EXTRACTION OF THE SQUARE ROOT.

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14. What is the mean proportional between 64 and 9? Ans. 24.

To find the side of a square, equal in area to any given superficies, extract the square root of the given area.

15. In a square plantation, containing 505521 trees, each six feet distant, what is the length of the side? Ans. 4266 feet.

16. A gentleman has two fields; the first contains 8 acres, 2 roods, 1 pole,—the second 6 acres, 2 roods; he wishes to exchange them for a square field; required the side of the square? Ans. 49 po. NOTE. Circles are to each other as the squares of their diameters.

17. A gentleman has two circular ponds in his pleasure grounds; the diameter of the one is 200 feet, and the other three times as large; what is its diameter? Ans. 346-4704.

18. A maltster has a kiln, 14 feet in diameter, which is too little by for his business, what is the diameter of one that will answer his purpose? Ans. 18.78 feet.

Given any two sides of a right-angled triangle, to find the other side.

The square of the hypothenuse, or longest side, is equal to the sum of the squares of the other two sides; therefore the difference of the squares of the hypothenuse, and either of the other sides, is the square of the remaining side.

19. A wall is 48 feet high, and a ditch before it is 36 feet wide, required the length of a ladder that will reach from the opposite side of the ditch to the top of the wall? Ans. 60 feet.

20. A line, of 205 feet in length, reaches from the top of a steeple to a point 140 feet from its foundation, what is the height of the steeple? Ans. 149.75 feet.

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EXTRACTION OF THE CUBE ROOT.

RULE 1. Divide the given number into periods of three figures each, beginning at the right hand in integers, and pointing toward the left. But in decimals, begin at the place of thousands, and point toward the right.

2. Find the greatest cube number, in the left-hand period, and place the root of that number as the first figure of the root sought subtract the number itself from the said period, and to the remainder bring down the next period for a dividend.

3. Find a divisor by multiplying the square of the part of the root found by 300, divide the dividend by it, and put the quotient figure for the next figure of the root.

4. Multiply the part of the root formerly found by the last figure placed in it, and this product by 30; place this last product under the divisor, and under this product write the square of the figure last placed in the root.

5. Multiply the sum of these three by the figure last placed in the root, and subtract the product from the dividend.

6. To the remainder bring down the next period for a new dividend, with which proceed as before.

Or add up the three last numbers, reckoning the middle one twice in the operation; to this sum annex two ciphers, the result will be the next trial-divisor, with which divide and complete as before.

Required the Cube Root of the following numbers?

To find the side of a cube, equal in solidity to any given solid, extract the cube root of its solid content.

11. The solid content of a globe is 32768 feet, what is the side of a cube of equal solidity? Ans.

32 feet.

12. Required the side of a cubical mound equal to one whose length is 840 feet, breadth 500 feet, and depth 340 feet? Ans. 522.68 feet.

13. Required the side of a cubical vessel which will contain 100 imperial gallons? Ans. 30-267 inches.

Similar solids are to each other as the cubes of their diameters, sides, &c.

14. A ball, 12 inches diameter, weighs 30 lb., what will a ball weigh whose diameter is 18 inches? Ans. 101 lb.

15. A ball, 16 inches diameter, weighs 56 lb., what will be the diameter of a ball which weighs 6 times as much? Ans. 29.07 inches.

16. The length of a stone is 10 feet, breadth 4 feet, and thickness 3 feet, required the dimensions of another 6 times as large? Ans. 18.17 feet long, 8.177 broad, 5.45 thick.

SLIDING RULE.

RULES for working on the Carpenter's Sliding Rule and on Gunter's Scale.

NOTE. The Rules are commonly marked with A on the Rule, B and C on the Slider, and D on the Girt or Square Line.

To multiply by the Sliding Rule.

RULE. Set 1 on B to one of the factors on A; then against the other factor on B, you have the product on A.

By Gunter's Scale.

RULE. Extend the compasses from the beginning of the line of numbers to one of the factors; the same distance shall reach from the other factor to the product.

To divide by the Sliding Rule.

RULE. Set the divisor on B to the dividend on A; then against 1 on B, you have the quotient on A.

By Gunter's Scale.

RULE. Extend the compasses from the dividend to the divisor; the same distance will reach from 1 to the quotient. Proportion by the Sliding Rule.

RULE. Set the first number on B to the second on A; then against the third number on B is the answer on A. By Gunter's Scale.

RULE. Extend the compasses from the first number to the second; that extent will reach in the same direction from the third number to the fourth.

Superficial Measure by the Sliding Rule.

RULE. Set the breadth in inches on B to 12 on A; then against the length in feet on A is the content on B, in square feet, &c.

By Gunter's Scale.

RULE. Extend the compasses from 12 to the breadth; that extent set the same way will reach from the length to the content.

Solid Measure by the Sliding Rule.

RULE. Set the length in feet on C to 12 on D, or girt line; then against the side of the square on D you have the answer on C.

By Gunter's Scale.

RULE. Extend the compasses from 12 to the side of the Square; that extent twice set will reach from the length to the answer.

To extract the Square Root by the Sliding Rule.

RULE. Set the middle division on C to 10 on D; then opposite to the given number on C you have the answer on D.

NOTE. If the given number consists of 2, 4, 6, &c. figures, the answer is to be found on the left hand of the line C; but if it consists of 3, 5, 7, &c. figures, it is found on the right hand of the line C.

To find a Mean Square by the Sliding Rule.

RULE. Multiply the breadth and depth together, and extract the square root of the product; the root is the mean square. Or set the breadth in inches on C to ditto on D ; then against the depth or thickness in inches on C you have the mean square in inches on D, or girt line.

By Gunter's Scale.

RULE. The middle point, between the breadth and depth, is the mean square.

DUODECIMALS, OR CROSS MULTIPLICATION, Is a rule by which artificers cast up the contents of their work.

RULE 1. Write the multiplier under the multiplicand, feet under feet, inches under inches, seconds under seconds, &c.

2. Multiply each denomination of the length by the feet of the breadth, beginning at the lowest, and place each product under that denomination of the multiplicand from which it arises, always carrying one for every 12.

3. Multiply by the inches, and set each product one place farther to the right hand.

4. Multiply by the seconds or parts, and set each product another place toward the right hand.

5. Proceed in this manner with all the rest of the denominations, and their sum will be the answer.

NOTE. Feet multiplied by feet give feet.

Feet multiplied by inches give inches.
Feet multiplied by seconds give seconds.
Inches multiplied by inches give seconds.
Inches multiplied by seconds give thirds.
Seconds multiplied by seconds give fourths.
12 Fourths 1 Third,

=

12 Thirds = 1 Second, or part,
12 Seconds = 1 Inch,

12 Inches = 1 Foot.

1. Multiply 6 feet, 3 inches, by 3 feet, 2 inches. Ans. 19 feet, 9 in. 6 s.

2. Multiply 4 feet, 5 inches, by 3 feet, 6 inches. Ans. 15 f. 5 in. 6 s.

3. Multiply 5 feet, 6 in. by 4 feet, 3 in. Ans. 23 f. 4 in. 6 s.

4. Multiply 6 feet, 6 in. by 3 feet, 8 in. Ans. 23 f. 10 in.

5. Multiply 24 feet, 3 in. by 16 feet, 7 in. Ans. 402 f. 1 in. 9 s.

6. Multiply 48 feet, 7 in. by 36 feet, 6 in. Ans. 1773 f. 3 in. 6 s.