Category: Departments & Disciplines

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The Quadrivium and the Stakes for Ordering the Mathematical Arts

 Fyodor Bronnikov, Pythagoreans’ Hymn to the Rising Sun, 1869. Oil on canvas.

Legend has it that Pythagoras sentenced the first person to discover irrational numbers, Hippasus of Metapontum (c.530-450 BC), to death. He was tossed overboard a ship to drown. Why? Pythagoras taught that number was the essence and cause of all things, and for Pythagoras and his followers, numbers meant integers. Hippasus’ discovery of irrational numbers appeared to undermine the very core of Pythagoras’ teachings about the numerical nature of the universe. The secret could not get out. Hippasus had to die.

Imaginary engraving of the philosopher Hippasus of Metapontum. From the 1580 book Illustrium philosophorum et sapientum effigies ab eorum numistatibus extractae, by Girolamo Olgiati. Reprinted 1583.

The existence of irrational numbers became a Pythagorean secret. They were called “unutterables” because in Greek, the ratio between two integers was called logos, and so, irrational numbers were called, alogos, which can be translated as either “irrational” or “not spoken.” The worry caused by this secret knowledge was somewhat alleviated by Eudoxus of Cnidos (408-355 BC) when he argued that the basis of reality was a ratio of magnitudes. In effect, Eudoxus made geometry replace arithmetic as the highest mathematical discipline, the foundation of all others. Geometry and arithmetic were hardly even separate disciplines at the time. This change of emphasis allowed Pythagorean teachings about the numeric nature of the universe to continue.

Philosophia et septem artes liberales (Philosophy and the Seven Liberal Arts), as illustrated in the Hortus deliciarum. The order of the arts here are: grammar, rhetoric, dialectic, music, arithmetic, geometry, and astronomy. A more detailed study of this image can be found here and here.

         The idea that the mathematical disciplines have some orderly relationship between each other is essential for understanding the medieval concept of “quadrivium.” While it is well known that the medieval liberal arts curriculum, at least in its ideal established by Boethius, taught that a student must study both the trivium and quadrivium before progressing to philosophy and theology, the exact nature and rationale for the quadrivium is often less understood. Lists of the arts comprising the quadrivium (arithmetic, geometry, astronomy, and music/harmony) are consistent, but the exact order for these lists can vary. While there is no doubt that sometimes there is truly no rationale for a given order of the mathematical arts, attention to the mathematical art considered the principle or highest can reveal at least three identifiable streams of quadrivial traditions coming from the ancient world (similar to Chenu’s identification of different kinds of Platonism): the Boethian, the Calcidean, and the Capellan. The mathematical art considered “principle” is the one closest to metaphysical reality of the universe and serves as the foundation for all other mathematical disciplines. While the problem of irrational numbers may not have been on the forefront of anyone’s mind in the Middle Ages…it was a closely guarded Pythagorean secret after all…the problem of the principle mathematical art, inherited from Pythagoreanism, was readily available in the source texts.

“Philosophy Presenting the Seven Liberal Arts to Boethius,” about 1460–1470, Coëtivy Master (Henri de Vulcop?), Ms. 42, leaf 2v (91.MS.11.2.verso), Getty Museum Collection.

Boethius (c.480-525) not only established the seven liberal arts as the traditional curriculum for the Middle Ages, but he also wrote treatises on all of the trivium as well as arithmetic, music, and geometry (the latter work is now lost).  He, coined the term, “quadrivium” in his attempt to translate the tessares methodoi (four methods) of the Neopythagorean, Nicomachus of Gerasa (c.60-120). Boethius’ own De institutione arithmetica largely draws upon the work of Nicomachus. Modern day history of mathematics textbooks often observe that Nicomachus’ work is one of the first to distinguish arithmetic and geometry as separate disciplines but that the actual quality of the mathematics contains basic errors. Unlike Euclid, Nicomachus doesn’t always give his proofs. Nicomachus presents arithmetic as the principle mathematical art and as a result, so does Boethius. While Boethius was unlikely to have gotten the problem of irrational numbers from Nicomachus because Nicomachus presents arithmetic as the highest mathematical art, Boethius adopts his fourfold division of the mathematical arts along with the belief that arithmetic was the principle mathematical art (De institutio arithmetica 1,1,8).

Image from Boethius’ De institutione arithmetica in British Library, Harley MS 549.

In his work on arithmetic, Boethius explains that the order of the quadrivium he offers (music, astronomy, geometry, and arithmetic) both reflects the true nature of the universe and is the proper pedagogical order for the study of mathematics as a preparation for philosophy. Progression through each of the arts trains the mind to move from sense perception to intelligible reality.

Philosophy Instructing Boethius on the Role of God. Coëtivy Master (Henri de Vulcop?), about 1460–1470. Ms. 42, leaf 3 (91.MS.11.3.recto), J. Paul Getty Museum.

This progression of the soul can be seen in the Consolation of Philosophy, where Boethius begins with music and is drawn to philosophy upward by means of astronomy, geometry, and finally arithmetic.

While Boethius’ highly influential order of the quadrivium was adopted by both Cassiodorus and Isidore, Calcidiuswrites very clearly in his commentary on Plato’s Timaeus that geometry is the foundation of all other mathematical arts (Commentum 2.32). His influence throughout the Middle Ages was also extensive. Calcidius’ translation and commentary of Plato’s Timaeus, was one of the only texts of Plato available throughout much of the Middle Ages. Although there were other translations of the Timaeus available, Calcidius’ commentary, as Reydams-Schils has demonstrated, was actually a very good introduction to Platonism as a whole because it was designed to introduce the reader to Platonic doctrine in a pedagogically sequenced way from mathematics to physics and then theology. Throughout the earlier Middle Ages, as Somfai has shown, the commentary was used to teach the quadrivium itself, and earlier versions contained numerous geometrical diagrams. While interest in his geometrical figures appears to fall out of favor in the twelfth century and in newer commentaries on the Timaeus, Nicholas of Cusa in the fourteenth century has both the old Calcidius’ commentaries and the newer commentaries, and geometry clearly plays a major role in his understanding of infinity and kinds of infinity.

Early tenth century manuscript of Calcidius’ Latin translation of Plato’s Timaeus from Italy. Reg. lat. 1308 fols. 21 verso – 22 recto medbio01 NAN.10.

The third line of quadrivial tradition can be found in Martianus Capella whose Marriage of Philology and Mercury, places music as the highest of the seven liberal arts, the culmination of his entire work. As Michael Masi has observed, this ordering can be found in many visual depictions of the quadrivium, including most famously, the Incarnation Portal at Chartres Cathedral, where arithmetic is paired with geometry as a mathematical study and music with astronomy as a study in harmony. While the complete reasons for this preference are too numerous to identify in a blog, there is a certain kind of Pythagorean logic even here. Music, for Pythagoras and his followers, was thought to be the best evidence for number being at the foundation of the universe. Even the movement of the stars and planets were considered to be one example of many kinds of music in the universe.

Charles Nègre (French, 1820 – 1880), photographer [Chartres Cathedral, Royal Portal (The Incarnation Portal), South Lateral Doorway], 1857. HeliogravureImage: 59.8 × 45.3 cm (23 9/16 × 17 13/16 in.),  Sheet: 71.6 × 55.1 cm (28 3/16 × 21 11/16 in.) The J. Paul Getty Museum, Los Angeles,  84.XM.692.4

The stakes for getting the order of the quadrivium right in the Middle Ages may not have risen to the level of murder (although that might make a nice monastic murder mystery written by Umberto Eco, Murder Most Irrational….). And yet, three sources for the quadrivial tradition in the Middle Ages did present the idea that the order of the mathematical arts reflects the most fundamental nature of the universe itself. Furthermore, this fundamental order of the universe has implications for the order of education in the mathematical arts. These stakes, the metaphysical order of the universe and of education, would still have been considered pretty high for most thinkers throughout the Middle Ages.

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.

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

Further Reading:

Albertson, David. Mathematical Theologies: Nicholas of Cusa and the Legacy of Thierry of Chartres. Oxford University Press, 2014. https://doi.org/10.1093/acprof:oso/9780199989737.001.0001.

Boethius. Boethian Number Theory: A Translation of the “De Institutione Arithmetica” with Introduction and Notes. Translated by Michael Masi. Studies in Classical Antiquity; v. 6. Amsterdam: Rodopi, 1983.

Boethius. The Consolation of Philosophy. Translated by Victor Watts. London: Penguin, 1999.

Burton, David M. The History of Mathematics: An Introduction. Dubuque, Iowa: Wm. C. Brown Publishers, 1988.

Caiazzo, Irene. “Teaching the Quadrivium in the Twelfth-Century Schools.” In A Companion to Twelfth-Century Schools, edited by Cédric Giraud, translated by Ignacio Duran, 88:180–202. Brill’s Companions to the Christian Tradition. Brill, 2019. https://doi.org/10.1163/9789004410138_010.

Calcidius. On Plato’s Timaeus. Dumbarton Oaks Medieval Library 41. Cambridge, Massachusetts; London, England: Harvard University Press, 2016.

Chenu, M. D. Nature, Man, and Society in the Twelfth Century: Essays on New Theological Perspectives in the Latin West. Chicago and London: University of Chicago Press, 1957.

Eco, Umberto. The Name of The Rose. Reprint edition. Boston: HarperVia, 2014.

Evans, Gillian R. “The Influence of Quadrivium Studies in the Eleventh- and Twelfth-Century Schools.” Journal of Medieval History 1, no. 2 (July 1975): 151–64.

Fassler, Margot E. The Virgin of Chartres: Making History through Liturgy and the Arts. Yale University Press, 2010.

Fournier, Michael. “Boethius and the Consolation of the Quadrivium.” Medievalia et Humanistica, no. 34 (2008): 1–21.

Gersh, Stephen. Middle Platonism and Neoplatonism: The Latin Tradition. 2 vols. Notre Dame: University of Notre Dame Press, 1986.

Martianus Capella. Martianus Capella and the Seven Liberal Arts. Translated by William Harris Stahl, Richard Johnson, and E.L. Burge. Vol. II: The Marriage of Philology and Mercury. 2 vols. Records of Western Civilization 84. Columbia University Press, 1992.

Masi, Michael. “Boethius and the Iconography of the Liberal Arts.” Latomus 33, no. 1 (January 1, 1974): 57–75.

Nicholas of Cusa. Nicholas of Cusa on Learned Ignorance: A Translation and an Appraisal of De Docta Ignorantia. Edited by Jasper Hopkins. Minneapolis: The Arthur Banning Press, 1985.

Oosterhoff, Richard. Making Mathematical Culture: University and Print in the Circle of Lefèvre d’Étaples. Oxford-Warburg Studies. Oxford: University Press, 2018. https://doi.org/10.1093/oso/9780198823520.001.0001.

Reydam-Schils, Gretchen. “Meta-Discourse: Plato’s Timaeus According to Calcidius.” Phronesis 52 (2007): 301–27.

Somfai, Anna. “Calcidius’ Commentary on Plato’s Timaeus and Its Place in the Commentary Tradition: The Concept of Analogia in Text and Diagrams.” Bulletin of the Institute of Classical Studies 47, no. Supplement_83_Part_1 (January 1, 2004): 203–20. https://doi.org/10.1111/j.2041-5370.2004.tb02303.x.

Somfai, Anna. “The Eleventh-Century Shift in the Reception of Plato’s Timaeus and Calcidius’ Commentary.” Journal of the Warburg and Courtauld Institutes 65 (2002): 1–21.

Stahl, William H. “The Quadrivium of Martianus Capella: Its Place in the Intellectual History of Western Europe.” In Arts libéraux et philosophie au moyen âge, 959–67. Actes du IVe Congrès internationl de philosophie médiévale. Montreal Paris, 1969.

Stahl, William Harris, Richard Johnson, and E.L. Burge. Martianus Capella and the Seven Liberal Arts. Vol. I: The Quadrivium of Martianus Capella. 2 vols. Records of Civilization, Sources and Studies 84. New York: Columbia University Press, 1971.

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Weigh Your Books! An Interview with Dr. Andrew Irving

This week, we’re revisiting the first published episode of “Meeting in the Middle Ages.” Back in 2022, we sat down with Dr. Andrew Irving, assistant professor of religion and heritage at the University of Groningen. We spoke to him about his journey to Medieval Studies, his work on the 11th century Uta Codex, why one should always weigh their books, and why liturgy is like a Wagnerian opera.

Dr. Irving’s story is one of a truly international scholar. A native of New Zealand who moved to the US to study for his PhD at Notre Dame, he now works in Europe on a broad range of medieval subjects. His stories of archival work highlight some of the unexpected challenges that researchers can face: limited access to resources, unconducive weather (it helps to examine books in “raking light”), or flat out denied permission to consult a manuscript. Traveling to another country to visit a library and examine its rarest materials can be intimidating, especially for young scholars. But Dr. Irving demonstrates that a personal connection and diligent preparation can pave the way for a smooth experience. It’s an instructive tale for young scholars, and sheds light on a part of scholarly work that may seem mysterious to the uninitiated.

Dr. Irving’s work is about place. His career has taken him all over the world, of course. But the place in which texts exist is also paramount. Through his work on texts like the Uta Codex, he provides some great examples of how a manuscript has to be considered in terms of its home. Where was it kept? What was the environment? What was the history of that home? Was it ever destroyed, raided, burned? If it was a written document, was it read aloud? To whom? Was it carried about? How much did it weigh? All of these questions must be asked to get at the truth of an object. A text has to be wrestled with on its own terms—in isolation—but this is only half the story. Each historical artifact is living history: it was created by someone for someone or something. We have to be prepared to engage with it in a multitude of ways. We must be historians, linguists, theologians, art historians, literary critics, and more. That is what it is to be a medievalist.

Thanks for listening. See you next time in the Middle Ages.

Will Beattie & Ben Pykare

MitMA-Irving-Interview-Transcript-2022Download
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Gendering the Harpy: Mythology, Medievalism, and Macabre Femininity

I have a fascination with the strange and obscure, and if I find oddities and curiosities during my travels that intersect with my medieval interests, even better. On a recent trip to Italy, I encountered a creature from both Greek mythology and medieval bestiaries at one of the most wonderfully macabre sites I’ve explored.

While on vacation in Rome this summer, I visited the Capuchin Crypt, an underground mausoleum containing an elaborate arrangement of human bones – lots and lots of bones. No one knows who designed the beautiful and haunting configurations comprised from the bones of approximately 3,700 bodies, presumably those belonging to Capuchin monks who sought refuge from religious persecution in France and perished while in Rome.

Unfortunately, photos are not allowed, and efforts to describe the intricacies and expanse of the design prove rather futile. Skulls and pelvic bones combine to create sculptures reminiscent of butterflies in the arches of doorways. Vertebrae dot and line the ceilings of the chambers like so many fresco tiles. Massive piles of assorted bones have been shaped into seats for carefully posed skeletons. Reviewing his experience, the Marquis de Sade rated the exhibit five stars by modern standards.

Inside one of the chambers of the Capuchin Crypt in Rome, Italy, courtesy of the Liturgical Arts Journal.

But the crypt is a 17th-century construction. It’s the museum that contains the medieval bits, and that’s where I noticed an early print book, dated to the 15th or 16th century, that clearly depicted a cockatrice and that the museum had identified as a harpy.[1] To be fair, the label included a question mark, indicating that the curator was unsure as to what kind of creature was on display.

Far less familiar than the harpy, the cockatrice is a legendary creature with a dragon’s body and a rooster’s head. The beast was believed to be hatched from a rooster’s egg incubated by either a serpent or a toad. Its first recorded mention in English appears in a Wycliffite bible dated 1382.[2]

Labeled as a koketrice in this medieval bestiary from England circa 1500, the creature combines a rooster’s head and feet with a dragon’s wings and tail. (Yale Center for British Art, Helmingham Herbal and Bestiary, folio 18v).

The cockatrice seems to have become synonymous with the basilisk in medieval bestiaries. [3] Most often, basilisks are depicted as a bird, typically a rooster, with a snake’s take. In some illustrations, the basilisk is all snake in terms of physical characteristics, though often with a crest reminiscent of a rooster’s head. The mythologies of the cockatrice and basilisk also share similar elements. As with the basilisk, it is fatal for a person to look the cockatrice in the eyes. Both creatures’ breath can also cause death according to folklore.

This medieval bestiary dated 1225-50 and produced in England portrays the basilisk as the king of serpents with lesser snakes paying homage. The creature exhibits mostly serpent features but retains the wings, legs, and crown of a cock. (Bodleian Library, MS Bodley 764, folio 93v).

A harpy, in contrast to the cockatrice, has a bird’s body with a human head and no serpent components. When I mentioned the mislabeling to the front desk staff, I was told that a historian had recently visited the museum and indicated the reverse but without additional explanation. I assured them that the rooster-headed serpent was—hands down—a cockatrice. Harpies have bird bodies, human heads, and zero snake parts. As imperatively, harpies are depicted as female.

Illustration of a harpy from Ulisse Aldrovandi’s Monstrorum Historia, Bologna, 1642, via World History Encyclopedia.

According to Greek mythology, harpyiai were winged female spirits thought to be embodied in sharp gusts of wind, and while certainly fearsome, they were not always so bestial. Known as the “hounds of Zeus,” the female entities were sent from Olympus to snatch people or objects from the earth. Sudden disappearances were, as a result, often attributed to the harpies.

In their earliest representations, harpies appeared as winged women, sometimes with the lower bodies of birds. They were vengeful creatures but not hideous in appearance. Writing between 750 and 650 BC, Hesiod describes harpies as winged maidens with beautiful hair, whom he praises for swiftness in flight that exceeds the speed of storms and birds. Homer, writing roughly around the same time, mentions a female harpy but says nothing derogatory about her looks.

By the end of the classical period, harpies had become monstrous portraits of femininity. They were birds with the heads of maidens, their faces visibly hungry, and had long claws extending from their hands. In the writings of Aeschylus around 500 BC, they are described as disgusting creatures with weeping eyes and foul breath. Virgil, in his Aeneid dated 30-19 BC, refers to them as bird-bodied and female-faced with talons for hands, whose faces reflect insatiable hunger and whose droppings are notably vile. These grotesque portrayals of the harpy—half woman, half monster—are the most well-known from classical mythology.

Harpies depicted as winged women take food from the table of the blind king Phineus on an Athenian vase from 480 BC housed at the J. Paul Getty Museum.

Interestingly, one mythographer did stick a rooster’s head on the otherwise female body of a harpy. Writing in Rome during the 1st century AD, Hyginus describes harpies as having feathered bodies, wings, and cocks’ heads and the arms, bellies, breasts, and genitals of a human woman.[4] Still, there are no serpent parts here to suggest that a medieval image of a cockatrice might instead be a harpy based on Hyginus’s design.

During the Middle Ages, harpies may not have been so distinctly gendered, at least in their encyclopedic cataloguing. Most representations in medieval bestiaries depict the creatures with bird bodies and female faces, but several manuscript illustrations appear androgynous and some even portray the harpy with a beard. The beard, however, may not be indicative of a male beast but instead emphasize the beastliness of the female creature.

Illumination of a harpy with facial feathers reminiscent of a beard from the medieval encyclopedia Der Naturen Bloeme, or The Flower of Nature, written in Middle Dutch and produced in Flanders circa 1350 (Koninklijke Bibliotheek, KA 16, folio 75r).

Furthermore, Ovid’s retelling of the Jason story in his Metamorphoses specifically mentions the harpies having the faces of virgin women. Written in the 9th century, Ovid’s collection of myths served as a source text for many medieval writers, including Dante Alighieri and Geoffrey Chaucer, and his treatment of the harpies suggests that their association with female monstrosity continued to resonate soundly during the period.

Engraving of the harpies in the Forest of the Suicides in reference to Dante Alighieri’s Inferno by French printmaker Gustave Doré (1832-83).

Turning to the etymology of the term, the first recorded instance of harpy in English actually appears in Chaucer’s Monk’s Tale around 1405.[5] The creatures are not specifically gendered; they are simply mentioned among the monsters defeated by Hercules, at which point the text reads, “He Arpies slow, the crueel bryddes felle” [“He slew the Harpies, the fierce cruel birds”] (2100).[6] Yet one cannot help but see the feminine slippage in the spelling of “bryd,” meaning both “bird” and “bride” in Middle English.[7] Indeed, the term harpy adopts a derogatory connotation in writing by the mid- to late 15th century.[8] The term cockatrice, too, took on a negative meaning specifically with respect to women by the mid-16th century, at which point it referred to a prostitute or a sexually promiscuous woman.[9]

Illumination of a harpy with a female face from the medieval encyclopedia Liber de natura rerum, or Book on the Nature of Things, written in Latin and produced in France during the 13th century (Bibliothéque Municipale de Valenciennes, MS 320, folio 86r).

While it’s possible that the harpy may have maintained some gender ambiguity during the medieval period, contemporary etymology and ideology has synonymized the harpy with femaleness but also, importantly, with power. The sheer number of times Hillary Clinton was called a “harpy” during her presidential campaign highlights how a powerful woman was characterized as not only threatening but also monstrous while pursuing a position historically deemed male domain.[10]

Harpies in medieval fantasy films are also perched at the intersection of femaleness and power, glorious in their might regardless of how monstrous their bodies may be. The Last Unicorn, a 1982 animated adaptation of Peter S. Beagle’s 1968 novel, provides a poignant example. Captured by a traveling circus, the titular character finds herself caged across from a harpy, the only authentic creature of legend in the menagerie apart from the unicorn herself.

In a magnificently ominous scene, the audience hears the harpy before they see her. A low growl grows to a raspy screech as the harpy appears on screen. She appears more bird than human, but her grotesque body is blatantly female with three elongated breasts visible beneath her beard and boar’s tusks. A knotted tree limb cracks from the strength of her talons, and her eyes glow red with rage when her captor approaches her cage. Once freed, she kills the old woman who boasted of keeping a harpy captive when no one else could.

In The Last Unicorn, the titular character recognizes the harpy as Celaeno, the same name given to one of the harpy sisters in the Greek story of Aeneas. The unicorn is freed from her cage under the cover of night, and she then proceeds to free her fellow immortal.

Considering the harpy’s history, it seems a shame to mistake her for any other creature from Greek mythology or medieval bestiaries. She has been such a fraught representation of both femininity and monstrosity, but she has also endured as a symbol of female ferocity. Even as her beauty eroded over the centuries, her power has not waned, and her macabre femininity has never ceased to inspire fear.

Emily McLemore
Ph.D. in English
University of Notre Dame

[1] Photos are prohibited in the museum, so I have no physical record of the image. I attempted to contact the Capuchin Museum regarding the object on display to acquire additional information, including the date and location of production, but received no response.

[2] “Cockatrice,” n. Oxford English Dictionary.

[3] “Basilisk,” The Medieval Bestiary.

[4] Fabulae from The Myths of Hyginus, translated and edited by Mary Grant.

[5] “Harpy,” n., def. 1, Oxford English Dictionary.

[6] Geoffrey Chaucer, The Monk’s Tale, The Canterbury Tales, Harvard’s Geoffrey Chaucer Website.

[7] “Brid” and “Brid(e,” n., Middle English Compendium, University of Michigan.

[8] “Harpy,” n., def. 2, Oxford English Dictionary.

[9] “Cockatrice,” n. def. 3, Oxford English Dictionary.

[10] For more on Greek mythology, female monstrosity, and contemporary resonance, I recommend Jess Zimmern’s Women and Other Monsters: Building a New Mythology (Beacon Press 2022).