viz. axb, or ab, and abe, are called single or simple quantities, as well as the factors, viz. a, b, c, from which they are produced, and the same is to be observed of the products, arising from the multiplication of any number of simple quantities.

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6. If an algebraical quantity consist of two terms, it is called a binomial, as a+b; if of three terms, a trinomial, as a+b+c; and if of four terms, a quadrinomial, as a+b+c+d; 25 and if there be more terms, it is called a multinomial, or polynomial; all of wch are compound quantities.

When a compound quantity is to be expressed as multiplied by a simple one, then we place the sign of multiplication between them, and draw a line over the compound quantity only; but when compound quantities are to be represented as multiplied together, then we draw a line over each of them, and connect them with a proper sign. Thus, axc denotes, that the compound quantity a+b is multiplied by the simple quantity c; so that if a were 10, b 6, and c 4, then would a+bxc be 10+6x4, or 16 into 4, which is 64; and a+bxc+d expresses the product of the compound quantities a+b and c+d multiplied together.

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

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, a cn b, which denotes the difference of a and b.

9. Powers of the same quantities or factors are the products of their multiplication: thus axa, or aa, denotes the square, or second power of the quantity, represented a; axa xa, or aaa, expresses the cube, or third power; and axaxa xa, 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 aa=axa=5×5= 25= the square of 5; aaa=axaxa=5×5×5=125= the cube VOL. I:

Ff

of 5; and aaaa=a×a×a×a=5×5×5×5=625= the fourth power of 5.

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

a2;

· a*,

&

11. These figures, which express the number of factors, that produce powers, are called their indis, or exponents; thus, 2 is the index or exponennt of a3is that of x3; 4 is that of x,&c.

But the exponent of the first power, though generally omitted, is unity, or 1; thus a1 signifies the same as a, namely, the first power of a; axa, the same as a1xa1, or a1+1, that is, q2, and a2xa is the same asxa1, or a2+1, or a3.

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,

a+b denotes the square or second power of a+b, considered as one quantity; a+b) the third power; a+b]*the fourth power, &c.

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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 b=2, then will a+b become 4+2, or 6; and

a+b=4+2=62=6x6=36, the square of 6; also a+

3

=4+2=63=6x6x6=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 fraction:

thus, represents the quantity arising by dividing a by e;

C

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

a+b

And denotes the quantity arising by dividing a+b by

a+b

ac; suppose a=12, b=6, and c=9, then will become

-C

called algebraic fractions; whereof the upper parts are called the numerators, and the lower the denominators: thus, a

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

of one term only, asa and a x, are called simple surds; and those consisting of several terms, as √ab+cd and ab+bc, compound surds.

16. When any quantity is to be taken more than once, the number is to be prefixed, which shows 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 is multiplied by 3, and the numeral coefficient is 3; also 5 √x2+a denotes, that the quantity x2+a2 is multiplied by 5, or taken 5 times.

When no number is prefixed, an unit or 1 is always understood to be the coefficient: thus, 1 is the coefficient of a or of x; for a signifies the same as 1a, and x the same as 1x, 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 ax3 denotes, that x'is to be taken a times, or as many times as there are units in a; and dy shows, that y is to be taken d times; so that the coefficient of axis a,

and that of dy is d.

Suppose a 6 and d=4, then will ax

1x

=6x2, and dy=4y. Again, x, or, denotes half of the

quantity x, and the coefficient of x is };

3x ix, or 4

so likewise

signifies of x, and the coefficient of x is 3.

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 √x2+a2 and 7 √x2+a2. But unlike quantities are those, which are expressed by different letters, or by the same letters under different powers thus 2ab, a3b, 2abc, 5ab2, 4x2, y, y3, and z2 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

2 a

ference. Thus, a±√ -b shows, that the quantity

a

4

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

19. A general exponent is one, that is denoted by a letter instead of a figure: thus, the quantity x" has a general exponent, namely, m, which universally denotes the mth power of the root x. Suppose m=2, then will x"=x2; if m=

20. This root, namely, a-b, 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 called 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 shows the

power, and the denominator the root: thus a signifies the same as a; and a+ats, the same as va+ab; likewise

3

a denotes the square of the cube root of the quantity a. Suppose a=64, then will a3=643=42-16; for the cube root of 64 is 4, and the square of 4 is 16.

Again +

ic root of a+b.

5

expresses the fifth power of the biquadratSuppose a 9 and b=7, then will

@ +6 =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=5, then will a"=a3, &c.

If n=4, then will

Moreover a+6 denotes the mth power of the nth root

m

of a+b. If m=3 and n=2, then will a+6|" =a+6|3⁄43, namely, the cube of the square root of the quantity a+b; and as

a" equals Va1, or′′va, so a+b|" =

-m

a+, namely, the nth

root of the mth power of a+b. So that the mth power of the nth root, and 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 x. Suppose x=2, then will x* =22=4; if x=3, then will x*=33=27.

ADDITION.

ADDITION, in Algebra, is connecting the quantities together by their proper signs, and uniting in simple terms such as are similar.

In addition there are three cases.