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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.
1 4 10 20 35 56 84 120 165 220 286 364 455 560 680 816 969 1110 1330 1540
1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 1.53 171 190) 210
1 5 14 30 55 91 140 204 295 385 506 650 819 1015 1240 1496 1785 2109 2470 2870
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 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.
The following algebraical expressions, deduced from the investigations of the sumns 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 +2x n + 1xn pile, is represented by
6 That serving to compute any square n + 1x2n + 1 x 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 x 30+ I
= 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 x 36 = 9455 shot or shells.
That serving to compute any obiong pile, is represented by 2n+1+3m x n + 1 x n
in which the letter n denotes 6
* Callipers are large compasses, with bowed shanks, serving to take the diameters of convex and concave bodies. The gunners' callipers consist of two thiy 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 persous employed about artillery ; but, except the measuring of the caliber of shot and cannon, and the measuring of saliant and re-entering ar gles, none of the articles, with which the callipers are usually filled, are essential to that instrument.
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 +1x = 23405 shot or shells.
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.
And they are stated
6 thus, 3:6 :: 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, ara, ar', ar", 'ars, &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 four quantities are in geometrical proportion, the product of the two extremes is equal to the product of the two means. As in these, 3, 6, 4, 8, where 3 x 8 = 6 x 4 = 24; and in these, d, ar, b, br, where a x br = ar x b=abr.
2. When four quantities are in geometrical proportion, the product of the means divided by either of the extremes gives the other extreme. Thus, if 3 : 6 :: 4 : 8, then 6 x 4
6 X 4 = 8, and = 3; also if a : ar :: 6 : br, then 3
abr br, or
And this is the foundation of the
br Rule of Three.
3. In any continued geometrical progression, the product of the two extremes, and that of any other two terms, equally distant from them, are equal to each other, or equal to the square of the middle term when there is an odd number of them. So, in the series 1, 2, 4, 8, 16, 32, 64, &c, it is 1 x 64 = 2 X 32 = 4 x 16 = 8 x 8 = 64.
4. In any continued geometrical series, the last term is equal to the first multiplied by such a power of the ratio as is denoted by 1 less than the number of terms. "Thus, in the series, 3, 6, 12, 24, 45, 96, &c, it is 3 x 25 96.
5. The sum of any series in geometrical progression, is found by multiplying the last term by the ratio, and dividing the difference of this product and the first term by the difference between 1 and the ratio. Thus, the sum of 3, 6,
192 x 2-3 12, 24, 48, 96, 192, is
= 384-3 = 381. And
2-1 the sum of n terms of the series a, ar, ar?, ar}, art, &c, to ar!" x g-a arn
gh agams, is
6. When four quantities, a, ar, b, br, or 2, 6, 4, 12, are proportional ; then any of the following forms of those quantities are also proportional, viz.
1. Directly :ar :: b : br; or 2:6 :: 4:12. 2. Inversely, ar: a :: br:b; or 6:2:: 12: 4. 3. Alternately, a : 6 :: ar : br; or 2:4 :: 6:12.
4. Compoundedly, 8:atar :: 6:5+ br; or 2:8::4:16.
::b: br; or 1:3 :: 4:12.
9. The numbers a, b, c, d, are in harmonical proportion, when a :d :: acs bicos d; or when their reciprocals I 1 1
are in arithmetical proportion.
1. Given the first term of a geometrical series 1, the ratio 2, and the number of terms 12; to find the sum of the series? First, 1 x 211
=1 x 2048, is the last term. 2048 x 2-1 4096-1 Then
=4095, the sum required. 2-1
1 2. Given the first term of a geometric series į, the ratio , and the number of terms 8; to find the sum of the series? First, } (!) = x = , is the last term. Then (-x) (1-) = (-765) + = x;
the sum required. 3. Required the sum of 12 terms of the series 1, 3, 9, 27, 81,&c.
Ans. 265720. 4. Required the sum of 12 terms of the series 1, }, }, IT, T, &c.
Ans. 265726 5. Required the sum of 100 terms of the series 1, 2, 4, 8, 16, 32, &c. Ans. 1267650600228229401496703205375.
See more of Geometrical Proportion in the Arithmetic.
An Equation is the expression of two equal quantities, with the sign of equality (=) placed between them. Thus, 10-4 = 6 is an equation, denoting the equality of the quantities 10 - 4 and 6.