The well established reputation and the high respectability of the authors from whom our selections have been made, renders it unnecessary for us to discuss their merits in order to secure a favorable reception of this. It will only be necessary for us, in the following pages, to preserve the same degree of accuracy and perspicuity in our digressions as characterise those works, and we shall have nothing to fear from the criticisms of scientific amateurs and mathematicians. CONTENTS. Definitions and General Propositions, CHAP. I.-General Principles and Illustrations, 131 Construction of Algebraical Quantities, Geometrical Questions, the modes of forming Equations therefrom, SPHERICAL GEOMETRY. SPHERICAL GEOMETRY. DEFINITIONS. 1. THE sphere is a solid terminated by a curve surface, all the points of which are equally distant from a point within, called the centre. 2. The radius of a sphere is a straight line, drawn from the centre to any point of the surface; the diameter, or axis, is a line passing through this centre, and terminated on both sides by the surface. All the radii of a sphere are equal; all the diameters are equal, and each double of the radius. 3. It will be shown (Prop. I.) that every section of the sphere, made by a plane, is a circle. This granted, a great circle is a section which passes through the centre; a small circle, one which does not pass through the centre. 4. A plane is tangent to a sphere, when their surfaces have but one point in common. 5. The pole of a circle of a sphere is a point in the surface equally distant from all the points in the circumference of this circle. 6. A spherical triangle is a portion of the surface of a sphere, bounded by three arcs of great circles. Those arcs, named the sides of the triangle, are always supposed to be each less than a semi-circumference. The angles which their planes form with each other, are the angles of the triangle. 7. A spherical triangle takes the name of right-angled, isosceles, equilateral, in the same cases as a rectilineal triangle. 8. A spherical polygon is a portion of the surface of a sphere, terminated by several arcs of great circles. 9. A lune is that portion of the surface of a sphere which is included between two great semicircles, meeting in a common diameter. 10. A spherical wedge, or ungula, is that portion of the solid sphere which is included between the same great semi-circles, and has the lune for its base. 11. A spherical pyramid is a portion of the solid sphere included between the planes of a solid angle, whose vertex is the centre. The base of the pyramid is the spherical polygon intercepted by the same planes. 12. A zone is the portion of the surface of the sphere included between two parallel planes, which form its bases. One of those planes may be tangent to the sphere; in which case, the zone has only a single base. 13. A spherical segment is the portion of the solid sphere included between two parallel planes which form its bases. One of these planes may be tangent to the sphere; in which case, the segment has only a single base. 14. The altitude of a zone or of a segment is the distance between the two parallel planes, which form the bases of the zone or segment. 15. Whilst the semicircle DAE (Def. 1.) revolving round its diameter DE, describes the sphere, any circular sector, as DCF or FCH, describes a solid, which is named a spherical sector. 16. The symbal. which occurs in this volume, is used to denote because; when applied in algebraic notation. |