The diagram on page 293 of Finnegans Wake (Figure 1) is a miniature of Joyce’s cyclic, overlapping universe. As such, it is representative of the Wake as a whole and particularly appropriate for the chapter of Nightly Lessons, in which the diagram provides the basis for Issy’s and the twins’ geometry homework. A graphic representation of contraries, it is, among other things, Joyce’s answer to the philosophical claim made by the Pythagoreans that men died because they could not join their beginnings to their ends. Given that this notion was reinterpreted by Yeats in A Vision, the diagram may also serve as Joyce’s good natured challenge to Yeats’s own geometrical gyres which, in Yeats’s words, attempted to put the serpent’s “tail in its mouth” (Figure 2).1
The figure’s overlapping circles illustrate the opposing and unifying relations at play in the Wake and the means by which ends and beginnings become one in a cosmological system driven by a ricorso. Wake readers are aware that the diagram serves many of the images and icons present in the text, from the twins in the womb, to the mobius strip or “Doublends Jined” of the text, to all the double Os (the “loos” and “loops” in the book), to the frontal and dorsal aspects of a body (male and female), to a map of Phoenix Park, to a portrait of the dreamer, to the process of meiosis, and so forth. In other words, this “lesson,” both for the children and for the reader, functions as a representation of many of the concepts present in the Wake that one would normally assume could not be joined but that, in the general overlapping portmanteau process Joyce employs, most exuberantly can be. By proving that such opposites might indeed be connected, Joyce recalls the philosophy and the geometry of Giordano Bruno, whose coincidencia oppositorum—the union of an action or nature [End Page 235] with its equal and opposite—forms the basis of one answer to the Pythagorean dilemma and underlies much of the Wake’s text.
In one of Issy’s footnotes referring to the diagram, Joyce alludes to The Metaphysical Foundations of Modern Physical Science by Edwin Burtt, which provides some clues as to the provenance of Joyce’s idea. In his chapter on pre-Copernican mathematics, Burtt glosses the tradition of the “geometry of the heavens,”2 citing Giordano Bruno, Nicholas of Cusa, and other precursors to the Scientific Revolution who believed in the mystical-transcendental properties of numbers and borrowed from Neo- Platonic and Pythagorean philosophy. Joyce was particularly fond of Bruno, having encountered him in his college days through J. Lewis McIntyre’s biography of the philosopher, which he reviewed. In that Bruno was fascinated by the idea of geometry as a means of expressing the unifying nature of monism, many of his diagrams in works such as De Monad, Numero, et Figura attempt to unify or to join the unjoinable (Figures 3 and 4). His depiction of androgyny (Figure 3), as Lucia Boldrini observes, is quite similar to Joyce’s diagram.3
If based on Bruno’s design, Wakean geometry is nevertheless much more than just a depiction of the theory of opposites. Joyce’s pictorial representation echoes the overlapping circles found in Burtt’s example of Copernican planetary motion, a diagram (Figure 5) whose content is derived from Copernicus’s De Revolutionibus. As an alternative to Aristotelic cosmology, Copernicus’s mathematical/astronomical model revived an interest in the tradition of the Pythagoreans (Burtt 40) and found a supporter in Bruno, who had become interested in Pythagorean concepts through the Neo-Platonic philosophy of Nicholas of Cusa.4
By referencing both Bruno and Copernicus (Figure 6), Joyce alludes to an entire astrological/astronomical tradition with roots in ancient philosophy (Figure 7). Therefore, the Wake diagram can be read as a summation and a joining of seemingly contrary philosophies and modes of thought—Copernicus’s heliocentric theory, Bruno monad, Cusa’s numerological metaphysics, and the Pythagorean centrality of numbers. By combining different versions of the same geometrical pattern of circles and triangles...