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This article considers how changes in the concept of number allow both poets and mathematicians in the early modern period to imagine and articulate concepts that resist referential signification. Specifically, it examines how both Shakespeare’s “The Phoenix and Turtle” and Robert Recorde’s The Whetstone of Witte employ hybridized lines possessing characteristics of both discrete and continuous types of quantity in order to render irrationality. Tracing the development of a formalized poetic “number line,” which understands verses as negotiating between aural, accentual-syllabic numbers and visual inscriptions, this article claims that “The Phoenix and Turtle” functions diagrammatically. In doing so, the poem engages habits of thought also required by Recorde’s practical manuals on mathematics, but does so in order to interrogate the limits of a rationalized, mathematical worldview.