In the Vatican manuscript of al-Hassar's *Kitab al Bayan*, this worked example of approximating a square root is followed by a further nine examples and exercises, only after which the colophon appears on fol. 76r (see Note 1). In the Christ Church manuscript, the order is reversed and the same nine examples and exercises appear immediately after the colophons, starting at the top of the left-hand column of fol. 31v and continuing up to fol. 33r (see **Figure 2**). They do not, however, appear in Suter’s 1901 translation nor, by implication, in the Gotha manuscript.

The presence of these items at the end of the Vatican and Christ Church manuscripts is anomalous. Judging by their subject matter, they really belong much earlier in the text. The first seven are arithmetical calculations involving fractions and belong in the relevant sections of Parts Two and Three. The subject of the eighth item is a long multiplication and that of the ninth is the technique of “casting out nines” used to check the result of a multiplication, both of which really belong under the heading “Multiplication” in Part One. A method for carrying out long multiplications does in fact appear under that heading but it is very different from the one presented here at the end of the Vatican and Christ Church manuscripts.

The method of long multiplication from Part One is the one that was employed when working with a sandboard and involves the deletion and rewriting of numerals at each step; this is easily done on such a device though it can be somewhat confusing and prone to errors. The worked example given is the multiplication of \(43\) by \(76\) and reads as follows (Note 2).

When it is said to you, multiply \(43\) by \(76.\) Put the \(43\) in one row and write the \(76\) in the row below, in such a way that the units column of the second number is under the tens column of the first, in the following manner: \[\genfrac{}{}{0pt}{}{\phantom{7}43}{76\phantom{3}}\]

Now multiply the last digit of the upper number with the first of the lower, i.e., \(4\) with \(7\); this gives \(28.\) Place the \(8\) above the \(7\) in the top line and the \(2\) to its left: \[\genfrac{}{}{0pt}{}{2843}{\phantom{2}76\phantom{3}}\]

Now multiply the same \(4\) by the \(6\) below it, which gives \(24\); superimpose the \(4\) on the upper line (which leaves it unchanged) and add the \(2\) to the \(8\) above the \(7,\) which gives \(10\); delete the \(8\) and put in its place the zero from the ten; add the \(2\) to this and the one from the ten which gives \(3\); delete the \(2\) and put the \(3\) in its place: \[\genfrac{}{}{0pt}{}{2843}{\phantom{2}76\phantom{3}}\rightarrow\genfrac{}{}{0pt}{}{2043}{\phantom{2}76\phantom{3}}\rightarrow\genfrac{}{}{0pt}{}{3043}{\phantom{2}76\phantom{3}}\]

Move the lower number one place to the right so that the \(6\) is under the \(3\) and the \(7\) under the \(4\); then multiply the \(3\) from the upper number by the \(7\) below which gives \(21\); add to this the \(4\) from the upper row which gives \(25\); now delete the \(4\) and set in its place the \(5\); put the \(2\) in the place of zero: \[\genfrac{}{}{0pt}{}{3043}{\phantom{30}76}\rightarrow\genfrac{}{}{0pt}{}{3253}{\phantom{32}76}\]

Multiply the \(3\) by the \(6\) below it which gives \(18\); replace the \(3\) with the \(8\) and add the \(1\) to the \(5\) which gives \(6\); delete the \(5\) and put the \(6\) in its place: \[\genfrac{}{}{0pt}{}{3253}{\phantom{32}76}\rightarrow\genfrac{}{}{0pt}{}{3268}{\phantom{32}76}\] So the result of the multiplication is \(3268.\)

This method became obsolete with the introduction of paper from the Islamic world into medieval Europe and, by the fifteenth century, when the Vatican and Christ Church manuscripts were written, there was clearly a need for a better technique (algorithm), especially for multiplying large numbers. (In fact, Suter (1901, p. 17) expresses surprise that there is no reference to the Lattice Multiplication or to any other method in the Gotha manuscript, attributing this to al-Hassar’s continued use of a sandboard or to copyists’ omissions.) Accordingly, the eighth item in the addendum to the Vatican and Christ Church manuscripts begins: “On the multiplication of integers by another method that is not from the book”: the “book” is presumably al-Hassar’s treatise. What follows is the now familiar pen and paper method of long multiplication. Two worked examples are given: squaring twenty-two, \(22\times 22 = 484,\) and multiplying four hundred and thirty-two by three hundred and twenty-three, \(432\times 323 = 139536.\)

This is as far as the Vatican manuscript goes, but the copyist of the Christ Church manuscript added an example of the lattice (gelosia or sieve) technique for the multiplication of large numbers; in this instance, to calculate the square of the number \(56742\) (see **Figure 9**).

**Figure 9.** The 5 x 5 grid on fol. 33r of the Christ Church manuscript for calculating the square of \(56742,\) annotated to show how the multiplication is carried out. The multiplicand is across the top of the lattice and the multiplier down the right side; in this example both are the same. A product is calculated for each cell by multiplying the digit at the top of the column and the digit at the right of the row: the tens digit of the product is placed above the diagonal that passes through the cell, and the units digit below. After filling all the cells, the digits in each diagonal are summed, starting from the bottom right cell and the units digit of the sum is entered below the adjacent column, as shown; if the sum is greater them ten, the tens are carried into the next diagonal (written outside the grid at the bottom of each diagonal). After summing all the diagonals, the answer, \(3219654564,\) is read off from top to bottom on the left and continuing from left to right below the grid. (The copyist explained and carried out the procedure correctly but, for some unexplained reason, he entered an incorrect answer, \(22106564,\) in the text.) (Image used by permission of Christ Church College Library)

Fibonacci is often credited with introducing this technique into Europe but this is incorrect. What he described in Chapter 3 of his *Liber Abaci* is a related technique known as “chessboard multiplication” that works differently. The cells are not divided diagonally and only the lower-order digit is entered in each cell. The earliest extant example of the lattice technique in Europe is in a 14th century Latin manuscript, *Tractatus de minutis philosophicis et vulgaribus (A Treatise on Small Measurements, Scientific and General; *see Note 3). It also appears in the earliest printed mathematics book, the *Treviso Arithmetic*, published in the town of that name in 1478, two years after the date of the Christ Church manuscript.

By the fifteenth century, the abacus and Roman numerals that had been in common use for more than a thousand years, were being replaced across Europe by the algorithm and Gobar-based numerals. The transition was slow in coming; the “abacists” would not surrender to the “algorists” without a fight (see **Figure 10**).

**Figure 10.** Abacist vs. Algorismist by Gregor Reisch, *Margarita philosophica,* Strasbourg, 1504. The woodcut shows Arithmetica observing an algorist and an abacist. She appears to favour the algorist; her dress is adorned with Gobar-based numerals and she is looking approvingly in his direction. (See Note 4. This image is provided courtesy of the Columbia University Libraries and may be used for instructional purposes; for all other uses, please contact the Columbia University Libraries. For five more images from the *Margarita philosophica* (or *Pearl of Wisdom*), see the article, “Mathematical Treasures – *Margarita philosophica* of Gregor Reisch,” here in *MAA Convergence*.)

The advantages of calculating with pen and paper were not always immediately apparent. The abacists’ archaic modes of doing arithmetic would, however, ultimately prove inadequate in the expanding mercantile economies of the Renaissance and this, together with the falling price of paper and the concomitant spread of printing, would ultimately lead to the triumph of the algorists in the sixteenth century.

**Note 1**. In the Vatican manuscript, there is also an appendix on the subject of the extraction of cube roots that is found in neither the Christ Church codex 189 nor Suter’s translation.

**Note 2.** This example occurs at Christ Church manuscript, fol. 3v and Vatican manuscript, fol. 7v. For another example of this method, see: *Episodes in the Mathematics of Medieval Islam,* by J.L. Berggren, Springer, New York (1986), p. 34.

**Note 3.** Bodleian Library, Oxford, MS Digby 190, fol. 75r.

**Note 4.** For an amusing demonstration of their relative advantages, see Richard Feynman’s “The Abacist versus the Algorist”: http://press.princeton.edu/chapters/i9662.pdf.