Make Aa and Bb equal in length to the given surface, and join AB and ab; then ABba is the required plan. It may be here remarked that the edge view of a circular surface will be a line equal in length to the diameter of the circle. For example, let it be required to find the plan of a circle, having its edge view given as in this problem (see figure, page 197). Let A'B' be its elevation. Then point C' will represent the centre of the circle, and the required plan will be found by taking any point C in the projector from C'. Then from Cas centre, and radius C' A' or C'B describe a circle which will be the required plan. Problem 7. To find the elevation of a line, its plan being given. Let AB be the plan of the given line. Draw A A' and BB' at right angles to xy, making A'a and B'b equal to the supposed height of A, B above the horizontal plane, and join A'B'; then A'B' is the elevation required. NOTE.—The line is inclined to both planes of projection as in Pr. 5. SECTION III, ELEMENTARY SOLIDS. The solids most commonly used to illustrate the principles of Solid Geometry are as follows: the cube, prism, pyramid, sphere, cone, and cylinder. (1.) “A cube is a solid figure contained by six equal squares” (Euc. XI., Def. 25). (2.) "A prism is a solid figure contained by plane figures, of which two that are opposite are equal, similar, and parallel to one another; and the others parallelograms” (Euc. XI., Def. 13). (3.) “A pyramid is a solid figure contained by planes that are constituted betwixt one plane and one point above it in which they meet” (Euc. XI., Def. 12). (4.) “A sphere is a solid figure described by the revolution of a semicircle about its diameter, which remains unmoved” (Euc. XI., Def. 14). (5.) “A cone is a solid figure described by the revolution of a right-angled triangle about one of the sides containing the right angle, which side remains fixed. If the fixed side be equal to the other side containing the right angle, the cone is called a right-angled cone ; if it be less than the other side, an obtuse-angled ; and if greater, an acuteangled cone" (Euc. XI., Def. 18). (6.) “A cylinder is a solid figure described by the revolution of a right-angled parallelogram about one of its sides which remains fixed” (Euc. XI., Def. 21). Problem 8. To find the plan of a cube, its elevation being given. Let A'B'C'D', &c., be the elevation of a cube. It is required to find its plan. From A' let fall a perpendicular to xy, and produce it from a' to 4, making a'A equal to the distance that A' is from the vertical plane. From B' draw a line at right angles to xy, and from A, with a radius equal to a side of the cube as A'a', cut the perpendicular in B. Join AB; it will be the plan of A'B'. Now from draw a line perpendicular to xy and produce it. From B, with a radius equal to b BÉ Ć a side of the cube, as B'b', cut it in C, then C will be the plan of C". Join BC; it will be the plan of B'C'. Again, drop a perpendicular from D to xy, and produce it. From C, with radius c'c describe an arc to cut it in 1. Join CD and DA to complete the required plan. The points a'd'V'c' being opposite to A'D'B'C'; their plans are exactly covered by the points A, D, B, C. Problem 9. The elevation of a cube being given, when one face is inclined to the ground at an angle of 60° and another face at an angle of 30°, to find its plan. Let A'B'C' D' be the elevation of the given cube. It is required to find its plan. From A' let fall a perpendicular A'A, and make Aa equal to the length of the line represented by the point A'; i.e., equal to A'B' or any side of the square A'B'C'D'. Through A and a, draw lines paral lel to xy, and from D, B' and C", drop perpendiculars intersecting these lines in D,d, B,b, and C,c; for as all the edges of a cube are equal, the lines of which the points B'C'D' are the vertical projections are equal to that expressed by A'; i.e., to Aa. NOTE.— The plan of the edge of the cube expressed by B' is shown by a dotted line, because it is not seen. Problem 10. Let ABCD be the plan of a cube. It is required to find its elevation. In this case, each corner of the square is the plan of one of the perpendicular edges of the cube, and ABCD is the plan of the upper surface also. From the points A, D, B, and C, draw projectors at right angles to xy; and at the points where these meet the ground line, we have the elevations of the four corners of the square. Continue the projectors through A', D, &c., making A'a', B'b', C'd, &c., equal in height to the edge of the given cube. a ď B c' A line then drawn through the points a', b', c', &c., will be parallel to xy, and will complete the required elevation. Problem 11. To draw the plan of a square prism, its elevation being given. B' B Let A'B'C'D' be the elevatior cf a square prism. It is required to |