SUBTRACTION.

Subtraction of algebraical quantities, is performed by changing all the signs of the quantity to be subtracted; that is to say, that +6 to be subtracted from a equal a-b, and that -b to be subtracted from a equal a+b; for, the result of the subtraction, joined to the quantity subtracted, must be equal to the other quantity; whence it follows, that if in the 1st example, to the difference a-b we add the quantity subtracted +6, we must have a for the result; in fact a−b+b=a; in like manner if, in the second example, to the difference a+b we add —b the result must be a, what happens in effect, for a+b—b=a, therefore, &c.

We plainly see, that if we had 5a, from which we wished to subtract 2a, instead of writing 5a-2a, we would simply write 3a.

Example. Let it be proposed to subtract 4a-b+c from 5a-3b-c+d: we must then change the signs of the quantity to be subtracted, which is here the first that presents itself, and write it after the second; which gives 5a-3b−c+d−4a+b➡c; this is the result of the subtraction; but we can simplify this result, by reducing the like quantities; and then we have a-2b-2c+d. We must never neglect making these kinds of reductions after addition or subtraction.

MULTIPLICATION.

There are three rules to be observed in multitiplication, that of the Signs, that of the Coefficients, and that of the letters.

The rule of multiplication for the Letters, is to write them one after another; but when a same letter is to

be taken several times as a factor in a monomial, it has been agreed upon to write this letter but once, with a little figure raised above it on the right hand, which expresses the number of times that this letter ought to be taken as factor; thus for example a. a. a, I write a3. this little figure placed above, is called exponent; we must take great care not to confound it with the coefficient.

If we had a Xa3, which is the same thing as aa Xaaa, we would then write a ; whence we see, that multiplication of like letters or quantities, is performed by adding together their exponents; thus a* bc3d ab2 c2=a5b3c5d, considering that unity is always supposed to be the exponent of every quantity which has none; for b is evidently the same thing as b1.

What we have just said, only regards a monomial to be multiplied by a monomial; but with a little reflection, we shall see clearly that its application is easy, when a polynomial is to be multiplied by a monomial; and that we shall not be more embarrassed, to multiply a polynomial by a polynomial; for all the parts of the multiplicand, being to be multiplied by each part of the multiplier, we see plainly that to form all the partial products which compose the total product, we shall never have but a monomial, to be multiplied by a monomial :

The rule of multiplication for the Coefficients, consists in multiplying them by one another, according to the rules of arithmetic. Thus 7ab × 9bd63ab2d.

The rule of multiplication for the Signs, requires a particular demonstration; we can form four different combinations with the signs + and

As to the first case, it stands in need of no demonstration; it is evident that +ax+b=+ab.

To demonstrate the second case, let us suppose we

have the binome b-ca; if we perform the multiplication without paying attention to the signs, we shall have the two products ab and ac; but in multiplying b by a, it is evident, that I have multiplied a quantity too great by c, for it was not b that I had to multiply, but b diminished by c; then as many times as I have taken b, so many times have I taken c too much; but I have taken b,a times too much, therefore from the product ab, we must subtract c taken a times, or ac and then write ab-ac; thus b-cxa-ab-ac, and -cx+a=~ ac, therefore -x+=−.

-

The third case falls under the second, because we can take indifferently either of the factors for multiplicand.

b

The fourth case is demonstrated thus. Let there be ab. c-d; let us first multiply a by cd, according to what we have just demonstrated, we shall have ac-ad; but it was not a that I had to multiply by c-d, but a diminished by ; thus, as many times as I have taken a, so many times have I taken too much; but I have taken a, c—d times, I must then subtract from it b taken c-d times, or bc-bd, and as to subtract, we must change the signs, I shall then have a-b -dac-ad—bc+bd; we see that the product of b by d=+bd; therefore--=+·

total products, 20a3-41a+b+50a3 b2 —45a2b3 +25ab*—6b*

DIVISION.

There are four rules to be observed in division, that of the Signs, that of the Coefficients, that of the Letters, and that of the Exponents.

The rule of Signs is easily found by the result of the signs of multiplication; in fact, since the divisor, multiplied by the quotient, must produce the dividend, it is evident,

Let us give out the reasoning for one case only, and let us take the second.

I say then, that — divided by gives less, for if I multiply the sign of the divisor by. the sign of the quotient, I shall have―, which is the sign of the dividend, therefore less ought to be the sign of the quotient.

1st. that

+

二十

+

2nd. that

3d. that

4th. that ==+

As to the rule of Coefficients, we must divide them by each other according to the principles of arithmetic ; then write the quotient instead of the dividend, and suppress the divisor, or write 1 in its place; but we must observe, that this division is performed only when it can be made without remainder; otherwise the 'quotient, being composed of two parts, the quantity would then be presented under a complicated form; in this case we only indicate the division, and write the numbers such as they are before the letters.