We assume, therefore, that the standard of symmetry, so estimated, is deduced from the simplest law that could have been conceived—the law that the angles of direction must all bear to some fixed angle the same simple relations which the different notes in a chord of music bear to the fundamental note; that is, relations expressed arithmetically by the smallest natural numbers. Thus the eye, being guided in its estimate by direction rather than by distance, just as the ear is guided by number of vibrations rather than by magnitude, both it and the ear convey simplicity and harmony to the mind without effort, and the mind with equal facility receives and appreciates them. _On the Rectilinear Forms and Proportions of Architecture._ As we are accustomed in all cases to refer direction to the horizontal and vertical lines, and as the meeting of these lines makes the right angle, it naturally constitutes the fundamental angle, by the harmonic division of which a system of proportion may be established, and the theory of symmetrical beauty, like that of music, rendered susceptible of exact reasoning. Let therefore the right angle be the fundamental angle, and let it be divided upon the quadrant of a circle into the harmonic parts already explained, thus:— Super- Sub- Sub- Sub- Semi-sub- Right tonic Mediant dominant Dominant mediant tonic tonic Tonic Angle. Angles. Angles. Angles. Angles. Angles. Angles. Angles. Angles. I. (1) (⁸⁄₉) (⁴⁄₅) (³⁄₄) (²⁄₃) (³⁄₅) (⁴⁄₇) (⁸⁄₁₅) (¹⁄₂) II. (¹⁄₂) (⁴⁄₉) (²⁄₅) (³⁄₈) (¹⁄₃) (³⁄₁₀) (²⁄₇) (⁴⁄₁₅) (¹⁄₄) III. (¹⁄₄) (²⁄₉) (¹⁄₅) (³⁄₁₆) (¹⁄₆) (³⁄₂₀) (¹⁄₇) (²⁄₁₅) (¹⁄₈) IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) (³⁄₃₂) (¹⁄₁₂) (³⁄₄₀) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆) In order that the analogy may be kept in view, I have given to the parts of each of these four scales the appropriate nomenclature of the notes which form the diatonic scale in music.