a°—5a3b+10a4b2—10a3b3+ 5a2ba— abs

--

ab5ab2-10a3b3+10a2ba_5ab3+b6 ao—6a3b+15a*b*—20a3b3+15a2ba—6ab3+bo,

power of a-b; and so of any other.

the sixth

But there is a rule, or theorem, given by Sir Isaac Newton, (demonstrated hereafter) whereby any power of a binomial a + b, or a—b, may be expressed in simple terms, without the trouble of those tedious multiplications required in the preceding operations; which is thus:

Let n denote any number at pleasure; then the nth power of a + b will be an + nan-1b +

n.n-1.n-2

n

1

N- 1.

2

4

5

And the nth power of a-b will be expressed in the very same manner, only the signs of the second, fourth, sixth, &c. terms where the odd powers of b are involved, must be negative.

An example or two will show the use of this general theorem.

First, then, let it be required to raise a + b to the third power. Here n, the index of the proposed power, being 3, the first term, a", of the general expression, is equal to a3; the second na1b 3a2b; the third N.N- -1

=

an-2b23ab; the fourth

63; and the fifth

=

n n - 1

N

4

.3

a-4b4, &c. nothing. Therefore the third power of a+b is truly expressed by a3+3a2b+3ab2+b3.

Again, let it be required to raise a+b to the sixth power. In which case the index, n, being 6, we shall, by proceeding as in the last example, have a = ao,

G

6

sequently a+b] = a + 6a3b + 15a4b2+20a3b3 + 15a2ba +6ab5+b; being the very same as was above determined by continual multiplication.

Lastly, let it be proposed to involve cc+xy to the fourth power.

Here a must stand for cc, b for xy, and n for 4; then by substituting these values, instead of a, b, and n, the general expression will become c + 40°xy + 6c1x2y2 + 4c2x3y3 + xy, the true value sought.

From the preceding operations, it may be observed, that the unciæ, or coefficients, increase till the indices of the two letters a and b become equal, or change values, and then return, or decrease again, according to the same order: therefore we need only find the coefficients of the first half of the terms in this manner; since, from these the rest are given.

SECTION VII.

Of Evolution.

EVOLUTION, or the extraction of roots, being directly the contrary to involution, or raising of powers, is performed by converse operations, viz. by the division of indices, as Involution was by their multiplication.

Thus the square root of x, by dividing the exponent by 2, is found to be a3; and the cube root of x, by dividing the exponent by 3, appears to be 2: moreover, the biquadratic root of a + x] will be a + x2; and the cube root of aa+xx will be aa+xx.

In the same manner, if the quantity given be a fraction, or consists of several factors multiplied together, its root will be extracted, by extracting the root of each particular factor.

Thus the square root of a2b2 will be ab; that of

will be aa—xx}; its cube root aa-xx]; and its biquadratic root, aa-xx]; and so of others. All which being nothing more than the converse of the operations in the preceding section, requires no other demonstration than what is there given.

Evolution of compound quantities is performed by the following rule.

First, place the several terms, whereof the given quantity is composed, in order, according to the dimensions of some letter therein, as shall be judged most commodious; then let the root of the first term be found, and placed in the quotient; which term being subtracted, let the first term of the remainder be brought down, and divided by twice the first term of the quotient, or by three times its square, or four times its cube, &c. according as the root to be extracted is a square, cubic, or biquadratic one, &c. and let the quantity thence arising be also wrote down in the quotient, and the whole be raised to the second, third, or fourth, &c. power, according to the aforesaid cases, respectively, and subtracted from the given quantity; and (if any thing remains) let the operation be repeated, by always dividing the first term of the remainder by the same divisor, found as above.

Suppose, for example, it were required to extract the square root of the compound quantity 2ax + a2 + x2 : then, having ranged the terms in order, according to the dimensions of the letter a, the given quantity will stand thus, a2 + 2ax + x2, and the root of its first term will be a; by the double of which I divide 2ax, (the first of the remaining terms) and add +x, the quantity thence arising to a (already found) and so have ax in the quotient; which being raised to the second power, and subtracted from the given quantity, nothing

remains; therefore a +x is the square root required. See the operation.

a2+2ax+x2 (a+x

2a) 2ax

a2+2ax+x2, second power of a+x.

0 0 0

In like manner, if the quantity a1-2a3x+3a2x2—2ax3 + be proposed to extract the square root thereof; the answer will come out a2-ax+x2, as appears by the pro

cess.

a1-2a3x+3a2x2—2ax3+x1 ( a2—ax+x2

2a2)-2a3x

a4 -2a3x+ a2x2, second power of a2—ax. 2a2) 2a2x2, first term of the remainder. a1—2a3x+3a2x2-2ax3+x1, square of a2-ax+x2.

Again, let it be required to extract the cube root of a3-6a2x+12ax2-8x3, and the work will stand thus: a3 —6a2x+12ax2—8x3 ( a—2x 3a2)-6a2x

a3 ́—6a2x+12ax2-8x3, cube of a-2x.

Lastly, let it be required to extract the biquadratic root of 16x96x3y+216x2y2-216xy3+81y, and the process will stand as follows:

16x4 —96x3y+216x2y2—216xy3+81y1 ( 2x—3y 32x3)—96x3y

16x96x3y+216x2y2—216xy3+81y1

And, in the same manner the root may be determined in any other case, where it is possible to be extracted; but if that cannot be done, or, after all, there is a remainder, then the root is to be expressed in the manner of a surd, according to what has been already shown. As to the truth of the preceding rule, it is too obvious to need a formal demonstration, every ope

ration being a proof of itself. I shall only add herc, that there are other rules besides that, for extracting the roots of compound quantities; which, sometimes, bring out the conclusions rather more expeditiously; but as these are confined to particular cases, and would take up a great deal of room to explain in a manner sufficiently clear and intelligible, it seemed more eligible to lay down the whole in one easy general method, than to discourage and retard the learner by a multiplicity of rules. However, as the extraction of the square root is much more necessary and useful than the rest, I shall here put down one single example thereof, wrought according to the common method of extracting the square root, in numbers: which I suppose the reader to be acquainted with, and which he will find more expeditious than the general rule explained above.

--

Examp. a*+4a3x+6a2x2+4ax3+x1 (a2 + 2ax + x2

THE reduction of fractional and radical quantities is of use in changing an expression to the most simple and commodious form it is capable of; and that, either by bringing it to its least terms, or all the members thereof (if it be compounded) to the same denomination.