Mathematical Treasure: Luca Pacioli’s Summa

The Italian mathematician Luca Pacioli published an influential compendium of the mathematics known at the time, titled Sūma de Arithmetica Geometria Proportioni & Proportionalita (Linda Hall Library call number QA32.P32 1494), in 1494. The book contains over 600 pages of densely packed type along with figures. The image below is the title page of the 1494 edition, published in Venice. There was also an edition published in 1523 in Toscolano. The 1494 volume at the Linda Hall Library has a vellum cover and handmade paper.

According to David Eugene Smith in Rara Arithmetica, “This volume, the first great general work on mathematics printed, includes treatises on arithmetic, algebra, and geometry, each being considered from a somewhat scientific rather than practical standpoint.” (p. 56)

The verso of folio 29 shows the start of two and a half pages of multiplication tables.

Later, on the verso of folio 72, there is a multiplication table resembling those that appear in modern elementary curricula.

Summa was the first printed book to contain a chart of how to represent numerals with fingers (verso of folio 36).

Pacioli is sometimes called the “Father of Accounting” since Summa may contain the first published explanation of double-entry bookkeeping. As with most of the material in Summa, Pacioli is not presenting original work. Double-entry bookkeeping had been in use in Venice for around two hundred years prior to the publication of Summa. Below is the first page of the section, namely the eleventh tractatus of the ninth distinction titled Particularis de pomputis et scripturis, about double-entry bookkeeping.

As influential as Pacioli’s Summa has been, the section on double-entry bookkeeping is the only part of the work with a readily available English translation.

A complete digital scan of Summa is available in the Linda Hall Library Digital Collections.

For more images of Pacioli’s Summa, visit the Convergence article, “Mathematical Treasures - Pacioli’s Summa," by Frank J. Swetz and Victor J. Katz.

Images in this article are courtesy of the Linda Hall Library of Science, Engineering & Technology and used with permission. The Linda Hall Library makes available all existing digital images from its collection that are in the public domain to be used for any purpose under the terms of a Creative Commons License CC by 4.0. The Library’s preferred credit line for all use is: “Courtesy of The Linda Hall Library of Science, Engineering & Technology.”

References

O'Connor, J. J., and E. F. Robertson. “Luca Pacioli.” MacTutor History of Mathematics archive. https://mathshistory.st-andrews.ac.uk/Biographies/Pacioli/.

Schemmann, Michael, editor. The Rules of Double-Entry Bookkeeping (English translation of Particularis de computis et scripturis by Luca Pacioli). IICPA Publications, 2010.

Smith, David Eugene. Rara Arithmetica: A Catalogue of the Arithmetics Written Before the Year 1601. 1908. Reprint, Cosimo Classics, 2007.

Index to Mathematical Treasures