[1] Witmer, Lightner:  ’Zur experimentellen Aesthetik einfacher
   raeumlicher Formverhaeltnisse,’ Phil.  Studien, 1893, IX., S.
   96-144, 209-263.

This criticism would apply to judgments of unequal division on either side the center of a horizontal line.  It would apply all the more to any general average of judgments including both sides, for, as we shall soon see, the judgments of individuals differ materially on the two sides, and this difference itself may demand its explanation.  And if we should include within this average, judgments above and below the center of a vertical line, we should have under one heading four distinct sets of averages, each of which, in the individual cases, might show important variations from the others, and therefore require some variation of explanation.  And yet that great leveller, the general average, has obliterated these vital differences, and is recorded as indicating the ’most pleasing proportion.’[3] That such an average falls near the golden section is immaterial.  Witmer himself, as we shall see,[4] does not set much store by this coincidence as a starting point for explanation, since he is averse to any mathematical interpretation, but he does consider the average in question representative of the most pleasing division.

   [2] op. cit., 212-215.

   [3] Witmer:  op. cit., S. 212-215.

   [4] op. cit., S. 262.

I shall now, before proceeding to the details of the experiment to be recorded, review, very briefly, former interpretative tendencies.  Zeising found that the golden section satisfied his demand for unity and infinity in the same beautiful object.[5] In the golden section, says Wundt,[6] there is a unity involving the whole; it is therefore more beautiful than symmetry, according to the aesthetic principle that that unification of spatial forms which occurs without marked effort, which, however, embraces the greater manifold, is the more pleasing.  But to me this manifold, to be aesthetic, must be a sensible manifold, and it is still a question whether the golden section set of relations has an actual correlate in sensations.  Witmer,[7] however, wrote, at the conclusion of his careful researches, that scientific aesthetics allows no more exact statement, in interpretation of the golden section, than that it forms ‘die rechte Mitte’ between a too great and a too small variety.  Nine years later, in 1902, he says[8] that the preference for proportion over symmetry is not a demand for an equality of ratios, but merely for greater variety, and that ’the amount of unlikeness or variety that is pleasing will depend upon the general character of the object, and upon the individual’s grade of intelligence and aesthetic taste.’  Kuelpe[9] sees in the golden section ’a special case of the constancy of the relative sensible discrimination, or of Weber’s law.’  The division of a line at the golden section produces ‘apparently equal differences’ between minor and major, and major and whole.  It is ’the pleasingness of apparently equal differences.’