Luca Pacioli

Luca Bartolomeo de Pacioli, also spelled Paccioli or Paciolo, was born around 1447 in the Tuscan town of Sansepolcro and died on 19 June 1517. He was an Italian mathematician, a Franciscan friar, and a collaborator with Leonardo da Vinci, making early contributions to what is now called accounting. Often called the father of accounting and bookkeeping, he was the first in continental Europe to publish a work on double-entry bookkeeping. He was also known as Luca di Borgo, after his birthplace, Borgo Sansepolcro.

Pacioli grew up in Sansepolcro, receiving an abbaco education—instruction in the local language rather than Latin, focused on merchant skills. His father was Bartolomeo Pacioli, but as a child he lived with the Befolci family. Around 1464, he moved to Venice, where he continued his studies while tutoring the three sons of a merchant. During this time, he wrote his first book, a treatise on arithmetic for those boys. He became a Franciscan friar between 1472 and 1475, earning the title Fra Luca. In 1475, he began teaching privately in Perugia, becoming the first chair in mathematics there in 1477, and wrote a comprehensive textbook in the vernacular for his students. He continued as a private tutor but was told to stop teaching at that level in Sansepolcro in 1491. His first major book, *Summa de arithmetica*, was published in Venice in 1494. In 1497, Duke Ludovico Sforza invited him to Milan, where he met, taught mathematics to, collaborated with, and lived alongside Leonardo da Vinci. When Louis XII seized Milan in 1499, Pacioli and Leonardo fled, and their paths likely separated around 1506. Pacioli died at about age 70 on 19 June 1517, probably in Sansepolcro, where he seems to have spent his final years.

Pacioli wrote several mathematical works. His manuscript *De viribus quantitatis* (Vatican Library, Lat. 3129) is a nearly 600-page textbook for his students at the University of Perugia, written between December 1477 and April 1478. It has 16 sections on merchant arithmetic—barter, exchange, profit, metal mixing, and algebra—though 25 pages on algebra are missing. A modern transcription and partial translation were published in 1996. His *Summa de arithmetica* (Venice, 1494) was a textbook for northern Italian schools, synthesizing the mathematical knowledge of his time. It contained the first printed algebra in the vernacular and one of the first published descriptions of the double-entry accounting system used by Venetian merchants. This system covered most of the modern accounting cycle: journals and ledgers, with the rule that debits must equal credits before sleep. His ledger included accounts for assets (receivables and inventories), liabilities, capital, income, and expenses—categories still used in balance sheets and income statements. He showed year-end closing entries and proposed a trial balance to prove a balanced ledger. The work also touched on accounting ethics and cost accounting, introduced the Rule of 72 (using an approximation of 100 times the natural log of 2, over a century before Napier and Briggs), and spread plus and minus symbols that became Renaissance mathematical standards. However, many exercises were copied without credit from Piero della Francesca’s earlier *Trattato d’abaco*. Another manuscript, *De viribus quantitatis* (University of Bologna, 1496–1508), is a treatise on mathematics and magic, containing the first reference to card tricks, along with guidance on juggling, fire-eating, and making coins dance. It also first noted that Leonardo da Vinci was left-handed. Divided into three sections—mathematical problems, puzzles and tricks, and a collection of proverbs and verses—it has been called the "Foundation of modern magic and numerical puzzles." Never published, it sat in Bologna’s archives, seen by few scholars until mathematician David Singmaster found a reference to it in a 19th-century manuscript. An English translation finally appeared in 2007. Pacioli also published a Latin translation of Euclid’s *Elements* in 1509, and *Divina proportione*, written in Milan between 1496 and 1498 and published in Venice in 1509. Two manuscript versions survive, one in Milan’s Biblioteca Ambrosiana and another in Geneva’s Bibliothèque Publique et Universitaire. This work explored mathematical and artistic proportion, especially the golden ratio and its architectural applications, but incorporated without credit a full translation of Piero della Francesca’s *De quinque corporibus regularibus*. Leonardo da Vinci drew the illustrations of regular solids for *Divina proportione*.