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4. When the given number consists of more than 4 places. Find, as above, the logarithm of the first 4 figures, then multiply the difference between this logarithm and the next greater in the table, by the remain. ing figures of the given number, and from the right hand of the prodoet cut off as many figures as you multiplied lry, adding the remaining figures to the logarithm already taken from the table; then will the sum be the decimal part of the logarithin sought before which write its proper indes, as before, Example.--Required the logarithm of 365154.6? Here logarithm of 965100

5.362412 And tab. diff. 119 X 54.6 = 64.976 or nearly

65 Hence, logarithm of 565154.6

= 5.362477 When the given number is a vulgar fraction.-Subtract the logarithm of the denominator from the logarithm of the numerator, and the remainiler will be the logarithm of the given fraction,

Ex. 1. Find the log. of to E.r. 2. Find the log, of 7 or
Here log. of 5 = 0.098970 Here log, of 29 = 1.462398
And loy. of 16 = 1.204120 And log. of 4 = 0.002003
Hence, log. of 16 = 1.1918.50

Hence, log. of 71 = 0·860333 To find the natural number answering to any given logarithin.-Look in the different columns for the decimal part of the given logarithm; but if you cannot find it exactly, take the next less tabular logarithm, and in a line with the log. found in the column on the left marked N, you have three figures of the number sought, and at the top of the column in which the log. is, you bave one figure more, which annex to the other three; then subtract the tabular log. from the given log, and annex two cyphers to the remainder, divide the result by the tabular difference, and annex the quotient to the four figures already found. In placing the decimal point, it must be remembered, that the number of integer places in the natural number sought, is one more than the index of the given logarithm. Thus an index of 1 requires two whole numbers, and of 2 three whole numbers.

Ex. Find the natural number answering to the logarithm 3.562477.
Here the giveu logarithm

562477
And the next less in the table 562412 its nat, num. = 3651.00
And

6500 + 119

54 Hence, the natural number required

.. = 3651.54 LOGARITHMICAL ARITHMETIC. Ist. The sum of the logarithms of two or more numbers is equal to the logarithms of the product of these numbers;

2d. The difference of the logarithms of two numbers is equal to the logarithm of the quoticut of these numbers;

3d. The logarithm of any power of a number is equal to the logarithm of the root multiplied by the index of the power, and

4th. The logarithm of any root of a number is equal to the logarithm of that number divided by the index of the root. On these four properties the following rules are founded. TABLES OF LOGARITHMIC SINES, TANGENTS,

SECANTS, &c. The Logarithmic Sines, Tangents, Secants, &c. are the logarithms.

=

of the natural numbers which express the measure of the sine, tangent,or secant of the corresponding arc, the radius being 10,000,000,000, the logarithm of which is 10.

PROB. 1.-To find the logarithmic sine, tengent, secant, &c. of any number of degrees and minutes.

Rule. When the number of degrees is less than 45o, find them at the top of some page, and opposite to the minutes on the left-hand, under the words sine, tangent, or secant, respectively, you have the logarithm required.

2. When the number of degrees is above 45°, and less than 90°, find them at the bottom of the page, then opposite to the minutes in the righthand columns, and above the words sine, tangent, or secant, respectively, you bave the logarithm required.

3. When the number of degrees is between 90° and 180°, take their supplement to 180° ; when between 180° and 270°, diminish them by 1800 ; when between 270° and 360°, take their complement to 360° ; and find the logarithm of the remainder as before. Otherwise, for the log. sine or tangent of an arc between 90° and 180°, or between 270o and 360°, take out the log. co-sine, or log. co-tangent of the excess of the arc above 90° or 270° ; for log. co-sine, or log. co-tangent, of an arc above 90° or 270o, take ont the log. sine, or log. tangent, of the excess of the arc above 90° or 270o. But for the Jog. sine and log. tangent, &c. of an arc between 180° and 270°, take the log. sine and log. tangent, &c. of the excess of the are above 180.

EXAMPLES 1. Of 250.45' the sine is 9.687935, and the tangent 9.683356. 2. Of 30°.19' the secant is 10.063864, and the co-sine 9. 936136. 3. Of 650.35 the sine is 9.959310, and the co-tangent 9.657028. 4. Of 740:20' the co-secant is 10.016412, and the tangent 10.552130. 5. Of 129o.10' the sine is 9.886477, that is, either the sine of 500.50', or

the co-sine of 390.10 6. Of 300°.30' the tangent is 10.229832, that is, either the tangent of 590.30',

or the co-tangent of 300,30'. 7. Of 220°.18' the side is 9.810763, that is, the sine of 40°.18'.

PROB. 2.- To find the logarithmic sine, tangent, or secant, &c. of any number of degrees, minutes, and seconds.

Rule. Find, as before, the logarithm for the given degrees and minutes; then, multiply the tabular difference taken from the column marked D, by the given number of seconds; from the product cut off two decimal places, and add to the logarithm already found, the figures which remain, then will the sum be the logarithm for the degrees, mi. nutes, and seconds, required.

Ex. Find the logarithmic sine of 27°.186.42".
Here logarithmic sine of 27'.18' is

= 9.661481
And the tabular difference = 408 X 42

Hence, the logarithmic sine of 270.18'.42" = 9.661652 When the logarithmic sine, tangent, or secant, of an angle under 3° is wanted, it may be found by the following

Rule. From the common logarithm of the number of seconds in the given angle, subtract the logarithm of the seconds in the degrees and minutes next lower, add the remainder to the logarithmic sine, tangent, or secant, of the degrees and minutes; the sum will be the logarithmic side, tangent, or sccant, required.

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Er. Find the logarithmic sine of 19.5'.20".
Here 19.5'.20" = 39204 ............ log.

593286
And 10.5' = 3900" ...

log.

591065 Their difference is

9221 To ukich, add the logarithmic sine of 10.5 . = 6.276614 Hence, the logarithmic sine of 10.5'.20".

= 8.278855 PROB. 3.-To find the degrees and minutes, answering to any given logarithmic sine, tangent, or secant.

Rule. Find the nearest logarithm to that given in the proper column; sf the title be at the top of the column, you have the number of degrees at the top of the page, and the minutes in the column on the lefthand; but, should the title be at the bottom of the column, you bare the degrees at the bottom of the page, and the minutes in the column on the right-hand.

EXAMPLES. 1. The log. sine..

9.457584 answers to *60.40'. 2. The log. tangent ........10.535401 answers to 730.45'.

The log. secant...... • 10.038598 answers to 239.48'. *. The log. co-sine

9.806106 answers to 50°.11'. PROB. 4.–To find the degrees, minutes, and seconds, answering to any given log. sine, tangent, secant, &c.

Rule. Find, as before, in the proper column, the degrees and minutes answering to the next less logarithm, which subtract from the given logarithm ; apues two cyphers to the remainder, and divide by the tabular difference ; then the quotient will be the seconds to be annexed to the degrees and minutes already found.

Er. Find the degrees, minutes, and seconds, corresponding to the log sine 9 647367. Flere given logarithm

9.647367 And the next less in the table .. 9.647 239 = sine of 260.21'. O Also

.......12800 - 425 = Hence, the given log. 9.647367 is = sine of

200.21.30 When the angle is under 3o, it will be more accurate to work by the following

Rule. Find the degrees and minutes corresponding to the next less logarithm ; which subtract from the given logarithm, and to the remainder add the common logarithm of the number of seconds in the degrecs and minutes found; then will the sum be the logarithm of the vumber of seconds in the angle required.

Ex. Find the degrees, minutes, and seconds, corresponding to the log. tangent 8.254527. Here given logarithm

= 8.254597 And the next less in the tables = tangent 1o.1'.0". -8.249102 Their difference is

5425 To which, add 10.1'.0" = 3660"

log.

3.563181 Hence, 1o.1'.46" — 3706" ........ log.

9.108906 Thus, the log. tangent 8.251527 corresponds to 1'.1'. 46."

Note.-The method of finding the natural sine or co-sine of any angle, from the Table of Natural Sines, is exactly similar to that of fioding the log. sine, &c. from the Tables of Logarithmic Sines, &c. But, it using them, it slould be remembered to multiply and divide as in other whole numbers.

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TABLE

CONTAINING THE

LOGARITHMS OF NUMBERS,

FROM 1 to 10,000.

TABLE I.

LOGARITHMS OF NUMBERS.

100

2.000000

No: 1

Lng. 1.000000 N. I Log. IN. I Log. | N | Log. I N. 1 Log. 1 0.000000 26 1.414973

26 1.414973 511.707570 76 1.880814 20.301030 27 1.431364 52 1.716003 77 1.886491 3 0.477121 28 1.447158

28 1.447158 53 1.724276 78 1.892095 4 0.602060 29 1.4623981 54 1.732391 79 1.897627 5 0.698970 30 1.477121 55 1.740363 80 1.903090

6 0.778151 31 1.491362 56 1.748188 81 1.908485
7 0.845098 32 1.505150
32 1.505150 57 1.755875

57 1.755875 82 1.913814 8 0.903090 33 1.518514 58 1.763428 83 1.919078 9 0.954243 34 1.531479 59 1.770852 84 1.924279 10 1.000000 35 1.544068 60 1.778151 85 1.929419|

11 1.041393 36 1.556302 61 1.785330 86 1.934498| 12 1.079181 37 1.568202 62 1.792392 87 1.939519 13 1.113943 38 1.579784 63 1.799341 88) 1.944483 14 1.146128 39 1.591065 64 1.806180 89 1.919390 15 1.176091 40 1.602060 65 1.812913 90 1.954243

16 1.204120 41 1.612784 66 1.819544 91 1.959041 171 1.230449 42 1.623249 67 1.826075 92 1.963788 13 1.255273 43 1.633468 68 1.832509 93 1.968483 19 1.278754 44 1.643453 69 1.838849 94 1.973128 20 1.301030 45 1.653213 70 1.845098

70 1.845098 95 1.977724

21 1.322219 46 1.662758 71 1.851258 96 1.982271 22 1.342423 47 1.672098 72 1.857332; 97 1.986772 23 1.361728 48 1.681241

48 1.681241 73 1.863323 981 1.991226 24 1.380211 49 1.690196 74 1.869232 99 1.993635

25 1.39794050 1.698970 75 1.875061 100 2.000000 N. 1 Log. N.1 . 1 | N. | Log. | N. I

| N | Log | N. 1 Log N. B. In the following part of the Table the Indices are omitted, as they are easily snpplied, being always, each of them, in the case of whole or mixed numbers, an unit less than the number of figures in the integral part of the corresponding natural number. If the number is a decimal, the index is negative, and is always an unit greater than the number of cyphers between the decimal point and the first significant figure of the decimal.

A

LOGARITHMS OF NUMBERS.

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No. 1000- 1600

Log.

000000 -204120 No 10 1

13
5 | 6 | 7 | 8 | 9 | Diff.

8 100 100000/000434 0008640013011001734.002166 0025980030290034600038911 452 101 4921 4751

51801 5609 6038 6466 6894 7321 7748 81741 428 102 8600 9026 9454 9876010300 010724011147011570011993,012415 424 103 012837,013259013680014100 4520 4940 5360 5779 6197 6615 420 104

7033 7451 7868 8284 8700 9116 9532 9917 020361 020775 416 105 021189021603022016 022428 022841 023252 023664024075 4486 4896 419 106 5306 5715 61 24 6533

6942 7350 7757 8164

8571

8978 408 107

9384 9789 0301950306001031004 031408 031812032216 032619 033021 404 108 033424 033826 4227 4628 5029 5430

5830 6229 6629 7028 400 109 7426 7825 8223 86201 9017

9414 9811040207 040602040998 397 110 041393041787042182 042575042969049362043755044148 044540044931 993 111

5323 5714 6105 6495 6885 7275) 7664 8053 8442 8830 S90 112

9218 9606 9993050380050766 0511520515380519240529091052694 386 119 05 3078 053463053846 4230 4613 4996) 5378 5760 6142 6524 583 114

6905 7286 7666 8046 8426 8805 9185 9569 9942060320 379 115 060698061075 061452 061829062206 062582062958063393063709 4083 376 i 116 4458 4832 5206 5580

5953 6326 6699

7443 7814 373 117 8186 8557 8928 9298 9668 070038070407 070776.071145|071514 370 118 071882072250 072617072985 073352 9718 4085 4151 4816 5182 366 119 5547 5912 6276

6649

7004 7368 7731 8094 8457 8819! 365 120 0791810795431079904 080266 080626 080987 081347081707 082067 082426 S60 121 1082785083144083503 3861 4219 4576 4934

5291 5647 6004 557 1 22 6360

6716 7071 7426 7781 8136 8490 8845 9198 9552 355 129 9905 090258 090611 090963091315091667092018|0923701092721093071 959 124 1093422 3779 4122 4471 4820 5169

5518 5866 6215 6562 349 125 6910 7257 7604 7951 8297 8644 8990 9385 96811000261 346 126 100370 100715101059101403 101747102090 102434 102777 105119 3462 843 127 3801 4146 4487 4828

5169

5510 5851 6191 6591 6870 341 128

7210 7549 7888 8227 8565 8909 9241 9578 9916110259 338 129 /110590 110926111262111598 111934112270112605 11 2940 113275 3609 385 130 113943114277114011 114944115278 115610115943116276 116608116940 332 131 7271 7603

7934 8265) 8595 8926 92561 9586 9915 120245 ss0 192 120574 120903121231121560121888 122216122544 122871 123198 3525 928

9852 4178 4504 4830 5156 5481 5806 6131 6456 6781 995 134 7105 7429 7752 8076 8399

8722 9045 9968 9690 130012 329 135 130834 130655 130977131298131619131999 192260/132580132900 3219 321 136 3539 3858 4177 4496 4814 5133 5451 5768 6086 6403 318 137 67211 7037 7954 7670 7987 8803 8618 8934 9249 9564 316 138 9879 140194 140508 140822141136 141450 141763 142076 142389|142702 S14 139 143015 3327 3639 3951 4265 4574

4885 5196 5507 5818 911 140 146128(146438 146748 147058 147367 147676 14:985 148294148603 148911 309 141 9219 9527 9835 150142 150449 150756 151063151370 151676 151982 307 142 152288 152594152900 3205

3510 3815 4119 4424 4728 50321 305 5336 5640 5943 6246 6549 6852) 7154 7457 7759 8061 SOS

8362 8664 8965 9266 9567 9868160168160468 160769 161068 800 145 161368 161667 161967 162266162564162863 3161 3460 3757 4055 295 146 4353 46501 4947 5244 5.5411 5898 6134 6430 6726 7622 295 147 7317 7613 7908 8203 8497 8792 9086

9380 9674 9968 294 148 170262170555170848171141171434171726 172019 172311 172603 172895 292 149 3186 3478 3769 *4'60 4351! 4641 4932 5222 5512 5802 290 150 176091 170381176670 176959 177248 177536 1778251781131178401178689

288 151 8977 9264 9552 9839 180126180413 180699 180986181272|181558 287 152 181844 182129182415182700 2985 3270 3554 S839 4123 4407 285 153 4601 4975 5259 5542 5325 6108 6391 6674 6956 72 91 283 154 7521 7808 8084 8366 8647 8928

9209 9490 9771 190051 231 155 190332190612190892191171191451|191730 192010192289 192567 2846 279 156 3125 3403 3631 3959

4237 4514 4792 5069 5346 5699 278 157 590) 6176 6452 6729 7005 7281

7556 7832 8107 8982 276 158 8657 8932 9206 9481 9755 200029 200303 200577 200850 201124 274 159 201397/201670 201 943202216202488) 2761 80331 9305 3577 3848 272 0

2

3
1 4 | 5 | 6

8
1

9

133

145 144

No.

| Dift.

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