4 Cubes: Hintonian Higher Space and its Thinking Subject

Charles Howard Hinton’s essay ‘What is the Fourth Dimension?’ was first published in the final issue of the ailing University Magazine in 1880.¹ Arriving shortly after Massey’s translation of Zöllner’s Transcendental Physics, it was read in this context: writing anonymously in The Nonconformist and Independent, the physicist William Barrett, a founding member of the Society for Psychical Research, noted Hinton’s essay and used it as a springboard for speculations on beings ‘superior to ourselves, but unknown to us, because living in space extended in the fourth dimension’.² Hinton’s essay lent itself to such readings. Grounded in geometry, working by analogy from two dimensions to three and then four, and referencing contemporary science, it announced itself clearly as a speculative piece:

It is the object of these pages to show that, by supposing away certain limitations of the fundamental conditions of existence as we know it, a state of being can be conceived with powers far transcending our own.³

The powers a four-dimensional being would possess would have been familiar to readers of Zöllner: ‘such a being would be able to make but a part of himself visible to us […] would suddenly appear as a complete and finite body and as suddenly disappear’ (SR, 25); it would have the ability to ‘get out of a closed box without going through the sides’ (SR, 27).

While Hinton employed the full analogical toolkit for imagining beings confined to lower-dimensioned spaces, a toolkit now tried and tested through repeated use in scientific journals, he offered considerable original insight on the subject. A striking metaphor of threads passing through a sheet of wax was developed to pose the question ‘is it possible to suppose that the movements and changes of material objects are the intersections with a three-dimensional space of a four-dimensional existence?’ (SR, 23). Hinton was concerned to bed the fourth dimension of space into physical models and drawn towards materialization. The ultimate focus of his thought experiments, though, was the human consciousness: ‘Why then, should not the four-dimensional beings be ourselves, and our successive states the passing of them through the three-dimensional space to which our consciousness is confined?’ (SR, 18).

‘What is the Fourth Dimension?’ was reprinted with an extra paragraph of material in 1883 in the magazine of the Cheltenham Ladies College and, despite its author’s social connections, it seems likely that this second iteration would have been the piece’s final printing were it not for the publication of the first edition of Flatland at the end of October 1884.⁴ Within weeks the publisher William Swan Sonnenschein was corresponding with Hinton, now science master at Uppingham College, and a pamphlet containing the first of a planned series of Scientific Romances was printed. In the hands of the new publisher What is the Fourth Dimension? was expanded and subtitled ‘Ghosts Explained’.

The text that expanded the first Sonnenschein edition of Hinton’s essay solidified his thesis, developing the idea that ‘the matter we know extending in three dimensions has also a small thickness in the fourth dimension’. This speculation was founded upon another: that gases might behave in four dimensions as liquids do in three, might ‘have a centre of attraction off in the fourth dimension’ (SR, 28). The conclusions drawn from such speculations were either that physical space was four-dimensional, but that humans were three-dimensional beings and appeared to four-dimensional beings as nothing but abstractions; or that we have an ‘infinitely minute’ four-dimensional existence (SR, 31).

The encroaching materiality of Hinton’s higher spatial thought is the arc around which this chapter orbits. Hinton repeatedly mediated higher space through matter, both in his theories, which constantly accounted for the physical implications of a fourth dimension or speculated a physical source, and in his practice, based on the manipulation and contemplation of a set of colour-coded (or named) cubes. Hinton was sufficiently conversant with the ethereal physics of his time, and the conceptual dissolution of matter into its atomic components, that he was able to weave his higher spatial theories into the electromagnetic spectrum. Over the course of his career he honed a key set of theories about the fourth dimension: that it was evidenced by the phenomena of electricity; that it existed at the macro and micro scales; that thinking it brought about material changes in the brain.

His body of work emerges as an immense force in shaping the idea of higher-dimensioned space. Where the Zöllner event nourished spiritualist appropriations of the fourth dimension, Hinton’s work prised open new channels. Linda Dalrymple Henderson treats Hinton as the chief popularizer of hyperspace philosophy.⁵ Henderson’s description is particularly useful because Hinton has often been mis-characterized as a writer of SF. This chapter outlines the scope of his project, a scope that Hinton himself described in A New Era of Thought (1888):

I propose a complete system of work, of which the volume on four space is the first installment. I shall bring forward a complete system of four-dimensional thought-mechanics, science, and art.⁶

The ambition of this claim is continuous with the visionary zeal that peaked in Hinton’s writing around the publication of A New Era but, as an overview of his thought shows, he did indeed attempt such an all-encompassing programme, even if the artistic aspect, beyond his own fiction, was left to the modern artists about whom Henderson writes. His interest was philosophical, but so too was it practical and informed by his approach to pedagogy.

It is telling that it took the work of an art historian to recuperate Hinton. Henderson reads Hinton as a ‘pioneer’, a popularizer whose work became a source, through mathematicians such as Jouffret and Theosophists such as C.W. Leadbeater, for Futurism, Cubism, Suprematism, and more. For Elizabeth Throesch it is a form of ‘“protomodernism” because […] his project is a transitional one’.⁷ Precisely the reasons Hinton’s work does not fit into the literary canon—its hybridity and materiality—are the reasons it is more amenable to art history, a discipline more used to dealing with objects, even if they lurk within texts. Henderson also treats occultists with an equanimity that has not always been afforded them in literary studies. The influence of Hinton’s work today can be felt most keenly in the slightly embarrassing esoteric sections of bookshops: as an important source for both Rudolph Steiner and P.D. Uspensky, Hinton has been assured a legacy in this section of the publishing market.⁸

After outlining Hinton’s work and its salient characteristics, this chapter focuses on the practical system at its heart which it recreates through archival and textual sources. Reconstituting the cubes he designed and described, the very things with which he thought the fourth dimension, we can ascertain how the author conceived them, how they were used, how read, and how we might think them. This work will involve the voicing of key sources for Hinton’s approach to these objects before recounting their cultural trajectory. In closing I will consider again the mediations between space and matter that loop through this book and which in the case of Charles Hinton were informed by his father’s published philosophy.

Also informed by the work of James Hinton was the idea of ‘casting out the self’, a voiding of spatial subjectivity that was an essential element in the use of the cubes. James Hinton inspired the altruistic currents that swirled beneath the more visionary aspects of Charles Hinton’s project and provided a unique intellectual and social platform for such work. Before turning to the son, what of the father?

James Hinton split his life between practice as an aural surgeon and writing as a philosopher. From 1858 to 1862 he gave up practice to write, publishing Man and his Dwelling Place (1859), writing The Mystery of Pain (1866), and contributing numerous articles to Cornhill Magazine which became the basis for Life in Nature (1862) and Thoughts on Health (1871).

Man and his Dwelling Place outlined a perceived lack in man’s knowledge of the universe: ‘The defective state of man causes our feelings not to correspond with the truth of things; so that we can only understand aright either ourselves or the world by remembering that man is wanting in life.’⁹  Life in Nature clarified these ideas:

Not only are the organic and inorganic worlds, which seem to be so different, truly one, exhibiting the same forces, powers, and laws; but life itself, or that which we have called so, appears as a mere result of chemical and mechanical agencies, into the effects of which its most distinct phenomena are resolved. We find no special power which we can call by that name.¹⁰

Process and systems of relations became of central importance, the flux and continuity between the organic and inorganic, man and the world. Shadworth Hodgson, writing after his friend’s death, cast this in terms of mind and matter:

The operations which have mind and those which have matter for their field are parts of one system of operations; and just because they are parts of a single whole do they recall and seem to repeat each other when each kind is separately examined.¹¹

Hodgson read Hinton’s view of nature as an extension of the Romantic philosophy of Coleridge and identified later encounters with German idealism as the source for an appreciation of the phenomenal and an alignment of the noumenal with the spiritual. Hinton wrote: ‘This physical world, known to be an appearance (or phenomenon,) is the appearance of that spiritual world which we also know.’¹² Hinton argued that instinct and the emotions, more attuned to the noumenal world, should be allowed to guide thought: ‘Our heart, in a word, asserts the true; science reveals to us the apparent.’¹³

Also writing after Hinton’s death, Henry Havelock Ellis recorded that Hinton referred to his philosophical position as ‘actualism’ and emphasized the moral tone that inflected his ideas with the emotional and instinctive in his later writing: ‘Hinton, when he began philosophising, firmly believed in an absolute which might be known; not indeed known intellectually but through the moral sense.’¹⁴

This moral philosophy, incipient in his first work, dominated his later thought. It began to surface in The Mystery of Pain, which set out to offer consolation to all those who had suffered. Hinton argued that pain was its own reward if viewed as sacrifice:

These facts are evident in human life even as it is: that man is framed for joy in sacrifice; that until it can be made his joy, sacrifice must be his torment, for it can never be banished; that without the willing acceptance of sacrifice, no end is really answered in human life, no satisfaction that is worthy of humanity achieved.¹⁵

As Thomas Dixon has noted, Hinton claimed that the causal relationships of the physical world illustrated the inevitability of self-sacrifice. Ellis quoted a passage from Hinton’s unpublished manuscripts, written around 1858:

How that idea of self-sacrifice (as the source of all life) is involved in the correlation of forces! ‘Each force merging itself as the force it produces becomes developed,’ says Grove. This is the very fact of creation, the exact statement of that self limit which is creative action. And this is the phenomenal, the ‘instinctive’ view of Nature.¹⁶

Ellis explained that the notion of ‘self-sacrifice’ was later supplanted by the idea of ‘service’ to account also for the role of pleasure. He glossed this modification: ‘By sacrifice he had meant the willing acceptance of pain, all thought of self being cast out; by service he now meant the acceptance of pleasure also, the thought being still not on the self; that is to say the acceptance of all things, either pleasure or pain, that served.’¹⁷

James Hinton’s work is gradually being recovered by Victorian scholarship. Thomas Dixon has highlighted the significance of his contribution to the naturalization of altruism in the second half of the nineteenth century. Hinton wrote to a friend: ‘The word altruistic I borrow from Comte. Is it not a capital word? I am resolved to naturalise it. We want it. It is the antithesis to “self”; self-being = deadness; altruistic being = life; and so on.’¹⁸ Dixon observes how altruism fitted in to his broader system: ‘He used altruistic to describe not just philanthropy, nor just the benevolent instincts so named by Comte, but the connectedness and relatedness of all natural phenomena, and the possession of “a consciousness co-extensive with humanity”.’¹⁹

Seth Koven, another scholar who has worked to recover James Hinton, blames the ‘quicksand of sexual scandal, based wholly on unsubstantiated rumour’ that came in the wake of Charles’s 1886 conviction for bigamy as a reason for the almost complete erasure of James Hinton’s influence on his contemporaries from the cultural record of the period.²⁰ This reputational damage is certainly significant for both generations of Hinton; for Charles it had geographical repercussions, sending him into exile at a time when his work was gaining a foothold in Britain.

A significant inheritance from his father that had bolstered this burgeoning public life was a network of intellectual and social connections. Old family connections between the Hinton family and the Nettleship family from Nonconformist circles in Oxford had been maintained, providing a significant philosophical and educational matrix for Charles. Richard Lewis Nettleship was a tutor of philosophy at Balliol, Charles’s Oxford college, and literary executor of the idealist T.H. Green; he was also formerly head-boy at Uppingham, where Charles later found employment, and a correspondent and friend of the headmaster Edward Thring; Richard’s brother, John Trivett Nettleship, the landscape painter and friend of the Pre-Raphaelites, who cited James Hinton in his study of Robert Browning, later married Charles’s sister Adaline.

Beyond the family were those who were, like Havelock Ellis, drawn to James Hinton’s philosophy. Seth Koven writes of the ‘dense networks of discipleship and affiliation surrounding [James] Hinton’ that included key figures in philanthropic and social purity movements and drove discussions of sexuality and sexual reform. Well documented in scholarly studies, and significant with reference to Charles, is the membership of the Men and Women’s Club. Henry Havelock Ellis, who had made contact with the Hintons immediately on his arrival in England and became friend and confidant to Charles, was the conduit. The South African novelist Olive Schreiner records frequent visits to the Hintons in her correspondence with Ellis. Caroline Haddon, James’s sister-in-law, and Margaret, his wife, were also attendees at these meetings, as they were at the Fellowship of the New Life, the group that became the socialist Fabian Society.²¹ James Hinton’s biographer, the social reformer Ellice Hopkins, Ellis’s lesbian wife Edith Ellis Lees, Arnold Toynbee, Roden Noel, and Charles Ashbee all produced work directly inspired by James Hinton’s writing.²²

While James Hinton’s work as a ‘philanthropic hedonist’, as Koven describes him, drew sexual reformers and progressives to his work, his books also met with the appreciation of figures more central to late Victorian culture. Having impressed Tennyson, James Hinton was invited to be a founder member of the Metaphysical Society, alongside William Gladstone, Thomas Huxley, John Tyndall, Henry Sidgwick, Walter Bagehot, W.K. Clifford, and, later, John Ruskin, with whom he became friends.²³

In his history of the Society, Alan Willard Brown argues that the foundation of the Society for Psychical Research in 1882 by Sidgwick ‘reflects the failure of the Metaphysical Society to bridge the gulf between the institutionist and empiricist positions and the admitted inability of the scientists to relate “the facts of consciousness” to their material hypotheses’.²⁴ Charles Howard Hinton’s work clearly continues aspects of this endeavour independently of the SPR. Brown also comments on the establishment in 1876 by Metaphysical Society member George Croom Robertson of the journal Mind:

And it is true that at the very time when the great popular reviews were beginning to exclude the more profound philosophical and scientific papers and turn to more popularly based articles, Mind appeared in answer to a need. As the Metaphysical Society declined, its most philosophically minded members turned increasingly to Mind rather than to the Nineteenth Century; and this, too, was a sign of the times.²⁵

Mind is perhaps the most important context: even though James died in 1876 before he could contribute, he was obituarized in the journal and all of Howard’s Scientific Romances were subsequently reviewed in its pages, with doffs of the cap to their author’s lineage. Its serious treatment of ‘mental philosophy’, a nexus between emergent psychology and philosophies of thought, mirrors a similar nexus in Charles’s higher space philosophy and his development of neurological ideas in A New Era of Thought. It also provides further connections, between Charles and Shadworth Hodgson, James Hinton’s friend and a contributor from the first issue, and later William James, a contributor from 1879 with whom Charles corresponded when he was living in the USA.

The tradition Charles Hinton inherited from his father was resistant to the processes characterized by Bruno Latour as purifications of nature from the social. James Hinton argued for a society and culture permeated by nature and natural processes. Under the guises of philosopher and surgeon he worked to complicate late Victorian social and scientific orthodoxy.

Charles’s interest in higher geometry began as he was building a public career as a young schoolmaster and scholar. In 1878 his paper ‘On the Co-ordination of Space’ was read at the Physical Society. William Crookes’s Chemical News summarized:

If a cubical space be divided into 27 numbered cubes, and each of these be again subdivided in the same way, and so on, the position of any point within the initial cube can be expressed by a reference to the numbers of the several cubes in which it is placed, and the more this series of numbers is extended, the more accurately is its position defined.²⁶

No copy of his first monograph, Science Notebook (1884), is known to survive, but reviews describe a well-received engagement in debates over non-Euclideanism that seems derived from the same system:

The author does not presuppose continuous elements as has been generally done, but only sets of points equally distributed in two dimensions, which, merely for the sake of convenience, are connected by straight lines […] The practical advantages of this new method in the form in which it is now published are purely educational, though it is wholly based on the principles just mentioned.²⁷

Later that year, the success of Flatland provided the impetus for Swan Sonnenschein’s interest in republishing Hinton’s twice-round-the-block essay ‘What is the Fourth Dimension?’. Correspondence from the publisher to Hinton began in November 1884. A specific publishing context in which the success of Flatland made pieces of fringe scholarly or pedagogical interest commercially viable contributed to the expansion of higher-dimensional thought.

Sonnenschein was canny and innovative, publishing an eclectic list that later included the first English translations of both Marx and Freud.²⁸ The son of a German mathematics teacher, he built his list in the early years (c.1878–82) around books for children, educational texts, and translations of German-language books, such as Grimm’s Teutonic Myths. Although Sonnenschein described himself as a liberal, he was closely connected socially to a number of Fabians and socialists, publishing, as well as Capital, 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 predecessors by George Allen and Unwin in the 1950s, 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.’²⁹ This business acumen was developed in the turbulent book market of this period. Alexis Weedon writes in her analysis of Victorian book publishing:

When the economic interdependence between novel publishers and the libraries began to fail in Britain in the 1880s, a newly competitive market-place arose. Shrewd publishers looked to their strengths and developed innovative publishing strategies to exploit them […] careful price structuring and timing of the release of each edition was crucial for them to sustain revenue and reap the full economic potential of the work […] publishers were able to capitalize on the cost savings of the more efficient printing technologies and cheaper raw materials by marketing the text in a range of formats.³⁰

One such format, the part issue, became a viable tool for publishers looking to bring a work to market rapidly to capitalize on favourable conditions. As the entry for ‘Serials and the Nineteenth Century Publishing Industry’ in the Dictionary of Nineteenth-century Journalism notes: ‘The principal motivations underlying the rise of serial publications were speed and economy.’³¹ The timing, pamphlet format, and re-editing of Hinton’s essay for publication by Swan Sonnenschein in November 1884 suggest a rapid commercial response to Flatland.

Swan Sonnenschein had a clear view of its readers, advertising in popular scientific journals such as R.A. Proctor’s Knowledge, and had the commercial nous to engage as many audiences as possible for its output.³² The new subtitle to Hinton’s essay, ‘Ghosts Explained’, was the work of the publisher and piggybacked on the currency of higher spatial ideas in spiritualist and occult circles. Like this subtitle, the term Scientific Romance had not been used by Hinton before his essay was brought out by Swan Sonnenschein and this phrase is yet more indicative of commercial expediency.

The reviewer of Hinton’s pamphlet for Nature had already read Flatland, and discussed Hinton’s ideas in relation to Abbott’s fictional narrative, noting that ‘these ideas are coming to the front again’.³³ It was not only readers who noted the relationship: the work of the two writers was also 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)

Abbott returned the compliment 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.³⁴

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. There is a slight echo of the extra-textual Hinton in the text of Flatland that adds support to the idea of the cubes as the central element to Charles’s work and sits comfortably within Flatland’s playful satire. To see it we also have to have already pictured Hinton’s preferred method of visualizing space, which he had not published by 1884, but which he had certainly been using in the classroom at Uppingham: that of working with sets of nine one-inch cubes. At the beginning of Section 15 of Flatland, 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.’

(F, 52–3)

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: ‘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, but the echo of this central aspect of Hintonian practice—flattened—is there. Abbott’s pencil sketch grasped the core of Hinton’s project: material objects deployed in a pedagogical context and mediating between geometry and space.

At the beginning of 1885 Hinton presented a refined version of his system of co-ordinate relations, now called ‘poiographs’ after Sir William Hamilton’s ‘hodographs’, to the Physical Society. Nature’s review suggests either a different approach or a different perception of Hinton: ‘As a result of a process of metaphysical reasoning, Mr. Hinton has come to the conclusion that relations holding about “number” should be extended to space.’³⁵ In 1886 he was commissioned to contribute an essay on the fourth dimension that ran to double the length of the entry on ‘Psychical Research, and the Society for’, to Hazell’s Cyclopaedia [sic], a publication that purported to provide ‘up-to-date information on such subjects as are now, or are likely to be, in the mind of the public’.³⁶

Meanwhile, Swan Sonnenschein began to publish a series of essays in pamphlet form as Scientific Romances. The second Romance, The Persian King, was a physical parable describing the system of governance in an isolated Persian valley, in which pain and pleasure assumed by a ruler and distributed among citizens was laid out on a thermodynamic basis, with accompanying calculations provided in a second part:

When the king wished to start a being on the train of activity he divided its apathy into pleasure and pain. The pleasure be connected with one act which we will call A. The pain he associated with another act which we will call B.

[…]

The sensation in the first A was 1000, in the first B it was 998, giving a disappearance of 2. In the second A it was 980, and in C, which starts concurrently with the second A, it was not 20 as might have been expected, but 16, giving a loss of 4.³⁷

Mind, as sympathetic a reviewer to Hinton’s concerns as existed, described it as ‘somewhat less effective’ than its predecessor, and as a piece of narrative fiction it was hamstrung by its didactic aims.³⁸ Considered as part of Hinton’s larger programme, however, it was both consistent with his teaching and an expansion of his higher spatial project, inserting his work into thermodynamic discourse and hybridizing it with moral elements from his father’s late philosophy of ‘service’. Bruce Clarke argues that The Persian King ‘seeks moral asylum from the materialism of the second law in the conceptual haven of the spatial fourth dimension’.³⁹  The Persian King illustrates, too, Hinton’s tendency towards allegory, and his debt to one allegory in particular: the character of Demiourgos, a creator controlling the valley from a distinct, if not higher, space, signalling the Platonic Demiurge of the Timaeus.

The physical concerns of The Persian King were extended in ‘A Plane World’, a post-Flatland meditation on congruence in two dimensions, and ‘A Picture of Our Universe’, which outlined two theories extended in his later work. The first considered congruence with regard to spiral twists, the mutual cancellation of opposite twists, and suggested a fourth-dimensional rather than ethereal explanation for the phenomena of electricity:

Thus if we suppose that in the minute motions which go on about us there is the possibility of moving in a four-dimensional way, then it is perfectly legitimate to assume that in a medium which cannot be twisted, but which is elastic, a twist calls up a real image twist. And thus the assumptions which we have made as the basis of an electrical theory are justified on the assumption of a four-dimensional space, are untenable except on that supposition.⁴⁰

The second theory was worked out in an appendix:

For suppose the aether, instead of being perfectly smooth, to be corrugated, and to have all manner of definite marks and furrows. Then the earth, coming in its course round the sun on this corrugated surface, would behave exactly like the phonograph behaves […] Corresponding to each of the marks in the aether there would be a movement of matter, and the consistency and laws of the movements of matter would depend on the predetermined disposition of the furrows and indentations of the solid surface along which it slips […] Thus matter may be entirely passive, and the history of nations, stories of kings, down to the smallest details in the life of individuals, be phonographed out according to predetermined marks in the aether. In that case a man would, as to his material body, correspond to certain portions of matter; as to his actions and thoughts he would be a complicated set of furrows in the aether.⁴¹

Bruce Clarke routes his investigations of ‘Hinton’s groovy phonographic ether’ through Friedrich Kittler’s theoretical writing, arguing: ‘In the transformation of the ether medium into the spatial and temporal fourth dimensions, mediated reality is metamorphosed into art forms—imaginary realms and symbolic structures.’⁴² We might adapt this to recognize the grooved ether as another physical model, a form into which the immaterial is compressed.

The grooved ether model was reprised in the full-length book A New Era of Thought (1888) to argue for the existence of both a material and an eternal ethereal body for any organism, and an ‘essential unity of the race’:

We find an organism which is not so absolutely separated from the surrounding organisms—an organism which is part of the aether, and which is linked to other aethereal organisms by its very substance—an organism between which and others there exists a unity incapable of being broken, and a common life which is rather marked than revealed by the matter which passes over it.

(ANE, 64)

A New Era not only drew together the theoretical work of Hinton’s Romances, but was the book in which he explicitly advanced the ethical, and therefore social, theories implicitly stated in those works. His writing found a new tone, less speculative, more visionary, as he approached the content that dominated the book: ‘And then those instincts which humanity feels with a secret impulse to be sacred and higher than any temporary good will be justified—or fulfilled’ (ANE, 94).

Charles paid direct, but anonymous, tribute to his father in A New Era of Thought, referring to ‘one with whose thought I have been very familiar, and to which I return again, after having abandoned it for the purely materialistic views which seem forced on us by the facts of science’. He summarized what he saw as the most significant element of his father’s thought:

He looked for a time when, driven from all thoughts of our own pain or pleasure, good or evil, we should say, in view of the miseries of our fellow-creatures, Let me be anyhow, use my body and my mind in any way, so that I serve.

(ANE, 72)

A New Era created a hinge between the ethical and the spatial by equating the absenting of the egoistic self from altruistic activity with the need to remove ‘self-elements’, the impositions of the corporeal self, from thought of absolute space: ‘Thus altruism, or the sacrifice of egoism to others, is followed by a truer egoism, or assertion of self’ (ANE, 27). The altruism and egoism binary might be equated with space in terms of the space occupied by the self: altruism is the voiding of the self from space; egoism the occupation of all of space to the exclusion of others.

Also extended from James Hinton’s work on the continuity of the organic and inorganic, mind and matter, was a striking theory of the action of higher-dimensional thought on the brain. Charles argued that ‘it is by a structure in the brain that [the human being] apprehends nature, not immediately’. What we perceive are ‘models and representations’ in ‘minute portions of matter’ in the brain, portions ‘beyond the power of the microscope in their minuteness’ (ANE, 48). These ‘brain molecules’ do not, however, directly mimic external matter:

It may be that these brain molecules have the power of four-dimensional movement, and that they can go through four-dimensional movements and form four-dimensional structures. If so, there is a practical way of learning the movements of the very small particles of matter—by observing, not what we can see, but what we can think.

(ANE, 49)

This oscillating neurological materiality incorporates the fugitive materiality of the sub-microscopic, locating Hinton’s ethereal fourth dimension within the brain, and the brain within it.

A New Era of Thought was completed just as his bigamous marriage was discovered. Hinton had married Maud Florence under the assumed name of John Weldon. There is evidence that certain of his friends and family had known since 1884 of his mistress, with whom he had twins, but the discovery of the affair by his wife led to a trial and conviction for bigamy and Hinton’s subsequent departure from England to Japan. He wrote of it to his publisher in intellectual terms:

I have had to give up everything and go through disgrace such as rarely falls to anyone’s lot. But still, although I have lost all outward things I have got on the right side of life. In the book which you have of mine lie the steps of my reasoning.⁴³

While in Japan he worked first at a mission before being recruited as the headmaster of the Victoria Public School, a school established by the British expatriate community to commemorate Victoria’s jubilee. Records from this period of his life are scarce, and he published no work, but remained interested in both literature and space, briefly hosting a young Lafcadio Hearn and devising cubic climbing structures from bamboo for his sons.⁴⁴ While he was abroad Swan Sonnenschein published two more Scientific Romances, left by Charles with his editors: ‘On the Education of the Imagination’, treated in detail below, and ‘Many Dimensions’, an answer to the question pre-empted by A New Philosophy, ‘if four dimensions, why not five, or six, or seven?’. Hinton employed a familiar homiletic tone, relating the myth of elephants supporting the universe on a turtle’s back, a well-worn illustration of infinite regression, before slipping into his own regressive reveries about the printed page:

And yet, looking at the same printed papers, being curious, and looking deeper and deeper into them with a microscope, I have seen that in splodgy ink stroke and dull fibrous texture, each part was definite, exact, absolutely so far and no farther, punctiliously correct; and deeper and deeper lying a wealth of form, a rich variety and amplitude of shapes, that in a moment leapt higher than my wildest dreams could conceive.⁴⁵

In 1895, two years after his arrival in the USA and assumption of a teaching post at Princeton, Hinton published Stella and An Unfinished Communication. The preface announced these pieces as artistic productions continuous with his broader project:

In the following pages an attempt has been made to dwell upon the wider bearing of conceptions which, whatever their origin, have found more definite expression in the speculations of modern mathematicians than at any other time […] Just as the study of the minute or the very large requires microscopes, telescopes, and other apparatus, so for the study of the Higher World we need to form within our minds the instrument of observation, the intuition of higher space, the perception of higher matter. Armed thus, we press on into that path wherein all that is higher is more real, hoping to elucidate the dark sayings of bright faith.⁴⁶

These novellas, then, were presented as types of ‘apparatus’ for the development of higher-dimensional thought. Stella was an invisible woman narrative that literalized honesty and openness as corporeal transparency, progressing the vision of higher spatial altruism first essayed in A New Era. Stella’s guardian has experimented upon her to make the ‘coefficient of refraction of the body […] equal to one’. ‘But why should he?,’ asks Stella’s paramour: ‘Don’t you see, Hugh, being is being for others. Michael used to say that true life begins with giving up’ (S, 35). The theories of Michael Graham, the guardian, voiced by Stella and left in a journal, described Hinton’s modulating eternal return:

‘If you feel eternity you will know that you are never separated from any one with whom you have ever been. You come to a different part of yourself each day, and think the part that is separated in time is gone. But in eternity it is always there.

[…]

If you felt it you would know that you are always living in your whole life, that it is always changing, though with your eyes you can only see the part you are in now.’

(S, 30)

The narrative of ‘An Unfinished Communication’ explicated this model. A young man, wondering through a poor part of New York, sees a sign advertising lessons with an ‘Unlearner’. Intrigued, he tracks the tutor down to a seaside village. At an initial encounter he claims that he is shackled by his past. The Unlearner responds: ‘But have you ever lived? For life is where man takes up the work of nature and forms a net-work of close personal knowledge, linking each to each, preparing that body in which the soul of man lives’ (S, 120). A series of stories are told by different characters, and the narrator is directed on to a further village, where he stays with the locals for several days, learning the allegories of their lives. As he makes the journey home the narrator is overwhelmed by the tide and, while drowning, has visions in which he passes between several different first-person consciousnesses:

And a new consciousness comes over me. I see that, like everything else in Nature, our lives are altering, developing, our whole lives in every event and circumstance. I see my life suddenly transformed from the pitiful thing it is. I see that it is changing—the whole of it.

(S, 174)

In America, Hinton’s work became more focused on popularization and professionalization. In 1897 he achieved some fame by inventing and patenting a cannon for firing baseballs at practising batsmen: even this was conceived in terms of the twists that could be imparted upon the ball. He gave a paper on his higher spatial thought to the Philosophical Society of Washington, moved to jobs at the University of Minneapolis and The Nautical Almanac, before settling at the Patent Office in 1904. Essays in Harper’s magazine discussed his baseball cannon and developed the ideas he had spoken about before the Washington Society, first tentatively sketched in ‘A Picture of Our Universe’, that electrical current could be represented as ‘a vortex sheet whose edge meets the aether along the wire of the circuit’.⁴⁷ The mathematical workings behind this idea were demonstrated in an article for the Proceedings of the Royal Irish Academy.

In that year The Fourth Dimension was published by Swan Sonnenschein, a summary of his work so far that did indeed reveal a complete, flexible, and complex programme of ‘thought-mechanics’. It provided colour plates for his system of cubes; a two-part history of higher-dimensional thought, from classical philosophy to the ‘meta-geometry’ of Bolyai, Lobatchewsky, and Gauss; a ‘proof’ of his vortex-sheet theory of aethereal electrical phenomena; uses of his system of poiographs for testing logical assertions; an application of this system to Kant’s ‘Theory of Experience’; a detailed description of the tesseract, the simplest four-dimensional figure, obtained using said systems; detailed instructions for the use of the cubes; and a chapter combining a paper to the Washington Philosophical Society with his paper on Cayley. His earlier work was synthesized and drawn together, the visionary tone calmed and the facts presented as accessibly as possible.

One last work was published in the year of his death. Appropriately enough, An Episode of Flatland ended his writing career in tribute to the text that had given it such early impetus. The two-dimensional narrative of An Episode contains passages that finally approach Bruce Clarke’s ‘science fiction in utero’ description of Hinton’s work.⁴⁸ The idea, first read in A New Era of Thought, that thinking higher space would result in structural changes in the brain, comes about in Unaea when the inhabitants begin to realize the existence of the third dimension:

‘It is undoubtedly the fact,’ said the director, ‘that this new conception of existence has a marked influence on the power and scope of volition. For one thing when the children get to know that real existence has a dimension they cannot see with their bodily eyes, and has a richness of movement they cannot make with their limbs, they realise that they are beings of this higher kind, directing these extended bodies of a lower plane […] And thus I find that the very bodies of the children are undergoing modification.’⁴⁹

An Episode inserted recognizable Hintonian practical and theoretical work into the well-honed lower-dimensional setting to predict utopian advance:

But amongst those who learned by means of models, making visible and tangible the aspects and views of the higher reality, were some who sprang, with a kind of inner awakening, to the knowledge of the third dimension […] Thus the intimate knowledge of the third dimension was the key which unlocked the mystery of the minute.⁵⁰

While Hinton’s mathematical work was of limited consequence in scholarly terms, and his fictional output of limited popularity or effectiveness, his project was always conceived as something para-academic and not strictly literary: an entire system of ‘thought-mechanics’. The mechanical system that underwrote it was both highly influential and effective, a hardware and software coupling for the self-helping consciousness expansionist.

The first instalment in Hinton’s programme, ‘the means of educating’, was a system of cubes described in great detail in the second part of A New Era. Bruce Clarke admits: ‘The readable portions of his work are essentially prolegomena for and inducements to the further, impenetrable system of hyperspace-instruction.’ I want to try to penetrate this system, and to investigate what Clarke describes as ‘an arduous playfulness’.⁵¹

In the first chapter of A New Era of Thought Hinton describes the processes by which he came to the philosophy he outlines in this text:

And so in despair of being able to obtain any other kind of mental possession in the way of knowledge, I commenced to learn arrangements, and I took as the objects to be arranged certain artificial objects of a simple shape. I built up a block of cubes and giving each a name, I learnt a mass of them.

(ANE, 12)

He had already rehearsed this process in ‘Casting out the Self’, the fifth of his Romances, where he detailed how the mass he first learnt covered a cubic foot, and he could describe objects in space by referring to the names of the cubes they occupied in his mass. For public consumption this mass was resized to comprise either twenty-seven or eighty-one one-inch cubes. In Casting Hinton described the necessity of unlearning up and down and left and right in relation to arrangements, as these ‘self-elements’, ‘arising from the particular conditions under which I was placed’ (ANE, 209), did not give absolute knowledge of arrangement. He worked on relearning the cube turned on each of its sides and upside down, only realizing later in his studies the significance of the system of arrangements for understanding higher space: as ‘a kind of solid paper’ through which to imagine the sides and edges of four-dimensional shapes. Hinton invoked the speculated higher beings of What is the Fourth Dimension? to imagine their children: ‘and just as children on the earth gain their familiarity with space by means of bricks and blocks and toys, so these higher children must have their own simple objects wherewith they grow into familiarity with their complex world’ (ANE, 224). His cubes were just such a set of toys, and he went on to describe the process of casting out the self as ‘seeing as a higher child’ (ANE, 227).

On the Education of the Imagination, issued as a pamphlet in 1888 with a brief endnote by its editor, Herman John Falk, also deals with the cubes. Its endnote states 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’.⁵² A pedagogical essay, addressed to a fellow educator and referring throughout to a putative pupil, it established a broad theoretical basis.

Hinton wrote that the piece was inspired by a series of extracts from Johannes Kepler’s Cosmographicum Mysterium. In this text, subtitled ‘on the marvellous proportion of the celestial spheres, and on the true and particular causes of the number, size and periodic motions of the heavens, established by means of the five regular geometric solids’, Kepler argued that the known planets followed courses through the sky that corresponded in ratio to a nested agglomeration of the Platonic solids: a sphere containing a cube containing a sphere containing a tetrahedron, and so on. He considered the cube ‘the first solid in its class’ for nine geometric reasons, for example:

1. It alone is generated by its base […] 2. It alone can be resolved into homogeneous cubes with no prism […] 3. It alone faces in all directions, and extends in three directions at right angles.

Kepler’s final reason for the lofty position of the cube in the heavenly pantheon elevated the human subject and an embodied geometry: ‘For a man himself is like a cube, in which there are so to speak six regions: upper, lower, fore, hind, right, left.’⁵³

Hinton was particularly interested in Kepler’s remarks on ratio and the harmony of physical form, the premises for his attempts to arrange the solar system in a harmonic, Platonic fashion. Hinton’s interest in Kepler is continuous with his debt to the Timaeus, the source text for the Neoplatonic understanding of the natural universe, and its equation of each of the Platonic solids with each of the elements: air, water, fire, earth. Working with the cubes, then, was working with the Platonic basis for earth, a metaphorical grounding when Hinton found himself at an intellectual impasse.

Hinton focused on the cube as an exemplary object for exploring arrangement and form. In such use of the imagination with ‘the utmost precision of form’ he explicitly connected the generative, eidetic, spatial nature of both mathematical and literary form in the imagination, writing: ‘Each line of Dante, for instance, seems to call up a visible image and shape.’ As he probed the relationship between the sensations and the mind later fleshed out in A New Era, Hinton also cited Goethe: ‘Goethe tells us in his Farbenlehre, that, when he was studying plants, on shutting his eyes images of flowers would present themselves to him, perfectly distinct in every particular, and would arrange themselves in rosettes or other regular figures.’⁵⁴

He hoped to achieve similar results with his 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.’⁵⁵ 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.

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. He 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.⁵⁶

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. In their preface to A New Era Falk and Boole suggest using ‘Kindergarten cubes’ to follow the exercises. They see active engagement with three-dimensional objects as crucial, noting the limits of the two-dimensional page in dealing with higher space, and advocate a form of what we might now think of as kinaesthetic learning:

Indeed, we consider that printing, as a method of spreading space knowledge, is but a ‘pis aller’, and we would go back to that ancient and more fruitful method of the Greek geometers, and, while describing figures on the sand, or piling up pebbles in series, would communicate to others that spirit of learning and generalization begotten in our consciousness by continuous contact with facts, and only by continuous contact with facts vitally maintained.

(ANE, vi)

The use of the term ‘kindergarten’ makes clear another source for Hinton’s cubic system. The pedagogical theorization of the imagination in terms of Platonic solids is indebted to Friedrich Fröbel. The German educational theorist and crystallographer devised sets of children’s toys, called Gifts, in which cubes were one of the primary constituent parts used to encourage the exploration of form. When Falk and Boole refer in the preface to A New Era of Thought to kindergarten cubes, we should recuperate the etymological context: Fröbel’s term Kindergarten had been in use in English for barely thirty years and was still very much associated with its author.

The Society of Arts Educational Exhibition Collection of Prospectuses of 1854, a collection of catalogues for educational aids, from microscopes to chemicals, had advertised the Gifts for sale in England.⁵⁷ Joseph Payne’s Fröbel and the Kindergarten System of Elementary Education, ‘a lecture delivered at the college of Preceptors on the 25th of February, 1874’ and published later that year, described the objects:

The fourth, fifth and sixth gifts consist of the cube variously divided into solid paralellopipeds, or brick-shaped forms, and into smaller cubes and prisms. Observation is called on with increasing strictness, relativity appreciated, and the opportunity afforded for endless manifestations of constructiveness. And all the while impressions are forming in the mind which, in due time, will bear geometrical fruits, and fruits, too, of aesthetic culture. The dawning sense of the beautiful, as well as of the true, is beginning to gain consistency and power.⁵⁸

Fröbel’s Gifts were marketed and sold in Britain by, among others, Hinton’s publisher Swan Sonnenschein, who also published assorted translations and commentaries on Fröbel’s educational work.⁵⁹ When Hinton began using cubes in the classroom for thinking, this was a Fröbelian move; when he went into manufacturing his own cubes, there was precedent for the publisher for selling such things.

A New Era detailed the construction of a full set of cubes, including slabs and catalogue cubes, and provided a number of exercises to work with them. These had been assembled by Boole and conspicuously lack the playfulness suggested by Hinton throughout ‘On the Education’. In the hands of the adept student, the ludic roots of the system were occluded. Correspondence between Falk and Swan Sonnenschein over the models, which the editors had advised could be purchased through the publisher, shows the difficulties encountered in the realization of Hinton’s vision.

On 21 September 1888, some months after publication, William Swan Sonnenschein received an inquiry about the cubes. Sonnenschein wrote: ‘It would perhaps be as well, should this gentleman give an order for a set, to have two sets made, as it looks rather bad to have to admit that inquiries for them are unusual.’⁶⁰ Another inquiry was received in January of 1889, but it wasn’t until February that Falk provided the first sets to the publisher, who returned them, writing: ‘The workmanship of the cubes is so rough it would affect sales very badly.’⁶¹ It took Falk until November to source improved sets, with the price set at 17/6 for trade plus 20 per cent for public sales. The models sold very slowly but continued to pique interest. In 1903, Swan Sonnenschein wrote to Hinton, now resident in the USA and once again managing his own affairs: ‘Can you send me one set of your models which a lady resident in Nice is very anxious to purchase?’⁶² In 1904 a Mr Dyson returned his set.⁶³

The cubes are pedagogical objects, kindergarten toys, playthings but in the Hintonian imagination were pre-coded with Platonic atomism. They are very much building blocks; symbolic building blocks of matter, retooled as the building blocks of thought on Neoplatonist basis; an assumption that mathematical ratio and harmony are the fundamentals of the universe. The fact that Hinton’s system places a crucial emphasis on memorization is highly significant and also recalls Renaissance Neoplatonism. The art of memory had long recognized the connection of memory and space. In her gloss of the classical text on memory technique, Ad Herennium, Frances Yates records:

The artificial memory is established from places and images […] A locus is a place easily grasped by the memory, such as a house, an intercolumnar space, a corner, an arch, or the like. Images are forms, marks or simulacra [formae, notae, simulacra] of what we wish to remember. For instance, if we wish to recall the genus of a horse, or a lion, or an eagle, we must place their images on definite loci.⁶⁴

The most typical form of arrangement employed in the classical art of memory was therefore architectural: a sequence of memory rooms.

In the renaissance, Giordano Bruno travelled in Europe revealing the occult secrets of the Hermetic art of memory, collected in his De umbris idearum, prefaced by a passage in which Hermes Trismegistus himself handed a book to Bruno. Bruno’s ‘shadows of ideas’ were in the Platonic tradition: ‘shadows of reality which are nearer to reality than physical shadows in the real world’. Yates describes Bruno’s theory:

By imprinting on memory the images of superior agents, we shall know the things below from above; the lower things will arrange themselves in memory once we have arranged there the images of the higher things, which contain the reality of the lower things in a higher form.⁶⁵

In Yates’s account, for Bruno memory was a practical means to achieving transcendent knowledge: ‘When the contents of memory are unified there will begin to appear within the psyche (so this Hermetic memory artist believes) the vision of the One beyond the multiplicity of appearance.’⁶⁶ In his final work on memory, Bruno developed an architectural system of memory, based on a magical geometry. Twenty-seven atria and nine fields were each divided into nine places; thirty cubicles formed a parallel spatial organization.

The Englishman Robert Fludd propounded a similar system of fields, atria, and cubicles, distinguishing between an ars rotunda and an ars quadrata. The round art located magical, immaterial images: the square, corporeal things and objects. Fludd advocated using real places in which to arrange memory loci, and favoured theatres upon which to construct his mnemonic stage; in the drawings of these we see a quadratic arrangement.

The insistent relationship between space and memory had been known since classical times: memory is spatial and space mnemonic. Hinton’s arrangement of cubes to be memorized operated within the tradition of these Hermetic systems. It is difficult not to be struck by the parallel hopes of Bruno and Hinton for the results of their mnemonic projects: access to a higher knowledge.

How should these cubes be thought? They were material objects, childish, playful things, repurposed. Abacus or set square? From a contemporary perspective it is tempting to think of them as hardware, and the exercises their author detailed as software, for thinking space. They were a palpable failure as commodities, yet they were simultaneous with, or preceded, the models of four-dimensional projections and cross-sections that now reside in display cases as examples of mathematical models from the halcyon days of their production.⁶⁷

The system of cubes is certainly illustrative of Hinton’s modus operandi and his relative strengths as a thinker: he took a tactile, hands-on approach to abstraction; he materialized space to enable its thought. His descriptions in chapter three of What is the Fourth Dimension?, of thread passing through a thin sheet of wax, seem likely to be derived from sewing cards, such as his mother-in-law Mary Everest Boole had repurposed for geometric education with curve-stitching.⁶⁸ Hinton took everyday objects out of their everyday use to educate and in this way was a popularizer of no small skill. His insistence upon the effectiveness of his system was resolute:

And after a number of years of experiment which were entirely nugatory, I can now lay it down as a verifiable fact, that by taking the proper steps we can feel four-dimensional existence, that the human being somehow, and in some way, is not simply a three-dimensional being.

(ANE, 46)

Hinton’s cubes leave a cultural spoor of great interest that allows us to understand how they acted, what they did, and the impact they had. His first dedicated pupils—apart, perhaps, from A Square’s nephew—were his sisters-in-law. H.S.M. Coxeter, who was introduced to Mary Ellen’s younger sister, Alicia, by her nephew G.I. Taylor, and who also worked with the amateur mathematician towards the end of her life, relates the story:

He brought a lot of little wooden cubes and piled them up into shapes in his attempt to elucidate the four-dimensional hypercube, or tesseract. He set the three youngest girls the task of memorizing the arbitrary list of Latin words (Decus, Pulvis, etc.) by which he had named the little cubes. Lucy, being a child with a strong sense of duty, worked hard. Ethel found the whole project a meaningless bore and dropped out as soon as she was allowed to do so. But for Alice, age seventeen or eighteen, it was an inspiration, the mainspring of all her research.⁶⁹

Alicia’s age, in this account, dates these exercises to the late 1870s. The census of 1882 recorded her as staying with the family of her sister and brother-in-law in Uppingham. In 1887 she co-edited A New Era with Herman John Falk before moving to Liverpool in 1889 to work as his secretary. She met and married the actuary Walter Stott in 1890 and became a housewife and mother, but continued to work on the visualization of higher geometry, coining the term ‘polytope’ to describe a convex solid in four dimensions. She constructed three-dimensional models of their cross-sections and in the late 1890s, through the agency of her husband, photographs of these models were sent to Dutch geometer Pietr Schoute, who recognized similarities with his own work. He came to visit her in England and a decade-long collaboration saw papers published by Boole Stott in 1900, 1908, and 1910.

A set of her models and many of her diagrams reside at the University of Groningen, which awarded her an honorary doctorate in 1914 in honour of her work with Schoute. She returned to geometry late in life, collaborating with Coxeter, then at Cambridge University, from 1930: a further set of her models are held there. The diagrams accompanying Alicia’s work make evident the genealogy from her brother-in-law’s system, and the colouring of her models recalls the modified system of The Fourth Dimension. Her methods, developed outside formal education, produced original research. They reveal a continuation of Hinton’s work, expanding his focus on the cube to the other Platonic solids.

Alicia’s research was conducted in isolation, but other researchers in Europe and America were also producing models in the 1880s. W.I. Stringham, working at Johns Hopkins under J.J. Sylvester and alongside Simon Newcomb, rediscovered the six convex polytopes originally discovered by Ludwig Schläfli in 1858. In 1880 Stringham published a paper describing his research featuring images of what appear to be paper models of projections of these solids.⁷⁰ Viktor Schlegel built demonstration models of the projections of four-dimensional solids in 1883 which were exhibited at the 1884 meeting of the Society of German Naturalists. These were sold commercially through mathematical catalogues. Sets, in various states of repair, exist at the Smithsonian and the University of Göttingen.⁷¹ In 1900, Basil Wedmore, then a demonstrator at the Finsbury Technical College working under Silvanus P. Thompson, demonstrated models at a Friday evening discourse on ‘Transparency and Opacity’ at the Royal Institution. ‘Mr. Wedmore’s cute idea is this,’ reported the Leeds Mercury:

He says we can represent, say, a cube on a piece of paper by foreshortened squares, that is a three dimensional figure by two dimensional ones. Correspondingly we ought to be able to represent by foreshortened solid figures inside a three dimensional figure the appearance of one of fourth value.⁷²

The visual genealogy of two-dimensional illustrations of fourth-dimensional figures has been studied in depth by Linda Dalrymple Henderson. The plates with which she accompanies her text make her point eloquently: it is easy to see the visual echoes of Jouffret’s diagrams of the tesseract in work by Picasso.⁷³ What of three-dimensional representations of the fourth dimension? Without the need to do so much work, crossing a uni-dimensional representation gap rather than a bi-dimensional, models were a more readily accessible tool: they had not only a visual, but also a tactile efficacy. They were, however, far scarcer than print illustrations and their display was apparently restricted to universities or scientific societies. Where specialist mathematical texts might circulate outside specialist mathematical groupings, models did not.

Hinton’s mediation of four-dimensional space through material objects, developed in the classroom, enabled and legitimized the work of writers who insisted on the empirical reality of higher space, and provided a practical course for consciousness expansion: a practical course that produced intellectual results. His focus on the eidetic imagination fed into the concerns with visualization of esoteric belief systems. The hybridization of his practical course of four-dimensional instruction with his father’s altruistic ethical philosophy nourished these belief systems so profoundly that it is this hybrid whose influence can be discerned most keenly in New Age texts throughout the twentieth century. As Henderson has demonstrated, many of Hinton’s followers came to his work through Theosophy. The engagements of key Theosophists with Hinton’s work form the core of Chapter 5 but a handful merit mention in the context of this consideration of Hinton’s cubes.

First published in the Occult Review in 1914, Algernon Blackwood’s short story ‘A Victim of Higher Space’ makes mention of Hinton’s system. Blackwood was a committed occultist—contributor to the Theosophical magazine Lucifer, investigator alongside Frank Podmore of haunted houses, member of the Golden Dawn (and probably the Esoteric Section of the Theosophical Society)—and drew heavily on his occult reading and experiences for his fiction, turning several cases detailed in Podmore’s Phantasms of the Living into short stories.

Statically located in the flat of psychic detective John Silence and relating an interview with the improbably named Racine Mudge (shades of Browning’s medium Sludge, perhaps, and smudges of corporeal insubstantiality?), it enmeshes a range of Theosophical interests into an account of the experience of higher space explicitly sourced through Hintonian practice—indeed the use of the Hintonian term ‘higher space’ itself is the first indicator of this source.

Mudge has sought out John Silence in the hope that he might be able to assist him in combating his slipping ‘nolens volens’ into the world of higher dimensions—not just the fourth dimension but the proliferating higher dimensions of this ‘spiritual’ and ‘mythical state’. Indeed, Mudge is not at first visible in Silence’s waiting room. Looking through a spy-hole into the room, Silence at first sees only a thickening line:

Then suddenly, at the top of the line, and about on a level with the face of the clock, he saw a round luminous disc gazing steadily at him. It was a human eye, looking straight into his own, pressed there against the spy-hole […] Then, like someone moving out of deep shadow into light, he saw the figure of a man come sliding sideways into view, a whitish face following the eye, and the perpendicular line he had first observed broadening out and developing into the complete figure of a human being.⁷⁴

Telling Silence how he came to be in his current condition, to have contracted his ‘disease’, Mudge leads the doctor through a well-versed account of Hintonian higher space, referencing the progenitors of non-Euclideanism—‘the audacious speculations of Bolyai, the amazing theories of Gauss […] the breathless intuitions of Beltrami and Lobatchewsky’—before arriving at the ‘dreamer’ whose work allowed him to access higher space and some objects with which we are by now very familiar:

I procured the implements and the coloured blocks for practical experiment, and I followed the instructions carefully till I had arrived at a working conception of four-dimensional space. The tesseract, the figure whose boundaries are cubes, I knew by heart. That is to say, I knew it and saw it mentally, for my eye, of course, could never take in a new measurement, or my hands and feet handle it.⁷⁵

Through use of the ‘implements and the coloured blocks for practical experiment’, Mudge has fulfilled Hinton’s prediction and accessed the universal humanity the author theorized:

I reached sometimes a point of view whence all the great puzzle of the world became plain to me, and I understood what they call in the Yoga books ‘The Great Heresy of Separateness’; why all great teachers have urged the necessity of man loving his neighbour as himself; how men are all really one; and why the utter loss of self is necessary to salvation and the discovery of the true life of the soul.⁷⁶

The knowledge of unity, the attainment of the Theosophical and Hintonian dream of higher space, however, turn sour for Mudge as his knowledge of the higher spatial condition shift gears: ‘accidentally, as the result of my years of experiment, I one day slipped bodily into the next world, the world of four dimensions, yet without knowing precisely how I got there, or how I could get back again’.

The idea that once willed access to higher space had been achieved, unwilled or automatic regress would follow was not confined to fiction. In his column on mathematical puzzles in the Scientific American for July 1962, Martin Gardner gave a broad overview of four-dimensional geometry, establishing the dimensional analogy and working through the cross-sections and projections of the tesseract. When this article was reprinted as a chapter in his 1965 book Mathematical Carnival, Gardner included a sensational letter from Hiram Barton, a ‘consulting engineer of Etchingham, Surrey’:

A shudder ran down my spine when I read your reference to Hinton’s cubes. I nearly got hooked on them myself in the nineteen-twenties. Please believe me when I say that they are completely mind-destroying. The only person I ever met who had worked with them seriously was Francis Sedlak, a Czech neo-Hegelian Philosopher (he wrote a book called The Creation of Heaven and Earth) who lived in an Oneida-like community near Stroud, in Gloucestershire.

As you must know, the technique consists essentially in the sequential visualizing of the adjoint internal faces of the poly-colored unit cubes making up the larger cube. It is not difficult to acquire considerable facility in this, but the process is one of autohypnosis and, after a while, the sequences begin to parade themselves through one’s mind of their own accord. This is pleasurable, in a way, and it was not until I went to see Sedlak in 1929 that I realized the dangers of setting up an autonomous process in one’s own brain. For the record, the way out is to establish consciously a countersystem differing from the first in that the core cube shows different colored faces, but withdrawal is slow and I wouldn’t recommend anyone to play around with the cubes at all.⁷⁷

Strip away the sensational rhetoric—Barton’s use of the words ‘hooked’ and ‘withdrawal’ and the idea that this form of thought could be ‘mind-destroying’ clearly reference the risks of drug use and perhaps reflect the date of the composition of the letter—and some very curious concepts remain. Barton describes the process of using the cubes as one of ‘autohypnosis’. Facility in the process creates an ‘autonomous process’ in the brain. In Barton’s description it is not so much the subject who is doing the thinking as the object of thought. Despite the implicit analogies to psychedelic or narcotic drugs, the cubes are not ingested: rather, they are thought, contemplated, visualized, and memorized. They enter the mind only through the senses and yet in this account they seem to become in some way structural, seem to become part of the organ of thought itself.

The Francis Sedlak to whom Barton refers was, as well as philosopher and communal liver, a Theosophist, contributing frequent articles to the Theosophical Review from 1906 to 1908 and to The Theosophist in 1911–12. He contributed an article to Orage’s The New Age disputing Einstein’s Theory of Relativity on the grounds that Einstein was insensible to the dictates of ‘Pure Reason’. His partner in a ‘free union’, Nellie Shaw, wrote about their life together in the Whiteway Colony, including an account of Sedlak’s interest in the cubes that beds into the utopian Theosophical version of higher spatial thinking and balances out the sensational tone of Barton’s letter. I hope that readers will forgive me for quoting from this long-forgotten text at some length: it is unique as an objective, narrative account of the practice of using Hinton’s cubes:

Some readers may be acquainted with a book by C. Howard Hinton, entitled The Fourth Dimension, which contains a coloured diagram representing twenty-seven cubes of various colours. This idea was seized upon by Francis, who adapted it to his own ideas.

A box of children’s playing blocks was obtained and each one painted a different and nameable shade. So far as I am able to understand, the idea was to build up from the whole twenty-seven cubes one cube, each separate colour being in a particular relation to the next one, and then to gaze fixedly at it until the whole was mentally visualised. This accomplished, the cube was unbuilt and then rebuilt with a different combination of colour, and visualised mentally as before.

This amazing performance required hours of time at first, but gradually the speed quickened, until eventually it became focused upon the mind, and Francis was able to review the blocks in all their twenty-seven positions so swiftly, that it became almost like seeing the cube from all sides at once.

It will be realised that the changes of position were almost innumerable. At first a very hard laborious task, it became an absorbing occupation, to which was given every spare moment. Many persons, not understanding, looked on it as a most unproductive way of spending time. Others admired the wonderful patience, but could see no useful result.

Just as the would-be athlete twists and turns on the parallel bars, using time and energy to develop his muscles and gain strength which can be used later in any direction which he may desire, so Francis assumed that this power gained by practice in visualization, seeing mentally the block of cubes on all sides simultaneously, could also be used in any sphere and on any subject; in fact, it was ability to see through anything, and must eventually lead to clairvoyance.

This study of the cubes was followed intermittently, since it was not a mental exercise calling for philosophic reasoning or mental effort whatever. So, after devoting many months to the cubes and having an urge in another direction, Francis would drop them again for several years.

The extraordinary thing was that afterwards he could resume the practice without difficulty. He did not lose the power; indeed, he seemed to have a positive affection for these bits of wood, which he would tenderly dust and preserve.

Towards the end of his long and trying illness, when terrible coughing prevented him from sleeping at night, the long silent hours seemed interminable. On my enquiring one morning as to what sort of a night he had had, he said almost joyfully, ‘Oh, being awake does not trouble me now. I do the cubes, and the time flies.’ So I thanked God and blessed the cubes, for which had been found a utilitarian use at a most desperate psychological juncture. Power won cannot be lost, and will some day be utilised.⁷⁸

In this more detailed account we read of the repetitive and arduous nature of the practice of the cubes, its reliance upon memory, and how facility improved with practice. Sedlak is the most dedicated student of the cubes since Alicia, and his practice, while it does not ultimately lead to clairvoyant ability, gives him great comfort and pleasure as a meditative device, a process of unselfing.

The process of ‘casting out the self’, a form of subjective decentring, is morphological in the Goethean sense. Beyond the quotation taken directly from Farbenlehre, other sections of ‘On the Education’ are derived from Goethe’s Theory of Colour: Hinton cites the experiment of looking through a prism to examine rays of light, before coming to conclusions about the purpose of analogy and the generation of original thought that echo Goethe. Hinton writes:

Undoubtedly, every fresh structure must grow out of some previously existent one; and every idea must spring from others already in existence. This is felt by the consciousness as analogy. Thus, we see that imagination, which consists in calling up images and in superposing them, as it were, is a necessary factor in the process of thought; for, without this superposition or juxtaposition, it would be impossible to form analogies.⁷⁹

Goethe’s account of the imagination places similar weight on repetitive procedures and the catalytic power of analogy:

Imagination is first re-creative, repeating only the objects. Furthermore, it is productive by animating, developing, extending, transforming the objects. In addition, we can postulate a perceptive imagination which apprehends identities and similarities […] Here it becomes evident how desirable analogy is which carries the mind to many related points, so that its activity can unite again the homogenous and the homologous.⁸⁰

Henry Bortoft’s detailed account of Goethean ‘seeing’ describes elements that are yet closer to the practice of using the cubes as we have read described, particularly as related to the process of removing the ‘self-elements’ of up and down and left and right. Goethean seeing is also a process of imaginary inversion:

Observing the phenomenon in Goethe’s way requires us to look, as if the direction of seeing were reversed, going from ourselves towards the phenomenon instead of vice-versa. This is done by putting attention into seeing, so that we really do see what we are doing instead of just having a visual impression […] He would then repeat the observations he had made, but this time doing so entirely in his imagination without using the apparatus. He called this discipline Exakte sinnliche Phantasie, which can be translated ‘exact sensorial imagination’. In this case it would mean trying to visualize making the observations with the prism, and seeing the qualities of the different colours in the right order at a boundary as if we were producing them. This would then be transformed in imagination into an image of the colours with the boundary in the opposite orientation, and then transformed back again. This process can be repeated several times.⁸¹

When we read Goethe’s statement that ‘every new object, clearly seen, opens up a new organ of perception in us’ we can better understand Hinton’s account of neurological function in A New Era of Thought, a knottedness between thought and matter that makes for material change in the organ of thought.

We find profound correlation between thought and matter, such that the act of thought can bring the matter of the thinking thing into being, an extraordinary sensitivity of co-creation. In light of our consideration of Zöllner’s knots as quasi-objects, the cubes seem to be quasi-objects too: they bring into play a collective of higher-dimensional thinkers, clustered around and contemplating the cubes, passing them on and recommending them to others. The act of engaging with the cubes occasions a mode of thought inaccessible to those without them: they have enabled the development of a body of theory—Hinton’s higher spatial philosophy—that could not exist without them, and in this theory thinking the cubes alters the material structures of the brain. They make blurry the distinction between subject and object, thing and thought.

Michel Serres has repeatedly considered the mythical origins of geometry in the story of Thales of Miletus who, standing in the shadow of the pyramids, realizes that he can calculate their height by means of a ratio, knowing his own height and that of his own shadow. Reading this classical fable repeatedly, attending particularly to the formal organization of its central image, Serres wonders where this moment of inspiration comes from.

In the essay ‘Gnomon: The Beginnings of Geometry in Greece’, Serres focuses his attention on the pyramid, the thing that casts a shadow: ‘We do not really know why the shaft or pin is called a gnomon, but we do know that this word designates that which understands, decides, judges, interprets or distinguishes, the rule which makes knowledge possible.’⁸² The gnomon itself is a machine, argues Serres, producing automatic knowledge. It defines no position for an operator, inscribing its knowledge directly onto the sand. What does this mean for the human subject? Serres explains:

The world represents itself, is reflected in the face of the sundial and we take part in this event no more and no less than the post, for standing upright, we also cast shadows, or, as seated scribes, stylus in hand, we too leave lines. Modernity begins when this real world space is taken as a scene and this scene, controlled by the director, turns inside out—like the finger of a glove or a simple optical diagram—and plunges into the utopia of a knowing, inner, intimate subject.⁸³

The human subject becomes equivalent to the gnomon and its sublimation of the gnomon marks the emergence of the modern subject.

Serres goes on to trace the development of the gnomon. He describes the knowledge that it produces as ‘algorithmic’: series of numbers that can be plotted onto tables. This mode of knowledge production, he argues, has two components:

One which could be called mechanical and the other which could be called mnemonic. These can be described as the accumulation or recapitulation of the results of mechanical procedures or conditions of their repetition; the automaton and tables or dictionaries; hardware and software.⁸⁴

This gives us a useful working description of what the cubes are: both gnomonic and mnemonic. The cubes cast their shadows in the mind. They produce knowledge without us, within us, and through us. They are thinking things, things thought upon and things that think. And when we make fluid this relationship between subject and thinking thing, in Hintonian and Goethean tradition, we better understand Hinton’s conjecture that the brain must exist in four dimensions because we can think in four dimensions. We are encountering an example of ‘a delicate empiricism which makes itself utterly identical with the object, thereby becoming true theory’.

Serres goes on to distinguish between the gnomonic thinking of the pre-Socratics and the geometry of the post-Socratic thinkers. He considers the Meno and notes the distinction between the slave who can recall learned multiplication tables, a form of algorithmic thinking, and Socrates’s demonstrative thought. He routes through the apagogic disproof as the first demonstration: ‘To invent geometry and demonstration consists of filling the gaps of the gnomon, those of knowledge, of artificial intelligence, of algorithmic thought.’⁸⁵ The cube users, collected around and with the cubes, a community of quasi-subjects and quasi-objects, begin to fill out the gaps from the gnomon Hinton discerned and in so doing intuit not just an alternate geometry, but an alternate spatial and social imaginary. Hinton’s discovery might be the recognition of a knowledge hidden in three-dimensional shadows.

Hinton’s contribution to the idea of higher space is a catalysing hybridization. Working on a space that oscillated between empirical and ideal, he mediated this idea through the material. Bringing ideas from ethical philosophy into speculative mathematical and physical treatises, he produced work that provided a treasure trove of ideas for artists, fiction writers, satirists, and esoteric theorists. His work was ‘popular’ in its insertion into print contexts that aimed to democratize scientific knowledge, the potency of its hybridity for appropriation by and redeployment within popular social/religious movements, and its provision of a practical course of application.

The work of Hinton indicated a utopian possibility for higher space, a possibility while not endorsed by Abbott’s ambivalent satire, nevertheless gestured towards by dint of its socially and intellectually progressive mores. Bruce Clarke sees Hinton’s ‘desire not just to imagine but to inhabit the fourth dimension’ as a quest to escape from the entropic doom of thermodynamics, but reads the scopic regime of Hinton’s hyperspace as paranoid and suggests the manipulation of cubes as a defence against exterior control.⁸⁶ I tend towards the former observation: Hinton’s higher space can usefully be reconfigured as a fugitive space; a space escaping but also of escape. And perhaps it is here that its roots in the kindergarten are best understood: Hinton’s higher space is accessed through disciplined playfulness and inflected with liberation from adult concerns and morality. Its utopia is as much Neverland as Spaceland. Yet space, no matter how abstract, is inherently socio-political, as Flatland makes clear. Hinton’s escape into a space of thought operates as a bulwark, and cartographers and colonists follow. At the fin de siècle these concerns and an analogy between higher space and colonial space become clearer. A reading of the fictions of higher space demonstrating such features follows in Chapter 6.

The dual sense of both play and learning makes Hintonian theory an attractive source for an emergent generic form that has frequently been characterized as childish and somehow insufficiently serious. For the science fictions of H.G. Wells, who relentlessly portrayed his speculations as grounded in scientific materialism, the fourth dimension as theorized in the work of Hinton provided a switch through which to mediate the transcendental: a narrative sleight of hand with which to set loose the imagination into freer realms where individuals might slip between worlds and return altered. Writers of ‘weird’ fictions following Wells recognized and developed the non-human potential skulking in spaces extended beyond thought and apparently accessible only through a form of willed automatism such as Sedlak’s. As Kantian space was threatened, so was the central position of the human in the universe. Furthermore, as the subject was made permeable, so were possibilities suggested for psychological disturbances that registered this shifting of the ground beneath coherent, correlated subjectivity.

The years surrounding the publication of A New Era of Thought witnessed responses to or quotation of the idea of the fourth dimension in an array of cultural contexts and demonstrate the currency and popularity of the idea. On 14 January 1887 E.A. Hamilton Gordon gave a paper at the Science Schools Debating Society in South Kensington on the fourth dimension, citing an article by R.A. Proctor that first appeared in the Gentleman’s Magazine in 1880. When his paper was reprinted in the Science Schools Journal Hamilton Gordon also acknowledged Hinton but claimed not to have read the Romances prior to his speculations.⁸⁷ The following year a story by a student who had heard Hamilton Gordon’s paper was serialized in the same journal. ‘The Chronic Argonauts’, by H.G. Wells, although only the first of several versions of what would become The Time Machine, already contained the idea of a geometry of four dimensions, with time occupying the y-axis.⁸⁸

The poem ‘A Pure Hypothesis’ was published in May Kendall’s 1887 collection Dreams to Sell alongside comic verses that had appeared in Punch. Kendall extrapolated upwards where Abbott had extrapolated downwards: her poem imagines a lover in four-dimensioned space for whom the idea of a lower space is confounding and ‘unutterably wrong’:

He told us: ‘Science can conceive

A race whose feeble comprehension

Can’t be persuaded to believe

That there exists our Fourth Dimension,

Whom Time and Space for ever baulk;

But of these things being incomplete,

Whether upon their heads they walk

Or stand upon their feet—

We cannot tell, we do not know,

Imagination stops confounded;

We can but say “It may be so,”

To every theory propounded.’

Too glad were we in this our scheme

Of things, his notions to embrace,—

But—I have dreamed an awful dream

Of Three-dimensioned Space!⁸⁹

In February 1887 Oscar Wilde’s ‘Canterville Ghost’, serialized in Court and Society Review, playfully juxtaposed contemporary spiritualist obsessions with the older gothic tropes of the ghost story. Early in the story, the titular ghost is set upon by the children of the family who have moved into Canterville Chase: ‘Hastily adopting the Fourth dimension of Space as a means of escape, he vanished through the wainscoting, and the house became quite quiet.’⁹⁰ The family rapidly acclimatize to the ghost and write a letter to the SPR.

In 1888, H.P. Blavatsky’s The Secret Doctrine was published to a rapturous reception by the many members of the Theosophical Society, and the yet more numerous readers of esoteric journals. Over two pages Blavatsky addressed ‘the fashion of speculating on the attributes of the two, three, and four or more “dimensional Space;”’, paying particular attention to the theories of Zöllner. Inimical towards the very idea of space, she shifted emphasis onto matter, and argued that what was in fact under discussion was ‘a sixth characteristic of matter’, concluding with the prophecy ‘that in the progress of time—as the faculties of humanity are multiplied—so will the characteristics of matter be multiplied also’.⁹¹ Blavatsky’s account was brief, but its material emphasis echoed and amplified Hintonian higher space and the prophetic character of her writing chimed with Hinton’s tone in A New Era of Thought. The responses to higher space of her followers in the Theosophical Society and more or less loosely associated occultists and esoteric thinkers will be assessed in Chapter 5.