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Luca Pacioli

c. 1445 - 1517

Luca Pacioli

Luca Pacioli was an Italian mathematician and teacher, who published books in the Italian vernacular to popularise mathematics.

  • Nationality
  • Italian

  • Subject
  • Mathematics

  • Fields
  • accountancy, bookkeeping, probability, chess, geometry

  • Distinctions
  • Pacioli is popularly known as the 'Father of Accounting', and as the first author of a popular mathematics book.

  • Posts
  • Franciscan friar

    Professor at Pergia, Rome, Naples, Pisa, Venice, Milan, and Florence.

    Leonardo da Vinci received mathematics instruction from Pacioli

  • Publications
  • Pacioli and Leonardo da Vinci collaborated on De divina proportione, in which Pacioli investigates the Golden Ratio, and Leonardo make drawings of three-dimensional shapes bearing this proportion.

    Summa de Arithmetica, Geometria, Proportioni et Proportionalità 1487 (printed 1494), considered the first popular book by a mathematician, and was written in the Italian vernacular, as opposed to scholarly Latin. In this much referenced work, Pacioli presents the state of the art of mathematical knowledge of his time. He also describes the theses of the Leonardo Bonacci (later aka Fibonacci), including the first and second order equation solutions, based on Al-Chwarizmi. The book also contains the first complete presentation of the 'Venetian Method' of double-entry bookkeeping.

    Tractatus mathematicus ad discipulos perusinos a textbook for his students at Perugia University, where he taught 1477 - 1480. It contains treatises on merchant arithmetic, algebra, and subjects such as partitioning problems.

    De viribus quantitatis, 1496 - 1508, on mathematics of gambling and magic. It is referred to as 'a foundation of modern magic and numerical puzzles'. It was not published lost at the time, and remained abscure in archives till its recent discovery. An English translation was published in 2007.

    De ludo scacchorum (On the Game of Chess). It is thought Leonardo designed and drew the chess figures featured in this book. The book was not published and rediscovered only recently.

  • Laws
  • Rule of 72: applying an approximation for 100ln 2 well ahead of his time.

  • Theories
  • Pacioli published and made available to a broad audience the double-entry accounting, but probably did not invent it, as is often proposed. He popularised the terms of debit, credit, receivables, inventories, liabilities, capital income, expenses, and the use of the balance sheet and income statement, and year-end closure.

  • Equations
  • Pacioli investigated higher order equation solutions, and popularised previous little known mathematical works on this and other subjects. He proposed that a solution to cubic (third order) equations was possible, and his prediction bore fruit later in the 16th century through the work of his compatriots Scipione del Ferro and Nicolo von Brescia (aka 'Tartaglia').

Rhombicuboctahedron, an Archimedean solid drawn by Leonardo da Vinci for Luca Pacioli
Luca Pacioli
Luca Pacioli, 1445 - 1514, wrote the 'De divina Proportione', in a portrait attributed to Jacopo de Barbari, 1495

Pacioli was the teacher and then life-long friend of Leonardo da Vinci, who went on to provide sketches for his books. His mathematical works brought the work of previously little known mathematicians, such as Fibonacci, to a greater pubic awareness.

Read about history on
Luca Pacioli, 1445 - 1514, wrote the 'De divina Proportione', 1509, illustrated by Leonardo Da Vinci

With Leonardo he explored the Golden Ratio, which led to its popularization in architecture and art. Leonardo was inspired by Pacioli to draw the famous 'Vitruvian Man' on the basis of the Golden Ratio as described by Vitruvius, the ancient Roman architect.

Luca Pacioli
Luca Pacioli, 1445 - 1514, wrote the 'De divina Proportione', 1509, illustrated by Leonardo Da Vinci

Pacioli developed and taught accounting, and popularised the techniques used at the time, which are still standard in companies around the world. For his presentation of the double-entry accounting technique he is often referred to as 'The Father of Accounting', although he probably did not actually invent it, as such.

Luca Pacioli, 1445 - 1514, wrote the first book about chess

He was also interested in mathematical applications to chess, magic tricks (including card tricks), probability, and art. At least two of his books had been lost and rediscovered only recently.

The story of Pacioli and Leonardo's extraordinary collaboration is recounted in Andrew Bone's 2010 novel 'Vitruvian Boy'.

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