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TABLE I.

LOGARITHMS OF NUMBERS.

EXPLANATION.

LOGARITHM

OGARITHMS are a series of numbers so contrived, that the sum of the Logarithms of any two numbers, is the logarithm of the product of these numbers. Hence it is inferred, that if a rank, or series of numbers in arithmetical progression, be adapted to a series of numbers in geometrical progression, any term in the arithmetical progression will be the logarithm of the corresponding term in the geometrical progression.

This table contains the common logarithms of all the natural numbers from 0 to 10000, calculated to six decimal places ; such, ois ac count of their superior accuracy, being preferable to those, that are computed only to five places of decimals.

In this form, the logarithm of 1 is 0, of 10,1; of 100, 2; of 1000, 3 &c. Whence the logarithm of any term between 1 and 10, being greater than 0, but less than 1, is a proper fraction, and is expressed decimally. The logarithm of each term between 10 and 100, is 1, with a decimal fraction annexed ; the logarithm of each terin between 100 and 1000 is 2, with a decimal annexed, and so on. The integral part of the logarithm is called the Index, and the other the decimal part... Except in the first hundred logarithms of this Table, the Indexes are not printed, being so readily supplied by the operator from this general rule; the Index of a Logarithm is always one less than the number of figures contained in its corresponding natural number-exclusive of fractions, when there are any in that number.

The Index of the logarithın of a number, consisting in whole, or in parts, of integers, is affirmative ; but when the value of a number is less than unity, or 1, the index is negative, and is usually marked by the sign, - placed either before, or above the index. If the first significant figure of the decimal fraction be adjacent to the decimal point, the index is 1,- or its arithmetical complement 9 ; if there is one cipher between the decimal point and the first significant figure in the decimal, the index is . 2, or its arith. comp. 8; if two ciphers, the index is 3, or 7, and so on ; but the arithmetical complements, 9, 8, 7 &c. are rather more conveniently used in trigonometrical calculations.

А

The decimal parts of the logarithms of numbers, consisting of the same figures, are the same, whether the number be integral, fractional, or mixed : thus,

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N. B. The arithmetical complement of the logarithm of any number, is found by subtracting the given logarithm from that of the radius, or by subtracting each of its figures from 9, except the last, or right-hand figure, which is to be taken from 10. The arithmetical complement of an index is found by subtracting it from 10.

PROBLEM I.

to find the logarithm of any given number.

RULES.

1. If the number is under 100, its logarithm is found in the first page of the table, immediately opposite thereto.

Thus the Log. of 53, is 1.724276. 2. If the number consists of three figures, find it in the first column of the following part of the table, opposite to which, and under 0, is its logarithm.

Thus the Log, of 384 is 2.584331-prefixing the index 2, because the natural number contains 3 figures.

Again the log. of 65.7 is 1.817565-prefixing the index 1, because there are two figures only in the integral part of the given number.

3. If the given number contains four figures, the three first are to be found, as before, in the side column, and under the fourth at the top of the table is the logarithm required.

Thus the log. of 8735 is 3.941263—for against 873, the three first figures found in the left side column, and under 5, the fourth figure found at the top, stands the decimal part of the logarithm, viz.941263, to which prefixing the index, 3, because there are four figures in the natural number, the proper logarithm is obtained.

Again the logarithm of 37.68 is 1.576111-Here the decimal part of the logarithm is found, as before, for the four figures ; but the index is 1, because there are two integral places only in the natural number.

4. If the given number exceeds four figures, find the difference between the logarithms answering to the first four figures of the given number, and the next following logarithm ; multiply this difference by the remaining figures in the given number, point off as many figures to the right-hand as there are in the multiplier, and the remainder, add.

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ed to the logarithm, answering to the first four figures, will be the required logarithm, nearly.'

Thus; to find the logarithm of 738582 ; the log. of the first four figures, viz. 7385 .868350 the next greater logarithm

= 868409 Dif.

59 to be multiplied by the remaining figures

= 82

118 472

48|38

then to .868350
add

48

the sum 5.868398, with the proper index prefixed, is the required logarithm.

5. The logarithm of a vulgar-fraction is found by subtracting the logarithm of the denominator from that of the numerator ; and that of a mixed quantity is found by reducing it to an improper fraction, and proceeding as before. Thus to find the Logarithm of };

from the log. of 7 = 0.845098 subtract the log. of 8 = 0.903090

Remainder = 9.942008 = the required log.

PROBLEM II.

To find the number answering to any given logarithm.

RULES.

1. Find the next less logarithm to that given in the column marked o at the top, and continue the sight along that horizontal line, and a logarithm the same as that given, or very near it, will be found ; then the three first figures of the corresponding natural number will be found opposite thereto in the side column, and the fourth figure immediately above it, at the top of the page. If the index of the given logarithm is 3, the four figures thus found are integers; if the index is 2, the three first figures are integers, and the fourth is a decimal, and so on. Thus the log. 3.132580 gives the Nat. Numb. 1357 2.132580 gives

135.7 1.132580 gives

13.57 0.132580 gives

1.357 9.132580 gives

.1357 &c. 2. If the given logarithm

cannot be exactly found in the table, and if more than four figures be wanted in the corresponding natural pumber; then find the difference between the given and the next less logas

rithms, to which anpex as many ciphers as there aro figures required above four in the natural number; which divide by the difference between the next less, and next greater logarithms, and the quotient annexed to the four figures formerly found, will give the required natural number. Thus to find the natural number of the log, 4.828991;

the next less log. is .828982 which gives 6735 ; the next greater log. is 829046

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therefore 1.4 being annexed to 6735, the required natural number, 67351.4, is now obtained.

TABLE I.

LOGARITHMS OF NUMBERS.

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2

84

No. Log.

No. Log. No. Log. No. Log. No.1 Lo 0.000000 21 1.322219 41 3.612784 61 1.785330 81 1.908485 0.301030 22 1.342423 42 1.623249 62 1.792392

82

1.913814 3 0.477121 23 1.361728 43 1.63346863

1.799341

83

1.9 19078 0.602060 24 1.380211 44 1.643453 | 64 1.806 180

1.924279 5. 0.698970 25 1.397940 45 1.653213 65 3.812913 85 1.929419 6 0.778151 26 1.414973 46 1.662758 66 1.81954486

1.934498 70.845098 1.431364 47 1.67209867 1.826075 87

1.939519 8 0.903099 28 1.447158 48

1.681241 68 1.832509

88

1.944483 9 0.954243 29 1.462398 49 1.690196 69 1.838849 89

1.949390 1.000000 30 1.477121 50 1.698970 70 1.845098 90 1.954243 1.041393 31 1.491362 si 1.707570 71 1.851258 91 1.959041

1.079181 32 1.505150 1.716003 72 1.857332 92 13 I.113943 33 1.518514 53 1.724276 73 1.863323

93 1.968483 1.146128 34 1.531479 54 1.732394 74 1.869232 94 15 1.176091 35 1.544068 55 1.740363 75

75 1.875061

1.977724 16

1.204120 36 1.556302156 1.748188 76 1.880814 95 1.982271
1.230449 37 1.568202 57 1.755875 77 1.886491 97 1.986772

38 1.579784 1 58 1.76342878 1.892095 98 1.991226
19 1 1.278754 39 1.591065 59 1.770852 79 1.897627 1.995635
20
1.301030 40 1.60 2060 60 1.778151 80

1.903090 1100 2.000000

10

12

1.963788

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1.973128

95

17

18 1.295273

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100

No.

1 2 3 4 5 | 6 | 7 8 9 000000000434 000868 00 1301 001734/002166 002598.003029003460003891 101 1004321004751005180005609 006038006466 006894007321 00774810081741 102 008600'00g206 009451 009876 010300 010724'011147 011570011993 012415 103 012837 013259 013680014100 014520014940 015360 0157791016197 016615 104 1017033.017451017868018284 018700019116 019532 019947020361 020775 105 021189 021603 0220161022428 0228411023252 023664024075 024486024896 106 025306,025715 026124 026533 026942 0273 50 027757 028164 028571028978 107 3293841029789 030195030600 031004 03 1408 031812 032216 032619033021 108 033424 033826 034227 034628 035029 035430 035830 0362290366291037028 109 037426 037825'0382231038620 039017039414 039811 040207|040602040998 110 041393041787 042182 042575 042069 043362043755044148044540 044930 un 0453230457 14 046105046495 046885047275047664048053 048442048830 112 049218 049606.049993050380 050766 051152 051538 051924 052309052694 113 053078053463053846054230 054613054996,0553781055760056142 056524 114 056905 057286.0576661058046 058426058805:059185059563|059942060320 115 060698061075 061452 061829 062206062582062958063333063709064083 116 0644581064832 065206065580 065953 066326066699067071067443 067814 117 068186068557068928 069298 069668070038070407070770071145 071514 118 071882072250 072617 072985 073352073718074085 074457074816075182 119 075547 075912076276 076640 077004 077368,077731 078094 078457078819 120 079181 079543 079904 080266 080626 080987081347081707082067 082426 121 1082785083144 083503 083861 084219 084576 084934 085291 0856471086004 122 086360086716087071 087426 087781 08813608849010888450891981089552 123 089909 090258 090611090963'091315 091667 0920181292370 092721093071 124 1093422|093772 094122 094471 094820 095169 09551810958660962151096562 125 096910097257 097604/097951098297 098644 098990099335 099681 100026 126 100370 100715101059, 101403 101747 102000 102434 102777|103119 103462 127 103804 104146 104487 104828 105169 105510105851 106191 106531 106870 128 107210 107549 107888108227 108565108903 1092411095781109916110253 129 110590 110926 "11262111598101934112270112605112940113275 113609 130 "13943114277 114611 114944/115278 115610 115943 46276 116608 116940 131 117277 117603117934118265118595 118926119256119586119915120245 132 1205741120903121231121560121888122216122543122871123198|123525 133 123852124178|124504 124830125156125481 125806126131|126456 126781 134 127105127429 127752128076128399 128722129045 129368 129690 130012 135 130354 +30655130977 131298 131619 131939 132260132580132900 133219 136 133539 133858134177134496 134814135133 135451 135768136086136403 137 136721137037137354 137670 137987 138303 138618138934139249139564 138 139879140194 140508 140822 141136141450 141763 142076 142389142702 13943015143327 143639 143951 144263144574144885145196 145507 145818 140 146128 146438 146748147058 147367 147676 347985148294 148603 148911 141 149219149527 149835 150142150449 150756 151063/1513701151676 151982) 142152288152594 +52900 153205153510153815154119 154424 154728155032 143 155336155640155943|156246 156549 156852 157154157457157759 158061 144 158362 158664 158965 159266159567 159868160168160468160769 161068 145 61368161667 161967 162266162564 162863 163161 163460163757164055 146 164353 164650 164947 165244 1655411658384166134 166430166726 167022 147 167317 167613|167908168203 168497 168792 169086169380169674 169968 148 170262 1705551708481711411171434171726 172019172311172603172895 149 173186173478 173769 174060174351 1746411174932175222|175512 175802 150 176091 076381 176670 176959 177248 177536 177825178113178401 178689 151 178977 179264 179552 179839 180126180413180699 180986181272|181558 152 181844182129182415 182700182985 183270183554183839 184123 184407 153 184691184975185259 185542 183325186108186391 186674 186956187239 154 187521187803 188084 188366188647 1889281189209 189490 189771|190051 155 190332190612190892191171191451191730 192010192289 192567 192846 156 193125 19340319368|193959 194237 194514/194792195069 195346 195623 157 195900 196176 196452196729 197005 197281197556197832 198107 198382 158 198657|198932 199206199481|199755 200029 20030; 200577 200850 201124 159 201397 201670 201943202216 202488 202761 203033 203305 203577 203848 0 2

4 5 6 7 8 9

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