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ed to the logarithm, answering to the first four figures, will be the fe quired logarithm, nearly.

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

to be multiplied by the remaining figures.

.368350

= 868409

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= 82

118 472

48/38

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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. 3.152580 gives the Nat. Numb. 1357

2.132580 gives
1.132580 gives
0.132580 gives

9.132580 gives

135.7

13.57

1.357

.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 number; then find the difference between the given and the next less loga

rithm's, to which annex as many ciphers as there are figures required above four in the natural number; which divide by the difference be tween the next less, and next greater logarithnis, and the quotient an nexed 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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No. Log. No. Log. No. Log. No.

1.785330 81 1.908485 1.792392

Log. (No. Log

1.612784 61

1.623249 62

82 1.913814

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1.633468 63

1.799341

4

0.602060 24 1.380211 44

1.643453 64

[blocks in formation]

1.653213 65

83 1.919078 1.806180 84 1.924279 1.812913 85

1.929419

[blocks in formation]

0.845098 27

1.431364 47

1.672098 67

1.826075

0.903090 28

1.447158 48 1.681241 68

1.832509

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0.954243 29 1.462398 49 ΤΟ 1.000000 30 1.477121 50

1.690196 69

1.838849

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67 90

88

89

1.934498 1.939519 1.944483 1.949390 1.954243

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51 1.707570 1.505150 52 1.716003 72 1.518514 53 1.724276 73

14 1.146128 34 1.531479 54 1.732394 74 1.176091 35 1.544068 55 1.740363 75

1.204120 36

1.2

1.255273

I.

1.748188 76 571-755875 77

1.579784 58 1.763428 78 1.278754 39 1.591065 59 1.770852 79 40 1.602060 60 1.778151 80

1.869232 94

1.903090 100 2.000000

[blocks in formation]

1.973128

1.875061 95 1.977724

1.880814 96 1.982271 1.886491 97 1.986773

1.892095 98 1.991226 1.897627 99 1.995635

No.

100

101

2

3

4

5

6

7

8 9 000000000434 000868 001301 001734 002166 002598 003029 003460003891 004321004751005180 005609 006038 006466 006894 007321 007748 008174 102 008600 009206 009451 009876 010300010724 011147 011570 011993 014415 103 012837 013259 013680/014100 014520014940 015360 015779 016197 016615 104 017033 017451 017868 018284 018700 019116 019532 019947 020361 020775 105 021189 021603 022016 022428 022841 023252023664024075 024486 024896 106 025306 025715 026124 026533 026942|027350027757028164 028571 028978 107 029384 029789 030195 030600 031004 031408 031812 032216032619 033021 108 033424033826 034227 034628,035029 035430035830036229036629 037028 109 037426 037825 038223 038620 039017 039414 039811 040207 040602 040998 110 041393 041787042182|042575 042969 043362 043755044148044540044931 III 045323 045714 046105 046495 046885 047275 047664 048053 048442 048830 112 049218 049606,049993050380 050766 051152 051538051924 052309,052694 113 053078 053463 053846 054230 054613 054996055378 055760 056142 056524 114056905057286 057666 058046 058426 058805059185059563059942 0603 20 115 060698 061075 061452 061829 062206 062582 062958 063333 063709 064083 116 064458 064832065206 065580 065953 066326 066699|067071 067443 067814 117 068186 068557 068928 069298 069668 070038 070407 070776 071145 071514 118 071882 072250 072617|072985.073352 073718 074085 074451 074816075182 119 075547 075912 076276 076640 077004 077368 077731 078094 078457 078819 120 079181 079543 079904 080266 080626 080987 081347 081707 082067 082426 121 082785 083144 083503 083861 084219 084576 084934085291 085647086004 122 086360 086716 087071 087426 087781 088136 088490 088845 089198 089552 123 089905 090258 090611 090963 091315 091667 092018 092370 092721 093071 124 093422 093772 094122 094471 094820 095169 095518 095866 096215 096562 125 096910 097257 097604 097951 098297 098644 098990099335 099681 100026 126 100370 100715 101059 101403 101747 102090 102434 102777 103119 103462 127 103804 104146 104487 104828 105169 105510 105851|106191 106531 106870 128 107210 107549 107888 108227 108565 108903 109241 109578 109916110253 129 110590 110926 111262 111598 111934 112270 112605 112940 113275 113609 130 113943 114277 114611114944 115278115610 115943116276 116608 116940 131 117271 117603 117934 118265 118595 118926 119256 119586 119915120245 132 120574 120903 121231 121560121888 122216 122543 122871 123198 123525 133 123852 124178 124504124830 125156 125481 125806 126131 126456 126781 134 127105 127429 127752 128076 128399 128722 129045 129368 129690 130012 135130334 130655 130977 131298 131619 131939 132260132580132900 133219 136 133539 133858 134177 134496 134814135133 135451 135768 136086 136403 *137 136721 137037137354 137670 137987 138303 138618138934139249 139564 138 139879 140194 140508 140822 141136 141450 141763 142076 142389 142702 139 143015143327 143639 143951 144263 144574 144885145196 145507 145818 140 146128 146438 146748 147058 147367 147676 147985 148294 148603 148911 141 149219 149527 149835150142 150449 150756 151063 151370 151676 151982 142 152288 152594 152900 153205 153510153815 154119 154424154728 155032 143 155336 155640 155943 156246 156549 156852157154 157457 157759 158061 144 158362 158664 158965159266 159567 159868 160168 160468 160769 161068 145161368 161667 161967 162266 162564 162863| 163161 163460 163757 164055 146 164353 164650 164947 165284165541 165838 166134 166430 166726 167022 147 167317 167613 167908 168203 168497 168792 169086 169380|169674 169968 148 170262 170555170848171141 171434171726 172019 172311172603172895 149 173186 173478 173769174060 174351 174641 174932 175222|175512 175802 150 176091 176381 176670 176959 177248 177536 177825 178113 178401 178689 151 178977 179264 179552 179839 180126 180413 180699 180986 181272 181558 152181844 182129 182415 182700 182985 183270 183554 183839 184123 184407 153 184691 184975 185259 185542 185825 186 108 186391 186674 186956 187239 154 187521 187803 188084 188366 188647 188928 189209 189490 189771 190051 155190332 190612190892|191171|191451|191730192010 192289|192567|192846 156 193125 193403 19368 || 193959 194237 194514 194792 195069 195346 195623 157 195900 196176 196452 196729 197005 197281 197556 197832198107 198382 158 198657198932 199206|199481199755 200029 200303 200577 200850 201124 159 201397 201670 201943 202216 202488 202761 203033 203305 203577 203848 2

0

5

6

7

8

9

No. 0

2

3 4 | 5 | 6 | 7 | 8 9 160 2041 20 204391 204662 204933 205204 205475 205745 206016 206286 2065 161 206826 207095 207365, 207634 207903 208172 208441 208710208978 20914 162 209515 209783 210051 210318 210586 210853 211120211388 211654 21192 163 212188 212454 212720 21 2986 213252 213518 213783 214049 214314/2145 164 214844 215109 215373 215638 215902 216166 216430 216694 216957 21721 165 217484 217747 218010 218273 218535 218798 219060 219322 219584 21984 166 220108 220370 220631 220892 221153 221414 221675221936 222196 2224 167222716 222976 223236 223496 223755 224015|224274224533|224792 225051 168 225309 225568 225826 226084 226342 226600 226858227115 227372227632 169 227887 228144 228400 228657 228913 229170229426229682229938|23019 170 230449.230704|230960 231215 231470 231724 231979 232233 232488 232745 171 232996 233250 233504 233757 234011 234264 234517 234770 235023235276 172 235528 235781 236033 236285 236537 236789|237041|237292 237544 237795 173 238046 238297 238548 238799 239049 239299 239550 239800|240050|240300 174 240549 240799241048 241297 241546 241795|242044|242293 242541|242750|| 175 243038 243286 243534 243782 244030 244277 244524 244772 245019245166|| 176 245513 245759 246006 246252 246499 246745 246991 247236 247482 24772 177 247973 248219 248464 248709|248954|249198 249443 249687 249932250176 178 250420 250664 250908 251151 251395 251638 251881 252125|252367|252613 179 252853 253096 253338253580 253822 254064 254306 254548|254790255031 180 255273 255514 255755 255996|256236|256477 256718|256958257198 257439 181 257679 257918 258158 258398 258637 258877 259116 259355 259594 25983 182 260071 260310 260548 260787 261025 261263|261501|261738 261976 183 262451 262688 262925 263162 263399 263636 263873 264109 264345 264582 184 264818 265054 265290 265525 265761 265996 266232 266467 266702 266417 185 267172 267406 267641 267875 268110 268344 268578268832 269046|264174| 186 269513 269746 269980 270213 270446 270679 270912 2711447137727160 187 2718421272074 272306 272538 272770 273001 273233 273464 273696|273927 188 274158 274389 274620 274850 275081 275311 275542 275772 27600227615 189 276462 27669 276921 277151 277380 277609|277838 278067 278206 178525 190 278754 278982 279210279439 279667 279895 280123280351 28057828080 191 281033 281261 281488 281715 281942 282169 282395 282622 282849 1850 192 283301 283527 283753 283979 284205 284431 284656 284882 2851072853352 193 285557 285782 286007 286232 286456 286681 286905 287130287354287578 194 287802 288025 288249 288473 288696 288920 289143 289366289589289812 195 290035 290257 290480 290702 290925 291147 291369 291591 29181304204 196 292256 292478 292699 292920 293141 293363 293583 293804294025 1944 197 294466 294687 294907 295127 295347 295567 295787 296007 29622629644 198 296665 296884 297 104 297323 297542 297761 297979 298198 298416/198639 199 298853 299071 299289 299507 299725 299943 500161300378 300,95/300813 200 301030301247 301464 301681 301898 302114302331 302547 302764 302080 201303196 303412 303628 303844 304059 304275304491 304706304922 30512 202 305351 305566 305781 305996 306211 306425 306639 306854 307068 203 307496 307710 307924 308137 308351 308564 308778 308991 309204300417 204 309630 309843 310056|310268 310481|310693 310906 311118 311330 205 311754 311966 312177 312389 312600 312812 313023 313234 313445 31363 206 313867 314078 314289 314499 314710 314920315130 315340 315550315760 207 315970 31618c 316390 316599 316809 317018317227 3174361317645 317854 208 318063 318272 318481 318689 318898, 319106 319314 319522 3197303199 209 320146 320354 320562 320769 320977 321184 321391 321598 321805 322012 210 322219 322426 322633 322839 323046 323252 323458 323665,323871 3240 211 324282 324488 324694 324899 325105 325310 325516 325721 325926 32613 212 326336 326541326745 326950 3271551327359 327563 32776713279721328170 213 328380 328583 328787 328991 329194 329398 329601329805.330008 330211 214 330414 330617 330819 331022 331225 331427 331630 331832 332034 332230 215 332438 332640332842 333044 333246:333447333649 333850 33405 334255 217 336460 336660 336860 337060 337260 337459 337659 337858 338058 33825 218338456 338656338855 339054 339253 339451 339650 339849 340047 340246 219 349444 340642 340841 341039 341237 341435 341632 341830 342028 342225

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