[p 42 ] 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.

[p43] [Geometric diagram][Geometric diagram]
Fig.32.

If it is required to divide a circle into any number of given unequal parts (as in the points A, B, and C, Fig.33.), the shortest way is thus to raise vertical lines from A and B to the side of the perspective square XY, and then draw to the vanishing-point, cutting the perspective circle in a and b, the points required. Only notice that if any point, as A, is on the nearer side of the circle ABC, its representative point, a, must be on the nearer side of the circle abc; and if the point B is on the farther side of the circle ABC, b must be [p44] on the farther side of abc. If any point, as C, is so much in the lateral arc of the circle as not to be easily determinable by the vertical line, draw the horizontal CP, find the correspondent p in the side of the perspective square, and draw pc parallel to XY, cutting the perspective circle in c.

[Geometric diagram]
Fig.33.

COROLLARY.

It is obvious that if the points P', Q', R, etc., by which the circle is divided in Fig.32., be joined by right lines, the resulting figure will be a regular equilateral figure of twenty sides inscribed in the circle. And if the circle be divided into given unequal parts, and the points of division joined by right lines, the resulting figure will be an irregular polygon inscribed in the circle with sides of given length.

Thus any polygon, regular or irregular, inscribed in a circle, may be inscribed in position in a perspective circle.