The Literary World (November 14, 1884), 389--390.

Whoever the author of this remarkable mathematical allegory may be, his cleverly elaborated fancy is not only likely to create a present sensation in the thinking world, but also to find an abiding place in the classic domains of the great satires of history. The subject is too abstruse, and involves too many metaphysical speculations expressed in geometrical terms, to appeal to the multitude with the fascination exerted by the immortal allegories of Bunyan and Swift, whose simplicity is their strength; and it may, therefore, share the fate of the Fairy Queen and Rabelais' Romance in being reserved for the delectation of the favoured few. Its irony, though severe, is delicate, and its interpretation not always easy. It is cast in the historical and descriptive, rather than the dramatic form. These facts may limit the number of those capable of appreciating it, as may a certain likeness in it here and there to the precise and formal lessons demonstrated on a blackboard to a class of schoolboys. But older scholars and fellow-teachers will admire the perspicuity and skill with which these difficult and original lessons are taught. In working out the details of his humour the writer has aimed his shafts sometimes at the world of society, sometimes at that of politics, sometimes at that of religion. His references are now plain and palpable, now recondite and obscure. But about the broad drift of his parable there can be no mistake whatever. His allegory is in the chiefest of its aspects a magnificent protest against self-sufficiency and dogmatism; against cherishing the idea that we have, in reference to any matter of experience whatever, seen the end of all perfection; against all narrowness, bigotry, and intolerance in any region of supposed knowledge, whether that of scientific self-assurance on the one hand, or religious fanaticism on the other. In emphasizing this doctrine the writer has availed himself of the idea, mooted some years ago by a great mathematician, that there might be states of existence blessed with more dimensions than those with which we ourselves are acquainted. Our experience has only told us of three-length, breadth, and height. But why should there not be more? And in a world I which there are more, what vaster revelations of ourselves, our present life, and our surroundings might be possible than those which we at present possess! Now, no flight of fancy will enable us to jump into a world of four dimensions; but we can imagine the conditions of existence in a world, which has only two, or less than two; and this is what the author has done. We cannot add an additional faculty to our beings; but we can guess what it would be like to be deprived of some we already enjoy. We know not in what direction to grope for a sixth sense; but we can tell the loss we should sustain in being deprived of one of those,-say sight, -hitherto belonging to us. Flatland is the realm of the superficial, on or in the surface of which creatures animate and inanimate exist, without either rising above or sinking below it. They have length and breadth, but neither height nor depth. In such a state, there can be no solids but every thing and every being is a plane figure. Society there is divided into the well-known shapes that appear in the pages of Euclid. The word spoken in jest by the wit, who retorted on an enraged and abusive fish-woman with the remark that she was an isosceles triangle, would have been taken solemnly and understood literally in Flatland. The narrator of the story is a Square. His wife is a Straight Line. His father is a Triangle, his sons are Pentagons, and his grandson a Hexagon. Social distinctions in Flatland are determined by these shapes. Many sidedness is characteristic of the highest class, and angularity-in the sense of extreme acuteness of the vertex-of the lowest. At the aristocratic end of society are Polygons, so many cornered as to approach circles; at the plebeian extremity Isosceles Triangles, so narrow as to approach straight lines. The necessary conditions of his problem seem to us to have been ably anticipated by the author. The chemical wizardry by which Jules Verne makes is seem practicable to travel across Africa in a balloon pales before the mathematical and optical enchantments of this literary conjurer. In his Flatland, of course, there are no shadows, as with us in Spaceland; nor are there sun or stars visible to the inhabitants. Light is diffused from an unknown source, and so fierce have been the speculations as to its origin, that legal enactments have been passed to restrain such inquiries. But without sun, star, or shadow how could the people in Flatland tell their position; or how cold they distinguish one another? Although the writer solves these problems one after another with great ability, he yet states them so formally that we never can get rid altogether of the idea that we are going through an educational course rather than reading a story. For the answers to many of them we must refer to the volume itself. But here is an instance of

Recognition in Flatland

Feeling is, among our Women and lower classes --- about our upper classes I shall speak presently --- the principal test of recognition, at all events between strangers, and when the question is, not as to the individual, but as to the class. What therefore ``introduction'' is among the higher classes in Spaceland, that the process of ``feeling'' is with us. ``Permit me to ask you to feel and be felt by my friend Mr. So-and-so'' --- is still, among the more old-fashioned of our country gentlemen in districts remote from towns, the customary formula for a Flatland introduction. But in the towns, and among men of business, the words ``be felt by'' are omitted and the sentence is abbreviated to, ``Let me ask you to feel Mr. So-and-so''; although it is assumed, of course, that the ``feeling'' is to be reciprocal. Among our still more modern and dashing young gentlemen --- who are extremely averse to superfluous effort and supremely indifferent to the purity of their native language --- the formula is still further curtailed by the use of ``to feel'' in a technical sense, meaning, ``to recommend-for-the-purposes-of-feeling-and-being-felt''; and at this moment the ``slang'' of polite or fast society in the upper classes sanctions such a barbarism as ``Mr. Smith, permit me to feel Mr. Jones.''

Let not my Reader however suppose that ``feeling'' is with us the tedious process that it would be with you, or that we find it necessary to feel right round all the sides of every individual before we determine the class to which he belongs. Long practice and training, begun in the schools and continued in the experience of daily life, enable us to discriminate at once by the sense of touch, between the angles of an equal-sided Triangle, Square, and Pentagon; and I need not say that the brainless vertex of an acute angled Isosceles is obvious to the dullest touch. It is therefore not necessary, as a rule, to do more than feel a single angle of an individual; and this, once ascertained, tells us the class of the person whom we are addressing, unless indeed he belongs to the higher sections of the nobility. There the difficulty is much greater. Even a Master of Arts in our University of Wentbridge has been known to confuse a ten-sided with a twelve-sided Polygon; and there is hardly a Doctor of Science in or out of that famous University who could pretend to decide promptly and unhesitatingly between a twenty-sided and a twenty-four sided member of the Aristocracy.

Two matters claim our especial attention in this division of the allegory; the one is the condition of the women of Flatland, and the other the relation between the governing classes and the lower orders. The women, as being entirely destitute of brain power, which is always situated in one of the angles of the figure, have no angles at all, and are simply straight lines. But they are extremely dangerous lines, 'being, so to speak, all point, at least, at the two extremities', and capable of running into, and then wounding fatally any male creature that comes in their way. The statesmen of Flatland have had much work to cope with this perpetual source of danger; for the women are not only lethally pointed, but in such a shadowless realm they are practically invisible as well. Among other precautions, therefore, to prevent accidental injuries being inflicted by women in public, it has been ordered in some States that when in public they should always move their backs from right to left, 'so as to indicate their presence to those behind them'.

Fashion in Flatland.

The power of Fashion is also on our side. I pointed out that in some less civilized States no female is suffered to stand in any public place without swaying her back from right to left. This practice has been universal among ladies of any pretensions to breeding in all well-governed States, as far back as the memory of Figures can reach. It is considered a disgrace to any State that legislation should have to enforce what ought to be, and is in every respectable female, a natural instinct. The rhythmical and, if I may so say, well- modulated undulation of the back in our ladies of Circular rank is envied and imitated by the wife of a common Equilateral, who can achieve nothing beyond a mere monotonous swing, like the ticking of a pendulum; and the regular tick of the Equilateral is no less admired and copied by the wife of the progressive and aspiring Isosceles, in the females of whose family no ``back motion'' of any kind has become as yet a necessity of life. Hence, in every family of position and consideration, ``back motion'' is as prevalent as time itself; and the husbands and sons in these households enjoy immunity at least from invisible attacks.

Next in point of danger to the women are the lower orders, whose vertices are so extremely sharp that they can easily slaughter the less offensive Polygons; and clever, indeed, are the devices which the latter have adopted for keeping them down. On which side the author's political sympathies turn it is not easy to say. On the whole, however, he appears to have aristocratic leanings, and to approve of rough and ready methods of suppressing the base and criminal classes; though alive to, and with some contempt for, the cunning with which the 'upper circles' have contrived to hold their own by setting class against class, and doling our favours in quantity just sufficient to keep down rebellion.

But it is when we come to the second half of the allegory, wherein the dweller in Flatland is introduced to Spaceland, that we encounter the most powerful lesson of the littleness of human knowledge. The incredulity of ignorance and the unreceptive attitude of the human mind towards new truth are painted to perfection. The narrator is unable to believe in a land of three dimensions, although he has been amazed at the folly of the monarch of Lineland, who, knowing nothing of any state when there is more than one dimension, attempts to put to death the visitor from Flatland, who tells him of a place where there are two. But even this denizen of outer darkness is quite an enlightened potentate compared with the King of Pointland --- the abyss of no dimensions --- who is his own universe.

The It of Existence

He ceased; and there arose from the little buzzing creature a tiny, low, monotonous, but distinct tinkling, as from one of your Spaceland phonographs, from which I caught these words, ``Infinite beatitude of existence! It is; and there is none else beside It.''

``What,'' said I, ``does the puny creature mean by it'?'' ``He means himself,'' said the Sphere: ``have you not noticed before now, that babies and babyish people who cannot distinguish themselves from the world, speak of themselves in the Third Person? But hush!''

``It fills all Space,'' continued the little soliloquizing Creature, ``and what It fills, It is. What It thinks, that It utters; and what It utters, that It hears; and It itself is Thinker, Utterer, Hearer, Thought, Word, Audition; it is the One, and yet the All in All. Ah, the happiness ah, the happiness of Being!''

``Can you not startle the little thing out of its complacency?'' said I. ``Tell it what it really is, as you told me; reveal to it the narrow limitations of Pointland, and lead it up to something higher.'' ``That is no easy task,'' said my Master; ``try you.''

Hereon, raising my voice to the uttermost, I addressed the Point as follows:

``Silence, silence, contemptible Creature. You call yourself the All in All, but you are the Nothing: your so-called Universe is a mere speck in a Line, and a Line is a mere shadow as compared with --- '' ``Hush, hush, you have said enough,'' interrupted the Sphere, ``now listen, and mark the effect of your harangue on the King of Pointland.''

The lustre of the Monarch, who beamed more brightly than ever upon hearing my words, shewed clearly that he retained his complacency; and I had hardly ceased when he took up his strain again. ``Ah, the joy, ah, the joy of Thought! What can It not achieve by thinking! Its own Thought coming to Itself, suggestive of Its disparagement, thereby to enhance Its happiness! Sweet rebellion stirred up to result in triumph! Ah, the divine creative power of the All in One! Ah, the joy, the joy of Being!''

``You see,'' said my Teacher, ``how little your words have done. So far as the Monarch understands them at all, he accepts them as his own --- for he cannot conceive of any other except himself --- and plumes himself upon the variety of Its Thought' as an instance of creative Power. Let us leave this God of Pointland to the ignorant fruition of his omnipresence and omniscience: nothing that you or I can do can rescue him from his self-satisfaction.''

But this creature of Pointland is only the beginning of the scale of those who will not open their eyes to the light. As he of Pointland cannot bring himself to believe in the possibilities of Lineland; nor he of Lineland in those of Flatland; nor he of Flatland in those of Spaceland; so we of the latter region are prone to think that our own horizon is the limit of existence. But this story of surfaces and squares and cubes may well serve to shake out of their conceited complacency the whole race of dogmatists, whether they belong to the schools of philosophy, science, or religion.