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By George David Birkhoff

Each reader of this booklet will locate it thought-provoking. Its first charm is to the analytically-minded aesthetician; yet there are questions and implications some distance past any specialist's domain.

In bankruptcy 1, Professor Birkhoff provides the subsequent mathematical formula of the elemental challenge: "Within each one category of aesthetic gadgets, to outline the order O and the complexity C in order that their ratio M=O/C yields the cultured degree of any item within the class." In next chapters, this challenge is solved for varied uncomplicated sessions of aesthetic items in nice element. the writer distinguishes among "formal" and "connotative" institutions, and explains that "our consciousness can be directed virtually solely towards the formal facet of paintings, to which on my own the fundamental formulation of aesthetic degree could be quantitatively applied," yet with "no purpose of denying the transcendent significance of the connotative aspect in all inventive art."

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Our first and principal aim will be to effect an analysis in typical important cases of the utmost simplicity. From the vantage point so reached it will be possible to consider briefly more general questions. In following this program, there is of course no intention of denying the transcendent importance of the connotative side in all creative art. 17. A DIAGRAM The adjoining diagram with the attached legend may be of assistance in recalling the above analysis of the aesthetic experience and the basic aesthetic formula to which it leads.

In this way the question arises as to whence comes the indubitable special agreeableness of rectangles approximating that of the Golden Section. . * Ästhetik, vol. ι , pp. 66-67, m y translation. (29) AESTHETIC MEASURE The rectangles in question are just those in which the smaller dimension is decisively subordinated to the greater. . It has already been indicated above why the rectangle which approaches the square too closely pleases little. We term it awkward because of its ambiguity.

In consequence such symmetry about an inclined axis is to be looked upon as arising from the combined vertical symmetry and rotational symmetry, and so as logically dependent upon them. Hence symmetry about an inclined axis need not be considered in this case as a separate aesthetic factor. W h a t is the importance of symmetry about a horizontal axis when there is no vertical symmetry? This case is illustrated most simply by the pentagonal polygon (a) in Figure 8. In the first place it is clear that the symmetry about the horizontal axis is much more easily appraised by the eye than in any other direction except the vertical.

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