Measure of the volume of a prism inscribed in the cylinder.--The volume of a right cylinder is measured by the product of the area of its base by its height.--This is also true of any cylinder, right or oblique, of any base whatever. _Of the sphere._ Every section of the sphere, made by a plane, is a circle.--Great circles and small circles. In every spherical triangle any one side is less than the sum of the other two. The shortest path from one point to another, on the surface of the sphere, is the arc of a great circle which joins the two given points. The sum of the sides of a spherical triangle, or of any spherical polygon, is less than the circumference of a great circle. Poles of an arc of a great or small circle.--They serve to trace arcs of circles on the sphere. Every plane perpendicular to the extremity of a radius is tangent to the sphere. Measure of the angle of two arcs of great circles. Properties of the polar or supplementary triangle. Two spherical triangles situated on the same sphere, or on equal spheres, are equal in all their parts, 1º when they have an equal angle included between sides respectively equal; 2º when they have an equal side adjacent to two angles respectively equal; 3º when they are mutually equilateral; 4º when they are mutually equiangular. In these different cases the triangles may be equal, or merely symmetrical. The sum of the angles of any spherical triangle is less than six, and greater than two, right angles. The lune is to the surface of the sphere as the angle of that lune is to four right angles. Two symmetrical spherical triangles are equivalent in surface. The area of a spherical triangle is to that of the whole sphere as the excess of the sum of its angles above two right angles is to eight right angles. When a portion of a regular polygon, inscribed in the generating circle of the sphere, turns around the diameter of that circle, the convex area engendered is measured by the product of its height by the circumference of the circle inscribed in the generating polygon.--The volume of the corresponding polygonal sector is measured by the area thus described, multiplied by the third of the radius of the inscribed circle. The surface of a spherical zone is equal to the height of that zone multiplied by the circumference of a great circle.--The surface of the sphere is quadruple that of a great circle. Every spherical sector is measured by the zone which forms its base, multiplied by the third of the radius. The whole sphere is measured by its surface multiplied by the third of its radius.[3] [Footnote 3: Numerical examples on the areas and volumes of the round bodies, including the area of a spherical triangle, will be required by the examiners. The calculations will be made by logarithms.] III. ALGEBRA.