GEOMETRY. 331 



a sphere, is a part of its surface, cut off by a plane, or intercepted 

 between two parallel planes ; and the intercepted solid is called a 

 spherical segment. A spherical triangle, is a portion of the sur- 

 face of a sphere, bounded by three arcs of great circles, that is, 

 circles whose planes pass through the centre. 



The convex surface of a cylinder, is equal to the circumference of 

 its base, multiplied by its altitude : that of a cone, is equal to the 

 circumference of its base, multiplied by its slant height, or distance 

 from the vertex to any point of the circumference just named : and 

 the surface of a sphere, is equal to the product of its diameter by the 

 circumference of a great circle, that is, of the sphere itself. The 

 measure of the solidity of a cylinder, is the product of its base by its 

 altitude ; that of a cone, is the product of its base by one-third of its 

 altitude ; and that of the sphere is the product of its surface by one- 

 third of its radius. 



6. Descriptive Geometry, relates to the representation of geo- 

 metrical figures on planes, and the construction of graphical problems 

 thereby. It includes, therefore, the principles of perspective, and 

 Spherical Projections. If we suppose the eye to be placed at a very 

 great height, and looking vertically down upon an object situated 

 above a horizontal plane, then the object will hide a part of the plane, 

 of the same shape or outline, as that which the object itself presents 

 to the eye. This representation of the object on the plane, is called 

 its horizontal projection; and the plane itself is called the horizontal 

 plane of projection. In like manner, if we suppose the eye to be 

 placed in front of an object, and a vertical plane behind it, we may 

 have a vertical projection, of the object, on the vertical plane of pro- 

 jection. When these two planes of projection are both used, they 

 intersect each other in a line called the ground line : and if we sup- 

 pose one of them, with any projections made upon it, to be revolved 

 about the ground line as an axis, till it coincides with the other plane, 

 we shall then have both the projections of any object, on one and the 

 same plane; as that of the paper or drawing board. Plate VII., 

 Fig. 9, represents the horizontal plane GHIB, as revolved about the 

 ground line JIB, until it coincides with the vertical plane ACDB, 

 prolonged downwards to EF. 



The projection of any point, is another point, directly above or 

 below it, or else directly before or behind it ; and is found by drawing 

 a perpendicular from the given point to one or the other plane of 

 projection. The projection of a line, is another line, lying in one 

 or the other plane of projection ; and is found by joining the projec- 

 tions of two of its points on that plane. The position of any plane, 

 in space, is known, if we have its intersections with the two planes 

 of projection ; which intersections are called its traces. If a given 

 plane be revolved about one of its traces as an axis, until it coincides 

 with the plane of projection in which that trace lies, each point of 

 the given plane will revolve in a circular arc, and take the same rela- 

 tive position in the given plane, after the revolution, as it had before. 

 It is by an ingenious application of these and similar principles, that 

 lines, surfaces and solids may be delineated on a single plane, and 

 their dimensions or relations determined to a surprizing extent. In 



