. Duodecima.s.

Fractions whose denominators are 12, 144, 1728, &c. are called duodecimals; and the division and sub-division of the integer are understood without being expressed, as in decimals. The method of operating by this class of fractions is principally in use among artificers, in computing the contents of work, of which the dimensions were taken in feet, inches, and twelfths of an inch.

RULE.-Set down the two dimensions to be multiplied together, one under the other, so that feet shall stand under feet, inches under inches, &c. Multiply each term in the multiplicand, beginning at the lowest, by the feet in the multiplier, and set the result of each immediately under its corresponding term, observing to carry 1 for every 12, from the inches to the feet. In like manner, multiply all the multiplicand by the inches of the multiplier, and then by the twelfth parts, setting the result of each term one place removed to the right hand when the multiplier is inches, and two places when the parts become the multiplier. The sum of these partial products will be the answer required.

Or, instead of multiplying by the inches, &c. take such parts of the multiplicand as these are of a foot.

Here the 2 which stands in the second place does not denote square inches, but rectangles of an inch broad and a foot long, which are to be added to the square inches in the third place, so that (212) x 4 = 28 are the square inches, and the product is 38 square feet, 28 square inches

2. Multiply 35 f. 4 inc. into 12 f. 3 inc.

Here, again, the product is 435 square feet, + (3 × 12) + 11 inches, or 434 square feet, 47 square inches. And this manner of estimating the inches must be observed in all cases where two dimensions in feet and inches are thus multiplied together

SECTION IX.-Powers and Roots.

A power is a quantity produced by multiplying any given number, called the root or radix, a certain number of times con tinually by itself. The operation of thus raising powers is

called involution.

3=3 is the root, or 1st power of 3;

3 x3=329, is the 2d power, or square of 3 3x3x3=33=27, is the 3d power, or cube of 3 3x3x3x3=34-81, do. 4th power or biquadrate of 3

&c. &c. &c.

Table of the first Nine Powers of the first Nine Numbers.

8

So again, ×==square of ; ×==cube of ; &× =,biquadrate of; and so of others. Where it is evident, that while the powers of integers become successively larger and larger, the powers of pure or proper fractions become successively smaller and smaller.

Evolution.

Evolution, or the extraction of roots, is the reverse of involution.

Any power of a given number may be found exactly; but we cannot, conversely, find every root of a given number exactly. Thus, we know the square root of 4 exactly, being 2; but we cannot assign exactly the cube root of 4. So again, though we know the cube root of 8, viz. 2, we cannot exactly assign the square root of 8. But of 64 we can assign both the square root and the cube root, the former being 8, the latter 4. By means of decimals we can in all cases approximate the root to any proposed degree of exactness.

Those roots which only approximate are called surd roots, or surds, or irrational numbers; as √2, 5, 9, &c., while those which can be found exactly are called rational; as 9 =3, 125=5,† 16=2.

To extract the square root.

RULE. Divide the given number into periods of two figures each, by setting a point over the place of units, another over the place of hundreds, and so on over every second figure, both to the left hand in integers, and to the right hand in decimals.

Find the greatest square in the first period on the left hand, and set its root on the right hand of the given number, after the manner of a quotient figure in Division.

Subtract the square thus found from the said period, and to the remainder annex the two figures of the next following period, for a dividend.

Double the root above mentioned for a divisor, and find how often it is contained in the said dividend, exclusive of its righthand figure; and set that quotient figure both in the quotient and divisor.

Multiply the whole augmented diviser by this last quotient

figure, and subtract the product from the said dividend, bringing down to the next period of the given number for a new dividend.

Repeat the same process, viz. find another new divisor, by doubling all the figures now found in the root; from which, and the last dividend, find the next figure of the root as before; and so on through all the periods, to the last.*

Note. The best way of doubling the root, to form the new divisors, is by adding the last figure always to the last divisor, as appears in the following Examples.-Also, after the figures belonging to the given number are all exhausted, the operation may be continued into decimals at pleasure, by adding any number of periods of ciphers, two in each period.

The reason for separating the figures of the dividend into periods or portions of two places each, is, that the square of any single figure never consists of more than two places; the square of a number of two figures of not more than four places, and so on. So that there will be as many figures in the root as the given number contains periods so divided or parted off.

And the reason of the several steps in the operation appears from the algebraic form of the square of any number of terms, whether two or three, or more. Thus, 352= = 302 +2.30.5+52 or generally (a + b)2 = a2 + 2 a b + b2 = a2 + (2 a+b) b, the square of two terms; where it appears that a is the first term of the root, and b the second term; also a the first divisor, and the new divisor is 2 a +b, or double the first term increased by the second. And hence the manner of extraction is as in the rule.