3. Or as the first sum, is to the difference of the given, and assumed cube, to the difference of the roots, nearly. 4. Again, by using, in like manner, the cube of the root last found as a new assumed cube, another root will be obtained still nearer. Repeat this operation as often as necessary, using always the cube of the last-found root, for the assumed root. Example.-To find the cube root of 21035.8. By trials it will be found first, that the root lies between 20, and 30; and, secondly, between 27, and 28. Taking, therefore, 27, its cube is 19683, which will be the assumed cube. Then by No. 2 of the Rule, As 60401 8: 61754 6:: 27: 27 6047 the Root, nearly. Again for a second operation, the cube of this root is 21035 318645155832, and the process by No. 3 of the Rule will be 21035 318645, &c. As 63106-43729 : diff. 481355:: 27 6047 : : the diff ⚫000210560 consequently the root required is 27.604910560 TABLE OF SQUARES, CUBES, AND ROOTS. No. Sqr. Cube. Sqr. root. Cube root. No. Sqr. Cube. Sqr. root. Cube root. 8 1.4142136 1.259921 52 27 1 7320508 1.442250 53 64 2.0000000 1.587401 54 125 2.2360680 1.709976 55 216 2 4494897 1.817121 56 343 2 6457513 1.912933 57 123110. 1 1 1.0000000 1.000000 51 2601 132651 7 1414284 3 708430 4 9 4 16 5 25 36 7 49 8 64 58 9 81 10 100 2704 140608 7.2111026 3.732511 2809 148877 7-2801099 3.756286 2916 157464 7.3484692 3.779763 3025 166375 7.4161985 3.802953: 3136 175616 7.4893148 3.825862 3249 185193 7 5498344 3.848501 3364 195112 7.6157731 3.870877 59 3481 205379 7.6811457 3.892996 60 3600 216000 7.7459667 3.914867 61 3721 226981 7.8102497 3.936497 3844 238328 7 8740079 3.957892 3969 250047 7-9372539 3.979057; 4096 262144] 8.0000000 4.000000 4225 274625 8.0622577 4.020726 4356 287496 8 1240384 4 041240 4489 300763 8.1853528 4.061548 4624 314432 8.2462113 4.081656 4761 328509 8*3066239 4.101566 4900 343000) 8.3666003 4.121285 5041 357911 8.4261498 4.140818 5184 373248 8.4852814 4.160168 5329 389017 8.5440037 4.179339. 5476 405224 8 6023253 4.198336 5625 421875 8.6602540 4.217163 5776 438976 8.7177979] 4.235824 5929 456533 8.7749644 4.254321 6084 474552 8 8317609 4.272659 6241 493039 8.8881944 4 290841 6400 512000 8*9442719 4.308870 6561 531441 9.0000000 4.326749 6724 551368 9.0553851 4.344481 6889 571787 9.1104336 4.362071 7056 592704 9.1651514 4.379519 7225 614125 9.2195445 4.396830 7396 636056 9.2736185 4.414005 7569 658503 9-3273791 4.431047 7744 681472 9.3808315 4.447960 7921 704969 9.4339811 4 464745 8100 729000 9.4868330 4'481405 8281 753571 9.5393920 4.497942 8464 778688 9.5916630 4.514357 8649 804357 9.6436508 4.530655 8836 830584 9.6953597 4 546836 95 9025 857375 9.7467943 4.562903 96 9216 884736 9.7979590 4.578857 9409 912673 9.8488578 4.594701 9604 941192 9.8994949 4.610436 9801 970299 9.9498744 4.626065 512 2 8284271 2 000000 729 3 0000000 2.080084 1000 3.1622777 2.154435 1331 3 3166248 2 223980 1728 3 4641016 2 289428 2197 3 6055513 2 351335) 2744 3 7416574 2 410142 3375 3.8729833 2.466212 4096 4 0000000 2.519842 66 4913 4.1231056 2.571282 67 324 5832 4.2426407 2.620741 68 361 6859 4 3588989 2 668402 69 400 8000 4 4721360 2.714418 70 441 9261 4.5825757 2.758923 71 22 484 10648 4.6904158 2.802039) 72 23 529 12167 4.7958315 2.843867 73 24 576 13824 4.8989795 2.884499, 74 25 625 15625 5 0000000 2.924018 75 26 676 17576 5.0990195 2.962496 76 27 729 19683 5 1961524 3.000000] 77 28 784 21952 5.2915026 3 036589 78 29 841 24389 5.3851648 3.072317 30 900 27000 5 4772256 3.107232 80 31 961 29791 5.5677644 3.141381 81 32 1024 32768 5.6568542 3.174802 82 33 1089 35937 5 7445626 3.207534 83 34 1156 39304 5.8309519 3.239612 84 35 1225 42875 5 9160798 3.271066 85 36 1296 46656 6.0000000 3.301927 86 1369 50653 6 0827625 3 332222 87 38 1444 54872 6 1644140 3.361975 88 39 1521 59319 6 2449980 3-391211 89 40 1600 64000 6.3245553 3.419952 90 41 1681 68921 6.4031242 3.448217 91 42 1764 74088 6.4807407 3.476027 92 43 1849 79507 6.5574385 3.503398. 93 44 1936 85184 6.6332496 3.530348 45 2025 91125 6.7082039 3 556893 46 2116 97336 6.7823300 3.583048) 47 2209 103823 6.8556546 3.608826 97 48 2304 110592 6.9282032 3.634241 98 49 2401 117649 7.0000000 3.659306 99 2500 125000 7.0170678 3.684031 100 10000 1000000 10 0000000 4.611589 PILING OF SHOT, AND SHELL. Shot, and shells, are usually piled in horizontal courses, the base being either an equilateral triangle, a square, or a rectangle. The triangular, and square piles terminate each in a single ball, but the rectangular pile finishes in a row of balls. To find the number of balls in a complete pile. Rule. Add the three parallel edges together; then the product of one-third of that sum, and of the number of balls in the triangular face, will be the number sought. Note 1.-The parallel edges in a rectangular pile are the two rows in length at the base, and the upper ridge. In the square pile the same, except that the upper row is only a single ball. In the triangular pile, one side of the base, the single ball at top, and that at the back, are considered the parallel edges. Note 2.-The number of balls in the triangular face is found by multiplying half the number in the breadth at the base, by the number in the breadth at the base plus 1. Note 3.-In all piles, the breadth of the bottom is equal to the number of courses. In the oblong pile, the top row is one more than the difference between the length, and breadth of the bottom. Example.-To find the shot in a triangular pile, the bottom row consisting of 12 shot. Example.-To find the shot in a square pile, the bottom row con- sisting of 12 shot. Example.-To find the shot in an oblong pile, whose base consists of 18 shot in length, and 12 in breadth. The Number of balls in a Pile may be found by using the following formulæ, in which let the letter (L) denote the 'number in the bottom row, or the Length; and (B) the Breadth of the lowest course. By referring to the following Table, the Number of Shot in any Pile (whose base does not exceed 21) may readily be ascertained. Square pile.-Look for the number of shot in the base, in the first vertical column on the left hand, and also in the diagonal column; and at their angle of meeting will be found the content required. Thus, 20 base gives 2870. Triangular pile. -Look for the number in the base row in the diagonal column, and opposite to it will be found the content. Thus, 18 base gives 1140. Oblong pile.-Look for the number in the length of the base in the vertical column, and the breadth of the base in the diagonal column, and at their angle of meeting will be found the content required. Thus, 17 length, and 12 breadth, gives 1040, To find the number of balls in an Incomplete pile.-Compute the number in the pile considered as complete; also the number in the upper pile, or part wanting; and the difference between the two piles thus found will be the number in the frustrum, or incomplete pile. Table for computing the Content of any Pile, whose base row does not exceed 21 balls. 14 41 74 120 175 238 308 384 465 | 550 638 728 819 560 704 910 1015 770 806 884 1001 1120 1240 | 16 15 680 816 836 15 44 962 1092 | 1225 | 1360 1496| 17 969 92 150 220 301 392 492 600 715 16 47 902 1040 1183 1330 1480 1632 1785 18 1140 17 50 98 160 235 322 420 528 645 770 968 1118 1274 1435 1600 1768 1938 2109 19 1330 18 53 104 170 250 343 448 564 690 825 19 56 110 180 265 364 476 600 735 880 1034 1196 1365 1540 1720 19042091 2280 2470 20 59 116 190 280 385 504 636 780 935 1100 1274 1456 1645 1840 2040 22442451 2660 2870 21 62 122 200 295 406 532 672 825 990 1166 1352 1547 1750 1960 2176 2397 2622 2850 3080 3311 20 1540 21 1771 22 2024 |