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43. Extract the square root of .00032754.
44. Extract the square root of 2.3.
45. Extract the square root of .
46. Extract the square root of
47. Extract the square root of
45. Extract the square root of
49. Extract the square root of 113].
50. Extract the square root of 267

The square root of the product of any two numbers 18 a mean proportional between those numbers.

Thus, 4 is a mean proportional between 2 and 8; because 2:4=4: 8. But when four numbers are proporcionals, the product of the extremes is equal to the produet of the ineans; that is, the product of the two given numbers is equal to the square of the mean proportional.

51. Find a mean proportional between 4 and 256. 52. Find a mean proportional between 4 and 196. 53. Find a mean proportional between 2 and 12.5. 54 Find a mean proportional between 9.8 and 5. 55. Find a mean proportional between 25 and 121. 56 Find a mean proportional between 190.625 and 10 57 Find a mean proportional between 52 and 5435. 58 Find a mean proporti nal between į and 3. 59 Find a mean proportional between 12 and 147. 60. Find a mean proportional between 1 and 4 61. Find a mean proportional between 5 and 99.

62. Find a mean proportional between 4062237 and 828.

63. Find a mean proportional between .25 and 1. 64. Find a mean proportional between .1 and 310. 65. Find a mean proportional between .04 and .36. 66. Find a mean proportional between .09 and .49, 67. Find a mean proportional between 2 and .018.

When the square root of the product of the two given numbers cannot be extracted without a remainder, the mean proportionai is a surD, and may be approximated by the aid of decimals. 68. Find a mean proportional between 6 and 12.

69. Find a mean proportional between 25 and 14. 70. Find a mean proportional between 64 and 21. 71. Find a mean proportional between 46 and 55. - 72. Find a mean proportional between 5 and 81. 73. Find a mean proportional between 77 and 19.

74. A number of men spent 1 pound 7 shillings in company, which was just as many pence for each man, as there were men in the company. How many were there?

75. A company of men made a contribution for a charitable purpose; each man gave as many cents, as there were men in the company. The sum collected was 31 dollars 36 cents. How many men did the company consist of?

76. If you would plant 729 trees in a square, how many rows must you have, and how many trees in a row?

77. A certain regiment consists of 623 men. How many must be placed in rank and file, to form the reginient into a square?

78. It is required to lay out 40 acres of land in a square. Of what length must a side of the square be ?

79. It is required to lay out 20 acres of land in the form of a right angled parallelogram, which shall be twice as long as it is wide. What will be its length and breadth? (See page 162.)

80. It is required to lay out 30 acres of land in the form of a right angled parallelogram, ihe length of which shall be three times the width. How long and how wide will it be?

Perpendicular

Hypotenuse

A TRIANGLE is a figure having three sides and three angles. When one of the angles is such as would form one corner of

a square,

the figure is called a right-angled triangle, and the following propositions belong to it.

Base PROPOSITION lst. The square of the hypotenuse is equal to the sum of the squares of the other two sides.

PROPOSITION 2d. The square root of the sum of the squares of the base and perpendicular is equal to the hypotenuse.

PROPOSITION 3d. The square root of the difference of the squares of the hypotenuse and base is equal to the perpendicular

PROPOSITION 4th. The square root of the difference of the squares of the hypotenuse and perpendicular is equal to the base.

By observing the above propositions, when any two sides of a right-angled triangle are given, we may always find the remaining side. For example, suppose the base of the preceding figure to be 4 yards in length, and the perpendicular to be 3 yards in height; then the square of the base is 16 yards, and the square of the perpendicular 9 yards, and the sum of their squares is 25 yards. The square root of 25 yards is 5 yards, which is the length of the hypotenuse.

81. A certain castle, which is 45 feet high, is surrounded by a ditch, 60 feet broad. What must be the length of a ladder, to reach from the outside of the ditch to the top of the castle ?

82. A ladder 40 feet long, resting on the ground at the distance of 24 feet from ihe bottom of a straight tree, and leaning against the tree, just reaches to the first linb What is the length of the tree's trunk?

83. Two brothers lest their father's house, and went one, 64 miles due west, the other, 49 miles due north and purchased farms, on which they now live. How fa from each other do they reside?

84. James and George, flying a kite, were desirous of knowing how high it was. After some consideration, they perceived, that their knowledge of the square root, and of the properties of a right angled triangle, would enable them to ascertain the height. James held the line close to the ground, and George ran forward till he came directly under the kite; then measuring the distance from James to George, they found it to be 312 feet; and pulling in the kite, they found the length of line out, to be 520 feet, How high was the kite ?

85. A ladder, 40 feet long, was so placed in a street, As to reach a window 33 feet from the ground on one side, and when turned to the other side without changing the place of its foot, reached a window 21 feet high. The breadth of the street is required.

86. The distance between the lower ends of two equal rafters, in the different sides of a roof, is 32 feet, and the height of the ridge above the foot of the rafters is 12 feet. Find the length of a rafter.

Diagonal

A straight

straight line, drawn through the centre of a square, or through the centre of a rightangled parallelogram, from one angle to its opposite, is called a DIAGONAL; and this diagonal is the hypotenuse of both the right-angled triangles into which the square or parallelogram is thus divided.

87. A certain lot of land, lying in a square, contains 100 acres: at what distance from each other are the opposite corners ?

88. There is a square field containing 10 acres: wha! is the distance of the centre from either corner?.

A CIRCLE is a plane surface bounded by one curve line, called the circumference, every part of which is equally distant from the centre.

an der A straight line through the centre of a circle is called a diameter, and a straight line from the centre of a circle to the circumference is called a radius.

The areas of all circles are to one another, as the squares of their like dimensions. That is, the area of a greater circle is to the area of a less circle, as the square

Radium

RULE.

of the diameter of the greater to the square of the diameter of the less. Or thus, the area of the greater is to the area of the less, as the square of the circumference of the greater to the square of the circunference of the less.

Therefore, to find a circle, which shall contain 2, 3, 4, &c. times more or less space than a given circle, we have the following

Square one of the dimensions of the given circle, and, if the required circle be greater, multiply the square by the given ratio, then the square root of the product will be the like dimension of the requ'red circle; but, if the required circle be less than the given one, divide the square by the given ratio; then the square root of the quotient will be the similar dimension of the circle required.

S9. The diameter of a given circle is 11 inches: what is the diameter of a circle containing 9 times as much

space?

90. Find the diameter of a circle, which shall contain one fourth of the area of a circle of 42 feet diameter.

91. What must be the circumference of a circular pond, to contain 4 times as much surface, as a pond, of limite in circumference ?

92. Find the circumference of a pond which shall contain it part as much surface, as a pond of 13 miles circumference.

93. Find the diameter of a circle, which shall be 36 times as much in area, as a circle of 184 rods diameter.

The diameter of a circle is to the circumference in the ratio of 1 to 3.14159265, nearly: therefore, if we know the one, we can find the other. Thus, the circumference of a circle, the diameter of which is 8, is 3.14159265 X 8=25.1327412; the diameter of a circle, the circumference of which is 15.70796325, is 15.70796325 - 3. 1415926555.

To find the area of a circle, multiply the circumference by the radius, and divide the product by 2.

94. How many feet in length is the side of a square, equal in area to a circle of 36 feet diameter ?

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