Abbreviations, Glyphs, Symbols and Symbolic Calculation

algebra, and Arabic algebra in general, is purely rhetorical B with one exception: in the outgoing twelfth century, an algebraic symbolism was created in the Maghreb B that is, a notation where operations are performed directly on the level of the notation, not via expansion into verbal text. In general, early abbacus algebra was not influenced, but one feature may be inspired: operations with “formal fractions”, that is, fractions where the numerator and/or the denominator are polynomials, not numbers. In the beginning, these polynomials are expressed verbally, showing that symbolic syntax does not need a symbolic lexicon. Sometimes, however, and in the fifteenth century often though never systematically, powers of the unknown as well as addition, subtraction and root taking are expressed by means of letter-abbreviations or other glyphs. They are used as mere abbreviations in the running text, but also serve as symbols proper in formal fractions and in schemes serving addition, subtraction and multiplication of polynomials. When algebra was taken over in German lands in the later fifteenth century, it was quite eclectic, but very soon a coherent system was used systematically by virtually everybody (Schreyber’s notation by means of exponents, like Chuquet’s similar invention, was adopted by nobody else). The system was still used in Clavius’s Latin Algebra from 1609, and thus in the teaching of Jesuit schools. French algebraic writers had no shared system, illustrating that the were French algebraic writers in the sixteenth century but no French algebraic school or tradition.