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Moses ibn Tibbon’s Hebrew Translation of al-Hassar's Kitab al Bayan - Multiplication of Fractions

Jeremy I. Pfeffer (Hebrew University of Jerusalem)

The focus of Part Two of al-Hassar's Kitāb al-Bayān is on the multiplication of simple, mixed, complex and compound fractions (see Note 1). It comprises seventy-two sub-sections or headings (שערים), in the first of which the new notation by which a horizontal bar (vinculum) separates the numerator and denominator of a fraction is formally presented along with its basic usages and applications (see Note 2).

The first fraction is a half, followed by a third, a fourth, a fifth … and to depict a half, write a two and draw a line above it, and over the line write a one thus \(\frac{1}{2}\) … and for a third, a three and a one over it thus \(\frac{1}{3}\) … and for two thirds, write a two in place of the one \(\frac{2}{3},\) and so on.

All simple fractions whose denominators are ten or less, or a prime number greater than ten, are represented in this way; for example, a quarter, five sixths, an eleventh, two thirteenths, six nineteenths: \[ \frac{1}{4},\,\,\frac{5}{6},\,\,\frac{1}{11},\,\,\frac{2}{13},\,\,\frac{6}{19},\] respectively, in modern Arabic numerals. The Hindus (Indians) wrote the denominator under the numerator but without the horizontal bar. The horizontal bar first appeared in Europe in Fibonacci’s Liber Abaci (1202), an innovation that he took from Arab sources.

Al-Hassar next moves on to the symbolic representation of those simple fractions whose denominators are greater than ten and can be factorised. These are designated "by two names" – i.e., as a fraction of a fraction. For example: a twelfth (a half of a sixth),

  \[\frac{1\quad {\phantom x}}{2\quad 6}\implies \frac{0}{6}+\frac{1}{2\times 6}=\frac{1}{12}\]

(see Note 3); a twenty-eighth (a quarter of a seventh),

  \[\frac{1\quad {\phantom x}}{4\quad 7}\implies \frac{0}{7}+\frac{1}{4\times 7}=\frac{1}{28};\]

ten fifteenths or two thirds (three fifths and a third of a fifth),

  \[\frac{1\quad {3}}{3\quad 5}\implies \frac{3}{5}+\frac{1}{3\times 5}=\frac{10}{15}=\frac{2}{3};\]

twenty-one twenty-fourths or seven eighths (five sixths and a quarter of a sixth),

  \[\frac{1\quad {5}}{4\quad 6}\implies \frac{5}{6}+\frac{1}{4\times 6}=\frac{21}{24}=\frac{7}{8};\]

and forty-seven one hundred and forty-threes (four thirteenths and three elevenths of a thirteenth),

  \[\frac{3\,\,\,\,\,\,\, {4}}{11\,\,\,\,\, 13}\implies \frac{4}{13}+\frac{3}{11\times 13}=\frac{47}{143}.\]

Al-Hassar’s representation is similar for fractions designated by three or more names; i.e., a fraction of a fraction of a fraction, or a fraction of a fraction of a fraction of a fraction, and so on. For example,

  \[\frac{1\quad 5\quad 7\,\,\,\, \phantom{1} 9}{2\quad 6\quad 9\,\,\,\,\, {11}} \]

reads nine elevenths and seven ninths of an eleventh and five sixths of a ninth of an eleventh and half of a sixth of a ninth of an eleventh; i.e., ninety-seven parts of one hundred and eight: \[\frac{1\quad {5}\quad {7}\quad {9}}{2\quad 6\quad {9}\quad {11}}\implies \frac{9}{11}+\frac{7}{9\times 11}+\frac{5}{6\times 9\times 11}+\frac{1}{2\times 6\times 9\times 11}=\frac{1067}{1188}=\frac{97}{108}.\]

Turning to other applications of his new notation, al-Hassar enjoins that the terms in a row of unrelated simple fractions – for example, three quarters, four fifths, five sixths, six sevenths and ten elevenths – should be clearly separated from one another: this is realised in the Christ Church manuscript (fol. 7v) by means of vertical strokes,


and in the Vatican manuscript by blank spaces (see Note 4), \[\frac{10}{11}\,\,\frac{6}{7}\,\,\frac{5}{6}\,\,\frac{4}{5}\,\,\frac{3}{4}.\] Numbers comprising an integer and a simple or composite fraction (mixed fractions) are denoted with the integer to the right of the fraction. (Fibonacci followed this Arab practice of placing the fraction to the left of the integer.) For example (see Note 5), eight and two sixths is written with \(\frac{2}{6}\) to the left of \(8,\)

  \[\frac{2}{6}\,8\implies 8\,\frac{1}{3},\]

and one and a seventh and a third of a seventh is written with \(1\) to the right of the fraction,

  \[\frac{1\quad {1}}{3\quad 7}\,1\implies 1+\frac{1}{7}+\frac{1}{3}\cdot \frac{1}{7}=1\,\frac{4}{21}.\]

Conversely, when the fraction (simple or composite) is to the right of a number, it indicates taking that fraction of the number. For example, three fourths of a fifth of eight (three twentieths of eight) is written as follows:


\[8\,\frac{3\quad {0}}{4\quad 5}\implies \left(\frac{0}{5}+\frac{3}{4\times 5}\right)\times 8 ={\frac{3}{20}}\times 8=\frac{24}{20}=\frac{6}{5}.\]

Standardised notations such as the now familiar arithmetical signs (+, –, ×, ÷, etc.) only came into use in the late sixteenth and early seventeenth centuries with the spread of printed mathematical books. In their absence, arithmetical operations were often indicated by a juxtaposition. Thus, for example, the simple addition of two fractions is represented in al-Hassar’s treatise by placing their symbolic representations side by side:


A notation could, however, have more than just one usage. For example, the constituent fractions represented by the sequence of numerators and denominators in the composite fraction notation are linked together by the conjunction "and" (= plus). In this way, the composite fraction at left (below) is read (from right to left) as three fourths and four fifths of a fourth and five sixths of a fifth of a fourth: \[\frac{5\quad 4\quad 3}{6\quad 5\quad 4}\implies \frac{3}{4}+\frac{4}{4\times 5}+\frac{5}{4\times 5\times 6}.\]

Al-Hassar now adds that this notation can have an alternate usage; namely, that it can also be read as a sequence of simple fractions without the conjunction ‘and’ (+), such that each fraction is that part of the following fractions. Taking the above example, this gives (reading from right to left) three fourths of four fifths of five sixths; i.e., in modern notation, the product \[\frac{3}{4}\times \frac{4}{5}\times \frac{5}{6}.\] The Andalusian mathematician, al-Qalasadi (1412-1486), differentiated between the two usages by inserting a vertical line between the individual fractions in the latter (Suter 1901, p. 27): \[\frac{5\,\vert\, 4\,\vert\, 3}{6\,\vert\, 5\,\vert\, 4}.\] This latter usage is exercised in al-Hassar's Kitāb al-Bayān in sub-sections 60 to 69 of Part One: Christ Church manuscript fol. 16r and Vatican fols. 36v to 42v.

Note 1. This section occupies fols. 7r–18r in the Christ Church manuscript, 14r–42r in the Vatican manuscript, and pp. 23–28 in Suter’s translation.

Note 2. Sub-section 1 appears on fols. 7r and 7v in the Christ Church manuscript and fols. 14r and 14v in the Vatican manuscript.

Note 3. Once again, the symbol \(\implies\) indicates replacement of al-Hassar's representation of arithmetic operations by modern notation. All images of handwritten Hebrew text are used by permission of Christ Church College Library.

Note 4. Vatican manuscript, fol. 14v. There is some confusion here. Although the text in the Vatican manuscript has “…four fifths…,” the copyist entered a 2 under the 4 in the symbolic representation and not a 5. Furthermore, the words “six sevenths” do not appear in the text though the fraction is included in the symbolic representation. And to compound it all, neither the words nor the fraction appear in the Schoenberg manuscript (fol. 25v), which also has spaces between the terms.

Note 5. The two examples that follow occur in sub-sections 2 and 4, respectively: fol. 7v in the Christ Church manuscript and fols. 15r and 15v in the Vatican manuscript.

Jeremy I. Pfeffer (Hebrew University of Jerusalem), "Moses ibn Tibbon’s Hebrew Translation of al-Hassar's Kitab al Bayan - Multiplication of Fractions," Convergence (May 2017)