''Divina proportione'' (15th century Italian for ''Divine proportion''), later also called ''De divina proportione'' (converting the Italian title into a Latin one) is a book on mathematics written by Luca Pacioli and illustrated by

Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially res ...

, completed by February 9th, 1498 in

Milan Milan ( , , Lombard: ; it, Milano ) is a city in northern Italy, capital of Lombardy, and the second-most populous city proper in Italy after Rome. The city proper has a population of about 1.4 million, while its metropolitan city h ...

and first printed in 1509. Its subject was mathematical proportions (the title refers to the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

) and their applications to geometry, to visual art through perspective, and to 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. Some of its content was

plagiarised Plagiarism is the fraudulent representation of another person's language, thoughts, ideas, or expressions as one's own original work.From the 1995 '' Random House Compact Unabridged Dictionary'': use or close imitation of the language and though ...

from an earlier book by

Piero della Francesca Piero della Francesca (, also , ; – 12 October 1492), originally named Piero di Benedetto, was an Italian painter of the Early Renaissance. To contemporaries he was also known as a mathematician and geometer. Nowadays Piero della Francesca i ...

, ''

De quinque corporibus regularibus ''De quinque corporibus regularibus'' (sometimes called ''Libellus de quinque corporibus regularibus'') is a book on the geometry of polyhedra written in the 1480s or early 1490s by Italian painter and mathematician Piero della Francesca. It is ...

''.

Contents of the book

The book consists of three separate manuscripts, which Pacioli worked on between 1496 and 1498. He credits

Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Wester ...

as the main source for the mathematics he presents.

''Compendio divina proportione''

The first part, ''Compendio divina proportione'' (''Compendium on the Divine Proportion''), studies the

from a mathematical perspective (following the relevant work of

Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...

), giving mystical and religious meanings to this ratio, in seventy-one chapters. Pacioli points out that golden rectangles can be inscribed by an icosahedron, and in the fifth chapter, gives five reasons why the golden ratio should be referred to as the "Divine Proportion": #Its value represents divine simplicity. #Its definition invokes three lengths, symbolizing the Holy Trinity. #Its irrationality represents

God In monotheistic thought, God is usually viewed as the supreme being, creator, and principal object of faith. Swinburne, R.G. "God" in Honderich, Ted. (ed)''The Oxford Companion to Philosophy'', Oxford University Press, 1995. God is typically ...

's incomprehensibility. #Its

self-similarity __NOTOC__ In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically se ...

recalls God's omnipresence and invariability. #Its relation to the

dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...

, which represents the quintessence It also contains a discourse on the regular and semiregular polyhedra, as well as a discussion of the use of geometric perspective by painters such as

Melozzo da Forlì Melozzo da Forlì (c. 1438 – 8 November 1494) was an Italian Renaissance painter and architect. His fresco paintings are notable for the use of foreshortening. He was the most important member of the Forlì painting school. Biography ...

and

Marco Palmezzano Marco Palmezzano (1460–1539) was an Italian painter and architect, belonging to the Forlì painting school, who painted in a style recalling earlier Northern Renaissance models. He was mostly active near Forlì. Biography Palmezzano was ...

Pacioli De Divina Proportione Head Equilateral Triangle 1509

''Trattato dell'architettura''

The second part, ''Trattato dell'architettura'' (''Treatise on Architecture''), discusses the ideas of

Vitruvius Vitruvius (; c. 80–70 BC – after c. 15 BC) was a Roman architect and engineer during the 1st century BC, known for his multi-volume work entitled '' De architectura''. He originated the idea that all buildings should have three attribut ...

(from his '' De architectura'') on the application of mathematics to architecture in twenty chapters. The text compares the proportions of the human body to those of artificial structures, with examples from classical Greco-Roman architecture.

''Libellus in tres partiales divisus''

The third part, ''Libellus in tres partiales divisus'' (''Book divided into three parts''), is a translation into Italian of

's Latin book ''

'' 'On_[theFive_Regular_Solids''.html" ;"title="he.html" ;"title="'On [the">'On [theFive Regular Solids''">he.html" ;"title="'On [the">'On [theFive Regular Solids'' It does not credit della Francesca for this material, and 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. Because della Francesca's book had been lost, these accusations remained unsubstantiated until the 19th century, when a copy of della Francesca's book was found in the

Vatican Library The Vatican Apostolic Library ( la, Bibliotheca Apostolica Vaticana, it, Biblioteca Apostolica Vaticana), more commonly known as the Vatican Library or informally as the Vat, is the library of the Holy See, located in Vatican City. Formally es ...

and a comparison confirmed that Pacioli had copied it.

Illustrations

After these three parts are appended two sections of illustrations, the first showing twenty-three capital letters drawn with a ruler and compass by Pacioli and the second with some sixty illustrations in

woodcut Woodcut is a relief printing technique in printmaking. An artist carves an image into the surface of a block of wood—typically with gouges—leaving the printing parts level with the surface while removing the non-printing parts. Areas tha ...

after drawings by

. 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. Another collaboration between Pacioli and Leonardo existed: Pacioli planned a book of mathematics and

proverb A proverb (from la, proverbium) is a simple and insightful, traditional saying that expresses a perceived truth based on common sense or experience. Proverbs are often metaphorical and use formulaic language. A proverbial phrase or a proverbia ...

s called ''De Viribus Quantitatis'' (''The powers of numbers'') which Leonardo was to illustrate, but Pacioli died before he could publish it.

History

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 Ludovico Maria Sforza (; 27 July 1452 – 27 May 1508), also known as Ludovico il Moro (; "the Moor"). "Arbiter of Italy", according to the expression used by Guicciardini,

; this manuscript is now preserved in Switzerland at the

Bibliothèque de Genève The Bibliothèque de Genève (BGE, English: Geneva Library, Library of Geneva), founded in 1559, was known as ''Bibliothèque publique et universitaire'' (BPU, English: Public and University Library) from 1907 to 2006. It occupies different build ...

Geneva , neighboring_municipalities= Carouge, Chêne-Bougeries, Cologny, Lancy, Grand-Saconnex, Pregny-Chambésy, Vernier, Veyrier , website = https://www.geneve.ch/ Geneva ( ; french: Genève ) frp, Genèva ; german: link=no, Genf ; it, Ginevr ...

. A second copy was donated to Galeazzo da Sanseverino and now rests at the

Biblioteca Ambrosiana The Biblioteca Ambrosiana is a historic library in Milan, Italy, also housing the Pinacoteca Ambrosiana, the Ambrosian art gallery. Named after Ambrose, the patron saint of Milan, it was founded in 1609 by Cardinal Federico Borromeo, whose agen ...

in Milan. On 1 June 1509 the first printed edition was published in Venice by Paganino Paganini; it has since been reprinted several times. Fra Luca Pacioli Letter M 1509

The book was displayed as part of an exhibition in Milan between October 2005 and October 2006 together with the

Codex Atlanticus The Codex Atlanticus (Atlantic Codex) is a 12-volume, bound set of drawings and writings (in Italian) by Leonardo da Vinci, the largest single set. Its name indicates the large paper used to preserve original Leonardo notebook pages, which was us ...

. The "M" logo used by the

Metropolitan Museum of Art The Metropolitan Museum of Art of New York City, colloquially "the Met", is the largest art museum in the Americas. Its permanent collection contains over two million works, divided among 17 curatorial departments. The main building at 1000 ...

in New York was adapted from one in ''Divina proportione''.

References

Works cited

External links

Full text of original edition

Full text of 1509 edition

Title page of a reprint in Vienna, 1889

A video featuring a 1509 edition on display at Stevens Institute of Technology

Full text of original edition (1498) in English
{{DEFAULTSORT:De Divina Proportione 1509 books History of geometry History of mathematics Mathematics books Mathematics manuscripts Medieval literature Leonardo da Vinci