act.

Sec. 3. Glancing back to the fourteenth and fifteenth paragraphs of the
chapter on Infinity, in the second volume, and to the third and tenth of
the chapters on Unity, the reader will find that abstract beauty of form
is supposed to depend on continually varied curvatures of line and
surface, associated so as to produce an effect of some unity among
themselves, and opposed, in order to give them value, by more or less
straight or rugged lines.

The reader will, perhaps, here ask why, if both the straight and curved
lines are necessary, one should be considered more beautiful than the
other. Exactly as we consider light beautiful and darkness ugly, in the
abstract, though both are essential to all beauty. Darkness mingled with
color gives the delight of its depth or power; even pure blackness, in
spots or chequered patterns, is often exquisitely delightful; and yet we
do not therefore consider, in the abstract, blackness to be beautiful.

[Illustration: FIG. 90.]

Just in the same way straightness mingled with curvature, that is to
say, the close approximation of part of any curve to a straight line,
gives to such curve all its spring, power, and nobleness: and even
perfect straightness, limiting curves, or opposing them, is often
pleasurable: yet, in the abstract, straightness is always ugly, and
curvature always beautiful.

Thus, in the figure at the side, the eye will instantly prefer the
semicircle to the straight line; the trefoil (composed of three
semicircles) to the triangle; and the cinqfoil to the pentagon. The
mathematician may perhaps feel an opposite preference; but he must be
conscious that he does so under the influence of feelings quite
different from those with which he would admire (if he ever does admire)
a picture or statue; and that if he could free himself from those
associations, his judgment of the relative agreeableness of the forms
would be altered. He may rest assured that, by the natural instinct of
the eye and thought, the preference is given instantly, and always, to
the curved form; and that no human being of unprejudiced perceptions
would desire to substitute triangles for the ordinary shapes of clover
leaves, or pentagons for those of potentillas.

Sec. 4. All curvature, however, is not equally agreeable; but the
examination of the laws which render one curve more beautiful than
another, would, if carried out to any completeness, alone require a
volume. The following few exam