With regard to the study of objects without a natural or suggested middle line, as for instance sculpture, many types of architecture, landscapes, gardens, room-arrangements, etc., we may fitly consider it as a corollary to the study of asymmetrical objects with artificial limits which do suggest a middle.  If we find, by the study of them, that a system of substitution of psychological factors does obtain, the whole field can be covered by the theory already propounded, and its application extended to the minutest details.  The hypothesis, having been so far confirmed, may be then easily applied to the field of asymmetrical objects without a natural middle line.

The set of problems here suggested to the student of symmetry will not be fully followed out in this paper.  The experimental treatment of geometrical symmetry, the analysis of the completely symmetrical products of civilized art, and the analysis of all forms of asymmetry except asymmetry in pictures will be omitted.  If, however, the fact of an original aesthetic feeling for symmetry is established by the study of primitive art, and the theory of the balance of motor impulses through the substitution of factors is established by the experimental treatment of isolated elements, and further confirmed by the analysis of pictures, the general argument may be taken as sufficiently supported.  This paper, then, will contain three sections:  an introductory one on symmetry in primitive art, and two main sections, one on experiments in substitutional symmetry, and one on substitutional symmetry or balance in pictures.

II.  SYMMETRY IN PRIMITIVE ART.

The question which this section will attempt to answer is this:  Is there in primitive art an original and immediate aesthetic feeling for symmetry?  This question depends on two others which must precede it:  To what extent does symmetry actually appear in primitive art? and, How far must its presence be accounted for by other than aesthetic demands?

For the purpose of this inquiry the word primitive may be taken broadly as applying to the products of savage and half-savage peoples of to-day, as well as to those of prehistoric races.  The expression primitive art, also, requires a word of explanation.  The primitive man seldom makes purely ornamental objects, but, on the other hand, most of his articles of daily use have an ornamental character.  We have to consider primitive art, therefore, as represented in the form and ornamentation of all these objects, constituting practically an household inventory, with the addition of certain drawings and paintings which do not appear to serve a definite practical end.  These last, however, constitute only a small proportion of the material.