Take a cut-out ellipse of (¹⁄₃), whose greater axis is about the length of the body of the intended vase, place it with its lesser axis upon the line S P, and its greater axis upon the line D O, and trace the part _a b_ of its circumference upon the diagram. Place the same ellipse with one of its foci upon C, and its greater axis upon C F, and trace its circumference upon the diagram. Take a cut-out ellipse of (¹⁄₅), whose greater axis is nearly equal to that of the ellipse already used; place it with its greater axis upon M H, and its lesser axis upon L N, and trace its circumference upon the diagram. Make similar tracings upon the other side of A B, and the diagram is complete. In this, as in the other diagrams, the strong portions of the lines give the contour of the vase. The harmonic elements of this classical form, therefore, appear to be the right angle and its following parts:— Tonic. Dominant. Mediant. Submediant. (¹⁄₂) (¹⁄₃) (²⁄₅) (³⁄₁₀) (¹⁄₅) (¹⁄₁₀) My third example is that of— _An Ancient Grecian Vase of a Horizontal Composition._ This example belongs to the same class as the last, but it is of a horizontal composition. It was carefully drawn from the original in the museum of the Vatican by Tatham, in whose etchings it will be found with its ornamental decorations. The diagram of its harmonic elements may be constructed as follows:— [Sidenote: Plate XIV.] Let A B (Plate XIV.) represent the full height of the vase. Through B draw B D, making an angle of (²⁄₅) with the vertical. Through A draw A H, A L, and A C, making respectively the following angles, (¹⁄₅) with the vertical, (⁴⁄₉) with the vertical, and (³⁄₁₀) with the horizontal. These angles determine the horizontal lines H B, L N, and C F, which divide the vase into its parts, and the inscribing rectangle D G K O is (³⁄₈). This completes the rectilinear portion of the diagram. The ellipse by which the curvilinear portion is added is one of (¹⁄₅), the greater axis of which, at _a b_, as also at _c d_, makes an angle of (¹⁄₁₂) with the vertical, and the same axis at _e f_ an angle of (¹⁄₁₂) with the horizontal. The harmonic elements of this vase, therefore, appear to be:— Tonic. Dominant. Mediant. Submediant. Supertonic. The Right (¹⁄₁₂) (²⁄₅) (³⁄₁₀) (⁴⁄₉) Angle. (¹⁄₅) My remaining examples are those of— _Etruscan Vases._ Of these vases I give four examples, by which the simplicity of the method employed in applying the harmonic law will be apparent. [Sidenote: Plate XV.] The inscribing rectangle D G E K of fig. 1, Plate XV., is one of (³⁄₈), within which are arranged tracings from an ellipse of (³⁄₁₀), whose greater axis at _a b_ makes an angle of (¹⁄₁₂), at _c d_ an angle of (³⁄₁₀), and at _e f_ an angle of (³⁄₄), with the vertical. The harmonic elements of the contour of this vase, therefore, appear to be:—