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

In the last two sub-sections in Part Two, nos. 71 and 72 (see Note 1), al-Hassar introduces a novel way of representing the subtraction of a fraction, which may be one of the earliest instances of an arithmetical sign. His innovation was to employ the Arabic word إلا (illâ = except for) as what we would now term a minus sign in the representation of multiplications involving fractions (see Note 2); in the Hebrew versions, the Arabic word illâ is translated as אלא. Its usage was not, however, standardised and its import was modulated by how the various juxtapositions were read. For example, in sub-section 71:

On the Multiplication of a Fraction with a Stipulation (בתנאי). [See Note 3.] When it is said to you, multiply three fourths lacking (חסר) a sixth by four fifths except for (אלא) a third, write down the question in this form:

		\[\frac{1}{6}\,\,ill\hat{a}\,\,\frac{3}{4}\]
		\[\frac{1}{3}\,\,ill\hat{a}\,\,\frac{4}{5}\]

This can be read in two different ways (see Note 4):

(i.) Take three fourths minus a sixth and multiply it by four fifths minus a third: \[\left({\frac{3}{4}-\frac{1}{6}}\right)\times\left({\frac{4}{5}-\frac{1}{3}}\right)=\frac{49}{180}.\]

Al-Hassar gives the result for this reading as:

\[\frac{3\,\,\,3\,\,\,2}{5\,\,\,8\,\,\,9}\implies\frac{2}{9}+\frac{3}{8\times 9}+\frac{3}{5\times 8\times 9}=\frac{98}{360}=\frac{49}{180}.\]

(ii.) Take three fourths minus a sixth of three fourths and multiply the result by four fifths minus a third of four fifths:

\[\left({\frac{3}{4}-\frac{1}{6}\cdot\frac{3}{4}}\right)\times\left({\frac{4}{5}-\frac{1}{3}\cdot\frac{4}{5}}\right) =\frac{15}{24}\times\frac{8}{5}=\frac{8}{24}=\frac{1}{3}.\]

The second of the sub-sections, no. 72, is headed, “On the Multiplication of a Fraction of a Number with a Stipulation (בתנאי).” The worked example reads:

When it is said to you, multiply three fourths of five except for (אלא) a sixth by four fifths of three except for (אלא) a fourth, write down the question in this form:

		\[\frac{1}{6}\,\,ill\hat{a}\,\,5\frac{3}{4}\]
		\[\frac{1}{4}\,\,ill\hat{a}\,\,3\frac{4}{5}\]

This can be read in three different ways:

(i.) Take three fourths of five minus a sixth of five and multiply it by four fifths of three minus a fourth of three:

\[\left({\frac{3}{4}\cdot 5-\frac{1}{6}\cdot 5}\right)\times\left({\frac{4}{5}\cdot 3-\frac{1}{4}\cdot 3}\right)=4\frac{65}{80}.\]

Al-Hassar gives the result for this reading as:

\[\frac{0\,\,\,1\,\,\,8}{6\,\,\,8\,\,10}\,4\implies 4\frac{8}{10}+\frac{1}{8\times 10}=4\frac{65}{80}.\]

(ii.) Take three fourths of five minus a sixth of three fourths of five and multiply it by four fifths of three minus a fourth of four fifths of three:

\[\left({\frac{3}{4}\cdot 5-\frac{1}{6}\cdot \frac{3}{4}\cdot 5}\right)\times\left({\frac{4}{5}\cdot 3-\frac{1}{4}\cdot \frac{4}{5}\cdot 3}\right)=5\frac{5}{8}.\]

(iii.) Take three fourths of five minus a sixth and multiply it by four fifths of three minus a fourth:

\[\left({\frac{3}{4}\cdot 5-\frac{1}{6}}\right)\times\left({\frac{4}{5}\cdot 3-\frac{1}{4}}\right)=7\frac{169}{240}.\]

Al-Hassar concludes: “What we have said about the multiplication of fractions should suffice for any person who studies it attentively …. And may God, the guide, show us the way to what is right.”

Note 1. Fol. 18r in the Christ Church manuscript and fols. 41v-42v in the Vatican manuscript.

Note 2. The same Arabic word was used in a similar way some two hundred years later in the Miftāḥ al Ḥ̣isāb, written by the Persian astronomer and mathematician, Jamshīd al-Kāshī (c. 1380–1429) (Saidan, p. 424).

Note 3. “… mit Ausschliessung (istitnâ)” – i.e., “an exception” – in Suter’s translation (1901, p. 28).

Note 4. All images of handwritten Hebrew text are used by permission of Christ Church College Library. The symbol \(\implies\) will continue to indicate replacement by modern notation of al-Hassar's representation of arithmetic operations. Furthermore, we will continue to use the modern symbols, \(-,\times,\cdot,\) and \(=,\) along with parentheses, to clarify our translations.