History of Science

Language Selection

Breadcrumb Navigation


Medieval Mathematical Manuscripts

Menso Folkerts: "Mathematische Handschriften des Mittelalters", in: Einsichten. Forschung an der Ludwig-Maximilians-Universität München 1 (1996), 28-31.

During the last couple of decades there was a fundamental shift in the appraisal of medieval scientific accomplishments. The old notion of the „dark ages“ has been replaced with a view of the Middle Ages as preserver and developer of the heritage of Antiquity as well as pioneer for and model of modern conceptions. The first case consists largely in the modes and ways of tradition of Greek and (to a lesser extent) Indian mathematics to the latin Middle Ages. Greek mathematics was only handed down to a small extent by the Romans and through monestaries; of far greater importance were the translations and the development of Greek texts by the Arabs. These works were in turn partially translated into Latin in the twelth century and therefore accessible once more to the Occident. Arab scholars also translated and worked with mathematical problems and methods of eastern cultures (India, China).

However, medieval mathematics not only passed on adopted knowledge. It was also a model and pioneer with regards to modern conceptions and ideas. For instance, the teaching of Formal Latitudes, which was developed in Paris and Oxford in the 13th and 14th century, contains the idea of functional dependence and has, at least indirectly, influenced the development of the concept of functions in the 17th century; using infinite strings for calculation, an idea that also surged in this context, was also built upon during the 17th century in connection with infinitesimal calculus; scholastic ideas, e.g. the study of motion sequences, influenced the works of Galilei as well as other natural scientists in relation to evenly accelerated motion.

Additionally, medieval mathematics gives us important insights into the interdependency of mathematics and other sciences. Questions originating in natural philosophy (e.g. regarding the movement of bodies), which arose mainly out of Aristotle´s works, influenced the development of mathematics (e.g. the teachings of proportions), while mathematical problems, especially concerning Euclid´s Elements, gave reason to new studies philosophy; the discussion about the „contingency angle“ (between circular arc and tangent), initiated by Euclid, led to speculation about continuity, steadiness, and the infinite. The boom of astronomy in the 15th and 16th centuries is closely connected with the development of certain mathematical branches (trigonometry, goniometry). The efforts to create a new calender, the construction of mechanical clocks as well as the development of technology as such are in close connection with mathematics at this time in history.

Mathematics was subject of multiple changes during the Middle Ages. However, the reasons for this are not yet sufficiently known. Without doubt factors both internal and external to the sciences played a part during this process. What is special about the research conducted in Munich is its focus on sources within mathematics.

This may come a surprise, but many of the important sources regarding medieval mathematics are not yet indexed, much less analyzed. Despite the fact that since the end of the nineteenth century there has been intense research into the history of mathematics; and despite the fact that scince 1945 historians of science, first in the USA and the former USSR, later in European countries as well, increasingly dealt with medieval mathematics; and even though there has been some progress in this field, twenty years ago our knowledge about occidental medieval mathematics as a whole was not much larger than it was in 1920.

In order to fill this gap I (first at the university Oldenburg; since 1980 at the local institute for the history of natural sciences) made the request in 1977 of the Volkswagen Foundation for the project „Materialien zur Geschichte der europäischen Mathematik in Mittelalter und Renaissance“ (Materials for the History of European Mathematics in the Middle Ages and Rennaissance). Relying on international cooperation this project has as its goal to collect and edit all mathematical manuscripts written in western languages, so long as they were recorded before the 17th century. „Western language“ here means not only Latin, which more or less was the only scientific language during the Early and High Middle Ages, but also western European national languages that gained importance during the Late Middle Ages (with regard to mathematics this comes down to Italian, French, German, English and Spanish, but also Catalan, Provençal, and Islandic). In order to initially reduce the ammount of relevant material the term mathematics has here been taken as rather narrow; excluded were, among other things, astronomical manuscripts. This project has been supported by the VW-Foundation with 633.000 DM from 1977 to 1984.

The first step in this project consisted in identifying the relevant mathematical texts. There has hardly been done any groundwork in this area. The vast majority of mathematical texts is written down in manuscrips; hardly any text were printed in the 15th century. Today, there are western medieval manuscripts in over two thousand libraries around the globe. Useful printed catalogues for manuscripts only exist in some of these libraries; however, for many of the most important collections this is not the case (Paris, Vienna, BSB Munich, Berlin, Florence, Vatican). Because of this we not only searched through the libraries´ catalogues, but also undertook many trips to libraries in order to consult the finding aids that were not printed. On top of this the systematic review of scientific historical literature gave detailled information as well. In total, more than one thousand three hundred catalogue volumes and other works on collections of manuscripts with more than half a million descriptions of manuscriots have been revised. As a result, we could show that only about one percent of medieval manuscripts contain mathematical texts. Despite all this, their total number came down to about three thousand. This survey not only produced already widely known manuscripts, but also a large amount of mathematical texts that had not yet been accounted for in scientific literature.

The VW-Foundation´s generous founds allowed us to take microfilm copies of all relevant manuscripts and collect them into one central archive. This archive now basically contains, either as microfilm or as ordinary copy, all medieval manuscripts in western languages that contain mathematical texts. Over the years the archive was expanded and now contains more than five thousand manuscripts from over 350 libraries; many of them not only contain mathematical texts, but also deal with astronomy or the natural sciences. Additionally there are now also several hundred microfilms of Arab mathematical manuscripts incorporated into the collection. All of this combined makes this collection a one of its kind in the entire world.

In the years 1985 – 1989 a database was constructed within the framework of project financed by the DFG which collected both printed and unprinted descriptions of mathematical-scientific manuscripts. The basis for this database was the information gathered while collecting the mathematical texts. However, descriptions of much younger manuscripts where also collected, including 19th and 20th century lecutre notes. Since the catalogues´ information is not standardized, a special system for coding had to be developed which considers both the manuscripts´ sctructure as well as the historians´ demands. Of special importance (besides the arrays denoting the texts´ authors) are the arrays which show the beginning of the text (Inciptis). Since there were no standardized titles in the middle ages, copies of the same text are most easily identified by the (usually) identical Inciptis. The Munich based databank (International Computer Catalog of Medieval Scientific Manuscripts: ICCMSM) was conceptualized in cooperation with a similar database still under construction in the US and Canada (Benjamin Data Bank). At this point in time, it contains information about over 10,000 manuscripts.

The databank allows a detailed search along all coded data. This included but is not limited to the authors, titels, Incipits, and Explicits. In a later phase it is planned to publish some of the researches most interesting results as catalogues, for instance registries of names and works as well as indices of manuscripts dealing with special topics.

One of the most important aspects of this database is the ability to search for different copies of the same text, which is the basis for the critical edition of any text. The combination of author, title and Incipit facilitates this. Upon finding the different manuscripts of any given text, the microfilms allow researchers to check the catalogues´ details, which in turn makes travel to the manuscripts´ home libraries redundant.

Some of the most important results of this project are the finding of the hitherto unknown mathematical texts and copies of such as well as correcting the description of some manuscripts. A couple of results as well as the research based on them and carried out at the institute for the history of natural sciences will be described here:

  • It was shown that manuscripts dating to the first half of the 15th century in Berlin and Breslau contain two hitherto unkown translations into high and low German of the best known Latin text of this time: The Algorism of Johannes de Sacrobosco. Alongside one additional German translation of this text, these two manuscripts are among the earliest mathematical works in German. A dissertation (Gerhard Brey) dealing with this translation and providing an index of German medieval mathematical texts is close to being finished.
  • Translating and working with Euclid´s Elements is one of the core activities in medieval science. The database alongside the microfilm archive allowed us to structure the confusingly large amount of texts dealing with this topic [1]. Several hitherto unkown manuscripts were discovered in this process, including one anonymous commentary which had to have been available to Albertus Magnus for his work dealing with Euclid. A couple of these manuscripts were used during university lectures in the Artes Liberales faculty in the 14th and 15th centuries.
  • The most important Euclid-text in Latin before the end of the 13th century was the so-called „Version II“, which, up to now, was attributed to Adelard von Bath. The edition of this version was created in cooperation with Dr. H. L. L. Busard (Venlo), taking into account all 63 preserved manuscripts [2].
  • The most widespread arithmetic text during the early medieval period was the arithmetic of Boethius, which is part of the neo-pythagoreic tradition. Unlike his treatise on music it is not known of the arithemtic text, how many manuscripts dating to the Carolingian time are presevered. Using the database, about 250 such manuscripts were found, which will be compiled in a list in the future.
  • There is a number-theoretical script based on the arithmetic of Boethius dating to the first half of the 14th century which is sometimes attributed to Thomas Bradwardine, other times to Simon Bredon. Using database and archive it could be shown that the scripts of both authors are in fact identical and furthermore conform to a printed work dating to 1495. There are not only these three texts, however, but 15. A critical edition of this important text ist being prepared.
  • One of the most common board games during the medieval times was the so called „Rithmimachie“ which originated in the 11th century and was, at times, more popular than chess. Using the database over 20 manuscripts dealing with this game were uncovered. The results were made available to the historian Arno Borst for his edition of early texts on Rithmimachie [3]. At this point in time I have edited a version of Werinher vom Tegernsee dating to the second half of the 12th cenutry which takes a central position in the history of this game [4]. An edition of an Italian conversation about this game (of Benedetto Varchi, 1539) is currently being worked on.
  • There is only one Latin version of the oldest script about the use of Indian-Arab figures, written by al-Hwarizmi in about 820 in Arabic. Until now, there was only known an incomplete manuscript of this text. I found a second, complete version [5]; there is an edition in the works.
  • Surprisingly, there is no comprehensive collection of sources about western medieval mathematics; one still often has to use the “Vorlesungen über Geschichte der Mathematik” of Moritz Cantor, held in 1900, as well as tediously collect new publication results out of different organs of publication. Because of this, I am collaborating with Dr. Busard to create a handbook which will inform in chronological order about the scripts of the different authors. This book will also mention the manuscripts of each author which transmit their work.
The future use of this databank and microfilm archive is not limited to the History of Science. Classical Philology, Medieval Studies, Renaissance Studies and other disciplines can draw on this well of knowledge for inspiration.

[1] Folkerts: Euclid in Medieval Europe. Winnipeg 1989
[2] H. L. L. Busard, M. Folkerts: Robert of Chester's (?) Redaction of Euclid's Elements: the so-called Adelard II Version. Basel / Boston / Berlin 1992
[3] Das mittelalterliche Zahlenkampfspiel. Heidelberg 1986
[4] M. Folkerts: "Die 'Rithmachia' des Werinher von Tegernsee, in: Vestigia Mathematica. Studies in medieval and early modern mathematics in honour of H. L. L. Busard, Amsterdam / Atlanta 1993, S. 107-142
[5] M. Folkerts: "Eine neue Handschrift von al-Hwarizmis Arithmetik", in: Cosmographica et geographica. Festschrift für Heribert M. Nobis zum 70. Geburtstag, hg. v. B. Fritscher und G. Brey. 1. Halbband. München 1994, S. 181-193

M. Folkerts: "Materialien zur Geschichte der europäischen Mathematik in Mittelalter und Renaissance. Ein Projekt der Universitäten Oldenburg und München", in: Jahrbuch der historischen Forschung in der Bundesrepublik Deutschland, Berichtsjahr 1984, München usw. 1985, 36-41
M. Folkerts, A. Kühne, M. Segre: "Der Aufbau einer Datenbank für die Geschichte der europäischen Mathematik im Mittelalter und der Renaissance", in: Berichte zur Wissenschaftsgeschichte 11 (1988) 256-260
M. Folkerts, A. Kühne (Hrsg.): The Use of Computers in Cataloging Medieval and Renaissance Manuscripts. München 1990 (Algorismus, Heft 4)

Author: Prof. Dr. Menso Folkerts