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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 io 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.

Swot and Shells are generally piled in three different form-s, called triangular, square, or oblong piles, according as their base is either a triangle, a square, or a rectangle. Fig. I. C

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Fig. 2.

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ABCD, fig. 1, is a triangular pile,
EFGH, fig. 2, is a square pile.

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* 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 escert,

A triangular

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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 VIU.

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

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Total

120 shot

in the pile proposed.

QUESTION IX.
To find the shot of the square piie EFGHI, fig. 2, the bottom:
row Ef consisting of 8 shot.

SOLUTION
The bottom rw 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 forining
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 shos
in the proposed pile.

QUESTION X.
To find the shot of the oblong pile ABCDEF, fig. 3 ; in
which BF =
16, and BC = 7

SOLUTION.
The oblong pile proposed, consisting of the square pile
ABCD, whose bottom row is 7 sho:; besides 9 arithmetical
triangles or progressions, in which the first and last term, as
also the number of ternis, are known; it follows, that,
if to the contents of the square pile

140
we add the sum of the 9th progression

252
their total gives the contents required

392 shot.
REMARK I.
The shot in the triangular and the square piles, as alse
the shot in each horizontal course, may at once be ascer.
VOL. I.

tained

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tained by the following table : the vertical column A, contairis 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.

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1
5
14
30
55
91
140
204
285
385
506
650
819
1015
1240
1496
1785
2109
2470
2870

78 91 105 120 136 153 171 190 210

25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400

12 19 14 15 16

17

18 19 20

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

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