called positive, the motion in the opposite direction should be considered negative.

Suppose a ship, setting out from the equator, sails northward 50 miles, then southward 27 miles, then northward 15 miles, then southward again 22 miles, and we wish to determine the last position of the ship. If we call the northerly motion+, the whole may be expressed algebraically thus:

+50-27+15-22,

which reduces to +16. The positive sign of the result indicates that the ship was 16 miles north of the equator.

Suppose the same ship sails again 8 miles north, then 35 miles south, the whole may be expressed thus:

+50-27+15-22+8-35,

which reduces to -11. The negative sign of the result indicates that the ship was now 11 miles south of the equator.

In this example we have considered the northerly motion +, and the southerly motion; but we might with equal propriety have considered the southerly motion +, and the northerly motion. It is, however, indispensable that we adhere to the same system throughout, and retain the proper sign of the result, as this sign shows whether the ship was at any time north or south of the equator.

In the same manner, if we consider easterly motion +, westerly motion must be regarded as -, and vice versa. And generally, when quantities which are estimated in different directions enter into the same algebraic expression, those which are measured in one direction being treated as +, those which are measured in the opposite direction must be regarded as -.

So, also, in estimating a man's property, gains and losses being of an opposite character, must be affected with different signs. Suppose a man, with a property of 1000 dollars, loses 300 dollars, afterward gains 100, and then loses again 400 dollars, the whole may be expressed algebraically thus:

+1000-300+100-400,

which reduces to +400. The + sign of the result indicates that he has now 400 dollars remaining in his possession. Suppose he further gains 50 dollars and then loses 700 dollars The whole may now be expressed thus:

+1000-300+100-400+50-700,

which reduces to -250. The sign of the result indicates that his losses exceed the sum of all his gains and the property originally in his possession; in other words, he owes 250 dollars more than he can pay, or, in common language, he is 250 dollars worse than nothing.

This phraseology must not be regarded as wholly figurative; for, in algebra, a negative quantity standing alone is regarded as less than nothing; and of two negative quantities, that which is numerically the greatest is considered as the least; for if from the same number we subtract successively numbers larger and larger, the remainders must continually diminish. Take any number, 5 for example, and from it subtract successively 1, 2, 3, 4, 5, 6, 7, 8, 9, &c., we obtain

5-1, 5-2,5-3, 5-4, 5-5, 5-6, 5-7, 5-8, 5-9, &c., or reducing

4, 3, 2, 1, 0, -1, -2, -3, -4.

Whence we see that 1 should be regarded as smaller than nothing; -2 less than -1; -3 less than 2, &c.

EXAMPLES.

1. From 8x'y'-5xy+9a'm take 6x'y'-10xy+7a3m.

Ans.

2. From 15a3bx—19p3q+11axy - 25 take 17a3bx — 13p3q +9axy+7.

Ans.

3. From 10ax +19a'x'y - 6ab' + 16gm take 4ax-13a'x'y +7ab2+8gm.

Ans.

4. From 11a'y — 14amn +9gm+13a take 15a3y+7amn -12gm-8a3.

Ans.

5. From 13axy2—6my+16a2bc+8a3 take 9axy3+2my+19a2bc -4a3.

Ans.

6. From 17a'c — 11bm + 3xy' +14amn take 15a'c-21bm — 6xy2+8amn3.

Ans.

Ans.

Ans.

7. From 35am3 — 19bx+27y'—11av2 take 15am3 — 7bx+31y' -23av3. 8. From 12ab' - 3cx-4xy'-7abc take 8ab'-6cx-5xy' -12abc2.

·

9. From 40a + 7b3c — 6b3 + 3ax'y' take 12a' — 4b3c+5b3 +4ax'y'.

Ans.

Ans.

10. From 7ax-50amn-3by'+6m take 9ax1-2amn+3by'

-4m.

SECTION IV.

MULTIPLICATION.

(48.) Multiplication is repeating the multiplicand as many times as there are units in the multiplier.

When several quantities are to be multiplied together, the result will be the same in whatever order the multiplication is performed.

This may be demonstrated in the following manner:

Let unity be repeated five times upon a horizontal line, and let there be formed four such parallel lines.

Then it is plain that the number of units in the table is equal to the five units of the horizontal line, repeated as many times as there are units in a vertical column; that is, to the product of 5 by 4. But this sum is also equal to the four units of a vertical line repeated as many times as there are units in a horizontal line; that is, to the product of 4 by 5. Therefore, the product of 5 by 4 is equal to the product of 4 by 5. For the same reason, 2×3×4 is equal to 2×4×3, or 4×3×2, or 3×4×2, the product in each case being 24. So, also, if a, b, and c represent any three numbers, we shall have abc equal to bca or cab.

It is convenient to consider the subject of multiplication under three Cases.

CASE I.

(49.) When both the factors are monomials.

From Article 14, it appears that, in order to represent the multiplication of two monomials, such as 3abc and 5def, we may write these quantities in succession without interposing any sign, and we shall have

3abc5def.

But, according to the principle stated in the preceding article, this result may be written

3×5abcdef, or 15abcdef.

Hence we deduce the following

RULE.

Multiply the coefficients of the two terms together, and to the product annex all the different letters in succession.

From Article 48, it appears to be immaterial in what order the letters of a term are arranged; it is, however, generally most convenient to arrange them alphabetically.

(50.) We have seen in Art. 21, that when the same letter appears several times as a factor in a product, this is briefly expressed by means of an exponent. Thus, aaa is written a3, the number 3 showing that a enters three times as a factor. Hence, if the same letters are found in two monomials which are to be multiplied together, the expression for the product may be abbreviated by adding the exponents of the same letters. Thus, if we are to multiply a3 by a2, we find a equivalent to aaa, and a2 to aa. Therefore the product will be aaaaa, which may be written a', a result which we might have ohtained at once by adding together 3 and 2, the exponents of the common letter a.

Hence, since every factor of both multiplier and multiplicand must appear in the product, we have the following

RULE FOR THE EXPONENTS.

Powers of the same quantity may be multiplied by adding

their exponents.

(51.) When the multiplicand is a polynomial.

If a+b is to be multiplied by c, this implies that the sum of the units in a and b is to be repeated c times; that is, the units in b repeated c times must be added to the units in a repeated also c times. Hence we de duce the following

RULE.

Multiply each term of the multiplicand separately by the mul tiplier, and add together the products.

(52.) When both the factors are polynomials.

If a+b is to be multiplied by c+d, this implies that the quantity a+b is to be repeated as many times as there are units in the sum of c and d; that is, we are to multiply a+b by c and d successively, and add the partial products. Hence we deduce the following

RULE.

Multiply each term of the multiplicand by each term of the multiplier separately, and add together the products.