123. Can you place two hexagons so that one side of one hexagon may coincide with one side of the other?

124. Can you divide a circle into twelve equal sectors?

125. Can you place two octagons so that one side of one octagon may coincide with one side of the other?

You have divided a sector into two equal sectors, and an angle into two equal angles.

126. Can you divide a sector into four equal sectors, and an angle into four equal angles?

127. Can you make a rhombus, whose long diagonal shall be twice as long as the short one?

128. Can you make a regular dodecagon in a circle?

129. Can you show how many squares may be made to touch at one point?

You recollect that plane figure that has the fewest lines possible for its boundaries.

130. Of how few plane surfaces can you make a solid body?

A body which has four plane, equal, and similar surfaces, is called a tetrahedron.

131. Make a hollow tetrahedron of one piece of cardboard, and show on paper how you arrange the surfaces to fit each other, and give a sketch of the tetrahedron when made.

You know how to fit a square in a circle.

132. Can you fit a square around a circle? When two triangles have the angles of one respectively equal to the angles of the other, but the sides of the one longer or shorter respectively than the sides of the other, such triangles, though not equal, are said to be similar each to the other. Now you have made two triangles that are equal and similar.

133. Can you make two triangles that shall not be equal, and yet be similar?

134. Make a rhomboid, and divide it several ways into two figures that shall be equal to each other, and similar to each other, and write on each figure its appropriate name.

135. Make two equal and similar rhomboids, and divide one into two equal and similar trian

gles by means of one diagonal, and the other into two equal and similar triangles by means of the other diagonal.

136. Can you make two triangles that shall be equal to each other, and yet not similar?

137. Can you show that all triangles upon the same base and between the same parallels are equal to one another?

138. Can you place a circle, whose radius is 11⁄2 inch, so that its circumference may touch two points 4 inches asunder?

139. How many squares may be placed around one square to touch it?

140. Divide a rhombus into four equal and similar figures several ways, and write in each figure its proper name.

141. Show how many hexagons may be made to touch one point.

142. Show how many circles may be made. to touch one point without overlapping, and compare that number with the number of hexagons, the number of squares, and the number of equilateral triangles.

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When a body has six equal and similar surfaces it is called a hexahedron.

143. Make of one piece of card a hollow hexahedron. Show on paper how you arrange the surfaces so as to fold together, and give a sketch of the hexahedron when finished; and say what other names a hexahedron has.

144. Can you make a right-angled triangle, whose base shall be 4 and perpendicular 6?

In a right-angled triangle, the side which faces the right angle is called the hypothenuse.

145. Can you make a right-angled triangle, whose base shall be 4 and hypothenuse 6 ?

146. Can you make a rectangle, whose length shall be 5 and diagonal 6?

147. Divide a rectangle several ways into four equal and similar figures, and write upon each figure its proper name.

The term vertex means the crown, the top, the zenith; and yet the angle of an isosceles triangle which is contained by the equal sides is called the vertical angle, however such triangle may be placed; and the side opposite to such

angle is still called the base, although it may not happen to be the lowermost side.

148. Place in different positions four isosceles triangles, and point out the vertex of each.

149. Construct an isosceles triangle, whose base shall be 1 inch, and each of the equal sides 2 inches, and place on the opposite side of the base another of the same dimensions.

150. Can you invent a method of dividing a circle into four equal and similar parts, having other boundaries rather than the radii?

You have made a square, and placed an equilateral triangle on each of its sides.

151. Can you make an equilateral triangle, and place a square on each of its sides?

152. Can you fit a square inside a circle, and another outside, in such positions with regard to each other as shall show the ratio the inner one has to the outer?

153. Can you divide a hexagon into four equal and similar parts?

154. Can you divide a line into two such