EXTRACTING ROOTS OF ALL POWERS. 219

-A GENERAL RULE

FOR EXTRACTING THE ROOTS OF ALL POWERS.

1. PREPARE the number given for extraction, by pointing off from the units' place, as the root required directs.

2.

Find the first figure in the root, by the table of powers, which fubtract from the given number.

3. Bring down the firft figure in the next point to the remainder, and call it the dividend.

4.

Involve the root into the next inferior power to that which is given; multiply it by the given power, and call it the divifor.

5. Find a quotient figure by common divifion, and annex it to the root; then involve the whole root into the given power, and call that the fubtrahend.

6. Subtract that number from as many points of the given power as is brought down, beginning at the lowest place, and to the remainder bring down the first figure of the next point for a new div. idend.

7. Find a new divisor, and proceed in all respects as before.

3. What is the biquadrate root of 19987173376?

19987173376(376

81

108) 1188 dividend

1874161 fubtrahend

202612) 1245563 dividend

19987173376 subtrahend

3X 3X 3 X 4 108 Divifor

37X 37X 37X 37=1874161 Subtrahend 37X 37X 37X 4202612 Divifor

376X 376 X 376 X 376-19987173376 Subtrahend.

Anf. 376.

DUODECIMALS.

DUODECIMALS, or Crofs Multiplication, is a rule made ufe of in measuring and computing the dimenfions of the feveral parts of buildings; it is likewise used to find ships' tonnage and the contents of bales, cafes, &c.

Dimensions are taken in feet, inches, and parts.

Artificers' work is computed by different measures, viz. Glazing, and masons' flat work, by the foot;

Painting, paving, plaftering, &c. by the yard;

Partitioning, flooring, roofing, tiling, &c. by the fquare of 100 feet;
Brick-work, &c. by the rod of 16 feet, whofe fquare is 2724.
The contents of bales, cafes, &c, by the ton of 40 Cubic feet.
The tonnage of fhips, by the ton of 95 feet.

RULE FOR MULTIPLYING DUODECIMALLY.

1. Under the multiplicand write the corresponding denominations of the multiplier.

2. Multiply each term in the multiplicand, (beginning at the loweft) by the feet in the multiplier; write each refult under its respective term, obferving to carry an unit from each lower denomination to its fuperior.

3. In the fame manner, multiply the multiplicand by the inches in the multiplier, and write the refult of each term, one place more to the right hand of them, in the multiplicand.

4. Work in the fame manner with the other parts in the multiplier, fetting the result of each term two places to the right hand of thofe in the multiplicand, and fo on for thirds, fourths, &c.

5. Proceed in the like manner with all the reft of the denominations, and their fum will give the answer required.

2. Multiply 9 feet 6 inches by 4 feet 9 inches.

3. What is the price of a marble flab, whofe length is 5 feet 7 inches, and breadth 1 foot 10 inches, at I dollar per foot ? Anf. 10 dols. 23 cts.

4. There is a house with three tiers of windows, 3 in a tier, the height of the first tier is 7 feet 10 inches, of the second 6 feet 8 inches, and of the third 5 feet 4 inches, and the breadth of each is 3 feet 11 inches; what will the glazing come to, at 14d. per foot?

Anf. £.13 115. 10 d.

5. If a house measures within the walls 52 feet 8 inches in length, and 39 feet 6 inches in breadth, and the roof be of a true pitch, or the rafters of the breadth of the building, what will it come to, roofing at 10s. 6d. per fquare? Anf. £.12 125. 11 d.

APPLICATION OF DUODECIMALS.

To find how many cubic or folid fquare feet (in order to afcertain the freight) are contained in cafes, bales, &c. that is, how many cubic feet they will take up in a fhip.

EXAMPLES.

1. Suppose the dimenfions of a bale to be 7 feet 6 inches, 3 feet inches, and 1 foot 10 inches; what is the folid content ?

What is the freight of a bale containing 65 feet 9 inches, at

2.

15 dollars

per ton of 40 feet?

3. A merchant imports from London 6 bales of the following

dimensions, viz.

No. 1.

What are the folid contents, and how much will the freight amount

To find Ships' Tonnage by Carpenters'-Measure.

RULE. For fingle decked veffels, multiply the length, breadth at the main beam, and depth of the hold together, and divide the product by 95.

EXAMPLE.

What is the tonnage of a fingle decked veifel, whose length is 60 feet, breadth 20 feet, and depth 8 feet?