Theory of the measure of parallelopipedons Tangent planes to cylinders, and development of their surface Relations of the sides and angles of spherical triangles Theorems relating to the identity and symmetry of sph. triangles 12 Relation of ratio of an arc to the quadrant with the ratio of an Expression for an angle in terms of unit of arc and unit of length Surface of a lune, volume of a wedge, surface of a spherical tri- GEOMETRY. DEFINITIONS. GEOMETRY is the science of position and extension. 1. A Point is position without magnitude or dimensions. It has neither length, breadth, nor thick ness. 2. A Line has one dimension only, length. 3. A Surface or Superficies has extension in two dimensions, length and breadth; but is without thickness.* 4. A Body or Solid has three dimensions, length, breadth, and depth or thickness. 5. A Right Line, or Straight Line, is A one which has every where the same direction. B When the term Line is used in this work without an adjective, a Right Line is understood. A line is designated by two letters placed upon it. Thus we say the line AB. 6. A Broken Line is one which changes its direction at intervals. 7. A Curve, or Curve Line, is one which is continually changing its direction. 8. Parallel Lines are those which have the same direction.† * A surface may be boundless, and a line interminable. + Parallel lines are sometimes said to meet at an infinite distance; in other words, they never meet. This follows evidently from the definition. The case where one line is the prolongation of another, or others, which seems to be embraced in this definition, is to be excluded; for these lines, in the unrestricted sense of the term, form one and the same straight line. Or, when two parallel lines coincide, they become one and the same straight line. A |