Topological Poetics: Gerard Manley Hopkins, Nineteenth-Century Mathematics, and the Principle of Continuity

In the nineteenth century, an important bifurcation occurred in the way that mathematicians thought about geometrical form. Traditional Euclidean geometry is metrical or quantitative; it is concerned with measurement, and the equality or proportion of lines, areas, and angles. However, newer fields such as descriptive geometry and topology focused not on measurement or quantification, but instead on invariant properties of form which are preserved under different types of continuous transformation. In this article, I argue that a similar distinction between quantitative and qualitative approaches to form is also apparent in nineteenth-century poetry. The work of Gerard Manley Hopkins, in particular, embodies a topological poetics at a number of different levels, from his theory of inscape and use of sprung rhythm through to his flexible sonnet forms and interests in philology. Hopkins is especially attentive to the relationship between what he calls "chromatic" (continuous) and "diatonic" (discontinuous) principles of form, in a way that reflects some of the central concerns of topologists and descriptive geometers.¹ More broadly, I show that there are parallels between nineteenth-century debates about quantitative and accentual prosody and the division of geometry into quantitative (Euclidean) and descriptive or topological approaches. Both quantitative prosody and Euclidean geometry have roots in the classical tradition, but, over the course of the nineteenth century, geometers and prosodists alike begin to develop newer and more flexible versions of geometry, and methods of scansion. An understanding of topology, I argue, not only allows us to connect different aspects of Hopkins's poetic practice together in new ways, but also shows us how they fit into a broader intellectual history of thinking about form.

In an 1854 lecture, mathematician James Joseph Sylvester explains that geometry "resolves itself naturally into two great divisions."² The older and more traditional branch of geometry was Euclidean geometry, which, as Sylvester explains, is a "geometry of measurement" (CMP, 2:8). As an example of a "metrical or quantitative" geometrical proposition, Sylvester cites the fact any two triangles with the same base and height will have the same area (CMP, 2:8):

Figure 1.

The two triangles pictured above both have a base of the same length (b), and the same height, measured perpendicularly to the base (h). Thus, the two triangles have the same area (b × h ÷ 2).

In order to determine that the base, height, and area are the same in both triangles, we must be able to measure them—this proposition thus inherently requires us to consider measurement and quantity. However, a more modern approach to geometry eschewed measurement and quantification, instead studying the non-metrical properties of geometrical figures which remain invariant under certain types of continuous transformations. Sylvester calls this approach the "geometry of position," or "geometry descriptive or morphological" (CMP, 2:8). In this article, I will usually refer to this as "descriptive geometry," following Sylvester's usage; it is also sometimes known as "projective geometry," for reasons to be explained shortly.³ Sylvester cites Blaise Pascal's so-called Mystic Hexagram as an example of a purely descriptive, non-metrical geometrical theorem.⁴ To construct this hexagram, "take two straight lines in a plane, and draw at random [six] other straight lines traversing in a zigzag fashion between them," finishing back at the starting point, to obtain a "twisted hexagon" (CMP, 2:8). The intersections of the first and fourth lines, the second and fifth lines, and the third and sixth lines will all lie on a straight line:

Figure 2.

A representation of Pascal's Mystic Hexagram, as described by Sylvester.

At no point does this proposition mention the lengths of any of these lines, or the sizes of the angles or areas enclosed between them, either in absolute or relative terms. As Sylvester writes, "This is a purely descriptive proposition, it refers solely to position, and neither invokes nor involves the idea of magnitude" (CMP, 2:9). It is an example of a descriptive, not a metrical, way of thinking about...