Lattice Multiplication - History

History

Lattice multiplication has been used historically in many different cultures. It is not known where it arose first, nor whether it developed independently within more than one region of the world. The earliest recorded use of lattice multiplication:

- in Arab mathematics was by Ibn al-Banna' al-Marrakushi in his Talkhīṣ a‘māl al-ḥisāb, in the Maghreb in the late 13th century

- in European mathematics was by the unknown author of a Latin treatise in England, Tractatus de minutis philosophicis et vulgaribus, c. 1300

- in Chinese mathematics was by Wu Jing in his Jiuzhang suanfa bilei daquan, completed in 1450.

The mathematician and educator David Eugene Smith asserted that lattice multiplication was brought to Italy from the Middle East. This is reinforced by noting that the Arabic term for the method, shabakh, has the same meaning as the Italian term for the method, gelosia, namely, the metal grille or grating (lattice) for a window.

It is sometimes erroneously stated that lattice multiplication was described by Muḥammad ibn Mūsā al-Khwārizmī (Baghdad, c. 825) or by Fibonacci in his Liber Abaci (Italy, 1202, 1228). In fact, however, no use of lattice multiplication by either of these two authors has been found. In Chapter 3 of his Liber Abaci, Fibonacci does describe a related technique of multiplication by what he termed quadrilatero in forma scacherii (“rectangle in the form of a chessboard”). In this technique, the square cells are not subdivided diagonally; only the lowest-order digit is written in each cell, while any higher-order digit must be remembered or recorded elsewhere and then "carried” to be added to the next cell. This is in contrast to lattice multiplication, a distinctive feature of which is that the each cell of the rectangle has its own correct place for the carry digit; this also implies that the cells can be filled in in any order desired. Swetz compares and contrasts multiplication by gelosia (lattice), by scacherii (chessboard), and other tableau methods.

Other notable historical uses of lattice multiplication include:

- Jamshīd al-Kāshī’s Miftāḥ al-ḥisāb (Samarqand, 1427), in which the numerals used are sexagesimal (base 60), and the grid is turned 45 degrees to a “diamond” orientation

- the Arte dell’Abbaco, an anonymous text published in the Venetian dialect in 1478, often called the Treviso Arithmetic because it was printed in Treviso, just inland from Venice, Italy

- Luca Pacioli’s Summa de arithmetica (Venice, 1494)

- the Indian astronomer Gaṇeśa’s commentary on Bhāskara II’s Lilāvati (16th century).