Tonic. Dominant. Subdominants. Submediant. The Right (¹⁄₁₂) (³⁄₄) (³⁄₁₀) Angle. (³⁄₈) The inscribing rectangle L M N O of fig. 2 is that of (¹⁄₂), within which are arranged tracings from an ellipse of (¹⁄₃), whose greater axis, at _a b_ and _c d_ respectively, makes angles of (¹⁄₂) and (⁴⁄₉) with the horizontal, while that at _e f_ is in the horizontal line. The harmonic elements of the contour of this vase, therefore, appear to be:— Tonic. Dominant. Subtonic. (¹⁄₂) (¹⁄₃) (⁴⁄₉) [Sidenote: Plate XVI.] The inscribing rectangle P Q R S of fig. 1, Plate XVI., is one of (⁴⁄₉), within which are arranged tracings from an ellipse of (³⁄₈), whose greater axis, at _a b_, _c d_, and _e f_, makes respectively angles of (¹⁄₆) with the horizontal, (³⁄₅) and (⁴⁄₅) with the vertical. Its harmonic elements, therefore, appear to be:— Tonic. Dominant. Mediant. Supertonic. Subdominant. Submediant. The Right (¹⁄₆) (⁴⁄₅) (⁴⁄₉) (³⁄₈) (³⁄₅) Angle. The inscribing rectangle T U V X of fig. 2 is one of (⁴⁄₉), within which are arranged tracings from an ellipse of (³⁄₈) whose greater axis at _a b_ is in the vertical line, and at _c d_ makes an angle of (¹⁄₂). The harmonic elements of the contour of this vase, therefore, appear to be:— Tonic. Submediant. Supertonic. (¹⁄₂) (³⁄₈) (⁴⁄₉) These four Etruscan vases, the contours of which are thus reduced to the harmonic law of nature, are in the British Museum, and engravings of them are to be found in the well-known work of Mr Henry Moses, Plates 4, 6, 14, and 7, respectively, where they are represented with their appropriate decorations and colours. To these, I add two examples of— _Ancient Grecian Ornament._ I have elsewhere shewn[26] that the elliptic curve pervades the Parthenon from the entases of the column to the smallest moulding, and we need not, therefore, be surprised to find it employed in the construction of the only two ornaments belonging to that great work. [Sidenote: Plate XVII.] In the diagram (Plate XVII.), I endeavour to exhibit the geometric construction of the upper part of one of the ornamental apices, termed antefixæ, which surmounted the cornice of the Parthenon. The first ellipse employed is that of (¹⁄₃), whose greater axis _a b_ is in the vertical line; the second is also that of (¹⁄₃), whose greater axis _c d_ makes, with the vertical, an angle of (¹⁄₁₂); the third ellipse is the same with its major axis _e f_ in the vertical line. Through one of the foci of this ellipse at A the line A C is drawn, and upon the part of the circumference C _e_, the number of parts, 1, 2, 3, 4, 5, 6, 7, of which the surmounting part of this ornament is to consist, are set off. That part of the circumference of the ellipse whose larger axis is _c d_ is divided from _g_ to _c_ into a like number of parts. The third ellipse employed is one of (¹⁄₄).