INSCRIBED AND CIRCUMSCRIBED POLYGONS.

84. In a given circle to inscribe a triangle similar to a given tri-

angle

85. About a given circle to circumscribe a triangle similar to a

given triangle .

86. In a given circle to inscribe an equilateral triangle

87. About a given circle to circumscribe an equilateral triangle

88. In a given circle to inscribe a square

89. About a given circle to circumscribe a square

90. In a given circle to inscribe a regular octagon or figure of eight

equal sides

91. About a given circle to circumscribe a regular octagon

92. In a given circle to inscribe a regular pentagon or decagon

93. About a given circle to circumscribe a regular pentagon or

decagon

.

61

61

61

61

62

95. About a given circle to circumscribe a regular hexagon

96. To inscribe in a given circle a regular dodecagon, or figure of

twelve equal sides

97. About a given circle to circumscribe a regular dodecagon

98. In a given circle to inscribe a quindecagon, or figure of fifteen

equal sides

99. In a given circle to inscribe a polygon of any number of equal

sides

100. About a given circle to circumscribe a polygon of any number

of equal sides

.

101. To measure the curvature of curved lines

94. In a given circle to inscribe a regular hexagon, or figure of

six equal sides .

288

63

63

CHAPTER III.

CONIC SECTIONS.

102. Definitions.--Ellipse, parabola, hyperbola

. 68

THE ELLIPSE.

103. Definitions.-Major and minor axes, centre, focus, radius-

vector, eccentricity, tangent, normal.

69

104. To describe an ellipse of given length of axes by continuous

motion

70

PAGE

105. To find any number of points on the circumference of an ellipse

of given axes

106. To draw an ellipse by help of the circumscribed circle, the

length of the axes being given

107. To draw a tangent at any given point on an ellipse, by help of

the foci

108. To draw a tangent at any point of an ellipse, by help of the cir-

cumscribed circle

109. To draw a tangent to an ellipse from a given point on the major-

axis produced.

110. To draw the normal, or perpendicular to the tangent at any

given point on an ellipse

111. To draw tangents to an ellipse from any given point outside of

it

112. About a given ellipse to circumscribe a trapezium of which two

opposite vertices are given

113. To approximate to the outline of an ellipse by means of arcs of

circles

71

71

72

72

72

73

73

114. Definitions.-Diameter, conjugate-diameter, ordinate, supple-

mental chords .

75

115. A pair of conjugate-axes being given to find points on the

ellipse by means of the supplemental chords

116. In a given parallelogram to inscribe an ellipse.

117. Any pair of conjugate-diameters of an ellipse being given to

find the axes

118. To apply the ellipse to form the contour of architectural

mouldings

119. To apply the ellipse to form the contour of leaf ornaments

120. To apply the ellipse to form the entasis of a column

121. To apply the ellipse to form the outline of a Tudor arch.

122. A diameter of an ellipse and an ordinate being given, to find ́

the conjugate diameter

123. To inscribe an ellipse in a given trapezium

THE HYPERBOLA.

124. Definitions.-Hyperbola, vertex, axis, foci, centre, eccentri-

city, radius-vector, asymptotes, rectangular-hyperbola, latus-

rectum, abscissæ

126. To find any number of points upon the contour of a hyperbola,
whose vertex and foci are given .

87

127. To draw the asymptotes of a hyperbola, whose vertex and foci

are given.

87

PAGE

128. To find points on a hyperbola by means of the asymptotes

129. To draw any hyperbola by means of the rectangular hyper-

bola

130. To draw any hyperbola by means of a scale of parts

131. To find points on the contour of a rectangular hyperbola

132. To find points on the contour of any hyperbola by the ruler

only

133. To draw the tangent at any given point of a hyperbola .

134. To draw a tangent to a hyperbola from any point upon the

axis outside the curve

135. To draw a normal at any given point on a hyperbola

136. To find the foci of a given hyperbola

137. To approximate to the contour of a hyperbola by means of

arcs of circles .

138. To apply the hyperbola to form the entasis of a column.

139. Application of the hyperbola to form the outline of mouldings

and other architectural forms

95

95

THE PARABOLA.

140. Definitions.-Parabola, axis, vertex, latus-rectum, directrix,

radius-vector, abscissa, ordinate

97

141. To draw the parabola by continuous motion.

142. To find points on the contour of the parabola

143. To find the lengths of ordinates to a parabola by means of a

98

99

144. To draw a tangent at any point of the parabola

145. From a given point on the axis outside the parabola to draw

a tangent

101

146. To draw a normal at any given point on the parabola

147. To apply the parabola to form the entasis of a column

148. To find the focus of a given parabola

149. To apply the parabola to form the outline of a Tudor arch

150. To approximate to the contour of the parabola by arcs of

151. Application of the parabola to mouldings, architectural orna-

ments, and other purposes

102

103

104

THE CATENARY.

152. Definition of a catenary

106

153. To apply the catenary to form the entasis of a column

154. To apply the catenary to form the entasis of a spire

106

106

CHAPTER IV.

CURVES OF FLEXURE.

155. Definitions.-Curve of flexure, point of contrary-flexure,

harmonic-curve or curve of sines, lemniscate

THE HARMONIC-CURVE.

156. To find points on the contour of the harmonic-curve

PAGE

108

110

111

112

112

157. To find points on the harmonic-curve by a scale of parts

158. To draw the tangent at any point of the harmonic-curve

159. To draw the normal at any given point of the harmonic-curve.

160. To set out the harmonic-curve as a railway curve.

161. To apply the harmonic-curve to form the entasis of a column. 113

162. To apply the harmonic-curve to form the contour of an ogival

arch or pinnacle

114

163. To apply the harmonic-curve to mouldings and other archi-

tectural ornaments

115

THE LEMNISCATE.

164. To draw a lemniscate by continuous motion.

165. To find points on the contour of a lemniscate

116

117

166. To find the algebraical equation to the lemniscate

167. To draw a lemniscate of given dimensions

168. To draw a tangent to the lemniscate

169. To apply the lemniscate to form an ogival arch

120

170. To apply the lemniscate to mouldings and other architectural

ornament

121

CHAPTER V.

SPIRALS.

171. Definitions.-Pole, spiral, radius-vector, axis, involute of circle,

spiral of Archimedes, reciprocal-spiral, equiangular or

logarithmic spiral

THE INVOLUTE OF THE CIRCLE.

124

172. To draw the involute by continuous motion .

173. To find points upon the involute .

127

128

THE SPIRAL OF ARCHIMEDES.

174. To draw a spiral whose distance from the pole is directly

proportional to the angle made by the radius-vector with

the fixed axis.

175. To draw the tangent at various points of the foregoing spiral .

176. To find points on the spiral whose distance from the pole is

proportional to the square of the angle made by the radius-

vector with the fixed axis

177. To draw the tangents at various points of the foregoing spiral.

178. To find points on the spiral whose distance from the pole is

proportional to the cube of the angle made by the radius-

vector with the fixed axis

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129

131

131

133

134

179. To draw the tangents at various points of the foregoing spiral . 137

180. To apply the above spiral to form a volute of given di-

182. To find points on the spiral in which the length of the radius-

vector varies inversely as the angle of revolution from a

fixed axis

183. To draw tangents at various points on the reciprocal spiral

141

144

THE LITUUS.

184. To find points on the contour of the spiral in which the radius-

vector varies inversely as the square root of the angle of

revolution from a fixed axis

146

185. To draw tangents at various points of the foregoing spiral

186. To apply the lituus to form the outline of a console or double

148

volute

187. To find points upon a spiral in which the radius-vector varies

inversely as the cube-root of the angle of revolution from a

fixed axis

188. To draw tangents at various points of the foregoing spiral

THE EQUIANGULAR OR LOGARITHMIC SPIRAL.

189. To find points on the equiangular spiral, the depth and axis of

which are given

190. To draw tangents and normals at various points on the equi-

angular spiral.