among the infinite numbers of the higher curves. I believe that almost
all are beautiful in their own nature, and that their comparative beauty
depends on the constant quantities involved in their equations. Of this
point I shall speak hereafter at greater length.

Sec. 12. How by nature obtained.

The universal forces of nature, and the individual energies of the
matter submitted to them, are so appointed and balanced, that they are
continually bringing out curves of this kind in all visible forms, and
that circular lines become nearly impossible under any circumstances.
The gradual acceleration, for instance, of velocity, in streams that
descend from hill-sides, as it gradually increases their power of
erosion increases in the same gradual degree the rate of curvature in
the descent of the slope, until at a certain degree of steepness this
descent meets, and is concealed by the right line of the detritus. The
junction of this right line with the plain is again modified by the
farther bounding of the larger blocks, and by the successively
diminishing proportion of landslips caused by erosion at the bottom, so
that the whole line of the hill is one of curvature, first, gradually
increasing in rapidity to the maximum steepness of which the particular
rock is capable, and then decreasing in a decreasing ratio, until it
arrives at the plain level. This type of form, modified of course more
or less by the original boldness of the mountain, and dependent both on
its age, its constituent rock, and the circumstances of its exposure, is
yet in its general formula applicable to all. So the curves of all
things in motion, and of all organic forms, most rudely and simply in
the shell spirals, and in their most complicated development in the
muscular lines of the higher animals.

This influence of apparent proportion, a proportion, be it observed,
which has no reference to ultimate ends, but which is itself, seemingly,
the end and object of operation in many of the forces of nature, is
therefore at the root of all our delight in any beautiful form
whatsoever. For no form can be beautiful which is not composed of curves
whose unity is secured by relations of this kind.

Sec. 13. Apparent proportion in melodies of line.

Not only however in curvature, but in all associations of lines
whatsoever, it is desirable that there should be reciprocal relation,
and the eye is unhappy without perception of it. It is utterly vain to