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Mathematical Treasure: Borghi’s Best Commercial Arithmetic

Frank J. Swetz (The Pennsylvania State University)

Piero Borghi (1424-1494) was a Venetian reckoning master. His book, Qui comenza la nobel opera de arithmetic (1484), is considered the best commercial arithmetic of the 15th century. Actually, the book has no formal title; the title used in the preceding sentence is the opening phrase of the text itself, and can be translated as:

Here begins a noble work on arithmetic, in which all things concerning business practice are treated. . . .

Borghi stated that the book was written to “prepare merchants,” which was the purpose of most Italian arithmetics of this period. The text was written and published in Venice, the center of Eastern trade. Later, this book was simply called “Borghi’s Arithmetic,” and went through at least seventeen editions. A preliminary page gives the reader “a pep talk” about the pleasure and usefulness of arithmetic.

Preliminary page from Borghi's Arithmetic (1484).

The next page of the book gives a “Table of Abbreviations” to be used in various problems. It includes symbols for money, weight, liquid measure, etc.

Table of abbreviations from Borghi's Arithmetic (1484).

Finally, the text proper begins with the above proclamation and includes the author’s name, Piero Borgi [Borghi] of Venice.

Introduction from Borghi's Arithmetic (1484).

On page 6, the first “operation,” numeration, is considered and demonstrated with a “place-value chart.” Note that the largest number recorded is “a million of a million of a million,” or, for the modern reader, 1018.

Table of numbers from Borghi's Arithmetic (1484).

On this page, Borghi took his reader/student through a division exercise, 56789 ÷36, employing the “galley” algorithm. He carried out the process in seven steps, with explanations given and computation performed in the left margin.

Example of galley algorithm from Borghi's Arithmetic (1484).

In the section entitled “Del moltiplichar de rotti” [On multiplication of fractions], the author considered five cases or the application of five rules (riegola is the word for one rule), beginning with the product of two proper fractions, and progressing through the cases of the product of a whole number and a proper fraction, the product of a mixed number and a proper fraction, the product of two mixed numbers, and, finally, the product of three proper fractions. In each instance, the result is reduced to lowest terms.

 Multiplication of fractions from Borghi's Arithmetic (1484).Multiplication of fractions from Borghi's Arithmetic (1484).

One rule still relevant to merchants at the time was the “Rule of Barter,” where one good was traded for another. In the 15th century, there was still a scarcity of minted monies in Europe, thus barter facilitated commerce.

Mathematics of bartering from Borghi's Arithmetic (1484).Mathematics of bartering from Borghi's Arithmetic (1484).

This material is obtained through the courtesy of the University of California Libraries. A complete digital copy can be read on the UC Libraries’ Internet Archive.

Index to Mathematical Treasures

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Borghi’s Best Commercial Arithmetic," Convergence (May 2017)