Poet and mathematician Paul Valéry begins his essay on poetic form, Les Coquillages, by noting that “the mollusk exudes its shell,” letting the building material “seep through” the environing mantle to give it protective and coherent shape. Gaston Bachelard comments in The Poetics of Space that Valéry’s preoccupation is with “the mystery of form-giving life, the mystery of slow, continuous formation.” But I would add here that form-giving life is the necessary and symbiotic complement of life-giving form. As Aristotle put it in the Metaphysics, matter can never appear in its pure state but only insofar as it has been informed by the eidos, the idea or form. Malacology and the arts in general have much in common. Valéry continues: “As we say, a ‘sonnet,’ an ‘ode,’ a ‘sonata,’ or a ‘fugue,’ to designate well-defined forms, so we say a ‘conch,’ a ‘helmet,’ a ‘cameo’…all of them names of shells.” And in either case, the carapace unfolds from within and provides for the object’s longevity. Valéry’s poetic contemporary Francis Ponge brings out the point in a prose-poem entitled “Snails” in which he refers to the shell as “an intrinsic part of their being…a work of art, a monument. It endures longer than they do.”
But poets in particular, unlike mollusks and snails, do not, to quote Valéry again, “derive the material of their works from their own [physical] substance” but rather from “a specified application of their minds.” In other words, the poet as a divided being must strive artifically to be “natural,” duplicating at a higher or conscious level of aspiration the involuntary and effortless phenomenon itself. The beauty and intricacy of nature is diffracted through the intervening medium of the mind. This application of the precedent law or eidos, when successful, is what Valéry denominates as perfection in art, a realized expression of the desire manifest in human production for “the sureness of execution, the inner necessity, the indissoluble bond between form and material that are revealed to us in the humblest of shells.”
And in the humblest of poems too. Consider the terse, psalm-like lyric by Gerard Manley Hopkins, “Pied Beauty.”
Glory be to God for dappled things—
For skies of couple-colour as a brinded cow;
For rose-moles all in stipple upon trout that swim;
Fresh-firecoal chestnut-falls; finches’ wings;
Landscape plotted and pieced—fold, fallow and plough;
And all trades, their gear and tackle and trim.
All things counter, original, spare, strange;
Whatever is fickle, freckled (who knows how?)
With swift, slow; sweet, sour; adazzle, dim;
He fathers-forth whose beauty is past change:
Praise him.
Perhaps the first thing to remark is that “Pied Beauty” is a rather short paean of ten lines tagged by a hemistich and arranged as two grammatical sentences. If we examine it technically, we see that it falls into two syntactical units of thought of 6 and 4½ lines, the first part formulated as a string of concrete particulars and the second as a general or abstract statement commenting on the nature of these particulars and ending with a hortative. Considered mathematically, in consolidating a ratio of 6 to 4½, it mimes exactly the relation of octave to sestet, or 8 to 6, of the staple Petrarchan sonnet. (Actually, looked at in reverse, the ratio is .750 to 1, relatively close to the Fibonacci number.) Further, it conforms to the Petrarchan model not only structurally but also thematically insofar as the second section develops and qualifies the longer, antecedent passage—the Petrarchan mode of thinking. The template Hopkins has followed is plainly the canonical 14-line “Italian” sonnet but the result is what he calls a “curtal sonnet,” an abbreviated version of the original paradigm yet wholly consistent with it.
All this is obvious enough but the real question is: why? A little reflection on the subject he is addressing will make his purpose clear. Hopkins begins by giving glory to God for the mutable and variegated nature of the Creation, a world in constant motion swarming with intricate specifics in both its natural and human manifestations. But in the coda—“coda,” deriving from the Latin word for “tail,” suggests another reason why the poet called this kind of sonnet curtal or “curtailed,” and why Milton, for that matter, called his extended sonnets “caudated”—we are now exhorted to praise God not only for the iridescent diversity of the Creation but for the paradox that the Author of change is Himself “past change.” As in Spenser’s “Cantos of Mutability” from The Faery Queene, which fathers-forth the very poem we are reading, we come to recognize that the beauty of the changeable is both caused and transcended by the beauty of the permanent, that the temporal is subsumed in the eternal, and that we can appreciate or construe the latter only through the medium of the former. The formal structure of the poem is consequently seen to exist in strict analogical symmetry with its theme. Hopkins has renovated, modified, changed the Petrarchan sonnet, giving us a poem which at first reading appears to have no affiliation with it, but he has simultaneously preserved the mathematical ratio of octave to sestet, establishing the permanence of the poetic tradition in which individual poems may vary, complexify and proliferate. While altering the gear and tackle and trim of the sonnet, he has succeeded in retaining its formal unity and changelessness. This particular sonnet is to the natural world as the sonnet form itself is to the supersensual one. By making his point not only discursively but also formally, structurally, Hopkins grounds his argument in exemplification. The poet is playing with a theologicopoetic version of Group Theory, pursuing a topological study of the Creation in which invariants are conserved under transformations. This is the principle of whose “secret Presence” yet another mathematician-poet, author of the ground- breaking Algebra, writes in Rubáiyát 51,
…through Creation’s veins
Running Quicksilver-like eludes your pains;
Taking all shapes from low to highest; and
They change and perish all—but He remains.
Both the form and the theme of “Pied Beauty” would seem to radiate from Augustine’s gloss on the Creation in Epistles 5, which Hopkins no doubt had clearly in mind when he set about composing his poem: “God is…the unchanged Creator of all things that change…[Who] adds, abolishes, curtails, increases or diminishes” (italics mine). But the poem also gestures toward Aquinas’ Summa Theologica I, Q. 1, Art. 9 where we read “It is natural to man to attain to intellectual truths through sensible things, because all our knowledge originates in sense…For what God is not is clearer to us than what He is. Therefore similitudes drawn from things farthest away from God form within us a truer estimate that God is above whatever we may think of Him.” It points as well to the conceptual fount of these reflections in St. Paul, Romans 1: 20: “For the invisible things of him from the creation of the world are …understood by the things that are made, even his eternal power and Godhead.” This latter passage itself stems materially—or so it seems to me—from Psalm 148 whose paratactic language and peroration “Pied Beauty” recruits and modulates: “Fire, and hail; snow, and vapours; stormy wind… praise the name of the Lord.”
Thus even the way in which Hopkins assimilated his sources and allowed them to “seep through” parallels the dialectic of the one and the many which informs the poem. As a theologian relying on a variety of literary, doctrinal and scriptural texts, Hopkins attempted to clarify the singular relation between the temporal and the eternal, between the plurality of the world and its sole, unchanging Creator. His sources are duly pied although the burden of his argument is simple and indivisible. But as a poet, he played with formal computations to construct a quasi-mathematical artifice, a poem abiding faithfully by the law of numbers which generates identity in difference but also difference in identity. The poem seems “natural,” like an unpremeditated meditation, but, like Valéry’s mollusk, it unfolds according to an inner law of precedent commutations which confers stability upon it, illustrating what Valéry describes in his Cahiers as the “acceptance of exquisite shackles” and “the unfailing triumph of sacrifice.” (Or like Francis Ponge’s cupboard which holds out against gravity, the poem exhibits “A built-in resistance [which] favours its personal identity and form.”) Stability, however, is not predictability. Note how much more surprising the Hopkins sonnet is in its rendering of a traditional dynamic than the standard Platonic version of the theme, framed perhaps most memorably in Joachim du Bellay’s pretty but entirely predictable “Sonnet to Heavenly Beauty”:
And there in the most highest heaven shalt thou
Behold the very beauty, whereof now
Thou worshippest the shadow upon earth.
The only thing predictable about a good poem is the astonishment we feel as the law of numbers produces something entirely individual and personal within its generic envelope. In the sweeping elucidation of Oswald Spengler in The Decline of the West, “The great arts are, one and all, modes of interpretation by means of limits placed on number.” And again: “It is the style of a Soul that comes out in the world of numbers, and the world of numbers includes something more than the science thereof.” More particularly, it is this which, according to Spengler, imparts character, whether to an individual or a work of art: “the highest constancy in the essential with the maximum variability in the details” (italics mine).
Poetic form thus carries a mathematical signature and may be regarded as “natural” only in the degree to which it is internally structured. I am tempted to define the ideal poem as the square root of five metrical feet, minus one catalectic syllable, divided by two symmetrical stanzas—which would give us numerically the Golden Section and its Fibonacci limit. Such a definition, of course, is only a metaphorical antic meant to suggest the natural quest for aesthetic perfection and beauty. Yet it is precisely such internal structuration that permits both the insoluble mysteriousness and the bounded uniqueness of the thing made.
