Arithmetic in the Thought of Gerbert of Aurillac

by Marek Otisk (Author)
©2022 Monographs 242 Pages


The book deals with Gerbert of Aurillac (Pope Sylvester II) and his investigations in the field of theoretical (philosophy of numbers) and practical (counting) arithmetic. The book represents the comprehensive inquiry of both these aspects of arithmetic in his thought. The analysed sources are Gerbert’s so-called scientific letters written to his friends, colleagues or pupils, and also including some of his other texts. On this basis, attention is paid to arithmetic as the mother of all sciences and as the path to wisdom (e.g., the so-called Saltus Gerberti, relation between arithmetic and other disciplines of the quadrivium, etc.) and also to practical arithmetic (e.g., the introduction of Hindu-Arabic numerals and the re-introduction of a new form of abacus to the Latin Christian West).

Table Of Contents

  • Cover
  • Title
  • Copyright
  • About the author
  • About the book
  • This eBook can be cited
  • Table of Contents
  • 1. The Nature of Numbers: Saltus Gerberti (Letter to Constantine)
  • 2. Figurative Numbers and Geometry (Letter to Adelbold)
  • 3. Music and Harmony (Two Letters to Constantine)
  • 4. Astronomy and Timekeeping (Letter De sphaera, Richer’s Historia, and the horological letter to Adam)
  • 1. Tabula abaci, Decimal Positional Notation and Ghubar Numbers
  • 2. Regulae multiplicationis
  • 3. Regulae divisionis
  • Abbreviations
  • Primary Sources
  • Secondary Sources
  • List of Tables
  • List of Figures
  • Index locorum
  • Index of Personal Names (before 1700)
  • Index of Personal Names (after 1700)
  • Series index


Since the 19th century, multiple interpretative strategies have portrayed the long period of the European Latin Middle Ages as a consecutive succession of renaissances.1 Despite the vagueness of such a designation, and even although it is difficult to summarise the vast diversity of historical events it denotes, it has become – pace all justified criticism – the dominant interpretative model applied to medieval intellectual and cultural history.2 Various terms have emerged over time, including the Carolingian Renaissance (Karolingische Renaissance),3 and its phases,4 which were supposedly preceded by the renaissances of the 7th century in, for example, Visigoth Hispania or the Irish monasteries and monasteries on the British Isles, which directly influenced the Carolingian Renaissance.5 If we ←9 | 10→divide the Carolingian Renaissance into several stages, we may distinguish the era of Charlemagne (the last third of the 8th century and the start of the 9th century), followed by the period after Charlemagne (especially the reign of Louis the Pious and Charles the Bald in the 9th century), and then the so-called Renaissance of the 10th century or the Ottonian Renaissance (Ottonische Renaissance),6 which overlaps with the so-called Big Restoration after the year 1000,7 and encompasses mainly the second half of the 10th century and the first quarter of the 11th century.8 This is succeeded by the Renaissance of the 11th century,9 in which the foundations were laid for the (most famous) medieval Renaissance of the 12th century and for the paradigmatic constants that remained typical of the scholastic era of the Middle Ages.10←10 | 11→

In these interpretative models, the Early Middle Ages, formerly branded a dark era of European intellectual and cultural history,11 became the succession of gradual milestones that essentially evidence the continual development of the ancient, as well as Byzantine and Arabic, intellectual heritage in the Latin West. Although it is possible to identify different forms of overlapping cultural and intellectual influences with varying degrees of intensity throughout the entire Middle Ages, this book focuses on the period labelled the “Ottonian Renaissance” or “the Renaissance of the second half of the 10th century,” when we have evidence not only of the development of ancient intellectual heritage in the Latin West but also of numerous contacts with the Byzantine (e.g., with the imperial court) and Arabic (especially the Iberian peninsula) intellectual environments. This historical era of the European West owes its name to the Liudolfinger imperial dynasty, originally Saxon dukes and, later, kings of East Francia, who were crowned as Roman Emperors; the first emperor of this dynasty was Otto I (crowned Holy Roman Emperor by Pope John XII in 962) who was succeeded by his son Otto II (crowned by Pope John XIII in 967), his grandson Otto III (crowned by Pope Gregory V in 996), and his great-grandson Henry II, known as Saint Henry the Exuberant (crowned by Pope Benedict VIII in 1014).12

Among the most well-known contemporary figures with an intellectual (but also ecclesiastical, and often even political) background at the end of the 10th century and the beginning of the 11th century, it is possible to mention Notker of Liège († 1008, bishop from 972 and prince-bishop from 980 in Liège),13 Abbo of ←11 | 12→Fleury († 1004, abbot from 988 in Fleury),14 Gerbert of Aurillac († 1003, abbot of Bobbio from 982, illegitimate archbishop of Reims from 991, archbishop of Ravenna from 998, Pope Sylvester II from 999),15 Fulbert of Chartres († 1028, bishop of Chartres from 1006),16 or Hermann the Lame († 1054, a monk of Reichenau monastery from 1020).17 For all the above, it is the case that their ←12 | 13→scholarly and philosophical activities were focused mainly on the seven liberal arts. Their attention was variously divided between the sciences of the trivium (primarily dialectics, but grammar and rhetoric were not neglected) and the sciences of the quadrivium (i.e., arithmetic, geometry, music, and astronomy). This commitment is apparent in the contemporaries (and, to a certain extent, rivals) Abbo of Fleury and Gerbert of Aurillac – both were, among other things, authors of texts on dialectics (see, for instance, Abbo’s texts on categorical and hypothetical syllogism, although he also wrote about grammar etc.,18 or Gerbert’s work De rationali et ratione uti19) on mathematics, and astronomy (Abbo is the author of computistic treatises, a commentary on the Calculus of Victorius, and several astronomical texts,20 while Gerbert wrote, for example, several “scientific” letters to his friend Constantine of Fleury and Micy, including the famous Regula de abaco computi, or the construction manual De sphera, and several other texts, predominantly dealing with topics related to the arts of the quadrivium).21

This book primarily focuses on Gerbert of Aurillac (sometimes known as Gerbert “of Reims,” “of Bobbio,” or “of Ravenna,” and also known by his papal name Sylvester II). The book’s content summarises my systematic long-term interest in the period of pre-scholastic Latin thought, especially regarding the relationship between dialectics, metaphysics, natural philosophy, and theology in the 10th and 11th centuries. The topical focus of this book is directed at the ←13 | 14→first of the mathematical sciences: arithmetic, and its relationship to knowledge as a whole. At the same time, attention is, naturally, given to the way arithmetic is intertwined with the other sciences of the quadrivium. The sources used predominantly include Gerbert’s correspondence, which mirrors a live dialogue that can point towards the themes that contemporary scholars (mostly Gerbert’s disciples, friends, and colleagues) considered important or useful. Finally, the book does not ignore Gerbert’s other texts and the works of his contemporaries and successors, in which we can track Gerbert’s commitment to (especially) the field of arithmetic.


Given the fact that I am not a native English speaker, I am obliged to David Černín, Gary Frost, Igor Jelínek, Daniela Rywiková, and Světla Hanke Jarošová for their help with language aspects. Without their help, this book would not have existed in its current form. For invaluable discussions and advice, I am grateful to Daniel Špelda, Costantino Sigismondi, Seweryn Blandzi, and Richard Psík. Since many conclusions and parts of the text presented in this book have already been published as parts of my various studies (for more details, see the Bibliography), I would also like to thank all the anonymous reviewers who have provided priceless feedback. However, any shortcomings of this book are solely my own responsibility.


ISBN (Hardcover)
Publication date
2022 (January)
medieval mathematics theory of numbers philosophy of numbers medieval astronomy medieval timekeeping abacus
Berlin, Bern, Bruxelles, New York, Oxford, Warszawa, Wien, 2022. 242 pp., 47 fig. b/w, 16 tables.

Biographical notes

Marek Otisk (Author)

Marek Otisk is Professor at the Department of Philosophy, Faculty of Arts, University of Ostrava. His research interest is focused on medieval philosophy, especially the metaphysics, dialectics (logic), mathematics or astronomy(natural philosophy) in the 10th and 11th centuries (for example Gerbert of Aurillac or Anselm of Canterbury).


Title: Arithmetic in the Thought of Gerbert of Aurillac
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244 pages