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The artillerist's manual and British soldier's compendium

Chapter 55: PILING OF SHOT, AND SHELL.
By F. A. Griffiths
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About This Book

This work compiles practical instructions and technical specifications for artillery and infantry operations, encompassing small arms and percussion firearms, cartridge manufacture, gun carriages, charges, ranges, and gunnery principles. It details service and handling of field and heavy ordnance, battery organization and manoeuvres, naval gun and boat charges, the use of rockets and signalling lights, and procedures for landing seamen with field pieces. Additional material covers drill for riflemen and light infantry, camp and tent management, harness and veterinary guidance, stores and implements, fortification and bridging, and mathematical methods for surveying and estimating distances, supported by plates and diagrams.

TABLE OF SQUARES, CUBES, AND ROOTS.

No. Sqr. Cube. Sqr. root. Cube root. No. Sqr. Cube. Sqr. root. Cube root.
1 1 1 1·0000000 1·000000 51 2601 132651 7·1414284 3·708430
2 4 8 1·4142136 1·259921 52 2704 140608 7·2111026 3·732511
3 9 27 1·7320508 1·442250 53 2809 148877 7·2801099 3·756286
4 16 64 2·0000000 1·587401 54 2916 157464 7·3484692 3·779763
5 25 125 2·2360680 1·709976 55 3025 166375 7·4161985 3·802953
6 36 216 2·4494897 1·817121 56 3136 175616 7·4893148 3·825862
7 49 343 2·6457513 1·912933 57 3249 185193 7·5498344 3·848501
8 64 512 2·8284271 2·000000 58 3364 195112 7·6157731 3·870877
9 81 729 3·0000000 2·080084 59 3481 205379 7·6811457 3·892996
10 100 1000 3·1622777 2·154435 60 3600 216000 7·7459667 3·914867
11 121 1331 3·3166248 2·223980 61 3721 226981 7·8102497 3·936497
12 144 1728 3·4641016 2·289428 62 3844 238328 7·8740079 3·957892
13 169 2197 3·6055513 2·351335 63 3969 250047 7·9372539 3·979057
14 196 2744 3·7416574 2·410142 64 4096 262144 8·0000000 4·000000
15 225 3375 3·8729833 2·466212 65 4225 274625 8·0622577 4·020726
16 256 4096 4·0000000 2·519842 66 4356 287496 8·1240384 4·041240
17 289 4913 4·1231056 2·571282 67 4489 300763 8·1853528 4·061548
18 324 5832 4·2426407 2·620741 68 4624 314432 8·2462113 4·081656
19 361 6859 4·3588989 2·668402 69 4761 328509 8·3066239 4·101566
20 400 8000 4·4721360 2·714418 70 4900 343000 8·3666003 4·121285
21 441 9261 4·5825757 2·758923 71 5041 357911 8·4261498 4·140818
22 484 10648 4·6904158 2·802039 72 5184 373248 8·4852814 4·160168
23 529 12167 4·7958315 2·843867 73 5329 389017 8·5440037 4·179339
24 576 13824 4·8989795 2·884499 74 5476 405224 8·6023253 4·198336
25 625 15625 5·0000000 2·924018 75 5625 421875 8·6602540 4·217163
26 676 17576 5·0990195 2·962496 76 5776 438976 8·7177979 4·235824
27 729 19683 5·1961524 3·000000 77 5929 456533 8·7749644 4·254321
28 784 21952 5·2915026 3·036589 78 6084 474552 8·8317609 4·272659
29 841 24389 5·3851648 3·072317 79 6241 493039 8·8881944 4·290841
30 900 27000 5·4772256 3·107232 80 6400 512000 8·9442719 4·308870
31 961 29791 5·5677644 3·141381 81 6561 531441 9·0000000 4·326749
32 1024 32768 5·6568542 3·174802 82 6724 551368 9·0553851 4·344481
33 1089 35937 5·7445626 3·207534 83 6889 571787 9·1104336 4·362071
34 1156 39304 5·8309519 3·239612 84 7056 592704 9·1651514 4·379519
35 1225 42875 5·9160798 3·271066 85 7225 614125 9·2195445 4·396830
36 1296 46656 6·0000000 3·301927 86 7396 636056 9·2736185 4·414005
37 1369 50653 6·0827625 3·332222 87 7569 658503 9·3273791 4·431047
38 1444 54872 6·1644140 3·361975 88 7744 681472 9·3808315 4·447960
39 1521 59319 6·2449980 3·391211 89 7921 704969 9·4339811 4·464745
40 1600 64000 6·3245553 3·419952 90 8100 729000 9·4868330 4·481405
41 1681 68921 6·4031242 3·448217 91 8281 753571 9·5393920 4·497942
42 1764 74088 6·4807407 3·476027 92 8464 778688 9·5916630 4·514357
43 1849 79507 6·5574385 3·503398 93 8649 804357 9·6436508 4·530655
44 1936 85184 6·6332496 3·530348 94 8836 830584 9·6953597 4·546836
45 2025 91125 6·7082039 3·556893 95 9025 857375 9·7467943 4·562903
46 2116 97336 6·7823300 3·583048 96 9216 884736 9·7979590 4·578857
47 2209 103823 6·8556546 3·608826 97 9409 912673 9·8488578 4·594701
48 2304 110592 6·9282032 3·634241 98 9604 941192 9·8994949 4·610436
49 2401 117649 7·0000000 3·659306 99 9801 970299 9·9498744 4·626065
50 2500 125000 7·0170678 3·684031 100 10000 1000000 10·0000000 4·641589

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 one-third of the product 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.

Parallel { 12
edges. {   1 12 ÷ 2 =   6
{   1 12 + 1 = 13
Triangular face     78
3 ) 14 4⅔
4⅔ 312
52
Answer     364

Example.—To find the shot in a square pile, the bottom row consisting of 12 shot.

12
  1 12 ÷ 2 =   6
  1 12 + 1 = 13
78
3 ) 25 8⅓
8⅓ 624
26
Answer     650

Example.—To find the shot in an oblong pile, whose base consists of 18 shot in length, and 12 in breadth.

18 18 - 12 =   6
18 1
7 7
3 ) 43
14⅓ 12 ÷ 2 =   6
12 + 1 = 13
78
14⅓
312
78  
26
Answer     1118

Triangular pile.

Rule.—Multiply the base by the base plus 1, this product by the base plus 2, and divide by 6.

Square pile.

Rule.—Multiply the bottom row by the bottom row plus 1, and this product by twice the bottom row plus 1, and divide by 6.

Rectangular, or oblong pile.

Rule.—Multiply the breadth of the base by itself plus 1; and this product by three times the length of the base plus 1, minus the breadth of the base, and divide by 6.

In the following formulæ let the letter (L) denote the number in the bottom row, or the length; and (B) the breadth of the lowest course.

Triangular pile L × (L + 1) × (L + 2)
6
Square pile L × (L + 1) × (2L + 1)
6
Oblong pile B × (B + 1) × (3L + 1 - B)
6

The number of shot in any pile,

(whose base does not exceed 21) may readily be ascertained by referring to the following Table, page 284.

For the 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.

For the 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.

For the 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.