In the middle of the twelfth century, the system of ciphered arithmetic which we use today, and the decimal notation in Hindu numerals on which it is based, were introduced into Western Europe. Until the advent of the algorism, and in some circles for a considerable time thereafter, calculations were made using non-ciphered systems such as finger computation or the classical abacus. This abacus was a board or table divided into columns denoting decimal ranks (units, tens, hundreds etc.) in ascending order from right to left. The columns were usually crowned by arches; indeed, the technical term for a column in the abacus treatises is arcus. Each group of three columns was spanned by a second larger arch, thereby setting off the hundreds, thousands, millions etc., much as commas do in our modern notation. Calculation was carried out by shifting small pebbles (calculi) from column to column as quantities were grouped and transferred.^{1} However, some medieval abacus treatises describe the abacus as a board spread with chalk or dust on which figures were inscribed and erased in the course of calculations. Some scholars contend that this chalk table is not an abacus at all, but a literary reminiscence of the ancient geometer's drawing board.^{2} But the algorism technique of stroking out figures in long division may in fact be derived from an older practice of writing in and then erasing numbers on an ad hoc abacus table.^{3}

In antiquity and the early medieval period, there were no treatises on the use of the abacus. Like finger-reckoning, it was taught directly, as a practical reckoning skill. It had no connection to the academic discipline of arithmetic embodied in Boethius' De institutione arithmeticae. However, in the 10th and 11th centuries, schoolmasters interested in the quadrivium began to experiment with the abacus as a teaching aid. At the close of the 10th century, the first written descriptions of the instrument and rules for its use were produced by a group of scholars in northern France.

Gerbert of Aurillac's treatises were the most influential in demonstrating the exciting possibilities -- pedagogical, theoretical, and even recreational -- of the complex calculations this instrument made possible. Gerbert also innovated by replacing the cumbersome calculi with counters (apices) bearing symbols of the first nine digits. It is highly likely that these symbols were Hindu numerals in the west Arabic or Spanish form, and it is beyond question that these numerals were first introduced into the West in connection with the Gerbertian abacus.^{4} Hindu notation, combined with the decimal place schema furnished by the abacus table, are the conceptual prerequisites of the algorism, whose terms and operations reveal their continuity with the abacus tradition.^{5}

For over a century, the abacus dominated the Western mathematical imagination. The spate of treatises on its use testifies to a fascination with its potential for elaborate calculations, and to the liberating effect of this device on people accustomed to the constraints of Roman numerals. Computistical calculations were undoubtedly a major setting for the practice of complex mathematical operations in the centuries before the emergence of exchequers and commercial book-keeping. Besides Bede's instructions for finger-calculation in De temporum ratione 1, computists could turn to the handy multiplication tables or Calculus of the computist Victorius of Aquitaine, which appeared in a number of computus manuscripts (cf. the table on MS 17 fol. 34v). Abbo of Fleury's interest in the abacus is reflected in his commentary on Victorius' calculus.^{6} Many of the second generation of abacists, such as Hermannus Contractus and Gerlandus Compotista, were also computists.^{7} The Lotharingian Robert Losinga, the bishop of Hereford in the last decades of the eleventh century who wrote a critique of Dionysius Exiguus' chronology, is praised by William of Malmesbury as "expert in all the liberal arts, particularly well versed in the abacus, lunar computus and the course of the heavenly stars."^{8} Robert's treatise is preserved in numerous copies, among them Bodleian Library Auct. F.1.9 fols. 2v-12v; this manuscript also contains an anthology of abacus materials very similar to that in MS 17, as well as a section on mathematics in relation to computus (fols. 47r-52v).^{9} This latter text may be connected to Robert's lost abacus treatise.^{10} And when the algorism was introduced, its major impact was felt through its adaptation as a tool for computists by Alexander of Ville-Dieu's Carmen de algorismo.^{11}

Nonetheless, abacus treatises are somewhat rare in computus manuscripts of MS 17's type, where there is a prominent core of calendar, Paschal tables, and related tables and texts, surrounded by a halo of texts on related topics. Abacus materials are more likely to be found in compilations of a predominantly mathematical character, where the computus materials play a subordinate role. The Worcester anthology Bodleian Library Auct. F.1.9 contains an even more extensive tract of abacus materials than does MS 17, but the manuscript as a whole is essentially a collection of treatises on astronomy and mathematics, with some computus texts -- but no calendar or tables. A smaller group of abacus texts appears in Getty Museum Ludwig XII.5, but in the midst of a large unstructured aggregate of extracts and treatises.^{12}

MS 17's abacus treatise is in fact a substitute for a type of text more commonly found in manuscripts of this type: the star catalogue. Even its position on the heels of the cosmographical anthology corresponds to that of the star catalogues in other comparable codices.^{13} Star catalogues, besides enhancing the scientific and encyclopedic tone of the computus, also functioned as reference texts for telling time by night.^{14} It is interesting to observe that the Peterborough computus, though very closely related to MS 17, has no abacus materials, but a very extensive star catalogue. MS 17's preference for a mathematical anthology bespeaks its orientation towards Worcester, Gerlandus Compotista, and the Lotharingians;^{15} this is somewhat surprising in a manuscript of MS 17's rather conservative character.

The abacus treatises included in MS 17 constitute a historical survey of the development of this branch of mathematics from the days of Gerbert up to the very decade in which MS 17 was compiled. They include commentaries on Gerbert's rules, a passage by Heriger of Lobbes, fraction tables by Hermannus Contractus, Gerlandus' text book, and fragments from the mathematical school of Laon, then at its apogee under Master Ralph. The material is not as extensive or as demanding as that in the Worcester manuscript, but it is well organized, provides good coverage of the subject, and is beautifully presented. MS 17's entire mathematical anthology also represents a pedagogically conceived package comprising large scale classroom diagrams (the first abacus table, the last fraction table), smaller versions for reference or private study (the fraction tables and smaller abacus on fols. 48v-49r), a technical manual (the Ratio numerorum abaci compilation), and discursive treatises suitable for teaching (Gerlandus', and the second anonymous abacus text). Though a long tradition associated mathematics and computus, such a collection is unprecedented in computus manuscripts. The precise stages by which this material was assembled are, as yet, impossible to reconstruct.

4 Richer, Historia 3.4 (2.64): Gerbert had 1000 caracteres made from horn and inscribed with nouem numero notas omnem numerum significantes. That these notae were Hindu numerals is upheld by Frova 1974, 329 and Lindgren 1976, 11, 18-20. For a dissenting view, see Lemay 1977. On Gerbert's abacus in general, see Kurt Vogel, "L'Aritmetica e la geometria di Gerberto," in Gerberto: scienza, storia e mito, 577-596; J. Lennart Berggren, "Medieval Arithmetic: Arabic Texts and European Motivations," in Contreni and Casciani 2002, 356-358; Menso Folkerts, "FrÃ¼he Darstellungen des Gerbertschen Abakus," in Itinera mathematica 23-24; Burnett 2002.On the possibility of a second, south Italian route for the introduction of the numerals, see Gibson and Newton 1995.

5 Evans 1977a, 25 and Evans 1977b, 118 notes the persistence in algorism treatises of terms evoking the "placing" of numbers on an imaginary grid.

7 On Hermannus, see Cordoliani 1963. Gerlandus' abacus treatise is included in MS 17 (for link see top of this overview). Neither have been edited.

8 "omnium artium liberalium peritissimus abacum precipue et lunarem compotum et caelesium cursum astrorum rimatus": William of Malmesbury, Gesta regum, 195.

12 Cambridge St John's College I.15 originally contained a number of treatises on the abacus, as the late medieval table of contents on is front flyleaf indicates, but these are now missing from the volume. I have not had the opportunity to examine Hereford Cathedral O.i.vi, a contemporary of MS 17 and the Worcester MS Bodleian Library Auct. F.1.9: on the relationship of these three volumes, see Evans 1979, 86.

## 8. MATHEMATICS: fols. 41v-58r: OVERVIEW

This section of MS 17 comprises the following:

In the middle of the twelfth century, the system of ciphered arithmetic which we use today, and the decimal notation in Hindu numerals on which it is based, were introduced into Western Europe. Until the advent of the algorism, and in some circles for a considerable time thereafter, calculations were made using non-ciphered systems such as finger computation or the classical abacus. This abacus was a board or table divided into columns denoting decimal ranks (units, tens, hundreds etc.) in ascending order from right to left. The columns were usually crowned by arches; indeed, the technical term for a column in the abacus treatises is

arcus. Each group of three columns was spanned by a second larger arch, thereby setting off the hundreds, thousands, millions etc., much as commas do in our modern notation. Calculation was carried out by shifting small pebbles (calculi) from column to column as quantities were grouped and transferred.^{1}However, some medieval abacus treatises describe the abacus as a board spread with chalk or dust on which figures were inscribed and erased in the course of calculations. Some scholars contend that this chalk table is not an abacus at all, but a literary reminiscence of the ancient geometer's drawing board.^{2}But the algorism technique of stroking out figures in long division may in fact be derived from an older practice of writing in and then erasing numbers on anad hocabacus table.^{3}In antiquity and the early medieval period, there were no treatises on the use of the abacus. Like finger-reckoning, it was taught directly, as a practical reckoning skill. It had no connection to the academic discipline of arithmetic embodied in Boethius'

De institutione arithmeticae. However, in the 10th and 11th centuries, schoolmasters interested in the quadrivium began to experiment with the abacus as a teaching aid. At the close of the 10th century, the first written descriptions of the instrument and rules for its use were produced by a group of scholars in northern France.Gerbert of Aurillac's treatises were the most influential in demonstrating the exciting possibilities -- pedagogical, theoretical, and even recreational -- of the complex calculations this instrument made possible. Gerbert also innovated by replacing the cumbersome

calculiwith counters (apices) bearing symbols of the first nine digits. It is highly likely that these symbols were Hindu numerals in the west Arabic or Spanish form, and it is beyond question that these numerals were first introduced into the West in connection with the Gerbertian abacus.^{4}Hindu notation, combined with the decimal place schema furnished by the abacus table, are the conceptual prerequisites of the algorism, whose terms and operations reveal their continuity with the abacus tradition.^{5}For over a century, the abacus dominated the Western mathematical imagination. The spate of treatises on its use testifies to a fascination with its potential for elaborate calculations, and to the liberating effect of this device on people accustomed to the constraints of Roman numerals. Computistical calculations were undoubtedly a major setting for the practice of complex mathematical operations in the centuries before the emergence of exchequers and commercial book-keeping. Besides Bede's instructions for finger-calculation in

De temporum ratione1, computists could turn to the handy multiplication tables orCalculusof the computist Victorius of Aquitaine, which appeared in a number ofcomputusmanuscripts (cf.the table on MS 17 fol. 34v). Abbo of Fleury's interest in the abacus is reflected in his commentary on Victorius'calculus.^{6}Many of the second generation of abacists, such as Hermannus Contractus and Gerlandus Compotista, were also computists.^{7}The Lotharingian Robert Losinga, the bishop of Hereford in the last decades of the eleventh century who wrote a critique of Dionysius Exiguus' chronology, is praised by William of Malmesbury as "expert in all the liberal arts, particularly well versed in the abacus, lunarcomputusand the course of the heavenly stars."^{8}Robert's treatise is preserved in numerous copies, among them Bodleian Library Auct. F.1.9 fols. 2v-12v; this manuscript also contains an anthology of abacus materials very similar to that in MS 17, as well as a section on mathematics in relation tocomputus(fols. 47r-52v).^{9}This latter text may be connected to Robert's lost abacus treatise.^{10}And when the algorism was introduced, its major impact was felt through its adaptation as a tool for computists by Alexander of Ville-Dieu'sCarmen de algorismo.^{11}Nonetheless, abacus treatises are somewhat rare in

computusmanuscripts of MS 17's type, where there is a prominent core of calendar, Paschal tables, and related tables and texts, surrounded by a halo of texts on related topics. Abacus materials are more likely to be found in compilations of a predominantly mathematical character, where thecomputusmaterials play a subordinate role. The Worcester anthology Bodleian Library Auct. F.1.9 contains an even more extensive tract of abacus materials than does MS 17, but the manuscript as a whole is essentially a collection of treatises on astronomy and mathematics, with some computus texts -- but no calendar or tables. A smaller group of abacus texts appears in Getty Museum Ludwig XII.5, but in the midst of a large unstructured aggregate of extracts and treatises.^{12}MS 17's abacus treatise is in fact a substitute for a type of text more commonly found in manuscripts of this type: the star catalogue. Even its position on the heels of the cosmographical anthology corresponds to that of the star catalogues in other comparable codices.

^{13}Star catalogues, besides enhancing the scientific and encyclopedic tone of thecomputus, also functioned as reference texts for telling time by night.^{14}It is interesting to observe that the Peterboroughcomputus, though very closely related to MS 17, has no abacus materials, but a very extensive star catalogue. MS 17's preference for a mathematical anthology bespeaks its orientation towards Worcester, Gerlandus Compotista, and the Lotharingians;^{15}this is somewhat surprising in a manuscript of MS 17's rather conservative character.The abacus treatises included in MS 17 constitute a historical survey of the development of this branch of mathematics from the days of Gerbert up to the very decade in which MS 17 was compiled. They include commentaries on Gerbert's rules, a passage by Heriger of Lobbes, fraction tables by Hermannus Contractus, Gerlandus' text book, and fragments from the mathematical school of Laon, then at its apogee under Master Ralph. The material is not as extensive or as demanding as that in the Worcester manuscript, but it is well organized, provides good coverage of the subject, and is beautifully presented. MS 17's entire mathematical anthology also represents a pedagogically conceived package comprising large scale classroom diagrams (the first abacus table, the last fraction table), smaller versions for reference or private study (the fraction tables and smaller abacus on fols. 48v-49r), a technical manual (the

Ratio numerorum abacicompilation), and discursive treatises suitable for teaching (Gerlandus', and the second anonymous abacus text). Though a long tradition associated mathematics andcomputus, such a collection is unprecedented incomputusmanuscripts. The precise stages by which this material was assembled are, as yet, impossible to reconstruct.1 For a description of the procedures for calculating on an abacus see Pullen 1970; Menninger 1969, 315-331; Michael Mahoney, "Mathematics," in Lindberg 1978, 147-148; Frova 1974, 340-341 (on long division); and Guy Beaujouan in Taton 1957-1964, 1.222-224; Hart 2003, 402-401.

2 Beaujouan 1947, 309-312.

3 Menninger 1969, 330-331; Evans 1979, 71.

4 Richer,

Historia3.4 (2.64): Gerbert had 1000caracteresmade from horn and inscribed withnouem numero notas omnem numerum significantes. That thesenotaewere Hindu numerals is upheld by Frova 1974, 329 and Lindgren 1976, 11, 18-20. For a dissenting view, see Lemay 1977. On Gerbert's abacus in general, see Kurt Vogel, "L'Aritmetica e la geometria di Gerberto," inGerberto: scienza, storia e mito, 577-596; J. Lennart Berggren, "Medieval Arithmetic: Arabic Texts and European Motivations," in Contreni and Casciani 2002, 356-358; Menso Folkerts, "FrÃ¼he Darstellungen des Gerbertschen Abakus," inItinera mathematica23-24; Burnett 2002.On the possibility of a second, south Italian route for the introduction of the numerals, see Gibson and Newton 1995.5 Evans 1977a, 25 and Evans 1977b, 118 notes the persistence in algorism treatises of terms evoking the "placing" of numbers on an imaginary grid.

6 Abbo of Fleury,

Commentary on the Calculus of Victorius. See also Abbo's essay on the motion of the Moon on fols. 38v-39r of MS 17.7 On Hermannus, see Cordoliani 1963. Gerlandus' abacus treatise is included in MS 17 (for link see top of this overview). Neither have been edited.

8 "omnium artium liberalium peritissimus abacum precipue et lunarem compotum et caelesium cursum astrorum rimatus": William of Malmesbury,

Gesta regum, 195.9 On Robert of Hereford, see Haskins 1927, 333-335 and Stevenson 1907. For his work as a computist, see Cordoliani 1966.

10 Evans 1979, 74.

11 Mahoney in Lindberg 1978, 151. The

Carmen de algorismoappears in the Cerne AbbeycomputusMS Cambridge, Trinity College 0.2.45, pp. 23-31.12 Cambridge St John's College I.15 originally contained a number of treatises on the abacus, as the late medieval table of contents on is front flyleaf indicates, but these are now missing from the volume. I have not had the opportunity to examine Hereford Cathedral O.i.vi, a contemporary of MS 17 and the Worcester MS Bodleian Library Auct. F.1.9: on the relationship of these three volumes, see Evans 1979, 86.

13 Some examples: Bodleian Library Bodley 614 fols. 16r-35v contains an illustrated star catalogue embedded in a cosmographical anthology; in British Library Cotton Tiberius B.V fols. 32v-49v, Cicero's

Arateaheads the standard carolingian cosmographical anthology of excerpts from Pliny, Macrobius, Martianus and Bede; roughly the same pattern obtains in Munich CLM 210 fol.s 113v-121r, the Peterboroughcomputus(Tiberius C.I fols. 19r-36r), British Library Harley 2506 fols. 33r-48v, British Library Royal 13.A.XI fols. 105r-113r and Durham Hunter 100 fols. 60v-64v. In Glasgow Hunter 85, Hyginus star catalogue follows the astronomical essays of Abbo of Fleury and Walcher of Malvern on the astrolabe.14 McCluskey 1990; Poole (R.) 1914-1915.

15 Borst 1986, 312-313.