Several exercises will be required on this important problem.
I. It is required to draw a circular flat-bottomed dish narrower at
the bottom than the top; the vertical depth being given, and the
diameter at the top and bottom.
[Illustration: Fig. 65.]
Let _ab_, Fig. 65., be the diameter of the bottom, _ac_ the diameter
of the top, and _ad_ its vertical depth.
Take _AD_ in position equal to _ac_.
On _AD_ draw the square _ABCD_, and inscribe in it a circle.
Therefore, the circle so inscribed has the diameter of the top of the
dish.
From _A_ and _D_ let fall verticals, _AE_, _DH_, each equal to _ad_.
Join _EH_, and describe square _EFGH_, which accordingly will be equal
to the square _ABCD_, and be at the depth _ad_ beneath it.
Within the square _EFGH_ describe a square _IK_, whose diameter shall
be equal to _ab_.
Describe a circle within the square _IK_. Therefore the circle so
inscribed has its diameter equal to _ab_; and it is in the center of
the square _EFGH_, which is vertically beneath the square _ABCD_.
Therefore the circle in the square _IK_ represents the bottom of the
dish.
Now the two circles thus drawn will either intersect one another, or
they will not.
If they intersect one another, as in the figure, and they are below
the eye, part of the bottom of the dish is seen within it.
[Illustration: Fig. 66.]
To avoid confusion, let us take then two intersecting circles without
the inclosing squares, as in Fig. 66.
Draw right lines, _ab_, _cd_, touching both circles externally. Then
the parts of these lines which connect the circles are the sides of
the dish. They are drawn in Fig. 65. without any prolongations, but
the best way to construct them is as in Fig. 66.
If the circles do not intersect each other, the smaller must either be
within the larger or not within it.
If within the larger, the whole of the bottom of the dish is seen from
above, Fig. 67. _a_.
[Illustration: Fig. 67.]
If the smaller circle is not within the larger, none of the bottom is
seen inside the dish, _b_.
If the circles are above instead of beneath the eye, the bottom of the
dish is seen beneath it, _c_.
If one circle is above and another beneath the eye, neither the bottom
nor top of the dish is seen, _d_. Unless the object be very large, the
circles in this case will have little apparent curvature.
II. The preceding problem is simple, because the lines of the profile
of the object (_ab_ and _cd_, Fig. 66.) are straight. But if these
lines of profile are curved, the problem becomes much more complex:
once mastered, however, it leaves no farther difficulty in
perspective.
Let it be required to draw a flattish circular cup or vase, with a
given curve of profile.
The basis of construction is given in Fig. 68., half of it only being
drawn, in order that the eye may seize its lines easily.
[Illustration: Fig. 68.]
Two squares (of the required size) are first drawn, one above the
other, with a given vertical interval, _AC_, between them, and each is
divided into eight parts by its diameters and diagonals. In these
squares two circles are drawn; which are, therefore, of equal size,
and one above the other. Two smaller circles, also of equal size, are
drawn within these larger circles in the construction of the present
problem; more may be necessary in some, none at all in others.
It will be seen that the portions of the diagonals and diameters of
squares which are cut off between the circles represent radiating
planes, occupying the position of the spokes of a wheel.
Now let the line _AEB_, Fig. 69., be the profile of the vase or cup to
be drawn.
Inclose it in the rectangle _CD_, and if any portion of it is not
curved, as _AE_, cut off the curved portion by the vertical line _EF_,
so as to include it in the smaller rectangle _FD_.
Draw the rectangle _ACBD_ in position, and upon it construct two
squares, as they are constructed on the rectangle _ACD_ in Fig. 68.;
and complete the construction of Fig. 68., making the radius of its
large outer circles equal to _AD_, and of its small inner circles
equal to _AE_.
The planes which occupy the position of the wheel spokes will then
each represent a rectangle of the size of _FD_. The construction is
shown by the dotted lines in Fig. 69.; _c_ being the center of the
uppermost circle.
[Illustration: Fig. 69.]
Within each of the smaller rectangles between the circles, draw the
curve _EB_ in perspective, as in Fig. 69.
Draw the curve _xy_, touching and inclosing the curves in the
rectangles, and meeting the upper circle at _y_.[32]
Then _xy_ is the contour of the surface of the cup, and the upper
circle is its lip.
If the line _xy_ is long, it may be necessary to draw other rectangles
between the eight principal ones; and, if the curve of profile _AB_ is
complex or retorted, there may be several lines corresponding to _XY_,
inclosing the successive waves of the profile; and the outer curve
will then be an undulating or broken one.
[Illustration: Fig. 70.]
III. All branched ornamentation, forms of flowers, capitals of
columns, machicolations of round towers, and other such arrangements
of radiating curve, are resolvable by this problem, using more or
fewer interior circles according to the conditions of the curves.
Fig. 70. is an example of the construction of a circular group of
eight trefoils with curved stems. One outer or limiting circle is
drawn within the square _EDCF_, and the extremities of the trefoils
touch it at the extremities of its diagonals and diameters. A smaller
circle is at the vertical distance _BC_ below the larger, and _A_ is
the angle of the square within which the smaller circle is drawn; but
the square is not given, to avoid confusion. The stems of the trefoils
form drooping curves, arranged on the diagonals and diameters of the
smaller circle, which are dotted. But no perspective laws will do work
of this intricate kind so well as the hand and eye of a painter.
IV. There is one common construction, however, in which, singularly,
the hand and eye of the painter almost always fail, and that is the
fillet of any ordinary capital or base of a circular pillar (or any
similar form). It is rarely necessary in practice to draw such minor
details in perspective; yet the perspective laws which regulate them
should be understood, else the eye does not see their contours rightly
until it is very highly cultivated.
[Illustration: Fig. 71.]
Fig. 71. will show the law with sufficient clearness; it represents
the perspective construction of a fillet whose profile is a
semicircle, such as _FH_ in Fig. 60., seen above the eye. Only half
the pillar with half the fillet is drawn, to avoid confusion.
_Q_ is the center of the shaft.
_PQ_ the thickness of the fillet, sight-magnitude at the shaft’s
center.
Round _P_ a horizontal semicircle is drawn on the diameter of the
shaft _ab_.
Round _Q_ another horizontal semicircle is drawn on diameter _cd_.
These two semicircles are the upper and lower edges of the fillet.
Then diagonals and diameters are drawn as in Fig. 68., and, at their
extremities, semicircles in perspective, as in Fig. 69.
The letters _A_, _B_, _C_, _D_, and _E_, indicate the upper and
exterior angles of the rectangles in which these semicircles are to be
drawn; but the inner vertical line is not dotted in the rectangle at
_C_, as it would have confused itself with other lines.
Then the visible contour of the fillet is the line which incloses and
touches[33] all the semicircles. It disappears behind the shaft at the
point _H_, but I have drawn it through to the opposite extremity of
the diameter at _d_.
Turned upside down the figure shows the construction of a basic
fillet.
The capital of a Greek Doric pillar should be drawn frequently for
exercise on this fourteenth problem, the curve of its echinus being
exquisitely subtle, while the general contour is simple.
[32] This point coincides in the figure with the extremity of the
horizontal diameter, but only accidentally.
[33] The engraving is a little inaccurate; the inclosing line
should touch the dotted semicircles at _A_ and _B_. The student
should draw it on a large scale.