3x Xibrts 
 
 A book was presented to the Industrial Division 
 Library of Santa Barbara State College by 
 
 19 ^ry 
 
 Library Collection Sponsored by Pi SIGMA CHI
 
 I
 
 </
 
 THE ELEMENTS 
 
 OF 
 
 HYDRAULIC ENGINEERING 
 
 PREPARED FOR STUDENTS OF 
 
 THE INTERNATIONAL CORRESPONDENCE SCHOOLS 
 SCRANTON, PA. 
 
 Volume IV 
 
 TABLES AND FORMULAS 
 
 first Edition 
 
 SCRANTON 
 
 THE COLLIERY ENGINEER CO. 
 1808 
 
 441
 
 Entered according to the Act of Congress, in the year 1898, by THE COLLIERY 
 
 ENGINEER COMPANY, in the office of the Librarian of Congress, 
 
 at Washington. 
 
 BURR PRINTING HOUSE, 
 FRANKFORT AND JACOB STREETS, 
 
 NEW YORK.
 
 1 2 
 
 UNIVERSITY OF CALIFORNIA 
 ) 4- SANTA BARBARA COLLEGE LIBRARY 
 
 TABLES AND FORMULAS. 
 
 This volume contains all the principal Tables and 
 Formulas which are likely to be used by the student in 
 practice. They have been collected and placed in this 
 volume in order to make them convenient for ready refer- 
 ence, so that the student will not be obliged to hunt them 
 out in the preceding volumes. The number after each 
 formula is the same as the number following the same 
 formula in the text. '.
 
 4-4-3 
 
 
 TABLK 
 
 OP 
 
 COMMON LOGARITHMS 
 
 OK NUMBERS 
 
 From 1 to 1 0.OOO. 
 
 X. Log. 
 
 N. 
 
 Log. 
 
 N. 
 
 Log. 
 
 N. 
 
 Log. 
 
 N. 
 
 Log. 
 
 
 
 - 00 
 
 20 
 
 3^ 103 
 
 40 
 
 60 206 
 
 60 
 
 77 815 
 
 80 
 
 90 309 
 
 i 
 
 OO OOO 
 
 21 
 
 32 222 
 
 4i 
 
 61 278 
 
 61 
 
 78 533 
 
 81 
 
 90 849 
 
 2 
 
 30 103 
 
 22 
 
 34 242 
 
 42 
 
 62 325 
 
 62 
 
 79 239 
 
 82 
 
 91 381 
 
 3 
 
 47 712 
 
 23 
 
 36 173 
 
 43 
 
 63 347 
 
 63 
 
 79 934 
 
 83 
 
 91 908 
 
 4 
 
 60 206 
 
 24 
 
 38 021 
 
 44 
 
 64 345 
 
 64 
 
 80 618 
 
 84 
 
 92 428 
 
 5 
 
 69 897 
 
 25 
 
 39 794 
 
 45 
 
 65 321 
 
 65 
 
 81 291 
 
 85 
 
 92 942 
 
 6 
 
 77 8i5 
 
 26 
 
 4i 497 
 
 46 
 
 66 276 
 
 66 
 
 81 954 
 
 86 
 
 93 45o 
 
 7 
 
 84 510 
 
 27 
 
 43 136 
 
 47 
 
 67 210 
 
 67 
 
 82 607 
 
 8? 
 
 93 952 
 
 8 
 
 90 309 
 
 28 
 
 44 7i6 
 
 48 
 
 68 124 
 
 68 
 
 83 251 
 
 88 
 
 94 448 
 
 9 
 
 95 424 
 
 29 
 
 46 240 
 
 49 
 
 69 020 
 
 69 
 
 83 885 
 
 89 
 
 94 939 
 
 10 
 
 00 000 
 
 30 
 
 47 712 
 
 50 
 
 69 897 
 
 ro 
 
 84 5io 
 
 90 
 
 95 424 
 
 ii 
 
 04 139 
 
 3i 
 
 49 136 
 
 5i 
 
 70 757 
 
 71 
 
 85 126 
 
 91 
 
 95 904 
 
 12 
 
 07 918 
 
 32 
 
 50 515 
 
 52 
 
 71 600 
 
 72 
 
 85 733 
 
 92 
 
 96 379 
 
 13 
 
 " 394 
 
 33 
 
 5i 851 
 
 53 
 
 72 428 
 
 73 
 
 86 332 
 
 93 
 
 96 848 
 
 M 
 
 14 613 
 
 34 
 
 53 148 
 
 54 
 
 73 239 
 
 74 
 
 86 923 
 
 94 
 
 97 313 
 
 15 
 
 17 609 
 
 35 
 
 54 407 
 
 55 
 
 74 036 
 
 75 
 
 87 506 
 
 95 
 
 97 772 
 
 16 
 
 20 412 
 
 36 
 
 55 630 
 
 56 
 
 74 819 
 
 76 
 
 88 081 
 
 96 
 
 98 227 
 
 17 
 
 23 045 
 
 37 
 
 56 820 
 
 57 
 
 75 587 
 
 77 
 
 88 649 
 
 97 
 
 98 677 
 
 18 
 
 25 527 
 
 38 
 
 57 978 
 
 58 
 
 76 343 
 
 78 
 
 89 209 
 
 98 
 
 99 "3 
 
 19 
 
 27 875 
 
 39 
 
 59 106 
 
 59 
 
 77 085 
 
 79 
 
 89 763 
 
 99 
 
 99 564 
 
 20 
 
 30 103 
 
 40 
 
 60 206 
 
 60 
 
 77 8i5 
 
 80 
 
 90 309 
 
 100 
 
 OO OOO
 
 LOGARITHMS. 
 
 N. 
 
 L. o 
 
 i 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 P. 
 
 p. 
 
 
 100 
 
 oo ooo 
 
 043 
 
 087 
 
 130 
 
 173 
 
 217 
 
 260 
 
 303 
 
 346 
 
 389 
 
 
 
 
 
 101 
 
 432 
 
 475 
 
 518 
 
 56i 
 
 604 
 
 647 
 
 689 
 
 732 
 
 775 
 
 817 
 
 
 
 
 
 102 
 
 103 
 
 104 
 
 105 
 1 06 
 107 
 
 108 
 109 
 
 860 
 oi 284 
 
 703 
 
 02 Ilg 
 531 
 938 
 03 342 
 
 743 
 
 9^3 
 326 
 745 
 160 
 572 
 979 
 383 
 782 
 
 94S 
 368 
 787 
 
 202 
 
 612 
 *oig 
 423 
 
 822 
 
 988 
 410 
 828 
 243 
 653 
 *o6o 
 463 
 862 
 
 *030 
 452 
 870 
 284 
 694 
 
 *IOO 
 
 503 
 902 
 
 494 
 912 
 
 325 
 735 
 *i4i 
 543 
 941 
 
 *iiS 
 
 536 
 953 
 366 
 776 
 *iSi 
 583 
 981 
 
 A ibV 
 578 
 995 
 407 
 816 
 
 *222 
 
 623 
 
 *02I 
 
 "199 
 620 
 *036 
 449 
 
 857 
 
 *262 
 66 3 
 
 *o6o 
 
 242 
 662 
 *078 
 
 49 
 898 
 
 *302 
 
 703 
 
 *IOO 
 
 i 
 
 3 
 
 4 
 
 I 
 7 
 8 
 9 
 
 44 
 
 4-4 
 
 13.2 
 17.6 
 
 26'.4 
 30.8 
 35-2 
 39-6 
 
 43 
 
 s'i 
 
 12.9 
 17.2 
 
 25*8 
 
 30.1 
 
 34-4 
 38.7 
 
 42 
 
 4- 
 8. 
 
 12. 
 
 16. 
 
 25- 
 29. 
 33-6 
 37-8 
 
 110 
 
 04 139 
 
 179 
 
 218 
 
 258 
 
 297 
 
 336 
 
 376 
 
 415 
 
 454 
 
 493 
 
 
 
 
 
 in 
 
 112 
 H3 
 114 
 
 H5 
 
 116 
 117 
 118 
 119 
 
 532 
 922 
 05 308 
 690 
 
 06 070 
 446 
 819 
 07 188 
 
 555 
 
 57i 
 961 
 346 
 729 
 108 
 483 
 856 
 225 
 59i 
 
 610 
 999 
 385 
 767 
 145 
 52i 
 893 
 262 
 628 
 
 650 
 *o 3 8 
 423 
 805 
 183 
 558 
 930 
 298 
 664 
 
 689 
 
 *o?7 
 461 
 843 
 
 221 
 
 595 
 967 
 335 
 700 
 
 727 
 *H5 
 500 
 881 
 258 
 633 
 *oo 4 
 372 
 737 
 
 766 
 *I54 
 538 
 918 
 296 
 670 
 *o4i 
 408 
 773 
 
 80 5 
 *I 9 2 
 576 
 956 
 
 333 
 707 
 *07S 
 445 
 809 
 
 844 
 
 *2 3 I 
 
 614 
 
 994 
 371 
 
 744 
 *ii5 
 482 
 846 
 
 883 
 
 *26 9 
 
 652 
 
 *0 3 2 
 4 08 
 781 
 *I51 
 
 518 
 882 
 
 3 
 4 
 
 7 
 
 9 
 
 41 
 
 4.1 
 
 12.3 
 16.4 
 
 i : " 
 
 IG'.O 
 
 40 
 
 12.0 
 
 16.0 
 24.0 
 32.0 
 
 39 
 
 3-9 
 7 -8 
 
 15.6 
 19-5 
 23.4 
 27.3 
 31.2 
 35-1 
 
 120 
 
 918 
 
 954 
 
 990 
 
 *02 7 
 
 *o63 
 
 *99 
 
 *I35 
 
 *i 7 i 
 
 *20 7 
 
 *2 4 3 
 
 
 
 
 
 121 
 
 122 
 123 
 124 
 125 
 126 
 127 
 128 
 I2 9 
 
 08 279 
 636 
 991 
 09 342 
 691 
 10 037 
 380 
 721 
 it 059 
 
 314 
 
 672 
 
 *026 
 
 377 
 726 
 072 
 415 
 755 
 093 
 
 350 
 707 
 *o6i 
 412 
 760 
 106 
 449 
 789 
 126 
 
 386 
 
 743 
 *og6 
 447 
 795 
 140 
 483 
 823 
 1 60 
 
 422 
 
 778 
 
 *I32 
 
 482 
 830 
 175 
 517 
 857 
 193 
 
 458 
 814 
 *i6? 
 517 
 864 
 209 
 55i 
 890 
 227 
 
 493 
 849 
 
 *202 
 
 552 
 899 
 243 
 
 585 
 
 924 
 261 J 
 
 529 
 884 
 *2 3 7 
 587 
 934 
 278 
 619 
 958 
 294 
 
 565 
 92O 
 *272 
 621 
 9 68 
 3" 
 
 11 
 
 327 
 
 600 
 955 
 *3Q7 
 656 
 
 *00 3 
 
 346 
 687 
 
 *02 5 
 
 361 
 
 3 
 4 
 5 
 6 
 7 
 
 9 
 
 38 
 
 3-8 
 7.6 
 11.4 
 15-2 
 
 22^8 
 26.6 
 3-4 
 
 34-2 
 
 37 
 
 3-7 
 7-4 
 
 18.5 
 
 ag'6 
 33-3 
 
 36 
 
 3.6 
 7.2 
 
 14.4 
 
 21.6 
 
 25.2 
 28.8 
 32-4 
 
 130 
 
 394 
 
 428 
 
 461 
 
 494 
 
 528 
 
 56i 
 
 594 
 
 628 
 
 661 
 
 694 
 
 
 
 
 
 131 
 132 
 133 
 134 
 135 
 136 
 137 
 138 
 
 727 
 12 057 
 385 
 710 
 13 033 
 354 
 672 
 988 
 
 760 
 090 
 418 
 743 
 066 
 386 
 704 
 *oig 
 
 793 
 123 
 
 45o 
 775 
 098 
 418 
 735 
 *osi 
 
 826 
 156 
 483 
 808 
 130 
 450 
 767 
 
 *082 
 
 860 
 189 
 5i6 
 840 
 162 
 481 
 
 * 799 
 *H4 
 
 893 
 
 222 
 548 
 872 
 I 9 4 
 513 
 830 
 
 *I45 
 
 926 
 254 
 58i 
 905 
 226 
 545 
 862 
 *i76 
 
 959 
 287 
 613 
 
 937 
 
 258 
 577 
 893 
 
 *208 
 
 992 
 320 
 646 
 969 
 290 
 609 
 9 2 5 
 *239 
 
 *024 
 
 352 
 678 
 
 *OOI 
 
 322 
 
 640 
 
 956 
 
 *2?0 
 
 2 
 
 3 
 4 
 5 
 6 
 
 7 
 
 35 
 
 34 
 
 3-4 
 6.8 
 
 13-6 
 17.0 
 20.4 
 23.8 
 
 33 
 
 6.6 
 9-9 
 13.2 
 16.5 
 ig.8 
 g . 
 
 139 
 
 14 3oi 
 
 333 
 
 364 
 
 395 
 
 426 
 
 457 
 
 489 
 
 520 
 
 55i 
 
 582 
 
 9 
 
 3I-S 
 
 30.5 
 
 29.7 
 
 140 
 
 613 
 
 64 
 
 675 
 
 706 
 
 737 
 
 768 
 
 799 
 
 829 
 
 860 
 
 891 
 
 
 
 
 
 141 
 142 
 143 
 144 
 145 
 146 
 147 
 
 922 
 15 229 
 534 
 836 
 16 137 
 435 
 732 
 
 953 
 259 
 564 
 866 
 167 
 465 
 761 
 
 983 
 290 
 594 
 897 
 197 
 495 
 791 
 
 *OI 4 
 
 320 
 625 
 927 
 
 227 
 524 
 820 
 
 *<H5 
 35i 
 655 
 957 
 
 256 
 554 
 850 
 
 *076 
 38i 
 685 
 987 
 286 
 584 
 879 
 
 *io6 
 412 
 715 
 *oi 7 
 316 
 613 
 909 
 
 *I37 
 442 
 746 
 *047 
 346 
 643 
 938 
 
 *i68 
 473 
 776 
 
 *o 7 7 
 376 
 673 
 967 
 
 *igS 
 503 
 806 
 *io 7 
 406 
 702 
 997 
 
 2 
 
 3 
 4 
 5 
 6 
 
 32 
 
 i:: 
 
 Q.6 
 12.8 
 
 19.2 
 
 31 
 
 1:1 
 
 9-3 
 1.1 
 
 30 
 
 6.0 
 
 148 
 149 
 
 17 026 
 319 
 
 056 
 
 348 
 
 085 
 377 
 
 114 
 406 
 
 143 
 
 435 
 
 173 
 464 
 
 202 
 
 493 
 
 231 
 522 
 
 260 
 
 55i 
 
 289 
 58o 
 
 9 
 
 i&i 
 
 27.9 
 
 *4- 
 
 150 
 
 609 
 
 638 
 
 667 
 
 696 
 
 725 
 
 754 
 
 782 
 
 811 
 
 840 
 
 869 
 
 
 
 
 
 N. 
 
 L. o 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 p. 
 
 P. 

 
 LOGARITHMS. 
 
 N. 
 
 L. o | i 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 Sii 
 
 8 
 
 9 
 
 P. P. 
 
 150 
 
 J5 1 
 152 
 153 
 154 
 155 
 156 
 157 
 158 
 T 59 
 160 
 161 
 162 
 163 
 164 
 165 
 1 66 
 167 
 1 63 
 169 
 
 uo 
 
 -i 
 
 73 
 74 
 75 
 76 
 77 
 78 
 79 
 180 
 Si 
 82 
 183 
 184 
 185 
 186 
 187 
 188 
 189 
 
 190 
 
 191 
 192 
 IQ3 
 194 
 195 
 196 
 197 
 198 
 199 
 200 
 
 17 609 
 
 638 
 
 66? 
 
 955 
 241 
 526 
 808 
 089 
 368 
 645 
 921 
 194 
 
 696 
 
 984 
 270 
 554 
 837 
 117 
 396 
 673 
 948 
 
 222 
 
 725 
 
 754 
 
 782 
 
 840 
 
 869 
 
 2 
 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 I 
 9 
 
 7 
 8 
 9 
 
 3 
 
 4 
 5 
 6 
 7 
 8 
 9 
 
 29 
 
 I'.l 
 8-7 
 n.6 
 '4-5 
 17-4 
 20.3 
 
 ll.l 
 
 S 
 
 27 
 
 2. 
 
 5- 
 8. 
 
 i: 
 
 S 
 
 24.3 
 
 2 
 
 2 
 
 5 
 ; 7 
 
 12 
 
 15 
 
 '7 
 
 8 20 
 9 22 
 
 24 
 
 2.4 
 4.8 
 
 7-2 
 
 9.6 
 
 12.0 
 14.4 
 
 16.8 
 
 IT.l 
 
 22 
 
 2.2 
 
 u 
 
 8.8 
 '3-2 
 
 1 
 
 19.8 
 
 28 
 
 2.8 
 
 5-6 
 8. 
 
 16. 
 
 19. 
 
 25- 
 
 '& 
 j.1 
 
 II- 
 23.4-' 
 
 5 
 23 
 
 ::i 
 
 6.9 
 
 9.2 
 
 i! 
 
 18.4 
 
 ZQ.J 
 
 21 
 
 2.1 
 4.2 
 6. 3 
 
 8.4 
 
 !:I 
 14.7 
 
 16.8 
 18.9 
 
 898 
 18 184 
 
 469 
 752 
 19 033 
 312 
 
 590 
 
 866 
 
 20 140 
 
 926 
 213 
 498 
 780 
 06 1 
 340 
 618 
 893 
 167 
 
 *oi3 
 298 
 583 
 865 
 145 
 424 
 700 
 976 
 249 
 
 *0 4 I 
 
 327 
 611 
 893 
 173 
 451 
 728 
 *oo 3 
 276 
 
 *o7o 
 355 
 639 
 921 
 
 201 
 
 479 
 756 
 *030 
 303 
 
 *o 99 
 384 
 667 
 949 
 229 
 507 
 783 
 *o 5 8 
 330 
 
 *I2 7 
 412 
 696 
 
 977 
 257 
 535 
 811 
 *o85 
 358 
 
 *i 5 6 
 441 
 
 724 
 *oo 5 
 285 
 562 
 838 
 
 *II2 
 
 385 
 
 412 
 
 439 
 
 466 
 
 493 
 
 520 
 
 548 
 
 575 
 
 602 
 
 629 
 
 656 
 
 68 3 
 952 
 21 2ig 
 484 
 748 
 22 Oil 
 
 272 
 531 
 
 789 
 
 710 
 978 
 245 
 5ii 
 775 
 037 
 298 
 557 
 814 
 
 737 
 *oo 5 
 
 272 
 
 537 
 So r 
 063 
 324 
 
 583 
 840 
 
 763 
 
 *0 3 2 
 299 
 564 
 
 827 
 089 
 350 
 608 
 
 866 
 
 790 
 *059 
 
 325 
 59 
 854 
 H5 
 376 
 634 
 891 
 
 817 
 *oSs 
 352 
 617 
 880 
 141 
 401 
 660 
 9i7 
 
 844 
 
 *II2 
 
 378 
 643 
 906 
 167 
 427 
 
 686 
 943 
 
 871 
 *i 39 
 405 
 669 
 932 
 194 
 453 
 712 
 968 
 223 
 
 477 
 729 
 980 
 229 
 477 
 724 
 969 
 
 212 
 
 455 
 
 308 
 *i&5 
 43i 
 696 
 958 
 220 
 479 
 737 
 994 
 
 925 
 *I 9 2 
 
 458 
 722 
 985 
 246 
 
 505 
 
 763 
 
 *oig 
 
 23 045 
 
 070 
 
 096 121 
 
 147 
 
 172 
 
 198 
 
 249 
 502 
 754 
 *o&5 
 254 
 502 
 748 
 993 
 237 
 479 
 
 274 
 
 528 
 
 779 
 *030 
 
 279 
 527 
 773 
 *oi8 
 261 
 503 
 
 300 
 
 553 
 805 
 24 055 
 304 
 55i 
 797 
 25 042 
 285 
 
 325 
 578 
 830 
 080 
 329 
 576 
 822 
 066 
 310 
 55i 
 
 350 
 60 3 
 855 
 105 
 
 353 
 601 
 846 
 091 
 334 
 
 376 
 629 
 880 
 130 
 378 
 625 
 871 
 iis 
 358 
 
 401 
 
 654 
 905 
 155 
 403 
 650 
 895 
 139 
 382 
 
 426 
 679 
 930 
 1 80 
 428 
 674 
 920 
 164 
 406 
 
 452 
 704 
 955 
 204 
 452 
 699 
 944 
 188 
 431 
 
 527 
 
 575 
 
 600 
 840 
 079 
 
 316 
 
 553 
 788 
 
 *02I 
 254 
 485 
 715 
 
 624 
 
 648 
 
 672 
 
 696 
 935 
 174 
 411 
 
 647 
 881 
 *ii4 
 346 
 577 
 807 
 
 *o 3 5 
 
 720 
 
 959 
 198 
 
 435 
 670 
 905 
 *i38 
 370 
 600 
 830 
 *os8 
 
 ~2^T 
 
 5" 
 
 735 
 959 
 181 
 
 403 
 
 623 
 842 
 *o6o 
 
 744 
 
 768 
 26 007 
 245 
 482 
 7i/ 
 95i 
 27 184 
 416 
 646 
 
 792 
 
 031 
 269 
 505 
 74i 
 975 
 207 
 
 439 
 
 ^L 
 
 898 
 
 ~I26~ 
 
 353 
 578 
 803 
 026 
 248 
 469 
 688 
 J>07_ 
 125 
 
 816 
 
 055 
 293 
 529 
 764 
 998 
 231 
 462 
 692 
 
 864 
 
 102 
 340 
 576 
 
 811 
 *045 
 
 277 
 508 
 738 
 
 888 
 126 
 364 
 600 
 834 
 *o6S 
 300 
 531 
 761 
 
 912 
 150 
 
 387 
 623 
 
 858 
 
 *0 9 I 
 
 323 
 
 554 
 784 
 
 983 
 
 221 
 
 458 
 694 
 928 
 
 *i6i 
 393 
 623 
 852 
 *o8i 
 
 875 
 
 921 
 
 944 
 
 967 
 194 
 421 
 646 
 
 870 
 092 
 3 r 4 
 535 
 754 
 _973_ 
 190 
 
 989 
 217 
 443 
 668 
 892 
 "5 
 336 
 557 
 776 
 994 
 
 211 
 
 *OI2 
 
 28 103 
 330 
 556 
 780 
 29 003 
 226 
 447 
 667 
 885 
 
 149 
 
 375 
 601 
 825 
 048 
 270 
 
 49 1 
 710 
 
 929 
 
 171 
 
 398 
 623 
 847 
 070 
 292 
 513 
 732 
 ..9SL 
 1 68 
 
 240 
 466 
 6 9 I 
 914 
 137 
 
 358 
 579 
 798 
 *oi6 
 
 262 
 488 
 713' 
 937 
 159 
 380 
 601 
 820 
 *038 
 
 307 
 533 
 758 
 981 
 203 
 425 
 645 
 863 
 *o8i 
 
 30 103 
 
 146 
 
 233 
 
 255 
 
 276 
 
 298 
 
 N. 
 
 L. o 
 
 I j 2 
 
 3 
 
 4 
 
 5 1 6 
 
 7 | 8 
 
 9 
 
 P. P.
 
 LOGARITHMS. 
 
 N. 
 
 L. o 
 
 i 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 P 
 
 P. 
 
 200 
 
 30 103 
 
 125 
 
 146 
 
 1 68 
 
 190 
 
 211 
 
 233 
 
 255 
 
 276 
 
 298 
 
 
 
 
 201 
 202 
 203 
 204 
 205 
 206 
 207 
 2O8 
 20 9 
 
 320 
 535 
 750 
 963 
 3i 175 
 387 
 597 
 806 
 32 015 
 
 34i 
 557 
 771 
 984 
 197 
 408 
 618 
 827 
 035 
 
 363 
 
 5/8 
 792 
 *oo6 
 218 
 429 
 
 639 
 848 
 056 
 
 384 
 600 
 814 
 
 *027 
 
 239 
 450 
 660 
 
 869 
 
 077 
 
 406 
 621 
 835 
 *o 4 8 
 260 
 471 
 68 1 
 890 
 098 
 
 4 2S 
 643 
 856 
 
 *o6g 
 281 
 492 
 702 
 911 
 118 
 
 449 
 664 
 878 
 *ogi 
 302 
 513 
 723 
 931 
 139 
 
 47i 
 
 685 
 899 
 
 *II2 
 323 
 
 534 
 744 
 952 
 160 
 
 492 
 707 
 920 
 *I33 
 345 
 555 
 765 
 973 
 181 
 
 5M 
 728 
 942 
 *i54 
 366 
 576 
 785 
 994 
 
 201 
 
 3 
 
 4 
 5 
 6 
 7 
 8 
 9 
 
 i 
 i 
 
 2 21 
 
 .2 2.1 
 4 4-2 
 
 .6 6/3 
 .8 8.4 
 
 :! 5:2 
 
 ).8 18.9 
 
 210 
 
 222 
 
 243 
 
 263 
 
 284 
 
 305 
 
 325 
 
 346 
 
 366 
 
 387 
 
 4 08 
 
 
 
 
 211 
 212 
 213 
 214 
 215 
 
 216 
 
 217 
 
 218 
 
 2I 9 
 
 428 
 634 
 838 
 
 33 041 
 244 
 445 
 646 
 846 
 34 044 
 
 449 
 654 
 858 
 062 
 264 
 465 
 666 
 866 
 064 
 
 469 
 675 
 879 
 082 
 284 
 486 
 686 
 885 
 084 
 
 490 
 
 695 
 899 
 
 102 
 304 
 506 
 7 06 
 
 95 
 104 
 
 5io 
 715 
 919 
 
 122 
 325 
 526 
 726 
 
 925 
 124 
 
 531 
 736 
 940 
 143 
 345 
 546 
 746 
 945 
 143 
 
 552 
 756 
 960 
 163 
 365 
 566 
 766 
 965 
 163 
 
 572 
 777 
 980 
 183 
 385 
 586 
 786 
 985 
 183 
 
 593 
 797 
 
 *OOI 
 
 203 
 405 
 606 
 806 
 *oo 5 
 203 
 
 6I 3 
 
 818 
 
 *02I 
 22 4 
 425 
 626 
 826 
 *02 5 
 
 223 
 
 
 3 
 4 
 
 I 
 9 
 
 20 
 
 e!o 
 
 8.0 
 
 14.0 
 16.0 
 18.0 
 
 220 
 
 242 
 
 262 
 
 282 
 
 301 
 
 321 
 
 341 
 
 361 
 
 380 
 
 400 
 
 420 
 
 
 
 
 221 
 222 
 22 3 
 224 
 225 
 226 
 227 
 228 
 22 9 
 
 439 
 635 
 830 
 35 025 
 218 
 411 
 603 
 793 
 984 
 
 459 
 655 
 850 
 044 
 238 
 430 
 622 
 813 
 *oo 3 
 
 479 
 674 
 869 
 064 
 257 
 449 
 641 
 832 
 
 *02I 
 
 498 
 694 
 889 
 083 
 276 
 468 
 660 
 851 
 
 *0 4 
 
 518 
 713 
 908 
 IO2 
 
 295 
 
 488 
 
 679 
 
 870 
 
 *05 9 
 
 537 
 733 
 928 
 
 122 
 315 
 507 
 698 
 889 
 
 *o?8 
 
 557 
 753 
 947 
 141 
 334 
 526 
 717 
 908 
 *og 7 
 
 577 
 772 
 967 
 160 
 353 
 545 
 736 
 927 
 *ii6 
 
 596 
 792 
 986 
 1 80 
 372 
 564 
 755 
 946 
 *I3S 
 
 616 
 8n 
 *oo 5 
 199 
 392 
 583 
 774 
 965 
 *I54 
 
 
 3 
 4 
 
 I 
 9 
 
 19 
 
 9 
 .8 
 
 'I 
 5 
 i .4 
 1 -3 
 
 I .2 
 
 230 
 
 3 6 J 73 
 
 192 
 
 211 
 
 229 
 
 248 
 
 267 
 
 286 
 
 305 
 
 324 
 
 342 
 
 
 
 
 231 
 232 
 233 
 234 
 235 
 236 
 237 
 238 
 239 
 
 361 
 549 
 736 
 922 
 37 107 
 291 
 475 
 658 
 840 
 
 380 
 568 
 754 
 940 
 125 
 310 
 493 
 676 
 858 
 
 399 
 
 586 
 
 773 
 959 
 144 
 328 
 5" 
 694 
 876 
 
 418 
 605 
 
 791 
 
 977 
 162 
 346 
 530 
 712 
 894 
 
 436 
 624 
 810 
 996 
 181 
 365 
 548 
 73i 
 912 
 
 455 
 642 
 829 
 
 *OI 4 
 
 199 
 383 
 566 
 749 
 93i 
 
 474 
 661 
 847 
 *o 3 3 
 218 
 401 
 585 
 767 
 949 
 
 493 
 680 
 866 
 
 *0 5 I 
 
 236 
 420 
 603 
 
 785 
 967 
 
 5" 
 
 698 
 884 
 *o7o 
 254 
 438 
 621 
 803 
 985 
 
 530 
 717 
 903 
 
 *o8S 
 273 
 457 
 639 
 822 
 
 *00 3 
 
 
 3 
 4 
 
 i 
 I 
 
 9 
 
 18 
 
 .8 
 .6 
 4 
 
 .2 
 12^6 
 
 \l:l 
 
 240 
 
 38 021 
 
 39 
 
 057 
 
 075 
 
 093 
 
 112 
 
 130 
 
 148 
 
 1 66 
 
 184 
 
 
 
 
 241 
 242 
 243 
 244 
 245 
 246 
 24? 
 248 
 249 
 
 202 
 382 
 561 
 
 739 
 917 
 39 094 
 270 
 445 
 620 
 
 220 
 
 399 
 
 578 
 757 
 934 
 in 
 287 
 463 
 637 
 
 238 
 417 
 596 
 775 
 952 
 129 
 35 
 480 
 655 
 
 256 
 435 
 614 
 792 
 970 
 146 
 322 
 498 
 672 
 
 274 
 453 
 632 
 810 
 987 
 164 
 340 
 515 
 690 
 
 2 9 2 
 471 
 650 
 
 828 
 
 *oo 5 
 182 
 358 
 533 
 707 
 
 3io 
 489 
 668 
 846 
 
 *02 3 
 
 199 
 
 375 
 550 
 724 
 
 328 
 507 
 686 
 863 
 *04i 
 217 
 393 
 568 
 742 
 
 346 
 
 525 
 703 
 881 
 ^058 
 235 
 410 
 585 
 759 
 
 364 
 
 543 
 721 
 899 
 *076 
 
 252 
 428 
 602 
 
 777 
 
 
 i 
 
 3 
 4 
 
 1 
 7 
 8 
 9 
 
 17 
 
 '7 
 3-4 
 5-' 
 6.8 
 5 
 
 s 
 
 15-3 
 
 250 
 
 794 
 
 8n 
 
 829 
 
 846 
 
 863 
 
 SSi 
 
 898 
 
 915 
 
 933 
 
 950 
 
 
 
 
 N. 
 
 L. o 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 J 
 
 P. 
 
 p.
 
 LOGARITHMS. 
 
 N. 
 
 L. o 
 
 ' 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. 
 
 P. 
 
 250 
 
 39 794 
 
 811 
 
 829 
 
 846 
 
 863 
 
 881 
 
 898 
 
 9i5 
 
 933 
 
 95 
 
 
 
 251 
 252 
 253 
 254 
 255 
 256 
 257 
 258 
 259 
 
 967 
 40 140 
 312 
 483 
 654 
 824 
 993 
 41 162 
 330 
 
 985 
 157 
 329 
 500 
 671 
 841 
 
 *OIO 
 
 179 
 
 347 
 
 *002 
 175 
 
 346 
 
 518 
 
 688 
 
 858 
 
 *02? 
 
 196 
 363 
 
 *oi9 
 192 
 364 
 
 535 
 705 
 875 
 *<M4 
 
 212 
 380 
 
 *o 37 
 209 
 38i 
 552 
 722 
 892 
 
 *06! 
 
 229 
 
 397 
 
 *54 
 226 
 398 
 569 
 739 
 909 
 *078 
 246 
 414 
 
 *0 7 I 
 
 243 
 4i5 
 586 
 756 
 926 
 *o 95 
 263 
 430 
 
 *o88 
 261 
 432 
 603 
 773 
 943 
 *ni 
 280 
 447 
 
 *io6 
 
 278 
 
 449 
 620 
 
 79 
 960 
 
 *I28 
 
 296 
 
 464 
 
 *I2 3 
 
 295 
 466 
 637 
 
 807 
 
 976 
 
 *i45 
 313 
 481 
 
 2 
 
 3 
 4 
 
 I 
 7 
 8 
 9 
 
 18 
 
 3-6 
 5-4 
 7.2 
 
 Q 
 
 ll'.6 
 14.4 
 16.2 
 
 260 
 
 497 
 
 514 
 
 531 
 
 547 
 
 564 
 
 58i 
 
 597 
 
 614 
 
 631 
 
 647 
 
 
 
 261 
 262 
 263 
 264 
 265 
 266 
 267 
 268 
 269 
 
 664 
 830 
 996 
 42 160 
 325 
 488 
 651 
 8i3 
 975 
 
 68 1 
 847 
 
 *OI2 
 177 
 341 
 504 
 667 
 830 
 991 
 
 697 
 863 
 
 *02g 
 193 
 357 
 52i 
 684 
 846 
 *oo8 
 
 7*4 
 
 880 
 *045 
 
 2IO 
 
 374 
 
 537 
 700 
 862 
 
 *02 4 
 
 73i 
 896 
 
 *062 
 
 226 
 39 
 
 553 
 716 
 878 
 *040 
 
 747 
 9 T 3 
 *o 7 8 
 243 
 406 
 570 
 732 
 894 
 *os6 
 
 764 
 929 
 *o 9 5 
 259 
 423 
 586 
 
 749 
 * 911 
 
 *0?2 
 
 780 
 946 
 *m 
 275 
 439 
 602 
 765 
 927 
 *o88 
 
 797 
 963 
 
 *I2 7 
 
 292 
 
 455 
 619 
 781 
 943 
 *I04 
 
 814 
 
 979 
 *i 4 4 
 308 
 472 
 635 
 797 
 959 
 
 *I20' 
 
 3 
 4 
 
 ' I 
 
 9 
 
 17 
 
 1-7 
 3-4 
 
 1:1 
 
 ti'i 
 
 i:S 
 
 15-3 
 
 270 
 
 43 136 
 
 152 
 
 169 
 
 185 
 
 201 
 
 217 
 
 233 
 
 249 
 
 265 
 
 281 
 
 
 
 271 
 
 272 
 273 
 274 
 275 
 276 
 277 
 278 
 279 
 
 297 
 457 
 6x6 
 
 775 
 933 
 44 091 
 248 
 404 
 560 
 
 313 
 
 473 
 632 
 791 
 949 
 107 
 264 
 420 
 576 
 
 329 
 489 
 648 
 807 
 965 
 
 122 
 279 
 436 
 
 592 
 
 345 
 505 
 664 
 823 
 981 
 138 
 295 
 45i 
 607 
 
 361 
 
 521 
 680 
 
 838 
 
 996 
 154 
 311 
 
 467 
 623 
 
 377 
 537 
 696 
 
 854 
 
 *OI2 
 170 
 326 
 
 483 
 6 3 8 
 
 393 
 553 
 712 
 870 
 
 *028 
 
 185 
 342 
 498 
 
 654 
 
 409 
 569 
 727 
 886 
 *044 
 2QI 
 
 358 
 514 
 669 
 
 425 
 584 
 743 
 902 
 
 *059 
 217 
 
 373 
 529 
 685 
 
 441 
 600 
 
 759 
 917 
 
 *Q75 
 232 
 389 
 545 
 700 
 
 i 
 
 2 
 
 3 
 4 
 
 i 
 
 7 
 8 
 9 
 
 16 
 
 1.6 
 
 b 
 
 6. 4 
 
 8.0 
 9.6 
 
 12^8 
 
 14.4 
 
 280 
 
 716 
 
 731 
 
 747 
 
 762 
 
 778 
 
 793 
 
 809 
 
 824 
 
 840 
 
 855 
 
 
 
 281 
 282 
 283 
 284 
 285 
 286 
 287 
 288 
 289 
 
 871 
 45 025 
 179 
 332 
 484 
 637 
 788 
 
 939 
 46 090 
 
 886 
 040 
 194 
 347 
 500 
 652 
 803 
 954 
 105 
 
 902 
 056 
 209 
 362 
 5i5 
 667 
 818 
 969 
 
 120 
 
 917 
 071 
 
 225 
 378 
 530 
 682 
 
 834 
 984 
 135 
 
 932 
 086 
 240 
 
 393 
 545 
 697 
 849 
 
 *000 
 
 150 
 
 948 
 
 102 
 255 
 408 
 561 
 712 
 86 4 
 
 *ois 
 165 
 
 963 
 117 
 271 
 423 
 576 
 728 
 
 879 
 *030 
 1 80 
 
 979 
 133 
 286 
 
 439 
 59i 
 743 
 894 
 
 *045 
 195 
 
 994 
 148 
 301 
 454 
 606 
 758 
 909 
 *o6o 
 
 210 
 
 *OIO 
 
 163 
 317 
 469 
 621 
 773 
 924 
 *075 
 225 
 
 2 
 
 3 
 4 
 
 I 
 7 
 8 
 9 
 
 15 
 
 3- 
 
 4- y 
 
 7- 
 9- 
 
 !3- 
 
 290 
 
 240 
 
 255 
 
 270 
 
 285 
 
 300 
 
 315 
 
 330 
 
 345 
 
 359 
 
 374 
 
 ' 
 
 
 291 
 292 
 293 
 294 
 295 
 296 
 297 
 298 
 299 
 
 389 
 538 
 687 
 835 
 982 
 47 129 
 276 
 422 
 567 
 
 404 
 553 
 702 
 850 
 997 
 144 
 290 
 436 
 582 
 
 419 
 568 
 716 
 864 
 *OI2 
 159 
 305 
 451 
 596 
 
 434 
 
 583 
 731 
 879 
 
 *026 
 
 173 
 319 
 
 465 
 
 611 
 
 449 
 598 
 746 
 894 
 
 *0 4 I 
 
 188 
 334 
 480 
 625 
 
 464 
 613 
 761 
 909 
 *Q56 
 
 202 
 
 349 
 494 
 640 
 
 479 
 627 
 776 
 923 
 
 *0 7 
 
 217 
 
 363 
 
 509 
 
 654 
 
 494 
 642 
 790 
 938 
 *o85 
 232 
 378 
 524 
 669 
 
 509 
 657 
 805 
 953 
 
 *IOO 
 
 246 
 392 
 
 538 
 683 
 
 523 
 672 
 820 
 967 
 *H4 
 261 
 407 
 553 
 698 
 
 3 
 
 4 
 
 I 
 7 
 8 
 9 
 
 14 
 
 1.4 
 
 2.8 
 
 5-6 
 
 7.0 
 
 b 
 
 II. 2 
 12.6 
 
 300 
 
 712 
 
 727 
 
 741 
 
 756 
 
 770 
 
 784 
 
 799 
 
 813 
 
 828 
 
 842 
 
 
 
 N. 
 
 L. o 
 
 i 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. 
 
 P.
 
 LOGARITHMS. 
 
 N. 
 
 L. o 
 
 i 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. 
 
 P. 
 
 300 
 
 47 712 
 
 727 
 
 741 
 
 756 
 
 770 
 
 784 
 
 799 
 
 813 
 
 828 
 
 842 
 
 
 
 301 
 302 
 303 
 304 
 305 
 306 
 307 
 308 
 309 
 
 857 
 48 ooi 
 144 
 287 
 430 
 572 
 7M 
 855 
 996 
 
 871 
 015 
 
 159 
 302 
 
 444 
 
 586 
 728 
 869 
 
 *OIO 
 
 885 
 029 
 173 
 316 
 
 458 
 
 601 
 
 742 
 
 883 
 
 *024 
 
 900 
 
 044 
 187 
 330 
 473 
 615 
 756 
 897 
 *o 3 8 
 
 914 
 
 058 
 
 202 
 
 344 
 487 
 629 
 770 
 911 
 
 *052 
 
 9 2 9 
 0/3 
 216 
 359 
 5oi 
 643 
 785 
 926 
 *o66 
 
 943 
 087 
 230 
 373 
 515 
 657 
 799 
 940 
 *oSo 
 
 958 
 
 101 
 
 244 
 
 387 
 
 530 
 671 
 813 
 
 954 
 *o 94 
 
 972 
 116 
 
 259 
 401 
 
 544 
 686 
 827 
 968 
 *io8 
 
 986 
 130 
 273 
 416 
 558 
 700 
 841 
 982 
 
 *I22 
 
 2 
 
 3 
 4 
 
 15 
 
 i -5 
 
 4- 
 
 7. 
 9- 
 
 310 
 
 49 136 
 
 150 
 
 164 
 
 178 
 
 192 
 
 206 
 
 220 
 
 234 
 
 248 
 
 262 
 
 7 
 
 10. 
 
 3" 
 
 312 
 313 
 314 
 315 
 316 
 317 
 3i8 
 319 
 
 276 
 415 
 554 
 693 
 831 
 969 
 50 1 06 
 243 
 379 
 
 290 
 429 
 
 568 
 707 
 
 845 
 982 
 
 120 
 
 256 
 393 
 
 304 
 
 443 
 
 582 
 721 
 85<J 
 996 
 133 
 270 
 406 
 
 3i8 
 457 
 596 
 734 
 
 872 
 
 *OIO 
 
 147 
 
 284 
 420 
 
 332 
 471 
 
 610 
 
 748 
 886 
 
 *024 
 
 161 
 
 297 
 433 
 
 346 
 485 
 624 
 762 
 900 
 *037 
 i?4 
 3" 
 447 
 
 360 
 
 499 
 
 638 
 776 
 914 
 
 *0 5 I 
 
 188 
 325 
 461 
 
 374 
 513 
 651 
 790 
 927 
 *o65 
 202 
 338 
 474 
 
 388 
 527 
 665 
 803 
 941 
 *07 9 
 215 
 352 
 488 
 
 402 
 541 
 679 
 8l 7 
 
 955 
 
 *0 9 2 
 
 229 
 365 
 5oi 
 
 9 
 
 13- 
 14 
 
 2.8 
 
 4.2 
 
 5-6 
 
 320 
 
 515 
 
 529 
 
 542 
 
 556 
 
 569 
 
 583 
 
 596 
 
 610 
 
 623 
 
 637 
 
 
 l:\ 
 
 321 
 322 
 323 
 324 
 325 
 326 
 327 
 328 
 329 
 
 651 
 
 786 
 920 
 5i 055 
 188 
 322 
 455 
 587 
 720 
 
 664 
 799 
 
 934 
 068 
 
 2O2 
 
 335 
 468 
 601 
 733 
 
 678 
 813 
 947 
 08 1 
 215 
 348 
 481 
 614 
 746 
 
 691 
 826 
 961 
 095 
 228 
 362 
 495 
 627 
 759 
 
 705 
 840 
 974 
 108 
 242 
 375 
 508 
 640 
 772 
 
 718 
 853 
 987 
 
 121 
 
 255 
 388 
 521 
 
 654 
 
 786 
 
 732 
 866 
 
 *OOI 
 
 135 
 268 
 402 
 534 
 667 
 799 
 
 745 
 880 
 
 *OI 4 
 
 148 
 282 
 415 
 548 
 680 
 812 
 
 759 
 
 S<J3 
 
 *028 
 
 162 
 295 
 
 428 
 561 
 693 
 
 825 
 
 772 
 907 
 
 *0 4 I 
 
 175 
 
 308 
 
 441 
 
 574 
 706 
 
 838 
 
 9 
 
 2 
 
 3 
 
 Q.8 
 II. 2 
 12.6 
 
 13 
 
 I'l 
 
 3-9 
 
 330 
 
 851 
 
 865 
 
 878 
 
 891 
 
 904 
 
 917 
 
 930 
 
 943 
 
 957 
 
 970 
 
 4 
 
 
 33i 
 
 332 
 333 
 334 
 335 
 336 
 337 
 338 
 339 
 
 983 
 52 114 
 244 
 375 
 504 
 634 
 763 
 892 
 53 020 
 
 996 
 127 
 257 
 388 
 5W 
 647 
 776 
 905 
 033 
 
 ^009 
 140 
 270 
 401 
 530 
 660 
 789 
 917 
 046 
 
 *022 
 
 153 
 284 
 
 414 
 
 543 
 673 
 802 
 930 
 058 
 
 *o 3 5 
 1 66 
 297 
 427 
 556 
 686 
 8i5 
 943 
 071 
 
 *048 
 179 
 310 
 440 
 569 
 699 
 .827 
 956 
 084 
 
 *o6i 
 192 
 323 
 453 
 582 
 711 
 840 
 969 
 097 
 
 *075 
 205 
 336 
 466 
 595 
 724 
 853 
 982 
 no 
 
 *o88 
 218 
 349 
 479 
 608 
 
 737 
 866 
 994 
 
 122 
 
 *IOI 
 
 231 
 362 
 492 
 621 
 750 
 879 
 *oo7 
 
 135 
 
 6 
 7 
 8 
 9 
 
 \'l 
 
 Q.I 
 II.7 
 
 12 
 
 1.2 
 
 340 
 
 148 
 
 161 
 
 173 
 
 1 86 
 
 199 
 
 212 
 
 224 
 
 237 
 
 250 
 
 263 
 
 3 
 
 3-6 
 
 34i 
 342 
 343 
 344 
 345 
 346 
 347 
 348 
 349 
 
 275 
 403 
 529 
 656 
 782 
 908 
 54 033 
 158 
 283 
 
 288 
 415 
 542 
 668 
 794 
 920 
 045 
 170 
 295 
 
 301 
 
 428 
 555 
 681 
 807 
 933 
 058 
 183 
 307 
 
 3U 
 441 
 567 
 694 
 820 
 945 
 070 
 195 
 320 
 
 326 
 453 
 580 
 706 
 832 
 958 
 083 
 208 
 332 
 
 339 
 466 
 
 593 
 719 
 845 
 970 
 095 
 
 220 
 
 345 
 
 352 
 479 
 605 
 732 
 857 
 983 
 108 
 233 
 357 
 
 364 
 
 491 
 618 
 744 
 870 
 995 
 
 120 
 
 245 
 
 370 
 
 377 
 504 
 631 
 
 757 
 882 
 *oo8 
 133 
 258 
 382 
 
 39 
 517 
 643 
 769 
 895 
 
 *020 
 145 
 270 
 
 394 
 
 4 
 5 
 6 
 7 
 8 
 9 
 
 4.8 
 
 K 
 
 9.6 
 
 350 
 
 407 
 
 419 
 
 432 
 
 444 
 
 456 
 
 469 
 
 481 
 
 494 
 
 506 
 
 5i8 
 
 
 
 N. 
 
 L. o 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P 
 
 p.
 
 LOGARITHMS. 
 
 N. 
 
 L. o 
 
 i 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. 
 
 P. 
 
 350 
 
 54 407 
 
 419 
 
 432 
 
 444 
 
 456 
 
 469 
 
 481 
 
 494 
 
 506 
 
 518 
 
 
 
 35i 
 352 
 353 
 354 
 355 
 356 
 357 
 358 
 359 
 
 531 
 654 
 111 
 900 
 55 023 
 M5 
 267 
 388 
 509 
 
 543 
 667 
 790 
 913 
 035 
 157 
 279 
 400 
 522 
 
 555 
 679 
 802 
 925 
 047 
 169 
 291 
 413 
 534 
 
 568 
 
 691 
 814 
 937 
 060 
 182 
 303 
 425 
 546 
 
 580 
 704 
 
 827 
 
 949 
 072 
 194 
 
 315 
 437 
 
 558 
 
 593 
 716 
 
 839 
 962 
 084 
 206 
 328 
 449 
 570 
 
 605 
 728 
 851 
 974 
 096 
 218 
 340 
 461 
 582 
 
 617 
 741 
 864 
 986 
 108 
 230 
 352 
 473 
 594 
 
 630 
 753 
 876 
 998 
 
 121 
 242 
 364 
 485 
 606 
 
 888 
 *on 
 133 
 255 
 376 
 497 
 618 
 
 2 
 
 3 
 4 
 5 
 6 
 7 
 
 13 
 
 1:1 
 
 '3-q 
 
 5 
 
 9.1 
 
 360 
 
 630 
 
 642 
 
 654 
 
 666 
 
 678 
 
 691 
 
 703 
 
 715 
 
 727 
 
 739 
 
 8 
 
 10.4 
 
 361 
 362 
 363 
 364 
 365 
 366 
 367 
 368 
 369 
 
 871 
 991 
 56 no 
 229 
 
 348 
 467 
 585 
 703 
 
 763 
 ^883 
 
 122 
 241 
 360 
 
 478 
 
 597 
 714 
 
 775 
 895 
 *oi5 
 134 
 253 
 372 
 490 
 608 
 726 
 
 787 
 
 *02'7 
 
 146 
 
 265 
 384 
 
 502 
 
 620 
 
 738 
 
 799 
 
 158 
 277 
 396 
 
 750 
 
 811 
 
 *oso 
 170 
 289 
 407 
 526 
 644 
 761 
 
 823 
 943 
 
 *062 
 
 182 
 301 
 419 
 
 538 
 656 
 773 
 
 835 
 
 *o 55 
 194 
 312 
 
 549 
 667 
 
 785 
 
 8 47 
 967 
 
 *o86 
 205 
 324 
 443 
 56i 
 679 
 797 
 
 859 
 979 
 *og8 
 217 
 336 
 455 
 573 
 691 
 808 
 
 2 
 
 3 
 4 
 
 J 
 
 12 
 
 4-8 
 
 370 
 
 820 
 
 832 
 
 844 
 
 855 
 
 867 
 
 879 
 
 891 
 
 902 
 
 914 
 
 926 
 
 | 
 
 7.2 
 
 372 
 373 
 374 
 375 
 376 
 377 
 378 
 379 
 
 937 
 
 57 054 
 171 
 287 
 403 
 519 
 634 
 749 
 864 
 
 949 
 066 
 183 
 299 
 415 
 530 
 646 
 761 
 875 
 
 961 
 078 
 194 
 310 
 426 
 542 
 657 
 772 
 887 
 
 972 
 089 
 206 
 322 
 
 438 
 
 553 
 669 
 784 
 898 
 
 984 
 
 IOI 
 
 217 
 
 334 
 449 
 565 
 680 
 795 
 910 
 
 996 
 
 229 
 345 
 461 
 576 
 692 
 807 
 921 
 
 *oo8 
 124 
 241 
 
 -357 
 473 
 588 
 703 
 818 
 933 
 
 *oig 
 136 
 252 
 368 
 484 
 600 
 715 
 830 
 944 
 
 *0 3 I 
 
 148 
 264 
 380 
 496 
 611 
 726 
 841 
 955 
 
 159 
 276 
 392 
 507 
 623 
 738 
 852 
 967 
 
 7 
 8 
 9 
 
 8 
 
 10.8 
 
 380 
 
 978 
 
 990 
 
 *OOI 
 
 *oi 3 
 
 *02 4 
 
 *o 35 
 
 *047 
 
 *os8 
 
 
 081 
 
 3 
 
 3-3 
 
 382 
 383 
 384 
 385 
 386 
 387 
 388 
 389 
 
 58 092 
 206 
 320 
 433 
 546 
 659 
 771 
 883 
 995 
 
 104 
 
 213 
 
 33i 
 444 
 557 
 670 
 782 
 894 
 *oo6 
 
 "5 
 229 
 343 
 456 
 569 
 68 1 
 794 
 906 
 *oi7 
 
 127 
 240 
 354 
 467 
 580 
 692 
 805 
 917 
 
 *028 
 
 138 
 
 252 
 365 
 478 
 
 59 1 
 704 
 816 
 928 
 
 *0 4 
 
 149 
 263 
 377 
 490 
 602 
 715 
 827 
 
 * 939 
 
 161 
 
 274 
 388 
 SGI 
 614 
 726 
 838 
 950 
 
 *062 
 
 172 
 
 286 
 
 399 
 512 
 625 
 737 
 850 
 961 
 *073 
 
 184 
 297 
 410 
 524 
 636 
 
 749 
 861 
 
 * 973 
 
 195 
 
 309 
 422 
 535 
 647 
 760 
 872 
 984 
 *o 9 5 
 
 1 
 
 I 
 g 
 
 1:1 
 
 9-9 
 
 10 
 
 390 
 
 59 Io6 
 
 118 
 
 129 
 
 140 
 
 151 
 
 162 
 
 173 
 
 184 
 
 195 
 
 207 
 
 i 
 
 1.0 
 
 392 
 393 
 394 
 395 
 396 
 397 
 398 
 399 
 
 21 8 
 
 329 
 439 
 550 
 660 
 770 
 879 
 988 
 60 097 
 
 229 
 340 
 450 
 
 671 
 780 
 890 
 
 999 
 1 08 
 
 240 
 
 461 
 572 
 682 
 791 
 
 *OIO 
 
 119 
 
 251 
 
 362 
 
 47-' 
 58 3 
 693 
 802 
 912 
 
 *02I 
 130 
 
 262 
 373 
 483 
 594 
 704 
 813 
 923 
 
 *0 3 2 
 141 
 
 273 
 384 
 494 
 605 
 715 
 824 
 934 
 *o 43 
 152 
 
 284 
 
 395 
 506 
 616 
 726 
 835 
 945 
 *054 
 163 
 
 295 
 406 
 517 
 627 
 737 
 846 
 956 
 *o6 5 
 173 
 
 306 
 417 
 
 528 
 638 
 748. 
 857 
 966 
 
 184 
 
 428 
 
 539 
 649 
 
 759 
 868 
 977 
 *o86 
 195 
 
 3 
 4 
 
 1 
 7 
 8 
 9 
 
 
 400 
 
 206 
 
 217 
 
 228 
 
 239 
 
 240 
 
 260 
 
 271 
 
 282 
 
 293 
 
 34 
 
 
 
 N. 
 
 L. o 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 
 9 
 
 P. 
 
 P.
 
 LOGARITHMS. 
 
 N. 
 
 L. o 
 
 i 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 282 
 
 8 
 
 9 
 
 P. P. 
 
 400 
 
 60 206 
 
 217 
 
 325 
 433 
 54i 
 649 
 756 
 863 
 970 
 077 
 183 
 
 228 
 
 ~33&" 
 
 444 
 552 
 660 
 767 
 874 
 981 
 087 
 194 
 
 239 
 
 249 
 
 260 
 
 271 
 
 293 
 
 304 
 
 i 
 
 3 
 4 
 
 1 
 
 9 
 
 3 
 4 
 
 i 
 
 7 
 8 
 9 
 
 2 
 
 3 
 4 
 
 1 
 
 9 
 
 11 
 
 6.6 
 
 u 
 
 9.9 
 
 10 
 
 3.0' 
 I'o 
 
 ti 
 
 9 
 
 i'.3 
 2-7 
 3-6 
 4- 
 
 7- 
 
 401 
 402 
 403 
 404 
 405 
 406 
 407 
 408 
 409 
 410 
 411 
 412 
 413 
 414 
 415 
 416 
 417 
 418 
 419 
 420 
 421 
 422 
 423 
 424 
 425 
 426 
 427 
 428 
 429 
 430 
 
 43i 
 432 
 433 
 434 
 435 
 436 
 437 
 438 
 439 
 440 
 441 
 442 
 443 
 444 
 445 
 446 
 447 
 448 
 449 
 450 
 
 314 
 423 
 531 
 638 
 746 
 853 
 959 
 61 066 
 172 
 
 347 
 455 
 563 
 670 
 778 
 885 
 991 
 098 
 204 
 
 358 
 466 
 574 
 68 1 
 
 788 
 895 
 
 *002 
 I0g 
 215 
 
 369 
 
 477 
 584 
 692 
 
 799 
 906 
 *oi3 
 119 
 225 
 
 379 
 487 
 595 
 703 
 810 
 917 
 
 *023 
 
 130 
 
 236 
 
 390 
 498 
 606 
 7i3 
 821 
 927 
 *o 3 4 
 140 
 247 
 
 401 
 509 
 617 
 724 
 831 
 938 
 *045 
 151 
 257 
 363 
 469 
 574 
 679 
 784 
 888 
 
 993 
 097 
 
 201 
 304 
 
 412 
 520 
 627 
 
 735 
 842 
 949 
 *055 
 162 
 268 
 
 374 
 479 
 584 
 690 
 794 
 899 
 -003 
 107 
 
 211 
 315 
 
 4 I8 
 
 278 
 
 289 
 
 300 
 
 3io 
 
 321 
 426 
 532 
 637 
 742 
 8 47 
 951 
 055 
 159 
 263 
 
 33i 
 
 342 
 
 352 
 458 
 563 
 669 
 773 
 878 
 982 
 086 
 190 
 294 
 
 384 
 490 
 595 
 700 
 805 
 909 
 62 014 
 118 
 
 221 
 
 395 
 500 
 606 
 711 
 815 
 920 
 024 
 128 
 232 
 335 
 
 405 
 5" 
 616 
 721 
 826 
 930 
 034 
 138 
 242 
 
 416 
 521 
 627 
 731 
 836 
 941 
 045 
 149 
 252 
 356 
 
 437 
 542 
 648 
 752 
 857 
 962 
 066 
 170 
 273 
 
 448 
 553 
 658 
 763 
 868 
 972 
 076 
 1 80 
 284 
 
 325 
 
 346 
 
 366 
 
 377 
 
 387 
 
 397 
 
 408 
 
 428 
 531 
 634 
 
 737 
 839 
 941 
 63 043 
 144 
 246 
 
 439 
 542 
 644 
 747 
 849 
 951 
 053 
 155 
 256 
 
 449 
 552 
 655 
 757 
 859 
 961 
 063 
 165 
 266 
 
 459 
 562 
 665 
 767 
 870 
 972 
 073 
 175 
 276 
 
 469 
 
 572 
 675 
 778 
 880 
 9 82 
 083 
 18 5 
 286 
 
 480 
 583 
 685 
 788 
 890 
 992 
 094 
 195 
 296 
 
 490 
 593 
 696 
 798 
 900 
 
 *002 
 104 
 205 
 306 
 
 500 
 603 
 706 
 808 
 910 
 
 *OI2 
 114 
 215 
 317 
 
 5" 
 613 
 716 
 818 
 921 
 
 *022 
 124 
 225 
 
 327 
 
 521 
 62 4 
 726 
 829 
 931 
 
 *o 33 
 134 
 236 
 J37. 
 438 
 
 347 
 
 357 
 
 367 
 
 377 
 
 387 
 488 
 589 
 689 
 
 789 
 88 9 
 9 88 
 088 
 187 
 286 
 
 397 
 
 407 
 508 
 60 9 
 709 
 80 9 
 909 
 
 *oo8 
 108 
 207 
 306 
 
 417 
 
 428 
 
 448 
 548 
 649 
 
 749 
 849 
 
 949 
 64 048 
 147 
 246 
 
 458 
 558 
 659 
 759 
 859 
 959 
 058 
 
 157 
 256 
 
 355 
 454 
 552 
 650 
 748 
 846 
 943 
 040 
 137 
 234 
 
 468 
 568 
 669 
 769 
 869 
 969 
 068 
 167 
 266 
 
 478 
 579 
 679 
 
 779 
 879 
 
 979 
 
 078 
 
 177 
 276 
 
 498 
 
 599 
 699 
 
 799 
 899 
 998 
 098 
 197 
 296 
 
 518 
 6lg 
 719 
 8lg 
 919 
 
 *oi8 
 118 
 217 
 316 
 
 528 
 629 
 729 
 829 
 
 929 
 *028 
 
 128 
 227 
 326 
 
 538 
 639 
 739 
 839 
 939 
 *o 3 8 
 137 
 237 
 335 
 434 
 532 
 631 
 729 
 826 
 924 
 
 *02I 
 
 118 
 215 
 312 
 
 345 
 
 365 
 464 
 562 
 660 
 758 
 856 
 953 
 050 
 147 
 244 
 
 375 
 473 
 572 
 670 
 768 
 865 
 963 
 060 
 157 
 254 
 
 385 
 
 _39S_ 
 493 
 59i 
 689 
 
 787 
 885 
 982 
 079 
 176 
 273 
 
 404 
 
 4^4 
 
 513 
 611 
 709 
 807 
 904 
 
 *002 
 Ogg 
 I 9 6 
 2 9 2 
 
 424 
 
 444 
 542 
 640 
 738 
 836 
 933 
 65 031 
 128 
 225 
 
 483 
 582 
 680 
 
 777 
 875 
 972 
 070 
 167 
 263 
 
 503 
 601 
 699 
 797 
 895 
 992 
 089 
 1 86 
 283 
 
 523 
 621 
 
 719 
 816 
 914 
 
 *OII 
 
 108 
 205 
 
 302 
 
 321 
 
 33i 
 
 34i 
 
 350 
 
 360 
 
 369 
 
 379 
 
 389 
 
 398 
 
 408 
 
 N. 
 
 L. o 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P.
 
 LOGARITHMS. 
 
 N. 
 
 L. o 
 
 i 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
 450 
 
 45i 
 452 
 453 
 454 
 455 
 456 
 457 
 458 
 459 
 460 
 461 
 462 
 463 
 464 
 465 
 466 
 467 
 468 
 469 
 
 4ro 
 
 471 
 472 
 
 473 
 474 
 475 
 476 
 477 
 478 
 479 
 480 
 481 
 482 
 483 
 484 
 485 
 486 
 487 
 488 
 489 
 490 
 491 
 492 
 493 
 494 
 495 
 496 
 497 
 498 
 499 
 500 
 
 65 321 
 
 331 
 
 341 
 
 35Q 
 447 
 543 
 639 
 734 
 830 
 925 
 
 *020 
 
 "5 
 
 2IO 
 
 360 
 
 369 
 
 379 
 475 
 57i 
 667 
 763 
 858 
 
 954 
 *04 9 
 
 143 
 238 
 
 _3S9_ 
 
 485 
 58i 
 677 
 772 
 868 
 963 
 *o 5 8 
 153 
 247 
 
 ^98_ 
 495 
 59i 
 686 
 782 
 877 
 973 
 *o68 
 162 
 257 
 
 408 
 
 I 
 g 
 
 2 
 
 3 
 4 
 
 i 
 
 7 
 
 8 
 9 
 
 3 
 4 
 5 
 6 
 
 I 
 9 
 
 10 
 
 |; 
 
 l0 
 
 9.0 
 9 
 
 13 
 
 2.7 
 
 3-6 
 4-5 
 5-4 
 6-3 
 7.2 
 
 8 
 
 1.6 
 2.4 
 3.2 
 
 4 !s 
 
 S-6 
 6.4 
 7.2 
 
 418 
 514 
 610 
 706 
 Soi 
 896 
 992 
 66 087 
 181 
 
 427 
 523 
 619 
 715 
 
 8n 
 906 
 
 *OOI 
 
 096 
 191 
 
 285 
 
 437 
 533 
 629 
 725 
 820 
 916 
 
 *OII 
 
 106 
 
 200 
 295 
 
 456 
 
 552 
 648 
 
 744 
 839 
 935 
 *O3O 
 124 
 219 
 314 
 408 
 502 
 
 783 
 876 
 969 
 062 
 154 
 
 466 
 562 
 658 
 753 
 849 
 944 
 *039 
 134 
 229 
 
 323 
 417 
 5" 
 605 
 699 
 792 
 885 
 978 
 071 
 164 
 
 504 
 600 
 696 
 792 
 887 
 982 
 *077 
 172 
 266 
 
 276 
 
 304 
 
 332 
 
 342 
 
 351 
 
 361 
 
 370 
 464 
 558 
 652 
 745 
 839 
 932 
 67 025 
 H7 
 
 380 
 474 
 567 
 66 1 
 
 755 
 848 
 941 
 034 
 127 
 
 389 
 483 
 
 577 
 671 
 764 
 857 
 950 
 043 
 136 
 228 
 321 
 413 
 504 
 596 
 688 
 
 779 
 870 
 961 
 052 
 
 398 
 492 
 586 
 680 
 
 773 
 867 
 960 
 052 
 145 
 237 
 
 427 
 521 
 614 
 708 
 801 
 894 
 987 
 080 
 173 
 
 436 
 530 
 624 
 717 
 811 
 904 
 997 
 089 
 182 
 
 445 
 539 
 633 
 727 
 820 
 
 $ 
 
 099 
 191 
 
 455 
 549 
 642 
 736 
 829 
 922 
 *ois 
 108 
 
 2OI 
 
 210 
 
 219 
 
 247 
 
 256 
 
 265 
 
 274 
 
 284 
 
 293 
 
 302 
 
 394 
 486 
 578 
 669 
 761 
 852 
 943 
 63 034 
 
 3ii 
 403 
 
 495 
 587 
 679 
 770 
 861 
 952 
 043 
 
 330 
 422 
 5U 
 605 
 697 
 788 
 879 
 970 
 06 1 
 
 339 
 431 
 523 
 614 
 706 
 797 
 888 
 
 979 
 070 
 
 348 
 440 
 532 
 624 
 715 
 806 
 897 
 988 
 079 
 169 
 260 
 350 
 440 
 529 
 619 
 708 
 797 
 886 
 975 
 064 
 
 357 
 449 
 54i 
 633 
 724 
 
 8i5 
 906 
 997 
 
 088 
 
 367 
 459 
 550 
 642 
 733 
 825 
 916 
 *oo6 
 097 
 
 376 
 468 
 560 
 651 
 742 
 834 
 925 
 *oi5 
 106 
 
 385 
 
 477 
 569 
 660 
 752 
 843 
 934 
 
 *024 
 
 "5 
 
 124 
 
 133 
 
 142 
 
 151 
 
 160 
 
 251 
 341 
 431 
 520 
 610 
 699 
 789 
 878 
 
 966 
 055 
 
 178 
 
 187 
 
 196 
 287 
 377 
 467 
 556 
 646 
 735 
 824 
 9!3 
 
 *002 
 090 
 
 205 
 
 215 
 305 
 395 
 485 
 574 
 664 
 753 
 842 
 93i 
 
 224 
 314 
 404 
 
 494 
 583 
 673 
 762 
 851 
 940 
 
 233 
 323 
 413 
 502 
 592 
 68 1 
 771 
 860 
 949 
 
 242 
 332 
 422 
 5" 
 601 
 690 
 780 
 869 
 958 
 
 269 
 359 
 449 
 
 538 
 628 
 717 
 806 
 895 
 984 
 
 278 
 368 
 458 
 547 
 637 
 726 
 815 
 904 
 993 
 
 296 
 386 
 476 
 565 
 655 
 744 
 833 
 922 
 
 *OII 
 
 099 
 
 69 020 
 
 028 
 
 037 
 
 046 
 
 073 
 
 082 
 
 108 
 
 197 
 
 285 
 
 373 
 461 
 548 
 636 
 723 
 810 
 
 117 
 205 
 
 3! 
 
 469 
 
 557 
 644 
 732 
 819 
 
 126 
 214 
 302 
 390 
 478 
 566 
 653 
 740 
 827 
 
 135 
 223 
 3" 
 399 
 
 487 
 574 
 662 
 
 749 
 
 836 
 
 144 
 
 232 
 320 
 408 
 496 
 583 
 671 
 758 
 845 
 932 
 
 152 
 241 
 329 
 417 
 
 504 
 592 
 679 
 767 
 854 
 940 
 
 161 
 249 
 
 338 
 425 
 513 
 601 
 
 688 
 775 
 862 
 
 949 
 
 170 
 
 258 
 346 
 434 
 522 
 609 
 697 
 784 
 871 
 958 
 
 179 
 267 
 
 355 
 443 
 531 
 618 
 
 705 
 793 
 880 
 
 188 
 276 
 364 
 452 
 539 
 627 
 7M 
 801 
 888 
 
 8Q7 
 
 906 914 
 
 923 
 
 966 
 
 975 
 
 N. 
 
 L. o 
 
 I 
 
 2 
 
 3 
 
 4 
 
 s 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P.
 
 10 
 
 LOGARITHMS. 
 
 N. 
 
 L. o 
 
 i 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. 
 
 P. 
 
 500 
 
 69 8 97 
 
 906 
 
 914 
 
 923 
 
 932 
 
 940 
 
 949 
 
 958 
 
 966 
 
 975 
 
 
 
 501 
 502 
 503 
 504 
 505 
 506 
 507 
 508 
 509 
 
 984 
 70 070 
 157 
 243 
 329 
 415 
 5oi 
 586 
 672 
 
 992 
 079 
 165 
 252 
 338 
 424 
 509 
 595 
 680 
 
 *OOI 
 
 088 
 174 
 260 
 
 346 
 
 432 
 
 518 
 603 
 
 689 
 
 *OIO 
 
 096 
 183 
 269 
 
 355 
 441 
 526 
 612 
 697 
 
 *oi8 
 105 
 191 
 
 278 
 364 
 449 
 535 
 621 
 706 
 
 *027 
 
 114 
 
 2OO 
 286 
 372 
 
 458 
 
 544 
 629 
 714 
 
 *03& 
 
 122 
 
 209 
 
 295 
 38r 
 467 
 552 
 638 
 723 
 
 *<H4 
 131 
 217 
 
 303 
 389 
 475 
 56i 
 646 
 73i 
 
 *053 
 140 
 226 
 312 
 398 
 484 
 569 
 655 
 740 
 
 *062 
 
 148 
 234 
 321 
 406 
 492 
 578 
 663 
 749 
 
 3 
 
 9 
 
 o.g 
 1.8 
 2.7 
 
 510 
 
 757 
 
 766 
 
 774 
 
 783 
 
 791 
 
 800 
 
 808 
 
 817 
 
 825 
 
 834 
 
 4 
 
 3.6 
 
 5" 
 512 
 513 
 514 
 515 
 5i6 
 5i7 
 518 
 519 
 
 842 
 927 
 
 71 012 
 096 
 
 181 
 
 265 
 
 349 
 433 
 517 
 
 851 
 935 
 020 
 105 
 189 
 273 
 357 
 441 
 525 
 
 859 
 944 
 029 
 H3 
 198 
 282 
 366 
 450 
 533 
 
 868 
 952 
 037 
 
 122 
 206 
 290 
 
 374 
 458 
 542 
 
 876 
 961 
 046 
 130 
 214 
 299 
 383 
 466 
 550 
 
 885 
 969 
 054 
 139 
 223 
 307 
 39i 
 475 
 559 
 
 893 
 978 
 063 
 M7 
 231 
 315 
 399 
 483 
 567 
 
 902 
 986 
 071 
 
 155 
 240 
 324 
 408 
 492 
 575" 
 
 910 
 995 
 079 
 164 
 248 
 332 
 416 
 500 
 584 
 
 919 
 
 *00 3 
 
 088 
 172 
 257 
 34i 
 425 
 508 
 59 2 
 
 6 
 7 
 8 
 9 
 
 7- 
 8. 
 
 520 
 
 600 
 
 609 
 
 617 
 
 625 
 
 634 
 
 642 
 
 650 
 
 659 
 
 667 
 
 675 
 
 
 
 521 
 522 
 523 
 524 
 525 
 526 
 527 
 528 
 529 
 
 684 
 767 
 850 
 933 
 72 016 
 099 
 181 
 263 
 346 
 
 692 
 775 
 858 
 941 
 024 
 107 
 189 
 272 
 354 
 
 700 
 
 784 
 867 
 950 
 032 
 H5 
 198 
 280 
 362 
 
 709 
 792 
 875 
 958 
 041 
 123 
 206 
 288 
 370 
 
 717 
 800 
 883 
 966 
 049 
 132 
 214 
 296 
 378 
 
 725 
 809 
 892 
 975 
 057 
 140 
 
 222 
 304 
 
 387 
 
 734 
 817 
 900 
 983 
 066 
 148 
 230 
 3*3 
 395 
 
 742 
 825 
 908 
 991 
 074 
 156 
 239 
 321 
 403 
 
 750 
 834 
 917 
 
 999 
 082 
 165 
 
 247 
 
 329 
 411 
 
 759 
 842 
 925 
 *oo8 
 090 
 173 
 255 
 337 
 419 
 
 i 
 a 
 3 
 4 
 
 I 
 
 I 
 9 
 
 8 
 
 '.6 
 4_ 
 
 5^6 ' 
 6.4 
 7.2 
 
 530 
 
 428 
 
 436 
 
 444 
 
 452 
 
 460 
 
 469 
 
 477 
 
 485 
 
 493 
 
 SGI 
 
 
 
 53i 
 532 
 533 
 534 
 535 
 536 
 537 
 533 
 539 
 
 509 
 591 
 673 
 754 
 835 
 916 
 997 
 73 078 
 159 
 
 5i8 
 599 
 681 
 762 
 843 
 925 
 *oo6 
 086 
 167 
 
 526 
 607 
 689 
 770 
 852 
 933 
 *oi4 
 094 
 175 
 
 534 
 616 
 697 
 779 
 860 
 941 
 
 *022 
 I O2 
 183 
 
 542 
 624 
 705 
 787 
 868 
 
 949 
 *030 
 in 
 
 IQI 
 
 550 
 6 3 2 
 713 
 
 795 
 876 
 ,957 
 *038 
 119 
 199 
 
 558 
 640 
 722 
 803 
 884 
 965 
 *046 
 127 
 207 
 
 567 
 648 
 730 
 811 
 892 
 973 
 *054 
 135 
 215 
 
 575 
 656 
 738 
 819 
 900 
 981 
 
 *062 
 
 143 
 223 
 
 583 
 665 
 746 
 827 
 908 
 989 
 *o7o 
 151 
 231 
 
 3 
 
 7 
 
 540 
 
 239 
 
 247 
 
 255 
 
 263 
 
 272 
 
 280 
 
 288 
 
 296 
 
 304 
 
 312 
 
 4 
 
 
 54i 
 
 542 
 543 
 544 
 545 
 546 
 547 
 548 
 549 
 
 320 
 400 
 480 
 56o 
 640 
 719 
 
 799 
 
 878 
 
 957 
 
 328 
 408 
 488 
 568 
 648 
 727 
 807 
 886 
 965 
 
 336 
 416 
 496 
 576 
 656 
 735 
 815 
 894 
 973 
 
 344 
 424 
 504 
 584 
 664 
 
 743 
 823 
 902 
 981 
 
 352 
 432 
 512 
 592 
 672 
 751 
 830 
 910 
 
 989 
 
 360 
 44 
 520 
 600 
 679 
 759 
 838 
 918 
 997 
 
 368 
 448 
 528 
 608 
 687 
 767 
 846 
 926 
 *oos 
 
 376 
 456 
 536 
 616 
 695 
 775 
 854 
 933 
 *oi3 
 
 384 
 
 464 
 544 
 624 
 703 
 783 
 862 
 941 
 
 *020 
 
 392 
 472 
 552 
 632 
 711 
 791 
 870 
 949 
 
 *028 
 
 6 
 
 I 
 9 
 
 :1 
 
 6. 3 
 
 550 
 
 74 036 
 
 044 
 
 052 
 
 060 
 
 068 
 
 076 
 
 084 
 
 092 
 
 099 
 
 107 
 
 
 
 N. 
 
 L. o 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. 
 
 p.
 
 LOGARITHMS. 
 
 11 
 
 N. 
 
 L. o 
 
 i 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
 550 
 
 74 036 
 
 044 
 
 052 
 
 060 
 
 068 
 
 076 
 
 084 
 
 092 
 
 099 
 
 107 
 
 
 55i 
 552 
 553 
 554 
 555 
 556 
 557 
 553 
 559 
 
 H5 
 194 
 273 
 351 
 429 
 507 
 586 
 663 
 74 1 
 
 123 
 
 202 
 
 280 
 
 359 
 437 
 515 
 593 
 6/1 
 749 
 
 131 
 2IO 
 
 288 
 367 
 
 445 
 523 
 601 
 679 
 
 757 
 
 139 
 218 
 296 
 374 
 453 
 53i 
 609 
 687 
 764 
 
 i47 
 
 225 
 304 
 382 
 461 
 
 539 
 617 
 
 695 
 
 772 
 
 155 
 233 
 312 
 
 390 
 
 468 
 
 547 
 624 
 702 
 780 
 
 162 
 241 
 
 320 
 
 398 
 476 
 554 
 632 
 710 
 788 
 
 170 
 249 
 327 
 406 
 484 
 562 
 640 
 718 
 796 
 
 178 
 257 
 335 
 414 
 492 
 570 
 648 
 726 
 803 
 
 186 
 
 265 
 343 
 421 
 500 
 
 578 
 656 
 
 733 
 811 
 
 
 
 560 
 
 819 
 
 827 
 
 834 
 
 842 
 
 850 
 
 858 
 
 865 
 
 873 
 
 881 
 
 889 
 
 g 
 
 561 
 562 
 563 
 564 
 565 
 566 
 567 
 568 
 569 
 
 896 
 974 
 75 051 
 
 128 
 205 
 282 
 358 
 435 
 Sir 
 
 904 
 981 
 
 059 
 136 
 213 
 289 
 366 
 442 
 519 
 
 912 
 
 989 
 066 
 143 
 220 
 297 
 374 
 450 
 526 
 
 920 
 997 
 074 
 151 
 
 228 
 35 
 38i 
 458 
 534 
 
 927 
 *oos 
 082 
 159 
 236 
 312 
 389 
 465 
 542 
 
 935 
 
 *OI2 
 089 
 
 1 66 
 243 
 320 
 
 397 
 473 
 549 
 
 * 943 
 
 *O2O 
 9 7 
 174 
 251 
 328 
 404 
 48l 
 
 557 
 
 95o 
 
 *028 
 
 105 
 182 
 259 
 
 335 
 412 
 488 
 565 
 
 958 
 *o 3 5 
 
 "3 
 189 
 266 
 343 
 420 
 496 
 572 
 
 966 
 *043 
 
 120 
 197 
 274 
 
 351 
 427 
 504 
 580 
 
 i 0.8 
 
 2 1.6 
 
 3 2.4 
 4 3-2 
 5 4- 
 6 4.8 
 7 5-6 
 8 6.4 
 9 7-2 
 
 5?0 
 
 587 
 
 595 
 
 603 
 
 6ro 
 
 618 
 
 626 
 
 633 
 
 641 
 
 648 
 
 6 5 6 
 
 
 57i 
 572 
 573 
 574 
 575 
 576 
 577 
 573 
 579 
 
 664 
 740 
 
 8i5 
 
 967 
 76 042 
 118 
 193 
 
 268 
 
 671 
 747 
 823 
 899 
 974 
 050 
 125 
 200 
 275 
 
 679 
 755 
 831 
 906 
 982 
 057 
 133 
 208 
 283 
 
 686 
 762 
 838 
 914 
 989 
 065 
 140 
 215 
 290 
 
 694 
 770 
 846 
 921 
 997 
 072 
 148 
 223 
 298 
 
 702 
 778 
 853 
 929 
 *oos 
 080 
 155 
 230 
 305 
 
 % 
 
 861 
 937 
 
 *OI2 
 
 087 
 163 
 238 
 313 
 
 717 
 793 
 868 
 944 
 
 *020 
 
 095 
 170 
 245 
 320 
 
 724 
 800 
 876 
 952 
 
 *027 
 
 103 
 
 178 
 253 
 
 328 
 
 732 
 808 
 88 4 
 
 959 
 *035 
 no 
 185 
 260 
 335 
 
 
 580 
 
 343 
 
 350 
 
 35? 
 
 365 
 
 373 
 
 380 
 
 388 
 
 395 
 
 403 
 
 410 
 
 
 58i 
 
 582 
 583 
 584 
 585 
 586 
 587 
 588 
 589 
 
 418 
 492 
 567 
 641 
 716 
 790 
 864 
 938 
 77 012 
 
 425 
 500 
 574 
 649 
 723 
 797 
 871 
 945 
 019 
 
 433 
 507 
 582 
 656 
 730 
 805 
 879 
 953 
 026 
 
 440 
 515 
 589 
 664 
 738 
 812 
 886 
 960 
 034 
 
 448 
 522 
 597 
 671 
 745 
 819 
 893 
 967 
 041 
 
 455 
 530 
 604 
 678 
 753 
 827 
 901 
 975 
 048 
 
 462 
 
 & 
 
 686 
 760 
 
 834 
 908 
 982 
 056 
 
 470 
 
 619 
 693 
 768 
 842 
 916 
 989 
 063 
 
 477 
 552 
 626 
 701 
 775 
 849 
 923 
 997 
 070 
 
 485 
 559 
 634 
 708 
 782 
 856 
 930 
 
 *00 4 
 
 078 
 
 r 
 
 I -7 
 2 .4 
 
 4 !8 
 1 * 
 
 I 'I 
 
 590 
 
 085 
 
 "93 
 
 100 
 
 107 
 
 "5 
 
 122 
 
 129 
 
 137 
 
 144 
 
 151 
 
 I 1:5 
 
 59i 
 592 
 593 
 594 
 595 
 596 
 597 
 =98 
 599 
 
 159 
 232 
 305 
 379 
 452 
 525 
 597 
 670 
 743 
 
 1 66 
 240 
 313 
 
 386 
 
 459 
 532 
 605 
 677 
 750 
 
 173 
 247 
 320 
 393 
 466 
 
 539 
 612 
 685 
 
 757 
 
 181 
 254 
 327 
 401 
 474 
 546 
 619 
 692 
 764 
 
 188 
 262 
 335 
 408 
 481 
 554 
 627 
 699 
 772 
 
 195 
 269 
 342 
 415 
 488 
 561 
 634 
 706 
 
 779 
 
 203 
 276 
 
 349 
 422 
 
 495 
 568 
 641 
 
 714 
 
 786 
 
 210 
 283 
 
 357 
 430 
 503 
 576 
 648 
 721 
 793 
 
 217 
 291 
 364 
 437 
 5io 
 583 
 656 
 728 
 80 1 
 
 225 
 298 
 371 
 444 
 517 
 590 
 663 
 735 
 808 
 
 
 600 
 
 815 
 
 822 
 
 830 
 
 837 
 
 844 
 
 851 
 
 859 
 
 866 
 
 873 
 
 880 
 
 
 N. 
 
 L. o 
 
 i 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P.P.
 
 LOGARITHMS. 
 
 N. 
 
 L. o 
 
 i 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. 
 
 P. 
 
 600 
 
 77 815 
 
 822 
 
 830 
 
 837 
 
 844 
 
 851 
 
 859 
 
 866 
 
 873 
 
 880 
 
 
 
 601 
 602 
 603 
 604 
 605 
 606 
 607 
 608 
 609 
 
 887 
 960 
 78 032 
 104 
 176 
 247 
 3i9 
 390 
 462 
 
 895 
 967 
 
 039 
 in 
 183 
 254 
 326 
 398 
 469 
 
 9 02 
 
 974 
 046 
 118 
 190 
 262 
 333 
 405 
 476 
 
 909 
 981 
 053 
 125 
 i97 
 269 
 340 
 412 
 483 
 
 916 
 988 
 061 
 132 
 204 
 276 
 347 
 419 
 490 
 
 924 
 996 
 068 
 140 
 
 211 
 
 283 
 
 355 
 426 
 497 
 
 931 
 *oo3 
 075 
 147 
 219 
 290 
 362 
 433 
 504 
 
 938 
 
 *OIO 
 
 082 
 154 
 226 
 297 
 
 369 
 440 
 512 
 
 945 
 *oi7 
 089 
 161 
 233 
 305 
 376 
 447 
 519 
 
 952 
 
 *02 5 
 
 097 
 168 
 240 
 312 
 
 383 
 455 
 526 
 
 3 
 4 
 
 8 
 
 1.6 
 2.4 
 
 3-2 
 
 610 
 
 533 
 
 540 
 
 547 
 
 554 
 
 56i 
 
 569 
 
 576 
 
 583 
 
 59 
 
 597 
 
 S 
 6 
 
 4 8 
 
 611 
 612 
 613 
 614 
 615 
 616 
 617 
 618 
 619 
 
 604 
 675 
 746 
 817 
 888 
 958 
 79 029 
 099 
 169 
 
 611 
 
 682 
 753 
 824 
 895 
 965 
 036 
 106 
 176 
 
 618 
 689 
 760 
 831 
 902 
 972 
 043 
 H3 
 183 
 
 625 
 
 $ 
 
 838 
 909 
 
 979 
 050 
 
 120 
 
 190 
 
 633 
 704 
 774 
 845 
 916 
 986 
 057 
 127 
 197 
 
 640 
 711 
 78i 
 852 
 923 
 993 
 064 
 134 
 204 
 
 647 
 718 
 789 
 859 
 930 
 
 *000 
 
 071 
 
 141 
 
 211 
 
 654 
 725 
 796 
 866 
 937 
 *oo 7 
 078 
 148 
 218 
 
 66 1 
 732 
 803 
 873 
 944 
 *oi4 
 085 
 155 
 225 
 
 668 
 
 739 
 810 
 880 
 95i 
 
 *02I 
 9 2 
 162 
 232 
 
 I 
 9 
 
 5-6 
 6.4 
 7.2 
 
 620 
 
 239 
 
 246 
 
 253 
 
 260 
 
 267 
 
 274 
 
 28l 
 
 288 
 
 295 
 
 302 
 
 
 
 621 
 622 
 623 
 624 
 625 
 626 
 627 
 628 
 629 
 
 309 
 
 379 
 449 
 5i8 
 
 588 
 657 
 727 
 796 
 865 
 
 316 
 
 386 
 456 
 525' 
 595 
 664 
 734 
 803 
 872 
 
 323 
 393 
 463 
 S3 2 
 602 
 671 
 74i 
 810 
 879 
 
 330 
 400 
 470 
 
 529 
 609 
 678 
 748 
 817 
 886 
 
 337 
 407 
 477 
 546 
 616 
 685 
 754 
 824 
 893 
 
 344 
 414 
 
 484 
 553 
 623 
 692 
 761 
 831 
 900 
 
 351 
 421 
 491 
 560 
 630 
 6 99 
 768 
 837 
 9 06 
 
 358 
 428 
 498 
 567 
 637 
 706 
 775 
 844 
 913 
 
 365 
 435 
 505 
 574 
 644 
 713 
 782 
 851 
 920 
 
 372 
 
 442 
 
 5" 
 
 58i 
 650 
 720 
 789 
 858 
 927 
 
 I 
 2 
 
 3 
 4 
 
 i 
 
 7 
 8 
 9 
 
 7 
 
 1.4 
 
 2.1 
 
 3-5 
 4.2 
 4-9 
 5-6 
 6-3 
 
 630 
 
 934 
 
 941 
 
 948 
 
 955 
 
 962 
 
 969 
 
 975 
 
 982 
 
 983 
 
 996 
 
 
 
 631 
 632 
 633 
 634 
 635 
 636 
 637 
 638 
 639 
 
 80 003 
 072 
 140 
 209 
 277 
 346 
 414 
 482 
 550 
 
 OIO 
 
 079 
 
 147 
 216 
 284 
 353 
 421 
 489 
 557 
 
 017 
 
 085 
 154 
 223 
 291 
 
 359 
 428 
 496 
 564 
 
 024 
 092 
 161 
 229 
 298 
 366 
 434 
 502 
 570 
 
 030 
 099 
 1 68 
 236 
 305 
 373 
 441 
 509 
 577 
 
 037 
 106 
 r?5 
 243 
 312 
 380 
 448 
 5i6 
 584 
 
 044 
 U3 
 182 
 250 
 3i8 
 387 
 455 
 523 
 59i 
 
 051 
 
 120 
 
 188 
 257 
 325 
 393 
 462 
 530 
 598 
 
 058 
 127 
 195 
 264 
 332 
 400 
 468 
 
 536 
 604 
 
 065 
 134 
 
 202 
 271 
 
 339 
 407 
 
 475 
 543 
 611 
 
 
 6 
 
 640 
 
 618 
 
 625 
 
 632 
 
 638 
 
 645 
 
 652 
 
 659 
 
 665 
 
 672 
 
 679 
 
 
 2.4 
 
 641 
 642 
 643 
 644 
 645 
 646 
 647 
 648 
 649 
 
 686 
 
 754 
 821 
 889 
 956 
 81 023 
 090 
 158 
 224 
 
 693 
 760 
 828 
 895 
 963 
 030 
 097 
 164 
 231 
 
 699 
 767 
 835 
 902 
 969 
 037 
 104 
 171 
 238 
 
 706 
 774 
 841 
 909 
 976 
 043 
 in 
 178 
 245 
 
 713 
 781 
 848 
 916 
 983 
 050 
 117 
 184 
 251 
 
 720 
 787 
 855 
 922 
 990 
 057 
 124 
 191 
 258 
 
 726 
 794 
 862 
 929 
 996 
 064 
 131 
 198 
 265 
 
 733 
 80 1 
 868 
 936 
 *oo3 
 070 
 137 
 204 
 271 
 
 740 
 808 
 875 
 943 
 
 *OIO 
 
 077 
 144 
 
 211 
 
 2 7 8 
 
 747 
 814 
 882 
 
 949 
 *oi7 
 084 
 151 
 218 
 285 
 
 7 
 8 
 9 
 
 3 3 :6 
 4.2 
 4.8 
 
 5-4 
 
 650 
 
 291 
 
 298 
 
 305 
 
 3" 
 
 3i8 
 
 325 
 
 33i 
 
 338 
 
 345 
 
 35i 
 
 
 
 N. 
 
 L. o 
 
 i 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P 
 
 P.
 
 LOGARITHMS. 
 
 13 
 
 N. 
 
 L. o 
 
 i 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
 650 
 
 651 
 652 
 653 
 654 
 655 
 656 
 657 
 658 
 
 659 
 660 
 
 661 
 662 
 663 
 
 664 
 665 
 666 
 667 
 668 
 669 
 
 6TO 
 
 671 
 672 
 673 
 674 
 675 
 676 
 677 
 678 
 679 
 680 
 681 
 682 
 683 
 684 
 685 
 686 
 687 
 688 
 689 
 
 690 
 
 691 
 692 
 693 
 694 
 695 
 696 
 697 
 698 
 699 
 
 roo 
 
 81 291 
 
 298 
 
 305 
 
 3" 
 
 318 
 
 325 
 
 331 
 
 338 
 
 345 
 
 35i 
 
 r 
 
 1-4 
 
 2.1 
 
 2.8 
 
 3-5 
 4.2 
 4-9 
 5-6 
 6.3 
 
 6 
 
 i 0.6 
 
 2 1.2 
 
 3 1.8 
 4 2-4. 
 
 1 r.6 
 7 4-2 
 8 4.8 
 9 5-4- 
 
 358 
 425 
 491 
 558 
 624 
 690 
 757 
 823 
 889 
 
 365 
 43i 
 498 
 564 
 631 
 6 9 7 
 763 
 829 
 895 
 
 371 
 438 
 505 
 571 
 637 
 704 
 770 
 836 
 9 02 
 
 378 
 445 
 5" 
 578 
 644 
 710 
 776 
 842 
 908 
 
 385 
 451 
 5i8 
 584 
 651 
 717 
 783 
 849 
 _9i5_ 
 981 
 
 391 
 
 458 
 
 525 
 591 
 657 
 723 
 790 
 856 
 
 _91 
 
 987 
 
 053 
 119 
 
 184 
 
 249 
 315 
 
 380 
 
 445 
 5io 
 575 
 640 
 
 398 
 465 
 53i 
 598 
 664 
 73o 
 796 
 862 
 928 
 
 405 
 47i 
 538 
 604 
 671 
 737 
 803 
 869 
 935 
 
 411 
 478 
 544 
 611 
 677 
 743 
 809 
 875 
 941 
 
 418 
 485 
 55i- 
 617 
 68 4: 
 750 
 816 
 882 
 948 
 
 954 
 
 961 
 
 968 
 
 974 
 
 994 
 
 *ooo 
 
 *oo 7 
 
 *oi4 
 
 82 020 
 086 
 151 
 217 
 282 
 347 
 413 
 478 
 543 
 
 027 
 092 
 158 
 223 
 289 
 354 
 419 
 484 
 549 
 
 033 
 99 
 164 
 230 
 295 
 360 
 426 
 
 49 1 
 556 
 
 040 
 105 
 
 171 
 236 
 302 
 367 
 432 
 497 
 562 
 
 046 
 
 112 
 
 I 7 8 
 243 
 308 
 
 373 
 439 
 504 
 569 
 
 060 
 125 
 191 
 256 
 321 
 387 
 452 
 517 
 5&2 
 
 066 
 132 
 197 
 263 
 
 328 
 393 
 458 
 523 
 588 
 
 073 
 138 
 204 
 269 
 334 
 400 
 465 
 530 
 595 
 
 079 
 145 
 
 210 
 2 7 6 
 341 
 4 06 
 471 
 536 
 
 601 
 666 
 
 607 
 
 614 
 
 620 
 
 627 
 
 633 
 
 646 
 
 653 
 
 659 
 
 672 
 737 
 802 
 866 
 930 
 995 
 83 059 
 123 
 187 
 
 679 
 743 
 
 808 
 872 
 937 
 
 *OOI 
 
 065 
 129 
 193 
 
 685 
 750 
 814 
 879 
 943 
 *oo8 
 072 
 136 
 200 
 
 692 
 756 
 821 
 885 
 950 
 *oi4 
 078 
 142 
 206 
 
 698 
 763 
 827 
 892 
 956 
 
 *020 
 08 5 
 149 
 213 
 
 705 
 769 
 834 
 898 
 963 
 
 *02? 
 
 091 
 155 
 219 
 
 711 
 776 
 840 
 905 
 969 
 *o 33 
 097 
 161 
 225 
 
 718 
 782 
 847 
 911 
 975 
 *040 
 104 
 1 68 
 232 
 
 724 
 789 
 853 
 918 
 982 
 *046 
 no 
 174 
 238 
 
 730 
 795 
 860 
 924 
 988- 
 
 *052 
 
 117 
 
 181 
 245 
 
 251 
 
 257 
 
 264 
 
 270 
 
 276 
 340 
 404 
 467 
 531 
 
 594 
 6 5 8 
 721 
 784 
 847 
 910 
 
 973 
 036 
 098 
 161 
 223 
 286 
 348 
 410 
 473 
 
 283 
 
 289 
 
 296 
 
 302 
 
 jo8_ 
 372 
 436 
 499 
 563 
 626 
 689 
 753 
 816 
 879 
 
 315 
 378 
 442 
 506 
 569 
 632 
 696 
 
 759 
 822 
 
 321 
 385 
 448 
 5" 
 575 
 639 
 702 
 765 
 828 
 
 327 
 391 
 455 
 5i8 
 582 
 645 
 708 
 771 
 835 
 897 
 
 334 
 398 
 461 
 525 
 588 
 651 
 715 
 778 
 841 
 
 347 
 410 
 474 
 537 
 601 
 664 
 727 
 790 
 853 
 916 
 
 353 
 417 
 480" 
 544 
 607 
 670 
 734 
 797 
 860 
 
 923 
 
 359 
 423 
 487 
 550 
 613 
 677 
 740 
 803 
 866 
 
 366 
 429 
 493 
 556 
 620 
 683 
 746 
 809 
 872 
 
 885 
 
 891 
 
 904 
 
 929 
 
 935 
 
 942 
 
 948 
 84 on 
 073 
 136 
 198 
 261 
 323 
 386 
 448 
 
 954 
 017 
 080 
 142 
 205 
 267 
 330 
 392 
 454 
 5i6 
 
 960 
 023 
 086 
 148 
 
 211 
 273 
 336 
 398 
 460 
 
 522 
 
 967 
 029 
 092 
 155 
 217 
 280 
 342 
 404 
 466 
 528 
 
 979 
 042 
 105 
 167 
 230 
 292 
 354 
 417 
 479 
 
 985 
 048 
 in 
 173 
 236 
 298 
 361 
 423 
 485 
 
 992 
 055 
 117 
 1 80 
 242 
 305 
 367 
 429 
 _49L 
 553 
 
 998 
 06 r 
 123 
 186 
 248 
 3" 
 373 
 435 
 497 
 559 
 
 *oo 4 
 067 
 130 
 192 
 255 
 317 
 379 
 442 
 504 
 
 5io 
 
 535 
 
 541 
 
 547 
 
 566 
 
 N. 
 
 L. o 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P.
 
 14 
 
 LOGARITHMS. 
 
 N. 
 
 L. o 
 
 i 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P 
 
 p. 
 
 roo 
 
 84 510 
 
 516 
 
 522 
 
 528 
 
 535 
 
 541 
 
 547 
 
 553 
 
 559 
 
 566 
 
 
 
 RRRRRRRRR 
 
 572 
 634 
 696 
 757 
 819 
 880 
 942 
 8-3 003 
 ,065 
 
 578 
 640 
 702 
 763 
 825 
 887 
 948 
 009 
 071 
 
 584 
 646 
 7 08 
 770 
 831 
 893 
 
 954 
 016 
 077 
 
 590 
 652 
 
 776 
 837 
 899 
 960 
 
 022 
 08 3 
 
 597 
 658 
 720 
 782 
 844 
 905 
 967 
 028 
 089 
 
 60 3 
 
 665 
 726 
 788 
 850 
 911 
 973 
 034 
 095 
 
 609 
 671 
 733 
 794 
 856 
 917 
 
 979 
 040 
 
 101 
 
 615 
 
 6 77 
 
 739 
 800 
 862 
 924 
 985 
 046 
 107 
 
 621 
 683 
 745 
 807 
 868 
 930 
 991 
 052 
 114 
 
 628 
 689 
 
 813 
 874 
 936 
 997 
 058 
 
 120 
 
 2 
 
 3 
 
 r 
 
 -4 
 
 no 
 
 126 
 
 132 
 
 138 
 
 144 
 
 150 
 
 156 
 
 163 
 
 169 
 
 175 
 
 181 
 
 4 
 
 2.8 
 
 711 
 712 
 713 
 714 
 715 
 
 717 
 718 
 719 
 
 187 
 248 
 309 
 370 
 431 
 491 
 552 
 
 Cl2 
 673 
 
 193 
 254 
 315 
 
 376 
 
 437 
 497 
 558 
 618 
 679 
 
 199 
 260 
 321 
 382 
 443 
 503 
 564 
 625 
 685 
 
 205 
 266 
 327 
 
 388 
 
 449 
 509 
 570 
 631 
 691 
 
 211 
 
 272 
 
 333 
 394 
 
 455 
 
 576 
 637 
 6 9 7 
 
 217 
 278 
 
 339 
 400 
 461 
 522 
 582 
 643 
 703 
 
 224 
 285 
 345 
 406 
 467 
 528 
 588 
 649 
 709 
 
 230 
 291 
 
 352 
 412 
 
 473 
 534 
 594 
 655 
 715 
 
 236 
 297 
 358 
 418 
 479 
 540 
 600 
 661 
 721 
 
 242 
 303 
 364 
 425 
 485 
 546 
 606 
 667 
 727 
 
 6 
 7 
 8 
 9 
 
 4.2 
 4.9 
 5.6 
 6-3 
 
 r2o 
 
 733 
 
 739 
 
 745 
 
 751 
 
 757 
 
 763 
 
 769 
 
 775 
 
 78i 
 
 788 
 
 
 
 721 
 
 722 
 723 
 724 
 
 725 
 726 
 727 
 
 728 
 
 729 
 
 794 
 854 
 914 
 
 974 
 86 034 
 094 
 153 
 213 
 273 
 
 800 
 860 
 920 
 980 
 040 
 
 IOO 
 
 159 
 219 
 
 279 
 
 806 
 866 
 926 
 986 
 046 
 106 
 165 
 225 
 285 
 
 812 
 872 
 932 
 992 
 052 
 
 112 
 171 
 231 
 291 
 
 818 
 878 
 938 
 998 
 058 
 118 
 177 
 237 
 297 
 
 824 
 884 
 ^944 
 
 064 
 124 
 183 
 243 
 303 
 
 830 
 890 
 950 
 
 *OIO 
 
 070 
 130 
 189 
 
 249 
 308 
 
 836 
 896 
 956 
 *oi6 
 076 
 136 
 195 
 255 
 
 842 
 902 
 962 
 
 *022 
 O82 
 141 
 201 
 26l 
 32O 
 
 848 
 908 
 968 
 
 *028 
 
 088 
 147 
 207 
 
 267 
 
 326 
 
 2 
 
 3 
 4 
 
 I 
 I 
 
 Q 
 
 1.2 
 2-4 
 
 36 
 
 4.2 
 4.8 
 5-4 
 
 rao 
 
 332 
 
 338 
 
 344 
 
 350 
 
 356 
 
 362 
 
 368 
 
 374 
 
 380 
 
 386 
 
 
 
 731 
 
 732 
 
 733 
 734 
 735 
 736 
 737 
 738 
 739 
 
 392 
 45i 
 
 570 
 629 
 
 688 
 747 
 806 
 864 
 
 398 
 -457 
 
 576 
 635 
 694 
 
 753 
 812 
 870 
 
 404 
 463 
 
 522 
 
 641 
 700 
 759 
 
 817 
 876 
 
 410 
 469 
 528 
 587 
 646 
 705 
 7 64 
 823 
 882 
 
 415 
 475 
 534 
 593 
 652 
 711 
 770 
 829 
 888 
 
 421 
 481 
 540 
 599 
 658 
 717 
 776 
 835 
 894 
 
 427 
 487 
 546 
 605 
 664 
 723 
 782 
 841 
 900 
 
 433 
 493 
 552 
 611 
 670 
 
 729 
 788 
 847 
 906 
 
 439 
 499 
 558 
 617 
 676 
 735 
 794 
 853 
 911 
 
 445 
 504 
 564 
 623 
 682 
 741 
 800 
 859 
 917 
 
 3 
 
 5 
 
 740 
 
 923 
 
 929 
 
 935 
 
 Q4I 
 
 947 
 
 953 
 
 958 
 
 964 
 
 970 
 
 976 
 
 4 
 
 2 - 
 
 741 
 742 
 743 
 744 
 745 
 746 
 747 
 748 
 749 
 
 982 
 87 040 
 099 
 157 
 216 
 274 
 332 
 390 
 448 
 
 988 
 046 
 
 221 
 280 
 338 
 396 
 
 454 
 
 994 
 052 
 in 
 169 
 227 
 286 
 344 
 402 
 460 
 
 999 
 058 
 116 
 175 
 233 
 291 
 
 349 
 408 
 466 
 
 *oo 5 
 064 
 
 122 
 
 181 
 
 239 
 297 
 
 355 
 413 
 
 *OII 
 
 070 
 128 
 1 86 
 245 
 303 
 361 
 419 
 477 
 
 *oi? 
 075 
 134 
 192 
 251 
 309 
 367 
 425 
 483 
 
 *02 3 
 
 08 1 
 140 
 
 198 
 256 
 
 315 
 
 373 
 489 
 
 *02g 
 087 
 146 
 204 
 262 
 320 
 379 
 437 
 495 
 
 093 
 
 210 
 268 
 326 
 384 
 442 
 500 
 
 6 
 7 
 8 
 9 
 
 3- 
 4.0 
 4-5 
 
 rso 
 
 506 
 
 512 
 
 5i8 
 
 523 
 
 529 
 
 535 
 
 54i 
 
 547 
 
 552 
 
 558 
 
 
 
 N. 
 
 L. o 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. 
 
 P.
 
 LOGARITHMS. 
 
 15 
 
 N. 
 
 L. o 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
 rso 
 
 87 506 
 
 512 
 
 518 
 
 523 
 
 529 
 
 535 
 
 54i 
 
 547 
 
 552 
 
 558 
 
 
 751 
 
 752 
 
 753 
 754 
 755 
 756 
 757 
 758 
 759 
 
 564 
 622 
 679 
 
 737 
 795 
 852 
 910 
 967 
 88 024 
 
 570 
 628 
 685 
 
 743 
 800 
 858 
 915 
 973 
 030 
 
 576 
 633 
 6 9 I 
 
 749 
 806 
 864 
 921 
 978 
 036 
 
 581 
 639 
 
 697 
 
 754 
 812 
 869 
 927 
 984 
 041 
 
 587 
 645 
 703 
 760 
 818 
 875 
 933 
 990 
 047 
 
 593 
 651 
 708 
 766 
 823 
 881 
 938 
 996 
 053 
 
 599 
 656 
 714 
 772 
 829 
 887 
 944 
 
 *OOI 
 
 058 
 
 604 
 662 
 720 
 
 777 
 835 
 892 
 950 
 
 *00 7 
 
 064 
 
 610 
 
 668 
 726 
 783 
 841 
 898 
 955 
 *oi3 
 070 
 
 616 
 674 
 731 
 789 
 846 
 904 
 961 
 *oi8 
 076 
 
 
 r60 
 
 08 1 
 
 087 
 
 093 
 
 098 
 
 104 
 
 no 
 
 116 
 
 121 
 
 127 
 
 133 
 
 
 761 
 762 
 763 
 764 
 765 
 766 
 767 
 768 
 769 
 
 138 
 195 
 252 
 39 
 366 
 423 
 480 
 536 
 593 
 
 144 
 
 201 
 
 258 
 315 
 372 
 429 
 485 
 542 
 598 
 
 150 
 207 
 264 
 321 
 377 
 434 
 491 
 547 
 604 
 
 156 
 213 
 270 
 326 
 383 
 440 
 497 
 553 
 610 
 
 161 
 
 218 
 275 
 332 
 389 
 446 
 502 
 559 
 615 
 
 167 
 224 
 281 
 338 
 395 
 45i 
 508 
 564 
 621 
 
 173 
 230 
 
 287 
 343 
 400 
 457 
 513 
 570 
 627 
 
 178 
 235 
 2 9 2 
 
 349 
 
 406 
 
 463 
 519 
 
 5/6 
 632 
 
 184 
 241 
 298 
 355 
 412 
 468 
 525 
 58i 
 638 
 
 190 
 
 247 
 304 
 360 
 417 
 474 
 530 
 587 
 643 
 
 6 
 
 3 '^8 
 4 2.4 
 5 3- 
 6 3.6 
 
 I ::S 
 
 9 5-4 
 
 rro 
 
 649 
 
 655 
 
 660 
 
 666 
 
 672 
 
 677 
 
 683 
 
 689 
 
 694 
 
 700 
 
 
 771 
 
 772 
 
 773 
 774 
 775 
 776 
 777 
 778 
 779 
 
 705 
 762 
 8x8 
 
 874 
 930 
 986 
 89 042 
 098 
 i54 
 
 711 
 
 76 7 
 824 
 880 
 936 
 99 2 
 048 
 104 
 159 
 
 717 
 
 773 
 829 
 885 
 941 
 997 
 053 
 109 
 165 
 
 722 
 779 
 835 
 891 
 947 
 *oo3 
 
 059 
 H5 
 170 
 
 728 
 784 
 840 
 897 
 953 
 
 *00 9 
 
 064 
 
 120 
 I 7 6 
 
 734 
 790 
 846 
 902 
 958 
 *oi4 
 070 
 126 
 182 
 
 739 
 795 
 852 
 908 
 964 
 
 *020 
 076 
 .131 
 
 I8 7 
 
 745 
 801 
 857 
 913 
 969 
 
 *02 5 
 
 08 1 
 137 
 193 
 
 750 
 807 
 863 
 919 
 975 
 031 
 087 
 143 
 198 
 
 756 
 812 
 868 
 925 
 981 
 *o 37 
 092 
 148 
 204 
 
 
 rso 
 
 209 
 
 215 
 
 221 
 
 226 
 
 232 
 
 237 
 
 243 
 
 248 
 
 254 
 
 260 
 
 
 7 8i 
 782 
 783 
 784 
 785 
 7 86 
 787 
 788 
 789 
 
 265 
 321 
 376 
 432 
 4 P 7 
 542 
 597 
 653 
 708 
 
 271 
 326 
 
 382 
 
 437 
 492 
 548 
 603 
 658 
 713 
 
 276 
 332 
 
 387 
 
 443 
 498 
 553 
 609 
 664 
 719 
 
 282 
 337 
 393 
 448 
 504 
 559 
 614 
 669 
 724 
 
 28 7 
 
 343 
 
 39 8 
 454 
 509 
 564 
 620 
 675 
 73 
 
 293 
 
 348 
 404 
 459 
 515 
 570 
 625 
 680 
 735 
 
 2 9 8 
 
 354 
 409 
 465 
 520 
 575 
 631 
 686 
 741 
 
 304 
 
 360 
 
 415 
 470 
 
 526 
 
 581 
 
 636 
 
 691 
 
 746 
 
 310 
 365 
 421 
 476 
 531 
 586 
 642 
 6 9 7 
 752 
 
 315 
 37i 
 426 
 481 
 537 
 592 
 647 
 702 
 757 
 
 5 
 
 3 * 
 
 S 2. 
 6 3- 
 
 1 \-l 
 
 9 4-5 
 
 r90 
 
 763 
 
 768 
 
 774 
 
 779 
 
 785 
 
 790 
 
 796 
 
 801 
 
 807 
 
 812 
 
 
 791 
 792 
 793 
 794 
 795 
 796 
 797 
 798 
 799 
 
 818 
 873 
 927 
 982 
 90 037 
 091 
 146 
 
 200 
 255 
 
 823 
 878 
 933 
 988 
 042 
 097 
 151 
 206 
 260 
 
 829 
 883 
 938 
 993 
 048 
 
 102 
 157 
 211 
 266 
 
 834 
 889 
 944 
 998 
 053 
 108 
 162 
 217 
 271 
 
 840 
 894 
 949 
 *oo4 
 
 059 
 H3 
 168 
 
 222 
 276 
 
 845 
 900 
 
 * 955 
 *OO9 
 
 064 
 119 
 173 
 
 227 
 282 
 
 851 
 905 
 
 9 6 
 *oi5 
 069 
 124 
 179 
 233 
 287 
 
 856 
 911 
 
 * 966 
 
 *020 
 075 
 I2 9 
 I8 4 
 2 3 8 
 293 
 
 862 
 916 
 971 
 
 *026 
 
 080 
 135 
 
 189 
 
 244 
 
 298 
 
 867 
 922 
 977 
 *03i 
 086 
 140 
 195 
 249 
 304 
 
 
 800 
 
 -309 
 
 3H 
 
 320 
 
 325 
 
 33t 
 
 336 
 
 342 
 
 347 
 
 352 
 
 358 
 
 
 N. 
 
 L. 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P.
 
 16 
 
 LOGARITHMS. 
 
 N. 
 
 L. o 
 
 i 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
 800 
 
 9 o 309 
 
 3U 
 
 320 
 
 325 
 
 33i 
 
 336 
 
 342 
 
 347 
 
 352 
 
 358 
 
 
 80 1 
 802 
 803 
 804 
 805 
 806 
 807 
 808 
 809 
 
 363 
 417 
 472 
 526 
 580 
 634 
 687 
 741 
 795 
 
 369 
 423 
 477 
 53i 
 585 
 639 
 693 
 747 
 800 
 
 374 
 
 428 
 482 
 536 
 59 
 644 
 698 
 752 
 806 
 
 380 
 434 
 488 
 542 
 596 
 650 
 703 
 757 
 811 
 
 385 
 439 
 493 
 547 
 601 
 655 
 709 
 763 
 816 
 
 390 
 445 
 499 
 553 
 607 
 660 
 7H 
 768 
 822 
 
 396 
 45o 
 504 
 558 
 612 
 666 
 720 
 773 
 827 
 
 401 
 
 455 
 509 
 563 
 617 
 671 
 725 
 779 
 832 
 
 407 
 461 
 
 515 
 569 
 623 
 677 
 730 
 784 
 838 
 
 412 
 
 466 
 520 
 574 
 628 
 682 
 736 
 789 
 843 
 
 
 810 
 
 849 
 
 854 
 
 859 
 
 865 
 
 870 
 
 875 
 
 881 
 
 886 
 
 891 
 
 897 
 
 
 811 
 812 
 813 
 814 
 8i5 
 816 
 817 
 818 
 819 
 
 902 
 956 
 91 009 
 062 
 116 
 169 
 
 222 
 
 275 
 328 
 
 907 
 961 
 014 
 068 
 
 121 
 174 
 228 
 28l 
 
 334 
 
 913 
 966 
 020 
 073 
 126 
 1 80 
 233 
 286 
 
 339 
 
 918 
 972 
 025 
 078 
 132 
 185 
 238 
 291 
 344 
 
 924 
 977 
 030 
 084 
 137 
 190 
 243 
 297 
 350 
 
 929 
 982 
 036 
 089 
 142 
 196 
 249 
 302 
 355 
 
 934 
 988 
 041 
 094 
 148 
 
 2OI 
 254 
 
 307 
 360 
 
 940 
 993 
 046 
 
 100 
 
 153 
 206 
 
 259 
 312 
 365 
 
 945 
 998 
 052 
 105 
 
 158 
 
 212 
 
 265 
 318 
 371 
 
 950 
 *oo4 
 057 
 no 
 
 164 
 
 217 
 
 270 
 323 
 
 376 
 
 6 
 
 2 1.2 
 
 3 1.8 
 4 2.4 
 
 1 l6 
 
 7 4-2 
 8 4.8 
 9 5-4 
 
 820 
 
 381 
 
 387 
 
 392 
 
 397 
 
 403 
 
 408 
 
 413 
 
 418 
 
 424 
 
 429 
 
 
 821 
 822 
 823 
 824 
 825 
 826 
 827 
 828 
 829 
 
 434 
 487 
 540 
 593 
 645 
 698 
 751 
 803 
 855 
 
 440 
 492 
 545 
 598 
 65i 
 703 
 756 
 808 
 861 
 
 445 
 498 
 551 
 603 
 656 
 709 
 761 
 814 
 866 
 
 450 
 503 
 556 
 609 
 661 
 7H 
 766 
 819 
 871 
 
 455 
 508 
 56i 
 614 
 .666 
 719 
 772 
 824 
 876 
 
 461 
 514 
 566 
 619 
 672 
 724 
 777 
 829 
 882 
 
 466 
 519 
 572 
 624 
 677 
 730 
 782 
 834 
 887 
 
 471 
 524 
 
 577 
 630 
 682 
 735 
 787 
 840 
 892 
 
 477 
 529 
 582 
 635 
 687 
 740 
 793 
 845 
 897 
 
 482 
 535 
 587 
 640 
 693 
 745 
 798 
 850 
 003 
 
 
 830 
 
 908 
 
 913 
 
 918 
 
 924 
 
 929 
 
 934 
 
 939 
 
 944 
 
 950 
 
 955 
 
 
 831 
 832 
 833 
 834 
 835 
 836 
 837 
 838 
 
 839 
 
 960 
 
 92 012 
 065 
 II? 
 I6 9 
 221 
 273 
 324 
 376 
 
 965 
 018 
 070 
 
 122 
 
 !?4 
 226 
 278 
 330 
 
 381 
 
 971 
 023 
 075 
 127 
 
 179 
 231 
 
 283 
 335 
 387 
 
 976 
 028 
 080 
 
 IS 
 
 236 
 
 288 
 340 
 392 
 
 981 
 033 
 085 
 137 
 189 
 241 
 293 
 345 
 397 
 
 986 
 038 
 091 
 143 
 195 
 247 
 298 
 350 
 402 
 
 991 
 044 
 096 
 148 
 200 
 252 
 304 
 355 
 407 
 
 997 
 049 
 
 IOI 
 
 153 
 205 
 257 
 309 
 361 
 412 
 
 *002 
 054 
 1 06 
 158 
 210 
 262 
 3M 
 
 306 
 418 
 
 *oo7 
 
 059 
 in 
 163 
 215 
 267 
 
 319 
 371 
 423 
 
 * 
 
 2. 
 
 : 2. 
 
 I: 
 
 g 4- 
 
 840 
 
 428 
 
 433 
 
 438 
 
 443 
 
 449 
 
 454 
 
 459 
 
 464 
 
 469 
 
 474 
 
 
 841 
 842 
 843 
 844 
 845 
 846 
 847 
 848 
 849 
 
 480 
 531 
 583 
 634 
 
 686 
 737 
 788 
 840 
 891 
 
 485 
 536 
 588 
 
 639 
 691 
 742 
 793 
 
 845 
 896 
 
 49 
 542 
 593 
 645 
 696 
 747 
 799 
 850 
 901 
 
 495 
 547 
 598 
 650 
 701 
 752 
 804 
 855 
 906 
 
 500 
 
 552 
 603 
 655 
 706 
 758 
 809 
 860 
 911 
 
 505 
 557 
 609 
 660 
 711 
 763 
 814 
 865 
 916 
 
 5" 
 562 
 614 
 665 
 716 
 768 
 819 
 870 
 921 
 
 5i6 
 567 
 619 
 670 
 722 
 773 
 824 
 875 
 927 
 
 52i 
 572 
 624 
 675 
 727 
 778 
 829 
 881 
 932 
 
 526 
 
 578 
 629 
 681 
 732 
 783 
 834 
 886 
 937 
 
 
 850 
 
 942 
 
 947 
 
 952 
 
 957 
 
 962 
 
 967 
 
 973 
 
 978 
 
 983 
 
 <)** 
 
 
 N. 
 
 L. o 
 
 ; 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 1 
 
 9 
 
 P. P.
 
 LOGARITHMS. 
 
 17 
 
 N. 
 
 L. o 
 
 i 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. 
 
 P. 
 
 850 
 
 92 942 
 
 947 
 
 952 
 
 957 
 
 962 
 
 967 
 
 973 
 
 978 
 
 983 
 
 988 
 
 
 
 851 
 852 
 853 
 854 
 855 
 856 
 857 
 858 
 
 859 
 
 993 
 93 044 
 095 
 146 
 197 
 247 
 298 
 349 
 399 
 
 998 
 049 
 
 100 
 
 151 
 
 202 
 252 
 303 
 
 354 
 404 
 
 *00 3 
 
 054 
 105 
 156 
 207 
 
 258 
 308 
 
 359 
 409 
 
 *oo8 
 
 059 
 no 
 161 
 
 212 
 
 263 
 
 313 
 
 364 
 
 414 
 
 *oi 3 
 064 
 
 1 66 
 217 
 268 
 
 369 
 420 
 
 *oi8 
 069 
 
 120 
 
 171 
 
 222 
 273 
 323 
 
 374 
 
 425 
 
 *024 
 
 075 
 125 
 
 176 
 
 227 
 
 278 
 328 
 
 379 
 43 
 
 080 
 131 
 
 181 
 232 
 283 
 334 
 384 
 435 
 
 *034 
 085 
 136 
 186 
 237 
 288 
 339 
 389 
 440 
 
 *039 
 090 
 141 
 192 
 242 
 293 
 344 
 394 
 445 
 
 2 
 
 3 
 
 6 
 
 1.2 
 
 860 
 
 450 
 
 455 
 
 460 
 
 465 
 
 470 
 
 475 
 
 480 
 
 485 
 
 49 
 
 495 
 
 4 
 5 
 
 2.4 
 
 861 
 862 
 863 
 864 
 865 
 866 
 867 
 868 
 869 
 
 500 
 551 
 60 1 
 651 
 702 
 752 
 802 
 852 
 902 
 
 505 
 556 
 606 
 656 
 707 
 757 
 807 
 857 
 907 
 
 56i 
 611 
 66 1 
 712 
 762 
 812 
 862 
 
 515 
 
 566 
 
 616 
 666 
 717 
 767 
 8i7 
 867 
 917 
 
 520 
 
 621 
 671 
 
 722 
 772 
 822 
 872 
 922 
 
 526 
 576 
 626 
 676 
 727 
 777 
 827 
 877 
 927 
 
 631 
 682 
 732 
 
 782 
 832 
 882 
 932 
 
 536 
 586 
 636 
 687 
 737 
 787 
 837 
 887 
 937 
 
 541 
 59 1 
 641 
 692 
 742 
 792 
 842 
 892 
 942 
 
 546 
 596 
 646 
 697 
 747 
 797 
 847 
 897 
 947 
 
 6 
 7 
 8 
 9 
 
 4-2 
 
 5-4 
 
 sro 
 
 952 
 
 957 
 
 962 
 
 967 
 
 972 
 
 977 
 
 982 
 
 987 
 
 992 
 
 997 
 
 
 
 8?2 
 
 873 
 874 
 
 875 
 
 876 
 
 877 
 
 878 
 879 
 
 94 002 
 052 
 
 IOI 
 
 201 
 250 
 300 
 
 349 
 399 
 
 007 
 057 
 106 
 156 
 206 
 255 
 305 
 354 
 404 
 
 012 
 062 
 III 
 
 161 
 
 211 
 260 
 310 
 
 359 
 409 
 
 017 
 067 
 116 
 1 66 
 216 
 265 
 315 
 364 
 414 
 
 022 
 072 
 121 
 
 221 
 270 
 320 
 369 
 419 
 
 027 
 077 
 126 
 176 
 226 
 275 
 325 
 374 
 424 
 
 032 
 082 
 
 181 
 231 
 280 
 330 
 379 
 429 
 
 037 
 086 
 136 
 1 86 
 236 
 285 
 335 
 384 
 433 
 
 042 
 091 
 141 
 191 
 240 
 290 
 340 
 389 
 438 
 
 047 
 096 
 146 
 196 
 245 
 295 
 345 
 394 
 443 
 
 a 
 3 
 4 
 
 i 
 I 
 
 9 
 
 5. 
 
 10- ' ) 
 
 '5 y 
 
 2 0. 
 *J> 
 
 3 5 
 4 o 
 4 5 
 
 880 
 
 448 
 
 453 
 
 458 
 
 463 
 
 4 68 
 
 473 
 
 478 
 
 483 
 
 488 
 
 493 
 
 
 
 881 
 882 
 883 
 884 
 885 
 886 
 887 
 888 
 889 
 
 49 s 
 547 
 596 
 645 
 694 
 743 
 792 
 841 
 890 
 
 503 
 552 
 601 
 650 
 6c; 
 748 
 797 
 846 
 895 
 
 507 
 557 
 606 
 655 
 704 
 753 
 802 
 851 
 900 
 
 512 
 562 
 611 
 660 
 709 
 758 
 807 
 856 
 905 
 
 517 
 567 
 
 616 
 665 
 
 763 
 8'* 
 861 
 910 
 
 522 
 
 621 
 670 
 719 
 
 768 
 ,817 
 $66 
 915 
 
 527 
 5/6 
 626 
 675 
 724 
 773 
 822 
 871 
 919 
 
 532 
 
 630 
 680 
 729 
 
 778 
 827 
 876 
 924 
 
 537 
 586 
 635 
 685 
 734 
 783 
 832 
 880 
 929 
 
 542 
 
 640 
 
 689 
 738 
 787 
 836 
 885 
 934 
 
 2 
 
 3 
 
 4 
 
 1.2 
 
 890 
 
 939 
 
 944 
 
 949 
 
 954 
 
 959 
 
 963 
 
 968 
 
 973 
 
 978 
 
 9 8 3 
 
 4 
 
 1.6 
 
 891 
 892 
 893 
 894 
 895 
 896 
 897 
 898 
 899 
 
 988 
 95 036 
 085 
 134 
 182 
 231 
 279 
 328 
 376 
 
 993 
 041 
 090 
 139 
 187 
 236 
 284 
 332 
 
 998 
 046 
 095 
 143 
 192 
 240 
 289 
 337 
 386 
 
 *002 
 
 051 
 
 100 
 
 148 
 197 
 245 
 294 
 
 342 
 390 
 
 056 
 105 
 153 
 
 202 
 
 250 
 299 
 
 347 
 395 
 
 *OI2 
 06 1 
 109 
 158 
 207 
 255 
 303 
 352 
 400 
 
 *oi 7 
 066 
 114 
 163 
 
 211 
 260 
 308 
 
 357 
 405 
 
 *022 
 
 071 
 119 
 168 
 216 
 265 
 313 
 361 
 410 
 
 *027 
 
 075 
 
 124 
 173 
 
 221 
 270 
 318 
 366 
 415 
 
 *0 3 2 
 O80 
 129 
 177 
 226 
 274 
 323 
 371 
 419 
 
 6 
 7 
 8 
 9 
 
 1:3 
 
 1:2 
 
 900 
 
 424 
 
 429 
 
 434 
 
 439 
 
 444 
 
 44 8 
 
 453 
 
 458 
 
 463 
 
 468 
 
 
 
 N. 
 
 L. o 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. 
 
 p.
 
 18 
 
 LOGARITHMS. 
 
 N. 
 
 L. o 
 
 i 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
 900 
 
 95 424 
 
 429 
 
 434 
 
 439 
 
 444 
 
 448 
 
 453 
 
 458 
 
 463 
 
 468 
 
 
 901 
 902 
 903 
 904 
 95 
 906 
 907 
 908 
 909 
 
 472 
 52i 
 569 
 617 
 665 
 713 
 761 
 809 
 856 
 
 477 
 525 
 574 
 622 
 670 
 718 
 766 
 813 
 861 
 
 482 
 530 
 578 
 626 
 674 
 722 
 770 
 818 
 866 
 
 487 
 535 
 583 
 631 
 679 
 727 
 775 
 823 
 871 
 
 492 
 540 
 588 
 636 
 684 
 732 
 780 
 828 
 875 
 
 497 
 
 545 
 593 
 641 
 689 
 
 737 
 785 
 832 
 880 
 
 501 
 550 
 598 
 646 
 694 
 742 
 789 
 837 
 885 
 
 506 
 554 
 602 
 650 
 698 
 746 
 794 
 842 
 890 
 
 5" 
 
 559 
 607 
 
 655 
 703 
 75i 
 799 
 
 847 
 895 
 
 5i6 
 564 
 612 
 660 
 708 
 756 
 804 
 852 
 899 
 
 
 910 
 
 904 
 
 909 
 
 914 
 
 918 
 
 923 
 
 928 
 
 933 
 
 938 
 
 942 
 
 947 
 
 5 
 
 911 
 912 
 
 913 
 914 
 915 
 916 
 917 
 918 
 919 
 
 952 
 999 
 96 047 
 
 095 
 142 
 190 
 237 
 284 
 332 
 
 957 
 *oo4 
 052 
 099 
 147 
 194 
 242 
 289 
 336 
 
 961 
 *oog 
 057 
 104 
 152 
 199 
 246 
 294 
 34i 
 
 966 
 *oi 4 
 061 
 109 
 156 
 204 
 251 
 298 
 346 
 
 971 
 *org 
 066 
 114 
 161 
 209 
 256 
 303 
 350 
 
 976 
 
 *02 3 
 
 071 
 118 
 1 66 
 213 
 261 
 308 
 355 
 
 980 
 
 *028 
 
 076 
 123 
 i?i 
 218 
 265 
 313 
 360 
 
 985 
 *o 33 
 080 
 128 
 175 
 223 
 270 
 317 
 365 
 
 990 
 *038 
 085 
 133 
 1 80 
 227 
 275 
 322 
 369 
 
 995 
 
 *042 
 
 090 
 137 
 185 
 232 
 280 
 327 
 374 
 
 a i. 
 3 * 
 4 2- 
 
 5 2- 
 
 6 3- 
 
 8 4' 
 9 4- 
 
 920 
 
 379 
 
 384 
 
 388 
 
 393 
 
 398 
 
 402 
 
 407 
 
 412 
 
 417 
 
 421 
 
 
 921 
 922 
 923 
 924 
 925 
 926 
 927 
 928 
 929 
 
 426 
 473 
 520 
 567 
 614 
 661 
 708 
 755 
 802 
 
 43i 
 478 
 525 
 572 
 619 
 666 
 713 
 759 
 806 
 
 435 
 483 
 530 
 577 
 624 
 670 
 717 
 764 
 8n 
 
 440 
 487 
 534 
 58i 
 628 
 675 
 722 
 769 
 816 
 
 445 
 492 
 
 539 
 586 
 
 633 
 680 
 727 
 774 
 820 
 
 450 
 497 
 544 
 591 
 638 
 685 
 731 
 778 
 825 
 
 454 
 5oi 
 548 
 595 
 642 
 689 
 736 
 783 
 830 
 
 459 
 506 
 553 
 600 
 647 
 694 
 74i 
 788 
 834 
 
 464 
 5" 
 558 
 605 
 652 
 699 
 745 
 792 
 839 
 
 468 
 515 
 562 
 609 
 656 
 703 
 750 
 797 
 844 
 
 
 930 
 
 848 
 
 853 
 
 858 
 
 862 
 
 867 
 
 872 
 
 876 
 
 881 
 
 886 
 
 890 
 
 
 93i 
 932 
 933 
 934 
 935 
 936 
 937 
 938 
 939 
 
 8 9 5 
 942 
 988 
 97 035 
 08 1 
 128 
 174 
 
 220 
 267 
 
 900 
 946 
 993 
 039 
 086 
 132 
 179 
 225 
 271 
 
 904 
 95i 
 997 
 044 
 090 
 137 
 183 
 230 
 276 
 
 909 
 
 956 
 
 *002 
 049 
 095 
 142 
 
 188 
 234 
 280 
 
 914 
 
 960 
 *oo 7 
 053 
 
 100 
 
 146 
 192 
 239 
 
 285 
 
 918 
 965 
 
 *OII 
 
 058 
 104 
 151 
 197 
 243 
 290 
 
 923 
 970 
 *oi6 
 063 
 109 
 155 
 
 202 
 2 4 8 
 294 
 
 928 
 974 
 
 *02I 
 067 
 114 
 
 160 
 
 206 
 253 
 299 
 
 932 
 
 979 
 
 *025 
 
 072 
 118 
 165 
 
 211 
 
 257 
 304 
 
 937 
 984 
 *030 
 077 
 123 
 169 
 216 
 262 
 308 
 
 4 
 i 0.4 
 
 2 0.8 
 
 I ^ 
 1 - 
 
 940 
 
 313 
 
 317 
 
 322 
 
 327 
 
 331 
 
 336 
 
 340 
 
 345 
 
 350 
 
 354 
 
 8 3-J 
 
 9 3- 6 
 
 941 
 942 
 943 
 944 
 945 
 946 
 947 
 948 
 
 949 
 
 359 
 405 
 45i 
 497 
 543 
 589 
 
 727 
 
 364 
 410 
 456 
 502 
 548 
 594 
 640 
 685 
 731 
 
 368 
 414 
 460 
 506 
 552 
 598 
 644 
 690 
 736 
 
 373 
 419 
 465 
 5ii 
 557 
 603 
 649 
 695 
 740 
 
 377 
 424 
 470 
 5i6 
 562 
 607 
 653 
 699 
 745 
 
 382 
 428 
 474 
 520 
 566 
 612 
 658 
 704 
 749 
 
 387 
 
 433 
 479 
 525 
 571 
 617 
 663 
 708 
 754 
 
 39i 
 437 
 483 
 529 
 575 
 621 
 667 
 713 
 759 
 
 396 
 442 
 488 
 
 534 
 580 
 626 
 672 
 717 
 763 
 
 400 
 447 
 493 
 539 
 585 
 630 
 676 
 722 
 768 
 
 
 950 
 
 772 
 
 777 
 
 782 
 
 786 
 
 791 
 
 795 
 
 800 
 
 804 
 
 809 
 
 8i3 
 
 
 N. 
 
 L. o 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P.
 
 LOGARITHMS. 
 
 19 
 
 N. 
 
 L. o 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
 950 
 
 97 772 
 
 777 
 
 782 
 
 786 
 
 791 
 
 795 
 
 800 
 
 804 
 
 809 
 
 813 
 
 
 95i 
 952 
 953 
 954 
 955 
 956 
 957 
 953 
 959 
 
 818 
 864 
 909 
 955 
 98 ooo 
 046 
 
 091 
 137 
 182 
 
 823 
 868 
 914 
 
 959 
 005 
 050 
 096 
 141 
 186 
 
 827 
 873 
 9 I8 
 964 
 00 9 
 055 
 100 
 
 146 
 191 
 
 832 
 877 
 923 
 968 
 014 
 059 
 105 
 150 
 i95 
 
 836 
 882 
 928 
 973 
 019 
 064 
 log 
 155 
 
 200 
 
 8 4I 
 886 
 932 
 978 
 023 
 068 
 114 
 
 159 
 204 
 
 845 
 891 
 937 
 982 
 028 
 073 
 118 
 164 
 209 
 
 850 
 896 
 941 
 987 
 032 
 078 
 123 
 168 
 214 
 
 855 
 900 
 946 
 991 
 037 
 082 
 127 
 i?3 
 218 
 
 859 
 905 
 95o 
 996 
 041 
 087 
 132 
 i?7 
 223 
 
 
 960 
 
 227 
 
 232 
 
 236 
 
 241 
 
 245 
 
 250 
 
 254 
 
 259 
 
 263 
 
 268 
 
 
 961 
 962 
 963 
 964 
 965 
 966 
 967 
 968 
 969 
 
 272 
 
 318 
 363 
 
 408 
 453 
 498 
 543 
 588 
 632 
 
 277 
 322 
 367 
 412 
 457 
 502 
 547 
 592 
 637 
 
 281 
 327 
 372 
 417 
 
 462 
 
 507 
 
 552 
 
 597 
 641 
 
 286 
 33i 
 376 
 421 
 466 
 5" 
 556 
 60 1 
 646 
 
 2 9 
 336 
 381 
 426 
 471 
 5 I6 
 561 
 60 5 
 650 
 
 295 
 340 
 385 
 430 
 475 
 520 
 565 
 610 
 655 
 
 299 
 345 
 390 
 435 
 480 
 525 
 570 
 614 
 659 
 
 304 
 349 
 394 
 439 
 484 
 529 
 574 
 619 
 664 
 
 308 
 354 
 399 
 444 
 489 
 534 
 579 
 623 
 668 
 
 313 
 358 
 403 
 448 
 493 
 538 
 583 
 628 
 673 
 
 5 
 
 2 I. 
 
 3 i- 
 
 i i: 
 
 I I: 
 
 9 4- 
 
 970 
 
 677 
 
 682 
 
 686 
 
 691 
 
 6 9 5 
 
 700 
 
 704 
 
 709 
 
 713 
 
 717 
 
 
 97i 
 972 
 973 
 974 
 975 
 976 
 977 
 978 
 
 979 
 
 722 
 767 
 8n 
 856 
 900 
 945 
 989 
 99 034 
 078 
 
 726 
 771 
 816 
 860 
 905 
 949 
 994 
 038 
 083 
 
 731 
 776 
 820 
 865 
 909 
 954 
 998 
 043 
 087 
 
 735 
 780 
 825 
 869 
 914 
 958 
 *oo 3 
 047 
 092 
 
 740 
 784 
 829 
 874 
 9 I8 
 963 
 
 *oo 7 
 052 
 096 
 
 744 
 789 
 834 
 878 
 923 
 967 
 
 *OI2 
 056 
 IOO 
 
 749 
 793 
 838 
 883 
 927 
 972 
 *oi6 
 06 1 
 105 
 
 753 
 798 
 843 
 887 
 932 
 976 
 
 *02I 
 06 5 
 I0 9 
 
 758 
 802 
 847 
 892 
 936 
 981 
 
 *025 
 
 069 
 
 114 
 
 762 
 807 
 851 
 896 
 941 
 985 
 
 *O2g 
 
 074 
 
 118 
 
 
 980 
 
 123 
 
 127 
 
 131 
 
 136 
 
 140 
 
 145 
 
 149 
 
 154 
 
 158 
 
 162 
 
 
 981 
 982 
 
 983 
 984 
 9 8 5 
 986 
 987 
 988 
 989 
 
 167 
 
 211 
 
 255 
 300 
 
 344 
 388 
 432 
 476 
 520 
 
 i?i 
 216 
 260 
 304 
 
 348 
 392 
 436 
 480 
 
 524 
 
 176 
 220 
 
 264 
 308 
 352 
 396 
 
 441 
 
 484 
 528 
 
 1 80 
 224 
 269 
 
 313 
 357 
 401 
 445 
 489 
 533 
 
 185 
 229 
 273 
 317 
 361 
 45 
 449 
 493 
 537 
 
 l8g 
 233 
 277 
 322 
 366 
 410 
 
 454 
 498 
 542 
 
 193 
 238 
 282 
 326 
 370 
 414 
 458 
 502 
 546 
 
 198 
 242 
 286 
 330 
 
 374 
 419 
 463 
 506 
 550 
 
 202 
 247 
 2 9 I 
 
 335 
 379 
 423 
 467 
 Sir 
 555 
 
 207 
 251 
 295 
 339 
 383 
 427 
 471 
 515 
 559 
 
 4 
 
 2 ois 
 
 \ \'l 
 5 !<> 
 
 6 2.4 
 7 2.8 
 8 3.2 
 9 3-6 
 
 990 
 
 564 
 
 568 
 
 572 
 
 577 
 
 58i 
 
 585 
 
 590 
 
 594 
 
 599 
 
 603 
 
 
 991 
 992 
 993 
 994 
 995 
 996 
 997 
 998 
 
 999 
 
 607 
 651 
 695 
 739 
 782 
 826 
 870 
 913 
 957 
 
 612 
 656 
 699 
 743 
 787 
 830 
 874 
 917 
 961 
 
 616 
 660 
 704 
 747 
 791 
 835 
 878 
 922 
 965 
 
 621 
 664 
 708 
 752 
 795 
 839 
 883 
 926 
 970 
 
 625 
 669 
 712 
 756 
 800 
 843 
 887 
 930 
 974 
 
 629 
 673 
 717 
 760 
 804 
 848 
 891 
 935 
 978 
 
 634 
 677 
 721 
 765 
 808 
 852 
 896 
 939 
 983 
 
 638 
 682 
 726 
 769 
 8i3 
 856 
 900 
 944 
 987 
 
 642 
 686 
 730 
 774 
 817 
 861 
 904 
 948 
 991 
 
 647 
 691 
 734 
 778 
 822 
 865 
 909 
 952 
 996 
 
 
 1000 
 
 00 000 
 
 004 
 
 009 
 
 013 
 
 017 
 
 022 
 
 026 
 
 030 
 
 035 
 
 039 
 
 
 N. 
 
 L. o 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P.
 
 TABLES 
 
 OF 
 
 NATURAL SINES, COSINES, 
 
 TANGENTS, 
 AND COTANGENTS 
 
 GIVING THE VALUES OF THE FUNCTIONS FOR 
 ALL DEGREES AND MINUTES FROM 
 
 O TO QO
 
 NATURAL SINES AND COSINES. 
 
 
 
 
 
 I 
 
 
 
 2 
 
 
 
 3 
 
 
 4 
 
 
 
 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 
 
 .00000 
 
 .00029 
 
 
 01745 
 .01774 
 
 .99985 
 .99984 
 
 .03490 
 03519 
 
 99939 
 .99938 
 
 05234 
 .05263 
 
 .99863 
 .99861 
 
 .06976 
 .07005 
 
 99756 
 99754 
 
 Go 
 
 TO 
 
 
 .00058 
 
 
 .01803 
 
 .99984 
 
 .03548 
 
 99937 
 
 .05292 
 
 .99^.60 
 
 .07034 
 
 99752 
 
 5 
 
 
 .00087 
 
 
 .01832 
 
 99983 
 
 3577 
 
 .99936 
 
 .05321 
 
 .99858 
 
 07063 
 
 .99750 
 
 =,7 
 
 
 .00116 
 
 
 .01862 
 
 .99983 
 
 .03606 
 
 99935 
 
 05350 
 
 99857 
 
 .07092 
 
 99748 
 
 56 
 
 
 .00145 
 
 
 .01891 
 
 .99982 
 
 03635 
 
 99934 
 
 05379 
 
 99855 
 
 .07121 
 
 .99746 
 
 55 
 
 
 .00175 
 
 
 .01920 
 
 .99982 
 
 .03664 
 
 99933 
 
 .05408 
 
 .99854 
 
 .07150 
 
 -99744 
 
 54 
 
 
 .00204 
 
 
 .01949 
 
 .99981 
 
 03693 
 
 99932 
 
 05437 
 
 99852 
 
 .07179 
 
 .99742 
 
 53 
 
 
 .00233 
 
 
 .01978 
 
 .99980 
 
 .03723 
 
 9993 1 
 
 .05466 
 
 .99851 
 
 .07208 
 
 99740 
 
 52 
 
 
 .00262 
 
 
 .02007 
 
 .99980 
 
 .03752 
 
 .99930 
 
 05495 
 
 .99849 
 
 .07237 
 
 99738 
 
 51 
 
 1 
 
 .00291 
 
 
 
 99979 
 
 03781 
 
 99929 
 
 .05524 
 
 99847 
 
 .07266 
 
 .99736 
 
 50 
 
 , 
 
 .00320 
 
 .99999 
 
 .02065 
 
 99979 
 
 .03810 
 
 99927 
 
 05553 
 
 .99846 
 
 .07295 
 
 99734 
 
 4" 
 
 I 
 
 .00349 
 
 .99999 
 
 .02094 
 
 .99978 
 
 .03839 
 
 .99926 
 
 .05582 
 
 .99844 
 
 .07324 
 
 9973 1 
 
 4S 
 
 I 
 
 .00378 
 
 .99999 
 
 .02123 
 
 99977 
 
 .03868 
 
 99925 
 
 .05611 
 
 .99842 
 
 7353 
 
 -99729 
 
 47 
 
 I. 
 
 .00407 
 
 .99999 
 
 .02152 
 
 99977 
 
 .03897 
 
 99924 
 
 .05640 
 
 .99841 
 
 .07382 
 
 .99727 
 
 4" 
 
 I : 
 
 .00436 
 
 
 .02181 
 
 .99976 
 
 .03926 
 
 .99923 
 
 .05669 
 
 99839 
 
 .07411 
 
 99725 
 
 45 
 
 I 
 
 .00465 
 
 .99999 
 
 .O22II 
 
 .99976 
 
 03955 
 
 .99922 
 
 .05698 
 
 99838 
 
 .07440 
 
 .99723 
 
 44 
 
 1? 
 
 .00495 
 
 .99999 
 
 .02240 
 
 99975 
 
 .03984 
 
 .99921 
 
 05727 
 
 .99836 
 
 .07469 
 
 .99721 
 
 43 
 
 18 
 
 .00524 
 
 .99999 
 
 .02269 
 
 99974 
 
 .04013 
 
 .99919 
 
 
 09834 
 
 .07498 
 
 .99719 
 
 42 
 
 '9 
 
 00553 
 
 .99998 
 
 .02298 
 
 99974 
 
 .04042 
 
 .99918 
 
 .05785 
 
 99833 
 
 07527 
 
 .99716 
 
 4 1 
 
 
 .00582 
 
 .99998 
 
 .02327 
 
 99973 
 
 .04071 
 
 .99917 
 
 .05814 
 
 .99831 
 
 07556 
 
 .09714 
 
 4'> 
 
 21 
 
 .00611 
 
 .99908 
 
 .02356 
 
 .99972 
 
 .04100 
 
 .99916 
 
 05844 
 
 .99829 
 
 07585 
 
 .99712 
 
 39 
 
 22 
 
 .00640 
 
 .99998 
 
 02385 
 
 .99972 
 
 .04129 
 
 99915 
 
 05873 
 
 .99827 
 
 .07614 
 
 .99710 
 
 j8 
 
 23 
 
 .00665 
 
 .99998 
 
 .02414 
 
 .99971 
 
 .04159 
 
 .99913 
 
 .05902 
 
 .99826 
 
 .07643 
 
 .99708 
 
 37 
 
 24 
 
 .00698 
 
 .99998 
 
 .02443 
 
 .99970 
 
 .04188 
 
 .99912 
 
 Q593 1 
 
 .99824 
 
 .07672 
 
 99705 
 
 3<- 
 
 25 
 
 .00727 
 
 .09997 
 
 .02472 
 
 .99969 
 
 .04217 
 
 .99911 
 
 .05960 
 
 .99822 
 
 .07701 
 
 99703 
 
 35 
 
 26 
 
 .00756 
 
 .99997 
 
 .02501 
 
 .99969 
 
 .04246 
 
 .99910 
 
 .05989 
 
 .99821 
 
 .07730 
 
 .09701 
 
 34 
 
 27 
 
 .00785 
 
 .99997 
 
 .02530 
 
 .99968 
 
 .04275 
 
 .99909 
 
 .06018 
 
 .99819 
 
 07759 
 
 .99699 
 
 33 
 
 29 
 
 .00814 
 .00844 
 
 .99997 
 .99996 
 
 .02560 
 .02589 
 
 .90967 
 .99966 
 
 .04304 
 04333 
 
 .99907 
 .99906 
 
 .06047 
 .06076 
 
 .99817 
 .99815 
 
 .07788 
 .07817 
 
 .99696 
 .99694 
 
 32 
 3 1 
 
 3 
 
 .00873 
 
 .99996 
 
 .026X8 
 
 .99966 
 
 .04362 
 
 99905 
 
 .06105 
 
 .99813 
 
 .07846 
 
 .99692 
 
 3 
 
 31 
 
 32 
 
 .00902 
 .00931 
 
 .99996 
 .99996 
 
 .026 4 7 
 .02676 
 
 .99965 
 99964 
 
 .04391 
 .04420 
 
 .99904 
 .99902 
 
 .06134 
 .06163 
 
 .99812 
 .99810 
 
 07875 
 .07904 
 
 .99689 
 .99687 
 
 3 
 
 33 
 
 .00960 
 
 99995 
 
 .02705 
 
 99963 
 
 .04449 
 
 .99901 
 
 .06192 
 
 .99808 
 
 07933 
 
 99685 
 
 27 
 
 34 
 
 .00989 
 
 99995 
 
 02734 
 
 99963 
 
 .04478 
 
 .09000 
 
 .06221 
 
 .99806 
 
 .07962 
 
 .99683 
 
 20 
 
 P 
 
 .01047 
 
 99995 
 99995 
 
 .02763 
 .02792 
 
 .99962 
 .99961 
 
 .04507 
 04536 
 
 .99898 
 99897 
 
 .06250 
 .06279 
 
 .99804 
 .99803 
 
 .07991 
 .08020 
 
 .99680 
 .99678 
 
 25 
 24 
 
 57 
 
 .01076 
 
 .99994 
 
 .O282I 
 
 .99960 
 
 04565 
 
 .99896 
 
 .06308 
 
 .99801 
 
 .08049 
 
 .99676 
 
 23 
 
 38 
 
 .01105 
 
 99994 
 
 .02850 
 
 99959 
 
 04594 
 
 99894 
 
 06337 
 
 99799 
 
 .08078 
 
 .99673 
 
 
 39 
 
 .01134 
 
 99994 
 
 .02879 
 
 .99959 
 
 .04623 
 
 99893 
 
 .06366 
 
 99797 
 
 .08.07 
 
 .99671 
 
 21 
 
 
 .01164 
 
 99993 
 
 
 99958 
 
 04653 
 
 .99892 
 
 .06395 
 
 99795 
 
 .08,36 
 
 .99668 
 
 20 
 
 4i 
 
 .01193 
 
 99993 
 
 .02938 
 
 99957 
 
 .04682 
 
 .99890 
 
 .06424 
 
 99793 
 
 .08165 
 
 .99666 
 
 9 
 
 42 
 
 .01222 
 
 99993 
 
 .02967 
 
 99956 
 
 .04711 
 
 .99889 
 
 .06453 
 
 99792 
 
 .-08194 
 
 .99664 
 
 a 
 
 43 
 
 .OI25I 
 
 .99992 
 
 .02996 
 
 99955 
 
 .04740 
 
 .99888 
 
 .06482 
 
 .99790 
 
 .08223 
 
 .99661 
 
 7 
 
 44 
 
 .01280 
 
 99992 
 
 .03025 
 
 99954 
 
 04769 
 
 .99886 
 
 .06511 
 
 .99788 
 
 .08252 
 
 .99650 
 
 (i 
 
 45 
 
 01309 
 
 .99991 
 
 03054 
 
 99953 
 
 .04798 
 
 99885 
 
 .06540 
 
 .99786 
 
 .08281 
 
 99657 
 
 5 
 
 4 f > 
 
 .0 33 8 
 
 99991 
 
 03083 
 
 99952 
 
 .04827 
 
 .99883 
 
 .06569 
 
 .99784 
 
 .08310 
 
 99654 
 
 4 
 
 47 
 
 .0 367 
 
 .99991 
 
 
 .99952 
 
 .04856 
 
 .99882 
 
 .06598 
 
 .99782 
 
 08339 
 
 .99652 
 
 i 
 
 48 
 
 .0 396 
 
 99990 
 
 .03141 
 
 99951 
 
 .04885 
 
 .99881 
 
 .06627 
 
 .99780 
 
 .08368 
 
 .99649 
 
 
 
 
 .0 425 
 
 99990 
 
 .03170 
 
 .99950 
 
 .04914 
 
 .99879 
 
 .06656 
 
 99778 
 
 .08397 
 
 99647 
 
 i 
 
 50 
 
 .0 454 
 
 .99989 
 
 .03109 
 
 99949 
 
 .04943 
 
 .99878 
 
 .06685 
 
 .99776 
 
 .08426 
 
 99644 
 
 
 
 Si 
 
 0483 
 
 .09989 
 
 .03228 
 
 .99948 
 
 .04972 
 
 99876 
 
 .06714 
 
 99774 
 
 08455 
 
 .99642 
 
 g 
 
 52 
 
 o 5'3 
 
 .99989 
 
 03257 
 
 99947 
 
 .05001 
 
 99875 
 
 .06743 
 
 .99772 
 
 .08484 
 
 .99639 
 
 
 
 53 
 54 
 
 .0 57! 
 
 .99988 
 .99988 
 
 .03286 
 .03316 
 
 .99946 
 99945 
 
 .05030 
 05059 
 
 99873 
 .99872 
 
 .06773 
 .06802 
 
 .99770 
 .99768 
 
 .08513 
 .08542 
 
 99637 
 99D35 
 
 7 
 6 
 
 55 
 
 .0 600 
 
 .99987 
 
 03345 
 
 .99944 
 
 .05088 
 
 .99870 
 
 .06831 
 
 .99766 
 
 .08571 
 
 .99632 
 
 5 
 
 5* 
 
 .0 629 
 
 99987 
 
 03374 
 
 99943 
 
 .05117 
 
 .99869 
 
 .06860 
 
 .99764 
 
 .08600 
 
 .99630 
 
 4 
 
 =57 
 
 .0 658 
 
 .99986 
 
 
 .99942 
 
 .05146 
 
 .99867 
 
 .06889 
 
 .99762 
 
 .08629 
 
 .99627 
 
 3 
 
 58 
 
 .0687 
 
 .99986 
 
 03432 
 
 .99941 
 
 05175 
 
 .99866 
 
 .06918 
 
 .99760 
 
 .08658 
 
 .99625 
 
 2 
 
 59 
 60 
 
 .0 716 
 
 .99985 
 .99985 
 
 .03461 
 .03490 
 
 .99940 
 99939 
 
 .05205 
 .05234 
 
 .90864 
 .99863 
 
 .06947 
 .06976 
 
 99758 
 99756 
 
 .08687 
 .08716 
 
 .09622 
 .99619 
 
 ,', 
 
 , 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 , 
 
 
 8 
 
 f 
 
 8 
 
 \ 
 
 8 
 
 f 
 
 8 
 
 
 8 
 
 ,0 

 
 NATURAL SINES AND COSINES. 
 
 
 5 
 
 
 
 6 
 
 
 
 7 
 
 
 
 8 
 
 
 
 9 
 
 o 
 
 
 
 Sine 
 
 Cosine 
 
 S ne 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 
 11 
 
 .08716 
 .08745 
 
 .99619 
 .99617 
 
 10453 
 . 10482 
 
 99452 
 
 99449 
 
 .12187 
 .12216 
 
 99255 
 99251 
 
 .13917 
 .13946 
 
 .99027 
 .99023 
 
 15643 
 .15672 
 
 .98769 
 .98764 
 
 60 
 
 2 
 
 .08774 
 
 .99614 
 
 .10511 
 
 99446 
 
 .12245 
 
 .99248 
 
 13975 
 
 .99019 
 
 .15701 
 
 .98760 
 
 S<"> 
 
 3 
 
 .08803 
 
 .99612 
 
 10540 
 
 99443 
 
 12274 
 
 99244 
 
 .14004 
 
 .99015 
 
 1573 
 
 98755 
 
 ^7 
 
 4 
 
 .08831 
 
 .99609 
 
 .10569 
 
 .99440 
 
 .12302 
 
 . 99240 
 
 14033 
 
 .99011 
 
 15758 
 
 98751 
 
 56 
 
 
 .08860 
 
 .99607 
 
 10597 
 
 99437 
 
 .12331 
 
 99237 
 
 .1406! 
 
 .99006 
 
 15787 
 
 .98746 
 
 ss 
 
 6 
 
 .08889 
 
 .99604 
 
 .10626 
 
 99434 
 
 .12360 
 
 99233 
 
 .14000 
 
 .99002 
 
 .15816 
 
 98741 
 
 54 
 
 7 
 
 .08918 
 
 .99602 
 
 .10655 
 
 9943 1 
 
 12389 
 
 .09230 
 
 
 .98998 
 
 15845 
 
 98737 
 
 5 \ 
 
 8 
 
 .08947 
 
 99599 
 
 . 10684 
 
 .99428 
 
 .12418 
 
 .99226 
 
 .14148 
 
 98994 
 
 15873 
 
 98732 
 
 =,-'- 
 
 g 
 
 .08976 
 
 .99596 
 
 .10713 
 
 .99424 
 
 12447 
 
 .99222 
 
 .14177 
 
 98990 
 
 .15902 
 
 .98728 
 
 =,' 
 
 10 
 
 .09005 
 
 99594 
 
 . 10742 
 
 .99421 
 
 .12476 
 
 .99219 
 
 . 14205 
 
 .98986 
 
 15931 
 
 .98723 
 
 
 ,, 
 
 09034 
 
 99591 
 
 .10771 
 
 .99418 
 
 . 12504 
 
 99215 
 
 14234 
 
 .98982 
 
 "5959 
 
 .98718 
 
 49 
 
 12 
 
 .09063 
 
 .99588 
 
 .10800 
 
 .99415 
 
 12533 
 
 .99211 
 
 .14263 
 
 .98978 
 
 . 15988 
 
 98714 
 
 48 
 
 13 
 
 .09092 
 
 .99586 
 
 . 10829 
 
 .99412 
 
 . I25 62 
 
 .99208 
 
 .14292 
 
 .98973 
 
 . 16017 
 
 .98709 
 
 47 
 
 
 .09121 
 
 99583 
 
 . 10858 
 
 .99409 
 
 12591 
 
 .99204 
 
 14320 
 
 .98969 
 
 .16046 
 
 .98704 
 
 4" 
 
 IS 
 
 09150 
 
 .99580 
 
 . 10887 
 
 .99406 
 
 . 12620 
 
 .99200 
 
 14349 
 
 .98965 
 
 .16074 
 
 .98700 
 
 41 
 
 If, 
 
 .09179 
 
 99578 
 
 . 10916 
 
 .99402 
 
 . 12649 
 
 .99197 
 
 14378 
 
 .9896! 
 
 .16103 
 
 98695 
 
 44 
 
 17 
 
 .09208 
 
 99575 
 
 .10945 
 
 99399 
 
 .12678 
 
 99193 
 
 .14407 
 
 98957 
 
 .16132 
 
 .98690 
 
 4 i 
 
 t8 
 
 .09237 
 
 99572 
 
 10973 
 
 .99396 
 
 .12706 
 
 .99189 
 
 .,4436 
 
 .98953 
 
 .16160 
 
 .98686 
 
 
 19 
 
 .09266 
 
 99570 
 
 . IOO2 
 
 99393 
 
 12735 
 
 .99186 
 
 .14464 
 
 .98948 
 
 .16189 
 
 .98681 
 
 4' 
 
 20 
 
 .09295 
 
 99567 
 
 . 1031 
 
 .99390 
 
 .12764 
 
 .99182 
 
 .14493 
 
 .98944 
 
 .16218 
 
 .98676 
 
 40 
 
 22 
 
 09324 
 09353 
 
 .99564 
 .99562 
 
 . 1O6O 
 . 08 9 
 
 .99386 
 99383 
 
 "793 
 
 .12822 
 
 .99178 
 
 99*75 
 
 14522 
 M55i 
 
 .98940 
 .98936 
 
 .16246 
 .16275 
 
 .98671 
 .98667 
 
 a 
 
 23 
 
 .09382 
 
 99559 
 
 . 118 
 
 .90380 
 
 .12851 
 
 .99171 
 
 .14580 
 
 .98931 
 
 16304 
 
 .98662 
 
 37 
 
 24 
 
 .09411 
 
 .90556 
 
 M7 
 
 99377 
 
 . 12880 
 
 .99167 
 
 .14608 
 
 .98927 
 
 J 6333 
 
 .98657 
 
 36 
 
 
 .09440 
 
 99553 
 
 . 176 
 
 99374 
 
 .12908 
 
 .99163 
 
 14637 
 
 .98923 
 
 .16361 
 
 .98652 
 
 35 
 
 2f> 
 
 .09469 
 
 99551 
 
 205 
 
 99370 
 
 12937 
 
 .99160 
 
 .14666 
 
 .98919 
 
 .16390 
 
 .98648 
 
 34 
 
 27 
 
 .09498 
 09527 
 
 99548 
 99545 
 
 : 203 
 
 09367 
 99364 
 
 .!2 9 66 
 
 12995 
 
 .99156 
 99152 
 
 .14695 
 .14723 
 
 .98914 
 .98910 
 
 .16419 
 16447 
 
 .98643 
 .98638 
 
 33 
 
 32 
 
 ag 
 
 09556 
 
 99542 
 
 . 291 
 
 .99360 
 
 .13024 
 
 .99148 
 
 14752 
 
 .98906 
 
 .!6 47 6 
 
 98633 
 
 n 
 
 3" 
 
 09585 
 
 .99540 
 
 . 320 
 
 99357 
 
 13053 
 
 .99144 
 
 .14781 
 
 .98902 
 
 . 16505 
 
 .98629 
 
 3 
 
 3 1 
 
 .09614 
 
 99537 
 
 349 
 
 99354 
 
 .13081 
 
 .99141 
 
 .14810 
 
 .98897 
 
 16533 
 
 .98624 
 
 20 
 
 32 
 
 .09642 
 
 99534 
 
 378 
 
 99351 
 
 .13110 
 
 99137 
 
 .14838 
 
 .98893 
 
 . 16562 
 
 .98619 
 
 2.S 
 
 33 
 
 .09671 
 
 99531 
 
 407 
 
 99347 
 
 13139 
 
 99133 
 
 .14867 
 
 98889 
 
 . 16591 
 
 .98614 
 
 '-'-7 
 
 34 
 
 .09700 
 
 .99528 
 
 436 
 
 99344 
 
 .13168 
 
 .99129 
 
 .14896 
 
 
 . 16620 
 
 .98609 
 
 26 
 
 35 
 
 .09729 
 
 .99526 
 
 465 
 
 99341 
 
 13197 
 
 .99125 
 
 14925 
 
 !g888o 
 
 .16648 
 
 .98604 
 
 as 
 
 36 
 
 .09758 
 
 99523 
 
 494 
 
 99337 
 
 .13226 
 
 .99122 
 
 M954 
 
 .98876 
 
 . 16677 
 
 .98600 
 
 
 37 
 
 .09787 
 
 .99520 
 
 523 
 
 99334 
 
 13254 
 
 .99118 
 
 . 14982 
 
 .98871 
 
 .16706 
 
 98595 
 
 23 
 
 39 
 
 .09816 
 .09845 
 
 99517 
 .99514 
 
 
 99331 
 .99327 
 
 .13283 
 
 .13312 
 
 .99114 
 .99110 
 
 .15011 
 . 15040 
 
 .98867 
 .98863 
 
 '.% 
 
 98590 
 98585 
 
 22 
 21 
 
 40 
 
 .09874 
 
 995" 
 
 .' 609* 
 
 99324 
 
 13341 
 
 .99106 
 
 15069 
 
 .98858 
 
 . 16792 
 
 .98580 
 
 30 
 
 41 
 
 .09903 
 
 .99508 
 
 638 
 
 .99320 
 
 13370 
 
 .99102 
 
 15097 
 
 .98854 
 
 .16820 
 
 98575 
 
 I" 
 
 42 
 
 .09932 
 
 .99506 
 
 . 667 
 
 99317 
 
 13399 
 
 .99098 
 
 15126 
 
 .98849 
 
 . 16849 
 
 .98570 
 
 i 
 
 43 
 
 .09961 
 
 99503 
 
 . 696 
 
 993M 
 
 - 13427 
 
 .99094 
 
 iS'55 
 
 .98845 
 
 .16878 
 
 98565 
 
 7 
 
 44 
 
 .09990 
 
 .99500 
 
 725 
 
 .99310 
 
 13456 
 
 .99091 
 
 .15184 
 
 .98841 
 
 .16906 
 
 98561 
 
 (, 
 
 45 
 
 
 99497 
 
 754 
 
 99307 
 
 13485 
 
 .99087 
 
 .15212 
 
 .98836 
 
 . 16935 
 
 98556 
 
 9 
 
 *6 
 
 .10048 
 
 99494 
 
 783 
 
 09303 
 
 13514 
 
 99083 
 
 .15241 
 
 98832 
 
 .16964 
 
 98551 
 
 4 
 
 47 
 
 .10077 
 
 .99491 
 
 . 812 
 
 .99300 
 
 13543 
 
 .99079 
 
 .15270 
 
 
 .16992 
 
 98546 
 
 3 
 
 ,,S 
 
 .10106 
 
 .99488 
 
 . 840 
 
 99297 
 
 13572 
 
 9975 
 
 15299 
 
 .98823 
 
 .17021 
 
 .98541 
 
 a 
 
 49 
 
 .10135 
 
 .99485 
 
 . 869 
 
 .99293 
 
 .13600 
 
 .99071 
 
 i53 2 7 
 
 .98818 
 
 .17050 
 
 .98536 
 
 i 
 
 5" 
 
 . 10164 
 
 .99482 
 
 . 898 
 
 .99290 
 
 . 13629 
 
 99067 
 
 15356 
 
 .98814 
 
 .17078 
 
 08531 
 
 " 
 
 5I 
 
 .10192 
 
 99479 
 
 927 
 
 .99286 
 
 .13658 
 
 .99063 
 
 .15385 
 
 .98809 
 
 .17107 
 
 98526 
 
 
 
 52 
 
 . I022I 
 
 99476 
 
 956 
 
 99283 
 
 13687 
 
 99059 
 
 i54M 
 
 .98805 
 
 .1 .36 
 
 .98521 
 
 B 
 
 53 
 
 . 10250 
 
 99473 
 
 985 
 
 .99279 
 
 .137,6 
 
 9955 
 
 15442 
 
 .98800 
 
 .1 16 4 
 
 .98516 
 
 7 
 
 54 
 
 . 10279 
 
 99470 
 
 
 .99276 
 
 13744 
 
 .99051 
 
 15471 
 
 .98796 
 
 i 193 
 
 .98511 
 
 6 
 
 55 
 
 .10308 
 
 99467 
 
 043 
 
 .99272 
 
 13773 
 
 .99047 
 
 .15500 
 
 .98791 
 
 .1 222 
 
 .98506 
 
 5 
 
 56 
 
 10337 
 
 99464 
 
 . 071 
 
 .99269 
 
 .! 3 802 
 
 .99043 
 
 .15529 
 
 .98787 
 
 .1 2 5 
 
 .98501 
 
 4 
 
 57 
 
 .10366 
 
 .99461 
 
 IOO 
 
 .99265 
 
 . 13831 
 
 .99039 
 
 15557 
 
 .98782 
 
 I 279 
 
 .98496 
 
 .-i 
 
 58 
 
 I03Q5 
 
 99458 
 
 129 
 
 .99262 
 
 .13860 
 
 .99035 
 
 15586 
 
 98778 
 
 I 38 
 
 .98491 
 
 2 
 
 59 
 
 . 10424 
 
 99455 
 
 158 
 
 .99258 
 
 . 13889 
 
 99031 
 
 15615 
 
 98773 
 
 17336 
 
 .98486 
 
 I 
 
 60 
 
 10453 
 
 99452 
 
 187 
 
 .99255 
 
 13917 
 
 .90027 
 
 .15643 
 
 98769 
 
 .17365 
 
 .98481 
 
 Q 
 
 , 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 , 
 
 
 8, 
 
 f 
 
 8. 
 
 5 
 
 8 
 
 2 
 
 8 
 
 : 
 
 8< 
 
 5 

 
 NATURAL SINES AND COSINES. 
 
 
 I 
 
 o 
 
 I 
 
 i 
 
 I 
 
 2 
 
 1 
 
 3 
 
 i 
 
 4 
 
 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 
 
 
 17365 
 17393 
 
 .98481 
 .98476 
 
 . 908. 
 
 . 9109 
 
 .98 63 
 
 98 57 
 
 .20791 
 .20820 
 
 97815 
 .97809 
 
 .22495 
 22523 
 
 97 37 
 97 3 
 
 .24192 
 . 24220 
 
 .97030. 
 97023 
 
 Go 
 
 5V 
 
 3 
 
 .17422 
 '7451 
 
 98471 
 .98466 
 
 . 9138 
 9167 
 
 .98 S2 
 
 .98 46 
 
 .20848 
 .20877 
 
 .97803 
 
 97797 
 
 22552 
 .22580 
 
 97 24 
 97 i? 
 
 .24249 
 24277 
 
 97015 
 .97008 
 
 58 
 57 
 
 4 
 
 17479 
 
 .98461 
 
 9'95 
 
 .98 40 
 
 .20905 
 
 97791 
 
 .22608 
 
 97 'I 
 
 24305 
 
 .97001 
 
 
 5 
 
 .17508 
 
 98455 
 
 9224 
 
 98 35 
 
 20933 
 
 .97784 
 
 .22637 
 
 .97404 
 
 24333 
 
 .96994 
 
 55 
 
 6 
 
 '7537 
 
 .08450 
 
 9252 
 
 .98 29 
 
 . 20962 
 
 .97778 
 
 . 22665 
 
 97398 
 
 .24362 
 
 .96987 
 
 54 
 
 7 
 
 17565 
 
 .98445 
 
 . 9281 
 
 .98 24 
 
 .20990 
 
 .97772 
 
 .22693 
 
 97391 
 
 .24390 
 
 .96980 
 
 53 
 
 8 
 
 17594 
 
 .98440 
 
 9309 
 
 
 . 1019 
 
 .97766 
 
 .22722 
 
 .97384 
 
 .24418 
 
 96973 
 
 52 
 
 
 
 .17623 
 
 98435 
 
 9338 
 
 ! 9 8 12 
 
 i47 
 
 .97760 
 
 .22750 
 
 97378 
 
 .24446 
 
 .96966 
 
 
 
 .17651 
 
 .98430 
 
 . 9366 
 
 .98 07 
 
 I0 7 6 
 
 97754 
 
 .22778 
 
 97371 
 
 .24474 
 
 .96959 
 
 5" 
 
 , 
 
 .17680 
 
 .98425 
 
 9395 
 
 .98101 
 
 . 1104 
 
 .97748 
 
 .228O7 
 
 97365 
 
 24503 
 
 .96952 
 
 4" 
 
 12 
 
 .17708 
 
 .98420 
 
 9423 
 
 .98096 
 
 "32 
 
 97742 
 
 .22835 
 
 97358 
 
 24531 
 
 .96945 
 
 48 
 
 13 
 
 . 17737 
 
 .98414 
 
 9452 
 
 .98090 
 
 . 1161 
 
 97735 
 
 .22863 
 
 97351 
 
 24559 
 
 .96037 
 
 47 
 
 '4 
 
 . 17766 
 
 .98409 
 
 . 9481 
 
 .98084 
 
 . 1189 
 
 .97729 
 
 .22892 
 
 97345 
 
 24587 
 
 963o 
 
 46 
 
 5 
 
 '7794 
 
 .98404 
 
 . 9509 
 
 .98079 
 
 , 1218 
 
 97723 
 
 .22920 
 
 97338 
 
 .24615 
 
 96923 
 
 45 
 
 6 
 
 .17823 
 
 .98399 
 
 9538 
 
 .98073 
 
 . 1246 
 
 97717 
 
 .22948 
 
 97331 
 
 . 24644 
 
 .96916 
 
 44 
 
 7 
 
 .17852 
 
 .98394 
 
 . 9566 
 
 .98067 
 
 1275 
 
 9771.1 
 
 .22977 
 
 973*5- 
 
 .23672 
 
 .96909 
 
 4i 
 
 8 
 
 .17880 
 
 .98389 
 
 9595 
 
 .98061 
 
 1303 
 
 97705 
 
 23005 
 
 97318 
 
 754700 
 
 .96902 
 
 42 
 
 9 
 
 .17909 
 
 98383 
 
 . 9623 
 
 .98056 
 
 '33' 
 
 .97698 
 
 23033 
 
 973" 
 
 .24728 
 
 .96894 
 
 4' 
 
 20 
 
 T 7937 
 
 -98378 
 
 . 9652 
 
 .98050 
 
 '360 
 
 .97692 
 
 .23062 
 
 97304 
 
 24756 
 
 .96887 
 
 4" 
 
 21 
 
 .17966 
 
 98373 
 
 . 9680 
 
 .98044 
 
 1388 
 
 .97686 
 
 .23090 
 
 .97298 
 
 .24784 
 
 .96880 
 
 30 
 
 22 
 
 '7995 
 
 .98368 
 
 979 
 
 .98039 
 
 '4'7 
 
 .97680 
 
 .23118 
 
 .97291 
 
 .24813 
 
 .96873 
 
 
 23 
 
 24 
 
 .18023 
 .18052 
 
 .98362 
 98357 
 
 9737 
 . 9766 
 
 ,98033 
 .98027 
 
 '445 
 '474 
 
 97673 
 .97667 
 
 23146 
 23175 
 
 97284 
 .97278 
 
 .24841 
 .24869 
 
 .96866 
 .96858 
 
 37 
 
 2 5 
 
 . 18081 
 
 98352 
 
 9794 
 
 .98021 
 
 1502 
 
 .97661 
 
 .23203 
 
 97271 
 
 .24897 
 
 .96851 
 
 35 
 
 26 
 
 .18109 
 
 98347 
 
 9823 
 
 .98016 
 
 1530 
 
 97655 
 
 .23231 
 
 .97264 
 
 .24925 
 
 .96844 
 
 34 
 
 2 7 
 
 .18138 
 
 98341 
 
 . 9851 
 
 .98010 
 
 '559 
 
 .97648 
 
 .23260 
 
 97257 
 
 24954 
 
 .96837 
 
 
 
 .18166 
 
 98336 
 
 . 9880 
 
 .98004 
 
 1587 
 
 .97642 
 
 .23288 
 
 97251 
 
 .24982 
 
 .96829 
 
 3* 
 
 99 
 
 . 18195 
 
 9833' 
 
 . 9908 
 
 .97998 
 
 . 1616 
 
 .97636 
 
 .23316 
 
 97244 
 
 .25010 
 
 .96822 
 
 31 
 
 JO 
 
 .18224 
 
 98325 
 
 9937 
 
 .97992 
 
 . 1644 
 
 .97630 
 
 23345 
 
 97237 
 
 .25038 
 
 .96815 
 
 30 
 
 .31 
 
 .18252 
 .18281 
 
 .98320 
 
 .98315 
 
 9965 
 
 97987 
 .07981 
 
 . 1672 
 
 .97623 
 
 23373 
 
 97 3 
 
 .25066 
 
 .96807 
 .96800 
 
 M 
 
 33 
 
 .18309 
 
 .98310 
 
 ! 20022 
 
 97975 
 
 '729 
 
 .97611 
 
 23429 
 
 97 '7 
 
 .25122 
 
 96793 
 
 27 
 
 34 
 
 '8338 
 
 .98304 
 
 .20051 
 
 97969 
 
 '758 
 
 .97604 
 
 .23458 
 
 97 10 
 
 25151 
 
 .96786 
 
 
 P 
 
 .18367 
 18395 
 
 .98299 
 .08294 
 
 .20079 
 .20108 
 
 .97963 
 97958 
 
 . 1786 
 . 1814 
 
 -97598 
 97592 
 
 .23486 
 23514 
 
 97 3 
 97 Q 6 
 
 25' 79 
 .25207 
 
 96778 
 .96771 
 
 25 
 
 37 
 
 .18424 
 
 .98288 
 
 .20136 
 
 97952 
 
 1843 
 
 97585 
 
 23542 
 
 97 89 
 
 25235 
 
 .96764 
 
 M 
 
 3B 
 
 .18452 
 
 .98283 
 
 .20165 
 
 .97946 
 
 . 1871 
 
 97579 
 
 2357' 
 
 97 82 
 
 25263 
 
 .96756 
 
 
 30 
 40 
 
 .1848, 
 .18509 
 
 98277 
 .98272 
 
 .20193 
 . 20222 
 
 .97940 
 97934 
 
 . 1899 
 . 1928 
 
 97566 
 
 23599 
 .23627 
 
 97 76 
 97 69 
 
 .25291 
 .25320 
 
 90749 
 .96742 
 
 to 
 
 4' 
 
 18538 
 
 .98267 
 
 .20250 
 
 .97928 
 
 1956 
 
 .97560 
 
 .23656 
 
 .97 62 
 
 25348 
 
 96734 
 
 HJ 
 
 2 
 
 18567 
 
 .98261 
 
 .20279 
 
 .97922 
 
 . 1985 
 
 97553 
 
 23684 
 
 97 55 
 
 25376 
 
 .96727 
 
 18 
 
 i 
 
 -18595 
 
 .98256 
 
 20307 
 
 .97916 
 
 2013 
 
 97547 
 
 .23712 
 
 97 48 
 
 .25404 
 
 96719 
 
 17 
 
 4 
 
 .18624 
 
 .98250 
 
 20336 
 
 97910 
 
 . 2041 
 
 97541 
 
 23740 
 
 97 4' 
 
 25432 
 
 .96712 
 
 6 
 
 5 
 
 .18652 
 
 .98245 
 
 .20364 
 
 97905 
 
 . 2070 
 
 97534 
 
 .23769 
 
 97 34 
 
 .25460 
 
 .96705 
 
 5 
 
 6 
 
 .18681 
 
 .98240 
 
 20393 
 
 .97899 
 
 . 2098 
 
 97528 
 
 .23797 
 
 97 27 
 
 .25488 
 
 .96697 
 
 4 
 
 7 
 
 .18710 
 
 98234 
 
 .2O421 
 
 97893 
 
 . 2126 
 
 97521 
 
 .23825 
 
 97 20 
 
 25516 
 
 .96690 
 
 3 
 
 8 
 
 .18738 
 
 .98229 
 
 . 20450 
 
 .97887 
 
 ^155 
 
 97515 
 
 23853 
 
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 98 74 
 
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 26 
 
 NATURAL SINES AND COSINES. 
 
 
 I 
 
 5 
 
 I 
 
 5 
 
 i 
 
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 I 
 
 8 
 
 I 
 
 9 
 
 
 
 Sine 
 
 Cosine 
 
 Sine 
 
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 96593 
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 27564 
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 25938 
 
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 32722 
 
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 29571 
 
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 29599 
 
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 47 
 
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 32969 
 
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 16 
 
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 28039 
 
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 94952 
 
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 43 
 
 18 
 
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 42 
 
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 94933 
 
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 40 
 
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 94351 
 
 39 
 
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 38 
 
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 95433 
 
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 94897 
 
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 37 
 
 
 .26556 
 
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 31565 
 
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 94322 
 
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 95415 
 
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 33244 
 
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 35 
 
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 33271 
 
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 34 
 
 27 
 
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 95398 
 
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 33 
 
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 28346 
 28374 
 
 95898 
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 30043 
 
 95389 
 9538o 
 
 3*675 
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 33326 
 
 33353 
 
 .94284 
 94274 
 
 38 
 
 31 
 
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 95372 
 
 3 I 73 
 
 94832 
 
 3338* 
 
 .94264 
 
 3 
 
 31 
 
 .26752 
 
 96355 
 
 .28429 
 
 .95874 
 
 .30098 
 
 95363 
 
 3*758 
 
 .94823 
 
 .33408 
 
 94254 
 
 29 
 
 3* 
 
 .26780 
 
 96347 
 
 28457 
 
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 95354 
 
 i 31786 
 
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 33436 
 
 94245 
 
 28 
 
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 .26808 
 
 .96340 
 
 28485 
 
 95857 
 
 30*54 
 
 95345 
 
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 .94805 
 
 33463 
 
 94235 
 
 27 
 
 34 
 
 .26836 
 
 96332 
 
 28513 
 
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 95337 
 
 3*84* 
 
 94795 
 
 33490 
 
 94225 
 
 26 
 
 35 
 
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 335*8 
 
 94215 
 
 25 
 
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 30237 
 
 953*9 
 
 .31896 
 
 94777 
 
 33545 
 
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 37 
 
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 28597 
 
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 33573 
 
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 39 
 
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 33627 
 
 94176 
 
 21 
 
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 95799 
 
 30348 
 
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 33655 
 
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 20 
 
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 27032 
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 4 
 
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 28875 
 
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 49 
 
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 28903 
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 30570 
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 32227 
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 D 

 
 NATURAL SINES AND COSINES. 
 
 
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 92707 
 
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 35973 
 
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 54 
 
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 7 
 
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 37209 
 
 92784 
 
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 6 
 
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 37326 
 
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 5 
 
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 35728 
 
 .93400 
 
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 .38966 
 
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 .40567 
 
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 4 
 
 57 
 
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 3 
 
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 35782 
 
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 59 
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 .358.0 
 35837 
 
 93368 
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 , 
 
 
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 9 
 
 6 
 
 3 
 
 6 
 
 7 
 
 6 
 
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 5 

 
 28 
 
 NATURAL SINES AND COSINES. 
 
 
 2C 
 
 
 21 
 
 
 2 
 
 1 
 
 2 
 
 
 2( 
 
 ) 
 
 
 Sine 
 
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 Sine 
 
 Cosine 
 
 Sine 
 
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 87391 
 
 6 
 
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 48634 
 
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 48659 
 
 87363 
 
 8 
 
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 87349 
 
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 .88028 
 
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 .48811 
 
 .87278 
 
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 .42631 
 
 90458 
 
 .44203 
 
 .89700 
 
 45762 
 
 .88915 
 
 .47306 
 
 '.88103 
 
 .48837 
 
 .87264 
 
 J j 
 
 .42657 
 
 . 90446 
 
 44229 
 
 .89687 
 
 45787 
 
 .88902 
 
 47332 
 
 
 .48862 
 
 .87250 
 
 10 
 
 .42683 
 
 90433 
 
 44255 
 
 .89674 
 
 458i3 
 
 .88888 
 
 .47358 
 
 .88075 
 
 .48888 
 
 87235 
 
 3 
 
 .42709 
 .42736 
 
 .90421 
 .90408 
 
 .44281 
 44307 
 
 .89662 
 .89649 
 
 45839 
 .45865 
 
 .88875 
 
 47383 
 .47409 
 
 .88062 
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 .48913 
 
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 .87221 
 .87207 
 
 19 
 
 .42762 
 
 .90396 
 
 44333 
 
 .89636 
 
 45891 
 
 .88848 
 
 47434 
 
 .88034 
 
 .48964 
 
 87193 
 
 20 
 
 .42788 
 
 .90383 
 
 44359 
 
 89623 
 
 459^7 
 
 .88835 
 
 .47460 
 
 .88020 
 
 .48989 
 
 .87178 
 
 21 
 
 . 28 I5 
 
 90371 
 
 44385 
 
 .89610 
 
 45942 
 
 .88822 
 
 .47486 
 
 .88006 
 
 .49014 
 
 .87164 
 
 22 
 
 . 2841 
 
 90358 
 
 .44411 
 
 89597 
 
 .45968 
 
 . 88808 
 
 475" 
 
 87993 
 
 .49040 
 
 .87:50 
 
 *3 
 
 . 2867 
 
 .90346 
 
 44437 
 
 .89584 
 
 45994 
 
 .88795 
 
 47537 
 
 .87979 
 
 .49065 
 
 87136 
 
 24 
 
 . 2894 
 
 90334 
 
 .44464 
 
 89571 
 
 .46020 
 
 .88782 
 
 47562 
 
 .87965 
 
 .49090 
 
 .87121 
 
 25 
 
 2920 
 
 .90321 
 
 . 44490 
 
 89558 
 
 .46046 
 
 .88768 
 
 .47588 
 
 .87951 
 
 .49116 
 
 .87107 
 
 26 
 
 . 2946 
 
 .00309 
 
 .44516 
 
 89545 
 
 .46072 
 
 88755 
 
 .47614 
 
 87937 
 
 .49141 
 
 .87093 
 
 27 
 
 42972 
 
 .90296 
 
 44542 
 
 .89532 
 
 .46097 
 
 .88741 
 
 47639 
 
 .87923 
 
 .49166 
 
 .87079 
 
 28 
 
 .42999 
 
 .90284 
 
 .44568 
 
 .89519 
 
 46123 
 
 .88728 
 
 47665 
 
 
 .49192 
 
 .87064 
 
 29 
 
 .43025 
 
 .90271 
 
 .44594 
 
 .89506 
 
 .46149 
 
 .887.5 
 
 .47690 
 
 .87896 
 
 .49217 
 
 -87050 
 
 3" 
 
 43051 
 
 .90259 
 
 .44620 
 
 .89493 
 
 46175 
 
 .88701 
 
 .47716 
 
 .87882 
 
 .49242 
 
 .87036 
 
 3i 
 
 43077 
 
 . 90246 
 
 .44646 
 
 .89480 
 
 .46201 
 
 88688 
 
 47741 
 
 .87868 
 
 .49268 
 
 .87021 
 
 32 
 
 .43104 
 
 .90233 
 
 44672 
 
 .89467 
 
 .46226 
 
 .88674 
 
 47767 
 
 .87854 
 
 49293 
 
 .87007 
 
 33 
 
 43 T 3 
 
 .90221 
 
 .44698 
 
 .89454 
 
 .46252 
 
 .88661 
 
 47793 
 
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 .49318 
 
 .86993 
 
 34 
 
 43!56 
 
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 44724 
 
 .89441 
 
 .46278 
 
 .88647 
 
 .47818 
 
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 49344 
 
 .86978 
 
 3 
 
 43182 
 .43209 
 
 .90196 
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 . 44750 
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 .46304 
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 .88634 
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 49369 
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 37 
 
 43235 
 
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 .44802 
 
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 .88607 
 
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 .87784 
 
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 86935 
 
 38 
 39 
 
 43261 
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 .90158 
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 .44828 
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 89389 
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 .47920 
 
 47946 
 
 .87770 
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 49445 
 
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 40 
 
 433'3 
 
 .90133 
 
 .44880 
 
 .89363 
 
 46433 
 
 .88566 
 
 47971 
 
 87743 
 
 49495 
 
 .86892 
 
 41 
 
 4334 
 
 .90120 
 
 .44906 
 
 .89350 
 
 46458 
 
 88553 
 
 47997 
 
 .87729 
 
 49521 
 
 .86878 
 
 42 
 
 4336 
 
 .90108 
 
 44932 
 
 89337 
 
 .46484 
 
 .88539 
 
 .48022 
 
 87715 
 
 .49546 
 
 .86863 
 
 43 
 
 .43392 
 
 .90095 
 
 .44958 
 
 .89324 
 
 .46510 
 
 .88526 
 
 .48048 
 
 .87701 
 
 4957' 
 
 .86849 
 
 44 
 
 .43418 
 
 .93082 
 
 .44984 
 
 .893:1 
 
 .46536 
 
 .88512 
 
 .48073 
 
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 .49596 
 
 .86834 
 
 45 
 
 43445 
 
 .90070 
 
 .45010 
 
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 .88499 
 
 .48099 
 
 87673 
 
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 .86820 
 
 46 
 
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 .89285 
 
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 .88485 
 
 .48124 
 
 87659 
 
 .49647 
 
 .86805 
 
 47 
 
 43497 
 
 .90045 
 
 .45062 
 
 .89272 
 
 46613 
 
 .88472 
 
 .48150 
 
 .87645 
 
 .49672 
 
 .86791 
 
 48 
 
 43523 
 
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 .45088 
 
 .89259 
 
 46639 
 
 .88458 
 
 48i75 
 
 .87631 
 
 49697 
 
 .86777 
 
 49 
 
 43549 
 
 .90019 
 
 45H4 
 
 .89245 
 
 .46664 
 
 .88445 
 
 .48201 
 
 .8 7 6i 7 
 
 49723 
 
 .86762 
 
 5'-' 
 
 43575 
 
 .90007 
 
 .45140 
 
 .89232 
 
 .46690 
 
 
 
 .87603 
 
 .49748 
 
 .86748 
 
 51 
 
 .43602 
 
 .89994 
 
 .45166 
 
 .89219 
 
 .46716 
 
 .88417 
 
 .48252 
 
 .87589 
 
 49773 
 
 .86733 
 
 53 
 
 .43628 
 
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 45 J 92 
 
 .89206 
 
 .46742 
 
 .88404 
 
 48277 
 
 87575 
 
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 53 
 
 .43654 
 
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 .89193 
 
 46767 
 
 .88390 
 
 48303 
 
 .87561 
 
 .49824 
 
 .86704 
 
 54 
 
 .43680 
 
 .89956 
 
 45243 
 
 .89 80 
 
 46793 
 
 88377 
 
 .48328 
 
 .87546 
 
 .49849 
 
 .86690 
 
 55 
 
 .43706 
 
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 .45269 
 
 .89 67 
 
 .46819 
 
 .88363 
 
 48354 
 
 87532 
 
 .49874 
 
 .86675 
 
 56 
 
 43733 
 
 .89930 
 
 45295 
 
 .89 53 
 
 .46844 
 
 .88349 
 
 48379 
 
 .87518 
 
 .49899 
 
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 57 
 
 43759 
 
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 88336 
 
 48405 
 
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 58 
 
 43785 
 
 .89905 
 
 45347 
 
 .89 27 
 
 .46896 
 
 .88322 
 
 .48430 
 
 .87490 
 
 .49950 
 
 .86632 
 
 59 
 
 .43811 
 
 .89892 
 
 45373 
 
 .89 14 
 
 .46921 
 
 .88308 
 
 48456 
 
 .87476 
 
 49975 
 
 .86617 
 
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 .89879 
 
 45399 
 
 .890, 
 
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 .88295 
 
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 .87462 
 
 .50000 
 
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 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 L 
 
 6 
 
 4 
 
 6 
 
 3 
 
 6 
 
 2 
 
 6 
 
 1 
 
 6 
 
 O
 
 NATURAL SINES AND COSINES. 
 
 
 3< 
 
 ) 
 
 3 1 
 
 
 32 
 
 
 
 32 
 
 
 
 2>A 
 
 
 
 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
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 Sine 
 
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 Sine 
 
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 .50000 
 .50025 
 
 .86603 
 .86588 
 
 .51504 
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 85717 
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 52992 
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 54464 
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 55919 
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 .50050 
 
 .86573 
 
 51554 
 
 85687 
 
 53041 
 
 .84774 
 
 545^3 
 
 .83835 
 
 55968 
 
 .82871 
 
 
 
 .50076 
 
 .86559 
 
 51579 
 
 .85672 
 
 .53066 
 
 84759 
 
 54537 
 
 .83819 
 
 55992 
 
 .82855 
 
 57 
 
 
 .50101 
 
 -86544 
 
 
 85657 
 
 53091 
 
 84743 
 
 5456i 
 
 .83804 
 
 .56016 
 
 .82839 
 
 56 
 
 
 .50126 
 
 .86530 
 
 .51628 
 
 .85642 
 
 53"5 
 
 .84728 
 
 54586 
 
 .83788 
 
 .56040 
 
 .82822 
 
 55 
 
 
 .50151 
 
 .86515 
 
 51653 
 
 .85627 
 
 53 I 40 
 
 .84712 
 
 .54610 
 
 83772 
 
 .56064 
 
 .82806 
 
 54 
 
 
 .50176 
 
 .86501 
 
 .51678 
 
 .85612 
 
 53164 
 
 .84697 
 
 54635 
 
 83756 
 
 .56088 
 
 .82790 
 
 S3 
 
 
 .50201 
 
 .86486 
 
 51703 
 
 85597 
 
 
 .84681 
 
 54659 
 
 .83740 
 
 .56112 
 
 82773 
 
 52 
 
 
 .50227 
 
 .86471 
 
 .51728 
 
 .85582 
 
 53214 
 
 .84666 
 
 54683 
 
 .83724 
 
 56136 
 
 82757 
 
 SI 
 
 I 
 
 .50252 
 
 86457 
 
 
 85567 
 
 53238 
 
 .84650 
 
 .54708 
 
 83708 
 
 .56160 
 
 .82741 
 
 3" 
 
 12 
 
 13 
 
 50277 
 .50302 
 .50327 
 
 .86442 
 .86427 
 .86413 
 
 51778 
 .51803 
 .51828 
 
 85551 
 
 85536 
 .85521 
 
 53263 
 .53288 
 53312 
 
 .8463 
 .8461 
 .8460 
 
 54732 
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 .56184 
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 a 
 
 47 
 
 
 50352 
 
 .86398 
 
 .51852 
 
 .85506 
 
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 .8458 
 
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 83645 
 
 .56256 
 
 82675 
 
 46 
 
 15 
 
 50377 
 
 .86384 
 
 51877 
 
 .85491 
 
 53361 
 
 8457 
 
 54829 
 
 .83629 
 
 .56280 
 
 .82659 
 
 45 
 
 16 
 
 .50403 
 
 .86369 
 
 5 1902 
 
 .85476 
 
 .53386 
 
 8455 
 
 54854 
 
 .8 3 6 I3 
 
 .56305 
 
 .82643 
 
 44 
 
 17 
 
 .50428 
 
 .86354 
 
 .51927 
 
 .85461 
 
 534" 
 
 8454 
 
 .54878 
 
 83597 
 
 56329 
 
 .82626 
 
 43 
 
 18 
 
 5453 
 
 .86340 
 
 51952 
 
 .85446 
 
 53435 
 
 84526 
 
 .54902 
 
 83581 
 
 56353 
 
 .82610 
 
 42 
 
 19 
 
 .50478 
 
 .86325 
 
 51977 
 
 85431 
 
 5346o 
 
 .84511 
 
 54927 
 
 83565 
 
 .56377 
 
 .82593 
 
 4' 
 
 20 
 
 50503 
 
 .86310 
 
 .52002 
 
 .85416 
 
 53484 
 
 .84495 
 
 54951 
 
 83549 
 
 56401 
 
 82577 
 
 4 
 
 21 
 
 50528 
 
 .86295 
 
 .52026 
 
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 53509 
 
 .84480 
 
 54975 
 
 83533 
 
 56425 
 
 .82561 
 
 35 
 
 22 
 
 50553 
 
 
 52051 
 
 85385 
 
 53534 
 
 .84464 
 
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 83517 
 
 .56449 
 
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 38 
 
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 53558 
 
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 55024 
 
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 56473 
 
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 37 
 
 24 
 
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 .86251 
 
 .52101 
 
 85355 
 
 53583 
 
 84433 
 
 .55048 
 
 .83485 
 
 56497 
 
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 25 
 
 .50628 
 
 .86237 
 
 .52126 
 
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 53607 
 
 84417 
 
 55072 
 
 .83469 
 
 56521 
 
 .82495 
 
 35 
 
 26 
 
 27 
 
 50654 
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 .86222 
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 52151 
 52175 
 
 85325 
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 53632 
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 .84402 
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 55097 
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 83453 
 83437 
 
 56545 
 56569 
 
 .82478 
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 34 
 33 
 
 28 
 
 . 50704 
 
 .86 92 
 
 .52200 
 
 .85294 
 
 .53681 
 
 84370 
 
 55145 
 
 .83421 
 
 56593 
 
 .82446 
 
 & 
 
 29 
 
 .50729 
 
 .86 78 
 
 .52225 
 
 .85279 
 
 53705 
 
 84355 
 
 .55169 
 
 .83405 
 
 .56617 
 
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 31 
 
 
 50754 
 
 .86 63 
 
 .52250 
 
 .85264 
 
 53730 
 
 .84339 
 
 55194 
 
 .83389 
 
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 3 o 
 
 3 1 
 
 50779 
 
 .8648 
 
 52275 
 
 85249 
 
 53754 
 
 84324 
 
 55218 
 
 83373 
 
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 82396 
 
 20 
 
 32 
 
 33 
 
 .50804 
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 86 33 
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 52299 
 .52324 
 
 85234 
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 53779 
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 .84308 
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 55242 
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 83356 
 83340 
 
 .56689 
 56713 
 
 .82380 
 .82363 
 
 28 
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 34 
 
 35 
 
 .50854 
 50879 
 
 .86 04 
 .86089 
 
 52349 
 52374 
 
 85203 
 .85188 
 
 .53828 
 53853 
 
 .84277 
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 55291 
 55315 
 
 83324 
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 .82347 
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 gfi 
 S 
 
 36 
 
 .50904 
 
 .86074 
 
 52399 
 
 85173 
 
 53877 
 
 .84245 
 
 55339 
 
 .83292 
 
 .56784 
 
 82314 
 
 24 
 
 37 
 
 50929 
 
 .86059 
 
 52423 
 
 85157 
 
 .53902 
 
 84230 
 
 55363 
 
 83276 
 
 ; 56808 
 
 .82297 
 
 23 
 
 38 
 
 50954 
 
 .86045 
 
 .52448 
 
 85142 
 
 .53926 
 
 .84214 
 
 55388 
 
 .83260 
 
 56832 
 
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 39 
 
 50979 
 
 .86030 
 
 52473 
 
 
 53951 
 
 .84198 
 
 55412 
 
 .83244 
 
 56856 
 
 .82264 
 
 21 
 
 4 
 
 .51004 
 
 .86015 
 
 .52498 
 
 .85112 
 
 53975 
 
 .84182 
 
 55436 
 
 .83228 
 
 .56880 
 
 .82248 
 
 20 
 
 4i 
 
 .51029 
 
 .86000 
 
 .52522 
 
 .85006 
 
 .54000 
 
 .84167 
 
 .55460 
 
 .83212 
 
 .56904 
 
 .82231 
 
 *9 
 
 42 
 
 5 I0 54 
 
 85985 
 
 52547 
 
 .85081 
 
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 .84.51 
 
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 10 
 
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 84135 
 
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 85956 
 
 52597 
 
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 54073 
 
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 55533 
 
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 56976 
 
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 16 
 
 45 
 
 5 129 
 
 .85941 
 
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 85035 
 
 54097 
 
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 55557 
 
 83M7 
 
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 15 
 
 46 
 
 5 i54 
 
 85926 
 
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 M 
 
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 5 i79 
 
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 85005 
 
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 30 
 
 NATURAL SINES AND COSINES. 
 
 
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 NATURAL SINES AND COSINES. 
 
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 73135 
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 60 
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 64323 
 
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 76548 
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 .65672 
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 754 I 4 
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 73076 
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 69529 
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 1873 
 1853 
 
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 64390 
 
 .76511 
 
 .65716 
 
 75375 
 
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 4217 
 
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 65738 
 
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 64435 
 
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 75299 
 
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 72957 
 
 69654 
 
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 .68412 
 
 72937 
 
 69675 
 
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 72717 
 
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 1508 
 
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 67387 
 
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 1386 
 
 33 
 
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 28 
 
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 27 
 
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 23 
 
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 73570 
 
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 72377 
 
 70257 
 
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 22 
 
 39 
 
 .65144 
 
 75870 
 
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 74722 
 
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 75851 
 
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 NATURAL TANGENTS AND COTANGENTS. 
 
 
 C 
 
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 2 
 
 
 
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 ^ 
 
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 Tang 
 
 Cotang 
 
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 57-2900 
 
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 28.6363 
 
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 19.0811 
 
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 14.3007 
 
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 3437-75 
 
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 56.356 
 
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 28.3994 
 
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 14.2411 
 
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 1718.87 
 
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 55.4415 
 
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 28. 1664 
 
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 18.87,1 
 
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 14.1821 
 
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 1145.92 
 
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 54-5613 
 
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 27.9372 
 
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 859.436 
 
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 53.7086 
 
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 27.7117 
 
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 18.6656 
 
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 14.0655 
 
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 687.549 
 
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 52.8821 
 
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 27.4899 
 
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 18.5645 
 
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 14.0079 
 
 6 
 
 .00175 
 
 572-957 
 
 .01920 
 
 52.0807 
 
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 27.2715 
 
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 18.4645 
 
 .07168 
 
 -13-9507 
 
 7 
 
 
 491.106 
 
 .01949 
 
 51.3032 
 
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 27.0566 
 
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 18-3655 
 
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 13.8940 
 
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 9 
 
 .00262 
 
 429.718 
 381.971 
 
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 50.5485 
 49.8157 
 
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 26.8450 
 26.6367 
 
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 18.2677 
 18.1708 
 
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 13-8378 
 13.782, 
 
 10 
 
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 343-774 
 
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 49.1039 
 
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 26.4316 
 
 05533 
 
 18.0750 
 
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 .00320 
 
 312.521 
 
 .02066 
 
 48.4121 
 
 .03812 
 
 26.2296 
 
 .05562 
 
 17 9802 
 
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 13.6719 
 
 12 
 
 .00349 
 
 286.478 
 
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 47-7395 
 
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 26.0307 
 
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 17-8863 
 
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 13.6174 
 
 13 
 
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 64.441 
 
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 47.0853 
 
 03871 
 
 25.8348 
 
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 17.7934 
 
 07373 
 
 13-5634 
 
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 .00407 
 
 45-552 
 
 .02153 
 
 46.4489 
 
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 25.6418 
 
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 17.7015 
 
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 13.5098 
 
 11 
 
 .00436 
 .00465 
 
 29.182 
 14.858 
 
 .02211 
 
 45-8294 
 45.2261 
 
 .03929 
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 25-4517 
 25.2644 
 
 .05678 
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 17.6106 
 
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 13-4566 
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 17 
 
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 44.6386 
 
 
 25.0798 
 
 5737 
 
 17-4314 
 
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 13-3515 
 
 18 
 
 .00524 
 
 90.984 
 
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 44.0661 
 
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 24.8978 
 
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 17-3432 
 
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 00553 
 
 80.932 
 
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 43-5o8i 
 
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 24.7185 
 
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 17-2558 
 
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 13.2480 
 
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 71-885 
 
 .02328 
 
 42.9641 
 
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 24-5418 
 
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 17-1693 
 
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 13.1969 
 
 21 
 
 .00611 
 
 63.700 
 
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 42-4335 
 
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 24-3675 
 
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 17.0837 
 
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 13-1461 
 
 22 
 
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 56.259 
 
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 41.9158 
 
 04133 
 
 24-1957 
 
 .05883 
 
 16.9990 
 
 .07636 
 
 13.0958 
 
 23 
 
 .00669 
 
 49.465 
 
 .02415 
 
 41.4106 
 
 .04162 
 
 24.0263 
 
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 16.9150 
 
 07665 
 
 13.0458 
 
 24 
 
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 43-237 
 
 
 40.9174 
 
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 23-8593 
 
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 12.9962 
 
 25 
 
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 37-507 
 
 02473 
 
 40.4358 
 
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 23.6945 
 
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 16.7496 
 
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 12.9469 
 
 26 
 
 .00756 
 
 32.219 
 
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 39.9655 
 
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 23.5321 
 
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 16.6681 
 
 7753 
 
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 27 
 
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 27.321 
 
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 39.5059 
 
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 23.3718 
 
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 12.8496 
 
 8 
 
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 22-774 
 
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 39.0568 
 
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 23.2137 
 
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 12.8014 
 
 29 
 
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 18.540 
 
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 38.6177 
 
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 23-0577 
 
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 12.7536 
 
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 .00873 
 
 14.589 
 
 .02619 
 
 38.1885 
 
 .04366 
 
 22.9038 
 
 .06116 
 
 16.3499 
 
 .07870 
 
 12.7062 
 
 3 1 
 
 ,00902 
 
 110.892 
 
 .02648 
 
 37.7686 
 
 .04395 
 
 22.7519 
 
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 ,6.2722 
 
 .07899 
 
 12.659, 
 
 3 2 
 
 .00931 
 
 107.426 
 
 .02677 
 
 37-3579 
 
 .04424 
 
 
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 16.1952 
 
 .07929 
 
 ,2.6,24 
 
 33 
 
 .00960 
 
 104.171 
 
 .02706 
 
 36.9560 
 
 .04454 
 
 22.4541 
 
 . 06204 
 
 16. 1190 
 
 .07958 
 
 12.5660 
 
 34 
 
 .00989 
 
 101.107 
 
 02735 
 
 36.5627 
 
 .04483 
 
 22.3081 
 
 .06233 
 
 16.0435 
 
 .07987 
 
 12.5199 
 
 35 
 
 
 98.2179 
 
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 36.1776 
 
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 22. 1640 
 
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 15-9687 
 
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 12.4742 
 
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 35.8006 
 
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 22 0217 
 
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 15.8945 
 
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 92-9085 
 
 
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 21.8813 
 
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 38 
 
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 35-0695 
 
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 21.7426 
 
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 T 5.7483 
 
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 12.339 
 
 39 
 
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 88^436 
 
 .02881 
 
 34.7I5I 
 
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 21.6056 
 
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 15.6762 
 
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 12.2946 
 
 
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 85.9398 
 
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 34-3678 
 
 .04658 
 
 21.4704 
 
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 15.6048 
 
 .08163 
 
 12.2505 
 
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 .01193 
 
 83.8435 
 
 81.8470 
 
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 34-0273 
 33.6935 
 
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 21.3369 
 21.2049 
 
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 15.5340 
 15-4638 
 
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 12.2067 
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 79-9434 
 
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 33-3662 
 
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 21.0747 
 
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 15-3943 
 
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 78.1263 
 
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 33-0452 
 
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 20.9460 
 
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 15-3254 
 
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 12.0772 
 
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 76.3900 
 
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 32.7303 
 
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 15-2571 
 
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 12.0346 
 
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 74-7292 
 
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 32-4213 
 
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 20:6932 
 
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 15-1893 
 
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 11.9923 
 
 47 
 
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 73- J 390 
 
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 20.5691 
 
 .06613 
 
 15.1222 
 
 .08368 
 
 11.9504 
 
 48 
 
 .01396 
 
 71-6151 
 
 .03143 
 
 31.8205 
 
 .04891 
 
 20.4465 
 
 .06642 
 
 15.0557 
 
 08397 
 
 11.9087 
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 49 
 
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 01455 
 
 68.7501 
 
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 31-2416 
 
 .04949 
 
 20.2056 
 
 .06700 
 
 14.9244 
 
 : 08456 
 
 i, .'8262 
 
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 .01484 
 
 67.4019 
 
 .03230 
 
 30.9599 
 
 .04978 
 
 20.0872 
 
 .06730 
 
 14-8596 
 
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 52 
 
 .01513 
 
 66.1055 
 
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 30.6833 
 
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 19.9702 
 
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 14-7954 
 
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 64.8580 
 
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 30.4116 
 
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 14-7317 
 
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 11.7045 
 
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 63.6567 
 
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 30.1446 
 
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 19.7403 
 
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 14.6685 
 
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 11.6645 
 
 
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 62.4992 
 61.3829 
 
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 29.8823 
 29.6245 
 
 05095 
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 19.6273 
 
 19.5156 
 
 .'06876 
 
 14.6059 
 14.5438 
 
 .08602 
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 11.6248 
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 57 
 
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 60.3058 
 
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 29-37I1 
 
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 .06905 
 
 14.4823 
 
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 11.5461 
 
 58 
 59 
 
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 59.2659 
 58.2612 
 
 03434 
 .03463 
 
 28:8771 
 
 .05182 
 
 19.2959 
 19.1879 
 
 .069-54 
 .06963 
 
 14.4212 
 14.3607 
 
 .08690 
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 11.5072 
 11.4685 
 
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 .01746 
 
 57.2900 
 
 .03402 
 
 28.6363 
 
 .05241 
 
 19.0811 
 
 .06993 
 
 14-3007 
 
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 11.4301 
 
 , 
 
 Cotang 
 
 Tang 
 
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 Cotang 
 
 Tang 
 
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 8( 
 
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 NATURAL TANGENTS AND COTANGENTS. 
 
 33 
 
 
 5 
 
 
 
 6 
 
 
 
 
 7 
 
 s 
 
 
 9 
 
 
 
 71 
 
 
 
 
 
 
 
 
 
 
 
 
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 Tang 
 
 Cotang 
 
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 11.3919 
 
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 9.5,436 
 9.4878, 
 
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 8.14435 
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 7.10038 
 
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 6.30189 
 
 60 
 
 59 
 
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 11.3540 
 
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 8.10536 
 
 4"3 
 
 7.08546 
 
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 6.29007 
 
 53 
 
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 9-435'S 
 
 .12367 
 
 8.08600 
 
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 7.07059 
 
 -15928 
 
 6.27829 
 
 57 
 
 4 
 
 .08866 
 
 ,1.2789 
 
 . 0628 
 
 9.40904 
 
 .12397 
 
 8.06674 
 
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 7-05579 
 
 
 6.26655 
 
 56 
 
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 .08895 
 
 11.2417 
 
 - 0657 
 
 9.38307 
 
 .12426 
 
 8.04756 
 
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 7.04105 
 
 . I 5988 
 
 6.25486 
 
 55 
 
 6 
 
 .08925 
 
 11.2048 
 
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 9-35724 
 
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 4232 
 
 7.02637 
 
 .l6ol7 
 
 6.24321 
 
 54 
 
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 9.33I55 
 
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 8.00948 
 
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 7-01174 
 
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 6.23160 
 
 53 
 
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 ii. ,3.6 
 
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 9-30599 
 
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 4291 
 
 6.99718 
 
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 6.22003 
 
 5-! 
 
 9 
 
 090,3 
 
 11.0954 
 
 0775 
 
 9.28058 
 
 12544 
 
 .97,76 
 
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 6.98268 
 
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 6.20851 
 
 5' 
 
 
 09042 
 
 11.0594 
 
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 9-25530 
 
 12574 
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 95302 
 93438 
 
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 6.96823 
 6.95385 
 
 .16137 
 
 6.19703 
 
 50 
 
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 09,01 
 .09130 
 
 0.9882 
 0.9529 
 
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 9.20516 
 
 .12003 
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 89734 
 
 4440 
 
 6-93952 
 6.92525 
 
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 6.18559 
 6.174,9 
 6.16283 
 
 40 
 
 48 
 
 47 
 
 M 
 
 .09159 
 
 0.9,78 
 
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 9-15554 
 
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 .87895 
 
 447 
 
 6.91104 
 
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 6.15151 
 
 46 
 
 1 6 
 
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 0:8483 
 
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 9.13093 
 9.10646 
 
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 4499 
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 6.88278 
 
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 6.14023 
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 45 
 44 
 
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 9.08211 
 
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 4559 
 
 6.86874 
 
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 6.11779 
 
 43 
 
 18 
 
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 0.7797 
 
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 9.05789 
 
 
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 6.85475 
 
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 6.10664 
 
 42 
 
 10 
 
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 0-7457 
 
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 9-03379 
 
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 6.84082 
 
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 6.09552 
 
 41 
 
 20 
 
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 0.71,9 
 
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 9.00983 
 
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 7735 
 
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 6.82694 
 
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 6.08444 
 
 40 
 
 21 
 
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 0.6783 
 
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 8.98598 
 
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 75254 
 
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 6.81812 
 
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 6.07340 
 
 39 
 
 22 
 
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 0.6450 
 
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 8.96227 
 
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 6.79936 
 
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 6.06240 
 
 38 
 
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 8.93867 
 
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 6.78564 
 
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 6.05,43 
 
 37 
 
 24 
 
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 8.91520 
 
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 6.04051 
 
 36 
 
 25 
 
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 8.89,85 
 
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 4796 
 
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 6.02962 
 
 35 
 
 20 
 
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 0.5136 
 
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 8.86862 
 
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 6.74483 
 
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 34 
 
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 8.8455, 
 
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 64732 
 
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 6-73133 
 
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 6.00797 
 
 33 
 
 28 
 
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 0.4491 
 
 335 
 
 8.82252 
 
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 6.71789 
 
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 5.99720 
 
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 364 
 
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 49'5 
 
 6.70450 
 
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 5-98646 
 
 31 
 
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 423 
 
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 29 
 
 32 
 
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 452 
 
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 13224 
 
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 8.7093, 
 
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 6.6383, 
 
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 54' 
 
 8.66482 
 
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 5.92283 
 
 
 36 
 
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 8.64275 
 
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 49465 
 
 
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 8.62078 
 
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 6.59921 
 
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 5.90191 
 
 
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 629 
 
 8.59893 
 
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 6.58627 
 
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 8-55555 
 
 13461 
 
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 6.56055 
 
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 10.0483 
 
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 777 
 
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 13580 
 
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 9.90211 
 
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 33190 
 
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 6.48456 
 
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 47 
 
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 9-87338 
 
 895 
 
 8.40705 
 
 .13669 
 
 .31600 
 
 545' 
 
 6.47206 
 
 17243 
 
 5-79944 
 
 
 48 
 
 0158 
 
 9.84482 
 
 
 8.38625 
 
 .13698 
 
 .30018 
 
 - 548' 
 
 6.45961 
 
 .17273 
 
 5 78938 
 
 
 49 
 
 . 0187 
 
 9.8,64, 
 
 954 
 
 8.36555 
 
 .13728 
 
 .28442 
 
 55" 
 
 6.44720 
 
 17303 
 
 5-77936 
 
 
 5" 
 
 .10216 
 
 9.78817 
 
 - 983 
 
 8.34496 
 
 '3758 
 
 .26873 
 
 5540 
 
 6.43484 
 
 '7333 
 
 5-76937 
 
 
 51 
 
 .10246 
 
 9.76009 
 
 - 013 
 
 8 . 32446 
 
 .13787 
 
 .25310 
 
 - 557 
 
 6.42253 
 
 .17363 
 
 5-7594' 
 
 
 52 
 
 '0275 
 
 9.73217 
 
 . 042 
 
 8.30406 
 
 .138,7 
 
 23754 
 
 . 5600 
 
 6.41026 
 
 '7393 
 
 5 . 74949 
 
 
 53 
 
 . 10305 
 
 9 . 7044 1 
 
 . 072 
 
 8.28376 
 
 .13846 
 
 . 22204 
 
 5630 
 
 6.39804 
 
 .17423 
 
 5-7396o 
 
 
 54 
 
 55 
 
 10334 
 0363 
 
 9.67680 
 9-64935 
 
 
 8-26355 
 8-24345 
 
 .13876 
 .13906 
 
 .20661 
 .19125 
 
 . 5660 
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 6.38587 
 6-37374 
 
 '7453 
 17483 
 
 5 72974 
 5.71992 
 
 
 5 
 
 I0 393 
 
 9.62205 
 
 : 160 
 
 8.22344 
 
 '3935 
 
 '7594 
 
 57'9 
 
 6.36165 
 
 I75I3 
 
 5-7IOI3 
 
 
 57 
 
 . 10422 
 
 9-5949 
 
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 8.20352 
 
 .'3965 
 
 .16071 
 
 5749 
 
 6.34961 
 
 '7543 
 
 5 . 70037 
 
 
 5 
 
 . 0452 
 
 9.56791 
 
 . 219 
 
 8.18370 
 
 '3995 
 
 '4553 
 
 5779 
 
 6.33761 
 
 '7573 
 
 5.69064 
 
 
 59 
 
 .10481 
 
 9-54'o6 
 
 249 
 
 8.16398 
 
 .14024 
 
 . 13042 
 
 . 5809 
 
 6.32566 
 
 .17603 
 
 5.68094 
 
 
 Go 
 
 .105,0 
 
 9.5M36 
 
 . 278 
 
 8.14435 
 
 .14054 
 
 7."537 
 
 5838 
 
 6.31375 
 
 17633 
 
 5.67128 
 
 
 , 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 f 
 
 
 8 
 
 4 
 
 8 
 
 3 
 
 8 
 
 2 
 
 8 
 
 t 
 
 8( 
 
 D 

 
 34 
 
 NATURAL TANGENTS AND COTANGENTS. 
 
 
 I 
 
 
 
 I 
 
 1 
 
 i 
 
 2 
 
 I 
 
 3 
 
 i 
 
 4 
 
 71 
 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 * i 
 
 
 
 :! 7 7 66 
 
 5-67128 
 5-66165 
 
 .19438 
 .19468 
 
 5-M455 
 5-13658 
 
 .21256 
 
 .2 286 
 
 4.70463 
 4.69791 
 
 .23087 
 .23117 
 
 33148 
 32573 
 
 24933 
 .24964 
 
 .00582 
 
 777 
 
 59 
 
 2 
 
 .1769 
 
 5.65205 
 
 -19498 
 
 5.12862 
 
 .2 3 ,6 
 
 4.69121 
 
 .23148 
 
 .32001 
 
 24995 
 
 .00086 
 
 58 
 
 3 
 
 .1772 
 
 5.64248 
 
 19529 
 
 5.12069 
 
 2 347 
 
 4-68452 
 
 23179 
 
 31430 
 
 .25026 
 
 .90592 
 
 57 
 
 4 
 
 1775 
 
 5-63295 
 
 
 5.11279 
 
 2 377 
 
 4.67786 
 
 .23209 
 
 .30860 
 
 .25056 
 
 .99099 
 
 56 
 
 5 
 
 .1778 
 
 5.62344 
 
 19589 
 
 5 10490 
 
 .2 4 08 
 
 4.67121 
 
 .23240 
 
 .30291 
 
 .25087 
 
 .98607 
 
 55 
 
 6 
 
 .17813 
 
 5.61307 
 
 .19619 
 
 5.09704 
 
 2 438 
 
 4.66458 
 
 .23271 
 
 29724 
 
 .25118 
 
 .981,7 
 
 54 
 
 7 
 
 17843 
 
 5.60452 
 
 .19649 
 
 5.08921 
 
 .2 469 
 
 4.65797 
 
 .23301 
 
 29159 
 
 .25,49 
 
 .97627 
 
 53 
 
 8 
 
 .17873 
 
 5-595" 
 
 . 19680 
 
 5.08139 
 
 .2 499 
 
 4.65138 
 
 23332 
 
 28595 
 
 .25,80 
 
 .97,39 
 
 52 
 
 9 
 
 17903 
 
 
 .197,0 
 
 5-07360 
 
 2 529 
 
 4.64480 
 
 23363 
 
 .28032 
 
 .252,1 
 
 -96651 
 
 51 
 
 JO 
 
 17933 
 
 5- '57638 
 
 .19740 
 
 5-06584 
 
 .2560 
 
 4-63825 
 
 23393 
 
 27471 
 
 .25242 
 
 3.96165 
 
 50 
 
 12 
 
 .17963 
 
 .17993 
 .18023 
 
 5.56706 
 
 5-55777 
 5.54851 
 
 .19770 
 .19801 
 .19831 
 
 5.05809 
 5-05037 
 5.04267 
 
 .2 590 
 .2 621 
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 4.6317, 
 4.625,8 
 4.6,868 
 
 23424 
 23455 
 23485 
 
 .26911 
 26352 
 25795 
 
 25273 
 .25304 
 25335 
 
 3-9568o 
 3.95,96 
 3-94713 
 
 9\ 
 
 47 
 
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 .18053 
 
 5-53927 
 
 .19861 
 
 5-03499 
 
 .2 682 
 
 4.6,2,9 
 
 .23516 
 
 25239 
 
 25366 
 
 3-94232 
 
 46 
 
 16 
 
 .18083 
 18113 
 
 5-53007 
 
 .1989, 
 
 5-02734 
 
 .2 712 
 
 4.60572 
 
 23547 
 
 24685 
 
 25397 
 
 3-9375' 
 
 45 
 
 17 
 
 -18-43 
 
 5-' 5 1*176 
 
 19952 
 
 5.01210 
 
 2 773 
 
 4-59283 
 
 2364.8 
 
 23580 
 
 25459 
 
 3-92793 
 
 43 
 
 18 
 
 .18173 
 
 5-50264 
 
 . 19982 
 
 5.00451 
 
 .2 804 
 
 4.5864, 
 
 23639 
 
 .23030 
 
 2549 
 
 3.92316 
 
 42 
 
 20 
 
 .18203 
 .18233 
 
 5.49356 
 5-48451 
 
 .20042 
 
 4.99695 
 4.98940 
 
 .2 8 34 
 .2864 
 
 4.5800, 
 4-57363 
 
 .23670 
 23700 
 
 .2248, 
 21933 
 
 25521 
 25552 
 
 3-91839 
 3.9,364 
 
 4' 
 
 40 
 
 2, 
 
 22 
 
 .18263 
 .18293 
 
 5.47548 
 5.46648 
 
 .20073 
 
 4.98188 
 4.97438 
 
 .2,895 
 21925 
 
 4.56726 
 4.5609, 
 
 23731 
 23762 
 
 .21387 
 
 .20842 
 
 25583 
 .256,4 
 
 3.90890 
 
 3.90417 
 
 P 
 
 23 
 
 .18323 
 
 5-45751 
 
 20133 
 
 4.96690 
 
 .21956 
 
 4-55458 
 
 23793 
 
 . 20298 
 
 .25645 
 
 3-89945 
 
 37 
 
 24 
 
 '8353 
 
 
 .20164 
 
 4-95945 
 
 .21986 
 
 4-54826 
 
 .23823 
 
 19756 
 
 .25676 
 
 3.89474 
 
 30 
 
 25 
 
 .18384 
 
 5 . 43966 
 
 .20194 
 
 4.95201 
 
 .22017 
 
 4.54,96 
 
 23854 
 
 19215 
 
 25707 
 
 3.89004 
 
 35 
 
 26 
 
 .18414 
 
 5-43077 
 
 
 4.94460 
 
 .22047 
 
 4-53568 
 
 23885 
 
 .18675 
 
 25738 
 
 3-88536 
 
 34 
 
 27 
 
 .18444 
 
 5.42192 
 
 .70254 
 
 4-93721 
 
 .220 7 8 
 
 4.5294, 
 
 23916 
 
 .18137 
 
 .25769 
 
 3.88068 
 
 33 
 
 28 
 
 .18474 
 
 5.41309 
 
 .2^285 
 
 4.92984 
 
 .22108 
 
 4.52316 
 
 23946 
 
 .17600 
 
 .25800 
 
 3.8760, 
 
 32 
 
 29 
 
 3 
 
 . 18504 
 .18534 
 
 5.40429 
 5-39552 
 
 20315 
 20345 
 
 4.92249 
 4.91516 
 
 -22139 
 .22169 
 
 4.51693 
 4.51071 
 
 23977 
 .24008 
 
 . .17064 
 16530 
 
 .25831 
 
 .25862 
 
 3.87136 
 3.86671 
 
 3 1 
 
 30 
 
 32 
 33 
 
 .18564 
 18594 
 .18624 
 
 5-38677 
 5-37805 
 5-36936 
 
 20376 
 .20406 
 .20436 
 
 4.90785 
 4.90056 
 4.89330 
 
 .2226 
 
 4.50451 
 4.49832 
 4-49215 
 
 24039 
 .24069 
 .24100 
 
 15997 
 15465 
 14934 
 
 25893 
 25924 
 25955 
 
 3-85745 
 3-85284 
 
 3 
 
 27 
 
 34 
 
 .18654 
 
 5.36070 
 
 .20466 
 
 4.88605 
 
 .2229 
 
 4.48600 
 
 24,31 
 
 14405 
 
 . 25986 
 
 {.84824 
 
 26 
 
 35 
 
 .18684 
 
 5.35206 
 
 .20497 
 
 4.87882 
 
 .2232 
 
 4-47986 
 
 .24162 
 
 13877 
 
 .260,7 
 
 3-84364 
 
 25 
 
 36 
 
 .18714 
 
 5-34345 
 
 .20527 
 
 4.87162 
 
 2235 
 
 4-47374 
 
 .24193 
 
 13350 
 
 .26048 
 
 3.83906 
 
 24 
 
 37 
 
 18745 
 
 5.33487 
 
 
 4.86444 
 
 2238 
 
 4 . 46764 
 
 .24223 
 
 .12825 
 
 .26079 
 
 3.83449 
 
 23 
 
 38 
 
 .18775 
 
 5-32631 
 
 . 20588 
 
 4-85727 
 
 .2241 
 
 4.46155 
 
 24254 
 
 .1230, 
 
 .26110 
 
 3.82992 
 
 22 
 
 39 
 
 .18805 
 
 5.3I778 
 
 .20618 
 
 4.85013 
 
 .2244 
 
 4-45548 
 
 .24285 
 
 .11778 
 
 .26,4, 
 
 3-82537 
 
 21 
 
 40 
 
 .18835 
 
 5-30928 
 
 .20648 
 
 4.84300 
 
 22475 
 
 4.44942 
 
 24316 
 
 -11256 
 
 .26,72 
 
 3-82083 
 
 20 
 
 41 
 
 . 18865 
 
 5.30080 
 
 .20679 
 
 4-8359 
 
 .22505 
 
 4.44338 
 
 24347 
 
 10736 
 
 .26203 
 
 3-81630 
 
 19 
 
 42 
 
 .18895 
 
 5-29235 
 
 .20709 
 
 4.82882 
 
 .22536 
 
 4-43735 
 
 24377 
 
 .10216 
 
 26235 
 
 3-81,77 
 
 18 
 
 43 
 
 .18925 
 
 5-28393 
 
 .20739 
 
 4.82175 
 
 .22567 
 
 4-43'34 
 
 .24408 
 
 .09699 
 
 .26266 
 
 3.80726 
 
 17 
 
 44 
 
 .18955 
 
 5-27553 
 
 .20770 
 
 4.81471 
 
 .22597 
 
 4.42534 
 
 24439 
 
 .09182 
 
 .26297 
 
 3.80276 
 
 ,6 
 
 45 
 
 . 18986 
 
 5-26715 
 
 .20800 
 
 4.80769 
 
 .22628 
 
 4.41936 
 
 2447 
 
 .08666 
 
 .26328 
 
 3.71,827 
 
 15 
 
 46 
 
 .19016 
 
 5-25880 
 
 .20830 
 
 4.80068 
 
 .22658 
 
 4.41340 
 
 24501 
 
 .08152 
 
 26359 
 
 3-79378 
 
 14 
 
 47 
 
 .19046 
 
 5.25048 
 
 .20861 
 
 4-79370 
 
 .22689 
 
 4.40745 
 
 24532 
 
 .07639 
 
 .26390 
 
 3-78931 
 
 
 48 
 
 .19076 
 
 5.24218 
 
 .20891 
 
 4.78673 
 
 .227,9 
 
 4.40152 
 
 .24562 
 
 .07,27 
 
 .26421 
 
 3.78485 
 
 12 
 
 49 
 
 .19106 
 
 5-23391 
 
 .20921 
 
 4.77978 
 
 .22750 
 
 4.3956o 
 
 24593 
 
 .06616 
 
 .26452 
 
 3.78040 
 
 II 
 
 50 
 
 .19136 
 
 5-22566 
 
 .20952 
 
 4.77286 
 
 .22781 
 
 4.38969 
 
 .24624 
 
 .06107 
 
 -26483 
 
 3-77595 
 
 1 
 
 5' 
 
 .19166 
 
 5.21744 
 
 .20982 
 
 4-76595 
 
 .22811 
 
 4-38381 
 
 .24655 
 
 05599 
 
 .265,5 
 
 3-77152 
 
 9 
 
 52 
 
 19197 
 
 5.20925 
 
 .21013 
 
 4.75906 
 
 .22842 
 
 4-37793 
 
 .24686 
 
 .05092 
 
 .26546 
 
 3.76709 
 
 8 
 
 53 
 
 .19227 
 
 5.20107 
 
 .21043 
 
 4.75219 
 
 .22872 
 
 4.37207 
 
 .24717 
 
 .04586 
 
 26577 
 
 3.76268 
 
 7 
 
 54 
 
 19257 
 
 5-I9293 
 
 .21073 
 
 4-74534 
 
 .22903 
 
 4-36623 
 
 24747 
 
 .04081 
 
 .26608 
 
 3-75828 
 
 6 
 
 
 .19287 
 
 5.18480 
 
 .21104 
 
 4-7385 1 
 
 22934 
 
 4.36040 
 
 -24778 
 
 03578 
 
 .26639 
 
 3-75388 
 
 
 
 19317 
 
 5-17671 
 
 .21134 
 
 4-73170 
 
 .22964 
 
 4-35459 
 
 .24809 
 
 .03076 
 
 .26670 
 
 3-7495 
 
 
 57 
 
 19347 
 
 5.16863 
 
 .21164 
 
 4.72490 
 
 22995 
 
 4 . 34879 
 
 .24840 
 
 "574 
 
 .26701 
 
 3-74512 
 
 
 58 
 
 19378 
 
 5.16058 
 
 .21195 
 
 4.71813 
 
 .23026 
 
 4-34300 
 
 .24871 
 
 .02074 
 
 -26733 
 
 3 74075 
 
 
 59 
 
 .19408 
 
 5-15256 
 
 .21225 
 
 4-71137 
 
 .23056 
 
 4-33723 
 
 .24902 
 
 .01576 
 
 .26764 
 
 3 73640 
 
 
 60 
 
 .19438 
 
 5-14455 
 
 .21256 
 
 4.70463 
 
 23087 
 
 4-33I48 
 
 24933 
 
 .0,078 
 
 26795 
 
 3-73205 
 
 
 t 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 , 
 
 
 7 
 
 f 
 
 7 
 
 8 
 
 7 
 
 7 
 
 7 
 
 6 
 
 7 
 
 5 

 
 NATURAL TANGENTS AND COTANGENTS. 
 
 
 I 
 
 5 
 
 I 
 
 
 I 
 
 7 
 
 I 
 
 8 
 
 I 
 
 ; 
 
 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 
 
 
 26795 
 
 3-73205 
 
 .28675 
 
 3.4874. 
 
 30573 
 
 3-27085 
 
 32492 
 
 3.07768 
 
 34433 
 
 .9042. 
 
 " 6 , 
 
 1 
 
 .26826 
 
 3.72771 
 
 . 8706 
 
 3-48359 
 
 .30605 
 
 3-26745 
 
 32524 
 
 3.07464 
 
 34465 
 
 .90147 
 
 59 
 
 2 
 
 .26857 
 
 3-72338 
 
 . 8738 
 
 3-47977 
 
 .30637 
 
 3 . 26406 
 
 32556 
 
 3.07.60 
 
 34498 
 
 .89873 
 
 
 3 
 
 .26888 
 
 3.7.907 
 
 . 8769 
 
 3-47596 
 
 .30669 
 
 3.26067 
 
 -32588 
 
 3.06857 
 
 34530 
 
 .89600 
 
 mm 
 
 4 
 
 .26920 
 
 3-71476 
 
 . 8800 
 
 3.47216 
 
 .30700 
 
 3.25729 
 
 .3262. 
 
 3-06554 
 
 34563 
 
 .89327 
 
 mA 
 
 5 
 
 26951 
 
 3.7.046 
 
 8832 
 
 3-46837 
 
 30732 
 
 
 32653 
 
 3.06252 
 
 34596 
 
 89055 
 
 55 
 
 6 
 
 .26982 
 
 3.70616 
 
 . 8864 
 
 3-46458; 
 
 30764 
 
 3-25055 
 
 32685 
 
 3.05950 
 
 34628 
 
 .88783 
 
 54 
 
 7 
 
 g 
 
 .27013 
 
 3.70188 
 3.69761 
 
 . 8895 
 
 3.46080 
 
 .30796 
 
 3-247-9 
 
 327-7 
 
 3-05649 
 
 .34661 
 
 .885.. 
 
 . 8824O 
 
 S3 
 
 9 
 
 -27044 
 .27076 
 
 3-69335 
 
 .28958 
 
 3.45327 
 
 .30860 
 
 3.24049 
 
 .32782 
 
 3*05349 
 3.05049 
 
 34693 
 .34726 
 
 87970 
 
 ?2 
 
 51 
 
 
 .27107 
 
 3.68909 
 
 .28993 
 
 3- 4495 - 
 
 3089. 
 
 3-237-4 
 
 .328.4 
 
 3-04749 
 
 34758 
 
 .87700 
 
 
 
 
 .27.38 
 
 3-68485 
 
 .2902. 
 
 3-44576 
 
 .30923 
 
 3-2338- 
 
 .32846 
 
 3-0445 
 
 3479- 
 
 .87430 
 
 M 
 
 12 
 
 .27169 
 
 3.6806. 
 
 29^53 
 
 3.44202 
 
 30955 
 
 3.23048 
 
 .32878 
 
 3.04152 
 
 34824 
 
 .8716. 
 
 V J 
 
 IJ 
 
 .2720. 
 
 3.67638 
 
 .29084 
 
 3.43829 
 
 .30987 
 
 3.227.5 
 
 .329-1 
 
 3.03854 
 
 34856 
 
 .86892 
 
 47 
 
 J 5 
 
 .27232 
 .27263 
 
 3-67217 
 3.66795 
 
 .29.16 
 .29.47 
 
 3-43456 
 3.43084 
 
 .3.0.9 
 .3.05. 
 
 3-22384 
 3.22053 
 
 32943 
 32975 
 
 3.03260 
 
 -34889 
 .34922 
 
 .86624 
 86356 
 
 45 
 
 1 6 
 
 27294 
 
 3.66375 
 
 .29.79 
 
 3-42713 
 
 .3.083 
 
 3.21722 
 
 .33007 
 
 3.02963 
 
 34954 
 
 .86089 
 
 44 
 
 17 
 
 27326 
 
 3-65957 
 
 .29210 
 
 3.42343 
 
 3-1-5 
 
 3.2.392 
 
 33040 
 
 3.02667 
 
 34987 
 
 .85822 
 
 45 
 
 18 
 
 27357 
 
 3-65538 
 
 .29242 
 
 3-4-973 
 
 3"47 
 
 3.21063 
 
 33072 
 
 3-02372 
 
 .35020 
 
 85555 
 
 42 
 
 19 
 
 27388 
 
 3-65-21 
 
 .29274 
 
 3.41604 
 
 .31,78 
 
 3.20734 
 
 33-04 
 
 3.02077 
 
 35052 
 
 .85289 
 
 4' 
 
 
 .274.9 
 
 3-64705 
 
 .29305 
 
 3.41236 
 
 .3.2.0 
 
 3.20406 
 
 33-36 
 
 3.0.783 
 
 35085 
 
 .85023 
 
 4'-' 
 
 2. 
 
 27451 
 
 3.64289 
 
 29337 
 
 3-40869 
 
 .3.242 
 
 3.20079 
 
 33-69 
 
 3-0.489 
 
 3S--8 
 
 .84758 
 
 59 
 
 22 
 
 .27482 
 
 3-63874 
 
 29368 
 
 3.40502 
 
 3-274 
 
 3- -9752 
 
 3320, 
 
 3.01,96 
 
 35150 
 
 .84494 
 
 .58 
 
 25 
 
 27513 
 
 3-6346, 
 
 .29400 
 
 3.40136 
 
 31306 
 
 3.19426 
 
 33233 
 
 3.00903 
 
 35-83 
 
 .84229 
 
 37 
 
 24 
 
 25 
 
 27545 
 27576 
 
 3.63048 
 3.62636 
 
 29432 
 29463 
 
 3-3977 1 
 3.39406 
 
 3-338 
 3-370 
 
 3.19100 
 3.I8775 
 
 .33266 
 33298 
 
 3.006,1 
 3.00319 
 
 .35216 
 35248 
 
 83965 
 .83702 
 
 3'-- 
 
 35 
 
 26 
 
 .27607 
 
 3.62224 
 
 29495 
 
 3 . 39042 
 
 .31402 
 
 3.18451 
 
 33330 
 
 3.00028 
 
 .35281 
 
 83439 
 
 34 
 
 3 
 
 27638 
 .27670 
 
 3.6.8.4 
 3.6.405 
 
 .29526 
 29558 
 
 3.38679 
 3-383I7 
 
 3-434 
 .3.466 
 
 3.18,27 
 3-17804 
 
 33363 
 33395 
 
 2.99738 
 2-99447 
 
 353-4 
 35346 
 
 83,76 
 
 33 
 
 52 
 
 2; 
 
 .27701 
 
 3.60996 
 
 .29590 
 
 3-37955 
 
 .31498 
 
 3.1748. 
 
 33427 
 
 2.99.58 
 
 35379 
 
 82653 
 
 31 
 
 
 27732 
 
 3.60588 
 
 .2962. 
 
 3-37594 
 
 31530 
 
 3--7-S9 
 
 .33460 
 
 2.98868 
 
 354-2 
 
 8239- 
 
 3 
 
 j, 
 
 .27764 
 
 3.60.8. 
 
 29653 
 
 3-37234 
 
 .3.562 
 
 3.16838 
 
 33492 
 
 .98580 
 
 35445 
 
 82.30 
 
 ag 
 
 3 2 
 
 27795 
 
 3-59775 
 
 .29685 
 
 3-36875 
 
 3-594 
 
 3.165-7 
 
 33524 
 
 .98292 
 
 35477 
 
 8.870 
 
 20 
 
 ^ ^ 
 
 .27826 
 
 3-5937 
 
 .297.6 
 
 3.36516 
 
 .31626 
 
 3.16.97 
 
 33557 
 
 .98004 
 
 355-0 
 
 8.6.0 
 
 27 
 
 34 
 
 .27858 
 
 308966 
 
 29748 
 
 3-36158 
 
 3-658 
 
 3-I5877 
 
 33589 
 
 977-7 
 
 35543 
 
 8-350 
 
 20 
 
 JS 
 
 .27889 
 
 3.58562 
 
 .29780 
 
 3.35800 
 
 .3.690 
 
 3-I5558 
 
 3362- 
 
 97430 
 
 35576 
 
 8.091 
 
 25 
 
 Jfi 
 
 .2792. 
 
 3.58.60 
 
 .298.1 
 
 3-35443 
 
 .31722 
 
 3.15240 
 
 33654 
 
 97-44 
 
 35608 
 
 80833 
 
 24 
 
 37 
 
 27952 
 
 3.57758 
 
 .29843 
 
 3-35087 
 
 3*754 
 
 3.14922 
 
 33686 
 
 .96858 
 
 .35641 
 
 80574 
 
 23 
 
 38 
 
 27983 
 
 3-57357 
 
 29875 
 
 3-34732 
 
 .3.786 
 
 3.14605 
 
 337-8 
 
 96573 
 
 35674 
 
 803.6 
 
 
 39 
 
 4' 
 
 128046 
 
 3-56957 
 3-56557 
 
 .29006 
 .29938 
 
 3-34377 
 3-34023 
 
 .318,8 
 .3-850 
 
 3.14288 
 3.I3972 
 
 33751 
 33783 
 
 .96288 
 .96004 
 
 35707 
 35740 
 
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 .28077 
 
 3.56.59 
 
 .29970 
 
 3-33670 
 
 .3,882 
 
 3.13656 
 
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 79545 
 
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 3-333I7 
 
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 3.13027 
 
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 3.52609 
 
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 3.30521 
 
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 3.10842 
 
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 3,09606 
 
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 2.76247 
 
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 3-28795 
 
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 3.09298 
 
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 3.50279 
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 2.75746 
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 3.49509 
 
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 3-27767 
 
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 2.75246 
 
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 59 
 
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 3-49I23 
 
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 3.27426 
 
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 3.08073 
 
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 2.74097 
 
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 3.48741 
 
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 3.27085 
 
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 3.07768 
 
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 NATURAL TANGENTS AND COTANGENTS. 
 
 
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 Tang 
 
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 36397 
 
 2. 4748 
 
 38386 
 
 2.60509 
 
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 2.47509 
 
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 2-35585 
 
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 2. 4499 
 
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 2.60283 
 
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 2.47302 
 
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 2-35395 
 
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 2. 4251 
 
 38453 
 
 2.60057 
 
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 2.47095 
 
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 2.35205 
 
 44593 
 
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 58 
 
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 36496 
 
 2. 4004 
 
 .38487 
 
 2.59831 
 
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 2.46888 
 
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 2-35015 
 
 .44627 
 
 24077 
 
 57 
 
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 .36529 
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 2. 3756 
 2. 3509 
 
 38520 
 38553 
 
 2.59606 
 2.59381 
 
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 2.46682 
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 42585 
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 2.34825 
 2-34636 
 
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 44697 
 
 3902 
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 56 
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 6 
 
 36595 
 
 2. 3263 
 
 38587 
 
 2.59156 
 
 .40606 
 
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 42654 
 
 2-34447 
 
 44732 
 
 3553 
 
 54 
 
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 2. 3017 
 
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 2.58932 
 
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 2.34258 
 
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 53 
 
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 2. 2771 
 
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 2.58708 
 
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 2. 2526 
 
 38687 
 
 2.58484 
 
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 45655 
 
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 2.33881 
 
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 36727 
 
 2. 228l 
 
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 2.58261 
 
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 2.33693 
 
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 36793 
 
 2. 2036 
 
 38754 
 
 58787 
 
 2:58038 
 
 40775 
 
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 .42826 
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 2.33505 
 
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 2. 1792 
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 2-57593 
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 .42894 
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 2.33130 
 2.32943 
 
 4497 
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 - 2337 
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 15 
 
 36892 
 
 2. 1062 
 
 .38888 
 
 2-57150 
 
 .40911 
 
 44433 
 
 .42963 
 
 2.32756 
 
 4504 
 
 1992 
 
 45 
 
 16 
 
 36925 
 
 2.70819 
 
 .38921 
 
 2.56928 
 
 .40945 
 
 .44230 
 
 .42998 
 
 2.32570 
 
 .4508 
 
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 44 
 
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 2.70577 
 
 38955 
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 2.56707 
 
 2 . 56487 
 
 .40979 
 
 44027 
 
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 2.32383 
 
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 2.7O335 
 2.70094 
 
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 2.56266 
 
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 43623 
 
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 2.32012 
 
 .4518 
 
 .21304 
 
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 37057 
 
 2.69853 
 2 60612 
 
 39055 
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 2.56046 
 
 2.55827 
 
 .41081 
 
 43422 
 
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 2.31826 
 
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 .21132 
 
 40 
 
 22 
 23 
 
 37 I 23 
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 2.09012 
 2.69371 
 2.69131 
 
 .39122 
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 2.55608 
 2.55389 
 
 4 149 
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 43019 
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 .43205 
 .43230 
 
 2.31456 
 2.31271 
 
 4529 
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 .20790 
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 38 
 37 
 
 24 
 
 37 I 9 
 
 2.68892 
 
 .39190 
 
 2.55170 
 
 4 217 
 
 .42618 
 
 43274 
 
 2.31086 
 
 45362 
 
 . 20449 
 
 36 
 
 25 
 
 37223 
 
 2.68653 
 
 .39223 
 
 2.54952 
 
 4 251 
 
 .42418 
 
 .43308 
 
 2.30902 
 
 -45397 
 
 .20278 
 
 35 
 
 26 
 
 37256 
 
 2.68414 
 
 39257 
 
 2-54734 
 
 4 285 
 
 .42218 
 
 43343 
 
 2.30718 
 
 45432 
 
 .20108 
 
 34 
 
 27 
 
 .37289 
 
 2.68175 
 
 3929 
 
 2.54516 
 
 4 319 
 
 .42019 
 
 43378 
 
 2.30534 
 
 45467 
 
 9938 
 
 33 
 
 28 
 
 37322 
 
 2.67937 
 
 39324 
 
 2.54299 
 
 4 353 
 
 .41819 
 
 .43412 
 
 2-3035 1 
 
 45502 
 
 9769 
 
 32 
 
 29 
 
 37355 
 
 2.67700 
 
 39357 
 
 2.54082 
 
 4 387 
 
 .41620 
 
 43447 
 
 2.30167 
 
 45538 
 
 9599 
 
 
 
 37388 
 
 2.67462 
 
 39391 
 
 2.53865 
 
 4 421 
 
 .41421 
 
 43481 
 
 2.29984 
 
 45573 
 
 9430 
 
 30 
 
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 37422 
 
 2.67225 
 
 39425 
 
 2.53648 
 
 4 455 
 
 2.41223 
 
 43Si6 
 
 2.29801 
 
 .45608 
 
 . 9261 
 
 29 
 
 32 
 
 37455 
 
 2.66989 
 
 .39458 
 
 2.53432 
 
 4 490 
 
 2.41025 
 
 4355 
 
 2.29619 
 
 45643 
 
 . 9092 
 
 28 
 
 33 
 
 .37488 
 
 2.66752 
 
 39492 
 
 2.53217 
 
 4 524 
 
 2.40827 
 
 43585 
 
 2.29437 
 
 .45678 
 
 8923 
 
 27 
 
 34 
 
 37521 
 
 2.66516 
 
 39526 
 
 2 . 53001 
 
 4 558 
 
 2.40629 
 
 .43620 
 
 2.20254 
 
 45713 
 
 - 8755 
 
 26 
 
 
 37554 
 
 2.66281 
 
 39559 
 
 2.52786 
 
 4 592 
 
 2.40432 
 
 43654 
 
 2.29073 
 
 .45748 
 
 - 8587 
 
 25 
 
 36 
 
 37588 
 
 2.66046 
 
 39593 
 
 2.5257I 
 
 . 626 
 
 2.40235 
 
 .43689 
 
 2.28891 
 
 45784 
 
 . 8419 
 
 24 
 
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 2.65811 
 
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 2.52357 
 
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 2.40038 
 
 43724 
 
 2.28710 
 
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 2.65576 
 
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 2.52142 
 
 . 1694 
 
 2.39841 
 
 43758 
 
 2.28528 
 
 .45854 
 
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 37687 
 
 2.65342 
 
 39694 
 
 2.51929 
 
 . 1728 
 1763 
 
 2.39645 
 
 43793 
 
 2.28348 
 
 .45889 
 
 79' 6 
 
 21 
 
 4 1 
 
 37754 
 
 2.64875 
 
 3976i 
 
 2.51502 
 
 41797 
 
 2.39253 
 
 .43862 
 
 2.27987 
 
 .45960 
 
 - 7582 
 
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 42 
 
 37787 
 
 2.64642 
 
 39795 
 
 2.51289 
 
 .4 831 
 
 2! 39058 
 
 .43897 
 
 2.27806 
 
 45995 
 
 
 18 
 
 43 
 
 .37820 
 
 2.64410 
 
 39829 
 
 2.51076 
 
 .4 865 
 
 2.38863 
 
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 2.27626 
 
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 17 
 
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 2.50864 
 
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 2.38668 
 
 .43966 
 
 2.27447 
 
 .46065 
 
 7083 
 
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 63945 
 
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 2.50652 
 
 4 933 
 
 2-38473 
 
 .44001 
 
 2.27267 
 
 .46101 
 
 69,7 
 
 15 
 
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 .39930 
 
 2.50440 
 
 .4968 
 
 2.38279 
 
 44036 
 
 2.27088 
 
 46136 
 
 675- 
 
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 63483 
 
 .39963 
 
 2.50229 
 
 .42002 
 
 2.38084 
 
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 2.26909 
 
 .46171 
 
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 13 
 
 
 .37986 
 
 .63252 
 
 39997 
 
 2.50018 
 
 .42036 
 
 2.37891 
 
 .44105 
 
 2.26730 
 
 .46206 
 
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 12 
 
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 . 38020 
 38053 
 
 .63021 
 .62791 
 
 .40031 
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 2.49807 
 
 2-49597 
 
 .42070 
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 37697 
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 44175 
 
 2.26552 
 2.26374 
 
 .46242 
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 .62561 
 
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 2.49386 
 
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 373 11 
 
 .44210 
 
 2.261^6 
 
 .46312 
 
 5925 
 
 9 
 
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 62332 
 
 .40132 
 
 2.49177 
 
 42173 
 
 -37"8 
 
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 2.26ol8 
 
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 8 
 
 53 
 
 
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 .40166 
 
 2.48967 
 
 .42207 
 
 .36925 
 
 4279 
 
 2.25840 
 
 .46383 
 
 5596 
 
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 54 
 
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 .40200 
 
 2.48758 
 
 .42242 
 
 36733 
 
 43>4 
 
 2.25663 
 
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 5432 
 
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 36541 
 
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 2.48340 
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 2.25309 
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 58 
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 38353 
 
 .60963 
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 40335 
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 2.4,7924 
 2.47716 
 
 42379 
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 35967 
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 2.24956 
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 42447 
 
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 Tang 
 
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 6 
 
 3 
 
 6 
 
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 6 
 
 7 
 
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 5 
 
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 NATURAL TANGENTS AND COTANGENTS. 
 
 87 
 
 
 2 
 
 
 2( 
 
 5 
 
 2 
 
 1 
 
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 2 
 
 f 
 
 
 
 Tang 
 
 Cotang 
 
 Tang 
 
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 2.14451 
 
 2.14288 
 
 48773 
 .48809 
 
 .05030 
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 50953 
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 53171 
 53208 
 
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 55431 
 55469 
 
 .80405 
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 59 
 
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 2. 4125 
 
 .48845 
 
 .04728 
 
 .5.026 
 
 95979 
 
 53246 
 
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 55507 
 
 .80158 
 
 
 3 
 
 46737 
 
 2. 3963 
 
 .48881 
 
 04577 
 
 .51063 
 
 .95838 
 
 53283 
 
 87677 
 
 55545 
 
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 57 
 
 
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 2. 380. 
 
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 53320 
 
 87546 
 
 55583 
 
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 5 
 
 .46808 
 
 2. 3639 
 
 48953 
 
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 .5"36 
 
 95557 
 
 53358 
 
 87415 
 
 5562. 
 
 79788 
 
 55 
 
 6 
 
 .46843 
 
 2- 3477 
 
 .48989 
 
 .04125 
 
 51173 
 
 95417 
 
 53395 
 
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 55659 
 
 79665 
 
 54 
 
 7 
 
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 2- 33l6 
 
 .49026 
 
 03975 
 
 
 95277 
 
 53432 
 
 .87.52 
 
 55697 
 
 79542 
 
 53 
 
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 .49062 
 
 .03825 
 
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 95137 
 
 53470 
 
 .87021 
 
 55736 
 
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 2. 2993 
 
 .49098 
 
 03675 
 
 .51283 
 
 94997 
 
 53507 
 
 .86891 
 
 55774 
 
 79296 
 
 51 
 
 
 .46985 
 
 2. 2832 
 
 49134 
 
 .03526 
 
 51319 
 
 .94858 
 
 53545 
 
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 .55812 
 
 79174 
 
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 2. 267. 
 
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 51356 
 
 .94718 
 
 53582 
 
 .86630 
 
 .55850 
 
 79051 
 
 49 
 
 12 
 
 .47056 
 
 2. 2511 
 
 .49206 
 
 .03227 
 
 5!393 
 
 94579 
 
 .53620 
 
 .86499 
 
 .55888 
 
 78929 
 
 3 
 
 13 
 
 .47092 
 
 2. 2350 
 
 .49242 
 
 .03078 
 
 SMS 
 
 .94440 
 
 53657 
 
 .86369 
 
 55926 
 
 .78807 
 
 . 7 
 
 T 4 
 
 .47128 
 
 2. 2190 
 
 49278 
 
 .02929 
 
 51467 
 
 94301 
 
 55694 
 
 86239 
 
 55964 
 
 78685 
 
 6 
 
 5 
 
 47-63 
 
 2. 20 3 
 
 49315 
 
 .02780 
 
 5!503 
 
 .94162 
 
 53732 
 
 .86109 
 
 .56003 
 
 78563 
 
 5 
 
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 2. ,8 7 I 
 
 49351 
 
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 51540 
 
 .94023 
 
 53769 
 
 85979 
 
 56041 
 
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 47234 
 
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 49387 
 
 .02483 
 
 51577 
 
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 .53807 
 
 85850 
 
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 78319 
 
 3 1 
 
 3 
 
 8 
 
 .47270 
 
 2- 1552 
 
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 02335 
 
 51614 
 
 93746 
 
 53844 
 
 .85720 
 
 .56.17 
 
 .78.98 
 
 
 g 
 
 4735 
 
 2. 1392 
 
 49459 
 
 .02.87 
 
 51651 
 
 .93608 
 
 .53882 
 
 85591 
 
 .56156 
 
 .78077 
 
 j 
 
 
 47341 
 
 2. 1233 
 
 49495 
 
 .02039 
 
 .51688 
 
 93470 
 
 53920 
 
 .85462 
 
 .56194 
 
 77955 
 
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 21 
 
 47377 
 
 2. 1075 
 
 49532 
 
 .01891 
 
 51724 
 
 93332 
 
 53957 
 
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 47412 
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 51798 
 
 93195 
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 53995 
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 .56270 
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 77713 
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 56347 
 
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 36 
 
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 2. 0442 
 
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 151872 
 
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 26 
 
 47555 
 
 2. 0284 
 
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 77230 
 
 34 : 
 
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 2. 0126 
 
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 33 
 
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 2.09969 
 
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 29 
 
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 56539 
 
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 52057 
 
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 56577 
 
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 2.08560 
 
 50-13 
 
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 54597 
 
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 52427 
 
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 75556 
 
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 50295 
 
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 38 
 
 NATURAL TANGENTS AND COTANGENTS. 
 
 
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 y 
 
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 60 
 
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 53693 
 
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 57 
 
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 53595 
 
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 56 
 
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 57929 
 
 72625 
 
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 59517 
 
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 53497 
 
 67663 
 
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 55 
 
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 72509 
 
 .60324 
 
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 59414 
 
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 54 
 
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 .58007 
 
 72393 
 
 .60364 
 
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 .62770 
 
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 .65231 
 
 .53302 
 
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 .47607 
 
 53 
 
 8 
 
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 52 
 
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 72163 
 
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 58279 
 
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 46 
 
 15 
 
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 64795 
 
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 . 58490 
 
 65563 
 
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 .68088 
 
 .46870 
 
 45 
 
 1 6 
 
 58357 
 
 7 358 
 
 .60721 
 
 .64687 
 
 63136 
 
 .58388 
 
 .65604 
 
 5 429 
 
 .68.30 
 
 .46778 
 
 44 
 
 17 
 
 .58396 
 
 7 244 
 
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 64579 
 
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 .65646 
 
 5 332 
 
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 43 
 
 1 8 
 
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 64363 
 
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 58513 
 
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 21 
 
 58552 
 
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 57879 
 
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 68343 
 
 .46320 
 
 39 
 
 22 
 
 58591 
 
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 38 
 
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 63934 
 
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 57676 
 
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 37 
 
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 57575 
 
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 36 
 
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 34 
 
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 33 
 
 28 
 
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 .63398 
 
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 5 275 
 
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 32 
 
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 31 
 
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 30 
 
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 .56868 
 
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 29 
 
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 56767 
 
 .66272 
 
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 .68814 
 
 45320 
 
 28 
 
 33 
 
 34 
 
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 .69428 
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 .45229 
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 27 
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 24 
 
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 10 
 
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 15 
 M 
 
 47 
 
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 49 
 
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 69545 
 
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 54972 
 
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 43703 
 
 10 
 
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 43614 
 
 9 
 
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 5977 
 
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 54675 
 
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 4 
 
 57 
 
 59967 
 
 66757 
 
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 54281 
 
 67324 
 
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 3 
 
 58 
 
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 54183 
 
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 .48442 
 
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 fc 
 
 .60046 
 .60086 
 
 66538 
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 .62446 
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 .60137 
 1.60033 
 
 .64899 
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 54085 
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 67451 
 
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 69977 
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 , 
 
 Cotang 
 
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 Tang 
 
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 5< 
 
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 5 
 
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 ) 

 
 NATURAL TANGENTS AND COTANGENTS. 
 
 39 
 
 
 3 
 
 5 
 
 3 
 
 5 
 
 3 
 
 7 
 
 3 
 
 3 
 
 3 
 
 9 
 
 
 
 Tang 
 
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 .70021 
 .70064 
 
 .428.5 
 .42726 
 
 72654 
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 37638 
 37554 
 
 75355 
 
 32704 
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 .78129 
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 27994 
 .27917 
 
 .80978 
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 23490 
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 to 
 
 59 
 
 
 .70.07 
 
 42638 
 
 72743 
 
 1-37470 
 
 75447 
 
 3 2 544 
 
 .78222 
 
 .2784. 
 
 .81075 
 
 23343 
 
 58 
 
 
 .70151 
 
 42550 
 
 .72788 
 
 37386 
 
 75492 
 
 32464 
 
 .78269 
 
 27764 
 
 
 .23270 
 
 57 
 
 
 70194 
 
 .42462 
 
 72832 
 
 .37302 
 
 75538 
 
 .32384 
 
 78316 
 
 .27688 
 
 .8lI7I 
 
 23.96 
 
 56 
 
 5 
 
 .70238 
 
 42374 
 
 .72877 
 
 .372,8 
 
 75584 
 
 32304 
 
 .78363 
 
 .276.1 
 
 .8l22O 
 
 .23.23 
 
 55 
 
 c 
 
 .70281 
 
 .42286 
 
 .72921 
 
 37134 
 
 75629 
 
 .32224 
 
 .784,0 
 
 27535 
 
 .8l268 
 
 23050 
 
 54 
 
 7 
 
 .70325 
 
 .42,98 
 
 .72966 
 
 37050 
 
 75675 
 
 .32144 
 
 78457 
 
 27458 
 
 .81316 
 
 .22977 
 
 53 
 
 8 
 
 .70368 
 
 .421,0 
 
 .73010 
 
 .36967 
 
 75721 
 
 .32064 
 
 78504 
 
 .27382 
 
 .81364 
 
 .22904 
 
 52 
 
 9 
 
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 73055 
 
 36883 
 
 75767 
 
 31984 
 
 .78551 
 
 27.306 
 
 .814.3 
 
 .2283, 
 
 5' 
 
 
 70455 
 
 41934 
 
 .73.00 
 
 .36800 
 
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 .3,904 
 
 .78598 
 
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 .22758 
 
 5 
 
 H 
 
 70499 
 
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 73144 
 
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 27153 
 
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 .22685 
 
 49 
 
 .2 
 
 .70542 
 
 41759 
 
 73189 
 
 .36633 
 
 .75904 
 
 31745 
 
 .78692 
 
 .27077 
 
 .81558 
 
 .22612 
 
 48 
 
 13 
 
 70586 
 
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 73234 
 
 
 75950 
 
 .3.666 
 
 78739 
 
 .2700. 
 
 .8,606 
 
 22539 
 
 47 
 
 
 .70629 
 
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 73278 
 
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 75996 
 
 31586 
 
 .78786 
 
 .26925 
 
 8,655 
 
 .22467 
 
 46 
 
 j^ 
 
 70673 
 
 4M97 
 
 73323 
 
 36383 
 
 .76042 
 
 31507 
 
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 45 
 
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 .707.7 
 
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 73368 
 
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 .31427 
 
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 81752 
 
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 44 
 
 17 
 
 .70760 
 
 .4.322 
 
 73413 
 
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 .76134 
 
 .31348 
 
 .78928 
 
 .26698 
 
 .81800 
 
 .22249 
 
 43 
 
 
 .70804 
 
 41235 
 
 73457 
 
 36134 
 
 .76180 
 
 .31269 
 
 78975 
 
 .26622 
 
 .81849 
 
 .22176 
 
 42 
 
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 73502 
 
 3605. 
 
 .76226 
 
 .31,90 
 
 .79022 
 
 .26546 
 
 .8,898 
 
 
 
 
 .70891 
 
 .41061 
 
 73547 
 
 .35968 
 
 .76272 
 
 .31110 
 
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 .8.946 
 
 .22031 
 
 4" 
 
 21 
 
 70935 
 
 40974 
 
 73592 
 
 .35885 
 
 .763.8 
 
 .31031 
 
 .79"7 
 
 26395 
 
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 2.959 
 
 39 
 
 22 
 
 .70979 
 
 .40887 
 
 73637 
 
 35802 
 
 76364 
 
 30952 
 
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 .263,9 
 
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 38 
 
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 7368. 
 
 35719 
 
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 30873 
 
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 .2.814 
 
 37 
 
 24 
 
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 73726 
 
 35637 
 
 76456. 
 
 30795 
 
 79259 
 
 .26.69 
 
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 36 
 
 25 
 
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 .40627 
 
 73771 
 
 35554 
 
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 .307.6 
 
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 .82.90 
 
 .21670 
 
 35 
 
 26 
 
 7"54 
 
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 .35472 
 
 76548 
 
 30637 
 
 79354 
 
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 -21598 
 
 34 
 
 27 
 
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 76594 
 
 30558 
 
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 25943 
 
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 33 
 
 28 
 
 .71242 
 
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 7396 
 
 35307 
 
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 79449 
 
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 2.454 
 
 .32 
 
 29 
 
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 73951 
 
 35224 
 
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 25792 
 
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 3 1 
 
 30 
 
 71329 
 
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 7396 
 
 35142 
 
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 30323 
 
 79544 
 
 25717 
 
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 31 
 
 71373 
 
 .40.09 
 
 .74041 
 
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 76779 
 
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 7959' 
 
 25642 
 
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 .21238 
 
 20 
 
 32 
 
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 79639 
 
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 28 
 
 33 
 
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 39936 
 
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 27 
 
 34 
 
 
 39850 
 
 4176 
 
 34814 
 
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 79734 
 
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 .2.023 
 
 26 
 
 35 
 
 71549 
 
 39764 
 
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 34732 
 
 76964 
 
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 25343 
 
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 25 
 
 36 
 
 71593 
 
 39679 
 
 4267 
 
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 29853 
 
 79829 
 
 .25268 
 
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 24 
 
 37 
 
 .71637 
 
 39593 
 
 4312 
 
 .34568 
 
 77057 
 
 29775 
 
 79877 
 
 25193 
 
 .82776 
 
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 3 
 
 3^ 
 
 .7.681 
 
 3957 
 
 4357 
 
 34487 
 
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 79924 
 
 251,8 
 
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 39 
 
 71725 
 
 39421 
 
 . 4402 
 
 3445 
 
 .77149 
 
 .296.8 
 
 79972 
 
 25044 
 
 .82874 
 
 .20665 
 
 i 
 
 40 
 
 71769 
 
 39336 
 
 4447 
 
 34323 
 
 77196 
 
 .29541 
 
 .80020 
 
 .24969 
 
 .82923 
 
 20593 
 
 a 
 
 4 1 
 
 71813 
 
 39250 
 
 4492 
 
 .34242 
 
 .77242 
 
 29463 
 
 .80067 
 
 24895 
 
 .82972 
 
 .20522 
 
 9 
 
 42 
 
 .7.857 
 
 39165 
 
 4538 
 
 34160 
 
 77289 
 
 29385 
 
 .80115 
 
 .24820 
 
 .83022 
 
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 8 
 
 43 
 44 
 
 .71901 
 .7.946 
 
 39079 
 38994 
 
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 34079 
 33998 
 
 77335 
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 29307 
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 24746 
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 20379 
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 I 
 
 45 
 
 .7,990 
 
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 4674 
 
 339i6 
 
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 24597 
 
 .83.69 
 
 .20237 
 
 5 
 
 46 
 
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 38824 
 
 4719 
 
 33835 
 
 77475 
 
 29074 
 
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 24523 
 
 .83218 
 
 .20.66 
 
 4 
 
 47 
 
 .72078 
 
 38738 
 
 4764 
 
 33754 
 
 77521 
 
 28997 
 
 .80354 
 
 24449 
 
 .83268 
 
 .20095 
 
 3 
 
 48 
 
 .72.22 
 
 38653 
 
 . 48.0 
 
 33673 
 
 77568 
 
 .28919 
 
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 24375 
 
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 2 
 
 49 
 
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 38568 
 
 4855 
 
 33592 
 
 77615 
 
 
 .80450 
 
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 .83366 
 
 10953 
 
 I 
 
 50 
 
 .722.1 
 
 38484 
 
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 335" 
 
 .77661 
 
 : 28764 
 
 .80498 
 
 .24227 
 
 83415 
 
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 10 
 
 51 
 
 72255 
 
 38399 
 
 4946 
 
 3343 
 
 .77708 
 
 .28687 
 
 .80546 
 
 24153 
 
 83465 
 
 .19811 
 
 9 
 
 52 
 
 .72299 
 
 38314 
 
 4991 
 
 33349 
 
 77754 
 
 .28610 
 
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 .24079 
 
 83514 
 
 .19740 
 
 8 
 
 53 
 
 72344 
 
 38229 
 
 5037 
 
 .33268 
 
 .77801 
 
 .28533 
 
 .80642 
 
 24005 
 
 83564 
 
 . 9669 
 
 7 
 
 54 
 
 .72388 
 
 .38.45 
 
 . 5082 
 
 33187 
 
 77848 
 
 .28456 
 
 .80690 
 
 23931 
 
 .83613 
 
 9599 
 
 6 
 
 P 
 
 .72432 
 72477 
 
 .38060 
 37976 
 
 75!73 
 
 33107 
 .33026 
 
 77895 
 77941 
 
 28379 
 
 80738 
 .80786 
 
 .23858 
 .23784 
 
 .83662 
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 9528 
 9457 
 
 5 
 4 
 
 57 
 
 7252. 
 
 37891 
 
 75219 
 
 .32946 
 
 .77988 
 
 .28225 
 
 .80834 
 
 .23710 
 
 .8376. 
 
 9387 
 
 3 
 
 58 
 
 72565 
 
 37807 
 
 75264 
 
 .32865 
 
 78035 
 
 .28.48 
 
 .80882 
 
 23637 
 
 .83811 
 
 93 X 6 
 
 2 
 
 59 
 
 .726,0 
 
 37722 
 
 7531 
 
 32785 
 
 .78082 
 
 .2807. 
 
 .80930 
 
 23563 
 
 .83860 
 
 9246 
 
 I 
 
 60 
 
 72654 
 
 37638 
 
 75355 
 
 32704 
 
 .78129 
 
 .27994 
 
 .80978 
 
 23490 
 
 .83910 
 
 . 9175 
 
 
 
 t 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 1 
 
 
 5' 
 
 t 
 
 5 
 
 3 
 
 5 
 
 2 
 
 5 
 
 L 
 
 5 
 
 D 

 
 40 
 
 NATURAL, TANGENTS AND COTANGENTS. 
 
 
 4 
 
 3 
 
 4 
 
 i 
 
 4 
 
 2 
 
 4 
 
 3 
 
 4 
 
 4 
 
 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 
 
 
 .839,0 
 
 19175 
 
 .86929 
 
 5037 
 
 .90040 
 
 .1,061 
 
 93252 
 
 07237 
 
 .96569 
 
 03553 
 
 60 
 
 ! 
 
 .83960 
 
 .19105 
 
 
 4969 
 
 .90093 
 
 . 10996 
 
 93306 
 
 .07174 
 
 .96625 
 
 03493 
 
 59 
 
 2 
 
 .84009 
 
 .19035 
 
 .87031 
 
 . 4902 
 
 .90,46 
 
 .10931 
 
 .93360 
 
 .07112 
 
 .9668, 
 
 3433 
 
 58 
 
 3 
 
 .84059 
 
 .18964 
 
 .87082 
 
 4834 
 
 .90,99 
 
 . 0867 
 
 934*5 
 
 .07049 
 
 .96738 
 
 03372 
 
 57 
 
 4 
 
 .84,08 
 
 . 18894 
 
 87133 
 
 4767 
 
 .9025, 
 
 . 0802 
 
 93469 
 
 .06987 
 
 .96794 
 
 .03312 
 
 56 
 
 5 
 
 .84,58 
 
 .18824 
 
 
 4699 
 
 .90304 
 
 0737 
 
 93524 
 
 .06925 
 
 .96850 
 
 .03252 
 
 55 
 
 6 
 7 
 
 .84208 
 .84258 
 
 18754 
 .18684 
 
 ! 87236 
 .87287 
 
 4632 
 4565 
 
 .90357 
 .90410 
 
 . 0607 
 
 93578 
 93633 
 
 .06862 
 .06800 
 
 .96907 
 .96963 
 
 .03192 
 .03132 
 
 54 
 53 
 
 
 84307 
 
 . ,86,4 
 
 87338 
 
 4498 
 
 .90463 
 
 0543 
 
 .93688 
 
 .06738 
 
 .97020 
 
 .03072 
 
 52 
 
 9 
 
 84357 
 
 .18544 
 
 .87389 
 
 4430 
 
 .90516 
 
 
 93742 
 
 .06676 
 
 .97076 
 
 .03012 
 
 5' 
 
 to 
 
 .84407 
 
 18474 
 
 .87441 
 
 4363 
 
 .90569 
 
 . 0414 
 
 93797 
 
 .06613 
 
 97*33 
 
 .02952 
 
 
 t , 
 
 .84457 
 
 . ,8404 
 
 .87492 
 
 . 4296 
 
 .90621 
 
 349 
 
 93852 
 
 .06551 
 
 .97189 
 
 .02892 
 
 49 
 
 12 
 
 
 .18334 
 
 87543 
 
 4229 
 
 .90674 
 
 
 .93906 
 
 .06489 
 
 97246 
 
 .02832 
 
 48 
 
 '3 
 
 84556 
 
 .18264 
 
 87595 
 
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 90727 
 
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 .93961 
 
 .06427 
 
 .97302 
 
 .02772 
 
 47 
 
 
 .84606 
 
 .18,94 
 
 .87646 
 
 4095 
 
 .9078, 
 
 . 0156 
 
 .940,6 
 
 .06365 
 
 97359 
 
 .02713 
 
 46 
 
 15 
 
 .84656 
 
 .18125 
 
 .87698 
 
 
 .90834 
 
 . 0091 
 
 94071 
 
 .06303 
 
 974*6 
 
 .02653 
 
 45 
 
 16 
 
 .84706 
 
 .,8055 
 
 .87749 
 
 396* 
 
 .90887 
 
 
 94*25 
 
 .06241 
 
 97472 
 
 02593 
 
 44 
 
 17 
 
 .84756 
 
 .17986 
 
 .87801 
 
 3894 
 
 .90940 
 
 .09963 
 
 .94,80 
 
 .06179 
 
 97529 
 
 02533 
 
 43 
 
 
 . 84806 
 
 .17916 
 
 .87852 
 
 3828 
 
 .90993 
 
 .09899 
 
 94235 
 
 .06,17 
 
 .97586 
 
 .02474 
 
 4 2 
 
 T 9 
 
 .84856 
 
 .17846 
 
 .87904 
 
 376, 
 
 .9,046 
 
 .09834 
 
 .94290 
 
 .06056 
 
 97643 
 
 .02414 
 
 4* 
 
 2.) 
 
 .84906 
 
 .17777 
 
 87955 
 
 . 3694 
 
 .9,099 
 
 .09770 
 
 94345 
 
 .05994 
 
 .97700 
 
 02355 
 
 40 
 
 21 
 22 
 
 .84956 
 .85006 
 
 .17708 
 .17638 
 
 : 88^9 
 
 . 3627 
 3561 
 
 9"53 
 .9,206 
 
 .09706 
 .09642 
 
 .94400 
 
 94455 
 
 05932 
 05870 
 
 97756 
 
 .02295 
 .02236 
 
 P 
 
 *3 
 
 85057 
 
 .17569 
 
 
 3494 
 
 .9, 59 
 
 .09578 
 
 .945,0 
 
 .05809 
 
 97870 
 
 .02176 
 
 37 
 
 4 
 
 .85,07 
 
 .17500 
 
 .88162 
 
 3428 
 
 .91 ,3 
 
 .09514 
 
 94565 
 
 05747 
 
 97927 
 
 .02117 
 
 36 
 
 
 85.57 
 .85207 
 
 .17430 
 .1736, 
 
 '.88265 
 
 3295 
 
 .0, 66 
 .91 ,9 
 
 .09450 
 .09386 
 
 .94676 
 
 .05685 
 .05624 
 
 .97984 
 .9804, 
 
 .01998 
 
 35 
 34 
 
 27 
 
 85257 
 
 .17292 
 
 .883,7 
 
 3228 
 
 9* 73 
 
 .09 3 22 
 
 9473* 
 
 .05562 
 
 .98098 
 
 *939 
 
 33 
 
 28 
 
 .85308 
 
 .17223 
 
 .88369 
 
 . 3l62 
 
 .91526 
 
 .09258 
 
 .94786 
 
 .05501 
 
 .O8l55 
 
 . 1879 
 
 
 29 
 
 85358 
 
 17154 
 
 .88421 
 
 . 3096 
 
 .9,580 
 
 .09195 
 
 .9484! 
 
 5439 
 
 .98213 
 
 . 1820 
 
 3" 
 
 30 
 
 .85408 
 
 .17085 
 
 .88473 
 
 . 3029 
 
 .91633 
 
 .09131 
 
 .94896 
 
 5378 
 
 .98270 
 
 '761 
 
 30 
 
 31 
 
 .85458 
 
 .17016 
 
 .88524 
 
 2963 
 
 .9,687 
 
 .09067 
 
 94952 
 
 053*7 
 
 .98327 
 
 . 1702 
 
 aq 
 
 32 
 
 .85509 
 
 16947 
 
 .88576 
 
 2897 
 
 .9,740 
 
 .09003 
 
 95007 
 
 05255 
 
 -98384 
 
 . 1642 
 
 23 
 
 53 
 
 85559 
 
 . 16878 
 
 .88628 
 
 . 2831 
 
 .9,794 
 
 .08940 
 
 .95062 
 
 .05194 
 
 .9844* 
 
 *58 3 
 
 27 
 
 34 
 
 .85609 
 
 .16809 
 
 .88680 
 
 2765 
 
 .9,847 
 
 .08876 
 
 .95118 
 
 5*33 
 
 .98499 
 
 .01524 
 
 26 
 
 35 
 
 .85660 
 
 .16741 
 
 .88732 
 
 . 2699 
 
 .91901 
 
 .08813 
 
 95*73 
 
 .05072 
 
 98556 
 
 .01465 
 
 25 
 
 36 
 
 .85710 
 
 . 16672 
 
 .88784 
 
 2633 
 
 9*955 
 
 .08749 
 
 95229 
 
 .05010 
 
 .98613 
 
 .01406 
 
 24 
 
 37 
 
 .8576, 
 
 .16603 
 
 .88836 
 
 2567 
 
 .92008 
 
 
 95284 
 
 .04949 
 
 .98671 
 
 01347 
 
 23 
 
 JB 
 
 .858,1 
 
 .36535 
 
 .88888 
 
 . 2501 
 
 .92062 
 
 ! 08622 
 
 9534 
 
 .04888 
 
 .98728 
 
 .0,288 
 
 22 
 
 39 
 
 .85862 
 
 .16466 
 
 .88940 
 
 2435 
 
 .921,6 
 
 .08559 
 
 95395 
 
 .04827 
 
 .98786 
 
 .01229 
 
 21 
 
 40 
 
 .85912 
 
 .16398 
 
 .88992 
 
 . 2369 
 
 .92170 
 
 .08496 
 
 95451 
 
 .04766 
 
 .98843 
 
 .01170 
 
 20 
 
 4 1 
 
 .85963 
 
 .16329 
 
 .89045 
 
 2303 
 
 .92224 
 
 .08432 
 
 95506 
 
 .04705 
 
 .98901 
 
 .01112 
 
 *') 
 
 42 
 
 .860,4 
 
 .16261 
 
 .89097 
 
 . 2238 
 
 92277 
 
 .08369 
 
 95562 
 
 .04644 
 
 .98958 
 
 .01053 
 
 ,8 
 
 13 
 
 .86064 
 
 .16,92 
 
 .89,49 
 
 . 2172 
 
 9233 1 
 
 .08306 
 
 .95618 
 
 04583 
 
 .990,6 
 
 .00994 
 
 '7 
 
 4 
 
 .86,15 
 
 .,6,24 
 
 .89201 
 
 . 2106 
 
 92385 
 
 .08243 
 
 95673 
 
 .04522 
 
 .99073 
 
 .00935 
 
 
 
 5 
 
 .86166 
 
 .16056 
 
 89253 
 
 . 2041 
 
 92439 
 
 .08,79 
 
 95729 
 
 .04461 
 
 99*3* 
 
 .00876 
 
 5 
 
 6 
 
 .86216 
 
 15987 
 
 .89306 
 
 975 
 
 92493 
 
 .08,16 
 
 95785 
 
 .04401 
 
 .99,89 
 
 .008,8 
 
 4 
 
 7 
 
 .86267 
 
 
 .89358 
 
 99 
 
 92547 
 
 .08053 
 
 .95841 
 
 .04340 
 
 .99247 
 
 .00759 
 
 3 
 
 8 
 
 .86318 
 
 .15851 
 
 .89410 
 
 844 
 
 .92601 
 
 .07990 
 
 95 8 97 
 
 .04279 
 
 .99304 
 
 .00701 
 
 
 9 
 
 . 86368 
 
 15783 
 
 .89463 
 
 778 
 
 92655 
 
 .07927 
 
 95952 
 
 .04218 
 
 .99362 
 
 .00642 
 
 i 
 
 5" 
 
 .86419 
 
 
 .895*5 
 
 7*3 
 
 .92709 
 
 .07864 
 
 .96008 
 
 04158 
 
 .99420 
 
 .00583 
 
 
 
 51 
 
 .86470 
 
 .15647 
 
 89567 
 
 . 1648 
 
 .92763 
 
 .07801 
 
 .96064 
 
 .04097 
 
 .99478 
 
 .00525 
 
 9 
 
 52 
 
 .86521 
 
 15579 
 
 .89620 
 
 58 
 
 .928,7 
 
 07738 
 
 .96120 
 
 .04036 
 
 .99536 
 
 .00467 
 
 8 
 
 53 
 
 .86572 
 
 .155,1 
 
 .89672 
 
 5' 
 
 .92872 
 
 .07676 
 
 .96176 
 
 .03976 
 
 99594 
 
 .00408 
 
 7 
 
 54 
 
 .86623 
 
 .15443 
 
 89725 
 
 45 
 
 .92926 
 
 .076,3 
 
 .96232 
 
 
 .99652 
 
 00350 
 
 6 
 
 55 
 
 .86674 
 
 I 5375 
 
 .89777 
 
 38 
 
 .92980 
 
 07550 
 
 
 03855 
 
 997*0 
 
 .00291 
 
 5 
 
 56 
 
 .86725 
 
 .15308 
 
 .89830 
 
 32 
 
 93034 
 
 .07487 
 
 96344 
 
 03794 
 
 .99768 
 
 .00233 
 
 4 
 
 57 
 
 .86776 
 
 .15240 
 
 .89883 
 
 . 256 
 
 .93088 
 
 .07425 
 
 .96400 
 
 03734 
 
 .99826 
 
 .00,75 
 
 3 
 
 
 .86827 
 
 .15172 
 
 89935 
 
 . ,91 
 
 93*43 
 
 07362 
 
 96457 
 
 03674 
 
 .99884 
 
 
 2 
 
 59 
 
 .86878 
 
 .15104 
 
 .89988 
 
 . ,26 
 
 93*97 
 
 .07299 
 
 965*3 
 
 .03613 
 
 .99942 
 
 .'00058 
 
 , 
 
 Go 
 
 .86929 
 
 
 .90040 
 
 . 061 
 
 93252 
 
 07237 
 
 96569 
 
 .03553 
 
 i .00000 
 
 .OOOOO 
 
 
 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 
 f 
 
 
 
 
 
 
 
 
 
 
 
 ' 
 
 
 4 
 
 9 
 
 4 
 
 8 
 
 4 
 
 7 
 
 4 
 
 6 
 
 4 
 
 5 

 
 TRAVERSE TABLES 
 
 OR 
 
 LATITUDES ^ DEPARTURES OF COURSES 
 
 CALCULATED TO 
 
 THREE DECIMAL PLACES 
 
 FOR 
 
 EACH QUARTER DEGREE OF BEARING
 
 LATITUDES AND DEPARTURES. 
 
 43 
 
 1 
 
 
 
 
 I 
 
 
 5 
 
 1 
 
 J 
 
 5 
 
 I 
 
 JL 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 i 
 
 
 
 .000 
 
 0.000 
 
 2.OOO 
 
 o.ooo 
 
 3.000 
 
 0.000 
 
 4-OOO 
 
 o.coo 
 
 5.OOO 
 
 90 
 
 o# 
 
 .000 
 
 0.004 
 
 2.OOO 
 
 0.009 
 
 3.000 
 
 0.013 
 
 4-OOO 
 
 0.017 
 
 5.OOO 
 
 89^ 
 
 oK 
 
 .000 
 
 0.009 
 
 2.00O 
 
 0.017 
 
 3.000 
 
 0.026 
 
 4.OOO 
 
 0.035 
 
 5.OOO 
 
 8 9 K 
 
 f< 
 
 .000 
 
 0.013 
 
 2.OOO 
 
 0.026 
 
 3.000 
 
 0.039 
 
 4.OOO 
 
 0.052 
 
 5.OOO 
 
 89# 
 
 
 I.OOO 
 
 0.017 
 
 2.OOO 
 
 0.035 
 
 3.000 
 
 0.052 
 
 3-999 
 
 0.070 
 
 4.999 
 
 89 
 
 i* 
 
 I.OOO 
 
 0.022 
 
 2.OOO 
 
 0.044 
 
 2-999 
 
 0.065 
 
 3-999 
 
 0.087 
 
 4-999 
 
 88 tf 
 
 l 'A 
 
 .000 
 
 O.O26 
 
 1-999 
 
 0.052 
 
 2.999 
 
 0.079 
 
 3-999 
 
 0.105 
 
 4.998 
 
 88^ 
 
 IJ? 
 
 .000 
 
 0.031 
 
 1-999 
 
 0.06 1 
 
 2-999 
 
 0.092 
 
 3-998 
 
 O.I22 
 
 4.998 
 
 88X 
 
 2 
 
 0.999 
 
 0.035 
 
 1-999 
 
 9.070 
 
 2.998 
 
 0.105 
 
 3-998 
 
 O.I4O 
 
 4-997 
 
 88 J 
 
 2X 
 
 0.999 
 
 0-039 
 
 1.998 
 
 0.079 
 
 2.998 
 
 0.118 
 
 3-997 
 
 0.157 
 
 4.996 
 
 87 X 
 
 2^ 
 
 0.999 
 
 0.044 
 
 1.998 
 
 0.087 
 
 2.997 
 
 0.131 
 
 3-996 
 
 0.174 
 
 4-995 
 
 87K 
 
 = * 
 
 0.999 
 
 0.048 
 
 1.998 
 
 0.096 
 
 2.997 
 
 0.144 
 
 3-995 
 
 O.I92 
 
 4-994 
 
 87^ 
 
 3 D 
 
 0.999 
 
 0.052 
 
 1.997 
 
 0.105 
 
 2.996 
 
 0.157 
 
 3-995 
 
 O.209 
 
 4-993 
 
 87 
 
 3^ 
 
 0.998 
 
 0-057 
 
 1-997 
 
 0.113 
 
 2-995 
 
 0.170 
 
 3-994 
 
 0.227 
 
 4.992 
 
 86 3/ 
 
 3K 
 
 0.998 
 
 O.06 1 
 
 1.996 
 
 0.122 
 
 2.994 
 
 0.183 
 
 3-993 
 
 0.244 
 
 4.991 
 
 86K 
 
 3% 
 
 0.998 
 
 O.065 
 
 1.996 
 
 O.I3I 
 
 2.994 
 
 0.196 
 
 3-991 
 
 0.262 
 
 4.989 
 
 86^ 
 
 4 
 
 0.998 
 
 O.O7O 
 
 1-995 
 
 O.I4O 
 
 2-993 
 
 0.209 
 
 3-990 
 
 0.279 
 
 4.988 
 
 86 
 
 4^ 
 
 0.997 
 
 0.074 
 
 1-995 
 
 0.148 
 
 2.992 
 
 0.222 
 
 3-989 
 
 0.296 
 
 4.986 
 
 85^ 
 
 4K 
 
 0.997 
 
 O.O78 
 
 1.994 
 
 0-157 
 
 2.991 
 
 0-235 
 
 3-988 
 
 0.314 
 
 4-985 
 
 85K 
 
 lM 
 
 0.997 
 
 0.083 
 
 1-993 
 
 o.i 66 
 
 2.990 
 
 0.248 
 
 3-986 
 
 0-331 
 
 4-983 
 
 85X 
 
 5 
 
 0.996 
 
 0.087 
 
 1.992 
 
 0.174 
 
 2.989 
 
 0.26l 
 
 3-985 
 
 0-349 
 
 4.981 
 
 85 
 
 5^ 
 
 0.996 
 
 0.092 
 
 1.992 
 
 0.183 
 
 2.987 
 
 0.275 
 
 3-983 
 
 0.366 
 
 4-979 
 
 84^ 
 
 5K 
 
 0-995 
 
 0.096 
 
 1.991 
 
 0.192 
 
 2.986 
 
 O.288 
 
 3-982 
 
 0-383 
 
 4-977 
 
 84^ 
 
 5%r 
 
 0-995 
 
 O.IOO 
 
 1.990 
 
 O.2OO 
 
 2.985 
 
 0.301 
 
 3-98o 
 
 0.401 
 
 4-975 
 
 84 X 
 
 6 
 
 0-995 
 
 0.105 
 
 1.989 
 
 O.2O9 
 
 2.984 
 
 0.314 
 
 3-978 
 
 0.418 
 
 4-973 
 
 84^ 
 
 6# 
 
 0.994' 
 
 0.109 
 
 1.988 
 
 0.218 
 
 2.982 
 
 0.327 
 
 3-9/6 
 
 0-435 
 
 4.970 
 
 83^ 
 
 6^ 
 
 0.994 
 
 0.113 
 
 1.987 
 
 O.226 
 
 2.981 
 
 0.340 
 
 3-974 
 
 0-453 
 
 4-968 
 
 83K 
 
 6V 
 
 0-993 
 
 0.118 
 
 1.986 
 
 0-235 
 
 2.979 
 
 0-353 
 
 3-972 
 
 0.470 
 
 4.965 
 
 83X 
 
 r 
 
 0-993 
 
 O.I22 
 
 1.985 
 
 0.244 
 
 2.978 
 
 0.366 
 
 3-970 
 
 0.487 
 
 4-963 
 
 83 
 
 7# 
 
 0.992 
 
 0.126 
 
 1.984 
 
 O.252 
 
 2.976 
 
 0-379 
 
 3-968 
 
 0.505 
 
 4-960 
 
 82^ 
 
 r/ 2 
 
 0.991 
 
 O.I3I 
 
 1.983 
 
 O.26l 
 
 2.974 
 
 0.392 
 
 3-966 
 
 0.522 
 
 4-957 
 
 82^ 
 
 iti 
 
 0.991 
 
 0.135 
 
 1.982 
 
 O.27O 
 
 2.973 
 
 0.405 
 
 3-963 
 
 0-539 
 
 4-954 
 
 82^ 
 
 8 
 
 0.990 
 
 0.139 
 
 1.981 
 
 O.278 
 
 2.971 
 
 0.418 
 
 3.961 
 
 0-557 
 
 4-951 
 
 82 
 
 8X 
 
 0.990 
 
 0.143 
 
 1.979 
 
 O.287 
 
 2.969 
 
 0.430 
 
 3-959 
 
 0-574 
 
 4.948 
 
 81^ 
 
 8^ 
 
 0.989 
 
 0.148 
 
 1.978 
 
 0.296 
 
 2.967 
 
 0-443 
 
 3-956 
 
 0.591 
 
 4-945 
 
 8i# 
 
 8^ 
 
 0.988 
 
 0.152 
 
 1.977 
 
 0.304 
 
 2.965 
 
 0.456 
 
 3-953 
 
 0.608 
 
 4-942 
 
 8ik 
 
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 0.988 
 
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 44 
 
 LATITUDES AND DEPARTURES. 
 
 
 
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 I
 
 LATITUDES AND DEPARTURES. 
 
 45 
 
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 0.899 
 
 0.438 
 
 .798 
 
 0.877 
 
 2.696 
 
 315 
 
 3-595 
 
 -753 
 
 4-494 
 
 64 U 
 
 ? 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
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 t 
 
 
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 5 
 
 - 
 
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 4 
 
 
 5 
 
 1
 
 40 
 
 LATITUDES AND DEPARTURES. 
 
 
 5 
 
 6 
 
 
 3 
 
 r 
 
 8 
 
 
 9 
 
 
 f 
 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 if 
 
 13 
 
 I.I25 
 
 .846 
 
 I.35O 
 
 6.821 
 
 1-575 
 
 7-795 
 
 I.Soo 
 
 8.769 
 
 2.025 
 
 77 
 
 
 1.146 
 
 .840 
 
 1-375 
 
 6.814 
 
 1.604 
 
 7.787 
 
 1.834 
 
 8.760 
 
 2.063 
 
 76* 
 
 1 3*A 
 
 1.167 
 
 834 
 
 1.401 
 
 6.807 
 
 -634 
 
 7-779 
 
 1.868 
 
 8-751 
 
 2. IOI 
 
 76^ 
 
 I3 3 / 
 
 I.I88 
 
 .828 
 
 1.426 
 
 6-799 
 
 .664 
 
 7-771 
 
 1.902 
 
 8.742 
 
 2.139 
 
 76/4 / 
 
 14 
 
 I.2IO 
 
 .822 
 
 1-452 
 
 6.792 
 
 -693 
 
 7.762 
 
 1-935 
 
 8-733 
 
 2.177 
 
 76 
 
 T 4* 
 
 I.23I 
 
 .815 
 
 1-477 
 
 6.785 
 
 .723 
 
 7-754 
 
 1.969 
 
 8.723 
 
 2.215 
 
 75 * 
 
 *4}4 
 
 1.252 
 
 5.809 
 
 1-502 
 
 6.777 
 
 753 
 
 7-745 
 
 2.003 
 
 8-713 
 
 2-253 
 
 75/4 
 
 ilM 
 
 1-273 
 
 5.802 
 
 1.528 
 
 6.769 
 
 .782 
 
 7-736 
 
 2.037 
 
 8.703 
 
 2.29! 
 
 75* 
 
 
 1.294 
 
 5-796 
 
 1-553 
 
 6.761 
 
 .812 
 
 7.727 
 
 2.071 
 
 8.693 
 
 2.329 
 
 75 
 
 J 5* 
 
 L3I5 
 
 5-789 
 
 1-578 
 
 6-754 
 
 .841 
 
 7.718 
 
 2.104 
 
 8.683 
 
 2.367 
 
 74* 
 
 i5/^ 
 
 L336 
 
 5-782 
 
 1.603 
 
 6-745 
 
 .871 
 
 7.709 
 
 2.138 
 
 8.673 
 
 2.405 
 
 74 1 A 
 
 15* 
 
 1-357 
 
 5-775 
 
 1.629 
 
 6-737 
 
 .900 
 
 7-700 
 
 2.172 
 
 8.662 
 
 2-443 
 
 74* 
 
 16 
 
 I-378 
 
 5-768 
 
 1.654 
 
 6.729 
 
 .929 
 
 7.690 
 
 2.205 
 
 8.651 
 
 2.481 
 
 74 3 
 
 16* 
 
 1-399 
 
 5.760 
 
 1.679 
 
 6.720 
 
 1-959 
 
 7.680 
 
 2.239 
 
 8.640 
 
 2.518 
 
 73* 
 
 i6} 
 
 1.420 
 
 5-753 
 
 1.704 
 
 6.712 
 
 1.988 
 
 7.671 
 
 2.272 
 
 8.629 
 
 2-556 
 
 73 X. 
 
 16* 
 
 1.441 
 
 5-745 
 
 1.729 
 
 6.703 
 
 2.017 
 
 7.661 
 
 2.306 
 
 8.618 
 
 2-594 
 
 73* 
 
 17 
 
 1.462 
 
 5-738 
 
 1-754 
 
 6.694 
 
 2.047 
 
 7.650 
 
 2-339 
 
 8.607 
 
 2.631 
 
 73 
 
 17* 
 
 1.483 
 
 5-730 
 
 1-779 
 
 6.685 
 
 2.076 
 
 7.640 
 
 2.372 
 
 8-595 
 
 2.669 
 
 72* 
 
 
 1.504 
 
 5-722 
 
 1.804 
 
 6.676 
 
 2.105 
 
 7-630 
 
 2.406 
 
 8-583 
 
 2.706 
 
 72^ 
 
 !y3/ 
 
 1.524 
 
 5-7H 
 
 1.829 
 
 6.667 
 
 2.134 
 
 7.619 
 
 2.439 
 
 8.572 
 
 2-744 
 
 7 2 * 
 
 18' 
 
 1-545 
 
 5.706 
 
 1.854 
 
 6.657 
 
 2.163 
 
 7.608 
 
 2.472 
 
 8.560 
 
 2.781 
 
 72 
 
 18* 
 
 1.566 
 
 5-698 
 
 1.879 
 
 6.648 
 
 2.192 
 
 7.598 
 
 2-505 
 
 8-547 
 
 2.818 
 
 7 1 * 
 
 
 1-587 
 
 5-690 
 
 1.904 
 
 ,6.638 
 
 2.221 
 
 7.587 
 
 2.538 
 
 8-535 
 
 2.856 
 
 71 y 2 
 
 18* 
 
 1.607 
 
 5.682 
 
 1.929 
 
 6.629 
 
 2.25O 
 
 7-575 
 
 2.572 
 
 8.522 
 
 2.893 
 
 7 1 * 
 
 19* 
 
 1.628 
 
 5-673 
 
 1-953 
 
 6.619 
 
 2.279 
 
 7-564 
 
 2.605 
 
 8.510 
 
 2.930 
 
 71 
 
 jgl^ 
 
 1.648 
 
 5-665 
 
 1.978 
 
 6.609 
 
 2.308 
 
 7-553 
 
 2.638 
 
 8-497 
 
 2.967 
 
 70* 
 
 !gl^ 
 
 1.669 
 
 5-656 
 
 2.003 
 
 6.598 
 
 2-337 
 
 7-541 
 
 2.670 
 
 8.484 
 
 3-004 
 
 70^ 
 
 19* 
 
 1.690 
 
 5-647 
 
 2.028 
 
 6.588 
 
 2.365 
 
 7-529 
 
 2.703 
 
 8.471 
 
 3.041 
 
 70* 
 
 20 
 
 1.710 
 
 5-638 
 
 2.052 
 
 6-578 
 
 2-394 
 
 7.518 
 
 2.736 
 
 8-457 
 
 3.078 
 
 70 
 
 20* 
 
 I-73I 
 
 5-629 
 
 2-077 
 
 6.567 
 
 2.423 
 
 7.506 
 
 2.769 
 
 8-444 
 
 3-"5 
 
 69* 
 
 20% 
 
 I-75I 
 
 5.620 
 
 2.IOI 
 
 6-557 
 
 2.451 
 
 7-493 
 
 2.802 
 
 8.430 
 
 3-152 
 
 69 j 
 
 203/ 
 
 1.771 
 
 5.611 
 
 2.126 
 
 6.546 
 
 2.480 
 
 7.481 
 
 2.834 
 
 8.416 
 
 3.189 
 
 69* 
 
 2T 
 
 1.792 
 
 5.601 
 
 2.150 
 
 6-535 
 
 2.509 
 
 7.469 
 
 2.867 
 
 8.402 
 
 3-225 
 
 69 
 
 21)4 
 
 i. 812 
 
 5-592 
 
 2-175 
 
 6.524 
 
 2-537 
 
 7.456 
 
 2.900 
 
 8.388 
 
 3.262 
 
 68* 
 
 21^2 
 
 1-833 
 
 5-582 
 
 2.199 
 
 6.513 
 
 2.566 
 
 7-443 
 
 2.932 
 
 8-374 
 
 3-299 
 
 68 y 2 
 
 21* 
 
 1-853 
 
 5-573 
 
 2.223 
 
 6.502 
 
 2-594 
 
 7-430 
 
 2.964 
 
 8-359 
 
 3-335 
 
 6 A 
 
 22 
 
 1-873 
 
 5-563 
 
 2.248 
 
 6.490 
 
 2.622 
 
 7-417 
 
 2-997 
 
 8-345 
 
 3-371 
 
 68 
 
 22^ 
 
 1.893 
 
 5-553 
 
 2.272 
 
 6-479 
 
 2.651 
 
 7-404 
 
 3.029 
 
 8-330 
 
 3-408 
 
 67* 
 
 22^ 
 
 I-9I3 
 
 5-543 
 
 2.296 
 
 6.467 
 
 2.679 
 
 7-39 1 
 
 3.061 
 
 8.315 
 
 3-444 
 
 67^ 
 
 22* 
 
 1-934 
 
 5-533 
 
 2.320 
 
 6-455 
 
 2.707 
 
 7-378 
 
 3-094 
 
 8.300 
 
 3-48o 
 
 67* 
 
 23 
 
 1-954 
 
 5-523 
 
 2-344 
 
 6.444 
 
 2-735 
 
 7-364 
 
 3.126 
 
 8.285 
 
 3-5I7 
 
 67 
 
 23* 
 
 1.974 
 
 5-513 
 
 2.368 
 
 6.432 
 
 2.763 
 
 7-350 
 
 3-158 
 
 8.269 
 
 3-553 
 
 66* 
 
 23^ 
 
 1.994 
 
 5-502 
 
 2.392 
 
 6.419 
 
 2.791 
 
 7-336 
 
 3.190 
 
 8.254 
 
 3-589 
 
 66 Yz 
 
 23* 
 
 2.014 
 
 5-492 
 
 2.416 
 
 6.407 
 
 2.819 
 
 7.322 
 
 3.222 
 
 8.238 
 
 3.625 
 
 66* 
 
 2V 
 
 2.034 
 
 5.481 
 
 2.440 
 
 6-395 
 
 2.847 
 
 7.308 
 
 3-254 
 
 8.222 
 
 3.661 
 
 66 
 
 2 4/4: 
 
 2.054 
 
 5-471 
 
 2.464 
 
 6.382 
 
 2.875 
 
 7-294 
 
 3.286 
 
 8.206 
 
 3-696 
 
 65* 
 
 24 X 
 
 2.073 
 
 5-460 
 
 2.488 
 
 6.370 
 
 2.903 
 
 7.280 
 
 3.318 
 
 8.190 
 
 3-732 
 
 65^ 
 
 243^ 
 
 2.093 
 
 5-449 
 
 2.512 
 
 6-357 
 
 2.931 
 
 7.265 
 
 3-349 
 
 8.173 
 
 3.768 
 
 65* 
 
 25 
 
 2.113 
 
 5-438 
 
 2-536 
 
 6-344 
 
 2.958 
 
 7.250 
 
 3-381 
 
 8.157 
 
 3-804 
 
 65 
 
 25* 
 
 2-133 
 
 5-427 
 
 2-559 
 
 6-331 
 
 2.986 
 
 7-236 
 
 3.4I3 
 
 8.140 
 
 3-839 
 
 64* 
 
 25^ 
 
 2-153 
 
 5.416 
 
 2.583 
 
 6.318 
 
 3.014 
 
 7.221 
 
 3-444 
 
 8.123 
 
 3.875 
 
 64^ 
 
 25* 
 
 2.172 
 
 5-404 
 
 2.607 
 
 6-305 
 
 3.041 
 
 7.206 
 
 3-476 
 
 8.106 
 
 3.910 
 
 64* 
 
 26 
 
 2. 192 
 
 5-393 
 
 2.630 
 
 6.292 
 
 3.069 
 
 7.190 
 
 3-507 
 
 8.089 
 
 3-945 
 
 64 
 
 W 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 a? 
 
 I 
 
 5 
 
 
 6 
 
 
 7 
 
 
 8 
 
 
 9 
 
 p
 
 LATITUDES AND DEPARTURES. 
 
 1 
 
 I 
 
 
 2 
 
 
 3 
 
 
 4 
 
 
 5 
 
 5 
 
 jL 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 
 
 26 3 
 
 0.899 
 
 0-438 
 
 798 
 
 0.877 
 
 2.696 
 
 315 
 
 3-595 
 
 1-753 
 
 4-494 
 
 64^ 
 
 26V 
 
 0.897 
 
 0.442 
 
 794 
 
 0.885 
 
 2.691 
 
 327 
 
 3-587 
 
 1.769 
 
 4.484 
 
 63V 
 
 26^ 
 
 0.895 
 
 0.446 
 
 790 
 
 0.892 
 
 2.685 
 
 339 
 
 3.580 
 
 1.785 
 
 4-475 
 
 63 K 
 
 26V 
 
 0.893 
 
 0.450 
 
 786 
 
 0.900 
 
 2.679 
 
 350 
 
 3-572 
 
 1.800 
 
 4.465 
 
 63 1 4 
 
 
 0.891 
 
 0-454 
 
 782 
 
 0.908 
 
 2-673 
 
 -362 
 
 3-564 
 
 1.816 
 
 4-455 
 
 63 
 
 27V 
 
 0.889 
 
 0.458 
 
 7-8 
 
 0.916 
 
 2.667 
 
 374 
 
 3-556 
 
 1.831 
 
 4-445 
 
 62V 
 
 27^ 
 
 0.887 
 
 0.462 
 
 774 
 
 0.923 
 
 2.661 
 
 .385 
 
 3-548 
 
 1.847 
 
 4-435 
 
 62^ 
 
 27V 
 
 0.885 
 
 0.466 
 
 .770 
 
 0.931 
 
 2.655 
 
 397 
 
 3-540 
 
 1.862 
 
 4-425 
 
 62V 
 
 28 
 
 0.883 
 
 0.469 
 
 .766 
 
 0-939 
 
 2.649 
 
 .408 
 
 3-532 
 
 1.878 
 
 4-4I5 
 
 62' 
 
 28V 
 
 0.881 
 
 0-473 
 
 .762 
 
 0-947 
 
 2.643 
 
 .420 
 
 3-524 
 
 1.893 
 
 4.404 
 
 61 V 
 
 28^ 
 
 0.879 
 
 0-477 
 
 -758 
 
 0-954 
 
 2.636 
 
 431 
 
 3-5I5 
 
 1.909 
 
 4-394 
 
 61^ 
 
 28 X 
 
 0.877 
 
 0.481 
 
 753 
 
 0.962 
 
 2.630 
 
 443 
 
 3-507 
 
 1.924 
 
 4-384 
 
 61 V 
 
 29 
 
 0.875 
 
 0.485 
 
 749 
 
 0.970 
 
 2.624 
 
 454 
 
 3-498 
 
 1-939 
 
 4-373 
 
 6V 
 
 29V 
 
 0.872 
 
 0.489 
 
 745 
 
 0-977 
 
 2.617 
 
 .466 
 
 3-490 
 
 1-954 
 
 4-362 
 
 6oV 
 
 29^ 
 
 0.870 
 
 0.492 
 
 .741 
 
 0.985 
 
 2.6II 
 
 477 
 
 3.481 
 
 1.970 
 
 4-352 
 
 
 29V 
 
 0.868 
 
 0.496 
 
 736 
 
 0.992 
 
 2.605 
 
 .489 
 
 3-473 
 
 1.985 
 
 4-341 
 
 6oV 
 
 30 
 
 0.866 
 
 0.500 
 
 732 
 
 .000 
 
 2.598 
 
 500 
 
 3.464 
 
 2.OOO 
 
 4-330 
 
 60 
 
 30 v 
 
 0.864 
 
 0.504 
 
 .728 
 
 .008 
 
 2.592 
 
 .511 
 
 3-455 
 
 2.015 
 
 4-3I9 
 
 59V 
 
 3/4 
 
 0.862 
 
 0.508 
 
 723 
 
 -015 
 
 2.585 
 
 523 
 
 3-447 
 
 2.030 
 
 4-308 
 
 59^ 
 
 30 V 
 
 0.859 
 
 0.511 
 
 .719 
 
 -023 
 
 2-578 
 
 534 
 
 3-438 
 
 2.045 
 
 4.297 
 
 59V 
 
 31 
 
 0.857 
 
 0-515 
 
 .714 
 
 .030 
 
 2-572 
 
 545 
 
 3-429 
 
 2.060 
 
 4.286 
 
 59 
 
 3 T V 
 
 0.855 
 
 0.519 
 
 .710 
 
 -038 
 
 2.565 
 
 556 
 
 3.420 
 
 2.075 
 
 4-275 
 
 58V 
 
 31^ 
 
 0.853 
 
 0.522 
 
 705 
 
 045 
 
 2.558 
 
 567 
 
 3-4" 
 
 2.090 
 
 4.263 
 
 58^ 
 
 *$ 
 
 0.850 
 0.848 
 
 0.526 
 0-530 
 
 .701 
 .696 
 
 -052 
 .060 
 
 2-551 
 2-544 
 
 579 
 -590 
 
 3.401 
 3-392 
 
 2.105 
 
 2.120 
 
 4-252 
 4.240 
 
 58V 
 58 
 
 32V 
 
 0.846 
 
 0-534 
 
 .691 
 
 .067 
 
 2-537 
 
 .601 
 
 
 2.134 
 
 4.229 
 
 57V 
 
 32^ 
 
 0-843 
 
 0-537 
 
 .687 
 
 075 
 
 2-530 
 
 .612 
 
 3-374 
 
 2-149 
 
 4-217 
 
 57*/2 
 
 32V 
 
 0.841 
 
 0.541 
 
 .682 
 
 .082 
 
 2-523 
 
 -623 
 
 3-364 
 
 2.164 
 
 4.205 
 
 57V 
 
 33 
 
 0.839 
 
 0-545 
 
 -677 
 
 .089 
 
 2.516 
 
 -634 
 
 3-355 
 
 2-179 
 
 4.193 
 
 sr 
 
 33V 
 
 0.836 
 
 0.548 
 
 -673 
 
 .097 
 
 2.509 
 
 -645 
 
 3-345 
 
 2.193 
 
 4.181 
 
 56V 
 
 33 l /2 
 
 0.834 
 
 0-552 
 
 .668 
 
 .104 
 
 2.502 
 
 -656 
 
 3-336 
 
 2.208 
 
 4.169 
 
 
 33% 
 
 0.831 
 
 0.556 
 
 -663 
 
 .111 
 
 2-494 
 
 .667 
 
 3-326 
 
 2.222 
 
 4-157 
 
 56V 
 
 34 
 
 0.829 
 
 0-559 
 
 -658 
 
 .118 
 
 2-487 
 
 .678 
 
 3-3i6 
 
 2-237 
 
 4.145 
 
 56 
 
 34V 
 
 0.827 
 
 0-563 
 
 -653 
 
 .126 
 
 2.480 
 
 .688 
 
 3-306 
 
 2.251 
 
 4-133 
 
 55V 
 
 
 0.824 
 
 0.566 
 
 .648 
 
 133 
 
 2.472 
 
 .699 
 
 3-297 
 
 2.266 
 
 4.121 
 
 
 343/J 
 
 0.822 
 
 0.570 
 
 -643 
 
 .140 
 
 2.465 
 
 .710 
 
 ! 3-287 
 
 2.280 
 
 4.108 
 
 55V 
 
 35 
 
 0.819 
 
 0-574 
 
 .638 
 
 .147 
 
 2-457 
 
 .721 
 
 : 3-277 
 
 2.294 
 
 4.096 
 
 55 
 
 35V 
 
 0.817 
 
 0-577 
 
 633 
 
 154 
 
 2.450 
 
 731 
 
 3.267 
 
 2.309 
 
 4-083 
 
 J54V 
 
 
 0.814 
 
 0.581 
 
 .628 
 
 .161 
 
 2.442 
 
 742 
 
 3-257 
 
 2.323 
 
 4.071 
 
 54^ 
 
 35 
 
 0.812 
 
 0.584 
 
 .623 
 
 .168 
 
 2-435 
 
 753 
 
 3.246 
 
 2-337 
 
 4-058 
 
 54V 
 
 36 
 
 0.809 
 
 0.588 
 
 .618 
 
 .176 
 
 2.427 
 
 763 
 
 3-236 
 
 2-351 
 
 4-045 
 
 54" 
 
 36V 
 
 0.806 
 
 0.591 
 
 -613 
 
 -183 
 
 2.419 
 
 774 
 
 3.226 
 
 2.365 
 
 4.032 
 
 53V 
 
 36 YT. 
 
 0.804 
 
 0-595 
 
 .608 
 
 .190 
 
 2.412 
 
 .784 
 
 3-215 
 
 2-379 
 
 4.019 
 
 
 36^ 
 
 0.801 
 
 0.598 
 
 .603 
 
 .197 
 
 2.404 
 
 795 
 
 3-205 
 
 2-393 
 
 4.006 
 
 53V 
 
 37 
 
 0.799 
 
 0.602 
 
 597 
 
 .204 
 
 2-396 
 
 -805 
 
 3-195 
 
 2.407 
 
 3-993 
 
 53= 
 
 37V 
 
 0.796 
 
 0.605 
 
 592 
 
 .211 
 
 2.388 
 
 .816 
 
 3.184 
 
 2.421 
 
 3-980 
 
 52V 
 
 
 0-793 
 
 0.609 
 
 -587 
 
 .218 
 
 2.380 
 
 .826 
 
 3-173 
 
 2-435 
 
 
 52^ 
 
 37 1| 
 
 0.791 
 
 0.612 
 
 -581 
 
 .224 
 
 2.372 
 
 -837 
 
 3-163 
 
 2-449 
 
 3-953 
 
 52V 
 
 38' 
 
 0.788 
 
 0.616 
 
 -576 
 
 .231 
 
 2-364 
 
 .847 
 
 3-152 
 
 2.463 
 
 3-940 
 
 52 
 
 38V 
 
 0.785 
 
 0.619 
 
 571 
 
 -2 3 8 
 
 2.356 
 
 -857 
 
 
 2.476 
 
 3-927 
 
 5 1 V 
 
 38^ 
 
 0.783 
 
 0.623 
 
 -565 
 
 245 
 
 2.348 
 
 .868 
 
 3-130 
 
 2.490 
 
 3-9I3 
 
 51^ 
 
 383^ 
 
 0.780 
 
 0.626 
 
 -560 
 
 -252 
 
 2.340 
 
 .878 
 
 3.120 
 
 2.504 
 
 3-899 
 
 5 1 V 
 
 39 
 
 0-777 
 
 0.629 
 
 554 
 
 259 
 
 2-331 
 
 .888 
 
 3.109 
 
 2-517 
 
 3.886 
 
 51 
 
 s* 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 IF 
 
 I 
 
 
 I 
 
 
 I 
 
 
 J 
 
 4 
 
 1 
 
 5 
 
 1
 
 48 
 
 LATITUDES AND DEPARTURES. 
 
 1 
 
 5 
 
 6 
 
 
 1 
 
 
 e 
 
 
 f. 
 
 1 
 
 2 
 
 OO. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 1 
 
 26" 
 
 2.192 
 
 5-393 
 
 2.630 
 
 6.292 
 
 3-069 
 
 7.190 
 
 3-507 
 
 8.089 
 
 3-945 
 
 64 
 
 26^ 
 
 2. 211 
 
 5-38I 
 
 2.654 
 
 6.278 
 
 3-096 
 
 7-175 
 
 3.538 
 
 8.072 
 
 3.981 
 
 633/ 
 
 26^ 
 
 2.231 
 
 5-370 
 
 2.677 
 
 6.265 
 
 3-123 
 
 7.IDO 
 
 3-570 
 
 8-054 
 
 4.016 
 
 63 K 
 
 263^ 
 
 2.25O 
 
 5-358 
 
 2.701 
 
 6.251 
 
 3-I5I 
 
 7-144 
 
 3.601 
 
 8.037 
 
 4.051 
 
 63 % 
 
 2<T 
 
 2.27O 
 
 5-346 
 
 2.724 
 
 6.237 
 
 3.178 
 
 7.128 
 
 3.632 
 
 8.019 
 
 4.086 
 
 63 
 
 27^ 
 
 2.289 
 
 5-334 
 
 2-747 
 
 6.223 
 
 3.205 
 
 7.112 
 
 3-663 
 
 8.001 
 
 4.121 
 
 6234" 
 
 
 2.309 
 
 5.322 
 
 2.770 
 
 6.209 
 
 3-232 
 
 7.096 
 
 
 7-983 
 
 4.156 
 
 62 y z 
 
 27* 
 
 2.328 
 
 5.310 
 
 2-794 
 
 6-195 
 
 3-259 
 
 7.080 
 
 3-725 
 
 7-965 
 
 4.190 
 
 62 tf 
 
 28" 
 
 2-347 
 
 5.298 
 
 2.817 
 
 6.181 
 
 3.286 
 
 7.064 
 
 
 7-947 
 
 4.225 
 
 62 
 
 28^ 
 
 2-367 
 
 5-285 
 
 2.840 
 
 6.166 
 
 3-3I3 
 
 7.047 
 
 3.787 
 
 7.928 
 
 4.260 
 
 6134- 
 
 28^ 
 
 2.386 
 
 5-273 
 
 2.863 
 
 6.152 
 
 3-340 
 
 7.031 
 
 3.817 
 
 7.909 
 
 4.294 
 
 61 y z 
 
 2834: 
 
 2.405 
 
 5.260 
 
 2.886 
 
 6-137 
 
 3-367 
 
 7.014 
 
 3-848 
 
 7.891 
 
 4-329 
 
 61 y[ 
 
 29 
 
 2.424 
 
 5-248 
 
 2.909 
 
 6.122 
 
 3-394 
 
 6-997 
 
 3.878 
 
 7.872 
 
 
 61 
 
 29^ 
 
 2-443 
 
 5-235 
 
 2.932 
 
 6.107 
 
 3.420 
 
 6.980 
 
 3.909 
 
 7-852 
 
 4-398 
 
 6034' 
 
 29^ 
 
 2.462 
 
 5-222 
 
 2-955 
 
 6.093 
 
 3-447 
 
 6.963 
 
 3-939 
 
 7.833 
 
 4-432 
 
 60^ 
 
 29 X 
 
 2.481 
 
 5-209 
 
 2-977 
 
 6.077 
 
 3-474 
 
 6.946 
 
 3-970 
 
 7.814 
 
 4.466 
 
 bo% 
 
 30 
 
 2.500 
 
 5.196 
 
 3.000 
 
 6.062 
 
 3.500 
 
 6.928 
 
 4.000 
 
 7-794 
 
 4-500 
 
 60 
 
 30^ 
 
 2.519 
 
 5-I83 
 
 3-023 
 
 6.047 
 
 3.526 
 
 6.911 
 
 4.030 
 
 7-775 
 
 4-534 
 
 59^ 
 
 30 y 2 
 
 2-538 
 
 5-170 
 
 3-045 
 
 6.031 
 
 3-553 
 
 6.893 
 
 4.060 
 
 7-755 
 
 4-568 
 
 
 30% 
 
 2.556 
 
 5.I56 
 
 3.068 
 
 6.016 
 
 3-579 
 
 6.875 
 
 4.090 
 
 7-735 
 
 4.602 
 
 59^ 
 
 31 
 
 2-575 
 
 5-143 
 
 3.090 
 
 6.000 
 
 3.605 
 
 6.857 
 
 4.120 
 
 7-7I5 
 
 4-635 
 
 59 
 
 3 I % 
 
 2-594 
 
 5.129 
 
 
 5.984 
 
 3-631 
 
 6.839 
 
 4.150 
 
 7.694 
 
 4-669 
 
 5334- 
 
 3 l *A 
 
 2.612 
 
 5-II6 
 
 3-135 
 
 5-968 
 
 3-657 
 
 6.821 
 
 4.180 
 
 7.674 
 
 4.702 
 
 58^ 
 
 31 x 
 
 2.631 
 
 5-102 
 
 3-157 
 
 5-952 
 
 3-683 
 
 6.803 
 
 4.210 
 
 7-653 
 
 4-736 
 
 5Stf 
 
 32 
 
 2.650 
 
 5.088 
 
 3-180 
 
 5.936 
 
 3-709 
 
 6.784 
 
 4-239 
 
 7.632 
 
 4.769 
 
 58 
 
 3 2 % 
 
 2.668 
 
 5-074 
 
 3.202 
 
 5-920 
 
 3-735 
 
 6.766 
 
 .269 
 
 7.612 
 
 4-802 
 
 573^ 
 
 32^ 
 
 2.686 
 
 5.060 
 
 3-224 
 
 5-904 
 
 3.761 
 
 6-747 
 
 .298 
 
 7-591 
 
 4-836 
 
 57^ 
 
 32 X 
 
 2.705 
 
 5-046 
 
 3.246 
 
 5.887 
 
 3-787 
 
 6.728 
 
 -328 
 
 7-569 
 
 4.869 
 
 S7% 
 
 33 
 
 2.723 
 
 5.032 
 
 3.268 
 
 5-87I 
 
 3-812 
 
 6.709 
 
 357 
 
 7-548 
 
 4.902 
 
 sr 
 
 33)4 
 
 2.741 
 
 5.018 
 
 3-290 
 
 5-854 
 
 3-838 
 
 6.690 
 
 4-386 
 
 7-527 
 
 4-935 
 
 5634- 
 
 33 X 
 
 2.760 
 
 5-003 
 
 3-312 
 
 5-837 
 
 3.864 
 
 6.671 
 
 4.416 
 
 7-505 
 
 4.967 
 
 56^ 
 
 33^ 
 
 2.778 
 
 4.989 
 
 3-333 
 
 5.820 
 
 3-889 
 
 6.652 
 
 4-445 
 
 7-483 
 
 5.000 
 
 56k 
 
 34 
 
 2.796 
 
 4-974 
 
 3-355 
 
 5-803 
 
 3-W4 
 
 6.632 
 
 4-474 
 
 7.461 
 
 5-033 
 
 56 
 
 34% 
 
 2.814 
 
 4.960 
 
 3-377 
 
 5.786 
 
 3-940 
 
 6.613 
 
 4.502 
 
 7-439 
 
 5-065 
 
 553^ 
 
 
 2.832 
 
 4-945 
 
 3-398 
 
 5-769 
 
 3-965 
 
 6-593 
 
 4-531 
 
 7-4I7 
 
 5-098 
 
 55^ 
 
 31*. 
 
 2.850 
 
 4-930 
 
 3-420 
 
 5-752 
 
 3-990 
 
 6-573 
 
 4-56o 
 
 7- 3') 5 
 
 5-130 
 
 55X 
 
 35 
 
 2.868 
 
 4-9 x 5 
 
 3-441 
 
 5-734 
 
 4.015 
 
 6-553 
 
 4-589 
 
 7-372 
 
 5.162 
 
 55 
 
 35% 
 
 2.886 
 
 4.900 
 
 3-463 
 
 5-716 
 
 4.040 
 
 6-533 
 
 4.617 
 
 7-350 
 
 5-194 
 
 5434' 
 
 
 2.904 
 
 4-885 
 
 3-484 
 
 5-699 
 
 4.065 
 
 6.513 
 
 4.646 
 
 7-327 
 
 5.226 
 
 54^ 
 
 35 3 /l 
 
 2.921 
 
 4.869 
 
 3-505 
 
 5-68i 
 
 4.090 
 
 6-493 
 
 4.674 
 
 7-304 
 
 5-258 
 
 54 % 
 
 36 C 
 
 2-939 
 
 4-854 
 
 
 5-663 
 
 4.115 
 
 6.472 
 
 4.702 
 
 7.281 
 
 5.290 
 
 54 
 
 3b% 
 
 2-957 
 
 4-839 
 
 3.548 
 
 
 4-139 
 
 6.452 
 
 4-730 
 
 7-258 
 
 5.322 
 
 53?4: 
 
 36X 
 
 2.974 
 
 4-823 
 
 3-569 
 
 5.627 
 
 4.164 
 
 6.431 
 
 4-759 
 
 7-235 
 
 5-353 
 
 53^ 
 
 36 3 4 
 
 2.992 
 
 4.808 
 
 3-590 
 
 5-609 
 
 4.188 
 
 6.410 
 
 4.787 
 
 7.211 
 
 5.385 
 
 
 
 3-oog 
 
 4.792 
 
 3.611 
 
 5-590 
 
 4-213 
 
 6.389 
 
 4.815 
 
 7.188 
 
 5.416 
 
 53 4 
 
 3~% 
 
 3.026 
 
 4.776 
 
 3-632 
 
 5-572 
 
 4-237 
 
 6.368 
 
 4.842 
 
 7.164 
 
 5-448 
 
 52 # 
 
 
 3-044 
 
 4.760 
 
 3-653 
 
 5-554 
 
 4-261 
 
 6-347 
 
 4.870 
 
 7.140 
 
 5-479 
 
 
 37% 
 
 3.061 
 
 4-744 
 
 3-673 
 
 5-535 
 
 4-2G6 
 
 6.326 
 
 4.898 
 
 7.116 
 
 5-510 
 
 52* 
 
 38 
 
 3.078 
 
 4-728 
 
 3-694 
 
 
 4.310 
 
 6.304 
 
 4-9-5 
 
 7.092 
 
 5-54t 
 
 52 J 
 
 38j^ 
 
 3.095 
 
 4.712 
 
 3-7I5 
 
 5-497 
 
 4-334 
 
 6.283 
 
 4-953 
 
 7.068 
 
 5-572 
 
 5 1 ^ 
 
 38^ 
 
 3-"3 
 
 4.696 
 
 3-735 
 
 5-478 
 
 4-358 
 
 6.261 
 
 4.980 
 
 7-043 
 
 5-603 
 
 51 y 2 
 
 38^ 
 
 3-130 
 
 4.679 
 
 3- 756 
 
 5-459 
 
 4.381 
 
 6.239 
 
 5.007 
 
 7.019 
 
 5-633 
 
 5 1 1 A 
 
 39 
 
 3-147 
 
 4-663 
 
 3-776 
 
 5-440 
 
 4-405 
 
 6.217 
 
 5-035 
 
 6-994 
 
 5-664 
 
 51 
 
 * 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 ~w 
 
 Jt 
 
 5 
 
 1 
 
 6 
 
 ' 
 
 7 
 
 
 3 
 
 
 9 
 
 i
 
 LATITUDES AND DEPARTURES. 
 
 49 
 
 
 
 
 
 > 
 
 i 
 
 J 
 
 
 1 
 
 5 
 
 g) 
 
 . 
 
 Lat, 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 I 
 
 39" 
 
 0.777 
 
 0.629 
 
 554 
 
 259 
 
 2-331 
 
 .888 
 
 3.109 
 
 2.5I7 
 
 3.886 
 
 51 
 
 39V 
 
 0.774 
 
 0.633 
 
 549 
 
 .265 
 
 2.323 
 
 .898 
 
 3-098 
 
 2-531 
 
 3.872 
 
 SoV 
 
 39^ 
 
 0.772 
 
 0.636 
 
 543 
 
 .272 
 
 2.315 
 
 .908 
 
 3.086 
 
 2-544 
 
 3.858 
 
 $0% 
 
 39V 
 
 0.769 
 
 0.639 
 
 -538 
 
 279 
 
 2.307 
 
 .918 
 
 3-075 
 
 2.558 
 
 3-844 
 
 
 40 
 
 0.766 
 
 0-643 
 
 532 
 
 .286 
 
 2.298 
 
 .928 
 
 3.064 
 
 2-571 
 
 3-830 
 
 50 
 
 4 o /4' 
 
 0.763 
 
 0.646 
 
 .526 
 
 .292 
 
 2.290 
 
 938 
 
 3-053 
 
 2.584 
 
 3-816 
 
 49V 
 
 40*4 
 
 0.760 
 
 0.649 
 
 521 
 
 .299 
 
 2.281 
 
 .948 
 
 3.042 
 
 2.598 
 
 3.802 
 
 49^ 
 
 4034; 
 
 0.758 
 
 0-653 
 
 515 
 
 .306 
 
 2.273 
 
 958 
 
 3-030 
 
 2.611 
 
 3.788 
 
 
 41 
 
 0.755 
 
 0.656 
 
 509 
 
 .312 
 
 2.264 
 
 .968 
 
 3.019 
 
 2.624 
 
 3-774 
 
 49 
 
 41* 
 
 0.752 
 
 0.659 
 
 504 
 
 319 
 
 2.256 
 
 .978 
 
 3-007 
 
 2.637 
 
 3-759 
 
 48v 
 
 
 0.749 
 
 0.663 
 
 .498 
 
 .325 
 
 2.247 
 
 .988 
 
 2.996 
 
 2.650 
 
 3-745 
 
 48^ 
 
 4i# 
 
 0.746 
 
 0.666 
 
 .492 
 
 332 
 
 2.238 
 
 .998 
 
 2.984 
 
 2.664 
 
 3-730 
 
 
 42 
 
 0-743 
 
 0.669 
 
 .486 
 
 .338 
 
 2.229 
 
 2.007 
 
 2-973 
 
 2.677 
 
 3.716 
 
 48 4 
 
 42V 
 
 0.740 
 
 0.672 
 
 .480 
 
 345 
 
 2.221 
 
 2.017 
 
 2.961 
 
 2.689 
 
 3.701 
 
 47V 
 
 42^ 
 
 0-737 
 
 0.676 
 
 475 
 
 351 
 
 2.212 
 
 2.027 
 
 2.949 
 
 2.702 
 
 3.686 
 
 47^ 
 
 42?4: 
 
 0-734 
 
 0.679 
 
 469 
 
 -358 
 
 2.2O3 
 
 2.036 
 
 2-937 
 
 2.715 
 
 3-672 
 
 47V 
 
 43 
 
 Q-73 1 
 
 0.682 
 
 463 
 
 364 
 
 2.194 
 
 2.046 
 
 2.925 
 
 2.728 
 
 3-657 
 
 41 
 
 43V 
 
 0.728 
 
 0.685 
 
 457 
 
 370 
 
 2.185 
 
 2.056 
 
 2.913 
 
 2.741 
 
 3-642 
 
 46V 
 
 43 K 
 
 0.725 
 
 0.688 
 
 451 
 
 377 
 
 2.176 
 
 2.065 
 
 2.901 
 
 2.753 
 
 3-627 
 
 
 43 3/ 
 
 0.722 
 
 0.692 
 
 445 
 
 383 
 
 2.167 
 
 2.075 
 
 2.889 
 
 2.766 
 
 3.612 
 
 4&V 
 
 44 
 
 0.719 
 
 0.695 
 
 439 
 
 389 
 
 2.158 
 
 2.084 
 
 2.877 
 
 2.779 
 
 3-597 
 
 46 
 
 44V 
 
 0.716 
 
 0.698 
 
 433 
 
 .396 
 
 2.149 
 
 2.093 
 
 2.865 
 
 2.791 
 
 
 45 X 
 
 44 ]/2 
 
 0.713 
 
 0.701 
 
 .427 
 
 .402 
 
 2.140 
 
 2.103 
 
 2-853 
 
 2.804 
 
 3-566 
 
 45^ 
 
 44% 
 
 0.710 
 
 0.704 
 
 .420 
 
 .408 
 
 2.I3I 
 
 2.II2 
 
 2.841 
 
 2.816 
 
 3-551 
 
 45V 
 
 45 
 
 0.707 
 
 0.707 
 
 .414 
 
 .414 
 
 2. 121 
 
 2. 121 
 
 2.828 
 
 2.828 
 
 3.536 
 
 45 
 
 B'ring 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 B'ring 
 
 1 
 
 5 
 
 ( 
 
 > 
 
 J 
 
 r 
 
 t 
 
 $ 
 
 < 
 
 > 
 
 QD 
 
 JL 
 
 Dep. 
 
 Lat, 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Jjj 
 
 39 
 
 3-147 
 
 4-663 
 
 3-776 
 
 5-440 
 
 4.405 
 
 6.217 
 
 5-035 
 
 6.994 
 
 5-664 
 
 "sF 
 
 39V 
 
 3.164 
 
 4.646 
 
 
 5.421 
 
 4.429 
 
 6.195 
 
 5.062 
 
 6.970 
 
 5-694 
 
 sov 
 
 39^ 
 
 3-i8o 
 
 4.630 
 
 3^8l6 
 
 5.401 
 
 4-453 
 
 6.173 
 
 5-089 
 
 6-945 
 
 5-725 
 
 50^ 
 
 39V 
 
 3-197 
 
 4.613 
 
 3.837 
 
 
 4.476 
 
 6.151 
 
 
 6.920 
 
 5-755 
 
 
 40 
 
 3.214 
 
 4.596 
 
 3-857 
 
 5-362 
 
 4-500 
 
 6.128 
 
 5,142 
 
 6.894 
 
 5.785 
 
 50 
 
 4V 
 
 3-231 
 
 4-579 
 
 3-877 
 
 5-343 
 
 4-523 
 
 6.106 
 
 5.169 
 
 6.869 
 
 5-815 
 
 49 X 
 
 4//J 
 
 3-247 
 
 4-562 
 
 
 5-323 
 
 4-546 
 
 6.083 
 
 5.196 
 
 6.844 
 
 5.845 
 
 49K 
 
 4") V 
 
 3.264 
 
 4-545 
 
 3-9I7 
 
 5-303 
 
 4-569 
 
 6.061 
 
 5-222 
 
 6.818 
 
 5.875 
 
 
 41 
 
 3.280 
 
 4-528 
 
 3-936 
 
 5-283 
 
 4-592 
 
 6.038 
 
 5-248 
 
 6.792 
 
 5-905 
 
 49 
 
 4 T V 
 
 3-297 
 
 4-5ii 
 
 3-956 
 
 5-263 
 
 4.615 
 
 6.015 
 
 5-275 
 
 6.767 
 
 5-934 
 
 4 8V 
 
 41^ 
 
 3-313 
 
 4.494 
 
 3-976 
 
 5-243 
 
 4-638 
 
 5-992 
 
 5-301 
 
 6.741 
 
 5.964 
 
 48^ 
 
 4134' 
 
 3-329 
 
 4.476 
 
 3-995 
 
 5-222 
 
 4.661 
 
 5-968 
 
 5-327 
 
 6-715 
 
 5-993 
 
 48'^ 
 
 42 
 
 
 4-459 
 
 4.015 
 
 5-202 
 
 4-684 
 
 5-945 
 
 5-353 
 
 6.688 
 
 6.022 
 
 48 
 
 42V 
 
 3' 362" 
 
 4-441 
 
 4-034 
 
 5.182 
 
 4./07 
 
 5-922 
 
 5-379 
 
 6.662 
 
 6.051 
 
 47% 
 
 4 2 /^ 
 
 3-37S 
 
 4.424 
 
 4-054 
 
 5.I6I 
 
 4.729 
 
 5-898 
 
 5-405 
 
 6.635 
 
 6.08O 
 
 4T/2 
 
 4234^ 
 
 3-394 
 
 4.406 
 
 4-073 
 
 5.140 
 
 4-752 
 
 5.875 
 
 5-430 
 
 6.609 
 
 6.IO9 
 
 47V 
 
 43" 
 
 3.410 
 
 4-388 
 
 4.092 
 
 5.119 
 
 4-774 
 
 5-851 
 
 5-456 
 
 6.582 
 
 6.138 
 
 4f 
 
 43V 
 
 3-426 
 
 4-3/0 
 
 4.111 
 
 5-099 
 
 4.796 
 
 5-827 
 
 5,481 
 
 6-555 
 
 6.167 
 
 46V 
 
 43K 
 
 3-442 
 
 4-352 
 
 4.130 
 
 5.078 
 
 4.818 
 
 5-803 
 
 5-507 
 
 6.528 
 
 6.195 
 
 
 4334: 
 
 3-458 
 
 4-334 
 
 4.149 
 
 5-057 
 
 4.841 
 
 5-779 
 
 5-532 
 
 6.501 
 
 6.224 
 
 4&V 
 
 44 J 
 
 3-473 
 
 4.316 
 
 4.168 
 
 5-035 
 
 4.863 
 
 5-755 
 
 5-557 
 
 6.474 
 
 6.252 
 
 46 
 
 44V 
 
 3-489 
 
 4.298 
 
 4.187 
 
 5.014 
 
 4.885 
 
 5-730 
 
 5.582 
 
 6.447 
 
 6.280 
 
 45V 
 
 44K 
 
 3-505 
 
 4.280 
 
 4.206 
 
 4-993 
 
 4.906 
 
 5-706 
 
 5.607 
 
 6.419 
 
 6.308 
 
 45^ 
 
 
 3-520 
 
 4.261 
 
 4.224 
 
 4.971 
 
 4.928 
 
 5.681 
 
 5.632 
 
 6.392 
 
 6.336 
 
 45V 
 
 4 45 4 
 
 3-536 
 
 4-243 
 
 4-243 
 
 4-950 
 
 4.950 
 
 5-657 
 
 5.657 
 
 6.364 
 
 6364 
 
 45 
 
 B'ring 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat 
 
 Dep. 
 
 Lat. 
 
 Dep 
 
 J..-U. 
 
 B'rinjr
 
 TABLES AND FORMULAS. 51 
 
 TABLE OF 
 
 HORIZONTAL DISTANCES AND DIFFERENCES 
 OF ELEVATION FOR STADIA MEASUREMENTS. 
 
 The formulas used in the computation of the following 
 tables furnish expressions for horizontal distances and 
 differences of elevation for stadia measurements with the 
 conditions that the stadia rod be held vertical and the stadia 
 wires be equidistant from the center wire. The formulas 
 used are as follows: For the horizontal distance 
 
 D = c cos n + a k cos 2 , (94.) Art. 1 3O1 . 
 
 in which D = the corrected distance ; c = the constant ; 
 a k = the stadia distance, and n the vertical angle. 
 
 For the difference of elevation, the following formula is 
 used: 
 
 E = c sin n + a k ^2f. (95.) Art. 1 3O1 . 
 
 For application of tables see Art. 13O1*
 
 TABLES AND FORMULAS. 
 
 53 
 
 HORIZONTAL DISTANCES AND DIFFERENCES 
 OF ELEVATION FOR STADIA MEASUREMENTS. 
 
 
 
 
 
 
 I 
 
 
 
 2 
 
 
 
 3 
 
 
 
 Minutes, 
 o' 
 
 2 
 
 4 . ... 
 
 Hor. 
 Dist. 
 
 IOO.OO 
 IOO.OO 
 IOO.OO 
 
 Diff. 
 Elev. 
 
 .00 
 
 .06 
 
 . 12 
 
 Hor. 
 
 Dist. 
 
 99-97 
 99-97 
 
 Diff. 
 Kiev. 
 
 1.74 
 i. so 
 
 1.86 
 
 Hor. 
 Dist. 
 
 99.88 
 99.87 
 
 00.87 
 
 Diff. 
 Elev. 
 
 3-49 
 
 3-55 
 3 60 
 
 Hor. 
 Dist. 
 
 99-73 
 99.72 
 99.71 
 
 Diff. 
 Elev. 
 
 5-23 
 5.28 
 
 5-34 
 
 6 
 8 
 
 10 
 
 12 
 14 
 
 IOO.OO 
 IOO.OO 
 100.00 
 
 IOO.OO 
 IOO.OO 
 
 17 
 23 
 .29 
 
 -35 
 .41 
 
 99.96 
 99.96 
 99.96 
 
 99.96 
 99-95 
 
 1.92 
 1.98 
 2.04 
 
 2.09 
 2.15 
 
 99.87 
 
 99-86 
 99.86 
 
 99-85 
 99-85 
 
 3-66 
 3-72 
 3-78 
 
 3-84 
 3-9 
 
 99.71 
 99.70 
 99.69 
 
 99.69 
 99.68 
 
 5-40 
 5-46 
 
 5-52 
 
 5-57 
 5.63 
 
 16 
 18 
 
 20 
 22 .... 
 
 IOO.OO 
 IOO.OO 
 IOO.OO 
 
 IOO.OO 
 
 47 
 
 52 
 58 
 
 .64 
 
 99-95 
 99-95 
 99-95 
 
 2.21 
 2.27 
 2-33 
 2.38 
 
 99.84 
 99.84 
 99-83 
 
 QQ 8s 
 
 3-95 
 4.01 
 4.07 
 
 4- I 3 
 
 99.68 
 99.67 
 99.66 
 
 99.66 
 
 5-69 
 
 5-75 
 5-80 
 
 5-86 
 
 24 
 26 
 28 
 
 100.00 
 
 99-99 
 99-99 
 
 .70 
 .76 
 .81 
 
 87 
 
 99-94 
 
 99-94 
 99-93 
 
 2-44 
 2.50 
 2.56 
 
 2 62 
 
 99.82 
 99.82 
 99.81 
 
 QQ 8 1 
 
 4.18 
 4.24 
 4-30 
 4.36 
 
 99-65 
 99.64 
 99-63 
 09.6^ 
 
 5-92 
 5-98 
 6.04 
 6.09 
 
 32 
 
 34 
 
 99-99 
 99-99 
 
 93 
 99 
 
 99-93 
 99-93 
 
 2.6 7 
 2.73 
 
 99.80 
 99.80 
 
 4.42 
 4.48 
 
 99.62 
 99.62 
 
 6.15 
 
 6.21 
 
 36 
 
 
 1.05 
 
 
 2.7Q 
 
 
 4-53 
 
 qq.6l 
 
 6 27 
 
 38 
 
 99-99 
 
 i. ii 
 
 99.92 
 
 2.85 
 
 
 4-59 
 
 90.60 
 
 6-33 
 
 40 
 
 99-99 
 
 1.16 
 
 99.92 
 
 2.gl 
 
 99.78 
 
 4-65 
 
 99-59 
 
 6.38 
 
 4 2 
 
 
 1.22 
 
 
 2. Q7 
 
 00.78 
 
 4- 7 1 
 
 99.59 
 
 6.44 
 
 44 .... 
 
 99-98 
 
 1.28 
 
 99.91 
 
 3-O2 
 
 00.77 
 
 4.76 
 
 99.58 
 
 6.50 
 
 46 
 
 99.98 
 
 1.34 
 
 99.90 
 
 3.08 
 
 00.77 
 
 4.82 
 
 99-57 
 
 6.56 
 
 48 
 50 
 
 99.98 
 99.98 
 
 1.40 
 
 i-45 
 
 99.90 
 99.90 
 
 3-14 
 3-20 
 
 99.76 
 99.76 
 
 4.88 
 4-94 
 
 99-56 
 99.56 
 
 6.6r 
 6.67 
 
 52 
 54 
 56 
 
 99.98 
 99.98 
 99-97 
 
 i-5i 
 i-57 
 1.63 
 
 99.89 
 99.89 
 99.89 
 
 3.26 
 3-31 
 3-37 
 
 99-75 
 99-74 
 99.74 
 
 4-99 
 5-05 
 5.11 
 
 99-55 
 99-54 
 99-53 
 
 6-73 
 6.78 
 6.84 
 
 58 
 60 
 
 99-97 
 99-97 
 
 1.69 
 1.74 
 
 99.88 
 99.88 
 
 3-43 
 3-49 
 
 99-73 
 99-73 
 
 5-i7 
 5-23 
 
 99-52 
 99-51 
 
 6.90 
 6.96 
 
 c= -75 
 
 75 
 
 .01 
 
 75 
 
 .02 
 
 75 
 
 03 
 
 75 
 
 05 
 
 C = 1. 00 
 
 1. 00 
 
 .01 
 
 I.OO 
 
 03 
 
 I.OO 
 
 .04 
 
 I.OO 
 
 .06 
 
 c = 1.25 
 
 1.25 
 
 .02 
 
 1-25 
 
 03 
 
 1.25 
 
 05 
 
 1-25 
 
 .08
 
 TABLES AND FORMULAS. 
 
 HORIZONTAL DISTANCES AND DIFFERENCES 
 OF ELEVATION FOR STADIA MEASUREMENTS. 
 
 
 4 C 
 
 > 
 
 5 
 
 3 
 
 6 
 
 ) 
 
 7 
 
 3 
 
 Minutes. 
 
 Hor. 
 Dist. 
 
 99.51 
 
 Diff, 
 Elev. 
 6.96 
 
 Hor. 
 Dist. 
 
 99.24 
 
 Diff. 
 Elev. 
 
 8.68 
 
 Hor. 
 Dist. 
 98.91 
 
 Diff. 
 Elev. 
 10.40 
 
 Hor. 
 Dist. 
 
 98.51 
 
 Diff. 
 Elev. 
 
 12.10 
 
 2 
 
 99-51 
 
 7.02 
 
 99.23 
 
 8.74 
 
 98.90 
 
 10.45 
 
 98.50 
 
 12.15 
 
 
 99- 50 
 
 7.07 
 
 99.22 
 
 8.80 
 
 98.88 
 
 10.51 
 
 98.48 
 
 12.21 
 
 6 
 8 
 
 99.49 
 99.48 
 99-47 
 
 7-13 
 7.19 
 
 7.25 
 
 99-21 
 99.20 
 99- *9 
 
 8.85 
 8.91 
 8.97 
 
 98.87 
 98.86 
 98.85 
 
 10.57 
 10.62 
 
 10.68 
 
 98.47 
 98.46 
 98.44 
 
 12.26 
 12.32 
 12.38 
 
 12 
 14 
 
 16 
 
 99.46 
 99.46 
 99-45 
 
 7-30 
 7-36 
 7.42 
 
 99.18 
 99.17 
 99.16 
 
 9-03 
 9.08 
 
 Q. 14 
 
 98.83 
 98.82 
 98.81 
 
 10.74 
 10.79 
 10.85 
 
 98-43 
 98.41 
 98.40 
 
 12.43 
 
 12.49 
 
 I2o5 
 
 18 
 
 99.44 
 
 7.48 
 
 99.15 
 
 g.2O 
 
 98.80 
 
 10.91 
 
 98.39 
 
 12. 60 
 
 20 
 
 22 
 24 
 
 99-43 
 99.42 
 99.41 
 
 7-53 
 
 7-59 
 7.65 
 
 99.14 
 
 99-13 
 99.11 
 
 9-25 
 9-31 
 
 9-37 
 
 98.78 
 
 98.77 
 98.76 
 
 10.96 
 
 11.02 
 II.08 
 
 98-37 
 
 98.36 
 98.34 
 
 12.66 
 
 12.72 
 12.77 
 
 26 
 
 99.40 
 
 7.71 
 
 99.10 
 
 9-43 
 
 98.74 
 
 11.13 
 
 98.33 
 
 12.83 
 
 28 
 
 99-39 
 
 7.76 
 
 
 Q 48 
 
 98.73 
 
 Il.lg 
 
 98.31 
 
 12.88 
 
 30 
 32 
 
 99-38 
 99.38 
 
 7.82 
 7.88 
 
 99.08 
 99.07 
 
 9-54 
 9.60 
 
 98.72 
 98.71 
 
 11.25 
 11.30 
 
 98.29 
 93.28 
 
 12.94 
 13-00 
 
 
 99-37 
 
 7-94 
 
 99.06 
 
 0.65 
 
 98.69 
 
 11.36 
 
 98.27 
 
 13.05 
 
 16 
 
 99.36 
 
 7-99 
 
 99.05 
 
 9.71 
 
 98.68 
 
 11.42 
 
 98.25 
 
 13.11 
 
 *8 
 
 99.35 
 
 8-05 
 
 99.04 
 
 9-77 
 
 98.67 
 
 11.47 
 
 98.24 
 
 I3-I7 
 
 40 
 
 99-34 
 
 8.ii 
 
 99.03 
 
 9.83 
 
 98.65 
 
 ii-53 
 
 98.22 
 
 13.22 
 
 42 
 
 99-33 
 
 8.17 
 
 99.01 
 
 9.88 
 
 98.64 
 
 11.59 
 
 98.20 
 
 13-28 
 
 44 
 
 99.32 
 
 8.22 
 
 99.00 
 
 9.94 
 
 98.63 
 
 11.64 
 
 98.19 
 
 *3 33 
 
 46 
 
 99.31 
 
 8.28 
 
 98.99 
 
 IO.OO 
 
 98.61 
 
 11.70 
 
 98.17 
 
 I 3-39 
 
 48 
 
 5 
 
 99-30 
 99.29 
 
 8-34 
 8.40 
 
 98.98 
 98.97 
 
 10.05 
 
 IO. II 
 
 98.60 
 
 98.58 
 
 11.76 
 11.81 
 
 98.16 
 98.14 
 
 13-45 
 I3-50 
 
 52 
 
 99.28 
 
 8.45 
 
 98.96 
 
 10.17 
 
 98.57 
 
 11.87 
 
 98.13 
 
 I3-56 
 
 54 
 
 99.27 
 
 8.51 
 
 98.94 
 
 IO. 22 
 
 98.56 
 
 11.93 
 
 98.11 
 
 i',6i 
 
 56 
 58 
 
 99.26 
 99-25 
 
 8-57 
 8.63 
 
 98.93 
 98.92 
 
 10.28 
 IO.34 
 
 98-54 
 98.53 
 
 11.98 
 12.04 
 
 98.10 
 98.08 
 
 13-67 
 13-73 
 
 60 
 
 
 8.68 
 
 98.91 
 
 IO.4O 
 
 98.51 
 
 12.10 
 
 98.06 
 
 I3-78 
 
 
 
 
 
 
 
 
 
 
 c= .75 
 
 75 
 
 .06 
 
 75 
 
 .07 
 
 75 
 
 .08 
 
 74 
 
 .10 
 
 c = i.oo 
 
 I.OO 
 
 .08 
 
 99 
 
 .09 
 
 99 
 
 .11 
 
 99 
 
 13 
 
 c = 1.25 
 
 1-25 
 
 .10 
 
 1.24 
 
 .11 
 
 1.24 
 
 .14 
 
 1.24 
 
 .16
 
 TABLES AND FORMULAS. 
 
 55 
 
 HORIZONTAL DISTANCES AND DIFFERENCES 
 OF ELEVATION FOR STADIA MEASUREMENTS. 
 
 
 8 
 
 3 
 
 9 
 
 3 
 
 1C 
 
 , 
 
 II 
 
 
 
 Minutes. 
 
 Hor. 
 Dist. 
 
 98.06 
 
 Diff. 
 Elev. 
 
 13.78 
 
 Hor. 
 Dist. 
 
 97-55 
 
 Diff. 
 Elev. 
 
 15.45 
 
 Hor. 
 Dist. 
 96.98 
 
 Diff. 
 Elev. 
 
 17.10 
 
 Hor. 
 Dist. 
 
 96.36 
 
 Dili: 
 Elev. 
 
 18 7*? 
 
 2 ... 
 
 98.05 
 
 13.84 
 
 97-53 
 
 15- 51 
 
 96.96 
 
 17. 16 
 
 96.34 
 
 18.78 
 
 4 
 6 
 8 
 
 98-03 
 98.01 
 98.00 
 
 13.89 
 13-95 
 14.01 
 
 97-52 
 97-50 
 97.48 
 
 15-56 
 15.62 
 15-67 
 
 96-94 
 96.92 
 96.90 
 
 17.21 
 17.26 
 17.32 
 
 96-32 
 96.29 
 96.27 
 
 18.84 
 18.89 
 
 18.95 
 
 
 97.98 
 
 14.06 
 
 97.46 
 
 15.73 
 
 96.88 
 
 17-37 
 
 96.25 
 
 
 12 
 
 97-97 
 97-95 
 
 14.12 
 14.17 
 
 97-44 
 97-43 
 
 15.78 
 15.84 
 
 96.86 
 06.84 
 
 17-43 
 17 48 
 
 96.23 
 96.21 
 
 19.05 
 
 16 
 
 97-93 
 
 14.23 
 
 97.41 
 
 15.89 
 
 96.82 
 
 17.54 
 
 96. 1 8 
 
 io 16 
 
 18 
 20 
 
 97.92 
 97.90 
 
 14.28 
 
 14-34 
 
 97-39 
 97-37 
 
 15-95 
 16.00 
 
 96.80 
 96.78 
 
 17-59 
 17-65 
 
 96.16 
 96.14 
 
 19.21 
 19.27 
 
 22 
 
 97.88 
 
 14.40 
 
 97-35 
 
 1 6. 06 
 
 96.76 
 
 17.70 
 
 96.12 
 
 iq 32 
 
 24 
 
 97.87 
 
 14.45 
 
 97-33 
 
 1 6. ii 
 
 96.74 
 
 17.76 
 
 96.09 
 
 19.38 
 
 26 
 
 97.85 
 
 14.51 
 
 97.31 
 
 16.17 
 
 96.72 
 
 17.81 
 
 96.07 
 
 19-43 
 
 28 
 
 97.83 
 
 14 56 
 
 97.29 
 
 l6 22 
 
 06 70 
 
 17.86 
 
 96.05 
 
 IQ 48 
 
 1O 
 
 97.82 
 
 14.62 
 
 97.28 
 
 16.28 
 
 06.68 
 
 17.92 
 
 96.03 
 
 IQ 54 
 
 32 
 
 34 
 
 97.80 
 97.78 
 
 14.67 
 14.73 
 
 97.26 
 97.24 
 
 16-33 
 
 16 39 
 
 96.66 
 96.64 
 
 17-97 
 18.03 
 
 96.00 
 05.08 
 
 19.59 
 
 36 
 
 97.76 
 
 14.79 
 
 97.22 
 
 16.44 
 
 96.62 
 
 18.08 
 
 95-96 
 
 IQ. 7O 
 
 38 
 
 97-75 
 
 14.84 
 
 97.20 
 
 16.50 
 
 96.60 
 
 18.14 
 
 95-93 
 
 19.75 
 
 40 
 42 ... 
 
 97-73 
 97.71 
 
 14.90 
 14.95 
 
 97.18 
 97.16 
 
 16-55 
 16.61 
 
 96-57 
 
 Q6. 55 
 
 18.19 
 18.24 
 
 95-91 
 95.89 
 
 Ig.SO 
 lg.86 
 
 44 
 
 97.69 
 
 15.01 
 
 97.14 
 
 1 6. 66 
 
 96. 53 
 
 18.30 
 
 95-86 
 
 ig.gl 
 
 46 
 
 48 
 
 97.68 
 07.66 
 
 15.06 
 1512 
 
 97.12 
 97- Io 
 
 16.72 
 16 77 
 
 96.51 
 
 18.35 
 18.41 
 
 95.84 
 95.82 
 
 19.96 
 2O O2 
 
 50 
 
 97.64 
 
 15-17 
 
 97.08 
 
 16.83 
 
 96.47 
 
 18.46 
 
 95-79 
 
 2O.O7 
 
 52 
 
 97.62 
 
 15.23 
 
 97.06 
 
 16.88 
 
 96.45 
 
 18.51 
 
 95-77 
 
 20. 1 2 
 
 54 
 
 97 61 
 
 15.28 
 
 97.04 
 
 1 6 04 
 
 06.42 
 
 18.57 
 
 95- 75 
 
 20. 1 8 
 
 56 .... 
 
 97-59 
 
 15.34 
 
 97.02 
 
 16.99 
 
 96.40 
 
 18.62 
 
 95.72 
 
 20.23 
 
 58 
 
 97- 57 
 
 15.40 
 
 97.00 
 
 17.05 
 
 96.38 
 
 18.68 
 
 95.70 
 
 20.28 
 
 60 
 
 97-55 
 
 15.45 
 
 96.98 
 
 17.10 
 
 96.36 
 
 18.73 
 
 95-68 
 
 20.34 
 
 
 
 
 
 
 
 
 
 
 C= .75 
 
 74 
 
 .11 
 
 74 
 
 .12 
 
 74 
 
 .14 
 
 73 
 
 15 
 
 C = I.OO 
 
 99 
 
 15 
 
 99 
 
 .16 
 
 .98 
 
 .18 
 
 .98 
 
 .20 
 
 C = 1.25 
 
 1.23 
 
 .18 
 
 1-23 
 
 .21 
 
 1-23 
 
 23 
 
 1.22 
 
 25
 
 TABLES AND FORMULAS. 
 
 HORIZONTAL DISTANCES AND DIFFERENCES 
 OF ELEVATION FOR STADIA MEASUREMENTS. 
 
 
 12 
 
 
 
 i; 
 
 
 
 I/ 
 
 \ 
 
 1C 
 
 
 Minutes. 
 
 o' 
 
 Hor. 
 Dist. 
 
 95.68 
 
 Diff. 
 Elev. 
 
 20.34 
 
 Hor. 
 Dist. 
 
 Diff. 
 Elev. 
 
 Hor. 
 Dist. 
 
 04 15 
 
 Diff. 
 
 Elev. 
 
 21 47 
 
 Hor. 
 Dist. 
 
 QO OQ 
 
 Diff. 
 Elev. 
 
 2 
 
 95.65 
 
 20.39 
 
 04.0! 
 
 
 Q4. 12 
 
 23.52 
 
 Q3.27 
 
 25 05 
 
 4 
 6 
 8 
 
 95-63 
 95.61 
 95-58 
 
 20.44 
 
 20.50 
 20.55 
 
 94.89 
 94.86 
 94.84 
 
 22.02 
 22.O8 
 22.13 
 
 94.09 
 94.07 
 94.04 
 
 23.58 
 23-63 
 23.68 
 
 93-24 
 93-21 
 93.18 
 
 25.10 
 25-15 
 25.20 
 
 10 
 
 95.56 
 
 20.60 
 
 04.81 
 
 
 04.01 
 
 2 3 73 
 
 93- *6 
 
 25 25 
 
 12 
 
 95.53 
 
 20.66 
 
 94- 79 
 
 22 2^ 
 
 93.98 
 
 23.78 
 
 93-13 
 
 25.30 
 
 14 
 
 95.51 
 
 20.71 
 
 94.76 
 
 22.28 
 
 93-95 
 
 23.83 
 
 93.10 
 
 25.35 
 
 16 
 
 95-49 
 
 20.76 
 
 94- 73 
 
 
 Q3 Q-l 
 
 23.88 
 
 93-07 
 
 25 40 
 
 18 
 
 95.46 
 
 20.81 
 
 94- 7 1 
 
 
 93- QO 
 
 21-03 
 
 93.04 
 
 25.45 
 
 20 
 22 
 
 95-44 
 95.41 
 
 20.87 
 20 92 
 
 94.68 
 94.66 
 
 22.44 
 
 93.87 
 
 QQ 84 
 
 23-99 
 
 93-oi 
 92.98 
 
 25-50 
 
 2 5 55 
 
 24 
 26 
 28 
 
 95-39 
 95-36 
 
 95-34 
 05-32 
 
 20.97 
 21.03 
 21.08 
 
 21 11 
 
 94-63 
 94.60 
 94-58 
 
 22.54 
 22.60 
 22.65 
 
 93.81 
 93-79 
 93-76 
 
 24.09 
 24.14 
 24.19 
 24 24 
 
 92.95 
 92.92 
 92.89 
 
 02.86 
 
 25.60 
 25-65 
 25.70 
 
 2 5 75 
 
 32 
 
 95-29 
 
 21. l8 
 
 94. 52 
 
 22 75 
 
 93-7 
 
 24.29 
 
 92.83 
 
 25.80 
 
 34 
 36 
 
 95-27 
 95.24 
 
 21.24 
 21.29 
 
 94-50 
 O4.47 
 
 22.80 
 22.85 
 
 93-67 
 93.65 
 
 24-34 
 24.39 
 
 92.80 
 92.77 
 
 25.85 
 25.90 
 
 38 
 
 95-22 
 
 21-34 
 
 
 
 93.62 
 
 24.44 
 
 92.74 
 
 2G 05 
 
 40 
 
 42 
 44 
 
 95-19 
 95-17 
 
 95.14 
 
 21-39 
 
 21-45 
 21.50 
 
 94.42 
 
 94-39 
 94-36 
 
 22.96 
 23.01 
 
 93-59 
 93-56 
 93-53 
 
 24-49 
 
 24-55 
 24.60 
 
 92.71 
 
 92.68 
 92.65 
 
 26.OO 
 
 26.05 
 
 26. 10 
 
 46 
 48 
 50 
 
 95.12 
 95-09 
 
 05.07 
 
 21-55 
 21.60 
 21.66 
 
 94-34 
 94-31 
 94.28 
 
 23.11 
 
 23.16 
 
 93-50 
 93-47 
 03 45 
 
 24-65 
 24.70 
 
 24. 75 
 
 92.62 
 
 92-59 
 92.56 
 
 26.15 
 26.20 
 26 25 
 
 52 
 
 54 
 
 95.04 
 
 05 O2 
 
 21.71 
 21.76 
 
 94.26 
 
 23.27 
 
 93-42 
 
 24.80 
 
 24.85 
 
 92.53 
 
 26.30 
 
 56 
 
 94-99 
 
 21. 8l 
 
 94.20 
 
 23.37 
 
 93.36 
 
 24.90 
 
 92.46 
 
 26.40 
 
 58 
 60 
 
 94-97 
 94-94 
 
 21.87 
 21.92 
 
 94.17 
 94-15 
 
 23-42 
 
 23-47 
 
 93-33 
 93-30 
 
 24-95 
 25.00 
 
 92-43 
 92.40 
 
 26.45 
 
 26.50 
 
 c= .75 
 
 73 
 
 .16 
 
 73 
 
 17 
 
 73 
 
 .19 
 
 72 
 
 .20 
 
 C i.oo 
 
 .98 
 
 .22 
 
 97 
 
 
 97 
 
 .25 
 
 .06 
 
 2 7 
 
 
 
 
 
 
 
 
 
 
 c = 1.25 
 
 1.22 
 
 .2? 
 
 1. 21 
 
 .29 
 
 I. 21 
 
 3i 
 
 I. 2O 
 
 34
 
 TABLES AND FORMULAS. 
 
 57 
 
 HORIZONTAL DISTANCES AND DIFFERENCES 
 OF ELEVATION FOR STADIA MEASUREMENTS. 
 
 
 id 
 
 
 
 *7 
 
 
 
 i 
 
 
 
 *S 
 
 
 Minutes. 
 
 o' 
 2 
 
 4 
 
 6 
 
 Hor. 
 Dist. 
 
 92.40 
 92-37 
 92-34 
 92.31 
 
 Diff, 
 Elev. 
 
 26.50 
 26.55 
 26.59 
 26.64 
 
 Hor. 
 Dist. 
 
 91-45 
 91.42 
 
 91-39 
 91.35 
 
 Diff. 
 Elev. 
 
 27.96 
 28.01 
 28.06 
 28.10 
 
 Hor. 
 Dist. 
 
 90.45 
 90.42 
 90.38 
 90.35 
 
 Diff. 
 Elev. 
 
 29-39 
 29.44 
 29.48 
 29.53 
 
 Hor. 
 Dist. 
 
 89.40 
 89.36 
 89-33 
 89.29 
 
 Diff. 
 Elev. 
 
 30.78 
 30-83 
 30.87 
 30.92 
 
 3 
 
 92.28 
 92.25 
 
 26.69 
 26.74 
 
 91.32 
 91.29 
 
 28.15 
 28.20 
 
 90.31 
 90.28 
 
 29.58 
 29.62 
 
 89.26 
 89.22 
 
 30.97 
 31.01 
 
 12 
 
 U 
 
 16 
 
 92.22 
 92.19 
 92.15 
 
 26.79 
 26.84 
 26.89 
 
 91.26 
 91.22 
 
 91.19 
 
 28.25 
 28.30 
 
 28.34 
 
 90.24 
 90.21 
 90.18 
 
 29.67 
 29.72 
 29.76 
 
 89.18 
 89.15 
 89.11 
 
 31.06 
 31.10 
 31. 15 
 
 18 
 
 2O 
 
 92.12 
 92.09 
 
 26.94 
 26.99 
 
 91.16 
 91.12 
 
 28.39 
 28.44 
 
 90.14 
 90. 1 1 
 
 29.81 
 29.86 
 
 89.08 
 89.04 
 
 31-19 
 31.24 
 
 
 92.06 
 
 27.04 
 
 91.09 
 
 28.49 
 
 90.07 
 
 29.90 
 
 89.00 
 
 31 28 
 
 24 
 
 92.03 
 
 27.09 
 
 91.06 
 
 28.54 
 
 90.04 
 
 29.95 
 
 88.96 
 
 3 1 33 
 
 26 
 
 92.00 
 
 27.13 
 
 91.02 
 
 28.58 
 
 90.00 
 
 30.00 
 
 88.93 
 
 31.38 
 
 2 8 
 
 91.97 
 
 27.18 
 
 90.99 
 
 28.63 
 
 89.97 
 
 30.04 
 
 88 80 
 
 
 
 91-93 
 
 27.23 
 
 90.96 
 
 28.68 
 
 89.93 
 
 30.09 
 
 88.86 
 
 
 
 91.90 
 
 27.28 
 
 90.92 
 
 28.73 
 
 89 90 
 
 30. 14 
 
 88 82 
 
 
 34 
 36 
 38 
 40 
 
 4 2 
 44 
 46 
 48 
 
 91.87 
 91.84 
 91.81 
 91.77 
 
 91-74 
 91.71 
 91.68 
 91.65 
 91.61 
 
 27-33 
 27-38 
 27-43 
 27-48 
 
 27-52 
 27-57 
 27.62 
 27.67 
 
 90.89 
 90.86 
 90.82 
 90.79 
 
 90.76 
 90.72 
 90.69 
 90.66 
 90.62 
 
 28.77 
 28.82 
 28.87 
 28.92 
 
 28.96 
 29.01 
 29.06 
 29.11 
 
 89.86 
 89-83 
 89.79 
 89.76 
 
 89.72 
 89.69 
 89-65 
 89.61 
 
 30.19 
 30.23 
 30.28 
 30.32 
 
 30.37 
 30.41 
 30.46 
 30.51 
 
 88.78 
 88-75 
 88.71 
 88.67 
 
 88.64 
 88.60 
 
 88.56 
 88-53 
 88 49 
 
 3I-56 
 31.60 
 31-65 
 31.69 
 
 31-74 
 31-78 
 31-83 
 31-87 
 
 52 
 54 
 56 
 
 58 
 
 91.58 
 91-55 
 91.52 
 91.48 
 
 27-77 
 27.81 
 27.86 
 
 90-59 
 
 9-55 
 90-52 
 90.48 
 
 29.20 
 29.25 
 29.30 
 
 89-54 
 89.51 
 89.47 
 
 80 44 
 
 30.60 
 30-65 
 30.69 
 
 30. 74 
 
 88.45 
 88.41 
 88.38 
 88 14 
 
 31.96 
 32.01 
 32-05 
 
 60 
 
 91.45 
 
 27.06 
 
 90.45 
 
 29.39 
 
 Su 40 
 
 30.78 
 
 88 30 
 
 
 
 
 
 
 
 
 
 
 
 c= -75 
 
 .72 
 
 .21 
 
 72 
 
 23 
 
 71 
 
 .24 
 
 7i 
 
 .25 
 
 C = 1. 00 
 
 .86 
 
 .28 
 
 -95 
 
 30 
 
 95 
 
 32 
 
 94 
 
 33 
 
 c = 1.25 
 
 1.20 
 
 35 
 
 1.19 
 
 38 
 
 | 1-19 
 
 .40 
 
 1.18 
 
 .42
 
 58 
 
 TABLES AND FORMULAS. 
 
 HORIZONTAL DISTANCES AND DIFFERENCES 
 OF ELEVATION FOR STADIA MEASUREMENTS, 
 
 
 2C 
 
 
 
 21 
 
 
 
 22 
 
 o 
 
 2 3 
 
 
 Minutes. 
 
 o' 
 2 
 
 Hor. 
 Dist. 
 
 88.30 
 88.26 
 
 Diff. 
 Elev. 
 32.14 
 
 32.18 
 
 j 
 
 Hor. 
 Dist. 
 
 87.16 
 87.12 
 
 Diff. 
 Elev. 
 33-46 
 
 33- 5 
 
 Hor. 
 Dist. 
 
 85-97 
 85.93 
 
 Diff. 
 Elev. 
 
 34-73 
 34.77 
 
 Hor. 
 Dist. 
 
 84-73 
 84.69 
 
 Diff. 
 Elev. 
 
 35-97 
 36.01 
 
 
 88.23 
 
 32.23 
 
 87.08 
 
 33 54 
 
 85.89 
 
 34.82 
 
 84.65 
 
 36.05 
 
 6 
 
 88.19 
 
 32.27 
 
 87.04 
 
 33. 59 
 
 85.85 
 
 -24.86 
 
 84 6 1 
 
 36.09 
 
 8 
 
 88.15 
 
 32.32 
 
 87.00 
 
 33.63 
 
 85.80 
 
 34.90 
 
 84. 57 
 
 36. 13 
 
 10 
 
 88.11 
 
 32.36 
 
 86.96 
 
 33-67 
 
 85.76 
 
 34-94 
 
 84.52 
 
 36.17 
 
 12 
 
 88.08 
 
 32.41 
 
 86.92 
 
 33 7 2 
 
 85.72 
 
 34.98 
 
 84.48 
 
 36.21 
 
 14 
 
 16 
 18 
 
 86.04 
 88.00 
 87-96 
 87.93 
 
 32.45 
 32-49 
 32-54 
 32.58 
 
 86.88 
 86.84 
 86.80 
 86.77 
 
 33-76 
 33-8o 
 33-84 
 TV 80 
 
 85.68 
 85-64 
 85.60 
 85.56 
 
 35.02 
 35-07 
 35-n 
 35- *5 
 
 84.44 
 84.40 
 84-35 
 84 ii 
 
 36.25 
 36.29 
 36.33 
 36 37 
 
 22 
 
 87.89 
 
 32.63 
 
 86.73 
 
 33-93 
 
 85.52 
 
 35.19 
 
 84.27 
 
 36.41 
 
 24 
 
 87.85 
 
 32.67 
 
 86.69 
 
 33-97 
 
 85.48 
 
 35.23 
 
 84.23 
 
 36-45 
 
 26 
 
 87.81 
 
 32.72 
 
 86.65 
 
 34.01 
 
 85.44 
 
 35.27 
 
 84 18 
 
 36 49 
 
 28 
 
 87.77 
 
 32.76 
 
 86. 6 1 
 
 34.06 
 
 85.40 
 
 35.31 
 
 84 14 
 
 36 53 
 
 
 87.74 
 
 32.80 
 
 86.57 
 
 34- Io 
 
 85.36 
 
 35.36 
 
 84.10 
 
 36. 57 
 
 32 
 
 34 
 36 
 38 
 
 87.70 
 87.66 
 87.62 
 87.58 
 
 32.85 
 32.89 
 32-93 
 32.98 
 
 86.53 
 86.49 
 86.45 
 86.41 
 
 34-M 
 
 34.18 
 34-23 
 34.27 
 
 85-31 
 
 85-27 
 85.23 
 85.19 
 
 35-40 
 35-44 
 
 35-48 
 35.52 
 
 84.06 
 84.01 
 
 83-97 
 83.93 
 
 36.61 
 36.65 
 36-69 
 36.73 
 
 40 
 
 42 
 44 
 46 
 48 
 
 87-54 
 
 87-51 
 87.47 
 87-43 
 87.39 
 
 33-02 
 
 33-0? 
 33-" 
 33-15 
 33.20 
 
 86.37 
 
 86-33 
 86.29 
 86.25 
 86.21 
 
 34-31 
 
 34-35 
 34-40 
 34-44 
 
 14 48 
 
 85-15 
 
 85.11 
 85.07 
 85.02 
 84.98 
 
 35-56 
 35-6o 
 35.64 
 35-68 
 35.72 
 
 83-89 
 
 83-84 
 83.80 
 83.76 
 83. 72 
 
 36.77 
 36.80 
 36-84 
 36.88 
 36.92 
 
 50 
 
 52 
 54 
 
 87-35 
 
 87-31 
 87.27 
 
 33-24 
 33-28 
 33-33 
 
 86.17 
 
 86.13 
 86.09 
 
 34-52 
 
 34-57 
 34.61 
 
 84-94 
 84.90 
 84.86 
 
 35.76 
 35.80 
 35.85 
 
 83.67 
 
 83-63 
 83.59 
 
 36.96 
 37-00 
 37-4 
 
 56 
 
 87.24 
 
 33-37 
 
 86.05 
 
 34.65 
 
 84.82 
 
 35-Sg 
 
 83.54 
 
 37.08 
 
 58 
 
 87.20 
 
 33.41 
 
 86. o i 
 
 34- 69 
 
 84.77 
 
 35-93 
 
 83.50 
 
 37.12 
 
 60 
 
 87.16 
 
 33-46 
 
 85.97 
 
 34 73 
 
 84.73 
 
 35-97 
 
 83.46 
 
 37.16 
 
 
 
 
 
 
 
 
 
 
 c= .75 
 
 .70 
 
 .26 
 
 .70 
 
 .27 
 
 .69 
 
 .29 
 
 .69 
 
 30 
 
 C = I.OO 
 
 94 
 
 35 
 
 93 
 
 37 
 
 .92 
 
 38 
 
 .92 
 
 .40 
 
 c 1.25 . 
 
 
 
 i 16 
 
 46 
 
 
 48 
 
 
 
 
 
 
 
 
 
 

 
 TABLES AND FORMULAS. 
 
 HORIZONTAL DISTANCES AND DIFFERENCES 
 OF ELEVATION FOR STADIA MEASUREMENTS. 
 
 
 24 
 
 
 
 25 
 
 
 2( 
 
 ) 
 
 27 
 
 
 
 Minutes, 
 o' 
 
 2 
 
 Hor. 
 Dlst, 
 
 83-46 
 83.41 
 
 Diff. 
 Elev. 
 
 37.16 
 37.20 
 
 Hor. 
 Dist. 
 
 82.14 
 82.09 
 
 Diff, 
 
 Elev. 
 
 38.30 
 38.34 
 
 Hor, 
 Dist. 
 
 80.78 
 80.74 
 
 Diff. 
 Elev, 
 
 39-40 
 39.44 
 
 Hor. 
 Dist. 
 
 79-39 
 79.34 
 
 Diff. 
 EleY. 
 
 40.45 
 40.49 
 
 
 83.37 
 
 37.23 
 
 82.05 
 
 38.38 
 
 80.69 
 
 39-47 
 
 79-3O 
 
 40. 52 
 
 6 
 8 
 
 83-33 
 83.28 
 
 37-27 
 37.31 
 
 82.01 
 81.96 
 
 38.41 
 38.45 
 
 80.65 
 80.60 
 
 39-51 
 39- 54 
 
 79-25 
 79.20 
 
 40-55 
 40.59 
 
 10 
 
 12 
 
 83.24 
 83.20 
 
 37-35 
 37.39 
 
 81.92 
 
 81.87 
 
 38-49 
 38.53 
 
 80.55 
 80.51 
 
 39-58 
 39.61 
 
 79-15 
 79.11 
 
 40.62 
 40.66 
 
 14 
 
 16 
 
 18 
 
 83-15 
 83.11 
 83.07 
 
 37-43 
 37-47 
 77 e.i 
 
 81.83 
 81.78 
 81.74 
 
 38.56 
 38.60 
 38 64 
 
 80.46 
 
 80.41 
 80.37 
 
 39-65 
 39-69 
 39-72 
 
 79.06 
 79-oi 
 78.96 
 
 40.69 
 40.72 
 40.76 
 
 20 
 
 83.02 
 
 37-54 
 
 81.69 
 
 38.67 
 
 80.32 
 
 39- 76 
 
 78.92 
 
 40.79 
 
 22 
 
 82.98 
 
 37.58 
 
 81.65 
 
 38.71 
 
 80.28 
 
 39-79 
 
 78.87 
 
 40.82 
 
 24 
 
 82.93 
 
 37.62 
 
 81.60 
 
 38.75 
 
 80.23 
 
 39.83 
 
 78.82 
 
 40.86 
 
 26 
 
 82.89 
 
 37.66 
 
 81.56 
 
 38.78 
 
 80. 1 8 
 
 39.86 
 
 78.77 
 
 40.89 
 
 2 8 
 
 82.85 
 
 37.70 
 
 81.51 
 
 38.82 
 
 80.14 
 
 39-90 
 
 78.73 
 
 40.92 
 
 30 
 
 82.80 
 
 37.74 
 
 81.47 
 
 38.86 
 
 80.09 
 
 39-93 
 
 78.68 
 
 40.96 
 
 72 
 
 82.76 
 
 37-77 
 
 81.42 
 
 38.89 
 
 80.04 
 
 30.07 
 
 78.63 
 
 40.99 
 
 34 
 
 82.72 
 
 37.81 
 
 81.38 
 
 38.93 
 
 80.00 
 
 40.00 
 
 78.58 
 
 41.02 
 
 36 
 38 
 
 82.67 
 82.63 
 82 58 
 
 37.85 
 37-89 
 
 8i-33 
 81.28 
 81.24 
 
 38.97 
 39.00 
 
 79-95 
 79.90 
 79.86 
 
 40.04 
 40.07 
 
 78.54 
 78.49 
 78.44 
 
 41.06 
 41.09 
 41.12 
 
 42 
 44 
 46 
 
 82.54 
 82.49 
 82.45 
 
 37-96 
 38.00 
 38.04 
 
 81.19 
 81.15 
 81.10 
 
 39.08 
 
 39-n 
 39.15 
 
 79.81 
 79.76 
 79.72 
 
 40.14 
 40.18 
 40.21 
 
 78.39 
 78.34 
 78.30 
 
 41.16 
 41.1.9 
 41.22 
 
 48 
 
 82.41 
 
 38 08 
 
 81.06 
 
 30. 1 8 
 
 79.67 
 
 40.24 
 
 78.25 
 
 41.26 
 
 50 
 
 5 2 
 
 82.36 
 82.32 
 
 38.11 
 38 15 
 
 81.01 
 80.97 
 
 39-22 
 39.26 
 
 79.62 
 70 58 
 
 40.28 
 40.31 
 
 78.20 
 78 15 
 
 41.29 
 41.32 
 
 54 
 56 
 58 
 60 
 
 82.27 
 82.23 
 82.18 
 82 14 
 
 38.19 
 38-23 
 38.26 
 
 Og OQ 
 
 80.92 
 
 80.87 
 80.83 
 80.78 
 
 39-29 
 39-33 
 39-36 
 
 79-53 
 79.48 
 79-44 
 70-30 
 
 40-35 
 40.38 
 40.42 
 40.45 
 
 78.10 
 78.06 
 
 78.01 
 
 41-35 
 41-39 
 41.42 
 41.45 
 
 
 
 
 
 
 
 
 
 
 c .75 
 
 .68 
 
 aj 
 
 .68 
 
 3 2 
 
 .67 
 
 33 
 
 .66 
 
 35 
 
 
 
 
 
 
 
 
 
 
 c = i.oo 
 
 .91 
 
 .41 
 
 .90 
 
 43 
 
 .89 
 
 45 
 
 .89 
 
 .46 
 
 c 1.25 
 
 1.14 
 
 52 
 
 I-I3 
 
 54 
 
 1. 12 
 
 56 
 
 i. ii 
 
 58
 
 TABLES AND FORMULAS. 
 
 HORIZONTAL DISTANCES AND DIFFERENCES 
 OF ELEVATION FOR STADIA MEASUREMENTS. 
 
 
 28 
 
 
 
 29 
 
 
 
 3^ 
 
 
 Minutes. 
 
 Hor. 
 Dist. 
 
 77.96 
 
 Diff. 
 Elev. 
 
 41-45 
 
 Hor. 
 Dist. 
 
 76.50 
 
 Diff. 
 Elev. 
 
 Hor. 
 Dist. 
 
 Diff. 
 Elev. 
 
 43 3O 
 
 2 
 
 77.91 
 
 41.48 
 
 76.45 
 
 42.43 
 
 74-95 
 
 43-33 
 
 
 77.86 
 
 41. 52 
 
 76.40 
 
 42 46 
 
 74.90 
 
 43.36 
 
 6 
 8 
 
 77.81 
 
 77-77 
 77.72 
 
 41-55 
 41.58 
 41.61 
 
 76-35 
 76.30 
 76.25 
 
 42-49 
 42-53 
 
 74-85 
 74.80 
 74-75 
 
 43-39 
 43-42 
 43-45 
 
 12 
 
 77.67 
 
 41.65 
 
 76.20 
 
 42. 50 
 
 74.70 
 
 43-47 
 
 14 
 16 
 
 77.62 
 77-57 
 
 41.68 
 41.71 
 
 76-15 
 76.10 
 
 42.62 
 
 42.65 
 
 74-65 
 74.60 
 
 43-50 
 43-53 
 
 18 
 
 77-52 
 77.48 
 
 41.74 
 41.77 
 
 76.05 
 76.00 
 
 42.68 
 
 74-55 
 74-49 
 
 43.56 
 43- 59 
 
 
 77.42 
 
 41.81 
 
 75-95 
 
 
 74-44 
 
 43 62 
 
 
 77.38 
 
 41.84 
 
 75-9O 
 
 
 74-39 
 
 43.65 
 
 2 6 
 
 77-33 
 
 41.87 
 
 75.85 
 
 42 80 
 
 74-34 
 
 43.67 
 
 2 8 
 
 77.28 
 
 41.90 
 
 75.80 
 
 4 8^ 
 
 74.29 
 
 43-7 
 
 30 
 
 77-23 
 77.18 
 
 41-93 
 41.97 
 
 75-75 
 75.70 
 
 42.86 
 
 42 80 
 
 74-24 
 74.19 
 
 43-73 
 43.76 
 
 
 77.13 
 
 42.00 
 
 75-65 
 
 42 Q2 
 
 74.14 
 
 43-79 
 
 36 
 38 
 40 
 
 42 
 
 77.09 
 77.04 
 76.99 
 
 76.94 
 
 42-03 
 42.06 
 42.09 
 
 42.12 
 
 75.60 
 
 75-55 
 75-50 
 
 75-45 
 
 42.95 
 42.98 
 
 43-oi 
 43.04 
 
 74.09 
 74.04 
 73-99 
 73-93 
 
 43.82 
 43-84 
 43-87 
 43.90 
 
 44 
 
 76.89 
 
 42.15 
 
 75-4O 
 
 43-07 
 
 73.88 
 
 43-93 
 
 46 
 48 
 50 . ... 
 
 76.84 
 76.79 
 76.74 
 
 42.19 
 42.22 
 42.25 
 
 75-35 
 75-30 
 75.25 
 
 43.10 
 43-13 
 
 43- !6 
 
 73-83 
 73.78 
 73-73 
 
 43-95 
 43-gS 
 44.01 
 
 52 
 
 76.69 
 
 42.28 
 
 75.20 
 
 4T 18 
 
 73-68 
 
 44.04 
 
 54 
 56 
 58 
 
 76.64 
 
 76.59 
 76.55 
 
 42.31 
 
 42-34 
 42.37 
 
 75-15 
 75-io 
 75.05 
 
 43.21 
 43-24 
 43.27 
 
 73-63 
 
 73-58 
 73.53 
 
 44.07 
 44.09 
 44.12 
 
 60 
 
 76.50 
 
 42.40 
 
 75.00 
 
 43-30 
 
 73-47 
 
 44-15 
 
 c= -75 
 
 .66 
 
 36 
 
 65 
 
 37 
 
 65 
 
 38 
 
 c = i. oo 
 
 .88 
 
 .48 
 
 .87 
 
 49 
 
 .86 
 
 51 
 
 c = 1.25 
 
 1. 10 
 
 .60 
 
 1.09 
 
 .62 
 
 1.08 
 
 .64
 
 TABLES AND FORMULAS. 61 
 
 TABLE OF 
 
 RADII AND CHORD AND TANGENT 
 DEFLECTIONS. 
 
 The formulas used in the computation of the following 
 tables are as follows: 
 
 For Radii, R = Jj^jy. (89.) Art. 1249. 
 For Chord Deflections, 
 
 d=^>. (92.) Art. 1255. 
 For Tangent Deflections, 
 
 tan deflection = -- (93.) Art. 1255.
 
 TABLES AND FORMULAS. 63 
 
 TABLE OF RADII AND DEFLECTIONS. 
 
 
 
 
 Tan- 
 
 
 
 
 Tan- 
 
 
 
 
 Tan- 
 
 De- 
 
 Radii. 
 
 Chord 
 Deflec- 
 
 gent 
 De- 
 
 De- 
 
 Radii. 
 
 Chord 
 Deflec- 
 
 gent 
 De- 
 
 De- 
 
 Radii. 
 
 Chord 
 Deflec- 
 
 s? 
 
 gree. 
 
 
 tion. 
 
 flec- 
 
 ^ree. 
 
 
 tion. 
 
 flec- 
 
 jree. 
 
 
 tion. 
 
 flec- 
 
 
 
 
 tion. 
 
 
 
 
 tion. 
 
 
 
 
 tion. 
 
 o 5 
 
 68754.94 
 
 MS 
 
 73 
 
 5 15 
 
 091.73 
 
 9. 160 
 
 4.580 
 
 10 50 
 
 529.67 
 
 18.880 
 
 9.440 
 
 10 
 
 34377-48 
 
 .291 
 
 .145 
 
 20 
 
 074-68 
 
 9-305 
 
 653 
 
 
 
 
 
 15 
 
 20 
 
 17188.76 
 
 .582 
 
 .291 
 
 30 
 
 042.14 
 
 9-596 
 
 .798 
 
 
 513-91 
 
 19. 169 
 19.459 
 
 9-585 
 9.729 
 
 25 
 
 13751.02 
 
 .727 
 
 364 
 
 35 
 
 O26.6O 
 
 9.741 
 
 .870 
 
 20 
 
 506-38 
 
 19-748 
 
 9.874 
 
 3 
 
 11459.19 
 
 873 
 
 .436 
 
 40 
 
 OII.5I 
 
 9.886 
 
 943 
 
 30 
 
 499.06 
 
 20.038 
 
 ro.oig 
 
 35 
 
 9822.18 
 
 
 509 
 
 45 
 
 ^6.87 
 
 10.031 
 
 .016 
 
 40 
 
 491.96 
 
 20.327 
 
 10.164 
 
 40 
 
 8594.41 
 
 1:164 
 
 .582 
 
 50 
 
 Q82.6 4 
 
 10.177 
 
 .088 
 
 50 
 
 485-05 
 
 20.616 
 
 10.308 
 
 45 
 
 7639-49 
 
 1.309 
 
 .654 
 
 55 
 
 068. 8r 
 
 10.322 
 
 .161 
 
 
 
 
 
 5 
 
 6875-55 
 
 1-454 
 
 .727 
 
 
 
 
 
 12 
 
 478-34 
 
 20.906 
 
 10.453 
 
 55 
 
 6250.51 
 
 i. 600 
 
 .800 
 
 6 o 
 
 955-37 
 
 10.467 
 
 234 
 
 10 
 
 471.8! 
 
 21.195 
 
 10.597 
 
 
 
 
 
 5 
 
 942.29 
 
 10.612 
 
 .306 
 
 2O 
 
 465-46 
 
 21.484 
 
 10.742 
 
 '1 O 
 
 5729.65 
 
 1-745 
 
 873 
 
 10 
 
 929-57 
 
 10.758 
 
 379 
 
 3 
 
 459.28 
 
 21.773 
 
 ,0.887 
 
 5 
 
 5288.92 
 
 1.891 
 
 945 
 
 15 
 
 917.19 
 
 10.903 
 
 .451 
 
 40 
 
 453-26 
 
 22.063 
 
 11.031 
 
 
 4911.15 
 
 2.036 
 
 .018 
 
 
 905-13 
 
 11.048 
 
 524 
 
 5 
 
 447.40 
 
 22.352 
 
 11.176 
 
 15 
 
 4583-75 
 
 2.182 
 
 .091 
 
 25 
 
 893-39 
 
 11.193 
 
 597 
 
 
 
 
 
 20 
 
 4297.28 
 
 2.327 
 
 .164 
 
 3 
 
 881.95 
 
 "339 
 
 .669 
 
 13 o 
 
 441.68 
 
 22.641 
 
 11.320 
 
 25 
 
 4044.51 
 
 2.472 
 
 .236 
 
 35 
 
 870.79 
 
 11.484 
 
 .742 
 
 
 436.12 
 
 22. 930 
 
 11.465 
 
 3 
 
 3819.83 
 
 Xfi T Q Q 
 
 n -6-j 
 
 .309 
 
 40 
 
 859.92 
 
 11.629 
 
 
 20 
 
 430.69 
 
 23.219 
 
 11.609 
 
 35 
 40 
 
 3OIO.OO 
 3437.87 
 
 2.7O3 
 2.900 
 
 454 
 
 5 
 
 838.97 
 
 11.910 
 
 .960 
 
 40 
 
 420.23 
 
 23 796 
 
 ".$ 
 
 45 
 
 3274-I7 
 
 3-054 
 
 .527 
 
 55 
 
 828.88 
 
 12.065 
 
 .032 
 
 5 
 
 415-19 
 
 24.085 
 
 12.043 
 
 5 
 
 3125.36 
 
 3.200 
 
 .600 
 
 
 
 
 
 
 
 
 
 55 
 
 29 9.4 
 
 3-345 
 
 -673 
 
 5 
 
 809.40 
 
 '2.355 
 
 .177 
 
 
 405-47 
 
 24-663 
 
 12.331 
 
 2 O 
 
 2864.93 
 
 3.490 
 
 745 
 
 10 
 
 800.00 
 
 12.500 
 
 .250 
 
 2O 
 
 400.78 
 
 24.95I 
 
 12.476 
 
 5 
 
 275 -35 
 
 3-636 
 
 
 15 
 
 790.81 
 
 12.645 
 
 323 
 
 3 
 
 396.20 
 
 25.240 
 
 12.620 
 
 10 
 
 15 
 
 2644.58 
 2546.64 
 
 3-781 
 
 3-927 
 
 '.891 
 .963 
 
 25 
 
 781.84 
 773.07 
 
 12.790 
 12.936 
 
 '.III 
 
 50 
 
 391-72 
 387-34 
 
 25.817 
 
 
 
 2455.70 
 
 4.072 
 
 -036 
 
 3 
 
 764.49 
 
 I3 .o8i 
 
 54 
 
 
 
 
 
 25 
 
 2371.04 
 
 4.218 
 
 
 35 
 
 756.10 
 
 13.226 
 
 6.613 
 
 15 o 
 
 383-06 
 
 26. 105 
 
 13-053 
 
 3 
 
 2292.01 
 
 4-363 
 
 . 181 
 
 40 
 
 747.89 
 
 13-371 
 
 6.685 
 
 IO 
 
 378.88 
 
 26.394 
 
 '3-197 
 
 35 
 
 2218.09 
 
 4.508 
 
 254 
 
 45 
 
 739.86 
 
 13-516 
 
 6.758 
 
 20 
 
 374-79 
 
 26.682 
 
 '3-341 
 
 40 
 
 2148.79 
 
 4-654 
 
 3 2 7 
 
 50 
 
 732.01 
 
 13.661 
 
 6.831 
 
 30 
 
 370.78 
 
 26.970 
 
 13-485 
 
 45 
 
 2083.68 
 
 4-799 
 
 
 55 
 
 724-31 
 
 13.806 
 
 6.90; 
 
 4 
 
 366.86 
 
 27.258 
 
 13.629 
 
 5 
 
 2022.41 
 
 4-945 
 
 472 
 
 
 
 
 
 50 
 
 363-02 
 
 27-547 
 
 13-773 
 
 55 
 
 1964.64 
 
 5.090 
 
 545 
 
 8 o 
 
 716.78 
 
 i3-95i 
 
 6.976 
 
 
 
 
 
 
 
 
 
 5 
 
 709.40 
 
 14.096 
 
 7.048 
 
 16 o 
 
 359-26 
 
 27.835 
 
 I3-9I7 
 
 3 
 
 1910.08 
 
 5-235 
 
 .618 
 
 10 
 
 702.18 
 
 14.241 
 
 
 IO 
 
 355-59 
 
 28.123 
 
 14.061 
 
 5 
 
 1858.47 
 
 5-38i 
 
 .690 
 
 15 
 
 695.09 
 
 14-387 
 
 193 
 
 20 
 
 35L98 
 
 28. 4 II 
 
 14.205 
 
 
 1809.57 
 
 5.526 
 
 763 
 
 
 688.16 
 
 14-532 
 
 .266 
 
 30 
 
 348.45 
 
 28.699 
 
 28 Q86 
 
 14-349 
 
 *5 
 
 1703. is 
 
 1719.12 
 
 5-817 
 
 .908 
 
 3 
 
 674.69 
 
 14.822 
 
 .411 
 
 50 
 
 344-99 
 341.60 
 
 29.274 
 
 '4-493 
 14-637 
 
 2 5 
 
 1677.20 
 
 5.962 
 
 .981 
 
 35 
 
 668.15 
 
 14.967 
 
 483 
 
 
 
 
 
 3 
 
 1637.28 
 
 6.108 
 
 54 
 
 40 
 
 661.74 
 
 15-112 
 
 556 
 
 I 7 
 
 338.27 
 
 29.562 
 
 20 . 850 
 
 14.781 
 
 35 
 40 
 
 1562.88 
 
 6.398 
 
 .199 
 
 5 
 
 649.27 
 
 15.402 
 
 .701 
 
 20 
 
 331.82 
 
 30-137 
 
 15-060 
 
 45 
 
 I528.I6 
 
 6-544 
 
 .272 
 
 55 
 
 643-22 
 
 iS-547 
 
 773 
 
 3 
 
 328.68 
 
 30-425 
 
 15.212 
 
 5 
 
 1494.95 
 
 6.689 
 
 345 
 
 
 
 
 
 40 
 
 325.60 
 
 30.712 
 
 I5-356 
 
 55 
 
 1463.16 
 
 6.835 
 
 .417 
 
 9 o 
 
 637.27 
 
 15-692 
 
 .846 
 
 5 
 
 322.59 
 
 31.000 
 
 15-500 
 
 
 
 
 
 5 
 
 631.44 
 
 15-837 
 
 .918 
 
 
 
 
 
 4 
 
 1432.69 
 
 6.980 
 
 .490 
 
 
 625.7! 
 
 15-982 
 
 7.991 
 
 18 o 
 
 319.62 
 
 31.287 
 
 15.643 
 
 5 
 
 1403.46 
 
 7.125 
 
 563 
 
 15 
 
 620.09 
 
 16.127 
 
 8.063 
 
 IO 
 
 316.71 
 
 31-574 
 
 15.787 
 
 10 
 
 I 375-4 
 
 7.271 
 
 635 
 
 20 
 
 614-56 
 
 16.272 
 
 8.13) 
 
 20 
 
 313-86 
 
 31.861 
 
 15.931 
 
 15 
 
 1348.45 
 
 7.416 
 
 .708 
 
 2 5 
 
 609.14 
 
 16.417 
 
 
 30 
 
 311.06 
 
 32-149 
 
 16.074 
 
 20 
 
 1322.53 
 
 7.56! 
 
 .781 
 
 30 
 
 603.80 
 
 16.562 
 
 8.281 
 
 4 
 
 308.30 
 
 32-436 
 
 16.218 
 
 25 
 
 1297.58 
 
 7.707 
 
 853 
 
 35 
 
 598.57 
 
 16.707 
 
 8-353 
 
 5 
 
 305.60 
 
 32-723 
 
 16.361 
 
 3 
 
 '273-57 
 
 7-852 
 
 .926 
 
 40 
 
 593.42 
 
 16.852 
 
 8.426 
 
 
 
 
 
 35 
 
 1250.42 
 
 7-997 
 
 999 
 
 45 
 
 588.36 
 
 16.996 
 
 8.498 
 
 19 o 
 
 302.94 
 
 33-010 
 
 16.505 
 
 
 1228.11 
 
 8.143 
 
 .071 
 
 5 
 
 583-38 
 
 17.141 
 
 8-571 
 
 
 300.33 
 
 33-296 
 
 16.648 
 
 45 
 
 1206.57 
 
 8.288 
 
 .144 
 
 55 
 
 578.49 
 
 17.286 
 
 8.643 
 
 20 
 
 297.77 
 
 33.583 
 
 16.792 
 
 5 
 
 1185.78 
 
 8-433 
 
 
 
 
 
 
 30 
 
 295-25 
 
 33-870 
 
 i6.935 
 
 55 
 
 1165.70 
 
 8-579 
 
 [289 
 
 10 o 
 
 573.69 
 
 17-431 
 
 8.716 
 
 4 
 
 292.77 
 
 34-157 
 
 ,7-078 
 
 
 
 
 
 IO 
 
 564-31 
 
 17.721 
 
 8.860 
 
 5 
 
 290.33 
 
 34-443 
 
 17.222 
 
 5 o 
 
 1146.28 
 
 8 724 
 
 362 
 
 20 
 
 555-23 
 
 18.011 
 
 9.005 
 
 
 
 
 
 5 
 
 1127.50 
 
 s.seg 
 
 435 
 
 30 
 
 546.44 
 
 i8. 3 > 
 
 9.150 
 
 20 
 
 287.94 
 
 34-730 
 
 17-365 
 
 10 
 
 1109.33 
 
 9.014 
 
 507 
 
 40 
 
 537-92 
 
 18.500 
 
 9-295 
 
 
 

 
 64 
 
 TABLES AND FORMULAS. 
 MOMENTS OF INERTIA. 
 
 Dotted Line Shows Position 
 of Neutral Axis. 
 
 td+t.b 
 
 bd - *, 
 
 /,/ af-csf,
 
 TABLES AND FORMULAS. 65 
 
 BENDING MOMENTS AND DEFLECTIONS. 
 
 Manner of Supporting 
 Beams. 
 
 Maximum 
 
 Bending 
 
 Moment, J/. 
 
 Maximum 
 Deflection, ^. 
 
 Remarks. 
 
 OQOQQQOQOOO 
 
 Cantilever, more than 
 one load. 
 
 Cantilever, uniform 
 load w Ib. per unit 
 of length. 
 
 48 El 
 
 Cantilever, load partly 
 uniform, partly con- 
 centrated. 
 
 Simple beam, load at 
 middle. 
 
 Simple beam, load at 
 some other point 
 than the middle. 
 
 Simple beam, uni- 
 formly loaded. 
 
 One end fixed, other 
 end supported, load 
 in the middle. 
 
 One end fixed, other 
 end supported, uni- 
 formly loaded. 
 
 Both ends fixed, load 
 
 in the middle. 
 
 Both ends fixed, uni- 
 formly loaded.
 
 TABLES AND FORMULAS. 
 
 SPECIFIC GRAVITIES AND WEIGHTS 
 PER CUBIC FOOT. 
 
 METALS. 
 
 Substance. 
 
 Specific 
 Gravity. 
 
 Weight per 
 Cubic Foot 
 in Pounds. 
 
 Osmium 
 
 
 i 4.77 c 
 
 Platinum 
 
 2 1 so 
 
 I 242.8 
 
 Gold 
 
 
 i 218 8 
 
 Mercury 
 Lead (cast) 
 
 13.60 
 
 850.0 
 
 7OQ 4. 
 
 Silver 
 
 10 ^o 
 
 6s6 i 
 
 Copper (cast) 
 Brass ... 
 
 8.79 
 8 28 
 
 549-4 
 
 C 2 2 8 
 
 Wrought Iron 
 
 7 68 
 
 480. o 
 
 Cast Iron 
 
 721 
 
 4 CO O 
 
 Steel 
 
 7 84 
 
 400 o 
 
 Tin (cast) 
 
 7 20 
 
 4C c 6 
 
 Zinc (cast) 
 
 6 86 
 
 428 8 
 
 Antimony 
 
 Aluminum 
 
 6.71 
 2 so 
 
 419.4 
 1^6^ 
 
 
 
 
 WOODS. 
 
 Substance. 
 
 Specific 
 Gravity. 
 
 Weight per 
 Cubic Foot 
 in Pounds. 
 
 Ash 
 
 84C 
 
 C2 80 
 
 Beech 
 
 .8^2 
 
 C7 2C. 
 
 Cedar 
 
 e6i 
 
 T. C 06 
 
 Cork 
 
 240 
 
 i c; oo 
 
 Ebony (American) 
 
 i. 2,2,1 
 
 82.10 
 
 Lignum- vitae 
 
 I 3, 27 
 
 82. 70 
 
 Maple 
 
 .7150 
 
 46.88 
 
 Oak (old) 
 
 I I 70 
 
 72, IO 
 
 Spruce ...... 
 
 . "JOO 
 
 21. 2*1 
 
 Pine (yellow) 
 
 660 
 
 41. 2O 
 
 Pine (white) 
 
 . t ? t \4 
 
 24.6O 
 
 Walnut 
 
 .671 
 
 41.90 
 
 

 
 TABLES AND FORMULAS. 
 LIQUIDS. 
 
 67 
 
 Substance. 
 
 Specific 
 Gravity. 
 
 Weight 
 per 
 Cubic Foot 
 in Pounds. 
 
 Acetic Acid 
 
 i 062 
 
 66 4 
 
 Nitric Acid 
 
 I 217 
 
 76.1 
 
 Sulphuric Acid 
 
 1.841 
 
 1 15. i 
 
 Muriatic Acid 
 
 I 2OO 
 
 75. 
 
 Alcohol 
 
 .800 
 
 tJO.O 
 
 Turpentine 
 
 .870 
 
 54-4 
 
 Sea. \Vater (ordinary) 
 
 I.O26 
 
 64.1 
 
 Milk 
 
 1.032 
 
 64.5 
 
 
 
 
 GASES. 
 
 At 32 F., and under a Pressure of One Atmosphere. 
 
 Substance. 
 
 Specific 
 Gravity. 
 
 Weight 
 per 
 Cubic Foot 
 in Pounds. 
 
 Atmospheric Air. . . ... 
 
 I OOOO 
 
 . 08073 
 
 Carbonic Acid 
 
 I. 52QO 
 
 . 12344 
 
 Carbonic Oxide 
 
 0674. 
 
 07810 
 
 Chlorine 
 
 2 4400 
 
 . 19700 
 
 Oxygen 
 
 I 1056 
 
 0802 ^ 
 
 Nitrogen 
 
 07 ^6 
 
 .07860 
 
 Smoke (bituminous coal) 
 
 IO2O 
 
 00815 
 
 Smoke (wood) . . . 
 
 0900 
 
 . 00727 
 
 *Steam at 212 F 
 
 4.7OO 
 
 07700 
 
 Hydrogen 
 
 0602 
 
 ooc so 
 
 
 
 
 * The specific gravity of steam at any temperature and pressure com- 
 pared with air at the same temperature and pressure is 0.622.
 
 (IS 
 
 TABLES AND FORMULAS. 
 MISCELLANEOUS. 
 
 Substance. 
 
 Specific 
 Gravity. 
 
 Weight 
 per 
 Cubic Foot 
 in Pounds. 
 
 
 400 
 
 2 ?O 
 
 Glass (average) 
 
 2 80 
 
 1 7 c 
 
 Chalk 
 
 2.78 
 
 174 
 
 Granite 
 
 2.6? 
 
 166 
 
 Marble 
 
 2. 70 
 
 169 
 
 Stone (common) 
 
 2. CT2 
 
 1^8 
 
 
 217 
 
 1 1 1 
 
 Soil (common) 
 
 i. 08 
 
 1 24 
 
 Clay 
 
 1.93 
 
 121 
 
 Brick 
 
 I. QO 
 
 118 
 
 Plaster Paris (average) 
 
 2.OO 
 
 12? 
 
 Sand 
 
 I 80 
 
 11^ 
 
 
 
 
 COEFFICIENTS FOR FLOW OF WATER. 
 
 DISCHARGE OF STANDARD ORIFICES. 
 
 COEFFICIENTS FOR CIRCULAR VERTICAL ORIFICES. 
 
 Head h 
 
 Diameter of Orifice in Feet. 
 
 in Feet. 
 
 O.O2 
 
 0.04 
 
 0.07 
 
 o. 10 
 
 o. 20 
 
 0.60 
 
 I. 00 
 
 0.4 
 
 
 0.637 
 
 0.624 
 
 0.618 
 
 
 
 
 0.6 
 
 - 6 55 
 
 .630 
 
 .618 
 
 .613 
 
 0.601 
 
 Q-593 
 
 
 0.8 
 
 .648 
 
 .626 
 
 615 
 
 .610 
 
 .601 
 
 594 
 
 0.590 
 
 I.O 
 
 644 
 
 .623 
 
 .612 
 
 .608 
 
 .600 
 
 595 
 
 591 
 
 i-5 
 
 637 
 
 .618 
 
 .608 
 
 .605 
 
 .600 
 
 59 6 
 
 593 
 
 2.0 
 
 .632 
 
 .614 
 
 .607 
 
 .604 
 
 599 
 
 597 
 
 595 
 
 2 -5 
 
 .629 
 
 .612 
 
 .605 
 
 .603 
 
 599 
 
 .598 
 
 59 6 
 
 3- 
 
 .627 
 
 .611 
 
 .604 
 
 .603 
 
 599 
 
 598 
 
 597 
 
 4.0 
 
 .623 
 
 .609 
 
 603 
 
 .602 
 
 599 
 
 597 
 
 59 6 
 
 6.0 
 
 .6.18 
 
 .607 
 
 .602 
 
 .600 
 
 598 
 
 597 
 
 59 6 
 
 8.0 
 
 .614 
 
 .605 
 
 .601 
 
 .600 
 
 598 
 
 .596 
 
 59 6 
 
 IO.O 
 
 .611 
 
 .603 
 
 599 
 
 598 
 
 597 
 
 59 6 
 
 595 
 
 20. o 
 
 .601 
 
 599 
 
 597 
 
 596 
 
 59 6 
 
 59 6 
 
 594 
 
 50.0 
 
 59 6 
 
 595 
 
 594 
 
 594 
 
 594 
 
 594 
 
 593 
 
 100. 
 
 593 
 
 592 
 
 592 
 
 592 
 
 592 
 
 592 
 
 592
 
 TABLES AND FORMULAS. 
 
 69 
 
 COEFFICIENTS FOR SQUARE VERTICAL ORIFICES. 
 
 Head // 
 in Feet. 
 
 Side of the Square in Feet. 
 
 O.O2 
 
 0.04 
 
 0.07 
 
 O. 10 
 
 o. 20 
 
 0.60 
 
 I.OO 
 
 0.4 
 
 
 0.643 
 
 0.628 
 
 0.621 
 
 
 
 
 0.6 
 
 0.660 
 
 .636 
 
 .623 
 
 .617 
 
 0.605 
 
 0.598 
 
 
 0.8 
 
 .652 
 
 .631 
 
 .620 
 
 .615 
 
 .605 
 
 .600 
 
 -597 
 
 I.O 
 
 .648 
 
 .628 
 
 .618 
 
 .613 
 
 .605 
 
 .601 
 
 599 
 
 i-5 
 
 .641 
 
 .622 
 
 .614 
 
 .610 
 
 .605 
 
 .602 
 
 .601 
 
 2.O 
 
 .637 
 
 .619 
 
 .612 
 
 .608 
 
 .605 
 
 .604 
 
 .602 
 
 2-5 
 
 .6 3 4 
 
 .617 
 
 .610 
 
 .607 
 
 .605 
 
 .604 
 
 .602 
 
 3- 
 
 .632 
 
 .616 
 
 .609 
 
 .607 
 
 .605 
 
 .604 
 
 .603 
 
 4.0 
 
 .628 
 
 .614 
 
 .608 
 
 .606 
 
 .605 
 
 .603 
 
 .602 
 
 6.0 
 
 .623 
 
 .612 
 
 .607 
 
 .605 
 
 .604 
 
 .603 
 
 .602 
 
 8.0 
 
 .619 
 
 .610 
 
 .606 
 
 .605 
 
 .604 
 
 .603 
 
 .602 
 
 IO.O 
 
 .6l6 
 
 .608 
 
 .605 
 
 .604 
 
 .603 
 
 .602 
 
 .601 
 
 20.0 
 
 .606 
 
 .604 
 
 .602 
 
 .602 
 
 .602 
 
 .601 
 
 .600 
 
 5O.O 
 
 .6O2 
 
 .601 
 
 .601 
 
 .600 
 
 .600 
 
 599 
 
 599 
 
 IOO.O 
 
 599 
 
 .598 
 
 598 
 
 598 
 
 598 
 
 598 
 
 598 
 
 COEFFICIENTS FOR RECTANGULAR ORIFICES 
 1 FOOT WIDE. 
 
 Head // 
 on Center 
 of Orifice 
 in Feet. 
 
 Depth of Orifice in Feet. 
 
 o. 125 
 
 0.25 
 
 o. 50 
 
 0.75 
 
 I.OO 
 
 1.50 
 
 2.OO 
 
 0.4 
 
 0.634 
 
 0.633 
 
 0.622 
 
 
 
 
 
 0.6 
 
 .633 
 
 .633 
 
 .619 
 
 0.614 
 
 
 
 
 0.8 
 
 .633 
 
 633 
 
 .618 
 
 .612 
 
 0.608 
 
 
 
 I.O 
 
 .632 
 
 .632 
 
 .618 
 
 .612 
 
 .606 
 
 0.626 
 
 
 J -5 
 
 .630 
 
 .631 
 
 .618 
 
 .611 
 
 .605 
 
 .626 
 
 0.628 
 
 2.0 
 
 .629 
 
 .630 
 
 .617 
 
 .611 
 
 .605 
 
 .624 
 
 .630 
 
 2 -5 
 
 .628 
 
 .628 
 
 .616 
 
 .611 
 
 .605 
 
 .616 
 
 .627 
 
 3- 
 
 .627 
 
 .627 
 
 .615 
 
 .610 
 
 .605 
 
 .614 
 
 .619 
 
 4.0 
 
 .624 
 
 .624 
 
 .614 
 
 .609 
 
 .605 
 
 .612 
 
 .6l6 
 
 6.0 
 
 .615 
 
 .615 
 
 .609 
 
 .604 
 
 .602 
 
 .606 
 
 .6lO 
 
 8.0 
 
 .609 
 
 .607 
 
 .603 
 
 .602 
 
 .601 
 
 .602 
 
 .604 
 
 IO.O 
 
 .606 
 
 .603 
 
 .601 
 
 .601 
 
 .601 
 
 .601 
 
 .602 
 
 20.0 
 
 
 
 
 .601 
 
 .601 
 
 .601 
 
 .602
 
 70 
 
 TABLES AND FORMULAS. 
 
 DISCHARGE OF WEIRS. 
 
 COEFFICIENTS FOR WEIRS WITH END CONTRACTIONS. 
 
 Effective 
 Head in 
 Feet. 
 
 Length of Weir in Feet. 
 
 0.66 
 
 i 
 
 2 
 
 3 
 
 5 
 
 10 
 
 19 
 
 O.I 
 
 0.632 
 
 0.639 
 
 0.646 
 
 0.652 
 
 - 6 53 
 
 - 6 55 
 
 0.656 
 
 0.15 
 
 .619 
 
 .625 
 
 .634 
 
 .638 
 
 .640 
 
 .641 
 
 .642 
 
 o. 20 
 
 .611 
 
 .618 
 
 .626 
 
 .630 
 
 .631 
 
 633 
 
 634 
 
 0.25 
 
 .605 
 
 .612 
 
 .621 
 
 .624 
 
 .626 
 
 .628 
 
 .629 
 
 0.30 
 
 .601 
 
 .608 
 
 .6l6 
 
 .619 
 
 .621 
 
 .624 
 
 .625 
 
 0.40 
 
 595 
 
 .601 
 
 .609 
 
 .613 
 
 .615 
 
 .618 
 
 .620 
 
 0.50 
 
 59 
 
 -59 6 
 
 .605 
 
 .608 
 
 .611 
 
 .615 
 
 .617 
 
 0.60 
 
 587 
 
 593 
 
 .6oi 
 
 .605 
 
 .608 
 
 .613 
 
 615 
 
 0.70 
 
 
 59 
 
 598 
 
 .603 
 
 .606 
 
 .612 
 
 .614 
 
 0.80 
 
 
 
 595 
 
 .600 
 
 .604 
 
 .611 
 
 .613 
 
 0.90 
 
 
 
 592 
 
 598 
 
 .603 
 
 .609 
 
 .612 
 
 I.OO 
 
 
 
 59 
 
 595 
 
 .601 
 
 .608 
 
 .611 
 
 I. 2 
 
 
 
 585 
 
 591 
 
 597 
 
 .605 
 
 .610 
 
 1.4 
 
 
 
 .580 
 
 587 
 
 594 
 
 .602 
 
 .609 
 
 1.6 
 
 
 
 
 .582 
 
 59 1 
 
 .600 
 
 .607 
 
 NOTE. The head given is the effective head, 
 velocity of approach is small, h is neglected. 
 
 When the 
 
 COEFFICIENTS FOR WEIRS WITHOUT END CONTRACTIONS. 
 
 Effective 
 
 Length of Weir in Feet. 
 
 neau irr 
 Feet. 
 
 1 9 
 
 IO 
 
 7 
 
 5 
 
 4 
 
 3 
 
 2 
 
 o. 10 
 
 0.657 
 
 0.658 
 
 0.658 
 
 0.659 
 
 
 
 
 0.15 
 
 643 
 
 .644 
 
 .645 
 
 645 
 
 0.647 
 
 0.649 
 
 0.652 
 
 o. 20 
 
 .635 
 
 .637 
 
 637 
 
 .638 
 
 .641 
 
 .642 
 
 645 
 
 0.25 
 
 .630 
 
 .632 
 
 .633 
 
 634 
 
 .636 
 
 .638 
 
 .6 4 I 
 
 0.30 
 
 .626 
 
 .628 
 
 .629 
 
 .631 
 
 633 
 
 .636 
 
 639 
 
 0.40 
 
 .621 
 
 .623 
 
 625 
 
 .628 
 
 .630 
 
 .633 
 
 .636 
 
 0.50 
 
 .619 
 
 .621 
 
 .624 
 
 .627 
 
 .630 
 
 633 
 
 .637 
 
 0.60 
 
 .618 
 
 .620 
 
 .623 
 
 .627 
 
 .630 
 
 634 
 
 .638 
 
 0.70 
 
 .618 
 
 .620 
 
 .624 
 
 .628 
 
 .631 
 
 635 
 
 .640 
 
 0.80 
 
 .618 
 
 .621 
 
 .625 
 
 .629 
 
 633 
 
 .637 
 
 643 
 
 0.90 
 
 .619 
 
 .622 
 
 .627 
 
 631 
 
 .635 
 
 6 39 
 
 .645 
 
 I.OO 
 
 .619 
 
 .624 
 
 .628 
 
 633 
 
 .637 
 
 .641 
 
 .648 
 
 I. 2 
 
 .620 
 
 .626 
 
 .632 
 
 .636 
 
 .641 
 
 .646 
 
 
 1.4 
 
 .622 
 
 .629 
 
 634 
 
 .640 
 
 .644 
 
 
 
 1.6 
 
 .623 
 
 .631 
 
 .637 
 
 .642 
 
 .647 
 
 
 
 NOTE. The head given is the effective head, //+-<//. When the 
 velocity of approach is small, h may be neglected.
 
 TABLES AND FORMULAS. 
 
 71 
 
 
 ffi 
 
 5 
 
 
 o o o o o o o o o o o o o o o o o o 
 
 OO-t-MCNwi-iO oco co" t^o" m -t^co M M 
 
 o o o o o o o o o o o o o o o o o o o 
 
 ini^coor^mmcoOcot^-or^OcoOOOOO 
 
 oooooooooooooooooooo 
 o o o o o o o o o o o o o o o o o o o o 
 o^ooooooooooooooooooooo 
 
 I 
 
 1 
 
 COC^C^dCIWC^MWC'IC^C^MMMh-IMMI-IMI-IM 
 
 j OOOOOOOOOOOOOOOOOOOOOO 
 
 o 
 
 lr>Mco'o > u->in^f-t-cON M ? O C?cc r^o in -r co co M 
 
 o* c? o o o o o q q q q q q q q q q q q q q q 
 
 q q q o o o o o q q o o q o q o o q q q q q 
 
 ? P^ CO CO g"^ g"g 'gg'g'g'gggg 2" "2 M"M w 1 M" 
 
 mOQmOOOinOmincocoOcoOOOOOOO 
 O ^00 in4?i Ocoo m-tcoco<N w 5 Oco r--O vn 
 
 o > ooo 1 o > o'o i o > ooqqqqqqooqqqq 
 
 O >n co co'o CO O CO Q co O O 
 
 S qqqi-ii-iwMcoinocoqconoqin__ 
 
 | ., II II II II II II II II II II II II II II H II II II II II H H 
 8 ^^'-^^---^coocjocoo^oo^co^^
 
 7-2 
 
 TABLES AND FORMULAS. 
 
 COEFFICIENTS FOR ANGULAR BENDS. 
 
 a" = angle of bend in degrees. 
 
 a = 
 
 10 
 .017 
 
 20 
 .046 
 
 40 
 139 
 
 60 
 3 6 4 
 
 80 
 74 
 
 90 
 .984 
 
 1 00 
 1.26 
 
 I IO 
 
 1.56 
 
 T2O 
 
 1.86 
 
 130 
 2.16 
 
 140 
 2-43 
 
 *5 
 
 C 
 
 2.81 
 
 COEFFICIENTS FOR CIRCULAR BENDS. 
 
 r = radius of pipe. R = radius of bend. 
 
 r 
 -R- 
 
 . i 
 
 . 2 
 
 3 
 
 4 
 
 5 
 
 .6 
 
 7 
 
 .8 
 
 9 
 
 i .0 
 
 c' = 
 
 I3 1 
 
 .138 
 
 158 
 
 .206 
 
 .294 
 
 .440 
 
 .661 
 
 977 
 
 i . 408 
 
 1.978 
 
 COEFFICIENTS FOR DARCY'S FORMULA. 
 
 Diameter 
 of Pipe 
 in Inches. 
 
 Coefficients for 
 Rough Pipes. 
 
 Coefficients for 
 Smooth Pipes. 
 
 3 
 
 0.00080 
 
 0.00040 
 
 4 
 
 .00076 
 
 .00038 
 
 6 
 
 .00072 
 
 .00036 
 
 8 
 
 .00068 
 
 .00034 
 
 10 
 
 .00066 
 
 .00033 
 
 12 
 
 .00066 
 
 .00033 
 
 14 
 
 .00065 
 
 .00033 
 
 16 
 
 .00064 
 
 .00032 
 
 24 
 
 .00064 
 
 .00032 
 
 3 
 
 .00063 
 
 .00032 
 
 36 
 
 .00062 
 
 .00031 
 
 48 
 
 .00062 
 
 .00031
 
 TABLES AND FORMULAS. 
 
 7:5 
 
 THE PROPERTIES OF SATURATED 
 STEAM. 
 
 c 
 
 'S 
 
 Quantities of Heat in British 
 Thermal Units. 
 
 "o 
 
 Volume. 
 
 3 "~J 
 
 ,q 
 
 
 I-S 
 
 
 V & n " 
 
 "3 
 
 N 
 
 S 
 
 S^ d 
 
 rt S3 
 
 ^ 
 
 "c3 "*"* O 
 
 ** 
 
 CO 
 
 o 
 
 S 
 
 S ^ '"^ 
 
 o- 
 
 S 
 
 ti^? 
 
 S . 
 
 <D 
 > 
 
 13 o 
 
 OQ w 
 
 % g 
 
 If 
 
 II 
 
 Sfis 
 
 !-> O 
 
 ts 
 
 
 
 03 
 
 *fl 
 
 u 
 
 5*S| 
 
 II 
 
 s, 
 
 s 
 
 0} 
 
 !|| 
 
 ' C -4- 
 
 9 
 
 S 
 HI 
 "3 
 
 'S 
 
 1 
 
 Pi 
 
 ~ 
 
 
 g^ 1 " 
 
 i 
 
 I 
 
 ^ 
 
 
 
 P^ c5"o 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 / 
 
 t 
 
 
 
 L 
 
 H 
 
 IP 
 
 F 
 
 * 
 
 I 
 
 IO2.OI8 
 
 70.040 
 
 1043.015 
 
 "I3-055 
 
 .003027 
 
 330.4 
 
 20623 
 
 2 
 
 I26.3O2 
 
 94.368 
 
 1026.094 
 
 1120.462 
 
 .005818 
 
 171.9 
 
 10730 
 
 3 
 
 141.654 
 
 109.764 
 
 1015.380 
 
 1125.144 
 
 .008522 
 
 H7-3 
 
 7325 
 
 4 
 
 153.122 
 
 121. 271 
 
 1007.370 
 
 1128.641 
 
 .011172 
 
 89.51 
 
 5588 
 
 5 
 
 162.370 
 
 I3 -563 
 
 1000.899 
 
 1131.462 
 
 .013781 
 
 72.56 
 
 453 
 
 6 
 
 170.173 
 
 138.401 
 
 995-441 
 
 1133.842 
 
 .016357 
 
 61.14 
 
 3816 
 
 7 
 
 176.945 
 
 I45-2I3 
 
 990.695 
 
 1135.908 .018908 
 
 52.89 
 
 332 
 
 8 
 
 182.952 
 
 151.253 
 
 986.485 
 
 1137.740.021436 
 
 46.65 
 
 2912 
 
 9 
 
 188.357 
 
 156. 699 
 
 982.690 
 
 1 139.389 
 
 .023944 
 
 41-77 
 
 2607 
 
 10 
 
 193.284 
 
 l6l.66o 
 
 979.232 
 
 1140.892 
 
 .026437 
 
 37.83 
 
 2361 
 
 1 1 
 
 197.814 
 
 166. 225 
 
 976.050 
 
 1142.275 
 
 .02891 i 
 
 34-59 
 
 2159 
 
 12 
 
 2O2.OI2 
 
 170.457 
 
 973.098 
 
 "43-555 
 
 .031376 
 
 31-87 
 
 1990 
 
 13 
 
 205.929 
 
 174.402 
 
 970.346 
 
 1144.748 
 
 .033828 
 
 29-56 
 
 1845 
 
 14 
 
 209.604 
 
 178.112 
 
 967-757 
 
 1145.869 
 
 .036265 
 
 27-58 
 
 1721 
 
 14.69 
 
 2I2.OOO 
 
 180.531 
 
 966.069 
 
 1 146.600 
 
 .037928 
 
 26.37 
 
 1646 
 
 15 
 
 213.067 
 
 181.608 
 
 965.318 
 
 1 146.926 
 
 .038688 
 
 25-85 
 
 1614 
 
 16 
 
 216.347 
 
 184.919 
 
 963.007 
 
 1 147.926 
 
 .041109 
 
 24-33 
 
 15*9 
 
 17 
 
 219.452 
 
 188.056 
 
 960.818 
 
 1148.874 
 
 0435 T 9 
 
 22.98 
 
 1434 
 
 1 8 
 
 222.424 
 
 191.058 
 
 958.721 
 
 1149.779 
 
 .045920 
 
 21.78 
 
 1359 
 
 19 
 
 225.255 
 
 193.918 
 
 956.725 
 
 1150.643 
 
 .048312 
 
 20.70 
 
 1292
 
 TABLES AND FORMULAS. 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 p 
 
 / 
 
 9 
 
 L 
 
 // 
 
 w 
 
 V 
 
 R 
 
 20 
 
 227.964 
 
 196.655 
 
 954.814 
 
 1 151.469 
 
 .050696 
 
 19-73 
 
 1231.0 
 
 22 
 
 233.069 
 
 201.817 
 
 951.209 
 
 1153.026 
 
 055446 
 
 18.04 
 
 1126.0 
 
 24 
 
 2 3 7 .80 3 
 
 206.610 
 
 947.861 
 
 1154.471 
 
 .060171 
 
 16.62 
 
 1038.0 
 
 26 
 
 242.225 
 
 21 1.089 
 
 944-73 
 
 1155.819 
 
 .064870 
 
 15-42 
 
 962.3 
 
 28 
 
 246.376 
 
 215.293 
 
 941.791 
 
 1157.084 
 
 069545 
 
 14.38 
 
 897.6 
 
 3 
 
 250.293 
 
 219.261 
 
 939.019 
 
 1158.280 
 
 .074201 
 
 13.48 
 
 841.3 
 
 32 
 
 254.002 
 
 223.021 
 
 936.389 
 
 1159.410 
 
 .078839 
 
 12.68 
 
 791.8 
 
 34 
 
 257.5 2 3 
 
 226.594 
 
 933-891 
 
 1 160.485 
 
 .083461 
 
 11.98 
 
 748.0 
 
 36 
 
 260.883 
 
 23O.OOI 
 
 931.508 
 
 1 161.509 
 
 .088067 
 
 11.36 
 
 708.8 
 
 33 
 
 264.093 
 
 233-26I 
 
 929.227 
 
 1162.488 
 
 .092657 
 
 10.79 
 
 673-7 
 
 40 
 
 267.168 
 
 236.386 
 
 927.040 
 
 1 163.426 
 
 .097231 
 
 10.28 
 
 642.0 
 
 42 
 
 270. 122 
 
 239-389 
 
 924.940 
 
 1164.329 
 
 .101794 
 
 9.826 
 
 6i3-3 
 
 44 
 
 272.965 
 
 242.275 
 
 922.919 
 
 1165.194 
 
 .106345 
 
 9-403 
 
 587.0 
 
 46 
 
 275.704 
 
 245.061 
 
 920.968 
 
 1166.029 
 
 .110884 
 
 9.018 
 
 563-0 
 
 48 
 
 278.348 
 
 247.752 
 
 919.084 
 
 1166.836 
 
 .115411 
 
 8.665 
 
 540-9 
 
 5 
 
 280.904 
 
 25 -355 
 
 917.260 
 
 1 167.615 
 
 .119927 
 
 8.338 
 
 520.5 
 
 5 2 
 
 283.381 
 
 252.875 
 
 9*5-494 
 
 1168.369 
 
 124433 
 
 8.037 
 
 5 OI -7 
 
 54 
 
 285.781 
 
 255-32I 
 
 913.781 
 
 1169.102 
 
 .128928 
 
 7-756 
 
 484.2 
 
 56 
 
 288.III 
 
 257-695 
 
 912.118 
 
 1169.813 
 
 133414 
 
 7.496 
 
 467.9 
 
 58 
 
 290.374 
 
 26O.OO2 
 
 910.501 
 
 1170.503 
 
 .137892 
 
 7.252 
 
 452.7 
 
 60 
 
 292-575 
 
 262.248 
 
 908.928 
 
 1171. 176 
 
 .142362 
 
 7.024 
 
 438.5 
 
 62 
 
 294.717 
 
 264.433 
 
 907.396 
 
 1171.829 
 
 .146824 
 
 6.811 
 
 425-2 
 
 64 
 
 296.805 
 
 266.566 
 
 905.900 
 
 1 172.466 
 
 .151277 
 
 6.610 
 
 412.6 
 
 66 
 
 298.842 
 
 268.644 
 
 904.443 
 
 1173.087 
 
 155721 
 
 6.422 
 
 400.8 
 
 68 
 
 300.831 
 
 270.674 
 
 903.020 
 
 1173.694 
 
 . 160157 
 
 6.244 
 
 389.8 
 
 70 
 
 302.774 
 
 272.657 
 
 901.629 
 
 1174.286 
 
 .164584 
 
 6.076 
 
 379-3 
 
 72 
 
 304.669 
 
 274-597 
 
 900. 269 
 
 1174.866 
 
 .169003 
 
 5-9 1 ? 
 
 369-4 
 
 74 
 
 306.526 
 
 276.493 
 
 898.938 
 
 H75-43 1 
 
 173417 
 
 5-767 
 
 360.0 
 
 76 
 
 308.344 
 
 278.350 
 
 897-635 
 
 1175-985 
 
 .177825 
 
 5.624 
 
 35 1 - 1 
 
 78 
 
 310.123 
 
 280.' 170 
 
 896.359 
 
 1176.529 
 
 . 182229 
 
 5.488 
 
 342.6 
 
 80 
 
 311.866 
 
 281.952 
 
 895.108 
 
 1177.060 
 
 .186627 
 
 5-358 
 
 334-5 
 
 82 
 
 3I3-576 
 
 283. 701 
 
 893.879 
 
 1177.580 
 
 . 191017 
 
 5-235 
 
 326.8 
 
 84 
 
 3 I 5- 2 5 
 
 285.414 
 
 892.677 
 
 1 1.78.091 
 
 .195401 
 
 5.118 
 
 3 r 9-5 
 
 86 
 
 316.893 
 
 287.096 
 
 891.496 
 
 1178.592 
 
 .199781 
 
 5.006 
 
 3 I2 -5 
 
 88 
 
 318.510 
 
 288.750 
 
 890-335 
 
 1179.085 
 
 204155 
 
 4.898 
 
 305-8
 
 TABLES AND FORMULAS. 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 p 
 
 t 
 
 f 
 
 L 
 
 H 
 
 W 
 
 V 
 
 R 
 
 90 
 
 320.094 
 
 29 -373 
 
 889.196 
 
 1179.569 
 
 . 208525 
 
 4.796 
 
 299-4 
 
 92 
 
 3 2I - 6 53 
 
 291.970 
 
 888.075 
 
 1 180.045 
 
 .212892 
 
 4.697 
 
 293.2 
 
 94 
 
 3 2 3- 18 3 
 
 293-539 
 
 886.972 
 
 1 180.511 
 
 .217253 
 
 4.603 
 
 287.3 
 
 96 
 
 324.688 
 
 295.083 
 
 885.887 
 
 1180.970 
 
 .221604 
 
 4.513 
 
 281.7 
 
 98 
 
 326. 169 
 
 296.601 
 
 884.821 
 
 1 181.422 
 
 225950 
 
 4.426 
 
 276.3 
 
 100 
 
 327.625 
 
 298.093 
 
 883.773 
 
 1181.866 
 
 -230293 
 
 4-342 
 
 271.1 
 
 I0 5 
 
 331.169 
 
 301.731 
 
 881.214 
 
 1 182.945 
 
 .241139 
 
 4-147 
 
 258.9 
 
 I IO 
 
 334-5 82 
 
 305.242 
 
 878.744 
 
 1183.986 
 
 .251947 
 
 3-969 
 
 247.8 
 
 1*5 
 
 337-874 
 
 308.621 
 
 876.371 
 
 1184.992 
 
 .262732 
 
 3.806 
 
 237-6 
 
 120 
 
 341.058 
 
 311.885 
 
 874.076 
 
 1185.961 
 
 .273500 
 
 3-656 
 
 228.3 
 
 125 
 
 344-I3 6 
 
 3i5-5* 
 
 871.848 
 
 1186.899 
 
 .284243 
 
 3-518 
 
 219.6 
 
 130 
 
 347.121 
 
 318.121 
 
 869.688 
 
 1187.809 
 
 .294961 
 
 3-390 
 
 21 1.6 
 
 135 
 
 35- OI 5 
 
 321.105 
 
 867.590 
 
 1188.695 
 
 305659 
 
 3.272 
 
 204.2 
 
 140 
 
 352-827 
 
 324.003 
 
 865.552 
 
 Il8 9-555 
 
 316338 
 
 3- J 6i 
 
 T 97-3 
 
 !45 
 
 355-5 6 2 
 
 326.823 
 
 863.567 
 
 i 190.390 
 
 .326998 
 
 3-058 
 
 190.9 
 
 '5 
 
 358-223 
 
 329.566 
 
 861.634 
 
 1191.200 
 
 337643 
 
 2.962 
 
 184.9 
 
 1 60 
 
 363-346 
 
 334-850 
 
 857.912 
 
 1192.762 
 
 .358886 
 
 .786 
 
 J 73-9 
 
 170 
 
 368.226 
 
 339-892 
 
 854-359 
 
 1194.251 
 
 .380071 
 
 .631 
 
 164.3 
 
 180 
 
 372.886 
 
 344.708 
 
 850.963 
 
 1195.671 
 
 .401201 
 
 493 
 
 'SS- 6 
 
 190 
 
 377-352 
 
 349-329 
 
 847.703 
 
 1197.032 
 
 .422280 
 
 .368 
 
 147.8 
 
 200 
 
 381.636 
 
 353.766 
 
 844.573 
 
 II 9 8 -339 
 
 4433 10 
 
 .256 
 
 140.8 
 
 2IO 
 
 385.759 
 
 358.041 
 
 841.556 
 
 IJ 99-597 
 
 464295 
 
 .154 
 
 J34-5 
 
 22O 
 
 389-736 
 
 362.168 
 
 838.642 
 
 1200.810 
 
 485237 
 
 .061 
 
 128.7 
 
 2.30 
 
 393-575 
 
 366. 152 
 
 835.828 
 
 1201.980 
 
 .506139 
 
 .976 
 
 123-3 
 
 2 4 
 
 397-285 
 
 370.008 
 
 833- 10 3 
 
 1203. in 
 
 527003 
 
 .898 
 
 118.5 
 
 250 
 
 400.883 
 
 373-75 
 
 830.459 
 
 1204. 209 
 
 .547831 
 
 .825 
 
 1 14.0 
 
 260 
 
 404.370 
 
 377-377 
 
 827.896 
 
 1205.273 
 
 .568626 
 
 -759 
 
 109.8 
 
 270 
 
 407.755 
 
 380.905 
 
 825.401 
 
 1206.306 
 
 58939 
 
 697 
 
 105.9 
 
 280 
 
 41 1.048 
 
 384.337 
 
 822.973 
 
 1207.310 
 
 .610124 
 
 639 
 
 102.3 
 
 290 
 
 414.250 
 
 387.677 
 
 820.609 
 
 1208.286 
 
 .630829 
 
 585 
 
 99.0 
 
 3 00 
 
 4i7-37i 
 
 390-933 
 
 818.305 
 
 1209. 238 
 
 .651506 
 
 -535 
 
 95- 8
 
 TABLES AND FORMULAS 
 
 MISCELLANEOUS TABLES. 
 
 STANDARD DIMENSIONS OF WROUGHT-IRON 
 STEAM, GAS, AND WATER PIPES. 
 
 Nominal 
 Diameter 
 in Inches. 
 
 Thickness 
 in Inches. 
 
 Actual 
 Internal 
 Diameter 
 in Inches. 
 
 Actual 
 External 
 Diameter 
 in Inches. 
 
 Threads 
 per Inch. 
 
 H 
 
 Pitch of 
 Threads. 
 
 I 
 
 .068 
 
 .270 
 
 45 
 
 27 
 
 37 
 
 i 
 
 .088 
 
 364 
 
 54 
 
 18 
 
 .056 
 
 1 
 
 .091 
 
 494 
 
 675 
 
 18 
 
 .056 
 
 t 
 
 .109 
 
 .623 
 
 .840 
 
 14 
 
 .071 
 
 1 
 
 113 
 
 .824 
 
 1.050 
 
 14 
 
 .071 
 
 i 
 
 134 
 
 1.048 
 
 I.3I5 
 
 "i 
 
 .087 
 
 I| 
 
 .140 
 
 1.380 
 
 i. 660 
 
 "i 
 
 .087 
 
 4 
 
 145 
 
 1.61 1 
 
 1.900 
 
 * 
 
 .087 
 
 2 
 
 154 
 
 2.067 
 
 2-375 
 
 i 
 
 .087 
 
 * 
 
 .204 
 
 2.468 
 
 2-875 
 
 8 
 
 .125 
 
 3 
 
 .217 
 
 3.061 
 
 3-5 
 
 8 
 
 .125 
 
 3i 
 
 .226 
 
 3-548 
 
 4.000 
 
 8 
 
 125 
 
 4 
 
 237 
 
 4.026 
 
 4.500 
 
 8 
 
 I2 5 
 
 4l 
 
 .247 
 
 4.508 
 
 5.000 
 
 8 
 
 '25 
 
 5 
 
 2 59 
 
 5-45 
 
 5-5 6 3 
 
 8 
 
 I2 5 
 
 6 
 
 .280' 
 
 6.065 
 
 6.625 
 
 8 
 
 .125 
 
 7 
 
 .301 
 
 7.023 
 
 7.625 
 
 8 
 
 .125 
 
 8 
 
 .322 
 
 7.982 
 
 8.625 
 
 8 
 
 - I2 5 
 
 9 
 
 344 
 
 9.001 
 
 9.688 
 
 8 
 
 .125 
 
 10 
 
 .366 
 
 10.019 
 
 10.750 
 
 8 
 
 .125
 
 TABLES AND FORMULAS. 
 
 STANDARD PIPE FLANGES. 
 
 Inside 
 Diam. 
 of 
 Pipe. 
 
 Thick- 
 ness of 
 Pipe. 
 
 Diam. 
 of 
 Bolts. 
 
 Length 
 of 
 Bolts. 
 
 No. of 
 Bolts. 
 
 Thick- 
 ness of 
 Flange. 
 
 Diam. of 
 Bolt 
 
 Circle. 
 
 Diam. 
 of 
 Flange. 
 
 2.O 
 
 .409 
 
 f 
 
 2.O 
 
 4 
 
 f 
 
 4-75 
 
 6.0 
 
 2 -5 
 
 .429 
 
 | 
 
 2.25 
 
 4 
 
 H 
 
 5-25 
 
 7.0 
 
 3- 
 
 .448 
 
 f 
 
 2-5 
 
 4 
 
 t 
 
 6.0 
 
 7-5 
 
 3-5 
 
 .466 
 
 f 
 
 2-5 
 
 4 
 
 H 
 
 6-5 
 
 8-5 
 
 4.0 
 
 .486 
 
 1 
 
 2-75 
 
 4 
 
 H 
 
 7-25 
 
 9.0 
 
 4-5 
 
 .498 
 
 f 
 
 3- 
 
 8 
 
 H 
 
 7-75 
 
 9-25 
 
 5 
 
 5 2 5 
 
 f 
 
 3-o 
 
 8 
 
 H 
 
 8-5 
 
 IO.O 
 
 6 
 
 -563 
 
 f 
 
 3- 
 
 8 
 
 i 
 
 9.625 
 
 I I.O 
 
 7 
 
 .600 
 
 t 
 
 3-25 
 
 8 
 
 *A 
 
 IO -75 
 
 I2 -5 
 
 8 
 
 639 
 
 f 
 
 3-5 
 
 8 
 
 4 
 
 11-75 
 
 13-5 
 
 9 
 
 .678 
 
 1 
 
 3-5 
 
 I 2 
 
 '* ' 
 
 13.0 
 
 'S- 
 
 10 
 
 713 
 
 1 
 
 3-625 
 
 12 
 
 ' ft 
 
 14-25 
 
 16.0 
 
 12 
 
 79 
 
 1 
 
 3-75 
 
 12 
 
 '1 
 
 16.5 
 
 19.0 
 
 14 
 
 .864 
 
 i 
 
 4-25 
 
 12 
 
 if 
 
 18.75 
 
 21.0 
 
 J 5 
 
 .904 
 
 i 
 
 4-25 
 
 16 
 
 4 
 
 20. o 
 
 22.25 
 
 16 
 
 .946 
 
 i 
 
 4-25 
 
 16 
 
 'A 
 
 21.25 
 
 23-5 
 
 18 
 
 i. 020 
 
 I* 
 
 4-75 
 
 16 
 
 ^ 
 
 22.75 
 
 25.0 
 
 20 
 
 1.090 
 
 *i 
 
 5- 
 
 20 
 
 'tt 
 
 25.0 
 
 27-5 
 
 22 
 
 1.180 
 
 'i 
 
 5-5 
 
 20 
 
 H 
 
 27-25 
 
 29-5 
 
 2 4 
 
 1.250 
 
 l 
 
 5-5 
 
 20 
 
 l 
 
 29-5 
 
 32.0 
 
 26 
 
 1.300 
 
 * 
 
 5-75 
 
 24 
 
 2 
 
 3i-75 
 
 34-25 
 
 28 
 
 1.380 
 
 i 
 
 6.0 
 
 28 
 
 *A 
 
 34-Q 
 
 36-5 
 
 3 
 
 1.480 
 
 If 
 
 6.25 
 
 28 
 
 2| 
 
 36.0 
 
 38.75 
 
 36 
 
 1.710 
 
 if 
 
 6-5 
 
 32 
 
 2f 
 
 42.75 
 
 45-75 
 
 42 
 
 1.870 
 
 4 
 
 7-25 
 
 36 
 
 *f 
 
 49-5 
 
 52-75 
 
 48 
 
 2. 170 
 
 4 
 
 7-75 
 
 44 
 
 2f 
 
 56.0 
 
 59-5
 
 TABLES AND FORMULAS. 
 
 SPECIFIC HEAT OF SUBSTANCES. 
 
 Substance. 
 
 Specific 
 Heat. 
 
 Substance. 
 
 Specific 
 Heat. 
 
 Water 
 
 I OOOO 
 
 Ice 
 
 
 Sulphur 
 Iron 
 Copper 
 
 . 2026 
 
 .1138 
 
 OQ 1 I 
 
 Steam (superheated) 
 Air 
 Oxygen 
 
 .4805 
 2375 
 
 2 I 7 ? 
 
 Silver 
 
 . CK 7O 
 
 Hydrogen 
 
 7 4000 
 
 Tin 
 
 oc62 
 
 Carbon monoxide 
 
 24.70 
 
 Mercury 
 
 Ot 1 7 
 
 Carbon dioxide 
 
 2 I 7O 
 
 Lead 
 
 O"? 14. 
 
 Nitrogen 
 
 24.^8 
 
 
 
 
 
 CONSTANTS FOR APPARENT CUT-OFFS USED 
 IN DETERMINING M. E. P. 
 
 Cut-off. 
 
 Constant. 
 
 Cut-off. 
 
 Constant. 
 
 Cut-off. 
 
 Constant. 
 
 % 
 
 .566 
 
 3/S 
 
 .771 
 
 2 /3 
 
 .917 
 
 Ys 
 
 .603 
 
 4 
 
 .789 
 
 7 
 
 .926 
 
 ti 
 
 6 59 
 
 /4 
 
 .847 
 
 % 
 
 937 
 
 3 
 
 .708 
 
 .6 
 
 895 
 
 .8 
 
 944 
 
 /3 
 
 743 
 
 5 /B 
 
 .904 
 
 7 A 
 
 95i 
 
 RIVETED JOINTS OF BOILERS. 
 
 Thick- 
 ness of 
 
 Diam- 
 
 Diam- 
 eter of 
 
 Pitch. 
 
 Efficiency of Joint. 
 
 Plate. 
 
 Rivet. 
 
 Hole. 
 
 d 
 
 Single. 
 
 Double. 
 
 Single, 
 
 Double. 
 
 X* 
 
 $/s" 
 
 U" 
 
 2" 
 
 3" 
 
 .66 
 
 77 
 
 IT" 
 
 -H" 
 
 y\" 
 
 2 iV" 
 
 3/8" 
 
 .64 
 
 .76 
 
 ys ff 
 
 y" 
 
 ii!" 
 
 2^" 
 
 3 1 A" 
 
 .62 
 
 75 
 
 rV" 
 
 H; 
 
 fa 
 
 2 iV" 
 
 3/s" 
 
 .60 
 
 74 
 
 1/2 " 
 
 7/8 " 
 
 
 
 2^" 
 
 3l/2/> 
 
 58 
 
 73
 
 TABLES AND FORMULAS. 79 
 
 POSITIONS OF ECCENTRIC RELATIVE TO CRANK. 
 
 
 Kind of 
 
 Kind of 
 Rocker- 
 
 Angle Between 
 Crank and 
 
 Position of 
 Eccentric Rela- 
 
 
 Valve. 
 
 Arm. 
 
 Eccentric. 
 
 tive to Crank. 
 
 I. ... 
 
 Direct. . . 
 
 Direct 
 
 90 -j- angle of 
 
 
 
 
 
 advance 
 
 Ahead of crank. 
 
 II... 
 
 Direct. . . 
 
 Reversing.. 
 
 90 angle of 
 
 
 
 
 
 advance 
 
 Behind crank. 
 
 III.. 
 
 Indirect . 
 
 Direct 
 
 90 angle of 
 
 
 
 
 
 advance 
 
 Behind crank. 
 
 IV... 
 
 Indirect. 
 
 Reversing.. 
 
 90 -f- angle of 
 
 
 
 
 
 advance 
 
 Ahead of crank. 
 
 DIAMETERS OF STEAM AND EXHAUST PIPES. 
 
 Diam. of cylinder. . . 
 
 10 
 
 12 
 
 14 
 
 16 
 
 18 
 
 20 
 
 22 
 
 24 
 
 26 
 
 28 
 
 3 
 
 Diam. of steam pipe . 
 
 3 
 
 z% 
 
 4 
 
 4^ 
 
 5 
 
 6 
 
 6 
 
 7 
 
 7 
 
 8 
 
 9 
 
 Diam. of exhaust pipe 
 
 3/2 
 
 4 
 
 5 
 
 6 
 
 6 
 
 7 
 
 8 
 
 9 
 
 9 
 
 9 
 
 10 
 
 PISTON SPEEDS OF STEAM ENGINES. 
 
 Ft. per min. 
 
 Small stationary engines 300 to 600. 
 
 Large stationary engines 600 to 1,000. 
 
 Corliss engines 400 to 750. 
 
 Locomotives 600 to 1,200. 
 
 RATIO OF GRATE AREA OF BOILER TO 
 HORSEPOWER. 
 
 Ratio. Average. 
 
 Plain cylindrical 5 to . 7 
 
 Flue 4 to .5 
 
 Multitubular 4 to .6 
 
 Water tube 3 
 
 Vertical 6 to . 7 
 
 Locomotive . . . .01 to .06 
 
 .6 
 
 45 
 
 5 
 
 3
 
 80 TABLES AND FORMULAS. 
 
 RATIO OF HEATING SURFACE TO GRATE AREA. 
 
 Plain cylindrical 1 2 to 15 
 
 Flue 20 to 25 
 
 Multitubular 25 to 35 
 
 Vertical 25 to 30 
 
 Water tube 35 to 40 
 
 Locomotive 50 to 100 
 
 RATIO OF HEATING SURFACE TO HORSEPOWER. 
 
 Plain cylindrical 6 to 10 
 
 Flue 8 to 12 
 
 Multitubular 14 to 18 
 
 Vertical 15 to 20 
 
 Water tube 10 to 12 
 
 Locomotive. . i to 2
 
 TABLES AND FORMULAS. 
 
 81 
 
 
 
 I 
 
 
 
 
 
 g 
 
 
 J^8S85'R^S^S;S 1 i:'8'&J?R^S 
 
 
 
 M 
 
 
 
 
 
 
 
 
 
 
 5 
 
 
 
 
 
 
 
 
 
 
 c 
 
 
 CON 1-1 O Ooot^vOin^coN M O O^cot^vSxn 
 
 
 
 g 
 
 
 
 
 
 
 
 
 
 
 
 g 
 
 
 
 
 
 
 
 
 
 H 
 
 
 
 
 
 
 JJj 
 
 
 
 
 
 2 
 
 
 
 s 
 
 
 
 
 
 i 
 
 
 1 
 
 
 k 
 
 K 
 
 3 
 
 c 
 
 "o 
 
 
 fa 
 
 
 O 
 
 5 
 
 
 
 1 
 
 M o Ooo r^o vn-3-cow M o ^co r>.\o u^rfco 
 
 
 
 
 
 
 
 
 
 r^ 
 
 
 M O O^oo t^vo "->Tt-tON M O Ooo r^%S I/-ITJ-CO 
 
 C 
 
 
 
 
 ^! 
 
 - 
 
 
 
 
 
 5 
 
 
 *-? 
 
 
 N Tj-COCOCOOJ N M M 1H O O O OONOOOOOOOO 
 
 S 8 ^^ IT'S M^^W 2 ^oo o'ooo 
 
 fa 
 
 
 
 
 
 
 
 
 
 1-1 O O^co r^-O iD-rfcow O O^oo t^-o */> ^ co w 
 
 
 
 
 
 
 
 
 
 
 O O Oco r-ir>Ttcow M o Ooo i^O m-^-cow 
 
 
 
 
 
 ..%..... 
 
 
 
 
 
 
 
 
 S, 
 
 
 Its : s'1 < lss2'lf?fll?i 
 
 
 
 
 
 
 
 
 vn 
 
 
 
 
 
 
 
 
 
 1 
 
 I] 
 
 cJ 
 jS 
 
 .oooooogooooo^ooogo 
 
 
 1 
 
 
 
 N
 
 82 
 
 TABLES AND FORMULAS. 
 
 saqouj 
 ui J9}a 
 
 -UJBIQ 
 
 00 
 
 _ 
 
 -t- 
 
 ^ 
 
 o 
 
 tf} 
 
 
 
 o> 
 
 N 
 
 90 
 
 rf 
 
 
 
 SO 
 
 M 
 
 00 
 
 Tj- 
 
 O 
 
 SO 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 S31JOUI 
 
 '9jBnbs 
 
 JO 3 P 1 S 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 o 
 00 
 
 SO 
 00 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 '5d ' b S 
 
 UI B3jy 
 
 IBn^oy 
 
 t^ 
 
 ~ 
 
 *f 
 
 30 
 
 - 
 
 
 
 
 
 
 
 g. 
 
 SO 
 
 - 
 
 oc 
 
 OO 
 
 rf 
 
 CO 
 
 ^ 
 
 N 
 
 "-! 
 
 
 
 
 
 
 
 
 
 * 
 
 - 
 
 - 
 
 esi 
 
 00 
 
 O 
 
 ^ 
 
 fO 
 
 
 
 -t 
 
 Os 
 
 4 
 
 IO 
 
 Id ' b S 
 
 UI B3J\T 
 
 SApoajjg 
 
 Os 
 
 t^ 
 
 
 
 o 
 
 yz 
 
 oo 
 
 U-> 
 
 "t 
 
 t^. 
 
 Tf 
 
 I/"; 
 
 ^0 
 
 t-. 
 
 Tj- 
 
 'l- 
 
 >n 
 
 * 
 
 ro 
 
 00 
 
 O 
 
 SO 
 
 Tj- 
 
 
 
 
 
 
 
 
 
 
 M 
 
 M 
 
 H 
 
 M 
 
 M 
 
 CM 
 
 to 
 
 Height of Chimney in Feet. 
 
 
 
 N 
 
 Commercial Horsepower. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 X 
 ON 
 
 oo 
 
 M 
 
 
 
 t^ 
 
 rO 
 so 
 
 ro 
 
 ON 
 00 
 
 SO 
 
 tsT 
 
 tC, 
 t>< 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 00 
 
 "51- 
 
 r~ 
 
 OO 
 Os 
 
 u-. 
 
 
 
 
 ro 
 
 10 
 
 
 r-~ 
 t^ 
 
 t^- 
 
 o 
 
 - 
 
 
 
 M 
 
 M 
 
 tsl 
 
 2 
 
 
 
 
 
 
 
 
 
 
 
 UO 
 
 u-j 
 
 
 
 SO 
 
 ON 
 
 Tj- 
 
 00 
 
 to 
 
 M 
 
 O 
 
 -i 
 
 00 
 
 Tj- 
 
 ON 
 
 to 
 
 
 
 SO 
 
 t^ 
 
 CC 
 
 w 
 
 M 
 
 M 
 
 - 
 
 w 
 
 >C 
 
 N 
 
 
 
 
 
 
 
 
 
 
 Os 
 
 X) 
 
 CO 
 
 <T) 
 
 IO 
 
 CO 
 
 O 
 
 
 
 Tt- 
 
 tO 
 
 Os 
 
 ^ 
 
 o 
 
 Tf- 
 
 Os 
 
 SO 
 Os 
 -t 
 
 
 1^ 
 
 C 
 
 TH 
 
 
 
 
 
 
 
 
 
 
 
 1^ 
 
 fSI 
 
 U") 
 
 SO 
 
 fO 
 
 N 
 
 t^. 
 
 Tj- 
 
 f^ 
 Os 
 lO 
 
 oo 
 
 M 
 1^ 
 
 SO 
 r^. 
 
 OO 
 
 a; 
 
 r^ 
 
 o 
 
 t 
 S) 
 
 IO 
 
 Tf- 
 
 sO 
 
 
 
 M 
 
 s 
 
 IN 
 
 I 
 
 
 
 
 
 
 
 M 
 
 oo 
 
 Os 
 N 
 
 X 
 10 
 
 M 
 
 00 
 -t 
 fO 
 
 Os 
 
 Tf 
 
 r(- 
 
 u-> 
 
 SO 
 u-j 
 
 -t 
 Os 
 SO 
 
 10 
 
 to 
 
 00 
 
 VO 
 
 Os 
 0\ 
 
 to 
 
 
 
 rf 
 
 Tf 
 
 to 
 
 to 
 
 10 
 
 
 
 
 
 
 
 re 
 
 -t 
 
 ^ 
 t^- 
 
 00 
 
 M 
 
 UO 
 
 Tj- 
 
 
 
 fD 
 rO 
 
 t^. 
 
 -t 
 
 Os 
 '^ 
 IO 
 
 CC 
 
 10 
 SO 
 
 M 
 Os 
 r~ 
 
 
 
 
 
 c 
 
 3C 
 
 
 
 tsi 
 
 so 
 
 00 
 
 o" 
 
 to 
 
 to 
 
 fi 
 
 o 
 
 ? 
 
 N 
 
 fO 
 
 si 
 
 * 
 
 10 
 
 
 >o 
 
 
 
 
 
 
 
 
 
 N 
 
 t^. 
 
 Tj- 
 
 00 
 
 10 
 
 00 
 
 t^ 
 
 
 
 
 UO 
 
 N 
 
 "0 
 
 fo 
 
 oo 
 
 Hi 
 
 \0 
 (M 
 
 
 
 
 
 
 
 
 
 
 i 
 
 u-> 
 
 00 
 
 to 
 
 rf 
 u-> 
 
 N 
 
 r^. 
 
 N 
 
 ON 
 
 u-> 
 
 Tj- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ro 
 N 
 
 rO 
 
 ON 
 rf 
 
 10 
 
 so 
 
 rt- 
 VI 
 
 
 
 
 
 
 
 
 
 
 
 

 
 TABLES AND FORMULAS 83 
 
 RULES AND FORMULAS. 
 
 FORMULAS USED IN ALGEBRA. 
 
 Let a and b be any two quantities, then, 
 
 (a + by = a + '*"b + ' (!) Art - 
 
 ( - ) = a" - lab + b\ (2.) Art. 432. 
 
 (0 + b}(a - b) = a 1 - V (3.) Art. 432. 
 
 ^) = (a+b)\ (4.) ' Art. 455. 
 
 ^) = (-^) 3 . (5.) Ait. 455. 
 
 ^ -b 9 =(a + b)(a - b}. (6.) Art. 462. 
 
 Let ax* -f- ^- l ' = ^ be any quadratic equation (it must be 
 borne in mind that b and c may be positive or negative) ; 
 
 then, 
 
 THE TRIGONOMETRIC FUNCTIONS. 
 Art. 754. 
 
 ,, . side opposite 
 
 Rule l.Stne =. . &- . 
 
 hypotenuse 
 
 Rule 2. Side opposite = hypotenuse x sine. 
 
 ~ . side adjacent 
 
 Rule 3. Cosine = . ^ . 
 
 hypotenuse 
 
 Rule 4. Side adjacent = hypotenuse X cosine.
 
 84 TABLES AND FORMULAS. 
 
 side opposite 
 Rule 5. Tangent = r-. ^V 
 
 side adjacent 
 
 Rule 6. Side opposite = side adjacent x tangent. 
 
 side adjacent 
 
 Rule 7. Cotangent = r-; = . 
 side opposite 
 
 Rule 8. Side adjacent = side opposite X cotangent 
 
 side opposite 
 
 Rule 9. Hypotenuse = v^ . 
 
 sine 
 
 side adjacent 
 
 Rule 1O. Hypotenuse = . 
 
 cosine 
 
 RULES FOR USING TABLE OF LOGARITHMS 
 OF NUMBERS. 
 Arts. 625-636. 
 
 I. To find the Characteristic. Fora number greater 
 than 1 the characteristic is one less than the number of in- 
 tegral places in the number. For a number wtiolly decimal 
 the characteristic is negative, and is numerically one greater 
 than the number of ciphers between the decimal point and the 
 first digit of the decimal. 
 
 II. To find the Logarithm of a Number not hav- 
 ing more than four figures. Find the first three sig- 
 nificant figures of the number whose logarithm is desired in 
 the left-hand column ; find the fourth figure in the column at 
 the top (or bottom) of the page, and in the column under (or 
 above'] this figure, and opposite the first three figures previously 
 found, will be the mantissa, or decimal part, of the logarithm. 
 
 The characteristic being found as described above, write it at 
 the left of the mantissa, and the resulting expression will be 
 the logarithm of the required number. 
 
 III. To find the Logarithm of a Number con- 
 sisting of five or more figures. 
 
 (a) If the number consists of more than five figures, and 
 tJie six tli figure is 5 or greater, increase t/ie fifth figure by 1, 
 and write ciphers in place of the sixth and remaining figures.
 
 TABLES AND FORMULAS. 85 
 
 (6) Find the mantissa corresponding to the logarithm of 
 the first four figures, and subtract tJiis mantissa from the 
 next greater mantissa in the table ; the remainder is the 
 difference. 
 
 (c) Find in the secondary table headed P. P. a column 
 headed by the same number as tJiat just found for the differ- 
 ence, and in this column opposite tJie number corresponding to 
 the fifth figure (or fifth figure increased by 1) of the given 
 number (this figure is always situated at the left of the 
 dividing line of the column] will be found the P. P. (propor- 
 tional part} for that number. The P. P. thus found is to be 
 added to t/ie mantissa found in (b}, and the result is the 
 mantissa of the logarithm of the given number, as nearly as 
 may be found with five-place tables. 
 
 IV. To find a Number whose Logarithm is 
 given. 
 
 (a) Consider the mantissa first. Glance along the 
 different columns of the table wJiicJi are Jieadcd O until the 
 first two figures of the mantissa are found. TJicn glance 
 doivn the same column until the third figure is found (or 1 
 less than the third figure}. Having found the first three 
 figures, glance to the right along the row in which they are 
 situated until the last three figures of the mantissa are found. 
 Then the number which heads the column in ivJiicli the last 
 three figures of the mantissa are found is the fourth figure 
 of t/ie required number, and iJic first tJiree figures lie in the 
 column headed N, and in the same row in which lie the last 
 three figures of the mantissa. 
 
 (b} If the mantissa cannot be found in the table, find the 
 mantissa wJiicJi is nearest to, but less than, the given mantissa, 
 and which call the next less mantissa. Subtract the next less 
 mantissa from the next greater mantissa in the table to obtain 
 the difference. Also subtract tJie next less mantissa from the 
 mantissa of the given logarithm, and call the remainder the 
 P. P. Looking in the secondary table headed P. P. for the 
 column headed by the difference just found, find the number 
 opposite the P. P. just found (or the P. P. corresponding most
 
 SO TABLES AND FORMULAS. 
 
 nearly to that just found] ; this number is the fifth figure of 
 the required number ; tJie fourth figure will be found at the 
 top of t/te column containing' the next less mantissa, and the 
 first three figures in the column headed X, and in the same 
 row which contains the next less mantissa. 
 
 (c) Having found the figures of the number as above 
 directed, locate the decimal point by tlie rules for the c/iarac- 
 t eristic, annexing ciphers to bring the number up to the re- 
 quired number of figures if t lie e/iaracteristic is greater than 4- 
 
 RULES FOR USING TRIGONOMETRIC TABLES. 
 
 Given, an angle, to find its sine, cosine, tangent, 
 and cotangent. 
 
 Rule 1 1. Find in the table the sine, cosine, tangent, or co- 
 tangent corresponding to the degrees and minutes of the angle. 
 
 For the seconds, find the difference of the values of the sine, 
 cosine, tangent, or cotangent taken from the table, between 
 whicli the seconds of the angle fall ; multiply this difference by 
 a fraction whose numerator is the number of seconds in the 
 given angle, and whose denominator is 60. 
 
 If sine or tangent, add this correction to the value first found; 
 if cosine or cotangent, subtract the correction. Art. 756. 
 
 Given, the sine, cosine, tangent, or cotangent to 
 find the angle corresponding. 
 
 To find the angle corresponding to a given sine, cosine, 
 tangent, or cotangent whose exact value is not contained in 
 the table : 
 
 Rule 12. Find the difference of the two numbers in the 
 table between whicli the given sine, cosine, tangent, or co- 
 tangent falls, and use the number of parts in this difference 
 as the denominator of a fraction. 
 
 Find the difference between the number belonging to the 
 smaller angle, and the given sine, cosine, tangent, or cotangent, 
 and use the number of parts in the difference jiist found as the 
 numerator of the fraction mentioned above. Multiply this 
 fraction by 60, and the result will be the mimbcr of seconds to 
 be added to the smaller angle. Art. 758.
 
 TABLES AND FORMULAS. 87 
 RULES FOR MENSURATION. 
 
 THE TRIANGLE. 
 
 Rule. TJie area of any triangle eqttals one-half t/ie 
 producr of the base and the altitude. Art. 766. 
 
 THE QUADRILATERAL. 
 
 To find the area of a parallelogram: 
 
 Rule. The area of any parallelogram equals the product 
 of the base and the altitude. Art. 111. 
 
 To find the area of a trapezoid : 
 
 Rule. The area of a trapezoid equals one-half the sum of 
 the parallel sides multiplied by the altitude. Art. 778. 
 
 To find the area of an irregular figure bounded by straight 
 lines: 
 
 Rule. Divide the figure into triangles, and find the area 
 of cac/i triangle separately. T/ie sum of the areas of all the 
 triangles u>ill be the area of the figure. Art. 779. 
 
 THE CIRCLE. 
 
 To find the circumference or diameter of a circle: 
 
 Rule. The circumference of a circle equals the diameter 
 multiplied by 3. 1416. Art. 78O. 
 
 Rule. The diameter of a circle equals the circumference 
 divided by 3.1416. Art. 78O. 
 
 To find the length of an arc of a circle: 
 
 Rule. The length of an arc of a circle equals the circum- 
 ference of the circle of which the are is apart multiplied by the 
 number of degrees in the arc, and divided by 360. Art. 781 . 
 
 To find the area of a circle: 
 
 Rule. Square the diameter, and multiply by . 7854- Art. 
 782. 
 
 Given, the area of a circle to find its diameter: 
 
 Rule. Divide the area by . 7854, and extract the square 
 root of the quotient. Art. 783.
 
 88 TABLES AND FORMULAS. 
 
 To find the area of a sector : 
 
 Rule. Divide the number of degrees in the arc of a sector 
 by 360. Multiply the result by the area of the circle of which 
 the sector is a part. Art. 784. 
 
 To find the area of a segment of a circle : 
 
 Rule. Draiv radii from the center of the circle to the 
 extremities of the arc of 'the segment ; find the area of the 
 sector thus formed, subtract from this the area of the- triangle 
 formed by the radii and the chord of the arc of the segment, 
 and the result is the area of the segment. Art. 785. 
 
 THE ELLIPSE. 
 
 To find the perimeter of an ellipse: 
 
 Rule. Multiply the major axis by 1.83, and t lie minor axis 
 by 1.315. The sum of the results will be the perimeter. Art. 
 788. 
 
 To find the exact area of an ellipse: 
 
 Rule. The area of an ellipse is equal to the product of its 
 two diameters multiplied by .7854- Art. 789. 
 
 To find the area of any plane figure : 
 
 Rule. The area of any plane figtire may be found by 
 dividing it into triangles, quadrilaterals, circles or parts of 
 circles, and ellipses, finding the area of each part separately, 
 and adding them together. Art. 79O. 
 
 THE PRISM AND CYLINDER. 
 
 To find the area of the convex surface of any right prism 
 or right cylinder: 
 
 Rule. Multiply the perimeter of the base by the altitude. 
 Art. 8O3. 
 
 To find the volume of a right prism or cylinder: 
 
 Rule. The volume of any right prism or cylinder equals 
 the area of the base multiplied by the altitude. Art. 8O4.
 
 TABLES AND FORMULAS. 89 
 
 THE PYRAMID AND CONE. 
 
 To find the area of a right pyramid or right cone: 
 Rule. The convex area of a rigJit pyrainid or cone equals 
 
 the perimeter of the base multiplied by one-half the slant 
 
 height. Art. 8O9. 
 
 To find the volume of a right pyramid or cone: 
 
 Rule. The volume of a rigJit pyramid or cone equals 2 'he area 
 
 of the base multiplied by one-third of the altitude. Art. 81O. 
 
 THE FRUSTUM OF A PYRAMID OR CONE. 
 
 To find the convex area of a frustum of a right pyramid 
 or right cone: 
 
 Rule. The convex area of a frustum of a right pyramid 
 or right cone equals one -half the sum of the perimeters of its 
 bases multiplied by the slant height of the frustum. Art. 
 814. 
 
 To find the volume of the frustum of a pyramid or cone: 
 
 Rule. Add the areas of the upper base, the lower base, and 
 the square root of the product of the areas of the two bases ; 
 multiply this sum by one-third of the altitude. Art. 815. 
 
 THE SPHERE. 
 
 To find the area of the surface of a sphere : 
 
 Rule. The area oftJie surface of a sphere equals the square 
 of the diameter multiplied by 3. 1416. Art. 817. 
 
 To find the volume of a sphere: 
 
 Rule. The volume of a sphere equals the cube of the diam 
 eter multiplied by .5236. Art. 818. 
 
 FORMULAS USED IN ELEMENTARY 
 MECHANICS. 
 
 UNIFORM MOTION. 
 
 Let 5 = the length of space passed over uniformly ; 
 
 / = the time occupied in passing over the space S; 
 V = the velocity.
 
 90 TABLES AND FORMULAS. 
 
 V= . (7.) Art. 859. 
 
 5= Vt. (8.) Art. 859. 
 t = y. (9.) Art. 859. 
 
 MASS, WEIGHT, AND GRAVITY. 
 
 If the mass of the body be represented by ///, its weight 
 by W, and the force of gravity at the place where the body 
 was weighed by g, we have 
 
 weight of body W 
 
 mass = -. -. rf-, or m = . (1O.) Art. 888. 
 
 force of gravity g 
 
 FORMULAS FOR GRAVITY PROBLEMS. 
 
 Let W= weight of body at the surface; 
 
 w = weight of a body at a given distance above or 
 
 below the surface ; 
 d distance between the center of the earth and the 
 
 center of the body ; 
 R = radius of the earth = 4,000 miles. 
 
 Formula for weight when the body is below the surface: 
 
 wR=dW. (11.) Art. 891. 
 
 Formula for weight when the body is above the surface: 
 wd*=WR\ (12.) Art. 891. 
 
 FALLING BODIES. 
 
 Let g = force of gravity = constant accelerating force due 
 
 to the attraction of the earth ; 
 / = number of seconds the body falls ; 
 v = velocity at the end of the time t; 
 JL distance that a body falls during the time t. 
 
 v-gt. (13.) Art. 896. 
 
 That is, the velocity acquired by a freely falling body at the 
 end of t seconds equals 32. 16 multiplied by the time in seconds 
 
 f = ~. (14.) Art. 896.
 
 TABLES AND FORMULAS. ( J1 
 
 That is, the number of seconds during which a body must 
 have fallen to acquire a given velocity equals the given velocity 
 in feet per second divided by 32. 16. 
 
 h = ^- (15.) Art. 896. 
 
 > 
 
 That is, the height from which a body must fall to acquire 
 a given velocity equals the square of the given velocity divided 
 by 2x32.16. 
 
 v-^^gh. (16.) Art. 896. 
 
 That is, the velocity that a body will acquire in falling 
 through a given height equals the square root of the product 
 of twice 32. 16 and the given height. 
 
 (17.) Art. 896. 
 
 That is, the distance a body will fall in a given time equals 
 32.16 -r- 2 multiplied by the square of the number of seconds. 
 
 t = \. (18.) Art. 896. 
 
 <5> 
 
 That is, the time it will take a body to fall through a given 
 height equals the square root of twice the height divided by 
 32. 16. 
 
 CENTRIFUGAL FORCE. 
 
 The value of the centrifugal force of any revolving body, 
 expressed in pounds, is 
 
 F = .00034 W R N*; (19.) Art. 9O3. 
 
 in which F = centrifugal force ; 
 
 W =. total weight of body in pounds; 
 
 R = radius, usually taken as the distance be- 
 tween the center of motion and the cen- 
 ter of gravity of the revolving body, in 
 feet; 
 
 N = number of revolutions per minute.
 
 92 TABLES AND FORMULAS. 
 
 THE CENTER OF GRAVITY OF TWO BODIES. 
 
 Let / = the distance between the centers of the bodies; 
 /, = the short arm ; 
 w = weight of small body ; 
 W= weight of large body. 
 
 THE EFFICIENCY OF A MACHINE. 
 
 Let F = the force applied to the machine ; 
 V = the velocity ratio of the machine ; 
 W= the weight actually lifted or equivalent resist- 
 
 ance overcome; 
 E = the efficiency of the machine ; 
 
 W 
 Then, E = -. (22.) Art. 95O. 
 
 WORK. 
 
 If the force necessary to overcome the resistance be repre- 
 sented by F, the space through which the resistance acts by 
 vS, and the work done by [7, then U F S. 
 
 \iW the weight of a body, and // = the height through 
 which it is raised, U = W h. Hence the work done 
 
 U=FS=Wh. (23.) Art. 953. 
 
 POWER. 
 
 The power of a machine may always be determined by 
 dividing the work done in foot-pounds by the time in minutes 
 required to do the work; i. e. , 
 
 Power = ^-. (24.) Art. 954. 
 
 KINETIC ENERGY. 
 
 Let W= the weight of the body in pounds; 
 v its velocity in feet per second ; 
 h = the height in feet through which the body must 
 fall to produce the velocity v; 
 
 W 
 m the mass of the body = . (See formula 1O.)
 
 TABLES AND FORMULAS. 93 
 
 The work necessary to raise a body through a height h is 
 Wh. The velocity produced in falling a height // is 
 
 v 4/2 gh, and // = . (See formulas 15 and 16.) 
 
 * v* w 
 
 Therefore, work = W h = W = \ x X v 1 = mv* 
 
 %g g 
 
 or ]Vh = %jnv*. (25.) Art. 957. 
 
 DENSITY. 
 
 The density of a body is its mass divided by its volume 
 in cubic feet. 
 
 Let D be the density; then the density of a body is 
 
 m W W 
 
 D = ^. Since m = _,/?= ~^~ (26.) Art. 962. 
 
 RULES AND FORMULAS USED IN HYDRAULICS. 
 
 PASCAL'S LAW. 
 
 Rule. -The pressure per unit of area exerted anywhere 
 upon a mass of liquid is transmitted undiminished in all 
 directions, and acts with the same force upon all surfaces in 
 a direction at right angles to those surfaces. Art. 968. 
 
 THE GENERAL LAW FOR THE DOWNWARD PRESSURE 
 UPON THE BOTTOM OF ANY VESSEL. 
 
 Rule. The pressure upon the bottom of a I'csscl containing 
 a fluid is independent of the shape of the -vessel, and is equal 
 to tJie weight of a prism of the fluid whose base has the same 
 area as the bottom of the vessel, and whose altitude is the 
 distance between the bottom and the upper surface of the fluid 
 pins t/ie pressure per unit of area upon the upper surface of 
 the fluid, multiplied by the area of the bottom of the vessel. 
 Art. 971. 
 
 GENERAL LAW FOR UPWARD PRESSURE. 
 
 Rule. The upward pressure on any submerged horizontal 
 surface equals the weight of a prism of the liquid whose 
 base has an area equal to the area of the submerged surface,
 
 94 TABLES AND FORMULAS. 
 
 and whose altitude is the distance between the submerged 
 surface and the upper surface of the liquid plus the pressure 
 per unit of area on the upper surface of the fluid, multiplied 
 by the area of the submerged surface. Art. 973. 
 
 GENERAL LAW FOR LATERAL PRESSURE. 
 
 Rule. The pressure upon any vertical surface due to tJie 
 weight of a liquid is equal to the weight of a prism of the 
 liquid whose base has the same area as the vertical surface, 
 and whose altitude is the depth of the center of gravity of 
 the vertical surface below the level of the liquid. 
 
 Any additional pressure is to be added, as in the previous 
 cases. Art. 975. 
 
 GENERAL LAW FOR PRESSURE. 
 
 Rule. The pressure exerted by a fluid in any direction 
 upon any surface is equal to the weight of a prism of the 
 fluid whose base is tJie projection of the surface at rig/it 
 angles to the direction considered, and whose height is the 
 depth of the center of gravity of the surface below the level 
 of the liqiiid. Art. 979. 
 
 SPECIFIC GRAVITY. 
 
 Let IV be the weight of the solid in air and W the weight 
 in water; then, W W = the weight of a volume of water 
 equal to the volume of the solid, and 
 
 W 
 Sp. Gr. = j^rjy,- (27.) Art. 982. 
 
 If the body be lighter than water, a piece of iron or other 
 heavy substance must be attached to it sufficiently heavy to 
 sink both. Then zvcigh both bodies in air and both in water. 
 
 Let W = weight of both bodies in air; 
 W = weight of both bodies in water; 
 w = weight of light body in air; 
 W^ = weight of heavy body in air; 
 W t = weight of heavy body in water.
 
 TABLES AND FORMULAS. 95 
 
 Then, the specific gravity of the light body is given by 
 
 s P- Gr - = ( ^-?r)-(n/-^)- (27 "- ) Art - 983 - 
 
 To find the specific gravity of a liquid: 
 
 Weigli an empty flask ; fill it with water, then weigh it, 
 and find the difference between the two results ; this will 
 equal the weight of the water. Then weigh the flask filled 
 ivit/i the liquid, and subtract t/ic weight of tJic flask ; the 
 result is the weight of a volume of the liquid equal to the 
 volume of the water. The weight of the liquid divided by 
 the weight of the water is the specific gravity of the liqiiid. 
 
 Let W = the weight of the flask and liquid; 
 IT' = the weight of the flask and water; 
 iv = the weight of the flask. 
 
 Then, Sp. Gr. = ^ ^- (276.) Art. 984. 
 
 FORMULAS FOR FLOW OF WATER. 
 
 MEAN VELOCITY. 
 
 Let Q = the quantity in cubic feet which passes any sec- 
 tion in 1 second; 
 
 A = the area of the section in square feet ; 
 
 v m = the mean velocity in feet per second. 
 Then, Q = Av m , (28*.) Art. 989. 
 
 and v %- (28*.) Art. 989. 
 
 VELOCITY OF EFFLUX FROM AX ORIFICE. 
 
 Let v the velocity of efflux in feet per second; 
 h = the head in feet on the orifice considered; 
 //, the head equivalent to a pressure/; 
 IV = the weight of the water in pounds flowing 
 through the aperture per second. 
 
 ixr .! 
 
 The kinetic energy of the issuing water = .
 
 96 TABLES AND FORMULAS. 
 
 The work the issuing water can do = W h. 
 
 Wh = -^-, or v = ^Tg~h. 
 * ' S 
 
 //, = -frrr, where h l is in feet, and / in pounds per 
 . 4o4: 
 
 square inch. 
 
 h = -=, where h is in feet, and p in pounds per 
 b*. O 
 
 square foot. 
 h-^-h^ the total head. 
 
 v = tf*g(h l + 7J). (29.) Art. 991. 
 
 If a is the area of a large orifice in the bottom of a small 
 vessel whose area is A, the velocity is 
 
 Art . 993. 
 
 THEORETICAL RANGE OF A JET. 
 
 Let // = head on center of orifice ; 
 
 y = vertical height of orifice above the surface where 
 
 the water strikes; 
 R = range. 
 
 Then, R = tfUTy. (3O.) Art. 992. 
 
 FLOW THROUGH ORIFICES. 
 
 Velocity of the Jet. 
 
 Let v = theoretical velocity ; 
 
 v' actual maximum velocity ; 
 
 c' = coefficient of velocity ; 
 
 h = head on center of orifice ; 
 
 g acceleration due to gravity 32.16. 
 
 'if = c' v = c' \fogh. (32.) Art. 994. 
 
 An average value of c' is .98.
 
 TABLES AND FORMULAS. 97 
 
 Discharge of an Orifice. 
 
 Let Q theoretical discharge; 
 Q' = actual discharge ; 
 a area of orifice ; 
 c" = coefficient of discharge ; 
 // = head on center of orifice; 
 g = acceleration dtie to gravity 32.16. 
 
 An average value of c" is .61. Then, 
 Q = c" Q = c"a 4/2^7: = . 61 a ^Tgli. (33.) Art. 994. 
 
 Discharge of Standard Orifices. 
 
 Let Q discharge in cubic feet per second; 
 
 d = diameter of a circular or length of a side of 
 
 a square orifice in feet ; 
 d' = depth of a rectangular orifice in feet; 
 /; = breacjtfi of a rectangular orifice in feet ; 
 Ji = head on the center of a circular or of a square 
 
 orifice in feet; 
 // a = head on the upper edge of a rectangular orifice 
 
 in feet; 
 // a = head on the lower edge of a rectangular orifice 
 
 in feet; 
 c = coefficient of discharge (see tables of Coefficients 
 
 of Discharge for Standard Orifices) ; 
 g = acceleration due to gravity = 32.16. 
 
 For a circular vertical orifice, 
 
 Q = . 7854 d*c 4/2^7;= 6.299 d* c \fJi. (340.) Art. 996. 
 For a square vertical orifice, 
 
 Q = cd* yTp = 8.02 c d n - \Hi. (340. ) Art. 997. 
 
 For a rectangular vertical orifice, 
 
 (34^.) Art. 998. 
 If the head h on the center of a rectangular vertical
 
 98 TABLES AND FORMULAS. 
 
 orifice is greater than 4 //, the discharge may be computed 
 by the formula 
 
 Q = c b d^gli = 8.Cr>fl><t 4//7. (34</. ) Art. 998. 
 
 For approximate computations the value of c that may be 
 used in formulas 34r and 34^/ is c .015. 
 Discharge of a Submerged Rectangular Orifice. 
 
 Let Q = discharge in cubic feet per second ; 
 b = breadth of orifice in feet ; 
 d = depth of orifice in feet ; 
 h o the difference in the level of the water on the 
 
 two sides of the orifice in feet ; 
 g acceleration due to gravity = 33.10. 
 Then, 
 Q = . 015 b d4/~Zgh = 4. 932 b d \/k . (34r. ) Art. 999. 
 
 THE DISCHARGE OF WEIRS. 
 
 Let / = length of the weir in feet; 
 H measured head in feet'; 
 v = velocity with which the water approaches the 
 
 weir in feet per second; 
 It = head equivalent to the velocity with which the 
 
 water approaches the weir in feet; 
 c = coefficient of discharge (see tables of Coefficients 
 
 of Discharge for Weirs) ; 
 
 Q = theoretical discharge in cubic feet per second ; 
 Q = actual discharge in cubic feet per second. 
 The theoretical discharge per second is 
 
 Q = %\frgl(H+/tf. (35*.) Art. 1006. 
 
 If there is no velocity of approach, this becomes 
 
 Q = \\/^glH l . (35*.) Art. 1OO6. 
 
 The actual discharge for weirs with end contractions is 
 given by the formulas 
 
 (36*. ) Art. 1OO6. 
 and 
 
 Art. 1OO6.
 
 TABLES AND FORMULAS. 09 
 
 For weirs without end contractions, the formulas are 
 Q> = c%\i*Tgl(H+ 1.4//) = 5.347 <:/(# + 1.4//)*, 
 
 (37 a.) Art. 1OO6. 
 
 and 0'=rf4/2]r///*= 5.347 <:///*. (376.) Art. 1OO6. 
 
 The velocity of approach is the mean velocity with 
 
 which the water flows through the canal leading to the 
 
 weir. If A is the area of the cross-section of the water in 
 
 this canal, we have v = ^-, from which we see that Q' must 
 
 be determined approximately by assuming i> 0, and then 
 use this value of Q' to find i<. V may also be measured 
 approximately by means of a float on the water in the canal 
 or stream. 
 
 v" 1 
 Having found v, we have the equivalent head // = - = 
 
 . 01555 if. (See Arts. 99O and 991.) Since v is small with 
 a properly constructed weir, it is usually neglected, unless 
 great accuracy is required. 
 
 FLOW OF WATER THROUGH PIPES. 
 
 Let / = length of pipe in feet ; 
 
 d diameter of pipe in feet; 
 
 d l diameter of pipe in inches ; 
 
 i' = mean velocity of flow through pipe in feet per 
 second ; 
 
 Ji = total head on outlet end of pipe in feet ; 
 
 //' head in feet equivalent to the velocity v; 
 
 /;'"=head in feet equivalent to the loss of pressure 
 at entrance to pipe; 
 
 // IV =head in feet equivalent to the loss in pressure 
 produced by friction in pipe ; 
 
 // v = head in feet equivalent to loss in pressure pro- 
 duced by angular bends in pipe; 
 
 // VI = head in feet equivalent to loss in pressure pro- 
 duced by circular bends in pipe; 
 
 f = a coefficient for loss of head due to friction (see 
 table of Coefficients f for Smooth Iron Pipes); 
 
 ;;/ = a coefficient for loss of head at entrance ; 
 
 n = number of bends in pipe;
 
 100 TABLES AND FORMULAS. 
 
 c = a coefficient for loss of head due to angular 
 
 bends (see table of Coefficients for Angular 
 
 Bends); 
 c 1 a coefficient for loss of head due to circular 
 
 bends (see table of Coefficients for Circular 
 
 Bends) ; 
 Q = quantity discharged by pipe in cubic feet per 
 
 second; 
 Q' = quantity discharged by pipe in gallons per 
 
 second ; 
 
 r = radius of pipe in feet ; 
 R = radius of circular bend in pipe in feet; 
 a = number of degrees of angular bend in pipe. 
 
 General Formulas. 
 
 Loss of head at entrance, 
 
 h'" = m/i" = m^-. (39.) Art. 1O2O. 
 
 Loss of head due to friction, 
 
 W.-fL. (400.) Art. 1021. 
 
 Loss of head due to angular bends, 
 
 h v = c^ (4O0.) Art. 1O23. 
 
 *g 
 
 Loss of head due to circular bends, 
 
 h^ c'^-. (4O^.) Art. 1O23. 
 
 Total head, 
 
 /fr It' | /^ r I *^ 1 ^^ 
 
 ~+f j- + m ^r+ nc '^T' ( 41rt -) Art. 1O24. 
 
 Velocity of flow, 
 
 c= / *s* 
 
 d 
 8.02 / (42.) Art. 1O24.
 
 TABLES AND FORMULAS. 101 
 
 If in . 5 and there are no sharp bends, 
 
 / * gk . = 8.02 / ^ ; (43.) Art. 1024e 
 
 d 
 and, when the diameter is in inches, 
 
 / 7T77 
 
 Art. 1O25. 
 
 Velocity Through Long Pipes. 
 
 When the diameter is in feet, 
 
 v = 8.02 \'j~. (44.) Art. 1O25. 
 
 When the diameter is in inches, 
 
 v = 2. 315 y '-jfj-. (44*.) Art. 1O25. 
 
 Head Required to Produce a Given Velocity. 
 
 General formula, 
 
 (45.) Art. 1026. 
 
 When the influence of bends is neglected and m has the 
 value .5, the formula is 
 
 333 T;2 - 45 ^- Art - 
 
 When the diameter is given in inches, 
 
 // = /l-^L-f. .0233 T' 2 . (45^.) Art. 1O26. 
 
 O. oD it 
 
 The Quantity Discharged from Pipes. 
 
 When the diameter is given in feet, the discharge in cubic 
 feet per second is 
 
 0=. 7854^7*. (46.) Art. 1O27. 
 
 Since one cubic foot contains 7.48 gallons, if the diameter 
 is in feet, we have
 
 102 TABLES AND FORMULAS. 
 
 Q' . 7854 d* v X 7. 48 gallons per second ; (46. ) 
 
 Art. 1O27. 
 
 and for the diameter in inches, 
 ' = .0408 d? v gallons per second. (46.) Art. 1O27. 
 
 The Diameter of Pipes. 
 
 With //, /, and d in feet and the quantity Q in cubic feet 
 per second, the formula for the diameter of a pipe without 
 sharp bends is 
 
 d- 0.479 (1.5^ + //)~ (47.) Art. 1O28. 
 
 In using this formula, take the approximate value of f as 
 .02, and compute an approximate value for c/, neglecting the 
 term 1.5 d in the second member of the formula. With 
 this value of d, find the value of v from the formula 
 
 v = and find the corresponding value of /"from the 
 
 . 7854 d 
 
 table of Coefficients for Pipes. 
 
 Repeat the computation for d by placing the approximate 
 values of d and f just found in the second member of the 
 formula. One or two repetitions of this process will give a 
 near approximation of d from which to select the pipe from 
 the standard market sizes. 
 
 For pipes whose length is more than 4,000 times their 
 diameter, the following formula may be used : 
 
 (4:7 a.) Art. 1O28. 
 
 FLOW OF WATER IN CONDUITS AND CHANNELS. 
 
 Let 5 = slope of a conduit or channel ; 
 h a given fall ; 
 
 / = distance in which the fall 1i occurs ; 
 p = wetted perimeter; 
 a area of water cross-section ; 
 r = hydraulic radius; 
 v = mean velocity of flow ; 
 Q quantity discharged;
 
 TABLEvS AND FORMULAS. 103 
 
 c a coefficient to be determined by Kutter's for- 
 mula; 
 
 n coefficient of roughness to be used in Kutter's 
 formula (see table of Coefficients of Roughness). 
 
 Formula for slope, 
 
 (48.) Art. 1032. 
 Hydraulic radius, 
 
 (49.) Art. 1O32. 
 
 Discharge, 
 
 Q = ar. Art. 1O32. 
 
 Mean velocity, 
 
 v = *4&& (50.) Art. 1033. 
 
 To find the value of c use Kutter's formula, 
 
 03 | * | - 00155 
 
 .. (51-) Art. 1033. 
 
 The value of n to be used in this formula is to be taken 
 from the following table to correspond with the character 
 of the channel : 
 
 VALUES OF THE COEFFICIENT OF ROUGHNESS. 
 
 For Use in Kutter's Formula. 
 Character of Channel. Value of n. 
 
 Clean, well-planed timber 009 
 
 Clean, smooth, glazed iron and stoneware pipes oio 
 
 Masonry smoothly plastered with cement on 
 
 Clean, smooth cast-iron pipe on 
 
 Ordinary cast-iron pipe 012 
 
 Unplaned timber 012 
 
 Selected sewer pipes, well laid and thoroughly flushed. .012 
 
 Rough iron pipes 013 
 
 Ordinary sewer pipes laid under usual conditions 013 
 
 Dressed masonry and well-laid brickwork 015
 
 104 TABLES AND FORMULAS. 
 
 Character of Channel. Value of n. 
 
 Good rubble masonry and ordinary rough or fouled 
 
 brickwork 017 
 
 Coarse rubble masonry 020 
 
 Gravel, compact and firm 020 
 
 Earth canals, well made and in good alinement 0225 
 
 Rivers and canals in moderately good order and per- 
 fectly free from stones and weeds 025 
 
 Rivers and canals in rather bad condition and some- 
 what obstructed by stones and weeds 030 
 
 Rivers and canals in bad condition, overgrown with 
 vegetation and strewn with stones and other 
 detritus, according to condition 035 to .050 
 
 FORMULAS USED IN PNEUMATICS. 
 
 PRESSURE, VOLUME, DENSITY, AND WEIGHT OF AIR 
 WHEN THE TEMPERATURE IS CONSTANT : 
 
 Mariotte's Law. The temperature remaining the same, 
 the volume of a given quantity of gas varies inversely as the 
 pressure. 
 
 Let/ = pressure for one position of the piston; 
 
 / t = pressure for any other position of the piston ; 
 v = volume corresponding to the pressure/; 
 v l = volume corresponding to the pressure/,. 
 
 Then, / v=p, v v (53.) Art. 1O49. 
 Let D be the density corresponding to the pressure / and 
 volume v, and Z>, be the density corresponding to the 
 pressure /, and volume i\ ; then, 
 
 / :D =A = A or / A=A A (54.) Art. 1O52. 
 and v:D l = ^ : Z>, or v D = ^ Z\. (55.) Art. 1O52. 
 
 Thus, let IV be the weight of a cubic foot of air or other gas, 
 whose volume is v, and pressure is/; let W^ be the weight 
 of a cubic foot when the volume is v lt and pressure is /,; 
 then, 
 
 / W^ =/, W. (56.) Art. 1052. 
 
 v W = v, W,. (57.) Art. 1O52.
 
 TABLES AND FORMULAS. 103 
 
 PRESSURE A1VD VOLUME OF A GAS "WITH VARIABLE 
 TEMPERATURE : 
 
 Gay-L/ussac's Law. If tJie pressure remains constant, 
 every increase of temperature of 1 F. produces in a given 
 quantity of gas an expansion of ^\-% of its volume at 32 F. 
 
 If the pressure remains constant it will also be found that 
 every decrease of temperature of 1 F. will cause a decrease 
 of j^ of the volume at 32 F. 
 
 Let v = original volume of gas ; 
 i\ = final volume of gas; 
 
 / temperature corresponding to volume v; 
 /j = temperature corresponding to volume v^ 
 
 Then > v * = v - (58 - } Art - 1054 - 
 
 That is, the volume of gas after heating (or cooling) equals 
 the original volume multiplied by 4.60 plus the final tempera- 
 ture divided by 460 plus the original temperature. 
 
 = the original tension; 
 = the corresponding temperature; 
 l final tension ; 
 /j = final temperature. 
 
 Then, A=/- (59.) Art. 1O55. 
 
 Let / = pressure in pounds per square inch ; 
 V= volume of air in cubic feet; 
 T = absolute temperature ; 
 W = weight in pounds. 
 
 Then, / V= .37052 T. (6O.) Art. 1O56. 
 
 If the weight of the air be greater or less than 1 pound, 
 the following formula must be used : 
 
 / F=. 37052 W T. (61.) Art. 1O57. 
 Let /,, f 7 ,, and 7", represent the pressure, volume, and 
 temperature of the same weight of air in another state; 
 then, 
 
 - = ^-. (62.) Art. 1O58.
 
 100 TABLES AND FORMULAS. 
 
 MIXTURE OF TWO GASES HAVING UNEQUAL VOLUMES 
 AND PRESSURES. 
 
 Let i' and / be the volume and pressure, respectively, of 
 one of the gases. 
 
 Let ?', and p^ be the volume and pressure, respectively, 
 of the other gas. 
 
 Let V and P be the volume and pressure, respectively, of 
 the mixture. Then, if the temperature remains the same, 
 
 VP=vp + v l p l . (63.) Art 1O62. 
 
 MIXTURE OF TWO VOLUMES OF AIR HAVING UNEQUAL 
 PRESSURES, VOLUMES, AND TEMPERATURES. 
 
 If a body of air having a temperature /,, a pressure/,, and 
 a volume i\ be mixed with another volume of air having a 
 temperature t^ a pressure / 2 , and a volume 7' 2 , to form a 
 volume V having a pressure P t and a temperature /, then, 
 either the new temperature /, the new volume V, or the new 
 pressure P may be found, if the other two quantities are 
 known, by the following formula, in which T^ T^ and T 
 
 (64.) Art. 1063. 
 
 FORMULAS USED IN STRENGTH OF 
 MATERIALS. 
 
 UNIT STRESS, UNIT STRAIN, AND COEFFICIENT OF 
 ELASTICITY. 
 
 Let P the total stress in pounds; 
 
 A = area, of cross-section in square inches; 
 5 = unit stress in pounds per square inch ; 
 
 / = length of body in inches ; 
 
 ^ = elongation in inches; 
 
 s = unit strain ; 
 
 R = coefficient of elasticity. 
 
 5 = , or P= A S. (65.) Art. 1 1O3.
 
 TABLES AND FORMULAS. 107 
 
 s=,ore=ts. (66.) Art. 11O4. 
 
 j, 
 
 Art. 1110. 
 
 STRBIVGTH OF PIPES AND CYLINDERS. 
 
 Let d inside diameter of pipe in inches; 
 / = length of pipe in inches; 
 p pressure in pounds per square inch; 
 P = total pressure; then, P = p Id; 
 t thickness of pipe; 
 5 = working strength of the material. 
 For longitudinal rupture 
 //</=2//S, or 
 
 / d= a / S. (68.) Art. 1 123. 
 
 For transverse rupture 
 
 pd=tS. (69.) Art. 1124. 
 
 Since, for longitudinal rupture, p d= 2 / 5, it is seen that 
 a cylinder is twice as strong against transverse rupture as 
 against longitudinal rupture. 
 
 For pipes and cylinders whose thickness is greater than 
 -1*5- of the radius, use the following formula, in which 
 r = the inner radius, and the other letters have the same 
 meaning as before. 
 
 / = -^- f (70.) Art. 1 1 25. 
 
 The following formula gives the collapsing pressure in Ib 
 per sq. in. for wrought-iron pipe: 
 
 / =9, GOO, 000 *-. (71.) Art. 1126. 
 
 MOMENT OF INERTIA, RESISTING MOMENT, AND BENDING 
 MOMENT OF BEAMS. 
 
 Let / moment of inertia; 
 A area of cross-section ; 
 r = radius of gyration;
 
 108 TABLES AND FORMULAS. 
 
 c = distance from neutral axis to outermost fiber; 
 5 4 = ultimate strength of flexure; 
 f = factor of safety ; 
 M bending moment. 
 
 I=Ar\ (72.) Art. 1154. 
 
 The resisting moment is given by the expression 
 
 S . S . / 
 
 A r = /, or 5 
 c c c 
 
 For the bending moment 
 
 M-$J- c , (73.) Art. 1156. 
 
 and, when a factor of safety is used, 
 
 M=^. (74.) Art. 1159. 
 
 DEFLECTION OF A BEAM. 
 
 Let a = a constant depending on the manner of loading 
 
 the beam and the condition of the ends; 
 s = the deflection; 
 E = the coefficient of elasticity; 
 / = the length of the beam in inches; 
 W = the total weight supported in pounds; 
 / = moment of inertia about the neutral axis. 
 
 W /* 
 s = a. (75.) Art. 1162. 
 
 STRENGTH OF COLUMNS. 
 
 Let W= load on a column; 
 
 5 2 = ultimate strength for compression; 
 
 A area of section of column ; 
 
 f factor of safety ; 
 
 / = length of column in inches; 
 
 g = a constant to be taken from table; 
 
 / = least moment of inertia of cross-section; 
 
 b length of longer side of a rectangular column;
 
 TABLES AND FORMULAS. 100 
 
 d = length of shorter side of a rectangular column, 
 
 or the diameter of a circular column; 
 c = length of one side of a square column. 
 
 Art. 1169. 
 
 For a circular column 
 
 / 
 
 * 1/- 3l83 ^/i ,/ 3183 H7V. 3183 Hy 16^1 
 S, ^ \ A / 
 
 (78.) Art. 1171. 
 For a rectangular column, assume d, then, 
 
 (79.) Art. 1171. 
 
 d 
 
 STRENGTH OF SHAFTS. 
 
 Let d = diameter of a round shaft, or side of a square 
 
 shaft, in inches; 
 
 c = a constant (see table of Constants for Shafting) ; 
 c t = a constant (see table of Constants for Shafting) ; 
 P a force applied at the end of a lever arm in 
 
 pounds; 
 
 r length of lever arm in inches; 
 H horsepower transmitted by shaft; 
 N = number of revolutions per minute; 
 k a constant (see table of Constants for Shafting) ; 
 , = a constant (see table of Constants for Shafting) ; 
 q = a constant (see table of Constants for Shafting) ; 
 g l a constant (see table of Constants for Shafting). 
 
 For all solid shafts below 11 inches in diameter use the 
 formula 
 
 c ^ ( 80 .) Art. 1173.
 
 110 
 
 TABLES AND FORMULAS. 
 
 If the diameter of a wrought-iron shaft is greater than 
 12.4", of a cast-iron shaft greater than 10.3", or of a steel 
 shaft greater than 13.6", use the following formula: 
 
 k l \-. (81.) Art. 1174. 
 
 For a hollow (round) shaft use formula 82 or 83. 
 p = q /^_pA 4 \ (82.) Art. 1 1 74. 
 
 or H=g l N^*~ d *\ (83.) Art. 1174. 
 
 CONSTANTS FOR SHAFTING. 
 
 VALUES OF c AND c, TO BE USED IN FORMULA 8O. 
 
 Material 
 
 
 
 c 
 
 
 
 Round. 
 
 Square. 
 
 Round. 
 
 Square. 
 
 Wrought Iron . 
 Cast Iron 
 
 .310 
 
 tC 7 
 
 .272 
 ^OQ 
 
 4.92 
 
 C. CO 
 
 4-3 1 
 4.89 
 
 Steel 
 
 .297 
 
 . 260 
 
 4.70 
 
 4.11 
 
 VALUES OF k, Jt,, q, AND q, TO BE USED IN FORMULAS 
 
 81, 82, AND 83. 
 
 Material. 
 
 k 
 
 *i 
 
 9 
 
 9i 
 
 Wrought Iron 
 Cast Iron 
 
 .0909 
 . 1 14"> 
 
 3.62 
 4."?6 
 
 i,335 
 669 
 
 .0212 
 .0106 
 
 Steel 
 
 0828 
 
 1 30 
 
 i 767 
 
 0280 
 
 
 
 
 
 
 STRENGTH OF ROPES AND CHAINS. 
 
 Let P= working or safe load in pounds; 
 C circumference of rope in inches; 
 d= diameter of the link of a chain in inches.
 
 TABLES AND FORMULAS. Ill 
 
 For manila ropes, hemp ropes, or tarred hemp ropes, 
 
 P=100C\ (84.) Art. 1175. 
 
 For iron wire rope of 7 strands, 19 wires to the strand, 
 
 P=GOOC\ (85.) Art. 1176. 
 
 For the best steel wire rope, 7 strands, 19 wires to the 
 strand, 
 
 />= 1,000 f. (86.) Art. 1176. 
 
 For open-link chains made from a good quality of wrought 
 iron, 
 
 P= 12,000 d\ (87.) Art. 1179. 
 
 and for stud-link chains, 
 
 P= 18,000 d*. (88.) Art. 1179. 
 
 FORMULAS USED IN SURVEYING. 
 
 RADIUS OF A CURVE. 
 
 To find the radius, the degree being given: 
 Let R the length of the required radius; 
 
 D = the deflection angle equal to one-half the degree 
 of the given curve. 
 
 Kf) 
 
 (89 ' } Ar 
 
 LENGTH OF SUB-CHORDS. 
 
 For curves of short radii : 
 
 Let C the length of the required chord; 
 
 R = the radius of the given curve; 
 
 D the deflection angle of the given curve, equal to 
 one-half its degree. 
 
 C=2^sinZ>. (9O.) Art. 125O. 
 
 LENGTH OF THE TANGENT OF A CURVE. 
 
 When the radius and intersection angle are given: 
 Let T = the length of the required tangent; 
 
 R = the radius of the given curve; 
 
 / = the intersection angle of the given curve. 
 
 r=A'tan|7. (91.) Art. 1251.
 
 112 TABLES AND FORMULAS. 
 
 CHORD DEFLECTION. 
 
 When the length of the chord and the radius are given : 
 Let d the required chord deflection ; 
 
 c = the length of the chord of the given curve; 
 
 R the radius of the given curve. 
 
 <*=. (92.) Art. 1255. 
 
 TANGENT DEFLECTION. 
 
 When the length of the tangent, or of its corresponding 
 chord, and the radius are given: 
 
 Let c = the length of the tangent or corresponding chord ; 
 R = the radius of the given curve. 
 
 tangent deflection ^. (93.) Art. 1255. 
 
 Or, find the chord deflection as in the preceding formula 
 and divide it by 2. The quotient is the required tangent 
 deflection. 
 
 STADIA MEASUREMENTS. 
 
 To find the horizontal distance between two given points, 
 the distance between them having been read with the stadia 
 and the vertical angle taken : 
 
 Let D = the corrected or horizontal distance ; 
 
 c = the constant ; 
 a k = the stadia distance ; 
 n = the vertical angle. 
 
 D c cos n + a k cos 3 . (94.) Art. 13O1. 
 
 To find the difference of elevation between two given 
 points in stadia work : 
 
 Let E = the required difference in elevation; 
 
 c = the constant ; 
 a k the stadia distance ; 
 n = the vertical angle. 
 
 (95.) Art. 13O1. 
 
 - 
 a
 
 TABLES AND FORMULAS. 113 
 
 BAROMETRICAL LEVELING. 
 
 To find the difference of elevation between two points 
 with the aneroid barometer: 
 
 Let Z the difference of elevation between the two given 
 
 stations; 
 h = the reading in inches of the barometer at the 
 
 lower station; 
 //"=the reading in inches of the barometer at the 
 
 higher station; 
 / and /' = the temperature (F. ) of the air at the two stations. 
 
 Z= (log // - log//) X 00,384.3 (l 
 
 + 
 
 (96.) Art. 1304. 
 
 RULES AND FORMULAS USED IN SURVEYING 
 AND MAPPING. 
 
 Rule for Balancing a Survey. As the sum of all 
 
 tJic courses is to any separate course, so is the whole difference 
 in latitude to the correction for that course. A similar 
 proportion corrects the departures. Art. 1315. 
 
 Rule for Double Longitudes. Tlic double longitude 
 of the first course is equal to its departure. 
 
 The double longitude of the second course is equal to the 
 double longitude of the first course plus the departure of that 
 course plus the departure of tJic second course. 
 
 The double longitude of the third course is equal to the 
 double longitude of the second course plus the departure of 
 that course plus the departure of the course itself. 
 
 The double longitude of any course is equal to the double 
 longitude of tJie preceding course plus the departure of that 
 course plus the departure of tJie course itself. 
 
 The double -longitude of the last course (as well as of the 
 first] is equal to its departure. This result, u'Jicn obtained 
 by the above rule, proves the accuracy of the calculation of the 
 double longitudes of all the preceding courses. Art. 1319.
 
 114 TABLES AND FORMULAS. 
 
 APPLICATION OF DOUBLE LONGITUDES TO FINDING 
 AREAS. 
 
 1. Prepare ten columns, and in the first three write the 
 stations, bearings, and distances, respectively. 
 
 2. Find the latitudes and departures of each course by the 
 Traverse Table, placing them in the four following columns. 
 
 3. Balance them by the above rule for balancing a survey, 
 correcting them in red ink. 
 
 4. Find the double longitudes by the rule for double longi- 
 tudes, with reference to a meridian passing tJirong]i tJie 
 extreme east or west station, and place them in the eighth 
 column. 
 
 5. Multiply the double longitude of each course by the cor- 
 rected latitude for that course, placing the north products in 
 the ninth column and the south products in the tenth column. 
 
 6.. Add the last two columns; subtract the smaller sum 
 from the larger, and divide the difference by 2. The quotient 
 will be the area required. Art. 1321. 
 
 AREAS OF IRREGULAR FIGURES. 
 
 Trapezoidal Rule. Divide t lie figure into any sufficient 
 number of equal parts by means of vertical lines called 
 ordi nates; add half the sum of the t^vo end ordinates to the 
 sum of all the other ordinates ; divide by the number of spaces 
 (that is, by one less than the number of ordinates} to obtain 
 the mean ordinate, and multiply t lie quotient by the length of 
 the section to obtain the area. 
 
 Simpson's Rule. Divide the length of the figure into 
 any even number of equal parts, at t/ie common distance D 
 apart, and draw ordinates tJirough the points of division. 
 Add together the length of the first and the last ordinates 
 and call the sum A ; add together the even ordinates and call 
 the stun B ; add together the odd ordinates, except the first 
 and the last, and call the sum C. 
 
 Then, area of figure - x D Art 1324.
 
 TABLES AND FORMULAS. 115 
 
 VOLUMES OF IRREGULAR SOLIDS. 
 
 To find the volume included between two parallel cross- 
 sections whose areas are known, 
 
 Let A = area of one section in square feet; 
 
 B = area of the other section in square feet; 
 
 C distance between the two sections in feet; 
 
 D = required volume in cubic feet. 
 Then, approximately, 
 
 D = ^-^-x C. (97.) Art. 1325. 
 
 The Prismoidal Formula. A more accurate fesult 
 than that given by the last formula is given by the 
 prismoidal formula. 
 
 Let A = area of one section in square feet; 
 
 B =. area of the other section in square feet ; 
 
 M= the area of the average or mean section in 
 
 square feet; 
 
 L = distance between the sections in feet ; 
 5 = the required volume in cubic feet. 
 
 ). (98.) Art. 1326. 
 
 LATITUDES AND DEPARTURES. 
 
 To find the latitude and departure of a course by means 
 of a table of sines and cosines, 
 
 Latitude distance x cosine bearing. (99.) Art. 1338. 
 Departure = distance X sin bearing. (1OO.) Art. 1338. 
 
 FORMULAS USED IN STEAM AND STEAM 
 ENGINES. 
 
 SPECIFIC HEAX. 
 
 W weight of body in pounds ; 
 / temperature before heat is applied ; 
 /, = temperature after heat is applied; 
 c = specific heat of body;
 
 116 TABLES AND FORMULAS. 
 
 U = number of B. T. U. required to raise temperature of 
 body from / to /,. 
 
 U=cW(t l -t}. (101.) Art. 1379. 
 
 TEMPERATURE OF MIXTURES. 
 
 ;>, iv ^ w t , . . . . = weights of the several substances, respect- 
 ively ; 
 
 <:, c l , c t , . . . . specific heats of the substances, respect- 
 ively; 
 
 ,/,,/,,.... = temperatures of the substances, respect- 
 ively ; 
 
 T = final temperature of mixture. 
 
 T _wcf + w^c l t l -\-w t c t t -{- . ...^ Art< 1383 , 
 
 Mixture of Steam and Water. 
 
 W = weight of steam in pounds; 
 
 w = weight of water in pounds; 
 
 t l = temperature of steam; 
 
 / = temperature of water; 
 
 T = final temperature of mixture; 
 
 L = latent heat of steam at the given temperature. 
 
 WORK DO1VE BY PISTON. 
 
 / = net pressure on piston in pounds per square inch 
 V = volume in cubic feet swept through by piston; 
 W= work done by moving piston. 
 
 W= 144/ V. (103.) Art. 1395. 
 
 REAL AND APPARENT CUT-OFF. 
 
 s = apparent cut-off; 
 
 k = real cut-off; 
 
 i = clearance expressed as a per cent, of the stroke. 
 
 =*-i (104.) Art. 1457.
 
 TABLES AND FORMULAS. 117 
 
 HORSEPOWER. 
 
 I. H. P. = indicated horsepower of engine; 
 
 P= mean effective pressure in pounds per sq. in. ; 
 
 A = area of piston in square inches; 
 
 L = length of stroke in feet; 
 
 N = number of strokes per minute. 
 
 PT A N 
 
 L H ' R = - 105 - Art ' 
 
 MEAN EFFECTIVE PRESSURE. 
 
 p = gauge pressure; 
 
 k constant depending upon cut-off (see table of 
 
 Constants used in determining M. E. P.); 
 M. E. P. = mean effective pressure. 
 
 M. E. P. = .9 [(/>+ 14. 7) - 17]. Art. 1496. 
 
 PISTON SPEED. 
 
 / = length of stroke in inches; 
 R number of revolutions per minute; 
 5 = piston speed in feet per minute. 
 
 / /? 
 S = . (1O6.) Art. 1497. 
 
 MECHANICAL EFFICIENCY OF ENGINE. 
 
 I. H. P. = indicated horsepower; 
 
 Friction H. P. = horsepower absorbed in overcoming fric- 
 tion of engine; 
 
 Net H. P.= I. H. P. Friction H. P. = horsepower avail- 
 able to perform useful work ; 
 
 E = efficiency of engine. 
 
 STEAM CONSUMPTION. 
 
 / distance between two points on the indicator dia- 
 gram, one on the expansion line, and the other on 
 the compression line, both being equally distant 
 from the vacuum line ;
 
 118 TABLES AND FORMULAS. 
 
 L = length of indicator diagram ; 
 
 a = absolute pressure of steam at the two points chosen 
 
 W '= weight of a cubic foot at pressure a; 
 
 Q = steam consumption in pounds per I. H. P. per hour. 
 
 . (107.) Art. 1507. 
 
 THERMAL EFFICIENCY OF ENGINE. 
 
 T^= absolute temperature of steam entering cylinder; 
 7" a = absolute temperature of steam leaving cylinder; 
 E = thermal efficiency. 
 
 E = ^ 7 ^. Art. 1512. 
 
 WATER REQUIRE!) BY CONDENSER. 
 
 /j = temperature of departing condensing water; 
 
 / a = temperature of entering condensing water; 
 
 /, = temperature of the condensed steam upon leaving 
 
 the condenser; 
 H= total heat of vaporization of one pound of steam at the 
 
 pressure of the exhaust (see steam table, column 5) ; 
 W = weight of water required to condense a pound of steam. 
 
 W= H ~+ 3 *. (108.) Art. 1520. 
 
 RATIO OF EXPANSION. 
 
 e = ratio of expansion in high-pressure cylinder ; 
 E = total ratio of expansion ; 
 
 v = volume of cylinder receiving steam from boiler ; 
 V volume of cylinder or cylinders exhausting into 
 atmosphere or condenser, 
 
 =~- (109.) Art. 1527.
 
 TABLES AND FORMULAS. 119 
 
 FORMULAS USED IN STEAM BOILERS. 
 
 AIR REQUIRED FOR COMBUSTION AND HEAT OF 
 COMBUSTION. 
 
 C percentage of carbon in a fuel expressed in parts of 
 
 a hundred; 
 // percentage of hydrogen in a fuel expressed in parts 
 
 of a hundred ; 
 A = cubic feet of air required to burn a pound of the fuel. 
 
 A = 1.5Z(C+3H). (110.) Art. 1546. 
 
 B = British thermal units produced by the combustion of 
 
 the fuel ; 
 U'= weight of water that can be evaporated by a pound 
 
 of the fuel. 
 
 (111.) Art. 1547. 
 
 STRENGTH OF BOILER SHELLS. 
 
 P gauge pressure of steam, pounds per square inch ; 
 
 d = diameter of shell in inches; 
 
 '/ length of shell in inches; 
 
 / = thickness of material ; 
 
 S safe stress in material: 9,000 Ib. for wrought iron;. 
 
 11,000 Ib. for steel; 
 
 F = total force tending to rupture the shell; 
 e = efficiency of joint (see table of Riveted Joints). 
 F=Pdl. (112.) Art. 1603. 
 
 (113.) Art. 1604. 
 
 d ' 
 
 HORSEPOWER OF BOILERS. 
 
 W= pounds of water evaporated per hour; 
 F factor of evaporation (see table of Factors of Evapo- 
 ration) ; 
 H = horsepower of boiler. 
 
 W F 
 H=~~. (114.) Art. 1618. 
 
 O4:. O
 
 120 TABLES AND FORMULAS. 
 
 THE SAFETY VALVE. 
 
 A = area of opening in valve-seat in square inches ; 
 
 p = blow-off pressure of valve; 
 
 a = power arm of lever valve; i.e., the distance of valve- 
 stem from fulcrum ; 
 
 d = weight arm of lever valve; i. e., the distance of 
 weight from fulcrum ; 
 
 H reading of spring scale, when the lever and valve are 
 attached to it, at the point where the valve-stem 
 joins the lever; 
 
 P = weight of ball hung on end of lever; 
 
 W= weight required in a dead-weight valve; 
 
 S = pounds of steam generated per hour. 
 
 W=pA. (115.) Art. 1621. 
 / = -?". (116.) Art. 1621. 
 
 /i 
 
 paA=Pd. Art. 1623. 
 (pA -H)a = P t 
 
 d= (pA-H}a 
 
 (117.) Art. 1624. 
 
 DRAFT PRESSURE OF CHIMNEY. 
 
 H = height of chimney in feet; 
 
 T a = absolute temperature of air; 
 
 T c = absolute temperature of escaping gases ; 
 
 / = draft pressure in inches of water. 
 
 (119.) Art. 1662. 
 
 6 7.9V 
 
 77?
 
 TABLES AND FORMULAS. 121 
 
 QUALITV OF STEAM (BARREL CALORIMETER). 
 
 W = weight of cold water in barrel; 
 
 w weight of mixture of steam and water run into 
 
 barrel; 
 / = temperature of steam corresponding to observed 
 
 pressure ; 
 
 t i = original temperature of cold water; 
 / 2 = temperature of cold water after steam is condensed ; 
 L latent heat of a pound of steam at the observed 
 
 pressure (see column 4, steam tables) ; 
 x = portion of weight w that is dry steam ; 
 
 Q = quality of steam = . 
 
 
 FORMULAS USED IN WATER-WHEELS. 
 
 THEORETICAL ENERGY OF A GIVEN HEAD AND WEIGHT 
 OF WATER. 
 
 Let h = available head ; 
 
 v = velocity the water would attain if it fell freely 
 
 through the height It; 
 W= weight of water; 
 
 g = acceleration due to force of gravity = 32.16; 
 K theoretical energy. 
 
 K= Wh = W-. (121.) Art. 1727. 
 
 THEORETICAL POWER. 
 
 Rule. To find t/ie theoretical horsepozver that a given 
 quantity of 'water will furnish, multiply the iveight of water 
 that falls in one second by the distance through which it falls, 
 and divide this product by 550 ; the quotient will be the 
 theoretical horsepower.
 
 122 TABLES AND FORMULAS. 
 
 Let H. P. = theoretical horsepower; 
 
 Q quantity of water falling in cubic feet per 
 
 second ; 
 H = total available fall in feet. 
 
 H. P. = Q X 6 ^ X H = .1136 QH. (122.) Art. 173O. 
 550 
 
 ENERGY OF A JET. 
 
 Let K = energy of the jet; 
 
 W = weight of water that flows from the orifice or 
 
 nozzle in one second ; 
 
 w = weight of a cubic foot of water 62.5 pounds; 
 a = area of the jet in square feet; 
 v = velocity of flow from the orifice in feet per 
 
 second ; 
 
 c = coefficient of velocity for the orifice; 
 h = head on the orifice in feet ; 
 g = acceleration due to gravity = 32.16. 
 
 K=W^--cWh. (123.) Art. 1731. 
 
 W=wav. (124.) Art. 1731. 
 
 K= *^ = cwavh (125.) Art. 1731. 
 
 PRESSURE DUE TO IMPACT AND REACTION OF A JET. 
 
 Let P pressure produced by the impact; 
 R = reaction of the jet; 
 W ' weight of water that flows from the orifice or 
 
 nozzle in one second ; 
 
 w= weight of a cubic foot of water = 62.5 pounds; 
 a = area of the jet in square feet; 
 v = velocity of flow from the orifice in feet per 
 
 second ; 
 
 c = coefficient of velocity for the orifice; 
 h = head on the orifice in feet ; 
 g = acceleration due to gravity 32.16.
 
 TABLES AND FORMULAS. 123 
 
 Pressure on a Vertical Surface. When the jet 
 impinges on a vertical surface the pressure is 
 
 P=wa- = Zcwah = W"-. (126.) Art. 1732. 
 
 o o 
 
 Reaction. The reaction of the jet is 
 R = P^wa = <lcwak= W-. (127.) Art. 1732. 
 
 o o 
 
 Pressure Produced by Change of Direction. 
 
 Let a = the angle between the original direction of the 
 jet and its direction after being deflected. 
 
 The pressure exerted on the deflecting surface in the 
 original direction of the jet is 
 
 P = (1 - cos a) W-. (128.) Art. 1734. 
 
 o 
 
 Pressure on a Hemispherical Cup. -When the jet 
 strikes into a hemispherical cup a= 180, and the pressure is 
 
 P = (l - cos 180) W- = 2 W-\ 
 g g 
 
 that is, the pressure is twice as great as the pressure pro- 
 duced iu lien the jet struck a surface at right angles to its 
 direction of motion. Art. 1734. 
 
 Effect When the Surface Is in Motion. 
 
 Let v'= the velocity with which the surface moves along 
 the line of motion of the jet. 
 
 The pressure on the surface is 
 
 P = (l - cosrt ) w(l ~~\ Z ^^-- (129.) Art. 1735. 
 If the surface is a hemispherical cup, the pressure is 
 P=. 0622 ~^(z/ -?/)'. (130.) Art. 1735. 
 
 The theoretical work of a Jet impinging in a moving 
 hemispherical cup is a maximum when the velocity of the cup
 
 124 TABLES AND FORMULAS. 
 
 is one-half tJie Telocity of t/ie jet, and it is equal to tJie 
 theoretical work that zvould be done by the energy due to the 
 velocity of the water. Art. 1 735. 
 
 EFFICIENCY. 
 
 Rule I. To fijid the amount of work or poivcr that can 
 be obtained from a given fall of ivatcr 'when the efficiency of 
 the motor is given, multiply the theoretical work or power by 
 the efficiency expressed as a decimal fraction, and the product 
 will give the available work or power. Art. 1 737. 
 
 Rule II. To Jin d the quantity of water required to fur- 
 nish a given amount of power with a given efficiency, divide 
 the theoretical quantity of water by t/te efficiency; the quotient 
 will be the quantity required. Art. 1 737. 
 
 OVERSHOT WATER-WHEELS. 
 
 Let H = total fall of water ; 
 
 v = velocity of circumference of wheel ; 
 
 v e = velocity with which water enters wheel ; 
 
 h = head required to produce velocity of entry v e \ 
 
 D outside diameter of wheel ; 
 
 N = number of revolutions of wheel; 
 
 Z ' = number of buckets; 
 
 b breadth of buckets; 
 
 d = depth of buckets; 
 
 Q quantity of water in cubic feet per second; 
 
 c = clearance between wheel and trough. 
 
 v e =l$vto%v. (131.) Art. 1 743. 
 
 //=l.l|C (132.) Art. 1743. 
 
 D=ff-(A + c). Art. 1 743. 
 N=lQ.l^. (133.) Art. 1743. 
 
 Z= IQDto 12 Z>. (134.) Art. 1743. 
 
 d 10 inches to 15 inches. (135.) Art. 1743. 
 
 =3Uo4. (136.) Art. 1743.
 
 TABLES AND FORMULAS. 125 
 
 BREAST WHEELS. 
 
 The following rules may be used for the principal dimen- 
 sions of a breast wheel: 
 
 Velocity of circumference of wheel v = 2 feet per second 
 to 8 feet per second. Velocity of entry v e = \%v to Zv. 
 
 Depth of floats d = 10 inches to 15 inches. Pitch of 
 floats t = d. 
 
 Diameter of wheel, about twice the total head. 
 
 Breadth of wheel, b = 14--^- to 2-p, where Q is in cubic 
 a v dv 
 
 feet per second, b and d in feet, and v in feet per second. 
 Art. 1749. 
 
 UNDERSHOT WHEELS. 
 
 Let H. P. = horsepower; 
 
 v = velocity of water in race in feet per second; 
 v i = velocity of circumference of wheel in feet per 
 
 second ; 
 Q = quantity of water flowing through race in 
 
 cubic feet per second ; 
 F area of the immersed portion of the float of a 
 
 paddle wheel in an unconfined current. 
 
 For a wheel in a confined race, 
 
 H. P. = .00215 (v-v,) 7', Q. (137.) Art. 1754. 
 
 For a simple paddle wheel in an unconfined current, 
 
 H. P. = .00282 (v - v^ v i\ F. (138.) Art. 1755. 
 
 FOTVCELET'S WHEEL. 
 
 Let H= total fall in feet; 
 
 Q = the quantity of water in feet per second; 
 
 D the outside diameter of the wheel in feet; 
 
 d = depth of floats in feet ; 
 
 d l = depth of water current entering the wheel in feet ; 
 
 v e = the velocity of the water current entering the 
 
 wheel in feet per second ; 
 b = breadth of the wheel and of the sluice in feet;
 
 126 
 
 TABLES AND FORMULAS. 
 
 v l velocity of circumference of wheel in feet per 
 
 second ; 
 
 R = radius of curvature of floats; 
 A = angle A O B (see Fig. 527, Art. 1 756) ; 
 u number of revolutions per minute of wheel; 
 n = number of floats in wheel. 
 
 H and Q must be determined by actual measurement; 
 the other dimensions may then be made as follows: 
 
 d = 
 
 d l \ foot to 1 foot ; 
 
 a =20 to 45 
 
 Art. 1757. 
 
 TURBINES. 
 
 In the rules and formulas used to determine the principal 
 dimensions of reaction turbines 
 
 Let Q = the available quantity of water in cubic feet per 
 
 second ; 
 
 h = the total available head on the wheel in feet ; 
 v e = the velocity of the flow from the guide buckets 
 
 in feet per second ; 
 i> r = the relative velocity of water entering the wheel 
 
 buckets in feet per second; 
 v = the relative velocity of flow from the wheel 
 
 buckets in feet per second; 
 v f = the absolute velocity of the water leaving the 
 
 wheel buckets; 
 
 v w = the velocity of the wheel buckets at entrance; 
 v' w = the velocity of the wheel buckets at discharge;
 
 TABLES AND FORMULAS. 12? 
 
 a the angle which the direction of outflow from 
 
 the guides makes with the radius in a radial- 
 flow turbine or with a perpendicular to the 
 
 direction of motion of the wheel buckets in an 
 
 axial-flow turbine; 
 a 1 = the angle which the relative direction of inflow 
 
 to the wheel makes with the same lines; 
 a y = the angle which the relative direction of flow 
 
 from the vanes makes with the same lines; 
 A = the effective outflow area of guide passages in 
 
 square feet; 
 A! = the effective inflow area of wheel passages in 
 
 square feet ; 
 A^ = the effective outflow area of wheel passages in 
 
 square feet; 
 A^ = sectional area of flow for draft tube in square 
 
 feet; 
 A t = effective outflow area of draft tube in square 
 
 feet; 
 
 N = the number of revolutions per minute ; 
 r = the mean radius of an axial turbine in feet; 
 r l = the radius of the wheel at inflow, in feet, for a 
 
 radial-flow turbine; 
 r 2 = the radius of the wheel at outflow, in feet, for a 
 
 radial-flow turbine; 
 
 g = acceleration due to force of gravity; 
 K l = a coefficient for finding the radius r or r, , from 
 
 the area A ; 
 
 k = a coefficient for finding the velocity f e ; 
 P = the pitch of the guide buckets; 
 P^ = the pitch of the wheel buckets; 
 Z = the number of guide buckets; 
 Z^ = the number of wheel buckets; 
 // o = the height of guide buckets in an axial-flow 
 
 turbine; 
 h\ = the height of wheel buckets in an axial-flow 
 
 turbine;
 
 128 TABLES AND FORMULAS. 
 
 x = the distance between the outflow ends of two 
 consecutive guide buckets, measured perpen- 
 dicular to the direction of flow ; 
 
 x t = the distance between the outflow ends of two 
 consecutive wheel buckets, measured perpen- 
 dicular to the direction of flow ; 
 
 / = the thickness of guide buckets near ends; 
 
 /j = the thickness of wheel buckets near ends; 
 
 s the part of the distance x that would be covered 
 by the inflow end of one wheel bucket, to be 
 measured in the same direction as .v; 
 
 e = the width of outflow ends of guide buckets ; 
 
 e^ =. the width of inflow ends of wheel buckets; 
 
 e^ =. the width of outflow ends of wheel buckets. 
 
 General Relations. The usual proportions and values 
 to be used in designing the different types of wheel are as 
 follows : 
 
 (a) For axial turbines using a large quantity of water 
 under a low head, where is greater than 16 square feet, 
 
 A = 70 to 66. 
 A^ = 70 to 66. 
 
 k = 1 to l\. 
 
 P = 10 inches to 12 inches. 
 
 ^ inch to f inch for cast iron. 
 
 inch to f inch for wrought iron. 
 
 (b] For axial turbines using a medium quantity of water 
 
 under medium head, where is greater than 2 and less 
 
 v e 
 
 than 16 square feet, 
 
 A = 75 to 70. P= -f to ^-_. 
 
 ^, = 7^ to 73. :3 ' 7 4 ' 5 
 
 " / = r = same as above. 
 
 A t = .67. 
 
 k = 1.25 to 1.5. h = // = T - to -f-. 
 
 4 4. o
 
 TABLES AND FORMULAS. 129 
 
 (r) For axial turbines using a small quantity of water 
 under a high head, where ^- is less than 2 square feet, 
 
 75 to 73 . p~ 4^ t g inches. 
 
 A t 77 to 74. / _ /^ _ same as above. 
 A", .07. r r 
 
 k 1. 5 to 2. ^ ~~ " iO t0 3 ' 
 
 (;/) For radial inward-flow turbines, where Q ranges 
 from 2.4 to 275 cubic feet per second, and h is from 3 feet 
 
 to 80 feet, 
 
 A = 80 to 66. 
 A 9 80 to 66. 
 r a = r, to 4 TV 
 A^ = 0.725 to 0.64. 
 k -0.75 to 1.75. 
 P 4^- inches to 12 inches. 
 / = ^, = same as for axial-flow turbines. 
 Z, = Z to . 7 Z. 
 
 (c) For radial outward-flow turbines, where Q ranges 
 from 2.5 to 350 cubic feet per second, and h ranges from 
 3 feet to 25 feet, 
 
 A = 75 to 66 and less. 
 A n _ = 80 to 60 and less. 
 
 KI = 0.725 to 0.64. 
 k = 1.5 to 2. 
 
 / = t l = same as for axial-flow turbines. 
 Z, == 1.2 Z to 1. 3 Z. Art. 1 782. 
 
 Velocity of Entrance. From the general relations 
 select a value of A" t to correspond with the type of wheel 
 and the conditions under which it works; then, 
 
 (139.) Art. 1783.
 
 130 TABLES AND FORMULAS. 
 
 Effective Area. From Q and v e the effective area A 
 of the passages from the guide buckets is computed from 
 the formula 
 
 A=j-. (140.) Art. 1783. 
 
 Radius. From this value of A the mean radius of a 
 parallel-flow wheel is computed from the formula 
 
 ) Art. 1783. 
 where k is a coefficient that depends on the relation between 
 <2 and k. (See general relations.) 
 
 For a radial-flow turbine, the radius of the wheel where 
 the water enters is given by the formula 
 
 r l = k^A. (1410.) Art. 1783. 
 
 where k depends on the style of wheel, whether outward 
 flow or inward flow. (See general relations.) 
 
 Revolutions. The number of revolutions per minute 
 is given by 
 
 = 9.549 (142*.) Art. 1783. 
 
 for axial-flow turbines, and for radial-flow turbines, 
 N= 9.549^'. (1420.) Art. 1783. 
 
 Number of Vanes. Having chosen the pitch P approxi- 
 mately to suit the given conditions, the number of guide 
 vanes for an axial-flow turbine is given by the formula 
 
 Z=^, (143.) Art. 1786. 
 
 and for a radial-flow turbine the number of guide vanes is 
 Z=^~^. (1430.) Art. 1786. 
 
 These formulas give approximate values for Z, and the 
 actual value is the nearest corresponding whole number. 
 The number of wheel vanes Z, for axial-flow turbines
 
 TABLES AND FORMULAS. 131 
 
 should always be greater than Z. For ordinary cases we 
 may take 
 
 Z,=Z+2. (144*.) Art. 1787. 
 
 For radial inward-flow turbines use the values 
 
 Z 1 = Zto.7Z, (1440.) Art. 1787. 
 
 and for radial outward-flow turbines 
 
 Z, = 1.2Zto 1.3Z. (144f. Art. 1787. 
 
 Pitch. The exact pitch for the guide vanes of axial-flow 
 wheels is now given by 
 
 P = ^L t (145.) Art. 1787. 
 
 
 
 and the pitch of wheel vanes by 
 
 P l = "^. (146*.) Art. 1787. 
 
 ^j 
 
 The pitch at the outflow ends of guide vanes for radial 
 flow turbines is 
 
 P = ^1. (1450.) Art. 1787. 
 
 For the inflow ends of the wheel vanes the pitch is 
 /^^p. (1460.) Art. 1787. 
 
 "Width of Vanes. Width of outflow end of guide 
 vanes, 
 
 The width c l of the inflow end of the wheel vanes is made 
 a little greater than e, usually 
 
 r E = ? + inch to f + f. (148.) Art. 1796. 
 
 Width of outflow end of wheel vanes, 
 
 f t = ^-^. ( 1 49.) Art. 1 796.
 
 132 TABLES AND FORMULAS. 
 
 FORMULAS USED IN HYDRAULIC MACHINERY. 
 
 SIZE OF AIR AND VACUUM CHAMBERS. 
 
 Let V = volume of piston displacement ; 
 V l = volume of air chamber; 
 V^~ volume of vacuum chamber. 
 
 For ordinary double-acting pumps working under moder- 
 ate pressures at ordinary speeds, 
 
 V^-^V. Art. 1885. 
 
 For pressures of 100 pounds per square inch and upwards, 
 or for high piston speeds, 
 
 V l = QV. Art. 1885. 
 For ordinary cases, make 
 
 F = iF. Art. 1889. 
 
 CALCULATIONS RELATING TO PUMPS. 
 
 Displacement. 
 
 Let D = displacement in cubic feet per minute; 
 
 d = diameter of piston or plunger in inches; 
 
 L =. length of stroke of piston or plunger in inches; 
 
 N number of discharge strokes made by piston or 
 
 plunger in 1 minute. 
 Then, D = .000455 d*LN. Art. 19O5. 
 
 Slip. 
 
 Let s = slip ; 
 
 D = displacement; 
 C actual discharge. 
 
 Then, s= ~ . Art. 19O9. 
 
 Head and Pressure. 
 
 Let H head in feet ; 
 
 P = pressure in pounds per square inch. 
 
 Art. 1914. 
 Art. 1915o
 
 TABLES AND FORMULAS. 133 
 
 Size of Piston or Plunger. 
 
 Let G = number of gallons discharged per minute; 
 
 5 = speed in feet per minute of piston or plunger; 
 
 d = diameter of piston or plunger in inches; 
 
 F = number of cubic feet discharged per minute. 
 Then, the theoretical diameter of piston or plunger is 
 
 d = 4. 95 V ^ = 13. 54 |/^-. Art. 1916. 
 
 If we add 25 per cent, to the required discharge to allow 
 for slip, the diameter of the piston or plunger will be 
 
 </= 5.535 V ^ = 15. 138 V^-. (152.) Art. 1916. 
 
 Discharge. 
 
 The theoretical discharge in cubic feet equals the displace- 
 ment. 
 
 The theoretical discharge in gallons per minute is 
 . Art. 1917. 
 
 If we make the same allowance for slip as was made in 
 formula 152, the discharge in gallons per minute is 
 G =. 03264 d*S; (153.) Art. 1917. 
 
 and in cubic feet per minute, 
 
 F = . 00436 d* S. Art. 1917. 
 
 Power. 
 
 Let H. P. = horsepower; 
 
 H = vertical height in feet from the surface of the 
 
 water in the well or sump to the center of the 
 
 outlet end of discharge pipe ; 
 G = discharge in gallons per minute; 
 F = discharge in cubic feet per minute. 
 
 The theoretical power is 
 
 H. P. = .000-254 G H = .0019 F H. Art. 1918. 
 
 If, for ordinary cases, the frictional resistances are
 
 134 TABLES AND FORMULAS. 
 
 assumed to be 50 per cent, of the power developed by the 
 engine, the power required is 
 
 H. P. = .00038 GH. (154.) Art. 1918. 
 
 To find the height to which a given power will raise a 
 given quantity of water, making the same allowance for 
 friction as in the last formula, 
 
 1919. 
 
 Size of Steam Cylinder. 
 
 Let S = steam piston speed ; 
 
 d = diameter of steam cylinder in inches; 
 
 r = ratio between length of stroke and diameter of 
 cylinder; 
 
 / = length of stroke in feet; 
 
 N = number of strokes per minute; 
 H. P. .= horsepower; 
 
 P = steam pressure in pounds per square inch. 
 
 Then, for simple direct-acting steam pumps, 
 
 rJ'N 
 
 Art. 1920. 
 
 Art 
 
 i 
 
 Having obtained the diameter of the steam, piston by 
 either of the above formulas, the stroke can be found by 
 multiplying the diameter by the value of the ratio r. When 
 formula 157 is used, the number of strokes can be found 
 by dividing the piston speed by the length of the stroke 
 in feet. 
 
 Sizes of Suction and Delivery Pipes. 
 
 For a velocity of 200 feet per minute in the suction pipe 
 and 400 feet per minute in the delivery pipe, 
 
 Let d^ = diameter of suction pipe; 
 d^ = diameter of delivery pipe; 
 G discharge in gallons per minute.
 
 TABLES AND FORMULAS. 135 
 
 = 4.95 Y~, or d l = .35 \/G. (158.) Art. 1921. 
 
 < = 4.95 f , or <f 9 = .25 \/G. (159.) Art. 1921. 
 
 The pipes may be made larger than the values calculated 
 by the above formulas, particularly the suction pipe, but it 
 is not good practice to make them any smaller. 
 
 DUTY OF A PUMP. 
 
 Old Standard. According to the old standard, the duty 
 
 of a pumping engine is the number of pounds of ivater raised 
 
 one foot high for each 100 pounds of coal burned in the boiler. 
 
 Let G = number of gallons discharged in a given period 
 
 h = total vertical distance in feet from the surface 
 
 of the water in the well, or other source of 
 
 supply, to the point of discharge; 
 
 W = the number of pounds of coal burned in the 
 
 given period ; 
 
 D = the duty in foot-pounds. 
 Then, 
 
 Standard Recommended by Committee of Ameri- 
 can Society of Mechanical Engineers. 
 
 The duty of a pumping engine is equal to the total 
 number of foot-pounds of work actually done by the pump, 
 divided by the total number of heat units in the steam used 
 by the pump, including the steam used by the condensers 
 (if any) and boiler feed, and this quotient multiplied by 
 1,000,000. 
 
 The number of foot-pounds of work done by the pump is 
 to be found as follows : A pressure gauge is attached to the 
 discharge pipe and a vacuum gauge to the suction pipe, 
 both as near the pump as convenient; then the pressure 
 against which the pump plunger works is equal to the 
 difference in the pressures shown by these two gauges plus
 
 136 TABLES AND FORMULAS. 
 
 the head due to the difference in level of the points in the 
 pipes to which they are attached; and the number of foot- 
 pounds is equal to the continued product of the net area of 
 the plunger (making allowance for piston rods), the length 
 of the plunger stroke in feet, the number of plunger strokes 
 made during the trial, and the pressure against which the 
 pump plungers work, as shown by the gauges. 
 
 The number of heat units furnished to the pump is the 
 number of British thermal units (B. T. U.) in the steam 
 from the boilers, and is to be determined by an evaporation 
 test of the boilers. If we let 
 
 A = the net area of the plunger in square inches; 
 
 P = the pressure in pounds per square inch indicated by 
 
 the gauge on the discharge pipe ; 
 
 p = the pressure in pounds per square inch corresponding 
 to the vacuum indicated by the gauge on the suction 
 pipe; 
 5 = the pressure in pounds per square inch corresponding 
 
 to the difference in level between the two gauges; 
 L = the average length of stroke of pump plunger in feet ; 
 N the total number of single strokes of plunger made 
 
 during the trial; 
 H = the total number of heat units consumed by the 
 
 engine during the trial; 
 W= the total number of foot-pounds of work done by the 
 
 pump during the trial; and 
 D = the duty. 
 Then,W=A(P/> + S)LN, (161.) Art. 1924. 
 
 W 
 and Z> = -, x 1,000,000 = 
 
 
 (162.) Art. 1924. 
 
 CALCULATIONS RELATING TO HYDRAULIC MACHINERY. 
 
 Relations Between Pressure and Size of Ram. 
 
 Let D = the diameter of a hydraulic piston or ram ; 
 
 W = the weight of the ram and attachments that 
 must be lifted by the water;
 
 TABLES AND FORMULAS. 137 
 
 p = the pressure of the water in pounds per square 
 
 inch; 
 
 /'"= the percentage of friction; 
 /-* = the net pressure exerted by the ram. 
 To find the net pressure exerted by a ram or plunger of a 
 hydraulic press, 
 
 P= .7854 X > 2 X/>X (I-T) - W. (163.) 
 
 Art 1969. 
 
 To find the pressure per square inch required to exert a 
 given net pressure when the diameter and weight of the 
 ram and the percentage of friction are given, use the for- 
 mula 
 
 p\ w 
 /=- ^r. (164.) Art. 1970. 
 
 To find the diameter of piston or ram required to exert 
 a given net pressure, use the formula 
 
 />==/- (165.) Art. 1971. 
 
 "Weight and Volume of Accumulators. 
 
 Let IT, = Aveight of accumulator ram; 
 
 ]l'\ load on accumulator ram; 
 
 /}, = diameter of accumulator ram ; 
 
 /, = maximum pressure per square inch in the accu- 
 mulator cylinder; 
 
 f t = minimum pressure per square inch in the accu- 
 mulator cylinder; 
 
 p = mean pressure per square inch in the accumu- 
 lator cylinder; 
 
 5 = stroke of accumulator ram ; 
 
 V = total volume of water displaced by accumulator 
 ram during the stroke S; 
 
 F = the percentage of friction. 
 
 To find the mean pressure / corresponding to a given 
 case, use the formula
 
 138 TABLES AND FORMULAS. 
 
 The maximum pressure is found by the formula 
 
 W 4- W 
 ?,= r- (167.) Art. 1973. 
 
 and the minimum pressure by 
 
 (168.) Art. 1973. 
 
 The "weight required to produce a given mean pressure 
 when diameter and weight of the ram are known may be 
 found from the following formula: 
 
 fF, = ,7854x/V X/- W;. (169.) Art. 1974. 
 
 The relations between the stroke, diameter, and volume 
 of an accumulator are given by the following formulas: 
 V = .7854 /VS. (17O.) Art. 1977. 
 
 777 
 Z\ = 1.128 f-^. (172.) Art. 1977. 
 
 o 
 
 In the above formulas, if 7) l and vS are in inches, the 
 volume will be given in cubic inches; and if D l and S are in 
 feet, V will be given in cubic feet. 
 
 FORMULAS USED IN WATER SUPPLY AND 
 DISTRIBUTION. 
 
 DIMENSIONS OF SPILLWAY OR O% 7 ERFLOW. 
 
 Let Z= length of lip of spillway in feet; 
 
 A area of watershed above dam in square miles; 
 D depth of notch of spillway in feet ; 
 Q =. cubic feet of water per second per square mile; 
 C = a constant depending on the character of the 
 
 dam and its surroundings and the area of the 
 
 watershed.
 
 TABLES AND FORMULAS. 139 
 
 Then, L = W^A. (173.) Art. 2O48. 
 
 C. (174.) Art. 2048. 
 
 16 
 
 If we assume Q = 64, which corresponds to a little over 
 41 million gallons per 24 hours, per square mile, and repre- 
 sents a very powerful freshet flow, although, perhaps, not 
 the maximum, formula 174 reduces to 
 
 D = \/~A + C. (175.) Art. 2O49. 
 
 MASONRY DAMS. 
 
 Let A = thickness of top of a trapezoidal dam in feet ; 
 B = thickness of base of dam in feet ; 
 C = a factor of safety against either sliding or 
 
 overturning; 
 D density (weight per cubic foot) of material of 
 
 which dam is built ; 
 
 H head of water pressing against the dam in feet ; 
 R = resistance of wall to sliding; 
 T = horizontal thrust in pounds on the dam, due to 
 
 the head//"; 
 
 MR = moment of resistance of dam against over- 
 turning by rotating about its outer toe; 
 M T moment of thrust about the outer toe of the 
 
 dam. 
 The thrust is 
 
 r=31.25# a , (176.) Art. 2063. 
 
 and the moment of thrust 
 
 MT=10A2H\ (177.) Art. 2O63. 
 
 The resistance of the wall to sliding is 
 
 R = 0.1! 5 AD. (178.) Art. 2O65. 
 
 The moment of resistance to overturning for a wall with 
 vertical sides is 
 
 7) //" 7' 2 
 
 - , (179.) Art. 2066.
 
 140 TABLES AND FORMULAS. 
 
 and for a trapezoidal wall 
 
 MR = ^(A B - ~- + /A ( 1 80.) Art. 2O66. 
 
 The relation between A, B, D, and H for a factor of 
 safety C against sliding is given by the formula 
 
 B = ^- D CH -A. (181.) Art. 2067. 
 
 For a factor of safety C against overturning, the breadth 
 of the base is given by the formula 
 
 = $ + 3A*-- (182.) Art. 2068. 
 
 JJ A 
 
 Average Dimensions. For practical values of A and 
 , a satisfactory value of B is 
 
 B = f H to \H. Art. 2O7O. 
 
 HIGH MASONRY DAMS. 
 
 Maximum Unit Stress on Base of Dam for 
 Unequally Distributed Load. 
 
 Let L = length of base of a section through the dam; 
 
 d = length of the shorter segment of this base; 
 L d length of the remaining segment; 
 W = the resultant of the weight of the section, or 
 
 the vertical component of this resultant; 
 P maximum unit stress. 
 
 There are three empirical formulas for the value of P t 
 which experience shows give satisfactory results; viz. : 
 
 P=^f(L-\.Zd), (183.) Art. 2072. 
 
 o w 
 P=~j, (1 84.) Art. 2072. 
 
 and P = K( ^~^. (185.) Art. 2O72. 
 
 Of these formulas, the last is probably the most satis- 
 factory.
 
 TABLES AND FORMULAS. 141 
 
 DARCY'S FORMULAS. 
 
 Let D = diameter of pipe in feet ; 
 H total head in feet; 
 L = total length in feet; 
 V = velocity of efflux in feet per second; 
 C = an experimental coefficient (see table of Coeffi 
 
 cients for Darcy's Formula) ; 
 
 Q = quantity discharged in cubic feet per second; 
 A = area of pipe in square feet ; 
 
 Ti- 
 ll head per 1,000 feet of length = -=-. 
 
 I ? 000 -Z-- 
 
 Fundamental Formulas for Long Pipes. 
 
 1. (186.) Art. 2O92. 
 
 (187.) Art. 2092. 
 
 Q--Air. (188.) Art. 2092. 
 
 /75~Tr 
 
 Q = 0.7854 D*VLL. (189.) Art. 2O92. 
 
 (190.) Art 2092. 
 
 Approximate Formulas for Rough Pipes 
 
 For pipes from 8 inches to 48 inches in diameter, 
 
 Art . 2094 . 
 
 (192.) Art. 2O94. 
 (193.) Art. 2094. 
 ( 1 94.) Art. 2094. 
 Art.2094.
 
 142 TABLES AND FORMULAS. 
 
 For pipes from 3 inches to 6 inches in diameter, 
 j2_ = 0.785. (196.) Art. 2O94. 
 
 0=0.894/^1. (197.) Art. 2094. 
 
 Formulas for Smooth Pipes. 
 
 Q=^%D'/i. (198.) Art. 2O95. 
 
 7)5=2. (199.) Art. 2O95. 
 (2=1.404/271 (2OO.) Art. 2O95. 
 
 General Relation Between Smooth and Rough 
 Pipes. 
 
 In general, the discharge tJirougJi a smooth pipe is 1.40 
 times t/iat through a rough pipe of the same diameter; and, 
 reciprocally, the discharge through a rough pipe is 0. 70 times 
 that through a smooth one of the same diameter. These 
 factors represent the practical limits between which the 
 extremes of smoothness and roughness can affect the flow. 
 Art. 2O95. 
 
 Formulas for Velocity. 
 
 For rough pipes of from 8 inches to 48 inches in diameter, 
 
 F=1.274/Z>1. (201.) Art. 2096. 
 
 For rough pipes of smaller diameter, 
 
 F=1.134/2>1 (2O2.) Art. 2O96. 
 
 For smooth pipes of large diameter, 
 
 V= 1.78 4/2?l. (2O3.) Art. 2O96. 
 
 For smooth pipes of small diameter, 
 
 F= 1.60 4/2)1. (2O4.) Art. 2O96.
 
 TABLES AND FORMULAS. 143 
 
 General Relation Between the Elements of Two 
 Pipes. 
 
 Let D, Q, L, //, and C be the respective elements of one 
 pipe and 77, Q', L ', //', and C' the similar elements of 
 another; then, 
 
 DHC'L'V* 
 
 D'H'CL V 
 If, as can usually be done, we make C C', we have 
 
 * = L (205.) Art. 2O97. 
 
 Also, = I- (206.) Art. 2O97. 
 
 If L and H equal, respectively, L' and //', 
 
 Tj = T ^T- (207.) Art. 2O98. 
 
 To find the number x of small pipes with the diameter D' 
 to replace a pipe whose diameter is D, 
 
 x = fX-1. (208.) Art. 2098. 
 
 COMPOUND PIPES. 
 
 To find the diameter of a simple pipe that will give the 
 same delivery as a given compound system: 
 
 Let D = diameter of the simple pipe; 
 L length of the simple pipe; 
 d, d' , d" , etc. = diameters of the respective sections 
 
 of the compound pipe; 
 /, /', /", etc. = lengths of the respective sections of 
 
 the compound pipe. 
 
 Then, 
 
 = - + * + "* + etc ' (209 Art ' 21 1 1 '
 
 144 TABLES AND FORMULAS. 
 
 PUMPING INTO MAINS. 
 
 Theoretical horsepower required to force a given quantity 
 of water into a main against a given pressure head: 
 
 Let H. P. = theoretical horsepower; 
 H pressure head in feet; 
 <2 = quantity of water in cubic feet per second. 
 
 H. P. = '. (210.) Art. 2117. 
 
 0.0 
 
 "WEIGHTS ANI> THICKNESS OF CAST-IRON PIPES. 
 
 Let W weight in pounds; 
 D diameter in inches; 
 T ~ thickness in inches; 
 L length in inches; 
 P = weight in long tons (2,240 pounds); 
 M = length in miles ; 
 
 W = approximate weight per foot in pounds; 
 H = total head in feet. 
 
 T) Tx L. (211.) Art. 2125. 
 
 T} T. (212.) Art. 2126. 
 
 P=Z5Jlf(D + T) T. (213.) Art. 2127. 
 
 T = 0. 00006 HD+ 0.01 337? + 0.290. (214.) Art. 2128. 
 
 DARCY'S FORMULAS FOR FLOW IN OPEN CHANNELS. 
 
 Let U = mean velocity of flow in feet per second ; 
 
 5 water section in square feet; 
 W P wet perimeter in feet; 
 
 R = mean hydraulic radius = -rrj-n't 
 
 I = slope of free water surface per foot of length = 
 
 total fall of surface divided by total length ; 
 D = interior diameter of a circular conduit in feet.
 
 TABLES AND FORMULAS. 145 
 
 For an ordinary tunnel or channel lined with well-laid 
 brick. 
 
 Art. 2,43. 
 
 For a circular brick-lined conduit running full, 
 
 FORMULAS USED I1V IRRIGATION. 
 
 APPROXIMATE DISCHARGE OF WEIRS. 
 
 Let / = length of notch in feet; 
 
 //= measured head on crest in feet; 
 Q discharge in cubic feet per second; 
 
 then, Q=^IH\ (217.) Art. 2163. 
 
 FLOW OF WATER THROUGH CONDUITS. 
 
 Let // = difference in level between the ends of the canal, 
 
 or any two cross-sections of the canal ; 
 / = horizontal length of that portion of the canal 
 included between the sections whose difference of 
 level is // ; 
 
 s = slope = the ratio y; 
 
 a area of the water cross-section; 
 p = wetted perimeter; 
 
 r = hydraulic radius the ratio ; 
 
 c ' = a coefficient depending on the nature ot the sur- 
 face of the conduit; 
 and i> = mean velocity of flow. 
 
 The laws for the resistance to flow may be expressed by 
 the relation Ji a = c' I p i? , from which we have the general 
 formula 
 
 v = <- X a - = \~ xsxr. (218.) Art. 21 73.
 
 140 TABLES AND FORMULAS. 
 
 By replacing \ , by the equivalent factor r, we have 
 v c\/rs, the same as formula 5O, Art. 1O33. 
 Formulas for Flow in Canals. 
 
 Canals with earthen banks, 
 
 Canals lined with dry stone, 
 
 (22O.) Art. 2183. 
 
 Canals lined with rubble masonry, 
 
 v r~s-^ 'iTT 7 "- (221.) Art. 2184. 
 
 Wooden flumes, 
 
 TIMBER FOR PLUMBS. 
 
 Let W= total load in pounds carried by any beam; 
 / = length of beam in inches; 
 b = breadth of beam in inches; 
 d = depth of beam in inches; 
 
 S maximum unit fiber stress in pounds per square 
 inch. 
 
 For a simple beam with a uniformly distributed load, 
 pr-fi^S. (222.) Art. 2189. 
 
 For a simple beam with a concentrated load at the middle, 
 W=% b --S. (223.) Art. 2190.
 
 TABLES AND FORMULAS. 
 
 147 
 
 For a beam with a concentrated load at a distance / : from 
 one support and / from the other, where / : -{- / 2 = /, 
 
 (224.) Art. 2191. 
 
 l.L 
 
 For a beam on which the load at one end is zero, with a 
 uniform increase in the load to the other end, 
 
 W=1.3~-S. (225.) Art. 2192. 
 
 SAFE WORKING STRESS S. 
 
 For good sound timber. 
 
 Kind of Timber. 
 
 Safe Working Stress. 
 
 Steady Load. 
 
 Variable Load. 
 
 Yellow Pine 
 White Oak 
 
 i, 800 
 1,350 
 1,250 
 1,200 
 
 1,100 
 
 1,200 
 1,000 
 900 
 
 800 
 
 Spruce 
 
 Hemlock 
 
 White Pine 
 
 
 TRUSSES. 
 
 Trussed Stringers. 
 
 Let L = span in inches; 
 
 H depth of truss in inches; 
 
 b = breadth of stringer in inches; 
 d = depth of stringer in inches; 
 W t = total uniformly distributed load in pounds; 
 5 = allowable unit fiber stress in stringer or strut; 
 S^ total stress in tie-rods; 
 
 h width of strut in inches; 
 
 t = thickness of strut in inches.
 
 148 TABLES AND FORMULAS. 
 
 Relation between W t and dimensions of stringer, 
 
 Stress in tie-rods, 
 
 Art. 2198. 
 
 Stress in strut, 
 
 W t = -\htS. (229.) Art. 2198. 
 
 The King-Rod Truss. 
 
 Let L length of span in inches; 
 
 W t = total uniformly distributed load in pounds; 
 W = total stress in each strut in pounds; 
 S s = safe unit stress in king-rods; 
 A = net sectional area of king-rods; 
 and H depth of truss in inches^; 
 
 (230.) Art. 2199. 
 and A=$^-. (231.) Art. 2199. 
 
 The Queen-Rod Truss. 
 
 Let S t maximum unit stress in tie-beam; 
 
 W = total uniformly distributed load in pounds; 
 L length of span in inches; 
 H = depth of span in inches; 
 b = breadth of tie-beam in inches; 
 d = depth of tie-beam in inches; 
 S c = total stress in upper chord member in pounds; 
 5 S = total stress in struts in pounds; 
 S q = allowable unit stress in queen-rods; 
 A = sectional area of queen-rods. 
 
 The maximum unit stress in tie-beam is 
 
 (232.) Art. 22OO.
 
 TABLES AND FORMULAS. N'.t 
 
 The total stress in the upper chord member is 
 S c = %^. (233.) Art. 22OO. 
 
 The total stress in each strut is 
 
 * 
 
 S.= nVi + - 6 V^-. (234.) Art. 2200. 
 
 The net area of each queen-rod is 
 
 A=^^-. (235.) Art. 22OO. 
 
 ^fl 
 
 The Howe Truss. 
 
 Let N t = the number of a tie, counting from the center; 
 S t = total stress in a tie ; 
 L, = length of a tie ; 
 L s = length of a strut; 
 5 S = total stress in a strut ; 
 A^ =: number of panels in truss from center to either 
 
 abutment; 
 n = number of panels from a given panel to the 
 
 nearer abutment ; 
 P = panel load in pounds ; 
 L p = length of a panel ; 
 
 S t< . = total stress in top chord of a given panel ; 
 S b( . = total stress in bottom chord of a given panel. 
 
 The stress in any tie whose number is N t is 
 
 S, = (iV t + i) P. (236.) Art. 22O2. 
 
 The stress in a strut is 
 
 5 S = *pS t . (237.) Art. 22O2. 
 
 **t 
 
 The top chord stress is 
 
 S te = nP(N-$ n} ^. (238.) Art. 22O2.
 
 150 TABLES AND FORMULAS. 
 
 The bottom chord stress is 
 
 REFUSAL OF PILES. 
 
 Let 5 = weight a pile will bear with safety; 
 
 W weight of hammer, in the same unit as S; 
 H height of fall of hammer in feet; 
 
 then, 5 = W H. (24O.) Art. 22O5.
 
 INDEX. 
 
 TABLES. PAGE 
 
 Common Logarithms . . 1-19 
 
 Natural Sines, Cosines, Tangents, 
 
 and Cotangents . . . 21-40 
 Traverse Tables, or Latitudes and 
 
 Departures of Courses . . 41-49 
 Horizontal Distances and Differ- 
 ences of Elevation for Stadia 
 Measurements . . . 51-60 
 Radii and Chord and Tangent De- 
 flections 61-63 
 
 Moments of Inertia .... 64 
 Bending Moments and Deflections 65 
 Specific Gravities and Weights per 
 
 Cubic Foot .... 66-68 
 Discharge of Standard Orifices 68-69 
 " " Weirs .... 70 
 
 Coefficients of Friction for Smooth 
 Cast or Wrought Iron 
 Pipes .... 
 for Angular Bends 
 " " Circular Bends 
 
 " " Darcy's Formula . 
 
 Properties of Saturated Steam 73- 
 Standard Dimensions of Wrought- 
 Iron Steam, Gas, and 
 Water Pipes ... 76 
 " Pipe Flanges ... 77 
 Specific Heat of Substances . . 78 
 Constants for Apparent Cut-Offs 
 
 Used in Determining M. E. P. . 78 
 Riveted Joints of Boilers ... 78 
 Positions of Eccentric Relative to 
 
 Crank 79 
 
 Diameters of Steam and Exhaust 
 
 Pipes 79 
 
 Piston Speeds of Steam Engines . 79 
 Ratio of Grate Area of Boiler to 
 
 Horsepower ... 79 
 " " Heating Surface to Grate 
 
 Area .... 80 
 
 " " Heating Surface to Horse- 
 power .... 80 
 Factors of Evaporation . . . 81 
 Size of Chimneys and Horsepower 
 of Boilers . . 82 
 
 RULES AND FORMULAS. PAGE 
 
 Formulas Used in Algebra . . 83 
 
 Trigonometric Functions . . 83 
 Rules for Using Tables of Loga- 
 rithms of Num- 
 bers . . 84-86 
 " " " Trigonometric 
 
 Tables . . 86 
 
 RULES FOR MENSURATION. 
 
 The Triangle 87 
 
 '* Quadrilateral .... 87 
 
 " Circle 87 
 
 " Ellipse 88 
 
 " Prism and Cylinder . . 88 
 
 " Pyramid and Cone . . 89 
 " Frustum of a Pyramid or 
 
 Cone 89 
 
 " Sphere 89 
 
 FORMULAS USED ix ELEMENTARY 
 
 MECHANICS. 
 
 Uniform Motion .... 89 
 
 Mass, Weight, and Gravity . . 90 
 
 Formulas for Gravity Problems 90 
 
 Falling Bodies .... 90 
 
 Centrifugal Force .... 91 
 
 Center of Gravity of Two Bodies 92 
 
 The Efficiency of a Machine . 92 
 
 Work ... . . 92 
 
 Power 92 
 
 Kinetic Energy . . . -92 
 
 Density 93 
 
 RULES AND FORMULAS USED IN 
 HYDRAULICS. 
 
 Pascal's Law 93 
 
 General Law for the Downward 
 Pressure Upon the Bottom of 
 
 Any Vessel 93 
 
 General Law for Upward Pres- 
 sure 93 
 
 General Law for Lateral Pressure 94 
 
 " " " Pressure . . 94 
 
 Specific Gravity .... 94 
 
 Mean Velocity .... 95 
 
 Velocity of Efflux from an Orifice 95
 
 RULES AND FORMULAS USED IN 
 
 HYDRAULICS Continued. PAGE 
 Theoretical Range of a Jet . . 96 
 Velocity of a Jet . . . .96 
 Discharge of an Orifice ... 97 
 " " Standard Orifices . 97 
 
 " " a Submerged Rect- 
 
 angular Orifice . 98 
 " Weirs ... 98 
 Flow of Water Through Pipes 99-102 
 " Water Through Conduits 
 
 and Channels . 102-104 
 Values of the Coefficient of Rough- 
 ness for Use in Kutter's For- 
 mula 103 
 
 FORMULAS USED IN PNEUMATICS. 
 Pressure, Volume, Density, and 
 Weight of Air When the Tem- 
 perature Is Constant . . .104 
 Mariotte's Law .... 104 
 Pressure and Volume of a Gas 
 
 with Variable Temperature . 105 
 Gay-Lussac's Law . . . .105 
 Mixture of Two Gases Having 
 
 Unequal Volumes and Pressures 106 
 Mixture of Two Volumes of Air 
 Having Unequal Pressures, Vol- 
 umes, and Temperatures . . 106 
 
 FORMULAS USED IN STRENGTH OF 
 
 MATERIALS. 
 
 Unit Stress, Unit Strain, and Co- 
 efficient of Elasticity . . .106 
 Strength of Pipes and Cylinders 107 
 Moment of Inertia, Resisting Mo- 
 ment, and Bending Moment of 
 
 Beams 107 
 
 Deflection of a Beam . . . 108 
 Strength of Columns . . . 108 
 ' ' " Shafts .... 109 
 Constants for Shafting . . .no 
 Strength of Ropes and Chains . no 
 
 FORMULAS USED IN SURVEYING. 
 
 Radius of a Curve . . . .in 
 
 Length of Subchords . . .in 
 
 Length of a Tangent of a Curve . in 
 
 Chord Deflection . . . .112 
 
 Tangent Deflection . . . 112 
 
 Stadia Measurements . . .112 
 
 Barometrical Leveling . . . 113 
 
 RULES AND FORMULAS USED IN 
 SURVEYING AND MAPPING. 
 
 Rule for Balancing a Survey . 113 
 " " Double Longitudes . 113 
 
 Application of Double Longi- 
 tudes to Finding Areas . . 114 
 
 RULES AND FORMULAS USED IN 
 SURVEYING AND MAPPING 
 
 Continued. PAGE 
 
 Trapezoidal Rule . . . .114 
 
 Simpson's Rule . . . .114 
 
 Volumes of Irregular Solids . 115 
 
 The Prismoidal Formula . . 115 
 
 Latitudes and Departures . . 115 
 
 FORMULAS USED IN STEAM AND 
 STEAM ENGINES. 
 
 Specific Heat 115 
 
 Temperature of Mixtures . . 116 
 
 Mixture of Steam and Water . 116 
 Work Done by Piston . . .116 
 
 Real and Apparent Cut-Off . . 116 
 
 Horsepower 117 
 
 Mean Effective Pressure . . 117 
 
 Piston Speed n 7 
 
 Mechanical Efficiency of Engine 
 Steam Consumption 
 Thermal Efficiency of Engine 
 
 Water Required by Condenser . n8 
 Ratio of Expansion . . .118 
 
 FORMULAS USED IN STEAM BOIL- 
 ERS. 
 Air "Required for Combustion 
 
 and Heat of Combustion . . 119 
 Strength of Boiler Shells . .119 
 Horsepower of Boilers . . . 119 
 
 Safety Valve J2 o 
 
 Draft Pressure of Chimney . . 120 
 Quality of Steam . . . .121 
 
 FORMULAS USED IN WATER-WHEELS. 
 
 Theoretical Energy of a Given 
 Head and Weight of Water . 121 
 
 Theoretical Power . . .121 
 
 Energy of a Jet . . . . 122 
 
 Pressure Due to Impact and Re- 
 action of a Jet 
 
 Efficiency 
 
 Overshot Water- Wheels 
 
 Breast Wheels 
 
 Undershot Wheels . 
 
 Poncelet's Wheel . 
 
 Turbines . 
 
 . 122 
 
 . 124 
 
 . 124 
 
 125 
 . 125 
 
 125 
 126-131 
 
 FORMULAS USED IN HYDRAULIC 
 MACHINERY. 
 
 Size of Air and Vacuum Cham- 
 bers 
 
 Displacement of Pumps 
 
 Slip 
 
 Head and Pressure 
 
 Size of Pump Piston or Plunger . 
 
 Discharge of Pumps 
 
 Power of Pumps ....
 
 FORMULAS USED IN HYDRAULIC 
 
 MACHIX ERY Con finued. PAGE 
 
 Size of Steam Cylinder for Pumps 134 
 Sizes of Suction and Delivery 
 
 Pipes 134 
 
 Duty of a Pump . . . .135 
 Relations Between Pressure and 
 
 Size of a Ram . . . .136 
 Weight and Volume of Accumu- 
 lators 
 
 FORMULAS USED IN WATER SUP- 
 PLY AND DISTRIBUTION. 
 Dimensions of Spillway or Over- 
 flow 
 
 Masonry Dams .... 
 High Masonry Dams . 
 Darcy's Formulas for Long Pipes 
 " " " Rough Pipes 
 
 " " Smooth Pipes 
 
 General Relations Between 
 Smooth and Rough Pipes 
 
 37 
 
 FORMULAS USED IN WATER SUP- 
 PLY AND DISTRIBUTION Cent. PAGE 
 
 Darcy's Formulas for Velocity . 142 
 
 General Relation Between Ele- 
 ments of Two Pipes . 
 
 Compound Pipes .... 
 
 Pumping Into Mains 
 
 Weights and Thickness of Cast- 
 iron Pipes 
 
 Darcy's Formulas for Flow in 
 Open Channels .... 
 
 FORMULAS USED IN IRRIGATION. 
 Approximate Discharge of Weirs 
 Flow of Water Through Conduits 
 " in Canals .... 
 Timber for Flumes 
 Trussed Stringers .... 
 King-Rod Truss .... 
 Queen-Rod Truss .... 
 
 Howe Truss 
 
 Refusal of Piles
 
 BRAN' 
 
 UNIVERSITY OF CALIFORNIA 
 
 Santa Barbara College Library 
 Santa Barbara, California 
 
 Return to desk from which borrowed. 
 This book is DUE on the last date stamped below. 
 
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