[ XCI]
How To Draw a Circle in Perspective Without a Geometrical Plan
| Fig. 169. |
| Fig. 170. |
| Fig. 171. |
Divide base AB into four equal parts. At B drop perpendicular Bn, making Bn equal to Bm, or one-fourth of base. Join mn and transfer this measurement to each side of d on base line; that is, make df and df· equal to mn. Draw fS and f·S, and the intersections of these lines with the diagonals of square will give us the four points o o o o.
The reason of this is that ff· is the measurement on the base AB of another square o o o o which is exactly half of the outer square. For if we inscribe a circle in a square and then inscribe a second square in that circle, this second square will be exactly half the area of the larger one; for its side will be equal to half the diagonal of the larger square, as can be seen by studying
the following figures. In Fig. 170, for instance, the side of small square K is half the diagonal of large square o.
In Fig. 171, CB represents half of diagonal EB of the outer square in which the circle is inscribed. By taking a fourth
of the base mB and drawing perpendicular mh we cut CB at h in two equal parts, Ch, hB. It will be seen that hB is equal to mn, one-quarter of the diagonal, so if we measure mn on each side of D we get ff· equal to CB, or half the diagonal. By drawing ff, f·f passing through the diagonals we get the four points o o o o through which to draw the smaller square. Without referring to geometry we can see at a glance by Fig. 172, where we have simply turned the square o o o o on its centre so that its angles touch the sides of the outer square, that it is exactly half of square ABEF, since each quarter of it, such as EoCo, is bisected by its diagonal oo.
| Fig. 172. | Fig. 173. |