LECTURE ON THE SOCIAL PROPERTIES OF A CONIC SECTION,
AND THE THEORY OF POLEMICAL MATHEMATICS.
Most Learned Professors and Students of this University,--From the
interest manifested in my first lecture, I conclude that my method of
investigation has not proved altogether unsatisfactory to you, and I
hope ere long to produce certain investigations which will probably
startle you, and revolutionize the current thought of the age. The
application of mathematics to the study of Social Science and Political
Government has curiously enough escaped the attention of those who ought
to be most conversant with these matters. I shall endeavour to prove in
the present lecture that the relations between individuals and the
Government are similar to those which mathematical knowledge would lead
us to postulate, and to explain on scientific principles the various
convulsions which sometimes agitate the social and political world.
Indeed, by this method we shall be able to prophesy the future of states
and nations, having given certain functions and peculiarities
appertaining to them, just as easily as we can foretell the exact day
and hour of an eclipse of the moon or sun. In order to do this, we must
first determine the _social properties of a conic section_.
For the benefit of the unlearned and ignorant, I will first state that a
cone is a solid figure described by the revolution of a right-angled
triangle about one of the sides containing the right angle, which
remains fixed. The fixed side is called the axis of the cone. Conic
sections are obtained by cutting the cone by planes. It may easily be
proved that if the angle between the cutting plane and the axis be equal
to the angle between the axis and the revolving side of the triangle
which generates the cone, the section described on the surface of the
cone is a parabola; if the former angle be greater than the latter, the
curve will be an ellipse; and if less, the section will be a hyperbola.
But the simplest conic section is, of course, a circle, which is formed
by a plane at right angles to the axis of the cone; and the simplest
circle is that formed by a plane passing through the apex of the cone.
All this is simple mathematics; and let beginners consult more
elementary treatises than this one to satisfy themselves on these
points. But if they will assume these things to be true, they will know
quite enough for our present purpose. The simplest conic section of all
has been proved to be a _point_. Now, this represents the simplest and
original form of society, a _single family_. 'It is not good for man to
be alone' was the first observation made by the wise Creator upon the
rational creature whom He had introduced into Paradise as its lord.
Marriage is the rudiment of all social life, from which all others
spring, out of which all others are developed. Around the parents'
knees soon cluster a group of children, and in their relation to each
other we discern the earliest forms of law and discipline--the bonds by
which society is held together. When the children grow up, separate
households are formed; and then the multiplication of families, the
congregating of men together for purposes of security and mutual
advantages in division of labour; and thus is gradually formed a state,
which is only the development of the family--the king representing the
parent, and ruling on the same principle.
Mathematically speaking, our plane no longer passes through the apex.
The point represented the single family; but keeping the plane
horizontal, we move it along the axis, the sections will become
_circles_, which represent mathematically the next simplest form of
society, where the centre is the seat of government, which is connected
with each individual member of the social circle by equal radii. The
social property of a circle is that of a monarchical government in its
purest and simplest form. The larger the circle becomes (_i.e._, the
further you move the plane from the apex), the greater the distance
between the individual and the monarch. Therefore, the more independent
the monarchy becomes, and the less influence do individuals possess over
the ruling power. Hence, we may infer that as years roll on, the
government will become more despotic; but the stability of the country
diminished, and probably some individual particle, when sufficiently
withdrawn from the attraction of the central head, will begin to revolve
on its own account, and spontaneously generate a government of its own.
We may, therefore, conclude from mathematical reasoning that an
unlimited monarchy, though advantageous for small states, is not a safe
form of government for a large or populous country, inasmuch as the
people do not derive much benefit from the sovereign; the mutual
attraction, which ought to exist in a flourishing state between the
ruler and the ruled, is weakened; and the isolation of the monarch
tends to make him still more despotic. As a practical example of the
truth of the foregoing statement, I may mention the present condition of
Russia, which shows that the result of an unlimited monarchy, in a large
and unwieldy social circle, is such as we should have reasonably
expected from mathematical investigations.
Invariably, under the circumstances which I have described, the country
will become disorganized; the sovereign will cease to have any power
over the people, and the country will become a chaos, without order,
influence, or power.
When the centre of a conic section moves along the axis of the curve to
infinity, banished by the mutual consent of the individual particles
which compose the curve, or the nation, a figure is formed, called a
_parabola_. This is the curve which the most erratic bodies in the
universe describe in space, as they rush along at a speed inconceivable
to human minds, and are supposed to produce all kinds of mischief and
injury to the worlds whose courses they wend their way among.
This curve, then, represents the position which the nation assumes when
the constituted monarchy, the centre of the system, has been _banished
to infinity_. A revolution has occurred; the monarch has been dethroned;
and it is not hard to see that the same erratic course which the comet
pursues in its flight, is observable with respect to the social system
which is represented by a parabola. We observe with eager scrutiny the
wanderings of these erratic comets. They appear suddenly with their
vapoury tails; sometimes they shine upon us with their soft, silvery
light, brilliant as another moon; sometimes they stand afar off in the
distant skies, and deign not to approach our steady-going earth, which
pursues its regular course day by day, and year by year. Then, after a
few days' coy inspection of our planet from different points of view,
they fly to other remote parts of the universe, and do not condescend to
show themselves again for a hundred years or so. Such is the erratic
conduct of a heavenly body whose course is regulated by a parabolic
curve.
We may look for similar eccentric behaviour on the part of a community,
nation, or state, whose centre is at infinity, whose constitution has
been violently disturbed, and whose monarchy is situated in the far-off
regions of unlimited space. The erratic course of Republican rule is
proverbial. There is no stability, no regularity. To-day we may observe
its brilliancy, which seems to laugh at and eclipse the sombre shining
of more steady and enduring worlds; but ere to-morrow's moon has risen,
it may have vanished into the regions of eternal night, and we look for
its bright shining light in the councils of the nations, but it has
ceased to shed its rays, and we are disappointed. Sometimes it is asked,
with fear and trembling: 'What would be the effect if our earth were to
come in contact with the tail of a comet? Should we be destroyed by the
collision, and our ponderous world cease to be?' But we are assured
that no such disastrous results would follow. We have already passed
through the tails of many comets, but we have not discovered any
inconvenient change in our ordinary mode of procedure. It is probable
that the comet's tail is composed of no solid substance.
We may therefore infer by analogy that a Republican State would not
offer any powerful resistance if it were to come into collision with a
nation possessing a more settled form of government. A shower of
meteoric stones, like passing fireworks, might take place; but beyond
that nothing would occur to excite the fear, or arouse the energies of
the more favoured nation. As an example of the weakness of a Republican
State I may mention France. There we see an industrious race of people,
endowed with many natural gifts and graces, a country rich and
productive; and yet, owing to the unsettled nature of its government,
all these natural advantages are neutralized; its course amongst the
nations is erratic in the extreme, a spectacle of feeble
administration; and it would offer no more resistance to a colliding
Power than the empty vacuum of a comet's tail. This example will
demonstrate to you the truth of our theory with regard to the
instability of a social system which is geometrically represented by a
parabolic curve.
We will now turn from this picture of insecurity and unrest to another
figure which possesses most advantageous social properties. I refer to
the ellipse. An ellipse is a curve formed by the section of a cone by a
plane surface inclined at an angle to the vertical axis of the cone,
greater than the angle between the axis and the generating line.
Now, this is a curve which possesses most attractive properties. It is
the curve which the earth and other planetary orbs describe around the
centre of the solar system, as if nature intended that we should take
this figure as a guide in choosing the most advantageous social system.
It possesses a centre, C, in view of all the particles which compose the
curve, and connected with them by close ties. It has two foci, S and
S', fixed points, by the aid of which we may trace the curve.
In the interpretation of this figure, the centre of the curve represents
the throne of monarchy. There is no tendency here to revolutionize the
State, to banish the ruling power, and institute a Republican form of
government; but inasmuch as we saw the weakness of an absolute monarchy
in large and populous States, as represented by the circle, the wisdom
of an elliptical social system has ordained that there shall be two
foci, or houses of representatives of the people, who shall assist in
regulating the progress of the nation. Here we have a limited monarchy;
the throne is supported by the representatives of the people; and the
nearer these foci of the nation are to the centre (_i.e._, in
mathematical language, the less the _eccentricity_ of the curve), the
more perfect the system becomes--the greater the happiness of the
community.
In cases where the _eccentricity_ becomes very great, the beauty of the
curve is destroyed, and ultimately the ellipse is merged into one
straight line. Most learned Professors, here we have a terrible warning
of the awful result of too much eccentricity. Whether we regard the life
of the nation or of the individual, let all bear in mind this alarming
fact, that eccentricity of thought, habit, or behaviour may result, as
in the case of this unfortunate ellipse, which once presented such fair
and promising proportions to the student's admiring gaze, in the
'sinister effacement of a man,' or the gradual absorption of a State
into an uninteresting thing 'which lies evenly between its extreme
points.'
The great examples of Bacon, of Milton, of Newton, of Locke, and of
others, happen to be directly opposed to the popular inference that
eccentricity and thoughtlessness of conduct are the necessary
accompaniments of talent, and the sure indications of genius. I am
indebted to Lacon for that reflection. You may point to Byron, or
Savage, or Rousseau, and say, 'Were not these eccentric people
talented?' 'Certainly,' I answer; 'but would they not have been better
and greater men if they had been less eccentric--if they had restrained
their caprice, and controlled their passions?' Do not imagine, my young
students of this university, that by being eccentric you will therefore
become great men and women of genius. The world will not give you credit
for being brilliant because you affect the extravagances which sometimes
accompany genius. Some of you ladies, I perceive, have adopted a
peculiar form of dress, half male, half female; or, to be more correct,
three-fourths male, and one-fourth female. Do not imagine that you will
thus attain to the highest honours in this university by your
eccentricity, unless your talents are hid beneath your short-cut hair,
and brains are working hard under your college head-gear. As well might
we expect to find that all females who wear sage-green and extravagant
aesthetic costumes are really born artists and future Royal Academicians.
It is apparent that many aspirers to fame and talent are eager to
exhibit their eccentricities to the gaze of the world, in order that
they may persuade the multitude that they possess the genius of which
eccentricity is falsely supposed to be the outward sign.
I may remark in passing that the eccentricity of a parabolic curve is
always _unity_. What does this prove? You will remember that a
Republican State is represented by a parabola. Therefore, however such a
nation may strive to alter its condition, and secure a settled form of
government, its eccentricity will always remain the same. It will always
be erratic, peculiar, unsettled; and this conclusion substantiates our
previous proposition with regard to the condition of a social system
represented by a parabola.
With regard to other advantages afforded by an elliptical social system,
we will defer the consideration of this important subject until my next
lecture.