In all conics there is a straight line called the 'directrix,' which represents in social or polemical science the laws of the nation, and plays a prominent part in the mutual relations of the individual particles. For instance, in the case of the parabola, the distance of any particle from the directrix is equal to its distance from the focus. From this we may conclude that if an individual deviates at all from the path which the laws (or, directrix) indicate, if he does not show true respect to the decrees of the focal government, and preserve the true position between them, directly he is found deviating from his course, he is quickly banished to a less enlightened sphere. In an ellipse there is less likelihood of his straying away from the course which the directrix points out, on account of the two-fold guidance which he receives from the two foci. The following curious problem may be noticed. If a parabola roll on another parabola, their vertices coinciding, the focus of the first traces out the directrix of the second. Here we come to the consideration of the international relationship of States. Two nations have the same form of government (in this example this form is Republican); their policies coincide: we may conclude from this proposition that the course which the government of one nation will pursue, will be that which is prescribed by the laws of the other. The subject of the contact of curves presents many interesting problems with reference to Polemical Science, and may be extended indefinitely. It is well known that there are different orders of contact, which are designated as the _first_, _second_, or _third_ order. This last order may be termed the 'marriage of curves,' cemented by the osculating circle, or 'wedding-ring;' and when two nations have contact of the third order, they have formed a very close alliance, and by calculation we can obtain the _radius of curvature_, or size of the wedding-ring, by means of which they may be united. The theory and nature of contact constitute a branch of our newly discovered science which we commend to the careful consideration of those who have undertaken the difficult and perplexing study of international law. Alas! too many States refuse this friendly contact, and, consequently, _cut_ each other, instead of blending in sweet accord. Their peace is at best an armed neutrality; and if they have contact of only the _first_ or _second_ order, we can prove mathematically that they are sure to intersect in some other point or points; and divergence of policy and disturbed relations are the results. Contact of the _third, or highest, order_ is the only safe position for two allied, or contiguous, States.