PILING OF BALLS. course employed in forming the square pile; it follows, that the sum of the squares of these roots will be the shot required; and the sum of the squares divided by 8, 7, 6, 5, 4, 3, 2, 1, being 204, expresses the shot in the proposed pile. To find the shot of the oblong pile ABCDEF, fig. QU. VIII. 3; in which BF = 16, and вC = 7. Solution. The oblong pile proposed, consisting of the square pile ABCD, whose bottom row is 7 shot; besides 9 arithmetical triangles or progressions, in which the first and last term, as also the number of terms, are known; it follows, that, if to the contents of the square pile their total gives the contents required REMARK I. 140 252 392 shot. The shot in the triangular and the square piles, as also the shot in each horizontal course, may at once be ascertained by the following table: the vertical column a contains the shot in the bottom row, from 1 to 40 inclusive; the column B contains the triangular numbers, or number of each course; the column c contains the sum of the triangular numbers, that is, the shot contained in a triangular pile, commonly called pyramidal numbers; the column D contains the square of the numbers of the column A, that is, the shot contained in each square horizontal course; and the column E contains the sum of these squares or shot in a square pile. Thus, the bottom row in a triangular pile, consisting of 19 shot, the contents will be 1330; and when of 19 in the square pile, 2470.-In the same manner, the contents either of a square or triangular pile being given, the shot in the bottom row may be easily ascertained. The contents of any oblong pile by the preceding table may be also with little trouble ascertained, the less side not exceeding 40 shot, nor the difference between the less and the greater side 40. Thus, to find the shot in an oblong pile, the less side being 15, and the greater 35, we are first to find the contents of the square pile, by means of which the oblong pile may be conceived to be formed; that is, we are to find the contents of a square pile, whose bottom row is 15 shot: which being 1240, we are, secondly, to add these 1240 to the product 2400 of the triangular number 120, answering to 15, the number expressing the bottom row of the arithmetical triangle, multiplied by 20, the number of those triangless and their sum, being 3640, expresses the number of shot the proposed oblong pile. REMARK II. The following algebraical expressions, deduced from the investigations of the sums of the powers of numbers in arithmetical progression, which are seen upon many gunners' callipers*, serve to compute with ease and expedition the shot or shells in any pile. That serving to compute any triangular (n+2) × (n+1)×n pile, is represented by 6 That serving to compute any square (n+1)× (2n+1)×n pile, is represented by 6 In each of these, the letter n represents the number in the bottom row: hence, in a triangular pile, the number in the bottom row being 30; then this pile will be (30+2)×(30+1) X 30 = = 4960 shot or shells. In a square pile, the number in the bottom row being also 30; then this pile will be (30+1) x (60+1) × 9455 shot or shells. × * Callipers are large compasses, with bowed shanks, serving to take the diameters of convex and concave bodies. The gunners' callipers consist of two thin rules or plates, which are moveable quite round a joint, by the plates folding one over the other: the length of each rule or plate is 6 inches, the breadth about 1 inch. It is usual to represent, on the plates, a variety of scales, tables, proportions, &c. such as are esteemed useful to be known by persons employed about artillery; but, except the measuring of the caliber of shot and cannon, and the measur ing of saliant and re-entering angles, none of the articles, with which the callipers are usually filled, are essential to that instrument. That serving to compute any oblong pile, is represented by (2n+1+3m) × (n + 1) × n in which the letter n denotes the number of courses, and the letter m the number of shot, less one, in the top row; hence, in an oblong pile the num ber of courses being 30, and the top row 31; this pile will be 60+1+90 x 30 +1 x 30=23405 shot or shells. REMARK III. One practical rule, of easy recollection, will include the three cases of the triangular, square, and rectangular, complete piles. Thus, recurring to the diagrams 1, 2, and 3, we shall have, balls in (BD + A + c) X (BF + BF +AE) X BDC = triar lar pile. Hence, for a general rule: add to the number of balls or shells in one side of the base, the numbers in its two paral lels at bottom and top (whether row or ball), the sum being multiplied by a third of the slant end or face, gives the number in the pile. GEOMETRICAL PROPORTION, AND PRO- GEOMETRICAL PROPORTION contemplates the relation of quantities considered as to what part or what multiple one is of another, or how often one contains, or is contained in, another. Of two quantities compared together, the first is called the Antecedent, and the second the Consequent, Their ratio is the quotient which arises from dividing the one by the other. Four Quantities are proportional, 'when the two couplets have equal ratios, or when the first is the same part or multiple of the second, as the third is of the fourth. Thus, 3, 6, 4, 8, and a, ar, b, br, are geometrical proportionals. ar br For = ៖ = 2, and = thus, 36 4: 8, &c. -- a b =r. And they are stated See the Arithmetic. Geometrical Progression is one in which the terms have all successively the same ratio; as 1, 2, 4, 8, 16, &c. where the common ratio is 2. The general and common property of a geometrical progression is, that the product of any two terms, or the square of any one single term, is equal to the product of every other two terms that are taken at an equal distance on both sides from the former. So of these terms, 1, 2, 4, 8, 16, 32, 64, &c. 1 X 64 = 2 × 32 = 4 × 16 = 8 X 8 64. In any geometrical progression, if a denote the least term, z the greatest term, r the common ratio, n the number of the terms, s the sum of the series, or all the terms; then any of these quantities may be found from the others, by means of these general values or equations, viz. When the series is infinite, then the least term a is nothing, and the sum s = In any increasing geometrical progression, or series beginning with 1, the 3d, 5th, 7th, &c. terms will be squares; the 4th, 7th, 10th, &c. cubes; and the 7th will be both a square and a cube. Thus, in the series 1, r, r2, p3, p2, p5, po, p7, r3, p3, &c. r2, r1, r3, r3, are squares; r3, r, ro, cubes ; and both a square and a cube. In a decreasing geometrical progression, the ratio, r, is a fraction, and then s=- a. If n be infinite, this becomes a 1 T 8= ;; a being the first term. |