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19. What is the square root of 52% ?
20. What is the square root of 95 ?
21. What is the square root of 363?
22. Extract the square root of 63.
23. Extract the square root of 85.
24. Required the square root of 2.

Ans. 7. Ans. 9. Ans. 19

Ans. 2.5298+. Ans. 2.9519+. Ans. 1.41421+.

APPLICATION OF THE SQUARE ROOT.

The following propositions are demonstrably true.

25. Circles are to each other, as the squares of their diameter. 26. In a right-angled triangle, the square of the longest side is equal to the sum of the squares of the other two sides.

27. In all similar triangles, that is, in all triangles whose colresponding angles are equal, the sides about the equal angles are in direct proportion to each other; that is, as the longest side of one triangle is to the longest side of the other, so is the corresponding side of the former triangle to the corresponding side of the latter.

28. If the diameter of a circle be multiplied by 3.14159, the product is the circumference.

29. If the square of the diameter of a circle be multiplied by .785398, the product is the area.

30. If the square root of half the square of the diameter of a circle be extracted, it is the side of an inscribed square.

31. If the area of a circle be divided by .785398, the quotient is the square of the diameter.

32. A certain general has an army of 141376 men. How many must he place in rank and file to form them into a square?

Ans. 376. 33. If the area of a circle be 1760 yards, how many feet must the side of a square measure to contain that quantity ? Ans. 125.857+-feet. 84. If the diameter of a round stick of timber be 24 inches, how large a square stick may be hewn from it? Ans. 16.97+inches.

35. I wish to set out an orchard of 2400 mulberry trees, so that the length shall be to the breadth as 3 to 2; and the distance of each tree, one from the other, 7 yards. How many trees must be in length, and how many in breadth; and how many square yards of ground do they stand on ?

Trees in length,

60

Trees in breadth, 40 Answer.
Square yards, 112749

36. If a lead pipe of an inch in diameter, will fill a cistern in 3 hours, what should be its diameter to fill it in 2 hours? Ans. .918+inches.

37. If a pipe 13 inches in diameter will fill a cistern in 50 minutes, how long would it require a pipe, that is 2 inches in diameter, to fill the same cistern ? Ans. 28m. 7 sec.

38. If a pipe 6 inches in diameter, will draw off a certain quantity of water in 4 hours; in what time would it take 3 pipes of four inches in diameter, to draw off twice the quantity?

Ans. 6 hours.

39. If a line, 144 feet long, will reach from the top of a fort to the opposite side of a river, that is 64 feet wide; what is the height of the fort ? Ans. 128.99+

40. A certain room is 20 feet long, 16 feet wide, and 12 feet high; how long must be a line to extend from one of the lower corners to an opposite upper corner? Ans. 28.28 feet. 41. Two ships sail from the same port; one goes due north, 128 miles, the other due east, 72 miles; how far are the ships from each other? Ans. 146.86+

42. There are two columns in the ruins of Persepolis, left standing upright; one is 70 feet above the plane, and the other 50; in a straight line between these, stands an ancient small statue, 5 feet in height; the head of which is 100 feet from the summit of the higher, and 80 feet from the top of the lower column. Required the distance between the tops of the two columns. Ans. 143.537+feet.

43. The height of a tree, growing in the centre of a circular island, 100 feet in diameter, is 160 feet; and a line extending from the top of it to the further shore is 400 feet. What is the breadth of the stream, provided the land on each side of the water be level? Ans. 316.6 feet.

44. A ladder 70 feet long is so planted as to reach a window 40 feet from the ground, on one side of the street, and without moving it at the foot, will reach a window 30 feet high on the other side; what is the breadth of the street?

Ans. 120.69+feet. 45. If an iron wire, inch in diameter, will sustain a weight of 450 pounds; what weight might be sustained by a wire an inch in diameter ? Ans. 45000 lbs.

46. Suppose a tree to stand on a horizontal plane 80 feet in height; at what height from the ground must it be cut off, so that the top of it may fall on a point 40 feet from the bottom of the tree, the end where it was cut off resting on the stump? Ans. 30 feet.

47. Four men, A, B, C, D, bought a grindstone, the diameter of which was 4 feet; they agreed that A should grind off his share first, and that each man should have it alternately until he had worn off his share; how much will each man grind off? Ans. A 3.22+, B 3.81+, C 4.97+, D 12 inches. 48. What is the length of a rope that must be tied to a horse's neck, that he may feed on an acre? Ans. 7.136+rods.

SECTION LIV.

EXTRACTION OF THE CUBE ROOT.

A CUBE is a regular body, with six equal sides.

A number is said to be cubed, when it is multiplied into its square.

To extract the cube root, is to find a number, which, multiplied into its square, will produce the given number.

RULE.

1. Separate the given number into periods of three figures each, by putting a point over the unit figure, and every third figure beyond the place of units.

2. Find by the table the greatest cube in the left hand period, and put its root in the quotient.

3. Subtract the cube thus found, from this period, and to the remainder bring down the next period; call this the dividend.

4. Multiply the square of the quotient by 300, calling it the triple square; multiply also the quotient by 30, calling it the triple quotient; the sum of these call the divisor.

5. Find how many times the divisor is contained in the dividend, and place the result in the quotient.

6. Multiply the triple square by the last quotient figure, and write the product under the dividend; multiply the square of the last quotient figure by the triple quotient, and place this product under the last; under all, set the cube of the last quotient figure, and call their sum the subtrahend.

7. Subtract the subtrahend, from the dividend and to the remainder bring down the next period for a new dividend, with which proceed as before, and so on, till the whole is completed.

NOTE. The same rule must be observed for continuing the operation, and pointing for decimals, as in the square root.

To illustrate this rule, let the pupil obtain the following appa

ratus.

1. A cubical block, whose side measures two inches; which let him call No. 1.

2. Three blocks, two inches square and half an inch thick. These let him call No. 2.

3. Three blocks, two inches long and half an inch square. These he should call No. 3.

4. One cubic block each side of which is half an inch in length. Call this No. 4.

We now consider the following question.

I have 46.656 cubic blocks of granite, which measure 1 foot on each side. With these I wish to erect a cubical monument. It is required to ascertain how many blocks or feet will be the length of one side of the monument.

It is evident, that the number of blocks will be equal to the cube root of 46.656. As the given number consists of five figures, its cube root will contain two places. For the cube of any number can never contain more than three times that number, and at least but two less. We therefore separate the given number into periods of three figures each, by putting a point over the unit figure, and every third figure beyond the place of units; thus 46.656. We find by the table of powers the greatest power in the left hand period, 46 (thousand) is 27 (thousand,) the root of which is 3. This root we write in the quotient, and as it will occupy the place of tens, its real value is 30. If this be considered the side of a cube, it will contain 27.000 cubic feet, 30X30X30= 27.000.

Let this cube be represented by our apparatus No. 1. From 46.656 we take 27.000 and there remain 19.656 cubic feet. That

OPERATION.

46.656(36 Ans.

27

2700)19656

16200

3240

216 19656

is, we should subtract the cube of the quotient from the first period, and to the remainder, we should bring down the next period. (See operation.) The cubic feet or blocks, that remain must be applied to three sides of the monument; for unless a cube be equally increased on three sides, it ceases to be a cube. To effect this, we must find the superficial contents of three sides of the cube, and with this we must divide the remaining number of cubic feet or blocks, and the quotient will show the thickness of the additions. As the length of a side is 30, the superficial contents will be 30X30=900; and this multiplied by 3, will give the whole surface of the three sides; thus, 900×3-2700. This is equivalent to multiplying the square of the quotient by 900; thus, 3X3X900 8100. *With this as a divisor, we inquire how many times it is contained in 19.656, and find it to be 6 times. (One or two units are generally to be allowed on account of the other deficiencies in enlarging the cube.) This 6 is the thickness of each of the three additions to the cube; and by multiplying their superficial contents by it, we have the amount of the additions to be made; thus, 2700×6=16200; that is, we multiply the triple square by the last quotient figure. (See operation.)

If we now apply our blocks of the apparatus No. 2. to the three sides of the block, No. 1, we shall find there are three other deficiencies to complete the cube, the length of which is equal to that

* It is not necessary, that the triple quotient should be added to form a divisor.

of the addition, 30; and the height and breadth of each are equal to the thickness of the additions, 6. To find the contents of these, we multiply the continued product of the length, breadth, and thickness of each by their number; thus, 30X6X6X3=3240. Or, which is the same thing, we multiply the triple quotient by the square of the last quotient figure; thus, 90X6X6-8240. Let the contents of these be represented by the apparatus No. 3, and let these, with No. 2, be applied to No. 1, and we shall find one other small deficiency to form a complete cube; the length, breadth, and thickness of which are equal to the thickness of the former additions, viz. 6. The contents of this may be found by multiplying its length, breadth, and thickness together; that is, by cubing the last quotient figure, 6×6×6=216. Let this last deficiency of the cube be supplied by adding the apparatus No. 4, to the former additions, and we shall find the cube is completed, and our cubical monument is finished; and that the contents of the first cube, together with the several additions to it, is equal to the number of cubical blocks, 46.656.

[blocks in formation]

46656 contents of the whole monument.

1 Required the cube root of 77808776.

[blocks in formation]
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