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Mathematical Treasure: The Arithmetic of Boethius

Frank J. Swetz (The Pennsylvania State University)

This image of a well-worn vellum manuscript page is from Boethius’ De arithmetica (ca. 1000). Anicius Manlius Severinus Boetius, commonly known as Boethius (ca. 480-524), was a Roman aristocrat, statesman, and scholar. Literate in Greek, Boethius was a translator and commentator on earlier Greek works and, as such, provided a vital intellectual link between the ancient classical world and the emerging Middle Ages. His De arithmetica is largely a translation of Nicomachus of Gerasa’s De institutione arithmetica libri duo (ca. 100). Boethius considered mathematics as consisting of four parts: arithmetic, music, geometry, and astronomy – the four subjects that formed the medieval quadrivium. Arithmetic, as the foundation of the other three, was the most important of these subjects. His De arithmetica consists of rather esoteric number theory involving complex categorizations of numbers. Modern scholarship shows how such number theory was useful in proportions involving music and architecture (Masi 1983).

The page shown on the right is from Chapter 11 of De arithmetica. It discusses “oddly-even” numbers. To generate such numbers, the reader is instructed to compile a table as shown, where the upper row is comprised of a sequence of the odd numbers: 3, 5, …, 15, and the second row below is formed by the “evenly-even” numbers: 4, 8, 16, …, 256. Then each term of the first sequence, in turn, is multiplied by all terms in the second sequence. The products resulting, 3 x 4 = 12, 3 x 8 = 24, …, 5 x 4 = 20, 5 x 8 = 40, …, and 15 x 256 = 3840, are the “oddly-even” numbers.

On these pages, Boethius discussed equality and inequality. He noted that equals bear the same name, such as denarius, cubit, or foot, whereas unequals are designated by different names, such as teacher and pupil or conqueror and conquered. The numerical listing at the bottom of the righthand page denotes the ten categories into which a comparison of unequals falls: the greater into five classes bearing such names as superpartient, and the less into five classes with similarly unusual names, such as multiple subsuperparticular. Boethian arithmetic books remained popular throughout the Middle Ages and well into the 16th century.


The author would like to thank Barnabas Hughes for his assistance with the Latin translation.


Michael Masi, Boethian Number Theory: A Translation of the De Institutione Arithmetica, Studies in Classical Antiquity, vol. 6, Rodopi, Amsterdam, 1983.

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.

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: The Arithmetic of Boethius," Loci (August 2014)