The Emergence of the Fourth Dimension: Higher Spatial Thinking in the Fin de Siècle by Mark Blacklock, and: Algebraic Art: Mathematical Formalism and Victorian Culture by Andrea K. Henderson (review)

The development of higher-dimensional and non-Euclidean geometries in the latter half of the nineteenth century called into question the intuitive correspondence between geometric description and the natural world. If mathematicians can produce coherent theorems and proofs treating four-dimensional or curved space, then what does that imply about the spaces they describe? Do these spaces really exist, and if so, does that mean that the flat, Euclidean space we feel ourselves to occupy is a fiction? While this representational crisis can arguably be traced back to René Descartes's invention of analytic geometry, which translates the axioms of Euclidean space into abstract algebraic formulas, it was the exotic new Victorian mathematics that brought these questions to the forefront of popular consciousness.

Two recent books trace the development of new mathematical models in the Victorian period: Mark Blacklock's The Emergence of the Fourth Dimension: Higher Spatial Thinking in the Fin de Siècle, and Andrea K. Henderson's Algebraic Art: Mathematical Formalism and Victorian Culture. Blacklock's book is an erudite and exhaustively researched history that crosshatches back and forth, in alternating chapters, between broader cultural responses to the development of n-dimensional Euclidean geometries and literary treatments of higher space. He lays the groundwork for the larger themes of the study with a deep dive into the philosophical pre-history of spatial thought in Immanuel Kant and in debates among Victorian science writers about the status of Kantian space, to which the development of n-dimensional geometries posed conceptual challenges. The book then turns to a discussion of spiritualism and the impact that n-dimensional geometries had on psychical research (and vice versa). As some spiritualists speculated, supernatural phenomena could be explained as the interventions of beings from higher dimensions into our three-dimensional world. Think of how shocked the two-dimensional narrator of Edwin A. Abbott's Flatland (1884) is when a sphere suddenly seems to appear out of nowhere when it intersects with his planar world by moving downward from the third dimension. Such intersections with four-dimensional beings, some spiritualists believed, might explain the appearance of ghosts and other seemingly supernatural phenomena in our own world. Their case is bolstered by the work of German astrophysicist-cum-spiritualist Johann Carl Friedrich Zöllner, who undertook a series of knot experiments with the medium Henry Slade that supposedly proved Slade's thesis that intelligent beings from higher dimensions intervene in our 3-D world. Subsequent chapters of Blacklock's book analyze Flatland at length; discuss the work of mathematician, fiction writer, and higher-dimension popularizer Charles Howard Hinton; consider the relationship between n-dimensional geometries and Theosophy; and examine treatments of higher space in late-Victorian and Edwardian fiction.

Blacklock's elegant study is at its best when diving into the voluminous and internecine debates between Victorian writers and researchers on seemingly abstruse geometrical questions (my brief chapter summaries do not do justice to the complexity and thoroughness of his discussions), and when drawing out the philosophical implications and broader cultural impact of the new geometrical theories. The final chapter on fiction will be of particular interest to literary scholars: it not only discusses the appearance of higher space as a fictional theme, but also speculates about the impact of n-dimensional thinking on the development of narrative technique and literary form at the fin de siècle. The chapters on spiritualism, Theosophy, and Hinton are fascinating and relatively autonomous studies that expand and complicate previous scholarship in the former two cases, and redress a relative dearth of scholarship in the latter. Surprisingly, the chapter on Abbott...