ITS PROPER EXTENT 212 Lines; Polygons; Polyhedrons 212 Not to be farther restricted 213 Improper Application of Analysis 214 Attempted Demonstrations of Axioms 216 GEOMETRY OF THE RIGHT LINE 217 GRAPHICAL SOLUTIONS 218 _Descriptive Geometry_ 220 ALGEBRAICAL SOLUTIONS 224 _Trigonometry_ 225 Two Methods of introducing Angles 226 1. By Arcs 226 2. By trigonometrical Lines 226 Advantages of the latter 226 Its Division of trigonometrical Questions 227 1. Relations between Angles and trigonometrical Lines 228 2. Relations between trigonometrical Lines and Sides 228 Increase of trigonometrical Lines 228 Study of the Relations between them 230 CHAPTER III. MODERN OR ANALYTICAL GEOMETRY Page THE ANALYTICAL REPRESENTATION OF FIGURES 232 Reduction of Figure to Position 233 Determination of the position of a Point 234 PLANE CURVES 237 Expression of Lines by Equations 237 Expression of Equations by Lines 238 Any change in the Line changes the Equation 240 Every "Definition" of a Line is an Equation 241 _Choice of Co-ordinates_ 245 Two different points of View 245 1. Representation of Lines by Equations 246 2. Representation of Equations by Lines 246 Superiority of the rectilinear System 248 Advantages of perpendicular Axes 249 SURFACES 251 Determination of a Point in Space 251 Expression of Surfaces by Equations 253 Expression of Equations by Surfaces 253 CURVES IN SPACE 255 Imperfections of Analytical Geometry 258 Relatively to Geometry 258 Relatively to Analysis 258 THE PHILOSOPHY OF MATHEMATICS. INTRODUCTION. GENERAL CONSIDERATIONS. Although Mathematical Science is the most ancient and the most perfect of all, yet the general idea which we ought to form of it has not yet been clearly determined. Its definition and its principal divisions have remained till now vague and uncertain. Indeed the plural name--"The Mathematics"--by which we commonly designate it, would alone suffice to indicate the want of unity in the common conception of it.