The (almost really) Complete Works of Lewis Carroll

Introduction.

‘It was a lovely Autumn evening, and the glorious effects of chromatic aberration were beginning to show themselves in the atmosphere as the earth revolved away from the great western luminary, when two lines might have been observed wending their weary way across a plane superficies. The elder of the two had by long practice acquired the art, so painful to young and impulsive loci, of lying evenly between his extreme points; but the younger, in her girlish impetuosity, was ever longing to diverge and become an hyperbola or some such romantic and boundless curve. They had lived and loved: fate and the intervening superficies had hitherto kept them asunder, but this was no longer to be: a line had intersected them, making the two interior angles together less than two right angles. It was a moment never to be forgotten, and, as they journeyed on, a whisper thrilled along the superficies in isochronous waves of sound, “Yes! We shall at length meet if continually produced!”’ (Jacobi’s Course of Mathematics, Chap. 1.)

We have commenced with the above quotation as a striking illustration of the advantage of introducing the human element into the hitherto barren region of Mathematics. Who shall say what germs of romance, hitherto unobserved, may not underlie the subject? Who can tell whether the parallelogram, which in our ignorance we have defined and drawn, and the whole of whose properties we profess to know, may not be all the while panting for exterior angles, sympathetic with the interior, or sullenly repining at the fact that it cannot be inscribed in a circle? What mathematician has ever pondered over an hyperbola, mangling the unfortunate curve with lines of intersection here and there, in his efforts to prove some property that perhaps after all is a mere calumny, who has not fancied at last that the ill-used locus was spreading out its asymptotes as a silent rebuke, or winking one focus at him in contemptuous pity?

In some such spirit as this we have compiled the following pages. Crude and hasty as they are, they yet exhibit some of the phenomena of light, or ‘enlightenment,’ considered as a force, more fully than has hitherto been attempted by other writers.

Chapter I. General Considerations.

Definitions

I

Plain Superficiality is the character of a speech, in which any two points being taken, the speaker is found to lie wholly with regard to those two points.

II

Plain Anger is the inclination of two voters to one another, who meet together, but whose views are not in the same direction.

III

When a Proctor, meeting another Proctor, makes the votes on one side equal to those on the other, the feeling entertained by each side is called Right Anger.

IV

When two parties, coming together, feel a Right Anger, each is said to be complementary to the other, (though, strictly speaking, this is very seldom the case).

V

Obtuse Anger is that which is greater than Right Anger.

Postulates

I

Let it be granted, that a speaker may digress from any one point to any other point.

II

That a finite argument, (i. e. one finished and disposed of,) may be produced to any extent in subsequent debates.

III

That a controversy may be raised about any question, and at any distance from that question.

Axioms

I

Men who go halves in the same (quart) are (generally) equal to another.

II

Men who take a double in the same (term) are equal to anything.

On Voting

The different methods of voting are as follows:

I

Alternando, as in the case of Mr. … who voted for and against Mr. Gladstone, alternate elections.

II

Invertendo, as was done by Mr. … who came all the way from Edinburgh to vote, handed in a blank voting-paper, and so went home rejoicing.

III

Componendo, as was done by Mr. … whose name appeared on both committees at once, whereby he got great praise from all men, by the space of one day.

IV

Dividendo, as in Mr. …’s case, who being sorely perplexed in his choice of candidates, voted for neither.

V

Convertendo, as was wonderfully exemplified by Messrs. … and … who held a long and fierce argument on the election, in which, at the end of two hours, each had vanquished and converted the other.

VI

Ex æquali in proportione perturbatå seu inordinatå, as in the election, when the result was for a long time equalised, and as it were held in the balance, by reason of those who had first voted on the one side seeking to pair off with those who had last arrived on the other side, and those who were last to vote on the one side being kept out by those who had first arrived on the other side, whereby, the entry to the Convocation House being blocked up, men could pass neither in nor out.

On Representation

Magnitudes are algebraically represented by letters, men by men of letters, and so on. The following are the principal systems of representation.

1. Cartesian: i. e. by means of ‘cartes.’ This system represents lines well, sometimes too well; but fails in representing points, particularly good points.

2. Polar: i. e. by means of the 2 poles, ‘North and South.’ This is a very uncertain system of representation, and one that cannot safely be depended upon.

3. Trilinear: i. e. by means of a line which takes 3 different courses. Such a line is usually expressed by three letters, as W.E.G.

That the principle of Representation was known to the ancients is abundantly exemplified by Thucydides, who tells us that the favourite cry of encouragement during a trireme race was that touching allusion to Polar Co-ordinates which is still heard during the races of our own time, ‘${\rho }_5$, ${\rho }_6$, $cos\varphi$, they’re gaining!’

Chapter II. Dynamics of a Particle.

Particles are logically divided according to Genius and Speeches.

Genius is the higher classification, and this, combined with Differentia, (i. e. difference of opinion,) produces Speeches. These again naturally divide themselves into three heads.

Particles belonging to the great order of Genius are called ‘able’ or ‘enlightened.’

Definitions

I

A Surd is a radical whose meaning cannot be exactly ascertained. This class comprises a very large number of particles.

II

Index indicates the degree, or power, to which a particle is raised. It consists of two letters, placed to the right of the symbol representing the particle. Thus, ‘A.A.’ signifies the 0th degree; ‘B.A.’ the 1st degree; and so on, till we reach ‘M.A.’ the 2nd degree (the intermediate letters indicating fractions of a degree); the last two usually employed being ‘R.A.’ (the reader need hardly be reminded of that beautiful line in The Princess ‘Go dress yourself, Dinah, like a gorgeous R.A.’) and ‘S.A.’ This last indicates the 360th degree, and denotes that the particle in question, (which is $\frac{1}{7}$th part of the function $\overline{E+R}$ ‘Essays and Reviews,’) has effected a complete revolution, and that the result = 0.

III

Moment is the product of the mass into the velocity. To discuss this subject fully, would lead us too far into the subject Vis Viva, and we must content ourselves with mentioning the fact that no moment is ever really lost, by fully enlightened Particles. It is scarcely necessary to quote the well-known passage:—‘Every moment, that can be snatched from academical duties, is devoted to furthering the cause of the popular Chancellor of the Exchequer.’—(Clarendon, History of the Great Rebellion.)

IV

A Couple consists of a moving particle, raised to the degree M.A., and combined with what is technically called a ‘better half.’ The following are the principal characteristics of a Couple: (1) It may be easily transferred from point to point. (2) Whatever force of translation was possessed by the uncombined particle, (and this is often considerable,) is wholly lost when the Couple is formed. (3) The two forces constituting the Couple habitually act in opposite directions.

On Differentiation

The effect of Differentiation on a Particle is very remarkable, the first Differential being frequently of a greater value than the original Particle, and the second of less enlightenment.

For example, let L = ‘Leader,’ S = ‘Saturday,’ and then L.S. = ‘Leader in the Saturday,’ (a particle of no assignable value). Differentiating once, we get L.S.D., a function of great value. Similarly it will be found that, by taking the second Differential of an enlightened Particle, (i. e. raising it to the Degree D.D.,) the enlightenment becomes rapidly less. The effect is much increased by the addition of a C: in this case the enlightenment often vanishes altogether, and the Particle becomes conservative.

It should be observed that, whenever the symbol L is used to denote ‘Leader,’ it must be affected with the sign ±: this serves to indicate that its action is sometimes positive and sometimes negative—some particles of this class having the property of drawing others after them, (as ‘a Leader of an army,’) and others of repelling them, (as ‘a Leader of the Times.’)

Propositions

Prop. I. Pr. To find the value of a given Examiner

Example. A takes in 10 books in the Final Examination, and gets a 3d Class: B takes in the Examiners, and gets a 2nd. Find the value of the Examiners in terms of books. Find also their value in terms in which no Examination is held.

Prop. II. Pr. To estimate Profit and Loss

Example. Given a Derby Prophet, who has sent 3 different winners to 3 different betting-men, and given that none of the three horses are placed. Find the total Loss incurred by the three men (α) in money, (β) in temper. Find also the Prophet. Is this latter generally possible?

Prop. III. Pr. To estimate the direction of a line.

Example. Prove that the definition of a line, according to Walton, coincides with that of Salmon, only that they begin at opposite ends. If such a line be divided by Frost’s method, find its value according to Price.

Prop. IV. Th.

The end, (i. e. ‘the product of the extremes,’) justifies (i. e. ‘is equal to’ see Latin ‘æquus,’) the means.

No example is appended to this Proposition, for obvious reasons.

Prop. V. Pr. To continue a given series

Example. A and B, who are respectively addicted to Fours and Fives, occupy the same set of rooms, which is always at Sixes and Sevens. Find the probable amount of reading done by A and B while the Eights are on.

We proceed to illustrate this hasty sketch of the Dynamics of a Parti-cle, by demonstrating the great Proposition on which the whole theory of Representation depends, namely:—“To remove a given Tangent from a given Circle, and to bring another given Line into contact with it.”

To work the following problem algebraically, it is best to let the circle be represented as referred to its two tangents, i. e. first to WEG, WH, and afterwards to WH, GH. When this is effected, it will be found most convenient to project WEG to infinity. The process is not given here in full, since it requires the introduction of many complicated determinants.

Prop. VI. Pr.

To remove a given Tangent from a given Circle, and to bring another given Line into contact with it.

Let UNIV be a Large Circle, whose centre is O, (V being, of course, placed at the top,) and let WGH be a triangle, two of whose sides, WEG and WH, are in contact with the circle, while GH (called ‘the base’ by liberal mathematicians,) is not in contact with it. (See Fig. 1.) It is required to destroy the contact of WEG, and to bring GH into contact instead.

Let I be the point of maximum illumination of the circle, and therefore E the point of maximum enlightenment of the triangle. (E of course varying perversely as the square of the distance from O.)

Let WH be fixed absolutely, and remain always in contact with the circle, and let the direction of OI be also fixed.

Now, so long as WEG preserves a perfectly straight course, GH cannot possibly come into contact with the circle, but if the force of illumination, acting along OI, cause it to bend (as in Fig. 2), a partial revolution on the part of WEG and GH is effected, WEG ceases to touch the circle, and GH is immediately brought into contact with it. Q.E.F.

The theory involved in the foregoing Proposition is at present much controverted, and its supporters are called upon to show what is the fixed point, or ‘locus standi,’ on which they propose to effect the necessary revolution. To make this clear, we must go to the original Greek, and remind our readers that the true point or ‘locus standi,’ is in this case ἄρδις, (or ἅρδις according to modern usage,) and therefore must not be assigned to WEG. In reply to this it is urged that, in a matter like the present, a single word cannot be considered a satisfactory explanation, such as ἁρδέως.

It should also be observed that the revolution here discussed is entirely the effect of enlightenment, since particles, when illuminated to such an extent as actually to become φώς, are always found to diverge more or less widely from each other; though undoubtedly the radical force of the word is ‘union’ or ‘friendly feeling.’ The reader will find in ‘Liddell and Scott’ a remarkable illustration of this, from which it appears to be an essential condition that the feeling should be entertained φοράδεν, and that the particle entertaining it should belong to the genus σκότος, and should therefore be, nominally at least, unenlightened.