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 examples will be enough to put the reader in the way of pursuing the subject for himself. [Illustration: FIG. 91.] Take any number of lines, _a b_, _b c_, _c d_, &c., Fig. 91, bearing any fixed proportion to each other. In this figure, _b c_ is one third longer than _a b_, and _c d_ than _b c_; and so on. Arrange them in succession, keeping the inclination, or angle, which each makes with the preceding one always the same. Then a curve drawn through the extremities of the lines will be a beautiful curve; for it is governed by consistent laws; every part of it is connected by those laws with every other, yet every part is different from every other; and the mode of its construction implies the possibility of its continuance to infinity; it would never return upon itself though prolonged for ever. These characters must be possessed by every perfectly beautiful curve. If we make the difference between the component or measuring lines less, as in Fig. 92, in which each line is longer than the preceding one only by a fifth, the curve will be more contracted and less beautiful. If we enlarge the difference, as in Fig. 93, in which each line is double the preceding one, the curve will suggest a more rapid proceeding into infinite space, and will be more beautiful. Of two curves, the same in other respects, that which suggests the quickest attainment of infinity is always the most beautiful. [Illustration: FIG. 92.] [Illustration: FIG. 93.]