These preliminary designs opened the way for richer combinations; but the subject affording such an immense field of variety, I confined myself to the narrowest limits, and to one oval disk of seven inches transverse diameter, from which seven different designs were shown on paper. The first had a variety of serpentine lines placed at random, all produced by the disk of the oval just named, and the confluent lines of two such, placed side by side, or end to end, _Fig._ 12; which oval disk was put upon the lines to prove the construction. These lines, without expressing or forming any sort of figure, exhibit a set of elegant curves, of varied quantities of convex and concave, with which our eye will be more pleased than any set of right lines similarly distributed, as in _Fig._ 13, which follows. [p006] [Illustration: _Fig._ 13.] [Illustration: _Fig._ 14.] Two other diagrams were placed before the company, each a circle of 12 ovals, from the same disk, revolved upon an axis, resting upon one end of the transverse diameter, (the length-ways of the oval,) which figure in the skeleton was a duodecagon. _Fig._ 14 is one of the diagrams; the ovals folding regularly over each other. By suppressing the continuity of the oval disk, where the lines would traverse, a very pleasing figure [p007] is created. It may be easily converted into foliage, and can be amazingly varied in principle, by having fewer ovals, and making them revolve upon an arm or continuation of a line from the transverse diameter. _Fig._ 15 is the same diagram, with all the oval lines described, which forms a figure of elegant intricacy; each member, or curvilinear subdivision, assumes a most agreeable shape: the whole, at the first sight, does not carry the evidence of being generated from the same disk. These agreeable figures may be varied to an extraordinary extent: the two that were presented were mere examples of some of the numerous changes that any given oval disk may create. [Illustration: _Fig._ 15.] [Illustration: _Fig._ 16.] [Illustration: _Fig._ 17.]