








































© Equinox Publishing Ltd. 2007, Unit 6, The Village, 101 Amies Street, London SW11 2JW

[Expositions 1.2 (2007) 241–246] Expositions (print) ISSN 1747-5368  
doi:10.1558/expo.v1i2.241 Expositions (online) ISSN 1747-5376 

Neglected Classic

In each issue of Expositions, we draw attention to a “neglected classic,” an 
important contribution to the humanities that has lost favor or fallen out 
of fashion. In particular, we spotlight works that foster interdisciplinary 
thinking. For our first issue, we present a review of a new scholarly edi-
tion of Edwin A. Abbott’s “Flatland: A Romance of Many Dimensions,” a 
beguiling piece of Victorian literature that addresses the connections—and 
conflicts—between the humanities and the sciences.

Edwin A. Abbott.  Flatland: A Romance of Many Dimensions.  Intro-
duction by Thomas Banchoff.  Princeton, NJ: Princeton University 
Press (Princeton Science Library), 2005.  136 pp.  $9.95 pb.   
ISBN 0691123667 

A.C. McCarthy, City University of New York Graduate Center

A classicist by training and an Anglican clergyman by profession, Ed-
win A. Abbott (1838–1926) published some fifty volumes, includ-
ing studies of Shakespeare, an edition of Francis Bacon’s Essays, and a 
number of theological works. He corresponded with George Eliot and 
was intensely interested in the cause of women’s education several dec-
ades before female students could take university degrees. As the head-
master of the prestigious City of London School from 1865–1889, 
Abbott initiated a cutting-edge curriculum that required the study of 
English literature and chemistry and offered instruction in advanced 
mathematics and shorthand. Yet, despite his involvement in the rich 
intellectual life of the second half of the nineteenth century, Abbott 
is best known today for a single novel, Flatland: A Romance of Many 
Dimensions.

First published anonymously in 1884 and continuously in print ever 
since, Abbott’s Flatland tells the story of A. Square, a member of the 
professional class in a two-dimensional reality. Through the interven-
tion of a mysterious Sphere, he journeys to the three-dimensional Spa-
celand and becomes convinced of the truth of the “Gospel of Three-
Dimensions”—and the possibility of even higher-dimensional spaces. 



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By narrating the efforts of the Sphere to convince A. Square of the 
reality of the third dimension through example and analogy, Abbott 
gives his three-dimensional readers a sense of the method by which 
they, too, might be able to infer other conceptions of reality that evade 
the traditional standards of scientific proof, human perception, and 
language itself. The analogical method of Flatland has proved flexible 
enough to serve as a touchstone for more than a century of scien-
tific investigations, and the book has something of a cult status among 
mathematicians. In his introduction to the recently reissued Princeton 
Science Library edition of Flatland, leading Abbott scholar Thomas 
Banchoff, a mathematics professor at Brown University, notes that, as 
today’s computer animation technology makes it possible to visually 
represent multi-dimensional objects, “there is no better introduction 
to the problem of dealing with higher-dimensional slicing phenomena 
than this book” (xvii). 

However, the mathematical usefulness of Flatland cannot fully ac-
count for its enduring popularity, nor explain the novel’s relevance in 
the context of the liberal arts. Victorian readers understood Flatland 
as a satire of the more repressive aspects of nineteenth-century British 
society, particularly its restrictions on women’s opportunities and the 
privileging of social status over ability. Anxiety about identity and the 
potential for deceit underlies the whole of Flatland’s politics, which 
provide for the control of the lower classes (represented as Isosceles 
triangles), restrictions on women (who are invariably straight lines in a 
world of triangles, squares, and polygons), and repression of Irregular 
shapes. Social interaction is a matter of life and death: 

If even the angle of a respectable Triangle in the middle class is not 
without its dangers; if to run against a Working Man involves a gash; 
if collision with an officer of the military class necessitates a serious 
wound; if a mere touch from the vertex of a Private Soldier brings with 
it danger of death;—what can it be to run against a Woman, except 
absolute and immediate destruction? (12)

Flatland is, moreover, a society that has traded aesthetic creativity and 
intellectual discovery for safety and security: the practice of painting 
was banned after the Chromatic Sedition threatened to make women 
visually indistinguishable from the priestly Circle class.  



Neglected Classic: Flatland: A Romance of Many Dimensions 243

© Equinox Publishing Ltd. 2007

The construction of Flatland society allows Abbott to foreground 
the disruptive power of higher dimensionality. Seemingly secure bi-
naries, such as inside/outside, are revealed to be dimensionally-bound 
constructions. Just as A. Square, a being of two dimensions, is able 
to view the “insides” of Lineland’s one-dimensional inhabitants, the 
Sphere’s three-dimensional perspective gives him access to Flatland’s 
private inner spaces. A. Square first reacts to the Sphere’s visitation as a 
kind of home invasion, and for all his talk about male “rationality” (as 
diametrically opposed to feminine “affections”), part of his resistance 
to the Sphere’s logical argumentation originates in a powerful feeling of 
violation. He is deeply offended when the Sphere touches him in the 
stomach—a feat that no two-dimensional being could perform on an-
other: “It seemed intolerable that I should endure existence subject to 
the arbitrary visitations of a Magician who could thus play tricks with 
one’s very stomach” (79). Visiting Spaceland proves that the Sphere is 
no magician, but it cannot allay the more deeply disturbing implication 
that higher dimensions of perception do not correlate with increased 
moral sensibility. When A. Square moves to worship the Sphere for 
his “omnividence”—a divine attribute in the teachings of Flatland’s 
theologians—the latter replies: “Is it so indeed? Then the very pick-
pockets and cut-throats of my country are to be worshipped by your 
wise men as being Gods: for there is not one of them that does not 
see as much as you see now” (82). Indeed, no advanced visualization 
techniques can dispel the deeply disturbing realization of the fragility 
of the distinction between inside and outside, human and divine. The 
spiritual shock of the inter-dimensional encounter is as much a part of 
Flatland’s story as what it has to teach us about scientific perception.

In his introduction, Banchoff contrasts Flatland’s first section with 
fictional travelogues told by visitors to exotic lands. “It is as if the story 
of Gulliver were told by the mayor of Lilliput or the adventures of Alice 
by the White Rabbit,” he writes (xxii). However, both Lilliput’s mayor 
and the white rabbit are characters with stable “insider” status in their 
communities. The situation of Abbott’s narrator is at once more compli-
cated and more poignant, for his experience in Spaceland isolates him 
from his countrymen both intellectually and physically. He is arrested 
and thrown in jail at the end of the novel for speaking publicly about 



244  Book Reviews

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the third dimension, and it is here that he remains, vainly attempting 
to hold on to his memories of Spaceland, reduced to the quixotic man-
tra, “Upward, not Northward” (103). A. Square’s imprisonment makes 
his often approving descriptions of the social arrangements of Flatland, 
even those he apparently repudiates in his confinement, particularly—
and tragically—ironic. Though the author’s introduction to the second 
edition (published just a month after the first) claims that A. Square 
has relaxed some of his more extreme views on women and the lower 
classes, he evidently persists in regarding himself as a unique case of 
wrongful imprisonment in a system that is otherwise sound.

The central sections of Banchoff’s introduction ably situate Flatland 
within the context of Victorian science as well as that of modern com-
puter animation techniques, which use the methods devised by Abbott 
and others as the basis for creating visual representations of objects in 
four dimensions and beyond. The clarity of his explanations of Flat-
land’s mathematics will be especially helpful to readers from humani-
ties disciplines, whose scientific education may have stopped at the 
undergraduate or high school level. He anticipates some of the confu-
sions that may arise from popular misconceptions of current science, 
such as the assumption that the fourth dimension is always time. For 
A. Square, Banchoff notes, time would be the third dimension, and

for beings whose universe has a spatial fourth dimension, the fifth 
dimension would be the one assigned to time, and in general we can 
expect that time will come in right after the dimensions that are re-
served for space. (xxvii)

The structure of Banchoff’s introduction, however, privileges the 
mathematical aspects of Flatland and treats them as largely independ-
ent of the novel’s other concerns. While it is certainly appropriate to 
dwell more fully on the former in an introduction aimed at an au-
dience with scientific interests, Banchoff perhaps overstates his point 
when he claims that the 

major reason for our interest in Flatland [is] that for the first time we 
can achieve some of the dreams of our ancestors a century ago and 
obtain direct visual experience of phenomena in a dimension higher 
than our own. (xxvi)



Neglected Classic: Flatland: A Romance of Many Dimensions 245

© Equinox Publishing Ltd. 2007

His examination of Flatland’s social satire, style, and historical context 
appear perfunctory next to the more fully realized section on “Flatland 
and New Modes of Visualization.” The latter section, for instance, con-
tains a number of references to other works on the dimensional anal-
ogy, but he does not provide any suggestions for further reading on 
other subjects, implying that the literary and social levels of Flatland 
are so self-evident as to need little examination. 

Banchoff provides a more in-depth discussion of the philosophical 
and historical sources of Flatland in his article “From Flatland to Hy-
pergraphics” (1990), which places Abbott more fully in his religious 
and intellectual milieu. A less simplistic treatment of Flatland’s rela-
tionship to nineteenth century culture appears in an article by Elliot 
Gilbert, who argues, “in Flatland we are dealing with an unassuming 
but insightful study of a persistent nineteenth-century preoccupation, 
the quest for new creative directions in a culture powerfully—if more 
and more uncomfortably—committed to history and tradition.” How-
ever, if Banchoff too quickly dismisses the complexities of Flatland’s so-
cial dimensions for modern readers, Gilbert tends towards the opposite 
extreme of marginalizing the science of the novel’s “quirky geometric 
fantasy” (Gilbert 1991, 396), a view of the novel obviously not shared 
by mathematicians. 

Perhaps the fullest and most balanced portrait of Abbott can be found 
in Ian Stewart’s introduction to and notes in The Annotated Flatland 
(2001). While he gives ample space to the mathematical concepts in the 
novel (and includes a number of detailed diagrams), Stewart presents 
Abbott as a consummate Victorian intellectual and clergyman, who was 
concerned not only about the search for scientific knowledge but about 
its spiritual consequences, an orientation that links Abbott to more ca-
nonical nineteenth-century authors such as Arnold and Tennyson. He 
draws upon Abbott’s other writings, particularly The Spirit on the Waters 
(1897), to illuminate the principles set forth in Flatland’s dedication, 
which wishes that we, the inhabitants of Spaceland,

May aspire yet higher and higher / To the Secrets of Four Five or 
even Six Dimensions / Thereby contributing / To the Enlargement 
of the Imagination / And the possible Development / Of that most 



246  Book Reviews

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rare and excellent Gift of Modesty / Among the superior races / Of 
Solid Humanity.

Banchoff, for his part, pays little attention to Abbott’s theology and 
perhaps overstates the uniqueness of his view that “the techniques of 
literary criticism developed by classical scholars should be used with 
the same vigor to examine the Scriptures” (xxi). At a time when science 
and religion are frequently portrayed as being mutually exclusive, if 
not actually hostile to each other, the marginal role given to Abbott’s 
theology represents a missed opportunity for a profitable discussion.  

Given the interdisciplinary range of Abbott’s own interests, it is clear 
that Flatland should be read at the intersection of science and the hu-
manities. Indeed, the most important reason for reading and discuss-
ing Flatland, particularly in an undergraduate classroom, lies not in 
what it teaches about dimensionality or Victorian culture specifically, 
but in the process of engaging the methods from across the liberal arts 
curriculum. The brevity and readability of Flatland, not to mention 
its social, mathematical, and political suggestiveness, makes it a quin-
tessentially “teachable” text, offering numerous possibilities for class-
room discussion. Though instructors will want to have Stewart’s more 
comprehensive edition on hand, the reprint of the Princeton edition 
provides an ideal starting point for undergraduate reading in multiple 
dimensions. Moreover, it gives humanities scholars in particular the 
opportunity to make the non-scientific possibilities of Flatland as com-
pelling as Banchoff has made its mathematics.

References
Banchoff, Thomas

1990 From Flatland to Hypergraphics: Interacting with Higher Dimensions. 
Interdisciplinary Science Reviews 14(4): 364–72.

Gilbert, Elliot L.
1991 “Upward, but not Northward”: Flatland and the Quest for the New. Eng-

lish Literature in Transition (1880–1920) 34: 391–404.

Stewart, Ian
2001 The Annotated Flatland.  Cambridge, MA: Perseus.


