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Mathematical Treasure: Italian Arithmetica Pratica

Frank J. Swetz (The Pennsylvania State University)

Entries from an anonymous Italian Arithmetica Pratica (1575) stress the use of “Hindu Arabic” numerals and their algorithms. The manuscript shown above was handwritten with a quill pen in humanistic cursive. It is unique and perhaps was intended for personal reference or for teaching notes in a reckoning school. The heading at the top of the lefthand page shown above denotes “Definitions of the operations [atti] of arithmetic.” Under the heading, the author began a discussion of the first operation, “enumeration” or the use of the new numbers. The ten basic figures of the Hindu-Arabic (or Indo-Arabic) system, the digits [digitti], are illustrated; then the articles [articolo] or multiples of 10 are considered; and finally the formation of composites [composito] from units and tens (e.g. 12, 53, 178, …) is explained. On the lefthand page, a table illustrates the use of place value, expanding numbers from units [numerero] up to centinara de mighara de mighara de milleoni, literally hundreds of thousands of thousands of millions. A standardization of larger numbers had not yet been adopted. The author then told the reader that this process continues to infinity. A noteworthy feature of this page is that on the largest number written, the author denoted place value groupings using a check mark.

On the lefthand page above, we find the conclusion of the previous discussion on a division operation. Little indication is given as to the method employed, but it seems very likely that it was the “a danda” algorithm then popular in Italy. Today we recognize this as our common downward technique of performing long division. Here, the author demonstrated a method of checking the correctness of the resulting quotient; namely, multiplication! The problem 5487 \(\div\) 385 had resulted in the answer 14 97/385, and, at the lower right, the author employed multiplication to validate the answer. The numbers contained within “crosses” are part of a “casting out nines” process also employed. On the following page, we find Section 9 of this work, “Galley Division.” In the lower right of this page we find 3464 \(\div\) 343, using the galley algorithm. The result is 10 with remainder 34. (For a more detailed examination of these techniques, see Frank Swetz, Capitalism and Arithmetic: The New Math of the 15th Century, pp. 211-221.)

The discussion continues on the following pages, shown above, where we see two demonstrations of galley division performed: 844849 \(\div\) 7564 and 949434 \(\div\) 3994. The answers – respectively, 111 remainder 5245 and 237 remainder 2856 – are validated by multiplication.

These images are provided courtesy of the Beinecke Rare Book and Manuscript Library, Yale University. You may use them in your classroom; all other uses require permission from the Beinecke Rare Book and Manuscript Library. The Mathematical Association of America is pleased to cooperate with the Beinecke Library and Yale University to make these images available to a larger audience.

Index to Mathematical Treasures

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Italian Arithmetica Pratica," Convergence (April 2014)