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984: Some Letters by Gerbert d’Aurillac Dealing with “Arabic” Mathematics and Astrology: Difference between revisions
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984: Some Letters by Gerbert d’Aurillac Dealing with “Arabic” Mathematics and Astrology (view source)
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Most important in this quote is Richer’s reference to “nine numerals which signify all possible numbers” (''novem numero notas'' ''omnem numerum significantes''). What Richer seems to suggest here, is that Gerbert did not use Roman numbers composed of one or more individual signs (e.g. the numbers 4, 6-9, i.e. IV, VI, VII, VIII, IX). Rather, he used individual signs to represent the basic numerals that make up the decimal system, i.e. one individual sign for each number from 1 to 9. Gerbert was not the first to substitute Roman numerals with alternative signs. So far, however, Greek letters had been used as substitutes.<ref name="ftn55">Folkerts, Names and Forms, p. 246, 248. </ref> As already mentioned above, the ''Codex Vigilanus'', a manuscript copied in the Catalonian monastery of Albelda in the year 976 contains the earliest depiction of Hindu-Arabic numerals in a Latin manuscript. These signs represent a western variant used on the Iberian Peninsula that are classified as “dust” (''ġubār'') numerals (See Fig. 1).<ref name="ftn56">See Kunitzsch, Transmission of Hindu-Arabic Numerals, pp. 11-14.</ref> Some of the names of the ''ġubār''-numerals are clearly of Arabic origin: this applies to the numbers four (''arbas'' < ''arbaʿ''), five (''quimas < ḫamsa''), and eight (''temenias < ṯamaniya'').<ref name="ftn57">Buddhue, Origin of Our Numerals, pp. 256-267; Ball, ''A Short History of Mathematics'', p. 115<nowiki>; Ľ Huillier, Regards sur la formation progressive, p. 544; Allard, Ľ influ</nowiki>ence des mathématiques arabes dans ľOccident médiéval, p. 200.</ref> | Most important in this quote is Richer’s reference to “nine numerals which signify all possible numbers” (''novem numero notas'' ''omnem numerum significantes''). What Richer seems to suggest here, is that Gerbert did not use Roman numbers composed of one or more individual signs (e.g. the numbers 4, 6-9, i.e. IV, VI, VII, VIII, IX). Rather, he used individual signs to represent the basic numerals that make up the decimal system, i.e. one individual sign for each number from 1 to 9. Gerbert was not the first to substitute Roman numerals with alternative signs. So far, however, Greek letters had been used as substitutes.<ref name="ftn55">Folkerts, Names and Forms, p. 246, 248. </ref> As already mentioned above, the ''Codex Vigilanus'', a manuscript copied in the Catalonian monastery of Albelda in the year 976 contains the earliest depiction of Hindu-Arabic numerals in a Latin manuscript. These signs represent a western variant used on the Iberian Peninsula that are classified as “dust” (''ġubār'') numerals (See Fig. 1).<ref name="ftn56">See Kunitzsch, Transmission of Hindu-Arabic Numerals, pp. 11-14.</ref> Some of the names of the ''ġubār''-numerals are clearly of Arabic origin: this applies to the numbers four (''arbas'' < ''arbaʿ''), five (''quimas < ḫamsa''), and eight (''temenias < ṯamaniya'').<ref name="ftn57">Buddhue, Origin of Our Numerals, pp. 256-267; Ball, ''A Short History of Mathematics'', p. 115<nowiki>; Ľ Huillier, Regards sur la formation progressive, p. 544; Allard, Ľ influ</nowiki>ence des mathématiques arabes dans ľOccident médiéval, p. 200.</ref> | ||
Figure 1: ''Codex conciliorum Albeldensis seu Vigilanus'', Madrid: El Escorial, MS D.I.2, fol. 12v.<ref name="ftn58">[https://upload.wikimedia.org/wikipedia/commons/3/3b/Codex_Vigilanus_Primeros_Numeros_Arabigos.jpg https://upload.wikimedia.org/wikipedia/commons/3/3b/Codex_Vigilanus_Primeros_Numeros_Arabigos.jpg]. For a printed reproduction, see Szpiech, From Mesopotamia to Madrid, p. 44. The text runs: “Scire debemus in Indos subtilissimum ingenium habere. et ceteras gentes eis in arithmetica et geometrica. et ceteris liberalibus disciplinis concedere. Et hoc manifestum est in nobem figuris quibus designant unumquemque gradum. Cuiuslibet gradus quarum hec sunt forma. 987654321”, i.e. “We should know that we find subtle inventiveness among the Indians and among several peoples, which they have given to them in arithmetics and geometry and other liberal disciplines. This becomes manifest in nine figures, each of which designate a specific [numeric] position. These are the forms of each position: 987654321.” Note that the list of numbers still lacks zero.</ref> | '''Figure 1''': ''Codex conciliorum Albeldensis seu Vigilanus'', Madrid: El Escorial, MS D.I.2, fol. 12v.<ref name="ftn58">[https://upload.wikimedia.org/wikipedia/commons/3/3b/Codex_Vigilanus_Primeros_Numeros_Arabigos.jpg https://upload.wikimedia.org/wikipedia/commons/3/3b/Codex_Vigilanus_Primeros_Numeros_Arabigos.jpg]. For a printed reproduction, see Szpiech, From Mesopotamia to Madrid, p. 44. The text runs: “Scire debemus in Indos subtilissimum ingenium habere. et ceteras gentes eis in arithmetica et geometrica. et ceteris liberalibus disciplinis concedere. Et hoc manifestum est in nobem figuris quibus designant unumquemque gradum. Cuiuslibet gradus quarum hec sunt forma. 987654321”, i.e. “We should know that we find subtle inventiveness among the Indians and among several peoples, which they have given to them in arithmetics and geometry and other liberal disciplines. This becomes manifest in nine figures, each of which designate a specific [numeric] position. These are the forms of each position: 987654321.” Note that the list of numbers still lacks zero.</ref> | ||
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Based on the research of Menso Folkerts, Charles Burnett, Paul Kunitzsch, and others<ref name="ftn59">Folkerts, Frühe Darstellungen, pp. 23-43; Folkerts, Names and Forms, pp. 245-265; Burnett, Abacus at Echternach, pp. 91-108; Kunitzsch, Transmission of Hindu-Arabic Numerals, pp. 3-22.</ref>, Marek Otisk has listed the early pictorial and textual evidence for ''ġubār''-numerals in Latin manuscripts produced east of the Iberian Peninsula. The earliest evidence dates from the late tenth and early eleventh centuries and includes# <div style="margin-left:0.635cm;margin-right:0cm;">Bernelinus’ ''Liber abaci'', which—because it refers to “pope Gerbert”—is dated to the years 999-1003. According to its author, it replicates Gerbert’s teaching on the abacus.<ref name="ftn60">Bernelinus, Liber abaci, in: Gerbert, ''Oeuvres'', ed. Olleris, p. 357; Gerbertus, ''Opera mathematica'', ed. Bubnov, p. 383.</ref> It contains an extensive description of the ''ġubār''-numbers and equates them to the Greek letters used hitherto as a substitute for Roman numerals.<ref name="ftn61">Bernelinus, Liber abaci, ed. Olleris, ''Oeuvres de Gerbert'', p. 361: “His igitur expeditis, ad ipsos caracteres veniamus, et quibus figuris praenotentur, ascribere properemus. Unitas, quae primus caracter dicitur, sic figuratur (…).”; Bernelin, ''Libre d’abaque'', ed. Bakhouche and Cassinet, p. 25. See Otisk, Descriptions and Images, pp. 17-19; Folkerts, Names and Forms of Numerals, pp. 246, 248.</ref> | Based on the research of Menso Folkerts, Charles Burnett, Paul Kunitzsch, and others<ref name="ftn59">Folkerts, Frühe Darstellungen, pp. 23-43; Folkerts, Names and Forms, pp. 245-265; Burnett, Abacus at Echternach, pp. 91-108; Kunitzsch, Transmission of Hindu-Arabic Numerals, pp. 3-22.</ref>, Marek Otisk has listed the early pictorial and textual evidence for ''ġubār''-numerals in Latin manuscripts produced east of the Iberian Peninsula. The earliest evidence dates from the late tenth and early eleventh centuries and includes | ||
# <div style="margin-left:0.635cm;margin-right:0cm;">Bernelinus’ ''Liber abaci'', which—because it refers to “pope Gerbert”—is dated to the years 999-1003. According to its author, it replicates Gerbert’s teaching on the abacus.<ref name="ftn60">Bernelinus, Liber abaci, in: Gerbert, ''Oeuvres'', ed. Olleris, p. 357; Gerbertus, ''Opera mathematica'', ed. Bubnov, p. 383.</ref> It contains an extensive description of the ''ġubār''-numbers and equates them to the Greek letters used hitherto as a substitute for Roman numerals.<ref name="ftn61">Bernelinus, Liber abaci, ed. Olleris, ''Oeuvres de Gerbert'', p. 361: “His igitur expeditis, ad ipsos caracteres veniamus, et quibus figuris praenotentur, ascribere properemus. Unitas, quae primus caracter dicitur, sic figuratur (…).”; Bernelin, ''Libre d’abaque'', ed. Bakhouche and Cassinet, p. 25. See Otisk, Descriptions and Images, pp. 17-19; Folkerts, Names and Forms of Numerals, pp. 246, 248.</ref> | |||
# <div style="margin-left:0.635cm;margin-right:0cm;">two eleventh-century manuscripts, which contain depictions of the abacus of Echternach, the latter dated to the 990s. These depictions contain ''ġubār''-numbers as well.<ref name="ftn62">Otisk, Descriptions and Images, pp. 19-20; Burnett, Abacus at Echternach, pp. 91-108.</ref> | # <div style="margin-left:0.635cm;margin-right:0cm;">two eleventh-century manuscripts, which contain depictions of the abacus of Echternach, the latter dated to the 990s. These depictions contain ''ġubār''-numbers as well.<ref name="ftn62">Otisk, Descriptions and Images, pp. 19-20; Burnett, Abacus at Echternach, pp. 91-108.</ref> | ||
# <div style="margin-left:0.635cm;margin-right:0cm;">a mathematical and computation manuscript from Bern from the late tenth century depicting an abacus. It does not contain ''ġubār''-numbers, but is very similar to the Echternach depictions. Moreover, it features the heading “Gerbertus Latio numeros abacique figuras”, i.e. “Gerbert [gave] the numbers and the figures of the abacus to Latium”, i.e. the Latin sphere.<ref name="ftn63">Otisk, Descriptions and Images, pp. 20-22; Folkerts, Frühe Darstellungen, p. 28.</ref> | # <div style="margin-left:0.635cm;margin-right:0cm;">a mathematical and computation manuscript from Bern from the late tenth century depicting an abacus. It does not contain ''ġubār''-numbers, but is very similar to the Echternach depictions. Moreover, it features the heading “Gerbertus Latio numeros abacique figuras”, i.e. “Gerbert [gave] the numbers and the figures of the abacus to Latium”, i.e. the Latin sphere.<ref name="ftn63">Otisk, Descriptions and Images, pp. 20-22; Folkerts, Frühe Darstellungen, p. 28.</ref> | ||