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Flatland’s critique of analogy is reminiscent of Thomas Reid, writing in 1767. Reid noted both the utility and frequency of analogical thinking, and the way in which it was particularly common in figuring thought itself as a material parallel to make clear the abstract: ‘The second, and the most common way in which men form their opinions concerning the mind and its operations we may call the way of analogy. There is nothing in the course of nature so singular, but we can find some resemblance, or at least some analogy, between it and other things with which we are acquainted. The mind naturally delights in hunting after such analogies, and attends to them with pleasure.’ Thomas Reid, An Inquiry into the Human Mind on the Principles of Common Sense, ed. by Derek R. Brookes (Edinburgh: Edinburgh University Press, 1997), p. 203.

Reid argued that philosophers were not immune from such a tendency and that until Descartes philosophy was liable to ‘materialize the mind and its faculties.’  (209) He doubted that many were capable of the rigorous reasoning required to reach concepts by other means: ‘If one attentively examines the systems of the ancient philosophers, either concerning the material world or concerning the mind, he will find them to be built solely upon the foundation of analogy.’ (204)

Intriguingly, in the same text from which these lines are taken Reid set out a thought experiment in which a race of two-dimensional beings he called Idomenians, confined to the surface of a sphere and having only the sense of sight, were unable to conceive of a three-dimensional geometry.

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I’ve just cut the below from my Flatland chapter because it doesn’t fit with where it’s going any more. There is plenty more to be said on Flatland’s evolutionary concerns beyond this narrow focus on Galton, but I really enjoyed writing this and watching the liberal theologian give the eugenicist a proper kicking. This also fleshes out a remark in the previous post.

One focused target of Flatland’s satire can be drawn out from the first part of the text. As a theologian, Abbott was likely to have become aware of Francis Galton during the ‘prayer-gauge debate’ of 1872 which Galton joined in the pages of the Contemporary Review: this debate over attempts to scientifically measure the effectiveness of prayer was, after all, the ‘sensation of the season’.[1]  As an educator, Abbott could scarcely have failed to have followed the prolific Galton’s pronouncements on nature over nurture throughout the 1870s when the statistician’s research privileged the hereditary transmission of mental and moral characteristics.

Galton was then, as now, most associated with the theory he would neologise in Inquiries into the Human Faculties (1883) as eugenics, ‘the science of improving stock’.[2] Rosemary Jann has noted ‘the voice of the eugenicist’ in Flatland and others have described the context for Flatland’s particular version of geometric evolution.[3] Listening closely for this voice and recording its utterances not only beds Flatland into contemporary social and intellectual concerns but also points to a direct identification of Galton.

A Square describes a society that is an evolutionary hierarchy. As he explains: ‘It is a Law of Nature with us that a male child shall have one more side than his father, so that each generation shall rise (as a rule) one step in the scale of development and nobility.’ (F, 7) At the lower echelons are female Flatlanders, straight lines, figures more one-dimensional than two-, with no interior angles to measure. The lower and middle classes are triangles: sharp isosceles are workmen and soldiers, the middle-class are equilaterals. The professional and gentlemanly classes are squares, such as the narrator, and pentagons. The nobility begin with hexagons and ascend through all polygons. At the very apex are the priestly class, circles, or at the very least figures with so many sides that they approximate circles.

A professional such as A Square feels pity and contempt for the ‘degraded condition’ (F, 8) of the Isosceles class who ‘can hardly be said to deserve the name of human figures, since they have not all their sides equal’ (F, 8). Fortunately for the Isosceles, a Lamarckian hereditary transmission of acquired characteristics means that focused self-improvement and careful selection of breeding partners combine to give a gradual increase in internal angles over the generations.

Lower even than Isosceles are Irregular figures who display no equality of sides. A Square informs us that ‘”Irregularity of Figure” means with us the same as, or more than, a combination of moral obliquity and criminality with you, and is treated accordingly’ (F, 24). This elision of moral and physical characteristics chimes directly with Galton’s study in Inquiries, in which he writes that ‘the innate moral and intellectual faculties are so closely bound up with the physical ones that these must be considered as well’ (IHF, 3). In Flatland, indeed, interior angle correlates directly with intellectual capacity: ‘the family brain was registered at only 58°.’ (F, 16)

Rosemary Jann has located the timbre of Galton’s arguments in A Square’s observation on ‘the extraordinary fecundity of the Criminal and Vagabond classes.’[4] In Inquiries Galton dealt with criminals and the insane in a brief chapter in which he gave his support to this popularly held Malthusian idea: ‘the criminal population […] is well-suited to flourish under half-savage conditions, being naturally both healthy and prolific’ (IHF, 43). It should be noted, though, that Galton diverged from Malthus’s conclusion that prudent men would check their fertility, arguing that the lower classes could not be relied upon to practice prudence.

Other eugenic motifs of Flatland appear remarkably prescient. The Sanitary and Social Board, responsible for certificating equilaterals, draws on the mid-century concern of social reformers with public hygiene and demographics to anticipate the concept of racial hygiene, coined by Alfred Ploetz in 1905, and taken up in eugenic discourse of the early twentieth century. The notion of eugenic certification itself anticipates with unerring accuracy the future trajectory of Galton’s thought: his unpublished utopian novel Kantsaywhere, discovered by Karl Pearson in his papers after his death, envisaged an even more advanced eugenic certification system in which those failing to achieve grading were segregated and prevented from reproducing.[5]

Most chilling are the stentorian tones of the eugenic principle in Flatland policy. Irregulars are frequently destroyed and ‘the diminution of the redundant Isosceles population [is] an object that every statesman in Flatland constantly keeps in view.’ (F, 17) For a reader familiar with Galton’s biography more personal attacks might have been discerned in Flatland’s text. Certainly, had Frances Galton been a Flatlander, his lot would have been unhappy. Galton had failed to gain a degree from Cambridge, having suffered a breakdown in the run-up to his exams. In Flatland

the condition of the unsuccessful minority is truly pitiable. Rejected from the higher class, they are also despised by the lower. They have neither the matured and systematically trained powers of the Polygonal Bachelors and Masters of Arts, nor yet the native precocity and mercurial versatility of the youthful Tradesman. The professions, the public services, are closed against them; and though in most States they are not actually debarred from marriage, yet they have the greatest difficulty in forming suitable alliances, as experience shews that the offspring of such unfortunate and ill-endowed parents is generally itself unfortunate, if not positively Irregular. It is from these specimens of the refuse of our Nobility that the great Tumults and Seditions of past ages have generally derived their leaders; and so great is the mischief thence arising that an increasing minority of our more progressive Statesmen are of opinion that true mercy would dictate their entire suppression, by enacting that all who fail to pass the Final Examination of the University should be either imprisoned for life, or extinguished by a painless death. (F, 22)


[1] The Prayer-Gauge Debate, ed. by John O’Means (Boston: Congregational Publishing Society, 1876), p. 3.

[2] Inquiries, 17

[3] Rosemary Jann, ‘Introduction’ in Flatland: A Romance of Many Dimensions (Oxford: Oxford University Press, 2006), p. xvii.

[4] Jann, ‘Introduction’, p. xvii.

[5] See Karl Pearson, Life, Letters and Labours of Francis Galton (1930), vol IIIA, pp. 414-424.

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I’m going back into Flatland so I’m justifying procrastination by looking through Flatland adaptations. I’ve been trying to get my mitts on the below version for some time featuring, as it does, the inimitable Dudley Moore as A Square, an unsurpassable piece of casting. The animation was by John Hubley, who was blacklisted for refusing to name names to the House Committee on Un-American Activities and is noticeably influenced by the geometric Modernist stylings of Russian animation. It is available on DVD from Documentary Educational Resources, but they charge an absurd $65, which for an 11-minute film seems a bit steep. I guess the Greek youtuber below felt the same, and shared a copy in the interests of unhindered education.

I post this first because it is charming and seems closer to the spirit of the novel than the 2007 film, Flatland: The Movie, which had a significantly larger budget: significantly large enough to cast Martin Sheen as A Square and Michael York as the Sphere. It’s an interesting proposition. The animation is whizzy and while the kind of scriptwriting that has a character declaring “Dude, you’re freaking me out!” makes me instinctively uncomfortable – for a start, it’ll age pretty badly – I wonder if its good intentions and intended audience don’t actually mirror those of Abbott. When it came out in 2007 I thought they’d missed the opportunity to do it in 3D, which would have made so much sense and brought alive the Spaceland sections, so I’m glad they’re remastering it for 3D Imax, no less. Dude! You really are freaking me out!

Despite claims from the producer that selling it directly made more money than he had made on other projects, I guess they ran out of budget for the website. Thomas Banchoff was involved as a consultant, so the geometry and Abbott scholarship are rock solid.

This production must have been enormously bad news for the producers of Flatland: The Film, which also came out in 2007, without the big names. I’m afraid I’m not able to feel too much pity for them, though, because it looks as if they’ve played rather loose with plot and while the animation shows a debt to Abbott’s original illustrations, I’m not sure there’s a lot to be gained from making Flatland a war movie (and the soundtrack?)

Flatland has clearly found a contemporary niche as an educational resource, a way of instructing dimensionality. This is interesting, and certainly part of the story, but as a cultural historian I would hate to leave it to the maths classroom, as valuable as it may be there. Flatland’s complexity and responsive immersion in late-Victorian intellectual life seem reduced by these later versions. Given the prominence of ideas related to Malthus, Lamarck and Galton it might just as well be a resource for teaching evolutionary narratives. Biologists! To the DVD player!

It is also worth observing that narrative is precisely its strength today: the fact that the geometry is embedded in a story is what makes it so attractive to teachers. I think this, too, leads is into more interesting avenues of thought, particularly relating to how and where Flatland fits into literary history. An intriguing essay by Mark McGurl observes that the critically commonplace descriptions of characters as flat and round, coined by E.M. Forster in Aspects of the Novel (1927) , surely owes to Flatland. (I’m uncertain about other claims McGurl makes, regarding Flatland’s position on class, but this is a great point). Flatland , I think, actively participates in and is aware of its place in the literary theory of the 1880s.

As I have argued before on this blog, higher space is also play, so it is appropriate that Flatland has inspired a number of recent computer game projects, which I’ll round up in my next post.

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Literary historical discussions of Flatland have frequently toyed with the relationship between its author Edwin Abbott Abbott and Charles Howard Hinton. There are a handful of highly suggestive connections. 1) The pair were mutually aware. Hinton praised Abbott but stressed the difference between the two in the introduction to his third romance, A Plane World, first published in the summer of 1886:

And I should have wished to be able to refer the reader altogether to that ingenious work, “Flatland.” But on turning over its pages again I find that the author has used his rare talent for a purpose foreign to the intent of our work. For evidently the physical conditions of life on the plane have not been his main object. He has used them as a setting wherein to place his satire and his lessons. (SR, 129)

Hinton was undercooking the debt slightly: ‘A Plane World’ may have had different intentions but its triangular characters and title didn’t really obscure the inspiration for his working in this way with this material. Abbott returned the acknowledgement in The Kernel and the Husk, a collection of theological essays published in 1887:

You know – or might know if you would read a little book recently published called Flatland, and still better, if you would study a very able and original work by Mr C. H. Hinton – that a being of Four Dimensions, if such there were, could come into our closed rooms without opening door or window, nay, could penetrate into, and inhabit, our bodies.[1]

A degree of social contact between the two writers has been noted. Specifically, Hinton’s colleague at Uppingham, Howard Candler, was a close friend of Abbott and, indeed, the dedicatee of Flatland. More tenuously, Hinton’s previous employer at Cheltenham Ladies College, the headmistress Dorothea Buss, had professional contact with Abbott.[2]

The Candler connection seems pretty suggestive to me. It’s not a reach to imagine old friends, both senior educators, gossiping about a new man at the school at which one of them teaches, especially if said new man is the son of a well-known and controversial man-of-letters and gave vent to his slightly unconventional views on space in the classroom.

Hinton’s On the Education of the Imagination, issued as a pamphlet in 1888 dealt with Hinton’s system of cubes and their use in the classroom. Its endnote by editor Herman John Falk stated that it was written ‘some years ago’ and ‘contains the germ of the work, which is more fully illustrated in his more recent writings, and thus in some respects forms a good introduction to them’.[3] A pedagogical essay, addressed to a fellow educator and referring throughout to a putative pupil, it established its theoretical basis in the work of Johannes Kepler before outlining a practical course of education: ‘The first step, then, in the cultivation of the imagination, is to give a child 27 cubes, and make him name each of them according to its place, as he puts them up.’ (OEI, 12-13)

Cube illustration from 'Casting Out the Self'
Illustration of a block of cubes from ‘Casting Out the Self’, p.208 of Scientific Romances. Despite being lifted from the earlier published essay, it illustrates the same system described in ‘On the Education of the Imagination’

The author warned against constricting rules, and encouraged exercises and games based on newly acquired spatial skill:

If, for instance, he is told to put a chair in (1), another in (2), and himself in (11), he is highly amused at having to seat himself in the second chair; and if then he is told to put his hat in (20) he will, after a little consideration, put it on his head. (OEI, 13)

Hinton remarked that he had also developed a form of cubical chess (!) although he confessed that none of his pupils were able to play it. The author referred to the experimental nature of the work he had undertaken with his pupils, and suggested that he had further research in mind:

Owing to the co-operation of several of my pupils, who devoted a good deal of their spare time to testing different suggestions, I have been able to work out the application of this method in several directions; and, when certain experiments on colour and sound are finished, I hope to give a detailed account of the various ways in which the method may be found serviceable. (OEI, 17)

It’s easy to see how Hinton’s lessons might have been quite entertaining. What ‘On the Education’ makes clear is the genesis of Hinton’s system of cubes in his teaching. It is devised with, and for, children, and playful elements are stressed.

Compare this to the beginning of Section 15 of Flatland, in which A. Square describes giving a domestic geometry lesson to his grandson, a hexagon:

Taking nine Squares, each an inch every way, I had put them together so as to make one large Square, with a side of three inches, and I had hence proved to my little Grandson that – though it was impossible for us to see the inside of the Square – yet we might ascertain the number of square inches in a Square by simply squaring the number of inches in the side: “and thus,” said I, “we know that three-to-the-second, or nine, represents the number of square inches in a Square whose side is three inches long.”

The hexagon is a bright student and extrapolates by analogy from this planar system to inquire about three to the third, much as Hinton hoped students of his cubic system would start thinking about four-dimensional space: ‘It must be that a Square of three inches every way, moving somehow parallel to itself (but I don’t see how) must make Something else (but I don’t see what) of three inches every way – and this must be represented by three-to-the-third.’ The passage is brief, as A. Square behaves in an un-Hintonian fashion and dismisses his grandson’s speculations.

This is, to my mind, a pretty clear sketch of Charles Howard Hinton and his spatial exercises as developed in the classroom at Uppingham. What conclusions can we draw from this? The temptation to read the whole of Flatland as a parody of Hinton as a dreamer and crackpot is very great: it wouldn’t, after all, be so unfair. Also, A. Square does come across as more rigorous than the Sphere in his attempts to extrapolate by analogy, a comparison that seems to accurately represent the single-minded vision of the young Hinton in pursuing and developing his system.

Where this gets interesting is if we pick up on the suggestion made by Smith, Berkove and Baker, that Flatland is a criticism of the misapplication of reasoning by analogy. They argue that Abbott was keen to critique what he saw as the over-extension of analogical reasoning of which Cardinal Newman, for one, was guilty, and what he saw as the tendency to obscure the linguistic roots of this rhetorical construction. They conclude: ‘Flatland is a cautionary tale about the dangers of the imagination when wrongly applied.’

This really is compelling if we line Hinton up with A. Square because its reliance upon the dimensional analogy is surely the greatest flaw in Hinton’s spectacularly generative work.

The question is, is this a flaw also blackboxed in the theoretical physics that gives us contemporary string theory? Are we still obscuring that rhetorical construction in our reach for higher dimensions of space? Or is the extention of Cliffordian physics, developed algebraically, exempt from this charge? Clifford himself reached for the dimensional analogy. I’d be interested to hear from any physicists out there, if there are any.


[1] Edwin A. Abbott, The Kernel and the Husk: letters on spiritual Christianity (London: Macmillan, 1886) p. 259.

[2] See Ian Stewart, The Annotated Flatland (Cambridge, MA: Perseus, 2002), p. Xxiii, and Thomas Banchoff, ‘From Flatland to Hypergraphics: Interacting with Higher Dimensions’, Interdisciplinary Science Reviews,15: 4 (1990) 364-372.

[3] ‘On the Education of the Imagination’, Scientific Romances Vol. 2 (London: Swan Sonnenschein, 1895), pp. Xx (first published 1888). ‘On the Education’ details researches carried out with male pupils: Hinton started teaching at Uppingham in 1880, so it must have been written after this date. A piece entitled ‘The Next Step in Education’ was discussed with his publisher from mid-1885. All further references to this essay are given in the body of the text after the abbreviation OEI.

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An overview of the secondary literature on Flatland should correctly begin with the various notices and reviews of the period. A number of these are admirably collected and posted by Thomas Banchoff here, so rather than commenting, I’ll allow the reviewers of the period to speak for themselves. The breadth of the publications in which notices were published demonstrates the immediate reach of the book, and the second flurry of reviews follows the publication of the third edition with a foreword by William Garnett, in 1926.

Despite the fact that it remained in print pretty much throughout the 20th century, the only writing on Flatland to be found in the middle of the century was Banesh Hoffman’s 1952 intro to the Dover edition, which has been reproduced several times since. In laudatory mode, Hoffmann situates the text in scientific history as a pre-relativity work: ‘In these days space-time and the fourth dimension are household words. But Flatland, with its vivid picture of one and two and three and more dimensions, was not conceived in the era of relativity. It was written some seventy years ago, when Einstein was a mere child and the idea of space-time lay almost a quarter of a century in the future.’ He describes it as ‘no trifling tale of science fiction. Its aim is to instruct, and it is written with subtle artistry.’

David W. Davies took a more literary historical approach in 1978, penning a brief biography of Abbott and citing the reviewers of the day and Abbott’s own article from The Contemporary Review in 1890, ‘Illusion in Religion’, to contextualise Flatland. Noting also Abbott’s engagement with Bacon and Cardinal Newman, Davies’s short intro is surely the wellspring for more recent criticism. Significantly, Davies closed by comparing Abbott’s application of ‘a mathematical way of thinking to literature’ to that of the OULIPO writer Harry Matthews, concluding: ‘Mathew’s permutations are for fun, and as the Boston Advertiser reviewer noted, that is the purpose of Flatland.’

Interest really began to increase in the 1980s, and I’d suggest a couple of reasons for this: the impending centenary of publication in 1884 and the advent of computing that could facilitate the rendering of four-dimensional images. This surge of interest was led by SF writers, publishers and the critics who had begun to found a scholarly response to the genre. So in 1982 Isaac Asimov contributed as dry an introduction as one might expect from the hardest of hard SF writers, praising Flatland as an essentially educational text, while Ray Bradbury’s 1983 introduction responded in far softer terms to the satirical intent of the novel. Rudy Rucker, meanwhile, commented on and responded to dimensional work by both Abbott and Hinton extensively in his fiction and non-fiction writing of this period, and will receive closer attention in a future post. For now I want to look at Bradbury’s response, which is rare in Flatland criticism by paying particular attention to the literary qualities of the work and taking considerable joy in them. It’s worth quoting a sizeable chunk of this:

Why has the book remained so popular for almost a hundred years? Because, like Mark Twain, Professor Abbott must have thought: I refuse to be serious about a serious subject. Churches brim with seriousness and snoozers snooze. Scientific conferences of one denomination or another drone on through endless and ungoiden afternoons and one chooses the catnap as against suicide. The only medicine is high spirits and good humor. Professor Abbott has both in tonic proportions. I cannot help feeling that those who shared his home with him while he was flattening his concepts to fit his pen must have heard quick bursts of laughter from his den when it suddenly struck him to write, for instance, those sections on ’feeling’ as a means of identification amongst the Flatlanders. There are serene and marvelous sexual under-and-overtones here perhaps more for us in this neo-barbarian age, than for those who inhabited the three-plus-one dimensions of 1884. Abbott, in other words, is able to play himself and win. Given the measurements of Flatland he moves out intuitively and with huge delight to ’feel’ his own creations, sum them up in shapes, and report back to us. We go with him, because it is not often we have such a guest, in our living room, so full of mathematical logic leaning into fun that we are quite content to shut our mouths and score his game.

The same year witnessed the publication of the first scholarly genre pre-history of science fiction by Darko Suvin. The progressive strain in Flatland was central to Suvin’s elevation of the text to lofty status within his pantheon of Victorian Science Fiction. For Suvin, Flatland is categorised as a ‘sophisticated alternative history’. He’s highly approving of ‘the first concrete account of a plebeian rebellion in UK SF’ (372) (despite its failure, which he doesn’t mention) and describes Abbott’s abstraction as ‘not unworthy of the fertile analytical abstractions of Darwin and Bacon (of whom he knew) or indeed of Marx (of whom he did not know).’ (370)

Roger Luckhurst’s dismissal of Suvin’s definition of SF as a literature of ‘cognitive estrangement’ – on the grounds that this definition is so ideologically conditioned that it fails to account for important texts that don’t conform to a Marxist perspective, and here’s a list of such texts by the man himself – should be noted, but need not concern us unduly here, because Flatland gets the thumbs up from Suvin and is therefore given considerable attention. (In Luckhurst’s account, mechanisation defines the genre of SF, which clearly excludes Flatland, and here it’s hard to disagree. As any fule no, Flatland is in fact math-fi, so perhaps we shouldn’t get too hung up on generic definitions.)

Suvin makes one particularly useful observation which I would like to stet, writing: ‘Cleverly adapting Carroll’s and Verne’s strategy of subsuming but transcending the juvenile reader, Abbott’s is in truth “A Romance of Many Dimensions”; in its thoroughgoing democratism, it is addressed to the best minds in the new reading public, issuing from the newly introduced obligatory primary schooling.’ (373) As has been noted numerous times subsequently (Jameson, Stableford, Luckhurst, among others), the idea that a new generation of lower middle class readers had been produced by the primary system by the 1880s has been pretty much put to bed, but let’s not lose the first part of Suvin’s point, that Flatland works for both younger and older readers. Secondary criticism since Suvin has tended to ignore this younger readership.

Such criticism has been more concerned with contextualising Flatland in the broader cultural field. In 1986 Rosemary Jann argued that ‘as part of Abbott’s wider commentary in the role of imagination in cognition, Flatland alludes to contemporary debate over the role of hypothesis in scientific discovery and the relationship between material proof and religious faith’ (473). For Jann, Flatland is a paean for ‘the progressive force of the imagination’ (486), and negotiates a middle way through inductive science, responding to debates over the unseen in the natural world, and dogmatic faith, allowing for a less absolute faith in the literal truth of the written scriptures.

Jann’s work informed Jonathan Smith, writing a decade later, in his chapter ‘Euclid Honourably Shelved’, which offers a Baconian reading of Abbott. Suvin’s comment (above) on Abbott being aware of Bacon doesn’t really do justice to the relationship: Abbott published two books on Bacon, and Smith charts the trajectory of his modulating opinion. In so doing he positions Flatland among the contemporary arguments for and against non-Euclidean geometry, working deftly with the detail of the primary material. Smith’s overview of secondary writing on Flatland is worth quoting:

When not treated as a joke, Flatland has tended to be approached in ways that divorce it from its cultural position in the debate over non-Euclideanism and its implications. Historically, literary critics have treated it as an early example of science fiction and fantasy, while scientists and mathematicians have used it as a clever way to introduce their students to concepts of dimensionality and non-Euclideanism. It has only been recently that the book has been brought back to the center of the study of Victorian culture, and it will be to further that movement that I approach the novel here. (191)

The party most guilty of treating Flatland as a joke in Smith’s account was Bertrand Russell, who addressed ‘metageometry’ in his Essay on the Foundations of Geometry, published in 1897 and based on his PhD thesis. It’s not strictly accurate, however, to say that Russell treated Flatland as a joke: he actually wrote that Abbott used the dimensional analogy as a joke, and this seemingly subtle difference is quite significant, I think.

Shortly after the publication of his book on Bacon in 1996, Smith contributed to an essay co-written with Berkove and Baker that responded to Rosemary Jann’s reading of Flatland’s ending as ambivalent by highlighting what the authors argued was an implied criticism of the theology of Cardinal Newman. Most interestingly for me, Smith et al. draw out from Flatland not an analogical inspiration, but rather an extended critique of misapplied analogy, of which they argue that Abbott believed Newman was guilty. They draw attention to English for English Readers, a textbook Abbott compiled with his friend J.R. Seeley, and its sections on analogy. English was aimed at the improving native reader and writer of English – the schoolboy – and its lessons on analogy, and the parent category of induction, are clear.

Analogy meaning Likeness. – Analogy meant originally an Equality of Ratios, or Proportion. It is sometimes, however, loosely used to represent not so much proportion, as the similarity and regularity of natural phenomena. (265) So far as it is an argument at all, [it] comes under the head of Induction. Otherwise it is not an argument, but a metaphorical illustration of an argument. (273)

What’s more, induction itself is unsound:

The Induction that proceeds from enumeration of instances to a general statement about a class […] is evidently an insecure method of proof […] It is based upon the principle of uniformity in nature, “what has been is and will be” […] Induction is always incomplete […] Thus all statements that result from merely enumerative induction are temporary and liable to correction. They may therefore be called provisional. (262-263)

Smith et al therefore turn Jann’s conclusion on its head: ‘Flatland is a cautionary tale about the dangers of the imagination when wrongly applied.’ (129-130)

Most recently K.G. Valente has done some serious digging and found an essay published in the City of London School Magazine in 1877 dealing with higher dimensioned space. Please click through the scans below to read ‘A New Phbilosophy’ for yourself.

It’s a striking find for a number of reasons: like Flatland it presents a humorous response to very recent writing on n-dimensions; it was published anonymously in the magazine of the school at which Abbott was headmaster during his tenure; and it also essays the sketching of a parodic belief system based on the reductio ad absurdum of the dimensional analogy. Is it Flatland in utero? It’s certainly a fantastic piece of writing for a schoolboy audience, ably demonstrating the euphoric headlong rush into error through rigid application of a woolly logic, eerily similar to that upon which Zollner was just embarking in Leipzig.

Valente wisely refrains from pointing the finger directly at Abbott (and how frustrating it must be not to be able to stand it up!) Of course, it could have been Abbott who authored the piece, but it could just as easily have been William Garnett, the headboy at City of London in Abbott’s third year as headmaster, who had by 1877 begun work as Clerk Maxwell’s assistant and would go on to be his biographer (Garnett would have been just as well versed in the contemporary discussions of n-dimensioned space taking place in The Academy, Nature, Mind and at the BAAS: Clerk-Maxwell alluded to it in his verse on a number of occasions in this period). Or it could simply be the work of an unusually bright pupil at City of London. We’re unlikely ever to know.

While I would follow Valente in leaving the question of authorship tantalisingly open (the beauty of blogging – I can have my cake and eat it by adding my own wild speculations before retreating rapidly!), I wouldn’t necessarily subscribe to his reasoning. Valente writes that

even with such satirical potential there is one element of the piece that would not be subjected to [Abbott’s] ridicule. Recall that the author identified serious limitations that materialism represented regarding religious contemplation; this is a conviction Abbott would not wish to mock. Although Jann noted ambivalent elements in Flatland (488), testimonials by students and peers strongly suggest that Abbott’s intellectual integrity made it difficult for him to condone ambiguity. Consequently, attributing “A New Philosophy” to him would require the problematic approbation of contradictory ideological juxtapositions as well as an explanation for their amelioration by 1884. (66)

But ‘A New Philosophy’ is monomaniacal. I can’t help but feel that while Valente acknowledges that ‘A New Philosophy’ is a humourous piece, he reads it with too straight a face, and as I’ve suggested two pars back, I think it’s intended as a piece of absurdist reasoning and is parodic to its bones: it could certainly contain ‘ideological juxtapositions’ to Abbott’s stated beliefs without entertaining any ambiguity. I also believe it has a specific satirical target in its sights, and, once again, I plan to develop that idea in a later post.

The trajectory of current criticism on Flatland, then, follows Smith’s aim to bring the text back to the ‘center of the study of Victorian culture’. While this shift in emphasis has produced some inspiring work and has rescued Flatland from ghettoisation as a sci-fi precursor text, it also, by bedding it so thoroughly into the contexts of religious or scholarly discussion of significance in this period, risks obscuring the anomalous nature of the text and the very source of its popularity. Not only was Flatland the only fiction published by the prolific Abbott in a thirty-year writing career, but it is also, like ‘A New Philosophy’, less than serious.

I’m likewise suspicious when the secondary criticism positions Flatland as an element in some kind of smooth-surfaced theological project on the part of Abbott. I think it’s important, particularly when attempting to recreate the ‘cultural position’ of the text, to hold in mind William Garnett’s description, in his preface to the third edition, of the book as a ‘jeu d’esprit’, and the comments of earlier respondents like Bradbury and Davies: Abbott was having fun with this book, and writing for a broader audience than that he habitually addressed. Russell recognised this, as did contemporary reviewers. There are certainly consistencies with his theological writings, as one would expect, but we must remember that what we are reading is not a manifesto (in the case of Flatland, at least: a manifesto is exactly what ‘A New Philosophy’ is, and a very good a priori spoof of the form it is too.)

Of course, to argue for a reading of Flatland responsive to its humourous intent is to create a rod for my own back – is satire necessarily actually funny? can intent be assumed? – but what I’m really advocating is the recuperation of elements of earlier criticism rather than the dismissal of fresher discoveries. It goes without saying that I think Smith’s work in contextualising Flatland in discussion of non-Euclideanism was entirely necessary; it’s important, too, to hold in mind Abbott’s position on miracles and the imagination; I’m particularly persuaded by the argument that Flatland critiques rigid and literalised analogical reasoning; and I’m totally thrilled by ‘A New Philosophy’. None of this stops Flatland being an amusing book written with a schoolboy audience in mind: this is a crucial point when it comes to the spread of the concepts of higher space. It’s important that we don’t ‘divorce it from its cultural position’ as a funny and popular book. And I also think it’s overstating the case to think of Flatland as an integrated part of a theological project when it is enmeshed in a complex matrix of contemporary social and cultural concerns and responsive to such a broad range of ideas as Galton’s eugenics, ‘plebeian rebellion’, geometrical pedagogy and the education of women.

So that’s a fairly breezy overview, which has doubtless missed some significant contributions, but it’s a reasonable launch-pad for some consideration of thee text itself, which I always seem to be threatening without ever actually doing…

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While on the subject of publishing contexts at the end of 1884, and before edging further into Flatland and dealing with content…

It has become customary to connect Flatland to the work of Charles Howard Hinton, and the connections between Edwin Abbott and the author of Scientific Romances have been explored in some detail by a number of writers (Banchoff, Stewart, Valente). Developing the case made by Banchoff in 1990, that ‘Hinton lies at the centre of a web of intellectual, mathematical and social influences’, Ian Stewart argues that ‘the similarities between Hinton’s 1880 article [‘What is the Fourth Dimension?’] and Flatland are far too great to be coincidence’ and that ‘the circumstantial evidence that they probably did meet – or that, at the very least, Abbott was strongly influenced by Hinton’s ideas – is considerable’.

Extrapolating the publishing history of Hinton’s work clarifies one such connection. Hinton’s ‘What is the Fourth Dimension?’ had, as noted, been first published in 1880, but to a very limited audience; indeed, to an audience so scant that it failed to sustain The University Magazine, the ailing journal in whose last number the essay appeared (originally the monthly Dublin University Magazine, The University Magazine had been renamed in 1878, and reduced frequency of publication from monthly to quarterly from June 1880, before finally closing at the end of 1880. Hinton’s mother-in-law Mary Boole had been a frequent contributor).

By the end of 1880 Charles Howard Hinton was working as assistant science master at Uppingham College (one of the connections made by Banchoff: Abbott’s lifelong friend Howard Candler, to whom Flatland was dedicated, was mathematics master at the same school). He was not a novice to publishing, having edited a collection of his father’s work, Chapters on the Art of Thinking and Other Essays, published by C.K. Paul & co in 1879, but ‘What is the Fourth Dimension?’ was his first published work under his own name.

It was reprinted in slightly expanded form in 1883 in the magazine of Cheltenham Ladies College, where the author had worked as assistant master from 1877 to 1880. Once again, it is safe to assume that the school magazine had a limited audience, although precise figures are not available. Stewart’s speculation that Edwin Abbott’s acquaintance with the headmistress Dorothea Buss in the 1880s was another potential point of contact between Abbott and Hinton seems more tenuous than the Candler link. What is clear from both the titles in which Hinton’s essay first appeared – a magazine hoping to appeal to a core student readership, and the magazine of a school – is that its author considered it a pedagogical piece. An instructional essay for students it is likely to have remained were it not for Abbott’s book.

The timing, format and re-editing of Hinton’s essay for publication by Swan Sonnenschein in November 1884 suggests very powerfully a commercial response to Flatland, whose first edition of 1,000 copies had been sold within a month of publication. What is the Fourth Dimension? (italics will henceforth be used to distinguish between the pamphlet and the collected essay) came hot on the heels of Abbott’s book as a part-issue, a format suggestive of a rapid publishing response: as the entry for ‘Serials and the Nineteenth Century Publishing Industry’ in the Dictionary of Nineteenth-century Journalism notes: ‘The principle motivations underlying the rise of serial publications were speed and economy.’ (Brake, Demoor eds, 2009: 567) There is also considerable evidence in the archives of Swan Sonnenschein that Hinton did not yet have enough completed work for a book.

Should there be any doubt concerning the opportunistic nature of the 1884 re-publication of Hinton’s essay on its third go round the block, its new title and subtitle surely settle them. It has been suggested by Rudy Rucker that the subtitle Ghosts Explained was added by the canny publisher, aware of the Zöllnerian hypothesis and its currency in spiritualist groupings. But surely the title of the series, Scientific Romances, is even more suggestive of commercial expediency? Hinton’s first ‘romance’, after all, was not even fiction, but a pedagogical exposition answering its own question in terms that only began to hint at the visionary hue of the psychological metaphysics that would follow. Stylistically, it owed more to the popular science writing of Tyndall than it did to Stevenson, but the content was evidently particularly amicable towards Flatland and the market was demonstrably keen on dimensional romances in November 1884.

It seems highly likely, then, that the chosen designation of ‘romance’ would have identified Hinton’s work to the readership to whom it was most likely to appeal: recalling Stevenson’s definitional account, a young (?), masculine, domestic (British) readership. The subject matter of geometry would further limit the audience to those educated in mathematics.

Darko Suvin’s obsessive historical materialist categorisations of the readerships of early SF precursor texts are interesting here, not because I would like to categorise Hinton’s work in such a way, but because in identifying a social proximity between the authors of proto-SF, scientific non-fiction and the readers of both, outside of mainstream circuits, he speaks directly to the textual hybridity of Hinton’s work: ‘Indications from the textual system point to one of those groups comprising mostly upper-middle and middle class males with special interest in politics, religion and public affairs in general. This is a circuit very close, perhaps even identical, to that of the bourgeois nonfiction reading – which would explain the intertextual closeness to SF of such nonfiction genres as the social blueprint, the political tract, the predictive essay, even the semi-religious apocalypse.’ (Suvin, 1983: 403)

This also, however, creates an interesting tension. I find myself wanting to argue that savvy publishing nouse helped to make the fourth dimension a subject of discussion in social groupings beyond specialist mathematicians and spiritualists.  If the readerships of texts such as Flatland and What is the Fourth Dimension? are as socially narrow as Suvin suggests, however, do they really introduce the arcania of higher space to a broader audience? I think the answer to that question probably lies, in part, elsewhere: it’s what these texts do with the subject, as well as to whom they tell it, that catalyses interest.

Finally, a word or two on that canny publisher, William Swan Sonnenschein. Sonnenschein built his list in the early years (ca. 1878-1882) around books for children, educational texts or theoretical work concerning education policy. There was also a focus on German language translations, such as Grimm’s Teutonic Myths. Both arose naturally from the publisher’s family background: his father was a German-born mathematics teacher. Although Sonnenschein described himself as a liberal, he was closely connected socially to a number of Fabians and socialists, publishing both the first English translation of Marx’s Capital and George Bernard Shaw’s Unsocial Socialist in 1887. (Stepniak, exiled Russian revolutionary, was apparently often to be encountered taking tea chez Sonnenschein).

The Swan Sonnenschein list also always included philosophy, and the publisher was a member of the first Ethical Society in the late 1880s. Commissioned to write a history of the firm’s precursors by George Allen and Unwin in the 1950s, the historian F. A. Mumby wrote: ‘Throughout his life Swan Sonnenschein was a remarkable blend of other-worldliness and business acumen; a man of wide erudition whose interests were quickly roused by the simplest human problems’. Combining education, mathematics, philosophy and literature, Swan Sonnenschein was a highly appropriate home for the esoteric and hybrid work of Hinton.

So, some further lines of research worth pursuing with regard to dimensional romance: its roots in pedagogy and a progressive, broadly socialist, political subtext. Onwards and upwards. Or, as Flatland has it, Upward, not Northward.

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Some initial thoughts on Flatland. So, to begin at the beginning with the title page…

Many dimensions

Many dimensions

Strange that this has occasioned so little critical comment. Iain Stewart’s excellent annotated edition of the text locates the Shakepearean quotes (Hamlet, Act I Scene v, the appearance of the ghost, and Titus Andronicus, Act III Scene i), both of which are fairly obviously puns and perhaps only tangentially connected to their context. I’m intrigued by the illustration – is it a map? – of a nebulous mass, perhaps fog, perhaps clouds.

It might well be a map. Flatland, ‘a Romance of Many Dimensions’, was published in October 1884. As such it arrived not terribly long into the ‘romantic revival’ of the 1880s, inaugurated, according to most accounts, the previous year, with Robert Louis Stevenson’s Treasure Island. Before that, in the launch issue of Longman’s Magazine in November 1882, Stevenson had given a theoretical outline of his fictional practice with ‘A Gossip on Romance’, advocating a robust, masculine, adventuring, fiction delivering a ‘kaleidoscopic dance of images’ and recalling books read in the ‘bright troubled period of boyhood’. Stevenson’s advocacy of romance has subsequently been read in opposition to Henry James’s championing of the interiorized, feminine and despicably foreign (!) realist novel.

This brief sketch is sufficient for now to give an idea of one aspect of the context into which Flatland arrived: while the descriptive term romance had been used in the title of many earlier nineteenth century novels, and even proto-SF novels – Edward Maitland’s An Historical Romance of the Future (1873) being an (the only?) example of the latter – when Abbott subtitled his book a ‘romance’, he connected it to a very current trend in fiction publishing. There was good reason for so doing: Treasure Island had been a bestseller. The inclusion on the title page of Flatland of a map would have underlined the connection to Treasure Island in particular.

Treasure Island

Treasure Island

So is it fog, or is it clouds? The text contains fog – common, apparently, in the temperate regions of Flatland – but the closing illustration repeats the nebulous illustration with more Shakespeare, this time from Prospero’s speech in The Tempest, Act IV, Scene i:

You do look, my son, in a moved sort,

As if you were dismay’d: be cheerful, sir.

Our revels now are ended. These our actors,

As I foretold you, were all spirits, and

Are melted into air, into thin air:

And, like the baseless fabric of this vision,

The cloud-capp’d towers, the gorgeous palaces,

The solemn temples, the great globe itself,

Yea, all which it inherit, shall dissolve,

And, like this insubstantial pageant faded,

Leave not a rack behind. We are such stuff

As dreams are made on; and our little life

Is rounded with a sleep.

The baseless fabric of vision

The baseless fabric of vision

Thin air, then, and clouds. And I’d suggest that the classical scholar Abbott may also have had in mind an earlier passage of satire, from Aristophanes’ The Birds. In the following exchange the tyrannical Pithetaerus passes judgement on the geometer Meton in his attempt to enter Cloudcuckooland:

(Enter METON, With surveying instruments.)

METON: I have come to you…

PITHETAERUS (interrupting): Yet another pest! What have you come to do? What’s your plan? What’s the purpose of your journey? Why these splendid buskins?

METON: I want to survey the plains of the air for you and to parcel them into lots.

PITHETAERUS: In the name of the gods, who are you?

METON: Who am I? Meton, known throughout Greece and at Colonus.

PITHETAERUS: What are these things?

METON: Tools for measuring the air. In truth, the spaces in the air have precisely the form of a furnace. With this bent ruler I draw a line from top to bottom; from one of its points I describe a circle with the compass. Do you understand?

PITHETAERUS: Not in the least.

METON: With the straight ruler I set to work to inscribe a square within this circle; in its centre will be the market-place, into which all the straight streets will lead, converging to this centre like a star, which, although only orbicular, sends forth its rays in a straight line from all sides.

PITHETAERUS: A regular Thales!

Tools for measuring the air, indeed! This prompts a number of lines of thought. A bone of contention in discussions over higher space concerned its imaginary as opposed to its empirical nature. As an algebraic and then a geometric theory – in other words, as a mathematical construct – higher space remained comfortably ideal. With interventions from physics and Zollner’s catastrophic/catalytic misreading of four-dimensional space, the waters became muddied – or perhaps better to write that the airs became fogged. Was physical space actually four-dimensional?

Higher space existed in the interstices between the ideal and the empirical, as did the emergent sciences of mind, in which perception of space was a primary site of conflict. Abbott’s cloud, then, is thought, imagination, the higher space of mind in which the higher space of geometry existed, an analogue noted by William Spottiswoode in his 1878 address to the BAAS: ‘Or once more, when space already filled with material substances is mentally peopled with immaterial beings, may not the imagination be regarded as having added a new element to the capacity of space, a fourth dimension of which there is no evidence in experimental fact?’

As for Pithetaerus’s remark on Thales, this suggest routing discussions through Michel Serres, whose two essays on the origins of geometry consider the case of Thales and the discovery of mathematical analogy. But I believe that would justify another post entirely and this one already needs more work!

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