THE SOCIAL PROPERTIES OF A CONIC SECTION,
AND THE THEORY OF POLEMICAL MATHEMATICS--(_continued_).
Most learned Professors and Students of this University,--You have
already gathered from my preceding lecture my method of procedure in the
investigation of the corresponding properties of curves and States. You
have perceived that we have here the elements of a new science, which
may be extended indefinitely, and applied to the various departments of
self-government and State control. This new science of polemical
mathematics is in itself an extension of the _principle of continuity_,
for the discovery of which Poncelet is so justly renowned. We can prove
by geometry that the properties of one figure may be derived from those
of another which corresponds to it; and the new science teaches us that
if we can represent, by projection or otherwise, a society of particles
or individuals on a plane surface, the properties of the State so
represented are analogous to the properties of the curve with which it
corresponds. It is only possible for me to touch upon the elements of
the science in these lectures, but I hope to arouse an interest in these
somewhat unusual complications and curious problems, that you may
hereafter make further discoveries in this unexplored region of
knowledge, and that the world may reap the benefit of your labours and
abstruse studies. I have already, in my previous lecture, touched upon
the social properties of the parabola, and examined the constitution of
erratic curves and eccentric nations. It is my intention to-day to speak
of similar problems which arise with reference to elliptical States.
But, first, let me answer an objection which may have occurred to your
minds. Am I wrong in my calculations in attributing too much to the
power and usefulness of forms of government? Does the well-being and
happiness of a nation depend on the government, or upon the individuals
who compose the nation? Most assuredly, I assert, they rest upon the
former. Men love their country when the good of every particular man is
comprehended in the public prosperity; they undertake hazard and labour
for the government when it is justly administered. When the welfare of
every citizen is the care of the ruling power, men do not spare their
persons or their purses for the sake of their country and the support of
their sovereign. But where selfish aims are manifest in Court or
Parliament, the people care not for State officials who are indifferent
to their country's weal; they become selfish too; Liberty hides her
head, and shakes off the dust of her feet ere she leaves that doomed
land, and the stability, welfare, and prosperity of that country cease.
I might refer you to many a stained page of national history in order to
prove this. Compare the closing chapters of the life of the Roman empire
with the record of the brave deeds of its ancient warriors and valorous
statesmen. Grecian preeminence and virtue died when liberty expired. I
agree with Sidney when he writes that it is absurd to impute this to the
change of times; for time changes nothing, and nothing was changed in
those times but the government, and that changed all things. These are
his words: 'As a man begets a man, and a beast a beast, that society of
men which constitutes a government upon the foundation of justice,
virtue, and the common good, will always have men to promote those ends;
and that which intends the advancement of one man's desires and vanity
will abound in those that will foment them.' I may not, therefore, be
altogether wrong in attributing the prosperity and well-being of a
nation to the form of government which it possesses.
We will now proceed to the consideration of the social advantages which
an elliptical State affords. This is the form of government and social
position which we, as a nation, at present enjoy; and from mathematical
considerations I am of opinion that it is the best, and hope that no
change will ever be made in our constitution. You may remember that I
have previously stated that an ellipse has a centre and two foci, in
view of all the particles which compose the curve, and connected with
them by close ties. The centre, in the projected figure, represents the
monarchy, which is limited; and the government is carried on by the aid
of the two houses of representatives of the people, depicted in the
projection by the two foci.
Now the social advantages of the ellipse are given by the fact that the
sum of the distances of any point from the foci is always constant. No
particle is left out in the cold; no one does not possess the advantages
of a social government. Though his distance may be far from the Upper
House, he has the advantage of nearness to the Lower, and _vice versa_.
The sum of the distances is constant. The extinction of one focus, the
House of Lords, for example, would create a complete disorganization of
the whole system: the other focus would set up a powerful magnetic
attraction, and a curious bulb-shaped curve would be evolved, very
different from the beautiful symmetrical form which the original figure
presented to the eye. The centre of the system would be disturbed; and
it is probable that ere long it would disappear along the axis and be
vanished to infinity. Thus the curve would become a parabola. This is
the alarming result of the extinction of one focus. Abolish the House of
Lords, and you will soon find that the Throne will be disturbed; the
State will become disorganized; the nation will become confused by the
magnetic force of the Lower House, uncounteracted by any other
attraction; and very soon a complete revolution of the whole system
will set in: the monarch will be dethroned, and a Republican form of
government, with all the eccentricities of a parabolic course, will take
the place of a more orderly and settled constitution. This is a plain
deduction from our mathematical investigations; and it behoves all our
statesmen, our philosophers and great men, our fellow-citizens and the
humblest artisans in our manufacturing towns, to weigh well this
alarming result of the abolition of that House which has been threatened
with destruction; and to ascertain for themselves the truths upon which
my proposition and reasoning rest.
I have already observed that the fact that the earth's orbit and that of
other planets are in the form of ellipses; that the curvature of the
earth is nearly the same, ought to guide us in choosing this particular
curve as a model of the projection of a complete and most advantageous
social system.
The circle described on the major axis of an ellipse, is called the
_auxiliary circle_, and affords much assistance in the investigation of
the properties of an ellipse. As we have already shown, the circle
represents the simplest form of monarchical government. Hence, if we
compare the form of government represented by an ellipse (_i.e._, such
as we now enjoy) with that of a system where the king is the only
governing power, we may obtain great assistance in solving complicated
political problems.
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.
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.'
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, 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.
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 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 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.'
3. The same opinions are often held by individuals in quite different
walks and classes of life. Let these individuals be represented by
points on an ellipse. Join these, and we have a system of parallel
chords. Draw a straight line through the middle points of these chords,
and lo! it will always pass through the centre. This shows that the
central thought of all people is directed to the sovereign--that
_loyalty_ is inherent in the hearts of those who recognise elliptical
laws.
I will conclude this lecture with a few remarks on the nature and
properties of the _radical axis_. This name was first given, I believe,
by M. Gaultier, of Tours, and for a full account of its nature I refer
you to the _Journal de l'Ecole Polytechnique_, xvi., 1813. The radical
axis of two circles is the line perpendicular to the line joining the
centres, from any point of which the tangents to the circles are equal.
Let us suppose that one circle becomes a point, and that this point is
situated on the circumference of the first circle. What is the result?
The radical axis becomes the tangent to the circle. Hence we may
conclude that in a social system of monarchical government the radical
axis is perpendicular to the line attaching the individual with the
monarch. Therefore we may conclude that the radical axis indicates a
tendency of particles, or individuals, to fly off at a tangent, at right
angles to the connecting-link between the individual and the king. When
any motion takes place, this is evident, and this tendency is called
centrifugal force. Sad is it for the State when this force is called
into play, and the radical axis is a standing menace to the stability of
States and nations. The only way to counteract its baneful, disturbing
influence is to increase the attraction of the monarch on the
individual, which nullifies the former force, and prevents further
mischief. This is the method which nature itself adopts in the motions
of the planetary worlds; the attraction of the sun prevents any
disturbance which might be caused in the course of the planets by the
action of centrifugal force, and nature suggests this plan for our
adoption. Increase the attraction of the Throne; rigidly connect each
individual by the strong chords of affection, advantage and utility with
the ruling power; and then, though the radical axis may be there, it
will cease to indicate any motion along it, it will not prevail over the
counteracting influence of loyalty, and the stability of the social
system and the happiness of the individuals will be the results.
'I would serve my King,
Serve him with all my fortune here at home,
And serve him with my person in the wars;
Watch for him, fight for him, bleed for him, die for him,
As every true-born subject ought.'
This, most noble professors, is the language of true patriotic loyalty.
Let the monarch be loved and loving, let the laws be just and equal,
happy will be the people, prosperous the realm. There are those who
counsel different things, and preach sedition and the breaking-up of
laws; but those who advocate such doctrines lack that judicial
mathematical training which we, students and professors of Girtham
College, have acquired. If polemical mathematics, the science of the
future, should become more widely studied; if its results were
disseminated far and wide; above all, if the proper position which women
ought to occupy in the counsels of the nation were assigned to them, we
should hear less of these wild schemes and foolish theories, and the
influence of women would tend greatly to promote the stability and
security of the State.
Why, let me ask, should woman be excluded from that position which is so
justly hers? from those duties which she can discharge so faithfully? It
has been said that if we wish to know the political and moral condition
of a State, we must ask what rank women hold in it. We are told that
women have more strength in their looks than men have in their laws.
Why, then, do men debar her from those fields of occupation wherein she
may labour for the nation's good, and use her influence, which they
acknowledge to be great, in those callings wherein she may most easily
benefit the State, and the country she so ardently loves?
At some future time I hope to speak more fully on this subject; and in
concluding this lecture, I will remark that English politics need a
leavening influence which will counteract the evil tendencies and
corrupt theories which, in spite of our advantageous social system, at
present exist; and this leavening influence will be best produced by
the admission of those into the counsels of the nation who are
acknowledged to have a benign and healthy influence--the women of
England. Let women have their proper share in the government of the
country, and I have no fear lest we shall preserve our elliptical
constitution, and all the advantages which we at present enjoy.
* * * * *
[Editorial Note.]--In the bundle of papers which contained the foregoing
lectures, some letters of great interest were found, which show that the
fame of the learned Lady Professor of Girtham College had already gone
abroad, and attracted the attention of the leading statesmen of the day.
It is to be regretted that the answers to these letters are not
forthcoming, as it might be proved from them that the science of
polemical mathematics has already influenced the minds of our
legislators in their conduct of affairs at home and abroad. The
following letter is of unique interest, and may be taken as evidence of
the favourable impression which this new science has made on the mind of
one of our greatest thinkers and statesmen:
Downing Street,
May, 18--
My dear Lady Professor,--The report of the amazing results of your
scientific researches has reached me, and I congratulate you most
heartily on the originality and acumen which you have displayed in your
investigations. A new light has dawned upon our country. Instead of
groping in the darkness of political warfare, ensnared by party ties and
jealousies, the statesmen of the future will be able to calculate and
determine the correct course with mathematical precision and perfect
accuracy. No one can dispute the truth of a proposition in Euclid, or
the genuineness of Newton's laws; and if your method enables men to
calculate and determine the correct political course of action, to solve
political problems as easily as exponential equations, why--then adieu
to the bickerings of party, the querulous complaints of the Opposition!
Nay, joy to the Ministry! There will be no Opposition! Our statesmen
will be able to guide the great ship of the State by means of charts
which know no error; and they will resemble an association of savants
met together to determine the exact moment of the transit of Venus, or
to examine the degree of density of a comet's tail.
This condition of Parliamentary procedure is much to be desired; you
have shown how such an ideal state of things may be obtained. In the
name of the Government I thank you for your endeavours on behalf of your
country's welfare, and look forward to a further development of your
admirably conceived system. As in the domain of ordinary science there
are complex questions which defy the acumen of the philosopher; so in
polemical science there may be questions which present the same
difficulties and complications. But as the first are daily yielding
before the persevering attacks of the mathematician, so I doubt not
polemical science will soon overcome the various problems which may
arise.
But it is mainly on my own account that I venture to address you. I
desire to consult you with regard to certain matters--political
complications--which have recently occupied the attention of Her
Majesty's Ministers. By the help of your new science, can you aid us
in our deliberations? Of course, I am writing to you in _strict
confidence_, and beg that you will keep this communication profoundly
secret. I fear that would be a hard task for many of your sex, who do
not possess your knowledge and powers of mind; but I have great
confidence in your discretion.
These are the problems which are presented to us for solution:
1. Some members of the Cabinet are secretly in favour of Protection, and
the country is rather stirred by the question. Can you, from your
knowledge of the contact of curves and nations, help us to determine
what course we ought to take with regard to Spain, for example? Are the
principles of Adam Smith mathematically correct?
2. I observe that England is represented mathematically by an ellipse.
Are we right in assuming that Ireland is a portion of that ellipse? Or,
on the other hand, in our chart of nations, must we describe that
troublesome country as a rotating parabola, or complex figure,
altogether outside our more favoured State?
3. Do you consider, from your minute observation of our social system,
that the form of our elliptical government is gradually undergoing a
change, and that a revolutionary parabolic tendency is observable in the
action of individual particles?
4. Is it not possible that the differences in the policy of the various
nations of Europe; the difficulties which beset the carrying out of
international law; the jealousies, quarrels, and rivalries of States
might disappear, if the same form of government (_i.e._, elliptical)
were adopted in each?
If you will kindly favour Her Majesty's Ministers with your opinion on
these questions, they will owe you a debt of gratitude, which they, as
representatives of the nation, will do their utmost to repay.
With every good wish for your further success in the regions of
polemical science,
I beg to remain,
My dear Lady Professor,
Your faithful servant,
+----------------------+
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[Editorial Note.]--The next letter is not of quite the same pleasing
nature as the foregoing, and shows that it is impossible to please
everyone, even if that happy consummation were desirable. This letter
was evidently called forth by some remarks which the learned Lady
Professor had made in her third lecture with reference to eccentricity
in dress. Our readers will recollect that the professor pointed out that
an extravagant 'bloomer' costume--half male, half female--was no more a
sign of genius than aesthetic dresses, always betokened the artist.[5]
This latter statement evidently gave great offence to the members of a
society which called itself the 'Aesthetic and Dress Improvement
Association,' and the following letter is the result of one of their
solemn conclaves:
Oscar Villa, South Kensington,
June, 18--.
The Secretary of the Aesthetic and Dress Improvement Association presents
his compliments to the Lady Professor of Girtham College, and begs to
contradict emphatically her statements with regard to a subject upon
which she is evidently in entire and lamentable ignorance, and to
protest against her aspersions upon the artistic studies of this and
kindred societies. He begs to state that true aesthetes are _not_
eccentric (they leave that to lady professors and her Philistine
followers); that to dress becomingly is one of the principal objects of
life, and that true greatness is achieved as much by the study of the
art of dress as by any other noble pursuit or graceful accomplishment.
Are not Horatio Postlethwaite, Leonara Saffronia Gillan, Vandyke
Smithson entitled to greatness? And yet their laurels have been won
solely by the art of dress. Perhaps the lady professor has never read
'Sartor Resartus'! In conclusion, he would ask the Lady Professor to
refrain from casting obloquy upon the work of the Association which he
has the honour to represent; to prevail upon her pupils to abandon the
unfeminine attire which some of them have assumed, contrary to the first
principles of art; to array themselves in flowing robes of sage-green
and other choice colours (patterns enclosed), and to study art, instead
of absurd mathematics, which no one can understand, and do no one any
good.
(Approved by the Committee of the Aesthetic and Dress
Improvement Association.)
June, 18--.
[Editorial Note.]--The next letter, written by a pupil of the Lady
Professor, requires no explanation, and speaks for itself.
Jesus College, Cambridge,
March, 18--.
My dear Tutor,
You will be glad to hear that after superhuman exertions I have at last
succeeded in passing my Little-go, and I am eternally grateful to you
for all you have done for me. I should never have got through if it had
not been for you. All the coaches in Cambridge would never have managed
it, but you drove me through in a canter. And why? I never could make up
my mind to work for them; but when I coached with you, you made me like
it. I almost revelled in the Binomial when you wrote it out for me; and
then I could not help listening to you; and you looked so grieved when I
would not learn, and made me feel such a brute; so somehow or other you
drove some mathematics into my head, and I pulled through. By-the-bye, I
think you must have tried the 'brain wave' dodge with the examiners, as
five out of the six propositions in Euclid, which you told me to get up
specially, were set! I wish I could read people's thoughts; can you read
mine? If I were a Don, or a Fellow, or something, I would advise the
University to have some lady professors like you to teach the men,
instead of some of these sleepy old tutors. It would be a great
improvement, and I am sure we should get through a great deal more work.
They have given me a place in the Jesus Eight, which I shall take now
that I am released from your professorial ban, and have time for rowing.
But I don't half like giving up mathematics. You see, I have grown fond
of the study. Do you think you could make a wrangler of me? At any rate,
I should like to come to your lectures again. May I?
Your Grateful Pupil.
* * *
[4] It is to be regretted that this letter has evidently fallen
into the hands of some autograph collector, who has ruthlessly
cut off the signature; but the reader will easily determine,
after careful perusal of the document, from whose pen it emanated.
[5] Cf. page 36.