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

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" [End Page 133] (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):

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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: [End Page 134]

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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...

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