With your permission I will add a few words to those I have already uttered with regard to the directrix. As necessary as the directrix is to the curve, so are the corresponding laws to the State. I will prove this fact by a few examples. English people have laws, and know how to obey them; therefore their numbers increase; they thrive and are prosperous. A friendly critic of another nation has said that the reason why Englishmen rule the world, is because they know how to obey. On the other hand, the gipsies have no laws; hence they become fewer and less powerful. What is the condition of all tribes and nations which are not governed by laws? They invariably remain poor and miserable. They are in want of a directrix; and if we could supplement the gift with foci and centre, they would soon emerge from their savage condition, and become more civilized. I have omitted to mention the hyperbolic form of government. The curve formed by the intersection of the surface of a cone with a plane will be a hyperbola, when the inclination of the cutting plane to the axis of the cone is less than the constant angle which the generating line forms with the axis. It is manifest that the plane will thus intersect the higher cone, and produce the figure which is known to mathematicians as the hyperbola. We may hence deduce the following property of the corresponding hyperbolic State. We take cognizance of that higher cone with which the mundane affairs of the lower cone are closely connected. As an example of this system we may mention the vast temporal rule and power of the Papal Throne, which formerly exercised such marvellous sway over the nations of Europe. By an appeal to a Higher Authority than that of earthly kings and potentates was this rule exercised; but its hyperbolic form is fast passing away, and degenerating into that of a circle with indefinitely small radius. We shall not, therefore, discuss the complex polemical problems which a hyperbolic State suggests. I will now mention a few problems which are easily capable of proof, and deduce from them the necessary conclusions which must follow when we apply our newly discovered principles of polemical science. 1. 'If from any point in a straight line a pair of tangents be drawn to an ellipse, the chords of contact will pass through a fixed point.'