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OUR laCURE MATHEMATIOAL TABLES OTTOMLEY

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IN MEMORIAM FLORIAN CAJO

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FOUR FIGURE MATHEMATICAL TABLES.

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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

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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 corresponding 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 principle 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 advantage, 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 trigonometrical 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 convenient way of writing down the given numbers and their logarithms. It is scarcely possible to overestimate the importance of strict adherence 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 forthcoming, 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

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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

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.

The text on this page is estimated to be only 25.84% accurate

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 3233 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

The text on this page is estimated to be only 24.49% accurate

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 98 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

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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-IN48 SQUARE ROOTS FROM 100 TO 999-9. 10 n 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 33 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 0 1 2 3 4 5 6 7 8 9 •1 -2 -3 •4 -5 -6 •7 -8 -9 lo-oo 10-05 10-10 10-15 10-20 10-25 10-30 10-34 10-39 10-44 oil 223 3 4 4 IO-49 IO-95 11-40 10-54 11-00 11-45 10-58 11-05 11-49 10:63 1 1 -09 "•53 10-68 11-14 11-58 10-72 II-18 11-62 10-77 11-22 11-66 10-82 11-27 11-70 10-86 11 -31 "-75 10-91 11-36 11-79 oil oil oil 223 223 223 3 4 4 3 4 4 3 3 4 11-83 12-25 12-65 11-87 12-29 12-69 11*92 12-33 12-73 11-96 12-37 12-77 I2-00 12-41 I2-8I 12-04 12-45 12-85 1 2 -08 12-49 12-88 12-12 12-53 12-92 1217 12-57 12-96 12-21 12-61 13-00 oil 0 I I 0 I 1 2 2 2 2 2 2 2 2 2 3 3 4 3 3 4 3 3 4 13-04 13-42 1378 13-08 13-45 13-82 13-11 13-49 13-86 13-15 13-53 13-89 13-19 13-56 13-93 13-23 13-60 13-96 13-27 13-64 14-00 13-30 13-67 14-04 13-34 13-71 14-07 13-38 13-75 14-11 oil 0 I I oil 2 2 2 I 2 2 12 2 3 3 3 3 3 3 3 3 3 14-14 14-18 14-21 14-25 14-28 14-32 14-35 14-39 14-42 14-46 oil I 2 2 233 14-49 14-83 15-17 14-53 14-87 15-20 14-56 14-90 15-23 14-59 14-93 15-26 14-63 14-97 15-30 14-66 15-00 15-33 14-70 15-03 15-36 14-73 15-07 15-39 14-76 1510 15-43 14-80 15-13 15-46 0 I I oil 0 I I 12 2 I 2 2 12 2 233 233 233 15-49 15-81 l6-I2 15-52 15-84 16-16 15-56 15-87 16-19 15-59 15-91 16-22 15-62 15-94 16-25 15-65 15-97 16-28 15-68 16-00 16-31 15-72 16-03 16-34 1575 1 6 -06 16-37 15-78 16-09 16-40 0 I I oil 0 I I 12 2 I 2 2 1 2 2 233 233 223 16-43 16-7^ 17-03 16-46 16-76 17-06 16-49 16-79 17-09 16-52 16-82 17-12 16-55 16-85 17-15 16-58 16-88 17-18 16-61 16-91 17-20 16-64 16-94 17-23 16-67 16-97 17-26 16-70 17-00 17-29 oil oil 0 I I 1 2 2 112 I I 2 223 223 223 17-32 17-35 17-38 17-41 17-44 17-46 17-49 17-52 17-55 17-58 0 1 I I I 2 223 i7'6i 17-89 18-17 17-64 17-92 18-19 17-66 17-94 18-22 17-69 17-97 18-25 17-72 1800 18-28 1775 18-03 18-30 17-78 18 -06 18-33 17-80 18-08 18-36 17-83 18-11 18-38 17-86 18-14 18-41 oil oil 0 I I 1 I 2 I I 2 I I 2 223 222 222 18-44 18-71 18-97 18-47 1873 19-00 18-49 iS-76 19-03 18-52 18-79 19-05 18-55 i8-8i 19-08 18-57 18-84 19-10 1 8 -60 18-87 19-13 18-63 18-89 19-16 18-65 18-92 19-18 18-68 18-95 19-21 0 I I oil 0 I I 112 I I 2 112 222 222 222 19-24 19-49 1975 1926 19-52 19-77 19-29 19-54 19-80 19-31 19-57 1982 19-34 19-50 19-85 19-36 19-62 19-87 19-39 19-65 19-90 19-42 19-67 19-92 19-44 19-70 19-95 19-47 19-72 1997 oil oil 0 I I 112 I I 2 1 I 2 222 222 222 20-00 20-02 20-05 20-07 20-10 20-12 20-15 20-17 20-20 20-22 001 I I I 222 20-25 20-49 20-74 20-27 20-52 20-76 20-30 20-54 20-78 20-32 20-57 20 -S I 20-35 20-59 20-83 20-37 20-62 20-S6 20-40 20-64 20-8S 20-42 20-66 20-90 20-45 20-69 20-93 20-47 20-71 20-95 0 0 I 001 0 0 1 I I I III I I I 222 222 222 20-98 21-21 21-45 2I*00 21-24 21-47 21 -02 21-26 21-49 21-05 21 -28 21-52 21-07 21-31 21-54 21-10 21-33 21-56 21-12 21-35 21-59 21-14 21-38 21-61 21-17 21-40 21-63 21-19 21-42 21-66 0 0 I 001 0 0 I III I I I I I I 222 222 222 21-68 21-91 22-14 21-70 21-93 22-l6 21-73 21-95 22-18 21-75 21-98 22-20 21-77 22-00 22-23 21-79 22-02 22-25 21-82 22-05 22-27 21-84 22-07 22-29 21-86 22-09 22-32 21-89 22-11 22-34 0 0 1 0 0 1 0 0 I I I 1 1 1 I I I I 222 222 222 22-36 22-38 22-41 22-43 22-45 22-47 22-49 22-52 22-54 22-56 001 I I 1 222 22-58 22 -So 23-02 22-61 22-83 23-04 22-63 22-85 23-07 22-65 22-87 23-09 22-67 22-89 23-11 22 69 22-91 23-13 22-72 22-93 23-15 22-74 22 96 23-17 22-76 22-98 23-19 22-78 23-00 23-22 0 0 I 0 0 I 001 I I I III 1 I I 222 222 222 23-24 23-26 23-28 23-30 23-32 23-35 23-37 23-39 23-41 23-43 0 0 I I I I I 2 2

rAcL SQUARE ROOTS FROM 100 TO 999*9. 49 66 56 67 68 69 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 0 1 2 3 4 6 6 7 8 9 -1 -2 -3 •4 -5 -6 •7 -8 9 23-45 23 06 23-4723-4923-52 23-6923-7123-73 23-54 2375 23-5623-58 23-7723-79 23-60 23-81 23-62,23-64 23-832385 0 0 I 0 0 I 122 122 23-87 24-08 24-29 23-9023-92 24-1024-12 24-31 24-33 23-94 24-15 24-35 23-96 23-98'24-oo 24-1724-1924-21 24-37 24-39 24-41 24-02 24-23 24-43 24 -04 24 -06 24-2524-27 24-45|24-47 0 0 I 0 0 I 0 0 1 1 2 2 I 2 2 122 24-49 24-5224-54 24-56 24-58 24-60 24-62 24-64 24-6624-68 0 0 I 122 24-70 24-90 251C 24-7224-74 24-92:24-94 25-1225-14 24-76 24-96 25-16 24-78 24-98 25-18 24-80 24 82 25-0025-02 25-2025-22 24-84 25-04 25-24 24-86^24 88 25 -06 25 08 25-2625-28 001 0 0 I 0 0 I 1 2 2 1 2 2 1 2 2 25-30 25-50 25-09 25-32 25-51 25-71 25-3425-36 25-5325-55 25-732575 25-38 25-57 25-77 25-40 25-59 25-79 25-42 25-61 25-81 25-44 25-63 25-83 25-46 25-65 25-85 25.48 25-07 25-87 0 0 1 0 0 I 001 1 2 2 I 2 2 I 2 2 25-88 2608 26-27 25-90 26-10 26 29 25-92 26-12 26-31 25-94 26-13 26-32 25-96 26-15 26-34 25-98 26-17 26-36 26-00 26-19 26-38 26-02 26-21 26-40 26-04 26-23 26-42 26-06 26-25 26-44 001 001 0 0 1 I 2 2 122 I 2 2 26-46 26-48 26-50 26-51 26-53 26-55 26-57 26-59 26-61 26-63 0 0 I I 2 2 26-65 26-83 27-02 26-66 26-85 2704 26-6S 26-87 27-06 26-70 26-89 27-07 26-72 26-91 27-09 26-74 26-93 27-11 26-76 26-94 27-13 26-78 26-96 27-15 26-80 26-98 27-17 26-81 27 00 27-18 001 0 0 I 0 0 I 1 I 2 1 1 2 1 1 2 27-20 27-39 27-57 27-22 27-40 27-59 27-24 27-42 27 60 27-26 27-44 27-62 27-28 27-46 27-64 27-29 27-48 27-66 27-31 27-50 27-68 27-33 2751 27-69 27-35 27-53 27-71 27-37 27-55 27-73 001 0 0 1 0 0 1 112 112 112 27 75 27-93 28-11 27-77 27-95 28-12 27-78 27-96 2814 27 So 27-98 28-16 2782 28-00 28-18 27-84 28-02 28-20 27 -86 28-04 28-21 27-87 2805 28-23 27 -89 28-07 28-25 27-91 28-09 28-27 0 0 I 0 0 I 0 0 I 1 1 2 1 1 2 1 I 2 28-28 28-30 28-32 28-34 28-35 28-37 28-39 28-41 28-43 28-44 0 0 I 1 I 2 28-46 28-64 28-81 28-48 28-65 28-S3 28-50 28-67 28-84 28-51 28-69 28-86 2S-53 28-71 28-88 28-55 28-72 28-90 28-57 28-74 28-91 28-58 2S-76 28-93 28-60 28-77 2S-95 28-62 28-79 28-97 001 0 0 I 001 1 I 2 1 I 2 112 28-98 I29-15 29-33 29 00 29-17 29-34 29-02 29-19 29-36 29-03 29-21 29-38 29-05 29-22 29-39 29-07 29-24 29-41 29-09 29-26 29-43 29-10 29-27 29-44 29-12 29-29 29-46 29-14 2931 29-48 001 0 0 I 001 1 1 2 1 1 2 1 I 2 29-50 2966 29-83 29-51 2968 29-85 29-53 29-70 29-87 29-55 29-72 29-88 29-56,29-58 297329-75 29-9029-92 29-60 29-77 29-93 29-61 29-78 29-95 29-63 29-80 29-97 29-65 29-82 29-98 0 0 1 001 0 0 1 112 112 I I 2 30-00 30-02 30-03 30-05 30-07 30-08 30-10 30-12 30-13 30-15 000 III 30-17 30-33 30-50 30-18 30-35 30-51 30-20 30-36 30-53 30-22 30-38 30-55 30-23 30-40 30-56 30*25 3a-4i 30-58 30-27 30-43 30-59 30-28 30-45 30-61 30-30 30-46 30-63 30-32 30-48 30-64 000 000 000 I I I I 1 I 1 I I 94 95 96 30-66 3082 30-98 30-68 30-84 31-00 30-6930-71 30-S530-S7 31-0231-03 30-72 3089 31-05 30-74 30-90 31-06 30-76 30-92 31-08 30-7730-79 30-94 30-95 31-1031-11 30-81 30-97 31-13 000 000 000 I I 1 I I 1 I 1 1 97 98 99 31-14 31-30 31-46 ] 31-16 31-32 31-48 31-1831-19 31-3431-35 3i-503i"5i 31-2131-2231-24 31 -3731 -38 31 -40 31-5331-5431-56 ! 31-2631-27 31-4231-43 31-5831-59 1 31-29 31-45 31-01 000 000 000 I 1 I I 1 1 I I I B ^t-v

C/lXAV 5° SQUARE ROOTS FROM I pop TO 9999. 10 o 1 2 3 4 5 6 7 8 9 123 456 7 8 9 31-62 31-7S 31-94 32-09 32-25 32-40 32-56 32-71 32-86 33-02 2 3 5 689 II 12 14 11 12 13 14 16 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 IT 52 63 64 33-17 34-64 36-06 33-3233-4733-62 3479,34-93 35-07 36-1936-3336-47 33-76 35-21 36-61 33-91 35-36 36-74 34-06 35-50 36-88 34-21 35-64 37-01 34*35 35-78 37-15 34-50 35-92 37-28 I 3 4 I 3 4 I 3 4 679 678 578 10 12 13 ion 13 10 II 12 37-42 38-73 40-00 37-5537-68 38-8638-99 40-12 40-25 37-82 39-12 40*37 37-95 39-24 40-50 38*08 39*37 40-62 38-21 39-50 40-74 38*34 39-62 40-87 38-47 3975 40-99 38-60 39S7 41-11 I 3 4 I 3 4 I 2 4 5 78 568 567 9 II 12 9 10 II 9 10 II 41-23 42-43 43-59 41-3541-47 42-54'42-66 43-7043-82 41-59 42-78 43-93 41-71 42*90 44-05 41*83 43-01 44-16 41-95 43-13 44*27 42-07 43-24 44-38 42*19 43*36 44*50 42-31 43-47 44-61 I 2 4 I 2 3 I 2 3 567 567 8 10 II 8 9 10 8 9 10 44-72 44-83 44*94 45-06 45°i7 45-28 45*39 45-50 45*6i 45-72 I 2 3 467 8 9 10 45-83 46-90 47-96 45-9346-04 47-01 47;^2 48-0648-17 46*15 47-22 48*27 46-26 47*33 48-37 46-37 47-43 48-48 46-48 47-54 48-58 46-58 47-64 48-68 46*69 4775 48-79 4680 47-85 48-S9 I 2 3 I 2 3 I 2 3 456 456 456 8 9 10 7 8 9 7 8 9 48-99 50-00 50-99 49-0949-19 50-1050-20 51-0951-19 49-30 50-30 51-28 49-40 50-40 51-38 49-50 50-50 51-48 4960 50-60 51-58 49-70 50-70 51-67 49 -So 50-79 51*77 49-90 50-89 51-87 I 2 3 I 2 3 1 2 3 456 4 5 6 456 7 8 9 7 8 9 7 8 9 51-96 52-92 53-85 52*06 53-01 53-94 52-15 53*10 54-04 52-25 53*20 54-13 52-35 53-29 54-22 52-44 53-39 54-31 52-54 53-48 54-41 52-63 53*57 54-50 52-73 53-67 54-59 52-82 53-76 54-68 I 2 3 I 2 3 I 2 3 4 5 6 456 4 5 5 7 8 9 7 7 8 678 54-77 54-8654-95 55-05 55-14 55-23 55-32 55-41 55-50 55-59 I 2 3 4 4 5 678 55-68 56-57 57-45 5577 55-86 56-6656-75 57-53;57-62 55-95 56-83 5771 56-04 56-92 57-79 56-12 57-01 57-88 56-21 57-10 57-97 56-30 57-18 58-05 56-39 57-27 58-14 56-48 57-36 58 •?2 I 2 3 I 2 3 I 2 3 3 4 5 3 4 5 3 4 5 678 678 678 58-31 59-16 60-00 58-4o|58-48 59-2559-33 60-08 6o-i 7 58-57 59-41 60*25 58-65 59-50 60-33 58-74 59-58 60-42 58-S2 59-67 60-50 58-91 59-75 60-58 58-99 59-83 6066 59-oS 5992 6075 I 2 3 122 I 2 2 3 4 5 3 4 5 3 4 5 678 678 6 7 7 60-S3 61-64 62-45 60*91, 60-99 61 -73 61 -81 62-53^62*61 61*07 61-89 62-69 61*16 61-97 62*77 61-24 62-05 62-85 61 -32 62 -1 3 62*93 61-40 62-21 63-01 61-48 62-29 63-09 61-56 62-37 63-17 122 122 I 2 2 3 4 5 3 4 5 3 4 5 6 7 7 667 667 63-25 63*3263*40 63-48 63-56 63-64 63-72 63-80 63 -87 63-95 122 3 4 5 667 64-03 64-81 65-57 64-11,64*19 64-88:64-96 65-6565-73 64-27 65-04 65-80 64-34 65-12 65-88 64*42 65*19 65-95 64*50 65-27 66-03 64-58 65-35 66-11 64-65 65-42 66-18 64-73 65-50 66-26 122 I 2 2 I 2 2 3 4 5 3 4 5 3 4 5 5 6 7 5 6 7 5 6 7 66-33 67-08 67-82 66-41 66-48 67*1667*23 67-9067-97 66-56 67-31 68-04 66-63 67-38 68-12 66-71 67-45 68-19 66-78 67-53 68-26 66-86 67-60 68-34 66-93 6/ -68 6841 67-01 67 75 68-48 I 2 2 I I 2 I I 2 3 4 5 3 4 4 3 4 4 5 5 7 5 6 7 5 6 7 6S-56 69-28 70-00 68-63 69-35 70-07 68*70 69-43 70*14 68-77 69-50 70-21 6S-85 69-57 70*29 68-92 69-64 70*36 68-99 69-71 70-43 69-07 69-79 70-50 69-14 69-86 70-57 69 21 69-93 70-64 I I 2 I I 2 I I 2 3 4 4 3 4 4 3 4 4 5 6 7 566 566 70-71 70-78 70-85 70-92 70-99 71*06 71-13 71-20 71*27 71-34 r I 2 3 4 4 566 71-41 72-11 72-80 73-48 71-48 72-18 72-87 71-55 72-25 72-94 71-62 72-32 73-01 71-6971-76 72-39^72-46 73-0873-14 71 -S3 72-53 73-21 71-90 72-59 73-28 71-97 72'-66 73*35 72-04 72-73 73-42 I I 2 I I 2 I I 2 3 4 4 3 3 4 3 3 4 566 566 5 5 6 73-55 73*62 73-69 73-76,73-82 73 -89 73-96 74-03 74-09 I I 2 3 3 4 5 5 6

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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

The text on this page is estimated to be only 25.47% accurate

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

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USEFUL FORMULAS AND NUMBERS. Binomial Theorem. (i±.)^ n.n :neI o . n.n\.n — . '^H&c. Hence, when ne is so small that its square and higher powers maybe 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 respectively ; and let T and t be the respective atmospheric temperatures 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, multiply by To convert hyperbolic into common logarithms, multiply 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, assuming 5 "67 as mean density, . 6'i4X 10" grammes Earth's mass (approximately), . 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 centimetre-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, 2323Silver, 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 Decomposition, 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 experiment.

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