[_Obs._ In drawing Fig. 31. the station-point has been taken much nearer the paper than is usually advisable, in order to show the character of the curve in a very distinct form. If the student turns the book so that _EG_ may be vertical, Fig. 31. will represent the construction for drawing a circle in a vertical plane, the sight-line being then of course parallel to _GL_; and the semicircles _ADB_, _ACB_, on each side of the diameter _AB_, will represent ordinary semicircular arches seen in perspective. In that case, if the book be held so that the line _EH_ is the top of the square, the upper semicircle will represent a semicircular arch, _above_ the eye, drawn in perspective. But if the book be held so that the line _GF_ is the top of the square, the upper semicircle will represent a semicircular arch, _below_ the eye, drawn in perspective. If the book be turned upside down, the figure will represent a circle drawn on the ceiling, or any other horizontal plane above the eye; and the construction is, of course, accurate in every case.] [20] 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. 85. [21] Or to its vanishing-point, if _CD_ has one. [22] Of course, by dividing the original rectangle into any number of equal rectangles, and dividing the perspective rectangle similarly, the curve may be approximately drawn without any trouble; but, when accuracy is required, the points should be fixed, as in the problem. PROBLEM XII. TO DIVIDE A CIRCLE DRAWN IN PERSPECTIVE INTO ANY GIVEN NUMBER OF EQUAL PARTS. Let _AB_, Fig. 32., be the circle drawn in perspective. It is required to divide it into a given number of equal parts; in this case, 20. Let _KAL_ be the semicircle used in the construction. Divide the semicircle _KAL_ into half the number of parts required; in this case, 10. Produce the line _EG_ laterally, as far as may be necessary. From _O_, the center of the semicircle _KAL_, draw radii through the points of division of the semicircle, _p_, _q_, _r_, etc., and produce them to cut the line _EG_ in _P_, _Q_, _R_, etc. From the points _PQR_ draw the lines _PP′_, _QQ′_, _RR′_, etc., through the center of the circle _AB_, each cutting the circle in two points of its circumference. Then these points divide the perspective circle as required. If from each of the points _p_, _q_, _r_, a vertical were raised to the line _EG_, as in Fig. 31., and from the point where it cut _EG_ a line were drawn to the vanishing-point, as _QQ′_ in Fig. 31., this line would also determine two of the points of division. [Illustration: Fig. 32.]