• 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):

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]

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 geometrical form.

Descriptive geometry largely grew out of the work of early nineteenth-century French mathematicians such as Gaspard Monge and Jean-Victor Poncelet, although elements of the subject are prefigured in the work of Girard Desargues and Pascal.5 In fact, however, its roots go back to the development of perspective drawing in the Renaissance; as John L. Bell writes, descriptive or "projective" geometry is a "rare example of a mathematical discipline whose origins lie entirely in art."6 William Barnes—now best remembered for his dialect poetry and philological work—published a treatise on perspective drawing in 1842 which employs methods similar to those used in projective geometry, demonstrating, for instance, how to draw the "perspective projection of a circle."7 [End Page 135]

The metrical properties of the figure on the right have been distorted by the angle of projection. It now appears as an ellipse, rather than a circle; its width is no longer equal to its height. Nevertheless, there are other aspects of the figure which remain constant—for instance, the intersections of the lines of the grid overlaying the circle on the left correspond to points depicted in the perspective projection. Projective geometry, essentially, concerns itself not with the metrical properties of a figure, but instead with properties which remain invariant under different types of continuous transformation, such as a change in the angle or distance of projection. This is the "principle of continuity," which, as Joan L. Richards writes, "asserted that theorems proved for one figure were equally true for correlative figures formed from the original by a continuous transformation."8

Another important mathematical paradigm for thinking about form in a non-quantitative fashion emerged in topology. While projective geometry was mostly concerned with continuous transformations based around projection and section, topology in the nineteenth century was principally studied in the context of knot theory. As P. G. Tait writes, "once a knot is made on a cord, and the free ends tied together, its nature remains unchangeable, so long as the continuity of the string is maintained, and is therefore totally independent of the actual or relative dimensions and form of any of its parts."9 When discussing the fundamental form of a knot, it is usual to reduce it to [End Page 136] its simplest configuration, that with the minimum number of crossings. For instance, the following diagram, given by Tait in his article "On Knots," has not been reduced to its simplest form.10

Both crossings in this knot are what Tait calls "nugatory"—they can be undone simply by twisting or pulling the knot.11 The following diagrams of knots, however, have been reduced to their simplest configurations. The first knot is a trefoil knot, with three crossings; the second has four crossings; and the third and fourth are two distinct varieties of knots with five crossings each:

No amount of twisting, pulling, or stretching will undo these knots, or transform the first knot into the second, the second into the third, or the third into the fourth. In order to undo any of these knots, or to transform one into another, we would first need to cut the cord—a discontinuous transformation, which is not permitted in topology. As Tait writes, topology deals with "those space-relations which are independent of measure, though not always of number"; it is a "qualitative" approach to geometry, in contradistinction to "ordinary geometry in which quantitative relations chiefly are treated."12 Like descriptive or projective geometry, topology is concerned with form, but not with [End Page 137] measurement—it relates to invariant properties of structure which persist throughout continuous transformations.

In Forms: Whole, Rhythm, Hierarchy, Network, Caroline Levine uses the term "affordance," taken from design theory, to designate the "potential uses or actions latent in materials and designs"—thus, for instance, "Glass affords transparency and brittleness. Steel affords strength, smoothness, hardness, and durability."13 A knot affords the possibility of a unique identity defined through configuration rather than through measurement; it also affords flexibility, stretchiness, and continuous transformation. In the rest of this article, I want to suggest that the approach to thinking about form that emerges in branches of nineteenth-century mathematics such as descriptive geometry and topology also manifests itself in poetic experimentation.14 Understanding the affordances of these approaches to mathematics can thus help us to better articulate exactly how these types of poetic forms work.

As Sylvester once argued, "The incongruity between advanced mathematics and verse composition is more apparent than real."15 An avid amateur poet, in addition to a renowned professional mathematician, Sylvester extended his distinction between "quantitative or metrical" and "descriptive or morphological" form to his poetic theory. In 1869, he addressed the British Association for the Advancement of Science with a polemical speech in defense of the beauty, interest, and importance of mathematics, extolling, in particular, the principle of Continuity as the "pole-star round which the mathematical firmament revolves, the central idea which pervades as a hidden spirit the whole corpus of mathematical doctrine."16 He republished this defense the following year alongside a rather idiosyncratic work entitled The Laws of Verse, on the grounds that the "great law of Continuity" was foundational to both mathematics and poetics (LV, 14–15). As Daniel Brown discusses in The Poetry of Victorian Scientists, Sylvester not only recognized the importance of continuity to both of these subjects, but made them "intimate through a shared terminology," drawing much of the vocabulary he uses to describe poetry from the language of mathematics.17 In The Laws of Verse, Sylvester divides versification into "Metric" (discontinuous) and "Synectic" (continuous) components (LV, 10).18 He dispatches the metrical aspects of verse ("Accent, Quantity, and Suspensions," LV, 10) fairly quickly, declaring that he subscribes to Edgar Allan Poe's principles as expressed in his "Rationale of Versification," that "the substratum of measure is time; that an accented syllable is a long syllable, and that an unaccented [End Page 138] syllable is a short one of varying degrees of duration, and that feet in modern metre are of equal length" (LV, 64–65). For Sylvester, meter, like Euclidean geometry, is therefore fundamentally quantitative.

However, just as he prefers descriptive to Euclidean geometry, Sylvester is also more interested in the "continuous" or "synectic" aspects of verse (LV, 10), and in particular what he calls "Phonetic Syzygy" (LV, 11).19 Sylvester had earlier used the term "syzygy" in a mathematical context, with reference to "the members of any group of functions, more than two in number, whose nullity is implied in the relation of double contact," before later importing the word into his poetics.20 In the context of poetry, as Sylvester's colleague at Johns Hopkins University, Sidney Lanier, explains, syzygy occurs when a line of verse exhibits a "succession of the same, or similar," consonant colors, "without reference to whether they occur at the beginning (alliterative letter), at the end (junction consonant) or in the body, of words."21 As Sylvester himself notes, "alliteration and rhyme … are under one point of view only extreme cases, or as we say in algebra, limiting forms of phonetic syzygy" (LV, 45–46n). Through phonetic syzygy, sounds are "regularly introduced and carried out of the verse-canvas: suspended, prepared, recalled, played with, as it were, before finally let go, concentrated, diffused, crossed, perplexed, and interlaced" (LV, 46n). For instance, to draw an example from his own verse, he declares that the line "Maidenhood unindovined" is "full of syzygy"; in this case, it contains a repetition of "d" and "n" sounds.22 Syzygy thus emphasizes the "continuous" properties of verse; without it, verse is "utterly flabby and limp, as void of backbone as a jelly-fish," and will "fail to make any continuous or solid impression on the minds or organs of the hearer" (LV, 46n). In Sylvester's Laws of Verse, therefore, we can see a deliberate attempt to map something like the split between Euclidean and descriptive geometry onto poetics. According to Sylvester, meter, which divides the line up into discrete feet of equal lengths, is both quantitative and discontinuous; on the other hand, the synectic properties of verse share, with descriptive geometry, a focus on continuity—not of space, but of "sound" and "mental impression" (LV, 15).23

Like Sylvester, Hopkins also used the principles of continuity and discontinuity to structure his ideas of form and aesthetics. In his Platonic dialogue, "On the Origin of Beauty," one of the characters describes the pattern of three dots in a triangle (∵)—which "mak[es] the sign of therefore because in Mathematics," and which is exemplified in the detail of Laura's dress in Dante Gabriel Rossetti's illustration for [End Page 139] "Goblin Market"—as an "extreme case of non-continuous or disjoined lines," noting the "contrast between the continuity, the absolutely symmetrical continuity, of the straight lines wh. are the sides of the suggested triangle, and the discontinuity, if I may use the word, the emphasised extreme discontinuity, of the three dots" (4:155).24

He goes on to establish a division between "transitional and abrupt" or "chromatic and diatonic" beauty (CW, 4:157). Hopkins makes similar distinctions between chromatic and diatonic principles in his essay "On the Signs of Health and Decay in the Arts," in which he writes that [End Page 140] beauty can be divided into two kinds, expressed variously as "abrupt and gradual," "parallelistic and continuous," "intervallary and chromatic," and "quantitative and qualitative" (CW, 4:108). Several critics have noted that Hopkins often seems to show a "distinct preference for diatonic beauty"—Jude V. Nixon, for instance, writes that Hopkins prefers "diatonicism to chromaticism, fixity to flux."25 The evidence most commonly cited for this claim is Hopkins's undergraduate essay, "The Probable Future of Metaphysics," in which he associates chromatism with evolution, the "ideas so rife now of a continuity without fixed points, not to say saltus {leaps} or breaks, of development in one chain of necessity, of species having no absolute types and following only accidentally fixed" (CW, 4:289):

To the prevalent philosophy and science nature is a string all the differences in which are really chromatic but certain places in it have become accidentally, one might say arbitr fixed and the series of fixed points becomes an arbitrary scale. The new Realism will maintain that in musical strings the roots of chords, to use technical wording, are mathematically fixed and give a standard by wh. to fix the all the notes of the appropriate scale: when points between these are sounded the ear is annoyed by a solecism, or to analyse mo deeper, the mind cannot grasp the notes of the scale and the intermediate sound in one conception; so also there are certain designs forms wh. have a great hold on the mind and are always reappearing and seem imperishable, such as the designs of Greek vases and lyres, the cone upon Indian shawls, the honeysuckle moulding, the fleur-de-lys, while every day we see designs both simple and elaborate wh. do not live and are at once forgotten; and some pictures we may long look at and never grasp or hold together, while the compositions composition of others strikes the mind with a conception of unity wh. is never dislodged: and these things are inexplicable on the theory of pure chromatism or continuity—the forms have in some sense or other an absolute existence. It may be maintainable then that species are fixed and to be fixed at only at definite distances in the string and that the developing principle will only act when the precise conditions are fulfilled. To ascertain these distances and to point out how they are to be mathematically or quasi-mathematically expressed will be one work of this metaphysic.

(CW, 4:289–90)

In this passage, Hopkins critiques the philosophy of chromatism, continuity, and flux, and instead suggests a "new Realism," which will recognize certain "imperishable" forms as being "mathematically fixed," like the roots of musical cords, anchored at "definite distances" along a string. Initially, this does seem like a rejection of the principle [End Page 141] of continuity which undergirds descriptive geometry or topology. As Brown notes, however, Hopkins does not dismiss chromatism entirely, only the theory of "pure chromatism":

He both insists that natural "type[s] or species" are like "the roots of [musical] chords … mathematically fixed" and delights in "Áll things counter, original, spare, strange" ["Pied Beauty," 7]. He accordingly sets himself the delicate task of reconciling the two principles in a synthesis that diminishes neither one for the sake of the other. In the midst of the Darwinian assault upon essentialism he seeks a new affirmation of form that acknowledges and is able to incorporate the fluidity of nature.26

The knot is an example of just such a form. Although a knot can be stretched, pulled, or otherwise subjected to continuous or "chromatic" transformations, it is still defined (or "mathematically fixed") by the number and configuration of its crossings in its simplest state. These crossings are not fixed "only at definite distances," like the notes of a musical chord on a string, but they do secure a certain "absolute existence" to that form, establishing each knot as a particular immutable type. The number of crossings is, moreover, a discrete property—it is not possible to have a knot with, say, four-and-a-half crossings. As such, the knot combines elements of both the chromatic and the diatonic.

The extent of Hopkins's familiarity with nineteenth-century mathematical developments such as descriptive geometry and topology is uncertain. Hopkins's interests in the sciences have been well documented by Brown, Nixon, Tom Zaniello, Gillian Beer, and Marie Banfield amongst others, while John Kerrigan has illuminated his engagement with mathematics and number.27 We know that he was familiar with some of P. G. Tait's work on physics, as he references Tait's 1884 book Light in a letter to Richard Watson Dixon in 1886 (CW, 2:799). It is also possible that he read Tait's obituary of Listing, containing a description of his work on topology, which appeared in Nature in February 1883, right in the middle of the period when Hopkins himself was publishing a series of letters in this journal, from November 1882 to October 1884 (see CW, 2:549).28 Sylvester's address to the British Association for the Advancement of Science (BAAS), which underscored the importance of the "Science of Continuity" to modern mathematics, had also been published in Nature under the title "A Plea for the Mathematician," in 1869–70.29 Although there is no positive record of Hopkins having read Sylvester's speech, we know that he did read other addresses delivered to the BAAS—he mentions [End Page 142] in an 1874 letter to his mother that he had read Tyndall's presidential address from that year (CW, 1:237).

Whether or not Hopkins was familiar with topology in a mathematical context, however, his journals and diaries indicate an interest in knots which predates even Tait's scientific work on the subject. His diary from 1864, for instance, contains several interesting knotted designs (CW, 3:207):

At first glance, these drawings might merely seem to resemble some of the other sketches of windows or tracery found in Hopkins's journals. Looking closely at these two particular images, however, it becomes apparent that the top design can be traced in one continuous loop, and that Hopkins has tried to maintain the continuity of the line; while the second drawing, although it has loose ends, has been carefully shaded to indicate that the lines it is composed of cross over and under each [End Page 143] other where they intersect. More importantly, just a few pages later the journal features a poem beginning "It was a hard thing to undo this knot":

It was a hard thing to undo this knot.The rainbow shines, but only in the thoughtOf him that looks. Yet not in that alone,For who makes rainbows by invention?And many standing round a waterfallSee one bow each, yet not the same to all,But each a hand's breadth further than the next.The sun on falling waters writes the textWhich yet is in the eye or in the thought.It was a hard thing to undo this knot.

(CW, 3:210)

Significantly, this poem ends by repeating the same line that it began with, as if knotted around on itself.

Hopkins's diaries and letters are littered with references to knots, often in relation to natural forms—buds, trees, and clouds. To give just a few examples, he writes of the "star knot" of the oak tree, the leaves of which "are rounded inwards and figure out ball-knots" (CW, 3:379); ashes "open[ing] their knots" (CW, 3:511); "knots" of beech and sycamore leaves (CW, 3:425; 3:526); a "knot or 'knoop'" of bluebell buds (CW, 3:510); the "knotted" ends of glaciers (CW, 3:444); "regularly cu curled knots" of clouds (CW, 3:504); and a sunset with "one or two knots of rosy cloud middled with purple" (CW, 3:485). In other places he uses imagery associated with coils of string, rope, or lace—he describes freshly caught mackerel fish making "scapes of motion, quite lik as strings do, nodes and all" (CW, 3:555); a lark's song as a "skein and coil" (CW, 2:552); and frequently refers to "ropes" of clouds (CW, 3:411, 507, 564, 601), or clouds "unravelling" in the wind (CW, 3:504). Hopkins's forms are constantly laced, knotted, roped, lashed, coiled, and wound.30

This sense of knottiness is closely tied to Hopkins's concept of inscape. Inscape refers to the "species or individually-distinctive style beauty of style" (CW, 2:835), the "design" or "pattern" of a thing's form:

as air, melody, is what strikes me most of all in music and design in painting, so design, pattern or what I am in the habit of calling 'inscape' is what I above all in aim at in poetry. Now it is the virtue of design, pattern, or inscape to be distinctive and it is the vice of distinctiveness to become queer.

(CW, 1:334) [End Page 144]

For Hopkins, "All the world is full of inscape" (CW, 3:549); it is a term that can be applied not only to poetry and other artworks, but also to objects in nature, including cloudscapes and landscapes. James Milroy observes that Hopkins has a set of favorite words which "constitute his basic vocabulary for describing inscape (skeined, rope, comb, rack, bow, and others)," a vocabulary which, notably, contains a number of words related to twisting, pulling, or other types of distortion.31 As J. Hillis Miller writes, for Hopkins,

Each object is held in being by a system of strands, ropes, or sinews, lines of force which reach everywhere from the center to the surface, organize the thing, and make it one. The unity of an object lies not so much in its exterior pattern or texture, as in the way every morsel of it is strung together and held in tension by an intertwined pattern of bones, muscles, and veins. Hopkins sees even the most apparently slack and unstructured objects, like clouds or water, as roped and corded together by a tense network of lines of energy.32

Inscape is specifically something that preserves its identity even when subject to continuous transformations. As Hopkins describes, regarding a group of chestnuts in motion: "When the wind tossed them they plunged and crossed one another without losing their inscape. (Observe that motion multiplies inscape only when inscape is discovered, otherwise it disfigures)" (CW, 3:489). Hopkins's parenthesis indicates that once an inscape has been caught or apprehended by the mind, it is capable of undergoing continuous changes without losing its identity. Cloudscapes, for instance, are frequently cited by Hopkins as possessing inscape, despite their mobility. In one journal entry he describes a scene featuring "streamers" of cloud, with "regularly cu curled knots springing … from fine stems, like foliation in wood or stone"—"[i]t changed beautiful changes," he writes, and "[u]nless you refresh the mind from time to time you cannot always remember or believe how deep the inscape in things goes is" (CW, 3:504). He describes elsewhere how "[a] beautiful instance of inscape sided on the slide, that is / one successive sidings of one siding is inscape, is seen in the behaviour of the flag flower from the shut bee bud to the full blowning" (CW, 3:513). The "successive sidings" of the flower as it unfurls from the bud are a type of continuous transformation, and "one inscape" is preserved throughout. And, just as the inscape of a tree or flower undergoes a transformation in the process of blooming, so too in the process of decay: [End Page 145]

the Horned Violet is a pretty thing, gracefully lashed. Even in withering the flower ran through beautiful inscapes by the screwing up of the petals into straight little barrels or tubes. It is not that inscape does not govern the behaviour of things in slack and decay as one can see even in the pining of the skin in the old and even in a skeleton but that horror prepossesses the mind, but in this case there was nothing in itself to shew even whether the flower were shutting or opening (CW, 3:513).

Inscape, for Hopkins, can be pulled, bent, stretched, or twisted without losing its fundamental nature, as long as this change occurs through a natural and continuous process.

However, while inscape is preserved throughout continuous transformations, discontinuous transformations disrupt it. While even a tree naturally dying or decaying can retain its inscape, the inscape of a tree is destroyed when felled by human agency:

The ash tree growing in the corner of the garden was felled. It was lopped first: I heard the sound and looking out and seeing it maimed there came at that moment a great pang and I wished to die and not to see the inscapes of the world destroyed any more (CW, 3:549)33

In "Binsey Poplars," likewise, a grove of aspens are "Áll félled, félled, are áll félled" (3)—the "Strókes of havoc únsélve / The sweet especial scene" (21–22).34 Hopkins compares this to puncturing an eyeball: "like this sleek and seeing ball / But a prick will make no eye at all" (14–15). In Hopkins's poetry, therefore, as in knot theory, while continuous transformations (stretching, twisting, pulling, and unfurling) can occur without irrevocably damaging the "invariant" properties which make up the inscape of a thing, discontinuous transformations, those which "Háck and rack" (11), destroy its form, just as cutting a knot destroys its topological identity.

The same distinction between continuous and discontinuous transformation can also be seen in the molding or shaping of a person. "Carrion Comfort" opens with the declaration: "Not, I'll not, carrion comfort, Despair, not feast on thee;/Not untwist—slack they may be—these last strands of man" (1–2). The self here is figured as a tangled form which can be twisted or slackened; these lines, moreover, include a pun on the word "knot," especially in the phrase "Not [knot] untwist." In "The Wreck of the Deutschland," Hopkins describes God as having "bóund bónes and véins" in him, "fástened" his flesh (5). When he implores God to help reshape this self, it is in terms of continuous [End Page 146] transformation—"Wring[ing]," "forg[ing]," "melt[ing]" (67, 74, 76). Although these words suggest a significant degree of violence, none of them describe God actually breaking, cutting, or puncturing a person, but rather stretching, molding, wringing, and reshaping them. On the other hand, the types of violence and reshaping that humans enact upon one another can be destructive, as we see in Hopkins's unfinished drama, St. Winifred's Well, in which the heroine's head is literally "[s]truck őff" or "sheared from her shoulders" by a "stroke [of] Caradoc's right arm" (B 3, 18, 2). The destruction, moreover, is not confined to Winifred; Caradoc also suffers: "I all my being have hacked | in half with hér neck" (B 60). This act of violence, of discontinuous transformation, destroys the integrity—literally and figuratively—both of the object and the actor of it.

In this instance, however, Winifred is raised from the dead, her head rejoined to her body, and a spring arises at the location of the miracle, to which sick pilgrims flock hoping to be healed. God, essentially, restores the continuity which is disrupted by Caradoc. The succession of cures which takes place at the well also forms a continuous chain; as Hopkins writes in his journal, after visiting the well in person, "The strong unfailing flow of the water and the chain of cures from year to year all these centuries took hold of my mind with wonder at the bounty of God in one of His saints … the spring in pleace leading back the thoughts to ^by^ its spring in time to its spring in eternity" (CW, 3:608). God's ability to secure continuity throughout the passage of time is, moreover, a major theme in "The Leaden Echo and the Golden Echo," which originated as a song for St. Winifred's Well. The Leaden Echo begins with the fear of aging, a process of natural and continuous change: "is there no frowning of these wrinkles, rankèd wrinkles deep, / Down? no waving off of these most mournful , still , sad and stealing messengers of grey?" (3–4). The Leaden Echo wonders if it might be possible to keep beauty back from vanishing away, through "bow or brooch or braid or brace, lace, latch or catch or key" (1); however, no mortal bow, braid, lace—or knot—can secure the flow of time. The Golden Echo, however, responds that "whatever's and passes of us, everything that's fresh and fast flӳing of us," that which seems to be "done with, undőne" (24) is in fact "fastened with the truth / To its őwn best being and its loveliness of youth" (28–29) by God, "beauty's self and beauty's giver" (35), who keeps it with "[f]onder a […] than we could have kept it" (44). [End Page 147]

For Hopkins, it is not just the inscapes of discrete objects or beings which have a knot-like form, but the entire fabric of reality. In "Duns Scotus' Oxford," Hopkins describes Scotus as "[o]f realty the rarest-veinèd unraveller," following this up with another knot pun: "a not / Rívalled insight" (12–13). A similar metaphor of unravelling is used in "Spelt from Sibyl's Leaves," which describes the earth as having "unbóund" her being in the twilight (5); likewise, Hopkins imagines the world, on judgment day, being wound off onto two separate spools:

Lét life, wáned, ah lét life        wíndOff hér once skéined stained véined varíety | upon, áll on twó        spools; párt, pen, páckNow her áll in twó flocks, twó folds—bláck, white; | ríght, wrong;        réckon but, réck but, míndBut thése two; wáre of a wórld where bút these | twó tell, éach        off the óther; of a ráckWhere, selfwrung, selfstrung, sheathe- and shelterless, |        thóughts agaínst thoughts ín groans grínd.

(10–14)

This passage, significantly, suggests that forms which seem to be knotted could, in fact, just be tangled, and ultimately able to be teased out.35

The unique structure of "Spelt from Sibyl's Leaves"—which Hopkins described as the "longest [sonnet] ever made" (CW, 2:841)—also provides an illustration of how Hopkins extends his topological understanding of form to the poetic forms of his works. Like a knot, Hopkins's sonnet forms can be contracted, stretched, and pulled in different directions without losing their identity. The sonnet conventionally consists of fourteen lines of iambic pentameter; Hopkins's sonnets, however, exhibit a great deal of flexibility in their line lengths. He reaches his maximum scope in the octameter lines of "Spelt from Sibyl's Leaves," but also uses hexameters in sonnets such as "Felix Randal," "Carrion Comfort," and "That Nature is a Heraclitean Fire." Hopkins's use of sprung rhythm and outriding feet also allows him to lengthen or compress individual lines even within sonnets nominally in pentameter. As W. H. Gardner writes, the "sonnet-mould received its first considerable stretching in the muscular proportions of The Windhover"; "Never had [the iambic decasyllable] been expanded by heating and then suddenly contracted by cooling as in: 'As a skáte's heel sweeps smóoth on a bów-bend: the húrl and glíding / Rebúffed the bíg wínd. My heárt in híding …' [5-6]."36 Notably, "Spelt from Sibyl's Leaves" has a marked caesura (or cut) dividing each line in two, which might initially seem to disrupt the topological continuity of the form. [End Page 148] However, given that the poem is as much about "dismembering" (7) and division as it is about unwinding, the caesura seems appropriate; the poem's form, with its "voluminous" (1), stretched-out lines split by caesurae, recreates the process that it describes.

Hopkins's topological approach to the sonnet form extends not only to the lengths of his lines, but also to their number. In "On the Origin of Beauty," he writes that a sonnet "must be made up of fourteen lines: if you were to take a line out, that wd. be an important loss to the structural unity" (CW, 4:151), and in a letter to Dixon, he gives a mathematical equation for the form of the "best sonnet": "(4 + 4) + (3 + 3) = 2.4 + 2.3 = 2(4 + 3) = 2.7 = 14" (CW, 1:476). And yet, a number of Hopkins's own sonnets notably do not follow these prescriptions; his "curtal sonnets," for instance, shrink the sonnet's length by precisely three-quarters:

Curtal-Sonnets … are constructed in proportions resembling those of the sonnet proper, namely 6 + 4 instead of 8 + 6, with however a halfline tailpiece (so that the equation is rather ).37

Curtal sonnets thus have ten-and-a-half, rather than fourteen lines. However, in the same letter in which he gives the equation of the "best sonnet," he acknowledges that there is no inherent "mystery" in the "cy^i^pher 14"; the "real characteristic" of the sonnet, he argues, is its division into the octave and sestet, and "what is not so marked off and moreover at least has not the octet again divided into quatrains is not to be called a sonnet at all" (CW, 1:476). Although they differ in line length from the conventional sonnet, Hopkins's curtal sonnets retain these "invariant" properties. They are divided into two parts in the proportion 4:3, and the first of these parts is again divided into two halves through an abc abc rhyme scheme. The curtal sonnet can thus be produced from the traditional sonnet form through a continuous transformation or compression—indeed, as Jennifer Ann Wagner observes, Hopkins's curtal sonnet "Peace" is written in alexandrines, and thus "contracts and expands the form at the same time."38

Hopkins's language and wordplay also has something of a topological nature, a focus on principles of continuity. He was "fascinated" by Welsh cynghanedd or, as he calls it, "consonant-chime," which involves an intricate interlacing of repeated consonantal sounds throughout a line of verse (CW, 2:551). In his lecture notes he describes the process of "vowelling off," or "changing of vowel down some scale or strain or keeping" (as opposed to "vowelling on," or assonance, which retains [End Page 149] the vowel sound), and admires the "beautifully rich combination" of alliteration, assonance, and "shothending" (skothending, or "final half-rhyme") found in Norse poetry.39 As Milroy writes, "Hopkins's lines are often organized in steps, in which a phonetic characteristic (say, alliteration), present but not necessarily dominant in the first step, is taken up by the second, which at the same time usually abandons the dominant principle of the first; the third then takes up a subsidiary phonetic effect of the second, again abandoning the dominant, and so on."40 For example:

Earnest, earthless, equal, attuneable, | vaulty, voluminous, … stupendous ("Spelt from Sibyl's Leaves," 1)

Thís Jack, jóke, poor pótsherd, | patch, matchwood, immortal diamond, / Is immortal diamond.

("That Nature is a Heraclitean Fire," 23–24)

bow or brooch or braid or brace, lace, latch or catch or key to keep ("The Leaden Echo," 1)

Milroy notes, moreover, that the steps in these series of words are often not only "phonetically," but also "semantically" linked, something which is clearly "foreshadowed in [Hopkins's] undergraduate etymological lists, where sound and sense are obviously considered to be related."41 For example:

Grind, gride, gird, grit, groat, grate, greet, er κρoύειν, crush, crash, κροτεîν etc.

Orig. meaning to strike, rub, particularly together.

(CW, 3:111)

Flick, fillip, flip, fleck, flake.

[…] Key to meaning of flick, fleck and flake is that of striking or cutting off the surface of a thing (CW, 3:126)

Each of these word lists is associated with a central idea or original meaning. The words in each group contain some similar sounds (fixed or invariant points), yet also incorporate room for change or difference, as if Hopkins were experimenting with the extent to which a word's meaning can remain stable, or at least part of the same family of meaning, while undergoing certain defined transformations.

In "Making Earnest of Game: G. M. Hopkins and Nonsense Poetry," David Sonstroem observes that Hopkins's technique "has much in common with the old word-game whose object is to progress from one [End Page 150] word to another of the same length through a series of words formed by changing only one letter of the previous word":

For example, one might move from "black" to "white" by means of the series, -slack-shack-shark-share-shale-whale-while-. Hopkins "wins" his game by letting accidents of his words carry him from black to white, from a pessimistic presentation of a problem to a reassuring answer:

This Jack, joke, poor potsherd, patch matchwood,                immortal diamond,                    Is immortal diamond.42

Lewis Carroll was particularly fond of these sorts of word games, including a variation which allowed for more than one letter to be changed in each step. He asks his readers, for instance, to "Change a CONSERVATIVE into a LIBERAL," subsequently publishing the following solution from a contributor.43

CONSERVATIVE(servati)observation(ation)deliberations(libera)LIBERAL44

Carroll, significantly, calls these types of word puzzles "syzygies"; as Brown writes, Carroll "appl[ies] Sylvester's mathematical relation of syzygy to relations between words."45 Like Sylvester's phonetic syzygies, these repeated sounds provide a thread of continuity which yokes the chain of words—or line of verse—together.

Hopkins and Sylvester have much in common in terms of their poetic theory—not least an insistence on using an idiosyncratic and often inadequately explained technical vocabulary.46 Indeed, Sylvester, who was immensely proud of his poetic coinages, singled out the words "Twin-lit, unshape, and whirls-in-one" as particular favorites, and it is impossible not to feel that these words have a strikingly Hopkins-esque sound to them (LV, 41). Hopkins's poetry, conversely, is replete with features which Sylvester would class as "synectic" or continuous. In addition to "syzygetic" techniques such as consonant-chime and the other more complex patterns described above, Hopkins also frequently uses alliteration, which (although a "limiting [form] of phonetic syzygy" LV, 46n) falls, according to Sylvester, more properly under the heading of "symptosis," another branch of the synectic, which also deals with [End Page 151] "rhymes, assonances (including alliterations, so called), and clashes (this last comprising as well agreeable reiterations, or congruences, as unpleasant ones, i.e., jangles or jars)" (LV, 11). As Sylvester notes, these types of techniques might initially seem to consist of "discreet matter" (LV, 11), and therefore to belong to the discontinuous, rather than the continuous, branch of versification; nevertheless, Sylvester argues that their effect is ultimately continuous, just as "in an iron shield or curtain or a trial target, the bolts and screws are rivets are separate, but serve to consolidate and bring into conjunction the plates, and to give cohesion and unity to the structure" (LV, 12). Both Hopkins and Sylvester share a fascination with the continuous and discontinuous—metric and synectic (Sylvester) or chromatic and diatonic (Hopkins)—elements of verse, and both see this split as a fundamental organizing principle of poetic theory, in the same way that Sylvester, as a mathematician, is able to identify it as a crucial division within nineteenth-century geometry.

In The Laws of Verse, Sylvester clearly identifies meter as belonging exclusively to the "discontinuous," as opposed to the "synectic" or "continuous," aspect of poetry. In the final part of this article, however, I suggest that, during this period, the way that meter itself is theorized essentially bifurcates along quantitative and non-quantitative lines. The nineteenth century witnessed both a resurgence of interest in quantitative systems of prosody, and, simultaneously, a developing fascination with accentual and alliterative rhythms.47 In many ways, these two competing strains of prosodic analysis mirror the split between quantitative Euclidean geometry, and topology or descriptive geometry. In particular, I argue that Hopkins's sprung rhythm provides a model for thinking about poetic rhythm in a topological fashion.

Quantitative prosody emerges from the study of classical poetry, just as quantitative geometry is associated with classical, Euclidean mathematics. Indeed, in some cases, the two converge. Richard Roe's 1801 Elements of English Metre suggests a connection to Euclid's Elements even in its title; Roe goes on to declare that "[m]etre consists of a succession of parts, in subordinate proportions, and within easily calculable limits: which parts, abstractedly considered, are those solely of time or duration."48 He then resorts to his "rule and compasses" in order to illustrate the duration of verse via extension in space: "With that I shall draw a straight line under each example, and with these shall divide it into spaces analogous to the parts intended to be measured; and so, on every occasion, shall frame a rule, or scale, for the reader's use."49 [End Page 152]

Roe thus literally uses the instruments of Euclidean geometry to measure English verse. John Ruskin's 1880 The Elements of English Prosody, like Roe's treatise, has a suggestively Euclidean title, and commences by emphasizing the etymological connection between meter and measurement, observing that the Greek "metron" or "measure" had been "adopted … to signify a measured portion of a verse."50 He then goes on to elaborate a system of musical scansion, representing each foot as a bar of verse, with the spondee (or "spondeus") corresponding to two minims, the iambus to a crotchet followed by a minim, and so forth.51 The length of these feet is also illustrated with reference to spatial measurement; Ruskin claims that the spondeus corresponds to the time that an average person would need to take two paces, while "holding himself well erect, and walking in regular time, so firmly that he could carry a vase of water on his head without spilling it or losing its balance," and that all other "divisions of time" should be based upon this measure.52

In the United States, Edgar Allan Poe's 1848 Rationale of Verse had likewise sought to establish an approach to prosody based on temporal equality. According to Poe, "nine tenths" of versification "appertain to the mathematics"; while acknowledging that, in English, syllables have no absolutely fixed time, he argues that "for the purposes of verse we suppose a long syllable equal to two short ones:—and the natural deviation from this relativeness we correct in perusal … if the relation does not exist of itself, we force it by emphasis, which can, of course, make any syllable as long as desired."53 Metrical substitutions are permitted only between feet when the "sum of [their] syllabic times" is equal."54 Poe's system, as noted above, is the one which Sylvester claims to subscribe to in The Laws of Verse. Lanier also employed a system of quantitative prosody in 1880 in The Science of English Verse, arguing that "[t]ime is the essential basis of rhythm," and that "every series of English sounds, whether prose or verse, suggests to the ear exact co-ordinations with reference to duration."55 Lanier writes, [End Page 153] however, that while in Greek and Latin verse quantity was "confined to a single proportion, namely, that of 1 to 2, all the sounds being divided into 'longs' and 'shorts', of which any 'long' was equal in time to two 'shorts'," English verse is "not limited to the single proportion, 1 to 2, but exists clearly in the further proportions of 1 to 3, 1 to 4, 1 to 5, and so on."56 Like Ruskin, Lanier uses musical notation to assist with his explanations.57

Coventry Patmore's Essay on English Metrical Law, first published in 1857, also argued for the temporal equality of the foot as a unit of verse but allowed somewhat more flexibility in quantifying the lengths of individual syllables. In this work, Patmore argues that verse should be "divided into equal or proportionate spaces," with the "ictus" or "beat" functioning "like a post in a chain railing," "mark[ing] the end of one space, and the commencement of another."58 Nevertheless, Patmore is interested principally in the duration of time which elapses between each metrical ictus, rather than the duration of individual syllables. Patmore concedes, moreover, that "the equality and proportion of metrical intervals between accent and accent is no more than general and approximate, and that expression in reading, as in singing or playing, admits, and even requires, frequent modifications, too insignificant or too subtle for notation, of the nominal equality of those spaces."59 Patmore's system, therefore, combines certain principles of isochrony or temporal equality with a greater degree of freedom than was usually permitted by classical quantitative prosody.

Alongside the quantitative approaches to versification outlined above, there was also a revitalization of interest in accentual or rhythmical systems of scansion. In his preface to "Christabel," Samuel Taylor Coleridge had outlined his system of "counting in each line the accents, not the syllables. Though the latter may vary from seven to twelve, yet in each line the accents will be found to be only four."60 Although Coleridge claims this to be a "new principle," interest in accentual rhythm was also fueled through attention to Old and Middle English alliterative, accentual verse.61 As Phelan notes, scholars such as Sharon Turner, Edwin Guest, and George P. Marsh often turned to the Venerable Bede's distinction between "Metrum" and "Rhythmus" to differentiate between these two traditions, in which, in Phelan's words, "'Metrum' is a system of versification based on quantity: 'Rhythmus' makes accent or ictus the dominant principle."62 Guest, in his 1838 History of English Rhythms, writes that while Sanskrit, Greek, and Latin "made time the index of their rhythm," the remainder of the Indo-European language family "adopted accent," and that accent is [End Page 154] the "sole principle" that governs English rhythms.63 George P. Marsh's lecture on "Alliteration, Line-Rhyme and Assonance" (which Hopkins draws upon heavily in his own lecture notes on "Rhythm and other structural parts of rhetoric—verse") likewise declares that the "poetry of the Anglo-Saxons was always rhythmical, but not always metrical," and that "Anglo-Saxon verse … would strike a modern hearer as merely an unmeasured and irregular recitative."64

Hopkins, significantly, names his prosodic innovation "sprung rhythm," suggesting that he considered it more closely allied to the tradition of native English "rhythmus" rather than classical "metrum"; he describes it, moreover, as having "existed in full force in Anglo saxon verse and in great beauty; in a degraded and doggrel shape in Piers Ploughman" (CW, 2:543). I suggest that sprung rhythm is essentially topological in its approach to verse form. In an 1878 letter to Dixon, Hopkins explains that sprung rhythm "consists in scanning by accents or stresses alone, without any account of the number of syllables, so that a foot may be consist of but one strong syllable or ^it may be^ many light and one strong" (CW, 1:317). Or, as he put it succinctly elsewhere, "one stress makes one foot, no matter how many or few the syllables" (CW, 1:345). This means that, in some cases, sprung rhythm results in two stressed syllables succeeding each other; in other cases, multiple unstressed syllables can intervene between each stress. According to these definitions, therefore, sprung rhythm is principally concerned only with the overall number of stressed positions per line, but not the number of syllables between them. In this sense, it shares some of the topological properties of the knot, which is defined by invariant features such as the number and configuration of its minimum crossings, rather than the length of string or cord between them. Notably, Hopkins describes sprung rhythm as having greater "flexibility" than running rhythm and refers to the unstressed syllables in sprung rhythm as the "slack."65

There are, however, certain points at which Hopkins describes sprung rhythm as having a quantitative component. In a letter to Everard Hopkins, for instance, he writes that sprung rhythm "mak[es] up by regularity, equality, of a larger unit (the foot merely) for ^in^equality in the less, the syllable" (CW, 2:748); to Robert Bridges, that "[s]ince the syllables in sprung rhythm are not counted, time or equality in strength is of a more importance than in common counted rhythm" (CW, 1:358). Again, in his "Author's Preface," he writes that "[i]n Sprung Rhythm, as in logaoedic rhythm generally, the feet are assumed to be equally long or strong and their seeming inequality [End Page 155] is made up by pause or stressing."66 These passages all suggest that Hopkins was indeed concerned with questions of quantity, or at least isochrony. As Meredith Martin observes, Hopkins's writings are full of "equivocation over, and revision of, what sprung rhythm was"; as a result, as Emily Harrington writes, "critics have attempted to co-opt Hopkins both for accounts of meter that are increasingly accentual and increasingly temporal."67 At the same time, Hopkins's undergraduate writings on aesthetics also revealed a tension between the chromatic and the diatonic, and yet, as we have seen, although he overtly seems to privilege the diatonic over the chromatic he nevertheless finds himself attracted to forms which make room for a degree of flexibility. Likewise, I suggest that although he does refer to principles of quantity in his writings about sprung rhythm, his prosody also contains more scope for flexibility than might initially appear from the descriptions above. As Elisabeth M. Schneider observes, Hopkins often comes "close to describing his verse feet as equal in duration, but each time, as if deliberately backing away, he inserts an alternative"—he describes his feet as "equally long or strong," [Schneider's italics] and the addition of "strength" allows for some ambiguity over whether his feet are actually temporally equivalent.68 Joseph Phelan, likewise, admits that Hopkins's description of sprung rhythm in his "Author's Preface" "looks remarkably like Patmore's isochronous intervals," but argues that Hopkins's verse "includes elements which simply cannot be made to fit into a purely isochronous schema."69 The most important of these is Hopkins's use of "outrides" or extra-metrical syllables: "one, two, or three slack syllables added to a foot and not counting in the nominal scanning," which he describes as a license "natural to Sprung Rhythm."70 Thus, while feet in sprung rhythm might "nominally" be equally long (or strong), in practice they deviate from this equality, as the "slack" portions of the foot can be extended through the use of outriding syllables. Hopkins once claimed, moreover, that "Spelt from Sibyl's Leaves" should be "almost sung: it is most carefully timed in tempo rubato" (CW, 2:842), a term which indicates "[a] temporary disregard for strict tempo to allow an expressive quickening or slowing, typically without altering the overall pace."71 Although Hopkins does sometimes describe sprung rhythm in ways which allude to quantitative or isochronous principles, the licenses he allows himself, such as outriding feet and tempo rubato, mean that in reality his rhythm admits of considerably more scope than would be allowed by a strict temporally quantitative system of scansion. [End Page 156]

A partial resolution of these difficulties may be found in an 1881 letter to Dixon on sprung rhythm, in which Hopkins distinguished between "its εἶναι {simply being} and its εὖ εἶναι {being well}, the writing it somehow and the writing it as it should be written" (CW, 1:413). When written "anyhow," he declares, "it is a matter of accent only, like common rhythm, and not of quantity at all," but it is a "shambling business and a corruption" (CW, 1:413); when written as it should be written, however, "great attention to quantity is necessary" (CW, 1:415). This passage suggests that, for Hopkins, sprung rhythm in its most essential form was fundamentally accentual rather than quantitative, but that quantitative principles could be applied within the broader framework of the accentual rhythm. In some ways, this reflects the relationship between metrical and descriptive geometry. Metrical or Euclidean geometry was often considered to be a special case of the more general descriptive geometry; in Arthur Cayley's words, "Metrical geometry is thus a part of descriptive geometry, and descriptive geometry is all geometry."72 From this point of view, as Richards writes, descriptive geometry was seen as the "more basic, hitherto hidden spatial structure lying behind … metrical construction."73 Likewise, I argue, sprung rhythm at its most basic is a "matter of accent only," a form of prosody akin to descriptive geometry; within this structure, however, it is possible to introduce a subordinate set of metrical principles if desired. At a foundational level, however, sprung rhythm, with its invariant points of stress and intervening slack, exhibits much of the topological flexibility of the knot.

I've argued, in this article, that Hopkins's idea of form was fundamentally topological. But was Hopkins himself consciously thinking about form in these terms? As he once wrote in a letter to Bridges, regarding similarities between Whitman's long lines and his own sprung rhythm, "In a matter like this to do a thing does not exist, unless it consci is not done unless it is wittingly and willingly done; to recognise the form you are employing and to mean it is everything" (CW, 2:544). Did Hopkins, then, recognize and mean the form he was using as a topological one, "wittingly and willingly"? It is certainly possible that Hopkins was familiar with topology as a branch of mathematics—as outlined above, he was familiar with at least some of Tait's work, and could have read his obituary of Johann Benedict Listing in Nature. He also frequently thought about poetic form in explicitly mathematical terms—he claimed that his work on the Dorian Measure "needs mathematics" (CW, 2:842), and intended to present it before a "physical and mathematical science club" (CW, 2:849); in 1887 he wrote to Patmore [End Page 157] that he was working on a "sizeable book" on meter, and that "[f]or the purpose of grounding the matter thoroughly [he was] subjecting the terms of geometry, line, surface, and solid and so on, many others to a searching examination" (CW, 2:883). However, although it might be difficult to establish conclusively whether Hopkins was aware of either descriptive geometry or topology (although neither would be unlikely), he was certainly and demonstrably thinking about knots, and about the principles of continuity and discontinuity. These are precisely the types of ideas that nineteenth-century mathematicians were thinking about in connection with topology and descriptive geometry. Although they approach the topic from different backgrounds and with different terminology, Hopkins, and mathematicians like Tait and Sylvester, are thinking about form in very similar ways, spurred on by the same fundamental inquiries and basic principles.

Likewise, nineteenth-century debates regarding quantitative and accentual prosody rehearse those surrounding quantitative and descriptive geometry. In some cases, we can chart a connection between these two disciplines through particular individuals, such as Sylvester, who was not only a poet and mathematician himself, but who was also connected to both Sidney Lanier and Matthew Arnold (to whom he dedicated The Laws of Verse). In concluding this article, however, I would also like to briefly sketch some of the broader historical factors which led to both prosodists and mathematicians becoming embroiled in similar debates about form.

For much of the nineteenth-century, mathematics in Britain was "not seen as a specialized, research subject but rather as a universal, educative one."74 William Whewell argued that the most crucial components of a liberal education were "permanent" studies, such as classical literature and Greek geometry, which taught the foundations of language and reasoning respectively; modern, "progressive" mathematics (like modern literature) cannot "take the place of the Permanent portions, in our Higher Education, without destroying the value of our system."75 Over the course of the century, however, these attitudes began to change. As Alice Jenkins describes,

Euclidean geometry had traditionally enjoyed enormously high prestige as the foundation not only for a scientific education but for any kind of systematic thought. … But by the early nineteenth century cracks were appearing in its status as groups with no geometrical training, and no belief in its relevance to their educational needs, denounced its prestige as a repressive outdated shibboleth. Even much more moderate writers, with vested interests in retaining geometry and other kinds of mathematical education at the heart of the British elite [End Page 158] educational system, offered suggestions for reforming its teaching, and for making it more usefully applicable to the needs of industrial and technological workers.76

Similar issues affected the teaching of poetry; as Jason David Hall writes,

By the mid-century, particularly in the aftermath of the Clarendon Commission's inquiry into public school curricula, Latin and Greek prosody began the long process of ceding their pride of place in this institutional matrix to English language and literature (often taught on principles inherited directly from the study of the classics) and other, more "useful" subjects, which gradually gained acceptance alongside or in place of the classical training that had asserted its dominance in British education for centuries.77

The terminology used to discuss English verse, as Martin notes, often continued to be drawn from classical prosody, and so seemed endowed with a sense of "cachet, elite knowledge"; however, "these terms were also suspect—Why use foreign names for something English? Why scan verses according to the classical system at all?"78 "Increasingly," Martin writes, during the late-nineteenth and early-twentieth centuries, "the foreign names for classical feet were called into question and students were taught to feel English poetry according to 'natural' accents (traced to an Anglo-Saxon past) divorced from the valueless and hegemonic classical system of iambs and trochees."79 While, at the beginning of the nineteenth century, both Euclidean geometry and classical prosody shared an elite status as the core of a liberal education, by the end of the century this began to be threatened by calls for more modern, practical, and "progressive" approaches to both mathematics and literature.

More broadly, the principle of "continuity" which pervaded descriptive geometry was also crucial to nineteenth-century evolutionary theory, which posited that different species all belonged to a continuous chain of evolution. The natural sciences began to compete with both mathematics and classical literature for their place within the educational curriculum, and both Sylvester and Arnold found it necessary to defend their disciplines against attacks by Thomas Henry Huxley. Sylvester particularly objects to Huxley's assertion that mathematics is an "almost purely deductive" science of the "same general nature" as the teaching of languages (cited in LV, 107). Huxley seems to consider that (in Sylvester's words) "the business of the mathematical student is from a limited number of propositions (bottled up and labelled ready for future use) to deduce any required result by a process of the same general [End Page 159] nature as a student of language employs in declining and conjugating his nouns and verbs" (LV, 107). Huxley's view is outdated, however; as Sylvester argues, "Induction and analogy are the special characteristics of modern mathematics, in which theorems have given place to theories, and no truth is regarded otherwise than as a link in an infinite chain" (LV, 118n). As Richards notes, "Unlike Euclidean geometry … projective methods were seen as scientific, closely akin to the inductive generalizing techniques of the natural sciences," which made them appealing to those who favored more "progressive" approaches to mathematics.80 Likewise, although the study of dead languages might be dismissed as the mere rote learning of conjugations and declensions, Sylvester considered philology to be shot through with the same "golden thread" of metamorphosis that he saw as central to modern geometry, arguing that the "relation between these two sciences is not perhaps so remote as may at first appear" (LV, 130n). Like descriptive geometry, comparative philology draws on inductive methodologies—the philologist must observe a number of different words and language systems in order to inductively determine their relationship to each other, and the laws by which they work.81 Max Müller, in Lectures on the Science of Language, describes the study of language and comparative philology as "one of the physical sciences."82 Both mathematics and the study of languages, therefore, evolve as a result of competition from the natural sciences, in ways which draw attention to the morphological and continuous elements of these subjects.

Mathematician William Spottiswoode once claimed that it is often possible to "trace under-currents of thought which having issued from a common source fertilise alike the mathematical and the nonmathematical world."83 The topological or morphological ideas of form which are suggested by knot theory and descriptive geometry are, I argue, one such shaping undercurrent of thought; we can see the "principle of continuity" manifested across a range of disciplines in the nineteenth century, from prosody and philology to evolutionary theory and mathematics. Hopkins and Sylvester are clearly both thinking rigorously about the relationship between the quantitative and the qualitative, the continuous and discontinuous, the metric and synectic, the chromatic and diatonic. The type of thinking that is involved in developing a new theory of meter and working out the problems within it, I argue, is not unlike that which goes into developing a new approach to geometry.84 Hopkins, essentially, produces a theory of topology in poetic form. [End Page 160]

Imogen Forbes-Macphail
University of California, Berkeley

## NOTES

I would like to thank Ian Duncan, Kent Puckett, Massimo Mazzotti, Andrea K. Henderson, and Kristin Hanson for their helpful feedback on draft versions of this paper.

1. Gerard Manley Hopkins, The Collected Works of Gerard Manley Hopkins, 8 vol., ed. Lesley Higgins, Michael F. Suarez, S. J., and others (Oxford: Oxford Univ. Press, 2006-), 4:157. Hereafter cited parenthetically by volume and page number and abbreviated CW.

2. James Joseph Sylvester, "A Probationary Lecture On Geometry," in The Collected Mathematical Papers of James Joseph Sylvester, ed. Henry F. Baker, 4 vol. (Cambridge: Cambridge Univ. Press, 1904–12), 2:8. Hereafter cited parenthetically by volume and page number and abbreviated CMP.

3. As Joan L. Richards writes, during the nineteenth century this type of geometry was "indiscriminately referred to as 'descriptive,' 'modern,' 'projective' or even 'the newer' geometry" (Mathematical Visions: The Pursuit of Geometry in Victorian England [Boston: Academic Press, 1988], 130–31).

4. The particular example given by Sylvester is often known as Pappus's theorem, of which Pascal's theorem is a generalization. See Morris Kline, Mathematical Thought from Ancient to Modern Times (New York: Oxford Univ. Press, 1972), 128 and 297–98.

5. For an account of nineteenth-century developments in projective geometry, see Richards, "Projective Geometry and Mathematical Science," in Mathematical Visions, 117–58.

6. John L. Bell, "Reflections on Mathematics and Aesthetics," Aisthesis 8.1 (2015): 165.

7. William Barnes, The Elements of Linear Perspective and The Projection of Shadows, Adapted to the Use of Mathematical and Drawing Classes and Private Students (London: Longman, Brown and Co., 1842), 28.

8. Richards, Mathematical Visions, 120.

9. Peter Guthrie Tait, "Johann Benedict Listing," in Scientific Papers, 2 vol. (Cambridge: Cambridge Univ. Press, 1898–1900), 2:82.

10. Tait, "On Knots," in Scientific Papers, 1:276.

11. Tait, "On Knots," 1:276.

12. Tait, "Listing's Topologie," in Scientific Papers, 2:85; Tait, "Johann Benedict Listing," 2:82.

13. Caroline Levine, Forms: Whole, Rhythm, Hierarchy, Network (Princeton: Princeton Univ. Press, 2015), 6.

14. There have already been several forays into the ways that topological ideas of form might have resonances outside of mathematics; see, for instance, Angus Fletcher, The Topological Imagination: Spheres, Edges and Islands (Cambridge: Harvard Univ. Press, 2016); Mark Blacklock, "Topology, Conjuring, and the Spiritualist Fourth Dimension" in The Emergence of the Fourth Dimension: Higher Spatial Thinking in the Fin de Siècle (Oxford: Oxford Univ. Press, 2018); Lee M. Johnson, "The Topologic Self of The Prelude" in Wordsworth's Metaphysical Verse: Geometry, Nature and Form (Toronto: Univ. of Toronto Press, 1982); Steven Connor, "Topologies: Michel Serres and the Shapes of Thought," Anglistik 15 (2004): 105–117; and Andrew Piper, "Reading's Refrain: From Bibliography to Topology," ELH 80.2 (2013): 373–99.

15. James Joseph Sylvester, Fliegende Blätter: Supplement to the Laws of Verse (London: Grant & Co., 1876), 5.

16. James Joseph Sylvester, "Inaugural Presidential Address" in The Laws of Verse or Principles of Versification Exemplified in Metrical Translations: Together with an Annotated Reprint of the Inaugural Presidential Address to the Mathematical and Physical Section of the British Association at Exeter (London: Longmans, Green, and Co., 1870), 124–25. Hereafter, all quotations from this volume (including the "Presidential Address," "The Laws of Verse," and the "Preface") will be cited parenthetically by page number and abbreviated LV.

17. Daniel Brown, The Poetry of Victorian Scientists: Style, Science and Nonsense (Cambridge: Cambridge Univ. Press, 2013), 207.

18. Sylvester also identifies a "Chromatic" principle which lies between the metric and synectic, but does little more than "occasionally glanc[e] at its existence" (LV, 10–11).

19. Sylvester once declared that "[t]he early study of Euclid made me a hater of geometry" (LV, 126) and that he would rejoice to see "Euclid honourably shelved or buried 'deeper than e'er plummet sounded' out of the schoolboy's reach" (LV, 120). According to Sylvester, the synectic has three principal branches, Anastomosis, Symptosis, and Phonetic Syzygy (LV, 11). Anastomosis relates to "the junction of words, the laying of them duly alongside one another … so as to provide for the easy transmission and flow of breath" (LV, 12–13). Symptosis involves features such as rhyme, assonance, and alliteration (LV, 11).

20. Sylvester, "On the Correlations of two Conics expressed by Indeterminate Coordinates," CMP, 1:132. See also Brown, The Poetry of Victorian Scientists, 213.

21. Sidney Lanier, The Science of English Verse and Essays on Music, ed. Paull Franklin Baum (Baltimore: The Johns Hopkins Press, 1945), 237.

22. Sylvester, Fliegende Blätter, 32, 33. This line comes from Sylvester's poem, "To Rosalind," which consists (in this iteration) of 268 lines each rhyming on the same sound. In Brown's words, "Akin to a mathematical variable, x or y, the unnamed subject of 'To Rosalind' is known only through her transformations, Sylvester's projections of her"; the poem itself, he argues, with its monochromatic rhyme scheme, "is projected on the two-dimensional plane of the page as a continuous line consisting of numerous subordinate lines, each of which bears a formal 'invariant' relation to the others through the rhyme scheme" (The Poetry of Victorian Scientists, 220, 223).

23. Brown, in fact, connects Sylvester's metrical theory not only to projective geometry, but also to differential calculus, describing syzygy as a "Calculus of Forms," a "measure of subtle and varying rates of change, the great raison d'être of calculus": "While [accentual and classical quantitative metrics] divide verse into fixed and arbitrary units, regular segments of lines, restricting it to simple Euclidean geometry and space, Sylvester's prosody brings to the fore the continuous principles of invariance and local metrics, which he images as free to bend and curve" (The Poetry of Victorian Scientists, 229).

24. Detail of Dante Gabriel Rossetti's illustration of "Goblin Market" from Christina Rossetti, Goblin Market and Other Poems, second edition (London and Cambridge: Macmillan and Co., 1865), frontispiece.

25. J. Hillis Miller, The Disappearance of God: Five Nineteenth-Century Writers (Urbana: Univ. of Illinois Press, 2000), 278. Jude V. Nixon, Gerard Manley Hopkins and His Contemporaries: Liddon, Newman, Darwin, and Pater (New York: Garland Publishing, Inc., 1994), 149.

26. Daniel Brown, Hopkins's Idealism: Philosophy, Physics, Poetry (Oxford: Clarendon Press, 1997), 90, 24.

27. See Brown, Hopkins's Idealism; Nixon, Gerard Manley Hopkins and his Contemporaries and "'Death blots black out': Thermodynamics and the Poetry of Gerard Manley Hopkins," Victorian Poetry 40.2 (2002): 131–55; Tom Zaniello, Hopkins in the Age of Darwin (Iowa City: Univ. of Iowa Press 1988); Gillian Beer, "Helmholtz, Tyndall, Gerard Manley Hopkins: Leaps of the Prepared Imagination" in Open Fields: Science in Cultural Encounter (Oxford: Clarendon Press, 1996), 242–72; Marie Banfield, "Darwinism, Doxology, and Energy Physics: The New Sciences, the Poetry and the Poetics of Gerard Manley Hopkins," Victorian Poetry 45.2 (2007): 175–94; and John Kerrigan "Writing numbers: Keats, Hopkins, and the history of chance," in Keats and History, ed. Nicholas Roe (Cambridge: Cambridge Univ. Press, 1995), 280–305.

28. As Gillian Beer observes, moreover, the first of these letters was prompted by an entry in the 9 November 1882 issue, which is found in close proximity to a selection from one of James Clerk Maxwell's poems—beginning with the line "My soul's an amphicheiral knot"—in which Maxwell satirizes some of Tait's speculations on science and religion in The Unseen Universe, which he co-authored with Balfour Stewart. See "The Life of Clerk Maxwell," Nature 27 (1882): 26–28; and Beer, "Helmholtz, Tyndall, Gerard Manley Hopkins," 256–57. For discussions of this poem, see Brown, "Science on Parnassus" in The Poetry of Victorian Scientists, 234–62; Blacklock, Emergence of the Fourth Dimension, 46–48; and Daniel S. Silver, "The Last Poem of James Clerk Maxwell," Notices of the AMS 55.10 (2008): 1266–70.

29. See James Joseph Sylvester, "A Plea for the Mathematician," parts 1 and 2, Nature 1 (30 December 1869): 237–39; and (6 January 1870): 261–63.

30. Another key term is "burl," which, as Brown notes, can be used either to refer to the bubbling of a spring or fountain, or to "knotted strands in fabric and knots in the grain of wood" (Hopkins's Idealism, 229). According to Brown, "Hopkins appears to abstract from these senses of the word a principle of determinate being which participates in a larger being, a unified knot-like configuration of being that is integral to the larger ocean (or fabric) of being." Brown situates Hopkins's work in the context of nineteenth-century energy physics and Faraday's field theory, suggesting that "[i]nstress can be represented according to the ontology of Faraday's theory as a discrete point, a knot or vortex, of energy in the field of being, and particular stresses, such as those of grace or the epistemological 'stem of stress,' as lines of force in this field" (Hopkins's Idealism, 238). While Brown's argument, in Hopkins's Idealism, concentrates principally on the context of energy physics, and mine on mathematics, it is worth noting that one of the main motivations for the surge of mathematical interest in topology in the 1870s and 1880s lay in what was known as the "vortex atom theory," developed by William Thompson and P. G. Tait, which suggested that what we call atoms were essentially knots or vortices in the ether (see Brown, The Poetry of Victorian Scientists, 171–72; and Blacklock, The Emergence of the Fourth Dimension, 44–45).

31. James Milroy, The Language of Gerard Manley Hopkins (London: Andre Deutsch, 1977), 155. See also Nixon: "In Hopkins, roped, laced, and fastening—the idea being that things are held together, fastened—are made possible by charges. … Hopkins's antonym to 'instress,' 'slack,' has such synonyms as 'untwist,' 'unfastening,' 'unbound,' 'unwound,' and 'dismembering'" ("Death blots black out," 144).

32. Miller, Disappearance of God, 288.

33. Compare CW, 3:520: "There is one notable dead tree in the N. ^N. W.^ corner of the nave, the inscape markedly holding its most simple and beautiful oneness the up from the ground through a graceful swerve below (I think) the spring of branches up to the top of the timber."

34. Gerard Manley Hopkins, The Poetical Works of Gerard Manley Hopkins, ed. Norman H. MacKenzie (Oxford: Clarendon Press, 1990). All citations of Hopkins's poetry are taken from this edition and cited parenthetically by line number.

35. Justin Tackett suggests that this image of unspooling was influenced by "phonographic thinking," in particular the idea of recording sound on a rotating cylinder: "Unlike the leaves of a codex, the cylinder permits linear text—be it words or grooves—to spiral continuously along its surface, end to end." He suggests, likewise, that "an organizing principle of 'Sibyl's Leaves' is continuity, of the kind afforded by the cylinder" ("Phonographic Hopkins: Sound, Cylinders, Silence, and 'Spelt from Sibyl's Leaves'," Victorian Poetry, 56.2 [2018]: 149, 153).

36. W. H. Gardner, Gerard Manley Hopkins (1844–1889): A Study of Poetic Idiosyncrasy in Relation to Poetic Tradition, 2 vol. (London: Oxford Univ. Press, 1958), 1:98.

37. Hopkins, "Author's Preface," Poetical Works, 117.

38. Jennifer Ann Wagner, A Moment's Monument: Revisionary Poetics and the Nineteenth-Century English Sonnet (Madison: Fairleigh Dickinson Univ. Press, 1996), 170.

39. Gerard Manley Hopkins, "Rhythm and Other Structural Parts of Verse—Rhetoric," in The Note-Books and Papers of Gerard Manley Hopkins, ed. Humphry House (London: Oxford Univ. Press, 1937), 243, 246.

40. Milroy, 148.

41. Milroy, 150.

42. David Sonstroem, "Making Earnest of Game: G. M. Hopkins and Nonsense Poetry," Modern Language Quarterly 28.2 (1967): 200–201.

43. "The Syzygies Columns from The Lady" [1891–92], from The Pamphlets of Lewis Carroll: Games, Puzzles & Related Pieces, ed. Christopher Morgan (New York: Lewis Carroll Society of North America, 2015), 222.

44. Carroll, Pamphlets, 231.

45. Brown, The Poetry of Victorian Scientists, 213.

46. Brown has also drawn a comparison between the ways that Hopkins and Sylvester insist their poems are "written to be read aloud, actively informed by breath as utterance," arguing that, although "coming to their analyses from different directions, with strong interests and complimentary aptitudes in science and poetry, both Sylvester and Hopkins treat words in poetry as physical objects with their own mass and momentum, which accordingly enter into dynamic relations with one another and resonate in the air" (The Poetry of Victorian Scientists, 232–33).

47. For further discussion of the relationship between accentual and quantitative theories of meter in the nineteenth century, see Joseph Phelan, The Music of Verse: Metrical Experiment in Nineteenth-Century Poetry (Hampshire: Palgrave Macmillan, 2012); Meredith Martin, The Rise and Fall of Meter: Poetry and English National Culture, 1860–1930 (Princeton: Princeton Univ. Press, 2012); Emily Harrington, "The Measure of Time: Rising and Falling in Victorian Meters," Literature Compass 4.1 (2007): 336–54; and Yopie Prins, "Victorian Meters," in The Cambridge Companion to Victorian Poetry, ed. Joseph Bristow (Cambridge: Cambridge Univ. Press, 2000), 89–113. As Martin notes, many of the terms used here are somewhat contentious: "Though the few scholars who notice the historical divisions in the study of English prosody often cast the debate in terms of 'accent' versus 'quantity,' or 'stress' versus 'time,' the very definitions of 'accent,' 'quantity,' 'stress,' and 'time' in English verse were dynamic, malleable, and shifted in specificity and abstraction depending on the intended audience" (94–95).

48. Richard Roe, The Elements of English Metre, Both in Prose and Verse, Illustrated, Under a Variety of Examples, by the Analogous Proportions of Annexed Lines, and by Other Occasional Marks (London: 1801), 1. See also Julia S. Carlson, Romantic Marks and Measures: Wordsworth's Poetry in Fields of Print (Philadelphia: Univ. of Pennsylvania Press, 2016), who reads Roe's treatise as partaking in the context of "contemporary cartographic culture" (4).

49. Roe, Elements of English Metre, 2; 20.

50. John Ruskin, Elements of English Prosody, For Use in St. George's Schools (Kent: George Allen, 1880), 2.

51. Ruskin, Elements of English Prosody, 4, 9.

52. Ruskin, 5.

53. Edgar Allan Poe, "The Rationale of Verse" in Essays and Reviews, ed. G. R. Thompson (New York: Literary Classics of the United States, 1984), 26, 35. It is also possible, according to Poe, for short syllables to be pronounced in "double quick time" (43).

54. Poe, "Rationale of Verse," 45.

55. Lanier, Science of English Verse, 53, 51.

56. Lanier, Science of English Verse, 55–56.

57. For more on nineteenth-century musical prosody, see Joseph Phelan, The Music of Verse; and Yopie Prins, "'Break, Break, Break' into Song," in Meter Matters, ed. Jason David Hall (Athens: Ohio Univ. Press, 2011), 105–34.

58. Coventry Patmore, Coventry Patmore's "Essay on English Metrical Law": A Critical Edition with a Commentary, ed. Sister Mary Augustine Roth (Washington, D. C.: The Catholic Univ. of America Press, 1961), 15.

59. Patmore, English Metrical Law, 21.

60. Samuel Taylor Coleridge, The Major Works, ed. H. J. Jackson (Oxford: Oxford Univ. Press, 2008), 69.

61. Coleridge, The Major Works, 69.

62. Phelan, Music of Verse, 103 (see also 96–97, 110–111). Although accentual rhythm and quantitative meter are often opposed, it should be noted that accentual poetry is sometimes subsumed into a system of isochronous scansion on the grounds that, while individual syllables might not conform to the highly regulated long and short quantities of classical prosody, the accented syllables themselves may demarcate equivalent intervals of time. Patmore, for instance, writes that the "great Gothic alliterating metre" is "one of the most scientifically perfect metres ever invented," obeying the law of "approximate equality of time between accent and accent" (Patmore, English Metrical Law, 32–33). However, as Phelan notes, "[w]here Patmore is determined to force alliterative verse to conform to the pattern of isochronous intervals," others, such as George P. Marsh, are "content to see it as a separate but related species of verse with its own internal logic" (Phelan, Music of Verse, 111).

63. Edwin Guest, A History of English Rhythms, 2 vol. (London: William Pickering, 1838), 1:2, 111; later published as a revised edition in 1882.

64. George P. Marsh, "Lecture XXV: Alliteration, Line-Rhyme and Assonance," in Lectures on the English Language (New York: Charles Scribner, 1860), 542, 544–45. See also Phelan, 110–11.

65. Hopkins, "Author's Preface," 116.

66. Hopkins, "Author's Preface," 116.

67. Meredith Martin, "Hopkins's Prosody" in Victorian Poetry 49.2 (2011): 2. Harrington, "The Measure of Time," 349.

68. Elisabeth M. Schneider, The Dragon in the Gate: Studies in the Poetry of G. M. Hopkins (Berkeley: Univ. of California Press, 1968), 65.

69. Phelan, The Music of Verse, 124.

70. Hopkins, "Author's Preface," 116–17.

71. OED, s.v., "rubato, adj. and n."

72. Arthur Cayley, "Sixth Memoir Upon Quantics," Philosophical Transactions of the Royal Society of London 149 (1859), 90.

73. Richards, Mathematical Visions, 130.

74. Richards, Mathematical Visions, 7.

75. William Whewell, Of a Liberal Education in General; and with Particular Reference to the Leading Studies of the University of Cambridge. Part I. Principles and Recent History (London: John W. Parker, 1850), 29. See also Richards, Mathematical Visions, 132–33.

76. Alice Jenkins, Space and the 'March of the Mind': Literature and the Physical Sciences in Britain 1815–1850 (Oxford: Oxford Univ. Press, 2007), 24–25.

77. Jason David Hall, Nineteenth-Century Verse and Technology: Machines of Meter (Basingstoke: Palgrave Macmillan, 2017), 63.

78. Martin, Rise and Fall of Meter, 10.

79. Martin, Rise and Fall of Meter, 98

80. Richards, Mathematical Visions, 138.

81. For the rise of comparative philology in nineteenth-century Britain and, in particular, its effect on Hopkins's work, see Cary H. Plotkin, The Tenth Muse: Victorian Philology and the Genesis of the Poetic Language of Gerard Manley Hopkins (Carbondale: Southern Illinois Univ. Press, 1989).

82. Max Müller, "Lecture I: The Science of Language one of the Physical Sciences," Lectures on The Science of Language (London: Longman, Green, Longman, and Roberts, 1862), 22.

83. William Spottiswoode, "Address," Report of the Forty-Eighth Meeting of the British Association for the Advancement of Science, Held at Dublin in August 1878 (London: John Murray, 1879), 17.

84. As Yopie Prins—drawing on the work of Simon Jarvis—writes in "What is Historical Poetics?," prosody can operate both as a kind of "cognition," or "thinking-through-making" (Jarvis), and as a form of "recognition," "thinking-through-reading"—modes of "thinking through (simultaneously about and in) verse" ("What is Historical Poetics?," Modern Language Quarterly 77.1 [2016]: 15, 18).

ISSN
1080-6547
Print ISSN
0013-8304
Pages
133-166
Launched on MUSE
2021-03-09
Open Access
No