The Romance of Mathematics / Being the Original Researches of a Lady Professor of Girtham College in Polemical Science, with some Account of the Social Properties of a Conic; Equations to Brain Waves; Social Forces; and the Laws of Political Motion.

by P. H. Ditchfield

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 [49] 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.

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 [50] 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 [51] 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.’

I will not trouble you with the proof of this proposition, as it is evident to all mathematicians, and can easily be demonstrated. But mark well the deductions, when we interpret this mathematical language in correct polemical terms. A State, through various convulsions of its own, [52] has merged into a condition represented by a straight line, having lost its symmetry, its beauty, its curvilinear proportion. An individual unhappily situated in this unfortunate community regards with longing eyes the prosperous condition of those who enjoy the social advantages of a settled form of government, and other blessings which accompany elliptical jurisdiction and laws. [Two tangents are drawn to an ellipse.] No matter where the individual may be in the unhappy envious straight line, the result of his reflection will be the same. Sympathetic chords are drawn, joining the points of contact of the tangents with the curve; they all pass through a fixed point. All these conclusions of the various individuals on the straight line will be the same. All are of opinion that the elliptical form is the best; and they mourn in secret over the sad events which have occurred in their own national life, their eccentricity, their lawlessness, when they see the advantages which their more staid and sober-minded neighbours so freely enjoy.

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2. The normal at any point of an ellipse bisects the angle between the focal distances of that point.

The normal is the perpendicular from the point on the major axis; it is the line of thought directed by the observance of just laws and rules. Hence this proposition shows that the individual citizen, when guided by sound judgment, regards with equal favour and entire approval the existence of both foci, or Houses of Legislature. He considers that both are necessary to his comfort, and the right regulation of the State’s welfare. He cares not for the abnormal condition of those who talk as if the existence of either House were unnecessary to his country’s weal, and bestows a pitying glance on those wandering lights, or disturbed erratic governments, which do not possess the advantages which from experience he has learned to love and to respect. No matter what his condition may be, the same opinions are held by all classes, all ranks and degrees; and if a self-opinionated particle think otherwise, he ought to be transferred to a less [54] enlightened sphere, and migrate to a parabolic state, or uninteresting straight line. And when he has changed his location, he will look back on his old home and old surroundings with longing eyes and an aching heart, thinking of the blessings he has lost by his own rash act. This can be proved mathematically. He looks for an ideal state of society, leaps after the shadow his fancy has depicted; and when he finds himself outside his former state, he looks back with longing eyes at the once-scorned focus. What is the focus of a perpendicular on the tangent of an ellipse from any external point? Can it not be proved to be a circle? That is to say, he will be more conservative than ever. He would like to return to a primitive form of government. Farewell to his wild schemes and revolutionary measures! Farewell to his disestablishments, abolitions, and suppressions! The throne and government have new attractions in his eyes; loyalty, a new feeling, asserts its benign influence; and if he could return [55] to his former position, his normal conduct would be straighter than ever, for by sad experience he has learned the value of those things which he once despised.

But we need not depend upon one proof alone. Exactly the same result may be obtained from the well-known proposition which states that ‘the angle between the tangent from any external point and the focal distance is equal to the angle between the other tangent and the focal distance.’