Category: Departments & Disciplines

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Mathematical “Small Things” and the Quadrivium

What did early mathematical education look like in the Middle Ages? As is commonly known, the ideal Liberal Arts curriculum of the Middle Ages featured both the Trivium (dedicated to the study of words) and Quadrivium (dedicated to the study of nature in the form of mathematical arts). The Trivium included Grammar, Logic, and Rhetoric. The Quadrivium consisted of Music, Astronomy, Geometry, and Arithmetic. These seven ways (viae) of liberal arts learning prepared students who studied them diligently to “comprehend everything that they read, elevat[ing] their understanding to all things and empower[ing] them to cut through the knots of all problems possible of solution” (John of Salisbury, Metalogicon I.12). Even today, the concept of a Liberal Arts education that prepares a student for life and whatever (foreseen and unforeseen) challenges lay aheadremains. And yet, for anyone who has educated a child, the idea of delaying mathematical education until the early teen years (which is when the formal Quadrivium was taught) seems completely impractical and misguided.

Unknown Miniaturist, French (active 1190s Paris). Bodleian Library, MS Laud Misc. 409 (Oxford Manuscript ) f.3v.

Did medieval educators really wait until students had a full understanding of the Trivium before introducing mathematics? The difficulty here comes in part from the lack of extensive knowledge of the curriculum of early childhood medieval education, including mathematical instruction. The institutions for learning changed over time and even geographic region. Early childhood education could take place in a home, in a monastery, or at a local cathedral school. Another difficulty may also be that our cultures mean slightly different things when we talk about the discipline of mathematics.

The “paper trail” for exactly what early childhood mathematical education might have looked like is not vast. But one tiny, but vivid, glimpse of what boyhood mathematical pursuits might have looked like can be seen in the writings of Hugh of St. Victor, an early twelfth century canon regular who wrote a book on the Liberal Arts called the Didascalicon. In this work, Hugh of St. Victor gives a rare view of his own early mathematical education:

I laid out pebbles for numbers, and I marked the pavement with black coals and by a model placed right before my eyes, I plainly showed what difference there is between an obtuse-angled, and an acute triangle. Whether or not an equilateral parallelogram would yield the same area as a square when two of its sides were multiplied together, I learned by walking both figures and measuring them with my feet. Often I kept watch outdoors through the winter nights like one of the fixed stars by which we measure time. Often I used to bring out my strings, stretched to their number on the wooden frame, both that I might note wih my ear the difference among the tones and that I might at the same time delight my soul with the sweetness of the sound. These were boyish pursuits…yet not without their utility for me, nor does my present knowledge of them lie heavy upon my stomach. (VI.3)

Hugh describes these activities as grounding him “in things small” so that he could “safely strive for all” later in life.

Notice how many of the activities mentioned by Hugh of St. Victor do not require a textbook at all, especially with a charismatic teacher, or in the case of Hugh’s own life, a particularly inquisitive child. Counting and the study of angles required only pebbles. The figuring of surface area required only the measurement of feet. An early acquaintance with the stars required actually going out to look at the night sky, even when it was cold, and the study of the relationship between musical notes came from literally fiddling around with a simple stringed instrument. To these activities, we might presumably add the common medieval practices of singing (cantus) and possibly dancing in set patterns. Or the calculating of times and seasons (computus). Or measurements of land and sea masses for commerce or geography. Or ratios for cooking. Many of these activities can be conveyed orally through constant interaction with numbers in the physical world. That is not to say that no formal study or book learning could or was be done in these areas, but the bulk of early mathematical learning did not need to take place in a school environment with a textbook. All that was needed was a student, the physical world, and a teacher with mathematical knowledge.

Christine and the Sybil pointing to a ladder from the heavens, from the Book of the Queen, France (Paris), c. 1410-1414, Harley MS 4431, f. 189v.

What Hugh recognized was that these mathematical activities, whether for play or practical application, were essential for what he and his contemporaries would have considered the formal discipline of mathematics as a liberal art (i.e. the Quadrivium), which would have taken place during the teenage years at higher level schools. Hugh distinguishes arts and disciplines in the following manner: “Knowledge can be called an art ‘when it comprises the rules and precepts of an art’ as it does in the study of how to write; knowledge can be called a discipline when it is said to be ‘full’ as it is in the ‘instructional’ science, or mathematics” (II.1).

A table used for Computus. Harley MS 3667 f 001v.

 In other words, the sorts of activities Hugh describes himself doing as a boy were not mathematical disciplines in his terminology. Instead, his boyish mathematical play was both pleasant at the time and useful as he grew up to study the mathematical disciplines. For this reason, Hugh praised such activity as best because it aids one’s movement “step by step” rather than “fall[ing] head over heels when [attempting] to make a great leap ahead” (VI.2). This learning process mirrors the original discovery of the disciplines themselves by humanity. As Hugh writes:

Such was the origin of all the arts; scanning them all, we find this true. Before there was grammar, men both wrote and spoke; before there was dialectic, they distinguished the true from the false by reasoning; before there was rhetoric, they discoursed upon civil laws; before there was arithmetic, there was knowledge of counting; before there was an art of music, they sang; before there was geometry, they measured fields; before there was astronomy, they marked off periods of time from the courses of the stars. But then came the arts, which, though they took their rise in usage, nonetheless excel it. (I.11)

Early childish mathematical play was not the Quadrivium, but Hugh considered it a necessary preparation for the later study of the Quadrivial arts. Just as Boethius argued in Institutio arithmetica 1,1,7 that the quadrivium provides steps (gradus) by which the mind is progressively illuminated and can raise itself from its immediate sensible circumstances to the certainty of intelligible truth, so Hugh argued that the humble mathematical play of childhood was one step on the way to learning the discipline of mathematics. Computus, stargazing, learning to sing, learning to dance, and making geometric shapes with pebbles—none of this was Quadrivium. These activities could be boyhood pursuits…or in some cases, ends in themselves practiced into adulthood, but activities of this sort were, in Hugh’s opinion, a necessary preparatory step for the Quadrivial disciplines.

Lesley-Anne Dyer Williams
Public Humanities Postdoctoral Fellow
Medieval Institute
University of Notre Dame

Lesley-Anne Dyer Williams is a Professor for Memoria College’s Masters of Arts in Great Books program and graduated with her doctorate from the University of Notre Dame’s Medieval Institute in 2012. She was also the founding director Liberal Arts Guild at LeTourneau University. Her research focuses upon twelfth-century Platonism and poetry, especially Thierry of Chartres and Bernard Silvestris.

Further Reading:

Hugh of St. Victor. The Didascalicon of Hugh of St. Victor:  A Medieval Guide to the Arts. Edited & translated by Jerome Taylor, Columbia University Press, 1991.

Jaeger, C. Stephen. The Envy of Angels: Cathedral Schools and Social Ideals in Medieval Europe, 950-1200. University of Pennsylvania Press, 1994.

John of Salisbury. Metalogicon. Translated by C.C.J. Webb, Clarendon Press, 1929.

Orme, Nicholas. Medieval Children. Yale University Press, 2001.

Orme, Nicholas. Medieval Schools. Yale University Press, 2006.

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Epithets, Epistles, and Erasmus, Oh my [most serene king of Britain]!

In his treatise on the writing of letters, the De Conscribendis epistolis,[1] Dutch humanist and prolific letter-writer Desiderius Erasmus emphasizes the importance of the opening section of the letter. Going wrong here in the salutatio, the writer will, he says,“…like a poor helmsman…run aground right in the harbour.”[2]  His advice is to keep things simple, adopting the “new” Ciceronian style:

I approve the simplicity of the ancients; I only wish that we could emulate it everywhere amid the corrupt practices of our age, so that we might greet one another by the mere mention of names, as in: ‘Pliny gives his Calvus greeting!’ What could be truer or simpler? When you hear a man’s name pronounced you hear all his good qualities in a nutshell.[3]

Here and throughout his treatise, Erasmus makes a point of attacking the “corrupt practices of our age,” that is to say, medieval dictaminal practice. Erasmus’ friend and correspondent, the younger scholar Juan Luis Vives, who wrote a De Conscribendis Epistolis of his own, agrees that in medieval practice the use of epithets had spiraled out of control. Vives suggests that they should instead be employed as sparingly as possible, their use restricted to legitimate titles derived from the office of the addressee: “senator, consul, quaestor, bishop, priest, curate.”[4] He underscores the fact that a badly placed epithet might tarnish honor instead of enhancing it, suggesting that a name like Erasmus of Rotterdam carries its own distinction with no need for titles:

Other titles, originating from a debased custom, produce laughter or annoyance rather than confer distinction. Is it not more flattering to be so highly thought of that there is no need of epithets, as in the case of Guillaume Bude, Erasmus of Rotterdam, or Thomas More? In the lustre of such names expressions like “most learned in both tongues,” “consummate theologian,” “gentleman of greatest renown” are superfluous.[5]

Detail, Two Studies of the Left Hand of Erasmus of Rotterdam; Study of the Right Hand Writing. Silverpoint, black crayon and red chalk on grey-primed paper, 20.6 × 15.3 cm, Louvre, Paris. Christian Müller; Stephan Kemperdick; Maryan Ainsworth; et al, Hans Holbein the Younger: The Basel Years, 1515–1532, Munich: Prestel, 2006. Erasmus holds a pointed italic quill, suited to the purposes of his humanistic hand.

Erasmus ostensibly agrees, directly criticizing the customary pleonastic manner of addressing royals and nobles:

The king of the French alone is called “most Christian,” the king of Spain alone “Catholic,” the king of England “most serene,” the emperor alone “ever august,” dukes “most illustrious,” other members of the lesser nobility “illustrious,” and others “most noble.” Who introduced this superstition about titles into the world? … By the constant repetition of phrases like “most reverend lordships,” “Catholic majesties,” and “magnificent fatherhoods” we fill up a large part of a letter, and ruin the gracefulness of the Latin tongue. I pardon those who use them against their will; I do not pardon those who devise them, or who insist upon them as a serious matter.[6]

Despite this ideal, as with so many rules and regulations meant to govern the rules of prose, humanistic or otherwise, theory does not always accord with practice. Vives certainly uses flattering epithets in his own letters, despite his counsel to the contrary.[7] But although Erasmus begins by praising the “mere mention of names,” in practice his treatise goes on to linger on the salutatio for nearly a dozen sections, recommending the use of essential titles and the use of an apt—but not sycophantic—epithet (he suggests over 100 as suitable).[8]

Moreover, for the 1515 dedication to his Senecae Lucubrationes, Erasmus composes a salutation that hardly appears to follow his own advice: “To the most distinguished Father D. Thomas Ruthall,” he writes, “Bishop of Durham, Secretary of State of the Most Serene King of Britain, Erasmus of Roterdam sends greeting.[9] Though he does give himself merely his two names Erasmus Roterodamus, he is not content to give his friend Ruthall the single epithet amplissimo—he goes on to add his titles as well as those of Ruthall’s master, the “Most Serene King of Britain.” Leaving aside the elevated diction Secretarius Magnus and the choice of Britanniae instead of Angliae, Erasmus appears to fall right into the very “superstition about titles” he criticized above: calling the king of England “most serene.” As Erasmus cannot be imagined to here use epithets “against [his] will, in our charity we must conclude that he simply does not “insist upon them as a serious matter.” Indeed, we might imagine this apparent “do as I say, not as I do” as part of a game among friends. A glance at the letters of another friend of Erasmus, fellow correspondent Sir Thomas More, reveals that More uses titles only rarely in his salutations—rare exceptions include the high-flung salutation of Henry (that same most serene king of Britain) as Britanniae Galliaeque Regi and a 1506 letter addressed to Regio apud Anglos Secretario, our very own Thomas Ruthall. In the year 1506, Erasmus was staying in More’s house at Bucklersbury, and the two were engaged in the translation of Lucian’s dialogues. More’s almost overly learned letter to Ruthall offers some “first fruits” of these Greek studies. More and Erasmus engage here in a game of language and words, breaking their own rules, offering their efforts to a mutual humanist friend they knew would delight in their linguistic play.

Despite living the most fruitful parts of his adult career after the conclusion of what is generally considered the medieval period, Desiderius Erasmus never really attempted to avoid or evade the Middle Ages. Both in his return to the Classics and his agitations for a new humanistic approach to writing and scholarship, Erasmus continues to engage with medieval thinkers and medieval ways of thinking. In responding to and helping drive the dramatic shift away from the centuries-old medieval dictaminal tradition designed for the mass production of documents essential to the court of every Christian kingdom to a humanistic model grown out of the fourteenth-century Renaissance and Francesco Petrarcha’s rediscovery of the personal letters of Cicero to his friend Atticus, Erasmus engages in a humanistic game that plays off of tension with the near medieval past.

Rebecca West, PhD
Literature Core Faculty
University of Dallas

[1] Erasmus was already writing an early version of this text for his student Robert Fisher by about 1498 (Epistularum scribendarum ratio).⁠ A pirated version of his treatise was published at Oxford in 1521 by Siberch,⁠ more or less forcing Erasmus to come out with an expanded, corrected official version in 1522. The standard edition of the treatise is Charles Fantazzi, ed., “On the Writing of Letters / De Conscribendis Epistolis,” in The Collected Works of Erasmus: Literary and Educational Writings, 3 and 4, by J. K. Sowards, ed., 25 (University of Toronto Press, 1985). Hereafter abbreviated as CWE 25/3.

[2] CWE 25/3:50. The ars dictaminis aimed at organizing the letter—a form largely meant for public declamation of official communications—according to standardized models following a set of rules derived from ancient Ciceronian oratory. Erasmus devotes significant portions of his treatise to the proper way to frame the opening of a letter, the portion corresponding to the salutatio and captatio benevolentiae of a letter written according to the terminology of the medieval dictaminalmodel.

[3] CWE 25/3:51

[4] Charles Fantazzi, ed., J.L. Vives: De Conscribendis Epistolis: Critical Edition with Introduction, Translation and Annotation, trans. Charles Fantazzi (Leiden: Brill, 1989), 47.

[5] Vives: De Conscribendis Epistolis, 47.

[6] CWE 25/3:61.

[7] See the introduction to Fantazzi’s edition for the use of flattering epithets in Vives’ own corpus of letters.

[8] CWE 25/3:50-62.

[9] Amplissimo patri D. Thomae Ruthallo Episcopo Dunelmensi Serenissimi Britanniae Regis Secretario Magno Erasmus Roterodamus S. D. Text from Elizabeth Frances Rogers, ed., The Correspondence of Sir Thomas More, (Princeton: University Press, 1947), letter 5.Translation from Clarence H. Miller, Leicester Bradner, Charles A. Lynch, and Revilo P. Oliver, eds., The Complete Works of St. Thomas More, Volume 3, Part II (New Haven: Yale University Press, 1984), 1:2–8.

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“A Cord Laid Tight Loosens Discord”: The Shifting Role of Precision in the Byzantine LandSurvey Tradition

Through my dissertation research on the middle Byzantine Maeander Valley in Western Asia Minor (modern Türkiye), I had become fascinated by an eleventh-century estate ledger, known as the Praktikon of Adam, and how Byzantine surveyors described landscapes in technical language.[1]  The boundary description or periorismos was composed of formulaic phrases describing the route taken needed for a surveyor to encircle a property.  Sometimes, but not always, these descriptions are accompanied by measurements, which can be rationalized from the Byzantine units into meters.  I am interested in mapping these boundary descriptions to compare them with the results of archaeological field surveys. 

This blog entry will be a short musing upon the shifting role of precision in the tradition of Byzantine land survey.  In our modern world, cartographic precision has become an unquestioned backdrop to how we view landscapes.  We rarely feel the need to justify spatial precision when representing a landscape on a map (i.e., Fig. 1).  Such a casual aesthetic commitment to cartographic precision has no counterpart among ancient and medieval representations of landscapes.  Therefore, any study of boundary descriptions must rest upon why such precision was necessary.  The presence (and absence) of that precision reveals the underlying motivations of the surveyors.  Such motivations must be considered when using these documents to understand Byzantine landscapes. 

The Casual Use of Cartography on a Mural for the City of Owego, New York.  Photo by author.

Surveying for Taxation

The original goal of Byzantine land survey was calculating the tax burden of a property.  Twelve Byzantine survey manuals survive, which were written to instruct new bureaucrats on how to survey the land and then use those measurements to calculate area.  The study of these textbooks provides a starting point for understanding how and why Byzantine surveyed the land.[2]  Surviving documents found in Byzantine archives, such as the Praktikon of Adam, show how the recommendations of these textbooks were or were not enacted.

The purpose of taxation prioritized the taking of measurements.  Land is measured with ropes (schoinioi or sokaria).  An illustration from a Byzantine Octateuch (Fig. 2) shows the survey of land in action.  I took the title of the blog entry from the Byzantine Greek written on this image: “A cord laid tight loosens discord” – akin to the English proverb “Good fences make good neighbors.”  The whole set of illustrations show a Byzantine twist on the delineation of land in the last ten chapters of the Book of Josuah in the Old Testament.  These ropes were not just a tool but also the most important unit of measurement.  Ropes are divided into fathoms (orgyia), based on two different criteria: the quality of the soil or the region of the empire.  The 10-fathom rope was the standard for high quality soils, while lesser soils were measured with a 12-fathom rope.  In Thrakesion (western Asia Minor), the 10-fathom rope was the standard for all soils. 

Two Byzantine Surveyors Measure Land with a Rope, Vat. Gr. 746, f. 461r. By permission of Biblioteca Apostolica Vaticana, with all rights reserved.

The desired product of the fiscal survey was calculating the area of many properties efficiently.  Therefore, the methods were less than geometrically sound.  Or as Jacques Lefort and his team wrote: “The geometric technique of the tax office is nevertheless very simple, since it consists, in a world implicitly conceived as almost everywhere orthogonal, in multiplying the length by the width.  It therefore owes nothing to geometric science and is in fact resolved by arithmetic, and even then, only by the art of multiplying well.”[3]  In other words, no field cannot be reduced to a rectangle (Fig. 3).  While scholars in the past attributed these imprecise methods to the decline of geometry in the Middle Ages, the Byzantines were capable of complicated geometry when it suited the task at hand.  Precision in individual measurements was not important for the tax office.

Fields were Simplified into Rectangles to Calculate Approximate Area.  Drawn by Author.

Finally, the tax surveyors were interested in a space, but not in a place.  The taxation surveys of fields are often unmoored from their landscapes.  The precise location of the field made little difference when calculating the tax burden.  Therefore, without other correlating data, the precision of the tax survey, while useful for maintaining an empire, provides little help to the archaeologist.

The Motivation of the Boundary Description

On the other hand, a boundary description represents a diverging motivation for land survey within the same tradition.  Not all have measurements, but when they did, the individual measurements appear to be more important than the whole.  The description is grounded in the specifics of the landscape.  The precision of individual measurements included often outstrips the needs of calculating area (Fig. 3).  Instead, precision correlates with the presence of properties owned by neighbors of the estate (Fig. 4).  Instead of taxation, the purpose of the precision of the boundary description appears to be related to the control of land.  Theft of land by unscrumptious neighbors was a growing problem from the late eleventh century through to the end of Byzantium.  The Praktikon of Adam reveals two such cases where theft was discovered.  Still, the former was recorded in a boundary description, while the latter still relied upon the taxation-based survey technique, which shows that both techniques could theoretically reveal theft even if the boundary description appears better equipped.

The Correlation between Neighbors and Precision in a Boundary Description.

While there is more research to be done, I am still convinced that the embeddedness of the boundary description within their respective settings and the incorporation of measurements into the descriptions can make these technical descriptions valuable comparanda to archaeological survey data in reconstructing the landscapes of the Byzantine world.

Tyler Wolford, PhD
Byzantine Studies Postdoctoral Fellowship
Medieval Institute
University of Notre Dame

[1] M. Nystazopoulou-Pelekidou, ed., Βυζαντινὰ ἔγγραφα τῆς Μονῆς Πάτμου. Volume 2. Δημοσίων λειτουργῶν. Athens: National Institute of Research, 1980, Document 50.

[2] J. Lefort, B. Bondoux, J.-Cl. Cheynet, J.-P. Grélois, V. Kravari.  Géométries du fisc byzantin.  Réalités Byzantines 4.  Paris: Éditions P. Lethielleux, 1991.

[3] Lefort et al., Géométries du fisc byzantine, 244.