of the compound quantities a+b and cd multiplied together.

7. When we would express, that one quantity, as a, is greater than another, as b, we write ab, or ab; and if we would express, that a is less than b, we write ab, or a 4b.

8. When we would express the difference between two quantities, as a and b, while it is unknown which is the greater of the two, we write them thus, ab, which de notes the difference of a and b.

9. Powers of the same quantities or factors are the products of their multiplication: thus aXa, or da, denotes the square, or second power, of the quantity represented by a ; axaxa, or aaa, expresses the cube, or third power; and aXaXaXa, or aaaa, denotes the biquadrate, or fourth power of a, &c.

And it is to be observed, that the quantity a is the root of all these powers. Suppose a=5, then will aaa×a 5×525 the square of 5; aaaa×a×a=5×5×5= 125 the cube of 5; and aaaaaXaXaXa➡5x5x5 X5625 the fourth power of 5.

10. Powers are likewise represented by placing above the root, to the right hand, a figure expressing the number of factors, that produce them. Thus, instead of aa, we write a; instead of aaa, we write a3; instead of aaaa, we write at, &c.

2

4

11. These figures, which express the number of factors, that produce powers, are called their indices, or exponents: thus, 2 is the index or exponent of a2; 3 is that of x3 ; 4 is that of x, &c.

But the exponent of the first power, though generally omitted, is unity, or 1; thus a signifies the same as a,

K K

namely,

namely, the first power of a; axa, the same as axd', or 11, that is, a", and axa is the same as a xa', ora'+', or a3.

α

24

12. In expressing powers of compound quantities, we usually draw a line over the given quantity, and at the end of the line place the exponent of the power. Thus,

2

a+b denotes the square or second power of a+b,' consid

ered as one quantity; at the third powers a+b the fourth power, &c.

And it may be observed, that the quantity a+b, called the first power of a+b, is the root of all these powers. Let a 4 and 62, then will a+b become 4+2, or 6;

2

and a+6)=4+2)=62=6×6=36, the square of 6

also a+b=4+2=63=6×6×6=216, the cube of 6.

13. The division of algebraic quantities is very frequently expressed by writing down the divisor under the dividend with a line between them, in the manner of a vulgar

a

fraction thus, represents the quantity arising by di

C

viding a by c; so that if a be 144 and 4, then will

ing by dividing a+b by a-c; suppose a 12, 66 and

called algebraic fractions; whereof the upper parts are call

ed

`ed the numerators, and the lower the denominators : thus, a

15. Quantities, to which the radical sign is applied, are called radical quantities, or surds; whereof those consisting

of one term only, as

andax, are called simple

surds; and those consisting of several terms, asab + cd

16. When any quantity is to be taken more than once, the number is to be prefixed, which shews how many times it is to be taken, and the number so prefixed is called. the numeral coefficient: thus, 2a signifies twice a, or a taken twice, and the numeral coefficient is 2; 3x signifies, that the quantity x is multiplied by 3, and the numeral coefficient is 3; also 5x+a* denotes, that the quanis multiplied by 5, or taken 5 times.

tity

24

+a

24

When no number is prefixed, an unit or 1 is always understood to be the coefficient: thus, I is the coefficient of a or of x; for a signifies the same as 1a, and x the same as Ix, since any quantity, multiplied by unity, is still the

same.

Moreover, if a and d be given quantities, and x and y required ones; then ax' denotes, that x is to be taken a times, or as many times as there are units, in a; and dy shews, that y is to be taken d times; so that the coefficient of ax is a, and that of dy is d: suppose a6 and d=4,

then

then will ax'6x, and dy4y. Again, x, or

Ix

de

2

notes the half of the quantity x, and the coefficient of

3x

is; so likewise x, or

, signifies of x, and the co

4

efficient of x is 4.

17. Like quantities are those, that are represented by the same letters under the same powers, or which differ only, in their coefficients; thus, 3a, 5a and a are like quantities, and the same is to be understood of the radicals

those, which are expressed by different letters, or by the same letters under different powers: thus 2ab, a1b, 2abc, 5ab, 4x, y, y and z* are all unlike quantities.

18. The double or ambiguous sign + signifies plus or minus the quantity, which immediately follows it, and being placed between two quantities, it denotes their sum, or dif

ference. Thus, a

4

b shews, that the quan

is to be added to, or subtracted from, a.

m

1

19. A general exponent is one, that is denoted by a letter instead of a figure: thus, the quantity a" has a general exponent, viz. m, which universally denotes the mth power of the root x. Suppose m2, then will x"=x ; if m 3, then will x"=x3; if m=4, then will "=x+, &c,

3

-m

In like manner, a -expresses the mth power

2

of a-b.

20. This root, viz. ab, is called a residual root, because its value is no more than the residue, remainder, or difference, of its terms a and b. It is likewise call

ed

ed a binomial, as well as a+b, because it is composed of two parts, connected together by the sign

21. A fraction, which expresses the root of a quantity, is also called an index, or exponent; the numerator shews

the power, and the denominator the root; thus a signifies

✔a+ab;

the same as a; and a+ab the same as a+ab; likewise a denotes the square of the cube root of the quantity a. Suppose a=64, then will a}=64†=4*= 16; for the cube root of 64 is 4, and the square of 4 is 16.

Again, a+ ratic root of a+b.

expresses the fifth power of the biquad

Suppose a 9 and 67, then will

5

2 + b | 7 = 9 +7 | * =16*=25 =32; for the biquadratic

root of 16 is 2, and the fifth power of 2 is 32.

Also, a

signifies the nth root of a. If n=4, then will aa; if n=5, then will ana, &c.

I

m

• Moreover, a+b" denotes the mth power of the nth root

L

m

of a+b. If m=3 and n=2, then will a+b=a+b2, namely, the cube of the square root of the quantity a+b;

I

n

m

n

and as aa equals ✔✅a, ora, √a, so a+b1 = √ a+bi", namely, the nth root of the mth power of a+b. So that the mth power of the nth root, or the nth root of the mth power, of a quantity are the very same in effect, though differently expressed.

22. An exponential quantity is a power, whose exponent is a variable quantity, as *. Suppose x=2, then will **=2=4; if x=3, then will x*33=27.

ADDITION.