creased by 2, and the constant logarithm 9. 221849: the sum (rejecting. 10 in the index) will be the logarithm of the required number of balls.

Example.

Required the number of balls in a triangular pile, each side of its

Note. The constant logarithm employed in this Problem is the arithmetical complement of the logarithm of 6, the established divisor for triangular, square, and rectangular piles of shot.

PROBLEM XIII,

To find the Number of Balls in a Square Pile,

RULE.

To the logarithm of the number of balls in one side of the bottom row, add the logarithm of that number increased by 1, the logarithm of twice the same number increased by 1, and the constant logarithm 9.221849: the sum (abáting 10 in the index) will be the logarithm of the required number of balls.

Example.

Required the number of balls in a square pile, each side of its base containing 30 balls?

Note. The constant logarithm used in this Rule, is the same as that

given in the Rule to Problem XII.

PROBLEM XIV.

To find the Number of Balls in a Rectangular Pile.

RULE.

From three times the number of balls contained in the length of the bottom row, subtract the number of balls, less by 1, contained in the breadth of that row; then, to the logarithm of the remainder, add the logarithm of the number of balls contained in the breadth of the bottom row, the logarithm of that number increased by 1, and the constant logarithm 9.221849: the sum (rejecting 10 in the index) will be the logarithm of the required number of balls.

Example.

Required the number of balls in a rectangular pile, which contains 46 balls in the base row of its longest side, and 15 balls in that of its shortest side?

Note.-The constant logarithm made use of in this Rule is the same

as that which is given in the Rule to Problem XII.

PROBLEM XV.

To find the Number of Balls in an incomplete Triangular Pile.

RULE.

Find the number of balls in the whole pile, considered as complete, by Problem XII., page 646; and find also, by the same Problem, the number of balls answering to the triangular pile, the side of whose base is represented by the number of shot in the side of the top course of the incomplete pile diminished by 1; then, the difference of the two results will be the number of shot remaining in the pile.

Example.

Required the number of shot in an incomplete triangular pile; each side of its bottom course containing 40 balls, and each side of its top course containing 20 balls?

To find the Number of Balls in the complete Pile.

Number of balls for the whole pile=11480 Log. = 4.059942

To find the Number of Balls deficient.

Balls in each side of top course = 20−1 = 19 Log.=1.278754 Diminished course, or 19, increased by 1 = 20 Log.=1.301030 Ditto, increased by 2 =

Constant log. =

Number of shot wanting =

22 Log.=1.322219

Now, 11480-1330=10150 is the number of shot in the incomplete

pile.

PROBLEM XVI.

To find the Number of Balls in an incomplete Square Pile.

RULE.

Find the number of balls in the whole pile, considered as complete, by Problem XIII., page 647; and find also, by the same Problem, the number of balls answering to the square pile, each side of whose base is represented by the number of shot in each side of the top course of the incomplete pile diminished by 1; then, the difference of the two results will be the number of shot remaining in the pile.

Example.

Required the number of shot in an incomplete square pile; each side of its bottom course containing 24 balls, and each side of its top course 8 balls?

To find the Number of Balls in the complete Pile.

Number of balls for the whole pile=4900 Log. = 3.690196

Now, 49001404760 is the number of shot in the incomplete

pile.

PROBLEM XVII.

To find the Number of Balls in an incomplete Rectangular Pile.

RULE.

Find the number of balls in the whole pile, considered as complete, by Problem XIV., page 648; and find also, by the same Problem, the number of balls answering to the rectangular pile, whose sides are represented by the respective sides of the top course of the incomplete pile, the number of shot in each side being diminished by 1; then, the difference of the two results will be the number of shot remaining in the pile.

Example.

Required the number of shot in an incomplete rectangular pile; the length of its bottom course being 40 balls, its breadth 20, and the length of its top course 29 balls, and its breadth 9?

To find the Number of Balls in the complete Pile.

Bottom course, 40 × 3 = 120

Numb. of balls for whole pile=7070 Log.=3.849419

Now, 7070-924 6146 is the number of shot in the incomplete pile.

Note. In triangular and square piles, the number of horizontal rows or courses is always equal to the number of balls in one side of the bottom row; and, in rectangular piles, the number of horizontal rows is equal to the number of balls in the breadth of the bottom row. In these piles, the number of balls in the top row or edge is always one more than the difference between the number of balls contained in the length and the breadth of the bottom row.

PROBLEM XVIII.

To find the Velocity of any Shot or Shell.

RULE

From the logarithm of twice the weight of the charge or powder, in pounds, subtract the logarithm of the weight of the shot: to half the remainder add the constant logarithm 3. 204120, and the sum (rejecting 5 in the index) will be the logarithm of the velocity in feet, or the number of feet which the shot or shell passes over in a second.

Example 1.

With what velocity will a 24-pounds ball be projected by 8 lbs. of powder ?