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 :: be: brc; or 2.3 : 6.53

α ar

3. Division, -:-::b: br; or 1: 3 :: 4:12.

:: 4 : 12.

9. The numbers a. b, c, d, are in harmonical proportion, when a : d:: a b c d; or when their reciprocals 1 1 1 1

a b c d

are in arithmetical proportion.

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, 1 X 211 = 1 X 2048, is the last term.

2. Given the first term of a geometrical series, the ratio

8; to find the sum of the series? 3, is the last term.

=

, and the number of terms First, () = X +18 Then. ( × 1) ÷ (1 3, the sum required.

=

5

38

3849

3. Required the sum of 12 terms of the series 1, 3, 9, 27, 31, &c.

Ans. 265720.

65720

4. Required the sum of 12 terms of the series 1, 4, 7, 27, 'T, &c. Ans. 3947395. 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 in 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 ax2 = = b, is a simple equation containing only one power of the unknown quantity x. But x2-2ax 62 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 pertorming 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 1.

WHEN known quantities are connected with the unknown byor; 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 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

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,

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.

is

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

4c3

4c2

:+36= ; and then transposing 3b, gives x————

--3b.

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.

x

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

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, if2r+ 10 = 8:

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

Lastly, dividing by 2, gives x = 27.

Also, if 3/3x+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 3r = 23;

Lastly, dividing by 3, gives x = 73.

RULE V.

WHEN that side of the equation which contains the unknown quantity is a complete to one, by rule 1, 2, or 3 power on both sides of the square root when it is a square power, or the cube root when it is a cube, &c.

power, or can easily be reduced
then extract the root of the said
equation; that is, extract the

Thus, if x2+8x + 16 = 36, or (x
Then by extracting the roots, it is
And by transposing 4, it is x = 6

+ 4)2 = 36:

+ 4
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 x2-6 = 24.

Then transposing 6, gives 3x2 = 30;
And multiplying by 4, gives 3x2 = 120 ;
Then dividing by 3, gives x2 = 40;
Lastly, extracting the root, gives x =

= 6;

40 6·324556;

RULE

RULE VI.

WHEN there is any analogy or proportion, it is to be changed into an equation, by multiplying the two extreme terms together, and the two means together, and making the one product equal to the other.

Thus, if 2x: 9 :: 3 : 5.

Then, mult. the extremes and means, gives 10x = 27 ;
And dividing by 10, gives x

And if x a:: 5b: 2c.

20.

1

Then mult. extremes and means gives cx
And multiplying by 2, gives 3cx = 10ab;

Lastly, dividing by 3c, gives x =

Also, if 10-x: x : 3: 1.

10ab

3c

=5ab;

Then mult. extremes and means, gives 10-x = 2x ;
And transposing x, gives 10 = 3x;
Lastly, dividing by 3, gives 31 = x.

RULE VII.

WHEN the same quantity is found on both sides of an equation, with the same sign, either plus or minus. it may be left out of both: and when every term in an equation is either multiplied or divided by the same quantity it may be struck out of them all.

Thus, if 3x + 2a = 2a + b :

Then by taking away 2a, it is 3x = b.
And, dividing by 3, it is x = b.

Also if there be 4ax + 6ab = 7ac.

Then striking out or dividing by a, gives 4x + 6b = 7c.
Then, by transposing 6b, it becomes 4x=7c-6b;

And then dividing by 4 gives x = Jc - 3b.

Again, if x = 4-}·

Then, taking away the 3, it becomes x =
And taking away the 3's, it is 2x = 10;
Lastly, dividing by 2 gives x = 5.

MISCELLANEOUS EXAMPLES.

1. Given 7x 18=4x+6; to find the value of x.
First, transposing 18 and 5x gives 3x = 24;

Then dividing by 3, gives x =

8.

2. Given