In examining the remains of the ornamental works of the ancient Greek artists, it appears highly probable that the harmony of their proportions and melody of their contour are equally the result of a systematised application of the same harmonic law. This probability not being fully elucidated in any of my former works, I will require to go into some detail on the present occasion. I take for my first illustration an unexceptionable example, viz.:— _The Portland Vase._ Although this beautiful specimen of ancient art was found about the middle of the sixteenth century, inclosed in a marble sarcophagus within a sepulchral chamber under the Monte del Grano, near Rome, and although the date of its production is unknown, yet its being a work of ancient Grecian art is undoubted; and the exquisite beauty of its form has been universally acknowledged, both during the time it remained in the palace of the Barberini family at Rome, and since it was added to the treasures of the British Museum. The forms and proportions of this gem of art appear to me to yield an obedience to the great harmonic law of nature, similar to that which I have instanced in the proportions and contour of the best specimens of ancient Grecian architecture. [Sidenote: Plate XII.] Let the line A B (Plate XII.) represent the full height of the vase. Through A draw A _a_, and through B draw B _b_ indefinitely, A _a_ making an angle of (¹⁄₂), and B _b_ an angle of (¹⁄₃), with the vertical. Through the point C, where A _a_ and B _b_ intersect one another, draw D C E vertical. Through A C and B respectively, draw A D, C F, and B E horizontal. Draw similar lines on the other side of A B, and the rectilinear portion of the diagram is complete. The curvilinear contour may be thus added:— Take a cut-out ellipse of (¹⁄₄), whose greater axis is equal to the line A B, and _1st._ Place it upon the diagram, so that its circumference may be tangential to the lines C E and C F, and its greater axis _m n_ may make an angle of (¹⁄₅) with the vertical, and trace its circumference. _2d._ Place it with its circumference tangential to that of the first at the point m, while its greater axis (of which _o p_ is a part) is in the horizontal, and trace the portion of its circumference _q o r_. _3d._ Place it with its circumference tangential to that of the above at _v_, while its greater axis (of which _u v_ is a part) makes an angle of (³⁄₁₀) with the vertical, and trace the portion of its circumference _s v t_. Thus the curvilinear contour of the body and neck are harmonically determined. The curve of the handle may be determined by the same ellipse placed so that its greater axis (of which _i k_ is a part) makes an angle of (¹⁄₆) with the vertical. Make similar tracings on the other side of A B, and the diagram is complete. The inscribing rectangle D G E K is that of (²⁄₅).