This study triangulates the three features of the final scene in the Summoner's Tale identified in my title. The relationship between two of these, farting and theology, is clear enough, since the final episode involves a parody of Pentecost. The scatological substitution of Thomas's fart for the mighty wind accompanying the descent of the Holy Spirit in Acts(FN2) has religious significance as a satiric image of friars' false claims to apostolic stature.(FN1) Also prominent in this scene is language loaded with academic, particularly logical/mathematical, meaning. The challenge to divide the fart equally is seen by the lord of the village as a "probleme" arising out of the "ymaginacioun" (III 2218-19) of the churl Thomas. Initially appearing so strange as to be a "question" heretofore unknown in "ars-metrike" (III 2222-23) and so difficult as to be "an inpossible" (III 2231), the problem requires "demonstracion" (III 2224) of a solution. Jankin then formulates his cartwheel proposal, claiming that it provides "preeve which that is demonstratif" (III 2272) that a fart can be "evene deled" (III 2249), and the people present judge him to have addressed the issue as "wel as Euclide [dide] or Ptholomee" (III 2289). 2 The Pentecostal parody does not require this academic register. I want to add to thinking about that register by focusing on the geometric aspects of the episode, apparent enough in Jankin's circular solution, in the repeated insistence that the fart must be "departed equally" (III 2237; see also III 2225, 2249, 2273), and most obviously in the comparisons to Euclid and Ptolemy. My argument is, first, that a particular intellectual development in fourteenth-century England contributed to Chaucer's staging of the Pentecostal parody within the framework of an academic parody, and, second, that this framework has effects beyond antifraternal satire, one of which is to invite reflection on efforts to measure or quantify abstract theological concerns.
    The intellectual development behind this episode is what John Murdoch calls "the near frenzy to measure everything imaginable." Murdoch and others have explored this concern among a group of logicians and philosophers at Oxford in the 1320s to 1340s known as the "Merton school" or the "Oxford calculators," the most important of whom is Thomas Bradwardine. The calculators and the people they influenced applied various "measure languages" (analytical terminology used to discuss such subjects as proportion, infinity and continuity, and local motion) not only to problems in logic and natural science, but also to philosophical and theological questions. For example, the mathematical distinction between the infinite and the finite was applied to the theological issue of distinguishing the love due God from that due one's fellow creatures, and as a means of demonstrating how there could be variation within species and yet incommensurability between species. The language of intension and remission of forms (acceleration/deceleration, or increase/decrease in qualities such as heat) was used to analyze questions of the movement of the will.(FN3)
    Euclid was important to anyone thinking along such lines. The clarity and explicitness of the Elements in its structure and its proofs made it the quintessential model for mathematical/scientific thinking and the presentation of demonstrative arguments. Book I begins with first principles (definitions, postulates, and common notions or axioms) and then works out propositions (problems and theorems) based on logical reasoning from those principles. Subsequent books introduce new definitions before presenting further propositions. The most popular medieval Latin versions of the Elements gave its methodological procedures even greater prominence.(FN4) Ptolemy's Almagest, the central text for medieval mathematical astronomy, confirmed Euclid's status, presenting itself as an applied version of the Elements, with extended geometric proofs used to demonstrate and rationalize astronomical motion. In his preface Ptolemy divides theoretical philosophy into theology, mathematics, and physics. He claims that the first and third are more "guesswork" than science -- theology because of its "ungraspable nature," physics because sublunar matter is unstable. Mathematics is the only way to "sure and unshakeable knowledge" because "its kind of proof proceeds by indisputable methods, namely arithmetic and geometry." Thus, in approaching his subject theoretically, he will describe astronomical movement "by means of proofs using geometrical method," which means using the Elements, whose principles and conclusions are appealed to throughout.(FN5) No wonder Euclid and Ptolemy are linked at the end of the Summoner's Tale. In a sense Jankin's solution to Thomas's division problem is even more Ptolemaic than Euclidean, since it brings geometric thinking to bear on physical data (though hypothetically, like the arguments per imaginationem of so many fourteenth-century thinkers) rather than dealing with the pure forms of geometry.
    Bradwardine's works include a Geometria speculativa, a sort of digest of the Elements for arts students, and other treatises on arithmetic and what might be called applied mathematics, notably De proportionibus velocitatum in motibus, which begins in Euclidean fashion with definitiones, suppositiones, and conclusiones and culminates with his famous function concerning local motion. Its preface states that investigating motion is the province of natural philosophy but requires a mathematical knowledge of proportions, and then makes a further claim, quoting Boethius: "whoever omits mathematical studies has destroyed the whole of philosophic knowledge."(FN6) Chaucer mentions Bradwardine in the Nun's Priest's Tale as one of many "clerkis" who "in scole" have sifted through arguments on questions of free will and predestination (VII 3234-50). He is referring to the enormous De causa dei, a treatise on God's causative role in all earthly experience, and in this theological work Bradwardine, acting on the belief just cited, did not omit mathematics.
    The early studies of De causa dei by Gordon Leff and Heiko Oberman considered its ideas solely in relation to contemporary theological views, but more recent scholarship has focused on its place within the intellectual context of the calculators and within the totality of Bradwardine's writing.(FN7) From this perspective we see that Euclid is not just one among many authors cited in De causa dei but the major inspiration for its structure. Just as the Geometria speculativa had begun with a concern for the principles of demonstration ("principia demonstrationis"), starting with definitions, and the De proportionibus had followed a similar logical sequence, so too De causa dei starts with two primary "suppositiones" followed by a large set of "correlationes" and then builds its arguments sequentially with reference to what has been previously assumed or demonstrated. Bradwardine's terminology here is not identical to that in Euclid or in his own mathematical works, but Marco Sbrozi has argued persuasively that De causa dei follows an essentially Euclidean structure, once one takes into account how its theology would affect the choice of a demonstrative procedure and language. For example, the treatise could not begin with definitions, since no finite human being can adequately define God, the extent of whose perfection and goodness is "simpliciter infinita," as Bradwardine says in elucidating his first suppositio.(FN8)
    Geometry offered not only a model for reasoning, but also a wealth of material that could be enlisted in arguments on specific topics in natural science, philosophy, or theology. One of these topics, the nature of continua, has some relevance to the Summoner's Tale because discussions of it include two ideas important to the mathematical resonance of the final scene: divisibility and incommensurability. Do continuous phenomena like lines, space, and time lack distinguishable constituents (thus being potentially infinitely divisible into smaller and smaller parts), or are they composed of a number of separate constituents, indivisibles, that cannot be further subdivided -- points that constitute lines, intervals that constitute space, instants that constitute time? Medieval discussion of this topic in natural philosophy was based on Book VI of Aristotle's Physics, but the calculators extended Aristotle's thinking beyond space and time into qualities like heat and cold.(FN9) In Bradwardine's contribution to this debate, De continuo, he identifies a number of positions taken on the nature of continua by ancient and contemporary thinkers: (1) continua are not made up of indivisibles ("athomis") but of endlessly divisible parts -- this is Aristotle's position, also his own and that of the vast majority of medieval natural philosophers; (2) they are made up of indivisible bodies; (3) they are made up of a finite number of indivisible points; (4) they are made up of an infinite number of indivisible conjoined points; (5) they are made up of an infinite number of indivisible points set one next to the other. This variety of views suggests something of the detailed speculation on continua taking place in fourteenth-century academic circles.(FN10)
    Continuists often used geometrical examples to refute the indivisibilist approach, as in the following argument, which became popular after its appearance in Duns Scotus. Imagine a square with a diagonal drawn through it, thus creating two isosceles right triangles sharing a hypotenuse. Then imagine drawing all the possible parallel lines from every point on the left side of the square to every point on the right. Continuists argued that if there were a certain number of indivisible points constituting each side, then every line drawn across the square would have to pass at one and only one point through the diagonal, which means that it would have the same number of points and thus the same length as the sides. But that consequence, of course, violates any common-sense observation of isosceles right triangles, and more importantly violates Elements I.47, the famous Pythagorean theorem, also known in later medieval England by its Latin nickname, dulcarnon, which Chaucer mentions in Troilus and Criseyde (III, 931, 933). "Dulcarnoun" proves that the length of the hypotenuse of any right triangle is the square root of the sum of the squares of the length of the two other sides. In the case of an isosceles right triangle, the diagonal's length is incommensurable with each side ([root]2:1), inexpressible by any ratio of single (and hence of indivisible) units. Thus the idea that a line contains some given number of indivisible points leads to a geometric impossibility and must be wrong.(FN11)
    Indivisibilists had a difficult time dealing with this argument, and one of them was John Wyclif. Norman Kretzmann has argued that Wyclif must have known he was taking a decidedly minority position in this philosophical debate, one that opposed not only Bradwardine, whom in other respects he admired, but also a generation of Oxonians, some of whom may have been his teachers. Yet he held his ground, offering various criticisms of the continuists, including an effort to refute their geometrically based incommensurability argument. Kretzmann thinks Wyclif's theology drove his reasoning here: God knows his creation completely and distinctly, knows precisely even what may appear infinite or incommensurable to human perception.(FN12)
    Wyclif's indivisibilist view of continua was adopted in a long vernacular Lollard sermon, where it is applied to human beings' relationship to God. The anonymous author of the Omnis plantacio (written ca. 1410) discusses how contemporary sects within the church (monks, canons, friars) have "departid" Christian unity. (That sweeping charge of the division of spiritual oneness, which both Lollards and their opponents hurled at each other, ultimately provides the broadest socio-religious context for the "departynge of the fart," as it is called in a rubric before Sum T, III 2243, in some manuscripts.) As a parallel to the "errours and heresies" introduced by these divisive groups, he points to "dyuerse sectis" among ancient philosophers and notes some of their incompatible teachings. He focuses particularly on the difference between indivisibilist Plato and continuist Aristotle in regard to whether lines and time are composed of "indiuisible" points and instants that stand next to each other without any intermediary, or whether lines and time always admit of large numbers of points or instants coming "bitwene" any two given points, no matter how close they are. He phrases the whole debate in the sort of language used by the indivisibilists and states emphatically that the Aristotelian position is wrong. Later he returns to the idea in the context of Jesus' reference to the "long praiers" of the hypocritical scribes and Pharisees (Matt. 23.14; Mark 12.40). An entity's length is measured by the distance between its ends: the "shortist" line has "two poyntis," the shortest time "two instantis." The two ends of a prayer are the human soul and God, and when they are "knytt togidir bi charite," without any sin "goynge bitwene," the prayer is spiritually short no matter how long it takes. If the human end of the prayer is distanced from God through sinfulness, it will never reach heaven. In this analogy the theology of prayer and philosophical indivisibilism seem tightly interlinked, and as Kretzmann says in regard to Wyclif, the theology seems to prompt the physics: an indivisibilist view that lines can ultimately be shortened to two points "togidir" seems more hospitable to the idea of an unmediated connection between a devout Christian and God than does a continuist view always admitting further divisions "bitwene" points.(FN13)
    The author of the Omnis plantacio clearly had a university background that not all Lollards had. Still, he seems to think that his example from natural science is not so recondite as to be beyond a vernacular audience, and its use suggests that at least some academic controversies might circulate beyond their original scholastic settings. Chaucer could have become aware of mathematical intrusions into philosophy and theology in various ways. He could have seen them in Bradwardine's De causa dei, which uses arithmetical and geometrical explanations and comparisons throughout, or in the work of Wyclif. In spite of his condemnation of "curiositates in scienciis mathematicis" and other academic vanities, Wyclif knew well such kinds of thinking, as Kretzmann has shown, and was quite willing to use his own logical/scientific knowledge to make theological points.(FN14) Chaucer could have easily learned about Oxford academic tendencies from his "philosophical" friend Ralph Strode (Tr, V, 1857). Before becoming a London lawyer, Strode was a fellow at Merton in 1359-60, during a period when intellectual trends established earlier were being continued if not impressively enriched. He wrote a treatise on logic about the same time that Wyclif did, and over a period of several years he and Wyclif engaged in some respectful disputation, in regard to which Wyclif's responses survive.(FN15) Other individuals in Chaucer's circle or among his civil service acquaintances could have had similar interests. Astronomical references in his poetry as well as the Treatise on the Astrolabe suggest at least some scientific inclination in a portion of his audience, and both the presence of "Lollard knights" at the royal court and Richard II's growing concern in the 1390s with enforcing orthodoxy could have prompted attention to the academic culture out of which Wyclif and Strode emerged.(FN16)
    However he acquired it, Chaucer shows elsewhere his alertness to details of academic training, notably in the Reeve's Tale: Symkyn jests that although his house is small, the clerks who need to spend the night there have "lerned art" and can make "argumentes" to turn twenty feet of space into an area a mile broad (I 4122-24). William F. Woods contends that this passage, rather than merely representing a "lewed" person's view of university education, suggests knowledge of some sophisticated thinking on the relationship of substance and size and the possibility of vastly smaller or larger worlds. Susan Yager notes that at the end of the story the "whit thyng" Symkyn's wife sees and then misinterprets (I 4301-2) probably alludes to the academic fondness for using examples of whiteness in arguments having to do with perception and cognition.(FN17) The end of the Summoner's Tale, I think, is a more extended development of a similar sort of knowledge. At this level the scene is a farcical playing out of a continuum problem. How does one "parte that wol nat departed be/To every man yliche" (III 2214-15)? That is, how does one evenly divide a continuous magnitude like "eir" (III 2234), in this case with its accompanying qualities of "soun" (III 2226, 2273) and "savour" (III 2226) or "stynk" (III 2274), especially when it is not only in motion but is diminishing -- "And evere it wasteth litel and litel awey" (III 2235)? In medieval terms, Thomas's problem entails a question not just about dividing a continuous quantity but also about the intension and remission of forms, what writers referred to as uniform or difform difformity, that is, changes in intensity or velocity that were either steady or variable.
    Thomas's fart-problem and Jankin's solution thus evoke not only Pentecost but also the fourteenth-century fashion of importing mathematical measurement and geometrical demonstration into discussions of both physical and metaphysical realities. What is the tone of this parody? What are its implications? I would suggest, as Peter Travis has, multiple possibilities.(FN18) As often noted, it certainly contributes to the Summoner's satirical treatment of Friar John, who earlier associates himself with learning (III 2108-14, 2186) yet now finds himself unable to solve a problem posed by a churl. It is also part of Chaucer's occasional play with technical academic terms, though it is hard to know how many members of his audience would have been responsive to such play, and in what way. The parody can be seen as a comic triumph of geometric intelligence, a celebration of the "ymaginacioun" of both the person who posed it and the person who solved it. Thomas, in the view of the lord's court, changes from "demonyak" to man of "subtiltee" and "heigh wit" (III 2240, 2290-92) once it is proven that his problem is not "an inpossible."(FN19) Jankin's practical cartwheel geometry assures equality of distribution among the convent (assuming that the movement of sound and smell is a uniform difformity), since it stations each friar equally distant from the source and from each other (except of course for John's special place), thus assuring the same "part," that is, portion, to all.(FN20) In effect, the squire does with the fart what Ptolemy often did with the cosmos and what Nicole Oresme, John Casali, and others did with the calculators' efforts to quantify qualities -- he approaches a real-world phenomenon in a way that allows for a two-dimensional geometric rationalization: the Euclidean foundation of the twelve-spoked wheel and its even spacing of the friars is a dodecagon inscribed within a circle.(FN21) This triumph is itself diversely interpretable, but it certainly entails, given socio-religious issues of the 1390s, a victory for lay judgment over clerical privilege, realized in large part, Fiona Somerset argues, by the "translation" of clerical academic discourse into the mouths of various lay speakers.(FN22)
    On the other hand, the relentless antifraternalism of the tale and the grossness of the Pentecostal parody link the geometry to the satire and make it readable as an absurd effort to apportion out, divide up, the corrupt physical dregs of an original spiritual unity and fullness. From that perspective, the scene may suggest, as Timothy D. O'Brien has argued, a more general skepticism toward theological discourse that tries to provide mathematical certainties in regard to matters of faith. I disagree with O'Brien's contention that the final scene is deeply critical of all its characters and their intellectual activity, but the Chaucerian position he stakes out seems consistent with the kind of Christian outlook the author demonstrates elsewhere, one more interested in personal behavior or piety than in abstract or intricate theological speculation. In that regard, Thomas's problem has a further dimension in that it evokes the controversy over transubstantiation, both in its intellectual content (dividing a unity into twelve) and in its academic register.(FN23)
    The debate over the nature of the eucharist must have made the use of close logical reasoning in regard to theological matters both more apparent and more provocative to an audience beyond the university. In particular, a popular Lollard position was that the orthodox view (the consecrated host possesses the accidents of bread and the substance of Christ) over-academicized an issue on which an ordinary, common-sense view was correct: if it looks like bread, breaks like bread, tastes like bread, and especially if Jesus called it bread, it is bread. Thus Pierce the Plowman's Crede ends by condemning the folly of fraternal "studye" and "disput[ing]" about the eucharist, which leaves the "maystres of dyvinitie" less truly faithful than many of the "lewede"; and William Thorpe dismisses the official explanation of transubstantiation as clerical "scole-mater" and "curious and so sotile sofestrie." Yet the same charge could be made against the lengthy and sometimes tendentious reasoning of Wyclif and his followers -- John Tissington calls his unorthodox opponents "isti subtiles Scripturae scrutatores." Rita Copeland, discussing such mutual name-calling, has pointed out that Wyclif and at least some of his followers were in fact academic "insiders" who liked to pose as "outsiders," an observation no doubt available also to a friend of Ralph Strode's.(FN24) Given a social context of public religious debate in which each side accused the other both of sophistical school-talk and of wrongly dividing what cannot be divided, it seems uncertain what if any partisan implications the eucharistic echoes at the end of the Summoner's Tale have. Recent criticism suggests that the secular solution has anticlerical (conceivably Wycliffite) force. But the fact that a manorial lord and squire in the hinterlands perform this funny, hyperbolic academicizing while the friar "maister" (III 2184-86) stands around in baffled outrage might also poke fun at the Lollard position, with its sharp insistence on common-sense lay understanding of the eucharist as opposed to university-inspired sophistry about it. Chaucer's scene comically reverses and thus undercuts the Wycliffite stereotypes. In sum, while the target of the Pentecostal parody seems reasonably clear, the target of the academic parody seems less so. Particularly in regard to the transubstantiation controversy, the scene suggests perhaps less a partisan than a more detached, independent view: a plague on both your housels.
    This complexity of implication is what makes the end of the Summoner's Tale so rich, not least in regard to Chaucer's relationship to a fourteenth-century mentality of measurement. Vitally concerned with accurate measurement in other contexts, notably the Treatise on the Astrolabe, here he seems to be playing in an elusive way -- critical in some respects, jovial in others -- with intellectual efforts to apportion and quantify. Some kinds of measurement, of course, can be made easily. "What is a ferthyng worth parted in twelve?," asks Friar John (III 1967), trying to get Thomas to donate money exclusively to his own convent. That question of course anticipates the climactic parting in twelve, perhaps in a more philosophical way than has been acknowledged. John's division question has an easy arithmetical answer: a farthing parted in twelve, even though no such coin existed for exchange purposes, is worth 1/48th of a penny. Thomas's division problem, as we have seen, is much more difficult and ultimately points to a unity that by dogma is indivisible. The farthing, the fart, the divine flatus of Acts 2 parodied by the fart, the Holy Spirit symbolized by that biblical wind -- here is a range from the material to the immaterial, the measurable to the immeasurable and incommensurable, that in the Summoner's Tale becomes a comic invitation to question how far principles, or ideologies, of rational measurement can be fruitfully applied.
    For Chaucer, I think, there are limits to that fruitfulness, and his own work, especially in the 1390s, suggests what they are: the Treatise on the Astrolabe is for measuring physical, not metaphysical, phenomena. But that sense of limitation does not (pace O'Brien) suggest an exclusively critical view of the impulse to measure that developed at Oxford several decades earlier. Bradwardine especially understood that mathematical rigor was both important and limited. His view is implicit in the structure and theology of De causa dei, and it is neatly summarized at the end of De proportionibus. He has completed his book, he says, "by the grace of that Mover from whom all motions proceed and between whom and the thing he moves there exists no proportion -- to whom be honor and glory as long as there is any motion."(FN25) God is beyond rational calculation -- he is incommensurable with the human intellect. Yet Bradwardine will try to measure measurable motion and proportion as best he can, and use mathematical ideas even to help prove that human beings cannot measure God. So too the combination of Pentecostal parody and geometric comedy at the end of the Summoner's Tale suggests a cautious but not gloomy attitude to what "heigh wit" can measure.
ADDED MATERIAL
    Cleveland State University Cleveland, Ohio ([email protected])

FOOTNOTES
1. I reaffirmed and explored the Pentecostal parody in "The End of The Summoner's Tale and the Uses of Pentecost," Studies in the Age of Chaucer 21 (1999): 209-45. John V. Fleming has developed the argument further in "The Pentecosts of Four Poets," in R. F. Yeager and Charlotte C. Morse, eds., Speaking Images: Essays in Honor of V. A. Kolve (Asheville, N.C., 2001), 111-41. In "Chaucer's Summoner's Tale: Flatulence, Blasphemy, and the Emperor's Clothes," Studies in Philology 104 (2007): 455-70, John Finlayson argues that the tale's biblical parodies may be more thoroughly anti-authoritarian than most recent criticism has acknowledged.
2. All Chaucer quotations are taken from The Riverside Chaucer, ed. Larry D. Benson, 3rd edn. (Boston, 1987). The academic register, often noted, is explored most thoroughly in Roy J. Pearcy, "Chaucer's 'An Impossible,'" Notes and Queries, n.s. 14 (1967): 322-25; and Timothy D. O'Brien, "'Ars-Metrik': Science, Satire, and Chaucer's Summoner," Mosaic 23 (1990): 1-22, which covers some of the same ground I do here.
3. The scholarship on the calculators is large. Useful starting points are William J. Courtenay, Schools and Scholars in Fourteenth-Century England (Princeton, 1987), 219-306; and Edith Dudley Sylla, "The Oxford Calculators," in Norman Kretzmann, Anthony Kenny, and Jan Pinborg, eds., The Cambridge History of Later Medieval Philosophy (Cambridge, U.K., 1982), 540-63. The calculators' two most famous mathematical achievements are the mean speed theorem and Bradwardine's function relating velocity, force, and resistance; for background, analysis, and relevant texts, see Marshall Clagett, The Science of Mechanics in the Middle Ages (Madison, 1959), 199-329, 421-503. John E. Murdoch's comment on the rage to measure appears in "From Social into Intellectual Factors: An Aspect of the Unitary Character of Late Medieval Learning," in John E. Murdoch and Edith Dudley Sylla, eds., The Cultural Context of Medieval Learning (Boston, 1975), 271-348, at 287. Among Murdoch's several substantial essays on this material, see also "Mathesis in philosophiam scholasticam introducta: The Rise and Development of the Application of Mathematics in Fourteenth Century Philosophy and Theology," in Arts libéraux et philosophie au moyen âge, Actes du Quatrième Congrès International de Philosophie Médiévale (Montreal, 1969), 215-54. Murdoch, Sylla, and Courtenay all stress that the calculators' measure languages developed not as mathematics per se, but within the academic disciplines of logic, natural philosophy, or theology (e.g., in regard to questions on the eternity of the world, the nature of angelic motion). For the socioeconomic context of the calculators' interest in quantification, see Joel Kaye, Economy and Nature in the Fourteenth Century: Money, Market Exchange, and the Emergence of Scientific Thought (Cambridge, U.K., 1998). In focusing on this fourteenth-century movement, I do not mean to imply that earlier thinkers did not use mathematics in support of theological reasoning, or to deny or ignore the earlier scientific and musical theories that inform Chaucer's treatment of sound in SumT and HF. On these, see Robert A. Pratt, "Albertus Magnus and the Problem of Sound and Odor in the Summoner's Tale," Philological Quarterly 57 (1978): 267-68; and Peter Travis, "Thirteen Ways of Listening to a Fart: Noise in Chaucer's Summoner's Tale," Exemplaria 16 (2004): 323-48, at 330-33, 342-43. On the influence of optics, see Linda Tarte Holley, Chaucer's Measuring Eye (Houston, 1990). But the principal contextual pressure here, as Pearcy and O'Brien have argued, is chronologically closer to home.
4. Euclid, The Thirteen Books of the Elements, 2nd edn., trans. Thomas L. Heath, 3 vols. (New York, 1956). On the text as a model for Aristotelian ideals of demonstrative science, see Sten Ebbesen, "Ancient Scholastic Logic as the Source of Medieval Scholastic Logic," in Kretzmann, Kenny, and Pinborg, eds., The Cambridge History, 101-27, at 115-16; and Eileen Serene, "Demonstrative Science," also in The Cambridge History, 496-518, at 497, 506. The textual history of the Elements is enormously complicated and still in the process of investigation. Murdoch has argued that the best-known Latin versions of Euclid in the later Middle Ages are more "didactic" than the Greek text, more concerned with making explicit the logical structure of the Elements and with linking the text to philosophical concerns like infinity and continuity. These features are of course consistent with the use of Euclid in universities. See John E. Murdoch, "The Medieval Euclid: Salient Aspects of the Translations of the Elements by Adelard of Bath and Campanus of Novara," Revue de synthèse, 3rd ser., nos. 49-52 (1968): 67-94, and "Transmission into Use: The Evidence of Marginalia in the Medieval Euclides latinus," Revue d'histoire des sciences 56 (2003): 369-82.
5. Claudius Ptolemy, Ptolemy's Almagest, trans. G. J. Toomer (Princeton, 1998), 36-38.
6. Thomas of Bradwardine His Tractatus de Proportionibus: Its Significance for the Development of Mathematical Physics, ed. and trans. H. Lamar Crosby, Jr. (Madison, 1955), 64-65: "Quisquis scientias mathematicales praetermiserit, constat eum omnem philosophiae perdisse doctrinam."
7. Marco Sbrozi, "Metodo matematico e pensiero teologico nel De causa Dei di Thomas Bradwardine," Studi medievali 31 (1990): 143-91; Edith Wilks Dolnikowski, Thomas Bradwardine: A View of Time and a Vision of Eternity in Fourteenth-Century Thought (Leyden, 1995); and George Molland, "Addressing Ancient Authority: Thomas Bradwardine and Prisca Sapientia," Annals of Science 53 (1996): 213-33.
8. Thomas Bradwardine, Geometria speculativa, ed. and trans. George Molland (Stuttgart, 1989); Thomas Bradwardine, De causa dei, ed. Henry Savile (London, 1618; repr. Frankfurt am Main, 1964), 1; and Sbrozi, "Metodo matematico," 163-79.
9. John E. Murdoch, "Infinity and Continuity," in Kretzmann, Kenny, and Pinborg, eds., The Cambridge History, 564-91; Edith Sylla, "Medieval Quantifications of Qualities: The 'Merton School,'" Archive for History of Exact Sciences 8 (1971): 9-39. For detailed studies over a range of periods, see Norman Kretzmann, ed., Infinity and Continuity in Ancient and Medieval Thought (Ithaca, N.Y., 1982). Nicole Oresme, who knew the work of the calculators, has much to say about continua and divisibility at the start of his translation/commentary on Aristotle's De caelo. To Aristotle's statement that continua are "divisible en parties touzjours divisibles," he adds that such a process is of two kinds: physical division, like splitting a log, which cannot proceed indefinitely without destroying substance, and conceptual division, like dividing the heavens into degrees, minutes, and smaller segments, which "par ymaginacion" can take place "sans cesser" (Le Livre du ciel et du monde, ed. and trans. Albert D. Menut and Alexander J. Denomy, C.S.B. [Madison, Wisc., 1968], 44-46). Or, as Chaucer says when describing the markings of the astrolabe, the degrees of the zodiac are divided into "mynutes," the minutes into "secundes, and so furth into smale fraccions infinite" (Astr, I.8.11-13).
10. Text in Murdoch, "Infinity and Continuity," 576n36; discussion, 577-80. On the De continuo and its relationship to De causa dei, see Edith Sylla, "Thomas Bradwardine's De continuo and the Structure of Fourteenth-Century Learning," in Edith Sylla and Michael McVaugh, eds., Texts and Contexts in Ancient and Medieval Science: Studies on the Occasion of John E. Murdoch's Seventieth Birthday (Leiden, 1997), 148-86.
11. Murdoch, "Infinity and Continuity," 579; and Olli Hallamaa, "Continuum, Infinity and Analysis in Theology," in Jan A. Aertsen and Andreas Speer, eds., Raum und Raumvorstellungen im Mittelalter (Berlin, 1998), 375-88, at 378. For an illustration of the argument and its attempted refutation, see John E. Murdoch and Edward A. Synan, "Two Questions on the Continuum: Walter Chatton (?), O. F. M. and Adam Wodeham, O. F. M.," Franciscan Studies 25 (1966): 212-88, at 255, 261-62, where the indivisibilist author, conceding that the diameter is "incommensurabilis" with the sides while evading the consequence of that fact, argues that at the point where each line drawn across the square crosses the diagonal, a "linea tortuosa" is created that would account for the diameter's greater length. More simply, continuists could appeal directly to Elements I.10, where a given straight line is bisected, thus enabling theoretically the division of any line ad infinitum. One of the Latin versions of Euclid even points out that this proposition is incompatible with the idea that a line is made up of indivisible points, since that idea would make any line with an odd number of points incapable of being bisected. See Murdoch, "Medieval Euclid," 91. The proportion (ratio) of diagonal to side of a square is for Bradwardine the principal example of an irrational proportion, which is by definition found only in "quantitatibus incommensurabilibus seu irrationabilibus" (incommensurable or irrational quantities); incommensurable quantities are those that have no "mensura communis" (common measure) (Bradwardine, De proportionibus, 66; compare Geometria speculativa, 100). Perhaps debate/misunderstanding on this subject in continuist/indivisibilist disputes prompted Nicole Oresme, in his twenty-one questions on the Elements, to give this important mathematical fact a prominent place: he asks in two separate questions "Utrum dyameter quadrati sit commensurabilis sue coste" (whether the diagonal of a square is commensurable with its side) and provides two different arguments that it is not; in the second he states as a corollary of the incommensurability that a continuum is endlessly divisible and not composed of finite divisibles (Quaestiones super geometriam Euclidis, ed. H. L. L. Busard [Leiden, 1961], 16-25, 92-106).
12. Norman Kretzmann, "Continua, Indivisibles, and Change in Wyclif's Logic of Scripture," in Anthony Kenny, ed., Wyclif in His Times (Oxford, 1986), 31-65 (refutation discussed on 48-50).
13. The Works of a Lollard Preacher, ed. Anne Hudson, EETS o.s. 317 (Oxford, 2001), 24-25, 136-37, and notes.
14. Johannis Wyclif, Opera minora, ed. J. Loserth (London, 1913), 441. The context of his condemnation of curiositas is a commentary on Matt. 23.8-10. Wyclif continues that the faithful should discern the "scientia et mensura" (knowledge and measure) that are useful for virtuous living and salvation and seek heavenly wisdom "secundum illam mensuram" (according to that measure/standard). The linking of mathematical curiosity and measurement may suggest that he was thinking specifically of the sort of work done by the calculators. For his own use of scientific learning in theological debate, see Heather Phillips, "John Wyclif and the Optics of the Eucharist," in Anne Hudson and Michael Wilks, eds., From Ockham to Wyclif (Oxford, 1987), 245-58; and Iohannis Wyclif, De eucharistia, ed. J. Loserth (London, 1892), esp. 232-68, where, in refuting the idea that Christ's physical body could be in two places at once, he launches into a variety of logical, mathematical, and scientific arguments, including the question of whether continua are composed of indivisibles. At the end of the treatise he notes that he has digressed into "difficultates huius materie scolastice" (the difficulties of this subject in an academic way) in order to expose error (324).
15. On the academic scene in these years, see Courtenay, Schools and Scholars, 332-48. On Strode, see Derek Pearsall, The Life of Geoffrey Chaucer (Oxford, 1992), 133-34; Courtenay, passim; and Rodney Delasanta, "Chaucer and Strode," Chaucer Review 26 (1991): 205-18. For the Strode-Wyclif material, see Williell R. Thomson, The Latin Writings of John Wyclyf: An Annotated Catalogue (Toronto, 1983), 227, 234-39.
16. On Richard's relationship to Lollardy, see Nigel Saul, Richard II (New Haven, 1997), 293-303. On the question of astronomical interest within Chaucer's circle, see the recent summary and assessment by Edgar Laird, "Chaucer and Friends: The Audience for the Treatise on the Astrolabe," Chaucer Review 41 (2007): 439-44, at 441-42. Richard's scientific interests, to judge from the manuscripts most closely associated with him, tended toward astrology, physiognomy, and geomancy. See Patricia Eberle, "Richard II and the Literary Arts," in Anthony Goodman and James L. Gillespie, eds., Richard II: The Art of Kingship (Oxford, 1999), 231-53, esp. 241-44.
17. William F. Woods, "Symkyn's Place in the Reeve's Tale," Chaucer Review 39 (2004): 17-40; and Susan Yager, "'A whit thyng in hir eye': Perception and Error in the Reeve's Tale," Chaucer Review 28 (1994): 393-404. For the medieval philosophical analysis of whiteness, in relation to BD, see Peter Travis, "White," Studies in the Age of Chaucer 22 (2000): 1-66, at 13-25.
18. Travis, "Thirteen Ways."
19. The language judging Thomas here may contain a further geometric allusion. A few Euclidean propositions were given nicknames, two of which, "dulcarnoun" and "flemyng of wrecches," Chaucer mentions in Tr, III, 929-38. These names appear to refer to their difficulty for students, implying the intellectual prowess of both Euclid and anyone who can understand the proof. So does another: some manuscripts identify the diagram illustrating Elements IV.10 as "figura demonis," and in one mathematical treatise it is called, interestingly, "figura demonis sive intellectis" (figure of the demon or the knowing one). I will discuss this possibility in a future article.
20. Of course, as John Fleming has pointed out ("The Pentecosts of Four Poets," 130), cartwheel spokes were not hollow, and the image is for iconographic and literary, not technological, purposes. But whether or not the spokes are imagined as hollow (III 2273-74 suggest they are), the envisaged placement of fraternal noses would assure equal distribution.
21. On the use of geometry to visualize some of the calculators' ideas about measurement, particularly Nicole Oresme's graphing efforts, see Clagett, Science of Mechanics, 331-416. No Euclidean proposition concerns inscribing a regular dodecagon, but there are such propositions in Book IV for other regular polygons, including the hexagon; the geometrical form of the cartwheel is easily constructed by inscribing a regular hexagon and then bisecting each of the resulting arcs. In the most popular version of the Elements in the later Middle Ages, Campanus of Novara adds a note at the end of Book IV making explicit how polygons can be inscribed following a bisecting procedure, such as one that would move from triangle to hexagon to dodecagon and beyond: "Quia igitur scimus inscribere triangulum equilaterum, sciemus per hoc etiam et exagonum et per exagonum duodecagonum" (thus knowing how to inscribe an equilateral triangle, through this [procedure] we may inscribe a hexagon, and from a hexagon inscribe a dodecagon; Campanus of Novara and Euclid's Elements, ed. H. L. L. Busard, 2 vols. [Stuttgart, 2005], 1:158).
22. Fiona Somerset, "'As just as is a squyre': The Politics of 'Lewed Translacion' in Chaucer's Summoner's Tale," Studies in the Age of Chaucer 21 (1999): 187-207. See also Larry Scanlon, Narrative, Authority, and Power: The Medieval Exemplum and the Chaucerian Tradition (Cambridge, U.K., 1994), 160-75. For the wider social background of vernacular "clergie" at this time, see Somerset's Clerical Discourse and Lay Audience in Late Medieval England (Cambridge, U.K., 1998). For a different view of the final scene's comedy, one that reads it as carnivalesque throughout, see James Andreas, "'Newe Science' from 'Olde Bokes': A Bakhtinian Approach to the Summoner's Tale," Chaucer Review 25 (1990): 138-51.
23. Firmly established by Fiona Somerset, "Here, There, and Everywhere? Wycliffite Conceptions of the Eucharist and Chaucer's 'Other' Lollard Joke," in Fiona Somerset, Jill C. Havens, and Derrick G. Pitard, eds., Lollards and Their Influence in Late Medieval England (Woodbridge, Suffolk, 2003), 127-38. Most of her contextual evidence comes from material very close to the likely date of the tale. The divisibility issue was a staple of debate on the eucharist from the start; an early critic of Wyclif, the Franciscan John Tissington, notes that the consecrated bread is "secundum substantiam indivisibilis, secundum speciem divisibilis" (indivisible in regard to its substance, divisible in regard to its appearance; W. W. Shirley, ed., Fasciculi zizaniorum [London, 1858], 152). Wyclif of course maintained that if that substantia was Christ's body "corporaliter" in the host, then it would be divisible; only because Christ's body is in the host "spiritualiter" can a person receive it "integrum" when the sacrament is broken (De eucharistia, 12-13). With such language having been in the air for years, it is hard to imagine that a charge to "parte that wol nat departed be" would not have evoked the transubstantiation wars. See also Mary Hayes, "Privy Speech: Sacred Silence, Dirty Secrets in the Summoner's Tale," Chaucer Review 40 (2006): 263-88, esp. 281-84. On the idea of transubstantiation as an impetus to late medieval scientific thinking per imaginationem, see Woods, "Symkyn's Place," 24-28.
24. Rita Copeland, "Sophistic, Spectrality, Iconoclasm," in Jeremy Dimmick, James Simpson, and Nicolette Zeeman, eds., Images, Idolatry, and Iconoclasm in Late Medieval England (New York, 2002), 112-30; she cites Thorpe on 126. Pierce the Plowman's Crede was quite possibly written about the same time as SumT and is similarly devoted in large part to letting friars' speech condemn itself. I cite from lines 817-27, in Helen Barr, ed., The Piers Plowman Tradition (London, 1993), 96. For John Tissington, see Fasciculi zizaniorum, 144.
25. Bradwardine, De proportionibus, 140-41: "cum illius motoris auxilio a quo motus cuncti procedunt, cuius ad suum mobile nulla proportio reperitur; cui sit