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Mathematical Treasures - Boethius's Arithmetic

Frank J. Swetz and Victor J. Katz


The Arithmetic of Boethius (480-524) dates from the early sixth century. This page is from a mansucript (Plimpton MS 165) that dates from approximately 1294, written on vellum. This page (folio 13) lists the powers of 2, 3, and 4 (at the bottom) and also has some other tables representing multiplication by some of these powers. Note that the forms of the figures are not always identical to the modern form.


This manuscript page (folio 15) contains illustrations of square and pentagonal numbers.

The four images below are of the same manuscript (Plimpton MS 165) of Boethius's Arithmetic as those above. They were added to this article in 2018. While the two images above were prepared specifically for the Mathematical Association of America (MAA) by the Rare Book and Manuscript Library of the Columbia University Libraries, those below are presented courtesy of the Columbia University Libraries via Digital Scriptorium.

Boethian arithmetics were quite popular during the Early Middle Ages. These were copies of the Arithmetic of Boethius, who lived from about 480 to 524 in what is now Italy. This illustrated French manuscript from 1294 (Plimpton MS 165) attests to its scribe’s craftsmanship. We begin with an image of the page of figurate numbers shown just above (folio 15) and its facing page (folio 14v). The facing page, not surprisingly, discusses and illustrates triangular numbers.

A “Table of Multiplication Facts” is rendered in red and green ink in folio 9, below.

The scribe's beautiful work is shown especially well in folio 12, below.

Much of the textual discussion concerns number properties, also demonstrated by the use of diagrams.

The images above have been obtained through the kind cooperation of the Rare Book and Manuscript Library of the Columbia University Libraries. These and more images may be accessed via Digital Scriptorium, a digital collection of medieval and early Renaissance manuscripts made available by a consortium of cooperating university libraries headed by the University of California, Berkeley.

Index to Mathematical Treasures

Frank J. Swetz and Victor J. Katz, "Mathematical Treasures - Boethius's Arithmetic," Convergence (January 2011)

Mathematical Treasures from the Smith and Plimpton Collections at Columbia University