* The ten foregoing examples of simple quantities being obvious, we pass by them; but shall illustrate the eleventh example, in order to the ready understanding of those, which follow. In the eleventh example, the compound quantity 2ax +4 being taken from the simple quantity 5ax2, the remainder is 3ax2-4, and it is plain, that the more there is taken from any number or quantity, the less will be left; and the less there is taken, the more will be left. Now, if only 2ax2 were taken from 5ax2, the remainder would be 3ax2; and consequently, if 2ax2+4, which is greater than 2ax2 by 4, be taken from 5ax2, the remainder will be less than 3ax2 by 4, that is, there will remain 3ax2 -4, as above. For by changing the sign of the quantity 2ax2 +4, and adding it to 5ax2, the sum is 5ax2-2ax2-4; but here the term-2ax2 destroys so much of 5ar2 as is equal to itself, and so 5ax2-2ax2-4 becomes equal to 3ax2-4, by the general rule for subtraction.

In multiplication of algebraic quantities there is one general rule for the signs; namely, when the signs of the fac tors are both affirmative or both negative, the product is affirmative; but if one of the factors be affirmative and the other negative, then the product is negative.*

* That like signs make +, and unlike signs, in the prod uct, may be shown thus.

1. When + a is to be multiplied by +b; it implies, that +a is to be taken as many times, as there are units in b; and since the sum of any number of affirmative terms is affirmative, it follows, that +ax+b makes + ab.

a

2. When two quantities are to be multiplied together; the result will be exactly the same, in whatever order they are placed; for a times b is the same as b times a ; and therefore,.when is to be multiplied by +b, or +6 bya, it is the same thing as taking a as many times as there are units in +b; and since

CASE I.

When both the factors are simple quantities.

RULE.

Multiply the coefficients of the two terms together, to the product annex all the letters of the terms, and prefix the proper sign.

the sum of any number of negative terms is negative, it follows, that-x+b, or +ax-b, makes or produces

ab.

3. When -a is to be multiplied by -b; here- —ɑ is to be subtracted as often as there are units in b; but subtracting negatives is the same as adding affirmatives, by the demonstration of the rule for subtraction; consequently the quotient is 6 times a, or tab.

Otherwise.

Since a-a=0, therefore a—a—b is also =0, because 0, multiplied by any quantity, is still 0; and since the first term of the product, or ax-b, ——ab, by the second case; therefore the last term of the product, or-a-b, must be +ab, to make the sum =0, or➡ab+ab=0; that is, ɑ X-b=+ab,

NOTE 1. To multiply any power by another of the same root; add the exponent of the multiplier to that of the multiplicand, and the sum will be the exponent of their product. Thus, the product of a3, multiplied into a3, is a3+3, or a3. That of x into x is x "+1.

That of "into x2 is x *+*.

m

That of x into x" is "+".

+2 into y is cy"+ar+r, or cy

And that of cy"+2 into y′′

or cynt. Again, the product of a+x", multiplied into a +x, is

a+x]

And that of x+y)" into x+y" is x+y

tr

This rule is equally applicable, when the exponents of any roots of the same quantity are fractional.

Thus, the product of a3, multiplied into a, is aa1 xa3=

Hence it appears, that, if a surd square root be multiplied into itself, the product will be rational; and if a surd cube. root be multiplied into itself, and that product into the same root, the product is rational. And in general, when the sum of the numerators of the exponents is divisible by the common denominator, without a remainder, the product will be rational.

5

3

3

543

8

Thus, a3xa3=a+=a+ =a3=a3.

Here the quantity a is reduced to a2, by actually dividing 8, the numerator of the exponent, by its denominator 4; and the sum of the exponents, considered merely as vulgar fractions, is +==2.

When the sum of the numerators and the denominator of the exponents admit of a common divisor greater than unity, then the exponent of the product may always be reduced, like a vulgar fraction, to lower terms, retaining still the same value.

Compound surds of the same quantity are multiplied in the same manner as simple ones.

2

1

Thus, a+a+3=a+a|3=a+x=a+x;

8

8

So likewise Va+xx ®√ a+x=ˆ√ a+x=a+x

And

? +ux √ a+x=√ a+x= a+x2.

And Va+xx√ a+x=a+x.

These examples show the grounds, on which the products of surds become rational.

NOTE 2. Different quantities under the same radical sign are multiplied together like rational quantities, only the product, if it do become rational, must stand under the same radical sign.

Thus, V7XV3=√7x3=√21.

vaxvb=vab.

3√7cx ×3√2y = 3√ 14cxy.

And "√/4dx √2cd=”√′8¢d2 = 8cd2\".

It may not be improper to observe, that unequal surds have sometimes a rational product.