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Mathematical Treasure: Spanish Arithmetic and Geometry

Frank J. Swetz (The Pennsylvania State University)

Juan de Ortega (1480-1568) was a Spanish Dominican priest and instructor of commercial arithmetic. His Tractado subtilissimo d’arismetica y de geometria, published in Barcelona in 1512, discussed commercial arithmetic and geometry. The four images shown immediately below are from the 1552 edition of this work published in Seville.


Ortega’s Tables of multiplication facts are interesting and employ mathematical principles not evident in other such tables of this period. In the first table, he began with the first nine counting numbers, each multiplying themselves thus providing their squares. The remainder of the table assumes a knowledge of the commutative property of multiplication to diminish the number of products listed for each successive set of products. For example, if 2 x 3 = 6 has been found, then 3 x 2 = 6 is not given for the next set of products. Ortega is noted today for his innovative technique for extracting the square root of a given number: the process is discussed later in this book.

Among the rules Ortega listed for merchants is “The Rule of Company” which shows how to divide profits or debts among a group or company of business partners.

The title page of Juan de Ortega’s Arte dela arismetica y ... de geometria, published in Barcelona in 1512, which seems to be the same book described above, depicts a master (or teacher) lecturing to his disciples (or students).

The title page of the 1563 edition can be seen here.

These images are obtained through the cooperation of the Hispanic Society of America Museum and Library.

Index to Mathematical Treasures

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Spanish Arithmetic and Geometry," Convergence (August 2017)