"The foregoing statements of general principles preclude the necessity of minute details in relation to particular curves. I shall at present consider those which do not return into themselves, so as to constitute the outlines of figures in the geometrical sense. Let us first select a semi-circumference, for example, that of a rainbow of maximum dimensions. In tracing it once, we employ three out of the four muscles. They are brought into action successively and rapidly, but not abruptly. All these circumstances are favorable to pleasure. Yet they are not conducive to it in the highest possible degree; for each muscle acts only once unless the examination be repeated; and in case of its repetition, the momentum of the eyeball is destroyed in stopping and reversing its motion. The waving line, as Hogarth's line of beauty, obviates the first difficulty. This ensures not only the successive action of different muscles, but a repetition of action in the same. If the line forms a number of equal waves, these repetitions will be proportional to the number of waves, and will alternately and totally relieve, at least two muscles, and allow, in the action of a third, regular remissions of intensity at equal intervals. We have proved then, that on this physiological theory, a semi-circumference possesses more of the elements of beauty than any straight line, and a regular-waved line more than either. These results are conformable to experience. If there is any difficulty in admitting this, it will vanish on comparing the ocular with other muscles. "Let us select a joint, which, in its spherical form, and the circular arrangement of its muscles, is analogous to the eye; for example, the shoulder joint. I think it will be uniformly found, that in the use of this joint, the curves most readily traced, are those of gentle and nearly equal curvature, and being such as are most easily traced by the eye, they would appear more beautiful than those drawn by the fingers with the same education. For example, let a man, without bending his wrist or elbow, draw various lines with a light stick or cane on the surface of snow: the lines most easily drawn (or most easily traced if already drawn), will be curves of considerable beauty, and nearly equal curvature; such as waved lines and spirals and looped curves. Circles and ellipses would also be among the figures with most facility and precision traced, and especially in cases of repeated tracing; but we are not at present considering figures in the proper geometrical sense of the term. In writing letters by the above method, a succession of 'e's, would be more readily drawn than a succession of 'i's, or a zigzag line with acute angles.