HOW TO DRAW A CIRCLE IN PERSPECTIVE WITHOUT A GEOMETRICAL PLAN
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_.
HOW TO DRAW A CIRCLE IN ANGULAR PERSPECTIVE
Let _ABCD_ be the oblique square. Produce _VA_ till it cuts the base line at _G_.
Take _mD_, the fourth of the base. Find _mn_ as in Fig. 171, measure it on each side of _E_, and so obtain _Ef_ and _Ef·_, and proceed to draw _fV_, _EV_, _f·V_ and the diagonals, whose intersections with these lines will give us the eight points through which to draw the circle. In fact the process is the same as in parallel perspective, only instead of making our divisions on the actual base _AD_ of the square, we make them on _GD_, the base line.