John Dee on geometry: Texts, teaching and the Euclidean tradition

Abstract

John Dee’s mathematical interests have principally been studied through his Mathematicall praeface to Henry Billingsley’s 1570 translation of Euclid’s Elements. The focus here is broadened to include the notes he added to Books X–XIII of the Elements. I argue that this additional material drew on a manuscript text, the Tyrocinium mathematicum, that Dee wrote a decade earlier, probably as tutor to the youthful Thomas Digges. Using new evidence for this now-lost work, as well as his notes on Euclid, makes it possible to clarify Dee’s approach to geometry. The contrasting positions adopted by his Parisian acquaintance Petrus Ramus also illuminate Dee’s geometrical choices and values. Unlike Ramus, Dee was not a pugnacious advocate of radical reform, yet he did look beyond the limits of Euclid’s geometry towards deeper disciplinary visions of knowledge. The first published work of his pupil Thomas Digges not only suggests how Dee shaped the younger man’s work but also reflects fresh light back on Dee’s own programme for a ‘more general art Mathematical.’

Introduction

In his Mathematicall praeface to the 1570 English Euclid, John Dee places great stress on the fundamental role of arithmetic and geometry. They lead upwards to the contemplations of pure intellect and downwards to productive engagement with the natural world, serving as the source and root of a whole spectrum of ‘arts mathematical derivative.’ Strangely, given the scholarship that has been devoted to the Mathematicall praeface and to Dee’s active pursuit of various practical mathematical arts, remarkably little attention has been given to his treatment of these two principal mathematical sciences.¹

In this paper I want to focus on Dee’s approach to geometry, both to identify his intellectual position and to assess his significance and impact. Twenty years on from its publication, my principal point of reference is Nicholas Clulee’s John Dee’s natural philosophy (1988). Faced with the terrifyingly broad field of Dee’s intellectual formation and disciplinary ambition, Clulee’s work represents a model of scholarship—integrating biography and the pursuit of patronage with the careful analysis of intellectual development and affiliation. He always seeks to pinpoint Dee’s sources with precision rather than constantly invoking a single overarching explanatory theme (whether Neoplatonism, Renaissance magic, Hermeticism, or any other all- encompassing interpretative term).

Clulee’s attention was centred on Dee’s natural philosophy and his position within the history of science. Although the Mathematicall praeface appears as one of his key texts, Dee’s mathematics enters the account primarily for the light it sheds on these central issues. Here I attempt a similar style of analysis, but take Dee’s geometry as worthy of study in its own right. This means giving close and serious attention to his writings, both surviving and lost, and establishing them within the chronological framework of Dee’s career, while also being alert to wider resonances and responses.

Textbook histories of mathematics have typically focused on the innovations of algebra as the central development of the Renaissance. Judged against transhistorical (or even ‘eternal’) criteria of mathematical significance, sixteenth-century geometry has been considered a static or even stagnant field.² Bennett (2002) has responded to ‘the baneful influence of timelessness’ in the history of Renaissance mathematics by shifting attention to practical geometry, to the vigorous world of instruments and practice. A full account of Dee’s geometry would have to address that broad field, to encompass his inventions and devices for astronomy, navigation, geography and much else. But the realm of texts and scholarship need not be entirely abandoned: far from the cliché of stagnation, geometry was in reality culturally dynamic and diverse. Ancient authorities were not only textually recovered and renewed, but also reworked, supplemented and even attacked. Rather than a placid backwater, geometry was a field pockmarked by active debate and controversy, most famously on the quadrature of the circle.³ Dee was familiar with most of these cross-currents, both through his extraordinarily rich library and his personal acquaintance with many of the leading mathematicians of Europe. By locating him within this contemporary mathematical landscape we can recover what was at stake in the pursuit and teaching of geometry, and see how his mathematical values encouraged the work of others.

Rather than dwelling exclusively on the familiar terrain of Dee’s Mathematicall praeface, I want to take as my point of departure the notes he added to Euclid’s Elements. In the Compendious rehearsal’s listing of the printed and manuscript works produced during his fifty years of study, Dee gave these additions a separate record from the entry for the Mathematicall praeface. He described them as ‘divers & many Annotations, and Inventions Mathematicall, added in sundry places of the foresaid English Euclide, after the tenth Booke of the same.’⁴ They are scattered throughout Books X to XIII and, while not adding up to a connected work, they do represent a significant investment of time.⁵ I argue that these comments drew on a manuscript text, the Tyrocinium mathematicum, that Dee wrote a decade earlier, probably as tutor to the youthful Thomas Digges. Using new evidence for this now-lost work together with his notes on Euclid makes it possible to clarify Dee’s approach to geometry, particularly when illuminated by the contrasting positions adopted by his Parisian acquaintance Petrus Ramus. Unlike Ramus, Dee was not a pugnacious advocate of radical reform, yet he did look beyond the limits of Euclid’s geometry towards deeper disciplinary visions of knowledge. The first published work of his pupil Thomas Digges not only suggests how Dee shaped the younger man’s work but also reflects fresh light back on Dee’s own programme for a ‘more general art Mathematical.’ Situating Dee in his contemporary context provides vital insight, but I conclude that in many respects Dee was continuing well- established traditions of medieval geometry, just as he did in natural philosophy.