Luca Pacioli - Mathematics

Mathematics

Pacioli published several works on mathematics, including:

• Tractatus mathematicus ad discipulos perusinos (Ms. Vatican Library, Lat. 3129), a textbook of almost 600 pages, dedicated to his students at the University of Perugia where Pacioli was teaching from 1477 to 1480. The manuscript was written between December 1477 and 29 April 1478. It contains 16 sections on merchant arithmetic, such as barter, exchange, profit, mixing metals, and algebra. One part of 25 pages is missing from the chapter on algebra. A modern transcription has been published by Calzoni and Cavazzoni (1996) as well as a partial translation of the chapter on partitioning problems (Heeffer, 2010).
• Summa de arithmetica, geometria, proportioni et proportionalità (Venice 1494), a textbook for use in the schools of Northern Italy. It was a synthesis of the mathematical knowledge of his time and contained the first printed work on algebra written in the vernacular (i.e. the spoken language of the day). It is also notable for including the first published description of the method of bookkeeping that Venetian merchants used during the Italian Renaissance, known as the double-entry accounting system.The system he published included most of the accounting cycle as we know it today. He described the use of journals and ledgers, and warned that a person should not go to sleep at night until the debits equalled the credits. His ledger had accounts for assets (including receivables and inventories), liabilities, capital, income, and expenses — the account categories that are reported on an organization's balance sheet and income statement, respectively. He demonstrated year-end closing entries and proposed that a trial balance be used to prove a balanced ledger. He is widely considered the "Father of Accounting". Also, his treatise touches on a wide range of related topics from accounting ethics to cost accounting. Remarkably, in the solution for one problem he presented in the book he used an approximation of 100*log 2, more than 100 years before Napier and Briggs.
• De viribus quantitatis (Ms. Università degli Studi di Bologna, 1496–1508), a treatise on mathematics and magic. Written between 1496 and 1508 it contains the first reference to card tricks as well as guidance on how to juggle, eat fire and make coins dance. It is the first work to note that Leonardo was left-handed. De viribus quantitatis is divided into three sections: mathematical problems, puzzles and tricks, and a collection of proverbs and verses. The book has been described as the "foundation of modern magic and numerical puzzles", but it was never published and sat in the archives of the University of Bologna, seen only by a small number of scholars since the Middle Ages. The book was rediscovered after David Singmaster, a mathematician, came across a reference to it in a 19th-century manuscript. An English translation was published for the first time in 2007.
• Geometry (1509), a Latin translation of Euclid's Elements.
• De divina proportione (written in Milan in 1496–98, published in Venice in 1509). Two versions of the original manuscript are extant, one in the Biblioteca Ambrosiana in Milan, the other in the Bibliothèque Publique et Universitaire in Geneva. The subject was mathematical and artistic proportion, especially the mathematics of the golden ratio and its application in architecture. Leonardo da Vinci drew the illustrations of the regular solids in De divina proportione while he lived with and took mathematics lessons from Pacioli. Leonardo's drawings are probably the first illustrations of skeletonic solids, which allowed an easy distinction between front and back. The work also discusses the use of perspective by painters such as Piero della Francesca, Melozzo da Forlì, and Marco Palmezzano. As a side note, the "M" logo used by the Metropolitan Museum of Art in New York City is taken from De divina proportione.