QUESTION IV. * A number of gabions being given to be placed in six xanks, one above the other, in such a manner as that each rank exceeding one another equally, the first may consist of 4 gabions, and the last of 9: What is the number of gabions in the six ranks; and what is the difference between each rank? Answer, the difference between the ranks will be 1, and the number of gabions in the six ranks will be 39. QUESTION V. Two detachments, distant from each other 37 leagues, and both designing to occupy an advantageous post equi-distant from each other's camp, set out at different times; the first detachment increasing every day's march 1 league and a half, and the second detachment increasing each day's march 2 leagues: both the detachments arrive at the same time; the first after 5 days' march, and the second after 4 days' march What is the number of leagues marched by each detachment each day? 7 The progression, 2, 3, 5% 6%, answers the conditions of the first detachment: and the progression 1, 3, 5, 7, answers the conditions of the second detachment. QUESTION VI. A deserter, in his flight, travelling at the rate of 8 leagues a day; and a detachment of dragoons being sent after him, with orders to march the first day only 2 leagues, the second 5 leagues, the third 8 leagues, and so on What is the number of days necessary for the detachment to overtake the deserter, and what will be the number of leagues marched before he is overtaken? Answer, 5 days are necessary to overtake him; and consequently 40 leagues will be the extent of the march. *Gabions are baskets, open at both ends, made of ozier twigs, and of a cylindrical form; those made use of at the trenches are 2 feet wide, and about 3 feet high; which, being filled with earth, serve as a shelter from the enemy's fire; and those made use of to construct batteries, are generally higher and broader. There is another sort of gabion, made use of to raise a low parapet; its height is from 1 to 2 feet, and 1 foot wide at top, but somewhat less at bottom, to give room for placing the muzzel of a firelock between them; these gabions serve instead of sand bags. A sand bag is generally made to contain about a cubical foot of earth. QUESTION QUESTION VII. A convoy * distant 35 leagues, having orders to join its camp, and to march at the rate of 5 leagues per day; its escort departing at the same time, with orders to march the first day only half a league, and the last day 9 leagues; and both the escort and convoy arriving at the same time: At what distance is the escort from the convoy at the end of each march? OF COMPUTING SHOT OR SHELLS IN A FINISHED PILE. SHOT and Shells are generally piled in three different forms, called triangular, square, or oblong piles, according as their base is either a triangle, a square, or a rectangle. Fig. 1. A D C By convoy is generally meant a supply of ammunition or provisions, conveyed to a town or army. The body of men that guard this supply is called escort. A triangular 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 contained 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 8 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, 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 of 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 BC = ? 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 alse the shot in each horizontal course, may at once be ascer VOL. I. Gg tained 2 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 |