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 625 men. many must be placed in rank and file, to form the regiment into a square?

How

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, the length of which shall be three times the width. How long and how wide will it be?

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.

Hypotenuse

Base

Perpendicular

PROPOSITION 1st. 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 ba 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 the bottom of a straight tree, and leaning against the tree, just reaches to the first limb What is the length of the tree's trunk?

83. Two brothers left their father's house, and went one, 64 miles due west, the other, 48 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.

A 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.

Diagonal.

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: what is the distance of the centre from either corner?

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

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 circumference 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—

RULE. 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 required 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.

89. 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 11⁄2 mile in circumference?

92. Find the circumference of a pond which shall contain part as much surface, as a pond of 131 miles

circumference.

93. Find the diameter of a circle, which shall be 36 times as much in area, as a circle of 18 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 25.1327412; the diameter of a circle, the circumference of which is 15.70796325, is 15.70796325÷3. 14159265- 5.

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?

95. Find the side of a square equal in area to a circle of 20 rods in diameter.

96. Find the diameter of a pond, that shall contain as much surface, as a pond of 6.986 miles circumference. 97. Find the length and breadth of a right-angled parallelogram, which shall be 4 times as long as it is wide, and equal in area to a circle of 43.9822971 rods circumference.

98. Find the circumference of a pond, which shall contain as much surface, as 9 ponds of of a mile diameter each.

XXX.

EXTRACTION OF THE CUBE ROOT.

A CUBE is represented by a solid block-like either of those annexed-with six plane surfaces; having its length, breadth, and height all equal. Consequently, the solid contents of a cube are found by multiplying one of its sides twice into itself. For this reason, the third power of any number is called a cube.

Therefore, if we multiply the square of a number by its root, we obtain a product, which is called a cube, or a

cubic number. For instance, 4 multiplied by 4 produces 16, which is the square of 4, as shown on one of the sides of this larger block; and 16 multiplied by 4 produces 64, which is the cube of 4, as shown by the whole of the larger block.

Thus the cube of any quantity is produced by multiplying the quantity by itself, and again multiplying the product by the original quantity. When the quantity to be