An application of the great harmonic law seems to prove that there is no object in nature in which the science of beauty is more clearly developed than in the human head and countenance, nor to the representations of which the same science is more easily applied; and it is to the mode in which this is done that the varieties of sex and character may be imparted to works of art. Having gone into full detail, and given ample illustrations in a former work,[18] it is unnecessary for me to enter upon that part of the subject in this _résumé_; but only to shew the typical structure of beauty by which this noble work of creation is distinguished. The angles which govern the form and proportions of the human head and countenance are, with the right angle, a series of seven, which, from the simplicity of their ratios to each other, are calculated to produce the most perfect concord. It consists of the right angle and its following parts— Tonic. Dominant. Mediant. Subtonic. (¹⁄₂) (¹⁄₃) (¹⁄₅) (¹⁄₇) (¹⁄₄) (¹⁄₆) These angles, and the figures which belong to them, are thus arranged:— [Sidenote: Plate IX.] The vertical line A B (Plate IX. fig. 2) represents the full length of the head and face. Taking this line as the greater axis of an ellipse of (¹⁄₃), such an ellipse is described around it. Through A the lines A G, A K, A L, A M, and A N, are drawn on each side of the line A B, making, with the vertical, respectively the angles of (¹⁄₃), (¹⁄₄), (¹⁄₅), (¹⁄₆), and (¹⁄₇). Through the points G, K, L, M, and N, where these straight lines meet the curved line of the ellipse, horizontal lines are drawn by which the following isosceles triangles are formed, A G G, A K K, A L L, A M M, and A N N. From the centre X of the equilateral triangle A G G the curvilinear figure of (¹⁄₂), viz., the circle, is described circumscribing that triangle. The curvilinear plane figures of (¹⁄₂) and (¹⁄₃), respectively, represent the solid bodies of which they are sections, viz., a sphere and a prolate spheroid. These bodies, from the manner in which they are here placed, are partially amalgamated, as shewn in figures 1 and 3 of the same plate, thus representing the form of the human head and countenance, both in their external appearance and osseous structure, more correctly than they could be represented by any other geometrical figures. Thus, the angles of (¹⁄₂) and (¹⁄₃) determine the typical form. From each of the points _u_ and _n_, where A M cuts G G on both sides of A B, a circle is described through the points _p_ and _q_, where A K cuts G G on both sides of A B, and with the same radius a circle is described from the point _a_, where K K cuts A B. The circles _u_ and _n_ determine the position and size of the eyeballs, and the circle _a_ the width of the nose, as also the horizontal width of the mouth.