As to the colonnade or vertical part, the line _a b_, which determines the three middle intercolumniations, makes an angle of (¹⁄₅); the line _c d_, which determines the two outer intercolumniations, makes an angle of (¹⁄₆); and the line _e f_, which determines the lesser diameter of the columns, makes an angle of (¹⁄₁₂) with the vertical. I need give no further details here, as my intention is to shew the simplicity of the method by which this theory may be reduced to practice, and because I have given in my other works ample details, in full illustration of the orthography of these two structures, especially the first.[10] The foregoing examples being both horizontal rectangular compositions, the proportions of their principal parts have necessarily been determined by lines drawn from the extremities of the base, making angles with the horizontal line, and forming thereby the diagonals of the various rectangles into which, in their leading features, they are necessarily resolved. But the example I am now about to give is of another character, being a vertical pyramidal composition, and consequently the proportions of its principal parts are determined by the angles which the oblique lines make with the vertical line representing the height of the elevation, and forming a series of isosceles triangles; for the isosceles triangle is the type of all pyramidal composition. This third example is the east end of Lincoln Cathedral, a Gothic structure, which is acknowledged to be one of the finest specimens of that style of architecture existing in this country. The angles which govern the proportions of this elevation are the following harmonic parts of the right angle:— Tonic. Dominant. Mediant. Subtonic. Supertonic. (¹⁄₂) (¹⁄₃) (¹⁄₅) (¹⁄₇) (²⁄₉) (¹⁄₄) (¹⁄₆) (¹⁄₁₀) (¹⁄₉) (¹⁄₁₂) [Sidenote: Plate IV.] In Plate IV. I give a diagram of the vertical, horizontal, and oblique lines, which compose the orthography of this beautiful elevation.