De divina proportione

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Woodcut from De divina proportione illustrating proportions of the human face.

De divina proportione (About the Divine Proportion) is a book on mathematics written by Luca Pacioli and illustrated by Leonardo da Vinci, composed around 1497 in Milan and first printed in 1509.[1] Its subject was mathematical proportions (the title refers to the golden ratio) and their applications to geometry, visual art and architecture. The clarity of the written material and Leonardo's excellent diagrams helped the book to achieve an impact beyond mathematical circles, popularizing contemporary geometric concepts and images.[2][3]

The Book[edit]

The first printed illustration of a rhombicuboctahedron, by Leonardo da Vinci, from De divina proportione.

The book consists of three individual manuscripts,[1] which Pacioli worked on between 1496 and 1498.

The first part, Compendio Divina Proportione, studies the golden ratio from a mathematical perspective (following the relevant work of Euclid) and explores its applications to various arts, in seventy-one chapters.[1] It also contains a discourse on the regular and semiregular polyhedra,[4][5] as well as a discussion of the use of geometric perspective by painters such as Piero della Francesca, Melozzo da Forlì, and Marco Palmezzano.

The second part discusses the ideas of Vitruvius on the application of mathematics to architecture in twenty chapters.

The third part, Libellus in tres partiales tractatus divisus, is mainly an Italian translation of Piero della Francesca's Latin writings On [the] Five Regular Solids ("De quinque corporibus regularibus")[1] and mathematical examples.[4] In 1550 Giorgio Vasari wrote a biography of della Francesca, in which he accused Pacioli of plagiarism and claimed that he stole della Francesca's work on perspective, on arithmetic and on geometry.[1]

After these three parts are appended two sections of illustrations, the first showing the capital letters drawn with a ruler and compass by Pacioli and the second with some sixty illustrations in woodcut after drawings by Leonardo da Vinci.[6] Leonardo drew the illustrations of the regular solids 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.


Letter M from De divina proportione

Pacioli produced three manuscripts of the treatise by different scribes. He gave the first copy with a dedication to the Duke of Milan, Ludovico il Moro; this manuscript is now preserved in Switzerland at the Bibliothèque de Genève in Geneva. A second copy was donated to Galeazzo da Sanseverino and now rests at the Biblioteca Ambrosiana in Milan. The third, which has gone missing, was given to Pier Soderini, the Gonfaloniere of Florence.[7]

On 1 June 1509 the first printed edition was published in Venice by Paganino Paganini.[8] The book has since then been reprinted several times. The book was displayed as part of an exhibition in Milan between October 2005 and October 2006 together with the Codex Atlanticus.[9]

The "M" logo used by the Metropolitan Museum of Art in New York is adapted from one in De divina proportione.[10]


  1. ^ a b c d e O'Connor, J J; Robertson, E F (July 1999). "Luca Pacioli". School of Mathematics and Statistics. University of St Andrews. Retrieved 15 January 2015. 
  2. ^ Hart, George W. "Luca Pacioli's Polyhedra". Virtual Polyhedra. Retrieved 23 January 2015. 
  3. ^ Hoechsmann, Klaus Hoechsmann (1 April 2001). "The Rose and the Nautilus". University of British Columbia. Retrieved 15 January 2015. 
  4. ^ a b Gardes, Michel (20 June 2001). "La Divine Proportion de Luca Pacioli" (in French). Académie de Poitiers. Retrieved 15 January 2015. 
  5. ^ Field, J F (1997). "Rediscovering the Archimedean polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler". Arch. Hist. Exact Sci. 50 (3-4): 241–289. 
  6. ^ "Divina proportione, after Leonardo da Vinci". The Collection Online. Metropolitan Museum of Art, New York. Retrieved 15 January 2015. 
  7. ^ Di Teodoro, Francesco Paolo (2014). "PACIOLI, Luca". Dizionario Biografico degli Italiani (in Italian) 80. Treccani. Retrieved 30 January 2015. 
  8. ^ Nuovo, Angela (2014). "PAGANINI, Paganino". Dizionario Biografico degli Italiani (in Italian) 80. Treccani. Retrieved 27 January 2015. 
  9. ^ "The Virtual Codex Atlanticus". Leonardo3. Retrieved 15 January 2015. 
  10. ^ "Renaissance 'M' Bookmark". The Met Store. Metropolitan Museum of Art, New York. Retrieved 15 January 2015. 

External links[edit]