FULLIC LIBRARY

317483

TILDEN FJ ATIONS.

100

Entered according to Act of Congress, in the year 1845, by
JAMES MUNROE AND COMPANY,

In the Clerk's Office of the District Court of the District of Massachusetts.

I HAVE carefully examined the manuscript of "An Introduction to Geometry," and think it admirably adapted to supply an important want in education. It is not a mere geometrical logic, but a natural and simple introduction to the Science of Form. By a beautiful and original series of inductive processes, it avoids tedious demonstrations, develops the taste for observation, which is so strong in the quick mind of youth, and leads the pupil to a real and practical knowledge of the truths of Geometry with a rapidity which would not have been anticipated. From these considerations, and from observing the strange neglect into which this science has fallen in our schools, I have strongly urged the publication of this excellent treatise, and think that its study should be insisted upon, as a valuable preliminary to a good education either at college or in business.

Section.

98. Two points determine the direction of a straight line,
99. What is meant by measuring. Units of measure,

100. In what manner the operation is performed,

102. Division of angles according to the nature of their sides,

103. When angles are equal,

104. Adjacent and vertical angles,

105. Of the angles made by the intersection of two parallel lines

112. When it may be inferred that two lines are parallel,

115. Method of measuring angles. The plumb. The level,

IV. FIGURES.

116. Definition of a figure. Of figures in general,

117. One side of a triangle less than the sum of the other two,

118. Sum of the angles of a triangle,

119.

to Corollaries from the preceding proposition,

123.

124. Value of one exterior angle of a triangle,

125. Value of the three exterior angles of a triangle,

129. The angles at the base of an isosceles triangle equal,

130. The angles formed by producing the equal sides of an isosceles

triangle are equal,

131. Value of each angle of an equilateral triangle,

132. To find the value of either angle of an isosceles triangle,

133. Proportions which may be demonstrated by erecting a perpen-

dicular on the middle of the base of an isosceles triangle,

134. The sides of a triangle which are opposite to equal angles are

equal,

135. Case of equal right-angled triangles,

136. Any point in a perpendicular to the middle of a straight line,

is at equal distances from extremities of that line,

137. Of lines drawn obliquely to a perpendicular,

138. Case of equal triangles,

139. Greater side of a triangle opposite to greater angle,

141. The converse,

83

84

84

84

84

85

Section.

Page.

89

146. A parallelogram divided by a diagonal into 2 equal triangles,

147. In a parallelogram the sides opposite to equal angles are

equal, .

148. Definition of equivalent figures,

149. Altitude of a figure,

150.

151.

Cases of equivalent parallelograms,

152. A triangle equivalent to half a parallelogram of equal base

and altitude,

153. Case of equivalent triangles,

154. Sum of the angles of a polygon,

155. Value of each angle of a regular polygon,

156. Value of each angle which may be formed at the middle of a

polygon by drawing straight lines to its vertices,

157. Value of the exterior angles of a polygon,

90

91

91

. 96

96

97

158. Table of the size of angles in a regular figure,

and surface,

159. Mensuration of a rectilineal figure. Superficial unit. Area

166. Equal angles at the centre of a circle intercept equal arcs, and
the chords of those arcs are equal,

167. A perpendicular to the middle of a chord bisects the arc and

172. Measure of an inscribed angle,

173.

to Value of angles inscribed in segments, .

176.

177. Value of the opposite angles of a quadrilateral,

178. Measure of angles made by a tangent and a chord,

179. Radius may be drawn six times as a chord,

180. To inscribe a regular hexagon,

. 102

. 102

. 102

187. Definition of a solid. Of solids in general,

188. Superficial contents of a prism,

106

107

sphere,

. 109

. 109

and volume,

195. Unit of solidity. Meaning of terms solidity, solid content,

196. A right prism; oblique prism; parallelopiped; rectangular

197. Solidity of a rectangular parallelopiped,

a cube,

any parallelopiped,

a triangular prism,

any prism,

a cylinder,

a triangular pyramid,

any pyramid,

a cone,

solids in general,

a sphere,

VI. MISCELLANEOUS PROPOSITIONS.

209. To find the distance between two inaccessible objects,

210.

distance between two objects, only one of which

can be reached,

211. To divide a triangle into two equivalent parts,

212. To transform a parallelogram into an equivalent triangle,

213. To construct a square 1 times as great as a given square,

214. To transform a triangle into an equivalent parallelogram,

215. Two diagonals in a parallelogram bisect each other,

Section.

216. Two diagonals in a square or lozenge cut each other at right

angles; in other parallelograms at oblique angles,

Page

119

119

217. Diagonals in a square or lozenge divide the angles into equal.
parts; in other parallelograms into unequal parts,

218. To divide a parallelogram into 2, 3, 4, or more equal parts, 119

219. A straight line drawn through the middle of a diagonal of a

parallelogram bisects the parallelogram,

220. Ratio of the square of the hypothenuse to the sum of the

squares of the other two sides of a right triangle,

221. Comparative magnitude of an inscribed and circumscribed

square,

222. To construct a square equivalent to two given squares,

120

120

121

121

121

122

. 123

. 123

228. To draw a tangent to a circle from a given point,

. 123

230. To bisect an angle made by two tangents,

. 123

. 124

233. To describe a segment which shall contain a given angle,
234. To construct a triangle, of which base, altitude, and an angle

235. Equal chords are at an equal distance from the centre,

236. Greater chord at least distance from the centre,

238. To describe a circle whose circumference shall pass through

two given points,

244. To inscribe a circle in a triangle,

245. To describe a circle whose circumference shall touch that

another circle, and shall also pass through a given point,