In each of the remaining seventyone sections of Part Two of alHassar's Kitāb alBayān, a different calculation involving the multiplication of a fraction (or fractions) is exemplified: an integer by a simple fraction; a composite fraction by a mixed fraction; a composite fraction by another composite fraction; a fraction of a mixed fraction (simple or composite) by an integer, a simple fraction, a composite fraction or another mixed fraction (simple or composite); and so on. Each of these “how to” worked examples starts with the words “When it is said to you (כשיאמר לך) …” In a number of instances, alHassar shows how to check the answer arrived at by the technique of “casting out” and some of the more advanced or complex examples are also followed by a scholium (פרק). Suter remarks: “These sections are the richest of all known examples of fractions in Arabic arithmetic books, so extensive that it appears tiring, unwieldy and confusing for the practitioner” (1901, p. 24). Indeed, at times the work reads almost like a manual or recipe book. Not surprisingly, there are numerous copyist errors in both Hebrew versions, many of which were later noticed and corrected in the margins.
The following are a representative sample of these worked examples.
Example 1. "On the multiplication of a fraction by an integer." (See Note 1.)
When it is said to you, multiply five sixths by ten. Place the five sixths on one row and the ten on the row below, in this way:


\[\genfrac{}{}{0pt}{}{\frac{5}{6}}{10}\]

Multiply the five above the six by the ten, \(5\times 10=50,\) and divide the result by six, \(50\div 6 = 8\frac{2}{6}.\) The answer is eight and two sixths.
Example 2. "On the multiplication of a composite fraction by an integer." (See Note 2.)
When it is said to you, multiply a fifth and a half of a fifth by twelve. Place the fifth and a half of a fifth on one row and the twelve on the row below, in this way:


\[\genfrac{}{}{0pt}{}{\frac{1\,\,1}{2\,\,5}}{12}\]

Multiply the one above the five by two, \(1\times 2 = 2,\) and add to this the one that is above the two: \(1+2=3.\) Multiply the three by twelve, \(3\times 12=36,\) and divide the result by the denominators; i.e., by two, \(36\div 2=18,\) followed by five, \(18\div 5 = 3\frac{3}{5}.\) The answer is three and three fifths.
Example 3. "On the multiplication of a simple fraction by another simple fraction." (See Note 3.)
When it is said to you, multiply seven eighths by nine tenths. Place the seven eighths on one row and the nine tenths on a row below, in this way:


\[\genfrac{}{}{0pt}{}{\frac{7}{8}}{\frac{9}{10}}\]

Multiply the seven above the eight by the nine above the ten, \(7\times 9 = 63,\) and divide the result by the denominators; i.e., by eight, \(63\div 8=7\frac{7}{8},\) followed by ten, \(7\frac{7}{8}\div 10 = \frac{7}{10}+\frac{7/8}{10}.\) The answer is seven tenths and seven eighths of a tenth:

\[\frac{7\,\,\quad {7}}{8\quad {10}}\implies \frac{7}{10}+\frac{7}{8\times {10}}=\frac{63}{80}.\]

Example 4. "On the multiplication of a composite fraction by a simple fraction." (See Note 4.)
When it is said to you, multiply six sevenths and a third of a seventh by eight ninths. Place the six sevenths and a third of a seventh on one row and the eight ninths on a row below, in this way.


\[\genfrac{}{}{0pt}{}{\frac{1\,\,\,6}{3\,\,\,7}}{\frac{8}{9}}\]

Multiply the six above the seven in the upper multiplicand by the three below the line, \(6\times 3=18,\) and add the product to the one above the line making nineteen. Multiply this by the eight in the lower multiplicand, \(19\times 8=152.\) Divide the one hundred and fiftytwo by the denominators of the two multiplicands; i.e., by three, \(152\div 3=50\frac{2}{3},\) followed by seven, \(50\frac{2}{3}\div 7 = 7+\frac{1\, 2/3}{7},\) followed by nine: \(\left(7+\frac{1\,2/3}{7}\right)\div 9=\frac{7}{9}+\frac{1/7}{9}+\frac{\frac{2/3}{7}}{9}.\) This gives seven ninths and a seventh of a ninth and twothirds of a seventh of a ninth:


$$\frac{2\,\,\,1\,\,\,7}{3\,\,\,7\,\,\,9}$$

Example 5. "On the multiplication of the sum of two fractions by an integer." (See Note 5.)
When it is said to you, multiply the sum of three quarters and four fifths by fifteen. Place the three quarters and four fifths in a row with the fifteen below them, in this form:


$$\genfrac{}{}{0pt}{}{\frac{4}{5}\,\,\,\frac{3}{4}}{15}$$

Starting with the row of fractions, multiply the three (the numerator) that is over the four by the five (the denominator) under the four \(\left(3\times 5=15\right)\) and the four (the numerator) that is over the five by the four (the denominator) under the three \(\left(4\times 4=16\right).\) Adding the two products gives thirtyone: \(15+16=31.\)
Multiplying thirtyone by fifteen gives four hundred and sixtyfive: \(31\times 15=465.\) Dividing this by the denominators of the two fractions, four followed by five, gives twentythree and one fifth and a quarter of a fifth: \(465\div 4=116\frac{1}{4}\) then \[116\frac{1}{4}\div 5 = 23\frac{1}{5}+\frac{1/4}{5}\impliedby\frac{1\,\,\,1}{4\,\,\,5}\,23.\]
Example 6. "On the multiplication of a fraction of one integer by a different fraction of another integer." (See Note 6.)
When it is said to you, multiply five sixths and half a sixth of eight by eight ninths and a fifth of a ninth of twelve:


\[8\,\frac{1\,\,\,5}{2\,\,\,6}\]



\[12\,\frac{1\,\,\,8}{5\,\,\,9}\]

Taking the top row first, multiply the five over the six by the two in the denominator, \(5\times 2 = 10\); add the product to the one above the line, \(10+ 1 = 11\); and multiply the sum by eight, \(11\times 8 = 88.\)
Moving to the second row, multiply the eight over the nine by the five in the denominator, \(8\times 5 = 40\); add the product to the one above the line, \(40+ 1 = 41\); and multiply the sum by twelve, \(41\times 12 = 492.\)
Multiply the eightyeight by the four hundred and ninetytwo, \(88\times 492 = 43296.\) Divide the fortythree thousand two hundred and ninetysix by the denominators of the two composite fractions, two, five, six, and nine, as follows: \[43296\div 2 = 21648\frac{0}{2}=21648,\quad 21648\div 5 = 4329\frac{3}{5},\] \[4329\frac{3}{5}\div 6 = 721\frac{3}{6}+\frac{3/5}{6},\] \[\left(721\frac{3}{6}+\frac{3/5}{6}\right)\div 9 = 80\frac{1}{9}+\frac{3/6}{9}+{\frac{\frac{3/5}{6}}{9}}.\]
This gives eighty and a ninth and three sixths of a ninth and three fifths of a sixth of a ninth and zero halves of a fifth of a sixth of a ninth [we did not carry the \(0\) through in the work above]:

\[\frac{0\,\,\,3\,\,\,3\,\,\,1}{2\,\,\,5\,\,\,6\,\,\,9}\,80\implies80\frac{48}{270}=80\frac{8}{45}.\]

Dividing by the denominators in the reverse order – nine, six, five and two – produces a different but equivalent composite fraction; i.e., one with the same value: \[\frac{6\,\,\,4\,\,\,1\,\,\,0}{9\,\,\,6\,\,\,5\,\,\,2}\,80\implies80\frac{96}{540}=80\frac{8}{45}.\]
Because of the different ways in which a juxtaposition could be understood – addition in the case of two fractions or taking that fraction of a number when the fraction (simple or composite) is to the right of the number – ambiguities could arise, especially where complex calculations were concerned. Example 7 illustrates this.
Example 7. "On the multiplication of a fraction and an integer and two simple fractions and a whole number and a fraction by a similar expression." (See Note 7.)
When it is said to you, multiply three fourths of five and a half and five sixths of three and two fifths by two thirds of four and a seventh and three eighths of two and three elevenths, write down the question in this form [reading from right to left]:

\[\frac{2}{5}\,3\,\frac{5}{6}\,\frac{1}{2}\,5\,\frac{3}{4}\] 

\[\frac{3}{11}\,2\,\frac{3}{8}\,\frac{1}{7}\,4\,\frac{2}{3}\] 
The answer is \[\frac{0\,\,\,6\,\,\,5\,\,\,5\,\,\,5\,\,\,1}{10\,\,7\,\,\,8\,\,\,8\,\,\,9\,\,11}\,25\implies25\frac{6593}{44352}=25.1486516955 \dots.\]
To arrive at this result, alHassar had read each of the symbolic representations as the sum of two parts, each a fraction of a mixed fraction. In modern notation, this gives:
\[{\left({\frac{3}{4}\times5\frac{1}{2}+\frac{5}{6}\times3\frac{2}{5}}\right)}\times{\left({\frac{2}{3}\times4\frac{1}{7}+\frac{3}{8}\times2\frac{3}{11}}\right)}=25.1486516955 \dots.\]
However, because of the different ways in which a juxtaposition can be understood other readings were also possible. AlHassar was clearly aware of this and so he continues:
This question can be read in different ways. For example, take the five sixths from the second part of [the expression in the top row] and join it to the first part, whereupon the first part becomes taking a fraction of an integer and two fractions. [See Note 8.] This gives: \[\frac{2}{5}\,3\,\frac{5}{6}\,\frac{1}{2}\,5\,\frac{3}{4}\implies\frac{3}{4}\left(5+\frac{1}{2}+\frac{5}{6}\right)+3\frac{2}{5}.\] Furthermore, the expression does not have to be read as the sum of two parts; it could just as well be read as the sum of three, four, five or even more parts.
In the scholium that follows, alHassar provides other examples of how this expression could be read; for instance, in two parts: \[\frac{2}{5}\,3\,\frac{5}{6}\,\frac{1}{2}\,5\,\frac{3}{4}\implies\frac{3}{4}\times 5+\left(\frac{1}{2}+\frac{5}{6}\right)\times3\frac{2}{5},\] or in three parts: \[\frac{2}{5}\,3\,\frac{5}{6}\,\frac{1}{2}\,5\,\frac{3}{4}\implies\frac{3}{4}+\left(5\frac{1}{2}+\frac{5}{6}\right)+3\frac{2}{5}\] or \[\frac{2}{5}\,3\,\frac{5}{6}\,\frac{1}{2}\,5\,\frac{3}{4}\implies\frac{3}{4}\times 5+\frac{1}{2}\times\frac{5}{6}+3\frac{2}{5}.\]
Arithmetical texts were all handwritten at the time which would only have added even more ambiguities to those already inherent in the absence of standardised signs and the reliance on juxtapositions. In alUqlīdisī’s treatise, groups of numbers are in some instances surrounded by lines apparently to separate them from the writing around them, but this is not systematically adhered to. Addition and multiplication are likewise indicated in places by the insertion of three dots, ∴ (the modern handwritten “therefore sign”), between the numbers, though, as often as not, they too are omitted (Saidan, p. 423).
Note 1. Subsection 2: fol. 7v in the Christ Church manuscript and fol. 14v in the Vatican manuscript. All images of handwritten Hebrew text are used by permission of Christ Church College Library. Throughout this webpage, the symbol \(\implies\) will indicate replacement by modern notation of alHassar's representation of arithmetic operations. Furthermore, we will use the modern symbols, \(+,\times,\div,\) and \(=,\) along with parentheses, to clarify our translations.
Note 2. Subsection 3: fol. 7v in the Christ Church manuscript and fol. 15r in the Vatican manuscript. There is a copyist error in the wording of the example in the Christ Church manuscript. It reads “multiply two fifths …” and not “a fifth” whereas the answer given requires that it be the latter. The symbolic representation of the calculation is, however, correct as is the text in the Vatican manuscript.
Note 3. Subsection 28: fol. 11r in the Christ Church manuscript and fol. 24v in the Vatican manuscript.
Note 4. Subsection 29: fol. 11r in the Christ Church manuscript and fol. 25r in the Vatican manuscript.
Note 5. Subsection 5: fol. 8r in the Christ Church manuscript and fol. 15v in the Vatican manuscript. A copyist error in the Christ Church manuscript gives the final answer in this example as “twentythree and two fifths and a quarter of a fifth.”
Note 6. Subsection 39: fol. 12v in the Christ Church manuscript and fol. 28r in the Vatican manuscript.
Note 7. Subsection 58: fol. 15v in the Christ Church manuscript and fol. 35r in the Vatican manuscript.
Note 8. AlHassar adds that a calculation of this type has already been exemplified in subsection fortyfour.