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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
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87. About a given circle to circumscribe an equilateral triangle
88. In a given circle to inscribe a square
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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
94. In a given circle to inscribe a regular hexagon, or figure of
six equal sides .
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
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97. About a given circle to circumscribe a regular dodecagon
98. In a given circle to inscribe a quindecagon, or figure of fifteen
64
99. In a given circle to inscribe a polygon of any number of equal
sides
65
100. About a given circle to circumscribe a polygon of any number
of equal sides
101. To measure the curvature of curved lines
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
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
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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
114. Definitions.-Diameter, conjugate-diameter, ordinate, supple- mental chords .
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
125. To draw a hyperbola by continuous motion
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.
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 .
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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
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THE PARABOLA.
140. Definitions.—Parabola, axis, vertex, latus-rectum, directrix,
radius-vector, abscissa, ordinate
141. To draw the parabola by continuous motion.
scale of parts
142. To find points on the contour of the parabola
143. To find the lengths of ordinates to a parabola by means of a
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
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
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
CHAPTER IV.
CURVES OF FLEXURE.
155. Definitions.-Curve of flexure, point of contrary-flexure,
harmonic-curve or curve of sines, lemniscate
THE HARMONIC-CURVE.
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110
111
156. To find points on the contour of the harmonic-curve
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. 112
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
112
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
166. To find the algebraical equation to the lemniscate
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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 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
129
175. To draw the tangent at various points of the foregoing spiral 131
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-
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-
mensions
181. Table of the squares and cubes of integers
THE RECIPROCAL SPIRAL.
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
141
183. To draw tangents at various points on the reciprocal spiral 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
149
151
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
154
190. To draw tangents and normals at various points on the equi-
angular spiral.
156