have y + zx, we know that the value of x is equal to the values of y and z added together. Thus, if y be 6, and z be 4, the value of x must be 10; for 6 + 4 = 10.

× This sign signifies multiplication; as, 3 × 4 = 12; that is, 3 multiplied by 4 is equal to 12. This character is often omitted when multiplication is implied; as in the expression x y, which is the product of a multiplied by y. Thus, if x = 5, and y = 3, x y15; for 3 x 5 = 15. But it is never omitted between two numbers which are to be multiplied. ÷This sign expresses that the quantity which precedes it is to be divided by that which follows it. Thus, 1243; that is, 12 divided by 4 is equal to 3.

But division is more frequently expressed in the form of a fraction; thus, 123, which may be read in the same manner.

So, too, xy, or, expresses the

divided by y. Thus, if x = 10, and y

10

for = 5.

A vinculum

is used to connect two or more quan

tities together. Thus, 4 × a + b implies that the sum

of a and b is to be multiplied by 4.

Suppose the

value of a to be 5, and of b to be 6; then,

4 × 5 + 6 = 44;

for 5611, and 4 × 11 = 44.

Again, a + b × c+d signifies that the sum of a + b is to be multiplied by the sum of c + d. Let a = 2, b3, c = 4, and d = 5; then,

2+3 × 4+ 5 = 5 × 9, or 45.

A parenthesis () is often used instead of a vinculum, to indicate that several quantities are to be taken together. Thus, 3 (x + y) expresses that the sum of x and y is to be taken three times. If the value of x be 6, and of y be 4,

3(x+y)=3 (6+4)= 3 x 10, or 30.

The several quantities under a vinculum, or included in a parenthesis, may be taken collectively, and regarded as a simple quantity, of which the number prefixed is the coefficient.

SECTION III.

Simple, Compound, Similar, Positive and Negative Quantities.

A Simple quantity consists of a single term, that is, of one letter or number, or of several letters joined together without the signor; as, x, 3 y, ab c, and xy, each of which is a simple quantity.

A Compound quantity consists of two or more simple quantities joined together by the sign + or —; as x + y, a b + 3 c, a b 7 +2 8x, each of which is a compound quantity.

A compound quantity, which consists of two terms only, as xy, or a b, is called a binomial. The latter expression, a—b, is also called a residual quantity, because it expresses the residue or remainder, after one of the terms has been taken from the other.

Similar quantities are such as differ only in their coefficients or signs. Thus, 3 a and 5 a are similar

quantities; so are 3 x y and 7 x y ; as also 2 a b and - 8 a b; and the compound quantities 3 a b + 4 x, and 5 a b 9 x.

All the quantities used in an algebraic calculation, are considered, in relation to each other, either as positive or negative.

A Positive quantity has the sign + prefixed to it; it is, in general, something to be added. When a positive quantity stands alone, as x, or is the first term of a compound quantity, as a + b, the sign is commonly omitted; but the sign + is always understood in such cases. Thus, x is the same as + x, and a + b the same as + a + b.

A Negative quantity is one to be subtracted, and always has the sign — prefixed to it. Thus, in the expression a- b, b is a negative quantity, because its value is to be subtracted from a.

As the subject of positive and negative quantities is very apt to perplex beginners, a few examples will be given, by way of illustration.

1. William has 12 apples, and gives 5 of them to Samuel. How many has he left?

In this question, 12, the number of apples which William had in the first place, is a positive quantity; and 5, which must be subtracted to obtain the answer, is a negative quantity. 12-5.

2. William has 12 apples, and Samuel gives him 5 How many has he then?

more.

Here, as 5 must be added to 12 to obtain the answer, it is a positive quantity. 12 +5.

3. A man bought a watch for 25 dollars, and sold

it again for 30 dollars. How much did he gain by the bargain?

To find his gain, we must subtract what he gave for the watch from the sum for which he sold it: 30 is, therefore, a positive, and 25 a negative quantity. 30 25.

4. A man sold a watch for 30 dollars, by which bargain he gained 5 dollars. What did the watch cost him?

Here, the gain must be subtracted from the price of the watch: 5 is, therefore, a negative quantity. 30

5.

5. A merchant went into trade with a certain sum, say a dollars; and, at the end of the year, he found himself worth b doilars. How much did he gain during the year?

We must subtract what he had at the beginning of the year, a dollars, from what he had at the end of it, b dollars, to ascertain his gain: a is, therefore, a negative quantity, and the state of his affairs may be expressed thus, ba.

In this question, if we suppose the merchant to have lost instead of gained by his business, it is evident that the value of b will be less than that of a, and we shall be required to subtract a greater number from a less, which is impossible. But it is perfectly easy to represent such a subtraction, as, for instance, 18-32; and hence it frequently happens in Algebra, that a negative quantity stands alone, as - x, when there is no quantity from which it is to be actually taken.

6. A man has in his possession 200 dollars, and owes debts to the amount of 500 dollars.

much is he worth?

How

Here, the moncy the man has is a positive quantity, and the amount of his debts, which is to be subtracted, is a negative quantity; therefore, the expression 200 500 will represent the state of his property. Now, if he pay off his debts, as far as his 200 dollars will go, there will still be $

300 left; that is, he will be 300 dollars worse than nothing.

In the last question, the amount of the debts might be regarded as the positive quantity; and then the opposite quantity, the money on hand, would be negative, and 500-200 would represent the amount of debts which the man could not pay.

It is evident, therefore, that positive and negative are merely relative terms, which are, in general, opposed to each other. In any calculation, whatever quantity is assumed as positive, all other quantities of a similar nature, or which tend to increase it, are also positive; and whatever quantities are opposed to it, in any way, or which serve to diminish it, are negative,