A triangle is a figure having three sides and three angles, when one of the angles would form a corner of a square the figure is called a right angled triangle.

The longest side of a right angled triangle is called the hypotenuse.

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

Hence the square root of the sum of the squares of the base and perpendicular is equal to the hypotenuse.

2d. The square root of the difference of the squares of the

hypotenuse and base is equal to the perpendicular. (See the diagram on last page.)

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

124. In a right angled triangle the base is 12 feet in length, and the perpendicular is 16 feet; what is the length of the hypotenuse? Ans. 20 feet.

125. The wall of a certain fortress is 18 feet high, and is surrounded by a ditch 24 feet wide; how long must a ladder be to reach from the outside of the ditch to the top of the wall? Ans. 30 feet.

126. 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? Ans. 32 feet.

127. A certain castle which is 45 feet high, is surrounded by a ditch 60 feet broad; what length must a ladder be to reach from the outside of the ditch to the top of the castle? Ans. 75 feet.

128. Two ships sail from the same port, one goes due south 84 miles, the other due east 112 miles; how far are they from each other? Ans. 140 miles. 129. 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; what is the length of a rafter? Ans. 20 feet.

130. If a hall be 48 feet long, and 36 feet wide, what is the distance between the opposite corners? Ans. 60 feet.

131. A line 27 yards long will exactly reach from the top of a fort to the opposite bank of a river, which is known to be 23 yards wide; what is the height of the fort?

Ans. 14.142+ yds.

132. Suppose a ladder 60 feet long, be so planted as to reach a window 37 feet from the ground on one side of the street, and without moving it at the foot will reach a window 23 feet high on the opposite side; how wide is the street? Ans. 102.649+ feet. 133. A certain room is 28 feet long, 24 feet wide, and 18 feet high; how long must a line be to extend from one of the lower corners to an opposite upper corner? Ans. 41.036+ feet.

NOTE. Take the square root of the sum of the squares of the three numbers.

134. The height of a tree growing in the centre of a circular island 100 feet in diameter, is 140 feet, and a line extending from the top of it to the further shore is 600 feet; what is the width of the stream, supposing the land on each side of the water to be level? Ans. 533.43+

135. A farmer wishes to set out a peach orchard, which shall contain 2400 trees, and to have the length of the orchard to the breadth in the ratio of 3 to 2; the distance between the trees is to be 7 yards, that is, from centre to centre; how many trees must there be in the length, and how many in the breadth of the orchard, and on how many square yards of ground will they stand?

Ans. 60 trees in length, 40 in breadth, stand on 112,749 sq. yds.

NOTE. Observe that if the length were made equal to the breadth the orchard would contain but 1600 trees, and as one corner tree will belong to both length and breadth, the number of spaces between the trees will be one less than the number of trees in a row.

The square root of the product of any two numbers is a mean proportional between those numbers; 9 is a mean proportional between 3 and 27; that is, 9 has the same ratio to 27 that 3 has to 9.

136. Find a mean proportional between 2 and 8. 137. Find a mean proportional between 4 and 16. 138. Find a mean proportional between 2 and 123. 139. Find a mean proportional between 2 and 24ž. 140. Find a mean proportional between 5 and 91.

Ans. 7.

141. Find a mean proportional between 16 and 144.

Ans. 48.

142. Find a mean proportional between 25 and 169.

Ans. 65.

143. Find a mean proportional between 9 and 64.

Ans. 24.

144. Find a mean proportional between 6 and 24.

Ans. 12.

145. Find a mean proportional between 7 and 28.

150. Find a mean proportional between 24 and 96.

Ans. 48.

151. Find a mean proportional between and 98.

Ans. 7.

152. Find a mean proportional between 12 and 147.

Ans. 42.

153. Find a mean proportional between 180.625 and 10. Ans. 42.5.

154. Find a mean proportional between 18 and 648.

155. Find a mean proportional between 17

156. Find a mean proportional between .04

Ans. 108. and 153.

Ans. 51. and .49. Ans. .14.

157. Find a mean proportional between .09 and .64.

Ans. .24.

158. Find a mean proportional between .05 and 12.8. Ans. .8.

EXTRACTION OF THE CUBE ROOT.

A cube is a solid body with six equal square sides, and containing equal angles.

A number is said to be cubed when it is raised to the 3d power, or is multiplied into its square.

The extraction of the cube root, is the finding of a number, which multiplied into its square, will produce the given

number. Thus, the cube root of 27 is 3, because 3 multiplied into its square (9) will produce 27; what is the cube root of 8, of 64, of 125, of 216, of 343, of 512, of 729?

Take a number of 2 figures, 24 for instance, and raise it to the 3d power, or cube it, writing each product in a separate column.

The left-hand column contains 8, the cube of 2, (the tens figure.) The next column contains 3 times the square of the tens figure (2) multiplied by the units figure, (4,) thus, 3 times 2x4=48. The next column to the right of this, contains 3 times the square of the units figure, (4,) multiplied by the tens figure, (2,) thus, 3 times 43×2=96. And the right-hand column contains the cube of the units figure, (4)3=64.

These numbers when fully expressed will be,

Complete cube=13824=(24)3.

The following proposition should be remembered by the learner:

The cube of any number of 2 figures contains the sum of the cube of the tens figure, 3 times the square of the tens figure multiplied by the units figure, 3 times the square of