




















Chapter Two:

Teaching the Liberal Arts in the Middle Ages: 

The Cathedral School at Reims in the Late Tenth Century
Abstract: In the late tenth century, Gerbert of Aurillac served as the lead teacher at the cathedral school in Reims.  The curriculum he developed and implemented was based on a combination of classical literature, encyclopedias written in late antiquity, and Greco-Roman teaching tools.  This curriculum preserved and transmitted classical learning to the schools of the High Middle Ages and contributed to a wider educational revival in northern France and Germany.

In the early Middle Ages, monastic and cathedral schools transmitted the gradeur of Rome and the glory of Greece to scholars, churchmen, and secular leaders. Even in the darkest century of the supposed Dark Ages, the tenth century, teachers preserved the knowledge of antiquity and transmitted it to generations of students. The cathedral school at Reims was one such center of learning; in the late tenth century, this school was one of the most famous schools in Western Europe. Students from across Western Europe flocked to the school at Reims to study the liberal arts under Gerbert of Aurillac, who taught at the school from 972-989. His curriculum, considered innovative by both Gerbert’s contemporaries and modern scholars, was based on classical and late antique literature and teaching tools, and was elaborated on in technical manuals and letters written by the schoolmaster himself.  A reconstruction of the curriculum, and the textual sources that informed it, demonstrates that the cathedral school at Reims preserved and transmitted classical literature from antiquity to the Middle Ages. 
The Curriculum of the Trivium
The curriculum of the trivium (the literary arts of grammar, rhetoric, and dialectic) was based on the mastery of classical and late antique texts. The study of these arts overlapped, as there was no rigid distinction between them; grammatical instruction reinforced the teaching of rhetoric and dialectic, and a student’s mastery of rhetoric and dialectic perfected his knowledge of grammar. At Reims, as at many other cathedral schools, this curriculum was divided between the grammarian (grammaticus) and the lead instructor (magister). Richer of Saint-Remi, a former student of Reims and one of our main sources for Gerbert’s instruction in the trivium, tells his readers that Gerbert studied logic (dialectics) under Gerannus, the grammarian at Reims, before Gerbert took over instruction that art.
In Reims, the grammaticus was Gerannus. Little is known of Gerannus, and Richer’s brevity implies that Gerannus’ instruction was both sufficient and ordinary. The standard education in grammar relied heavily on two major texts:  the Ars Minor of Donatus and Priscian’s Institutiones Grammatica. Donatus’ work, completed in the mid-fourth century and the main grammatical textbook throughout the Middle Ages, divided the components of speech into eight parts:  noun, pronoun, verb, adverb, participle, conjunction, preposition, and interjection. Each of the eight parts of speech was defined and then subdivided. Nouns, for example, were defined as “a part of speech which signifies with the case a person or a thing specifically or generally,” and were divided and explained based on gender (masculine, feminine, neuter, common) and case (nominative, genitive, dative, accusative, ablative, and vocative). Donatus defined verbs as “a part of speech with tense and person, without case, signifying ‘to perform some action’ or ‘to experience’ or neither,” and divided verbs between modes (indicative, imperative, optative, subjunctive, infinitive, impersonal), conjugations (first, second, third), and kinds (active, passive, neuter, deponent, common)(Chase 1926, 28 and 38). These examples of Donatus’ subdivision of nouns and verbs demonstrates the utility of his text in teaching the basic rules of Latin grammar, and likewise explains the medieval grammarians’ tendency to use it in teaching.


The second text probably used by Gerannus while teaching grammar was Priscian’s Institutiones Grammaticae, written in Constantinople in the early sixth century. Priscian, unlike Donatus, was not a native Latin speaker, and his work therefore resonated with students for whom Latin was not a mother tongue. The works of Priscian and Donatus were structurally similar. Both texts consisted of three parts: preliminary sections addressing sounds, letters, and syllables; a treatment of the eight parts of speech; and a concluding section in which Donatus deals with tropes in his text and Priscian addresses syntax. The overarching similarities, however, end there as Priscian’s work was not the brief and handy textbook summary that Donatus had created. It was, actually, a rather cumbersome work. It is most likely that Gerannus would have used Donatus as the main textbook in the teaching of grammar, and supplemented it with examples and expositions by Priscian and other authors as needed.

Grammatical instruction continued with classical and Late Antique texts to demonstrate the proper structure and use of written language (Riché 1976). Students were instructed to copy texts, to emulate them, and to adopt the language and style of classical authors such as Virgil, Terence, Cicero, and Horace. In addition to pagan literature, students were expected to learn the writings of church fathers such as Saint Augustine, Jerome, and Saint Ambrose, not only for the theological content of the works, but also for the literary style of their construction.  


Once Gerannus completed grammatical instruction, the cathedral school’s scholasticus, Gerbert, continued teaching the trivium with dialectic and rhetoric.  Dialectic was defined as “the science of disputing well” and taught students how to construct logical arguments (Gerbert 1996, 168B). Instruction in this art followed an established textual progression, and Richer provides substantial information on Gerbert’s pedagogy on this subject:
In the first place he explicated Porphyry’s Isagoge (that is “Introduction”) according to the translation of the rhetorician Victorinus and the commentary by Boethius.  Next he explained the book of Aristotle called the Categories (that is “Predications”) and he skillfully demonstrated the difficulty of the Peri hermeneias (“On Interpretation”).  After that he introduced students to the Topics (that is, the “Sources of Arguments”), as translated from Greek into Latin by Cicero and expounded in the six books of commentaries written by the consul Manlius Boethius (Richer 2011, 69-70).
The teaching of dialectics proceeded from the general to the specific, with Porphyry’s work introducing students to the topic, and then proceeding through several technical manuals and explanatory texts associated with those manuals.

Richer also mentions several untitled works used in teaching dialectic: “he [Gerbert] lectured on and effectively expounded four books on differential topics, two books on categorical syllogisms, three on hypothetical syllogisms, and one book each on definition and division” (Richer 2011, 71). These were likely Boethius’ logical works, which fall into three general categories: Latin translations of Greek works, commentaries, and original treatises. One was probably Boethius’ De Topicis Differentiis, a treatise on finding arguments based on Aristotle and Cicero.  The work on categorical syllogisms was likely On Categorical Syllogisms, the work on hypothetics may be On Hypothetical Syllogisms, and the work on division may be On Division.  The study of such dialectical textbooks established a theoretical framework for students’ analysis and construction of logical arguments, which was also the foundation of rhetoric. 

Gerbert’s dialectical treatise, though written after his teaching career at Reims ended, demonstrates his understanding of dialectics and reflects his pedagogy. The De Rationali et Ratione Uti was written for Emperor Otto III in 997 and was intended as an explanation of Boethius’ exposition of specific differences as discussed in Porphyry’s Introduction. De Rationali et Ratione Uti assumes an audience equipped to deal with terms used in a specifically dialectical context and well versed in the study of dialectic. The dedicatory letter of this tract mentions “many noble and learned scholars who used to be with us” but that “we saw none of them who explained any of these questions properly” (Gerbert 1961, 298). Thus, the tract was written specifically to counter a shortcoming in contemporary teaching, and assumed an audience who was both familiar with the technical language prevalent in dialectical debates, and who had a specific knowledge base derived from an education in the arts of the trivium.

De Rationali et Ratione Uti is the only place where Gerbert specifically defines dialectic:  “Dialectics is the science of disputing well” (Gerbert 1996, 168B). Gerbert’s use of dialectical terminology is based on Porphyry’s definitions of those terms, and his exposition on the text assumes his student’s diligent study of Porphyry. The question:  “how is ‘to reason’ predicated of ‘rational’ when ‘rational seems greater than ‘to reason’” is followed by the answer: 

Since what descends in a straight line from the most general to the specific, either through sub alternate genera or through differentiae, whether they are in substances, or established in accidents, they are such [that] the universally extended names and definitions of all superiors are received as sensible by the inferior: when it is a differentia placed under a living body, it takes the universally extended name and definition of the superiors; because each sensible and living body is a substance, and each attribute is a relation and a quality (Gerbert 1996, 160).

Gerbert continues through an extensive explanation, in which he both provides quotes from notable philosophers and dialecticians and creates his own examples employing characters such as Plato, Cicero, and Socrates.


If dialectic was the science of disputing well, then rhetoric was the art of presenting those disputations. Dialectic and rhetoric followed the same underlying principles, a notion clearly recognized by Late Antique authors like Cassiodorus Senator. Though some early medieval scholars, such as Hrabanus Maurus, associated rhetoric with preaching, Gerbert of Aurillac viewed rhetoric as having wider applications. One of Gerbert’s letters summarizes his views on the art of rhetoric: 

Since philosophy does not separate ways of conduct and ways of speaking, I have always added the fondness for speaking well to the fondness for living well, although it may be more excellent to live well than to speak well, and if one be freed from the cares of governing, the former is enough without the latter.  But to us, busied in affairs of state, both are necessary.  For speaking effectively to persuade and restraining the minds of angry persons from violence by smooth speech are both of the greatest usefulness (Gerbert 1961, 90).
Gerbert’s letter reveals his belief that artful and persuasive speaking was a necessary skill for secular and ecclesiastical administrators. The centrality of rhetoric in Gerbert’s educational program and his belief that eloquent speaking and writing were necessary skills for statesmen indicates that the primary goal of Gerbert’s instruction was to prepare his students for state service.
Proper instruction in rhetoric required a revisiting of the literature learned in the study of grammar to refresh students’ understanding of style and persuasion based on classical authors:
he [Gerbert] was concerned that they would not be able to learn the art of oratory without prior knowledge of the modes of expression, which are learned in the works of the poets.  For this reason he introduced them to the poets with whom he thought they ought to be made acquainted, expounding and teaching the poets Virgil, Statius, and Terence, the satirists Juvenal, Persius, and Horace, and the historian Lucan (Richer 2011, 71). 
Such authors used wit, allusion, irony and puns to express ideas eloquently, and students learned these literary models to emulate classical style and content.  
Upon completion of this refresher, Gerbert led students on to their study of oratory, which was based on Ciceronian rhetoric. Gerbert calls Cicero “the father of Roman eloquence”, and writes “both in business and in leisure I am the diligent follower of the precepts of Marcus Tullius [Cicero]” (Gerbert 1961, 127 and 196). Specifically, Ciceronian texts such as De Oratore and De Inventione figured prominently in the curriculum for rhetoric at Reims. For Cicero, and therefore for Gerbert, a proper rhetorical education was all encompassing, including a thorough general education, a cultural education in the mannerisms befitting a public speaker, and instruction in carriage and personal presentation. Quintilian, a Late Antique orator, echoed Cicero’s inclusion of history and poetry as a subject necessary for the cultivation of rhetorical abilities (Quintilian 2001).
Unfortunately, Richer does not specifically address Gerbert’s methods for teaching rhetoric.  Richer’s statement, that “they engaged in practice disputations, conducting themselves according to the princples of this art in such a way that they seemed to act without any art at all, which is the greatest achievement for an orator” (Richer 2011, 73) indicates that ability to argue was of the utmost importance to Gerbert. As a former student of Gerbert, Richer learned to emulate literary styles and possessed desire to persuade without betraying the rhetorical or literary sources that inform that argument. Like many medieval authors, Richer and Gerbert considered imitation the highest form of flattery, and modeled much of their writings on classical literature. The emulation of literary authorities is encouraged by one of Gerbert’s rhetorical sources, Quintilian, who praises imitation in his Institutio Oratoriae (Quintilian 2001). 

It is possible, however, to reconstruct components of Gerbert’s model for a rhetorical education based on a combination of his letters, the contents of his personal library, and his general regard for classical rhetoric. Gerbert’s personal library included several texts that would have been useful in teaching rhetoric. The basic textbook used in teaching rhetoric was Capella’s De nuptiis Philologiae et Mercurii (Gerbert’s copy is Bamberg MS Class.39).  Justin Lake has demonstrated the influence of Capella’s terminology on the writings of Gerbert’s student, Richer of Saint-Remi, indicating that the De nuptiis figured prominently in the teaching of rhetoric at Reims during Gerbert’s tenure (Lake 2008).
Gerbert valued the study of rhetoric so highly that he deliberately and purposefully sought out texts to enhance the study of rhetoric at Reims: “For speaking effectively to persuade and restraining the minds of angry persons from violence by smooth speech are both of the greatest usefulness. For this activity, which must be prepared beforehand, I am diligently forming a library” (Gerbert 1961, 90). This library likely contained a variety of classical and late antique works, including works of Roman comedy, poetry, rhetorical manuals, speeches and political documents, letter collections, sermons, theological tracts, the works of Church Fathers, Roman satires, etc. The textual references in Gerbert’s letters provide some insight into its’ probable contents, most of which were Ciceornian texts. Gerbert’s letters contain references to In Catalinam, Tusculanae Disputationes, De inventione, Topica, De senectute, De officiis, De amicitia and De oratore. Gerbert also requested manuscripts of Cicero’s work; Gerbert wrote letters to obtain the end of Pro rege deiotaro, Orationes, De re publica, In verrum, and the Rhetoricam ad Herrenium, which Gerbert (like many other Medieval thinkers) attributed to Cicero but modern scholars now believe was written by another unnamed author, pseudo-Cicero.  In addition to these Ciceronian texts, Gerbert’s letters reference to many other classical authors:  Virgil’s The Aeneid and Georgicon; Horace’s Odes, Ars Poetica, and Epistulae; Sallust’s Oratio in Tullium and Oratio in Catalinam; Terence’s Andria, Hecyra, and Heauton Timorumenos.  Christian authors also appear as sources in Gerbert’s letters: Augustine’s De Baptismo and Epistolae; Jerome’s Sermones; Ambrose’s De Fide; Hilary’s De Synodis; Gregory the Great’s Registrum and Homilia in Evangelia; and Dionysus Exiguus’ Codex Canonum Ecclesiasticorum are all cited or referenced in Gerbert’s surviving letters (Gerbert 1961). This wide variety of referenced texts implies that these manuscripts were available, or at least as excerpts or selections, to Gerbert and his students at Reims and were probably used in the curriculum on rhetoric and grammar.  
The two most important texts for studying rhetoric were Cicero’s De Oratore and De Inventione, both of which Gerbert knew and probably used as teaching texts. The De Oratore was fundamentally different from Cicero’s previous work on the same topic, the De Inventione, which Cicero completed around 88 BC and he later dismissed as immature. Most of the De Inventione’s influence on Gerbert’s teaching is likely related to the components of rhetoric, which Cicero divided in five parts:  invention (the discovery of arguments), disposition (the arrangement of speech into parts), style, memory, and delivery. Additionally, the disposition, or arrangement of speeches into parts, is further divided into seven parts, the exordium, narration, partition, confirmation, refutation, digression, and peroration (Cicero 1976). Gerbert’s ideas on rhetoric were most deeply influenced by the De Oratore, a text he possessed and is currently held in the Erlangen university library as MS Erlangen 380. Cicero’s conception of the ideal orator, as presented in the De Oratore, was therefore instrumental in informing Gerbert’s own position on the study of rhetoric and the development of rhetorical skill among his students. Cicero’s idea is summarized in the prologue of his work, in which he argued that skilled public speakers must blend various methods: they must appeal to the audience’s emotions; employ wit and humor appropriately and sparingly; demonstrate the culture and refinement of a gentleman; and draw upon various sources of information including history and law (Cicero 2001). Thus, for Cicero, and therefore for Gerbert, a proper rhetorical education was all encompassing, including a thorough general education, a cultural education in the mannerisms befitting a public speaker, and instruction in carriage and personal presentation. As such, the variety of classical and late antique references in Gerbert’s letter collection is hardly surprising, and mirrored the belief that the study of rhetoric required any student to be widely read and familiar with various authors and literary styles.
In addition to the late antique and classical literature likely read by the students at Reims, Gerbert condensed the most salient components of his ideal rhetorical education into a student handbook. While this handbook does not survive, Gerbert described it in one of his letters: 

I offer to noble scholars the pleasing fruits of the liberal disciplines to feed upon… [and] out of love for them I constructed a certain table of the rhetorical art, arranged on twenty-six sheets of parchment fastened together in the shape of an oblong, made of two sheets by thirteen sheets – a work truly wonderful for the ignorant, and useful for the studious for comprehending the fleeting and very obscure materials of the rhetoricians and for fixing them in the mind (Gerbert 1961, 140).  

Overall, the curriculum employed in the arts of the trivium at Reims demonstrates a thorough working knowledge of many classical and late antique texts on the part of the grammaticus and magister. The educational process transmitted these texts to each successive body of students, both in terms of the physical text (teaching the text required at least one copy present in the cathedral library), and in terms of its actual content and literary style. The study of the trivium at Reims was based on a corpus of classical and late antique literature, interpreted and expounded on, both orally and in prose, by the magister Gerbert. That body of literature was supplemented by additional texts penned by Gerbert to further elucidate the arts of the trivium. These additional materials supplemented and elaborated on the information provided in the classical texts, providing additional explanation on the classical and late antique texts that were the cornerstone of the curriculum in the trivium.
The Curriculum of the Quadrivium

Upon completion of Gerbert’s instruction in the trivium, he moved on to the subjects of the quadrivium. Richer’s account indicates that Gerbert’s progression of subjects in the trivium began with arithmetic, then moved to music, astronomy, and finishing with geometry (Richer 2011).  Richer’s historia devotes a a bit more space to Gerbert’s teaching on the quadrivium than he does to Gerbert’s teaching on the trivium, and especially focuses on Gerbert’s use of practical teaching tools. These tools, which Richer viewed as new and innovative, were actually based on classical and late antique tools that were reintroduced to European scholars in the tenth century.  

Gerbert’s progression of teaching mathematics began with arithmetic, and then proceeded to music, astronomy, and finally geometry. Arithmetic was the foundation for mathematical study as it enabled students to perform computations and learn number theory. The study of music, astronomy, and geometry all built upon principles learned during one’s study of arithmetic. Numerical ratios, for example, were introduced during the study of arithmetic but reappeared in music and geometry.  Additionally, tools used in mathematical instruction were applied to Gerbert’s teaching on multiple subjects. The abacus was useful for both basic arithmetical computation and the study of geometry.  


Richer describes Gerbert’s teaching of arithmetic only briefly, stating “I do not think it would be out of place to mention how much effort he expended on the mathematical arts.” (Richer 2011, 73). We know that Gerbert used the abacus in teaching numerical computation, though Richer describes the abacus with his discussion of Gerbert’s teaching of geometry. Richer described the abacus Gerbert used in great detail:

It was divided into twenty-seven columns, and nine symbols could be arranged upon it to signify any number.  He had a thousand markers with these symbols on them made out of horn.  These could be then taken in and out of the twenty seven columns of the abacus to represent the multiplication or division of any number… (Richer 2011, 83).
Richer goes to great lengths to describe Gerbert’s abacus, implying that the use of such a tool was somehow unique, but scholars recognize that the abacus was also known to Gerbert’s contemporaries Abbo of Fleury, Walter of Speyer, and Notker of Liège (Lattin 1933). Gerbert’s knowledge and use of the abacus included textual explanations of it, especially through his letters to former students. Gerbert’s explanation of abacus arithmetic was in response to specific questions, as his letter delves immediately into the response to Remi’s query. Remi’s original question does not survive, but Gerbert’s response indicates that the question centered on how to properly perform multiplication or addition with the abacus. Gerbert wrote: 

You adequately comprehended, indeed, how the tenth number measures itself.  For one times one is one.  But every number does not therefore measure itself because it is equal to itself, as you have written.  On the contrary, although one times four is four, four does not therefore measure four, but rather two.  For twice two are four.  

Moreover, the letter i which you have found written down under the ten times figure signifies ten unities, which separated into six and four, make a sesquialter ratio.  The same may be observed in the ratio three to two where unity is a difference (Gerbert 1961, 172).

In addition to such explanatory letters, Gerbert wrote a brief treatise on the abacus which does not survive. Richer mentioned this work:  “Whoever wishes to acquire a more complete knowledge of this body of knowledge should read the book that Gerbert wrote for Constantine the Grammarian” (Richer 2011, 83). The construction of such a handbook is consistent with his pedagogy on the trivium, where he used his own manual on rhetoric to assist his students learning. His handbook on the abacus, likely written sometime around 980, was a concise treatment of the terminology and rules for using it, written from his memory of reading a similar work, probably during his studies in Spain between 967 and 970. Gerbert himself referred to this work in a letter to Constantine of Fleury, which he dedicated to his former student (Gerbert 1961). The brevity of this work dissatisfied Gerbert’s contemporaries and led other scholars to write commentaries on this text around the year 1000 (Lattin 1933).  
For Gerbert, the most important text relating to arithmetic was Boethius’ De Arithmetica, and he owned two copies. Boethius’ work was a general textbook on the subject, progressing from the definition of a number, to the differences between even and odd digits, and finally to an exposition of the relationships between numbers (Boethius 1983). Boethius divided these relationships into five main types:  multiplex ratios, superparticular ratios, superpartient ratios, multiplex superparticular ratios, and multiplex superpartient ratios. Each of these categories is subdivided:  multiplex ratios consist of duplex (1:2, 2:4, 3:6, 4:8), triplex (1:3, 3:9, 4:12), and quadruple (1:4, 2:8, 3:12). Superparticular ratios are those in which one number is divisible by the other, with the remainder being a fraction of the smaller number. These are further divided into three types:  sesquialter superparticulars, where the remainder is ½ of the smaller number (2:3, 6:9); sesquitertian superparticulars, where the remainder is one-third of the smaller number (3:4, 6:8); and sesquiquartan superparticulars, where the remainder is ¼ of the smaller number (4:5, 12:15). In superpartient ratios, the remainder of the two numbers cannot divide evenly into either number (4:7, 5:9). Multiplex superparticular numbers are combinations of multiplex and superparticular ratios (3:8 is 3:6 plus 3:2); multiplex superpartient ratios combine multiplex and superpartient ratios (5:13 is 5:10 plus 5:3)(Boethius 1983). Gerbert’s combination of abacus arithmetic and Boethian number theory is quite consistent with his pedagogy in the arts of the trivium. In both cases, Gerbert fuses classical or late antique texts with practical exercises grounded in those texts.  
The exposition of mathematical ratios dovetailed with instruction in music, the second art of the quadrivium. Gerbert’s instruction in music was two-fold, addressing both the practicality of musical performance and the underlying numerical theory informing it. The theoretical instruction was based, again, on the works of Boethius.  Gerbert references Boethius’ De Musica on multiple occasions and quotes the text nearly verbatim in his letters, with clear references to the text itself:
The theory of superparticular numbers, briefly discussed in the second book of De Musica, is expounded more fully and by examples in the second chapter of the fourth book.  We shall copy these words here and proceed forthwith to discuss them: “If a superparticular ratio is multiplied by two, the result is neither a superparticular nor a multiple.  The ratio is said to be multiplied by two when the ratio is doubled so that as the first ratio is, so is the second” (Gerbert 1961, 42-43).

Gerbert quotes Boethius’ text again in the following letter:

In Chapter 21 of the Second Book of De Musica, Boethius states as follows: If from every superparticular, one subtracts a continuous superparticular ratio, that which is left, which is indeed less, is less than half of the ratio which has been subtracted: as in the sequialtera and sesquitertia.  Because the sequialtera is larger, let us subtract the sesquitertia from the sesquialtera.  There remains a sesquioctaval ratio, which, when doubled will not make a whole sesquitertian ratio, but is less by the distance that is found in a semitone.  But if the doubled sesquioctaval relationship is not a whole sesquitertia, a simple sesquioctave is not a whole half of a sesquitertian ratio (Gerbert 1961, 43)

The reiteration of these numerical ratios in the teaching of music reinforced Gerbert’s teaching of arithmetic, thereby reviewing the basic tenets of number theory and providing a solid base of knowledge for other studies in the quadrivium.
Richer’s account of Gerbert’s teaching, on the other hand, focuses on practical and hands-on instruction in musical theory: 

By demonstrating the different musical genera on the monochord, by separating their consonances or harmonies into whole tones and semitones, major thirds and quarter-tones, and by methodically breaking town these tones into their constituent sounds…(Richer 2011, 73-75).

The monochord was a musical tool, generally between three and four feet long, which used one string stretched over a board and marked to measure length with a mobile bridge separating sections of the string so they produced sounds independent of one another when plucked. Like Gerbert’s other teaching tools, the monochord was transmitted to the Middle Ages through classical and late antique texts. Boethius, for example, discusses the monochord at length in his De Musica, but Boethius’ explanation is derived from earlier sources including Pythagoras, Aristoxenus, Archytas, and Ptolemy (Smith 1984). Thus, the monochord, like the abacus was a classical tool adapted for medieval teaching at Reims.
While the monochord was designed to demonstrate the mathematical theory of music, Gerbert’s letters imply that he also taught musical performance. Students at Reims already had a basic knowledge of chant, so they were familiar with harmonies, tones, and the foundations for musical performance. One of Gerbert’s letters alludes to instruction in organ playing: “if anyone of you is moved by an interest in such things, as well as in the learning of music and the playing of organs, I will see to it that what I am unable to finish myself will be completed by Constantine of Fleury ...” (Gerbert 1961, 140). This letter advances Constantine of Fleury, one of Gerbert’s former students, as a capable successor to Gerbert in teaching the organ, implying that Gerbert had trained his students to play the organ.
Astronomy, like arithmetic and music, was both practical and theoretical. As a practical skill, knowledge of astronomy was necessary for churchmen who were responsible for dating Easter and other moveable feast days, and for determining monastic schedules because of the specific timing of daily prayers (Lindberg 2002). Richer does not identify the texts used in Gerbert’s instruction in astronomy, but a partial reconstruction of Gerbert’s sources is possible due to the survival of his personal library and the contents of his letters. Much of Gerbert’s classroom work was based on Pliny the Elder’s Natural History, and he owned a copy of the text. In addition to Pliny, Gerbert also possessed and therefore likely used a number of other texts. One such manuscript is Isidore of Seville’s De Natura Rerum, a text that was heavily drawn from the work of Pliny (Eastwood 2007).  Martianus Capella’s De Nuptiis was also central in Gerbert’s teaching of this art, though there is little evidence for how he used it in teaching. Gerbert’s letters contain only one explicit reference to Capella’s work as it relates to astronomy, discussed below. Martianus Capella claimed that the Earth was the center of the universe, and was circled by the sun, that Venus and Mercury circled the sun, and therefore always appeared close to one another. Martianus Capella’s model of the universe is closely related to Pliny the Elder’s, and Pliny’s Natural History may have served as one of Capella’s sources. 

Macrobius’ Commentary on the Dream of Scipio also figured into Gerbert’s teaching of astronomy. This text was based on Cicero’s narrative, Somnium Scipionis. In this narrative, the soul marvels at the cosmos and asks about the wondrous music it hears, only to discover that the sounds are the result of the movement of the heavenly spheres spaced at musical ratios. Gerbert certainly knew this text and interpretation of the universe because he requested a copy of Cicero’s De re publica and both Cicero’s dream and Macrobius’ work on it were Book VI of the De re publica. For Macrobius, the Earth was at the center of a spherical universe. This universe consisted of seven planetary spheres encircled by a sphere of stars which rotated from east to west. The motion of the planetary sphere caused the planets to move in a similar fashion, though they would also move west to east. Macrobius did not designate the order of the planets in this model of the universe because Cicero and Plato placed the sun in different locations. In addition to Macrobius’ Commentary on the Dream of Scipio, another text was probably used in teaching astronomy. This work was complimented by Boethius’ De Astronomia. Collectively, these works comprised the textual foundation for Gerbert’s educational program in astronomy.
Although Gerbert’s letters and his personal library provide the textual foundation for his teaching on astronomy, his former student Richer emphasized Gerbert’s use of practical instruments in studying astronomy. Richer’s account includes great detail on several of Gerbert’s teaching tools, including armillary spheres, moveable globes, observation tools, and horologia: “[Gerbert] made it [astronomy] comprehensible in marvelous fashion by employing certain instruments” (Richer 2011, 75).  Richer chooses to emphasize the practical tools Gerbert used in teaching rather than texts used to inform Gerbert’s ideas on astronomy. This decision indicates the novelty of these devices and demonstrates the interactive approach Gerbert took towards his teaching of astronomy. These practical devices include a variety of spheres used to demonstrate the orientation of the earth and the relation of the stars to it. Gerbert used a basic armillary sphere as a template for his teaching tools, and then altered similar spheres for his specific purposes. He used one sphere to teach the movement of constellations, one to demonstrate the division of the Earth, one to demonstrate the movement of the planets, and one identify the constellations and the stars which composed them  (Richer 2011).  These tools, used in the Roman and Late Antique worlds, was reintroduced to Western Europe by Gerbert. Richer described one of Gerbert’s armillary sphere thusly:

He fashioned a celestial globe out of a solid sphere of wood, using the smaller as a model of the larger.  He then tilted it in alignment with the two poles, adding the northern constellations to the upper half and the southern constellations to the lower one. He regulated their position by means of the circle that the Greeks call the horizon and the Latins the limitans or delimitans because it divides and separates the visible constellations from those that cannot be seen. 
 (Richer 2011, 75-79).

These armillary spheres were first created by the Hellenistic astronomer Claudius Ptolemy in his Almagest, written during the second century. Later writers would refer to these armillary spheres, including Martianus Capella, an author whose work Gerbert knew thoroughly.  Capella's discussion of astronomy has Astronomia describing spheres similar to those Gerbert constructed (Capella 1983).   This consistency indicates that Gerbert was reintroducing an object known in both the Roman and Late Antique periods. 
After using an armillary sphere to demonstrate the movement of the planets and the relationship between the planets and the zodiac, Gerbert moved on to a demonstration of the movement of the stars. This celestial phenomenon was again demonstrated through a sphere to track the movements of the heavens, but in combination with a sighting tube so that students could view a model of the heavens (Richer 2011).  This sphere demonstrated the placement and movement of the constellations of the zodiac around the earth, rather than the planets, but allowed the students to view both the constellations themselves, and models of those constellations which Gerbert constructed and attached to this sphere. The sighting tube which Gerbert attached to the sphere allowed students to easily compare the models with the actual constellations.  

Gerbert’s creation of these spheres and their incorporation into his teaching were an innovation into the teaching of astronomy which he shared with his students. In addition to armillary spheres, which were models used in classroom instruction, Gerbert also constructed hemispheres to assist with astronomical observations. In a letter to Constantine of Fleury in 978, Gerbert explains the construction of one of his hemispheres for making astronomical observations. Gerbert’s explanation contains step-by-step instructions for the construction of such a hemisphere and its use, including how to properly identify a reference point (Gerbert 1961).
Gerbert’s acquaintance with classical texts and his construction of models with moveable parts and visual aids follows the same teaching patterns he established in his work on rhetoric, music, and arithmetic. Astronomy was useful, as it allowed its practitioners to construct accurate calendars and calculate the dates of religious festivals. Knowledge of astronomy also allowed scholars and clerics to construct horologia. Some of Gerbert’s work on this was based on Martianus Capella’s De nuptiis; in one letter, Gerbert recounts Capella’s discussion on the increase in hours of sunlight during the course of the year.

So, Martianus in his Astronomy thinks that the increases of the hours [of sunlight] occur as follows: “One must know”, he says, “that the day increases from the shortest day so that in the first month it increases a twelfth of that time which is added in summer.  In the second month, a sixth.  In the third, a fourth, and in the fourth month, another fourth.  In the fifth, a sixth.  In the sixth, a twelfth (Gerbert 1961, 189).

Gerbert's rendition of Martianus Capella's discussion for the increasing length of the day according to the time of year demonstrates more than his knowledge of Capella's astronomy. This enabled Gerbert to apply astronomy to practical uses in his contemporary medieval world.  Calculations for the amount of daylight in a given time of year, determining the proper date of Easter, and constructing horologia were important skills for clerics as they enabled churchmen to perform their duties more efficiently and accurately. This practical component to the study of astronomy rendered Gerbert’s teaching as firmly grounded in the available Late Antique literature. Gerbert’s exact curriculum for the teaching of astronomy remains obscure, as Richer choose to discuss the tools Gerbert used rather than the texts.  

The final component of the curriculum at Reims was geometry, the fourth art of the quadrivium. The curriculum Gerbert used in the teaching of geometry is unclear. His letters contain few, if any, references that tell us what texts he used in teaching geometry. Richer’s discussion of Gerbert’s teaching of geometry is limited to his recitation of Gerbert’s use of the abacus, discussed above, and there is no mention of the texts used. Therefore, it is impossible to complete a reconstruction of Gerbert’s educational program in geometry. Although the textual foundation for Gerbert’s teaching remains unknown, Gerbert did author his own treatise on the subject, the De Geometria. This work was based on a Roman survey of geometry and was likely written in 983, therefore making it available to Gerbert’s students at Reims upon his return to the city. Gerbert’s work was exceedingly logical; he begins with the assertion that all bodies are composed of parts, and proceeds to a discussion of the composite parts (surfaces, lines, points, and angles). This discussion was explained in both written and visual form, and included illustrations, diagrams and expositions on finding the area of triangles.    
Conclusion
The liberal arts curriculum at the cathedral school of Reims fused classical and Late Antique learning into a cohesive program of study that was further elaborated on and explained through the treatises and letters of the cathedral school master. This curriculum was developed and implemented by Gerbert of Aurillac, the scholasticus of the cathedral school from 972-989.  Through the use of classical and late antique texts and teaching tools, considered innovative by his contemporaries and modern scholars, Gerbert contributed to the wider intellectual revival of the late tenth century and helped lay the foundation for subsequent scholastic developments. The liberal arts curriculum developed at the Reims cathedral school created a cohesive educational program that facilitated and encouraged the transmission of both literary works and physical objects from Roman and Late Antique society to the intellectuals of Europe in the High Middle Ages. Gerbert’s students revitalized various school communities throughout northern France, initiating a general upswing in education at the turn of the first millennium. As such, Gerbert’s school at Reims, and the wider network of cathedral schools in the late tenth and early eleventh centuries, served conduits for the transmission of both texts and objects from the ancient to medieval worlds and as the source of a wider educational flowering in the period.
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