The outline resulting from this diagram, not only is in perfect agreement with my recollection of the form, but with the measurements of the original given in the “Penny Cyclopædia;” of the accuracy of which there can be no doubt. They are stated thus:—“It is about ten inches in height, and beautifully curved from the top downwards; the diameter at the top being about three inches and a-half; at the neck or smallest part, two inches; at the largest (mid-height), seven inches; and at the bottom, five inches.” The harmonic elements of this beautiful form, therefore, appear to be the following parts of the right angle:— Tonic. Dominant. Mediant. Submediant. (¹⁄₂) (¹⁄₃) (¹⁄₅) (³⁄₁₀) (¹⁄₄) (¹⁄₆) When we reflect upon the variety of harmonic ellipses that may be described, and the innumerable positions in which they may be harmonically placed with respect to the horizontal and vertical lines, as well as upon the various modes in which their circumferences may be combined, the variety which may be introduced amongst such forms as the foregoing appears almost endless. My second example is that of— _An Ancient Grecian Marble Vase of a Vertical Composition._ I shall now proceed to another class of the ancient Greek vase, the form of which is of a more complex character. The specimen I have chosen for the first example of this class is one of those so correctly measured and beautifully delineated by Tatham in his unequalled work.[25] This vase is a work of ancient Grecian art in Parian marble, which he met with in the collection at the Villa Albani, near Rome. Its height is 4 ft. 4¹⁄₂ in. [Sidenote: Plate XIII.] The following is the formula by which I endeavour to develop its harmonic elements:— Let A B (Plate XIII.) represent the full height of this vase. Through B draw B D, making an angle of (¹⁄₅) with the vertical. Through D draw D O vertical, through A draw A C, making an angle of (²⁄₅); through B draw B L, making an angle of (¹⁄₂), and B S, making an angle of (³⁄₁₀), each with the vertical. Through A draw A D, through B draw B O, through L draw L N, through C draw C F, and through S draw S P, all horizontal. Through A draw A H, making an angle of (¹⁄₁₀) with the vertical, and through H draw H M vertical. Draw similar lines on the other side of A B, and the rectilinear portion of the diagram is complete, and its inscribing rectangle that of (³⁄₈). The curvilinear portion may thus be added—