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Edwin Abbott, edited by Lila Marz Harper
Broadview Editions
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Bill Wood
, on

Part mathematical exploration, part satire, and part fairy tale, Flatland: A Romance of Many Dimensions by Edwin Abbot has been around for more than a century and remains a standard in mathematics education. Mathematics does not have a lot of books like that. The Broadview Edition of the book combines the text with a variety of notes and essays that enhance the reading and study of this classic.

Flatland was published in 1884 and tells A Square’s story as a citizen of his two-dimensional world of Flatland, where the residents are polygons with an elaborate class structure based on regularity and number of sides. The first part of the book is A Square’s description of his society, doubly serving to allow the reader to think carefully about the limitations of dimension and to satirize Abbot’s own Victorian England. The second part documents A Square’s visits to one-dimensional Lineland and three-dimensional Spaceland, ultimately inviting the reader to consider by analogy a four-dimensional world.

To consider Flatland as satire one must have some understanding of Abbot’s time. This edition begins with a thorough, concisely written, and meticulously referenced introduction that takes the reader through Abbot’s life, career, and the issues of the day. Responses, sources, mathematical background, and influences are discussed in this introduction, with plenty of reference to the relevant source material found later in the appendices.

The text of Flatland (the second edition, with its updated preface) is faithfully reproduced and features plenty of footnotes that offer clarifications of language, discussions of influences, historical context, and other elaborations.

What sets this annotated edition apart from others is the inclusion of several appendices of additional material. The first appendix features contemporary reviews of Flatland. The next appendix includes a variety of sources, including excerpts from Plato’s allegory of the cave, Dostoevsky’s The Brothers Karamazov, and two contemporary essays on dimension. Later appendices include additional writings by Abbot and several essays about or influenced by Flatland. This material is not presented in isolation, as introductions and footnotes in each source connect it to the main text and the other sources.

The edition is an essential beginning for a scholarly study of Flatland. The inclusion of the source material in the appendices will be particularly useful to those wanting to teach the material with a philosophical or historical focus. Every mathematical consumer should give Flatland a look at some point and this affordable and thorough edition would be a fine one to have.

Bill Wood lives in Conway, Arkansas where he does a variety of mathematical things.

Appendix A: Contemporary Reviews

  1. The Oxford Magazine (5 November 1884)
  2. From The Literary World (14 November 1884)
  3. The Exchange with The Athenaeum (November-December 1884)
  4. The Architect (15 November 1884)
  5. R.Tucker, Nature (27 November 1884)
  6. New York Times (23 February 1885)
  7. From the New York Tribune (6 March 1885)
  8. Advertisement Run by Robert Brothers Publishers in The Literary World (21 March 1885)

Appendix B: Sources and Influences

  1. From Benjamin Jowett's Translation of Plato's Republic (1871)
  2. From Hermann von Helmholtz, "The Axioms of Geometry" (1870)
  3. From Fyodor Dostoyevsky, The Brothers Karamazov (1879-80)
  4. From C.H. Hinton, "What is the Fourth Dimension?" (1884)

Appendix C: Other Works by Abbott

  1. From The Kernel and the Husk (1886)
  2. From The Spirit on the Water: The Evolution of the Divine From the Human (1897)

Appendix D: The Influence of Flatland

  1. From A.T. Schofield, Another World (1905)
  2. From C.H. Hinton, The Fourth Dimension (1904)
  3. From C.H. Hinton, An Episode of Flatland: or How a Plane Folk Discovered the Third Dimension (1907)

Appendix E: Mathematical Background

  1. Macmillan's Catalog of Geometry Textbooks (1884)
  2. From Euclid's Elements
  3. The T.H. Huxley-J.J. Sylvester Debate (1869-77)