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 edgesin arectangular pileare the two rows in length at the base, and the upper ridge. In thesquare pilethe same, except that the upper row is only a single ball. In thetriangular 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 faceis found by multiplying half the number in the breadth at the base, by the number in the breadth at the baseplus1.
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.
Top view of a triangular pyramid
Example.—To find the shot in a square pile, the bottom row consisting of 12 shot.
Top view of a square pyramid
Example.—To find the shot in an oblong pile, whose base consists of 18 shot in length, and 12 in breadth.
Top view of a rectangular pyramid
Triangular pile.
Rule.—Multiply the base by the baseplus1, this product by the baseplus2, and divide by 6.
Square pile.
Rule.—Multiply the bottom row by the bottom rowplus1, and this product by twice the bottom rowplus1, and divide by 6.
Rectangular, or oblong pile.
Rule.—Multiply the breadth of the base by itselfplus1; and this product by three times the length of the baseplus1,minusthe 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.
The number of shot in any pile,
(whose base does not exceed 21) may readily be ascertained by referring tothe 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 diagonalcolumn; 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.