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Er. Find the logarithmic sine of 19.5'.20".
Here 10.5'.20" 39204

log.

503286 And 10.5' - 3900"

log.

591065 Their difference is ...

9221 To which, add the logarithmic sine of 10.5" .. = 6.27 6614 Hence, the logarithmic sine of 1o.5'.204

= 8.278895 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 left. hand; but, should the title be at the bottom of the column, you bare the degrees at the bottom of the page, and the ininutes in the column on the right-hand.

EXAMPLES. 1. The log. sine..

9.457584 answers to 169.40'. 2. The log. tangent .. 10.533401 answers to 73° 45'.

The log. secant.......... 10.038598 answers to 239.48'. 4. The log. co-sine ........ 9.806406 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'. 0"
Also

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

200.91'.so 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 number 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" = 8.449102 Their difference is

5445 To which, add 19.1.0" = 3660"

log.

3.563181 Hence, 19.1.46" = 3706"

log.

3.108406 Thus, the log. tangent 8.251527 corresponds to 1o.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, it using them, is slould be rmembered to multiply and divide as in other whole numbers.

..

....c

SO

A

TABLE

CONTAINING THE

LOGARITHMS OF NUMBERS,

FROM 1 to 10,000.

TABLE I.

LOGARITHMS OF NUMBERS,

100

2.000000

No: 1

Log. 1.000000 N. Log. I N. I Log. I N. I

| N | Log. TNI Log. 1 0.000000 26 1.414973 51 1.707570 76 1.880814 20.301030 27 1.431364

27 1.431364 52 1.716003 77 1.886491 3 0.477121 28 1.447158 53 1.721276 78 1.892095 4 0.602060 29 1.462398 54 1.732394 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 82 1.913814 8 0.903090 33 1.518514 58 1.763428 83 1.919078 9 0.9542413 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 1.1461281 39 1.591065 64 1.806180 89 1.949390 15 1.17609) 40 1.602060 65 1.812913

65 1.812913 90 1.954243

16 1.204120 41 1.612784 66 1.819544 17 1.23044942 1.62324967 1.826075 13 1.255273 43 1.633468 68 1.832509 19 1.278754 44 1.643453 69 1.838849 20 1.301030 45 1.653213 70 1.845098

91 1.959041 92 1.963788 93 1.968483 94 1.973128 95 1.977724

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21 1.322219 46 1.662758 71 1.351258 96 1.982271 22 1.342423 47 1.672098 72 1.857332 97 1.986772 23 1.361728 48 1.681241 73 1.863323 98 1.991226 24 1.380211 49 1.690196 74 1.869232 99 1.995635 25 1.397940 50 1.698970 75 1.875061 100 2.000000

Log. | N | Loz. 1 N. I Log. 1 N. I Log 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.

LOGARITHMS OF NUMBERS.

6894)

109

No. 1000- 1600

Log. 000000 204120 No 1 0

2 13 4 5 16 17 | 8 | 9 | Dif. 100 100000/000434 00086800130 11001734002166 0025980030290034600038911 492 101 4921 4751 5180 5609 6038 6466

7321 7748) 81741 428 102 8600 9026 9451 9876010300 010724011147 011570011993,01 2415 424 103 012837013259 013680014100 4520 4940 5360 5779 6197 661 51 420 104 7033 7451 7868 8284 8700

9116 9532 9917 020361 020775 416 105 021189 021603 022016 022428 022841 023252029664 024075 4486 4896 412 106 5306 5715 6124 6533 69-12 73501 7757 8164 8571 8978 408 107

9384 9789 030195 0306001031004 031408 031812032216 032619033021 404 109 033424033826 4227 46281 5029 5430 5830 6229 6629 7028 400

7426 7825 8223 86201 9017 9414 9811 040207 040602040998 397 110 041393 041787042182042575042969043362043755 044148 0445401044931 993 111 5323 5714 6105 6495 6885 7275 7664 8053 8442 8890 890 112 9218 9606 9993050380050766 051152 051598051924 0523091052694 986 119 053078 053463053846 4230 4613 49965378 57601 6142 6524 383 114

6905 7286 7666 8046 8426 8805 9185 9563 9942060320 379 115 060698061075061452 061829 062206 062582062958063393063709 4089 376 116 4458 4832 5206 5580 5953 6326 6699 7071 74431 7814 79 117 8186 8557 8928 9298 9668 070038 070407 070776071145071514 370 118 071882072250 072617072985 073352 9718 4085 4151

4816 5182 366 119 5547 5912 6276 664N 7004 7368 7731 8094 8457 8819 369 120 079181 0795431079904 080266 080626 080987081947 081707 082067082426 560 121 082785083144083503 3861 4219

4576 4934 5291 5647 6004 557 1 22 6360 6716 7071 7426 7781 8136 8490 8845 9198 9552 355 129 9905090258 090611 090963091315 091667 0920180923701092721093071 859 124 1093422 3779 4122 4471 4820 5169 5518 5866 62151 6562 349 125 6910 7257 7604 7951 8297) 8644 89901 9355 9681100026 346 126 100370 100715101059101403 101747102090 102434 102777|1081191 3462 343 127 3801 4146 4487 4828 5169 5510 5851 6191

6591 6870 341 128 7210 7549 7888 8227 8565 8909 9241 9578) 99161 10259 338 129 110590 110926111262111598111934 112270112605112940 113275 3609 335 130 113943 114277 114011114944115278 115610115948 116276 116608/116940 332 131 7271 7603 7934 8265 8595 8926) 9256 9586 9915 120245 SSO 132 120574 120903121231|121560121888 122216122544 122871 123198 3525 928 133 9852 4178 4504 4830 5156 5481 5806 6131 6456 6781 995 134 7105 7429 7752 8076 8399 8722 9045 9968 9690 130012 329 135 130834130655 130977131298 131619131939 192260/132580132900 3219 321 136 3.599 3858 4177 4496 4814 5133

5451 5768 6086 640S 918 137 6721 7037 7354 7670 7987 8803 8618 8934 9249 9564! 316 138

9879140194 140508 ! 40822141136 141450 141763 142076 142389 142702 914 139 143015 3327 3639 3951 4265 4574

4885 5196 5507 5818

911 140 1461281146438 146748 147058 147367 14767614.9851148294 148603 148911

S09 141 9219 9527 9835 150142 150449 150756 151063151370 151676 151982 307 142 152288|152594152900 3205 3510 3815 4119

4424 4728 509x 305 145 5336 5640 5943 6246 6549 6852 7154 7457 77591 8061 SO3 144

8362 8664 8965 9266 9567 9868 160168160468|160759 161068 800 145 161368 161667 161967 162266 162564 169863 9161 34601 3757 40551 299 146

4353) 46501 4947 5244 5541 5898 6134 64301 6726 7622 295 147 7317 7613 7908 8203 8497 8792 9086

9380 9674 99681 294 148 170262170555170818171141171434171726 172019 172311172603172895 292 149 3186 3478 3769 480 4351! 4641 4932 5222 5512 5802 290 150 176091170381 176670 176959 177248 177536 177825178113178401|178689 288 151 8977 9264 9552 9839 180126180413 180699 180986|181272181558 287 159 181844 182129182415182700 2985 3270

3554 S839 4123 4407) 285 153 4691 4975 5259 5542 5325 6108 6391 6674 6956 72 9 283 154 7521 7808 8084 83661 8647 8928 9209 9490 9771190051 281 155 190332 190612190892:91171191451|191730 192010192289|192567 2846 279 3125 31031 3681 3959 4237 4514 4792 5069

5346 5689 278 1 57 50001 6176 6452 6729 7005 7261 7556 7832 8107 8982 276 158 8657 8932 9206 9481 9755 200029 200303 200577 200850201124 974 159 201397/201670 201 943 202216202488 2761 8083 3305 35771 3848

272 No. 1 0

2 1 3
4 | 5 1 61 7

8 1

9

| Dift

156

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No. 1

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No. 1600- -2200

Log 204120 342423
O 1 I 1

3 14 15 16 17 ! 8 | 9 | Diff. 160 20412020439120466-2 204933|205204 205475 205745 206016|206286206556 271 161 6826 7095 7365 7634 7903 8172 8441 8710 8978 9247) 269 162 9515 9783210051 2103181210586 210853 211121211388 211654 211921| 267 163

2720 2986 3252 3518 8783 4049 4314 4579 266 164 4844 5109 5373

5638 5902 6160 64301 6694 6957 7221 264 165 7484 7747 8010

8279 8535 8798 9060 9322 9584 9846262 166 220108 220370 2206311220892 221153 221414 221675 221936 222196 222456 261 167 2716 2976 3236 3496 3755 4015 4274 4533 4792 5051 259 168 5309 5568 5826 6084 6942 6600 6858 7115 7872 7630) 258 169 7887 8144 8400 8657 8913 9170 9426

9682 9938 230193 256 170 230449230704 230960 231215 231470 231724 231979 232233232488232742 255 171 2996 3250 350+ 3757

4011 4264 4517 4770 5023 5276 253 172 5528) 5781 6033 6285 6537 6789 7041 7292 7544 7795) 252 173

8046) 8297 8548 8799 9049 9299 9550 9800 240050 240300 250 174 240549 240799 241048 241297241546241795 242044 242293 2541 27901 249 175 3038 3286 3534 3782 4030 4277 4524 4772 5019 5266 247 176 5513 5759 6006 6252 6499 6745 6991 7236 7482 7728 246 177 7973 8219 8464 8709 8954 9198 9443 9087 9932 250176 245 178 250420 250664250909 251151 251395 251638 251881 252125 252367 2610 243 179 2853 3096 3338 3580 3822 4064 4306 4548 4790 5031 242 180 255273255514255755 255996256236 256477 256718 256958 257198 257439 241 181 7679 7918 8158 8998 8637 8877 9116 9355 9594 9838 239 182 260071 260910 260548|260787|261025 261263 261501 261738 261976 262214 238 183 2451 2688 2925 3162 3399 3636 3873 4109 4345 4582 237 184 4818 5054 5290 5525 5761 5996 6232 6467 6702 69371 295 185 7172 7406 7641 7875 8110 8944 8580 8812 9046 92791 234 186 9513 9746 9980270213 270446 270679 2709122711442713771271609 233 187 271842 272074 272306 2538 2770 3001 3233 9464 9696 3927] 232 188 4158 4389 4620 4850 5081 5311 5542 5772 6002 6232 230 189 6462 6691 6921 7151 7380 7609 78981 8067 8296 8525 229 190 278754278982279210279439279667279895|280129 280351 280578 280806 228 191 1281033 281261281488 281715 281942 282169/ 2395 2622 2849 3075) 227 192 8901 $527

3753 3979 4205 44311 4656 4882 5107 5332 226 193 5557 5782 6007 6232 6456 6681 6905 7180 7354 7578 225 194 7802 8025 8249 8473 8696 8920 9143 9366 9589 9812 223 195 290035290257 290480 290702 290925 291147 291369 291 591 291813 292034 222 196 2256 2478 2699 2920 3141 3363 3583 3804 4025 4246 221 197 4466 4687 4907 5127 5947 5567 5787 6007 6226 6446 220 198 6665 6884 7104 7323 7542 7761 79791 8198 8416 8695 219 8853 9071 9289

9507 9725 9949 300161 300378 300595 300813 218 200 301030 301247 301464 301681 301898 302114302391 302547 302764 302980 217 201 3196 9412 3628 3844 4059 4275 4491 4706 4921 5136 216 202 5351 5566 5781 5996 6211 6425 6699 6854 7068 7282 214 203 74961 7710 7924 8137 8851 8564 8778

199

8991 9204 9417 218 204 9630 9843 310056 310268 310481 310693 310906311118311330311542 212 205 311754 311966 2177 2389 2600 2812 9023 9234 9445 3656 211 206 3867 4078 4289 4499 4710 4920 5130 5340 5550 5760 210 207 5970 6180 6390 6.599 6809 7018 7227 7456 7645 7854209 208 80631 8272 8481 8689 8898 9106 9814

9522) 9730 9938 208 209 320146 320354 320562 320769320977321184921391 321598 321805 322012 207 210 322219 322425|322633322839 323046 323252323458 323665323871 324077 206 211 4282 4488 4694 4899

5105 5910 55161 5 21 5926 61311 205 2:2 6356 6541 6745

6950 7155 7359

7563 7767 7972 8176 204 213 8380 8583 8787 8991 9194 9398 9601 9805 350008333211 203 214 330414 S30617 930819 33102233 225 331427 331630'331832 2034 2236 202 215 2438

2842 3044 9246 3447 9649 3850 4051 4253 202 216 4454

4655
4856 5056 5257 5453

5658 58 59 6059 6260 201 217 6460

6660 6860 7050 7260 7459 7659 79581 8058 8257' 200 218 8456 8656 8855

9054 9253 94511 96501 9849-347)47,340246 199 219 341).44 340512 3108 II '941039 341237341135'311639'34!£30 2038

222; 198 0 I 1 2 3 4 5 6 17 18

1 Diff. A 2

2640

No 1

LOGARITHMS OF NUMBERS.

No. 2200 2800

Log. 342423 -447158 No. 1 0 1 1 21 S 1 4 1 5 1 6 17 18 19 | Diff. 220 342429 342620 942817 345014313212 S43409 943606 S43802 343999344196 197 221 4992 4589 4785 4981 5178 5374 55701 5766 5962 6157 196 222 6353 6549 67441 6999 7135 7330 75251 7720 7915 8110 195 223 8305

8500 8694 8889 9083 9278 9472 9666 9860 350054 194 224 350248 350442 850636 350829 851023 351216351410 351603 351796 1989 193 225 2182 2375 2568 61 2954 3147 3399 3532 3724 3916 193 226 4108 4301 4493 4685 4876 5068 52601

5452 5643 5894 192 227

6026 6217 6408 6599 6790 6981 7172 7369 7554 7744 191 228 7935 8125 8316 8506 8696 8886

9076 9266 9456 9646 190 229 9835360025 360215 360404 360593 360783 360972361161 361350 561 539 189 230 561728 361917 362105362294 562482 362671362859 363018 363236363424 188 231 3612 9800 3988 4176 4363 4551 4739 4926 5113

5901! 188 232 5488 5675 5862 6049 6236 6423 6610

6796 6989 7169 187 293 7356 7542 7729 7915

8101

8287 8473 8659 8844 9030 186 234 9216 9401 9587 9772 9958370143 370328 370513 370698370883 185 235 971068 371253 371437 371622 371806 1991

2175 2360 2544

2728 184 236 2912 3096 3280 3464 3647 3831 4015

4198 4382 4565 184 237 4748 4932 5115 5298 5481

5664

5846

6029 6212 238 6577 6759 6942 7124 7306 7488 7670

7852 8094 239 8398 8580 8761 8943 9124

9306

9487 9668 240 380217 380392 380573 380754 380934 381115381296 381476 381656 381837 18! 241 2017 2197 2377 2557 2737

3097

3456

3277
242
3815

4174
9995
4353 4533

4891 5070 243 5606

5964
5785
6142 6921

6677 244

6856
7390

7746
7568
7929 8101

8694 8811 245 9166

8456 9343 9520 9697 946 390935 391112 391288 391464391641 247 2697 2873 3048 9224 3400

1993

2169 248 4452 4627 4802 4977 5152

3751

3926 249 6199

6548
6974
6722

5676
6896

5501
7245

7418 7592 7766 174

6994 183

8216 189 9849 380030 181

3636 180 5249 5428 179 7034 7212 178

8989 178

2945 2521 176 4101 4277 176 5850 6025 175

2796 2964 3132 168 4472

4639 4806 167

2917 4712 6499

8279 9875 390051 3 90228 390405 390582 390759 172

1817 3575 5326

7071 250 397940 398114 398287 398461 398634 398808 398981 399154 399327 599501 179 9847 400020 400192 400365 400598 400711 400883401056 401228 179

2261 3978 5688 7391 9087 24601

4137 260 414973 415140 415307 415474 415641 415808 415974 416141 41630841647:17

7472 9956 428191 420286 420451 4206 16 420731 420945_421110 42127542149 150

2426 3901 4064 5594 5697 7161 7324 8782 8944

7906 8964

251 9674
252 401 400 401573 1745

1917

2089 253 3120

S292
3464 8635 8807

2439 2605

2777 2949 172 254 4834) 5005 5175 5346

5517

4149

4320 4492 4663 171 255 65401 6710 6881 7051 7221

5858 6029

6199 6370 171 256 8240 8410 8579 8749 8918

7561
7731

7900 8070 170 257

9257

9426 9595 9764 169 258 411620 1788 1956 2124

2292 259 33001 3467 3635 3802

39701

2628

4305 261 6640 6807 6973 7189 262 8301 8467 8699 8798

7638 263

9295 264 421604 1768 1999 2097

2261 265 3246 3410 3573 3737

2590 4882 5045 5208 5371

4228 267 6511 6674 6896 6999

5860 268 8135 8297 8459 8621

7486 269 9752

9106 9268 9914 430075430236 430398430559 430720 430881 431042 491203 161 271 270 431364 431525 431685 431846 4320074321671432328 482488 432649 432809 160

2969 3129 3290 3450 279 4569 4728 4888 5048

3950 4090 4249 973 6169 6922 6481 6640

5526 274 7751 7909 8067

8226 275 9333 9491 9648 9806

8700 8859 276 440909 441066 441224 441381 441 538

99641440122 440279 440437 440594 440752 158 277 2480 2696 2793 2950

1852 2009 278 4045 4201 4957 4513

3419

3732) 279 5604 5760 5915 6071

6848 0 1 1 1 2 | 3

7804 7970 8155 166 9460 9625 9791 165

266

2754 2918 9082 164 4392 4555 4718 164 6023 6186 6349 163 7648 7811 7979 169

9429 9591 169

3610 3770 52071 5366 6800 6957 888418542

4409 160 5685 5844 6003 159 7116 7275 7433 7592 159

9017 9175 158

3106 4669 6226

1695 92631 4825 6982

2166 2329 157 9576

9888 157 5137 5293

5448 156 6692

7003) 155

No. 1

4981
6537

5

9 | Did

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