First edition of one of the world’s most famous books. The work is divided into five parts - three of text and two dedicated to illustrations. The first part studies the golden section from a mathematical viewpoint and explores its applications. It also describes a study on regular and semiregular polyhedrons and presents a discussion on the use of perspective by the painters Piero della Francesca, Marco Palmezzano and Melozza da Forlì. The second part is a treatise on architecture based on Vitruvian theories. Luca Pacioli (c. 1445-1517) highlights the mathematical applications in the treatise by Vitruvius and compares the proportions of the human body to those of ancient architecture. The third part is substantially the Italian translation of the famous mathematical treatise De quinque corporibus regularibus written by Piero della Francesca, which Pacioli claims to be his own work. This plagiarism was denounced by Vasari, resulting in lengthy disputes. The two final parts are dedicated entirely to illustrations. The first series features a beautiful alphabet of twenty-three capital letters drawn by Pacioli himself, while the second series features sixtyone woodcuts based on drawings of polyhedrons by his friend Leonardo da Vinci and three architectural plates. The final woodcut illustration in black and white represents “the genealogical tree of proportions”, copied from the one that appears in the Summa de arithmetica also by Pacioli, published in Venice in 1494. The tree, which would be taken up by Leonardo among others, shows the categories of proportionality (“Arithmetica”, “Propredicta”, “Armonica”) and their subcategories (“Discontina”, “Rationalis”, “Multiplex”, etc.) on twelve lines. 

Pacioli, who felt himself to be a debtor to Leonardo due to his “ineffable left hand” with which he produced the famous drawings of “flat”, “recessed”, “raised”, “solid” and “void” polychrome polyhedrons contained in De Divina Proportione, “formed a close relationship with Leonardo da Vinci when both were at the court of Lodovico il Moro in Milan. Pacioli’s treatises and Leonardo’s notebooks testify to their association. Pacioli knew Leonardo’s work during this period in great detail.”¹ In De Divina Proportione Pacioli explains the golden number with clarity and logic. Pacioli and Leonardo’s shared objective was to publish a treatise that described the world according to its geometric elements, that is to say according to divine proportion, also known as the golden number. The golden number is 1.618 (otherwise known to mathematicians as Φ or phi, from the initial of Phidias, the great Greek sculptor), which in a certain sense is defined as “the number of beauty”, or rather the number able to express the harmony and proportion between the parts making up every element. The formula, which became known as the “golden section” in the 19th century, indicated the “right method” or the “golden mean”, which did away with excesses in two opposing directions. Renaissance scholars consider De Divina Proporzione to be an essential and wonderful source of information on the interaction between science and art. This subject was so important that St Bonaventure wrote “there is no beauty without proportion” and “all things of beauty are in some way delightful”. Just as Galileo, when reflecting on the golden section, maintained that the book of nature is written in mathematical language. The golden number was discovered in ancient times. Indeed, the discovery was probably made in an empirical fashion during prehistory, but the term “golden number” is a modern one. It did not have a particular name among the Greeks, while Pacioli used the term “divina proportione” (divine proportion) and Kepler used “sectio divina” (divine section). Leonardo was the first to use the term “sectia aurea”, or golden section. Therefore, despite always having fascinated scholars, it was during the Renaissance that the golden number enjoyed its greatest success and was applied in many different ways, thanks to the strongly intertwined interests of mathematicians and artists that still persist today. 

Roberto Cena 

¹ M. Daly Davis 1977, p. 5.