PROBLEM XX. To find the hour of the night by a sun dial, when the moon shines on it. Find the moon's age, or the number of days elapsed since a change or full. Add of this number, being the time in hours corresponding to the moon's distance from the sun, to the time shewn by the moon on the dial, and the sum, or its excess above 12 when it exceeds that number, is the time required. With radii of convenient length, describe two concentric circles A, B; divide the circumference of each into 29 equal parts, and join the corresponding points. On another plane, which is circular, and its radius equal to CB, CB, describe an equinoctial solar dial, as, for example, the superior one D. NOTE. This dial has the same stile in the centre C, and its plane is st in the same position, as the equinoctial solar dial. The circle D is to be put on B concentric with it, and moveable about the centre. Then the XII o'clock line on D is to be set to the moon's age on the plane AB. END OF DIALING. SPHERICAL GEOMETRY. I. DEFINITIONS. SPHERICAL GEOMETRY is the doctrine of the Sphere, particularly of the circles described on it ; with the method of projecting the same on a plane, and measuring their arcs and angles when projected. 2. Circles of the sphere, whose planes pass through the centre, are called great circles, and all others small circles. 3. A straight line, drawn through the centre of any circle of the sphere perpendicular to its plane, and limited on both sides by the surface of the sphere, is called the axis of that circle. 4. The poles of a circle of the sphere are the extremities of its axis. 5. By the distance of two points on the surface of the sphere is meant the arc of a great circle, intercepted between them. 6. To project an object is to represent every point of it on the same plane, as it appears to the eye in a certain position. 7. That plane, on which the object is projected, is called the plane of Projection; and the point, where the eye is supposed to be situated, the projecting point. 8. The orthographic projection of the sphere is that, in which a great circle is assumed as the plane of projection, and a point at an infinite distance in the axis of it produced as the projecting point. 9. The stereographic projection of the sphere is that, in which a great circle is assumed as the plane of projection, and one of its poles as the projecting point. 10. The great circle, on the plane of which the projection is made, is called the primitive. 11. A direct circle is that, whose plane is directly opposite to the eye, or perpendicular to the axis of the eye, when directed to the centre of the primitive. 12. A right circle is that, whose plane is coincident with the axis of the eye. 13. An oblique circle is that, whose plane is oblique to the axis of the eye. 14. The line of measures of a circle of the sphere, is that diameter of the primitive, produced indefinitely, which is perpendicular to the line of common section of the circle and the primitive. NOTE. The projection or representation of any point is where the straight line, drawn from the projecting point through it, intersects the plane of projection. ORTHOGRAPHIC ORTHOGRAPHIC PROJECTION. PROBLEMI. To project a circle parallel to the primitive. Take the complement of its distance from the primitive, and set it from A to E; and with centre C, and radius CD perpendicular EF, describe the circle DG.* PROBLEM The following are laws of the orthographic projection. 1. The rays coming from the eye, being at an infinite distance, and making the projection, are parallel to each other, and perpendicular to the plane of projection. 2. A right line, perpendicular to the plane of projection, is projected into a point, where that line meets the said plane. 3. A right line, as AB, or CD, not perpendicular, but either parallel or oblique to the plane of the projection, is projected into a right line, as EF, or GH, and is always comprehended between the extreme perpendiculars AE and BF, or CG and DH. EG HF B 4. The projection of the right line AB is the greatest, when AB is parallel to the plane of the projection. 5. Hence |