forms present similarities of direction, simple and compound, and the characteristic beauty of many forms, both in nature and in art, is traceable in part to the prominence of some one element of direction. Thus the various charms of the forms of cedar and birch among trees, and of the Romanesque and Gothic among architectural styles, are partly due to the predominance of some characteristic feature of form, as the horizontal or drooping line, the rounded or pointed arch.
The sense of equality enters into geometry much more prominently than into visual art; yet it is is not excluded from the latter, it only appears in a more disguised way. All equalities of magnitude among lines, surfaces, etc., are, to speak with Fechner, above the threshold of enjoyment, and the study of art in all its branches shows how considerable this enjoyment is. Among the equalities to which the æsthetically cultivated eye is specially susceptible are those in change of direction, whether angular or curvilinear. In all regular rectilinear figures equality of angle is appreciated as well as that of linear magnitude. The beauty of uniform curves and of undulating lines rests in part on a feeling for this factor of regular and equal change.
That relations of proportion enter into beautiful form is allowed by all. A technically trained eye may recognize, and perhaps enjoy, simple numerical ratios among magnitudes in lines, etc., but this factor does not appear to enter, in a conscious way at least, into ordinary æsthetic appreciation of form. We hardly experience any addition of enjoyment in learning that the ratio of the axes of a pleasing oval is 2 : 1. So far as conscious reflection can tell us, our enjoyment of proportion rests on a vague estimation of one magnitude in relation to another. But, though this relation is not numerically appreciated, it is very exactly estimated. Our enjoyment of the subtile relations of linear magnitude which enter into the beauty of a refined face shows how delicate this quantitative appreciation really is.
It is to be observed, further, that this fine sense of proportion among the various parts of a visible form includes a recognition more or less distinct of an equality between relations of magnitude. And it is this fact which brings the sense of proportion under the head of a feeling for similarity and equality. This is plain enough in the case of all imitative forms. The recognition of a face by means of a miniature portrait is really an example of a very fine perception of equalities of relation, for it rests on a distinct appreciation of the relative linear magnitudes and distances of the several features, and on a perception of the identity of these relations with all changes in absolute magnitude.
It is hardly less certain that the sense of proportion in art, when not thus based on a knowledge of the relations of natural objects, really implies a like recognition of identity of quantitative relations. The enjoyment of a due proportion between the breadth and length of a column, or among the numerous details of a Gothic church, springs
