LOGARITHMS OF NUMBERS.

820 913814 913867 913920 913973 914026 914079 914131 914184 914237 914290 821 914343 914396 914449 914502 914555 914608 914660 914713 914766 914819 822 914872 914925 914977 915030 915083 915136 915189 915241 915294 915347 823 915400 915453 915505 915558 915611 915664 915716 915769915822 915874 824 915927 915980 916033 916085 916138 916191 916243 916296 916349 916401 825 916454 916507 916559 916612 916664 916717 916770 916822 916875 916927 826 916980 917033 917085 917138 917190 917243 917295 917348 917400 917453 827 917505 917558 917610 917663 917715 917768 917820 917873 917925 917978 828 918030918083 918135 918188 918240 918292 918345 918397 918450 918502 829 918555918607 918659 918712918764 918816 918869 918921 918973 919026 830 919078 919130 919183 919235 919287 919340 919392 919444 919496 919549 831 919601 919653 919705 919758 919810 919862 919914 919967 9200 19 920071 832 920123 920175 920228 920280 920332/920384 920436 920489 920541 920598 833 920645 920697 920749 920801 920853 920906 920958 9210:0 921062 921114 834 921166 921218 921270 921322 921374 921426 921478 921530 921582 921634 835 921686 921738 921790 921842 921894 921946 921998 922050 922102 922154 836 922206 922258 922310 922362 922414 922466 922518 922570 922622 922674 837 922725 922777 922829 922881 922933 922985 923037 923088 923140 923192 838 923244 923296 923348 923399 923451 923503 923555 923607 923658 923710 839 923762 923814 923865 923917 923969 924021 924072 924124 924176 924228 840 924279 924331 924383 924434 924486 924538 924589 924641 924693 924744 841 924796 924848 924899 924951 925002 925054 925106 925157 925209 925260) 842 925312 925364 925415 925467 925518 925570 925621 925673 925724 925776 843 925828 925879 925931 925982 926034 926085 926137 926188 926239 926291 844 926342 926394 926445 926497 926548 9266c0 926651 926702 926754 926805 845 926857 926908 926959 927011 927062 927114 927165 927216 927268 927319 8293 9283451 846 927 370 927422 927473 927524 927576 927627 927678 927730 927781 927832 847 927883 927935 927986 928037 928088 928140 928191/928242/9282 848 928396 928447 928498 928549 928601 928652 928703 928754 928805 928856 849 928908 928959 929010 929061 929112 929163 929214 929266 929317 929368 850 929419 929470 929521929572 929623 929674 929725 929776 929827 929878 851 929930 929981 930032 930083 930134 930185 930236 930287 930338930389 852 930440 930491 930541 930592 930643 930694 930745 930796 930847 930898) 853 930949 931000 931051 931102 931153 931203 931254 931305 931356 931407 854 931458 931509 931560 931610 931661 931712 931763 931814 931864 931915 855 931966 932017 932068 932118 932169 932220 932271 932321 932372 932423 856 932474 932524 932575 932626 932677 932727 932778 932829 932879 932930 857 932981 933031 933082 933133 933183 933234 933285 933335 933386 933437 858 933487 933538 933588 933639 933690 933740 933791 933841933892 933943 859 933993 934044 934094934145 934195 934246 934296 934347 934397 934448 860 934498 934549 934599 934650 934700 934751 934801 934852 934902 934953 861 935003 935054 935104 935154 935205 935255 935306 935356 935406 935457 862 935507 935558 935608 935658 935709 935759 935809 935860 935910 935960 863 936011 936061 936111 936162 936212 936262 936313 936363 936413 936463 864 936514 936564 936614 936664 936715 936765 936815 936865 936916 936966 865 937016 937066 937116 937167 937217 937267 937317 937367 937418 937468 866 937518 937568 937618 937668 937718 937769 937819 937869 937919 937969 867 938019 938069 938119 938169 938219 938269 938319 938370 938420 938470 868 938520 938570 938620 938670 938720 938770 938820 938870 938920 938970 869 939020 939970 939120 939170 939220 939270 939319 939369939419 939469 870 939519 939569 939619 939669 939719 939769 939819 939868 939918 939968 871 940018 940068 940118 940168 940218 940267 940317 940367 940417 940467 872 740516 940566 940616 940666 940716 940765 940815 940865 940915 940964 873 941014 941064 941114 941163 941213 941263 941313 941362 941412 941462 874 941511 941561 941611 941660 941710 941760 941809 941859 941909 941958 875 942008 942058 942107 942157 942206 942256 942306 942355 942405 942454 876 942504 942554 942603 942653 942702 942752 942801 942851 942900 942950 877 943000 943049 943099 943148 943198 943247 943297 943346 943396 943445 878 943494 943544 943593 943643 943692 943742 943791 943841 943890 943939 879 943989 944038194 8944137 944186 944236 944285 944335 944384 944433

880 944483 944532 944581 944631 944680 944729 944779|944828|944877|944927 881 944976 9450251945074 945124 945173 945222 945272945321 945370945419 882 945469 945518 945567 945616 945665 945715 945764 945813945862 945911 883 945961 946010 946059 946108 946157 946207 946256 946305 946354 946403 884 946452 946501 946550 946600 946649 946698 946747 946796 946845 946894 885 946943 946992 947041 947090947139 947189 947238 947287 947336 947385 886 947434 947483 947532 947581 947630 947679 947728 947777 947826 947875 887 947924 947973 948021 948070 948119 948168 948217 948266 948315 948364 888 948413 948462 948511 948560 948608 948657 948706 948755 948804 948853 889 9489029489511948999 949048 949097949146 949195 949244 949292 949341 890 949390 949439 949488 949536 949585 949634 949683 949731 949780949829| 891 949878 949926 949975 950024 950073 950121 950170950219 950267 950316 892 950365 950413950462 950511|950560 950608950657|950705950754 950803 893 950851950900950949 950997 951046 951095 951143 951192 951240 951289| 894 951337 951386 951435 951483 95532 951580951629 951677 951726 951774 895 951823 951872 951920 951969 952017 952066|952114 952163 952211 952259 896 952308 952356 952405 952453 952502 952550 952599 952647 952696 952744 897 1952792952841.952889 952938952986 953034 953083 953131 953180953228 898 953276 953325953373 953421 953470 953518 953566 953615 953663 953711 899 953760 953808 953856 953905 953953 954001 954049 954098 954146 954194 900 954242 954291 954339954387 954435954484 954532954580 954628 954677 901 954725 954773 954821 954869 954918 954966 955014955062955110955158 902 1955206.955255 955303 955351 955399 955447 955495|955543 955592955640 903 955688 955736 955784 955832 955880 955928 955976 956024 956072 956120 904 956168 956216 956264 956312 956361 956409 956457 956505 956553 956601 905 956649 956697 956745956792956840956888 956936 956984 957032 957080 906 957128 957176 957224 957272957320957368 957416957464 957511 957559 907 957607 957655 957703 957751 957799 957847 957894 957942 957990 958038 908 958086 958134 958181 958229 958277 958325 958373 958420 958468|958516 909 1958564 958612 958659 958707 958755 958803 958850958898 958946 958994 910 959041.959089 959137 959184 959232|959280 959328|959375 9594-31959471 911 959518 959566 959614959661 959709 959757 959804 9598521959900 959947 912 959995 960042 960090 960138 960185 960233 960281 960328 960376 960423 913 960471 960518 960566|960613 960661 960709 960756 9608041960851 960899 914 960946 960994 961041 961089 961136 961184 961231 961279 961326 961374 915 961421 961469 961516 961563 961611 961658 961706961753 961801 961848 916 961895 961943 961990 962038 962085 962132 962180962227 962275 962322 917 962369 962417 962464 962511|962559 962606 962653 962701 962748 962795 918 962843 962890 962937 962985|963032|963079 963126|963174 963221 963268 919 963315 963363 963410 963457 963504 963552 963599 963646 963693 963741 920 963788 963835963882 963929 963977 964024 964071 964118 964165 964212 921 964260 964307 964354 964401 964448 964495 964542 964590 964637 964684 922 964731 964778 964825 964872 964919 964966 965013 965060 965108 965155 923 965202 965249 965296 965343 965390 965437 965484 965531 965578 965625 924 965672 965719 965766 965813 9658601965907 965954 966001 966048 966095 925 966142 966189 966236 966283 966329 966376 966423 966470 966517966564| 926 966611 966658 966705 966752 966798 966845 966892 966939|966986|967033 927 967080 967127 967173 967220 967267 967314 967361 967408 967454 967501 928 967548 967595 967642 967688 967735 967782 967829 967875967922 967969 929 968016 968062 968109/968156 968203 968249 968296 968343 968389 958436 930 968483 968530 968576 968623 968670 968716 968763 968810 968856 968903 931 968950 968996 969043 969090 969136 9691831969229 969276 969323 969369 932 969416, 969462 969509 969556 969602 969649 969695 969742 969788 969835 933 969882 9699281969975 970021 970068 970114 970161 9702071970254|970300 934 970347 970393 970440 970486 970533 970579 970626 970672 970719 970765 935 970812 970858 970904 970951 970997 97 10441971090 971137 971183 971229 936 971276 971322 971369 971415 971461 971508 971554 971600, 971647 971693 937 971740 971786 971832 971879 971925 971971 972018 972064 972110 9721561 938 972203 972249 972295 972342 972388 972434 972480 972527|972573 972619) 939 972666 972712 972758 972804 972851 972897 972943 972989 973035 973082

2

3

940 973128 973174 973220 973266 973313 973359 973405 973451 973497 973543 941 973590 973636 973682 973728 973774 973820 973866 973913 973959 974005 942 974051 974097 974143 974189 974235 974281 974327 974373 974420 974466 943 974512 974558 974604 974650 974696 974742 974788 974834 974880474926 944 974972 975018 975064 975110 975156 975202 975248 975294 975340 975386 945 975432 975478 975524 975570 975616 975661 97570 975753 975799 975845 946 975891 975937 975983,976029 976075 976121 976166 976212 976258 976304 947 976350 976396 976442 976487 976533 976579 976625 976671 976717 976762 948 976808 976854 976900 976946 976991 977037 977083 977129 977175 977220 949 977266 977312 977358 977403 977449 977495 977541 977586 977632 977678 950 977724 977769 977815 977861 977906 977952 977998 978043 978089 978135 951 978180 978226 978272 978317 978363978409 978454 978500 978546 978591 952 978637 978683 978728 978774 978819 978865 978911 978956 979002 979047 953 979093 979138 979184 979230 979275 979321 979366 979412 979457 979503 954 979548 979594 979639 979685 979730 979776 979821 979867 979912 979958 955 980003 980049 980094 980140 980185 980231 980276 980322 980367 980412 956 9804589805031980549/980594 980640 980685 980730 980776 980821 980867 957 980912 980957 981003 981048 981093 981139 981184 981229951275 981320 958 981365 981411 981456 981501 981547 981592 981637 981683 981728 981773 959 981819 981364 981909 981954 982000 982045 982090 982135 982181 982226 960 982271982316 982362 982407 982452 982497 982543 982588 982633 982678 961 982723 982769 982814 982859 982904 982949 982994 983045 983085 983130 962 983175 983220 983265 983310 983356 983401 983446 983491983536 983581 963 983626 983671 983716 983762 983807 983852 983897 983942 983987 984032 964 984077 984122 984167 984212 984257 984302984347|984392 984437 984482 965 984527 984572 984617 984662 984707|984752 984797 984842 984887 984932 966 984977 985022 985067 985112 985157 985202 985247 985292 985337 985382 967 985426 985471 985516 985561 985606|985651 985696 985741 985786 985830 968 985875 985920 985965 986010 986055 986100 986144 986189 986234 986279 969 986324 986369 986413 986453 986503 986548 986593 986637 986682 986727 970 986772 986816 986861 986906 986951 986995 987040 987085 987130 987174 971 987219 987264 987309 987353 987398 987443 987487 987532 987577 987622 972 987666 987711 987756 987800 987845 987890 987934 987979 988024 988068 973 988113 988157 988202 988247 988291 988336 988381 988425 988470 988514 974 988559 988603 988648 988693 908737 988782 988826 988871 988915 988960 975 989005 989049 989094 989138 989183 989227 989272 989316 989361 989405 976 989450 989494 989539 989583 989628 989672 989717 989761 989806 989850 977 989895 989939 989983 990028 990072 990117 990161 990206 990250 990294 978 990339 990383 990428 990472 990516 990561 990605 990650 990694 990738 979 990783 990827 990871 990916 990960 991004 991049 991093 991137 991182 980 991226 991270 991315 991359 991403 991448 991492 991536 991580 991625 981 991659 991713 991757 991802 991846 991890 991934 991970 992023 992067 982 992111 992156 992200 992244 992288 992333,992377 992421992465 992509 983 992553 992598 992642 992686 992730992774 992818 992863 992907 992951| 984 992995 993039 993083 993127 993172 993216 993260 993304 993348 993392 985 993436 993480 993524 993568 993613 993657 993701 993745 993789 993833 986 993877 993921 993965 994009994053|994097 994141 994185 994229994273 987 994317 994361|994405|994449994493 994537 994581|994625|994669 994713 988 994757 994801994845 994889 994933 994977 995021 995064 995108 995152 989 995196 995240 995284 995328 995372 995416 995460 995 504 995547 995591 990 995635 995679 995723 995767:995811 995854 995898 995942 995986 996030 991 996074 996117 996161 996205 996249 996293 996336 996380 996424 996468 992 996512 996555 996599 996643 996687 996730 996774 996818 996862 996905 993 996949 996993|9970371997080 997124 997168 997212 997255 997299 997343 994 997386 997430 997474 997517 997561 997605 997648 997692 997736 997779 995 997823997867 997910 997954 997998 998041 998085 998128 998172998216) 996 998259 998303 998346 998390 998434 998477 998521 998564 998608 998652 997 998695 998739|998782|998826|998869998913 998956 999000 999043 999087 998 999130999174 999218 999261 999305 999348 999392 999435 999478 999522 909 999565 999609 999652 999696 999739 999783 999826 999870 699913 999957 7

2

3

4

5

6.

Q

TABLE 2.

Logarithmic Sines, Tangents, and Secants.

This table contains the logarithmic, or, as they are sometimes called, the artificial sines, tangents, and secants, to each degree and minute of the quadrant, with their complements or co-sines, co-tangents, and co-secants, to six places of figures besides the index.

To find the Logarithmic Sine, Co-Sine, &c. of any Number of Degrees and Minutes.

If the given degrees be under 45, they are to be taken from the top, and the minutes from the left side column, opposite to which in that column with the name of the logarithm at the top, will be found the required logarithm. But if the degrees be more than 45, they will be found at the bottom of the page, and the minutes in the right side column; likewise the name of the logarithm is to be taken from the bottom of the page.

When the given degrees exceed 90, they are to be subtracted from 180 degrees, and the logarithm of the remainder taken out as before. Or the logarithmic sine, tangent, &c. of degrees more than 90, is the logarithmic co-sine, co-tangent, &c. of their excess above 90 degrees.

To find the Degrees and Minutes nearest corresponding to a given Logd rithmic Sine, Co--sine, &c.

Look in the column marked at the top or bottom with the name of the given logarithm, and when the nearest to it is found, the corresponding degrees and minutes will be those required. observing that when the name is at the top of the column, the degrees are to be taken from the top and the minutes from the left side column, but if the name is at the bottom, the corresponding degrees will be there likeWise, and the minutes in the right side column.

EXAMPLES.

The degrees and minutes corresponding to the

log. sine 9.265390 are 100 37'

The logarithmic sines, &c. taken out to degrees and minutes only are in general sufficiently accurate, but in some of the more rigid astronomical calculations, it is frequently necessary to take them out to the nearest second; when this is the case they are to be found in the following manner :

To find the sine, tangent, &c. of an arch expressed in degrees, minutes and seconds.

RULE.

Find the sine, tangent, &c. answering to the given degree and minute, and also that answering to the next greater minute; multiply the difference between them by the given number of seconds, and divide the product by 60; then, the quotient added to the sine, tangent, &c. of the given degree and minute, or subtracted from the co-sine, co-tangent, &c. will give the quantity required, nearly.

If the arch be less than three degrees, it will be necessary to use the following rule

To the arithmetical complement of the given degrees and minutes reduced to seconds, add the logarithm of the given degrees, minutes, and seconds, reduced to seconds, and the log.-sine, tangent, &c. of the given degrees and minutes, the sum, rejecting 10 from the index, will be the log.sine, tangent, &c. of the proposed number of degrees, minutes, and seconds.

To find the degrees, minutes, and seconds, answering to a given logarithmic sine, tangent, c.

RULE.

Find the degrees minutes and seconds answering to the next less loga rithmic sine, tangent, &c. which subtract from that given; multiply the remainder by 60, and divide the product by the difference between the next less and next greater logarithms, and the quotient will be the seconds to be annexed to the degrees and minutes before found.

If the given logarithm is that of the sine or tangent of a small archthen, to the arithmetical complement of the next less logarithm in the tables, add the given logarithm, and the logarithm of the degrees and minutes, in seconds, answering to the next less logarithm, the sum, rejecting radius, will be the logarithm of the number of seconds in the required arch.