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 × 4= 24; and in these, a, ar, b, br, where a x brar x babr.

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 × 4 = 3; also if a: ar :: b: br, then

3 abr

a

=8, and

6 x 4

Rule of Three.

8

= a. And this is the foundation of the

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 × 32 = 4 × 16 = 8 x8 = 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 2596.

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 × 2-3

12, 24, 48, 96, 192, is

2 1

384-3381. And

the sum of n terms of the series a, ar, ar2, ar3, ar1, &c, to ar-xr-a ara a ar, is

r-1

1

a.

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.

4.

2. Inversely, ar: a ::
br: b ; or 6:2::12:
3. Alternately, a bar: br; or 2:4:: 6:12.

4. Com

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, 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 ad amb cond; or when their reciprocals

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 × 211 = 1 x 2048, is the last term.

2. Given the first term of a geometric series, the ratio , and the number of terms 8; to find the sum of the series? First, () = { × 728 = 74, is the last term. ×

Then (

849

3. Required the sum of 12 terms of the series 1, 3, 9, 27, $1, &c. Ans. 265720.

Ans. 265720 177147

4. Required the sum of 12 terms of the series 1, 3, 4, 77) IT, &C. 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.

SIMPLE EQUATIONS.

AN Equation is the expression of two equal quantities, with the sign of equality (=) placed between them. Thus, 10-46 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 ar2 = b, is a simple equation, containing only one power of the unknown quantity. But x2 – 2ax = b2 is a compound one.

GENERAL RULE.

1

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 name the particular operation to be performed on the equation in the last line, in order to produce the next form or state of the equation, in applying each of these rules, according as the particular form of the equation may require; applying them according to the

order

Thus, if .r+5=8; then transposing 5 gives r=8-5=3. And, if x-3+7=9; then transposing the 3 and 7, gives 93-75.

Also, if x

-

a + b cd: then by transposing a and b, it is abcd. x =

In like manner, if 5x−6 = 4x + 10, then by transposing 6 and 4r, it is 5x-4r 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 3r = 15; and then by dividing by 3, it is x = 5.

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

a

4c2

; and then transposing 3b, gives x = .36.

a

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.

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

3b+2cd:

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

- ad.

order in which they 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

RULE IV.

WHEN the unknown quantity is included in any root or surd: transpose the rest of the terms, if there be any, by Rule 1; then raise each side to such a power as is denoted by the index of the surd; viz. square each side when it is the square root; cube each side when it is the cube root; &c. which clears that radical.

Thus, if √x-3= 4; then transposing 3, gives √x=7; And squaring both sides gives x = 49.

And, if √2x + 10 = 8;

Then by squaring, it becomes 2r + 10 = 64 ;
And by transposing 10, it is 2x = 54;

Lastly, dividing by 2, gives x = 27.

Also, if 3/3r+4+3 = 6:

Then by transposing 3, it is 3/3x + 4 = 3;
And by cubing, it is 3x+4= 27; ·
Also, by transposing 4, it is 3x=23;
Lastly, dividing by 3, gives x = 73.

RULE V.

WHEN that side of the equation which contains the unknown quantity is a complete power, or can easily be reduced to one, by rule 1, 2, or 3: then extract the root of the said power on both sides of the equation; that is, extract the square root when it is a square power, or the cube root when it is a cube, &c.

Thus, if x2+8x + 16 36, or (x + 4)2 = 36:
Then by extracting the roots, it is x + 4 = 6;
And by transposing 4, it is x 6 − 4 = 2.

And if 3x2-1921 + 35.

Then, by transposing 19, it is 3x2 = 75;

And dividing by 3, gives x2 = 25;

And extracting the root, gives x = 5.

Also, if 2-6 24.

Then transposing 6, gives r2 = 30;

And multiplying by 4, gives 3x2 = 120;

Then dividing by 3, gives r2 = 40;

Lastly, extracting the root, gives x 40 6324555.

RULE