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Mathematical Treasure: Benedetto da Firenze’s Trattato d’abacho

Author(s): 
Frank J. Swetz (The Pennsylvania State University)

Benedetto da Firenze (1429–1479) is credited as the author of an illustrated manuscript, Trattato d’abacho, that was copied around 1480. Perhaps this work was based on his previous, unpublished manuscript. The practice of encasing computations in red ink is similar in both works, but a textual analysis would be needed to verify this premise.

First page of text:

First page of Benedetto da Firenze’s Trattato d’abacho (circa 1480).

Several pages of “multiplication facts” follow; here is an example:

Multiplication table from Benedetto da Firenze’s Trattato d’abacho (circa 1480).

Two algorithms for the multiplication of two multi-digit numbers are illustrated.

Multiplication algorithm from Benedetto da Firenze’s Trattato d’abacho (circa 1480).

Multiplication algorithm from Benedetto da Firenze’s Trattato d’abacho (circa 1480).

Examples of highlighted computations:

Example of worked problem from Benedetto da Firenze’s Trattato d’abacho (circa 1480).

Worked problems from Benedetto da Firenze’s Trattato d’abacho (circa 1480).

Worked problem from Benedetto da Firenze’s Trattato d’abacho (circa 1480).

A collection of illustrated problem situations is then presented for the reader’s practice. These are standard medieval problems which can be identified from the illustrations. The viewer may speculate as to the situations and the required mathematics to obtain solutions.

Real-world problem from Benedetto da Firenze’s Trattato d’abacho (circa 1480).

Real-world problem from Benedetto da Firenze’s Trattato d’abacho (circa 1480).

Real-world problem from Benedetto da Firenze’s Trattato d’abacho (circa 1480).

A discussion about convex polyhedra:

Diagrams of polyhedra from Benedetto da Firenze’s Trattato d’abacho (circa 1480).

A full digitization is available from Museo Galileo. See also this section of one of the museum’s online exhibits. Benedetto’s Trattato d’arismetricha (ca 1460) may also be viewed in Convergence.

Index to Mathematical Treasures

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Benedetto da Firenze’s Trattato d’abacho," Convergence (February 2023)