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Mathematical Treasures - Juan de Ortega's 16th-Century Arithmetic

Frank J. Swetz (The Pennsylvania State University)

Suma de arithmetica: ordinate per Johane de Ortega was a commercial arithmetic written by the Spanish Dominican friar Juan de Ortega (1480–1568). He was a teacher of commercial arithmetic who first published this work in 1512 in Barcelona under the title Tractado subtilissimo d’arismetica y de geometria. In 1515, the work was republished in Rome and Lyons under the title shown here, becoming the first commercial arithmetic to appear in France. It went through several more printings and experienced a wide distribution in Europe. In his last chapter, “Rules of Geometry,” Ortega approximated the square roots of fourteen numbers using an original method involving what we now call “Pell’s equation.”

Here is the Table of Contents for the Suma. Note the various specific rules that deal with trade and commerce, such as computing tare (tara, page 53); exchanging currencies (cambio, page 54); and forming a company (compagnie, pages 69 and 73). For a discussion of such rules, see the reference below.

On this page, the "Rule of Five” is demonstrated. Five knowns are provided and an unknown must be found through the use of proportion.​


Frank Swetz, Capitalism and Arithmetic: The New Math of the 15th Century, Open Court Publishing, 1987.

These images from its George Arthur Plimpton Collection are presented through the courtesy of the Columbia University Libraries.

See images from a 1552 edition of Juan de Ortega's Tractado subtilissimo d’arismetica y de geometria, together with the title page of a 1512 edition of the work, in Mathematical Treasure: Spanish Arithmetic and Geometry here in Convergence.

Index to Mathematical Treasures​

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasures - Juan de Ortega's 16th-Century Arithmetic," Convergence (February 2020)

Mathematical Treasures from the Smith and Plimpton Collections at Columbia University