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