A triangular pile is formed by the continual laying of triangular horizontal courses of shot one above another, in such a manner, as that the sides of these courses, called rows, decrease by unity from the bottom row to the top row, which ends always in 1 shot. A square pile is formed by the continual laying of square horizontal courses of shot one above another, in such a manner, as that the sides of these courses decrease by unity from the bottom to the top row, which ends also in 1 shot. In the triangular and the square piles, the sides or faces being equilateral triangles, the shot contamed in those faces form an arithmetical progression, having for first term unity, and for last term and number of terms, the shot contained in the bottom row; for the number of horizontal rows, or the number counted on one of the angles from the bottom to the top, is always equal to those counted on one side in the bottom: the sides or faces in either the triangular or square piles, are called arithmetical triangles; and the numbers contained in these, are called triangular numbers: ABC, fig. 1, EFG, fig. 2, are arithmetical triangles. The oblong pile may be conceived as formed from the square pile ABCD; to one side or face of which, as AD, a number of arithmetical triangles equal to the face have been added: and the number of arithmetical triangles added to the square pile, by means of which the oblong pile is formed, is always one less than the shot in the top row; or, which is the same, equal to the difference between the bottom row of the greater side and that of the lesser. QUESTION VIII. To find the shot in the triangular pile ABCD, fig. 1, the bottom row AB consisting of 8 shot. SOLUTION. The proposed pile consisting of 8 horizontal courses, each of which forms an equilateral triangle; that is, the shot contained in these being in an arithmetical progression, of which the first and last term, as also the number of terms, are known; it follows, that the sum of these particular courses, or of the & progressions, will be the shot contained in the proposed pile; then The To find the shot of the square pile EFGH, fig. 2, the bottom row EF consisting of 8 shot. SOLUTION. The bottom row containing 8 shot, and the second only 7; that is, the rows forming the progression, 8, 7, 6, 5, 4, 3, 2, 1, in which each of the terms being the square root of the shot contained in each separate square 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. QUESTION X. To find the shot of the oblong pile ABCDEF, fig. 3; in which BF 16, and EC = 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 ascer tained by the following table: the vertical column A, contains the shot in the bottom row, from 1 to 20 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 9 shot, the contents will be 165; and when of 9 in the square pile, 285.-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 20 shot, nor the difference between the less and the greater side 20. Thus, to find the shot in an oblong pile, the 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 triangles; and their sum, being 3640, expresses the number of shot in 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 ) " + 2 × n + 1 ×n pile, is represented by That serving to compute any square pile, is represented by 6 n + 1 x 2n + 1 x n 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 × 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 × 60 + 1 x 309455 shot or shells. That serving to compute any obiong pile, is represented by 2n+1+3m x n + 1 x n 6 in which the letter n denotes * 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 measuring of saliant and re-entering angles, none of the articles, with which the callipers are usually filled, are essential to that instrument. the the number of courses, and the letter m the number of shot, less one, in the top row: hence, in an oblong pile the number 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. GEOMETRICAL PROPORTION. 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 f = r. And they are stated a b thus, 36 :: 4 : 8, &c. Direct Proportion is when the same relation subsists between the first term and the second, as between the third and the fourth As in the terms above. But Reciprocal, or Inverse Proportion, is when one quantity increases in the same proportion as another diminishes: As in these, 3, 6, 8, 4; and these, a, ar, br, b. The Quantities are in geometrical progression, or continuous proportion, when every two terms have always the same ratio, or when the first has the same ratio to the second as the second to the third, and the third to the fourth, &c. Thus, 2, 4, 8, 16, 32, 64, &c, and a, ar, ar2, ar3, art, ar3, &c, are series in geometrical progression. The most useful part of geometrical proportion is contained in the following theorems; which are similar to those in Arithmetical Proportion, using multiplication for addition, &c. 1. When |