''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 Luca Bartolomeo de Pacioli, O.F.M. (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and an early contributor to the field now known as account ...

and illustrated by

Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 1452 - 2 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially rested o ...

, completed by February 9th, 1498 in

Milan Milan ( , , ; ) is a city in northern Italy, regional capital of Lombardy, the largest city in Italy by urban area and the List of cities in Italy, second-most-populous city proper in Italy after Rome. The city proper has a population of nea ...

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 summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...

) 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 from an earlier book by

Piero della Francesca Piero della Francesca ( , ; ; ; – 12 October 1492) was an Italian Renaissance painter, Italian painter, mathematician and List of geometers, geometer of the Early Renaissance, nowadays chiefly appreciated for his art. His painting is charact ...

, '' De quinque corporibus regularibus''.

Contents of the book

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

Fibonacci Leonardo Bonacci ( – ), commonly known as Fibonacci, was an Italians, Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, ''Fibonacci ...

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 (; ; 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 geometry that largely domina ...

), 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 In classical theistic and monotheistic theology, the doctrine of divine simplicity says that God is simple (without parts). God exists as one unified entity, with no distinct attributes; God's existence is identical to God's essence. Overview ...

. #Its definition invokes three lengths, symbolizing the

Holy Trinity The Trinity (, from 'threefold') is the Christian doctrine concerning the nature of God, which defines one God existing in three, , consubstantial divine persons: God the Father, God the Son (Jesus Christ) and God the Holy Spirit, three ...

. #Its

irrationality Irrationality is cognition, thinking, talking, or acting without rationality. Irrationality often has a negative connotation, as thinking and actions that are less useful or more illogical than other more rational alternatives. The concept o ...

represents God's incomprehensibility. #Its

self-similarity 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 self-similar ...

recalls God's

omnipresence Omnipresence or ubiquity is the property of being present anywhere and everywhere. The term omnipresence is most often used in a religious context as an attribute of a deity or supreme being, while the term ubiquity is generally used to describ ...

and invariability. #Its relation to the

dodecahedron In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...

, which represents the quintessence It also contains a discourse on the regular and

semiregular polyhedra In geometry, the term semiregular polyhedron (or semiregular polytope) is used variously by different authors. Definitions In its original definition, it is a polyhedron with regular polygonal faces, and a symmetry group which is transitive on ...

, as well as a discussion of the use of

geometric perspective Linear or point-projection perspective () is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, generally on a flat surface, of ...

by painters such as

Melozzo da Forlì Melozzo da Forlì ( – 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 Melozzo was s ...

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 ( ; ; –70 BC – after ) was a Roman architect and engineer during the 1st century BC, known for his multi-volume work titled . As the only treatise on architecture to survive from antiquity, it has been regarded since the Renaissan ...

(from his ''

De architectura (''On architecture'', published as ''Ten Books on Architecture'') is a treatise on architecture written by the Ancient Rome, Roman architect and military engineer Vitruvius, Marcus Vitruvius Pollio and dedicated to his patron, the emperor Caesa ...

'') 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 '' De quinque corporibus regularibus'' '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 (, ), more commonly known as the Vatican Library or informally as the Vat, is the library of the Holy See, located in Vatican City, and is the city-state's national library. It was formally established in 1475, alth ...

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 that ...

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 ) or an adage is a simple, traditional saying that expresses a perceived truth based on common sense or experience. Proverbs are often metaphorical and are an example of formulaic speech, formulaic language. A proverbial phrase ...

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; 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. 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 "M" logo used by the

Metropolitan Museum of Art The Metropolitan Museum of Art, colloquially referred to as the Met, is an Encyclopedic museum, encyclopedic art museum in New York City. By floor area, it is the List of largest museums, third-largest museum in the world and the List of larg ...

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 Medieval literature Leonardo da Vinci