The Project Gutenberg EBook of The Romance of Mathematics, by P. Hampson This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: 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. Author: P. Hampson Release Date: August 29, 2008 [EBook #26481] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK THE ROMANCE OF MATHEMATICS *** Produced by David Wilson and the Online Distributed Proofreading Team at https://www.pgdp.net (This file was produced from images generously made available by The Internet Archive/Canadian Libraries) 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. HAMPSON, M.A., ORIEL COLLEGE, OXFORD. LONDON: ELLIOT STOCK, 62, PATERNOSTER ROW. 1886. iii INTRODUCTION. The lectures, essays, and other matter contained in these pages have been discovered recently in a well-worn desk which was formerly the property of a Lady Professor of Girtham College; and as they contain some original thoughts and investigations, they have been considered worthy of publication. How they came into the possession of the present writer it is not his intention to disclose; but inasmuch as they seemed to his unscientific mind to contain some important discoveries which might be useful to the world, he determined to investigate thoroughly the contents of the mysterious desk, and make the public acquainted with its profound treasures. He found some documents which did not refer exactly to the subject of ‘Polemical Mathematics;’ but knowing the truth of the Hindoo proverb, ‘The words of the wise are precious, and never to be disregarded,’ and feeling sure that this Lady Professor of Girtham College was entitled to that appellation, he ventured to include them in this volume, and felt confident that in so doing he would be carrying out the intention of the Authoress, had she expressed any wishes on the subject. In fact, as he valued the interests of the State and his own peace of mind, he dared not withhold any particle of that which he conceived would confer a lasting benefit on mankind. Internal evidence seems to show that the earlier portion of the MS. was written during the period when the authoress was still in statu pupillari; but her learning was soon recognised by the Collegiate Authorities, and she was speedily elected to a Professorship. Her lectures were principally devoted to the abstruse subject of Scientific Politics, and are worthy of the attention of all those whose high duty it is to regulate the affairs of the State. The Editor has been able to gather from the varied contents of the desk some details of the Author’s life, which increase the interest which her words excite; and he ventures to hope that the public will appreciate the wisdom which created such a profound impression upon those whose high privilege it was to hear the lectures for the first time in the Hall of Girtham College. v vi vii CONTENTS. PAPER PAGE I. Some Remarks on Female Education: Cambridge Man’s Powers of Application.—Torturing Ingenuity of Examiners.—Slaying an Enemy.—‘Concentration.’—‘Tangential Action.’—‘Gravity’ 1 II. Lecture on the Theory of Brain Waves and the Transmigration and Potentiality of Mental Forces 15 III. The Social Properties of a Conic Section, and the Theory of Polemical Mathematics: ‘Circle.’—‘Parabola.’—‘Ellipse.’ ‘Eccentricity of Curves’ 25 IV. The Social Properties of a Conic Section (continued): ‘Ellipse.’—Most favoured State.—Alarming Result of Suppression of House of Lords.—Analogies of Nature.—Directrix.—Contact of Curves and States.—‘Hyperbola.’—Problems.—Radical Axis and Patriotism.— Extension of Franchise to Women.—Correspondence 39 V. Social Forces, with some Account of Polemical Kinematics: The Use of Imagination in Scientific Discovery.—Kinetic and Potential Energy.—Social Statics and Dynamics.—Attractive Forces.—Cohesion.— Formation of States.—Inertia.—Dr. Tyndall on Social Forces 71 VI. Social Forces (continued): Polemical Statics and Dynamics: ‘Personal Equation.’—Public Opinion, how calculated.—Impulsive Forces. —Friction.—Progress 89 VII. Laws of Political Motion: M. Auguste Comte on Political Science.—First Law of Motion.—The Biology of Politics.—Stages of Growth and Decay of States.—Doctrine of Nationality.—Doctrine of Independence.—Law of Morality.—Ignorance of Electors and Selfishness of Statesmen opposed to Action of Law.—Final ‘Reign of Law’ 101 VIII. The Principle of Polemical Cohesion: Centralization.—Co-operation of States.—Marriage.—Trade Unions.— International Law 115 Extracts from the Diary of the Lady Professor 125 Conclusion 129 ix x xi PAPER I. SOME REMARKS OF A GIRTHAM GIRL ON FEMALE EDUCATION. [This essay upon Female Education was evidently written when the future Professor of Girtham College was still in the lowlier condition of studentship, before she attained that eminence for which her talents so justly entitled her. Its unfinished condition tends to show that it was probably evolved during moments of relaxation from severer studies, without any idea of subsequent publication.] Oh, why should I be doomed to the degradation of bearing such a foolish appellation! A Girtham Girl! I suppose we have to thank that fiend of invention who is responsible for most of the titular foibles and follies of mankind—artful Alliteration. The two G’s, people imagine, run so well together; and it is wonderful that they do not append some other delectable title, such as ‘The Gushing Girl of Girtham,’ or ‘The Glaring Girl of Glittering Girtham.’ O Alliteration! Alliteration! what crimes have been wrought in thy name! Little dost thou think of the mischief thou hast done, flooding the world with meaningless titles and absurd phrases. How canst thou talk of ‘Lyrics of Loneliness,’ ‘Soliloquies of Song,’ ‘Pearls of the Peerage’? Why dost thou stay thine hand? We long for thee to enrich the world with ‘Dreams of a Dotard,’ the ‘Dog Doctor’s Daughters,’ and other kindred works. Exercise thine art on these works of transcendent merit, but cease to style thy humble, but rebellious, servant a Girtham Girl! But what’s in a name? Let the world’s tongue wag. I am a student, a hard-working, book- devouring, never-wearied student, who burns her midnight oil, and drinks the strong bohea, to keep her awake during the long hours of toil, like any Oxford or Cambridge undergraduate. I often wonder whether these mighty warriors in the lists—the class lists, I mean—really work half so hard as we poor unfortunate ‘Girls of Girtham.’ Now that I am writing in strict confidence, so that not even the walls can hear the scratchings of my pen, or understand the meaning of all this scribbling, I beg to state that I have my serious doubts upon the subject; and when last I attended a soirée of the Anthropological Society, sounds issued forth from the windows of the snug college rooms, which could not be taken as evidences of profound and undisturbed study. Sometimes I glance at the examination papers set for these hard-working students, in order that they may attain the glorious degree of B.A., and astonish their sisters, cousins, and aunts by the display of these magic letters and all-resplendent hood. And again I say in strict confidence that if this same glorious hood does not adorn the back of each individual son of Alma Mater, he ought to be ashamed of himself, and not to fail to assume a certain less dignified, but expressive, three-lettered qualification. But before those Tripos Papers I bow my head in humble adoration. They sometimes take my breath away even to read the terrible excruciating things, which seem to turn one’s brain round and round, and contort the muscles of one’s face, and stop the pulsation of one’s heart, when one tries to grasp the horrid things. Here is a fair example of the ingenuity of the hard-hearted examiners, who resemble the inquisitors presiding over the tortures of the rack, and giving the hateful machine just one turn more by way of bestowing a parting benediction on their miserable victims: ‘A uniform rod’ (it is a marvellous act of mercy that the examiner invented it uniform; it is strange that its thickness did not vary in some complicated manner, and become a veritable birch-rod!) ‘of length 2c, rests in stable equilibrium’ (stable! another act of leniency!), ‘with its lower end at the vertex of a cycloid whose plane is vertical’ (why not incline it at an angle of 30°?) ‘and vertex downwards, and passes through a small, smooth, fixed ring situated in the axis at a distance b from the vertex. Show that if the equilibrium be slightly disturbed, the rod will perform small oscillations with its lower end on the arc of the cycloid in the time 4π√ a{c² + 3(b - c)²} , 3g(b² - 4ac) where 2a is the length of the axis of the cycloid.’ A sweet pretty problem, truly! And there are hundreds of the same kind—birch-rods for every back! How the examiner must have rejoiced when he invented this diabolical rod, with its equilibrium, its oscillations, its cycloid, and other tormenting accessories. And yet, I suppose, before my days of studentship are over, I shall be called upon to attack some such impregnable fortresses of mathematics, when I hope to be declared equal to some twentieth wrangler, if I escape the misfortune of sharing a portion of the ‘wooden spoon.’ 1 2 3 4 5 Ah, you male sycophants! You would prevent us from competing with you; you would separate yourselves on your island of knowledge, and sink the punt which would bear us over to your privileged shore. Of all the twaddle—forgive me, male sycophants!—that the world has ever heard, I think the greatest is that which you have talked about female education. And the best of it is, you are so anxious about our welfare; you are so afraid that we should injure our health by overmuch mental exertion; you profess to think that our brains are not calculated to stand the strain of continued mental exercise; you think that competition is not good for the female mind; that we are too competitive by nature—too ambitious! Yes, we are so ambitious that we would enter the lists with those who are asked in Public Examinations to find the simple interest on £1,000 for 5 years at 6¼ per cent.; so ambitious that we would compete with those who are requested to disclose the first aorist middle of τυπτω. Oh, think of the mental strain involved in such questions! How it must ruin your health to find out how many times a wheel of radius 6 feet will turn round between York and London, a distance of 200 miles! It is quite wonderful how your brains, my dear male sycophants, can stand such fearful demands upon your intelligence and industry! But you are so kind to us, so afraid of our health! Really, we are much obliged to you. If you married one of us, or became our guardian, or left us a legacy, we should then recognise your interest in us, and be very grateful to you for your good advice. But as matters stand, we are quite capable of taking care of ourselves. We will promise not to work too hard, if you will promise not to weary us with your paternal jurisdiction. But, male sycophants, I want a word with you. Why do you object to our taking degrees, or going in for examinations in order to qualify ourselves for our duties in life? You need not speak out loud if you would rather not. Are you not just a little afraid that we might eclipse you? And it is not pleasant to be beaten by a woman, is it? And then you profess to think that we ought to be all housewives and cooks, and knitters of stockings, and sewers-on of our husbands’ buttons; but what if we have no husbands, no buttons to sew? And is it not a little selfish, my dear male sycophant, to wish to keep us all to yourself? to attend upon the wants of the lords of creation, who often distinguish themselves so much in the domain of science? Now, look me straight in the face (no shirking, sir!). Is it not jealousy—green-eyed, false- tongued jealousy—which saps your generous instincts, and makes you talk rubbish and nonsense about strains, and brains, and ambition, and the like? And if that is not hypocritical, I do not know what is. Well, good-day to you, male sycophant! I really have not time to indulge myself in scolding you any more. You are a good creature, no doubt; and when you have shown us what you can do, and can estimate the capacity of the female brain, and take a common-sense view of things, we will recognise your privilege to speak; and when I am the presiding genius of Girtham College, I will grant you the use of our hall for the purpose of lecturing to us on ‘Women’s Rights,’ or, as you may prefer to entitle your discourse, ‘Men’s Wrongs.’ * * * * * Oh, this is shameful! I really am very sorry. Here have I been wasting a good half-hour in dreaming, and slaying an imaginary enemy with envenomed words and frequent dabs of ink. If I cannot concentrate my mind more on these mathematical researches, I fear a dreadful ‘plough’ will harrow my feelings at the end of my sojourn in these halls of learning. Concentration! How many of our words and ideas and thoughts are derived from that primal fount of all arts and sciences—mathematics! Here is one which owes its origin to the mathematically trained mind of some early philological professor, who had learnt to apply his scientific knowledge to the enrichment of his native tongue. He quoted to himself the words of the Roman poet: ‘Ego cur, acquirere pauca Si possum, invideor, cum lingua Catonis et Ennî Sermonem patrium ditaverit, et nova rerum Nomina protulerit? Licuit, semperque licebit.’ His mind conceived endless figures of circles and ellipses scattered promiscuously over the page, defying the attempts of the student to reduce them to order. What must he do before he can apply his formulæ and equations, determine their areas, or describe their eccentric motion? He must reduce them to a common centre, and then he can proceed to calculate the abstruse problems in connection with the figures described. They may be the complex motions of double-star orbits, or the results of the impact of various projectiles on the tranquil surface of a pool. It matters not—the principle is the same; he must concentrate, and reduce to a common centre. 6 7 8 9 10 This is the great defect of those who have no accurate mathematical knowledge; they cannot concentrate their minds with the same degree of intensity upon the work which lies before them. Their thoughts fly off at a tangent, as mine do very often; but then I have not been classed yet in the Tripos; and, O male poetical sycophant, you may be right after all when you say: ‘O woman! in our hours of ease Uncertain, coy and hard to please, As variable as the noon-day shade.’ Yes, as variable as the most variable quantities x, y, z. I, a student of Girtham College, blush to own that my thoughts very often fly off at a tangent. ‘Fly off at a tangent!’ All hail to thee, most noble mathematical phrase! Here is another fine mathematical expression, plainly exemplifying the action of centrifugal force. The faster the wheel turns, the greater is the velocity of the discarded particles which fly off along the line, perpendicular to the radius of the circle. The world travels very fast now; the increased velocity of the transit of earthly bodies, the rate at which they live, the multiplicity of engagements, etc., have made the social world revolve so fast that the speed would have startled the torpid life of the last century. And what is the result? Men’s thoughts fly off at a tangent; they are unable to concentrate their minds on any given subject; they are content with hasty generalisms, with short magazine articles on important subjects, which really require large volumes and patient study to elucidate them fully. What we want to do is to increase the attractive force, in order to prevent this tangential motion—to increase the force of gravity. ‘Well,’ says the young lady who loves to revel in the ‘Ghastly Secret of the Moated Dungeon,’ or the ‘Mysteries of Footlight Fancy,’ ‘you are grave enough. Pray don’t increase your gravity!’ Thank you, gentle critic. I will, in turn, ask you one favour. Leave for once the ‘Mysteries of Footlight Fancy;’ seek to know no more ‘ghastly secrets,’ and increase your gravity—your mental weight; and hence your attraction in the eyes of all who are worth attracting will be marvellously increased, by understanding a little about Newton’s law of universal gravitation, and don’t fly off at a tangent. At the end of this portion of the MS. the editor of these papers discovered a photograph which, from subsequent inquiry, proved to be that of the accomplished authoress of the above reflections. The face is one of considerable beauty, with eyes as clear, steadfast, and open as the day. There is a degree of firmness about the mouth, but it is a sweet and pretty one notwithstanding; and a smile, half scornful, half playful, can be detected lurking about the corners of the lips, which do not seem altogether fitted for pronouncing hard mathematical terms and abstruse scientific problems. This photograph might have been the identical one which nearly brought an enamoured youth into grave difficulties by its secretion in the folds of his blotting-paper during examination. The said enamoured youth had evidently placed it there for the sake of its inspiring qualities; and it was said that all his hopes of gaining the hand of the fair original depended upon his passing that same examination. But the wakeful eye of a stern examiner had watched him as he turned again and again to consult the sweet face which beamed from beneath his blotting-paper; and he narrowly escaped expulsion from the Senate-house on the charge of ‘cribbing.’ Certainly he took a mean advantage of his fellow- sufferers, if this were the identical photograph, for it portrays a most inspiring face. Forgive us, lenient reader; one moment! There—thank you—we have done. And now we will proceed to disclose the researches and original problems which the MS. contains. Evidently the collegiate authorities were not slow in recognising the talents of the assiduous student, and elected her without much delay to a Professorship of Girtham. In this capacity the learned lady delivered several lectures, of which the second MS. contains the first of the series. 11 12 13 14 PAPER II. LECTURE ON THE THEORY OF BRAIN WAVES AND THE TRANSMIGRATION AND POTENTIALITY OF MENTAL FORCES. Professors and Students of the University of Girtham, my Lords, Ladies, and Gentlemen,—I have the honour to bring before you this evening some original conceptions and discoveries which have been formulated by me during my researches in the boundless field of mathematical knowledge; and though you may be inclined at first to pronounce them as somewhat hastily conceived hypotheses, I hope to be able to demonstrate the actual truth of the propositions which I shall now endeavour to enunciate. It is with some feelings of diffidence that I stand before so august an assembly as the present; and if I were not actually convinced of the accuracy of my calculations, I should never have presumed to appear before you in the character of a lecturer. But ‘Magna est veritas, et prævalebit.’ I cast aside maiden timidity; I clothe myself in the professorial robe which you have bestowed upon me, and sacrifice my own feelings on the altar of Truth. I have been engaged, as you are doubtless aware, for some years in the pursuit of mathematical research, exploring the mines of science, which have of late been worked very persistently, but often, like the black diamond mines, at a loss. Concurrently with these researches, I have speculated on the great social problems which perplex the minds of men, both individually and collectively. And I have come to the conclusion that the same laws hold good in both spheres of work; that methods of mathematical procedure are applicable to the grand social problems of the day and to the regulation of the mutual relations which exist between man and man. Take, for example, the Force of public opinion. Of what is it composed? It is the Resultant of all the forces which act upon that which is generally designated the ‘Social System.’ Public opinion is a compromise between the many elements which make up human society; and compromise is a purely mechanical affair, based on the principle of the Parallelogram of Forces. Sometimes disturbing forces exert their influence upon the action of Public Opinion, causing the system to swerve from its original course, and precipitating society into a course of conduct inconsistent with its former behaviour; and it is the duty of the Governing Body to eliminate as far as possible such disturbing forces, in order that society may pursue the even tenor of its way. Professors, we have one great problem to solve; and all questions social, political, scientific, or otherwise, are only fragments of that great problem. All truths are but different aspects of different applications of one and the same truth; and although they may appear opposed, they are not really so; and resemble lines which run in various directions, but lovingly meet in one centre. Now, let us take for our consideration the secret influence which men exert upon each other, apart from that produced by the power of speech (although that would come under the same general law). As mathematicians, you are aware that the undulatory theory of light and heat and sound are now accepted by scientific men as the only sure basis of accurate calculation. We know that the rays of light travel in waves, and the equation representing the waves is y = a sin2π (vt - r), r λ where y is the disturbance of the ether, a the initial amplitude, r the distance from the starting- point, λ the wave-length, and v the velocity of light. Sound and heat likewise have much the same form of equation. Now, I maintain that the waves of thought are governed by the same laws, and can be determined by an equation of the same form. You are aware that in all these equations a certain quantity denoted by λ appears, and varies for the different media through which the sound, or light, or heat passes, and which must be determined by experiment Now, in my equation for brain waves, the same quantity λ appears which must be determined by the same method—by experiment. But how is this to be done? After mature deliberation and much careful thought, I have discovered the method for finding λ. This method is mesmerism. We find the ratio of brain to brain—the relative strength which one bears to another; and then by an application of our formula we can actually determine the wave of thought, and read the minds of our fellow-creatures. An unbounded field for reflection and speculation is here suggested. Like all great discoveries, the elements of the problem have unconsciously been utilized by many who are unable to account for their method of procedure. For example, thought-readers, mesmerists, and the like, have unconsciously been working on this principle, although lack of mathematical training has prevented them from fully mastering the details of the problem. Hence in popular minds a kind of mystery has hung about the actions of such people, and excited the curiosity of mankind. 15 16 17 18 19 20 The development of this theory of brain waves may be of great practical utility to the world. It shows that great care ought to be exercised in the domain of thought, as well as that of speech. For example: A man has made a startling discovery, from which he expects to receive considerable worldly advantage. He would be careful not to disclose his discovery in speech to his acquaintances until his plans are sufficiently matured, lest they should impart it to the world, patent his device, and reap the reward. But while he is endeavouring to talk carelessly about it, the wave of thought may be travelling from brain to brain, suggesting the existence of the discovery; and if the conditions are favourable, and λ sufficiently small, it is possible that the idea itself may be conveyed. Of course the more complicated the discovery, the less likely would the wave convey the conception. Or suppose that one of the learned professorial body of our sister university should conceive an attachment for a lady-student of Girtham College (of course a very improbable supposition!), and the infatuated savant became somewhat jealous of another learned lecturer of the same college (another improbability!), the fact of his jealousy would be imparted to the latter by a wave of thought, and might cause considerable confusion in the serene course of love or science. The fact of the existence of the wave is indisputable. What do all the stories of impressions and double- sight teach us? How could the intelligence of the death of Professor Steele have been conveyed to his friend and fellow-student, Professor Tait—the one at Cambridge, the other at Edinburgh—were it not for the existence of some wave, which, like that of electricity, wings its rapid flight unobserved by human eyes? Are all the records of the Psychical Society only myths and legends bred of superstitious fancy? It were hard to suppose so. But if, gentlemen, and ladies especially, you wish to keep your secret discoveries to yourselves, watch over your thoughts as well as your words; for my researches prove, and the universal experience of mankind corroborates the fact, that some portion of your inmost thoughts and secret desires are understood by your neighbours (especially when λ is small!); that they travel along the waves which I have attempted to indicate; and if you would desire to extend your influence in the world, probe the secret instincts of mankind, and prevent yourself from being deceived and wronged—study the art and science of Brain Waves. The following verses of rather doubtful merit were found in connection with the previous MS. They were evidently written by a different hand; but inasmuch as they were deemed worthy of preservation by the learned owner of the sealed desk, we venture to publish them. They are closely connected with the previous lecture, and were evidently composed by an admirer of the fair lecturer who did not share her love for scientific research. Wavelet,1 wing thy airy flight; Let thine amplitude be great; Tell her all my thoughts to-night, How I long to know my fate. All the fields of Mathematics I have roamed at her decree; From Binomial and Quadratics, To the strange hyperbole.2 I have soared through Differential, Deeply drunk of Finite Boole;3 Though its breath is pestilential, Reeking of the hateful School. I have tried to shape a Conic, Vainly read the Calculus; But my feebleness is chronic, Morbus Mathematicus. All my curves are cardioidal; I confuse my x and ys, Which they say is suicidal; And my tutor vainly sighs. Wavelet, tell her how I love her, As she mounts her learned throne; And that love I hope may cover All the failings which I own. Wavelet, cry to her for pity; Bid her end this bitter woe; I might do something ‘in the city,’ But never pass my Little-go. 21 22 23 24 1 We presume this is addressed to an imaginary brain wave. 2 We observe here the dash of an indignant pen, and a substituted for e. But now the rhyme is spoiled. Gentle Muse, thou art sacrificed by the stern hand of Mathematical Truth! 3 Query: Does the writer refer to the learned treatise on Finite Differences by Professor Boole? PAPER III. 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 25 26 27 28 29 30 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 31 32 33 34 35 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 æsthetic 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. 36 37 38 PAPER IV. 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 versâ. 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 39 40 41 42 43 44 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...