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+90×30+1×20=23405 shot or shells.

PROPORTION AND GEOMETRICAL PROGRESSION.

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. For 2, and =r. And they are stated thus,

3: 6:4:8, &c.

ar

br a b

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.

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, ara, ar3, &c. are series in geometrical progression.

The most useful part of Proportion and Geometrical Proportion is contained in the following theorems.

1. When

1. When four quantities are in proportion, the product of the two extremes is equal to the product of the two means. As in these, 3, 6, 4, 8, where 3X8=6×4=24; and in these, a, ar, b, br, where axbr=ar Xb=abr.

2. When four quantities are in proportion, the product of the means divided by either of the extremes gives the other

this is the foundation of the Rule of Three.

br

3. If 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 1X64 =2X32=4X16=8X8=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, 48, 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 dif ference between 1 and the ratio. Thus, the sum of 3, 6, 12,

24, 48, 96, 192, is

192×2-3
2-1

=384-3= 381. And the

sum of n terms of the series, a, ar, ar3, ar3, ar1, &c. to

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, a: ar :: b: br; or 2:6 :: 4 : 12. 2. Inversely, ar: a :: br: b; or 6 : 2 :: 12: 4. 3. Alternately, a : b :: ar: br; or 2: 4 :: 6 : 12. 4. Compoundedly; a: a+ar:: b: b+br; or 2: 8: 4:16. 5. Dividedly, a : ar. -a:: b: br-b, or 2: 4 :: 4 : 8.

6. Mixed,

6. Mixed, ar+a: ar—a :: br+b: br—b; or 8:4:: 16: 8. 7. Multiplication, ac: arc;; bc: brc; or 2.3 : 6.3 ;; 4: 12.

9. The numbers a, b, c, d, are in harmonical proportion, when a dab: con d.

EXAMPLES.

1. Given the first term of a geometric series 1, the ratio 2, and the number of terms 12; to find the sum of the series; First, X2=1X2048, is the last term.

Then

2048X2-1 4096 1

2-1

1

=4095, the sum required.

2. Given the first term of a geometrical series, the ratio , and the number of terms 8; to find the sum of the series? First, 1× (1), is the last term.

8

Then, × }) ÷ (1-4)=(} − + })÷÷}=}{} × 7 = 1 } }, the sum required.

572

3. Required the sum of 12 terms of the series, 1, 3, 9, 27, 31, &c. Ans: 265720. 4. Required the sum of 12 terms of the series 1, 1, d. 24, ar, &c. Ans. $749. 5. Required the sum of 100 terms of the series 1, 2, 4, 8, 16, 32, &c. Ans. 1267650600228229401496703205375, See more of Proportion in the Arithmetic.

SIMPLE EQUATIONS.

AN Equation is the expression of two equal quantities, with the sign of equality (=) placed between them. Thus, 104-6 is an equation, denoting the equality of the quantities 10-4 and 6.

Equations

Equations, are either simple or compound. A Simple Equation, is that which contains only one power of the unknown quantity, without including different powers. Thus, x➡a=b+c, or ax2b, is a simple equation containing only one power of the unknown quantity x. But x2-2ux=b2 is a compound one.

GENERAL RULE.

Reduction of Equations, is the finding the value of the unknown quantity. And this consists in disengaging that quantity from the known ones; or in ordering the equation so, that the unknown letter or quantity may stand alone on one side of the equation, or of the mark of equality, without a co-efficient: and all the rest, or the known quantities, on the other side.-In general, the unknown quantity is disengaged from the known ones, by performing always the reverse operations. So if the known quantities are connected with it by + or addition, they must be subtracted; if by minus (-), or subtraction, they must be added; if by multiplication, we must divide by them; if by division, we must multiply; when it is in any power, we must extract the root; and when in any radical, we must raise it to the power. As in the following particular rules; which are founded on the general principle of performing equal operations on equal quantities; in which case it is evident that the results must still be equal, whether by equal additions, or subtractions, or multiplications, or divisions, or roots, or powers.

PARTICULAR RULE I.

WHEN known quantities are connected with the unknown by+or; transpose them to the other side of the equation, and change their signs. Which is only adding or subtracting the same quantities on both sides, in order to get all the unknown terms on one side of the equation, and all the known ones on the other side.*

Thus,

* Here it is earnestly recommended that the pupil be accustomed, at every line or step in the reduction of the equations, to naine the particular operation to be performed on the equation in the line, in order to produce the next form or state of the equation, in applying each of these rules, according as the particular forms of the equation may require; applying them according to the order in which they

Thus, if x+5=8; then transposing 5 gives x=8-5=3. And, if x-3+7=9; then transposing the 3 and 7, gives x=9+3-7=5.

Also, if x-a+b=cd: then by transposing a and b, it is x=a-b+cd.

In like manner, if 5x-6=4x+10, then by transposing 6 and 4x, it is 5x-4x=10+6, or x=16.

RULE II.

WHEN the unknown term is multiplied by any quantity; divide all the terms of the equation by it.

Thus, if ax-ab-4a; then dividing by a, gives x=b-4. And, if 3x+5=20; then first transposing 5 gives 3x=15; and then by dividing by 3, it is x=5.

In like manner, if ax+3ab=4c2; then by dividing by a, it

4c2

is x+36=- ; and then transposing 3b, gives x

α

4c2

α

-36.

RULE III.

WHEN the unknown term is divided by any quantity; we must then multiply all the terms of the equation by that divisor; which takes it away.

Thus, if=3+2: then mult. by 4, gives x=12+8=20.

then by mult. a, it gives x=3ab+2ac-ad.

are here placed; and beginning every line with the words Then by, as in the following specimens of Examples; which two words will always bring to his recollection, that he is to pronounce what particular operation he is to perform on the last line, in order to give the next; allotting always a single line for each operation, and ranging the equations neatly just under each other, in the several lines, as they are successively produced.

RULE