I remarked, that by adhering to regular proportional quantities of 1 and 2, 3 and 5, 2 and 5, 7 and 5, 7 and 2, &c., and using elliptic disks or curves, very great beauties are derived. [Illustration: _Fig._ 19.] [Illustration: _Fig._ 20.] A skeleton of the tazza in angles was drawn on a black painted board, together with oval disks placed upon those lines, which clearly demonstrated the whole system of the construction. The explanation of these various diagrams necessarily involved a circumstantial description of each created figure, which were thoroughly analysed. Quantity and variety were particularly dwelt upon, as absolutely necessary to the production of perfect beauty; equalities being unfriendly to that symmetry which accords with nature. Some other diagrams were drawn, to show the inelegant appearance of radiating lines from the concave or convex half of an oval or an ellipse, _Fig._ 19: but by drawing another convex half of an oval, and placing those lines as tangents, greater beauty was formed by the alternate changes and varieties of inclination of each tangent, _Fig._ 20. This was capable of an immediate adaptation to elegant vegetation; [p011] a few convex and concave elliptic curves added to each tangent, produced an ear of barley, or an ear of rye, the elegant construction of which, is rarely noticed in our remarks on nature, _Fig._ 21. [Illustration: _Fig._ 21.] The discussion on these various designs being concluded, some important compositions of three great and renowned painters were produced, to corroborate what had been advanced in support of the native beauty of the oval and ellipse. Raphael’s grand composition of the dispute on the Sacrament is in three grand oval curves. The Doctors of the Church on the ground plan are ranged in an oval convex line; and the heavenly Choirs engage two concave oval shapes of the same proportion, but of unequal quantities. This is also a proof of a composition of parts, bearing two to one. The facility of expressing such a composition, by being geometrical, is extremely easy. The second illustration was the Aurora, by Guido, of the Aldobrandini palace. This was pointed out to depend upon an oval curve, and continued curvilinear details: the striking beauty of this fine composition is owing to its great and simple elliptic curve, which includes the whole group; the attendant hours have the principle of radiating to a centre of the oval: thus harmonizing and uniting forms congenial both to principle and nature. The third grand composition was by Rubens, the Coronation ceremony of Mary de Medicis, one of the grand Luxemburg pictures.