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7. What is the square root of 4.372594 ? Ans. 2.091+ 8. What is the square root of .00103041 ? Ans. .0321. 9. What is the square root of 2704?

Ans. 10. What is the square root of 75?

Ans. .89802+ 11. What is the square root of 1?

Ans.. .2. What is the square root of 61?

Ans. 2) 13. What is the difference between 81 and 812?

Ans. 6552. 14. What is the square root of 334?

Ans. 15. What is the square root of ii?

Ans. .9574+ 16. What is the square root of 91225?

Ans. 30 17. What is the square root of 21 ?

Ans. 1.5. 18. What is the square root of 9980.01 ?

Ans. 99.9. 19. What is the square root of 2 ?

Ans. 1.414213. 20. What is the square root of 7447? Ans. 215.09756+ 21. What is the square root of .0000316969 ? Ans. .00563. 22. What is the square root of 964.5192360241 ?

Ans. 31.05671. 23. What is the square root of mo?

Ans. 12. 24. What is the 'square root of 2198 ? 25. What is the square root of 6.9169 ?

Ans. 2.63. 26. What is the square root of 106929 ?

Ans. 327. 27. What is the square root of is?

Ans. 4. 28. What is the square root of 43 ?

Ans. .7745. 29. What is the square root of 201?

Ans. 41

APPLICATION.
To find a mean proportional between two numbers.

RULE.

Multiply the given numbers together, and extract the square root of the product, which will be the mean proportional sought.. 30. What is the mean proportional between 24 and 96 ?

Ans. 96X24=18. To find the side of a square equal in area to any given superficies

whatever.

RULE.

Find the area, and the square root is the side of the square sought.

31. If the area of a circle be 184.125, what is the side of a square equal in area thereto?

Ans. 184.125=13.569+

32. A general has an army of 5625 men; how many inust hé place in rank and file to form them into a square ?

Ans., 5625=75. 33. Suppose that Napoleon, at the battle of Marengo, commanded an army of 256036 men ;

how many did he place in rank and file to form them into a solid square ? Ans. 506.

Having the area of a circle to find the diameter.

RULE.

Multiply the square root of the area by 1.12837, and the product will be the diameter. Or multiply the area by 1.2732 and take the square root of the product.

34. Required the diameter of a circle whose area is 82 feet 87 inches.

Ans. 10 feet 3.13 inches. 35. Admit a leaden pipe i of an inch in diameter will fill a cistern in 3 hours; I demand the diameter of another pipe which will fill the same cistern in 1 hour.

Thus i =.75 and .75 X.75=.5625; then 3 hours : 5625 :: 1 hour: 1.6875 and /1.6875=1.3 inches pearly, Ans.(inversely.)

36. If a circular pipe of 1.5 inches diameter fill a cistern in 5 hours, in what time would it be filled by one 3.5 inches diameter ?

Ans. 55 minutes, 4.8 seconds. 37. If 784 trees be planted in a square orchard, how many must be in a row ?

Ans. 28. The square of the longest side or hypotenuse of a right-angled triangle, is equal to the sum of the squares of the other two sides ; consequently, the difference of the square of the longest, and either of the others, is the square of the remaining side.

38. The wall of a fort is 17 feet high, which is surrounded by a ditch 20 feet in breadth ; required the length of a ladder 10 reach from the outside of the ditch to the top of the wall.

Ans. 17X17=289 : 20 x 20=400+-289=/689=26.2-+

Ladder 26.2+ feet. hypotenuse.

Wall 17 feet. perpendicular,

base. ditch 20 feet.

39. Two ships leave the same port, one sails due east 40 miles' and the other due north 50 miles; required the distance from each other.

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40. There is a church 48 feet in height from the ground te the eaves or roof, and the street is 64 feet in width ; what length of cord would it require to reach from the opposite side of the street to the eaves of the house?

Ans. 80 feet. 41. In a circle, whose area or superficial content is 4096 feet, I demand what will be the length of one side of the square containing the same number of feet.

Ans. 64 feet. 42. A certain square garden measures 4 rods on each side; what will be the length of one side of a garden containing 4 times as many square rods?

Ans. 8 rods. 43. If one side of a square piece of land measure 5 rods, what will be the side of one measuring 4 times as large ? 16 times as large ? 36 times as large ?

Ans. 10, 20, 30. 44. In a load of 120 cherry-boards of one inch in thickness, 27 inches in width, and 11.5 feet in length, how many square feet, and what is the value of the boards at D22 per M. ?

Ans. 3105 feet; value D68.31. 45. In a tract of land of 640 acres, required the length of one side ; and suppose the tract to contain 16 times as much, what would be the length of one side ? and suppose each rod valued at D75, what would be the amount ?

Ans. Ist, 320 poles, length of one side ; 2d, 1280 po., or 16. times as much ; 3d, D122880000 value.

46. One side of a square field is 400 rods in length; required the number of acres in the field.

Note. For examples in square and cub.measure, see appendix.

REVIEW.

What is evolution? What is the square root of a number? What is the cube root? When

you

wish to extract the square root of a number, what must first be done ? After you have counted off the given number into periods, how will you proceed? Repeat the rule. When the given number is a whole number, how many figures will there be in the root ?. When you can not get the exact root, what can be done?

How can you extract the root of a decimal fraction? What will you do with a whole number and decimal ? What is the rule ? How will you extract the square root of a vulgar fraction ? Repeat the rule? Give an example on the slate. What is the difference between a square and the square root ?

47. Required the square root of 9876.47921"? (This is given for a trial question.)

CUBE ROOT. The cube of a nunber is the product of that number multiplied into its square.

To extract the cube root, is to find a number which being multiplied into itself, and then into that product, will produce the given number.

ILLUSTRATION.

The solid called a cube, has its length, breadth, and thickness, all equal. As the number of solid feet, inches, &c., in a cube, are found by multiplying the length, height, and breadth, together-that is, by multiplying one side into itself twice, the third power of a number is called the cube of that number; thus a cubical or solid foot has 6 equal sides of 1 foot square, consequently it follows that a cubical foot contains 1728 solid or cubical inches, because 12x12=144 x 12=1728 inches : that is, a cubical foot or block of wood may be sawn into 1728 blocks, each of which will be a solid inch, which may be represented in the following manner :

5

2

3

4

6

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Figure 2 is a solid or cubical foot with 6 equal sides ; sup. pose this block be sawn into 12 pieces at the distance of 1 inch, it would make 12 pieces, as represented above, each 1 foot square, and 1 inch in thickness, as in figure 1 (S. R.); now, each of those 12 square pieces may be divided into 144 solid blocks : the 12 blocks divided would be 12 x12=144 inches for each of the 12 pieces, as represented by the 12 squares, which is equal to 1728 inches in a cubical or solid foot, and 12 is the cube root of 1728.

Thus, the root 12 is equal to 12 square feet of 1 inch in thickness (as 1, 2, 3, &c.), or 12 inches in length, breadth, and thickness, which is equal to 1 solid foot. Thus, 1 X 300=300 : 2 x30=60 : 2x2=4 60

1728(12 root.

.

1 364 divisor,

364) 728

728

4+

In the example above, I first point off the three figures from the right, and place the period over the 7; this leaves 1, or unity, for the first operation ; consequently the quotient figure must in this case be i, which I put in the quotient, and under 1 in the dividend; then bring down the next period (728); now seek for the divisor ; first multiply 1 by 300=300 for a trial divisor, then suppose 300 in 728, say 2, which put in the quotient; then 2 x 30=60; then 2x2=4 the square of 2; then add these several products together and you have the true divisor (364) which multiply by the last quotient figure (2) and the work is finished.

RULE I.

1. Point the given number into periods of three figures each, beginning at the right or units' place.

2. Find the greatest cube in the left-hand period, and sitract it therefrom, and set down the root in the quotient, and then to this remainder bring down and annex the next period for a dividend.

3. Square the quotient by multiplying it into itself, and multiply that product by 300 for a trial divisor; see how often it is contained in the dividend, and set the result in the quotient.

4. Multiply the figure last put in the quotient by the other figrires in the quotient, and that product by 30.

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