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)

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.