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Lattice multiplication, also known as the Italian method, Chinese method, Chinese lattice, gelosia multiplication, sieve multiplication, shabakh, diagonally or Venetian squares, is a method of multiplication that uses a lattice to multiply two multi-digit numbers. It is mathematically identical to the more commonly used long multiplication algorithm, but it breaks the process into smaller steps, which some practitioners find easier to use.
The method had already arisen by medieval times, and has been used for centuries in many different cultures. It is still being taught in certain curricula today.
As an example, consider the multiplication of 58 with 213. After writing the multiplicands on the sides, consider each cell, beginning with the top left cell. In this case, the column digit is 5 and the row digit is 2. Write their product, 10, in the cell, with the digit 1 above the diagonal and the digit 0 below the diagonal (see picture for Step 1).
If the simple product lacks a digit in the tens place, simply fill in the tens place with a 0. After all the cells are filled in this manner, the digits in each diagonal are summed, working from the bottom right diagonal to the top left. Each diagonal sum is written where the diagonal ends. If the sum contains more than one digit, the value of the tens place is carried into the next diagonal (see Step 2). Numbers are filled to the left and to the bottom of the grid, and the answer is the numbers read off down (on the left) and across (on the bottom). In the example shown, the result of the multiplication of 58 with 213 is 12354.
The mathematician and educator David Eugene Smith asserted that lattice multiplication was brought to Italy from the Middle East.Smith, David Eugene, History of Mathematics, Vol. 2, “Special Topics of Elementary Mathematics” (New York: Dover, 1968). 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).The original 1202 version of Liber Abaci is lost. The 1228 version was later published in its original Latin in Boncompagni, Baldassarre, Scritti di Leonardo Pisano, vol. 1 (Rome: Tipografia delle Scienze Matematiche e Fisiche, 1857); an English translation of the same was published by Sigler, Laurence E., Fibonacci’s Liber Abaci: A Translation into Modern English of Leonardo Pisano’s Book of Calculation (New York: Springer Verlag, 2002). 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 each cell of the rectangle has its own correct place for the carry digit; this also implies that the cells can be filled in any order desired. SwetzSwetz, Frank J., Capitalism and Arithmetic: The New Math of the 15th Century, Including the Full Text of the Treviso Arithmetic of 1478, Translated by David Eugene Smith (La Salle, IL: Open Court, 1987), pp. 205-209. compares and contrasts multiplication by gelosia (lattice), by scacherii (chessboard), and other tableau methods.
Other notable historical uses of lattice multiplication include:
The same principle described by Matrakçı Nasuh underlay the later development of the calculating rods known as Napier's bones (Scotland, 1617) and Genaille–Lucas rulers (France, late 1800s).
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