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 1759
1884 1818 1752
1878 iSii
1746
I I 2 I I 2 I I 2
3 3 4 3 3 4 3 3 4
5 5 6 5 5 6 5 5 6
1739 1675 1612
1733 1669 1605
1726 1662 1599
1720 1656 1593
I713 1649 1586
1707 1643
1580
1701 1637 1574
1694 1630 156S
1688 1624 1561
I68I I6I8 1555
112 I I 2 I I 2
3 3 4 3 3 4
3 3 4
456 4 5 6 456
1549
1543
1537
1530
1524
I5I8
1512
1506
1500
1494
I I 2
234
4 5 5
14S7 1427
1367
1481 1421 1361
1475 I415 1355
1469 1409 1349
1463 1403 1343
1457 1397 1337
145 1 1391 1331
1445 1385 1325
1439 1379 1319
1433 1373 I3I4
I I 2 I I 2 I I 2
234 234 234
4 5 5 4 5 5 4 5 5
1308 1249 1 1 92
1302 1244 1186
1296 1238 1 180
1290 1232
1175
1284 1226 1 169
1278
I22I II63
1273 1215 1158
1267 1209 1 1 52
1261 1203 1 146
1255
1 1 98 II4I
I I 2 I I 2 I I 2
233 233 2 3 .3
4 5 5 4 5 5 4 5 5
"35 1079 1024
1 129
1073 1018
II24 1068 IOI3
1118 1062 1007
III3
1057 1002
1 107 IO5I 0996
IIOI
1046 0991
1096 1040 09S5
1090
1035 0980
I0S5 1029
0975
I I 2 I I 2 I I 2
233
233 233
4 4 5 4 4 5 4 4 5
0969
0964
O95S
0953
0947
0942
0937
0931
0926
0921
112 233
4 4 5
0915
0862
'0809
0910
0857 0S04
0904 0851 0799
0899 0846 0794
0894 0S4I 0788
0888 0835 0783
0883 0830 0778
087S 0825
0773
0872 0S20 0768
0S67
0814 0762
I I 2
I I 2 I I 2
233 233 233
4 4 5 4 4 5 4 4 5
0757
0706
0655
0752 0701 0650
0747 0696 0645
0742 0691 0640
0737 0685
0635
0731 0680 0630
0726 0675 0625
0721 0670 0620
0716 0665 0615
07 1 1 0660 0610
I I 2 112 I I 2
233 233 233
4 4 5 4 4 5 4 4 5
,0605
;0555 0506
0600 0550 0501
0595 0545 0496
0590 0540 0491
0585
0535 0487
0580
0531 0482
0575 0526 0477
0570 0521 0472
0565 0516
0467
0560 0511
0462
0 I I
0 I I 0 I I
223 344 223 344 223 344
90
045S
0453
0448
0443
0438
0434
0429
0424
0419
0414
0 I I
0 I I 0 I I 0 I I
223344
91 92 93
94 95 96
97
98 99
0410 0362
03 1-5
0405
0357 0311
0400
0353 0306
0395 034S 0301
0391
0343 0297
0386
0339 0292
03S1
0334 0287
0376 0329 0283
0372
0325
0278
0367
0320 0273
223 223 223
3 4 4 3 4 4 3 4 4
0269 0223 0177
0264 021S 0173
0259 0214 0168
0255 0209
0164
0250 0205 0159
0246 0200 0155
0241
0195 0150
0237 0191 0146
0232 01S6 0141
0227
0IS2 0137
oil 0 I I 0 I I
223
2 2"^
223
3 4 4 3 4 4 3 4 4
0132 008S
0044
0128 00S3 0039
0123 0079 0035
0119
0074 0031
01 14
0070 0026
Olio
0066
0022
0106 0061 0017
OIOI
0057 0013
0097 0052 0009
0092 0048 0004
oil oil 0 I I
223
2 2'?
/•y ^ ■^
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
32I^
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
4064
I 2 3
4 5 6
678
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
4 5
Inf. Neg.
7-2419
5429
7190
8439
940S
•C200
0S70
1450
1961
21 41 62
82 103
8-2419 8-5428 8-71S8
2832 5640 7330
3210
5S42 7468
3558
6035 7602
3880 6220 7731
4179 6397 7857
4459 6567 7979
4723 6731 8098
4971 6889
S213
5206 7041 S326
8-8436 8-9403 9-0192
8543 9489 0264
8647
9573 0334
8749
9655 0403
8849
9736 0472
8946 9816 0539
9042
9894 0605
9135 9970 0670
9226 0046 0734
9315 0120 0797
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
CnX/>v
,<-
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
I I
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
So 79*44
78
79-37 80-0080-06,80
9775-03 6375-70 2976-35
76
77 78-23'78-29
-9477-01 -59,77-65
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'
0777 7277
•367S
1477-20 7877-S5
42
87 78
50,79 1280
78 79 80 ■
99,79 6279
25S0
-06 -69
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 I 2
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 I 2 I 2
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
'IoS2' 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
I 2
234 234
234
5 5 5 5 5 5
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- 83-
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 I 2
234 234 234
86-6086-66
9785
5685 14S6
85
85 86
8672^86-78
86-83
85
85 86'
85-21 85-79 86-37
85-26 85-85 86-43
S5 32 85-91 86-49
86 -89
86-95
S7-01
85-38 85-97 86-54
I 2 I 2 I 2
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 •8687 -4388
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 -66
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 I 2
233 233 233
4 5 4 5 4 5
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 •21 97*26
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
I 2 I 2 I 2
233 233 233
97-88
97-93
I 2
2 3
4 5 4 5 4 5
4 4 5
96 97 98
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
-2398-29 •749879
98-34 98-84
99-35
98-39 98-89 99-46
98-44 98-94 99-45
I 2 I 2 I I
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.
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