Fig. 67.

II. The preceding problem is simple, because the lines of the profile of the object (a b and c d, 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.

Fig. 68.

Two squares (of the required size) are first drawn, one above the other, with a given vertical interval, A c, 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 a E B, Fig. 69., be the profile of the vase or cup to be drawn.

Enclose it in the rectangle C D, and if any portion of it is not curved, as A E. cut off the curved portion by the vertical line E F, so as to include it in the smaller rectangle F D.

Draw the rectangle A C B D in position, and upon it construct two squares, as they are constructed on the rectangle A C D in Fig. 68.; and complete the construction of Fig. 68., making the radius of its large outer circles equal to A D, and of its small inner circles equal to A E.

The planes which occupy the position of the wheel-spokes will then each represent a rectangle of the size of F D. The construction is shown by the dotted lines in Fig. 69.; c being the centre of the uppermost circle.

Within each of the smaller rectangles between the circles, draw the curve EB in perspective, as in Fig. 69.

Fig.. 69.

Draw the curve x y, touching and enclosing the curves in the rectangles, and meeting the upper circle at y.1

Then x y is the contour of the surface of the cup, and the upper circle is its lip.

If the line x y is long, it may be necessary to draw other rectangles between the eight principal ones; and, if the curve of profile A B is complex or retorted, there may be several lines corresponding to x y, enclosing the successive waves of the profile; and the outer curve will then be an undulating or broken one.

III. All branched ornamentation, forms of flowers, capitals of columns, machicolations of round towers, and other

1 This point coincides in the figure with the extremity of the horizontal diameter, but only accidentally.

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 E D C F, and the extremities of the trefoils touch it at the extremities of its diagonals and diameters. A smaller circle is at the vertical distance B C 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

Fig. 7a 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.

Fig. 71. will show the law with sufficient clearness: it represents the perspective construction of a fillet whose profile is a semicircle, such as F H in Fig. 60., seen above the eye. Only half the pillar with half the fillet is drawn, to avoid confusion.

Q is the centre of the shaft.

P Q the thickness of the fillet, sight-magnitude at the shaft's centre.

Round P a horizontal semicircle is drawn on the diameter of the shaft a b.

Round Q another horizontal semicircle is drawn on diameter c d.

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.

Fig. 71.

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 encloses and touches1 all the semicircles. It disappears

1 The engraving is a little inaccurate; the enclosing line should touch the dotted semicircles at A and B. The student should draw it on a large scale.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.