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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 remaining figures of the given number, and from the right hand of the product cut off as many figures as you multiplied by, adding the remaining figures to the logarithm already taken from the table; then will the sum be the decimal part of the logarithm sought,before which write its proper index, as before.

Example.-Required the logarithm of 365154.6?

Here logarithm of 365100

And tab. diff. 119 x 54.6 64.976 or nearly
Hence, logarithm of 365154.6

5.562412

65

= 5.562477

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

Ex. 1. Find the log. of

Here log. of 5 0.698970
And log. of 16 = 1.204120

Hence, log. of = 1.494850

Ex. 2. Find the log. of 71 or 2
Here log. of 29 = 1.462398
And log. of 4 0.602062
Hence, log. of 740-860338

To find the natural number answering to any given logarithm.-Look in the different colunins 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 have 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 given logarithm .....
And the next less in the table

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562477

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562412 its nat. num.

3651.00

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Hence, the natural number required

LOGARITHMICAL ARITHMETIC.

1st. 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 quotient 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 45°, 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 have 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 180°; 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 270° 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 270°, take out the log. sine, or log. tangent, of the excess of the arc above 90° or 270°. But for the log. 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 25°.45' the sine is 9.687955, and the tangent 9.683356.

2. Of 30°.19′ the secant is 10.065864, and the co-sine 9, 936136.
3. Of 65.55' the sine is 9.959310, and the co-tangent 9.657028.
4. Of 74°.20′ the co-secant is 10.016442, and the tangent 10.552130.

5. Of 129.10' the sine is 9.889477, that is, either the sine of 50°.50', or the co-sine of 39°.10'

6. Of 300°.30′ the tangent is 10.229852, that is, either the tangent of 59°.30', or the co-tangent of 30°.30'.

7. Of 220°.18′ the sine 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, minutes, and seconds, required.

Ex. Find the logarithmic sine of 27°.18′.42′′.
Here logarithmic sine of 27".18′ is

And the tabular difference = 408 X 42 =

9.661481
171

Hence, the logarithmic sine of 27°.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 sine, tangent, or secant, required.

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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; of 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 have the degrees at the bottom of the page, and the minutes in the column on the right-hand.

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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; annex 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.

Ex. Find the degrees, minutes, and seconds, corresponding to the log sine 9 647367.

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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 degrees and minutes found; then will the sum be the logarithm of the number of seconds in the angle required.

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

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log.

Hence, 1°.1'.46" = 3706"

3.568906

Thus, the log. tangent 8.254527 corresponds to 19.1. 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 finding the log. sine, &c. from the Tables of Logarithmic Sines, &c. But, in using them, it should be remembered to multiply and divide as in other whole numbers.

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2 0.301030 27 1.431364 52 1.716003
3 0.477121 28 1.447158
4 0.602060 29 1.462398
5 0.693970 30 1.477121
30 1.477121

77 1.886491

53 1.724276

78 1.892095

54 1.732394

79 1.897627

55 1.740363

80 1.903090

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96 1.982271 72 1.857332 97 1.986772 73 1.863323 98 1.991226 74 1.869232 99 1-995635 75 1.875061 100 2.000000 Log. | N. I Log.

21 1.322219 46 1.662758 71 1.851258 22 1.342423 47 1.672098 23 1.361728 48 1.681241 24 1.380211 49 1.690196 25 1.397940 50 1.698970 N. I Log. N. I Log. N. I N. B. In the following part of the Table the Indices are omitted, as they are easily supplied, 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

No. 1000

No I O

4321

LOGARITHMS OF NUMBERS.

1600

1

2

3 1 4

-204120

Log. 0000005 16 | 7 | 8 | 9 Diff. 100 000000000434 000868 001301001734 002166 002598 003029003460003891 432 101 4751 5180 5609 6038 6466 6894 7321 7748 8174 428 102 8600 9026 9451 9876 010300010724011147 011570011993,012415 424 103 012837 013259 013680 014100 4520 4940 5360 5779 6197 6615 420 104 7033 7451 7868 8284 8700 9116 9532 9947 020361 020775 416 105 021189 021603 022016 022428 022841 023252 023664 024075 4486 4896 412 106 5306 5715 6124 6533 6942 7350 7757 8164 8571 8978 408 107 9384 9789 030195 030600 031004 031408 031812032216032619033021 404 108 033424033826 4227 4628 5029 5430 5830 6229 6629 7028 400 109 7426 7825 8223 8620 9017 9414 9811 040207 040602|040998 397 110 041393 041787 042182042575 042969 043362 043755 044148 044540044931 393 111 5323 5714 6105 6495 6885 7275 7664 8053 8442 8890 $90 112 9218 9606 9993050380 050766051152 051538051924 052309 052694 386 113 053078 053463 053846 4230 4613 4996 5378 5760 6142 6524 $83 114 6905 7286 7666 8046 8426 8805 9185 9563 9942 060320 379 115 060698061075061452061829062206 062582062958 063333 063709 4089 376 116 4458 4832 5206 5580 5953 6326 6699 7071 7443 7814 373 117 8186 8557 8928 9298 9668 070038 070407 070776 071145071514 370 118 071882 072250072617072985 073352 3718 4085 4151 4816 5182 366 119 5547 5912 6276 6640 7004 7368 7731 8094 8457 8819 363 120 079181 079543 079904 080266 080626 080987 081347 081707 082067 082426 360 121 082785 083144 083503 3861 4219 4576 4934 5291 5647 6004 $57 122 6360 6716 7071 7426 7781 8136 8490 8845 9198 9552 355 123 9905 090258 090611 090963 091315 091667 092018|092370|092721093071 352 124 093422 3779 4122 4471 4820 5169 5518 5866 6215 6562 349 125 6910 7257 7604 7951 8297 8644 8990 9355 9681 100026| 346 126 100370 100715 101059 101403 101747 102090 102434 102777 105119 3462 343 127 3801 4146 4487 4828 5169 5510 5851 6191 6591 6870 341 128 7210 7549 7888 8227 8565 8903 9241 9578 9916110259 338 129 110590 110926 111262 111598 111934 112270 112605112940 113275 3609 335 130 113943 114277114011114944115278 115610 115943 116276 116608 116940 332 131 7271 7603 7934 8265 8595 8926 9256 9586 9915120245 330 132 120574 120903121231 121560121888 122216 122544 122871|123198 3525 928 133 3852 4178 4504 4830 5156 5481 5806 6131 6456 6781 325 7105 7429 7752 8076 8399 8722 9045 9368 9690 130012 325 135 130334 130655 130977 131298 131619 131939 132260 132580|132900 3219 321 136 3539 3858 4177 4496 4814 5133 5451 5768 6086 6403 318 137 6721 7037 7354 7670 7987 8303 8618 8934 9249 9564 316 138 9879 140194 140508140822141136 141450 141763 142076142389|142702| $14 139 143015 3327 3639 39511 4263 4574 4885 5196 5507 5818 311

134

3205

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155 190332190612190892|191171|191451191730 192010|192289|192567 2846 279 156 3125 3403 3681 3959 4237 4514 4792 5069 5346 5629 278 157 5900 6176 6452 6729 7005 7281 7556 7832 8107 8382 276 158 8657 8932 9206 9481 9755 200029 200303 200577|200850|201124 274 159 201397 201670201943 202216 202488 2761 8033 3305 8577 3848 272 3

No. 0 | 1 | 2

4 1 56

7 1

8

9 Diff.

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