Fig. 26. B GF, HI, on the top of the square pillar, cutting each other in c. Therefore c is the centre of the square F G H I. (Prob. VIII. Cor. II.) Join C E, CA, C B. Then A B C E is the pyramid required. If the base of the pyramid is above the eye, as when a square spire is seen on the top of a church-tower, the con. struction will be as in Fig 27. Fig. 27. LET A B, Fig. 28., be the curve. Enclose it in a rectangle, C D E F. Fix the position of the point c or D, and draw the rectangle. (Problem VIII. Coroll. I.)* Or if the curve is in a vertical plane, Coroll. to Problem IX. As a rectangle may be drawn in any position round any given curve, its position with respect to the curve will in either case be regulated by convenience. See the Exercises on this Problem in the Appendix p. 115. LET A B, Fig. 6., be the given right line, joining the given points A and B. Let the direct, lateral, and vertical distances of the point A be T D, D C, and c A. Let the di. cct, lateral, and vertical distances of the point в be T D', D c', and c' B. Then, by Problem I., the position of the point ▲ on the plane of the picture is a. And similarly, the position of the point в on the plane of the picture is b. Join a b. Then a b is the line required. COROLLARY I. If the line A B is in a plane parallel to that of the picture, one end of the line A B must be at the same direct distance from the eye of the observer as the other. Therefore, in that case, D T is equal to D' T. Then the construction will be as in Fig. 7.; and the student will find experimentally that a bis now parallel to A B.* For by the construction A Tar:: BT: bт; and therefore the two triangles A B T, a b T, (having a common angle ▲ T B) are similar. |