I find it necessary in this place to go into some details regarding the specific character of the two latter figures, because the proper mode of describing these beautiful curves, and their high value in the practice of the architectural draughtsman and ornamental designer, seem as yet unknown. In proof of this assertion, I must again refer to Mr Penrose’s great work published by the “Society of Dilettanti.” At page 52 of that work it is observed, that “by whatever means an ellipse is to be constructed mechanically, it is a work of time (if not of absolute difficulty) so to arrange the foci, &c., as to produce an ellipse of any exact length and breadth which may be desired.” Now, this is far from being the case, for the method of arranging the foci of an ellipse of any given length and breadth is extremely simple, being as follows:— Let A B C (figure 1) be the length, and D B E the breadth of the desired ellipse. [Illustration: Fig. 1.] Take A B upon the compasses, and place the point of one leg upon E and the point of the other upon the line A B, it will meet it at F, which is one focus: keeping the point of the one leg upon E, remove the point of the other to the line B C, and it will meet it at G, which is the other focus. But, when the proportions of an ellipse are to be imparted by means of one of the harmonic angles, suppose the angle of (¹⁄₃), then the following is the process:— Let A B C (figure 2) represent the length of the intended ellipse. Through B draw B _e_ indefinitely, at right angles with A B C; through C draw the line C _f_ indefinitely, making, with B C, an angle of (¹⁄₃). Take B C upon the compasses, and place the point of one leg upon D where C f intersects B _e_, and the point of the other upon the line A B, it will meet it at F, which is one focus. Keeping the point of one leg still upon D, remove the point of the other to the line B C, and it will meet it at G, which is the other focus. [Illustration: Fig. 2.] The foci being in either case thus simply ascertained, the method of describing the curve on a small scale is equally simple. [Sidenote: Plate V.]