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 .368350

the next greater logarithm

868409

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

...

rithms, 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

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

TABLE I.

LOGARITHMS OF NUMBERS.

No. Log. No. Log. No. Log. No.

Log. No. Log

16 1.204120 36 1.556302 56 1.748188 76 17 1.230449 37 1.568202 57 1.755875 77 38 1.579784 58 1.763428 78

18

1.880814 96 1.982271
1.886491 97 1.986773
1.892095 98 1.991226
1.897627
99 1.995635
1.903090 100 2.000000

100 000000000434 000868 001301 001734 002166 002598 003029 003460003891 101 0043210047511005180005609 006038 006466006894 007321 007748 008174 102 008600009206 009451 009876 010300010724 011147 011570 011993 014415 103 012837 013259 013680014100 014520014940015360015779 016197 016615 104 017033 017451 017868 018284-018700 019116019532019947 020361 020775 105 021189/021603 022016 022428 022841023252 023664024075024486 024896 106 025306 025715 026124 026533-026942 027350027757 028164 028571 028978 107 029384/029789030195030600031004 031408031812032216032619 033021 108 033424033826 034227 034628,035029 035430035830036229 036629 037028 109 037426037825038223 038620039017039414039811 040207 040602040998 110 041393/041787042182 042575 042969 043362 043755044148 044540044931 111 045323 045714 046105 046495 046885 047275 047664 048053048442048830 112 049218 049606,049993050380 050766 051152051538051924 0523091052694 113 053078 053463053846 054230 054613 054996055378055760056142056524 0114 056905057286057666 058046 058426058805059185059563 059942 060320 115 060698 061075061452061829 062206 062582 062958 063333 063709 064083 116 064458 064832065206065580065953 066326 066699 067071 067443 067814 117 068186068557 068928 069298 069668 070038 070407 070776 071145 071514 118 071882 072250 072617072985.073352073718074085074451074816 075182 119 075547 075912 076276 076640077004 077368 077738078094078457 078819 120 079181 079543 079904 080266 080626 080987081347 081707082067082426 121 082785 083144083503083861 084219 084576084934085291 085647086004 122 086360 086716 087071087426 087781088136 088490 088845 089198089552 123 089905 090258 090611 090963 091315 091667 092018 092370 092721093071 124 093422 093772 094122 094471 094820095169 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 109916 110253 129 110590 110926 111262 111598 111934 112270 112605 112940 113275 113609 130 113943 114277 114611 114944 115278 115610 115943 116276 116608 116940 131 117271 117603 117934 118265 118595 118926 119256 119586 119915 120245 132 120574 120903 121231 121560 121888 122216 122543 122871 123198 123525 133 123852 124178 124504 124830 125156 125481 125806 126131 126456 126781 134 127105 127429 127752 128076 128399 128722 129045 129368 129690 130012 135 130334 130655 130977 131298 131619 131939 132260 132580 132900 133219 136 133539 133858 134177 134496 134814 135133 135451 135768 136086 136403 137 136721 137037 137354 137670 137987 138303 138618 138934 139249 139564 138 139879 140194 140508 140822 141136 141450 141763 142076 142389 142702 139 143015 143327 143639 143951 144263 144574 144885 145196 145507 145818 140 146128 146438 146748 147058 147367 147676 147985 148294 148603 148911 141 149219 149527 149835 150142 150449 150756 151063 151370 151676 151982 142 152288 152594 152900 153205 153510 153815 154119 154424 154728 155032 143 155336 155640 155943 156246 156549 156852 157154 157457 157759 158061 144 158362 158664 158965 159266 159567 159868 160168 160468 160769 161068 145 161368 161667 161967 162266 162564 162863 163161 163460 163757 164055 146 164353 164650 164947 165244 165541 165838 166134 166430 166726 167022 147 167317 167613 167908 168203 168497 168792 169086 169380 169674 169968 148 170262 170555 170848 171141 171434 171726 172019 172311 172603 172895 149 173186 173478 173769 174060 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 152 181844 182129 182415 182700 182985 183270 183554 183839 184123 184407 153 184691 184975 185259 185542 185825 186108 186391 186674 186956 187239 154 187521 187803 188084 188366 188647 188928 189209 189490 189771 190051 155 190332 190612 190892 191171 191451 191730 192010 192289 192567 192846 156 193125 193403 19368 1 193959 194237 194514 194792 195069 195346 195623 157 195900 196176 196452 196729 197005 197281 197556 197832 198107 198382 158 198657 198932 199206 199481199755 200029 200303 200577 200850 201124

159 201397 201670 201943 202216 202488 202761203033 203305 203577 203848

160 204120 204391 204662 204933 205204 205475 205745 206016 206286 20655

161 206826 207095 207365 207634,207903 208172 208441 208710 208978 20914 162 209515 209783 210051 210318 210586 210853 211120 211388 211654 211911 163 212188 212454 212720 212986 213252 213518 213783 214049 214314 2145 164 214844 215109 215373 215638 215902 216166 216430 216694 216957 217221 165 217484 217747 218010 218273 218535 218798 219060 219322 219584 219840 166 220108 220370 220631 220892 221153 221414 221675 221936 222196 22245 167 222716 222976 223236 223496 223755 224015224274 224533 22479222505 168 225309 225568 225826 226084 226342 226600 226858 227115 227372 227630 169 227887 228144 228400 228657 228913 229170 229426 229682 229938 230197 170 230449.230704 230960 231215 231470 231724 231979 232233 232488 232743 171 232996 233250 233504 233757 234011 234264 234517 234770 235023 235270 172 235528 235781236033 236285 236537 236789 237041 237292 237544 237795 173 238046 238297 238548 238799 239049 239299 239550 239800 240050 240300 174240549/240799 241048 241297 241546 241795 242044 242293 242541 242790 175 243038 243286 243534 243782 244030 244277 244524 244772 245019 245368 176 245513 245759 246006 246252 246499 246745 246991 247236 247482 24371 177 247973-248219 248464 248709 248954 249198 249443 249687/2499322501 178 250420/250664 250908 251151 251395 251638 251881252125 252367 252610 179 252853 253096 253338 253580 253822254064 254306 254548 254790 255030 180 255273 255514255755255996256236256477256718 256958 257198 257439 181 257679 257918 258158 258398 258637 258877 259116 259355 259594 25983 182 260071 260310 260548 260787 261025 261263 261501 261738 261976 261114 183 262451 262688 262925 263162263399 263636 263873 264109 264345 264588 184 264818 265054 265290 265525 265761 265996 266232 266467 266702 26693 185 267172 267406 267641 267875 268110 268344 268578 268812 269046 2691 186 269513 269746 269980 270213 270446 270679 270912 271144 274377 27160 187 271842 272074 272306 272538 272770 273001 273233273464 273696 273927 188 274158 274389 274620 274850 275081 275311 275542 275772 276002 27613189 276462 276691276921 277151 277380 277609 277838 278067 278206 278525 190 278754 278982 279210 279439 279667 279895 280123 280351 280578 28080 191 281033 281261 281488 281715 281942 282169 282395 282622 282849 1850 192 283301 283527 283753 283979 284205 284431 284656 284882 285107 28332 193 285557 285782 286007 286232286456 286681 286905 287130 287354 287578 194 287802 288025 288249 288473 288696 288920 289143 289366 289589289812 195 290035 290257 290480 290702 290925 291147 291369 291591 291813092034 196 292256 292478 292699 292920 293141 293363 293583 293804 294025 1944 197 294466 294687 294907 295127 295347 295567 295787 296007 296226 3964 198 296665 296884 297104 297323 297542 297761 297979 298198 2984161298635 199 298853 299071 299289 299507 299725 299943 300161300378 300,95/300813 200 3010301301247 301464 301681 301898 302114 302331 302547 302764 301080 201303196 303412 303628 303844 304059 304275 304491 304706 304923051 202 305351 305566 305781 305996 306211 306425 306639 306854-307068 307232 203 307496 307710 307924 308137 308351/308564 308778 308991309204 3094 204 309630 309843 310056 310268 310481 310693 310906 311118 31133031154 205 311754 311966 312177 312389 312600 312812 313023 383234 313445 3136 206 313867 314078 314289 314499 314710 34920 315130 315340 315550 315760 207 3159701316180 316390 316599 316809 317018 317227 3174361317645317854 208 318063318272 318481 318689 318898 319106 319314 319522319730 319918 209 320146 320354 320562 320769 320977 321184 321391 321598,321805 322012 210 322219 322426 322633 322839 323046 323252 323458 323665,3238713140 324282 324488 324694 324899 325105 325310 325516 325721 325926,326131 326336 326541 326745 326950 3271551327359 327563 3277671327972328176 213 328380 328583 328787 328991 329194 329398 329601 329805.330008 330211

211

212

214

215

330414 330617 330819

331022

331225

331630

332438 332640332842-333044 333246:333447 333649

216 334454 334856 335056 335257 335458 335658 217 336460 336660 336860 337060 337260 337459 337659

218 338456 338656 338855 339054 339253 339451 339650 219340444 340642 340841 341039 341237 341435 341632

0

1

2

3

4

5

6

332236 333850 33405 334253 335859 336059 336260 337858338058/338257

341830342028-342215

9

220 342423 342020 342817 343014 343212 343409 343606 343802 343999 344196 221 344392 344589 344785 344981 345178 345374 345570 345766 345962 346157 222 346353 346549 346744 346939 347135 347330 347525 347720 347915/348110 223 348305 348500 348694 348889 349083 349278 349472 349666 349860 350054 224 350248 350442 350636 350829 351023 351216 351410 351603 351796 351989 225 352182 352375 352568 352761 352954 353 47 353339 353532 353724 353916 226 354108 354301 354493 354685 354876 355068 355260 355452 355643 355834 227 356026 356217 356408 356599 356790 356981357172 357363 357554 357744 228 357935 358125 358316 358506 358696 358886 359076 359266 359456 359646 229 359835 360025 360215 360404 360593 360783 360972 361161 361350 361539 230 361728 361917 362105 362294 362482 362671 362859 363048 363236 363424 231 363612 363800 363988 364176 364363 364551 364739 364926 365113 365301 232 365488 365675 365862 366049 366236 366423 366610 366796 366983 367169 233 367356 367542 367729 367915 368101368287 368473 368659 368844 369030 234 369216 369401 369587 369772 369958 370143 370328 370513 370698 370883 239 371068 371253 374437 371622 371806 371991 372175 372360 372544 372728 236 372912 373096 373280 373464 373647 373831 374015374198 374382374565 237 374748 374932 375115 375298 375481 375664 375846 376029 376212 376394 238 376577 376759 376942 377124 377306 377488 377670 377852 378034 378216 239 378398 378580 378761 378943 379124 379306 379487 379668 379849 380030 240 380211 380392 380573 380754 380434 381115 381296 381476 381656 381837 241 382017 382197 382377 382557 382737 382917 383097 383277 383456 383636 242 383815 383995 384174 384353 384533 384712 384891 385070 385249 385428 243 385606 385785 385964386142 386321 386499 386677 386856 387034 387212 244 387390 387568 387746 387923 388101 388279 388456 38863438881138989 245 389166 389343 389520 389697 389875 390051 390228 390405 390582 390759 246 390935 391112 391288 391464 391641 391817 391993 392169 392345 392521 247 392697 392873 393048 393224 393400 393575 393751 393926 394101 394277 248 394452 394627 394802 394977 395152 395326 395501 395676 395850 396025 249 396199 396374 396548 396722 396896 397070 397245 397418397592 397766 250 397940 398114 398287 398461 398634 398808 398981399154 399327 399501 251 399674 399847 400020 400192 400365 400538 400711 400883 401056 401228 252 401400/401573 401745 401917 402089 402261 402433 402605 402777 402949 253 403120 403292 403464 403635 403807 403978 404149 404320 404492 404663 254 404834 405005 405175405346 405517 405688 405858 406029 406199 406370 255 406540 406710 406881 407051 407221 407391 407561 407731407900 408070 256 408240 408410 408579 408749 408918 409087 409257 409426 409595 409764 257 409933 410102 410271 410440 410608 410777 410946 411114 41128341145' 258 411620 411788 411956 412124 412292 412400 412628 412796 412964413132 259 413300 413467 413635 413802 413970 414137 414305 414472414639 414806 260 414973415140415307 415474 415641415808 415974 416141 416308 416474 261 416640 416807 416973 417139 417306 417472 417638 417804 417970418135 262 418301 418467418633 418798 418964 419129 419295 419460 419625 419791 263 419956 420121 420286 420451420616 420781 420945 421110 421275 421439 264 421604 421768 421933 422097 422261 422426 422590 422754 422918 423082 265 423246423410 423573 423737 423901 424064 424228 424392 424555 424718 266 424882425045 425208 425371 425534 425697 425860 426023 426186 426340 267 426511 426674 426836 426999 427161 427324 427486 427648 427811 427973 268 428135 428297 428459/428621 428782 428944 429106 429268 429429 429591 269 429752 429914 430075 430236 430398 430559 430720 430881 431042/431/03 270 431364 431525 431685 431846 432007 432167 432328 432488 432649 432804 271 432969/433129433290 433450 433610 433770 433930 434090 434249 434409 272 434569434728 434888 435048 435207 435366 435526 435685 435844 436003 273 436163 436322 436481 436640 436798 436957 437116 437275 437433 437592 274 437751430909 438067 438226 438384 438542 438700 438859 439017 439175 275-439333 #39491 439648 439806 439964 440122 440279 440437 440594 440757 276 440909 441066 441224 441381 441538 441605 441852 442009 442166 442323 277 442480 442636 442793 442950 443106 443263 443419443576 443732 44385 278 444045 444201 444357 444513 444669 444825 444981445137 445293 445448 279 445604 445760 445915 446071 446226 446382 446537 446692 446848 44700