Part Three of al-Hassar's Kitāb al-Bayān encompasses a variety of operations and computations involving fractions and combinations of integers and fractions. These include the transformation, addition and subtraction of fractions; summations of numerical series; examples of useful commercial and monetary calculations; the famous wheat and chessboard problem; algebraic type computations in which the value of an unknown quantity (šai in Arabic, translated as דבר in Hebrew and meaning ‘a thing’) is sought; and some novel methods for extracting exact and/or approximate square roots of integers and fractional numbers. Here, too, many of the worked examples are accompanied by a scholium (פרק).
Taking the designations in al-Hassar’s treatise as our guide, the text of Part Three can be viewed as comprising five chapters, each composed of headings and sub-headings: in each sub-heading, a specific type of calculation or problem is first delineated and then exemplified by one or more worked examples or solutions. For the source of the chapter or sub-section in the Christ Church, Vatican, and/or Gotha (as translated by Suter in 1901) manuscripts, see the table in Note 1.
3.1 Transforming Fractions (שער פריטה).
3.1.1. Changing a Fraction’s Denominator (five sub-headings).
3.2 Addition.
3.2.1. The Addition of Fractions (nine sub-headings).
3.2.2. Monetary Calculations (five sub-headings).
3.2.3. The Summation of Numerical Series (ten sub-headings).
3.2.4. The Article on Doubling (המאמר בכפול).
If a chessboard were to have grains of wheat placed upon each square such that one was placed on the first square, two on the second, four on the third, and so on; how many grains of wheat would there be on the board at the finish?
3.3 Subtraction (שער ההשלכה).
3.3.1. The Subtraction of Fractions (four sub-headings).
3.3.2. Monetary Calculations (seven sub-headings).
3.3.3. The Reed and Fish Problems.
(i) A reed, standing in the mud by a river bank, has a third of its length in the mud, a quarter in the water and 10 spans (units of length) showing above the water. How long is the reed? [Ans. 24 spans]
(ii) A reed, standing in the mud by a river bank, has a third of its length and two spans in the mud, a quarter of its length and three spans in the water and 10 spans (units of length) showing above the water. How long is the reed? [Ans. 36 spans]
(iii) If a fish’s head is a third of its weight, the tail a quarter and its middle weighs 10 pounds, how heavy is the fish? [Ans. 24 pounds]
3.4 Division (שער החלוקה).
3.4.1. Division of a Smaller Number by a Larger One (twenty-two sub-headings).
3.4.2. Division of a Larger Number by a Smaller One (twenty-six sub-headings).
3.4.3. The Augmenting (חיתום) of Fractions.
Know that this section is of great assistance in the whole of arithmetic and especially in algebra (Hebrew: חיתום; Arabic: el-ğebr; see Note 2).
If it is said to you, by how much must a third be augmented in order to become one? In other words, by what number must a third be multiplied to make one? Answer: By one divided by a third.
3.4.4. The Reduction (ירידה) of a Fraction.
Know that Reduction (Hebrew: ירידה; Arabic: el-ḥatt) is the reverse of Augmenting.
If it is said to you, by how much must one be reduced in order to become a half? In other words, by what must one be multiplied to make a half? Answer: By a half divided by one.
3.5 Extracting the Roots of Integers and Fractions.
3.5.1. Finding the Roots of Integers and Fractions that have Exact (Rational) Square Roots.
3.5.2. Finding the Approximate Roots of Integers and Fractions that do not have Exact (Rational) Square Roots.
A full and detailed description of all the computations is far beyond the remit of this article. Suffice to say, that although al-Hassar employs his new composite notation throughout, the underlying mathematics is not new. Indeed, the topics and the order in which they appear are little changed from that in earlier Arabic arithmetical texts.
The last worked example before the colophons on fol. 31v of the Christ Church codex is “Find the square root of three sevenths.” Al-Hassar's instructions, using modern notation, are as follows:
Multiply \(\frac{3}{7}\) by the square of its denominator: \(49\cdot\frac{3}{7}=21;\) then take the square root of \(21\) which is approximately \(4\frac{3}{5}\) and divide it by the square root of \(49:\,\,4\frac{3}{5}\div 7=\frac{23}{35}.\) The result is a fairly close approximation to the square root of \(\frac{3}{7}.\) (See Note 3.)
The method used here is typical of those al-Hassar employs to find approximate values for irrational square roots. It is also the last item in Suter’s German translation of the Gotha manuscript and is, presumably, where both al-Hassar’s original treatise and Moses ibn Tibbon’s Hebrew translation ended. However, neither the Vatican nor the Christ Church manuscript actually ends at this point.
Note 1. Sections of Part Three of al-Hassar's Kitāb al-Bayān, as outlined above, and their manuscript sources.
| 3.1 | Christ Church, fols. 18v to 19r; Vatican, fols. 42r to 44r; Suter’s Drittes Kapitel, pp. 28-29. |
| 3.2 | Suter’s Viertes Kapitel, pp. 29-34. There is, however, no break or new heading at this point in the Hebrew texts. |
| 3.2.1 | Christ Church, fols. 19r to 20r; Vatican, fols. 44r to 46r; Suter, p. 29. |
| 3.2.2 | Christ Church, fols. 20r to 21r; Vatican fols. 46r to 48v; Suter, pp. 29-31. |
| 3.2.3 | Christ Church fols. 21r to 22v; Vatican fols. 48v to 52r; Suter pp. 31-34. |
| 3.2.4 | Christ Church fol. 22v; Vatican fol. 52r. |
| 3.3 | Suter’s Fünftes Kapitel pp. 34-35. |
| 3.3.1 | Christ Church fols. 22v to 23r; Vatican fols. 52r to 54r. There are 12 sub-sections. |
| 3.3.2 | Christ Church fols. 23r to 24r; Vatican fols. 52r to 54r. |
| 3.3.3 | Christ Church fol. 24r; Vatican fol. 55v. |
| 3.4 | Suter’s Sechtes Kapitel pp. 35-37. |
| 3.4.1 | Christ Church fols. 24r to 26v; Vatican fols. 56r to 61v. The headings of the 22 sub-sections are listed at the end of the section on fols. 26v and 61v, respectively. The text of sub-sections 16 to 21 is missing, however, in the Christ Church manuscript; an entry in the left-hand margin of fol. 26r opposite where they should appear reads בכאן חסר (there are [items] missing here). |
| 3.4.2 | Christ Church fols. 26v to 29r; Vatican fols. 61v to 67v. |
| 3.4.3 | Christ Church fol. 29v; Vatican fols. 68v to 69v; Suter p. 36. |
| 3.4.4 | Christ Church fol. 29v; Vatican fol. 69v; Suter pp. 36-37. |
| 3.5 | Suter’s Siebentes Kapitel pp. 37-39. |
| 3.5.1 | Christ Church fols. 30r to 31r; Vatican fols. 69v to 72r; Suter p. 37. |
| 3.5.2 | Christ Church fols. 31r to 31v; Vatican fols. 72r to 73r; Suter pp. 37-39. |
Note 2. Depending on the context, the Hebrew word חיתום can mean Algebra or, as in the present instance, refer to a procedure for augmenting a fraction. In all the instances, the corresponding Arabic word is el-ğebr.
Note 3. For another example see: Friedrich Katscher's article, "Extracting Square Roots Made Easy: A Little Known Medieval Method," and especially the section, “Al-Hassar’s Description of the Method,” here in MAA Convergence. The example Katscher describes is on fol. 31r of the Christ Church manuscript and fol. 73r of the Vatican text.