Change each + into, and each into +, in the subtrahend, or suppose them to be thus changed; then proceed as in addition, and the sum will be the true remainder.

EXAMPLES.

*This rule is founded on the consideration, that addition and subtraction are opposite to each other in their nature and operation, as are the signs + and by which they are expressed and represented. And since to unite a negative with a positive quantity of the same kind has the effect of diminishing it, or subducting an equal positive quantity from it; therefore to subtract a positive, which is the opposite of uniting or adding, is to add the equal negative quantity. In like manner, to subtract a negative quantity is the same in effect, as to add or unite an equal positive quantity. So that, by changing the sign of a quantity from + or from to, its nature is changed from a subductive to an additive quantity; and any quantity is in effect subtracted by barely changing its sign.

"to

*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 tak en from the simple quantity 5ax', the remainder 3ax2-4, and it is plain, that the more there is taken from any number or quan.

is

tity, the less will be left; and the less there is taken, the more will be left. Now, if only zax were taken from 5ax2, the remainder would be 3ax; and consequently, if zax2+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 24x2+4, and adding it to 5ax2, the sum is 5ax2—2ax2—4 ; but here the term2ax destroys so much of 5ax2 as is equal to itself, and 30 5ax2—2ax2-4 becomes equal to 3ax2-4, by the general rulę for subtraction.

Rem. 3/ax-5a —3√a2+b2 +5a √x+x2-y2

In multiplication of algebraic quantities there is one general rule for the signs; namely, when the signs of the factors 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 uct, may be shewn thus:

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.

a is

1. When +a is to be multipled by +b; it implies, that 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 fax+b makes

ab.

2. When two quantities are to be multiplied together; the result will be exactly the same, in whatever order they are placed; for 4 times b is the same as b times a; and therefore, when -a is to be multiplied by +b, or +b by -a, it is the same thing as taking -a as many times as there are units in +; and since the sum of any number of negative terms is negative, it follows, that -a×+b, or +ax-b, makes or produces —ab.

3. When -a is to be multiplied by -b; here -a 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 +ab.

Otherwise.

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 a multiplied into a3 is a5t3, or a3. That of " into x is "41.

That of " into x is +*.

That of x into x" is x+".

And that of car into y

is cytartu➡r, or cy2nt?.

Again, the product of a+x multiplied into a+x is

And that of x+yl into x+yl is x+yl

ntr

This

Otherwise. Since a-ao, therefore a-ax-b is also =0, beause o multiplied by any quantity is still ; and since the first term of the product, or a×—b, ab, by the second case; therefore the last term of the product, or -ax-b, must be +ab, tò make the sum ©, or —ab+ab=o; that is, aX—b➡+ab.