Luca Pacioli

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Portrait of Luca Paciolis, painted by Jacopo de Barbari , 1495
500 lire coin with a portrait of Luca Pacioli (1994)

Luca Pacioli (* around 1445 in Borgo San Sepolcro , Tuscany ; † 1514 or 1517 in Rome) was an Italian mathematician and Franciscan . He is known in economics because in 1494 he was the first to completely describe double- entry bookkeeping .


From 1477 Pacioli was a professor in Perugia , Rome , Naples , Pisa , Venice , Milan and Florence . He probably studied mathematics with Piero della Francesca . After his death in 1492, he owned at least two of his three mathematical writings, which he processed in his own works, especially the treatise on the abacus (probably written before 1460). The younger Leonardo da Vinci received mathematics lessons from him, was friends with him and stimulated Pacioli to write his treatise on the golden ratio , published in 1509 , which was also illustrated by Leonardo da Vinci. He argues that perspectivity makes painting a mathematical discipline like music. Da Vinci's well-known representation of humans in squares and circles (homo ad quadratum et circulum) is interpreted in relation to the striving of (finite) humans to learn about (infinite) God. Albrecht Dürer was influenced by the work. The human body, especially the human hand, but also architectural harmony and that of the letters of Roman inscriptions is considered to be "divine proportion".

Pacioli's impact is based primarily on his book Summa de Arithmetica, Geometria, Proportioni et Proportionalità (completed in 1487 and printed in 1494, 2nd edition in 1523), which is probably the first printed book by a mathematician and was used accordingly. It offers few independent ideas, but bundles the most important mathematical knowledge on the basis of u. a. of Liber Abbaci by Leonardo Fibonacci , 1202, 2. A. 1228, which had handed down the solutions to first and second order equations from Al-Chwarizmi , but this was hardly known until Pacioli took it up again. Pacioli also uses the works of Jordanus Nemorarius , 1236, and Johannes de Sacrobosco , 1256. This knowledge was taught at the Italian abacus schools for merchants of the time. The book contains the first complete account of the “Venetian method” ( double-entry bookkeeping ), as it was probably used by the Venetians and Medici at the time . However, he also mentions the use of the same in the books of the Genoese tax officials and, contrary to many opinions, is not the inventor of double-entry bookkeeping. This is probably the dealer Benedetto Cotrugli from Ragusa . The book was translated into many languages ​​and copied by other authors, which helped double-entry bookkeeping to its breakthrough.

An illustration by Leonardo da Vinci (1509) for the work De Divina Proportione by Luca Pacioli

Pacioli found no solution for cubic equations and closes his work with the assertion that in the state of the art these are as little generally solvable as the squaring of the circle ; However, there is reason to hope that solutions will be found in the future. So it was indeed, also inspired by these remarks, first of all with the solutions that Scipione del Ferro and Nicolo of Brescia (called “Tartaglia”) found. Pacioli also discusses the problem of division .

Around 1500 Pacioli wrote, presumably together with Leonardo da Vinci, a book about the game of chess , which was often quoted in the specialist literature: De ludo scacchorum , called Schifanoia . The famous manuscript was considered lost until the book historian Duilio Contin discovered the only known copy in the library of the Palazzo Coronini Cronberg Foundation in Gorizia in 2006 .

A work De viribus quantitatis contains mathematical jokes and tricks as well as explanations for magic tricks . These trick instructions, which are mostly based on the use of scientific knowledge, are considered to be the oldest writing on magic. In 1509 in Venice he also published an improved version of the text by Johannes Campanus von Novara's translation of Euclid's Elements .


  • De divina proportione , Venice 1509
  • Summa de arithmetica, geometria, proportioni et proportionalità , Venice 1494 (reprinted by Enrico Giusti, Abrizzi, Venice 1994, ISBN 88-317-6008-4 )
  • De viribus quantitatis , 1508 (first important collection of mathematical riddles and tricks)
  • De ludo scacchorum detto Schifanoia , manuscript. Facsimile reprint: Aboca Museum di Sansepolcro, 2007

Editions and translations

  • Giuseppina Masotti Biggiogero (ed.): De divina proportione (= Fontes ambrosiani , vol. 31). Scriptorium, Sassari 1998; English translation: Divine Proportion , Norwalk, CT 2001; Italian translation by Andrea Masini in: Arnaldo Bruschi u. a. (Ed.): Scritti rinascimentali di architettura , Milan 1978, pp. 23-244.
  • Augusto Marinoni, Maria Garlaschi Peirani (eds.): Luca Pacioli: De viribus quantitatis. Duck Raccolta Vinciana, Milano 1997


  • A. Agostini: Il "De viribus quantitatis" di Luca Pacioli , in: Periodico di matematiche 4/4 (1924), pp. 165-192
  • RG Brown, KS Johnston: Pacioli on Accounting , McGraw-Hill, New York 1965.
  • Argante Ciocci: Luca Pacioli e la matematizzazione del sapere nel Rinascimento , Collana di storia della scienza 3, Caccucci, Bari 2003, ISBN 88-8422-269-9 .
  • Diego D'Elia, Duilio Contin, Attilio Bartoli Langeli, Enzo Mattesini, Alessandro Sanvito: Gli scacchi di Luca Pacioli . Aboca Museum Edizioni, Sansepolcro 2007, ISBN 978-88-95642-16-1 .
  • R. Franci, L. Toti Rigatelli: Towards a history of algebra from Leonardo of Pisa to Luca Pacioli , in: Janus 72 / 1-3 (1985), pp. 17-82.
  • Enrico Giusti, Carlo Maccagni (eds.): Luca Pacioli e la matematica del Rinascimento . Giunti, Florence 1994, ISBN 88-09-20470-0 .
  • Richard H. Macve: Pacioli's Lecacy , in: TA Lee, A. Bishop, RH Parker (eds.): Accounting History from the Renaissance to the Present . A Remembrance of Luca Pacioli, Garland, New York 1996, pp. 3-30.
  • Stanley Morison: Fra Luca de Pacioli of Borgo S. Sepolcro . Kraus / Grolier Club, New York 1969.
  • Maria Paola Negri: Luca Pacioli e Daniele Gaetani . Scienze matematiche e retorica nel Rinascimento, in: Annali Bibl. Statale e Libreria Civica di Cremona 45/4 (1994), pp. 111-136.
  • Luca Parisoli: Volontarismo e diritto soggettivo . La nascita medievale di una teoria dei diritti nella scolastica francescana. Con prefazione di Andrea Padovani, Bibliotheca seraphico-capuccina 58, Istituto Storico dei Cappuccini, Rome 1999.
  • Orazio Puletti: Fra Luca Pacioli e le sue opere , Viterbo 1955, 2. A. EMI, Bologna 1997.
  • R. Emmett Taylor: No Royal Road : Luca Pacioli and His Times. University of North Carolina Press, Chapel Hill 1942.
  • Grahame Thompson: Early Double-Entry Bookkeeping and the Rhetoric of Accounting Calculation , in: Anthony G. Hopwood, Peter Miller (eds.): Accounting as a Social and Institutional Practice, Cambridge University Press, Cambridge 1994, pp. 40-66.
  • Grahame Thompson: Is Accounting Rhetorical? Methodology, Luca Pacioli and Printing, in: Accounting, Organizations, and Society 16 / 5-6 (1991), pp. 573-580.
  • Jane Gleeson-White: Debit and Credit: Double-entry Bookkeeping and the Emergence of Modern Capitalism . Velcro-Cotta 2015.
  • SA Jayawardene: Pacioli, Luca . In: Charles Coulston Gillispie (Ed.): Dictionary of Scientific Biography . tape 10 : SG Navashin - W. Piso . Charles Scribner's Sons, New York 1974, p. 269-272 .

Web links

Commons : Luca Pacioli  - collection of images, videos and audio files

Individual evidence

  1. De divina proportione, ed. C. Winterberg, Sources for Art History NR 2, Vienna 1889, p. 129.