46
]
31-16
31-32
31-48
31-1831-19
31-3431-35
3i-503i"5i
31-2131-2231-24
31 -3731 -38 31 -40
31-5331-5431-56
!
31-2631-27
31-4231-43
31-5831-59
1
31-29
31-45
31-01
000
000
000
I 1 I
I 1 1
I I I
B ^t-v
C/lXAV
5°
SQUARE
ROOTS FROM
I pop
TO 9999.
10
o
1
2
3
4
5
6
7
8
9
123
456
7 8 9
31-62
31-7S
31-94
32-09
32-25
32-40
32-56
32-71
32-86
33-02
2 3 5
689
II 12 14
11
12
13
14
16
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
IT
52
63
64
33-17
34-64
36-06
33-3233-4733-62
3479,34-93 35-07
36-1936-3336-47
33-76
35-21
36-61
33-91
35-36
36-74
34-06
35-50
36-88
34-21
35-64
37-01
34*35
35-78
37-15
34-50
35-92
37-28
I 3 4
I 3 4
I 3 4
679
678
578
10 12 13
ion 13
10 II 12
37-42
38-73
40-00
37-5537-68
38-8638-99
40-12 40-25
37-82
39-12
40*37
37-95
39-24
40-50
38*08
39*37
40-62
38-21
39-50
40-74
38*34
39-62
40-87
38-47
3975
40-99
38-60
39S7
41-11
I 3 4
I 3 4
I 2 4
5 78
568
567
9 II 12
9 10 II
9 10 II
41-23
42-43
43-59
41-3541-47
42-54'42-66
43-7043-82
41-59
42-78
43-93
41-71
42*90
44-05
41*83
43-01
44-16
41-95
43-13
44*27
42-07
43-24
44-38
42*19
43*36
44*50
42-31
43-47
44-61
I 2 4
I 2 3
I 2 3
567
567
8 10 II
8 9 10
8 9 10
44-72
44-83 44*94
45-06
45°i7
45-28
45*39
45-50
45*6i
45-72
I 2 3
467
8 9 10
45-83
46-90
47-96
45-9346-04
47-01 47;^2
48-0648-17
46*15
47-22
48*27
46-26
47*33
48-37
46-37
47-43
48-48
46-48
47-54
48-58
46-58
47-64
48-68
46*69
4775
48-79
4680
47-85
48-S9
I 2 3
I 2 3
I 2 3
456
456
456
8 9 10
7 8 9
7 8 9
48-99
50-00
50-99
49-0949-19
50-1050-20
51-0951-19
49-30
50-30
51-28
49-40
50-40
51-38
49-50
50-50
51-48
4960
50-60
51-58
49-70
50-70
51-67
49 -So
50-79
51*77
49-90
50-89
51-87
I 2 3
I 2 3
1 2 3
456
4 5 6
456
7 8 9
7 8 9
7 8 9
51-96
52-92
53-85
52*06
53-01
53-94
52-15
53*10
54-04
52-25
53*20
54-13
52-35
53-29
54-22
52-44
53-39
54-31
52-54
53-48
54-41
52-63
53*57
54-50
52-73
53-67
54-59
52-82
53-76
54-68
I 2 3
I 2 3
I 2 3
4 5 6
456
4 5 5
7 8 9
7 7 8
678
54-77
54-8654-95
55-05
55-14
55-23
55-32
55-41
55-50
55-59
I 2 3
4 4 5
678
55-68
56-57
57-45
5577 55-86
56-6656-75
57-53;57-62
55-95
56-83
5771
56-04
56-92
57-79
56-12
57-01
57-88
56-21
57-10
57-97
56-30
57-18
58-05
56-39
57-27
58-14
56-48
57-36
58 •?2
I 2 3
I 2 3
I 2 3
3 4 5
3 4 5
3 4 5
678
678
678
58-31
59-16
60-00
58-4o|58-48
59-2559-33
60-08 6o-i 7
58-57
59-41
60*25
58-65
59-50
60-33
58-74
59-58
60-42
58-S2
59-67
60-50
58-91
59-75
60-58
58-99
59-83
6066
59-oS
5992
6075
I 2 3
122
I 2 2
3 4 5
3 4 5
3 4 5
678
678
6 7 7
60-S3
61-64
62-45
60*91, 60-99
61 -73 61 -81
62-53^62*61
61*07
61-89
62-69
61*16
61-97
62*77
61-24
62-05
62-85
61 -32
62 -1 3
62*93
61-40
62-21
63-01
61-48
62-29
63-09
61-56
62-37
63-17
122
122
I 2 2
3 4 5
3 4 5
3 4 5
6 7 7
667
667
63-25
63*3263*40
63-48
63-56
63-64
63-72
63-80
63 -87
63-95
122
3 4 5
667
64-03
64-81
65-57
64-11,64*19
64-88:64-96
65-6565-73
64-27
65-04
65-80
64-34
65-12
65-88
64*42
65*19
65-95
64*50
65-27
66-03
64-58
65-35
66-11
64-65
65-42
66-18
64-73
65-50
66-26
122
I 2 2
I 2 2
3 4 5
3 4 5
3 4 5
5 6 7
5 6 7
5 6 7
66-33
67-08
67-82
66-41 66-48
67*1667*23
67-9067-97
66-56
67-31
68-04
66-63
67-38
68-12
66-71
67-45
68-19
66-78
67-53
68-26
66-86
67-60
68-34
66-93
6/ -68
6841
67-01
67 75
68-48
I 2 2
I I 2
I I 2
3 4 5
3 4 4
3 4 4
5 5 7
5 6 7
5 6 7
6S-56
69-28
70-00
68-63
69-35
70-07
68*70
69-43
70*14
68-77
69-50
70-21
6S-85
69-57
70*29
68-92
69-64
70*36
68-99
69-71
70-43
69-07
69-79
70-50
69-14
69-86
70-57
69 21
69-93
70-64
I I 2
I I 2
I I 2
3 4 4
3 4 4
3 4 4
5 6 7
566
566
70-71
70-78
70-85
70-92
70-99
71*06
71-13
71-20
71*27
71-34
r I 2
3 4 4
566
71-41
72-11
72-80
73-48
71-48
72-18
72-87
71-55
72-25
72-94
71-62
72-32
73-01
71-6971-76
72-39^72-46
73-0873-14
71 -S3
72-53
73-21
71-90
72-59
73-28
71-97
72'-66
73*35
72-04
72-73
73-42
I I 2
I I 2
I I 2
3 4 4
3 3 4
3 3 4
566
566
5 5 6
73-55
73*62
73-69
73-76,73-82
73 -89
73-96
74-03
74-09
I I 2
3 3 4
5 5 6
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SQUARE ROOTS from iooo to 9999.
51
65
0
74-16
74"23 74*30 74'36
74*43
74-50
6
74-57
74-6;
8
9 123 456 789
74-70
74-77
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3 3 4
5 5 6
56
57
58
59
60
61
62
63
64
74-83
75-50
76-16
74-9074
75-5675'
76-22 76'
76-81
77-46
78-10
76-88
77-52
78-17
747s
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79*44
78
79-37
80-0080-06,80
9775-03
6375-70
2976-35
76
77
78-23'78-29
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75
75
76;
77'
77
78
1075-17
7675-83
4276-49
75-23
75-89
76-55
75-30
75-96
76-62
75-37
76-03
76-68
75-43
76*09
76-75
78'
79'
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50,79
1280
78
79
80 ■
99,79
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78-49
77-27
77-91
78-55
79-12
79-75
80-37
79-18
79-81
80-44
77-33
77-97
78-61
77-40
78-04
78-68
79-25
79-87
80-50
79-3
79-94
80-56
I 2
I 2
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3 3 4
3 3 4
3 3 4
I 2
I 2
I 2
3 3 4
3 3 4
3 3 4
I 2
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3 3 4
3 3 4
234
5 6
5 6
5 6
5 6
5 6
5 6
5 6
5 6
5 6
65
80-62
80-6880-7580-81
3681
98,82
5882
80-S780-93
80-99
81 -06
81-12 81-18
I 2
2344 5 5
66
67
68
81 -2
81-85
82-46
48
1-3081
8i-9i[8i
82-5282
42
04
64
81
82
82
49,81
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7082'
5581-61
1682-22
76
82-8:
81-67
82-28
82-89
81-73
82-34
82-95
81-79
82-40
83-01
I 2
I 2
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234
234
234
5 5
5 5
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69
70
71
72
73
74
75
83-07
83-67
84-26
•3-1383
13-7383
84-3284-38
84-85
85-44
86-02
■1983
■7983
84-91 84
85-5085
86-0886
84
37
96
84-5084-56
90
83-
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83 -43
84-02
84-62
83-49
84-08
84-68
j 55 83-61
84-14 84-20
84-73 84-79
I 2
I 2
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234
234
234
86-6086-66
9785
5685
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86-83
85
85
86'
85-21
85-79
86-37
85-26
85-85
86-43
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85-91
86-49
86 -89
86-95
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85-38
85-97
86-54
I 2
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233
233
233
87-06 87-12
I 2
5 5
5 5
5 5
5 5
5 5
5 5
4 5 5
76
77
78
87-18
87-75
88-32
87-24
87-81
88-37
•2987
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87
87
87
88
88
87-52
88-09
88-66
87-58
88-15
88-71
87-64
88 -20
88-77
8769
88 -26
88-83
I 2
I 2
I 2
233
233
233
S 5
4 5
4 5
79
80
81
88-8888-9488-9989-05
89-4489-5089-5589-61
90-00 90-06 90-1 1 90-17
89
89
90
85
89
90'
8q-22
89-78
90-33
89-27
89-83
90-39
89-33
89-89
90-44
89-39
89-94
90-50
I 2
I 2
I 2
233
233
233
4 5
4 5
4 5
82
83
84
85
90-55 90*61 90*66 90*72
9i-i09i*i69i-2ij9i-27
91-65917191-7691*82
90 '
91
91
•83
•38
■92
90-88
91-43
91-98
90-94
91-49
92-03
90-99
91-54
92*09
91-05
91-60
92*14
I 2
I 2
I 2
92-20 92-25'92-3o 92-36
92-41
92-47
92-52
92-57
92-63
92-68
I 2
233
233
233
4 5
4 5
4 5
4 4 5
86
87,
88
92
93-27
93-81
7492
7992
93-33,93
93-8693
■8492*90
■38I93-43
■9193-97
92
93
94
93-01
93-54
94*07
93-06
93-59
94-13
93*11
93-65
94-18
93-17
93-70
94-23
93-22
93-75
94-29
I 2
I 2
I 2
233
233
233
4 5
4 5
4 5
89
90
91
94-34194-39
94-87
95-39
94-92
95-45
94-45,94-50
94-97,95-03
95-5095-55
94'
95
95
■60
■13
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94-66
95-18
95-71
94-71
95-24
95-76
94-76
95-29
9581
94-8
95-34
95-86
I 2
I 2
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233
233
233
4 5
4 5
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92
93
94
95
95-92
96*44
96-95
95-9796-0296*07
96*4996*54196-59
97*01 97*0697*11
96'
96
97
•1896*23
7096*75
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96*28
96-80
97-31
97-47
97-5297-57,97-62
97-67
97-72
97-78
97-83
96-33
96-85
97-37
96-38
96-90
97-4
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233
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97-88
97-93
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97
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97-98 98-03:98 -oS;98-i;
98-49
98-99
98-5498-5998-64
99*0599-1099*15
98
98
99
98
98
99*25199*30
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98-34
98-84
99-35
98-39
98-89
99-46
98-44
98-94
99-45
I 2
I 2
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233
233
223
99 !]99-5099-5599-6o99-65 99-7099*7599-80 99-85 99-90
99-95
4 5
4 5
4 4
o I I 2 2 3 3 4 4
1J2
52
RECIPROCALS
OF
NUMBERS
FROM 1000 TO 9999
•
10
n
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
0
1
2
3
4
6
6
7
8
9
12 3
4 5 6
7 8 9
0*0010000
9901
9804
9709
9615
9524
9434
9346
9259
9174
91827
36 45 55
64 73 82
0*0009091
0*0008333
0*0007692
9009
8264
7634
8929
8197
7576
S850
8130
7519
8772
8065
7463
8696
8000
7407
8621
7937
7353
8547
7874
7299
8475
7813
7246
8403
7752
7194
81523
6 13 19
5 II 16
30 38 45
26 32 38
22 27 33
536168
455158
38 44 49
0*0007143
0*0006667
0*0006250
7092
6623
621 1
7042
6579
6173
6993
6536
6135
6944
6494
609S
6897
6452
6061
6849
6410
6024
6803
6369
5988
6757
6329
5952
6711
6289
5917
5 10 14
4 813
4 711
19 24 29
17 21 25
15 18 22
333843
29 33 38
262933
0*0005882
0*0005556
0*0005263
5848
5525
5236
5814
5495
5208
5780
5464
5181
5747
5435
5155
5714
5405
512S
5682
5376
5102
5650
5348
5076
5618
5319
5051
5587
5291
5025
3 6 10
369
3 5 8
13 16 20
12 15 17
II 13 16
23 26 29
20 23 26
18 21 24
0*0005000
4975
4950
4926
4902
4878
4854
4831
4808
47S5
2 5 7
10 12 14
17 1921
0*0004762
0*0004545
0*0004348
4739
4525
4329
4717
4505
4310
4695
4484
4292
4673
4464
4274
4651
4444
4255
4630
4425
4237
460S
4405
4219
45S7
4386
4202
4566
4367
4184
2 4 7
246
245
9 II 13
8 10 12
7 9 II
15 1 7 20
14 16 18
13 14 16
0*0004167
0*0004000
0*0003846
4149
3984
3S31
4132
3968
3817
4115
3953
3802
4098
3937
3788
4082
3922
3774
4065
3906
3759
4049
3891
3745
4032
3876
3731
4016
3861
3717
2 3 5
235
I 3 4
7 8 10
689
678
12 13 15
II 12 14
10 II 13
0*0003704
0*0003571
0*0003448
3690
3559
3436
3676
3546
3425
3663
3534
3413
3650
3521
3401
3636
3509
3390
3623
3497
3378
3610
3484
3367
3597
3472
3356
3584
3460
3344
I 3 4
I 2 4
123
5 7 8
5 6 7
5 6 7
9 II 12
9 10 II
8 9 10
0*0003333
3322
3311
3300
3289
3279
326S
3257
3247
3236
I 2 3
456
7 910
0*0003226
0*0003125
0-0003030
3215
3115
3021
3205
3106
3012
319s
3096
3003
3185
3086
2994
3175
3077
2985
3165
3067
2976
3155
305S
2967
3145
3049
2959
3135
3040
2950
I 2 3
I 2 3
I 2 3
4 5 6
456
4 4 5
7 8 9
7 8 9
678
0-0002941
0-0002857
0-0002778
2933
2849
2770
2924
2841
2762
2915
2833
2755
2907
2825
2747
2S99
2817
2740
2890
2809
2732
28S2
2801
2725
2874
2793
2717
2865
2786
2710
I 2 3
I 2 2
I 2 2
3 4 5
3 4 5
3 4 5
678
667
5 6 7
0-0002703
0-0002632
0-0002564
2695
2625
2558
26S8
2618
2551
2681
261 1
2545
2674
2604
2538
2667
2597
2532
2660
2591
2525
2653
2584
2519
2646
2577
2513
2639
2571
2506
I I 2
I I 2
I I 2
3 4 4
3 3 4
3 3 4
566
5 5 6
456
0-0002500
2494
2488
2481
2475
2469
2463
2457
2451
2445
I I 2
234
4 5 5
0-0002439
0-0002381
0-0002326
2433
2375
2320
2427
2370
2315
2421
2364
2309
2415
2358
2304
2410
2353
2299
2404
2347
2294
2398
2342
22S8
2392
2336
22S3
2387
2331
2278
I I 2
I I 2
I I 2
233
233
233
4 5 5
4 4 5
4 4 5
0-0002273
0-0002222
0 '0002 1 74
2268
2217
2169
2262
2212
2165
2257
2208
2160
2252
2203
2155
2247
2198
2151
2242
2193
2146
2237
2188
2141
2232
2183
2137
2227
2179
2132
I I 2
oil
0 I I
233
223
223
4 4 5
3 4 4
3 4 4
0-0002128
O-OOO20S3
0*0002041
2123
2079
2037
2119
2075
2033
2114
2070
2028
2110
2066
2024
2105
2062
2020
2101
2058
2016
2096
2053
2012
2092
2049
2008
2088
2045
2004
0 I I
0 I I
oil
223
223
222
3 4 4
3 3 4
3 3 4
0*0002000
1996
1992
1988
1984
1980
1976
1972
1969
1965
0 I I
222
3 3 4
0*0001961
0*0001923
0*0001887
1957
1919
1883
1953
1916
1880
1949
1912
1876
1946
1908
1873
1942
1905
1869
1938
1901
1866
1934
1898
1S62
1931
1894
1859
1927
1890
1855
0 I I
0 I I
0 I -1
222
122
I '2 • 2
3 3 3
3 3 3
233
0*0001852
1848
1845
1842
1838
1835
1832
1828
1825
1821
0 1 I
I 12 '2
2?3 3I
N.B. — Three zeros follow the decimal point in the reciprocal of any four figure whole number except
■ the number 1000.
Note. — Numbers in difference columns to be subtracted, not added. — Sec Ritleu
RECIPROCALS OF NUMBERS from
1000 TO
9999.
S3
O
1
2 3
4
5
6
7
8
9
123
456
789
55
o-oooiSiS
1815
1S12
1808
1805
1802
1799
1795
1792
17S9
0 I I
I 2 2
233
56
57
58
59
60
61
62 1
63
64
65
; 0-0001786
0-0001754
0-0001724
I7S3
I751
I72I
1779
1748
1718
1776
1745
1715
1773
1742
1712
1770
1739
1709
1767
1736
1706
1764
1733
1704
1761
1730
1701
1757
1727
1698
0 I I
0 I I
0 I I
122
I 2 2
I I 2
233
223
223
0-0001695
0-0001667
0-0001639
1692
1664
1637
16S9
1661
1634
1686
1658
163I
16S4
1656
1629
i68i
1653
1626
1678
1650
1623
1675
1647
1621
1672
1645
1618
1669
1642
1616
0 I I
0 I I
oil
I I 2
I I 2
I I 2
223
223
222
0-0001613
0-0001587
0-0001563
1610
1585
1560
1 60S
15S2
1558
1605
15S0
1555
1603
1577
1553
1600
1575
1550
1597
1572
1548
1595
1570
1546
1592
1567
1543
1590
1565
1541
0 I I
GDI
0 0 I
112
222
222
222
0-0001538
1536
1534
1531
1529
1527
1524
1522
1520
1517
0 0 I
222
66
67
68
69
70
71
0-0001515
0-0001493
0-000147 1
I513
1490
I46S
1511
1 488
1466
1508
i486
1464
1506
1484
1462
1504
1481
1460
1502
1479
1458
1499
1477
1456
1497
1475
1453
1495
1473
145 1
001
001
0 0 I
222
222
222
0-0001449
0-0001429
0-0001408
1447
1427
1406
1445
1425
1404
1443
1422
1403
1441
1420
1401
1439
1418
1399
1437
1416
1397
1435
1414
1395
1433
1412
1393
1431
1410
1391
0 0 I
0 0 I
001
222
I 2 2
122
72
73
74
75
76
77
78
0-0001389
0-OC01370
0-0001351
1387
1368
1350
13S5
1366
134S
I3S3
1364
1346
13S1
1362
1344
1379
1361
1342
1377
1359
1340
1376
1357
1339
1374
1355
1337
1372
1353
1335
0 0 I
0 0 I
0 0 I
122
122
112
0-0001333
1332
1330
I32S
1326
1325
1323
1321
1319
131S
001
I I 2
0-0001316
0-0001299
0-0C012S2
I314
1297
1280
1312
1295
1279
I31I
1294
1277
1309
1292
1276
1307
1290
1274
1289
1272
1304
1287
1271
1302
1285
1269
1300
1284
1267
001
000
000
112
79
80
81
82
83
84
85
0-0001266
0-0001250
0-0001235
1264
1248
1233
1263
1247
1232
1 26 1
1245
1230
1259
1244
1229
125S
1242
1227
1256
1241
1225
1255
1239
1224
1253
1238
1222
1252
1236
1221
000
000
000
0 -0001220
0-0001205
0-0001190
I218
1203
I1S9
1217
1202
1 188
I215
1200
1 186
1214
"99
1185
1212
1198
1 183
1211
1 196
1182
1209
"95
1181
1208
"93
"79
1206
1192
1178
000
000
000
0-0001176
"75
1174
II72
1171
1 1 70
1 168
1167
1 166
1 164
000
86
87
88
0-0001163
0-0001149
0-0001136
1161
1 148
"35
1 160
1 147
"34
"59
"45
"33
"57
"44
1131
1 1 56
"43
1 130
"55
1 142
1129
"53
1 140
1127
1152
"39
1 126
"51
1138
1125
000
000
000
89
90
91
0-000 1 1 24
o-oooiiii
0-0001099
1122
mo
1098
1121
1 109
1096
1 120
1 107
1095
1119
1 106
1094
1117
1105
1093
1116
1 104
1092
"15
1 103
109 1
1114
IIOI
1089
1112
1 100
1088
000
000
000
0 I I
92
93
94
0-0001087
0-OCOI075
0-0001064
1086
1074
1063
10S5
1073
1062
1083
1072
1060
1082
107 1
1059
1081
1070
1058
1080
1068
1057
1079
1067
1056
1078
1066
1055
1076
1065
1054
000
000
000
0 I I
0 I I
0 I I
95
0-0001053
1052
1050
1049
1048
1047
1046
1045
1044
1043
000
0 I I
96
97
98
99
0-OCOI042
0-000 1 03 1
0-000I020
104 1
1030
1019
1040
1029
1018
1038
1028
1017
1037
1027
1016
1036
1026
1015
1035
1025
1014
1034
1024
1013
1033
1022
1012
1032
1021
lOII
000
000
000
oil
0 I I
0 I I
O'OOOIOIO
1009
looS
1007
1006
1005
1004
1003
1002
lOOI
000
001
I I I
tV .
D. — Three zer..
3S fullo
w die
Jeclma
1 point
in the
recipr
ocal of
any fo
ur figi
ire whole
number <
;xcept
tlie number 1000.
Note. — Numbers in difference columns to be subtracted, not added.— ^<7^ KuUs,
54
NEPERIAN OR
HYPERBOLIC
LOGARITHMS.
10
11
1-2
1-3
1-4
1-5
1-6
1-7
1-8
1-9
2 0
2-1
2-2
2 3
2-4:
2-5
2-6
2-7
2-8
2-9
3 0
31
3-2
3-3
3-4
3-5
3-6
3-7
3-8
3-9
4-0
4 1
4-2
4-3
4-4
4-5
4-6
4-7
4-8
4-9
5-0
5-1
5-2
5-3
5-4
0
1
2
3
4
5
6
7
8
9
12 3
4 6 6
7 8 9
o-oooo
0100
0198
0296
0392
048S
0583
0677
0770
0862
10 19 29
38 48 57
67 76 86
0-0953
0-1S23
0-2624
0-3365
0-4055
0-4700
1044
1906
2700
"33
19S9
2776
1222
2070
2852
1310
2151
2927
1398
2231
3001
1484
2311
3075
1570
2390
3148
1655
2469
3221
1740
2546
3293
9 17 26
8 1624
7 15 22
35 44 52
32 40 48
30 37 44
61 70 78
56 64 72
52 59 67
3436
4121
4762
3507
4187
4S24
3577
4253
4886
3646
4318
4947
3716 3784
43831 4447
5008 5068
3853
4511
5128
3920
4574
5188
3988
4637
5247
7 14 21
6 13 19
6 12 18
28 35 41
26 32 39
24 30 36
48 55 62
45 52 58
42 48 55
0-5306
0-5878
0-6419
5365
5933
6471
5423
5988
6523
5481
6043
6575
5539
6098
6627
5596
6152
6678
5653
6206
6729
5710
6259
6780
5766
6313
6831
5822
6366
688 1
6 II 17
5 II 16
5 10 15
23 29 34
22 27 32
20 26 31
40 46 51
38 43 49
36 41 46
0-6931
6981
7031
7080
7129
7178
7227
7275
7324
7372
5 1015
20 24 29
34 39 44
0-7419
0-7885
0-8329
7467
7930
8372
7514
7975
8416
7561
8020
8459
7608
8065
8502
7655
8109
8544
7701
8154
8587
7747
8198
8629
7793
8242
S671
7839
82S6
8713
5 9 14
4 9 13
4 9 13
19 23 28
18 22 27
17 21 26
33 37 42
31 36 40
30 34 38
0-8755
0-9163
0-9555
8796
9203
9594
8838
9243
9632
8879
9282
9670
8920
9322
9708
8961
9361
9746
9002
9400
9783
9042
9439
9821
9083
9478
9858
9123
9517
9895
4 8 12
4 8 12
4 8 II
16 20 24
16 20 24
15 19 23
29.33 37
27 31 35
26 30 34
0-9933
I -0296
I -0647
9969
0332
0682
0006
0367
0716
0043
0403
0750
0080
0438
0784
01 16
0473
0818
6152
0508
0852
01 88
0543
0S86
0225
0578
0919
0260
0613
0953
4 7 II
4 7 II
3 7 10
15 18 22
14 18 21
14 17 20
25 29 33
25 28 32
24 27 31
I -0986
1019
1053
1086
1119
1151
1 184
1217
1249
1282
3 7 10
13 16 20
23 26 30
1-1314
1-1632
I -1939
1346
1663
1969
1378
1694
2000
1410
1725
2030
1442
1756
2060
1474
1787
2090
1506
1817
2119
1537
1848
2149
1569
1878
2179
1600
1909
2208
3 6 10
369
369
13 16 19
12 15 18
12 15 18
22. 25 29
21, 25 28
21 24 27
1-2238
1-2528
I -2809
2267
2556
2837
2296
2585
2865
2326
2613
2892
2355
2641
2920
2384
2669
2947
2413
2698
2975
2442
2726
3002
2470
2754
3029
2499
2782
3056
369
3 6 8
3 5 8
12 15 17
II 14 17
II 14 16
20 23 26
20 22 25
19 22 25
I -30S3
1-3350
I -3610
3110
3376
3635
3137
3403
3661
3164
3429
3686
3191
3455
3712
3218
3481
3737
3244
3507
3762
3271
3533
3788
3297
3558
3813
3324
3584
383^
3 5 8
3 5 8
3 5 8
II 13 16
10 13 16
10 13 15
19 21 24
18 21 23
18 20 23
1-3863
3888
3913
3938
3962
3987
4012
4036
4061
4085
2- 5 7
10 12 15
17 20 22
I -41 10
1-4351
1-4586
4134
4375
4609
4159
4398
4633
4183
4422
4656
4207
4446
4679
4231
4469
4702
4255
4493
4725
4279
4516
4748
4303
4540
4770
4327
4563
4793
2 5 7
2 5 7
2 5 7
10 12 14
9 12 14
9 12 14
17 19 22
16 19 21
16 18 21
I -4816
I -5041
1-5261
4839
5063
5282
4861
5085
5304
4884
5107
5326
4907
5129
5347
4929
5151
5369
4951
5173
5390
4974
5195
5412
4996
5217
5433
5019
5239
5454
2 5 7
247
246
9 II 14
9 II 13
9 II 13
16 18 20
15 18 20
15 17 19
I -5476
1-5686
1-5892
5497
5707
5913
551S
5728
5933
5539
5748
5953
5S6o
5769
5974
5581
5790
5994
5602
5810
6014
5623
5831
6034
5644
5851
6054
5665
5872
6074
246
246
246
8 II 13
8 10 12
8 10 12
15 17 19
14 16 19
14 16 18
I -6094
61 14
6134
6154
6174
6194
6214
6233
6253
6273
246
8 10 12
14 16 18
1-6292
I -6487
I -6677
6312
6506
6696
6882
6332
6525
6715
6351
6544
6734
6371
6563
6752
6390
6582
6771
6409
6601
6790
6429 i 6448
6620 6639
6808 6827
6467
6658
6845
246
246
246
8 10 12
8 10 II
7 9 II
14 16 iS
13 15 17
13 15 17
I -6864
6901' 6919
6938
6956 6974
6993' 701 1
7029
2 4 5
7 9 ii'i3 15 i6|
TABLE OF NEPERIAN LOGARITHMS OF lo+n
n
I
2
■^
J
4
5
6
7
8
9
log 10"
2-3026
4*6052
6-9078
9*2103
11-5129
13*8155
i6'iiSi
18-4207
20-7233
NEPERIAN
OR
HYPERBOLIC
LOGARITHMS. 55
5-5
5-6
5-7
5-8
5-9!
— i
6-0,
6-11
6-2
6-3
6-4
6-5
6-6
6-7
6-8
6-9
7 0
7 1
7-2
7-3
7-4
7-5
7-6
7-7
7-8
7-9
8 0
8 1
8-2
8-3
8-4
8-5
8-6
8-7
8-8
8-9
9 0
9 1
9-2
9-3
9-4
9-5
9-6
0
1
2
3
4
5
6
7
8
9
12 3
45 6
7 8 9
I -7047
1-7228
7066
7246
70S4
7263
7102
7281
7120
7299
7138
7317
7156
7334
7174
7352
7192
7370
7210
73S7
245
2 4 5
7 9 II
7 9 II
13 14 16
12 14 16
I -7405
17579
1-7750
7422
7596
7766
7440
7613
7783
7457
7630
7800
7475
7647
7817
7492
7664
7834
7509
7681
7851
7527
7699
7867
7544
7716
7884
7561
7733
7901
235
2 3 5
2 3 5
7 9 10
7 9 10
7 8 10
12 14 16
12 14 15
12 13 15
I -7918
7934
7951
7967
7984
8001
S017
8034
8050
8066
2 3 5
7 8 ioii2 13 15 1
I -8083
1-S245
1-8405
8099
8262
8421
8116
8278
8437
8132
8294
8453
8148
8310
8469
8165
8326
8485
8181
8342
8500
8197
8358
8516
8213
8374
8532
8229
8390
8547
235
2 3 5
235
6 8 10
6 8 10
689
II 13 15
II 13 14
II 13 14
1-8563
I -8718
I -887 1
8579
8733
8886
8594
8749
8901
8610
8764
8916
8625
8779
8931
8641
8795
8946
8656
8810
8961
8672
8825
8976
8687
8840
8991
8703
8856
9006
2 3 5
2 3 5
23s
689
689
689
II 12 14
11 12 14
II 12 14
I -902 1
1-9169
1-9315
9036
9184
9330
9051
9199
9344
9066
9213
9359
9081
922S
9373
9095
9242
9387
9110
9257
9402
9125
9272
9416
9140
9286
9430
9155
9301
9445
I 3 4
I 3 4
I 3 4
679
679
679
10 12 13
10 12 13
10 12 13
I -9459
9473
948S
9502
9516
9530
9544
9559
9573
9587
I 3 4
679
10 II 13
I -9601
I -9741
I -9879
9615
9755
9892
9629
9769
9906
9643
9782
9920
9657
9796
9933
9671
9810
9947
9685
9824
9961
9699
9838
9974
9713
9851
9988
9727
9865
0001
1 3 4
I 3 4
I 3 4
678
678
5 7 8
10 II 13
10 II 12
I'o II 12
2-0015
2-0149
2-0281
0028
0162
0295
0042
0176
0308
0055
0189
0321
0069
0202
0334
00S2
0215
0347
0096
0229
0360
0109
0242
0373
0122
0255
0386
0136
0268
0399
I 3 4
I 3 4
I 3 4
5 7 8
5 7 8
5 7 8
9 II 12
9 II 12
9 10 12
2-0412
2-0541
2 -0669
0425
0554
0681
0438
0567
0694
0451
05S0
0707
0464
0592
0719
0477
0605
0732
0490
061 8
0744
0503
0631
0757
0516
0643
0769
0528
0656
0782
I 3 4
I 3 4
I 3 4
568
568
5 6 8
9 10 12
9 10 12
9 10 II
2-0794
0807
0819
0832
0844
0857
0869
0882
0894
0906
I 3 4
5 6 8^
9 10 II
2-0919
2 -104 1
2-1163
0931
1054
1175
0943
1066
1 187
0956
1078
1 199
0968
1090
1211
0980
1 102
1223
0992
1114
1235
1005
1126
1247
1017
1138
1258
1029
1 1 50
1270
I 2 4
I 2 4
I 2 4
5 6 7
5 6 7
5 6 7
9 10 II
9 10 II
8 10 II
2-1282
2-1401
2-1518
1294
1412
1529
1306
1424
1541
1318
1436
1552
1330
1448
1564
1342
1459
1576
1353
1471
15S7
1365
1483
1599
1377
1494
1610
1389
1506
1622
I 2 4
I 2 4
I 2 3
5 6 7
5 6 7
5 6 7
8 10 II
8 9 II
8 9 10
•2-1633
2-1748
2-1861
1645
1759
1872
1656
1770
1883
1668
1782
1894
1679
1793
1905
1691
1804
1917
1702
1815
1928
1713
1827
1939
1725
183S
1950
1736
1849
1961
I 2 3
1 2 3
I 2 3
5 6 7
5 6 7
4 6 7
8 9 10
8 9 10
8 9 10
2-1972
19S3
1994
2006
2017
2028
2039
2050
2061
2072
I 2 3
467
8 9 10
2 -2083
2-2192
2 -2300
2094
2203
2311
2105
2214
2322
2116
2225
2332
2127
2235
2343
2138
2246
2354
2148
2257
2364
2159
2268
2375
2170
2279
2386
2181
2289
2396
I 2 3
I 2 3
I 2 3
4 5 7
4 5 6
456
8 9 10
8- 9 10
7 9 10
2-2407
2-2513
2-2618
2418
2523
2628
2428
2534
2638
2439
2544
2649
2450
2555
2659
2460
2565
2670
2471
2576
2680
2481
2586
2690
2492
2597
2701
2502
2607
2711
I 2 3
I 2 3
I 2 3
456
456
456
7 8 10
7 8 9
7 8 9
9-7
9-8
9-9
2-2721
2-2824
2-2925
2732
2834
2935
2742
2844
2946
2752
2854
2956
2762
2865
2966
2773
2875
2976
2783
2885
29S6
2793
2895
2996
2803
2905
3006
2814
2915
3016
I 2 3
I 2 3
I 2 3
456
456
456
7 8 9
7 8 9
7 8 9
TABLE OF
NEPERIAN LOGARITHMS
OF 10
-n.
n
I
2
3
4
5
6
7
8
9
log^io-"
3 '6974
5-3948
7 -0922
io-7897jl2-487i
14-1845
T7-8819
1^9-5793
2I -2767
56
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USEFUL FORMULAS AND NUMBERS.
Binomial Theorem.
(i±.)^
n.n
:ne-
I o . n.n-
\.n — .
'^H&c.
Hence, when ne is so small that its square and higher powers may-
be neglected, (i ±d')" N i ± ne.
Examples —
c='o\ ; (i + *oi)2= i-q:
(i + -oi)J=^roo5; (H-'oi)-i ==0-9967.
Barometric Formula. — Let P and p be the atmospheric pressures
observed by the barometer at the lower and upper stations respec-
tively ; and let T and t be the respective atmospheric tempera-
tures on the Fahrenheit scale ; then H, being the difference of
levels in feet,
r+/-64\
//= 60360 {log P- log ;^} (^
1 +
986
6 = 271828.
2-30258.
Base of Hyperbolic or Neperian Logarithms, ....
To convert common into hyperbolic logarithms, multi-
ply by
To convert hyperbolic into common logarithms, multi-
ply by 0-43429.
Ratio of circumference of circle to diameter, ^ = 3-14159.
Number of degrees in one radian (the unit angle, which is the angle
subtended by arc equal to radius), 57°'295S = 57° 17' 45" = 206265".
T=3"i4i59
€ = 271828
Metre in inches,
Foot in centimetres.
Mile in kilometres,
Gramme in grains.
Pound in grammes,
Kilogramme in pounds
British ton in French tons
of 1000 kilos.,
Litre in cubic inches
Cubic inch in cubic centimetres, 16-3866
Cubic foot in cubic centimetres, 28316-0
39"37043
30 "4797
1-6093
IS '43235
453 '593
2-2046
1-016
61-0253
57
Logarithm.
o^"497iS
0-43429
Cubic inch of distilled
water at 4°C,
Cubic foot of water at
4°C, ....
Cubic inch of mercury
ato°C,
Do. do.,
Litre of dry air at o°C,
76o'"-"'- pressure,
Cubic foot ,, ,,
Density of mercury,
252-89 grains,
62-43 ^t)S.
3439 grains.
-4913 lbs.
I "2932 grms.
565-1 grains.
13 •596.
58
I centim. gramme .......
I metre kilogramme ......
I ft. lb
I ft. poundal (independent of^.) ....
I joule ( I watt for i second) .....
I horse power. .......
I watt (rate of working of i volt through i ohm, or of i volt carrying i ampere)
= io~ ergs per second.
981 ergs.
9 81 X 10''.
1 3 '56 X 106 ergs.
421390 ergs.
10" ergs.
7 '46 X io9 ergs per sec.
Earth's mean radius, 6*371 x lo^centiras
Earth's mean radius
(approx.), . . 21 X 10* feet.
Mass of earth, as-
suming 5 "67 as
mean density, . 6'i4X 10" grammes
Earth's mass (ap-
proximately), . i3'5 X 10-* lbs. =
6 X 10'-' tons.
Seconds pendulum at
Greenwich, . . 39'i39 inches =
99'4i4 centims.
Gravity of i pound
at Greenwich, . 32*i9i poundals.
Gravity of i pound
mass in lat. 55° 35'
(approximately thai
of Edinburgh or
Glasgow), . . 32'2 poundals.
Gravity of i gramme
in same latitude, . 981 "424 dynes.
Mass of moon, . 1/
Distance of moon
from earth,
Sun's radius.
Mass of sun.
Distance of sun
from earth.
Distance of sun
from earth,
! I "5 of earth's mass
3'8 X 10'" centims.
697 X 10* centims.
324000 earth's mass.
I "498 X 10^' centims.
93'i X 10^ miles.
Mass in grammes which
concentrated at a
point I centimetre
distant from a point at
which another equal
mass is concentrated
would attract it with a
force of I dyne, 3928 grammes.
Same where the foot,
pound, and poundal
are units of length,
mass, and force, 31,075 lbs.
Height of Homogeneous atmosphere at Greenwich at c°C, 26,210 ft.=
7*988 X 10^ centims.
Newtonian velocity of sound in air at c°C, . . 9i8"5 feet per second,
= 27996 centims. per second.
True velocity at /^C = 33240 J i + '00366 / centims. per second.
Joule's Equivalent. 777'2 Greenwich foot-pounds of work will raise
I lb. pure water from 60° to 61° Fahrenheit.
This is equivalent to 1399 ft.-lbs. per pound degree centigrade, — or
41*84 X 10'' ergs per gramme degree centigrade, — or 42600 centi-
metre-grammes per gramme degree.
Latent heat of water, 79*25. Latent heat of steam at ico", 537
Specific heat of air pressure constant, 0*237 ;
sp. heat of air pres. const. ^ - . ,
sp. heat of air vol. const.
I litre of hydrogen at o°C and 760 mm. pressure weighs 0*0896 gm.
Density of hydrogen compared with air =0*0693=—^ — .
I4'43
59
Conductivity of heat. Quantity, in gramme-water-centigrade units ;
conducted per second ; per square centimetre of area ; per degree,
per centimetre of thickness, of difference of temperature of two
sides of plate.
Copper, "996
Iron, 15 to "19
Stone, 01 to 005
Velocity of light in vacuum = 3*004 x iqI" centims. per second.
Mean wave length 5"3 x lo"^ centim.
One electromagnetic unit=3X lo^** electrostatic units of electricity
I B. A. Unit = -9866 Ohm. i Ohm=roi358 B. A. Unit.
Resistance of 100 metres of pure annealed round wire, i mm. in
diameter, at o°C, Copper, .... 2"028 ohms.
12-34
11-50
4i'i7
1-07 volt.
■ I "95 „
1-435 volt.
i-43S[i--ooo77(/-i5)]
J>
»
Iron,.
M
)>
Platinum, .
»
))
Platinoid, .
Electro-motive fo
rce of I Daniell's cell, .
1)
>)
I Grove's cell,
V
»
Standard Clark cell at 15
'C,
5>
))
One volt through one ohm (i watt) generates per second
thermal unit (gramme-water-centigrade).
Electro-Chemical Decomposition. —
4-184
of a
Element.
Atomic Weight.
Chem. Equivalent
Hydrogen,
I
I
Potassium,
39-03
39-03
Sodium,
23-
23-
Silver,
1077
107-7
Copper,-*
63-35
31-68
Zinc,
64-88
32-44
Lead,
206-4
103-2
O.Kygen,
15-96
7-98
Chlorine,
35-37
35-37
Electrolytic Decom-
position, Grammes per
second, per ampere.
•00001038
•0004051
•0002387
'OOII18
•0003290
•0003367
•00 1 07 I
■00008283
■000367 1
' For cathode surface of 50 sq. centims. per ampere, the quantity of copper deposited per
ampere per second is "0:03287 gms. For increments in cathode surface subtract i-i6th p.o.
P''r 50 sq. centims. The numbers given for silver and copper are the results of direct experi-
ment.
GLASGOW : PRINTED AT THE UNIVERSITY PRESS
BY ROBERT MACLEHOSE AND CO. LTD.
YD 09882
14 DAY USE
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