them; as, x, 3a2, -4ab, &c. A negative quantity is of an opposite nature to a positive one, with respect to addition and subtraction; the condition of its determination being such, that it must be subtracted when a positive quantity would be added, and the reverse.

22. Also quantities have different denominations, according to the number of terms (connected by the signsor) of which they consist; thus, a, 3b, -4ad, &c., quantities consisting of one term, are called simple quantities, or monomials; a+x; a quantity consisting of two terms, a binomial; a—x is sometimes called a residual quantity. A trinomial is a quantity consisting of three terms; as, a+2x-3y; a quadrinomial of four; as, a−b+3x -4y; and a polynomial, or multinomial, consists of an indefinite number of terms. Quantities consisting of more than one term may be called compound quantities.

23. Quantities the signs of which are all positive, or all negative, are said to have like signs; thus, +3a, +4x, +5ab, have like signs; also, -4a, -3b, -4dc: When some are positive, and others negative they have unlike signs; thus, the quantities +3a and -5ab have unlike signs; also, the quantities -3xx, +3a2x; and the quantities -b, +b.

24. If the quotients of two pairs of numbers are equal, the numbers are proportional, and the first is to the second, as the third to the fourth; and any quantities, expressed by such numbers, are also proportional; thus, if

α C

-

b d

then, a is to b as c to d.

The abbreviation of the proportion; a:b::c: d and it is sometimes written a b= =c d; if a=8,

b=4, c=12, and d=6; then, 6:12.

8 12

-= =2, and 8: 4: 4 6

25. A term, is any part or member of a compound quantity, which is separated from the rest by the sign and ; thus, a and b are the terms of a+b; and 3a, -2b, and +5ad, are the terms of the compound quantity 3a-2b+5ad. In like the terms of a product, fraction, or proportion, are the several parts or quantities of which they are composed; thus, a and b are the terms of ab, or of; and a, b, c, d, are the terms of the b

manner,

a

proportion ab::c: d.

26. A measure, or divisor, of any quantity, is that which is contained in it some exact number of times; thus, 4 is a measure of 12, and 7 is a measure of 35a 35a, because

7

=5.

27. A prime number, is that which has no exact divisor, except itself, or unity; 2, 3, 5, 7, 11, &c. and the intervening numbers; 4, 6, 8, &c. are composite numbers (Art. 13).

28. Commensurable numbers, or quantities, are such as have a common measure; thus, 6 and 3, Sab and 4ab, are commensurable quantities; the common divisors being 2 and 4; also, 4ax2 and 5ax are commensurable, the common divisor being ax.

29. Also, two or more numbers are said to be prime to each other, when they have no common measure or divisor, except unity; as 3 and 5, 7 and 9, 11 and 13, &c.

30. A multiple of any quantity, is that which is some exact number of times that quantity; thus, 12 is a multiple of 4; and 15a is a multiple of 3a,

15a

because =5.

3a

31. The reciprocal of a quantity is that quantity

inverted or unity divided by it. Thus, the recipro

a

1

α b

cal of a, or of is the reciprocal of is and

α

α -b a+b

the reciprocal of is

a+b a-b

b a

1

a2

,

32. The reciprocal of the powers and roots of quantities, is frequently written with a negative index or exponent; thus, the reciprocal of a2 = may be written a-2; the reciprocal of (a+x)3 = , may be written (a+x); but this method (a+x) 3' of notation requires some farther explanation, which will be given in a subsequentt part of the work.

1

33. A function of one or more quantities, is. an expression into which those quantities enter in any manner whatever, either combined, or not, with known quantities; thus, a+2x, ax+3ax2, 5ax1— 3a, &c. are functions of r; and 3ax+xya, 2(x2+ 5xy), &c. are functions of x and y.

34. When quantities are connected by the sign of equality, the expression itself is called an equation; thus, a+b=c+d, means that the quantities a and b, are equal to the quantities c and d; and this is called an equation; it is divided into two members by the sign of equality, a+b is the first, and c+d, the second member of the equation.

35. In algebraical operations the word therefore, or consequently, often occurs. To express this word, the sign. is generally made use of; thus, a=b, therefore, a+c=b+c; is expressed •.• a+c=b+c.

Also is the sign of infinity; signifying that the quantity standing before it, is of an unlimited value, or greater than any quantity that can be assigned. 36. The signs + and give a kind of quality

or affection to the quantities to which they are annexed. As all those terms which have the sign +prefixed to them, are to be added (Art. 4), and those quantities which have the sign-prefixed to them, are to be subtracted, (Art. 5), from the terms which precede them; the former has a tendency to increase, and the latter to diminish, the quantities with which they are combined; thus, the compound quantity, a-x, will therefore be positive or negative, according to the effect which it produces upon some third quantity b; if a be greater than x, then, (since a is added, and b subtracted) b+a-x is >b; but if a be less than x; then, b+a -x is <b.

In the first place, let a=10, x=6, and b=8; then ba-x=8+10-6, which is > 8; since 10 -6-4, a positive quantity; therefore, a-x is positive. Next, let a=12, x=14, and 6=20; then b+ a-x=20+12-14, which is <20; since 12-14— -2, a negative quantity; therefore a-x is negative. In like manner, it may be shown that the expression a-b+c-d is positive or negative according as ac is> or <b+d; and so of all compound quantities whatever.

37. The use of these several signs, symbols, and abbreviations, may be exemplified in the following

manner:

EXAMPLES.

EXAMPLE 1. In the algebraic expression a+b+ c-d, let a=8, b=7, c=4, and d=6; then, a+b+c-d=8+7+4-6=19-6=13.

Ex. 2. In the expression ab+ax-by, let a=5, b=4, x=8, and y=12; then, to find its value, we have ab+ax-by=5×4+5x8-4X12

20+40-43 =60-48=12.

Ex. 3. What is the value of

x=5, y=10, and b=6?

3ax+2y, where a=4,

" a+b

Here 3ax+2y=3X4X5+2×10=60+20=80, and a+b=4+6=!0;

Ex. 4.. What is the value of a2+2ab-c+d, when a=6, b=5, c=4, and d=1?

Aus. 93.

Ex. 5. What is the value of ab+ce-bd, when a=8, b=7, c=6, d=5, and e!? Ans. 27.

Ex. 6. In the expression a+by, let a=5, b=3,

6+x

a=5,b=3,

x=7, and y=5; What is its numerical value?

Ex. 7. In the expression

ax2+b2 bx-a2. C

Ans. 5.

let a=3,

b=5, c=2, x=6; What is its numerical value?

Ans. 7.

Ex. 8. What is the value of a2 ×(a+b)—2abe, where a=6, b=5, and c=4? Ans. 156.

Ex. 9. There is a certain algebraic expression consisting of three terms connected together by the sign plus; the first term of it arises from multiplying three times the square of a by the quantity b; the second is the product of a, b, and c; and the third is two-thirds of the product of a and b. Required the expression in algebraic writing, and its numerical value, where a=4, b=3, and c=2? Ans. 176.

DEFINITIONS.

38. A proposition, is some truth advanced, which is to be demonstrated, or proved; or something