Author: Medieval Studies Research Blog

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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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“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.

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Melusine: The Myth, the Woman, the Legend

I recently gave a guest lecture during Dr. Megan Hall’s fall 2024 course entitled “Witches, Warriors, and Wonder Women: Women, Power, and Writing in History.” To prepare for my visit, students read excerpts from Jean d’Arras’s Melusine; Or, the Noble History of Lusignan as translated and edited by Donald Maddox and Sara Sturm-Maddox (Penn State UP, 2012). The story of Melusine, for those not in the know, is steeped in folklore. Various accounts of the tale involve a fairy woman named Melusine who takes on a half-serpent form from the waist down during part of the week and must hide this from her lover lest they suffer the consequences! Jean’s version of the tale, which dates to 1393, offers an account of the founding of the Lusignan dynasty and how Melusine, a half-fairy woman cursed to assume a half-serpent form on Saturdays, played a major role in its establishment and prestige. The House of Lusignan in its heyday counted Crusader kings among their ranks. My goals were to get Dr. Hall’s students thinking about the representation of non-humanness, the ways in which Melusine wields and exercises power, the significance of relating a historical family’s lineage to a fairy founder, and the truth claims that Jean makes. Certainly a tall order for our hour and fifteen minutes together, but trust me: the students rose to the occasion!

Tour Mélusine was built at the end of the 12th century/beginning of the 13th century to support the fortified town of Vouvant in western France. It is a vestige from when members of the House of Lusignan built a castle in the area.
Legend has it that Melusine herself built the tower in a single night. (Right, close up) Note the weather vane at the top! It’s shaped like Melusine’s half-serpent form.

The text’s prologue immediately draws you into Jean’s narrative web. I find it striking how he claims to weave together various sources and reconciles them with his Latin Christian faith so that he can then go on and discuss Melusine. He references Aristotle and Saint Paul’s Epistle to the Romans in his discussion of marvels. Yet, he writes that “even those well versed in science can hear or see things they cannot believe but which are nonetheless true. I mention these matters because of the marvels that occur in the story I am about to tell you, as it pleased God my Creator and at the behest” of his patron, John, Duke of Berry (20). The marvels that Jean references, of course, are the ones associated with fairy magic and power. Fairies—and Jean references Gervase of Tilbury’s account of fairies for this portion of the prologue—can take on the form of beautiful human women. They can marry human men and even bear children with them, but these men must make promises to their fairy wives and uphold them. These promises can range from a prohibition from seeing the fairy wife nude to never seeing her in childbed.

According to Jean’s summary of Gervase, “[a]s long as these men kept their promises, they increased in rank and prosperity, but at the moment they broke them, they lost the women, and their fortunes slowly declined” (21). In a few short paragraphs, Jean not only tries to bolster the authority of his tale with these references to notable predecessors, but he also builds the world of fairies for his audience. Melusine then is but one example of how the marvel of fairies can operate, leading to some major consequences. As a class, we tried to make sense of Jean’s claims and how he reconciles fanciful fairy tales with what seemed like major authoritative sources. Surely, there must be something special about fairies—why else would Jean spend so much time insisting on their existence and the truth of the tale he is about to share? Furthermore, why would a powerful family like the Lusignans want to connect their family line to a fairy? After discussing the prologue, we were ready to tackle the rest of the text.

Raymond walks in on his wife, Melusine, in her bath and discovers she has the lower body of a serpent. Illustration from the Jean d’Arras work, Le livre de Mélusine (The Book of Melusine), 1478.

The tale unfolds as Jean recounts Melusine’s first encounter with her soon-to-be husband, Raymondin. Melusine’s enchanting beauty causes Raymondin to fall in love at first sight. The two marry under one important condition: that he never attempt to see her on Saturdays. Ever! Unbeknownst to Raymondin, on Saturdays Melusine keeps away from him and hides the fact that her lower half takes on a serpentine form. For years, they enjoy a prosperous marriage. They have numerous sons together, and the majority of them, according to Jean, go on to be rulers of Cyprus, Armenia, and more. Melusine takes on the role of master planner and architect. She builds fortresses and advises her husband on how to increase his wealth and prestige. This marital bliss, however, comes to an end when Raymondin spies on Melusine and discovers her Saturday secret. Though he keeps it to himself for some time, in a moment of anger he reveals knowledge of her weekly transformation and weaponizes it against her in an argument. Since he breaks his promise to her, Melusine takes leave of him. When his death nears, she returns in the form of a dragon.

Dr. Hall’s students truly impressed me with their thoughtful engagement with the text. We had conversations about female agency and power. I asked them to think about narratives parallels in the text and the role of curses and magic across similar events. I pointed out how when fairies and humans reproduce, their progeny bear remarkable physical features ranging from gigantism to having an unusual number of eyes. We pondered what a prestigious family like the Lusignans would gain from claiming a half-fairy woman as a major progenitor and have her powers be a major explanation for their wealth and prestige. I had so much fun diving in the text with them. It was clear to me that they had plenty to say about the ways in which the text represents various interpersonal dynamics. Melusine’s fairy qualities add to her allure and ability to influence those around her, for better or worse.

As our short time together came to an end, I asked them to think about Melusine’s legacy and afterlives. The legend of Melusine has endured and continued to captivate over the centuries. The major coffee chain Starbucks, for instance, has a rendition of Melusine as its logo (though the company’s own lore obscures this link!). For the jazz fans reading, please know that there is an incredible musician by the name of Cécile McLorin Salvant, whose seventh solo album, Mélusine, was released in March 2023. In the description for the album on the website Bandcamp[1] , Salvant unpacks the significance of the Melusine story and how it resonates with her.

Album art for Cécile McLorin Salvant’s Mélusine (2023).

What I find striking about Salvant’s reflections on the Melusine story is her reading of how significant gazes are. She shares that the tale is “also the story of the destructive power of the gaze. Raymondin’s sword pierces a hole into [Mélusine’s] iron door. His gaze does too. The gaze is transformative and combustible. She sees that he is secretly seeing her. Her secret is revealed. This double gaze turns her into a dragon.” Dr. Hall’s students definitely picked up on the power of the gaze but also recalled that it is a power that had to be weaponized before it could transform. In other words, when Raymondin first sees Melusine’s true form, he keeps his transgression to himself. His gaze is a breach of trust, and Melusine knows that her husband spied on her but decides to forgive him because he maintains her secret. However, when he becomes enraged, his anger causes him to lose all discretion. Raymondin angrily reveals that he knows about Melusine’s weekly transformations and resents her. The students recognized that it was precisely this resentment that made the revelation of the secret so powerful.

Another powerful way that Salvant relates to the Melusine story is through the idea of hybridity. Salvant was born in the United States to a French mother and a Haitian father. She spent ample time studying music in Aix-en-Provence, France. She is intimately familiar with negotiating various languages and cultures. When discussing the album Mélusine, she says it is “partly about that feeling of being a hybrid, a mixture of different cultures.” Though we did not get ample time to discuss Salvant’s feeling of hybridity, I do love how she draws a connection between her experiences and that of the legendary Melusine, who also had to navigate different cultures and human/non-human experiences. I find it beautiful that the tale of Melusine endures after all this time and can inspire people to explore the intersections of their own identity and reflect on how they experience the world.

While students were packing up their bags and heading out the door, one student lingered behind and wanted to keep the conversation going. This student wanted to talk about how Melusine’s representation of fairies as having conflict with God struck them; in their culture, fairies and even gnomes are guardians and protectors. The enthusiasm that the student exuded was infectious! And reflecting on this moment now, this just speaks even more to the allure not just of Melusine but of the magical realm at large: a space of play and imagination, sure, but also of power relations, of fidelity, of exploring identity, and so much more.

Anne Le, Ph.D.
Public Humanities Postdoctoral Fellow
Medieval Institute
University of Notre Dame

https://cecilemclorinsalvant.bandcamp.com/album/m-lusine